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--- abstract: 'We show that a recently proposed Rudin–Shapiro-like sequence, with balanced weights, has purely singular continuous diffraction spectrum, in contrast to the well-known Rudin–Shapiro sequence whose diffraction is absolutely continuous. This answers a question that had been raised about this new sequence.' address: 'School of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, United Kingdom Email addresses: {lax.chan,uwe.grimm}@open.ac.uk' author: - Lax Chan - Uwe Grimm title: 'Spectrum of a Rudin–Shapiro-like sequence' --- Introduction ============ Substitution dynamical systems are widely used as toy models for aperiodic phenomena in one dimension [@PF02]. Crystallographers are interested in the diffraction spectrum of these systems because it provides information about the structure of a material [@BG12]. Dworkin [@SD93] showed that the diffraction spectrum is related to part of the dynamical spectrum, which is the spectrum of a unitary operator acting on a Hilbert space, as induced by the shift action. For recent developments regarding the relation between diffraction and dynamical spectra, we refer to the review [@BL16] and references therein. The Rudin–Shapiro (RS) sequence [@Sha51; @Rud59] (in its (balanced) binary version with values in $\{\pm 1\}$) is a rare example of a substitution-based system with purely absolutely continuous diffraction spectrum (while its dynamical spectrum is mixed, containing the dyadic integers as its pure point part); see [@AS03] for background. A ‘Rudin–Shapiro-like’ (RSL) sequence was recently introduced and analyzed in [@PNR15]. It is defined as $$\label{eq:rsl} \textnormal{RSL}(n)\, =\, (-1)^{\textnormal{inv}^{}_2(n)},$$ where $\textnormal{inv}^{}_{2}(n)$ counts the number of occurences of $10$ (‘inversions’) as a scattered subsequence in the binary representation of $n$. In [@PNR15], it is shown that this sequence exhibits some similar properties as the Rudin–Shapiro sequence. In particular, this concerns the partial sums $\varSigma(N):=\sum_{0\leq n\leq N}\textnormal{RSL}(N)$, which are shown to have the form $\varSigma(N)=\sqrt{N}\,G(\log_{4}N)$, where $G$ is a function that oscillates periodically between $\sqrt{3}/3$ and $\sqrt{2}$. At the end of [@PNR15], the question is raised whether this similarity between the two sequences extends to the property that $$\label{eq:ineq} \sup_{\theta\in{{\mathbb{R}}}}\left|\sum_{n<N}RSL(n)\, e^{2\pi in\theta}\right| \, \le\, C\, N^{\frac{1}{2}},$$ which is satisfied by the Rudin–Shapiro sequence [@AL91], and which is linked to the purely absolutely continuous diffraction measure of the balanced RS sequence. In what follows, we are going to employ a recent algorithm by Bartlett [@AB14] to show that the Rudin–Shapiro-like sequence has purely singular continuous diffraction spectrum, pointing to a big structural difference to the Rudin–Shapiro sequence. In particular, this will imply that Equation  does *not* hold for the Rudin–Shapiro-like sequence. A sketch of Bartlett’s algorithm ================================ By generalizing and developing previous work of Queffélec [@MQ10], Bartlett [@AB14] provides an algorithm that characterizes the spectrum of an aperiodic, constant length substitution $S$ on ${{\mathbb{Z}}}^{d}$. It describes the Fourier coefficients of mutually singular measures of pure type, giving rise to the maximal spectral type. Here, we can only give a brief sketch of Bartlett’s algorithm, concentrating on the case of dimension $d=1$. We assume that the substitution system is primitive. We first compute the instruction matrices (or digit matrices) $R_j$, where $j\in[0,q)$ and $q$ is the length of the substitution (which will be $q=2$ in our case). These matrices encode the letters that appear at the $j$-th position of the image of the substitution system; we shall show this for the explicit example of the Rudin–Shapiro-like sequence below. The substitution matrix $M_S$ is given by the sum of the instruction matrices. Due to primitivity, the Perron–Frobenius theorem [@BG13 Thm. 2.2] ensures that the eigenvector to the leading eigenvalue of $M_S$ can be chosen to have positive entries only. We denote this vector, after normalizing it to be a probability vector, by $u$. Note that $u=(u_{{\gamma}})_{{\gamma}\in\mathcal{A}}$ determines a point counting measure as it counts how frequently each letter ${\gamma}$ in the alphabet $\mathcal{A}$ appears asymptotically. One then applies the following lemma [@JP86] to verify aperiodicity. Another property that is used is the so-called height of the substitution $S$, which can be calculated using [@MQ10 Def. 6.1]. A primitive $q$-substitution $S$ which is one-to-one on $\mathcal{A}$ is aperiodic if and only if $S$ has a letter with at least two distinct neighbourhoods. Bartlett’s algorithm employs the bi-substitution of the substitution $S$, which is defined as follows. Let $S$ be a $q$-substitution on the alphabet $\mathcal{A}$. The substitution product $S\otimes S$ is a $q$-substitution on $\mathcal{A}\mathcal{A}$ (the alphabet formed by all pairs of letters in $\mathcal{A}$) with configuration $R\otimes R$ whose $j$-th instruction is the map $$(R\otimes R)_{j}\!:\, \mathcal{A}\mathcal{A}\longrightarrow \mathcal{A}\mathcal{A}\quad\text{with}\quad (R\otimes R)_{j}\!:\,{\alpha}{\gamma}\longmapsto R_j({\alpha})R_j({\gamma}).$$ The substitution $S\otimes S$ is called the *bi-substitution* of $S$. The Fourier coefficients $\widehat{{\Sigma}}$ of the correlation measures ${\Sigma}$ can then be obtained using following theorem of Bartlett [@AB14]. \[thm:AB1\] Let $S$ be an aperiodic $q$-substitution on $\mathcal{A}$. Then, for $p\in{{\mathbb{N}}}$, we have $$\widehat{{\Sigma}}(k)\, =\, \frac{1}{q^{p}}\sum_{j\in[0,q^{p})}R_{j}^{p}\otimes R_{j+k}^{p}\,\widehat{{\Sigma}}\lfloor j+k\rfloor_{p} \, =\, \lim_{n\to\infty}\frac{1}{q^{n}}\sum_{j\in[0,q^{n})}R_{j}^{n}\otimes R_{j+k}^{n}\,\widehat{{\Sigma}}(0),$$ where $\lfloor j+k\rfloor_{p}$ is the quotient of $j+k$ under division modulo $q^{p}$. Here $R_j\otimes R_{j+k}$ is the Kronecker product of the instruction matrices at position $j$ and $j+k$. Together with the above theorem and Michel’s lemma [@AB14 Thm. 2.1], we have $$\widehat{{\Sigma}}(0)\, =\, \sum_{{\gamma}\in\mathcal{A}}u\cdot e_{{\gamma}{\gamma}},$$ where in general $e_{\alpha\beta}$ is the standard unit vector in ${{\mathbb{C}}}^{\mathcal{A}^{2}}$ corresponding to the word $\alpha\beta$. Define the $p$-th carry set to be $\Delta_p(k):=\{j\in [0,q^{p}):j+k\neq [0,q^{p})\}$. As a consequence of the above theorem, we have the following expression, $$\label{equation:1} \widehat{{\Sigma}}(1)\, =\, \left(qI-\sum_{j\in\Delta_1(1)}R_j\otimes R_{j+1}\right)^{-1} \sum_{j\notin\Delta_1(1)} R_j\otimes R_{j+1}\, \widehat{{\Sigma}}(0).$$ We then use the following proposition [@AB14 Prop. 2.2] to compute the bi-substitution and to partition the alphabet into its ergodic classes and a transient part. \[prop:AB2\] Let $S$ be a substitution of constant length on $\mathcal{A}$. Then there is an integer $h>0$ and a partition of the alphabet $\mathcal{A}=E_1\sqcup\cdots\sqcup E_k\sqcup T$ so that 1. $S^{h}\!:\, E_{j}\to E_{j}^{+}$ is primitive for each $1\leq j\leq K$, 2. ${\gamma}\in T$ implies $S^{h}({\gamma})\notin T^{+}$, where $\sqcup$ denotes the disjoint union, $E_j$ its ergodic classes and $T$ the transient part. $E_{j}^{+}$ and $T^{+}$ are the words formed by elements of the ergodic classes and transient part, respectively. We define the *spectral hull* $K(S)$ of a $q$-substitution to be $$K(S)\, :=\, \{v\in{{\mathbb{C}}}^{\mathcal{A}^{2}}: C_{S}^{t}v=qv \text{ and } v\geq 0\},$$ and denote the *extreme rays* of $K(S)$ by $K^{*}$. Here, $C_S=\sum_{j}R_j\otimes R_j$, the sum of the Kronecker product of the instruction matrices at each position $j$. Using the following lemma of Bartlett [@AB14] and enforcing strong semi-positivity, we obtain the extreme rays $K^{*}$ of the spectral hull $K(S)$. Here, we use the notation $\vec{E}:=\sum_{{\gamma}\delta\in E}e_{{\gamma}\delta}\in{{\mathbb{C}}}^{\mathcal{A}^{2}}$. \[lem:AB3\] A vector $v\in\mathbb{C}^{\mathcal{A}^{2}}$ satisfies $v\in K(S)$ if and only if $$v\, =\, V+P_{T}(QI-P_{T}C_{S}^{t})^{-1}P_{T}C_S^{t}V \quad\text{and}\quad v\geq 0,$$ where $V=\sum_{j}w_j\vec{E}_{j}$ with $w_j\in{{\mathbb{C}}}$, and where $P_{T}$ is the standard projection onto the transient pairs $T$ of $\mathcal{A}^{2}$. Finally, the *maximal spectral type* is given by $$\label{eq:mst} \sigma_{\textnormal{max}}\, \sim\, \omega_q*\sum_{w\in K^{*}}{\lambda}_{w},$$ where $\omega_q$ is a probability measure supported by the $q$-adic roots of unity. For each $w\in K^{*}$, we compute $$\widehat{{{\lambda}}_{w}}(k)\, =\, w\widehat{{\Sigma}}(k).$$ If $\widehat{{{\lambda}}_{w}}(k)$ is periodic in $k$, then ${\lambda}_{w}$ is a pure point measure, if $\widehat{{\lambda}_{w}}(k)=0$ for all $k\neq 0$, then ${\lambda}_w$ is Lebesgue measure. Otherwise, ${\lambda}_{w}$ is purely singular continuous. Thus, the maximal spectral type is completely characterized by this algorithm. The Rudin–Shapiro-like sequence =============================== The Rudin–Shapiro-like sequence of [@PNR15] can be described by the following substitution rule $$\label{eq:subst} S^{}_{\text{RSL}^{}}\!:\; 0\mapsto 01,\quad 1\mapsto 20,\quad 2\mapsto 13, \quad 3\mapsto 32,$$ on four letters. This is similar to the Rudin–Shapiro case, where the binary sequence is also obtained from a four-letter substitution rule, after applying a reduction map. We apply the recoding $0,1\to +1$ and $2,3\to -1$. Both letters $\pm 1$ then are equally frequent, so we are in the balanced weight case. In the remaining of this article, we are going to apply Bartlett’s algorithm to prove the following result. The (balanced weight) sequence $S^{}_{\text{RSL}^{}}$ has purely singular continuous diffraction spectrum. The instruction matrices and the substitution matrix can be read off from the substitution rule of Equation  and are given by $$R_0= \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 1 \end{pmatrix},\quad R_1= \begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ 0 & 0 & 1 & 0 \end{pmatrix} \quad\textnormal{and}\quad M_{\textnormal{RSL}}^{}= \begin{pmatrix} 1 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0\\ 0 & 1 & 0 & 1\\ 0 & 0 & 1 & 1 \end{pmatrix}.$$ As $M_{\textnormal{RSL}}^{3}\gg 0$, the substitution is primitive. The third iterate of the seed $0$ is $01201301$, which shows that the letter $0$ can be preceded by $2$ or by $3$, and that the letter $1$ can be followed by either $2$ or by $3$. Hence both $0$ and $1$ have two distinct neighbourhoods and, by Pansiot’s Lemma, the sequence is aperiodic. In accordance with the Perron–Frobenius theorem, we find ${\lambda}_{\textnormal{PF}}=2$ and $u=\frac{1}{4}(1,1,1,1)$ for the eigenvalue and statistically normalized eigenvector of $M_{\textnormal{RSL}}^{}$. By applying Theorem \[thm:AB1\], we obtain $\widehat{{\Sigma}}(0)=\frac{1}{4}\sum_{{\alpha}\in\mathcal{A}}e_{{\alpha}{\alpha}}$. As we are dealing with a length two substitution, we have $\Delta_{1}(1)=\{1\}$. Using Equation , we find that $$\widehat{{\Sigma}}(1)\, =\, \left(0,\frac{1}{6},0,\frac{1}{12},0,0,\frac{1}{12}, \frac{1}{6},\frac{1}{6},\frac{1}{12},0,0,\frac{1}{12},0, \frac{1}{6},0\right).$$ We then proceed to compute $\widehat{{\Sigma}}(k)$ for any $k\geq 2$. By using Proposition \[prop:AB2\], we calculate the ergodic decomposition of the bi-substitution $S_{\textnormal{RSL}^{}}\otimes S_{\textnormal{RSL}^{}}$ to obtain $$E_1=\{00,11,22,33\}, \quad E_2=\{03,12,21,30\}, \quad E_3=\{01,02,10,13,20,23,31,32\}$$ as the ergodic classes. In our case, the transient part turns out to be empty. Note that $E_1$ and $E_2$ contain exactly the same elements as the two corresponding ergodic classes of the Rudin–Shapiro sequence. Using Lemma \[lem:AB3\], and taking into account that we have an empty transient part $P_T^{}=0$, it follows that $$v\, =\,\left(\begin{matrix} w_1 & w_3 & w_3 & w_2 \\ w_3 & w_1 & w_2 & w_3\\ w_3 & w_2 & w_1 & w_3\\ w_2 & w_3 & w_3 & w_1 \end{matrix} \right).$$ We then diagonalize the matrix $v$, $$v_d\, =\, \left(\begin{matrix} w_2+w_1+2w_3 & 0 & 0 & 0\\ 0 & w_2+w_1-2w_3 & 0 & 0\\ 0 & 0 & -w_2+w_1 & 0\\ 0 & 0 & 0 & -w_2+w_1 \end{matrix}\right).$$ Setting $w_1=1$, strong semi-positivity is equivalent to $w_{2}$ and $w_{3}$ satisfying the following three inequalities, $$1-w_2\geq 0,\quad 1+w_2+2w_3\geq 0,\quad 1+w_2-2w_3\geq 0.$$ The extreme points are given by the solutions $(w_1,w_2,w_3)=(1,1,1)$, $(w_1,w_2,w_3)=(1,1,-1)$ or $(w_1,w_2,w_3)=(1,-1,0)$. Thus, the extremal rays are $$\begin{aligned} v_1=&(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1), \\ v_2=&(1,-1,-1,1,-1,1,1,-1,-1,1,1,-1,1,-1,-1,1)\mbox { and }\\ v_3=&(1,0,0,-1,0,1,-1,0,0,-1,1,0,-1,0,0,1).\end{aligned}$$ As usual, ${\lambda}_{v_1}=\delta_{0}$ which gives rise to the pure point component, via Equation . Using the previously computed values of $\widehat{{\Sigma}}(k)$, one checks that $\widehat{{\lambda}_{v_2}}(k)$ and $\widehat{{\lambda}_{v_3}}(k)$ do not vanish at all positions $k\neq 0$, which proves that there are no absolutely continuous components. One can then easily verify that the substitution system is of trivial height, therefore the pure point component is entirely supported by the Dirac measure $\delta_{0}$. The other two measures are neither absolutely continuous nor show the necessary periodicity to contribute to the pure point part. By Dekking’s theorem [@AB14 Thm. 5.6], we thus conclude that the other two measures have to be singular continuous. Thus, we have a purely singular continuous diffraction spectrum in the balanced weight case (in which the pure point component is extinguished). If we assumed that the Rudin–Shapiro-like sequence satisfied the inequality , it would imply that the diffraction spectrum was absolutely continuous, as a consequence of the following result [@MQ10 Prop. 4.9]. If $\sigma$ is the unique correlation measure of the sequence $\gamma$, $\sigma$ is the weak-$*$ limit point of the sequence of absolute continuous measures $R_N\cdot m$, where $m$ is the Haar measure and $R_N=\frac{1}{N}\left|\sum_{n<N}\gamma(n)e^{2\pi in\theta}\right|^{2}$, Let us denote $\zeta_N=R_N\cdot m$ and suppose weak convergence to a limit $\zeta$. Assuming that Equation  holds, it follows that $\zeta(g)\leq C\int g\ dm$, which implies absolute continuity. Hence, it follows from the singular diffraction that the inequality does *not* hold for the Rudin–Shapiro-like sequence. Comparison with the Rudin–Shapiro sequence ========================================== Let us close with a brief comparison with the Rudin–Shapiro sequence. The following result about the Rudin–Shapiro sequence is well known; see [@BG13 Ch. 10.2] and references therein for background and details. The Rudin–Shapiro sequence (with balanced weights) has purely absolute continuous diffraction spectrum. We refer the readers to [@AB14 Ex. 5.8] to see how Bartlett’s algorithm can be employed to show the above result. Both the RS sequence and the RSL sequence are based on (four-letter) substitutions of constant length $q=2$ (and a subsequent reduction to a balanced two-letter sequence), and superficially looks quite similar, including sharing the behaviour of partial sums that we mentioned earlier. The ergodic classes $E_1$ and $E_2$ of both substitutions contain exactly the same elements. The elements that form the transient part of the Rudin–Shapiro sequence are exactly the same elements that form the third ergodic class of the Rudin–Shapiro-like sequence. However, the values obtained from the Fourier transform of the correlation measures differ between these two systems. Hence, we have two structurally different systems that exhibit a similar arithmetic structure. Bartlett’s algorithm indicates that it may be quite difficult to construct substitution-based sequences with absolutely continuous diffraction spectrum, because it requires $\widehat{{\lambda}_{v}}(k)$ to vanish for *all* $k\ne 0$ for one of the extremal rays. Intuitively, this is the case because any non-trivial correlation will give rise to long-range correlations due to the built-in self-similarity of the substitution-based sequence. Generically, this property will not be fulfilled, so one should expect singular continuous spectra to dominate, which is indeed what is observed. A notable exception is provided by substitution sequences based on Hadamard matrices [@Fra03]. The authors would like to thank Michael Baake and Ian Short for many helpful discussions and comments on improving this paper, to Alan Bartlett on explaining his paper and Jean-Paul Allouche for sharing his preprint [@JA16]. The first author is supported by the Open University PhD studentship. [99]{} N. Pytheas Fogg, *Substitutions in Dynamics, Arithmetics and Combinatorics*, Lecture Notes in Mathematics Vol.  1794 (Springer, Berlin, 2002). M. Baake, U. Grimm, Mathematical diffraction of aperiodic structures, *Chem. Soc. Rev.* **41** (2012) 6821–6843. S. Dworkin, Spectral theory and [X]{}-ray diffraction, *J. Math. Phys.* **34** (1993) 2965–2967. M. Baake, D. Lenz, Spectral notions of aperiodic order, *Preprint* arXiv:1601.06629. H. Shapiro, *Extremal Problems for Polynomials and Power Series*, Masters thesis (MIT, Boston, 1951). W. Rudin, Some theorems on fourier coefficients, *Proc. Amer. Math. Soc.* **10** (1959) 855–859. J.-P. Allouche, J. Shallit, *Automatic Sequences* (Cambridge University Press, Cambridge, 2003). P. Lafrance, N. Rampersad, R. Yee, Some properties of a [R]{}udin-[S]{}hapiro-like sequence, *Adv. in Appl. Math.* **63** (2015) 19–40. J.-P. Allouche, P. Liardet, Generalized [R]{}udin-[S]{}hapiro sequences, *Acta Arith.* **60** (1991) 1–27. A. Bartlett, Spectral theory of $\mathbb{Z}^{d}$ substitutions, *Ergod. Th. & Dynam. Syst.* (to appear, *Preprint* arXiv:1410.8106). M. Queff[é]{}lec, *Substitution Dynamical Systems—Spectral Analysis*, 2nd Ed., Lecture Notes in Mathematics Vol.  1294 (Springer, Berlin, 2010). M. Baake, U. Grimm, *Aperiodic Order. Vol. 1. A Mathematical Invitation*, Encyclopedia of Mathematics and its Applications Vol. 149 (Cambridge University Press, Cambridge, 2013). J.-J. Pansiot, Decidability of periodicity for infinite words, *RAIRO Inform.  Théor. Appl.* **20** (1986) 43–46. N. Frank, Substitution sequences in $\mathbb{Z}^{d}$ with a non-simple [L]{}ebesgue component in the spectrum, *Ergod. Th. & Dynam. Syst.* **23** (2003) 519–532. J.-P. Allouche, On a [G]{}olay–[S]{}hapiro-like sequence, *Preprint*.

--- abstract: 'We present the spatially-resolved near-infrared (2.5–5.0 $\micron$) spectra of the edge-on starburst galaxy NGC 253 obtained with the Infrared Camera onboard $AKARI$. Near the center of the galaxy, we clearly detect the absorption features of interstellar ices ($\mathrm{H_2O}$: 3.05 $\micron$, $\mathrm{CO_2}$: 4.27 $\micron$, and $\mathrm{XCN}$: 4.62 $\micron$) and the emission of polycyclic aromatic hydrocarbons (PAHs) at 3.29 $\micron$ and hydrogen recombination line Br$\alpha$ at 4.05 $\micron$. We find that the distributions of the ices differ from those of the PAH and gas. We calculate the column densities of the ices and derive the abundance ratios of $N(\mathrm{CO_2})/N(\mathrm{H_2O}) = 0.17 \pm 0.05$. They are similar to those obtained around the massive young stellar objects in our Galaxy ($0.17 \pm 0.03$), although much stronger interstellar radiation field and higher dust temperature are expected near the center of NGC 253.' author: - 'Mitsuyoshi Yamagishi, Hidehiro Kaneda, Daisuke Ishihara, Shinki Oyabu, Takashi Onaka, Takashi Shimonishi, and Toyoaki Suzuki' title: '$AKARI$ Near–Infrared Spectroscopic Observations of Interstellar Ices in Edge-on Starburst Galaxy NGC 253' --- Introduction ============ The 2.5–5.0 $\micron$ near-infrared (NIR) spectra of the interstellar media in galaxies are dominated by various emission and absorption features. For example, the absorption of various ice species (solid-state molecules, e.g. $\mathrm{H_2O}$: 3.05 $\micron$, $\mathrm{CO_2}$: 4.27 $\micron$, $\mathrm{XCN}$: 4.62 $\micron$, and $\mathrm{CO}$: 4.67 $\micron$), as well as the emission of polycyclic aromatic hydrocarbons (PAHs) at 3.29 $\micron$ and hydrogen recombination lines such as Br$\alpha$ at 4.05 $\micron$, are included in the NIR regime. In particular, ices are important to understand interstellar chemistry, since the absorption profiles of ices are known to be sensitive to the chemical composition and the temperature of dust grains (e.g. Pontoppidan et al. 2008; Zasowski et al. 2009). Ices around young stellar objects (YSOs) in our Galaxy and the Large Magellanic Cloud (LMC) have been studied well until now (e.g. Gerakines et al. 1999; Gibb et al. 2004). Shimonishi et al. (2008, 2010) showed that the abundance ratios $N(\mathrm{CO_2})/N(\mathrm{H_2O})$ around massive YSOs in the LMC (0.36 $\pm$ 0.09) are significantly higher than those in our Galaxy (0.17 $\pm$ 0.03; Gerakines et al. 1999; Gibb et al. 2004). Ices are also detected in Galactic quiescent molecular clouds; Whittet et al. (2007) reported that they show the abundance ratios of 0.18 $\pm$ 0.04. Ices in nearby galaxies, however, have not been studied well; there are only a few reports about the detection of ices. Sturm et al. (2000) reported the first detection of $\mathrm{H_2O}$ ice absorption in the NIR and mid-infrared (MIR) spectra of NGC 253 and M 82 with the $ISO$ SWS. Following the detection of the $\mathrm{H_2O}$ ice, the detection of the $\mathrm{CO_2}$, XCN and CO ices was reported in the nucleus of NGC 4945 (Spoon et al. 2000, 2003). However spatially-resolved study about ices has not been conducted yet except for the L- and M-band study of the circumnuclear $10\arcsec$ region of NGC 4945 by Spoon et al. (2003). NGC 253 is a well-studied starburst galaxy at a distance of 3.5 Mpc (Rekola et al. 2005), which has a large inclination angle ($\sim 80^\circ$). Due to the high inclination angle, we can gain high column densities along the line of sight. Hence, it is relatively easy to detect various absorption features, if any, from NGC 253. The kinematic center of NGC 253 is a compact radio source at a wavelength of 2 cm, TH2, while the peak of NIR emission is spatially separated from the TH2 by $4\arcsec$ (see Fig. \[region\]). The NIR peak is thought to be a young super star cluster (Keto et al. 1999, Kornei & McCrady 2009). In Fig.\[region\], prominent dust lanes are visible on the north and the south-west side of the NIR peak. Kuno et al. (2007) presented the integrated $\mathrm{^{12}CO}$ map of NGC 253 with the beam size of $15\arcsec$. In the CO map (Fig.\[region\]), there is no apparent structure corresponding to the NIR dust lane. The central activity of the galaxy is known to be strong enough to produce prominent X-ray (Dahlem et al. 1998) and H$\alpha$ (Hoopes et al. 1996) as well as large-scale HI plumes (Boomsma et al. 2005). Moreover, $AKARI$ clearly detected far-infrared dust outflow from the galactic disk (Kaneda et al. 2009b), Tacconi-Garman et al. (2005) showed the distribution of PAH 3.3 $\micron$ emission for the central region of NGC 253 by using the narrow-band images with the VLT. In this letter, we present the NIR (2.5–5.0 $\micron$) spectra of NGC 253 obtained with the Infrared Camera (IRC; Onaka et al. 2007) on board the $AKARI$ satellite (Murakami et al. 2007). The spectra clearly show the absorption features of the $\mathrm{H_2O}$ and $\mathrm{CO_2}$ ices. Based upon the spectra, we discuss the interstellar chemical condition in NGC 253. Observations and Data Reduction =============================== The NIR spectroscopic observations were performed as part of the $AKARI$ mission program “ISM in our Galaxy and Nearby galaxies” (ISMGN; Kaneda et al. 2009a) in the $AKARI$ post-helium phase (phase 3). The observations were carried out on December 21 2009. To obtain 2.5–5.0 $\micron$ spectra, we used a grism spectroscopic mode (R $\sim$ 120) with the slit of $5 \arcsec \times 48 \arcsec$ for its width and length, respectively (Ohyama et al. 2007). Figure \[region\] shows the slit positions of the observations and the regions from which we created the spectra. We observed two regions in NGC 253, the north and south sides of the NIR peak (Observation ID: 1422187, 1422196). To avoid saturation effects, each region was selected not to cover the NIR peak. We observed each region two times to improve data quality. The basic spectral analysis was performed by using the standard IDL pipeline prepared for reducing phase 3 data with a newly calibrated spectral response curve[^1]. In addition to the basic pipeline process, we applied the following custom procedures to improve S/Ns for each spectrum: before creating a spectrum, we removed hot pixels from the three array images, where pixel intensities are replaced by the median values of contiguous 8 pixels, and then we obtained three spectra for the same region by integrating pixel intensities over the spatial scale of $7.5 \arcsec$ along the direction of the slit length. Next, we combined the two spectra by calculating a median value of 6 pixels, where 3 pixels in the direction of wavelength per spectrum were considered for the calculation. Standard deviations were then adopted as flux errors. Finally, we applied smoothing with a boxcar kernel of 3 pixels ($\sim 0.03$ $\micron$) in the direction of wavelength. We neglected the background of each spectrum since signals in a region $5 \arcmin$ away from the center of NGC 253 are about a hundred times smaller than those of the center. Result ====== The obtained spectra are shown in Fig \[spectra\]. The surface brightness of the spectra is different from region to region; the S1 and N1 spectra show the highest surface brightness for each slit aperture, which monotonically decreases toward the N5 and S5 spectra. The slopes of the spectra also change from the N1 and S1 to the N5 and S5 spectra. Several strong features are detected in the spectra; PAH emission at 3.3 $\micron$, hydrogen recombination line Br$\alpha$ at 4.05 $\micron$, and the absorption of ices. The absorption features of the $\mathrm{H_2O}$ ice centered at 3.05 $\micron$ and the $\mathrm{CO_2}$ ice at 4.27 $\micron$ are detected in all the spectra. Some spectra also show the absorption feature of $\mathrm{XCN}$ ice at 4.62 $\micron$ and the pure rotational line of molecular hydrogen $\mathrm{H_2S(9)}$ at 4.69 $\micron$. With $ISO$, Sturm et al. (2000) reported only the detection of the $\mathrm{H_2O}$ ice, and hence this is the first detection of the $\mathrm{CO_2}$ and XCN ices in NGC 253. The PAH emission at 3.3 $\micron$ is also detected in all the spectra. The spatial distribution of the PAH 3.3 $\micron$ emission to the south-west direction from the NIR peak is at least 2.5 times wider than that shown in Tacconi-Garman et al. (2004) owing to high sensitivity in the space observations. To obtain continuum spectra, we fit the continuum regions at 2.65–2.70 $\micron$, 3.60–3.70 $\micron$, 4.10–4.15 $\micron$, 4.35–4.45 $\micron$, and 4.85–4.95 $\micron$ by a fourth order polynomial. The best-fit continuum curve for each spectrum is shown in Fig \[spectra\]. We divide the original spectra by the continuum spectra to derive the optical depth spectra (Fig \[tau\]). To calculate the column densities of the $\mathrm{H_2O}$ ice, we fit a Gaussian profile to the optical depth spectra (Fig \[tau\]). Since the present spectra resolve the absorption feature of the $\mathrm{H_2O}$ ice, we can measure a true optical depth. However we have to consider the contribution of the PAH emission at 3.3 $\micron$ and its sub-features at 3.4 and 3.5 $\micron$ to the absorption of the $\mathrm{H_2O}$ ice. Hence Lorentzian profiles are included in the model fitting for the 3.3 and 3.4 $\micron$ features and a Gaussian for the 3.5 $\micron$ feature to fit the range of 2.65–3.65 $\micron$. We first fitted the S1 spectrum, determined the widths of the Gaussian and Lorentzian profiles, and then applied the same widths to the other spectra. In the spectral fitting, the centers of Gaussian and Lorentzian profiles are fixed at 3.05, 3.29, 3.42, and 3.50 $\micron$ for the $\mathrm{H_2O}$ ice, 3.3, 3.4, and 3.5 $\micron$ features, respectively. The result of fitting to one of the optical depth spectra is shown in Fig \[tau\]. The derived optical depths of the $\mathrm{H_2O}$ ice in the N1 and the S1 region ($\tau = 0.23 \pm 0.2$ for both regions) are consistent with that previously measured by $ISO$ ($\tau \sim 0.25$; Sturm et al. 2000) within the errors; the S1, S2, N1, and N2 regions overlap with the slit aperture of Sturm et al. (2000). We derive the column density, $N$, from the equation $$N = \int \tau d\nu/A,$$ where $A$, $\tau$, and $\nu$ are the band strength of each ice feature measured in a laboratory, an optical depth, and a wavenumber, respectively. The band strength of $2.0 \times 10^{18}$ $\mathrm{cm~molecule^{-1}}$ is used for the $\mathrm{H_2O}$ ice (Gerakines et al. 1995). On the other hand, the present spectra cannot resolve the absorption feature of the $\mathrm{CO_2}$ ice. We, however, applied the method of integrating the optical depth spectra rather than a curve-of-growth method since the equivalent width of $\mathrm{CO_2}$ ice absorption is very small ($\sim$ 0.01 $\micron$). Hence we use a Gaussian profile to fit each optical depth spectrum of the $\mathrm{CO_2}$ ice and used the fitting range of 4.20–4.35 $\micron$. In the above equation, we use the band strength of $7.6 \times 10^{17}$ $\mathrm{cm~molecule^{-1}}$ (Gerakines et al. 1995). The systematic error of each column density is estimated to be 15 % for the $\mathrm{H_2O}$ ice and 10 % for the $\mathrm{CO_2}$ ice. The errors are evaluated by changing the above-defined continuum regions with small shifts of $\pm 0.05$ $\micron$ for both ices. The derived column densities of the $\mathrm{H_2O}$ and $\mathrm{CO_2}$ ice, the integrated line intensities of PAH 3.3 $\micron$, Br$\alpha$, and $\mathrm{H_2S(9)}$, and the surface brightness at 2.7 $\micron$ and 4.9 $\micron$ are shown in Fig \[tau\]. The surface brightness at 2.7 $\micron$ and 4.9 $\micron$ is the median values over the wavelength ranges of 2.65 to 2.75 $\micron$ and 4.85 to 4.95 $\micron$, respectively. In Fig \[tau\], the spatial profiles of the ices are different from the other features; the integrated line intensity and the surface brightness have peaks at the N1 and S1 region and decrease rapidly to the N5 and S5 region except the integrated intensity of the $\mathrm{H_2S(9)}$, while the column densities of the ices show much smaller changes from region to region. In addition, on the south side, the ices show peaks in the S3 and S4 regions far from the NIR peak. These profiles suggest that the absorbers responsible for the ice features are more widely distributed than the line and the continuum emitters. The profiles of the $\mathrm{H_2O}$ and $\mathrm{CO_2}$ ices are similar to each other, suggesting a good correlation between the column densities of the $\mathrm{H_2O}$ and $\mathrm{CO_2}$ ice, although the phase of the gas dominating the spectral features differs from region to region. The N1 and N2 spectra show weaker continuum emission than the S1 and S2 spectra due to the presence of the NIR dust lane on the north side of the NIR peak (Fig.\[region\]). In the S3 and S4 regions, another prominent dust lane is visible, which presumably contributes to the larger column densities of the ices. Thus some of the ices responsible for the observed absorption are likely to be associated with the dust lanes, while the others are not. On the other hand, the distribution of the CO emission does not show clear spatial correspondence with those of the ice absorptions and the dust lanes. The CO map in Fig.\[region\] reveals a more centrally-concentrated distribution, which does not have a local maximum around the sub-apertures of S3 and S4, although the beam size of the CO map ($\sim$ $15\arcsec$) is somewhat larger than the spatial scale of the sub-apertures ($\sim$ $7\arcsec$). Therefore a majority of the CO molecular clouds do not significantly contribute to the observed absorptions due to the ices. In Fig \[ratio\], we show the correlation plot of the derived column densities of the $\mathrm{H_2O}$ and $\mathrm{CO_2}$ ices, which shows a linear correlation on both sides of the NIR peak. There is no systematic difference in the relation between the north and south regions. From the slope of the best-fit line to the data, the averaged $\mathrm{CO_2}$/$\mathrm{H_2O}$ ice ratio is calculated to be $0.17 \pm 0.05$. The ratio is similar to that obtained from the Galactic massive YSOs ($0.17 \pm 0.03$, Gerakines et al. 1999; Gibb et al. 2004). In our observation, we detect the superposition of ices in various kinds of clouds present along the line of sight, while the observations of the Galactic YSOs trace the chemical environment of individual star-forming clouds. Therefore it is interesting that these observations show similar $\mathrm{CO_2}$/$\mathrm{H_2O}$ ice ratios despite the different situations. In the above calculation, we assume the following geometry: the continuum emissions are in the background and absorbed by the ices with the covering fraction of 100 % for each sub-aperture, while the PAH 3.3 $\micron$ emission and its sub-features are distributed in the foreground of the ices. We also calculate the column densities in the case that the covering fraction of the ice features is 50 %. Then, the column densities change to 4.8–16.8 $\times$ $10^{17}$ $\mathrm{cm^{-2}}$ and 1.1–3.1 $\times$ $10^{17}$ $\mathrm{cm^{-2}}$ for the $\mathrm{H_2O}$ and $\mathrm{CO_2}$ ice, respectively, and the $\mathrm{CO_2}$/$\mathrm{H_2O}$ ice ratio of 0.17 $\pm$ 0.05, the same as above, is obtained. However, if the covering fraction is small, there is a possibility that the observed broad profile of the $\mathrm{H_2O}$ ice feature might be saturated. We therefore compare the obtained optical depth spectra of the $\mathrm{H_2O}$ ice with those obtained in the laboratory from the Leiden Molecular Astrophysics database (Ehrenfreund et al. 1996). We use the laboratory profile of the pure $\mathrm{H_2O}$ ice at 10 K. As a result, we do not find any significant difference between both optical depth spectra. Therefore, it is unlikely that the observed profiles of the $\mathrm{H_2O}$ ice are saturated. Moreover, we also calculate the column densities of the $\mathrm{H_2O}$ ice in the case that the PAH 3.3 $\micron$ emission and its sub-features are distributed in the background of the ices. Then we derive the slightly ($\sim$ 2 %) smaller column densities of the $\mathrm{H_2O}$ ice than the above calculation. Thus our results do not significantly depend on the assumed geometry. On the other hand, Shimonishi et al. (2008, 2010) showed that the $\mathrm{CO_2}$/$\mathrm{H_2O}$ ice abundance ratios of the massive YSOs in the LMC ($0.36 \pm 0.9$) are significantly higher than those in our Galaxy. Shimonishi et al. (2008, 2010) interpreted that the high ratio is caused by the relative increase of the $\mathrm{CO_2}$ ice possibly due to either strong interstellar ultraviolet (UV) photon irradiation to $\mathrm{H_2O}$-$\mathrm{CO}$ binary ice mixtures (e.g. Watanabe et al. 2007) or relatively high dust temperatures (Bergin, Neufeld, & Melnick 1999; Ruffle & Herbst 2001). In the center of NGC 253, nuclear starburst has occurred (Dudley & Wynn-Williams 1999), which indicates the existence of strong UV radiation field. The detection of XCN ice at 4.62 $\micron$ is indicative of strong UV irradiation (Bernstein, Sandford, & Allamandola 2000; Spoon et al. 2003). The slopes of the NIR continuum spectra suggest the presence of hot dust. Therefore our result suggests that intense interstellar UV radiation field and high dust temperatures are not important factors to determine the ice abundance ratio. Metallicity in NGC 253 is known to be close to a solar value, $Z \sim 1 Z_\odot$ (Webster & Smith 1983), while $Z \sim 0.3 Z_\odot$ for the LMC (Luck et al. 1998). Therefore, the interstellar metallicity might be an important chemical condition to affect the ice abundance ratio. Conclusions =========== With $AKARI$, we have performed the NIR (2.5–5.0 $\micron$) spectroscopic observation of the central region of the edge-on starburst galaxy NGC 253. We clearly detect the absorption features of the $\mathrm{H_2O}$, $\mathrm{CO_2}$, and XCN ices in addition to the PAH 3.3 $\micron$ feature and its sub-features at 3.4–3.5 $\micron$, the hydrogen recombination line Br$\alpha$ at 4.05 $\micron$, the molecular hydrogen pure-rotational line $\mathrm{H_2S(9)}$ at 4.69 $\micron$, and hot dust continuum emission. We for the first time obtain the spatial variations of the ice absorption features for nearby galaxies. We find that the ices have different distributions from PAH, ionized gas, molecular gas, and hot dust. We evaluate the column densities of the $\mathrm{H_2O}$ and $\mathrm{CO_2}$ ices and derive their abundance ratios, $N(\mathrm{CO_2})$/$N(\mathrm{H_2O})$, of $0.17 \pm 0.05$. The obtained ratios are very close to those observed for massive YSOs in our Galaxy. However they are significantly lower than those in the LMC where strong UV radiation and high dust temperatures are expected but not as much as in the central region of NGC 253. Therefore we conclude that intense interstellar UV radiation field and high dust temperatures are not important factors to determine the ice abundance ratio. We would like to thank all the members of the $AKARI$ project for their intensive efforts. We also express many thanks to the anonymous referee for the useful comments. This work is based on observations with $AKARI$, a JAXA project with the participation of ESA. Bergin, E. A., Neufeld, D. A., & Melnick, G. J. 1999, , 510, L145 Bernstein, M. P., Sandford, S. A, & Allamandola, L. J. 2000, , 542, 894 Boomsma, R., Oosterloo, T. A., Fraternali, F., van der Hulst, J. M., & Sancisi, R. 2005, , 431, 65 Dahlem, M., Weaver, K. A., & Heckman, T. M. 1998, , 118, 401 Dudley, C. C., & Wynn-Williams, C. G. 1999, , 304, 549 Ehrenfreund, P., Boogert, A. C. A., Gerakines, P. A., Jansen, D. J., Schutte, W. A., Tielens, A. G. G. M., & van Dishoeck, E. F. 1996, , 315, L341 Gerakines, P. A. et al. 1999, , 522, 357 Gerakines, P. A., Schutte, W. A., Greenberg, J. M., & van Dishoeck, E. F. 1995, , 296, 810 Gibb, E. L., Whittet, D. C. B., Boogert, A. C. A., & Tielens, A. G. G. M. 2004, , 151, 35 Hoopes, C. G., Walterbos, R. A. M., & Greenwalt, B. E. 1996, , 112, 1429 Kaneda, H., Koo, B. C., Onaka, T., & Takahashi, H. 2009a, AdSpR, 44, 1038 Kaneda, H., Yamagishi, M., Suzuki, T., & Onaka, T. 2009b, , 698, L125 Keto, E., Hora, J. L., Fazio, G. G., Hoffmann, W., & Deutsch, L. 1999, , 518, 183 Kornei, K. A., & McCrady, N. 2009, , 697, 1180 Kuno, N., et al. 2007, , 59, 117 Luck, R. E., Moffett, T. J., Barnes, T. G., III, & Gieren, W. P. 1998, , 115, 605 Murakami, H., et al. 2007, , 59, S369 Ohyama, Y., et al. 2007, , 59, S411 Onaka, T., et al. 2007, , 59, S401 Pontoppidan, K. M., et al. 2008, , 678, 1005 Rekola, R., Richer, M. G., McCall, M. L., Valtonen, M. J., Kotilainen, J. K., & Flynn, C. 2005, , 361, 330 Ruffle, D. P., & Herbst, E. 2001, , 324, 1054 Shimonishi, T., Onaka, T., Kato, D., Sakon, I., Ita, Y., Kawamura, A., & Kaneda, H. 2008, , 686, L99 Shimonishi, T., Onaka, T., Kato, D., Sakon, I., Ita, Y., Kawamura, A., & Kaneda, H. 2010, , 514, A12 Spoon, H. W. W., Koornneef, J., Moorwood, A. F. M., Lutz, D., & Tielens, A. G. G. M. 2000, , 357, 898 Spoon, H. W. W., Moorwood, A. F. M., Pontoppidan, K. M., Cami, J., Kregel, M., Lutz, D., & Tielens, A. G. G. M. 2003, , 402, 499 Sturm, E., Lutz, D., Verma, A., Netzer, H., Sternberg, A., Moorwood, A. F. M., Oliva, E., & Genzel, R. 2002, , 393, 821 Tacconi-Garman, L. E., Sturm, E., Lehnert, M., Lutz, D., Davies, R. I., & Moorwood, A. F. M. 2005, , 432, 91 Watanabe, N., Mouri, O., Nagaoka, A., Chigai, T., Kouchi, A., & Pirronello, V. 2007, , 668, 1001 Webster, B. L., & Smith, M. G. 1983, , 204, 743 Whittet, D. C. B., Shenoy, S. S., Bergin, E. A., Chiar, J. E., Gerakines, P. A., Gibb, E. L., Melnick, G. J., & Neufeld, D. A. 2007, , 655, 332 Zasowski, G., Kemper, F., Watson, D. M., Furlan, E., Bohac, C. J., Hull, C., & Green, J. D. 2009, , 694, 459 [^1]: http://www.ir.isas.jaxa.jp/ASTRO-F/Observation/

--- abstract: 'The Gravity Probe B (GP-B) experiment is complete and the results are in agreement with the predictions of general relativity (GR) for both the geodetic precession, 6.6 arcsec/yr to about 0.3%, and the Lense-Thirring precession, 39 marcsec to about 19%. This note is concerned with the theoretical basis for the predictions. The predictions depend on three elements of gravity theory, firstly that macroscopic gravity is described by a metric theory such as general relativity, secondly that the Lense-Thirring metric provides an approximate description of the gravitational field of the spinning earth, and thirdly that the spin axis of a gyroscope is parallel displaced in spacetime, which gives its equation of motion. We look at each of these three elements to show how each is solidly based on previous experiments and well-tested theory. The agreement of GP-B with theory strengthens our belief that all three elements are correct and increases our confidence in applying GR to astrophysical phenomena. Conversely, if GP-B had not verified the predictions a major theoretical quandary would have occurred.' author: - | Ronald J. Adler\*\ Hansen Experimental Physics Laboratory, Gravity Probe B Mission,\ Stanford University, Stanford, California 94309,\ and\ Department of Physics and Astronomy,\ San Francisco State University, San Francisco, California,\ and\ Kavli Institute for Particle Astrophysics and Cosmology,\ Stanford University, Stanford California 94035 94132 date: ' May 21, 2014' title: 'The three-fold theoretical basis of the Gravity Probe B gyro precession calculation' --- \*electronic mail address: adler@relgyro.stanford.edu or gyroron@gmail.com Introduction ============ After 47 years the Gravity Probe B (GP-B) experiment is complete.[@1; @2] The data analysis was more demanding than expected, due largely to complicating classical effects, for example electric charge on the rotors and housing as discussed at length in other papers in this volume.[@3; @4] The bottom line is that the predictions of general relativity (GR) for the geodetic effect are confirmed to about 0.3% and for the Lense-Thirring (LT) effect to about 19%. In this paper we will be concerned with what that experimental confirmation implies for gravity theory in general and in particular for GR.[@5; @6; @7] Our aim in this paper is to focus on how the prediction of the gyro precessions come about and what assumptions are needed, and thus to what extent the experiment verifies theory, in particular GR. Three key elements enter the calculation of the precession. The first is the most fundamental, that macroscopic gravity is described by a geometric theory, and specifically a metric theory.[@8; @9; @10; @11; @12] The second key element is that the specific metric for a nearly spherical spinning body, such as the earth, is the approximate one found in 1918 by Lense and Thirring using linearized GR.[@13; @14; @15; @16] The third key element is that the spin vector of a gyro is parallel displaced in spacetime, which implies that the equation of motion for the spin is that its covariant derivative is zero.[@17; @18] We will focus on analyzing how well founded are these elements. Our discussion will not be exhaustive since the literature contains many variations on the theme. Thus, unfortunately, we cannot reference many interesting and important theoretical papers on the subject. Just a few are listed in the references.[@19; @20; @21; @22; @23; @24; @25] There are of course many small corrections to the precession calculation, due for example to the multipole moments of the real earth, rather than the idealized spherical earth, due to the presence of the sun and moon, etc.[@14; @15; @26] There are also small corrections to the geodesic motion of spinning test bodies that are relevant to the equivalence principle, which we discuss in sec.5. Throughout this paper we will make use of appropriate approximations to gravity theory since the field of the earth is quite weak, and we will also make use of the fact that the earth and the gyro move at low velocity. As in most theory papers we will use units in which $c=1$. Metric theory in general ======================== It has been standard lore since the formulation of GR that gravity is described by a metric theory.[@8; @9; @10] The most obvious motivation for this assumption is the so-called weak equivalence principle (EP), or more accurately the “universality of free fall" for test bodies in a gravitational field. A metric theory provides an obvious elegant explanation for why the trajectories of test bodies in a gravitational field are independent of their masses and also various internal properties. The EP has been tested to impressive accuracy, better than about $10^{-12}$.[@5] This may be improved to $10^{-15}$ in an upcoming free-fall satellite experiment, perhaps to $10^{-15}$ by future atomic beam interferometry, and hopefully to $10^{-18}$ in a more accurate satellite experiment in the more distant future.[@27; @28; @29; @30] That $10^{-18}$ estimate seems to be the present anticipated limit. Various authors, notably Jordan and later Brans and Dicke, have suggested that a scalar field should be added to the description of gravity.[@31; @32; @33; @34] Some authors are of the opinion that string theory motivates such a modification, but there is as yet no experimental evidence to support string theory and no experimental evidence for a scalar field in gravity theory.[@35] Will discusses both scalar tensor theory and its experimental tests from the PPN perspective.[@36] In summary, so far the evidence is that a pure metric theory is adequate to describe macroscopic gravity, but the question remains interesting and open to experiment. Experimental status of GR and the Schwarzschild metric in a nutshell ==================================================================== This section will be a shamelessly short and over-simplified summary of parts of the book and arxiv paper by Will, leading to the conclusion that the Schwarzschild metric of GR has been quite well tested by observation and experiment.[@5] Unfortunately all of the evidence involves weak fields and rather low velocities, and there are as yet no precision tests of strong gravity; observations of black holes may lead to such tests in the future by studying, for example, the motion of material near the surfaces of black holes.[@37; @5] The Òclassical testsÓ of GR, the gravitational red shift, the orbit of Mercury and the deflection of light by the sun, are all based on the Schwarzschild metric, obtained in 1916, which describes the metric field of a spherically symmetric non-spinning body.[@38; @5; @12] In the standard coordinates the metric is $$\label{1} ds^2=(1-2m/r_s)dt^2-(1-2m/r_s)^{-1}dr_s^2-r^2_sd\theta ^2 -r_s^2\sin^2\theta d\varphi ^2 ,$$ where $m$ is termed the geometric mass; $M$ is the mass of the body and $G$ is NewtonÕs constant. In the so-called isotropic coordinates, which are convenient for comparison with observation, the metric is,[@39] $$\begin{aligned} \label{2} ds^2= \frac{(1-m/2r)^2} {(1+m/2r)^2}-(1+m/2r)^4d\vec{r}^2 \end{aligned}$$ $$\label{2a} =\left( 1-\frac{2m}{r}+\frac{2m^2}{r^2}+ ...\right)dt^2-\left( 1+\frac{2m}{r}+ \frac{3m^2}{2r^2}...\right)d\vec{r}^2 .$$ The power series expansion in the last line is useful and valid for distances far from the central body where $m/r<<1$. Eddington re-expressed (2) in terms of 3 dimensionless parameters, $\alpha,\beta,\gamma$, as [@40] $$\label{3} ds^2=\left( 1-\alpha\frac{2m}{r}+\beta\frac{2m^2}{r^2}+ ...\right)dt^2-\left( 1+\gamma\frac{2m}{r}+ ...\right)d\vec{r}^2 .$$ The parameter $\alpha$ is a measure of the distortion of time due to gravity, but the way in which it enters the metric makes it impossible to separate from NewtonÕs constant $G$, and as a result it may be taken to be 1; we will retain it only as a bookkeeping device, as we will discuss below. The parameter $\beta$ is a measure of the nonlinearity of time distortion effects; $\gamma$ is a measure of the distortion of space to first order. In GR all the parameters are equal to unity, $\alpha=\beta=\gamma$. The quadratic term in the spatial part of the metric (2) is not yet measurable and does not appear in (3), nor do any other higher order terms. The Eddington form (3) of the metric can be viewed in two ways. The first is as a bookkeeping device to see how various physical predictions depend on properties of GR; for example the precession of the orbit of mercury depends on the combination $\beta+\gamma$ so we may say that the nonlinearity of time distortion and linear space distortion are being tested. The GPB experiment measured $\gamma$ as we will discuss below. The second point of view of (3) is that the parameterization could describe a metric theory other than GR, and is thus more general. The parameterized post Newtonian (PPN) theory of Nordtvedt, Will and others carries this viewpoint to a high level of generality and sophistication with the use of about 9 parameters that can be tested experimentally.[@5; @6; @7] Moreover the PPN approach involves an expansion in powers of $m/r$ and $1/c$ and often provides clear intuitive understanding of physical effects, analogous to Newtonian theory.[@41] As we have noted, previous observations and experiments in the solar system and observations of pulsar systems are in agreement with GR, but all involve weak fields, even the pulsar systems. As a measure of the accuracy of such tests the Eddington parameters $\beta,\gamma$, which are predicted to be 1 by GR, are found from various observations to be $|\gamma-1|< 2.3\times10^{-5}$ and $|\beta-1| < 8\times10^{-5}$.[@5] Thus the approximate Schwarzschild metric (3) is well verified, and the Eddington parameters are very close to 1. The Lense-Thirring metric from several points of view ===================================================== The Schwarzschild metric in (1) and the approximation (3) only describe the metric exterior to a spherically symmetric non-spinning body and therefore allow us to calculate only the geodetic part of the gyro precession. The LT part of the gyro precession depends on a generalization of the Schwarzschild metric due to the spin of the source. For the earth, which is not very massive and spins slowly, the modification is quite small, making its effect on the gyro excedingly difficult to detect. The metric for the exterior of a spinning spherical body was first obtained by Lense and Thirring in 1918 using linearized GR.[@13] They worked to lowest order in the gravitational fields and velocities and obtained a metric that we may write in spherical coordinates as $$\label{4} ds^2=(1-2m/r)dt^2-(1+2m/r) d\vec{r}^2+2\left(\frac{2GJ}{r}\right)\sin^2\theta d\varphi dt,$$ where $J$ is the angular momentum of the spinning source body. A more general version of the LT metric can also be obtained using the so-called gravito-electromagnetic (GEM) approximation, which applies for weak fields and slowly moving bodies, and in which many equations are similar to those of classical electrodynamics.[@14; @15] In the GEM approximation the metric may be written as $$\label{5} ds^2=(1+2\phi)dt^2-(1-2\phi) d\vec{r}^2+2(\vec{h} \cdot d\vec{r} )dt .$$ Here $\phi$ is the Newtonian potential outside of the body and $\vec{h}$ is called the gravito-magnetic 3-vector potential, analogous to the 3-vector potential of electrodynamics; $\phi$ and $\vec{h}$ may be defined as $$\label{6} \phi(\vec{r})=-G\int \frac{\rho(\vec{r'})d^3 r'}{|\vec{r} - \vec{r'}|} , \; \vec{h}(\vec{r})=4G\int \frac{\rho(\vec{r'})\vec{v}(\vec{r}')d^3 r'}{|\vec{r} - \vec{r'}|} .$$ There is one approach to the LT metric that we believe is worth further discussion because it rests on a solid semi-empirical basis and is thus nearly independent of theory. [@14; @15] This approach depends on three well founded assumptions that are motivated by experiment and established theory. It also assumes the weak fields and low velocities appropriate for GP-B, and makes clear why the LT metric depends (to an excellent approximation) only on the Eddington parameters $\alpha =1$ and $\gamma$ and not on any independent new parameter related to the spin and gravito-magnetism. First, the metric for the exterior of a small spherical body, essentially a point mass, is given by the approximate expression (3) for the Schwarzschild geometry, which is well verified by experiment as we have discussed. Second, the metric for such a body in motion is given by a Lorentz transformation of (3) in accord with basic relativity theory. Third, due to the weakness the fields, the metric for many such small bodies or point masses is a superposition of the individual metric for each body, analogous to the superposition of potentials in Newtonian theory. We will review the logic of the derivation in some detail. We begin with the Eddington form (3) for a point mass at rest and apply a Lorentz transformation in the $x$ direction, to first order in velocity $v$, $$\label{7} t_s=t-vx , \; x_s=x-vt ,$$ and obtain the metric for a slowly moving point mass, to lowest order in $m/r$ and velocity $v$, $$\label{8} ds^2=\left(1-\alpha\frac{2m}{r}\right)dt^2-\left(1+\gamma\frac{2m}{r}\right)d\vec{r}^2+(\alpha + \gamma)\left(\frac{4m}{r}\right)(\vec{v}\cdot d\vec{r})dt.$$ The generalization to motion in any direction $\vec v$ is obvious from (8). Since the fields are assumed to be weak we superpose the fields of many such point masses just as in Newtonian theory, using the recipes $$\label{9} \frac{-GM}{r} \rightarrow-G\int \frac{\rho(\vec{r'})d^3 r'}{|\vec{r} - \vec{r'}|}=\phi(\vec{r}) , \; \frac{4GM\vec{v}}{r} \rightarrow4G\int \frac{\rho(\vec{r'})\vec{v}(\vec{r}')d^3 r'}{|\vec{r} - \vec{r'}|} =\vec{h}(\vec{r}) .$$ These are exactly the same functions that occur in (6) so we obtain $$\label{10} ds^2=(1+2\alpha \phi)dt^2-(1-2\gamma \phi) d\vec{r}^2+(\alpha + \gamma)(\vec{h} \cdot d\vec{r} )dt ,$$ which is the same as (5) but includes the Eddington parameters. Our derivation produced no new parameters in the last expression, and the effects of gravito-magnetism are parameterized by $\alpha+\gamma=1+\gamma$. Within the broader context of the PPN formalism there is another parameter that could be included in the above discussion, called $\alpha_1$, which is related to the possible existence of a preferred inertial reference frame. It would entail adding $\alpha_1/4$ to $\gamma$ in (8), but observations constrain $\alpha_1$ to be less than $10^{-4}$ so it is not relevant to GP-B and we will not include it here.[@42; @43; @6] We note that the expression (5) is obviously not limited to a spherical body; it presumes only weak fields and low velocities. From (5) the multipole corrections for a slightly non-spherical body such as the earth have also been worked out.[@15] For a nearly spherical body such as the earth the metric (10) may also be conveniently written in spherical coordinates, again to lowest order in $m/r$, as $$\label{11} ds^2=(1-2\alpha m/r)dt^2-(1+2 \gamma m/r) d\vec{r}^2+(\alpha+\gamma)\left(\frac{2GJ}{r}\right)\sin^2\theta d\varphi dt,$$ which is the LT form of the metric (4), but with Eddington parameters included. The LT metric has been obtained in various other ways. For example it can be derived by an expansion from the exact Kerr metric for a spinning black hole.[@16] The equation of motion for the spin =================================== The precession of a gyro in GR and other metric theories is an extraordinary effect: in Newtonian theory there is no analog. For example a gyro in a uniform Newtonian force field does not precess. This is one of the reasons that the GPB experiment is particularly interesting to theorists. Moreover the precession is a so-called Machian effect: the presence of the rotating earth has an effect on determining the local inertial frame, in sharp contrast to Newton’s absolute space. In the context of GR and similar theories the behavior of the gyro spin has been studied in a number of ways, but we will focus on only two of them and briefly mention a third.[@17; @18; @19; @20; @21; @22; @23; @24; @25; @44; @42] The conclusion is that the spin four-vector $S^{\mu}$ is parallel displaced along its trajectory in spacetime, which gives a simple equation for the gyro precession. This conclusion is independent of theoretical details and not limited to GR. Our first argument is based on simplicity and general covariance, and makes the equation of motion intuitively obvious. Let us first consider a general affine space, in which there is a law of parallel displacement using coefficients of affine connection. The space need not even have a metric. In such a space there is only one privileged or special curve, a geodesic; the geodesic may be defined as that curve for which the tangent vector (or four-velocity) $u^{\mu}=dx^{\mu}/ds=\dot{x}^{\mu}$ is displaced parallel to itself along the curve, or that the curve is Òparallel to itself.Ó This implies that the covariant derivative of $u^{\mu}$ along the curve is zero, or $$\label{12} Du^{\mu}/Ds=du^{\mu}/ds+\Gamma^{\mu}_{\omega \sigma} u^{\omega}u^{\sigma}=0 \; \; or \; \; \ddot{x}+\Gamma^{\mu}_{\omega \sigma} \dot{x}^{\omega} \dot{x}^{\sigma}=0.$$ In the most general case the arc length may be replaced by any invariant parameter.[@45] We can use similar reasoning to heuristically motivate an equation for the gyro spin $S^{\mu}$. First we note that in the rest frame of the gyro the four-velocity and spin vector are $$\label{13} u^{\mu}=(1,0,0,0) \; , \; S^{\mu}=(0,\vec{S}) \; , \; S^{\mu}u_{\mu}=0 , \; (gyro\; rest\; frame) \; ,$$ and since $S^{\mu}u_{\mu}=0$ is a covariant expression it holds in any frame. It is a well-known property of parallel displacement that that if two vectors are parallel displaced together then their inner product does not change; it thus becomes natural to demand that the spin $S^{\mu}$ be parallel displaced along the geodesic path of the gyro, along with the four-vector velocity. Then the spin equation of motion and the orthogonality condition are $$\label{14} DS^{\mu}/Ds=dS^{\mu}/ds+\Gamma^{\mu}_{\omega \sigma} u^{\omega}S^{\sigma}=0 \; , S^{\mu}u_{\mu}=0\; .$$ We stress that parallel displacement is a sufficient but not a necessary condition that the inner product $S^{\mu}u_{\mu}$ remains zero along the trajectory. Note also that (14) is a rather general result and does not depend on any particular theory of gravity, but of course its application to a particular problem will use affine connections which do depend on the specific theory. We can also phrase the argument in terms of general principles. In sec. 2 we mentioned the weak equivalence principle (EP) or universality of free fall, which states that the trajectories of test bodies in a gravitational field are independent of their masses and various internal properties. Thus in a freely falling lab or reference frame test bodies behave as if there were no gravitational field present. The phenomenon has become familiar in television broadcasts from orbiting spacecraft. Conversely, in an accelerated lab or reference frame test bodies behave as if there were a gravitational field present. Einstein proposed an extended version of the equivalence principle, called the Einstein equivalence principle (EEP) that assumes “complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system." The EEP includes nongravitational phenomena, such as electromagnetism, as well as gravitational phenomena. It leads to a Òprinciple of general covarianceÓ that has proven to be very powerful in formulating nongravitational physical laws, such as Maxwellian electrodynamics in the presence of a gravitational field. To use the principle of general covariance one writes an equation that is known to be correct in the absence of gravity, and takes it to be true for a freely falling reference frame in which there are no effects of gravity, according to the EEP. Then if the equation is expressed in generally covariant form it must also be correct in any reference frame.[@17; @12] It is easy to apply the ideas of the EEP and general covariance to the equation of motion of the gyro and to its spin vector, as elucidated clearly by Weinberg.[@17] In a space with no gravitational field it is obvious that both the four-vector velocity $u^{\mu}$ of the gyro and the spin vector $S^{\mu}$ should be constant, or $du^{\mu}/ds=0$ and $dS^{\mu} / ds=0$. According to the principle of general covariance the correct generalization of these equations in a gravitational field is obtained simply by replacing the Lorentz metric by the general Riemannian metric and the ordinary derivative by a covariant derivative, making the equations generally covariant. The result is (12) and (14). Our second argument was given by Papapetrou and is much more physical.[@18] His argument does not depend explicitly on the field equations of GR, but on the conservation of energy momentum, expressed as the tensor equation $(T^{\mu \nu})_{;\nu}=0$; the conservation equation does of course follow from the field equations of GR. Papapetrou analyzed a small ball of material, making few assumptions about its internal structure, and derived the correct geodesic equation of motion as a first approximation by ignoring various three-space moments of the ball. (He called it the monopole approximation.) He then took account of internal structure and motion of the material in the ball to lowest order in the size of the ball, including first moments in three-space, and thereby obtained an equation of motion for its second rank anti-symmetric spin tensor $\tilde{S}^{\mu \nu}$; that tensor is defined as $$\label{15} \tilde{S}^{\mu \nu}=\int dV(\delta x^{\mu} u^{\nu}-\delta x^{\nu} u^{\mu})\rho \;,$$ where the integral is over the three-space volume of the ball, $\delta x^{\mu}$ is the position in the ball relative to its center of mass, $u^{\nu}$ is the four-velocity of the ball material, and $\rho$ is its density. The equation of motion for the spin tensor that he obtained is $$\label{16} D\tilde{S}^{\mu \nu}/Ds+u_{\rho} (u^{\nu} D\tilde{S}^{\rho \mu}/Ds-u^{\mu} D\tilde{S}^{\rho \nu}/Ds)=0 \; .$$ We still need to relate the spin vector $S^{\mu}$ to the spin tensor $\tilde{S}^{\mu \nu}$ and also relate the equations that they obey, that is (14) and (16). The spin vector has three independent components and the tensor has six, but only three of them determine the angular position of the gyro. We can find the desired relation by using the low velocity limit as a guide. The tensor $\tilde{S}^{\mu \nu}$ is antisymmetric, the time displacement on a spatial surface is $\delta x^0=0$, and $u^0 \approx c =1$. Hence $\tilde{S}^{\mu \nu}$ is approximately $$\label{17} \tilde{S}^{00}=0 \;, \tilde{S}^{0j}=\int dV \rho \delta x^j = 0\; , \tilde{S}^{ij}=\int dV \rho( \delta x^i v^j - \delta x^j v^i) .$$ The second relation in (17) follows since $\delta x^j$ is measured from the center of mass of the body. Thus the spatial part of the spin tensor is the familiar angular momentum tensor of three-dimensional mechanics. The three-vector angular momentum is related to it by the well-known equation $$\label{18} S^i=(1/2)\epsilon_{ijk} \tilde{S}^{jk} .$$ What we now need is a covariant generalization of (18) to relate $S^{\mu}$ to $\tilde{S}^{\mu \nu}$. A moment’s thought provides an answer, which is $$\label{19} S^{\mu}=(-1/2)u_\rho {e^{\rho \mu}}_{\alpha \beta} \tilde{S}^{\alpha \beta} , \; e_{\alpha \beta \gamma \delta}\equiv \sqrt{-g} \epsilon_{\alpha \beta \gamma \delta} ,$$ where $\epsilon_{\alpha \beta \gamma \delta}$ is the usual Levi-Cevita alternating symbol and $e_{\alpha \beta \gamma \delta}$ is the Levi-Cevita tensor. [@24; @46] Equation (19) is clearly a generally covariant expression. Moreover it is obvious from the antisymmetry of the Levi-Cevita tensor that $$\label{20} S^{\mu}u_{\mu}=0 ,$$ which is the same as the orthogonality relation in (14). Finally we can obtain the parallel displacement relation (14) from Papapetrou’s equation (16). To do this we first note that the Levi-Cevita tensor has a zero covariant derivative, as does the four-velocity vector $u^{\mu}$ along a geodesic.[@46] Then from (16) the covariant derivative of the vector $S^{\mu}$ is $$\begin{aligned} \label{21} DS^{\mu}/Ds=(-1/2)u_\rho {e^{\rho \mu}}_{\alpha \beta} D\tilde{S}^{\alpha \beta}/Ds \end{aligned}$$ $$\label{21a} =(-1/2)u_\rho {e^{\rho \mu}}_{\alpha \beta} (u_\sigma u^\beta D\tilde{S}^{\alpha \sigma}/Ds- u_\sigma u^\alpha D\tilde{S}^{\beta \sigma}/Ds)=0 .$$ Thus the Papapetrou analysis leads to the same equation we obtained previously; the parallel displaced spin vector equation (14) obtained from general principles also follows from a more detailed “nuts and bolts" analysis The spin equation (14) implies an important fact about the gyro precession since it is homogeneous in the spin $S^\mu$. The vector $S^\mu$ and tensor $\tilde{S}^{\mu \nu}$ clearly depend on the rotation rate of the gyro, which is clear from the definition in (15). But since the spin equation is homogeneous the angular precession is independent of the magnitude of $S^\mu$, so the gyro spin velocity is, in principle, irrelevant and has no effect on the precession; $S^\mu$ merely serves to define a direction in space. Of course in the real world of experiments the spin velocity may be very important in the accurate measurement of the precession. Papapetrou noted another fact of interest, that a spinning body does not follow a geodesic exactly, as in (12), but deviates a little due to the interaction between spin, orbital angular momentum and curvature. His equation giving the modified geodesic is the following $$\label{22} \frac{D}{Ds}(mu^\alpha + u_\beta \frac{D\tilde S^{\alpha \beta}}{Ds}) + \frac{1}{2} \tilde S^{\mu \nu} u^\sigma {R^\alpha}_{\nu \sigma \mu} =0 .$$ As might be expected the extra terms in (22) are far too small to be relevant for the GP-B experiment, or any solar system experiment envisioned at present. In reference [@24] Will notes that there is some disagreement about the result (22), and gives further references; he also estimates such effects to be well below $10^{-20} g$. However it is clear that in principle the motion of a body depends on its spin and internal structure, so the EP or universality of free fall cannot be an exact principle but only an extraordinarily accurate approximation. That is, GR transcends the EP; to quote Nordtvedt “Principles in physics are for when you have no theory." Furthermore spin effects such as displayed in (22) may be large for some astronomical systems, such as black holes or neutron stars in close orbit. The gravitational radiation emitted by such bodies during their final inspiral may allow observation of the spin effects, as indicated by numerical GR simulations.[@5] We mention in passing one other interesting approach to the LT gyro spin theory. Murphy, Nordtvedt and Turyshev have used a PPN approach to derive the LT gyro precession, in agreement with the one we give here. The virtue of their quasi-Newtonian derivation is that it shows how the LT gyro precession results from the gravito-magnetic acceleration of each moving point mass in the rotating gyro.[@42] They also include the $\alpha_1$ parameter as mentioned previously, which is known to be small from previous observations and has a negligible contribution. One extension of GR theory, that in principle could affect the spin equation, involves the concept of torsion; in GR the affine connections are symmetric in the lower indices, but if they are allowed to have an antisymmetric part the result is a more general theory than GR, called Einstein-Cartan theory, which involves the concept of torsion.[@47; @48] Many theorists believe torsion should be included in gravity theory, for example to accommodate the spin of particles, although no experiments indicate such a need.[@49] Moreover other authors have developed the quantum theory of spin 1/2 particles interacting with gravity without the use of torsion, so torsion appears to be neither observed nor needed for theoretical consistency.[@50; @51] Of course that does not prove it does not exist in nature. In summary, the general spin equation (14) appears to be well founded on both mathematical and physical grounds. Solving the spin equation for GP-B ================================== From the LT metric in (10) and the general spin equation (14) it is straightforward although slightly tedious to calculate the precession of the GP-B gyro in its polar orbit. The gravitational field of the earth is weak so that the expansion of the metric to order $m/r$ is adequate, and the rotational velocity of the earth and the orbital velocity of the satellite are small, so we need only work to first order in $v$. Also we will assume a perfectly circular polar orbit with the gyro spin in the orbital plane. See fig.1a and also fig.1 of the overview paper by Everitt in this volume. We will briefly sketch the calculation following references \[15\] and \[17\]. The first step of the calculation is to obtain the affine connections from the LT metric and substitute them into the spin equations (14). This yields the following equation for the space part of the spin vector, written in index notation, $$\label{23} \dot{S}^i=\left[ \gamma V^i (\phi_{,k} S^k ) + \gamma S^i (\phi_{,k }V^k ) - (\alpha + \gamma)\phi_{,i} (S^k V^k)\right] + (1/4)(\alpha + \gamma)(h_{i,k} - h_{k,i})S^k.$$ Here $V^k$ is the 3-vector velocity of the gyro. In three-vector notation (23) is $$\label{24} \dot {\vec S}=\left[ \gamma \vec V (\nabla{\phi} \cdot \vec S ) + \gamma \vec S (\nabla{\phi} \cdot \vec V) - (\alpha + \gamma) \nabla \phi (\vec S \cdot \vec V)\right] + (1/4)(\alpha + \gamma)(\nabla \times \vec h ) \times \vec S.$$ The terms that contain the Newtonian potential $\phi$ contribute to the geodetic precession, and those that contain the gravito-magnetic potential $\vec h$ contribute to the LT precession. Next we split the square bracket in (24) containing the potential $\phi$ into two parts, anti-symmetric and symmetric in the pair of vectors $\vec V$ and $\nabla \phi$, and write it as $$\begin{aligned} \label{25} (\gamma + \alpha /2)\left[ \vec V (\nabla{\phi} \cdot \vec S)- \nabla \phi (\vec S \cdot \vec V )\right ] + \left \{ \gamma \vec S (\nabla \phi \cdot \vec V) -\alpha /2 [ \vec V (\nabla \phi \cdot \vec S ) + \nabla \phi (\vec S \cdot \vec V) ] \right \}\end{aligned}$$ $$\label{25a} =(\gamma + \alpha /2)(\nabla \phi \times \vec V ) \times \vec S + \left \{ \gamma \vec S (\nabla \phi \cdot \vec V) -\alpha /2 [ \vec V (\nabla \phi \cdot \vec S ) + \nabla \phi (\vec S \cdot \vec V) ] \right \}$$ The curly bracket in (25), which is symmetric in $\vec V$ and $\nabla \phi $, averages to zero over a circular orbit. More generally, in the Newtonian approximation $\dot{ \vec V}=-\nabla \phi $, and also the change in the spin $\vec S$ is extremely slow; these two facts allow us to express the curly bracket as a time derivative, so it must average to zero over general orbits. We will henceforth ignore it. Then (24) simplifies to $$\label{26} \dot {\vec S}=\left[ (\gamma + \alpha /2)(\nabla \phi \times \vec V) + (1/4)(\alpha + \gamma )\nabla \times \vec h \right] \times \vec S.$$ To make (26) beautiful we define two vector fields, a geodetic vector field a gravito-magnetic vector field, in terms of the Newtonian potential and the gravito-magnetic vector potential, as $$\label{27} \vec \Omega_G =(\gamma + \alpha /2)(\nabla \phi \times \vec V) \; , \; \vec \Omega_{LT }=(1/4)(\gamma + \alpha)\nabla \times \vec h .$$ Both fields are independent of time for the spherical spinning earth. Then (26) becomes $$\label{28} \dot {\vec S}=\left( \vec \Omega_G + \vec \Omega_{LT}\right) \times \vec S,$$ which we recognize as the classical equation for a precessing gyro. Since that problem is quite well known our problem is nearly solved. For the GP-B gyro the precession is extremely slow, so the spin does not change appreciably over the course of many orbits, and we may write the change in $\vec S$ in time $\Delta t$ as $$\label{29} \Delta \vec S=\Delta \vec S_G + \Delta \vec S_{LT}= (\vec \Omega_G \times \vec S) \Delta t +(\vec \Omega_{LT} \times \vec S) \Delta t ,$$ with $\vec S$ treated as a constant. The last expression in (29) defines the geodetic and LT drifts, which are linear in time. Consider the geodetic term of (29) first, by far the larger part. For a circular orbit the gravitational force and the velocity are perpendicular, and the geodetic field is thus perpendicular to the orbit plane. The geodetic vector and its magnitude are $$\label{30} \vec \Omega_G =(\gamma + \alpha /2)\left (\frac{GM}{r^2} \right) (\hat r \times \vec V) \; , \Omega_G =(\gamma + \alpha /2)\left (\frac{GMV}{r^2} \right) .$$ The various vector directions are shown in fig.1. The geodetic precession is in the plane of the orbit. The LT precession depends on the gravito-magnetic field $\vec \Omega_{LT}$, which varies with position in the orbit. The gravito-magnetic vector potential $\vec h$ of the spinning earth can be calculated in the same way as the vector potential of a spinning ball of charge in electrodynamics. The results for $\vec h$ and $\vec \Omega_{LT}$ are $$\label{31} \vec h =\left ( \frac{2G}{r^3} \right )(\vec r \times \vec J )\;, \; \vec \Omega_{LT} = (1/2)(\gamma + \alpha)G \left ( \frac{ \vec J}{r^3} - \frac {3 \vec r}{r^5} (\vec r \cdot \vec J) \right ) ,$$ where $\vec J$ is the angular momentum of the earth. The gravito-magnetic field has exactly the same shape as a magnetic dipole field, as might be expected. Since the gyro precesses so slowly we need only average $\vec \Omega_{LT}$ over an orbit to obtain the LT precession, $$\label{32} \left\langle \vec \Omega_{LT} \right \rangle = (1/2)(\gamma + \alpha) \frac{G \vec J}{2 r^3} ,$$and the magnitude of this is the LT precession. The precession is perpendicular to the orbit plane as shown in fig.1. Our final results for the precessions are given in (29) to (32) with directions shown in fig.1. It is important to emphasize that for the polar orbit the geodetic and LT precessions are perpendicular; the LT precession is very much smaller than the geodetic and would not be measurable if the two were not accurately perpendicular. We will not discuss small and subtle corrections to the basic precessions, such as the effects of the earth multipole moments, the presence of the moon and the sun, variations in the spacecraft altitude and orbital orientation, etc. These are covered in the references and the other papers in this volume.[@44; @15] Summary and further comments ============================ This work has focused on the bases for the theoretical predictions of the gyro precession in the GP-B experiment. We have at most only mentioned some of the interesting subtleties and small corrections to the predictions, such as the effect of the sun and the quadrupole moment of the earth, which are covered in the references. We have not found any of the suggestions that the standard results are substantially wrong convincing enough to discuss them; this seems well justified by the experimental results. Some small but possibly interesting and well-founded modifications to the basic predictions might concern a scalar field component added to gravity theory or a torsion related addition to the equation of motion for the spin. The GP-B results indicate that neither of these is presently needed, at least within the accuracy of the experiment. Acknowledgements ================ The GP-B theory group has provided interesting comments and criticisms on the theory behind the experiment. In particular we thank Robert Wagoner, Paul Worden, and Francis Everitt for patient discussions of the experimental situation, the PPN formalism, and references to the literature. We thank Kenneth Nordtvedt for correspondence and his notes regarding the equivalence principle. In particular his comment that “Principles in physics are for when you have no theory" is particularly relevant to our discussion in sec.5. Finally, we thank Cliff Will for technical comments and updates on the references. [99]{} C. W. F. Everitt et. al. Phys. Rev. Lett. 106, 221101 (2011). An online overview of the experiment is at [$<$http://einstein.stanford.edu$>$]{}. C. W. F. Everitt, see paper $1$ in this CQG volume. A. S. Silbergleit, see paper $2$ in this CQG volume. C. M. Will, *Theory and experiment in gravitational physics, revised edition* (Cambridge University Press, Cambridge UK, 1993); for updated material, including parameter values, see $<$[http://arxiv.org/pdf/1403.7377v1.pdf]{}$>$. For a quick informal overview of the experimental situation see the Wikipedia article $<$[http://en.wikipedia.org/wiki/Tests\_of\_general\_relativity]{}$>$ An early journal reference on the PPN formalism for testing gravity theory is C. M. Will and K. Nordtvedt, Astrophys. J., 177, 757, (1972). A. Einstein, “The foundations of the general theory of relativity" in *The principle of relativity*, Dover edition, reprinted from Annalen der Physik, 1916. C. W. Misner, K. S. Thorne, and J. A. Wheeler, *Gravitation* (W. H. Freeman, San Francisco, 1973). See chapter 1 on basic ideas of geometrodynamics. S. Weinberg, *Gravitation and Cosmology* (Wiley and Sons, New York, 1972). See chapters 1.1 and 1.2. See chapter 3 of reference \[5\] R. J. Adler, M. Bazin, and M. M. Schiffer, *Introduction to general relativity, 2nd ed.* (McGraw Hill, New York, 1965). See the introduction. J. Lense and H. Thirring, Phys. Z. [**19**]{}:156 (1918). R. J. Adler, GRG [**31**]{}, (1999). R. J. Adler and A. S. Silbergleit, Int. J. Th. Phys. [**39**]{}, 1287 (2000). See chapter 7.7 of reference \[12\]. See chapters 5.1 and 9.6 of reference \[10\]. Chapter 9.6 contains a sign error that does not affect the answer. A. Papapetrou, Proc. R. Soc. Lond. A 1951, 209, (doi:10.1098/rspa.1051.0200) See chapter 40.7 of reference \[9\]. Note that MTW use different PPN parameters, as discussed in reference \[6\]. K. S. Thorne on gravito-magnetism in J. D. Fairbank, B. S. Deaver Jr., C. W. F. Everitt, and P. F. Michelson, *Near Zero* (W. H. Freeman, New York, 1988). L. I. Schiff, Am. J. Phys. 28, 340, 1960. L. I. Schiff, ÒComparison of theory and observation in general relativity,Ó in *Relativity Theory and astrophysics, I Relativity and Cosmology*, Jurgen Ehlers editor, (American Mathematical Society, Providence RI, 1976). H. C. Ohanian and R. Ruffini, *Gravitation and Spacetime*, (Norton, New York, 1976). See chapter 7.8. See chapter 6.5 of reference \[5\]. See chapter 9.6 of reference \[10\]. See J. Breakwell on corrections to the precession in reference \[20\]. See chapters 2 and 8 of reference \[5\]. Microscope website is $<$[http://sci2.esa.int/Microscope/microscope\_ao\_final.pdf]{}$>$. For a discussion of the atomic beam experiment see S. Dimopoulos, P. W. Graham, J. M. Hogan, and M. A. Kasevich, Phys. Rev. Lett. 98, 111102 (2007). For a discussion of a possible future highly accurate space test of the equivalence principle (STEP) see the article by J. Mester in $<$http://einstein.stanford.edu$>$. P. Jordan, Schwerkraft und Weltall, Vieweg (Braunschweig, 1955), Projective Relativity, is the first paper on Jordan-Brans-Dicke (JBD) theories. C.H. Brans and R.H. Dicke, Phys. Rev. [**124**]{}: 925, 1961. For a discussion of theories with more than one scalar field see R. Wagoner, Phys. Rev. D1(812): 3209, 2004. For a textbook discussion of the Brans-Dicke version of scalar tensor gravity see chapter 11.5 of reference \[12\]. T. Damour and A. M. Polyakov, ÒString theory and gravity,Ó a book in the ArXiv, $<$[http://arxiv.org/abs/gr-qc/9411069v1]{}$>$. See chapter 5.3 of reference \[5\]. M. Kramer et al in $<$http://arxiv.org/abs/astro-ph/0409379$>$; J. McClintock, F. Narayan, J. Steiner in $<$http://arxiv.org/pdf/1303.1583.pdf$>$ . K. Schwarzschild, Sitzber. Preuss. Acad. Wissen. Berlin, p. 189, 1916. See chapter 6 of reference \[12\]. A. S. Eddington, *The mathematical theory of relativity* (Cambridge University Press, New York, 1988) See p. 105 on the parameters. See chapter 9 of reference \[10\]. T. W. Murphy Jr., K. Nordtvedt, and S. G. Turyshev, arXiv:gr-qc/0702028v1 2007. See chapter 40.7 of reference \[9\], which uses a different set of PPN parameters. G. E. Pugh, WSEG research memorandum number 11 (Weapons system evaluation group), Pentagon, Washington D. C., November 12, 1969. See chapter 2.3 of reference \[12\]. See chapter 3.5 of reference \[12\]. H. Kleinert, \[gr-qc\] arxiv:1005,1460 (2010). Kleinert discusses an equivalence between curvature and torsion. A. Trautman, ÒEinstein-Cartan Theory.Ó In Encyclopedia of Mathematical Physics, edited by J.-P. Francoise, G.L. Naber and S. T. Tsou, (Elsevier, Oxford UK, 2006) A. Randono, \[gr-qc\] arxiv:1010.5822 (2010). This is a review of modern Einstein-Cartan type theories. R. J. Adler, P. Chen, and E. Varani, Phys. Rev. D, 85, 025026, 2012. H. R. Pagels, Ann. Phys. 31, 64, 1965. ![(A) The orbital and spin orientation vectors. (B) Vectors associated with geodetic precession. (C) Vectors associated with LT precession[]{data-label="Fig.1"}](Fig1.pdf){width="4in"}

--- bibliography: - 'mice.bib' --- 00a-Top-matter/00a-Top-matter 99-Styles/MICE-defs 00b-Abstract/00b-Abstract

--- abstract: 'The advent of modern computers has added an increased emphasis on channeling computational power and statistical methods into digital humanities. Including increased statistical rigor in history poses unique challenges due to the inherent uncertainties of word-of-mouth and poorly recorded data. African genealogies form an important such example, both in terms of individual ancestries and broader historical context in the absence of written records. Our project aims to bridge the lack of accurate maps of Africa during the trans-Atlantic slave trade with the personalized question of where *within* Africa an individual slave may have hailed. We approach this question with a two part mathematical model informed by two primary sets of data. We begin with a conflict intensity surface which can generate capture locations of theoretical slaves, and accompany this with a Markov decision process which models the transport of these slaves through existing cities to the coastal areas. Ultimately, we can use this two-step approach of providing capture locations to a historical trade network in a simulative fashion to generate and visualize the conditional probability of a slave coming from a certain spatial region given they were sold at a certain port. This is a data-driven visual answer to the research question of where the slaves departing these ports originated. [Keywords: Kriging; Markov decision process; Gaussian process; Kernel Density Estimation; Oyo; African Diaspora; Translatlantic slave trade; digital humanities;]{}' bibliography: - 'Oyobibliography.bib' --- [ **Mapping the uncertainty of 19th century West African slave origins using a Markov decision process model.** ]{} [ **Zachary Mullen and Ashton Wiens and Eric Vance and Henry Lovejoy** ]{} [****]{} Introduction ============ The study of colonial empires is dominated by incomplete data and missing records. Despite this difficulty, there is significant interest in tracing the forced diaspora of African peoples via slavery [@lovejoywebsite]. Many black organizations in the modern Americas can trace their origins to the cultural unity required in overcoming the struggles of their subjugation [@chambers2012]. To date, much of the work in understanding this cultural genesis has focused on genealogy and literary interpretation, but the forced relocation of a predominately illiterate population leads to significant shortcomings in availability of written history. As a result, modern understanding of the exodus often lacks comprehensive regional descriptions of the socio-political climate within Africa that enabled the internal slave trade then exploited by colonial European powers. The growing field of digital humanities attempts to expand upon logocentric analyses of African history with modern methods in text mining, linguistic analysis, and machine learning [@lovejoytalk2017]. These GIS and geospatial methods [@knowles2008placing] have been employed heavily in World War II and Holocaust studies [@knowles2014geographies]. One West African slave-trading state was the Oyo empire, which peaked in the late 18th century, culminating in a rapid decline over a series of crises and invasions around the 1820s. During these conflicts, slavers regularly departed from the coast of the Oyo empire and bordering West African states, and many of these voyages are well documented by the slave traders. In addition to ship logs, a handful anecdotal accounts of individual slave movements from the collapse of the Oyo empire have been reconstructed from written and oral records [@kelley2016origins; @kelley2016voyage] (maybe cite slavebiographies.org). Recent work has emphasized integrating the collapse of the Oyo empire into the digital humanities, including the creation of detailed maps on the shifting borders of the collapsing empire [@lovejoy2013redrawing]. One current question regarding the collapse of the Oyo empire is exploring the logistics and detailed movements of the internal slave trade and how those systems actually filled the ships leaving the West African coast. In many cases, the state-controlled ships have accurate passenger counts, ports of arrival, and ports of origin, and modern genealogical explorations can often trace ancestries to those specific ships. However, little historical evidence explicitly connects the passenger logs - where available - and the movements of the ships to the politics of inland Africa at the time. Questions of ancestry often dead-end at these transit points despite the work and literature documenting the internal conflicts during the Oyo collapse. We attempt to expand on the understanding of the internal slave trade of the Oyo Empire by synthesizing spatial mathematical models onto conflict maps and conjoining them with models for decision processes governing inland slave movements. The first question is one of using discrete events such as recorded dates of battles or towns destroyed to create a model for the location and intensities of conflict. We use spatial smoothing on recorded conflict events to create a continuous density map of the warring regions, augmenting the existing maps of shifting borders by an accompanying picture of which cities and regions in the empire were most likely locations for slavers to capture individuals. We couple this map of conflict regions with a Markov decision process for the Oyo region’s internal slave trading network. We view adjacent or nearby cities as a connected network, and the Markov decision process attempts to ask: “what are the likely movement paths” of slaves captured until their eventual sales and departures via ship or into the trans-Sahara region. The goal is to provide a functional and descriptive model for the most likely inland origin locations of slaves given a known year and port of origin. As a result, the conflict map and slaver decision process models combine to answer this: we use the conflict map to generate annual maps of likely locations slaves were captured, then pass them into the trading network to determine where slaves captured at those locations would be most likely to leave the region. The resulting counts allow for the inverse question as well; e.g. “for all slaves leaving Lagos in 1824, from which conflict regions did they originate?” This allows our analysis to bridge the process-focused models that stay true to historical narrative with the ends-oriented goals of a genealogist, who may wish to reverse-engineer the historical origin stories. We hope for our exploration to be applicable and available to historians in both other regions of the African diaspora and to studies of other instances of forced transit, such as the Holocaust or the relocation of American indigenous peoples. Data {#S:Data} ==== **Section in Progress pending collaboration: Describe the conflict data - what historical accounts were used?** We have several geopolitical data sets, describing the trade routes and conflicts that were we think were present during the collapse of the kingdom of Oyo from approximately 1816-1836 near modern day Togo, Benin, and western Nigeria. The data are shown in \[f:1\]. For each year, we also have approximations of the total number of slaves departing the region as a whole and specific trading ports. The data were collected in \[Mapping the Collapse of Oyo\]. The conflict data is a table where each row describes a 2D spatial location where a conflict occurred. There are variables describing the start year and end year, as well as the intensity of the conflict. The intensity was encoded as a categorical variable with four levels: 0 means a city is founded, 1 means a city is rebuilt, 5 means a city is attacked, and 10 means a city is destroyed. We did not use the founded/rebuilt city data. Similarly, we have a list of cities with spatial coordinates and the years the city existed (dependent on being destroyed or rebuilt). To infer the trade network among these places, we relied on the map \[fig:1816TradeMap\]. This map has been informed by both available historical records from the time (**CITATION MISSING**) and geographic ease of transit between cities. This representation adds a layer of detail and geopolitical information on top of those published in prior works, such as [@lovejoy2013redrawing]. We encoded the relationships (edges) between the nodes of this graph into an adjacency matrix, describing which cities are connected. An adjacency matrix $A$ for a set of locations (nodes) $s_1, \cdots, s_n$ is of dimension $n \times n$. An entry $A_{ij}$ is nonzero (usually 1) if there is a connection starting at $s_i$ and ending at $s_j$. This formulation describes a directed graphical structure. If the edges are undirected, then $A_{ij} = A_{ji}$ and so $A$ is symmetric. We use this adjacency matrix to construct the probability transition matrix needed in the Markov Decision Process, described in Section \[SS:MDP\]. The third data set we have is the port total data: for each year, the total number of slaves leaving each port was estimated using digitally transcribed hand-written ship logs. Some of the estimates are assigned to an unknown port. This data was not used in formulation of any models we develop in this paper, but we used it as validation data to tune parameters in the model. Finally, we have shapefile data with prominent geographical features that existed in the region during the historical period. In particular, we include bodies of water in plots which are relevant to identifying the boundaries of the various states. **The data was downloaded from ....**. Several bodies of water which were created since the historical period were removed from the data set. ![Map of Trade in Oyo, 1816[]{data-label="fig:1816TradeMap"}](figures/1816TradeMap.png){width="0.95\linewidth"} Model ===== Mapping Conflicts {#SS:krig} ----------------- The historical narrative surrounding the fall of the Oyo empire is one of borders collapsing inwards from the independence of Dahomey and lost conflicts to Ilorin and Ijebu. While the borders of the resulting countries are relatively well known, the question of slave origins requires unpacking the conflicts themselves to determine the regions within the greater Oyo area most impacted by each conflict. The data available provides conflicts and a measure for their intensity, with battles marked as less intense than the complete destruction of towns. However, a conflict between nations does not unfold over just the sites of the major battles and events: armies mobilize, raid, and occupy villages throughout contested regions. To account for this behavior not having explicit historical accounts, we decided to create a continuous spatial map of the conflict borders given the discrete time and place data available. ### Gaussian Process for Origin Locations We model our conflicts as arising from a spatially correlated Gaussian Process with an underlying M’tern covariance. Notationally, if we have conflict intensity measures $\boldsymbol{Y}$ observed at 2-D spatial locations $\boldsymbol{S}=\{s_1, s_2, \dots s_n\}$, the Gaussian process considers $\boldsymbol{Y}$ to be a single draw from a multivariate normal on $\mathbb{R}^n$. This corresponds to a log-density of $$f(\boldsymbol{Y}) \propto -\log\det\left(\Sigma+\tau^2I\right)-\boldsymbol{Y}^T \left(\Sigma+\tau^2I\right)^{-1} \boldsymbol{Y}$$ where $\Sigma_{i,j}$ is given by the Mátern covariance: $ \Sigma_{i,j}=\sigma^2 k(s_i,s_j)+\mathbbm{1}_{\{i=j\}}(\tau^2)$ for $k(s_i,s_j)=\frac{2^{1-\nu}}{\Gamma(\nu)} \left(a\|s_1-s_2\|\right)^\nu K_\nu(a\|s_1-s_2\|)$ with $K_\nu$ a modified Bessel function of the second kind of order $\nu$ and $\|\cdot\|$ denoting Euclidean distance. The formal kriging estimator fills in a map of the Oyo region at chosen resolution by taking each desired locations $s_0$ on the fine grid and computing $$\hat{\boldsymbol{Y}}(s_0)=\sigma^2 k(s_0,\vec{s}) \left(\Sigma+\tau^2I\right)^{-1} \boldsymbol{Y}$$ ### Alternatives Considered There are a variety of alternative mathematical options available to bridge a set of discrete spatial observations (sites of conflict) into a smoother continuous map. These include smoothing over observed conflicts onto other locations via splines or the above kriging estimators, treating conflicts as draws from an underlying density and using a kernel density estimator (KDE) to recover that density, or viewing conflict sites as actualizations of an inhomogeneous Poisson point process and estimating the associated intensity function. Of these options, we chose to use a classical Kriging estimator of an underlying Gaussian process for a few reasons. These include: - A typical Kriging formulation views the data as part of a demeaned autocorrelated process, where the assumption of zero mean rapidly pushes the surface to zero-valued when far from observations [@cressie1992statistics]. This corresponds to the idea that the conflicts and attacked towns themselves were the predominant sources of slaves at the time. - The parameters available to Kriging covariance models are both flexible and can be interpreted in the units of the data. Specifically, the variance parameter (or sill) is a scaling of the relative importance of minor/major conflicts and the range parameter measures the distances from observed conflicts at which the overall region of conflict exists. - Where a classical kernel density estimate is symmetric due to its equal weighting of observations-as-kernels, the choice of surface smoothness parameter in a correlated Gaussian Process allows for ridge-like structures that approximate the shifting borders of a conflict region. Figure \[fig:krigmodel\] shows the flexibility of the Matérn in capturing a shifting border of conflict and contrasts against a kernel density estimate. The leftmost plot shows a kernel density estimate, where the greater *count* of conflicts in the northeastern area generates a corresponding increased conflict intensity. The kriging maps on the other hand show the shapes of the conflicts: the parameters $\nu$ and $a$ have some interaction and different combinations can lead to very similar shapes of conflict borders, with differences largely captures in the magnitude of the range parameters. A larger range - or smaller $a$ - leads to maps with a larger region of uncertainty and non-zero conflict, as shown in the breadth of the yellow region in the middle map compared to the right-most map. ![Left to Right: 1825 conflict map via a kernel density estimate with $h=2$; krig with $\nu=.5$ and $a=5/2$; krig with $\nu=3$ and $a=6$[]{data-label="fig:krigmodel"}](figures/1825conflictkde.pdf "fig:"){width="0.3\linewidth"} ![Left to Right: 1825 conflict map via a kernel density estimate with $h=2$; krig with $\nu=.5$ and $a=5/2$; krig with $\nu=3$ and $a=6$[]{data-label="fig:krigmodel"}](figures/1825conflictnotsmooth.pdf "fig:"){width="0.3\linewidth"} ![Left to Right: 1825 conflict map via a kernel density estimate with $h=2$; krig with $\nu=.5$ and $a=5/2$; krig with $\nu=3$ and $a=6$[]{data-label="fig:krigmodel"}](figures/1825conflictsmooth.pdf "fig:"){width="0.3\linewidth"} ### Estimation Rather than provide for fully unconstrained estimation of the 4 spatial covariance parameters, we chose to tune the spatial smoothness according to the heuristics of the underlying problem. In general, we want to force ranges to be small enough that regions of high correlation are trapped within the same topographical and population areas as the observed conflicts: we found that a $10km$ range accomplished this. Smoothness was similarly fixed at a 4.5 times differentiable Matern. While this is fairly smooth in the context of dense spatial data, our observations are quite sparse, and higher smoothness helps ensure the ridge-like structure that mimics a shifting border to reflect the ebb and flow of borders in a region of conventional warfare. Lower smoothness would enforce a rapid decay to zero away from observations, and would push our model closer to one found from kernel density estimation, as it would result in conflicts being modeled as small, radially symmetric, disconnected, additive kernels about the observed locations. For the variance parameters of sill and nugget, we fit them via variogram with Cressie weights [@cressie1985fitting] to the 1828 data set, then used those parameters to fill in the remaining years. Using these fixed Matérn covariance parameters are all that is required to perform spatial kriging at any desired location. In the classical kriging sense, this surface would exist in the units of the $\boldsymbol{Y}$: the 2-valued marker for intensity of battle at a village. However, we can also view the resulting surface as an implied probability density function, where the higher points of the ridge near conflict locations represent regions of increased probability of slave capture. By using the kriging estimator to fill in a high-resolution surface over the Oyo region, we then create an annual empirical cumulative density function by dividing the surface by its overall integral. For each such surface we can simulate sample slave origin locations via direct inversion. Trading Network for Slave Transport {#SS:MDP} ----------------------------------- The map in figure \[fig:1816TradeMap\] displays cities within and around Oyo connected by the most probable trade routes at the time. Such a depiction naturally translates to a graph-based approach to capturing the economics of the region. From the map, we construct a transition matrix of valid city-to-city movements, and classify a handful of cities where slaves could depart the region: Lagos, Porto Novo, and Ouidah for Atlantic departure; Abomey and Benin City for departure to the neighboring coastal states; and Djougou, Kalama, Bussa, Ogudu, Tsaragi, and Ogodo for departure into the trans-Saharan slave trade. Rather than directly assign probabilities to the flow of slaves in the region, we turn to a finite horizon Markov decision process to mirror the calculus of slavers at the time. Markov decision processes (MDPs) can be used for sequential decision making but have been underutilized in the social sciences [@boucherie2017markov]. ### A Markov Decision Process A Markov decision process describes the partially deterministic and partially stochastic movement of an agent through a network in discrete time. The agent’s actions at each state are chosen based on the rewards and costs associated with reaching a state in the network, but the actual event that takes place is probabilistic. Formally, a Markov decision process consists of a 5-tuple $(S, A, P_a, R_a, \gamma)$. $S$ is a finite set of states in a network, often spatially located. $A$ are the actions an agent can take from any given state $s \in S$. $S$ and $A$ can also be thought of as the nodes and edges in a network, respectively. In our case, $S$ corresponds to the cities in the trade network, and $A$ are the valid routes in the trade network a slaver can take; the same set of trade nodes discussed in section \[S:Data\]. For an action $a \in A$ taken in state $s$, we must define the probability of actually reaching state $s'$ for all states in $S$. Thus, for each action $a$, we must define $P(s_{t+1}' \, | \, s_{t}, a)$. Similarly, we must provide the expected immediate reward/cost incurred after moving from $s$ to $s'$ via action $a$, which we write as $R(s_{t+1}' \, | \, s_{t}, a)$. The idea is that the MDP will designate a best possible route, but a slaver might choose a slightly different one. The difference between the optimal route and the chosen route is reflected in the probabilities $P$, whereas the rewards that determined the best route are saved in $R$. For our purposes, $R$ includes both negative values that represent cost of movement and positive values that correspond to getting a slave to a point-of-sale. The end result is a “best route” determined by the model for a slaver to reach a point-of-sale, but some chances of deviations along that route to account for the slavers’ personal preferences and/or their imperfect information. The MDP solves the problem of finding an optimal policy $\pi(s)$ whose value specifies the action $a$ to take by the agent at state $s$. The function $\pi$ is found by maximizing some function of the random rewards. Most often this is the expected discounted rewards: $$\sum^{\infty}_{t=0} {\gamma^t R_{a_t} (s_t, s_{t+1})}$$ The discount factor $\gamma \in [0, 1)$ allows the rewards incurred in later time steps to be downweighted. The discount factor is fixed at $\gamma=1$ because we found it unnecessary in modeling the historical application. Many algorithms have been developed to solve this optimization problem, e.g. using linear or dynamic programming. We use the policy iteration algorithm implemented in the `R` package `MDPtoolbox` [@MDPtoolbox]. Prior to running the MDP, we augment the adjacency matrix $A$ implied by \[fig:1816TradeMap\] by adding an additional row or state corresponding to each point-of-sale city. Movement into these added states represents a sale, and holds the postive contents of the rewards $R$. This flexibility allows some amount of transit between our sale locations - in particular to reflect canoe and naval traffic along the coastal lakes - in order to not view arrival at a port as an inherent end state of the process. Instead, caravans are free to move until the optimal reward is reached, which may include moving along coasts in the presence of unequal sale rewards. ### MDP Heuristics Similar to the choice of kriging, some *ad hoc* decision-making between modeling options is merited. We harbor multiple criteria for the model for slave transit. For one, we require sufficient noise or stochasticity to allow for slaves captured in similar locations to deviate in port of departure simply by chance, thought of historically as the personal preferences and knowledge of slavers. In addition, we require a model with the ability to downweight probabilities of transit based on conflict intensities: in general, slavers would be incentivized to avoid areas of conflict, and as much of the slave transit in the Oyo is state-sanctioned, avoiding regions of conflict also serves as a proxy for slavers tending to avoid the shifting international borders and stay within their preferred home countries. Finally, it would be ideal for a method to allow for some sale locations to be preferred to others *a priori*, whether that preference is informed by volume of trade or the gradual West-to-East blockade of West African slave ports by the British Navy during the period in question. This MDP formulation allows for considerable flexibility in meeting our criteria for a transit model. - In an MDP, an optimal choice is calculated, but the underlying Markov chain allows for pseudo-random (non-“optimal”) movement from step-to-step. With a chosen probability, slavers may choose to traverse a sub-optimal route $a$ out of their city/state $s$, possibly including remaining at state $s$ for an additional time-step. - Flexibility in rewards can account for both individual slaver preference and the broader temporal shift from the Western Oyo ports and Dahomey to the Eastern ports and Benin. Setting all point-of-sale rewards to be equal asks the question “what the least resistance route to *any* point-of-sale,” whereas varying the reward vector allows for individual slavers to balance preferred or higher revenue sale locations with the implied costs of a longer journey or a journey through regions of conflict. - The cost-benefits formulation of MDP reward maximization can also easily be adjusted to downweight specific movements. In our case, we explicitly make movement through conflict regions less desirable. More generally, this formulation could be expanded to include disincentives to cross borders, venture through certain terrains, etc. Each of these are in addition to the distance-based cost terms we initialize the model with. ### MDP Parameter Tuning Because we have no information on the price of slaves as a function of point of sale, we choose as a baseline a variant of the Markov decision process with equal expected rewards for each absorbing state. Transitions along each edge incurred a cost proportional to $D*(1+C)$, where D is the length of that edge and $C$ was the maximum of the conflict kriging predictor along that edge, scaled to an annual maximum of $C=3$. This allows both the transition chains and the slave origin locations to vary with conflict. To illustrate this effect, consider the example of a simple MDP seen in figure \[fig:ToyMDP\]. In this case, we observe the shortest route being taken in the absence of conflict, but a longer route circumventing the conflict when the intense area of conflict would have overlapped one of the routes taken. ![Example of MDP decision chain for a start in S3 with an absorbing state in S1 under no conflict (yellow) or conflict (green)[]{data-label="fig:ToyMDP"}](figures/mdp.pdf){width="0.45\linewidth"} Figure \[fig:DecisionMaps\] depicts the decision processes for 1825 and 1826 as sets of arrows connecting each city in the trading network. In particular, note the difference in decisions made in in the regions around Abeokuta and the label for Ibadan (founded shortly after this conflict). In general, more trade is flowing northbound in 1826, but we see less traffic through Oyo and Ogodo, instead seeing increases in paths through Ilorin and Kaiama. Some of this traffic is also deflected away from an eventual port of departure in Porto Novo. ![Example of MDP decision chains for 1825 (left) and 1826 (right)[]{data-label="fig:DecisionMaps"}](figures/1825arrow.pdf "fig:"){width="0.45\linewidth"} ![Example of MDP decision chains for 1825 (left) and 1826 (right)[]{data-label="fig:DecisionMaps"}](figures/1826arrow.pdf "fig:"){width="0.45\linewidth"} We can combine the kriging conflict estimator with the MDP to simulate the capture of a slave at a specific location and the resulting movement of the slave through the trade network to a point-of-sale. Use of unequal rewards to vary these simulative results is discussed further in sections \[SS:FinalMaps\] and \[S:Validation\]. Maps of Location Given Point-of-Sale {#SS:FinalMaps} ------------------------------------ Once the smoothed conflict estimator and MDP process are implemented, repeated simulation of slaves can be passed to the MDP with either varying or identical reward vectors. This allows for us to create a large sample of slaves and their eventual points-of-sale. These can be used to describe the ultimate goals: what were the eventual points-of-sale of slaves and also from what conflict and locations did slaves who departed from specific ports originate? ### Large-Scale Simulation To gain origin and departure information from the models in \[SS:krig\] and \[SS:MDP\], we generate many slaves from direct inversion of the conflict estimator cumulative density functions. Then, for each such slave, we generate a random reward vector from a distribution specifying the end reward for selling a slave at any given absorbing state in the network. Then, we fit an MDP for each individual slave and reward vector pair, resulting in an optimal policy and eventual path of motion for each individual slave. Ultimately, the routing suggested by an MDP with equal rewards at each of point-of-sale is deterministic: every slave caravan of an identical origin location would choose the same optimal route given the annual conflict map. This doesn’t correspond well to the underlying historical narrative: many routes are nearly identical in terms of distance and slavers may have personal connections and preferences leading them to prefer certain routes to others. Incorporating a random reward vector represents each slaver’s knowledge of the conflicts and rewards present, which allows for a non-deterministic result to the question of where a slave will be sold *given* their capture location. Figure \[fig:rewardvar\] depicts simulated slaves from the 1832 version of the model with different randomness in the rewards. With no randomness the boundaries between colors and eventual points-of-sale are strict, whereas the figures shows increasing uncertainty when routes are nearly equivalent in terms of base cost-to-rewards. Due to the ability to simulate both capture locations and decision processes for any desired number of points, this allows us to generate these mappings to arbitrary probabilistic precision. ![Simulated Slave Origins colored by their points-of-sale. Left to Right: increasing variance in rewards[]{data-label="fig:rewardvar"}](figures/rr1.pdf "fig:"){width="0.3\linewidth"} ![Simulated Slave Origins colored by their points-of-sale. Left to Right: increasing variance in rewards[]{data-label="fig:rewardvar"}](figures/rr3.pdf "fig:"){width="0.3\linewidth"} ![Simulated Slave Origins colored by their points-of-sale. Left to Right: increasing variance in rewards[]{data-label="fig:rewardvar"}](figures/rr5.pdf "fig:"){width="0.3\linewidth"} ### Kernel Density Smoothing For Maps One question posed by historians is how to integrate this information - at this point a large collection of origin points encoded by their point-of-departure - into more cleanly interpreted spatial maps. As historians and genealogists often know the port of departure of slaves, we can use the repeated samples of simulated data to figure out such intensity maps for the slaves that left a given port on a given year. Because the maps will be a set of points corresponds to individual slaves, we are tasked with a similar question as in creating the conflict estimator: how can we provide a continuous image or heat map for slave origin locations given the simulated slaves? For this we use a simple kernel density estimator, which creates a small, radially decaying kernel function at each simulated slave leaving from a specific port. The addition of each such kernel function for every slave at a given port gives a heat map for slave origin given port of departure. Formally, the kernel density estimate takes a radially symmetric function $K(r)$ and estimates the regional heat map $\hat{f}$ via the weighted sum $$\hat{f}(x)=\frac{1}{nh}\sum_{i=1}^n K\left(\frac{|x-x_i|}{h}\right)$$ where $x_1, x_2, \dots x_n$ are the $n$ slave locations for the slaves departing from the port in question. We choose the multivariate normal as the radial function $K$, as is often convention. In general, a kernel density estimate requires only tuning one parameter: the bandwidth $h$ that determines the distance/width of the kernel function centered on each simulated slave. The `R` function `kde2d` in package `MASS` implements the multivariate normal kernel density as its default, and is employed here [@MASS]. While a kernel density function can be sensitive to the number $n$ of points employed, our simulated-based model allows us to simulate any arbitrary amount of spatial samples and construct the resulting kernel density estimator to desired precision. In the applet mentioned in section \[S:App\] we allow $h$ to vary from $.5-2$km for a sample of 10,000 simulated slaves and find this provides an appealing map; a larger simulated sample would in turn allow for smaller bandwidths. ### End-Result Parameter Tuning {#SS:Validation} As laid out so far, our model includes a considerable amount of parameters with no mathematical optimization strategy. These include the spatial covariance parameters of the kriging estimator, the relative increase in cost of movement to pass through conflict, and the variance in the point-of-sale rewards. There exist two sources of data that allow for tuning the model to optimize these selections: ship total estimates for ships leaving the southern coast of Oyo and ship ledgers of *names* of slaves as recorded by the colonizing states. As presently available to us, the passenger counts on each known ship are considered accurate. However, as figure \[fig:MissingShips\] denotes, a considerable amount of slave traffic was not recorded. Many ships that were recorded in our data set also have no specified ports of departure. As a result, a considerable amount of geospatial data is missing, and if that data was biased in any way - whether by the British blockades or some other administrative correlation - drawing geospatial conclusions about the within-Africa geospatial data would inherit these biases. ![Estimated Versus Recorded Trans-Atlantic Slave Departures from the Bight of Benin[]{data-label="fig:MissingShips"}](figures/MissingShips.png){width="0.6\linewidth"} A more recent second stream of data comes from parsing the transcriptions of slave names from known ship logs. For many ships leaving the area, the Portuguese slave ships recorded each slaves name and attempted to transliterate it into Portuguese. Recent efforts by historians have begun to translate these names, and placed them into the native tongues of the Bight of Benin. This allows us to take a few of the ships and have a new kind of mapping: one of general cultural and linguistic origins. If we take our simulated maps of slave origins, these can be graded and scored against the linguistic data by observing the exact proportions of our simulated data coming from each region. Such a score specifically suggests a $\chi^2$ optimization, where for each set of parameters we can generate a goodness-of-fit measure that best ensures our simulated departures on each ship most closely match the observed data. To date, very few ships have been fully transliterated, but a couple such examples coming from 1832 are shown in Figure \[fig:Linguistic\]. Here, linguistic distributions are shown as categorical data which must be scored against the plotted distributions. One advantage of the $\chi^2$ is that each ship added to the linguistic data records can be independently scored as a $\chi^2$, and their additive score is $\chi^2$ as well. ![Top: Model and Linguistic Data for a ship leaving Lagos, 1832; Bottom: for Ouidah[]{data-label="fig:Linguistic"}](figures/1832LagosModel.png "fig:"){width="0.45\linewidth"} ![Top: Model and Linguistic Data for a ship leaving Lagos, 1832; Bottom: for Ouidah[]{data-label="fig:Linguistic"}](figures/1832Lagos.png "fig:"){width="0.45\linewidth"}\ ![Top: Model and Linguistic Data for a ship leaving Lagos, 1832; Bottom: for Ouidah[]{data-label="fig:Linguistic"}](figures/1832OuidahModel.png "fig:"){width="0.45\linewidth"} ![Top: Model and Linguistic Data for a ship leaving Lagos, 1832; Bottom: for Ouidah[]{data-label="fig:Linguistic"}](figures/1832Ouidah.png "fig:"){width="0.45\linewidth"} Model Summary ------------- A summary of the model: 1. Take space-time locations for conflict and, for each year, create an estimate for the conflict intensity map that represents the shifting border of the wars involved. This includes some parameters that are viewed as *fixed*. Simulate slave capture locations out of this heat map. 2. Create a trade map designating roads, cities, and common caravan routes at the time. Specify a few cities to be valid points-of-sale in this map. 3. For each and every slave in step 1, create a Markov Decision Process out of the matrix in step 2 by pairing it with a (randomized) reward vector. Record each slaves point of capture and point of sale. This includes at least two flexible parameters: the cost-of-movement through conflict $C$ and the variance of the rewards for each sale location. 4. Aggregate the origin points and points of sale and score the model - either by historians’ debate or a $\chi^2$ as linguistic data become available. Optimize all flexible parameters. Interactive Web Application {#S:App} =========================== To make this research more widely available to a general audience, we created an interactive web application using the `Shiny` package in the `R` programming language. The user can select a year and one or more points-of-sale, and the application generates and displays a conditional probability map showing the most likely region of capture based on our simplified model. The app can also display the yearly conflict data as discrete points, a heatmap of the estimated intensity surface, or a contour plot. Furthermore, the annual approximate state borders [@lovejoy2013redrawing] and trade network informing the MDP can be overlayed. We have run our model independently for each year from 1816-1836 with the annual trade network and reward vectors changing over time, reflecting the historical narrative. For each year, we generate 10,000 capture locations and record the spatial coordinates, the initial location in the network, and the point-of-sale. We use Kriging on these annual data using the methods in this paper to produce an annual conditional probability surface. For each year, we save the conditional probability Kriging surface, the conflict point data, the KDE conflict intensity surface, the trade network, and the state border shapefiles, which are all the data sets required to host the app. Our web application is easy to use and freely hosted at `website.com`. It allows a general audience to interactively explore the history of West African slave trade by visualizing the data and models used in this paper. Note that we do not claim that these maps display the historical truth, but rather the results from a model which provide an approximation of the truth. Conclusions and Future Work =========================== The Markov decision process framework allows for considerable tuning. While we choose rewards for each slave point of sale that are on average equal between locations, scaling the rewards according to the port departure totals to account for the West-to-East blockading of the region by the British Empire would shift the decision processes of the slavers accordingly. Our model could also be adapted to account for the time variance in the process and the lack or precision in observed conflict dates. One option would be adding positive spatio-temporal correlation from one year’s conflict map to the next. Another option adds a time delay to each step along the Markov decision process, allowing for recalculations as the conflict shifts each year. A couple additional sources of validation for potential future work merit discussion. First, genetic databases and genealogical tracking have become much more powerful in recent years, and we look forward to an increase in the availability of such data. In particular, if descendents of passengers of any known ships where to compare their genetics to the current genetic mapping of the Bight of Benin areas, we could begin to rapidly improve on the model validation. A second broader issue our model begins to cover is the difference between traffic through the trans-Saharan and Niger areas. To date, historians have little understanding or discussion of the amount of slave traffic that moved north out of the coastal regions, and if our model is able to withstand critique for its treatment of the coastal areas, the estimates the MDP-based model gives for northward flow could help provide initial estimates for this movement. A final source of tuning and validation would be to fit similar models to similar historical situations with better availability of data. Mapping continuous conflict borders from discrete city observations could be done for nearly any conventional war fought in the $18^{th}$ or $19^{th}$ centuries. Forced transit situations in the Holocaust did not originate from conventional battles as in Oyo, but have considerably better data due to the relative recency, and could be used to better tune the decision process and exit location models. We look forward to seeing the expansion of mathematical models in creating both maps in the presence of uncertainty and making those tools available in a non-proprietary form to the public and academic genealogists.

--- abstract: 'We obtain an improvement of the Beckner’s inequality $\| f\|^{2}_{2} -\|f\|^{2}_{p} \leq (2-p) \| \nabla f\|_{2}^{2}$ valid for $p \in [1,2]$ and the Gaussian measure. Our improvement is essential for the intermediate case $p \in (1,2)$, and moreover, we find the natural extension of the inequality for any real $p$.' address: - 'Department of Mathematics, Kent State University, Kent, OH 44240' - 'Department of Mathematics, Michigan State University' author: - Paata Ivanisvili and Alexander Volberg title: 'Improving Beckner’s bound via Hermite functions' --- [^1] Introduction ============ The history of the problem -------------------------- The Poincaré inequality [@JN] for the standard Gaussian measure $d\gamma_{n} = \frac{e^{-|x|^{2}/2}}{\sqrt{(2\pi)^{n}}}dx$ states that $$\begin{aligned} \label{poincare} \int_{\mathbb{R}^{n}} f^{2} d\gamma_{n} - \left(\int_{\mathbb{R}^{n}} f d\gamma_{n}\right)^{2} \leq \int_{\mathbb{R}^{n}} | \nabla f |^{2} d\gamma_{n}\end{aligned}$$ for any smooth bounded function $f :\mathbb{R}^{n} \to \mathbb{R}$. Later William Beckner [@WB] generalized (\[poincare\]) for any real power $p$, $1 \leq p \leq 2$ as follows $$\begin{aligned} \label{beckner} \int_{\mathbb{R}^{n}} f^{p} d\gamma_{n} - \left(\int_{\mathbb{R}^{n}} f d\gamma_{n}\right)^{p} \leq \frac{p(p-1)}{2}\int_{\mathbb{R}^{n}} f^{p-2}| \nabla f |^{2} d\gamma_{n}\end{aligned}$$ for any smooth bounded $f : \mathbb{R}^{n} \to (0,\infty)$. We caution the reader that in [@WB] inequality (\[beckner\]) was formulated in a slightly different but equivalent form (see Theorem 1, inequality (3) in [@WB]). It should be also mentioned that in case $p=2$ inequality (\[beckner\]) does coincide with (\[poincare\]) for all $f \geq 0$ but it does not imply the Poincaré inequality for the functions taking the negative values, especially when $\int_{\mathbb{R}^{n}} f d \gamma_{n}=0$. If $p \to 1+$ then (\[beckner\]) provides us with log-Sobolev inequality (see [@WB]). In general, the constant $\frac{p(p-1)}{2}$ is sharp in the right hand side of (\[beckner\]) as it can be seen for $n=1$ on the test functions $f(x)=e^{\varepsilon x}$ by sending $\varepsilon \to 0$. Later Beckner’s inequality (\[beckner\]) was studied by many mathematicians for different measures, in different settings and for different spaces as well. For possible references we refer the reader to [@ABD; @ALS; @BCR1; @BCR2; @BR1; @Bob1; @Bob2; @BBL; @Chaf; @WFY; @RK; @KO]. An analysis done in [@IV] indicates that the right hand side (RHS) of (\[beckner\]) can be improved. In the present paper we address this issue: what is the precise estimate of the difference given in the left hand side (LHS) of (\[beckner\]), and whether the requirement $p \in [1,2]$ can be avoided by slightly changing the RHS of (\[beckner\]). We give complete answers to these questions. For example, if $p=\frac{3}{2}$ we will obtain an improvement in Beckner’s inequality (\[beckner\]) $$\begin{aligned} &\int_{\mathbb{R}^{n}} f^{3/2} d\gamma_{n} - \left(\int_{\mathbb{R}^{n}} f d\gamma_{n}\right)^{3/2} \leq \label{b3/2}\\ &\int_{\mathbb{R}^{n}}\left( f^{3/2} - \frac{1}{\sqrt{2}}(2f-\sqrt{f^{2}+ |\nabla f|^{2}})\sqrt{f+\sqrt{f^{2}+|\nabla f|^{2}}} \right)d\gamma_{n}. \nonumber\end{aligned}$$ The LHS of (\[b3/2\]) coincides with the LHS of (\[beckner\]) for $p=3/2$, but the RHS of (\[b3/2\]) is strictly smaller than the RHS in (\[beckner\]). Indeed, notice that we have the following *pointwise* inequality $$\begin{aligned} \label{impr1} x^{3/2} - \frac{1}{\sqrt{2}}(2x-\sqrt{x^{2}+ y^{2}})\sqrt{x+\sqrt{x^{2}+y^{2}}} \leq \frac{3}{8} x^{-1/2}y^{2} \quad \text{for all} \quad x, y \geq 0,\end{aligned}$$ which follows from the homogeneity, i.e., take $x=1$. As one can see the improvement of Beckner’s inequality (\[beckner\]) is essential. Indeed, if $y \to \infty$ then the RHS of (\[impr1\]) increases as $y^{2}$ whereas the LHS of (\[impr1\]) increases as $y^{3/2}$. Also notice that if $x \to 0$ then the difference in (\[impr1\]) tends to infinity. The only place where the quantities in (\[impr1\]) are comparable is when $y/x \to 0$. Main results ------------ Let $k$ be a real parameter. Let $H_{k}(x)$ be the Hermite function such that it satisfies the Hermite differential equation $$\begin{aligned} \label{hermite} H_{k}''-xH_{k}'+k H_{k}=0, \quad x \in \mathbb{R},\end{aligned}$$ and which grows relatively slowly $H_{k}(x) = x^{k}+o(x^{k})$ as $x \to +\infty$. If $k$ is a nonnegative integer then $H_{k}$ is the *probabilists’* Hermite polynomial of degree $k$ with the leading coefficient $1$, for example, $H_{0}(x)=1, H_{1}(x)=x, H_{2}(x)=x^{2}-1$ etc. In general, for arbitrary $k \in \mathbb{R}$ one should think that $H_{k}$ is the analytic extension of the Hermite polynomials in $k$ (existence and many other properties will be mentioned in Section \[prf\]). For $k \in \mathbb{R}$ let $R_{k}$ be the rightmost zero of $H_{k}(x)$ (see Lemma \[EMMlemma\]). If $k \leq 0$ then we set $R_{k}=-\infty$. Define $F_{k}(x)$ as follows $$\begin{aligned} \label{bdef} F_{k}\left(\left| \frac{H'_{k}(q)}{H_{k}(q)} \right|\right) = \frac{H_{k+1}(q)}{H^{1+\frac{1}{k}}_{k}(q)} \quad \text{for} \quad q \in (R_{k}, \infty).\end{aligned}$$ We will see in the next section $F_{k} \in C^{2}([0,\infty))$ is well-defined and $F_{k}(0)=1$. Moreover, if $k > -1$ then $F_{k}$ will be decreasing concave function, and if $k<-1$ then $F_{k}$ will be increasing convex function. One may observe that $$\begin{aligned} F_{1}(y)=1-y^{2}; \quad F_{2}(y)=\frac{1}{\sqrt{2}}(2-\sqrt{1+ y^{2}})\sqrt{1+\sqrt{1+y^{2}}}. \end{aligned}$$ If $k=0$ then definition (\[bdef\]) should be understood in the limiting sense as follows $$\begin{aligned} F_{\exp}(H_{-1}(q)) = q\exp\left( \alpha - \int_{1}^{q} H_{-1}(s)ds\right) \quad \text{for all} \quad q \in \mathbb{R}, \end{aligned}$$ where $$\begin{aligned} \label{kap} \alpha = \int_{1}^{\infty}\left(H_{-1}(s)-\frac{1}{s}\right)ds \approx - 0.266\ldots . \end{aligned}$$ \[better\] For any $p \in \mathbb{R} \setminus [0,1]$ and any smooth bounded $f\geq 0$ with $\int_{\mathbb{R}^{n}} f^{p} d\gamma_{n}<\infty$ we have $$\begin{aligned} \label{our} \int_{\mathbb{R}^{n}}f^{p} F_{\frac{1}{p-1}}\left(\frac{|\nabla f|}{f}\right) d\gamma_{n} \leq \left( \int_{\mathbb{R}^{n}} f d\gamma_{n} \right)^{p}.\end{aligned}$$ The inequality is reversed if $p \in (0,1)$. The theorem improves Beckner’s inequality (\[beckner\]). This will follow by taking the first two nonzero Taylor terms of $F_{\frac{1}{p-1}}(t)$ as its lower estimate. \[prob\] We have pointwise improvement in Beckner’s inequality (\[beckner\]), i.e., $$\begin{aligned} \label{bbeckner} 1-\frac{p(p-1)}{2}t^{2} \leq F_{\frac{1}{p-1}}\left(t\right) \quad \text{for all} \quad t \geq 0, \; p \in (1,2].\end{aligned}$$ The improvement will be essential when $t \to \infty$. For example, it will become clear in the next section that as $t \to \infty$ we have $$\begin{aligned} &F_{\frac{1}{p-1}}(t) \sim -t^{p} \left( H'_{\frac{1}{p-1}}(R_{\frac{1}{p-1}})\right)^{1-p} \quad \text{for} \quad p>1; \label{a1}\\ &F_{\frac{1}{p-1}}(t) \sim \left( \frac{p}{1-p}\right) \left(\frac{e^{t^{2}/2} \, \sqrt{2 \pi}}{t \Gamma(\frac{1}{1-p}) }\right)^{1-p} \quad \text{for} \quad p<1, \quad p \neq 0. \label{a2}\end{aligned}$$ Our theorem interpolates several inequalities. If $p\to 1+$ then (\[our\]) gives log-Sobolev inequality. If $p=2$ then (\[our\]) provides us with Poincaré inequality. If $p\to \pm \infty$ then we obtain *$e$-Sobolev inequality*: \[e-sob\] For any smooth bounded $f$ we have $$\begin{aligned} \int_{\mathbb{R}^{n}} \exp(f)\, F_{\exp}(|\nabla f|) d\gamma_{n} \leq \exp\left( \int_{\mathbb{R}^{n}} f d\gamma_{n}\right). \end{aligned}$$ Finally if $p \to 0$ we obtain *negative log-Sobolev* inequality: \[nlog\] For any smooth bounded $f\geq 0$ with $\int_{\mathbb{R}^{n}} \ln f d\gamma_{n}>-\infty$ we have $$\begin{aligned} \int_{\mathbb{R}^{n}} - \ln f d\gamma_{n} + \ln \left(\int_{\mathbb{R}^{n}} f d\gamma_{n} \right) \leq \int_{\mathbb{R}^{n}}- F_{-\ln}\left(\frac{|\nabla f|}{f}\right)d\gamma_{n}\end{aligned}$$ where $F_{-\ln}(t)$ is defined as follows $$\begin{aligned} F_{-\ln} \left( \frac{H_{-2}(x)}{H_{-1}(x)}\right) = \int_{1}^{x}H_{-1}(s)ds-c + \ln H_{-1}(x), \quad x \in \mathbb{R}.\end{aligned}$$ It is worth mentioning that the current paper provides with estimates of $\Phi$-entropy (see [@Chaf]): $$\begin{aligned} \boldsymbol{\mathrm{Ent}}_{\gamma_{n}}^{\Phi}(f) \stackrel{\mathrm{def}}{=} \int_{\mathbb{R}^{n}}\Phi(f)d\gamma_{n} - \Phi\left(\int_{\mathbb{R}^{n}} f d\gamma_{n} \right) \end{aligned}$$ for the following fundamental examples: $$\begin{aligned} &\Phi(x) = x^{p} \quad \text{for} \quad p \in \mathbb{R}\setminus [0,1] \quad \text{Theorem~\ref{better}};\\ &\Phi(x) = -x^{p} \quad \text{for} \quad p \in (0,1) \quad \text{Theorem~\ref{better}};\\ &\Phi(x) = e^{x}, \quad \text{Corollary~\ref{e-sob}, or $p \to \pm \infty$ in Theorem~\ref{better}};\\ &\Phi(x) = -\ln x, \quad \text{Corollary~\ref{nlog}, or $p \to 0$ in Theorem~\ref{better}};\\ &\Phi(x) = x\ln x, \quad \text{$p \to 1$ in Theorem~\ref{better}}. \end{aligned}$$ The proof of the theorem {#prf} ======================== The proof of the theorem amounts to check that the real valued function $$\begin{aligned} \label{f1} M(x,y)=x^{p}F_{k}\left( \frac{y}{x} \right)\end{aligned}$$ defined on $[\varepsilon,\infty)\times [0, \infty)$ for any $\varepsilon>0$ obeys necessary smoothness condition, it has a boundary condition $M(x,0)=x^{p}$ and it satisfies the following partial differential inequality $$\begin{aligned} \label{matrica} \begin{pmatrix} M_{xx}+\frac{M_{y}}{y} & M_{xy}\\ M_{xy} & M_{yy} \end{pmatrix} \leq 0,\end{aligned}$$ with reversed inequality in (\[matrica\]) if $p\in (0,1)$. Then by Theorem 1 in [@IV] we obtain that $$\begin{aligned} \int_{\mathbb{R}^{n}} f^{p}F_{k}\left( \frac{|\nabla f|}{f}\right) d\gamma_{n} = \int_{\mathbb{R}^{n}} M(f, |\nabla f| )d\gamma_{n} \leq M\left(\int_{\mathbb{R}^{n}}f d\gamma_{n},0 \right) = \left( \int_{\mathbb{R}^{n}} fd\gamma_{n}\right)^{p} \end{aligned}$$ for any smooth bounded $f \geq \varepsilon$ which is the statement of the theorem we want to prove (except we need to justify the passage to the limit $\varepsilon \to 0$ and this will be done later). Notice that the inequality is reversed if $p \in (0,1)$, indeed, in this case we should work with $-M(x,y)$ instead of $M(x,y)$. Next we will need some tools regarding the Hermite functions $H_{k}$. Properties of Hermite functions ------------------------------- $H_{k}$ can be defined (see [@HO]) by $$\begin{aligned} \label{hyp1} H_{k}(x) = -\frac{2^{-k/2} \sin (\pi k)\; \Gamma(k+1)}{2\pi} \sum_{n=0}^{\infty} \frac{\Gamma((n-k)/2)}{n!} (-x \sqrt{2} )^{n},\end{aligned}$$ or in terms of the confluent hypergeometric functions (see [@LD]) by $$\begin{aligned} H_{k}(x) = &\sqrt{\frac{2^{k}}{\pi}}\left[\cos\left( \frac{\pi k}{2}\right) \, \Gamma\left( \frac{k+1}{2}\right)\, {}_{1}F_{1}\left(-\frac{k}{2}, \frac{1}{2}; \frac{x^{2}}{2}\right)\right. \label{hyp2} \\ &+\left. t\sqrt{2} \sin \left( \frac{\pi k}{2} \right) \, \Gamma\left( \frac{k}{2}+1\right)\, {}_{1}F_{1}\left(\frac{1-k}{2}, \frac{3}{2}; \frac{x^{2}}{2}\right) \right].\nonumber\end{aligned}$$ If $k$ is a nonnegative integer then one should understand (\[hyp1\]) and (\[hyp2\]) in the limiting sense. Notice the following recurrence properties:$$\begin{aligned} &H'_{k}(x) = k H_{k-1}(x); \label{pirveli}\\ &H_{k+1}(x) = x H_{k}(x) - H'_{k}(x). \label{meore}\end{aligned}$$ These properties follow from (\[hyp1\]) and the fact that $\Gamma(z+1)=z\Gamma(z)$. We also notice that $$\begin{aligned} H_{k}(x) := e^{x^{2}/4}D_{k}(x),\end{aligned}$$ where $D_{k}(x)$ is the *parabolic cylinder function*, i.e., it is the solution of the equation $$\begin{aligned} D''_{k}+\left( k+\frac{1}{2}-\frac{x^{2}}{4}\right)D_{k}=0. \end{aligned}$$ Since $H_{k}(x)$ is an entire function in $x$ and $k$ (see [@temme] for the parabolic cylinder function) sometimes it will be convenient to write $H(x,k)$ instead of $H_{k}(x)$. The precise asymptotic for $x \to +\infty$, $x>0$ and any $k \in \mathbb{R}$ is given as follows $$\begin{aligned} \label{loran} H_{k}(x) \sim x^{k} \cdot \sum_{n=0}^{\infty} (-1)^{n} \frac{(-k)_{2n}}{n! (2x^{2})^{n}}.\end{aligned}$$ Here $(a)_{n}=1$ if $n=0$ and $(a)_{n}=a(a+1)\ldots(a+n-1)$ if $n>0$. When $x \to -\infty$ we have $$\begin{aligned} \label{loran-} H_{k}(x) \sim |x|^{k}\cos(k \pi) \sum_{n=0}^{\infty} (-1)^{n} \frac{(-k)_{2n}}{n! (2x^{2})^{n}} + \frac{\sqrt{2 \pi}}{\Gamma(-k)} |x|^{-k-1}e^{x^{2}/2}\sum_{n=0}^{\infty} \frac{(1+k)_{2n}}{n! (2x^{2})^{n}}. \end{aligned}$$ We refer the reader to [@temme; @NIST]. For instance, for (\[loran\]) we can use the asymptotic formula (12.9.1) in [@NIST] for the parabolic cylinder function. To verify (\[loran-\]) we can express $H_{k}(-x)$ as a linear combination of two parabolic cylinder functions but having argument $x$ instead of $-x$ (see (12.2.15) in [@NIST]), and then we can use (12.9.1) and (12.9.2) in [@NIST]. Next we will need the result of Elbert–Muldoon [@EMM1] which describes the behavior of the real zeros of $H_{k}(x)$ for any real $k$. \[EMMlemma\] For $k\leq 0$, $H_{k}(x)$ has no real zeros, and it is positive on the real axis. For $n<k \leq n+1$, $n=0,1,\ldots, $ $H_{k}(x)$ has $n+1$ real zeros. Each zero is increasing function of $k$ on its interval of definition. The proof of the lemma is Theorem 3.1 in [@EMM1]. It is explained in the paper that as $k$ passes through each nonnegative integer $n$ a new leftmost zero appears at $-\infty$ while the right-most zero passes through the largest zero of $H_{k}(x)$. More precise information about the asymptotic behavior of the zeros as $k \to \infty$ can be found in [@EMM2]. Further we will need Turán’s inequality for $H_{k}$ for any real $k$. \[turl\] We have the following Turán’s inequality: $$\begin{aligned} H_{k}^{2}(x)-H_{k-1}(x)H_{k+1}(x)> 0 \quad \text{for all} \quad k \in \mathbb{R}, \; x \geq L_{k} \label{tu1} \end{aligned}$$ where $L_{k}$ denotes the leftmost zero of $H_{k}$. If $k\leq 0$ then $L_{k}=-\infty$. The lemma is known as Turán’s inequality when $k$ is a nonnegative integer. Unfortunately we could not find the reference in the case when $k$ is different from a positive integer therefore we decided to include the proof of the lemma. The following is borrowed from [@MT]. Take $f(x) = e^{-\frac{x^{2}}{2}}(H_{k}^{2}(x)-H_{k-1}(x)H_{k+1}(x))$. Asymptotic formulas (\[loran\]) and (\[loran-\]) imply that $$\begin{aligned} &\lim_{x \to +\infty} f(x) =0 \quad \text{for all} \quad k \in \mathbb{R};\nonumber\\ &f(x) \sim \sqrt{2 \pi} |x| >0\quad \text{for} \quad x \to -\infty, \quad k =0;\nonumber\\ &f(x) \sim \frac{2\pi e^{x^{2}/2}}{\Gamma(-k)\Gamma(-k+1)}|x|^{-2k-2} \quad \text{for} \quad x \to -\infty, \quad k \notin \{0\}\cup \mathbb{N}. \label{atinf}\end{aligned}$$ On the other hand notice that $$\begin{aligned} \label{der} f'(x) = -e^{-\frac{x^{2}}{2}} H_{k}H_{k-1}.\end{aligned}$$ If $k\leq 0$ then by Lemma \[EMMlemma\] $f'<0$, and because of the conditions $f(-\infty)=+\infty$ and $f(\infty)=0$ we obtain that $f>0$ on $\mathbb{R}$. To verify the statement for $k>0$ we notice that $$\begin{aligned} \label{der1} f''(x) = e^{-\frac{x^{2}}{2}}(H_{k}^{2} -k H_{k-1}^{2}).\end{aligned}$$ Now we notice that if $H_{k}(c)=0$ then $H_{k-1}(c)\neq 0$. Indeed, assume contrary $H_{k-1}(c)=0$. Then by (\[pirveli\]) we have $H'_{k}(c)=0$ and by (\[hermite\]) we obtain $H''_{k}(c)=0$, and again taking derivative in (\[pirveli\]) we obtain that $H_{k-2}(c)=0$. Repeating this process we obtain that $H_{k-N}(c)=0$ for any large integer $N>0$. But this contradicts to Lemma \[EMMlemma\]. Thus by (\[der\]) and (\[der1\]) we obtain that $c$ is a point of the local minimum of $f$ if and only if $ H_{k-1}(c)=0$. Then $f(c) = e^{-x^{2}/2}H_{k}^{2}(c)>0$. Finally we obtain that $f : [L_{k}, \infty) \to \mathbb{R}$ is positive on its local minimum points, $f(\infty)=0$ and $f(L_{k}) >0$ (because $H_{k-1}, H_{k+1}$ have opposite signs at zeros of $H_{k}$ by (\[meore\])). Therefore $f > 0$ on $[L_{k}, \infty) \to \mathbb{R}$ and the lemma is proved. If $k \in \mathbb{N}$ then $H_{k}$ is the probabilists’ Hermite polynomial of degree $k$, so $f(x)$ will be even and inequality (\[tu1\]) will hold for all $x \in \mathbb{R}$ which confirms the classical Turán’s inequality. However, if $k>0$ but $k \notin \mathbb{N}$ then (\[tu1\]) fails when $x \to -\infty$ (see (\[atinf\])). Finally the next corollary together with Lemma \[EMMlemma\] implies that $\left|\frac{H'_{k}}{H_{k}}\right|=\mathrm{sign}(k) \frac{H'_{k}(q)}{H_{k}(q)} $ is positive and decreasing for $q \in (R_{k}, \infty)$ and $k\in \mathbb{R}\setminus\{0\}$. \[cor\] For any $x \ge L_{k}$ and any $k \in \mathbb{R}\setminus\{0\}$ we have $$\begin{aligned} \mathrm{sign}[(H'_{k})^{2}-H_{k}H''_{k}]=\mathrm{sign}(k).\end{aligned}$$ The proof follows from Lemma \[turl\] and the following identity $$\begin{aligned} \label{identity1} k(H_{k}^{2}-H_{k-1}H_{k+1}) = (H'_{k})^{2}-H_{k}H''_{k}\end{aligned}$$ from (\[hermite\]), (\[pirveli\]) and (\[meore\]). Checking the partial differential inequality -------------------------------------------- Let $p=1+\frac{1}{k}$. Further we assume $k \neq 0, -1$. Define $F=F_{k}$ as in the introduction: $$\begin{aligned} \label{opr} F(t) = \frac{H_{k+1}(q)}{H_{k}^{1+1/k}(q)} \quad \text{where} \quad \left| \frac{H'_{k}(q)}{H_{k}(q)}\right|=t, \quad q \in (R_{k}, \infty), \quad t \in (0, \infty).\end{aligned}$$ Notice that by Corollary \[cor\] function $\left| \frac{H'_{k}(q)}{H_{k}(q)}\right|=\mathrm{sign}(k) \frac{H'_{k}(q)}{H_{k}(q)}$ is positive decreasing in $q$ for $q \in (R_{k}, \infty)$, moreover by (\[loran\]) we have $\frac{H'_{k}(q)}{H_{k}(q)} \sim \frac{k}{q}$ when $q \to +\infty$. From the same asymptotic formulas it follows that when $t \to 0+$ we have $$\begin{aligned} F(t)= 1-\frac{p(p-1)}{2}\, t^{2} +O(t^{4}).\end{aligned}$$ Therefore $F$ is well-defined function and $F \in C^{2}([0,\infty))$. Take a positive $\varepsilon>0$ and define $M(x,y)$ as in (\[f1\]): $$\begin{aligned} \label{k1} M(x,y) :=x^{p} F\left(\frac{y}{x}\right) \quad \text{for} \quad y\geq 0, \quad x > \varepsilon >0.\end{aligned}$$ Clearly $M(x,\sqrt{y}) \in C^{2}([\varepsilon, \infty)\times[0, \infty))$. By Theorem 1 in [@IV] we have inequality $$\begin{aligned} \label{ivo1} \int_{\mathbb{R}^{n}} M(f,|\nabla f|) d\gamma_{n} \leq M\left(\int_{\mathbb{R}^{n}}f d\gamma_{n}, 0 \right)\end{aligned}$$ for all smooth bounded $f \geq \varepsilon$ if (\[matrica\]) holds. In terms of $F$ (see (\[k1\])) condition (\[matrica\]) takes the form $$\begin{aligned} &tFF''p(p-1)+F'F''-t(p-1)^{2}(F')^{2} \geq 0 \qquad \text{i.e., the determinant of (\ref{matrica}) is nonnegative}\label{determinant}\\ & F'' (t+t^{3})+F'(2t^{2}+1-2pt^{2})+Fp(p-1)t\leq 0 \qquad \text{i.e., the trace of (\ref{matrica}) is nonpositive} \label{trace}\end{aligned}$$ where $t = \frac{y}{x}$ is the argument of $F$. In fact we will show that we have equality in (\[determinant\]) instead of inequality therefore the sign of (\[matrica\]) will depend on the sign of trace (\[trace\]). We will see that inequality (\[trace\]) will be reversed for $p \in (0,1)$. From (\[opr\]), (\[identity1\]) and (\[tu1\]) we obtain $$\begin{aligned} &F'(t) =-\frac{k+1}{|k|} \frac{1}{H_{k}^{1/k}};\label{odin}\\ &F''(t) = \frac{F'}{|k|}\cdot \frac{H_{k}H_{k-1}}{H_{k}^{2}-H_{k+1}H_{k-1}};\label{dva}\\ &F(t) = -\frac{|k|}{k+1} \frac{H_{k+1}}{H_{k}}\, F'. \label{tri}\end{aligned}$$ If we plug (\[dva\]) and (\[tri\]) into (\[determinant\]) we obtain that the left hand side of (\[determinant\]) is zero. If we plug (\[dva\]) and (\[tri\]) into (\[trace\]) we obtain $$\begin{aligned} \text{LHS of}\; (\ref{trace}) = \left[ \frac{(kH_{k-1}^{2}-H_{k}^{2}+H_{k-1}H_{k+1})^{2}+H_{k-1}^{2}H_{k}^{2}}{H_{k}^{2}(H_{k}^{2}-H_{k+1}H_{k-1})}\right]\, F'. \end{aligned}$$ Thus the sign of LHS of (\[trace\]) coincides with the sign of $F'$ which coincides with $\mathrm{sign}(-(k+1))$. The condition $p \in \mathbb{R}\setminus [0,1]$ implies that $k>-1$ and therefore (\[matrica\]) holds. The condition $p \in (0,1)$ implies that $k<-1$ and therefore inequality in (\[matrica\]) is reversed. Thus we have obtained (\[ivo1\]) for smooth bounded functions $f \geq \varepsilon$. Next we claim that for an arbitrary smooth bounded $f\geq 0$ with $\int_{\mathbb{R}^{n}}f^{p} d\gamma_{n}<\infty$ we can apply the inequality to $f_{\varepsilon}:= f +\varepsilon$ and send $\varepsilon$ to $0$ in (\[our\]). Indeed, it follows from (\[bdef\]) and (\[loran\]) that as $t \to \infty$ we have $$\begin{aligned} &F(t) \sim -t^{1+\frac{1}{k}} (H'_{k}(R_{k}))^{-\frac{1}{k}} \quad \text{for} \quad k>0;\\ &F(t) \sim \mathrm{sign}(-1-k) \left(\frac{e^{t^{2}/2} \, \sqrt{2 \pi}}{t \Gamma(-1-k) }\right)^{-\frac{1}{k}} |1+k|^{1+\frac{1}{k}}\quad \text{for} \quad k<0, \quad k \neq -1. \end{aligned}$$ Thus for $p>1$ (i.e., $k>0$) the claim about the limit follows from the estimate $|F(t)| \leq C_{1}+ C_{2} t^{p}$ together with the Lebesgue dominated convergence theorem. If $p <0$ (i.e., $k \in (-1,0)$) we rewrite (\[our\]) in a standard way as follows $$\begin{aligned} \label{lim} \int_{\mathbb{R}^{n}} f_{\varepsilon}^{p} d\gamma_{n} - \left( \int_{\mathbb{R}^{n}} f_{\varepsilon} d\gamma_{n} \right)^{p} \leq \int_{\mathbb{R}^{n}}f_{\varepsilon}^{p}\left(1- F\left(\frac{|\nabla f |}{f_{\varepsilon}}\right)\right) d\gamma_{n}.\end{aligned}$$ Since $f$ is bounded, $f\geq 0$ and $\int_{\mathbb{R}^{n}} f^{p} d\gamma_{n} <\infty$ there is no issue with the left hand side of (\[lim\]) when $\varepsilon \to 0$. For the right hand side of (\[lim\]) we notice that the function $x^{p}(1-F(y/x))$ is nonnegative and decreasing in $x$ then the claim follows from the monotone convergence theorem. The non negativity follows from the observation that $F(0)=1$ and $F'<0$ (see (\[odin\]) where we have $k>-1$). The monotonicity follows from (\[bdef\]), (\[odin\]), (\[pirveli\]) and the straightforward computations $$\begin{aligned} \label{hh4} \frac{\partial }{\partial x} \left(x^{p}(1-F(y/x)) \right)=x^{p-1}\left(p-pF(t)+tF'(t) \right)=x^{p-1} p \left[1-\frac{q}{H_{k}^{\frac{1}{k}}(q)} \right],\end{aligned}$$ where $|k| \frac{H_{k-1}(q)}{H_{k}(q)}=t=\frac{y}{x}$ and $q\in (R_{k}, \infty)$. The last expression in (\[hh4\]) is negative because $$\begin{aligned} 1 \geq F(t) = \frac{H_{k+1}}{H_{k}^{1+\frac{1}{k}}}=\frac{qH_{k}-kH_{k-1}}{H_{k}^{1+\frac{1}{k}}} >\frac{q}{H_{k}^{\frac{1}{k}}}. \end{aligned}$$ Finally if $p \in (0,1)$ (i.e., $k <-1$) we have the opposite inequality in (\[lim\]). In this case the situation is absolutely the same as for $k \in (-1,0)$ except now we should consider the function $x^{p}(F(y/x)-1)$ which is nonnegative and decreasing in $x$ (see (\[hh4\])). This finishes the proof of the theorem. Now let us show Proposition \[prob\]. Since $F(0)=1$ it is enough to show a stronger inequality, namely $F'+p(p-1)t \geq 0$. From (\[odin\]) and the fact that $k\geq 1$ (since $p \in [1,2]$) we obtain that it is enough to show the following inequality $$\begin{aligned} -\frac{p}{H_{k}^{1/k}}+p(p-1)\frac{H'_{k}}{H_{k}}\geq 0 \quad \text{for all} \quad k \geq 1, \; q \in (R_{k}, \infty). \end{aligned}$$ Using (\[pirveli\]) and $p=1+\frac{1}{k}$ we notice that the inequality can be rewritten as follows $1>\frac{H_{k}(q)}{H_{k-1}^{\frac{k}{k-1}}(q)}$ for all $q \in (R_{k}, \infty)$. To verify the last inequality we remind that $F(0)=1$ and $F'(t) <0$. Therefore $F(t)\leq 1$. We recall the definition of $F(t)$ (see (\[opr\])). It follows that $1\geq F = \frac{H_{k+1}}{H_{k}^{1+1/k}}$ for all $k>0$. The last inequality is the same as $$\begin{aligned} 1>\frac{H_{k}(q)}{H_{k-1}^{\frac{k}{k-1}}(q)} \quad \text{for all} \quad q \in (R_{k}, \infty), \quad k \geq 1.\end{aligned}$$ This finishes the proof of the theorem. Proof of Corollary \[e-sob\] and Corollary \[nlog\]: ---------------------------------------------------- Notice that as $t \to 0$ we have $$\begin{aligned} F_{\exp}(y) = 1-\frac{y^{2}}{2}+O(y^{4}) \quad \text{and} \quad F_{-\ln}(y) = -\frac{y^{2}}{2}+O(y^{4}). \end{aligned}$$ There are two ways to obtain the corollaries. ### The first way: One can check that $$\begin{aligned} &M_{\exp}(x,y)=e^{x}F_{\exp}(y), \quad M_{\exp}(x,0)=e^{x}, \quad M_{\exp}(x,\sqrt{y})\in C^{2}(\mathbb{R}\times \mathbb{R}_{+}); \\ &M_{-\ln}(x,y)=-\ln(x)+F_{-\ln}\left(\frac{y}{x}\right), \quad M_{-\ln }(x,0)=-\ln x, \quad x>0, \end{aligned}$$ and $M_{-\ln}(x,\sqrt{y}) \in C^{2}([\varepsilon, \infty) \times \mathbb{R}^{+})$ for any $\varepsilon>0$. By straightforward computations we notice that if we set $\psi(q) = \alpha - \int_{1}^{q} H_{-1}(s)ds$ then using the identity $1=qH_{-1}(q)+H_{-2}(q)$ we obtain $$\begin{aligned} F_{\exp}(H_{-1}) = q e^{\psi}, \quad F'_{\exp}(H_{-1})=-e^{\psi} \quad \text{and} \quad F''_{\exp}(H_{-1}) = - \frac{H_{-1}}{H_{-2}}.\end{aligned}$$ Similarly we compute that $$\begin{aligned} F'_{-\ln}\left(\frac{H_{-2}}{H_{-1}}\right)=-H_{-1}\quad \text{and} \quad F''_{-\ln}\left(\frac{H_{-2}}{H_{-1}}\right)=- \frac{H_{-2}H_{-1}^{2}}{H_{-1}^{2}-H_{-2}}.\end{aligned}$$ Next one notices that $M_{\exp}$ and $M_{-\ln}$ satisfy (\[matrica\]) (in fact the determinant of (\[matrica\]) is zero). Then by Theorem 1 in [@IV] we obtain the corollaries. The passage to the limit for $M_{-\ln}(x,y)$ when $\varepsilon \to 0$ follows from the monotone convergence theorem. Indeed, we notice that $-F_{-\ln}(y/x) \geq 0$ is decreasing in $x$. We apply Corollary \[nlog\] to $f_{\varepsilon}=f+\varepsilon$ and send $\varepsilon \to 0$. ### The second way: We will obtain the corollaries as a limiting case of Theorem \[better\]. Indeed, to verify Corollary \[e-sob\] let $f^{p}=e^{g}$ in (\[our\]). Then (\[our\]) takes the form $$\begin{aligned} \label{ee-sob} \int_{\mathbb{R}^{n}}e^{g} F_{\frac{1}{p-1}}\left( \frac{|\nabla g|}{p}\right)d\gamma_{n} \leq \left( \int_{\mathbb{R}^{n}} e^{g/p} d\gamma_{n}\right)^{p}.\end{aligned}$$ Now we take $p \to \infty$. The RHS of (\[ee-sob\]) tends to $\exp(\int_{\mathbb{R}^{n}} g d\gamma_{n})$. For the LHS of (\[ee-sob\]) we should compute the limit $$\begin{aligned} F_{\exp}(t) :=\lim_{p \to \infty} F_{\frac{1}{p-1}}\left( \frac{t}{p}\right) = \lim_{p \to \infty} F_{\frac{1}{p-1}}\left( \frac{t}{p-1}\right) = \lim_{k \to 0+} F_{k}(tk). \end{aligned}$$ It is clear that $F_{\exp}(0)=1$. Next if we take $k \to 0+$ in (\[bdef\]) we obtain $$\begin{aligned} \lim_{k \to 0+} F_{k}\left(\left| \frac{H'_{k}}{H_{k}} \right| \right) = \lim_{k \to 0+} F_{k}\left(k \frac{H_{k-1}}{H_{k}} \right) = \lim_{k \to 0+} F_{k}\left(k \frac{H_{-1}}{H_{0}} \right) = F_{\exp}(H_{-1})\end{aligned}$$ On the other hand for the RHS of (\[bdef\]) we have $$\begin{aligned} \lim_{k \to 0+} \frac{H_{k+1}(q)}{H_{k}^{1+\frac{1}{k}}} = q \lim_{k \to 0+} H_{k}^{-1/k}.\end{aligned}$$ Here we have used $H_{0}(q)=1$ and $H_{1}(q)=q$. Thus it remains to find $\lim_{k \to 0+}H_{k}^{-1/k}$. Notice that $H(x,k):=H_{k}(x)$ is an entire function in $x$ and $k$ (see [@temme] for the Parabolic cylinder function). If we take derivative in $k$ of (\[pirveli\]) we obtain $H_{xk}(x,k)=H(x,k-1)+kH_{k}(x,k)$ (here subindices denote partial derivatives). Now taking $k=0$ we obtain $H_{xk}(x,0)=H(x,-1)$. Thus $H_{k}(x,0)$ is an antiderivative of $H(x,-1)=H_{-1}$. So $$\begin{aligned} \lim_{k \to 0+} H_{k}^{-1/k}= \lim_{k \to 0+} \exp\left( -\frac{1}{k}\ln (1+kH_{k}(x,0)+o(k))\right) = \exp\left(- \int H_{-1}(s) ds \right).\end{aligned}$$ Finally we obtain $$\begin{aligned} \label{hh1} F_{\exp}(H_{-1}(q)) = q \exp \left(C - \int_{1}^{q}H_{-1} \right)\end{aligned}$$ In order to satisfy the condition $F_{\exp}(0)=1$ the constant $c$ must be chosen as follows $C= \int_{1}^{\infty}(H_{-1}-\frac{1}{s})ds$ (indeed send $q \to \infty$ in (\[hh1\])). This finishes the proof of Corollary \[e-sob\]. It is worth mentioning that we have also obtained (see (\[kap\])) $$\begin{aligned} H_{k}(x,0)=\int_{1}^{x}H_{-1}(s)ds - \alpha.\end{aligned}$$ To verify Corollary \[nlog\] let $F(x,k):=F_{k}(x)$. Let $F_{k}(x,k)$ denotes the partial derivative in $k$ of $F(x,k)$. If we send $p \to 0, p<0$ in (\[our\]) and compare the terms of order $p$ we obtain $$\begin{aligned} \int_{\mathbb{R}^{n}}\left( \ln f - F_{k}\left(\frac{|\nabla f|}{f}, -1\right) \right)d\gamma_{n} \geq \ln \left(\int_{\mathbb{R}^{n}} f d\gamma_{n} \right)\end{aligned}$$ It remains to find the function $F_{k}(x,-1)$. Let us equate terms of order $(k+1)$ as $k \to -1, k<-1$ in the following equality $$\begin{aligned} F\left(\frac{H_{x}(x,k)}{H(x,k)},k \right) = \frac{H(x,k+1)}{H(x,k)^{1+\frac{1}{k}}}. \end{aligned}$$ The straightforward computation shows that $$\begin{aligned} F_{k}\left( \frac{H_{-2}(x)} {H_{-1}(x)},-1\right) = H_{k}(x,0)+\ln H_{-1}(x)=\int_{1}^{x}H_{-1}(s)ds - \alpha + \ln H_{-1}(x)\end{aligned}$$ where $$\begin{aligned} \alpha=\int_{1}^{\infty}\left( H_{-1}(s) - \frac{1}{s}\right)ds.\end{aligned}$$ Concluding remarks ================== The reader may wander how we guessed the choice (\[f1\]). Of course it was not a random guess. Function (\[f1\]) is the best possible in the sense that the determinant of (\[matrica\]) is identically zero $$\begin{aligned} \label{monge} &M_{yy}(M_{xx}+\frac{M_{y}}{y})-M_{xy}^{2}=0,\\ &M(x,0)=x^{p} \quad \text{for} \quad x \geq 0. \nonumber\end{aligned}$$ Initially this was the way we started looking for $M(x,y)$ as the solution of the Monge–Ampère equation with a drift (\[monge\]). By a proper change of variables the equation reduces to the backwards heat equation (see [@IV] for more details where the connection with R. Bryant, Ph. Griffiths theory of exterior differential systems was exploited) $$\begin{aligned} &u_{xx}+u_{t}=0, \label{h1}\\ &u(x,0)=C x^{\frac{p}{p-1}} \label{h2} \quad \text{for} \quad x\geq 0. \end{aligned}$$ One can notice that the Hermite polynomials do satisfy (\[h1\]) and (\[h2\]) when $\frac{p}{p-1}$ is a positive integer. In general, one should invoke Hermite functions and this is the reason of appearance of these functions in our theorem. Another possibility is to assume that $M(x,y)$ should be homogeneous of degree $p$ which enforces $M$ to have the form (\[k1\]) for some $F$. Next setting $h=\frac{F}{F'}$ and further by a subtle change of variables one obtains Hermite differential equation (\[hermite\]). Nevertheless, for the formal proof of Theorem \[better\] we do not need to go through the details. We have $M(x,y)$ defined by (\[f1\]) and we just need to check that it satisfies the desired properties. The fact that $M(x,y)$ (see (\[f1\])) satisfies (\[matrica\]) makes it possible to extend Theorem \[better\] in a semigroup setting for uniformly log-concave measures. Indeed, let $d\mu = e^{-U}dx$ where $\mathrm{Hess}\, U \geq R\cdot Id,$ $R>0$. Let $L=\Delta - \nabla U \cdot \nabla$, and let $P_{t} = e^{tL}$ be the semigroup with generator $L$ (see [@IV; @BGL]). For any $p \in \mathbb{R}\setminus [0,1]$ and any smooth bounded $f\geq0$ with $\int_{\mathbb{R}^{n}} f^{p} d\mu <\infty$ we have $$\begin{aligned} P_{t} \left[f^{p} F_{\frac{1}{p-1}}\left(\frac{| \nabla f|}{f\sqrt{R}}\right) \right] \leq (P_{t} f)^{p} F_{\frac{1}{p-1}}\left(\frac{| \nabla P_{t} f|}{P_{t} f \sqrt{R}}\right). \end{aligned}$$ The inequality is reversed if $p\in (0,1)$. Notice that $\tilde{M}(x,y) = M(x,\frac{y}{\sqrt{R}})$ satisfies (\[matrica\]). Now it remains to use inequality (2.3) from [@IV]. Next by taking $t \to \infty$ and using the fact that $| \nabla P_{t} f | \leq e^{-tR} P_{t} | \nabla f |$ we obtain the following corollary Let $d\mu =e^{-U}dx$ where $\mathrm{Hess}\, U \geq R \cdot Id$ for some $R>0$. For any $p \in \mathbb{R}\setminus [0,1]$ and any smooth bounded $f \geq 0$ with $\int_{\mathbb{R}^{n}} f^{p} d\mu<\infty$ we have $$\begin{aligned} \int_{\mathbb{R}^{n}} f^{p} F_{\frac{1}{p-1}}\left(\frac{| \nabla f|}{f\sqrt{R}}\right) d\mu \leq \left( \int_{\mathbb{R}^{n}} f d\mu \right)^{p}. \end{aligned}$$ The inequality is reversed if $p \in (0,1)$. See Corollary 1 in [@IV]. The limiting cases of these inequalities when $p \to \pm \infty$ and $p \to 0$ should be understood in the sense of functions $M_{\exp}$ and $M_{-\ln}$ as in Corollary \[e-sob\] and Corollary \[nlog\]. Finally we would like to mention that having characterization (\[matrica\]) of functional inequalities (\[ivo1\]) makes approach to the problem (\[our\]) systematic. Very similar *local estimates* happen to rule some *global inequalities*. We refer the reader to our recent papers on this subject [@IV01; @IV02; @IV03]. Acknowledgements {#acknowledgements .unnumbered} ---------------- We are very grateful to Robert Bryant who suggested a change of variables in (\[determinant\]). [9]{} A. Arnold, J. P. Bartier, J. Dolbeault, *Interpolation between logarithmic Sobolev and Poincaré inequalities*, Commun. Math. Sci. **5** (2007) 971–979. P. D. Pelo, A. Lanconelli, A. I. Stan, *An extension of the Beckner’s type Poincaré inequality to convolution measures on abstract Wiener spaces*, arXiv: 1409.5861 D. Bakry, I. Gentil, M. Ledoux, *Analysis and Geometry of Markov Diffusion Operators*, Grundlehren der Mathematischen Wissenschaften **348**. Springer, Cham. F. Barthe, P. Cattiaux, C. Roberto, *Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry*, (to appear in Revista Mat. 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--- abstract: 'We developed a new direct-tree hybrid $N$-body algorithm for fully self-consistent $N$-body simulations of star clusters in their parent galaxies. In such simulations, star clusters need high accuracy, while galaxies need a fast scheme because of the large number of the particles required to model it. In our new algorithm, the internal motion of the star cluster is calculated accurately using the direct Hermite scheme with individual timesteps and all other motions are calculated using the tree code with second-order leapfrog integrator. The direct and tree schemes are combined using an extension of the mixed variable symplectic (MVS) scheme. Thus, the Hamiltonian corresponding to everything other than the internal motion of the star cluster is integrated with the leapfrog, which is symplectic. Using this algorithm, we performed fully self-consistent $N$-body simulations of star clusters in their parent galaxy. The internal and orbital evolutions of the star cluster agreed well with those obtained using the direct scheme. We also performed fully self-consistent $N$-body simulation for large-$N$ models ($N=2\times 10^6$). In this case, the calculation speed was seven times faster than what would be if the direct scheme was used.' author: - 'Michiko <span style="font-variant:small-caps;">Fujii</span>' - 'Masaki <span style="font-variant:small-caps;">Iwasawa</span>, Yoko <span style="font-variant:small-caps;">Funato</span>' - 'Junichiro <span style="font-variant:small-caps;">Makino</span>' title: 'BRIDGE: A Direct-tree Hybrid $N$-body Algorithm for Fully Self-consistent Simulations of Star Clusters and their Parent Galaxies' --- Introduction ============ Accuracy and Performance ======================== Summary and Discussion ====================== The authors thanks Piet Hut for useful comments and the name of the hybrid scheme, Keigo Nitadori and Ataru Tanikawa for fruitful discussions, and the referee, Simon F. Portegies Zwart, for useful comments on the manuscript. M. F. is financially supported by Research Fellowships of the Japan Society for the Promotion of Science (JSPS) for Young Scientists. This research is partially supported by the Special Coordination Fund for Promoting Science and Technology (GRAPE-DR project), Ministry of Education, Culture, Sports, Science and Technology, Japan. Part of calculations were done using the GRAPE system at the Center for Computational Astrophysics (CfCA) of the National Astronomical Observatory of Japan.

--- abstract: 'In this paper, we investigate the capacity of the Gaussian two-hop full-duplex (FD) relay channel with residual self-interference. This channel is comprised of a source, an FD relay, and a destination, where a direct source-destination link does not exist and the FD relay is impaired by residual self-interference. We adopt the worst-case linear self-interference model with respect to the channel capacity, and model the residual self-interference as a Gaussian random variable whose variance depends on the amplitude of the transmit symbol of the relay. For this channel, we derive the capacity and propose an explicit capacity-achieving coding scheme. Thereby, we show that the optimal input distribution at the source is Gaussian and its variance depends on the amplitude of the transmit symbol of the relay. On the other hand, the optimal input distribution at the relay is discrete or Gaussian, where the latter case occurs only when the relay-destination link is the bottleneck link. The derived capacity converges to the capacity of the two-hop ideal FD relay channel without self-interference and to the capacity of the two-hop half-duplex (HD) relay channel in the limiting cases when the residual self-interference is zero and infinite, respectively. Our numerical results show that significant performance gains are achieved with the proposed capacity-achieving coding scheme compared to the achievable rates of conventional HD relaying and/or conventional FD relaying.' author: - 'Nikola Zlatanov, Erik Sippel, Vahid Jamali, and Robert Schober [^1] [^2] [^3]' bibliography: - 'litdab.bib' nocite: '[@5089955; @Choi:2010; @5961159; @5985554; @Jain_2011; @6177689; @6280258; @6353396; @Bharadia:2013:FDR:2486001.2486033; @6542771; @6523998; @6702851; @6736751; @6656015; @6782415; @6832592; @6862895; @6832471; @6832464; @6832439; @7105647; @7024120; @7051286; @7390828; @6736751; @7182305]' title: 'Capacity of the Gaussian Two-Hop Full-Duplex Relay Channel with Residual Self-Interference ' --- Introduction ============ In wireless communications, relays are employed in order to increase the data rate between a source and a destination. The resulting three-node channel is known as the relay channel [@cover]. If the distance between the source and the destination is very large or there is heavy blockage, then the relay channel can be modeled without a source-destination link, which leads to the so called two-hop relay channel. For the relay channel, there are two different modes of operation for the relay, namely, the full-duplex (FD) mode and the half-duplex (HD) mode. In the FD mode, the relay transmits and receives at the same time and in the same frequency band. As a result, FD relays are impaired by self-interference, which is the interference caused by the relay’s transmit signal to the relay’s received signal. Latest advances in hardware design have shown that the self-interference of an FD node can be suppressed significantly, see [@5089955]-[@7182305], which has led to an enormous interest in FD communication. For example, [@Bharadia:2013:FDR:2486001.2486033] reported that self-interference suppression of 110 dB is possible in certain scenarios. On the other hand, in the HD mode, the relay transmits and receives in the same frequency band but in different time slots or in the same time slot but in different frequency bands. As a result, HD relays completely avoid self-interference. However, since an HD relay transmits and receives only in half of the time/frequency resources compared to an FD relay, the achievable rate of the two-hop HD relay channel may be significantly lower than that of the two-hop FD relay channel. Information-theoretic analyses of the capacity of the two-hop HD relay channel were provided in [@kramer2004models], [@zlatanov2014capacity-globecom]. Thereby, it was shown that the capacity of the two-hop HD relay channel is achieved when the HD relay switches between reception and transmission in a symbol-by-symbol manner and not in a codeword-by-codeword manner, as is done in conventional HD relaying [@1435648]. Moreover, in order to achieve the capacity, the HD relay has to encode information into the silent symbol created when the relay receives [@zlatanov2014capacity-globecom]. For the Gaussian two-hop HD relay channel without fading, it was shown in [@zlatanov2014capacity-globecom] that the optimal input distribution at the relay is discrete and includes the zero (i.e., silent) symbol. On the other hand, the source transmits using a Gaussian input distribution when the relay transmits the zero (i.e., silent) symbol and is silent otherwise. The capacity of the Gaussian two-hop FD relay channel with ideal FD relaying without residual self-interference was derived in [@cover]. However, in practice, canceling the residual self-interference completely is not possible due to limitations in channel estimation precision and imperfections in the transceiver design [@6832464]. As a result, the residual self-interference has to be taken into account when investigating the capacity of the two-hop FD relay channel. Despite the considerable body of work on FD relaying, see e.g. [@5961159; @5985554; @6280258; @6862895; @7390828], the capacity of the two-hop FD relay channel with residual self-interference has not been explicitly characterized yet. As a result, for this channel, only achievable rates are known which are strictly smaller than the capacity. Therefore, in this paper, we study the capacity of the two-hop FD relay channel with residual self-interference for the case when the source-relay and relay-destination links are additive white Gaussian noise (AWGN) channels. In general, the statistics of the residual self-interference depend on the employed hardware configuration and the adopted self-interference suppression schemes. As a result, different hardware configurations and different self-interference suppression schemes may lead to different statistical properties of the residual self-interference, and thereby, to different capacities for the considered relay channel. An upper bound on the capacity of the two-hop FD relay channel with residual self-interference is given in in [@cover] and is obtained by assuming zero residual self-interference. Hence, the objective of this paper is to derive a lower bound on the capacity of this channel valid for any linear residual self-interference model. To this end, we consider the worst-case linear self-interference model with respect to the capacity, and thereby, we obtain the desired lower bound on the capacity for any other type of linear residual self-interference. For the worst-case, the linear residual self-interference is modeled as a conditionally Gaussian distributed random variable (RV) whose variance depends on the amplitude of the symbol transmitted by the relay. For this relay channel, we derive the corresponding capacity and propose an explicit coding scheme which achieves the capacity. We show that the FD relay has to operate in the decode-and-forward (DF) mode to achieve the capacity, i.e., it has to decode each codeword received from the source and then transmit the decoded information to the destination in the next time slot, while simultaneously receiving. Moreover, we show that the optimal input distribution at the relay is discrete or Gaussian, where the latter case occurs only when the relay-destination link is the bottleneck link. On the other hand, the capacity-achieving input distribution at the source is Gaussian and its variance depends on the amplitude of the symbol transmitted by the relay, i.e., the average power of the source’s transmit symbol depends on the amplitude of the relay’s transmit symbol. In particular, the smaller the amplitude of the relay’s transmit symbol is, the higher the average power of the source’s transmit symbol should be since, in that case, the residual self-interference is small with high probability. On the other hand, if the amplitude of the relay’s transmit symbol is very large and exceeds some threshold, the chance for very strong residual self-interference is high and the source should remain silent and conserve its energy for other symbol intervals with weaker residual self-interference. We show that the derived capacity converges to the capacity of the two-hop ideal FD relay channel without self-interference [@cover] and to the capacity of the two-hop HD relay channel [@zlatanov2014capacity-globecom] in the limiting cases when the residual self-interference is zero and infinite, respectively. Our numerical results reveal that significant performance gains are achieved with the proposed capacity-achieving coding scheme compared to the achievable rates of conventional HD relaying and/or conventional FD relaying. This paper is organized as follows. In Section \[Sec2\], we present the models for the channel and the residual self-interference. In Section \[Sec3\], we present the capacity of the considered channel and propose an explicit capacity-achieving coding scheme. Numerical examples are provided in Section \[Sec-Num\], and Section \[con\] concludes the paper. System Model {#Sec2} ============ In the following, we introduce the models for the two-hop FD relay channel and the residual self-interference. Channel Model ------------- We assume a two-hop FD relay channel comprised of a source, an FD relay, and a destination, where a direct source-destination link does not exist. We assume that the source-relay and the relay-destination links are AWGN channels, and that the FD relay is impaired by residual self-interference. In symbol interval $i$, let $X_S[i]$ and $X_R[i]$ denote RVs which model the transmit symbols at the source and the relay, respectively, let $\hat Y_R[i]$ and $\hat Y_D[i]$ denote RVs which model the received symbols at the relay and the destination, respectively, and let $\hat N_R[i]$ and $\hat N_D[i]$ denote RVs which model the AWGNs at the relay and the destination, respectively. We assume that $\hat N_R[i]\sim\mathcal{N}(0,\hat\sigma_R^2)$ and $\hat N_D[i]\sim\mathcal{N}(0,\hat \sigma_D^2)$, $\forall i$, where $\mathcal{N}(\mu, \sigma^2)$ denotes a Gaussian distribution with mean $\mu$ and variance $\sigma^2$. Moreover, let $h_{SR}$ and $h_{RD}$ denote the channel gains of the source-relay and relay-destination channels, respectively, which are assumed to be constant during all symbol intervals, i.e., fading[^4] is not considered. In addition, let $\hat I[i]$ denote the RV which models the residual self-interference at the FD relay that remains in symbol interval $i$ after analog and digital self-interference cancelation. Using the notations defined above, the input-output relations describing the considered relay channel in symbol interval $i$ are given as $$\begin{aligned} \hat Y_R[i]&=h_{SR} X_S[i]+ \hat I[i]+\hat N_R[i]\label{r1}\\ \hat Y_D[i]&=h_{RD} X_R[i]+\hat N_D[i].\label{r2}\end{aligned}$$ Furthermore, an average “per-node” power constraint is assumed, i.e., $$\begin{aligned} E\{X_{\beta}^2[i]\} &= \lim\limits_{n\to \infty}\frac{1}{n}\sum_{k=1}^{n}X_{\beta}^2[k] \le P_{\beta},\quad\beta\in\{S,R\}, \label{con3a}\end{aligned}$$ where $E\{\cdot\}$ denotes statistical expectation, and $P_S$ and $P_R$ are the average power constraints at the source and the relay, respectively. Residual Self-Interference Model {#sec_2-2} -------------------------------- Assuming narrow-band signals such that the channels can be modelled as frequency flat, a general model for the residual self-interference at the FD relay in symbol interval $i$, $\hat I[i]$, is given by [@Bharadia:2013:FDR:2486001.2486033] $$\begin{aligned} \label{r1_eq_1} \hat I[i]=\sum_{m=1}^M h_{RR,m}[i] \big(X_R[i]\big)^m,\end{aligned}$$ where $M\leq \infty$ is an integer and $ h_{RR,m}[i]$ is the residual self-interference channel between the transmitter-end and the receiver-end of the FD relay through which symbol $\big(X_R[i]\big)^m$ arrives at the receiver-end. Moreover, for $m=1$, $\big(X_R[i]\big)^m$ is the linear component of the residual self-interference, and for $m\geq 2$, $\big(X_R[i]\big)^m$ is a nonlinear component of the residual self-interference. As shown in [@Bharadia:2013:FDR:2486001.2486033], only the terms for which $m$ is odd in (\[r1\_eq\_1\]) carry non-negligible energy while the remaining terms for which $m$ is even can be ignored. Moreover, as observed in [@Bharadia:2013:FDR:2486001.2486033], the higher order terms in (\[r1\_eq\_1\]) carry significantly less energy than the lower order terms, i.e., the term for $m=5$ carries significantly less energy than the term for $m=3$, and the term for $m=3$ carries significantly less energy than the term for $m=1$. As a result, in this paper, we adopt the first order approximation of the residual self-interference in (\[r1\_eq\_1\]), i.e., $\hat I[i]$ is modeled as $$\begin{aligned} \label{r1_eq_2} \hat I[i]\approx h_{RR}[i] X_R[i],\end{aligned}$$ where $ h_{RR}[i]= h_{RR,1}[i]$ is used for simplicity of notation. Obviously, the residual self-interference model in (\[r1\_eq\_2\]) takes into account only the linear component of the residual self-interference and assumes that the nonlinear components can be neglected. Such a linear model for the residual self-interference is particularly justified for relays with low average transmit powers [@6353396]. The residual self-interference channel gain in (\[r1\_eq\_2\]), $ h_{RR}[i]$, is time-varying even when fading is not present, see e.g. [@5985554; @6353396; @6782415; @6832439]. The variations of the residual self-interference channel gain, $ h_{RR}[i]$, are due to the cumulative effects of various distortions originating from noise, carrier frequency offset, oscillator phase noise, analog-to-digital/digital-to-analog (AD/DA) conversion imperfections, I/Q imbalance, imperfect channel estimation, etc., see [@5985554; @6353396; @6782415; @6832439]. These distortions[^5] have a significant impact on the residual self-interference channel gain due to the very small distance between the transmitter-end and the receiver-end of the self-interference channel. Moreover, the variations of the residual self-interference channel gain, $ h_{RR}[i]$, are random and thereby cannot be accurately estimated at the FD node [@5985554; @6353396; @6782415; @6832439]. The statistical properties of these variations are dependent on the employed hardware configuration and the adopted self-interference suppression schemes. In [@6353396], $ h_{RR}[i]$ is assumed to be constant during one codeword comprised of many symbols. Thereby, the residual self-interference model in [@6353396] models only the long-term, i.e., codeword-by-codeword, statistical properties of the residual self-interference. However, the symbol-by-symbol variations of $ h_{RR}[i]$ are not captured by the model in [@6353396] since they are averaged out. Nevertheless, for a meaningful information-theoretical analysis, the statistics of the symbol-by-symbol variations of $ h_{RR}[i]$ are needed. On the other hand, the statistics of the variations of $ h_{RR}[i]$ affect the capacity of the considered relay channel. In this paper, we derive the capacity of the considered relay channel for the worst-case linear residual self-interference model, which yields a lower bound for the capacity for any other linear residual self-interference model. To derive the worst-case linear residual self-interference model in terms of capacity, we insert (\[r1\_eq\_2\]) into (\[r1\]), and obtain the received symbol at the relay in symbol interval $i$ as $$\begin{aligned} \hat Y_R[i] = h_{SR} X_S[i] + h_{RR} X_R[i] + \hat N_R[i]. \label{r1v1a} \end{aligned}$$ Now, since in general $h_{RR}[i]$ can have a non-zero mean, without loss of generality, we can write $h_{RR}[i]$ as $$\begin{aligned} h_{RR}[i] &= \bar h_{RR} + \hat h_{RR}[i] , \label{eq_eq_x-h1}\end{aligned}$$ where $\bar h_{RR}$ is the mean of $h_{RR}[i]$, i.e., $\bar h_{RR}=E\{h_{RR}[i]\}$ and $\hat h_{RR}[i]=h_{RR}[i]-\bar h_{RR}$ is the remaining zero-mean random component of $h_{RR}[i]$. Inserting (\[eq\_eq\_x-h1\]) into (\[r1v1a\]), we obtain the received symbol at the relay in symbol interval $i$ as $$\begin{aligned} \hat Y_R[i]&= h_{SR} X_S[i] + \bar h_{RR} X_R[i]+ \hat h_{RR}[i] X_R[i] + \hat N_R[i]. \label{r1v1} \end{aligned}$$ Given sufficient time, the relay can estimate any mean in its received symbols arbitrarily accurately, see [@841172]. Thereby, given sufficient time, the relay can estimate the deterministic component of the residual self-interference channel gain $\bar h_{RR}$. Moreover, since $X_R[i]$ models the desired transmit symbol at the relay, and since the relay knows which symbol it wants to transmits, the outcome of $X_R[i]$, denoted by $x_R[i]$, is known in each symbol interval $i$. As a result, the relay knows $\bar h_{RR} X_R[i]$ and thereby it can subtract $\bar h_{RR} X_R[i]$ from the received symbol $\hat Y_R[i]$ in (\[r1v1\]). Consequently, we obtain a new received symbol at the relay in symbol interval $i$, denoted by $\tilde Y_R[i]$, as $$\begin{aligned} \tilde Y_R[i]&= h_{SR} X_S[i] + \hat h_{RR}[i] X_R[i] + \hat N_R[i] . \label{r1v3a} \end{aligned}$$ Now, assuming that the relay transmits symbol $x_R[i]$ in symbol interval $i$, i.e., $X_R[i]=x_R[i]$, from (\[r1v3a\]) we conclude that the relay “sees” the following additive impairment in symbol interval $i$ $$\begin{aligned} \label{eq_r1-eq3} \hat h_{RR}[i] x_R[i] + \hat N_R[i]. \end{aligned}$$ Consequently, from (\[eq\_r1-eq3\]), we conclude that the worst-case scenario with respect to the capacity is if (\[eq\_r1-eq3\]) is zero-mean independent and identically distributed (i.i.d.) Gaussian RV[^6], which is possible only if $ \hat h_{RR}[i] $ is a zero-mean i.i.d. Gaussian RV. Hence, modeling the linear residual self-interference channel gain, $ h_{RR}[i]$, as an i.i.d. Gaussian RV constitutes the worst-case linear residual self-interference model, and thereby, leads to a lower bound on the capacity for any other distribution of $ h_{RR}[i]$. Considering the developed worst-case linear residual self-interference model, in the rest of this paper, we assume $ \hat h_{RR}[i]\sim\mathcal{N}(0,\hat\alpha) $, where $\hat \alpha$ is the variance of $\hat h_{RR}[i]$. Since the average power of the linear residual self-interference at the relay is $\hat \alpha E\{X_R^2[i]\}$, $\hat \alpha$ can be interpreted as a self-interference amplification factor, i.e., $1/\hat \alpha$ is a self-interference suppression factor. Simplified Input-Output Relations for the Considered Relay Channel ------------------------------------------------------------------ To simplify the input-output relation in (\[r1v3a\]), we divide the received symbol $\tilde Y_R[i]$ by $h_{SR}$ and thereby obtain a new received symbol at the relay in symbol interval $i$, denoted by $Y_R[i]$, and given by $$\begin{aligned} Y_R[i]&= X_S[i] + \frac{ \hat h_{RR}[i] }{h_{SR}} X_R[i] +\frac{\hat N_R[i]}{h_{SR}} = X_S[i] + I[i] + N_R[i], \label{r1v3} \end{aligned}$$ where $$\begin{aligned} \label{eq_asd} I[i]= \frac{\hat h_{RR}[i] }{h_{SR}} X_R[i] \end{aligned}$$ is the normalized residual self-interference at the relay and $ N_R[i]= \hat N_R[i] /h_{SR}$ is the normalized noise at the relay distributed according to $N_R[i]\sim\mathcal{N}(0, \sigma_R^2)$, where $\sigma_R^2= \hat\sigma_R^2/h_{SR}^2$. The normalized residual self-interference, $ I[i]$, is dependent on the transmit symbol at the relay, $X_R[i]$, and, conditioned on $X_R[i]$, it has the same type of distribution as the random component of the self-interference channel gain, $\hat h_{RR}[i]$, i.e., an i.i.d. Gaussian distribution. Let $\alpha$ be defined as $\alpha=\hat \alpha/h_{SR}^2$, which can be interpreted as the normalized self-interference amplification factor. Using $\alpha$ and assuming that the transmit symbol at the relay in symbol interval $i$ is $X_R[i]=x_R[i]$, the distribution of the normalized residual self-interference, $ I[i]$, can be written as $$\begin{aligned} \label{eq_sdxx2} I[i]\sim\mathcal{N}(0, \alpha x_R^2[i]),\; \textrm{ if } \; X_R[i]=x_R[i].\end{aligned}$$ To obtain also a normalized received symbol at the destination, we normalize $\hat Y_D[i]$ in (\[r2\]) by $h_{RD}$, which yields $$\begin{aligned} Y_D[i]&= X_R[i]+N_D[i].\label{r2a}\end{aligned}$$ In (\[r2a\]), $N_D[i]$ is the normalized noise power at the destination distributed as $N_D[i] = \mathcal{N}\left(0, \sigma_D^2 \right) $, where $ \sigma_D^2 = \hat \sigma_D^2/h^2_{RD} $. Now, instead of deriving the capacity of the considered relay channel using the input-output relations in (\[r1\]) and (\[r2\]), we can derive the capacity using an equivalent relay channel defined by the input-output relations in (\[r1v3\]) and (\[r2a\]), respectively, where, in symbol interval $i$, $X_S[i]$ and $X_R[i]$ are the inputs at source and relay, respectively, $Y_R[i]$ and $Y_D[i]$ are the outputs at relay and destination, respectively, $N_R[i]$ and $N_D[i]$ are AWGNs with variances $\sigma_R^2= \hat \sigma_R^2/h_{SR}^2$ and $\sigma_D^2=\hat \sigma_D^2/h_{RD}^2$, respectively, and $ I[i]$ is the residual self-interference with conditional distribution given by (\[eq\_sdxx2\]), which is a function of the normalized self-interference amplification factor $ \alpha$. Capacity {#Sec3} ======== In this section, we study the capacity of the considered Gaussian two-hop FD relay channel with residual self-interference. Derivation of the Capacity -------------------------- To derive the capacity of the considered relay channel, we first assume that RVs $X_S$ and $X_R$, which model the transmit symbols at source and relay for any symbol interval $i$, take values $x_S$ and $x_R$ from sets $\mathcal{X}_S$ and $\mathcal{X}_R$, respectively. Now, since the considered relay channel belongs to the class of memoryless degraded relay channels defined in [@cover], its capacity is given by [@cover Theorem 1] $$\begin{aligned} \label{con2} C=\max_{p(x_S,x_R)\in\mathcal{P}}&~\min\big\{I(X_S;Y_R|X_R),I(X_R;Y_D)\big\} \nonumber\\ \textrm{Subject to} \textrm{ C1: }& E\{X_S^2\}\leq P_S\nonumber\\ \textrm{ C2: }& E\{X_R^2\}\leq P_R,\end{aligned}$$ where $\mathcal{P}$ is a set which contains all valid distributions. In order to obtain the capacity in (\[con2\]), we need to find the optimal joint input distribution, $p(x_S,x_R)$, which maximizes the $\min\{\cdot\}$ function in (\[con2\]) and satisfies constraints C1 and C2. To this end, note that $p(x_S,x_R)$ can be written equivalently as $ p(x_S,x_R)=p(x_S|x_R)p(x_R). $ Using this relation, we can represent the maximization in (\[con2\]) equivalently as two nested maximizations, one with respect to $p(x_S|x_R)$ for a fixed $p(x_R)$, and the other one with respect to $p(x_R)$. Thereby, we can write the capacity in (\[con2\]) equivalently as $$\begin{aligned} C=\max_{p(x_R)\in\mathcal{P}}~\max_{p(x_S|x_R)\in\mathcal{P}} ~ & \mathrm{min}\big\{I(X_S;Y_R|X_R),I(X_R;Y_D)\big\}\nonumber\\ \textrm{Subject to} \textrm{ C1: }& E\{X_S^2\}\leq P_S\nonumber\\ \textrm{ C2: }& E\{X_R^2\}\leq P_R.\label{con2a}\end{aligned}$$ Now, since in the $\min\{\cdot\}$ function in (\[con2a\]) only $I(X_S;Y_R|X_R)$ is dependent on the distribution $p(x_S|x_R)$, whereas $I(X_R;Y_D)$ does not depend on $p(x_S|x_R)$, we can write the capacity expression in (\[con2a\]) equivalently as $$\begin{aligned} C=\max_{p(x_R)\in\mathcal{P}} &~ \mathrm{min}\left\{\max_{p(x_S|x_R)\in\mathcal{P}}I(X_S;Y_R|X_R),I(X_R;Y_D)\right\}\nonumber\\ \textrm{Subject to} \textrm{ C1: }& E\{X_S^2\}\leq P_S\nonumber\\ \textrm{ C2: }& E\{X_R^2\}\leq P_R. \label{con2b}\end{aligned}$$ Hence, to obtain the capacity of the considered relay channel, we first need to find the conditional input distribution at the source, $p(x_S|x_R)$, which maximizes $I(X_S;Y_R|X_R)$ such that constraint C1 holds. Next, we need to find the optimal input distribution at the relay, $p(x_R)$, which maximizes the $\min\{\cdot\}$ expression in (\[con2b\]) such that constraints C1 and C2 hold. Optimal Input Distribution at the Source $p^*(x_S|x_R)$ ------------------------------------------------------- The optimal input distribution at the source which achieves the capacity in (\[con2b\]), denoted by $p^*(x_S|x_R)$, is given in the following theorem. \[Theo1\] The optimal input distribution at the source $p^*(x_S|x_R)$, which achieves the capacity of the considered relay channel in (\[con2b\]), is the zero-mean Gaussian distribution with variance $P_S(x_R)$ given by $$\begin{aligned} \label{P1} P_S(x_R)=\alpha\max\{0,x_{\rm th}^2- x_R^2\}, \end{aligned}$$ where $x_{\rm th}$ is a positive threshold constant found as follows. If $p(x_R)$ is a discrete distribution, $x_{\rm th}$ is found as the solution of the following identity $$\begin{gathered} \sum_{x_R\in \mathcal{X}_R} \alpha \max\{0,x_{\rm th}^2- x_R^2\} p(x_R) = P_S, \label{36a} \end{gathered}$$ and the corresponding $\max\limits_{p(x_S|x_R)\in\mathcal{P}} I(X_S;Y_R|X_R)$ is obtained as $$\begin{aligned} \label{eq_1-dis} \max_{p(x_S|x_R)\in\mathcal{P}} I(X_S;Y_R|X_R) = \sum\limits_{x_R\in \mathcal{X}_R} \frac{1}{2} \log_2\left(1+\frac{\alpha \max\{0,\;x_{\rm th}^2- x_R^2\}}{\sigma_R^2+\alpha x_R^2}\right) p(x_R) . \end{aligned}$$ Otherwise, if $p(x_R)$ is a continuous distribution, the sums in (\[36a\]) and (\[eq\_1-dis\]) have to be replaced by integrals. Please refer to Appendix A. From Theorem \[Theo1\], we can see that the source should perform power allocation in a symbol-by-symbol manner. In particular, the average power of the source’s transmit symbols, $P_S(x_R)$, given by (\[P1\]), depends on the amplitude of the transmit symbol at the relay, $|x_R|$. The lower the amplitude of the transmit symbol of the relay is, the higher the average power of the source’s transmit symbols should be since, in that case, there is a high probability for weak residual self-interference. Conversely, the higher the amplitude of the transmit symbol of the relay is, the lower the average power of the source’s transmit symbols should be since, in that case, there is a high probability for strong residual self-interference. If the amplitude of the transmit symbol of the relay exceeds the threshold $ x_{\rm th}$, the chance for very strong residual self-interference becomes too high, and the source remains silent to conserve energy for the cases when the residual self-interference is weaker. On the other hand, from the relay’s perspective, the relay transmits high-amplitude symbols, i.e., symbols which have an amplitude which exceeds the threshold $x_{\rm th}$, only when the source is silent as such high amplitude symbols cause strong residual self-interference. Optimal Input Distribution at the Relay ${p^*(x_R)}$ ---------------------------------------------------- The optimal input distribution at the relay, denoted by $p^*(x_R)$, which achieves the capacity of the considered relay channel is given in the following theorem. \[theo\_2\] If condition $$\begin{aligned} \log_2\left(1+\frac{ P_R}{\sigma_D^2}\right) \leq \displaystyle\int\limits_{-x_{\rm th}}^{x_{\rm th}} \log_2\left(1+ \frac{\alpha(x_{\rm th}^2-x_R^2)}{\sigma_R^2+\alpha x_R^2}\right) \frac{1}{\sqrt{2\pi P_R}} e^{-\frac{x_R^2}{2 P_R}} dx_R \label{39}\end{aligned}$$ holds, where the amplitude threshold $x_{\rm th}$ is found from $$\begin{aligned} \sqrt{\frac{2 P_R }{\pi}} \alpha x_{\rm th} \exp\left(-\frac{x_{\rm th}^2}{2 P_R }\right) +\alpha(x_{\rm th}^2 - P_R)\mathrm{erf}\left(\frac{x_{\rm th}}{\sqrt{2 P_R}}\right)=P_S,\label{40}\end{aligned}$$ with $\mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^2}dt$, then the optimal input distribution at the relay, $p^*(x_R)$, is the zero-mean Gaussian distribution with variance $ P_R$ and the corresponding capacity of the considered relay channel is given by $$\begin{aligned} C=\frac{1}{2}\log_2\left(1+\frac{ P_R}{\sigma_D^2}\right). \label{cap_1}\end{aligned}$$ Otherwise, if condition (\[39\]) does not hold, then the optimal input distribution at the relay, $p^*(x_R)$, is discrete and symmetric with respect to $x_R=0$. Furthermore, the capacity and the optimal input distribution at the relay, $p^*(x_R)$, can be found by solving the following concave optimization problem $$\begin{aligned} C=\max_{p(x_R)\in\mathcal{P}} & \sum_{x_R\in \mathcal{X}_R} \frac{1}{2} \log_2\left(1+\frac{\alpha \max\{0,x_{\rm th}^2- x_R^2\}}{\sigma_R^2+\alpha x_R^2}\right) p(x_R)\nonumber\\ \textrm{Subject to} \textrm{ C1: }& \sum_{x_R\in \mathcal{X}_R} \frac{1}{2} \log_2\left(1+\frac{\alpha \max\{0,x_{\rm th}^2- x_R^2\}}{\sigma_R^2+\alpha x_R^2}\right) p(x_R) \leq I(X_R;Y_D)\nonumber\\ \textrm{ C2: }& \sum_{x_R\in \mathcal{X}_R} x_R^2 p(x_R)\leq P_R\nonumber\\ \textrm{ C3: }& \sum_{x_R\in \mathcal{X}_R} \alpha \max\{0,x_{\rm th}^2 - x_R^2\} p(x_R) = P_S. \label{cap_2}\end{aligned}$$ Moreover, solving (\[cap\_2\]) reveals that constraint C1 has to hold with equality and that $p^*(x_R)$ has the following discrete form $$\begin{aligned} p^*(x_R)=p_{R,0}\delta(x_R)+\sum_{j=1}^{J}\frac{1}{2}p_{R,j}(\delta(x_R-x_{R,j})+\delta(x_R+x_{R,j})),\label{n10} \end{aligned}$$ where $p_{R,j}\in[0,1]$ is the probability that $X_R=x_{R,j}$ will occur, where $x_{R,j}>0$ and $\sum_{j=0}^{J}p_{R,j}=1$ hold. With $p^*(x_R)$ as in (\[n10\]), the capacity has the following general form $$\begin{aligned} \label{cap_2a} C= \frac{p_{R,0}}{2} \log_2\left(1+\frac{\alpha x_{\rm th}^2}{\sigma_R^2} \right) + \sum_{j=1}^{J} \frac{p_{R,j}}{2} \log_2\left(1+\frac{\alpha \max\{0,x_{\rm th}^2- x_{R,j}^2\}}{\sigma_R^2+\alpha x_{R,j}^2}\right) .\end{aligned}$$ Please refer to Appendix B. From Theorem \[theo\_2\], we can draw the following conclusions. If condition (\[39\]) holds, then the relay-destination channel is the bottleneck link. In particular, even if the relay transmits with a zero-mean Gaussian distribution, which achieves the capacity of the relay-destination channel, the capacity of the relay-destination channel is still smaller than the mutual information (i.e., data rate) of the source-relay channel. Otherwise, if condition (\[39\]) does not hold, then the optimal input distribution at the relay, $p^*(x_R)$, is always discrete and symmetric with respect to $x_R=0$. Moreover, in this case, the mutual informations of the source-relay and relay-destination channels have to be equal, i.e., $I(X_S;Y_R|X_R)\big|_{p(x_R)=p^*(x_R)}=I(X_R;Y_D)\big|_{p(x_R)=p^*(x_R)}$ has to hold. In addition, we note that constraint C2 in (\[cap\_2\]) does not always have to hold with equality, i.e., in certain cases it is optimal for the relay to reduce its average transmit power. In particular, if the relay-destination channel is very strong compared to the source-relay channel, then, by reducing the average transmit power of the relay, we reduce the average power of the residual self-interference in the source-relay channel, and thereby improve the quality of the source-relay channel. We note that this phenomenon was first observed in [@5089955] and [@5961159], where it was shown that, in certain cases, it is beneficial for FD relays to not transmit with the maximum available average power. However, even if the average transmit power of the relay is reduced, $I(X_S;Y_R|X_R)\big|_{p(x_R)=p^*(x_R)}=I(X_R;Y_D)\big|_{p(x_R)=p^*(x_R)}$ still has to hold, for the data rates of the source-relay and the relay-destination channels to be equal. From (\[40\]), it can be observed that threshold $x_{\rm th}$ is inversely proportional to the normalized self-interference amplification factor $\alpha$. In other words, the smaller $\alpha$ is, the larger $x_{\rm th}$ becomes. In the limit, when $\alpha\to 0$, we have $x_{\rm th}\to\infty$. This is expected since for smaller $\alpha$, the average power of the residual self-interference also becomes smaller, which allows the source to transmit more frequently. If $\alpha\to 0$ the residual self-interference tends to zero. Consequently, the source should never be silent, i.e., $x_{\rm th}\to\infty$, which is in line with the optimal behavior of the source for the case of ideal FD relaying without residual self-interference described in [@cover]. On the other hand, inserting the solution for $x_{\rm th}$ from (\[40\]) into (\[39\]), and then evaluating (\[39\]), it can be observed that the right hand-side of (\[39\]) is a strictly decreasing function of $\alpha$. This is expected since larger $\alpha$ result in a residual self-interference with larger average power and thereby in a smaller achievable rate on the source-relay channel. Achievability of the Capacity ----------------------------- The source wants to transmit message $W$ to the destination, which is drawn uniformly from a message set $\{1,2,...,2^{nR}\}$ and carries $nR$ bits of information, where $n\to\infty$. To this end, the transmission time is split into $B+1$ time slots and each time slot is comprised of $k$ symbol intervals, where $B\to\infty$ and $k\to\infty$. Moreover, message $W$ is split into $B$ messages, denoted by $w(1),...,w(B)$, where each $w(b)$, for $b=1,...,B$, carries $kR$ bits of information. Each of these messages is to be transmitted in a different time slot. In particular, in time slot one, the source sends message $w(1)$ during $k$ symbol intervals to the relay and the relay is silent. In time slot $b$, for $b=2,...,B$, source and relay send messages $w(b)$ and $w(b-1)$ to relay and destination during $k$ symbol intervals, respectively. In time slot $B+1$, the relay sends message $w(B)$ to the destination during $k$ symbol intervals and the source is silent. Hence, in the first time slot, the relay is silent since it does not have information to transmit, and in time slot $B+1$, the source is silent since it has no more information to transmit. In time slots $2$ to $B$, both source and relay transmit. During the $B+1$ time slots, the channel is used $k(B+1)$ times to send $nR=BkR$ bits of information, leading to an overall information rate of $$\begin{aligned} \lim_{B\to\infty} \lim_{k\to\infty} \frac{ Bk R}{k(B+1)}=R \;\;\textrm{ bits/symbol}.\end{aligned}$$ A detailed description of the proposed coding scheme for each time slot is given in the following, where we explain the rates, codebooks, encoding, and decoding used for transmission. We note that the proposed achievability scheme requires all three nodes to have full channel state information (CSI) of the source-relay and relay-destination channels as well as knowledge of the self-interference suppression factor $1/\hat\alpha$. *Rates:* The transmission rate of both source and relay is denoted by $R$ and given by $$R=C -\epsilon ,\label{self-interferenceeq_r_2}$$ where $C$ is given in Theorem \[theo\_2\] and $\epsilon>0$ is an arbitrarily small number. *Codebooks:* We have two codebooks, namely, the source’s transmission codebook and the relay’s transmission codebook. The source’s transmission codebook is generated by mapping each possible binary sequence comprised of $k R$ bits, where $R$ is given by (\[self-interferenceeq\_r\_2\]), to a codeword $\mathbf{ x}_{S}$ comprised of $k p_T$ symbols, where $p_T$ is the following probability $$\begin{aligned} \label{eq_pt} p_T={\rm Pr}\left\{|x_R|< x_{\rm th }\right\}.\end{aligned}$$ Hence, $p_T$ is the probability that the relay will transmit a symbol with an amplitude which is smaller than the threshold $x_{\rm th}$. In other words, $p_T$ is the fraction of symbols in the relay’s codeword which have an amplitude which is smaller than the threshold $x_{\rm th}$. The symbols in each codeword $\mathbf{ x}_{S}$ are generated independently according to the zero-mean *unit variance* Gaussian distribution. Since in total there are $2^{k R}$ possible binary sequences comprised of $k R $ bits, with this mapping, we generate $2^{k R }$ codewords $\mathbf{ x}_{S}$ each comprised of $k p_T$ symbols. These $2^{k R }$ codewords form the source’s transmission codebook, which we denote by $\mathcal{ C}_{S}$. On the other hand, the relay’s transmission codebook is generated by mapping each possible binary sequence comprised of $k R $ bits, where $R $ is given by (\[self-interferenceeq\_r\_2\]), to a transmission codeword $\mathbf{x}_R$ comprised of $k$ symbols. The symbols in each codeword $\mathbf{x}_R$ are generated independently according to the optimal distribution $p^*(x_R)$ given in Theorem \[theo\_2\]. The $2^{k R}$ codewords $\mathbf{x}_R$ form the relay’s transmission codebook denoted by $\mathcal{C}_R$. The two codebooks are known at all three nodes. Moreover, the power allocation policy at the source, $P_S(x_R)$, given in (\[P1\]), is assumed to be known at source and relay. We note that the source’s codewords, $\mathbf{x}_S$, are shorter than the relay’s codewords, $\mathbf{x}_R,$ since the source is silent in $1-p_T$ fraction of the symbol intervals because of the expected strong interference in those symbol intervals. Since the relay transmits during the symbol intervals for which the source is silent, its codewords are longer than the codewords of the source. Note that if the silent symbols of the source are taken into account and counted as part of the source’s codeword, then both codewords will have the same length. *Encoding, Transmission, and Decoding:* In the first time slot, the source maps $w(1)$ to the appropriate codeword $\mathbf{ x}_{S}(1)$ from its codebook $\mathcal{ C}_{S}$. Then, codeword $\mathbf{ x}_{S}(1)$ is transmitted to the relay, where each symbol of $\mathbf{ x}_{S}(1)$ is amplified by $\sqrt{P_S(x_R=0)}$, where $P_S(x_R)$ is given in (\[P1\]). On the other hand, the relay is scheduled to always receive and be silent (i.e., to set its transmit symbol to zero) during the first time slot. However, knowing that the codeword transmitted by the source in the first time slot, $\mathbf{ x}_{S}(1)$, is comprised of $k p_T$ symbols, the relay constructs the received codeword, denoted by $\mathbf{ y}_{R}(1)$, only from the first $k p_T$ received symbols. \[lem\_1\] The codeword $\mathbf{ x}_{S}(1)$ sent in the first time slot can be decoded successfully from the codeword received at the relay, $\mathbf{ y}_{R}(1)$, using a typical decoder [@cover2012elements] since $R $ satisfies $$\begin{aligned} \label{eq_d_1aa} R < \max_{p(x_{S}|x_R=0)} I\big(X_{S}; Y_{R}| X_R=0\big) p_T = \frac{1}{2} \log_2\left(1+\frac{\alpha x_{\rm th}^2 }{\sigma_R^2 }\right) p_T .\end{aligned}$$ Please refer to Appendix \[app\_5\]. In time slots $b=2,...,B$, the encoding, transmission, and decoding are performed as follows. In time slots $b=2,...,B$, the source and the relay map $w(b)$ and $w(b-1)$ to the appropriate codewords $\mathbf{ x}_S (b)$ and $\mathbf{x}_R(b)$ from codebooks $\mathcal{ C}_{S }$ and $\mathcal{C}_{R}$, respectively. Note that the source also knows $\mathbf{x}_R(b)$ since $\mathbf{x}_R(b)$ was generated from $w(b-1)$ which the source transmitted in the previous (i.e., the $(b-1)$-th) time slot. As a result, both source and relay know the symbols in $\mathbf{x}_R(b)$ and can determine whether their amplitudes are smaller or larger than the threshold $x_{\rm th}$. Hence, if the amplitude of the first symbol in codeword $\mathbf{x}_R(b)$ is smaller than the threshold $x_{\rm th}$, then, in the first symbol interval of time slot $b$, the source transmits the first symbol from codeword $\mathbf{ x}_{S}(b)$ amplified by $\sqrt{P_S(x_{R,1})}$, where $x_{R,1}$ is the first symbol in relay’s codeword $\mathbf{x}_R(b)$ and $P_S(x_R)$ is given by (\[P1\]). Otherwise, if the amplitude of the first symbol in codeword $\mathbf{x}_R(b)$ is larger than threshold $x_{\rm th}$, then the source is silent. The same procedure is performed for the $j$-th symbol interval in time slot $b$, for $j=1,...,k$. In particular, if the amplitude of the $j$-th symbol in codeword $\mathbf{x}_R(b)$ is smaller than threshold $x_{\rm th}$, then in the $j$-th symbol interval of time slot $b$, the source transmits its next untransmitted symbol from codeword $\mathbf{ x}_{S}(b)$ amplified by $\sqrt{P_S(x_{R,j})}$, where $x_{R,j}$ is the $j$-th symbol in relay’s codeword $\mathbf{x}_R(b)$. Otherwise, if the amplitude of the $j$-th symbol in codeword $\mathbf{x}_R(b)$ is larger than threshold $x_{\rm th}$, then for the $j$-th symbol interval of time slot $b$, the source is silent. On the other hand, the relay transmits all symbols from $\mathbf{x}_R(b)$ while simultaneously receiving. Let $\mathbf{\hat y}_R(b)$ denote the received codeword at the relay in time slot $b$. Then, the relay discards those symbols from the received codeword, $\mathbf{\hat y}_R(b)$, for which the corresponding symbols in $\mathbf{x}_R(b)$ have amplitudes which exceed threshold $x_{\rm th}$, and only collects the symbols in $\mathbf{\hat y}_R(b)$ for which the corresponding symbols in $\mathbf{x}_R(b)$ have amplitudes which are smaller than $x_{\rm th}$. The symbols collected from $\mathbf{\hat y}_R(b)$ constitute the relay’s information-carrying received codeword, denoted by $\mathbf{ y}_{R}(b)$, which is used for decoding. \[lema\_2\] The codewords $\mathbf{ x}_{S}(b)$ sent in time slots $b=2,\dots,B$ can be decoded successfully at the relay from the corresponding received codewords $\mathbf{ y}_{R}(b)$, respectively, using a jointly typical decoder since $R$ satisfies $$\begin{aligned} R <\sum_{ x_R\in\mathcal{X}_R }\max_{p(x_{S}|x_R)} I\big(X_{S}; Y_{R}| X_R=x_R\big) p^*(x_R) = \sum\limits_{ x_R\in\mathcal{X}_R } \frac{1}{2} \log_2\left(1+\frac{\alpha \max\{0,\;x_{\rm th}^2- x_R^2\}}{\sigma_R^2+\alpha x_R^2}\right) p^*(x_R) . \label{self-interferenceeq_r_2b}\end{aligned}$$ Please refer to Appendix \[app\_4\]. On the other hand, the destination listens during the entire time slot $b$ and receives a codeword denoted by $\mathbf{y}_D(b)$. By following the “standard" method for analyzing the probability of error for rates smaller than the capacity, given in [@cover2012elements Sec. 7.7], it can be shown in a straightforward manner that the destination can successfully decode $\mathbf{x}_R(b)$ from the received codeword $\mathbf{y}_D(b)$, and thereby obtain $w(b-1)$, since rate $R$ satisfies $$\begin{aligned} R <I(X_R; Y_{D})\big|_{p(x_R)=p^*(x_R)}, \label{self-interferenceeq_r_2a}\end{aligned}$$ where $I(X_R; Y_{D})$ is given in Theorem \[theo\_2\]. In the last (i.e., the $(B+1)$-th) time slot, the source is silent and the relay transmits $w(B)$ by mapping it to the corresponding codeword $\mathbf{x}_R(B+1)$ from codebook $\mathcal{C}_{R}$. The relay transmits all symbols in codeword $\mathbf{x}_R(B+1)$ to the destination during time slot $B+1$. The destination can decode the received codeword in time slot $ B+1 $ successfully, since (\[self-interferenceeq\_r\_2a\]) holds. Finally, since both relay and destination can decode their respective codewords in each time slot, the entire message $W$ can be decoded successfully at the destination at the end of the $(B+1)$-th time slot. A block diagram of the proposed coding scheme is presented in Fig. \[Fig:Channel\]. In particular, in Fig. \[Fig:Channel\], we show schematically the encoding, transmission, and decoding at source, relay, and destination. The flow of encoding/decoding in Fig. \[Fig:Channel\] is as follows. Messages $w(b-1)$ and $w(b)$ are encoded into $\mathbf{x}_R(b)$ and $ \mathbf{ x}_{S}(b)$ at the source using the encoders $\mathrm{C}_R$ and $\mathrm{ C}_{S}$, respectively. Then, an inserter $\mathrm{In}$ is used to create a vector $\mathbf{\hat x}_{S}(b)$ by inserting the symbols of $\mathbf{ x}_{S}(b)$ into the positions of $\mathbf{\hat x}_{S}(b)$ for which the corresponding elements of $\mathbf{x}_R(b)$ have amplitudes smaller than $x_{\rm th}$ and setting all other symbols in $\mathbf{\hat x}_{S}(b)$ to zero. Hence, vector $\mathbf{\hat x}_{S}(b)$ is identical to codeword $\mathbf{ x}_{S}(b)$ except for the added silent (i.e., zero) symbols generated at the source. The source then transmits $\mathbf{\hat x}_{S}(b)$ and the relay receives the corresponding codeword $\mathbf{\hat y}_{R}(b)$. Simultaneously, the relay encodes $w(b-1)$ into $\mathbf{x}_{R}(b)$ using encoder $\mathrm{C}_R$ and transmits it to the destination, which receives codeword $\mathbf{y}_{D}(b)$. Next, using $\mathbf{x}_R(b)$, the relay constructs $\mathbf{ y}_{R}(b)$ from $\mathbf{\hat y}_{R}(b)$ by selecting only those symbols from $\mathbf{\hat y}_{R}(b)$ for which the corresponding symbols in $\mathbf{x}_R(b)$ have amplitudes smaller than $x_{\rm th}$. Using decoder $\mathrm{D}_R$, the relay then decodes $\mathbf{ y}_{R}(b)$ into $w(b)$ and stores the decoded bits in its buffer $\mathrm{Q}$. On the other hand, the destination decodes $\mathbf{y}_{D}(b)$ into $w(b-1)$ using decoder $\mathrm{D}_D$. Analytical Expression for Tight Lower Bound on the Capacity {#sec_num_1} ----------------------------------------------------------- For the non-trivial case when condition (\[39\]) does not hold, i.e., the relay-destination link is not the bottleneck, the capacity of the Gaussian two-hop FD relay channel with residual self-interference is given in the form of an optimization problem, cf. (\[cap\_2\]), which is not suitable for analysis. As a result, in this subsection, we propose a suboptimal input distribution at the relay, which yields an analytical expression for a lower bound on the capacity, derived in Theorem \[theo\_2\]. Our numerical results show that this lower bound is tight, at least for the considered numerical examples, cf. Fig. \[fig\_1\_new\]. In particular, we propose that the relay uses the following input distribution $$\begin{aligned} \label{eq_r_in_app} p(x_R)=p_{\rm B}(x_R)=q \frac{1}{\sqrt{2\pi p_R/q}}\exp\left(-\frac{x_R^2}{2p_R/q}\right) + (1-q) \delta(x_R),\end{aligned}$$ where the value of $p_R$ is optimized in the range $p_R\leq P_R$ in order for the rate to be maximized. Hence, with probability $q$, the relay transmits a symbol from a zero-mean Gaussian distribution with variance $p_R/q$, and is silent with probability $1-q$. Since the relay transmits only in $q$ fraction of the time, the average transmit power when the relay transmits is set to $p_R/q$ in order for the average transmit power during the entire transmission time to be $ p_R$. Now, with the input distribution $p_{\rm B}(x_R)$ in (\[eq\_r\_in\_app\]), we obtain the mutual information of the source-relay channel as $$\begin{aligned} \label{eq_Isr-app} & \max_{p(x_S|x_R)\in\mathcal{P}} I(X_S;Y_R|X_R) \bigg|_{p(x_R)=p_{\rm B}(x_R)}\\ &= q \int\limits_{-x_{\rm th}}^{x_{\rm th}} \hspace{-2mm} \frac{1}{2} \log_2\left(1+ \frac{\alpha(x_{\rm th}^2-x_R^2)}{\sigma_R^2+\alpha x_R^2}\right) \frac{1}{\sqrt{2\pi p_R/q}}\exp\left(-\frac{x_R^2}{2p_R/q}\right) dx_R + (1-q) \frac{1}{2} \log_2\left(1+ \frac{\alpha x_{\rm th}^2}{\sigma_R^2}\right), \nonumber\end{aligned}$$ and the mutual information of the relay-destination channel as $$\begin{aligned} \label{eq_Ird-app} &I(X_R;Y_D) \bigg|_{p(x_R)=p_{\rm B}(x_R)} \nonumber\\ &=- \int_{-\infty}^\infty \Bigg[ q \frac{1}{\sqrt{2\pi (p_R/q+\sigma_D^2)}} \exp\left( -\frac{y_D^2}{2 (p_R/q+\sigma_D^2)} \right) +(1-q) \frac{1}{\sqrt{2\pi \sigma_D^2}} \exp\left( -\frac{y_D^2}{2 \sigma_D^2} \right) \Bigg] \nonumber\\ &\quad\times \log_2\Bigg( q \frac{1}{\sqrt{2\pi (p_R/q+\sigma_D^2)}} \exp\left( -\frac{y_D^2}{2 (p_R/q+\sigma_D^2)} \right) +(1-q) \frac{1}{\sqrt{2\pi \sigma_D^2}} \exp\left( -\frac{y_D^2}{2 \sigma_D^2} \right) \Bigg) \nonumber\\ &~-\frac{1}{2}\log_2(2\pi e \sigma_D^2).\end{aligned}$$ The threshold $x_{\rm th}$ in (\[eq\_Isr-app\]) and the probability $q$ in (\[eq\_Isr-app\]) and (\[eq\_Ird-app\]) are found from the following system of two equations $$\begin{aligned} \label{36b-1} \left\{ \begin{array}{ll} &q\left(\sqrt{\frac{2 p_R/q }{\pi}} \alpha x_{\rm th} \exp\left(-\frac{x_{\rm th}^2}{2 p_R/q }\right) +\alpha(x_{\rm th}^2 - p_R/q)\mathrm{erf}\left(\frac{x_{\rm th}}{\sqrt{2 p_R/q }}\right)\right)+ (1-q)\alpha x_{\rm th}^2 = P_S \\ % &\max\limits_{p(x_S|x_R)\in\mathcal{P}} I(X_S;Y_R|X_R) \bigg|_{p(x_R)=p_{\rm B}(x_R)} =I(X_R;Y_D) \bigg|_{p(x_R)=p_{B}(x_R)}. \end{array} \right.\end{aligned}$$ Thereby, $x_{\rm th}$ and $q$ are obtained as a function of $p_R$. Now, the achievable rate with the suboptimal input distribution $p_{\rm B}(x_R)$ is found by inserting $x_{\rm th}$ and $q$ found from (\[36b-1\]) into (\[eq\_Isr-app\]) or (\[eq\_Ird-app\]), and then maximizing (\[eq\_Isr-app\]) or (\[eq\_Ird-app\]) with respect to $p_R$ such that $p_R\leq P_R$ holds. Numerical Evaluation {#Sec-Num} ==================== In this section, we numerically evaluate the capacity of the considered two-hop FD relay channel with self-interference and compare it to several benchmark schemes. To this end, we first provide the system parameters, introduce benchmark schemes, and then present the numerical results. System Parameters {#set_par} ----------------- We compute the channel gains of the source-relay ($SR$) and relay-destination ($RD$) links using the standard path loss model $$\begin{aligned} \label{eq_h1} h_{L}^2=\left(\frac{c }{f_{c} 4\pi}\right)^2 d_{L}^{-\gamma},\;\; \textrm{for }L\in\{SR,RD\},\end{aligned}$$ where $c$ is the speed of light, $f_c$ is the carrier frequency, $d_L$ is the distance between the transmitter and the receiver of link $L$, and $\gamma$ is the path loss exponent. For the numerical examples in this section, we assume $\gamma=3$, $d_{SR}=500$m, and $d_{RD}=500$m or $d_{RD}=300$m. Moreover, we assume a carrier frequency of $f_c=2.4$ GHz. The transmit bandwidth is assumed to be $200$ kHz. Furthermore, we assume that the noise power per Hz is $-170$ dBm, which for $200$ kHz leads to a total noise power of $2\times 10^{-15}$ Watt. Finally, the normalized self-interference amplification factor, $\alpha$, is computed as $\alpha=\hat \alpha/h_{SR}^2$, where $\hat \alpha$ is the self-interference amplification factor. For our numerical results, we will assume that the self-interference amplification factor $\hat \alpha$ ranges from $-110$ dB to $-140$ dB, hence, the self-interference suppression factor, $1/\hat \alpha$, ranges from $110$ dB to $140$ dB. We note that self-interference suppression schemes that suppress the self-interference by up to $110$ dB in certain scenarios are already available today [@Bharadia:2013:FDR:2486001.2486033]. Given the current research efforts and the steady advancement of technology, suppression factors of up to 140 dB in ceratin scenarios might be possible in the near future. Benchmark Schemes {#Bench} ----------------- *Benchmark Scheme 1 (Ideal FD Transmission without Residual Self-Interference):* The idealized case is when the relay can cancel all of its residual self-interference. For this case, the capacity of the Gaussian two-hop FD relay channel without self-interference is given in [@cover] as $$\begin{aligned} \label{eq_c_fd_ideal} C_{\rm FD,Ideal}=\min\left\{\frac{1}{2}\log_2\left(1+\frac{P_S}{\sigma_R^2}\right) , \frac{1}{2}\log_2\left(1+\frac{P_R}{\sigma_D^2}\right) \right \}.\end{aligned}$$ The optimal input distributions at source and relay are zero-mean Gaussian with variances $P_S$ and $P_R$, respectively. *Benchmark Scheme 2 (Conventional FD Transmission with Self-Interference):* The conventional FD relaying scheme for the case when the relay suffers from residual self-interference uses the same input distributions at source and relay as in the ideal case when the relay does not suffer from residual self-interference, i.e., the input distributions at source and relay are zero-mean Gaussian with variances $P_S$ and $p_R$, respectively, where the relay’s transmit power, $p_R$, is optimized in the range $p_R\leq P_R$ such that the achieved rate is maximized. Thereby, the achieved rate is given by $$\begin{aligned} \label{eq_r_fd_conv} R_{\rm FD,Conv}=\max_{p_R\leq P_R}\min\Bigg\{\hspace{-1mm}&\int_{-\infty}^\infty \hspace{-1mm} \frac{1}{2}\hspace{-0.5mm}\log_2\hspace{-1mm}\left(1+\frac{P_S}{\sigma_R^2+\alpha x_R^2}\right)\hspace{-1mm} \frac{e^{-x_R^2/(2 p_R)}}{\sqrt{2\pi p_R}} d x_R\; ;\; \frac{1}{2}\log_2\left(1+\frac{p_R}{\sigma_D^2}\right) \bigg \}.\end{aligned}$$ *Benchmark Scheme 3 (Optimal HD Transmission):* The capacity of the Gaussian two-hop HD relay channel was derived in [@zlatanov2014capacity-globecom], but can also be directly obtained from Theorem \[theo\_2\] by letting $\alpha\to\infty$. This capacity can be obtained numerically and will be denoted by $C_{\rm HD}$. In this case, the optimal input distribution at the relay is discrete. On the other hand, the source transmits using a Gaussian input distribution with constant variance. Moreover, the source transmits only when the relay is silent, i.e., only when the relay transmits the symbol zero, otherwise, the source is silent. Since both source and relay are silent in fractions of the time, the average powers at source and relay for HD relaying are adjusted such that they are equal to the average powers at source and relay for FD relaying, respectively. *Benchmark Scheme 4 (Conventional HD Transmission):* The conventional HD relaying scheme uses zero-mean Gaussian distributions with variances $P_S$ and $P_R$ at source and relay, respectively. However, compared to the optimal HD transmission in [@zlatanov2014capacity-globecom], in conventional HD transmission, the relay alternates between receiving and transmitting in a codeword-by-codeword manner. As a result, the achieved rate is given by [@1435648] $$\begin{aligned} \label{eq_r_hd_conv} R_{\rm HD,Conv}=\max_{t}\min\Bigg\{ \frac{1-t}{2}\log_2\left(1+\frac{P_S/(1-t)}{\sigma_R^2}\right); \frac{t}{2}\log_2\left(1+\frac{P_R/t}{\sigma_D^2}\right)\Bigg\}.\end{aligned}$$ In (\[eq\_r\_hd\_conv\]), since source and relay transmit only in $(1-t)$ and $t$ fraction of the time, the average powers at source and relay are adjusted such that they are equal to the average powers at source and relay for FD relaying, respectively. We note that Benchmark Schemes 1-4 employ DF relaying. We do not consider the rate achieved with amplified-and-forward (AF) relaying because it was shown in [@cover] that the optimal mode of operation for relays in terms of rate for the class of degraded relay channels, which the investigated two-hop relay channel belongs to, is the DF mode. This means that for the considered two-hop relay channel, the rate achieved with AF relaying will be equal to or smaller than that achieved with DF relaying. Numerical Results {#NumRes} ----------------- In this subsection, we denote the capacity of the considered FD relay channel, obtained from Theorem \[theo\_2\], by $C_{\rm FD}$. In Fig. \[Dis1\], we plot the optimal input distribution at the relay, $p^*(x_R)$, for $d_{SR}= d_{RD}=500$m, $P_S=P_R=25$ dBm, and a self-interference suppression factor of $1/\hat \alpha=130$ dB. As can be seen from Fig. \[Dis1\], the relay is silent in $40$% of the time, and the source transmits only when $|x_R|<x_{\rm th}=0.9312$. Hence, similar to optimal HD relaying in [@zlatanov2014capacity-globecom], shutting down the transmitter at the relay in a symbol-by-symbol manner is important for achieving the capacity. This means that in a fraction of the transmission time, the FD relay is silent and effectively works as an HD relay. However, in contrast to optimal HD relaying where the source transmits only when the relay is silent, i.e., only when $x_R=0$ occurs, in FD relaying, the source has more opportunities to transmit since it can transmit also when the relay transmits a symbol whose amplitude is smaller than $x_{\rm th}$, i.e., when $- x_{\rm th} \leq x_R\leq x_{\rm th}$ holds. For the example in Fig. \[Dis1\], the source transmits $96$ % of the time. ![Optimal input distribution at the relay, $p^*(x_R)$, for $d_{SR}= d_{RD}=500$m, $P_S=P_R=25$ dBm, and self-interference suppression factor, $1/\hat\alpha=130$ dB.[]{data-label="Dis1"}](fig_new_5){width="5in"} ![Comparison of the derived capacity with the rates of the benchmark schemes as a function of the source and relay transmit powers $P_S=P_R$ in dBm for a self-interference suppression factor, $1/\hat\alpha= 130$ dB.[]{data-label="fig_1_new"}](fig_new_1){width="5in"} ![Comparison of the derived capacity with the rates of the benchmark schemes as a function of the source and relay transmit powers $P_S=P_R$ in dBm for a self-interference suppression factor, $1/\hat\alpha= 120$ dB.[]{data-label="fig_2_new"}](fig_new_2){width="5in"} ![Capacity gain of optimal FD relaying compared to optimal HD relaying as a function of the self-interference suppression factor, $1/\hat\alpha$, for different average transmit powers at source and relay $P_S$ and $P_R$.[]{data-label="fig_3_new"}](fig_new_3){width="5in"} ![Comparison of the derived capacity with the capacities achieved with ideal FD and optimal HD relaying as a function of the source’s average transmit power $P_S$ in dBm for a fixed transmit power at the relay of $P_R=25$ dBm, and for different self-interference (SI) suppression factors, $1/\hat \alpha$.[]{data-label="fig_4_new"}](fig_new_4){width="5in"} In Fig. \[fig\_1\_new\], we compare the capacity of the considered FD relay channel, $C_{\rm FD}$, with the achievable rate for the suboptimal input distribution, $p_{\rm B}(x_R)$, given in Section \[sec\_num\_1\], denoted by $R_{\rm FD,B}$, the capacity achieved with ideal FD relaying without residual self-interference, $C_{\rm FD, Ideal}$, cf. Benchmark Scheme 1, the rate achieved with conventional FD relaying, $R_{\rm FD,Conv}$, cf. Benchmark Scheme 2, the capacity of the two-hop HD relay channel, $C_{\rm HD}$, cf. Benchmark Scheme 3, and the rate achieved with conventional HD relaying, $R_{\rm HD,Conv}$, cf. Benchmark Scheme 4, for $d_{SR}= d_{RD}=500$m and a self-interference suppression factor, $1/\hat\alpha$, of 130 dB as a function of the average source and relay transmit powers $P_S=P_R$. The figure shows that indeed the achievable rate with the suboptimal input distribution given in Section \[sec\_num\_1\], $R_{\rm FD,B}$, is a tight lower bound on the capacity $C_{\rm FD}$. Hence, this rate can be used for analytical analysis instead of the actual capacity rate, which is hard to analyze. In addition, the figure shows that for $P_S=P_R> 20$ dBm, the derived capacity $C_{\rm FD}$ achieves around $1.5$ dB power gain with respect to the rate achieved with conventional FD relaying, $R_{\rm FD,Conv}$, using Benchmark Scheme 2. Also, for $P_S=P_R> 20$ dBm, the the derived capacity $C_{\rm FD}$ achieves around 5 dB power gain compared to capacity of the two-hop HD relay channel, $C_{\rm HD}$, and around 10 dB power gain compared to rate achieved with conventional HD relaying, $R_{\rm HD,Conv}$. In Fig. \[fig\_2\_new\], the same parameters as for Fig. \[fig\_1\_new\] are adopted, except that a self-interference suppression factor, $1/\hat\alpha$, of 120 dB is assumed. Thereby, Fig. \[fig\_2\_new\] shows that for $P_S=P_R> 20$ dBm, the derived capacity $C_{\rm FD}$ achieves around $2$ dB gain with respect to capacity of the two-hop HD relay channel, $C_{\rm HD}$, and around 6 dB gain with respect to the rates achieved with conventional FD relaying, $R_{\rm FD,Conv}$, and conventional HD relaying, $R_{\rm HD,Conv}$. In this example, we can see that, due to the strong residual self-interference, the rate achieved with convectional FD relaying is considerably lower than the derived capacity of FD relaying and even the capacity of HD relaying. From Figs. \[fig\_1\_new\] and \[fig\_2\_new\], we observe that the multiplexing gain of the derived capacity is $1/2$, i.e., the same as the value for the HD case. In fact, when $P_S=P_R$, the derived capacity for FD relaying achieves only several dB power gain compared to the capacity for HD relaying. This means that for the adopted worst-case linear residual self-interference model, a self-interference suppression factor of 130 dB is too small to yield a multiplexing gain of 1 in the considered range of $P_S=P_R$. Intuitively, this happens since for $P_S=P_R> 15$ dBm, the average power of the residual self-interference, $\hat \alpha E\{ X_R^2[i]\}$, exceeds the average power of the Gaussian noise. In fact, in general, for a fixed self-interference suppression factor $1/\hat \alpha$ and $P_S=P_R\to\infty$, the power of the residual self-interference at the relay also becomes infinite. As a result, the corresponding multiplexing gain is limited to $1/2$. In Fig. \[fig\_3\_new\], we show the capacity gain of the two-hop FD relay channel compared to the two-hop HD relay channel as a function of the self-interference suppression factor, $1/\hat\alpha$, for different average transmit powers at source and relay $P_S=P_R$ and $d_{SR}= d_{RD}=500$m[^7]. As can be seen from Fig. \[fig\_3\_new\], for a self-interference suppression factor of 120 dB, we obtain only a 5 percent capacity increase for FD relaying compared to HD relaying. In contrast, for a self-interference suppression factor of 130 dB, we obtain around 10 to 15 percent increase in capacity depending on the average transmit power. A 50 percent increase in capacity is possible if $P_S$ and $P_R$ are larger than 25 dBm and the self-interference suppression factor is larger than 150 dB. However, such large self-interference suppression factors might be difficult the realize in practice. In Fig. \[fig\_4\_new\], we compare the capacity of the considered FD relay channel, $C_{\rm FD}$, with the capacities of the ideal FD relay channel without self-interference, $C_{\rm FD, Ideal}$, and the HD relay channel, $C_{\rm HD}$, for $d_{SR}= 500$m, $d_{RD}= 300$m, and $P_R=25$ dBm as a function of the average transmit power at the source, $P_S$. This models a practical scenario where the transmission from a source, e.g. a base station, is supported by a dedicated low-power FD relay. Different self-interference suppression factors are considered for this scenario. For this example, since the relay transmit power is fixed, the capacity of the relay-destination channel is also fixed to around $1.84$ Mbps. As a result, the capacity of the considered relay channel cannot surpass $1.84$ Mbps. In addition, it can be observed from Fig. \[fig\_4\_new\] that the derived capacity of the considered FD relay channel, $C_{\rm FD}$, is significantly larger than the capacity of the HD relay channel, $C_{\rm HD}$ when the transmit power at the source is larger than 30 dBm. For example, for 1.5 Mbps, the power gains are approximately 30 dB, 25 dB, 20 dB, and 15 dB compared to HD relaying for self-interference suppression factors of 140 dB, 130 dB, 120 dB, and 110 dB, respectively. This numerical example shows the benefits of using a dedicated low-power FD relay to support a high-power base station. Conclusion {#con} ========== We studied the capacity of the Gaussian two-hop FD relay channel with linear residual self-interference. For this channel, we considered the worst-case linear residual self-interference model, and thereby, obtained a capacity which constitutes a lower bound on the capacity for any other linear residual self-interference model. We showed that the capacity is achieved by a zero-mean Gaussian input distribution at the source whose variance depends on the amplitude of the transmit symbols at the relay. On the other hand, the optimal input distribution at the relay is Gaussian only when the relay-destination link is the bottleneck link. Otherwise, the optimal input distribution at the relay is discrete. Our numerical results show that significant performance gains are achieved with the proposed capacity-achieving coding scheme compared to the achievable rates of conventional HD and/or FD relaying. In addition, we proposed a suboptimal input distribution at the relay, which, for the presented numerical examples, achieves rates that are close to the capacity achieved with the optimal input distribution at the relay. Proof of Theorem \[Theo1\] {#app_1} -------------------------- We first assume that $p(x_R)$ is discrete. In addition, we assume that $p(x_S|x_R)$ is a continuous distribution, which will turn out to be a valid assumption. Now, from (\[con2b\]), the corresponding maximization problem with respect to $p(x_S|x_R)$ is given by $$\begin{aligned} \label{app_1-eq_1} \max\limits_{p(x_S|x_R)\in\mathcal{P}} & \sum\limits_{x_R\in\mathcal{X}_R} I(X_S;Y_R|X_R=x_R) p(x_R)\nonumber\\ \textrm{Subject to} \textrm{ C1: }& \sum\limits_{x_R\in\mathcal{X}_R}\left[ \int_{x_S} x_S^2 p(x_S|x_R) dx_S\right] p(x_R)\leq P_S.\end{aligned}$$ Since $I(X_S;Y_R|X_R=x_R)$ is the mutual information of a Gaussian AWGN channel with noise power $\sigma_R^2+\alpha x_R^2$, cf. (\[r1v3\]), the optimal distribution $p(x_S|x_R)$ that maximizes $I(X_S;Y_R|X_R=x_R)$ is the zero-mean Gaussian distribution with variance $P_S(x_R)$. The variance $P_S(x_R)$ has to satisfy constraint C1 in (\[app\_1-eq\_1\]). Hence, to find the variance $P_S(x_R)$, we first substitute $p(x_S|x_R)$ in (\[app\_1-eq\_1\]) with the zero-mean Gaussian distribution with variance $P_S(x_R)$. Thereby, we obtain the following optimization problem $$\begin{aligned} \max_{P_S(x_R)}&~\sum_{x_R\in\mathcal{X}_R} \frac{1}{2} \log_2\left(1+\frac{P_S(x_R)}{\sigma_R^2+\alpha x_R^2}\right)p(x_R ) \nonumber\\ \textrm{Subject to} \textrm{ C1:}& ~ \sum_{x_R\in\mathcal{X}_R} P_S(x_R) p(x_R) \leq P_S\nonumber\\ \textrm{ C2:}& ~ P_S(x_R) \geq 0, \; \forall x_R.\label{eq_3}\end{aligned}$$ Since (\[eq\_3\]) is a concave optimization problem, it can be solved in a straightforward manner using the Lagrangian method, which results in (\[P1\]). In (\[P1\]), $x_{\rm{th}}$ is a Lagrange multiplier which has to be set such that constraint C1 in (\[eq\_3\]) holds with equality. Inserting (\[P1\]) into constraint C1 in (\[eq\_3\]), we obtain (\[36a\]). Whereas, inserting $P_S(x_R)$ in (\[P1\]) into the objective function in (\[eq\_3\]), we obtain (\[eq\_1-dis\]). Following a similar procedure as above for the case when $p(x_R)$ is assumed to be continuous, we arrive at the same solution for $P_S(x_R)$ and $\max\limits_{p(x_S|x_R)\in\mathcal{P}} I(X_S;Y_R|X_R)$ as in (\[P1\]) and (\[eq\_1-dis\]), respectively, but with the sums replaced by integrals. This concludes the proof. Proof of Theorem \[theo\_2\] {#app_2} ---------------------------- Assuming $p(x_R)$ is discrete, the corresponding capacity expression for the $p(x_S|x_R)$ given in Theorem \[Theo1\] is given by $$\begin{aligned} C=&\max_{p(x_R)\in\mathcal{P}} ~ \mathrm{min}\left\{ \sum_{x_R\in \mathcal{X}_R}\frac{1}{2}\log_2\left(1+\frac{\alpha \max\{0,x_{\rm th}^2 - x_R^2\}}{\sigma_R^2+\alpha x_R^2}\right)p(x_R) ,I(X_R;Y_D)\right\}\nonumber\\ \textrm{Subject to} \textrm{ C1: }& \sum_{x_R\in \mathcal{X}_R} \alpha \max\{0,x_{\rm th}^2 - x_R^2\} = P_S\nonumber\\ \textrm{ C2: }& \sum_{x_R\in \mathcal{X}_R} x_R^2 p(x_R) \leq P_R \label{con2c}\end{aligned}$$ Using its epigraph form, the optimization problem in (\[con2c\]) can be equivalently represented as $$\begin{aligned} \label{MPR2} \begin{array}{rl} {\underset{p(x_R),\; u}{\rm{Maximize: }}}& u \\ {\rm{Subject\;\; to }}\;\; {\rm C1:}&\; u- \sum\limits_{x_R\in \mathcal{X}_R}\frac{1}{2}\log_2\left(1+\frac{\alpha\max\{0,x_{\rm th}^2 - x_R^2\}}{\sigma_R^2+\alpha x_R^2}\right)p(x_R) \leq 0 \\ {\rm C2:}&\; u- I(X_R;Y_D) \leq 0 \\ {\rm C3:}&\; \sum\limits_{x_R\in \mathcal{X}_R} \alpha \max\{0,x_{\rm th}^2 - x_R^2\}p(x_R) = P_S \\ {\rm C4:}&\; \sum\limits_{x_R\in \mathcal{X}_R} x_R^2 p(x_R) \leq P_R \\ {\rm C5:}&\; \sum\limits_{x_R\in \mathcal{X}_R} p(x_R) -1=0. \end{array}\end{aligned}$$ In the optimization problem (\[MPR2\]), constraint C2 is convex with respect to $p(x_R)$, and constraints C1, C3, C4, and C5 are affine with respect to $p(x_R)$. Hence, the optimization problem in (\[MPR2\]) is a concave optimization problem and can be solved using the Lagrangian method. The Lagrangian function of the optimization problem in (\[MPR2\]) is given by $$\begin{aligned} \label{eq_1} &L= u-\xi_1 \left(u- \sum\limits_{x_R\in \mathcal{X}_R}\frac{1}{2}\log_2\left(1+\frac{\alpha \max\{0,x_{\rm th}^2 - x_R^2\}}{\sigma_R^2+\alpha x_R^2}\right)p(x_R) \right) - \xi_2 \left( u- I(X_R;Y_D) \right) \nonumber\\ &-\lambda_1 \left(\sum\limits_{x_R\in \mathcal{X}_R} \alpha \max\{0,x_{\rm th}^2 - x_R^2\} p(x_R) - P_S \right) - \lambda_2 \left( \sum\limits_{x_R\in \mathcal{X}_R} x_R^2 p(x_R) - P_R \right) -\nu \left( \sum\limits_{x_R\in \mathcal{X}_R} p(x_R) -1 \right),\end{aligned}$$ where $\xi_1$, $\xi_2$, $\lambda_1$, $\lambda_2$, and $\nu$ are Lagrange multipliers corresponding to constraints C1, C2, C3, C4, and C5, respectively. Due to the KKT conditions, the following has to hold \[eq\_2\] $$\begin{aligned} &\xi_1 \left(u- \sum\limits_{x_R\in \mathcal{X}_R}\frac{1}{2}\log_2\left(1+\frac{\alpha \max\{0,x_{\rm th}^2 - x_R^2\}}{\sigma_R^2+\alpha x_R^2}\right)p(x_R) \right) =0,\;\;\; \xi_1\geq 0\label{eq_2a}\\ & \xi_2 \left( u- I(X_R;Y_D) \right) =0, \;\;\; \xi_2\geq 0\label{eq_2b}\\ &\lambda_1 \left(\sum\limits_{x_R\in \mathcal{X}_R} \alpha \max\{0,x_{\rm th}^2 - x_R^2\} p(x_R) - P_S \right) =0\label{eq_2c}\\ & \lambda_2 \left( \sum\limits_{x_R\in \mathcal{X}_R} x_R^2 p(x_R) - P_R \right) =0,\;\;\; \lambda_2\geq 0,\label{eq_2d}\\ & \nu \left( \sum\limits_{x_R\in \mathcal{X}_R} p(x_R) -1 \right) =0 .\label{eq_2e}\end{aligned}$$ Differentiating $L$ with respect to $u$, we obtain that $\xi_1=1-\xi_2=\xi$ has to hold, where $0\leq \xi\leq 1$. Inserting this into (\[eq\_1\]), then differentiating with respect to $p(x_R)$, and equating the result to zero we obtain the following $$\begin{aligned} \label{eq_1aa} & \xi \frac{1}{2}\log_2\left(1+\frac{\alpha \max\{0,x_{\rm th}^2 - x_R^2\}}{\sigma_R^2+\alpha x_R^2}\right) + (1-\xi) I'(X_R;Y_D) \nonumber\\ &-\lambda_1 \alpha \max\{0,x_{\rm th}^2 - x_R^2\} - \lambda_2 x_R^2 -\nu =0,\end{aligned}$$ where $I'(X_R;Y_D)=\partial I (X_R;Y_D)/\partial p(x_R)$. We note that there are three possible solutions for (\[eq\_1aa\]) depending on whether $\xi= 1$, $\xi=0$, or $0<\xi< 1$, respectively. In the following, we analyze these three cases. *Case 1:* Let us assume that $\xi=1$ holds. Then, from (\[eq\_2\]), we obtain that $$\begin{aligned} \label{eq_8} u< I(X_R;Y_D) \textrm{ and } u =\sum\limits_{x_R\in \mathcal{X}_R}\frac{1}{2}\log_2\left(1+\frac{\alpha \max\{0,x_{\rm th}^2 - x_R^2\}}{\sigma_R^2+\alpha x_R^2}\right)p(x_R) ,\end{aligned}$$ which means that for the optimal $p(x_R)$ the following holds $$\begin{aligned} \label{eq_9} I(X_R;Y_D)\bigg|_{p(x_R)=p^*(x_R)}>\sum\limits_{x_R\in \mathcal{X}_R}\frac{1}{2}\log_2\left(1+\frac{\alpha \max\{0,x_{\rm th}^2 - x_R^2\}}{\sigma_R^2+\alpha x_R^2}\right)p^*(x_R).\end{aligned}$$ The optimal $p^*(x_R)$ in this case has to maximize the right hand side of (\[eq\_9\]), i.e., $$\begin{aligned} \label{eq_f} \sum\limits_{x_R\in \mathcal{X}_R}\frac{1}{2}\log_2\left(1+\frac{\alpha \max\{0,x_{\rm th}^2 - x_R^2\}}{\sigma_R^2+\alpha x_R^2}\right)p(x_R).\end{aligned}$$ It turns out that the optimal $p(x_R)$ which maximizes (\[eq\_f\]) is $p^*(x_R)=\delta(x_R)$, i.e., the relay is always silent and never transmits. However, if we insert $p^*(x_R)=\delta(x_R)$ in $I(X_R;Y_D)$ in (\[eq\_9\]), we obtain the following contradiction $$\begin{aligned} \label{eq_9a} I(X_R;Y_D)\bigg|_{p(x_R)=\delta(x_R)} =0 > \frac{1}{2}\log_2(1+P_S/\sigma_R^2) >0 .\end{aligned}$$ Hence, $\xi=1$ is not possible. The only remaining possibilities are $\xi=0$ and $0<\xi<1$. *Case 2:* Let us assume that $\xi=0$ holds. Then, from (\[eq\_2\]), we obtain that $$\begin{aligned} \label{eq_4} u= I(X_R;Y_D) \textrm{ and } u< \sum\limits_{x_R\in \mathcal{X}_R}\frac{1}{2}\log_2\left(1+\frac{\alpha \max\{0,x_{\rm th}^2 - x_R^2\}}{\sigma_R^2+\alpha x_R^2}\right)p(x_R) ,\end{aligned}$$ has to hold, which means that for the optimal $p(x_R)$ the following holds $$\begin{aligned} \label{eq_5} I(X_R;Y_D)\bigg|_{p(x_R)=p^*(x_R)}< \sum\limits_{x_R\in \mathcal{X}_R}\frac{1}{2}\log_2\left(1+\frac{\alpha \max\{0,x_{\rm th}^2 - x_R^2\}}{\sigma_R^2+\alpha x_R^2}\right)p^*(x_R) .\end{aligned}$$ The optimal $p(x_R)$ in this case is the one which maximizes the left hand side of (\[eq\_5\]), i.e., maximizes $I(X_R;Y_D)$. Since the relay-destination link is an AWGN channel, $I(X_R;Y_D)$ is maximized for $p^*(x_R)$ being the zero-mean Gaussian distribution with variance $P_R$. As a result, the capacity is given by $$\begin{aligned} \label{eq_6} I(X_R;Y_D)\bigg|_{p(x_R)=p^*(x_R)} =\frac{1}{2} \log_2\left(1+\frac{P_R}{\sigma_D^2}\right).\end{aligned}$$ Hence, (\[eq\_6\]) is the capacity if and only if (iff) after substituting $p^*(x_R)$ with the zero-mean Gaussian distribution with variance $P_R$, (\[eq\_5\]) holds, i.e., (\[39\]) holds. *Case 3:* Let us assume that $0<\xi<1$. Then, from (\[eq\_2\]), we obtain that $$\begin{aligned} \label{eq_10} u= I(X_R;Y_D) \textrm{ and } u= \sum\limits_{x_R\in \mathcal{X}_R}\frac{1}{2}\log_2\left(1+\frac{\alpha \max\{0,x_{\rm th}^2 - x_R^2\}}{\sigma_R^2+\alpha x_R^2}\right)p(x_R) ,\end{aligned}$$ which means that for the optimal $p(x_R)$, the following holds $$\begin{aligned} \label{eq_11} I(X_R;Y_D)\bigg|_{p(x_R)=p^*(x_R)} =\sum\limits_{x_R\in \mathcal{X}_R}\frac{1}{2}\log_2\left(1+\frac{\alpha \max\{0,x_{\rm th}^2 - x_R^2\}}{\sigma_R^2+\alpha x_R^2}\right)p(x_R).\end{aligned}$$ For $0<\xi<1$, we can find the optimal distribution $p^*(x_R)$ as the solution of (\[eq\_1aa\]). To this end, we need to compute $I'(X_R,Y_D)$. Since for the AWGN channel, $I(X_R;Y_D)=H(Y_D)-H(Y_D|X_R)$, where $H(Y_D|X_R)=\frac{1}{2}\log_2(2\pi e \sigma_D^2)$ hold, we obtain that $I'(X_R;Y_D)=H'(Y_D)$. On the other hand, $H'(Y_D)$ for the AWGN channel is found as $$\begin{aligned} \label{eq_13} H'(Y_D)=-\int_{-\infty}^\infty \frac{1}{\sqrt{2\pi \sigma_D^2}} \exp\left(-\frac{(y_D-x_R)^2}{2\sigma_D^2}\right) \log_2(p(y_D)) dy_D -\frac{1}{\ln(2)}.\end{aligned}$$ Inserting (\[eq\_13\]) into (\[eq\_1aa\]), we obtain $$\begin{aligned} \label{eq_14} & \xi \frac{1}{2}\log_2\left(1+\frac{\alpha \max\{0,x_{\rm th}^2 - x_R^2\}}{\sigma_R^2+\alpha x_R^2}\right) - (1-\xi) \int_{-\infty}^\infty \frac{1}{\sqrt{2\pi \sigma_D^2}} \exp\left(-\frac{(y_D-x_R)^2}{2\sigma_D^2}\right) \log_2(p(y_D)) dy_D \nonumber\\ &- (1-\xi)\frac{1}{\ln(2)} -\lambda_1 \alpha \max\{0,x_{\rm th}^2 - x_R^2\} - \lambda_2 x_R^2 -\nu =0.\end{aligned}$$ Hence, the optimal $p(x_R)$ has to produce a $p(y_D)$ for which (\[eq\_14\]) holds. In Appendix \[app\_3\], we prove that (\[eq\_14\]) cannot hold if $p(x_R)$ is a continuous distribution and that (\[eq\_14\]) can hold if $p(x_R)$ is a discrete distribution since then it has to hold only for the discrete values $x_R\in\mathcal{X}_R$. Although we derived (\[eq\_1aa\]) assuming that $p(x_R)$ was discrete, we would have arrived at the same result if we had assumed that $p(x_R)$ was a continuous distribution. To do so, we first would have to replace the sums in the optimization problem in (\[MPR2\]) with integrals with respect to $x_R$. Next, in order to obtain the stationary points of the corresponding Lagrangian function, instead of the ordinary derivative, we would have to take the functional derivative and equate it to zero. This again would have led to the identity in (\[eq\_1aa\]). Hence, the conclusions drawn from the Lagrangian and (\[eq\_1aa\]) are also valid when $p(x_R)$ is a continuous distribution. Proof That $p(x_R)$ is Discrete When $0<\xi<1$ {#app_3} ---------------------------------------------- This proof is based on the proof for the discreteness of a distribution given in [@6193208]. Furthermore, similar to [@6193208], to simplify the derivation of the proof, we set $\sigma_D^2=1$. First, we decompose the integral in (\[eq\_14\]) using Hermitian polynomials. To this end, we define $$\begin{aligned} \log_2(p(y_D))=\sum_{m=0}^{\infty}c_mH_m(y_D)\label{c13}, \end{aligned}$$ where the $c_m$, $\forall m$, are constants and $H_m(y_D)$, $\forall m$, are Hermitian polynomials, see [@6193208]. Note that $\mathrm{ln}(p(y_D))$ is square integrable with respect to $e^{-\frac{y_D^2}{2}}$ and hence can be decomposed using a Fourier-Hermite series decomposition, see [@6193208]. Then, the integral in (\[eq\_13\]) with $\sigma_D^2=1$, can be written as $$\begin{aligned} &\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-\frac{(y_D-x_R)^2}{2}}\log_2(p(y_D))dy_D = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{\frac{y_D^2}{2}}e^{(-\frac{x_R^2}{2}+x_Ry_D)}\log_2(p(y_D))dy_D\notag\\ &\overset{(a)}{=}\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{\frac{y_D^2}{2}}\sum_{n=0}^{\infty}H_n(y_D)\frac{x_R^n}{n!}\sum_{m=0}^{\infty}c_mH_m(y_D)dy_D \overset{(b)}{=} \sum_{m=0}^{\infty}c_m x_R^m,\label{c14} \end{aligned}$$ where $(a)$ is obtained by inserting (\[c13\]) and using the generating function of Hermitian polynomials, given by $$\begin{aligned} e^{(-\frac{t^2}{2}+tx)}=\sum_{m=0}^{\infty}H_m(x)\frac{t^m}{m!}.\label{d2}\end{aligned}$$ Furthermore, $(b)$ in (\[c14\]) follows since Hermitian polynomials are orthogonal with respect to the weight function $\omega(x)=e^{-\frac{x^2}{2}}$, i.e., $$\begin{aligned} \int_{-\infty}^{\infty}H_m(x)H_n(x)\omega(x)dx=\begin{cases} m!\sqrt{2\pi} &\textrm{if } m=n\\ 0 & \textrm{otherwise} \end{cases}\label{d1}\end{aligned}$$ holds. By inserting (\[c14\]) into (\[eq\_14\]), we obtain $$\begin{aligned} (1-\xi) \sum_{m=0}^{\infty}c_m x_R^m & = \xi \frac{1}{2}\log_2\left(1+\frac{\alpha \max\{0,x_{\rm th}^2 - x_R^2\}}{\sigma_R^2+\alpha x_R^2}\right) -\lambda_1 \alpha \max\{0,x_{\rm th}^2 - x_R^2\} - \lambda_2 x_R^2 \nonumber\\ & - (1-\xi)\frac{1}{\ln(2)} -\nu .\label{n1} \end{aligned}$$ Now, we have two cases for $|x_R|$, one when $|x_R|\geq x_{\rm th}$ and the other one when $|x_R|< x_{\rm th}$. Also, we have two cases for $\lambda_2$, one when $\lambda_2>0$ (constraint C2 in (\[con2c\]) holds with equality) and the other are when $\lambda_2=0$ (constraint C2 in (\[con2c\]) does not hold with equality). The resulting four cases are analyzed in the following. *Case 1:* If $|x_R|\geq x_{\rm th}$ and $\lambda_2>0$ hold, then (\[n1\]) simplifies to $$\begin{aligned} \sum_{m=0}^{\infty}c_m x_R^m & = -\frac{ \lambda_2}{ 1-\xi} x_R^2 - \frac{1}{\ln(2)} -\frac{\nu}{1-\xi} .\label{n1a} \end{aligned}$$ Comparing the exponents in , we obtain $$\begin{aligned} \label{eq_coef_1} c_0 =- \frac{1}{\ln(2)} -\frac{\nu}{1-\xi};\quad c_1=0;\quad c_2 =\frac{\lambda_2}{1-\xi};\quad c_n =0,\quad \forall n>2. \end{aligned}$$ Inserting (\[eq\_coef\_1\]) into (\[c13\]), we obtain $p(y_D)$ as $$\begin{aligned} p(y_D)= e^{\ln(2)(c_0 H_0(y_D)+c_2 H_2(y_D))}\overset{(a)}{=}e^{\ln(2)(c_0-c_2)}e^{\ln(2) c_2y_D^2},\label{n2} \end{aligned}$$ where $(a)$ follows from $H_0(y_D)=1$ and $H_2(y_D)=y_D^2-1$. This solution for $p(y_D)$ can be a valid probability density function only for $c_2<0$, which yields a Gaussian distribution for $p(y_D)$. Now, for $Y_D$ to be Gaussian distributed and $Y_D=X_R+N_D$ to hold, where $N_D$ is also Gaussian distributed, $p(x_R)$ also has to be Gaussian distributed. However, since the Gaussian distribution is unbounded in $x_R$, the Gaussian distribution $p(x_R)$ cannot hold only in the domain $|x_R|\geq x_{\rm th}$ but has to hold in the entire domain $|x_R|\leq \infty$. Hence, we have to see whether a Gaussian $p(x_R)$ is also optimal for $|x_R|<x_{\rm th}$. If we obtain that $p(x_R)$ is not Gaussian for $|x_R|<x_{\rm th}$, then $p(x_R)$ can only be discrete in the domain $|x_R|\geq x_{\rm th}$ for any $x_{\rm th}>0$. *Case 2:* If $|x_R|< x_{\rm th}$ and $\lambda_2>0$ hold, (\[n1\]) simplifies to $$\begin{aligned} \sum_{m=0}^{\infty}c_m x_R^m & = \frac{ \xi}{1-\xi} \frac{1}{2}\log_2\left(1+\frac{\alpha (x_{\rm th}^2 - x_R^2)}{\sigma_R^2+\alpha x_R^2}\right) -\frac{ \lambda_1\alpha+\lambda_2}{1-\xi} x_R^2 -\frac{1}{1-\xi}(1/\ln(2)+\nu +\lambda_1 \alpha x_{\rm th}^2) .\label{n2a} \end{aligned}$$ We now represent the $\log_2(\cdot)$ function in (\[n2a\]) using a Taylor series expansion as $$\begin{aligned} \label{eq_ts} \log_2\left(1+\frac{\alpha (x_{\rm th}^2 - x_R^2)}{\sigma_R^2+\alpha x_R^2}\right) = \sum_{n=0}^{\infty} (-1)^n a_n x_R^{2n}, \end{aligned}$$ where $a_n>0$, $\forall n$, and the exact (positive) values of these coefficients are not important for this proof. Inserting (\[eq\_ts\]) into (\[n2a\]), we obtain $$\begin{aligned} \sum_{m=0}^{\infty}c_m x_R^m & = \frac{ \xi}{1-\xi} \frac{1}{2\ln(2)} \sum_{n=0}^{\infty} (-1)^n a_n x_R^{2n} -\frac{ \lambda_1\alpha+\lambda_2}{1-\xi} x_R^2 -\frac{1}{1-\xi}(1/\ln(2)+\nu +\lambda_1 \alpha x_{\rm th}^2) .\label{n2b} \end{aligned}$$ Comparing the exponents on the left hand side and the right hand side of (\[n2b\]), we can find $c_m$ as $$\begin{aligned} \label{eq_coef_2} c_m=\left\{ \begin{array}{ll} \frac{ \xi}{1-\xi} \frac{1}{2\ln(2)} a_0 -\frac{1}{1-\xi}(1/\ln(2)+\nu +\lambda_1 \alpha x_{\rm th}^2) & \textrm{ if } m=0\\ 0 & \textrm{ if } m \textrm{ is odd}\\ \frac{ \xi}{1-\xi} \frac{1}{2\ln(2)} \ (-1) a_n x_R^{2} -\frac{ \lambda_1\alpha+\lambda_2}{1-\xi} x_R^2 & \textrm{ if } m=2\\ \frac{ \xi}{1-\xi} \frac{1}{2\ln(2)} (-1)^{m/2} a_{m/2} x_R^{m} & \textrm{ if } m>2 \textrm{ and } m \textrm{ is even }\\ \end{array} \right. \end{aligned}$$ Inserting (\[eq\_coef\_2\]) into (\[c13\]), we obtain $p(y_D)$ as $$\begin{aligned} p(y_D)=e^{\ln(2)\sum_{m=0}^{\infty}c_{2m}H_{2m}(y_D) } &=e^{\ln(2)\sum_{n=0}^{\infty}q_ny_D^{2n}}=\prod_{n=0}^{\infty}e^{\ln(2) q_ny_D^{2n}},\label{c15} \end{aligned}$$ where $q_n$ are known non-zero constants. Now, since $q_n>0$ for some $n\to\infty$, $p(y_D)$ in (\[c15\]) cannot be a valid distribution since $p(y_D)$ becomes unbounded. As a result, $p(x_R)$ has to be discrete in the domain $|x_R|<x_{\rm th}$. Consequently, $p(x_R)$ also has to be discrete in the domain $|x_R|\geq x_{\rm th}$. This concludes the proof for the case when $\lambda_2>0$. Following a similar procedure for $\lambda_2=0$ as for the case when $\lambda_2>0$, we obtain that again $p(x_R)$ has to be discrete in the entire domain of $x_R$. On the other hand, $p^*(x_R)$ has to be symmetrical with respect to $x_R=0$. To prove this, assume that we have an unsymmetrical $p(x_R)$, denoted by $p_{u}(x_R)$, with only one unsymmetrical mass point $x_{Ru}$ which has probability $p_{Ru}$. Now, let us construct a new, symmetrical $p(x_R)$, denoted by $p_{s}(x_R)$, by making $p_{u}(x_R)$ symmetrical. In particular, in $p_{u}(x_R)$, we first reduce the probability of the mass point $x_{Ru}$ to $p_{Ru}/2$. Next, we add the mass point $-x_{Ru}$ to $p_{u}(x_R)$ and set its probability to $p_{Ru}/2$. Now, it is clear that the average power of the relay is identical for both $p_{u}(x_R)$ and $p_{s}(x_R)$. On the other hand, by making $p(x_R)$ symmetrical, we have increased the entropy of $X_R$, i.e., $H(X_R)|_{p(x_R)=p_{u}(x_R)}\leq H(X_R)|_{p(x_R)=p_{s}(x_R)}$ holds. Consequently, we have increased the differential entropy of $Y_D$, i.e., $h(Y_D)|_{p(x_R)=p_{u}(x_R)}\leq h(Y_D)|_{p(x_R)=p_{s}(x_R)}$ holds. Now, since for the AWGN channel $h(Y_D|X_R)$ is independent of $p(x_R)$, it follows that $I(X_R;Y_D)|_{p(x_R)=p_{u}(x_R)}\leq I(X_R;Y_D)|_{p(x_R)=p_{s}(x_R)}$ holds. This concludes the proof of the symmetry of $p^*(x_R)$. Proof of Lemma \[lem\_1\] {#app_5} ------------------------- Here, we only prove the non-trivial case when (\[39\]) does not hold. The trivial case is identical to the case without self-interference and its achievability is shown [@cover]. Let us assume that condition (\[39\]) does not hold. Then, according to Theorem \[Theo1\], $p(x_R)$ is discrete and the capacity $C$ is given in (\[cap\_2\]). Moreover, for the considered coding scheme, $R$ satisfies the following $$\begin{aligned} \label{app_3-eq_1} R&<C=\max_{p(x_{S}|x_R)} I\big(X_{S}; Y_{R}| X_R\big) = \sum_{x_R\in\mathcal{X}_R}\max_{p(x_{S}|x_R)} I\big(X_{S}; Y_{R}| X_R=x_R\big) p^*(x_R) \nonumber\\ &\stackrel{(a)}{=} \sum_{\substack{x_R\in\mathcal{X}_R\\|x_R|<x_{\rm th}}}\max_{p(x_{S}|x_R)} I\big(X_{S}; Y_{R}| X_R=x_R\big) p^*(x_R) \stackrel{(b)}{\leq} \sum_{\substack{x_R\in\mathcal{X}_R\\|x_R|<x_{\rm th}}}\max_{p(x_{S}|x_R)} I\big(X_{S}; Y_{R}| X_R=0\big) p^*(x_R) \nonumber\\ & = \max_{p(x_{S}|x_R)} I\big(X_{S}; Y_{R}| X_R=0\big) \sum_{\substack{x_R\in\mathcal{X}_R\\|x_R|<x_{\rm th}}} p^*(x_R) =\max_{p(x_{S}|x_R)} I\big(X_{S}; Y_{R}| X_R=0\big) p_T \nonumber\\ & \stackrel{(c)}{=} \frac{1}{2} \log_2\left(1+\frac{\alpha x_{\rm th}^2 }{\sigma_R^2 }\right) p_T ,\end{aligned}$$ where $(a)$ follows since for the considered coding scheme the source is silent when $|x_R|\geq x_{\rm th}$ and as a result $I\big(X_{S}; Y_{R}| X_R=x_R\big)=0$ for $|x_R|\geq x_{\rm th}$, $(b)$ follows since, for the considered relay channel, $I\big(X_{S}; Y_{R}| X_R=x_R\big)$ is maximized for $x_R=0$, because in that case there is no residual self-interference at the relay, and $(c)$ follows from (\[eq\_1-dis\]). Now, note that for the considered coding scheme in time slot $1$, the source-relay channel can be seen as an AWGN channel with a fixed channel gain $\sqrt{P_S(x_R=0)}=\sqrt{\alpha} x_{\rm th}$ and AWGN with variance $\sigma_R^2$ which is used $k p_T$ times. Hence, any codeword selected uniformly from a codebook comprised of $2^{kR}$ Gaussian distributed codewords, where each codeword is comprised of $k p_T$ symbols, with $k\to\infty$ and $R$ satisfying $$\begin{aligned} \label{eq_ss} kR/(kp_T)< \frac{1}{2} \log_2\left(1+\frac{\alpha x_{\rm th}^2 }{\sigma_R^2 }\right), \end{aligned}$$ can be successfully decoded at the relay using a jointly-typical decoder, see [@cover2012elements]. Noting that the proposed coding scheme satisfies the properties outlined above, we can conclude that the codeword transmitted in time slot 1 can be decoded successfully at the relay. Proof of Lemma \[lema\_2\] {#app_4} -------------------------- Again, we only prove the non-trivial case when (\[39\]) does not hold. In time slot $b$, for $2\leq b\leq N$, the source-relay channel can be seen equivalently as an AWGN channel with states $X_R$, where a different state $X_R=x_R$ produces a different channel gain and a different noise variance. In particular, for channel state $X_R=x_R$, the channel gain of the equivalent AWGN channel is $\sqrt{P_S(x_R)}$ and the variance of the AWGN is $\sigma_R^2+\alpha x_R^2$. Moreover, for this equivalent AWGN channel with states, the source has to transmit unit-variance symbols in order for the average power constraint of the original source-relay channel, given by $E\{X_S^2\}\leq P_S$, to be satisfied. Furthermore, for the equivalent AWGN channel with states, note that both source (i.e., transmitter) and relay (i.e., receiver) have CSI in each channel use and thereby know that the channel gain and the noise variance in channel use $j$ will be $\sqrt{P_S(x_{R,j})}$ and $\sigma_R^2+\alpha x_{R,j}^2$, respectively. Now, instead of deriving a capacity-achieving coding scheme for the original source-relay channel, we can find equivalently a capacity-achieving coding scheme for the equivalent AWGN channel[^8] with states. To this end, we will first find the capacity of an “auxiliary AWGN channel”, using results which are already available in the literature. Then, we will modify the capacity-achieving coding scheme of the “auxiliary AWGN channel” in order to obtain a capacity-achieving coding scheme for the equivalent AWGN channel with states. The “auxiliary AWGN channel” is identical to the equivalent AWGN channel but without CSI at the source (i.e., transmitter). The channel coding scheme that achieves the capacity of the “auxiliary AWGN channel” in $k\to\infty$ channel uses is the following, see [@782125; @720551] for proof. The codebook is comprised of $2^{kR}$ codewords, where each codeword is comprised of $k$ symbols and each symbol is generated independently according to the zero-mean unit-variance Gaussian distribution. Moreover, the parameter $R$ of the channel code has to satisfy $$\begin{aligned} \label{eq__v1} R &< \max_{\substack{p(x'_{S}|x_R)\\E\{X_S'^2\}= 1}} I(X_S';Y_D|X_R)\Big|_{p(x_R)=p^*(x_R)} = \sum_{x_R\in\mathcal{X}_R}\max_{\substack{p(x'_{S}|x_R)\\E\{X_S'^2\}= 1}} I\big(X_{S}'; Y_{R}| X_R=x_R\big) p^*(x_R) \nonumber\\ &\stackrel{(a)}{=} \sum\limits_{ x_R\in\mathcal{X}_R } \frac{1}{2} \log_2\left(1+\frac{P_S(x_R)}{\sigma_R^2+\alpha x_R^2}\right) p^*(x_R) ,\nonumber\\ &\stackrel{(b)}{=} \sum\limits_{ x_R\in\mathcal{X}_R } \frac{1}{2} \log_2\left(1+\frac{\alpha \max\{0,\;x_{\rm th}^2- x_R^2\}}{\sigma_R^2+\alpha x_R^2}\right) p^*(x_R) ,\end{aligned}$$ where $X_S'$ is the input at the source of the “auxiliary AWGN channel”, $(a)$ follows due to the unit-variance constraint $E\{X_S'^2\}= 1$ and since for each state $X_R=x_R$ the channel is AWGN with channel gain $\sqrt{P_S(x_{R,j})}$ and noise variance $\sigma_R^2+\alpha x_{R,j}^2$, and $(b)$ follows from (\[P1\]). Any codeword selected uniformly from this codebook and transmitted in $k$ channel uses can be successfully decoded at the relay (i.e., receiver) using a jointly typical decoder, see [@782125; @720551; @cover2012elements]. Now, for the “auxiliary AWGN channel” note that the source transmits a symbol during all $k$ channel uses. Hence, the source transmits a symbol during channel uses for which the channel gain is zero, i.e., $\sqrt{P_S(x_R)}=0$ holds. Obviously, the symbols transmitted when $\sqrt{P_S(x_R)}=0$ do not reach the relay due to the zero channel gain, i.e., the relay receives only noise during these channel uses. Now, for the equivalent AWGN channel, we can use the same coding scheme as for the “auxiliary AWGN channel”, but, since in this case the source has CSI, the source can choose not to transmit during a channel use for which $\sqrt{P_S(x_R)}=0$ holds. Moreover, since the source has knowledge that $\sqrt{P_S(x_{R,j})}>0$ holds in a $p_T$ fraction out of the $k$ channel uses, the source can reduce the length of the codewords from $k$ to $k p_T$. Thereby, the channel code for the equivalent AWGN channel has a codebook comprised of $2^{kR}$ Gaussian distributed codewords, where each codeword is comprised of $k p_T$ symbols. Moreover, for the equivalent AWGN channel, the source is silent for states for which $\sqrt{P_S(x_R)}=0$ holds, i.e., $|x_R|\geq x_{\rm th}$ holds, and transmits a symbol from the selected codeword only when $\sqrt{P_S(x_R)}>0$, i.e., $|x_R|< x_{\rm th}$ holds, which is exactly the proposed scheme. Hence, for the proposed scheme, we can conclude that the codeword transmitted in time slot $b$, for $2\leq b\leq N$, can be decoded successfully at the relay. [^1]: This work was accepted in part for presentation at IEEE Globecom 2016 [@C_FD_SI_conf]. [^2]: N. Zlatanov is with the Department of Electrical and Computer Systems Engineering, Monash University, Melbourne, VIC 3800, Australia (e-mail: nikola.zlatanov@monash.edu). [^3]: E. Sippel, V. Jamali, and R. Schober are with the Friedrich-Alexander University of Erlangen-Nürnberg, Institute for Digital Communications, D-91058 Erlangen, Germany (e-mails: (erik.sippel@fau.de, vahid.jamali@fau.de robert.schober@fau.de). [^4]: As customary for capacity analysis, see e.g. [@cover2012elements], as a first step we do not consider fading and assume real-valued channel inputs and outputs. The generalization to a complex-valued signal model is relatively straightforward [@TSE05]. On the other hand, the generalization to the case of fading is considerably more involved. For example, considering the achievability scheme for HD relays in [@BA-relaying-adaptive-rate], we expect that when fading is present, both HD and FD relays have to perform buffering in order to achieve the capacity. However, the corresponding detailed analysis is beyond the scope of this paper and presents an interesting topic for future research. [^5]: We note that similar distortions are also present in the source-relay and relay-destination channels. However, due to the large distance between transmitter and receiver, the impact of these distortions on the channel gains $h_{SR}$ and $h_{RD}$ is negligible. [^6]: This is because a Gaussian RV has the highest uncertainty (i.e., entropy) among all possible RVs for a given second moment [@cover2012elements]. [^7]: In Fig. 5, for certain values of $1/\hat \alpha$, the capacity gain decreases as $P_S=P_R$ increases. This is because in this range of $1/\hat \alpha$, the capacity achieved with HD relaying increases faster with $P_S=P_R$ than the capacity achieved with FD relaying. [^8]: The capacity-achieving coding scheme of the original source-relay channel can be obtained straightforwardly from the equivalent AWGN channel with states. In particular, the only modification is that the source has to multiply the transmitted symbol in channel use $j$ by $\sqrt{P_S(x_{R,j})}$.

--- abstract: 'In 1996, Shi [@shi] generalized the ${\epsilon}$-regularity theorem of Schoen and Uhlenbeck [@su] to energy-minimizing harmonic maps from a domain equipped with a Riemannian metric of class $L^{\infty}$. In the present work we prove a compactness result for such energy-minimizing maps. As an application, we combine our result with Shi’s theorem to give an improved bound on the Hausdorff dimension of the singular set, assuming that the map has bounded energy at all scales. This last assumption can be removed when the target manifold is simply-connected.' address: 'Department of Mathematics, Stanford University, Stanford, CA 94305' author: - Da Rong Cheng bibliography: - 'compactness.bib' title: 'A Compactness Result for Energy-minimizing Harmonic Maps with Rough Domain Metric' ---

--- abstract: 'The temporal characterization of ultrafast laser pulses has become a cornerstone capability of ultrafast optics laboratories and is routine both for optimizing laser pulse duration and designing custom fields. Beyond pure temporal characterization, spatio-temporal characterization provides a more complete measurement of the spatially-varying temporal properties of a laser pulse. These so-called spatio-temporal couplings (STCs) are generally nonseparable chromatic aberrations that can be induced by very common optical elements – for example diffraction gratings and thick lenses or prisms made from dispersive material. In this tutorial we introduce STCs and a detailed understanding of their behavior in order to have a background knowledge, but also to inform the design of characterization devices. We then overview a broad range of spatio-temporal characterization techniques with a view to mention most techniques, but also to provide greater details on a few chosen methods. The goal is to provide a reference and a comparison of various techniques for newcomers to the field. Lastly, we discuss nuances of analysis and visualization of spatio-temporal data, which is an often underappreciated and non-trivial part of ultrafast pulse characterization.' address: 'LIDYL, CEA, CNRS, Universit[é]{} Paris-Saclay, CEA Saclay, 91 191 Gif-sur-Yvette, France' author: - 'Spencer W. Jolly, Olivier Gobert, and Fabien Qu[é]{}r[é]{}' bibliography: - 'biblo\_tutorial.bib' --- originally from 2 March 2020, on arXiv 8 July 2020 Introduction {#sec:intro} ============ The frequency dependence of the spatial properties of a broadband light beam or of the optical response of a system is known as chromatism, and has been discussed for decades in many different fields of classical optics. In photography for example, chromatism of the imaging lens affects the ability to properly image an object illuminated by ambient white light, because slightly different images are produced for each color of the incident light. Due to the time-frequency uncertainty principle, ultrashort laser beams necessarily have significant spectral widths, and can therefore also be affected by chromatism. As for any other broadband light source, this impacts the spatial properties of the beam: if a chromatic ultrashort laser beam is focused by a perfect optic, its different frequency components are focused differently, resulting in a degradation of the spatial concentration of the laser light at focus. Yet, compared to incoherent broadband light, chromatism has further consequences for this peculiar type of light sources, now in the time domain: if the spectral properties (in amplitude and phase) of the laser beam are position-dependent, then by Fourier-transformation its temporal properties vary in space too. Such a dependence is known as a spatio-temporal coupling (STC), and implies that chromatism not only affects the concentration of light energy in space, but also its bunching in time, which is the key feature of ultrashort lasers. Properly assessing the impact of chromatism on ultrashort lasers therefore requires specific measurement methods, which give access to the full *spatio-temporal* structure of these beams. Developing such a spatio-temporal metrology, up to the point where it becomes part of the standard characterization routine of ultrashort lasers, is essential because STCs can have highly detrimental effects on the performance of these lasers. As is clear from previous qualitative analyses, they often have the effect of increasing the pulse duration and reducing intensity in focus [@bourassin-bouchet11], but can also have more complex yet very relevant effects, for example on pulse contrast [@li17; @li18-1]. On the other hand, STCs also provide extremely powerful ways of controlling the properties of light beams and therefore laser-matter interaction processes. Examples include optimization of non-colinear sum- or difference-frequency generation [@martinez89; @maznev98; @huangS-W12; @gobert14]), broadband THz generation [@stepanov03; @fulop14], isolated attosecond pulse generation by the attosecond lighthouse effect [@vincenti12; @wheeler12; @kim13; @quere14; @auguste16], improved non-linear microscopy using spatio-temporal focusing [@DURST20081796], and even laser machining [@sun18; @wangP18; @liQ19]. There is a broad collection of purely temporal laser diagnostics [@stibenz06; @walmsley09], which are meant to characterize the evolution of the electric field of a laser pulse in time. These measurements are generally either an average over a given aperture of the pulse, or essentially done at a single point (i.e. a small aperture), and therefore the result is only the local electric field resolved in time. These techniques include frequency-resolved optical gating (FROG) [@kane93; @trebino97; @oshea01; @bates10], spectral phase interferometry for direct electric-field reconstruction (SPIDER) [@iaconis98; @gallmann99; @mairesse05; @radunsky07; @mahieu15], self-referenced spectral interferometry (SRSI, WIZZLER device) [@oksenhendler10; @moulet10; @trisorio12; @oksenhendler12], and D-Scan [@miranda11; @loriot13] among others. The devices and techniques to characterize a laser pulse spatio-temporally are often related to these purely temporal techniques, but also can employ completely separate schemes. Although not a pre-requisite, prior knowledge of temporal measurement techniques for ultrashort pulses will facilitate the reading of this tutorial. Extensive reviews, tutorials or even courses can be found in various past works [@Monmayrant_2010; @dorrer19]. This tutorial aims not to review the entire field of spatio-temporal metrology, especially since there has been an extremely comprehensive review done very recently [@dorrer19]. In contrast, it aims to introduce spatio-temporal couplings and a large range of techniques to diagnose them, in a manner to guide those without significant experience on this topic. We hope that scientists can use this tutorial to determine how to most simply and correctly diagnose or control spatio-temporal couplings in their specific situation. Section \[sec:concepts\] is mostly devoted to defining STCs in a pedagogical way, and introducing the characteristics of the most basic and common couplings. We finish this section by first touching upon techniques that require a minimal amount of specialized equipment, but may not be able to measure arbitrary STCs. In sections \[sec:spatial\] and \[sec:frequency\] we will then expand to more complete and advanced techniques, which are intended to determine the complete spatio-temporal structure of ultrashort laser beams. This ideally requires sampling a field in a three-dimensional space (two spatial coordinates, and time or frequency). This can be considered as one of the main difficulties of STC metrology, since the main light sensors available to date are cameras, which only have two dimensions. This problem has often been circumvented by resolving one spatial dimension only, obviously at the cost of a significant and potentially highly detrimental loss of information. Many present techniques are actually affected by this limitation, but will nonetheless be discussed in this tutorial due to their importance in the development of this field. Spatio-temporal or spatio-spectral metrology uses in general one of two methodologies: resolving a complete temporal or spectral characterization method in one (or more) spatial dimension(s) (’spatially-resolved spectral measurements’), or resolving the amplitude and phase of a spatial measurement at multiple frequencies (’frequency-resolved spatial measurements’). Although the separation based on these definitions can sometimes be difficult to distinguish, the two sections on ’complete’ techniques will be delineated according to our interpretation of these descriptions. The outcome of a complete measurement is a three-dimensional complex matrix describing the $E$-field of the laser beam in space-time or space-frequency. Interpreting and exploiting such a measurement result is far from straightforward, and the visualization and analysis of such datasets can therefore be considered as another significant difficulty of STC metrology. Specific tools have been developed over the last few years, and are summarized in the final section of this tutorial. Key concepts of spatio-temporal couplings and their metrology {#sec:concepts} ============================================================= Before discussing specific advanced methods to characterize the spatio-temporal properties of ultrashort laser pulses, it is necessary to understand exactly what STCs are, the implications on the beam properties in different parameter spaces, and the first very simple steps one might take to diagnose the presence of STCs, at least qualitatively. This is necessary to understand the capabilities of a given measurement device, i.e. it is crucial to understand what forms low- or high-order STCs may take at the measurement position. This is also helpful to finally analyze the result of any complete or incomplete measurement. The goal of any characterization device is to measure as completely as possible the 3-dimensional electric field of an ultrashort laser pulse $E$ in space and time $E(x,y,t)$, or in space and frequency $\hat{E}(x,y,\omega)$ (for the sake of simplicity, we will assume throughout this paper that the field is linearly-polarized, with the same polarization direction all across the beam). The quantities $E$ and $\hat{E}$ are related to each other by the one dimensional Fourier transform from time to frequency. We use $x$ and $y$ as the transverse dimensions, where the beam is propagating along $z$. Because a fully-characterized beam can be numerically propagated to any $z$, we are interested in the measurement of $E$ at only one $z$ that depends on the characterization device in use. In each case the field is composed of an amplitude term and a phase term, i.e.: $E(x,y,t)=\sqrt{I(x,y,t)}e^{i\phi(x,y,t)}$ and $\hat{E}(x,y,\omega)=\sqrt{\hat{I}(x,y,\omega)}e^{i\hat{\phi}(x,y,\omega)}$, where we will sometimes refer to the intensity $\hat{I}(x,y,\omega)$ or the amplitude $\hat{A}(x,y,\omega)=\sqrt{\hat{I}(x,y,\omega)}$. We will use these notations for the rest of the tutorial. The function $\hat{\phi}(x,y,\omega)$ is the ’spatio-spectral phase’, a crucial quantity for the properties of ultrashort laser beams. Much of the complexity of understanding and measuring STCs is actually concentrated in this function. It is closely related to the simple spectral phase $\hat{\phi}(\omega)$ provided by usual temporal measurement devices, in that $\hat{\phi}(\omega)$ is either the value of $\hat{\phi}(x,y,\omega)$ at a test position $(x_0, y_0)$, or a spatial average of this function over $x$ and $y$. Just as in the case of temporal metrology, it is a much simpler problem to measure only the spatio-spectral amplitude or intensity, but measuring both the amplitude and phase is more challenging and will be the topic of the two next sections of this tutorial. We feel that it is important to note finally that the term ’spatio-temporal’ and other similar versions of the term are often used to refer to the combination of measurements that are simply spatial and temporal. This has sometimes been the case in pulse characterization, but is much more often the case in fields that are more far-afield such as microscopy or spectroscopy, where it is less common to also have temporal information in the first place. Our definition is much stronger, i.e. in this tutorial a spatio-temporal measurement is not just the addition of a spatial measurement device and a temporal measurement device, but it is the measurement of the full spatio-temporal field (whether there are STCs present or not). The general concept of spatio-temporal coupling {#sec:concepts_general} ----------------------------------------------- For the purposes of this report we define the basic concept of what a spatio-temporal coupling actually is, in the most simple terms possible. That is: a spatio-temporal coupling is any property of an ultrashort laser pulse that results in the inability to describe the electric field of the laser pulse as a product of functions in space and time. Mathematically, if a beam has STCs, then the following statement is true: $$\label{eq:STC} E(x,y,t)\neq f(x,y)\times g(t) \quad \forall \quad f(x,y), g(t) .$$ In such a case, a similar inequality holds for $\hat{E}(x,y,\omega)$, since it is related to $E(x,y,t)$ by a simple Fourier transform with respect to time. In other words, as mentioned in the introduction, a beam with spatio-temporal couplings also has spatio-spectral couplings (i.e. chromatism), and we will often use these terms interchangeably. In fact the representation in frequency space is often the more convenient one to analyze the beam properties. An example of a nonseparable beam can be seen in a sketch in Fig. \[fig:STC\_concept\], where the beam in panel (a) has no STCs, and the beam in panel (b) does. The example with no STCs is perfectly described by separable functions $f(x,y)$ and $g(t)$ in space and time respectively. For the example in Fig. \[fig:STC\_concept\](b) there is both a varying arrival time of the pulse with the transverse dimension and some transverse variation in the temporal width. To account for the former effect, one may naively describe the field now in terms of $g(t-\tau_0(x))$ with $\gamma$ according to the magnitude of the tilt of the pulse. This is quite simple and potentially valid, but would still result in the full field $E(x,y,t)$ no longer being separable. ![Basic concept of STCs. Both panels show a sketch of the spatio-temporal electric field of an ultrashort laser beam. In (a), this a beam without STCs, where the full electric field can be expressed by separable functions, and the local pulse duration $\tau_0$ is valid globally. In (b), the beam has significant STCs, where the field is no longer separable and the local duration $\tau_0$ is different than the global duration $\tau_G$. The carrier wave here is a sketch and not meant to be to scale.[]{data-label="fig:STC_concept"}](STC_concept.pdf){width="83mm"} This distinction is simple to see when the mathematical descriptions of the fields are compared. We consider a Gaussian beam in space and time for convenience. If $r^2=x^2+y^2$ and the beam has a spatial width $w$, temporal width $\tau_0$, and central frequency $\omega_0$, then the case of Fig. \[fig:STC\_concept\](a) is written simply as $$\label{eq:GaussNoSTC} E_{1a}\propto e^{-r^2/w^2}e^{-t^2/\tau_0^2}e^{i\omega_0 t} .$$ This case is clearly separable. If the pulse has the properties shown in Fig. \[fig:STC\_concept\](b), then the field is written as $$\label{eq:GaussYesSTC} E_{1b}\propto e^{-r^2/w^2}e^{-(t-\tau_0(x))^2/\tau(r)^2}e^{i\omega_0 t} .$$ This is non-separable. As mentioned, this non-separability also has implications on the description of the electric field in frequency and space, but it then takes a different specific form, as will be further discussed in Section \[sec:concepts\_manifestations\]. Beyond having an impact on the mathematical description of the electric field, the presence of an STC will also affect measurable parameters. The most obvious is the temporal duration, which in the presence of some STCs could have spatial variation. It is not in the case of all STCs that the local duration will vary in space, but it is true that with any STC there will be a difference between the local pulse duration and the global pulse duration [@bourassin-bouchet11], referred to as $\tau_0$ and $\tau_G$ respectively in Fig. \[fig:STC\_concept\]. This generally results in a decrease in the peak intensity, and sometimes a varied spatial distribution of the different frequencies within the beam. The next few sections will discuss the nuances of the previous statements and classify a few of the well-known STCs. {width="171mm"} Introduction to low-order couplings {#sec:concepts_low-order} ----------------------------------- Here we introduce some common-place low-order STCs, which provide highly instructive examples. The term low-order refers to STCs where the field variations in space-time or space-frequency can be described by low-order polynomials of position coordinates and time/frequency, and that are therefore more likely to occur. For more complete analysis, we urge the reader to reference significant past work on describing and reviewing this topic [@akturk05; @gabolde07; @akturk10], and work that has gone over alternative matrix-based formalism specifically designed to describe dispersive optical systems [@kostenbauder90; @lin95; @marcus16]. The most prevalent and lowest-order STC is pulse-front tilt (PFT), where the duration of the beam is constant in space, but the arrival time varies linearly with one spatial dimension (Fig. \[fig:STC\_simple\](a)). With PFT, both the wavefront and the pulse-front (describing the location of the electromagnetic energy, i.e. the pulse envelope) of the beam are perfectly flat, but they are constantly at an angle to each other. In other words, the pulse front is tilted with respect to the propagation direction of the pulse. The next most common STC is pulse-front curvature (PFC), where the duration is still constant in space, but the arrival time now varies quadratically with the radial position (Fig. \[fig:STC\_simple\](b)). These two canonical STCs, PFT and PFC, can be caused by very simple and commonplace optical elements. For example, PFT can be induced via propagation through a wedged prism of any dispersive optical material (which includes glasses, the most ubiquitous optical materials), as seen in Fig. \[fig:STC\_simple\](a). The portion of the beam traveling though the thin part of the prism has traversed less material, so then the accumulated group-delay is less than that of the part of the beam passing through the thick portion of the prism. Because the thickness of the prism linearly depends on the transverse dimension in the plane of the page, then the accumulated group-delay will depend linearly on position as well, and this results in the rotation of the pulse front after propagation through the prism. For the same reason, the phase fronts also get tilted, but they do so according to the phase refractive index rather than the group refractive index. The output beam will have PFT if these two rotation angles are different, which occurs if the phase velocity $v_p$ is different than the group velocity $v_g$, i.e. the medium is dispersive. At higher orders, the output beam can also exhibit a spatially-dependent spectral chirp due to the different encountered thicknesses of glass. This generally has negligible impact compared to the PFT. PFC has a similar commonplace source, which is simple chromatic singlet lenses [@bor88; @bor89-1], as seen in Fig. \[fig:STC\_simple\](b). From the temporal point of view the portion of the beam at the outer edge of the lens will accrue less group delay than the center of the beam, resulting in a radially-varying arrival time. If the medium is dispersive ($v_p\neq v_g$) then the curvature of the pulse-front will be different than that for the phase-front after such a lens. Manifestations of couplings {#sec:concepts_manifestations} --------------------------- This section focuses on expanding the descriptions from the previous sections to be more quantitative and to describe couplings in different domains that are relevant for characterization devices and experiments. The main domains we will consider are the near-field (NF) where the beam is collimated (e.g. the output of a laser system), and the far-field (FF) where the beam is at a focus (e.g. where experiments are generally performed), both in time and frequency. The NF and FF of course have broader definitions in classical optics, but for simplicity we will refer to only these two planes as NF and FF throughout this tutorial. These NF and FF spaces defined in this way are related via the principles of Fourier optics, so that the NF is related to the FF by a two dimensional spatial Fourier transform and a coordinate change depending on the focal length ($x_\textrm{FF}=k_x \lambda f/2\pi$) [@doi:10.1002/0471213748.ch4]. Therefore the FF is technically equivalent to the $(k_x,k_y)$ reciprocal space at the NF plane (see Refs. [@akturk05; @akturk10]). However, from the authors’ point of view, considering different physical planes separated by propagation and focusing enables a simpler understanding and is physically more relevant than analyzing the field in different mathematical spaces. We must note that the previously mentioned Gaussian generalization [@akturk05; @akturk10] also provides important insight into the behavior of different couplings in different domains, and previous work has also gone into great detail specifically on the effect of couplings on the pulse duration in focus [@bourassin-bouchet11]. A beam that has no STCs can be described in a simple fashion in all four of the relevant domains (NF and FF, both in time and frequency). A beam with a Gaussian spatial distribution, flat wavefront, and a Gaussian temporal envelope in the NF will have the same temporal envelope in the FF and a Gaussian spatial distribution with a waist determined by the focusing conditions. The description in frequency will be similarly straightforward, regardless of if there is non-zero spectral phase. This is essentially the propagated and/or Fourier-transformed results of Eq. (\[eq:GaussNoSTC\]), where the Gaussian nature allows for analytical representations in all cases. We will take this example of a Gaussian STC-free beam as the baseline, which is pictured in all four domains, with amplitude and phase, in the top row (denoted with (a)) of Fig. \[fig:all\_couplings\]. The NF in frequency is in the pair of panels (i), the NF in time is in panels (ii), the FF in frequency is in (iii), and the FF in time is in (iv). We choose to describe a beam having a Fourier-limited duration $\tau_0$ and central wavelength $\omega_0$ such that $\omega_0\tau_0=10$ (i.e. the beam technically must be very broadband, but we do this to be able to visualize the carrier frequency in time). We plot in normalized units in order to ease the visualization, where for simplicity the NF and FF have a characteristic width noted as $w$ in both cases. In this section we will describe mostly the canonical couplings of PFT and PFC as well as one example beyond that, but we must stress that the final goal of any characterization device is to measure arbitrary couplings. Therefore the importance of this section is to develop the knowledge of how couplings manifest themselve in different spaces. The first very important intuition is for comparing the properties of a beam in time and in frequency domains, related by a 1D Fourier transform. To this end, in the cases of PFT (Fig. \[fig:all\_couplings\](b)(i)) and PFC (Fig. \[fig:all\_couplings\](c)(i)), we can first use a simple physical analysis of the optical systems that induce these couplings, before turning to a more formal description. As discussed in the previous section, PFT can be induced by a prism made of a dispersive glass. As is well-known, such a prism induces angular dispersion (AD), i.e. it results in different propagation directions at the prism output for the different frequency components. We can therefore expect PFT (time-domain description) to be equivalent to a frequency-dependent wavefront tilt (frequency-domain description). Similarly, PFC can be induced by a chromatic lens, which is known to induce a different wavefront curvature for the different frequency components (CC for chromatic curvature). We can therefore expect PFC (time-domain description) to be equivalent to a frequency-dependent wavefront curvature (frequency-domain description). We insist on the fact that PFT and AD (or similarly PFC and CC) correspond to the description of the very same beam, but considered in different spaces. Because of this equivalence, these canonical STCs will be referred to as AD/PFT and CC/PFC in the rest of this work. {width="171mm"} These correspondences between the time- and frequency-domain descriptions can of course be derived mathematically. A general derivation for arbitrary pulse-front distortions is provided in  \[sec:appendixA\] of this tutorial. This simple calculation shows that when considered in the frequency domain, a beam with AD/PFT is characterized by a spatio-spectral phase $\hat{\phi}(x,y,\omega)=\gamma x (\omega-\omega_0)$, where $\gamma$ represents the magnitude of the AD/PFT. This phase is plotted in Fig. \[fig:all\_couplings\](b)(i) and can be understood in two ways. One the one hand, this can be considered as a phase varying linearly in frequency, with a slope $\partial \hat{\phi}/\partial \omega$ (corresponding to a delay in the time domain) that varies linearly with position: this describes PFT. On the other hand, this can be considered as a phase varying linearly in position (i.e. a wavefront tilt), with a slope that varies linearly with frequency: this describes AD. Similarly, PFC is described by a spatio-spectral phase $\hat{\phi}(r,\omega)=\alpha r^2 (\omega-\omega_0)$, where $\alpha$ represents the magnitude of the CC/PFC. This is plotted in Fig. \[fig:all\_couplings\](c)(i), and can either be considered as a linear spectral phase with a slope varying quadratically with position (PFC), or as a quadratic spatial phase (wavefront curvature) varying linearly with frequency (CC). We have now emphasized multiple times that for a beam with STC, different frequencies have different spatial properties, and this has been nicely illustrated by the previous discussion on AD/PFT and CC/PFC. As result, when a beam affected by chromatism propagates, the different frequency components evolve differently. The beam’s spatio-spectral properties and spatio-temporal properties therefore change upon propagation. We now illustrate this point by considering the FF properties of beams that initially have AD/PFT and CC/PFC in the NF. To this end, we start from the frequency-domain description of these beams in the NF. For a beam with AD/PFT, the different frequencies have different wavefront tilt. Therefore, in the FF they must have a varying best-focus position in the transverse dimension. This is displayed in Fig. \[fig:all\_couplings\](b)(iii), and is known as ’transverse spatial chirp’. As a result of the transformation of the beam upon propagation, the temporal structure of the beam in the FF is also very different from that in the NF. In time, the pulse at focus no longer has any PFT, but has a longer local duration corresponding to the global duration in the NF. The focal spot is spatially larger than that of the perfect reference beam, since different frequencies are focused at different transverse positions. Finally, the spatio-temporal phase has a peculiar structure referred to as wavefront rotation (see the phase map of Fig. \[fig:all\_couplings\](b)(iv)). At negative times the spatial phase is tilted in one direction, and over time it changes to finally tilt in the opposite direction at positive times. This describes the fact that the propagation direction of light rotates in time on the scale of the pulse temporal envelope. This effect has interesting applications in high-intensity optics [@quere14], in particular for the generation of isolated attosecond pulses. For a beam with CC/PFC, the different frequencies have a different wavefront curvature in the NF. Therefore, in the FF they must have a varying best-focus position, now along the longitudinal dimension. At a single longitudinal position this manifests as a varying beam size according to frequency, and a spatio-spectral phase that represents the Guoy phase for each frequency. This is seen in panel (c)(iii) of Fig. \[fig:all\_couplings\]. The pulse in time at focus has a more complex amplitude profile, with a longer duration on-axis and a duration and arrival time that vary with the radial coordinate. Beyond the visualization just presented in Fig. \[fig:all\_couplings\], which had a flat spectral phase for at least one position in the beam, it could be such that a beam with either AD or CC were significantly chirped everywhere in space. This would not change much the spatio-spectral picture, since this simply corresponds to the addition of a spatially-homogeneous spectral phase, but would drastically change the picture in time. Because of this, pulse-front tilt and pulse-front curvature are only strictly proper names for these two couplings with no global chirp, and therefore angular dispersion or chromatic curvature are in a sense more general terms. To illustrate more complex cases, the fourth STC we look at (Fig. \[fig:all\_couplings\](d)) is a simple extension of the previous two couplings, where the spatio-spectral phase in the NF is now quadractic in frequency rather than linear (Fig. \[fig:all\_couplings\](d)(i)), i.e. $\hat{\phi}(x,y,\omega)=\zeta x (\omega-\omega_0)^2$. This can be understood as a transversely-varying linear temporal chirp, which in time corresponds to a transversely-varying pulse duration. This is also equivalent to the different colors having a wavefront tilt that varies quadratically with the frequency offset. In the focus this frequency-varying tilt manifests as a quadratically-varying best-focus position in the transverse dimension (Fig. \[fig:all\_couplings\](d)(iii)). In time at focus the pulse amplitude is quite complex, but the temporal phase no longer exhibits any chirp, because the chirps of different signs in the NF average-out at focus. The summaries above and the quantitative visualizations in Fig. \[fig:all\_couplings\] are on one hand relatively simple, and of low-order, but on the other hand can be quite difficult to digest in one sitting. However, understanding the difference between the spatio-spectral phases employed and the reasoning behind the relationships between time and frequency and also NF and FF is key to understanding STCs. This is true both of low-order STCs and those of arbitrary nature. All four of the cases in Fig. \[fig:all\_couplings\] have unique effects in focus and also in time on the collimated beam, but have identical spatio-spectral amplitudes in the NF. In the NF, it is *only* the spatio-spectral phases that differentiate them. From a practical point of view this makes sense, since we often imagine STCs being induced on the collimated beam in the form of chromatic phase aberrations, but in the general case we must also be open to more complex field configurations. In the next section, we briefly discuss some physically relevant cases which involve more complex couplings in the spectral domain. Examples of more complex couplings {#sec:concepts_complex} ---------------------------------- A simple example of pure amplitude coupling in the spectral domain and NF is the case of a beam that has been compressed by a single-pass grating compressor. In this case, the beam central frequency varies linearly with the transverse position, i.e. it has transverse chirp in the NF. If an overall temporal chirp is applied to such a beam, for instance by moving one of the gratings in the compressor, then the combination of these two effects obviously results in a tilt between the pulse-front and the wavefront [@akturk04] (Fig. \[fig:PFT\_akturk04\]). This is the same temporal intensity effect as in the ’standard’ AD/PFT, yet with a field configuration that is actually different both in the NF and FF. Hence this example is very instructive from the point of view of STC metrology, since a measurement of only the spatio-temporal intensity would not provide information on the full nature of the beam. ![A transversely varying central frequency and a spatially-homogeneous linear temporal chirp (quadratic spectral phase), shown in the top row, produce pulse-front tilt in time, shown in the bottom row. However, the field is different than the ’standard’ AD/PFT, despite the PFT in both cases. The color scale for phase goes from $-2\pi$ to $2\pi$.[]{data-label="fig:PFT_akturk04"}](PFT_akturk04.png){width="83mm"} It is important to stress as well that amplitude couplings can spontaneously arise when pulses are very broadband, even in the simple case of a freely propagating beam, due to the chromatic character of diffraction and propagation [@feng98; @porras02]. An example is the case of a broadband cavity operating with Kerr-lens mode-locking, where the Rayleigh range is fixed to be the same for all frequencies. This results in a beam-size that varies according to $\sqrt{\lambda}$ [@cundiff96]. This type of effect can become very significant when pulses approach the few-cycle limit, affecting even the Gouy phase and central frequency through a focus [@porras09; @hoff17]. Amplitude couplings can also easily occur in the misalignment of non-collinear OPAs [@harth18]. Beyond low-order couplings, high order couplings can have a myriad of sources, for example due to changes in laser gain medium [@tamer18] or temporal gain dynamics in highly-saturated Joule-level amplifiers [@jeandet19]. Simple or incomplete measurement techniques {#sec:concepts_measurements} ------------------------------------------- Before discussing advanced techniques, it is useful to discuss some experimentally simple techniques that can determine whether certain couplings are present, although not necessarily precisely their magnitudes. This is useful since these methods generally require very little specific or expensive devices and are therefore very accessible, and also apply to many of the real-world scenarios that scientists may encounter. The most well-known of these simple techniques is to diagnose the focus of an ultrashort laser beam with the full spectrum and compare to that with a narrow central part of the spectrum. In practice, this can be achieved using an appropriate band-pass spectral filter, placed in front of the sensor used to measure the focal spot profile. Referencing Fig. \[fig:all\_couplings\] can already hint that for both AD/PFT and CC/PFC in the NF, the effects of the couplings should be easily visible at the focus (FF). For AD/PFT the focus will be extended in one direction, i.e. elliptical, but will be round with only a narrow part of the original spectrum. Similarly a beam with CC/PFC will be larger than the expected diffraction-limited spot size in focus, but will get closer to this expectation with only a narrow part of the spectrum. In both cases an achromatic aberration may at first be suspected, for example astigmatism causing an elliptical focus, but the different nature of the focus with a narrower spectrum can make it possible to distinguish between chromatic and achromatic aberrations. ![An example of simple diagnostic of STCs, applied to two different cases. (a) and (b) are from the same focused beam with PFT, but (b) has a band-pass filter in front of the camera, while (a) is a measurement with the full beam spectrum. (c) and (d) compare the same types of measurements, now for a beam with PFC. In these two examples, the focus with the band-pass filter added shows the high quality of the focus, but the focus with the full spectrum reveals the coupling. Both sets of data are from different 800nm laser systems with large spectra, which have different focusing conditions. (c) and (d) are adapted from Ref. [@jolly20-1] The Optical Society.[]{data-label="fig:STC_focus_exp"}](STC_focus_exp.pdf){width="83mm"} Experimental examples of this for AD/PFT and CC/PFC in the NF (with different lasers and focusing conditions) can be seen in Fig. \[fig:STC\_focus\_exp\]. Due to transverse chirp, the elliptical focus obtained with the full spectrum in Fig. \[fig:STC\_focus\_exp\](a) is revealed to be round with only the central part of the spectrum in Fig. \[fig:STC\_focus\_exp\](b). Due to longitudinal chirp, the large beam in Fig. \[fig:STC\_focus\_exp\](c) is revealed to be smaller and more round with only the central part of the spectrum in Fig. \[fig:STC\_focus\_exp\](d). In this latter case of CC/PFC the beam with the entire spectrum (Fig. \[fig:STC\_focus\_exp\](c)) is more complex since the NF had a flat-top profile, so the frequencies not at best focus do not have a simple spatial distribution. Although this technique does show the presence of a coupling, only after a complex convolution of the spectrum and the measured profiles with and without the band-pass filter could one expect to quantify the coupling. Still, it can be a very useful technique due to the simplicity. This is why, when the source of the coupling is known and it is a simple step to tune the value, such a method can be very practical and useful. For example, minimizing the ellipticity of a beam with the full spectrum can be an indirect measure for minimizing AD/PFT when using a prism or grating compressor, where it is known that AD/PFT is very easily induced by misalignement. However, when the situation is more complex it cannot give much information, especially when the spectrum contains many features or there are a combination of multiple STCs and/or achromatic focusing aberrations. A further advancement of such a measurement was undertaken on a 100TW laser system and produced meaningful results, which was simply using an imaging spectrometer to spectrally resolve the focal spot along both spatial axes [@kahaly14]. Further extensions of this simple approach would consist in spatially scanning the beam in two dimensions with a fiber spectrometer, or scanning in one dimension with an imaging spectrometer, in order to reconstruct the full spatio-spectral amplitude. Although useful, the weakness of such measurements is that they do not provide any information on the spatio-spectral phase in the NF or FF. The measurement results of Ref. [@kahaly14], shown in Fig. \[fig:kahaly14\], provide an interesting illustration of this limitation. A curved spatio-spectral amplitude was observed in the FF, qualitatively similar to the case of the last coupling of Fig. \[fig:all\_couplings\](d). Yet, since the spatio-spectral phase remained unknown, there was no way to experimentally verify that the measured beam distortion was actually due to a spatially-varying temporal chirp in the NF. As a consequence, unambiguously identifying the nature and physical origin of this distortion in the laser system turned out to be impossible. ![Measurements of the spectrally-resolved focal spot profile along two slices in the FF of a 100TW laser beam (a). The slice shown in (b) has a quasi-parabolic dependence of central frequency on position, where the slice in (c) shows no significant STC. Results are taken from Ref. [@kahaly14] with permission.[]{data-label="fig:kahaly14"}](kahaly14.pdf){width="83mm"} There are many other examples of such ’incomplete’ measurements, of varying complexity, which produce results that are not complete representations of the pulse electric field. These include interferometric measurement of radial group-delay [@bor89-2; @netz00], extensions of single-shot autocorrelation to measure pulse-front tilt [@pretzler00; @sacks01; @akturk03; @figueira19] or pulse-front curvature [@wu16], multiple-slit spatio-temporal interferometry [@li18-2; @li19], There are more advanced diffractive methods that can do similar analysis, using a structured diffraction grating, referred to as “chromatic diversity” [@bahk18], or a measurement of angular chirp simultaneously in both spatial dimensions [@osvay05; @borzsonyi13]. There are also methods that are interested in only the temporal intensity profile (including the absolute intensity magnitude), for example the Temporally-Resolved Intensity Contouring (TRIC) technique [@haffa19]. Although of interest, such methods will not be discussed further in this tutorial. Spatially-resolved spectral measurement techniques {#sec:spatial} ================================================== We first address techniques that we deem are spatially-resolved spectral/temporal measurements, i.e. measurements that resolve the spectrum and spectral phase, extended to one or more spatial dimensions in order to resolve STCs. It is very important to emphasize that, maybe counter-intuitively, simply adding spatial resolution without caution to one of the usual techniques for purely temporal measurements, for instance by scanning the measurement device over space, actually does not provide a full spatio-temporal characterization of a laser beam. This is due to the fact that these techniques generally only measure the components of the spectral phase that affect the pulse duration and shape, but are blind to those that are constant or linear in frequency—which respectively correspond to the Carrier-Envelope relative Phase, and to the pulse arrival time. Therefore a device scanned across space would be able to detect the spatially varying envelope (due either to a change in nonlinear components of the spectral phase or to a change in the spectral width), but not something as simple as AD/PFT or CC/PFC, where the pulse shape does not vary at all spatially. ![Sketch explaining the necessity of spatial-temporal characterization. Since the temporal characterization device (“Device”) in this case is blind to spatial variations of the carrier phase and absolute arrival time of the true pulse in (a), the pulse-front tilt cannot be resolved in the reconstructed pulse shown in (b).[]{data-label="fig:STC_need"}](STC_need.pdf){width="83mm"} This important idea is illustrated via a sketch in Figure \[fig:STC\_need\], where the rastering of the device can resolve the more nuanced fluctuations in pulse length, but not the pulse-front tilt. Of course this may already be a useful amount of information, for example in pulse broadening in a plasma [@zair07; @beaurepaire16], but it is not a complete measurement. Furthermore, in practice the rastering process is itself limited due to the large number of measurements necessary to have a high resolution, especially if the measurement is performed on both transverse dimensions. We outline three different types of measurements that rely on spatially-resolving various methods of pulse characterization. Some of them make it possible to avoid the previous issue, while this can be a very tricky problem for others. The techniques are all based on forms of interferometry, so in every case the unknown beam needs to interfere either with a known reference, or with itself (so-called self-referenced interferometric techniques). One of the key challenges of this category of techniques is precisely to find ways to generate an appropriate reference beam. These techniques can be differentiated mainly by the type of reference used, and the method for resolving the measurement spatially and spectrally. These include: self-referenced techniques such as SPIDER or SRSI, resolved on an imaging spectrometer (section \[sec:SPIDER\], ’established techniques extended to spatial dimensions’); spectral interferometry raster-scanned over the spatial extent of a beam, which can be either externally-referenced or be referenced to a single point on the unknown beam (section \[sec:SEA-TADPOLE\], ’spatially-resolved spectral interferometry); and self-referenced Fourier-transform interferometry, where a spatially-extended reference is made from some central portion of the unknown beam and spectral resolution is obtained via Fourier-Transform spectroscopy (section \[sec:TERMITES\], ’spatially-resolved Fourier-transform interferometry’). Established techniques extended to spatial dimensions {#sec:SPIDER} ----------------------------------------------------- {width="171mm"} In this section we will discuss established techniques extended to spatial dimensions, including mainly the SPIDER and SRSI techniques. In each case the technique is expanded to one spatial dimension only, which already limits the information it provides. Regarding which STCs are accessible by this class of technique, there is some ambiguity in the literature, but we will provide our perspective here. SPIDER is a self-referenced interferometric technique for spectral/temporal measurements, where the reference beam consists of a spectrally-sheared version of the test pulse (TP) to be characterized. Implemented with a 1D spectrometer, this provides the local spectral amplitude and phase of the field in one shot [@iaconis98]. An obvious extension of this technique consists in rather using a 2D imaging spectrometer to obtain spatial resolution along one spatial axis. Historically, this has been one of the first approaches implemented for STC measurements  [@gallmann01], and this is why we discuss it first. We will however show that all spatially-resolved versions of SPIDER are affected by the limitation illustrated in Fig. \[fig:STC\_need\], and explain how only a more sophisticated measurement scheme can circumvent this limitation  [@dorrer02-3; @dorrer02-1; @dorrer02-2]. In order to understand the subtle issues involved in the generalization of SPIDER to STC measurements, it is useful to provide a more detailed description of the technique. To this end, we will focus on the particular implementation called SEA-F-SPIDER [@witting09-1], sketched in Fig. \[fig:SEA-SPIDER\], both because this is the latest and most advanced version, and because it has been used for the spatio-temporal characterization of few-cycle near-infrared and mid-infrared sources  [@witting12; @austin16; @witting16; @witting18]. For a review of the historical development of SPIDER and its different versions, and of the numerous practical advantages of SEA-F-SPIDER we refer the reader to Ref. [@Monmayrant_2010]. In any SPIDER device, the key operation is to create two replicas of the TP, which are sheared in frequency by a fraction of the TP’s spectral width. This is typically achieved by performing sum frequency generation of this test pulse in a second-order nonlinear crystal, with two quasi-monochromatic waves of slightly different frequencies. In SEA-F-SPIDER, these two waves are generated by producing two ancillary beams, obtained by passing two samples of the TP through separate narrowband spectral filters, placed at slightly different angles. For STC measurements, these ancillae can also be spatially filtered to avoid spatio-temporal distortions of the TP in the frequency conversion process  [@witting09-1; @wyatt11]. The technique then consists in comparing these two replicas, which is achieved by spectrally-resolved interferometry. As sketched in Fig. \[fig:SEA-SPIDER\], in SEA-SPIDER, the two replicas are recombined at an angle on the entrance slit on an imaging spectomerer, creating spatial fringes which encode the difference in spatio-spectral phase between the two beams. The mathematical expression of the interferogram measured in this scheme is: $$\begin{aligned} \begin{split} &S\left(x,\omega\right)=|\hat{E}\left(x,\omega\right)|^2+|\hat{E}\left(x,\omega-\Omega\right)|^2 \\ &+2|\hat{E}\left(x,\omega\right)\hat{E}\left(x,\omega-\Omega\right)| \\ &\times\cos\left[\hat{\phi}\left(x,\omega\right)-\hat{\phi}\left(x,\omega-\Omega\right)-k_x x\right] \label{eq:SEA-SPIDER}, \end{split}\end{aligned}$$ with $\Omega$ the induced spectral shear, and $k_x$ the transverse wave vector difference between the two beams. A straightforward processing, based on Fourier transformations and filtering and commonly used in interferometry, makes it possible to get the phase $\Delta \varphi(x,\omega)=\hat{\phi}\left(x,\omega\right)-\hat{\phi}\left(x,\omega-\Omega\right)-k_x x$ from $S\left(x,\omega\right)$, provided $k_x$ is large enough. The next step, common to all implementations of SPIDER and called the calibration procedure, is to eliminate the phase term responsible for the interference fringes, which in the case of SEA-SPIDER is $-k_x x$. This phase term can be determined and then subtracted by performing a measurement of $S\left(x,\omega\right)$ without the spectral shear [@kosik05], leading to $\Delta \varphi(x,\omega)=-k_x x$. In SEA-F-SPIDER, this is achieved by simply rotating one of the spectral filters, so that both ancillae have the same frequency and no shear is induced between the two interfering replicas of the TP. We emphasize that this calibration step is crucial for SPIDER, and has to be performed with great care to make sure that no extra phase term is introduced in the procedure. This can become particularly tricky for STC measurements: in Ref. [@wyatt09], it was found that the SFG process of the TP with the spectrally-sheared ancillae in a non-collinear geometry introduces an extra phase term compared to the calibration configuration. This extra phase term corresponds to a spurious angular dispersion, and needs to be corrected for to get meaningful results. After this calibration step, and assuming no extra phase term has been introduced, one gets $\hat{\phi}\left(x,\omega\right)-\hat{\phi}\left(x,\omega-\Omega\right)\approx \Omega \;\partial \hat{\phi}\left(x,\omega\right)/\partial \omega$. The spatio-spectral phase $\hat{\phi}\left(x,\omega\right)$ can then be obtained by an integration with respect to frequency, obviously up to an unknown frequency-independent phase term. In other words, this type of technique provides no information on the frequency-average wavefront of the beam. As we now explain, this unknown phase term prevents the determination of certain types of STC, such as pulse front distortions *in the measurement plane*. To demonstrate this point, we consider the simple case of a beam with AD/PFT in the measurement plane. As explained in section \[sec:concepts\_manifestations\] and demonstrated in  \[sec:appendixA\], such a beam is described in the spatio-spectral domain by a phase $\hat{\phi}\left(x,\omega\right)=\gamma (\omega-\omega_0) x$, with $\omega_0$ the central frequency of the beam. A SPIDER measurement will then provide $\partial \hat{\phi}\left(x,\omega\right)/\partial \omega= \gamma x$, whatever its specific implementation and assuming a perfect calibration procedure. It is then tempting to conclude that information about this coupling is indeed obtained, but this conclusion omits an essential point. As explained in  \[sec:appendixA\], a perfect STC-free beam that propagates at an angle (here with respect to optical axis of the SPIDER device) is described by a phase $\hat{\phi}\left(x,\omega\right)=\gamma \omega x$, which only differs from the phase of the beam with AD/PFT by a term independent of frequency. For such a beam, a SPIDER device also measures $\partial \hat{\phi}\left(x,\omega\right)/\partial \omega= \gamma x$. This shows that SPIDER cannot distinguish between a beam with AD/PFT and a simple tilted beam, due to the fact that it retrieves the phase up to an unknown frequency-independent term. Following the same reasoning, it is clear that it cannot either distinguish a beam with PFC from a converging or diverging beam. This is precisely the type of limitation illustrated in Fig. \[fig:STC\_need\]. Although this is a conceptual limitation of this technique, it is worth stressing that in most practical cases, this is probably not a significant shortcoming. Indeed, in SPIDER it is generally the laser field in the plane of the SFG crystal that is characterized. This corresponds to the far-field of the laser beam, while couplings such as PFT, PFC or other pulse front distortions typically occur in the NF and the nature of STCs change from NF to FF as explained in section \[sec:concepts\_manifestations\]. Techniques like SEA-F-SPIDER can therefore still provide very useful information on STC at focus. The measurement at focus of the peculiar PFT resulting from the combination of transverse spatial chirp and temporal chirp has for instance indeed been demonstrated [@witting16]. Overcoming this general limitation of the technique is possible, however, by combining a spatially-resolved SPIDER with a spectrally-resolved spatial-shearing interferometer [@dorrer02-3], in order to be able to measure not only $\partial\hat{\phi}/\partial\omega$ but also $\partial\hat{\phi}/\partial{x}$ [@dorrer02-1; @dorrer02-2]. This scheme is sometimes called 2D-SPIDER. From these two measurements it is possible to reconstruct the entire spatio-spectral phase along one spatial dimension. Considering again the comparison of a beam with AD/PFT and a tilted beam, these two cases can now be distinguished, since $\partial\hat{\phi}/\partial{x}=\gamma (\omega-\omega_0)$ in the first case, while $\partial\hat{\phi}/\partial{x}=\gamma \omega$ in the second. The technique is potentially single shot, but requires a rather complex optical set-up and a precise and very careful calibration, and has therefore not been in common use so far. SRSI [@oksenhendler10; @moulet10; @trisorio12] is another technique which is commonly used to determine the electric field of a pulse in time. In this technique, a nonlinear $\chi^{(3)}$ effect (cross-polarized wave generation, XPW [@minkovski04]) generates a temporally-filtered replica of the input TP, which is then used as a reference pulse for interferometric measurements. The key idea is that this reference pulse, which is spectrally broader than the TP, can be considered to have a nearly flat spectral phase. Imperfections of this reference pulse can be taken into account through the use of an iterative algorithm [@oksenhendler10]. In order to be able to obtain spatially-resolved measurement and potentially STCs, a tilt between the TP and the reference is implemented (SRSI-ETE) [@oksenhendler17], and the interferogram is measured with an imaging spectrometer. The interferogram obtained in this case at the sampled position $y_0$, with $k_x$ and $\tau$ representing the angle and temporal delay respectively, is given by: $$\begin{aligned} \begin{split} &S\left(x,\omega\right)=|\hat{E}\left(x,\omega\right)|^2+|\hat{E}_{\textrm{XPW}}\left(x,\omega\right)|^2 \\ &+2|\hat{E}\left(x,\omega\right)\hat{E}_{\textrm{XPW}}\left(x,\omega\right)| \\ &\times\cos\left[\hat{\phi}\left(x,\omega\right)-\hat{\phi}_{\textrm{XPW}}\left(x,\omega\right)-k_x x-\omega\tau\right] . \end{split}\end{aligned}$$ If the spectral phase of the reference is properly filtered by the XPW process, and with a proper calibration of $k_x x + \omega\tau$, this interferogram can be used to retrieve $\hat{\phi}\left(x,\omega\right)$. Yet, this technique should suffer from the same ambiguities as the SPIDER technique, again related to the issue emphasized in Fig. \[fig:STC\_need\]: the XPW process used to generate the reference does not filter the constant and linear terms of the spectral phase, corresponding in space-time to wave front and pulse front. As a result, the SRSI-ETE technique should not be able to resolve pulse-front distortions such as AD/PFT and CC/PFC. We note however that Ref. [@oksenhendler17] claimed a measurement of the former, which might have been possible in this specific implementation because the XPW process was carried out in the FF, and the SRSI-ETE measurement in the NF. The form of PFT due to the combination of spatial and temporal chirps would be resolved, since it is due to second-order spectral phase that would be filtered on the reference by the XPW process. There other examples of devices expanding upon the FROG technique, ImXFROG [@eilenberger13] and HcFROG [@mehta14]. However they have had very limited use and therefore fall outside of the scope of this tutorial. All the techniques described here have their own merits, but also tend to be experimentally complicated and require very precise calibration. Indeed, for various ambiguities it is not completely clear experimentally what the limits of the techniques are, and because they are self-referenced it is difficult to set a threshold for when the calibration has been done properly. So one should take great consideration when choosing if a technique in this section is suitable for one’s application. Spatially-resolved spectral interferometry {#sec:SEA-TADPOLE} ------------------------------------------ {width="171mm"} Spectral interferometry between an unknown pulse and a *known reference pulse* can provide the full spectral amplitude and phase information of the unknown pulse, and is a component of many standard temporal pulse characterization devices that have matured significantly to this point in time. We will outline devices that involve spatially-resolving spectral interferometry measurements in order to reconstruct the 3-D electric field, which are referred to as SEA-TADPOLE, STARFISH, and RED-SEA-TADPOLE. These are very different from the SEA-SPIDER and SRSI-ETE methods, in that they have no spectral or spatial shear, and therefore require an additional spectral phase measurement to reconstruct the complete spatio-temporal electric field, and spatial resolution is obtained by scanning the beam with an optical fiber. The original TADPOLE technique was developed to temporally characterize very weak pulses using a combination of a FROG measurement for a reference beam, and a single spectral interferogram between the reference and an unknown pulse [@fittinghoff96]. SEA-TADPOLE was then developed as a variant where the unknown and reference pulses are collected by monomode fibers, and then compared by spatio-spectral interferometry [@meshulach97] between the two separate fiber outputs [@bowlan06]. This was then naturally extended to spatio-temporal metrology by scanning one of the two fibers’ input across the spatial extent of the unknown beam, to measure its spatially-resolved spectral amplitude and phase [@bowlan07] through comparison with the fixed reference beam. The first measurements with this technique could achieve quite high spectral and spatial resolutions [@bowlan08]. The STARFISH technique [@alonso10; @alonso12; @alonso13] is essentially a simple variant of the same technique, which simplifies the experimental implementation by rather using standard fiber spectrometers and a fiber coupler, and relying on pure spectral interferometry, thus avoiding the need for a 2D spectrometer. In most practical cases, the reference beam is obtained from a part of the unknown beam itself, which is then separately characterized in time. We emphasize that this reference must cover the full spectral extent of the unknown beam, which can be an issue for the characterization of beams with strong inhomogeneities in spectral amplitude. A critical experimental issue is related to phase fluctuations in the monomode fibers, due e.g. to vibrations or temperature fluctuations. These are particularly difficult to avoid in these measurement schemes as at least one of the fibers needs to be scanned spatially—which necessarily implies a minimum amount of deformation. The main effect will generally be a randomly-varying overall phase term induced on the spectral phase of the test pulse as the beam is scanned. This implies that the wave front of the beam cannot be retrieved, and hence than STC such as PFT, PFC or any other pulse front distortions cannot be measured. A solution to this issue has been proposed and demonstrated in Ref. [@bowlan12], where SEA-TADPOLE measurements were done in multiple $z$ planes, and a standard phase retrieval algorithm (see section \[sec:phase\_retrieval\]) was then applied to the measurement results to retrieve the wavefront. This however requires scanning the beam transversely (in principle in 2D) with the fiber in *multiple planes*. The SEA-TADPOLE results cited here were generally applied on a focused beam or a beam near the focus, which is why the phase-retrieval technique proposed in Ref. [@bowlan12] was possible. However, if the beam to be measured has a pointing jitter such that fluctuations in the focus position become a significant fraction of the focal spot size, sampling this focus by scanning an optical fiber obviously becomes meaningless. Such a situation typically occurs on high-power ultrashort lasers. It is then necessary to rather scan the beam in the near-field, where the spatial extent is significantly larger. With a larger scanning range, the issues of phase fluctuations in the fiber, and of the stability and accuracy of the rastering stage and of the whole interferometer become much more critical. An evolution to the SEA-TADPOLE device, named RED-SEA-TADPOLE [@gallet14-2], was developed to solve these concerns and thus be able to apply this type of techniques to the collimated beams of e.g. high-power ultrashort lasers. The RED-SEA-TADPOLE device, shown in Fig. \[fig:SEA-TADPOLE\](a), uses a second reference beam which needs to fulfill stringent conditions (detailed in Ref. [@gallet14-2]), the main ones being that it must be in a spectral range different from that of the test beam, and that it has to be free of STCs. This reference beam is collected by the two fibers together with the unknown beam. The purpose of this spectrally distinct reference is to be able to independently measure imperfections in the spatial scanning and fluctuations in the fibers and to subtract them from the final measurement of the unknown beam. Essentially, since the spectrally distinct reference is homogeneous and free of STCs, any distortion of this beam retrieved by the measurement must be due to the stage wobbling or the fiber fluctuations. The production of a suitable spatially-extended reference beam with sufficient photon flux is however far from trivial. This was done in Ref. [@gallet14-2] with an expanded photonic-crystal fiber-based supercontinuum source and a band-pass filter outside of the band of the unknown laser pulse spectrum. So with the RED-SEA-TADPOLE device a large collimated beam can now be measured with all scanning imperfections accounted for. In Ref. [@gallet14-2] this was successfully utilized to measure both AD/PFT and CC/PFC due to a detuned grating compressor and a chromatic lens respectively. However, there remains one important issue that is relevant to SEA-TADPOLE, STARFISH, and RED-SEA-TADPOLE (despite the improvements in the latter). That is, due to ambiguities in phase-unwrapping, the spatial scanning resolution necessary to truly resolve an unknown pulse in the nearfield are very stringent. This phase unwrapping ambiguity is shown in Fig. \[fig:SEA-TADPOLE\](b-c). If the measured pulse is simply tilted with respect to the plane of the rastering, then the true phase across one spatial cut will be linear with a slope depending on the tilt (shown in the dashed lines). This is relevant because in practice it is extremely difficult to align the plane of a 2D stage exactly with the plane perpendicular with the laser propagation. The raw measured phase, which is necessarily wrapped, will require unwrapping to resolve this tilt. When the beam is scanned with high resolution as in Fig. \[fig:SEA-TADPOLE\](b), there is no ambiguity in the phase unwrapping, so the result is correct. However, when the scanning resolution is much lower as in Fig. \[fig:SEA-TADPOLE\](c), the wrapped phase is no longer unambiguous, because it is not known a priori how many $2\pi$ phase jumps occur between each measurement point. When the phase is unwrapped there is a high likelyhood of producing an incorrect unwrapped phase. This is a general issue in phase unwrapping, but in the context of RED-SEA-TADPOLE combined with a large beam in the NF, this implies that rastering in both tranverse dimension requires millions of sample points. This issue means that for practical purposed SEA-TADPOLE on a collimated beam is often just too inconvenient to implement. However, for beams in-focus it is still a reasonable solution, because in such a configuration having a very high spatial sampling is practically achievable [@bowlan08]. We however note that in all measurements performed with these techniques so far, the beam was in practice sampled along one spatial direction only, due to the burden of finely scanning the fiber tip in 2D. As a last point, it is theoretically possible to create a multiplexed version of SEA-TADPOLE for measurements in the near field, to reduce or eliminate the need for scanning in 2D, referred to as MUFFIN in Ref. [@gallet14-1]. But in this case the same difficulties regarding the phase-jumps in space exist, and the experimental complexity of adding fibers and the lack of high spatial resolution in the final data make it not very attractive in the end, and in fact it was never implemented for the characterization of STCs. Spatially-resolved Fourier-transform spectroscopy {#sec:TERMITES} ------------------------------------------------- A major drawback of the previous techniques is that the unknown beam needs to be scanned spatially in 2D to measure the full 3D spatio-temporal or spatio-spectral field. It would obviously be more straightforward to directly resolve the two transverse spatial coordinates of the beam on a camera. But one then needs to find a way to resolve the third coordinate, i.e. to get spectral resolution. We now explain how this can be achieved by exploiting Fourier-transform spectroscopy (FTS), leading to different techniques where only one physical parameter—a temporal delay- now needs to be scanned, described in this section and later in section \[sec:phase\_retrieval\]. FTS is a powerful method to resolve the spectrum of an unknown beam by measuring the evolving signal on a photodiode as the unknown beam temporally interferes with a delayed copy of itself. The resulting interferometic trace, composed of the signal measured at all of the scanned delays $\tau$, contains the first-order autocorrelation function of the field, and can be Fourier-transformed to frequency in order to directly produce the spectral intensity $\hat{I}(\omega)$, when selecting only the spectral information at the positive frequency peak. More explicitly: $$\hat{I}(\omega)=\left\lbrace \mathcal{F}_{\tau\rightarrow\omega}\left[\int \left|E(t)+E(t-\tau)\right|^2 dt\right]\right\rbrace_{+\omega} ,\label{eq:FTS}$$ which is essentially the Wiener-Khinchin theorem [@doi:10.1002/0471213748.ch11]. Since the beams interfering are copies of each other, the resulting beam has a zero spectral phase regardless of the phase on the input beam. This has the benefit that the spectral phase of the beam to be measured does not matter, but of course the downside is that FTS cannot resolve the spectral phase. Since FTS is a scanning method there are significant implications of shot-to-shot fluctuations and noise on the resulting spectrum [@dorrer00]. This makes FTS generally less preferred to simple fiber-coupled grating spectrometers (when available) to measure the 1-D spectrum, but in the case of full spatio-temporal characterization it has found a new application. FTS can indeed be spatially-resolved quite easily, just by resolving the interfering beams on a camera. When the interfering beams are exact copies of each other, this provides a straightforward way to measure the spatio-spectral intensity. However, there is no phase information just as in the 1-D case. A self-referenced version of spatially-resolved FTS that can resolve the spatio-spectral phase was developed simultaneously and independently in  [@miranda14] and  [@gallet-thesis], and later named TERMITES [@pariente16], which will be the focus of the rest of this section. TERMITES has been used successfully with varying specific experimental layouts for measurements on a Terawatt laser [@pariente16], a Petawatt laser [@jeandet19], ultrafast optical vortices [@miranda17], and a femtosecond OPCPA used for high repetition-rate high-order harmonic generation [@harth18]. In each case it could resolve both standard STCs and more complex couplings in both amplitude and phase. The TERMITES technique involves interfering the unknown beam with a spatial portion of itself, which has been expanded to overlap across the whole spatial extent of the beam. The key idea is that this expanded portion of the original beam can be used as a reference, because it comes from a small enough portion of the original beam to be free from STCs. Since the reference is then not strictly an exact copy of the unknown beam, the spatially-resolved FTS will have both amplitude and phase information, as will be explained more precisely below. Comparing TERMITES to SEA-TADPOLE, the reference in TERMITES has been expanded spatially so that on a single 2D image there is interference at all points, eliminating the need to scan in 2D that was present in SEA-TADPOLE. {width="171mm"} An example of experimental implementation of TERMITES using a modified Michelson interferometer with one curved end-mirror is shown in Fig. \[fig:TERMITES\_1\](a). The overlapping beam and reference are either viewed directly on a camera chip, or put on to a scattering screen which is then imaged using a viewing objective and camera. The FTS is performed by stepping through the delay $\tau$ with steps small enough to beat the Shannon limit. Therefore significantly sub-cycle accuracy on the delay stage is essential: for characterizing 800nm wavelength beams we have employed steps of 150nm using a piezo-electric stage with fluctuations on the 10nm level [@pariente16]. In the end a TERMITES measurement produces a 3-D matrix of interferometric signals in $x$, $y$, and $\tau$. This data represents the spatially-resolved interferences of the unknown beam and the reference, with the addition of a large curvature value due to the fact that the reference is diverging. The first analysis steps involve Fourier-transforming from $\tau$ to $\omega$, selecting only relevant positive frequencies, and subtracting the known curvature. These steps are shown in Fig. \[fig:TERMITES\_1\](b-d). After these steps there remains the “cross-spectral density” $\hat{s}(x,y,\omega)$, which corresponds to the product of the complex spectral amplitude of the unknown beam, and the conjugate of the complex spectral amplitude of the reference beam. Algorithmically this corresponds to $$\begin{aligned} \label{termites-equation} \hat{s}(x,y,\omega)&=\hat{E}(x,y,\omega)\hat{E}_R^*(x,y,\omega) \\ &=A(x,y,\omega)A_R(x,y,\omega)e^{\hat{\phi}(x,y,\omega)-\hat{\phi}_R(x,y,\omega)} ,\end{aligned}$$ where the curvature of the reference beam has already been removed. At each point of the beam, one can thus get the difference in spectral phase $\hat{\phi}(x,y,\omega)-\hat{\phi}_R(x,y,\omega)$ between the unknown and reference beams. This is why we include TERMITES in the category of spatially-resolved spectral measurement techniques. The final step of the data processing leading to the reconstruction of the unknown beam depends on the way TERMITES is implemented. In the first version utilized in Lund [@miranda14], the reference came from a very small portion of the original beam, which was moreover spatially filtered before interfering with the unknown beam. The reference beam can then be considered as originating from a point source, such that it can be assumed to be free from STC, i.e. $\hat{\phi}_R(x,y,\omega)=\hat{\phi}_R^0(\omega)$ (and similarly for the spatio-spectral amplitude). Since TERMITES is self-referenced, it is blind to any spatially homogeneous spectral phase of the unknown beam. Therefore, to reconstruct the field in the spatio-temporal domain, a single temporal measurement is still necessary, either on the reference beam or at any single point of the unknown beam (see section \[sec:phase-stitching\], phase stitching). Note however that even without this final measurement, all pure spatio-spectral effects are resolved. In the implementation presented in Fig. \[fig:TERMITES\_1\](a), the reference rather comes from some finite central portion of the unknown beam, due to the practical constraints imposed by the application to a laser beam of large diameter. For instance, in the version used in Ref. [@pariente16], the reference came from the central half of the beam. This scheme is then similar to radial-shearing interferometry [@doi:10.1002/9780470135976.ch5], and the reference may itself still contain STCs. Based on the fact that it originates from a sub-pupil of the unknown beam, an iterative algorithm can be applied to the spectral amplitude and phase provided by Eq. (\[termites-equation\]), to eliminate the contribution of the reference and reconstruct the complex spatio-spectral field of the unknown beam. As before, a temporal measurement at a single point of the beam is still required to determine the field in space-time. There are strict requirements on the camera properties in a TERMITES device. Due to the varying angle between the reference and unknown beam, the interference fringes increase in spatial frequency towards the outer part of the beam (see example images in Fig. \[fig:TERMITES\_1\](b)). The input beam diameter of the collimated beam $D$, the focal length $f$ of the convex mirror for the reference, and the fraction $\beta$ of the unknown beam diameter used to produce the reference fix the pixel size required to resolve the fringes at the outer part of the beam. If we say at the edge of the beam $p$ pixels are required for each fringe, then the linear number of pixels needed across the beam is $N=pD^2\beta/2\lambda_0|f|=pD^2(1-\beta)/2\lambda_0 L$, which leads to a total image size of at least $N^2$ pixels (see supplementary material of Ref. [@pariente16], and note the geometric constraint of $\beta=|f|/(|f|+L)$, where L is related to the total path length of the device). This can lead to requirements of tens of megapixels for beam diameters of a few centimeters, having a big impact on the data size of the final measurement (routinely many Gigabytes for a single measurement). A nuanced distinction regarding the required pixels, which is not discussed in previous work, is that the true constraint is on the size of the pixels such that the camera signal is not averaging over a significant portion of a spatial interference fringe. If a camera chip was decimated so that fewer total pixels were chosen, but the pixel size was still small enough, then the proper signal would still be measured (although with lower resolution), now with a reduced data transfer time and smaller final data size. Additionally, the pixel size at the center of the beam could theoretically be much larger, since the constraint on pixel size is only an important constraint near the edge of the beam. But in reality both of these cases would require more advanced analysis or expensive hardware, and have not been presently investigated. There is also a geometry of TERMITES where rather than the unknown beam staying collimated and the reference being diverging, the curved mirror can have a positive focal length and is used to reflect the unknown beam, such that the reference now remains collimated while the unknown beam converges. In this case the reference is still larger than the unknown beam on the camera, but the overall size of the relevant image is smaller. This is shown in Figure \[fig:TERMITES\_1\](e). The advantage of such a setup is that the size of the camera chip can be much smaller. If $\beta$ is still defined as the ratio of the unknown beam size to the reference beam size on the camera, a simple calculation shows that the requirement for the linear number of pixels across the beam in this configuration is $N=pD^2\beta/2\lambda_0|f|=pD^2\beta(1-\beta)/2\lambda_0 L$. So essentially, in this converging setup with the same device size determined by $L$, the number of pixels across the beam can be reduced by a factor of $\beta$ compared to the diverging setup, resulting in a total image with $\beta^2$ fewer total pixels. Due to the fixed geometrical relationship between $\beta$, $f$, and $L$ the focal length of the converging mirror must be longer to have a device with the same $\beta$ and $L$ ($\beta=(f-L)/f$ in the converging case). The most important parameter when designing a TERMITES device is the fraction $\beta$ of the unknown beam diameter used to produce the reference. For example, a smaller $\beta$ will make the iterative algorithm more likely to converge to the true physical result since the reference would be from a smaller portion of the unknown beam, and therefore more likely to be free of STCs. But according to the previous discussion, a small $\beta$ will also result in more stringent geometric constraints, potentially making the setup untenable in size or price. There are also limits on $\beta$ since the best signal-to-noise ratio in the computed complex spectrum is when there is perfect fringe contrast. However, since the reference is diverging this requires the beamsplitter in the interferometer to be different than 50:50. As $\beta$ become smaller this becomes a bigger issue and one must eventually compromise with non-ideal fringe contrast. We now emphasize certain advantages and limitations of the TERMITES technique. Because of the strict requirements on pixel size the device will generally have a very good spatial resolution. This totally eliminates the phase unwrapping ambiguity previously encountered with SEA-TADPOLE and its variants, and will also provide TERMITES with remarkable ability to resolve very fine spatial features, as demonstrated in Ref.[@jeandet19]. On the other hand, it also causes the initial data files to be in the tens of Gigabytes. From a physical point of view, the spectrum of the whole beam can only be resolved properly if the reference is as spectrally broad or broader than the unknown beam. So, if the center of the unknown beam where the reference is taken from has a narrower spectrum than the outer portions, then the resulting spectrum after all analysis steps will be narrower than in reality. The TERMITES technique (and FTS in general) requires taking camera images based on many successive laser shots. Because of this any fluctuations of parameters can have an effect on the final result: intensity fluctuations can be accounted for using a portion of the camera image that is not interfering or by using the integrated signal on the detector, but fluctuations in spectrum, pointing, wavefront, or the STCs on the pulse could have a significant effect. Additionally, since delay steps are not perfect, fluctuations in the delay above a certain level can cause a degradation of the retrieved spectrum [@dorrer00]. Despite this inherent shortcoming of being multi-shot, the few measurements on very large laser systems that tend to have non-trivial fluctuations have been successful [@jeandet19; @pariente16]. ![The SEA-TERMITES technique is a single-shot version of TERMITES whereby an imaging spectrometer resolves the cross-spectral density $\hat{s}$ in only one spatial dimension. The idea was proposed in Ref. [@gallet-thesis], with the visualization style in this figure adopted from Ref. [@pariente16].[]{data-label="fig:TERMITES_2"}](SEA-TERMITES.pdf){width="83mm"} Category Technique(s) Spatial dim. Spectral information obtained by Single-shot? Complete? Type of reference beam -------------------------------------------- --------------------------------- ---------------- ---------------------------------- -------------- -------------------- -------------------------------------- Extension of established techniques 2D-SPIDER, SEA-SPIDER, SRSI-ETE 1D Spectrometer Yes No Shearing of unknown beam, XPW effect Spatially-resolved spectral-interferometry SEA-TADPOLE, STARFISH 2D, usually 1D Spectrometer No Yes, but difficult Independent or part of unknown beam Spatially-resolved FTS TERMITES 2D FTS No Yes Part of unknown beam As a last discussion related to the TERMITES principle, there exists a version of TERMITES that is single-shot, but only resolved in one spatial dimension. We refer to this as SEA-TERMITES [@gallet-thesis], which is pictured in Fig. \[fig:TERMITES\_2\]. The concept is simple, rather than scanning over $\tau$ the device can be set up at a single delay and an imaging spectrometer can be installed at the location of the screen or camera. Then a single-shot measurement will produce the TERMITES data at one spatial slice via spectral interferometry. Following this measurement a similar analysis procedure must be followed (except that the data is already resolved in frequency) with vastly decreased data size and therefore speed, with the price of loosing information in one spatial dimension. In the case of TERMITES the success of the iterative algorithm requires that the measured slice is precisely that going through the mutual center of the reference and the unknown beam, and cylindrical symmetry must be assumed. This same equivalence between spectral interferometry and FTS will be seen again briefly in the case of the INSIGHT device, with the same trade-off of being single-shot, but losing one dimension of information and having restrictions on the symmetry of the beam. As a conclusion of this section, the main properties of the different techniques discussed in this section are summarized in Table \[tab:table1\]. Frequency-resolved spatial measurement techniques {#sec:frequency} ================================================= The second major classification of characterization methods corresponds to the techniques that can be referred to as frequency-resolved spatial measurements. This mostly takes the form of techniques that frequency resolve a measurement that is generally used to determine the spatial amplitude and phase profiles of a beam, but will also include more advanced examples that are either similar in philosophy or in methods. This sometimes results in loss of one spatial coordinate, or is made at the cost of resolution. The measurement of the laser wavefront is important for many experiments involving focused laser beams, especially those operating at high-intensity. Measurement of a laser wavefront generally uses one of the following three approaches: 1- measuring the local slope of the wavefront, e.g. using an array of micro-lenses to create an array of foci whose position depends on the local wavefront (Shack-Hartmann method), 2- interferometry, either externally-referenced or self-referenced (as in the common four-wave lateral shearing interferometry [@Primot:93; @chanteloup04]), 3- phase-retrieval algorithms applied to amplitude-only measurements in different $z$ planes  [@10.1117/12.472377]. Essentially, the techniques described in this section are derived from one of these approaches. Extensions of established wavefront sensing techniques {#sec:wavefront} ------------------------------------------------------ What we call here “direct” spatial phase measurements correspond to industry-standard wavefront measurements, which in this section are expanded to include the frequency dimension. We include here both Shack-Hartmann and Four-wave shearing interferometry, although the latter is also related to a following subsection that discusses interferometric methods. Wavefront autocorrelation was already attempted early on [@grunwald03], where autocorrelation was performed on the sub-foci of an all-reflective Shack-Hartmann device. However, this result was not expanded upon in the literature, so we rather focus on other methods that have been detailed more recently. A first measurement involving a wavefront measurement at a small number of frequencies and propagation calculations [@hauri05] has confirmed that resolving the wavefront spectrally is indeed a valid method for reconstructing or approximating the entire electric field, providing a good foundation for these techniques. A further result resolved in frequency, combined with spectral phase stitching (see section \[sec:phase-stitching\]), was the Shackled-FROG technique [@rubino09]. This technique, visualized in Fig. \[fig:HAMSTER\](a) is the combination of an imaging spectrometer with a Shack-Hartmann wavefront sensor, as well as a single FROG measurement. Essentially the Shack-Hartmann wavefront sensor is placed at the image plane of an imaging spectrometer, producing an array of sub-foci that are resolved in frequency along the dispersive axis of the grating, and resolved in one spatial dimension. This measurement results in the spatio-spectral phase $\hat{\phi}_S(x,\omega)$ at one sampled $y_0$ position, up to an unknown spatially homogeneous spectral phase. When combined with a FROG measurement, in this case at one point on the same spatial slice $y_0$, the total spatio-spectral phase can be reconstructed. The intensity of the sub-foci can lead to a proxy measurement for the spectral intensity, which means that the spatio-spectral electric field is fully known along one spatial axis. ![(a) Shackled-FROG technique, based on the schematic from Ref. [@rubino09]. (b) HAMSTER technique, based on the schematic from Ref. [@cousin12].[]{data-label="fig:HAMSTER"}](HAMSTER.pdf){width="83mm"} The HAMSTER technique [@cousin12], pictured in Fig. \[fig:HAMSTER\](b) also uses a Shack-Hartman device, but keeps both spatial dimensions. This is accomplished by using an acouto-optic dispersive filter (AOPDF) before the wavefront sensor to select narrow regions of the original spectrum. After making multiple such measurements the spatio-spectral phase $\hat{\phi}_S(x,\omega)$ is known, again without pure spectral phase knowledge. A local FROG measurement can lead to the reconstruction of the full spatio-spectral phase, and again the intensity of the sub-foci on the Shack-Hartman device leads to knowledge of the spatio-spectral amplitude. This device then is capable of measuring the full spatio-spectral electric field, but has a few restrictions. In particular, in order to select the narrow spectral regions without adverse effects, the AOPDF must be behind any amplifiers. Because the AOPDF generally has a very small aperture, this limits the type of beams that can be measured without prior demagnification. Taking a more direct approach, a Shack-Hartman device has been used in combination with various band-pass and long-pass filters (shown in Fig. \[fig:wavefront\](a)) to assess the wavefront of broadband laser sources, such as a white-light continuum [@hauri05; @kueny18] . This method is conceptually similar to the HAMSTER technique, but potentially cheaper to implement and is only limited spatially by the aperture of the sensor and the filters. However, it is difficult to design filters that have a narrow transmission bandwidth, so the wavefront must be constructed via a very small number of frequencies as in Ref. [@hauri05] or a set of measurements that had overlapping spectra as in Ref.[@kueny18]. This could lead to errors if the transmission of the filters is not well-known or the spectrum is highly modulated, and will regardless lead to a relatively low spectral resolution. It must also be taken into account that the spectral filters may themselves impart wavefront imperfections. From this point of view the problem of spatio-spectral measurement is somewhat shifted to filter calibration in this technique. There is also a device in development that uses a single filter rotated or translated to shift its transmission [@ranc19], mitigating some of the mentioned limitations. Rather than filtering the incoming beam and then measuring with a wavefront sensor, it has also been shown that one can use a multi-spectral camera, combined with either a Hartmann mask or a checkerboard mask (for four-wave lateral shearing interferometry) [@dorrer18] (shown in Fig. \[fig:wavefront\](b)). This approach is elegant and simple, but suffers from a very low spatial resolution, low wavelength resolution, and requires a multi-spectral camera that is generally quite expensive. For example, two common pixel patterns are shown in the inset of Fig. \[fig:wavefront\](b) that only have two or three colors, and reduce the resolution of the CCD by a factor of 4. However, the rapid industrial progress in so-called “snapshot” multi-spectral imaging techniques [@hagen13] could be adapted for the spatio-temporal characterization of ultrashort beams. This application of multi-spectral or hyper-spectral cameras could be transformative. {width="171mm"} An example of a snapshot multi-spectral imaging technique was deployed to rapidly characterize a scattering medium [@boniface19] (shown Fig. \[fig:wavefront\](c)) and could be adapted for pulse characterization. This technique used the combination of a lens array and a tilted grating to create an array of sub-foci on a 2D sensor, intimately related to the field of integral field spectroscopy common to astronomy [@allington-smith06]. Because the grating was tilted relative to the axis of the lens array, the dispersed sub-foci do not overlap. With a proper patterning of the lens array, the ability to pack a given spectrum on the 2D sensor can be maximized, although there is still a limitation on the bandwidth of the pulse to be measured. It could be that other methods applied to study scattering media [@liX19], or the multi-spectral properties of scattering media or multi-mode fibers themselves [@xiong20; @xiong19-2; @ziv20] could be used to characterize ultrashort pulses, although for now it is highly speculative. Frequency-resolved detection of the wavefront of the trains of attosecond pulses produced via high-harmonics generation has been done in various implementations [@frumker09; @austin11; @freisem18; @dacasa19], but we emphasize that the discrete nature of the high-harmonic spectra makes it a significantly different problem than for a single ultrafast pulse with a continuous spectrum. Iterative phase retrieval with frequency resolution {#sec:phase_retrieval} --------------------------------------------------- In this section, we look at measurement methods that rely on iterative phase retrieval algorithms. This type of approach is well known for wavefront measurements of monochromatic beams. The principle is to measure the spatial intensity profile of a beam –which is straightforward to do using a camera– at multiple $z$ planes separated by known distances. The evolution of this profile as the beam propagates obviously depends on the phase properties of the beam. Iterative algorithms such as the Gerchberg-Saxton one have thus been developed to extract this information from a few measured profiles [@gonsalves79; @fienup82; @matsuoka00]. Such phase-retrieval methods are already used in concert with deformable mirrors in order to optimize the *on-target* focal spot of high-power lasers [@pharao; @oasys; @beamtuner]. However, directly applying this type of approach to a broadband laser beam is doomed to fail if this beam is affected by significant chromatic effects. Indeed, in such a case, the measured intensity profile is the incoherent sum of multiple and potentially different intensity profiles associated to each frequency, $I(x,y)=\sum_{\omega}{I(x,y,\omega)}$, which each evolve in their own way along the propagation axis. Such effects are not taken into account in the iterative phase retrieval algorithm, which will then either poorly converge, or converge to a wrong solution. However, if the intensity profile of the beam is known at each frequency, then this approach can be safely applied independently to all frequencies of the beam. This is the approach discussed in this section, which can be implemented in different ways. ![CROAK technique, based on the schematic from Ref. [@bragheri08], but improved to have less complicated steps.[]{data-label="fig:CROAK"}](CROAK.pdf){width="83mm"} The first implementation of such a phase-retrieval technique for spatio-temporal measurements was termed the CROAK technique, standing for Complete Retrieval of the Optical Amplitude and phase using the $(k,\omega)$ spectrum [@bragheri08]. This method is detailed in Figure \[fig:CROAK\] with a simplified geometry compared to that in the original reference. The technique requires three steps. The first step, in Fig. \[fig:CROAK\](a) is to measure the spectral phase at a well-known position via any method (FROG in this example). The second step, also in Fig. \[fig:CROAK\](a) involves measuring the spatially resolved spectrum of the fundamental beam to be measured in the near-field along one axis using an imaging spectrometer (including the point where the FROG measurement was done). And lastly, the third step requires measuring the spatially resolved spectrum directly in the focus of a lens with a known focal length (and without aberrations) with the same imaging spectrometer, shown in Fig. \[fig:CROAK\](b). The combination of the latter two steps allows for reconstruction of the one-dimensional spatial phase at each frequency using phase retrieval algorithms, within certain limits and requiring certain assumptions. When combined with the spectral phase measurements of the first step this could produce the complete E-field in one spatial dimension and in either frequency or time. There are many issues with this method, however. Firstly, the measurement of the spatio-spectral amplitude in the near-field and at the far-field must be done on exactly the same slice of the beam, and any aberrations on the beam which lack cylindrical symmetry about the axis of the spectrometer slit could cause significant errors. And secondly, since the spatio-spectral amplitude is measured in the near-field and then at the far-field, there are very tight restrictions on either the size of the near-field beam, the focal length, or the number and size of the pixels in the imaging spectrometer. {width="171mm"} A significant improvement in spatio-temporal measurement using phase retrieval is the INSIGHT technique [@borot18]. Rather than using an imaging spectrometer, the INSIGHT technique resolves the spectral amplitude at each point of the measured beam via spatially-resolved Fourier-transform interferometry around the focus. Knowing the spectrum at each point of the beam, one can obviously determine the spatial profile at each frequency, which is the information required for proper phase-retrieval on broadband beams. The approach is implemented by splitting the beam near its focus into two copies as shown in Fig. \[fig:INSIGHT\](a), and resolving the interference of these two copies on a standard CCD camera. This has the advantage of resolving the spatial properties in two dimensions. The FTS is performed by taking camera images as the delay $\tau$ is stepped through with sub-cycle accuracy (Fig. \[fig:INSIGHT\](b)), and the spatially-resolved spectrum is calculated via taking the Fourier transform with respect to $\tau$ and selecting only the positive frequencies (Fig. \[fig:INSIGHT\](c)). This procedure removes the large and expensive imaging spectrometer device when compared to CROAK and resolves the second spatial dimension, but requires a delay stage capable of delay steps of a fraction of a wavelength as was the case for the TERMITES technique. Performing this temporal scan for one $z$ plane already provides the beam spatio-spectral amplitude. In order to obtain the spatio-spectral phase, the INSIGHT technique requires this spatio-spectral amplitude at multiple planes, just as with the CROAK technique. In order to allow for a better convergence of the phase-retrieval algorithm the FTS is repeated at two additional planes out-of-focus (at $\pm\delta{z}$). Once the spatio-spectral intensity is found around the focus (Fig. \[fig:INSIGHT\](d)) and out-of-focus (Fig. \[fig:INSIGHT\](e)) the phase-retrieval algorithm is done at each frequency to compute the spatio-spectral phase (Fig. \[fig:INSIGHT\](f)). Finally with a single measurement of the spectral phase a one position ($x_0$,$y_0$) the spatio-temporal electric field can be computed (Fig. \[fig:INSIGHT\](g)). Since INSIGHT is done in-focus it allows for the optics of the measurement device to be very small and lightweight, and also allows the measurement to be done in exactly the location of eventual experiments. The out-of-focus planes are generally measured at $\delta{z}\approx3-10 z_R$, so the camera properties can be optimized to have high-resolution. However, the camera chip should be significantly larger than the beam focus so that high spatial frequencies in the near-field can still be resolved. The INSIGHT device was used successfully for measurements on Terawatt [@borot18] and Petawatt lasers [@jeandet19]. With the addition of a second camera looking at the leak-through of one interferometer arm (shown in Fig. \[fig:INSIGHT\](a)) pointing fluctuations can be measured for each laser shot and numerically corrected, which significantly increases the fidelity of the computed spatially-resolved spectrum [@borot18]. This is especially important for measurement on high-power and low rep-rate systems, where pointing fluctuations can be significant. A birefringent delay line [@harvey04; @brida12] has recently been used for hyperspectral imaging [@perri19] (spatially-resolved Fourier-spectroscopy without phase information), and we have recently implemented this scheme in an INSIGHT device as well [@jolly_prisms]. Finally, using an imaging spectrometer near the focus of the INSIGHT device could essentially be a single-shot version that is resolved in only one spatial dimension (similar to the single-shot version of TERMITES, SEA-TERMITES), and would be very similar to steps 2 and 3 of the CROAK technique (but done around the focus). However, when compared to SEA-TERMITES, operating very close to the focus would still allow for the optics to be small and lightweight. But we consider the loss of one spatial dimension and the demand for cylindrical symmetry and measurement of the exact same spatial slice to be severe downsides of such a modified version of INSIGHT. If suitable hyperspectral cameras eventually become available, it might be possible to implement a single-shot version of INSIGHT that does not suffer from these limitations, by fitting multiple replicas of the beam into the camera’s chip, to measure the spectrally-resolved spatial intensity profile of the beam in multiple $z$ planes in a single-shot. Implementation of this idea, however, is far from straightforward. Interferometric Techniques {#sec:interferometric} -------------------------- Techniques of this class rely on the interference of the unknown beam with a second beam, considered as a reference, which can either be obtained from the unknown beam itself (self-referenced interferometry), or be an independent perfectly characterized beam. A self-referenced interferometric technique commonly used for the spatial characterization of laser beams is spatial shearing interferometry. This is the spatial analogous of the SPIDER technique, where the unknown beam is interfered with a spatially-sheared replica of itself. The resulting interference pattern can be used to determined the spatial derivative of the spatial phase. This technique has been extended for the spatio-spectral characterization of ultrashort laser beams, leading to a technique called “spectrally-resolved spatial-shearing interferometry” [@dorrer02-3]. In this technique, a Michelson interferometer is used to generate, on the slit of an imaging spectrometer (oriented along the $x$ axis at the sampling point $y_0$), two beams separated by a delay $\tau$ , with an angle offset represented by $k_x$ and with a spatial shear $X$. The interferogram signal $S$ generated can then be written as follows: $$\begin{aligned} \label{eq:dorrer} \begin{split} &S\left(x,\omega\right)=|\hat{E}\left(x,\omega\right)|^2+|\hat{E}\left(x-X,\omega\right)|^2 \\ &+2|\hat{E}\left(x,\omega\right)\hat{E}\left(x-X,\omega\right)| \\ &\times\cos\left[\hat{\phi}\left(x,\omega\right)-\hat{\phi}\left(x-X,\omega\right)-k_x x-\omega\tau\right] . \end{split}\end{aligned}$$ By nulling the shear $X$, it is possible to calibrate for the term $-k_x x-\omega\tau$. After adding the shear again, one has access to the phase difference $\hat{\phi}\left(x,\omega \right)-\hat{\phi}\left(x-X,\omega\right)$, proportional directly to $\partial\hat{\phi}\left(x,\omega\right)/\partial{x}$. From this data, we can reconstruct the phase $\hat{\phi}_S\left(x,\omega\right)=\hat{\phi}\left(x,\omega\right)+\alpha(\omega)$ with $\alpha$ being an arbitrary function of $\omega$. This gives the spatio-spectral phase up to an unknown overall spectral phase, and does not directly give the spectral amplitude. Essentially it is a measurement of the one-dimensional wavefront (i.e. along $x$) resolved in frequency. This combined with a spatially-resolved SPIDER is the 2D-SPIDER technique referenced earlier [@dorrer02-1]. We note that in principle, by sweeping the position $y$ of the beam on the slit of the spectrometer and then by rotating the beam by 90${}^{\circ}$ around the $z$ axis and by measuring the interferogram at a position $x_0$, the full STC phase $\hat{\phi}_S(x,y,\omega)$ could be obtained (with still an ambiguity of an arbitrary function of $\omega$). However, this is very difficult to do in practice and in fact has not been demonstrated. As we have seen already, diffraction, including of course the standard linear diffraction grating, is useful for separating the frequencies for doing spatio-spectral measurements. This has so far been to essentially use one dimension of a 2D detector to resolve the frequency while the other dimension remains for one spatial axis (as in SEA-SPIDER [@witting16], Shackled-FROG [@rubino09], CROAK [@bragheri08]). However, we saw already briefly in Fig. \[fig:wavefront\](c) [@boniface19] that a cleverly oriented grating can orient the frequency information in such a way that the 2D detector has continuous 2D spatial information along with discrete frequency information. Indeed, this shifts the difficulty from the sensor to the analysis. The STRIPED-FISH technique, which is complete and single-shot, uses a tilted 2D diffraction grating and a frequency filter to create spatially-separated profiles at discrete frequencies on a standard 2D CCD chip [@gabolde06; @gabolde08; @guang14; @guang15]. The interference of these dispersed intensity profiles with a *spatio-temporally perfect* reference, or a reference that has been *perfectly characterized in space-time*, in principle allows for measurement of the full spatio-spectral amplitude and phase, and hence for the reconstruction of the full spatio-temporal field. This technique is summarized in Fig. \[fig:STRIPED-FISH\]. ![STRIPED-FISH technique. The setup in (a) requires both the unknown beam and a characterized reference, a 2D diffraction grating, an interferometric bandpass filter (IBPF), and a 2D CCD detector. The analysis in (b) involves the standard spatial FFT, but also requires segmenting and overlaying the discrete frequency components. Images modified with permission from Refs. [@gabolde06; @gabolde08] The Optical Society.[]{data-label="fig:STRIPED-FISH"}](STRIPED-FISH.pdf){width="84mm"} Category Technique(s) Spatial dim. Spectral information obtained by Single-shot? Complete? ------------------------------------------------------- --------------------------------------------------------- ------------------- -------------------------------------------------------- -------------- ----------- -- Extension of established wavefront sensing techniques HAMSTER, Shackled FROG, hyperspectral wavefront sensors 1D or 2D Spectrometer/grating or filter or multispectral camera Yes Yes Spectrally-resolved phase retrieval CROAK, INSIGHT 2D (INSIGHT only) Spectrometer or FTS Not yet Yes Interferometry STRIPED FISH 2D Filter Yes Yes As shown in Fig. \[fig:STRIPED-FISH\](a) the reference and unknown beams are incident with an angle $\alpha$ on a 2D diffraction grating that is tilted by an angle $\varphi$ in a plane perpendicular to the propagation direction of the unknown beam. The grating produces many diffraction orders that are of course diffracted at larger angles from that of the incident beams. An interferometric bandpass filter (IBPF) is placed after the grating, tilted at an angle $\beta$, yet relative to a different plane (see Fig. \[fig:STRIPED-FISH\](a)). Because it is an interferometric filter the transmitted spectrum varies with incidence angle on the filter. Since different diffraction orders of the grating have a different angle of incidence, this causes each order to be filtered to a different bandpass wavelength. The result is a mosaic of spots on the 2D CCD sensor which correspond to the different frequency components of the incident beams. Since the unknown beam and reference beam have a relative angle $\alpha$, this produces spatial interference fringes on each spot, which allows one to extract phase information as well. The analysis steps involve the standard spatial FFT to calculate the amplitude and phase, segmenting the acquired image so that each frequency component can be analyzed independently, and finally stacking the information properly to produce the 3D amplitude and phase (see Fig. \[fig:STRIPED-FISH\](b)). The strength of STRIPED-FISH is that it is both a complete 3D technique, and single-shot—a significant advantage over techniques like INSIGHT and TERMITES for instance, especially for lasers that have low repetition-rates or fluctuate from shot to shot. Yet, it has significant drawbacks and limitations. The main limitation is one of principle: it requires a reference beam that covers the full spectrum of the unknown beam, and either has no STCs, or has been fully characterized in space-time. In most real-life cases, this reference needs to be produced from the unknown beam itself, which actually makes STRIPED FISH a self-referenced technique. In practice, this means that this technique is mostly suited to the measurement of the spatio-temporal effects induced by an optical system on a laser beam. A pick-off of this beam prior to this optical system can then be used as a reference, provided this system does not increase or shift the beam spectral content—i.e. this system should be linear. In terms of performance, in order to pack all of the interferograms for all frequencies on the 2D detector, there are strong limits on either the size of the detector, the size of the beam, the number of frequencies, or the spatial resolution. Finally, accurately calibrating the wavelength of each diffraction order, properly stacking the different diffraction orders, and finally producing the correct spatio-spectral phase all require a very careful calibration for each unique device and a very robust analysis algorithm. The STRIPED-FISH technique has for instance been used to measure the ultrafast lighthouse effect [@guang16] and beams from a multi-mode fiber [@guang17], where a pick-off on the input beam prior to the optical system under investigation was used as a reference. But due to the previous limitations, this technique has not been in common use so far, despite its conceptual elegance and its complete and single-shot character. In particular, because of the difficulty of producing a reliable reference, it has not yet been used to directly characterize spatio-temporally the output of a complex laser system, to the best of our knowledge. As a conclusion of this section, the properties of the main techniques discussed in this section are summarized in Table \[tab:table2\]. Analysis and visualization {#sec:analysis_visualization} ========================== The results of a 2-D or 3-D spatio-temporal or spatio-spectral measurement in general consist of a large matrix of complex numbers describing the laser field. For example, in the case of TERMITES or INSIGHT measurements this could be a 3-D complex-valued matrix of a size above 300$\times$300$\times$50 pixels (${x}\times{y}\times\omega$, 4.5 million points). With such a large set of complex data in more than two dimensions, it is not always straightforward to analyze, to extract meaningful physical information, or to visualize the results of a successful measurement. We will outline some methods to analyze the data produced from these measurements with various goals in mind, and in each case will also provide examples of how to effectively visualize the data and the individual analysis steps. Phase-stitching {#sec:phase-stitching} --------------- We start with a post-processing treatment of the measurement data that in practice is often the very first step of the analysis. For many of the techniques described in this tutorial, we have seen that the spatio-spectral phase is actually measured up to an unknown overall spectral phase, that equally applies to all points of the beam. This includes TERMITES, INSIGHT, and direct wavefront measurements with filters or multi-spectral cameras. Therefore these techniques by themselves should be more rigorously called spatio-spectral—rather than spatio-temporal—characterization devices. For other techniques such as SEA-TADPOLE and STRIPED-FISH, only when the technique is done with a suitably characterized reference can the measured spatio-spectral phase be considered complete. All spatio-spectral couplings are still resolved well in every case, and as we will see below, a lot can be said about the beam properties even with this remaining indeterminacy. But without knowledge of the overall spectral phase, the actual spatio-temporal field cannot be calculated. ![For some measurement techniques, phase stitching is necessary to have the proper phase relationship between frequencies, and be able to calculate the field in the spatio-temporal domain by a Fourier transformation. A measurement technique may be blind to spectral phase as in (a), but produce spatial phase results at many frequencies. A single measurement of spectral phase (b), in this case at $x=2$mm, will fix the phase relation and produce the correct spatio-spectral phase at all positions. This figure is courtesy of A. Jeandet.[]{data-label="fig:phase-stitching"}](phase-stitching-tutorial.pdf){width="83mm"} In all cases, a measurement of the spectral phase at a known single point in space can resolve this issue. This single measurement gives the relationship between the retrieved spatial phase maps at different frequencies. Using this measurement one can do “phase-stitching” to transform the data from the spatio-spectral device, with the data at different frequencies being essentially independent of each other, to data having the complete *physical* spatio-spectral phase. This phase-stitching procedure is illustrated in Fig. \[fig:phase-stitching\] for 2D data (one spatial coordinate only), but the concept applies equally well for 3D data. Mathematically, this phase stitching operation consists in applying the following transformation to the measured spatio-spectral phase $\hat{\phi}_{meas}(x,y,\omega)$ (displayed in Fig. \[fig:phase-stitching\](a)), to obtain the physical spatio-spectral phase $\hat{\phi}(x,y,\omega)$ (displayed in Fig. \[fig:phase-stitching\](b)): $$\hat{\phi}(x,y,\omega)=\hat{\phi}_{meas}(x,y,\omega)-\hat{\phi}_{meas}(x_0,y_0,\omega)+\varphi(\omega),$$ where $\varphi(\omega)$ is the spectral phase measured at a given point $(x_0,y_0)$ of the unknown beam (green line in Fig. \[fig:phase-stitching\](b)), using a temporal measurement device such as a SPIDER, FROG, or D-scan for instance. Because performing a local measurement of the spectral phase is much easier on an unfocused beam, this procedure is generally applied to the spatio-spectral phase in the NF. In such a case, when the spatio-spectral measurement is performed in the FF (like in INSIGHT), the measured field needs to be numerically propagated from the FF to the NF before applying phase stitching. Calculating the magnitude of low-order couplings {#sec:analysis_low} ------------------------------------------------ One of the key steps when analyzing measured data from a spatio-temporal characterization device is generally estimating the magnitude of the lowest-order couplings, or that of the couplings expected to be present based on the nature of the source. This is an essential step because the most common couplings are also typically those of lowest-orders. Returning to the canonical couplings of AD/PFT and CC/PFC, we can find a straightforward way to calculate the magnitude of these couplings using the phase data of the 3-D matrix. If we consider the NF, these couplings are only concerning the phase, so we reference the reconstructed spatio-spectral phase $\hat{\phi}(x,y,\omega)$ or $\hat{\phi}(r,\omega)$. We can find the AD/PFT via the following relation $$\gamma_x=\frac{\partial}{\partial\omega}\frac{\partial \hat{\phi}(x,y,\omega)}{\partial x} ,$$ and we can find the CC/PFC via the following relation $$\alpha=\frac{1}{2}\frac{\partial}{\partial\omega}\frac{\partial^2 \hat{\phi}(r,\omega)}{\partial r^2} .$$ In order to be insensitive to the spectral phase of the measured pulse, the spatial derivate must be performed first. This is important especially for measurements that do not have a pure spectral phase measurement included (i.e. without spectral phase stitching). ![Examples of low-order couplings analysis for (a) AD/PFT and (b) CC/PFC. A slice of the spatio-spectral phase at $y=0$ is shown in the left column. lineouts of this phase at discrete frequencies $\omega_i$ are shown in the central column, which have varying linear or quadratic coefficients for $x$, $c(\omega)$, if the coupling is AD/PFT or CC/PFC respectively. Fitting a linear curve to $c(\omega)$ in each case, as shown in the right column, can result in the magnitude of the coupling. In the case of CC/PFC the behavior is the same in terms of the radial coordinate $r$, but a slice of $x$ at $y=0$ is shown here for simplicity.[]{data-label="fig:analysis_1"}](Low_analysis.png){width="83mm"} These are simple relationships that make it easy to calculate these STCs when the spatio-spectral phase is a known function, but of course when the phase is represented not as a continuous function, but rather as a discrete data set, derivatives cannot be taken as such. In practice, to find the magnitude of these couplings, the spatial phase at each frequency should rather be fit to a polynomial in space. Then the relevant coefficients at each frequency (either linear in position for AD/PFT or quadratic in radius for CC/PFC) should be fit to a polynomial in frequency. The linear component of this polynomial in frequency is the magnitude of the coupling. More explicitly for AD/PFT: $$\hat{\phi}(x,y,\omega)=c_1(\omega)\times x ,\quad c_1(\omega)=\gamma_x\times (\omega-\omega_0) ,$$ and for CC/PFC: $$\hat{\phi}(r,\omega)=c_2(\omega)\times r^2 ,\quad c_2(\omega)=\alpha\times (\omega-\omega_0) ,$$ where in each case $c_i(\omega)$ and the subsequent coupling (either $\gamma$ or $\alpha$) are found via a least-squares regression. Figure \[fig:analysis\_1\] shows this procedure for both AD/PFT and CC/PFC. It is very important to do these analysis steps on phase data, only within the spectral region where there is significant intensity. Most measurement devices will produce random or highly irregular phase data outside of the real spectral region of the measured beam, which must be ignored because it would negatively influence any fitting. Addressing the magnitude of arbitrary phase couplings {#sec:analysis_high} ----------------------------------------------------- Beyond low-order couplings addressed in the previous section, it may be that higher-order phase couplings are expected, or that it is clear there are some effects not of low-order. And besides expectations, in general it can be tedious to fit individual polynomials to 3-D data. Furthermore, when looking at the 3-D data resulting from a spatio-temporal or spatio-spectral measurement, it cannot always be clear whether the high-order aberrations that are present are chromatic or not. Therefore it is necessary to have some type of general way to address this, especially if there is no particular expectation or prediction (which is often the case in the real world). We borrow a standard technique from monochromatic wavefront analysis, and propose to utilize frequency-resolved Zernike polynomials to describe the general phase aberrations present. This was introduced and implemented with great utility in recent work [@borot18; @jeandet19]. The Zernike polynomials are a way to represent an arbitrary function over the unit disk via a set of orthogonal polynomials [@zernike34]. These polynomials can be used to represent the spatial phase of a laser beam over a defined pupil [@born99], which corresponds to the area where there is significant intensity. Without going into detail, we simply remind that the Zernike polynomials $Z_n^m$ generally have two indices $m$ and $n$ (with $\left|m\right|\le n$) that correspond to the azimuthal and radial degrees of freedom respectively, where $m$ can be negative, but $n$ is limited to the natural numbers. When the phase map is decomposed onto this basis of Zernike polynomials, the result is a list of constants $C_n^m$ corresponding to the amplitude of each polynomial. Algorithmically this is much simpler than fitting arbitrary polynomials to the spatial data, since it amounts to decomposing a known function on a complete orthonormal basis set. We emphasize that these coefficients can be divided by $k=2 \pi /\lambda$, with $\lambda$ the wavelength of the beam under consideration, such that they describe the actual physical distance of displacement/deformation of the wavefront across the beam. Although this is not necessarily the standard practice in wavefront sensing, it is preferable to use this normalization of the coefficients for the analysis of chromatic effects discussed below, and this is what we will assume in the rest of this section. The extension of the Zernike polynomials to include frequency is quite straightforward. At each frequency $\omega$, the spatial phase map is decomposed on the basis of Zernike polynomials, leading to coefficients $C_n^m(\omega)$ that can now depend on frequency. When a given term does depend on frequency, it can then be concluded that there is a chromatic effect on that Zernike component. For instance, for a beam with PFT, at least one of the coefficients $C_1^{\pm1}$ will vary with frequency - while they would be exactly constant for a tilted beam (provided the normalization mentioned above is used). Such a picture is quite powerful, since it intuitively can show the chromatic nature of different phase aberrations. The intuition of the various aberrations (defocus, astigmatism, etc.) can be utilized to attempt to understand the data that now has the additional dimension of frequency. An example of this data for a beam having mostly CC/PFC in shown in Fig. \[fig:Zernike\](a). {width="171mm"} We now explain how to relate these coefficients to different coupling parameters introduced earlier in this tutorial for the canonical couplings AD/PFT and CC/PFT (see section \[sec:concepts\_manifestations\]). For calculating the magnitude of the AD/PFT coefficient $\gamma$, the tilt Zernike terms $C_1^{\pm 1}(\omega)$ first need to be related to the frequency-varying wavefront tilt $\theta$ (in direction $x$ or $y$), through the following relationship: $$\theta_{x,y}(\omega)\approx\frac{d}{R_p}=\frac{2 C_1^{\pm 1}(\omega)}{ R_p} \label{eq:ZernikeTilt} \\$$ where $R_p$ is the pupil radius used for the Zernike computation, and $d$ is the displacement of the wavefront at a given frequency at the edge of the unit disc defined by the pupil. Note that this relationship assumes that the Zernike modes are normalized to have a modulus of $\pi$ over the unit disc **(really?)**. The AD/PFT coefficient $\gamma$ is then directly related to the linear slope of $\theta$ via: $$\gamma_{x,y}=\frac{\omega_0}{c}\frac{\partial\theta_{x,y}}{\partial\omega}\Big|_{\omega_0} \label{eq:ZernikePFT},$$ with $\omega_0$ the central frequency of the pulse. Following the same reasoning, the CC/PFC coefficient $\alpha$ can be deduced from the frequency-varying Zernike terms for defocus $C_2^0(\omega)$, by relating both quantities to the frequency-resolved wavefront curvature $1/R(\omega)$: $$\begin{aligned} \frac{1}{R(\omega)}&=\frac{2d}{d^2 + R_p^2}\approx\frac{2d}{R_p^2}=\frac{4\sqrt{3} C_2^0}{ R_p^2} \label{eq:ZernikeDefocus} \\ \alpha&=\frac{\omega_0}{2c}\frac{\partial(1/R(\omega))}{\partial\omega}\Big|_{\omega_0} \label{eq:ZernikePFC},\end{aligned}$$ where $d$ is the same as before. A schematic of this is shown in Fig. \[fig:Zernike\](b)–(c). The phase maps at three frequencies in Fig. \[fig:Zernike\](b) show qualitatively the varying curvature, but the linear fit to the frequency-dependence of $1/R(\omega)$ in Fig. \[fig:Zernike\](c) produces the clear quantitative value of the CC/PFC based on Eq. (\[eq:ZernikePFC\]). Both of these examples are essentially identical to the straightforward approach outlined in the previous section for low-order couplings, but require a different set of steps. Depending on one’s priorities and capabilities either method should result in the same quantitative result. The obvious advantages of the Zernike polynomial method are that the higher-order aberrations are generated for free, a similar analysis can be done for the chromatic nature of these higher-order aberrations, and all of the technical considerations of fitting at low or high orders are not relevant. However, it is important to realize that choosing an appropriate pupil for the Zernike calculations is very important for calculating the correct result. Assessing the total effect of couplings {#sec:analysis_total} --------------------------------------- As a complementary step to quantifying the magnitude of specific couplings, which have identified causes or effects, it is useful to quantify the overall impact of all couplings present. There are methods to quantify the various effects of phase and amplitude couplings separately or together, and again they depend on the application. In some optical setups, for example a NOPA [@harth18] or a multi-pass cell for pulse compression [@weitenberg17; @lavenu18], there may be a significant effect on the homogeneity of the spectral amplitude when the system is not properly aligned. So in these cases a quantity can be used to assess this level of homogeneity when optimizing. This assessment of spectral homogeneity over the spectrum can be defined for example by the spectral overlap integral $V$ [@weitenberg17; @lavenu18] $$V(r)=\frac{\left[\int\sqrt{I(\lambda,r)\times I(\lambda,r=0)}d\lambda\right]^2}{\left(\int I(\lambda,r)d\lambda\right)\times\left(\int I(\lambda,r=0)d\lambda\right)}.$$ This integral is essentially comparing the spectrum at off-axis positions to the spectrum on-axis. As the spectrum becomes more homogeneous this integral will approach 1 at every transverse position. An example of this calculation is shown in Fig. \[fig:analysis\_3\]. ![A perfect pulse (a), with the spectral amplitude shown on the left and the calculated (perfect) overlap integral shown on the right. An example pulse with spectral amplitude on the left in (b) having a decreasing spectral width with increasing $y$, with the spectral overlap integral calculated the right, showing a decrease away from the axis.[]{data-label="fig:analysis_3"}](total_analysis1.png){width="83mm"} Of course in many applications the phase is important, and in general the phase on the NF beam (which is more often analyzed) will have a larger effect than the amplitude on desired parameters, such as in-focus pulse duration or peak intensity. So in addition to looking at the spectral amplitude, various spatio-temporal Strehl ratios can quantify the impact of spatio-temporal phase distortions [@pariente16; @jeandet19]. The beam measured has a spatio-temporal or spatio-spectral amplitude $A(x,y,\omega)$ and phase $\hat{\phi}(x,y,\omega)$. The commonly known Strehl ratio ($\textrm{SR}_\textrm{WFS}$, associated with standard wavefront sensors) quantifies the effect of the frequency-averaged wavefront on the focusing of the frequency-averaged beam profile. This is usually performed on data that is already averaged (via measurement on a CCD camera), i.e. $\textrm{SR}_\textrm{WFS}=I\left[\overline{A}(x,y)e^{i\overline{\phi}(x,y)}\right]/I\left[\overline{A}(x,y)\right]$, where the upper bar symbol indicates an average over frequency. We use $I[ ]$ to denote the calculation of the focused intensity of a given beam. With knowledge of the full 3D intensity and phase, more nuanced versions of this quantity can be calculated as we now show. Although there are many possible definitions of a spatio-temporal Strehl ratio, we will focus on only a few versions to demonstrate the concept, which have been used in the previous works [@pariente16; @jeandet19]. The Strehl ratio assessing the impact of all phase distortions both chromatic and not, termed $\textrm{SR}_\textrm{Full}$, compares the fully measured beam with a beam having zero phase at every frequency $$\textrm{SR}_\textrm{Full}=\frac{I\left[\hat{A}(x,y,\omega)e^{i\hat{\phi}(x,y,\omega)}\right]}{I\left[A(x,y,\omega)\right]}.$$ When $\textrm{SR}_\textrm{Full}$ is less than one, it represents the departure in focused intensity from the perfect case of the fully measured beam. It should be the representation of the physically existing pulse intensity. Note that with this definition, the value of $\textrm{SR}_\textrm{Full}$ also depends on the spectral phase of the beam -which not the case with the usual, spatial-only, definition of the Strehl ratio of laser beams: indeed, a chirped laser pulse, even without any spatio-temporal coupling (i.e. the chirp is spatially homogeneous), necessarily has $\textrm{SR}_\textrm{Full}<1$. The Strehl ratio assessing the impact of only the chromatic phase distortions, which we call $\textrm{SR}_\textrm{STC}$, compares the measured beam with the frequency-averaged wavefront subtracted to a beam having zero phase at each frequency $$\textrm{SR}_\textrm{STC}=\frac{I\left[\hat{A}(x,y,\omega)e^{i\left(\hat{\phi}(x,y,\omega)-\overline{\phi}(x,y)\right)}\right]}{I\left[A(x,y,\omega)\right]}.$$ The physical case that $\textrm{SR}_\textrm{STC}$ describes is that where a deformable mirror was implemented perfectly so as to remove all non-chromatic wavefront distortions, but all chromatic aberrations still remain. This is useful if it is known that achromatic aberrations exist (and are either impossible to remove or not necessary to remove at that moment), and one wants to assess the impact of STCs only. A simple example is shown in Fig. \[fig:analysis\_4\]. ![Visualization of the Strehl ratio calculations for one simple case. In (a) a perfect beam, having flat phase, is defined to have an intensity of 1 in the far-field. In (b) a beam with AD/PFT in the nearfield produces a beam with a lower intensity in the far-field, which would correspond to a calculation of $\textrm{SR}_\textrm{Full}=\textrm{SR}_\textrm{STC}=0.35$.[]{data-label="fig:analysis_4"}](total_analysis2.png){width="83mm"} Note that if there are no chromatic phase distortions then $\textrm{SR}_\textrm{Full}=\textrm{SR}_\textrm{STC}$. These definitions of $\textrm{SR}_\textrm{Full}$ and $\textrm{SR}_\textrm{STC}$ were used on measurements of the BELLA PW system in recent work [@jeandet19]. Lastly, If one wants to quantify the effect of all spatio-temporal distortions, both in phase and amplitude, then a mixed ratio can be calculated as was done in [@pariente16]. In that case it was calculated by comparing the measured beam to a beam with zero phase in space and frequency, and also with the amplitude replaced by the average in space and the average in frequency, i.e. $\textrm{SR}_\textrm{mixed}=I\left[\hat{A}(x,y,\omega)e^{i\left(\hat{\phi}(x,y,\omega)\right)}\right]/I\left[\overline{A}(x,y)\overline{A}(\omega)\right]$. However, we believe that looking at the impact of phase and amplitude effects separately generally provides more insight. Visualization {#sec:visualization} ------------- Due to the complexity and multi-dimensional nature of spatio-temporal couplings, visualization is an important issue [@rhodes17; @li18-1]. Even after a successful measurement using one of the devices described, it is not trivial to properly discern the couplings present, nor is it simple to properly communicate the magnitude of the couplings. Therefore visualization is crucial to both assess initial measurements to guide the analysis priorities, but also to communicate the impact after analysis has taken place. We showcase a few examples of visualization options in Figure \[fig:visualization\] (taken from Ref. [@borot18]). These methods are: spatial properties visualized at discrete frequencies (Fig. \[fig:visualization\](a)), spectral and/or temporal properties visualized at discrete spatial coordinates (Fig. \[fig:visualization\](b)) and, as already discussed, the frequency-resolved Zernike coefficients visualized in a 3-D format (Fig. \[fig:Zernike\](a)). {width="150mm"} The method used for visualization depends strongly on the desired knowledge. If one is applying a certain spatio-temporal coupling in order to induce a given mechanism at the experimental focus, then visualizations exactly as in Fig. \[fig:visualization\](a) or (b) are likely most helpful. This is because they give a direct visualization of certain properties where they are important. However, if one is a laser physicist looking to remove undesired spatio-temporal couplings, then the same views as in Fig. \[fig:visualization\](a) or (b) may be desired, but on the collimated beam rather than at the focus. This is because the bulk of most laser amplifiers and the optics that may induce undesired couplings act on the collimated beam (although likely at increasing diameter throughout a laser chain). Because of this the views in the near-field may provide more direct input into the source of undesired couplings. The last important view was already shown in Fig. \[fig:Zernike\](a), the 3D view of the frequency-resolved Zernike polynomials, is likely useful to all scientists. This is because, although the Zernike coefficients are calculated based on the near-field beam, their nature also provides direct input in to the manifestation of any given chromatic effect in the focus. For this reason the frequency-resolved Zernike coefficients may be the most universal and helpful view of all. Since they show no amplitude information, other views will always be necessary as a complement. The data shown in Fig. \[fig:Zernike\](a) is for a beam with CC/PFC (i.e. a linear slope in frequency of the focus) to contrast with that already shown in Ref. [@borot18]. There are more compact methods to visualize spatio-temporal couplings. One example is where rather than a color scale representing the amplitude or intensity, it corresponds to the local instantaneous frequency. This is especially relevant for couplings where the spatio-spectral amplitude changes with propagation. A systematic review of visualization using this method along with many examples was presented in Ref. [@rhodes17]. Additionally, three dimensional stationary views of pulse intensity can be made with quadrants cut out (see many examples in Ref. [@li18-1]) or with constant intensity contours (see for example Fig. 7 of Ref. [@borot18]). Beyond the stationary views discussed, it is often useful and instructive to use movies to achieve multiple objectives. This includes: 1) panning or rotating a fixed 3-D plot in order to have a more immediate sense of the 3-D presentation (as in Supplementary movies 1 and 3 of Ref. [@pariente16], or Visualization 2 of Ref. [@borot18]), 2) showing a 2-D map and stepping through a third parameter (time or frequency) as the movie progresses (as in Supplementary movie 2 of Ref. [@pariente16]), or movies 1 and 2 of Ref. [@jeandet19]), or 3) visualizing a 2-D or 3-D property as the beam is numerically propagated through space where the movie steps through time or propagation distance (as in Visualization 1 of Ref. [@borot18], or movie 3 of Ref. [@jeandet19]). Although movies are not necessarily as useful as stationary plots within scientific journal articles, they are becoming better integrated in certain journals and their use is becoming more prevalent. Moreover, movies are an extremely useful tool for analysis for a scientist when interpreting results, so for the reader of this tutorial they could be important. Conclusion {#sec:conclusion} ========== In this tutorial we have introduced spatio-temporal couplings in a detailed fashion and reviewed techniques ranging from the simple to the complex for characterizing ultrashort laser pulses completely. This included very simple qualitative techniques, established temporal characterization methods extended to include one or all spatial dimensions, and advanced methods using a variety of techniques. The fact that this work is a tutorial was especially stressed in the order of introducing techniques and the level of detail included for a small number of them. From this point of view, it should not be treated as a full review of STC characterization (of which there is a good recent example [@dorrer19]). In addition to some past results or techniques that have not been discussed, there are many up-and-coming techniques which may prove to be integral to making spatio-temporal characterization more widespread in the community. For example, the STRIPED-FISH technique is the only technique employed for ultrashort pulses that is single-shot, although it requires a reference. A reference-free single-shot method that is more simple to implement experimentally is the grand challenge of this field. Indeed, techniques such as TERMITES and INSIGHT are functioning well and on the road to becoming available products for the community, but they are still methods that require scanning over many independent pulses. It may be that intuition from the mature but separate world of hyperspectral imaging [@hagen13] may provide innovation for pulse characterization if they are improved to handle the broad spectrum of ultrashort pulses, or even via the field of imaging through scattering media [@boniface19; @liX19]. The methods exposed in this paper for visible and near-infrared can be used as inspiration for characterization of sources in other wavelength ranges. This includes the much shorter wavelengths in attosecond pulse (see Ref. [@dacasa19]) and the longer wavelengths of a growing number of mid- and far-infrared ultrafast sources. This is important because, for example, attosecond pulses generated from gases are considered to often have extreme levels of spatio-temporal couplings depending on the precise generation parameters [@wikmark19]. The chief difficulty in developing devices for these exotic wavelengths is generally the components: optics such as mirrors, beamsplitters, and filters are commonplace for near-infrared sources, but can be quite bulky, expensive, or perform worse for extreme wavelengths. Even beyond sources of different wavelengths, ultrafast vector beams — beams with a spatially-varying polarization — add a completely new challenge to characterization. There are some solutions in development [@alonso19], and this will surely become a very active area. Most of the examples of either simulated or real STCs in this tutorial were simple in nature, mostly in order to clearly demonstrate the concept. These simple STCs have many applications as discussed in the introduction. However, there are many exotic STCs or exotic scenarios where STCs can be an avenue for fine control of physical mechanisms. These mechanisms include: Simultaneous space-and-time focusing caused by focusing a beam with spatial chirp in the nearfield [@zhu05; @durfee12; @heF14]; Spatio-temporal light springs [@pariente15], relevant potentially for laser-plasma acceleration [@vieira18], and extended to the attosecond regime [@porras19-3]; A “Flying Focus” in the focus of a beam with longitudinal chromatism and temporal chirp [@sainte-marie17; @froula18; @jolly20-1] for Raman amplification [@turnbull18-1], ionization waves of arbitrary velocity [@palastro18; @turnbull18-2], or photon acceleration [@howard19]; spatial chirp or chromatic focusing for high harmonic generation in gases [@hernandez-garcia16; @holgado17]; steering of beams in laser-plasma acceleration due to pulse-front tilt [@popp10; @thevenet19-2; @mittelberger19] and the effect on the polarization of betatron radiation [@schnell13]; circumventing intrinsic limits of laser-plasma acceleration [@debus19]; pulse-front tilt for dielectric laser acceleration [@plettner08; @wei17]; THz beams with tilted pulse-front for traveling-wave electron acceleration [@walsh17]; In-band noise filtering of high-power lasers [@wangJ18]; Diffraction-free space-time wave packets [@kondakci16; @kondakci17; @kondakci19-1; @bhaduri19-1; @bhaduri19-2; @kondakci19-2; @yessenov19-1; @yessenov19-2], among many others. The recent activity in designing new spatio-temporal characterization devices and the wealth of applications in ultrafast physics underscores the importance of the field. With many Terawatt and Petawatt lasers coming online across the world [@danson19], and pulses with few-cycle duration becoming ever more commonplace, the increase in familiarity with the concepts in this tutorial is paramount for the community to successfully utilize these sources and to characterize and troublehoot their spatio-temporal performance. Acknowledgements {#acknowledgements .unnumbered} ================ We acknowledge Antoine Jeandet for general discussions and for supplying the phase-stiching figure. calculation of the spatio-spectral phase of beams with pulse-front distortions {#sec:appendixA} ============================================================================== We consider a beam of central frequency $\omega_0$ whose spatio-temporal field is described by a function of the following form: $$\nonumber E(x,t)=f(x) g[t-t_0(x)] e^{i \omega_0 t}$$ This corresponds to a beam whose temporal profile $g(t)$ is invariant in space, but whose arrival time $t_0(x)$ depends on the position $x$ in the beam—while the wave front at the carrier frequency, described by the last term of the equation, is assumed to be flat and normal to the $z$ axis. The term $f(x)$ describes the spatial envelope of the beam, and will be omitted in all following calculations, as it appears as an overall factor in all equations. For simplicity, we restrict the analysis to one transverse spatial dimension only, but it can easily be generalized to two transverse coordinates. The spatio-spectral description of this beam is obtained by performing a Fourier-transformation with respect to $t$. To carry out this transformation, we first rewrite the previous equation as: $$\nonumber E(x,t)=[g(t) \otimes \delta[t-t_0(x)] ] \times e^{i \omega_0 t},$$ where $\otimes$ is the symbol for the convolution product. The Fourier-transform of this field is then given by: $$\begin{aligned} \nonumber \hat{E}(x,\omega)&=[\hat{g}(\omega) \times FT\left\{\delta[t-t_0(x)]\right\} ] \otimes FT\left\{e^{i \omega_0 t}\right\} \\ &=[\hat{g}(\omega) \times e^{i\omega t_0(x)}] \otimes \delta(\omega-\omega_0) \nonumber\\ &=e^{i\delta\omega \: t_0(x)}\hat{g}(\delta \omega), \nonumber\end{aligned}$$ where $\hat{g}(\omega)$ is the Fourier-transform of $g(t)$, and $\delta \omega=\omega-\omega_0$ is the frequency offset from the central frequency $\omega_0$. This equation shows that a pure pulse front distortion $t_0(x)$ in the time domain is entirely described in the spectral domain by a spatio-spectral phase $\hat{\phi}(x,\omega)=\delta \omega \: t_0(x)$. In the case of pulse front tilt, we have $t_0(x)=\gamma x$ leading to $\hat{\phi}(x,\omega)=\gamma \: \delta \omega \: x$. In the case of pulse front curvature, $t_0(x)=\alpha x^2$ leading to $\hat{\phi}(x,\omega)=\alpha \:\delta \omega \: x^2$. These expressions of the spatio-spectral phase are the ones discussed in section \[sec:concepts\_manifestations\] of the main text. We now provide a more detailed discussion of these two cases, which is actually very useful to understand some of the subtleties of STCs and their metrology. We first analyze the mathematical differences between a beam with PFT, and a perfect STC-free beam propagating at an angle with the $z$ axis. In the later case, the field writes: $$\begin{aligned} \nonumber E(x,t)&=g(t-\gamma x) e^{i \omega_0 (t-\gamma x)} \\ &=[g(t)e^{i \omega_0 t}]\otimes \delta(t-\gamma x). \nonumber\end{aligned}$$ When going to the spectral domain, a calculation similar to the previous one leads to: $$\hat{E}(x,\omega)= e^{i \gamma x \omega}\hat{g}(\delta \omega). \nonumber$$ This shows that in the spatio-spectral domain, the only difference between a beam with PFT, and a perfect tilted beam, lies in a subtle difference in the spatio spectral phase: in the former case, $\hat{\phi}(x,\omega)=\gamma \delta \omega x$, while in the later, $\hat{\phi}(x,\omega)=\gamma \omega x$. This has important consequences for STC metrology: a measurement method can only distinguish these two types of beams, and hence detect PFT, if it can differentiate these two types of spatio-spectral phases. A similar analysis shows that in the spatio-spectral domain, a perfect STC-free curved beam (e.g. a perfect beam just after a perfect focusing optic) is described by the spatio-spectral phase $\hat{\phi}(x,\omega)=\alpha \omega x^2$. This again only slightly differs from the case of a beam with PFC, where $\hat{\phi}(x,\omega)=\alpha \delta \omega x^2$. This last case actually leads to an interesting question. Let us consider again a perfect beam without STC, which goes through a perfect focusing optic. Just after this optic, the field has the form: $$E(x,t)=g(t-\alpha x^2) e^{i \omega_0 (t-\alpha x^2)}. \nonumber$$ The beam wave front and pulse front are both curved by the same amount. This form of field is not separable as a product of a function of time and a function of space. According to the definition of section \[sec:concepts\_general\] (Eq. (\[eq:STC\])), this would imply that such a beam presents STC. Yet, intuitively, one would not consider this beam as suffering from STC—or equivalently chromatic aberrations. The point of view of the authors is that this apparent contradiction is a weakness or flaw of the present commonly-used definition of STC, which will have to be clarified by further theoretical work. One potential solution would be to consider that a beam has no STC when there exists a transformation $t'=h(t,x)$ such that $E(x,t')$ can be decomposed as $E(x,t')=f(x)g(t')$. With such a definition, a perfect curved beam would then be free of STC, since the transformation $t'=t-\alpha x^2$ makes it separable. Whether this definition makes sense in a more general case remains on open question. References {#references .unnumbered} ==========

-5mm   [and]{}  [Paul A. Pearce]{}\ [*Department of Mathematics and Statistics, University of Melbourne*]{}\ [*Parkville, Victoria 3010, Australia*]{}\ [J.Rasmussen@ms.unimelb.edu.au]{}     [P.Pearce@ms.unimelb.edu.au]{} [**[Abstract]{}**]{} .4cm Two-dimensional critical percolation is the member ${\cal LM}(2,3)$ of the infinite series of Yang-Baxter integrable logarithmic minimal models ${\cal LM}(p,p')$. We consider the continuum scaling limit of this lattice model as a ‘rational’ logarithmic conformal field theory with extended ${\cal W}={\cal W}_{2,3}$ symmetry and use a lattice approach on a strip to study the fundamental fusion rules in this extended picture. We find that the representation content of the ensuing closed fusion algebra contains 26 ${\cal W}$-indecomposable representations with 8 rank-1 representations, 14 rank-2 representations and 4 rank-3 representations. We identify these representations with suitable limits of Yang-Baxter integrable boundary conditions on the lattice and obtain their associated ${\cal W}$-extended characters. The latter decompose as finite non-negative sums of ${\cal W}$-irreducible characters of which 13 are required. Implementation of fusion on the lattice allows us to read off the fusion rules governing the fusion algebra of the 26 representations and to construct an explicit Cayley table. The closure of these representations among themselves under fusion is remarkable confirmation of the proposed extended symmetry. Introduction {#SectionIntroduction} ============ The study of percolation [@BroadHamm57; @Kesten82; @Grimmet89; @Stauffer92] as a lattice model has a long history [@Saleur87; @DuplantierSaleur87; @SaleurSUSY92]. In this paper, it is convenient to regard two-dimensional critical percolation as the member ${\cal LM}(2,3)$ of the infinite series of Yang-Baxter integrable logarithmic minimal models ${\cal LM}(p,p')$ [@PRZ]. It is a well-established principle that two-dimensional lattice systems in general [@Cardy87] and percolation in particular [@LPS94; @Cardy01] are conformally invariant in the continuum scaling limit. Our lattice approach to studying these conformal field theories is predicated on the supposition that, in the continuum scaling limit, a transfer matrix with prescribed boundary conditions gives rise to a representation of the Virasoro algebra. Different boundary conditions naturally lead to different representations which can be of different types — reducible or irreducible, decomposable or indecomposable. We further assume that, if in addition, the boundary conditions respect the symmetry of a larger conformal algebra ${\cal W}$, then the continuum scaling limit of the transfer matrix will yield a representation of the extended algebra ${\cal W}$. Notwithstanding the fact that critical percolation is one of the very few systems which has been rigorously shown [@Smirnov01] to be conformally invariant in the continuum scaling limit, the study of critical percolation as a Conformal Field Theory (CFT) is not so well advanced. In large part, this is because critical percolation [@Cardy92; @Gurarie93; @Cardy99; @GuLu99; @FFHST02; @GuLu04; @MathieuRidout07], like critical dense polymers ${\cal LM}(1,2)$ [@Gennes; @Cloizeaux; @Saleur87b; @Duplantier; @PR07] or symplectic fermions [@Kausch95; @Kausch00], is a prototypical [*logarithmic*]{} CFT. The properties [@Flohr03; @Gaberdiel03; @Kawai03] of logarithmic CFTs differ dramatically from the familiar properties of [*rational*]{} CFTs. In particular, they are non-rational and non-unitary with a countably infinite number of scaling fields. Unlike rational CFTs, whose field or representation content consists entirely of [*irreducible*]{} Virasoro representations, logarithmic CFTs admit [*reducible yet indecomposable*]{} representations [@Roh96] of the Virasoro algebra. These representations, some of which are accompanied by non-trivial Jordan-cell structures for the Virasoro dilatation generator $L_0$, play an essential role and are in fact characteristic Recently, Virasoro fusion rules have been proposed [@GabKausch96; @EberleF06; @RS07; @RP0706; @RP0707] for all the augmented minimal or logarithmic minimal models ${\cal LM}(p,p')$. Interestingly, it was found that only indecomposable representations of rank 1, 2 or 3 appear corresponding to Jordan cells of dimension 1, 2 or 3 respectively. However, a central question of much current interest [@Flohr96; @GK9606; @FG05; @GR06] is whether an extended symmetry algebra ${\cal W}$ exists for these logarithmic theories. Such a symmetry should allow the countably [*infinite*]{} number of Virasoro representations to be reorganized into a [*finite*]{} number of extended ${\cal W}$-representations which close under fusion. In the case of the logarithmic minimal models ${\cal LM}(1,p)$, the existence of such an extended ${\cal W}$-symmetry and the associated fusion rules are by now well established [@GK9606; @FHST03; @FGST05; @GR07; @GTipunin07; @PRR08]. By stark contrast, although there are strong indications [@FGST06a; @FGST06b] that there exists a ${\cal W}_{p,p'}$ symmetry algebra for general augmented minimal models, very little is known about the ${\cal W}$-extended fusion rules for the ${\cal LM}(p,p')$ models with $p\ge 2$. In this paper, we use a lattice approach on a strip, generalizing the approach of [@PRR08], to obtain fusion rules of critical percolation ${\cal LM}(2,3)$ in the extended symmetry picture. In [@PRR08], it was shown that in fact symplectic fermions is just critical dense polymers ${\cal LM}(1,2)$ viewed in the extended picture. Likewise in the case of critical percolation, the extended picture is described by the [*same*]{} lattice model as the Virasoro picture. We nevertheless find it useful to distinguish between the two pictures by denoting the extended picture ${\cal WLM}(2,3)$ and thus reserve the notation ${\cal LM}(2,3)$ for critical percolation in the non-extended Virasoro picture. A similar distinction applies to the entire infinite series of logarithmic minimal models. We intend to discuss these ${\cal W}$-extended models, which we denote by ${\cal WLM}(p,p')$, elsewhere. The ${\cal W}$-extended fusion rules we obtain for critical percolation are based on the [*fundamental*]{} fusion algebra in the Virasoro picture [@RP0706; @RP0707] which is a subset of the [*full*]{} fusion algebra. The latter remains to be determined and may eventually yield a larger ${\cal W}$-extended fusion algebra than the one presented here. The layout of this paper is as follows. In Section 2, we review the Virasoro fusion rules for critical percolation [@RP0706]. In Section 3, we summarize the ${\cal W}$-representation content consisting of 26 ${\cal W}$-indecomposable representations with 8 rank-1 representations, 14 rank-2 representations and 4 rank-3 representations and present their associated extended characters. The latter decompose as finite non-negative sums of ${\cal W}$-irreducible characters of which 13 are required. These are all identified. Lastly, in this section, we present the explicit Cayley table of the fundamental ${\cal W}$-extended fusion rules obtained by implementing fusion on the lattice. In Section 4, we identify the ${\cal W}$-extended representations with suitable limits of Yang-Baxter integrable boundary conditions on the lattice and give details of their construction and properties. We conclude with a short discussion. Throughout, we use the notation $\mathbb{Z}_{n,m}=\mathbb{Z}\cap[n,m]$, with $n,m\in\mathbb{Z}$, to denote the set of integers from $n$ to $m$, both included, and denote an $n$-fold fusion of the representation $A$ with itself by A\^[n]{}=\_n Critical Percolation ${\cal LM}(2,3)$ ===================================== Logarithmic minimal model ${\cal LM}(p,p')$ ------------------------------------------- A logarithmic minimal model ${\cal LM}(p,p')$ is defined [@PRZ] for every coprime pair of positive integers $p<p'$. The model ${\cal LM}(p,p')$ has central charge c =  1-6 \[c\] and conformal weights \_[r,s]{} = ,r,s \[D\] The fundamental fusion algebra $\big\langle(2,1),(1,2)\big\rangle_{p,p'}$ [@RP0706; @RP0707] of the logarithmic minimal model ${\cal LM}(p,p')$ is generated by the two fundamental Kac representations $(2,1)$ and $(1,2)$ and contains a countably infinite number of inequivalent, indecomposable representations of rank 1, 2 or 3. For $r,s\in\mathbb{N}$, the character of the Kac representation $(r,s)$ is \_[r,s]{}(q) = (1-q\^[rs]{})  = (q\^[(rp’-sp)\^2/4pp’]{}-q\^[(rp’+sp)\^2/4pp’]{}) \[chikac\] where the Dedekind eta function is given by (q) = q\^ \_[n=1]{}\^(1-q\^n) \[eta\] Such a representation is of rank 1 and is irreducible if $r\in\mathbb{Z}_{1,p}$ and $s\in p'\mathbb{N}$ or if $r\in p\mathbb{N}$ and $s\in\mathbb{Z}_{1,p'}$. It is a reducible yet indecomposable representation if $r\in\mathbb{Z}_{1,p-1}$ and $s\in\mathbb{Z}_{1,p'-1}$, while it is a fully reducible representation if $r\in p\mathbb{N}$ and $s\in p'\mathbb{N}$ where (kp,k’p’) = (k’p,kp’) = \_[j=|k-k’|+1,  2]{}\^[k+k’-1]{}(jp,p’) =  \_[j=|k-k’|+1,  2]{}\^[k+k’-1]{}(p,jp’) \[kpkp\] These are the only Kac representations appearing in the fundamental fusion algebra. The characters of the reducible yet indecomposable Kac representations just mentioned can be written as sums of two irreducible Virasoro characters \_[r,s]{}(q) = \_[r,s]{}(q)+\_[2p-r,s]{}(q) = \_[r,s]{}(q)+\_[r,2p’-s]{}(q), r\_[1,p-1]{},s\_[1,p’-1]{} In general and with $r_0\in\mathbb{Z}_{1,p-1}$, $s_0\in\mathbb{Z}_{1,p'-1}$ and $k\in\mathbb{N}-1$, the irreducible Virasoro characters read [@FSZ] \_[r\_0+kp,s\_0]{}(q)&=&K\_[2pp’,(r\_0+kp)p’-s\_0p;k]{}(q)-K\_[2pp’,(r\_0+kp)p’+s\_0p;k]{}(q)\_[r\_0+(k+1)p,p’]{}(q)&=& (q\^[(kp+r\_0)\^2p’/4p]{}-q\^[((k+2)p-r\_0)\^2p’/4p]{}) \_[(k+1)p,s\_0]{}(q)&=& (q\^[((k+1)p’-s\_0)\^2p/4p’]{}-q\^[((k+1)p’+s\_0)\^2p/4p’]{}) \_[(k+1)p,p’]{}(q)&=& (q\^[k\^2pp’/4]{}-q\^[(k+2)\^2pp’/4]{}) \[laq\] where $K_{n,\nu;k}(q)$ is defined as K\_[n,;k]{}(q) = \_[j\_[1,k]{}]{}q\^[(-jn)\^2/2n]{} \[Kk\] For $r\in\mathbb{Z}_{1,p}$, $s\in\mathbb{Z}_{1,p'}$, $a\in\mathbb{Z}_{1,p-1}$, $b\in\mathbb{Z}_{1,p'-1}$ and $k\in\mathbb{N}$, the representations denoted by $\R_{kp,s}^{a,0}$ and $\R_{r,kp'}^{0,b}$ are indecomposable representations of rank 2, while $\R_{kp,p'}^{a,b}\equiv\R_{p,kp'}^{a,b}$ is an indecomposable representation of rank 3. Their characters read (q)&=& (1-\_[k,1]{}\_[s,p’]{})\_[kp-a,s]{}(q)+2\_[kp+a,s]{}(q)+\_[(k+2)p-a,s]{}(q) (q)&=& (1-\_[k,1]{}\_[r,p]{})\_[r,kp’-b]{}(q)+2\_[r,kp’+b]{}(q)+\_[r,(k+2)p’-b]{}(q) (q)&=& (1-\_[k,1]{})\_[(k-1)p-a,b]{}(q)+2\_[(k-1)p+a,b]{}(q) +2(1-\_[k,1]{})\_[kp-a,p’-b]{}(q) &+&4\_[kp+a,p’-b]{}(q)+(2-\_[k,1]{})\_[(k+1)p-a,b]{}(q) +2\_[(k+1)p+a,b]{}(q) &+&2\_[(k+2)p-a,p’-b]{}(q)+\_[(k+3)p-a,b]{}(q) &=& (1-\_[k,1]{})\_[a,(k-1)p’-b]{}(q)+2\_[a,(k-1)p’+b]{}(q) +2(1-\_[k,1]{})\_[p-a,kp’-b]{}(q) &+&4\_[p-a,kp’+b]{}(q)+(2-\_[k,1]{})\_[a,(k+1)p’-b]{}(q) +2\_[a,(k+1)p’+b]{}(q) &+&2\_[p-a,(k+2)p’-b]{}(q)+\_[a,(k+3)p’-b]{}(q) \[chiR\] For $a\in\mathbb{Z}_{0,p-1}$, $b\in\mathbb{Z}_{0,p'-1}$ and $k,k'\in\mathbb{N}$, a decomposition similar to (\[kpkp\]) applies to the higher-rank [*decomposable*]{} representations $\R_{kp,k'p'}^{a,b}$ as we have \_[kp,k’p’]{}\^[a,b]{} = \_[k’p,kp’]{}\^[a,b]{}  = \_[j=|k-k’|+1,  2]{}\^[k+k’-1]{}\_[jp,p’]{}\^[a,b]{}  = \_[j=|k-k’|+1,  2]{}\^[k+k’-1]{}\_[p,jp’]{}\^[a,b]{} Here we have introduced the convenient notation \_[r,s]{}\^[0,0]{}  (r,s) Fusion in the fundamental fusion algebra $\big\langle(2,1),(1,2)\big\rangle_{p,p'}$ decomposes into ‘horizontal’ and ‘vertical’ components. With $a\in\mathbb{Z}_{0,p-1}$, $b\in\mathbb{Z}_{0,p'-1}$ and $k\in\mathbb{N}$, we thus have \_[p,kp’]{}\^[a,b]{} = \_[p,1]{}\^[a,0]{}\_[1,kp’]{}\^[0,b]{} = \_[kp,1]{}\^[a,0]{}\_[1,p’]{}\^[0,b]{} \[decomp\] The Kac representation $(1,1)$ is the identity of the fundamental fusion algebra. For $p>1$, this is a reducible yet indecomposable representation, while for $p=1$, it is an irreducible representation. Below, we summarize the fusion rules in the case of critical percolation ${\cal LM}(2,3)$. The associated extended Kac table is given in Figure \[KacTable\]. Fundamental fusion algebra of ${\cal LM}(2,3)$ ---------------------------------------------- The fundamental fusion algebra $\big\langle(2,1),(1,2)\big\rangle_{2,3}$ is generated by the irreducible Kac representation $(2,1)$ and the reducible yet indecomposable Kac representation $(1,2)$ and contains a variety of representations (2,1), (1,2)\_[2,3]{} = (1,1), (1,2), (2k,s), (r,3k), \_[2k,s]{}\^[1,0]{}, \_[r,3k]{}\^[0,b]{}, \_[2k,3]{}\^[1,b]{}\_[2,3]{} \[A2112\] where $r,b\in\mathbb{Z}_{1,2}$, $s\in\mathbb{Z}_{1,3}$ and $k\in\mathbb{N}$. The representations $(2k,3)\equiv(2,3k)$ are listed twice and it is recalled that $\R_{2k,3}^{1,0}\equiv\R_{2,3k}^{1,0}$, $\R_{2k,3}^{0,b}\equiv\R_{2,3k}^{0,b}$ and $\R_{2k,3}^{1,b}\equiv\R_{2,3k}^{1,b}$. As already mentioned, the reducible yet indecomposable Kac representation $(1,1)$ is the identity of the fundamental fusion algebra (1,1)A = A where $A$ is any of the representations listed in (\[A2112\]). Thanks to the decomposition illustrated in (\[decomp\]), the fundamental fusion algebra follows from a straightforward merge of the horizontal and vertical components. To appreciate this, we follow [@RP0706] and let $A_{r,s}=\bar{a}_{r,1}\otimes\ a_{1,s}$, $B_{r',s'}=\bar{b}_{r',1}\otimes\ b_{1,s'}$, $\bar{a}_{r,1}\otimes\bar{b}_{r',1}=\bigoplus_{r''}\bar{c}_{r'',1}$ and $a_{1,s}\otimes b_{1,s'}=\bigoplus_{s''}c_{1,s''}$. Our fusion prescription now yields A\_[r,s]{}B\_[r’,s’]{}&=&(|[a]{}\_[r,1]{}a\_[1,s]{}) (|[b]{}\_[r’,1]{}b\_[1,s’]{})  = (|[a]{}\_[r,1]{}|[b]{}\_[r’,1]{}) (a\_[1,s]{}b\_[1,s’]{}) &=&(\_[r”]{}|[c]{}\_[r”,1]{})(\_[s”]{}c\_[1,s”]{})  = \_[r”,s”]{}C\_[r”,s”]{} \[rs\] where $C_{r'',s''}=\bar{c}_{r'',1}\otimes c_{1,s''}$. In order to describe the component fusion algebras explicitly, we introduce the Kronecker delta combinations [@RP0706] &=& 2-\_[j,|k-k’|]{}-\_[j,k+k’]{} &=& 4-3\_[j,|k-k’|-1]{}-2\_[j,|k-k’|]{}-\_[j,|k-k’|+1]{} -\_[j,k+k’-1]{}-2\_[j,k+k’]{}-3\_[j,k+k’+1]{} \[d24\] where $k,k'\in\mathbb{N}$. The horizontal fusion algebra (2,1)\_[2,3]{} = (2k,1),\_[2k,1]{}\^[1,0]{}\_[2,3]{} then reads (2k,1)(2k’,1)&=&\_[j=|k-k’|+1,  2]{}\^[k+k’-1]{} \_[2j,1]{}\^[1,0]{}(2k,1)\_[2k’,1]{}\^[1,0]{}&=&\_[j=|k-k’|]{}\^[k+k’]{} (2j,1) \_[2k,1]{}\^[1,0]{}\_[2k’,1]{}\^[1,0]{}&=&\_[j=|k-k’|]{}\^[k+k’]{} \_[2j,1]{}\^[1,0]{} \[fusion21\] while the vertical fusion algebra (1,2)\_[2,3]{}  = (1,1),(1,2),(1,3k),\_[1,3k]{}\^[0,1]{},\_[1,3k]{}\^[0,2]{}\_[2,3]{} \[A12\] reads (1,1)A&=&A (1,2)(1,2)&=&(1,1)(1,3) (1,2)(1,3k)&=&\_[1,3k]{}\^[0,1]{} (1,2)\_[1,3k]{}\^[0,1]{}&=&\_[1,3k]{}\^[0,2]{}2(1,3k) (1,2)\_[1,3k]{}\^[0,2]{}&=&\_[1,3k]{}\^[0,1]{}(1,3(k-1))(1,3(k+1)) (1,3k)(1,3k’)&=& \_[j=|k-k’|+1,  2]{}\^[k+k’-1]{}(\_[1,3j]{}\^[0,2]{}(1,3j)) (1,3k)\_[1,3k’]{}\^[0,1]{}&=& (\_[j=|k-k’|+1,  2]{}\^[k+k’-1]{}2\_[1,3j]{}\^[0,1]{}) (\_[j=|k-k’|,  2]{}\^[k+k’]{}(1,3j)) (1,3k)\_[1,3k’]{}\^[0,2]{}&=& (\_[j=|k-k’|,  2]{}\^[k+k’]{}\_[1,3j]{}\^[0,1]{}) (\_[j=|k-k’|+1,  2]{}\^[k+k’-1]{}2(1,3j)) \_[1,3k]{}\^[0,1]{}\_[1,3k’]{}\^[0,1]{}&=&  (\_[j=|k-k’|,  2]{}\^[k+k’]{}\_[1,3j]{}\^[0,1]{}) (\_[j=|k-k’|+1,  2]{}\^[k+k’-1]{}(2\_[1,3j]{}\^[0,2]{}4(1,3j))) \_[1,3k]{}\^[0,1]{}\_[1,3k’]{}\^[0,2]{}&=& (\_[j=|k-k’|+1,  2]{}\^[k+k’-1]{}2\_[1,3j]{}\^[0,1]{}) (\_[j=|k-k’|,  2]{}\^[k+k’]{}(\_[1,3j]{}\^[0,2]{} 2(1,3j))) \_[1,3k]{}\^[0,2]{}R\_[1,3k’]{}\^[0,2]{}&=& (\_[j=|k-k’|,  2]{}\^[k+k’]{}\_[1,3j]{}\^[0,1]{}) (\_[j=|k-k’|+1,  2]{}\^[k+k’-1]{}2\_[1,3j]{}\^[0,2]{})&& (\_[j=|k-k’|-1,  2]{}\^[k+k’+1]{} (1,3j)) \[fusion12\] where $A$ is any of the representations listed in (\[A12\]). To illustrate the merge of the two components, we conclude this discussion of critical percolation in the Virasoro picture ${\cal LM}(2,3)$ by considering the fusion \_[2k,3]{}\^[1,1]{}\_[2k’,3]{}\^[1,1]{}&=&(\_[2k,1]{}\^[1,0]{}\_[1,3]{}\^[0,1]{}) (\_[2k’,1]{}\^[1,0]{}\_[1,3]{}\^[0,1]{})  = (\_[2k,1]{}\^[1,0]{}\_[2k’,1]{}\^[1,0]{}) (\_[1,3]{}\^[0,1]{}\_[1,3]{}\^[0,1]{}) &=&(\_[j=|k-k’|]{}\^[k+k’]{}\_[2j,1]{}\^[1,0]{}) (\_[1,6]{}\^[0,1]{}2\_[1,3]{}\^[0,2]{}4(1,3)) &=&(\_[j=|k-k’|-1]{}\^[k+k’+1]{}\_[2j,3]{}\^[1,1]{}) (\_[j=|k-k’|]{}\^[k+k’]{}(2\_[2j,3]{}\^[1,2]{}4\_[2j,3]{}\^[1,0]{})) \[ex3311\] ${\cal W}$-Extended Critical Percolation ${\cal WLM}(2,3)$ ========================================================== In this section, we summarize our findings in the extended picture for the representation content, their characters and their closed fusion algebra. Unless otherwise specified, we let $\kappa,r,b\in\mathbb{Z}_{1,2}$, $s\in\mathbb{Z}_{1,3}$ and $k,k'\in\mathbb{N}$ in the following. Summary of representation content --------------------------------- We have the 8 ${\cal W}$-indecomposable rank-1 representations {,}     \[8r1\] where $\ketw{2,3}$ is listed twice, the 14 ${\cal W}$-indecomposable rank-2 representations {, } \[14r2\] and the 4 ${\cal W}$-indecomposable rank-3 representations {,,}     \[4r3\] Here we are asserting that these ${\cal W}$-representations are indeed ${\cal W}$-indecomposable. In terms of Virasoro-indecomposable representations, the ${\cal W}$-indecomposable rank-1 representations decompose as &=&\_[k]{}(2k-2+)(2(2k-2+),s) &=&\_[k]{}(2k-2+)(r,3(2k-2+)) where the two expressions for $\ketw{2,3}$ agree and where    Likewise, the ${\cal W}$-indecomposable rank-2 representations decompose as &=& \_[k]{}(2k-2+)\_[2(2k-2+),s]{}\^[1,0]{} &=&\_[k]{}(2k-2+)\_[r,3(2k-2+)]{}\^[0,b]{} \[Rr2\] Finally, the ${\cal W}$-indecomposable rank-3 representations decompose as &=&\_[k]{}(2k-2+)\_[2(2k-2+),3]{}\^[1,b]{} &=&\_[k]{}(2k-2+)\_[2,3(2k-2+)]{}\^[1,b]{} \[Rr3\] where the two expressions for $\ketw{\R_{2,3}^{1,b}}$ agree and where    Summary of ${\cal W}$-extended characters ----------------------------------------- The characters of the ${\cal W}$-indecomposable rank-1 representations read \_[2,s]{}(q) = \_[k]{}(2k-2+)\_[2(2k-2+),s]{}(q), \_[r,3]{}(q) = \_[k]{}(2k-2+)\_[r,3(2k-2+)]{}(q) \[chihch\] where it is recalled that $\ketw{4,3}\equiv\ketw{2,6}$. The characters of the ${\cal W}$-indecomposable rank-2 representations read (q) &=&\_[,1]{}{1-\_[s,3]{}}+\_[k]{}4k\_[4k+1,s]{}(q) +\_[k]{}(4k-2)\_[4k-1,s]{}(q) (q) &=&\_[,1]{}{1-\_[r,2]{}} +\_[k]{}(4k+2-2)\_[r,6k+6-3-b]{}(q) &&+\_[k]{}(4k-4+2)\_[r,6k-6+3+b]{}(q) We note the character identities (q) = (q), (q) = (q) \[chid1\] and the character relations (q) = 1+(q), (q) = 1+(q) \[chrel1\] and (q)+(q)  = (q)+(q) \[chrelsum\] The characters of the ${\cal W}$-indecomposable rank-3 representations read (q) = 2+\_[k]{}4k\_[2k+1,b]{}(q) +\_[k]{}8k\_[4k+1,3-b]{}(q)+\_[k]{}(8k-4) \_[4k-1,3-b]{}(q) \[chitRr3\] and are seen to be independent of $\kappa$. As we will discuss below, the dependence on $\kappa$ manifests itself in the distinct Jordan-cell and general embedding structures of $\ketw{\R_{2\kappa,3}^{1,b}}$ for different $\kappa,b\in\mathbb{Z}_{1,2}$. Likewise, the ${\cal W}$-indecomposable rank-2 representations appearing in (\[chid1\]) have distinct embedding structures. We also have ${\cal W}$-extended characters of the various [*subfactors*]{} of the ${\cal W}$-indecomposable representations \_[0]{}(q)&=& 1 \_[1]{}(q)&=& \_[k]{}(2k-1)\_[4k-1,2]{}(q) =  \_[k]{}(2k-1)\_[1,6k-2]{}(q) &=&\_[k]{} k\^2(q\^[(12k-7)\^2/24]{}-q\^[(12k+1)\^2/24]{}) \_[2]{}(q)&=& \_[k]{}(2k-1)\_[4k-1,1]{}(q) =  \_[k]{}(2k-1)\_[1,6k-1]{}(q) &=&\_[k]{} k\^2(q\^[(12k-5)\^2/24]{}-q\^[(12k-1)\^2/24]{}) \_[5]{}(q)&=& \_[k]{}2k\_[4k+1,2]{}(q) =  \_[k]{}2k\_[1,6k+1]{}(q) &=&\_[k]{} k(k+1)(q\^[(12k-1)\^2/24]{}-q\^[(12k+7)\^2/24]{}) \_[7]{}(q)&=& \_[k]{}2k\_[4k+1,1]{}(q) =  \_[k]{}2k\_[1,6k+2]{}(q) &=&\_[k]{} k(k+1)(q\^[(12k+1)\^2/24]{}-q\^[(12k+5)\^2/24]{}) \[chih\] Here we have used the notation $\chih_{\D}(q)$, where $\D$ is the conformal dimension of the corresponding representation, and some of the identities &&\_[1,6k+2]{} = \_[4k+1,1]{},\_[1,6k+1]{} = \_[4k+1,2]{},\_[1,6k-1]{} = \_[4k-1,1]{}, \_[1,6k-2]{} = \_[4k-1,2]{} &&\_[2,6k+2]{} = \_[4k,1]{},\_[2,6k+1]{} = \_[4k,2]{}, \_[2,6k-1]{} = \_[4k-2,1]{}, \_[2,6k-2]{} = \_[4k-2,2]{} &&\_[1,3k]{} = \_[2k+1,3]{},\_[2,3k]{} = \_[2k,3]{} Similarly, written as $\chih_{\D}(q)$, the 8 independent characters in (\[chihch\]) read &&\_(q) = \_[k]{}(2k-1)\_[4k-1,3]{}(q)  = \_[k]{}(2k-1)\_[1,6k-3]{}(q)  =  \_[k]{} (2k-1)q\^[3(4k-3)\^2/8]{} &&\_(q) = \_[k]{}2k\_[4k+1,3]{}(q)  = \_[k]{}2k\_[1,6k]{}(q)  =  \_[k]{} 2kq\^[3(4k-1)\^2/8]{} \[chihsum1\] and &&\_(q) = \_[k]{}(2k-1)\_[4k-2,2]{}(q)  = \_[k]{}(2k-1)\_[2,6k-2]{}(q)  =  \_[k]{} (2k-1)q\^[(6k-5)\^2/6]{} &&\_(q) = \_[k]{}(2k-1)\_[4k-2,1]{}(q)  = \_[k]{}(2k-1)\_[2,6k-1]{}(q)  =  \_[k]{} (2k-1)q\^[(6k-4)\^2/6]{} &&\_(q) = \_[k]{}2k\_[4k,2]{}(q)  = \_[k]{}2k\_[2,6k+1]{}(q)  =  \_[k]{} 2kq\^[(6k-2)\^2/6]{} &&\_(q) = \_[k]{}2k\_[4k,1]{}(q)  = \_[k]{}2k\_[2,6k+2]{}(q)  =  \_[k]{} 2kq\^[(6k-1)\^2/6]{} \[chihsum2\] and &&\_[-]{}(q) = \_[k]{}(2k-1)\_[4k-2,3]{}(q)  = \_[k]{}(2k-1)\_[2,6k-3]{}(q)  =  \_[k]{} (2k-1)q\^[(6k-6)\^2/6]{} &&\_(q) = \_[k]{}2k\_[4k,3]{}(q)  = \_[k]{}2k\_[2,6k]{}(q)  =  \_[k]{} 2kq\^[(6k-3)\^2/6]{} \[chihsum3\] We believe that the 5 characters in (\[chih\]) and the 8 characters in (\[chihsum1\]) through (\[chihsum3\]) correspond to ${\cal W}$-[*irreducible*]{} representations. This yields a total of 13 ${\cal W}$-irreducible representations. In terms of these irreducible characters, we have the decompositions (q)&=& (q)  = 2\_(q)+2\_(q) (q)&=&(q)  = 2\_(q)+2\_(q) (q)&=&(q)  =  2\_(q)+2\_(q) \[chiR2irrrat\] and (q)&=&1+(q)  = 1+2\_[2]{}(q)+2\_[7]{}(q) (q)&=&1+(q)  = 1+2\_[1]{}(q)+2\_[5]{}(q) (q)&=&1+(q)  = 1+2\_[1]{}(q)+2\_[7]{}(q) (q)&=&1+(q)  = 1+2\_[2]{}(q)+2\_[5]{}(q) \[chiR2irr\] and (q) = 2+4\_[1]{}(q)+4\_[2]{}(q)+4\_[5]{}(q)+4\_[7]{}(q) \[chiR3irr\] The ${\cal W}$-irreducible representations whose characters are given by (\[chih\]) are denoted below by $\ketw{\D}$. Sometimes, we extend this practice to the ${\cal W}$-irreducible representations (\[chihch\]) as well. We refer to the finite Kac table in Figure \[FiniteKacTable\] for a natural organization of the conformal weights of the 13 ${\cal W}$-irreducible representations. Letting $\chit_{r,s}(q)$ denote the character of the Kac representation $(r,s)$ where $r,s\in\mathbb{N}$, we have \_0(q)&=&\_[1,1]{}(q)-\_[k]{}(\_[4k-1,1]{}(q)-\_[4k+1,1]{}(q))  = \_[1,1]{}(q)-\_[k]{}(\_[1,6k-1]{}(q)-\_[1,6k+1]{}(q)) \_1(q)&=&\_[k]{}k\^2(\_[4k-1,2]{}(q)-\_[4k+1,2]{}(q))  = \_[k]{}k\^2(\_[1,6k-2]{}(q)-\_[1,6k+2]{}(q)) \_2(q)&=&\_[k]{}k\^2(\_[4k-1,1]{}(q)-\_[4k+1,1]{}(q))  = \_[k]{}k\^2(\_[1,6k-1]{}(q)-\_[1,6k+1]{}(q))\ \_5(q)&=&\_[k]{}k(k+1)(\_[4k+1,2]{}(q)-\_[4(k+1)-1,2]{}(q))  = \_[k]{}k(k+1)(\_[1,6k+1]{}(q)-\_[1,6(k+1)-1]{}(q)) \_7(q)&=&\_[k]{}k(k+1)(\_[4k+1,1]{}(q)-\_[4(k+1)-1,1]{}(q))  = \_[k]{}k(k+1)(\_[1,6k+2]{}(q)-\_[1,6(k+1)-2]{}(q)) \[chihKac\] Since the Kac representations appearing in (\[chihch\]) and (\[chihsum1\]) through (\[chihsum3\]) are [*irreducible*]{} Virasoro representations themselves, we have \_[2(2k-2+),s]{}(q) = \_[2(2k-2+),s]{}(q), \_[r,3(2k-2+)]{}(q) = \_[r,3(2k-2+)]{}(q) and hence \_[2,s]{}(q) = \_[k]{}(2k-2+)\_[2(2k-2+),s]{}(q), \_[r,3]{}(q) = \_[k]{}(2k-2+)\_[r,3(2k-2+)]{}(q) \[RussianKacTable\] ### Theta forms The characters of the 13 ${\cal W}$-irreducible representations agree with those of [@FGST06b].Ê In particular, they admit the expressions given there in terms of theta functions \_[,k]{}(q,z) = \_[j+]{} q\^[kj\^2]{} z\^[k j]{},|q|&lt;1,z,k, and theta-constants \_[,k]{}(q) = \_[,k]{}(q,1),\_[,k]{}\^[(m)]{}(q)  = (z)\^m\_[,k]{}(q,z)|\_[z=1]{},m Introducing the abbreviations \_(q) = \_[,pp’]{}(q),’\_(q) = \_[,pp’]{}\^[(1)]{}(q), ”\_(q) = \^[(2)]{}\_[,pp’]{}(q) the theta forms are \_[r,s]{}(q)&=&(\_[sp-rp’]{}(q)-\_[sp+rp’]{}(q)), r\_[1,p-1]{},s\_[1,p’-1]{}, sp+rp’pp’\ \^+\_[r,s]{}(q)&=&(”\_[sp+rp’]{}(q) -”\_[sp-rp’]{}(q)-(sp+rp’)’\_[sp+rp’]{}(q)+(sp-rp’)’\_[sp-rp’]{}(q)\ &&+\_[sp+rp’]{}(q)-\_[sp-rp’]{}(q)), r\_[1,p]{},s\_[1,p’]{}\ \^-\_[r,s]{}(q)&=&(”\_[pp’-sp-rp’]{}(q) -”\_[pp’+sp-rp’]{}(q)+(sp+rp’)’\_[pp’-sp-rp’]{}(q)\ &&+(sp-rp’)’\_[pp’+sp-rp’]{}(q)+\_[pp’-sp-rp’]{}(q)\ &&-\_[pp’+sp-rp’]{}(q)), r\_[1,p]{},s\_[1,p’]{} where the Dedekind eta function is defined in (\[eta\]). As the notation suggests, these are believed to be the theta forms relevant in the case of general $p,p'$ [@FGST06b]. It is noted that the theta form $\chih_{r,s}(q)$ is identical to the well-known irreducible Virasoro character $\chit_{\D_{r,s}}(q)=\ch_{r,s}(q)$. The precise relations between our ${\cal W}$-irreducible characters and the theta forms for $p=2$ and $p'=3$ are \_0(q) = \_[1,1]{}(q) = 1, [ll]{} \_1(q) = \^+\_[1,2]{}(q), &\_5(q) = \^-\_[1,2]{}(q)\ \_2(q) = \^+\_[1,1]{}(q), &\_7(q) = \^-\_[1,1]{}(q) \ \_[2,s]{}(q) =  \^+\_[2,s]{}(q), &=1\ \^-\_[2,s]{}(q), &=2 \_[r,3]{}(q) =  \^+\_[r,3]{}(q), &=1\ \^-\_[r,3]{}(q), &=2 The ${\cal W}$-irreducible characters $\chih_{2,3}(q)$ and $\chih_{4,3}(q)=\chih_{2,6}(q)$ are listed twice. A compact version of the Kac table in Figure \[FiniteKacTable\] is given in Figure \[RussianKacTable\]. Embedding diagrams and Jordan-cell structures --------------------------------------------- We conjecture that every ${\cal W}$-indecomposable rank-2 representation has an embedding pattern of one of the types \[E\] where the horizontal arrows indicate the non-diagonal action of the Virasoro mode $L_0$. Specifically, we conjecture that the 14 ${\cal W}$-indecomposable rank-2 representations (\[14r2\]) enjoy the embedding patterns &&\~(2,7;0) ,\~(7,2), \~(,) ,\~(,) &&\~(1,5;0) ,\~(5,1), \~(,) ,\~(,) &&\~(1,7;0) ,\~(7,1), \~(,) ,\~(,) &&\~(2,5;0) ,\~(5,2) We can encode the Jordan-cell structure of a ${\cal W}$-indecomposable rank-2 representation in its character by introducing the matrix \_2 =  1&1\ 0&1 Its trace is simply $\mathrm{Tr}(\mathcal{J}_2)=2$ but can be used to indicate the presence of Jordan cells of rank 2. By (\_2)(\_[r,s]{}(q)+\_[r’,s’]{}(q))+2\_[r”,s”]{}(q) we thus mean a sum of 6 irreducible characters where a Jordan cell of rank 2 is formed between every pair of matching states in the 2 modules labelled by $r,s$ and between every pair of matching states in the 2 modules labelled by $r',s'$ while no state in the modules labelled by $r'',s''$ is part of a non-trivial Jordan cell. The characters of the ${\cal W}$-indecomposable rank-2 representations then read (q)&=&\_[,1]{}{1-\_[s,3]{}} +2\_[k]{}(2k+1-)\_[4k+3-2,s]{}(q) && +(\_2)\_[k]{}(2k-2+)\_[4k-3+2,s]{}(q) (q)&=&\_[,1]{}{1-\_[r,2]{}} +2\_[k]{}(2k+1-)\_[r,6k+6-3-b]{}(q) && +(\_2)\_[k]{}(2k-2+)\_[r,6k-6+3+b]{}(q) These refined character expressions demonstrate the inequivalence of the various representations despite the character identities (\[chid1\]). The relations (\[chrelsum\]) are valid for the refined characters as well, whereas the relations (\[chrel1\]) are not. We note that the refined character expressions contain enough information to distinguish between the different rank-2 representations. That is, the distinctions can be made by solely emphasizing the Jordan-cell structures without further reference to the complete embedding patterns. Similar refinements of the rank-3 characters are possible (see below) but not required to demonstrate inequivalence of the associated ${\cal W}$-indecomposable rank-3 representations. Indeed, it suffices to focus on the presence of rank-3 Jordan cells to which end we introduce the matrix \_3 =  1&1&0\ 0&1&1\ 0&0&1 with trace $\mathrm{Tr}(\mathcal{J}_3)=3$. Ignoring Jordan cells of rank 2 all together, the ‘semi-refined’ characters of the ${\cal W}$-indecomposable rank-3 representations then read (q)&=&2+4\_[k]{}k\_[2k+1,b]{}(q) +4\_[k]{}(2k+1-)\_[4k+3-2,3-b]{}(q) &&+{(\_3)+1}\_[k]{}(2k-2+) \_[4k-3+2,3-b]{}(q) With $\kappa,b\in\mathbb{Z}_{1,2}$, these 4 semi-refined characters correspond to 4 distinct representations despite the character identities implicit in (\[chitRr3\]). We conclude this discussion of embedding patterns by conjecturing that the ${\cal W}$-indecomposable rank-3 representations also have embedding structures described by the patterns in (\[E\]). Specifically, we conjecture that \~(,) \~(,) \[RE\] where the ${\cal W}$-irreducible representations $\ketw{\D_h}$ and $\ketw{\D_v}$ have been replaced by ${\cal W}$-indecomposable rank-2 representations. It is noted that each of the 4 rank-3 representations is thus proposed to be viewable in two different ways. This corresponds to viewing it as an indecomposable ‘vertical’ combination of ‘horizontal’ rank-2 representations $\ketw{\R^{1,0}}$ or as an indecomposable ‘horizontal’ combination of ‘vertical’ rank-2 representations $\ketw{\R^{0,b}}$. Converting the two rank-2 Jordan cells linked by a horizontal arrow into a rank-3 and a rank-1 Jordan cell, we finally arrive at the announced refined characters (q)&=&(\_2)\_0(q) +{(\_3)+1}\_1(q) +4\_2(q)+2(\_2)\_5(q)+2(\_2)\_7(q) (q)&=&2\_0(q)+2(\_2)\_1(q) +2(\_2)\_2(q)+{(\_3)+1}\_5(q) +4\_7(q) (q)&=&(\_2)\_0(q)+4\_1(q) +{(\_3)+1}\_2(q)+2(\_2)\_5(q) +2(\_2)\_7(q) (q)&=&2\_0(q)+2(\_2)\_1(q) +2(\_2)\_2(q)+4\_5(q) +{(\_3)+1}\_7(q) here expressed explicitly in terms of the ${\cal W}$-irreducible characters (\[chih\]). Summary of ${\cal W}$-extended fusion algebra --------------------------------------------- We denote the fusion product in the ${\cal W}$-extended picture by $\fus$ and reserve the symbol $\otimes$ for the fusion product in the Virasoro picture. A summary of the fusion algebra of critical percolation in the ${\cal W}$-extended picture ${\cal WLM}(2,3)$ is given in the following. It is both associative and commutative. To compactify the results a bit, we introduce the following linear combinations \_s = 22,&& \^[1,0]{}\_s = 22 \^[0,b]{} = 22,&& \^[1,b]{} = 22 \^0 = 4\_32\^[0,1]{}2\^[0,2]{},&& \^1 = 4\^[1,0]{}\_32\^[1,1]{}2\^[1,2]{} and \_[,3’]{}\^[0,b]{} = 2 2, \_[2,3]{}\^[1,b]{} = 2 2 where it is recalled that $\ketw{2,6}\equiv\ketw{4,3}$ and where mn = ,m,n The fusion rules are listed in the tables in Figure \[Cayleyr1r1\] through Figure \[Cayleyr3r3\]. They are easily combined to form a complete Cayley table as indicated in Figure \[SchematicCayley\]. $$\renewcommand{\arraystretch}{1.5} \begin{array}{c||ccc} \hat\otimes&\mathrm{rank}\ 1&\mathrm{rank}\ 2&\mathrm{rank}\ 3 \\[4pt] \hline \hline \rule{0pt}{14pt} \mathrm{rank}\ 1 &F_{\ref{Cayleyr1r1}}&U_{\ref{Cayleyr1r23}}^T&L_{\ref{Cayleyr1r23}}^T \\[4pt] \mathrm{rank}\ 2 &U_{\ref{Cayleyr1r23}}&\big(U_{\ref{Cayleyr2r23a}}|U_{\ref{Cayleyr2r23b}}\big) &\big(L_{\ref{Cayleyr2r23a}}|L_{\ref{Cayleyr2r23b}}\big)^{\! T} \\[4pt] \mathrm{rank}\ 3 &L_{\ref{Cayleyr1r23}}&\big(L_{\ref{Cayleyr2r23a}}|L_{\ref{Cayleyr2r23b}}\big)&F_{\ref{Cayleyr3r3}} \end{array}$$ $$\renewcommand{\arraystretch}{1.5} \begin{array}{c||cc|cc|cc|cc} \hat\otimes&\ketw{2,1}&\ketw{4,1}&\ketw{2,2}&\ketw{4,2}&\ketw{1,3}&\ketw{1,6}&\ketw{2,3}&\ketw{4,3} \\[4pt] \hline \hline \rule{0pt}{14pt} \ketw{2,1}&\ketw{\R_{2,1}^{1,0}}&\ketw{\R_{4,1}^{1,0}}&\ketw{\R_{2,2}^{1,0}}&\ketw{\R_{4,2}^{1,0}} &\ketw{2,3}&\ketw{4,3}&\ketw{\R_{2,3}^{1,0}}&\ketw{\R_{4,3}^{1,0}} \\[4pt] \ketw{4,1}&\ketw{\R_{4,1}^{1,0}}&\ketw{\R_{2,1}^{1,0}}&\ketw{\R_{4,2}^{1,0}} &\ketw{\R_{2,2}^{1,0}}&\ketw{4,3}&\ketw{2,3} &\ketw{\R_{4,3}^{1,0}}&\ketw{\R_{2,3}^{1,0}} \\[4pt] \hline \rule{0pt}{14pt} \ketw{2,2}&\ketw{\R_{2,2}^{1,0}}&\ketw{\R_{4,2}^{1,0}} &\ketw{\R_{2,1}^{1,0}}\oplus\ketw{\R_{2,3}^{1,0}}&\ketw{\R_{4,1}^{1,0}}\oplus\ketw{\R_{4,3}^{1,0}} &\ketw{\R_{2,3}^{0,1}}&\ketw{\R_{2,6}^{0,1}}&\ketw{\R_{2,3}^{1,1}}&\ketw{\R_{4,3}^{1,1}} \\[4pt] \ketw{4,2}&\ketw{\R_{4,2}^{1,0}}&\ketw{\R_{2,2}^{1,0}} &\ketw{\R_{4,1}^{1,0}}\oplus\ketw{\R_{4,3}^{1,0}}&\ketw{\R_{2,1}^{1,0}}\oplus\ketw{\R_{2,3}^{1,0}} &\ketw{\R_{2,6}^{0,1}}&\ketw{\R_{2,3}^{0,1}}&\ketw{\R_{4,3}^{1,1}}&\ketw{\R_{2,3}^{1,1}} \\[4pt] \hline \rule{0pt}{14pt} \ketw{1,3}&\ketw{2,3}&\ketw{4,3}&\ketw{\R_{2,3}^{0,1}}&\ketw{\R_{2,6}^{0,1}} &\ketw{1,3}\oplus\ketw{\R_{1,3}^{0,2}}&\ketw{1,6}\oplus\ketw{\R_{1,6}^{0,2}} &\ketw{2,3}\oplus\ketw{\R_{2,3}^{0,2}}&\ketw{4,3}\oplus\ketw{\R_{2,6}^{0,2}} \\[4pt] \ketw{1,6}&\ketw{4,3}&\ketw{2,3}&\ketw{\R_{2,6}^{0,1}}&\ketw{\R_{2,3}^{0,1}} &\ketw{1,6}\oplus\ketw{\R_{1,6}^{0,2}}&\ketw{1,3}\oplus\ketw{\R_{1,3}^{0,2}} &\ketw{4,3}\oplus\ketw{\R_{2,6}^{0,2}}&\ketw{2,3}\oplus\ketw{\R_{2,3}^{0,2}} \\[4pt] \hline \rule{0pt}{14pt} \ketw{2,3}&\ketw{\R_{2,3}^{1,0}}&\ketw{\R_{4,3}^{1,0}}&\ketw{\R_{2,3}^{1,1}} &\ketw{\R_{4,3}^{1,1}}&\ketw{2,3}\oplus\ketw{\R_{2,3}^{0,2}} &\ketw{4,3}\oplus\ketw{\R_{2,6}^{0,2}} &\ketw{\R_{2,3}^{1,0}}\oplus\ketw{\R_{2,3}^{1,2}} &\ketw{\R_{4,3}^{1,0}}\oplus\ketw{\R_{4,3}^{1,2}} \\[4pt] \ketw{4,3}&\ketw{\R_{4,3}^{1,0}}&\ketw{\R_{2,3}^{1,0}}&\ketw{\R_{4,3}^{1,1}}&\ketw{\R_{2,3}^{1,1}} &\ketw{4,3}\oplus\ketw{\R_{2,6}^{0,2}}&\ketw{2,3}\oplus\ketw{\R_{2,3}^{0,2}} &\ketw{\R_{4,3}^{1,0}}\oplus\ketw{\R_{4,3}^{1,2}}&\ketw{\R_{2,3}^{1,0}}\oplus\ketw{\R_{2,3}^{1,2}} \end{array}$$ $$\renewcommand{\arraystretch}{1.5} \begin{array}{c||cc|cc|cc|cc} \hat\otimes&\ketw{2,1}&\ketw{4,1}&\ketw{2,2}&\ketw{4,2}&\ketw{1,3}&\ketw{1,6}&\ketw{2,3}&\ketw{4,3} \\[4pt] \hline \hline \rule{0pt}{14pt} \ketw{\R_{2,1}^{1,0}}&\Cc_1&\Cc_1&\Cc_2&\Cc_2 &\ketw{\R_{2,3}^{1,0}}&\ketw{\R_{4,3}^{1,0}}&\Cc_3&\Cc_3 \\[4pt] \ketw{\R_{4,1}^{1,0}}&\Cc_1&\Cc_1&\Cc_2&\Cc_2 &\ketw{\R_{4,3}^{1,0}}&\ketw{\R_{2,3}^{1,0}}&\Cc_3&\Cc_3 \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{2,2}^{1,0}}&\Cc_2&\Cc_2&\Cc_1\oplus\Cc_3&\Cc_1\oplus\Cc_3 &\ketw{\R_{2,3}^{1,1}}&\ketw{\R_{4,3}^{1,1}}&\Cc^{0,1}&\Cc^{0,1} \\[4pt] \ketw{\R_{4,2}^{1,0}}&\Cc_2&\Cc_2&\Cc_1\oplus\Cc_3&\Cc_1\oplus\Cc_3 &\ketw{\R_{4,3}^{1,1}}&\ketw{\R_{2,3}^{1,1}}&\Cc^{0,1}&\Cc^{0,1} \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{2,3}^{1,0}}&\Cc_3&\Cc_3&\Cc^{0,1}&\Cc^{0,1} &\ketw{\R_{2,3}^{1,0}}\oplus\ketw{\R_{2,3}^{1,2}}&\ketw{\R_{4,3}^{1,0}}\oplus\ketw{\R_{4,3}^{1,2}} &\Cc_3\oplus\Cc^{0,2}&\Cc_3\oplus\Cc^{0,2} \\[4pt] \ketw{\R_{4,3}^{1,0}}&\Cc_3&\Cc_3&\Cc^{0,1}&\Cc^{0,1} &\ketw{\R_{4,3}^{1,0}}\oplus\ketw{\R_{4,3}^{1,2}}&\ketw{\R_{2,3}^{1,0}}\oplus\ketw{\R_{2,3}^{1,2}} &\Cc_3\oplus\Cc^{0,2}&\Cc_3\oplus\Cc^{0,2} \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{1,3}^{0,1}}&\ketw{\R_{2,3}^{0,1}}&\ketw{\R_{2,6}^{0,1}} &2\ketw{2,3}\oplus\ketw{\R_{2,3}^{0,2}}&2\ketw{4,3}\oplus\ketw{\R_{2,6}^{0,2}} &\Dc_{1,3}^{0,1}&\Dc_{1,6}^{0,1} &\Dc_{2,3}^{0,1}&\Dc_{2,6}^{0,1} \\[4pt] \ketw{\R_{1,6}^{0,1}}&\ketw{\R_{2,6}^{0,1}}&\ketw{\R_{2,3}^{0,1}} &2\ketw{4,3}\oplus\ketw{\R_{2,6}^{0,2}}&2\ketw{2,3}\oplus\ketw{\R_{2,3}^{0,2}} &\Dc_{1,6}^{0,1}&\Dc_{1,3}^{0,1} &\Dc_{2,6}^{0,1}&\Dc_{2,3}^{0,1} \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{1,3}^{0,2}}&\ketw{\R_{2,3}^{0,2}}&\ketw{\R_{2,6}^{0,2}} &2\ketw{4,3}\oplus\ketw{\R_{2,3}^{0,1}}&2\ketw{2,3}\oplus\ketw{\R_{2,6}^{0,1}} &\Dc_{1,6}^{0,1}&\Dc_{1,3}^{0,1} &\Dc_{2,6}^{0,1}&\Dc_{2,3}^{0,1} \\[4pt] \ketw{\R_{1,6}^{0,2}}&\ketw{\R_{2,6}^{0,2}}&\ketw{\R_{2,3}^{0,2}} &2\ketw{2,3}\oplus\ketw{\R_{2,6}^{0,1}}&2\ketw{4,3}\oplus\ketw{\R_{2,3}^{0,1}} &\Dc_{1,3}^{0,1}&\Dc_{1,6}^{0,1} &\Dc_{2,3}^{0,1}&\Dc_{2,6}^{0,1} \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{2,3}^{0,1}}&\ketw{\R_{2,3}^{1,1}}&\ketw{\R_{4,3}^{1,1}} &2\ketw{\R_{2,3}^{1,0}}\oplus\ketw{\R_{2,3}^{1,2}}&2\ketw{\R_{4,3}^{1,0}}\oplus\ketw{\R_{4,3}^{1,2}} &\Dc_{2,3}^{0,1}&\Dc_{2,6}^{0,1} &\Dc_{2,3}^{1,1}&\Dc_{4,3}^{1,1} \\[4pt] \ketw{\R_{2,6}^{0,1}}&\ketw{\R_{4,3}^{1,1}}&\ketw{\R_{2,3}^{1,1}} &2\ketw{\R_{4,3}^{1,0}}\oplus\ketw{\R_{4,3}^{1,2}}&2\ketw{\R_{2,3}^{1,0}}\oplus\ketw{\R_{2,3}^{1,2}} &\Dc_{2,6}^{0,1}&\Dc_{2,3}^{0,1} &\Dc_{4,3}^{1,1}&\Dc_{2,3}^{1,1} \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{2,3}^{0,2}}&\ketw{\R_{2,3}^{1,2}}&\ketw{\R_{4,3}^{1,2}} &2\ketw{\R_{4,3}^{1,0}}\oplus\ketw{\R_{2,3}^{1,1}}&2\ketw{\R_{2,3}^{1,0}}\oplus\ketw{\R_{4,3}^{1,1}} &\Dc_{2,6}^{0,1}&\Dc_{2,3}^{0,1} &\Dc_{4,3}^{1,1}&\Dc_{2,3}^{1,1} \\[4pt] \ketw{\R_{2,6}^{0,2}}&\ketw{\R_{4,3}^{1,2}}&\ketw{\R_{2,3}^{1,2}} &2\ketw{\R_{2,3}^{1,0}}\oplus\ketw{\R_{4,3}^{1,1}}&2\ketw{\R_{4,3}^{1,0}}\oplus\ketw{\R_{2,3}^{1,1}} &\Dc_{2,3}^{0,1}&\Dc_{2,6}^{0,1} &\Dc_{2,3}^{1,1}&\Dc_{4,3}^{1,1} \\[4pt] \hline\hline \rule{0pt}{14pt} \ketw{\R_{2,3}^{1,1}}&\Cc^{0,1}&\Cc^{0,1}&2\Cc_3\oplus\Cc^{0,2}&2\Cc_3\oplus\Cc^{0,2} &\Dc_{2,3}^{1,1}&\Dc_{4,3}^{1,1} &2\Cc_3\oplus2\Cc^{0,1}&2\Cc_3\oplus2\Cc^{0,1} \\[4pt] \ketw{\R_{4,3}^{1,1}}&\Cc^{0,1}&\Cc^{0,1}&2\Cc_3\oplus\Cc^{0,2}&2\Cc_3\oplus\Cc^{0,2} &\Dc_{4,3}^{1,1}&\Dc_{2,3}^{1,1} &2\Cc_3\oplus2\Cc^{0,1}&2\Cc_3\oplus2\Cc^{0,1} \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{2,3}^{1,2}}&\Cc^{0,2}&\Cc^{0,2}&2\Cc_3\oplus\Cc^{0,1}&2\Cc_3\oplus\Cc^{0,1} &\Dc_{4,3}^{1,1}&\Dc_{2,3}^{1,1} &2\Cc_3\oplus2\Cc^{0,1}&2\Cc_3\oplus2\Cc^{0,1} \\[4pt] \ketw{\R_{4,3}^{1,2}}&\Cc^{0,2}&\Cc^{0,2}&2\Cc_3\oplus\Cc^{0,1}&2\Cc_3\oplus\Cc^{0,1} &\Dc_{2,3}^{1,1}&\Dc_{4,3}^{1,1} &2\Cc_3\oplus2\Cc^{0,1}&2\Cc_3\oplus2\Cc^{0,1} \end{array}$$ $$\renewcommand{\arraystretch}{1.5} \begin{array}{c||cc|cc|cc} \hat\otimes&\ketw{\R_{2,1}^{1,0}}&\ketw{\R_{4,1}^{1,0}}&\ketw{\R_{2,2}^{1,0}} &\ketw{\R_{4,2}^{1,0}}&\ketw{\R_{2,3}^{1,0}}&\ketw{\R_{4,3}^{1,0}} \\[4pt] \hline \hline \rule{0pt}{14pt} \ketw{\R_{2,1}^{1,0}}&\Cc^{1,0}_1&\Cc^{1,0}_1&\Cc^{1,0}_2 &\Cc^{1,0}_2&\Cc^{1,0}_3&\Cc^{1,0}_3 \\[4pt] \ketw{\R_{4,1}^{1,0}}&\Cc^{1,0}_1&\Cc^{1,0}_1&\Cc^{1,0}_2 &\Cc^{1,0}_2&\Cc^{1,0}_3&\Cc^{1,0}_3 \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{2,2}^{1,0}}&\Cc^{1,0}_2&\Cc^{1,0}_2&\Cc^{1,0}_1\oplus\Cc^{1,0}_3 &\Cc^{1,0}_1\oplus\Cc^{1,0}_3&\Cc^{1,1}&\Cc^{1,1} \\[4pt] \ketw{\R_{4,2}^{1,0}}&\Cc^{1,0}_2&\Cc^{1,0}_2&\Cc^{1,0}_1\oplus\Cc^{1,0}_3 &\Cc^{1,0}_1\oplus\Cc^{1,0}_3&\Cc^{1,1}&\Cc^{1,1} \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{2,3}^{1,0}}&\Cc^{1,0}_3&\Cc^{1,0}_3&\Cc^{1,1} &\Cc^{1,1}&\Cc^{1,0}_3\oplus\Cc^{1,2}&\Cc^{1,0}_3\oplus\Cc^{1,2} \\[4pt] \ketw{\R_{4,3}^{1,0}}&\Cc^{1,0}_3&\Cc^{1,0}_3&\Cc^{1,1} &\Cc^{1,1}&\Cc^{1,0}_3\oplus\Cc^{1,2}&\Cc^{1,0}_3\oplus\Cc^{1,2} \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{1,3}^{0,1}}&\ketw{\R_{2,3}^{1,1}}&\ketw{\R_{4,3}^{1,1}} &2\ketw{\R_{2,3}^{1,0}}\oplus\ketw{\R_{2,3}^{1,2}} &2\ketw{\R_{4,3}^{1,0}}\oplus\ketw{\R_{4,3}^{1,2}}&\Dc_{2,3}^{1,1}&\Dc_{4,3}^{1,1} \\[4pt] \ketw{\R_{1,6}^{0,1}}&\ketw{\R_{4,3}^{1,1}}&\ketw{\R_{2,3}^{1,1}} &2\ketw{\R_{4,3}^{1,0}}\oplus\ketw{\R_{4,3}^{1,2}} &2\ketw{\R_{2,3}^{1,0}}\oplus\ketw{\R_{2,3}^{1,2}}&\Dc_{4,3}^{1,1}&\Dc_{2,3}^{1,1} \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{1,3}^{0,2}}&\ketw{\R_{2,3}^{1,2}}&\ketw{\R_{4,3}^{1,2}} &2\ketw{\R_{4,3}^{1,0}}\oplus\ketw{\R_{2,3}^{1,1}} &2\ketw{\R_{2,3}^{1,0}}\oplus\ketw{\R_{4,3}^{1,1}}&\Dc_{4,3}^{1,1}&\Dc_{2,3}^{1,1} \\[4pt] \ketw{\R_{1,6}^{0,2}}&\ketw{\R_{4,3}^{1,2}}&\ketw{\R_{2,3}^{1,2}} &2\ketw{\R_{2,3}^{1,0}}\oplus\ketw{\R_{4,3}^{1,1}} &2\ketw{\R_{4,3}^{1,0}}\oplus\ketw{\R_{2,3}^{1,1}}&\Dc_{2,3}^{1,1}&\Dc_{4,3}^{1,1} \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{2,3}^{0,1}}&\Cc^{0,1}&\Cc^{0,1}&2\Cc_3\oplus\Cc^{0,2} &2\Cc_3\oplus\Cc^{0,2}&2\Cc_3\oplus2\Cc^{0,1}&2\Cc_3\oplus2\Cc^{0,1} \\[4pt] \ketw{\R_{2,6}^{0,1}}&\Cc^{0,1}&\Cc^{0,1}&2\Cc_3\oplus\Cc^{0,2} &2\Cc_3\oplus\Cc^{0,2}&2\Cc_3\oplus2\Cc^{0,1}&2\Cc_3\oplus2\Cc^{0,1} \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{2,3}^{0,2}}&\Cc^{0,2}&\Cc^{0,2}&2\Cc_3\oplus\Cc^{0,1} &2\Cc_3\oplus\Cc^{0,1}&2\Cc_3\oplus2\Cc^{0,1}&2\Cc_3\oplus2\Cc^{0,1} \\[4pt] \ketw{\R_{2,6}^{0,2}}&\Cc^{0,2}&\Cc^{0,2}&2\Cc_3\oplus\Cc^{0,1} &2\Cc_3\oplus\Cc^{0,1}&2\Cc_3\oplus2\Cc^{0,1}&2\Cc_3\oplus2\Cc^{0,1} \\[4pt] \hline\hline \rule{0pt}{14pt} \ketw{\R_{2,3}^{1,1}}&\Cc^{1,1}&\Cc^{1,1}&2\Cc^{1,0}_3\oplus\Cc^{1,2} &2\Cc^{1,0}_3\oplus\Cc^{1,2}&2\Cc^{1,0}_3\oplus2\Cc^{1,1}&2\Cc^{1,0}_3\oplus2\Cc^{1,1} \\[4pt] \ketw{\R_{4,3}^{1,1}}&\Cc^{1,1}&\Cc^{1,1}&2\Cc^{1,0}_3\oplus\Cc^{1,2} &2\Cc^{1,0}_3\oplus\Cc^{1,2}&2\Cc^{1,0}_3\oplus2\Cc^{1,1}&2\Cc^{1,0}_3\oplus2\Cc^{1,1} \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{2,3}^{1,2}}&\Cc^{1,2}&\Cc^{1,2}&2\Cc^{1,0}_3\oplus\Cc^{1,1} &2\Cc^{1,0}_3\oplus\Cc^{1,1}&2\Cc^{1,0}_3\oplus2\Cc^{1,1}&2\Cc^{1,0}_3\oplus2\Cc^{1,1} \\[4pt] \ketw{\R_{4,3}^{1,2}}&\Cc^{1,2}&\Cc^{1,2}&2\Cc^{1,0}_3\oplus\Cc^{1,1} &2\Cc^{1,0}_3\oplus\Cc^{1,1}&2\Cc^{1,0}_3\oplus2\Cc^{1,1}&2\Cc^{1,0}_3\oplus2\Cc^{1,1} \end{array}$$ $$\renewcommand{\arraystretch}{1.5} \begin{array}{c||cc|cc|cc|cc} \hat\otimes&\ketw{\R_{1,3}^{0,1}}&\ketw{\R_{1,6}^{0,1}}&\ketw{\R_{1,3}^{0,2}}&\ketw{\R_{1,6}^{0,2}} &\ketw{\R_{2,3}^{0,1}}&\ketw{\R_{2,6}^{0,1}} &\ketw{\R_{2,3}^{0,2}}&\ketw{\R_{2,6}^{0,2}} \\[4pt] \hline \hline \rule{0pt}{14pt} \ketw{\R_{2,1}^{1,0}}&\ketw{\R_{2,3}^{1,1}}&\ketw{\R_{4,3}^{1,1}} &\ketw{\R_{2,3}^{1,2}} &\ketw{\R_{4,3}^{1,2}}&\Cc^{0,1} &\Cc^{0,1}&\Cc^{0,2}&\Cc^{0,2} \\[4pt] \ketw{\R_{4,1}^{1,0}}&\ketw{\R_{4,3}^{1,1}}&\ketw{\R_{2,3}^{1,1}} &\ketw{\R_{4,3}^{1,2}} &\ketw{\R_{2,3}^{1,2}}&\Cc^{0,1} &\Cc^{0,1}&\Cc^{0,2}&\Cc^{0,2} \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{2,2}^{1,0}}&2\ketw{\R_{2,3}^{1,0}}\oplus\ketw{\R_{2,3}^{1,2}} &2\ketw{\R_{4,3}^{1,0}}\oplus\ketw{\R_{4,3}^{1,2}} &2\ketw{\R_{4,3}^{1,0}}\oplus\ketw{\R_{2,3}^{1,1}} &2\ketw{\R_{2,3}^{1,0}}\oplus\ketw{\R_{4,3}^{1,1}}&2\Cc_3\oplus\Cc^{0,2} &2\Cc_3\oplus\Cc^{0,2} &2\Cc_3\oplus\Cc^{0,1}&2\Cc_3\oplus\Cc^{0,1} \\[4pt] \ketw{\R_{4,2}^{1,0}}&2\ketw{\R_{4,3}^{1,0}}\oplus\ketw{\R_{4,3}^{1,2}} &2\ketw{\R_{2,3}^{1,0}}\oplus\ketw{\R_{2,3}^{1,2}} &2\ketw{\R_{2,3}^{1,0}}\oplus\ketw{\R_{4,3}^{1,1}} &2\ketw{\R_{4,3}^{1,0}}\oplus\ketw{\R_{2,3}^{1,1}}&2\Cc_3\oplus\Cc^{0,2} &2\Cc_3\oplus\Cc^{0,2} &2\Cc_3\oplus\Cc^{0,1}&2\Cc_3\oplus\Cc^{0,1} \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{2,3}^{1,0}}&\Dc_{2,3}^{1,1}&\Dc_{4,3}^{1,1} &\Dc_{4,3}^{1,1} &\Dc_{2,3}^{1,1}&2\Cc_3\oplus2\Cc^{0,1} &2\Cc_3\oplus2\Cc^{0,1} &2\Cc_3\oplus2\Cc^{0,1}&2\Cc_3\oplus2\Cc^{0,1} \\[4pt] \ketw{\R_{4,3}^{1,0}}&\Dc_{4,3}^{1,1}&\Dc_{2,3}^{1,1}&\Dc_{2,3}^{1,1} &\Dc_{4,3}^{1,1}&2\Cc_3\oplus2\Cc^{0,1} &2\Cc_3\oplus2\Cc^{0,1} &2\Cc_3\oplus2\Cc^{0,1}&2\Cc_3\oplus2\Cc^{0,1} \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{1,3}^{0,1}}&\Dc_{1,6}^{0,1}\oplus\Dc_{1,3}^{0,2} &\Dc_{1,3}^{0,1}\oplus\Dc_{1,6}^{0,2} &\Dc_{1,3}^{0,1}\oplus\Dc_{1,6}^{0,2} &\Dc_{1,6}^{0,1}\oplus\Dc_{1,3}^{0,2}&\Dc_{2,6}^{0,1}\oplus\Dc_{2,3}^{0,2} &\Dc_{2,3}^{0,1}\oplus\Dc_{2,6}^{0,2} &\Dc_{2,3}^{0,1}\oplus\Dc_{2,6}^{0,2} &\Dc_{2,6}^{0,1}\oplus\Dc_{2,3}^{0,2} \\[4pt] \ketw{\R_{1,6}^{0,1}}&\Dc_{1,3}^{0,1}\oplus\Dc_{1,6}^{0,2} &\Dc_{1,6}^{0,1}\oplus\Dc_{1,3}^{0,2}&\Dc_{1,6}^{0,1}\oplus\Dc_{1,3}^{0,2} &\Dc_{1,3}^{0,1}\oplus\Dc_{1,6}^{0,2}&\Dc_{2,3}^{0,1}\oplus\Dc_{2,6}^{0,2} &\Dc_{2,6}^{0,1}\oplus\Dc_{2,3}^{0,2} &\Dc_{2,6}^{0,1}\oplus\Dc_{2,3}^{0,2} &\Dc_{2,3}^{0,1}\oplus\Dc_{2,6}^{0,2} \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{1,3}^{0,2}}&\Dc_{1,3}^{0,1}\oplus\Dc_{1,6}^{0,2} &\Dc_{1,6}^{0,1}\oplus\Dc_{1,3}^{0,2}&\Dc_{1,6}^{0,1}\oplus\Dc_{1,3}^{0,2} &\Dc_{1,3}^{0,1}\oplus\Dc_{1,6}^{0,2}&\Dc_{2,3}^{0,1}\oplus\Dc_{2,6}^{0,2} &\Dc_{2,6}^{0,1}\oplus\Dc_{2,3}^{0,2} &\Dc_{2,6}^{0,1}\oplus\Dc_{2,3}^{0,2} &\Dc_{2,3}^{0,1}\oplus\Dc_{2,6}^{0,2} \\[4pt] \ketw{\R_{1,6}^{0,2}}&\Dc_{1,6}^{0,1}\oplus\Dc_{1,3}^{0,2} &\Dc_{1,3}^{0,1}\oplus\Dc_{1,6}^{0,2}&\Dc_{1,3}^{0,1}\oplus\Dc_{1,6}^{0,2} &\Dc_{1,6}^{0,1}\oplus\Dc_{1,3}^{0,2}&\Dc_{2,6}^{0,1}\oplus\Dc_{2,3}^{0,2} &\Dc_{2,3}^{0,1}\oplus\Dc_{2,6}^{0,2} &\Dc_{2,3}^{0,1}\oplus\Dc_{2,6}^{0,2} &\Dc_{2,6}^{0,1}\oplus\Dc_{2,3}^{0,2} \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{2,3}^{0,1}}&\Dc_{2,6}^{0,1}\oplus\Dc_{2,3}^{0,2} &\Dc_{2,3}^{0,1}\oplus\Dc_{2,6}^{0,2}&\Dc_{2,3}^{0,1}\oplus\Dc_{2,6}^{0,2} &\Dc_{2,6}^{0,1}\oplus\Dc_{2,3}^{0,2}&\Dc_{4,3}^{1,1}\oplus\Dc_{2,3}^{1,2} &\Dc_{2,3}^{1,1}\oplus\Dc_{4,3}^{1,2} &\Dc_{2,3}^{1,1}\oplus\Dc_{4,3}^{1,2} &\Dc_{4,3}^{1,1}\oplus\Dc_{2,3}^{1,2} \\[4pt] \ketw{\R_{2,6}^{0,1}}&\Dc_{2,3}^{0,1}\oplus\Dc_{2,6}^{0,2} &\Dc_{2,6}^{0,1}\oplus\Dc_{2,3}^{0,2}&\Dc_{2,6}^{0,1}\oplus\Dc_{2,3}^{0,2} &\Dc_{2,3}^{0,1}\oplus\Dc_{2,6}^{0,2}&\Dc_{2,3}^{1,1}\oplus\Dc_{4,3}^{1,2} &\Dc_{4,3}^{1,1}\oplus\Dc_{2,3}^{1,2} &\Dc_{4,3}^{1,1}\oplus\Dc_{2,3}^{1,2} &\Dc_{2,3}^{1,1}\oplus\Dc_{4,3}^{1,2} \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{2,3}^{0,2}}&\Dc_{2,3}^{0,1}\oplus\Dc_{2,6}^{0,2} &\Dc_{2,6}^{0,1}\oplus\Dc_{2,3}^{0,2}&\Dc_{2,6}^{0,1}\oplus\Dc_{2,3}^{0,2} &\Dc_{2,3}^{0,1}\oplus\Dc_{2,6}^{0,2}&\Dc_{2,3}^{1,1}\oplus\Dc_{4,3}^{1,2} &\Dc_{4,3}^{1,1}\oplus\Dc_{2,3}^{1,2} &\Dc_{4,3}^{1,1}\oplus\Dc_{2,3}^{1,2} &\Dc_{2,3}^{1,1}\oplus\Dc_{4,3}^{1,2} \\[4pt] \ketw{\R_{2,6}^{0,2}}&\Dc_{2,6}^{0,1}\oplus\Dc_{2,3}^{0,2} &\Dc_{2,3}^{0,1}\oplus\Dc_{2,6}^{0,2}&\Dc_{2,3}^{0,1}\oplus\Dc_{2,6}^{0,2} &\Dc_{2,6}^{0,1}\oplus\Dc_{2,3}^{0,2}&\Dc_{4,3}^{1,1}\oplus\Dc_{2,3}^{1,2} &\Dc_{2,3}^{1,1}\oplus\Dc_{4,3}^{1,2} &\Dc_{2,3}^{1,1}\oplus\Dc_{4,3}^{1,2} &\Dc_{4,3}^{1,1}\oplus\Dc_{2,3}^{1,2} \\[4pt] \hline\hline \rule{0pt}{14pt} \ketw{\R_{2,3}^{1,1}}&\Dc_{4,3}^{1,1}\oplus\Dc_{2,3}^{1,2} &\Dc_{2,3}^{1,1}\oplus\Dc_{4,3}^{1,2}&\Dc_{2,3}^{1,1}\oplus\Dc_{4,3}^{1,2} &\Dc_{4,3}^{1,1}\oplus\Dc_{2,3}^{1,2}&\hat\Cc^0&\hat\Cc^0&\hat\Cc^0&\hat\Cc^0 \\[4pt] \ketw{\R_{4,3}^{1,1}}&\Dc_{2,3}^{1,1}\oplus\Dc_{4,3}^{1,2} &\Dc_{4,3}^{1,1}\oplus\Dc_{2,3}^{1,2}&\Dc_{4,3}^{1,1}\oplus\Dc_{2,3}^{1,2} &\Dc_{2,3}^{1,1}\oplus\Dc_{4,3}^{1,2}&\hat\Cc^0&\hat\Cc^0&\hat\Cc^0&\hat\Cc^0 \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{2,3}^{1,2}}&\Dc_{2,3}^{1,1}\oplus\Dc_{4,3}^{1,2} &\Dc_{4,3}^{1,1}\oplus\Dc_{2,3}^{1,2} &\Dc_{4,3}^{1,1}\oplus\Dc_{2,3}^{1,2} &\Dc_{2,3}^{1,1}\oplus\Dc_{4,3}^{1,2}&\hat\Cc^0&\hat\Cc^0&\hat\Cc^0&\hat\Cc^0 \\[4pt] \ketw{\R_{4,3}^{1,2}}&\Dc_{4,3}^{1,1}\oplus\Dc_{2,3}^{1,2} &\Dc_{2,3}^{1,1}\oplus\Dc_{4,3}^{1,2}&\Dc_{2,3}^{1,1}\oplus\Dc_{4,3}^{1,2} &\Dc_{4,3}^{1,1}\oplus\Dc_{2,3}^{1,2}&\hat\Cc^0&\hat\Cc^0&\hat\Cc^0&\hat\Cc^0 \end{array}$$ $$\renewcommand{\arraystretch}{1.5} \begin{array}{c||cc|cc} \hat\otimes&\ketw{\R_{2,3}^{1,1}}&\ketw{\R_{4,3}^{1,1}}&\ketw{\R_{2,3}^{1,2}}&\ketw{\R_{4,3}^{1,2}} \\[4pt] \hline \hline \rule{0pt}{14pt} \ketw{\R_{2,3}^{1,1}} &\hat\Cc^1&\hat\Cc^1&\hat\Cc^1&\hat\Cc^1 \\[4pt] \ketw{\R_{4,3}^{1,1}} &\hat\Cc^1&\hat\Cc^1&\hat\Cc^1&\hat\Cc^1 \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{2,3}^{1,2}} &\hat\Cc^1&\hat\Cc^1&\hat\Cc^1&\hat\Cc^1 \\[4pt] \ketw{\R_{4,3}^{1,2}} &\hat\Cc^1&\hat\Cc^1&\hat\Cc^1&\hat\Cc^1 \end{array}$$ All entries of the Cayley table of the fusions of ${\cal W}$-indecomposable rank-3 representations provided in Figure \[Cayleyr3r3\] are given by \^1 =  884 44 4 It is noted that the fusion algebra just listed does not contain an identity. We will discuss this further in Section \[SectionDiscussion\]. Lattice Realization of ${\cal WLM}(2,3)$ {#SectionLattice} ======================================== In [@PRR08], we used the infinite series of logarithmic minimal lattice models ${\cal LM}(1,p)$ to obtain ${\cal W}$-extended fusion rules applicable in the extended pictures ${\cal WLM}(1,p)$. A crucial ingredient was the construction of a ${\cal W}$-invariant identity representation $\ketw{1,1}$ defined as the infinite limit of a triple fusion of Virasoro-irreducible Kac representations in ${\cal LM}(1,p)$. On the other hand, as indicated above and further discussed in Section \[SectionDiscussion\], there is no obvious natural candidate for an identity in the lattice realization of ${\cal WLM}(2,3)$. It nevertheless turns out fruitful to adopt the use of infinite limits of triple fusions of Virasoro-irreducible Kac representations. This also allows us to identify the various ${\cal W}$-representations with suitable limits of Yang-Baxter integrable boundary conditions on the lattice. Firmly based on the lattice-realization of the fundamental fusion algebra of ${\cal LM}(2,3)$, our fusion prescription for ${\cal WLM}(2,3)$ yields a commutative and associative fusion algebra. Horizontal component -------------------- Working in the [*fundamental*]{} fusion algebra of critical percolation ${\cal LM}(2,3)$, as opposed to the less understood but larger [*full*]{} fusion algebra [@RP0706; @RP0707], the only horizontal Kac representations at our disposal are $\{(2k,1);\ k\in\mathbb{N}\}$. It is noted that these are all Virasoro-irreducible representations. There are many possible triple fusions to consider of which the following one offers fairly straightforward access to the ${\cal W}$-extended horizontal component \_[n]{}(4n,1)\^[3]{} = \_[k]{}2k(2k,1)  = 2(\_[k]{}(2k-1)(4k-2,1)) 2(\_[k]{}2k(4k,1)) \[4n\] Indeed, we now assert that this limit corresponds to the following direct sum of four ${\cal W}$-indecomposable representations 22:= \_[n]{}(4n,1)\^[3]{} \[def4n\] whose decompositions in terms of Virasoro-irreducible representations read  = \_[k]{}(2k-1)(4k-2,1),  = \_[k]{}2k(4k,1) \[2141\] Since the participating Virasoro representations all are of rank 1, the ${\cal W}$-indecomposable representations $\ketw{2,1}$ and $\ketw{4,1}$ themselves are of rank 1. Without going into details, this separation or disentanglement of the triple fusion into four ${\cal W}$-indecomposable representations can be made manifest from the lattice by separating the set of link states accordingly. Since no non-trivial Jordan cells are formed between the representations on the right-hand side of (\[4n\]), selecting the link states associated to either $\ketw{2,1}$ or $\ketw{4,1}$ is a valid procedure. When non-trivial Jordan cells are involved, on the other hand, such a selection may affect the distribution and ranks of the cells and hence would not be valid. Having identified $\ketw{2,1}$ and $\ketw{4,1}$, we now define the ${\cal W}$-indecomposable rank-2 representations := (2,1), := (2,1) \[R2121\] Their decompositions into Virasoro-indecomposable rank-2 representations are given in (\[Rr2\]). Of importance for the evaluation of fusion products below, we note that the ${\cal W}$-indecomposable representations (\[2141\]) and (\[R2121\]) have the stability properties (4n,1) = 2n, (4n,1) = 4n4n and (2,1) = (2,1) =  22 As we will see in the following, there are many more such properties, but this list suffices for now. From the lattice, we define the ${\cal W}$-extended fusion product $\hat\otimes$ by {22}:=  \_[n]{}()\^3(4n,1)\^[3]{} and obtain the fusions given in Figure \[hor1\] where \_2 = , \_ =  $$\renewcommand{\arraystretch}{1.5} \begin{array}{c||cc|cc} \hat\otimes&\ketw{2,1}&\ketw{4,1}&\ketw{\R_{2,1}^{1,0}}&\ketw{\R_{4,1}^{1,0}} \\[4pt] \hline \hline \rule{0pt}{14pt} \ketw{2,1}&\ketw{\R_{2,1}^{1,0}}&\ketw{\R_{4,1}^{1,0}}&2\Ac_2&2\Ac_2 \\[4pt] \ketw{4,1}&\ketw{\R_{4,1}^{1,0}}&\ketw{\R_{2,1}^{1,0}}&2\Ac_2&2\Ac_2 \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{2,1}^{1,0}}&2\Ac_2&2\Ac_2&2\Ac_\R&2\Ac_\R \\[4pt] \ketw{\R_{4,1}^{1,0}}&2\Ac_2&2\Ac_2&2\Ac_\R&2\Ac_\R \\[4pt] \end{array}$$ To appreciate this, we consider the two cases $A=(2,1)$ and $A=(4,1)$ and find &&{}  = \_[n]{}()\^3(4n,1)\^[3]{} &=&\_[n]{}()\^2(4n,1)\^[2]{}  = \_[n]{}()(4n,1){} &=& \[214121\] and likewise {} =  \[214141\] We are still faced with the task of disentangling these results since the identification of the individual fusions such as $\ketw{2,1}\fus\ketw{2,1}$ is ambiguous at this point. However, since (4k-2,1)(4k’-2,1) = \_[j=|k-k’|+1]{}\^[k+k’-1]{}\_[4j-2,1]{}\^[1,0]{} and with the Virasoro decomposition of $\ketw{2,1}$ in (\[2141\]) in mind, it follows that the Virasoro decomposition of the fusion $\ketw{2,1}\fus\ketw{2,1}$ only involves rank-2 representations of the form $\ketw{\R_{4j-2,1}^{1,0}}$. Initially comparing this with (\[214121\]) and subsequently with (\[214141\]), we conclude that  =  = ,  =  In order to complete the Cayley table in Figure \[hor1\], we also need to evaluate fusions like  = (2,1)  = (2,1) = 22 and  = (2,1)()  = 22 The remaining fusions follow similarly. Additional representations are obtained by fusing the ones above by the simple vertical (Virasoro-indecomposable) Kac representations $(1,2)$ and $(1,3)$. We thus define the rank-1 representations := (1,s)  = \_[k]{}(2k-2+)(2(2k-2+),s),s\_[2,3]{} \[2s\] and the rank-2 representations := (1,s)  = \_[k]{}(2k-2+)\_[2(2k-2+),s]{}\^[1,0]{} ,s\_[2,3]{} \[R2s\] Having ventured into the bulk part of the Kac table, we note the stability properties (1,6n-3) = (2n-1),&& (1,6n) = 2n (1,6n-3) = (2n-1),&& (1,6n) = 2n Vertical component ------------------ The vertical component is developed and described in much the same way as the horizontal component above. From the lattice, we choose to consider \_[n]{}(1,6n-3)\^[3]{}&=&\_[k]{}(2k-1)(2k-1,1) &=&3(\_[k]{}(2k-1)(1,6k-3)) 2(\_[k]{}2k\_[1,6k]{}\^[0,1]{}) (\_[k]{}(2k-1)\_[1,6k-3]{}\^[0,2]{}) \[6nm3\] Care has to be taken when disentangling this result in order to identify the ${\cal W}$-extended representations involved. First, we observe that the conformal weights of the Virasoro representations in the first sum all have rational part $1/3$ while the Virasoro representations in the second and third sums all have integer conformal weights. This allows us to separate the first sum from the other two and we have  = \_[k]{}(2k-1)(1,6k-3) Now, fusing this with $(1,3)$ gives (1,3) = \_[k]{}(2k-1)((1,6k-3)\_[1,6k-3]{}\^[0,2]{}) Having separated $\ketw{1,3}$ from this, we naturally identify the remaining part of the sum as the ${\cal W}$-extended rank-2 representation  = \_[k]{}(2k-1)\_[1,6k-3]{}\^[0,2]{} The second sum in (\[6nm3\]) can now be isolated and is identified as the ${\cal W}$-extended rank-2 representation  = \_[k]{}2k\_[1,6k]{}\^[0,1]{} We thus assert that the limit of the triple fusion in (\[6nm3\]) corresponds to the following sum of 6 ${\cal W}$-indecomposable representations 32:= \_[n]{}(1,6n-3)\^[3]{} \[def6nm3\] Here we emphasize a difference between the horizontal and vertical components. In the horizontal case, we could perform the disentanglement in (\[def4n\]) explicitly from the lattice by choosing the set of link states appropriately. As already indicated in the discussion following (\[def4n\]) and (\[2141\]), this is not necessarily possible when non-trivial Jordan cells are present. One is faced with similar but more transparent complications in the Virasoro picture as well where the indecomposable rank-2 representations $\R_{1,3k}^{0,2}$ cannot be constructed individually from the lattice but only in combination with the Kac representations $(1,3k)$. To illustrate this, let us consider (1,3)(1,3) = (1,3)\_[1,3]{}\^[0,2]{}, (q) = \_[1,1]{}(q)+\_[1,5]{}(q) \[131313\] The Kac representations $(1,1)$, $(1,3)$ and $(1,5)$ are constructed by allowing exactly 0, 2 or 4 defects, respectively, to propagate through the bulk of the lattice, while the fusion $(1,3)\otimes(1,3)$ is evaluated by allowing 0, 2 or 4 defects to propagate through the bulk of the lattice. In the latter case, pairs of defects can be annihilated thus yielding a block-[*triangular*]{} matrix realization of the transfer fusion matrix. This block-triangularity may give rise to non-trivial Jordan cells as it does in the fusion $(1,3)\otimes(1,3)$. With reference to (\[131313\]), it is now tempting to regard the indecomposable representation $\R_{1,3}^{0,2}$ as the result of allowing 0 or 4 defects to propagate through the bulk. Since defects could be annihilated in quadruples, this would indeed give rise to a block-triangular matrix. However, it turns out that no non-trivial Jordan cells are formed in this case implying that this choice of boundary condition simply corresponds to the [*direct*]{} sum of the two indecomposable rank-1 representations $(1,1)$ and $(1,5)$. As already mentioned, this phenomenon carries over to the ${\cal W}$-extended picture where the limiting process, though, obscures the clarity of the Virasoro example just discussed. To continue, we could apply the analysis based on (\[6nm3\]) above to the infinite limit of the triple fusion of $(1,6n)$ with itself. Alternatively, we simply define the ${\cal W}$-extended rank-2 representation := (1,2) = \_[k]{}(2k-1)\_[1,6k-3]{}\^[0,1]{} and disentangle the fusions (1,2)&=&2(\_[k]{}2k(1,6k)) (\_[k]{}(2k-1)\_[1,6k-3]{}\^[0,1]{}) (1,2)&=&2(\_[k]{}2k(1,6k)) (\_[k]{}2k\_[1,6k]{}\^[0,2]{}) to identify  = \_[k]{}2k(1,6k) and subsequently  = \_[k]{}2k\_[1,6k]{}\^[0,2]{} We thus have the stability properties (1,2)  = 2 with further stability properties reading (1,6n-3) &=&(2n-1){} (1,6n-3) &=&2(2n-1){} In accordance with horizontal fusion, we use {32}&=& \_[n]{}()\^3(1,6n-3)\^[3]{} &=&\_[n]{}()\^3(1,6n-3)\^[3]{} when evaluating vertical fusions of ${\cal W}$-representations. With the abbreviations \_[3]{} = , \_[3]{} =  and in much the same way as for the horizontal component, this yields the fusion rules in Figure \[ver1a\]. $$\renewcommand{\arraystretch}{1.5} \begin{array}{c||cc|cc|cc} \hat\otimes&\ketw{1,3}&\ketw{1,6}&\ketw{\R_{1,6}^{0,1}} &\ketw{\R_{1,3}^{0,2}}&\ketw{\R_{1,3}^{0,1}}&\ketw{\R_{1,6}^{0,2}} \\[4pt] \hline \hline \rule{0pt}{14pt} \ketw{1,3}&\Ac_3&\Ac_6&2\Bc_3&2\Bc_3&2\Bc_6&2\Bc_6 \\[4pt] \ketw{1,6}&\Ac_6&\Ac_3&2\Bc_6&2\Bc_6&2\Bc_3&2\Bc_3 \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{1,6}^{0,1}}&2\Bc_3&2\Bc_6&2\Ac_3\oplus2\Bc_3&2\Ac_3\oplus2\Bc_3 &2\Ac_6\oplus2\Bc_6&2\Ac_6\oplus2\Bc_6 \\[4pt] \ketw{\R_{1,3}^{0,2}}&2\Bc_3&2\Bc_6&2\Ac_3\oplus2\Bc_3&2\Ac_3\oplus2\Bc_3 &2\Ac_6\oplus2\Bc_6&2\Ac_6\oplus2\Bc_6 \\[4pt] \hline \rule{0pt}{14pt} \ketw{\R_{1,3}^{0,1}}&2\Bc_6&2\Bc_3 &2\Ac_6\oplus2\Bc_6&2\Ac_6\oplus2\Bc_6&2\Ac_3\oplus2\Bc_3&2\Ac_3\oplus2\Bc_3 \\[4pt] \ketw{\R_{1,6}^{0,2}}&2\Bc_6&2\Bc_3 &2\Ac_6\oplus2\Bc_6&2\Ac_6\oplus2\Bc_6&2\Ac_3\oplus2\Bc_3&2\Ac_3\oplus2\Bc_3 \end{array}$$ Further following the analysis of the horizontal component, we introduce the rank-1 representations := (2,1)  = \_[k]{}(2k-2+)(2,3(2k-2+)) As required by consistency of notation, the representation $\ketw{2,3}$ defined in (\[2s\]) must agree with this expression, and indeed it does since \_[k]{}(2k-1)(4k-2,3) = \_[k]{}(2k-1)(2,6k-3) It is likewise noted that    since \_[k]{}2k(2,6k) = \_[k]{}2k(4k,3) We also introduce the rank-2 representations := (2,1)  = \_[k]{}(2k-2+)\_[2,3(2k-2+)]{}\^[0,b]{} Combination of the two components --------------------------------- Our notation implies that AB Particularly useful such relations are part of the stability properties (1,3)&=& = (2,1) (1,3)&=& = (4,1) (1,6)&=& = (2,1) (1,6)&=& = (4,1) \[2113\] and \_[1,3]{}\^[0,b]{}&=&  = (2,1) \_[1,3]{}\^[0,b]{}&=&  = (4,1) \_[1,6]{}\^[0,b]{}&=&  = (2,1) \_[1,6]{}\^[0,b]{}&=&  = (4,1) \[21R\] To illustrate the derivation of these, we assume (\[2113\]) when considering the first equality in the third line in (\[21R\]) \_[1,6]{}\^[0,1]{}&=&(1,6)(1,2)  = 2(2,1)(1,2)  = 2(2,1) &=&2 The ${\cal W}$-indecomposable rank-3 representations can be defined by := \_[1,3]{}\^[0,b]{} or equivalently through  =   = \_[2,1]{}\^[1,0]{} They have the stability properties (1,2)  = 2 To get started with the evaluation of combined fusions, we consider && {32}  = \_[n]{}()\^3(1,6n-3)\^[3]{} &=&\_[n]{}()\^2(1,6n-3)\^[2]{}  = \_[n]{}()\^2(1,6n-3)\^[2]{}(1,3) &=&(1,3)\^[3]{}  = {3(1,3)2\_[1,6]{}\^[0,1]{}\_[1,3]{}\^[0,2]{}} &=&32 Since the multiplicities appearing in the decomposition of the fusion $\ketw{2,1}\fus\{3\ketw{1,3}\}$ must be divisible by 3, we find that  =  Using a similar argument, we then deduce that  =  and finally  =  We subsequently find  = (1,2) = (1,2) =  and hence &=&{(1,2) }  = (1,2) &=&(2,1){2} &=& The remaining fusions follow similarly or by simple applications of commutativity and associativity. Indeed, in our final example, we assume that all fusions but the ones between two rank-3 representations have been examined. Thus using commutativity, associativity and the fusion rules appearing in Figure \[Cayleyr1r1\] through Figure \[Cayleyr2r23b\], we consider the fusion &=&  = {\_[4,3]{}\^[1,1]{}\_[2,3]{}\^[1,2]{}} &=&{42 2} &=&4\_3\^[1,0]{}2\^[1,1]{}2\^[1,2]{} which is recognized as $\hat\Cc^1$, cf. Figure \[Cayleyr3r3\]. Self-consistency of our fusion prescription requires that the evaluation of a given fusion product based on (\[def4n\]) must yield the same result as the evaluation of the same fusion product based on (\[6nm3\]), when both methods are applicable. This can be verified explicitly and stems from the fact that the stability properties (\[2113\]) and (\[21R\]) ensure that &&\_[n]{}()\^3(4n,1)\^[3]{} {32} &=&{22} {32} &=& \_[n]{}()\^3{22} (1,6n-3)\^[3]{} Fusion subalgebras ------------------ It is noted that there are many fusion subalgebras. We have already encountered two of them, namely the horizontal and vertical fusion algebras whose Cayley tables are given in Figure \[hor1\] and Figure \[ver1a\], respectively. A noteworthy six-dimensional fusion subalgebra is (,1),(,1),(1,),(1,)  = (,1),(,1),(1,),(1,),(,),(,)\[EO\] It is generated by the four ${\cal W}$-representations (,1):= \_[n]{}(4n,1)\^[3]{} = 2\_2, (,1):= (2,1)(,1) = \_ \[E1\] where it is noted that $\lim_{n\to\infty}(4n-2,1)^{\otimes3}=\lim_{n\to\infty}(4n,1)^{\otimes3}$, and (1,):= \_[n]{}(1,6n)\^[3]{} = \_62\_6, (1,):= \_[n]{}(1,6n-3)\^[3]{} = \_32\_3 \[1E\] The remaining two representations are defined by &&(,):= (,1)(1,) =  \_[\_[1,2]{},b\_[0,2]{}]{}(6-2b) &&(,):= (,1)(1,) =  \_[\_[1,2]{},b\_[0,2]{}]{}(3-b) where $\ketw{\R_{2,3\kappa}^{0,0}}\equiv\ketw{2,3\kappa}$, and are seen to arise also in the fusions (,1)(1,) = (,),(,1)(1,) = (,) The Cayley table of the complete fusion subalgebra (\[EO\]) is given in Figure \[CayleyEO\]. A virtue of this fusion subalgebra is that it does [*not*]{} rely on any disentangling procedure. $$\renewcommand{\arraystretch}{1.5} \begin{array}{c||cc|cc|cc} \hat\otimes&(\Ec,1)&(\Oc,1)&(1,\Ec)&(1,\Oc)&(\Ec,\Ec)&(\Oc,\Oc)\\[4pt] \hline \hline \rule{0pt}{14pt} (\Ec,1)&8(\Oc,1)&8(\Ec,1)&(\Ec,\Ec)&(\Ec,\Ec)&8(\Oc,\Oc)&8(\Ec,\Ec)\\[4pt] (\Oc,1)&8(\Ec,1)&8(\Oc,1)&(\Oc,\Oc)&(\Oc,\Oc)&8(\Ec,\Ec)&8(\Oc,\Oc)\\[4pt] \hline \rule{0pt}{14pt} (1,\Ec)&(\Ec,\Ec)&(\Oc,\Oc)&27(1,\Oc)&27(1,\Ec)&27(\Ec,\Ec)&27(\Oc,\Oc)\\[4pt] (1,\Oc)&(\Ec,\Ec)&(\Oc,\Oc)&27(1,\Ec)&27(1,\Oc)&27(\Ec,\Ec)&27(\Oc,\Oc)\\[4pt] \hline \rule{0pt}{14pt} (\Ec,\Ec)&8(\Oc,\Oc)&8(\Ec,\Ec)&27(\Ec,\Ec)&27(\Ec,\Ec)&216(\Oc,\Oc)&216(\Ec,\Ec)\\[4pt] (\Oc,\Oc)&8(\Ec,\Ec)&8(\Oc,\Oc)&27(\Oc,\Oc)&27(\Oc,\Oc)&216(\Ec,\Ec)&216(\Oc,\Oc) \end{array}$$ Discussion {#SectionDiscussion} ========== Two-dimensional critical percolation, with central charge $c=0$, is viewed as the member ${\cal LM}(2,3)$ of the infinite series of Yang-Baxter integrable logarithmic minimal models ${\cal LM}(p,p')$ [@PRZ]. As in the rational case [@BP01], the Yang-Baxter integrable boundary conditions give insight into the conformal boundary conditions [@BPPZ00] in the continuum scaling limit as well as the fusion of their associated representations. This enabled us in [@PRZ] to construct integrable boundary conditions labelled by $(r,s)$ and corresponding to so-called Kac representations with conformal weights in an infinitely extended Kac table (Figure \[KacTable\]). Moreover, from the lattice implementation of fusion, we obtained [@RP0706] the closed fusion algebra generated by these Kac representations finding that indecomposable representations of ranks 1, 2 and 3 are generated by the fusion process. Although there is a countable infinity of representations, the ensuing fusion rules are quasi-rational in the sense of Nahm [@Nahm94], that is, the fusion of any two representations decomposes into a finite sum of representations. This is the relevant picture in the case where the conformal algebra is the Virasoro algebra. Of course, there is no claim, in the context of this logarithmic CFT, that the representations generated in this picture exhaust all of the representations associated with conformal boundary conditions. This picture is in stark contrast to the context of rational CFTs where all representations decompose into direct sums of a finite number of irreducible representations. In this paper, we have reconsidered critical percolation (or more precisely the ${\cal LM}(2,3)$ lattice model) in the continuum scaling limit to expose its nature as a ‘rational’ logarithmic CFT with respect to the extended conformal algebra ${\cal W}={\cal W}_{2,3}$ [@FGST06b]. Under the extended symmetry, the infinity of Virasoro representations are reorganized into a finite number of ${\cal W}$-representations. Following the approach of [@PRR08], we construct new solutions of the boundary Yang-Baxter equation which, in a suitable limit, correspond to these representations. Specifically, with respect to a suitably defined ${\cal W}$-fusion, we find that the representation content of the ensuing closed fusion algebra is [*finite*]{} containing 26 ${\cal W}$-indecomposable representations with 8 rank-1 representations, 14 rank-2 representations and 4 rank-3 representations. We have also identified their associated ${\cal W}$-extended characters which decompose as finite non-negative sums of 13 ${\cal W}$-irreducible characters. Implementation of fusion on the lattice has allowed us to read off the fusion rules governing the fusion algebra of the 26 representations and to construct an explicit Cayley table. The closure of these representations among themselves under fusion is remarkable confirmation of the proposed extended symmetry. A somewhat surprising feature of our closed ${\cal W}$-extended fusion algebra of ${\cal WLM}(2,3)$ is that there appears to be no natural identity $\Ic_{\cal W}$ expressed in terms of the fundamental Virasoro fusion algebra and with respect to the fusion multiplication $\hat\otimes$. Since the Kac representation $(1,1)$ is the identity of the fundamental fusion algebra itself, it may be tempting to include it in the spectrum and identify it with $\Ic_{\cal W}$. However, we have {22}\_[W]{} = \_[n]{}()\^3(4n,1)\^[3]{}(1,1) = 0 demonstrating that this simple extension fails. We find it natural, though, to expect that one can extend our fusion algebra of ${\cal WLM}(2,3)$ by working with the [*full*]{} Virasoro fusion algebra. We hope to discuss this and re-address the identity question elsewhere. .5cm Acknowledgments {#acknowledgments .unnumbered} =============== .1cm This work is supported by the Australian Research Council (ARC). 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--- abstract: 'A real $2$-elementary K3 surfaces of type $((3,1,1),- {\mathrm{id}})$ yields a real anti-bicanonical curve $s \cup A^\prime_1$ (disjoint union) on the $4$-th real Hirzebruch surface ${\mathbb{F}}_4$ where $s$ is the exceptional section of ${\mathbb{F}}_4$ and the real curve $A^\prime_1$ has one real double point. (See Section \[RealK3-311\] below.) We give a criterion (Proposition \[criterion\]) which determines whether the real double point is degenerate or not. One direction of the assertion of Proposition \[criterion\] has already been proved in Lemma 4.6 in the preceding paper [@SaitoSachiko2015]. In this paper we prove the inverse direction.' address: | Department of Mathematics Education, Asahikawa Campus,\ Hokkaido University of Education, Asahikawa, JAPAN author: - Sachiko Saito title: 'On real anti-bicanonical curves with one double point on the $4$-th real Hirzebruch surface. II' --- Introduction:  Review of real $2$-elementary K3 surfaces {#real_2-elementary K3} ======================================================== Real $2$-elementary K3 surfaces ------------------------------- In this paper we mainly discuss K3 surfaces $X$ with a non-symplectic holomorphic involution $\tau$. We often call them [*$2$-elementary K3 surfaces*]{} $(X,\tau)$ ([@Nikulin81], [@AlexeevNikulin2006], [@NikulinSaito05], [@NikulinSaito07], [@SaitoSachiko2015], and e.t.c.). Note that every K3 surface with a non-symplectic holomorphic involution is algebraic. Hence, it has hyperplane sections. We say that a triple $(X,\tau,\varphi)$ is a [*real*]{} K3 surface with non-symplectic holomorphic involution (or [*real*]{} $2$-elementary K3 surface) if\   [(1)]{} $(X,\tau)$ is a K3 surface $X$ with a non-symplectic holomorphic involution $\tau$,\   [(2)]{} $\varphi$ is an anti-holomorphic involution on $X$, and\   [(3)]{} $\varphi \circ \tau = \tau \circ \varphi$. For a $2$-elementary K3 surface $(X,\tau)$, let $${H_2}_+(X, {\mathbb{Z}})$$ denote the fixed part of $\tau_* : H_2(X, {\mathbb{Z}}) \to H_2(X, {\mathbb{Z}})$. It is well-known that $H_2(X, {\mathbb{Z}})$ is an even unimodular lattice of signature $(3,19)$. ${H_2}_+(X, {\mathbb{Z}})$ is a primitive hyperbolic $2$-elementary sublattice of $H_2(X, {\mathbb{Z}})$. Note that $${H_2}_+(X, {\mathbb{Z}}) \subset {\mathop\mathrm{Pic}}(X),$$ where ${\mathop\mathrm{Pic}}(X)$ denotes the Picard lattice of $X$. ——————————————————————————————————————————– [ partially supported by JSPS Grant-in-Aid for Challenging Exploratory Research 25610001 (2013/4 — 2016/3).\ [*2010 AMS Mathematics Subject Classification*]{}:  14J28, 14P25, 14J10.]{} Let ${\mathbb{L}}_{K3}$ be an even unimodular lattice of signature $(3,19)$ and fix it. Note that the isometry class of ${\mathbb{L}}_{K3}$ is unique. Let $$S \ \ (\subset {\mathbb{L}}_{K3})$$ be a primitive hyperbolic $2$-elementary sublattice of ${\mathbb{L}}_{K3}$. We set $r(S) := {\mathop\mathrm{rank}}S$.   The non-negative integer $a(S)$ is defined by $S^\ast /S \cong ({\mathbb{Z}}/2{\mathbb{Z}})^{a(S)}$. We define the “parity" $\delta (S)$ of $S$ as follows. $$\delta (S) := \left\{ \begin{array}{cl} 0 &\ \ \ \mbox{if}\ z \cdot \sigma (z) \equiv 0 {\ \mathrm{mod}\ }2 \ \ (\forall z \in {\mathbb{L}}_{K3})\\ 1 &\ \ \ \mbox{otherwise,} \end{array} \right.$$ where $\sigma : {\mathbb{L}}_{K3} \to {\mathbb{L}}_{K3}$ is the unique integral involution whose fixed part is $S$. It is known that the triplet $(r(S),a(S),\delta(S))$ determines the isometry class of the lattice $S$ ([@Nikulin81]). Moreover, if $S$ and $S^\prime$ are isometric primitive hyperbolic $2$-elementary sublattices of the K3 lattice ${\mathbb{L}}_{K3}$, then there exists an ambient automorphism $f$ of ${\mathbb{L}}_{K3}$ such that $f(S^\prime) = S$ ([@AlexeevNikulin2006], [@Nikulin79]). We fix a half cone $$V^+(S)$$ of the cone $$V(S):= \{ x \in S \otimes {\mathbb{R}}\ |\ x^2 > 0\}.$$ Moreover, we fix a fundamental subdivision $$\Delta(S)=\Delta(S)_+\cup -\Delta(S)_+$$ of all elements with square $-2$ in $S$. This is equivalent to fixing a fundamental (closed) chamber (see [@NikulinSaito05]) $${\mathcal{M}}\ \ \ \ (\subset V^+(S))$$ for the group $W^{(-2)}(S)$ generated by reflections in all elements with square $(-2)$ in $S$. Note that ${\mathcal{M}}$ and $\Delta(S)_+$ define each other by the condition ${\mathcal{M}}\cdot \Delta(S)_+ \ge 0$. Let $\theta$ be an integral involution of $S$. \[real\_2-elementary K3\_S\_theta\] We say that $(X,\tau,\varphi)$ is a real $2$-elementary K3 surface [*of type $(S,\theta)$*]{} if there exists an isometry (so-called “marking" later) $$\alpha : H_2(X, {\mathbb{Z}}) \cong {\mathbb{L}}_{K3}$$ such that $\alpha({H_2}_+(X, {\mathbb{Z}})) = S$ and the following diagram commutes: $$\begin{CD} {H_2}_+(X, {\mathbb{Z}}) @> {\alpha}>>S\\ @V{\varphi_*}VV @VV{\theta}V\\ {H_2}_+(X, {\mathbb{Z}}) @> {\alpha}>>S . \end{CD}$$ \[marked\_real\_K3\] We define that a [*marked real $2$-elementary K3 surface of type $(S,\theta)$*]{} is a pair $$((X,\tau,\varphi),\ \alpha)$$ of a real $2$-elementary K3 surface $(X,\tau,\varphi)$ of type $(S,\theta)$ (Definition \[real\_2-elementary K3\_S\_theta\] above) and an isometry, which is called [*marking*]{}, $$\alpha : H_2(X, {\mathbb{Z}}) \cong {\mathbb{L}}_{K3}$$ such that - $\alpha({H_2}_+(X, {\mathbb{Z}})) = S$,\ - $\alpha \circ \varphi_* = \theta \circ \alpha \ \ \ \text{on} \ {H_2}_+(X, {\mathbb{Z}})$,\ - $\alpha_{{\mathbb{R}}}^{-1}(V^+(S))$ contains a hyperplane section of $X$, where $\alpha_{{\mathbb{R}}}$ stands for the real extension of $\alpha$, and\ - the set $\alpha^{-1}(\Delta(S)_+)$ contains only effective classes of $X$. Note that ([@NikulinSaito05]) for any $(X,\tau)$, we can take $\alpha$ such that $\alpha_{{\mathbb{R}}}^{-1}(V^+(S))$ contains a hyperplane section of $X$. Integral involutions of ${\mathbb{L}}_{K3}$ of type $(S,\theta)$ ---------------------------------------------------------------- Let $S$ be a hyperbolic $2$-elementary sublattice of ${\mathbb{L}}_{K3}$ and $\theta : S \to S$ be an integral involution (as above). Let $\psi : {\mathbb{L}}_{K3} \to {\mathbb{L}}_{K3}$ be an integral involution of the lattice ${\mathbb{L}}_{K3}$ such that the following diagram commutes: $$\begin{array}{rcl} S & \subset & {\mathbb{L}}_{K3} \\ \theta \ \downarrow & & \downarrow \ \psi \\ S & \subset & {\mathbb{L}}_{K3} . \end{array}$$ We call such a pair $({\mathbb{L}}_{K3},\psi)$ (or $\psi$ itself) an [*integral involution of ${\mathbb{L}}_{K3}$ of type $(S,\theta)$*]{}. Let $((X,\tau,\varphi),\ \alpha)$ be a marked real $2$-elementary K3 surface of type $(S,\theta)$ as above. If we set $$\psi := \alpha \circ \varphi_* \circ \alpha^{-1} : {\mathbb{L}}_{K3} \to {\mathbb{L}}_{K3},$$ then we have $\psi(S) = S$, and $\psi (x) = \theta (x)$ for every $x \in S$. Hence, $({\mathbb{L}}_{K3},\psi)$ is an integral involution of ${\mathbb{L}}_{K3}$ of type $(S,\theta)$. We call the integral involution $({\mathbb{L}}_{K3},\psi)$ of type $(S,\theta)$ [*the associated integral involution*]{} with a marked real $2$-elementary K3 surface $((X,\tau,\varphi),\ \alpha)$ of type $(S,\theta)$ if the following diagram commutes: $$\begin{CD} H_2(X, {\mathbb{Z}}) @>{\alpha}>>{\mathbb{L}}_{K3} \\ @V{\varphi_*}VV @VV{\psi}V\\ H_2(X, {\mathbb{Z}}) @>{\alpha}>>{\mathbb{L}}_{K3} . \end{CD}$$ Let $\Delta(S,L)^{(-4)}$ be the set of all elements $\delta_1$ in $S$ such that $\delta_1^2=-4$ and there exists $\delta_2\in S^\perp_L$ such that $(\delta_2)^2=-4$ and $\delta=(\delta_1+\delta_2)/2\in L$. Let $W^{(-4)}(S,L)$ be the subgroup of $O(S)$ generated by reflections in all elements in $\Delta(S,L)^{(-4)}$, and $W^{(-4)}(S,L)_{\mathcal{M}}$ be the stabilizer subgroup of ${\mathcal{M}}$ in $W^{(-4)}(S,L)$. We define the subgroup $G$ to be generated by reflections $s_{\delta_1}$ in all elements $\delta_1\in \Delta(S,L)^{(-4)}$ which are contained either in $S_+$ or in $S_-$ and satisfy $(s_{\delta_1})_{{\mathbb{R}}}({\mathcal{M}})={\mathcal{M}}$, where $s_{\delta_1}$ denotes the reflection at the orthogonal hyperplane ${\delta_1}^{\perp}$ on $S$, $(s_{\delta_1})_{{\mathbb{R}}}$ stands for the real extension of $s_{\delta_1}$, and we set $S_{\pm} := \{ x\in S \,|\, \theta(x) = \pm x \}$. Then $G$ is a subgroup of $W^{(-4)}(S,L)_{\mathcal{M}}$. Let $({\mathbb{L}}_{K3},\psi_1)$ and $({\mathbb{L}}_{K3},\psi_2)$ be two integral involutions of ${\mathbb{L}}_{K3}$ of type $(S,\theta)$. We define that an [*isometry with respect to the group*]{} $G$ from $({\mathbb{L}}_{K3},\psi_1)$ to $({\mathbb{L}}_{K3},\psi_2)$ is an isometry $f: {\mathbb{L}}_{K3} \to {\mathbb{L}}_{K3}$ such that $f(S)=S$, ${f|}_S \in G$, and the following diagram commutes: $$\begin{CD} {\mathbb{L}}_{K3} @>{f}>>{\mathbb{L}}_{K3} \\ @VV\psi_1 V@V\psi_2 VV\\ {\mathbb{L}}_{K3} @>{f}>>{\mathbb{L}}_{K3} . \end{CD}$$ We say that two integral involutions $({\mathbb{L}}_{K3},\psi_1)$ and $({\mathbb{L}}_{K3},\psi_2)$ of type $(S,\theta)$ are [*isometric with respect to the group $G$*]{} if there exists an isometry with respect to the group $G$ from $({\mathbb{L}}_{K3},\psi_1)$ to $({\mathbb{L}}_{K3},\psi_2)$. By an [*automorphism*]{} of an integral involution $({\mathbb{L}}_{K3},\psi)$ of type $(S,\theta)$ [*with respect to the group $G$*]{} we mean an isometry with respect to the group $G$ from $({\mathbb{L}}_{K3},\psi)$ to itself. Namely, an isometry $f: {\mathbb{L}}_{K3} \to {\mathbb{L}}_{K3}$ which satisfies that $\psi \circ f = f \circ \psi,\ f(S)=S \ \text{and} \ {f|}_S \in G$. \[analytic-iso\] We say that two marked real $2$-elementary K3 surfaces $((X,\tau,\varphi),\ \alpha)$ and $((X^\prime,\tau^\prime,\varphi^\prime),\alpha^\prime)$ of type $(S,\theta)$ are [*analytically isomorphic with respect to the group*]{} $G$ if there exists an analytic isomorphism $f : X \to X^\prime$ such that $f \circ \tau = \tau^\prime \circ f$, $f \circ \varphi = \varphi^\prime \circ f$ and $\alpha^\prime \circ f_* \circ \alpha^{-1}|S \in G $. Period domains {#period domains section} -------------- Now let us [**fix**]{} an integral involution $$({\mathbb{L}}_{K3},\psi)$$ of type $(S,\theta)$ throughout this subsection. We follow the formulations of period domains of marked real $2$-elementary K3 surfaces (see Itenberg [@Itenberg92] and Nikulin-Saito [@NikulinSaito05]). We set $$\Omega_\psi := \{ \omega \ (\in {\mathbb{L}}_{K3}\otimes {\mathbb{C}}) \ |\ \omega \cdot \omega =0, \ \omega \cdot \overline{\omega} >0, \ \omega \cdot S =0, \ \psi_{{\mathbb{C}}}(\omega)=\overline{\omega} \}/{\mathbb{R}}^{\times} .$$ Let $((X,\tau,\varphi),\ \alpha)$ be a marked real $2$-elementary K3 surface of type $(S,\theta)$ satisfying $$\alpha \circ \varphi_* \circ \alpha^{-1} = \psi , \footnote{All marked real $2$-elementary K3 surfaces whose associated integral involutions are \underline{isometric} to $({\mathbb{L}}_{K3},\psi)$ with respect to $G$ satisfy $\alpha \circ \varphi_* \circ \alpha^{-1} = \psi$ if we change their markings appropriately (see \cite{SaitoSachiko2015}).}$$ and let $H \ \ (\subset H_2(X,{\mathbb{C}}))$ be the Poincare dual of $H^{2,0}(X)$. The $1$-dimensional subspace $\alpha_{{\mathbb{C}}}(H)$ of ${\mathbb{L}}_{K3}\otimes {\mathbb{C}}$ is regarded as an element of $\Omega_\psi$. We call $\alpha_{{\mathbb{C}}}(H)$ the [*period*]{} of a marked real $2$-elementary K3 surface $((X,\tau,\varphi),\ \alpha)$ of type $(S,\theta)$ satisfying $\alpha \circ \varphi_* \circ \alpha^{-1} = \psi$. We say that a point $[\omega] \ (\in \Omega_\psi)$ is [*equivalent*]{} to a point $[\omega^\prime] \ (\in \Omega_\psi)$ if $[\omega^\prime] = f_{{\mathbb{C}}}([\omega])$ for an automorphism $f$ of $({\mathbb{L}}_{K3},\psi)$ of type $(S,\theta)$ with respect to the group $G$. \[equivalence-lemma\] If a point $[\omega] \ (\in \Omega_\psi)$ is equivalent to $[\omega^\prime] \ (\in \Omega_\psi)$ and $[\omega]$ is the period of some marked real $2$-elementary K3 surface $((X,\tau,\varphi),\ \alpha)$ of type $(S,\theta)$ satisfying $\alpha \circ \varphi_* \circ \alpha^{-1} = \psi$, then $[\omega^\prime]$ is also the period of a marked real $2$-elementary K3 surface $((X,\tau,\varphi),\ \alpha^\prime)$ of type $(S,\theta)$ satisfying $(\alpha^\prime) \circ \varphi_* \circ (\alpha^\prime)^{-1} = \psi$ where $\alpha^\prime$ is another marking of $(X,\tau,\varphi)$. Using the global Torelli theorem, if two periods are equivalent, then corresponding marked real $2$-elementary K3 surfaces are analytically isomorphic (see Definition \[analytic-iso\]). The converse is also true. The domain $\Omega_\psi$ has two connected components which are interchanged by $-\psi$. Since $-\psi$ is an automorphism of $({\mathbb{L}}_{K3},\psi)$ with respect to the group $G$, by Lemma \[equivalence-lemma\], it is enough to investigate the quotient space $$\Omega_\psi /-\psi .$$ Now we set $${\mathbb{L}}^\psi := \{ x \in {\mathbb{L}}_{K3}\ |\ \psi (x) = x \},\ \ {\mathbb{L}}_\psi := \{ x \in {\mathbb{L}}_{K3}\ |\ \psi (x) = - x \} .$$ We restrict ourselves to the case $$S \subset {\mathbb{L}}_\psi,\ \ \text{i.e.,}\ \ \theta = - {\mathrm{id}},$$ where “${\mathrm{id}}$" stands for the identity map on $S$. We set $${\mathbb{L}}_{-,S} := {\mathbb{L}}_\psi \cap S^{\perp}.$$ For $[\omega] \in \Omega_\psi$  ($\omega \in {\mathbb{L}}_{K3} \otimes {\mathbb{C}}$), we consider the orthogonal decomposition $\omega = \omega_+ + i\,\omega_- \ \ (\omega_+ \in {\mathbb{L}}^\psi \otimes {\mathbb{R}},\ \omega_- \in {\mathbb{L}}_\psi \otimes {\mathbb{R}})$. Then we have $\omega_- \in {\mathbb{L}}_{-,S}\otimes {\mathbb{R}}$ and $\omega_+^2 = \omega_-^2 > 0$. Hence, both ${\mathbb{L}}^\psi$ and ${\mathbb{L}}_{-,S}$ are hyperbolic lattices. We set $V({\mathbb{L}}^\psi):= \{ x \in {\mathbb{L}}^\psi \otimes {\mathbb{R}}\ |\ x^2 > 0\}$. $V({\mathbb{L}}^\psi)$ has two connected components. Let $V^+({\mathbb{L}}^\psi)$ be one of those (half cone). Let ${\mathcal{L}}_+$ denote the set of all rays (half lines) through ${\mathbf{0}}$ in $V^+({\mathbb{L}}^\psi)$. (${\mathcal{L}}_+$ is called the [*hyperbolic*]{} (or [*Lobachevsky*]{}) space obtained from ${\mathbb{L}}^\psi$. ) We define the hyperbolic space ${\mathcal{L}}_{-,S}$ obtained from ${\mathbb{L}}_{-,S}$ in the same way. Then we have the following identification: $$\label{domain-quotient} \Omega_\psi /-\psi \ = \ {\mathcal{L}}_+ \times {\mathcal{L}}_{-,S}\ \ \ \ (\mbox{a direct product}).$$ (${\mathcal{D}}{\mathbb{R}}$)-nondegenerate marked real $2$-elementary K3 surfaces ---------------------------------------------------------------------------------- We say that an element $x (\neq 0) \in {\mathop\mathrm{Pic}}(X)\otimes {\mathbb{R}}$ is [*nef*]{} (for $X$) if $x \cdot C \ge 0$ for every effective curve $C$ in ${\mathop\mathrm{Pic}}(X)$. \[da-degenerate\] [**(i)**]{} We say that a marked $2$-elementary K3 surface $((X,\tau),\ \alpha)$ of type $S$ is [*(${\mathcal{D}}$)-degenerate*]{} if there exists an element $x_0 \in \alpha_{{\mathbb{R}}}^{-1}({\mathcal{M}})$ which is [**not**]{} nef. Namely, $x_0 \cdot C < 0$ for an effective curve $C$ in ${\mathop\mathrm{Pic}}(X)$. This condition is equivalent (see [@Nikulin86], [@AlexeevNikulin2006]) to the existence of an irreducible $(-2)$-curve on the quotient surface $Y:=X/\tau$. And this condition is also equivalent to the existence of an element $\delta \in {\mathop\mathrm{Pic}}(X)$ with $\delta^2=-2$ such that $\delta=(\delta_1+\delta_2)/2$ where $\delta_1\in \alpha^{-1}(S), \ \ \delta_2\in \alpha^{-1}(S)^\perp_{{\mathop\mathrm{Pic}}(X)}, \ \ \text{and} \ \ \delta_1^2=\delta_2^2 = -4$. [**(ii)**]{} We say that a marked real $2$-elementary K3 surface $((X,\tau,\varphi),\ \alpha)$ of type $(S,\theta)$ is [*(${\mathcal{D}}{\mathbb{R}}$)-degenerate*]{} if there exists a “real" element $x_0 \in \alpha_{{\mathbb{R}}}^{-1}(S_- \cap {\mathcal{M}})$ which is [**not**]{} nef, where we set $S_{\pm} := \{ x\in S \,|\, \theta(x) = \pm x \}$. This condition is equivalent to the existence of an element $\delta \in {\mathop\mathrm{Pic}}(X)$ with $\delta^2=-2$ such that $\delta=(\delta_1+\delta_2)/2$ where $\delta_1\in \alpha^{-1}(S),\ \ \delta_2\in \alpha^{-1}(S)^\perp_{{\mathop\mathrm{Pic}}(X)}\ \ \text{and} \ \ \delta_1^2=\delta_2^2=-4,$ and $\delta_1$ is orthogonal to an element $x \in \alpha_{{\mathbb{R}}}^{-1}(S_-\cap \text{int}({\mathcal{M}}))$, where $\text{int}({\mathcal{M}})$ denote the interior part of ${\mathcal{M}}$, i.e., the polyhedron ${\mathcal{M}}$ without its faces. Considering associated integral involutions, we have: \[theorem2005moduli\] The natural map gives a bijective correspondence between the connected components of the period domain of (${\mathcal{D}}{\mathbb{R}}$)-nondegenerate marked real $2$-elementary K3 surfaces of type $(S,\theta)$ and the isometry classes with respect to $G$ of integral involutions of ${\mathbb{L}}_{K3}$ of type $(S,\theta)$ such that the fixed part ${\mathbb{L}}^\psi$ of $\psi$ is hyperbolic. Real $2$-elementary K3 surfaces of type $((3,1,1),- {\mathrm{id}})$ {#RealK3-311} =================================================================== Now we fix a sublattice $S$ of the K3 lattice ${\mathbb{L}}_{K3}$ with the invariants $$(r(S),a(S),\delta(S)) = (3,1,1).$$ We consider $2$-elementary K3 surfaces of type $S \ (\cong (3,1,1))$ ([@NikulinSaito07],[@SaitoSachiko2015]). We quote the following results from Alexeev and Nikulin [@AlexeevNikulin2006]. See also [@NikulinSaito07] and [@SaitoSachiko2015]. Let $(X,\tau)$ be a $2$-elementary K3 surface of type $S \cong (3,1,1)$. Let $A := X^{\tau}$ be the fixed point set (nonsingular complex curve) of $\tau$. Then we have $$A = A_0 \cup A_1\ \ \mbox{(disjoint union)},$$ where $A_0$ is a nonsingular rational curve ($\cong {\mathbb{P}}^1$), and  $A_1$ is a nonsingular curve of genus $9$. $(X,\tau)$ has a structure of an elliptic pencil $|E+F|$, and $\tau$ is the inversion map of the group structure of the elliptic pencil with the zero section $A_0$. The unique reducible fiber $E+F$ having the following properties: (i) : $E$ is a nonsingular rational curve ($\cong {\mathbb{P}}^1$) and $E\cdot A_0=1$. (ii) : $E\cdot F=2$,  $F^2=-2$,  $F\cdot A_0=0$,  and   $F$ is either\ a nonsingular rational curve (“[*type IIa case*]{}"), or\ the union of two nonsingular rational curves $F^\prime$ and $F^{\prime\prime}$ which are conjugate by $\tau$, $F^\prime \cdot F^{\prime\prime} = 1$ (“[*type IIb case*]{}"). (iii) : The classes $[A_0]$, $[E]$ and $[F]$ generate the lattice ${H_2}_+(X, {\mathbb{Z}}) \ (\cong S)$. Moreover, $A_1 \cdot E = 1,\ \ A_1 \cdot F = 2$. The Gram matrix of the lattice ${H_2}_+(X, {\mathbb{Z}})$ with respect to the basis $[E],\ [F]$ and $[A_0]$ is $$\begin{array}{crrr} & {[E]} & {[F]} & {[A_0]} \\ {[E]} & -2 & 2 & 1 \\ {[F]} & 2 & -2 & 0 \\ {[A_0]} & 1 & 0 & -2 . \end{array}$$ Then we have an orthogonal decomposition $${H_2}_+(X, {\mathbb{Z}}) = {\mathbb{Z}}([A_0],\ [E]+[F]) \oplus {\mathbb{Z}}([F]) ,$$ where the subgroups ${\mathbb{Z}}([A_0],\ [E]+[F])$ and ${\mathbb{Z}}([F])$ are isometric to the hyperbolic plane and $\langle -2 \rangle$ respectively. We now consider the quotient surface $Y := X/\tau$ (so-called “DPN surface" ([@NikulinSaito05])) and let $\pi : X \to Y$ be the quotient map. We define the curves on $Y$ as follows: $$e:=\pi(E) \ \ \ \text{and} \ \ \ f:= \pi(F) .$$ If $F$ is the union of two nonsingular rational curves $F^\prime$ and $F^{\prime\prime}$ which are conjugate by $\tau$ and $F^\prime \cdot F^{\prime\prime} = 1$, then we have $f = \pi(F) = \pi(F^\prime \cup F^{\prime\prime}) = \pi(F^\prime) = \pi(F^{\prime\prime}).$ We use the same symbols $A_0$ and $A_1$ for their images in $Y$ by $\pi$. Then, the Picard group ${\mathop\mathrm{Pic}}(Y)$ of $Y$ is generated by the classes $[e]$, $[f]$ and $[A_0]$. The Gram matrix of ${\mathop\mathrm{Pic}}(Y)$ with respect to the basis $[e],\ [f]$ and $[A_0]$ is $$\begin{array}{crrr} & {[e]} & {[f]} & {[A_0]} \\ {[e]} & -1 & 1 & 1 \\ {[f]} & 1 & -1 & 0 \\ {[A_0]} & 1 & 0 & -4 . \end{array}$$ We have $A_1\cdot e = 1 \ \ \ \mbox{and} \ \ \ A_1\cdot f = 2.$ We next contract the exceptional curve $f = \pi(F)$ to a point. Then we get a blow up $${\mathrm{bl}}: Y \to {\mathbb{F}}_4,$$ where ${\mathbb{F}}_4$ is the $4$-th Hirzebruch surface. (See [@NikulinSaito07].) We set $s:={\mathrm{bl}}(A_0),\ A^\prime_1 := {\mathrm{bl}}(A_1),\ c:={\mathrm{bl}}(e)$. Then $s$ is the exceptional section of ${\mathbb{F}}_4$ with $s^2 = -4$, and $c$ is a fiber of the fibration ${\mathbb{F}}_4\to s$ with $c^2=0$. We have $${\mathrm{bl}}(A) = {\mathrm{bl}}(A_0) + \, {\mathrm{bl}}(A_1) = s + A^\prime_1 \ \ \in |-2K_{{\mathbb{F}}_4}|.$$ Namely, ${\mathrm{bl}}(A)$ is . Since $s \cdot A^\prime_1 = 0$, $A^\prime_1$ does not intersect the section $s$. Since $-2K_{{\mathbb{F}}_4} \sim 12c+4s$, we have $A^\prime_1 \ \ \in |12c+3s|$. For any real $2$-elementary K3 surface $(X,\tau,\varphi)$ of type $(S,\theta)$ with $S \cong (3,1,1)$, we have $$\theta = - {\mathrm{id}},$$ and $$G = \{ {\mathrm{id}}\}.$$ It is known that all real $2$-elementary K3 surfaces of type $((3,1,1), - {\mathrm{id}})$ are (${\mathcal{D}}{\mathbb{R}}$)-non-degenerate. \[the real double point\] $F$ is a nonsingular rational curve if and only if $A_1$ intersects $f$ in two distinct points, and $F$ is a union of two nonsingular rational curves if and only if $A_1$ touches $f$. If $A_1$ intersects with $f$ at two distinct points, then they are real points or non-real conjugate points. In the former case the real double point of $A^\prime_1$ is a real node, and in the latter case it is a real isolated point. Anyway the real double point of $A^\prime_1$ is nondegenerate. If $A_1$ touches to $f$ in $Y$, then the real double point is a real cusp (a degenerate double point). Since all real $2$-elementary K3 surfaces of type $((3,1,1), - {\mathrm{id}})$ are (${\mathcal{D}}{\mathbb{R}}$)-non-degenerate, by Theorem \[theorem2005moduli\], the connected components of the period domain of marked real $2$-elementary K3 surfaces of type $((3,1,1), - {\mathrm{id}})$ and the isometry classes with respect to $G = \{ {\mathrm{id}}\}$ of integral involutions of ${\mathbb{L}}_{K3}$ of type $((3,1,1), - {\mathrm{id}})$ such that the fixed part ${\mathbb{L}}^\psi$ of $\psi$ is hyperbolic are in bijective correspondence. However, as is written in Remark 8 in Subsection 2.4 in [@NikulinSaito07], it is possible that an isometry class of integral involutions of ${\mathbb{L}}_{K3}$ of type $((3,1,1), - {\mathrm{id}})$ corresponds to [**both type IIa case and type IIb case**]{}. In other words, such isometry classes with respect to $G$ (in the sense of Theorem \[theorem2005moduli\]) cannot distinguish [**degenerate and nondegenerate double points**]{} of the curves $A^\prime_1$ on ${\mathbb{F}}_4$ (Recall Remark \[the real double point\]). Therefore, we would like to determine whether the real double point of the curve $A^\prime_1$ on ${\mathbb{F}}_4$ is degenerate or not. See also Lemma 4.6 and Problem 1 in [@SaitoSachiko2015]. In order to do this, we define more strict markings of real $2$-elementary K3 surfaces of type $((3,1,1),- {\mathrm{id}})$. Compare the following Definition \[marked311\] to the Definition \[marked\_real\_K3\] above. Let $ {\mathcal{E}},\ \ {\mathcal{F}},\ \ \mbox{and}\ \ {\mathcal{A}}$ be generators of $S$ with the Gram matrix $$\begin{array}{crrr} & {{\mathcal{E}}} & {{\mathcal{F}}} & {{\mathcal{A}}} \\ {{\mathcal{E}}} & -2 & 2 & 1 \ \\ {{\mathcal{F}}} & 2 & -2 & 0 \ \\ {{\mathcal{A}}} & 1 & 0 & -2 \end{array}$$ where ${\mathcal{M}}\cdot {\mathcal{E}}\ge 0$,  ${\mathcal{M}}\cdot {\mathcal{F}}\ge 0$, and   ${\mathcal{M}}\cdot {\mathcal{A}}\ge 0$. And we set $${\mathbb{U}}:= {\mathbb{Z}}({\mathcal{A}},\ {\mathcal{E}}+ {\mathcal{F}}).$$ ${\mathbb{U}}$ is isometric to the hyperbolic plane, and $$S = {\mathbb{U}}\oplus {\mathbb{Z}}({\mathcal{F}})$$ is an orthogonal decomposition of $S$. \[marked311\] We define that a [*marked real $2$-elementary K3 surfaces $((X,\tau,\varphi), \alpha)$ of type $((3,1,1),- {\mathrm{id}})$*]{} is a pair of a real $2$-elementary K3 surface $(X,\tau,\varphi)$ of type $((3,1,1),- {\mathrm{id}})$ (Definition \[real\_2-elementary K3\_S\_theta\]) and a marking (isometry) $$\alpha : H_2(X, {\mathbb{Z}}) \cong {\mathbb{L}}_{K3}$$ such that - $\alpha({H_2}_+(X, {\mathbb{Z}})) = S$, - $\alpha \circ \varphi_* = \theta \circ \alpha \ \text{on} \ {H_2}_+(X, {\mathbb{Z}})$, - $\alpha_{{\mathbb{R}}}^{-1}(V^+(S))$ contains a hyperplane section of $X$, - the set $\alpha^{-1}(\Delta(S)_+)$ contains only classes of effective curves of $X$, - $\alpha([A_0]) = {\mathcal{A}}$, $\alpha([E]) = {\mathcal{E}}$, and $\alpha([F])={\mathcal{F}}$. Note that every real $2$-elementary K3 surface $(X,\tau,\varphi)$ of type $((3,1,1),- {\mathrm{id}})$ has such a marking $\alpha$. We give a criterion for the unique double point of a real anti-bicanonical curve ${\mathrm{bl}}(A) = s + A^\prime_1 $ on ${\mathbb{F}}_4$ with one real double point on $A^\prime_1$ to be nondegenerate. Let us consider and fix an integral involution $$({\mathbb{L}}_{K3},\psi)$$ of type $((3,1,1),- {\mathrm{id}})$ again, and consider the period domain in Subsection \[period domains section\] $$\Omega_\psi /-\psi \ = \ {\mathcal{L}}_+ \times {\mathcal{L}}_{-,S} \ .$$ \[criterion\] Let $[[\omega]]$ be a point [^1] in $\Omega_\psi /-\psi$. Then, the real double point of the curve $A^\prime_1$ on ${\mathbb{F}}_4$ (Remark \[the real double point\]) which corresponds to $((X,\tau,\varphi), \alpha)$ is nondegenerate if and only if there are no ${\mathbf{v}}\ (\neq \pm {\mathcal{F}})$ in ${\mathbb{L}}_{K3}$ satisfying: $${\mathbf{v}}\cdot \omega = 0,\ \ {\mathbf{v}}\cdot {\mathbb{U}}=0,\ \ \text{and}\ \ {\mathbf{v}}^2 = -2.$$ The ($\Longleftarrow$) direction has been proved in Lemma 4.6 of [@SaitoSachiko2015]. Here we prove the ($\Longrightarrow$) direction. Assume that the double point of the real anti-bicanonical curve ${\mathrm{bl}}(A)$ on ${\mathbb{F}}_4$ is nondegenerate. Then $F$ is irreducible ([@NikulinSaito07], [@SaitoSachiko2015]). Suppose that there [**exists**]{} a ${\mathbf{v}}\ (\neq \pm {\mathcal{F}})$ in ${\mathbb{L}}_{K3}$ satisfying that ${\mathbf{v}}\cdot \omega = 0,\ {\mathbf{v}}\cdot {\mathbb{U}}=0$, and ${\mathbf{v}}^2 = -2$. Then $-{\mathbf{v}}$ has the same properties. By Riemann-Roch Theorem, we have $l(\alpha^{-1}({\mathbf{v}}))+l(-\alpha^{-1}({\mathbf{v}})) \geq \alpha^{-1}({\mathbf{v}})^2/2 +2 = 1$. Hence, $\alpha^{-1}({\mathbf{v}})$ or $-\alpha^{-1}({\mathbf{v}})$ is effective. (See [@AlexeevNikulin2006], p.23 or [@BHPV], p.311.) Hence, we may assume $\alpha^{-1}({\mathbf{v}})$ is an effective class. Then, $\tau^*(\alpha^{-1}({\mathbf{v}}))$ is also effective. We have $\tau^*(\alpha^{-1}({\mathbf{v}}))^2 = -2$, $\tau^*(\alpha^{-1}({\mathbf{v}}))\cdot [A_0] = 0,\ \tau^*(\alpha^{-1}({\mathbf{v}}))\cdot ([E]+[F])=0$ and $\alpha(\tau^*(\alpha^{-1}({\mathbf{v}})))\cdot \omega = 0$ since $\tau^*$ is non-symplectic. Hence, $\alpha(\tau^*(\alpha^{-1}({\mathbf{v}})))$ has the same properties as ${\mathbf{v}}$. Since $\alpha(\tau^*(\alpha^{-1}({\mathbf{v}}))) + {\mathbf{v}}\in S$, it is contained in ${\mathbb{Z}}({\mathcal{F}})$. There exists an integer $n$ such that $$\alpha(\tau^*(\alpha^{-1}({\mathbf{v}}))) + {\mathbf{v}}= n{\mathcal{F}}.$$ Let us consider a hyperplane section class $h \ \in {\mathop\mathrm{Pic}}X$, which is ample. Since $\tau^*(\alpha^{-1}({\mathbf{v}}))$ and $\alpha^{-1}({\mathbf{v}})$ are effective, we have $\tau^*(\alpha^{-1}({\mathbf{v}}))\cdot h >0$ and $\alpha^{-1}({\mathbf{v}})\cdot h >0$. (Nakai’s criterion) We have $\tau^*(\alpha^{-1}({\mathbf{v}})) + \alpha^{-1}({\mathbf{v}}) \neq 0$. Hence, we have $n \neq 0$. Moreover, since $\tau^*(\alpha^{-1}({\mathbf{v}})) + \alpha^{-1}({\mathbf{v}})$ and $[F] (= \alpha^{-1}({\mathcal{F}}))$ are effective, we have . Actually, if $n<0$, then $0=(\tau^*(\alpha^{-1}({\mathbf{v}})) + \alpha^{-1}({\mathbf{v}}) + (-n)\alpha^{-1}({\mathcal{F}}))\cdot h >0$. This is a contradiction. If $\tau^*(\alpha^{-1}({\mathbf{v}})) = \alpha^{-1}({\mathbf{v}})$, then $(\tau^*(\alpha^{-1}({\mathbf{v}})) + \alpha^{-1}({\mathbf{v}}))^2 = (2\alpha^{-1}({\mathbf{v}}))^2 = -8$. On the other hand, $(\tau^*(\alpha^{-1}({\mathbf{v}})) + \alpha^{-1}({\mathbf{v}}))^2 = (n[F])^2 = -2n^2$. Hence, $n=2$. Thus we have ${\mathbf{v}}= {\mathcal{F}}$. This is a contradiction. Thus we have $\tau^*(\alpha^{-1}({\mathbf{v}})) \neq \alpha^{-1}({\mathbf{v}})$. Let $\alpha^{-1}({\mathbf{v}})=\displaystyle \sum_i m_i {\mathbf{v}}_i$, where $m_i$ are positive integers and ${\mathbf{v}}_i$ are irreducible effective classes. Obviously we have $\alpha({\mathbf{v}}_i) \cdot \omega = 0$ for any $i$. If ${\mathbf{v}}_i = [E]$ or $[F]$, then ${\mathbf{v}}_i \cdot ([E]+[F]) = 0$. And if ${\mathbf{v}}_i \neq [E]$ and $\neq [F]$, then we have ${\mathbf{v}}_i \cdot ([E]+[F]) \geq 0$ for any $i$. Here $E$ is irreducible, and moreover, $F$ is irreducible by the assumption. Since $\alpha^{-1}({\mathbf{v}})\cdot ([E]+[F])=0$, we have ${\mathbf{v}}_i \cdot ([E]+[F]) = 0$ for any $i$. Since $[A_0] \cdot ([E]+[F]) = 1$, we see ${\mathbf{v}}_i \neq [A_0]$ for any $i$. Thus we also have ${\mathbf{v}}_i \cdot [A_0] \geq 0$. Since $\alpha^{-1}({\mathbf{v}})\cdot [A_0]=0$, we have ${\mathbf{v}}_i \cdot [A_0] = 0$ for any $i$. Thus, we have $\alpha({\mathbf{v}}_i) \cdot {\mathbb{U}}= 0$ for any $i$. Since ${\mathbb{U}}$ is of signature $(1,1)$, we have $({\mathbf{v}}_i)^2 <0$ for any $i$ by the Hodge index theorem. For such ${\mathbf{v}}_i$’s, we have ${\mathbf{v}}_i^2 = -2$, and ${\mathbf{v}}_i$’s are nonsingular rational ($\cong {\mathbb{P}}^1$). Suppose $\alpha({\mathbf{v}}_i) = {\mathcal{F}}$ for some $i$, say $i=1$. Namely, ${\mathbf{v}}_1 = [F]$. Let ${\mathbf{v}}^\prime := \alpha^{-1}({\mathbf{v}}) - m_1 [F]$. Since $\tau^*(\alpha^{-1}({\mathbf{v}})) \neq \alpha^{-1}({\mathbf{v}})$ (see above), we have ${\mathbf{v}}^\prime \neq 0$. Thus, there exists a ${\mathbf{v}}_i (\neq \pm [F])$ in $H_2(X,{\mathbb{Z}})$ such that ${\mathbf{v}}_i \cong {\mathbb{P}}^1$ (irreducible), ${\mathbf{v}}_i^2 = -2$, $\alpha({\mathbf{v}}_i) \cdot \omega = 0$, and $\alpha({\mathbf{v}}_i) \cdot {\mathbb{U}}= 0$. Eventually we can choose ${\mathbf{v}}$ such that $\alpha^{-1}({\mathbf{v}})$ is an irreducible class. Since $\tau^*(\alpha^{-1}({\mathbf{v}})) \neq \alpha^{-1}({\mathbf{v}})$, $\tau^*(\alpha^{-1}({\mathbf{v}}))$ and $\alpha^{-1}({\mathbf{v}})$ are represented by different irreducible curves respectively. Hence, we have $\tau^*(\alpha^{-1}({\mathbf{v}})) \cdot \alpha^{-1}({\mathbf{v}}) \geq 0$. Since $(\alpha(\tau^*(\alpha^{-1}({\mathbf{v}}))) + {\mathbf{v}})^2 = (-2) + (-2) + 2(\tau^*(\alpha^{-1}({\mathbf{v}})) \cdot \alpha^{-1}({\mathbf{v}})) = -2n^2$, we have $2-n^2 \geq 0$. Thus, we have $n=1$. Namely, we have $${\mathcal{F}}= {\mathbf{v}}+ \alpha(\tau^*(\alpha^{-1}({\mathbf{v}}))),$$ i.e., $[F] = \alpha^{-1}({\mathbf{v}}) + \tau^*(\alpha^{-1}({\mathbf{v}}))$. Let $F^\prime$ be an irreducible curve representing $\alpha^{-1}({\mathbf{v}})$, and we set $F^{\prime \prime} := \tau^*(F^\prime)$. Thus $F^\prime + F^{\prime \prime}$ represents the class $[F]$. Hence, there exists a marked real $2$-elementary K3 surface corresponding to the period $[\omega]$ which has $E + F^\prime + F^{\prime \prime}$ as the unique reducible fiber of its elliptic fibration. Conversely, every marked real $2$-elementary K3 surface corresponding to the period $[\omega]$ has $E + F^\prime + F^{\prime \prime}$ as the unique reducible fiber of its elliptic fibration. Hence, we have $F= F^\prime + F^{\prime \prime}$. This contradicts the assumption that $F$ is irreducible. This completes the proof of Proposition \[criterion\]. For curves of degree $6$ with one double point on ${\mathbb{R}}P^2$, the corresponding criterion to Proposition \[criterion\] is written on the top of p. 281 in Itenberg’s paper [@Itenberg92]. As is written in [@SaitoSachiko2015], we now get the precise image ($\subset {\mathcal{L}}_+ \times {\mathcal{L}}_{-,S}$) of the period map on the set of all marked real $2$-elementary K3 surfaces $((X,\tau,\varphi), \alpha)$ of type $((3,1,1),- {\mathrm{id}})$ for which the real double points of the curves $A^\prime_1$ on ${\mathbb{F}}_4$ are [**nondegenerate**]{}. Thus we are able to continue the interesting arguments as in [@Itenberg92]. [**Acknowledgments.**]{} The author would like to thank Professor Hisanori Ohashi for his helpful suggestions and comments, Professor Ilia Itenberg for his inspiring papers, and Professor Viacheslav Nikulin for his constant encouragement. [99]{} V.A. Alexeev, V.V. Nikulin, [ *Del Pezzo and K3 Surfaces,*]{} MSJ Memoirs, 2006. W.P.Barth, K. Hulek, C.A.M.Peters, and A.Van de Ven, [*Compact Complex Surfaces,*]{} Springer, 2004. I. Itenberg, [ *Curves of degree $6$ with one non-degenerate double point and groups of monodromy of non-singular curves,* ]{} Real Algebraic Geometry, Proceedings, Rennes 1991, Lecture Notes in Math., [**1524**]{}, Springer, (1992), 267–288. V.V. Nikulin, [ *Integral symmetric bilinear forms and some of their geometric applications,* ]{} Math. USSR Izv., [**14**]{} (1980), 103–167. V.V. Nikulin, [ *On the quotient groups of the automorphism groups of hyperbolic forms by the subgroups generated by $2$-reflections, Algebraic-geometric applications,* ]{} J. Soviet Math., [**22**]{} (1983), 1401–1476. V.V. Nikulin, [*Discrete reflection groups in Lobachevsky spaces and algebraic surfaces,* ]{} Proc. Intern. Congr. Math. Vol. 1 (Berkeley, 1986) (Providence, RI), Amer. Math. Soc., (1987), 654–671. V.V. Nikulin, Sachiko Saito, [ *Real K3 surfaces with non-symplectic involution and applications,*]{} Proc. London Math. Soc., [**90**]{} (2005), 591–654. V.V. Nikulin, Sachiko Saito, [ *Real K3 surfaces with non-symplectic involution and applications. II,*]{} Proc. London Math. Soc., [**95**]{} (2007), 20–48. Sachiko Saito, [ *On real anti-bicanonical curves with one double point on the 4-th real Hirzebruch surface,*]{} Journal of Singularities [**11**]{} (2015), 1–32. [^1]: By the surjectivity of the period map of marked K3 surfaces ([@BHPV], p.339), $[\omega]$ is the period of a marked real $2$-elementary K3 surface $((X,\tau,\varphi), \alpha)$ of type $((3,1,1),- {\mathrm{id}})$ satisfying $\alpha \circ \varphi_* \circ \alpha^{-1} = \psi$.

harvmac Atsuo Kuniba$^{1,}$ [^1] [e-mail: kuniba@math.sci.kyushu-u.ac.jp]{} , Tomoki Nakanishi$^{2,}$ [^2] [e-mail: nakanisi@string.harvard.edu]{} [^3][Permanent Address: Department of Mathematics, Nagoya University, Nagoya 464 Japan]{}, Junji Suzuki$^{3,}$ [^4] [e-mail: jsuzuki@tansei.cc.u-tokyo.ac.jp]{} 0.5cm $^1$Department of Mathematics, Kyushu University Fukuoka 812 JAPAN $^2$Lyman Laboratory of Physics, Harvard University Cambridge, MA 02138 USA $^3$Institute of Physics, University of Tokyo, Komaba Meguro-ku, Tokyo 153 JAPAN .3in Abstract. We propose a new $q$-series formula for a character of parafermion conformal field theories associated to arbitrary non-twisted affine Lie algebra $\widehat{g}$. We show its natural origin from a thermodynamic Bethe ansatz analysis including chemical potentials. 1\. Introduction Recently new aspects in conformal field theories (CFTs) are being recognized through studies of thermodynamic limit of integrable models such as 1d quantum spin chains and $(1+1)$d factorized scattering systems. In these analysis, the Rogers dilogarithm function plays a key role that connects thermodynamic quantities in those models to the CFT data, most notably, central charges and scaling dimensions. For example, the following conjecture emerged ,,, from the restricted solid-on-solid (RSOS) type ,,spin chains: where the lhs is the central charge $c_{\rm PF}$ of the parafermion (PF) CFT ,associated to an affine Lie algebra $\widehat{g}$ with rank $r$, level $\ell$ and dual Coxeter number $g^\vee$. (See  for a generalization of  including the scaling dimensions.) The set $G$ is given by (5) and $f^{(a)}_m$ is the unique solution to the simultaneous algebraic equation in the range $0<f^{(a)}_m <1$, with the notations specified later. Needless to say, the equation of such form as well as the appearance of the dilogarithm are reflecting rich structures encoded in the integrable models. Eq. is thereby connecting the two fundamental ingredients; the CFT data  on the lhs which is of affine Lie algebraic origin and the intricate formula on the rhs occurring from thermodynamics of the integrable models. The purpose of this Letter is to put forward such a connection even further based on the thermodynamic Bethe ansatz (TBA) ,,,,,. We shall propose a new $q-$series formula for a PF character, which is essentially equivalent to a string function  of any non-twisted affine Lie algebra $\widehat{g}$ at any level $\ell \in {\bf Z}_{\ge 1}$. It has a surprisingly simple form and seems to reveal an interesting structure of the PF modules. When $q \rightarrow 1^-$, the $q-$series formula leads to by comparing the asymptotics on both sides with the method of . Thus our new proposal (9) may be viewed as a “lift" of  to a PF character formula in the sense of ,. More importantly, we point out that the $q-$series formula arises naturally from the spectra of the TBA-originated effective central charge  involving dilogarithms. The key is to observe a one to one correspondence between the independent states in the Hilbert space of the PF CFT and the ways of analytic continuations of the dilogarithm. The idea provides a new insight toward a structural correspondence between CFTs and TBA hence its presentation also consists of our main aim in this Letter. We remark that for the special case $\widehat{g} = A^{(1)}_1$, our $q-$series formula coincides with that in . [**2. New $q-$series formula** ]{} Let $g$ denote one of the classical simple Lie algebras $A_r (r \ge 1), B_r (r \ge 2), C_r (r \ge 1), D_r (r \ge 4), E_{6,7,8}, F_4$ and $G_2$. We write $r = {\rm rank }\, g$ and $\widehat{g}$ to mean the non-twisted affinization of $g$ . Let $\Delta$, $\Delta_+$, $\Pi$, $h$, $(\cdot | \cdot)$ denote the root system, the set of positive roots, the set of the simple roots, the Cartan subalgebra, the invariant form on $g$, respectively. The spaces $h$ and $h^*$ are identified via the form $(\cdot | \cdot)$. We employ the normalization $\vert$long root$\vert^2=2$ and set $t_a=2/(\alpha_a|\alpha_a), \, \alpha_a^{\vee} = t_a \alpha_a$ for each simple root $\alpha_a$, where the nodes $1 \le a \le r$ on the Dynkin diagram are enumerated according to . The root lattice $Q=\bigoplus {\bf Z}\alpha_a$, the coroot lattice $Q^{\vee}=\bigoplus {\bf Z}\alpha_a^{\vee}$ and the weight lattice $P=(Q^\vee)^*$ are as usual. We find it convenient to label the weights of $\widehat{g}$ (mod null root) by its projection onto the classical part $P$. Throughout the Letter we fix an integer $\ell \in {\bf Z}_{\ge 1}$ and put $\ell_a = t_a \ell$ and following , . Let $L^\Lambda$ denote the integrable $\widehat{g}$-module having a level $\ell$ dominant integral weight $\Lambda$ as the highest weight . One can fit the action of the (homogeneous) Heisenberg algebra $\widehat{a}$ of rank $r$ on $L^{\Lambda}$ ,. The algebra $\widehat{a}$ has a basis $\{ a^x_{n} | x \in \Pi, n \in {\bf Z} \}\cup\{id\}$. The irreducible module $\Omega^\Lambda$ of PF algebra is isomorphic to the subspace of $L^\Lambda$ consisting of the vectors $v$ such that The space admits the weight space decomposition The PF currents $\psi^\alpha_n$ ($\alpha \in \Delta$), which commute with the operators $a^x_{\pm n} \, (n \in {\bf Z}_{\ge 1})$, map the elements in $\Omega^\Lambda_\lambda$ into another sector $\Omega^\Lambda_{\lambda+\alpha}$. The character of $\lambda$-weight sector $\Omega^\Lambda_\lambda$ (with variable $q$) is given by , where $c^\Lambda_\lambda(q)$ is a string function of $\widehat{g}$ at level $\ell$ and $\eta(q)$ is the Dedekind eta function. The string function is by definition the character of the (graded) $\lambda$-weight subspace of $L^\Lambda$, which is of fundamental importance. So far its explicit formula is not known for general $\widehat{g}$ and $\ell$ although several expressions are available in some cases ,,. Let $\bar\Omega^{\Lambda}$ be the quotient of the space $\Omega^{\Lambda}$ by the identification $\Omega^{\Lambda}_\lambda \sim \Omega^{\Lambda}_{\lambda+\ell Q^{\vee}}$, and the Hilbert space of the chiral half of the PF CFT corresponds to the direct sum of $\bar\Omega^{\Lambda}$’s. From now on we shall exclusively consider [*the vacuum module*]{} $\Omega^{0}$ case and propose the following character formula for each $\lambda$-sector ($\lambda \in Q$): Here the summation in runs over the vectors under the indicated restriction $\lambda(\bn )\equiv \lambda \mod \ell Q^\vee$ with which is compatible with the invariance property $c^{\Lambda}_{\lambda}= c^{\Lambda}_{\lambda +\ell Q^{\vee}}$. Under the above restriction, it can be easily shown that the rhs of contains only non-negative integer powers of $q$ up to an overall factor $q^p$ with $p \equiv -{c_{\rm PF} \over 24}-{\vert \lambda \vert^2 \over 2\ell}$ mod ${\bf Z}$. The character of the space $\bar\Omega^{0}$ is now given by the same formula but [*without*]{} any restriction on the $\bn-$sum other than . At present, a proof is not known for for general $\widehat{g}$ and $\ell$. However one can verify several cases directly and observe a wealth of consistency as we shall see below. For $\widehat{g} = A^{(1)}_1$, some generalizations into different directions have also been conjectured in ,. Firstly,  is indeed valid for $(\widehat{g},\ell) = (A^{(1)}_1,{\rm general})$ as it coincides  with the formula in . So is the case $(\widehat{g},\ell) = (B^{(1)}_r,1)$ with $r$ general, where one can actually compute the $\bn-$sum by means of eq.(2.2.6) in  and compare the result with that in . The case $(\widehat{g},\ell) = (G^{(1)}_2,1)$ can also be proved since the $q-$series  then reduces to that for $(A^{(1)}_1,3)$ (cf.). Not to mention, is trivially ture for $\widehat{g} = A^{(1)}_r, D^{(1)}_r$ and $E^{(1)}_{6,7,8}$ with $\ell = 1$ when the PF module becomes 1 dimensional. Secondly, we have generated the low power terms in by computer and checked agreements with the known results on the string functions for $(\widehat{g},\ell) = (A^{(1)}_{2,3},2,3)$ , $(C^{(1)}_{3,4},1),(F^{(1)}_4,1)$ and $(E^{(1)}_8,2)$ ,. For instance in the last example, the rhs of with $\lambda = \Lambda_1$ yields $q^{3/16}(1 + 29q + 288q^2 + 1878q^3 + \cdots)$. This agrees with the $E^{(1)}_8$ level 2 result $b^{2\Lambda_0}_{\Lambda_1}$ given by eq.(4.4.3a) and Proposition 4.4.1(e) in as an order 9 polynomial of the Virasoro characters. ($\Lambda_i$ denotes the $i-$th fundamental weight.) One may substitute  and into the character formula under the principal specialization of $z$. We have then checked that the resulting $q-$series for $\ch(L^0)$ indeed fulfills the known factorization property  up to some power for many examples including $(\widehat{g},\ell) = (A^{(1)}_{2,3},3), (B^{(1)}_3,2,3), (D^{(1)}_4,2,3),$ $ (F^{(1)}_4,2)$ and $(G^{(1)}_2,2)$. Thirdly, if is true, then must hold by comparing the leading powers on both sides. Here $n^0_\lambda$ is the minimum eigenvalue of the Virasoro operator $L_0$ in the $\lambda$-weight subspace of $L^0$ and is equal to the minimum number of roots to express $\lambda$ as their sum if it is possible within $\ell$ roots. Our quadratic form ${\cal K}(\bn)$ has the consistent property to it since is valid for any positive root $\lambda$, which is a special case of . Finally, we remark that is also consistent in that it leads to the dilogarithm conjecture by comparing the asymptotics on both sides as $q \rightarrow 1^-$. To see this, we firstly note that the leading divergence of the lhs in is $({\bar q})^{-c_{\rm PF}/24} \, ({\bar q} = e^{-2\pi i/\tau})$ when $q = e^{2\pi i \tau} \rightarrow 1^-$ . As for the rhs, one can apply the argument in , to get a crude estimate $({\bar q})^{-\sum L(f^{(a)}_m)/4\pi^2}$, from which follows. In particular, arises essentially from the “saddle-point condition" $q^{n^{(a)}_m} = 1 - f^{(a)}_m$ with respect to $n^{(a)}_m$. Before closing this section, let us discuss how our $q$-series will indicate a basis structure in the PF module in the light of the earlier works ,. The space $\Omega^{0}$ is certainly spanned by the vectors where $v_0$ is the highest weight vector and $T^{\gamma}$ is the translation isomorphism $T^{\gamma}: \Omega^{0}_{\lambda} \rightarrow \Omega^{0}_{\lambda+\gamma}$. Furthermore by introducing a lexicographic ordering in this set, we can choose Poincaré-Birkhoff-Witt type vectors among them as a spanning set of $\Omega^0$. To illustrate the idea let us take the example $(\widehat{g},\ell) = (A^{(1)}_2,2)$ with $\lambda = \alpha_1 + \alpha_2 \in \Delta$ and consider the $n^{(1)}_1=n^{(2)}_1=1$ term ${q^{1/2} \over (q)_1(q)_1}$ in (apart from $q^{-c_{\rm PF}/24}$). In view of and the restriction $\lambda(\bn )\equiv \lambda \mod \ell Q^\vee$, it corresponds to the character of the subspace of $\Omega^{0}_{\alpha_1 + \alpha_2}$ spanned by the vectors since their contributions amount to it as Here the prefactor $q^{-1/2}$ comes from $-|\alpha_1 + \alpha_2|^2/2\ell =-1/2$. In general non-trivial relations exist among the operators $\prod_i \psi^{\beta_i}_{-k_i}$ if $(\sum \beta_i|\Lambda_j) \geq \ell$ for some fundamental weight $\Lambda_j$, hence one must eliminate some spanning vectors to get a real basis. We leave it as an interesting future problem. [**3. Origin from TBA**]{} Our proposal for the PF character has stemmed from an analysis based on the TBA type integral equation in which represents interacting “pseudo particles" with energy $\epsilon^{(a)}_m(u)$ labeled by $(a,m) \in G$. Here, ${\cal M}_a > 0$, $\pi i D^{(a)}_m$ and $R$ are independent of the rapidity $u$ and stand for the mass, the chemical potential (cf. ) and the system size corresponding to the inverse temperature in TBA. The integration kernel $\Psi^{m k}_{a b}(u)$ decays rapidly when $\vert u \vert \rightarrow \infty$ and has been specified in eq.(18) of . Here we will not need its explicit form but the properties Eq. has a similar form to many earlier examples of the TBA equations ,,,,, and is a candidate describing a massive deformation of the level $\ell$ $\widehat{g}$ PF CFT by a certain relevant operator . Actually, one can apply the standard TBA technique to show that the free energy has the ultraviolet (UV) asymptotics Eq. is a characteristic behavior of the CFT with the central charge $c$ ,. In the derivation, we have used  and put $\epsilon^{(a)}_{m,+}(u) = \epsilon^{(a)}_m(u + \log{2 \over R})$ and passed to the limit $R \rightarrow 0$ firstly to deduce $\epsilon^{(a)}_{m,+}(+\infty) = +\infty$ from the assumption ${\cal M}_a > 0$. We have also set $f^{(a)}_m = \bigl(1 + \hbox{exp}(\epsilon^{(a)}_{m,+}(-\infty))\bigr)^{-1}$, which is natural since the simplest branch choice $\log f^{(a)}_m, \log(1-f^{(a)}_m) \in {\bf R}$ for all $(a,m) \in G$ in then yields $D^{(a)}_m = 0$ hence the ground state value $c = c_{\rm PF}$ by means of the dilogarithm conjecture . However, one may allow various branches in  and thereby introduces non-trivial chemical potentials and possibly extracts the excitation spectra as argued in ,,,. To be more precise and systematic, we introduce the universal covering space ${\cal R}$ of ${\bf C}\setminus \{ 0,1 \}$ and the covering map $\ti{i}:{\cal R} \rightarrow {\bf C}\setminus \{ 0,1 \}$, which specifies analytic continuations of the dilogarithm. The effective central charge $c$ is then to be understood as a function on the set of the points $(\ti{f}\am)_{(a,m) \in G}$ on ${\cal R}^{|G|}$ such that $\ti{i}(\ti{f}\am)=f\am$, i.e., as introduced in eq.(11) of  (with $z=0$ therein). Here, ${\cal S}$ denotes the collection $({\cal C}_{a,m})_{(a,m) \in G}$ of the contours ${\cal C}_{a,m}$ from an arbitrary base point to $0 < f^{(a)}_m < 1$ in ${\bf C}\setminus \{ 0,1 \}$ specifying the point $\ti{f}\am$ on ${\cal R}$. We warn readers that ${\cal C}_{a,m}$ here does [*not*]{} mean the integration contour as opposed to the convention in . We fix the branch $-\pi < {\rm Im}(\log(\cdot)) \le \pi$ in hence $L(x)$ has the cuts $(-\infty, 0]$ and $[1, +\infty)$ on the complex $x-$plane. The $\ti{L}(\cdot)$ and $\Log(\cdot)$ in ,stand for the analytic continuations of $L(\cdot)$ and $\log(\cdot)$ to ${\cal R}$, respectively. Because $c({\cal S})$ actually depends only on the homotopy classes of the contours ${\cal C}_{a,m}$, we shall parametrize them by the integers $\xi^{(a)}_{m,j},\eta^{(a)}_{m,j}\, (j \ge 1)$ as ${\cal C}_{a,m} = {\cal C}[f^{(a)}_m \vert \xi^{(a)}_{m,1},\xi^{(a)}_{m,2}, \ldots \vert \eta^{(a)}_{m,1},\eta^{(a)}_{m,2}, \ldots]$, wherein the notation ${\cal C}[f \vert \xi_1,\xi_2, \ldots \vert \eta_1,\eta_2, \ldots]$ signifies the contour going from the base point to $f$ as follows (Fig.1). It firstly goes across the cut $[1,+\infty)$ for $\eta_1$ times then crosses the other cut $(-\infty, 0]$ for $\xi_1$ times then $[1,+\infty)$ again for $\eta_2$ times, $(-\infty, 0]$ for $\xi_2$ times and so on before approaching $f$ finally. Here intersections have been counted as $+1$ when the contour goes across the cut $(-\infty, 0]$ (resp. $[1,+\infty)$) from the upper (resp. lower) half plane to the lower (resp. upper) and $-1$ if opposite. We call $\xi_j$ and $\eta_j$ the winding numbers and assume that they are all zero for $j$ sufficiently large. From these definitions one deduces the formulas which make the dependences on the contour ${\cal C} = {\cal C}[f \vert \xi_1,\xi_2, \ldots \vert \eta_1,\eta_2, \ldots]$ explicit. The collection ${\cal S}$ of the contours is now equivalently represented by the collection of winding numbers $(\xi^{(a)}_{m,j}, \eta^{(a)}_{m,j})_{(a,m) \in G, j \ge 1}$. By applying , to , and using $\log\bigl(f^{(a)}_m\bigr) = \sum_{(b,k) \in G}K^{m\, k}_{a\, b} \log\bigl(1-f^{(b)}_k\bigr)$ from , one can split the $c({\cal S})$ into ${\cal S}-$dependent and independent parts. The latter turns our to be $c_{\rm PF}$ due to and we get [^5][Using this opportunity we remark that eqs.(9b) and (12d) (with $z=0$) in are erroneous and should be corrected as eqs.(29) and (31) here, respectively.]{} where ${\cal K}({\bf n})$ is defined in and the vector $\bn =( n^{(a)}_m )_{(a,m) \in G} \in {\bf Z}^{|G|}$ is specified from ${\cal S}$ by The formulas - describe the ${\cal S}-$dependence of the effective central charge manifestly. Let us now investigate the spectra of the $c({\cal S})$ when ${\cal S}$ consists of those contours that intersect the cuts $[1, +\infty)$ and $(-\infty, 0]$ always from the lower half plane to the upper with the former crossed firstly if ever. Respecting , such an ${\cal S}$ is a collection of ${\cal C}_{a,m}$ $((a,m) \in G)$ parametrized as for some $n^{(a)}_m \ge 0$ and $\xi^{(a)}_{m,1},\ldots,\xi^{(a)}_{m,n^{(a)}_m} \le 0$. Denote by ${\cal O}$ the totality of such ${\cal S}$’s. Then from -, one can compute the spectra of the effective central charge as follows. According to , the last expression is nothing but the character ${\rm ch}(\bar\Omega^0)$. In this way, the spectra of the effective central charge leads to the PF CFT character itself. This extends our earlier observation in  (with $\Lambda = 0$) further toward a structural correspondence between CFTs and TBA. Namely, the independent states in the Hilbert space $\bar\Omega^0$ are in one to one correspondence to the lifts $( \ti{f}\am )_{(a,m) \in G} \in {\cal R}^{|G|}$ of the point $( f\am )_{(a,m) \in G}$ parametrized by the set $\cO$. [**4. Discussions**]{} We have seen that the whole excitation spectra in the PF vacuum module $\Omega^0$ is obtainable from the UV free energy (or the effective central charge $c(\cS)$) by a certain analytic continuation procedure. It implies that the ground state energy also possesses informations on the excitations since the latters correspond to just different branches of the former. It will be interesting to seek such a phenomenon in a wider class of models in 2d statistical mechanics and quantum field theories. As for our examples in this Letter, there are at least two routes to possibly explain this phenomenon. The first one is to interpret $c(\cS)$ as the expectation value of the symmetry operator under which the corresponding excited state becomes the leading . Though this argument has been applied almost exclusively to some primary excitations, one may generalize it by including descendant operators to accommodate the full spectra. The second route is to regard $c(\cS)$ as representing finite-size corrections , to various transfer matrix eigenvalues of critical RSOS-type ,, spin chains. In such analyses one treats the integral equations similar to  originated from the actual functional relations  among the row to row transfer matrices as done in  for $\widehat{g} = A^{(1)}_1$. There, non trivial branch choices have indeed been observed to yield various eigenvalues. Thus our prescription here might be related to such an approach by using the $U_q(\widehat{g})$ functional relation in . The $q-$series formula  and the computation in the previous section concern the UV limit $R \rightarrow 0$ where the TBA just starts to deform the CFT. So in principle they should allow a “continuous deformation" in some sense which will exhibit rich integrability structures away from criticality. Finally, though we have considered only the vacuum module case in this Letter, it is natural to expect similar formulas to for general PF modules in the light of the result in . We hope to report them in our forthcoming paper . 0.7cm We would like to thank I. Cherednik, P. Fendley, E. Frenkel, S. Hosono and K. Mimachi for valuable discussions. T. N. would like to thank Prof. C. Vafa for his kind hospitality. This work is supported in part by JSPS fellowship, NSF grants PHY-87-14654, PHY-89-57162 and Packard fellowship. 0.7cm [**Figure Captions**]{} Fig.1 An example of a contour from a base point $z_0$ to a point $f \in (0,1)$ in ${\bf C}\setminus \{ 0,1 \}$. Its homotopy type is parametrized as ${\cal C}[f| -2, 0, \dots |2, 1, 0, \dots]$ or also as ${\cal C}[f| 0, -2, 0, \dots | 1, 1, 1, 0, \dots]$. [^1]: $^a$ [^2]: $^b$ [^3]: [^4]: $^c$ [^5]: $^1$

--- abstract: 'We report the study of far-IR sizes of submillimeter galaxies (SMGs) in relation to their dust-obscured star formation rate (SFR) and active galactic nuclei (AGN) presence, determined using mid-IR photometry. We determined the millimeter-wave ($\lambda_{\rm obs}=1100\,\mu$m) sizes of 69 ALMA-identified SMGs, selected with $\geq10$$\sigma$ confidence on ALMA images ($F_{\rm 1100 \mu m}=1.7$–7.4mJy). We found that all the SMGs are located above an avoidance region in the millimeter size-flux plane, as expected by the Eddington limit for star formation. In order to understand what drives the different millimeter-wave sizes in SMGs, we investigated the relation between millimeter-wave size and AGN fraction for 25 of our SMGs at $z=1$–3. We found that the SMGs for which the mid-IR emission is dominated by star formation or AGN have extended millimeter-sizes, with respective median $R_{\rm c,e} = 1.6^{+0.34}_{-0.21}$ and 1.5$^{+0.93}_{-0.24}$kpc. Instead, the SMGs for which the mid-IR emission corresponds to star-forming/AGN composites have more compact millimeter-wave sizes, with median $R_{\rm c,e}=1.0^{+0.20}_{-0.20}$kpc. The relation between millimeter-wave size and AGN fraction suggests that this size may be related to the evolutionary stage of the SMG. The very compact sizes for composite star-forming/AGN systems could be explained by supermassive black holes growing rapidly during the SMG coalescing, star-formation phase.' author: - Soh Ikarashi - 'KarinaI. Caputi' - Kouji Ohta - 'R.J. Ivison' - 'ClaudiaD. P. Lagos' - Laura Bisigello - Bunyo Hatsukade - Itziar Aretxaga - 'JamesS. Dunlop' - 'DavidH. Hughes' - Daisuke Iono - Takuma Izumi - Nobunari Kashikawa - Yusei Koyama - Ryohei Kawabe - Kotaro Kohno - Kentaro Motohara - Kouichiro Nakanishi - Yoichi Tamura - Hideki Umehata - 'GrantW. Wilson' - Kiyoto Yabe - 'MinS. Yun' title: 'Very compact millimeter sizes for composite star-forming/AGN submillimeter galaxies' --- Introduction ============ The morphology and size of star-forming regions in submillimeter galaxies (SMGs) are important properties with which we can address the nature of their prodigious, dust-obscured star formation, and consequently the formation and evolution of the most massive galaxies. The Atacama Large Millimeter/submillimeter Array (ALMA) is enabling astronomers to image high-redshift SMGs with angular resolutions of $\lesssim0''$.3. Some ALMA studies have reported effective radii ($R_{\rm e}$) of $\sim0.3$–3kpc [e.g. @ika15; @sim15; @hod16]. These radii are small compared with what astronomers expected from studies of SMG sizes based on radio continuum and CO emission [e.g. @tac06; @big08; @ivi11]. These new results represent a new milestone in our understanding of star formation in SMGs, suggesting that these galaxies plausibly evolve to compact quiescent galaxies [e.g. @tof14; @ika15; @sim15]. As a next step, it would be useful to test the hypothesis that SMGs are connected to the formation of the most massive galaxies, being triggered by major mergers, and then evolving into compact quiescent galaxies via quenching in a QSO phase [e.g. @san88; @hop08; @tof14]. The compact submillimeter sizes of SMGs, including recent reports of the existence of subkilopersec-scale starburst cores [@ion16; @ika17; @ote17], suggests that the intense star-formation activity might be quenched by active galactic nuclei (AGN), as observed in some luminous QSOs [e.g. @mai12; @car16]. The link between SMGs and QSOs is still unclear, though. However, previous X-ray [e.g. @ale05; @wan13] and mid-IR [e.g. @ivi04; @cop10; @ser10] studies indicate that some SMGs do harbor AGN. In this letter, we report a millimeter-wave size study of 69 ALMA-identified AzTEC SMGs. Firstly, we study the empirical relation between the ALMA continuum flux densities and the millimeter-wave sizes of SMGs. Secondly, we investigate the relationship between millimeter-wave sizes and the presence of AGN in SMGs at $z=1$–3, as determined from mid-IR data. We adopt throughout a cosmology with $H_{\rm 0}=70$kms$^{-1}$Mpc$^{-1}$, $\Omega_{\rm M}=0.3$, and $\Omega_{\rm \Lambda}=0.7$. ALMA Observations and samples ============================= The sample used in this paper comes from our ALMA 1100-$\mu$m continuum imaging survey of 144 bright AzTEC/ASTE sources with $F_{\rm 1100 \mu m,\,AzTEC}\geq 2.4$mJy in the Subaru/[*XMM-Newton*]{} Deep Field [SXDF; @fur08]. The SXDF survey was conducted in the ALMA Cycles 2 and 3 (2013.1.00781, 2015.1.00442.S: PI. Hatsukade; B.Hatsukadeetal.2017, in preparation). The ALMA observations in Cycle 2 were carried out with the array configurations C34-5 and C34-7, with 37–38 working 12-m antennas covering up to a $uv$ distance of $\sim 1500$k$\lambda$. In Cycle 3, the observations were executed in array configuration C40-4, covering up to a $uv$ distance of $\sim 1000$k$\lambda$. On-source integration times per source in each cycle were 0.6min. The typical synthesized beam size for our ALMA continuum images is $\sim 0.''30 \times 0.''23$ ($\rm PA \sim 56^{\circ}$), after combining the Cycle 2 and 3 data. The average r.m.s. noise level is 120$\mu$Jybeam$^{-1}$. The images were generated with Briggs weighting, using a robust parameter of 0.3. The ALMA continuum maps yielded 70 ALMA-identified AzTEC SMGs (hereafter ASXDF SMGs) with $S_{\rm peak}/N\geq 10$ detections, suitable for reliable ALMA millimeter-wave size measurements [e.g. @ika15]. We removed one lensed SMG [ASXDF1100.001; @ika11], leaving 69 SMGs. ALMA fluxes were re-measured in tapered ALMA images with a synthesized beam of $\sim0{''}.6$, which is larger than the measured mm-wave sizes of SMGs in this paper, using the IMFIT task in CASA. For 51 ASXDF SMGs, we obtained well-constrained photometric redshifts, with a median error $\delta z= 0.13\pm0.02$, based on the individual 1-$\sigma$ errors estimated by [*Le Phare*]{} [e.g. @ilb06] in spectral energy distribution (SED) model fitting using the $B$, $V$, $Rc$, $i'$, $z'$, $J$, $H$, $Ks$, 3.6 and 4.5$\mu$m data (S.Ikarashi et al. 2017, in preparation). The remaining SMGs lie outside the coverage of the optical/near-IR images, or have individual 1-$\sigma$ errors of $>1$. Photometric and spectroscopic redshifts from the literature are listed in Table \[tbl-1\]. ALMA millimeter-wave source size measurements ============================================= We measured millimeter-wave sizes as circularized effective radii ($R_{\rm c,e}$) for the 69 ASXDF SMGs with ALMA visibility data, in the same manner as @ika15. We used $uv$-distance versus amplitude plots (hereafter $uv$-amp plots) for our measurements. Although the ALMA data cover $uv$ distances up to $\sim 1500$k$\lambda$, we used only data at $\leq 500$k$\lambda$, which corresponds to a scale of $\sim0.''2$. Adopting this cutoff for the longest $uv$ distance is the equivalent of smoothing with a larger size kernel in the image plane. We aim to mitigate the effects of possible clumpy structures in the size measurements and to measure $R_{\rm c,e}$ robustly. For the sources detected with $\geq10\sigma$ in the ALMA Cycle-2 images alone, we measured their sizes using only Cycle-2 data, to avoid effects due to any systematic absolute flux calibration offsets between our Cycle 2 and 3 data [^1]. We measured sizes by fitting a Gaussian model to the observed data in the $uv$-amp plots. When we measure the size, the other sources ($\geq5\sigma$) in each ALMA image were removed from the visibility data based on simple source properties derived by IMFIT task. In order to estimate possible systematics in the size measurements, we injected mock sources into ALMA noise visibility images, generated from the actual ALMA data as in @ika15. Briefly we injected a symmetric Gaussian component for a range of source sizes and flux densities that cover the putative parameter range of our ASXDF sources with uniform probability. As tested in @ika15, our method can measure circularized effective radii correctly even if a source has an asymmetric Gaussian profile. We corrected our raw measured sizes based on the results of the simulations for the data used in this paper. As a result, we obtained ALMA millimeter-wave sizes of 0$''.08$–0$''.68$ (FWHM) for the 69 ASXDF SMGs. Note that ASXDF1100.009.1 has two distinct millimeter-wave components with a separation of $\sim$0$''$.6, sharing a host galaxy at $z_{\rm spec}=0.9$. Relation between millimeter sizes and fluxes ============================================ Fig. \[fig:sizeflux\] (left panel) shows all 69 ASXDF SMGs in an ALMA 1100-$\mu$m vs. millimeter-wave size plot. Additionally, we plot 13 ALMA-identified, fainter SMGs at $z\gtrsim 3$ from @ika15. ASXDF SMGs are absent from the top-left and the bottom-right corners of this plot. The expected source selection limit for $\geq10\sigma$ continuum detection based on simple Gaussian models explains the absence of SMGs in the top-left corner. The bottom-right corner, instead, is free from any such selection biases, so the absence of SMGs requires an explanation. The absence of SMGs in the bottom-right corner of Fig.  \[fig:sizeflux\] can be interpreted as the influence of Eddington-limited star formation [@mur05]. According to @you08, which reported pioneering studies of maximum star formation in bright SMGs, a maximum star-formation rate is given by $$SFR_{\rm max} = 480\sigma^2_{400}D_{\rm kpc}\,\kappa^{-1}_{100} M_{\odot} yr^{-1},$$ where $D_{\rm kpc}$ is the characteristic physical scale of the starburst region in kpc, $\sigma_{\rm 400}$ is the line-of-sight gas velocity dispersion in units of 400kms$^{-1}$, and $\kappa_{\rm 100}$ is the dust opacity in units of 100cm$^2$g$^{-1}$. Here we adopt a Chabrier initial mass function [@cha03]; $\kappa_{\rm 100}=1$, as in @you08; and a median gas velocity dispersion of 510kms$^{-1}$ from CO line observations of SCUBA SMGs [@bot13]. We also adopt 2$\times$ FWHM or 4$\times R_{\rm c,e}$, which is expected to include 94% of the total light, as $D_{\rm kpc}$. The derived $SFR_{\rm max}$ was corrected with this factor of 0.94. In order to plot the relation between SFR and physical scale described by Equation 1 on Fig. \[fig:sizeflux\] (the left panel), we assume a fixed redshift $z=2$. The conversion factors from ALMA fluxes to SFRs were derived by bootstrapping given a dust temperature ($T_d$) distribution for lensed 1.3mm-selected galaxies [@wei13] and an SED library with $T_d$ information compiled in @swi14. For these assumptions, we obtain a possible range of Eddington-limited star formation rates. For a more direct comparison of the millimeter fluxes and sizes of SMGs with Eddington-limited star formation, we re-plot 51 of the 69 SMGs at $z=0.7$–6.8 with optical/near-IR photometric or spectroscopic redshifts on the SFR–physical size plane (Fig. \[fig:sizeflux\], right panel). The SFRs are derived from $F_{\rm 1100 \mu m}$, given the range of possible dust temperatures $T_d$ and SEDs noted above. We assume that the AGN contribution to the submillimeter flux is negligible [see references in @ros12]. In order to visualize the coverage of the size-SFR plane produced by the large SFR uncertainties (due to the unknown dust SED temperatures), we show the full SFR probability density distribution (rather than a single value) for each SMG. The results in both panels of Fig. \[fig:sizeflux\] show that the SMGs avoid the SFR region around the Eddington limit, suggesting that the minimum possible millimeter-wave sizes for bright SMGs are given by the Eddington limited star formation. The empirical relation between flux and size can explain the apparent discrepancy between the reported (sub)millimeter-wave (median) sizes of $0.''20^{+0.''03}_{-0.''05}$ by @ika15 and $0.''3\pm0.''04$ by @sim15. Given $F_{\rm 870 \mu m}/F_{\rm 1100 \mu m}=2$ for conversion of the observed fluxes, @sim15 covered $F_{\rm 1100 \mu m} \gtrsim 2.5$mJy. In this regime, our ASXDF SMGs have a median size of$0.''31^{+0.''03}_{-0.''03}$. \[fig:sizeflux\] Relation between millimeter sizes and AGN {#sec:agn} ========================================= We present our studies of the connection between the millimeter-wave sizes and AGN in SMGs, based on a mid-IR AGN diagnostic. We consider 25 ALMA-identified SMGs with $1<z_{\rm phot\,or\,spec}<3$, which are detected in all IRAC and MIPS 24$\mu$m images. All SMGs here have redshift information and a single component at $\sim$0$''$.2 resolution. More than 15 out of the 25 are located above $4\times$ the main sequence at $z\sim2$ in the stellar mass vs. SFR plane (Fig. \[fig:masssfr\]), indicating that the majority of the sample are starbursts [@bis18]. Note that among the 29 SMGs with $z=1$–3, four are not considered in our analysis: two SMGs are not detected at 24$\mu$m and the other two are blended in the IRAC maps. Mid-IR AGN diagnostic --------------------- A 4.5$\mu$m/8$\mu$m/24$\mu$m color-color plot has often been used as an AGN diagnostic for high-redshift, dusty infrared-luminous galaxies, such as SMGs and DOGs at $z\sim2$ [e.g. @ivi04; @ivi07; @pop08a; @pop08b]. We refer the reader to @kir15, who presented a detailed study of mid-IR SED evolution versus AGN fraction for high-$z$ galaxies. Empirical SED templates (top left panel in Fig. \[fig:sizeagn\]) suggest that high-redshift galaxies dominated by star formation or AGN in mid-IR light can be segregated from each other in the mid-IR color-color plane. The position of our 25 SMGs in this color-color plot shows that some of them do not follow either the model tracks for star-formation-dominated or AGN-dominated galaxies. We generated the expected mid-IR colors of galaxies that are a composite of SF and AGN by combining SEDs of SF and AGN with various SF/AGN ratios. This ‘toy’ color prediction reproduces the colors of ‘composite SMGs’ which are likely to be dominated by neither an AGN nor a starburst in the mid-IR (top right panel in Fig. \[fig:sizeagn\]). We divided the 25 SMGs into four sub-groups based on their 4.5/8/24-$\mu$m colors: star-forming, composite, AGN-dominant and ‘no class’. The criteria are: - $F_{\rm 8 \mu m}/F_{\rm 4.5 \mu m}<1.15$ $\bigwedge$ $F_{\rm 24 \mu m}/F_{\rm 8 \mu m}\geq 5$ (star-forming) - $F_{\rm 8 \mu m}/F_{\rm 4.5 \mu m}\geq1.15$ $\bigwedge$ $F_{\rm 24 \mu m}/F_{\rm 8 \mu m}\geq 5$ (composite) - $F_{\rm 8 \mu m}/F_{\rm 4.5 \mu m}\geq1.50$ $\bigwedge$ $F_{\rm 24 \mu m}/F_{\rm 8 \mu m}< 5$ (AGN) - $F_{\rm 8 \mu m}/F_{\rm 4.5 \mu m}<1.50$ $\bigwedge$ $F_{\rm 24 \mu m}/F_{\rm 8 \mu m}< 5$ (no class). The model colors (top, Fig. \[fig:sizeagn\]) indicate that the SMGs categorized as ‘no class’ could be in the star-forming or composite classes. Due to this ambiguity, we consider the ‘no class’ separately. Note that, In our diagnostic, the star-forming class and AGN dominant class are defined first. We choose $F_{\rm 8.0 \mu m}/F_{\rm 4.5 \mu m}=$1.15 as criterion for separation, as this ensures that all galaxies that satisfy neither an AGN criteria by @don12 nor another criteria by @ste05 also lie on the star-forming region of the colour-colour diagram. The predicted 24$\mu$m/8$\mu$m color evolution with redshift, as derived by public empirical mid-IR SED templates for high-$z$ star-forming galaxies, composite galaxies, and AGN dominant galaxies [@kir15], are shown along with our sample SMGs (bottom left, Fig. \[fig:sizeagn\]). For these templates, the respective mid-IR AGN fractions of each sample are $<$20, 20–80, and $\geq$80%. In this plot we averaged the public SEDs in each AGN class, after scaling all fluxes at $\lambda_{\rm rest}=8$$\mu$m. The predictions based on the Kirkpatrick et al. SED templates suggest that our criteria for 24$\mu$m/8$\mu$m color can work to select an AGN-dominant class, and show that our composite type is expected to have typically AGN fractions of around $\sim$50%, consistently with our ’toy’ models. Results ------- In the millimeter-wave physical size vs. SFR plot (bottom right panel in Fig. \[fig:sizeagn\]), all SMGs with composite mid-IR components are evidently more compact and located closer to the Eddington limit than the other SMGs with star-forming dominant or AGN dominant mid-IR components. The respective median $R_{\rm c,e}$ for the SMGs classified as star-forming dominant, composites, and AGN dominant are 1.6$^{+0.34}_{-0.21}$, 1.0$^{+0.20}_{-0.20}$, and 1.5$^{+0.93}_{-0.24}$kpc. The size difference between the SMGs with composite and star-forming mid-IR components, and the difference between the SMGs with composite and AGN-dominant mid-IR components are real, with a significance level of $>99$%, according to the Kolmogorov-Smirnov test. This indicates that the composite type galaxies are characterized by more compact star-forming regions than those of the star-forming or AGN-dominant types. We also explored the relation between size and stellar mass in our sample and found that the size differences are not a consequence of different stellar masses. Composite SMGs are the most compact of the three types, even at fixed stellar mass. None of our ALMA-identified AzTEC SMGs are detected in the existing [*XMM-Newton*]{} X-ray maps [@ued08], probably because these maps are too shallow. Nevertheless, we can compare our results with the sizes derived for the host galaxies of five high-$z$, X-ray-selected AGN ($L_{\rm 2-8keV}=10^{42.1-43.6}$ergs$^{-1}$) by @har16. These authors reported a size distribution for their AGN hosts similar to the SMG sizes in @sim15. The most X-ray luminous source in their sample (with $L_{\rm 2-8keV}=10^{43.6}$ergs$^{-1}$) has an extended size, and the remaining four ($L_{\rm 2-8keV}=10^{42.1-43.4}$ergs$^{-1}$) have compact sizes, which are comparable to those of our composite type here (Fig. \[fig:sizeagn\], bottom right). AGN growth during a very compact star-forming phase? ---------------------------------------------------- The very compact millimeter-wave sizes of the SMGs with composite mid-IR components suggest that a central supermassive black hole could be growing in a compact and coalescing star-forming phase, which is consistent with the predictions of @spr05 for galaxy major mergers. The extended millimeter-wave sizes of the SMGs of the star-forming dominant class can be explained by a mid-stage merger as seen in, e.g., VV114 [@sai15]. Actually ASXDF1100.055.1 with the star-forming dominant class shows merger-like near-IR morphology (Fig.\[fig:hst\]). Instead, the extended sizes of the SMGs with the AGN-dominant class are puzzling. In line with the evolutionary scenarios of, e.g., @san88 [@hop08; @tof14] that SMGs evolve into QSOs, these extended sizes may be explained by positive AGN feedback by a growing supermassive black hole in the phase of star-formation quenching, as it is suggested by simulations for luminous AGN/QSOs [e.g. @ish12; @zub13] and considered for some luminous QSOs [e.g. @car16]. In fact, ASXDF1100.057.1 with the AGN dominant class has a QSO-like near-IR morphology (Fig.\[fig:hst\]). However, no significant near-IR morphological difference between AGN-host and non-AGN-host galaxies, that are not submillimeter selected, is reported [e.g. @koc12]. The extended submillimeter sizes in our SMGs may come from the nature of their host galaxies. [ l c c c c c c c c c ]{} ID & R.A. & Dec. &SNR& $F_{\rm 1100 \mu m}$ & $z_{\rm photo}$ & SFR & mm-wave size & mm-wave size & AGN\ & (J2000) & (J2000) & & (mJy) & & ($M_{\odot}$yr$^{-1}$) & (FWHM; arcsec) & ($R_{\rm c,e}$; kpc) & (mid-IR)\ ASXDF1100.002.1 & 2:17:30.63 & -4:59:36.8 & 15 & 4.81$\pm$0.43 & 3.3$^{+0.07}_{-0.87}$ & 990$^{+720}_{-340}$ & 0.42$^{+0.06}_{-0.02}$ & 1.6$^{+0.2}_{-0.1}$ &\ ASXDF1100.004.1 & 2:18:05.65 & -5:10:49.7 & 14 & 4.39$\pm$0.56 & 3.5$^{+0.35}_{-0.16}$ & 880$^{+420}_{-290}$ & 0.40$^{+0.06}_{-0.04}$ & 1.5$^{+0.2}_{-0.1}$ &\ ASXDF1100.005.1 & 2:17:30.45 & -5:19:22.5 & 25 & 7.24$\pm$0.45 & 0.7$^{+0.01}_{-0.01}$ & 1200$^{+990}_{-420}$ & 0.34$^{+0.04}_{-0.02}$ & 1.2$^{+0.1}_{-0.1}$ &\ ASXDF1100.006.1 & 2:17:27.32 & -5:06:42.8 & 10 & 5.11$\pm$0.50 & 4.5$^{+0.18}_{-0.15}$ & 930$^{+340}_{-330}$ & 0.68$^{+0.06}_{-0.06}$ & 2.2$^{+0.2}_{-0.2}$ &\ ASXDF1100.007.1 & 2:18:03.01 & -5:28:42.0 & 20 & 6.26$\pm$0.53 & 3.2$^{+0.28}_{-0.22}$ & 1300$^{+930}_{-450}$ & 0.32$^{+0.04}_{-0.02}$ & 1.2$^{+0.1}_{-0.1}$ &\ ASXDF1100.008.1 & 2:16:47.93 & -5:01:29.9 & 12 & 6.45$\pm$0.59 & 2.2$^{+0.02}_{-0.08}$ & 1500$^{+950}_{-460}$ & 0.62$^{+0.06}_{-0.06}$ & 2.6$^{+0.2}_{-0.2}$ & AGN\ ASXDF1100.009.1A & 2:17:42.11 & -4:56:27.6 & 19 & 4.68$\pm$0.40 &(0.5)$^a$ & 550$^{+430}_{-190}$ & 0.30$^{+0.02}_{-0.04}$ & 0.9$^{+0.1}_{-0.1}$ &\ ASXDF1100.009.1B & 2:17:42.16 & -4:56:28.5 & 11 & 1.16$\pm$0.12 &(0.5)$^a$ & 140$^{+110}_{-50}$ & 0.10$^{+0.08}_{-0.06}$ & 0.6$^{+0.5}_{-0.4}$ &\ ASXDF1100.011.1 & 2:17:50.59 & -5:30:59.2 & 13 & 4.22$\pm$0.41 & 5.5$^{+0.08}_{-0.63}$ & 730$^{+440}_{-260}$ & 0.38$^{+0.04}_{-0.04}$ & 1.1$^{+0.1}_{-0.1}$ &\ ASXDF1100.014.1$^{\dagger}$ & 2:17:29.77 & -5:03:18.6 & 11 & 3.12$\pm$0.17 & 2.2$^{+0.04}_{-0.03}$ & 690$^{+270}_{-210}$ & 0.50$^{+0.06}_{-0.08}$ & 2.1$^{+0.2}_{-0.3}$ & SF\ ASXDF1100.016.1 & 2:16:41.11 & -5:03:51.4 & 19 & 4.79$\pm$0.35 & 5.0$^{+0.54}_{-0.06}$ & 850$^{+390}_{-240}$ & 0.24$^{+0.02}_{-0.04}$ & 0.8$^{+0.1}_{-0.1}$ &\ ASXDF1100.018.1 & 2:18:13.83 & -4:57:43.5 & 14 & 3.47$\pm$0.32 & 1.7$^{+0.09}_{-0.02}$ & 850$^{+650}_{-280}$ & 0.26$^{+0.04}_{-0.04}$ & 1.1$^{+0.2}_{-0.2}$ & NO\ ASXDF1100.020.1$^{\bullet}$ & 2:18:23.73 & -5:11:38.5 & 13 & 4.94$\pm$0.43 & 2.7$^{+0.01}_{-0.01}$ & 1100$^{+460}_{-380}$ & 0.30$^{+0.04}_{-0.02}$ & 1.2$^{+0.2}_{-0.1}$ &\ ASXDF1100.021.1 & 2:18:16.49 & -4:55:08.8 & 16 & 4.03$\pm$0.28 & 2.3$^{+0.03}_{-0.04}$ & 920$^{+720}_{-310}$ & 0.28$^{+0.02}_{-0.04}$ & 1.1$^{+0.1}_{-0.2}$ & COM\ ASXDF1100.022.1 & 2:18:42.68 & -4:59:32.1 & 15 & 3.09$\pm$0.31 & 2.3$^{+0.01}_{-0.06}$ & 710$^{+550}_{-240}$ & 0.20$^{+0.04}_{-0.04}$ & 0.8$^{+0.2}_{-0.2}$ & COM\ ASXDF1100.023.2 & 2:18:20.40 & -5:31:43.2 & 10 & 2.17$\pm$0.27 & 2.5$^{+0.10}_{-0.12}$ & 480$^{+350}_{-160}$ & 0.16$^{+0.10}_{-0.06}$ & 0.6$^{+0.4}_{-0.2}$ &\ ASXDF1100.025.2$^{\dagger}$ & 2:17:32.59 & -4:50:26.4 & 13 & 2.34$\pm$0.12 & 3.4$^{+0.16}_{-0.07}$ & 470$^{+320}_{-150}$ & 0.34$^{+0.06}_{-0.04}$ & 1.3$^{+0.2}_{-0.1}$ &\ ASXDF1100.029.1$^{\dagger}$ & 2:17:20.80 & -4:49:49.5 & 11 & 2.67$\pm$0.21 & 2.8$^{+0.16}_{-0.17}$ & 570$^{+360}_{-180}$ & 0.46$^{+0.08}_{-0.10}$ & 1.8$^{+0.3}_{-0.4}$ & AGN\ ASXDF1100.031.1$^{\dagger}$ & 2:17:37.24 & -4:47:53.0 & 13 & 2.09$\pm$0.15 & 2.5$^{+0.18}_{-0.12}$ & 480$^{+380}_{-170}$ & 0.28$^{+0.04}_{-0.06}$ & 1.1$^{+0.2}_{-0.2}$ & COM\ ASXDF1100.033.1 & 2:18:03.56 & -4:55:27.3 & 15 & 4.86$\pm$0.33 & (2.6)$^c$ & 1100$^{+860}_{-350}$ & 0.34$^{+0.04}_{-0.02}$ & 1.4$^{+0.2}_{-0.1}$ & COM\ ASXDF1100.034.1 & 2:17:59.32 & -5:05:04.6 & 11 & 2.84$\pm$0.32 & (1.6)$^b$& 680$^{+640}_{-220}$ & 0.16$^{+0.08}_{-0.06}$ & 0.7$^{+0.3}_{-0.3}$ &\ ASXDF1100.035.1$^{\dagger,\bullet}$ & 2:17:35.37 & -5:28:37.3 & 12 & 2.09$\pm$0.12 & 2.7$^{+0.07}_{-0.11}$ & 450$^{+360}_{-150}$ & 0.52$^{+0.08}_{-0.08}$ & 2.1$^{+0.3}_{-0.3}$ &\ ASXDF1100.041.1 & 2:17:53.87 & -5:26:35.7 & 10 & 2.91$\pm$0.29 & 0.8$^{+0.00}_{-0.00}$ & 520$^{+260}_{-180}$ & 0.42$^{+0.06}_{-0.10}$ & 1.6$^{+0.2}_{-0.4}$ &\ ASXDF1100.042.1 & 2:18:38.29 & -5:03:18.3 & 12 & 3.26$\pm$0.40 & 3.2$^{+0.02}_{-0.01}$ & 680$^{+440}_{-240}$ & 0.42$^{+0.04}_{-0.06}$ & 1.6$^{+0.1}_{-0.2}$ &\ ASXDF1100.044.1 & 2:17:45.85 & -5:00:56.7 & 12 & 1.93$\pm$0.26 & 6.8$^{+0.20}_{-0.72}$ & 330$^{+210}_{-84}$ & 0.09$^{+0.07}_{-0.05}$ & 0.2$^{+0.2}_{-0.1}$ &\ ASXDF1100.046.1 & 2:17:13.34 & -4:58:57.4 & 16 & 4.00$\pm$0.32 & 3.5$^{+0.01}_{-0.10}$ & 810$^{+620}_{-280}$ & 0.28$^{+0.04}_{-0.04}$ & 1.0$^{+0.1}_{-0.1}$ &\ ASXDF1100.047.1$^{\dagger}$ & 2:17:56.73 & -4:52:39.0 & 11 & 2.25$\pm$0.17 & 2.2$^{+0.01}_{-0.02}$ & 500$^{+400}_{-160}$ & 0.40$^{+0.08}_{-0.06}$ & 1.6$^{+0.3}_{-0.2}$ & SF\ ASXDF1100.048.1$^{\dagger}$ & 2:17:46.16 & -4:47:47.2 & 14 & 2.55$\pm$0.11 & 2.5$^{+0.21}_{-0.12}$ & 570$^{+460}_{-200}$ & 0.40$^{+0.06}_{-0.04}$ & 1.6$^{+0.2}_{-0.2}$ & NO\ ASXDF1100.050.1$^{\star}$ & 2:18:22.30 & -5:07:37.0 & 11 & 3.32$\pm$0.40 & 3.0$^{+0.15}_{-0.15}$ & 700$^{+360}_{-240}$ & 0.24$^{+0.08}_{-0.08}$ & 0.9$^{+0.3}_{-0.3}$ &\ ASXDF1100.051.1$^{\dagger}$ & 2:18:23.96 & -5:32:07.8 & 12 & 2.63$\pm$0.23 & 0.7$^{+0.00}_{-0.04}$ & 430$^{+270}_{-150}$ & 0.08$^{+0.06}_{-0.04}$ & 0.3$^{+0.2}_{-0.1}$ &\ ASXDF1100.051.2$^{\dagger}$ & 2:18:24.59 & -5:31:48.5 & 11 & 2.88$\pm$0.23 & 4.7$^{+0.24}_{-0.15}$ & 520$^{+270}_{-160}$ & 0.30$^{+0.10}_{-0.06}$ & 1.0$^{+0.3}_{-0.2}$ &\ ASXDF1100.052.1$^{\dagger}$ & 2:17:33.17 & -5:01:54.5 & 11 & 2.05$\pm$0.14 & 2.8$^{+0.25}_{-0.65}$ & 440$^{+340}_{-150}$ & 0.34$^{+0.04}_{-0.06}$ & 1.3$^{+0.2}_{-0.2}$ & AGN\ ASXDF1100.055.1$^{\dagger}$ & 2:17:20.03 & -5:13:05.8 & 13 & 2.54$\pm$0.15 & 2.1$^{+0.02}_{-0.24}$ & 570$^{+290}_{-180}$ & 0.34$^{+0.06}_{-0.06}$ & 1.4$^{+0.2}_{-0.2}$ & SF\ ASXDF1100.057.1 & 2:17:32.41 & -5:12:50.9 & 12 & 3.54$\pm$0.38 & 1.9$^{+0.04}_{-0.11}$ & 820$^{+360}_{-260}$ & 0.34$^{+0.04}_{-0.06}$ & 1.4$^{+0.2}_{-0.3}$ & AGN\ ASXDF1100.076.1 & 2:16:41.04 & -5:01:12.5 & 13 & 4.13$\pm$0.55 & 4.8$^{+0.13}_{-0.41}$ & 750$^{+550}_{-230}$ & 0.34$^{+0.04}_{-0.06}$ & 1.1$^{+0.1}_{-0.2}$ &\ ASXDF1100.077.1$^{\dagger}$ & 2:18:11.00 & -4:49:51.9 & 12 & 1.69$\pm$0.20 & 4.1$^{+0.02}_{-0.12}$ & 320$^{+190}_{-110}$ & 0.22$^{+0.08}_{-0.08}$ & 0.8$^{+0.3}_{-0.3}$ &\ ASXDF1100.089.1 & 2:18:10.64 & -5:34:53.6 & 21 & 4.73$\pm$0.30 & 5.4$^{+0.11}_{-0.09}$ & 830$^{+600}_{-200}$ & 0.24$^{+0.04}_{-0.02}$ & 0.7$^{+0.1}_{-0.1}$ &\ ASXDF1100.095.1$^{\dagger}$ & 2:17:12.97 & -5:14:12.2 & 10 & 1.91$\pm$0.19 & 2.2$^{+0.11}_{-0.08}$ & 440$^{+320}_{-150}$ & 0.32$^{+0.08}_{-0.08}$ & 1.3$^{+0.3}_{-0.3}$ & AGN\ ASXDF1100.100.1 & 2:17:53.25 & -4:49:51.5 & 13 & 2.84$\pm$0.29 & 2.2$^{+0.16}_{-0.08}$ & 670$^{+550}_{-210}$ & 0.24$^{+0.04}_{-0.04}$ & 1.0$^{+0.2}_{-0.2}$ & COM\ ASXDF1100.105.1 & 2:18:02.86 & -5:00:31.6 & 13 & 2.86$\pm$0.30 & (1.1)$^b$ & 630$^{+460}_{-220}$ & 0.24$^{+0.06}_{-0.08}$ & 1.0$^{+0.2}_{-0.3}$ & COM\ ASXDF1100.107.1$^{\dagger}$ & 2:18:07.85 & -5:25:49.3 & 11 & 1.67$\pm$0.16 & 4.6$^{+0.18}_{-0.86}$ & 310$^{+190}_{-80}$ & 0.34$^{+0.06}_{-0.06}$ & 1.1$^{+0.2}_{-0.2}$ &\ ASXDF1100.115.1 & 2:16:59.42 & -5:10:55.8 & 12 & 4.23$\pm$0.33 & (0.6)$^a$ & 600$^{+500}_{-220}$ & 0.50$^{+0.06}_{-0.06}$ & 1.7$^{+0.2}_{-0.2}$ &\ ASXDF1100.134.1 & 2:17:54.80 & -5:23:23.8 & 15 & 3.27$\pm$0.27 & 2.5$^{+0.16}_{-0.05}$ & 740$^{+500}_{-260}$ & 0.24$^{+0.06}_{-0.04}$ & 1.0$^{+0.2}_{-0.2}$ & COM\ ASXDF1100.156.1 & 2:16:38.33 & -5:01:21.5 & 11 & 3.33$\pm$0.31 & 1.8$^{+0.04}_{-0.10}$ & 810$^{+630}_{-260}$ & 0.34$^{+0.06}_{-0.06}$ & 1.4$^{+0.3}_{-0.3}$ & SF\ ASXDF1100.188.1$^{\dagger,\star}$ & 2:16:41.94 & -5:07:04.3 & 10 & 2.42$\pm$0.18 & 2.6$^{+0.28}_{-0.20}$ & 530$^{+450}_{-180}$ & 0.22$^{+0.10}_{-0.08}$ & 0.9$^{+0.4}_{-0.3}$ &\ ASXDF1100.203.1$^{\dagger}$ & 2:18:23.15 & -5:27:02.0 & 11 & 1.90$\pm$0.12 & 2.5$^{+0.03}_{-0.15}$ & 440$^{+330}_{-150}$ & 0.34$^{+0.10}_{-0.10}$ & 1.4$^{+0.4}_{-0.4}$ & NO\ ASXDF1100.227.1 & 2:17:44.27 & -5:20:08.6 & 24 & 7.42$\pm$0.57 & 3.7$^{+0.35}_{-0.14}$ & 1400$^{+760}_{-510}$ & 0.34$^{+0.02}_{-0.02}$ & 1.2$^{+0.1}_{-0.1}$ &\ ASXDF1100.228.1 & 2:18:09.66 & -5:18:43.1 & 12 & 3.11$\pm$0.34 & 1.9$^{+0.05}_{-0.14}$ & 740$^{+610}_{-240}$ & 0.38$^{+0.06}_{-0.06}$ & 1.6$^{+0.3}_{-0.2}$ & SF\ ASXDF1100.229.1 & 2:18:18.84 & -4:50:29.9 & 11 & 3.60$\pm$0.36 & 2.3$^{+0.05}_{-0.11}$ & 820$^{+620}_{-270}$ & 0.26$^{+0.06}_{-0.08}$ & 1.1$^{+0.2}_{-0.3}$ & COM\ ASXDF1100.235.1 & 2:17:36.00 & -5:20:34.4 & 13 & 4.64$\pm$0.40 & 2.3$^{+0.04}_{-0.14}$ & 1100$^{+820}_{-370}$ & 0.26$^{+0.06}_{-0.04}$ & 1.1$^{+0.2}_{-0.2}$ & COM\ ASXDF1100.236.1$^{\dagger}$ & 2:17:21.54 & -5:19:07.7 & 11 & 1.65$\pm$0.14 & 2.4$^{+0.02}_{-0.02}$ & 370$^{+250}_{-120}$ & 0.15$^{+0.09}_{-0.09}$ & 0.6$^{+0.4}_{-0.4}$ & COM\ ASXDF1100.247.1$^{\dagger}$ & 2:16:33.85 & -5:02:42.7 & 11 & 1.87$\pm$0.18 & 2.6$^{+0.11}_{-0.14}$ & 410$^{+260}_{-140}$ & 0.24$^{+0.08}_{-0.10}$ & 1.0$^{+0.3}_{-0.4}$ & COM\ ASXDF1100.003.1$^{\dagger}$ & 2:16:44.48 & -5:02:21.6 &15 &2.85$\pm$0.13 && &0.36$^{+0.04}_{-0.04}$ &&\ ASXDF1100.010.1 & 2:17:39.79 & -5:29:19.2 &24 &5.94$\pm$0.37 && &0.28$^{+0.02}_{-0.02}$ &&\ ASXDF1100.026.1$^{\dagger}$ & 2:17:42.55 & -5:29:00.3 &11 &1.69$\pm$0.17 && & 0.18$^{+0.06}_{-0.12}$&&\ ASXDF1100.040.1 & 2:17:55.24 & -5:06:45.1 &15 &3.14$\pm$0.35 && &0.20$^{+0.06}_{-0.04}$&&\ ASXDF1100.053.1 & 2:16:48.20 & -4:58:59.6 &10 &4.02$\pm$0.51 && &0.42$^{+0.06}_{-0.06}$ &&\ ASXDF1100.054.1 & 2:17:15.41 & -4:57:55.6 &11 &4.12$\pm$0.38 && &0.38$^{+0.06}_{-0.06}$ &&\ ASXDF1100.068.1 & 2:17:42.17 & -5:25:46.8 &12 &3.24$\pm$0.30 && &0.24$^{+0.04}_{-0.06}$&&\ ASXDF1100.070.1$^{\dagger}$ & 2:18:46.15 & -5:04:12.5 &12 &2.17$\pm$0.13 && &0.30$^{+0.04}_{-0.06}$&&\ ASXDF1100.074.1 & 2:18:33.31 & -4:58:07.0 &10 &2.77$\pm$0.33 && &0.32$^{+0.06}_{-0.06}$ &&\ ASXDF1100.097.1 & 2:18:18.54 & -5:34:34.7 &11 &2.53$\pm$0.26 && &0.20$^{+0.08}_{-0.06}$ & &\ ASXDF1100.097.2$^{\dagger}$ & 2:18:17.61 & -5:34:27.9 &10 &2.14$\pm$0.26 && &0.32$^{+0.08}_{-0.10}$& &\ ASXDF1100.133.1 & 2:18:05.51 & -5:35:46.5 &11 &2.25$\pm$0.26 && &0.08$^{+0.08}_{-0.04}$ & &\ ASXDF1100.161.1$^{\dagger}$ & 2:18:13.76 & -5:37:27.3 &12 &2.68$\pm$0.20 && &0.44$^{+0.06}_{-0.06}$ & &\ ASXDF1100.168.1 & 2:18:04.37 & -5:34:03.5 &11 &1.79$\pm$0.21 && &0.16$^{+0.08}_{-0.06}$& &\ ASXDF1100.213.1$^{\dagger}$ & 2:18:44.02 & -5:35:31.3 &12 &2.90$\pm$0.28 && &0.16$^{+0.08}_{-0.08}$& &\ ASXDF1100.231.1 & 2:17:59.65 & -4:46:49.8 &12 &2.88$\pm$0.36 && &0.28$^{+0.08}_{-0.08}$ &&\ ASXDF1100.243.1$^{\dagger}$ & 2:16:50.43 & -5:10:16.2 &10 &2.09$\pm$0.20 && &0.37$^{+0.09}_{-0.11}$&&\ ASXDF1100.252.1 & 2:17:05.65 & -5:15:04.9 &12 &2.62$\pm$0.25 && &0.24$^{+0.06}_{-0.08}$ &&\ \ \ \ \ \ \ \ \ Alexander, D. M., Bauer, F. E., Chapman, S. C., et al. 2005, , 632, 736 Banerji, M., Chapman, S. C., Smail, I., et al. 2011, , 418, 1071 Biggs, A. D., & Ivison, R. J. 2008, , 385, 893 Bisigello, L., Caputi, K. 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Therefore, we corrected the primary flux calibration for this effect.

--- abstract: 'For an arbitrary Hilbert space-valued Ornstein-Uhlenbeck process we construct the Ornstein-Uhlenbeck Bridge connecting a starting point $x$ and an endpoint $y$ that belongs to a certain linear subspace of full measure. We derive also a stochastic evolution equation satisfied by the OU Bridge and study its basic properties. The OU Bridge is then used to investigate the Markov transition semigroup associated to a nonlinear stochastic evolution equation with additive noise. We provide an explicit formula for the transition density and study its regularity. Given the Strong Feller property and the existence of an invariant measure we show that the transition semigroup maps $L^p$ functions into continuous functions. We also show that transition operators are $q$-summing for some $q>p>1$, in particular of Hilbert-Schmidt type.' address: - 'School of Mathematics, The University of New South Wales, Sydney 2052, Australia' - | Institute of Mathematics, Academy of Sciences of Czech Republic\ Žitn' a 25, 11567 Praha 1, Czech Republic author: - 'B. Goldys' - 'B. Maslowski' title: THE ORNSTEIN UHLENBECK BRIDGE AND APPLICATIONS TO MARKOV SEMIGROUPS --- [^1] Introduction ============ Let $\left(Z_t^x\right)$ be an Ornstein-Uhlenbeck process on a separable Hilbert space $H$. By this we mean that $\left(Z_t^x\right )$ is a solution to a linear stochastic evolution equation $$\left\{\begin{array}{l} dZ_t^x=AZ_t^xdt+\sqrt {Q}dW_t,\\ Z_0^x=x\in H.\end{array} \right.\label{01}$$ In the above equation $\left(W_t\right)$ is a standard cylindrical Wiener process defined on a certain stochastic basis $\left(\Omega ,\mathcal{F},\left(\mathcal{F}_t\right),\mathbb{P}\right )$ and $Q=Q^{*}\ge 0$ is a bounded operator on $H$. We assume that the operator $\left(A,\mbox{\rm dom} (A)\right)$ is a generator of a $C_0$-semigroup $\left(S_t\right)$ on $H$. Under the assumptions given below the solution to (\[01\]) is defined by the formula $$Z_t^x=S_tx+\int_0^tS_{t-s}\sqrt QdW_s.\label{02}$$ The aim of this paper is to study the basic properties of the Ornstein-Uhlenbeck Bridge (sometimes called a Pinned Ornstein-Uhlenbeck process) $\left(\hat Z_t^{x,y}\right)$ associated to the Ornstein-Uhlenbeck process $\left(Z_t^x\right)$ and its applications. Let us recall informally, that this process is defined via the formula $$\mathbb P\left(\left.Z_t^x\in B\right|Z_T^x=y\right)=\mathbb P\left (\hat Z_t^{x,y}\in B\right),\quad t<T,$$ where $x,y\in H$ and $B\subset H$ is a Borel set. Intuitively, it is an Ornstein-Uhlenbeck process “conditioned to go from $x$ at time $t=0$ to $y$ at time $t=T$” (a rigorous definition is given in Section 2, cf. Def. \[OUB\]). The importance of various types of bridge processes in the theory of finite dimensional diffusions is well recognised, see for example [@yor]. In infinite dimensional framework this concept was developed in [@simao1] in order to study regularity of transition semigroup of certain linear and nonlinear diffusions on Hilbert space. In [@masi1] and [@masi2] an Ornstein-Uhlenbeck Bridge is introduced in order to obtain lower estimates on the transition kernel of some semilinear stochastic evolution equations. Those estimates provide a powerful tool to study exponential ergodicity and $V$-uniform ergodicity for such equations. In particular, they allowed us to obtain in [@den] explicit estimates of the rate of exponential convergence to the invariant measure. In the present paper the OU Bridge is studied under much more general conditions and in more detail. We provide also further applications of the OU Bridge to the analysis of transition densities and the regularity of associated Markov semigroups. Regularity of Strongly Feller transition semigroups was studied in [@furman] (see also references therein). We use methods completely different from [@furman] and obtain stronger results but for bounded drifts only while the aforementioned paper allows linearly growing drifts. Closely related results for semigroups that are not strongly Feller may be found in [@ania]. For the regularity of strongly Feller semigroups associated to the OU process we refer to [@reg]. Let us describe the contents of this paper. In Section 2 we provide, for the reader’s convenience, some relevant facts about linear measurable mappings and conditional distributions of Hilbert space valued Gaussian random vectors. Then we give a definition of the OU Bridge and some basic results on OU processes and OU Bridges. Some of the technical results from [@den] that are needed in the sequel are stated without proof and others (Lemma \[vt1a\], Proposition \[ou0\] and Lemma \[lh1\]) are reproved under more general conditions. In Section 3, a stochastic equation for the OU Bridge is derived. A new Brownian Motion adapted to the filtration of the Ornstein Uhlenbeck Bridge is obtained and then it is shown that the Bridge process is a unique mild (and weak) solution of a linear nonhomogeneous stochastic evolution equation with singular coefficients. Section 4 is devoted to applications of the previous results to semilinear stochastic equations; continuity of Markov transition densities (with respect to the Gaussian invariant measure $\nu $ that is an invariant measure with respect to the OU process) is proved (Theorem \[dens\] and Remark \[iny\]), the Markov semigroup is shown to map the space $L^p(H,\nu )$, $p>1$, into the space of continuous functions on $H$ (Theorem \[lp\]) and is also shown to be Hilbert-Schmidt on $L^2(H, \nu )$ and $q$-summing (in particular, compact) as a mapping $L^p(H, \nu ) \to L^q(H, \nu )$ even if $q>p$ provided the gap between $q$ and $p$ is not too large (Theorem \[HSS\]). At the end of the section the results are illustrated in the case of one-dimensional semilinear stochastic parabolic equation (Example \[example\]) in which case all conditions imposed in the paper are verified or specified. ACKNOWLEDGEMENT. The authors are grateful to Jan Seidler for his valuable comments and suggestions. Preliminaries on OU Processes and Bridges ========================================= In this section we collect, for the reader convenience, some properties of infinite-dimensional OU processes and Gaussian random variables which will be useful in the paper. We also define the OU Bridge and recall some known results that will be useful in the sequel. Measurable Linear Mappings -------------------------- Let $H$ be a real separable Hilbert space and let $\mu =N(0,C)$ be a centered Gaussian measure on $H$ with the covariance operator $C$ such that $\overline {\mbox{\rm im}(C)}=H$. The space $H_C=\mathrm{im}\left(C^{1/2}\right)$ endowed with the norm $|x|_ C=\left|C^{-1/2}x\right|$ can be identified as the Reproducing Kernel Hilbert Space of the measure $\mu$. In the sequel we will denote by $\left\{e_n:n\ge 1\right \}$ the eigenbasis of $C$ and by $\left\{c_n:n\ge 1\right\}$ the corresponding set of eigenvalues: $$Ce_n=c_ne_n,\quad n\ge 1.$$ For any $h\in H$ we define $$\phi_n(x)=\sum_{k=1}^n\frac 1{\sqrt {c_k}}\left\langle h,e_k\right \rangle\left\langle x,e_k\right\rangle ,\quad x\in H.$$ The following two lemmas are well known (see e.g. [@den]): \[fiha\] The sequence $\left(\phi_n\right)$ converges in $L^2(H,\mu )$ to a limit $ \phi$ and $$\int_H\left|\phi (x)\right|^2\mu (dx)=|h|^2.$$ Moreover, there exists a measurable linear space $\mathcal{M}_h\subset H$, such that $\mu\left(\mathcal{M}_h\right)= 1$, $\phi$ is linear on $\mathcal M_h$ and $$\phi (x)=\lim_{n\to\infty}\phi_n(x),\quad x\in\mathcal M_h.\label{limit}$$ We will use the notation $\phi (x)=\left\langle h,C^{-1/2}x\right \rangle$. Let $H_1$ be another real separable Hilbert space and let $T:H\to H_1$ be a bounded operator. The Hilbert-Schmidt norm of $T$ will be denoted by $\left\|T\right\|_{HS}$. Let $$\tilde {T}_nx=\sum_{k=1}^n\frac 1{\sqrt {c_k}}\left\langle x,e_ k\right\rangle Te_k,\quad x\in H.$$ \[tmeas\] Let $T:H\to H_1$ be a Hilbert-Schmidt operator. Then the sequence $\left(\tilde T_n\right)$ converges in $L^2\left(H,\mu ; H_1\right)$ to a limit $\tilde {T}$ and $$\int_H\left|\tilde T(x)\right|^2_{H_1}\mu (dx)=\left\|T\right\|_{ HS}^2.$$ Moreover, there exists a measurable linear space $\mathcal M_T\subset H$, such that $\mu\left(\mathcal M_T\right)= 1$, $\tilde {T}$ is linear on $\mathcal M_T$ and $$\tilde {T}(x)=\lim_{n\to\infty}\tilde {T}_nx,\quad x\in\mathcal M_T.\label{limit1}$$ We will use the notation $TC^{-1/2}x := \tilde T(x)$. The above procedure is specified in the following Lemma (the proof of which may be found in [@den]) to operator-valued functions: \[integral\] Let $K(t,s):H\to H$ be an operator-valued, strongly measurable function, such that for each $a\in (0,T)$ $$\int_0^a\int_0^a\left\|K(t,s)\right\|^2_{HS}dsd t<\infty .\label{double}$$ Then the following holds. \(a) There exists a Borel set $B\subset [0,T]^2$ of full Lebesgue measure such that the measurable linear mapping $K(t,s)C^{-1/2}$ is well defined for all $(s,t)\in B$. \(b) There exists a measurable mapping $f:[0,T)^2\times H\to H$ and a measurable linear space $\mathcal M\subset H$ of full measure such that $f(t,s,y)=K(t,s)C^{-1/2}y$ for $y\in \mathcal M$ and for each $a<T$ $$\int_0^a\left|f(t,s,y)\right|ds<\infty$$ for almost all $t\in [0,T]$. We will use the notation $K(t,s)C^{-1/2}y := f(t,s,y)$. Conditional Distributions ------------------------- Let $H_1$ and $H_2$ be two real separable Hilbert spaces and let $(X,Y)\in H_1\times H_2$ be a Gaussian vector with mean values $$m_X=\mathbb EX,\quad\mbox{\rm and}\quad m_Y=\mathbb EY.$$ The covariance operator of $X$ is determined by the equation $$\mathbb E\left\langle X-m_X,h\right\rangle\left\langle X-m_X,k\right\rangle =\left\langle C_Xh,k\right\rangle ,\quad h,k \in H_1,\label{con1}$$ and a similar condition determines the covariance $C_Y$ of $Y$. The covariance operator $C_{XY}:H_1\to H_2$ is defined by the condition $$\left\langle C_{XY}h,k\right\rangle =\mathbb E\left\langle X-m_ X,h\right\rangle\left\langle Y-m_Y,k\right\rangle ,\quad h\in H_1 ,k\in H_2,$$ and then $C_{XY}^{*}=C_{YX}$.\ For a linear closable operator $G$ on $H$ the closure of $G$ will be denoted by $\overline {G}$. The next theorem is well known, see for example [@mandelbaum] \[tcond\] Assume that $C_X$ is injective. Then the following holds. \(a) We have $$\mbox{\rm im}\left(C_{YX}\right)\subset\mbox{\rm im}\left (C_X^{1/2}\right),\label{con2}$$ the operator $T=C_X^{-1/2}C_{YX}$ is of Hilbert-Schmidt type on $H$ and $T^{*}=\overline {C_{XY}C_X^{-1/2}}$. \(b) We have $$\mathbb E\left(Y|X\right)=m_Y+T^{*}C_X^{-1/2}\left(X-m_X\right) ,\quad\mathbb P^X-a.s.$$ \(c) The conditional distribution of $Y$ given $X$ is Gaussian $N\left(\mathbb E\left(Y|X\right),C_{Y|X}\right)$, where $$C_{Y|X}=C_Y-T^{*}T.$$ Moreover, the random variables $T^{*}C_X^{-1/2}X$ and $\left(Y-T^{*}C_X^{-1/2}X\right)$ are independent. Some Properties of the Ornstein-Uhlenbeck Process ------------------------------------------------- The following hypothesis is a standing assumption for the rest of the paper. \[hou\] For every $t>0$ $$\int_0^t\left\|S_sQ^{1/2}\right\|^2_{HS}ds<\infty ,\label{hs1}$$ and $$\overline {\mbox{\rm im}\left(Q_t\right)}=H,\label{hs2}$$ where, in view of (\[hs1\]) $$\label{ouqt} Q_t=\int_0^tS_sQS_s^*ds.$$ is a well defined trace class operator. It is well known that if Hypothesis \[hou\] holds then the process (\[02\]) is a well defined $H$-valued, Gaussian and Markov process, see [@dz1]. Let $\mu$ denote the probability law of the process $\left\{Z_t^0:t\in [0 ,1]\right\}$ that is concentrated on $L^2(0,T;H)$ and let $\mathcal L:L^2(0,T;H)\to C(0,T;H)$ be defined by the formula $$\label{rkhs} \mathcal Lu(t)=\int_0^tS_{t-s}Q^{1/2}u(s)ds.$$ Note that, cf. [@dz1], $\mathrm{im}(\mathcal L)=RKHS(\mu )$ (the Reproducing Kernel Hilbert Space of the measure $\mu$). We will use the notation $\mu_t^x$ for the Gaussian measure $N\left (S_tx,Q_t\right)$ and $\mu_t$ for $\mu_t^0$. By the properties of Gaussian distribution $\mu_t^x$ is the probability distribution of a random variable $Z_t^x$ and we set $Z_t = Z^0_t$. In the rest of this subsection we give several statements on properties of the family of covariance operators $\left\{ Q_t:\, t\le T\right\} $ that will be useful later. The definition of $Q_t$ given in (\[ouqt\]) yields immediately a simple identity that will be frequently used: $$\label{qtd} Q_T=Q_t+S_tQ_{T-t}S_t^*,\quad t\le T.$$ \[lqt\] We have $$\mathrm{im}\left(Q_t^{1/2}\right)\subset\mathrm{im}\left(Q_T^{1/2}\right),\quad t\le T,$$ hence the operator $U_t=Q_T^{-1/2}Q_t^{1/2}$ is bounded on $H$ for every $t\le T$ and $\left\|U_t\right\|\le 1$. Moreover, $U_t^*=\overline{Q_t^{1/2}Q_T^{-1/2}}$, the closure of the operator $Q_t^{1/2}Q_T^{-1/2}$ defined on the domain $\mathrm{im}\left(Q_T^{1/2}\right)$. From the definition of the covariance operators $Q_t$ it follows that $|Q_tx|^2 \le |Q_Tx|^2$ for each $x\in H$ and $0\le t\le T$ and the conclusion easily follows. \[vt1\] (a) The operator $V_t=Q_T^{-1/2}S_{T-t}Q_t^{1/2}$ is well defined and bounded on $H$ and $$\left\|V_t\right\|\le 1,\quad t\in (0,T).\label{nonexp}$$ Moreover, $$\lim_{t\to T}V_t^{*}x=\lim_{t\to T}V_tx=x,\quad x \in H.\label{c0}$$ (b) For any $t\in [0,T]$ $$Q_{T-t}=Q_T^{1/2}\left(I-V_tV_t^{*}\right)Q_T^{1/ 2}.\label{contr1}$$ The inequality (\[nonexp\]) has been proved in [@neerven], the convergence (\[c0\]) in [@den]. Part (b) follows immediately from (\[qtd\]). Under a slightly stronger condition we show that the inequality (\[nonexp\]) is sharp, more precisely, we have \[vt1a\] The following conditions are equivalent: \(a) For any $t\in (0,T]$ $$\label{hh2} \mbox{\rm im}\left(Q_t^{1/2}\right)=\mbox{\rm im}\left(Q_T^{1/2}\right ).$$ (b) $\mathrm{im}\left(U_t\right)$ is dense in $H$ for each $t\in (0,T)$. \(c) We have $$\label{contr} \left\|V_t\right\| <1,\quad t \in (0,T).$$ Obviously (a) implies (b). To prove that (b) implies (c) note first that (\[qtd\]) yields $$\left|Q_{T-t}^{1/2}x\right|^2=\left|Q_T^{1/2}x\right|^2-\left|V_t^*Q_T^{1/2}x\right|^2,$$ hence putting $y=Q_T^{1/2}x$ we obtain $$\left|Q_{T-t}^{1/2}Q_T^{-1/2}y\right|^2=\left|y\right|^2-\left|V_t^*y\right|^2.$$ Assume that $\left\|V_t^*\right\|=1$ for a certain $t\in(0,T)$. Since $\mathrm{im}\left(Q_T^{1/2}\right)$ is dense in $H$, there exists a sequence $y_n\in\mathrm{im}\left(Q_T^{1/2}\right)$, such that $\left|y_n\right|=1$ and $\left|V_t^*y_n\right|\to 1$. Therefore, $$\label{lim0} \lim_{n\to\infty}\left|Q_{T-t}^{1/2}Q_T^{-1/2}y_n\right|^2=\lim_{n\to\infty}\left(1-\left|V_t^*y_n\right|^2\right)=0.$$ Let $y_{n_k}$ be a subsequence converging weakly to $y\in H$. Since $$\mathrm{im}\left(Q_{T-t}^{1/2}\right)\subset\mathrm{im}\left(Q_T^{1/2}\right),\quad t\le T,$$ and $$\left(Q_T^{-1/2}Q_{T-t}^{1/2}\right)^*=\overline{Q_{T-t}^{1/2}Q_T^{-1/2}},$$ we find that $$Q_{T-t}^{1/2}Q_T^{-1/2}y_{n_k}\to \overline{Q_{T-t}^{1/2}Q_T^{-1/2}}y,\quad\mathrm{weakly},$$ and by (\[lim0\]) we obtain $\overline{Q_{T-t}^{1/2}Q_T^{-1/2}}y=0$ and since $\left|V_t^*y\right|=1$ we obtain $y\neq 0$. It follows that the range of the operator $Q_T^{-1/2}Q_{T-t}^{1/2}$ is not dense in $H$, which shows that (b) implies (c). Finally, assume that (c) holds. Then (\[contr1\]) and Proposition B1 in [@dz1] yield $$\mathrm{im}\left(Q_{T-t}^{1/2}\right)=\mathrm{im}\left(Q_T^{1/2}\left(I-V_tV_t^*\right)^{1/2}\right).$$ Since $\left\|V_t\right\|<1$, the operator $I-V_tV_t^*:H\to H$ is an isomorphism, hence $$\mathrm{im}\left(Q_{T-t}^{1/2}\right)=\mathrm{im}\left(Q_T^{1/2}\right),\quad t<T,$$ and (a) follows. \[remh2\] Necessary and sufficient conditions for (\[hh2\]) to hold are not known but it was proved to be satisfied in the following cases. \(a) If $$\mathrm{im}\left(S_t\right)\subset\mathrm{im}\left(Q_t^{1/2}\right), \quad t>0,$$ then (\[hh2\]) holds. It is known that the above condition is equivalent to the strong Feller property of the OU transition semigroup $R_t\phi(x)=\mathbb E\phi\left(Z_t^x\right)$, see [@dz1] for details. \(b) Assume that the process $\left(Z_t^x\right)$ admits a nondegenerate invariant measure $\nu$ and $\mathrm{im}(Q)$ is dense in $H$. Let $H_Q=\mathrm{im}\left(Q^{1/2}\right)$ be endowed with the norm $|x|_Q=\left|Q^{-1/2}x\right|$. Assume that $H_Q$ is invariant for the semigroup $\left(S_t\right)$ and its restriction to $H_Q$ is a $C_0$-semigroup in $H_Q$. Then (\[hh2\]) holds, see [@acta]. These assumptions are satisfied for any process $\left(Z_t^x\right)$ with the transition semigroup analytic in $L^2(H,\nu )$, in particular they are satisfied for any reversible OU process. We define the operator $B: Q_T^{1/2}(H) \to L^2(0,T;H)$, $$Bx(t)=Q^{1/2}S_{T-t}^{*}Q_T^{-1/2}x,\quad t\in [0,T],\quad x\in Q_T^{1/2}(H).$$ The following simple Lemma has been proved in [@den]: \[ania\] (a) The operator $B$ with the domain $\mathrm{dom}(B)=Q_T^{1/2}(H)$ extends to a bounded operator (still denoted by $B$) $B:H\to L^2( 0,T;H)$. Moreover, $$|Bx|_{L^2(0,T;H)}=|x|_H,\quad x\in H.$$ \(b) Seting $$H\ni x\to\mathcal Kx(t)=K_tx\in L^2(0,T;H),\label{dk}$$ where $$K_t=Q_t^{1/2}V_t^{*},\label{kt}$$ we have $\mathcal K=\mathcal LB$. In particular the operator $\mathcal K:H\to C(0,T;H)$ is bounded. Fundamentals on OU Bridge ------------------------- In the present subsection we give the definition and some basic properties of the OU Bridge. Since $V_t^{*}=\overline {Q_t^{1/2}S_{T-t}^{*}Q_T^{-1/2}}$ is bounded, the operator $K_t$ is of Hilbert-Schmidt type on $H$ for each $t\in [0,T)$. Also, $\mathcal K : H\to L^2(0,T;H)$ is Hilbert-Schmidt.\ Note that if $K_t$ is defined by (\[kt\]) then, in view of Lemma \[tmeas\], the measurable function $K_tQ_T^{-1/2}$ is well defined for each $t\in [0,T]$. We will start from the definition of the process $(\hat Z_t )$, $$\hat {Z}_t=Z_t-K_tQ_T^{-1/2}Z_T,\quad t\in [0,1),\quad\mbox{\rm and} \quad\hat {Z}_1=0.$$ \[ou0\] (a) An $H$-valued Gaussian process $\left(\hat Z_t\right)$ is independent of $ Z_T$. \(b) The covariance operator $\hat {Q}_t$ of $\hat {Z}_t$ is given by $$\hat {Q}_t=Q_t^{1/2}\left(I-V_t^{*}V_t\right)Q_t^{ 1/2}.\label{covp}$$ (c) The process $\left(\hat Z_t\right)$ is mean-square continuous on $[0,T]$. \(d) If, moreover, one of the equivalent conditions (a)-(c) of Lemma \[vt1a\] holds then $$\mbox{\rm im}\left(\hat Q_t^{1/2}\right)=\mbox{\rm im}\left (Q_t^{1/2}\right),\quad t\in (0,T).\label{im12}$$ Theorem \[tcond\] yields immediately (a) since $\hat {Z}_t=Z_t - \mathbb E\left(\left.Z_t\right|Z_T\right)$. Invoking (c) of Theorem \[tcond\]with $C_X=Q_T$, $C_Y=Q_t$ and $T^{*}=K_t$ and (\[kt\]) we obtain $$\hat {Q}_t=Q_t-K_tK_t^{*}=Q_t^{1/2}\left(I-V_t^{*}V_t\right)Q_t^{ 1/2},\quad t<T.$$ Using (\[nonexp\]) we find easily that $$\lim_{t\to 0}\mbox{\rm tr}\left(\hat Q_t\right)=0 .\label{c00}$$ To prove that $$\lim_{t\to T}\mbox{\rm tr}\left(\hat Q_t\right)=0 ,\label{c1}$$ we note first that $$\mbox{\rm tr}\left(\hat Q_t\right)=\mbox{\rm tr}\left(\left(I-V_ t^{*}V_t\right)\left(Q_t-Q_T\right)\right)+\mbox{\rm tr}\left(\left (I-V_t^{*}V_t\right)Q_T\right).$$ Next, it is easy to see that $$0\le\lim_{t\to T}\mbox{\rm tr}\left(\left(I-V_t^{ *}V_t\right)\left(Q_T-Q_t\right)\right)\le\lim_{t\to T}\mbox{\rm tr}\left (Q_T-Q_t\right)=0.\label{z0}$$ Finally, $$\mbox{\rm tr}\left(\left(I-V_t^{*}V_t\right)Q_T\right)=\mbox{\rm tr}\left (Q_T\right)-\mbox{\rm tr}\left(V_tQ_TV_t^{*}\right)$$ $$=\mathrm{tr}\left(Q_T\right)-\sum_{k=1}^{\infty}\left|Q_T^{1/2} V_t^{*}e_k\right|^2,$$ where $\left\{e_k:k\ge 1\right\}$ is a CONS in $H$. Therefore, $$\lim_{t\to T}\mbox{\rm tr}\left(\left(I-V_t^{*}V_ t\right)Q_T\right)=0\label{z1}$$ by Lemma \[vt1\] and the Dominated Convergence Theorem. Combining (\[z0\]) and (\[z1\]) we obtain (\[c1\]) and, consequently, (c). Part (d) follows immediately from Lemma \[vt1a\] and (\[covp\]). \[CL\] The conditional distribution of the process $(Z^x_t)$ in the space $H_2 = L^2(0,T;H)$ given $Z^x_T$ is $N(\lambda , \overline{Q})$, where $$\lambda (t) = S_t x + K_tQ^{-1/2}_T Z_T, \label{X1}$$ $$\overline{Q} = \tilde Q - \mathcal{K} \mathcal {K}^*, \label{X2}$$ where $\tilde Q$ is the covariance operator of the process $(Z^x_t)$ in $H_2$, $\tilde{Q}: H\to H_2$, $$[\tilde Q y] (t) = \int_0^t R(t,s)y(s)ds,\quad y\in H_2,$$ and $$R(t,s)z = \int_0^s S_{t-r}QS^*_{s-r}z dr,\quad z\in H,\quad 0\le s\le t \le T,$$ and $\mathcal{K} : H\to H_1$ is defined in (\[dk\]). We use Theorem \[tcond\] with $H_1= H$, $H_2= L^2(0,T;H)$, $X=Z^x_t$, $Y=(Z^x_t)$, $C_X=Q_T$, and $C_Y=\tilde Q$. By the definition of the covariance $C_{XY}$, $$\left\langle C_{XY}k, h \right\rangle _{L^2(0,T;H)} = \mathbf{E} \left\langle Z^x_T,k\right\rangle\left\langle Z^x,h\right\rangle _{L^2(0,T,H)}, \quad k\in H_1,\, h\in H_2,$$ it is easy to compute $[C_{XY} k](t) = Q_tS^*_{T-t}k,\, t\in [0,T]$. Hence we have $T^* = \overline{C_{XY}C^{-1/2}_X} = \mathcal{K}$ and $T:H_2 \to H_1,\, Ty = \mathcal{K}^* y = \int_0^T K_t^*y(t) dt$. By Theorem \[tcond\] we have that $$\overline{Q} = C_Y - T^* T = \tilde Q - \mathcal{K} \mathcal {K}^*,$$ and $$\lambda (t) = \mathbb{E} (Z_t^x|Z_T^x) = \mathbb{E} (S_tx + Z_t | Z^x_T) =\mathbb{E}(S_tx +\hat Z_t + K_tQ^{-1/2}_tZ_T|Z^x_T)$$ which yields $\lambda (t) = S_tx + K_tQ^{-1/2}_T Z_T$, because $\hat Z_t$ and $Z^x_T$ are stochastically independent, hence (\[X1\]) and (\[X2\]) hold true. Recall that $\mu _T$ denotes the probability law of $Z_T$ on $H$. \[tp2\] There exists a Borel subspace $\mathcal M\subset H$ such that $\mu_T(\mathcal M)=1$ and for all $x\in H$ and $y\in S_Tx+\mathcal M$ the $H$-valued Gaussian process $$\hat {Z}_t^{x,y}=Z_t^x- \mathcal{K}Q_T^{-1/2}\left(Z_T^x-y\right ),\label{p0}$$ is well defined with paths in $L^2(0,T;H)$ and $$\hat {Z}_t^{x,y}=S_tx- \mathcal{K}Q_T^{-1/2}\left(S_Tx-y\right)+\hat {Z}_ t,\quad\mathbb P-a.s.\label{p1}$$ By Lemma \[tmeas\] we can choose a measurable linear space $\mathcal M$ such that $\mathcal{K}Q_T^{-1/2}$ is linear on $\mathcal M$ and $\mu_T\left(\mathcal M\right)=1$. Therefore, $\mathcal{K}Q_T^{-1/2}\left(Z_T^x-y\right)$ is well defined for any $y\in S_Tx+\mathcal M$ and (\[p1\]) holds. \[repr\] Let $\Phi :L^2(0,T;H)\to\mathbb R$ be a Borel mapping such that $$\mathbb E\left|\Phi\left(Z^x\right)\right|<\infty .$$ Then $$\label{DOU} \mathbb E\left(\left.\Phi\left(Z^x\right)\right|Z_T^x=y\right)=\mathbb E\Phi\left(\hat Z^{x,y}\right),\quad\mu_T^x-a.e.$$ where the left-hand side of (\[DOU\]) is defined as a function $g_\Phi = g_\Phi (y) \in L^1(H,\mu_T^x)$ such that $\mathbb{E}(\Phi (Z^x)|Z^x_T) = g_\Phi (Z^x_T)\ \mathbb{P}$-a.s. We have to show that $$\mathbb{E}(\Phi (Z^x)|Z^x_T) = \mathbb{E} (\Phi (\hat Z^{x,y}))|_{Z^x_T = y}\quad \mathbb{P}-a.s.$$ By Proposition \[CL\] we have $$\label{conde1} \mathbb{E}(\Phi (Z^x)|Z^x_T)= \int_H \Phi (z) N(\lambda , \overline{Q}) (dz) \quad \mathbb{P}-a.s.,$$ where $\lambda$ and $\overline{Q}$ are defined by (\[X1\]) and (\[X2\]), respectively. On the other hand, the covariance operator $\hat Q$ of the process $\hat Z^{x,y}_t$ in $H_2$ is by (\[p1\]) the same as the one of $\hat Z_t$. Since $Z_t= \hat Z_t + K_tQ_T^{-1/2}Z_T$ and the summands on the right-hand side are independent random variables, we obtain $\tilde Q = \hat Q +\mathcal{K} \mathcal {K}^*$, that is, $\hat Q = \overline{Q}$. Also, we have $$\mathbb{E} \hat Z^{x,y}_t =S_tx - \mathcal{K}Q_T^{-1/2}(S_Tx - y),$$ and therefore $$\mathbb{E} (\Phi (\hat Z^{x,y}))|_{Z^x_T = y} = \int_H \Phi (z) N(S_tx - K_t Q_T^{-1/2}(S_Tx -y), \overline{Q})(dz)|_{Z^x_T =y} =\int_H \Phi (z) N(\lambda, \overline{Q})(dz)$$ $\mathbb{P}-a.s.$, which together with (\[conde1\]) concludes the proof. \[OUB\] Given $x,\ y\in H$ and an $H$-valued OU process $(Z_t^x)$, a process $(\hat Z_t^{x,y})$ satisfying (\[DOU\]) is called an Ornstein-Uhlenbeck Bridge (connecting points $x$ at time $t=0$ and $y$ at time $t=T$). The probability law of the process $(\hat Z_t^{x,y})$ in the space $L^2(0,T;H)$ will be denoted by $\hat{\mu}^{x,y}$. Thus we have shown that the OU Bridge may be written in the form (\[p0\]) or (\[p1\]) and its probability law $\hat{\mu}^{x,y}$ is $N(\gamma ,\overline Q)$ where $\gamma (t) = \mathbb{E}[\lambda (t)| Z^x_T =y] = S_tx - K_tQ^{-1/2}_T(S_Tx-y)$ $\mu _T^x - a.e.$ The following Theorem has been proved in [@den] : \[conc\] Let $\mathcal E$ be a Banach space such that $\mu (\mathcal E)=1$. Then $\hat{\mu}^{0,y}(\mathcal E)=1$ for $y\in\mathcal M$. SDE associated to the OU Bridge =============================== In the sequel we will need the following \[h1\] For any $t>0$ $$\mbox{\rm im}\left(S_tQ^{1/2}\right)\subset\mbox{\rm im}\left (Q_t^{1/2}\right).\label{im}$$ \[r1\] Condition (\[im\]) is satisfied in some important cases. \(a) If the process $(Z^x_t)$ is strong Feller then $\mathrm{im}\left(S_t\right)\subset\mathrm{im}\left(Q_t^{1/2}\right )$ and therefore (\[im\]) holds. \(b) Let $H_Q=Q^{1/2}(H)$ be endowed with the norm $|x|_Q=\left|Q^{ -1/2}x\right|$, where $Q$ is assumed to be nondegenerate. Assume that $ S_tH_Q\subset H_Q$ for all $t\ge 0$ and $\left(S_t\right)$ restricted to $ H_Q$ is a $C_0$-semigroup. It was proved in [@acta] that in this case $ S_t(H)\subset Q_t^{1/2}(H)$ for all $t>0$ and there exists $c>0$ such that $$\left\|Q_t^{-1/2}S_tQ^{1/2}\right\|\le\frac c{\sqrt {t}},\quad t>0.$$ Assume additionally that the process $(Z^x_t)$ admits a Gaussian invariant measure $\nu$. Then, cf. [@acta], $\left(S_t\right)$ is a $C_0$-semigroup on $H_Q$ if the transition semigroup of the process $(Z^x_t)$ is analytic on $L^2(H,\nu )$, in particular this holds for a symmetric Ornstein-Uhlenbeck process. Explicit conditions for the analyticity and symmetry of the transition semigroup of the process $(Z^x_t)$ in $L^2(H,\nu )$ may be found in [@acta] and [@symm]. \[lh1\] Assume that Hypothesis \[h1\] holds. Then the function $$t\to\left|Q_t^{-1/2}S_tQ^{1/2}h\right|,$$ is nonincreasing on $(0,\infty )$ for each $h\in H$. By Lemma \[vt1\] we have $$\left\|Q_{t+s}^{-1/2}S_tQ_s^{1/2}\right\|\le 1.\label{jan1}$$ By assumption the operator $Q_{t+s}^{-1/2}S_{t+s}Q^{1/2}$ is well defined and bounded and $S_sQ^{1/2}h\in\mathrm{im}\left(Q_s^{1/2}\right)$. Therefore, by (\[jan1\]) $$\left|Q_{t+s}^{-1/2}S_{s+t}Q^{1/2}h\right|=\left|Q_{t+s}^{-1/2} S_tQ_s^{1/2}Q_s^{-1/2}S_sQ^{1/2}h\right|$$ $$\le\left|Q_s^{-1/2}S_sQ^{1/2}h\right|,$$ and (b) follows. Let $$Y_u=\int_u^TS_{T-s}Q^{1/2}dW_s,\quad u\le T.$$ Since the operator-valued function $t\to Q_t$ is continuous in the weak operator topology and all the operators $Q_t$ are compact for $t>0$, there exists a measurable choice of eigenvectors $\left\{e_k(t):k\ge 1\right\}$ and eigenvalues $\left \{\lambda_k(t):k\ge 1\right\}$. For each $n\ge 1$ we define a process $$X^n_u=\sum_{k=1}^n\frac 1{\sqrt{\lambda_k(T-u)}}\left\langle Y_u,e_k(T-u)\right\rangle F_u^{*}e_k(T-u),$$ where $F_u=Q_{T-u}^{-1/2}S_{T-u}Q^{1/2}$. \[yu\] There exists a measurable stochastic process $\left(X_u\right)$ defined on $[0,T)$ such that for each $ a<T$ $$\lim_{n\to\infty}\mathbb E\int_0^a\left|X_u^n-X_u\right |^2du=0.\label{eps}$$ and for each $h\in H$ and $a<T$ the series $$\left\langle X_u,h\right\rangle =\sum_{k=1}^{\infty}\frac 1{\sqrt {\lambda_k(T-u)}}\left\langle Y_u,e_k(T-u)\right\rangle\left \langle e_k(T-u),F_uh\right\rangle\label{fu}$$ converges in $L^2(0,a)$ in mean square. Moreover, if $0\le u\le v<T$ then for all $h,k\in H$ $$\mathbb E\left\langle X_u,h\right\rangle\left\langle X_v,k\right\rangle =\left\langle F_uh,Q_{T-u}^{-1/2}Q_{T-v}^{1/2} F_vk\right\rangle ,\label{covx}$$ where the operator $Q_{T-u}^{-1/2}Q_{T-v}^{1/2}$ is bounded. For $u\le v\le T$ $$\mathbb E\left\langle Y_u,h\right\rangle\left\langle Y_v,k\right\rangle =\left\langle Q_{T-v}h,k\right\rangle ,\quad h ,k\in H.\label{q}$$ Therefore $$\mathbb E\left\langle X_u^n-X_u^m,h\right\rangle^2=\sum_{j=m+1}^ n\frac 1{\lambda_k(T-u)}\mathbb E\left\langle Y_u,e_k(T-u)\right\rangle^ 2\left\langle e_k(T-u),F_uh\right\rangle^2$$ $$=\sum_{j=m+1}^n\left\langle e_k(T-u),F_uh\right\rangle^ 2\underset{n,m\to\infty}\longrightarrow 0,\label{koszi}$$ hence the process $$\left\langle X_u,h\right\rangle =\sum_{k=1}^{\infty}\frac 1{\sqrt { \lambda_k(T-u)}}\left\langle Y_u,e_k(T-u)\right\rangle\left\langle e_k(T-u),F_uh\right\rangle =\left\langle Q_{T-u}^{-1/2}Y_u,F_uh\right \rangle$$ is well defined for each $h\in H$ and $u<T$. For $u,v$ such that $0<u\le v<T$ we have $$\mathrm{im}\left(Q_{T-v}^{1/2}\right)\subset\mathrm{ im}\left(Q_{T-u}^{1/2}\right).\label{im1}$$ Let $P_n$ is an orthogonal projection on $\mbox{\rm lin}\left\{e_k(T-v):k\le n\right\}$ and $F_u^n=P_nF_u$. Then $ Q_{T-u}^{-1/2}F^n_u$ is bounded on $H$. Let $$X_u^n=\left(Q_{T-u}^{-1/2}F_u^n\right)^{*}Y_u.$$ By (\[q\]) $$\mathbb E\left\langle X_u^n,h\right\rangle\left\langle X_v^nk\right \rangle =\left\langle Q_{T-v}Q_{T-u}^{-1/2}F_u^nh,Q_{T-v}^{-1/2}F_ v^nk\right\rangle$$ $$=\left\langle F_u^nh,Q_{T-u}^{-1/2}Q_{T-v}^{1/2}F_v^nk\right\rangle .$$ By (\[im1\]) the operator $Q_{T-u}^{-1/2}Q_{T-v}^{1/2}$ is bounded and therefore $$\mathbb E\left\langle Q_{T-u}^{-1/2}Y_u,F_uh\right\rangle\left\langle Q_{T-v}^{-1/2}Y_v,F_vk\right\rangle =\lim_{n\to\infty}\mathbb E\left \langle X_u^n,h\right\rangle\left\langle X_v^n,k\right\rangle$$ $$=\left\langle F_uh,Q_{T-u}^{-1/2}Q_{T-v}^{1/2}F_vk\right\rangle .$$ It follows from (\[covx\]) that $$\mathbb E\left\langle X_u,h\right\rangle^2=\left|F_uh\right|^2,$$ and by Lemma \[lh1\] we obtain for $u\le a$ $$\mathbb E\left\langle X_u^n,h\right\rangle^2\le\mathbb E\left\langle X_u,h\right\rangle^2\le |h|^2\left\|F_{T-a}\right\|^2.$$ Then (\[koszi\]) and the Dominated Convergence Theorem yield $$\lim_{n,m\to\infty}\int_0^a\sup_{|h|\le 1}\mathbb E\left\langle X_u^n-X_u^m,h\right\rangle^2du=0.$$ As a consequence we find that (\[eps\]) holds for any $a\in (0,T)$. By Lemma \[lh1\] a cylindrical process $$I_t=\int_0^tF_u^{*}Q_{T-u}^{-1/2}Y_udu$$ is well defined, that is for any $h\in H$ the real-valued process $$\left\langle I_t,h\right\rangle =\int_0^t\left\langle Q_{T-u}^{-1/2}Y_u,F_uh\right\rangle du$$ is well defined for all $t<T$. \[wiener\] The cylindrical process $$\zeta_t=W_t-\int_0^tF_u^{*}Q_{T-u}^{-1/2}Y_udu,\quad t\le T,$$ is a standard cylindrical Wiener process on $H$. The proof of this Lemma is omitted; it is a word by word repetition of the proof of Lemma 4.7 in [@den] if we use Lemmas \[lh1\] and \[yu\] above. \[weak1\] For all $t<T$ $$\mathbb E\int_0^t\left|S_{t-s}Q^{1/2}F_s^{*}Q_{T- s}^{-1/2}S_{T-s}\hat Z_s\right|^2ds<\infty ,\label{pa}$$ and $$\hat {Z}_t=-\int_0^tS_{t-s}Q^{1/2}F_s^{*}Q_{T-s}^{ -1/2}S_{T-s}\hat {Z}_sds+\int_0^tS_{t-s}Q^{1/2}d\zeta_s,\quad\mathbb P-a.s.\label{main}$$ We will show first that the operator $Q_{T-s}^{-1/2}S_{T-s}\hat {Q}_sS_{T-s}^{*}Q_{T-s}^{-1/2}$ is bounded. Let $ h,k\in H$. Then by Proposition \[ou0\]  and (\[qtd\]) we obtain $$\left\langle S_{T-s}\hat Q_sS_{T-s}^{*}h,k\right\rangle =\left\langle S_{T-s}Q_sS_{T-s}^{*}h,k\right\rangle -\left\langle S_{T-s}Q_s^{1 /2}V_s^{*}V_sQ_s^{1/2}S_{T-s}^{*}h,k\right\rangle$$ $$=\left\langle\left(Q_T-Q_{T-s}\right)h,k\right\rangle -\left\langle Q_T^{-1/2}S_{T-s}Q_sS_{T-s}^{*}h,Q_T^{-1/2}S_{T-s}Q_sS_{T-s}^{*}k\right \rangle$$ $$=\left\langle\left(Q_T-Q_{T-s}\right)h,k\right\rangle -\left\langle Q_T^{-1/2}\left(Q_T-Q_{T-s}\right)h,Q_T^{-1/2}\left(Q_T-Q_{T-s}\right )k\right\rangle$$ $$=\left\langle\left(Q_T-Q_{T-s}\right)h,k\right\rangle -\left\langle\left (Q_T-Q_{T-s}\right)Q_T^{-1}\left(Q_T-Q_{T-s}\right)h,k\right\rangle$$ $$=\left\langle\left(Q_T-Q_{T-s}\right)h,k\right\rangle -\left\langle\left (Q_T-Q_{T-s}\right)\left(I-Q_T^{-1}Q_{T-s}\right)h,k\right\rangle$$ $$=\left\langle\left(Q_{T-s}-Q_{T-s}Q_T^{-1}Q_{T-s}\right)h,k\right \rangle =\left\langle Q_{T-s}^{1/2}\left(I-Q_{T-s}^{1/2}Q_T^{-1}Q_{ T-s}^{1/2}\right)Q_{T-s}^{1/2}h,k\right\rangle .$$ Since the operator $Q_{T-s}^{1/2}Q_T^{-1}Q_{T-s}^{1/2}$ is bounded for $ s<T$ we find that the operator $$T_s=Q_{T-s}^{-1/2}S_{T-s}\hat {Q}_sS_{T-s}^{*}Q_{ T-s}^{-1/2}=I-Q_{T-s}^{1/2}Q_T^{-1}Q_{T-s}^{1/2}\label{ts}$$ is bounded as well . Therefore, for $s\le T-\epsilon$ Lemma \[lh1\] and (\[ts\]) yield $$\mathbb E\left|S_{t-s}Q^{1/2}F_s^{*}\left(Q_{T-s}^{-1/2}S_{T-s} \hat Z_s\right)\right|^2$$ $$=\left\|S_{t-s}Q^{1/2}F_s^{*}\left(Q_{T-s}^{-1/2}S_{T-s}\hat Q_ s^{1/2}\right)\right\|_{HS}^2\le\left\|S_{t-s}Q^{1/2}\right\|_{HS}^ 2\left\|F_s\right\|^2\left\|T_s^{1/2}\right\|$$ $$\le\left\|S_{t-s}Q^{1/2}\right\|_{HS}^2\left\|F_{T-\epsilon}\right \|^2,$$ which completes the proof of (\[pa\]). As a byproduct of the argument given above we proved also that the process $Q_{T-s}^{-1/2}S_{T-s}\hat {Z}_s$ is well defined for all $ s\le T$. Now, we are ready to prove (\[main\]). By Lemma \[wiener\] we have $$\hat {Z}_t=Z_t-K_tQ_T^{-1/2}Z_T$$ $$=\int_0^tS_{t-s}Q^{1/2}d\zeta_s+\int_0^tS_{t-s}Q^{1/2}F_s^{*}Q_{ 1-s}^{-1/2}Y_sds-K_tQ_T^{-1/2}Z_T,$$ and since $$Y_s=Z_T-S_{T-s}Z_s=Z_T-S_{T-s}K_sQ_T^{-1/2}Z_T-S_{T-s}\hat {Z}_ s,$$ we find that $$\hat {Z}_t=\int_0^tS_{t-s}Q^{1/2}d\zeta_s-\int_0^tS_{t-s}Q^{1/2} F_s^{*}Q_{T-s}^{-1/2}S_{T-}{}_s\hat {Z}_sds$$ $$+\int_0^tS_{t-s}Q^{1/2}F_s^{*}Q_{T-s}^{-1/2}\left(Z_T-S_{T-s}K_ sQ_T^{-1/2}Z_T\right)ds-K_tQ_T^{-1/2}Z_T.$$ It remains to show that $$\int_0^tS_{t-s}Q^{1/2}F_s^{*}Q_{T-s}^{-1/2}\left( Z_T-S_{T-s}K_sQ_T^{-1/2}Z_T\right)ds-K_tQ_T^{-1/2}Z_T=0.\label{00}$$ To this end note first that $$K_tQ_T^{-1/2}Z_T=\left(\int_0^tS_{t-s}Q^{1/2}F_s^{ *}ds\right)Q_T^{-1/2}Z_T,\label{k1}$$ and $$S_{T-t}K_tQ_T^{-1/2}Z_T=\left(\int_0^tS_{T-s}Q^{1/2}F_s^{*}ds\right )Q_T^{-1/2}Z_T$$ $$=\left(Q_T-Q_{T-t}\right)Q_T^{-1}Z_T=Z_T-Q_{T-t}Q_T^{-1}Z_T,$$ and thereby $$Z_T-S_{T-t}K_tQ_T^{-1/2}Z_T=Q_{T-t}Q_T^{-1}Z_T.\label{k2}$$ Finally, (\[k2\]) and the definition of $F_s^{*}$ give $$\int_0^tS_{t-s}Q^{1/2}F_s^{*}Q_{T-s}^{-1/2}\left(Z_T-S_{T-s}K_s Q_T^{-1/2}Z_T\right)ds$$ $$=\int_0^tS_{t-s}Q^{1/2}F_s^{*}Q_{T-s}^{-1/2}Q_{T-s}Q_T^{-1}Z_Td s=\left(\int_0^tS_{t-s}Q^{1/2}F_s^{*}ds\right)Q_T^{-1/2}Z_T,$$ and (\[00\]) follows from (\[k1\]). We will consider now the general case of the bridge $\left(\hat Z_t^{x,y}\right)$ connecting points $x\in H$ and $y$. We will impose the stronger condition (\[hh2\]) which is now formulated as a separate hypothesis: \[h2\] For any $t\in (0,T]$ $$\mbox{\rm im}\left(Q_t^{1/2}\right)=\mbox{\rm im}\left(Q_T^{1/2}\right ).$$ For $y\in H_1 := \mbox{\rm im}(Q_T^{1/2})$ we define $$Ny(t)=\int_0^tS_{t-s}Q^{1/2}F_s^{*}Q_{T-s}^{-1/2}yds,\quad t\le T-\epsilon .$$ \[vy\] Assume that Hypotheses \[h1\] and \[h2\] hold. Then the following holds. \(a) The operator $N:H_1\to L^2\left(0,T-\epsilon ;H\right)$ is Hilbert-Schmidt. \(b) For any $x\in H$ and $y\in\mathcal M$ $$\hat {Z}_t^{x,y}=S_tx-\int_0^tS_{t-s}Q^{1/2}F_s^{*}Q_{T-s}^{-1/ 2}S_{T-s}\hat {Z}_s^{x,y}ds+\int_0^tS_{t-s}Q^{1/2}d\zeta_s$$ $$+\int_0^tS_{t-s}Q^{1/2}F_s^{*}Q_{T-s}^{-1/2}yds.\label{b2}$$ Recall that by Lemma \[ania\] (b) we have $\mathcal{K} =\mathcal{L} B$, hence for $z\in\mathcal M$ $$K_tQ_T^{-1/2}z=\int_0^tS_{t-s}Q^{1/2}B_sQ_T^{-1/2} zds.\label{ky}$$ Next, for $s\le T-\epsilon$ $$\sup_{s\le T-\epsilon}\left\|Q_{T-s}^{-1/2}Q_T^{1/2}\right\|=\left \|Q_{\epsilon}^{-1/2}Q_T^{1/2}\right\|<\infty ,$$ and invoking Lemma \[lh1\] we find that $$\left\|NQ_T^{1/2}\right\|_{HS}^2\le\int_0^{T-\epsilon}\left\|\int_ 0^tS_{t-s}Q^{1/2}F_s^{*}Q_{T-s}^{-1/2}Q_T^{1/2}ds\right\|_{HS}^2d t$$ $$\le\left(\int_0^T\left\|S_sQ^{1/2}\right\|^2_{HS}ds\right)\left (\int_0^{T-\epsilon}\left\|F_s^{*}Q_{T-s}^{-1/2}Q_T^{1/2}\right\|^ 2ds\right)$$ $$\le\left\|F_{T-\epsilon}\right\|^2\left\|Q_{\epsilon}^{-1/2}Q_T^{ 1/2}\right\|^2\left(\int_0^T\left\|S_sQ^{1/2}\right\|^2_{HS}ds\right )<\infty .$$ Therefore, the measurable function $$y\to\int_0^tS_{t-s}Q^{1/2}F_s^{*}Q_{T-s}^{-1/2}yds,$$ is well defined. We are ready now for the proof of (\[b2\]). Let $x,y\in\mbox{\rm im}\left(Q_T^{1/2}\right)$. Then Hypothesis \[h2\] yields $S_Tx\in\mbox{\rm im}\left(Q_T^{1/2}\right)$, hence $y\in\mathcal M$. By (\[p1\]) we have $$\hat {Z}_t^{x,y}=\hat {Z}_t+S_tx-K_tQ_T^{-1/2}\left(S_Tx-y\right ),$$ and Theorem \[weak1\] yields $$\hat {Z}_t^{x,y}=S_tx-K_tQ_T^{-1/2}S_Tx+K_tQ_T^{-1/2}y$$ $$-\int_0^tS_{t-s}Q^{1/2}F_s^{*}Q_{T-s}^{-1/2}S_{T-s}\hat {Z}_sds +\int_0^tS_{t-s}Q^{1/2}d\zeta_s$$ $$=S_tx-K_tQ_T^{-1/2}S_Tx+K_tQ_T^{-1/2}y$$ $$-\int_0^tS_{t-s}Q^{1/2}F_s^{*}Q_{T-s}^{-1/2}S_{T-s}\left(\hat Z_ s^{x,y}-S_sx+K_sQ_T^{-1/2}S_Tx-K_sQ_T^{-1/2}y\right)ds+\int_0^tS_{ t-s}Q^{1/2}d\zeta_s$$ $$=-K_tQ_T^{-1/2}S_Tx+\int_0^tS_{t-s}Q^{1/2}F_s^{*}Q_{T-s}^{-1/2} S_{T-s}\left(S_s-K_sQ_T^{-1/2}S_T\right)xds$$ $$+K_tQ_T^{-1/2}y+\int_0^tS_{t-s}Q^{1/2}F_s^{*}Q_{T-s}^{-1/2}S_{T -s}K_sQ_T^{-1/2}yds$$ $$+S_tx-\int_0^tS_{t-s}Q^{1/2}F_s^{*}Q_{T-s}^{-1/2}S_{T-s}\hat {Z}_ s^{x,y}ds+\int_0^tS_{t-s}Q^{1/2}d\zeta_s$$ $$=:H_tx+G_ty+S_tx-\int_0^tS_{t-s}Q^{1/2}F_s^{*}Q_{T -s}^{-1/2}S_{T-s}\hat {Z}_s^{x,y}ds+\int_0^tS_{t-s}Q^{1/2}d\zeta_ s.\label{hg}$$ We will show first that $$G_ty=\int_0^tS_{t-s}Q^{1/2}F_s^{*}Q_{T-s}^{-1/2}y ds.\label{gt}$$ For $y\in\mbox{\rm im}\left(Q_T^{1/2}\right)$ $$S_{T-t}K_ty=\int_0^tS_{t-s}QS_{T-s}^{*}Q_T^{-1/2}yds=\int_0^tS_{ T-s}QS_{T-s}^{*}Q_T^{-1/2}yds$$ $$=\left(Q_T-Q_{T-t}\right)Q_T^{-1/2}y,\label{s1}$$ and therefore $$F_s^{*}Q_{T-s}^{-1/2}S_{T-s}K_sy=F_s^{*}Q_{T-s}^{-1/2}Q_T^{1/2} y-F_s^{*}Q_{T-s}^{1/2}Q_T^{-1/2}y$$ $$=F_s^{*}Q_{T-s}^{-1/2}Q_T^{1/2}y-Q^{1/2}S_{T-s}^{*}Q_T^{-1/2}y.$$ Hence, taking Lemma \[ania\] (b) into account we find that $$G_ty=K_tQ_T^{-1/2}y+\int_0^tS_{t-s}Q^{1/2}F_s^{*}Q_{T-s}^{-1/2} S_{T-s}K_sQ_T^{-1/2}yds$$ $$=K_tQ_T^{-1/2}y+\int_0^tS_{t-s}Q^{1/2}F_s^{*}Q_{T-s}^{-1/2}yds- K_tQ_T^{-1/2}y,$$ and (\[gt\]) follows. Next, we claim that for $x\in\mbox{\rm im}\left(Q_T^{1/2}\right)$ $$H_tx=0.\label{ht}$$ Indeed, using (\[s1\]) we obtain $$H_tx=-K_tQ_T^{-1/2}S_Tx+\int_0^tS_{t-s}Q^{1/2}F_s^{*}Q_{T-s}^{- 1/2}S_{T-s}\left(S_s-K_sQ_T^{-1/2}S_T\right)xds$$ $$=-K_tQ_T^{-1/2}x+\int_0^tS_{t-s}Q^{1/2}F_s^{*}Q_{T-s}^{-1/2}S_T xds-\int_0^tS_{t-s}Q^{1/2}F_s^{*}Q_{T-s}^{-1/2}S_{T-s}K_sQ_T^{-1/ 2}S_Txds$$ $$=-K_tQ_T^{-1/2}x+\int_0^tS_{t-s}Q^{1/2}F_s^{*}Q_{T-s}^{-1/2}S_T xds$$ $$-\int_0^tS_{t-s}Q^{1/2}F_s^{*}Q_{T-s}^{-1/2}\left(Q_T-Q_{T-t}\right )Q_T^{-1/2}S_Txds=0,$$ which yields (\[ht\]) for $x\in\mbox{\rm im}\left(Q_T^{1/2}\right )$ and therefore for all $x\in H$. Finally, combining (\[hg\]), (\[gt\]) and (\[ht\]) we obtain (\[b2\]). \[col1\] Assume Hypotheses \[h1\] and \[h2\]. Then for each $t<T$, and $h\in\mbox{\rm dom}\left(A^{*}\right)$ and all $x\in H$ and $ y\in\mathcal M$ $$\left\langle\hat Z_t^{x,y},h\right\rangle =\left\langle x,h\right \rangle +\int_0^t\left\langle\hat Z_s^{x,y},A^{*}h\right\rangle d s-\int_0^t\left\langle F_s^{*}Q_{T-s}^{-1/2}S_{T-s}\hat Z_s^{x,y} ,Q^{1/2}h\right\rangle ds$$ $$+\int_0^t\left\langle F_s^{*}Q_{T-s}^{-1/2}y,Q^{1/2}h\right\rangle ds+\left\langle\zeta_t,Q^{1/2}h\right\rangle .$$ On any interval $\left[0,T_0\right]$ with $T_0<T$ and for any $y\in\mathcal M$ the functions $$s\to Q^{1/2}F_s^*Q_{T-s}^{-1/2}S_{T-s}\hat{Z}^{x,y}_s\quad\mathrm{and}\quad s\to Q^{1/2}F_s^*Q_{T-s}^{-1/2}y$$ are $\mathbb P$-a.s. Bochner integrable by Theorem \[vy\] and therefore standard results about the equivalence of weak and strong solutions of deterministic and stochastic evolution equations can be applied to prove the corollary, see for example [@ball] for deterministic and [@ania-mild], [@on] for stochastic versions. Applications to Semilinear Equations ==================================== In this Section, transition densities and Markov semigroups defined by semilinear stochastic equations are studied using the OU Bridge. Throughout the Section we assume (beside (\[hou\])) that the OU process $(Z^x_t)$ is strongly Feller, that is, the condition $$\label{SFP} \mathrm{im}(S_t) \subset \mathrm{im} (Q_t^{1/2}),\quad t\in (0,T),$$ is satisfied. Note that (\[SFP\]) trivially implies the preceding Hypotheses \[h1\] and \[h2\] (or (\[hh2\])). Let $(\mathcal{P}, \| .\| _{var})$ denote the space of probability measures on the Borel sets of $H$ endowed with the metric of total variation. We start from a simple proposition where some continuity properties of the OU Bridge are given. \[cont\] (a) For each $t\in (0,T), y\in \mathcal{M} $, where $\mathcal M$ has been defined in Proposition \[tp2\], the mappings $$\label{co1} x\mapsto \hat Z^{x,y}_t(\omega ), \quad H \to H,$$ $$\label{co2} x\mapsto \hat Z^{x,y}(\omega ), \quad H \to L^2(0,T;H),$$ are continuous for $\mathbb{P}$-almost all $\omega \in \Omega $, and the mapping $$\label{co3} x \mapsto \hat \mu ^{x,y}_t,\quad H \to (\mathcal{P}, \| . \|_{var}),$$ is continuous. \(b) If, moreover, for each $t\in(0,T)$ we have $\overline{K_tQ^{-1/2}_T} \in \mathcal{L}(\hat H, H)$, where $\hat H$ is a separable Banach space continuously embedded into $H$, then the mapping $y\mapsto \hat Z^{x,y}_t(\omega )$ is $\hat H \to H$ $\mathbb{P}$- a.s. continuous. Similarly, if $$\overline{\mathcal{K} Q_T^{-1/2}} \in \mathcal{L}(\hat H, L^2(0,T;H))$$ then $\mathcal{M} \supset \hat H$ and the mapping $y \mapsto \hat Z^{x,y}(\omega )$ is $\mathbb{P}$-a.s. $\hat H\to L^2(0,T;H))$ continuous. \(a) By (\[SFP\]) we have that $S_Tx \in \mathrm{im} (Q_T^{1/2})$ for each $x\in H$ and hence $S_Tx\in \mathcal{M}$ by construction of $\mathcal{M}$, hence $y\in \mathcal{M}$. Furthermore, (\[SFP\]) implies that the mappings $\mathcal{K}Q_T^{-1/2}S_T$ and $K_tQ^{-1/2}_TS_T,\, t\in (0,T]$, are in $\mathcal{L}(H,L^2(0,T;H))$ and $\mathcal{L}(H)$, respectively, and (\[co1\]) and (\[co2\]) follow by (\[p1\]). To show (\[co3\]) we recall Proposition \[ou0\] and Lemma \[vt1a\] , by which we have $\mathrm{im}(\hat Q_t^{1/2}) = \mathrm{im} (Q_t^{1/2})$. Hence the measures $(\hat \mu ^{x,y}_T), x\in H,$ are equivalent and $$\label{psi} \psi^y(t,x,z)=\frac {d\mu^{x,y}_t}{d\mu^{0,y}_t}(z)=\exp\left(-\frac 12\left|Q_t^{-1/2}S_tx\right|^2+\frac 12\left|Q_T^{-1/2}S_Tx\right |^2+\left\langle Q_t^{-1/2}z,Q_t^{-1/2}S_tx\right\rangle\right).$$ Indeed, by the Cameron-Martin formula we have $$\psi^y(t,x,z)=\exp\left(-\frac 12\left|\hat Q_t^{-1/2}m\right|^ 2+\left\langle\hat Q_t^{-1/2}z,\hat Q_t^{-1/2}m\right\rangle\right ),$$ where $m=Q_t^{1/2}\left(I-V_t^{*}V_t\right)Q_t^{-1/2}S_tx$. Then using (\[covp\]) we get (\[psi\]) and the assertion easily follows. The proof of part (b) is completely analogous. \[psi1\] (a) The equivalent form of the density (\[psi\]) is $$\psi^y(t,x,z)=\exp\left(-\frac 12\left|\left(I-V_t^{*}V_t\right )^{1/2}Q_t^{-1/2}S_tx\right|^2+\left\langle Q_t^{-1/2}z,Q_t^{-1/2} S_tx\right\rangle\right).$$ (b) Note that the OU Bridge $(\hat Z^{x,y}_t)$ satisfies the SDE (\[b2\]) which defines an (inhomogeneous) Markov process on the interval $(0,T)$. By (\[co3\]) this process is strongly Feller. Now consider a stochastic semilinear evolution equation of the form $$\label{SEM} dX_t = AX_tdt + F(X_t)dt + \sqrt{Q}dW_t,\quad X_0=x\in H$$ where $A$, $W_t$ and $Q$ are as before and $F: H\to H$ is a nonlinear continuous mapping. Suppose that $\mathrm{im}(F) \subset \mathrm{im} (Q^{1/2})$ and set $G:= Q^{-1/2}F$. \[GIR\] The mapping $G: H\to H$ is bounded and continuous. Now we formulate technical assumptions on the linear part of the equation. For simplicity of presentation, it is stated in the form that is verifiable in examples and includes all assumptions made previously in the paper. \[h4\] Assume either \(i) $\mathrm{dim} H <\infty $ or \(ii) There exist $\alpha \in (0,1)$ and $\beta < \frac{1+\alpha }{2}$ such that $$\int_0^{T_0} t^{-\alpha} \| S_t Q^{1/2}\| ^2_{HS} dt <\infty\quad and$$ $$\| Q_t^{-1/2}S_t\| \le \dfrac{c}{t^{\beta}},\quad t\in (0,T_0),$$ for some $c>0$ and $T_0>0$. Conditions from (ii) are often used in the theory of stochastic equations and have been widely studied (cf.[@dz1] or [@den], see also the Example below). Note that Hypothesis \[h4\] (ii) implies all previous assumptions made in the paper on the linear part of the equation (\[SEM\]) (i.e., all except for Hypothesis \[GIR\]). It is well known (see e.g. [@on] ) that under Hypotheses \[GIR\] and \[h4\] the equation (\[SEM\]) defines an $H$-valued Markov process induced by the mild formula $$\label{MF} X_t = S_tx + \int_0^t S_{t-r}F(X_r)dr + \int_0^t S_{t-r}\sqrt{Q}d\widetilde W_r, \quad t\ge 0,$$ where $\widetilde W_t$ is a standard cylindrical Wiener process on $H$ defined on a suitable probability space. Finally, we assume that the OU process defined by the linear equation (\[01\]) has an invariant measure $\nu$ that will be used as a reference measure. This is equivalent to the condition $$\label{ea1} \sup_{t>0} tr (Q_t) < \infty.$$ If (\[ea1\]) holds then $\nu$ is a centered Gaussian measure with the covariance operator $$Q_{\infty}=\int_0^{\infty}S_tQS_t^*dt.$$ Moreover, it has been shown in [@fock] that $S_tQ_{\infty}^{1/2}(H)\subset Q_{\infty}^{1/2}(H)$ and the family of operators $$S_0(t)=Q_{\infty}^{-1/2}S_tQ_{\infty}^{1/2},\quad t\ge 0,$$ defines a $C_0$-semigroup of contractions on $H$. Moreover, if part (ii) of Hypothesis \[h4\] holds then $\left\|S_0(t)\right\|<1$ for all $t>0$. Denote by $(P_t)$ the transition Markov semigroup defined by the equation (\[SEM\]) and set $$P(t,x, \Gamma )= P_t1_\Gamma (x),\, x\in H,\, t>0$$ and $\Gamma$ Borel sets in $H$, and $$d(t,x,y) = \frac{P(t,x,dy)}{\nu (dy)}.$$ It is standard to see that the density $d$ exists, because Girsanov Theorem may be used to show the equivalence of measures $P(t,x,dy) \sim \mu ^x_t$, and $\mu ^x_t \sim \nu$ by (\[SFP\]) (see e.g. [@den]). \[dens\] Let Hypotheses \[GIR\], \[h4\] and (\[ea1\]) be satisfied and let $T>0$ be fixed. Then for $\nu$-almost all $y\in H$ the mapping $x \mapsto d(T,x,y)$ is continuous on $H$. \[lp\] Let Hypotheses \[GIR\], \[h4\] and (\[ea1\]) be satisfied. Then for $p>1$, $T>0$, we have $$P_T(L^p(H,\nu )) \subset \mathcal{C} (H),$$ that is, the semigroup $(P_t)$ maps the space $L^p(H,\nu )$ into the space of continuous functions on $H$. For $p,q>1$ we introduce the notation $$\left\|P_t\right\|_{p,q}=\left(\int_H\left(\int_Hd^{p'}(t,x,y)\nu(dy)\right)^{q/p'}\nu(dx)\right)^{1/q},$$ where $p'=\frac{p}{p-1}$. Note that $\left\|P_t\right\|_{2,2}$ is a Hilbert-Schmidt norm of $P_t$. Moreover, if $\left\|P_t\right\|_{p,q}<\infty$ then the operator $P_t:L^p(H,\nu)\to L^q(H,\nu)$ is compact. Under assumptions more general than ours necessary and sufficient conditions were given in [@ania] for boundedness of the operator $P_t:L^p(H,\nu)\to L^q(H,\nu)$. In the theorem below we use different arguments based on the formula for transition densities to show that a stronger property holds: $\left\|P_t\right\|_{p,q}<\infty$. \[HSS\] Let Hypotheses \[GIR\], \[h4\] and (\[ea1\]) be satisfied. Then for any fixed $T>0$ and $q>0$ satisfying $$q<1+\frac{p-1}{\left\|S_0(T)\right\|^2}$$ we have $\left\|P_T\right\|_{p,q}<\infty$. In particular, the operator $P_T:L^p(H,\nu)\to L^q(H,\nu)$ is $q$-summing and $P_T$ is Hilbert-Schmidt in the space $L^2(H,\nu )$. By the above mentioned equivalence of probabilities we may write $$\label{y1} d(T,x,y) =\frac{P(T,x,dy)}{\mu^x_T(dy)}\cdot \frac{\mu^x_T(dy)}{\mu^0_T(dy)}\cdot \frac{\mu^0_T(dy)}{\nu(dy)}$$ $$=: h(T,x,y)\cdot g(T,x,y)\cdot k(T,y),$$ where $k$ does not depend on $x$, $g$ is given by the Cameron-Martin formula $$\label{y1a} g(T,x,y) = \exp \{\left\langle x, \overline{S^*_TQ^{-1/2}_T} Q^{-1/2}_T y\right\rangle - \frac{1}{2}|Q^{-1/2}_TS_Tx|^2\}$$ for $\nu$-almost all $y\in H$, and $h$ may be expressed by means of the OU Bridge $(\hat Z^{x,y}_t)$, $$\label{y2} h(T,x,y) = \mathbb{E} \exp \{ \rho (\hat Z^{x,y} ) - \int_0^T \left\langle G(\hat Z^{x,y}_s), B_1(s) \hat Z_s + B_2(s) x - B_3(s)y \right\rangle ds \}$$ (cf.[@den], Theorem 5.2), where $$\rho (\hat Z^{x,y}) = \int_0^T \left\langle G(\hat Z^{x,y}_s), dW_s\right\rangle - \frac{1}{2}\int_0^T |G(\hat Z^{x,y}_s)|^2ds$$ and $$B_1(s) = (Q^{-1/2}_{T-s}S_{T-s}Q^{1/2})^* Q^{-1/2}_{T-s}S_{T-s},$$ $$B_2(s) = (Q^{-1/2}_TS_{T-s}Q^{1/2})^*Q_T^{-1/2}S_T,$$ $$B_3(s)y =(Q^{-1/2}_TS_{T-s}Q^{1/2})^*Q_T^{-1/2}y,\quad y\in \mathrm{im}\left(Q_T^{1/2}\right).$$ From Lemma \[ania\] it follows that $$\label{y3} \int_0^T |B_2(s)x|^2 ds = |Q^{-1/2}_TS_Tx|^2,\quad x\in H,$$ and by [@den], Proposition 4.9, we have that $$\label{y4} \mathbb{E} \int_0^T |B_1(s)\hat Z_t| ds <\infty$$ and $$\label{y5} \int_0^T |B_3(s)y|ds <\infty$$ for $\nu$- almost all $y \in \mathcal{M}$ (with no loss of generality we may assume that (\[y5\]) holds for all $y\in\mathcal{M},\, \nu (\mathcal{M}) =1$). The proofs of Theorems \[dens\], \[lp\] and \[HSS\] are based on the following technical lemma: \[est\] Given $T>0$ and $q\in [0,\infty)$, there exists a constant $k_q>0$ such that $$\label{y6} \begin{aligned} h_q(T,x,y):&= \mathbb{E}\exp\{ q(\rho (\hat Z^{x,y} ) - \int_0^T \left\langle G(\hat Z^{x,y}_s), B_1(s) \hat Z_s + B_2(s) x - B_3(s)y \right\rangle ds )\}\\ &\le k_q \exp \{k_q (|x| + \int_0^T |B_3(s)y|ds )\} \end{aligned}$$ for all $x\in H$ and $y\in \mathcal{M}$, in particular, $$h(t,x,y) \le k_1 \exp \{k_1 (|x| + \int_0^T |B_3(s)y|ds)\}.$$ By the Cauchy inequality we have $$\label{y7} h_q(T,x,y) \le (\mathbb{E} \exp \{2q\rho (\hat Z^{x,y}\})^{1/2}$$ $$\times (\mathbb{E} \exp \{2q(\int_0^T |\left\langle G(\hat Z^{x,y}_s), B_1(s) \hat Z_s + B_2(s) x - B_3(s)y \right\rangle |ds)\} )^{1/2}$$ and since the process $s \mapsto G(\hat Z^{x,y}_s)$ is bounded the first expectation on the right-hand side of (\[y7\]) is bounded (uniformly w.r.t. $x$ and $y$). By (\[y3\]) and (\[y5\]) we thus have $$h_q(T,x,y) \le C_q( \mathbb{E} \exp \{ C_q \int_0^T (|B_1(s)\hat Z_s| + |B_2(s)x| + |B_3(s)y|)ds\})^{1/2}$$ $$\le \tilde C_q \exp \{ \tilde C_q (|Q_t^{-1/2}S_Tx| + \int_0^T |B_3(s)y|ds)\} (\mathbb{E} \exp \{ \tilde C_q \int_0^T |B_1(s) \hat Z_s|ds \} )^{1/2}$$ for some $C_q,\, \tilde C_q$, and (\[y6\]) follows by (\[y4\]) and the Fernique inequality. [*Proof of Theorem \[dens\].*]{} Without loss of generality (dropping, if necessary, a set of $\nu$-measure zero) we may suppose that $g(T,x,y)$ and $k(T,y)$ are defined for all $y\in \mathcal{M}$. By (\[y1a\]) we have that the mapping $x\mapsto g(T,x,y)k(T,y)$ is continuous, so we only have to prove continuity of the mapping $x\mapsto h(T,x,y),\, y\in \mathcal{M},\, T>0$. Let $x_n \to x_0$ in $H$. First we show (possibly, for a subsequence) that $$\label{y8} \lim _{n\to \infty} \exp \{ \rho (\hat Z^{x_n,y} ) - \int_0^T \left\langle G(\hat Z^{x_n,y}_s), B_1(s) \hat Z_s + B_2(s) x_n - B_3(s)y \right\rangle ds\}$$ $$= \exp \{ \rho (\hat Z^{x_0,y} ) - \int_0^T \left\langle G(\hat Z^{x_0,y}_s), B_1(s) \hat Z_s + B_2(s) x_0 - B_3(s)y \right\rangle ds\}$$ $\mathbb{P}$-a.s. We have $$\label{y9} \begin{aligned} &\int_0^T \left|\left\langle G(\hat Z^{x_n,y}_s), B_1(s) \hat Z_s + B_2(s) x_n - B_3(s)y \right\rangle - \int_0^T \left\langle G(\hat Z^{x_0,y}_s), B_1(s) \hat Z_s + B_2(s) x_0 - B_3(s)y \right\rangle\right| ds\\ &\le \int_0^T |G(\hat Z^{x_n,y}_s) - G(\hat Z^{x_0,y}_s)| (|B_1(s)\hat Z_s| + |B_2(s)x_0| + |B_3(s)y|)ds\\ &+ \int_0^T \left|G(\hat Z^{x_0,y}_s)|\cdot |B_2(s)(x_n-x_0)\right|ds, \end{aligned}$$ which tends to zero by continuity and boundedness of $G$, (\[y3\]) and Dominated Convergence Theorem. Also, we have $$\mathbb{E} |\rho (\hat Z^{x_n,y}) - \rho (\hat Z^{x_0,y})| \le C \left(\left(\mathbb{E} \int_0^T |G(\hat Z^{x_n,y}_s) - G(\hat Z^{x_0,y}_s)|^2 ds\right)^{1/2}\right.$$ $$\left. + \mathbb{E} \int_0^T |G(\hat Z^{x_n,y}_s) - G(\hat Z^{x_0,y}_s)|^2 ds \right),$$ which again tends to zero by Dominated Convergence Theorem, so there is a subsequence converging $\mathbb{P}$-a.s. Taking into account (\[y9\]) we obtain (\[y8\]). By (\[y6\]) (used, for instance, with $q=2$) the random variables on the left-hand side of (\[y8\]) are integrable uniformly in $n$, hence the convergence in (\[y8\]) holds also in the space $L^1(\Omega )$ and, consequently, we obtain $h(T,x_n,y) \to h(T,x_0,y)$. Since we may choose a subsequence with this property from an arbitrary sequence $x_n \to x_0$, the convergence takes place for the whole sequence. [*Proof of Theorem \[lp\].*]{} Let $T>0,\, \phi \in L^p(H, \nu)$ and $x_n \to x_0$ in $H$. Then $$|P_T\phi (x_n) - P_T \phi (x_0) | \le \int_H |\phi (y)| |d(T,x_n)-d(T,x_0,y)| \nu(dy)$$ $$\le (\int_H|\phi |^pd\nu )^{1/p} (\int_H |d(T,x_n, y) - d(T,x_0,y)|^{p'} \nu (dy))^{1/p'},$$ so by Theorem \[dens\] it suffices to show that $$\label{y10} \int_H (d(T,x_n,y))^q \nu (dy) < c_q,\quad q\in (1,\infty),$$ where $c_q$ does not depend on $n$. The same property (uniform boundedness in arbitrary $L^q(H,\nu )$) has been shown for Gaussian densities $g(T,x_n,\cdot )$ and $k(T,\cdot )$ in [@reg], so we only have to show (\[y10\]) where $d(T,x_n,y)$ is replaced by $h(T,x_n,y)$. However, by Lemma \[est\] and H" older inequality we have $$\label{y11} \int_H (h(T,x_n,y))^q \nu (dy) \le \int_H h_q(T,x_n,y) \nu (dy)$$ $$\le k_q \exp \{k_q |x_n|\} \int_H \exp \{\int_0^T |B_3(s)y|ds\}\nu (dy) <c_q$$ where $c_q$ does not depend on $n$, since the sequence $x_n$ is obviously bounded and $$\int_H \exp \{\int_0^T |B_3(s)y|ds\}\nu (dy) <\infty$$ by (\[y5\]), (\[SFP\]) and the Fernique inequality. [*Proof of Theorem \[HSS\].*]{} We can rewrite (\[y1\]) in the form $$d(T,x,Y)=h(T,x,y)H(T,x,y),$$ where $$H(T,x,y)=\frac{\mu_T^x(dy)}{\nu(dy)}.$$ Invoking the Hölder inequality we obtain $$\label{1015} \begin{aligned} \left\|P_T\phi\right\|_{p,q}^q&=\int_H\left(\int_H hH\phi\nu(dy)\right)^q\nu(dx)\\ &\le \int_H\left(\left(\int_Hh^{p'}H^{p'}\nu(dy)\right)^{1/p'}\left(\int_H|\phi |^p\nu(dy)\right)^{1/p}\right)^q\nu(dx)\\ &=\|\phi\|_p^q\int_H\left(\int_Hh^{p'}H^{p'}\nu(dy)\right)^{q/p'}\nu(dx). \end{aligned}$$ It remains to show that $$\label{1029} K=\int_H\left(\int_Hh^{p'}H^{p'}\nu(dy)\right)^{q/p'}\nu(dx)<\infty .$$ Indeed, using successively the Hölder equality we obtain for any $r>1$ $$\label{637} \begin{aligned} K&\le \int_H\left(\int_Hh^{p'r'}\nu(dy)\right)^{q/p'r'}\left(\int_HH^{p'r}\nu(dy)\right)^{q/p'r}\nu(dx)\\ &\le \left(\int_H\left(\int_Hh^{p'r'}\nu(dy)\right)^{q/p'}\nu(dx)\right)^{1/r'}\left(\int_H\left(H^{p'r}\nu(dy)\right)^{q/p'}\nu(dx)\right)^{1/r}. \end{aligned}$$ It was shown in [@reg] that $$\label{655} \int_H\left(H^{a'}\nu(dy)\right)^{b/a'}\nu(dx)<\infty ,$$ for any $a,b\ge 1$, such that $$\label{813} b\le 1+\frac{a-1}{\left\|S_0(T)\right\|^2}.$$ Putting $$a=\frac{p'r}{p'r-1}\quad\mathrm{and}\quad b=qr,$$ we find that there exists $r>1$ such that (\[813\]) holds. Therefore, for such an $r$ $$\label{816} \int_H\left(H^{p'r}\nu(dy)\right)^{q/p'}\nu(dx)=\int_H\left(H^{a'}\nu(dy)\right)^{b/a'}\nu(dx)<\infty .$$ Next, we need to show that $$\label{817} \int_H\left(\int_Hh^{p'r'}\nu(dy)\right)^{q/p'}\nu(dx)<\infty .$$ To prove (\[817\]) we note that if $\frac{q}{p'}\ge 1$ then $$\int_H\left(\int_Hh^{p'r'}\nu(dy)\right)^{q/p'}\nu(dx)\le \int_H\int_Hh^{r'q}\nu(dy)\nu(dx)$$ However, using Lemma \[est\] for $\tilde q =r'q$ we have $$\int_H \int_H (h(T,x,y))^{\tilde q} \nu (dx) \nu (dy) \le \int_H \int_H h_{\tilde q}(T,x,y) \nu (dx) \nu (dy)$$ $$\le \int_H \int_H k_{\tilde q} \exp \{k_{\tilde q} (|x| + \int_0^T |B_3(s)y|ds )\}\nu (dx) \nu (dy)$$ $$\le k_{\tilde q} \int_H \exp \{k_{\tilde q} |x|\} \nu (dx) \int_H \exp \{ k_{\tilde q} \int_0^T |B_3(s)y|ds )\} \nu (dy)$$ $$= k_{\tilde q} \mathbb{E} e^{k_{\tilde q} |\tilde Z|}\cdot \mathbb{E} \exp \{k_{\tilde q} \int_0^T |B_3(s)\tilde Z|ds \}$$ where $\tilde Z$ is an arbitrary random variable with probability distribution $\nu $. By (\[y5\]), (\[SFP\]) and the Fernique inequality we conclude that (\[817\]) holds true. The proof of (\[817\]) for the case when $\frac{q}{p'}<1$ is even simpler and is omitted. \[iny\] There is a natural question whether the transition density is regular (continuous) “in $y$”, that is, whether the mapping $y \mapsto d(T,x,y)$ is continuous, at least on a certain subspace $\hat H \subset H$) of full measure. In the Gaussian case the formulas for the density may be used to conclude that if $\overline{S_T^*Q^{-1}_T} \in \mathcal{L}(\hat H,H)$ then $y\to g(T,x,y)$ is continuous on $\hat H$ for all $T>0$ and $x\in H$ (cf. the Cameron-Martin formula (\[y1a\])). A similar well-known formula for $k(T,y)$ (see e.g. [@reg]) yields $\hat H\to H$ continuity of the mapping $y\mapsto k(T,y)$ provided $$\label{y13} C(T) := \overline{Q_{\infty}^{-1/2}(I-S_0(T)S^*_0(T))^{-1}S_0(T)S_0^*(T)Q_{\infty}^{-1/2}} \in \mathcal{L}(\hat H,H)$$ where $S_0(T) = Q_{\infty}^{-1/2}S_TQ_{\infty}^{1/2}$. Following the proof of Theorem \[dens\] we can easily see that the remaining factor, the function $h(T,x,y)$ is continuous in $y\in\hat{H}$ if the mapping $y \to \hat Z^{x,y}_t$ is $\hat H \to H$ a.s. continuous (which by Proposition \[cont\] (b) happens if $\overline{K_tQ_T^{-1/2}} \in \mathcal{L}(\hat H,H)$) and $$\label{B3} B_3 \in \mathcal{L}(\hat H, L^1(0,T;H)).$$ We are able to verify these additional conditions in some important cases (supposing that the standing assumptions of this Section (\[GIR\]), (\[h4\]) and (\[ea1\]) are satisfied). \(a) All three conditions are satisfied if $\mathrm{dim}{H}<\infty$. \(b) In the commutative case the first two conditions are satisfied with $\hat{H}=H$. However, condition (\[B3\]) is not satisfied with $\hat H=H$ even in simple infinite - dimensional situations and a smaller space $\hat H$ must be considered (cf. Example \[example\] below for details). \[example\] Consider the semilinear stochastic heat equation $$\label{HEQ} \frac{\partial u}{\partial t}(t,\xi ) = \frac{\partial ^2 u}{\partial \xi ^2} (t,\xi ) + f(u(t,\xi )) + \eta (t,\xi ), \quad (t, \xi )\in \mathbb{R}_+ \times (0,1),$$ with an initial condition and Dirichlet boundary conditions $$\label{HEQ2} u(0, \xi)= x(\xi ),\quad u(t,0)=u(t,1)=0,\quad t\ge 0,\, \xi \in (0,1)$$ where $f: \mathbb{R} \to \mathbb{R}$ is bounded and continuous and $\eta $ denoted formally a space-dependent white noise. As well known (see e.g. [@dz1] for fundamentals on the theory of stochastic evolution equations) the system (\[HEQ\]) - (\[HEQ2\]) may be understood as an equation of the form (\[SEM\]) in the space $H = L^2(0,1)$ where $A= \frac{\partial ^2}{\partial \xi ^2}$, $\mathrm{dom} (A) = H^1_0(0,1)\cap H^2(0,1)$, $F: H \to H$, $F(y)(\xi) := f(y(\xi ))$, $ y\in H$, $\xi \in (0,1)$, and $\sqrt{Q}$ is a bounded operator on $H=L^2(0,1)$. We assume that the operator $Q$ is boundedly invertible on $H$, (i.e., the noise is nondegenerate). Then Hypothesis \[GIR\] is obviously satisfied and Hypothesis \[h4\] (ii) is satisfied with $\beta = \frac{1}{2}$ and arbitrary $\alpha \in (0, \frac{1}{2})$ (cf.[@den], Example 9.2 and references therein). Thus the conclusions of Theorems \[dens\], \[HSS\] and \[lp\] hold true in the present example. As far as continuity of the transition density “in the variable $y$” is concerned (cf. Remark \[iny\] ), the problem is more difficult and we only can verify our conditions in the diagonal (commutative) case. Denote by $(e_n)$ and $(\alpha _n)$ the orthonormal basis in $H$ consisting of eigenvectors of the operator $-A$ and its corresponding eigenvalues (so we have $\alpha _n >0$, $ \alpha _n \sim n^2$), and assume that $Q$ commutes with $A$, that is, $$Qe_n = \lambda _n e_n, \quad 0< \inf \lambda _n \le \sup \lambda _n <\infty.$$ Then it is easy to compute eigenvalue expansions of all operators that are needed in Remark \[iny\]. We have $$\label{y14} K_tQ_T^{-1}e_n = \frac{1-e^{-2\alpha _n t}}{1-e^{-2\alpha _n T}}e^{-\alpha _n (T-t)} e_n,$$ $$\label{y15} Q_T^{-1}S^*_T e_n= 2e^{-\alpha _n T} \frac{\alpha _n}{\lambda _n}(1-e^{-\alpha _n T})^{-1} e_n,$$ $$\label{y15a} C(T) e_n= 2e^{-2\alpha _n T} \frac{\alpha _n}{\lambda _n}(1-e^{-2\alpha _n T})^{-1} e_n,$$ $$\label{y16} B_3(s) e_n = 2e^{-\alpha _n (T-s)} \frac{\alpha _n}{\sqrt{\lambda _n}}(1-e^{-\alpha _n T})^{-1} e_n.$$ Obviously, all operators given in (\[y14\]) and (\[y15a\]) are in $\mathcal{L} (H)$, but it is easy to see that $\| B_3(s) \| \sim \frac{1}{T-s}$, so $B_3$ is not an element of $\mathcal{L} (H,L^1(0,T;H))$ and we do not obtain the continuity in $y$ in the norm of $H$. However, taking $\hat H = \mathrm{dom} ((-A)^\delta )$ endowed with the graph norm for any $\delta >0$ (which coincides with a suitable Sobolev-Slobodetskii space) we may easily check that the condition (\[B3\]) is satisfied and we may conclude that the mapping $y\mapsto d(T,x,y)$ is $\hat H\to H$ continuous. In the present case it is also easy to write equation (\[b2\]) for the OU Bridge that splits into a sequence of independent one-dimensional equations for particular coordinates $\hat z_n^{x,y}(t) := \left\langle \hat Z^{x,y}_t,e_n\right\rangle$. We obtain $$d\hat z_n^{x,y}(t) = [-\alpha _n\hat z_n^{x,y}(t)-2\alpha_ne^{-\alpha_n(T-t)}(1-e^{-2\alpha_n (T-t)})^{-1} (e^{-\alpha_n(T-t)}\hat z_n^{x,y}(t) -y_n)]dt + \sqrt{\lambda_n}d \zeta_n(t)$$ for $t\in(0,T)$ with the initial condition $$\hat z_n^{x,y}(0) = x_n,$$ where $x_n = \left\langle x, e_n \right\rangle$, $y_n = \left\langle y, e_n \right\rangle$ and $\zeta _n(t) = \left\langle \zeta _t, e_n \right\rangle $. Here we do not have to assume that the eigenvalues $\alpha_n$ are all negative, only $\alpha_n \neq 0$. If $\alpha_n =0$ for some $n$ the corresponding equation takes the form $$d\hat z_n^{x,y}(t) = \frac{y_n-\hat z_n^{x,y}(t)}{T-t}dt + \sqrt{\lambda_n}d \zeta_n(t),\quad t\in(0,T),$$ which is a well-known equation for a one-dimensional Brownian Bridge. [99]{} Ball J. M.: Strongly continuous semigroups, weak solutions, and the variation of constants formula *Proc. Amer. Math. Soc.* 63 (1977), 370-373 Chojnowska-Michalik A.: Stochastic differential equations in Hilbert spaces, in: Probability theory (Papers, VIIth Semester, Stefan Banach Internat. Math. Center, Warsaw, 1976), pp. 53-74, Banach Center Publ., 5, PWN, Warsaw, 1979 Chojnowska-Michalik A.: Transition semigroups for stochastic semilinear equations on Hilbert spaces *Dissertationes Math.* 396 (2001) Chojnowska-Michalik A. and Goldys B.: Nonsymmetric Ornstein-Uhlenbeck semigroup as second quantized operator. *J. Math. 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--- abstract: 'Notwithstanding the big efforts devoted to the investigation of the mechanisms responsible for the high-energy ($E>100$ MeV) $\gamma-$ray emission in active galactic nuclei (AGN), the definite answer is still missing. The X-ray energy band ($0.4-10$ keV) is crucial for this type of study, since both synchrotron and inverse Compton emission can contribute to the formation of the continuum. Within an ongoing project aimed at the investigation of the $\gamma-$ray emission mechanism acting in the AGN detected by the EGRET telescope onboard CGRO, we firstly focused on the sources for which X-ray and optical/UV data are available in the *XMM-Newton* public archive. The preliminary results are outlined here.' author: - 'L. Foschini' - 'G. Ghisellini' - 'C.M. Raiteri' - 'F. Tavecchio' - 'M. Villata' - 'M. Dadina' - 'G. Di Cocco' - 'G. Malaguti' - 'L. Maraschi' - 'E. Pian' - 'G. Tagliaferri' title: 'The $XMM-Newton$ view of $\gamma-$ray loud active nuclei' --- Introduction ============ The discovery of $\gamma-$ray loud AGN dates back to the dawn of $\gamma-$ray astronomy, when the European satellite *COS-B* ($1975-1982$) detected photons in the $50-500$ MeV range from 3C273 (Swanenburg et al. 1978). However, 3C273 remained the only AGN detected by *COS-B*. A breakthrough in this research field came later with the Energetic Gamma Ray Experiment Telescope (EGRET) on board the *Compton Gamma-Ray Observatory* (CGRO, 1991-2000). The third catalog of point sources contains $271$ sources detected at energies greater than $100$ MeV and $93$ of them are identified with blazars ($66$ at high confidence and $27$ at low confidence), and $1$ with the nearby radiogalaxy Centaurus A (Hartman et al. 1999). Therefore, EGRET discovered that the blazar type AGN are the primary source of high-energy cosmic $\gamma-$rays (von Montigny et al. 1995). Later on, Ghisellini et al. (1998) and Fossati et al. (1998) proposed a unified scheme for $\gamma-$ray loud blazars, based on their physical properties (see, however, Padovani et al. 2003). Specifically, the blazars are classified according to a sequence going from BL Lac to flat-spectrum radio quasar depending on the increase of the observed luminosity, which in turn leads to a decrease of the synchrotron and inverse Compton peak frequencies, and an increase of the ratio between the emitted radiation at low and high frequencies. In other words, the spectral energy distribution (SED) of blazars is typically composed of two peaks, one due to synchrotron emission and the other to inverse Compton radiation. Low luminosity blazars have the synchrotron peak in the UV-soft X-ray energy band and therefore are “high-energy peaked” (HBL). As the synchrotron peak shifts to low energies (near infrared, “low-energy peaked”, LBL), the luminosity increases and the X-ray emission can be due to synchrotron or inverse Compton or a mixture of both. For the Flat-Spectrum Radio-Quasars (FSRQ), the blazars with the highest luminosity, the synchrotron peak is in the far infrared and the X-ray emission is due to inverse Compton. Moreover, the two-peaks SED is a dynamic picture of the blazar behaviour: indeed, these AGN are characterized by strong flares during which the SED can change dramatically. The X-ray energy band can therefore be crucial to understand the blazars behaviour and to improve the knowledge of high-energy emission. Sample selection and data analysis ================================== To investigate the X-ray and optical/UV characteristics of $\gamma-$ray loud AGN in order to search for specific issues conducive to the $\gamma-$ray loudness, we cross correlated the $3^{\rm rd}$ EGRET Catalog (Hartman et al. 1999), updated with the identifications performed to date, with the public observations available in the *XMM-Newton* Science Archive to search for spatial coincidences within $10'$ of the boresight of the EPIC camera. Fourteen AGN have been found (Table 1) as of April $14^{\rm th}$, 2005, for a total of $43$ observations. For three of them there are several observations available: 15 for 3C $273$, 6 for Mkn $421$, 9 for PKS $2155-304$. The data from $6$ sources of the present sample are analyzed here for the first time and, among them, one has never been observed in X-rays before (PKS $1406-706$). Data from the EPIC camera (MOS, Turner et al. 2001; PN, Strüder et al. 2001) and the Optical Monitor (Mason et al. 2001) have been analyzed with `XMM SAS 6.1` and `HEASoft 6.0`, together with the latest calibration files available at April $14^{\rm th}$, 2005, and by following the standard procedures described in Snowden et al. (2004). In addition, the Optical Monitor makes it possible to have optical/UV data simultaneous to X-ray for most of the selected sources, with the only exception of PKS $0521-365$, Mkn $421$, and Cen A. 3EG Counterpart Type$^{\mathrm{*}}$ Redshift -------------- ---------------- --------------------- ------------------------- J$0222+4253$ $0219+428$ LBL $0.444$ J$0237+1635$ AO $0235+164$ LBL $0.94$ J$0530-3626$ PKS $0521-365$ FSRQ $0.05534$ J$0721+7120$ S5 $0716+714$ LBL $>0.3$ J$0845+7049$ S5 $0836+710$ FSRQ $2.172$ J$1104+3809$ Mkn $421$ HBL $0.03002$ J$1134-1530$ PKS $1127-145$ FSRQ $1.184$ J$1222+2841$ ON $231$ LBL $0.102$ J$1229+0210$ 3C $273$ FSRQ $0.15834$ J$1324-4314$ Cen A RG $0.00182^{\mathrm{**}}$ J$1339-1419$ PKS $1334-127$ FSRQ $0.539$ J$1409-0745$ PKS $1406-076$ FSRQ $1.494$ J$1621+8203$ NGC $6251$ RG $0.0247$ J$2158-3023$ PKS $2155-304$ HBL $0.116$ : Main characteristics of the observed AGN. \[tab:host\] Main Results ============ The main findings of this study can be summarized as follows: \(i) the EGRET blazars studied here have spectral characteristics in agreement with the unified sequence of Ghisellini et al. (1998) and Fossati et al. (1998); \(ii) no evident characteristics conducive to the $\gamma-$ray loudness have been found: the photon indices are generally consistent with what is expected for this type of sources, with FSRQ that are harder than BL Lac; there are hints of some differences in the photon indices when compared with other larger catalogs (e.g. *BeppoSAX* Giommi et al. 2002), particularly for FSRQ: the sources best fit with a simple power law model show a harder photon index ($1.39\pm 0.09$ vs $1.59\pm 0.05$); however, the statistics is too poor to make firm conclusions (3 sources vs 26 in the *BeppoSAX* catalog); \(iii) three sources show Damped Lyman $\alpha$ systems along the line of sight (AO $0235+164$, PKS $1127-145$, S5 $0836+710$), but it is not clear if the intervening galaxies can generate gravitational effects altering the characteristics of the blazars so to enhance the $\gamma-$ray loudness; \(iv) no evidence of peculiar X-ray spectral features has been found, except for the emission lines of the iron complex in Cen A. More details of the analysis will be available in Foschini et al. (2005). Acknowledgments {#acknowledgments .unnumbered} =============== This work is based on public observations obtained with *XMM–Newton*, an ESA science mission with instruments and contributions directly funded by ESA Member States and the USA (NASA). This work was partly supported by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00321 and by the Italian Space Agency (ASI). Foschini L. et al., 2005, submitted Fossati G. et al., 1998, MNRAS 299, 433 Ghisellini G. et al., 1998, MNRAS 301, 451 Giommi P. et al., 2002, astro-ph/0209596 Hartman R.C. et al., 1999, ApJS 123, 79 Mason K.O. et al., 2001, A&A 365, L36 Padovani P. et al., 2003, ApJ 588, 128 Snowden S. et al., 2004. An introduction to XMM-Newton data analysis. Version 2.01, 23 July 2004. Strüder L. et al., 2001, A&A 365, L18 Swanenburg B.N. et al., 1978, Nature 275, 298 Turner M.J. et al., 2001, A&A 365, L27 von Montigny C. et al., 1995, ApJ 440, 525

--- abstract: 'Periodic control systems used in spacecrafts and automotives are usually period-driven and can be decomposed into different modes with each mode representing a system state observed from outside. Such systems may also involve intensive computing in their modes. Despite the fact that such control systems are widely used in the above-mentioned safety-critical embedded domains, there is lack of domain-specific formal modelling languages for such systems in the relevant industry. To address this problem, we propose a formal visual modeling framework called as a concise and precise way to specify and analyze such systems. To capture the temporal properties of periodic control systems, we provide, along with , a property specification language based on interval logic for the description of concrete temporal requirements the engineers are concerned with. The statistical model checking technique can then be used to verify the models against desired properties. To demonstrate the viability of our approach, we have applied our modelling framework to some real life case studies from industry and helped detect two design defects for some spacecraft control systems.' author: - | Zheng Wang$^{1, 5}$, Geguang Pu$^{1}$, Shenchao Qin$^{2}$, Jianwen Li$^{1}$,\ Kim G. Larsen$^{3}$, Jan Madsen$^{4}$, Bin Gu$^{5}$, Jifeng He$^{1}$ bibliography: - 'main.bib' title: | : A Mode Diagram Modeling Framework for\ Periodic Control Systems --- $^{1}$ [<wangzheng@sei.ecnu.edu.cn>]{}, [<ggpu@sei.ecnu.edu.cn>]{},\ Shanghai Key Laboratory of Trustworthy Computing,\ East China Normal University\ $^{2}$ [<s.qin@tees.ac.uk>]{}, University of Teesside\ $^{3}$ [<kgl@cs.aau.dk>]{}, Aalborg University of Denmark\ $^{4}$ [<jan@imm.dtu.dk>]{}, Technical University of Denmark\ $^{5}$ [<gubin88@yahoo.com.cn>]{}, Beijing Institute of Control Engineering intro.tex lang.tex property.tex verification.tex experiment.tex related.tex

--- abstract: 'The concept of homology, originally developed as a useful tool in algebraic topology, has by now become pervasive in quite different branches of mathematics. The notion particularly carries over quite naturally to the setup of measure-preserving transformations arising from various group actions or, equivalently, the setup of stationary sequences considered in this paper. Our main result provides a sharp criterion which determines (and rules out) when two stationary processes belong to the same *null-homology equivalence class*. We also discuss some concrete cases where the notion of null-homology turns up in a relevant manner.' address: - | G. Alsmeyer\ Inst. Math. Stochastics,\ Department of Mathematics\ and Computer Science\ University of Münster\ Orléans-Ring 10, D-48149\ Münster, Germany\ \ - | C. Mukherjee\ Inst. Math. Stochastics,\ Department of Mathematics\ and Computer Science\ University of Münster\ Orléans-Ring 10, D-48149\ Münster, Germany\ author: - - title: 'On null-homology and stationary sequences' --- Introduction and motivation {#sec:intro} =========================== Homology is a notion that arises in various branches of mathematics. It was originally developed in algebraic topology in order to associate a sequence of algebraic objects. A typical fundamental question is the following: When does a $n$-cycle of a (simplical) complex form the boundary of a $(n+1)$-chain, or equivalently, when is its fundamental class a boundary for the singular homology? If such a requirement is fulfilled, the cycle is said to be *homologous to $0$* or *null-homologous*. In the present article, we provide a suitable criterion for null-homology in a different context, namely measure-preserving transformations arising from natural group actions on any complete and separable metric space. To formulate the question precisely, we recall some basic definitions. Let $\bX= (X_{n})_{n\in \Z}$ be a sequence of random variables defined on a probability space with underlying probability measure $\Prob$ and such that the $X_{n}$’s take values in a complete separable metric space $\mathscr S$. Note that $\bX$ forms a stationary stochastic process if, for all $n\in\N$ and $m\in \Z$, $$\Prob\big((X_{1},\dots, X_{n}) \in \cdot\big)\ =\ \Prob\big((X_{m+1},\dots, X_{m+n})\in\cdot\big).$$ In other words, the joint law of $(X_{1},\dots, X_{n})$ for any $n$ coincides with the law of any of its “shifts" under the action of the additive group $\Z$ on the space of doubly-infinite sequences $\mathscr S^{\Z}$. There is a natural notion of *homology*, first coined by Lalley [@L86 p. 197] in this setup, that arises from the group action. Indeed, given any stationary sequence $\bX$ and measurable functions $F, G\colon \mathscr S^\Z\to \R^{d}$, we say that $F$ is *homologous* to $G$ (with respect to $\bX$ and $\Prob$) and write $F\sim G$ if there exists a function $\xi: \mathscr S^\Z\to \R^{d}$ such that $$\label{eq:F,G homologous} F(X_{0}) - G(X_{0})\ =\ \xi (X_{1})- \xi(X_{0})\quad\Prob\text{-a.s.}$$ Then $\sim$ is an equivalence relation, and if $F\sim 0$, thus $$\label{eq:F null-homologous} F(X_{0})\ =\ \xi (X_{1})- \xi(X_{0})\quad\Prob\text{-a.s.},$$ we say that $F$ is *null-homologous*. Now observe that, given any stationary process $\bX$ and a null-homologous function $F$, the process $(F(X_{n}))_{n\in\Z}$ is not only also stationary but in fact the incremental sequence of another stationary process, viz. $(\xi(X_{n}))_{n\in\Z}$. In view of this, the converse question which stationary processes are of this “incremental” type and therefore allowing a representation with respect to a null-homologous function appears to be natural. The main goal of the present article is to provide a sharp criterion for this fundamental property which is of interest for various reasons as will also be explained. Indeed, mere tightness of the partial sums $S_{n}=X_{1}+\cdots+X_{n}$, $n\in\N$, associated with the stationary process $\bX$ turns out to be the necessary and sufficient condition, see Theorem \[thm:main result\]. The proof, which does not even require ergodicity, is quite simple and relies on the construction of some *commutative maps* in a proper setup and an application of Schauder’s fixed point theorem. To put our work into context, we first discuss some concrete cases where null-homology turns up in a relevant way. Markov random walks ------------------- In the [@L86], Lalley considered random walks with increments from a fairly general class of stationary sequences, albeit restricted to the [*integrable set up*]{}, see Remark \[rem-Lalley\]. As a main result, he proved a Blackwell-type renewal theorem for which it was necessary to rule out a certain “lattice-type" behavior which is intimately connected to the notion of null-homology. In the following, we give a brief introduction of this notion within the framework of *Markov random walks* which are also called *Markov-additive processes* and indeed comprise random walks with stationary increments as explained below. Let $(\cS,\fS)$ be an arbitrary measurable space and $\cB(\R^{m})$ the Borel $\sigma$-field on $\R^{m}$ for $m{\geqslant}1$. Suppose that $(M_{n},X_{n})_{n{\geqslant}0}$ is a *Markov-modulated sequence* of $\cS\times\R^{d}$-valued random variables, where $\cS\times\R^{d}$ is endowed with the product $\sigma$-field $\fS\otimes\cB(\R^{d})$. This means that $X_{0},X_{1},\ldots$ are conditionally independent given the *driving chain* $(M_{n})_{n{\geqslant}0}$ and $$\begin{aligned} \Prob(X_{i}\in B_{i},\,0{\leqslant}k{\leqslant}n|M_{j}=s_{j},\,j{\geqslant}0)\ =\ P_{0}(s_{0},B_{0})\prod_{i=1}^{n}P((s_{i-1},s_{i}),B_{i})\end{aligned}$$ for all $n\in\N_{0}$, $s_{0},\ldots,s_{n}\in\cS$, measurable $B_{0},\ldots,B_{n}\subset\R^{d}$ and suitable kernels $P_{0}$ and $P$ which describe the conditional laws of $X_{0}$ given $M_{0}$ and of $X_{n}$ given $(M_{n-1},M_{n})$ for $n{\geqslant}1$, respectively. We make the additional assumption that $(M_{n})_{n{\geqslant}0}$ *is ergodic with unique stationary distribution $\mu$*. Defining $S_{0}:=0$ and $$S_{n}\,:=\,\sum_{i=1}^{n}X_{i},\quad n=1,2,\ldots,$$ the bivariate sequence $(M_{n},S_{n})_{n{\geqslant}0}$ and also $(S_{n})_{n{\geqslant}0}$ are called *Markov random walk (MRW)* and $(M_{n})_{n{\geqslant}0}$ its *driving* or *modulating chain*. For our purposes, it is enough to study these objects in stationary regime, that is, under $\Prob_{\mu}:=\int_{\cS}\Prob(\cdot|M_{0}=s)\,\mu(\mathrm d s)$. We may then further assume the existence of a doubly infinite stationary extension $(M_{n},X_{n})_{n\in\Z}$ with associated doubly infinite random walk $$\begin{aligned} S_{n}\ =\ \begin{cases} \sum_{i=1}^{n}X_{i}&\text{if }n{\geqslant}1,\\ 0,&\text{if }n=0,\\ -\sum_{i=n+1}^{0}X_{i}&\text{if }n<0. \end{cases}\end{aligned}$$ In this context, both $(M_{n},S_{n})_{n\in\Z}$ and $(M_{n},X_{n})_{n\in\Z}$ are called *null-homologous* if there exists a measurable function $\xi:\cS\to\R^{d}$ such that $$\begin{gathered} X_{n}\ =\ \xi(M_{n})-\xi(M_{n-1})\quad\Prob_{\mu}\text{-a.s.}\label{eq:null-hom X} \shortintertext{and thus} S_{n}\ =\ \xi(M_{n})-\xi (M_{0})\quad\Prob_{\mu}\text{-a.s.}\label{eq:null-hom S}\end{gathered}$$ for all $n\in\Z$. The reader should note that the latter implies the stationarity of $(S_{n}+\xi(M_{0}))_{n\in\Z}$ and thus the “almost stationarity” of the random walk $(S_{n})_{n\in\Z}$ itself, in particular its tightness. Now let $(X_{n})_{n\in\Z}$ be any doubly infinite stationary sequence of $\R^{m}$-valued random variables and put $$M_{n}\ :=\ (X_{i})_{i{\leqslant}n}$$ for $n\in\Z$. Observe that $(M_{n})_{n\in\Z}$ constitutes a stationary Markov chain (ergodic iff $(X_{n})_{n\in\Z}$ is ergodic) and $(M_{n},X_{n})_{n\in\Z}$ a Markov-modulated sequence. This shows that null-homology for stationary processes may indeed be viewed as a special instance of the very same notion within the framework of Markov-modulation under stationarity. Null-homology arises also quite naturally in connection with the lattice-type of MRW’s. As before, let $(M_{n})_{n{\geqslant}0}$ be ergodic with unique stationary law $\mu$. Following Shurenkov [@S84], the MRW $(M_{n},S_{n})_{n{\geqslant}0}$ is called $d$-arithmetic if $d$ is the maximal positive number such that $$\Prob_{\mu}\big(X_{1}\in \xi(M_{1})-\xi(M_{0})+d\Z\big)\ =\ 1$$ for a suitable function $\xi:\cS\to [0,d)$, called *shift function*. If no such $d$ exists, it is called nonarithmetic. Equivalently, $(M_{n},S_{n})_{n{\geqslant}0}$ is $d$-arithmetic if $d>0$ is the maximal number such that $(M_{n},X_{n}-X_{n}')_{n\in\Z}$ is Markov-modulated and null-homologous for a sequence of $d\Z$-valued random variables $(X_{n}')_{n\in\Z}$. Namely, with $\xi$ denoting the shift function, $$X_{n}'\,:=\,X_{n}-\xi(M_{n})+\xi(M_{n-1})$$ for $n\in\Z$. Stochastic homogenization ------------------------- The notion of a *corrector* plays an important rôle in the context of *stochastic homogenization* of a random media. We will describe the setup and how null-homology comes into play for a particular instance of a random walk in random environment (in the reversible setup) known as the *random conductance model*. Let $$E_d\ =\ \big\{(x,y)\colon |x-y|=1, \, x,y \in \Z^{d}\big\}$$ be the set of nearest neighbor bonds in $\Z^{d}$ and $\Omega= [a,b]^{E_d}$ for any two fixed numbers $0<a<b$. We assume that $\Omega$ is equipped with the product $\sigma$-field $\mathcal B$ and carries a probability measure $\Prob$. For simplicity, we also assume that the canonical coordinates are i.i.d. variables under $\Prob$. Note that any $x\in\Z^{d}$ acts on $(\Omega,\mathcal B,\Prob)$ as a $\Prob$-preserving and ergodic transformation $\tau_{x}$, defined as the canonical translation $$\Omega\ \ni\ \omega(\cdot)\,\mapsto\,\omega(x+\cdot).$$ For any $\omega\in \Omega$, we then have a Markov chain $(S_{n})_{n{\geqslant}0}$ on $\Z^{d}$ under a family of probability measures $(P^{\pi,\omega}_{x})_{x\in\Z^{d}}$ such that $P^{\pi,\omega}_{x}(S_{0}=x)=1$ and transition probabilities are given by $$\label{pi} \begin{aligned} &P_{x,\omega}(S_{n+1}=y+e|S_{n}=y)\ =\ \pi_\omega(y,y+e)\\ &\hspace{.3cm}:=\ \frac {\omega((y,y+e))}{\sum_{|e^\prime|=1} \omega((y,y+e^\prime))}\ =\ \pi_{\tau_{y}\omega}(0,e) \end{aligned}$$ for any $e$ with $|e|=1$ and $x\in\Z^{d}$. Furthermore, the sequence $\widehat \omega_{n}\stackrel{\mathrm{def}}=\tau_{S_{n}}\omega$ for $n{\geqslant}0$, with initial state $\omega$ and taking values in the “environment space" $\Omega$, is also a Markov chain with transition kernel $\widehat\Pi$ defined by $$(\widehat \Pi f)(\omega)\ :=\ \sum_{|e|=1} \pi_\omega(0,e) f(\tau_{e} \omega)$$ for all bounded and measurable $f$. It is called the *environment seen from the moving particle*, or simply the *environmental process*, and particularly useful in the following scenario: Suppose there is a probability density $\phi\in L^1(\Prob)$ (i.e. $\phi{\geqslant}0$ and $\int \phi\,\mathrm d\Prob=1$) such that $\mathrm d\Q=\phi\,\mathrm d\Prob$ is $\widehat\Pi$-invariant, i.e. $$\begin{gathered} \label{invariance} \langle \widehat \Pi f, \phi\rangle_{L^{2}(\Prob)}=\langle f, \phi\rangle_{L^{2}(\Prob)}, \intertext{for all bounded and measurable $f$ or, equivalently,} L^\star\phi\,=\,0\quad\mbox{if }L=\mathrm{Id}-\widehat\Pi.\nonumber\end{gathered}$$ It can be shown, see [@PV81; @K85; @KV86] and also [@BS02 Theorem 1.2], that such an invariant density $\phi$ if it exists is necessarily unique. Moreover, $\Prob$ and $\Q$ are then equivalent measures and $(\widehat\omega_{n})_{n{\geqslant}0}$ an ergodic process in equilibrium (under initial law $\Q$). In the random conductance model with transition probabilities , the invariant density $\phi$ can easily be found by reversibility (solving the detailed balance equations), viz. $$\phi(\omega)\ =\ \frac 1C \sum_{|e|=1} \omega((0,e)),\quad\text{where}\quad C\ =\ \int\sum_{|e|=1}\omega((0,e))\ \Prob(\mathrm d\omega).$$ Reversibility further implies that $\widehat\Pi$ is self-adjoint on $L^{2}(\Q)$, that is $$\label{reversible} \langle f, \widehat\Pi g\rangle_{L^{2}(\Q)}\ =\ \langle\widehat\Pi f,g\rangle_{L^{2}(\Q)}$$ for all bounded and measurable functions $f,g$. Returning to the Markov chain $(S_{n})_{n{\geqslant}0}$ under $P_{0,\omega}$, the ergodicity of $(\widehat\omega_{n})_{n{\geqslant}0}$ fairly easily provides a strong law of large numbers, viz. $S_{n}/n \to 0$ $P_{0,\omega}$-a.s. for $\Prob$-almost all $\omega$. To see this, let $$\mathrm d(x,\omega)\ =\ E^{\pi,\omega}_{x}[S_{1}-S_{0}]\ =\ \sum_{|e|=1}e\pi_{\omega}(x,x+e)\ =\ \mathrm d(0,\tau_{x}\omega)$$ denote the *local drift at $x$* under $P_{0,\omega}$. As $(\widehat\omega_{n})_{n{\geqslant}0}$ is ergodic under initial law $\Q$, Birkhoff’s ergodic theorem implies $$n^{-1} \sum_{j=0}^{n-1}\mathrm d(S_{j},\omega)\ =\ n^{-1} \sum_{j=0}^{n-1} \mathrm d(0,\widehat\omega_{j})\ \xrightarrow{n\to\infty}\ \mathbb \int\mathrm d(0,\omega')\ \Q(\mathrm d\omega')\ =\ 0$$ for $\Q$-almost all and thus $\Prob$-almost all $\omega$ (as $\Prob,\Q$ are equivalent), the right-hand side being 0 by reversibility (recall ) and the definition of $\Q$. Now observe that $Z_{n}= S_{n}-S_{0}-\sum_{j=0}^{n-1} \mathrm d(S_{j},\omega)$, $n{\geqslant}0$, is a $P_{0,\omega}$-martingale with bounded (uniformly in $\omega$) increments and therefore satisfies, by the Azuma-Hoeffding inequality, $$P_{0,\omega}\big(n^{-1}Z_{n}{\geqslant}n^{-\frac{1}{2}+\eps}\big)\ {\leqslant}\ \exp(-Cn^{2\eps})$$ for any $\eps>0$ and some $C>0$ (not depending on $\omega$). Finally, by an appeal to the Borel-Cantelli lemma, we infer that $Z_{n}/n\to 0$ holds $\Prob$-a.s., and since $\sum_{j=0}^{n-1} \mathrm d(S_{j},\omega)=o(n)$ a.s., it follows that $S_{n}/n\to 0$ $\Prob$-a.s., too. As will be explained next, stochastic homogenization comes into play when turning to the more ambitious aim of deriving an almost sure central limit theorem (or an invariance principle) for the distribution (under the quenched measure $P^{\pi,\omega}_0$) of the random walk $(S_{n})_{n{\geqslant}0}$, and it leads to the notion of a *corrector*. Note that the local drift $\mathrm d$ is bounded and therefore particularly $\in L^{2}(\Prob)$. For any fixed $\eps>0$, let $g_{\eps}\in L^{2}(\Prob)$ be a solution to the *Poisson equation* $$((1+\eps)\textrm{Id}-\widehat\Pi)g_{\eps}\ =\ \mathrm d$$ The solution is well-defined and in fact given by the Neumann series $$g_{\eps}\ =\ d\ +\ \sum_{n{\geqslant}1} \frac{\wh{\Pi}^{n}d}{(1+\eps)^{n}}.$$ Putting $G_{\eps}(\omega,e):=(\nabla_e g_{\eps})\omega)=g_{\eps}(\tau_e\omega)-g_{\eps}(\omega)$ for any $e$ with $|e|=1$, we then have the result $$G_{\eps}(\cdot,e)\circ \tau_{x}\ \xrightarrow{L^{2}(\Prob)}\ G(\cdot,e)\circ\tau_{x}$$ for any $x\in\R^{d}$, see [@KV86 Theorem 1.3], where $G$ is a (divergence free) *gradient field*, i.e., it satisfies the *closed loop condition* $$\label{closed loop condition} \sum_{j=1}^{n}G(\tau_{x_{j}}\omega, {x_{j+1}-x_{j}})\ =\ 0\quad\text{a.s.}$$ for any closed path $x_{0}\to x_{1}\to\dots\to x_{n}=x_{0}$ in $\R^{d}$. The last property allows us to define the *corrector* corresponding to $G$ as $$\label{corrector} V_{G}(x,\omega)\,:=\,\sum_{j=1}^{n} G(\tau_{x_{j}}\omega, {x_{j+1}-x_{j}})$$ along any path $0\to x_{1}\to\ldots\to x_{n-1}\to x_{n}=x$, the particular choice of the path being irrelevant because of . It also follows that $V_{G}$ has stationary and $L^{2}$-bounded gradient in the sense that $$\begin{gathered} V_{G}(x,\omega)-V_{G}(y,\omega)\ =\ V_{G}(x-y,\tau_y\omega)\quad\text{for all }x,y\in\Z^{d} \shortintertext{and} \sup_{x\in\Z^{d}}\|V_{G}(x+e,\cdot)-V_{G}(x,\cdot)\|_{L^{2}(\Prob)}\,<\,C,\end{gathered}$$ respectively. Even more importantly, the mapping $x\mapsto V_{G}(x,\omega)+x$ is harmonic with respect to the transition probabilities for $\Prob$-almost all $\omega$. This means that $(S_{n}+V_G(S_{n},\cdot))_{n{\geqslant}0}$ is a martingale with respect to $P^{\pi,\omega}_0$ so that the corrector $V_{G}$ expresses the “distance" (or the *deformation*) of the martingale from the random walk $(S_{n})_{n{\geqslant}0}$ itself. One can show that the contribution of this deformation grows at most sub-linearly at large distances (i.e. $\sup_{|x|\leq n} n^{-1} V_{G}(x,\cdot)\xrightarrow{n\to\infty} 0$ a.s.) whence, by the martingale central limit theorem, the laws $P^{\pi,\omega}_{0}(S_{n}/\sqrt n \in \cdot)$ converge weakly to a Gaussian law for almost every $\omega$, see [@SS04; @BB07; @MP07] for a detailed recount of the substantial progress made in this direction. In order to finally make a connection with the notion of null-homology, let us note that the result just mentioned does not rule out the possibility that the corrector grows stochastically to infinity. Namely, although the gradient of $V_{G}$ is stationary and thus tight as pointed out above, the latter property may naturally fail for $V_{G}$ itself. On the other hand, a tight corrector means that the above martingale is just a “negligible” perturbation of the random walk $(S_{n})_{n{\geqslant}0}$ itself which is a much stronger statement than the above central limit theorem. Our main result, Theorem \[thm:main result\] below, establishes, as a further information, the equivalence of this property with the null-homology of its stationary gradient, under no extra assumptions. Now, for the random conductance model in dimension $d{\geqslant}3$, the tightness of $V_{G}$ has indeed been shown to hold, see [@GO15] and [@AKM17]. The main result =============== We proceed with a description of the setup that allows us to define null homology in terms of probability measures rather than random variables. This appears to be more convenient to state and prove our main result. Without any loss of generality, we work with $\mathscr S= \R^{d}$ and write $\Omega=(\R^{d})^{\otimes\Z}$ for the space of doubly infinite sequences ${\bf x}=(x_{n})_{n\in\Z}$ endowed with the Borel $\sigma$-field and $T:\Omega\to\Omega$ the (left) shift operator on $\Omega$, viz. $${\bf x}\ =\ (\ldots,x_{-1},x_{0},x_{1},\ldots)\ \mapsto\ (\ldots,x_{0},x_{1},x_{2},\ldots).$$ The coordinate mappings on $\Omega$ are denoted $X_{n}$ for $n\in\Z$, and we let $S_{n}$ be the mapping ${\bf x}\mapsto s_{n}$ on $\Omega$ for $n\in\Z$, where $$\begin{aligned} s_{n}\ =\ \begin{cases} \hfill x_{1}+\ldots+x_{n}&\text{if }n{\geqslant}1,\\ \hfill 0,&\text{if }n=0,\\ -(x_{-n+1}+\ldots+x_{0})&\text{if }n{\leqslant}-1. \end{cases}\end{aligned}$$ So $(X_{n})_{n\in\Z}$ forms a stationary sequence with associated random walk $(S_{n})_{n\in\Z}$ under any $T$-invariant probability measure on $\Omega$. Next, we denote by $\cM(\Omega)$ the locally convex vector space of finite signed measures on $\Omega$ endowed with the topology of weak convergence and further by $\cM_{T}(\Omega)$ its subsets of $T$-invariant probability measures. Defining the map $D:\Omega\to\Omega$ by $${\bf x}\ \mapsto\ T{\bf x}-{\bf x}\ =\ (\ldots,x_{0}-x_{-1},x_{1}-x_{0},x_{2}-x_{1},\ldots),$$ we obviously have that $\Prob\in\cM_{T}(\Omega)$ implies $\Prob D^{-1}=\Prob(D\in\cdot)\in\cM_{T}(\Omega)$. Null homology for elements of $\cM_T(\Omega)$ can now be defined as follows. Any $T$-invariant probability measure $\Prob\in\cM_T(\Omega)$ is called null-homologous if $\Prob=\Q D^{-1}$ for some $\Q\in\cM_T(\Omega)$. Plainly, null homology of $\Prob$ is equivalent to the null homology of $(X_{n})_{n\in\Z}$ under $\Prob$. The subsequent Theorem \[thm:main result\] provides a characterization of this property in terms of the laws of the $S_{n}$ under $\Prob$, thus $\{\Prob S_{n}^{-1}:n\in\Z\}$. \[thm:main result\] Given any $\Prob\in\cM_T(\Omega)$, the following assertions are equivalent: - $\Prob$ is null-homologous. - $\{\Prob S_{n}^{-1}:n{\geqslant}0\}$ is tight. - $\{\Prob S_{-n}^{-1}:n{\geqslant}0\}$ is tight. \[rem-Lalley\]In [@L86 Proposition 6], Lalley provided a criterion for null-homology within the subclass of integrable stationary sequences, called *$L^{1}$-null-homology*. It enabled him to rule out a certain lattice-type behavior for the derivation of a renewal theorem for certain stationary processes. In fact, he showed that $L^{1}$-null-homology is equivalent to the $L^1$-boundedness of the partial sums of the stationary sequence, i.e., of the associated random walk. Naturally, this is a much stronger requirement than the tightness appearing in our theorem above. Also, the proof of our result, which is based on an application of the Schauder fixed-point theorem in an appropriate context, differs entirely from the arguments used in [@L86]. Note that our criterion for null-homology holds for any $T$-invariant measure $\Prob\in \cM_T(\Omega)$ and is not restricted to the the ergodic ones, i.e., extremal points of $\cM_T(\Omega)$. In the above context of $T$-invariant probability measures, we say that $\Prob\in\cM_T(\Omega)$ as well as the coordinate sequence $(X_{n})_{n\in\Z}$ (under $\Prob$) are *$L^{p}$-null-homologous* if they are null-homologous and $\Erw|X_{0}|^{p}<\infty$. Before giving the proof of Theorem \[thm:main result\], we provide as an immediate consequence the following corollary which characterizes *$L^{p}$-null-homology* for any $p>0$ and particularly comprises Lalley’s results for $p=1$ and $p=2$. Given any $\Prob\in\cM_T(\Omega)$, the following assertions are equivalent for any $p>0$: - $\Prob$ is $L^{p}$-null-homologous. - $(S_{n})_{n{\geqslant}0}$ is $L^{p}$-bounded. - $(S_{-n})_{n{\geqslant}0}$ is $L^{p}$-bounded. Obviously, it suffices to show that (b) implies (a). To this end, we consider the bivariate mappings $$\begin{gathered} \Lambda_{k}:\Omega\to\Omega\times\Omega,\\ {\bf x}\ \mapsto\ \big(T^{k}{\bf x},{\bf x}+\ldots+T^{k-1}{\bf x}\big)\ =\ \big((x_{n+k})_{n\in\Z},(x_{n}+\ldots+x_{n+k-1})_{n\in\Z}\big)\end{gathered}$$ for $k\in\N$ and point out that (b) entails the tightness of the family $$\cP\ =\ \{\Prob\Lambda_{k}^{-1}:k\in\N\}.$$ We can lift the shift $T$ as well as the projections $X_{n}$ in a canonical way to mappings on $\Omega\times\Omega$ and, by slight abuse of notation, may call these mappings again $T$ and $X_{n}$. The projections on the $y$-components, namely $({\bf x},{\bf y})\mapsto y_{n}$ if ${\bf y}=(y_{k})_{k\in\Z}$, are denoted $Y_{n}$ for $n\in\Z$. Then the $T$-invariance of $\Prob$ implies the very same for the elements of $\cP$. Now let $\cD$ be the closed convex hull of all weak limit points of $\cP$ which forms a compact convex subset of $\cM(\Omega\times\Omega)$. Consider the map $$S:\Omega\times\Omega\to\Omega\times\Omega,\quad \big({\bf x},{\bf y}\big)\ \mapsto\ \big(T{\bf x},{\bf x}+{\bf y}\big)$$ which is linear, continuous, commutes with $T$, i.e. $S\circ T=T\circ S$, and satisfies further $S\circ\Lambda_{n}=\Lambda_{n+1}$, thus $\Gamma_{n}S^{-1}=\Gamma_{n+1}$ for all $n\in\N$, where $\Gamma_{n}:=\Prob\Lambda_{n}^{-1}$. Then the last property entails that the set $\cD$ is $S$-invariant which in turn, by invoking Schauder’s fixed point theorem, allows us to conclude that $S$ has a fixed point, say $\Gamma$, in $\cD$. This means that $\Gamma S^{-1}=\Gamma$ or, equivalently, that $\Gamma$ is $S$-invariant. Finally, by considering the map $$G=(X_{0},Y_{0}):\Omega\times\Omega\to\R^{d}\times\R^{d},\quad \big({\bf x},{\bf y}\big)\ \mapsto\ (x_{0},y_{0})$$ we have that $(X_{n}',Y_{n}'):=G\circ S^{n}=(X_{n},Y_{n}\circ S^{n})$, $n{\geqslant}0$, is stationary under $\Lambda$ and satisfies: - $(X_{n}')_{n{\geqslant}0}=(X_{n})_{n{\geqslant}0}$ and has law $\Prob$ under $\Lambda$, because this is the case under any element of $\cD$. - $Y_{n+1}'=Y_{0}+X_{0}+\ldots+X_{n}=Y_{n}'+X_{n}'$, thus $$X_{n}\ =\ X_{n}'\ =\ Y_{n+1}'-Y_{n}'$$ for all $n{\geqslant}0$. Since $(Y_{n}')_{n{\geqslant}0}$ is stationary under $\Lambda$, (a) follows. [WWW98]{} , [T. Kuusi]{} and [J.-C. Mourrat.]{} , 2017 and [M. Biskup]{}. Quenched invariance principle for random walk on percolation clusters. , [**137**]{}, 1-2, 83-120, 2007. and [A-S. Sznitman]{} and [F. Otto.]{} arXiv:1510.08290, 2015. and [S.R.S. Varadhan]{}. Limit theorem for additive functionals of reversible Markov chains and application to simple exclusions. , 1-19, 1986. , , (Russian math surveys), [**40**]{}, 73-145, 1985. . . , [**72**]{}, Issue 2, pp 195-213, 1986. and [A. Piatnitski]{}. Quenched invariance principle for random walks on percolation clusters. , 463, 2287-2307, 2007. and [S.R.S. Varadhan]{}. . In: Random fields, Vol. I, II (Esztergom, 1979), volume 27 of Colloq. Math. Soc. Janos Bolyai, pages 835-873. North-Holland, Amsterdam, 1981. and [A. S. Sznitman]{}. Quenched invariance principles for walks on clusters of percolation or among random conductances. , [**129**]{}, 219-244, 2004. . . [*Theory Probab. Appl.*]{}, [**29**]{}, [247–265]{}, 1984.

--- abstract: 'The swampland distance conjecture (SDC) addresses the ability of effective field theory to describe distant points in moduli space. It is natural to ask whether there is a local version of the SDC: is it possible to construct local excitations in an EFT that sample extreme regions of moduli space? In many cases such excitations exhibit horizons or instabilities, suggesting that there are bounds on the size and structure of field excitations that can be achieved in EFT. Static bubbles in ordinary Kaluza-Klein theory provide a simple class of examples: the KK radius goes to zero on a smooth surface, locally probing an infinite distance point, and the bubbles are classically unstable against radial perturbations. However, it is also possible to stabilize KK bubbles at the classical level by adding flux. We study the impact of imposing the Weak Gravity Conjecture (WGC) on these solutions, finding that a rapid pair production instability arises in the presence of charged matter with $q/m\gtrsim 1$. We also analyze 4d electrically charged dilatonic black holes. Small curvature at the horizon imposes a bound $\log(M_{BH})\gtrsim |\Delta\phi|$, independent of the WGC, and the bound can be strengthened if the particle satisfying the WGC is sufficiently light. We conjecture that quantum gravity in asymptotically flat space requires a general bound on large localized moduli space excursions of the form $ |\Delta\phi|\lesssim |\log(R\Lambda)|$, where $R$ is the size of the minimal region enclosing the excitation and $\Lambda^{-1}$ is the short-distance cutoff on local EFT. The bound is qualitatively saturated by the dilatonic black holes and Kaluza-Klein monopoles.' bibliography: - 'dipole\_refs.bib' --- [  \ ]{} 2.2cm [**Transplanckian Censorship\ and the Local Swampland Distance Conjecture** ]{} 1.4cm [ Patrick Draper$^{(a)}$ and Szilard Farkas]{}\ \ 1.5cm 1.0 cm Introduction ============ It is of interest to try to determine ways in which low-energy physics is constrained by consistent embedding in a quantum theory of gravity. A number of conjectures have focused on the properties of moduli spaces. For example, the swampland distance conjecture (SDC) [@swampland2] states that homogeneous motion over large distances in any large moduli space results in a tower of exponentially light states descending below the cutoff of the initial EFT. A simple example of a gravitational theory with a large moduli space is ordinary Kaluza-Klein (KK) theory. The energy of the KK spacetime ${\cal R}^{D,1}\times S^1$ does not depend on the size $R$ of the circle, and the invariant distance between two points in the moduli space is $\int dR/R$, which diverges logarithmically as the circle size goes to zero or infinity. If one changes the asymptotic value of the modulus in this theory, a tower of states – either KK states or wound string states – becomes light. These ideas are conceptually clear, and there has been considerable recent investigation of the SDC (see, for example, [@Klaewer:2016kiy; @Blumenhagen:2017cxt; @Palti:2017elp; @Hebecker:2017lxm; @Grimm:2018ohb; @Heidenreich:2018kpg; @Hebecker:2018vxz; @Scalisi:2018eaz; @Palti:2019pca; @Lust:2019zwm]). However, since a given EFT corresponds to fixed asymptotic values of the moduli, it is natural to pose a complementary question: is there a local version of the SDC? In other words, is there any limitation on localized excitations that sample distant regions of moduli space? Such obstructions might arise in a different way than the appearance of a tower of light states. In fact, a number of other rather disparate classical and semiclassical examples of this “transplanckian censorship" phenomenon are known [@tbanks; @ArkaniHamed:2007js; @nicolis; @draperetal; @Draper:2018lyw].[^1] For example, in a 4d massless scalar field theory minimally coupled to gravity, static, spherically symmetric excursions of the scalar in regions of subplanckian curvature are bounded by $\calO(1)$ in Planck units [@nicolis]. However, this theory can also be realized as the dimensional reduction of the 5d KK theory. In the KK theory there are solutions known as KK bubbles that sample all the way to $R=0$ in a local region of low 5d curvature. KK bubbles are thus a concrete example of a localized excitation sampling an infinite distance in moduli space, and it is of interest to examine their properties in more detail. Informally, KK bubbles describe spherical holes of size $\rho_0$ in asymptotically KK space. Expanding bubbles can nucleate nonperturbatively [@BON], and the description of this process as tunneling under an energy barrier was elucidated in [@brillhorowitz]. One might already take Witten’s bubble of nothing as an indication of the inconsistency of the theory. However, the lifetime of the ordinary KK vacuum can be exponentially long, and there are other static bubble excitations that exhibit more dramatic behaviors. Static “Schwarzschild" bubble solutions were first found in [@sorkin; @GP], along with a larger family of static “Kerr" solutions. Near the wall of a KK bubble, the circle radius $R$ goes to zero, smoothly truncating the spacetime the physical radius $\rho_0$. The geometries therefore have the interesting property that they sample points separated by an infinite proper distance in moduli space in well-localized, low-curvature regions of physical space. From the perspective of dimensional reduction, the KK scalar diverges on the surface of the bubbles. These solutions are thus a natural laboratory for the questions raised above.[^2] It turns out that all of these static bubbles are classically unstable. The instability of the static, asymptotically flat Schwarzschild bubble was demonstrated in [@grossperryyaffe] and given a mechanical interpretation in [@brillhorowitz]: this bubble sits at the top of the potential “hill" under which Witten’s bubble mediates tunneling. (The static bubble is therefore also responsible for topology change at high temperature [@brownthermal], analogous to a sphaleron in gauge theories.) Similarly, the asymptotically flat Kerr bubbles were shown to be unstable in [@Draper:2019zbb], with an equivalent relationship to a known tunneling process [@dowkeretal]. It was suggested in [@Draper:2019zbb] that the classically instabilities of the Schwarzschild and Kerr bubbles should be thought of as a pathology of the type described above: distant points in moduli space are “hidden" behind an instability. It is also known, however, that KK bubbles can be perturbatively stabilized by embedding them in spacetimes with different asymptotics, or, in asymptotically flat space, by wrapping them in flux. In the latter case, explicit examples of bubble geometries stabilized by 3-form flux were found in [@Gibbons:1994vm; @Horowitz:2005vp]. These spacetimes do not appear to be particularly theoretically exotic, and so it is curious that they do not seem to exhibit horizons or instabilities. In a different context, it has recently been shown in Refs. [@Crisford:2017zpi; @Crisford:2017gsb; @Horowitz:2019eum] that potential counterexamples to [*cosmic*]{} censorship can be avoided by imposing the weak gravity conjecture (WGC) [@wgc]. In short, the proposed counterexamples involve electromagnetic fields, and when charged scalar fields satisfying $q/m>1$ are added, the solutions are unstable against scalar perturbations. (Scalar fields are used to facilitate a classical analysis; fermions are expected to perform a similar function, but a more complicated treatment is required.) We will apply the idea of [@Crisford:2017zpi; @Crisford:2017gsb; @Horowitz:2019eum] to the perturbatively-stable charged KK bubble spacetimes of [@Horowitz:2005vp] and argue that a new instability arises in the presence of charged matter satisfying the WGC. Charged objects are wound strings, and one can screen some of the bubble’s charge by throwing oppositely charged strings into it. For sufficiently large $q/m$, we might expect that the vacuum will become unstable against rapid Schwinger production near the bubble wall. We study this question with a toy model in the dimensionally reduced theory, where the lowest wound string modes are represented by a massive charged scalar field coupled to ordinary electromagnetic flux. We show that in this model the negatively-charged ground state energy drops below $-m$ for $q/m\gtrsim 1$, signaling an instability against pair creation, and we argue that discharge rate is typically much faster than the tunneling rate to larger expanding bubbles. This suggests that the WGC can play a similar role in the censorship of infinite localized field excursions. Another interesting class of geometries is provided by charged black holes with large moduli variations outside the horizon. We estimate the discharge rate of 4d charged dilatonic black holes of [@Garfinkle:1990qj]. In these geometries, the size of the dilaton excursion from infinity to the horizon is controlled by the charge of the black hole, diverging in the extremal limit. Low curvature in the region of the large excursion requires $|\Delta\phi|\lesssim \log(M)$. For sub-extremal black holes the excursion is finite, and we find that the discharge rate is fast if the WGC is satisfied by a light particle of mass $m\ll1/M$. For sufficiently large black holes $M\gtrsim {\rm max}(e^{|\Delta\phi|},1/m)$ the rate is slow. We conclude with a loose conjecture: quantum gravity in asymptotically flat space requires a general bound on large localized moduli space excursions of the form $ |\Delta\phi|\lesssim |\log(R\Lambda)|$, where $R$ is the size of the minimal region enclosing the excitation and $\Lambda^{-1}$ is a short-distance cutoff. Both neutral and charged KK bubbles have finite $R$ and infinite excursions, but are strongly unstable. Dilatonic black holes in a controlled EFT also satisfy the bound. KK monopoles provide another example: they are stable and sample an infinite distance in moduli space, but only at a single point, so the visible excursion is limited by the short-distance cutoff. Charged KK Bubbles ================== We begin by discussing a representative class of Kaluza-Klein bubbles perturbatively stabilized by flux. We then add matter of mass $m$ and charge $q$ to the system and demonstrate the existence of a rapid pair-production instability for $q/m\gtrsim 1$. Classical Solutions ------------------- A number of static charged bubble solutions were obtained in [@Gibbons:1994vm; @Horowitz:2005vp]. In [@Horowitz:2005vp], a 6D bubble stabilized by electric and magnetic 3-form flux was constructed from a family of 5D zero-momentum initial data characterizing bubbles of different sizes. This method is particularly convenient for assessing bubble stability against radial perturbations. We review this construction here, simplifying to case of purely electric 3-form flux. A family of five-dimensional spatial metrics is given by $$\begin{aligned} ds^2_{spatial}=U(\rho)d\chi^2+\frac{d\rho^2}{U(\rho) h(\rho)}+\rho^2d\Omega_3 \label{eq:spatial}\end{aligned}$$ with $$\begin{aligned} U(\rho)\equiv 1-\frac{\rho_0^2}{\rho^2}. \label{eq:U}\end{aligned}$$ The function $h$ will be determined by the Hamiltonian constraint, and the periodicity of the KK circle at infinity $\chi\sim\chi+L$ will be determined by smoothness of the metric at $\rho=\rho_0$. We now add electric 3-form flux to the bubbles, $C=\frac{Q_0}{2\pi^2}(\star \epsilon_3)$, where $\epsilon_3$ is the volume element of the spatial $S^3$. Concretely, the field strength is $$\begin{aligned} C_{\rho t\chi}=\frac{NQ_0}{2\pi^2 \sqrt{h}\rho^3}\;, \label{eq:cN}\end{aligned}$$ where $N$ is the lapse function. The Hamiltonian constraint is then[^3] $$\begin{aligned} {}^5R=\frac{Q^2}{\rho^6}\;,\end{aligned}$$ where $Q\equiv Q_0/(2\pi^2M_6^2)$ and $M_6=(8\pi G)^{-1/4}$ is the 6D Planck scale. We find $$\begin{aligned} h(\rho)\equiv1+\frac{b}{3\rho^2-2\rho_0^2}-\frac{Q^2}{4\rho_0^2\rho^2} \label{eq:h}\end{aligned}$$ where $b$ is an arbitrary constant. To make the geometry smooth everywhere, we impose periodicity $\chi\sim\chi+L$ on the KK circle, where $$\begin{aligned} L=\frac{2\pi \rho_0}{\left(1+\frac{b}{\rho_0^2}-\frac{Q^2}{4\rho_0^4}\right)^{1/2}}\;. \label{eq:b}\end{aligned}$$ With this periodicity, space ends on a smooth cap at $\rho=\rho_0$. ![ The energy (Eq. (\[eq:Mbubble\])) of a one-parameter family of initial data labeled by bubble radius $\rho_0$. From bottom to top, curves correspond to $Q=0,\dots,Q=Q_{max}$. The most stable bubbles lie at small $Q$; for $Q>Q_{\max}$, no stable solution exists. []{data-label="fig:Mrho0"}](Mrho0.pdf){width="0.5\linewidth"} Thus far we have a family of charged bubble initial data. For a given charge $Q$ and asymptotic circle size $L$, it is a one-parameter family of bubbles labeled by the radius $\rho_0$. The energy of the family is $$\begin{aligned} M=\pi^2 LM_6^4\left(\frac{Q^2}{2\rho_0^2}+2\rho_0^2-\frac{4\pi^2\rho_0^4}{L^2}\right)\;. \label{eq:Mbubble}\end{aligned}$$ There is a stable local minimum of the energy $M(\rho_0)$ for all $Q$, $L$ such that $$\begin{aligned} Q<Q_{max}=\frac{L^2}{3\pi^2\sqrt{3}}\;. \label{eq:stab}\end{aligned}$$ There is also an unstable maximum at larger $\rho_0$. For $Q>Q_{max}$, there are no stationary points. In Fig. \[fig:Mrho0\] we illustrate the mass of the initial data as a function of bubble radius $\rho_0$ for $0\leq Q\leq Q_{max}$. Clearly even the perturbatively stable bubble at the local minimum is unstable against tunneling to larger radii. At $Q= Q_{max}$, the barrier disappears completely. On the other hand, for small $Q$, the barrier grows, and the rate for this transition may become extremely slow. In the next section we will show that a new, rapid instability arises in this limit. In general the local minimum is a root of a cubic, equivalent to $b=0$ in Eq. (\[eq:b\]). For small $Q$ it simplifies to $$\begin{aligned} \rho_0^2\approx Q/2.\end{aligned}$$ It is straightforward to verify that the local minimum is a static solution to the full Einstein equations with spacetime metric $$\begin{aligned} ds^2=-h(\rho)dt^2+U(\rho)d\chi^2+\frac{d\rho^2}{U(\rho) h(\rho)}+\rho^2d\Omega_3\; \label{eq:dsfull}\end{aligned}$$ with $U$ and $h$ given by Eqs. (\[eq:U\]), (\[eq:h\]) and $b=0$. The mass of this static bubble is $$\begin{aligned} M_{min}=\pi^2 LM_6^4\left(\frac{3Q^2}{4\rho_0^2}+\rho_0^2\right),\end{aligned}$$ where $L$ is given by Eq. (\[eq:b\]) evaluated at $b=0$. The field strength (\[eq:cN\]) surrounding the static bubble simplifies to $$\begin{aligned} C_{\rho t\chi}=\frac{Q_0}{2\pi^2 \rho^3}\;. \label{eq:cN}\end{aligned}$$ The point in moduli space where the size $R$ of the KK circle vanishes lies an infinite proper distance $\int dR/R$ away from any point of finite circle size. $R=0$ is sampled locally on the wall of KK bubbles, since $V\rightarrow 0$ as $\rho\rightarrow\rho_0$, while $R=L$ at spatial infinity. Typically, static neutral bubbles of nothing in asymptotically flat space have a single unstable mode, corresponding to perturbations of the bubble radius. The solution (\[eq:dsfull\]), lying at a local minimum of the energy, is perturbatively stabilized by the flux. It disappears for $Q=0$, leaving only the perturbatively unstable point corresponding to an ordinary neutral static bubble. We also see from the Hamiltonian constraint that for small $Q/L^2$, curvatures near the bubble are of order $1/Q$. Therefore there is a minimum $Q$, controlled by the cutoff, for which we can study this geometry classically. Adding Charged Matter --------------------- We would like to study the stability of the flux-stabilized bubble against the introduction of probe charges. In the 6D description, charged objects are wound strings. To simplify the analysis, we consider only the lowest states of a string with winding number one and zero KK excitations. Formally, we can dimensionally reduce over the circle to obtain a 5D geometry with ordinary electromagnetic flux, and we introduce a massive scalar particle with charge $q$ to represent the wound string. Near the bubble wall the scalar mass decreases with the radion. In this toy model we can study the single-particle ground states of positive and negative charge as a function of $q$. For $|q|/m\gtrsim 1$, the electrostatic potential energy of a negatively charged particle near the bubble wall is sufficient to compensate for its rest mass energy at infinity. The naïve vacuum in the zero-charge sector is then unstable against spontaneous pair creation, and in the subsequent section we argue that the bubble discharge rate is unsuppressed. We begin by parametrizing the 6D spacetime (\[eq:dsfull\]) as $$\begin{aligned} ds^2=G_{\mu\nu}dx^\mu dx^\nu+Vd\chi^2\; \label{eq:fivemetric}\end{aligned}$$ where $V=U(\rho)$ by comparison with (\[eq:spatial\]). The dimensional reduction of the three-form flux gives rise to an ordinary 5D electric field, and we choose a gauge where the field arises from a scalar potential vanishing at infinity, $$\begin{aligned} A_t=\frac{\sqrt{L}Q_0}{4\pi^2\rho^2}\;. \label{eq:At}\end{aligned}$$ Dimensionally reducing the worldsheet action for a wound string of tension $T$ with no $\chi$ excitations, the corresponding worldline action for the free particle is $$\begin{aligned} m \int d\tau \sqrt{V}\sqrt{-G_{\mu\nu}\partial_\tau x^\mu\partial_\tau x^\nu}\end{aligned}$$ where $m=TL$ for $L\gg1/\sqrt T$. Therefore, in our toy model we introduce a charged scalar with action $$\begin{aligned} S[\Phi]=\int d^5x\sqrt{-G}\left(-G^{\mu\nu}(D_\mu\Phi)^*(D_\nu\Phi)-m^2 V |\Phi|^2\right),\end{aligned}$$ where $D_\mu=\partial_\mu+i q A_\mu$ for a particle of charge $q$. Here $q$ has mass dimension $-1/2$. We treat $\Phi$ as a probe, neglecting backreaction on the metric and gauge field. The Klein-Gordon equation for $\Phi$ is $$\begin{aligned} ({\cal D}^2-m^2 V)\Phi =(\Box+2iqA^t\partial_t-q^2 A^t A_t -m^2 V )\Phi=0\;, \label{eq:KGfull}\end{aligned}$$ and the energy is $$\begin{aligned} E&=\int d^4 x \sqrt{-G}\left[-G^{tt} |\partial_t\Phi|^2+G^{ii} |\partial_i\Phi|^2+\Phi^*\left(q^2G^{tt}A_t^2 +m^2 V\right)\Phi\right]\;. \label{eq:energy}\end{aligned}$$ The kinetic, mass, and gradient terms in the energy are positive, although the mass term is suppressed by a factor of $V$ near the bubble wall. The $A_t^2$ term, which arises from the last term in the gauge-invariant charge density $J_t=i \Phi^* \partial_t\Phi-i \partial_t\Phi^* \Phi-2 qA_t|\Phi|^2$, is negative, indicating that small perturbations can lower the energy if this term dominates. Note, however, that the “potential energy operator" $-\nabla^2+q^2G^{tt}A_t^2 +m^2 V$ differs from the fluctuation operator appearing in the equation of motion by a term $ -2iqA^t\partial_t$. Thus negative energy perturbations do not immediately imply complex frequencies or the exponentially growing modes characteristic of classical instabilities. We will look for $s$-wave solutions to Eq. (\[eq:KGfull\]). Setting $\Phi=\phi(\rho) e^{i\omega t}$ we obtain $$\begin{aligned} h U \phi '' + \left(U h'+\frac{h U'}{2}+\frac{3 h U}{\rho }\right)\phi '+ \left(\frac{\left(q A_t+\omega \right){}^2}{h}-m^2 U\right)\phi=0\;. \label{KGeq}\end{aligned}$$ This equation admits bound states of finite Klein-Gordon norm (charge). The energy and charge of a bound mode is $$\begin{aligned} E_\phi&=\int d^4 x \sqrt{-G}G^{tt}\phi^*\left(-2\omega^2-2qA_t\omega\right)\phi\nonumber\\ \calQ_\phi&=-q\int d^4 x \sqrt{-G}G^{tt}J_t=-\frac{q}{\omega} E_\phi\;. \label{eq:charge}\end{aligned}$$ Here we have used the equation of motion and assumed Dirichlet conditions at $\rho=\rho_0$, which is sufficient to prevent energy or charge flux through the bubble wall and simplifies the analysis of modes. Below we will relax this condition and allow charged matter to fall into the bubble, annihilating some of the bubble charge. Now we can study the single-particle ground states in the charge $\pm$ sectors as a function of the charge to mass ratio, $$\begin{aligned} w\equiv q M_5^{3/2}/m.\end{aligned}$$ Here $M_5$ is the 5d Planck scale, $M_5^3=LM_6^4$. Before proceeding, we note that the analysis of Eq. (\[KGeq\]) in the static background (\[eq:spatial\]),(\[eq:dsfull\]) is only meaningful for the following hierarchies of mass scales: $$\begin{aligned} M\gg M_5\gg m\gg\sqrt{m/L} \gg Q^{-1/2}\gg L^{-1}\;. \label{eq:hierarchy}\end{aligned}$$ The first and second inequalities allow us to treat the bubble as a fixed classical background on which the single particle states are a perturbation. The third inequality allows us to set the wound string mass to $m=TL\gg\sqrt{T}$, and the fourth imposes the requirement that the spacetime curvature near the bubble wall is below the string scale $\sqrt{T}$. The final inequality arises from Eq. (\[eq:stab\]) and the requirement that tunneling transitions to larger bubbles are suppressed (cf. Fig. \[fig:Mrho0\]). Subsequently we will mostly work in 5d Planck units, $M_5=1$. ![Ground state frequencies obtained by numerical solution of Eq. (\[KGeq\]). Normalized to $m$, the frequencies depend only on the combinations $q/m$, $Q/Q_{max}$, and $mL$. We take a point where $mL=100$ and $Q/Q_{max}\approx 1/4$, scanning over $q/m$ in $5d$ Planck units. The upper solid line (blue) shows $\omega_0$, the ground state frequency (energy) in the charge $-q$ sector. The lower solid line (red) shows the ground state frequency (-energy) in the charge $+q$ sector. The dashed line denotes the charge $+q$ continuum. For $q=0$, $\omega_0\sim \sqrt{T}\ll m$. For small positive $q$, $\omega_0$ crosses zero. At some $q/m\sim\calO(1)$, $\omega_0$ crosses into the positive-charge continuum. At this point the energy of an oppositely-charged pair vanishes, and the system becomes unstable against spontaneous pair creation. []{data-label="fig:numerics"}](omega_m_1000_L_1000_Q_25.pdf){width="0.6\linewidth"} Since the metric functions (\[eq:U\]), (\[eq:h\]) are somewhat involved and the exact solution for $\rho_0$ is the root of a cubic, it is simplest to perform detailed analysis numerically. Normalized to $m$, the eigenfrequencies depend only on the dimensionless ratios $w$, $Q/Q_{max}\ll1$, and $mL\gg 1$. For typical sets of parameters we can solve (\[KGeq\]) numerically for the bound states. In Fig. \[fig:numerics\] we show the frequencies of the charge $\pm q$ ground states as a function of $q$. We see that in this example, $\omega\ll m$ for $q=0$ and decreases approximately linearly as $q$ is increased. These properties are straightforward to understand physically. For $q=0$ and vanishing energy flux through the bubble wall, the $s-$wave spectrum includes a discrete set of normalizable bound modes of $|\pm\omega|<m$. The lowest modes have $|\omega|\sim \sqrt{T}$, reflecting the fact that wound strings near the cigar tip are close to becoming unwound (and indeed would become unwound with any asymmetric perturbation). In the hierarchy (\[eq:hierarchy\]), these modes are deeply bound, $|\omega|\ll m$. For small positive $q$, the spectrum shifts downward. Positive frequencies, corresponding to negative charges, become more tightly bound, while their negative frequency counterparts shift toward $\omega=-m$. Since the modes are deeply bound, we can approximate the electrostatic potential energy term in the Klein-Gordon equation by its value at the bubble wall, $qA_t(\rho_0)$. Then the bound mode frequencies decrease as $$\begin{aligned} \frac{d\omega}{dq}\approx -\frac{Q}{2\rho_0^2}\approx -1.\end{aligned}$$ Denoting the lowest mode in the negative charge sector by $\omega_0$, for some small value of $q$, the ground state energy $\omega_0\rightarrow 0$. At this point the negative charge state has binding energy that completely cancels its asymptotic rest mass. However, because of charge conservation, this is not yet enough to indicate an instability in the charge-zero sector. As $q$ is increased further the ground state energy drops below zero. Once $q\sim m$, we find $\omega_0\rightarrow -m$. At this point the bound state now has sufficiently negative energy to compensate for the rest mass of an additional positive charge at infinity. For larger $q$, $\omega_0$ cannot simply fall below $-m$. These solutions are unbound and correspond to positive-charge scattering states. This situation is identical to the physics of high-$Z$ “over-critical" nuclei [@Rafelski:1976ts]. It is energetically favorable to spontaneously produce opposite-charge pairs, and the naïve zero-particle ground state is unstable. Discharge Rate -------------- Heuristically, the discharge process can be thought of as pair creation near the bubble wall. The negative charge annihilates some of the bubble charge, while the positive charge tunnels out and escapes to infinity. The whole process must conserve energy, so the energy $\epsilon_+$ of the escaping positive charge must be compensated by the reduction in mass of the bubble, $$\begin{aligned} \epsilon_+=-\Delta M = \frac{qQ}{2\rho_0^2}\end{aligned}$$ assuming $q\ll Q$ in 5D Planck units. ($\Delta M$ is also equal to the classical energy of a negative charge at rest at the bubble wall, but we do not need this interpretation.) If the escaping “positron" encounters a potential barrier, then the discharge rate is proportional to a tunneling exponent which can be straightforwardly estimated in the WKB approximation. The worldline action for the positively-charged state moving in the radial direction $\rho(t)$ is $$\begin{aligned} S=\int dt\left(-m\sqrt{V}\sqrt{-G_{tt}-G_{\rho\rho}\dot\rho^2}-\frac{qQ}{2\rho^2}\right)\;.\end{aligned}$$ The Hamiltonian is $$\begin{aligned} H&=\frac{-m\sqrt{V}G_{tt}}{\sqrt{-G_{tt}-G_{\rho\rho}\dot\rho^2}}+\frac{qQ}{2\rho^2}\nonumber\\ &=\sqrt{-G_{tt}(G^{\rho\rho}\pi^2+m^2V)}+\frac{qQ}{2\rho^2}\end{aligned}$$ where in the second line we have written the Hamiltonian in terms of the canonical momentum $\pi(t)$. In the WKB approximation, the tunneling amplitude through classically forbidden regions is proportional to $$\begin{aligned} \Gamma\sim\exp{(i\int_{\rho_-}^{\rho_+} \pi d\rho )} \label{eq:wkbexp}\end{aligned}$$ where $$\begin{aligned} \pi=\sqrt{-G_{\rho\rho}\left(G^{tt}\left(\epsilon_+-\frac{qQ}{2\rho^2}\right)^2-m^2V\right)}, \label{eq:wkbpi}\end{aligned}$$ and $\rho_\pm$ are the classical turning points. In classically forbidden regions the integral is complex, and the factor (\[eq:wkbexp\]) suppresses the tunneling rate. The turning points are located at $\rho_-=0$ and $$\begin{aligned} \rho_+=\frac{Q\rho_0\sqrt{q^2-m^2}}{\sqrt{q^2Q^2-4m^2\rho_0^4}}\end{aligned}$$ Since $Q< 2\rho_0^2$, the outer turning point is only finite (and therefore the pair production rate is only nonzero) if $$\begin{aligned} q\gtrsim m \frac{2\rho_0^2}{Q}\approx m\;.\end{aligned}$$ If this inequality is satisfied, the WKB integral gives $$\begin{aligned} \int_0^{\rho_+} \pi d\rho \approx i\sqrt{\rho_0^2-\frac{Q^2}{4\rho_0^2}}\left(-m+\frac{qQ}{2\rho_0^2} {\rm tanh}^{-1} \left(\frac{2\rho_0^2}{q Q}m\right)\right)\;. \label{eq:intwkbexp}\end{aligned}$$ Let us first examine this exponent in the limit $Q/L^2\ll 1$. This is the limit in which the bubble is most stable against tunneling to a larger, expanding bubble. Expanding (\[eq:intwkbexp\]) in $Q/L^2$, we find $$\begin{aligned} \Gamma\sim \exp\left({-\frac{\pi m Q (-m+q~{\rm tanh}^{-1}(m/q)}{m L}}\right)\sim \exp\left[{-\pi \left(\frac{Q}{L^2}\right) \left(\frac{m^2}{q^2}\right)(mL)}\right] \label{eq:wkbrate}\end{aligned}$$ where in the last step we have approximated $q/m\gtrsim{\rm~few}$. For comparison, in the small $Q$ limit the decay rate into a larger, expanding bubble should be well-approximated by the decay of the ordinary KK vacuum into neutral bubbles. (For larger $Q$, the rate will be faster.) The rate for this process is of order $$\begin{aligned} \Gamma_{\rm Witten}\sim e^{-M_6^4 L^4}\sim e^{-M_5^3 L^3}. \end{aligned}$$ For $q/m\gtrsim 1$ and recalling the hierarchies in Eq. (\[eq:hierarchy\]), the discharge rate (\[eq:wkbrate\]) satisfies $$\begin{aligned} \Gamma> e^{-mL}\gg \Gamma_{\rm Witten}\;.\end{aligned}$$ We cannot take $Q$ smaller than $L/m$ if we want to keep curvatures everywhere below the string scale. In this limit, the discharge rate becomes $$\begin{aligned} \Gamma\sim \exp{\left(-\frac{\pi m^2}{q^2}\right)}\;.\end{aligned}$$ In other words, when the decay rate into expanding bubbles is minimized, the discharge process is unsuppressed, if the WGC is satisfied. We can also assess the WKB exponent numerically for other values of the parameters. In Fig. \[fig:WKBexp\] we show the exponent for $L/m<Q<Q_{max}$ and $mL=10^3$ as a function of $q/m$. The exponent is typically not large if $q/m\gtrsim{\rm few}$. Determining the endpoint of the discharge process requires consideration of backreaction effects. Qualitatively, we expect the bubble to collapse into a black string. It is remarkable that in the limit where one instability is made small, a new one appears with large rate. ![ The WKB exponent in Eq. (\[eq:intwkbexp\]) for $mL=10^3$ as a function of $q/m$. From bottom to top the curves correspond to $Q=(L/m,\dots,Q_{max})$. Higher curves (larger $Q$) have increasingly suppressed discharge rates, but are increasingly unstable against tunneling to expanding bubbles. (The top curve is completely unstable.) We see that when the WGC is satisfied by a modest amount the exponent is generally small and the discharge rate is fast. []{data-label="fig:WKBexp"}](WKBexp.pdf){width="0.5\linewidth"} Charged Dilatonic Black Holes ============================= The KK bubbles considered above sample an infinite distance in moduli space. We can also consider charged dilatonic black hole spacetimes in which the dilaton excursion is finite but can be made arbitrarily large [@Garfinkle:1990qj]. The electrically charged solutions of [@Garfinkle:1990qj] are given by the dilaton, electric field, and Einstein frame metric: $$\begin{aligned} \phi=-\phi_0+\frac{1}{2}\log\left(1-\frac{Q^2 e^{-2\phi_0}}{Mr}\right)\;,\end{aligned}$$ $$\begin{aligned} F_{tr}=E=\frac{e^{-2\phi_0}Q}{r^2}\;,\end{aligned}$$ $$\begin{aligned} g_{\mu\nu}dx^\mu dx^\nu=-f\, dt^2 + f^{-1}\, dr^2+r\left(r-\frac{Q^2 e^{-2\phi_0}}{M}\right)(d\theta^2+\sin^2\theta d\phi^2)\;,\end{aligned}$$ where $f=1-2M/r$ and the dilaton Lagrangian is $2(\nabla \phi)^2+e^{-2\phi}F^2$. The extremal limit is $Q_{ext}=\sqrt{2}e^{\phi_0}M$ and there is a horizon at $r_+=2M$. Let us define an extremality parameter $y=Q/Q_{ext}$. In terms of $y$ the dilaton excursion between infinity and $r_+$ is $$\begin{aligned} \Delta\phi=\frac{1}{2}\log(1-y^2)\;.\end{aligned}$$ We see that $|\Delta\phi|$ is finite and large for near-extremal black holes, and $\Delta\phi\rightarrow-\infty$ in the extremal limit. The Maxwell term in this theory is $e^{-2\phi}F^2$, and it is convenient to canonically normalize it at infinity. We define $\hat F=e^{\phi_0}F$ and $\hat Q=e^{-\phi_0}Q$ such that $\hat E=\hat Q/r^2$ and the extremal limit is $\hat Q_{ext}=\sqrt{2}M$. The Hawking temperature is $$\begin{aligned} T=\frac{1}{8\pi M}\;.\end{aligned}$$ Despite the finite temperature, the Kretschmann scalar behaves as $M^{-4} e^{-4\Delta\phi}$ near the horizon for $|\Delta\phi|\gg 1$. Subplanckian curvatures thus imply $|\Delta\phi|\lesssim\log(M)$. Now we introduce a charged particle of mass $m$ and charge $q$ coupled to the gauge field $\hat A_t=\hat Q/r$. The black hole may be hot ($m/T\lesssim 1$) or cold ($m/T\gtrsim 1$). For simplicity we assume a minimally coupled charged scalar field, which is meaningful if $m\ll 1$. In the hot case, the emission probability for charges $\pm q$ is proportional to a Boltzmann factor of the form $$\begin{aligned} e^{-\frac{1}{T}\left(m\pm\frac{qQ}{r_+}\right)}= e^{-\frac{m}{T}\left(1\pm\frac{q}{\sqrt{2}m}(1-e^{2\Delta\phi})\right)}.\end{aligned}$$ If $|\Delta\phi|\gg 1$, the discharge rate is fast if the WGC is satisfied and the black hole is hot. Now we consider the case $m/T\gg 1$. Here the discharge rate is governed by the Schwinger process, and the rate exponent can be determined by barrier penetration arguments for a mode of frequency equal to the electrostatic potential energy at the horizon, $\omega_+=-qQ/r_+$ [@Gibbons:1975kk]. For $\omega_+$ to be a scattering state, we must have $q>\sqrt{2}m$ near extremality. The Klein-Gordon equation for the $s$-wave mode of frequency $\omega_+$ is $$\begin{aligned} \Phi''(r)+W\Phi(r)=0\;,\;\;\;\;\;W=\frac{q^2}{2}-\frac{m^2r}{r-2M}+\frac{(Me^{2\Delta\phi})^2}{(r-2M)^2(r-2M+2Me^{2\Delta\phi})^2}\;.\end{aligned}$$ Here we have put the equation in normal form and taken $e^{2\Delta\phi}\ll 1$. The barrier $W<0$ extends approximately from $r\sim 2M+e^{2\Delta\phi}M \equiv \alpha$ to $r\sim 2M\left(1+ \frac{2m^2}{q^2-2m^2}\right)\equiv \beta$. In the WKB approximation, the barrier penetration factor is $$\begin{aligned} e^{-2\int_\alpha^\beta \sqrt{W}dr}.\end{aligned}$$ We can approximately evaluate the WKB integral by splitting it into regions where the last two terms in $W$ dominate ($\equiv W_{23}$, valid near $\alpha$) and where the first two terms in $W$ dominate ($\equiv W_{12}$, valid near $\beta$). The two regions overlap where the first and third terms are of similar order, near $r\sim 2M+1/\sqrt{2}q$. Putting the pieces together and keeping only the dominant terms, we find the production rate is of order $$\begin{aligned} \Gamma_{Einstein} \sim e^{-2\int_\alpha^\gamma \sqrt{W_{23}}dr-2\int_\gamma^\beta \sqrt{W_{12}}dr}\approx e^{-\frac{2\sqrt{2}\pi M m^2}{\sqrt{q^2-2m^2}}}\;. \label{eq:einsteinrate}\end{aligned}$$ This is similar to the Schwinger exponent $\pi m^2/q\hat E$ arising from a constant field of magnitude $\hat E=\hat Q/r_+^2$. For a large, cold black hole, the rate is exponentially small for $q\sim m$. We conclude that a large field excursion $|\Delta\phi|\gg1$ requires a large source. Exponentially large sources are required to control curvature invariants at the horizon. In the presence of a light particle of mass $m$ satisfying the weak gravity conjecture, it is also possible for the black hole to rapidly discharge. Combining the requirements of slow discharge and low curvature, we obtain the bound $$\begin{aligned} M\gg {\rm max}(e^{|\Delta\phi|},1/m).\end{aligned}$$ This is reminiscent of other indications that large localized field excursions can be sustained around exponentially large sources [@nicolis]. Other similar dilatonic black hole solutions can be obtained, including different dilaton couplings $e^{-2a\phi}F^2$ and general dyonic charges [@Ivashchuk:1999jd; @Abishev:2015pqa] (see also [@Loges:2019jzs] for a recent analysis in the context of the WGC).[^4] In some cases simple analytic solutions are known. In the magnetic case, similar results are obtained. In the dyonic case, it is possible to have a finite dilaton excursion in the extremal limit. However, the curvature at the horizon is still controlled by the mass of the black hole and the amplitude of the dilaton excursion, in such a way that analogous bounds of the form $|\Delta\phi|<\log(M)$ still hold. Discussion ========== Both neutral and charged KK bubbles sample infinite distances in moduli space in finite spatial regions with size $R$ of order the bubble radius. Neutral bubbles are classically unstable, and we have argued that charged bubbles are destabilized in the presence of charged matter with $q/m\gtrsim 1$. Dilatonic black holes in a controlled EFT have a finite excursion $ |\Delta\phi|\lesssim |\log(M_{BH})|$. We have not discussed KK monopoles [@sorkin; @GP], but they provide another interesting example. They are stable and sample an infinite distance in moduli space, but only at a single point. The KK scalar modulus diverges as $$\begin{aligned} |\Delta \phi| \sim |\log(r/L)|\end{aligned}$$ near the center of the monopole. Access to distance scales shorter than $L$ is required to see the excursion, but in principle this is permissable since we do not insist on dimensional reduction. The accessible excursion is ultimately limited by the short-distance cutoff on the semiclassical KK theory, $|\Delta \phi| \lesssim |\log(\Lambda L)|$. Motivated by these examples, we conclude with a loose conjecture: quantum gravity in asymptotically flat space requires a general bound on localized, (meta)stable moduli space excursions of the form $$\begin{aligned} |\Delta\phi|\lesssim |\log(R\Lambda)|\end{aligned}$$ where $\phi$ parametrizes the modulus, $R$ is a scale characterizing the minimal region enclosing the excitation, and $\Lambda^{-1}$ is a short-distance cutoff on local quantum field theory. In a sense, the swampland distance conjecture applies to the limit $R\rightarrow\infty$, where the moduli are moved everywhere in space, and the bound is trivially satisfied. This bound is consistent with the Newtonian analysis of [@nicolis], but we have seen that it is less trivial in general relativity, and requires the WGC in some cases. There also appears to be another connection with the SDC: large excursions are typically confined near surfaces or points, rather than being spread over finite bubble volumes. Consequently, access to the excursion requires access to short distance scales. In the KK examples, this also implies access to a tower of KK states scaling exponentially with the observable excursion. [**Acknowledgements:**]{} PD thanks B. Heidenreich, G. Horowitz, M. Montero, M. Reece, G. Shiu, and I. Valenzuela for useful discussions. This work was supported by NSF grant PHY-1719642. [^1]: This notion of transplanckian censorship is distinct from the recent conjectures in [@Bedroya:2019snp; @Bedroya:2019tba]. [^2]: Casimir energies lift the moduli space in nonsupersymmetric KK theories. As usual we assume that the classical bubble solutions provide useful approximations to solutions in theories with moduli stabilized by additional fluxes or other objects. Ref. [@Dine:2004uw], for example, found that neutral bubble solutions persist after adding simple stabilizing potentials. [^3]: The matter Lagrangian is normalized as $-\frac{1}{12}\int d^6x\, \sqrt{-g_6}C^2$. [^4]: We thank Gary Shiu for bringing these solutions to our attention.

--- author: - Roman Klokov - Edmond Boyer - Jakob Verbeek title: | Discrete Point Flow Networks\ for Efficient Point Cloud Generation ---

--- author: - 'A.E.Shalyt-Margolin[^1] and A.Ya.Tregubovich [^2]' title: Deformed Density Matrix and Generalized Uncertainty Relation in Thermodynamics --- \ [**Abstract**]{}\ [A generalization of the thermodynamic uncertainty relations is proposed. It is done by introducing of an additional term proportional to the interior energy into the standard thermodynamic uncertainty relation that leads to existence of the lower limit of inverse temperature. The authors are of the opinion that the approach proposed may lead to proof of these relations. To this end, the statistical mechanics deformation at Planck scale. The statistical mechanics deformation is constructed by analogy to the earlier quantum mechanical results. As previously, the primary object is a density matrix, but now the statistical one. The obtained deformed object is referred to as a statistical density pro-matrix. This object is explicitly described, and it is demonstrated that there is a complete analogy in the construction and properties of quantum mechanics and statistical density matrices at Plank scale (i.e. density pro-matrices). It is shown that an ordinary statistical density matrix occurs in the low-temperature limit at temperatures much lower than the Plank’s. The associated deformation of a canonical Gibbs distribution is given explicitly.]{} Introduction ============ In this paper generalization of the thermodynamic uncertainty relations is proposed. It is done by introducing of an additional term proportional to the interior energy into the standard thermodynamic uncertainty relation that leads to existence of the lower limit of inverse temperature. Consequently, statistical mechanics at Planck scale should be deformed. As is known, at Planck scale Quantum Mechanics (QM) undergoes variation: it should be subjected to deformation also. This is realized due to the presence of the Generalized Uncertainty Relations (GUR) and hence the fundamental length [@r1],[@r2]. The deformation in Quantum Mechanics at Planck scale takes different paths: commutator deformation (Heisenberg’s algebra deformation) [@r4],[@r5] or density matrix deformation [@r7], [@r8]. In the present work the second approach is extended by the authors to the Statistical Mechanics at Plank scale. To this end, a deformed statistical density matrix, also called a statistical density pro-matrix, is constructed as a complete analog to the deformed quantum mechanics matrix. In Quantum Mechanics with fundamental length (QMFL) the deformation parameter was represented by the value $\alpha=l_{min}^{2}/x^{2}$ where $x$ is the scale, whereas in case of the Statistical Mechanics this value will be $\tau = T^{2}/T^{2}_{max}$ where $T_{max}$ is a maximum temperature of the order of the Planck’s. Existence of $T_{max}$ follows from (GUR) for the “energy - time” pair. The limitations on the parameter variation interval are the same. In this way it is demonstrated that there exists a complete analogy in the construction and properties of quantum mechanics and statistical density matrices at Planck scale (density pro-matrices). It should be noted that an ordinary statistical density matrix appears in the low-temperature limit (at temperatures much lower than the Planck’s). The associated deformation of a canonical Gibbs distribution is described explicitly. Generalized Uncertainty Relation in\ Thermodynamics ==================================== It is well known that in thermodynamics an inequality for the pair interior energy - inverse temperature, which is completely analogous to the standard uncertainty relation in quantum mechanics [@r9] can be written down [@r10] – [@r12]. The only (but essential) difference of this inequality from the quantum mechanical one is that the main quadratic fluctuation is defined by means of classical partition function rather than by quantum mechanical expectation values. In the last 14 - 15 years a lot of papers appeared in which the usual momentum-coordinate uncertainty relation has been modified at very high energies of order Planck energy $E_p$ [@r1]–[@castro1]. In this note we propose simple reasons for modifying the thermodynamic uncertainty relation at Planck energies. This modification results in existence of the minimal possible main quadratic fluctuation of the inverse temperature. Of course we assume that all the thermodynamic quantities used are properly defined so that they have physical sense at such high energies. We start with usual Heisenberg uncertainty relations [@r9] for momentum - coordinate: $$\label{U1} \Delta x\geq\frac{\hbar}{\Delta p}.$$ It was shown that at the Planck scale a high-energy term must appear: $$\label{U2} \Delta x\geq\frac{\hbar}{\Delta p}+ \alpha^{\prime} L_{p}^2\frac{\triangle p}{\hbar}$$ where $L_{p}$ is the Planck length $L_{p}^2 = G\hbar /c^3 \simeq 1,6\;10^{-35}m$ and $\alpha^{\prime}$ is a constant. In [@r3] this term is derived from the string theory, in [@r1] it follows from the simple estimates of Newtonian gravity and quantum mechanics, in [@r4] it comes from the black hole physics, other methods can also be used [@r5],[@r6]. Relation (\[U2\]) is quadratic in $\Delta p$ $$\label{U4} \alpha^{\prime} L_{p}^2\, ({\Delta p})^2 - \hbar\,\Delta x\Delta p + \hbar^2 \leq0$$ and therefore leads to the fundamental length $$\label{U5} \Delta x_{min}=2 \surd \alpha^{\prime} L_{p}$$ Using relations (\[U2\]) it is easy to obtain a similar relation for the energy - time pair. Indeed (\[U2\]) gives $$\label{U6} \frac{\Delta x}{c}\geq\frac{\hbar}{\Delta p c }+\alpha^{\prime} L_{p}^2\,\frac{\Delta p}{c \hbar},$$ then $$\label{U7} \Delta t\geq\frac{\hbar}{\Delta E}+\alpha^{\prime}\frac{L_{p}^2}{c^2}\,\frac{\Delta p c}{\hbar}=\frac{\hbar}{\Delta E}+\alpha^{\prime}t_{p}^2\,\frac{\Delta E}{ \hbar}.$$ where the smallness of $L_p$ is taken into account so that the difference between $\Delta E$ and $\Delta (pc)$ can be neglected and $t_{p}$ is the Planck time $t_{p}=L_p/c=\sqrt{G\hbar/c^5}\simeq 0,54\;10^{-43}sec$. Inequality (\[U7\]) gives analogously to (\[U2\]) the lower boundary for time $\Delta t\geq2t_{p}$ determining the fundamental time $$\label{U10b} \Delta t_{min}=2\sqrt{\alpha^{\prime}}t_{p}$$ Thus, the inequalities discussed can be rewritten in a standard form $$\label{U11b} \left\{ \begin{array}{ll} \Delta x & \geq\frac{\displaystyle\hbar}{\displaystyle\Delta p}+\alpha^{\prime} \left(\frac{\displaystyle\Delta p}{\displaystyle P_{pl}}\right)\, \frac{\displaystyle\hbar}{\displaystyle P_{pl}} \\ & \\ \Delta t & \geq\frac{\displaystyle\hbar}{\displaystyle\Delta E}+\alpha^{\prime} \left(\frac{\displaystyle\Delta E}{\displaystyle E_{p}}\right)\, \frac{\displaystyle\hbar}{\displaystyle E_{p}} \end{array} \right.$$ where $P_{pl}=E_p/c=\sqrt{\hbar c^3/G}$. Now we consider the thermodynamics uncertainty relations between the inverse temperature and interior energy of a macroscopic ensemble $$\label{U12} \Delta \frac{1}{T}\geq\frac{k}{\Delta U}.$$ where $k$ is the Boltzmann constant.\ N.Bohr [@r10] and W.Heisenberg [@r11] first pointed out that such kind of uncertainty principle should take place in thermodynamics. The thermodynamic uncertainty relations (\[U12\]) were proved by many authors and in various ways [@r12]. Therefore their validity does not raise any doubts. Nevertheless, relation (\[U12\]) was proved in view of the standard model of the infinite-capacity heat bath encompassing the ensemble. But it is obvious from the above inequalities that at very high energies the capacity of the heat bath can no longer to be assumed infinite at the Planck scale. Indeed, the total energy of the pair heat bath - ensemble may be arbitrary large but finite merely as the universe is born at a finite energy. Hence the quantity that can be interpreted as the temperature of the ensemble must have the upper limit and so does its main quadratic deviation. In other words the quantity $\Delta (1/T)$ must be bounded from below. But in this case an additional term should be introduced into (\[U12\]) $$\label{U12a} \Delta \frac{1}{T}\geq\frac{k}{\Delta U} + \eta\,\Delta U$$ where $\eta$ is a coefficient. Dimension and symmetry reasons give $$\eta \sim \frac{k}{E_p^2}\enskip or\enskip \eta = \alpha^{\prime} \frac{k}{E_p^2}$$ As in the previous cases inequality (\[U12a\]) leads to the fundamental (inverse) temperature. $$\label{U15} T_{max}=\frac{\hbar}{2\surd \alpha^{\prime}t_{p} k}=\frac{\hbar}{\Delta t_{min} k}, \quad \beta_{min} = {1\over kT_{max}} = \frac{\Delta t_{min}}{\hbar}$$ It should be noted that the same conclusion about the existence of the maximal temperature in Nature can be made also considering black hole evaporation [@castro2].\ Thus, we obtain the system of generalized uncertainty relations in a symmetric form $$\label{U17} \left\{ \begin{array}{lll} \Delta x & \geq & \frac{\displaystyle\hbar}{\displaystyle\Delta p}+ \alpha^{\prime} \left(\frac{\displaystyle\Delta p}{\displaystyle P_{pl}}\right)\, \frac{\displaystyle\hbar}{\displaystyle P_{pl}}+... \\ & & \\ \Delta t & \geq & \frac{\displaystyle\hbar}{\displaystyle\Delta E}+\alpha^{\prime} \left(\frac{\displaystyle\Delta E}{\displaystyle E_{p}}\right)\, \frac{\displaystyle\hbar}{\displaystyle E_{p}}+...\\ & & \\ \Delta \frac{\displaystyle 1}{\displaystyle T}& \geq & \frac{\displaystyle k}{\displaystyle\Delta U}+\alpha^{\prime} \left(\frac{\displaystyle\Delta U}{\displaystyle E_{p}}\right)\, \frac{\displaystyle k}{\displaystyle E_{p}}+... \end{array} \right.$$ or in the equivalent form $$\label{U18} \left\{ \begin{array}{lll} \Delta x & \geq & \frac{\displaystyle\hbar}{\displaystyle\Delta p}+\alpha^{\prime} L_{p}^2\,\frac{\displaystyle\Delta p}{\displaystyle\hbar}+... \\ & & \\ \Delta t & \geq & \frac{\displaystyle\hbar}{\displaystyle\Delta E}+\alpha^{\prime} t_{p}^2\,\frac{\displaystyle\Delta E}{\displaystyle\hbar}+... \\ & & \\ \Delta \frac{\displaystyle 1}{\displaystyle T} & \geq & \frac{\displaystyle k}{\displaystyle\Delta U}+\alpha^{\prime} \frac{\displaystyle 1}{\displaystyle T_{p}^2}\, \frac{\displaystyle\Delta U}{\displaystyle k}+... \end{array} \right.$$ where the dots mean the existence of higher order corrections as in [@r21]. Here $T_{p}$ is the Planck temperature: $T_{p}=\frac{E_{p}}{k}$.\ In the conclusion of this section we would like to note that the restriction on the heat bath made above turns the equilibrium partition function to be non-Gibbsian [@r13].\ Note that the last inequality is symmetrical to the second one with respect to the substitution [@r15]\ $$t\mapsto\frac{1}{T}, \hbar\mapsto k,\triangle E\mapsto \triangle U .$$ However this observation can by no means be regarded as a rigorous proof of the generalized uncertainty relation in thermodynamics.\ There is a reason to believe that a rigorous justification for the last (thermodynamic) inequalities in systems (\[U17\]) and (\[U18\]) may be made by means of a certain deformation of Gibbs distribution.\ Let us outline the main aspects of above-considered deformation. In our opinion it could be obtained as the result of density-matrix deformation in Statistical Mechanics (see [@r16], Section 2, Paragraph 3): $$\label{U19} \rho=\sum_{n}\omega_{n}|\varphi_{n}><\varphi_{n}|,$$ where probability is given by\ $$\omega_{n}=\frac{1}{Q}\exp(-\beta E_{n}).$$\ Deformation of density matrix $\rho$ (\[U19\]) can be carried out similarly to deformation of density matrix (density pro-matrix) in Quantum Mechanics at Planck’s scale (see [@r7],[@r8]). Proceeding with this analogy density matrix $\rho$ in (\[U19\]) should be changed by $\rho(\tau)$, where $\tau$ is a parameter of deformation. Deformed density matrix must fulfill the condition $\rho(\tau)\approx\rho$ when $T\ll T_{p}$. By analogy with [@r7],[@r8], only probabilities $\omega_{n}$ are subject of deformation in (\[U19\]), changing by $\omega_{n}(\tau)$ and correspondingly deformed statistical density matrix is $$\label{U20} \rho(\tau)=\sum_{n}\omega_{n}(\tau)|\varphi_{n}><\varphi_{n}|.$$ This approach in our opinion could give us the possibility to obtain Deformed Canonical Distribution as well as a rigorous proof of thermodynamical general uncertainty relations. In section 4 the construction of such a deformed statistical mechanics at Planck scale is demonstrated. However, first it seems expedient to outline briefly the principal features of the corresponding deformation in QM. Deformation of Quantum-Mechanics Density Matrix at Planck Scale =============================================================== In this section the principal features of QMFL construction with the use of the density matrix deformation are briefly outlined [@r8]. As mentioned above, for the fundamental deformation parameter we use $\alpha = l_{min}^{2 }/x^{2 }$ where $x$ is the scale. In contrast with [@r8], for the deformation parameter we use $\alpha$ rather than $\beta$ to avoid confusion, since quite a distinct value is denoted by $\beta$ in Statistical Mechanics:$\beta=1/kT$.\ [**Definition 1.**]{} [**(Quantum Mechanics with Fundamental Length)**]{}\ Any system in QMFL is described by a density pro-matrix of the form $${\bf \rho(\alpha)=\sum_{i}\omega_{i}(\alpha)|i><i|},$$ where 1. $0<\alpha\leq1/4$; 2. The vectors $|i>$ form a full orthonormal system; 3. $\omega_{i}(\alpha)\geq 0$ and for all $i$ the finite limit $\lim\limits_{\alpha\rightarrow 0}\omega_{i}(\alpha)=\omega_{i}$ exists; 4. $Sp[\rho(\alpha)]=\sum_{i}\omega_{i}(\alpha)<1$, $\sum_{i}\omega_{i}=1$; 5. For every operator $B$ and any $\alpha$ there is a mean operator $B$ depending on $\alpha$:\ $$<B>_{\alpha}=\sum_{i}\omega_{i}(\alpha)<i|B|i>.$$ Finally, in order that our definition 1 agree with the result of section 2, the following condition must be fulfilled: $$\label{U1b} Sp[\rho(\alpha)]-Sp^{2}[\rho(\alpha)]\approx\alpha.$$ Hence we can find the value for $Sp[\rho(\alpha)]$ satisfying the condition of definition 1: $$\label{U2b} Sp[\rho(\alpha)]\approx\frac{1}{2}+\sqrt{\frac{1}{4}-\alpha}.$$ According to point 5), $<1>_{\alpha}=Sp[\rho(\alpha)]$. Therefore for any scalar quantity $f$ we have $<f>_{\alpha}=f Sp[\rho(\alpha)]$. In particular, the mean value $<[x_{\mu},p_{\nu}]>_{\alpha}$ is equal to\ $$<[x_{\mu},p_{\nu}]>_{\alpha}= i\hbar\delta_{\mu,\nu} Sp[\rho(\alpha)]$$\ We denote the limit $\lim\limits_{\alpha\rightarrow 0}\rho(\alpha)=\rho$ as the density matrix. Evidently, in the limit $\alpha\rightarrow 0$ we return to QM. As follows from definition 1, $<(j><j)>_{\alpha}=\omega_{j}(\alpha)$, from whence the completeness condition by $\alpha$ is\ $<(\sum_{i}|i><i|)>_{\alpha}=<1>_{\alpha}=Sp[\rho(\alpha)]$. The norm of any vector $|\psi>$ assigned to $\alpha$ can be defined as\ $<\psi|\psi>_{\alpha}=<\psi|(\sum_{i}|i><i|)_{\alpha}|\psi> =<\psi|(1)_{\alpha}|\psi>=<\psi|\psi> Sp[\rho(\alpha)]$, where $<\psi|\psi>$ is the norm in QM, i.e. for $\alpha\rightarrow 0$. Similarly, the described theory may be interpreted using a probabilistic approach, however requiring replacement of $\rho$ by $\rho(\alpha)$ in all formulae. It should be noted: 1. The above limit covers both Quantum and Classical Mechanics. Indeed, since $\alpha\sim L_{p}^{2 }/x^{2 }=G \hbar/c^3 x^{2}$, we obtain: 1. $(\hbar \neq 0,x\rightarrow \infty)\Rightarrow(\alpha\rightarrow 0)$ for QM; 2. $(\hbar\rightarrow 0,x\rightarrow \infty)\Rightarrow(\alpha\rightarrow 0)$ for Classical Mechanics; 2. As a matter of fact, the deformation parameter $\alpha$ should assume the value $0<\alpha\leq1$. However, as seen from (\[U2b\]), $Sp[\rho(\alpha)]$ is well defined only for $0<\alpha\leq1/4$, i.e. for $x=il_{min}$ and $i\geq 2$ we have no problems at all. At the point, where $x=l_{min}$, there is a singularity related to complex values assumed by $Sp[\rho(\alpha)]$ , i.e. to the impossibility of obtaining a diagonalized density pro-matrix at this point over the field of real numbers. For this reason definition 1 has no sense at the point $x=l_{min}$. 3. We consider possible solutions for (\[U1\]). For instance, one of the solutions of (\[U1\]), at least to the first order in $\alpha$, is $$\rho^{*}(\alpha)=\sum_{i}\alpha_{i} exp(-\alpha)|i><i|,$$ where all $\alpha_{i}>0$ are independent of $\alpha$ and their sum is equal to 1. In this way $Sp[\rho^{*}(\alpha)]=exp(-\alpha)$. Indeed, we can easily verify that $$\label{U3} Sp[\rho^{*}(\alpha)]-Sp^{2}[\rho^{*}(\alpha)]=\alpha+O(\alpha^{2}).$$ Note that in the momentum representation $\alpha\sim p^{2}/p^{2}_{pl}$, where $p_{pl}$ is the Planck momentum. When present in matrix elements, $exp(-\alpha)$ can damp the contribution of great momenta in a perturbation theory. 4. It is clear that within the proposed description the states with a unit probability, i.e. pure states, can appear only in the limit $\alpha\rightarrow 0$, when all $\omega_{i}(\alpha)$ except for one are equal to zero or when they tend to zero at this limit. In our treatment pure state are states, which can be represented in the form $|\psi><\psi|$, where $<\psi|\psi>=1$. 5. We suppose that all the definitions concerning a density matrix can be transferred to the above-mentioned deformation of Quantum Mechanics (QMFL) through changing the density matrix $\rho$ by the density pro-matrix $\rho(\alpha)$ and subsequent passage to the low energy limit $\alpha\rightarrow 0$. Specifically, for statistical entropy we have $$\label{U4b} S_{\alpha}=-Sp[\rho(\alpha)\ln(\rho(\alpha))].$$ The quantity of $S_{\alpha}$ seems never to be equal to zero as $\ln(\rho(\alpha))\neq 0$ and hence $S_{\alpha}$ may be equal to zero at the limit $\alpha\rightarrow 0$ only. Some Implications: 1. If we carry out measurement on the pre-determined scale, it is impossible to regard the density pro-matrix as a density matrix with an accuracy better than particular limit $\sim10^{-66+2n}$, where $10^{-n}$ is the measuring scale. In the majority of known cases this is sufficient to consider the density pro-matrix as a density matrix. But on Planck’s scale, where the quantum gravitational effects and Plank energy levels cannot be neglected, the difference between $\rho(\alpha)$ and $\rho$ should be taken into consideration. 2. Proceeding from the above, on Planck’s scale the notion of Wave Function of the Universe (as introduced in [@r17]) has no sense, and quantum gravitation effects in this case should be described with the help of density pro-matrix $\rho(\alpha)$ only. 3. Since density pro-matrix $\rho(\alpha)$ depends on the measuring scale, evolution of the Universe within the inflation model paradigm [@r18] is not a unitary process, or otherwise the probabilities $p_{i}=\omega_{i}(\alpha)$ would be preserved. Deformation of Statistical Density Matrix ========================================= It follows that we have a maximum energy of the order of Planck’s from an inequality (\[U7\]):\ $$E_{max}\sim E_{p}$$\ Proceeding to the Statistical Mechanics, we further assume that an internal energy of any ensemble U could not be in excess of $E_{max}$ and hence temperature $T$ could not be in excess of $T_{max}=E_{max}/k \sim T_{p}$. Let us consider density matrix in Statistical Mechanics : $$\label{U8} \rho_{stat}=\sum_{n}\omega_{n}|\varphi_{n}><\varphi_{n}|,$$ where the probabilities are given by\ $$\omega_{n}=\frac{1}{Q}\exp(-\beta E_{n})$$ and\ $$Q=\sum_{n}\exp(-\beta E_{n})$$\ Then for a canonical Gibbs ensemble the value $$\label{U9} \overline{\Delta(1/T)^{2}}=Sp[\rho_{stat}(\frac{1}{T})^{2}] -Sp^{2}[\rho_{stat}(\frac{1}{T})],$$ is always equal to zero, and this follows from the fact that $Sp[\rho_{stat}]=1$. However, for very high temperatures $T\gg0$ we have $\Delta (1/T)^{2}\approx 1/T^{2}\geq 1/T_{max}^{2}$. Thus, for $T\gg0$ a statistical density matrix $\rho_{stat}$ should be deformed so that in the general case $$\label{U10} Sp[\rho_{stat}(\frac{1}{T})^{2}]-Sp^{2}[\rho_{stat}(\frac{1}{T})] \approx \frac{1}{T_{max}^{2}},$$ or $$\label{U11} Sp[\rho_{stat}]-Sp^{2}[\rho_{stat}] \approx \frac{T^{2}}{T_{max}^{2}},$$ In this way $\rho_{stat}$ at very high $T\gg 0$ becomes dependent on the parameter $\tau = T^{2}/T_{max}^{2}$, i.e. in the most general case\ $$\rho_{stat}=\rho_{stat}(\tau)$$ and $$Sp[\rho_{stat}(\tau)]<1$$\ and for $\tau\ll 1$ we have $\rho_{stat}(\tau)\approx\rho_{stat}$ (formula (\[U8\])) .\ This situation is identical to the case associated with the deformation parameter $\alpha = l_{min}^{2 }/x^{2}$ of QMFL given in section 3. That is the condition $Sp[\rho_{stat}(\tau)]<1$ has an apparent physical meaning when: 1. At temperatures close to $T_{max}$ some portion of information about the ensemble is inaccessible in accordance with the probability that is less than unity, i.e. incomplete probability. 2. And vice versa, the longer is the distance from $T_{max}$ (i.e. when approximating the usual temperatures), the greater is the bulk of information and the closer is the complete probability to unity. Therefore similar to the introduction of the deformed quantum-mechanics density matrix in section 3 of [@r8] and in previous section of this paper,we give the following\ [**Definition 2.**]{} [**(Deformation of Statistical Mechanics)**]{}\ Deformation of Gibbs distribution valid for temperatures on the order of the Planck’s $T_{p}$ is described by deformation of a statistical density matrix (statistical density pro-matrix) of the form\ $${\bf \rho_{stat}(\tau)=\sum_{n}\omega_{n}(\tau)|\varphi_{n}><\varphi_{n}|}$$ having the deformation parameter $\tau = T^{2}/T_{max}^{2}$, where 1. $0<\tau \leq 1/4$; 2. The vectors $|\varphi_{n}>$ form a full orthonormal system; 3. $\omega_{n}(\tau)\geq 0$ and for all $n$ at $\tau \ll 1$ we obtain $\omega_{n}(\tau)\approx \omega_{n}=\frac{1}{Q}\exp(-\beta E_{n})$ In particular, $\lim\limits_{T_{max}\rightarrow \infty (\tau\rightarrow 0)}\omega_{n}(\tau)=\omega_{n}$ 4. $Sp[\rho_{stat}(\tau)]=\sum_{n}\omega_{n}(\tau)<1$, $\sum_{n}\omega_{n}=1$; 5. For every operator $B$ and any $\tau$ there is a mean operator $B$ depending on $\tau$\ $$<B>_{\tau}=\sum_{n}\omega_{n}(\tau)<n|B|n>.$$ Finally, in order that our Definition 2 agree with the formula (\[U11\]), the following condition must be fulfilled: $$\label{U12b} Sp[\rho_{stat}(\tau)]-Sp^{2}[\rho_{stat}(\tau)]\approx \tau.$$ Hence we can find the value for $Sp[\rho_{stat}(\tau)]$ satisfying the condition of Definition 2 (similar to Definition 1): $$\label{U13} Sp[\rho_{stat}(\tau)]\approx\frac{1}{2}+\sqrt{\frac{1}{4}-\tau}.$$ It should be noted: 1. The condition $\tau \ll 1$ means that $T\ll T_{max}$ either $T_{max}=\infty$ or both in accordance with a normal Statistical Mechanics and canonical Gibbs distribution (\[U8\]) 2. Similar to QMFL in Definition 1, where the deformation parameter $\alpha$ should assume the value $0<\alpha\leq1/4$. As seen from (\[U13\]), here $Sp[\rho_{stat}(\tau)]$ is well defined only for $0<\tau\leq1/4$. This means that the feature occurring in QMFL at the point of the fundamental length $x=l_{min}$ in the case under consideration is associated with the fact that [**highest measurable temperature of the ensemble is always**]{} ${\bf T\leq \frac{1}{2}T_{max}}$. 3. The constructed deformation contains all four fundamental constants: $G,\hbar,c,k$ as $T_{max}=\varsigma T_{p}$,where $\varsigma$ is the denumerable function of $\alpha^{\prime}$ (\[U2\])and $T_{p}$, in its turn, contains all the above-mentioned constants. 4. Again similar to QMFL, as a possible solution for (\[U12\]) we have an exponential ansatz\ $$\rho_{stat}^{*}(\tau)=\sum_{n}\omega_{n}(\tau)|n><n|=\sum_{n} exp(-\tau) \omega_{n}|n><n|$$\ $$\label{U14} Sp[\rho_{stat}^{*}(\tau)]-Sp^{2}[\rho_{stat}^{*}(\tau)]=\tau+O(\tau^{2}).$$ In such a way with the use of an exponential ansatz (\[U14\]) the deformation of a canonical Gibbs distribution at Planck scale (up to factor $1/Q$) takes an elegant and completed form: $$\label{U15b} {\bf \omega_{n}(\tau)=exp(-\tau)\omega_{n}= exp(-\frac{T^{2}} {T_{max}^{2}}-\beta E_{n})}$$ where $T_{max}= \varsigma T_{p}$ Conclusion ========== It has been demonstrated that a nature of deformations in Quantum and Statistical Mechanics at Plank scale is essentially identical. Still further studies are required to look into variations of the formulae for entropy and other quantities in this deformed Statistical Mechanics. Of particular interest is the problem of a rigorous proof for the Generalized Uncertainty Relations (GUR) in Thermodynamics (section 2 of the present paper and [@r14],[@r19]) as a complete analog of the corresponding relations in Quantum Mechanics [@r1], [@r3; @r4; @r5; @r6], in turn necessitating the deformation of Gibbs distribution. The present paper as an integration of [@r19],[@r20]is aimed at the solution of this problem. [99]{} R.J.Adler and D.I.Santiago,On Gravity and the Uncertainty Principle, Mod.Phys.Lett.A14(1999)1371\[gr-qc/9904026\] L.Garay,Quantum Gravity and Minimum Length Int.J.Mod.Phys.A.A10(1995)145\[gr-qc/9403008\] G.Veneziano,A stringly nature needs just two constant Europhys.Lett.2(1986)199;D.Amati,M.Ciafaloni and G.Veneziano,Can spacetime be probed below the string size? 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--- abstract: 'The processes of neutrino production of electron–positron pairs, $\nu \bar\nu \to e^- e^+$ and $\nu \to \nu e^- e^+$, in a magnetic field of arbitrary strength, where electrons and positrons can be created in the states corresponding to excited Landau levels, are analysed. The results can be applied for calculating the efficiency of the electron–positron plasma production by neutrinos in the conditions of the Kerr black hole accretion disc considered by experts as the most possible source of a short cosmological gamma burst.' address: - '$^1$ Division of Theoretical Physics, Department of Physics, Yaroslavl State P G Demidov University, Sovietskaya 14, 150000 Yaroslavl, Russia' - '$^2$ Division of Physics, Yaroslavl Higher Military School of Air Defence, Moskovskiy Prosp. 28, 150001 Yaroslavl, Russia' author: - 'A V Kuznetsov$^1$, D A Rumyantsev$^{1,\ast}$ and V N Savin$^{2,\ast \ast}$' title: 'Neutrino processes $\nu \bar\nu \to e^- e^+$ and $\nu \to \nu e^- e^+$ in a strong magnetic field' --- Introduction ============ An intense electromagnetic field makes possible the processes which are forbidden in a vacuum such as the neutrino production of an electron–positron pair $\nu \to \nu e^- e^+$. The list of papers devoted to an analysis of this process and the collection of the results obtained could be found e.g. in [@KM_Book_2013]. In most cases, calculations of this kind were made either in the crossed field approximation, or in the limit of a superstrong field much greater than the critical value of $B_e = m_e^2/e \simeq 4.41\times 10^{13}$ G (we use natural units $c = \hbar = k_{\rm{B}} = 1$), when the electrons and positrons are born in states corresponding to the ground Landau level. However, there exist physical situations of the so-called moderately strong magnetic field, $p_\perp^2 \ge e B \gg m_e^2$, when electrons and positrons mainly occupy the ground Landau level, however, a noticeable fraction may be produced at the next levels. The indicated hierarchy of physical parameters corresponds to the conditions of the Kerr black hole accretion disk, regarded by experts as the most likely source of a short cosmological gamma-ray burst. The disc is a source of copious neutrinos and anti-neutrinos, which partially annihilate above the disc and turn into $e^{\mp}$ pairs, $\nu \bar\nu \to e^- e^+$. This process was proposed and investigated in many details (for the list of references see e.g. [@Beloborodov:2011; @Kuznetsov:2014]) as a possible mechanism for creating relativistic, $e^{\mp}$-dominated jets that could power observed gamma-ray bursts. In [@Beloborodov:2011], in addition to $\nu \bar\nu$ annihilation, the contribution of the magnetic field-induced process $\nu \to \nu e^- e^+$ to the neutrino energy deposition rate around the black hole was also included for the first time. The authors [@Beloborodov:2011] concluded in part, that the process $\nu \to \nu e^- e^+$ could dominate over the basic process $\nu \bar\nu \to e^- e^+$. They used the result for the energy deposition rate in the process $\nu \to \nu e^- e^+$ obtained in [@Kuznetsov:1997a; @Kuznetsov:1997b] in the crossed field limit, while in those physical conditions ($B$ to 180 $B_e$, $E_\nu$ to 25 MeV) the approximation of a crossed field is poorly applicable (as well as the approximation of a superstrong field when $e^-e^+$ are created in the ground Landau level). The next Landau levels can be also excited, as we have shown in our paper [@Kuznetsov:2014]. Furthermore, the authors [@Beloborodov:2011] considered the process $\nu \bar\nu \to e^- e^+$ without taking account of the magnetic field influence. Thus, the aim of this paper is the study of the processes $\nu \bar\nu \to e^- e^+$ and $\nu \to \nu e^- e^+$ in the physical conditions of the moderately strong magnetic field, where the electrons and positrons would be born in the states corresponding to the excited Landau levels. Possible astrophysical applications are discussed. Neutrino process $\nu \to \nu e^- e^+$ in a strong magnetic field ================================================================= The total probability of the process $\nu \to \nu e^-_{(n)} e^+_{(\ell)}$, when the electron and the positron are created in the $n$th and $\ell$th Landau levels, is, in a general case, the sum of the probabilities of the four polarization channels: $$\label{eq:Wtot} W_{n \ell} = W^{--}_{n \ell} + W^{-+}_{n \ell} + W^{+-}_{n \ell} + W^{++}_{n \ell} \, .$$ For each of the channels, the differential probability over the final neutrino momentum per unit time, after integration over the momenta of the electron and positron, is reduced to one nontrivial integral: $$\begin{aligned} {\mathrm{d}}W^{s s'}_{n \ell} = \frac{\beta \, {\mathrm{d}}^3 P'}{(2 \pi)^4 16 E E'} \, \int \, \frac{{\mathrm{d}}p_z}{\varepsilon_n \, \varepsilon'_{\ell}} \, \delta(\varepsilon_n + \varepsilon'_{\ell} - q_0) \, |{\cal M}_{n \ell}^{s s'}|^2 \, , \label{eq:dw2} \end{aligned}$$ where $\varepsilon_n = \sqrt{M_n^2 + p_z^2}$, $M_n = \sqrt{m_e^2 + 2 \beta n}$, $\beta = e B$. The energy of the initial neutrino should exceed a certain threshold value. In the reference frame, where the momentum of the initial neutrino directed at an angle $\theta$ to the magnetic field, the threshold energy is given by: $$E \, \sin \theta \ge M_{n} + M_{\ell} \, . \label{eq:condE}$$ Some details of calculations can be found in our paper [@Kuznetsov:2014]. The probability of the $\nu \to \nu e^- e^+$ process defines its partial contribution into the neutrino opacity of the medium. The estimation of the neutrino mean free path with respect to this process gives the result which is too large [@KM_Book_2013] compared with the typical size of any compact astrophysical object, where a strong magnetic field could exist. However, a mean free path does not exhaust the neutrino physics in a medium. In astrophysical applications, we could consider the values that probably are more essential, namely, the mean values of the neutrino energy and momentum losses, caused by the influence of an external magnetic field. These values can be described by the four-vector of losses $Q^{\alpha}$, $$Q^\alpha \, = \, E \int q^\alpha \, {\mathrm{d}}W = - E \, ({\cal I}, {\bf F}) \,. \label{eq:Q0}$$ where $q$ is the difference of the momenta of the initial and final neutrinos, $q = P - P'$, ${\mathrm{d}}W$ is the total differential probability of the process. The zeroth component of $Q^{\alpha}$ is connected with the mean energy lost by a neutrino per unit time due to the process considered, ${\cal I} = {\mathrm{d}}E/{\mathrm{d}}t$. The space components of the four-vector (\[eq:Q0\]) are similarly connected with the mean neutrino momentum loss per unit time, ${\bf F} = {\mathrm{d}}{\bf P}/{\mathrm{d}}t$. It should be noted that the four-vector of losses $Q^{\alpha}$ can be used for evaluating the integral effect of neutrinos on plasma in the conditions of not very dense plasma, where an one-interaction approximation of a neutrino with plasma is valid. In [@Beloborodov:2011], the formula for the energy deposition rate was taken, which was calculated in the crossed field limit [@Kuznetsov:1997a; @Kuznetsov:1997b]. However, in the region of the physical parameters used in [@Beloborodov:2011] ($B$ to 180 $B_e$, $E_\nu$ to 25 MeV), the approximation of a crossed field is poorly applicable, as well as the approximation of a superstrong field when $e^- e^+$ are created in the ground Landau level. The contribution of the next Landau levels which can be also excited, should be taken into account. In [@Kuznetsov:2014], the results are presented of our calculation of the mean neutrino energy losses caused by the process $\nu \to \nu e^- e^+$ in a moderately strong magnetic field, i.e. in the conditions of the Kerr black hole accretion disk. It was shown that the crossed field limit gives the overstated result which is in orders of magnitude greater than the sum of the contributions of lower excited Landau levels. On the other hand, the results with $e^- e^+$ created at the ground Landau level give the main contribution to the energy deposition rate, and almost exhaust it at $B = 180 B_e$. This would mean that the conclusion [@Beloborodov:2011] that the contribution of the process $\nu \to \nu e^- e^+$ to the efficiency of the electron-positron plasma production by neutrino exceeds the contribution of the annihilation channel $\nu \bar\nu \to e^- e^+$, and that the first process dominates the energy deposition rate, does not have a sufficient basis. A new analysis of the efficiency of energy deposition by neutrinos through both processes, $\nu \bar\nu \to e^- e^+$ and $\nu \to \nu e^- e^+$, in a hyper-accretion disc around a black hole should be performed, with taking account of our results [@Kuznetsov:2014] for the process $\nu \to \nu e^- e^+$. The strong magnetic field influence on the process $\nu \bar\nu \to e^- e^+$ ============================================================================ The local energy-momentum deposition rate due to the process $\nu \bar\nu \to e^- e^+$ is defined by the equation [@Birkl:2007]: $$Q^\alpha_{\nu \bar\nu} = \int \frac{{\mathrm{d}}^3 p}{(2 \pi)^3} \, f_{\nu} (p) \int \frac{{\mathrm{d}}^3 p'}{(2 \pi)^3} \, f_{\bar\nu} (p') \, \left( p^\alpha + p'^\alpha \right) \frac{(p p')}{E E'} \, \sigma (\nu \bar\nu \to e^- e^+) \,, \label{eq:Q1}$$ where $\sigma$ is the cross-section of the process, $p$ and $p'$ are the four-momenta of the neutrino and antineutrino, $f_{\nu} (p)$ and $f_{\bar\nu} (p')$ are the local distribution functions depending on the distribution functions at the surface of the black hole accretion disc, and on the details of propagation. In a strong magnetic field, the cross-section takes the form: $$\sigma (\nu \bar\nu \to e^- e^+) = \sum\limits_{f=e,\mu,\tau} \, \sum\limits_{n, \ell} \, \sigma (\nu_f {\bar\nu}_f \to e^-_{(n)} e^+_{(\ell)}) \,, \label{eq:sigmaB}$$ where the upper limit of summation over $n, \ell$ is defined by the condition $(M_n + M_{\ell})^2 \le ( p + p' )_{\mprl}^2$, ($q_{\mprl}^2=q_0^2-q_z^2$, if $z$ is along $\bf B$). Unlike the cross-section in vacuum where it depends on the Mandelstam parameter $S$ only, the cross-section in a magnetic field depends on the set of kinematic variables, e.g.: energies $E, E'$, two polar angles and one azimuth angle. In figure \[fig:function180\], we take for the sake of illustration the case $E=E'$, and take certain angles. The dependence is presented of $\sigma$ (solid line) and $\sigma_{\rm vac}$ (dashed line) on $E$. The cross-section has a peculiar “sawtooth” profile due to the square-root singularities [@Klepikov:1954], which is similar to the profile of the process $\gamma^\ast \to e^- e^+$ width in a strong field [@Daugherty:1983; @Baier:2007]. After averaging over small intervals $E \pm \Delta E$, the dependence becomes smoother. It can be seen that in calculations of the energy-momentum deposition rate by integration over the neutrino and antineutrino momenta, the field influence appears to be inessential. ![Cross-section of the process $\nu \bar\nu \to e^- e^+$ in the case $E=E'$ and for fixed angles: the energy dependence in the field (solid line) and in vacuum (dashed line); $\sigma_0 = {4 \, G_{\mathrm{F}}^2 \, m_e^2}/{\pi}$ is the so-called typical weak cross-section.[]{data-label="fig:function180"}](sigma-180){width="90.00000%"} Conclusions =========== - The processes $\nu \to \nu e^- e^+$ and $\nu \bar\nu \to e^- e^+$ are investigated in the magnetic field of an arbitrary strength, when $e^- e^+$ can be produced in the excited Landau levels. - The neutrino energy losses due to the process $\nu \to \nu e^- e^+$ are calculated. The results should be used for calculations of the efficiency of the $e^- e^+$ plasma production by neutrinos in the conditions of the Kerr black hole accretion disk. In these conditions, the crossed field limit gives the overstated result which is in orders of magnitude greater than the sum over the lower Landau levels. - The cross-section of the process $\nu \bar\nu \to e^- e^+$ in a strong field, has a peculiar “sawtooth” profile, which is close to the vacuum cross-section after averaging. In calculations of the energy-momentum deposition rate by integration over the neutrino and antineutrino momenta, the field influence appears to be inessential. The study was performed with the support by the Project No. 92 within the base part of the State Assignment for the Yaroslavl University Scientific Research, and was supported in part by the Russian Foundation for Basic Research (Project No. ). References {#references .unnumbered} ========== [9]{} Kuznetsov A and Mikheev N 2013 [*Electroweak Processes in External Active Media*]{} (Berlin, Heidelberg: Springer-Verlag) Zalamea I and Beloborodov A M 2011 [*Mon. Not. R. Astron. Soc.*]{} [**410**]{} 2302 Kuznetsov A V, Rumyantsev D A and Savin V N 2014 [*Int. J. Mod. Phys.*]{} A [**29**]{} 1450136 Kuznetsov A V and Mikheev N V 1997 [*Phys. Lett.*]{} B [**394**]{} 123 Kuznetsov A V and Mikheev N V 1997 [*Phys. At. Nucl.*]{} [**60**]{} 1865 (Original Russian text: [*Yad. Fiz.*]{} [**60**]{} 2038) Birkl R, Aloy M A, Janka H-Th and M[" u]{}ller E 2007 [*Astron. Astrophys.*]{} [**463**]{} 51 Klepikov N P 1954 (In Russian) [*Zh. Eksp. Teor. Fiz.*]{} [**26**]{} 19 Daugherty J K and Harding A K 1983 [*Astrophys. J.*]{} [**273**]{} 761 Baier V N and Katkov V M 2007 [*Phys. Rev.*]{} D [**75**]{} 073009

--- abstract: 'The evolution of infrastructure networks such as roads and streets are of utmost importance to understand the evolution of urban systems. However, datasets describing these spatial objects are rare and sparse. The database presented here represents the road network at the french national level described in the historical map of Cassini in the $18^{th}$ century. The digitization of this historical map is based on a collaborative methodology that we describe in detail. This dataset can be used for a variety of interdisciplinary studies, covering multiple spatial resolutions and ranging from history, geography, urban economics to network science.' author: - 'Julien Perret^1[\*]{}^, Maurizio Gribaudi^2^, Marc Barthelemy^3,4^' title: 'Roads and cities of $18^{th}$ century France' --- 1\. COGIT, IGN. 73 avenue de Paris, 94165 Saint-Mande Cedex, France. 2. LaDéHiS, EHESS. 190-198 avenue de France, 75013 Paris, France. 3. IPhT, CEA. Orme-des-Merisiers, 91191 Gif-sur-Yvette, France. 4. CAMS, EHESS. 190-198 Avenue de France, 75013 Paris, France.\ [\*]{} Corresponding author (julien.perret@gmail.com) Background & Summary {#background-summary .unnumbered} ==================== ![Part of the Cassini map of Paris and its digitization. The map is produced by EHESS, CNRS and BnF [@cassini] and can be freely accessed by web service [@geoportail].[]{data-label="fig:paris"}](paris_map_digitized){width="\textwidth"} Triggered by recent, powerful digitization techniques, there is a huge interest in historical data, in particular when they allow to track temporal changes at different spatial scales. Such projects comprise for example the NYPL initiative [@NYPL], the digitization of the road network of a region in Italy [@Strano2012], of Paris over 200 years [@Barthelemy2013], and the digitization of ancient French forests [@Dupouey2007; @Vallauri2012]. New historical datasets extracted from maps allow researchers to study the time evolution of urban systems, to extract stylized facts, and for the first time to test theoretical ideas and models. Historical datasets of road networks allow to study territorial evolutions at different scales and to build tools to accurately answer theoretical questions. In particular, one can ask about the impact of the road network on subsequent urbanization, the correlation between the location of an entity (such as a city, town, etc.) and socio-economical indicators such as population or importance in the trade network, immigration, etc. More generally, such historical datasets are of interest to a wide variety of scientists comprising historians, geographers, mathematicians, archeologists, geo-historians, geomaticians, and computer scientists [@Masucci2013; @Wang2015; @Gribaudi2014; @Porta2014]. The digitization of historical sources is usually done locally by researchers for their immediate research needs without sharing their work and results with others. In contrast, we believe that it is essential to build a platform to share our work, but also to have a collective control over the production process of the data, its transformation and its analysis. Operations such as scanning, georeferencing and digitization of historical sources imply several and delicate choices that should be documentated and tracked. Historical sources might have deformations originating from aging. Their georeferencing carries its own deformations which have to be minimized in order for the sources to remain legible. Our approach consists in taking these geometric displacements into account after the digitization process using spatial data matching tools [@Walter1999] to find corresponding entities in consecutive data sets. Such tools should allow researchers to control and take into account the imperfections of the data throughout their analysis [@Olteanu2008]. This way, we can reduce the impact of the georeferencing in the matching process and the analysis. Furthermore, opendata and open source tools provide the scientific community with the ability to control, track and reproduce the results at every stage. With these ideas in mind, we developed a collaborative way to digitize the Cassini map of the 18th century (see Figure \[fig:paris\] for a visualization of a small subset of the map and the corresponding digitized data). This map is the first one that restitutes with geometrical precision the entire French territory in the second half of the eighteenth century at a scale of 1/86 000. First conceived in the late $17^{th}$ century, this work was made possible by the development of geodesic triangulation techniques and their generalization. The determination of the Paris meridian and the establishment of a single framework for all triangulations of France (1744) provided the reference needed for putting together several local maps [@Maraldi1744]. In 1747 César-François Cassini de Thury was formally commissioned by Louis XV to draw the entire map of France showing the entire kingdom but also finer details. Cassini and his engineers divided the French territory in a grid of 180 rectangles with a size of about 80 km $\times$ 50 km which lead to as many maps printed on sheets of size 104 cm $\times$ 73 cm. Due to financial difficulties, the Revolution and regime changes, the constitution of this map was delayed and it is not before 1815 that the last sheets were released, under the direction of Jean-Dominique Cassini, son of César-François. The maps that serve as a basis for our work is the digital copy of the so-called “Marie-Antoinette” version, commissioned in 1780 by the queen. These maps were completed, corrected and updated in the subsequent years. For example, the map of the Paris region which was initially drawn between 1749 and 1755, and published the first time in 1756, displays clear signs of corrections made during the post-revolution period with the introduction of administrative divisions created during the Republic in 1790. An important part of the project was therefore to analyze each sheet, to give a precise date of its drawing and to provide an assessment of its accuracy. This was done by comparing different printed and dated versions, and many minutes and notes from the National Institute of Geographic and Forest Information (IGN) archives. The main work was however (see Methods) to analyze and vectorize a large number of features of the Cassini map such as roads, water networks, towns and villages, forest and crops, industrial and administrative structures. The digitized data have been made available on a dedicated geo-historical portail [@geohistoricaldatawebsite]. These different features put together under a digital form give us a detailed picture of the french territory in the second half of the eighteenth century. Methods {#methods .unnumbered} ======= The digitization of the Cassini maps and, in particular, of its road network, was achieved in a collaborative way using a shared PostgreSQL [@postgresql] database and its spatial extension PostGIS [@postgis]. GIS editing tools such as Quantum GIS [@qgis] were used to remotely digitize the objects using a WMTS (Web Tile Map Service) layer provided by IGN [@cassini] as background. Details on the methods used to produce the georeferenced map are available on a dedicated website [@cassiniwebsite]. This way, several operators have been able to digitize data simultaneously on the same database. In order to provide consistent data records, data specifications were proposed as a result of an important collaborative work. Nevertheless, as the specifications were enhanced during the digitization process, local variations in the capture of several attributes might be found (the attribute “bordered” was added after a few months of digitization for instance). Further work will focus on the consistency of the data (both for attributes and geometries). An important aspect of the Cassini dataset is the fact that the Cassini map was not homogeneously drawn (different sheets might show different levels of detail as seen in Figure \[fig:french\_network\]) or conceived as a road map [@Pelletier2002]. Hence, one has to be careful when studying the road network extracted from it [@Bonin2014]. Specifically, the road network inside most cities was not drawn in the map. An automatic process is therefore proposed to create so-called “fictive” edges inside cities allowing to link all roads leading them. As shown in Figure \[fig:city\_fictive\_edge\], a node representing the city is created at its centroid (or rather at the centroid of the geometry representing its boundary in the map) and edges are created to connect this node to the edges ending in the city. Furthermore, in order to speed up the digitizing process, some roads have been captured as continuous strokes rather than by topological road segments: some users digitized entire roads instead of stopping the capture at each road intersection. We therefore use the PostGIS topology engine [@postgistopology] to convert the digitized strokes into a topological network. This process uses a distance threshold to merge points closer than the given threshold and thus allows for the correction of minor shifts between points and a second threshold for to collect all nodes in the neighboorhood of a city. The thresholds used in the current export are 10 meters and 20 meters respectively. The digitized roads and cities are also provided in the export and the code for the topological export is available [@cassinitopology]. ![The digitized $18^{th}$ century french road network.[]{data-label="fig:french_network"}](french_network){width="\textwidth"} ![Construction of “fictive” edges in cities: the digitized edges of the road network connected to the city are linked by the created edges (in blue).[]{data-label="fig:city_fictive_edge"}](city_fictive_edge){width="\textwidth"} Data Records {#data-records .unnumbered} ============ The data records contain the roads and cities as captured (the names of the attributes have been translated though) and the topological nodes, edges and faces. We propose five shapefiles (which each actually refer to four files with .shp, .dbf, .shx and .prj extentions) and two CSV files containing simplified versions of the nodes and edges. The dataset is stored at the Harvard Dataverse (Data Citation 1). Roads (france\_cassini\_roads.shp) {#roads-france_cassini_roads.shp .unnumbered} ---------------------------------- This file contains the roads represented in the Cassini maps. It includes the following attributes: - [*id:*]{} the (unique) identifier for each road segment (integer); - [*geometry:*]{} the geometry of the segment (linestring) in RGF93 / Lambert-93 (EPSG:2154). - [*type:*]{} the type of road or connexion as represented in the map: either “red”, “white”, “trail”, “forest”, “bridge”, “ferry” or “gap”. These values refer respectively to main roads, secondary roads, trails, forest trails, bridges, tubs, and shifts between sheets (string). - [*name:*]{} the name of the segment when it has one (string). - [*uncertain:*]{} whether the nature of the segment is difficult to clearly identify in the map (boolean). - [*bordered:*]{} whether the segment is bordered by trees (boolean). - [*comments:*]{} comments left by our contributors when the object raises specific questions (string). Cities (france\_cassini\_cities.shp) {#cities-france_cassini_cities.shp .unnumbered} ------------------------------------ This file describes some of the main types of land use identifiable in the maps. - [*id:*]{} the (unique) identifier for each object (integer). - [*geometry:*]{} the geometry of the object (multipolygon) in RGF93 / Lambert-93 (EPSG:2154). - [*type:*]{} the type of object: “city”, “town”, “domain”, “fort” (string), respectively for cities, towns, domains and forts. - [*name:*]{} the name of the land element when it has one (string). - [*fortified:*]{} is the city fortified? (boolean). Can only be true if the type is “city”. - [*comments:*]{} comments left by our contributors when the object raises specific questions (string). Topological Nodes (node.shp) {#topological-nodes-node.shp .unnumbered} ---------------------------- - [*id:*]{} the (unique) identifier for each object (integer). - [*geom:*]{} the geometry of the object (point) in RGF93 / Lambert-93 (EPSG:2154). - [*city\_id:*]{} identifier of the city it lies in (from france\_cassini\_cities.shp) - [*city\_name:*]{} the name of the city (from france\_cassini\_cities.shp) - [*city\_type:*]{} the type of the city (from france\_cassini\_cities.shp) - [*component:*]{} the identifier of the connected component the node belongs to (integer) Topological Edges (edge.shp) {#topological-edges-edge.shp .unnumbered} ---------------------------- Edges are not oriented so the start and end nodes are arbitrary. Nevertheless, they are consistent with the order of the points in the geometry of the edge (the start node position is the first point of the geometry of the edge). When the edge is built from a road, it holds the identifier of this road. Its type is also given for convenience but is recoverable by join (combining the Edge table with the type from the roads table by using the common identifier road\_id). Note that “fictive” edges do not hold such an identifier. Furthermore, in cases where multiple roads are merged into the same edge, the identifier is arbitrary. - [*id:*]{} the (unique) identifier for each object (integer). - [*geom:*]{} the geometry of the object (linestring) in RGF93 / Lambert-93 (EPSG:2154). - [*start\_node:*]{} identifier of the initial node of the edge (from node.shp) - [*end\_node:*]{} identifier of the final node of the edge (from node.shp) - [*road\_id:*]{} identifier of the road it stems from (from france\_cassini\_roads.shp) - [*road\_type:*]{} type of the road(from france\_cassini\_roads.shp) - [*length:*]{} length of the edge (meters) - [*component:*]{} the identifier of the connected component the edge belongs to (integer) Topological Faces (face.shp) {#topological-faces-face.shp .unnumbered} ---------------------------- As the resulting network is a planar graph (*i.e.* a graph that can be embedded in the plane), the faces (*i.e.* the regions bounded by edges) are also provided. - [*id:*]{} the (unique) identifier for each object (integer). - [*geom:*]{} the geometry of the object (polygon) in RGF93 / Lambert-93 (EPSG:2154). Simplified Topological Nodes (node.csv) {#simplified-topological-nodes-node.csv .unnumbered} --------------------------------------- This file contains the same nodes as node.shp but in a different easily accessible format. The position of the roads is given in lat/long. - [*id:*]{} the (unique) identifier for each object (integer) - [*lat:*]{} the latitude of the node in WGS 84 (EPSG:4326) - [*long:*]{} the longitude of the node in WGS 84 (EPSG:4326) - [*city\_id:*]{} the identifier of the city it lies in (from france\_cassini\_cities.shp) - [*city\_name:*]{} the name of the city (from france\_cassini\_cities.shp) - [*city\_type:*]{} the type of the city (from france\_cassini\_cities.shp) Simplified Topological Edges (edge.csv) {#simplified-topological-edges-edge.csv .unnumbered} --------------------------------------- This file contains the same edges as edge.shp without the geometry. It is therefore a simplified version. The length of the edge is the cartesian 2D length of the geometry (a linestring, i.e. a sequence of line segments) from edge.shp computed using PostGIS funtion *ST\_Length*. - [*id:*]{} the (unique) identifier for each object (integer) - [*start\_node:*]{} identifier of the initial node of the edge (from node.shp) - [*end\_node:*]{} identifier of the final node of the edge (from node.shp) - [*road\_id:*]{} identifier of the road it stems from (from france\_cassini\_roads.shp) - [*road\_type:*]{} type of the road(from france\_cassini\_roads.shp) - [*length:*]{} length of the edge (meters) Technical Validation {#technical-validation .unnumbered} ==================== Topological Validation {#topological-validation .unnumbered} ---------------------- The topology created using PostGIS Topology is first validated by the same tool and the provided function *ValidateTopology* without error. This function checks for several errors including crossing edges, and mismatching edge/node topology. Furthermore, we compute the number of input edges corresponding to the edges of the final network. This allows us to identify the duplicated edges, i.e. the edges in the final network which correspond to multiple edges in the input data. These duplicated edges usually correspond to digitization errors and are used to manually validate the digitized data. The latest version (V5) of the topology does not contain any duplicated edge. Connected Components Validation {#connected-components-validation .unnumbered} ------------------------------- The second validation consists in computing and analysing the connected components of the network. Indeed, such a road network should essentially be connected and small connected components are unlikely (they would mean small ’islands’ disconnected from the rest of the network). Our network contains 1274 connected components. The largest component is about 110,000 kilometers in length (more than 96% of the total length of the network) whereas the smallest is about 100 meters. Figure \[fig:components\] shows the three largest connected components in the network. Note that the second largest component is at the very edge of the map (in Germany) and is not visually connected to the network in the map. Finally, the third largest component is the Jersey island. Other large components represent other islands but also forests which paths are represented (and thus digitized) but rarely connected to the road network. The smallest components represent isolated features such as bridges. They can also correspond to digitization errors and the connected components can be used as a tool for data correction. ![The three largest connected components of the network.[]{data-label="fig:components"}](connected_components_composition){width="80.00000%"} Collaborative Validation {#collaborative-validation .unnumbered} ------------------------ Our third validation method is still ongoing work. It was inspired by the “Building Inspector” [@buildinginspector], developped by NYPL and used for the validation of buildings automatically vectorized from insurance maps. With the help of NYPL, we adapted this tool to collaboratively validate and correct our digitized data. The resulting application, “L’Arpenteur Topographe” [@arpenteurtopographe] is being tested on the digitized cities. The code of the application (from NYPL and our contributions) is available online [@arpenteurtopographe-code]. Further tests should be carried out on other objects in the future. Further work will also focus on better handling the interaction between the collaborative digitization process (using desktop or online GIS tools) and the collaborative validation, correction and enrichment processes such as in “L’Arpenteur Topographe”. Acknowledgements {#acknowledgements .unnumbered} ================ The digitization of the Cassini maps is the result of the collective work of the following group of colleagues as much as it is the work of the authors (in alphabetical order) : N. Abadie (IGN), S. Baciocchi (EHESS), C. Bertelli (Charta s.r.l.), O. Bonin (IFSTTAR), P. Bordin (Geospective), B. Costes (IGN), P. Cristofoli (EHESS), B. Dumenieu (IGN/EHESS), J. Gravier (Geographie-Cités), J.-P. Hubert (IFSTTAR), P.-A. Le Ny (Le Ny Conseil), E. Mermet (EHESS), C. Motte (EHESS), M. Pardoen (EHESS), A.-M. Raimond (IGN), S. Robert (EHESS), M.-C. Vouloir (EHESS). Author Contributions {#author-contributions .unnumbered} ==================== J.P. took care of the construction of the database and collaborative tools, initiated the project and wrote the paper. M.G. is responsable for the historical dimension, initiated the project and wrote the paper. M.B. initiated the project and wrote the paper. Competing financial interests {#competing-financial-interests .unnumbered} ============================= The author(s) declare no competing financial interests. [1]{} url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , & , ** (). , **, (). , **, (). , , , & , , **, **** (). , , & , , **, **** (). , , , , , & , , **, **** (). , , & , **, Rapport WWF/INRA, Marseille (). , & , , **, ****(8) (). , & , , **, ****(4), (). , **, (). , , , & , , **, ****(16), (). & , , **, **** (). & , , **, (). & , **, Paris, Delisle (). , **, (). , **, (). , **, (). , **, (). , **, (). & , **, (). , , , , ed., (). , **, (). , **, (). , **, (). , **, (). & , **, (). Data Citations {#datacitation1 .unnumbered} ============== 1\. Perret, J., Gribaudi, M., Barthelemy, M., Abadie, N., Baciocchi, S., Bertelli, C., Bonin, O., Bordin, P., Costes, B., Cristofoli, P., Dumenieu, B., Gravier, J., Hubert, J.-P., Le Ny, P.-A., Mermet, E., Motte, C., Pardoen, M., Raimond, A.-M., Robert, S. & Vouloir, M.-C., *The 18th century Cassini roads and cities dataset*, http://dx.doi.org/10.7910/DVN/28674, Harvard Dataverse, V5 (2015).

--- abstract: 'In this extended abstract, we present a simple approach to convergence on term graphs that allows us to unify term graph rewriting and infinitary term rewriting. This approach is based on a partial order and a metric on term graphs. These structures arise as straightforward generalisations of the corresponding structures used in infinitary term rewriting. We compare our simple approach to a more complicated approach that we developed earlier and show that this new approach is superior in many ways. The only unfavourable property that we were able to identify, viz. failure of full correspondence between weak metric and partial order convergence, is rectified by adopting a strong convergence discipline.' author: - Patrick Bahr bibliography: - 'compact.bib' title: 'Convergence in Infinitary Term Graph Rewriting Systems is Simple (Extended Abstract)[^1]' --- Introduction {#sec:introduction} ============ In *infinitary term rewriting* [@kennaway03book] we study infinite terms and infinite rewrite sequences. Typically, this extension to infinite structures is formalised by an ultrametric on terms, which yields infinite terms by metric completion and provides a notion of convergence to give meaning to infinite rewrite sequences. In this paper we extend infinitary term rewriting to term graphs. In addition to the metric approach, we also consider the partial order approach to infinitary term rewriting [@bahr10rta2] and generalise it to the setting of term graphs. One of the motivations for studying infinitary term rewriting is its relation to *non-strict evaluation*, which is used in programming languages such as Haskell [@marlow10haskell]. Non-strict evaluation defers the evaluation of an expression until it is “needed” and thereby allows us to deal with conceptually infinite data structures and computations. For example, the function `from` defined below constructs for each number $n$ the infinite list of consecutive numbers starting from $n$: from(n) = n :: from(s(n)) This construction is only conceptual and only results in a terminating computation if it is used in a context where only finitely many elements of the list are “needed”. Infinitary term rewriting provides us with an explicit limit construction to witness the outcome of an infinite computation as it is, for example, induced by `from`. After translating the above function definition to a term rewrite rule $\mathit{from}(x) \to x \cons \mathit{from}(s(x))$, we may derive an infinite rewrite sequence $$\mathit{from}(0) \to 0 \cons \mathit{from}(s(0)) \to 0 \cons s(0) \cons \mathit{from}(s(s(0))) \to \dots$$ which converges to the infinite term $0 \cons s(0) \cons s(s(0)) \cons \dots$, which represents the infinite list of numbers $0, 1, 2, \dots$ – as intuitively expected. Non-strict evaluation is rarely found in isolation, though. Usually, it is implemented as lazy evaluation [@henderson76popl], which complements a non-strict evaluation strategy with *sharing*. The latter avoids duplication of subexpressions by using pointers instead of copying. For example, the function `from` above duplicates its argument `n` – it occurs twice on the right-hand side of the defining equation. A lazy evaluator simulates this duplication by inserting two pointers pointing to the actual argument. While infinitary term rewriting is used to model the non-strictness of lazy evaluation, term graph rewriting models the sharing part of it. By endowing term graph rewriting with a notion of convergence like in infinitary term rewriting, we aim to unify the two formalisms into one calculus, thus allowing us to model both aspects within the same calculus. #### Contributions & Outline {#sec:contributions} At first we recall the basic notions of infinitary term rewriting (Section \[sec:infin-term-rewr\]). Afterwards, we construct a metric and a partial order on term graphs and show that both are suitable as a basis for notions of convergence in term graph rewriting (Section \[sec:graphs-term-graphs\]). Based on these structures we introduce notions of convergence (weak and strong variants) for term graph rewriting and show correspondences between metric-based and partial order-based convergence (Section \[sec:weak-convergence\] and \[sec:strong-convergence\]). We then present soundness and completeness properties of the resulting infinitary term graph rewriting calculi w.r.t. infinitary term rewriting (Section \[sec:soundness\]). Lastly, we compare our calculi with previous approaches (Section \[sec:concluding-remarks\]). Infinitary Term Rewriting {#sec:infin-term-rewr} ========================= Before starting with the development of infinitary *term graph* rewriting, we recall the basic notions of infinitary *term* rewriting. Rewrite sequences in infinitary rewriting, also called *reductions*, are sequences of the form $(\phi_\iota)_{\iota<\alpha}$, where each $\phi_\iota$ is a rewrite step from a term $t_\iota$ to $t_{\iota+1}$ in a term rewriting system (TRS) $\calR$, denoted $\phi_\iota\fcolon t_\iota \to[\calR] t_{\iota+1}$. The length $\alpha$ of such a sequence can be an arbitrary ordinal. For example, the infinite reduction indicated in Section \[sec:introduction\] is the sequence $(\phi^\mathrm{f}_i\fcolon t^\mathrm{f}_i \to[\calR^\mathrm{f}] t^\mathrm{f}_{i+1})_{i<\omega}$, where $t^\mathrm{f}_i = 0 \cons \dots \cons s^{i-1}(0) \cons \mathit{from}(s^i(0))$ for all $i<\omega$ and $\calR^\mathrm{f}$ is the TRS consisting of the single rule $\mathit{from}(x) \to x \cons \mathit{from}(s(x))$. Metric Convergence {#sec:metric-convergence} ------------------ The above definition of reductions ensures that consecutive rewrite steps are “compatible”, i.e. the result term of the $\iota$-th step, viz. $t_{\iota+1}$, is the start term of the $(\iota+1)$-st step. However, this definition does not relate the start terms of steps at limit ordinal positions to the terms that preceded it. For example, we can extend the abovementioned reduction $(\phi^\mathrm{f}_i)_{i<\omega}$ of length $\omega$, to a reduction $(\phi^\mathrm{f}_i)_{i<\omega+1}$ of length $\omega +1$ using any reduction step $\phi^\mathrm{f}_\omega$, e.g. $\phi^\mathrm{f}_\omega\fcolon \mathit{from}(0) \to 0 \cons \mathit{from}(s(0))$. In our informal notation this reduction $(\phi^\mathrm{f}_i)_{i<\omega+1}$ reads as follows: $$\mathit{from}(0) \to 0 \cons \mathit{from}(s(0)) \to 0 \cons s(0) \cons \mathit{from}(s(s(0))) \to \quad \dots\quad \mathit{from}(0) \to 0 \cons \mathit{from}(s(0))$$ Intuitively, this does not make sense since the sequence of terms that precedes the last step intuitively converge to the term $0 \cons s(0) \cons s(s(0)) \cons \dots$, but not $\mathit{from}(0)$. In infinitary term rewriting such reductions are ruled out by a notion of convergence and a notion of continuity that follows from it. Typically, this notion of convergence is derived from a metric $\dd$ on the set of (finite and infinite) terms $\iterms$: $\dd(s,t) = 0$ if $s = t$, and $\dd(s,t) = 2^{-d}$ otherwise, where $d$ is the minimal depth at which $s$ and $t$ differ. Using this metric, we may also construct the set of (finite and infinite) terms $\iterms$ by *metric completion* of the metric space $(\terms,\dd)$ of finite terms. The mode of convergence in the metric space $(\iterms,\dd)$ is the basis for the notion of *weak $\mrs$-convergence* of reductions: a reduction $S = (\phi_\iota\fcolon t_\iota \to[\calR] t_{\iota+1})_{\iota<\alpha}$ is *weakly $\mrs$-continuous* if $\lim_{\iota\limto\lambda} t_\iota = t_\lambda$ for all limit ordinals $\lambda < \alpha$; it *weakly $\mrs$-converges* to a term $t$, denoted $S\fcolon t_0 \wmato[\calR] t$, if it is weakly $\mrs$-continuous and $\lim_{\iota\limto\wsuc\alpha} t_\iota = t$, where $\wsuc\alpha$ is the length of the underlying sequence of terms $(t_\iota)_{\iota<\wsuc\alpha}$. For example, the reduction $(\phi^\mathrm{f}_i)_{i<\omega}$ weakly $\mrs$-converges to the term $0 \cons s(0) \cons s(s(0)) \cons \dots$; but the sequence $(\phi^\mathrm{f}_i)_{i<\omega+1}$ does not weakly $\mrs$-converge, it is not even weakly $\mrs$-continuous as $\lim_{\iota\limto\omega}t^\mathrm{f}_\iota$ is not $\mathit{from}(0)$. Weak $\mrs$-convergence is quite a general notion of convergence. For example, given a rewrite rule $a \to a$, we may derive the reduction $a \to a \to \dots$, which weakly $\mrs$-converges to $a$ even though the rule $a \to a$ is applied again and again at the same position. This generality causes many desired properties to break, such as unique normal form properties and compression [@kennaway95ic]. That is why Kennaway et al. [@kennaway95ic] introduced *strong $\mrs$-convergence*, which in addition requires that the depth at which rewrite steps take place tends to infinity as one approaches a limit ordinal: Let $S = (\phi_\iota\fcolon t_\iota \to[\pi_\iota] t_{\iota+1})_{\iota<\alpha}$ be a reduction, where each $\pi_\iota$ indicates the position at which the step $\phi_\iota$ takes place and $\len{\pi_\iota}$ denotes the length of the position $\pi_\iota$. The reduction $S$ is said to be *strongly $\mrs$-continuous* (resp. *strongly $\mrs$-converge* to $t$, denoted $S\fcolon t_0 \mato t$) if it is weakly $\mrs$-continuous (resp. weakly $\mrs$-converges to $t$) and if $(\len{\pi_\iota})_{\iota<\lambda}$ tends to infinity for all limit ordinals $\lambda < \alpha$ (resp. $\lambda \le \alpha$). For example, the reduction $(\phi^\mathrm{f}_i)_{i<\omega}$ also strongly $\mrs$-converges to the term $0 \cons s(0) \cons s(s(0)) \cons \dots$. On the other hand, the reduction $a \to a \to \dots$ indicated above weakly $\mrs$-converges to $a$, but it does not strongly $\mrs$-converge to $a$. Partial Order Convergence {#sec:part-order-conv} ------------------------- Alternatively to the metric approach illustrated in Section \[sec:metric-convergence\], convergence can also be formalised using a partial order $\lebot$ on terms. The idea to use this partial order for infinitary rewriting goes back to Corradini [@corradini93tapsoft]. The signature $\Sigma$ is extended to the signature $\Sigma_\bot$ by adding a fresh constant symbol $\bot$. When dealing with terms in $\ipterms$, we call terms that do not contain the symbol $\bot$, i.e. terms that are contained in $\iterms$, *total*. We define $s \lebot t$ iff $s$ can be obtained from $t$ by replacing some subterm occurrences in $t$ by $\bot$. Interpreting the term $\bot$ as denoting “undefined”, $\lebot$ can be read as “is less defined than”. The pair $(\ipterms,\lebot)$ is known to form a *complete semilattice* [@goguen77jacm], i.e. it has a least element $\bot$, each directed set $D$ in $(\ipterms,\lebot)$ has a *least upper bound* (*lub*) $\Lub D$, and every *non-empty* set $B$ in $(\ipterms,\lebot)$ has *greatest lower bound* (*glb*) $\Glb B$. In particular, this means that for any sequence $(t_\iota)_{\iota<\alpha}$ in $(\ipterms,\lebot)$, its *limit inferior*, defined by $\liminf_{\iota \limto \alpha}t_\iota = \Lub_{\beta<\alpha} \left(\Glb_{\beta \le \iota < \alpha} t_\iota\right)$, exists. In the same way that the limit in the metric space gives rise to weak $\mrs$-continuity/-convergence, the limit inferior gives rise to *weak $\prs$-continuity* and *weak $\prs$-convergence*; simply replace $\lim$ by $\liminf$. We write $S\fcolon t_0 \wpato t$ if a reduction $S$ starting with term $t_0$ weakly $\prs$-converges to $t$. The defining difference between the two approaches is that $\prs$-continuous reductions always $\prs$-converge. The reason for that lies in the complete semilattice structure of $(\ipterms,\lebot)$, which guarantees that the limit inferior always exists (in contrast to the limit in a metric space). The definition of the strong variant of $\prs$-convergence is a bit different from the one of $\mrs$-convergence, but it follows the same idea: a reduction $(\phi_i\fcolon t_i\to[\pi_i] t_{i+1})_{i<\omega}$ weakly $\mrs$-converges iff the minimal depth $d_i$ at which two consecutive terms $t_i, t_{i+1}$ differ tends to infinity. The strong variant of $\mrs$-convergence is a conservative approximation of this condition; it requires $\len{\pi_i}$ to tend to infinity. This approximation is conservative since $\len{\pi_i} \le d_i$; differences between consecutive terms can only occur below the position at which a rewrite rule was applied. In the partial order approach we can make this approximation more precise since we have the whole term structure at our disposal instead of only the measure provided by the metric $\dd$. In the case of $\mrs$-convergence, we replaced the actual depth of a minimal difference $d_i$ with its conservative under-approximation $\len{\pi_i}$. For $\prs$-convergence, we replace the glb $t_i \glb t_{i+1}$, which intuitively represents the common information shared by $t_i$ and $t_{i+1}$, with the conservative under-approximation $\substAtPos{t_i}{\pi_i}{\bot}$, which replaces the redex at position $\pi_i$ in $t_i$ with $\bot$. This term $\substAtPos{t_i}{\pi_i}{\bot}$ – called the *reduction context* of the step $\phi_i\fcolon t_i \to t_{i+1}$ – is a lower bound of $t_i$ and $t_{i+1}$ w.r.t. $\lebot$ and is, thus, also smaller than $t_i \glb t_{i+1}$. The definition of strong $\prs$-convergence is obtained from the definition of weak $\prs$-convergence by replacing $\liminf_{\iota\limto\lambda} t_\iota$ with $\liminf_{\iota\limto\lambda} \substAtPos{t_\iota}{\pi_\iota}{\bot}$. A reduction $S = (\phi_\iota\fcolon t_\iota \to[\pi_\iota] t_{\iota+1})_{\iota<\alpha}$ is called *strongly $\prs$-continuous* if $\liminf_{\iota\limto\lambda} \substAtPos{t_i}{\pi_i}{\bot} = t_\lambda$ for all limit ordinals $\lambda < \alpha$; it *strongly $\prs$-converges* to $t$, denoted $S\fcolon t_0 \pato t$, if it is strongly $\prs$-continuous and either $\liminf_{\iota\limto\alpha} \substAtPos{t_i}{\pi_i}{\bot} = t$ in case $\alpha$ is a limit ordinal, or $t = t_{\alpha+1}$ otherwise. The previously mentioned reduction $(\phi^\mathrm{f}_i)_{i<\omega}$ both strongly and weakly $\prs$-converges to the infinite term $0 \cons s(0) \cons s(s(0)) \cons \dots$ – like in the metric approach. However, while the reduction $a \to a \to \dots$ does not strongly $\mrs$-converge, it strongly $\prs$-converges to the term $\bot$. The partial order approach has some advantages over the metric approach. As explained above, every $\prs$-continuous reduction is also $\prs$-convergent. Moreover, strong $\prs$-convergence has some properties such as infinitary normalisation and infinitary confluence of orthogonal systems [@bahr10rta2] that are not enjoyed by strong $\mrs$-convergence. Interestingly, however, the partial order-based notions of convergence are merely conservative extensions of the metric-based ones: \[thr:strongExt\] For every reduction $S$ in a TRS, the following equivalences hold: $S\fcolon s \wmato t$ iff $S\fcolon s \wpato t$ in $\iterms$.\[item:strongExtI\] $S\fcolon s \mato t$ iff $S\fcolon s \pato t$ in $\iterms$.\[item:strongExtII\] The phrase “in $\iterms$” means that all terms in $S$ are total (including $t$). That is, if restricted to total terms, $\mrs$- and $\prs$-convergence coincide. Graphs and Term Graphs {#sec:graphs-term-graphs} ====================== In this section, we present our notion of term graphs and generalise the metric $\dd$ and the partial order $\lebot$ from terms to term graphs. Our notion of graphs and term graphs is largely taken from Barendregt et al. [@barendregt87parle]. \[def:graph\] A *graph* over signature $\Sigma$ is a triple $g = (N,\glab,\gsuc)$ consisting of a set $N$ (of *nodes*), a *labelling function* $\glab\fcolon N \funto \Sigma$, and a *successor function* $\gsuc\fcolon N \funto N^*$ such that $\len{\gsuc(n)} = \srank{\glab(n)}$ for each node $n\in N$, i.e. a node labelled with a $k$-ary symbol has precisely $k$ successors. If $\gsuc(n) = \seq{n_0,\dots,n_{k-1}}$, then we write $\gsuc_{i}(n)$ for $n_i$. The successor function $\gsuc$ defines, for each node $n$, directed edges from $n$ to $\gsuc_i(n)$. A path from a node $m$ to a node $n$ is a finite sequence $\seq{e_0,\dots,e_l}$ of numbers such that $n=\gsuc_{e_l}(\dots \gsuc_{e_0}(m))$, i.e. $n$ is reached from $m$ by taking the $e_0$-th edge, then the $e_1$-th edge etc. \[def:tgraph\] A *term graph* $g$ over $\Sigma$ is a tuple $(N,\glab,\gsuc,r)$ consisting of an *underlying* graph $(N,\glab,\gsuc)$ over $\Sigma$ whose nodes are all reachable from the *root node* $r\in N$. The class of all term graphs over $\Sigma$ is denoted $\itgraphs$. A *position* of $n \in N$ in $g$ is a path in the underlying graph of $g$ from $r$ to $n$. The set of all positions of $n$ in $g$ is denoted $\nodePos{g}{n}$. The *depth* of $n$ in $g$, denoted $\depth{g}{n}$, is the minimum of the lengths of the positions of $n$ in $g$, i.e. $\depth{g}{n} = \min \setcom{\len{\pi}}{\pi \in \nodePos{g}{n}}$. The term graph $g$ is called a *term tree* if each node in $g$ has exactly one position. We use the notation $N^{g}$, $\glab^{g}$, $\gsuc^{g}$ and $r^{g}$ to refer to the respective components $N$,$\glab$, $\gsuc$ and $r$ of $g$. Given a graph or a term graph $h$ and a node $n$ in $h$, we write $\subgraph{h}{n}$ to denote the sub-term graph of $h$ rooted in $n$. The notion of homomorphisms is crucial for dealing with term graphs. For greater flexibility, we will parametrise this notion by a set of constant symbols $\Delta$ for which the homomorphism condition is suspended. This will allow us to deal with variables and partiality appropriately. \[def:D-hom\] Let $\Sigma$ be a signature, $\Delta\subseteq \Sigma^{(0)}$, and $g,h \in \itgraphs$. A *$\Delta$-homomorphism* $\phi$ from $g$ to $h$, denoted $\phi\fcolon g \homto_\Delta h$, is a function $\phi\fcolon N^g \funto N^h$ with $\phi(r^g) = r^h$ that satisfies the following equations for all for all $n \in N^g$ with $\glab^g(n) \nin \Delta$: $$\begin{aligned} \glab^g(n) &= \glab^h(\phi(n)) \tag{labelling}\\ \phi(\gsuc^g_i(n)) &= \gsuc^h_i(\phi(n)) \quad \text{ for all } 0 \le i < \srank{\glab^g(n)} \tag{successor} \end{aligned}$$ Note that, for $\Delta = \emptyset$, we get the usual notion of homomorphisms on term graphs (e.g. Barendsen [@barendsen03book]) and from that the notion of isomorphisms. The nodes labelled with symbols in $\Delta$ can be thought of as holes in the term graphs that can be filled with other term graphs. We do not want to distinguish between isomorphic term graphs. Therefore, we use a well-known trick [@plump99hggcbgt] to obtain canonical representatives of isomorphism classes of term graphs. \[def:canTgraph\] A term graph $g$ is called *canonical* if $n = \nodePos{g}{n}$ holds for each $n \in N^g$. That is, each node is the set of its positions in the term graph. The set of all (finite) canonical term graphs over $\Sigma$ is denoted $\ictgraphs$ (resp. $\ctgraphs$). For each term graph $h \in \ictgraphs$, its *canonical representative* $\canon{h}$ is obtained from $h$ by replacing each node $n$ in $h$ by $\nodePos{h}{n}$. This construction indeed yields a canonical representation of isomorphism classes. More precisely: $g \isom \canon g$ for all $g\in\itgraphs$, and $g \isom h$ iff $\canon g = \canon h$ for all $g,h \in \itgraphs$. We consider the set of terms $\iterms$ as the subset of canonical term trees of $\ictgraphs$. With this correspondence in mind, we can define the *unravelling* of a term graph $g$ as the unique term $\unrav g$ such that there is a homomorphism $\phi\fcolon \unrav g \homto g$. For example, $g_0$ from Figure \[fig:convWeird\] is the unravelling of $g_1$, and $h_0$ and $g_\omega$ from Figure \[fig:fixedPointComb\] both unravel to the infinite term $@(f,@(f,\dots))$. Term graphs that unravel to the same term are called *bisimilar*. A Simple Partial Order on Term Graphs {#sec:simple-partial-order} ------------------------------------- In this section, we want to establish a partial order suitable for formalising convergence of sequences of canonical term graphs similarly to weak $\prs$-convergence on terms. Weak $\prs$-convergence on term rewriting systems is based on the partial order $\lebot$ on $\ipterms$, which instantiates occurrences of $\bot$ from left to right, i.e. $s \lebot t$ iff $t$ is obtained by replacing occurrences of $\bot$ in $s$ by arbitrary terms in $\ipterms$. Analogously, we consider the class of *partial term graphs* simply as term graphs over the signature $\Sigma_\bot = \Sigma \uplus \set{\bot}$. In order to generalise the partial order $\lebot$ to term graphs, we need to formalise the instantiation of occurrences of $\bot$ in term graphs. For this purpose, we shall use $\Delta$-homomorphisms with $\Delta=\set\bot$, or $\bot$-homomorphisms for short. A $\bot$-homomorphism $\phi\colon g \to_\bot h$ maps each node in $g$ to a node in $h$ while “preserving its structure”. Except for nodes labelled $\bot$ this also includes preserving the labelling. This exception to the homomorphism condition allows the $\bot$-homomorphism $\phi$ to instantiate each $\bot$-node in $g$ with an arbitrary node in $h$. Using $\bot$-homomorphisms, we arrive at the following definition for our simple partial order $\lebots$ on term graphs: For each $g,h \in \ipctgraphs$, define $g \lebots h$ iff there is some $\phi\fcolon g \homto_\bot h$. One can verify that $\lebots$ indeed generalises the partial order $\lebot$ on terms. Considering terms as canonical term trees, we obtain the following characterisation of $\lebot$ on terms $s,t\in \ipterms$: $$s \lebot t \iff \text{ there is a $\bot$-homomorphism } \phi\fcolon s \homto_\bot t.$$ The first important result for $\lebots$ is that the semilattice structure that we already had for $\lebot$ is preserved by this generalisation: \[thr:complSemilattice\] The partially ordered set $(\ipctgraphs,\lebots)$ forms a complete semilattice. For terms, we already know that the set of (potentially infinite) terms can be constructed by forming the *ideal completion* of the partially ordered set $(\pterms,\lebot)$ of finite terms [@berry77popl]. More precisely, the ideal completion of $(\pterms,\lebot)$ is order isomorphic to $(\ipterms,\lebot)$. An analogous result can be shown for term graphs: \[thr:idealCompletion\] The ideal completion of $(\pctgraphs, \lebots)$ is order isomorphic to $(\ipctgraphs,\lebots)$. A Simple Metric on Term Graphs {#sec:simple-metric-term} ------------------------------ Next, we shall generalise the metric $\dd$ from terms to term graphs. To achieve this, we need to formalise what it means for two term graphs to coincide up to a certain depth, so that we can reformulate the definition of the metric $\dd$ for term graphs. To this end, we follow the same idea that the original definition of $\dd$ on terms from Arnold and Nivat [@arnold80fi] was based on. In particular, we introduce a truncation construction that cuts off nodes below a certain depth: \[def:trunca\] Let $g \in \iptgraphs$ and $d \le \omega$. The *simple truncation* $\truncs{g}{d}$ of $g$ at $d$ is the term graph defined as follows: $$\begin{aligned} N^{\truncs{g}{d}} &= \setcom{n \in N^g}{\depth{g}{n} \le d} & r^{\truncs{g}{d}} &= r^g \\ \glab^{\truncs{g}{d}}(n) &= \begin{cases} \glab^g(n) &\text{if } \depth{g}{n} < d \\ \bot &\text{if } \depth{g}{n} = d \end{cases} & \gsuc^{\truncs{g}{d}}(n) &= \begin{cases} \gsuc^g(n) &\text{ if }\depth{g}{n} < d\\ \emptyseq &\text{ if }\depth{g}{n} = d \end{cases} \end{aligned}$$ The definition of the simple metric $\dds$ follows straightforwardly: The *simple distance* $\dds\fcolon \ictgraphs \times \ictgraphs \to \realsnn$ is defined as follows: $$\begin{gathered} \dds(g,h) = \begin{cases} 0&\text{if } g = h\\ 2^{-d}&\text{if } g \neq h \text{ and } d =\max\setcom{e<\omega}{\truncs{g}{e}\isom\truncs{h}{e}} \end{cases} \end{gathered}$$ Again, we can verify that $\dds$ generalises $\dd$. In particular, we can show that our notion of truncation coincides with that of Arnold and Nivat [@arnold80fi] if restricted to terms. As desired, this generalisation retains the complete ultrametric space structure: \[thr:smetricComplete\] The pair $(\ictgraphs,\dds)$ forms a complete ultrametric space. The metric space analogue to ideal completion is metric completion. On terms, we already know that we can construct the set of (potentially infinite) terms $\iterms$ by metric completion of the metric space $(\terms,\dd)$ of finite terms [@barr93tcs]. More precisely, the metric completion of $(\terms,\dd)$ is the metric space $(\iterms,\dd)$. This property generalises to term graphs as well: \[thr:metricCompletion\] The metric completion of $(\ctgraphs,\dds)$ is the metric space $(\ictgraphs,\dds)$. (r1) [$f$]{} child[ node (n1) [$c$]{} ]{} child[ node (n2) [$c$]{} ]{}; (r2) [$f$]{} child [ node (n2) [$c$]{} edge from parent\[transparent\] ]{}; (r2) edge \[bend right=25\] (n2) edge \[bend left=25\] (n2); (r1) – (r2); (r3) [$f$]{} child[ node (n1) [$c$]{} ]{} child[ node (n2) [$c$]{} ]{}; (r2) – (r3); (r4) [$f$]{} child [ node (n2) [$c$]{} edge from parent\[transparent\] ]{}; (r4) edge \[bend right=25\] (n2) edge \[bend left=25\] (n2); (r3) – (r4); (r5) [$f$]{} child[ node (n1) [$c$]{} ]{} child[ node (n2) [$c$]{} ]{}; (r4) – (r5); ; ; ; ; ; Infinitary Term Graph Rewriting {#sec:infin-term-graph} =============================== In this paper, we adopt the term graph rewriting framework of Barendregt et al. [@barendregt87parle]. In order to represent placeholders in rewrite rules, we use variables – in a manner much similar to term rewrite rules. To this end, we consider a signature $\Sigma_\calV = \Sigma\uplus\calV$ that extends the signature $\Sigma$ with a set $\calV$ of nullary variable symbols. Given a signature $\Sigma$, a *term graph rule* $\rho$ over $\Sigma$ is a triple $(g,l,r)$ where $g$ is a graph over $\Sigma_\calV$ and $l,r \in N^g$ such that all nodes in $g$ are reachable from $l$ or $r$. We write $\lhs\rho$ resp. $\rhs\rho$ to denote the left- resp. right-hand side of $\rho$, i.e. the term graph $\subgraph{g}{l}$ resp. $\subgraph{g}{r}$. Additionally, we require that for each variable $v\in\calV$ there is at most one node $n$ in $g$ labelled $v$, and we have that $n \neq l$ and that $n$ is reachable from $l$ in $g$. A *term graph rewriting system (GRS)* $\calR$ is a pair $(\Sigma,R)$ with $\Sigma$ a signature and $R$ a set of term graph rules over $\Sigma$. The notion of unravelling straightforwardly extends to term graph rules: the *unravelling* of a term graph rule $\rho$, denoted $\unrav{\rho}$, is the term rule $\unrav{\rho_l} \to \unrav{\rho_r}$. The unravelling of a GRS $\calR=(\Sigma,R)$, denoted $\unrav{\calR}$, is the TRS $(\Sigma,\setcom{\unrav{\rho}}{\rho\in R})$. \[ex:fixedPointCombRules\] Figure \[fig:fixedPointCombA\] shows two term graph rules which both unravel to the term rule $\rho\fcolon @(Y, x) \to @(x,@(Y,x))$ that defines the fixed point combinator $Y$. Note that sharing of nodes is used both to refer to variables in the left-hand side from the right-hand side and in order to simulate duplication. Without going into all details of the construction, we describe the application of a rewrite rule $\rho$ with root nodes $l$ and $r$ to a term graph $g$ in four steps: at first a suitable sub-term graph of $g$ rooted in some node $n$ of $g$ is *matched* against the left-hand side of $\rho$. This matching amounts to finding a $\calV$-homomorphism $\phi$ from the left-hand side $\lhs\rho$ to $\subgraph{g}{n}$, the *redex*. The $\calV$-homomorphism $\phi$ allows us to instantiate variables in the rule with sub-term graphs of the redex. In the second step, nodes and edges in $\rho$ that are not in $\lhs\rho$ are copied into $g$, such that each edge pointing to a node $m$ in $\lhs\rho$ is redirected to $\phi(m)$. In the next step, all edges pointing to the root $n$ of the redex are redirected to the root $n'$ of the *contractum*, which is either $r$ or $\phi(r)$, depending on whether $r$ has been copied into $g$ or not (because it is reachable from $l$ in $\rho$). Finally, all nodes not reachable from the root of (the now modified version of) $g$ are removed. With $h$ the result of the above construction, we obtain a *pre-reduction step* $\psi\fcolon g \preto[n] h$ from $g$ to $h$. The definition of term graph rewriting in the form of pre-reduction steps is very operational. While this style is beneficial for implementing a rewriting system, it is problematic for reasoning on term graphs modulo isomorphism, which is necessary for introducing notions of convergence. However, one can easily see that the construction of the result term graph of a pre-reduction step is invariant under isomorphism, which justifies the following definition of reduction steps: Let $\calR = (\Sigma,R)$ be GRS, $\rho \in R$ and $g,h \in \ictgraphs$ with $n \in N^g$ and $m\in N^h$. A tuple $\phi = (g,n,h)$ is called a *reduction step*, written $\phi\fcolon g \to[n] h$, if there is a pre-reduction step $\phi'\fcolon g' \preto[n'] h'$ with $\canon{g'} = g$, $\canon{h'} = h$, and $n = \nodePos{g'}{n'}$. We also write $\phi\fcolon g \to[\calR] h$ to indicate $\calR$. In other words, a reduction step is a canonicalised pre-reduction step. Figure \[fig:fixedPointCombB\] and Figure \[fig:fixedPointCombC\] illustrate some (pre-)reduction steps induced by the rules $\rho_1$ respectively $\rho_2$ shown in Figure \[fig:fixedPointCombA\]. Weak Convergence {#sec:weak-convergence} ---------------- In analogy to infinitary term rewriting, we employ the partial order $\lebots$ and the metric $\dds$ for the purpose of defining convergence of transfinite term graph reductions. Let $\calR = (\Sigma,R)$ be a GRS. (i) Let $S = (g_\iota \to_\calR g_{\iota+1})_{\iota < \alpha}$ be a reduction in $\calR$. $S$ is *weakly $\mrs$-continuous* in $\calR$ if $\lim_{\iota\limto\lambda} g_\iota = g_\lambda$ for each limit ordinal $\lambda < \alpha$. $S$ *weakly $\mrs$-converges* to $g \in \ictgraphs$ in $\calR$, written $S\fcolon g_0 \wmato[\calR] g$, if it is weakly $\mrs$-continuous and $\lim_{\iota\limto\wsuc\alpha} g_\iota = g$. (ii) Let $\calR_\bot$ be the GRS $(\Sigma_\bot, R)$ over the extended signature $\Sigma_\bot$ and $S = (g_\iota \to[\calR_\bot] g_{\iota+1})_{\iota < \alpha}$ a reduction in $\calR_\bot$. $S$ is *weakly $\prs$-continuous* in $\calR$ if $\liminf_{\iota<\lambda} g_i = g_\lambda$ for each limit ordinal $\lambda < \alpha$. $S$ *weakly $\prs$-converges* to $g\in\ipctgraphs$ in $\calR$, written $S\fcolon g_0 \wpato[\calR] g$, if it is weakly $\prs$-continuous and $\liminf_{\iota<\wsuc\alpha} g_i = g$. \[ex:fixedPointCombWeak\] Figure \[fig:fixedPointCombC\] illustrates an infinite reduction derived from the rule $\rho_1$ in Figure \[fig:fixedPointCombA\]. Since $\truncs{g_i}{(i+1)} \isom \truncs{g_\omega}{(i+1)}$ for all $i < \omega$, we have that $\lim_{i \limto \omega} g_i = g_\omega$, which means that the reduction weakly $\mrs$-converges to the term graph $g_\omega$. Moreover, since each node in $g_\omega$ eventually appears in a term graph in $(g_i)_{i<\omega}$ and remains stable afterwards, we have $\liminf_{i\limto\omega}g_\iota = g_\omega$. Consequently, the reduction also weakly $\prs$-converges to $g_\omega$. Recall that weak $\prs$-convergence for TRSs is a conservative extension of weak $\mrs$-convergence (cf.Theorem \[thr:strongExt\]). The key property that makes this possible is that for each sequence $(t_\iota)_{\iota<\alpha}$ in $\iterms$, we have that $\lim_{\iota\limto\alpha} t_\iota = \liminf_{\iota\limto\alpha} t_\iota$ whenever $(t_\iota)_{\iota<\alpha}$ converges, or $\liminf_{\iota\limto\alpha} t_\iota$ is a total term. Sadly, this is not the case for the metric space and the partial order on term graphs: the sequence of term graphs depicted in Figure \[fig:convWeird\] has a total term graph as its limit inferior, viz. $g_\omega$, although it does not converge in the metric space. In fact, since the sequence in Figure \[fig:convWeird\] alternates between two distinct term graphs, it does not converge in any Hausdorff space, i.e. in particular, it does not converge in any metric space. This example shows that we cannot hope to generalise the compatibility property that we have for terms: even if a sequence of total term graphs has a total term graph as its limit inferior, it might not converge. However, the converse direction of the correspondence does hold true: \[thr:limLiminf\] If $(g_\iota)_{\iota<\alpha}$ converges, then $\lim_{\iota\limto\alpha} g_\iota = \liminf_{\iota\limto\alpha} g_\iota$. From this property, we obtain the following relation between weak $\mrs$- and $\prs$-convergence: Let $S$ be a reduction in a GRS $\calR$. $\text{If}\quad S\fcolon g \wmato[\calR] h \qquad \text{then} \qquad S\fcolon g \wpato[\calR] h.$ As indicated above, weak $\mrs$-convergence is not the total fragment of weak $\prs$-convergence as it is the case for TRSs, i.e. the converse of the above implication does not hold in general: \[ex:rulesWeird\] There is a GRS that yields the reduction shown in Figure \[fig:convWeird\], which weakly $\prs$-converges to $g_\omega$ but is not weakly $\mrs$-convergent. This reduction can be produced by alternately applying the rules $\rho_1,\rho_2$, where the left hand side of both rules and the right-hand side of $\rho_1$ is $g_0$, and the right-hand side of $\rho_2$ is $g_1$. Strong Convergence {#sec:strong-convergence} ------------------ The idea of strong convergence is to conservatively approximate the convergence behaviour somewhat independently from the actual rewrite rules that are applied. Strong $\mrs$-convergence in TRSs requires that the depth of the redexes tends to infinity thereby assuming that anything at the depth of the redex or below is potentially affected by a reduction step. Strong $\prs$-convergence, on the other hand, uses a better approximation that only assumes that the redex is affected by a reduction step – not however other subterms at the same depth. To this end strong $\prs$-convergence uses a notion of reduction contexts – essentially the term minus the redex – for the formation of limits. The following definition provides the construction for the notion of reduction contexts that we shall use for term graph rewriting: \[def:truncl\] Let $g \in \iptgraphs$ and $n \in N^g$. The *local truncation* of $g$ at $n$, denoted $\truncl{g}{n}$, is obtained from $g$ by labelling $n$ with $\bot$ and removing all outgoing edges from $n$ as well as all nodes that thus become unreachable from the root. \[prop:stepContext\] Given a reduction step $g \to[n] h$, we have $\truncl{g}{n} \lebots g, h$. This means that the local truncation at the root of the redex is preserved by reduction steps and is therefore an adequate notion of reduction context for strong $\prs$-convergence [@bahr10rta]. Using this construction we can define strong $\prs$-convergence on term graphs analogously to strong $\prs$-convergence on terms. For strong $\mrs$-convergence, we simply take the same notion of depth that we already used for the definition of the simple truncation $\truncs{g}{d}$ and thus the simple metric $\dds$. Let $\calR = (\Sigma,R)$ be a GRS. (i) The *reduction context* $c$ of a graph reduction step $\phi\fcolon g \to[n] h$ is the term graph $\canon{\truncl{g}{n}}$. We write $\phi\fcolon g \to[c] h$ to indicate the reduction context of a graph reduction step. (ii) Let $S = (g_\iota \to[n_\iota] g_{\iota+1})_{\iota<\alpha}$ be a reduction in $\calR$. $S$ is *strongly $\mrs$-continuous* in $\calR$ if $\lim_{\iota \limto \lambda} g_\iota = g_\lambda$ and $(\depth{g_\iota}{n_\iota})_{\iota<\lambda}$ tends to infinity for each limit ordinal $\lambda < \alpha$. $S$ *strongly $\mrs$-converges* to $g$ in $\calR$, denoted $S\fcolon g_0 \mato[\calR] g$, if it is strongly $\mrs$-continuous and either $S$ is closed with $g = g_\alpha$ or $S$ is open with $g = \lim_{\iota \limto \alpha} g_\iota$ and $(\depth{g_\iota}{n_\iota})_{\iota<\alpha}$ tending to infinity. (iii) Let $S = (g_\iota \to[c_\iota] g_{\iota+1})_{\iota<\alpha}$ be a reduction in $\calR_\bot=(\Sigma_\bot,R)$. $S$ is *strongly $\prs$-continuous* in $\calR$ if $\liminf_{\iota \limto \lambda} c_\iota = g_\lambda$ for each limit ordinal $\lambda < \alpha$. $S$ *strongly $\prs$-converges* to $g$ in $\calR$, denoted $S\fcolon g_0 \pato[\calR] g$, if it is strongly $\prs$-continuous and either $S$ is closed with $g = g_\alpha$ or $S$ is open with $g = \liminf_{\iota \limto \alpha} c_\iota$. As explained in Example \[ex:fixedPointCombWeak\], the reduction in Figure \[fig:fixedPointCombC\] both weakly $\mrs$- and $\prs$-converges to $g_\omega$. Because contraction takes place at increasingly large depth, the reduction also strongly $\mrs$-converges to $g_\omega$. Moreover, since each node in $g_\omega$ eventually appears also in the sequence of reduction contexts $(c_i)_{i<\omega}$ of the reduction and remains stable afterwards, we have that $\liminf_{i\limto\omega}c_i = g_\omega$. Consequently, the reduction also strongly $\prs$-converges to $g_\omega$. Remarkably, one of the advantages of the strong variant of convergence is that we regain the correspondence between $\mrs$- and $\prs$-convergence that we know from infinitary term rewriting: \[thr:graphExt\] Let $\calR$ be a GRS and $S$ a reduction in $\calR_\bot$. We then have that $S\fcolon g \mato[\calR] h$ $S\fcolon g \pato[\calR] h$ in $\ictgraphs$. In particular, the GRS given in Example \[ex:rulesWeird\] that induces the reduction depicted in Figure \[fig:convWeird\] does not provide a counterexample for the “if” direction of the above equivalence – in contrast to weak convergence. The reduction in Figure \[fig:convWeird\] does not strongly $\mrs$-converge but it does strongly $\prs$-converge to the term graph $\bot$, which is in accordance with Theorem \[thr:graphExt\] above. Soundness and Completeness {#sec:soundness} -------------------------- In order to assess the value of the modes of convergence on term graphs that we introduced in this paper, we need to compare them to the well-established counterparts on terms. Ideally, we would like to see a strong connection between converging reductions in a GRS $\calR$ and converging reductions in the TRS $\unrav{\calR}$ in the form of soundness and completeness properties. For example, for $\mrs$-convergence we want to see that $g \wmato[\calR] h$ implies $\unrav g \wmato[\unrav{\calR}] \unrav h$ – i.e. soundness – and vice versa that $\unrav g \wmato[\unrav{\calR}] t$ implies $g \wmato[\calR] h$ with $\unrav h = t$ – i.e. completeness. Completeness is already an issue for finitary rewriting [@kennaway94toplas]: a single term graph redex may correspond to several term redexes due to sharing. Hence, contracting a term graph redex may correspond to several term rewriting steps, which may be performed independently. In the context of weak convergence, also soundness becomes an issue. The underlying reason for this issue is similar to the phenomenon explained above: a single term graph rewrite step may represent several term rewriting steps, i.e. $g \to[\calR] h$ implies $\unrav g \fto+[\unrav\calR]\unrav h$.[^2] When we have a converging term graph reduction $(\phi_\iota\fcolon g_\iota \to g_{\iota+1})_{\iota<\alpha}$, we know that the underlying sequence of term graphs $(g_\iota)_{\iota<\wsuc\alpha}$ converges. However, the corresponding term reduction does not necessarily produce the sequence $(\unrav{g_\iota})_{\iota<\wsuc\alpha}$ but may intersperse the sequence $(\unrav{g_\iota})_{\iota<\wsuc\alpha}$ with additional intermediate terms, which might change the convergence behaviour. While we cannot prove soundness for weak convergence due to the abovementioned problems, we can show that the underlying modes of convergence are sound in the sense that convergence is preserved under unravelling. \[thr:unravLim\] (i) $\lim_{\iota\limto\alpha} g_\iota = g$ implies $\lim_{\iota\limto\alpha} \unrav{g_\iota} = \unrav g$ for every sequence $(g_\iota)_{\iota<\alpha}$ in $(\ictgraphs,\dds)$. (ii) $\unrav{\liminf_{\iota\limto\alpha}g_\iota} = \liminf_{\iota\limto\alpha}\unrav{g_\iota}$ for every sequence $(g_\iota)_{\iota<\alpha}$ in $(\ipctgraphs,\lebots)$. Note that the above theorem in fact implies soundness of the modes of convergence on term graphs with the modes of convergence on terms since both $\dds$ and $\lebots$ specialise to $\dd$ respectively $\lebot$ if restricted to term trees. However, we can observe that strong convergence is more well-behaved than weak convergence. It is possible to prove soundness and completeness properties for strong $\prs$-convergence: \[thr:pConvSoundCompl\] Let $\calR$ be a left-finite GRS. (i) If $\calR$ is left-linear and $g \pato[\calR] h$, then $\unrav{g} \pato[\unrav{\calR}] \unrav{h}$. (ii) If $\calR$ is orthogonal and $\unrav{g} \pato[\unrav\calR] t$, then there are reductions $g \pato[\calR] h$ and $t \pato[\unrav\calR] \unrav{h}$. Note that the above completeness property is not the one that one would initially expect, namely $\unrav g \pato[\unrav{\calR}] t$ implies $g \pato[\calR] h$ with $\unrav h = t$. But this general completeness property is known to already fail for finitary term graph rewriting [@kennaway94toplas]. The soundness and completeness properties above have an important practical implication: GRSs that only differ in their sharing, i.e.they unravel to the same TRS, will produce the same results, i.e. the same normal forms up to bisimilarity. GRSs with more sharing may, however, reach a result with fewer steps. This can be observed in Figure \[fig:fixedPointComb\], which depicts two rules $\rho_1, \rho_2$ that unravel to the same term rule. Rule $\rho_1$ reaches $g_\omega$ in $\omega$ steps whereas $\rho_2$ reaches a term graph $h_0$, which is bisimilar to $g_\omega$, in one step. The situation for strong $\mrs$-convergence is not the same as for strong $\prs$-convergence. While we do have soundness under the same preconditions, i.e. $g \mato[\calR] h$ implies $\unrav{g} \mato[\unrav{\calR}] \unrav{h}$, the completeness property we have seen in Theorem \[thr:pConvSoundCompl\] fails. This behaviour was already recognised by Kennaway et al. [@kennaway94toplas]. Nevertheless, we can find a weaker form of completeness that is restricted to normalising reductions: Given an orthogonal, left-finite GRS $\calR$ that is normalising w.r.t. strongly $\mrs$-converging reductions, we find for each normalising reduction $\unrav{g} \mato[\unrav\calR] t$ a reduction $g \mato[\calR] h$ such that $t = \unrav{h}$. Concluding Remarks {#sec:concluding-remarks} ================== We have devised two independently defined but closely related infinitary calculi of term graph rewriting. This is not the first proposal for infinitary term graph rewriting calculi; in previous work [@bahr12lmcs] we presented a so-called *rigid* approach based on a metric and a partial order different from the structures presented here. There are several arguments why the simple approach presented in this paper is superior to the rigid approach. First of all it is simpler. The rigid metric and partial order have been carefully crafted in order to obtain a correspondence result in the style of Theorem \[thr:strongExt\] for weak convergence on term graphs. This correspondence result of the rigid approach is not fully matched by the simple approach that we presented here, but we do regain the full correspondence by moving to strong convergence. Secondly, the rigid approach is very restrictive, disallowing many reductions that are intuitively convergent. For example, in the rigid approach the reduction depicted in Figure \[fig:fixedPointCombC\], would not $\prs$-converge (weakly or strongly) to the term graph $g_\omega$ as intuitively expected but instead to the term graph obtained from $g_\omega$ by replacing $f$ with $\bot$. Moreover, this sequence would not $\mrs$-converge (weakly or strongly) at all. Lastly, as a consequence of the restrictive nature of the rigid approach, the completion constructions of the underlying metric and partial order do not yield the full set of term graphs – in contrast to our findings here in Theorem \[thr:idealCompletion\] and \[thr:metricCompletion\]. Unfortunately, we do not have solid soundness or completeness results for weak convergence apart from the preservation of convergence under unravelling and the metric/ideal completion construction of the set of term graphs. However, as we have shown, this shortcoming is again addressed by moving to strong convergence. [^1]: The full version of this paper will appear in Mathematical Structures in Computer Science [@bahr13mscs]. [^2]: If the term graph $g$ is cyclic, the corresponding term reduction may even be infinite.

--- abstract: 'We present a combined x-ray diffraction and infrared spectroscopy study on the phase behavior and molecular dynamics of n-hexadecanol in its bulk state and confined in an array of aligned nanochannels of 8 nm diameter in mesoporous silicon. Under confinement the transition temperatures between the liquid, the rotator R$_{II}$ and the crystalline C phase are lowered by approximately 20 K. While bulk n-hexadecanol exhibits at low temperatures a polycrystalline mixture of orthorhombic $\beta$- and monoclinic $\gamma$-forms, geometrical confinement favors the more simple $\beta$-form: only crystallites are formed, where the chain axis are parallel to the layer normal. However, the $\gamma$-form, in which the chain axis are tilted with respect to the layer normal, is entirely suppressed. The $\beta$-crystallites form bi-layers, that are not randomly orientated in the pores. The molecules are arranged with their long axis perpendicular to the long channel axis. With regard to the molecular dynamics, we were able to show that confinement does not affect the inner-molecular dynamics of the CH$_2$ scissor vibration and to evaluate the inter-molecular force constants in the C phase.' author: - | R. Berwanger$^1$, A. Henschel$^2$, K. Knorr$^2$, P. Huber$^2$, and R. Pelster$^1$\ Universit[ä]{}t des Saarlandes, $^1$FR 7.2 Experimentalphysik & $^2$FR 7.3 Technische Physik,\ 66041 Saarbr[ü]{}cken, Germany title: | Phase transitions and molecular dynamics of n-hexadecanol\ confined in silicon nanochannels --- Introduction ============ The physical properties of condensed matter spatially confined in pores or channels of a few nanometer in diameter can differ markedly from the behavior in the bulk state. In particular, phase transitions can be entirely suppressed or significantly altered in comparison to their bulk counterparts [@Gelb1999; @AlbaSim2006; @Christenson2001; @Knorr2008]. Also the dynamics of condensed matter confined in mesopores, most prominently in the vicinity of glass transitions [@Koppensteiner2008; @Scheidler2000; @Kremer1999; @Jackson1991; @Barut98; @Pelster99prb; @Daoukaki98prb; @Pissis98; @Schranz2007; @Frick2003], can be affected markedly. Intimately related to these changes in the phase transition phenomenology the architectural principles of molecular solids can substantially differ in the spatially confined state from the bulk state. This depends, however, sensitively on the complexity of the building blocks. For simple van-der-Waals systems, such as Ar and N$_2$, a remarkable robustness of the bulk structures has been found for the solid state under confinement [@Huber1998; @Wallacher2001; @Knorr2003]. By contrast, the structural properties of pore fillings built out of more complex building blocks, such as linear hydrocarbons [@Huber2006; @Henschel2007; @Montenegro2003; @Xie2008; @Valliulin2006] or liquid crystals [@Crawford1996; @Kityk2008] are very susceptible to confinement on the meso- and nanoscale. For example, a quenching of the lamellar ordering of molecular crystals of n-alkanes has been observed in tortuous silica mesopores of Vycor [@Huber2004]. However, in tubular channels of mesoporous silicon this building principle of hydrocarbon molecular crystals survives, albeit a peculiar texture has been observed for the pore confined solids [@Henschel2007]: The long axes of the molecules and thus the stacking direction of the lamellae are oriented perpendicular to the long axis of the pores. Here we present an experimental study on a medium-length, linear alcohol C$_{16}$H$_{33}$OH, a representative of the 1-alcohol series, imbibed in mesoporous silicon. We explore the phase behavior of the confined alcohol by a combination of x-ray diffraction and infrared spectroscopy measurements. As we shall demonstrate, we profit in those experiments both from the parallel alignment of the silicon channels and from the transparency of the silicon host in the infrared region. Experimental ============ The porous silicon samples used in this study were prepared by electrochemical etching of a heavily p-doped (100) silicon wafer [^1] with a current density of 13 $\frac{mA}{cm^2}$ in a solution composed of HF, ethanol and H$_{2}$O (1:3:1 per volume) [@Lehmann1991; @Zhang2000; @Cullis1997]. These conditions led to a parallel arrangement of non-interconnected channels oriented with their long axes along the $<$100$>$ crystallographic direction of silicon, which coincides with the normal of the wafer surface. After the porous layer had reached the desired thickness of 70 microns, the anodization current was increased by a factor of ten with the result that the porous layer was released from the bulk wafer underneath. Using nitrogen sorption isotherms at $T=77$ K, we determined a porosity of 60% and a mean channel diameter of 8 nm. The single crystalline character of the matrix was checked by x-ray diffraction. Transmission electron micrographs of channel cross sections indicate polygonal, rough channel perimeters rather than circular, smooth circumferences [@Gruener2008]. The matrix both for the infrared spectroscopy and the x-ray measurements were filled completely via capillary action (spontaneous imbibition) with liquefied C$_{16}$H$_{33}$OH [@Huber2007]. Bulk excess material at the surface was removed by paper tissues.\ Infrared spectra in a range of wavenumbers $\overline{\nu}$ from 4000 to 800 cm$^{-1}$ with a resolution of 1 cm$^{-1}$ were measured with a Fourier Transform Spectrometer (FTIR Perkin Elmer System 2000). This range corresponds to frequencies from $3\cdot10^{13}$ Hz to $1.2\cdot10^{14}$ Hz (wavelengths from 10 $\mu$m to 2.5 $\mu$m). For both the bulk material and the filled porous samples the same sample holder was used, i. e. a copper cell with two transparent KBr windows. In the confinement experiments the long channel axes were oriented parallel to the beam axis, i. e. perpendicular to the electric field vector. The sample holder was placed into a cryostat (a closed cycle refrigerator CTI cryogenics, Model 22) allowing us to vary the temperature from 50 to 340 K. The temperature was controlled with a LakeShore 340 temperature controller with a precision of $\pm 0.25$ K. All IR-spectra that we show in the following were measured during cooling [**(typical cooling rates were of the order of 0.5 K/min)**]{}. Heating scans show the same behavior except for the transition temperatures, which are some degrees higher (see below).\ For the x-ray measurements the sample was mounted on a frame in a sample cell consisting of a Peltier cooled base plate and a Be cap. The cell was filled with He gas for better thermal contact. The Be cap sits in a vacuum chamber, the outer jacket of which has Mylar windows allowing the passage of the x-rays over a wide range of scattering angles $\theta$ within the scattering plane (see Fig. \[realRaum1\]). But the set-up allowed practically no tilt with respect to the scattering plane. The temperature was controlled by a LakeShore 330 over an accessible range from 245 K up to 370 K. The measurements were carried out on a two-circle x-ray diffractometer with graphite monochromatized CuK$_{\rm \alpha}$ radiation emanating from a rotating anode. The porous sheet was mounted perpendicular to the scattering plane. The two angles that could be varied were the detector angle $2\theta$ and the rotation angle $\omega$ about the normal of the scattering plane. The samples were studied as a function of temperature by performing several $\Phi$-scans. In this paper we concentrate on radial $2\theta$-$\omega$-Scans in reflection geometry, i.e. along q$_{\rm p}$ with $\Phi$=0°, and in transmission geometry, i.e. along q$_{\rm s}$ with $\Phi$=90° (see Fig. \[realRaum1\]). ![\[realRaum1\] ](realraum5.eps) Structure of bulk n-hexadecanol {#sec:bulkstruct} ------------------------------- n-Hexadecanol, C$_{16}$H$_{33}$OH, is an almost rod-like molecule with a length of 22  and a width of 4 . The C-atoms of the backbone are in an all-trans-configuration so that they are located in a plane [@Huber2004]. At low temperatures n-alcohols form bi-layered crystals in two possible modifications: the so-called $\gamma$-form, i. e. a monoclinic structure as sketched in Fig. \[fig:bulkstructure\_cryst\] ($C_{2h}^{6}-A2/a$ [@Metivaud2005; @Abrahamsson1960]), or the so-called $\beta$-form, i. e. an orthorhombic structure as sketched in Fig. \[fig:confstructure\_cryst\] [@Tasumi1964]. In the $\gamma$-form, the molecules include an angle of $122$° with the layer plane. Within the layers, they are close-packed in a quasi-hexagonal 2D array, described by the rectangular in-plane lattice parameters $a$ and $b$ (according to Ref. [@Abrahamsson1960] $a=7.42$  and $b=4.93$  holds, so that $a/b=1.5$). There are two different alternating orientations for the C-C-plane of the backbone leading to a herringbone structure (see Fig. \[fig:bulkstructure\_cryst\]b). The $\beta$-form exhibits an identical orientational order of the backbone, but the molecules’ axes remain perpendicular to the layers as sketched in Fig. \[fig:confstructure\_cryst\] [@Tasumi1964]. In addition, gauche- and trans-conformation of the CO-bond alternate with molecules in this phase, while they are in an all-trans configuration in the $\gamma$-form. In general, the $\gamma$-form dominates at low temperatures for the even alcohols, while the $\beta$-form is more frequent in odd n-alcohols [@Ventola2002; @Tasumi1964]. For n-hexadecanol both the orthorhombic $\beta$-form [@Tasumi1964] and the monoclinic $\gamma$-form [@Metivaud2005; @Abrahamsson1960] are reported. Depending on the preparation conditions it is possible to obtain a polycristalline mixture of the monoclinic $\gamma$- and the orthorhombic $\beta$-form [@Ventola2002]. ![\[fig:bulkstructure\_cryst\] $\gamma$-form of the crystalline low temperature phase of bulk C$_{16}$H$_{33}$OH ($T \le 310$ K). The structure is monoclinic. The left sketch shows the orientation of the molecules with respect to the layer normal $n$, the right sketch the in-plane arrangement, i. e. a projection of the backbones into the a-b-plane. Compare with the $\beta$-form sketched in Fig. \[fig:confstructure\_cryst\]](kristallinmonoklin.eps) ![\[fig:confstructure\_cryst\] $\beta$-form of the crystalline low temperature phase of C$_{16}$H$_{33}$OH. In contrast to the $\gamma$-form (see Fig. \[fig:bulkstructure\_cryst\]a), the long chain axes are not tilted but parallel to the layer normal $n$, i. e. the structure is orthorhombic. Bulk C$_{16}$H$_{33}$OH can exhibit a polycrystalline mixture of $\gamma$- and $\beta$-form (see Sec. \[sec:bulkstruct\]). Confinement into nanopores leads to the $\beta$-form (see below, Sec. \[sec:confstruct\]).](kristallinortho.eps) Upon heating, the crystalline phase undergoes a transition into a so-called Rotator-(II)-phase $R_{II}$, which is schematically depicted in Fig. \[fig:bulkstructure\_rot\] [^2]. This phase has a hexagonal in-plane arrangement with the $c$-direction perpendicular to the cell base. The hexagonal arrangement can be indexed with an orthorhombic cell with a ratio of rectangular basal lattice parameters of $a/b=\sqrt{3}$  [@Sirota1996]. On a microscopic level the change in the center of mass lattice from the low-temperature crystalline phase to the rotator phase can be attributed to the onset of hindered rotations of the molecules about their long axes between six equivalent positions (the stars in Fig. \[fig:bulkstructure\_rot\]b). Further heating above 322 K leads to the liquid state [@Sirota1996]. ![Structure of bulk n-hexadecanol in the Rotator-(II)-phase (for $310 \le T \le 322$ K), a hexagonal arrangement. The right picture shows the perfect hexagonal lattice in the a-b-plane. Confined C$_{16}$H$_{33}$OH exhibits the same structure in its rotator phase, but in a different temperature range (see below, Table \[tab:MeltingPoints\]). \[fig:bulkstructure\_rot\]](rotatorortho.eps) Results ======= Structure of confined n-hexadecanol {#sec:confstruct} ----------------------------------- We have determined structures, phase sequences and transition temperatures of n-C$_{16}$H$_{33}$OH confined in mesoporous silicon by x-ray diffractometry. The upper panel in Fig. \[realRaum\] shows diffraction patterns along q$_{\rm p}$ at selected temperatures while cooling. The appearance of a broad Bragg peak at $2 \theta \simeq 21$° indicates solidifaction. Its position is compatible with the leading hexagonal in-plane reflection of the $R_{II}$ phase. Upon further cooling a second peak at $2 \theta \simeq 24$° shows up. This change in the diffraction pattern indicates an uniaxial deformation of the hexagonal lattice. Both reflections can be mapped on a 2D rectangular mesh characteristic of an uniaxially deformed hexagonal cell. The overall resulting pattern is, however, incompatible with the monoclinic structure of the low temperature bulk crystalline phase. Additionally to the q$_{\rm p}$-scans, we performed also scans for a variety of additional orientations of the scattering vector with regard to the long axis of the channels. These patterns differ markedly, which is indicative of a strong texture of the pore confined cystallized alcohol. It is no powder in the crystallographic sense. In particular, there are strong in-plane reflections and no layering reflections for scans along q$_{\rm p}$, while the q$_{\rm s}$-scans for the same sample show at least very weak reflections characteristic of a bi-layer stacking and only very weak leading in-plane reflections (see Fig. \[realRaum\]). An analysis of the width of the layering reflections yields a coherence length of 7($\pm1.5$)nm. As discussed in more detail in Refs. [@Henschel2007] and [@Henschel2008], the overall picture which emerges from these results can be summed up as follows: the alcohol molecules form orthorhombic structures with a bilayer-stacking direction along the $c$-direction. Within the bilayers (the a-b-plane), the molecules’ backbones are untilted with regard to the stacking direction and the backbones are orientationally either fully ordered (in a herringbone fashion) or partially ordered, as known from the R$_I$ phase of n-alkanes. The superlattice reflection characteristic of the full, herringbone type orientational ordering has been searched for and could weakly be detected at low temperatures. The degree of uniaxial deformation of the hexagonal center of mass cell, quantified by the deviation of the ratio $a/b$ from its value in the hexagonal phase ($\sqrt{3}$), also indicates a full orientational ordered state (see Table I, [@Abrahamsson1960]). Thus, the diffraction data are compatible with the bulk $\beta$ modification discussed above. This conclusion is also supported by an analysis of the infrared spectroscopy data sets presented below. More importantly, the peculiar dependency of the diffraction patterns on the orientation of the q-vector with regard to the silicon host indicate that the bi-layer stacking direction is perpendicular to the long axis of the channels and, consequently, that the long axis of the molecules is oriented perpendicular to the long axis of the channels (see Fig. \[Porenschnitt\]). At first glance, this finding may appear somewhat counter-intuitive. Albeit it can be understood as resulting from the crystallization process in a strongly anistropic, capillary-like confined liquid [@Henschel2007; @Steinhart2006]. It is a well established principle in single crystal growth that in narrow capillaries the fastest growing crystallization direction prevails over other directions and propagates along the long axes of capillaries [@Palibin1933]. For layered molecular crystals of rod-like building blocks this direction is an in-plane direction, which is perpendicular to the long axis of the rods. If this direction is aligned parallel to the silicon nanochannels due to the crystallization process, it dictates a perpendicular arrangement of the molecules’ long axes with regard to the long channel axis, in agreement with the diffraction results presented here. --------- -------- ------------ -------- cryst. R$_{II}$ cryst. a \[Å\] 7.42 8.35 7.33 b \[Å\] 4.93 4.82 5.04 a/b 1.51 $\sqrt{3}$ 1.45 d \[Å\] 8.91 9.64 8.90 --------- -------- ------------ -------- : \[tab:ab\] Lattice parameters a and b of bulk and confined C$_{16}$H$_{33}$OH and the diagonal $d=\sqrt{a^2+b^2}$ of the subcell (see Fig. \[fig:lattice\]). The confined data result from our x-ray measurements and the bulk data are taken from the literature [@Abrahamsson1960]. ![\[realRaum\] ](diffraktogramme.eps) The temperature dependent diffraction study allows us to gain additional information on the relative stability of the different nanochannel confined phases. In Table \[tab:MeltingPoints\] we display the phase transition temperatures of confined C$_{16}$H$_{33}$OH as inferred from the appearance or disappaerance of characteristic Bragg peaks. There is a hysteresis between heating and cooling for both the fluid-R$_{II}$- and the R$_{II}$-C-transition (8 K and 3 K, respectively). Compared to the bulk data (see also Tab. \[tab:MeltingPoints\]), the transition temperatures of pore confined C$_{16}$H$_{33}$OH are lowered. On cooling, the lowering is of the order of $\Delta T$= 18 K for the fluid-$R_{II}$-transition and $\Delta T$= 26 K for the $R_{II}$-C-transition. This observation is analogous to phase transitions shifts reported for other pore condensates [@Christenson2001; @AlbaSim2006]. Furthermore, the temperature range of the confined $R_{II}$ phase, 14 K upon cooling and 19 K upon heating, is larger than that of the bulk material (12 K). Obviously, confinement stabilizes the orientational disordered $R_{II}$ phase, similarly as has been found for n-alkanes [@Henschel2007] and for other orientational disordered, plastic phases under spatial confinement [@Knorr2008]. fluid - R$_{II}$ R$_{II}$ - C fluid - R$_{II}$ R$_{II}$ - C -------------------- ------------------ -------------- ------------------ -------------- confined (cooling) 304 291 confined (heating) 312 293 bulk 322 310 : \[tab:MeltingPoints\] ![\[Porenschnitt\] ](streugeometrie1.eps) ![(color online). Temperature dependence of the area per molecule $A$ for C$_{16}$H$_{33}$OH confined in porous silicon.[]{data-label="fig:GitterparameterFlaeche"}](flaeche.eps) Since the pores were completely filled at higher temperatures, when hexadecanol is in its liquid state, the pore filling at low temperatures does not consist only of bi-layer crystals: the change of volume at the R$_{II}$-C phase transition is about 10% (see Fig. \[fig:GitterparameterFlaeche\]), so that there are voids and/or molecules that are not part of a bi-layer crystal. However, our experiments do not give us information about their spatial arrangement. Molecular dynamics ------------------ The dynamics of bulk-C$_{16}$H$_{33}$OH has already been investigated in IR-measurements in the past [@Metivaud2005; @Tasumi1964]. In order to show later on how the molecular dynamics is affected by spatial confinement on the nm-scale, we display some of our bulk spectra in the following. Here we focus on two characteristic vibrations, the OH-stretching and the CH$_2$-scissoring vibration.\ Figs. \[fig:OHspectrum\]a) and \[fig:OHTemp\]a) show the bulk spectra of the OH-stretching-band in the respective phases (compare with Figs. \[fig:bulkstructure\_cryst\]-\[fig:bulkstructure\_rot\]). In the liquid state (above 322 K) the peak maximum is located at about 3345 cm$^{-1}$. A decrease of temperature below 321 K yields a shift of the peak position to about 3325 cm$^{-1}$ indicating the molecular rearrangement in the $R_{II}$ phase. A further decrease of temperature below 310 K results in a splitting into two peaks at approximately 3310 cm$^{-1}$ and 3220 cm$^{-1}$. Confined C$_{16}$H$_{33}$OH shows a different behavior. There is only one peak in the whole temperature range, the position of which changes reflecting the transition between liquid phase and $R_{II}$ phase as well as between $R_{II}$ phase and C phase (see Figs. \[fig:OHspectrum\]b and \[fig:OHTemp\]b). The fact that the OH-band of bulk C$_{16}$H$_{33}$OH splits at low temperatures while no splitting is observed under confinement confirms the structural differences already observed in the x-ray experiment. For example, Tasumi et. al have studied bulk alcohols C$_n$H$_{2n+1}$OH from $n=11-37$ using infrared spectroscopy [@Tasumi1964], Ventòla et al. alcohols with $n=17-20$ [@Ventola2002]. Those alcohols showing at low temperatures (C phase) the monoclinic $\gamma$-form, such as C$_{16}$H$_{33}$OH, exhibit the splitting of the OH-band, while those that take the orthorhombic $\beta$-form show a single peak. This is due to differences in the spatial arrangement of the hydrogen bonds as well as in the distances of neighboring O-atoms: in the crystalline $\gamma$-form, where the molecule axis are tilted (see Fig. \[fig:bulkstructure\_cryst\]), the molecules show an all trans conformation, and the intra-layer O-distance ($\simeq 2.74$ A) differs from the inter-layer O-distance ($\simeq 2.69$ A). However, in the orthorhombic $\beta$-form (Fig. \[fig:confstructure\_cryst\]) trans- and gauche-molecules alternate and the intra-layer O-distance ($2.73$ A) nearly equals the inter-layer O-distance ($2.72$ A), so that the splitting is suppressed [@Tasumi1964]. Therefore, the observed OH-band splitting for bulk C$_{16}$H$_{33}$OH shows the presence of the $\gamma$-form. Either the whole bulk material exhibits the $\gamma$-form or there is a mixture of $\gamma$- and $\beta$-crystallites. The latter case is frequently observed [@Tasumi1964; @Ventola2002]: in fact, in the range of wavenumbers from 1150 cm$^{-1}$ to 950 cm$^{-1}$, where C-C stretching vibrations are visible, we see indications for a superposition of both forms (not shown). On the other hand, pore confined C$_{16}$H$_{33}$OH shows no OH-band-splitting at low temperatures. This reflects that the molecular arrangement doesn’t transform in the monoclinic $\gamma$-form but remains in an orthorhombic structure, i. e. only the $\beta$-form is present (compare Figs. \[fig:bulkstructure\_cryst\] and \[fig:confstructure\_cryst\]). This result is in agreement with the x-ray data presented above. [**Upon cooling, both the bulk and the confined hexadecanol pass from an hexagonal $R_{II}$- phase into a crystalline phase. The bulk material undergoes a stronger structural change, i. e. there is a mixture of the orthorombic $\beta$- and the monoclinic $\gamma$-form. The latter one is suppressed under confinement, so that only the $\beta$-form remains, which is quite similar to the hexagonal structure of the R$_{II}$-phase:**]{} the fact that the crystallites have to fit into nanopores of irregular shape might favor the geometrically more simple $\beta$-form [@Christenson2001; @Morishige2000](see Fig. \[Porenschnitt\]). ![(a) IR spectrum in the OH - stretching range for bulk C$_{16}$H$_{33}$OH. At lower temperatures the peak shifts to lower wavenumbers and then splits into two peaks. (b) Spectrum for confined C$_{16}$H$_{33}$OH, where no splitting is visible. \[fig:OHspectrum\]](ohspektrum.eps) ![(a) Wavenumber $\omega/2\pi c$ of the OH - stretching peak vs temperature for bulk C$_{16}$H$_{33}$OH \[compare with Fig. \[fig:OHspectrum\]a)\]. Three different phases are visible: a) above 321 K, b) from 310 to 321 K, where the peak position appears at lower wavenumbers and c) below 310 K where the peak splits up into two peaks. (b) Wavenumber of the OH - stretching peak for confined C$_{16}$H$_{33}$OH (compare with Fig. \[fig:OHspectrum\]b). The transition between the C and the R$_{II}$ phase seems to be smeared in a range around $T=291 \pm 5$ K. The R$_{II}$-liquid transition does not affect the OH-stretching. \[fig:OHTemp\]](ohpeakpos2.eps) Now let us turn towards the scissor-vibration of the $\text{CH}_2$ groups (bending mode) that will give us information about inner-molecular and inter-molecular force constants. The spectra are shown in Fig. \[fig:CHspectrum\]. At first, we want to discuss the bulk material. At high temperatures (liquid state) a superposition of two peaks at 1467 cm$^{-1}$ and 1460 cm$^{-1}$ is observed. In the intermediate temperature range ($R_{II}$ phase; see Fig. \[fig:bulkstructure\_rot\]) the intensity of the peak labeld “1” increases strongly. At low temperatures (C phase; see Fig. \[fig:bulkstructure\_cryst\]) this band splits up into two peaks. The latter transition can be clearly seen in Figs. \[fig:CHTemp\]a) and \[fig:CHInt\]a), where we display the peak positions and intensities as a function of temperature. The results are similar to those obtained for the bulk state of n-paraffines, that apart from the missing OH-group are similar in their structure, i. e. that have the same CH$_2$-backbone [@Snyder1961]. In IR-spectra only one CH$_2$-scissoring-band is observed at high temperatures, i. e.intra-molecular interactions of the CH$_2$-groups are too small to lead to a series of distinct peaks. The band splitting at low temperatures has been attributed to inter-molecular interactions (see Ref. [@Snyder1961] and text below). Qualitatively, a behavior similar to that of the bulk state is observed for confined C$_{16}$H$_{33}$OH (see Fig. \[fig:CHspectrum\]b). In the high-temperature liquid phase two overlapping peaks are visible. The stronger one, i. e. that at higher wavenumbers, undergoes an increase in intensity at about 304 K (see Fig. \[fig:CHInt\]b), indicating the transition from the liquid phase to the $R_{II}$ phase, while the secondary peak at lower wavenumbers gets weaker and finally disappears. At the second transition temperature of $T=291$ K the remaining strong peak splits (see also Fig. \[fig:CHTemp\]b). The separation is not as distinct as for bulk material. These transition temperatures, $T=304$ K and $T=291$ K (see Figs. \[fig:CHTemp\]b and \[fig:CHInt\]b), agree well with those obtained via x-ray measurements (compare with Table \[tab:MeltingPoints\]).\ ![(a) IR spectrum showing the $\text{CH}_2$ scissor-vibration for bulk C$_{16}$H$_{33}$OH at various temperatures. (b) IR spectrum showing the $\text{CH}_2$ scissor-vibration of C$_{16}$H$_{33}$OH confined in mesoporous Si at various temperatures. \[fig:CHspectrum\]](scherspektrum.eps) ![Wavenumber of the CH - scissor peak vs temperature for (a) bulk C$_{16}$H$_{33}$OH and (b) confined C$_{16}$H$_{33}$OH (the peak labels refer to Fig. \[fig:CHspectrum\] ). \[fig:CHTemp\] ](scherpeakpos.eps) ![Integrated intensity of the CH - scissor peak vs temperature for (a) bulk C$_{16}$H$_{33}$OH and (b) confined C$_{16}$H$_{33}$OH (the peak labels refer to Fig. \[fig:CHspectrum\] ). \[fig:CHInt\] ](scherint.eps) ------------------------------ ------------ ------------ ------------ ------------ liquid $R_{II}$ liquid $R_{II}$ scissor \[cm$^{-1}$\] 1467 1467 1467 1467 sym. stretch \[cm$^{-1}$\] 2854 2851 2854 2851 assym. stretch \[cm$^{-1}$\] 2927 2921 2924 2918 $f_d$ \[N/m\] 455 453 454 452 $f_\alpha$ \[N/m\] 56 $\pm$ 1 56 $\pm$ 1 57 $\pm$ 1 57 $\pm$ 1 ------------------------------ ------------ ------------ ------------ ------------ : \[tab:ergebnis1\] Wavenumbers $\overline{\nu}=\omega/(2\pi c)$ (with $\omega$ being the angular frequency and $c$ the speed of light) and resulting stretching and bending force constants in the liquid and $R_{II}$ phase of bulk and confined C$_{16}$H$_{33}$OH. $f_\alpha$ has been evaluated using both the Eq. (\[eq:alpha1\]) and the Eq. (\[eq:alpha2\]). The difference yields the specified uncertainty. $f_\alpha$ are in units of $N/m$ (see Eq. (\[eq:pot2\]) in Appendix A and Ref. [@Meister1946]). To get $f_\alpha$ in units $Nm/rad^{2}$ one has to multiply $f_\alpha$ with d$^2$, where $d=1,09 \cdot 10^{-10}$ m is the CH bond length. In the following we want to analyze the dynamics of the CH$_2$-groups [**in order to check whether it is affected by the geometric confinement, e. g. by an interaction with the pore surfaces, by the limited number of neighboring molecules (finite-size-effects) or by structural changes**]{}. In a first approximation we can assume that it is not affected by the stretching of the OH - groups. On the one hand, there is the scissor vibration, where the angle $\alpha$ between the two CH-bonds oscillates around its equilibrium value $\alpha=109.47$° (see Fig. \[fig:moleculeCH2\]). In addition, symmetric and asymmetric stretching vibrations of the CH-bonds are observable (for the values see Table \[tab:ergebnis1\]). Let $f_\alpha$ and $f_d$ denote the respective force constants. These can be calculated from the measured vibration frequencies using Eqs. (\[eq:fd\])-(\[eq:alpha2\]) (see Appendix A; the difference in calculating $f_\alpha$ via Eq. (\[eq:alpha1\]) or Eq. (\[eq:alpha2\]) is below 3.5% confirming that the inner-molecular coupling terms can be neglected). Table \[tab:ergebnis1\] shows the results for the liquid and the $R_{II}$ phase. Neither the phase transition liquid $\rightarrow$ R$_{II}$ nor geometrical confinement does markely affect the innermolecular constants. ![$\text{CH}_2$ molecules with C-H bondlength d, H-C-H angle $\alpha$ and the resulting inner force constants $f_d$ and $f_{\alpha}$\[fig:moleculeCH2\] ](ch2.eps)  In the $R_{II}$ phase the molecules rotate about their long axis, so that the primitive cell consists of only one molecule per layer (see Fig. \[fig:bulkstructure\_rot\]). Therefore, no splitting is observed. But in the C phase (below 310 K for bulk and below 291 K for confined C$_{16}$H$_{33}$OH), where the molecules are arranged in a herringbone structure, there are two molecules per layer in the primitive cell. So the symmetry of the arrangement allows a splitting of the scissoring band and obviously the molecular interactions are sufficiently strong that we are able to observe a double peak (see above, Figs. \[fig:CHspectrum\] and \[fig:CHTemp\]). The strength of interaction depends on the distances between neighboring H-atoms of adjacent chains and can be analyzed using a formalism developed by Snyder (see Ref. [@Snyder1961] and Appendix B). In Fig. \[fig:lattice\] we have sketched the orthorhombic lattice of the crystalline C$_{16}$H$_{33}$OH subcell (a view on the a-b-plane perpendicular to the molecules axis). In what follows we restrict ourselves to this $\beta$-form, that is characteristic for confined C$_{16}$H$_{33}$OH (a quantitative analysis of bulk C$_{16}$H$_{33}$OH is difficult due to the superposition of $\beta$- and $\gamma$-form). Assuming that the inner force constant $f_\alpha$ does not change at the phase transition, the intermolecular force constants $f_{3,j}$ can be evaluated from the observed splitting of the scissor band as described in Appendix B (see Eq. (\[eq:fij\])). The values needed are the lattice parameters (see Table \[tab:ab\]) and the herringbone angle $\zeta$ between the projection of the backbone and the $a$-axis (see Fig. \[fig:lattice\]). The latter one is determined via Eq. (\[eq:Winkel\]) and the measured intensities of the two CH$_2$-scissoring-peaks. For confined C$_{16}$H$_{33}$OH we have $I_a = 13.29$ and $I_b = 22.23$ yielding an angle of $\zeta = 37.7$° (see Fig. \[fig:CHInt\]b for $T=245$ K). We display the intermolecular force constants in Table \[tab:ergebnis4\]. For comparison, we also list literature values for an alkane, C$_{23}$H$_{48}$ at 90 K, which have been evaluated in the same way [@Snyder1961]. This alkane and C$_{16}$H$_{33}$OH exhibit a similar structure: The backbones of the molecules consist of the same CH$_2$-units and both take the $\beta$-form at low-temperatures. In addition, also the values of the lattice constants for C$_{23}$H$_{48}$, $a=7.45$ A and $b=4.96$ A, are close to those of C$_{16}$H$_{33}$OH (see Tab. \[tab:ab\]). Due to this structural similarity the intermolecular distances listed in Table \[tab:ergebnis4\] are similar, however, the respective force constants differ slightly by 10 to 20%. This is mainly due to the orientation of CH$_2$-groups (the projection of the backbones on the a-b-plane) characterized by the herringbone angle $\zeta$. For C$_{16}$H$_{33}$OH $\zeta=37.7$° holds, for the alkane $\zeta=42$°. This difference is probably due to the presence of polar OH-groups in C$_{16}$H$_{33}$OH that are strongly interacting and thus have an impact on the molecular orientation. The above comparison confirms once again that confined C$_{16}$H$_{33}$OH takes the $\beta$-form in contrast to the bulk material ($\gamma$- and $\beta$-form). In order to assess the validity of our analysis, we also calculate the theoretical band splitting of the CH$_2$-scissoring vibration and compare it with the measured values. Using the values from Table \[tab:ergebnis4\] as well as Eqs. (\[eq:splitting\]) and (\[eq:Gab\]), we get a theoretical value of $\Delta\overline{\nu}_{calc}=8.1$ cm$^{-1}$ for confined C$_{16}$H$_{33}$OH at $T = 245$ K. The measured band splitting is $\Delta\overline{\nu}_{meas}=7.8$ cm$^{-1}$. Therefore, the experimental data is in good agreement with the theory. Summary ======= We have studied the structure and molecular dynamics of n-hexadecanol confined in nanochannels of mesoporous silicon and of bulk n-hexadecanol in their respective phases (in the order of decreasing temperature: liquid, rotator R$_{II}$ and C). For this purpose we have performed x-ray and infrared-measurements. The transition-temperatures for confined C$_{16}$H$_{33}$OH are lower than for bulk C$_{16}$H$_{33}$OH ($\Delta T \simeq 20$ K, see Table \[tab:MeltingPoints\]). In addition, under confinement the phase transitions are smeared, probably due to a distribution of pore diameters. Geometrical confinement does not affect the innermolecular force constants of the CH$_2$-scissoring vibration (see Table \[tab:ergebnis1\]) but has an impact on the molecular arrangement. The R$_{II}$ phase of both bulk and confined hexadecanol is characterized by an orthorhombic subcell, where the chain axis are parallel to the layer normal (see Fig. \[fig:bulkstructure\_rot\]). However, in the low-temperature C phase there is a fundamental structural difference. While bulk C$_{16}$H$_{33}$OH exhibits a polycrystalline mixture of $\beta$- and $\gamma$-forms (see Figs. \[fig:bulkstructure\_cryst\] and \[fig:confstructure\_cryst\]), geometrical confinement favors a phase closely related to the $\beta$-form: only crystallites with an orthorhombic subcell are formed, where the chain axes are parallel to the bi-layer normal. However, the $\gamma$-form having a monoclinic subcell, in which the chain axis are tilted with respect to the layer normal, is suppressed. A reason for this might be the irregular shape of the nanochannels, into which the crystallites have to fit, favoring the formation of the geometrically more simple and less bulky form [@Christenson2001; @Morishige2000] (see Fig. \[Porenschnitt\]). Since only the pure $\beta$-form is present under confinement, we were able to evaluate the inter-molecular force constants of the CH$_2$-scissor vibration. 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Meister and F. F. Cleveland, Am. J. Physics **14**, 13 (1946). E. B. Wilson Jr., J. Chem. Physics **7**, 1047 (1939). H. Siebert, Anwendungen der Schwingungsspektroskopie in der anorganischen Chemie, Springer-Verlag, Berlin (1966). M. Tasumi, T. Shimanouchi, J. Chem. Phys. **43**, 1245 (1965). R. S. Stein, J. Chem. Phys. **23**, 734 (1955). D. A. Dows, J. Chem. Phys. **32**, 1342 (1960). J. deBoer, Physica **9**, 363 (1942). Appendix A {#appendix-a .unnumbered} ========== In this section we show how the innermolecular force constants of the CH$_2$-groups can be evaluated using three characteristic vibration frequencies, that are easily measured: the scissor vibration as well as the symmetric and asymmetric CH-bond stretching. For this purpose we apply the Wilson FG - matrix method [@Wilson1939]. We use the notation of Meister and Cleveland for the similar $\text{H}_2\text{O}$ molecule [@Meister1946] and perform the calculations in the same way.\ Fig. \[fig:moleculeCH2\] shows a single $\text{CH}_2$ - molecule. In the following we will neglect the influence of the neighboring molecules on this one. $d=1.09$ A is the length of the C - H bond and $\alpha = 109.47$° the angle between the two C - H bonds [@Abrahamsson1960]. This kind of molecule belongs to the $C_{2_{\nu}}$ point group. This means, there are two vibrations of type $A_1$ (symmetric stretching and bending vibration) and one vibration of type $B_2$ (asymmetric stretching vibration). The internal coordinates of this molecule are $\Delta d_1$, $\Delta d_2$ and $\Delta\alpha$. $\Delta d_1$ and $\Delta d_2$ mean changes in the bond length of the two C - H - bonds and $\Delta\alpha$ changes in the angle between the two bonds. Therefore we get three symmetry coordinates, two for $A_1$ and one for $B_2$. If we assume d being the equilibrium C - H bond length, then we obtain for the three symmetry coordinates: $$R_1=\sqrt{\frac{1}{2}}\Delta d_1+\sqrt{\frac{1}{2}}\Delta d_2\\$$ $$R_2=\Delta\alpha \cdot d$$ $$R_3=\sqrt{\frac{1}{2}}\Delta d_1-\sqrt{\frac{1}{2}}\Delta d_2$$\ Now, we have to calculate the **F** matrix, related to the potential energy, and the **G** matrix related to the kinetic energy. The potential energy can be written as $$2V=\sum f_{ik}r_ir_k \label{eq:pot1}$$ and with the internal coordinates $$\begin{split} 2V=&f_d\left[\left(\Delta d_1\right)^2+\left(\Delta d_2\right)^2\right]\\ &+f_{\alpha}(d\Delta\alpha)^2+2f_{d\alpha}\left(\Delta d_1+\Delta d_2\right)\left(d\Delta\alpha\right)\\ &+2f_{dd}\left(\Delta d_1\right)\left(\Delta d_2\right) \label{eq:pot2} \end{split}$$ Now we set $d_1=d_2=d$ and write Eq. (\[eq:pot2\]) as $$2V=\sum F_{jl}R_jR_l$$ with $F_{jl}=F_{lj}$. In matrix form, Eqs. (\[eq:pot1\]) and (\[eq:pot2\]) become $$2V = \textbf{r}'\textbf{fr} \label{eq:vmatrix1}$$ and $$2V=\textbf{R}'\textbf{FR} \label{eq:vmatrix2}$$ **r**$'$ and **R**$'$ are the transposes of **r** and **R**. With Eqs. (\[eq:vmatrix1\]) + (\[eq:vmatrix2\]) $$\textbf{r}'\textbf{fr}=\textbf{R}'\textbf{FR}$$ The $R_i$’s are linear combinations of the $r_i$’s $$\begin{split} R_i&=\sum_{k}U_{ik}r_k\\ \textbf{R}&=\textbf{Ur} \end{split}$$ Since the $R_i$’s are orthogonal and normalized, then $\textbf{U}^{-1}=\textbf{U}'$ and $$\begin{aligned} \textbf{r}=\textbf{U}'\textbf{R}\\ \textbf{r}'=(\textbf{U}'\textbf{R})'=\textbf{R}'\textbf{U}\end{aligned}$$ This means with Eqn. (10) $$\begin{aligned} \textbf{R}'(\textbf{UfU}')\textbf{R}=\textbf{R}'\textbf{FR}\\ \textbf{F}=\textbf{UfU}'\end{aligned}$$ The **F** matrix is [c|ccc]{} & d\_1 & d\_2 &\ d\_1 & f\_d & f\_[dd]{} & df\_[d]{}\ d\_2 & f\_[dd]{} & f\_d & df\_[d]{}\ & df\_[d]{} & df\_[d]{} & d\^2f\_ The **U** matrix for type $A_1$ is\ [c|ccc]{} A\_1 & d\_1 & d\_2 &\ R\_1 & & & 0\ R\_2 & 0 & 0 & 1 and for $B_2$ [c|ccc]{} B\_2 & d\_1 & d\_2 &\ R\_3 & & - & 0 So, for the type $A_1$ the **F** matrix is\ $$\textbf{F}_{A_1} = \textbf{UfU}'= \left(\begin{array}{cc} F_{11} & F_{12}\\ F_{21} & F_{22} \end{array}\right) = \left( \begin{array}{cc} f_d + f_{dd} & \sqrt{2}df_{\alpha}\\ \sqrt{2}df_{\alpha} & d^2f_{\alpha} \end{array}\right)$$ and for the $B_2$ type $$\textbf{F}_{B_2}=(F_{33})=(f_d-f_{dd})$$ The exact derivation of the $\textbf{G}$ matrix shouldn’t be shown here. It can be gleaned by Meister and Cleveland [@Meister1946]. Only the most important steps shall be explained here.\ If only non-degenerate vibrations are present, the elements of the kinetic energy matrix can be written as $$G_{jl}=\sum_{p}\mu_pg_p\textbf{S}_j^{(t)}\textbf{S}_l^{(t)}$$ where j and l refer to symmetry coordinates used in determining the **S** vector, $p$ refer to a set of equivalent atoms, a typical one of the set being t. $\mu_p$ is the reciprocal of the mass of the typical atom $t_p$ and $g_p$ is the number of equivalent atoms in the $p$th set. The **S** vector is given by $$\textbf{S}_j^{(t)}=\sum_{k}U_{jk}s_{kt}$$ where $j$, $U_jk$ and $\sum_k$ have the same meaning as above. $s_{kt}$ can be expressed in terms of unit vectors along the chemical bonds and depends on the changes in the bond length or the angle between the bonds. So, the **G** matrix for the $A_1$ vibration type has the form $$\begin{split} \textbf{G}_{A_1}&= \left(\begin{array}{cc} G_{11} & G_{12}\\ G_{21} & G_{22} \end{array}\right)\\ &=\left(\begin{array}{cc} \mu_H+\mu_C(1+\cos\alpha) & -\frac{\mu_C\sqrt(2)\sin\alpha}{d}\\ -\frac{\mu_C\sqrt(2)\sin\alpha}{d} & \frac{2\mu_H+\mu_C(1-\cos\alpha)}{d} \end{array}\right) \end{split}$$ and for the $B_2$ vibration type $$\textbf{G}_{B_2} (G_{33})=(\mu_H+\mu_C(1-\cos\alpha))$$ To determine the frequencies, one has to solve the equation $$\left|\textbf{GF}-\lambda\textbf{E}\right|=0 \qquad$$ where $\lambda = \omega^2= (\overline{\nu} 2 \pi c)^2$ denotes the square of the angular frequency. For the $A_1$ type one gets the equation $$\begin{split} \lambda^2&-\lambda(F_{11}G_{11}+2F_{12}G_{12}+F_{22}G_{22})\\ &+\begin{array}{|cc|} F_{11}&F_{12}\\ F_{21}&F_{22} \end{array} \cdot \begin{array}{|cc|} G_{11}&G_{12}\\ G_{21}&G_{22} \end{array}=0 \label{eq:loesung1} \end{split}$$ and for the $B_2$ type $$\lambda_3-F_{33}G_{33}=0 \label{eq:loesung2}$$ Eq. (\[eq:loesung1\]) can be separated with the Vieta expression [@Siebert1966]. Inserting the terms for the $F_{ij}$ and $G_{ij}$, we obtain $$\begin{aligned} \begin{aligned} \lambda_1+\lambda_2 & = (f_d+f_{dd})[\mu_C(1+\cos\alpha)+\mu_H]\\ & +2f_{\alpha}[\mu_C(1-\cos\alpha)+\mu_H]-4f_{d\alpha}\mu_C\sin\alpha \label{l1+l2} \\ \lambda_1\cdot\lambda_2 &= [(f_d+f_{dd})f_{\alpha}-2f_{d\alpha}^2]2\mu_H(2\mu_C+\mu_H) \end{aligned} \label{lamb1lamb2}\end{aligned}$$ For Eq. (\[eq:loesung2\]) one obtains $$\lambda_3=(f_d-f_{dd})[\mu_C(1-\cos\alpha)+\mu_H] \label{lamb3}$$ Neglecting the coupling constants $f_{dd}$ and $f_{d\alpha}$ allows to evaluate the innermolecular force constants using the measured wave numbers, $\overline{\nu}_{d,sym}=\sqrt{\lambda_1}/(2\pi c)$, $\overline{\nu}_\alpha=\sqrt{\lambda_2}/(2\pi c)$ and $\overline{\nu}_{d,asym}=\sqrt{\lambda_3}/(2\pi c)$. Then Eq. (\[lamb3\]) yields $$f_d= (2\pi c)^2 \cdot \frac{\overline{\nu}_{d, asym}^2}{\mu_C(1-\cos\alpha)+\mu_H}\label{eq:fd}$$ Inserting this result into Eq. (\[lamb1lamb2\]) yields $$\label{eq:alpha1} f_{\alpha}= (2\pi c)^2 \cdot \frac{\overline{\nu}_{d,sym}^2\cdot \overline{\nu}_\alpha^2} {\overline{\nu}_{d,asym}^2} \cdot \frac{\mu_C(1-\cos\alpha)+\mu_H}{2\mu_H(2\mu_C+\mu_H)}$$ There is a second possibility to evaluate $f_\alpha$, i. e. by inserting Eq. (\[eq:fd\]) into Eq. (\[l1+l2\]). This yields $$f_{\alpha}= (2\pi c)^2 \cdot \frac{\overline{\nu}_{d,sym}^2 + \overline{\nu}_\alpha^2- \overline{\nu}_{d,asym}^2 \cdot \frac{\mu_C(1+\cos\alpha)+\mu_H}{\mu_C(1-\cos\alpha)+\mu_H}}{2(\mu_C(1-\cos\alpha)+\mu_H)} \label{eq:alpha2}$$ Taking the measured wavenumbers listed in Table \[tab:ergebnis1\] and the average angle between the CH-bonds, $\alpha=109.4$°, as well as the masses of the atoms, $1/\mu_C=12u$ and $1/\mu_H=1u$ ($u=1.6606 \cdot 10^{-27}$ kg), Eqs. (\[eq:fd\])-(\[eq:alpha2\]) yield the force constants listed in Table \[tab:ergebnis1\]. The difference in calculating $f_\alpha$ via Eq. (\[eq:alpha1\]) or Eq. (\[eq:alpha2\]) is below 3.1% confirming that the inner-molecular coupling terms can be neglected. Appendix B {#appendix-b .unnumbered} ========== What follows is a summary of Snyder’s derivation of the intermolecular force constants between the CH$_2$ groups of neighboring molecules that gives rise to a splitting of the scissor band at low temperatures [@Snyder1961]. We show how this formalism can be applied to C$_{16}$H$_{33}$OH. An alternative description can be found in Ref. [@Tasumi1965]. In Fig. \[fig:lattice\] we display a hexagonal subcell of C$_{16}$H$_{33}$OH. While Stein [@Stein1955] has taken only one pair of neighboring $\text{CH}_2$ into account to calculate the splitting of rocking and scissoring bands, Snyder has shown that more pairs have to be included. When we consider the distances of H-atoms from the H-atom no. 3 (see Fig. \[fig:lattice\]), then all atoms except no. $2'$, $6'$, 5 and 6 have distances larger then 3.7 Å. The internal coordinates $\alpha_i$ are always half of the angle between the C - H bonds of a $\text{CH}_2$ molecule. Solid circles are H-atoms in the same plane, dashed circles H-atoms in a plane above or below. ![Lateral view at the long axis of the C$_{16}$H$_{33}$OH chain, x is the projection of the C - C distance on the c-axis of the crystal lattice \[fig:Kette\]](kette.eps) \ Now, we want to write the positions of these five H-atoms as a vector. Fig. \[fig:Kette\] shows the lateral view of a part of the C$_{16}$H$_{33}$OH chain. With values from Abrahamsson [@Abrahamsson1960] for $l_{CC}=1.545$ A and $\alpha_{CCC}=110.4$°, we can calculate the distance of the a-b-plane to the corresponding plane above or below with $$\nonumber x=l_{CC}\sin(\frac{\alpha_{CCC}}{2})=1.2687\,\text{\AA}$$ Assuming that the hydrogen in the central plane has the $c$ component 0, the hydrogen in the plane above has the component $c=1.2687$ Å. ![Lateral view at the long axis of the C$_{16}$H$_{33}$OH chain, d is the projection of half of the C - C distance on the a-b-plane of the subcell. \[fig:seitlich\]](seitenansicht.eps) \ The projection of the C - C bond in the a-b-plane is according to Fig. \[fig:seitlich\] $$\nonumber d=\frac{l_{CC}}{2}\cos(\frac{\alpha_{CCC}}{2})=0.4409\,\text{\AA}$$ Taking the point **0** for the lower left edge of the ab - plane the five atoms have the coordinates: $$\nonumber \begin{split} H_3&= \left(\begin{array}{c} \frac{a}{2}-d\cos(\zeta)-l\cos(\alpha_3-\zeta)\\ \frac{b}{2}+d\sin(\zeta)-l\sin(\alpha_3-\zeta)\\ 0 \end{array}\right)\\ \nonumber H_{2'}&= \left(\begin{array}{c} -d\cos(\zeta)+l\sin(\alpha_{2'}-\frac{\pi}{2}+\zeta)\\ b-d\sin(\zeta)-l\cos(\alpha_{2'}-\frac{\pi}{2}+\zeta)\\ 0 \end{array}\right)\\ \nonumber H_{5}&= \left(\begin{array}{c} d\cos(\zeta)-l\sin(\alpha_{5}-\frac{\pi}{2}+\zeta)\\ d\sin(\zeta)+l\cos(\alpha_{5}-\frac{\pi}{2}+\zeta)\\ 1.2674 \end{array}\right)\\ \nonumber H_{6}&= \left(\begin{array}{c} d\cos(\zeta)+l\cos(\alpha_6-\zeta)\\ d\sin(\zeta)-l\sin(\alpha_6-\zeta)\\ 1.2674 \end{array}\right)\\ \nonumber H_{6'}&= \left(\begin{array}{c} d\cos(\zeta)+l\cos(\alpha_{6'}-\zeta)\\ b+d\sin(\zeta)-l\cos(\alpha_{6'}-\zeta)\\ 1.2674 \end{array}\right) \end{split}$$ with $a$ and $b$ being the lattice constants of the crystalline phase. We get the distances between the atom 3 and the other ones (see Fig. \[fig:lattice\]) with $$r_{3j}=\left|H_3-H_j\right| \label{eq:distances}$$ where $j=2', 6', 5, 6$ is. The herringbone angle $\zeta$ (see the upper left corner of Fig. \[fig:lattice\]) can be determined with the relation [@Snyder1961] $$\frac{I_a}{I_b}=\tan^2\zeta \label{eq:Winkel}$$ where $I_a$ is the integrated intensity of the scissoring mode, which is polarized in the a direction (higher mode at 1473 cm$^{-1}$) and $I_b$ the one of the mode, which is polarized in the $b$ direction (lower mode at 1462 cm$^{-1}$).\ With the distances of two hydrogen atoms $H_3$ and $H_j$ (=$r_{3,j}$) \[in our case 3 denotes the central H-atom (see Fig. \[fig:lattice\]) and $j=2',5,6,6'$ the neighboring H-atoms that interact\] we obtain the intermolecular force constants $f_{3,j}$: $$\begin{aligned} f_{3,j} & = & \frac{\partial^2 V_{HH}}{\partial\alpha_3\partial\alpha_j} \nonumber \\ & = & \left(\frac{\partial^2 V_{HH}}{\partial r^2}\right)_{r_{3j}}\left(\frac{\partial r}{\partial \alpha_3}\right)\left(\frac{\partial r}{\partial \alpha_j}\right) \label{eq:fij}\end{aligned}$$ where $V_{HH}$ is the hydrogen repulsion potential introduced by Dows [@Dows1960]: $$V_{HH}=1.2\cdot 10^{-10}e^{-3.52r} \label{eq:potential}$$ with r in Å. The values of $(\partial^2 V_{HH}/\partial r^2)_{r_{ij}}$ are obtained from Eq. (\[eq:beta\]). $$\beta=\frac{\partial^2 V_{HH}}{\partial r^2}=1.486848\cdot 10^{-9}e^{-3.52r} \label{eq:beta}$$ with $\beta$ in $\frac{ergs}{\text{\AA}^2}=10^{16}\frac{dyne}{cm}=10^{13}\frac{N}{m}$. The measured intensity ratio (Eq. (\[eq:Winkel\])) allows us to calculate the distances $r_{ij}$ as well as the partial derivatives $\partial r/\partial \alpha_i$ (see Eq. (\[eq:distances\]) and above). Finally, by knowing the intermolecular force constants $f_{3,j}$ from Eq. (\[eq:beta\]) we can evaluate the band splitting of the scissoring vibration [@Snyder1961]. For the angular frequencies $$\overline{\nu}_1^2 - \overline{\nu}_2^2 = \left( \frac{1}{2 \pi c}\right)^2 \cdot \underbrace{G_a^B\cdot \left\{ 2f_{3,2'}-2(f_{3,6}+f_{3,6'})-4f_{3,5} \right\}}_{\Delta\lambda^B} \label{eq:splitting}$$ holds [^3] with $$G_a^B =\frac{4}{3}Q_R^2\mu_C +Q_r^2\mu_H \qquad \label{eq:Gab}$$ Here $1/\mu_C=12u$ and $1/\mu_H=1u$ denote the masses of the atoms ($u=1.6606 \cdot 10^{-27}$ kg), $1/Q_R=1.545 \cdot 10^{-10}$ m the C-C distance and $1/Q_r=1.09 \cdot 10^{-10}$ m the C-H distance, so that $G_a^B=(0.88825/u)$ Å$^{-2}= 5.349 \cdot 10^{46}$ (N m)$^{-1}$s$^{-2}$. For the band splitting of the wavenumbers we thus obtain $$\begin{split} \Delta \overline{\nu}&=\overline{\nu}_1 - \overline{\nu}_2\\ &= \left( \frac{1}{2 \pi c}\right)^2 \cdot \frac{ G_a^B \cdot \left\{ 2f_{3,2'}-2(f_{3,6}+f_{3,6'})-4f_{3,5} \right\} }{ \overline{\nu}_1 + \overline{\nu}_2} \end{split}$$ Inserting the values of Table \[tab:ergebnis4\], i. e.$(2f_{3,2'}-2(f_{3,6}+f_{3,6'})-4f_{3,5})=15.784 \cdot 10^{-21}$ Nm for confined C$_{16}$H$_{33}$OH, as well as the respective wave numbers, that we take from Fig. \[fig:CHTemp\] at low temperatures (labels 1a and 1b: $\overline{\nu}_1 + \overline{\nu}_2 \simeq 2 \cdot 146700$ 1/m) we get $\Delta\overline{\nu}=810$ m $^{-1}\equiv 8.1$ cm$^{-1}$ for confined C$_{16}$H$_{33}$OH. [^1]: producer: SiMat, Landsberg, Germany; specific conductivity: $\rho=0.01-0.025$ $\Omega$cm. [^2]: For several alkanes, there also exists a Rotator-(I)-Phase $R_I$, where the molecules switch between two equal positions. [^3]: In Snyder’s general theory the force constants for the scissoring vibration are denominated as $f_a^3$ (= $f_{3,2'}$), $f_b^2$ (= $f_{3,6}+f_{3,6'}$) and $f_b^3$ (= $f_{3,5}$).

--- abstract: | Wigner’s irreducible positive energy representations of the Poincaré group are often used to give additional justifications for the Lagrangian quantization formalism of standard QFT. Here we study another more recent aspect. We explain in this paper modular concepts by which we are able to construct the local operator algebras for all standard positive energy representations directly i.e. without going through field coordinatizations. In this way the artificial emphasis on Lagrangian field coordinates is avoided from the very beginning. These new concepts allow to treat also those cases of  “exceptional” Wigner representations associated with anyons and the famous Wigner “spin tower”which have remained inaccessible to Lagrangian quantization. Together with the d=1+1 factorizing models (whose modular construction has been studied previously), they form an interesting family of theories with a rich vacuum-polarization structure (but no on shell real particle creation) to which the modular methods can be applied for their explicit construction. We explain and illustrate the algebraic strategy of this construction. We also comment on possibilities of formulating the Wigner theory in a setting of a noncommutative spacetime substrate. This is potentially interesting in connection with recent unitarity- and Lorentz invariance- preserving results of the special nonlocality caused by this kind of noncommutativity. author: - | Lucio Fassarella and Bert Schroer[^1]\ CBPF, Rua Dr. Xavier Sigaud, 150, 22290-180 Rio de Janeiro - RJ, Brazil\ email Fassarel@cbpf.br, Schroer@cbpf.br date: December 2001 title: Wigner Particle Theory and Local Quantum Physics --- The setting of the problem ========================== The algebraic framework of local quantum physics shares with the standard textbook approach to QFT the same physical principles but differs in concepts and tools used for their implementation. Whereas the standard approach is based on “field-coordinatizations” in terms of pointlike fields (without which the canonical- or functional integral- quantization is hardly conceivable), the algebraic framework permits to formulate local quantum physics directly in terms of a net of local operator algebras i.e. without the intervention of the rather singular pointlike field coordinates whose indiscriminate use is the potential source of ultraviolet divergencies. Among the many advantages is the fact that the somewhat artistic[^2] standard scheme is replaced by a conceptually better balanced setting. The advantages of such an approach [@Haag][@Bu][@Bu-Ha] were in the eyes of many particle physicist offset by its constructive weaknesses of which even its protagonists (who used it mainly for structural investigations as TCP, Spin&Statistics and alike) were well aware [@Bu-Ha]. In particular even those formulations of renormalized perturbation theory which were closest in spirit to the algebraic approach namely the causal perturbation theory and its recent refinements [@Due-Fre] uses a coordinatization of algebras in terms of fields at some stage. The underlying “Bogoliubov-axiomatics” [@Tod] in terms of an off-shell generating “S-matrix” S(g) suffers apparently from the same ultraviolet limitations as any other pointlike field formulation. However there are signs of change which are not only a consequence of the lack of promised success of many popular attempts in post standard model particle theory. Rather it is also becoming slowly but steadily clear that the times of constructive nonperturbative weakness of the algebraic approach (AQFT) are passing and the significant conceptual investments are beginning to bear fruits for the actual construction of models. The constructive aspects of these gains are presently most clearly visible in situations in which there is no real (on-shell) particle creation but for which, different from free field theories, the vacuum-polarization structure remains very rich. It is not possible in those models to locally generate one-particle states from the vacuum without accompanying vacuum-polarization clouds. Besides the well-known d=1+1 factorizing models, this includes the QFTs associated with exceptional Wigner representations i.e. d=1+2 “anyonic” spin and the d=1+3 “spin towers” (Wigner’s famous exceptional zero mass representations with an infinite number of interlinked helicity states). In both cases the absence of compact localization renders the theories more noncommutative and in turn less accessible to Lagrangian quantization methods. The main content of this paper deals with constructive aspects of such models. The historical roots of the algebraic approach date back to the 1939 famous Wigner paper [@Wig] whose aim was to obtain an intrinsic conceptual understanding of particles avoiding the ambiguous wave equation method and the closely related Lagrangian quantization so that a physical equivalence of different Lagrangian descriptions could be easily recognized. In fact it was precisely this fundamental intrinsic appeal and the unicity of Wigner’s approach that some authors felt compelled to present this theory as a kind of additional partial justification for the the Lagrangian (canonical- or functional-) quantization [@Wei]. Since the late 50s there has been a dream about a royal path into nonperturbative particle physics which starts from Wigner’s representation-theoretic particle setting and introduces interactions in a maximally intrinsic and invariant way i.e. by using concepts which avoid doing computations in terms of the standard singular field coordinationations and lean instead on the unitary and crossing symmetric scattering operator and the associated spaces of formfactors. It is well-known that this dream in its original form failed, and that some of the old ideas were re-processed and entered string theory via Veneziano’s dual model. In the following we will show that certain aspects of that old folklore (which certainly does not include that of a “Theory of Everything”), if enriched with new concepts, can have successful applications for the above mentioned class of models. According to Wigner, particles should be described by irreducible positive energy representation of the Poincaré group. In fact they are the indecomposable building blocks of those multi-localized asymptotically stable objects in terms of which each state can be interpreted and measured in counter-coincidence arrangements in the large time limit. This raises the question what localization properties particles should be expected to have, and which positive energy representations permit what kind of localization. There are two localization concepts. One is the “Born-localization” taken over from Schroedinger theory which is based on probabilities and associated projectors projecting onto compactly supported subspaces of spatially localized wave functions at a fixed time (which in the relativistic context also bears the name “Newton-Wigner” localization). The incompatibility of this localization with relativistic covariance and Einstein causality was already noted and analyzed by its protagonists [@N-W]. Covariance as well as macro-causality are however satisfied in the asymptotic region and therefore the covariance and the cluster separability of the Moeller operators and the S-matrix are not effected by the use of this less than perfect quantum mechanical localization. On the other hand there exists a fully relativitic covariant localization which is intimately related to the characteristic causality- and vacuum polarization- properties of QFT; in the standard formulation of QFT it is that localization which is encoded in the position of the dense subspace obtained by applying smeared fields (with a fixed test function support) to the vacuum. Since in the field-free formulation of local quantum physics this localization turns out to be inexorably linked to the Tomita-Takesaki modular theory of operator algebras, it will be shortly referred to as “modular localization”. Its physical content is less obvious and its consequences are less intuitive and therefore we will take some care in its presentation. In fact the remaining part of this introductory section is used to contrast the Newton-Wigner localization with the modular localization. This facilitates the understanding of both concepts. The use of Wigner’s group theory based particle concept for the formulation of what has been called[^3] “direct interactions” in relativistic mutiparticle systems can be nicely illustrated by briefly recalling the arguments which led to this relativistic form of macro-causal quantum mechanics. Bakamjian and Thomas [@BT] observed as far back as 1953 that it is possible to introduce an interaction into the tensor product space describing two Wigner particles by keeping the additive form of the total momentum $\vec{P}$, its canonical conjugate $\vec{X}$ and the total angular momentum $\vec{J}$ and by implementing interactions through an additive change of the invariant free mass operator $M_{0}$ by an interaction $v$ (with only a dependence on the relative c.m. coordinates $\vec{p}_{rel}$) which then leads to a modification of the 2-particle Hamiltonian $H$ with a resulting change of the boost $\vec{K}$ according to $$\begin{aligned} M & =M_{0}+v,\,\,M_{0}=2\sqrt{\vec{p}_{rel}^{2}+m^{2}}\\ H & =\sqrt{\vec{P}^{2}+M^{2}}\nonumber\\ \vec{K} & =\frac{1}{2}(H\vec{X}+\vec{X}H)-\vec{J}\times\vec{P}(M+H)^{-1}\nonumber\end{aligned}$$ The commutation relations of the Poincaré generators are maintained, provided the interaction operator $v$ commutes with $\vec{P},\vec{X}$ and $\vec{J}.$ For short range interactions the validity of the time-dependent scattering theory is easily established and the Moeller operators $\Omega _{\pm}(H,H_{0})$ and the $S$-matrix $S(H,H_{0})$ are Poincaré invariant in the sense of independence on the L-frame $$O(H,H_{0})=O(M,M_{0}),\,\,O=\Omega_{\pm},S$$ and they also fulfill the cluster separability$$s-\lim_{\delta\rightarrow\infty}O(H,H_{0})T(\delta)\rightarrow\mathbf{1}$$ where the $T$ operation applied to a 2-particle vector separates the particle by an additional spatial distance $\delta.$ The subtle differences to the non-relativistic case begin to show up for 3 particles [@Coester]. Rather than adding the two-particle interactions one has to first form the mass operators of the e.g. 1-2 pair with particle 3 as a spectator and define the 1-2 pair-interaction operator in the 3-particle system $$\begin{aligned} M(12,3) & =\left( \left( \sqrt{M(12)^{2}+\vec{p}_{12}^{2}}+\sqrt {m^{2}+\vec{p}_{3}^{2}}\right) ^{2}-\left( \vec{p}_{12}+\vec{p}_{3}\right) ^{2}\right) ^{\frac{1}{2}}\\ V^{(3)}(12) & \equiv M(12,3)-M(1,2,3),\,\ M(1,2,3)\equiv M_{0}(123)\nonumber\end{aligned}$$ where the notation speaks for itself (the additive operators carry a subscript labeling and the superscript in the interaction $V^{(3)}(12)$ operators remind us that the interaction of the two particles within a 3-particle system is not identical to the original two-particle $v\equiv V^{(2)}(12)$ operator in the two-particle system). Defining in this way $V^{(3)}(ij)$ for all pairs, the 3-particle mass operator and the corresponding Hamiltonian are given by $$\begin{aligned} M(123) & =M_{0}(123)+\sum_{i<j}V_{{}}^{(3)}(ij)\\ H(123) & =\sqrt{M(123)^{2}+p_{123}^{2}}\nonumber\end{aligned}$$ and lead to a L-invariant and cluster-separable 3-particle Moeller operator and S-matrix, where the latter property is expressed as a strong operator limit $$\begin{aligned} S(123) & \equiv S(H(123),H_{0}(123))=S(M(123),M_{0}(123))\\ & s\text{-}\lim_{\delta\rightarrow\infty}S(123)T(\delta_{13},\delta _{23})=S(12)\times\mathbf{1}\nonumber\end{aligned}$$ with the formulae for other clusterings being obvious. By iteration and the use of the framework of rearrangement collision theory (which introduces an auxiliary Hilbert space of bound fragments), this can be generalized to n-particles including bound states [@C-P]. As in nonrelativistic scattering theory, there are many different relativistic direct particle interactions which lead to the same S-matrix. As Sokolov showed, this freedom to modify off-shell operators (e.g. $H,\vec{K}$ as functions of the single particle variables $\vec{p}_{i},\vec{x}_{i},\vec {j}_{i}$ and the interaction $v$) may be used to construct to each system of the above kind a “scattering-equivalent” system in which the interaction-dependent generators $H,\vec{K}$ restricted to the images of the fragment spaces become the sum of cluster Hamiltonians (or boosts) with interactions between clusters being switched off [@C-P]. Using these interaction-dependent equivalence transformations, the cluster separability can be made manifest. It is also possible to couple channels in order to describe particle creation, but this channel coupling “by hand” does not define a natural mechanism for interaction-induced vacuum polarization. Even though such direct interaction models between relativistic particles can hardly have fundamental significance, their very existence as relativistic theories (i.e. consistent with the physically indispensible macro-causality) help us rethink the position of micro-causal and local versus nonlocal but still macro-causal relativistic theories. Since our intuition on theses matters is notoriously unreliable and ridden by prejudices, it is very useful to have such illustrations. This is of particular interest in connection with recent attempts to implement nonlocality through noncommutativity of the spacetime substrate (see the last section). But even some old piece of QFT folklore, which claimed that the construction of unitary relativistic invariant and cluster-separable S-matrices can only be achieved through local QFT, are rendered incorrect. It turns out that if one adds crossing symmetry to the list of S-matrix properties it is possible to show that if the on-shell S-matrix originates at all from a local QFT, it determines its local system of operator algebras uniquely [@S-matrix]. This unicity of local algebras is of course the only kind of uniqueness which one can expect since individual fields are analogous to coordinates in differential geometry (in the sense that passing to another locally related field cannot change the S-matrix). The new concept which implements the desired crossing property and also insures the principle of “nuclear democracy“[^4] (both properties are not compatible with the above relativistic QM) is modular localization. In contrast to the quantum mechanical Newton Wigner localization, it is not based on projection operators (which project on quantum mechanical subspaces of wave functions with support properties) but rather is reflected in the Einstein causal behavior of expectation values of local variables in modular localized state vectors. Modular localization in fact relates off-shell causality, interaction-induced vacuum polarization and on-shell crossing in an inexorable manner and in particular furnishes the appropriate setting for causal propagation properties (see next section). Since it allows to give a completely intrinsic definition of interactions in terms of the vacuum polarization clouds which accompany locally generated one-particle states without reference to field coordinates or Lagrangians, one expects that it serves as a constructive tool for nonperturbative investigations. This is borne out for those models considered in this paper. It is interesting to note that both localizations are preempted in the Wigner theory. Used in the Bakajian-Thomas-Coester spirit of QM of relativistic particles with the Newton-Wigner localization, it leads to relativistic invariant scattering operators which obey cluster separability properties and hence are in perfect harmony with macro-causality. On the other hand used as a starting point of modular localization one can directly pass to the system of local operator algebras and relate the notion of interaction (and exceptional statistics) inexorably with micro-causality and vacuum polarization clouds which accompany the local creation of one particle states. Perhaps the conceptually most surprising fact is the totally different nature of the local algebras from quantum mechanical algebras. In the second section we will present the modular localization structure of the standard halfinteger spin Wigner representation in the first subsection and that of the exceptional (anyonic, spin towers) representations in the second subsection. The subject of the third section is the functorial construction of the local operator algebras associated with the modular subspaces of the standard Wigner representations. The vacuum polarization aspects of localized particle creation operators associated with exceptional Wigner representations are treated in the fourth section. In section 5 we explain our strategy for the construction of theories which have no real particle creation but (different from free fields) come with a rich vacuum polarization structure in the context of d=1+1 factorizing models. Apart from the issue of anyons, the most interesting and unexplored case of QFTs related to positive energy Wigner representations is certainly that of the massless d=1+3 “Wigner spin towers”. This case is in several aspects reminiscent of structures of string theory. It naturally combines all (even, odd, supersymmetric) helicities into one indecomposable object. If it would be possible to introduce interactions into this tower structure, then the standard argument that any consistent interacting object which contains spin 2 must also contain an (at least a quasiclassical) Einstein-Hilbert action (which is used by string theorist in order to link strings with gravity) applies as well here [^5]. Recently there has been some interest in the problem whether the Wigner particle structure can be consistent with a noncommutative structure of spacetime where the minimal consistency is the validity of macro-causality. We will have some comments in the last section. Modular aspects of positive energy Wigner representations ========================================================= In this in the next subsection we will briefly sketch how one obtains the interaction-free local operator algebras directly from the Wigner particle theory without passing through pointlike fields. The first step is to show that there exist a relativistic localization which is different from the non-covariant Newton-Wigner localization. The standard case: halfinteger spin ----------------------------------- For simplicity we start from the Hilbert space of complex momentum space wave function of the irreducible $(m,s=0)$ representation for a neutral (selfconjugate) scalar particle. In this case we only need to remind the reader of published results [@1997][@AP][@BGL][@JMP]. $$\begin{aligned} & H_{Wig}=\left\{ \psi(p)|\int\left| \psi(p)\right| ^{2}\frac{d^{3}p}{2\sqrt{p^{2}+m^{2}}}<\infty\right\} \\ & \left( \frak{u}(\Lambda,a)\psi\right) (p)=e^{ipa}\psi(\Lambda ^{-1}p)\nonumber\end{aligned}$$ For the construction of the real subspace $H_{R}(W_{0})$ of the standard $t$-$z$ wedge $W_{0}=(z>\left| t\right| ,x,y$ arbitrary) we use the $z-t$ Lorentz boost $\Lambda_{z-t}(\chi)\equiv\Lambda_{W_{0}}(\chi)$$$\Lambda_{W_{0}}(\chi):\left( \begin{array} [c]{c}t\\ z \end{array} \right) \rightarrow\left( \begin{array} [c]{cc}\cosh\chi & -\sinh\chi\\ -\sinh\chi & \cosh\chi \end{array} \right) \left( \begin{array} [c]{c}t\\ z \end{array} \right)$$ which acts on $H_{Wig}$ as a unitary group of operators $\frak{u}(\chi)\equiv$ $\frak{u}(\Lambda_{z-t}(\chi),0)$ and the $z$-$t$ reflection $r:$ ($z,t)\rightarrow(-z$,$-t)$ which, since it involves time reflection, is implemented on Wigner wave functions by an unti-unitary operator $\frak{u}(r)$ [@JMP][@BGL]. One then forms (by the standard functional calculus) the unbounded[^6] “analytic continuation” in the rapidity $\frak{u}(\chi\rightarrow i\chi)$ which leads to unbounded positive operators. Using a notation which harmonizes with that of the modular theory (see appendix A), we define the following operators in $H_{Wig}$ $$\begin{aligned} & \frak{s}=\frak{\ \frak{j}}\delta^{\frac{1}{2}}\label{pol}\\ \frak{\ \frak{j}} & =\frak{u}(r)\nonumber\\ & \delta^{it}=\frak{u}(\chi=-2\pi t)\nonumber\\ & \left( \frak{s}\psi\right) (p)=\psi(-p)^{\ast} \label{s}$$ Note that all the operators are functional-analytically extended geometrically defined objects within the Wigner theory; in particular the last line is the action of an unbounded involutive $\frak{s}$ on Wigner wave functions which involves complex conjugation as well as an “analytic continuation” into the negative mass shell. Note that $\frak{u}(r)$ is apart from a $\pi$-rotation around the x-axis the one-particle version of the TCP operator. The last formula for $\frak{s}$ would look the same even if we would have started from another wedge $W\neq W_{0}.$ This is quite deceiving since physicists are not accustomed to consider the domain of definition as an integral part of the definition of the operator. If the formula would describe a bounded operator the formula would define the operator uniquely but in the case at hand $dom\frak{s\equiv}dom\frak{s}_{W_{0}}\neq dom\frak{s}_{W}$ for $W_{0}\neq W$ since the domains of $\delta_{W_{0}}$ and $\delta_{W}$ are quite different; in fact the geometric positions of the different $W^{\prime}s$ can be recovered from the $\frak{s}^{\prime}s$. All Tomita S-operators are only different in their domains but not in their formal appearance; this makes modular theory a very treacherous subject. The content of (\[pol\]) is nothing but an adaptation of the spatial version of the Bisognano-Wichmann theorem to the Wigner one-particle theory [@JMP][@BGL]. The former is in turn a special case of Rieffel’s and van Daele’s spatial generalization [@Rieffel] of the operator-algebraic Tomita-Takesaki modular theory (see appendix A). Since the antiunitary $t$-$z$ reflection commutes with the $t$-$z$ boost $\delta^{it}$, it inverts the unbounded ($\delta^{i})^{-i}=\delta$ i.e. $\frak{j}\delta=\delta^{-1}\frak{j.}$ As a result of this commutation relation, the unbounded antilinear operator $\frak{s}$ is involutive on its domain of definition i.e. $\frak{s}^{2}\subset1$ so that it may be used to define a real subspace (closed in the real sense i.e. its complexification is not closed) as explained in the appendix. The definition of $H_{R}(W_{0})$ is in terms of +1 eigenvectors of $\frak{s}$ $$\begin{aligned} H_{R}(W_{0}) & =clos\left\{ \psi\in H_{Wig}|\,\frak{s}\psi=\psi\right\} \\ & =clos\left\{ \psi+\frak{s}\psi|\,\psi\in dom\frak{s}\right\} \nonumber\\ \frak{si\psi} & =\frak{-i\psi},\text{ }\psi\in H_{R}(W_{0})\nonumber\end{aligned}$$ The +1 eigenvalue condition is equivalent to analyticity of $\delta^{it}\psi$ in $i\pi<Imt<0$ (and continuity on the boundary) together with a reality property relating the two boundary values on this strip. The localization in the opposite wedge i.e. the $H_{R}(W^{opp})$ subspace turns out to correspond to the symplectic (or real orthogonal) complement of $H_{R}(W)$ in $H_{Wig}$ i.e.$$\operatorname{Im}(\psi,H_{R}(W_{0}))=0\Leftrightarrow\psi\in H_{R}(W_{0}^{opp})\equiv\frak{j}H_{R}(W_{0})$$ One furthermore finds the following properties for the subspaces called “standardness” $$\begin{aligned} & H_{R}(W_{0})+iH_{R}(W_{0})\,\,is\,\,dense\,\,in\,\,H_{Wig}\\ & H_{R}(W_{0})\cap iH_{R}(W_{0})=\left\{ 0\right\} \nonumber\end{aligned}$$ For completeness we sketch the proof. The closedness of the densely defined $\frak{s}$ leads to the following decomposition of the domain $dom\frak{s}$$$\begin{aligned} dom(\frak{s}) & =\left\{ \psi\in H_{Wig}|\,\psi=\frac{1}{2}\left( \psi+\frak{s\psi}\right) +\frac{i}{2}\left( \psi-\frak{s\psi}\right) \right\} \\ & =H_{R}(W_{0})+iH_{R}(W_{0})\nonumber\end{aligned}$$ On the other hand from $\psi\in H_{R}(W_{0})\cap iH_{R}(W_{0})$ one obtains $$\begin{aligned} \psi & =\frak{s}\psi\\ i\psi & =\frak{si\psi=-is\psi=-i\psi}\nonumber\end{aligned}$$ from which $\psi=0$ follows. In the appendix it was shown that vice versa the standardness of a real subspace $H_{R}$ leads to the modular objects $\frak{j},\delta$ and $\frak{s}$. Since the Poincaré group acts transitively on the $W^{\prime}s$ and carries the $W_{0}$-affiliated $\frak{u}(\Lambda_{W_{0}}(\chi)),\frak{u}(r_{W_{0}})$ into the corresponding $W$-affiliated L-boosts and reflections, the subspaces $H_{R}(W)\,$have the following covariance properties $$\begin{aligned} \frak{u}(\Lambda,a)H_{R}(W_{0}) & =H_{R}(W=\Lambda W_{0}+a)\\ \frak{s}_{W} & =\frak{u}(\Lambda,a)s_{W_{0}}\frak{u}(\Lambda,a)^{-1}\nonumber\end{aligned}$$ Having arrived at the wedge localization spaces, one may construct localization spaces for smaller spacetime regions by forming intersections over all wedges containing this region $\mathcal{O}$ $$H_{R}(\mathcal{O})=\bigcap_{W\supset\mathcal{O}}H_{R}(W) \label{int}$$ These spaces are again standard and covariant. They have their own “pre-modular” (see the appendix on the spatial theory, the true Tomita modular operators appear in the next section) object $\frak{s}_{\mathcal{O}}$ $\frak{\ }$and the radial and angular part $\delta_{\mathcal{O}}$ and $\frak{j}_{\mathcal{O}}$ in their polar decomposition (\[pol\]), but this time their action cannot be described in terms of spacetime diffeomorphisms since for massive particles the action is not implemented by a geometric transformation in Minkowski space. To be more precise, the action of $\delta_{\mathcal{O}}^{it}$ is only local in the sense that $H_{R}(\mathcal{O})$ and its symplectic complement $H_{R}(\mathcal{O})^{\prime }=H_{R}(\mathcal{O}^{\prime})$ are transformed onto themselves (whereas $\frak{j}$ interchanges the original subspace with its symplectic complement), but for massive Wigner particles there is no geometric modular transformation (in the massless case there is a modular diffeomorphism of the compactified Minkowski space). Nevertheless the modular transformations $\delta _{\mathcal{O}}^{it}$ for $\mathcal{O}$ running through all double cones and wedges (which are double cones “at infinity”) generate the action of an infinite dimensional Lie group. Except for the finite parametric Poincaré group (or conformal group in the case of zero mass particles) the action is partially “fuzzy” i.e. not implementable by a diffeomorphism on Minkowski spacetime but still being the product of modular group action where each factor respects the causal closure (causal “horizon”) of a region $\mathcal{O}$ (more precisely: it is asymptotically gemometric near the horizon). The emergence of these *fuzzy acting Lie groups is a pure quantum phenomenon*; there is no analog in classical physics. They describe hidden symmetries [@SW1][@S-W3] which the Lagrangian formalism does not expose. Note also that the modular formalism characterizes the localization of subspaces. In fact for the present $(m,s=0)$ Wigner representations the spaces $H_{R}(\mathcal{O})$ have a simple description in terms of Fourier transforms of spacetime-localized test functions. In the selfconjugate case one finds $$H_{R}(\mathcal{O})=rclos\left\{ \psi=E_{m}\tilde{f}\,|f\in\mathcal{D}(\mathcal{O}),f=f^{\ast}\right\} \label{supp}$$ where the closure is taken within the real subspace i.e. one imposes the reality condition $f=f^{\ast}$ in the mass-shell restriction corresponding to a projector $E_{m}$ acting on the Fourier transform i.e. ($E_{m}\tilde {f})(p)=\left( E_{m}\tilde{f}\right) ^{\ast}(-p),\,p^{2}=m^{2},p_{0}>0.$ This space may also be characterized in terms of a closure of a space of entire functions with a Pailey-Wiener asymptotic behaviour. From these representations (\[int\]\[supp\]) it is fairly easy to conclude that the inclusion-preserving maps $\mathcal{O}\rightarrow H_{R}(\mathcal{O})$ are maps between orthocomplemented lattices of causally closed regions (with the complement being the causal disjoint) and modulare localized real subspaces (with the simplectic or real orthogonal complement). In particular one finds $H_{R}(\mathcal{O}_{1}\cap\mathcal{O}_{2})=H_{R}(\mathcal{O}_{1})\cap H_{R}(\mathcal{O}_{2}).$ The complement of this relation is called the additivity property which is an indispensible requirement if the Global is obtained by piecing together the Local. The dense subspace $H(W)=H_{R}(W)+iH_{R}(W)$ of $H_{Wig}$ changes its position within $H_{Wig}$ together with $W.$ If one would close it in the topology of $H_{Wig}$ one would loose all this subtle geometric information encoded in the $\frak{s}$-domains. One must change the topology in such a way that the dense subspace $H(W)$ becomes a Hilbert space in its own right. This is achieved in terms of the graph norm of $\frak{s}_{W}$ (for the characterization of the $H_{R}(\mathcal{O})$ in terms of test function (\[supp\]) one did not need the $\frak{s}$-operator $$\left( \psi,\psi\right) _{G\frak{s}}\equiv\left( \psi,\psi\right) +\left( \frak{s}\psi,\frak{s}\psi\right) <\infty$$ This topology is simply an algebraic way of characterizing a Hilbert space which consists of localized vectors only. It is easy to write down a modified measure in which the $\frak{s}$ becomes a bounded operator $$\begin{aligned} \left( \psi,\psi\right) _{ther} & =\int\psi^{\ast}(\theta,p_{\perp})\frac{1}{\delta-1}\psi(\theta,p_{\perp})d\theta\\ \psi(\theta,p_{\perp}) & =\psi(p),\,\,p=(m_{eff}\cosh\theta,p_{\perp },m_{eff}\sinh\theta)\nonumber\end{aligned}$$ Clearly $\delta=\frak{s}^{\ast}\frak{s}$ and $1+\delta$ are bounded in this norm. Defining the Fourier transform $$f(\theta)=\frac{1}{\sqrt{2\pi}}\int\tilde{f}(\kappa)e^{i\kappa\theta}$$ The modification takes on the appearance of a *thermal* Bose factor at temperature $T=2\pi$ with the role of the Hamiltonian being played by the Lorentz boost generator $K$ in $\delta=e^{-2\pi K}$ (which is the reason for using the subscript $ther).$ In fact the Wigner one-particle theory preempts the fact that the associated free field theory in the vacuum state restricted to the wedge becomes thermal i.e. satisfies the KMS condition and the thermal inner product becomes related to the two-point-function of that wedge restricted QFT. We have taken a wedge because then the modular Hamiltonian K has a geometric interpretation in terms of the L-boost, but the modular Hamiltonian always exists; if not in a geometric sense then as a fuzzy transformation which fixes the localization region and its causal complement. Hence for any causally closed spacetime region $\mathcal{O}$ and its nontrivial causal complement $\mathcal{O}^{\prime}$ there exists such a thermally closed Hilbert space of localized vectors and for the wedge $W$ this preempts the Unruh-Hawking effect associated with the geometric Lorentz boost playing the role of a Hamiltonian (in case of $(m=0,s=$halfinteger$)$ representations this also holds for double cones since they are conformally equivalent to wedges). After having obtained some understanding of modular localization it is helpful to highlight the difference between N-W and modular localization by a concrete illustration. Consider the energy momentum density in a one-particle wave function of the form $\psi_{f}=E_{m}f\in H_{R}(\mathcal{O})\,$ where $suppf\subset\mathcal{O}$, $f$ real$$\begin{aligned} t_{\mu\nu}(x,\psi) & =\partial_{\mu}\psi_{f}(x)\partial_{\nu}\psi _{f}(x)+\frac{1}{2}g_{\mu\nu}(m^{2}\psi_{f}(x)^{2}-\partial^{\nu}\psi _{f}(x)\partial_{\nu}\psi_{f}(x))\label{exp}\\ & =\left\langle f,c\left| :T_{\mu\nu}(x):\right| f,c\right\rangle ,\,\,\left| f,c\right\rangle \equiv W(f)\left| 0\right\rangle \,\,\nonumber\end{aligned}$$ where on the right hand side we used the standard field theoretic expression for the expectation value of the energy-momentum density in a coherent state obtained by applying the Weyl operator corresponding to the test function $f$ to the vacuum. Since $\psi_{f}(x)=\int\Delta(x-y,m)f(y)d^{4}y$ we see that the one-particle expectation (\[exp\]) complies with Einstein causality (no superluminal propagation outside the causal influence region of $\mathcal{O}),$ but there is no way to affiliate a projector with the subspace $H_{R}(\mathcal{O})$ or with coherent states (the real projectors appearing in the appendix are really unbounded operators in the complex sense). We also notice that as a result of the analytic properties of the wave function in momentum space the expectation value has crossing properties, i.e. it can be analytically continued to a matrix element of T between the vacuum and a modular localized two-particle two-particle state. This follows either by explicit computation or by using the KMS property on the field theoretic interpretation of the expectation value. A more detailed investigation shows that the appearance of this crossing (vacuum polarization) structure and the absence of localizing projectors are inexorably related. This property of the positive energy Wigner representations preempts a generic property of local quantum physics: *relativistic localization cannot be described in terms of (complex) subspaces and projectors, rather this must be done in terms of expectation values of local observables in modular localized states which belong to real subspaces.* The use of the inappropriate localization concept is the prime reason why there have been many misleading papers on “superluminal propagation” in which Fermi’s result that the classical relativistic propagation inside the forward light cone continues to hold in relativistic QFT was called into question (for a detailed critical account see [@Bu-Yng]). On the more formal mathematical level this absence of localizing projectors is connected to the absence of pure states and minimal projectors in the local operator algebras. The standard framework of QM and the concepts of “quantum computation” simply do not apply to the local operator algebras since the latter are of von Neumann type $III_{1}$ hyperfinite operator algebras and not of the quantum mechanical type $I$. Therefore it is a bit misleading to say that local quantum physics is just QM with the nonrelativistic Galilei group replaced by Poincaré symmetry; these two requirements would lead to the relativistic QM mentioned in the previous section whereas QFT is characterized by micro-causality of observables and modular localization of states. To avoid any misunderstanding, projectors in compact causally closed local regions $\mathcal{O}$ of course exist, but they necessarily describe fuzzy (non sharp) localization within $\mathcal{O}$ [@Hor] and the vacuum is necessarily a highly entangled temperture state if restriceted via this projector (in QM spatial restrictions only create isotopic representations i.e. enhanced multiplicities but do not cause genuine entanglement or thermal behavior). It is interesting that the two different localization concepts have aroused passionate discussions in philosophical circles as evidenced e.g. from bellicose sounding title as “Reeh-Schlieder defeats Newton-Wigner” in [@Halvor]. As it should be clear from our presentation particle physics finds both very useful, the first for causal (non-superluminal) propagation and the second for scattering theory where only asymptotic covariance and causality is required. After having made pedagogical use of the simplicity of the scalar neutral case in order to preempt some consequences of the modular aspects of QFT on the level of the Wigner one-particle theory, it is now easy to add the modifications which one has to make for charged scalar particles and those with nonzero spin. The Wigner representation of the connected part of a Poincaré group describes only one particle, so in order to incorporate the antipartice which has identical Poincaré properties one just doubles the Wigner space and defines the $\frak{j}$ and the $\frak{s}$ as follows (still spin-less) $$\begin{aligned} & \left( \frak{j}\psi\right) (p)=\psi^{c}(rp),\,\,\left( \frak{s}\psi\right) (p)=\psi^{c}(-p)\\ \psi(p) & =\left( \begin{array} [c]{c}\psi_{1}(p)\\ \psi_{2}(p) \end{array} \right) ,\,\psi^{c}(p)=\left( \begin{array} [c]{c}\psi_{2}(p)^{\ast}\\ \psi_{1}(p)^{\ast}\end{array} \right) \nonumber\\ \psi^{c}(p) & =C\psi(p)^{\ast},\,\,C=\left( \begin{array} [c]{cc}0 & 1\\ 1 & 0 \end{array} \right) \nonumber\end{aligned}$$ It is then easy to see that $\frak{s}$ has a polar decomposition as before in terms of $j$ and a Lorentz boost $\frak{s}=\frak{j\delta.}$ The real subspaces resulting from closed +1 eigenstates of $\frak{s}$ are $$H_{R}(W)=rclos\left\{ \psi(p)+\psi^{c}(-p)|\,\psi\in dom\frak{s}\right\}$$ where the real closure is taken with respect to real linear combinations. Again the subspaces $H_{R}(\mathcal{O})$ defined by intersection as in (\[int\]) admit a representation in terms of real closures of (mass shell projected, two-component, C-conjugation-invariant) $\mathcal{O}$-supported test function spaces as in (\[supp\]). However it would be misleading to conclude from this spinless example that modular localization in positive energy Wigner representations theory is always quite that simple. For nontrivial halfinteger spin massive particles the 2s+1 component wave function transform according to $$\begin{aligned} \left( \frak{u}(\tilde{\Lambda},a)\psi\right) (p) & =e^{iap}D^{(s)}(\tilde{R}(\Lambda,p))\psi(\Lambda^{-1}p)\label{Wigner}\\ \tilde{R}(\Lambda,p) & =\alpha(L(p))\alpha(\Lambda)\alpha(L^{-1}(\Lambda^{-1}p))\nonumber\\ \alpha(L(p)) & =\sqrt{\frac{p^{\mu}\sigma_{\mu}}{m}}\nonumber\end{aligned}$$ Here $\alpha$ denotes the SL(2.C) covering (transformation of undotted fundamental spinors) and $\tilde{R}(\Lambda,p)\,$ is an element of the (covering of the) “little group” which is the fixed point subgroup[^7] of the chosen reference vector $p_{R}=(m,0,0,0)$ on the $(m>0,s)$ orbit. $L(p)$ is the chosen family of boosts which transform $p_{R}$ into a generic $p$ on the orbit. The fixed point group for the case at hand is the quantum mechanical rotation group i.e. $\tilde{R}(\Lambda,p)\in SU(2)$ and the $D$-operators are representation matrices $D^{(s)}$ of $SU(2)$ obtaines by symmetrizing the 2s-fold SU(2) tensor products. For $s=\frac{n}{2},n\,$odd$,$ the Wigner matrices $\tilde{R}(\Lambda_{W_{0}}(-2\pi t),p)$ enter the definition of the operator $\frak{s}$ and they generally produce a square-root cut in the analytic strip region. As a representative case of halfinteger spin we consider the case of a selfdual massive $s=\frac{1}{2}$ particle. The fact that the $SU(2)$ Wigner rotation is only pseudo-real i.e. that the conjugate representation (although being $i\sigma_{2}$-equivalent to the defining one, there is no equivalence transformation which makes them identical) forces us to double order deal with selfconjugate Wigner transformation matrices$$\begin{aligned} \psi_{d} & :=\frac{1}{2}\left( \begin{array} [c]{cc}1 & 1\\ -i & i \end{array} \right) \left( \begin{array} [c]{c}\psi_{1}\\ i\sigma_{2}\psi_{2}\end{array} \right) ,\,\\ \psi_{d} & \rightarrow D_{d}\psi_{d},\,\,D_{d}=\left( \begin{array} [c]{cc}\operatorname{Re}D & \operatorname{Im}D\\ -\operatorname{Im}D & \operatorname{Re}D \end{array} \right) \nonumber\end{aligned}$$ where $D$ denote the original $SU()$-valued Wigner transformation matrices. Therefore the representation space will be represented by $4\times2$ component spinor$$\Psi(p)=\left( \begin{array} [c]{c}\psi_{d}^{(1)}(p)\\ \psi_{d}^{(2)}(p) \end{array} \right) \overset{C}{\longrightarrow}\Psi^{C}(p)=\left( \begin{array} [c]{c}\psi_{d}^{(2)}(p)\\ \psi_{d}^{(1)}(p) \end{array} \right)$$ so that the definition for the spatial Tomita operator$$\begin{aligned} \frak{s}\Psi(p) & =\Psi^{C}(-p)\\ H_{R}(W) & =\left\{ \Psi(p)|\,\frak{s}\Psi(p)=\Psi(p)\right\} \curvearrowright\psi_{d}^{(1)}(p)=\psi_{d}^{(2)}(-p)^{\ast}\nonumber\end{aligned}$$ complies with the conjugacy properties of the Wigner transformations. For selfconjugate (Majorana) particles one has in addition $\psi_{1}=\psi_{2}.$ The original Wigner transformation $D$ (\[Wigner\]) contains the t-dependent 2$\times2$ matrix which in Pauli matrix notation reads $$\frac{1}{\sqrt{m}}\left( \cosh2\pi t\cdot p^{0}\mathbf{1}-\sinh2\pi t\cdot p^{1}\sigma_{1}+p^{2}\sigma_{2}+p^{3}\sigma_{3}\right) ^{\frac{1}{2}}$$ which in the analytic continuation $t\rightarrow z$ develops a square root cut in the would-be analytic strip $-i\pi<z<0.$ This square root cut in $D_{d}$ complicates the description of the domain $dom\frak{s.}$ The only way to retain strip analyticity in the presence of the Wigner transformation law is to have a compensating singularity in the transformed wave function $\Psi(\Lambda_{W_{0}}(-2\pi t)p)$ as t is continued into the strip. This is achieved by factorizing the Wigner wave function in terms of intertwiners $\alpha$. Let us make the following ansatz for the original 2-component Wigner wave function$$\begin{aligned} & \psi(p)=\alpha(L(p))\left( E_{m}\Phi\right) (p)\\ & \alpha(L(p))=\sqrt{\frac{p^{\mu}\sigma_{\mu}}{m}}\nonumber\\ & \tilde{R}(\Lambda,p)\alpha(L(\Lambda^{-1}p))=\alpha(L(p))\alpha (\Lambda)\nonumber\end{aligned}$$ where in the last line we wrote the intertwining relation for the intertwining matrix $\alpha(L(p)).$ $\Phi_{\alpha}(x)\in\mathcal{D}(W_{0}),\alpha=1,2$ is a two-component space of test functions with support in the standard wedge $W_{0}.$ Such test functions whose associated Fourier transformed wave functions projected onto the mass shell $\left( E_{m}\Phi\right) (p)$ obviously fulfill the strip analyticity are interpreted as (undotted) spinors i.e. they are equipped with the transformation law $$\Phi(x)\rightarrow\alpha(\Lambda)\Phi(\Lambda x),\,\alpha(\Lambda)\in SL(2,C)$$ The covariant (undotted) spinorial transformation law[^8] changes the support in a geometric way. As a consequence of  group theory, the spinor wave function defined by (with $E_{m}$ a mass shell projector as before and $\frak{u}(p)$ intertwiner matrix $\frak{u}(p)=$ transforms according to Wigner as $$\psi(p)\rightarrow\alpha(\tilde{R}(\Lambda,p))\frak{u}(\Lambda^{-1}p)(E_{m}\Phi)(\Lambda^{-1}p)=\frak{u}(p)\alpha(\Lambda)\psi(\Lambda^{-1}p)$$ where in the second line we wrote the intertwining law of $\frak{u}(p)=\alpha(L(p))$ of which the first line is a consequence. We see that the product Ansatz $\psi=uE_{m}\Phi$ solves the problem of the strip analyticity since the $\frak{u}(p)$ factor develops a square root cut which compensates that of the Wigner rotation and $E_{m}\Phi$ is analytic from the wedge localization of $\Phi$. The test function space provides a dense set in $H_{R}(W)$ so by adding limits, one obtains all of $H_{R}(W)$ i.e. all the full +1 eigenspace of $\frak{s.}$ In fact this Ansatz avoids the occurance of singular pre-factor for any causally complete localization region $\mathcal{O};$ in the compact case the closure of the test function space turns out to be a space of entire functions with an appropriate Pailey-Wiener-Schwartz asymptotic behaviour reflecting the size of the double cones $\mathcal{O}.$ Although our analyticity discussion was done on the original Wigner representation, it immediatly carries over to the doubled version which we have used for the construction of the real modular subspaces $H_{R}(W).$ Again $H(W)=H_{R}(W)+iH_{R}(W)$ will be dense in $H_{Wig}$ for the same reason as in the cases before. To obtain the solution for arbitrary halfinteger spin one only has to use symmetrized tensor representations of $SL(2,C)$ and its $SU(2)$ subgroup. If we now try to represent our $\frak{s}$-operator as $j\Delta^{\frac{1}{2}}$in terms of geometrically defined reflections and boosts we encounter a surprise; the geometrically defined object is different by a phase factor $i.$ This factor results from the analytically continued Wigner rotation in the boost parameter for all halfinteger spins. The only way to compensate it consistent with the polar decomposition is to say that the $j$ deviates from the geometric $j_{0}$ by a phase factor $t$$$j=tj_{0},\,\,t=i$$ It turns out that this also happens for the exeptional Wigner representations; for d=1+2 anyons one obtains a phase factor related to the spin of the anyon whereas for the d$>$1+3 spin towers $t$ is an operator in the infinite tower space related to the analytically continued infinite dimensional Wigner matrix. These cases are characteized by the failure of compact modular localization (see below). The modular localization in the massless case is similar as long as the helicity stays finite (trivially represented Euclidean “translations”) is similar. The concrete determination of the $\Lambda,p$-dependent $\tilde{R}$ requires a selection of a family of boosts i.e. of Lorentz transformations $\tilde{L}(p)$ which relate the reference vector $p_{R}$ uniquely a general $p$ on the respective orbit. The natural choice for the associated $2\times2$ matrices in case of d=1+3 is (we use $\alpha$ for the $SL(2,C)$ representation) $$\alpha(\tilde{L}(p)_{0})=\frac{1}{\sqrt{p_{0}+p_{3}}}\left( \begin{array} [c]{cc}p_{0}+p_{3} & p_{1}-ip_{2}\\ 0 & 1 \end{array} \right) ,\,m=0 \label{prev}$$ with the associated little groups being $SU(2)$ or for m=0 $\tilde{E}(2)\,$ (the 2-fold covering of the 2-dim. Euclidean group) $$\tilde{E}(2):\left( \begin{array} [c]{cc}e^{i\frac{\varphi}{2}} & z=a+ib\\ 0 & e^{-i\frac{\varphi}{2}}\end{array} \right) ,\,\,\,m=0$$ For the standard (halfinteger helicity) massless representations the “z-translations” are mapped into the identity. As a result of the projection property of the reference vector there exists a projected form of the intertwining relation ($\alpha(\tilde{L}(p))$ as in (\[prev\])) $$\begin{aligned} & \textsl{p}_{R}\tilde{R}(\Lambda,p)=\textsl{p}_{R}\tilde{R}(\Lambda ,p)_{11}\\ & \tilde{R}(\Lambda,p)=\alpha(\tilde{L}(p))\tilde{\Lambda}\alpha(\tilde {L}^{-1}(\Lambda^{-1}p))\nonumber\end{aligned}$$ This projection allows to incorporate the one-component formalism into the SL(2,C) matrix formalism. In fact this embedding permits to use the same mass independent $W$-supported test function spaces as before, one only has to replace the $E_{m}$ projectors by projectors on the zero mass orbit. Again the definition of $\frak{j}$ generally demands a further doubling of the test function. At the end one obtains a representation of modular localization spaces $H_{R}(W)$ (and more generally $H_{R}(\mathcal{O})$ for double cones $\mathcal{O}$) in terms of $W$ or $\mathcal{O}$ supported spinorial test function spaces whose nontriviality is secured by the classical Schwartz distribution theory. It is easy to see that the modular formalism also works for halfinteger spin in d=1+2 dimensions. In this case one can work with the same $2\times2$ matrix model, we only have to restrict $SL(2,R)$ to $SL(2,R)\simeq SU(1,1)$ which is conveniently done by omitting the $\sigma_{2}$ Pauli matrix. Choosing again the rest frame reference vector we obtain $$\begin{aligned} \tilde{L}(p) & =+\sqrt{\frac{p^{\mu}\sigma_{\mu}}{m}},\,m>0,\text{ }\sigma_{2}\text{ }omitted\\ \tilde{L}(p) & =\frac{1}{\sqrt{p_{0}+p_{3}}}\left( \begin{array} [c]{cc}p_{0}+p_{3} & p_{1}\\ 0 & 1 \end{array} \right) ,\,m=0\nonumber\end{aligned}$$ with the little group $G_{l}$ being the abelian rotation or the abelian “translation” group respectively. $$\begin{aligned} \textsl{gp}_{R}\textsl{g}^{\ast} & =\textsl{p}_{R}\\ G_{l} & :\textsl{g}=\left( \begin{array} [c]{cc}\cos\frac{1}{2}\Omega & \sin\frac{1}{2}\Omega\\ \sin\frac{1}{2}\Omega & \cos\frac{1}{2}\Omega \end{array} \right) ,\,m>0\nonumber\\ G_{l} & :\textsl{g}=\left( \begin{array} [c]{cc}1 & b\\ 0 & 1 \end{array} \right) ,\,m=0\nonumber\end{aligned}$$ In order to preserve the analogy in the representations, we take halfinteger spin representations in the first case and trivial representation of the little group in the massless case. Whereas the massless case has a modular wedge structure like the scalar case, the modular structure of the (m,s) case is solved by a u-intertwiner as in the previous d=1+3 case. We have and will continue to refer to these representations with finite (half)integer finite spin as “standard”. Their modular localization spaces $H_{R}(\mathcal{O})$ can be described in terms of classical $\mathcal{O}$-supported test functions. The remaining cases, here called “exceptional”, will be treated in the next subsection. They include the d=1+2 “anyonic” spin of massive particles as well as massless cases with faithful representations of the little group in any spacetime dimension $d\geq1+2.$ For $d\geq1+3$ they are identical to the famous Wigner spin towers where infinitely many spins (like in a dynamical string) are combined in one irreducible representation. We will see that for these exceptional representations the best possible modular localization is noncompact and generally not susceptible to a classical description in terms of support properties of functions. This preempts the more noncommutative properties of the associated QFTs which are outside of Lagrangian quantization. Exceptional cases: anyons and infinite “spin towers” ---------------------------------------------------- The special role of d=1+2 spacetime dimensions for the existence of braid group statistics is due to the fact that the universal covering is infinite sheeted and not two-fold as considered in the previous section. The fastest way to obtain a parametrization of the latter is to use the Bargmann [@Barg] parametrization$$\left\{ \left( \gamma,\omega\right) |\,\gamma\in\mathbb{C},\left| \gamma\right| <1,\,\omega\in\mathbb{R}\right\}$$ for the two-fold matrix covering $$\frac{1}{\sqrt{1-\gamma\bar{\gamma}}}\left( \begin{array} [c]{cc}e^{i\frac{\omega}{2}} & \gamma e^{i\frac{\omega}{2}}\\ \bar{\gamma}e^{-i\frac{\omega}{2}} & e^{-i\frac{\omega}{2}}\end{array} \right)$$ It is then easy to abstract the multiplication law for the universal covering from this matrix model $$\begin{aligned} & \left( \gamma_{2},\omega_{2}\right) \left( \gamma_{1},\omega_{1}\right) =(\gamma_{3},\omega_{3})\\ & \gamma_{3}=\frac{\left( \gamma_{1}+\gamma_{2}e^{-i\frac{\omega}{2}_{1}}\right) }{\left( 1+\gamma_{2}\bar{\gamma}_{1}e^{-\frac{\omega}{2}_{1}}\right) }\nonumber\\ e^{i\frac{\omega_{3}}{2}} & =e^{i\frac{\omega_{1}+\omega_{2}}{2}}\left( \frac{1+\gamma_{2}\bar{\gamma}_{1}e^{-i\frac{\omega}{2}_{1}}}{1+\gamma _{2}\gamma_{1}e^{i\frac{\omega}{2}_{1}}}\right) ^{\frac{1}{2}}\nonumber\end{aligned}$$ &gt;From these composition laws one may obtain the irreducible transformation law of a (m,s)Wigner wave functions in terms of a one-component representation involving a Wigner phase $\varphi((\gamma,\omega),p)$ But there are some quite interesting and physically potentially important positive energy representations for which the above covariantization does not work and the $H_{R}(\mathcal{O})$ do not have such a geometric description i.e. the modular localization is more ”quantum” than geometric. These exceptional representations include $d=1+2$ spin$\neq$halfinteger anyons and the still somewhat mysterious $d\geq1+3$ massless “infinite spin-tower” (called “continuous spin” by Wigner, unfortunately a somewhat misleading name). These are the cases which also resist Lagrangian quantization attempts. However the modular localization method reveal for the first time that those representations do not allow a compact (with pointlike as limiting case) localization in fact these cases are only consistent with a noncompact modular localization which extends to infinity. The associated multiparticle spaces do not have the structure of a Fock space and the localized operators describing creation and annihilation are too noncommutative for a Lagrangian quantization interpretation. Before we look at those special cases let us note that the localization in wedges and in certain special intersection of two wedges is a general property of all positive energy representations of $\mathcal{P}_{+}$. The above proof of standardness of the $\frak{s}$ operator only uses general properties of the boost and the r reflection which are evidently true in each positive energy representation of the extended Poincaré group $\widetilde{\mathcal{P}}_{+}.$ A bit more tricky is the nontriviality of the following intersected spaces (Guido and Longo [@GL]) Let W$_{1}$ and W be orthogonal wedges (in the sense of orthogonality of their spacelike edges) and define $W_{2}=\Lambda _{W}(-2\pi t)W_{1}.$ Then $H(W_{1}\cap W_{2})\equiv H_{R}(W_{1}\cap W_{2})+iH_{R}(W_{1}\cap W_{2})$ is dense in the positive energy representation space $\mathcal{P}_{+}.$ The size of the intersection decreases with increasing t. It is conic with apex at the origin, but it does not look like a spacelike cone since it contains lightlike rays (for t$\rightarrow\infty$ its core is a lightlike string). &gt;From the assumptions one obtains a geometric expression for $\frak{s}_{2}\frak{s}_{1}$$$\frak{s}_{2}\frak{s}_{1}=\Delta_{W_{2}}^{it}\Delta_{W_{1}}^{-\frac{1}{2}}\Delta_{W_{2}}^{it}\Delta_{W_{1}}^{\frac{1}{2}}$$ where we used the orthogonality assumption via $\frak{j}_{W_{1}}\Delta_{W_{2}}^{it}$ $\frak{j}_{W_{1}}=\Delta_{W_{2}}^{-it}.$ The claimed density is equivalent to the denseness of the subspace: $$\left\{ \psi|\,\frak{s}_{2}\frak{s}_{1}\psi=\psi\right\} \Leftrightarrow \left\{ \psi|\,\Delta_{W_{1}}^{-\frac{1}{2}}\Delta_{W_{2}}^{it}\Delta_{W_{1}}^{\frac{1}{2}}\psi=\Delta_{W_{2}}^{-it}\psi\right\}$$ but according to a theorem in [@GL] this is a consequence of the denseness of the domain of $\Delta_{W_{1}}^{-\frac{1}{2}}\Delta_{W_{2}}^{it}\Delta_{W_{1}}^{\frac{1}{2}}$ which holds for every unitary representation of SL(2,R) which, as easily shown, is the group generated by the two orthogonal wedges. Before this theorem will be applied to the localization of the exceptional Wigner representation it is instructive to recall the argument for the lack of compact localization in these cases. Any localization beyond those of group theoretical origin requires the construction of at least partial intertwiners. Before we comment on this let us first show that in the cases of d=1+2 anyonic and d=1+3 infinite spin a compact localization is impossible (which also shows that there are no intertwiners in the previous sense). The typical causally closed simply connected compact region has the form of a double cone i.e. the intersection of the upper light cone with the lower one. Since in terms of wedges one needs infinitely many intersections, we will prove the even the larger region of the intersection of two wedges (which is infinite in transverse direction) has a trivial $H_{R}.$ In order to compute the action of $\frak{s}$ we use the Wigner cocycle (\[Wigner\]) for the t-x boost $\Lambda_{W_{0}}$ $$\begin{aligned} e^{is\Omega(\Lambda_{W_{0},},p)} & =\left( \frac{1-\gamma(p)\gamma _{t}+\left( \gamma_{t}-\gamma(p)\right) \overline{\gamma(\Lambda_{W_{0}}(-t)p)}}{c.c.}\right) ^{s}\\ & =u(p)u(\Lambda_{W_{0}}(-t)p),\,\,u(p)\equiv(\frac{p_{0}-p_{1}+m+ip_{2}}{p_{0}-p_{1}+m-ip_{2}})^{s}\nonumber\end{aligned}$$ This formula results by specialization from the following formula for the action of the L-group on one-component massive Wigner wave functions [@Mu-S][@Mund] $$\begin{aligned} & \left( \frak{u}\psi\right) (p,s)=e^{is\Omega(\tilde{R}(\Lambda,p))}\psi(\Lambda^{-1}p)\\ & e^{is\Omega(\Lambda(\omega,\gamma),p)}=e^{is\frac{\omega}{2}}\left( \frac{1-\gamma(p)\bar{\gamma}e^{-i\frac{\omega}{2}}+(\gamma-\gamma (p)\bar{\gamma}e^{-i\frac{\omega}{2}})\bar{\gamma}(\Lambda(\gamma,\omega )^{-1}p))}{c.c.}\right) ^{s}$$ and a similar phase factor for the massless case with a faithful little group representation. In case of the d=1+3 massless spin-tower representation this is more tricky. One finds $$\begin{aligned} \left( \frak{u}(\Lambda,a)\psi\right) (p) & =e^{iap}V_{\Xi,\pm}(\tilde {R}(\Lambda,p))\psi(\Lambda^{-1}p)\\ \left( V_{\Xi,\pm}(\Lambda_{z,\varphi})f\right) (\theta) & =\left\{ \begin{array} [c]{c}\left\{ \exp i(\Xi\left| z\right| \cos(\arg z-\vartheta))\right\} f(\vartheta-\varphi)\\ \left\{ \exp i(\Xi\left| z\right| \cos(\arg z-\vartheta)+\frac{1}{2}\varphi)\right\} f(\vartheta-\varphi) \end{array} \right. \nonumber\end{aligned}$$ with the + sign corresponding to an integer valued spin tower. In this case the infinite component wave function $\psi(p)$ is a square integrable map from the momentum space mass shell to functions with values in the L$_{2}$ space on the circle (in which the noncompact $\tilde{E}(2)$ group is irreducibly represented by the last formula). $\Xi$ is an invariant (Euclidean “mass”) of the $\tilde{E}(2)$ $\,$representation$.$ Scaling the $\Xi$ to one and introducing a “spin basis” (discrete Fourier-basis) $e^{in\varphi}$, the $V_{\Xi,\pm}(\Lambda_{\varphi})$ becomes diagonal and the translational part $V_{\Xi,\pm}(\Lambda_{z})$ can be written in terms of Bessel functions $$V_{\Xi,\pm}(\Lambda_{z})_{n,m}=\left( \frac{z}{\left| z\right| }\right) ^{n-m}J_{n-m}(\Xi\left| z\right| )$$ &gt;From this one can study the analyticity behavior needed for the modular localization. The following theorem may is easily established The d=1+2 representations with s$\neq$halfinteger and the d=1+3 Wigner spin tower representations do not allow a compact double cone localization. For the spin tower this was already suggested by an ancient No-Go theorem of Yngvason [@Yng] who showed that there is an incompatibility with the Wightman setting. We will prove in fact the slightly stronger statement that the space $H_{R}(W\cap W_{a}^{^{\prime}})$ which describes the intersection of a wedge with its translated opposite (which has still a noncompact transversal extension) is trivial. This implies a fortiori the triviality of compact double cone intersections. The common origin of the weaker localization properties for the exceptional positive energy representations is the fact that the analytical continuation of the wave function to the opposite boundary of the strip (which combines together with the action of the charge-conjugating geometric involution to a would be $\frak{s}$) has in addition a matrix part (a phase factor for d=1+2) which has to be cancelled by a compensating modification of the involution part $$\frak{j=tj}_{geo}$$ The $t$, which in the case of the spin-tower is a complicated operator in the representation space of the little group, is the preempted field theoretic twist operator T whose presence shows up in commutation relations of spacelike (noncompactly) localized operators (braid group statistics in case of d=1+2). According to the second last theorem the localization in the noncompact intersection of two wedges in a selected relative position (where the second one results from applying an “orthogonal” boost to the first) is always possible for all positive energy representations in all spacetime dimensions. But only in d=1+2 this amounts to a spacelike cone localization (with a semiinfinite spacelike string as a core). In that case one knows that plektonic situations do not allow for a better localization. However there is a problem with the application of that theorem to anyons since it refers to the representation of the Poincare group in $d\geq3$ spacetime but not to its covering $\mathcal{\tilde{P}}_{+}$ in $d=3$ which would be needed for the case of anyons. Fortunately Mund has found a direct construction of spacelike cone $C$ localized subspaces $H_{R}(C)$ in terms of a partial intertwiner $u(p)$ and subspace of of doubled test functions $\Phi$ with supports in spacelike cones. If one starts from the standard $x$-$t$ wedge and wants to localize in cones which contain the negative y-axis then Mund’s localization formula and his partial $u$ (to be distinguished from the previous $\frak{u}$) are$$u(p)E_{m}\Phi,\,\,u(p)=(\frac{p^{0}-p^{1}}{m})^{s}(\frac{p_{0}-p_{1}+m+ip_{2}}{p_{0}-p_{1}+m-ip_{2}})^{s} \label{spread}$$ For spacelike cones along other axis the form of the partial intertwiner changes. Running through all $C$-localized test functions the formula describes a dense set of spacelike cone-localized Wigner wave function only for those spacelike cones which contain the negative y-axis after apex($C$) has been shifted to the origin (which includes the standard $x$-$t$ wedge as a limiting case). He then shows an interesting “spreading” mechanism namely that if one chooses a better localized function with compacr support in that region, the effect of the partial intertwiner “ is to radially extend the support to spacelike infinity. The anyonic spin Wigner representation can be encoded into many infinite dimensional covariant representations [@Mu-S] (also appendix), but this does not improve the localization since infinite dimensional covariant transformation matrices, unlike finite dimensional ones, are not entire functions of the group parameters. For d=1+3 the intersection region has at its core a 2-dimensional spacelike half-plane. There is good reason to believe that this is really the optimally possible localization for the spin-tower representation. The argument is based on converting this representation into the factorizing form $uE_{m}f$ where $u$ is the infinite dimensional intertwiner from the covariant representation (appendix) to the Wigner representation. The best analytic behavior which the unitary representation theory of the L-group (necessarily infinite dimensional) can contribute to modular localization seems to be that of the above Guido-Longo theorem. Whereas for the standard representations the support of the classical test function multiplets determine the best localization region (because the finite dimensional representations of the Lorentz group are entire analytic functions), the exceptional representations spread any test function localization which tries to go beyond those which pass through the intertwiner. This goes hand in hand with a worsening of the spacelike commutativity properties in the associated operator algebras. Therefore in the case in which the modular localization cannot be encoded into the support property of a test function multiplet, we often use the word “quantum localization”.  These are the cases which cannot not be described as a quantized classical structure or in terms of Euclidean functional integrals. As will be shown in the next section the QFT associated with such particles do not allow sub-wedge PFGs i.e. better than wedge-localized operators which applied to the vacuum create one-particle states free of vacuum polarization. Whereas in standard Boson/Fermion systems (halfinteger spin representations) the vacuum polarization is caused by the interaction (this can be used to define the intrinsic meaning of interaction for such systems), the sub-wedge vacuum polarization phenomenon associated with the QFT of the exceptional Wigner representations is of a more kinematical kind; it occurs in those other cases already without interaction; the polarization clouds are simply there to sustain e.g. the anyonic spin&statistics connection. From Wigner representations to the associated local quantum physics =================================================================== In the following we will show that such net of operator algebras of free particles with halfinteger spin/helicity can be directly constructed from the net of modular localized subspaces in standard Wigner representations. For integral spin $s$ one defines with the help of the Weyl functor $Weyl(\cdot)$ the local von Neumann algebras [@Sch1][@BGL] generated from the Weyl operators as $$\mathcal{A}(W):=alg\left\{ Weyl(f)|f\in H_{R}(W)\right\} \label{Weyl}$$ a process which is sometimes misleadingly called “second quantization”. These Weyl generators have the following formal appearance in terms of Wigner (momentum space) creation and annihilation operators and modular localized wave functions $$\begin{aligned} & H_{R}(W)\overset{\Gamma}{\rightarrow}Weyl:\,\,f\rightarrow Weyl(f)=e^{iA(f)}\\ & A(f)=\sum_{s_{3}=-s}^{s}\int(a^{\ast}(p,s_{3})f_{s_{3}}(p)+b^{\ast}(p,s_{3})f_{s_{3}}^{\ast}(-p)+h.c.)\frac{d^{3}p}{2\omega}\nonumber\end{aligned}$$ It is helpful to interprete the operator $A(f)$ as an inner product$$A(f)=\int\left( \begin{array} [c]{cc}a^{\ast}(p) & b^{\ast}(p) \end{array} \right) \left( \begin{array} [c]{c}f(p)\\ f^{\ast}(-p) \end{array} \right) \frac{d^{3}p}{2\omega}+h.c \label{Segal}$$ of an operator bra with a ket vector of a 2$\times(2s+1)$ eigenfunction of $\frak{s}$ representing a vector in $H_{R}(W).$ The formula refers only to objects in the Wigner theory; covariant fields or wave functions do not enter here. Unlike those covariant objects, the Weyl functor is uniquely related to the (m,s) Wigner representation. The special hermitian combination entering the exponent of the Weyl functor is sometimes called the I. Segal operator [@Segal]. The local net $\left\{ \mathcal{A}(\mathcal{O})\right\} _{\mathcal{O}\in\mathcal{K}}$ may be obtained in two ways, either one first constructs the spaces $H_{R}(\mathcal{O})$ via (\[int\]) and then applies the Weyl functor, or one first constructs the net of wedge algebras (\[Weyl\]) and then intersects the algebras according to $$\mathcal{A(O)}=\bigcap_{W\supset\mathcal{O}}A(W)$$ The proof of the net properties follows from the well-known theorem that the Weyl functor relates the orthocomplemented lattice of real subspaces of $H_{Wig}$ (with the complement $H_{R}^{\prime}$ of $H_{R\text{ }}$being defined in the symplectic sense of the imaginary part of the inner product in $H_{Wig})$ to von Neumann subalgebras $\mathcal{A}(H_{R})\subset \mathcal{B}(H_{Fock})$ This functorial mapping $\Gamma$ also maps the above pre-modular operators into those of the Tomita-Takesaki modular theory $$J,\Delta,S\frak{=\Gamma(\ \frak{j},\delta,s)}$$ Whereas the pre-modular operators of the spatial theory (denoted by small letters) act on the Wigner space, the modular operators $J,\Delta$ have an $Ad$ action ($AdUA\equiv UAU^{\ast}$) on von Neumann algebras in Fock space which makes them objects of the Tomita-Takesaki modular theory $$\begin{aligned} & SA\Omega=A^{\ast}\Omega,\,S=J\Delta^{\frac{1}{2}}\\ & Ad\Delta^{it}\mathcal{A}=\mathcal{A}\nonumber\\ & AdJ\mathcal{A}=\mathcal{A}^{\prime}\nonumber\end{aligned}$$ The operator $S$ is that of Tomita i.e. the unbounded densely defined normal operator which maps the dense set $\left\{ A\Omega|\,A\in\mathcal{A}(W)\right\} $ via $A\Omega\rightarrow A^{\ast}\Omega$ into itself and gives $J$ and $\Delta^{\frac{1}{2}}$ by polar decomposition. The nontrivial miraculous properties of this decomposition are the existence of an automorphism $\sigma_{\omega}(t)=Ad\Delta^{it}$ which propagates operators within $\mathcal{A}$ and only depends on the state $\omega$ (and not on the implementing vector $\Omega)$ and a that of an antiunitary involution $J$ which maps $\mathcal{A}$ onto its commutant $\mathcal{A}^{\prime}.$ The theorem of Tomita assures that these objects exist in general if $\Omega$ is a cyclic and separating vector with respect to $\mathcal{A}.$ An important thermal aspect of the Tomita-Takesaki modular theory is the validity of the Kubo-Martin-Schwinger (KMS) boundary condition [@Haag] $$\omega(\sigma_{t-i}(A)B)=\omega(B\sigma_{t}(A)),\,\,A,B\in\mathcal{A} \label{KMS}$$ i.e. the existence of an analytic function $F(z)\equiv\omega(\sigma_{z}(A)B)$ holomorphic in the strip $-1<Imz<0$ and continuous on the boundary with $F(t-i)=\omega(B\sigma_{t}(A))$ or briefly (\[KMS\])$.$ The fact that the modular theory applied to the wedge algebra has a geometric aspect (with $J$ equal to the TCP operator times a spatial rotation and $\Delta^{it}=U(\Lambda_{W}(2\pi t))$) is not limited to the interaction-free theory [@Haag]. These formulas are identical to the standard thermal KMS property of a temperature state $\omega$ in the thermodynamic limit if one formally sets the inverse temperature $\beta=\frac{1}{kT}$ equal to $\beta=-1.\,$This thermal aspect is related to the Unruh-Hawking effect of quantum matter enclosed behind event/causal horizons. For halfinteger spin, the Weyl functor has to be replaced by the Clifford functor $R$. In the previous section we already noted that there exists a mismatch between the geometric and the spatial complement which led to the incorporation of an additional phase factor $i$ into the definition of $\frak{j}.$ A Clifford algebra is associated to a real Hilbert space $H_{R}$ with generators $$\begin{aligned} & R:\mathcal{S}(\mathbb{R}^{4})\rightarrow B(H_{R})\\ & \left( f,g\right) _{R}=\operatorname{Re}\left( f,g\right) \nonumber\end{aligned}$$ where the real inner product is written as the real part of a complex one. One sets $$R^{2}(f)=(f,f)_{R}\mathbb{I}$$ or $$\{R(f),R(g)\}=2(f,g)_{R}\mathbb{I}$$ where $\mathcal{S}(\mathbb{R}^{4})$ is the Schwartz space of test functions over $\mathbb{R}^{4}$ and $B(H_{R})$ is the space of bounded operators over $H_{R}$. These $R(f)$’s generates $Cliff(H_{R})$ as polynomials of $R$’s. The norm is uniquely fixed by the algebraic relation, e.g. $$||R(f)||^{2}=|R(f)^{\ast}R(f)||-||R^{2}(f)||=||f||_{R}$$ and similarly for all polynomials, i.e., on all $Cliff(H_{R})$. The norm closure of the Clifford algebra is sometimes called $CAR(H_{R})$ (canonical anti-commutation) C$^{\ast}$-algebra. It is unique (always up to C$^{\ast}$-isomorphisms) and has no ideals. This Clifford map may be used as the analog of the Weyl functor in the case of halfinteger spin. It turns out to be more useful to work with a alternative version of $CAR$ which is due to Araki: the selfdual $CAR$-algebra. In that description, the reality condition is implemented via a antiunitary involution $\Gamma$ inside the larger complex Hilbert space $H$. Now $$\begin{aligned} f & \longrightarrow B(f)\\ B(f)^{\ast} & =B(\Gamma f)\nonumber\\ \{B^{\ast}(f),B(g)\} & =(f,g)\mathbb{I}\nonumber\end{aligned}$$ is a complex linear map of $H$ into generators a normed \*-algebra whose closure is by definition the C\*-algebra $CAR(K,\Gamma)$. The previous Clifford functor results from the selfadjoint objects $B(\Gamma f)=B(f)$ or $\Gamma f=f.$ In physical terms $\Gamma$ is the charge conjugation operation $C$ which enters the definition of the $\frak{s}$-operator. The functor maps this spatial modular object into an operator of the Clifford algebra; the analog of (\[Segal\]) is $$f\in H_{R}(W)\rightarrow R(f)=\Psi\cdot f+h.c.$$ where, as explained in section 2.2, the Wigner wave function $f\in H_{R}(W)$ interpreted as a $4\times(2s+1)$ component column vector and $\Psi$ is a bra vector of Wigner creation and annihilation operators. As a consequence of the presence of a twist factor in the spatial involution $j=tj_{geo}$ one obtains a twist operator in the algebraic involution $J$ $$\begin{aligned} S & =J\Delta^{\frac{1}{2}},\,\,J=TJ_{geo}\\ T & =\frac{1-iU(2\pi)}{1-i}=\left\{ \begin{array} [c]{c}1\,\,on\text{\thinspace\thinspace}even\\ i\,\,on\,\,odd \end{array} \right. \nonumber\\ SA\Omega & =A^{\ast}\Omega,\,\,A\in\mathcal{A}(W)=a\lg\left\{ B(f)|\,f\in H_{R}(W)\right\} \nonumber\end{aligned}$$ The presence of the twist operator (which is one on the even and $i$ on the odd subspaces of $H_{Fock}$) accounts for the difference between the von Neumann commutant $\mathcal{A}(W)^{\prime}$ and the geometric opposite $\mathcal{A}(W^{\prime}).$ The bosonic CCR (Weyl) and the fermionic CAR (Clifford) local operator algebras are the only ones which permit a functorial interpretation in terms of a “quantization” of classical function algebras. In the next section we will take notice of the fact that they are also the only QFTs which possess sub-wedge-localized PFGs. In the case of d=1+2 anyonic spin representations the presence of a plektonic twist has the more radical consequences. Whereas the fermionic twist is still compatible with the existence of PFGs and free fields in Fock space, the twist associated with genuine braid group statistics causes the presence of vacuum polarization for any sub-wedge localization region. The same consequences hold for the spin tower representations. . Our special case at hand, in which the algebras and the modular objects are constructed functorially from the Wigner theory, suggest that the modular structure for wedge algebras may always have a geometrical significance associated with a fundamental physical interpretation in any QFT. This is indeed true, and within the Wightman framework this was established by Bisognano and Wichmann [@Haag]. In the general case of an interacting theory in d=1+3 with compact localization (which according to the DHR theory is necessarily a theory of interacting Bosons/Fermions) the substitute for a missing functor between a spatial and an algebraic version of modular theory is the modular map between a real subspace of the full Hilbert space $H$ and a local subalgebra of algebra of all operators $B(H).$ In a theory with asymptotic completeness i.e. with a Fock space incoming (outgoing) particle structure $H=H_{Fock}$ the scattering operator $S_{scat}$ turns out to play the role of a relative modular invariant between the wedge algebra of the free incoming operators and that of the genuine interacting situation $$\begin{aligned} J & =J_{0}S_{scat}\\ S & =S_{0}S_{scat}$$ This relation follows directly by rewriting the TCP transformation of the S-matrix and the use of the relation of $J$ with the TCP operator. The computation of the real subspaces $H_{R}(W)\in H_{Fock}$ requires diagonalization of the S-matrix. The difficult step about which presently nothing is known is the passing from these subspaces to wedge-subalgebras whose selfadjoint part applied to the vacuum generate these subspaces. Although it is encouraging that the solution of the inverse problem $S_{scat}\rightarrow\,\left\{ \mathcal{A}(\mathcal{O})\right\} _{\mathcal{O}\in\mathcal{K}}$ is unique [@S-matrix], a general formalism which takes care of the existence part of the problem is not known apart from some special but very interesting cases which will be presented in the next section. Connes has developed a theory involving detailed properties of the natural modular cones $\mathcal{P}_{\mathcal{A}(W),\Omega}$ which are affiliated with a single standard pair $(\mathcal{A}(W),\Omega)$ (the net structure is not used) but it is not clear how to relate his facial conditions on these cones to properties of local quantum physics. As a matter of fact even in the case of standard Wigner representations it is not clear how one could obtain the modular algebraic structure if one would be limited to the Connes method [@Connes] without the functorial relation. For these reasons the modular based approach which tries to use the twist/S-matrix factor in $J=J_{0}T$ respectively $J=J_{0}S_{scat}$ for the determination of the algebraic structure of $\mathcal{A}(W)$ and subsequently computes the net $\,\left\{ \mathcal{A}(\mathcal{O})\right\} _{\mathcal{O}\in\mathcal{K}}$ by forming intersections is presently limited to theories which permit only vitual but no real particle creation. Besides the exeptional Wigner representation (anyons, spin towers) which lead to a twist and changed spacelike commutation relations, the only standard (bosonic, fermionic) interacting theories are the $S_{scat}=S_{el}$ models of the d=1+1 bootstrap-formfactor setting (factorizing models). For those readers who are familiar with Weinberg’s method of passing from Wigner representation to covariant pointlike free fields, it may be helpful to add a remark which shows the connection to the modular approach. For writing covariant free fields in the (m,s) Fock space $$\begin{aligned} \psi^{\lbrack A,\dot{B}]}(x) & =\frac{1}{(2\pi)^{3/2}}\int\{e^{-ipx}\sum_{s_{3}}u(p_{1},s_{3})a(p_{1},s_{3})+\label{field}\\ & +e^{ipx}\sum_{s_{s}}v(p_{1},s_{3})b^{\ast}(p_{1},s_{3})\}\frac{d^{3}p}{2\omega}\nonumber\end{aligned}$$ where $a^{\#},b^{\#}$ are creation/annihilation opertors of Wigner (m,s) particles and $\psi^{\lbrack A,\dot{B}]}$ are covariant dotted/undotted fields in the SL(2,C) spinor formalism, it is only necessary to find intertwiners $$u(p)D^{(s)}(\tilde{R}(\tilde{\Lambda},p))=D^{[A,\dot{B}]}(\tilde{\Lambda })u(\tilde{\Lambda}^{-1}p) \label{1}$$ between the Wigner $D^{(s)}(\tilde{R}(\tilde{\Lambda},p))$ and the covariant $D^{[A,\dot{B}]}(\tilde{\Lambda})$ and these exist for all $A,\dot{B}$ which relative to the given s obey $$\mid A-\dot{B}\mid\leq s\leq A+\dot{B} \label{2}$$ For each of these infinitely many values $(A,\dot{B})$ there exists a rectangular ($2A+1)(2\dot{B}+1)\times(2s+1)$ intertwining matrix $u(p).$ Its explicit construction using Clebsch-Gordan methods can be found in Weinberg’s book [@Wei]. Analogously there exist antiparticle (opposite charge) intertwiners $v(p)$: $D^{(s)\ast}(R(\Lambda,p)\longrightarrow D^{[A,\dot{B}]}(\Lambda)$. All of these mathematically different fields in the same Fock space describe the same physical reality; they are just the linear part of a huge local equivalence class and they do not exhaust the full “Borchers class” which consists of all Wick-ordered polynomials of the $\psi^{\lbrack A,\dot{B}]}.$ They generate the same net of local operator algebras and in turn furnish the singular coordinatizations. Free fields for which the full content of formula (\[field\]) can be described by the totality of all solutions of an Euler-Lagrange equation exist for each (m,s) but are very rare (example Rarita-Schwinger for s=$\frac{3}{2}$). It is a misconception that they are needed for physical reason. The causal perturbation theory can be done in any of those field coordinates and that one needs Euler-Lagrange fields in the setting of Euclidean functional integrals is an indication that differential geometric requirements and quantum physical ones do not always go into the same direction. On the other hand our modular method for the construction of localized spaces and algebras use only the minimal intertwiners which are described by square $\left( 2s+1\right) \times\left( 2s+1\right) $ matrices. Without their use there would be no purely analytic characterization of the domain of the modular Tomita S-operator. Vacuum polarization and breakdown of functorial relations ========================================================= The functorial relation of the previous section between Wigner subspaces and operator algebras are strictly limited to the standard halfinteger spin representations for which generating pointlike free fields exist. The noncompactly localizable exceptional Wigner representations (anyonic spin, faithful spin-tower representations of the massless little group) as well as interacting theories involving standard (halfinteger spin/helicity) particles do not permit a direct functorial relations between wave function spaces and operator algebras. In order to understand the physical mechanism which prevents a functorial relation it is instructive to look directly to the operators algebras. Given an operator algebra $\mathcal{A}(\mathcal{O})$ localized in a causally closed region $\mathcal{O}$ with a nontrivial causal complement $\mathcal{O}^{\prime }$ (so that ($\mathcal{A}(\mathcal{O}),\Omega$) is standard pair) we may ask whether this algebra admits a “polarization-free-generator” (PFG) namely an affiliated possibly unbounded closed operator $G$ such that $\Omega$ is in the domain of $G,G^{\ast}$ and $G\Omega$ and $G^{\ast}\Omega$ are vectors in $E_{m}H$ with $E_{m}$ projector on the one-particle space. It turns out that if one admits very crude localizations as that in wedges then one can reconcile the standardness of the pair $(\mathcal{A}(W),\Omega)$ (i.e. physically the unique $A\Omega\leftrightarrow A\in\mathcal{A}(W)$ relationship) with the absense of polarization clouds caused by localization. For convenience of the reader we recall the abstract theorem from modular theory whose adaptation to the local quantum physical situation at hand will supply the existence of wedge-affiliated PFGs. An interesting situation emerges if these operators which always generate a dense one-particle subspace also generate an algebra of unbounded operators which is affiliated to a corresponding von Neumann algebra $\mathcal{A}(\mathcal{O}).$ For causally complete sub-wedge regions $\mathcal{O}$ such a situation inevitably leads to interaction-free theories i.e. the local algebras generated by ordinary free fields are the only $\mathcal{A}(\mathcal{O})$-affiliated PFGs. Such a situation is achieved by domain restrictions on the (generally unbounded) PFGs. Without any further domain restriction on these (generally unbounded) operators it would be difficult to imagine a constructive use of PFGs. Before studying PFGs it is helpful to remind the reader of the following theorem of general modular theory. Let S be the modular operator of a general standard pair $(\mathcal{A},\Omega)$ and let $\Phi$ be a vector in the domain of S. There exists a unique closed operator F affiliated with $F$ (notation $F\eta\mathcal{A)}$ which together with F$^{\ast}$ has the reference state $\Omega$ in its domain and satisfies $$F\Omega=\Phi,\,\,F^{\ast}\Omega=S\Phi\label{abstract}$$ A proof of this and the following theorem can be found in [@BBS]. For the special field theoretic case $(\mathcal{A}(W),\Omega),$ the domain of $S$ which agrees with that of $\Delta^{\frac{1}{2}}=e^{\pi K},K=$ boost generator has evidently a dense intersection $\mathcal{D}^{(1)}=H^{(1)}\cap\mathcal{D}_{\Delta^{\frac{1}{2}}}$ with the one-particle space $H^{(1)}=E_{m}H.$ Hence the operator $F$ for $\Phi^{(1)}\in\mathcal{D}^{(1)}$ is a PFG $G$ as previously defined. However the abstract theorem contains no information on whether the domain properties admit a repeated use of PFGs similar to smeared fields in the Wightman setting, nor does it provide any clew about the position of a $domG$ relative to scattering states. Without such a physically motivated input, wedge-supported PFGs would not be useful. An interesting situation is encountered if one requires the $G$ to be tempered. Intuitively speaking this means that $G(x)=U(x)GU(x)^{\ast}$ has a Fourier transform as needed if one wants to use PFGs in scattering theory. If one in addition assumes that the wedge algebras to which the PFGs are affiliated are of the are of the standard Bose/Fermi type i.e. $\mathcal{A}(W^{\prime})=\mathcal{A}(W)^{\prime}$ or the twisted Fermi commutant $\mathcal{A}(W)^{tw}$, one finds PFGs for the wedge localization always region exist, but the assumption that they are tempered leads to a purely elastic scattering matrix $S_{scat}=S_{el},$ whereas in d$>$1+1 is only consistent with $S_{scat}=1$. Together with the recently obtained statement about the uniqueness of the inverse problem in the modular setting of AQFT [@S-matrix] one finally arrives at the interaction-free nature in the technical sense that the PFGs can be described in terms of free Bose/Fermi fields. The nonexistence of PFGs in interacting theories for causally completed localization regions smaller than wedges (i.e. intersections of two or more wedges) can be proven directly i.e. without invoking scattering theory PFGs localized in smaller than wedge regions are (smeared) free fields. The presence of interactions requires the presence of vacuum polarization in all state vectors created by applying operators affiliated with causally closed smaller wedge regions. The proof of this theorem is an extension of the ancient theorem [@St-Wi] that pointlike covariant fields which permit a frequency decomposition (with the negative frequency part annihilating the vacuum) and commute/anticommute for spacelike distances are necessarily free fields in the standard sense. The frequency decomposition structure follows from the PFG assumption and the fact that in a given wedge one can find PFGs whose localization is spacelike disjoint is sufficient for the analytic part of the argument to still go through, i.e. the pointlike nature in the old proof is not necessary to show that the (anti)commutator of two spacelike disjoint localized PFGs is a c-number (which only deviates from the Pauli-Jordan commutator by its lack of covariance). The most interesting aspect of this theorem is the inexorable relation between interactions and the presence of vacuum polarization which for the first time leads to a completely intrinsic definition of interactions which is not based on the use of Lagrangians and particular field coordinates. This poses the interesting question how the shape of localization region (e.g. size of double cone) and the type of interaction is related with the form of the vacuum polarization clouds which necessarily accompany a one-particle state. We will have some comments in the next section. As Mund has recently shown, this theorem has an interesting extension to d=1+2 QFT with braid group (anyon) statistics. ([@M]) There are no PFGs affiliated to field algebras localized in spacelike cones with anyonic commutation relations i.e. sub-wedge localized fields obeying braid group commutation relations applied to the vacuum are always accompanied by vacuum polarization clouds. Even in the absence of any genuine interactions this vacuum polarization is necessary to sustain the braid group statistics and maintain the spin-statistics relation. This poses the interesting question whether quantum mechanics is compatible with a nonrelativistic limit of braid group statistics. The nonexistence of vacuum polarization-free locally (sub-wedge) generated one particle states suggests that as long as one maintains the spin-statistics connection throughout the nonrelativistic limit procedure, the result will preserve the vacuum polarization contributions and hence one will end up with nonrelativistic field theory instead of quantum mechanics[^9]. Using the concept of PFGs one can also formulate this limitation of quantum mechanics in a more provocative way by saying that (using the generally accepted fact that QFT is more fundamental than QM) QM owes its physical relevance to the fact that the permutation group (Boson/Fermion) statistics permits sub-wedge localized PFGs (free fields which create one particle states without vacuum polarization admixture) whereas the more general braidgroup statistics does not. Another problem which even in the Wigner setting of noninteracting particles is interesting and has not yet been fully understood is the pre-modular theory for disconnected or topologically nontrivial regions e.g. in the simplest case for disjoint double intervals of the massless $s=\frac{1}{2}$ chiral model on the circle. Such situations give rise to nongeometric (fuzzy) “quantum symmetries” of purely modular origin without a classical counterpart. Construction of models via modular localization =============================================== Since up to date more work had been done on the modular construction of d=1+1 factorizing models, we will first illustrate our strategy in that case and then make some comments of how we expect our approach to work in the case of higher dimensional d=1+2 anyons and $d\geq1+3$ spin towers. The construction consists basically of two steps, first one classifies the possible algebraic structures of tempered wedge-localized PFGs and then one computes the vacuum polarization clouds of the operators belonging to the double cone intersections. Let us confine ourself to the simplest model which we may associate with a massive selfconjugate scalar particle. If there would be no interactions the appropriate theorem of the previous section would only leave the free field which is a PFG for any localization$$\begin{aligned} A(x) & =\frac{1}{\sqrt{2\pi}}\int\left( e^{-ip(\theta)x}a(\theta )+e^{ip(\theta)x}a^{\ast}(\theta)\right) d\theta\\ A(f) & =\int A(x)\hat{f}(x)d^{2}x=\frac{1}{\sqrt{2\pi}}\int_{C}a(\theta)f(\theta)d\theta,\text{ }supp\hat{f}\in W\nonumber\\ p(\theta) & =m(\cosh\theta,\sinh\theta)\nonumber\end{aligned}$$ where in order to put into evidence that the mass shell only carries one parameter, we have used the rapidity parametrization in which the plane wave factor is an entire function in the complex extension of $\theta$ with $p(\theta-i\pi)=-p(\theta).$ The last formula for the smeared field with the localization in the right wedge has been written to introduce a useful notation; the integral extends over the upper and lower conture $C:\theta$ and $\theta-i\pi,-\infty<\theta<\infty$ where the Fourier transform $f(\theta)$ is analytic and integrable in the strip which $C$ encloses as a result of its x-space test function support property. Knowing that tempered PFGs only permit elastic scattering (see previous section), we make the “nonlocal” Ansatz $$\begin{aligned} G(x) & =\frac{1}{\sqrt{2\pi}}\int\left( e^{-ipx}Z(\theta)+e^{ipx}Z^{\ast }(\theta)\right) d\theta\label{PFG}\\ G(\tilde{f}) & =\frac{1}{\sqrt{2\pi}}\int_{C}Z(\theta)f(\theta )d\theta\nonumber\end{aligned}$$ where the $Zs$ are defined on the incoming n-particle vectors by the following formula for the action of $Z^{\ast}(\theta)$ for the rapidity-ordering $\theta_{i}>\theta>\theta_{i+1},\,\,\theta_{1}>\theta_{2}>...>\theta_{n}$ $$\begin{aligned} & Z^{\ast}(\theta)a^{\ast}(\theta_{1})...a^{\ast}(\theta_{i})...a^{\ast }(\theta_{n})\Omega=\label{bound}\\ & S(\theta-\theta_{1})...S(\theta-\theta_{i})a^{\ast}(\theta_{1})...a^{\ast }(\theta_{i})a^{\ast}(\theta)...a^{\ast}(\theta_{n})\Omega\nonumber\\ & +contr.\,from\text{ }bound\,states\nonumber\end{aligned}$$ In the absence of bound states (which we assume in the following) this amounts to the commutation relations[^10] $$\begin{aligned} Z^{\ast}(\theta)Z^{\ast}(\theta^{\prime}) & =S(\theta-\theta^{\prime })Z^{\ast}(\theta^{\prime})Z^{\ast}(\theta),\,\theta<\theta^{\prime}\label{ab}\\ Z(\theta)Z^{\ast}(\theta^{\prime}) & =S(\theta^{\prime}-\theta)Z^{\ast }(\theta^{\prime})Z(\theta)+\delta(\theta-\theta^{\prime})\nonumber\end{aligned}$$ where the structure functions $S$ must be unitary in order that the $Z$-algebra be a $^{\ast}$-algebra. It is easy to show that the domains of the $Zs$ are identical to free field domains. We still have to show that our “nonlocal” $Gs$ are wedge localized. According to modular theory for this we have to show the validity of the KMS condition. It is very gratifying that the KMS condition for the requirement that the $G(\tilde{f})$ $supp\tilde {f}\subset W$ are affiliated with the algebra $\mathcal{A}(W)$ is equivalent with the crossing property of the $S.$ The PFG’s with the above algebraic structure for the Z’s are wedge-localized if and only if the structure coefficients $S(\theta)$ in (\[ab\]) are meromorphic functions which fulfill crossing symmetry in the physical $\theta $-strip i.e. the requirement of wedge localization converts the Z-algebra into a Zamolodchikov-Faddeev algebra. Improving the support of the wedge-localized test function in $G(\hat{f})$ by choosing the support of $\hat{f}$ in a double cone well inside the wedge does not improve $locG(\hat{f}),$ it is still spread over the entire wedge. This is similar to the spreading property of (\[spread\]) and certainly very different from the behavior of smeared pointlike fields. By forming an intersection of two oppositely oriented wedge algebras one can compute the double cone algebra or rather (since the control of operator domains has not yet been accomplished) the spaces of double-cone localized bilinear forms (form factors of would be operators). The most general operator $A$ in $\mathcal{A}(W)$ is a LSZ-type power series in the Wick-ordered $Zs$ $$\begin{aligned} A & =\sum\frac{1}{n!}\int_{C}...\int_{C}a_{n}(\theta_{1},...\theta _{n}):Z(\theta_{1})...Z(\theta_{n}):d\theta_{1}...d\theta_{n}\label{series}\\ A & \in\mathcal{A}_{bil}(W) \label{bil}$$ with strip-analytic coefficient functions $a_{n}$ which are related to the matrix elements of $A$ between incoming ket and outgoing bra multiparticle state vectors (formfactors). The integration path $C$ consists of the real axis (associated with annihilation operators and the line $Im\theta=-i\pi.$ Writing such power series without paying attention to domains of operators means that we are only dealing with these objects (as in the LSZ formalism) as bilinear forms (\[bil\]) or formfactors whose operator status still has to be settled. Now we come to the second step of our algebraic construction, the computation of double cone algebras. The space of bilinear forms which have their localization in double cones are characterized by their relative commutance (this formulation has to be changed for Fermions or more general objects) with shifted generators $A^{(a)}(f)\equiv U(a)A(f)U^{\ast}(a)$$$\begin{aligned} \left[ A,A^{(a)}(f)\right] & =0,\,\forall f\,\,suppf\subset W\,\label{inter}\\ A & \subset\mathcal{A}_{bil}(C_{a})\nonumber\end{aligned}$$ where the subscript indicates that we are dealing with spaces of bilinear forms (formfactors of would-be operators localized in $C_{a}$) and not yet with unbounded operators and their affiliated von Neumann algebras. This relative commutant relation [@JMP] on the level of bilinear forms is nothing but the famous ”kinematical pole relations” which relate the even $a_{n}$ to the residuum of a certain pole in the $a_{n+2}$ meromorphic functions. The structure of these equations is the same as that for the formfactors of pointlike fields; but whereas the latter lead (after splitting off common factors [@Kar] which are independent of the chosen field in the same superselection sector) to polynomial expressions with a hard to control asymptotic behavior, the $a_{n}$ of the double cone localized bilinear forms are solutions which have better asymptotic behavior controlled by the Pailey-Wiener-Schwartz theorem. We will not discuss here the problem of how this improvement can be used in order to convert the bilinear forms into genuine operators. Although we think that this is largely a technical problem which does not require new concepts, the operator control of the second step is of course important in order to convince our constructivist friends that modular methods really do provide a rich family of nontrivial d=1+1 models. We hope to be able to say more in future work. The extension to the general factorizing d=1+1 models should be obvious. One introduces multi-component $Zs$ with matrix-valued structure functions $S.\,$ The contour deformation from the original integral to the “crossed” contour which is necessary to establish the KMS conditions in the presence of boundstate poles in the physical $\theta$-strip compensates those pole contributions against the boundstate contributions in the state vector Ansatz (\[bound\]) [@JMP]. The fact that the structure matrix $S(\theta -\theta^{\prime})$ is the 2-particle matrix element of the elastic S-matrix of the constructed algebraic net of double cone algebras is not used in this construction. Of the two aspects of an S-matrix in local quantum physics namely the large time LSZ (or Haag-Ruelle) scattering aspect and that of the S-matrix as a relative modular invariant of the wedge algebra we only utilized the latter. As a side remark we add that the $Z^{\#}$ operators are conceptually somewhere between the free incoming and the interacting Heisenberg operators in the following sense: whereas any particle state in the theory contributes to the structure of the Fock space and has its own incoming creation/annihilation operator, the $Z^{\#}$ operators are (despite the rather rough wedge localization properties of their spacetime related PFGs $G$) similar to charge-carrying local Heisenberg operators in the sense that all other operators belonging to particles whose charge is obtained by fusing that of $Z$ and $Z^{\ast}$ are functions of $Z$ [@Zi]$.$ The particle-field duality which holds for free fields becomes already incalidated by the interacting wedge-localized PFG $G$ before one gets to the double-cone-localized operators. Let us finally make some qualitative remarks about a possible adaptation of the above two-step processs to the higher dimensional exceptional Wigner cases. Since their are many wedges, one uses a $\theta$-ordering with respect to the standard wedge as in [@BBS]. Then the nongeometrical nature of the twist modification $\frak{t}$ of the spatial $\frak{j}$ operator in the Wigner representation leads to a field-theoretic twist operator $T$ which is the analog of the $S_{el}$ operator in the previous discussion. This $T$ is responsible for the modification similar to (\[ab\]), but this time with piecewise constant structure constants in the $Z$-analogs which still refer to the standard wedge ($R$-operators acting on the tower indices in case of spin towers). With other words the wedge formalism with respect to the standard wedge is like a tensor product formalism i.e. the n-“particle” states are analog to n-fold tensor products in a Fock space. The mismatch between the algebraic commutant and the geometric opposite of the wedge algebra is responsible for a drastic modification of the Bisognano-Wichmann theorem and leads to braid commutation relations between wedge and opposite wedge operators. The next step namely the formation of the intersection is analog to the previous case except that instead of a lightlike translation we now have to take the orthogonal wedge intersection as in section 2.2. The intersection naturally has to be taken with respect to the twisted relative commutant. It is expected to build up a rich vacuum polarization structure for the d=1+2 massive anyons as well as for the spin towers. The impossibility of a compact localization in the case of the exceptional Wigner representation places them out of reach by Lagrangian quantization methods. The charge-carrying PFG operators corresponding to the wedge-localized subspaces as well as their best localized intersections are more “noncommutative” than those for standard QFT and the worsening of the best possible localization is inexorably interwoven with the increasing spacelike noncommutativity. This kind of noncommutativity should however be kept apart from the noncommutativity of spacetime itself whose consistency with the Wigner representation theory will be briefly mentioned in the subsequent last section. Outlook ======= In the past the power of Wigner’s representation theory has been somewhat underestimated. As a completely intrinsic relativistic quantum theory which stands on its own feet (i.e. it does not depend on any classical quantization parallelism and thus gives quantum theory its deserved dominating position) it was used in order to back up the Lagrangian quantization procedure [@Wei], but thanks to its modular localization structure it is capable to do much more and shed new light also on problems which remained outside Lagrangian quantization and perturbation theory. This includes problems where, contrary to free fields, no PFG operator (one which creates a pure one-particle state without a vacuum polarization admixture) for sub-wedge regions exist, but where wedge-localized algebras still have tempered generators as d=1+1 factorizing models d=1+2 “free” anyons and “free” Wigner spin towers. It should however be mentioned that the braid group statistics particles refered to as anyons associated to d=1+2 continuous spin Wigner representations in this particular way (i.e. by extending the one-particle twist to multiparticle states with abelian phase composition) do not exhaust all possibilities of plektonic statistcs. Since conformal theories in any dimensions (even beyond chiral theories) are “almost free” (in the sense that the only structure which distinguishes them from free massless theories is the spectrum of anomalous dimension which is related to an algebraic braid-like structure in timelike direction [@braid]), we believe that they also can be classified and constructed by modular methods. This leaves the question of how to deal with interacting massive theories which have in addition to vacuum polarization real (on shell) particle creation. For such models PFG generators of wedge algebras are (as a result of their non-temperedness) too singular objects. One either must hope to find different (non-PFG) generators, or use other modular methods [@Hor] related to holographically defined modular inclusions or modular intersections. For example holographic lightfront methods are based on the observation that the full content of a d-dimensional QFT can be encoded into d-1 copies of one abstract chiral theory whose relative placement in the Hilbert space of the d-dimensional theory carries the information. What remains to be done is to characterize the kind of chiral theory and its relative positions in a constructively manageable way. Another insufficiently understood problem is the physical significance of the infinitely many modular symmetry groups which (beyond the Poincaré or conformal symmetry groups which leave the vacuum invariant) act in a fuzzy way within the localization regions and in their causal complements [@fuzzy]. An educated guess would be that they are related to the nature of the vacuum polarization clouds which local operators in that region generate from the vacuum. It is an interesting (and in recent years again fashionable) question whether besides the macro-causal relativistic quantum mechanics mentioned in the introduction and the micro-causal local quantum physics there are other relativistic non-micro causal quantum theories[^11] which permit at least the physical notion of scattering and which unlike the the relativistic mechanics preserve some of the vacuum polarization properties especially those which are necessary to keep the TCP theorem (so that the existence of antiparticles is an inexorably consequence) address the question of localization (string theory presently does not; if the localization discussed there would have the fundamental quantum significance as the one used in this paper then string theory would be a special kind of AQFT). All attempts to obtain ultraviolet improved renormalizable theories naturally after the discovery of renormalization) by allowing nonlocal interactions, starting from the Kristensen-Moeller-Bloch [@M-K][@Bloch] replacement of pointlike Lagrangian interactions by formfactors and the Lee-Wick complex pole modification [@L-W] of Feynman rules up to some of the recent noncommutative spacetime failed. Even if Lorentz invariance and unitarity (including the optical theorem) could have been maintained in those proposals, the main reason for original motivation namely ultraviolet convergence was not borne out [@Bloch]. Of course even without this motivation it would be very interesting to know if there are any “physically viable” nonlocal relatistic theories at all. By this we mean the survival of the physically indispensible macro-causality[^12]. For the relativistic particle theory mentioned in the introduction this macro-causality was insured via the cluster-separability properties of the S-matrix. more than 50 years of history on this issue has taught time and again that the naive idea that a mild modification of pointlike Lagrangian interactions will still retain macro-causality turns out to be wrong under closer scrutiny. The general message is that the notion of a mild violation of micro-causality (i.e. maintaining macro-causality) within the standard framework is an extremely delicate concept [@cau]. In more recent times Doplicher Fredenhagen and Roberts [@DFR] discovered a Bohr-Rosenfeld like argument which uses a quasiclassical interpretation of the Einstein field equation (coupled with a requirement of absence of measurement-caused black holes which would trap photons) and leads to uncertainty relations of spacetime. Although the initiating idea was very conservative, the authors were nevertheless led to quite drastic conceptual changes since the localization indexing of field theoretic observables is now done in terms of noncommutative spacetime in which points correspond to pure states on a quantum mechanical spacetime substrate on which the Poincaré group acts. They found a model which saturate their commutation relations and they started to study QFTs over this new structure. In more recent times it was realized [@BDF], that when one recast such models into the setting of Yang-Feldman perturbation theory with nonlocal interactions, many problems which appear if one does not rethink the formalism but just copies old perturbative recipes from the standard case [@Filk] (as violation of L-invariance and unitarity, which have their origin in the fact that in the new context Feynman $i\varepsilon$ prescription is not the same as time-ordering)  disappear and the only conceptual problem which remains is an appropriate form of macro-causality. Interestingly enough, these were precisely the techniques used in the first post renormalization investigations of nonlocal interactions [@M-K]. So there seems to be at least some hope that those specific nonlocalities caused by those models whose lowest nontrivial perturbative order is discussed in [@BDF] are exempt from the historical lessons. This would be a theory to which the Wigner approach is applicable and the Fock space structure is maintained but with different localization concepts. It would be very interesting indeed if besides the two mentioned relativistic theories build on different localization concept treated in this article there could exist a theory of Wigner particles interacting on noncommutative spacetime in a possibly macro-causal way and uphold the significant gains concerning the TCP structure and antipartices which are so inexorably linked to vacuum polarization. Such a quest on a fundamental level should not be confused with the phenomenological use of the language of noncommutative geometry for certain conventional Schroedinger systems involving constant magnetic fields [@Douglas] since in those cases the localization concepts of the Schroedinger theory are in no way affected by the observation that one may write the system in terms of different dynamical variables. In this context it is worthwhile to remember that the full local (anti)commutativity is not used in e.g. the derivation of the TCP theorem [@St-Wi]. Using the present terminology the TCP property is in fact known to be equivalent to wedge localization. However the question of whether a modular wedge localization is possible in the context of the correctly formulated noncommutative L-invariant and unitary models [@DFR][@BDF]  may well have a positive answer [@Fre]. This point is certainly worthwhile to return to in future work. It is very regrettable that such conceptually subtle points[^13] seem to go unnoticed in the new globalized way of doing particle physics [@Douglas]. It seems that the ability of recognizing conceptually relevant points, which has been the hallmark of part of 20 century physics, has been lost in the semantic efforts of attaching physical-sounding words to mathematical inventions. It is well-known to quantum field theorist with some historical awareness that the role of causality and localization was almost never appreciated/understood by most mathematicians. This has a long tradition. A good illustration is the impressive scientific curriculum of Irvine Segal, one of the outstanding pioneers of the algebraic approach. If in those papers localization concepts would have been treated with the same depth and care as global mathematical aspects of AQFT, quantum field theory probably would have undergone a more rapid development and we would have been spared the many differential geometric traps and pitfalls, including the banalization of Euclidean methods. **Acknowledgements:** One of the authors (B.S.) is indebted to Wolfhardt Zimmermann for some pleasant exchanges of reminiscences on conceptual problems of QFT of the 50s and 60s, as well as for related references. B.S. is also indebted to Sergio Doplicher and Klaus Fredenhagen for an explanation of the actual status of their 1995 work. Finally the authors would like to thank Fritz Coester for some valuable email information which influenced the content of the introduction. Appendices ========== Here we have collected some mathematical details for the convenience of the reader. Appendix A: The abstract spatial modular theory ----------------------------------------------- Suppose we have a “standard” spatial modular situation i.e. a closed real subspace $H_{R}$ of a complex Hilbert space $H$ such that $H_{R}\cap iH_{R}=\left\{ 0\right\} $ and the complex space $H_{D}\equiv H_{R}+iH_{R}$ is dense in $H.\,$ Let $e_{R}$ and $e_{I}$ be the projectors onto $H_{R}$ and $iH_{R}$ and define operators $$t_{\pm}\equiv\frac{1}{2}(e_{R}\pm e_{I})$$ Because of the reality restriction the two operators have very different conjugation properties, $t_{+}$ turns out to be positive $0<t_{+}<\mathbf{1}$, but $t_{-}$ is antilinear. These properties follow by inspection through the use of the projection- and reality-properties. There are also some easily derived quadratic relations between involving the projectors and $t\pm$$$\begin{aligned} e_{R,I}t_{+} & =t_{+}(1-e_{I,R})\\ t_{+}t_{-} & =t_{-}(1-t_{+})\nonumber\\ t_{+}^{2} & =t_{-}(1-t_{-})\nonumber\end{aligned}$$ ([@Rieffel]) In the previous setting there exist modular objects[^14] $J$, $\Delta$ and $S=$ $\frak{j}\Delta^{\frac{1}{2}}$ which reproduce $H_{R}$ as the +1 eigenvalue real subspace of $S$. They are related to the previous operators by $$\begin{aligned} t_{-} & =J\left| t_{-}\right| \\ \Delta^{it} & =\left( 1-t_{+}\right) ^{it}t_{+}^{-it}$$ The proof consists in showing the commutation relation $J\Delta^{it}=\Delta^{it}$ $J$ ($\curvearrowright$ $J\Delta=\Delta^{-1}$ $J$ since $J$ is antiunitary) which establishes the dense involutive nature $S^{2}\subset1$ of $S$ by using the previous identities. It is not difficult to show that $0$ is not in the point spectrum of $\Delta^{it}.$ If $H_{R}$ is standard, then $iH_{R},$ $H_{R}^{\perp}$ and $iH_{R}^{\perp}$ are standard. Here the orthogonality $\perp$ refers to the real inner product $Re(\psi,\varphi).$ Furthermore the $J$ acts on $H_{R}$ as $$JH_{R}=iH_{R}^{\perp}$$ We leave the simple proofs to the reader (or look up the previous reference [@Rieffel]). The orthogonality concept is often expressed in the physics literature by $iH_{R}^{\perp}=H_{R}^{symp\perp}$ referring to symplectic orthogonality in the sense of $Im(\psi,\varphi).$ There is also a more direct analytic characterization of $\Delta$ and $J$ (spatial KMS condition) The functions f(t)=$\Delta^{it}\psi,$ $\psi\in H_{R}$ permits an holomorphic continuation f(z) holomorphic in the strip -$\frac {1}{2}\pi<\operatorname{Im}z<0,$ continuous and bounded on the real axis and fulfilling $f(t-\frac{1}{2}i)=$ $Jf(t)$ which relates the two boundaries. The two commuting operators $\Delta^{it}$ and j are uniquely determined by these analytic properties i.e. H$_{R}$ does not admit different modular objects. Another important concept in the spatial modular theory is “modular inclusion” (analogous to Wiesbrock) A inclusion of a standard real subspace $K_{R}$ into a standard space $K_{R}\subset H_{R}$ is called “modular” if the modular unitary $\Delta_{H_{R}}^{it}$ of $H_{R}$ compresses $K_{R}$ for one sign of t $$\Delta_{H_{R}}^{it}K_{R}\subset K_{R}\,\,\,\,t<0$$ If necessary one adds a -sign i.e. if the modular inclusion happens for t$>$0 one calls it a $-$modular inclusion. The modular group of a modular inclusion i.e. $\Delta_{K_{R}}^{it}$ together with $\Delta_{H_{R}}^{it}$ generate a unitary representation of the two-parametric affine group of the line. The proof consists in observing that the positive operator $\Delta_{K_{R}}-\Delta_{H_{R}}\geq0$ is essentially selfadjoint. Hence we can define the unitary group $$U(a)=e^{i\frac{1}{2\pi}a\overline{(\Delta_{K_{R}}-\Delta_{H_{R}})}}$$ The following commutation relation $$\begin{aligned} \Delta_{H_{R}}^{it}U(a)\Delta_{H_{R}}^{-it} & =U(e^{\pm2\pi t}a)\\ J_{H_{R}}U(a)J_{H_{R}} & =U(-a)\nonumber\end{aligned}$$ and several other relations between $\Delta_{H_{R}}^{it},\Delta_{K_{R}}^{it}, $ $J_{H_{R}},$ $J_{K_{R}},U(a).$ The above relations are the Dilation-Translation relations of the 1-dim. affine group. It would be interesting to generalize this to the modular intersection relation in which case one expects to generate the SL(2,R) group. The actual situation in physics is opposite: from group representation theory of certain noncompact groups $\pi(G)$ one obtains candidates for $\Delta^{it}$ and $J$ from which one passes to $S$ and $H_{R}.$ In the case of the Poincaré or conformal group the boosts or proper conformal transformations in positive energy representations lead to the above situation. The representations do not have to be irreducible; the representation space of a full QFT is also in the application range of the spatial modular theory. If the positive energy representation space is the Fockspace over a one-particle Wigner space, the existence of the CCR (Weyl) or CAR functor maps the spatial modular theory into operator-algebraic modular theory of Tomita and Takesaki. In general such a step is not possible. Connes has given conditions on the spatial theory which lead to the operator-algebraic theory. They involve the facial structure of positive cones associated with the space $H_{R}.$ Up to now it has not been possible to use them for constructions in QFT. The existing ideas of combining the spatial theory of particles with the Haag-Kastler framework of spacetime localized operator algebras uses the following 2 facts - The wedge algebra $\mathcal{A}(W)$ has known modular objects $$\begin{aligned} \Delta^{it} & =U(\Lambda_{W}(-2\pi t))\\ J & =S_{scat}J_{0}\nonumber\end{aligned}$$ Whereas the wedge affiliated L-boost (in fact all P$_{+}^{\uparrow}$ transformations) is the same as that of the interacting or free incoming/outgoing theory, the interaction shows up in those reflections which involve time inversion as $J.$ In the latter case the scattering operator $S_{scat}$ intervenes in the relation between the incoming (interaction-free) $J_{0}$ and its Heisenberg counterpart $J.$ In the case of interaction free theories the $J_{0}$ contains in addition to the geometric reflection (basically the TCP) a “twist” operator which is particularly simple in the case of Fermions. - The wedge algebra $\mathcal{A}(W)$ has PFG-generators. In certain cases these generators have nice (tempered) properties which makes them useful in explicit constructions. Two such cases (beyond the standard free fields) are the interacting d=1+1 factorizing models and the free anyonic and Wigner spin-tower representations in both cases the PFG property is lost (vacuum polarization is present) for sub-wedge algebras. In the last two Wigner cases the presence of the twist requires this, only the fermionic twist in the case of $S_{scat}=1$ is consistent with having PFGs for all localizations. Appendix B: Infinite dimensional covariant representations ---------------------------------------------------------- In terms of the little group generators relative to the fixed vector $\frac {1}{2}(1,0,0,1)$ the Pauli-Lubanski operators has the form $$W_{\mu}=-\frac{1}{2}\varepsilon_{\mu\nu\sigma\tau}J^{\nu\sigma}P^{\tau}=\frac{1}{2}(M_{3},\Pi_{1},\Pi_{2},M_{3})$$ where $M_{3}$ is the 3-component of the angular momentum and $\Pi_{i}$ are the two components of the Euclidean translations which together make up the infinitesimal generators of $\tilde{E}(2).$ An representation of the little group can be given in any of the Gelfand at al. irreducible representation spaces of the homogeneous Lorentz group. These consist of homogeneous functions of two complex variables $\zeta=$($\zeta_{1},\zeta_{2}$) which are square integrable with respect to the following measure $$\begin{aligned} d\mu(\zeta) & =\frac{1}{4\pi}\left( \frac{i}{2}\right) ^{2}d^{2}\zeta d^{2}\bar{\zeta}\delta(\frac{1}{2}\zeta\textsl{q}\zeta^{\ast}-1),\,\textsl{q=}\sigma^{\mu}q_{\mu},\,\,q^{2}=0,q_{0}>0\\ \left( f,g\right) & =\int d\mu(\zeta)\bar{f}(\zeta)\,g(\zeta),\,\,\,f(\rho e^{i\alpha}\zeta)=\rho^{2(c-1)}e^{2il_{0}\alpha}f(\zeta),\,\,\lambda_{0}=0,\pm\frac{1}{2},\pm1,..,c=i\nu,\nonumber\end{aligned}$$ The inner product is independent of the choice of the lightlike vector $q$ if $c=i\nu$ because the integrand has total homogeneous degree -4 and on functions $F(\rho\zeta)=\rho^{-4}F(\zeta)$ with this degree the integral is q-independent. This family of unitary irreducible representations $\chi=\left[ \lambda_{0},c=i\nu\right] $ for $-\infty<\nu<\infty$ of SL(2,C) is called the *principal series* representation. Another such family, the *supplementary series* $\chi=\left[ \lambda_{0},c\right] ,\,\,-1<c $ $<1$ contains an additional integral operator $K(\zeta,\eta)$$$\begin{aligned} \left( f,g\right) & =\int d\mu(\zeta)\bar{f}(\zeta)\,\int K(\zeta ,\eta)g(\eta)\\ K(\zeta,\eta) & =N^{-1}\left( \eta\varepsilon\zeta\right) ^{-l_{0}-c-1}\overline{\left( \eta\varepsilon\zeta\right) }^{-l_{0}-c-1}\nonumber\end{aligned}$$ We now define basisvectors in the above representation spaces which carry a representation of the little group $$\begin{aligned} \left( \Pi_{1}^{2}+\Pi_{2}^{2}\right) f_{\lambda}^{\chi,\rho}(\zeta) & =\rho^{2}f_{\lambda}^{\chi,\rho}(\zeta),\,\,M_{3}f_{\lambda}^{\chi,\rho}(\zeta)=-\lambda f_{\lambda}^{\chi,\rho}(\zeta)\\ \left( U(\tilde{E})f_{\lambda}^{\chi,\rho}\right) (\zeta) & =\sum _{\lambda^{\prime}}f_{\lambda^{\prime}}^{\chi,\rho}(\zeta\tilde{E})d_{\lambda^{\prime},\lambda}(\tilde{E})\nonumber\\ f_{\lambda}^{\chi,\rho}(\zeta) & =\left| \zeta_{2}\right| ^{2c-2}e^{-i\lambda\phi}J_{l_{0}-\lambda}(2\rho\left| z\right| )e^{il_{0}\alpha },\,\,\phi\nonumber\end{aligned}$$ In a similar way, the d=1+2 anyonic representations may be rewritten in terms of infinite dimensional covariant representations. It has been shown [@Mu-S] that the following family of covariant unitary representations of $\mathcal{\tilde{P}}_{3}^{\uparrow}$ are useful in the covariant description of the (m,s) Wigner representation $$\begin{aligned} \left( U(a,(\gamma,\omega))\psi\right) (p,z) & =e^{ipa}\tau_{h,\sigma }((\gamma,\omega);z)\psi(\Lambda(\gamma,\omega)^{-1}p,(\gamma,\omega )^{-1}z)\,\,\\ \tau_{h,\sigma}((\gamma,\omega);z) & =e^{-i\omega h}\left( \frac {1+z\bar{\gamma}}{1+z^{-1}\gamma}\right) ^{h}\left( 1+z\bar{\gamma}\right) ^{-1-2\sigma}\left( 1+\left| \gamma\right| \right) ^{\frac{1}{2}+\sigma}\\ (\gamma,\omega)\cdot z & =e^{-i\omega}\frac{z-\gamma e^{i\omega}}{1-z\bar{\gamma}e^{-i\omega}}$$ Here the $\tau$ are Bargmann’s principle series representations of $\widetilde{SL(2,R)}$ acting on the covering of the circle with the circular coordinate being $z$, $\left| z\right| =1.$ The last formula is the action of the Moebius group on the circle.$\,$The wave functions $\psi(p,z)\,$ in this formula are from $L^{2}(p\in H_{m}^{\uparrow},z=e^{i\varphi};\frac {dp}{2p_{0}},d\varphi)$ and in the range $-\frac{1}{2}<h\leq\frac{1}{2},\,\sigma\in iR$ the action is unitary. It has been shown that this covariant representation can be decomposed into a direct sum of Wigner representations $(m,s=k-h).$ $k\in\mathbb{Z}$. [99]{} R. Haag, *Local Quantum Physics*, Springer Verlag 1992 D. Buchholz, *Current trends in axiomatic field theory*, hep-th/9811233 D. Buchholz and R. Haag, JMP **41**, (2000) 3674 M. Duetsch and K. Fredenhagen, Commun.Math.Phys. **219**, (2001) 5 and prior literature therein N. N. Bogoliubov, A. A. Logunov, A. I. Oksak, and I. T. Todorov, *General principles of quantum field theory*, Dordrecht: Kluwer 1990 E. P. Wigner, Ann. Math. **40**, (1939) 149 T. D. Newton and E. P. Wigner, Rev. Mod. Phys. **21**, (1949) 400 B. Bakamjian and L. H. Thomas, Phys. Rev. **92**, (1953) 1300 F. Coester, Helv. Phys. Acta **38**, (1965) 7 F. Coester and W. N. Polyzou, Phys. Rev. D **26**, (1982) 1348 and references therein B. Schroer, *Uniqueness of Inverse Scattering Problem in Local Quantum Physics*, hep-th/0106066 B. Schroer, Nucl. 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Math. **48**, (1947) 568 R. F. Streater and A. S. Wightman, PCT, Spin&Statistics and all That, Benjamin 1964 M. Reed and B. Simon, *Methods of Modern Mathematical Physics*, Academic, San Diego, 1980; the original reference is I. Segal, Trans. Amer.Math. Soc. **81**, (1956) 106 A. Connes, Ann. Inst. Fourier **126**, (1974) 121 D. Guido and R. Longo, Commun. Math. Phys. **172**, (1995) 517 J. Yngvason, Commun. math. phys. **18**, (1970) 195 J. Mund, FU-Berlin PhD-Thesis (1998), see forthcoming paper entitled *Localization of Massive Particles with ”Any” Spin in d=2+1* H.J. Borchers, D. Buchholz and B. Schroer, Commun. Math. Phys. **219**, (2001) 125, hep-th/0003243 J. Mund, Lett. Math. Phys. **43** (1998) 319 J. M Leinaas and J. Myrheim, Il Nuovo Cim. **37 b**, (1977) 1 B. Schroer, J. Math. Phys. **41**, (2000) 3801 and ealier papers of the author quoted therein W. Zimmermann, Nuov. Cim. **10**, (1958) 995 J. Mund and R. Schrader, in Proceedings of the Conference on Advances in Dynamical Systems and Quantum Physics, (Capri Italy) ed. S. Albeverio, R. Figari, E. Orlandi and A. Teta, World Scientific Singapore 1995 pp.235-328 B. Schroer, Phys. Lett. **B506**, (2001) 337, J. Phys. A: Math. Gen **34**, (2001) 3689 L. Fassarella and B. Schroer, *The Fuzzy Analog of Chiral Diffeomorphisms in higher dimensional Quantum Field Theories*, hep-th/0106064 E. H. Lieb and M. Loss, *Stability of a Model of Relativistic Quantum Electrodynamics*, math-ph/0109002 P. Kristensen and C. Moeller, Dan. Mat. Fys. Medd. 27, no. 7 (1952) T. D. Lee and G. C. Wick, Phys. Rev. **D2**, (1970) 1033 C. Bloch, Dan. Mat. Fys. Medd. 27, no. 8 (1952) G. C. Marques and J. A. Swieca, Nucl. Phys. **B43**, (1972) 205 B. Schroer, JPA **32**, (1999) 5937 S. Doplicher, K. Fredenhagen and J. E. Roberts, Commun. Math. Phys. **172**, (1995) 187, see also S. Doplicher, *Spacetime and Fields, a Quantum Texture*, hep-th/0105251 D. Bahns, S. Doplicher, K. Fredenhagen and Gh. Piacitelli, in preparation Th. Filk, Phys. Rev. Lett. **B 376**, (1996) 53 K. Fredenhagen, private communication M. R. Douglas and N. A. Nekrasov, *Noncommutative Field Theory*, hep-th/0110071 [^1]: work supported by CNPq [^2]: The postulated canonical or functional representation requirement is known to get lost in the course of the calculations and the physical (renormalized) result only satisfies the more general causality/locality properties. [^3]: This name was chosen in [@Coester] in order to distinguish it from the field-mediated interactions of standard QFT. [^4]: Every particle may be interpreted as bound of all others whose fused charge is the same. An explicit illustration is furnished by the bootstrap properties of d=1+1 factorizing S-matrices [@Kar]. [^5]: In this connection it appears somewhat ironic that the infinite spin tower Wigner representation is often dismissed as “not used by nature” without having investigated its physical potential. [^6]: The unboundedness is of crucial importance since the domain of definition is the only distinguishing property of the involution (\[s\]) into which geometric properties (causally closed regions in Minkowski space) are encoded. [^7]: We will use the letter $R$ even in the massless case when the little group becomes the noncompact Euclidean group. [^8]: Since here we have to distinguish between undotted and dotted spinors, we use the notation $\alpha(\Lambda)$ and $\beta(\Lambda)=\overline{\alpha(\Lambda)}$ instead of the previous $\tilde{\Lambda}.$ [^9]: The Leinaas-Myrheim geometrical arguments [@L-M] do not take into account the true spin-statistics connection. [^10]: In the presence of bound states such commutation relations only hold after applying suitable projection operators. [^11]: A recent paper by Lieb and Loss [@Lieb] contains an interesting attempt to combine relativistic QM with local quantum field theory. To make this model fully cluster separable (macro-causal) one probably has to combine the localization properties of relativistic quantum mechanics with those of modular localization for the photon field. [^12]: In case of formfactor modifications of pointlike interaction vertices this was shown in [@Bloch] and in case of the Feynman rule modifications by complex poles in [@Swie]. [^13]: The claim in [@Douglas] that ”noncommutativity of the space-time coordinates generally conflicts with Lorentz invariance” contradicts the results of the 1995 seminal paper [@DFR] and a fortiori the forthcoming explicit perturbative model calculations in [@BDF]. [^14]: In the physical application the Hilbert space can be representation space of the Poincaré group which carries an irreducible positive energy representation or the bigger Fock space of (free or incoming) multi-particle states. In order to have a uniform notation we use (different from section 2) big letters for the modular objects and the transformations, i.e. $S,J,$ $\Delta,U(a,\Lambda).$

--- abstract: 'We derive the momentum space dynamic equations and state functions for one dimensional quantum walks by using linear systems and Lie group theory. The momentum space provides an analytic capability similar to that contributed by the z transform in discrete systems theory. The state functions at each time step are expressed as a simple sum of three Chebyshev polynomials. The functions provide an analytic expression for the development of the walks with time.' author: - Ian Fuss - 'Langford B. White and Sanjeev Naguleswaran' - 'Peter J. Sherman' bibliography: - 'D:/DPOLP/documents/bib/quant.bib' title: Momentum Dynamics of One Dimensional Quantum Walks --- Introduction ============ The study of quantum walks has received considerable attention since the introductory papers on the subject, such as [@aharanov00; @Kempe03] and references therein. In this paper, we develop an analytic approach to study the properties of these walks based on a momentum space representation. This paper is structured such that in Section 2 of the paper the momentum space dynamic equations for one dimensional quantum walks are derived via the Z transform of the position space dynamic equations and its representation of the discrete Fourier transform when Z lies on the unit circle. An exponential form of of the momentum space time operator is derived in section 3 by using the group theory of $SU(2)$ and a matrix inner product space. The exponential form allows a simple analytic calculation of the time evolution operator for arbitrary time intervals. This is used in Section 4 to obtain analytic expressions for the momentum space wave functions of quantum walks at arbitrary times. These wave functions are expressed quite simply in terms of Chebyshev Polynomials of the second kind. Some plots of the momentum space probability densities for different parameter values and times are provided in section 5. The conclusions are summarised in Section 6. Momentum Space Dynamic Equations ================================ For a given $\psi(0,0)$ we consider the evolution of a quantum state $\psi(t,x)\in C^{2}$ for discrete times $t\ge0$ on a line $x\in Z.$ The dynamics of the state then evolve according to the difference equations,$$\begin{aligned} & \psi_{0}(t,x)=e^{i\alpha}[a\psi_{0}(t-1,x-1)+b\psi_{1}(t-1,x-1)],\nonumber \\ & \psi_{1}(t,x)=e^{i\alpha}[-b^{*}\psi_{0}(t-1,x+1)+a^{*}\psi_{1}(t-1,x+1)],\label{eq:momdy}\end{aligned}$$ where $|a|^{2}+|b|^{2}=1$ and $\alpha\in R$. Taking two-dimensional $Z$ transforms of these equations yields$$\begin{aligned} & \psi_{0}(z_{1},z_{2})=e^{i\alpha}z_{1}^{-1}z_{2}^{-1}[a\psi_{0}(z_{1},z_{2})+b\psi_{1}(z_{1},z_{2})\nonumber \\ & \psi_{1}(z_{1},z_{2})=e^{i\alpha}z_{1}^{-1}z_{2}^{-1}[-b^{*}\psi_{0}(z_{1},z_{2})+a^{*}\psi_{1}(z_{1},z_{2}).\end{aligned}$$ Thus the transfer matrix for the system is$$B(z_{1},z_{2})=e^{i\alpha}z_{1}^{-1}\left[\begin{array}{cc} az_{2}^{-1} & bz_{2}^{-1}\\ -b^{\ast}z_{2} & a^{\ast}z_{2}\end{array}\right]$$ therefore, for any iteration (time) index $n$, the quantum walk state $\ \Psi(n,x)$ has transform $x\leftrightarrow z$ $$\Psi(n,x)\leftrightarrow e^{in\alpha}C^{n}(z)\Psi(0,0),$$ where $C(z)$ is the matrix polynomial$$C(z)=\left[\begin{array}{cc} az^{-1} & bz^{-1}\\ -b*z & a*z\end{array}\right].$$ It should be noted that $C$ is paraunitary, that is $C^{-1}(z)=C^{T}(1/z).$ In particular this implies that $C(z)$ is unitary on $|z|=1.$ Further we note that $detC(e^{ip)})=1$ and hence the matrix $$S(p)=C(e^{ip})\label{eq:6}$$ is unimodular. The Fourier transform $x\leftrightarrow p$ is$$\Psi(n,x)\leftrightarrow e^{in\alpha}S^{n}(p)\Psi(0,0).$$ Thus by choosing Planck’s constant $\hbar=1,$ the momentum space representation of the quantum walk state vector $\phi(n,p)$ evolves as$$\phi(n,p)=e^{in\alpha}S^{n}(p)\phi(0,p),\label{eq:8}$$ where$$\phi(0,p)=\psi(0,0)=\left[\begin{array}{c} \psi_{0}(0,0)\\ \psi_{1}(0,0)\end{array}\right].\label{eq:9}$$ Thus the time evolution operator in the momentum space is a $2\times2$ matrix polynomial. Hence, the momentum space equations are much more amenable to analysis than those in position space. Exponentiation of the Time Evolution Operator ============================================= The unimodular matrix $S(p)$ can be written in exponential form as$$S(p)=Exp(i\theta(p)\overrightarrow{c}(p).\overrightarrow{\sigma})\label{eq10}$$ where $\theta$ and $\overrightarrow{c}$ are real functions of $p$ and the matrix vector $\overrightarrow{\sigma}$ has Pauli matrix components [@merz]$$\sigma_{1}=\left[\begin{array}{cc} 0 & 1\\ 1 & 0\end{array}\right],$$ $$\sigma_{2}=\left[\begin{array}{cc} 0 & -i\\ i & 0\end{array}\right]$$ and$$\sigma_{3}=\left[\begin{array}{cc} 1 & 0\\ 0 & -1\end{array}\right].\label{eq:11}$$ The inner product$$(A,B)=\frac{1}{2}Tr(AB)$$ defined on the vector space of $2\times2$ unitary matrices gives an inner product space. The set of matrices $\{ I,\sigma_{1},\sigma_{2,}\sigma_{3}\},$ provide an ortho-normal basis for this space. The coefficients of the matrices can be evaluated by taking the inner product of both sides of (\[eq10\]) $$(\sigma_{i},S(p))=(\sigma_{i},Exp(i\theta(p)\overrightarrow{c}(p).\overrightarrow{\sigma})$$ with each of the matrices $\sigma_{i}.$ In doing this we note that a generalised de-Moivre principle gives$$Exp(i\theta\overrightarrow{c}.\overrightarrow{\sigma})=Icos(\theta)+i\overrightarrow{c}.\overrightarrow{\sigma}sin(\theta),$$ where the $p$ dpendence has been suppressed for simplicity. Hence,$$(I,Exp(i\theta\overrightarrow{c}.\overrightarrow{\sigma}))=cos(\theta)\label{eq13}$$ and$$(\sigma_{j},Exp(i\theta\overrightarrow{c}.\overrightarrow{\sigma}))=ic_{j}sin(\theta).\label{eq14}$$ The equivalent coefficients for $S(p)$ can be obtained by defining $$a=cos(\beta)e^{-i\gamma},$$ $$b=sin(\beta)e^{-i\delta}.\label{eq:define_ab}$$ Substituting in (\[eq:6\]) gives$$S(p)=\left[\begin{array}{cc} cos(\beta)e^{-i(p+\gamma)} & sin(\beta)e^{-i(p+\delta)}\\ -sin(\beta)e^{i(p+\delta)} & cos(\beta)e^{i(p+\gamma)}\end{array}\right].$$ These expressions can be simplified by setting $p'=p+\gamma$ and $p''=p+\delta$. Using de Moivre’s principle once again we obtain the transition matrix coefficients$$\begin{aligned} & (I,S(p))=cos(\beta)cos(p'),\nonumber \\ & (\sigma_{1},S(p))=-isin(\beta)sin(p''),\nonumber \\ & (\sigma_{2},S(p))=isin(\beta)cos(p''),\nonumber \\ & (\sigma_{3},S(p))=-icos(\beta)sin(p').\label{eq17}\end{aligned}$$ Comparing coefficients in equations (\[eq13\]) and (\[eq14\]) with those of (\[eq17\]) we obtain$$cos(\theta)=cos(\beta)cos(p'),$$ $$c_{1}sin(\theta)=-sin(\beta)sin(p''),$$ $$c_{2}sin(\theta)=sin(\beta)cos(p''),$$ $$c_{3}sin(\theta)=-cos(\beta)sin(p').\label{eq20}$$ Momentum Space State Functions ============================== A dynamic equation for momentum space state functions was given in (\[eq:8\]). The exponentiation of the operator in (\[eq10\]) enables us to write the powers of the evolution operator as$$S^{n}(p)=Exp(in\theta\overrightarrow{c}.\overrightarrow{\sigma})=Icos(n\theta)+i\overrightarrow{c}.\overrightarrow{\sigma}sin(n\theta).$$ The trigonometric expressions in the above equation can be expressed in terms of the Chebyshev polynomials $T_{n}$ and $U_{n}$ as [@Arf]$$cos(n\theta)=T_{n}(cos(\theta))$$ and$$sin(n\theta)=U_{n-1}(cos(\theta))sin(\theta).$$ Using these expressions and writing the dot product as a sum of components (\[eq:11\]) becomes$$S^{n}(p)=T_{n}(cos(\theta))I+iU_{n-1}(cos(\theta))\sum_{i=1}^{3}c_{i}sin(\theta)\sigma_{i}.$$ The equalities of (\[eq20\]) enable us to rewrite this as$$S^{n}(p)=T_{n}(cos(\beta)cos(p'))I-iU_{n-1}(cos(\beta)cos(p'))[sin(\beta)sin(p'')\sigma_{1}-sin(\beta)cos(p'')\sigma_{2}+cos(\beta)sin(p')\sigma_{3}]$$ Using the Pauli matrices the matrix polynomial$$\begin{aligned} & S^{n}(p)=\left[\begin{array}{cc} T_{n}(cos(\beta)cos(p') & U_{n-1}(cos(\beta)cos(p'))sin(\beta)cos(p'')\\ -U_{n-1}(cos(\beta)cos(p'))sin(\beta)cos(p'') & T_{n}(cos(\beta)cos(p'))\end{array}\right]\nonumber \\ - & i\left[\begin{array}{cc} U_{n-1}(cos(\beta)cos(p'))cos(\beta)sin(p') & U_{n-1}(cos(\beta)cos(p'))sin(\beta)sin(p'')\\ U_{n-1}(cos(\beta)cos(p'))sin(\beta)sin(p'') & -U_{n-1}(cos(\beta)cos(p'))cos(\beta)sin(p')\end{array}\right]\label{eq25}\end{aligned}$$ is obtained. The evolution of the quantum walk in momentum space representation given in (\[eq:8\] )can also be expressed as $$\phi(n,p)e^{-in\alpha}=S^{n}(p)\phi(0,p).\label{eq:21}$$ (\[eq25\]) and (\[eq:9\]) enable this expression to be written as $$\begin{aligned} & \phi_{0}(n,p)e^{-in\alpha}=[T_{n}(cos(\beta)cos(p'))-iU_{n-1}(cos(\beta)cos(p'))cos(\beta)sin(p')]\Psi_{0}(0,0)\nonumber \\ & +[U_{n-1}(cos(\beta)cos(p'))sin(\beta)cos(p'')-iU_{n-1}(cos(\beta)cos(p'))sin(\beta)sin(p'')]\Psi_{1}(0,0)\end{aligned}$$ $$\begin{aligned} & \phi_{1}(n,p)e^{-in\alpha}=-[U_{n-1}(cos(\beta)cos(p'))sin(\beta)cos(p'')+iU_{n-1}(cos(\beta)cos(p'))sin(\beta)sin(p'')]\Psi_{0}(0,0)\nonumber \\ & +T_{n}(cos(\beta)cos(p'))+iU_{n-1}(cos(\beta)cos(p'))cos(\beta)sin(p')]\Psi_{1}(0,0)\end{aligned}$$ By using the relation$$T_{n}(x)=U_{n}(x)-xU_{n-1}(x)$$ this can be written as $$\begin{aligned} & \phi_{0}(n,p)e^{-in\alpha}=[U_{n}(cos(\beta)cos(p'))-U_{n-1}(cos(\beta)cos(p'))cos(\beta)[cos(p')+isin(p')]]\Psi_{0}(0,0)\nonumber \\ & +[[U_{n-1}(cos(\beta)cos(p'))sin(\beta)[cos(p'')-sin(p'')]]\Psi_{1}(0,0)\end{aligned}$$ $$\begin{aligned} & \phi_{1}(n,p)e^{-in\alpha}=-[U_{n-1}(cos(\beta)cos(p'))sin(\beta)[cos(p'')+isin(p'')]]\Psi_{0}(0,0)\nonumber \\ & +[U_{n}(cos(\beta)cos(p'))-U_{n-1}(cos(\beta)cos(p'))cos(\beta)[cos(p')+isin(p')]]\Psi_{1}(0,0).\end{aligned}$$ Inverting the de Moivre formula and moving the global phase term to the right hand side gives the analytic expressions$$\begin{aligned} & \phi_{0}(n,p)=e^{in\alpha}[U_{n}(cos(\beta)cos(p'))-U_{n-1}(cos(\beta)cos(p'))cos(\beta)e^{ip}]\Psi_{0}(0,0)\nonumber \\ & +e^{in\alpha}[U_{n-1}(cos(\beta)cos(p'))sin(\beta)e^{-ip}]\Psi_{1}(0,0)\end{aligned}$$ $$\begin{aligned} & \phi_{1}(n,p)=-e^{in\alpha}[U_{n-1}(cos(\beta)cos(p'))e^{ip}]\Psi_{0}(0,0)\nonumber \\ & +e^{in\alpha}[U_{n}(cos(\beta)cos(p'))-U_{n-1}(cos(\beta)cos(p'))cos(\beta)e^{-ip}]\Psi_{1}(0,0)\end{aligned}$$ for the general momentum space state functions for a one dimensional quantum walk at time n. Momentum Space Densities ======================== The denisity $|\phi_{0}(p:t)|^{2}$ for $\alpha=\gamma=\delta=0,$ $\Psi_{0}(0,0)=1$ and $\Psi_{1}(0,0)=0$ is plotted in figures \[cap:Momentum-Space-Density1\], \[cap:Momentum-Space-Density2\], \[cap:Momentum-Space-Density3\] for $\beta=\frac{\pi}{8},\frac{\pi}{4}$ and $\frac{3\pi}{8}$ and for times $t=10,30,50,70.$ When $\beta$ is fixed the dominant feature of the time series is an increase in oscillation frequency with time. This corresponds to the increase in support of the position space densities with time. The effect of increasing $\beta$ is to trade a decrease in the constant component of the density function for an increase in the oscillatory component. This corresponds to a shift in the position space of probability density from the zero region of the walk to the outer edges of the walk. ![\[cap:Momentum-Space-Density1\]Momentum Space Density functions for $\beta=\frac{\pi}{8}$](paperfig1) ![\[cap:Momentum-Space-Density2\]Momentum Space Density functions for $\beta=\frac{\pi}{4}$](paperfig2) ![\[cap:Momentum-Space-Density3\]Momentum Space Density functions for $\beta=\frac{3\pi}{8}$](paperfig3) The sequences shows that the densities converge to a limit as time increases. They also illustrate the fact that the momentum space is an attractive representation in which to derive this limit because the domain of the wave functions is constant, $p\in[-\pi,\pi].$ This is in contrast to the real space where the domain expands with time. Conclusions =========== It has been shown that the momentum space dynamic equations for a quantum walk can be derived using a z transform of the position space equations for the dynamic walk. An exponential representation of the momentum space time evolution operator was derived by using Lie group theory. This enabled the calculation of general momentum space wave functions in terms of Chebyshev polynomials. Some simple calculations of the momentum space probability densities illustrate the convergence of the momentum wave functions to a limit as time increases.

--- abstract: 'The Jahn-Teller problems of C$_{60}$ anions involving $t_{1g}$ next lowest unoccupied molecular orbital (NLUMO) were theoretically investigated. The orbital vibronic coupling parameters for the $t_{1g}$ orbitals were derived from the Kohn-Sham orbital levels with hybrid B3LYP functional by the frozen phonon approach. With the use of these coupling parameters, the vibronic states of the first excited C$_{60}^-$ were calculated, and were analyzed. The dynamical Jahn-Teller stabilization energy of the first excited C$_{60}^-$ is stronger than that of the ground electronic states, resulting in two times larger splitting of vibronic levels than those of the ground state C$_{60}^-$. The present coupling parameters prompt us to understand more about the excited C$_{60}$.' author: - Zhishuo Huang - Dan Liu title: 'Dynamical Jahn-Teller effect in the first excited C$_{60}^-$' --- Introduction ============ Highly symmetric C$_{60}$ exhibits complex Jahn-Teller dynamics characterized by orbital-vibration entanglement in various charged and excited states [@Chancey1997; @Bersuker2006; @Dunn2015]. Among these states, negatively charged C$_{60}$ is one of the most interesting cases because it often serves as a building brick of materials [@Gunnarsson2004; @Capone; @Alloul; @Kamaras; @Takabayashi; @Nomura2016; @Otsuka]. In order to comprehend thoroughly the role of the building brick, many properties of negatively charged C$_{60}$ should be understood clearly, especially about JT effect involved properties. Though JT effect, including dynamic JT effect, of C$_{60}$ anions have been intensively investigated [@Auerbach1994; @Manini1994; @Yu1994; @Dunn1995; @Gunnarsson1995; @OBrien1996; @Tosatti1996; @Yu1997; @Manini1998; @Sookhun2003; @Dunn2005; @Tomita2005; @Hands2008; @Frederiksen2008; @Iwahara2010; @Dunn2012; @Klupp2012; @Stochkel2013; @Ponzellini; @Kundu2015; @Iwahara2018; @Liu2018a; @Liu2018b; @Matsuda2018], it is only last years that the actual situation in the ground electronic states of C$_{60}^{n-}$ molecule $(n = 1-5)$ has been established with accurate coupling parameters, which showed the importance of dynamic JT effect [@Liu2018a; @Liu2018b]. So far, the works about the dynamic JT effect in negatively charged C$_{60}$ have been almost always in the ground electronic configuration populating only the lowest unoccupied molecular orbitals, which is the $t_{1u}$ orbital. However, to our knowledge, neither the vibronic coupling parameters for excited electronic configuration, say $t_{1g}$ next lowest unoccupied molecular orbial (NLUMO), nor the relevant JT effect has been theoretically investigated much. While it is believed that the nature of excited C$_{60}$ anions involving the next lowest unoccupied molecular orbital is of fundamental importance to interpret absorption spectra of isolated C$_{60}^-$ [@Kato1991; @Kato1993; @Kodama1994; @Kondo1995; @Kwon2001; @Kwon2002; @Tomita2005; @Stochkel2013; @Watariguchi2016], electron transfer process of fullerene [@ET1; @ET2], and excitation spectra of alkali-doped fullerides [@Knupfer1997; @Chibotaru1999; @Chibotaru2000], and the JT effect involving the NLUMO must be significant in highly alkali doped [@Knupfer1997] and alkali-earth/rare-earth doped fullerides [@Chen1999; @Margadonna2000; @Iwasa2003; @Li2003; @He2005; @Akada2006; @Heguri2010]. Furthermore, it might also be important [@Nava2018] in recently reported light induced superconductivity of alkali-doped fullerides [@Mitrano2016; @Cantaluppi2017]. Recently, bound excited states of C$_{60}^-$ have been theoretically investigated [@EX1; @EX2; @EX3; @EX4; @EX5], and the stability of the first excited ${}^2T_{1g}$ states of C$_{60}^-$ has been confirmed, nevertheless, the vibronic problem has not been investigated. In this work, we address the dynamical JT effect of first excited C$_{60}^-$ anion populating the $t_{1g}$ NLUMO. The vibronic coupling parameters are derived from the data obtained by density functional theory (DFT) calculations with hybrid B3LYP exchange-correlation functional. Using these coupling parameters, the vibronic states are obtained by numerically diagonalizing the dynamical JT Hamiltonian matrix, and are analyzed. Jahn-Teller Effect ================== Model Hamiltonian {#Sec:H} ----------------- The $t_{1g}$ next LUMO of neutral C$_{60}$ with $I_h$ symmetry is triply degenerate and separated from the other orbital levels [@Chancey1997]. According to the selection rule, the $t_{1g}$ orbitals couple to totally symmetric $a_g$ and five-fold degenerate $h_g$ representation as in the case of $t_{1u}$ orbitals [@Jahn1937]: $$\begin{aligned} [t_{1g} \otimes t_{1g}] = a_g \oplus h_g. \label{Eq:selection}\end{aligned}$$ In this work, we take the equilibrium structure of C$_{60}$ as the reference. Therefore, besides the $h_g$ modes, the vibronic couplings to the $a_g$ modes are nonzero. The linear vibronic Hamiltonian of C$_{60}^-$ in the first excited electronic $(t_{1g}^1)$ configuration resembles to that for the ground $t_{1u}^1$ electronic configuration [@OBrien1969; @Auerbach1994; @OBrien1996; @Chancey1997]: $$\begin{aligned} H &=& H_a + H_h, \label{Eq:H} \\ H_a &=& \frac{1}{2} \left( p_a^2 + \omega_a^2 q_{a}^2 \right) + V_a q_{a}, \label{Eq:Ha} \\ H_h &=& \sum_{\gamma = \theta, \epsilon, \xi, \eta, \zeta} \frac{1}{2}\left(p_{h\gamma}^2 + \omega_h^2 q_{h \gamma}^2\right) \nonumber\\ &&+ V_h \begin{pmatrix} \frac{1}{2} q_{h\theta} - \frac{\sqrt{3}}{2} q_{h\epsilon} & \frac{\sqrt{3}}{2} q_{h\zeta} & \frac{\sqrt{3}}{2} q_{h\eta} \\ \frac{\sqrt{3}}{2} q_{h\zeta} & \frac{1}{2} q_{h\theta} + \frac{\sqrt{3}}{2} q_{h\epsilon} & \frac{\sqrt{3}}{2} q_{h\xi} \\ \frac{\sqrt{3}}{2} q_{h\eta} & \frac{\sqrt{3}}{2} q_{h\xi} & -q_{h\theta} \\ \end{pmatrix}. \label{Eq:Hh}\end{aligned}$$ Here, $q_{\Gamma\gamma}$ and $p_{\Gamma\gamma}$ ($\gamma = \theta, \epsilon, \xi, \eta, \zeta$ for $\Gamma = h$) are mass-weighted normal coordinates and conjugate momenta, respectively, $\omega_\Gamma$ is frequency, and $V_\Gamma$ the vibronic coupling parameters. The basis of the marix is in the order of $|T_{1g}x\rangle$, $|T_{1g}y\rangle$, $|T_{1g}z\rangle$. The representation for normal coordinates and conjugate momenta possess the symmetry of real $d$-type ($(2z^2-x^2-y^2)/\sqrt{6}$, $(x^2-y^2)/\sqrt{2}$, $\sqrt{2}yz$, $\sqrt{2}zx$, $\sqrt{2}xy$), as they are in consistent with the original and most used representation [@OBrien1969; @Auerbach1994; @Manini1994; @Obrien1996; @Chancey1997]. The bases are different from those ($Q$) of some previous work [@Dunn1995]. The relation between them are $$\begin{aligned} \begin{split} q_\theta =\sqrt{\frac{3}{8}} Q_\theta + \sqrt{\frac{5}{8}} Q_\epsilon,\\ q_\epsilon=\sqrt{\frac{3}{8}} Q_\theta - \sqrt{\frac{5}{8}} Q_\epsilon. \end{split}\end{aligned}$$ In the above equation, the indices $g$ or $u$ indicating the parity and the indices $\mu$ distinguishing the frequencies are omitted for simplicity. They are added when necessary for the discussion. Adiabatic potential energy surface ---------------------------------- The model Hamiltonians for the ground electronic configuration and the first excited configuration are the same. Therefore, many electronic properties of the ground and the first excited electronic configurations are common too. The depth of the adiabatic potential energy surface (APES) with respect to the reference structure is given by [@OBrien1969] $$\begin{aligned} U_\text{min} &=& -E_a - E_\text{JT} \nonumber\\ &=& -\frac{V_a^2}{2\omega_a^2} - \frac{V_h^2}{2\omega_h^2}, \label{Eq:Umin}\end{aligned}$$ with $$\begin{aligned} q_{a,0} = -\frac{V_a}{\omega_a^2}, \quad \left|\bm{q}_{h,0} \right| = \frac{V_h}{\omega_h^2},\end{aligned}$$ where $E_a$ and $E_\text{JT}$ are the first and the second terms in the last expression in Eq. (\[Eq:Umin\]), respectively, and $\bm{q}_{h}$ is the list of $q_{h\gamma}$. The APES has two-dimensional continuous trough [@OBrien1969], suggesting the presence of SO(3) symmetry [@OBrien1971; @Pooler1980]. Vibronic states --------------- As in the case of the JT problem for the ground electronic configuration [@OBrien1971; @Romestain1971; @Pooler1980], the vibronic angular momenta $\hat{\bm{J}}$ also exist in the first excited state [@Chancey1997]: $$\begin{aligned} [\hat{H}_h, \hat{\bm{J}}^2] = [\hat{H}_h, \hat{J}_z] = 0. \label{Eq:symmetry}\end{aligned}$$ Therefore, the eignestates of $\hat{H}$ (vibronic states) are expressed by $J$, $M_J$, and principal quantum number $\alpha$, $$\begin{aligned} \hat{H}_h|\alpha JM_J\rangle &=& E_{\alpha J} |\alpha JM_J\rangle. \label{Eq:vibronicproblem}\end{aligned}$$ The analytical treatments of the vibronic states in the strong limit of vibronic coupling [@OBrien1969; @OBrien1971; @Auerbach1994; @OBrien1996; @Iwahara2018] and weak coupling limit [@Manini1994] have been discussed much. Nevertheless, for the quantitative description of C$_{60}$ ions, only numerical approach can provide accurate description. For numerical calculations, it is convenient to expand the vibornic states as $$\begin{aligned} |\alpha J M_J \rangle &=& \sum_\gamma \sum_{\bm{n}_h} |T_{1g} \gamma\rangle \otimes |\bm{n} \rangle C_{\gamma \bm{n}; \alpha J M_J}.\end{aligned}$$ Here, $\bm{n}_h = (n_{h\theta}, n_{h\epsilon}, n_{h\xi}, n_{h\eta}, n_{h\zeta})$ is the set of vibrational quantum numbers of the Harmonic oscillation part of Eq. (\[Eq:Hh\]). Such an expansion using the direct products of the electronic states and the eigenstates of harmonic oscillator has been developed long time ago [@Longuet-Higgins1958] and has been routinely used to study dynamical JT problems including fullerene anion [@OBrien1971; @Auerbach1994; @Gunnarsson1995; @OBrien1996; @Iwahara2010; @Iwahara2013; @Ponzellini; @Liu2018a]. In the present calculations, the vibrational basis is truncated as $$\begin{aligned} 0 \le n_{h(\mu)\gamma}, \quad \sum_{\mu\gamma} n_{h(\mu)\gamma} \le 7,\end{aligned}$$ because the dimension of the Hamiltonian matrix rapidly increases. To take account of the eight sets of $h_g$ modes in real C$_{60}$, $\mu$ is added in the condition. For the diagonalization of the vibronic Hamiltonian (\[Eq:Hh\]), Lanczos algorithm was applied [@Pooler1984]. Results ======= Orbital vibronic coupling parameters ------------------------------------ ![ The JT splitting of the NLUMO levels with respect to $q_{h_g(8)\epsilon}$ deformation (in atomic unit). The black points and gray lines indicate the DFT values and model energy, respectively. []{data-label="Fig:V"}](Vh8.eps){width="8cm"} ---------- ------- ------------------- ------------ ------------ ------------ ------------ ------------ ------------ $J$ $\Gamma$ $\mu$ $\omega_{\Gamma}$ $V_\Gamma$ $g_\Gamma$ $E_\Gamma$ $V_\Gamma$ $g_\Gamma$ $E_\Gamma$ $a_g$ 1 496 $-0.449$ $-0.418$ 5.38 $-0.264$ $-0.245$ 1.849 2 1470 $-2.480$ $-0.452$ 18.66 $-2.380$ $-0.422$ 16.543 $h_g$ 1 273 $-0.406$ $-0.926$ 14.50 0.192 0.455 3.415 2 437 $-0.476$ $-0.536$ 7.78 0.450 0.503 6.886 3 710 $-1.061$ $-0.577$ 14.64 0.754 0.396 7.069 4 774 $-0.594$ $-0.284$ 3.86 0.554 0.259 3.256 5 1099 $-0.498$ $-0.141$ 1.35 0.766 0.209 3.038 6 1250 $-1.664$ $-0.387$ 11.61 0.578 0.132 1.360 7 1428 0.125 0.024 0.05 2.099 0.394 13.867 8 1575 $-2.113$ $-0.348$ 11.79 2.043 0.326 10.592 ---------- ------- ------------------- ------------ ------------ ------------ ------------ ------------ ------------ The orbital vibronic coupling parameters are defined by the gradients of the $t_{1g}$ NLUMO level: $$\begin{aligned} v_a = \left.\frac{\partial \epsilon_{t_{1g}z}}{\partial q_a}\right|_{\bm{q} = \bm{0}}, \quad v_h = -\left.\frac{\partial \epsilon_{t_{1g}z}}{\partial q_{h\theta}}\right|_{\bm{q} = \bm{0}},\end{aligned}$$ where $\bm{q}$ is the set of all normal coordinates. In the present case, the vibronic coupling parameters $V_\Gamma$ correspond to the orbital vibronic coupling parameters $v_\Gamma$: $$\begin{aligned} V_\Gamma = v_\Gamma,\end{aligned}$$ in a good approximation because of the very small mixing of the orbitals under JT deformation. The vibronic coupling parameters are derived by fitting the model potential to the gradients of NLUMO levels calculated in Ref. . The calculations were done using DFT calculations with hybrid B3LYP functional, because, indicated by the previous studies, B3LYP could give closer parameters to the experimental data[@Iwahara2013; @Matsuda2018], and has a good agreement wit GW approximation calculations [@Faber2011]. The derived coupling parameters are listed in Table \[Table:V\], and one of the fittings are shown in Fig. \[Fig:V\] (see for the others Supplemental Materials). The stabilization energies in the first excited electron configuration are $E_a = 24.04$ and $E_h = 65.58$ meV, which are by 30.7 % and 32.5% larger than the stabilization energies of $a_{g}$ and $h_{g}$ mode for the ground configuration, respectively. Moreover, almost all the vibronic coupling parameters for the $h$ modes in the $T_{1g}$ state are opposite compared with the case for the $T_{1u}$ state. The difference in sign indicates that the relative displacements in the ground and excited electronic states are large, and hence, the vibronic progression under the transition ${}^2T_{1g} \leftarrow {}^2T_{1u}$ tends to be stronger than that under the photoelectron spectra of C$_{60}^-$. This tendency is seen in experimental absorption spectra of C$_{60}^-$ [@Kondo1995; @Tomita2005] and photoelectron spectra of C$_{60}^-$ [@Wang2005; @Huang2014]. Vibronic states --------------- The ground vibronic levels (termed by $J$=1) for the $T_{1u}$ (LUMO) and $T_{1g}$ (NLUMO) electornic states are $-96.5$ and $-113.8$ meV, respectively. Previous study shows that for LUMO, the contributions from the static and the dynamic JT effect to the ground energy is almost the same[@Liu2018a], but this is the not the same situation for NLUMO, as is shown in Table. \[Table:Energy\_contrib\]. The ratio of the dynamical contribution to the static contribution is smaller in NLUMO case than in LUMO case, which is consistent [@Auerbach1994; @OBrien1996] with the stronger vibronic coupling in NLUMO than in LUMO. Orbital E$_{total}$ E$_{static}$ E$_{dynamic}$ Ratio ----------------- ------------- -------------- --------------- ------- NLUMO $-113.8$ $-65.6$ $-48.2$ 0.74 LUMO[@Liu2018a] $-96.46$ $-50.3$ $-46.2$ 0.92 : Contributions to the ground vibronic energy (E$_{total}$) of NLUMO and LUMO of C$_{60}^{-}$. E$_{static}$, and E$_{dynamic}$ represent the static JT and dynamic JT stabilization energies. Ratio refers the ratio between E$_{static}$ and E$_{dynamic}$ (E$_{dynamic}$/E$_{static}$).[]{data-label="Table:Energy_contrib"} The low-energy vibronic levels are shown in Fig. \[Fig:Vibronic\_E\_level\] (see also Table. \[Table:Energy\_level\]) for NLUMO, and LUMO, and are compared with the vibrational levels of neutral C$_{60}$. For easy comparision, the energies of ground states of LUMO, NLUMO and the lowest vibrational state were shifted to a same level (0 meV). The group of the first excited vibronic levels ($J = 3,2,1$) split more in NLUMO than in LUMO, as expected from the stronger vibronic coupling in the former: The splitting of the former, 13.3 meV, is about two times larger than that of LUMO (4.4 meV). Such splitting may be observed as fine structure in e.g. high-resolution absorption spectra of C$_{60}^-$. ![Low-lying vibronic levels with respect to NLUMO and LUMO of C$_{60}^{-}$. The numbers next to the energy levels are $J$ with the numbers in the parenthesis are the degeneracy.[]{data-label="Fig:Vibronic_E_level"}](Vibronic_E_level.eps){width="8cm"} $J$ NLUMO LUMO ----- --- ------------ ----------- 1 $-113.815$ $-96.469$ 2 $-74.589$ $-60.753$ 3 $-57.370$ $-38.126$ 4 $-47.520$ $-29.757$ 5 $-40.736$ $-26.841$ 6 $-31.168$ $-11.395$ 7 $-25.493$ $-8.099$ 8 - $-5.411$ 9 - $-4.123$ 1 $-78.132$ $-61.918$ 2 $-58.955$ $-40.873$ 3 $-46.127$ $-29.004$ 4 $-31.122$ $-12.071$ 5 $-26.748$ $-8.264$ 6 - $-6.155$ 7 - $-4.265$ 1 $-87.854$ $-65.135$ 2 $-63.252$ $-46.786$ 3 $-55.282$ $-31.703$ 4 $-40.633$ $-25.806$ 5 $-33.815$ $-14.620$ 6 $-32.593$ $-12.575$ 7 $-25.581$ $-8.417$ 8 - $-3.843$ 9 - $-1.607$ 1 $-46.772$ $-28.549$ 2 $-34.314$ $-14.108$ 3 $-27.326$ $-7.724$ 1 $-60.505$ $-33.554$ 2 $-36.593$ $-14.985$ 3 $-27.281$ $ -$ : The vibronic energy levels with respect to NLUMO and LUMO of C$_{60}^{-}$ (meV). The data for LUMO are taken Ref. , and the number in the parentheses indicate $J$.[]{data-label="Table:Energy_level"} Conclusion ========== In this work, the vibronic states of the first excited C$_{60}^-$ were calculated based on the orbital vibronic coupling parameters in the $t_{1g}$ orbital, which is compared with that in $t_{1u}$. The results for the $t_{1g}^1$ configuration showed much stronger dynamic JT stabilization than that for the $t_{1u}^-$ configuration, which induces greater splitting group of the first excited vibronic states too. Combining the present vibronic coupling constant and those from Ref. [\[1.3\]]{}, it is possible to investigate the luminescence spectra[@Akimoto2002] involving $t_{1g}$ NLUMO. The authors thank Naoya Iwahara for his help with numerical calculations. They also gratefully acknowledge funding by the China Scholarship Council. [75]{}ifxundefined \[1\][ ifx[\#1]{} ]{}ifnum \[1\][ \#1firstoftwo secondoftwo ]{}ifx \[1\][ \#1firstoftwo secondoftwo ]{}““\#1””@noop \[0\][secondoftwo]{}sanitize@url \[0\][‘\ 12‘\$12 ‘&12‘\#12‘12‘\_12‘%12]{}@startlink\[1\]@endlink\[0\]@bib@innerbibempty [**](https://press.princeton.edu/titles/6243.html) (, , ) @noop [**]{} (, , ) [****, ()](\doibase http://dx.doi.org/10.1016/j.chemphys.2015.05.016) [**](https://www.worldscientific.com/worldscibooks/10.1142/5404) (, , ) @noop [****,  ()]{} [****,  ()](\doibase 10.1051/epjconf/20122300015) @noop [****,  ()]{} @noop [****,  ()]{} [****,  ()](http://stacks.iop.org/0953-8984/28/i=15/a=153001) [**** (), 10.3390/cryst8030115](\doibase 10.3390/cryst8030115) [****,  ()](\doibase 10.1103/PhysRevB.49.12998) [****,  ()](\doibase 10.1103/PhysRevB.49.13008) [****, ()](\doibase 10.1103/PhysRevB.50.5016) [****,  ()](\doibase 10.1103/PhysRevB.52.5996) [****,  ()](\doibase 10.1103/PhysRevLett.74.1875) [****, ()](\doibase 10.1103/PhysRevB.53.3775) [****,  ()](\doibase 10.1103/PhysRevB.54.17184) [****, ()](\doibase https://doi.org/10.1016/S0379-6779(97)81163-0),  [****,  ()](\doibase 10.1103/PhysRevB.58.782) [****,  ()](\doibase 10.1103/PhysRevB.68.235403) [****,  ()](\doibase 10.1103/PhysRevB.71.115411) [****,  ()](\doibase 10.1103/PhysRevLett.94.053002) [****,  ()](\doibase 10.1103/PhysRevB.77.115445) [****,  ()](\doibase 10.1103/PhysRevB.78.233401) [****,  ()](\doibase 10.1103/PhysRevB.82.245409) [****,  ()](\doibase 10.1088/1367-2630/14/8/083038) [****,  ()](\doibase 10.1038/ncomms1910) [****,  ()](http://scitation.aip.org/content/aip/journal/jcp/139/16/10.1063/1.4826097) **, @noop [Master’s thesis]{},  () [****,  ()](\doibase 10.1021/jp5126409) [****,  ()](\doibase 10.1103/PhysRevB.97.075413) [****,  ()](\doibase 10.1103/PhysRevB.97.115412) [****,  ()](\doibase 10.1103/PhysRevB.98.035402) [****,  ()](\doibase 10.1103/PhysRevB.98.165410) [****, ()](\doibase https://doi.org/10.1016/0009-2614(91)85147-O) [****,  ()](\doibase 10.1016/0009-2614(93)87142-P) [****,  ()](\doibase 10.1021/j100093a002) [****,  ()](\doibase https://doi.org/10.1016/0009-2614(95)00287-E) [****,  ()](\doibase 10.1021/jp0043625) [****, ()](\doibase 10.1140/e10053-002-0008-5) [****,  ()](\doibase 10.2116/analsci.32.463) [****,  ()](\doibase 10.1021/jp983512x) [****,  ()](\doibase 10.1039/C5CP05254H) [****,  ()](\doibase 10.1103/PhysRevLett.79.2714) in @noop [**]{} (, ) pp.  [****,  ()](\doibase 10.1063/1.1342459) [****,  ()](\doibase 10.1103/PhysRevB.60.4351) [****, ()](\doibase 10.1021/cm000330i) [****,  ()](\doibase 10.1088/0953-8984/15/13/202) [****,  ()](\doibase 10.1103/PhysRevB.68.165417) [****,  ()](\doibase 10.1103/PhysRevB.71.085404) [****,  ()](\doibase 10.1103/PhysRevB.73.094509) [****, ()](\doibase https://doi.org/10.1016/j.cplett.2010.02.069) [****,  ()](\doibase doi:10.1038/nphys4288) [****,  ()](https://www.nature.com/articles/nature16522) [****,  ()](\doibase 10.1038/s41567-018-0134-8) [****,  ()](\doibase 10.1021/jz4018514) [****,  ()](\doibase 10.1039/C4CP01447B) [****,  ()](\doibase 10.1021/jp412813m) [****,  ()](\doibase 10.1080/00268976.2015.1060367) [****,  ()](\doibase 10.1039/C6CP00667A) [****,  ()](\doibase 10.1098/rspa.1937.0142) [****,  ()](\doibase 10.1103/PhysRev.187.407) [****,  ()](\doibase 10.1088/0022-3719/4/16/017) [****,  ()](http://stacks.iop.org/0022-3719/13/i=6/a=013) [****,  ()](\doibase 10.1103/PhysRevB.4.4611) [****, ](\doibase http://doi.org/10.1098/rspa.1958.0022) [****,  ()](\doibase 10.1103/PhysRevLett.111.056401) “,” in @noop [**]{},  (, , ) Chap. , p.  [****, ()](\doibase https://doi.org/10.1016/0009-2614(91)90312-W) [****,  ()](\doibase 10.1103/PhysRevB.84.155104) [****,  ()](http://scitation.aip.org/content/aip/journal/jcp/123/5/10.1063/1.1998787) [****,  ()](http://scitation.aip.org/content/aip/journal/jcp/140/22/10.1063/1.4881421) @noop [ ()]{} [****,  ()](\doibase 10.1143/JPSJ.71.630)

--- abstract: 'An ideal in a polynomial ring encodes a system of linear partial differential equations with constant coefficients. Primary decomposition organizes the solutions to the PDE. This paper develops a novel structure theory for primary ideals in this context. We characterize primary ideals in terms of PDE, punctual Hilbert schemes, and the join construction, and we present an explicit algorithm for computing Noetherian operators.' author: - 'Yairon Cid-Ruiz, Roser Homs and Bernd Sturmfels' title: Primary Ideals and their Differential Equations --- Introduction {#sec1} ============ In his 1938 article [@GROBNER_MATH_ANN] on the foundations of algebraic geometry, Gröbner introduced differential operators to characterize membership in a polynomial ideal. He derived such characterizations for ideals that are prime or primary to a rational maximal ideal [@GROBNER_BOOK_AG_2 pages 174-178]. In a 1952 lecture [@GROBNER_LIEGE §1] he suggested that the same program can be carried out for any primary ideal. Gröbner was particularly interested in algorithmic solutions to this problem. Substantial contributions in this subject area were made by analysts. In the 1960s, Ehrenpreis [@EHRENPREIS] stated his *Fundamental Principle* on solutions to linear partial differential equations (PDE) with complex constant coefficients. A main step was the characterization of primary ideals by differential operators. But, he incorrectly claimed that operators with constant coefficients suffice. Using Example (\[exam:Palamodov\]) below, Palamodov [@PALAMODOV] pointed out the error, and he gave a correct proof by introducing the representation by [*Noetherian operators*]{}. Details on the Ehrenpreis-Palamodov Fundamental Principle can also be found in [@BJORK; @HORMANDER]. The ball returned to algebra in 1978 when Brumfiel published the little-known paper [@BRUMFIEL_DIFF_PRIM]. In 1999, Oberst [@OBERST_NOETH_OPS] extended Palamodov’s Noetherian operators to polynomial rings over arbitrary fields. In 2007, Damiano, Sabadini and Struppa [@DAMIANO] gave a computational approach. A general theory for Noetherian commutative rings was developed recently in [@NOETH_OPS]. Building on this, the present article develops a theory of primary ideals as envisioned by Gröbner. We now introduce a running example that serves to illustrate our title and results. The following prime ideal of codimension $c=2$ in $n=4$ variables is familiar to many algebraists: $$\label{eq:twistedcubic1} P \quad = \quad \langle\, x_1^2-x_2 x_3,\, x_1 x_2 - x_3 x_4, x_2^2 - x_1 x_4 \,\rangle \quad \subset \quad {\mathbb{C} }[x_1,x_2,x_3,x_4].$$ This ideal defines the (affine cone over the) [*twisted cubic curve*]{} $\,V(P) = \bigl\{ \,(s^2t, s t^2,s^3, t^3) \,:\, s,t \in {\mathbb{C} }\,\bigr\}$; see [@CUBIC_LITTLE]. We identify the polynomials in (\[eq:twistedcubic1\]) with PDE with constant coefficients by setting $x_i = \partial_{z_i}$. Solving these PDE means describing all functions $\psi(z_1,z_2,z_3,z_4)$ with $$\label{eq:twistedcubic2} \frac{\partial^2 \psi}{\partial z_1^2} = \frac{\partial^2 \psi}{\partial z_2 \partial z_3} \qquad {\rm and} \qquad \frac{\partial^2 \psi}{\partial z_1 \partial z_2} = \frac{\partial^2 \psi}{\partial z_3 \partial z_4} \qquad {\rm and} \qquad \frac{\partial^2 \psi}{\partial z_2^2} = \frac{\partial^2 \psi}{\partial z_1 \partial z_4}.$$ Results in analysis ensure that every solution comes from a measure $\mu$ on the $(s,t)$-plane: $$\label{eq:twistedcubic4} \psi(z_1,z_2,z_3,z_4) \,\,\,\, = \,\,\, \int {\rm exp} \bigl( z_1 s^2 t \,+\, z_2 s t^2 \,+\,z_3 s^3 \,+\, z_4 t^3 \bigr) \,\mu(s,t) \,{\rm d}s \,{\rm d}t .$$ For instance, if $\mu$ is the Dirac measure at the point $(2,3)$ then $\psi = {\rm exp}( 12 z_1 + 18 z_2 + 8 z_3 + 27 z_4)$. Thus, the functions $\psi$ are simply an analytic encoding of the affine surface $V(P) \subset {\mathbb{C} }^4$. The situation becomes interesting when we consider a non-reduced scheme structure on our surface. Algebraically, this means replacing the prime $P$ by a $P$-primary ideal. We use differential operators to give compact representations of $P$-primary ideals $Q$. For instance, $$\label{eq:twistedcubic6} \begin{matrix} Q \,\,=\, \, \bigl\{ \,f \in {\mathbb{C} }[x_1,x_2,x_3,x_4]\,:\, A_i \bullet f \in P \,\,\,{\rm for} \,\,\, i=1,2,3\, \bigr\}, \smallskip \\ {\rm where} \quad \,A_1 \,=\, 1\,,\;\, A_2\,=\, \partial_{x_1} \,\,\,{\rm and} \,\,\,A_3 \,=\, \partial_{x_1}^2 \,-\, 2 \,x_2\,\partial_{x_2} . \quad \end{matrix}$$ Here $\bullet$ means applying a differential operator to a function. Note that a prime ideal is always represented by just one Noetherian operator $A_1=1$. We can encode (\[eq:twistedcubic6\]) by the ideal $$\label{eq:magic} \!\! \bigl\langle u_1^2 - u_2 u_3, u_1 u_2 -u_3 u_4, u_2^2-u_1 u_4, \, x_1-u_1-y_1, x_2-u_2-y_2, x_3-u_3, x_4-u_4, \, \underline{ y_1^3, \, y_2 + u_2 \,y_1^2} \bigr\rangle.$$ The minimal generators of $Q$ are obtained from (\[eq:magic\]) by eliminating $\{u_1,u_2,u_3,u_4,y_1,y_2\}$: $$\begin{small} \begin{matrix} Q \,\, = & \!\!\! \bigl\langle\, 3 x_1^2 x_2^2-x_2^3 x_3-x_1^3 x_4-3 x_1 x_2 x_3 x_4+2 x_3^2 x_4^2\,,\,\, 3 x_1^3 x_2 x_4-3 x_1 x_2^2 x_3 x_4-3 x_1^2 x_3 x_4^2+3 x_2 x_3^2 x_4^2\\ & +2 x_2^3 -2 x_3 x_4^2\,,\,\, 3 x_2^4 x_3-6 x_1 x_2^2 x_3 x_4+3 x_1^2 x_3 x_4^2+x_2^3-x_3 x_4^2\,,\,\, 4 x_1 x_2^3 x_3+x_1^4 x_4-6 x_1^2 x_2 x_3 x_4\\ & -3 x_2^2 x_3^2 x_4+4 x_1 x_3^2 x_4^2\,,\,\, x_2^5-x_1 x_2^3 x_4-x_2^2 x_3 x_4^2+x_1 x_3 x_4^3\,,\,\, x_1 x_2^4-x_2^3 x_3 x_4-x_1 x_2 x_3 x_4^2+x_3^2 x_4^3\,,\\ & x_1^4 x_2-x_2^3 x_3^2-2 x_1^3 x_3 x_4+2 x_1 x_2 x_3^2 x_4\,,\,\, x_1^5-4 x_1^3 x_2 x_3+3 x_1 x_2^2 x_3^2+2 x_1^2 x_3^2 x_4-2 x_2 x_3^3 x_4\,,\\ & 3 x_1^4 x_4^2-6 x_1^2 x_2 x_3 x_4^2+3 x_2^2 x_3^2 x_4^2+4 x_2^4-4 x_2 x_3 x_4^2\,,\,\, x_2^3 x_3^2 x_4+x_1^3 x_3 x_4^2-3 x_1 x_2 x_3^2 x_4^2+x_3^3 x_4^3 \\ & +x_1 x_2^3-x_1 x_3 x_4^2\,, 3 x_1^4 x_3 x_4-6 x_1^2 x_2 x_3^2 x_4+3 x_2^2 x_3^3 x_4+2 x_1^3 x_2 +6 x_1 x_2^2 x_3-6 x_1^2 x_3 x_4-2 x_2 x_3^2 x_4\,,\\ & 4 x_2^3 x_3^3+4 x_1^3 x_3^2 x_4-12 x_1 x_2 x_3^3 x_4+4 x_3^4 x_4^2 -x_1^4+6 x_1^2 x_2 x_3+3 x_2^2 x_3^2-8 x_1 x_3^2 x_4\, \bigr\rangle. \end{matrix} \end{small}$$ As in (\[eq:twistedcubic1\]) and (\[eq:twistedcubic2\]), we can view $Q$ as a system of PDE by setting $x_i = \partial_{z_i}$. Its solutions are $$\label{eq:twistedcubic5} \begin{matrix} \!\! \psi(z_1,z_2,z_3,z_4) \, = \,\, \sum_{i=1}^3 \int \! B_i(z_1,z_2,s,t) \cdot {\rm exp} \bigl( z_1 s^2 t + z_2 s t^2 +z_3 s^3 + z_4 t^3 \bigr) \,\mu_i(s,t) \,{\rm d}s \,{\rm d}t , \smallskip \\ \qquad {\rm where} \quad B_1\,=\, 1\,,\;\, B_2 = z_1 \,\,\, {\rm and} \,\,\, B_3 \,=\, z_1^2 - 2 st^2 z_2 , \end{matrix}$$ for suitable measures $\mu_1,\mu_2,\mu_3$ on the $(s,t)$-plane ${\mathbb{C} }^2$. Note that $Q$ has multiplicity $3$ over $P$. The title of this paper refers to two ways of associating differential equations to a primary ideal in a polynomial ring. First, we use PDE with polynomial coefficients, namely Noetherian operators $A_i$ as in (\[eq:twistedcubic6\]), to give a compact encoding of $Q$. Second, we can interpret $Q$ itself as a system of PDE with constant coefficients, with solutions represented by [*Noetherian multipliers*]{} $B_i$ as in (\[eq:twistedcubic5\]). The dual roles played by the $A_i$ and $B_i$ is one of our main themes. This paper is organized as follows. In Section \[sec2\] we present characterizations of primary ideals in terms of punctual Hilbert schemes and Weyl-Noether modules. The former offers a parametrization of all $P$-primary ideals of a given multiplicity, and the latter establishes the links to differential equations. In Section \[sec3\] we turn to the Ehrenpreis-Palamodov Fundamental Principle. We present a self-contained proof of the algebraic part, and we introduce algorithms for computing Noetherian operators. In Sections \[sec4\] and \[sec6\] we prove the results stated in Section \[sec2\]. Section \[sec5\] reviews differential operators in commutative algebra and supplies tools for our proofs. In Section \[sec7\] we study the join construction for primary ideals, which offers a new perspective on ideals that are similar to symbolic powers. Finally, in Section \[sec8\] we establish a connection to numerical algebraic geometry. We propose a definition of [*numerical primary decomposition*]{} that puts a focus on the representation of primary ideals. Characterizing Primary Ideals {#sec2} ============================= Irreducible varieties and their prime ideals are the basic building blocks in algebraic geometry. Solving systems of polynomial equations means extracting the associated primes from the system, and to subsequently study their irreducible varieties. However, if the given ideal is not radical then we seek the primary decomposition and not just the associated primes. We wish to gain a precise understanding of the primary ideals that make up the given scheme. We furnish a representation theorem for primary ideals in a polynomial ring, extending the familiar case of zero-dimensional ideals (Macaulay’s inverse system [@Grobner37]). This combines a characterization via differential operators with a parametrization from a Hilbert scheme. Fix a field ${\mathbb{K}}$ of characteristic zero and a prime ideal $P$ of codimension $c$ in the polynomial ring $R = {\mathbb{K}}[x_1,\ldots,x_n]$. We write ${\mathbb{F}}$ for the field of fractions of the integral domain $R/P$. \[thm:main\] The following four sets of objects are in a natural bijective correspondence: (a) $P$-primary ideals $Q$ in $R$ of multiplicity $m$ over $P$, (b) points in the punctual Hilbert scheme $\,{\rm Hilb}^m({\mathbb{F}}[[y_1,\ldots,y_c]])$, (c) $m$-dimensional ${\mathbb{F}}$-subspaces of $\,{\mathbb{F}}[z_1,\ldots,z_c]$ that are closed under differentiation, (d) $m$-dimensional ${\mathbb{F}}$-subspaces of the Weyl-Noether module $ {\mathbb{F}}\,\otimes_R \,D_{n,c}$ that are $R$-bi-modules. Moreover, any basis of the ${\mathbb{F}}$-subspace in part (d) can be lifted to Noetherian operators $A_1,\ldots,A_m$ in the relative Weyl algebra $D_{n,c}$ that represent the ideal $Q$ in part (a) as in (\[eq:fromAtoQ\]). The purpose of this section is to define and explain all the concepts in Theorem \[thm:main\]. Our aim is to state the promised bijections as explicitly as possible. The proof of Theorem \[thm:main\] will be divided into smaller pieces and given in Sections \[sec4\] and \[sec6\]. The encoding of $Q$ by Noetherian operators $A_i$ will be explained in Section \[sec3\]. We already saw an example in (\[eq:twistedcubic6\]). The Weyl-Noether module in part (d) is our stage for the PDE that portray primary ideals. We begin by returning to Gröbner, whose 1937 article [@Grobner37] interpreted Macaulay’s inverse system as solutions to linear PDE. He considered the special case when $P = \langle x_1,\ldots,x_n \rangle $ is the maximal irrelevant ideal, so we have $c=n$ and ${\mathbb{F}}= {\mathbb{K}}$. The geometric intuition invoked in [@GROBNER_LIEGE §1] is captured by the punctual Hilbert scheme $\,{\rm Hilb}^m({\mathbb{K}}[[y_1,\ldots,y_n]])$, whose points are precisely the $P$-primary ideals of colength $m$. This zero-dimensional case is familiar to most commutative algebraists, especially the readers of [@MOURRAIN_DUALITY]. Here, parts (c) and (d) of Theorem \[thm:main\] refer to the $m$-dimensional ${\mathbb{K}}$-vector space of polynomial solutions to the PDE. The general case of higher-dimensional primary ideals $Q$ was of great interest to Gröbner. In his 1952 Liège lecture [@GROBNER_LIEGE], he points to Severi [@Severi], and he writes: [*En ce sense la variété algébraique correspondante à un idéal primaire $Q$ pour l’idéal premier $P$ consiste en les points ordinaires de la variété $\,V(P)$ et en certain nombre $m$ des points infinitesiment voisins, c’est-à-dire dans $m$ conditions différentielles ajoutées à chaque point de la variété $\,V(P)$. Le nombre $m$ de ces conditions différentielles est égal à la longueur de l’idéal primaire $Q$.*]{} But Gröbner was never able to complete the program himself, in spite of the optimism he still expressed in his 1970 textbook [@GROBNER_BOOK_AG_2]. After the detailed treatment of Macaulay’s inverse systems for zero-dimensional ideals, he proclaims: [*Es dürfte auch nicht schwer sein den oben angegebenen Formalismus auf mehrdimensionale Primärideale auszudehnen*]{} [@GROBNER_BOOK_AG_2 page 178]. The issue was finally resolved by the theory of Ehrenpreis-Palamodov [@EHRENPREIS; @PALAMODOV], presented in Section \[sec3\], and the subsequent developments [@BRUMFIEL_DIFF_PRIM; @NOETH_OPS; @DAMIANO; @OBERST_NOETH_OPS] we discussed in the Introduction. Theorem \[thm:main\] is our main contribution. We regard this as a definitive result on primary ideals in $R$. It captures the geometric spirit of Gröbner and Severi, as it relates their “infinitely near points” directly to current advances in numerical algebraic geometry (Section \[sec8\]). Two essential ingredients in Theorem \[thm:main\] are the function field ${\mathbb{F}}$ and the Weyl-Noether module $\, {\mathbb{F}}\otimes_R D_{n,c}$. We start our technical discussion with some insights into these objects. By [*Noether normalization*]{}, after a linear change of coordinates, the quotient ring $R/P$ is a finitely generated module over the polynomial subring ${\mathbb{K}}[x_{c+1},\ldots,x_n]$. This implies that ${\mathbb{F}}$ is algebraic over the field ${\mathbb{K}}(x_{c+1},\ldots,x_n)$, a purely transcendental extension of ${\mathbb{K}}$. Clear notation is very important for this article. This is why multiple letters $x,y,z,u$ are used to denote variables and differential operators. Elements in ${\mathbb{F}}$ are represented as fractions of polynomials in ${\mathbb{K}}[u_1,\dots,u_n]$, where $u_i$ denotes the residue class of $x_i$ modulo $P$. Whenever the number $n$ of variables is clear from the context, we use the multi-index notation $\mathbf{u}^\alpha=u_1^{\alpha_1}\cdots u_n^{\alpha_n}$. Elements $a(\mathbf{u})/b(\mathbf{u})$ of the field ${\mathbb{F}}$ can be uniquely represented by taking $a(\mathbf{u})$ and $b(\mathbf{u})$ coprime and in normal form with respect to a Gröbner basis of $P$. Arithmetic in ${\mathbb{F}}$ is performed via this Gröbner basis. The $R$-module structure of ${\mathbb{F}}$ is given by $\,\mathbf{x}^\alpha\cdot a(\mathbf{u})/b(\mathbf{u})=\mathbf{u}^\alpha a(\mathbf{u})/b(\mathbf{u})$. Alternatively, from the perspective of numerical algebraic geometry, a better approach to arithmetic in ${\mathbb{F}}$ is to work with generic points, obtained by realizing $R/P$ as a subring of a suitable field of functions on $V(P)$. In our running example, that suitable field could be $\,{\mathbb{K}}(s/t,t^3)$. It contains $R/P$ as the subring ${\mathbb{K}}[s^2 t, st^2, s^3, t^3]$. The [*relative Weyl algebra*]{} $D_{n,c} = {\mathbb{K}}\langle x_1,\ldots,x_n, \partial_{x_1},\dots,\partial_{x_c}\rangle$ is the ${\mathbb{K}}$-algebra on $n{+}c$ generators $x_1,\ldots,x_n,\partial_{x_1},\ldots,\partial_{x_c}$ that commute except for $\partial_{x_i}x_i=x_i\partial_{x_i}+1$. This is a subalgebra of the usual Weyl algebra, so $D_{n,c}$ is non-commutative. Its elements are linear differential operators with polynomial coefficients, where derivatives occur with respect to the first $c$ variables. The set $\left\lbrace x_1^{\alpha_1}\cdots x_n^{\alpha_n}{\partial_{x_1}}^{\!\!\!\!\beta_1}\cdots {\partial_{x_c}}^{\!\!\!\!\beta_c}:(\alpha,\beta)\in\mathbb{N}^n\times\mathbb{N}^c\right\rbrace$ is a ${\mathbb{K}}$-basis of $D_{n,c}$. We define the [*Weyl-Noether module*]{} of the affine variety $V(P)$ to be the tensor product $$\label{eq:relativeweyl} {\mathbb{F}}\,\otimes_R\,D_{n,c} \,\, = \,\, {\mathbb{F}}\,\otimes_R \, R \langle \partial_{x_1},\ldots,\partial_{x_c}\rangle.$$ Since ${\mathbb{F}}$ is the field of fractions of the integral domain $R/P$, it is clearly an $R$-module. Note that the relative Weyl algebra $D_{n,c} = R\langle \partial_{x_1},\dots,\partial_{x_c}\rangle$ is non-commutative, and it has two distinct $R$-module structures: it is a left $R$-module and it is a right $R$-module. In the tensor product (\[eq:relativeweyl\]), for convenience of notation, we mean the left $R$-module structure on $D_{n,c}$. Later, in Remark \[rem\_isom\_restrict\_Weyl\_mod\], we shall give an intrinsic description of ${\mathbb{F}}\otimes_R D_{n,c}$ with differential operators. By construction, the Weyl-Noether module (\[eq:relativeweyl\]) has both right and left $R$-module structures. The action by $R$ on the left is easy to write using the standard ${\mathbb{K}}$-basis above: $$\label{eq:leftaction} \mathbf{x}^\alpha \cdot \biggl( \frac{a(\mathbf{u})}{b(\mathbf{u})} \,\otimes_R\, \partial_{\mathbf{x}}^\beta\ \biggr) \,\,\, = \,\,\, \frac{\mathbf{u}^{\alpha} a(\mathbf{u})}{b(\mathbf{u})} \,\otimes_R\, \partial_{\mathbf{x}}^\beta.$$ For the action on the right we need the commutation identities in the Weyl algebra: $$\partial_{\mathbf{x}}^\beta \mathbf{x}^\alpha\,\,=\,\,\sum_{\gamma, \delta} \lambda_{\gamma,\delta} \, \mathbf{x}^\gamma \partial_\mathbf{x}^\delta.$$ Here $\lambda_{\gamma,\delta}$ are the positive integers derived in [@SatStu Problem 4]. With this, the right action is $$\label{eq:rightaction} \biggl( \frac{a(\mathbf{u})}{b(\mathbf{u})} \,\otimes_R \, \partial_{\mathbf{x}}^\beta \biggr) \cdot \mathbf{x}^\alpha \, \,\, = \,\,\, \frac{a(\mathbf{u})}{b(\mathbf{u})} \, \otimes_R \, \partial_{\mathbf{x}}^\beta \mathbf{x}^\alpha \,\,\,=\,\,\, \sum_{\gamma,\delta} \lambda_{\gamma,\delta} \, \frac{\mathbf{u}^{\gamma}a(\mathbf{u})}{b(\mathbf{u})} \, \otimes_R \, \partial_\mathbf{x}^\delta.$$ This means that the requirement to be an $R$-bi-module in Theorem \[thm:main\] (d) is very strong. From the action (\[eq:leftaction\]) we deduce that $\,{\mathbb{F}}\,\otimes_R\,D_{n,c}\,$ is a left ${\mathbb{F}}$-vector space with basis $\,\left\lbrace 1 \otimes_R \partial_{\mathbf{x}}^\beta: \beta\in\mathbb{N}^c\right\rbrace$, so we could also write ${\mathbb{F}}\langle \partial_{x_1},\dots,\partial_{x_c}\rangle$ for (\[eq:relativeweyl\]). However, we prefer the previous notation because it highlights that there are two distinct structures. The Weyl-Noether module is a left ${\mathbb{F}}$-vector space via (\[eq:leftaction\]) and it is a right $R$-module via (\[eq:rightaction\]). It is not a right ${\mathbb{F}}$-vector space because the right $R$-action is not compatible with passing to $R/P$: Fix the maximal ideal $P=\langle x_1, \ldots,x_n\rangle$ so that ${\mathbb{F}}={\mathbb{K}}$ and $c=n$. Since $\overline{x_j} = 0 \in R/P$, we have $\,x_j \cdot \left(1 \otimes_R \partial_{x_j}\right) = 0 \,$ and hence $\,\left(1 \otimes_R \partial_{x_j}\right) \cdot x_j = 1 \otimes_R 1$ holds in $\,{\mathbb{F}}\,\otimes_R\,D_{n,c}$. This shows that there is no right ${\mathbb{F}}$-action on the Weyl-Noether module $\,{\mathbb{F}}\,\otimes_R\,D_{n,c}$. We now come to our parameter space in part (b), namely the punctual Hilbert scheme $$\label{eq:hilbertscheme} {\rm Hilb}^m \bigl( \,{\mathbb{F}}[[y_1,\ldots,y_c]] \,\bigr).$$ This is a quasiprojective scheme over the function field ${\mathbb{F}}$. Its classical points are ideals of colength $m$ in the local ring ${\mathbb{F}}[[y_1,\ldots,y_c]]$. By Cohen’s Structure Theorem, this ring is the completion of $R_P$, the localization of $R$ at the prime $P$. To connect parts (a) and (b), we recall that the multiplicity $m$ of a primary ideal $Q$ over its prime $P = \sqrt{Q}$ is the length of the artinian local ring $\,R_P/Q R_P$. In symbols, using the command [degree]{} in [Macaulay2]{} [@MAC2], $$m \,\,=\,\, {\rm length}\bigl( R_P/QR_P \bigr) \,\,=\,\, \frac{{\tt degree}(Q)}{{\tt degree}(P)}.$$ The punctual Hilbert scheme (\[eq:hilbertscheme\]) is familiar to algebraic geometers, but its structure is very complicated when $c \geq 3$. We refer to Iarrobino’s article [@IARROBINO_HILB] as a point of entry. While the punctual Hilbert scheme is trivial for $c=1$, Briançon [@Bri77] undertook a detailed study for $c=2$. He showed that $\,{\rm Hilb}^m \bigl( {\mathbb{F}}[[y_1,y_2]] \bigr)\,$ is smooth and irreducible of dimension $m-1$. A dense subset is given by the $(m-1)$-dimensional family of $\langle y_1,y_2 \rangle$-primary ideals of the form $$\label{eq:HSfamily} \qquad \bigl\langle \,\, y_1^m\,,\, \,y_2 \,+\, a_1 y_1 + a_2 y_1^2 + \cdots + a_{m-1} y_1^{m-1} \, \bigr\rangle, \qquad {\rm where}\,\,a_1,a_2,\ldots, a_{m-1} \in {\mathbb{F}}.$$ For instance, for $m=3$, the Hilbert scheme (\[eq:hilbertscheme\]) is a surface over ${\mathbb{F}}$. Each of its points encodes a scheme structure of multiplicity $3$ on the variety $V(P)$. This is the generic point on $V(P)$ together with two “infinitely near points”, in the language of Gröbner and Severi. To see that the family (\[eq:HSfamily\]) is a proper subset of $\,{\rm Hilb}^m \bigl( {\mathbb{F}}[[y_1,y_2]] \bigr)$, we consider the points $$\qquad \langle\, y_1^3 \,, \,y_2 + \epsilon^{-1} y_1^2 \,\rangle\,\, =\,\, \langle \,y_1^2 + \epsilon y_2\, , \,y_1y_2\, ,\,y_2^2 \,\rangle \quad \in \,\,\, {\rm Hilb}^3 \bigl( {\mathbb{F}}[[y_1,y_2]] \bigr).$$ For $\epsilon \in {\mathbb{F}}\backslash \{0\}$, this $\langle y_1,y_2 \rangle $-primary ideal is in the family (\[eq:HSfamily\]), but for $\epsilon = 0$ it is not. In the zero-dimensional case, when $P = \langle x_1,\ldots,x_n \rangle$, the correspondences in Theorem \[thm:main\] are well-known since the 1930’s. Wolfgang Gröbner tells us: [*Die noch verbleibende Aufgabe, die Integrale eines Primärideals aus denjenigen für das zugehörige Primideal abzuleiten, wollen wir hier wenigstens für null-dimensionale Primärideale allgemein lösen*]{} [@Grobner39 page 272]. In our current understanding, the $P$-primary ideals are points in ${\rm Hilb}^m({\mathbb{K}}[[y_1,\dots,y_n]])$, subspaces closed by differentiation are Macaulay’s inverse systems, and these account for polynomial solutions to linear PDE with constant coefficients [@MOURRAIN_DUALITY; @STURMFELS_SOLVING]. The idea behind Theorem \[thm:main\] is to reduce the study of arbitrary primary ideals in $R = {\mathbb{K}}[x_1,\ldots,x_n]$ to a zero-dimensional setting over the function field ${\mathbb{F}}$. Recall that coordinates were chosen so that $R/P$ is finite over ${\mathbb{K}}[x_{c+1},\ldots,x_n]$. We define the inclusion map $$\label{eq_map_gamma} \gamma:R \hookrightarrow {\mathbb{F}}[y_1,\dots,y_c]\, , \qquad \begin{matrix} x_i &\mapsto & y_i+u_i, & \!\!\!\!\! \mbox{ for }1\leq i\leq c,\\ x_j & \mapsto & u_j,& \quad \mbox{ for }c+1\leq j\leq n, \end{matrix}$$ where $u_i$ denotes the class of $x_i$ in ${\mathbb{F}}$, for $1\leq i\leq n$. With this, we can give an explicit description of the correspondence between the objects in parts (a) and (b) of Theorem \[thm:main\]: $$\label{eq:corr12} \begin{array}{ccc} \left\lbrace\begin{array}{c} \mbox{$P$-primary ideals of $R$}\\ \mbox{with multiplicity $m$ over $P$} \end{array}\right\rbrace & \longleftrightarrow & \left\lbrace\begin{array}{c} \mbox{points in }{\rm Hilb}^m({\mathbb{F}}[[y_1,\ldots,y_c]])\\ \end{array}\right\rbrace\\ Q & \longrightarrow & I=\langle y_1,\dots,y_c\rangle^m+\gamma( Q){\mathbb{F}}[y_1,\dots,y_c]\\ Q\,=\,\gamma^{-1}(I) & \longleftarrow & I. \end{array}$$ \[Bij:1,2\] Fix $P $ and $Q$ as in the Introduction, with $n=4$, $m=3$, $c=2$, where $R/P$ is finite over ${\mathbb{C} }[x_3,x_4]$. The primary ideal $Q$ corresponds to a point in ${\rm Hilb}^3({\mathbb{F}}[[y_1,y_2]])$. See [@Bri77 Section IV.2] for a detailed description of points in the Hilbert scheme of degree 3 in two variables. The bijection in (\[eq:corr12\]) gives us the following point in the punctual Hilbert scheme: $$\label{eq:punctual2} I \,\,=\,\,\langle y_2^2,y_1y_2,y_1^2+u_2^{\,-1}y_2\rangle \,\,\,\subset \,\,{\mathbb{F}}[[y_1,y_2]].$$ Note that this ideal is also generated by $y_1^3$ and $y_2+u_2y_1^2$, as in (\[eq:magic\]). The bijection between (b) and (c) is Macaulay’s duality between zero-dimensional ideals in a power series ring and finite-dimensional subspaces in a polynomial ring that are closed under differentiation. To interpret polynomials in $I$ as PDE, we replace $y_i$ by $\partial_{z_i}$. So, by slight abuse of notation, we shall write ${\mathbb{F}}[[y_1,\ldots,y_c]]$ and ${\mathbb{F}}[[\partial_{z_1}, \ldots, \partial_{z_c}]]$ interchangeably. With this, the [*inverse system*]{} of a zero-dimensional ideal $I$ in the local ring ${\mathbb{F}}[[\partial_{z_1},\dots,\partial_{z_c}]]$, denoted by $I^\perp$, is the ${\mathbb{F}}$-vector space of solutions $\,\left\lbrace F\in {\mathbb{F}}[z_1,\ldots,z_c]: f\bullet F=0 \mbox{ for all }f\in I\right\rbrace$. Inverse systems furnish an explicit bijection between items (b) and (c) in Theorem \[thm:main\]: $$\label{eq:HilbBijection} \begin{array}{ccc} \left\lbrace\begin{array}{c} \mbox{points in }{\rm Hilb}^m\left({\mathbb{F}}[[\partial_{z_1},\ldots,\partial_{z_c}]]\right)\\ \end{array}\right\rbrace & \longleftrightarrow & \left\lbrace\begin{array}{c} \mbox{$m$-dimensional ${\mathbb{F}}$-subspaces} \\ \mbox{of $ {\mathbb{F}}[z_1,\dots,z_c]$} \\ \mbox{closed under differentiation} \end{array}\right\rbrace\\ I & \longrightarrow & V=I^\perp\\ I={\rm Ann}_{{\mathbb{F}}[[\partial_{z_1},\dots,\partial_{z_c}]]}(V) & \longleftarrow & V.\\ \end{array}$$ Setting $y_i=\partial_{z_i}$, the ideal in Example \[Bij:1,2\] is $I=\langle\partial_{z_2}^2,\partial_{z_1}\partial_{z_2},\partial_{z_1}^2+u_2^{\, -1}\partial_{z_2}\rangle\subset {\mathbb{F}}[[\partial_{z_1},\partial_{z_2}]]$. Note that $\,z_1^2-2u_2z_2\,$ belongs to the inverse system $I^\perp$ because this polynomial is annihilated by all operators in $I$. Applying the differential operators $\partial_{z_1}$ and $\partial_{z_1}^2$ to $B_3=z_1^2-2u_2z_2$ we obtain an ${\mathbb{F}}$-basis of the inverse system: $B_1=1$, $B_2=z_1$ and $B_3$. Moreover, $I^\perp$ is generated by $B_3$ as an ${\mathbb{F}}[[\partial_{z_1},\partial_{z_2}]]$-module. Hence $I$ is a Gorenstein ideal. The correspondence between items (c) and (d) in Theorem \[thm:main\] links generators of the inverse system of $I$ with Noetherian operators for $Q$. These will be discussed in depth in Section \[sec3\]. Suppose we are given an ${\mathbb{F}}$-basis $\{B_1,\ldots,B_m\}$ of the inverse system $I^\perp$ in (c). After clearing denominators, we can write $B_i(\mathbf{u},\mathbf{z})=\sum_{\vert\alpha\vert\leq m}\lambda_\alpha(\mathbf{u})\mathbf{z}^\alpha$ where $\lambda_\alpha(\mathbf{u})$ is a polynomial in $R$ that represents a residue class modulo $P$. We now replace the unknown $z_i$ in these polynomials with the differential operator $\partial_{x_i}$. This gives the Noetherian operators $$\label{eq:resultingDO} \qquad A_i(\mathbf{x},\partial_{x_1},\dots,\partial_{x_c})\,\,\,=\,\,\,\sum_{\vert\alpha\vert\leq m} \lambda_\alpha(\mathbf{x}) \partial_{x_1}^{\alpha_1}\cdots \partial_{x_c}^{\alpha_c} \qquad {\rm for} \,\,\, i =1,\ldots,m.$$ The transition from the $B_i$’s to the $A_i$’s is invertible, giving the bijection between (c) and (d). \[ex:noeth\] Consider the ideal $Q$ in (\[eq:twistedcubic6\]) and $I$ in (\[eq:punctual2\]). From the generators $B_1(u,z)=1$, $B_2(u,z)=z_1$ and $B_3(u,z)=z_1^2-2u_2z_2$ of the inverse system $I^\perp$ in ${\mathbb{F}}[z_1,z_2]$, we obtain the three Noetherian operators $A_1=1$, $A_2=\partial_{x_1}$ and $A_3=\partial_{x_1}^2-2x_2\partial_{x_2}$ that encode $Q$. Note that $A_3$ alone does not determine $Q$, although $B_3$ is enough to generate the inverse system. An Algebraic View on Ehrenpreis-Palamodov {#sec3} ========================================= In this section we derive the Noetherian differential operators that are central to the Fundamental Principle of Ehrenpreis [@EHRENPREIS] and Palamodov [@PALAMODOV]. In particular, we present a practical algorithm that computes these operators for arbitrary primary ideals in a polynomial ring over a field ${\mathbb{K}}$ of characteristic zero. Our approach extends the algebraic theory in [@BRUMFIEL_DIFF_PRIM; @NOETH_OPS; @OBERST_NOETH_OPS] and the first algorithmic steps taken in [@DAMIANO; @STURMFELS_SOLVING]. For analytic aspects of the Ehrenpreis-Palamodov Theorem we refer to [@EHRENPREIS; @PALAMODOV] and to the books by Björk [@BJORK] and Hörmander [@HORMANDER]. Our point of departure is a prime ideal $P$ in the polynomial ring $R = {\mathbb{K}}[x_1,\ldots,x_n]$. We are interested in $P$-primary ideals. Later on we shall interpret these ideals as systems of linear PDE, by replacing each variable $x_i$ by a differential operator $ \partial_{z_i} = \partial / \partial z_i$. First, however, we take a different path, aimed to turn part (d) in Theorem \[thm:main\] into an algorithm. After applying Noether normalization, $R/P$ is a finitely generated ${\mathbb{K}}[x_{c+1},\ldots,x_n$\]-module, where $c={\rm codim}(P)$. The relative Weyl algebra $D_{n,c} = {\mathbb{K}}\langle x_1,\ldots,x_n,\partial_{x_1},\ldots,\partial_{x_c} \rangle$ consists of linear differential operators with polynomial coefficients, where only derivatives for the first $c$ variables appear. Every operator $A = A(\mathbf{x}, \partial_\mathbf{x})$ in $D_{n,c}$ is a unique ${\mathbb{K}}$-linear combination of [*normal monomials*]{} $\,\mathbf{x}^\alpha \partial_\mathbf{x}^\beta = x_1^{\alpha_1} \cdots x_n^{\alpha_n} \partial_{x_1}^{\beta_1} \cdots \partial_{x_c}^{\beta_c}$, where $\alpha \in \mathbb{N}^n$, $\beta \in {\mathbb{N}}^c$. We write $A \bullet f$ for the natural action of $D_{n,c}$ on polynomials $f \in R$., which is defined by $$x_i \bullet f = x_i \cdot f \quad {\rm and} \quad \partial_{x_i} \bullet f = \partial f / \partial x_i .$$ Suppose we are given $A_1,\ldots,A_m$ in the relative Weyl algebra $D_{n,c}$. This specifies $$\label{eq:fromAtoQ} Q \,\,=\,\, \big\{\, f \in R \,: A_l \bullet f \in P \;\text{ for } \, l = 1,2,\ldots,m \,\big\}.$$ The set $Q$ is a ${\mathbb{K}}$-vector space. However, in general, the subspace $Q$ is not an ideal in $R$. \[ex:leftright\] Fix $n=m=2$, $P = \langle x_1,x_2 \rangle$ and $A_1 = \partial_{x_1}$. If $A_2 = \partial_{x_2}$ then $Q $ is the space of polynomials $f$ in ${\mathbb{K}}[x_1,x_2]$ such that $x_1$ and $x_2$ do not appear in the expansion of $f$. That space is not an ideal. However, if $A_2 = 1$ then the formula (\[eq:fromAtoQ\]) gives the ideal $\,Q = \langle x_1^2, x_2 \rangle$. \[rem:contains\_power\] The space $Q$ always contains a power of $P$. Namely, if $k$ is the maximal order among the operators $A_i$ then $P^{k+1} \subseteq Q$. This follows from the product rule of calculus. We next present a necessary and sufficient condition for $m$ operators in $D_{n,c}$ to specify a primary ideal via (\[eq:fromAtoQ\]). We abbreviate $S= {\mathbb{K}}(x_{c+1},\ldots,x_n)[x_1,\ldots,x_c]$. The point in (\[eq:leftequalsright\]) below is that the relative Weyl algebra $D_{n,c}$ is both a left $R$-module and a right $R$-module. \[thm:leftequalsright\] The space $Q$ is a $P$-primary ideal in the polynomial ring $R$ if and only if $$\label{eq:leftequalsright} \qquad A_i \cdot x_j\,\,\in \,\, S\cdot \{A_1,\ldots,A_m\} \,+\, PS \cdot D_{n,c} \qquad \text{for}\,\, \, i =1,\ldots,m \,\,\text{and} \,\,j =1,\ldots,n.$$ In Example \[ex:leftright\] with $\{A_1,A_2\} = \{\partial_{x_1},\partial_{x_2} \}$ we have $ R = S$. Here $Q$ is not an ideal, and (\[eq:leftequalsright\]) fails indeed for $i=j=1$. To see this, one checks that $ \partial_{x_1} x_1 \not\in R\cdot \{\partial_{x_1},\partial_{x_2}\} + \langle x_1,x_2 \rangle D_{2,2}$. It would be desirable to turn the criterion in Theorem \[thm:leftequalsright\] into a general practical algorithm. Suppose (\[eq:leftequalsright\]) holds and let $f \in Q$. By hypothesis, there exist $\,h_1,\ldots,h_m \in S$ such that $\,A_i x_j \,=\, \sum_{k=1}^m h_k \,A_k \,$ modulo $ PS\cdot D_{n,c}$. Since $A_k \bullet f \in P$, we see that $A_i \bullet (x_j f) = (A_i \,x_j) \bullet f$ lies in $P$ for all $i,j$, and hence $x_j f \in Q$. Thus, $Q$ is an ideal. Next we show that $Q$ is $P$-primary, by the following direct argument. Let $f,g \in R$ such that $f \cdot g \in Q$ and $g \not\in Q$. We claim that $f \in P$. We select an operator $A$ of minimal order among those inside $S\cdot \{A_1,\ldots,A_m\} + PS \cdot D_{n,c}$ that satisfy $A \bullet g \not\in PS$. The element $\,A \bullet (fg) \,=\, f \cdot (A \bullet g) \,+\, (A f - f A) \bullet g \,$ lies in $PS$. The commutator $A f - f A$ is a differential operator of order smaller than that of $A$. By (\[eq:leftequalsright\]), it is inside $S\cdot \{A_1,\ldots,A_m\} + PS \cdot D_{n,c}$. This ensures that $(A f - f A) \bullet g$ is in $PS$. We conclude that $f \cdot (A \bullet g) \in PS$. But, we know that $A \bullet g$ is not in $PS$, and hence $f$ is in the prime ideal $P$. Remark \[rem:contains\_power\] ensures that $\sqrt{Q}$ contains $ P$. Our argument shows that $Q$ is primary with $\sqrt{Q} = P$. The if-direction follows. For the only-if-direction we utilize the isomorphism in Remark \[rem\_isom\_restrict\_Weyl\_mod\] and Lemma \[lem\_prim\_ideal\_implies\_bimod\]. The condition (\[eq:leftequalsright\]) is equivalent to the bi-module condition in Lemma \[lem\_prim\_ideal\_implies\_bimod\]. The following result is the key algebraic ingredient in the Ehrenpreis-Palamodov theory. \[thm\_Noeth\_ops\] For every $P$-primary ideal $Q$ of multiplicity $m$ over $P$, there exist operators $A_1,\ldots,A_m$ in the relative Weyl algebra $D_{n,c}$ such that (\[eq:fromAtoQ\]) holds. Theorem \[thm\_Noeth\_ops\] follows from Theorem \[thm:main\], to be proved in the next three sections. Indeed, if we are given a $P$-primary ideal $Q$ of multiplicity $m$ over $P$, then $Q$ specifies an $m$-dimensional $R$-bi-module inside the ${\mathbb{F}}$-vector space ${\mathbb{F}}\otimes_R D_{n,c}$. We choose elements $A_1,\ldots,A_m$ in $D_{n,c}$ whose images form an ${\mathbb{F}}$-basis for that $R$-bi-module. These operators satisfy (\[eq:fromAtoQ\]). Following Palamodov [@PALAMODOV], we call $A_1,\ldots,A_m$ the [*Noetherian operators*]{} that encode the primary ideal $Q$. It is an essential feature that these are linear differential operators with polynomial coefficients. Operators with constant coefficients do not suffice. In other words, the Weyl algebra is essential in describing primary ideals. This key point is due to Palamodov. It had been overlooked initially by Gröbner and Ehrenpreis. For instance, consider the ideal $Q$ for $n=4, m=3$ in the Introduction. Three Noetherian operators $A_1,A_2,A_3$ are given in (\[eq:twistedcubic6\]), and it is instructive to verify condition (\[eq:leftequalsright\]). Algorithms for passing back and forth between Noetherian operators and ideal generators of $Q$ will be presented later in this section. Our problem is to solve a homogeneous system of linear PDE with constant coefficients. This is given by the generators of a primary ideal $Q$ in ${\mathbb{K}}[x_1,\ldots,x_n]$, where $x_j$ stands for the differential operator $\partial_{z_j} = \partial / \partial z_j$ with respect to a new unknown $z_j$. Our aim is to characterize all sufficiently differentiable functions $\psi(z_1,\ldots,z_n)$ that are solutions to these PDE. This characterization is the content of the Ehrenpreis-Palamodov Theorem, to be stated below. Note that, if we are given an arbitrary system $J \subset R$ of such PDE then we can reduce to the case discussed here by computing a primary decomposition of the ideal $J$. For the analytic aspects that follow, we work over the field ${\mathbb{K}}= {\mathbb{C} }$ of complex numbers. Suppose $Q = \langle p_1,p_2,\ldots,p_r \rangle$, where $p_k = p_k(\mathbf{x})$. The PDE we need to solve take the form: $$\label{eq:mustsolvethis} \qquad p_k(\partial_\mathbf{z})\bullet \psi(\mathbf{z})\,\, =\,\, 0 \qquad \text{ for } k=1,2,\ldots,r.$$ Let $\mathcal{K} \subset {\mathbb{R}}^n$ be a compact convex set. We seek all functions $\psi(\mathbf{z})$ in $\, C^\infty(\mathcal{K})\,$ that satisfy (\[eq:mustsolvethis\]). Here we also use vector notation, namely $\mathbf{z} = (z_1,\ldots,z_n)$ and $\partial_\mathbf{z} = (\partial_{z_1},\ldots,\partial_{z_n})$. According to Theorem \[thm\_Noeth\_ops\], there exist Noetherian operators $A_1(\mathbf{x},\partial_\mathbf{x}),\ldots,A_m(\mathbf{x},\partial_\mathbf{x})$ which encode the primary ideal $Q$ in the sense of (\[eq:fromAtoQ\]). In symbols, $\, A_l( \mathbf{x}, \partial_\mathbf{x}) \bullet f \in P\,$ for all $l$. Each $A_l$ is an element in the relative Weyl algebra $D_{n,c}$, given as a unique ${\mathbb{C} }$-linear combination of normal monomials $\,\mathbf{x}^\alpha \partial_\mathbf{x}^\beta $. This is important since $D_{n,c}$ is non-commutative. We now replace $\partial_\mathbf{x}$ by $\mathbf{z}$ in the normal monomials. This results in commutative polynomials $$\label{eq:thisresults} B_l (\mathbf{x},\mathbf{z}) \,\, := \,\, A_l(\mathbf{x},\partial_\mathbf{x})|_{\partial_{x_1} \mapsto z_1,\ldots, \partial_{x_c} \mapsto z_c} \qquad {\rm for} \quad l=1,2,\ldots,m .$$ We call $B_1,\ldots,B_m$ the [*Noetherian multipliers*]{} of the primary ideal $Q$. These are polynomial in $n+c$ variables, obtained by reinterpreting the Noetherian (differential) operators. Note that $B_1,\ldots,B_m$ span the inverse system in Theorem \[thm:main\] (c) when viewed inside ${\mathbb{F}}[z_1,\ldots,z_c]$. The Noetherian operators and Noetherian multipliers in the Introduction are $$\label{eq:NoetMult} \begin{matrix} A_1 \,=\, 1\,,\;\, A_2\,=\, \partial_{x_1} \,\,\,{\rm and} \,\,\,A_3 \,=\, \partial_{x_1}^2 \,-\, 2 \,x_2\,\partial_{x_2}, \\ B_1 \,=\, 1\,,\;\,\, B_2\,=\, z_1\, \,\,\,{\rm and}\, \,\,\,B_3 \,=\, z_1^2 \,-\, 2 \,x_2\,z_2. \qquad \end{matrix}$$ We note that this is consistent with (\[eq:twistedcubic5\]) because $\,x_2 = st^2\,$ holds on the variety $\,V(P)$. Here is now the celebrated result on solutions to linear PDE with constant coefficients: \[thm:Palamodov\_Ehrenpreis\] Fix the system (\[eq:mustsolvethis\]) of PDE given by the $P$-primary ideal $Q$. Any solution $\psi$ in $C^\infty(\mathcal{K})$ has an integral representation $$\label{eq:anysolution} \psi(\mathbf{z}) \,\,\,= \,\,\, \sum_{l=1}^m\,\int_{V(P)} \!\! B_l\left(\mathbf{x},\mathbf{z}\right) \exp\left( \mathbf{x}^t \,\mathbf{z} \right) d\mu_l(\mathbf{x})$$ for suitable measures $\mu_l$ supported in $V(P)$. And, conversely, all such functions are solutions. We follow the conventions used in analysis (cf. [@BJORK Chapter 8]) and we write our system in terms of the differential operators $D_{z_j} = -i\partial_{z_j}$, where $i=\sqrt{-1}$. We can account for this in the Noetherian multipliers by replacing $\mathbf{x}$ with $-i \mathbf{x}$. It is shown in [@BJORK Theorem 1.3, page 339] that any solution in $C^\infty(\mathcal{K})$ to the system (\[eq:mustsolvethis\]) can be written as $$\psi(\mathbf{z}) \,\,=\,\, \sum_{l=1}^m\int_{V(P)} B_l\left(-i\mathbf{x},\mathbf{z}\right) \exp\left(-i \mathbf{x}^t \, \mathbf{z} \right) d\mu_l(\mathbf{x}).$$ We can now change variables, by incorporating the multiplication with $-i$ into the measures, to get the formula (\[eq:anysolution\]). Conversely, to see that any such integral $\psi(\mathbf{z})$ is a solution to the PDE (\[eq:mustsolvethis\]) given by $Q$, we differentiate under the integral sign and use the Fourier transform. Consider the system of PDE determined by the ideal $Q$ in the Introduction. The Noetherian multipliers in (\[eq:NoetMult\]) furnish integral representations for all of its solutions: $$\psi(\mathbf{z}) \,\,= \,\, \int_{V(P)} \!\!\!\! \exp\left(\mathbf{x}^t \mathbf{z}\right) d\mu_1(\mathbf{x}) \,+\, \int_{V(P)}\!\!\!\! z_1 \exp\left(\mathbf{x}^t \mathbf{z}\right) d\mu_2(\mathbf{x}) \, +\, \int_{V(P)} \!\!\!\! (z_1^2-2x_2 z_2) \exp\left(\mathbf{x}^t \mathbf{z}\right) d\mu_3(\mathbf{x}).$$ Here $\mu_1,\mu_2,\mu_3$ are measures supported on the variety $\,V(P) = \bigl\{ \,(s^2t, s t^2,s^3, t^3) \,:\, s,t \in {\mathbb{C} }\,\bigr\}$. The assertion in (\[eq:twistedcubic5\]) is obtained by pulling the integrals back to the $(s,t)$-plane via the parametrization of $V(P)$. This replaces the measures $\mu_i$ by their pull-backs to that plane. We next present two algorithms for Theorem \[thm\_Noeth\_ops\]. The first is for computing Noetherian operators from the generators of $Q$, and the second for going in the reverse direction. A key ingredient is the map $\gamma$ in (\[eq\_map\_gamma\]) which we encode in the ideal $$\label{eq:weencode} \big\langle \,x_1-y_1-u_1,\,\ldots\,,\,x_c-y_c-u_c\,,\,\, x_{c+1}-u_{c+1}\,, \,\ldots\,,\,x_n-u_n \,\bigr\rangle.$$ This technique was used for encoding the differential operators in our running example in (\[eq:magic\]). \[alg:forward\]\ [Input:]{} Generators $p_1,p_2,\ldots,p_r$ of a $P$-primary ideal $Q$ in $R ={\mathbb{K}}[x_1,\ldots,x_n]$.\ [Output:]{} Elements $A_1,A_2,\ldots,A_m $ in the relative Weyl algebra $D_{n,c}$ that satisfy (\[eq:fromAtoQ\]).\ 1. Compute polynomials in ${\mathbb{F}}[y_1,\ldots,y_c]$ that generate the zero-dimensional ideal $I$ in (\[eq:corr12\]).\ 2. Using linear algebra over ${\mathbb{F}}$, compute a basis $\{B_1,\ldots,B_m\}$ for the inverse system $I^\perp$.\ 3. Lift each $B_i(\mathbf{u},\mathbf{z})$ to obtain the Noetherian multipliers $B_i(\mathbf{x},\mathbf{z})$.\ 4. Replace $\mathbf{z}$ by $\partial_\mathbf{x}$ to get the Noetherian operators $A_i(\mathbf{x},\partial_\mathbf{x}) $ in (\[eq:resultingDO\]). \[alg:backward\]\ [Input:]{} Elements $A_1,A_2,\ldots,A_m $ in the relative Weyl algebra $D_{n,c}$ that satisfy (\[eq:leftequalsright\]).\ [Output:]{} Generators $p_1,p_2,\ldots,p_r$ of a $P$-primary ideal $Q$ that is defined as in (\[eq:fromAtoQ\]).\ 1. In each $A_i(\mathbf{x},\partial_\mathbf{x})$ replace $\partial_\mathbf{x}$ by $\mathbf{z}$ to obtain the $m$ Noetherian multipliers $B_i(\mathbf{x},\mathbf{z})$ in (\[eq:thisresults\]).\ 2. Replace $\mathbf{x}$ by $\mathbf{u}$ to obtain an ${\mathbb{F}}$-basis $\{B_1,\ldots,B_m\}$ for the inverse system $I^\perp$.\ 3. Using $\,{\mathbb{F}}$-linear algebra in ${\mathbb{F}}[y_1,\ldots,y_c]$, find generators for the zero-dimensional ideal $I$.\ 4. Add the ideal $I$ to (\[eq:weencode\]) and eliminate $\{y_1,\ldots,y_c,u_1,\ldots,u_n\}$ to obtain generators of $Q$. We implemented both of these algorithms in [Macaulay2]{}. The code is made available at <https://software.mis.mpg.de>. We hope to develop this further into a [Macaulay2]{} package. We close this section by presenting a new example that explains the algorithms. To illustrate Algorithm \[alg:forward\], let $n = 4$ and fix the prime $P = \langle x_1,x_2,x_3 \rangle$ that defines a line in $4$-space ${\mathbb{K}}^4$. The following ideal is $P$-primary of multiplicity $m = 4$: $$Q \,\,=\,\, \bigl\langle\, x_1^2, \,x_1 x_2,\, x_1 x_3, \,x_1 x_4-x_3^2+x_1, \, x_3^2 x_4-x_2^2, \,x_3^2 x_4-x_3^2-x_2 x_3+2 x_1 \,\bigr\rangle .$$ In Step 1 we replace $x_1,x_2,x_3$ by $y_1,y_2,y_3$ and $x_4$ by $u_4$ to get a zero-dimensional ideal $I$ in ${\mathbb{F}}[y_1,y_2,y_3]$, where ${\mathbb{F}}= {\mathbb{K}}(u_4)$. Note that $I$ contains $\langle y_1,y_2,y_3 \rangle^4$. To check that $I$ is a point in ${\rm Hilb}^4({\mathbb{F}}[[y_1,y_2,y_3]])$, we exhibit a flat deformation to the square of the maximal ideal: $$I \,\,=\,\,\bigl\langle\, y_1^2\,,\,y_1 y_2\,,\, y_1 y_3\, ,\, y_2^2 -(u_4^2+u_4) \,y_1\,,\, y_2 y_3 - (u_4^2 + 1) \,y_1\,,\, y_3^2 - (u_4+1)\, y_1\,\bigr\rangle.$$ The inverse system $I^\perp$ lives in ${\mathbb{F}}[z_1,z_2,z_3]$. It is the $4$-dimensional ${\mathbb{F}}$-vector space with basis $$B_1 \,=\, (u_4^2+u_4) z_2^2 + 2 (u_4^2+1)z_2 z_3 + (u_4+1) z_3^2 +2z_1\,,\, B_2 = z_2\,,\, B_3 = z_3\,, \,B_4 = 1.$$ Note that this space is closed under differentiation. The Noetherian operators in Step 4 are $$A_1 \,=\, (x_4^2+x_4) \partial_{x_2}^2 + 2 (x_4^2+1) \partial_{x_2} \partial_{x_3} + (x_4+1) \partial_{x_3}^2 +2 \partial_{x_1},\,\, A_2 = \partial_{x_2\,}, \,\, A_3 = \partial_{x_3} \,, \,\, A_4 = 1 .$$ We can now check that these four operators in $D_{4,3}$ represent the given primary ideal: $$Q \,\,= \,\,\bigl\{\,f \in {\mathbb{K}}[x_1,x_2,x_3,x_4]:\, A_i \bullet f\in\langle x_1,x_2,x_3\rangle \hbox{ for $i=1,2,3,4$} \,\bigr\} .$$ Reversing this entire computation is the point of Algorithm \[alg:backward\]. Starting from the operators $A_1,A_2,A_3,A_4$, we compute the polynomials $B_1,B_2,B_3,B_4$ in ${\mathbb{F}}[z_1,z_2,z_3]$, which span the inverse system $I^\perp$. In Step 3, we find generators of the ideal $I$ in ${\mathbb{F}}[y_1,y_2,y_3$\]. And, finally, from this one obtains generators of $Q$ by the elimination process described in Step 4. Hilbert Schemes and Inverse Systems {#sec4} =================================== In this section we provide a proof of the bijections between parts (a), (b) and (c) of Theorem \[thm:main\]. Here the key players are punctual Hilbert schemes and Macaulay’s inverse systems. We retain the notation from Sections \[sec2\] and \[sec3\], and we write ${\mathfrak{p}}=PS$ for the extension of our prime ideal $P$ in $R = {\mathbb{K}}[x_1,\ldots,x_c,x_{c+1},\ldots,x_n]$ to $S={\mathbb{K}}(x_{c+1},\ldots,x_n)[x_1,\ldots,x_c]$. By Noether Normalization, we assume that ${\mathbb{K}}[x_{c+1},\ldots,x_n] \hookrightarrow R/P$ is an integral extension, and this implies that ${\mathfrak{p}}$ is a maximal ideal in $S$. Our first goal is to parametrize $P$-primary ideals of fixed multiplicity $m$ over $P$ by the punctual Hilbert scheme ${\rm Hilb}^m\bigl({\mathbb{F}}[[y_1,\ldots,y_c]]\bigr)$. A special role is played by the inclusion map $\gamma:R \hookrightarrow {\mathbb{F}}[y_1,\dots,y_c]$ in (\[eq\_map\_gamma\]). This induces an inclusion $\,\gamma_S : S \hookrightarrow {\mathbb{F}}[y_1,\ldots,y_c]$, also given by $\,x_i \mapsto y_i+u_i $ for $i \leq c\,$ and $\,x_j \mapsto u_j$ for $j > c$. \[rem\_local\_primary\_ideals\] Since ${\mathbb{K}}[x_{c+1},\ldots,x_n] \cap P=0$, the canonical map $R \hookrightarrow S$ gives a bijection between $P$-primary ideals and ${\mathfrak{p}}$-primary ideals (see, e.g., [@MATSUMURA Theorem 4.1]). The maximal irrelevant ideal in ${\mathbb{F}}[y_1,\dots,y_c]$ is denoted by ${\mathcal{M}}=\langle y_1,\ldots,y_c\rangle$. For any $f(\mathbf{x})=f(x_1,\ldots,x_n) \in P$, we have $f(\mathbf{u})=f(u_1,\ldots,u_n)=0$ in $ {\mathbb{F}}$. A Taylor expansion yields $$f(\mathbf{u}+\mathbf{y}) \,\,=\,\,f(u_1+y_1,\ldots,u_c+y_c,u_{c+1},\ldots,u_n) \,\,=\,\, \sum_{\substack{\lambda \in {\mathbb{N}}^c\\\lvert\lambda \rvert > 0}} \frac{\partial^{\lvert\lambda\rvert} f}{\partial_{x_1}^{\lambda_1}\cdots\partial_{x_c}^{\lambda_c}}(\mathbf{u})\,\mathbf{y}^\lambda.$$ This shows that $\gamma(P) \subseteq {\mathcal{M}}$, and therefore $\gamma_S( {\mathfrak{p}}) \subseteq {\mathcal{M}}$. The next proposition establishes a bijection between ${\mathfrak{p}}$-primary ideals containing ${\mathfrak{p}}^m$ and ${\mathcal{M}}$-primary ideals containing ${\mathcal{M}}^m$. \[prop\_corespondence\_primary\_ideals\] For all $m \ge 1$, the inclusion $\gamma_S$ induces the isomorphism of local rings $$S/{\mathfrak{p}}^m \,\,\xrightarrow{\cong}\,\, {\mathbb{F}}[y_1,\ldots,y_c]/{\mathcal{M}}^m.$$ This result has also appeared in [@BRUMFIEL_DIFF_PRIM Proposition 4.1] and [@NOETH_OPS Proposition 3.9]. In these sources it was assumed that ${\mathbb{K}}$ is a perfect field. This holds here since ${\rm char}({\mathbb{K}}) = 0$. \[rem\_ident\_Hilb\] (i) Any ideal of colength $m$ in ${\mathbb{F}}[[y_1,\ldots,y_c]]$ contains the ideal $\langle y_1,\ldots,y_c \rangle^m $. Therefore, ${\rm Hilb}^m\bigl({\mathbb{F}}[[y_1,\ldots,y_c]]\bigr)$ can be identified with $\,{\rm Hilb}^m\bigl({\mathbb{F}}[[y_1,\ldots,y_c]]/ \langle y_1,\ldots,y_c \rangle^m \bigr). $ (ii) Any $ \langle y_1,\ldots,y_c\rangle$-primary ideal of colength $m$ in the polynomial ring ${\mathbb{F}}[y_1,\ldots,y_c]$ contains the ideal $\langle y_1,\ldots,y_c \rangle^m \subset {\mathbb{F}}[y_1,\ldots,y_c]$. For all $m>0$, we have the natural isomorphism $$\frac{{\mathbb{F}}[[y_1,\ldots,y_c]]}{{ \langle y_1,\ldots,y_c \rangle}^m} \,\,\cong \,\,\frac{{\mathbb{F}}[y_1,\ldots,y_c]}{{ \langle y_1,\ldots,y_c \rangle }^m}.$$ Therefore, the $\langle y_1,\ldots,y_c \rangle$-primary ideals of colength $m$ in ${\mathbb{F}}[y_1,\ldots,y_c]$ are parametrized by ${\rm Hilb}^m \bigl( {\mathbb{F}}[[y_1,\ldots,y_c]] \bigr)$. From now on, $\langle y_1,\ldots,y_c \rangle $-primary ideals in the polynomial ring ${\mathbb{F}}[y_1,\ldots,y_c]$ will automatically be identified with ideals in the power series ring ${\mathbb{F}}[[y_1,\ldots,y_c]]$. Now we are ready to prove the correspondence between parts (a) and (b) in Theorem \[thm:main\]. \[thm:param\_primary\] As asserted in (\[eq:corr12\]), there is a bijective correspondence $$\begin{array}{ccc} \left\lbrace\begin{array}{c} \mbox{$P$-primary ideals of $R$}\\ \mbox{with multiplicity $m$ over $P$} \end{array}\right\rbrace & \longleftrightarrow & \left\lbrace\begin{array}{c} \mbox{points in }{\rm Hilb}^m({\mathbb{F}}[[y_1,\ldots,y_c]])\\ \end{array}\right\rbrace\\ Q & \longrightarrow & I=\langle y_1,\dots,y_c\rangle^m+\gamma( Q){\mathbb{F}}[y_1,\dots,y_c]\\ Q=\gamma^{-1}(I) & \longleftarrow & I. \end{array}$$ The canonical map $R \hookrightarrow S$ gives a bijection between $P$-primary ideals and ${\mathfrak{p}}$-primary ideals (Remark \[rem\_local\_primary\_ideals\]). Also, for any $P$-primary ideal $Q \subset R$ we have $R_P/QR_P \cong S_{{\mathfrak{p}}}/Q S_{{\mathfrak{p}}}$. So, nothing is changed if we take $S$ and ${\mathfrak{p}}$ instead of $R$ and $P$. We have the commutative diagram \(m) \[matrix of math nodes,row sep=3em,column sep=9em,minimum width=1.7em, text height=1.5ex, text depth=0.25ex\] [ S & \[\]\ S/\^[m]{} & \[\]/\^[m]{}.\ ]{}; (m-1-1) edge node \[above\] [$\gamma_S$]{} (m-1-2) (m-2-1) edge node \[above\] [$\cong$]{} (m-2-2) ; (m-1-1) – (m-2-1); (m-1-2) – (m-2-2); The map in the bottom row is the isomorphism in Proposition \[prop\_corespondence\_primary\_ideals\]. This gives an inclusion-preserving bijection between ${\mathfrak{p}}$-primary ideals containing ${\mathfrak{p}}^m$ and ${\mathcal{M}}$-primary ideals containing ${\mathcal{M}}^{m}$, in particular, colength does not change under this correspondence. In explicit terms, the ${\mathcal{M}}$-primary ideal $I$ corresponding to a ${\mathfrak{p}}$-primary ideal $QS \supseteq {\mathfrak{p}}^m$ is $$I \,\,=\,\, {\mathcal{M}}^{m} \,+\, \gamma_S(QS)\big({\mathbb{F}}[\mathbf{y}]\big).$$ And, the ${\mathfrak{p}}$-primary ideal $QS$ corresponding to an ${\mathcal{M}}$-primary ideal $I \supseteq {\mathcal{M}}^m$ is given by $$QS \,\,=\,\, \gamma_S^{-1}(I).$$ Finally, the result now follows from Remark \[rem\_ident\_Hilb\]. We next show the correspondence between parts (b) and (c) in Theorem \[thm:main\]. This follows from the usual Macaulay duality. Although this argument is well-known, we will need a short discussion to later connect parts (c) and (d) of Theorem \[thm:main\]. Consider the injective hull $E=E_{{\mathbb{F}}[[y_1,\ldots,y_c]]}({\mathbb{F}})$ of the residue field ${\mathbb{F}}\cong {\mathbb{F}}[[y_1,\ldots,y_c]]/\langle y_1,\ldots,y_c \rangle$ of ${\mathbb{F}}[[y_1,\ldots,y_c]]$. Since ${\mathbb{F}}[[y_1,\ldots,y_c]]$ is a formal power series ring, this equals the module of inverse polynomials: $$\label{eq_isom_E_inv_sys} E \,\,\cong \,\, {\mathbb{F}}[y_1^{-1},\ldots,y_c^{-1}].$$ For a derivation see e.g. [@Brodmann_Sharp_local_cohom Lemma 11.2.3, Example 13.5.3] or [@BRUNS_HERZOG Theorem 3.5.8]. Consider the polynomial ring ${\mathbb{F}}[z_1,\ldots,z_c]$ as an ${\mathbb{F}}[[y_1,\ldots,y_c]]$-module by setting that $y_i$ acts on ${\mathbb{F}}[z_1,\ldots,z_c]$ as $\partial_{z_i}$, that is, $y_i \cdot F = \partial_{z_i} \bullet F$ for any $F \in {\mathbb{F}}[z_1,\ldots,z_c]$. Since the field ${\mathbb{F}}$ has characteristic zero, we have the following isomorphism of ${\mathbb{F}}[[y_1,\ldots,y_c]]$-modules $$\label{eq_isom_inv_sys_char_zero} {\mathbb{F}}[y_1^{-1},\ldots,y_c^{-1}] \;\xrightarrow{\cong}\; {\mathbb{F}}[z_1,\ldots,z_c], \quad \frac{1}{\mathbf{y}^{\alpha}} \;\mapsto\; \frac{\mathbf{z}^\alpha}{\alpha!}.$$ Now, Macaulay’s duality is simply performed via Matlis duality. We use ${\left(-\right)}^\vee$ to denote Matlis dual ${\left(-\right)}^\vee={{\normalfont\text{Hom}}}_{{\mathbb{F}}[[y_1,\ldots,y_c]]}\left(-,E\right)$. This is a contravariant exact functor which establishes an anti-equivalence between the full-subcategories of artinian ${\mathbb{F}}[[y_1,\ldots,y_c]]$-modules and finitely generated ${\mathbb{F}}[[y_1,\ldots,y_c]]$-modules (see, e.g., [@BRUNS_HERZOG Theorem 3.2.13]). For any zero-dimensional ideal $I $ in the power series ring $ {\mathbb{F}}[[y_1,\ldots,y_c]]$, the isomorphisms (\[eq\_isom\_E\_inv\_sys\]) and (\[eq\_isom\_inv\_sys\_char\_zero\]) together with Matlis duality yield the following identifications: $$I^{\perp} \,\,=\,\, \left\lbrace F\in {\mathbb{F}}[z_1,\ldots,z_c]: f\bullet F=0 \mbox{ for all } f\in I\right\rbrace \,\,\cong \,\, (0 :_{E} I) \,\,\cong \,\, \bigl({\mathbb{F}}[[y_1,\ldots,y_c]]/I \bigr)^\vee.$$ On the other hand, consider any ${\mathbb{F}}[[y_1,\ldots,y_c]]$-submodule $V$ of $ {\mathbb{F}}[z_1,\ldots,z_c] \cong E$. Then $V$ is an ${\mathbb{F}}$-subspace of ${\mathbb{F}}[z_1,\ldots,z_c]$ that is closed by differentiation, as $y_i$ is identified with the operator $\partial_{z_i}$. Again, the isomorphisms (\[eq\_isom\_E\_inv\_sys\]) and (\[eq\_isom\_inv\_sys\_char\_zero\]) with Matlis duality give identifications $${\rm Ann}_{{\mathbb{F}}[[\partial_{z_1},\dots,\partial_{z_c}]]}(V) \,\, \cong \,\, {\rm Ann}_{{\mathbb{F}}[[y_1,\ldots,y_c]]}(V) \, \, \cong \,\, {\bigl( E /V \bigr)}^{\vee} \, \,\subset \,\, {\mathbb{F}}[[y_1,\ldots,y_c]].$$ Hence, from the above discussions, we get the connection between (b) and (c) in Theorem \[thm:main\]. \[thm:Macaulay\_dual\] As asserted in (\[eq:HilbBijection\]), there is a bijective correspondence $$\begin{array}{ccc} \left\lbrace\begin{array}{c} \mbox{points in }{\rm Hilb}^m\left({\mathbb{F}}[[\partial_{z_1},\ldots,\partial_{z_c}]]\right)\\ \end{array}\right\rbrace & \longleftrightarrow & \left\lbrace\begin{array}{c} \mbox{$m$-dimensional ${\mathbb{F}}$-subspaces of}\\ \mbox{${\mathbb{F}}[z_1,\dots,z_c]$ closed by differentiation} \end{array}\right\rbrace\\ I & \longrightarrow & V=I^\perp\\ I={\rm Ann}_{{\mathbb{F}}[[\partial_{z_1},\dots,\partial_{z_c}]]}(V) & \longleftarrow & V.\\ \end{array}$$ Differential Operators Revisited {#sec5} ================================ In this section we review basic material on differential operators in commutative algebra. This is used in Section \[sec6\] to complete the proof of Theorem \[thm:main\]. Even though the Noetherian operators $A_i$ live in the Weyl algebra, we need the abstract perspective to link them to the Weyl-Noether module (\[eq:relativeweyl\]). As before, ${\mathbb{K}}$ is a field of characteristic zero and $R={\mathbb{K}}[x_1,\ldots,x_n]$. For two $R$-modules $M$ and $N$, we regard ${{\normalfont\text{Hom}}}_{\mathbb{K}}(M, N)$ as an $(R\otimes_{\mathbb{K}}R)$-module, by setting $$\left((r \otimes_{\mathbb{K}}s) \delta\right)(w)\,\, =\,\, r \delta(sw) \quad \text{ for all }\,\, \delta \in {{\normalfont\text{Hom}}}_{\mathbb{K}}(M, N), \; w \in M,\; r,s \in R.$$ This is equivalent to saying that ${{\normalfont\text{Hom}}}_{\mathbb{K}}(M,N)$ is an $R$-bi-module, where the action on the left is given by post-composing $(r \cdot \delta)(w)=r\delta(w)$ and the action on the right is given by pre-composing $(\delta \cdot s)(w)=\delta(sw)$, for all $\delta \in {{\normalfont\text{Hom}}}_{\mathbb{K}}(M, N), \; w \in M,\; r,s \in R$. We use the bracket notation $[\delta,r](w) = \delta(rw)-r\delta(w)$ for all $\delta \in {{\normalfont\text{Hom}}}_{\mathbb{K}}(M, N)$, $r \in R$ and $w \in M$. \[nota\_T\_mod\_struct\] We write $T=R \otimes_{\mathbb{K}}R = {\mathbb{K}}[x_1, \ldots, x_n, y_1, \ldots, y_n]$ as a polynomial ring in $2n$ variables, where $x_i$ represents $x_i \otimes_{\mathbb{K}}1$ and $y_i$ represents $1 \otimes_{\mathbb{K}}x_i - x_i \otimes_{\mathbb{K}}1$. The action of $T$ on ${{\normalfont\text{Hom}}}_{\mathbb{K}}(M,N)$ is thus given as follows. For all $\delta \in {{\normalfont\text{Hom}}}_{\mathbb{K}}(M, N)$ and $w \in M$, we have $$(x_i \cdot \delta) (w) = x_i \delta(w) \quad \text{ and } \quad (y_i \cdot \delta) (w) = \delta(x_i w) - x_i\delta(w) = \left[\delta,x_i\right](w) \qquad {\rm for} \,\,i=1,\ldots,n.$$ Any $T$-module is regarded as an $R$-module via the canonical map $R \hookrightarrow T, x_i \mapsto x_i $. Thus, any $T$-module is given an $R$-module structure by using the left factor $R\otimes_{\mathbb{K}}1 \subset T = R \otimes_{\mathbb{K}}R$. The ${\mathbb{K}}$-linear differential operators form a $T$-submodule of ${{\normalfont\text{Hom}}}_{\mathbb{K}}(M, N)$, defined as follows. \[def\_diff\_ops\] Let $M, N$ be $R$-modules. The *$m$-th order ${\mathbb{K}}$-linear differential operators ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(M, N) \subseteq {{\normalfont\text{Hom}}}_{\mathbb{K}}(M, N)$ from $M$ to $N$* form a $T$-module that is defined inductively by: (i) ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{0}(M,N) \,:=\, {{\normalfont\text{Hom}}}_R(M,N)$. (ii) ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m}(M, N) \,:= \, \big\lbrace \delta \in {{\normalfont\text{Hom}}}_{\mathbb{K}}(M,N) : \,[\delta, r] \in {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(M, N) \,\text{ for all }\, r \in R \big\rbrace$. The set of all *${\mathbb{K}}$-linear differential operators from $M$ to $N$* is the $T$-module $${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(M, N) \,\,:=\,\, \bigcup_{m=0}^\infty {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(M,N).$$ Subsets $\mathcal{E} \subseteq {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(M, N)$ are viewed as differential equations. Their solutions spaces are $$\label{eq:solE} {\rm Sol}(\mathcal{E}) \,\,:= \,\,\big\lbrace w \in M : \delta(w) = 0 \text{ for all } \delta \in \mathcal{E} \big\rbrace \,\,= \,\,\bigcap_{\delta \in \mathcal{E} } {\rm Ker}(\delta).$$ Following the approach in [@NOETH_OPS Section 2], we now introduce the module of principal parts. By construction, the ideal $ \Delta_{R/{\mathbb{K}}} = \langle y_1, \ldots,y_n \rangle$ in $T$ is the kernel of the multiplication map $$T = R \otimes_{\mathbb{K}}R \,\rightarrow \,R\,, \quad r \otimes_{\mathbb{K}}s \,\mapsto\, rs.$$ Let $M$ be an $R$-module. The module of *$m$-th principal parts of $M$* equals $$P_{R/{\mathbb{K}}}^m(M) \,\,:=\,\, \frac{R \otimes_{\mathbb{K}}M}{\Delta_{R/{\mathbb{K}}}^{m+1} \left(R \otimes_{\mathbb{K}}M\right)}.$$ This is a $T$-module. It comes with the natural map $\,d^m : \,M \rightarrow P_{R/{\mathbb{K}}}^m(M),\, w \mapsto \overline{1 \otimes_{\mathbb{K}}w}$. In the special case $M=R$ we abbreviate $\,P_{R/{\mathbb{K}}}^m \,:=\, P_{R/{\mathbb{K}}}^m(R)=T/\Delta_{R/{\mathbb{K}}}^{m+1}$, and the map becomes $$\label{eq_univ_diff} d^m : R \rightarrow P_{R/{\mathbb{K}}}^m, \;\;x_i \, \mapsto \,\overline{1 \otimes_{\mathbb{K}}x_i} \,=\, \overline{x_i+y_i} .$$ The following proposition offers a fundamental characterization of differential operators. \[prop\_represen\_diff\_opp\] Let $m\ge 0$ and let $M, N$ be $R$-modules. Then, the following map is an isomorphism of $R$-modules: $$\begin{aligned} {\left(d^m\right)}^*\, :\, {{\normalfont\text{Hom}}}_R\left(P_{R/{\mathbb{K}}}^m(M), N\right) &\,\,\xrightarrow{\cong} \,\,{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(M, N), \\ \varphi \quad &\,\,\mapsto \,\quad \varphi \circ d^m. \end{aligned}$$ This is a very general result for commutative rings $R$. What we are interested in here is the polynomial ring $R = {\mathbb{K}}[x_1,\ldots,x_n]$ over a field ${\mathbb{K}}$ of characteristic zero. In this case, the $R$-module $P_{R/{\mathbb{K}}}^m=T/\Delta_{R/{\mathbb{K}}}^{m+1}$ is free, and a basis is given by $\bf y$-monomials of degree at most $m$: $$\label{eq_direct_sum_Prin} P_{R/{\mathbb{K}}}^m \,\,\,= \,\,\bigoplus_{\lvert \alpha \rvert \le m} R\mathbf{y}^\alpha \quad = \bigoplus_{\alpha_1 + \cdots + \alpha_r \le m} \!\!\!\! R y_1^{\alpha_1}\cdots y_n^{\alpha_n}.$$ Proposition \[prop\_represen\_diff\_opp\] implies that $\,{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R, R)\, \cong\, {{\normalfont\text{Hom}}}_R\bigl(P_{R/{\mathbb{K}}}^m, R\bigr)\,$ is a free $R$-module with basis $$\label{eq_basis_diff_ops} \big\{ {(y_1^{\alpha_1}\cdots y_n^{\alpha_n})}^* \circ d^m : \alpha_1 + \cdots + \alpha_n \le m \big\}.$$ For any polynomial $f(\mathbf{x})$ in $ R$, the operator $d^m$ in (\[eq\_univ\_diff\]) computes the Taylor expansion $$d^m(f(\mathbf{x})) \,\,= \,\,f(1\otimes_{\mathbb{K}}\mathbf{x}) \,\,= \,\,f(\mathbf{x}+\mathbf{y}) \,\,=\,\, \sum_{\lambda \in {\mathbb{N}}^n} \left(D_\mathbf{x}^\lambda f\right)\!(\mathbf{x})\,\mathbf{y}^\lambda,$$ where $\,D_\mathbf{x}^{\lambda}:R\rightarrow R\,$ is the differential operator we all know from calculus: $$D_\mathbf{x}^{\lambda}\,\, =\,\, \frac{1}{\lambda!}\partial_\mathbf{x}^\lambda \,\,= \,\, \frac{1}{\lambda_1!\cdots \lambda_n!}\partial_{x^1}^{\lambda_1}\cdots \partial_{x^n}^{\lambda_n}.$$ For any $\alpha \in {\mathbb{N}}^n$ we thus have $\, \left({(\mathbf{y}^\alpha)}^* \circ d^m\right)(f(\mathbf{x})) = \left(D_\mathbf{x}^\alpha f\right)(\mathbf{x})$. The equation (\[eq\_basis\_diff\_ops\]) implies $${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R)\,\, =\,\,\bigoplus_{\lvert \alpha \rvert \le m} R D_\mathbf{x}^\alpha \,\,=\,\, \bigoplus_{\lvert \alpha \rvert \le m} R \partial_\mathbf{x}^\alpha.$$ By letting $m$ go to infinity, we now recover the Weyl algebra in its well-known role: \[lem\_Weyl\_as\_diff\_ops\] $\,{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(R, R)$ coincides with the Weyl algebra ${\mathbb{K}}\langle x_1,\ldots,x_n,\partial_{x_1},\ldots,\partial_{x_n} \rangle$. Let $J $ be an ideal in $R= {\mathbb{K}}[x_1,\ldots,x_n]$. The canonical projection $\pi : R \rightarrow R/J$ induces a natural map of differential operators. This is the following homomorphism of $T$-modules: $$\label{eq:pimap} {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(\pi) \,: \,{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R, R) \,\rightarrow \, {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R, R/J), \quad \delta \,\mapsto\, \pi \circ \delta.$$ \[lem\_diff\_ops\_R/J\] We have the following explicit description of the objects in (\[eq:pimap\]): (i) ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R, R/J)$ is a free $R/J$-module with direct summands decomposition $$\qquad {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R/J) \,\,= \,\,\bigoplus_{\lvert \alpha \rvert \le m} (R/J) \overline{D_\mathbf{x}^\alpha}, \qquad {\rm where}\,\,\, \overline{D_\mathbf{x}^\alpha} = \pi \circ D_\mathbf{x}^\alpha.$$ (ii) The map ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(\pi)$ is surjective. Explicitly, any differential operator $$\epsilon \,\,=\, \sum_{\lvert \alpha \rvert \le m} \overline{r_\alpha} \overline{D_\mathbf{x}^\alpha} \,\in\, {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R/J), \quad \text{ where } r_\alpha \in R,$$ can be lifted to an operator $\,\delta=\sum_{\lvert \alpha \rvert \le m} r_\alpha D_\mathbf{x}^\alpha \,\in\, {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R)\,$ with $\,\epsilon = {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(\pi)(\delta)$. $(i)$ From Proposition \[prop\_represen\_diff\_opp\] and the Hom-tensor adjunction we obtain the isomorphisms $$\begin{aligned} \label{eq_isoms_Diff_R/J} \begin{split} {{\normalfont\text{Hom}}}_{R/J}\left(R/J \otimes_R P_{R/{\mathbb{K}}}^m, R/J\right) & \,\,\cong \,\, {{\normalfont\text{Hom}}}_R\left(P_{R/{\mathbb{K}}}^m, {{\normalfont\text{Hom}}}_{R/J}\left(R/J, R/J\right)\right)\\ & \,\,\cong \,\,{{\normalfont\text{Hom}}}_R\left(P_{R/{\mathbb{K}}}^m, R/J\right)\\ &\,\,\cong \,\,{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R/J). \end{split} \end{aligned}$$ The isomorphism from the first row to the second row in (\[eq\_isoms\_Diff\_R/J\]) is given by $$\psi \in {{\normalfont\text{Hom}}}_{R/J}\left(R/J \otimes_R P_{R/{\mathbb{K}}}^m, R/J\right) \;\;\mapsto \;\;\psi\circ h_m \in {{\normalfont\text{Hom}}}_R\left(P_{R/{\mathbb{K}}}^m, R/J\right),$$ where $\,h_m\,$ is the canonical map $\, P_{R/{\mathbb{K}}}^m \rightarrow R/J \otimes_R P_{R/{\mathbb{K}}}^m$. Therefore, the isomorphism from the first to the third row in (\[eq\_isoms\_Diff\_R/J\]) is given explicitly as $ \,\psi\, \mapsto \, \psi \circ h_m \circ d^m $. By using equation (\[eq\_direct\_sum\_Prin\]) we get that $R/J \otimes_R P_{R/{\mathbb{K}}}^m$ is a free $R/J$-module with direct summands decomposition $$R/J \otimes_R P_{R/{\mathbb{K}}}^m \,\,= \,\, \bigoplus_{\lvert \alpha \rvert \le m} (R/J)\mathbf{y}^\alpha.$$ Our explicit isomorphism for (\[eq\_isoms\_Diff\_R/J\]) shows that ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R/J)$ is a free $R/J$-module with basis $$\big\{ {(y_1^{\alpha_1}\cdots y_n^{\alpha_n})}^* \circ h_m \circ d^m : \alpha_1 + \cdots + \alpha_n \le m \big\}.$$ Now, for any polynomial $f(\mathbf{x})$ in $ R$, we obtain the equations $$\begin{aligned} \label{eq_diff_opp_z_alpha} \begin{split} \left({(\mathbf{y}^\alpha)}^* \circ h_m \circ d^m\right)(f(\mathbf{x})) &\,\, = \,\,\left({(\mathbf{y}^\alpha)}^* \circ h_m\right)\left(\sum_{\lambda \in {\mathbb{N}}^n} \left(D_\mathbf{x}^\lambda f\right)(\mathbf{x})\mathbf{y}^\lambda\right)\\ &\,\,=\,\, \left({(\mathbf{y}^\alpha)}^*\right)\left(\sum_{\lambda \in {\mathbb{N}}^n} \pi\left(\left(D_\mathbf{x}^\lambda f\right)(\mathbf{x})\right)\mathbf{y}^\lambda\right) \,\,=\,\,\,\pi\big(\left(D_\mathbf{x}^\alpha f\right)(\mathbf{x})\big). \end{split} \end{aligned}$$ This implies that the operators $\overline{D_\mathbf{x}^\alpha}=\pi \circ D_\mathbf{x}^\alpha$ with $\lvert\alpha\rvert \le m$ give a basis of ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R, R/J)$. Part $(ii)$ follows directly from part $(i)$. This concludes the proof of Lemma \[lem\_diff\_ops\_R/J\]. Since $R$ is a polynomial ring, the process of lifting differential operators is easy and explicit. However, the surjectivity of ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(\pi)$ is a subtle property, and it is not always satisfied over more general types of rings. For an illustration see [@NOETH_OPS Example 5.2]. Proof of the Representation Theorem {#sec6} =================================== We here finish the proof of Theorem \[thm:main\] by connecting part (d) with parts (a), (b), and (c). The section is divided into two subsections. In the first one, we treat the zero-dimensional situation, where $c=n$. In the second one, we use Noether normalization and the results on differential operators in Section \[sec5\] to reduce the general case to the zero-dimensional case. The zero-dimensional case ------------------------- We here restrict ourselves to ideals in $R= {\mathbb{K}}[x_1,\ldots,x_n]$ that are primary to a maximal ideal $P$. Hence $c=n$ and ${\mathbb{F}}=R/P$. Since the base field ${\mathbb{K}}$ is assumed to have characteristic zero, an adaptation of Gröbner’s classical approach via Macaulay’s inverse system will be valid. Using the notation $\,T=R\otimes_{\mathbb{K}}R={\mathbb{K}}[x_1,\ldots,x_n,y_1,\ldots,y_n]\,$ from Section \[sec5\], we now have $$\label{eq:FFRT} {\mathbb{F}}\otimes_R T \,\,=\,\, {\mathbb{F}}\otimes_R \left(R\otimes_{\mathbb{K}}R\right) \,\,\cong\,\, R/P \otimes_{\mathbb{K}}{\mathbb{K}}[y_1,\ldots,y_n] \,\,\cong\,\, {\mathbb{F}}[y_1, \ldots, y_n].$$ This endows ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m}\left(R, {\mathbb{F}}\right)$ with the structure of an ${\mathbb{F}}[y_1,\ldots,y_n]$-module. Applying Lemma \[lem\_diff\_ops\_R/J\] with $J=P$, we see that ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m}\left(R, {\mathbb{F}}\right)$ is a finite-dimensional ${\mathbb{F}}$-vector space. In the sequel, the irrelevant maximal ideal $ {\mathcal{M}}= \langle y_1,\ldots,y_n \rangle \subset {\mathbb{F}}[y_1,\ldots,y_n] $ will play an important role. This ideal is also given as ${\mathcal{M}}=\Delta_{R/{\mathbb{K}}} \left( {\mathbb{F}}\otimes_R T \right)$. For any $m \ge 0$ we identify $$\frac{{\mathbb{F}}[y_1,\ldots,y_n]}{{\mathcal{M}}^{m+1}} \,\,\,=\,\, \bigoplus_{\vert\alpha\rvert \le m} {\mathbb{F}}\mathbf{y}^{\alpha}.$$ For any ${\mathbb{F}}[y_1,\ldots,y_n]$-module $M$, the ${\mathbb{F}}$-dual ${{\normalfont\text{Hom}}}_{{\mathbb{F}}}(M, {\mathbb{F}})$ is naturally an ${\mathbb{F}}[y_1,\ldots,y_n]$-module as follows: if $\psi \in {{\normalfont\text{Hom}}}_{{\mathbb{F}}}(M, {\mathbb{F}})$, then $y_i \cdot \psi$ is the ${\mathbb{F}}$-linear map $\,\psi(y_i \cdot -) : w \in M \mapsto \psi(y_iw) \in {\mathbb{F}}$. The next result relates submodules of ${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m}\left(R, {\mathbb{F}}\right)$ to ${\mathcal{M}}$-primary ideals in ${\mathbb{F}}[y_1,\ldots,y_n]$. \[lem\_descrip\_diff\_opp\] The following statements hold for all positive integers $m$: (i) We have an isomorphism of ${\mathbb{F}}[y_1,\ldots,y_n]$-modules $${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}\left(R, {\mathbb{F}}\right)\,\, \cong \,\, {{\normalfont\text{Hom}}}_{{\mathbb{F}}}\Bigg(\frac{{\mathbb{F}}[y_1,\ldots,y_n]}{{{\mathcal{M}}}^{m}}, {\mathbb{F}}\Bigg).$$ (ii) The following map gives a bijective correspondence between ${\mathcal{M}}$-primary ideals $I $ in $ {\mathbb{F}}[y_1,\ldots,y_n]$ that contain ${\mathcal{M}}^{m}$ and ${\mathbb{F}}[y_1,\ldots,y_n]$-submodules of $\,{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}\left(R,{\mathbb{F}}\right)$: $$\label{eq:ImapHom} I \;\mapsto\; {{\normalfont\text{Hom}}}_{{\mathbb{F}}}\left(\frac{{\mathbb{F}}[y_1,\ldots,y_n]}{I}, {\mathbb{F}}\right).$$ (iii) Let $\mathcal{E} \subseteq {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,{\mathbb{F}})$ be the image under (\[eq:ImapHom\]) of an ${\mathcal{M}}$-primary ideal $I \supseteq {\mathcal{M}}^{m}$. Then $${\rm Sol}(\mathcal{E}) \,\,=\,\, \gamma^{-1}(I),$$ with notation as in (\[eq:solE\]), where $\gamma$ is the inclusion $ R \hookrightarrow {\mathbb{F}}[y_1,\ldots,y_n], x_i \mapsto y_i{+}u_i$ in (\[eq\_map\_gamma\]). This is essentially [@NOETH_OPS Lemma 3.8]. We provide a proof for the sake of completeness. $(i)$ Since ${\mathbb{F}}=R/P$, from equation (\[eq\_isoms\_Diff\_R/J\]) we obtain the isomorphism $${{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R, {\mathbb{F}})\,\, \cong \,\,{{\normalfont\text{Hom}}}_{{\mathbb{F}}}\bigl({\mathbb{F}}\otimes_R P_{R/{\mathbb{K}}}^{m-1}, {\mathbb{F}}\bigr).$$ Thus, the result follows from the fact that $\,{\mathbb{F}}\otimes_R P_{R/{\mathbb{K}}}^{m-1} \,\cong\, {\mathbb{F}}[\mathbf{y}]/ {\mathcal{M}}^{m} $. $(ii)$ Since ${\mathbb{F}}[\mathbf{y}]/{\mathcal{M}}^{m}$ is a finite-dimensional vector space over ${\mathbb{F}}$, the functor ${{\normalfont\text{Hom}}}_{{\mathbb{F}}}\left(-,{\mathbb{F}}\right)$ gives a bijection between quotients of ${\mathbb{F}}[\mathbf{y}]/{\mathcal{M}}^{m}$ and ${\mathbb{F}}[\mathbf{y}]$-submodules of ${{{\normalfont\text{Hom}}}_{{\mathbb{F}}}\left(\frac{{\mathbb{F}}[\mathbf{y}]}{{\mathcal{M}}^{m}},{\mathbb{F}}\right)}$. So, the claim follows from $(i)$. $(iii)$ By assumption, $\,\mathcal{E} \,=\,{{{\normalfont\text{Hom}}}_{{\mathbb{F}}}\left(\frac{{\mathbb{F}}[\mathbf{y}]}{I},{\mathbb{F}}\right)} \,$ is in $\,{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,{\mathbb{F}})$. Consider the canonical map $ \,\Phi_I : \frac{{\mathbb{F}}[\mathbf{y}]}{{\mathcal{M}}^{m}} \twoheadrightarrow \frac{{\mathbb{F}}[\mathbf{y}]}{I}\,$ given by the $\mathcal{M}$-primary ideal $I \supseteq {\mathcal{M}}^{m}$. From the isomorphism (\[eq\_isoms\_Diff\_R/J\]) we get $${\rm Sol}(\mathcal{E}) \,\,=\,\, \bigl\{ f \in R : \left(\psi \circ \Phi_I \circ h_{m-1} \circ d^{m-1}\right)(f)=0 \text{ for all } \psi \in {{{\normalfont\text{Hom}}}_{{\mathbb{F}}}\left({\mathbb{F}}[\mathbf{y}]/I ,{\mathbb{F}}\right)} \bigr\}.$$ The composition $\Phi_I \circ h_{m-1} \circ d^{m-1}$ coincides with the map $\,R \mapsto {\mathbb{F}}[\mathbf{y}]/ I,\, x_i \mapsto \overline{y_i+u_i}$. Hence $$\begin{aligned} {\rm Sol}(\mathcal{E}) &\,\,=\,\, \bigl\{\,f \in R : \psi\bigl(\,\overline{f( \mathbf{y}+\mathbf{u}})\, \bigr)=0\, \text{ for all } \,\psi \in {{{\normalfont\text{Hom}}}_{{\mathbb{F}}}\left({\mathbb{F}}[\mathbf{y}]/ I ,{\mathbb{F}}\right)}\bigr\} \\ &\,\,=\,\, \bigl\{\, f \in R : f\bigl(\mathbf{y}+\mathbf{u}\bigr) \in I \, \bigr\} \,\,=\,\, \gamma^{-1}(I). \end{aligned}$$ This completes the proof of Proposition \[lem\_descrip\_diff\_opp\]. Next, under the assumption of $P$ being maximal, we relate part (d) with the other parts in Theorem \[thm:main\]. By Definition \[def\_diff\_ops\] and Lemma \[lem\_Weyl\_as\_diff\_ops\], the Weyl-Noether module has the filtration $${\mathbb{F}}\,\otimes_R \, R\langle\partial_{x_1},\ldots,\partial_{x_n}\rangle \,\;= \;\, {\mathbb{F}}\, \otimes_R \, \biggl(\lim\limits_{\substack{\longrightarrow\\m}} {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R) \biggr) \,\;\cong \; \,\lim\limits_{\substack{\longrightarrow\\m}}\Big({\mathbb{F}}\otimes_R {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R)\Big).$$ Applying Lemma \[lem\_diff\_ops\_R/J\] with $J=P$ gives ${\mathbb{F}}\otimes_R {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,R) \cong {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^m(R,{\mathbb{F}}) \cong \bigoplus_{\vert\alpha\rvert\le m}{\mathbb{F}}\overline{\partial_{\mathbf{x}}^\alpha}$. This gives rise to the following isomorphisms of ${\mathbb{F}}$-vector spaces: $$\label{eq_isom_relWeyl_diff} {\mathbb{F}}\,\otimes_R \, R\langle\partial_{x_1},\ldots,\partial_{x_n}\rangle \;\cong \; {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(R,{\mathbb{F}}) \;\cong \; \bigoplus_{ \alpha \in {\mathbb{N}}^n} {\mathbb{F}}\overline{\partial_\mathbf{x}^\alpha}.$$ When the Weyl-Noether module was introduced in (\[eq:relativeweyl\]), we gave a purely algebro-symbolic treatment and we noticed that an ${\mathbb{F}}$-basis is given by $\,\left\lbrace 1 \otimes_R \partial_{\bf x}^\alpha: \alpha \in{\mathbb{N}}^n\right\rbrace$. Now, with the isomorphism (\[eq\_isom\_relWeyl\_diff\]), the elements $1 \otimes_R \partial_{\mathbf{x}}^\alpha $ are seen as the differential operators $\overline{\partial_{\mathbf{x}}^\alpha} \in {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(R, {\mathbb{F}})$. The following map is an isomorphism of ${\mathbb{F}}$-vector spaces: $$\label{eq_map_omega} \omega : {\mathbb{F}}[z_1,\ldots,z_n] \;\rightarrow\; {\mathbb{F}}\,\otimes_R \, R\langle\partial_{x_1},\ldots,\partial_{x_n}\rangle\;\cong\;{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(R,{\mathbb{F}}), \quad \mathbf{z}^\alpha \mapsto \partial_{\mathbf{x}}^\alpha.$$ From (\[eq:FFRT\]) and Notation \[nota\_T\_mod\_struct\] we get the following actions. For any $\alpha \in {\mathbb{N}}^n$ and $1 \le i \le n$, $$\label{eq_deriv_z_bracket_partial} \partial_{z_i} \bullet \mathbf{z}^\alpha \,=\, \alpha_iz_1^{\alpha_1}\cdots z_i^{\alpha_i-1}\cdots z_n^{\alpha_n} \quad\text{ and }\quad y_i\cdot\partial_{\mathbf{x}}^\alpha \,=\,\left[\partial_{\mathbf{x}}^\alpha,x_i\right]\,=\, \alpha_i\partial_{x_1}^{\alpha_1}\cdots \partial_{x_i}^{\alpha_i-1}\cdots \partial_{x_n}^{\alpha_n}.$$ Hence the map $\omega$ in (\[eq\_map\_omega\]) gives a bijection between ${\mathbb{F}}$-vector subspaces of ${\mathbb{F}}[z_1,\ldots,z_n]$ closed under differentiation and ${\mathbb{F}}[y_1,\ldots,y_n]$-submodules of ${\mathbb{F}}\otimes_R R\langle\partial_{x_1},\ldots,\partial_{x_n}\rangle$. The latter structure as a submodule is equivalent to being an $R$-bi-submodule of the Weyl-Noether module. \[lem\_prim\_ideal\_implies\_bimod\] Let $\mathcal{E}$ be a finite dimensional ${\mathbb{F}}$-vector subspace of $ {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(R,{\mathbb{F}})$. If $\,Q = {\rm Sol}(\mathcal{E})\,$ is a $P$-primary ideal in $R = {\mathbb{K}}[x_1,\ldots,x_n]$ then $\mathcal{E}$ is an $R$-bi-module. Fix $m \in {\mathbb{N}}$ such that $Q \supseteq P^m$ and $\mathcal{E} \subseteq {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,{\mathbb{F}})$. The map $\gamma$ in (\[eq\_map\_gamma\]) defines the ideal $I={\mathcal{M}}^m+\gamma( Q){\mathbb{F}}[y_1,\dots,y_n]$. Let $\mathcal{E}^\prime \subseteq {{\normalfont\text{Hom}}}_{\mathbb{F}}\left(\frac{{\mathbb{F}}[\mathbf{y}]}{{\mathcal{M}}^m},{\mathbb{F}}\right)$ be the ${\mathbb{F}}$-vector subspace coming from $\mathcal{E}$ under the isomorphism of Proposition \[lem\_descrip\_diff\_opp\]$(i)$. The hypothesis $\,Q={\rm Sol}(\mathcal{E})\,$ implies $$\label{eq_functionals_sols} I/{\mathcal{M}}^m\,\, = \,\,\bigl\{ \,w \in {\mathbb{F}}[\mathbf{y}]/ {\mathcal{M}}^m \,:\, \delta(w)=0 \,\text{ for all } \,\delta \in \mathcal{E}^\prime\, \bigr\}.$$ Dualizing the inclusion $\mathcal{E}^\prime \hookrightarrow {{\normalfont\text{Hom}}}_{\mathbb{F}}\left( {\mathbb{F}}[\mathbf{y}]/{\mathcal{M}}^m,{\mathbb{F}}\right)$ we get the short exact sequence $$\label{eq_dualize_short_E} 0 \,\,\rightarrow \,\,Z \,\,\rightarrow\,\, {\mathbb{F}}[\mathbf{y}] / {\mathcal{M}}^m \,\, \rightarrow \,\,{{\normalfont\text{Hom}}}_{\mathbb{F}}(\mathcal{E}^\prime, {\mathbb{F}})\,\, \rightarrow \,\, 0,$$ where $Z=\Big\lbrace w \in \frac{{\mathbb{F}}[\mathbf{y}]}{{\mathcal{M}}^m} \,:\, \delta(w)=0 \text{ for all } \delta \in \mathcal{E}^\prime\Big\rbrace$. Therefore, equations (\[eq\_functionals\_sols\]) and (\[eq\_dualize\_short\_E\]) yield the isomorphism $\,{{\normalfont\text{Hom}}}_{\mathbb{F}}(\mathcal{E}^\prime,{\mathbb{F}}) \cong {\mathbb{F}}[\mathbf{y}]/I$, and we conclude that $\mathcal{E} \cong \mathcal{E}^\prime$ is an $R$-bi-module. Finally, to complete the proof of Theorem \[thm:main\], it will suffice to prove the following. \[thm\_noeth\_ops\_zero\_dim\] Let $P$ be a maximal ideal in $R = {\mathbb{K}}[x_1,\ldots,x_n]$, and let $Q \subset R$ be a $P$-primary ideal of multiplicity $m$ over $P$. Then $\,Q = {\rm Sol}(\mathcal{E})$, where $\mathcal{E}$ is obtained by the following steps: (i) As in Theorem \[thm:param\_primary\], set $\,I=\langle y_1,\dots,y_n\rangle^m+\gamma( Q){\mathbb{F}}[y_1,\dots,y_n]$. (ii) As in Theorem \[thm:Macaulay\_dual\], set $\,V = I^\perp \,\subset\, {\mathbb{F}}[z_1,\ldots,z_n]$. (iii) Using the map $\omega$ in (\[eq\_map\_omega\]), set $\,\,\mathcal{E}=\omega(V) \;\subset \;{\mathbb{F}}\otimes_R R\langle\partial_{x_1},\ldots,\partial_{x_n}\rangle\;\cong\;{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(R,{\mathbb{F}})$. We claim that the correspondence in Proposition \[lem\_descrip\_diff\_opp\]$(ii)$ gives $$\mathcal{E} \,\,\cong\, \,{{\normalfont\text{Hom}}}_{{\mathbb{F}}}\bigl( {\mathbb{F}}[\mathbf{y}] / I, {\mathbb{F}}\bigr) \,\, \,\hookrightarrow\, \,\,{{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,{\mathbb{F}}).$$ The isomorphism (\[eq\_isom\_inv\_sys\_char\_zero\]) implies that $V\cong V^\prime = \left(0:_{{\mathbb{F}}\left[\mathbf{y^{-1}}\right]} I\right)$. Since $I \supseteq {\mathcal{M}}^m$, it follows that $V^\prime \subseteq \left(0:_{{\mathbb{F}}\left[\mathbf{y^{-1}}\right]} {\mathcal{M}}^m\right)$. For each $0 \le j < m$, there is a perfect pairing $$\label{eq_perf_pairing} {\left[\frac{{\mathbb{F}}[\mathbf{y}]}{{\mathcal{M}}^m}\right]}_j \;\otimes_{\mathbb{F}}\; {\left[\left(0:_{{\mathbb{F}}\left[\mathbf{y^{-1}}\right]} {\mathcal{M}}^m\right)\right]}_{-j} \;\rightarrow \;{\mathbb{F}}, \quad \mathbf{y}^\alpha \otimes_{\mathbb{F}}\frac{1}{\mathbf{y}^\beta} \,\mapsto\, \mathbf{y}^\alpha\cdot\frac{1}{\mathbf{y}^\beta} = \begin{cases} 1 \;\;\text{ if } \alpha = \beta\\ 0 \;\;\text{ otherwise}, \end{cases}$$ where $\vert\alpha\rvert=\vert\beta\rvert=j$, induced by the usual multiplication. Hence, we get the isomorphisms $$\label{eq_isom_inv_sys_diff_ops} \left(0:_{{\mathbb{F}}\left[\mathbf{y^{-1}}\right]} {\mathcal{M}}^m\right) \;\cong\; {{\normalfont\text{Hom}}}_{\mathbb{F}}\left(\frac{{\mathbb{F}}[\mathbf{y}]}{{\mathcal{M}}^m},{\mathbb{F}}\right) \;\cong\; {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,{\mathbb{F}}) .$$ The second isomorphism follows from Proposition \[lem\_descrip\_diff\_opp\]$(i)$. The Hom-tensor adjunction yields $$\label{eq_isom_V_prime_dual_quot_I} \!\! \!\! V^\prime = \left(0 :_{\left(0:_{{\mathbb{F}}\left[\mathbf{y^{-1}}\right]} {\mathcal{M}}^m\right)} I\right) \cong\;{{\normalfont\text{Hom}}}_{{\mathbb{F}}[[\mathbf{y}]]}\left(\frac{{\mathbb{F}}[\mathbf{y}]}{I}, {{\normalfont\text{Hom}}}_{\mathbb{F}}\left( \frac{{\mathbb{F}}[\mathbf{y}]}{{\mathcal{M}}^m},{\mathbb{F}}\right) \right) \cong\; {{\normalfont\text{Hom}}}_{{\mathbb{F}}}\left(\frac{{\mathbb{F}}[\mathbf{y}]}{I}, {\mathbb{F}}\right).$$ The isomorphism $V^\prime \cong {{\normalfont\text{Hom}}}_{\mathbb{F}}\left(\frac{{\mathbb{F}}[\mathbf{y}]}{I},{\mathbb{F}}\right)$ also follows from the duality in [@EISEN_COMM Proposition 21.4]. By the isomorphism (\[eq\_isom\_inv\_sys\_char\_zero\]) and the map $\omega$ in (\[eq\_map\_omega\]), $\mathcal{E}$ can be obtained from $V^\prime$ via the map $$V^\prime \xrightarrow{\cong} \mathcal{E}, \quad \frac{1}{\mathbf{y}^\alpha} \mapsto \frac{1}{\alpha!}\partial_{\mathbf{x}}^\alpha.$$ On the other hand, by (\[eq\_diff\_opp\_z\_alpha\]), (\[eq\_perf\_pairing\]) and (\[eq\_isom\_inv\_sys\_diff\_ops\]), the dual monomial ${(\mathbf{y}^\alpha)}^* \in {{\normalfont\text{Hom}}}_{\mathbb{F}}\left(\frac{{\mathbb{F}}[y_1,\ldots,y_n]}{{\mathcal{M}}^m},{\mathbb{F}}\right)$ is identified with the inverted monomial $\frac{1}{\mathbf{y}^\alpha} \in {\mathbb{F}}[\mathbf{y^{-1}}]$ and with the differential operator $\overline{D_\mathbf{x}^\alpha} = \frac{1}{\alpha!}\overline{\partial_{\mathbf{x}}^\alpha}\in {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,{\mathbb{F}})$. Therefore, the isomorphisms in (\[eq\_isom\_V\_prime\_dual\_quot\_I\]) imply that $\mathcal{E}$ is indeed determined by $I$ via the correspondence in Proposition \[lem\_descrip\_diff\_opp\]$(ii)$. After this identification, Proposition \[lem\_descrip\_diff\_opp\]$(iii)$ and Theorem \[thm:param\_primary\] imply that $${\rm Sol}(\mathcal{E})\, =\, \gamma^{-1}(I) \,=\,Q.$$ This completes the proof of Theorem \[thm\_noeth\_ops\_zero\_dim\], and we obtain Theorem \[thm:main\] for $P$ maximal. The general case ---------------- In this subsection, we complete the proof of Theorem \[thm:main\]. As before, $R={\mathbb{K}}[x_1,\ldots,x_n]$, ${\rm char}({\mathbb{K}}) = 0$, and $P \subset R$ is a prime ideal of height $c$. We use the notation from Section \[sec4\], where $S={\mathbb{K}}(x_{c+1},\ldots,x_n)[x_1,\ldots,x_c]$ and ${\mathfrak{p}}= PS$. By Noether normalization, ${\mathbb{K}}[x_{c+1},\ldots,x_n] \hookrightarrow R/P$ is an integral extension. The ideal ${\mathfrak{p}}\subset S $ is maximal and ${\mathbb{F}}= S/{\mathfrak{p}}$. The following remarks will allow us to derive Theorem \[thm:main\] from Theorems \[thm:param\_primary\], \[thm:Macaulay\_dual\] and \[thm\_noeth\_ops\_zero\_dim\]. \[rem\_clear\_fractions\_diff\_ops\] By Lemma \[lem\_diff\_ops\_R/J\], any operator $A^\prime\in {{\normalfont\text{Diff}}}_{S/{\mathbb{K}}(x_{c+1},\ldots,x_n)}^{m-1}(S,S/{\mathfrak{p}})$ can be written as $$A^\prime \,\,\,= \sum_{\substack{\beta \in {\mathbb{N}}^c\\ \lvert \beta \rvert \le m-1}} \overline{h_\beta}\,\, \overline{\partial_{x_1}^{\beta_1}\cdots\partial_{x_c}^{\beta_c}} \quad \text{ for some }\,\, h_\beta \in S.$$ We choose $h \in {\mathbb{K}}[x_{c+1},\ldots,x_n]$ such that $h\cdot h_\beta \in R$ for all $\beta$. Hence, we can consider $$A \,\,\, = \sum_{\substack{\beta \in {\mathbb{N}}^c\\ \lvert \beta \rvert \le m-1}} \overline{h\cdot h_\beta}\, \overline{\partial_{x_1}^{\beta_1}\cdots\partial_{x_c}^{\beta_c}} \;\in \; {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R/P).$$ This differential operator satisfies $\,{\rm Sol}(A)={\rm Sol}(A^\prime) \cap R$. \[rem\_lift\_diff\_ops\_sol\] Let $A^\prime = \sum_{\lvert \alpha \rvert \le m-1} \overline{r_\alpha} \overline{\partial_{\mathbf{x}}^\alpha} \,\in\, {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R/P)$ be a differential operator. By Lemma \[lem\_diff\_ops\_R/J\], we can lift this to $A = \sum_{\lvert \alpha \rvert \le m-1} r_\alpha \partial_\mathbf{x}^\alpha \,\in\, {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R)$. Then, it follows that $${\rm Sol}(A^\prime)=\lbrace f \in R : A \bullet f \in P \rbrace .$$ The next remark describes the Weyl-Noether module in terms of differential operators. \[rem\_isom\_restrict\_Weyl\_mod\] We have the following isomorphisms $$\begin{aligned} {\mathbb{F}}\otimes_R D_{n,c} \,\,=\,\, {\mathbb{F}}\otimes_R R\langle\partial_{x_1},\ldots,\partial_{x_c} \rangle \,& \,\cong\,\, {\mathbb{F}}\otimes_S \left( S \otimes_R R\langle \partial_{x_1},\ldots,\partial_{x_c}\rangle\right)\\ & \,\cong \,\, {\mathbb{F}}\otimes_S S\langle \partial_{x_1},\ldots,\partial_{x_c}\rangle\\ & \,\cong\,\, {{\normalfont\text{Diff}}}_{S/{\mathbb{K}}(x_{c+1},\ldots,x_n)}\left(S,{\mathbb{F}}\right). \end{aligned}$$ The last isomorphism follows from (\[eq\_isom\_relWeyl\_diff\]) by applying this to the polynomial ring $S={\mathbb{K}}(x_{c+1},\ldots,x_n)[x_1,\ldots,x_c]$ and the maximal ideal ${\mathfrak{p}}= PS$ in $S$. The correspondences between parts (a), (b) and (c) have already been established in Theorems \[thm:param\_primary\] and \[thm:Macaulay\_dual\]. Using Remark \[rem\_isom\_restrict\_Weyl\_mod\], we identify the Weyl-Noether module $\,{\mathbb{F}}\otimes_R D_{n,c} \,$ with $\, {{\normalfont\text{Diff}}}_{S/{\mathbb{K}}(x_{c+1},\ldots,x_n)}\left(S,{\mathbb{F}}\right)$. As in (\[eq\_map\_omega\]), we consider the map $$\begin{aligned} \label{eq_map_omega_S} \begin{split} \omega_S \,:\, {\mathbb{F}}[z_1,\ldots,z_c] \;&\rightarrow\; {\mathbb{F}}\,\otimes_R \, D_{n,c}\;\cong\;{{\normalfont\text{Diff}}}_{S/{\mathbb{K}}(x_{c+1},\ldots,x_n)}(S,{\mathbb{F}})\\ z_1^{\alpha_1}\cdots z_c^{\alpha_c} \;&\mapsto\; \partial_{x_1}^{\alpha_1}\cdots\partial_{x_c}^{\alpha_c}, \end{split} \end{aligned}$$ but now applied to the polynomial ring $S={\mathbb{K}}(x_{c+1},\ldots,x_n)[x_1,\ldots,x_c]$ and its maximal ideal ${\mathfrak{p}}\subset S$. This map $\omega_S$ yields the correspondence between parts (c) and (d), that is, between $m$-dimensional ${\mathbb{F}}$-vector subspaces of ${\mathbb{F}}[z_1,\ldots,z_c]$ that are closed under differentiation and $m$-dimensional ${\mathbb{F}}$-vector subspaces of ${\mathbb{F}}\otimes_R D_{n,c}$ that are $R$-bi-modules under the action (\[eq\_deriv\_z\_bracket\_partial\]). It remains to show that a basis of an ${\mathbb{F}}$-vector subspace in part (d) can be lifted to a set of Noetherian operators for the $P$-primary ideal in part (a). For that, let $Q$ be a $P$-primary ideal with multiplicity $m$ over $P$, and set $I=\gamma(Q)$, $V = I^\perp$ and $\mathcal{E} = \omega_S(V)$, by using Theorem \[thm:param\_primary\], Theorem \[thm:Macaulay\_dual\] and (\[eq\_map\_omega\_S\]), respectively. Then, Theorem \[thm\_noeth\_ops\_zero\_dim\] implies that, for any basis $A_1^{\prime\prime},\ldots,A_m^{\prime\prime}$ of the ${\mathbb{F}}$-vector subspace $\mathcal{E} \subset {{\normalfont\text{Diff}}}_{S/{\mathbb{K}}(x_{c+1},\ldots,x_n)}(S,{\mathbb{F}})$, we get the equality $ QS = {\rm Sol}(A_1^{\prime\prime},\ldots,A_m^{\prime\prime}). $ From Remark \[rem\_clear\_fractions\_diff\_ops\], we can choose differential operators $$A_i^\prime \,\,=\,\, \sum_{\alpha \in {\mathbb{N}}^c} \overline{r_{i,\alpha}} \,\, \overline{\partial_{x_1}^{\alpha_1}\cdots\partial_{x_c}^{\alpha_c}} \,\in\, {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}(R,R/P), \quad \text{ where } 1\le i \le m \text{ and } r_{i,\alpha} \in R,$$ such that $Q={\rm Sol}(A_1^\prime,\ldots,A_m^\prime)$. Finally, by Remark \[rem\_lift\_diff\_ops\_sol\], the lifted differential operators $$A_i \,\,=\,\, \sum_{\alpha \in {\mathbb{N}}^c} r_{i,\alpha} \partial_{x_1}^{\alpha_1}\cdots\partial_{x_c}^{\alpha_c} \,\in\, D_{n,c} $$ are Noetherian operators for $Q$, which means that (\[eq:fromAtoQ\]) holds. This completes the proof. Symbolic Powers and other Joins {#sec7} =============================== The symbolic power of an ideal is a fundamental construction in commutative algebra. We here work in the polynomial ring $R={\mathbb{K}}[x_1,\ldots,x_n]$ over a field ${\mathbb{K}}$ of characteristic zero, with irrelevant maximal ideal $\mathfrak{m} = \langle x_1,\ldots,x_n \rangle$. The $r$-th [*symbolic power*]{} of an ideal $J$ in $R$ equals $$J^{(r)} \,\,\,\,:= \,\,\bigcap_{{\mathfrak{p}}\in {\rm Ass}(J)} \!\! J^r R_{\mathfrak{p}}\cap R.$$ Hence, if $P$ is a prime ideal in $R$ then $P^{(r)}$ is the $P$-primary component of the usual power $P^r$. If $\,{\rm codim}(P) = c\,$ then the primary ideal $P^{(r)}$ has multiplicity $m = \binom{c+r-1}{c}$ over $P$, and in Theorem \[thm:param\_primary\] it is represented by the zero-dimensional ideal $\,I = \langle y_1,\ldots,y_c \rangle^r\, \subset \, {\mathbb{F}}[y_1,\ldots,y_c]$. Our point of departure in this section is a formula due to Sullivant [@SULLIVANT_SYMB Proposition 2.8]: $$\label{eq:sulli1} J^{(r)} \,\, = \,\, J \star \mathfrak{m}^r .$$ Here, $J$ is any radical ideal in $R$, and $\star$ denotes the join of ideals. This is a reformulation of the [*Zariski-Nagata Theorem*]{} which expresses the symbolic power via differential equations: $$\label{eq:sulli2} J^{(r)} \,\, = \,\, \, \biggl\{ \,f \,\in\, R \,\,\,\bigg\vert \,\,\, \frac{\partial^{i_1+i_2+\cdots+i_n} f}{ \partial x_1^{i_1} \partial x_2^{i_2} \cdots \partial x_n^{i_n} } \,\in\, J \quad \text{whenever}\, \,\,i_1+i_2 + \cdots + i_n < r \, \biggr\}.$$ The goal of this section is to generalize the equivalence between (\[eq:sulli1\]) and (\[eq:sulli2\]). We construct $P$-primary ideals by means of joins and we relate this to the results seen in earlier sections. If $J$ and $K$ are ideals in $R$, then their *join* is the new ideal $$J \star K \,\,\,:=\,\,\, \Big( J(\mathbf{v}) \,+\, K(\mathbf{w}) \,+\, \langle x_i - v_i - w_i : 1 \le i \le n \rangle \Big) \,\,\cap \,\,R,$$ where $J(\mathbf{v})$ is the ideal $J$ with new variables $v_i$ substituted for $x_i$ and $K(\mathbf{w})$ is the ideal $K$ with $w_i$ substituted for $x_i$. The parenthesized ideal lives in a polynomial ring in $3n$ variables. \[rem\_kernel\_map\_join\] Following Simis and Ulrich [@SIMIS_ULRICH_JOIN], the join $J \star K$ equals the kernel of the map $$\begin{aligned} R \;&\,\rightarrow \,\; \frac{{\mathbb{K}}[v_1,\ldots,v_n,w_1,\ldots,w_n]}{J(\mathbf{v})+K(\mathbf{w})} \,\;\;\xleftrightarrow{\cong}\; R/J \otimes_{\mathbb{K}}R/K\\ x_i \;&\, \mapsto \,\,\;\; \overline{v_i}+\overline{w_i} \qquad\qquad\qquad\qquad\leftrightarrow \;\;\overline{x_i} \otimes_{\mathbb{K}}1 + 1 \otimes_{\mathbb{K}}\overline{x_i}. \end{aligned}$$ Hence, the quotient $\,R/\left(J\star K\right)\,$ can be identified with a subring of $\,R/J \otimes_{\mathbb{K}}R/K$. The following result summarizes a few basic properties of the join construction. \[prop\_properties\_join\] Let $J$ and $K$ be ideals in $R$. Then, the following statements hold: (i) If $J = J_1 \cap J_2$, where $J_1,J_2 \subset R$ are ideals, then $J \star K = (J_1 \star K) \cap (J_2 \star K)$. (ii) $\sqrt{J \star K} = \sqrt{J} \star \sqrt{K}$; in particular, $J \star K$ is radical when $J$ and $K$ are. (iii) Suppose that ${\mathbb{K}}$ is algebraically closed. If $P_1$ and $P_2$ are prime ideals, then $P_1 \star P_2$ is a prime ideal. If $J$ and $K$ are primary ideals, then $J \star K$ is a primary ideal. (iv) If $M$ is an $\mathfrak{m}$-primary ideal, then $P \star M$ is a $P$-primary ideal. This is an adaptation of [@SIMIS_ULRICH_JOIN Proposition 1.2] for non-necessarily homogeneous ideals. $(i)$ The join distributes over intersections by [@SULLIVANT_SYMB Lemma 2.6]. $(ii)$ The ring $R/\sqrt{J} \otimes_{\mathbb{K}}R/\sqrt{K}$ is reduced by [@GORTZ_WEDHORN Corollary 5.57]. As the kernel of the map $R/J\otimes_{\mathbb{K}}R/K \twoheadrightarrow R/\sqrt{J} \otimes_{\mathbb{K}}R/\sqrt{K}$ is nilpotent, the claim follows from Remark \[rem\_kernel\_map\_join\]. $(iii)$ Since ${\mathbb{K}}$ is algebraically closed, $R/P_1 \otimes_{\mathbb{K}}R/P_2$ is an integral domain [@GORTZ_WEDHORN Lemma 4.23]. By Remark \[rem\_kernel\_map\_join\], $R/(P_1 \star P_2)$ is a subring of this domain. Thus, $P_1 \star P_2$ is a prime ideal. Suppose ${\rm Ass}(R/J)=\{P_1\}$ and ${\rm Ass}(R/K)=\{P_2\}$. From [@MATSUMURA Theorem 23.2] we infer $$\label{eq_equality_ass_primes} {\rm Ass}(R/J \otimes_{\mathbb{K}}R/K)\,\, = \,\, {\rm Ass}(R/P_1 \otimes_{\mathbb{K}}R/P_2).$$ We already saw that $R/P_1 \otimes_{\mathbb{K}}R/P_2$ is an integral domain. Therefore, $R/J \otimes_{\mathbb{K}}R/K$ has only one associated prime, and hence its subring $R/(J \star K)$ has only one associated prime. $(iv)$ The equality in (\[eq\_equality\_ass\_primes\]) is valid for any field. From this we get ${\rm Ass}(R/P \otimes_{\mathbb{K}}R/M) = {\rm Ass}(R/P \otimes_{\mathbb{K}}R/\mathfrak{m}) = \{ P \star \mathfrak{m} \} = \{P\}$. We hence conclude $\,{\rm Ass}(R / (P \star M))= \{P\}$. In Proposition \[prop\_properties\_join\] (iii) we need the hypothesis that ${\mathbb{K}}$ is algebraically closed. If ${\mathbb{K}}= {\mathbb{R}}$ then $P_1 = \langle x_1^2+1, x_2 \rangle$ and $P_2 = \langle x_1 , x_2^2+1 \rangle $ are prime but their join is not primary: $$P_1 \star P_2 \,\, = \,\, \langle x_1^2+1, x_2^2+1 \rangle \,\, =\,\, \langle x_1 - x_2 , x_2^2+1 \rangle \,\, \cap \,\, \langle x_1 + x_2 , x_2^2+1 \rangle .$$ In what follows we focus on the $P$-primary ideals $Q = P \star M$ in Proposition \[prop\_properties\_join\] (iv). These will be characterized by differential equations derived from the $\mathfrak{m}$-primary ideal $M$. Let $M$ be an $\mathfrak{m}$-primary ideal. We shall encode $M$ by a system $\mathfrak{A}(M)$ of linear PDE with constant coefficients. This is computed by the performing following steps: (i) Interpret $M$ as PDE by replacing the variables $x_i$ with $\partial_{z_i}$ for $i=1,\ldots,n$. (ii) Compute the inverse system $M^\perp=\left\lbrace F\in {\mathbb{K}}[z_1,\ldots,z_n]: f\bullet F=0 \mbox{ for all }f\in M\right\rbrace$. (iii) Let $\mathfrak{A}(M) \subset {\mathbb{K}}[\partial_{x_1},\ldots,\partial_{x_n}]$ be the image of $M^\perp$ under the map $\mathbf{z}^\alpha \mapsto \partial_{\mathbf{x}}^\alpha$. We say that the ${\mathbb{K}}$-subspace $\mathfrak{A}(M)$ comprises the *differential operators associated to $M$*. \[rem\_joins\_props\] (i) The space $\mathfrak{A}(M)$ is closed under taking brackets as in (\[eq\_deriv\_z\_bracket\_partial\]) and Theorem \[thm:Macaulay\_dual\]. \(ii) For any $r \ge 1$, we have $ \mathfrak{A}\left(\mathfrak{m}^r\right) = \bigoplus_{\lvert \alpha \rvert \le r-1} {\mathbb{K}}\, \partial_{\mathbf{x}}^\alpha $. Thus, $\mathfrak{A}(\mathfrak{m}^r)$ comprises the differential operators used in the Zariski-Nagata formula for symbolic powers; see (\[eq:sulli2\]) and [@EISEN_COMM §3.9]. The following result is a generalization of the classical Zariski-Nagata Theorem, to ideals obtained with the join construction. Of main interest is the situation when $J=P$ is prime. \[thm:ourZN\] Let $J$ be any ideal in $R = {\mathbb{K}}[x_1,\ldots,x_n]$ and let $M$ be an $\mathfrak{m}$-primary ideal. (i) The join of $J$ and $M$ equals $J \star M \;=\; \big\lbrace f \in R : A \bullet f \in J \; \text{ for all }\; A \in \mathfrak{A}(M) \big\rbrace$. (ii) If $J$ is radical and $r \in {\mathbb{N}}$ then $\, J^{(r)} \,=\, J \star \mathfrak{m}^r \,=\, \big\lbrace f \in R : \partial_{\mathbf{x}}^\alpha \bullet f \in J \; \text{ for all }\; \lvert \alpha \rvert \le r-1 \big\rbrace $. Let $n=4,c=2$, fix the prime ideal $P$ in (\[eq:twistedcubic1\]), and consider the $\mathfrak{m}$-primary ideal $ M = \langle x_1^2,x_2^2,x_3^2,x_4^2 \rangle$. The join $Q = P \star M$ is a $P$-primary ideal of multiplicity $m=11$. It is minimally generated by eight octics such as $\,x_1^8-4 x_1^6 x_2 x_3+6 x_1^4 x_2^2 x_3^2-4x_1^2 x_2^3 x_3^3+x_2^4 x_3^4$. The differential equations from $\mathfrak{A}(M)$ are simply the squarefree partial derivatives, so that $$\label{eq:repQ1} Q \,\,\,= \,\,\,\biggl\{ \,f \,\in\, R \,\,\,\bigg\vert \,\,\, \frac{\partial^{i_1+i_2+i_3+i_4} f}{ \partial x_1^{i_1} \partial x_2^{i_2} \partial x_3^{i_3} \partial x_4^{i_4} } \,\in\, P \quad \text{whenever}\, \,\,i_1,i_2,i_3,i_4 \in \{0,1\}\, \biggr\}.$$ This should be compared to the representation by Noetherian operators found in Algorithm \[alg:forward\]. In Step 1, we obtain the ideal $I = \langle y_1^4, u_2 y_1^3 y_2 - u_3 y_1 y_2^3, 3 u_1 y_1^2 y_2^2 - 5 u_3 y_1 y_2^3, y_2^4 \rangle$. The inverse system $I^\perp$ in Step 2 is the $11$-dimensional subspace of ${\mathbb{F}}[y_1,y_2]$ spanned by $$B({\bf u},{\bf z}) \,\,\,= \,\,\, 2 u_1 u_3 \,z_1^3 z_2\, +\, 5 u_2 u_3 \,z_1^2 z_2^2 \,+\, 2 u_1 u_2 \,z_1 z_2^3$$ together with all ten monomials $z_1^{j_1} z_2^{j_2} $ of degree $j_1+j_2\leq 3$. From Steps 3 and 4 we obtain $$A({\bf x},\partial_{\bf x}) \,\, = \,\, 2 x_1 x_3 \partial_{x_1}^3 \partial_{x_2 } + 5 x_2 x_3 \partial_{x_1}^2 \partial_{x_2}^2 + 2 x_1 x_2 \partial_{x_1} \partial_{x_2}^3 ,$$ and this gives rise to the following alternative representation of $Q$ by differential equations: $$\label{eq:repQ2} Q \,\,\,= \,\,\,\biggl\{ \,f \,\in\, R \,\,\,\bigg\vert \,\,\, A \bullet f \in P \,\,\, \,{\rm and} \,\,\, \frac{\partial^{j_1+j_2} f}{ \partial x_1^{j_1} \partial x_2^{j_2} } \,\in\, P \quad \text{whenever}\, \,j_1 + j_2 \leq 3 \, \biggr\}.$$ The two representations (\[eq:repQ1\]) and (\[eq:repQ2\]) differ in two fundamental ways. The operators in (\[eq:repQ1\]) have constant coefficients but differentiation involves all four variables. In (\[eq:repQ2\]) we are using an operator from $D_{4,2}$ with polynomial coefficients but we differentiate only two variables. The next example shows that not every primary ideal arises from the join construction. \[exam:Palamodov\] Let $n = 3$ and $c = 2$, and consider the primary ideal $Q = \langle x_1^2, x_2^2, x_1 - x_2x_3 \rangle$ with $P = \sqrt{Q}=\langle x_1, x_2 \rangle$. From [@BJORK Proposition 4.8 and Example 4.9, page 352] we know that $Q$ cannot be described by differential operators with constant coefficients only. Theorem \[thm:ourZN\] (i) implies that $Q$ does not arise from the join construction, i.e. we cannot find an $\mathfrak{m}$-primary ideal $M$ such that $Q = P \star M$. On the other hand, Algorithm \[alg:forward\] applied to $Q$ gives the two Noetherian operators $A_1 = 1, A_2 = x_3\partial_{x_1} + \partial_{x_2}$. $(i)$ We use the notation and results from Section \[sec5\]. We begin by fixing an integer $m$ such that $\mathfrak{m}^m \subseteq M$. In (\[eq\_isoms\_Diff\_R/J\]) we obtained the explicit isomorphism $$\label{eq_join_isom_diff_ops} {{\normalfont\text{Hom}}}_{R/J}\left(R/J \otimes_R P_{R/{\mathbb{K}}}^{m-1}, R/J\right) \xrightarrow{\cong} {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R/J), \quad \psi \mapsto \psi \circ h_{m-1} \circ d^{m-1}$$ where $h_{m-1}$ is the canonical map $ P_{R/{\mathbb{K}}}^{m-1} \rightarrow R/J \otimes_R P_{R/{\mathbb{K}}}^{m-1}$ and $d^{m-1}$ is the map in (\[eq\_univ\_diff\]). Setting $\,T=R \otimes_{\mathbb{K}}R ={\mathbb{K}}[x_1, \ldots, x_n, y_1, \ldots, y_n]\,$ as in Section \[sec5\], we have the following isomorphisms: $$\label{eq_isom_tensor_prods_join} R/J \otimes_R P_{R/{\mathbb{K}}}^{m-1} \;\cong\; \frac{T}{J(\mathbf{x}) \,+ \, \mathfrak{m}^m(\mathbf{y})} \;\cong\; R/J \otimes_{\mathbb{K}}R/\mathfrak{m}^m.$$ Recall that this ${\mathbb{K}}$-vector space is considered as an $R$-module via the left factor $R/J \otimes_{\mathbb{K}}1$. Using (\[eq\_join\_isom\_diff\_ops\]) and (\[eq\_isom\_tensor\_prods\_join\]), the surjection $\,R/J \otimes_{\mathbb{K}}R/\mathfrak{m}^m \twoheadrightarrow R/J \otimes_{\mathbb{K}}R/M\,$ induces the inclusion $$\label{eq_inclusion_join_diffs} {{\normalfont\text{Hom}}}_{R/J}\left(R/J \otimes_{\mathbb{K}}R/M, R/J\right) \,\,\hookrightarrow\,\, {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R/J).$$ Since $ R/J \otimes_{\mathbb{K}}R/M$ is a finitely generated free $R/J$-module, we have $$\big\lbrace w \in R/J \otimes_{\mathbb{K}}R/M : \psi(w)=0 \;\text{ for all }\; \psi \in {{\normalfont\text{Hom}}}_{R/J}\left(R/J \otimes_{\mathbb{K}}R/M, R/J\right) \big\rbrace \;=\; \{0\}.$$ Let $\,\mathcal{E} \subseteq {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R/J)\,$ denote the image of (\[eq\_inclusion\_join\_diffs\]). So, the isomorphism (\[eq\_join\_isom\_diff\_ops\]) implies $${\rm Sol}(\mathcal{E}) = {\rm Ker}\left(\overline{d^{m-1}}\right), \;\text{ where }\; \overline{d^{m-1}} : R \rightarrow R/J \otimes_{\mathbb{K}}R/M, \;\; x_i \mapsto \overline{x_i} \otimes_{\mathbb{K}}1 + 1 \otimes_{\mathbb{K}}\overline{x_i}.$$ Therefore, Remark \[rem\_kernel\_map\_join\] yields that ${\rm Sol}(\mathcal{E})=J \star M$. By [@MATSUMURA Theorem 7.11], the inclusion (\[eq\_inclusion\_join\_diffs\]) can be written equivalently as $$\begin{aligned} {{\normalfont\text{Hom}}}_{R/J}\left(R/J \otimes_{\mathbb{K}}R/M, R/J\right) \,\, \cong \,\, R/J &\otimes_{\mathbb{K}}{{\normalfont\text{Hom}}}_{\mathbb{K}}(R/M,{\mathbb{K}})\\ & \hookrightarrow\; R/J \otimes_{\mathbb{K}}{{\normalfont\text{Hom}}}_{\mathbb{K}}(R/\mathfrak{m}^m,{\mathbb{K}}) \,\, \cong \,\, {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R/J).\end{aligned}$$ The Hom-tensor adjunction and the perfect pairing in (\[eq\_perf\_pairing\]) give the following isomorphisms: $${{\normalfont\text{Hom}}}_{\mathbb{K}}\left(R/M,{\mathbb{K}}\right) \,\,\cong\,\, {{\normalfont\text{Hom}}}_R\left(R/M, {{\normalfont\text{Hom}}}_{{\mathbb{K}}}\left(R/\mathfrak{m}^m,{\mathbb{K}}\right)\right) \,\,\cong\,\, \left(0 :_{{\mathbb{K}}[\mathbf{x^{-1}}]} M\right).$$ Then, by arguments almost verbatim to those used in the proof of Theorem \[thm\_noeth\_ops\_zero\_dim\], we find that $\,\mathcal{E} \subseteq {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R, R/J)\,$ is a finitely generated free $R/J$-module, and it is generated by $\,\bigl\{ \,\overline{A} \,:\, A \in \mathfrak{A}(M) \subset {\mathbb{K}}[\partial_{\mathbf{x}}] \cap {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R)\bigr\} \subset {{\normalfont\text{Diff}}}_{R/{\mathbb{K}}}^{m-1}(R,R/J)$. Summing up, we conclude $$J \star M \,\,=\,\, {\rm Sol}(\mathcal{E}) \,\,=\,\, \big\lbrace f \in R : A \bullet f \in J \;\text{ for all }\; A \in \mathfrak{A}(M) \big\rbrace.$$ $(ii)$ Since $J$ is radical, $J=P_1 \cap \cdots \cap P_k$ for some prime ideals $P_j \subset R$, and so we have $J^{(r)}=P_1^{(r)} \cap \cdots \cap P_k^{(r)}$. Proposition \[prop\_properties\_join\]$(i)$ implies $J \star \mathfrak{m}^r = \left(P_1 \star \mathfrak{m}^r\right) \cap \cdots \cap \left(P_k \star \mathfrak{m}^r \right)$. Therefore, to finish the proof, it suffices to consider the case where $J=P$ is a prime ideal. The Zariski-Nagata Theorem implies $\, P^{(r)} = \big\lbrace f \in R : \partial_{\mathbf{x}}^\alpha \bullet f \in J \; \text{ for all }\; \lvert \alpha \rvert \le r-1 \big\rbrace $. The conclusion now follows from part $(i)$ applied to $M=\mathfrak{m}^r$. This establishes Theorem \[thm:ourZN\]. Decomposition and Fusion in a Numerical Future {#sec8} ============================================== This closing section takes the perspective of applied and computational mathematics. We consider a system of polynomial equations over the complex numbers ${\mathbb{C} }$, viewed as an ideal $I$ in the polynomial ring $R = {\mathbb{C} }[x_1,\ldots,x_n]$. This ideal has a minimal primary decomposition $$\label{eq:primdeco} I \,\,\, = \,\,\, Q_1 \,\cap \, Q_2 \,\cap \, \cdots \,\cap \, Q_s .$$ Each associated prime $P_i = \sqrt{Q_i}$ defines an irreducible variety $X_i = V(P_i)$ in ${\mathbb{C} }^n$. Solving the equations means identifying the varieties $X_i$ corresponding to the associated primes $P_i$. Computing the primary decomposition (\[eq:primdeco\]) from generators of $I$ thus refines the problem of solving polynomial systems. Algorithms for this task are a well-developed subject in computer algebra [@DECKER]. However, most studies focus on the irreducible components $X_i$ and the associated primes $P_i$, and they pay less attention to the primary ideals $Q_i$ themselves. The past decade has seen significant advances in numerical algebraic geometry [@BATES], and this has led to the design of numerical techniques for primary decomposition [@KRONE; @LEYKIN]. A paramount ingredient is the identification of all minimal primes $P_i$ from the generators of $I$. Algorithms and implementations for this are now well-established; see e.g. [@BATES Chapter 8]. In the output, each irreducible variety $X_i$ is represented by a finite [*witness set*]{} of the form $X_i \cap L_i$, where $L_i$ is a general affine-linear subspace of dimension $c_i = {\rm codim}(X_i)$ in ${\mathbb{C} }^n$. Numerical identification of embedded primes $P_i$ is more subtle. This topic was pioneered by Krone and Leykin [@KRONE; @LEYKIN] who proposed algorithms based on a technique known as [*inflation*]{}. However, the concluding paragraph in [@KRONE] indicates that more work is needed. Furthermore, their articles do not address the description of the primary ideals $Q_i$ in (\[eq:primdeco\]). The following definitions pave the way for future numerical algorithms. By Theorem \[thm:main\], each primary ideal $Q_i$ is encoded by a pair $(X_i,\mathfrak{A}_i$) where $\mathfrak{A}_i$ is an $m_i$-dimensional ${\mathbb{F}}_i$-vector subspace of ${\mathbb{F}}_i \otimes_R D_{n,c_i}$, where $X_i = V(P_i) = V(Q_i)$ and ${\mathbb{F}}_i $ denotes the field of fractions of $R/P_i$. The numerical representation of the prime ideal $P_i$ or the associated function field ${\mathbb{F}}_i$ is the same as that of $X_i$, namely it is simply a witness set as in [@BATES Chapter 8]. The space $\mathfrak{A}_i$ provides a set of Noetherian operators $A_{ij}({\bf x},\partial_{\bf x})$ for $Q_i$, where $j=1,2,\ldots,m_i$. We propose to use (\[eq:fromAtoQ\]) as the numerical encoding of primary ideals in future algorithms: $$\label{eq:fromAtoQnum} Q_i \,\, = \,\, \{ \, f \in R \,: \, A_{ij} \bullet f \,\,\text{vanishes on} \,\, X_i \, \,\text{for all} \,\, j \,\}.$$ Here $A_{ij}$ is an element in the relative Weyl algebra $D_{n,c_i}$ and its coefficients are given in floating point arithmetic. Likewise, the vanishing condition in (\[eq:fromAtoQnum\]) is meant to be inexact. \[def:numprimdec\] Given an ideal $I$ in $R = {\mathbb{C} }[x_1,\ldots,x_n]$, we define a [*numerical primary decomposition*]{} of $I$ to be a list $(X_1,\mathfrak{A}_1), \ldots, (X_s,\mathfrak{A}_s)$ of representations of primary ideals, where the $X_i$ are precisely the irreducible varieties that are associated to $I$, and we have $$\label{eq:fusionI} I \,\, = \,\, \{ \, f \in R \,: \, A \bullet f \,\,\,\text{vanishes on}\, \, X_i\, \,\,\text{for all $A \in \mathfrak{A}_i $ and all $i=1,\ldots,s$} \,\}.$$ By an abuse of notation, here each $\mathfrak{A}_i$ is also identified with an appropriate finite subset of Noetherian operators for $Q_i$. If $X_i$ is a geometric component then this subset is simply obtained from a basis of the relevant $R$-bi-module in part (d) of Theorem \[thm:main\], and its cardinality is the multiplicity $m_i$ of the primary ideal $Q_i$. However, if $X_i$ is an embedded component, say $X_i \subset X_j$, then we may use a subset of cardinality strictly less than $m_i$. Let $n=2$ and $I = \langle x_1^3, x_1^2 x_2^2\rangle\, = \, \langle x_1^2 \rangle \,\cap \, \langle x_1^3, x_2^2 \rangle$. A numerical primary decomposition consists of $(X_1,\mathfrak{A}_1)$ and $(X_2,\mathfrak{A}_2)$, where $X_1 $ is the $x_2$-axis with $\mathfrak{A}_1 = \{ 1, \partial_1 \}$, and $X_2 = \{(0,0)\}$ with $\mathfrak{A}_2 = \{ \partial_1^2 \partial_2, \partial_1^2 , \partial_1 \partial_2, \partial_2 \}$. Note that $|\mathfrak{A}_2|=4 < 6 = m_2 = {\rm mult}(\langle x_1^3,x_2^2\rangle) $. The computation of a numerical primary decomposition should be carried out by combining existing methods for numerical irreducible decomposition [@BATES; @KRONE; @LEYKIN] with an appropriate adaptation of Algorithm \[alg:forward\]. For each associated irreducible variety $X_i$ one must identify the inverse system in Step 2 using linear algebra over the function field ${\mathbb{F}}_i$ of the component $X_i$. Linear algebra over ${\mathbb{F}}_i$ is to be carried out not from equations but from the witness set alone. One task that arises naturally in this setting is the converse to primary decomposition. This process, which we propose to call [*primary fusion*]{}, amounts to combining a finite collection of primary ideals by their intersection. Let $Q_1$ and $Q_2$ be primary ideals in $R$, encoded by pairs $(X_1,\mathfrak{A}_1)$ and $(X_2,\mathfrak{A}_2)$ as above. Here $\mathfrak{A}_i$ is an $R$-bi-submodule of ${\mathbb{F}}_i \otimes_R D_{n,c_i}$. The first case to consider is when the underlying varieties agree, so $X_1 = X_2$ with $c = c_1 = c_2$. If $Q_1$ and $Q_2$ are $P$-primary ideals, then $Q_1 \cap Q_2$ is also $P$-primary. Its bi-module of Noetherian operators in ${\mathbb{F}}\otimes_R D_{n,c}$ is $\,\mathfrak{A}_1 + \mathfrak{A}_2$. This is the primary fusion. Next consider the situation when $P_1 = \sqrt{Q_1}$ and $P_2 = \sqrt{Q_2}$ are distinct. Suppose first that there is no containment between the varieties $X_1 $ and $X_2$. We certify this from their witness sets. In this case, the primary fusion of $(X_1,\mathfrak{A}_1)$ and $(X_2,\mathfrak{A}_2)$ is the union of their representations by Noetherian operators, that is, the primary fusion is (\[eq:fusionI\]) with $s=2$. The most interesting case arises when $X_1 \subset X_2$. The codimensions satisfy $c_1 > c_2$ and coordinates are chosen so that the Noether normalizations are compatible. Note that $\mathfrak{A}_1 \subset {\mathbb{F}}_1 \otimes_R D_{n,c_1} $ and $\mathfrak{A}_2 \subset {\mathbb{F}}_2 \otimes_R D_{n,c_2} $. We wish to replace the ${\mathbb{F}}_1$-vector space $\mathfrak{A}_1$ by a proper subspace in order to turn (\[eq:fusionI\]) into a minimal representation for $Q_1 \cap Q_2$. It would be desirable to develop an algorithm for doing this in practice, not just for two components but for an arbitrary number $s$ of numerically represented primary ideals $(X_i,\mathfrak{A}_i)$. \[prob:fusion\] Develop a practical numerical method for primary fusion in ${\mathbb{C} }[x_1,\ldots,x_n]$. The numerical solution of partial differential equations is a vast area whose importance for the sciences and engineering can hardly be overestimated. In this paper we explored one special aspect, namely systems of homogeneous linear PDE on ${\mathbb{C} }^n$ with constant coefficients. Such PDE are polynomials in the operators $ \partial_{z_1}, \ldots, \partial_{z_n}$, and their solutions are functions $\psi(z_1,\ldots,z_n)$. We seek numerical algorithms for computing and manipulating these $\psi({\bf z})$ via their integral representations (\[eq:anysolution\]), promised to us by Ehrenpreis [@EHRENPREIS] and Palamodov [@PALAMODOV]. These should go well beyond the zero-dimensional case, studied by Gröbner in the 1930’s. The given PDE form an ideal $I$ in the polynomial ring $R$. We view this input and the desired output in the spirit of numerical algebraic geometry [@BATES]. Exploiting the primary decomposition (\[eq:primdeco\]), our task is to numerically compute the objects of Theorem \[thm:Palamodov\_Ehrenpreis\] for each primary ideal $Q_i$. The varieties $X_i$ are given by witness sets. These need to be enhanced by measures $\mu_{ij}$ for the integral representation (\[eq:anysolution\]). The key algebraic objects are the Noetherian multipliers $B_{ij}({\bf x},{\bf z})$. Their construction is described in Step 3 of Algorithm \[alg:forward\], but this must now be done in a numerical setting. Moreover, to combine solutions $\psi_1({\bf z})$ and $\psi_2({\bf z})$ whose supports are nested, say $X_1 \subset X_2$, we also need primary fusion (Problem \[prob:fusion\]). The problem of solving linear PDE with constant coefficients was discussed in [@STURMFELS_SOLVING Chapter 10]. The author of [@STURMFELS_SOLVING] worked out several nice examples, like the one of page 144, but he was unable to go further, because he lacked the necessary tools from commutative algebra. Overcoming that barrier is precisely the contribution of the present paper. We here develop the tools from commutative algebra that were needed to advance [@STURMFELS_SOLVING Chapter 10]. Theorem \[thm:main\] offers a new characterization of primary ideals and their differential equations. This leads to Algorithms \[alg:forward\] and \[alg:backward\], and these lay the foundation for future development of the Ehrenpreis-Palamodov Fundamental Principle within numerical algebraic geometry. 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--- abstract: | In cyber-physical systems, software may control safety-significant operations. This report discusses a method to structure software testing to measure the statistical confidence that algorithms are true to their intended design. The subject matter appears in two main parts: theory, which shows the relationship between discrete systems theory, software, and the actuated automaton; and application, which discusses safety demonstration and indemnification, a safety assurance metric. The recommended form of statistical testing involves sampling algorithmic behavior in a specific area of safety risk known as a hazard. When this sample is random, it is known as a safety demonstration. It provides evidence for indemnification, a statistic expressing an assured upper bound for accident probability. The method obtains results efficiently from practical sample sizes. Keywords: software, safety, hazard, demonstration, operational profile, automata, confidence, statistics author: - '\' title: Software Safety Demonstration and Indemnification --- \[section\] \[theorem\][Lemma]{} \[theorem\][Corollary]{} \[theorem\][Conjecture]{} \[theorem\][Fact]{} \[theorem\][Definition]{} Prologue ======== Copyright {#S:COPYRIGHT} --------- This document may be freely copied or modified in accordance with the Creative Commons Attribution license[^1]. Executive summary {#S:EXECUTIVE_SUMMARY} ----------------- In systems of integrated hardware and software, the intangible nature of software raises the question of fitness in roles bearing safety risk. Such a safety risk in software, known as a hazard, is a region of code involving safety constraints (requirements) necessitating some degree of verification. Hazards are identified and monitored by safety engineers, and possess hypothetical (threatened) frequency and severity ratings. During its development, potentially hazardous software merits not only rigorously controlled general engineering process, but also quantitative assurance of hazards within particular products. ### Approach {#S:APPROACH} The topic of this essay is assuring the interplay between safety constraints (requirements) and software control. Software is appreciated as a branching process whose permutations are intractably numerous to test exhaustively. Barring exhaustive testing, statistical verification remains an option. The degree of statistical verification will be expressed as residual risk, a contravariant quantity. A software item’s total risk has many constituents. For instance, any software communicating with an operator runs human factors risk. Statistical safety risk, one constituent of total risk, focuses on hazardous code. Code is potentially hazardous if its statistical risk (numerical product of frequency of execution, probability of error, and expected safety loss per error) is sufficiently high. The subject matter results from applying standard mathematics to a well-known (but cloudy) problem. It is organized according to a mathematized version of the Joint Software Systems Safety Engineering Handbook[@DD10] of the United States Department of Defense (2010). This mathematization affords a deeper structural view of safety engineering. This view inspires a unification of that document’s risk management goals, and exerts commonality against its disparate hardware and software risk disciplines. ### Synopsis {#S:SYNOPSIS} A hazard in software is a region of code involving safety requirements, whose logical correctness is essential to *safe* operation (hazards do not embrace all forms of software error). This condition motivates some degree of formal verification of correctness. Hazards are measured according to their statistical risk, which is the numeric product of three factors associated with a software point. First is the point’s frequency of execution. Second is the probability of encountering error during execution of a code trajectory that reaches the point. Third is the point’s severity, its safety consequence (loss) per error. Software safety assurance may be accomplished via management of statistical risk. It is organized into two phases. First is guestimation, which uses expert opinion to yield a rough ranking of hazard risks, based on the three constituents. The subject of this essay is the approximation phase which follows, producing refined post-development risk for each hazard. This refinement, following hardware practice, is known as a residual risk. Data for calculation of each residual risk is drawn from a collection of specially constructed tests called a demonstration. Because error is associated with software sequences rather than points, demonstrations exercise a variety of approaching trajectories. Each demonstration does produce a maximum-likelihood estimate of the probability of walking into error, but this figure isn’t useful because it is usually zero. Define the indifference upper bound as the upper bound at 50% confidence, so the odds of underestimation balance those of overestimation. The indifference upper bound yields unbiased assurance. Indemnification is the risk level assured by the indifference upper bound on proportion failing some test of a demonstration. The indifference upper bound, which is non-zero, functionally replaces the maximum-likelihood estimate. Owing to its definition as a confidence upper bound, indemnification is also a quality assurance metric on completeness of safety testing relative to risk level. This essay proposes a re-unification of hardware and software risk, prescribing that statistical risk become the common standard bearer. ### Significance {#S:SIGNIFICANCE} Profound difference exists between this essay’s proposal and current standards such as MIL-STD-882E and its companion Joint Software Systems Safety Engineering Handbook. Present adherents of MIL-STD-882E must break new procedural ground if they intend to evaluate statistical risk. The protocol confuses statistical assurance with other techniques for design vetting. Perhaps in an effort to encompass both, the standard’s analysis describes a hierarchy for software based on safety impact: potential human intervention, redundancy, or level of safety responsibility. This protocol’s measure is a hierarchy of discrete categories rather than a continuum variable. It may enable some types of analysis, but it renders statistical risk assessment impossible. These standards modify the definition of risk, preferring to introduce separate risk concepts for hardware and software. According to the military standard, statistical risk exists only for hardware, and is consequently lost for software. This essay proposes a re-introduction of statistical risk to software, with the result that hardware and software risks become interchangeable in meaning. Apologies {#S:APOLOGY} --------- This essay is not rendered to academic standards of quality; it benefits from no formal literature search and was written in isolation. The experienced reader may find terms in nonstandard context. The author has strived to maintain consistency, but admits deficiency in standardization of terminology. The author apologizes for resulting inconvenience. The author also apologizes that the concepts discussed here are nascent. Difficult engineering must be accomplished before a mature technology is available for commercialization. The author features mathematics centrally,[^2] presuming undergraduate background and providing necessary computer science. This approach risks estranging many worthy engineering readers; however, a mathematical foundation is necessary. This essay serves that need. Informal introduction --------------------- ### Hall’s definitions {#S:SYSTEMS} The concept of *system* is intuitively obvious but describing its analytical properties is tricky. A famous example appeared in Hall’s 1962 treatise on systems engineering methodology [@aH62 p. 60ff]. Hall proposes succinct definitions of the terms *system* and *environment*: - A system is a set of objects with relationships between the objects and between their attributes. - For a given system, the environment is the set of all objects outside the system: (1) a change in whose attributes affect the system and (2) whose attributes are changed by the behavior of the system. These definitions allow a component to belong either to the system or the environment, because Hall’s definitions are ambiguous (different phraseology is used that is actually equivalent). Our regimen modifies Hall’s historical definitions to remove ambiguity; systems will be regarded as all-inclusive. From the standpoint of relevant influences, there simply is no “outside” influence. We clarify that a system is characterized as a sequence of stimulus and response. Below “component” is a synonym for “object.” These descriptions still suffer some circularity: - A system is the set of *all* components having attributes, changes to which affect the system’s response. - The environment is the set of all components *inside* the system whose attributes are *not* affected by the system’s response. In summary, the environment affects the system’s response, but the system response does not affect the environment’s attributes. Factors outside the system may influence the environment’s attributes. ### Classification {#S:INTRO_CLASSIFICATION} In a system, the terms mechanism, construct, and model have specially differentiated meaning. - *Mechanisms* are abstractions, not necessarily separable, whose structure emulates all behaviors of a given phenomenon. - *Constructs* are isolatable substructures of a mechanism, for examining particular behaviors. - *Models* interpret a behavior of a construct in terms of alternate infrastructure. *Exempli gratia*, hardware and software are mechanisms and operational profiles are constructs, while safety risk is a model. A description of major mechanisms, constructs, and models follows. #### Hardware mechanism {#S:INTRO_HARDWARE} The dynamics of hardware components is portrayed as constrained real time trajectories over a state space. A trajectory is a mapping from time into state space. A constraint relation is an alternative expression for what is familiar as an equation or inequality of state; it is merely a substitute for an equivalent equation. It is characteristic of systems that at any time, intersecting constraints delimit apparently independent choices so that just one is valid. Interacting constraints endow hardware with capabilities. Constraints can be classified according to their engineering significance. A violated safety constraint jeopardizes life, health, equipment, or surroundings. #### Software mechanism {#S:INTRO_SOFTWARE} Theoretical investigations of discrete reactive systems[^3] (or software) can be accomplished using a simple substitute for programming languages: the automaton. Automata are purely mechanistic structures positioned in the machine/language spectrum somewhere between the Turing machine and the Gurevich abstract state language (ASM). Beside its adequacy for examining theory, automata avoid selection of a preferred programming language, which would unnecessarily particularize concepts intended to be general. Automata perform work in discrete units called steps. A sequence of steps is further known as a walk. This essay presents the actuated automaton, a variant form whose work is deterministic conditional sequencing and application of instructions. Instructions are represented by mathematical morphisms, collectively known as functionalities. The order of these functionalities is governed by the actuated automaton through its state. Iteration of an actuated automaton emulates an operating program. #### Reactive mechanism {#S:INTRO_REACTIVE} Reactive systems characteristically need some means to transfer external stimuli. The reactive mechanism contains structures enabling the hardware and software mechanisms to inter-operate cohesively. The nature of time differs between the two; time is a continuum in hardware while it is discrete in software. The clock synchronization permits integration by specifying an ordered cross-reference between discrete and real time. Remaining is need for inter-mechanism communication. Two forms exist: - Sensors convey information about the hardware environment to the software mechanism. Using the clock synchronization, a real-time trajectory is sampled into a sequence of events. - Transducers map a point of the software state into a trajectory in hardware. This trajectory is called a control. #### Cone construct {#S:INTRO_CONE} The actuated automaton has a generalized inverse called the converse. Through reiteration, the converse constructs a partially ordered set (poset) of effects and potential causes. This poset is not linear because an effect may have more than one preceding cause. A cone is the result of decomposing the poset into constituent chains called reverse walks. Viewed as forward walks (reversing the reverse walks), these chains are ordinary sequences of causes and consequent effects. The collection of forward chains converges to a point known as the crux, while the cone diverges from the same point. One subcomponent of a cone is its edge, which is the collection of steps radially opposite the crux. #### Operational profile construct {#S:INTRO_OP_PROFILE} As previously mentioned, automata accomplish work in units called steps. An operational profile is a measure of a step’s excitation probability relative to a reference set of steps. The [reference set of steps]{} itself historically represented a software usage pattern, so it sought to resemble the natural mix of functionalities in deployed software. This idea is abstracted to a potentially purposeless reference set, but the software usage pattern remains important. From the usage pattern, along with the automaton’s static logic, arises the very notion of probability. An operational profile may be applied to the edge of a cone. #### Safety risk model {#S:INTRO_RISK} Accidents occur haphazardly with varying frequency and severity. In the context of software, risk expresses the potential impact of algorithmic design errors. Since its true extent is unknown, software safety risk is expressed as a statistical hypothesis. The compound Poisson process is a model simulating discrete event-based losses that accumulate with passing time. It offers the advantage of independent parameterization of the loss’ intensity (frequency) and severity. Indemnification is statistical assurance of software safety. ### Principle of emergence {#S:PRINCIPLE_OF_EMERGENCE} Emergence [@wW14e] is a broad principle of physics describing a process whereby larger entities possessing a property arise through interactions between simpler entities that themselves do not exhibit the property. Particularized to software testing, the [principle of (weak) emergence]{} is that erroneous software can do no actual harm until certain of its values emerge from the realm of digital logic into a physical subsystem. This principle inquires both into mechanics of transduction, and how transducible values come into being. The automaton of the software mechanism answers the latter question. If software hazard is to be evaluated starting at points of transduction and proceeding backwards through internal logic, then the automaton must support reverse inference – meaning reversed in computational order, from final conclusion to possible premise (see §\[S:INTRO\_CONE\]). CHOICE FORK {#S:CHOICE_FORK} ----------- Chapter 2 (Discrete Systems Theory) details relationships between systems theory and automata. From a mathematical standpoint the material is necessary, but there are readers for whom this chapter would duplicate existing knowledge. After verifying their understanding of operational profiles, section \[S:OPERATIONAL\_PROFILE\_SECTION\], they are invited to skip forward to Chapter 3. Chapter 2 is summarized here to decide whether to skip it. Software is described using a triad of structures: the process, the procedure, and the path; not all are independent. Rudiments underlying these structures consist of ensembles and Cartesian products. Walks, the actuated automaton, converse automata, reverse walks, and cones follow. Those desiring detailed introduction to fundamentals may access Appendix \[Ch:GROUNDWORK\], which reviews groundwork and notation used here. Its highlights include that an ensemble is a mapping from a set of stimuli into a set of responses. Ensembles are denoted by uppercase Greek letters such as $\Psi$. The general Cartesian product of an ensemble, called a choice space, is denoted ${\prod{\Psi}}$. Discrete systems theory ======================= Discrete systems theory (software) is identified with the actuated automaton. Process ------- Chains of stimulus and response characterize reactive discrete systems. In this chain, successive links are not independent: the response effected in one link feeds forward into the stimulus of the following link. For instance, in a system of cog-wheels and escapements, gear train movement accomplished in one stage of operation becomes input to the next. A formalism called a process captures this notion of sequential inheritance. We assemble processes from a simple unit called the frame, which is two-part structure consisting of starting and ending conditions. A process is a sequence of frames such that the starting condition of each frame subsumes the ending condition of its predecessor frame. Interpreted in systems language, a frame’s starting condition is a stimulus and its ending condition is a response. Current response re-appears as part of future stimulus. Definitions of *ensemble* and related basic concepts appear in Groundwork, Appendix \[S:ENSEMBLE\] ff. \[D:BASIS\] The pair of ensembles $\langle \Psi, \Phi \rangle$ is a *basis* if $\Phi \subseteq \Psi$. It is necessary to represent states (variables) which are used but not set – so-called [volatile]{} variables. For example, such variables can hold the transient values of sensors. The remainder $\Psi \setminus \Phi$ is the generating ensemble of volatile variables (see terminology following definition \[D:CHOICE\_SPACE\]). \[D:FRAME\_SPACE\] The *frame space* ${\mathbf{F}}$ of basis $\langle \Psi, \Phi \rangle$ is the set ${\prod{\Psi}} \times {\prod{\Phi}}$. A member ${\mathbf{f}} \in {\mathbf{F}}$ is a *frame*. Let ${\mathbf{f}} = (\psi, \phi) \in {\prod{\Psi}} \times {\prod{\Phi}}$ be a frame. The choice $\psi \in {\prod{\Psi}}$ is the frame’s starting condition (abscissa) and $\phi \in {\prod{\Phi}}$ is the frame’s ending condition (ordinate). Two frames may be related such that the ending condition of one frame is embedded within the next frame’s starting condition. This stipulation is conveniently expressed as a mapping restriction: \[D:CONJOINT\] Let $\langle \Psi, \Phi \rangle$ be a basis with frames ${\mathbf{f}} = (\psi, \phi)$, ${\mathbf{f}}\,' = (\psi', \phi') \in {\prod{\Psi}} \times {\prod{\Phi}}$. Frame ${\mathbf{f}}$ *conjoins* frame ${\mathbf{f}}\,'$ if ${{\psi'}\negmedspace\mid\negmedspace{{{\operatorname{dom}{\Phi}}}}} = \phi$. \[N:SEQUENCE\_NOTATION\] A sequence in a set $S$ is some mapping $\sigma \colon {\mathbb{N}}\to S$ – that is, $\sigma \in S^{\mathbb{N}}$. The anonymous sequence convention allows reference to a sequence using the compound symbol $\lbrace s_n \rbrace$, understanding $s \in S$. Formally, the symbol $s_i$ denotes that term $(i, s_i) \in \lbrace s_n \rbrace$. The convention is clumsy expressing functional notation; for instance $s_i = \lbrace s_n \rbrace(i)$ means $i \stackrel{\lbrace s_n\!\rbrace}{\mapsto} s_i$. \[D:SUCCESIVELY\_CONJOINT\] Let $\langle \Psi, \Phi \rangle$ be a basis with sequence of frames $\lbrace {\mathbf{f}}_n \rbrace \colon {\mathbb{N}}\to {\prod{\Psi}} \times {\prod{\Phi}}$. The sequence is *successively conjoint* if ${\mathbf{f}}_i$ conjoins ${\mathbf{f}}_{i+1}$ for each $i \geq 1$. \[D:PROCESS\] With $\langle \Psi, \Phi \rangle$ a basis, a *process* is a successively conjoint sequence of frames ${\mathbb{N}}\to {\prod{\Psi}} \times {\prod{\Phi}}$. \[D:ABSCISSA\_PROJECTION\] Let $\langle \Psi, \Phi \rangle$ be a basis with frame space ${\mathbf{F}} = {\prod{\Psi}} \times {\prod{\Phi}}$. Define the *abscissa* projection ${{\operatorname{absc}{}}}: {\mathbf{F}} \to {\prod{\Psi}}$ by $(\psi, \phi) \stackrel{{{\operatorname{absc}{}}}}{\mapsto} \psi$. Define the *ordinate* projection ${{\operatorname{ord}{}}}: {\mathbf{F}} \to {\prod{\Phi}}$ by $(\psi, \phi) \stackrel{{{\operatorname{ord}{}}}}{\mapsto} \phi$. \[D:PERSISTENT\_VOLATILE\_COMPONENTS\] Let $\langle \Psi, \Phi \rangle$ be a basis with persistent-volatile partition $\Psi = \Phi\Xi$ (see appendix §\[S:STATE\_EVENT\_PRTN\]). Suppose ${\mathbf{f}}$ is a frame in ${\prod{\Psi}} \times {\prod{\Phi}}$. The *reactive* state of frame ${\mathbf{f}}$ is $\psi = \phi \xi = {{\operatorname{absc}{{\mathbf{f}}}}}$. The *event* or *volatile* excitation state of frame ${\mathbf{f}}$ is $\xi = {{({{\operatorname{absc}{{\mathbf{f}})}}}}\negmedspace\mid\negmedspace{{{\operatorname{dom}{\Xi}}}}}$. Similarly, the *persistent* state of frame ${\mathbf{f}}$ is $\phi = {{({{\operatorname{absc}{{\mathbf{f}})}}}}\negmedspace\mid\negmedspace{{{\operatorname{dom}{\Phi}}}}}$. Process concepts interpret into systems language. The reactive space ${\prod{\Psi}}$ contains the system stimulus. Sequential conjointness allows circumstantial interpretation of the choice space ${\prod{\Phi}}$. It is the system’s response in the context of the frame ending condition. To place ${\prod{\Phi}}$ in context of the frame’s reactive state, the Cartesian product ${\prod{\Phi}} = \prod ({{\Psi}\negmedspace\mid\negmedspace{{{\operatorname{dom}{\Phi}}}}}) = (\thinspace\prod\Psi) \mid {{\operatorname{dom}{\Phi}}}$ \[by theorem \[T:CHC\_RSTR\_EQ\_RSTR\_CHC\]\] is the persistent state space. Using this nomenclature, sequential conjointness is summarized that each frame’s response becomes the next frame’s persistent state, symbolically ${{\operatorname{ord}{{\mathbf{f}}_{\,i}}}} = {{{{\operatorname{absc}{{\mathbf{f}}_{\,i+1}}}}}\negmedspace\mid\negmedspace{{{\operatorname{dom}{\Phi}}}}}$. Procedure --------- The procedure is useful to portray a process frame as a transformation from the stimulus space to the response space. To distinguish such transformations from other mappings, we use the special term [functionality]{} and stipulate that the collection of functionalities is a finite set called a catalog. The term [catalog]{} will later be applied to resource sets identified with an automaton. ### Functionality {#S:FUNCTIONALITY} The functionality generalizes the frame. If ${\mathbf{f}}_i = (\psi_i, \phi_i)$ is the $i^{\text{th}}$ process frame, this concept permits writing $\phi_i = {{\mathit{f}}}_i(\psi_i)$, where ${\mathit{f}}_i$ is some functionality belonging to catalog ${\mathscr{F}}$. \[D:FUNCTIONALITY\] A *functionality* is a mapping whose domain and codomain are choice spaces (definition \[D:CHOICE\_SPACE\]), with the codomain a subspace (definition \[D:SUBSPACE\]) of the domain. \[L:BASIS\_FUNCTIONALITY\] Let $\langle \Psi, \Phi \rangle$ be a basis. Any mapping ${\mathit{f}} \colon {\prod{\Psi}} \to {\prod{\Phi}}$ is a functionality. As a basis, definition \[D:BASIS\] establishes that $\Psi$ and $\Phi$ are ensembles with $\Phi \subseteq \Psi$. Since $\Psi$ and $\Phi$ are ensembles, definition \[D:CHOICE\_SPACE\] asserts that ${\prod{\Psi}}$ and ${\prod{\Phi}}$ are choice spaces. Theorem \[T:SUBSET\_IFF\_SUBSPACE\] provides that ${\prod{\Phi}}$ is a subspace of ${\prod{\Psi}}$ because $\Phi \subseteq \Psi$. By virtue of ${\mathit{f}} \in {{\prod{\Phi}}}^{{\prod{\Psi}}}$, then ${\mathit{f}} \colon {\prod{\Psi}} \to {\prod{\Phi}}$ is a mapping from one choice space to another, which is a subspace of the first. These conditions satisfy the premises of definition \[D:FUNCTIONALITY\]. In its programming sense, the term [function]{} will not be used here. A mathematical functionality differs from a software function; functionalities lack arguments. By virtue of its calling protocol, a programming function is effectively a class of functionalities. ### Procedure {#S:PROCEDURE} Procedures are sequences in a finite set of functionalities: \[D:CATALOG\_OF\_FUNCTIONALITY\] Let $\langle \Psi, \Phi \rangle$ be a basis. A finite subset ${\mathscr{F}} \subseteq {{\prod{\Phi}}}^{{\prod{\Psi}}}$ is a *catalog* of functionality. \[D:PROCEDURE\] Let $\langle \Psi, \Phi \rangle$ be a basis and ${\mathscr{F}} \subseteq {{\prod{\Phi}}}^{{\prod{\Psi}}}$ be a catalog of functionality. A *procedure* is a sequence $\lbrace {\mathit{f}}_n \rbrace \colon {\mathbb{N}}\to {\mathscr{F}}$. After noting the functionality’s successful generalization of the frame, the next question is whether the procedure correspondingly abstracts the process. We find that not all processes are [computable]{} as procedures based on a finite number of functionalities. ### Covering {#S:COVERING} The relation holding between frame ${\mathbf{f}} \in {\prod{\Psi}} \times {\prod{\Phi}}$ and functionality ${\mathit{f}} \colon {\prod{\Psi}} \to {\prod{\Phi}}$ is membership: either ${\mathbf{f}} \in {\mathit{f}}$ or ${\mathbf{f}} \notin {\mathit{f}}$. \[D:BASIC\_COVERING\] Let ${\mathbf{f}}$ be a frame and ${\mathit{f}}$ be a functionality. The functionality *covers* the frame if ${\mathbf{f}} \in {\mathit{f}}$ (that is, ${\mathbf{f}} = (\psi, \phi) = (\psi, {\mathit{f}}(\psi))$). \[D:COVERING\_PROCEDURE\] Let $\lbrace {\mathbf{f}}_n \rbrace$ be a sequence of frames and $\lbrace {\mathit{f}}_n \rbrace$ be a procedure. The procedure *covers* the sequence of frames if ${\mathbf{f}}_i \in {\mathit{f}}_i$ for each $i \geq 1$ (that is, ${\mathbf{f}}_i = (\psi_i, \phi_i) = (\psi_i, {\mathit{f}}_i(\psi_i))$). Any procedure covers some process. \[T:PROCEDURE\_COVER\_PROCESS\] Let $\langle \Psi, \Phi \rangle$ be a basis with persistent-volatile partition $\Psi = \Phi\Xi$ and catalog of functionality ${\mathscr{F}}$. Suppose $\lbrace {\mathit{f}}_n \rbrace \colon {\mathbb{N}}\to {\mathscr{F}}$ is a procedure. For each choice of persistent state $\varphi \in {\prod{\Phi}}$ and volatile excitation sequence $\lbrace \xi_n \rbrace \in {(\,{\prod{\Xi}})}^{{\mathbb{N}}}$ there is a process $\lbrace {\mathbf{f}}_n \rbrace \colon {\mathbb{N}}\to {\prod{\Psi}} \times {\prod{\Phi}}$ such that the procedure covers the process. The persistent state $\varphi$ and volatile excitation $\lbrace \xi_n \rbrace$ are given. Inductively define sequence $\lbrace \phi_n \rbrace$ using base clause $\phi_1 = \varphi$ and recursive clause $\phi_{i+1} = {\mathit{f}}_i(\phi_i\xi_i)$ for $i \ge 1$ (dyadic notation, definition \[D:DYADIC\_ENSEMBLE\_PRODUCT\] ff). We first show that the sequence $\lbrace \phi_n \rbrace$ lies in ${\prod{\Phi}}$. Through $\varphi$, definition \[D:PERSISTENT\_VOLATILE\_COMPONENTS\] provides that $\phi_1 \in {\prod{\Phi}}$. For the iterative part, the hypothesis $\phi_i \in {\prod{\Phi}}$ implies that $\phi_i\xi_i \in {\prod{\Psi}}$ by Theorem \[T:DYADIC\_CHOICE\_PROD\]. Since ${\mathit{f}}_i$ maps ${\prod{\Psi}}$ to ${\prod{\Phi}}$, then ${\mathit{f}}_i(\phi_i\xi_i) = \phi_{i+1} \in {\prod{\Phi}}$. This chain of implication concludes that $(\phi_i \in {\prod{\Phi}}) \Rightarrow (\phi_{i+1} \in {\prod{\Phi}})$. Use $\lbrace \phi_n \rbrace$ and $\lbrace \xi_n \rbrace$ to define another sequence $\lbrace {\mathbf{f}}_n \rbrace$ by setting ${\mathbf{f}}_i = (\phi_i\xi_i, \phi_{i+1})$. Since $\phi_i\xi_i \in {\prod{\Psi}}$ and $\phi_{i+1} \in {\prod{\Phi}}$, then $(\phi_i\xi_i, \phi_{i+1}) \in {\prod{\Psi}} \times {\prod{\Phi}}$ so $\lbrace {\mathbf{f}}_n \rbrace$ is a sequence of frames. Note that ${\mathbf{f}}_i = (\phi_i\xi_i, \phi_{i+1})$ and ${\mathbf{f}}_{i+1} = (\phi_{i+1}\xi_{i+1}, \phi_{i+2})$. In this case, the sequence is successively conjoint (definitions \[D:CONJOINT\] and \[D:SUCCESIVELY\_CONJOINT\]) because ${{\phi_{i+1}\xi_{i+1}}\negmedspace\mid\negmedspace{{{\operatorname{dom}{\Phi}}}}} = \phi_{i+1}$. As a successively conjoint sequence of frames, $\lbrace {\mathbf{f}}_n \rbrace$ is a process (definition \[D:PROCESS\]). Since ${\mathbf{f}}_i = (\phi_i\xi_i, \phi_{i+1})$, and by construction $\phi_{i+1} = {\mathit{f}}_i(\phi_i\xi_i)$, then ${\mathbf{f}}_i = (\phi_i\xi_i, {\mathit{f}}_i(\phi_i\xi_i))$ and ${\mathbf{f}}_i \in {\mathit{f}}_i$. By definition \[D:COVERING\_PROCEDURE\], procedure $\lbrace {\mathit{f}}_n \rbrace$ covers process $\lbrace {\mathbf{f}}_n \rbrace$. ### Uncoverable process {#S:UNCOVERABLE_PROCESS} Although any procedure does cover some process, some processes have no covering procedure. See §\[S:UNCOVERABLE\_PROCESS\_APPENDIX\]. Automata -------- The algorithm is conceived as a method to solve problems using a network of mechanistic steps consisting of decisions and contingent actions. An automaton is a formal machine whose architecture of states and transitions concretizes some aspects of the algorithm. The deterministic finite automaton (DFA, see example in §\[S:DFA\]) is a simple structure describing transit-based behavior. However, the DFA leaves unexplained the working mechanism underlying transitions. The DFA’s definition can be modified to effect closer alignment with our notion of software algorithm. The result is the actuated automaton, which mechanizes logic using structure analogous to programming language. An informal analogy between an automaton and a programming language will be proposed at the end of this section. ### Locus {#S:LOCUS} A formalization of the algorithm’s stepwise network of decisions and actions requires some means of indicating one’s place in the overall method, to track what is current and what is next. We provide this in the form of a set of loci, which serve as labels for the [locations]{} implicit in a program or algorithm. \[D:CATALOG\_OF\_LOCI\] A catalog of *loci* is a finite non-empty set $\Lambda$, each member $\lambda$ of which is called a locus. Let $\Lambda$ be a catalog of loci. A *path* is a sequence $\lbrace \lambda_n \rbrace \colon {\mathbb{N}}\to \Lambda$. ### Summary {#S:THREE_P} We now identify the components of a walk, the three fundamental P’s: - A path is a sequence in the catalog of loci $\Lambda$. - A process is a conjoint sequence in the frame space ${\mathbf{F}} = {\prod{\Psi}} \times {\prod{\Phi}}$. - A procedure is a covering sequence in the catalog of functionality ${\mathscr{F}}$. ### Auxiliary mechanisms {#S:AUXILIARY_MECHANISMS} \[D:LOCUS\_ACTUATOR\] Let ${\mathscr{F}}$ be a catalog of functionality on basis $\langle \Psi, \Phi\rangle$. An *actuator* ${\mathsf{a}}$ is a mapping ${\mathsf{a}} \colon {\prod{\Psi}} \to {\mathscr{F}}$ from the process stimulus space to the catalog of functionality. In other words, any ${\mathsf{a}} \in {\mathscr{F}}^{{\prod{\Psi}}}$ is an actuator. \[D:LOCUS\_CATALOG\_OF\_ACTUATION\] A catalog of *actuation* is a non-empty finite set ${\mathsf{A}} \subseteq {\mathscr{F}}^{{\prod{\Psi}}}$ of actuators. With each locus is associated exactly one designated actuator: \[D:LOCUS\_ACTUATOR\_DESIGNATION\] Let the *locator* be a surjective mapping $\ell \colon \Lambda \to {\mathsf{A}}$. To permit each actuator to be located, it is prerequisite that ${\lvert{{\mathsf{A}}}\rvert} \leq {\lvert{\Lambda}\rvert}$ (the number of actuators is less than or equal to the number of loci). \[D:JUMP\_FUNCTION\] Let $\Lambda$ be a catalog of loci. The *jump* function is a mapping $\Delta \colon \Lambda \times {\prod{\Psi}} \to \Lambda$. ### Actuated automaton {#S:ACTUATED_AUTOMATON} \[D:ACTUATED\_AUTOMATON\] The structure ${\mathfrak{A}} = \langle \Psi, \Phi, {\mathscr{F}}\!, {\mathsf{A}}, \Lambda, \ell, \Delta \rangle$ of an *actuated* automaton consists of seven synchronized catalogs: a basis $\langle \Psi, \Phi \rangle$ with catalog of functionality ${\mathscr{F}}$, catalog of actuation ${\mathsf{A}}$, catalog of loci $\Lambda$, locator function $\ell$, and jump function $\Delta$. Automata exist in many varieties. Since the actuated automaton occupies the entire present scope of interest, we forgo mandatory use of the qualifier [actuated.]{} ### Programming language analogy {#S:LANGUAGE_ANALOGY} The automaton $\langle \Psi, \Phi, {\mathscr{F}}\!, {\mathsf{A}}, \Lambda, \ell, \Delta \rangle$ of definition \[D:ACTUATED\_AUTOMATON\] resembles an algorithm written in an elementary programming language. The following comprise the analogy: - the generating ensemble $\Phi$ of the persistent state space represents ordinary program variables; - the remainder ensemble $\Psi \setminus \Phi$ represents volatile external inputs, such as sensors; - functionalities of the catalog ${\mathscr{F}}$ represent blocks of program assignment statements; - actuators in ${\mathsf{A}}$ implement if-then-elsif-else decisions, - the jump function $\Delta$ is a [goto]{} indicating the next point of execution, and - loci in $\Lambda$ are labels serving as [goto-able]{} points of execution. Iteration --------- While leaving undefined the notion of a step in an algorithm, we do formalize it for the automaton. Identifying a step space leads to defining iterative operators, and hence to iteration. ### Step space {#S:STEP_SPACE} The step space underlies automata. It is formed by augmenting a catalog of loci to the building blocks of processes and procedures, namely frames and functionalities. \[D:STEP\_SPACE\] Suppose $\Lambda$ is a catalog of loci. Let basis $\langle \Psi, \Phi\rangle$ underly frame space ${\mathbf{F}} = {\prod{\Psi}} \times {\prod{\Phi}}$ and catalog of functionality ${\mathscr{F}} \subseteq {{\prod{\Phi}}}^{{\prod{\Psi}}}$. A *step space* ${\mathbb{S}}$ is the Cartesian product ${\mathbb{S}} = \Lambda \times {\mathscr{F}} \times {\mathbf{F}}$. The volatile excitation, whose generating ensemble is $\Psi \setminus \Phi$, is intrinsically part of the definition of step space. Lest this implicit fact be forgotten, we shall adopt explicit but redundant notation as reminders. \[D:STEP\_SPACE\_PROJECTION\] Let ${\mathit{s}} = (\lambda,{\mathit{f}},{\mathbf{f}})$ be a member of step space ${\mathbb{S}} = \Lambda \times {\mathscr{F}} \times {\mathbf{F}}$. Define the *locus* projection $\mho_\Lambda \colon {\mathbb{S}} \to \Lambda$ by setting $\mho_\Lambda(\lambda,{\mathit{f}},{\mathbf{f}}) = \lambda$. Similarly define the *frame* and *functionality* projections by $\mho_{\mathbf{F}}(\lambda,{\mathit{f}},{\mathbf{f}}) = {\mathbf{f}}$ and $\mho_{\mathscr{F}}(\lambda,{\mathit{f}},{\mathbf{f}}) = {\mathit{f}}$ respectively. Definition \[D:CONJOINT\] considers the covering relation between a frame and a functionality. *Consistency* is the same principle applied to the context of a step: \[D:CONSISTENT\_STEP\] Let ${\mathit{s}} = (\lambda,{\mathit{f}},{\mathbf{f}})$ be a member of step space ${\mathbb{S}} = \Lambda \times {\mathscr{F}} \times {\mathbf{F}}$. The step ${\mathit{s}}$ is *consistent* if $\mho_{\mathbf{F}}({\mathit{s}}) = {\mathbf{f}} \in {\mathit{f}} = \mho_{\mathscr{F}}({\mathit{s}})$ (that is, its frame is a member of its functionality). #### Sequence projection {#S:SEQUENCE_PROJECTION} \[D:STEP\_SPACE\_WALK\] Let ${\mathbb{S}} = \Lambda \times {\mathscr{F}} \times {\mathbf{F}}$ be a step space and ${\mathscr{I}}$ be a denumerable index set. A *walk* is a sequence ${\mathscr{I}} \to {\mathbb{S}}$ of steps (usually ${\mathscr{I}}$ will be the natural numbers ${\mathbb{N}}$). We revisit the three fundamental P’s (path, procedure, and process – §\[S:THREE\_P\]). A walk in step space decomposes into these three sequences: $\textrm{walk}_{\,i} = (\textrm{path}_{\,i}, \textrm{process}_{\,i}, \textrm{procedure}_{\,i})$. This triple is not logically independent; being so, shorter characterizations of step space exist. However we retain the present representation, favoring its three-element formulation, which covers all possibilities in simple fashion. [\[Extended projection\]]{} \[D:EXTENDED\_PROJECTION\] Let $\lbrace {\mathit{s}}_n \rbrace$ be a walk in step space ${\mathbb{S}} = \Lambda \times {\mathscr{F}} \times {\mathbf{F}}$ and let ${\mathscr{I}}$ be a denumerable index set (usually ${\mathbb{N}}$). Use the locus projection $\mho_\Lambda \colon {\mathbb{S}} \to \Lambda$ of definition \[D:STEP\_SPACE\_PROJECTION\] to construct the sequential *path* projection $\overline{\mho}_\Lambda \colon {\mathbb{S}}^{\mathscr{I}} \to \Lambda^{\mathscr{I}}$ via setting $\overline{\mho}_\Lambda(\lbrace {\mathit{s}}_n \rbrace) = \lbrace (i,\mho_\Lambda({\mathit{s}})) \colon (i,{\mathit{s}}) \in \lbrace {\mathit{s}}_n \rbrace \rbrace$. Similarly define the sequential *process* and *procedure* projections $\overline{\mho}_{\mathbf{F}} \colon {\mathbb{S}}^{\mathscr{I}} \to {\mathbf{F}}^{\mathscr{I}}$ and $\overline{\mho}_{\mathscr{F}} \colon {\mathbb{S}}^{\mathscr{I}} \to {\mathscr{F}}^{\mathscr{I}}$. With $\lbrace x_n \rbrace \colon {\mathscr{I}} \to X$ a sequence in some set $X$, we alternatively denote the sequence’s ${{i}^{\text{th}}}$ term by $x_i = \lbrace x_n \rbrace(i)$. \[L:RELATED\_PROJECTION\] Let $\langle \Psi, \Phi \rangle$ and locus set $\Lambda$ be the bases for step space ${\mathbb{S}} = \Lambda \times {\mathscr{F}} \times {\mathbf{F}}$. Let $\lbrace {\mathit{s}}_n \rbrace$ be a walk ${\mathscr{I}} \to {\mathbb{S}}$. For each $i \in {\mathscr{I}}$, $(\overline{\mho}_\Lambda(\lbrace {\mathit{s}}_n \rbrace))(i) = \mho_\Lambda(\lbrace {\mathit{s}}_n \rbrace(i))$; that is, the ${{i}^{\text{th}}}$ term of the sequential path projection equals the locus projection of the ${{i}^{\text{th}}}$ step. Analogous assertions are true of the remaining sequential projections: $(\overline{\mho}_{\mathbf{F}}(\lbrace {\mathit{s}}_n \rbrace))(i) = \mho_{\mathbf{F}}(\lbrace {\mathit{s}}_n \rbrace(i))$ and $(\overline{\mho}_{\mathscr{F}}(\lbrace {\mathit{s}}_n \rbrace))(i) = \mho_{\mathscr{F}}(\lbrace {\mathit{s}}_n \rbrace(i))$. By definition \[D:EXTENDED\_PROJECTION\], the sequential path projection $\overline{\mho}_\Lambda \colon {\mathbb{S}}^{\mathscr{I}} \to \Lambda^{\mathscr{I}}$ is $\overline{\mho}_\Lambda(\lbrace {\mathit{s}}_n \rbrace) = \lbrace (i,\mho_\Lambda({\mathit{s}})) \colon (i,{\mathit{s}}) \in \lbrace {\mathit{s}}_n \rbrace \rbrace$. The $i^\text{th}$ term of $\overline{\mho}_\Lambda(\lbrace {\mathit{s}}_n \rbrace)$ is $(\overline{\mho}_\Lambda(\lbrace {\mathit{s}}_n \rbrace))(i)$. From set builder notation we observe that the $i^\text{th}$ term of the expression $\lbrace (i,\mho_\Lambda({\mathit{s}})) \colon (i,{\mathit{s}}) \in \lbrace {\mathit{s}}_n \rbrace \rbrace$ is $\mho_\Lambda({\mathit{s}}_i) = \mho_\Lambda(\lbrace {\mathit{s}}_n \rbrace(i))$. Since the sequences are equal, then each of their corresponding terms are equal: $(\overline{\mho}_\Lambda(\lbrace {\mathit{s}}_n \rbrace))(i) = \mho_\Lambda(\lbrace {\mathit{s}}_n \rbrace(i))$. Demonstration is similar for the other two sequential projections. ### Iterative operators {#S:ITERATIVE_OPERATOR} \[D:ITERATIVE\_OPERATOR\] Let ${\mathbb{S}}$ be a step space with basis $\langle \Psi, \Phi \rangle$. Suppose the volatile excitation space $\Psi \setminus \Phi$ is non-empty. A *iterative* operator is a mapping $V \colon {\mathbb{S}} \to {\mathbb{S}}$. #### Disambiguation An iterative operator maps a step space into itself. One element of a step space is a reactive state space, having persistent and volatile components. A functionality maps a reactive state space into the persistent subset of itself. #### Walk of iterative operator {#S:ITERATIVE_OPERATOR_WALK} \[D:ITERATIVE\_OPERATOR\_WALK\] Let $\langle \Psi, \Phi \rangle$ be a basis with step space ${\mathbb{S}} = \Lambda \times {\mathscr{F}} \times (\,{\prod{\Psi}} \times {\prod{\Phi}})$. Suppose $V \colon {\mathbb{S}} \to {\mathbb{S}}$ is an iterative operator, step ${\mathit{s}} \in {\mathbb{S}}$, and $\lbrace \xi_n \rbrace \in (\,{\prod{\Xi}})^{\mathbb{N}}$ is an volatile excitation sequence. Define inductively a sequence of steps by setting ${\mathit{s}}_1 = {\mathit{s}}$ and ${\mathit{s}}_{i+1} = V({\mathit{s}}_i)$ for each $i \ge 1$. The walk of ${\mathit{s}} \in {\mathbb{S}}$ under $V$, assuming sequence of volatile excitation $\lbrace \xi_n \rbrace$, is the walk $\lbrace {\mathit{s}}_n \rbrace$. An iterative operator’s $i^\text{th}$ *iteration* is its walk’s ${(i+1)}^\text{th}$ term. ### Automaton-induced iterative operators {#S:INDUCED_ITERATIVE_OPERATOR} Iteration of automata guarantees properties not necessarily enjoyed by other classes of iterative operators: automata generate consistent steps having conjoint processes. #### Automaton as transformation \[D:ITERATIVE\_TRANSFORM\] Let $\langle \Psi, \Phi \rangle$ be a basis with persistent-volatile partition $\Psi = \Phi\Xi$ and step space ${\mathbb{S}} = \Lambda \times {\mathscr{F}} \times (\,{\prod{\Psi}} \times {\prod{\Phi}})$. Let ${\mathfrak{A}} = \langle \Psi, \Phi, {\mathscr{F}}\!, {\mathsf{A}}, \Lambda, \ell, \Delta \rangle$ be an actuated automaton. Let $\xi \in {\prod{\Xi}}$ be an event stimulus and $(\lambda, {\mathit{f}}, {\mathbf{f}}) = (\lambda, {\mathit{f}}, (\psi, \phi)) \in {\mathbb{S}}$ be a step. The transform $T_{{\mathfrak{A}}}$ induced by ${\mathfrak{A}}$ is $$(\lambda, {\mathit{f}}, (\psi, \phi)) \stackrel{{\mathfrak{A}}}{\mapsto} (\lambda', {\mathit{f}}', (\psi', \phi')),$$ where $$\begin{aligned} \lambda' &= \Delta(\lambda, \psi).&\quad\text{[next locus]}\\ {\mathit{f}}' &= (\ell(\Delta(\lambda, \psi)))({\mathit{f}}(\psi) \xi')&\quad\text{[next functionality]} \\ {\mathbf{f}}\,' &= (\psi', \phi') = ([{\mathit{f}}(\psi) \xi'], [(\ell(\Delta(\lambda, \psi)))({\mathit{f}}(\psi) \xi')]([{\mathit{f}}(\psi) \xi']))&\quad\text{[next frame]}\end{aligned}$$ A partial unfolding of these expressions’ generators clarifies the roles of components in overall mechanism: 1. Current reactive state is $\psi = \phi \xi$. 2. Current locus is $\lambda$. 3. Current actuator is ${\mathsf{a}} = \ell(\lambda)$. 4. Current functionality is ${\mathit{f}} = {\mathsf{a}}(\psi) = (\ell(\lambda))(\psi)$. 5. Current frame is ${\mathbf{f}} = (\psi, {\mathit{f}}(\psi))$. 6. Current step is ${\mathit{s}} = (\lambda, (\psi, (\ell(\lambda))(\psi), {\mathit{f}}(\psi)))$. 7. Next reactive state is $\psi' = \phi' \xi' = {\mathit{f}}(\psi) \xi'$ (by conjointness). 8. Next locus is found through the jump function: $\lambda' = \Delta(\lambda, \psi)$. 9. Next actuator is ${\mathsf{a}}' = \ell(\lambda')$, 10. ${\mathsf{a}}' = \ell(\Delta(\lambda, \psi))$. 11. Next functionality is ${\mathit{f}}\,' = {\mathsf{a}}'(\psi')$, 12. ${\mathit{f}}' = [\ell(\Delta(\lambda, \psi))](\psi')$, 13. ${\mathit{f}}' = (\ell(\Delta(\lambda, \psi)))({\mathit{f}}(\psi) \xi')$. 14. Next frame is ${\mathbf{f}}\,' = (\psi', {\mathit{f'}}(\psi'))$, 15. ${\mathbf{f}}\,' = ([{\mathit{f}}(\psi) \xi'], {\mathit{f'}}([{\mathit{f}}(\psi) \xi']))$, 16. ${\mathbf{f}}\,' = ([{\mathit{f}}(\psi) \xi'], [(\ell(\Delta(\lambda, \psi)))({\mathit{f}}(\psi) \xi')]([{\mathit{f}}(\psi) \xi']))$. 17. Next step is ${\mathit{s}}' = (\Delta(\lambda, \psi), {\mathsf{a}}'(\psi'), (\psi', {\mathit{f'}}(\psi')))$. #### Automaton as iterative operator \[T:AUTOMATON\_OPERATOR\] Let $\langle \Psi, \Phi \rangle$ be a basis with persistent-volatile partition $\Psi = \Phi\Xi$ and step space ${\mathbb{S}}$. Let ${\mathfrak{A}}$ be an automaton. The transform $T_{\mathfrak{A}} \colon {\mathbb{S}} \to {\mathbb{S}}$ induced by ${\mathfrak{A}}$ is an iterative operator (that is, if ${\mathit{s}} \in {\mathbb{S}}$, then $T_{{\mathfrak{A}}}({\mathit{s}}) \in {\mathbb{S}}$). By definition \[D:ACTUATED\_AUTOMATON\], automaton ${\mathfrak{A}}$ consists of components $\langle \Psi, \Phi, {\mathscr{F}}\!, {\mathsf{A}}, \Lambda, \ell, \Delta \rangle$. Other premises are that the iterative operator’s domain is ${\mathbb{S}}$, the persistent-volatile partition $\Psi = \Phi\Xi$, and that $\xi \in {\prod{\Xi}}$. Suppose $\xi \in {\prod{\Xi}}$ and ${\mathit{s}} \in {\mathbb{S}}$. By definition \[D:STEP\_SPACE\] of step space, there exist locus $\lambda \in \Lambda$, functionality ${\mathit{f}} \in {\mathscr{F}} \subseteq {{\prod{\Phi}}}^{{\prod{\Psi}}}$, and frame ${\mathbf{f}} = (\psi, \phi) \in {\prod{\Psi}} \times {\prod{\Phi}}$ such that ${\mathit{s}} = (\lambda,(\psi, \phi), {\mathit{f}})$. Definition \[D:ITERATIVE\_TRANSFORM\] calls for application $(\ell(\lambda))(\phi)$ to define succeeding functionality ${\mathit{f}}'$. Definition \[D:LOCUS\_ACTUATOR\_DESIGNATION\] specifies $\ell \colon \Lambda \to {\mathsf{A}}$, so ${\mathsf{a}} = \ell(\lambda) \in {\mathsf{A}}$ is an actuator. Definition \[D:LOCUS\_ACTUATOR\] specifies ${\mathsf{a}} \colon {\prod{\Phi}} \to {\mathscr{F}}$, so application ${\mathit{f}}' = (\ell(\lambda))(\phi) = {\mathsf{a}}(\phi)$ is a functionality in ${\mathscr{F}}$. Definition \[D:ITERATIVE\_TRANSFORM\] next calls for evaluating $(\psi', \phi') = (\phi\xi, {\mathit{f}}'(\phi\xi))$ as the succeeding frame ${\mathbf{f}}\,'$. By premise $\xi \in {\prod{\Xi}}$. By virtue of its origin as a frame ordinate, $\phi \in {\prod{\Phi}}$. Since $\Psi = \Phi\Xi$, then $\phi\xi$ is a valid dyadic product and $\psi' = \phi\xi \in {\prod{\Psi}}$. Since definition \[D:CATALOG\_OF\_FUNCTIONALITY\] asserts ${\mathscr{F}} \subseteq {{\prod{\Phi}}}^{{\prod{\Psi}}}$ and ${\mathit{f}}' \in {\mathscr{F}}$, then ${\mathit{f}}' \colon {\prod{\Psi}} \to {\prod{\Phi}}$. With $\phi\xi \in {\prod{\Psi}}$, then $\phi' = {\mathit{f}}'(\phi\xi) \in {\prod{\Phi}}$. Hence ${\mathbf{f}}\,' = (\psi', \phi') \in {\prod{\Psi}} \times {\prod{\Phi}}$, and ${\mathbf{f}}\,'$ is a frame. Finally, definition \[D:ITERATIVE\_TRANSFORM\] calls for the succeeding locus as $\lambda' = \Delta(\lambda, \psi)$. Definition \[D:JUMP\_FUNCTION\] specifies the jump function as a mapping $\Delta \colon \Lambda \times {\prod{\Psi}} \to \Lambda$. It is established above that $\lambda \in \Lambda$ and $\psi \in {\prod{\Psi}}$, so $\lambda' = \Delta(\lambda, \psi) \in \Lambda$ is a locus. With locus $\lambda' \in \Lambda$, functionality ${\mathit{f}}' \in {\mathscr{F}}$, and frame ${\mathbf{f}}\,' = (\psi', \phi') \in {\prod{\Psi}} \times {\prod{\Phi}}$, we then summarize that ${\mathit{s}}' = (\lambda', {\mathit{f}}', (\psi', \phi')) \in {\mathbb{S}}$ is a step, and conclude that transform $T_{{\mathfrak{A}}}$ is an iterative operator ${\mathbb{S}} \to {\mathbb{S}}$. Application of the iterative operator $T_{{\mathfrak{A}}}$ induced by automaton ${\mathfrak{A}}$ is denoted ${\mathit{s}}' = {\mathfrak{A}}({\mathit{s}})$. Let ${\mathbb{S}}$ be a step space with automaton ${\mathfrak{A}}$ inducing iterative operator $T_{{\mathfrak{A}}}$. The walk of ${\mathit{s}} \in {\mathbb{S}}$ under $T_{{\mathfrak{A}}}$ assuming sequence of volatile excitation $\lbrace \xi_n \rbrace$ is denoted ${\mathfrak{A}}^{\mathbb{N}}_{\lbrace \xi_n \rbrace}({\mathit{s}})$. The notation ${\mathfrak{A}}^{\mathbb{N}}_{\lbrace \xi_n \rbrace}({\mathit{s}})$ is a reminder of the important role of the sequence of volatile excitations $\lbrace \xi_n \rbrace$. While each $\xi$ is entirely determined by initial frame $\psi$ within step ${\mathit{s}}$, this notation emphasizes that the volatile excitations are essentially free variables. Persistent variables are bound. #### Automaton iterative properties \[T:AUTOMATON\_ITERATE\_CONJOINT\] Let ${\mathfrak{A}}$ be an automaton and ${\mathbb{S}}$ be a step space with persistent-volatile partition $\Psi = \Phi\Xi$. Suppose step ${\mathit{s}} \in {\mathbb{S}}$ and event $\xi \in \Xi$. Frame $\mho_{\mathbf{F}}({\mathit{s}})$ conjoins frame $\mho_{\mathbf{F}}({\mathfrak{A}}_\xi({\mathit{s}}))$. By hypothesis $\xi \in {\prod{\Xi}}$ and ${\mathit{s}} \in {\mathbb{S}}$. By definition \[D:STEP\_SPACE\] of step space, there exist locus $\lambda \in \Lambda$, frame ${\mathbf{f}} = (\psi, \phi) \in {\prod{\Psi}} \times {\prod{\Phi}}$, and functionality ${\mathit{f}} \in {\mathscr{F}}$ such that ${\mathit{s}} = (\lambda, {\mathit{f}}, (\psi, \phi))$. Definition \[D:STEP\_SPACE\_PROJECTION\] establishes that $\mho_{\mathbf{F}}({\mathit{s}}) = (\psi, \phi)$. By Theorem \[T:AUTOMATON\_OPERATOR\] the automaton induces an iterative operator, so there exists ${\mathfrak{A}}_\xi( {\mathit{s}}) = (\lambda', {\mathit{f}}',(\psi', \phi')) \in {\mathbb{S}}$. Again by definition \[D:STEP\_SPACE\_PROJECTION\], $\mho_{\mathbf{F}}({\mathfrak{A}}_\xi( {\mathit{s}})) = (\psi', \phi')$. Definition \[D:ITERATIVE\_TRANSFORM\] evaluates ${\mathbf{f}}\,' = (\psi', \phi') = (\phi\xi, {\mathit{f}}'(\phi\xi))$ as the succeeding frame. Definition \[D:CONJOINT\] asserts that frame $(\psi, \phi)$ conjoins frame $(\psi', \phi')$ if ${{\psi'}\negmedspace\mid\negmedspace{{{\operatorname{dom}{\Phi}}}}} = \phi$. Here $\psi' = \phi\xi$, so ${{\phi\xi}\negmedspace\mid\negmedspace{{{\operatorname{dom}{\Phi}}}}} = \phi$ by virtue of persistent-volatile partition $\Psi = \Phi\Xi$. Thus we conclude $(\psi, \phi)$ conjoins $(\psi', \phi')$. Since $\mho_{\mathbf{F}}({\mathit{s}}) = (\psi, \phi)$, $(\psi, \phi)$ conjoins $(\psi', \phi')$, and $(\psi', \phi') = \mho_{\mathbf{F}}({\mathfrak{A}}_\xi( {\mathit{s}}))$, then by transitivity $\mho_{\mathbf{F}}({\mathit{s}})$ conjoins $\mho_{\mathbf{F}}({\mathfrak{A}}_\xi( {\mathit{s}}))$. \[T:AUTOMATON\_WALK\_PROCESS\] Let ${\mathfrak{A}}$ be an automaton and ${\mathbb{S}}$ be a step space with step ${\mathit{s}} \in {\mathbb{S}}$ and volatile excitation $\lbrace \xi_n \rbrace$. Sequential process projection $\overline{\mho}_{\mathbf{F}}({\mathfrak{A}}^{\mathbb{N}}_{\lbrace \xi_n \rbrace}({\mathit{s}}))$ is indeed a process per definition \[D:PROCESS\]. Definition \[D:PROCESS\] asserts that a process is a successively conjoint sequence of frames. To show contradiction, hypothesize that the frame sequence $\overline{\mho}_{\mathbf{F}}({\mathfrak{A}}^{\mathbb{N}}_{\lbrace \xi_n \rbrace}({\mathit{s}}))$ is *not* a process. Then there is some index $i$ such that frame $\mho_{\mathbf{F}}({\mathit{s}}_i)$ does not conjoin frame $\mho_{\mathbf{F}}({\mathit{s}}_{i+1})$. Let ${\mathit{s}}_i$ be the $i^\text{th}$ step of walk ${\mathfrak{A}}^{\mathbb{N}}_{\lbrace \xi_n \rbrace}({\mathit{s}})$. From definition \[D:ITERATIVE\_OPERATOR\_WALK\], the succeeding step is ${\mathit{s}}_{i+1} = {\mathfrak{A}}_{\xi_i}({\mathit{s}}_i)$. By Theorem \[T:AUTOMATON\_ITERATE\_CONJOINT\], frame $\mho_{\mathbf{F}}({\mathit{s}}_i)$ conjoins frame $\mho_{\mathbf{F}}({\mathfrak{A}}_{\xi_i}({\mathit{s}}_i))$. This contradicts the conclusion drawn from the hypothesis that the frame sequence is not a process, so $\overline{\mho}_{\mathbf{F}}({\mathfrak{A}}^{\mathbb{N}}_{\lbrace \xi_n \rbrace}({\mathit{s}}))$ *is* a process. Be a step consistent or not, in automaton-based iteration that step’s successor is consistent. \[T:AUTOMATON\_ITERATE\_CONSISTENT\] Let ${\mathfrak{A}}$ be an automaton and ${\mathbb{S}}$ be a step space with persistent-volatile partition $\Psi = \Phi\Xi$. Suppose step ${\mathit{s}} \in {\mathbb{S}}$ and event $\xi \in \Xi$. Step ${\mathfrak{A}}_\xi({\mathit{s}})$ is consistent. By hypothesis $\xi \in {\prod{\Xi}}$ and ${\mathit{s}} \in {\mathbb{S}}$. By definition \[D:STEP\_SPACE\] of step space, there exist locus $\lambda \in \Lambda$, frame ${\mathbf{f}} = (\psi, \phi) \in {\prod{\Psi}} \times {\prod{\Phi}}$, and functionality ${\mathit{f}} \in {\mathscr{F}}$ such that ${\mathit{s}} = (\lambda, (\psi, \phi), {\mathit{f}})$. By Theorem \[T:AUTOMATON\_OPERATOR\] the automaton induces an iterative operator, so there exists ${\mathfrak{A}}_\xi({\mathit{s}}) = (\lambda', (\psi', \phi'), {\mathit{f}}') \in {\mathbb{S}}$. By definition \[D:STEP\_SPACE\_PROJECTION\], $\mho_{\mathbf{F}}({\mathfrak{A}}_\xi({\mathit{s}})) = {\mathbf{f}}\,' = (\psi', \phi')$ and $\mho_{\mathscr{F}}({\mathfrak{A}}_\xi({\mathit{s}})) = {\mathit{f}}'$ Definition \[D:ITERATIVE\_TRANSFORM\] evaluates ${\mathbf{f}}\,' = (\psi', \phi') = (\phi\xi, {\mathit{f}}'(\phi\xi))$ as the succeeding frame. This complex relation separates into simple conditions $\psi' = \phi\xi$ and $\phi' = {\mathit{f}}'(\phi\xi) = {\mathit{f}}'(\psi')$. The assertion ${\mathbf{f}}\,' = (\psi', \phi') = (\psi', {\mathit{f}}'(\psi'))$ bears the same meaning as ${\mathbf{f}}\,' = (\psi', \phi') \in {\mathit{f}}'$. Definition \[D:CONSISTENT\_STEP\] states that step ${\mathfrak{A}}_\xi({\mathit{s}})$ is *consistent* if $\mho_{\mathbf{F}}({\mathfrak{A}}_\xi({\mathit{s}})) = {\mathbf{f}}\,' \in {\mathit{f}}' = \mho_{\mathscr{F}}({\mathfrak{A}}_\xi({\mathit{s}}))$, which is here satisfied. \[T:AUTOMATON\_WALK\_PROCEDURE\] Let ${\mathfrak{A}}$ be an automaton and ${\mathbb{S}}$ be a step space with consistent step ${\mathit{s}} \in {\mathbb{S}}$ and volatile excitation $\lbrace \xi_n \rbrace$. Suppose $\lbrace {\mathit{s}}_n \rbrace$ is the walk ${\mathfrak{A}}^{\mathbb{N}}_{\lbrace \xi_n \rbrace}({\mathit{s}})$. Sequential procedure projection $\overline{\mho}_{\mathscr{F}}(\lbrace {\mathit{s}}_n \rbrace)$ covers sequential process projection $\overline{\mho}_{\mathbf{F}}(\lbrace {\mathit{s}}_n \rbrace)$. We have the premises that step ${\mathit{s}} \in {\mathbb{S}}$ is consistent, that $\lbrace \xi_n \rbrace$ is an excitation, that ${\mathfrak{A}}$ is an automaton, and that $\lbrace {\mathit{s}}_n \rbrace$ is the walk ${\mathfrak{A}}^{\mathbb{N}}_{\lbrace \xi_n \rbrace}({\mathit{s}})$. We temporarily suppress the repetitive lengthy expression ${\mathfrak{A}}^{\mathbb{N}}_{\lbrace \xi_n \rbrace}({\mathit{s}})$ through the abbreviation ${\mathfrak{A}}^{\mathbb{N}}$. Induction demonstrates that each step of walk ${\mathfrak{A}}^{\mathbb{N}}$ is consistent. For the base clause, the case $i = 1$ is true by hypothesis, since the initial step ${\mathfrak{A}}^{\mathbb{N}}(1) = {\mathit{s}}$ is presumed consistent. For the recursive clause, definition \[D:ITERATIVE\_OPERATOR\_WALK\] provides that ${\mathit{s}}_{i+1} = {\mathfrak{A}}(\xi_{i}, {\mathit{s}}_i)$ for each $i \ge 1$. By theorem \[T:AUTOMATON\_ITERATE\_CONSISTENT\], step ${\mathfrak{A}}(\xi_{i}, {\mathit{s}}_i) = {\mathfrak{A}}^{\mathbb{N}}(i+1)$ is consistent. By the axiom of induction, step ${\mathfrak{A}}^{\mathbb{N}}(i)$ is consistent for each $i \ge 1$. By definition \[D:CONSISTENT\_STEP\] and the conclusion that step ${\mathfrak{A}}^{\mathbb{N}}(i)$ is consistent for each $i \ge 1$, it follows that $\mho_{\mathbf{F}}({\mathfrak{A}}^{\mathbb{N}}(i)) \in \mho_{\mathscr{F}}({\mathfrak{A}}^{\mathbb{N}}(i))$, also for each $i \ge 1$. Substituting ${\mathfrak{A}}^{\mathbb{N}}= \lbrace {\mathit{s}}_n \rbrace$ into lemma \[L:RELATED\_PROJECTION\] yields frame $(\overline{\mho}_{\mathbf{F}}({\mathfrak{A}}^{\mathbb{N}}))(i) = \mho_{\mathbf{F}}({\mathfrak{A}}^{\mathbb{N}}(i))$ and functionality $(\overline{\mho}_{\mathscr{F}}({\mathfrak{A}}^{\mathbb{N}}))(i) = \mho_{\mathscr{F}}({\mathfrak{A}}^{\mathbb{N}}(i))$. From equality it then follows that $(\overline{\mho}_{\mathbf{F}}({\mathfrak{A}}^{\mathbb{N}}))(i) \in (\overline{\mho}_{\mathscr{F}}({\mathfrak{A}}^{\mathbb{N}}))(i)$ for each $i \ge 1$. In simple language, the $i^\text{th}$ term of the process projection is a member of the $i^\text{th}$ term of the procedure projection. This satisfies the requirement of definition \[D:COVERING\_PROCEDURE\] that the procedure covers the sequence of frames: ${\mathbf{f}}_i \in {\mathit{f}}_i$ for each $i \geq 1$. Reverse inference ----------------- The construction of automata provides that steps unfold in sequential fashion – that is, the next step becomes known after completing the current step. Consequently automata inherit an intrinsic [forward]{} orientation. It is also reasonable to inquire what may have occurred in previous steps. This question is the motivation for reverse inference, which considers automata operating backwards. ### Iterative converse {#S:ITERATIVE_CONVERSE} Let $\langle \Psi, \Phi \rangle$ be a basis with persistent-volatile partition $\Psi = \Phi\Xi$ and step space ${\mathbb{S}} = \Lambda \times {\mathscr{F}} \times ({\prod{\Psi}} \times {\prod{\Phi}})$. Suppose step ${\mathit{s}} = (\lambda, {\mathit{f}}, (\psi, \phi)) \in {\mathbb{S}}$, whence the volatile excitation of ${\mathit{s}}$ is $\xi = {{\psi}\negmedspace\mid\negmedspace{{{\operatorname{dom}{\Xi}}}}}$. \[D:VOLATILE\_CONVERSE\] Let $\langle \Psi, \Phi \rangle$ be a basis with step space ${\mathbb{S}}$ and iterative operator $V \colon {\mathbb{S}} \to {\mathbb{S}}$. The *iterative* converse of $V$ is the mapping $\tilde{V} \colon {\mathbb{S}} \to {\mathscr{P}({{\mathbb{S}}})}$ defined by $$\tilde{V}({\mathit{s}}) = \lbrace \tilde{{\mathit{s}}} \in {\mathbb{S}} \colon V(\tilde{{\mathit{s}}}) = {\mathit{s}} \rbrace,$$ where ${\mathscr{P}({{\mathbb{S}}})}$ denotes the power set of ${\mathbb{S}}$ (in other words, the iterative converse is a mapping from a step to a set of steps). ### Converse actuated automaton {#S:CONVERSE_AUTOMATON} An actuated automaton ${\mathfrak{A}} = \langle \Psi, \Phi, {\mathscr{F}}\!, {\mathsf{A}}, \Lambda, \ell, \Delta \rangle$ induces an iterative operator on step space. Reverse inference is identifying all immediate predecessor steps ${{\mathit{s}}_{n-1}}$ such that ${\mathit{s}}_n = {\mathfrak{A}}({\mathit{s}}_{n-1})$. We have obviously $$\tilde{{\mathfrak{A}}}({\mathit{s}}) = \lbrace \tilde{{\mathit{s}}} \in {\mathbb{S}} \;\colon\; {\mathfrak{A}}(\tilde{{\mathit{s}}}) = {\mathit{s}} \rbrace.$$ Although referring informally to the converse $\tilde{{\mathfrak{A}}}$ of an automaton ${\mathfrak{A}}$, speaking precisely we have defined the converse $\tilde{{\mathit{s}}}$ of a *step* ${\mathit{s}} \in {\mathbb{S}}$ within the step space associated with that automaton. A system of equations ensues with subscripted variables known and held constant, and unsubscripted variables free. Roots of the system represent discrete solutions. Let us look carefully at the further case that ${\mathit{s}} = (\lambda, {\mathit{f}}, {\mathbf{f}}) = (\lambda, {\mathit{f}}, (\psi, \phi)) \in \Lambda \times {\mathscr{F}} \times {\mathbf{F}}$. $$\begin{aligned} \tilde{{\mathfrak{A}}}(\lambda_0, {\mathit{f}}_0, {\mathbf{f}}_0) &= \lbrace (\lambda, {\mathit{f}}, {\mathbf{f}}) \in {\mathbb{S}} \;\colon\; {\mathfrak{A}}(\lambda, {\mathit{f}}, {\mathbf{f}}) = (\lambda_0, {\mathit{f}}_0, {\mathbf{f}}_0) \rbrace \\ \tilde{{\mathfrak{A}}}(\lambda_0, {\mathit{f}}_0, (\psi_0, \phi_0)) &= \lbrace (\lambda, {\mathit{f}}, (\psi, \phi)) \in {\mathbb{S}} \;\colon\; {\mathfrak{A}}(\lambda, {\mathit{f}}, (\psi, \phi)) = (\lambda_0, {\mathit{f}}_0, (\psi_0, \phi_0)) \rbrace.\end{aligned}$$ ### Constraining equations {#S:CONSTRAINING_EQUATIONS} State transition in an actuated automaton is built in three successive phases: locus state, functionality state, and frame state. Definition \[D:ITERATIVE\_TRANSFORM\] presents rules governing forward state transition in the form of three equations, portraying current state as known and unknown future state as uniquely determined by formulas. This sense can be reversed, with current state known and feasible past states represented as unknowns. The automaton-induced *forward* transformation ${\mathfrak{A}} \colon {\mathbb{S}} \to {\mathbb{S}}$ has been set (definition \[D:ITERATIVE\_TRANSFORM\]) as $$\begin{aligned} \lambda' &= \Delta(\lambda, \psi),\\ {\mathit{f}}' &= (\ell(\Delta(\lambda, \psi)))({\mathit{f}}(\psi) \xi'),\\ {\mathbf{f}}\,' &= (\psi', \phi') = ([{\mathit{f}}(\psi) \xi'], [(\ell(\Delta(\lambda, \psi)))({\mathit{f}}(\psi) \xi')]([{\mathit{f}}(\psi) \xi'])).\end{aligned}$$ The respective governing *backwards* transformations are $$\begin{aligned} \lambda_0 &= \Delta(\lambda, \psi),\\ {\mathit{f}}_0 &= (\ell(\Delta(\lambda, \psi)))({\mathit{f}}(\psi) \xi_0),\\ {\mathbf{f}}_0 &= (\psi_0, \phi_0) = ([{\mathit{f}}(\psi) \xi_0], [(\ell(\Delta(\lambda, \psi)))({\mathit{f}}(\psi) \xi_0)]([{\mathit{f}}(\psi) \xi_0])).\end{aligned}$$ Due to conjointness, $\psi_0 = {\mathit{f}}(\psi) \xi_0 = \phi_0 \xi_0$ throughout: $$\begin{aligned} \lambda_0 &= \Delta(\lambda, \psi),\\ {\mathit{f}}_0 &= (\ell(\Delta(\lambda, \psi)))(\psi_0),\\ {\mathbf{f}}_0 &= (\psi_0, \phi_0) = (\psi_0, [(\ell(\Delta(\lambda, \psi)))(\psi_0)](\psi_0)])).\end{aligned}$$ Substituting $\lambda_0 = \Delta(\lambda, \psi)$ and ${\mathsf{a}}_0 = \ell(\lambda_0)$ into the last two formulas, $$\begin{aligned} \lambda_0 &= \Delta(\lambda, \psi),\\ {\mathit{f}}_0 &= (\ell(\lambda_0)(\psi_0) = {\mathsf{a}}_0(\psi_0),\\ {\mathbf{f}}_0 &= (\psi_0, \phi_0) = (\psi_0, [(\ell(\lambda_0))(\psi_0)](\psi_0)])) = (\psi_0, [{\mathsf{a}}_0(\psi_0)](\psi_0)])).\end{aligned}$$ Finally, substituting ${\mathit{f}}_0 = {\mathsf{a}}_0(\psi_0)$ into the last two formulas, $$\begin{aligned} \lambda_0 &= \Delta(\lambda, \psi),\\ {\mathit{f}}_0 &= {\mathit{f}}_0,\\ {\mathbf{f}}_0 &= (\psi_0, {\mathit{f}}_0(\psi_0)) = {\mathbf{f}}_0.\end{aligned}$$ Only the first of these is a [real]{} constraint; the others are identities. ### Solution set {#S:SOLUTION_SET} A feasible set for an equation is the collection of values that satisfy the equation. Producing the solution for a system of constraining equations involves intersecting the individual feasible sets $Q_i$. The general solution set for the system of equations is $Q = \bigcap_i Q_i$. The case of the converse automaton $\tilde{{\mathfrak{A}}}$ consists of three constraining equations (§\[S:CONSTRAINING\_EQUATIONS\]). First of these is $\lambda_0 = \Delta(\lambda, \psi)$. Let ${\mathbb{S}} = \Lambda \times {\mathscr{F}} \times (\;{\prod{\Psi}} \times {\prod{\Phi}})$ be a fully elaborated step space. In set-builder notation the feasible set is $Q_1 = \lbrace (\lambda, {\mathit{f}}, (\psi, \phi)) \in {\mathbb{S}} \colon \lambda_0 = \Delta(\lambda, \psi) \rbrace$. The other constraints are identities, which all members of ${\mathbb{S}}$ satisfy – in other words, $Q_2 = Q_3 = {\mathbb{S}}$. The general solution is $Q = Q_1 \cap Q_2 \cap Q_3 = Q_1 \cap {\mathbb{S}} \cap {\mathbb{S}} = Q_1$, or $Q = \lbrace (\lambda, {\mathit{f}}, (\psi, \phi)) \in {\mathbb{S}} \colon \lambda_0 = \Delta(\lambda, \psi) \rbrace$. An understandable abuse of terminology says that the solution set is $\lambda_0 = \Delta(\lambda, \psi)$, which is technically a constraining equation. Cone {#S:CONE_SECTION} ---- A cone is a construct prepared with the iterative converse of an actuated automaton. It consists of all finite backwards walks converging to a given point. The term [cone]{} is more ideologic than geometric. ### Description {#S:CONE_DESCRIPTION} The actuated automaton possesses a non-deterministic[^4] converse relation. See §\[S:ITERATIVE\_CONVERSE\]. A collection of reverse walks is realized through repetitive re-application of the converse, converging to a designated *crux* step. These iterative chains may be localized (trimmed to finite length) by enforcing some stopping criterion. This construction results in the cone, a structured set of possible localized walks eventually leading to the crux step. The starting points of such walks are known as *precursor* steps of the crux step. ### Inductive generation {#S:CONE_GENERATION} Let ${\mathbb{S}}$ be a step space containing crux step ${\mathit{s}}_{\text{crux}}$. Suppose $V \colon {\mathbb{S}} \to {\mathbb{S}}$ is an iterative operator with converse $\tilde{V} \colon {\mathbb{S}} \to {\mathscr{P}({{\mathbb{S}}})}$. (base clause) Let base protoset $G_\heartsuit^{(0)} = \lbrace {\mathit{s}}_\text{crux} \rbrace$ be the $0^{\text{th}}$ predecessor generation of ${\mathit{s}}_{\text{crux}}$. (inductive clause) The ${(n+1)}^{\text{st}}$ generation predecessors are defined in terms of the $n^{\text{th}}$ generation: $$G_\heartsuit^{(n+1)} = \bigcup_{{\mathit{s}} \in G_\heartsuit^{(n)}} \tilde{V}({\mathit{s}}).$$ This definition places protoset $G_\heartsuit^{(1)} = \tilde{V}({\mathit{s}}_{\text{crux}})$. For a discussion of protoset $G_\heartsuit^{(n)}$ in context of the Cartesian product, see Appendix §\[D:PROTOSET\]. ### Partial order {#S:CONE_ORDER} Membership in a converse iterative operator induces a partial ordering: Let ${\mathbb{S}}$ be a step space with converse iterative operator $\tilde{V} \colon {\mathbb{S}} \to {\mathscr{P}({{\mathbb{S}}})}$. If ${\mathit{s}}' \in \tilde{V}({\mathit{s}})$, then ${\mathit{s}}'$ *precedes* ${\mathit{s}}$, written ${\mathit{s}}' \prec {\mathit{s}}$. ### Predecessor walk {#S:CONE_WALK} A predecessor walk begins at step ${\mathit{s}}_0 = {\mathit{s}}_{\text{crux}}$ and proceeds backwards, indexing through the negative integers. In this case we abuse the proper sense of the term [sequence]{} by permitting an indexing not being the natural numbers. Let ${\mathbb{S}}$ be a step space. A localized predecessor walk starting with step ${\mathit{s}}_0 = {\mathit{s}}_{\text{crux}}$ is a *finite* sequence in step space such that ${\mathit{s}}_{i-1} \prec {\mathit{s}}_i$ for every $i \leq 0$. For example in the case $i = -2$, we have ${\mathit{s}}_{-3} \prec {\mathit{s}}_{-2}$. Since a localized predecessor walk ${\mathit{w}}$ is a *finite* sequence of steps, then it has a finite number of terms which run in index $i$ from $-(n-1) \leq i \leq 0$, where $n = {\lvert{{\mathit{w}}}\rvert} < \infty$ is the number of steps in ${\mathit{w}}$. \[S:COMPLETE\_CONE\_WALK\] A set ${\mathbb{W}}$ of localized predecessor walks, all starting at ${\mathit{w}}_0 = {\mathit{s}}_{\text{crux}}$, is *complete* if\ $$\forall({\mathit{w}} \in {\mathbb{W}}) \forall(-({\lvert{{\mathit{w}}}\rvert}-2) \leq i \leq 0) \forall({\mathit{s}} \in \tilde{V}({\mathit{w}}_{i})) \exists({\mathit{e}} \in {\mathbb{W}}) : {\mathit{w}}_{i} = {\mathit{e}}_{i} \wedge {\mathit{s}} = {\mathit{e}}_{i-1}.$$ Completeness assures combinatorial diversity. An algorithm equivalent to the above one-liner is:\ \ $ \text{for each }{\mathit{w}} \text{ in }{\mathbb{W}} \text{ begin}\\ \text{\quad\quad \# number of steps in localized walk is } {\lvert{{\mathit{w}}}\rvert}\\ \text{\quad\quad \# abused index runs between 0 [for start step } {\mathit{w}}_{\text{crux}} \text{] and } -({\lvert{{\mathit{w}}}\rvert}-1) \text{ [last predecessor step]}\\ \text{\quad\quad \# no iteration through } i = -({\lvert{{\mathit{w}}}\rvert}-1) \text{ because then } {\mathit{s}} \text{ == } {\mathit{e}}_{i-1} \text{ below would be undefined}\\ \text{\quad\quad for }i = 0 \text{ downto } -({\lvert{{\mathit{w}}}\rvert}-2) \text{ begin}\\ \text{\quad\quad\quad\quad for each }{\mathit{s}} \text{ in } \tilde{V}({\mathit{w}}_{i}) \text{ begin}\\ \text{\quad\quad\quad\quad\quad\quad if there is a member }{\mathit{e}} \in {\mathbb{W}} \text{ such that } {\mathit{w}}_{i} \text{ == } {\mathit{e}}_{i} \text{ and }{\mathit{s}} \text{ == } {\mathit{e}}_{i-1} \text{ then}\\ \text{\quad\quad\quad\quad\quad\quad\quad\quad answer = TRUE}\\ \text{\quad\quad\quad\quad\quad\quad else}\\ \text{\quad\quad\quad\quad\quad\quad\quad\quad answer = FALSE}\\ \text{\quad\quad\quad\quad\quad\quad if answer == FALSE then return FALSE}\\ \text{\quad\quad\quad\quad end}\\ \text{\quad\quad end}\\ \text{end}\\ \text{return TRUE}\\ $ \[S:DEPENDENT\_CONE\_WALK\] Let ${\mathit{w}}$ and ${\mathit{w}}'$ be localized predecessor walks starting at ${\mathit{w}}_0 = {\mathit{w}}_0' = {\mathit{s}}_{\text{crux}}$. Suppose the length of ${\mathit{w}}$ is ${\lvert{{\mathit{w}}}\rvert} = n$ and the length of ${\mathit{w}}'$ is ${\lvert{{\mathit{w}}'}\rvert} = m$, with $m \leq n$. If ${\mathit{w}}'(i) = {\mathit{w}}(i)$ for every $-(m-1) \leq i \leq 0$, then ${\mathit{w}}$ and ${\mathit{w}}'$ are *dependent* with ${\mathit{w}}'$ *dispensable*. \[S:INDEPENDENT\_CONE\_WALK\] Let ${\mathbb{W}}$ be a set of localized predecessor walks starting at ${\mathit{s}}_0 = {\mathit{s}}_{\text{crux}}$. The set is *independent* if it contains no dispensable member. ### Cone {#S:CONE_CONE_WALK} \[S:CONE\_DEFINITION\] A *cone* ${\mathcal{C}}$ is a complete independent set of localized predecessor walks starting at ${\mathit{s}}_{\text{crux}}$. [\[Stopping rule\]]{} We avoid the specificity of various stopping criteria (§\[S:CONE\_DESCRIPTION\]) by introducing the equivalent but arbitrary notion of localization. \[S:CONE\_EDGE\_STEP\] Let ${\mathcal{C}}$ be a cone with ${\mathit{w}} \in {\mathcal{C}}$ a member localized predecessor walk. Suppose $n = {\lvert{{\mathit{w}}}\rvert}$ is the number of steps in ${\mathit{w}}$. The terminus ${\mathit{w}}(-(n-1)) = {\mathit{w}}_{-(n-1)}$ is the *edge step* of walk ${\mathit{w}}$. \[S:CONE\_EDGE\] Let ${\mathcal{C}}$ be a cone. Its *edge*, written ${{\operatorname{edge}{{\mathcal{C}}}}}$, is the collection of edge steps of all member localized predecessor walks. \[S:ACYLIC\_CONE\] An *acyclic* cone has no cycle (loop) in its path projection $\overline{\mho}_\Lambda$ (see §\[D:EXTENDED\_PROJECTION\]). \[T:ACYLIC\_CONE\_CORRESPONDENCE\] The acyclic cone ${\mathcal{C}}$ and ${{\operatorname{edge}{{\mathcal{C}}}}}$ are in one-to-one correspondence via the edge step relation of a localized predecessor walk. Assume the opposite: there are different localized predecessor walks with the same edge step. Let ${\mathit{u}}$ and ${\mathit{v}}$ be two different walks with common edge step ${\mathit{s}}_\text{common}$. Suppose ${\lvert{{\mathit{u}}}\rvert} = m$ and ${\lvert{{\mathit{v}}}\rvert} = n$, so the indexes of ${\mathit{s}}_\text{common}$ are $-(m-1)$ and $-(n-1)$ respectively. We assert that if ${\mathit{u}}_{-(m-1)+i} = {\mathit{v}}_{-(n-1)+i}$ for some $0 \leq i$, then ${\mathit{u}}_{-(m-1)+(i+1)} = {\mathit{v}}_{-(n-1)+(i+1)}$. Suppose sequencing is governed by an actuated automaton ${\mathfrak{A}}$. So sequenced, the next step in predecessor walk ${\mathit{u}}$ is ${\mathit{u}}_{-(m-1)+(i+1)} = {\mathfrak{A}}({\mathit{u}}_{-(m-1)+i})$. Similarly, the next step in ${\mathit{v}}$ is ${\mathit{v}}_{-(n-1)+(i+1)} = {\mathfrak{A}}({\mathit{v}}_{-(n-1)+i})$. But if ${\mathit{u}}_{-(m-1)+i} = {\mathit{v}}_{-(n-1)+i}$, then ${\mathfrak{A}}({\mathit{u}}_{-(m-1)+i}) = {\mathfrak{A}}({\mathit{v}}_{-(n-1)+i}) = {\mathfrak{A}}({\mathit{s}})$. By transitivity of equality, ${\mathit{u}}_{-(m-1)+(i+1)} = {\mathfrak{A}}({\mathit{s}}) = {\mathit{v}}_{-(n-1)+(i+1)}$. Without loss of generality suppose $m \leq n$. Then ${\mathit{u}}_{-(m-1)+i} = {\mathit{v}}_{-(n-1)+i}$ is true for $i = 0, 1, \;...\; m-1$. At $i = m - 1$ we have ${\mathit{s}}_{\text{crux}} = {\mathit{u}}_0 = {\mathit{v}}_{-(n-1)+(m-1)} = {\mathit{v}}_{-(n-m)}$. Since ${\mathit{v}}$ is a localized predecessor walk of a cone, then ${\mathit{v}}_0 = {\mathit{s}}_{\text{crux}}$. But ${\mathit{v}}_0 = {\mathit{s}}_{\text{crux}} = {\mathit{v}}_{-(n-m)}$. Because the cone is assumed acyclic, ${\mathit{v}}_0$ and ${\mathit{v}}_{-(n-m)}$ must then be the same identical step – that is, $m = n$. Here the assumption of two different local predecessor walks with the same edge step leads to the contradiction that both are indeed the same identical walk. This means local predecessor walks within an acyclic cone ${\mathcal{C}}$ are in one-to-one correspondence with ${{\operatorname{edge}{{\mathcal{C}}}}}$ via the edge step relation. Operational profile {#S:OPERATIONAL_PROFILE_SECTION} ------------------- An operational profile is a limit of the cumulative history of software execution ratios under normal operations (which is troublesome to define). This section frequently uses the compound idiom that $\{x_n\}$ represents an anonymous sequence of objects of the same type as $x$. That is, if $X$ is the set of all $x_i$, then $\{x_n\} \colon {\mathbb{N}}\to X$. ### Musa’s operational profile {#S:MUSA_OP_PROFILE} Musa et al intended operational profiles as a tool for analysis of software reliability. A notion of the operational profile appeared in their pioneering exposition [@jM87]. This reference gives a definition in terms of the program’s higher purpose, as reflected in run types. Consequently an operational profile is the set of run types that the program can execute along with the probabilities they will occur. One can easily envision its extension to smaller program units. ### Extended operational profile {#S:EXTENDED_OP_PROFILE} We shall extend run types into steps, the elementary quantum of automata. This detaches the operational profile concept from notions such as run types which are part of human understanding rather than algorithmic structure (however, the idea pops up elsewhere). Despite appearances, this extension is not so large – the only addition is a method for counting step events. ### Counting {#S:COUNTING} Let $\{{\mathit{s}}_n\}$ be a walk (infinite sequence of steps) and let $Z$ be an arbitrary reference set of steps (members of step space ${\mathbb{S}} = \Lambda \times {\mathscr{F}} \times {\mathbf{F}}$). Simply summarized, $N_Z(\{{\mathit{s}}_n\},k)$ denotes the number of occurrences of any member of $Z$ before or at the ${{k}^{\text{th}}}$ automaton step. Details follow for those interested. When the ${{i}^{\text{th}}}$ step of the walk is a member of $Z$ (${{\mathit{s}}_i} \in Z$), then $\{{\mathit{s}}_n\}$ is said to *arrive* at $i$. \[D:ARRIVAL\] An *arrival* function is a sequence $\varphi \colon \{1,2,\cdots\} \to \{n_1,n_2,\cdots\}$ mapping each arrival, as identified by its ordinal occurrence number $i$, into its frame sequence number $n_i$. The arrival function assumes the natural order, that is, $i < j$ implies $n_i < n_j$. A related function counts *how many* arrivals occur within a given interval: \[D:COUNTING\] Suppose $\{{\mathit{s}}_n\}$ is a walk and $Z$ is a set of steps. Let $\varphi$ be an arrival function. The *counting* function $N_Z \colon {\mathbb{S}}^{{\mathbb{N}}} \times {\mathbb{Z}^+}\to {\mathbb{Z}^+}$ induced by $\varphi$ is $$N_Z(\{{\mathit{s}}_n\}, k) = \max_{\varphi(i) \leq k} \: i ,$$ and for completeness set $\max(\varnothing) = 0$. ### Normal operations {#S:NORMAL_OPERATIONS} The idea behind [normal operations]{} is a long program run following a software usage pattern. Interaction with the environment affects software behavior, which is ultimately transmitted through response to changing volatile variables. Normal operations calls for a run of figuratively unbounded duration during which software experiences the usage pattern’s variation of volatile stimulus, in response to which possibly unbalanced service is demanded from its inventory of functions. Systems engineering often augments what is here the automaton’s step poset with a transition network of modes. These modes symbolically encapsulate enabled or disabled capabilities. However, even though this augmentation facilitates visualization of behaviors, it fails to be mathematically definitive[^5]. #### Orbit {#S:ORBIT} We have not defined normal operations, but every example would certainly constitute a walk (sequence of steps). A special walk illustrating normal operations (as described above) will be termed here an *orbit*. Without formal definition, use of this term sacrifices rigor. The actuated automaton governs pure step transition logic, but an orbit also reflects a usage pattern. #### Limit conjecture {#S:LIMIT_CONJECTURE} Orbits may differ in specific sequence and content, but they have the same limit ratios. We consider a case drawn with the counting procedure of §\[S:COUNTING\]. \[T:LIMIT\_RATIO\] For different orbits ${\mathit{o}} = \{{\mathit{s}}_n\}$ and ${\mathit{o}}' = \{{\mathit{s}}'_n\}$ having the same usage pattern, $$\lim_{\;k \to \infty} \frac{N_U(\{{\mathit{s}}_n\}, k)}{N_Z(\{{\mathit{s}}_n\}, k)} = \lim_{\;k \to \infty} \frac{N_U(\{{\mathit{s}}'_n\}, k)}{N_Z(\{{\mathit{s}}'_n\}, k)}$$ for sets of steps $\varnothing \neq U \subseteq Z \subseteq {\mathbb{S}}$. ### Types of operational profile {#S:TYPES_OP_PROFILE} A relative operational profile is the conditional probability that a step in an actuated automaton’s orbit coincides with a particular member of the reference set, given that it agrees with the reference set. We consider one other: an absolute operational profile is the time rate at which a particular step of an orbit coincides with any member of the reference set. ### Relative operational profile {#S:RELATIVE_OP_PROFILE} Let ${\mathit{o}} = \{{\mathit{s}}_n\}$ be an orbit. Suppose $z \in Z \subset {\mathbb{S}}$ is a step of the reference set. Software encounters $N_{\{z\}}(\{{\mathit{s}}_n\}, k)$ instances of steps satisfying $\lbrace {\mathit{s}}_n \rbrace(i) = {\mathit{s}}_i = z$ during the first $k$ automaton steps. In the same execution there are $N_Z(\{{\mathit{s}}_n\}, k)$ instances of ${\mathit{s}}_i \in Z$. In the frequentist [@wW14fp] school of interpreting probability, $$P(z \mid Z) = \lim_{\;k \to \infty} \frac{N_{\{z\}}(\{{\mathit{s}}_n\}, k)}{N_Z(\{{\mathit{s}}_n\}, k)}$$ represents the conditional probability of occurrence of $z$, given that $Z$ occurs. By Conjecture \[T:LIMIT\_RATIO\], every orbit (of the same usage pattern) yields the same relative operational profile. A relative operational profile is an arbitrary set $Z$ of steps, along with each step’s conditional probability of execution. In other words, a relative operational profile is a mapping $\mathcal{O} \colon Z \to [0,1]$ having total measure 1. ### Absolute operational profile {#S:ABSOLUTE_OP_PROFILE} Let ${\mathit{o}} = \{{\mathit{s}}_n\}$ be an orbit. An absolute operational profile is the probability $P(Z)$ with which an orbit (of some usage pattern) coincides with any step of the reference set $Z$. As before, this probability is the limiting ratio of two counting functions. Its numerator contains $N_Z(\{{\mathit{s}}_n\}, k)$, the same count as appears in the denominator of the relative operational profile. In its denominator is the counting function of all possible steps, namely $N_{{\mathbb{S}}}(\{{\mathit{s}}_n\}, k)$, where ${\mathbb{S}}$ is the space of all steps. Thus the ratio of counting functions of reference set $Z$ to the entire space ${\mathbb{S}}$ is $\frac{N_Z(\{{\mathit{s}}_n\}, k)}{N_{{\mathbb{S}}}(\{{\mathit{s}}_n\}, k)} = \frac{N_Z(\{{\mathit{s}}_n\}, k)}{k}$, and the absolute operational profile (of collection $Z$) is $$P(Z) = \lim_{\;k \to \infty} \frac{N_Z(\{{\mathit{s}}_n\}, k)}{k}.$$ ### Conversion into a rate {#S:ABSOLUTE_OP_PROFILE_RATE} Section \[S:INTRO\_REACTIVE\] mentions the synchronization function, a cross-reference between discrete and real time. During each step, an amount of real time appropriate for a software system emulating the automaton’s step is added to the time consumption budget. Let $Z$ be the usual arbitrary reference collection of steps and ${\mathit{o}} = \{{\mathit{s}}_n\}$ be an orbit. These two provide a set of events and a sequence of steps in which to count the events’ arrivals. The synchronization records discrete pairs $(i, t_i)$, where $i$ is the index of the automaton step and $t_k$ is the total elapsed time after $k$ steps. Call this mapping the synchronization function, having the formalism $\sync \colon {\mathbb{S}}^{\mathbb{N}}\times {\mathbb{N}}\to {\mathbb{R}^+}$, along with assumed starting point $\sync(\{{\mathit{s}}_n\}, 0) = 0$. Let the sequence index of each step be the discrete analog of time. Of course, this has the effect that discrete software time will not hold proportional to hardware real time. The approximate real time required by execution of step ${\mathit{s}} = (\lambda, {\mathit{f}}, {\mathbf{f}})$ is $\tau({\mathit{f}})$ – that is, elapsed real time is taken as a function of the executing functionality. \[D:SYNC\_APPOX\] For orbit ${\mathit{o}} = \{{\mathit{s}}_n\}$, approximate time elapsed during the first $k$ steps accumulates to $$\sync(\{{\mathit{s}}_n\}, k) = \sum_{i=1}^k \tau({\mathit{f}}_i)) = \sum_{i=1}^k \tau(\mho_{{\mathscr{F}}}({\mathit{s}}_i)) = t_k.$$ A theorem to avoid creating dependency on specific orbits is in order. Inability to define normal operations leads instead to conjecture, expressing such need: \[T:TIME\_RATIO\] For different orbits ${\mathit{o}} = \{{\mathit{s}}_n\}$ and ${\mathit{o}}' = \{{\mathit{s}}'_n\}$ having the same usage pattern, $$\lim_{\;k \to \infty} \frac{N_Z(\{{\mathit{s}}_n\}, k)}{{\sync}(\{{\mathit{s}}_n\}, k)} = \lim_{\;k \to \infty} \frac{N_Z(\{{\mathit{s}}'_n\}, k)}{{\sync}(\{{\mathit{s}}'_n\}, k))}$$ for a reference set of steps $\varnothing \neq Z \subseteq {\mathbb{S}}$. The synchronization function allows expression of the absolute operational profile as an intensity (rate or quasi-frequency). The *counting norm* is written using the double bar notation ${\Vert{\cdot}\Vert}$: $${\Vert{Z}\Vert} = \lim_{\;k \to \infty} \frac{N_Z(\{{\mathit{s}}_n\}, k)}{\sync(\{{\mathit{s}}_n\}, k)}.$$ The absolute operational profile is properly a subadditive seminorm on sets of steps. As the limiting ratio of two counts in the natural numbers, the norm is positive. The norm is a seminorm because for some nonempty set $Z$ it may be true that ${\Vert{Z}\Vert} = 0$ (if the usage pattern does not activate any member of the reference set). This norm is subadditive because for any other set $S$, $N_{Z \cup S}(\{{\mathit{s}}_n\}, k) \leq N_Z(\{{\mathit{s}}_n\}, k) + N_S(\{{\mathit{s}}_n\}, k)$. It follows that ${\Vert{Z \cup S}\Vert} \leq {\Vert{Z}\Vert} + {\Vert{S}\Vert}$. Application =========== The safety demonstration furnishes data for the indemnification statistic, which originates in the compound Poisson random process. Reliability demonstration ------------------------- A reliability demonstration is a structured random experiment carrying controlled statistical uncertainty and providing [hard]{} evidence against potential liability. ### Safety demonstration {#S:SAFETY_DEMONSRATION} In software safety analysis, a hazard is a region of code bearing potential harmful side effects if incorrectly implemented. A safety demonstration is a special type of reliability demonstration posed to exercise a hazard. Here the region is presumed to be an acyclic cone, with the hazard located at its crux. The crux is a point of software/hardware transduction, illustrating the principle of emergence[^6] (see §\[S:PRINCIPLE\_OF\_EMERGENCE\]). To oversimplify, a safety demonstration is a random sample from such a region (acyclic cone). The complete story is not so simple, because the cone is not a probabilistic structure; it possesses no probability to support randomness. As a probabilistic structure, the operational profile (§\[S:RELATIVE\_OP\_PROFILE\]) permits random sampling from its reference set, regardless of its higher level meaning. As the edge of a cone is a set, it can become a relative operational profile’s reference set. Thus we tie an operational profile to a cone’s edge. Let $\mathcal{O} \colon {{\operatorname{edge}{{\mathcal{C}}}}} \to [0,1]$ be a relative operational profile on the edge of cone ${\mathcal{C}}$. At this stage we have the ability to draw a random sample from ${{\operatorname{edge}{{\mathcal{C}}}}}$. Theorem \[T:ACYLIC\_CONE\_CORRESPONDENCE\] asserts that an acyclic cone ${\mathcal{C}}$ and ${{\operatorname{edge}{{\mathcal{C}}}}}$ are in one-to-one correspondence via the edge step relation of a localized predecessor walk. Equivalent to the one-to-one correspondence is the bijection $\mathbf{b} = \{({{\operatorname{edge}{{\mathit{w}}}}},{\mathit{w}}) \colon {\mathit{w}} \in {\mathcal{C}}\}$. For ${\mathit{e}} \in {{\operatorname{edge}{{\mathcal{C}}}}}$, $\mathbf{b}({\mathit{e}})$ is the bijectively corresponding localized predecessor walk. We now bijectively associate the random edge event ${\mathit{e}} = \mathbf{b}^{-1}({\mathit{w}})$ with the localized predecessor walk ${\mathit{w}}$: $\mathcal{O}' = \{ (\mathcal{O}(\mathbf{b}^{-1}({\mathit{w}})), {\mathit{w}}) \colon {\mathit{w}} \in {\mathcal{C}} \}$. With probability inherited from an operational profile, we can speak validly of a random sample from a cone. ### Tests {#S:TESTS} The last piece of the safety demonstration story is converting local predecessor walks into tests. Localized predecessor walks are finite walks existing in confusion-prone backwards time. One may skip this section unless he wishes the detail of converting backward to forward walks. The *test* function reverses and re-indexes localized predecessor walks into conventional sequences. \[D:TEST\] Let ${\mathit{w}}$ be a localized predecessor walk of $n = {\lvert{{\mathit{w}}}\rvert}$ steps, indexed from $0$ down to $-(n-1)$. Define the *test* function $\test({\mathit{w}}) = \widetilde{{\mathit{w}}}$ according to formula $\widetilde{{\mathit{w}}}_i = {\mathit{w}}_{i - n}$ for $i = 1, 2, \cdots , n$. Assuming that a localized predecessor walk is indexed from $0$ down to $-(n-1)$, its corresponding test will be indexed from $1$ to $n$. In sense of direction, the localized predecessor walk traverses steps from ${\mathit{s}}_{\text{crux}}$ to ${\mathit{s}}_{\text{edge}}$, while the corresponding test traverses steps from ${\mathit{s}}_{\text{edge}}$ to ${\mathit{s}}_{\text{crux}}$. Suppose ${\mathcal{C}}$ is an acyclic cone and ${\mathbb{W}} \subset {\mathcal{C}}$ is a (unique) set of localized predecessor walks. If $\widetilde{{\mathbb{W}}} = \test({\mathbb{W}})$ is its converted set of reversed and re-indexed tests, then ${\mathbb{W}}$ and $\widetilde{{\mathbb{W}}}$ are in one-to-one correspondence. By virtue of construction, $\test$ is already a mapping. Remaining to show is that $\test$ is additionally a bijection. Let ${\mathit{u}}$ and ${\mathit{v}}$ be localized predecessor walks and ${\mathit{x}}$ be a finite walk. As hypothesis set ${\mathit{x}} = \test({\mathit{u}}) = \test({\mathit{v}})$. These sequences cannot be equal unless they possess the same number of terms, $n = {\lvert{{\mathit{x}}}\rvert} = {\lvert{\test({\mathit{u}})}\rvert} = {\lvert{\test({\mathit{v}})}\rvert}$. Since transformation $\test$ preserves the number of steps (from Definition \[D:TEST\] ${\lvert{{\mathit{w}}}\rvert} = {\lvert{\test({\mathit{w}})}\rvert}$), then $n = {\lvert{{\mathit{x}}}\rvert} = {\lvert{{\mathit{u}}}\rvert} = {\lvert{{\mathit{v}}}\rvert}$. Again invoking Definition \[D:TEST\] on the first part of the hypothesis, we write ${\mathit{x}}_i = {\mathit{u}}_{i - n}$. The second part similarly yields ${\mathit{x}}_i = {\mathit{v}}_{i - n}$. By equating the two parts, we now have ${\mathit{u}}_{i - n} = {\mathit{v}}_{i - n}$ for each $i$. In other words, the two localized predecessor walks are actually the same walk: ${\mathit{u}} = {\mathit{v}}$. Thus $\test$ is a bijection. #### Volatile variables preservation {#S:VOLATILE_VARIABLES} The danger in reversed thinking about tests is inadvertently conceptualizing volatile variables as free. This is untrue, as the volatile variables at any stage of a predecessor chain are fixed, and the [next]{} stage considers the set of what previous conditions may have led to the current stage. Thus, predecessor walks are chains of a poset of steps, which include the settings of volatile variables. One must be mindful to reproduce all volatile stimuli of the localized predecessor walk in its analogous test. ### Outcome {#S:OUTCOME} The outcome of a test, pass or fail, will be regarded as a Bernoulli random event, $P_\rho = \rho^n {(1 - \rho)}^{1 - n}$, for $n = 1$ (pass) or $n = 0$ (fail). These probabilities are statistically independent of the bias involved with drawing the sample from the operational profile. This bias affects the origin of discovered failures, but not how many failures are found. In other words, the total statistical power of the sampling plan is not affected by sampling bias. Sums of independent Bernoulli random variables are binomial. That is, the probability of finding $n$ failures collectively among $N$ sample items is binomial, $\binom{N}{n} \rho^n {(1 - \rho)}^{N - n}$. ### Physics {#S:PHYSICS} In the real world, tests pass or fail depending on whether the information transduced at step ${\mathit{s}}_\text{crux}$ meets all safety constraints. Such engineering requirements are varied, ultimately involving position, timing, voltage, insulation, dimensional tolerance, toxicity, temperature, mechanical shielding, luminosity, and hydrostatic pressure – just to name a few areas. Review of a test offers a last chance to discover a missed constraint (requirement). Another possibility is that the chain of precursor events should actually lead to a different conclusion. Transduced values potentially control the status of any safety concern. Tests simply pass or fail, but evaluation of why a test passes or fails can become nontrivial, requiring collaboration between mechanical, software, and system safety engineers. ### Statistics {#S:STATISTICS} Some statistical error originates in inference from random sample to [unknown]{} population (parametric family of probability distributions on a measurable space). Just one distribution is true, while the others are false. An assertion separating the parameterization into two decision units is called a hypothesis. One decision unit is traditionally designated null, while the other is called alternate. The true distribution belongs either to the null or alternative decision units. Each sample item either passes or fails its associated test (see §\[S:PHYSICS\]). Within the entire cone ${\mathcal{C}}$, suppose the proportion of tests that fail is $\rho$. This proportion is subsequently realized approximately through a random sample. Regardless of the sample size, since the application is to safety, the only cases of interest will be when the number of failures is zero. Other cases, implying need for reliability growth, are treated in the literature, particularly [@jM87]. We now examine the case defined by drawing a random sample of size $N$ from ${{\operatorname{edge}{{\mathcal{C}}}}}$ and allowing $n = 0$ failures in the associated tests from cone ${\mathcal{C}}$. The null decision unit contains the probability distribution $P_0(\text{pass}) = 1$ and $P_0(\text{fail}) = 0$. The alternate decision unit is the set of probability distributions $P_\rho$ having $0 < \rho \leq 1$. Hypothesis evaluation entails two types of error, known as $\alpha$ and $\beta$ error. #### False rejection ($\alpha$ error) {#S:FALSE_REJECTION} The first is false rejection of the null decision unit, with associated measurement error $\alpha$. The sampling plan can reject only if finds an error, so this sampling plan is incapable of false rejection. Thus $\alpha \equiv 0$. #### False acceptance ($\beta$ error) {#S:FALSE_ACCEPTANCE} The second is false acceptance of the null decision unit, with associated measurement error $\beta$. We experience false acceptance when $0 < \rho$ but the sample contains no failures. Under the binomial model, the probability of observing a random sample of size $N$ with $n$ failures collectively is $\binom{N}{n} \rho^n {(1 - \rho)}^{N - n}$. Proceeding to our case of interest, $n = 0$, we have $\binom{N}{n} \rho^n {(1 - \rho)}^{N - n} \bigm|_{n=0} = {(1 - \rho)}^{N}$. This expression is the probability that random samples of size $N$ from a source of characteristic failure proportion $\rho$ will be accepted. #### Power function {#S:POWER_FUNCTION} It is confusing to reason in terms of contravariant[^7] attributes. In our case we formulate probability of rejection as an increasing function of $\rho$, a measure of the population’s undesirability. The probability that random samples will be properly rejected is the previous expression’s complement: $$K_{N,0}(\rho) = 1 - {(1 - \rho)}^{N} = 1 - \beta.$$ This non-contravariant result is known as the power function of sample size $N$, tolerating zero $(0)$ failures. The graph of the power function always increases, starting at $0$ for $\rho = 0$ and ending at $1$ for $\rho = 1$. Just how fast this function increases in its midrange is determined by the sample size $N$. With sample size one $(N = 1)$, $K_{1,0}(\rho) = \rho$. $N$ $K_{N,0}(.001)$ $K_{N,0}(.01)$ $K_{N,0}(.05)$ $K_{N,0}(.10)$ $K_{N,0}(.50)$ $K_{N,0}(.90)$ ------- ----------------- ---------------- ---------------- ---------------- ---------------- ---------------- 1 .0010 .0100 .0500 .1000 .5000 .9000 5 .0050 .0490 .2262 .4095 .9688 1.0000 10 .0100 .0956 .4013 .6513 .9990 1.0000 15 .0149 .1399 .5367 .7941 1.0000 1.0000 20 .0198 .1821 .6415 .8784 1.0000 1.0000 30 .0296 .2603 .7854 .9576 1.0000 1.0000 50 .0488 .3950 .9231 .9948 1.0000 1.0000 100 .0952 .6340 .9941 1.0000 1.0000 1.0000 200 .1814 .8660 1.0000 1.0000 1.0000 1.0000 500 .3936 .9934 1.0000 1.0000 1.0000 1.0000 1000 .6323 1.0000 1.0000 1.0000 1.0000 1.0000 2000 .8648 1.0000 1.0000 1.0000 1.0000 1.0000 5000 .9933 1.0000 1.0000 1.0000 1.0000 1.0000 10000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 : Family of power functions (probability of rejection)[]{data-label="Ta:POWER_FUNCTION"} Within this family $\beta = {(1 - \rho)}^{N} = 1 - K_{N,0}(\rho)$. One is initially dismayed by this sketch of the family of power functions; it suggests that high degrees of assurance are unobtainable through random sampling using practical sample sizes. However, reasonable performance useful for coarser screening is very possible. Detecting a defective population of 10 percent with a probability of approximately 90% requires only 20 sample items. ### Sampling philosophy {#S:SAMPLING_PHILOSOPHY} Our safety demonstration sampling technique contrasts two assurance philosophies – software reliability versus software correctness. The software reliability perspective involves a separate operational profile on ${{\operatorname{edge}{{\mathcal{C}}}}}$, whereas software correctness examines only the structure within cone ${\mathcal{C}}$. The operational profile asserts the importance of relative excitational intensity to safety analysis. An accident that occurs more frequently is worse than an accident that happens less frequently, given that they are of comparable severity. This safety factor is ignored under software correctness alone. Modeling accidents ------------------ Accidents are diverse in effect and mechanism, including injury, death, or damage either to equipment or environment. Since the causality of accidents is temporarily unknown, they manifest an apparent nature of unpredictability or randomness. However, under emulation as a stochastic process, the exact timing of accidents *is* truly a random phenomenon rather than causal. Nevertheless, it has proven useful to compare well-understood summary statistics of stochastic processes with those of deterministic but unknown physical processes. ### Compound Poisson process {#S:CPP} Today’s prevalent safety model for the occurrence of accidents is the compound Poisson[^8] process. This model captures accidents’ two dominant attributes: rate of occurrence (intensity) and scalar measure of loss (severity). With some exceptions, neither the timing nor severity of one software accident affects another. The compound Poisson process (CPP) is appropriate to model accidents of this nature. As stochastic processes are models rather than mechanisms, deriving their properties involves somewhat out-of-scope mathematics. The interested reader can immediately find greater detail in Wikipedia online articles: [@wW_Poisson_distribution], [@wW_Poisson_process], [@wW_total_expectation], [@wW_Compound_Poisson_process], and [@wW_Cumulant]. Relevant theorems will be documented here simply as facts. ### Poisson processes {#S:POISSON_PROCESSES} We will consider three variants of basic stochastic process: the ordinary Poisson process, the compound Poisson process, and the intermittent compound Poisson process. #### Ordinary Poisson process {#S:INTERMITTENT_POISSON} (Ordinary) Poisson processes are characterized simply by their rate or intensity: - its fundamental rate $\lambda$, which is the expected number of arrivals per unit time. Let $\lambda$ be the rate of a Poisson process. The probability of experiencing $k$ arrivals in a time interval $t$ units long is $$P_\lambda(k) = e^{-\lambda t}\frac{{(\lambda t)}^k}{k!}.$$ #### Compound Poisson process {#S:INTERMITTENT_POISSON} A compound Poisson process is characterized by two rates: - its fundamental rate $\lambda$ as before, and - its rate of loss $L$, which is a random variable invoked once for each arrival. Let $\lambda$ be the rate and $L$ be the loss random variable of a compound Poisson process. The expectation of the compound process for a time interval $t$ units long is $$\begin{aligned} {1} \E (\text{compound Poisson}) &= \lambda t \cdot \E (L)\\ &= \lambda t \cdot \mu_L.\end{aligned}$$ \[D:STATISTICAL\_RISK\] The statistical *risk*, written $h$, of a compound Poisson process is the time derivative of its expectation in a duration of length $t$; that is $$h = \frac{d}{dt} \E(\text{compound Poisson}) = \frac{d}{dt} (\lambda t \cdot \mu_L) = \lambda \mu_L,$$ which is the product of its rate $\lambda$ and its expected loss $\mu_L$. #### Intermittent compound Poisson process {#S:INTERMITTENT_POISSON} A variation of the CPP is the intermittent compound Poisson process, which is intermittently on or off with expected durations $\E(\text{on}) = \mu_\text{on}$ and $\E(\text{off}) = \mu_\text{off}$. An intermittent compound Poisson process (ICPP) is characterized by three rates: - its fundamental rate $\lambda$ as before, and - its rate of loss $L$, also as before, - alternating durations of random lengths $\tau_\text{on}$ and $\tau_\text{off}$. Random variables $\tau_\text{on}$ and $\tau_\text{off}$ converge to $\mu_\text{on}$ and $\mu_\text{off}$ in the limit. The *idle* ratio of a intermittent compound Poisson process is $\iota = \frac{\mu_\text{off}}{\mu_\text{on} + \mu_\text{off}}$. Let $\lambda$ be the rate, $L$ be the loss random variable, and $\iota$ be the idle ratio of an intermittent compound Poisson process. The expectation of the ICPP for a time interval $t$ units long is $$\begin{aligned} {1} \E (\text{intermittent compound Poisson}) &= (1 - \iota) \cdot \lambda t \cdot \E(L) \\ &= (1 - \iota) \cdot \lambda t \cdot \mu_L.\end{aligned}$$ The statistical risk of an ICPP is $$\begin{aligned} {1} h &= \frac{d}{dt} \E(\text{intermittent compound Poisson}) \\ &= \frac{d}{dt} ((1 - \iota) \cdot \lambda t \cdot \mu_L + \iota \cdot 0 t \cdot 0) \\ &= (1 - \iota) \lambda \mu_L.\end{aligned}$$ Indemnification {#S:INDEMNIFICATION_FORMULA} --------------- Hypothesize that a software hazard is emulated by a compound Poisson process (CPP) having intensity $\lambda$ and expected loss $\mu_L$. Suppose further that the actual control mechanism is a cone convergent to the software point of exhibition of the hazard. We wish to consider statistical evidence that the hazard’s hypothetical description via the stochastic process is consistent with its mechanism as revealed by safety demonstration. ### Unification {#S:UNIFICATION} Before undertaking the question of whether test data supports a hypothetical stochastic process, we must establish the theoretical conditions under which equality is expected. #### Fundaments of the model The compound Poisson process is a model stochastic process for occurrence of accidents. This model is used in safety analysis to quantify the occurrence and losses of accidents without considering their causes. MIL-STD-882 (see Appendix \[S:MIL-STD-882\]) is an important example. In a time interval of duration $t$, accidents converge stochastically in rate to expectation $\lambda t$ and in mean loss to $\mu_L$. This means an intensity of $\lambda$ accidents per time unit. #### Fundaments of the mechanism The actuated automaton is a mechanism representing software. When extended by the principle of emergence (§\[S:PRINCIPLE\_OF\_EMERGENCE\]) and the constructs of the operational profiles (§\[S:OPERATIONAL\_PROFILE\_SECTION\]) and cones (§\[S:CONE\_SECTION\]), it becomes capable of representing precursor conditions for software accidents. Let ${\Vert{{{\operatorname{edge}{{\mathcal{C}}}}}}\Vert}$ (see §\[S:ABSOLUTE\_OP\_PROFILE\_RATE\]) be the rate-based absolute operational profile of the edge of an acyclic cone ${\mathcal{C}}$. Since a member of ${{\operatorname{edge}{{\mathcal{C}}}}}$ is executed at the average intensity of ${{\Vert{{{\operatorname{edge}{{\mathcal{C}}}}}}\Vert}}$, then so is the cone’s step of convergence ${\mathit{s}}_\text{crux}$. Let $\rho$ be the proportion of failing tests (localized predecessor walks). Under that supposition, failures occur at the intensity of $\rho \cdot {\Vert{{{\operatorname{edge}{{\mathcal{C}}}}}}\Vert}$. The definition of ${\Vert{{{\operatorname{edge}{{\mathcal{C}}}}}}\Vert}$, through the internal function $\text{sync}(\cdot)$, allows for the passage of time in the proper duration. #### Uniting mechanism and model We presume that one failing test equals one accident. The cone’s step of convergence is considered to be the point of exhibition of a hazard whenever safety constraints are not met. This mechanism may be separately equated to the intensity (not the rate of loss) of the compound Poisson process: $$\lambda = \rho \cdot {\Vert{{{\operatorname{edge}{{\mathcal{C}}}}}}\Vert}.$$ This equation places a property of the model on the left and properties of the mechanism on the right. The execution rate of the edge of an acyclic cone numerically equals the execution rate of the (set containing the) cone’s crux. The cone’s definitional status (as a complete independent set of localized predecessor walks ending at ${\mathit{s}}_{\text{crux}}$) causes this. Symbolically, $${\Vert{{{\operatorname{edge}{{\mathcal{C}}}}}}\Vert} = {\Vert{\lbrace{\mathit{s}}_{\text{crux}}\rbrace}\Vert}.$$ ### Evidence {#S:EVIDENCE} We propose that the same data used earlier for indemnification testing be re-used in a slightly different statistical context. Recall that an indemnification test has $N$ items among which are zero failures, where each item is a localized predecessor walk, and a cone is a structured collection of localized predecessor walks of an automaton. We wish to measure the amount of information in an indemnification sample to explain the phenomenon that larger samples justify more precise estimates than smaller samples. We refer to this information as measuring the *weight of evidence*. This situation differs from the familiar problem of finding the maximum likelihood estimator. #### Method of indifference {#S:INDIFFERENCE_POWER_FUNCTION} The power function of sample size $N$, tolerating zero $(0)$ failures, is $K_{N,0}(\rho) = 1 - {(1 - \rho)}^{N}$ (see §\[S:POWER\_FUNCTION\]). It measures the probability of rejection as a function of $\rho$. Each power function $K_{N,0}(\rho) = 1 - {(1 - \rho)}^{N}$ is characterized by its indifference proportion, which is defined as the proportion at which rejection and acceptance become equally likely (that is, $K_{N,0}(\hat{\rho}_\text{\,I}) = {^1\!\!/\!_2} = 1 - K_{N,0}(\hat{\rho}_\text{\,I})$). With only modest algebra, the analytic expression for the indifference proportion may be derived from the power function $K_{N,0}(\rho)$; it is $$\hat{\rho}_\text{\,I} = 1 - \sqrt[N]{{^1\!\!/\!_2}}. $$ Below is a numerical tabulation of the previous formula: $N$ $\hat{\rho}_{\,\text{indifference}}$ ------- -------------------------------------- 1 .50000 5 .12945 10 .06697 15 .04516 20 .03406 30 .02284 50 .01377 100 .00691 200 .00346 500 .00139 1000 .00069 2000 .00035 5000 .00014 10000 .00007 : Indifference proportion[]{data-label="Ta:INDIFFERENCE_PROPORTION"} #### Indemnification formula {#S:UPPER_BOUND} The indemnification formula provides a statistical upper bound on hazard intensity. Indemnification data may be expressed as an equivalent statistical upper bound on hazard intensity. This differs fundamentally from estimating the intensity of a hazard. By a statistically [guaranteed]{} hazard intensity, we mean an upper bound such that the true hazard intensity is likely to fall beneath this level with known confidence (probability). Suppose we choose ${^1\!\!/\!_2}$ as the known confidence. The indifference proportion $\hat{\rho}_\text{\,I} = 1 - \sqrt[N]{{^1\!\!/\!_2}}$ then has a second interpretation as an upper bound with confidence ${^1\!\!/\!_2}$. For any $\rho \leq \hat{\rho}_\text{\,I}$, it is true that power function $P_{N,0}(\rho) \leq {^1\!\!/\!_2}$, so $\hat{\rho}_\text{\,I}$ is an upper bound of confidence ${^1\!\!/\!_2}$. To convert from the size of the indemnification sample into its equivalent upper bound hazard intensity, find the indifference proportion $\hat{\rho}_{\,\text{I}} = 1 - \sqrt[N]{{^1\!\!/\!_2}}$. Check the sample physics. This amounts to analysis of the originating cone ${\mathcal{C}}$, which is the point of exhibition of a hazard whenever safety constraints are not met. The cone’s edge has an absolute operational profile expressed as a rate. This quantity is the counting norm of the cone’s edge. We have shown that the probable upper bound of the hazard intensity is proportional to the indifference proportion, with constant of proportionality furnished by the counting norm of the cone’s edge. The indemnification formula is: $$\hat{\lambda}_{\,\text{I}} = \hat{\rho}_{\,\text{I}} \cdot {\Vert{{{\operatorname{edge}{{\mathcal{C}}}}}}\Vert} = \hat{\rho}_{\,\text{I}} \cdot {\Vert{\lbrace{\mathit{s}}_{\text{crux}}\rbrace}\Vert}.$$ Epilogue ======== From previous discussion two structures of system safety emerge: the safety demonstration and indemnification, its measure of assurance. Opinion follows. Programmatic fit {#S:PROGRAMMATIC_FIT} ---------------- Safety demonstration and indemnification merge smoothly into today’s programmatic picture. Early in the development cycle, safety engineers provide [ballpark]{} quantifications of the threat of hazards[@DD12], expressed as intensity and severity. These numbers are often educated guesses: a mixture of circumstance, intuition, similar design, and history. At that stage, the process is without supporting evidence. Later in the development cycle, assuming a program of structured testing has been followed, statistical evidence is available in the form of a safety demonstration. These data are expressed as a statistical upper bound on each software hazard’s intensity – that is, an indemnification – and used as evidence of correct operation. This additional step frees the safety engineer from having to re-asses his original estimate using the same shallow method as the original guesstimate. Increasingly, automatic static syntax analyzers satisfy need for overall code robustness. However, exclusive reliance on these analyzers would result in a software safety engineering shortcoming, because they do not always detect code defects that have valid syntax (that is, syntactically valid but wrong algorithm). Commercialization {#S:COMMERCIALIZATION} ----------------- Difficult work remains before safety demonstration and indemnification can be supported as mature commercial technology. The role of the actuated automaton must be replaced by a real-world programming language. Present theory restricting tests (predecessor walks) to acyclic cones may require generalization to cyclic cones to achieve broader range. Safety demonstration demands the ability to produce approximate operational profiles from which can be drawn pseudo-random samples. Spin-offs from similar technologies may be possible; static analyzers are one example. Criticism of MIL-STD-882 {#S:CRITICISM} ------------------------ Statistical risk is a model (see §\[S:INTRO\_RISK\]) emulating the threat of accidents. Users of MIL-STD-882 are familiar with statistical risk as an accident model for hardware. Software is properly deterministic and therefore non-stochastic, but it’s successfully approximated with the same Poisson stochastic process as hardware. The rationale for this approximation is to apply a useful Poisson mathematical assurance property; thus assurance depends on the validity of the Poisson process as an approximation to the actuated automaton. The Standard introduces a risk-like scale replacing statistical risk for software. For purpose of this chapter, we call this replacement scale the *design* risk. The discussion of chapters 2 and 3 conclude that, from the standpoint of statistical assurance, there is no justification for the presently differing versions of the term *risk* between hardware and software. The state of software engineering is mixed science and sophisticated art. In the current Standard, art has somewhat overtaken science; for software, the concept of statistical risk has been abandoned. One subsequently loses the ability to measure and assure risk uniformly. Design risk and statistical risk do share a severity axis, but the similarity ends there. Statistical risk has another axis composed of a numerical product, the frequency of execution times the probability of error. What we have called design risk also possesses another axis, but it is a categorical scale arranged in decreasing order of the design safety importance of the software’s functionality. This results in an error of omission in MIL-STD-882 with serious consequence. Because the frequency of execution of a software point is not well-correlated with its functionality’s design safety importance, statistical risk does not correlate well with design risk. Because the Standard assigns statistical risk to hardware and design risk to software, and due to lack of correlation between the two, there is no way to rank the relative importance of hazards of mixed type. Loss of the ability to compare risks of all hazards is a flagrant omission. Under correct physics, risks of multiple hazards are additive. This is not the case under MIL-STD-882. The formal sense of *assurance* is lost by these definitional variants. Being quantitatively assured requires a limit value on proportion or mean deviation and a statement of statistical confidence for this limit; the Standard clearly lacks this characteristic. Properly assurance is a numeric quantity associated with statistical control of risk, not an engineering activity to further psychological confidence. Despite that its developers may express great confidence in the methodology, software built under the Standard is not quantitatively assured. Repair of MIL-STD-882 {#S:REPAIR} --------------------- Rehabilitation of MIL-STD-882 is straightforward. It must be amended to contain an engineering introduction to statistical risk for software, including allied procedures. This subject matter is covered here in mathematical language, but should be presented differently for engineers’ consumption. The revised Standard should distinguish between formal assurance and design confidence, and classify what procedures support either. Generally, the concerns associated with design risk align with developmental software engineering, while those of statistical risk align with responsibilities of system safety engineering, part of systems engineering. Software and Systems Engineering should not duplicate each other’s efforts. Groundwork {#Ch:GROUNDWORK} ========== This appendix examines ensembles, a fundamental structure in the theory of systems which formalizes the notion of stimulus and response. From this start, discussion proceeds into the Cartesian product, choice spaces and subspaces, and partitions of choice spaces into persistent and volatile components. Dyadic notation is introduced. Ensemble {#S:ENSEMBLE} -------- An ensemble is a special form of a more general structure known as a *family*. We employ nomenclature abridged from Halmos [@pH74 p. 34] as follows: Let $I$ and $X$ be non-empty sets, and $\varphi \colon I \to X$ be a mapping. Each element $i \in I$ is an *index*, while $I$ itself is an *index set*. The mapping $\varphi$ is a *family*; its co-domain $X$ is an *indexed set*. An ordered pair $(i,x)$ belonging to the family is a *term*, whose value $x = \varphi(i)\in X$ is often denoted $\varphi_i$. The family $\varphi$ itself is routinely but abusively denoted $\{\varphi_n\}$. This notation is a compound idiom.[^9] Especially in the case of sequences over a set $G$, the symbol $\{g_n\}$ signifies the mapping $\{ 1\mapsto g_1, \; 2\mapsto g_2, \; \ldots \;\}$. \[D:ENSEMBLE\] An *ensemble* is a non-empty family. \[D:CONSTANT\_VARIABLE\] Let $\Psi$ be an ensemble, with ${{\operatorname{ran}{\Psi}}}$ its range. If ${\lvert{{{\operatorname{ran}{\Psi}}}}\rvert} = 1$, the ensemble is *constant*; otherwise it is *variable*. Since physical systems possess only a finite number of attributes, the scope of practical interest is limited to ensembles having finite-dimensional index sets. A constant ensemble (definition \[D:CONSTANT\_VARIABLE\]) is also referenced under the historically colorful name *Hobson’s*[^10] choice. Using the word *choice* in its everyday sense, a Hobson’s choice is oxymoronic: a free choice in which only one option is offered, with gist [Only one choice is no choice.]{} Let $\Psi$ be an ensemble. For term $(i,P) \in \Psi$, we denote $P = \Psi_i$. Ensemble arithmetic {#S:ENSEMBLE_ARITHMETIC} ------------------- \[D:DISJOINT\_ENSEMBLES\] Ensembles $\Psi$ and $\Phi$ are *disjoint* if ${{\operatorname{dom}{\Psi}}}\thickspace\cap\thickspace{{\operatorname{dom}{\Phi}}} = \varnothing$ (that is, if their index sets have no member in common). \[D:COMPLEMENTARY\_ENSEMBLES\] Ensembles $\Psi$ and $\Phi$ are *complementary* with respect to a third ensemble $\Upsilon$ if they are disjoint and $\Psi \cup \Phi = \Upsilon$. Regarding ensembles $\Psi$ and $\Phi$, we write $\Phi \subseteq \Psi$ to express that $\Phi$ is contained in $\Psi$, following ordinary set theory that term $(i,P) \in \Phi$ implies $(i,P) \in \Psi$. \[D:ENSEMBLE\_DIFFERENCE\] Let $\Psi$ and $\Phi$ be ensembles such that $\Phi \subseteq \Psi$. In classification of difference between $\Psi$ and $\Phi$, $\Psi$ is the *minuend*, $\Phi$ is the *subtrahend*, and the set difference[^11] $\Psi \setminus \Phi$ is the *remainder*. \[L:DISJOINT\_AND\_COMPLEMENTARY\] Let $\Psi$ and $\Phi$ be ensembles such that $\Phi \subseteq \Psi$. The subtrahend $\Phi$ and remainder $\Psi \setminus \Phi$ are disjoint and complementary with respect to $\Psi$. Since $\Phi \subseteq \Psi$, definition \[D:ENSEMBLE\_DIFFERENCE\] applies. The minuend is $\Psi$, the subtrahend is $\Phi$, and the remainder is $\Psi \setminus \Phi$. Definition \[D:DISJOINT\_ENSEMBLES\] asserts that two ensembles $\Theta$ and $\Upsilon$ are disjoint if ${{\operatorname{dom}{\Theta}}}\thickspace\cap\thickspace{{\operatorname{dom}{\Upsilon}}} = \varnothing$. As hypothesis presume the lemma’s antithesis, namely that the subtrahend $\Phi$ and the remainder $\Psi \setminus \Phi$ are not disjoint. This implies that there exists some $i$ such that $i \in {{\operatorname{dom}{\Phi}}}\thickspace\cap\thickspace{{\operatorname{dom}{(\Psi \setminus \Phi)}}}$. For this $i$ to exist in the intersection of the domains of two mappings, there must be both a term $(i,P) \in \Phi$ and a term $(i,Q) \in (\Psi \setminus \Phi)$. Because $(i,P) \in \Phi$ and the lemma’s premise states that $\Phi \subseteq \Psi$, then $(i,P) \in \Psi$. Since term $(i,Q) \in (\Psi \setminus \Phi)$, then $(i,Q) \in \Psi$ and $(i,Q) \notin \Phi$. Since $\Psi$ is a mapping, then $(i,P) \in \Psi$ and $(i,Q) \in \Psi$ together imply $P = Q$. With $P = Q$ and $(i,Q) \notin \Phi$, then also $(i,P) \notin \Phi$. However, the immediately preceding conclusion that $(i,P) \notin \Phi$ contradicts the earlier inference that term $(i,P) \in \Phi$ must exist if $i$ is a member of the intersection of the domains. Since the presumption that the subtrahend and remainder are not disjoint arrives at a contradiction, then subtrahend $\Phi$ and remainder $\Psi \setminus \Phi$ are indeed disjoint. The ensembles are complementary with respect to $\Psi$ by definition \[D:COMPLEMENTARY\_ENSEMBLES\] because they are disjoint and $\Phi \cup (\Psi \setminus \Phi) = \Psi$. Choice space {#S:CHOICE_SPACE} ------------ Informally, a choice space is the totality of all possible combinations of variables’ values within a given ensemble. The general Cartesian product formalizes this notion. ### General Cartesian product {#S:GCP} \[D:PROTOSET\] Let $\Psi$ be an ensemble. By definition \[D:ENSEMBLE\], each member of ${{\operatorname{ran}{\Psi}}}$ is itself a non-empty set. The *proto-set* $\Psi_\heartsuit$ is the union of all such sets: $$\Psi_\heartsuit = \bigcup_{R \in\: {{\operatorname{ran}{\Psi}}}} R.$$ Definition \[D:PROTOSET\] expresses a relation between the ensemble $\Psi$’s indexed sets and the proto-set, namely that for each $i \in {{\operatorname{dom}{\Psi}}}$, $\Psi(i) \subseteq \Psi_\heartsuit$. An alternative portrayal uses the power set, claiming $\Psi(i) \in {\mathscr{P}({\Psi_\heartsuit})}$. \[D:PROTOSPACE\] Let $\Psi$ be an ensemble with proto-set $\Psi_\heartsuit$. The *proto-space* of $\Psi$ is the set ${\Psi_\heartsuit}^{{{\operatorname{dom}{\Psi}}}}$, where ${\Psi_\heartsuit}^{{{\operatorname{dom}{\Psi}}}}$ is the set of all mappings ${{\operatorname{dom}{\Psi}}} \to \Psi_\heartsuit$. \[D:CHOICE\] Let $\Psi$ be an ensemble with proto-set $\Psi_\heartsuit$. Let $\chi \colon {{\operatorname{dom}{\Psi}}} \to \Psi_\heartsuit$ be a mapping. If $\chi$ satisfies $\chi(i) \in \Psi(i)$ for each $i \in {{\operatorname{dom}{\Psi}}}$, then $\chi$ is a *choice* mapping of $\Psi$. \[D:CHOICE\_SPACE\] The set of all choice mappings of ensemble $\Psi$ is the *choice space* (or general Cartesian product) $\prod\Psi$. Through the general Cartesian product, an ensemble *generates* a choice space. For brevity we refer to a point in a choice space (that is, a choice mapping) simply as a *choice*. \[T:PROTOSPACE\_INCLUDES\_CHOICESPACE\] The proto-space ${\Psi_\heartsuit}^{{{\operatorname{dom}{\Psi}}}}$ of ensemble $\Psi$ includes its choice space $\prod\Psi$ (that is, $\prod\Psi \subseteq {\Psi_\heartsuit}^{{{\operatorname{dom}{\Psi}}}}$). Suppose $\chi \in \prod\Psi$. By definition \[D:CHOICE\], $\chi$ is a mapping ${{\operatorname{dom}{\Psi}}} \to \Psi_\heartsuit$. Then, by definition \[D:PROTOSPACE\], $\chi \in {\Psi_\heartsuit}^{{{\operatorname{dom}{\Psi}}}}$. Thus, any member of $\prod\Psi$ is also a member of ${\Psi_\heartsuit}^{{{\operatorname{dom}{\Psi}}}}$. From this we conclude $\prod\Psi \subseteq {\Psi_\heartsuit}^{{{\operatorname{dom}{\Psi}}}}$. \[T:ENSEMBLE\_UNIQ\_SPACE\] An ensemble generates one unique choice space: let $\Theta$ and $\Phi$ be two ensembles generating choice spaces $\prod\Theta$ and $\prod\Phi$ respectively. If $\Theta = \Phi$, then $\prod\Theta = \prod\Phi$. To show the contrapositive, suppose $\prod\Theta \not= \prod\Phi$. This premise can be true if either A: there is a choice $\alpha \in \prod\Theta$ such that $\alpha \notin \prod\Phi$, or if B: there is a choice $\beta \in \prod\Phi$ such that $\beta \notin \prod\Theta$. In case A, $\alpha$ is a choice of $\Theta$ – that is, for each $k \in {{\operatorname{dom}{\Theta}}}$, $\alpha(k) \in \Theta(k)$. For hypothesis, assume $\Theta = \Phi$, so that ${{\operatorname{dom}{\Theta}}} = {{\operatorname{dom}{\Theta}}}$. Then for each $k \in {{\operatorname{dom}{\Theta}}}$, $\alpha(k) \in \Phi(k)$, since by equality hypothesis $\Theta(k) = \Phi(k)$ and $\alpha(k) \in \Theta(k)$. This means $\alpha$ is a choice of $\Phi$, that is, $\alpha \in \prod\Phi$. However, this conclusion contradicts the second part of the premises for case A, namely that $\alpha \notin \prod\Phi$. Thus the hypothesis $\Theta = \Phi$ is false, so $\Theta \not= \Phi$. The argument for case B is the same as A, except reversing the roles of $\Theta$ and $\Phi$. Together, cases A and B show that $\prod\Theta \not= \prod\Phi$ implies $\Theta \not= \Phi$. Applying the contrapositive principle[^12] gives $\prod\Theta = \prod\Phi$ if $\Theta = \Phi$. \[T:SPACE\_UNIQ\_ENSEMBLE\] Any choice space has one unique generating ensemble: let $\Theta$ and $\Phi$ be two ensembles generating choice spaces $\prod\Theta$ and $\prod\Phi$ respectively. If $\prod\Theta = \prod\Phi$, then $\Theta = \Phi$. To show the contrapositive, suppose $\Theta \not= \Phi$. From this premise there must exist either A: $(i,P) \in \Theta$ such that $(i,P) \notin \Phi$, or B: $(j,Q) \in \Phi$ such that $(j,Q) \notin \Theta$. The first case A decomposes into two sub-cases: either A1: $i \in S$ and $i \in S'$, or A2: $i \in S$ and $i \notin S'$. In sub-case A1, we must have $\Theta(i) \not= \Phi(i)$ to support $\Theta \not= \Phi$. For this there must be either A1a: there is $u \in \Theta(i)$ such that $u \notin \Phi(i)$, or A1b: there is $v \in \Phi(i)$ such that $v \notin \Theta(i)$. In sub-sub-case A1a, there must exist by definitions \[D:CHOICE\_SPACE\] and \[D:CHOICE\], a choice $\alpha \in \prod\Theta$ such that $\alpha(i) = u$. However, there can be no choice $\beta \in \prod\Phi$ such that $\beta(i) = u$, since $u \notin \Phi(i)$. Therefore $\alpha \notin \prod\Phi$, with the consequence that $\prod\Theta \not= \prod\Phi$. In sub-sub-case A1b there is $v \in \Phi(i)$ such that $v \notin \Theta(i)$. Similarly to A1a, there exists $\beta \in \prod\Phi$ such that $\beta(i) = v$, but no $\alpha \in \prod\Theta$ such that $\alpha(i) = v$, again concluding that $\prod\Theta \not= \prod\Phi$. In sub-case A2, we have $i \in {{\operatorname{dom}{\Theta}}}$ and $i \notin {{\operatorname{dom}{\Phi}}}$. By definitions \[D:CHOICE\_SPACE\] and \[D:CHOICE\], for any $x \in \Theta(i)$, there exists choice $\alpha_x \in \prod\Theta$ such that $\alpha_x(i) = x$. However, there is no choice $\beta \in \prod\Phi$ such that $\beta(i) = x$ because $i \notin {{\operatorname{dom}{\Phi}}} = {{\operatorname{dom}{\beta}}}$. This concludes $\prod\Theta \not= \prod\Phi$ for sub-case A2. The proofs for subordinate cases of B are the same as A, reversing the roles of $\Theta$ and $\Phi$. Taken together, these sub-case analyses show that exhaustively $\Theta \not= \Phi$ implies $\prod\Theta \not= \prod\Phi$. By contraposition, $\Theta = \Phi$ if $\prod\Theta = \prod\Phi$. \[T:SPACE\_BIJECTION\_ENSEMBLE\] Choice spaces and generating ensembles are in one-to-one correspondence via the general Cartesian product. Theorem \[T:ENSEMBLE\_UNIQ\_SPACE\] asserts that any ensemble generates only one unique choice space, and \[T:SPACE\_UNIQ\_ENSEMBLE\] asserts that any choice space has only one unique generating ensemble. \[N:CARTESIAN\_PRODUCT\_INVERSE\] By Theorem \[T:SPACE\_BIJECTION\_ENSEMBLE\], the Cartesian product is invertible, providing a mapping from choice spaces to generating ensembles. There is need for a symbol designating this inverse. In preference to $\prod^{-1}$, for present purpose we borrow the coproduct symbol $\coprod$ from another field, since there is no danger of confusion. \[D:FINITE\_CHOICE\_SPACE\] A *finite* choice space is a choice space (definition \[D:CHOICE\_SPACE\]) whose generating ensemble has a finite index set. ### Dyadic Operators {#S:DYADIC_OPERATORS} This section presents binary operators on ensembles and choice spaces whose intended usage will value notational compactness. To this end these operators will be displayed through dyadic notation, which indicates an operator application implicitly by simple juxtaposition of the operator’s arguments, forgoing explicit rendering of the operation’s symbol as a prefix, infix, or suffix. For reason of denotational style, these operators will be called products despite that they suggest sums. \[D:DYADIC\_ENSEMBLE\_PRODUCT\] Let $\Theta$ and $\Phi$ be disjoint (definition \[D:DISJOINT\_ENSEMBLES\]) ensembles. The dyadic product $(\Theta,\Phi) \mapsto \Theta\Phi$ is $$\Theta\Phi = \Theta \cup \Phi.$$ An immediate consequence of this definition is commutativity $\Theta\Phi = \Phi\Theta$, since $\Theta \cup \Phi = \Phi \cup \Theta$. \[T:DYADIC\_PRODUCT\_IS\_ENSEMBLE\] Let $\Theta$ and $\Phi$ be disjoint ensembles. Their dyadic product $\Upsilon = \Theta\Phi$ is an ensemble with domain $({{\operatorname{dom}{\Theta}}} \cup {{\operatorname{dom}{\Phi}}})$ and range $({{\operatorname{ran}{\Theta}}} \cup {{\operatorname{ran}{\Phi}}})$. Suppose $i \in {{\operatorname{dom}{\Theta\Phi}}}$. By definition \[D:DYADIC\_ENSEMBLE\_PRODUCT\], $\Theta\Phi = \Theta \cup \Phi$. From this it follows that $i \in {{\operatorname{dom}{\Theta}}} \cup {{\operatorname{dom}{\Phi}}}$. Therefore, $i \in {{\operatorname{dom}{\Theta}}}$, $i \in {{\operatorname{dom}{\Phi}}}$, or both. The stipulation that $\Theta$ and $\Phi$ be disjoint entails ${{\operatorname{dom}{\Theta}}} \cap {{\operatorname{dom}{\Phi}}} = \varnothing$ (definition \[D:DISJOINT\_ENSEMBLES\]). That stipulation eliminates the possibility of both memberships, leaving two feasible cases: either A) $i \in {{\operatorname{dom}{\Theta}}}$ and $i \notin {{\operatorname{dom}{\Phi}}}$, or B) $i \notin {{\operatorname{dom}{\Theta}}}$ and $i \in {{\operatorname{dom}{\Phi}}}$. In case A, there exists $(i, \Theta_i) \in \Theta$, so $(i, \Theta_i) \in \Upsilon = \Theta \cup \Phi$. Furthermore, since $i \notin {{\operatorname{dom}{\Phi}}}$, $\Upsilon_i = \Theta_i$ is well-defined. Since by definition \[D:ENSEMBLE\] each member of ${{\operatorname{ran}{\Theta}}}$ is a non-empty set, then equivocally $\Upsilon_i = \Theta_i$ is a non-empty set. Argumentation for case B, supporting that $\Upsilon_i = \Phi_i$ is well-defined and that $\Upsilon_i$ is a non-empty set, is obtained by interchanging the roles of $\Theta$ and $\Phi$ in case A. Over all possibilities, $\Upsilon = \Theta \cup \Phi$ is well-defined as a mapping (family) and each element of its range is a non-empty set. Therefore the dyadic product of two disjoint ensembles is itself an ensemble. The sub-case analysis establishes ${{\operatorname{dom}{\Theta\Phi}}} \subseteq ({{\operatorname{dom}{\Theta}}} \cup {{\operatorname{dom}{\Phi}}})$. For the converse, first assume $i \in {{\operatorname{dom}{\Theta}}}$. Then there is some $P$ such that $(i,P) \in \Theta$. From this it follows that $(i,P) \in \Theta \cup \Phi$, so $i \in {{\operatorname{dom}{\Theta\Phi}}}$ and ${{\operatorname{dom}{\Theta}}} \subseteq {{\operatorname{dom}{\Theta\Phi}}}$. Similarly, ${{\operatorname{dom}{\Phi}}} \subseteq {{\operatorname{dom}{\Theta\Phi}}}$. From these two inclusions it follows that $({{\operatorname{dom}{\Theta}}} \cup {{\operatorname{dom}{\Phi}}}) \subseteq {{\operatorname{dom}{\Theta\Phi}}}$. With the previous result that ${{\operatorname{dom}{\Theta\Phi}}} \subseteq ({{\operatorname{dom}{\Theta}}} \cup {{\operatorname{dom}{\Phi}}})$, we conclude ${{\operatorname{dom}{\Theta\Phi}}} = ({{\operatorname{dom}{\Theta}}} \cup {{\operatorname{dom}{\Phi}}})$. The sub-case analysis shows also that for $(i, P) \in \Theta\Phi$, either $P \in {{\operatorname{ran}{\Theta}}}$ or $P \in {{\operatorname{ran}{\Phi}}}$. This implies that ${{\operatorname{ran}{\Theta\Phi}}} \subseteq {{\operatorname{ran}{\Theta}}} \medspace\cup\medspace {{\operatorname{ran}{\Phi}}}$. To demonstrate the converse, assume $P \in {{\operatorname{ran}{\Theta}}}$. Then there exists $(i,P) \in \Theta$, and by definition \[D:DYADIC\_ENSEMBLE\_PRODUCT\], $(i,P) \in \Theta\Phi$. From this we conclude that ${{\operatorname{ran}{\Theta}}} \subseteq {{\operatorname{ran}{\Theta\Phi}}}$. With modest changes as above we conclude ${{\operatorname{ran}{\Phi}}} \subseteq {{\operatorname{ran}{\Theta\Phi}}}$. This establishes the converse ${{\operatorname{ran}{\Theta}}} \medspace\cup\medspace {{\operatorname{ran}{\Phi}}} \subseteq {{\operatorname{ran}{\Theta\Phi}}}$. Finally, with both ${{\operatorname{ran}{\Theta\Phi}}} \subseteq {{\operatorname{ran}{\Theta}}} \medspace\cup\medspace {{\operatorname{ran}{\Phi}}}$ and its converse, ${{\operatorname{ran}{\Theta\Phi}}} = {{\operatorname{ran}{\Theta}}} \medspace\cup\medspace {{\operatorname{ran}{\Phi}}}$. \[D:DYADIC\_SPACE\_PRODUCT\] Let $\Theta$ and $\Phi$ be disjoint ensembles. The dyadic space product of $\prod\Theta$ and $\prod\Phi$ is $$\prod \Theta \prod \Phi = \prod \Theta\Phi$$ (that is, with $\Upsilon = \Theta\Phi$, the set of all $\Upsilon$-choices). \[D:DYADIC\_CHOICE\_PROD\] Let $\Theta$ and $\Phi$ be disjoint ensembles generating choice spaces $\prod\Theta$ and $\prod\Phi$. Suppose $\alpha \in \prod\Theta$ and $\beta \in \prod\Phi$ are choices. Their dyadic *choice* product $(\alpha,\beta) \mapsto \alpha\beta$ is $$\alpha\beta = \alpha \cup \beta.$$ \[T:DYADIC\_CHOICE\_PROD\] Let $\Theta$ and $\Phi$ be disjoint ensembles. The dyadic choice product is a bijection $$\prod\Theta \times \prod\Phi \leftrightarrow \prod\Theta\Phi.$$ Suppose $(\theta, \phi) \in \prod\Theta \times \prod\Phi$, whence it follows that choices $\theta \in \prod\Theta$ and $\phi \in \prod\Phi$. Consider the dyadic products $\theta\phi = \theta \cup \phi$ and $\Theta\Phi = \Theta \cup \Phi$. Since ${{\operatorname{dom}{\theta}}} = {{\operatorname{dom}{\Theta}}} = T$ and ${{\operatorname{dom}{\phi}}} = {{\operatorname{dom}{\Phi}}} = P$, then ${{\operatorname{dom}{\theta\phi}}} = T \cup P = {{\operatorname{dom}{\Theta\Phi}}}$. Let $i \in {{\operatorname{dom}{\theta\phi}}} = T \cup P$. Since $\Theta$ and $\Phi$ are disjoint, then exactly one of two cases holds: either $i \in T$ and $i \notin P$, or $i \notin T$ and $i \in P$. In the first case, $i \in T$, $\theta\phi(i) = \theta(i)$ and $\Theta\Phi(i) = \Theta(i)$. Since $\theta$ is a $\Theta$-choice, then $\theta(i) \in \Theta(i)$. But since $\theta\phi(i) = \theta(i)$ and $\Theta\Phi(i) = \Theta(i)$, then $\theta\phi(i) \in \Theta\Phi(i)$ for any $i \in T$. The second case is similar and leads to $\theta\phi(i) \in \Theta\Phi(i)$ for any $i \in P$. From these two cases we conclude that $\theta\phi(i) \in \Theta\Phi(i)$ for any $i \in T \cup P = {{\operatorname{dom}{\theta\phi}}}$, and that $\theta\phi \in \prod\Theta\Phi$. The preceding establishes that the choice product is a relation between $\prod\Theta \times \prod\Phi$ and $\prod\Theta\Phi$. For this relation to be a mapping, it must yet be established that any member of the domain is related to exactly one member of the co-domain. Again with $(\theta, \phi) \in \prod\Theta \times \prod\Phi$, suppose $\alpha \in \prod\Theta\Phi$ and $\beta \in \prod\Theta\Phi$ are $\Theta\Phi$-choices such that $(\theta, \phi) \mapsto \alpha$ and $(\theta, \phi) \mapsto \beta$. The hypothesis $\alpha \not= \beta$ now leads to the contradiction $\theta \cup \phi \not= \theta \cup \phi$, so the hypothesis is false and $\alpha = \beta$. Therefore the dyadic choice product is a mapping. To be bijective, this mapping must be injective and surjective. To assess injection, let $\theta$, $\theta' \in \Theta$ and $\phi$, $\phi' \in \prod\Phi$, and hypothesize $\theta\phi = \theta'\phi'$. With $\theta\phi = \theta'\phi'$, then the restrictions $\theta\phi\mid{{\operatorname{dom}{\Theta}}} = \theta'\phi'\mid{{\operatorname{dom}{\Theta}}}$. By \[D:DYADIC\_CHOICE\_PROD\], $\theta\phi\mid{{\operatorname{dom}{\Theta}}} = \theta$ and $\theta'\phi'\mid{{\operatorname{dom}{\Theta}}} = \theta'$. This establishes $\theta = \theta'$ in $\prod\Theta$. A similar approach, using restriction by ${{\operatorname{dom}{\Phi}}}$, establishes $\phi = \phi'$ in $\prod\Phi$. Since the assumption $\theta\phi = \theta'\phi'$ implies $\theta = \theta'$ and $\phi = \phi'$, then the dyadic choice product is injective. To assess surjection, consider a general $\gamma \in \prod\Theta\Phi$. Define $\theta = \gamma \mid {{\operatorname{dom}{\Theta}}}$. By definition \[D:DYADIC\_ENSEMBLE\_PRODUCT\], the non-empty sets $\Theta(i) = \Theta\Phi(i)$ for $i \in {{\operatorname{dom}{\Theta}}}$. Since $\theta(i) \in \Theta\Phi(i)$ by construction, and $\Theta(i) = \Theta\Phi(i)$, then $\theta(i) \in \Theta(i)$ for $i \in {{\operatorname{dom}{\Theta}}}$ – that is, $\theta$ is a choice in $\prod\Theta$. A similar tactic shows the existence of $\phi \in \prod\Phi$. Construct $\phi = \gamma \mid {{\operatorname{dom}{\Phi}}}$. For each $i \in {{\operatorname{dom}{\Phi}}}$, the non-empty sets $\Phi(i) = \Theta\Phi(i)$ by definition \[D:DYADIC\_ENSEMBLE\_PRODUCT\]. Since by construction $\phi(i) \in \Theta\Phi(i)$, and $\Phi(i) = \Theta\Phi(i)$, then $\phi(i) \in \Phi(i)$ for $i \in {{\operatorname{dom}{\Phi}}}$ – that is, $\phi$ is a choice in $\prod\Phi$. It is thus established that for any $\gamma \in \prod\Theta\Phi$, there exists $\theta \in \prod\Theta$ and $\phi \in \prod\Phi$ such that $\gamma = \theta\phi$. Therefore the dyadic choice product is surjective. Since the dyadic choice product is both injective and surjective, then it is a bijection $\prod\Theta \times \prod\Phi \leftrightarrow \prod\Theta\Phi$. ### Choice subspaces {#S:CHOICE_SUBSPACE} Any mapping, including a choice mapping, may be restricted to subsets of its domain. \[D:SUBCHOICE\] Let $\Psi$ be an ensemble, and let $R \subseteq {{\operatorname{dom}{\Psi}}}$ be a subset of its index set. Suppose $\chi \in \prod\Psi$ is a choice. A *subchoice* $\chi \mid R$ is the ordinary mapping restriction of $\chi$ to its domain subset $R$. In the above, degenerate case $R = \varnothing$ yields $\chi \mid R = \varnothing$. \[D:SUBSPACE\] Let $\Psi$ be an ensemble. For each $R \subseteq {{\operatorname{dom}{\Psi}}}$, the *subspace* $(\thinspace\prod\Psi) \mid R$ is the set of subchoices $\lbrace\thinspace\chi \mid R \medspace \colon \chi \in \prod\Psi\thinspace\rbrace$. \[T:CHC\_RSTR\_EQ\_RSTR\_CHC\] Let ensemble $\Psi$ generate choice space $\prod \Psi$, and let $R \subseteq {{\operatorname{dom}{\Psi}}}$ be a subset of its index set. The restriction of the choice space equals the choice space of the restriction: $$(\thinspace\prod\Psi) \mid R = \prod ({{\Psi}\negmedspace\mid\negmedspace{R}}).$$ Suppose $\xi \in (\thinspace\prod\Psi) \mid R$. By definition \[D:SUBCHOICE\], there exists $\chi \in \prod\Psi$ such that $\xi = \chi \mid R$. By definition of Cartesian product, for each $i \in {{\operatorname{dom}{\Psi}}}$, $\chi(i) \in \Psi(i)$. Since $R \subseteq {{\operatorname{dom}{\Psi}}}$, then for each $r \in R$, $\xi(r) \in \Psi(r)$. Consider ${{\Psi}\negmedspace\mid\negmedspace{R}}$, for which ${{\operatorname{dom}{({{\Psi}\negmedspace\mid\negmedspace{R}})}}} = R$. By definition of restriction, for $r \in R$, $({{\Psi}\negmedspace\mid\negmedspace{R}})(r) = \Psi(r)$. Since $\xi(r) \in \Psi(r)$ and $\Psi(r) = ({{\Psi}\negmedspace\mid\negmedspace{R}})(r)$, then for any $r \in R$, $\xi(r) \in ({{\Psi}\negmedspace\mid\negmedspace{R}})(r)$ – that is, $\xi$ is a choice of ${{\Psi}\negmedspace\mid\negmedspace{R}}$. From the preceding, $\xi \in (\thinspace\prod\Psi) \mid R$ implies $\xi \in \prod ({{\Psi}\negmedspace\mid\negmedspace{R}})$, or $(\thinspace\prod\Psi) \mid R \subseteq \prod ({{\Psi}\negmedspace\mid\negmedspace{R}})$. Next suppose $\xi \in \prod ({{\Psi}\negmedspace\mid\negmedspace{R}})$. Then, by definitions \[D:CHOICE\] and \[D:CHOICE\_SPACE\] covering Cartesian products, for each $r \in R$, $\xi(r) \in ({{\Psi}\negmedspace\mid\negmedspace{R}})(r)$. The ensemble $\Psi$ coincides with its restriction ${{\Psi}\negmedspace\mid\negmedspace{R}}$ on $R$. A restatement of this is $({{\Psi}\negmedspace\mid\negmedspace{R}})(r) = \Psi(r)$ for $r \in R$. Substituting $\Psi(r)$ for $({{\Psi}\negmedspace\mid\negmedspace{R}})(r)$ yields $\xi(r) \in \Psi(r)$ for each $r \in R$. From this it follows that $\xi \in (\thinspace\prod\Psi) \mid R$, with the further implication that $\prod ({{\Psi}\negmedspace\mid\negmedspace{R}}) \subseteq (\thinspace\prod\Psi) \mid R$. We conclude equality $(\thinspace\prod\Psi) \mid R = \prod ({{\Psi}\negmedspace\mid\negmedspace{R}})$ after establishing that each of these two sets is a subset of the other. \[L:SUBSPACE\_SUBSET\] Let $\Psi$ and $\Phi$ be ensembles. If ${\prod{\Phi}}$ is a subspace of ${\prod{\Psi}}$, then $\Phi \subseteq \Psi$. Let ${\prod{\Phi}}$ be a subspace of ${\prod{\Psi}}$. By definition \[D:SUBSPACE\], there exists $R \subseteq {{\operatorname{dom}{\Psi}}}$ such that ${\prod{\Phi}} = (\thinspace{\prod{\Psi}}) \mid R$. By Theorem \[T:CHC\_RSTR\_EQ\_RSTR\_CHC\], the restriction of the choice space equals the choice space of the restriction: $(\thinspace\prod\Psi) \mid R = \prod ({{\Psi}\negmedspace\mid\negmedspace{R}})$. Transitivity of equality implies ${\prod{\Phi}} = \prod ({{\Psi}\negmedspace\mid\negmedspace{R}})$. Then, by Theorem \[T:SPACE\_UNIQ\_ENSEMBLE\] (invertibility of the Cartesian product), $\Phi = {{\Psi}\negmedspace\mid\negmedspace{R}}$. Suppose term $(i,P) \in \Phi$. Since $\Phi = {{\Psi}\negmedspace\mid\negmedspace{R}}$, then $(i,P) \in {{\Psi}\negmedspace\mid\negmedspace{R}}$. By the definition of restriction, this implies both $(i,P) \in \Psi$ and $i \in R$. Since $(i,P) \in \Phi$ implies $(i,P) \in \Psi$, we conclude $\Phi \subseteq \Psi$. \[L:SUBSET\_RESTRICTION\] Let $\Psi$ and $\Phi$ be ensembles. If $\Phi \subseteq \Psi$ and $R = {{\operatorname{dom}{\Phi}}}$, then $\Phi = {{\Psi}\negmedspace\mid\negmedspace{R}}$. Consider $(i,P) \in {{\Psi}\negmedspace\mid\negmedspace{R}}$. It then follows from the definition of restriction that $(i,P) \in \Psi$ and $i \in R$. But $R = {{\operatorname{dom}{\Phi}}}$, so $i \in {{\operatorname{dom}{\Phi}}}$. This implies there exists $(i,Q) \in \Phi$. Since $\Phi \subseteq \Psi$, then $(i,Q) \in \Psi$. Since $\Psi$ is a mapping, then $(i,P) \in \Psi$ and $(i,Q) \in \Psi$ implies $P = Q$. From $P = Q$ and $(i,Q) \in \Phi$, we infer that $(i,P) \in \Phi$. Thus $(i,P) \in {{\Psi}\negmedspace\mid\negmedspace{R}}$ implies $(i,P) \in \Phi$, so ${{\Psi}\negmedspace\mid\negmedspace{R}} \subseteq \Phi$. Next suppose $(i,P) \in \Phi$. From this it follows that $i \in R = {{\operatorname{dom}{\Phi}}}$. From the premises $(i,P) \in \Phi$ and $\Phi \subseteq \Psi$ we conclude $(i,P) \in \Psi$. Together $(i,P) \in \Psi$ and $i \in R$ imply that $(i,P) \in {{\Psi}\negmedspace\mid\negmedspace{R}}$. Thus $(i,P) \in \Phi$ implies $(i,P) \in {{\Psi}\negmedspace\mid\negmedspace{R}}$, so $\Phi \subseteq {{\Psi}\negmedspace\mid\negmedspace{R}}$. From ${{\Psi}\negmedspace\mid\negmedspace{R}} \subseteq \Phi$ and $\Phi \subseteq {{\Psi}\negmedspace\mid\negmedspace{R}}$ we infer $\Phi = {{\Psi}\negmedspace\mid\negmedspace{R}}$. \[L:SUBSET\_SUBSPACE\] Let $\Psi$ and $\Phi$ be ensembles. If $\Phi \subseteq \Psi$, then ${\prod{\Phi}}$ is a subspace of ${\prod{\Psi}}$. Set $R = {{\operatorname{dom}{\Phi}}}$. Since $\Phi \subseteq \Psi$ by hypothesis, then by applying lemma \[L:SUBSET\_RESTRICTION\] we infer $\Phi = {{\Psi}\negmedspace\mid\negmedspace{R}}$. With this equality and Theorem \[T:SPACE\_UNIQ\_ENSEMBLE\] (invertibility of the Cartesian product), we have ${\prod{\Phi}} = \prod ({{\Psi}\negmedspace\mid\negmedspace{R}})$. Theorem \[T:CHC\_RSTR\_EQ\_RSTR\_CHC\] asserts that the restriction of the choice space equals the choice space of the restriction: $(\thinspace\prod\Psi) \mid R = \prod ({{\Psi}\negmedspace\mid\negmedspace{R}})$. Transitivity of equality implies ${\prod{\Phi}} = (\thinspace\prod\Psi) \mid R$. This last equality is exactly the premise of definition \[D:SUBSPACE\]: ${\prod{\Phi}}$ is a subspace of ${\prod{\Psi}}$. \[T:SUBSET\_IFF\_SUBSPACE\] Let $\Psi$ and $\Phi$ be ensembles. ${\prod{\Phi}}$ is a subspace of ${\prod{\Psi}}$ if and only if $\Phi \subseteq \Psi$. Lemma \[L:SUBSPACE\_SUBSET\] asserts that if ${\prod{\Phi}}$ is a subspace of ${\prod{\Psi}}$, then $\Phi \subseteq \Psi$. Lemma \[L:SUBSET\_SUBSPACE\] asserts that if $\Phi \subseteq \Psi$, then ${\prod{\Phi}}$ is a subspace of ${\prod{\Psi}}$. This pair of converse implications establishes the biconditional. \[L:ENSEMBLE\_PROD\_SUBSETS\] If $\Upsilon$, $\Psi$, and $\Phi$ are ensembles such that $\Upsilon = \Psi\Phi$, then $\Psi \subseteq \Upsilon$ and $\Phi \subseteq \Upsilon$. Since $\Upsilon$ is the dyadic product of $\Psi$ and $\Phi$, then by definition \[D:DYADIC\_ENSEMBLE\_PRODUCT\], $\Psi$ and $\Phi$ are disjoint ensembles and $\Upsilon = \Psi \cup \Phi$. Suppose $i \in {{\operatorname{dom}{\Upsilon}}} = {{\operatorname{dom}{(\Psi \cup \Phi)}}}$. Through definition \[D:DISJOINT\_ENSEMBLES\], disjointness entails that ${{\operatorname{dom}{\Psi}}}\thickspace\cap\thickspace{{\operatorname{dom}{\Phi}}} = \varnothing$. Thus, if $i \in {{\operatorname{dom}{\Upsilon}}}$, exactly one of two cases hold: either A: $i \in {{\operatorname{dom}{\Psi}}}$ and $i \notin {{\operatorname{dom}{\Phi}}}$, or B: $i \in {{\operatorname{dom}{\Phi}}}$ and $i \notin {{\operatorname{dom}{\Psi}}}$. Assume case A, that $i \in {{\operatorname{dom}{\Psi}}}$ and $i \notin {{\operatorname{dom}{\Phi}}}$. With $\Upsilon = \Psi \cup \Phi$, it follows from the definition of set union that for any $i \in {{\operatorname{dom}{\Psi}}}$, $(i,P) \in \Psi$ implies $(i,P) \in \Upsilon$ – that is, $\Psi \subseteq \Upsilon$. For case B, similar argument leads to $\Phi \subseteq \Upsilon$. \[C:ENSEMBLE\_PROD\_MEMBERS\] If $\Upsilon$, $\Psi$, and $\Phi$ are ensembles such that $\Upsilon = \Psi\Phi$, then $\Psi(i) = \Upsilon(i)$ for $i \in {{\operatorname{dom}{\Psi}}}$, and $\Phi(j) = \Upsilon(j)$ for $j \in {{\operatorname{dom}{\Phi}}}$. Under identical premises, lemma \[L:ENSEMBLE\_PROD\_SUBSETS\] provides $\Psi \subseteq \Upsilon$ and $\Phi \subseteq \Upsilon$. Suppose $i \in {{\operatorname{dom}{\Psi}}}$. If $(i,P) \in \Psi$, then $(i,P) \in \Upsilon$ since $\Psi \subseteq \Upsilon$. The notation $\Psi(i) = \Upsilon(i)$ (both equaling $P$) is equivalent. A similar argument demonstrates $\Phi(j) = \Upsilon(j)$ for $j \in {{\operatorname{dom}{\Phi}}}$. \[T:ENSEMBLE\_PROD\_CHOICE\_PROD\] Let $\Upsilon$, $\Psi$, and $\Phi$ be ensembles such that $\Upsilon = \Psi\Phi$. For each $\upsilon \in {\prod{\Upsilon}}$, there exist unique $\psi \in {\prod{\Psi}}$ and $\phi \in {\prod{\Psi}}$ such that $\upsilon = \psi\phi$. Suppose $\upsilon \in {\prod{\Upsilon}}$. Since any choice has the same domain as its generating ensemble, ${{\operatorname{dom}{\Upsilon}}} = {{\operatorname{dom}{\upsilon}}}$. Theorem \[T:DYADIC\_PRODUCT\_IS\_ENSEMBLE\] states that ${{\operatorname{dom}{\Upsilon}}} = {{\operatorname{dom}{\Psi}}} \cup {{\operatorname{dom}{\Phi}}}$, from which transitivity of equality provides ${{\operatorname{dom}{\upsilon}}} = {{\operatorname{dom}{\Psi}}} \cup {{\operatorname{dom}{\Phi}}}$. From lemma \[L:ENSEMBLE\_PROD\_SUBSETS\] we conclude $\Psi \subseteq \Upsilon$ and $\Phi \subseteq \Upsilon$. Since these relations hold for entire ensembles, then the same is true of the ensembles’ domains: ${{\operatorname{dom}{\Psi}}} \subseteq {{\operatorname{dom}{\Upsilon}}}$ and ${{\operatorname{dom}{\Phi}}} \subseteq {{\operatorname{dom}{\Upsilon}}}$. By substitution, the previous result ${{\operatorname{dom}{\Upsilon}}} = {{\operatorname{dom}{\upsilon}}}$ then establishes that ${{\operatorname{dom}{\Psi}}} \subseteq {{\operatorname{dom}{\upsilon}}}$ and ${{\operatorname{dom}{\Phi}}} \subseteq {{\operatorname{dom}{\upsilon}}}$. The inclusion ${{\operatorname{dom}{\Psi}}} \subseteq {{\operatorname{dom}{\upsilon}}}$ ensures that the restriction $\psi = {{\upsilon}\negmedspace\mid\negmedspace{{{\operatorname{dom}{\Psi}}}}}$ is well-defined. Similarly $\phi = {{\upsilon}\negmedspace\mid\negmedspace{{{\operatorname{dom}{\Phi}}}}}$ is also well-defined. We next focus on the restriction $\psi$ constructed above, seeking to demonstrate that it is also a member of the choice space ${\prod{\Psi}}$. Suppose term $(i, p) \in \psi$. Since $\psi = {{\upsilon}\negmedspace\mid\negmedspace{{{\operatorname{dom}{\Psi}}}}}$, then both $(i, p) \in \upsilon$ and $i \in {{\operatorname{dom}{\Psi}}}$. Since $\upsilon \in {\prod{\Upsilon}}$ by hypothesis, definition \[D:CHOICE\] demands that $p \in \Upsilon(i)$ whenever $(i, p) \in \upsilon$. Corollary \[C:ENSEMBLE\_PROD\_MEMBERS\] asserts $\Psi(i) = \Upsilon(i)$ for $i \in {{\operatorname{dom}{\Psi}}}$. Since $p \in \Upsilon(i)$ and $\Upsilon(i) = \Psi(i)$ then $p \in \Psi(i)$. Thus for any $(i,p) \in \psi$, it follows that $p \in \Psi(i)$. This means that $\psi$ is a choice of $\Psi$ by definition \[D:CHOICE\] – that is, $\psi \in {\prod{\Psi}}$ by definition \[D:CHOICE\_SPACE\]. Similar reasoning establishes that the other restriction $\phi$ is a member of ${\prod{\Phi}}$. The unique $\psi \in {\prod{\Psi}}$ and $\phi \in {\prod{\Psi}}$ such that $\upsilon = \psi\phi$ are expressed by the restrictions $\psi = {{\upsilon}\negmedspace\mid\negmedspace{{{\operatorname{dom}{\Psi}}}}}$ and $\phi = {{\upsilon}\negmedspace\mid\negmedspace{{{\operatorname{dom}{\Phi}}}}}$. Persistent-volatile partition {#S:STATE_EVENT_PRTN} ----------------------------- Definition \[D:BASIS\] asserts that basis $\langle \Psi, \Phi \rangle$ is comprised of two ensembles satisfying $\Phi \subseteq \Psi$. This allows partitioning terms of $\Psi$ into two sets: those terms that are members of both $\Psi$ and $\Phi$, and those terms that are members of $\Psi$ but not $\Phi$. The ensemble difference terminology of definition \[D:ENSEMBLE\_DIFFERENCE\] poses the minuend, subtrahend, and remainder of this basis as respectively $\Psi$, $\Phi$, and $\Psi \setminus \Phi$. This partition is important when interpreted as systems theory. The minuend $\Psi$ generates a choice space ${\prod{\Psi}}$ called the *stimulus* space. The subtrahend $\Phi$ generates the *persistent* (alternatively *response*) space ${\prod{\Phi}}$. The remainder $\Psi \setminus \Phi$ generates the *event* space ${\prod{(\Psi \setminus \Phi)}}$. \[T:STATE\_EVENT\_SPACES\] Let $\langle \Psi, \Phi \rangle$ be a basis. The persistent $(\Phi)$ and volatile $(\Upsilon = \Psi \setminus \Phi)$ generating ensembles are disjoint and complementary with respect to the generating ensemble $\Psi$ of the stimulus space. Expressed in dyadic product, $$\Psi = \Phi\Upsilon$$ Since $\langle \Psi, \Phi \rangle$ is a basis, then $\Phi \subseteq \Psi$ by definition \[D:BASIS\]. With that result and by lemma \[L:DISJOINT\_AND\_COMPLEMENTARY\], the subtrahend $\Phi$ and remainder $\Psi \setminus \Phi$ are disjoint and complementary with respect to $\Psi$. The dyadic product recapitulates these results. Since $\Psi \setminus \Phi$ and $\Phi$ are disjoint the dyadic product $[\Psi \setminus \Phi][\Phi]$ is well-defined. By definition \[D:DYADIC\_ENSEMBLE\_PRODUCT\], $[\Psi \setminus \Phi][\Phi] = (\Psi \setminus \Phi) \cup \Phi = \Psi$. The case $\Phi = \varnothing$ does not occur naturally in systems theory because no proper system is unresponsive to all possible stimuli. When $\Psi = \Phi$, the basis has no event space through which to receive transient external stimuli. Uncoverable processes {#S:UNCOVERABLE_PROCESS_APPENDIX} ===================== Although any procedure does cover some process, some processes have no covering procedure. This disparity arises naturally through limiting the quantity of distinct functionalities participating in a procedure. Here the constraining mechanism is the catalog of functionality, whose membership must be finite. This stricture’s rationale is to emulate software, which is presumed to possess finite functionality. Uncoverability of a process entails more than failure of definition \[D:COVERING\_PROCEDURE\] in the case of a particular procedure; uncoverability implies failure for *any* procedure constructed from a given catalog of functionality. With process $\lbrace {\mathbf{f}}_n \rbrace \colon {\mathbb{N}}\to {\prod{\Psi}} \times {\prod{\Phi}}$ and catalog of functionality ${\mathscr{F}}$, uncoverability requires an $i \in {\mathbb{N}}$ and term ${\mathbf{f}}_i = {\mathbf{f}}$ such that ${\mathbf{f}} \notin {\mathit{f}}$ for each ${\mathit{f}} \in {\mathscr{F}}$. Pigeonhole principle -------------------- The pigeonhole principle can verify uncoverability, but not coverability. Suppose two frames have the same abscissa but different ordinates. No single functionality can cover both frames, since functionalities are mappings. In more general analogy, let distinct equi-abscissa frames be pigeons, while functionalities be pigeonholes. If more than $N$ pigeons occupy $N$ pigeonholes, then some pigeonhole contains more than one pigeon, which is not allowed. Underpigeonholing ----------------- The *frame set* derived from the initial segment of length $k$ of process $\lbrace {\mathbf{f}}_n \rbrace$ is the set ${\mathbf{F}} = \lbrace {\mathbf{f}} \colon {\mathbf{f}} = {\mathbf{f}}_i \;\text{and}\; i \le k\rbrace$. This set’s $\psi$-homogeneous subset contains only those frames having initial condition (abscissa) $\psi \in {\prod{\Psi}}$. The corresponding end-condition set consists of those frames’ ordinates. Obviously the $\psi$-homogeneous end-condition set must have cardinality $\bigl\lvert\lbrace \phi \colon (\psi, \phi) = {\mathbf{f}}_i \;\text{and}\; i \le k\rbrace\bigr\rvert \le k$. The limit supremum (respecting initial segment length and homogeneity choice) presents the process’ worst case scenario. \[D:UNDERPIGEONHOLE\] Let $\langle \Psi, \Phi \rangle$ be a basis for process $\lbrace {\mathbf{f}}_n \rbrace \colon {\mathbb{N}}\to {\prod{\Psi}} \times {\prod{\Phi}}$ and catalog of functionality ${\mathscr{F}}$. Catalog ${\mathscr{F}}$ *under-pigeonholes* process $\lbrace {\mathbf{f}}_n \rbrace$ if $${\lvert{{\mathscr{F}}}\rvert} < \lim_{n \to \infty} \biggl(\sup_{\psi \in {\prod{\Psi}}} \biggl( \bigl\lvert \lbrace \phi \colon (\psi, \phi) = {\mathbf{f}}_i \;\text{and}\; i \le n\rbrace \bigr\rvert \biggr)\biggr).$$ Whenever the limit supremum fails to converge, no procedure based on a (finite) catalog can cover the process. MIL-STD-882 and the CPP {#S:MIL-STD-882} ======================= MIL-STD-882 is the United States Department of Defense Standard Practice for System Safety. Revision E became effective May 11, 2012. In preference to *accident*, this standard prefers the term *mishap*, which it defines as [an event or series of events resulting in unintentional death, injury, occupational illness, damage to or loss of equipment or property, or damage to the environment.]{} Its safety risk assessment method uses the compound Poisson process (CPP) to represent the timing and severity of mishaps. MIL-STD-882E partitions compound Poisson processes into a lattice of categories and levels that covers the range of interest. The *category* is a variable which, in an explicit range \[1-4\], expresses the expectation $(\mu_L)$ of the CPP loss random variable $L$. The *level* is a variable which, in an explicit range \[A-F\], expresses the rate or intensity $\lambda$ of the CPP. The system of categories and levels agrees with the limits of discernibility of human intuition. Two different compound Poisson processes having the same category and level are indeed different but in practice are indistinguishable. This characteristic imposes a logarithmic organization on the categories and levels. -------------- ---------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Description Severity Mishap Result Criteria Category Catastrophic 1 Could result in one or more of the following: death, permanent total disability, irreversible significant environmental impact, or monetary loss equal to or exceeding \$10M. Critical 2 Could result in one or more of the following: permanent partial disability, injuries or occupational illness that may result in hospitalization of at least three personnel, reversible significant environmental impact, or monetary loss equal to or exceeding \$1M but less than \$10M. Marginal 3 Could result in one or more of the following: injury or occupational illness resulting in one or more lost work day(s), reversible moderate environmental impact, or monetary loss equal to or exceeding \$100K but less than \$1M. Negligible 4 Could result in one or more of the following: injury or occupational illness not resulting in a lost work day, minimal environmental impact, or monetary loss less than \$100K. -------------- ---------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- : MIL-STD-882E Severity Categories[]{data-label="Ta:SEVERITY_CATEGORY"} Description Level Specific Individual Item Fleet or Inventory ------------- ------- ------------------------------------------------------------------------------------------ ---------------------------------------------------- Frequent A Likely to occur often in the life of an item. Continuously experienced. Probable B Likely to occur often in the life of an item. Will occur frequently. Occasional C Likely to occur sometime in the life of an item. Will occur several times. Remote D Unlikely, but possible to occur in the life of an item. Unlikely, but can reasonably be expected to occur. Improbable E So unlikely, it can be assumed occurrence may not be experienced in the life of an item. Unlikely to occur, but possible. Eliminated F : MIL-STD-882E Probability Levels[]{data-label="Ta:PROBABILITY_LEVELS"} Table 2 above is a qualitative description of levels. Table 3 below, appearing in MIL-STD-882E Appendix A, outlines certain pitfalls in accomplishing the same task quantitatively. Numerical expression of the intensity or rate of occurrence is generally preferable to mere qualitative phrasing. For quantitative description, the intensity is the ratio of mishaps (numerator) to some measure of exposure (denominator). Description Level Individual Item Fleet/Inventory\* Quantitative ------------- ------- ----------------------------------------------------------------------------------------- --------------------------------------------------- --------------------------------------------------------------------------------------- Frequent A Likely to occur often in the life of an item Continuously experienced. Probability of occurrence greater than or equal to $10^{-1}$. Probable B Will occur several times in the life of an item Will occur frequently. Probability of occurrence less than $10^{-1}$ but greater than or equal to $10^{-2}$. Occasional C Likely to occur sometime in the life of an item Will occur several times. Probability of occurrence less than $10^{-2}$ but greater than or equal to $10^{-3}$. Remote D Unlikely, but possible to occur in the life of an item Unlikely but can reasonably be expected to occur. Probability of occurrence less than $10^{-3}$ but greater than or equal to $10^{-6}$. Improbable E So unlikely, it can be assumed occurrence may not be experienced in the life of an item Unlikely to occur, but possible. Probability of occurrence less than $10^{-6}$. Eliminated F : MIL-STD-882E Example Probability Levels []{data-label="Ta:EXAMPLE_PROBABILITY_LEVELS"} The false hegemony of a single intuitively understood measure of exposure will now be examined. We will find that, however well-intended, Table 3 lacks essential explanation. Without that, it is an oversimplification. [Natural]{} measures of exposure must embrace a variety of units, some examples of which are: the life of one item, number of missile firings, flight hours, miles driven, or years of service. For example, an exposure measure of miles driven is expected silently to exclude substantial periods when the system is out of use. Similar would be any situation-based measure of exposure having a sizable portion of time spent in unused status (time not counted). This topic appeared in §\[S:INTERMITTENT\_POISSON\], the intermittant compound Poisson process. The natural unit of exposure can be tuned to the culture of a particular hazard. However, lacking conversion capability, this freedom of choice leads to the problem of a system composed of a heterogeneous plethora of non-comparable exposure units. What is behind this incomparability? Natural units are important but incomplete – MIL-STD-882E needs additional factors to paint a full quantitative picture. There is need for conversion of various natural units into a single common standard unit, so that comparison involves only observation of magnitudes, without pondering the meaning of different units. This is particularly important in the cases of many ambiguous references to [life.]{} Suppose we arbitrarily standardize time duration at one year. We then define a conversion factor $p$, which means that $p$ years constitute a life. A measure $\iota$ quantifies what fraction of time the system’s mission is inactive or idle. A conversion factor for remaining units must be established; without specifying what units remain to be converted, we can say that the unit conversion calculus of elementary physics results in some linear coefficient $\kappa$. With $N$ a natural exposure unit and $U$ a standard measure, what we have stated so far is summarized in the following form: $$U = \frac{\kappa \cdot (1 - \iota)}{p} \cdot N .$$ Standard units measure statistical risk as resulting from exposure to an intermittent compound Poisson process. These standard units may not be a proper exposure, but measure the exposure expected in a year’s duration. For this reason we celebrate the importance of the role of pure natural units; it is important to understand risk as proportionate to exposure. To understand this importance, imagine yourself as the one exposed to a transient but intense hazard. But that does not imply the dismissal of statistical risk as a concern; it is also part of the risk analysis picture to consider how much risk exposure occurs within a given duration. This is the role of the standard unit. Another complicating factor is the use of the term [level]{} itself. A level is a designator for a class of possibly intermittent indistinguishable probability distributions. Rather than being clear about this, MIL-STD-882E equivocates greatly in Table 3, confusing this designator with a literal probability statement. Only after full quantitative analysis is completed ($p$, $\iota$, and $\kappa$ known) can definite statements concerning probability be asserted. It is insufficient to mandate vague documentation of [all numerical definitions of probability used in risk assessments]{} without further guidance. Table 4 below is a categorical rendering of the hyperbola of statistical risk. Definition \[D:STATISTICAL\_RISK\] asserts $h = \lambda \mu_L$. Excepting the administrative level [Eliminated]{}, this cross-tabulation presents the level $(\lambda)$ on the vertical axis and the category $(\mu_L)$ along the horizontal axis. For each combination of level and category, another categorical variable[^13] represents the statistical risk $h = \lambda \mu_L$ with values: High, Serious, Medium, and Low. ---------------- -------------- ---------- ---------- ------------ SEVERITY / Catastrophic Critical Marginal Negligible PROBABILITY (1) (2) (3) (4) Frequent (A) High High Serious Medium Probable (B) High High Serious Medium Occasional (C) High Serious Medium Low Remote (D) Serious Medium Medium Low Improbable (E) Medium Medium Medium Low Eliminated (F) ---------------- -------------- ---------- ---------- ------------ : MIL-STD-882E Risk Assessment Matrix[]{data-label="Ta:RISK_ASSESSMENT_MATRIX"} This table suffers the same ambiguity as in Table 3. MIL-STD-882’s definitions are clearly inadequate for quantitative analysis. Through equivocation, exposure to an intermittent compound Poisson process is regarded as not different than exposure to a compound Poisson process, despite that the difference becomes obvious through the linear factor $(1 - \iota)$. MIL-STD-882 is an evolving document in its fifth major revision; let us hope these ambiguities are resolved in the future. Other approaches to automata ============================ Deterministic finite automaton {#S:DFA} ------------------------------ This depiction of the deterministic finite automaton [@wW11autmaton] appears in Wikipedia: > An *automaton* is represented formally by the 5-tuple $\langle Q, \Sigma, \delta, q_0, A \rangle$, where: > > - $Q$ is a finite set of *states*. > > - $\Sigma$ is a finite set of *symbols*, called the *alphabet* of the automaton. > > - $\delta$ is the *transition function*, that is, $\delta \colon Q \times \Sigma \to Q$. > > - $q_0$ is the *start state*, that is, the state which the automaton is *in* when no input has been processed yet, where $q_0 \in Q$. > > - $A$ is a set of states of $Q$ (i.e. $A \subseteq Q$) called *accept states*. > An approach for engineers is found in [@jH79]. [20]{} Benjamin S. Blanchard and Wolter J. Fabrycky, Systems Engineering and Analysis, 3rd edition, Prentice Hall, 1998 Arthur D. Hall, A Methodology For Systems Engineering, (New York: Van Nostrand Reinhold Company), 1962 John D. Musa and Anthony Iannino and Kazuhira Okumoto, [*Software reliability - measurement, prediction, application*]{}, McGraw-Hill, 1987 Hopcroft, J. E. and Ullman, J. D., [*Introduction to Automata Theory, Languages and Computation*]{}, Second Edition, Addison-Wesley, 1979 Paul R. Halmos, Naive Set Theory, Springer-Verlag New York, 1974 United States Department of Defense, Joint Software Systems Safety Engineering Handbook, Internet, http://www.acq.osd.mil/se/docs/Joint-SW-Systems-Safety-Engineering-Handbook.pdf, retrieved 2014-09-19 United States Department of Defense, MIL-STD-882E (Standard Practice for System Safety), Internet, http://www.assistdocs.com/search/search\_basic.cfm, retrieved 2012-05-13T18:01Z Wikipedia, Internet, http://en.wikipedia.org/wiki/Automata\_theory, retrieved July 18, 2011 Wikipedia, Internet, http://en.wikipedia.org/wiki/Emergence, retrieved June 13, 2014 Wikipedia, Internet, http://en.wikipedia.org/wiki/Frequentist\_probability, retrieved March 10, 2014 Wikipedia, Internet, http://en.wikipedia.org/wiki/Poisson\_distribution, retrieved May 20, 2013 Wikipedia, Internet, http://en.wikipedia.org/wiki/Poisson\_process, retrieved May 20, 2013 Wikipedia, Internet, http://en.wikipedia.org/wiki/Law\_of\_total\_expectation, retrieved May 20, 2013 Wikipedia, Internet, http://en.wikipedia.org/wiki/Compound\_Poisson\_process, retrieved May 20, 2013 Wikipedia, Internet, http://en.wikipedia.org/wiki/Cumulant, retrieved May 24, 2013 [^1]: http://creativecommons.org/licenses/by/3.0/ [^2]: The author is a retired software safety engineer, not a mathematician. [^3]: A reactive system responds to its environment, or external stimuli. [^4]: That is, the *converse* is not generally a pointwise invertible mapping as suggested by the term *inverse*. [^5]: The author is unaware of a proper definition. [^6]: software causes no harm until erroneous values transduce the boundary between software and hardware. [^7]: One increasing, the other decreasing [^8]: After Siméon Denis Poisson, mathematician and physicist, 1781 – 1840 [^9]: Its symbol is a composite of other notational devices that have no separate meaning. [^10]: Hobson was proprietor of a livery. He was noted for offering his customers their choice of any horse, as long as that horse was in the first unoccupied stall. [^11]: The set difference $A \setminus B$ is not conventionally restricted to $B \subseteq A$, as is stipulated here. [^12]: one form of which is $(\neg B \Rightarrow \neg A)\Leftrightarrow(A \Rightarrow B)$ [^13]: Not to be confused with the categorical variable named [category]{}

--- author: - 'Samaneh Abbasi-Sureshjani' - Jiong Zhang - Remco Duits - Bart ter Haar Romeny bibliography: - 'manuscript.bib' date: 'Received: date / Accepted: date' title: 'Retrieving challenging vessel connections in retinal images by line co-occurrence statistics' ---

--- abstract: 'In this paper we refine a weighted partition identity of Alladi. We write formulas for generating functions for the number of partitions grouped with respect to a partition statistic other than the norm. We tie our weighted results and the different statistic with the crank of a partition. In particular, we prove that the number of partitions into even number of distinct parts whose odd-indexed parts’ sum is $n$ is equal to the number of partitions of $n$ with non-negative crank.' address: 'Department of Mathematics, University of Florida, 358 Little Hall, Gainesville FL 32611, USA' author: - Ali Kemal Uncu title: 'Weighted Rogers–Ramanujan Partitions and Dyson Crank' --- \#1[10mu([@font mod]{}\#1)]{} Introduction and Notations ========================== A *partition* is a finite sequence $\pi=(\lambda_1,\lambda_2,\dots)$ of decreasing (not necessarily strict) positive integers. The elements of the sequence $\pi$ are called the *parts* of the partition $\pi$. We define the *norm* of a partition $\pi$ as the sum of all its parts, $\lambda_1+\lambda_2+\dots$, and this will be denoted as $|\pi|$. As an example, there are 5 partitions, $(4),\ (3,1),\ (2,2),\ (2,1,1),\ (1,1,1,1)$, with norm equal to 4. For an integer $n$, we will use *partitions of $n$* to denote the set of all the partitions with norm $n$. Abiding the general convention, we accept the empty sequence as a partition, and it is the unique partition of 0. The norm of partitions is one of the the most natural statistics. There are finitely many partitions with a fixed norm. This makes the norm a great candidate for indexing generating functions. The theory of partitions is primarily concerned with the relationship between the sizes of different sets of partitions where elements from both sets have the same norm. One early example is due to Euler [@Theory; @of; @Partitions]. \[EulerTHM\] The number of partitions of $n$ into distinct parts is the same as the number of partitions of $n$ into odd parts. Theorem \[EulerTHM\] and many other theorems of the same spirit utilizes generating functions in their proofs. Let $A$ be a set of partitions, and let $p_A(n)$ be the number of partitions in $A$ with norm $n$. Then $$\label{GENFUNC}\sum_{\pi\in A} 1 \cdot q^{|\pi|} = \sum_{n\geq 0} p_A(n) q^n$$ is the generating function for the number of partitions with the same norm from the set $A$ written in two separate combinatorial ways, abstract and enumerative respectively. Here it is clear that every partition $\pi \in A$ makes a contribution of one to the $q^{|\pi|}$ term. We would like to introduce four classically studied sets of partitions. i. Let ${\mathcal{U}}$ be the set of all (unrestricted) partitions. ii. Let ${{\mathcal{D}}}$ be the set of all partitions into distinct parts. iii. Let ${{\mathcal{R}}}{{\mathcal{R}}}_1$ be the set of all partitions with difference between parts $\geq 2$. iv. Let ${{\mathcal{R}}}{{\mathcal{R}}}_2$ be the set of all partitions with difference between parts $\geq 2$ where parts are $>1$. These listed sets are nested: ${{\mathcal{R}}}{{\mathcal{R}}}_2\subset{{\mathcal{R}}}{{\mathcal{R}}}_1\subset {{\mathcal{D}}}\subset{\mathcal{U}}$. The generating functions for the number of partitions from these sets are extensively studied in the literature. One can generalize the classical approach of writing abstract generating functions with respect to the norm by attaching weights in the place of 1. In 1997, Alladi [@AlladiWeighted] inquired about the existence and identification of a weight $\omega_S(\pi)$ on a set of partitions $S$ so that $$\label{GF_function_abstract}\sum_{\pi\in S} \omega_S(\pi)q^{|\pi|} = \sum_{\pi\in T} q^{|\pi|}$$ for some set of partitions $T$ that contains $S$. He proved the interesting result, which exemplifies the existence of solutions of : \[Alladi\_weighted\_sum\] Let $\nu(\pi)$ denote the number of parts of $\pi$. Then $$\label{omega_12}\sum_{\pi\in {{\mathcal{R}}}{{\mathcal{R}}}_1} \omega_{1,2}(\pi) q^{|\pi|} = \sum_{\pi\in U} q^{|\pi|}$$where $$\omega_{1,2}(\pi) := \lambda_{\nu(\pi)}\cdot\prod_{i=1}^{\nu(\pi)-1} (\lambda_{i}-\lambda_{i+1}-1),$$ and weight of the empty sequence is considered to be the empty product, and is set equal to 1. Similar weighted identities and their interesting applications have been discussed [@AlladiWeighted], [@AlladiBerkovich], and [@AlladiBerkovich2]. It should be noted that the relation $T\subset S$ in is of little interest. In this case one can define the weight $\omega_S(\pi)$ to be the indicator function $$\omega_S(\pi) := \left\{ \begin{array}{ll} 1, &\text{if }\pi\in T,\\ 0,&\text{otherwise.} \end{array} \right.$$ Our main motivation lies in the similar question to the one of Alladi’s. We would like to identify statistics $\Lambda$ such that for sets of partitions $S\subset T$ we have $$\label{GF_abstract_EXP_Weight}\sum_{\pi\in S}q^{\Lambda(\pi)}=\sum_{\pi\in T} q^{|\pi|}.$$Later we prove the following result: \[AliTHM1\] $$\sum_{\pi\in {{\mathcal{D}}}} q^{\mathcal{O}(\pi)} = \sum_{\pi \in {\mathcal{U}}} q^{|\pi|},$$ where $\mathcal{O}(\pi) := \lambda_1 + \lambda_3+\dots$, the sum of the odd indexed parts, for a partition $\pi=(\lambda_1,\lambda_2,\dots)$. Similar to the problem of identifying weights, the case $T \subset S$ is trivial since one can formally pick $$\Lambda(\pi)=\left\{ \begin{array}{ll} |\pi|, &\text{if }\pi\in T,\\ \infty, &\text{otherwise,} \end{array} \right.$$ where we assume $|q|<1$. For $i\in\{1,2\}$, identifying the weights $\omega_i(\pi)$, the partition statistics $\Lambda_i$, and sets of partitions (or vector partitions) $S$ and $T$ that satisfy $$\label{Constant_Weights_Def}\sum_{\pi\in S}\omega_1(\pi) q^{\Lambda_1(\pi)} = \sum_{\pi\in T} \omega_2(\pi)q^{\Lambda_2(\pi)} \\$$ is an enveloping generalization of the mentioned questions related with and . This general question reduces to the classical combinatorial study of partition identities for $\omega_i(\pi)\equiv 1$ and $\Lambda_i(\pi)\equiv |\pi|$ with sets of partitions $S$ and $T$. One example of this particular case is Theorem \[EulerTHM\]. In Section \[Section2\] we define $q$-Pochhammer symbols, and the Ferrers diagrams. We also remark some well-known results for completeness of the paper. Section \[Section3\] has the refinement and a proof of Theorem \[Alladi\_weighted\_sum\]. The crank of a partition and its relation with both the weighted identities and different partition statistics is given in Section \[Section4\]. Section \[Section5\] is devoted for a short excursion of writing generating functions with respect to the partition statistics, sum of the odd-indexed parts of a partition. Some Basics of Partition Theory {#Section2} =============================== The Ferrers diagram of a partition $\pi=(\lambda_1,\lambda_2,\dots)$ is a graphical representation of the parts of $\pi$, [@Theory; @of; @Partitions], where we put $\lambda_i$ many dots in the integral coordinates on the $i$-th row from the top of the diagram to represent this part. Two examples of such representations are the Ferrers diagrams of $(4,4,2,1,1)$ and $(5,3,2,2)$ respectively: [cc]{} (1,-5.1) grid (4.5,-1); (0.9,-5.1) rectangle (6.5,-0.9); plot(,[(-0-1\*)/1]{}); (1,-1) circle (1.5pt); (2,-1) circle (1.5pt); (3,-1) circle (1.5pt); (4,-1) circle (1.5pt); (1,-2) circle (1.5pt); (2,-2) circle (1.5pt); (3,-2) circle (1.5pt); (4,-2) circle (1.5pt); (1,-3) circle (1.5pt); (2,-3) circle (1.5pt); (1,-4) circle (1.5pt); (1,-5) circle (1.5pt); (6,-3) node\[anchor=center\] [,]{}; & (1,-5.1) grid (5.1,-1); (0.9,-5.1) rectangle (6.1,-0.9); plot(,[(-0-1\*)/1]{}); (1,-1) circle (1.5pt); (2,-1) circle (1.5pt); (3,-1) circle (1.5pt); (4,-1) circle (1.5pt); (1,-2) circle (1.5pt); (2,-2) circle (1.5pt); (3,-2) circle (1.5pt); (1,-3) circle (1.5pt); (2,-3) circle (1.5pt); (1,-4) circle (1.5pt); (2,-4) circle (1.5pt); (5,-1) circle (1.5pt); (6,-3) node\[anchor=center\] [.]{}; We note that taking the symmetric images of points in the Ferrers diagram over the (main) diagonal line gives us a Ferrers diagram of a partition. Two partitions whose Ferrers diagrams that are related by symmetry over the main diagonal are said to be *conjugate* of each other. In our example $(4,4,2,1,1)$ and $(5,3,2,2)$ are conjugate partitions. One should also stress that the partitions can be identified by their Ferrers diagrams and vice versa. From now on we will be using the notation $\pi$ for a partition or a partition’s Ferrers diagram interchangeably. For the product representations of generating functions of interest we define the $q$-Pochhammer symbols [@Theory; @of; @Partitions]. Let $L$, and $k$ be non-negative integers. The $q$-Pochhammer symbol is $$\begin{aligned} (a)_L:=(a;q)_L &:= \prod_{n=0}^{L-1}(1-aq^n).\\ \intertext{Some abbreviations of the notation we are going to use are} (a_1,a_2,\dots,a_k;q)_L &:= (a_1;q)_L(a_2;q)_L\dots (a_k;q)_L,\\ (a;q)_\infty &:= \lim_{L\rightarrow\infty} (a;q)_L,\text{ where }|q|<1.\end{aligned}$$ With these definitions we can write explicit formulas of the generating functions on the defined sets in multiple ways. $$\begin{aligned} \label{GF_unrestricted_pts}\sum_{\pi\in {\mathcal{U}}} q^{|\pi|} &=\sum_{n\geq 0} \frac{q^{n^2}}{(q)^2_n}=\frac{1}{(q)_\infty},\\ \label{GF_distinct_pts}\sum_{\pi\in {{\mathcal{D}}}} q^{|\pi|} &=(-q)_\infty = \frac{1}{(q;q^2)_\infty},\\ \label{RR1_identity}\sum_{\pi\in {{\mathcal{R}}}{{\mathcal{R}}}_1} q^{|\pi|} &= \sum_{n\geq0} \frac{q^{n^2}}{(q)_n} = \frac{1}{(q,q^4;q^5)_\infty},\\ \label{RR2_identity}\sum_{\pi\in {{\mathcal{R}}}{{\mathcal{R}}}_2} q^{|\pi|} &= \sum_{n\geq0} \frac{q^{n^2+n}}{(q)_n} = \frac{1}{(q^2,q^3;q^5)_\infty}.\end{aligned}$$ Second equality of is the analytic version of Theorem \[EulerTHM\]. The extreme right equality of and are the well-celebrated Rogers–Ramanujan identities. Let $\widehat{C}_1$ (and $\widehat{C}_2$) be the set of all the partitions into parts $\equiv \pm1\mod{5}$ (and $\equiv \pm 2\mod{5}$). Hence, for $i\in\{1,2\}$ $$\label{RR_combinatorics_example} \sum_{\pi\in {{\mathcal{R}}}{{\mathcal{R}}}_i} q^{|\pi|} = \sum_{\pi\in \widehat{C}_i} q^{|\pi|}. \\$$ This is a classical example of . Moreover, can be equivalently stated in its enumerative form [@Theory; @of; @Partitions]: \[RR\_combinatorial\_THM\]Let $n$ be any non-negative integer. The number of partitions of $n$ into parts with minimal distance of 2 between consecutive parts (where the smallest part $\not=1$) is equal to the number of partitions of $n$ into parts $\equiv \pm1\mod{5}$ (and $\equiv \pm 2\mod{5}$). Partition Identities Involving Weights {#Section3} ====================================== Let $M$, $k-1$, and $m$ be non-negative integers. Let ${\mathcal{P}}_M(k,m)$ be the set of partitions into exactly $M$ parts with the smallest part $\geq k$ and the gap between consecutive parts $\geq m$. We let ${\mathcal{P}}_0(k,m)$ be the set that contains the partition of zero, the empty sequence. The sets ${\mathcal{P}}_M(k,m)$ are mutually disjoint for distinct integers $M$ and satisfy the properties ${\mathcal{P}}_M(k,m+1)\subset {\mathcal{P}}_M(k,m)$ and ${\mathcal{P}}_M(k+1,m)\subset {\mathcal{P}}_M(k,m)$. For brevity we define the short-hand notation $$\label{larger_than_or_equal_to_M_Sets}{\mathcal{P}}_{\leq M} (k,m) = \bigcup_{l=0}^M {\mathcal{P}}_l(k,m).$$ Clearly we have ${\mathcal{U}}=\lim_{M\rightarrow\infty}{\mathcal{P}}_{\leq M}(1,0)$, ${{\mathcal{D}}}=\lim_{M\rightarrow\infty}{\mathcal{P}}_{\leq M}(1,1)$, and ${{\mathcal{R}}}{{\mathcal{R}}}_i = \lim_{M\rightarrow\infty}{\mathcal{P}}_{\leq M} (i,2)$ for $i\in\{1,2\}$. \[Finite\_Weighted\_GF\] For a partition $\pi = (\lambda_1,\lambda_2,\dots)$, $$\label{GF_M_parts_general_distance_smallest_part}\sum_{\pi\in {\mathcal{P}}_M(k,m)} \omega_{k,m}(\pi) q^{|\pi|} = \frac{q^{m{M\choose2}+kM}}{(q)_M^2},$$ where $$\label{General_Weight} \omega_{k,m}(\pi) :=(\lambda_{M}+1-k)\cdot\prod_{i=1}^{M-1} (\lambda_{i}-\lambda_{i+1}+1-m).$$ The $M=0$ case is obvious. Let $M$ and $k$ be positive and $m$ be non-negative. A combinatorially interpretation of $q^{m{M\choose 2}}$ is that it is the generating function for the partition $\pi_1:=((M-1)m,(M-2)m,\dots,2m,m)$. The partition $\pi_1$ is into $M-1$ distinct parts when $m$ and $M-1$ are non-zero and it is the empty partition otherwise. The $q^{kM}$ term is interpreted as the partition $\pi_2$ into $M$ parts each equal to $k$. Point-wise addition of two partitions can be defined as putting the $i$-th rows of the Ferrers diagrams back to back. The empty partition is the identity element of the defined point-wise addition. This operation on partitions yields new partitions. Point-wise addition of $\pi_1$ and $\pi_2$ gives $\pi^* := ((M-1)m+k,(M-2)m+k,\dots,2m+k,m+k,k)$. The partition $\pi^*$ has smallest norm satisfying the properties of $P_M(k,m)$. We consider this smallest partition to be colorless. As a generating function, $$\label{1/(q)^2_M}\frac{1}{(q)^2_M}$$ keeps count of the partitions into $\leq M$ parts where every column in the Ferrers diagram can come in one of two colors counted disregarding the order of these colors. Adding a partition $\pi'$ counted by with $\pi^*$ point-wise we get a partition $\pi\ = (\lambda_1,\lambda_2,\dots, \lambda_{M-1},\lambda_M) \in {\mathcal{P}}_M(k,m)$. There are $(\lambda_1-\lambda_2 + 1-m)$ many color combinations for the colored portion of the first part of $\pi$, $\lambda_1$, $(\lambda_2-\lambda_3+1-m)$ for colored portion of $\lambda_2$, and so on. The colored piece of $\lambda_M$ comes in $\lambda_M+1-k$ many possible color combinations disregarding the order of colors. Therefore, there is a total of $$(\lambda_{M}+1-k)\cdot\prod_{i=1}^{M-1} (\lambda_{i}-\lambda_{i+1}+1-m)$$ possibilities for the partition $\pi\in{\mathcal{P}}_M(k,m)$ that are counted by the generating function on the right-hand side of . \[Corollary\_of\_Finite\_Weighted\_GF\] For $\label{general_weight}\omega_{k,m}(\pi) $ as in $$\label{Corollary_of_Finite_Weighted_GF_EQN} \sum_{\pi\in {\mathcal{P}}_{\leq M}(k,m)} \omega_{k,m}(\pi) q^{|\pi|} = \sum_{i=0}^{M} \frac{q^{m{i\choose2}+ki}}{(q)_i^2}.$$ Letting $M\rightarrow\infty$ with $(k,m)=(1,2)$ in Corollary \[Corollary\_of\_Finite\_Weighted\_GF\] proves Theorem \[Alladi\_weighted\_sum\] by the first equality of . We also note that Theorem \[Finite\_Weighted\_GF\] is the refinement of the finite analogue of Theorem \[Alladi\_weighted\_sum\], [@AlladiWeighted Thorem 3], which connects the Durfee square sizes and the number of parts of partitions of the first Rogers–Ramanujan type, ${{\mathcal{R}}}{{\mathcal{R}}}_1$. Letting $(k,m)=(1,2)$ in Theorem \[Finite\_Weighted\_GF\] proves [@AlladiWeighted Thorem 3]. An interesting connection with the classical study of partitions comes from the choice of $(k,m)=(2,2)$ and letting $M\rightarrow\infty$ in Corollary \[Corollary\_of\_Finite\_Weighted\_GF\]. Connections with the crank {#Section4} ========================== In 1988, Andrews and Garvan [@FrankCrank] found an explanation of the crank of an ordinary partition $\pi$. Explicitly, the crank of a partition is defined as $$cr(\pi):=\left\{ \begin{array}{cl} \text{largest part of }\pi, &\text{if }1\text{ is not a part of }\pi,\\ \#\text{ of parts larger than }\#\text{ of }1\text{s} - \#\text{ of }1\text{s in }\pi, & \text{otherwise}. \end{array}\right.$$ Let ${\mathcal{C}}_{=M}$, ${\mathcal{C}}_{\leq M}$, and ${\mathcal{C}}_{\geq M}$ be the sets of partitions with crank $=M$, $\leq M$, and $\geq M$ respectively. We have \[RR2\_THM\] $$\label{omega_22}\sum_{\pi\in {{\mathcal{R}}}{{\mathcal{R}}}_2} \omega_{2,2}(\pi) q^{|\pi|} = \sum_{\pi \in {\mathcal{C}}_{\geq 0}} q^{|\pi|}.$$ Let $(k,m)=(2,2)$ and $M\rightarrow\infty$ in . Comparison between [@Auluck (3)] and [@Auluck (11)] with the use of the right-hand side equation of shows $$\sum_{i\geq 0} \frac{q^{i^2+i}}{(q)_i^2} = \frac{1}{(q)_\infty} \sum_{i\geq 0} (-1)^{i} q^{i+1\choose 2}.$$ The right-hand side of the above line is the summation over $k\geq 0$ of the Dyson’s equation for a fixed crank $k$, [@BerkovichGarvanCrank (3.1)]. This yields Theorem \[RR2\_THM\]. It is not clear that for $n\geq 2$ the number of partitions of $n$ with positive crank is the same as the number of partitions of $n$ with negative crank. A combinatorial proof of this phenomenon as well as refinements of the fixed crank’s generating functions can be found in [@BerkovichGarvanCrank]. We would like to point out that for $k\geq 2$ the weights $\omega_{k,m}(\pi)$ take the identical values on the sets ${\mathcal{P}}_M(k-1,m)$ and ${\mathcal{P}}_M(k,m)$. Therefore, by taking the difference of $\omega_{1,2}$ and $\omega_{2,2}$ on the set ${{\mathcal{R}}}{{\mathcal{R}}}_1$ one can show \[Tilde\_Weights\_THEOREM\] Let $\nu(\pi)$ denote the number of parts of $\pi$. Then $$\begin{aligned} \label{negative_crank}\sum_{\pi\in {{\mathcal{R}}}{{\mathcal{R}}}_1} \tilde{\omega}_{1}(\pi) q^{|\pi|} &= \sum_{\pi \in {\mathcal{C}}_{\leq -1}} q^{|\pi|} = q+\sum_{\pi \in {\mathcal{C}}_{\geq 1}} q^{|\pi|} ,\\ \label{zero_crank}\sum_{\pi\in {{\mathcal{R}}}{{\mathcal{R}}}_1} \tilde{\omega}_{2}(\pi) q^{|\pi|} &= -q + \sum_{\pi \in {\mathcal{C}}_{=0}} q^{|\pi|} , \end{aligned}$$ where $$\label{Tilde_Weights}\begin{array}{ccc}\displaystyle \tilde{\omega}_{1}(\pi) = \prod_{i=1}^{\nu(\pi)-1} (\lambda_{i}-\lambda_{i+1}-1) &\text{and}& \displaystyle\tilde{\omega}_{2}(\pi) = (\lambda_{\nu(\pi)}-2)\cdot\prod_{i=1}^{\nu(\pi)-1} (\lambda_{i}-\lambda_{i+1}-1). \end{array}$$ Replacing ${{\mathcal{R}}}{{\mathcal{R}}}_2$ with ${{\mathcal{R}}}{{\mathcal{R}}}_1$ on the left-hand side of and subtracting this from side-by-side proves . The second equality of is due to [@BerkovichGarvanCrank]. Difference of and shows . Let ${{\mathcal{D}}}_l$ to be the subset of ${{\mathcal{D}}}$ that consists of the partitions into exactly $l$ distinct parts. Recall that $\mathcal{O}(\pi) := \lambda_1 + \lambda_3+\dots$, the sum of the odd indexed parts, for a partition $\pi=(\lambda_1,\lambda_2,\dots)$. \[Weight\_Change\_Odd\_Indexed\_Parts\_with\_number\_of\_parts\_known\] For $l$ a non-negative number and $v\in\{0,1\}$, we have $$\label{Odd_Index_Part_Weights}\sum_{\pi\in {{\mathcal{D}}}_{2l+v}} q^{\mathcal{O}(\pi)} = \sum_{\pi \in {\mathcal{P}}_{l+v}(2-v,2)} \left[(1-v)\omega_{2,2}(\pi)+v\tilde{\omega}_1(\pi)\right] q^{|\pi|}.$$ Let $\pi = (\lambda_1,\lambda_2,\dots,\lambda_{2l+v})$ be a partition in $D_{2l+v}$. Consider the projection mapping $\textbf{P}_{2l+v}:{{\mathcal{D}}}_{2l+v} \rightarrow {\mathcal{P}}_{l+v}(2-v,2)$ as $\textbf{P}_{2l+v}(\pi) = (\lambda_1^*,\lambda_2^*,\dots,\lambda_{l+v}^*)=(\lambda_1,\lambda_3,\dots, \lambda_{2l+(-1)^{v+1}})$. Therefore, $\mathcal{O}(\pi) = |\textbf{P}_{2l+v}(\pi)|$. The number of pre-images of a partition must be counted for the verification of . Given $\textbf{P}_{2l+v}(\pi)$, there are $(\lambda_1^*-\lambda_2^*-1)=(\lambda_1-\lambda_3-1)$ possible $\lambda_2$’s in the pre-image, $(\lambda_2^*-\lambda_3^*-1)=(\lambda_3-\lambda_5-1)$ possible $\lambda_4$’s in the pre-image, and so on. Hence, the total number of possible $\pi \in {{\mathcal{D}}}_{2l+v}$ that would project to $\textbf{P}_{2l+v}(\pi) \in {\mathcal{P}}_{l+v}(2-v,2)$ is $$\prod_{i=1}^{l+v-1} (\lambda_{2l-1}-\lambda_{2l+1}-1)\times \text{``the number of possibilities for the smallest part."}$$ Depending on $v$ the number of possibilities for the smallest part changes. If $v=1$, then $\lambda_{l+1}^*=\lambda_{2l+1}$ is the smallest part. There is only one possibility for the smallest part, which makes the weight of $\textbf{P}_{2l+1}(\pi)$ to be $\tilde{\omega}_1(\pi)$ for a $\pi\in D_{2l+1}$. If $v=0$, then $\lambda_{l}^* = \lambda_{2l-1}$ is the second smallest part of the pre-image $\pi$. Hence, there are $(\lambda_{2l-1}-1)$ possibilities for the non-zero smallest part $\lambda_{2l}$ of $\pi$, which shows that the weight of $\textbf{P}_{2l}(\pi)$ is $\omega_{2,2}(\pi)$ for a $\pi\in D_{2l}$. Observe that Theorem \[Weight\_Change\_Odd\_Indexed\_Parts\_with\_number\_of\_parts\_known\] connects the generating function for the number of partitions with the same sum of the odd-indexed parts with all three theorems: Theorem \[Alladi\_weighted\_sum\], Theorem \[RR2\_THM\] and Theorem \[Tilde\_Weights\_THEOREM\]. For any partition $\pi\in{{\mathcal{R}}}{{\mathcal{R}}}_1$, $w_{1,2}(\pi) =\omega_{2,2}(\pi)+\tilde{\omega}_1(\pi)$ by and . Therefore, summing the left-hand side of over all $l$ and $v$ yields $$\label{proof_of_Ali_THM1}\sum_{\pi\in {{\mathcal{D}}}} q^{\mathcal{O}(\pi)}={\displaystyle\sum}_{l,v\geq 0}\ \sum_{\pi\in {{\mathcal{D}}}_{2l+v}} q^{\mathcal{O}(\pi)} = \sum_{\pi\in {{\mathcal{R}}}{{\mathcal{R}}}_1} \omega_{1,2}(\pi) q^{|\pi|}.$$ This shows Theorem \[AliTHM1\] as a corollary of Theorem \[Weight\_Change\_Odd\_Indexed\_Parts\_with\_number\_of\_parts\_known\] using Theorem \[Alladi\_weighted\_sum\]. Similarly letting $v=0$ in and summing over non-negative $l$ gives $$\label{Half_of_RR2_Crank_result}\sum_{\pi\in {{\mathcal{D}}}_e} q^{\mathcal{O}(\pi)}=\sum_{l\geq 0}\ \sum_{\pi\in {{\mathcal{D}}}_{2l}} q^{\mathcal{O}(\pi)} = \sum_{\pi\in {{\mathcal{R}}}{{\mathcal{R}}}_2} \omega_{2,2}(\pi) q^{|\pi|},$$ of Theorem \[RR2\_THM\], where ${{\mathcal{D}}}_e$ is the set of partitions into even number of distinct parts. Similarly $v=1$ gives the connection of this partition statistic with . Combinatorially, the equation with Theorem \[RR2\_THM\] gives: The number of partitions into even number of distinct parts whose odd-indexed parts’ sum is $n$ is equal to the number of partitions of $n$ with non-negative crank The proof of Theorem \[Weight\_Change\_Odd\_Indexed\_Parts\_with\_number\_of\_parts\_known\] shows that replacing the norm with a partition statistic in the generating function (such as $\mathcal{O}(\pi)$) may be related with generating function for the weighted count of partitions. Changing statistics in itself is an interesting question. Moreover, as exemplified in and , the study of writing new generating functions with respect to different statistics instead of the norm would also yield non-trivial examples of . Generating Functions with respect to the Sum of Odd-indexed Parts {#Section5} ================================================================= We want to remind the reader that the convenience of writing the generating functions for the number of partitions with respect to the norm comes from there being a finite number of partitions having the prescribed norm. The same applies for $\mathcal{O}(\pi)$. There are finite number of partitions $\pi$ with $\mathcal{O}(\pi)=n$. This is not necessarily true for all the partition statistics. An analogue of the statistics $\mathcal{O}(\pi)$ is an example of this observation. Let ${{\mathcal{E}}}(\pi):=\lambda_2+\lambda_4+\dots$, the sum of all the even-indexed parts, for $\pi= (\lambda_1,\lambda_2,\dots)$, then all partitions $\pi_i = (\lambda_2+i,\lambda_2,\dots)$ for $i\in \mathbb{Z}_{{>0}}$ satisfy ${{\mathcal{E}}}(\pi_i)={{\mathcal{E}}}(\pi_j)$, $\forall i,\ j\in \mathbb{Z}_{{>0}}$. Writing generating functions with some natural partition statistics such as $\mathcal{O}$ and ${{\mathcal{E}}}$ can be studied directly from the the results of [@BerkovichUncu2] ,[@Boulet], and [@Masao]. One decorates the Ferrers diagrams by writing on the dots on the rows. On odd-indexed rows one puts alternating $a$ and $b$’s starting from an $a$ and does the same with $c$ and $d$ for the even-indexed rows starting from a $c$. We call these *four-decorated* Ferrers diagrams. In 2006, Boulet [@Boulet] found explicit formulas for the generating functions for four-decorated Ferrers diagrams from the sets ${\mathcal{U}}$ and ${{\mathcal{D}}}$. Let $\Phi(a,b,c,d)$ be the generating function for the weighted four-decorated Ferrers diagrams and $\Psi(a,b,c,d)$ be the generating function for the weighted four-decorated Ferrers diagrams that has distinct row sizes. Abstractly $$\Psi(a,b,c,d) := \sum_{\pi\in {{\mathcal{D}}}} \omega_\pi(a,b,c,d),\text{ and }\ \Phi(a,b,c,d) := \sum_{\pi\in {\mathcal{U}}} \omega_\pi(a,b,c,d),$$ where $$\omega_\pi (a,b,c,d):=a^{\#\text{ of }a's}b^{\#\text{ of }b's}c^{\#\text{ of } c's}d^{\#\text{ of } d's}.$$ Explicit formulas of these generating functions are given by the following theorem. \[Boulet\_THM\]For variables $a$, $b$, $c$, and $d$ and $Q:=abcd$, we have$$\Psi(a,b,c,d) :=\frac{(-a,-abc;Q)_\infty}{(ab;Q)_\infty},\text{ and } \Phi(a,b,c,d) := \frac{(-a,-abc;Q)_\infty}{(ab,ac,Q;Q)_\infty}.$$ Theorem \[Boulet\_THM\] provides another direct proof of Theorem \[AliTHM1\]: $\Psi(q,q,1,1,)$. Similarly $\Phi(q,q,1,1)$, and $\Psi(q,1,q,1)$ yield \[AliTHM2\] $$\sum_{\pi\in {\mathcal{U}}} q^{\mathcal{O}(\pi)} = \frac{1}{(q)_\infty^2},\ \text{ and }\ \label{corollary_part_2}\sum_{\pi\in {{\mathcal{D}}}} q^{\mathcal{O}(\pi')} = (-q)^2_\infty,$$ where $\pi'$ is the conjugate of the partition $\pi$. Theorem \[AliTHM2\] can easily be translated to combinatorial results using vector partitions. Some non-trivial examples of coming from Theorem \[AliTHM2\] similar to and are as follows: \[COR\_ALI2\] Let $par(n)$ be the parity of $n$, $par(n)=0$ if $n$ even and $1$ otherwise. Let ${{\mathcal{K}}}$ be the set of partitions with gaps between parts and the smallest part both $\leq 2$. Let $\pi = (\lambda_1,\lambda_2,\dots)$, then $$\sum_{\pi\in {\mathcal{U}}} q^{\mathcal{O}(\pi)} =\sum_{\pi\in {\mathcal{U}}}\omega_{0,0}(\pi) q^{|\pi|} ,\ \text{ and }\ \sum_{\pi\in {{\mathcal{D}}}} q^{\mathcal{O}(\pi')} =\sum_{\pi\in {{\mathcal{K}}}} \hat{\omega}_1(\pi) q^{|\pi|},$$ where $$\omega_{0,0} (\pi) := (\lambda_{\nu(\pi)}+1)\cdot \prod_{i=1}^{\nu(\pi)-1} (\lambda_i-\lambda_{i+1}+1),\ \text{ and }\ \hat{\omega}_1(\pi) := 2^{par(\lambda_{\nu(\pi)})}\cdot \prod_{i=1}^{\nu(\pi)-1} 2^{par(\lambda_i - \lambda_{i+1})}.$$ Mark that $\omega_{0,0}(\pi)$ fits the definition . For uniform definition of a partition and the definition of ${\mathcal{P}}_{M}(k,m)$ sake, we have ignored this case in the prior conversation. We can also replace $\mathcal{O}(\pi')$ with ${{\mathcal{E}}}(\pi')$ on the set ${{\mathcal{D}}}$. This change would introduce a factor of 2 on the right-hand side of the respective identities of Theorem \[AliTHM2\] and Corollary \[COR\_ALI2\]. Conclusion {#Section7} ========== It appears that the approach of writing generating functions with statistics other than the norm offers a wide variety of questions. One of these questions is identifying a statistic and a set of partitions for given weighted count. There are many examples of Theorem \[Alladi\_weighted\_sum\] like weighted partition identities. Having a way of connecting these type of identities with the generating functions written with respect to a partition statistics would expand the horizons of this study. The crank’s appearance in this study is fortunate but not unexpected. The author would like to recall that Theorem \[RR2\_THM\] can been presented as a non-trivial example of . $$\sum_{\pi\in {{{\mathcal{D}}}}_{e}} q^{\mathcal{O}(\pi)} = \sum_{\pi \in {\mathcal{C}}_{\geq 0}} q^{|\pi|},$$ where $\widehat{{{\mathcal{D}}}}_{e}$ is the set of partitions into even number of distinct parts. In a similar fashion, Theorem \[Tilde\_Weights\_THEOREM\] can also be represented in the partition statistic $\mathcal{O}$ as Let ${{\mathcal{D}}}_o$ denote the set of partitions into odd number of distinct parts. Then, $$\begin{aligned} \sum_{\pi\in {{{\mathcal{D}}}}_{o}} q^{\mathcal{O}(\pi)} &= \sum_{\pi \in {\mathcal{C}}_{\leq -1}} q^{|\pi|}, \intertext{and} \sum_{\pi\in {{{\mathcal{D}}}}} (-1)^{\nu(\pi)}q^{\mathcal{O}(\pi)} &= -q+\sum_{\pi \in {\mathcal{C}}_{= 0}} q^{|\pi|},\end{aligned}$$ where $\nu(\pi)$ represents the number of parts of the partition $\pi$. We note that [@BerkovichUncu2], and [@Masao] makes it possible to refine the results involving the partition statistic $\mathcal{O}$ by imposing bounds on the number of parts and the largest part of partitions. Moreover, there are many more fundamental statistics of partitions similar to $\mathcal{O}$, and ${{\mathcal{E}}}$. It would be of interest to see results and weights related to the rank of a partition and similar known classical partition statistics. The author is planning on addressing these observations in the future. Acknowledgement =============== The author would like to thank George E. Andrews and Alexander Berkovich for their guidance. The author would also like to thank Alexander Berkovich, Jeramiah Hocutt, Frank Patane, and John Pfeilsticker for their helpful comments on the manuscript. [99]{} K. Alladi, *Partition identities involving gaps and weights*, Trans. Amer. Math. Soc. **349** (1997), no. 12, 5001-5019. K. Alladi, and A. Berkovich, *Göllnitz-Gordon partitions with weights and parity conditions*. Zeta functions, topology and quantum physics, Dev. Math., **14** (2005), 1–17, K. Alladi, and A. Berkovich, *New weighted Rogers-Ramanujan partition theorems and their implications*, Trans. Amer. Math. Soc. **354** (2002), no. 7, 2557-2577. G. E. Andrews, *The theory of partitions*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1998. Reprint of the 1976 original. MR1634067 (99c:11126) G. E. Andrews, F. G. Garvan *Dyson’s crank of a partition*, Bull. Amer. Math. Soc. (N.S.) **18** (1988), no. 2, 167-171. F. C. Auluck, *On some new types of partitions associated with generalized Ferrers graphs*, Proc. Cambridge Philos., **47** (1951), 679-686. A. Berkovich, and F. G. Garvan, *Some observations on Dyson’s New Symmetries of Partitions*, **100** (2002), no. 1, 61-93. A. Berkovich, and A. K. Uncu, *On partitions with fixed number of even-indexed and odd-indexed odd parts*, arXiv:1510.07301 \[math.NT\]. C. E. Boulet, *A four-parameter partition identity*, Ramanujan J. **12** (2006), no. 3, 315-320. M. Ishikawa, and J. Zeng *The Andrews-Stanley partition function and Al-Salam-Chihara polynomials*, Discrete Math. **309** (2009), no. 1, 151-175.

--- abstract: 'We investigate the lazy states, entangled states and discordant states for 2-qubit systems. We show that many lazy states are discordant, many lazy states are entangled, and many mixed entangled states are not lazy. With these investigations, we provide a laziness-discord-entanglement hierarchy diagram about 2-qubit quantum correlations.' author: - Jianwei Xu title: 'Lazy states, discordant states and entangled states for 2-qubit systems' --- Introduction ============ Quantum correlation is one of the most striking features of quantum theory. Entanglement is the most famous kind of quantum correlation, and leads to powerful applications [@Horodecki2009]. Discord is another kind of quantum correlation, which captures more correlation than entanglement in the sense that a disentangled state may have no zero discord [@Modi2012]. Due to the theoretical and applicational interests, discord has been extensively studied [@Modi2012] and still in active research (for examples see [@Rulli2011; @Xu2013; @Chi2013; @Liu2013]). A bipartite state is called lazy, if the entropy rate of one subsystem is zero under any coupling to the other subsystem. Necessary and sufficient conditions have recently been established for a state to be lazy [@Rosario2011], and it was shown that almost all states are pretty lazy [@Hutter2012]. It is shown that a maximally entangled pure state is lazy[@Ferraro2010]. This indicates that the correlation described by lazy states is not the same by entanglement. So we are interested to clarify the question that, whether there are many lazy states which are entangled, and whether there are many entangled states which are lazy. This paper answers this question for the 2-qubit case. This paper is organized as follows. In Section 2, we briefly review the definitions of entangled states, discordant states and lazy states. In Section 3, we establish a necessary and sufficient condition for 2-qubit lazy states. In Section 4, we show that there are many 2-qubit lazy states which are discordant states. In Section 5, we show that there are many disentangled states which are not lazy. In Section 6, we show that there are many 2-qubit mixed lazy states which are entangled. In section 7, we briefly summary this paper by providing a laziness-discord-entanglement hierarchy diagram to characterize the bipartite quantum correlations. Entangled states, discordant states, lazy states ================================================ We briefly review the definitions about entangled states, discordant states and lazy states. Finite-dimensional quantum systems $A$ and $B$ are described by the Hilbert spaces $H^{A}$ and $H^{B}$ respectively, the composite system $AB$ is then described by the Hilbert space $H^{A}\otimes H^{B}$. Let $n_{A}=\dim H^{A}$, $n_{B}=\dim H^{B}$. A state $\rho ^{AB}$ is called a disentangled state (or separable state) if it can be written in the form$$\begin{gathered} \rho ^{AB}=\sum_{i}p_{i}\rho _{i}^{A}\otimes \rho _{i}^{B},\end{gathered}$$ where $p_{i}\geq 0,\sum_{i}p_{i}=1,\{\rho _{i}^{A}\}_{i}$ are density operators on $H^{A}$, $\{\rho _{i}^{B}\}_{i}$ are density operators on $% H^{B}. $If $\rho ^{AB}$ is disentangled we then say $E(\rho ^{AB})=0.$ A state $\rho ^{AB}$ is called a zero-discord state with respect to $A$ if it can be written in the form$$\begin{gathered} \rho ^{AB}=\sum_{i=1}^{n_{A}}p_{i}|\psi _{i}^{A}\rangle \langle \psi _{i}^{A}|\otimes \rho _{i}^{B},\end{gathered}$$ where $p_{i}\geq 0,\sum_{i}p_{i}=1,\{|\psi _{i}^{A}\rangle \}_{i}$ is an orthonormal basis for $H^{A}$, $\{\rho _{i}^{B}\}_{i}$ are density operators on $H^{B}. $If $\rho ^{AB}$ is in the form Eq.(2) we then say $D_{A}(\rho ^{AB})=0.$ Evidently, $$\begin{gathered} D_{A}(\rho ^{AB})=0 \ ^{\Rightarrow } _{\nLeftarrow } \ E(\rho ^{AB})=0.\end{gathered}$$ A state $\rho ^{AB}$ is called a lazy state with respect to $A$ if [@Rosario2011] $$\begin{gathered} C_{A}(\rho ^{AB})=[\rho ^{AB},\rho ^{A}\otimes I^{B}]=0,\end{gathered}$$ where $\rho ^{A}=tr_{B}\rho ^{AB}$, $I^{B}$ is the identity operator on $% H^{B}.$ An important physical interpretation of lazy states is that the entropy rate of $A$ is zero in the time evolution under any coupling to $B,$ $$\begin{gathered} C_{A}(\rho ^{AB}(t))=0\Leftrightarrow \frac{d}{dt}tr_{A}[\rho ^{A}(t)\log _{2}\rho ^{A}(t)]=0\text{.}\end{gathered}$$ $D_{A}(\rho ^{AB})=0$ and $C_{A}(\rho ^{AB})=0$ has the inclusion relation below [@Ferraro2010] $$\begin{gathered} D_{A}(\rho ^{AB})=0 \ ^{\Rightarrow} _{\nLeftarrow} \ C_{A}(\rho ^{AB})=0.\end{gathered}$$ Maximal pure entangled states are the examples of $C_{A}(\rho ^{AB})=0$ but $% D_{A}(\rho ^{AB})\neq 0$ [@Ferraro2010]. The direct product states have the form $$\begin{gathered} \rho ^{AB}=\rho ^{A}\otimes \rho ^{B},\end{gathered}$$ they are obviously zero-discord states. The form of 2-qubit lazy states =============================== Any 2-qubit state can be written in the form [@Fano1983] $$\begin{gathered} \rho ^{AB}=\frac{1}{4}(I\otimes I+\sum_{i=1}^{3}x_{i}\sigma _{i}\otimes I+\sum_{j=1}^{3}y_{j}I\otimes \sigma _{j} \notag \\ +\sum_{i,j=1}^{3}T_{ij}\sigma _{i}\otimes \sigma _{j}),\end{gathered}$$ where $I$ is the two-dimensional identity operator,$\{\sigma _{i}\}_{i=1}^{3} $ are Pauli operators, $\{x_{i}\}_{i=1}^{3},\{y_{j}% \}_{j=1}^{3},\{T_{ij}\}_{i,j=1}^{3},$ are all real numbers satisfying some conditions (we will explore these conditions when we need them) to ensure the positivity of $\rho ^{AB}$, $\rho ^{A}$ and $\rho ^{B}$. We often omit $% I $ for simplicity without any confusion. $ $ **Proposition 1.** The 2-qubit state $\rho ^{AB}$ in Eq.(8) is lazy if and only if $$\begin{gathered} \{x_{i}\}_{i=1}^{3} // \{T_{ij}\}_{i=1}^{3} \text{ for }j=1,2,3.\end{gathered}$$ $ $ **Proof.** For state in Eq.(8), $$\begin{gathered} \rho ^{A}=\frac{1}{2}(I+\sum_{k=1}^{3}x_{k}\sigma _{k}\otimes I), \\ [\rho ^{AB},\rho ^{A}]=\frac{1}{8}\sum_{ijk=1}^{3}T_{ij}x_{k}[\sigma _{i}\otimes \sigma _{j},\sigma _{k}\otimes I] \notag \\ =\frac{1}{8}\sum_{ijk=1}^{3}T_{ij}x_{k}[\sigma _{i},\sigma _{k}]\otimes \sigma _{j} \notag \\ =\frac{i}{4}\sum_{ijkl=1}^{3}T_{ij}x_{k}\varepsilon _{ikl}\sigma _{l}\otimes \sigma _{j}.\end{gathered}$$ In the last line, $\varepsilon _{ikl}$ is the permutation symbol. Let $[\rho ^{AB},\rho ^{A}]=0,$ then $$\begin{gathered} \sum_{ik=1}^{3}T_{ij}x_{k}\varepsilon _{ikl}=0,\end{gathered}$$ this evidently leads to Eq.(9). $\square $ Lazy but diacordant 2-qubit states ================================== It is easy to check that $C_{A}(\rho ^{AB})=0$ defined in Eq.(4) is invariant under locally unitary transformations for arbitrary $n_{A}$ and $n_{B}$. Under locally unitary transformations, any 2-qubit state in Eq.(8) can be written in the form [@Luo2008] $$\begin{gathered} \rho ^{AB}=\frac{1}{4}(I\otimes I+\sum_{i=1}^{3}x_{i}\sigma _{i}\otimes I+\sum_{j=1}^{3}y_{j}I\otimes \sigma _{j} \notag \\ +\sum_{i=1}^{3}\lambda _{i}\sigma _{i}\otimes \sigma _{i}),\end{gathered}$$ where $0\leq \lambda _{1}\leq \lambda _{2}\leq \lambda _{3}$ being the singular values of $\{T_{ij}\}_{ij}$ in Eq.(8). Note that $% \{x_{i}\}_{i=1}^{3},\{y_{j}\}_{j=1}^{3}$ in Eq.(9) are not the same with in Eq.(8). We now look for the conditions such that $D_{A}(\rho ^{AB})=0.$ Suppose $D_{A}(\rho ^{AB})=0$, then according to Eq.(2), there exists real vector $% \overrightarrow{n}=\{n_{1},n_{2},n_{3}\}$ with $% n_{1}^{2}+n_{2}^{2}+n_{3}^{2}=1$ such that $$\begin{gathered} \rho ^{AB}=\Pi _{0}\otimes I\rho ^{AB}\Pi _{0}\otimes I+\Pi _{1}\otimes I\rho ^{AB}\Pi _{1}\otimes I,\end{gathered}$$ with $$\begin{gathered} \Pi _{0}=\frac{1}{2}(I+\overrightarrow{n}\cdot \overrightarrow{\sigma }), \\ \Pi _{1}=\frac{1}{2}(I-\overrightarrow{n}\cdot \overrightarrow{\sigma }).\end{gathered}$$ It can be check that $$\begin{gathered} \Pi _{0}\sigma _{i}\Pi _{0}+\Pi _{1}\sigma _{i}\Pi _{1}=n_{i} \overrightarrow{n}\cdot \overrightarrow{\sigma }.\end{gathered}$$ Then Eq.(14) becomes $$\begin{gathered} \rho ^{AB}=\frac{1}{4}(I\otimes I+\sum_{i=1}^{3}x_{i}n_{i}\overrightarrow{n}% \cdot \overrightarrow{\sigma }\otimes I \notag \\ +\sum_{j=1}^{3}y_{j}I\otimes \sigma _{j}+\sum_{i=1}^{3}\lambda _{i}n_{i}\overrightarrow{n}\cdot \overrightarrow{% \sigma }\otimes \sigma _{i}) \notag \\ =\frac{1}{4}(I\otimes I+\sum_{ij=1}^{3}x_{i}n_{i}n_{j}\sigma _{j}\otimes I \notag \\ +\sum_{j=1}^{3}y_{j}I\otimes \sigma _{j}+\sum_{ij=1}^{3}\lambda _{i}n_{i}n_{j}\sigma _{j}\otimes \sigma _{i}).\end{gathered}$$ Comparing to Eq.(13), then for $j=1,2,3,$ $$\begin{gathered} \sum_{i=1}^{3}x_{i}n_{i}n_{j}=x_{j}\Rightarrow \overrightarrow{n}//% \overrightarrow{x}, \\ \lambda _{i}n_{i}n_{j}=\delta _{ij}\lambda _{j}=\delta _{ij}\lambda _{i} \Rightarrow \lambda _{i}=0 \ \text{or} \ n_{i}=\pm 1.\end{gathered}$$ (i).If $\lambda _{1}=\lambda _{2}=\lambda _{3}=0,$ let $\overrightarrow{n}//% \overrightarrow{x},$ then $D_{A}(\rho ^{AB})=0$. (ii).If $0=\lambda _{1}=\lambda _{2}<\lambda _{3}=0,$ then $\overrightarrow{n}% =(0,0,\pm 1),$ to satisfy $\overrightarrow{n}//\overrightarrow{x},$ we see that only when $% \overrightarrow{x}=(0,0,x_{3})$ we have $D_{A}(\rho ^{AB})=0$. (iii).If $0=\lambda _{1}<\lambda _{2}<\lambda _{3}=0,$ then Eq.(20) can not be satisfied, so $\rho ^{AB}$ is discordant. (iv).If $0<\lambda _{1}<\lambda _{2}<\lambda _{3}=0,$ then Eq.(20) can not be satisfied, so $\rho ^{AB}$ is discordant. Comparing with Proposition 1, we then get Proposition 2 below. $ $ **Proposition 2.** A 2-qubit state in Eq.(13) is lazy but discordant if and only if $\overrightarrow{x}=0$ and $0<\lambda _{2}<\lambda _{3}$. $ $ Since any locally unitary transformation keeps $\overrightarrow{x}=0$ invariant in Eq.(8), then we rewrite Proposition 2 as Proposition 2$'$ below. $ $ **Proposition 2$'$.** A 2-qubit state in Eq.(8) is lazy but discordant if and only if $\overrightarrow{x}=0$ and the matrix $\{T_{ij}\}_{ij}$ have at least two positive singular values. $ $ We make a note that some constraints about $\{y_{j}\}_{j=1}^{3},\lambda _{1},\lambda _{2},\lambda _{3}$ are required to guarantee the positivity of $% \rho ^{AB},\rho ^{A}$, $\rho ^{B}$ in Proposition 2$.$These constraints are rather complex since there are so many parameters. To show there indeed exist many states described in Proposition 2, we choose some special states. For the state $$\begin{gathered} \rho ^{AB}=\frac{1}{4}(I\otimes I+\sum_{j=1}^{3}y_{j}I\otimes \sigma _{j}+\sum_{i=1}^{3}\lambda _{i}\sigma _{i}\otimes \sigma _{i}),\end{gathered}$$ where $0\leq \lambda _{1}\leq \lambda _{2},0<\lambda _{2}<\lambda _{3},$ we have $\rho ^{A}=I$,and $$\begin{gathered} \rho ^{B}=\frac{1}{2}(I+\sum_{j=1}^{3}y_{j}\sigma _{j}).\end{gathered}$$ $\rho ^{B}$ is positive then $$\begin{gathered} \sum_{j=1}^{3}y_{j}^{2}\leq 1.\end{gathered}$$ Let $y_{2}=y_{3}=\lambda _{1}=0,$ then the four eigenvalues of $\rho ^{AB}$ in Eq.(21) are $$\begin{gathered} \frac{1}{4}(1\pm \sqrt{y_{1}^{2}+(\lambda _{3}\pm \lambda _{2})^{2}}).\end{gathered}$$ These eigenvalues are all nonnegtive then we need $$\begin{gathered} 0<\lambda _{2}<\lambda _{3}, \\ y_{1}^{2}+(\lambda _{3}+\lambda _{2})^{2}\leq 1.\end{gathered}$$ There are many triples $\{y_{1},\lambda _{3},\lambda _{2}\}$ satisfy Eqs.(25,26), then the corresponding states in Eq.(21) are lazy but discordant states. Some disentangled but not lazy 2-qubit states ============================================= To show there exist many 2-qubit states which are disentangled but not lazy, we consider the states of the form $$\begin{gathered} \rho ^{AB}=p|\psi _{1}^{A}\rangle \langle \psi _{1}^{A}|\otimes \rho _{1}^{B}+(1-p)|\psi _{2}^{A}\rangle \langle \psi _{2}^{A}|\otimes \rho _{2}^{B},\end{gathered}$$ where $p\in (0,1),\{|\psi _{i}^{A}\rangle \}_{i=1}^{2}$ are normalized states in $H^{A}$ but not necessarily orthogonal,$\{\rho _{i}^{B}\}_{i=1}^{2} $ are density operators on $H^{B}.$ Note that $p=0$ or $% p=1$ leads to direct product states, so we do not consider such cases. Under locally unitary transformations, we let $$\begin{gathered} |\psi _{1}^{A}\rangle \langle \psi _{1}^{A}|=\frac{I+(0,0,1)\cdot \overrightarrow{\sigma }}{2}, \\ |\psi _{2}^{A}\rangle \langle \psi _{2}^{A}|=% \frac{I+(\sin \alpha ,0,\cos \alpha )\cdot \overrightarrow{\sigma }}{2}, \\ \rho _{1}^{B}=\frac{I+a(0,0,1)\cdot \overrightarrow{\sigma }}{2}, \\ \rho _{2}^{B}=\frac{I+b(\sin \beta ,0,\cos \beta )\cdot \overrightarrow{\sigma }}{% 2},\end{gathered}$$ where $\alpha ,\beta \in \lbrack 0,\pi ],a,b\in \lbrack 0,1].$ Some special states can be apparently specified. (v).$\alpha =0,\rho ^{AB}$ in Eq.(27) are direct product states; (vi).$\alpha =\pi ,\rho ^{AB}$ in Eq.(27) are zero-discord states; (vii).$a=b=0,\rho ^{AB}$ in Eq.(27) are direct product states. Now we consider the cases excluding (v), (vi), (vii) above. Taking Eqs.(28-31) into Eq.(27), and using the notations in Eq.(8), we get $$\begin{gathered} \overrightarrow{x}=((1-p)\sin \alpha ,0,p+(1-p)\cos \alpha ), \\ \{T_{i1}\}_{i}=(b(1-p)\sin \alpha \sin \beta ,0,b(1-p)\cos \alpha \sin \beta ), \\ \{T_{i2}\}_{i}=(0,0,0), \\ \{T_{i3}\}_{i}=(b(1-p)\sin \alpha \cos \beta ,0,ap+b(1-p)\cos \alpha \cos \beta ).\end{gathered}$$ From Proposition 1, $\rho ^{AB}$ in Eq.(27) is lazy if and only if $% \overrightarrow{x}//\{T_{i1}\}_{i}$ and $\overrightarrow{x}//\{T_{i3}\}_{i}. $ Since $x_{1}=(1-p)\sin \alpha \neq 0,$ then $\overrightarrow{x}//\{T_{i1}\}_{i}$ and $\overrightarrow{x}//\{T_{i3}\}_{i}$ lead to $$\begin{gathered} b\sin \beta =0. \\ a=b\cos \beta .\end{gathered}$$ Eqs.(36,37) together correspond to direct product states since $\rho _{1}^{B}=\rho _{2}^{B} $. Otherwise, there are many states violate Eq.(36) or Eq.(37), so they are not lazy states. $ $ **Proposition 3**. 2-qubit disentangled state $\rho ^{AB}$ in Eq.(27), is a direct product state when $|\psi _{1}^{A}\rangle =|\psi _{2}^{A}\rangle $ or $\rho _{1}^{B}=\rho _{2}^{B}$, is a zero-discord state when $\langle \psi _{1}^{A}|\psi _{2}^{A}\rangle =0$. Otherwise, $\rho ^{AB}$ is not lazy. Some lazy but entangled states ============================== We know that a bipartite pure state is lazy only if under locally unitary transformations it can be written in the form [@Rosario2011] $|\psi^{AB}\rangle=\frac{1}{\sqrt{s}}\sum_{i=1}^{s}|\psi^{A}_{i}\rangle|\psi^{B}_{i}\rangle$, where $\{|\psi _{i}^{A}\rangle \}_{i}$ are orthonormal sets in $H^{A}$, $\{|\psi _{i}^{B}\rangle \}_{i}$ are orthonormal sets in $H^{B}$, $s\leq \min \{n_{A},n_{B}\}$. When $s= \min \{n_{A},n_{B}\}$ it is maximally entangled state. In this section we look for more 2-qubit mixed states which are lazy but entangled. From Proposition 1, we know the following 2-qubit Bell-diagonal states are lazy $$\begin{gathered} \rho ^{AB}=\frac{1}{4}(I\otimes I+\sum_{i=1}^{3}\lambda _{i}\sigma _{i}\otimes \sigma _{i}),\end{gathered}$$ where $\{\lambda _{i}\}_{i=1}^{3}$ are real numbers satisfying some constraints to ensure the positivity of $\rho ^{AB}.$ In this section, for convenience, we do not assume $\{\lambda _{i}\}_{i=1}^{3}$ are all nonnegative. We represent the states in Eq.(38) in the $(\lambda _{1},\lambda _{2},\lambda _{3})$ space. The eigenvalues of $\rho ^{AB}$ in Eq.(38) are $$\begin{gathered} \frac{1}{4}\{1-\lambda _{1}+\lambda _{2}+\lambda _{3},1+\lambda _{1}-\lambda _{2}+\lambda _{3}, \notag \\ 1+\lambda _{1}+\lambda _{2}-\lambda _{3},1-\lambda _{1}-\lambda _{2}-\lambda _{3}\}.\end{gathered}$$ Then the positivity of $\rho ^{AB}$ requires that $\{\lambda _{i}\}_{i=1}^{3} $ are in the tetrahedron (with its boundary) with the vertices $(-1,-1,-1),(-1,1,1),(1,-1,1),(1,1,-1)$ in the $(\lambda _{1},\lambda _{2},\lambda _{3})$ space [@Horodecki1996]. Disentangled states in Eq.(38) are in the octahedron (with its boundary) with the vertices $(\pm 1,0,0),(0,\pm 1,0),(0,0,\pm 1)$ [@Horodecki1996]. From Proposition 2, we know the zero-discord states in Eq.(38) are only three line segments $(\lambda _{1},0,0)$ with $\lambda _{1}\in \lbrack -1,1], (0,\lambda _{2},0)$ with $\lambda _{2}\in \lbrack -1,1]$, $(0,0,\lambda _{3})$ with $\lambda _{3}\in \lbrack -1,1].$ Then the states in the tetrahedron (with its boundary) but not in the octahedron (with its boundary) are lazy but entangled. Among these, only the states at the vertices of tetrahedron are (maximally entangled) pure states. Summary: a hierarchy diagram ============================ We explored some 2-qubit states, showed that many states are lazy but discordant, many states are lazy but entangled, and many states are disentangled but not lazy. With these investigations, we can surely give a hierarchy diagram (Figure 1) of 2-qubit states, including lazy states, disentangled states and zero-discord states. 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--- author: - 'Á. Kóspál' - 'P. Ábrahám' - 'J. A. Acosta-Pulido' - 'M. J. Arévalo Morales' - 'M. I. Carnerero' - 'E. Elek' - 'J. Kelemen' - 'M. Kun' - 'A. Pál' - 'R. Szakáts' - 'K. Vida' bibliography: - 'paper.bib' date: 'Received date; accepted date' title: | The outburst and nature of two young eruptive stars\ in the North America/Pelican Nebula Complex --- [In order to determine the true nature of these two objects, we started an optical and near-infrared monitoring program, and complemented our data with archival observations and data from the literature.]{} [We plot and analyze pre-outburst and outburst spectral energy distributions (SEDs), multi-filter light curves, and color-color diagrams.]{} [The quiescent SED of HBC722 is consistent with that of a slightly reddened normal TTauri-type star. The source brightened monotonically in about two months, and the SED obtained during maximum brightness indicates the appearance of a hot, single-temperature blackbody. The current fading rate implies that the star will return to quiescence in about a year, questioning its classification as a bone fide FUor. The quiescent SED of VSXJ205126.1+440523 looks like that of a highly embedded Class I source. The outburst of this source happened more gradually, but reached an unprecedentedly high amplitude. At 2.5 months after the peak, its light curves show a deep minimum, when the object was close to its pre-outburst optical brightness. Further monitoring indicates that it is still far from being quiescent.]{} [The shape of the light curves, as well as the bolometric luminosities and accretion rates suggest that these objects do not fit into the classic FUor group. Although HBC722 exhibit all spectral characteristics of a bona fide FUor, its luminosity and accretion rate is too low, and its timescale is too fast compared to classical FUors. VSXJ205126.1+440523 seems to be an example where quick extinction changes modulate the light curve.]{} Introduction ============ In August 2010, two new young eruptive star candidates were discovered in the North America/Pelican Nebula Complex (distance: 550pc, @straizys1989). HBC722 (also known as LkH$\alpha$188G4 and PTF10qpf) brightened by $\Delta$R=3.3mag between 2010 May and August [@semkov2010a]. VSXJ205126.1+440523 (also known as IRAS20496+4354 and PTF10nvg) brightened by 1.8mag in unfiltered light between 2009 December and 2010 June, but Digitized Sky Survey plates show that it had been several magnitudes fainter in quiescence [@itagaki2010; @munari2010]. @semkov2010 and @miller2010 provided light curves and spectroscopy for HBC722 and concluded that we witness a bona fide outburst of a FUor-type object. FUors, named after the prototype object FUOrionis, brighten by up to 5mag at optical wavelengths and may stay in the high state for decades. @covey2010 presented light curves and spectroscopy for VSXJ205126.1+440523, and found that in many respects this object is different from FUors or EXors (the latter being another class of eruptive YSOs, named after the prototype EXLup, which flares up by 1-5mag in every few years and stay bright for several months). Currently only about two dozens of young eruptive stars (FUors and EXors) are known, thus the two new outbursts announced in August 2010 are noteworthy events. Should they turn out to be accretion-powered eruptions, their detailed study may contribute to the understanding of these important phases of early stellar evolution. In this paper we present an optical and infrared view of the two eruptive star candidates. Using archival and new data, we characterize their circumstellar environment and compare them with those of some better studied FUors and EXors. We present new optical and near-infrared photometric data points taken during the outburst, which indicate that HBC722 already passed its peak brightness and started a monotonous fading with a steady fading rate, while neither the brightening, nor the fading of VSXJ205126.1+440523 was monotonous. Observations and data reduction =============================== #### Optical observations. (B)VRI-band images were obtained between 19 September 2010 and 2 January 2011 with three telescopes: the 60/90/180cm (aperture diameter/primary mirror diameter/focal length) Schmidt telescope of the Konkoly Observatory (Hungary), the 1m (primary mirror diameter) RCC telescope of the Konkoly Observatory, and the 80cm (primary mirror diameter) IAC-80 telescope of the Teide Observatory in the Canary Islands (Spain). The Konkoly Schmidt telescope is equipped with a 4096$\times$4096 pixel Apogee Alta U16 CCD camera (pixel scale: 1.03$''$), and a Bessel BV(RI)$_{\rm C}$ filter set. The 1m RCC is equipped with a 1300$\times$1340 pixel Roper Scientific WersArray:1300B CCD camera (pixel scale: 0.306$''$), and a Bessel UBV(RI)$_{\rm C}$ filter set. The Teide IAC-80 telescope is equipped with a 2048$\times$2048 pixel Spectral Instruments E2V 42-40 back-illuminated CCD camera ‘CAMELOT’ (pixel scale: 0.304$''$), and a Johnson-Bessel UBV(RI)$_{\rm J}$ filter set. The images were reduced in IDL following the standard processing steps of bias subtraction and flat-fielding. On each night, for each target, images were obtained in blocks of 3 or 5 frames per filter. Aperture photometry for the target and other field stars were performed on each image using IDL’s *cntrd* and *aper* procedures. Since HBC722 is surrounded by a reflection nebula, in order to be consistent with the photometry of @semkov2010, we used the same apertures: an aperture radius of 4$''$ and a sky annulus between 13$''$ and 19$''$. For HBC722, instrumental magnitudes were transformed to the standard system using the 8 brightest stars (from star ‘A’ to star ‘H’) from the comparison sequence given in @semkov2010. For each image we fitted the difference of the instrumental and the standard magnitudes of the comparison stars as a function of the V$-$I color, and used this relationship to convert the instrumental magnitudes of our target to standard magnitudes. For VSXJ205126.1+440523, we observed the standard field NGC6823 with the Schmidt telescope during the photometric night 23/24 September 2010, and calibrated 6 comparison stars in the vicinity of our target. A finding chart and the standard magnitudes of the comparison stars can be seen in Fig. \[fig:compstars\] and in Tab. \[tab:compstars\] in the Online Material. The conversion of instrumental to standard magnitudes was done the same way as for HBC722. Similarly to the comparison stars of HBC722, we cannot exclude that the comparison stars of VSXJ205126.1+440523 might be variables on longer timescales, although they were constant within the measurement uncertainties during our observing period. The resulting photometry for our two targets is presented in Tabs. \[tab:phot\_vsxj20\] and \[tab:phot\_hbc722\] in the Online Material. We note that the R and I filters on the two telescopes are different, which may introduce a systematic difference in the magnitudes obtained with the different telescopes. However, in our experience, this difference is less than 0.05mag [@kospal2010]. Since the observed brightness variations of our targets are much larger than 0.05mag, this possible difference in the filter systems does not affect our analysis and conclusions. #### Near-infrared observations. JHK$_{\rm S}$ images were obtained with the 1.52m Telescopio Carlos Sanchez (TCS, Teide Observatory, Spain) between 19 September and 19 November 2010, using the 256$\,{\times}\,$256 pixel Nicmos 3 detector CAIN-3 with the wide field optics (pixel scale: 1$''$). Observations were performed in a 5-point dither pattern in order to enable proper sky subtraction. The images were reduced using a modified version of [*caindr*]{}, an iraf data reduction package written by J. Acosta-Pulido[^1]. Data reduction steps included sky subtraction, flat-fielding, and the co-addition of all frames taken with the same filter. The sky image was obtained as the median combination of all frames, masking regions occupied by bright sources. The final image was produced using the standard “shift-and-add” technique, including rejection of outlier pixels. The instrumental magnitudes of the target and all good-quality 2MASS stars in the field were extracted using aperture photometry in IDL. For the photometric calibration we used the Two Micron All Sky Survey (2MASS) catalog [@cutri2003]. We determined the offset between the instrumental and the 2MASS magnitudes by averaging typically 20-30 stars, using [*biweight\_mean*]{}, an outlier-resistant averaging method. The resulting photometry of our two targets is presented in Tabs. \[tab:phot\_vsxj20\] and \[tab:phot\_hbc722\] in the Online Material. We obtained additional near-infrared photometry using archival images from the UKIRT InfraRed Deep Sky Surveys (UKIDSS). These images were taken with the Wide Field Camera on the 3.8m diameter UKIRT in 2006, and they are part of Data Release 8. Aperture photometry and calibration were executed in the same way as for the TCS data. Results and analysis ==================== HBC722 ------ HBC722 is part of a small cluster of young stars called “LkH$\alpha$188 group” by @cohen1979. In their naming convention, HBC722 is called LkH$\alpha$ 188 G4. The whole cluster is located in a dark cloud separating the North America and the Pelican Nebulae [@straizys1989]. In outburst, the star is surrounded by a compact, asymmetric reflection nebula, which is well visible at optical wavelengths [@miller2010], but also discernible in our J band images. #### Light curve. The brightening of HBC722 is well documented in @semkov2010 and @miller2010. Between August 2009 and July 2010, the star gradually brightened by 1mag in the R band, then between July and August 2010, it brightened by another 3mag, reaching a maximum brightness at the end of August 2010. We have been monitoring the star since September 2010. Our data confirm that the star reached is maximal brightness, and is currently gradually fading (Fig. \[fig:light\_hbc722\]). Between 20 September 2010 and 3 December 2010, HBC722 decreased its brightness by 0.55, 0.54, 0.47, 0.37, 0.32, and 0.21mag in the B, V, R, I, J, H, and K$_{\rm S}$ bands, respectively, indicating that the source has become slightly redder. Fitting a line to the data points between these two epochs gives fading rates of 0.34, 0.25, 0.25, 0.21, 0.16, 0.13, and 0.07 mag/month in the B, V, R, I, J, H, and K$_{\rm S}$ band, respectively. Our last optical data points taken on 2 January 2011 fit into this trend. Assuming that the linear fading continues with these rates, and taking into account the pre-outburst optical fluxes observed by @semkov2010 around July 2009, we estimate that the source will return to quiescence some time between fall 2011 and spring 2012. The two pre-outburst JHK$_{\rm S}$ data sets (2MASS from June 2000 and UKIDSS from July 2006) agree within 0.1mag, indicating a rather constant pre-outburst near-infrared brightness. Considering the relatively slow near-infrared fading rates, HBC722 may exhibit higher than quiescent near-infrared fluxes even until fall 2013. Thus, current fading rates indicate that the outburst of HBC722 would last approximately 2 or 3 years. In order to put into context the outburst history of HBC722, in Fig. \[fig:context\] we compare its light curve with those of other young eruptive objects. The recent extreme outburst of EXLup exhibited a faster onset, more peaked maximum, and a fading which was initially very similar to that of HBC722. However, EXLup showed two deep minima before it went back to quiescence some 8 months after its peak brightness. The light curve of V1057Cyg, the FUor having the fastest known outburst and fading, is still much slower than that of HBC722. The light curve of V1647Ori, a source often classified as an intermediate-type between FUors and EXors, initially displayed a relatively slow fading, then suddenly went back to quiescence, thus had an approximately 2.5-year-long outburst. If HBC722 continues the linear fading it currently displays, the predicted outburst length will be remarkably similar to that of V1647Ori. This comparison implies that HBC722 is different from the classical FUors and is more similar to EXors, or intermediate objects between FUors and EXors. ![Light curves of HBC722. For better visibility, the B, V, I, J, H, and K$_{\rm S}$ light curves were shifted along the y axis by the values indicated in the figure. Filled dots are from this work, plus signs are from @semkov2010 and @miller2010.[]{data-label="fig:light_hbc722"}](light_hbc722.ps){width="\columnwidth"} ![Light curves of our targets and those of different young eruptive stars. [*Triangles:*]{} photographic light curve of V1057Cyg from @gieseking1973; [*stars:*]{} I$_{\rm C}$ light curve of V1647Ori from @acosta2007; [*squares:*]{} visual light curve of EXLup during its extreme outburst in 2008 [@abraham2009]. For HBC722 and VSXJ205126.1+440523 we plotted the R-band light curves. The data for V1057Cyg, V1647Ori, and EXLup were shifted along the y axis for better visibility and also along the x axis so that the peak brightness is approximately at the same position for all stars.[]{data-label="fig:context"}](ilight_long.ps){width="\columnwidth"} #### Spectral energy distribution. Fig. \[fig:sed\_hbc722\] shows the pre-outburst and outburst SEDs of HBC722. Pre-outburst data are from @miller2010 and references therein, while the outburst data are from this work. In this figure, we also overplotted with gray shading the typical SED of a TTauri-type star in the Taurus star-forming region [@dalessio1999; @furlan2006], scaled to the H-band data point, and reddened by A$_{\rm V}$=3.36mag [@cohen1979]. The outburst photometry indicates that a hot continuum is added to the quiescent SED. The B, V, R, I, J, H, and K$_{\rm S}$ points indicate a blackbody-like spectrum at all epochs during the outburst. We could fit these points with a single temperature blackbody and obtained a temperature of 4000K (using A$_{\rm V}$=3.36mag). With the assumption that the SED is similar to the Taurus median above 24$\,\mu$m, we calculated a pre-outburst bolometric luminosity of 0.85L$_{\odot}$ by integrating the de-reddened SED between 0.44 and 200$\,\mu$m. The outburst bolometric luminosity can be similarly calculated, but due to the lack of mid-infrared data points, we can either assume a blackbody shape until 10$\,\mu$m and assume that the SED did not change above 10$\,\mu$m, or assume that the SED changed self-similarly in the whole 2$-$200$\,\mu$m range. The former approach give L$_{\rm bol}$=8.7L$_{\odot}$, the latter, L$_{\rm bol}$=12L$_{\odot}$. The true outburst luminosity is probably between these two values. ![Spectral energy distribution of HBC722. Filled dots are pre-outburst data from @miller2010, while open circles are outburst data (this work). The gray shaded area indicates the median SED of TTauri stars in the Taurus star-forming region with spectral types between K5 and M2 (data below 1.25$\,\mu$m and above 40$\,\mu$m are from @dalessio1999, data between 1.25 and 40$\,\mu$m are from @furlan2006).[]{data-label="fig:sed_hbc722"}](sed_hbc722.ps){width="\columnwidth"} VSXJ205126.1+440523 ------------------- VSXJ205126.1+440523 is situated in an isolated molecular cloud located about 15$'$ southeast of the Pelican Nebula molecular cloud [@bally2003]. The eastern rim of this small cloud is well visible in the \[SII\] and H$\alpha$ images of @bally2003. This morphology suggests that VSXJ205126.1+440523 sits on the tip of a column of dense material, out of which it had been born. Apart from the H$\alpha$ emission from the rim, no extended emission seems to be associated with the source, not even in outburst. @bally2003 discovered several Herbig-Haro objects in this area, and claim that one of them, HH569, is possibly driven by VSXJ205126.1+440523. #### Light curve. In Fig. \[fig:light\_vsxj20\] we plotted the light curves of VSXJ205126.1+440523. @covey2010 reported the source to be between R=18–19.25mag in mid-2009. After that, it brightened by $\approx$6mag, reaching its maximal brightness in August 2010. Since then, the source started fading, and by November 2010, it has nearly reached its mid-2009 optical brightness, then it brightened again. The light curves show that neither the brightening, nor the fading was monotonous. Although the near-infrared light curves are not as well-sampled as the optical ones, they delineate similar trends but with smaller amplitudes. We note that the R=19.25mag reported by @covey2010 may not be the true quiescent brightness of the source, since the source was $\approx$20mag in the POSS2 red plate taken in 1990 [@itagaki2010]. The comparison of the UKIDSS and TCS photometry indicates that the source brightened by $\Delta$J=7.9mag, $\Delta$H=6.7mag, and $\Delta$K$_{\rm S}$=4.8mag between July 2006 and September 2010. We note that the source was K$_{\rm S}$=13.15mag in 2006, but it was not visible in the K$_{\rm S}$ band in the 2MASS images taken in October 2000. Since the 2MASS PSC is complete down to K$_{\rm S}$=14.3 [@cutri2003], the source must have brightened at least 1.15mag between 2000 and 2006, making the true K$_{\rm S}$-band magnitude change at least 5.8mag. The comparison of the light curve of VSXJ205126.1+440523 with other young eruptive stars in Fig. \[fig:context\] indicates that this source is different from all the other sources plotted, although the brightening and fading rates are most similar to those of EXLup. Especially remarkable is the deep minimum of VSXJ205126.1+440523 in November 2010, which is similar to the minima displayed by EXLup shortly before the end of the eruption. ![Light curves of VSXJ205126.1+440523. Filled dots are from this work, plus signs are from @covey2010 and from Seiichiro Kiyota and Hiroyuki Maehara (vsnet, http://tech.dir.groups.yahoo.com/group/vsnet-recent-fuori/messages), crosses are visual estimates by @itagaki2010.[]{data-label="fig:light_vsxj20"}](light_vsxj20.ps){width="\columnwidth"} #### Spectral energy distribution. In Fig. \[fig:sed\_vsxj20\] we compiled a pre-outburst SED using data from the UKIDSS database (this work), the MSX6C Infrared Point Source Catalog [@egan2003], the AKARI/IRC mid-infrared all-sky survey [@ishihara2010], and Spitzer data [@covey2010 and references therein]. Out of these data points, the UKIDSS and the Spitzer are quasi-simultaneous (all obtained between June and August 2006), while the MSX data are from 1996-1997, and the AKARI from 2006-2007. This SED should be analysed with caution, considering the K$_{\rm S}$-band variability mentioned above. The outburst SED contains optical and near-infrared photometry we obtained on 20/23 September 2010 and 16/17 November 2010. We note that by September, the source was already $\approx$2mag fainter in R-band than at maximal brightness some 20 days earlier. The shape of the SED and the fact that the source in quiescence was practically invisible (the only pre-outburst image where the source is visible at optical wavelengths is the POSS2 red plate in Fig. \[fig:map\_vsxj20\]) suggest that the source is highly extincted. However, dereddening its colors does not make it fall onto the TTauri locus (Fig. \[fig:tcd\]). Correcting for a reddening of A$_{\rm V}$=17...20mag would result in a J$-$H color typical for TTauri stars, but its H$-$K$_{\rm S}$ color would still be too red. The reason for the strange near-infrared colors of VSXJ205126.1+440523 may be partly interstellar reddening caused by the small cloud in which the source is embedded and whose outlines are visible in Fig. \[fig:map\_vsxj20\], partly circumstellar reddening by an envelope or thick disk. The relative importance of these two effects is not known, thus we do not attempt to correct for interstellar reddening, and calculate a bolometric luminosity of 14.7L$_{\odot}$ by simply integrating the quiescent SED from 1.25 to 200$\,\mu$m. We calculate an outburst luminosity of 22L$_{\odot}$ similarly, assuming that the SED did not change above 10$\,\mu$m, and using the September 2010 data points in Fig. \[fig:sed\_vsxj20\]. ![Spectral energy distribution of VSXJ205126.1+440523. Filled dots are pre-outburst data from various dates, while open circles are outburst data (see text).[]{data-label="fig:sed_vsxj20"}](sed_vsxj20.ps){width="\columnwidth"} ![Near-infrared color-color diagram. The main-sequence is marked by a thick solid line, the giant branch with dotted line [@koornneef1983], the reddening path with dashed lines [@cardelli1989], and the TTauri locus with dash-dotted line [@meyer1997]. [*Open symbols:*]{} quiescent colors; [*filled symbols:*]{} outburst colors. Source of data: @kenyon1991 for V1057Cyg, @acosta2007 for V1647Ori, 2MASS PSC and @juhasz2010 for EXLup, and this work for HBC722 and VSXJ205126.1+440523.[]{data-label="fig:tcd"}](tcd.ps){width="\columnwidth"} Discussion ========== The nature of the sources in quiescence --------------------------------------- The SED of HBC722 seems to be consistent with that of a slightly reddened TTauri star, both regarding the optical–near-infrared part of the SED and the 24$\,\mu$m photometric point. This conclusion is in accordance with the claim of @miller2010 that HBC722 is a Class II object where the central star is a K7-type star. It is noteworthy, however, that between 3.6 and 8$\,\mu$m there is an excess emission in the SED compared to the Taurus median. In this respect, the source somewhat resembles DRTau, a highly accreting TTauri star, suggesting that HBC722 might be a highly accreting TTauri star even in quiescence. The fact that the quiescent optical spectrum, taken by @cohen1979 exhibits an unusually prominent H$\alpha$ emission with an equivalent width of 100Å supports this idea. VSXJ205126.1+440523 seems to be a much more reddened source. Its quiescent bolometric luminosity ($\approx$15L$_{\odot}$) is significantly larger than that of HBC722, indicating a somewhat higher mass. The quiescent SED in Fig. \[fig:sed\_vsxj20\] probably represents a moderately reddened Class I source. The findings of @covey2010, who determined 6mag$<$A$_{V}{<}$12.4mag from the ratio of H emission lines observed in the outburst spectrum, supports this idea. This scenario requires a dense envelope in the system, but the lack of a reflection nebula around this source suggests the lack of an extended envelope. However, high interstellar extinction may render the scattered light invisible. The presence of a related Herbig-Haro object also advocates for the Class I scenario. The outburst mechanism ---------------------- The comparison of the quiescent and the outburst SEDs (Fig. \[fig:sed\_hbc722\]) suggests that the brightening of HBC722 can be interpreted as the appearance of a hot continuum. The data points can be fitted with a temperature of $\approx$4000K, somewhat less than what one would expect from an accretion outburst, where ionized material is present. Note however that the temperature may be higher if the extinction towards the system is higher. It is noteworthy that the outburst excess can be described in the optical–near-infrared regime by a single temperature blackbody rather than a disk-like emission reflecting a radial (usually outwards decreasing) temperature profile. The fading of HBC722 is mostly color-independent in the near-infrared regime (Fig. \[fig:tcd\]), while optical colors are becoming slightly redder. This may indicate that the hot continuum is both fading and cooling. The reason for the brightening of VSXJ205126.1+440523 is more enigmatic. In this case the excess emission in outburst is not a single-temperature radiation, but seems to have a temperature distribution at all epochs (Fig. \[fig:light\_vsxj20\]). The amplitude of the outburst in the J, H and K$_{\rm S}$ bands is larger than any brightening observed so far for YSOs. At first glance, one may think that the brightening was due to suddenly decreased extinction. However, in this case the source should have moved along the reddening path in the J$-$H vs. H$-$K$_{\rm S}$ diagram (Fig. \[fig:tcd\]) which was not the case during the rising part of the lightcurve. JHK$_{\rm S}$ photometry obtained after maximum brightness indicate that the fading of VSXJ205126.1+440523 initially happened along the same path as the brightening, suggesting that whatever was the cause of the flux changes, it was a reversible process. Changing accretion is an appealing idea because it would explain the presence of the Herbig-Haro object possibly ejected from the source during a previous outburst (enhanced accretion is often accompanied by enhanced mass outflow). It also suggests that the outburst activity of VSXJ205126.1+440523 might be repetitive. However, photometry obtained in November 2010 indicates a deep minimum in Fig. \[fig:light\_vsxj20\]. At the same time, the source moved along the reddening path in Fig. \[fig:tcd\]. Both the color changes and the brightness changes in the J, H, and K$_{\rm S}$ bands are consistent with an extinction increase of A$_{\rm V}$=9mag. Thus, it may be possible that the deep minimum in November 2010 was caused by a dust condensation effect (similar to what happened to V1515Cyg in 1980 [@kenyon1991], or an eclipse by dust clumps in an almost edge-on disk system (similarly to what causes deep optical minima of the UX Orionis-type stars, see e.g. @eaton1995). Classification as a FUor outburst --------------------------------- When discovered, both sources were announced as FUor candidates. Extensive analyses presented in @semkov2010 and @miller2010 indicated that HBC722 can be considered as a bona fide FUor. On the other hand, @covey2010 concluded that VSXJ205126.1+440523 does not appear to belong to either the FUor or the EXor class. FUors are usually bright objects in outburst with luminosities of a few hundred L$_{\odot}$ [@hk96]. The immense radiation is related to the increased accretion rate, which can reach values up to 10$^{-4}$M$_{\odot}$/yr. However, the outburst luminosities of our objects are only in the order of 10-20L$_{\odot}$. Assuming that the luminosity excess in eruption is all due to the release of accretion energy, the computed accretion rate for HBC722 (assuming a stellar mass of 0.5M$_{\odot}$ and a radius of 3R$_{\odot}$) is 10$^{-6}$M$_{\odot}$/yr. @covey2010 determined an accretion rate of 2.5$\times$10$^{-7}$M$_{\odot}$/yr for VSXJ205126.1+440523. Both of these values are well below the typical value for classical FUors. The brightening and fading rates for both of our sources are also too fast compared to classical FUor light curves. In the case of HBC722 we made a comparison with several young eruptive stars and found a mismatch with the prototype FUor V1057Cyg but more similarities with the light curves of EXLup, the prototype of EXors, and V1647Ori, an object often classified as an intermediate object between FUors and EXors. The slow brightening of VSXJ205126.1+440523 is not unheard of (both FUOri and V1057Cyg had a rise-time of about 1 year, for other sources the estimates range between 3 and 20 years, @bell1994), but the fading is far too rapid. According to our light curve (Fig. \[fig:light\_vsxj20\]), 2.5 months after peak brightness VSXJ205126.1+440523 dimmed by about 5mag in V-band. This is closer to the typical timescales of EXor flare-ups than that of bona fide FUor outbursts (decades to centuries). The non-monotonous fading of this source also resembles the light curve of the recent outburst of EXLup (Fig. \[fig:context\]). The moderate resolution near-infrared spectrum obtained in outburst by @covey2010 also shows similarities to that of EXLup [@kospal2011]. It is remarkable that HBC722 possesses all spectral characteristics of bona fide FUors but its luminosity is an order of magnitude lower, and its fading timescale is much faster. This suggests that the physical mechanism which is behind the FUor-type eruptions should also work with lower accretion rates although probably producing shorter outbursts. This conclusion questions the thermal instability model of @bell1994, who suggested the existence of a threshold mass accretion rate from the outer to the inner circumstellar disk. Matter can pile up at the inner edge of the disk and fall onto the stellar surface following a sudden thermal instability only if the quiescent accretion rate is higher than this threshold value. Thus, it seems that in the regime of low luminosity outbursts ($\approx$10L$_{\odot}$), both FUor-like eruptions (when the source exhibits all spectral characteristics of FUors like HBC722) and EXor-like events (when the source exhibits a typical TTauri spectrum with emission lines and CO bandhead emission, somewhat similar to VSXJ205126.1+440523) can occur. If both HBC722 and VSXJ205126.1+440523 are indeed young eruptive stars, one might conclude that the class of young eruptive stars is even more diverse than what was thought before. This work is based in part on observations made with the Telescopio Carlos Sanchez operated on the island of Tenerife by the Instituto de Astrofísica de Canarias in the Observatorio del Teide. The authors wish to thank the telescope manager A. Oscoz, support astronomer P. Montañes, and telescope operators R. Martí, and M. Díaz for their help during the observations. This work is based in part on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. This work is based in part on data obtained as part of the UKIRT Infrared Deep Sky Survey. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the NASA and the National Science Foundation. The research of Á.K. is supported by the Nederlands Organization for Scientific Research. [^1]: for more details on the [*caindr*]{} package, see http://www.iac.es/telescopes/cain/cain\_eng.html.

--- bibliography: - 'database.bib' --- \ \ \ \ Introduction {#intro .unnumbered} ============ In this work I will construct certain general bundles $\langle\mathfrak{M},\rho,X\rangle$ and $\langle\mathfrak{B},\eta,X\rangle$ of Hausdorff locally convex spaces associated to a given Banach bundle $\langle\mathfrak{E},\pi,X\rangle$. Then I will present conditions ensuring the existence of bounded selections $\mathcal{U}\in \Gamma^{x_{\infty}}(\rho)$ and $\mathcal{P}\in \Gamma^{x_{\infty}}(\eta)$ both continuous at a point $x_{\infty}\in X$, such that $\mathcal{U}(x)$ is a $C_{0}-$semigroup of contractions on $\mathfrak{E}_{x}$ and $\mathcal{P}(x)$ is a spectral projector of the infinitesimal generator of the semigroup $\mathcal{U}(x)$, for every $x\in X$. In a subsequent paper I shall produce examples of the general results presented here. Here ${\mathfrak{W}} \doteqdot {\left\langle {\mathfrak{M}},\rho,X{\right\rangle}}$ and ${\left\langle {\mathfrak{B}},\eta,X{\right\rangle}}$ are special kind of bundles of Hausdorff locally convex spaces (bundle of $\Omega-$spaces) while ${\mathfrak{V}} \doteqdot {\left\langle {\mathfrak{E}},\pi,X{\right\rangle}}$ is a suitable Banach bundle such that the common base space $X$ is a metrizable space. Moreover for all $x\in X$ the stalk ${\mathfrak{M}}_{x} \doteqdot \overset{-1}{\rho}(x)$ is a topological subspace of the space ${\mathcal{C}_{c} \left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})\right)}$ with the topology of compact convergence, of all continuous maps defined on ${\mathbb {R}}^{+}$ and with values in ${\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})$, and the stalk ${\mathfrak{B}}_{x} \doteqdot \overset{-1}{\eta}(x)$ is a topological subspace of ${\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})$. Here ${\mathfrak{E}}_{x} \doteqdot \overset{-1}{\pi}(x)$, while ${\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})$, is the space, of all linear bounded maps on ${\mathfrak{E}}_{x}$ with the topology of uniform convergence over the subsets of $S_{x}\subset Bounded({\mathfrak{E}}_{x})$ which depends, for all $x\in X$, on the same subspace ${\mathcal{E}}\subseteq\Gamma(\pi)$. Finally $\rho:{\mathfrak{M}}\to X$, $\eta:{\mathfrak{B}}\to X$, and $\pi:{\mathfrak{E}}\to X$ are the projection maps of the respecive bundles and $\Gamma^{x_{\infty}}(\rho)$ is the class of all bounded selections, i.e. maps belongings to the set $\prod_{x\in X}{\mathfrak{M}}_{x}$ continuous at $x_{\infty}$ with respect to the topology on the bundle space ${\mathfrak{M}}$, similarly for $\Gamma^{x_{\infty}}(\eta)$. **A fundamental remark is that the continuity at $x_{\infty}$ of ${\mathcal{U}}$ and ${\mathcal{P}}$ derives by a sort of continuity at the same point of the selection ${\mathcal{T}}$ of the graphs of the infinitesimal generators of the semigroups ${\mathcal{U}}$**, where this sort of continuity has to be understood in the following sense. For every $x\in X$ let ${\mathcal{T}}(x)$ be the graph of the infinitesimal generator $T_{x}$ of the semigroup ${\mathcal{U}}(x)$, then $$\label{19240703} \begin{cases} {\mathcal{T}}(x_{\infty}) = \left\{ \phi(x_{\infty}) \mid \phi \in \Phi \right\} \\ \Phi \subseteq \Gamma^{x_{\infty}}(\pi_{{\mathbf{E}}^{\oplus}}) \\ (\forall x\in X) (\forall \phi\in\Phi) (\phi(x)\in{\mathcal{T}}(x)), \end{cases}$$ where $\Gamma^{x_{\infty}}(\pi_{{\mathbf{E}}^{\oplus}})$ is the class of all bounded selections of the direct sum of bundles ${\mathfrak{V}}\oplus{\mathfrak{V}}$ which are continuous at $x_{\infty}$. Hence for any $v\in Dom(T_{x_{\infty}})$ there exits a bounded selection $\phi$ of ${\mathfrak{V}}\oplus{\mathfrak{V}}$ such that $$\label{16490703} \begin{cases} (v,T_{x_{\infty}}v) = \lim_{x\to x_{\infty}} (\phi_{1}(x),\phi_{2}(x)) \\ (\phi_{1}(x),\phi_{2}(x)) \in Graph(T_{x}), \forall x\in X-\{x_{\infty}\}, \end{cases}$$ where the limit is with respect to the topology on the bundle space of ${\mathfrak{V}}\oplus{\mathfrak{V}}$ [^1]. $\left(\Theta,{\mathcal{E}}\right)-$structure.\ Relation between the topologies on ${\mathfrak{M}}$ and ${\mathfrak{B}}$ and that on ${\mathfrak{E}}$. {#leftthetamathcaleright-structure.-relation-between-the-topologies-on-mathfrakm-and-mathfrakb-and-that-on-mathfrake. .unnumbered} ------------------------------------------------------------------------------------------------------- **The main general strategy for obtaining the continuity at $x_{\infty}$ of ${\mathcal{U}}$ and ${\mathcal{P}}$, it is to correlate the topologies on the bundles spaces involved, among others those on ${\mathfrak{M}}$ and ${\mathfrak{B}}$, with that on the space ${\mathfrak{E}}$.** Due to this fact it is clear that in this work the construction of the right structures has a prominent role. It is a well-known fact the relative freedom of choice of the topology on the bundle space of any bundle of $\Omega-$spaces. More exactly the possibility of choosing a linear subspace, which is the entire space if $X$ is compact, of the space of all (global) sections of the bundle, i.e. the space of all everywhere defined bounded continuous selections, see [@gie Theorem $5.9$]. This freedom of choice allows the construction of examples of some of the cited correlations of topologies. Without entering in the definition of the topology of a bundle of $\Omega-$space, we can appreciate how much important it is to choose the “right” set of all sections (in symbols $\Gamma(\zeta)$) of a general bundle ${\left\langle {\mathfrak{Q}},\zeta,X{\right\rangle}}$ of $\Omega-$space, by the following simple but fundamental result, Corollary \[28111707\]. Let $f\in\prod_{x\in X}^{b}{\mathfrak{Q}}_{x}$ be any bounded selection and $x_{\infty}\in X$ such that there exists a section $\sigma\in\Gamma(\zeta)$ such that $\sigma(x_{\infty})=f(x_{\infty})$. Then by setting $f\in\Gamma^{x_{\infty}}(\zeta)$ iff $f$ is bounded and continuous at $x_{\infty}$ we have $$\label{21040503} f\in\Gamma^{x_{\infty}}(\zeta) \Leftrightarrow (\forall j\in J) (\lim_{z\to x_{\infty}} \nu_{j}^{z}(f(z)-\sigma(z))=0),$$ where $J$ is a set such that $\{\nu_{j}^{z}\mid j\in J\}$ is a fundamental set of seminorms of the locally convex space ${\mathfrak{Q}}_{z}\doteqdot\overset{-1}{\zeta}(z)$ for all $z\in X$. About the problem of establishing if there are sections intersecting $f$ in $x_{\infty}$, we can use an important result of the theory of Banach bundles, stating that any Banach bundle over a locally compact base space is “full”, i.e. for any point of the bundle space there exists a section passing on it. For more general bundle of $\Omega-$space we can use the freedom before mentioned. *The criterium I used for determining the correlations between ${\mathfrak{M}}$ (resp.${\mathfrak{B}}$) and ${\mathfrak{E}}$ is that of extending to a general bundle two properties of the topology of the space ${\mathcal{C}_{c} \left(Y,{\mathcal{L}}_{s}(Z)\right)}$*. Here $Z$ is a normed space, $S$ is a class of bounded subsets of $Z$, ${\mathcal{L}}_{s}(Z)$ is the space of all linear continuous maps on $Z$ with the pointwise topology, finally ${\mathcal{C}_{c} \left(Y,{\mathcal{L}}_{s}(Z)\right)}$ is the space of all continuous maps on $Y$ with values in ${\mathcal{L}}_{s}(Z)$ with the topology of uniform convergence over the compact subsets of $Y$. In order to simplify the notations we here shall consider $Z$ as a Banach space thus ${\mathcal{L}}_{S}(Z)=B_{s}(Z)$, i.e. the space of all bounded linear operators on $Z$ with the strong operator topology. Let $X$ be a compact space, $Y$ a topological space $$\begin{aligned} {\mathcal{M}} \doteqdot & \{ F\in{\mathcal{C}_{b} \left(X,{\mathcal{C}_{c} \left(Y,B_{s}(Z)\right)}\right)} \mid (\forall K\in Comp(Y)) \\ & (C(F,K)\doteqdot \sup_{(x,s)\in X\times K} \|F(x)(s)\|_{B(Z)}<\infty) \} \\ {\mathbf{M}}_{x} & \doteqdot{\overline}{\{ F(x)\mid F\in{\mathcal{M}} \}} \end{aligned}$$ Let denote by ${\mathfrak{V}}\doteqdot{\left\langle {\mathfrak{E}},\pi,X{\right\rangle}}$ the trivial bundle with constant stalk $Z$ so $\Gamma(\pi) \simeq{\mathcal{C}_{b} \left(X,Z\right)}$, set $$\label{18252606} \begin{cases} {\mathcal{A}}_{x} \doteqdot\{\mu_{(v,x)}^{K} \mid K\in Comp(Y), v\in\Gamma(\pi) \}, \\ \mu_{(v,x)}^{K}: {\mathbf{M}}_{x}\ni G\mapsto \sup_{s\in K} \|G(s)v(x)\|, \\ {\mathbf{M}} \doteqdot\{ {\left\langle {\mathbf{M}}_{x},{\mathcal{A}}_{x}{\right\rangle}} \}_{x\in X}. \end{cases}$$ Then by using Lemma \[22312406\] and the cited [@gie Theorem $5.9$] we can construct a bundle of $\Omega-$space say ${\mathfrak{V}}({\mathbf{M}},{\mathcal{M}})$ whose stalk at $x$ is the locally convex space ${\left\langle {\mathbf{M}}_{x},{\mathcal{A}}_{x}{\right\rangle}}$ and whose space of bounded continuous sections $\Gamma(\pi_{{\mathbf{M}}})$ is such that $\Gamma(\pi_{{\mathbf{M}}})\simeq{\mathcal{M}}$. Let $f\in\prod_{x\in\ X}{\mathbf{M}}_{x}$ be such that $(\forall K\in Comp(Y)) (\sup_{(x,s)\in X\times K} \|f(x)(s)\|_{B(Z)}<\infty)$ then (see Thm. \[22372406\]) $(1) \Leftrightarrow (2) \Leftrightarrow (3)$ with 1. $(\forall K\in Comp(Y)) (\forall v\in\Gamma(\pi))$ $$(\lim_{x\to x_{\infty}} \sup_{s\in K} \| f(x)(s)v(x) - f(x_{\infty})(s)v(x) \| =0);$$ 2. $f\in\Gamma^{x_{\infty}}(\pi_{{\mathbf{M}}})$; 3. $f:X\to{\mathcal{C}_{c} \left(Y,B_{s}(Z)\right)}$ continuous at $x_{\infty}$. Moreover (see Thm. \[22372406\]) if $Y$ is locally compact for all $t\in Y$ $$\label{17150703} \Gamma(\pi_{{\mathbf{M}}})_{t} \bullet \Gamma(\pi) \subseteq \Gamma(\pi).$$ *Therefore we constructed two bundles ${\mathfrak{V}}\doteqdot{\left\langle {\mathfrak{E}},\pi,X{\right\rangle}}$ and ${\mathfrak{V}}({\mathbf{M}},{\mathcal{M}})$ whose topologies are $(I)$ stalkwise related by $\{{\mathcal{A}}_{x}\}_{x\in X}$ in and for which hold $(1)\Leftrightarrow(2)$ and $(II)$ globally related by . Finally $\Gamma^{x_{\infty}}(\pi_{{\mathbf{M}}})$ coincide with the subset of all maps $f:X\to{\mathcal{C}_{c} \left(Y,B_{s}(Z)\right)}$ continuous at $x_{\infty}$ such that $(\forall K\in Comp(Y)) (\sup_{(x,s)\in X\times K} \|f(x)(s)\|_{B(Z)}<\infty)$.* The natural generalization of the mentioned property $(I)$ leads to the concept of **$\left(\Theta,{\mathcal{E}}\right)-$structure**, see Definition \[10282712\] and Lemma \[15482712\], while the generalization of the property $(II)$ leads to that of **compatible** $\left(\Theta,{\mathcal{E}}\right)-$structure, see Definition \[10282712\]. A similar and more important global correlation between ${\mathfrak{M}}$ and ${\mathfrak{E}}$, this time for the case in which the topology on each stalk ${\mathfrak{M}}_{x}$ is that of the pointwise convergence instead of the compact convergence, is that encoded in in the definition of “invariant” $\left(\Theta,{\mathcal{E}},\mu\right)-$structures, see Definition \[17161902\], for a similar definition see Definition \[10282712\]. This conclude the discussion about the relationship between the topologies on ${\mathfrak{M}}$ and ${\mathfrak{E}}$, in particular between those on ${\mathfrak{B}}$ and ${\mathfrak{E}}$ [^2] Kurtz’s General Approximation Theorem {#kurtzs-general-approximation-theorem .unnumbered} --------------------------------------- Briefly I shall recall what here has to be understood as a classical stability problem in order to better explain how to generalize it through the language of bundles. The classical stability problem could be so described: fixed a Banach space $Z$ find a sequence $\{S_{n}:D_{n}\subseteq Z\to Z\}$ of possibly unbounded linear operators in $Z$ and a sequence $\{P_{n}\}\subset B(Z)$ where $P_{n}$ is a continuous spectral projector of $S_{n}$ for $n\in{\mathbb{N}}$, such that if $(A)$ : there exists an operator $S:D\subset Z\to Z$ such that $S=\lim_{n\to\infty}S_{n}$ with respect to a suitable topology or in any other generalized sense, $(B)$ : such that there exists a spectral projector $P\in B(Z)$ of $S$ such that $P=\lim_{n\to\infty}P_{n}$ with respect to the strong operator topology. Here a spectral projector of an operator $S$ in a Banach space is a continuous projector associated to a closed $S-$invariant subspace $Z_{0}$ such that $ \sigma(S{\upharpoonright}Z_{0}) \subset \sigma(S) $, where $\sigma(T)$ is the spectrum of the operator $T$. In Ch $IV$ [@kato] there are many stability theorems in which the previous limit of operators $S_{n}$ has to be understood with respect to the metric induced by the “gap” between the corresponding closed graphs. Moreover there are stability theorems even for operators defined in different spaces, obtained by using the concept of *Transition Operators* introduced by Victor Burenkov, see for expample [@bl1], [@bl2] and [@bll], or the results obtained by Massimo Lanza de Cristoforis and Pier Domenico Lamberti by using functional analytic approaches, see for examples [@l1], [@l2], [@ll]. If one try to generalize the classical stability problem to the case in which $Z$ is replaced by any sequence $\{Z_{n}\}$ of Banach spaces and $S_{n}$ is defined in $Z_{n}$ for all $n$, then he would face the following difficulty: how to adapt the definition of the gap given by Kato to the case of a sequence of different spaces, more in general in which sense to understand the convergence of operators defined in different spaces. A first step toward the generalization to the case of different spaces of the classical stability problem is the following result Thomas G. Kurtz, [@kurtz]. \[$2.1.$ of [@kurtz]\] \[16250603\] For each $n$, let $U_{n}(t)$ be a strongly continuous contraction semigroup defined on $L_{n}$ with the infinitesimal operator $A_{n}$. Let $A=ex-\lim_{n\to\infty}A_{n}$. Then there exists a strongly continuous semigroup $U(t)$ on $L$ such that $\lim_{n\to\infty}U_{n}(t)Q_{n}f=U(t)f$ for all $f\in L$ and $t\in{\mathbb {R}}^{+}$ if and only if the domain $D(A)$ is dense and the range $R(\lambda_{0}-A)$ of $\lambda_{0}-A$ is dense in $L$ for some $\lambda_{0}>0$. If the above conditions hold $A$ is the infinitesimal generator of $U$ and we have $$\label{19100603} \lim_{n\to\infty}\sup_{0\leq s\leq t} \| U_{n}(s)Q_{n}f- Q_{n}U(s)f \|_{n}=0,$$ for every $f\in L$ and $t\in{\mathbb {R}}^{+}$. Here ${\left\langle L,\|\cdot\|{\right\rangle}}$ is a Banach space, $\{{\left\langle L_{n},\|\cdot\|_{n}{\right\rangle}}\}_{n\in{\mathbb{N}}}$ is a sequence of Banach spaces, $\{Q_{n}\in B(L,L_{n})\}_{n\in{\mathbb{N}}}$ such that $\lim_{n\to\infty}\|Q_{n}f\|_{n}=\|f\|$ for all $f\in L$. Let $f\in L$ and $\{f_{n}\}_{n\in{\mathbb{N}}}$ such that $f_{n}\in L_{n}$ for every $n\in{\mathbb{N}}$, thus he set [^3] $$\label{16590603} f=\lim_{n\to\infty}f_{n} \Leftrightarrow \lim_{n\to\infty} \|f_{n}-Q_{n}f\|_{n}=0.$$ Moreover if $A_{n}:Dom(A_{n})\subseteq L_{n}\to L_{n}$ he defined $$\label{16500603} \begin{cases} Graph(ex-lim_{n\to\infty}A_{n}) \doteqdot \{ \lim_{n\in{\mathbb{N}}} s_{0}(n) \mid s_{0}\in\Phi_{0} \} \\ \begin{aligned} & \Phi_{0} \doteqdot \{ (f_{n},A_{n}f_{n})_{n\in{\mathbb{N}}} \in (Z\times Z)^{{\mathbb{N}}} \mid \\ & (\forall n\in{\mathbb{N}}) (f_{n}\in Dom(A_{n})) \wedge (\exists\, \lim_{n\in{\mathbb{N}}}( f_{n},A_{n}f_{n})) \}, \end{aligned} \end{cases} \tag{Gr}$$ where $(f,g)=\lim_{n\in{\mathbb{N}}} (f_{n},A_{n}f_{n})) $ iff $f=\lim_{n\in{\mathbb{N}}}f_{n}$ and $g=\lim_{n\in{\mathbb{N}}}A_{n}f_{n}$ and all these limits are those defined in . Whenever $Graph(ex-lim_{n\to\infty}A_{n})$ is a graph in $L$ he denoted by $ex-lim_{n\to\infty}A_{n}$ the corresponding operator in $L$. The Kurtz’s approach, just now described, did not make use of the bundle theory, and, except when imposing stronger assumptions, it cannot be implemented in a bundle of $\Omega-$spaces’ approach. The following consideration results fundamental for understanding the strategy behind this work. There is a strong resemblance of with . More importantly **if the topology on ${\mathfrak{M}}$ and that on ${\mathfrak{E}}$ are related by a $\left(\Theta,{\mathcal{E}}\right)-$structure (for a simple example see ) then there exists a “resemblance” of the selection convergence with the convergence of the sequence of semigroups $\{U_{n}\}_{n\in{\mathbb{N}}}$ to the semigroup $U$**. [^4] I used the word resemble due to the difficulty to build a couple of reasonable Kurtz’ bundles, i.e. two bundles of $\Omega-$spaces ${\left\langle {\mathfrak{E}},\pi,X{\right\rangle}}$ and ${\left\langle {\mathfrak{M}},\rho,X{\right\rangle}}$ such that $X$ is the Alexandroff compactification of ${\mathbb{N}}$ and , hold. In any case it is possible under strong assumptions, see Setion \[17572301\]. Although the difficulty of constructing a couple of Kurtz’s bundles, *the recognition of the before mentioned two resemblances, were sufficient to push me in investigating the possibility of extending the Kurtz’s Theorem \[16250603\] in the general framework of bundles of $\Omega-$spaces, by using $\left(\Theta,{\mathcal{E}}\right)-$structure.* Direct sum of Bundles of $\Omega-$spaces and (Pre)Graph Sections {#direct-sum-of-bundles-of-omega-spaces-and-pregraph-sections .unnumbered} ------------------------------------------------------------------ It should now be clear that in the way of extending the Kurtz’s Theorem I am replacing the sequence of Banach spaces $\{L_{n}\}_{n\in{\mathbb{N}}\cup\{\infty\}}$ and $\{{\mathcal{C}_{c} \left({\mathbb {R}}^{+},B_{s}(L_{n)}\right)} \}_{n\in{\mathbb{N}}\cup\{\infty\}}$, where $L_{\infty}\doteqdot L$, with a Banach bundle ${\mathfrak{E}}$ and the bundle of $\Omega-$spaces ${\mathfrak{M}}$ respectively, while the Kurtz’ convergences and with convergences of selections on the bundles spaces ${\mathfrak{M}}$ and ${\mathfrak{E}}$ respectively. In this view definition has to be replaced by that of Pre-Graph sections, see Definition \[16161212bis\] (essentially ), while the case in which $Graph(ex-lim_{n\to\infty}A_{n})$ is a graph in $L$ with that of Graph sections, see Definition \[12432110bis\]. Hence it arises as a natural question which topology has to be selected for the bundle space of ${\mathfrak{V}}\oplus{\mathfrak{V}}$. An essential tool used in the definition of $Graph(ex-lim_{n\to\infty}A_{n})$ in is that of convergence of a sequence $(f_{n},A_{n}f_{n})$ in the direct sum of the spaces $L_{n}\oplus L_{n}$, given by construction as the convergence of both the sequences in $L_{n}$ in the meaning of . It is exactly this factorization property the property which I want to preserve when selecting the “right” topology on the bundle space of ${\mathfrak{V}}\oplus{\mathfrak{V}}$. It is a well-known result the solution of this problem in the special case of Banach bundles. I generalize this result for a finite direct sum of bundles of $\Omega-$spaces, by constructing in Theorem \[16322110\] a family of seminorms on the direct sum of Hausdorff locally convex spaces which is fundamental for any of the following equivalent topologies: the direct sum top. the $lc$-direct sum top., the box top. and most importantly the product topology. The result that the fundamental set is direct along with Lemma \[19420111\] allow to define the direct sum of bundles of $\Omega-$spaces as given in Definition \[12422110\]. Finally the fundamental result that the topology on each stalk is the product topology, (fact encripted in ) the choice given in of the subset of section of the direct sum of bundles and the general convergence criterium in , allow to show the claimed factorization property in Corollary \[17571212\]. I.e. any selection of the direct sum $\bigoplus_{i=1}^{n}{\mathfrak{E}}_{i}$ of bundles is continuous at a point if and only if is continuous at the same point its projection selection of ${\mathfrak{E}}_{i}$ for every $i=1,...,n$. Semigroup Approximation Theorem. {#semigroup-approximation-theorem. .unnumbered} ---------------------------------- Roughly speaking ${\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}\in \Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}$ iff ${\mathcal{T}}(x)$ is the graph of the infinitesimal generator $T_{x}$ of a $C_{0}-$semigroup ${\mathcal{U}}(x)$ on ${\mathfrak{E}}_{x}$, for all $x\in X$, holds and $${\mathcal{U}}\in\Gamma^{x_{\infty}}(\rho).$$ Thus, according the discussed way of extending the Kurtz’ theorem which I intend to perform in this work, to find an element in the class $\Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}$ means to find an extension of Theorem \[16250603\]. In the first main result of this work, **Theorem \[17301812b\]**, has been constructed an element of the class $\Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}$. Laplace Duality Properties {#laplace-duality-properties .unnumbered} ---------------------------- There are essentially two strong hypothesis to be satisfied in Theorem \[17301812b\]. In constructing a model for hypothesis $[ii]$ one obtains **Corollary \[21343012\]**. In any case the most important one is hypothesis $[i]$, i.e. the assumption that the $\left(\Theta,{\mathcal{E}}\right)-$structure ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ has the Laplace duality property, see Definition \[16401812b\]. Roughly speaking the full Laplace duality property means that the natural action of $\prod_{x\in X} {\mathcal{L}}({\mathfrak{E}}_{x}) $ over $\prod_{x\in X}{\mathfrak{E}}_{x}$, induces, by restriction, an action over $\Gamma(\pi)$ of the Laplace trasform of $\Gamma(\rho)$. More exactly $$\label{Laplace} \tag{${\mathbf{LD}}$} \begin{cases} (\forall\lambda>0) \left({\mathfrak{L}}(\Gamma(\rho)) (\cdot)(\lambda) \bullet \Gamma(\pi) \subseteq \Gamma(\pi)\right) \\ {\mathfrak{L}}(F)(x)(\lambda) \doteqdot \int_{0}^{\infty} e^{-\lambda s} F(x)(s)\,ds \doteq \int_{{\mathbb {R}}^{+}} F(x)(s)\,d\mu_{\lambda}(s). \end{cases}$$ See Def. \[15531102\], Def. \[15062301\] and Def. \[16401812b\]. The implicit assumption is that for all $x\in X$ and $\lambda>0$ $${\mathfrak{M}}_{x} \subseteq {\mathfrak{L}}_{1}({\mathbb {R}}^{+}, {\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x}),\mu_{\lambda}),$$ where $\mu_{\lambda}$ is the Laplace measure associated to $\lambda$ and ${\mathfrak{L}}_{1}({\mathbb {R}}^{+}, {\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x}),\mu_{\lambda})$ is the space of all $\mu_{\lambda}-$integrable maps with values in the locally convex space ${\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})$. Among others informations in Prop. \[18390901\] (an application of a result in [@SilInt]) there are reasonable conditions ensuring the previous inclusion. I investigated in Section \[13320803\] a strategy for constructing classes having the full Laplace duality property. Although I worked in this section in a wide generality, here I shall present the applications of the results in the case of interest for the present introduction. Firstly we note that by construction $$\Gamma(\pi) \subset \prod_{x\in X}{\mathfrak{E}}_{x},$$ hence the most natural (duality) action to consider over $\Gamma(\pi)$ is the restriction on it of the “standard” [^5] action of $${\mathcal{L}}\left(\prod_{x\in X}{\mathfrak{E}}_{x}\right).$$ Secondly we recall that the Laplace duality property is described in terms of the action restricted over $\Gamma(\pi)$ of a subspace of $\prod_{x\in X} {\mathfrak{L}}_{1}({\mathbb {R}}^{+}, {\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x});\mu_{\lambda})$. Therefore the idea is to construct a suitable **locally convex space ${\mathfrak{G}}$** and a linear map $\Psi$ such that $$\label{18250803} \begin{cases} {\mathfrak{G}} \subset {\mathcal{L}}\left(\prod_{x\in X}{\mathfrak{E}}_{x}\right) \, \text{ as linear spaces} \\ \Psi: {\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda}) \to \prod_{x\in X} {\mathfrak{L}}_{1}({\mathbb {R}}^{+}, {\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x});\mu_{\lambda}), \end{cases}$$ and most importantly such that the following relation between the two actions holds for all ${\overline}{F} \in{\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$, $x\in X$, $\lambda>0$ and $v\in\Gamma(\pi)$ $$\label{18100803} {\left\langle \int \Psi({\overline}{F})(x)(s) \,d\mu_{\lambda}(s),v(x){\right\rangle}}_{x} = {\left\langle \int{\overline}{F}(s)\,d\mu_{\lambda}(s),v{\right\rangle}}(x),$$ see **Corollary \[15111901\]**. Here ${\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$ is the space of all $\mu_{\lambda}-$integrable maps on ${\mathbb {R}}^{+}$ and at values in the locally convex space ${\mathfrak{G}}$, while for any linear space $E$ we denote by ${\left\langle \cdot,\cdot{\right\rangle}}: End(E) \times E \to E$ the standard duality. **It is exactly by that we can rewrite as a duality problem**. More exactly if $\exists\,{\mathcal{F}}\subset \bigcap_{\lambda>0} {\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$ such that $\Psi({\mathcal{F}})=\Gamma(\rho)$ then $$\label{17280803a} {\mathbf{LD}} \Leftrightarrow (\forall\lambda>0) ({\left\langle {\mathcal{A}}_{\lambda},\Gamma(\pi){\right\rangle}} \subseteq\Gamma(\pi)),$$ where for all $\lambda>0$ $$\label{17280803b} {\mathcal{A}}_{\lambda} \doteqdot \left\{ \int{\overline}{F}(s)\,d\mu_{\lambda}(s) \mid{\overline}{F}\in{\mathcal{F}} \right\} \subset {\mathcal{L}}\left(\prod_{x\in X}{\mathfrak{E}}_{x}\right).$$ *There are two advantage of decoding the problem of finding the full Laplace duality property into the invariance problem . Firstly is a classical problem of invariance of a subspace of a linear topological space for the standard action of a subspace of the space of all linear continuous operators on it. Secondly in has involved through the definition of ${\mathcal{A}}_{\lambda}$ the space ${\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$, while in has involved the much more complicate space $\prod_{x\in X} \bigcap_{\lambda>0} {\mathfrak{L}}_{1}({\mathbb {R}}^{+}, {\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x});\mu_{\lambda})$ indeed $\Gamma(\rho)$ belongs to it.* The crucial idea behind Definition \[10221801\] of the space ${\mathfrak{G}}$ is the use of the concept of **locally convex final topology**. Indeed the well-known property of this topology allows in Lemma \[11011501\], to ensure that for all $v\in\Gamma(\pi)$ the evaluation map $$\label{18050803} {\mathfrak{G}}\ni A\mapsto A v \in \prod_{x\in X}{\mathfrak{E}}_{x}\, \text{ is continuous.}$$ And is essentially a consequence of . Although we are mainly interested to the equality , there is an important result strictly determined by the locally convex final topology on ${\mathfrak{G}}$. Namely Theorem \[15251401\] ensures that holds the second statement in and that for all ${\overline}{F}\in{\mathfrak{L}}_{1} ({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$ $$\int\Pr_{x}(\Psi({\overline}{F}))(s) \,d\mu_{\lambda}(s) = \Pr_{x} \circ \left[ \int{\overline}{F}(s)\,d\mu_{\lambda}(s) \right] \circ\imath_{x}.$$ The class $\Delta {\left\langle {\mathfrak{V}},{\mathfrak{D}},\Theta,{\mathcal{E}}{\right\rangle}}$.\ Projection Approximation Theorem. {#the-class-deltaleftlangle-mathfrakvmathfrakdthetamathcalerightrangle.-projection-approximation-theorem. .unnumbered} ----------------------------------------------------------------------------------- In this section we shall discuss the main result of this work namely **Theorem \[13020103\]**, ensuring the existence of an element ${\left\langle {\mathcal{T}},\Phi,x_{\infty}{\right\rangle}}$ of the class $\Delta {\left\langle {\mathfrak{V}},{\mathfrak{D}},\Theta,{\mathcal{E}}{\right\rangle}}$ such that $T_{x}$ is the infinitesimal generator of a $C_{0}-$semigroup of contractions on ${\mathfrak{E}}_{x}$ and such that there exists a selection $$\label{17321002} \boxed{ {\mathcal{P}} \in \Gamma^{x_{\infty}}(\eta),}$$ satisfying and such that ${\mathcal{P}}(x)$ is a spectral projector of $T_{x}$ for all $x\in X$, where $T_{x}$ is the operator whose graph is ${\mathcal{T}}(x)$. We shall start with the definition of the class $\Delta {\left\langle {\mathfrak{V}},{\mathfrak{D}},\Theta,{\mathcal{E}}{\right\rangle}}$, given in Definition \[15312011bis\]. Roughly speaking given a $\left(\Theta,{\mathcal{E}}\right)-$structure ${\left\langle {\mathfrak{V}},{\mathfrak{D}},X,\{pt\}{\right\rangle}}$ and denoted ${\mathfrak{D}} \doteqdot {\left\langle {\mathfrak{B}},\eta,X{\right\rangle}}$, we say that ${\left\langle {\mathcal{T}},\Phi,x_{\infty}{\right\rangle}} \in \Delta {\left\langle {\mathfrak{V}},{\mathfrak{D}},\Theta,{\mathcal{E}}{\right\rangle}}$ iff for all $x\in X$ the set ${\mathcal{T}}(x)$ is a graph in ${\mathfrak{E}}_{x}$, holds and there exists a selection ${\mathcal{P}} \in \Gamma^{x_{\infty}}(\eta)$ continuous at $x_{\infty}$ such that ${\mathcal{P}}(x)$ is a projector on ${\mathfrak{E}}_{x}$ and for all $x\in X$ $$\label{20020803} {\mathcal{P}}(x) T_{x} \subseteq T_{x} {\mathcal{P}}(x),$$ where $T_{x}$ is the operator in ${\mathfrak{E}}_{x}$ whose graph is ${\mathcal{T}}(x)$. In others words ${\left\langle {\mathcal{T}},\Phi,x_{\infty}{\right\rangle}} \in \Delta {\left\langle {\mathfrak{V}},{\mathfrak{D}},\Theta,{\mathcal{E}}{\right\rangle}}$ iff ${\mathcal{T}}$ is a selection of graphs in ${\mathfrak{E}}$ continuous at $x_{\infty}$ in the meaning of and such that there exists a selection ${\mathcal{P}}$ of projectors on ${\mathfrak{E}}$ continuous at $x_{\infty}$ such that *${\mathcal{P}}$ commutes with ${\mathcal{T}}$* in the meaning of . Notice that implies for all $x\in X$ that the following $$T_{x} {\mathcal{P}}(x) Dom(T_{x}) \subseteq {\mathcal{P}}(x){\mathfrak{E}}_{x}$$ is satisfied by any spectral projector $P_{x}$ of the opearor $T_{x}$, indeed by definition for them we have $T_{x} P_{x} {\mathfrak{E}}_{x} \subseteq P_{x}{\mathfrak{E}}_{x}$. Viceversa whenever $T_{x}$ is the infinitesimal generator of a $C_{0}-$semigroup ${\mathcal{W}}_{T}(x)$ of contractions on ${\mathfrak{E}}_{x}$, the most important case in this work, it results that is the property satisfied by all the spectral projectors of the form $${\mathcal{P}}(x) \doteqdot - \frac{1}{2\pi i} \int_{\Gamma} R(-T_{x};\zeta)\, d\zeta,$$ where $R(-T_{x};\zeta)$ is the resolvent map of the operator $-T_{x}$ and $\Gamma$ is a suitable closed curve on the complex plane. Hence we can consider the commutation in as the defining property of what we here consider as the “interesting” bundle ${\mathcal{P}}$ of spectral projectors associated to ${\mathcal{T}}$. ### Proof of Theorem \[13020103\]. *Invariant* $\left(\Theta,{\mathcal{E}},\mu\right)-$ structures. {#proof-of-theorem-13020103.-invariant-leftthetamathcalemuright--structures. .unnumbered} Let us describe the principle steps and new structures required on proving Theorem \[13020103\]. The first property involved in showing is that of an invariant $\left(\Theta,{\mathcal{E}}\right)-$structure, see Definition \[10282712\]. The characteristich property of an invariant $\left(\Theta,{\mathcal{E}}\right)-$ structure ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ is the following one $$\label{17481003} \left\{ F\in\prod_{z\in X}^{b} {\mathfrak{M}}_{z} \mid (\forall t\in Y) (F_{t} \bullet {\mathcal{E}}(\Theta) \subseteq \Gamma(\pi)) \right\} = \Gamma(\rho),$$ where ${\mathfrak{W}} \doteqdot {\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$ and $F_{t}(x)\doteqdot F(x)(t)$. The first reason of intoducing the concept of invariant structure is that the global relation implies the following corresponding local one for any $x_{\infty}\in X$, see Lemma \[14452602\] $$\label{17481003loc} \left\{ F\in\prod_{z\in X}{\mathfrak{M}}_{z} \mid F \bullet \Gamma^{x_{\infty}}(\pi) \subseteq \Gamma^{x_{\infty}}(\pi) \right\} \subseteq \Gamma^{x_{\infty}}(\rho).$$ Hence if we show that $$\label{18271003} {\mathcal{P}} \bullet \Gamma^{x_{\infty}}(\pi) \subseteq \Gamma^{x_{\infty}}(\pi)$$ the follows by for the special case $Y=\{pt\}$. The presence of the uniform convergence over compact subsets of $Y$ may drastically restrict the class of invariant $\left(\Theta,{\mathcal{E}}\right)-$ structures. In order to deal with this problem we introduced the concept of $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,{\mathbb {R}}^{+}{\right\rangle}}$, with ${\mathfrak{Q}} \doteqdot {\left\langle {\left\langle {\mathfrak{H}},\gamma{\right\rangle}},\xi,X,{\mathfrak{R}}{\right\rangle}}$, Definition \[17161902\]. It is essentially the same definition, given for $\left(\Theta,{\mathcal{E}}\right)-$ structures but with the following differences $$\label{20081003} \begin{cases} {\mathfrak{H}}_{x} \subseteq {\mathfrak{L}}_{1} \left(Y,{\mathcal{L}}_{S_{x}} ({\mathfrak{E}}_{x}),\mu\right) \\ {\mathfrak{H}}_{x} \text{ with the pointwise topology}. \end{cases}$$ Thus on each stalk ${\mathfrak{H}}_{x}$ the topology is that of pointwise convergence on $Y$ instead of that of compact convergence. Thus we have an *invariant* $\left(\Theta,{\mathcal{E}},\mu\right)-$ structures iff: $$\label{14451103} \left\{ F\in\prod_{z\in X}{\mathfrak{H}}_{z} \mid (\forall t\in Y) (F_{t} \bullet {\mathcal{E}}(\Theta) \subseteq \Gamma(\pi)) \right\} = \Gamma(\xi),$$ The second reason to be interested to the the invariant $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure, resides in the fact that it is a necessary assumption in order to have the following implication, see Corollary \[21152602\] and Remark \[12500303\] $$\label{17401003} {\mathcal{W}}_{T} \in \Gamma^{x_{\infty}}(\xi) \Rightarrow {\mathcal{P}} \bullet \Gamma^{x_{\infty}}(\pi) \subseteq \Gamma^{x_{\infty}}(\pi).$$ But as we know the first main result Theorem \[17301812b\] states that $ {\mathcal{W}}_{T} \in \Gamma^{x_{\infty}}(\rho)$ for a $\left(\Theta,{\mathcal{E}}\right)-$ structure which we know to be not a $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure. Hence we need a way of connecting the two types of structures. This is performed by the definition of the $\left(\Theta,{\mathcal{E}},\mu\right)-$structure ${\left\langle {\mathfrak{V}}, {\mathfrak{V}}({\mathbf{M}}^{\mu},\Gamma(\rho)),X,Y{\right\rangle}}$ underlying ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$, see Def, \[18072802\]. In view of the property which we are looking for, namely , we have to maintain the vicinity of the original and the underlying structure. This is performed by using [@gie Theorem $5.9$] for constructing bundles with a given subspace of continuous sections. In this way the space $\Gamma(\rho)$ of continuous sections of ${\mathfrak{W}}$ will be a subspace of the space $\Gamma (\pi_{{\mathbf{M}}^{\mu}})$ of all continuous sections of the underlying bundle ${\mathfrak{V}}({\mathbf{M}}^{\mu},\Gamma(\rho))$, (the equality if $X$ is compact). Thus we have, see Proposition \[23332802\], $$\label{17501003} \Gamma^{x_{\infty}}(\rho) \subseteq \Gamma^{x_{\infty}} (\pi_{{\mathbf{M}}^{\mu}}).$$ Finally by Theorem \[17301812b\] we know that ${\mathcal{W}}_{T} \in\Gamma^{x_{\infty}}(\rho)$ therefore , and , follows by and . In addition to invariant $\left(\Theta,{\mathcal{E}},\mu\right)-$ structures, for obtaining we need another concept, namely that of a $\mu-$related couple ${\left\langle {\mathfrak{V}},{\mathfrak{Z}}{\right\rangle}}$, Definition \[15492502\] and a bundle type generalization of the Lebesgue Theorem, see Theorem \[15101701\]. Summary of the main results ans structures {#summary-of-the-main-results-ans-structures .unnumbered} ============================================ The main results of this work are the following ones 1. An explicit construction of a direct fundamental set of seminorms of the topological direct sum of a finite family of Hausdorff locally convex spaces, (Theorem \[16322110\]); 2. Characterization of selections of ${\mathfrak{W}}$ continuous at a point when ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ is a $\left(\Theta,{\mathcal{E}}\right)-$structure, (Lemma \[15482712\]); 3. Construction of a $\left(\Theta,{\mathcal{E}}\right)-$structure ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ and characterization of a subclass of $\Gamma^{x_{\infty}}(\rho)$ when ${\mathfrak{V}}$ is trivial, (Theorem \[22372406\]) 4. Construction of an element in the class $\Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}$, (**Theorem \[17301812b\]**, Corollary \[21343012\]); 5. Conditions in order to satisfy the bounded equicontinuity of which in hypothesis $(ii)$ of Theorem \[17301812b\] (Corollary \[21343012\]); 6. Conditions in order to have (Proposition \[18390901\]); 7. The technical Lemma \[11011501\] and Theorem \[15251401\]; 8. **Theorem \[15332203\]** and Corollaries \[15111901\] and \[18491004\]; 9. ${\mathcal{K}}-$Uniform Convergence **Theorem \[10581004\]**; 10. Consequence of being an ${\left\langle \nu,\eta,E,Z,T{\right\rangle}}$ invariant set $V$ with respect to ${\mathcal{F}}$ (Proposition \[18310302\]); 11. Construction of a class $\Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{D}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}$ by using an ${\left\langle \nu,\eta,{\mathfrak{G}},K(\Gamma),{\mathbb {R}}^{+}{\right\rangle}}$ invariant set $V$ with respect to $\{{\overline}{F}_{T}\}$ (Corollary \[13511102\]); 12. A bundle version of the Lebesgue theorem for a $\mu-$related couple ${\left\langle {\mathfrak{V}},{\mathfrak{Z}}{\right\rangle}}$ (Theroem \[15101701\]); 13. Technical Lemmas \[12151902\] and \[14452602\]; 14. Corollary \[15262502\] 15. Construction of a selection of spectral projectors continuous at a point given a selection of semigroups continuous at the same point (Corollary \[21152602\] and Remark \[12500303\]); 16. The Main result of this work is the construction of an element in the class $\Delta {\left\langle {\mathfrak{V}},{\mathfrak{D}},\Theta,{\mathcal{E}}{\right\rangle}}$ (**Theorem \[13020103\]**). The main structures defined in this works are the following ones 1. Direct sum of bundles of $\Omega-$spaces (Definition \[12422110\]); 2. (Invariant) $\left(\Theta,{\mathcal{E}}\right)-$structure ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$, (**Definition \[10282712\]**); 3. Graph section ${\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}$, (**Definition \[12432110bis\]**); 4. Class $\Delta{\left\langle {\mathfrak{V}},{\mathfrak{D}},\Theta,{\mathcal{E}}{\right\rangle}}$, (**Definition \[15312011bis\]**); 5. Class $\Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}$, (**Definition \[19490412bis\]**); 6. Class $\Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{D}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}$; 7. ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ with the Laplace duality property, (Definition \[16401812b\]); 8. ${\mathbf{U}}-$Spaces (**Definition \[14302503\]**); 9. The locally convex space ${\mathfrak{G}}$ (**Definition \[10221801\]**); 10. ${\left\langle \nu,\eta,E,Z,T{\right\rangle}}$ invariant set $V$ with respect to ${\mathcal{F}}$ (Definition \[20030202\]); 11. $\mu-$related couple ${\left\langle {\mathfrak{V}},{\mathfrak{Z}}{\right\rangle}}$ (Definition \[15492502\]); 12. (Invariant) $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,Y{\right\rangle}}$ (Definition \[17161902\]); 13. $\left(\Theta,{\mathcal{E}}\right)-$ structure ${\left\langle {\mathfrak{V}},{\mathfrak{V}}({\mathbf{M}},\Gamma(\xi)),X,Y{\right\rangle}}$ underlying a $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,Y{\right\rangle}}$ (**Definition \[18072802\]**). Direct Sum ========== Notations {#notat} --------- Let $E$ be a topological vector space, ${\left\langle {\mathcal{L}}(E),\tau{\right\rangle}}$ the linear space of all continuous linear maps on $E$ with the topology $\tau$ compatible with the linear structure. Thus ${\mathbf{U}}({\left\langle {\mathcal{L}}(E),\tau{\right\rangle}})$ is the class of all continuous semigroup morphisms defined on ${\mathbb {R}}^{+}$ and with values in ${\mathcal{L}}_{\tau}(E)$, moreover if $\|\cdot\|$ is any seminorm on ${\mathcal{L}}(E)$ (not necessarly continuous with respect to $\tau$) we set ${\mathbf{U}}_{\|\cdot\|} ({\left\langle {\mathcal{L}}(E),\tau{\right\rangle}})$ as the subset of all $U\in{\mathbf{U}}({\left\langle {\mathcal{L}}(E),\tau{\right\rangle}})$ such that $\|U(s)\|\leq 1$, for all $s\in{\mathbb {R}}^{+}$. Finally we set ${\mathbf{U}}_{is} ({\left\langle {\mathcal{L}}(E),\tau{\right\rangle}})$ as the subset of all ${\mathbf{U}}({\left\langle {\mathcal{L}}(E),\tau{\right\rangle}})$ such that there exists a fundamental set $\Gamma$ of seminorms on $E$ such that $U(s)$ is an isometry with respect to any element in $\Gamma$, for all $s\in{\mathbb {R}}^{+}$. We use throughout this work the notations of [@gie] and often when referring to Banach bundles those of [@fell]. In particular $ {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},p,X,{\mathfrak{N}}{\right\rangle}} $ or simply $ {\left\langle {\mathfrak{E}},p,X{\right\rangle}} $, whenever $\tau$ and ${\mathfrak{N}}$ are known, is a bundle of $\Omega-$spaces ($1.5.$ of [@gie]), where we denote by $\tau$ the topology on ${\mathfrak{E}}$ while with $ {\mathfrak{N}}\doteqdot \{ \nu_{j}\mid j\in J \} $ the directed set of seminorms on ${\mathfrak{E}}$ ($1.3.$ of [@gie]). Thus we set $ {\mathfrak{N}}_{x} \doteqdot \{ \nu_{j}^{x} \mid j\in J \} $ with $ \nu_{j}^{x} \doteqdot \nu_{j}{\upharpoonright}{\mathfrak{E}}_{x} $ for all $x\in X$ and $j\in J$. Moreover for any $U\subseteq X$ we shall call the *Space of Sections of ${\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},p,X,{\mathfrak{N}}{\right\rangle}}$ on $U$* the linear space $\Gamma_{U}(p)$ of all continuous bounded selections of $p$ defined on $U$, namely [^6] $$\Gamma_{U}(p) \doteqdot {\mathcal{C}_{} \left(U,{\mathfrak{E}}\right)} \bigcap \prod_{x\in U}^{b} {\left\langle {\mathfrak{E}}_{x},{\mathfrak{N}}_{x}{\right\rangle}}$$ where $ {\mathfrak{E}}_{x} \doteqdot \overset{-1}{p}(x) $, $$\prod_{x\in U}^{b} {\left\langle {\mathfrak{E}}_{x},{\mathfrak{N}}_{x}{\right\rangle}} \doteqdot \left\{ \sigma\in \prod_{x\in U} {\mathfrak{E}}_{x} \mid \sup_{x\in U} \nu_{j}^{x} (\sigma(x)) <\infty \right\},$$ where $ {\mathcal{C}_{} \left(U,{\mathfrak{E}}\right)} $ is the linear space of all continuous maps $f:U\to{\mathfrak{E}}$. Let $U\subseteq X$ and $x\in U$ set $$\Gamma_{U}^{x}(p) \doteqdot \left\{ f\in \prod_{x\in U}^{b} {\left\langle {\mathfrak{E}}_{x},{\mathfrak{N}}_{x}{\right\rangle}} \mid f\text{ is continuous at $x$} \right\}.$$ So $\Gamma_{U}(p) = \bigcap_{x\in U} \Gamma_{U}^{x}(p)$. We set $\Gamma(p) \doteqdot \Gamma_{X}(p)$ and $\Gamma^{x}(p) \doteqdot \Gamma_{X}^{x}(p)$ for any $x\in X$. The definition of trivial bundle of $\Omega-$spaces is given in $1.8.$ of [@gie]. All vector spaces are assumed to be over ${\mathbb{K}}\in\{{\mathbb {R}},{\mathbb {C}}\}$, Hlcs is for Hausdorff locally convex spaces. We say that $ {\mathbf{V}} \doteqdot \{{\left\langle V_{x},{\mathcal{A}}_{x}{\right\rangle}}\}_{x\in X} $ is a *nice* family of Hlcs if $\{V_{x}\}_{x\in X}$ is a family of Hlcs such that $\exists\,J$ for which $\forall x\in X$ the set $ {\mathcal{A}}_{x} \doteqdot \{ \mu_{j}^{x} \}_{j\in J} $ is a directed [^7] family of seminorms on $V_{x}$ generating the lct on it. For any family of seminorms $\Gamma$ on a vector space $V$ we shall define the *directed family of seminorms associated to $\Gamma$* the following set $ \{ \sup F \mid F\in{\mathcal{P}}_{\omega}(\Gamma) \} $ with the standard order relation of pointwise order on $ {\mathbb {R}}^{V} $. Given two lcs $E$ and $F$ we denote by $ {\mathcal{L}}(E,F) $ the linear space of all linear and continuous maps on $E$ with values in $F$, and set $ {\mathcal{L}}(E) \doteqdot {\mathcal{L}}(E,E) $, moreover by $ \Pr(E) \doteqdot \{ P\in{\mathcal{L}}(E) \mid P\circ P =P \} $ we denote the class of all projectors on $E$. Let $S$ be a class of bounded subsets of a lcs $E$, thus $ {\mathcal{L}}_{S}(E) $ denotes the lcs whose underlining linear space is ${\mathcal{L}}(E)$ and whose lct is that of uniform convergence over the subsets in $S$. When $E$ is a normed space and $S$ is the class of all finite parts of $E$, then ${\mathcal{L}}_{S}(E)$ will be denoted by $B_{s}(E)$, while $B(E)$ denotes ${\mathcal{L}}(E)$ with the usual norm topology. Let $X,Y$ be two topological spaces then ${\mathcal{C}_{} \left(X,Y\right)}$ is the set of all continuous maps on $X$ valued in $Y$, while ${\mathcal{C}_{c} \left(X,Y\right)}$ is the topological space of all continuous maps on $X$ valued in $Y$ with the topology of uniform convergence over the compact subsets of $Y$. If $Y$ is a uniform space then ${\mathcal{C}}^{b}(X,Y)$ is the space of all bounded maps in ${\mathcal{C}_{} \left(X,Y\right)}$, while ${\mathcal{C}}_{c}^{b}(X,Y) = {\mathcal{C}_{c} \left(X,Y\right)}\cap {\mathcal{C}}^{b}(X,Y)$. If $E$ is a lcs then ${\mathcal{C}_{c} \left(X,E\right)}$ is a lcs, while if $E$ is a Hlcs and $Comp(X)$ is a covering of $X$, for example if $X$ is a locally compact space, then ${\mathcal{C}_{c} \left(X,E\right)}$ is a Hlcs. Let $Y$ be a locally compact space, $\mu\in Radon(Y)$ and $E\in Hlcs$, then ${\mathfrak{L}}_{1}(Y,E,\mu)$ denotes the linear space of all scalarly essentially $\mu-$integrable maps $f:Y\to E$ such that its integral belongs to $E$, see [@IntBourb Ch. $6$], while $Meas(Y,{\mathfrak{E}}_{x},\mu)$ denotes the linear space of all $\mu-$measurable maps $f:Y\to E$. If $S$ is any set then ${\mathcal{P}}_{\omega}(S)$ denotes the class of all finite subsets of $S$. Finally “u.s.c.” is for upper semicontinuous. Finally we shall give the following \[15012602\] Let $ {\mathfrak{A}} \doteqdot {\left\langle {\left\langle {\mathfrak{B}},\tau{\right\rangle}},\xi,X,{\mathfrak{N}}{\right\rangle}} $ be a bundle of $\Omega-$spaces, $x\in X$ and $Q,S$ subsets of $\prod_{z\in X}{\mathfrak{B}}_{z}$. Thus we set $$Q_{S}^{x} \doteqdot \{ H\in Q\mid (\exists\,F\in S) (H(x)=F(x)) \},$$ and $Q_{\diamond}^{x} \doteqdot Q_{\Gamma(\xi)}^{x}$, while $\Gamma_{S}^{x}(\xi) \doteqdot (\Gamma^{x}(\xi))_{S}^{x}$ and $\Gamma_{\diamond}^{x}(\xi) \doteqdot (\Gamma^{x}(\xi))_{\diamond}^{x}$. Direct Sum of Bundles $\Omega-$spaces --------------------------------------- ### Standard construction of Bundles of $\Omega-$spaces \[ $FM(3)-FM(4)$ ([@gie] $\S 5$) and $FM(3^{*})-FM(4)$ \] \[17471910s\] Let $ {\mathbf{V}} \doteqdot \{{\left\langle V_{x},{\mathcal{A}}_{x}{\right\rangle}}\}_{x\in X} $ be a nice family of Hlcs with $ {\mathcal{A}}_{x} \doteqdot \{\mu_{j}^{x} \}_{j\in J} $ for all $x\in X$; we say that ${\mathcal{G}}$ satisfies $FM(3)-FM(4)$ with respect to ${\mathbf{V}}$ (see [@gie] $\S 5$) if $ {\mathcal{G}} \subseteq \prod_{x\in X}^{\infty} {\left\langle V_{x},{\mathcal{A}}_{x}{\right\rangle}} \doteqdot \left\{ f\in \prod_{x\in X} V_{x} \mid (\forall j\in J) (\sup_{x\in X} \mu_{j}^{x}(f(x)) <\infty) \right\} $ and $FM(3)$ : $ \{f(x)\mid f\in {\mathcal{G}}\} $ is dense in $V_{x}$ for all $x\in X$; $FM(4)$ : $ X\ni x \mapsto \mu_{j}^{x}(f(x)) $ is u.s.c. $\forall j\in J$ and $\forall f\in{\mathcal{G}}$. Now we shall introduce a stronger condition namely we say that $ {\mathcal{G}} $ satisfies $FM(3^{*})-FM(4)$ with respect to ${\mathbf{V}}$ if $FM(3^{*})$ and $FM(4)$ hold where $$\label{15012310} (\forall x\in X) (\{f(x)\mid f\in {\mathcal{G}}\} = V_{x}). \tag*{$FM(3^{*})$}$$ \[21132406\] Let $ {\mathbf{V}}_{k} \doteqdot \{{\left\langle V_{x},{\mathcal{A}}_{x}^{k}{\right\rangle}}\}_{x\in X}$, with $k=1,2$, be two nice families of Hlcs such that ${\mathcal{A}}_{x}^{1}$ and ${\mathcal{A}}_{x}^{2}$ generate the **same** locally convex topology on $V_{x}$. Notice that ${\mathcal{G}}$ satisfies $FM(3)-FM(4)$ with respect to ${\mathbf{V}}_{1}$ **doesn’t imply** that ${\mathcal{G}}$ satisfies $FM(3)-FM(4)$ with respect to ${\mathbf{V}}_{2}$. Say ${\mathcal{A}}^{k}=\{\mu^{j_{k}}\mid j_{k}\in J_{k}\}$, then a necessary condition for which the fact described happens is the following one $(\exists\, X_{0}\subseteq X) (\exists\, k\in\{1,2\}) (\exists j_{s}:X_{0}\times J_{k} \to J_{s}) (\forall x\in X_{0}) (\forall j_{k}\in J_{k}) (\exists\,C>0) (\forall v\in V_{x})$ $$\mu_{j_{k}}^{x}(v)\leq C \mu_{j_{s}(x,j_{k})}^{x}(v).$$ In other words the index for some inequalities relating the seminorms on ${\mathcal{A}}^{k}$ with those on ${\mathcal{A}}^{s}$, depends on $x$. \[18111610\] Let $ {\mathbf{V}}' \doteqdot \{{\left\langle V_{x},{\mathcal{A}}_{x}'{\right\rangle}}\}_{x\in X} $ be a family of Hlcs where $ {\mathcal{A}}_{x}' \doteqdot \{\mu_{j_{x}}^{x} \}_{j_{x}\in J_{x}} $ is a directed family of seminorms on $V_{x}$ generating the lct on it, for all $x\in X$. Then we set $$\begin{cases} J \doteqdot \bigcup_{x\in X}\{x\}\times J_{x}, \\ \mu_{(y,j_{y})}^{x} \doteqdot \begin{cases} \mu_{j_{x}}^{x},\, x=y \\ {\mathbf{0}},\, x\neq y \end{cases} \forall x,y\in X, \forall j_{y}\in J_{y}, \\ {\mathcal{A}}_{x} \doteqdot \{\mu_{j}^{x} \}_{j\in J},\, \forall x\in X. \end{cases}$$ Moreover by setting $$\prod_{x\in X}^{\infty} {\left\langle V_{x},{\mathcal{A}}_{x}'{\right\rangle}} \doteqdot \left\{ f\in \prod_{x\in X} V_{x} \mid \left( \forall {\overline}{j}\in\prod_{x\in X}J_{x} \right) \left(\sup_{x\in X} \mu_{{\overline}{j}(x)}^{x}(f(x)) <\infty\right) \right\},$$ we say that $ {\mathcal{G}} \subseteq \prod_{x\in X}^{\infty} {\left\langle V_{x},{\mathcal{A}}_{x}'{\right\rangle}} $ satisfies $FM(3)-FM(4')$ with respect to ${\mathbf{V}}'$ if $FM(3)$ and $FM(4')$ hold where $$\label{17311811} \left(\forall{\overline}{j}\in J\right) \left(\forall f\in{\mathcal{G}}\right) \left(X\ni x \mapsto \mu_{{\overline}{j}(x)}^{x}(f(x)) \text{is u.s.c.}\right). \tag*{$FM(4')$}$$ \[18111744\] Definition \[18111610\] ensures the possibility of associating a nice family of Hlcs to any family of Hlcs. Namely let $ {\mathbf{V}}' \doteqdot \{{\left\langle V_{x},{\mathcal{A}}_{x}'{\right\rangle}}\}_{x\in X} $ be a family of Hlcs where $ {\mathcal{A}}_{x}' \doteqdot \{\mu_{j_{x}}^{x} \}_{j_{x}\in J_{x}} $ is a directed family of seminorms on $V_{x}$ generating the lct on it, for all $x\in X$. Then $ {\mathbf{V}} \doteqdot \{{\left\langle V_{x},{\mathcal{A}}_{x}{\right\rangle}}\}_{x\in X} $ is a nice family of Hlcs, called the *nice family of Hlcs associated to ${\mathbf{V}}'$*. Indeed ${\mathcal{A}}_{x}$ generates the lct on $V_{x}$ moreover it is trivially directed. Moreover $$\prod_{x\in X}^{\infty} {\left\langle V_{x},{\mathcal{A}}_{x}'{\right\rangle}} = \prod_{x\in X}^{\infty} {\left\langle V_{x},{\mathcal{A}}_{x}{\right\rangle}}.$$ and $ {\mathcal{G}} $ satisfies $FM(3)-FM(4')$ with respect to ${\mathbf{V}}'$ if and only if it satisfies $FM(3)-FM(4)$ with respect to ${\mathbf{V}}$. \[$5.2-5.3$ of [@gie]\] \[17471910Ba\] Let $ {\mathbf{E}} = \{ {\left\langle {\mathbf{E}}_{x},{\mathfrak{N}}_{x}{\right\rangle}} \}_{x\in X} $ be a nice family of Hlcs with $ {\mathfrak{N}}_{x} \doteqdot \{\nu^{x}_{j}\mid j\in J\} $ for all $x\in X$. Moreover let ${\mathcal{E}}$ satisfy $FM(3)-FM(4)$ with respect to $ {\mathbf{E}} $. Now we shall apply to ${\mathbf{E}}$ and ${\mathcal{E}}$ the general procedure described in $5.2-5.3$ of [@gie] for constructing bundles of $\Omega-$spaces. We define $${\mathfrak{V}}({\mathbf{E}},{\mathcal{E}})$$ to be the *Bundle generated by the couple ${\left\langle {\mathbf{E}},{\mathcal{E}}{\right\rangle}}$* , if 1. $ {\mathfrak{V}}({\mathbf{E}},{\mathcal{E}}) = {\left\langle {\left\langle {\mathfrak{E}}({\mathbf{E}}),\tau({\mathbf{E}},{\mathcal{E}}){\right\rangle}},\pi_{{\mathbf{E}}},X,{\mathfrak{N}}{\right\rangle}} $; 2. $ {\mathfrak{E}}({\mathbf{E}}) \doteqdot \bigcup_{x\in X} \{x\} \times {\mathbf{E}}_{x} $, $ \pi_{{\mathbf{E}}}:{\mathfrak{E}}({\mathbf{E}}) \ni (x,v) \mapsto x \in X $. 3. $ {\mathfrak{N}}=\{\nu_{j}\mid j\in J\} $, with $ \nu_{j}:{\mathfrak{E}} \ni (x,v) \mapsto \nu_{j}^{x}(v) $; 4. $ \tau({\mathbf{E}},{\mathcal{E}}) $ is the topology on ${\mathfrak{E}}$ [^8] such that for all $ (x,v)\in {\mathfrak{E}}({\mathbf{E}}) $ $${\mathcal{I}}_{(x,v)}^{\tau({\mathbf{E}},{\mathcal{E}})} \doteqdot {\mathfrak{F}_{{\mathcal{B}}_{{\mathbf{E}}}((x,v))}^{{\mathfrak{E}}({\mathbf{E}})}}$$ is the neighbourhood’s filter of $(x,v)$ with respect to it. Here ${\mathfrak{F}_{{\mathcal{B}}((x,v))}^{{\mathfrak{E}}({\mathbf{E}})}}$ is the filter on ${\mathfrak{E}}({\mathbf{E}})$ generated by the following filter’s base $$\begin{aligned} {1} {\mathcal{B}}_{{\mathbf{E}}}((x,v)) \doteqdot \{ T_{{\mathbf{E}}}(U,\sigma,{\varepsilon},j) & \mid U\in Open(X), \sigma\in{\mathcal{E}}, {\varepsilon}>0, j\in J \\ & \mid x\in U, \nu_{j}^{x}(v-\sigma(x)) <{\varepsilon}\},\end{aligned}$$ where $$\label{14332310a} T_{{\mathbf{E}}}(U,\sigma,{\varepsilon},j) \doteqdot \left\{ (y,w)\in{\mathfrak{E}}({\mathbf{E}}) \mid y\in U, \nu_{j}^{y}(w-\sigma(y)) <{\varepsilon}\right\};$$ What is important in this construction is the fact that ${\mathcal{E}}$ is canonically isomorphic to a linear subspace of $\Gamma(\pi_{{\mathbf{E}}})$ indeed \[17150312\] Let $ {\mathbf{E}} = \{ {\left\langle {\mathbf{E}}_{x},{\mathfrak{N}}_{x}{\right\rangle}} \}_{x\in X} $ be a nice family of Hlcs with $ {\mathfrak{N}}_{x} \doteqdot \{\nu^{x}_{j}\mid j\in J\} $ for all $x\in X$. Moreover let ${\mathcal{E}}$ satisfy $FM(3)-FM(4)$ with respect to $ {\mathbf{E}} $, and ${\mathfrak{V}}({\mathbf{E}},{\mathcal{E}})$ be the bundle generated by the couple $ {\left\langle {\mathbf{E}},{\mathcal{E}}{\right\rangle}} $. Thus according Prop. $5.8$ of [@gie] we have 1. $ {\mathfrak{V}}({\mathbf{E}},{\mathcal{E}}) $ is a bundle of $\Omega-$spaces; 2. with the notations of Definition \[17471910Ba\] ${\mathfrak{V}}({\mathbf{E}},{\mathcal{E}})$ is such that 1. ${\left\langle {\mathfrak{E}}({\mathbf{E}})_{x},\tau({\mathbf{E}},{\mathcal{E}}){\right\rangle}}$ as topological vector space is isomorphic to ${\left\langle {\mathbf{E}}_{x},{\mathfrak{N}}_{x}{\right\rangle}}$ for all $ x\in X $; 2. $ {\mathcal{E}} $ is canonically isomorphic [^9] to a linear subspace of $\Gamma(\pi_{{\mathbf{E}}})$ and if $X$ is compact and ${\mathbf{E}}$ is a function module, see [@gie 5.1.], then $ {\mathcal{E}} \simeq \Gamma(\pi_{{\mathbf{E}}}) $. \[15412610\] Let ${\mathbf{E}}$ be a nice family of Hlcs and let ${\mathcal{E}}$ satisfy $FM(3-4)$ with respect to ${\mathbf{E}}$. Thus for all $U\in Open(X)$, $\sigma\in{\mathcal{E}}$, ${\varepsilon}>0$, $j\in J$ $$T_{{\mathbf{E}}}(U,\sigma,{\varepsilon},j) = \bigcup_{y\in U} B_{{\mathbf{E}}_{y},j,{\varepsilon}}(\sigma(y))$$ where for all $ s \in {\mathbf{E}}_{y} $ $$B_{{\mathbf{E}}_{y},j,{\varepsilon}}(s) \doteqdot \left\{ (y,w) \in{\mathfrak{E}}({\mathbf{E}})_{y} \mid \nu_{j}^{y} \left(w-s\right) <{\varepsilon}\right\}.$$ In others words $T_{{\mathbf{E}}}(U,\sigma,{\varepsilon},j)$ is the $\sigma-$ deformed cilinder of radius ${\varepsilon}$, which justifies the name of ${\varepsilon}-$tube. ### Characterizations of Neighbourhood’s filters and Sections of Bundles of $\Omega-$spaces The following are simple but very useful characterizations of the convergence and of a section in a bundle of $\Omega-$spaces. \[28111555\] Let ${\mathfrak{V}} = {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}$ be a bundle of $\Omega-$spaces where $ {\mathfrak{N}}\doteqdot \{ \nu_{j}\mid j\in J \} $. Moreover let $b\in{\mathfrak{E}}$ and $\{b_{\alpha}\}_{\alpha\in D}$ a net in ${\mathfrak{E}}$. Then $(1) \Leftarrow (2) \Leftarrow (3) \Leftrightarrow (4)$ where 1. $\lim_{\alpha\in D}b_{\alpha}=b$; 2. $ (\exists\,U\in Op(X)\mid U\ni\pi(b)) (\exists\,\sigma\in\Gamma_{U}(\pi)) (\sigma\circ\pi(b)=b) $ such that $ \lim_{\alpha\in D}\pi(b_{\alpha}) =\pi(b) $ and $ (\forall j\in J) (\lim_{\alpha\in D} \nu_{j}(b_{\alpha}-\sigma(\pi(b_{\alpha}))) =0) $; 3. $ (\exists\,U'\in Op(X)\mid U'\ni\pi(b)) (\exists\,\sigma'\in\Gamma_{U}(\pi)\mid \sigma'\circ\pi(b)=b) $ and $ (\forall U\in Op(X)\mid U\ni\pi(b)) (\forall\sigma\in\Gamma_{U}(\pi)\mid \sigma\circ\pi(b)=b) $ we have $ \lim_{\alpha\in D}\pi(b_{\alpha}) =\pi(b) $ and $ (\forall j\in J) (\lim_{\alpha\in D} \nu_{j}(b_{\alpha}-\sigma(\pi(b_{\alpha}))) =0) $; 4. $ (\exists\,U'\in Op(X)\mid U'\ni\pi(b)) (\exists\,\sigma'\in\Gamma_{U}(\pi)) (\sigma'\circ\pi(b)=b) $ and $\lim_{\alpha\in D}b_{\alpha}=b$. Moreover if ${\mathfrak{V}}$ is locally full then $(1)\Leftrightarrow(4)$. Of course $(3)\rightarrow(2)$. $(2)$ is equivalent to say that $ (\exists\,U\in Op(X)\mid U\ni\pi(b)) (\exists\,\sigma\in\Gamma_{U}(\pi)) (\sigma\circ\pi(b)=b) $ such that $ (\forall V\in Op(X)\mid \pi(b)\in V\subseteq U) (\exists\,\alpha(V)\in D) (\forall\alpha\geq\alpha(V)) (\pi(b_{\alpha}) \in V) $ and $ (\forall j\in J) (\forall{\varepsilon}>0) (\exists\,\alpha(V)\in D) (\forall\alpha\geq\alpha(j,{\varepsilon})) (\nu_{j}(b_{\alpha}-\sigma(\pi(b_{\alpha}))) <{\varepsilon}) $. Set $ \alpha(V,j,{\varepsilon})\in D $ such that $ \alpha(V,j,{\varepsilon}) \geq \alpha(V), \alpha(j,{\varepsilon}) $ which there exists $D$ being directed, thus we have $ (\forall V\in Op(X)\mid \pi(b)\in V\subseteq U) (\forall j\in J) (\forall{\varepsilon}>0) (\exists\,\alpha(V,j,{\varepsilon})\in D) $ such that $(\forall\alpha\geq\alpha(V,j,{\varepsilon})) (\nu_{j}(b_{\alpha}-\sigma(\pi(b_{\alpha}))) <{\varepsilon})$ and $\pi(b_{\alpha}) \in V$. Thus $(1)$ follows by applying $1.5.\,VII$ of [@gie]. Finally by applying $1.5.\,VII$ of [@gie] $(4)$ (respectively $(1)$ if ${\mathfrak{V}}$ is locally full) is equivalent to $ (\exists\,U'\in Op(X)\mid U'\ni\pi(b)) (\exists\,\sigma'\in\Gamma_{U}(\pi)) (\sigma'\circ\pi(b)=b) $ and $ (\forall\sigma\in\Gamma_{U}(\pi)\mid \sigma\circ\pi(b)=b) (\forall j\in J) (\forall{\varepsilon}>0) (\forall V\in Op(X)\mid \pi(b)\in V\subseteq U) (\exists\,{\overline}{\alpha}\in D) (\forall\alpha\geq{\overline}{\alpha}) $ we have $ \pi(b_{\alpha}) \in V $ and $ \nu_{j}(b_{\alpha}-\sigma(\pi(b_{\alpha}))) <{\varepsilon}$ which is $(3)$. Although the following is a simple consequence of the previous result, we give to it the status of Theorem due to its extraordinary importance and use in the whole this work. \[15380512\] Let ${\mathfrak{V}}= {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}$ be a bundle of $\Omega-$spaces, $W\subseteq X$ and indicate $ {\mathfrak{N}}= \{ \nu_{j} \mid j\in J \} $. Moreover let $f\in{\mathfrak{E}}^{W}$, $x_{\infty}\in W$. Then $ (1) \Leftarrow (2) \Leftrightarrow (3) \Leftarrow (4) \Leftrightarrow (5) $ where 1. $f$ is continuous in $x_{\infty}$; 2. $ (\exists\,U\in Op(X)\mid U\ni x_{\infty}) (\exists\,\sigma\in\Gamma_{U}(\pi)) (\sigma(x_{\infty}) =f(x_{\infty}))$ such that $ \nu_{j}\circ(f-\sigma\circ\pi\circ f): W\cap U\to{\mathbb {R}}$ and $\pi\circ f:W\to X$ are continuous in $x_{\infty}$ for all $j\in J$; 3. $\pi\circ f:W\to X$ is continuous in $x_{\infty}$ and $ (\exists\,U\in Op(X)\mid U\ni x_{\infty}) (\exists\,\sigma\in\Gamma_{U}(\pi)) (\sigma(x_{\infty}) =f(x_{\infty}))$ such that $$(\forall j\in J) (\lim_{ y\to x_{\infty},y\in W\cap U} \nu_{j}(f(y)-\sigma \circ\pi\circ f(y))=0);$$ 4. $ (\exists\,U'\in Op(X)\mid U'\ni x_{\infty}) (\exists\,\sigma'\in\Gamma_{U}(\pi)) (\sigma'(x_{\infty}) =f(x_{\infty}))$ and $ (\forall U\in Op(X)\mid U\ni x_{\infty}) (\forall\sigma\in\Gamma_{U}(\pi) \mid \sigma(x_{\infty}) =f(x_{\infty}))$ we have $ \nu_{j}\circ(f-\sigma): W\cap U\to{\mathbb {R}}$ and $\pi\circ f:W\to X$ are continuous in $x_{\infty}$ for all $j\in J$; 5. $\pi\circ f:W\to X$ is continuous in $x_{\infty}$ and $ (\exists\,U'\in Op(X)\mid U'\ni x_{\infty}) (\exists\,\sigma'\in\Gamma_{U}(\pi)) (\sigma'(x_{\infty}) =f(x_{\infty}))$ and $ (\forall U\in Op(X)\mid U\ni x_{\infty}) (\forall\sigma\in\Gamma_{U}(\pi) \mid \sigma(x_{\infty}) =f(x_{\infty}))$ we have $$(\forall j\in J) (\lim_{ y\to x_{\infty},y\in W\cap U} \nu_{j}(f(y)-\sigma\circ\pi\circ f(y))=0);$$ 6. $ (\exists\,U'\in Op(X)\mid U'\ni x_{\infty}) (\exists\,\sigma'\in\Gamma_{U}(\pi)) (\sigma'(x_{\infty}) =f(x_{\infty}))$ and $f$ is continuous at $x_{\infty}$. Moreover if ${\mathfrak{V}}$ is locally full then $(1)\Leftrightarrow(6)$ and if it is full we can choose $U=X$ and $U'=X$. $(1)$ is equivalent to say that for each net $ \{x_{\alpha}\}_{\alpha\in D} \subset W $ such that $\lim_{\alpha\in D}x_{\alpha}= x_{\infty}$ in $W$, we have $\lim_{\alpha\in D}f(x_{\alpha}) =f( x_{\infty})$ in ${\mathfrak{E}}$. Similarly $(2)$ is equivalent to say that for each net $ \{x_{\alpha}\}_{\alpha\in D} \subset W $ such that $\lim_{\alpha\in D}x_{\alpha}= x_{\infty}$ in $W$, we have $ \lim_{\alpha\in D}\pi\circ f(x_{\alpha}) = \pi\circ f(x_{\infty}) $ and $ (\forall j\in J) (\lim_{\alpha\in D} \nu_{j}\circ(f-\sigma\circ\pi\circ f) (x_{\alpha}) = \nu_{j}\circ(f-\sigma\circ\pi\circ f) (x_{\infty})) $. Thus $(1) \Leftarrow (2)$ follows by the corresponding one in Proposition \[28111555\] with the positions $ (\forall\alpha\in D) (b_{\alpha}\doteqdot f(x_{\alpha})) $ and $b\doteqdot f(x_{\infty})$. Similarly $(1)\Leftarrow(5)$ follows by $(1)\Leftarrow(3)$ of Proposition \[28111555\]. Finally $(5)\Rightarrow(6)$ follows by $(5)\Rightarrow(1)$, while if $(6)$ is true then $\pi\circ f$ is continuous at $x_{\infty}$ indeed $\pi$ is continuous, then $(5)$ follows by the implication $(4)\Rightarrow(3)$ of Proposition \[28111555\] with the positions $ (\forall\alpha\in D) (b_{\alpha}\doteqdot f(x_{\alpha})) $ and $b\doteqdot f(x_{\infty})$. \[28111707\] Let ${\mathfrak{V}} = {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}$ be a bundle of $\Omega-$spaces, $W\subseteq X$ and indicate $ {\mathfrak{N}}= \{ \nu_{j}\mid j\in J \} $. Moreover let $f\in\prod_{x\in W}{\mathfrak{E}}_{x}$ and $x_{\infty}\in W$. Then $ (1) \Leftarrow (2) \Leftrightarrow (3) \Leftarrow (4) \Leftrightarrow (5) \Leftrightarrow (6) $ where 1. $f$ is continuous in $x_{\infty}$; 2. $(\exists\,U\in Op(X)\mid U\ni x_{\infty}) (\exists\,\sigma\in\Gamma_{U}(\pi)) (\sigma(x_{\infty}) =f(x_{\infty}))$ such that $ \nu_{j}\circ(f-\sigma): W\cap U\to{\mathbb {R}}$ is continuous in $x_{\infty}$ for all $j\in J$; 3. $(\exists\,U\in Op(X)\mid U\ni x_{\infty}) (\exists\,\sigma\in\Gamma_{U}(\pi)) (\sigma(x_{\infty}) =f(x_{\infty}))$ such that $$(\forall j\in J) (\lim_{ y\to x_{\infty},y\in W\cap U} \nu_{j}(f(y)-\sigma(y))=0);$$ 4. $(\exists\,U'\in Op(X)\mid U'\ni x_{\infty}) (\exists\,\sigma'\in\Gamma_{U}(\pi)) (\sigma'(x_{\infty}) =f(x_{\infty}))$ and $ (\forall U\in Op(X)\mid U\ni x_{\infty}) (\forall\sigma\in\Gamma_{U}(\pi) \mid \sigma(x_{\infty}) =f(x_{\infty}))$ we have that $ \nu_{j}\circ(f-\sigma): W\cap U\to{\mathbb {R}}$ is continuous in $x_{\infty}$ for all $j\in J$; 5. $(\exists\,U'\in Op(X)\mid U'\ni x_{\infty}) (\exists\,\sigma'\in\Gamma_{U}(\pi)) (\sigma'(x_{\infty}) =f(x_{\infty}))$ and $ (\forall U\in Op(X)\mid U\ni x_{\infty}) (\forall\sigma\in\Gamma_{U}(\pi) \mid \sigma(x_{\infty}) =f(x_{\infty}))$ we have $$(\forall j\in J) (\lim_{ y\to x_{\infty},y\in W\cap U} \nu_{j}(f(y)-\sigma(y))=0).$$ 6. $(\exists\,U'\in Op(X)\mid U'\ni x_{\infty}) (\exists\,\sigma'\in\Gamma_{U}(\pi)) (\sigma'(x_{\infty}) =f(x_{\infty}))$ and $f$ is continuous at $x_{\infty}$ If ${\mathfrak{V}}$ is locally full then $(1) \Leftrightarrow (6)$ and if it is full we can choose $U=X$ and $U'=X$. By Theorem \[15380512\] and $\pi\circ f=Id$. \[16572003\] Let ${\mathfrak{V}}$ be full and such that there exists a linear space $E$ such that for all $x\in X$ there exists a linear subspace $E_{x}\subseteq E$ such that ${\mathfrak{E}}_{x} = \{x\}\times E_{x}$, and that [^10] $$\{ X\ni x \mapsto (x,v)\in{\mathfrak{E}}_{x} \mid v\in \bigcap_{x\in X} E_{x} \} \subset \Gamma(\pi),$$ If $f_{0}\in\prod_{x\in X}E_{x}$ and $f\in\prod_{x\in X}{\mathfrak{E}}_{x}$ such that $f(x)=(x,f_{0}(x))$ for all $x\in X$ and $f_{0}(x_{\infty})\in \bigcap_{x\in X} E_{x}$, then $(1) \Leftrightarrow(2) \Leftrightarrow(3)$, where 1. $f$ is continuous at $x_{\infty}$ 2. $(\exists\,U\in Op(X)\mid U\ni x_{\infty}) (\exists\,\sigma\in{\mathcal{C}_{b} \left(U,E\right)}) (\sigma(x_{\infty}) =f(x_{\infty}))$ such that for all $j\in J$ $$\lim_{z\to x_{\infty},z\in W\cap U} \nu_{j}^{z}(f(z)-\sigma(z))=0;$$ 3. for all $j\in J$ $$\lim_{z\to x_{\infty},z\in W\cap U} \nu_{j}^{z}((z,f_{0}(z))- (z,f(x_{\infty})))=0.$$ By Corollary \[28111707\] $(1)\Leftrightarrow(2)$. Let $(3)$ hold then $(2)$ is true by setting $\sigma=\tau_{f(x_{\infty})}{\upharpoonright}U$. Let $(2)$ hold then $ \nu_{j}^{z}((z,f_{0}(z))- (z,f(x_{\infty}))) \leq \nu_{j}^{z}((z,f_{0}(z))- \sigma(z)) + \nu_{j}^{z}(\sigma(z)- \tau_{f(x_{\infty})}(z))$, thus $(3)$ follows by $(2)$ and by Corollary \[28111707\] applied to the continuous map $\tau_{f(x_{\infty})}{\upharpoonright}U$. \[21492812\] Let ${\mathfrak{V}} \doteqdot {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}$ be a bundle of $\Omega-$spaces, $W\subseteq X$ and indicate $ {\mathfrak{N}}= \{ \nu_{j}\mid j\in J \} $. Moreover let $f,g\in\prod_{x\in W}{\mathfrak{E}}_{x}$ and $x_{\infty}\in W$. Then if ${\mathfrak{V}}$ locally full or $\nu_{j}$ is continuous $\forall j\in J$, then $ (1) \rightarrow (2) $ where 1. $f(x_{\infty})=g(x_{\infty})$ and $f$ and $g$ are continuous in $x_{\infty}$; 2. $ (\exists\,U\in Op(X)\mid x_{\infty}\in U) $ such that $$(\forall j\in J) (\lim_{ y\to x_{\infty},y\in W\cap U} \nu_{j}(f(y)-g(y))=0).$$ Moreover if ${\mathfrak{V}}$ is full we can choose $U=X$. The statement is trivial in the case of continuiuty of all the $\nu_{j}$. Whereas if ${\mathfrak{V}}$ is locally full by $(1)\rightarrow(5)$ of Corollary \[28111707\] we have $(\exists\,U\in Op(X)) (\exists\,\sigma\in\Gamma_{U}(\pi)) (\sigma(x_{\infty}) =f(x_{\infty}) =g(x_{\infty}))$ such that $$(\forall j\in J) (\lim_{y\to x_{\infty},y\in W\cap U} \nu_{j}(f(y)-\sigma(y)) = \lim_{ y\to x_{\infty},y\in W\cap U} \nu_{j}(g(y)-\sigma(y)) =0).$$ Therefore $$\lim_{y\to x_{\infty},y\in W\cap U} \nu_{j}(f(y)-g(y)) \leq \lim_{y\to x_{\infty},y\in W\cap U} \nu_{j}(f(y)-\sigma(y)) + \lim_{y\to x_{\infty},y\in W\cap U} \nu_{j}(g(y)-\sigma(y)) =0.$$ \[281117010\] Let $ {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}} $ be a bundle of $\Omega-$spaces, $W\in Op(X)$ and indicate $ {\mathfrak{N}}= \{ \nu_{j}\mid j\in J \} $. Moreover let $f\in\prod_{x\in W}^{b}{\mathfrak{E}}_{x}$. Then $ (1) \Leftarrow (2) \Leftrightarrow (3) \Leftarrow (4) \Leftrightarrow (5) $ where 1. $f\in\Gamma_{W}(\pi)$; 2. $$(\forall x\in\ W) (\exists\,U_{x}\in Op(X)\mid U_{x}\ni x) (\exists\,\sigma_{x}\in\Gamma_{U_{x}}(\pi)) (\sigma_{x}(x) =f(x))$$ such that $ \nu_{j} \circ(f-\sigma_{x}) $ is continuous in $x$, $\forall j\in J$; 3. $$(\forall x\in\ W) (\exists\,U_{x}\in Op(X)\mid U_{x}\ni x) (\exists\,\sigma_{x}\in\Gamma_{U_{x}}(\pi)) (\sigma_{x}(x) =f(x))$$ such that $ (\forall j\in J) (\lim_{ y\to x,y\in W\cap U_{x}} \nu_{j}(f(y)-\sigma_{x}(y)) = 0)$; 4. $$(\forall x\in\ W) (\exists\,U_{x}'\in Op(X)\mid U_{x}'\ni x) (\exists\,\sigma_{x}'\in\Gamma_{U_{x}}(\pi)) (\sigma_{x}'(x) =f(x))$$ and $$(\forall\,U_{x}\in Op(X)\mid U_{x}\ni x) (\forall\sigma_{x}\in\Gamma_{U_{x}}(\pi) \mid \sigma_{x}(x) =f(x))$$ we have that $ \nu_{j} \circ(f-\sigma_{x}) $ is continuous in $x$ for all $x\in W$ and $j\in J$; 5. $$(\forall x\in\ W) (\exists\,U_{x}'\in Op(X)\mid U_{x}'\ni x) (\exists\,\sigma_{x}'\in\Gamma_{U_{x}}(\pi)) (\sigma_{x}'(x) =f(x))$$ and $$(\forall x\in\ W) (\forall\,U_{x}\in Op(X)\mid U_{x}\ni x) (\forall\sigma_{x}\in\Gamma_{U_{x}}(\pi) \mid \sigma_{x}(x) =f(x))$$ we have $(\forall j\in J) (\lim_{ y\to x,y\in W\cap U_{x}} \nu_{j}(f(y)-\sigma_{x}(y)) = 0)$. By Corollary \[28111707\]. In the case in which the bundle is locally full we can give some useful characterizations of the Neighbourhood’s filter ${\mathcal{I}}_{\alpha}^{\tau}$ of any point $\alpha$ in the bundle space ${\left\langle {\mathfrak{E}},\tau{\right\rangle}}$. \[${\varepsilon}-$Tubes\] \[12592310\] Let $ {\mathfrak{P}} \doteqdot {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},p,X,{\mathfrak{N}}{\right\rangle}} $ be a locally full bundle of $\Omega-$spaces, and let us denote $ {\mathfrak{N}}\doteqdot \{ \nu_{j} \mid j\in J \} $. Set $$\begin{cases} {\mathcal{K}}^{loc} \doteqdot \prod_{\alpha\in{\mathfrak{E}}} {\mathcal{K}}_{\alpha}^{loc} \\ {\mathcal{K}}_{\alpha}^{loc} \doteqdot \left\{ (U,\sigma_{U}) \mid U\in Op(X), \sigma_{U}\in\Gamma_{U}(p) \mid p(\alpha)\in U, \sigma_{U}(p(\alpha))=\alpha \right\}. \end{cases}$$ Moreover $\forall\alpha\in{\mathfrak{E}}$ and $\forall{\mathfrak{l}}\in{\mathcal{K}}^{loc}$ set $$\begin{cases} {\mathcal{B}}_{{\mathfrak{l}}}^{loc}(\alpha) \doteqdot \left\{ T^{loc} (V,{\mathfrak{l}}_{2}(\alpha),{\varepsilon},j) \mid V\in Op(X), {\varepsilon}>0, j\in J \mid p(\alpha) \in V \subseteq {\mathfrak{l}}_{1}(\alpha) \right\}, \\ T^{loc}(U,\sigma_{U},{\varepsilon},j) \doteqdot \left\{ \beta\in{\mathfrak{E}} \mid p(\beta)\in U, \nu_{j}(\beta-\sigma_{U}(p(\beta))) <{\varepsilon}\right\}, \end{cases}$$ $ (\forall U\in Op(X)) (\forall j\in J) (\forall{\varepsilon}>0) (\forall\sigma_{U}\in\Gamma_{U}(p)) $. If $ {\mathfrak{P}} $ is a full bundle then we can set $$\begin{cases} {\mathcal{K}} \doteqdot \prod_{\alpha\in{\mathfrak{E}}} {\mathcal{K}}_{\alpha} \\ {\mathcal{K}}_{\alpha} \doteqdot \left\{ (U,\sigma) \mid U\in Op(X), \sigma\in\Gamma(p) \mid p(\alpha)\in U, \sigma(p(\alpha))=\alpha \right\}. \end{cases}$$ Moreover $\forall\alpha\in{\mathfrak{E}}$ and $\forall{\mathfrak{l}}\in{\mathcal{K}}$ set $$\begin{cases} {\mathcal{B}}_{{\mathfrak{l}}}(\alpha) \doteqdot \left\{ T(V,{\mathfrak{l}}_{2}(\alpha),{\varepsilon},j) \mid V\in Op(X), {\varepsilon}>0, j\in J \mid p(\alpha) \in V \subseteq {\mathfrak{l}}_{1}(\alpha) \right\}, \\ T(U,\sigma,{\varepsilon},j) \doteqdot T^{loc}(U,\sigma{\upharpoonright}U,{\varepsilon},j), \end{cases}$$ $ (\forall U\in Op(X)) (\forall j\in J) (\forall{\varepsilon}>0) (\forall\sigma\in\Gamma(p)) $. Any set $ T^{loc}(U,\sigma{\upharpoonright}U,{\varepsilon},j) $ for a fixed ${\varepsilon}>0$ is called *${\varepsilon}-$Tube*. \[15512610\] Notice that $ (\forall U\in Op(X)) (\forall j\in J) (\forall{\varepsilon}>0) (\forall\sigma_{U}\in\Gamma_{U}(p)) $ $$T^{loc}(U,\sigma_{U},{\varepsilon},j) = \bigcup_{y\in U} B_{{\mathfrak{E}}_{y},j,{\varepsilon}}(\sigma_{U}(y))$$ where for all $ \gamma \in {\mathfrak{E}}_{y} $ $$B_{{\mathfrak{E}}_{y},j,{\varepsilon}}(\gamma) \doteqdot \left\{ \beta\in{\mathfrak{E}}_{y} \mid \nu_{j}^{y}\left(\beta-\gamma\right) <{\varepsilon}\right\}.$$ \[ Neighbourhood’s filter ${\mathcal{I}}_{\alpha}^{\tau}$ \] \[11372310\] Let $ {\mathfrak{P}} \doteqdot {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},p,X,{\mathfrak{N}}{\right\rangle}} $ be a bundle of $\Omega-$spaces 1. if ${\mathfrak{P}}$ is locally full $\forall\alpha\in{\mathfrak{E}}$ and $\forall{\mathfrak{l}}\in{\mathcal{K}}^{loc}$ the class ${\mathcal{B}}_{{\mathfrak{l}}}^{loc}(\alpha)$ is a basis of a filter moreover $${\mathfrak{F}_{{\mathcal{B}}_{{\mathfrak{l}}}^{loc}(\alpha)}^{{\mathfrak{E}}}} = {\mathcal{I}}_{\alpha}^{\tau};$$ 2. if ${\mathfrak{P}}$ is full or locally full over a completely regular space then $\forall\alpha\in{\mathfrak{E}}$ and $\forall{\mathfrak{l}}\in{\mathcal{K}}$ the class ${\mathcal{B}}_{{\mathfrak{l}}}(\alpha)$ is a basis of a filter moreover $${\mathfrak{F}_{{\mathcal{B}}_{{\mathfrak{l}}}(\alpha)}^{{\mathfrak{E}}}} = {\mathcal{I}}_{\alpha}^{\tau}.$$ Here ${\mathcal{I}}_{\alpha}^{\tau}$ is the neighbourhood’s filter of $\alpha$ in the topological space ${\left\langle {\mathfrak{E}},\tau{\right\rangle}}$. Statement $(1)$ follows by applying $1.5.\,VII$ of [@gie], while statement $(2)$ follows by statement $(1)$ and the fact that for all $U\in Op(X)$ and $\sigma\in\Gamma(p)$ we have $ \sigma{\upharpoonright}U \in\Gamma_{U}(p) $ and $ T(U,\sigma,{\varepsilon},j) \doteqdot T^{loc}(U,\sigma{\upharpoonright}U,{\varepsilon},j), $ for all $j\in J$ and ${\varepsilon}>0$. \[12312410\] Let $ {\mathbf{E}} \doteqdot \{{\left\langle {\mathbf{E}}_{x},{\mathfrak{N}}_{x}{\right\rangle}}\}_{x\in X} $ be a nice family of Hlcs with $ {\mathfrak{N}}_{x} \doteqdot \{ \nu_{j\in J}^{x} \} $ for all $x\in X$. Moreover let $ {\mathcal{E}} $ satisfy $FM(3^{*})-FM(4)$ with respect to ${\mathbf{E}}$. Set as usual $ {\mathfrak{E}}({\mathbf{E}}) \doteqdot \bigcup_{x\in X} \{x\} \times {\mathbf{E}}_{x} $ and $$\begin{cases} {\mathcal{K}}^{{\mathcal{E}}} \doteqdot \prod_{(x,v)\in{\mathfrak{E}}} {\mathcal{K}}_{(x,v)}^{{\mathcal{E}}} \\ {\mathcal{K}}_{(x,v)}^{{\mathcal{E}}} \doteqdot \left\{ (U,f) \mid U\in Op(X), f \in{\mathcal{E}} \mid x\in U, f(x)=v \right\}. \end{cases}$$ Moreover $ \forall(x,v)\in{\mathfrak{E}}({\mathbf{E}}) $ and $\forall{\mathfrak{l}}\in{\mathcal{K}}^{{\mathcal{E}}}$ set $$\label{15022310} {\mathcal{B}}_{{\mathfrak{l}}}^{{\mathcal{E}}}((x,v)) = \left\{ T_{{\mathbf{E}}}(V,{\mathfrak{l}}_{2}((x,v)),{\varepsilon},j) \mid {\varepsilon}>0, j\in J, V\in Op(X) \mid x \in V \subseteq {\mathfrak{l}}_{1}((x,v)) \right\}.$$ Here $T_{{\mathbf{E}}}(V,{\mathfrak{l}}_{2}((x,v)),{\varepsilon},j)$ has been defined in . \[ Neighbourhood’s filter ${\mathcal{I}}_{(x,v)}^{\tau({\mathbf{E}},{\mathcal{E}})}$ \] \[14272310\] Assume notations in Definition \[17471910Ba\] and in Definition \[12312410\]. Let $ {\mathbf{E}} \doteqdot \{{\left\langle E_{x},{\mathfrak{N}}_{x}{\right\rangle}}\}_{x\in X} $ be a nice family of Hlcs with $ {\mathfrak{N}}_{x} \doteqdot \{ \nu_{j\in J}^{x} \} $ for all $x\in X$, moreover let $ {\mathcal{E}} $ satisfy $FM(3^{*})-FM(4)$ with respect to ${\mathbf{E}}$. Then $ {\mathfrak{V}}({\mathbf{E}},{\mathcal{E}}) \doteqdot {\left\langle {\left\langle {\mathfrak{E}}({\mathbf{E}}),\tau({\mathbf{E}},{\mathcal{E}}){\right\rangle}} ,\pi_{{\mathbf{E}}},X,{\mathfrak{N}}{\right\rangle}} $ is a full bundle of $\Omega-$spaces and $\forall(x,v)\in{\mathfrak{E}}({\mathbf{E}})$ $${\mathfrak{F}_{{\mathcal{B}}_{{\mathfrak{l}}}^{{\mathcal{E}}}((x,v))}^{{\mathfrak{E}}({\mathbf{E}})}} = {\mathcal{I}}_{(x,v)}^{\tau({\mathbf{E}},{\mathcal{E}})}.$$ Here ${\mathcal{I}}_{(x,v)}^{\tau({\mathbf{E}},{\mathcal{E}})}$ is the neighbourhood’s filter of $(x,v)$ in the topological space $ {\left\langle {\mathfrak{E}}({\mathbf{E}}),\tau({\mathbf{E}},{\mathcal{E}}){\right\rangle}} $. By Theorem $5.9.$ of [@gie] ${\mathcal{E}}$ and $\Gamma(p_{1})$ are canonically isomorphic as linear spaces, so ${\mathfrak{V}}({\mathbf{E}},{\mathcal{E}})$ is full by \[15012310\]. The statement hence follows by statement $(2)$ of Corollary \[11372310\]. The following corollaries provide conditions under which the topologies over two bundle spaces are equal. \[15582210\] Let $ {\left\langle {\left\langle {\mathfrak{E}},\tau_{k}{\right\rangle}},p_{k},X,{\mathfrak{N}}_{k}{\right\rangle}} $ be a full bundle of $\Omega-$spaces or a locally full bundle over a completely regular space $X$, for $k=1,2$. If there exists a map $p$ such that $p=p_{1}=p_{2}$ (equality as maps) and $ \Gamma(p_{1}) = \Gamma(p_{2}) $ then $ \tau_{1} = \tau_{2} $. By statement $(2)$ of Corollary \[11372310\]. \[15592210\] Let us assume the hypothesis and notations in Corollary \[14272310\], moreover let $ {\mathfrak{P}}_{2} \doteqdot {\left\langle {\left\langle E,\tau_{2}{\right\rangle}},p_{2},X,{\mathfrak{N}}_{2}{\right\rangle}}$ be a bundle of $\Omega-$spaces and a map $p$ such that $p=\pi_{{\mathbf{E}}}=p_{2}$ as maps. Thus if the following conditions are satisfied 1. $X$ is compact; 2. $ {\mathcal{E}} $ and $\Gamma(p_{2})$ are canonically isomorphic as linear spaces then $\tau({\mathbf{E}},{\mathcal{E}})=\tau_{2}$. By Theorem $5.9.$ of [@gie] ${\mathcal{E}}$ and $ \Gamma(\pi_{{\mathbf{E}}}) $ are canonically isomorphic as linear spaces if $X$ is compact, so $\Gamma(\pi_{{\mathbf{E}}})=\Gamma(p_{2})$. Moreover \[15012310\] and the shown fact that ${\mathcal{E}}$ and $\Gamma(\pi_{{\mathbf{E}}})$ are canonically isomorphic ensure that ${\mathfrak{V}}({\mathbf{E}},{\mathcal{E}})$ is a full bundle, thus it is so ${\mathfrak{P}}_{2}$ by the equality $\Gamma(\pi_{{\mathbf{E}}})=\Gamma(p_{2})$. Hence the statement follows by Corollary \[15582210\]. ### Direct Sum of Bundles of $\Omega-$spaces {#directsum} \[[@jar]\] \[18111755\] Let $\{E_{i}\}_{i\in I}$ a family of lcs. Then we denote by $\tau_{0}$, $\tau_{b}$, $\tau_{l}$, $ \tau_{{\mathfrak{l}}} $ respectively the topology on $\bigoplus_{i\in I}E_{i}$ induced by the product topology on $\prod_{i\in I}E_{i}$, that induced by the box topology on $\prod_{i\in I}E_{i}$ (see [@jar]), the direct sum topology, Ch. $4$, $\S 3$ of [@jar], finally the lc-direct sum topology Ch. $6$, $\S 6$ of [@jar]. \[16322110\] Let ${\left\langle E_{i},\nu_{i}{\right\rangle}}_{i=1}^{n}$ a finite family of lcs where $\nu_{i}=\{\nu_{i,l_{i}}\mid l_{i}\in L_{i}\}$ is a fundamental directed set of seminorms of $E_{i}$. Let us set for all $i=1,...,n$, $l_{i}\in L_{i}$ and $ \rho\in \prod_{i=1}^{n} L_{i} $ $$\begin{cases} \hat{\nu}_{i,l_{i}} \doteqdot \nu_{i,l_{i}} \circ \Pr_{i} \\ \hat{\mu_{\rho}} \doteqdot \sum_{i=1}^{n} \hat{\nu}_{i,l_{i}}, \end{cases}$$ where $ \Pr_{i}:\prod_{k=1}^{n}E_{k} \ni x \mapsto x_{i} \in E_{i} $. Then $ \hat{\mu} \doteqdot \{\hat{\mu}_{\rho} \mid\rho\in\prod_{i=1}^{n}L_{i}\} $ is a directed set of seminorms on $ \bigoplus_{i=1}^{n} E_{i} $, moreover by setting $$\begin{cases} {\mathcal{B}}({\mathbf{0}}) \doteqdot \{ W_{{\varepsilon}}^{\rho} \mid {\varepsilon},\rho\in\prod_{i=1}^{n} L_{i} \} \\ W_{{\varepsilon}}^{\rho} \doteqdot \{ x\in \bigoplus_{i=1}^{n} E_{i} \mid \hat{\mu}_{\rho}(x) <{\varepsilon}\}, \end{cases}$$ we have that $ {\mathcal{B}}({\mathbf{0}}) $ is a base of a filter on $ \bigoplus_{i=1}^{n} E_{i} $ in addition $${\mathfrak{F}_{{\mathcal{B}}({\mathbf{0}})}^{\bigoplus_{i=1}^{n}E_{i}}} = {\mathcal{I}}_{{\mathbf{0}}}^{\tau},$$ where $\tau$ is the unique locally convex topology on $ \bigoplus_{i=1}^{n} E_{i} $ generated by $\hat{\mu}$ and ${\mathcal{I}}_{{\mathbf{0}}}^{\tau}$ is the neighbourhood’s filter of ${\mathbf{0}}$ with respect to the topology $\tau$. Finally with the notations of Definition \[18111755\] we have $ \tau = \tau_{0} = \tau_{b} = \tau_{l} = \tau_{{\mathfrak{l}}} $. Only in this proof we set $I\doteqdot\{1,...,n\}$, $L\doteqdot\prod_{i\in I}L_{i}$ and $ E^{\oplus} \doteqdot \bigoplus_{i=1}^{n} E_{i} $. Due to the fact that $n<\infty$ we know that $\prod_{i=1}^{n}E_{i}=E^{\oplus}$ so by [@jar] $\S 4.3.$ the set $ \{\prod_{i=1}^{n}U_{i}\mid U_{i}\in{\mathfrak{(}}U)_{i}\} $ is a ${\mathbf{0}}-$basis for the box topology on $E^{\oplus}$ if ${\mathfrak{U}}_{i}$ is a ${\mathbf{0}}-$basis for the topology on $E_{i}$. Moreover $\nu_{i}$ is directed so by $II.3$ of [@BourTVS] we can choose $$\label{17442110} {\mathfrak{U}}_{i} = \{ V(\nu_{i,l_{i}},{\varepsilon}) \doteqdot \{x_{i}\in E_{i}\mid\nu_{i,l_{i}}(x_{i})<{\varepsilon}\} \mid {\varepsilon}>0, l_{i}\in L_{i} \}.$$ Thus if we set $$\label{18052110} \begin{cases} {\mathcal{B}}_{1}({\mathbf{0}}) \doteqdot \{ U_{\eta}^{\rho}\mid\eta\in({\mathbb {R}}_{0}^{+})^{n}, \rho\in L \} \\ U_{\eta}^{\rho} \doteqdot \{ x\in E^{\oplus} \mid (\forall i\in I) (\hat{\nu}_{i,\rho_{i}}(x)<\eta_{i}) \}, \end{cases}$$ then ${\mathcal{B}}_{1}({\mathbf{0}})$ is is a ${\mathbf{0}}-$basis for the topology $\tau_{0}$. Moreover $ U_{{\varepsilon}}^{\rho} = \bigcap_{i=1}^{n} V(\hat{\nu}_{i,\rho_{i}}\eta_{i}) $ so if we set $${\mathcal{G}}({\mathbf{0}}) \doteqdot \left\{ \bigcap_{s\in M} V(\hat{\nu}_{s}{\varepsilon}_{M}(s)) \mid M\in {\mathcal{P}}_{\omega} \left(\bigcup_{i\in I}\{i\}\times L_{i}\right) {\varepsilon}_{M}:M\to{\mathbb {R}}_{0}^{+} \right\},$$ then by $ {\mathcal{B}}_{1}({\mathbf{0}}) \subseteq {\mathcal{G}}({\mathbf{0}}) $. Moreover by applying $II.3$ of [@BourTVS], ${\mathcal{G}}({\mathbf{0}})$ is a basis of a filter thus $${\mathfrak{F}_{{\mathcal{B}}_{1}({\mathbf{0}})}^{E^{\oplus}}} \subseteq {\mathfrak{F}_{{\mathcal{G}}({\mathbf{0}})}^{E^{\oplus}}}.$$ Now for all $M \in {\mathcal{P}}_{\omega} \left(\bigcup_{i\in I}\{i\}\times L_{i}\right) $ we have $ M=\bigcup_{i\in I}M_{i} $ with $ M_{i}\doteqdot M\cap(\{i\}\times L_{i}) = \{i\}\times Q_{i} $ for some $Q_{i}\in{\mathcal{P}}_{\omega}(L_{i})$. Hence $ \forall M \in {\mathcal{P}}_{\omega} \left(\bigcup_{i\in I}\{i\}\times L_{i}\right) $ and $\forall{\varepsilon}_{M}:M\to{\mathbb {R}}_{0}^{+}$ $$\begin{aligned} T\doteqdot \bigcap_{s\in M} V(\hat{\nu}_{s},{\varepsilon}_{M}(s)) & = \bigcap_{i\in I} \bigcap_{s\in M_{i}} V(\hat{\nu}_{s},{\varepsilon}_{M}(s)) \\ & = \bigcap_{i\in I} \bigcap_{l_{i}\in Q_{i}} \{ x\in E^{\oplus} \mid x_{i}\in V(\nu_{i,l_{i}},{\varepsilon}_{M}(i,l_{i})) \\ & = \bigcap_{i\in I} \left\{ x\in E^{\oplus} \mid x_{i}\in \bigcap_{l_{i}\in Q_{i}} V(\nu_{i,l_{i}},{\varepsilon}_{M}(i,l_{i})) \right\}. \end{aligned}$$ Moreover we know that ${\mathfrak{U}}_{i}$ is a basis of a filter on $E_{i}$ thus for aa $i\in I$ there exists $\lambda_{i}>0$ and $k_{i}\in L_{i}$ such that $$V(\nu_{i,k_{i}},\lambda_{i}) \subseteq \bigcap_{l_{i}\in Q_{i}} V(\nu_{i,l_{i}},{\varepsilon}_{M}(i,l_{i})),$$ hence $$\begin{aligned} {\mathcal{G}}({\mathbf{0}}) \ni T & \supseteq \bigcap_{i\in I} \{ x\in E^{\oplus} \mid x_{i}\in V(\nu_{i,k_{i}},\lambda_{i}) \} \\ & = \bigcap_{i\in I} V(\hat{\nu}_{i,k_{i}},\lambda_{i}) \in {\mathcal{B}}_{1}({\mathbf{0}}). \end{aligned}$$ Therefore by a well-known property of filters $ {\mathfrak{F}_{{\mathcal{G}}({\mathbf{0}})}^{E^{\oplus}}} \subseteq {\mathfrak{F}_{{\mathcal{B}}_{1}({\mathbf{0}})}^{E^{\oplus}}} $ then $$\label{18062110} {\mathfrak{F}_{{\mathcal{G}}({\mathbf{0}})}^{E^{\oplus}}} = {\mathfrak{F}_{{\mathcal{B}}_{1}({\mathbf{0}})}^{E^{\oplus}}}.$$ By applying $II.3$ of [@BourTVS] we know that ${\mathfrak{F}_{{\mathcal{G}}({\mathbf{0}})}^{E^{\oplus}}}$ is the ${\mathbf{0}}-$neighbourhood’s filter with respect to the locally convex topology generated by the family of seminorms $ \{\nu_{s}\mid s\in \bigcup_{i\in I}\{i\}\times L_{i} \} $ thus by and $$\label{18112110} \left\{\nu_{s}\mid s\in \bigcup_{i\in I}\{i\}\times L_{i} \right\} \text{ is a fss for $\tau_{0}$},$$ where $fss$ is for fundamental system of seminorms. Now $\hat{\mu}$ is a set of seminorms on $E^{\oplus}$. Let $\rho^{1},\rho^{2}\in L$ then by the hypothesis that $\nu_{i}$ is directed, for all $i\in I$ there exists $\rho_{i}\in L_{i}$ such that $\rho_{i}\geq\rho^{1},\rho^{2}$ thus $\hat{\mu}_{\rho} \geq \hat{\mu}_{\rho^{1}}, \hat{\mu}_{\rho^{2}} $, hence $\hat{\mu}$ is directed. Therefore setting $$\begin{cases} {\mathcal{B}}({\mathbf{0}}) \doteqdot \{ W_{{\varepsilon}}^{\rho} \mid {\varepsilon}>0, \rho\in L \} \\ W_{{\varepsilon}}^{\rho} \doteqdot \{ x\in E^{\oplus} \mid \hat{\mu}_{\rho}(x) <{\varepsilon}\} \end{cases}$$ by applying $II.3$ of [@BourTVS] $$\label{18242110} {\mathcal{B}}({\mathbf{0}}) \text{ is the ${\mathbf{0}}-$basis for the topology gen. by $\hat{\mu}$. }$$ Now $ (\forall(k,l_{k})\in \bigcup_{i\in I}\{i\}\times L_{i}) (\exists\,\rho\in L) (\hat{\nu}_{k,l_{k}}\leq a\hat{\mu}_{\rho}) $ indeed keep any $\rho$ s.t. $\rho(k)=l_{k}$. While $ (\forall\rho\in I) (m\in{\mathbb{N}}) (\exists\,s_{1},...,s_{m}\in \bigcup_{i\in I}\{i\}\times L_{i}) (\exists\,a>0) (\hat{\mu}_{\rho}\leq a \sup_{r}\hat{\nu}_{s_{r}}) $ indeed it is sufficient ot set $m=n$, $ a=n $ and $ s_{i} = (i,\rho_{i}) $ for all $i\in I$. Therefore by applying Corollary $1$ $II.7$ of [@BourTVS] and by and we have that $\hat{\mu}$ is a fundamental directedset of seminorms for the topology $\tau_{0}$ hence the part of the statement concerning $\tau_{0}$. By Prop. $2$, $\S 3$, Ch $4$ of [@jar] we know that $\tau_{0} = \tau_{b} = \tau_{l} $. Finally $ \tau_{{\mathfrak{l}}} = \tau_{l} $ by the fact that $ \tau_{{\mathfrak{l}}} $ is the finest locally convex topology among those which are coarser than $\tau_{l}$, $\S 6$, Ch $6$ of [@jar], and the just now shown fact that $\tau_{l}$ is locally convex being equal to $\tau_{0}$ which is generated by $\hat{\mu}$. Now we shall apply the result obtained in the previous proposition, in order to extend to the case of bundles of $\Omega-$spaces what is a standard construction in the Banach bundles case. \[12422110\] Let $ \{ {\mathfrak{V}}_{i} \doteqdot {\left\langle {\left\langle {\mathfrak{E}}_{i},\tau_{i}{\right\rangle}},\pi_{i},X,{\mathfrak{N}}_{i}{\right\rangle}} \}_{i=1}^{n} $ be a family of bunldes of $\Omega-$spaces, let us denote $ {\mathfrak{N}}_{i} = \{ \nu_{i,l_{i}} \mid l_{i}\in L_{i} \} $ and $ {\mathfrak{N}}_{i}^{x} = \{ \nu_{i,l_{i}}^{x} \doteqdot \nu_{i,l_{i}}{\upharpoonright}\left({\mathfrak{E}}_{i}\right)_{x} \mid l_{i}\in L_{i} \} $, with $ \left({\mathfrak{E}}_{i}\right)_{x} \doteqdot \overset{-1}{\pi_{i}}(x) $ for all $i=1,...,n$. Set 1. $ {\mathbf{E}}_{x}^{\oplus} \doteqdot \bigoplus_{i=1}^{n}\left({\mathfrak{E}}_{i}\right)_{x} $; 2. $ {\mathfrak{n}}_{x}^{\oplus} \doteqdot \{ \hat{\mu}_{\rho}^{x} \mid \rho \in \prod_{i=1}^{n} L_{i} \} $, where $$\label{18260603} \boxed{ \hat{\mu}_{\rho}^{x} = \sum_{i=1}^{n} \hat{\nu}_{i,\rho_{i}}^{x}; }$$ 3. $ {\mathcal{E}}^{\oplus} $ is the linear subspace of $ \prod_{x\in X} {\mathbf{E}}_{x}^{\oplus} $ generated by the following set $$\label{18270603} \bigcup_{i=1}^{n} \tilde{\Gamma}(\pi_{i}).$$ Here $ \Pr_{i}^{x}: {\mathbf{E}}_{x}^{\oplus} \ni x \mapsto x(i) \in \left({\mathfrak{E}}_{i}\right)_{x} $ while $ \hat{\nu}_{i,\rho_{i}}^{x} = \nu_{i,\rho_{i}}^{x} \circ \Pr_{i}^{x} $ and $ I_{i}^{x}: \left({\mathfrak{E}}_{i}\right)_{x} \to {\mathbf{E}}_{x}^{\oplus} $ is the canonical inclusion, i.e. $ \Pr_{j}^{x} \circ I_{i}^{x} = \delta_{i,j} \, Id^{x} $, finally $ \tilde{\Gamma}(\pi_{i}) \doteqdot \{ \tilde{f} \mid f\in \Gamma(\pi_{i}) \} $, with $ \tilde{f}(x) \doteqdot I_{i}^{x} (f(x)) $. Notice that $ \{ {\left\langle \left({\mathfrak{E}}_{i}\right)_{x},{\mathfrak{N}}_{i}^{x}{\right\rangle}} \}_{i=1}^{n} $ for all $x\in X$ is a family of Hlcs where ${\mathfrak{N}}_{i}^{x}$ is a directed family of seminorms defining the topology on $ \left({\mathfrak{E}}_{i}\right)_{x} $, for all $i=1,...,n$, moreover by Lemma \[19420111\] ${\mathcal{E}}^{\oplus}$ satisfies $FM(3)-FM(4)$ with respect to ${\mathbf{E}}^{\oplus}$. Finally by applying Theorem \[16322110\] we have that ${\mathfrak{n}}_{x}^{\oplus}$ is a *directed* set of seminorms on ${\mathbf{E}}_{x}^{\oplus}$ [^11] so for what before said we can state that the couple $$\label{21142110} {\left\langle {\mathbf{E}}^{\oplus},{\mathcal{E}}^{\oplus}{\right\rangle}},$$ where $${\mathbf{E}}^{\oplus} \doteqdot \left\{ {\left\langle {\mathbf{E}}_{x}^{\oplus},{\mathfrak{n}}_{x}^{\oplus}{\right\rangle}} \right\}_{x\in X},$$ satisfies the requirements of Proposition $5.8.$ of [@gie]. Therefore generates a bundle of $\Omega-$spaces which according the notations in Definition is \[17471910Ba\] $$\label{14442410} \bigoplus_{i=1}^{n} {\mathfrak{V}}_{i} \doteqdot {\mathfrak{V}}({\mathbf{E}}^{\oplus},{\mathcal{E}}^{\oplus})$$ and called the *bundle direct sum of the family $ \{ {\mathfrak{V}}_{i} \}_{i=1}^{n} $*. \[22550110\] Note that Theorem \[16322110\] shows much more then the directness of the set of seminorms ${\mathfrak{n}}_{x}^{\oplus}$, indeed it proves that ${\mathfrak{n}}_{x}^{\oplus}$ induces on ${\mathbf{E}}_{x}^{\oplus}$ the product topology. \[19420111\] ${\mathcal{E}}^{\oplus}$ satisfies $FM(3)-FM(4)$ with respect to ${\mathbf{E}}^{\oplus}$. $ I_{i}^{x} $ is a bijective map onto its range whose inverse is $ \Pr_{i}^{x} {\upharpoonright}Range(I_{i}^{x}) $. Moreover by definition of the product topology $\Pr_{i}^{x}$ is continuous with respect to the topology $\tau_{0}^{i}$ on $Range(I_{i}^{x})$ induced by $\tau_{0}$ [@BourGT Ch.1], while $I_{i}^{x}$ is continuous with respect to $\tau_{0}^{i}$ by [@jar $\S$ 4.3 Pr.1] and the definition of $\tau_{l}$. Hence by Theorem \[16322110\] $ I_{i}^{x} $ is an isomorphism of the tvs’s $ {\left\langle \left({\mathfrak{E}}_{i}\right)_{x},{\mathfrak{N}}_{i}^{x}{\right\rangle}} $ and $ I_{i}^{x} \left( \left({\mathfrak{E}}_{i}\right)_{x} \right) $ as subspace of $ {\left\langle {\mathbf{E}}_{x}^{\oplus},{\mathfrak{n}}_{x}^{\oplus}{\right\rangle}} $. Moreover [@gie 1.5.III] and [@gie 1.6.viii] [^12] we deduce that $ \{ \sigma(x)\mid \sigma\in\Gamma(\pi_{i}) \} $ is dense in $ {\left\langle \left({\mathfrak{E}}_{i}\right)_{x},{\mathfrak{N}}_{i}^{x}{\right\rangle}} $. Therefore $\forall i=1,...,n$ and $\forall x\in X$ $$\label{22170111} \{ I_{i}^{x}(\sigma(x))\mid \sigma\in\Gamma(\pi_{i}) \} \text{ is dense in $ I_{i}^{x} \left( \left({\mathfrak{E}}_{i}\right)_{x} \right) $. }$$ where $ I_{i}^{x} \left( \left({\mathfrak{E}}_{i}\right)_{x} \right) $ has to be intended as topological vector subspace of $ {\left\langle {\mathbf{E}}_{x}^{\oplus},{\mathfrak{n}}_{x}^{\oplus}{\right\rangle}} $. So by the continuity of the sum on $ {\left\langle {\mathbf{E}}_{x}^{\oplus},{\mathfrak{n}}_{x}^{\oplus}{\right\rangle}} $ and the fact that ${\mathbf{E}}_{x}^{\oplus}$ is generated as linear space by the set $ \bigcup_{i=1}^{n} I_{i}^{x} \left( \left( {\mathfrak{E}}_{i}\right)_{x}\right) $ we can state $\forall x\in X$ that $$\label{22050111} \{ F(x) \mid F \in {\mathcal{E}}^{\oplus} \} \text{ is dense in $ {\left\langle {\mathbf{E}}_{x}^{\oplus},{\mathfrak{n}}_{x}^{\oplus}{\right\rangle}}$. }$$ Namely by $$(\forall v\in{\mathfrak{E}}^{\oplus}) (\forall i=1,...,n) (\exists\, \{ \sigma_{\alpha_{i}} \}_{\alpha_{i}\in D_{i}} \text{ net } \subset \Gamma(\pi_{i}) )$$ such that $$\begin{aligned} v & = \sum_{i=1}^{n}I_{i}^{x}(\Pr_{i}^{x}(v)) = \sum_{i=1}^{n} \lim_{\alpha_{i}\in D_{i}} I_{i}^{x}(\sigma_{\alpha_{i}}(x)) \\ & = \sum_{i=1}^{n} \lim_{\alpha\in D}w_{\alpha}^{i}(x) = \lim_{\alpha\in D} \sum_{i=1}^{n} w_{\alpha}^{i}(x) \\ & = \lim_{\alpha\in D} \sum_{i=1}^{n} I_{i}^{x}(\sigma_{\alpha(i)}(x)), \end{aligned}$$ where $ D \doteqdot \prod_{i=1}^{n} D_{i} $ while $ w_{\alpha}^{i}(x) \doteqdot I_{i}^{x}(\sigma_{\alpha(i)}(x)) $ for all $\alpha\in D$. Moreover $\forall\alpha\in D$ $$\left( X\ni x\mapsto \sum_{i=1}^{n} I_{i}^{x}(\sigma_{\alpha(i)}(x)) \right) \in {\mathcal{E}}^{\oplus}$$ then and $FM(3)$ follow. Finally $FM(4)$ follows by [@gie 1.6.iii] applied to any $\sigma_{i}\in\Gamma(\pi_{i})$ for all $i=1,...,n$ indeed $\forall\sigma_{i}\in\Gamma(\pi_{i})$ $$\hat{ \nu_{i,\rho_{i}}^{x} } (\tilde{\sigma}_{i}(x)) = \nu_{i,\rho_{i}}^{x} \circ \Pr_{i}^{x} \circ I_{i}^{x} \circ \sigma_{i}(x) = \nu_{i,\rho_{i}}^{x} \circ \sigma_{i}(x).$$ \[21262110\] By Definition \[17471910Ba\] and $$\bigoplus_{i=1}^{n} {\mathfrak{V}}_{i} = {\left\langle {\left\langle {\mathfrak{E}}({\mathbf{E}}^{\oplus}),\tau({\mathbf{E}}^{\oplus},{\mathcal{E}}^{\oplus}){\right\rangle}},\pi_{{\mathbf{E}}^{\oplus}},X,{\mathfrak{n}}^{\oplus}{\right\rangle}}$$ where 1. $ {\mathfrak{E}}({\mathbf{E}}^{\oplus}) \doteqdot \bigcup_{x\in X} \{x\} \times {\mathbf{E}}_{x}^{\oplus} $, $ \pi_{{\mathbf{E}}^{\oplus}}: {\mathfrak{E}}({\mathbf{E}}^{\oplus}) \ni (x,v) \mapsto x \in X $. 2. $ {\mathfrak{n}}^{\oplus} = \{ \hat{\mu}_{\rho}: \mid \rho \in \prod_{i=1}^{n} L_{i} \} $, with $ \hat{\mu}_{\rho}: {\mathfrak{E}}({\mathbf{E}}^{\oplus}) \ni (x,v) \mapsto \hat{\mu}_{\rho}^{x}(v) $; 3. $\tau({\mathbf{E}}^{\oplus},{\mathcal{E}}^{\oplus})$ is the topology on ${\mathfrak{E}}({\mathbf{E}}^{\oplus})$ such that for all $ (x,v)\in {\mathfrak{E}}({\mathbf{E}}^{\oplus}) $ $${\mathcal{I}}_{(x,v)}^{{\mathfrak{E}}({\mathbf{E}}^{\oplus})} \doteqdot {\mathfrak{F}_{{\mathcal{B}}((x,v))}^{{\mathfrak{E}}({\mathbf{E}}^{\oplus})}}$$ is the neighbourhood’s filter of $(x,v)$ with respect to it. Here ${\mathfrak{F}_{{\mathcal{B}}((x,v))}^{{\mathfrak{E}}({\mathbf{E}}^{\oplus})}}$ is the filter on ${\mathfrak{E}}({\mathbf{E}}^{\oplus})$ generated by the following filter’s base $$\begin{aligned} {1} {\mathcal{B}}^{\oplus}((x,v)) \doteqdot \{ T_{{\mathbf{E}}^{\oplus}}(U,\sigma,{\varepsilon},\rho) & \mid U\in Open(X), \sigma\in{\mathcal{E}}^{\oplus}, {\varepsilon}>0, \rho \in \prod_{i=1}^{n} L_{i} \\ & \mid x\in U, \hat{\mu}_{\rho}^{x}(v-\sigma(x)) <{\varepsilon}\},\end{aligned}$$ where $$T_{{\mathbf{E}}^{\oplus}}(U,\sigma,{\varepsilon},\rho) \doteqdot \left\{ (y,w)\in{\mathfrak{E}}({\mathbf{E}}^{\oplus}) \mid y\in U, \hat{\mu}_{\rho}^{y}(w-\sigma(y)) <{\varepsilon}\right\};$$ Finally the following is a useful characterization of continuous maps valued in ${\mathfrak{E}}({\mathbf{E}}^{\oplus})$. \[17571212\] Let $ \{ {\mathfrak{V}}_{i} \doteqdot {\left\langle {\left\langle {\mathfrak{E}}_{i},\tau_{i}{\right\rangle}},\pi_{i},X,{\mathfrak{N}}_{i}{\right\rangle}} \}_{i=1}^{n} $ be a family of bunldes of $\Omega-$spaces, $f\in{\mathfrak{E}}({\mathbf{E}}^{\oplus})^{X}$ and $x\in X$. Then $f$ is continuous in $x$ if and only if $f_{0}^{i}:X\to{\mathfrak{E}}_{i}$ is continuous in $x$ for all $i=1,...,n$, where $ f_{0} \in \left( \bigcup_{z\in X} {\mathbf{E}}_{z}^{\oplus} \right)^{X} $ such that $\forall z\in X$ $ f(z)=(z,f_{0}(z)) $ and $$f_{0}^{i}(z) \doteqdot \Pr_{i}^{\pi_{{\mathbf{E}}^{\oplus}} (f(z))} \circ f_{0}(z).$$ In particular $f\in\Gamma(\pi_{{\mathbf{E}}^{\oplus}})$ if and only if $\left( X\ni z\mapsto \Pr_{i}^{z}\circ f_{0}(z) \in({\mathfrak{E}}_{i})_{z} \right) \in\Gamma(\pi_{i})$, for all $i=1,...,n$. By $(1) \Leftrightarrow (5)$ in Theorem \[15380512\] applied to $\bigcup_{i=1}^{n} \tilde{\Gamma}(\pi_{i})$. \[16392601\] Let $ \{ {\mathfrak{V}}_{i} \doteqdot {\left\langle {\left\langle {\mathfrak{E}}_{i},\tau_{i}{\right\rangle}},\pi_{i},X,{\mathfrak{N}}_{i}{\right\rangle}} \}_{i=1}^{n} $ be a family of bunldes of $\Omega-$spaces, and ${\mathfrak{V}}({\mathbf{E}}^{\oplus},{\mathcal{E}}^{\oplus})$ the bundle direct sum of the family $\{ {\mathfrak{V}}_{i} \}_{i=1}^{n}$. By construction we have that $\Gamma(\pi_{{\mathbf{E}}^{\oplus}}) \subset \prod_{x\in X} \{x\}\times {\mathbf{E}}_{x}^{\oplus}$. In what follows, except contrary mention, we convein to consider with abuse of language in the obvious manner $$\Gamma(\pi_{{\mathbf{E}}^{\oplus}}) \subset \prod_{x\in X} \bigoplus_{i=1}^{n}\left({\mathfrak{E}}_{i}\right)_{x}.$$ Similarly for $\Gamma^{x}(\pi_{{\mathbf{E}}^{\oplus}})$ for any $x\in X$. Moreover in the case in which for any $i=1,...,n$ we have ${\mathfrak{V}}_{i} ={\mathfrak{V}}({\mathbf{E_{i}}},{\mathcal{E_{i}}})$, with obvious meaning of the symbols we consider $$\Gamma(\pi_{{\mathbf{E}}^{\oplus}}) \subset \prod_{x\in X} \bigoplus_{i=1}^{n}\left({\mathbf{E}}_{i}\right)_{x}.$$ Main Claim ========== Map pre-bundle ${\mathcal{M}}$ relative to a map system --------------------------------------------------------- \[Map Systems\] \[17471910\] ${\left\langle X,{\mathbf{E}},{\mathcal{S}}{\right\rangle}}$ is a *Map System* if 1. $X$ is a set; 2. $ {\mathbf{E}} = \{ {\left\langle {\mathbf{E}}_{x},{\mathfrak{N}}_{x}{\right\rangle}} \}_{x\in X} $ is a nice family of Hlcs with $ {\mathfrak{N}}_{x} \doteqdot \{\nu^{x}_{j}\mid j\in J\} $ for all $x\in X$; 3. $ (\exists L\ne\emptyset) ({\mathcal{S}} = \{S_{x}\}_{x\in X}) $ where $ S_{x} \doteqdot \{ B_{l}^{x} \mid l\in L \} \subseteq Bounded({\mathbf{E}}_{x}) $ and $ \bigcup_{l\in L} B_{l}^{x} $ is total in ${\mathbf{E}}_{x}$ for all $x\in X$. \[ Map pre-bundle \] \[17471910A\] We say that $ {\mathbf{M}} $ is a *Map pre-bundle relative to ${\left\langle X,Y,{\mathbf{E}},{\mathcal{S}}{\right\rangle}}$* if 1. ${\left\langle X,{\mathbf{E}},{\mathcal{S}}{\right\rangle}}$ is a map system; 2. $ {\mathbf{M}} = \{ {\left\langle {\mathbf{M}}_{x},{\mathfrak{R}}_{x}{\right\rangle}} \}_{x\in X} $ is a nice family of Hlcs; 3. $Y$ is a Hausdorff topological space and $\forall x\in X$ $$\boxed{ \begin{aligned} {\mathbf{M}}_{x} & \subseteq {\mathcal{C}_{c} \left(Y,{\mathcal{L}}_{S_{x}}({\mathbf{E}}_{x})\right)}; \\ {\mathfrak{R}}_{x} & = \left\{ \sup_{(K,j,l)\in{\mathcal{O}}} q_{(K,j,l)}^{x} {\upharpoonright}{\mathbf{M}}_{x} \mid {\mathcal{O}}\in{\mathcal{P}}_{\omega}\left(Comp(Y) \times J\times L\right) \right\}. \end{aligned} }$$ Here we recall that ${\mathcal{P}}_{\omega}(A)$ is the class of all finite parts of the set $A$, ${\mathcal{L}}_{S_{x}}({\mathbf{E}}_{x})$, for all $x\in X$, is the $lcs$ of all continuous linear maps ${\mathcal{L}}({\mathbf{E}}_{x})$ on ${\mathbf{E}}_{x}$ with the topology of uniform convergence over the sets in $S_{x}$, hence its topology is generated by the following set of seminorms $$\label{21532606} \left\{ p_{j,l}^{x}: {\mathcal{L}}({\mathbf{E}}_{x}) \ni \phi \mapsto \sup_{v\in B_{l}^{x}} \nu_{j}^{x}(\phi(v)) \mid l\in L, j\in J \right\}.$$ Thus by the totality hypothesis and by [@BourTVS Prop. $3$, $III.15$] ${\mathcal{L}}_{S_{x}}({\mathbf{E}}_{x})$ is Hausdorff. Finally for all $(K,j,l) \in Comp(Y) \times J \times L$ we set $$\label{21552606} q_{(K,j,l)}^{x}: {\mathcal{C}_{c} \left(Y,{\mathcal{L}}_{S_{x}}({\mathbf{E}}_{x})\right)} \ni f \mapsto \sup_{t\in K} p_{j,l}^{x} (f(t))$$ By the fact that $\{t\}$ is compact for all $t\in Y$ we have that $\bigcup_{K\in Comp(Y)}K=Y$ thus by the shown fact that ${\mathcal{L}}_{S_{x}}({\mathbf{E}}_{x})$ is Hausdorff we deduce by [@BourGT Prp. $(1)$, $\S 1.2$, Ch $10$] that ${\mathcal{C}_{c} \left(Y,{\mathcal{L}}_{S_{x}}({\mathbf{E}}_{x})\right)}$ is Hausdorff. Moreover by [@BourGT Def. $(1)$, $\S 1.1$, Ch $10$] and by the fact that is a $f.s.s.$ on ${\mathcal{L}}_{S_{x}}({\mathbf{E}}_{x})$, we can deduce that $\left\{ \sup_{(K,j,l)\in{\mathcal{O}}} q_{(K,j,l)}^{x} \mid {\mathcal{O}} \in{\mathcal{P}}_{\omega} \left( Comp(Y) \times J \times L \right) \right\}$ is a *direct* $f.s.s.$ on ${\mathcal{C}_{c} \left(Y,{\mathcal{L}}_{S_{x}}({\mathbf{E}}_{x})\right)}$. Hence ${\left\langle {\mathbf{M}}_{x},{\mathfrak{R}}_{x}{\right\rangle}}$ is a topological vector subspace of ${\mathcal{C}_{c} \left(Y,{\mathcal{L}}_{S_{x}}({\mathbf{E}}_{x})\right)}$ so it is Hausdorff, hence by the construction of ${\mathfrak{R}}_{x}$ we can state that $\{ {\left\langle {\mathbf{M}}_{x},{\mathfrak{R}}_{x}{\right\rangle}} \}_{x\in X}$ is a nice family of Hlcs in agreement with request $(2)$ in Def. \[17471910A\]. For the sake of completeness and the large use of it in all the work, we shall use Definition \[17471910Ba\] for giving in the following remark the explicit form of ${\mathfrak{V}}({\mathbf{M}},{\mathcal{M}})$. \[17471910B\] Let $ {\mathbf{M}} = \{ {\left\langle {\mathbf{M}}_{x},{\mathfrak{R}}_{x}{\right\rangle}} \}_{x\in X} $ be a map pre-bundle relative to $ {\left\langle X,Y,{\mathbf{E}}={\left\langle {\mathbf{E}}_{x},{\mathfrak{N}}_{x}{\right\rangle}}_{x\in X}, {\mathcal{S}}{\right\rangle}} $, moreover let ${\mathcal{M}}$ satisfy $FM(3)-FM(4)$ with respect to $ {\mathbf{M}} $. Let us denote $ {\mathfrak{N}}_{x} = \{ \nu_{j}^{x} \mid j\in J \} $ for all $x\in X$ and use the notations in Definition \[17471910A\]. Thus for the bundle ${\mathfrak{V}}({\mathbf{M}},{\mathcal{M}})$ generated by the couple ${\left\langle {\mathbf{M}},{\mathcal{M}}{\right\rangle}}$ we have 1. $ {\mathfrak{V}}({\mathbf{M}},{\mathcal{M}}) = {\left\langle {\left\langle {\mathfrak{E}}({\mathbf{M}}),\tau({\mathbf{M}},{\mathcal{M}}){\right\rangle}} ,\pi_{{\mathbf{M}}},X,{\mathfrak{R}}{\right\rangle}} $; 2. $ {\mathfrak{E}}({\mathbf{M}}) \doteqdot \bigcup_{x\in X} \{x\} \times {\mathbf{M}}_{x} $, $ \pi_{{\mathbf{M}}} :{\mathfrak{E}}({\mathbf{M}}) \ni (x,f) \mapsto x \in X $; 3. $ {\mathfrak{R}} = \left\{ \sup_{(K,j,l)\in{\mathcal{O}}} q_{(K,j,l)} \mid {\mathcal{O}} \in{\mathcal{P}}_{\omega} \left( Comp(Y) \times J \times L \right) \right\}$, with $ q_{(K,j,l)}: {\mathfrak{E}}({\mathbf{M}}) \ni (x,f) \mapsto q_{(K,j,l)}^{x}(f)$; 4. $\tau({\mathbf{M}},{\mathcal{M}})$ is the topology on $ {\mathfrak{E}}({\mathbf{M}}) $ such that for all $(x,f)\in{\mathfrak{E}}({\mathbf{M}})$ $${\mathcal{I}}_{(x,f)}^{{\mathfrak{E}}({\mathbf{M}})} \doteqdot {\mathfrak{F}_{{\mathcal{B}}_{{\mathbf{M}}}((x,f))}^{{\mathfrak{E}}({\mathbf{M}})}}$$ is the neighbourhood’s filter of $(x,f)$ with respect to it. Here ${\mathfrak{F}_{{\mathcal{B}}_{{\mathbf{M}}}((x,f))}^{{\mathfrak{E}}({\mathbf{M}})}}$ is the filter on ${\mathfrak{E}}({\mathbf{M}})$ generated by the following filter’s base $$\begin{aligned} {1} & {\mathcal{B}}_{{\mathbf{M}}}((x,f)) \doteqdot \{ T_{{\mathbf{M}}} \left( U,\sigma,{\varepsilon}, {\mathcal{O}} \right) \mid U\in Open(X), \sigma\in{\mathcal{M}}, {\varepsilon}>0,\\ & {\mathcal{O}} \in{\mathcal{P}}_{\omega} \left( Comp(Y) \times J \times L \right) \mid x\in U, \sup_{(K,j,l)\in{\mathcal{O}}} q_{(K,j,l)}^{x} (f-\sigma(x)) <{\varepsilon}\},\end{aligned}$$ where $ \forall U\in Open(X), \sigma\in{\mathcal{M}}, {\varepsilon}>0 $ and $\forall{\mathcal{O}} \in{\mathcal{P}}_{\omega} \left( Comp(Y) \times J \times L \right)$ $$T_{{\mathbf{M}}} \left( U,\sigma,{\varepsilon}, {\mathcal{O}} \right) \doteqdot \left\{ (y,g)\in{\mathfrak{E}}({\mathbf{M}}) \mid y\in U, \sup_{(K,j,l)\in{\mathcal{O}}} q_{(K,j,l)}^{y} (g-\sigma(y)) <{\varepsilon}\right\}.$$ In Remark \[17471910B\] we gave explicitly the neighbourhood’ filters for the topology $\tau({\mathbf{M}},{\mathcal{M}})$. A simpler form for this filter can be obtained with additional requirement as we showed in Corollary \[14272310\]. \[18070512\] Let $ {\mathbf{M}} = \{ {\left\langle {\mathbf{M}}_{x},{\mathfrak{R}}_{x}{\right\rangle}} \}_{x\in X} $ be a map pre-bundle relative to $ {\left\langle X,Y,{\mathbf{E}}={\left\langle {\mathbf{E}}_{x},{\mathfrak{N}}_{x}{\right\rangle}}_{x\in X}, {\mathcal{S}}{\right\rangle}} $, moreover let ${\mathcal{M}}$ satisfy $FM(3)-FM(4)$ with respect to $ {\mathbf{M}} $. Thus by Remark \[15412610\] $ \forall U\in Open(X), \sigma\in{\mathcal{M}}, {\varepsilon}>0 $ and $\forall {\mathcal{O}}\in{\mathcal{P}}_{\omega}\left(Comp(Y) \times J\times L\right)$ $$T_{{\mathbf{M}}} (U,\sigma,{\varepsilon}, {\mathcal{O}}) = \bigcup_{y\in U} B_{{\mathbf{M}}_{y}, {\mathcal{O}},{\varepsilon}}(\sigma(y))$$ where for all $ s \in {\mathbf{M}}_{y} $ $$B_{{\mathbf{M}}_{y}, {\mathcal{O}},{\varepsilon}}(s) \doteqdot \left\{ (y,f)\in{\mathfrak{E}}({\mathbf{M}})_{y} \mid \sup_{(K,j,l)\in{\mathcal{O}}} q_{(K,j,l)}^{y} \left(f-s\right) <{\varepsilon}\right\}.$$ By applying Remark \[17150312\] we have the following \[16422010\] Let ${\mathbf{M}}$ be a map pre-bundle relative to $ {\left\langle X,Y,{\mathbf{E}},{\mathcal{S}}{\right\rangle}} $, moreover let ${\mathcal{M}}$ satisfy $FM(3)-FM(4)$ with respect to ${\mathbf{M}}$. Thus 1. ${\mathfrak{V}}({\mathbf{M}},{\mathcal{M}})$ is a bundle of $\Omega-$spaces; 2. with the notations of Definition \[17471910B\] ${\mathfrak{V}}({\mathbf{M}},{\mathcal{M}})$ is such that 1. ${\left\langle {\mathfrak{E}}({\mathbf{M}})_{x},\tau({\mathbf{M}},{\mathcal{M}}){\right\rangle}}$ as topological vector space is isomorphic to ${\left\langle {\mathbf{M}}_{x},{\mathfrak{R}}_{x}{\right\rangle}}$ for all $x\in X$; 2. ${\mathcal{M}}$ is canonically isomorphic with a linear subspace of $\Gamma(\pi_{{\mathbf{M}}})$ and if $X$ is compact $ {\mathcal{M}} \simeq \Gamma(\pi_{{\mathbf{M}}}) $. $\left(\Theta,{\mathcal{E}}\right)-$structures General case {#01121953} ------------------------------------------------------------- In Definition \[10282712\] we show how to generalize the topology of uniform convergence to the case of a bundle ${\left\langle {\mathfrak{M}},\rho,X{\right\rangle}}$ of $\Omega-$spaces, such that $\{{\mathfrak{M}}_{x}\}_{x\in X}$ is a map pre-bundle relative to ${\left\langle X,Y,\{{\mathfrak{E}}_{x}\}_{x\in X},{\mathcal{S}}{\right\rangle}}$, where ${\left\langle {\mathfrak{E}},\pi,X{\right\rangle}}$ is a bundle ${\left\langle {\mathfrak{M}},\rho,X{\right\rangle}}$ of $\Omega-$spaces. The aim is to correlate the topology on ${\mathfrak{M}}$ with that on ${\mathfrak{E}}$ in order to generalize the correlation established in the introduction for the trivial bundle case. \[15531102\] $$(\bullet): \prod_{x\in X}({\mathfrak{E}}_{x})^{{\mathfrak{E}}_{x}} \times \prod_{x\in X}{\mathfrak{E}}_{x} \to \prod_{x\in X}{\mathfrak{E}}_{x}$$ such that for all $F\in \prod_{x\in X}({\mathfrak{E}}_{x})^{{\mathfrak{E}}_{x}}, v\in\prod_{x\in X}{\mathfrak{E}}_{x}$ we have $$(F\bullet w)(x) \doteqdot F(x)(w(x)).$$ \[ **$\left(\Theta,{\mathcal{E}}\right)-$structures**\] \[10282712\] We say that ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ is a *$\left(\Theta,{\mathcal{E}}\right)-$structure* if 1. $ {\mathfrak{V}} \doteqdot {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}} $ is a bundle of $\Omega-$spaces; 2. ${\mathcal{E}}\subseteq\Gamma(\pi)$; 3. $\Theta\subseteq \prod_{x\in X} Bounded({\mathfrak{E}}_{x})$; 4. $\forall B\in\Theta$ 1. ${\mathbf{D}}(B,{\mathcal{E}})\ne\emptyset$; 2. $\bigcup_{B\in\Theta}{\mathcal{B}}_{B}^{x}$ is total in ${\mathfrak{E}}_{x}$ for all $x\in X$; 5. ${\mathfrak{W}} \doteqdot {\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$ is a bundle of $\Omega-$spaces such that $\{{\left\langle {\mathfrak{M}}_{x},{\mathfrak{R}}_{x}{\right\rangle}}\}_{x\in X}$ is a map pre-bundle relative to ${\left\langle X,Y,\{{\left\langle {\mathfrak{E}}_{x},{\mathfrak{N}}_{x}{\right\rangle}} \}_{x\in X},{\mathcal{S}}{\right\rangle}}$. Here ${\mathcal{S}} \doteqdot \{S_{x}\}_{x\in X}$ and $(\forall B\in\Theta)(\forall x\in X)$ $$\label{11232712} \boxed{ \begin{cases} {\mathbf{D}}(B,{\mathcal{E}}) \doteqdot {\mathcal{E}} \cap \left(\prod_{x\in X}B_{x}\right) \\ {\mathcal{B}}_{B}^{x} \doteqdot \{v(x)\mid v\in{\mathbf{D}}(B,{\mathcal{E}})\} \} \\ S_{x} \doteqdot \{{\mathcal{B}}_{B}^{x} \mid B\in\Theta\}. \end{cases} }$$ Moreover ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ is an *invariant $\left(\Theta,{\mathcal{E}}\right)-$ structure* if it is a $\left(\Theta,{\mathcal{E}}\right)-$ structure such that $$\label{18112502pre} \left\{ F\in\prod_{z\in X}^{b} {\mathfrak{M}}_{z} \mid (\forall t\in Y) (F_{t} \bullet {\mathcal{E}}(\Theta) \subseteq \Gamma(\pi)) \right\} = \Gamma(\rho).$$ Finally ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ is a *compatible $\left(\Theta,{\mathcal{E}}\right)-$structure* if it is a $\left(\Theta,{\mathcal{E}}\right)-$structure such that for all $t\in Y$ $$\label{18011202} \Gamma(\rho)_{t} \bullet {\mathcal{E}}(\Theta) \subseteq \Gamma(\pi).$$ Here $${\mathcal{E}}(\Theta) \doteqdot \bigcup_{B\in\Theta} {\mathbf{D}}(B,{\mathcal{E}}),$$ and $S_{t} \doteqdot \{ F_{t}\mid F\in S \}$ and $F_{t} \in\prod_{x\in X} {\mathcal{L}}({\mathfrak{E}}_{x})$ such that $F_{t}(x)\doteqdot F(x)(t)$, for all $S\subseteq\prod_{x\in X} {\mathcal{L}}({\mathfrak{E}}_{x})^{Y}$ $t\in Y$, and $F\in S$. \[22442606\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure. Then for all $x\in X$ $$\label{15073006} {\mathfrak{R}}_{x}= \left\{ \sup_{(K,j,B)\in O} q_{(K,j,B)}^{x}{\upharpoonright}{\mathfrak{M}}_{x} \mid O\in{\mathcal{P}}_{\omega} \left(Comp(Y)\times J\times \Theta\right) \right\}$$ where by using the notations of Def. \[10282712\] we set ${\mathfrak{N}}=\{\nu_{j}^{x}\mid j\in J\}$ and for all $K\in Comp(Y)$,$ j\in J$, $B\in\Theta$ $$\label{22542906} \boxed{ q_{(K,j,B)}^{x}: {\mathcal{C}_{c} \left(Y,{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})\right)} \ni f_{x} \mapsto \sup_{t\in K}\sup_{v\in{\mathbf{D}}(B,{\mathcal{E}})} \nu_{j}^{x}\left(f_{x}(t)v(x)\right) }$$ \[21022406\] Let ${\mathfrak{V}} \doteqdot {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}$ be a bundle, $ {\mathbf{M}} = \{ {\left\langle {\mathbf{M}}_{x},{\mathfrak{R}}_{x}{\right\rangle}} \}_{x\in X} $ a map pre-bundle relative to ${\left\langle X,Y,\{{\left\langle {\mathfrak{E}}_{x},{\mathfrak{N}}_{x}{\right\rangle}} \}_{x\in X},{\mathcal{S}}{\right\rangle}}$ and ${\mathcal{M}}$ satisfy $FM(3)-FM(4)$ with respect to $ {\mathbf{M}} $. Then Rmk. \[17471910B\] allows us to construct ${\mathfrak{W}}$ satisfying the condition $(5)$ in Def. \[10282712\]. About the problem of satisfying $FM(3)-FM(4)$ recall the very important Rmk. \[21132406\]. The following characterization of ${\mathcal{U}}\in\Gamma_{U}^{x_{\infty}}(\rho)$ will be very important in the sequel. \[15482712\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure, $x_{\infty}\in W\subseteq X$ and ${\mathcal{U}}\in\prod_{x\in W}^{b}{\mathfrak{M}}_{x}$. By using the notations in Definition \[10282712\] we have $ (1) \Leftarrow (2) \Leftarrow (3) \Leftrightarrow (4) $ moreover if ${\mathfrak{W}}$ is locally full $(1) \Leftrightarrow (2) \Leftrightarrow (3) \Leftrightarrow (4) $, finally if ${\mathfrak{W}}$ is full we can choose $U=X$ in $(2)$ and $U'=X$ in $(3)$ and $(4)$. Here 1. ${\mathcal{U}}\in\Gamma_{W}^{x_{\infty}}(\rho)$; 2. $ (\exists\,U\in Op(X)\mid U\ni x_{\infty}) (\exists\,F\in\Gamma_{U}(\rho)) (F(x_{\infty}) ={\mathcal{U}}(x_{\infty})) $ such that $ (\forall j\in J) (\forall K\in Compact(Y)) (\forall B\in\Theta) $ $$\label{02022912} \boxed{ \lim_{z\to x_{\infty},z\in W\cap U} \sup_{t\in K} \sup_{v\in{\mathbf{D}}(B,{\mathcal{E}})} \nu_{j}\left({\mathcal{U}}(z)(t)v(z)-F(z)(t)v(z) \right)=0;}$$ 3. $ (\exists\,U'\in Op(X)\mid U'\ni x_{\infty}) (\exists\,F\in\Gamma_{U'}(\rho)) (F(x_{\infty}) ={\mathcal{U}}(x_{\infty})) $ and $ (\forall U\in Op(X)\mid U\ni x_{\infty}) (\forall\,F\in\Gamma_{U}(\rho) \mid F(x_{\infty}) ={\mathcal{U}}(x_{\infty})) $ we have $(\forall j\in J) (\forall K\in Compact(Y)) (\forall B\in\Theta)$; 4. $ (\exists\,U'\in Op(X)\mid U'\ni x_{\infty}) (\exists\,F\in\Gamma_{U'}(\rho)) (F(x_{\infty}) ={\mathcal{U}}(x_{\infty})) $ and ${\mathcal{U}}\in\Gamma_{W}^{x_{\infty}}(\rho)$. Apply Corollary \[28111707\] to Definition \[17471910A\]. \[11121102\] Let us assume the hypotheses of Lemma \[15482712\] and that ${\mathfrak{W}}$ is full. Moreover let $B\in\Theta$ and $v\in {\mathbf{D}}(B,{\mathcal{E}})$. Then $(1) \Rightarrow (2)$, where 1. ${\mathcal{U}}\in\Gamma_{W}^{x_{\infty}}(\rho)$ and $\exists\,F\in\Gamma(\rho)$ such that $F(x_{\infty}) ={\mathcal{U}}(x_{\infty})$ and $(\forall t\in Y) (F(\cdot)(t) \bullet v\in\Gamma(\pi))$; 2. $(\forall t\in X) ({\mathcal{U}}(\cdot)(t)\bullet v\in \Gamma_{W}^{\infty}(\pi))$. By the position $(1)$ and by the implication $(1) \Rightarrow (3)$ of Lemma \[15482712\] and by the fact that the union of all compact subsets of $Y$ is $Y$, being locally compact, we deduce that $(\exists\,F\in\Gamma(\rho)) (F(x_{\infty}) ={\mathcal{U}}(x_{\infty}))$ such that $(\forall j\in J)(\forall t\in Y) (\forall B\in\Theta)$ and $\forall v\in{\mathbf{D}}(B,{\mathcal{E}})$ $$\begin{cases} \lim_{z\to x_{\infty},z\in W} \nu_{j} \left( {\mathcal{U}}(z)(t)v(z)-F(z)(t)v(z) \right) =0, \\ F(\cdot)(t)\bullet v \in\Gamma(\pi). \end{cases}$$ Thus the statement follows by implication $(3)\Rightarrow(1)$ of Corollary \[28111707\]. In general condition $(1)$ is much more stronger than $(2)$. Let us conclude this section with two results constructing a $\left(\Theta,{\mathcal{E}}\right)-$structure and describing $\Gamma^{x_{\infty}}(\rho)$ when ${\mathfrak{V}}$ is trivial. \[22312406\] Let $Z$ be a normed space $X,Y$ be two topological spaces. Set for all $x\in X$ and $v\in{\mathcal{C}_{b} \left(X,Z\right)}$ $$\begin{cases} \begin{aligned} {\mathcal{M}} \doteqdot & \{ F\in{\mathcal{C}_{b} \left(X,{\mathcal{C}_{c} \left(Y,{\mathcal{L}}_{s}(Z)\right)}\right)} \mid (\forall K\in Comp(Y)) \\ & (C(F,K)\doteqdot \sup_{(x,s)\in X\times K} \|F(x)(s)\|_{B(Z)}<\infty) \}, \end{aligned} \\ {\mathbf{M}}_{x} \doteqdot{\overline}{\{ F(x)\mid F\in{\mathcal{M}} \}}, \\ \mu_{(v,x)}^{K}: {\mathbf{M}}_{x}\ni G\mapsto \sup_{s\in K} \|G(s)v(x)\|, \\ {\mathcal{A}}_{x} \doteqdot\{\mu_{(w,x)}^{K} \mid K\in Comp(Y), w\in{\mathcal{C}_{b} \left(X,Z\right)} \}, \\ {\mathbf{M}} \doteqdot\{ {\left\langle {\mathbf{M}}_{x},{\mathcal{A}}_{x}{\right\rangle}}\}_{x\in X}. \end{cases}$$ closure in ${\mathcal{C}_{c} \left(Y,B_{s}(Z)\right)}$. Then ${\mathcal{M}}$ satisfies $FM3-FM4$ with respect to ${\mathbf{M}}$ $FM(3)$ is true by construction, let $v\in{\mathcal{C}_{b} \left(X,Z\right)}$, $K\in Comp(Y)$, $F\in{\mathcal{M}}$, then $$\sup_{x\in X} \mu_{(v,x)}^{K}(F(x)) \leq \sup_{(x,s)\in X\times K} \|F(x)(s)\|_{B(Z)} \sup_{x\in X} \|v(x)\| <\infty.$$ For all $x,x_{0}\in X$ $$\label{15472506} \mu_{(v,x)}^{K}(F(x)) \leq C \|v(x)-v(x_{0})\| + \sup_{s\in K} \|F(x)(s)v(x_{0})\|,$$ where $C \doteqdot \sup_{(x,s)\in X\times K} \|F(x)(s)\|_{B(Z)}$. Moreover the map ${\mathcal{C}_{c} \left(Y,B_{s}(Z)\right)} \ni f \mapsto \sup_{s\in K}\|f(s)w\| \in {\mathbb {R}}^{+}$, for all $w\in Z$ is a continuous seminorm, hence by the continuity of $F$ also the map $X\ni x\mapsto \sup_{s\in K}\|F(x)(s)w\| \in {\mathbb {R}}^{+}$ is continuous. So by $\varlimsup_{x\to x_{0}} \mu_{(v,x)}^{K}(F(x)) \leq \sup_{s\in K} \|F(x_{0})(s)v(x_{0})\| = \mu_{(v,x_{0})}^{K}(F(x_{0}))$, and by [@BourGT $(15)$, $\S 5.6$] we have $$\varlimsup_{x\to x_{0}} \mu_{(v,x)}^{K}(F(x)) = \mu_{(v,x_{0})}^{K}(F(x_{0})).$$ Therefore by [@BourGT $(13)$, $\S 5.6$], [@BourGT Prp. $3$, $\S 6.2$], and the fact that any map $g$ is $u.s.c.$ at a point iff $-g$ is $l.s.c.$, we can state that $X\ni x\mapsto \mu_{(v,x)}^{K}(F(x))$ is $u.s.c.$ at $x_{0}$ for all $x_{0}\in X$, hence it is $u.s.c.$, which is the $FM(4)$ condition. By Lemma \[22312406\] and Def. \[17471910Ba\] we can construct the bundle ${\mathfrak{V}}({\mathbf{M}},{\mathcal{M}})$ generated by the couple ${\left\langle {\mathbf{M}},{\mathcal{M}}{\right\rangle}}$. In the following result I will construct a $\left(\Theta,{\mathcal{E}}\right)-$structure and describe a large subclass of $\Gamma^{x_{\infty}}(\rho)$. \[22372406\] Let us assume the notations and hypotheses of Lemma \[22312406\], let ${\mathfrak{V}}$ be the trivial Banach bundle with constant stalk $Z$ and set $\Theta \doteqdot \left\{ B_{v} \mid v\in{\mathcal{C}_{b} \left(X,Z\right)} \right\}$. Then 1. ${\left\langle {\mathfrak{V}},{\mathfrak{V}}({\mathbf{M}},{\mathcal{M}}),X,Y{\right\rangle}}$ is a $(\Theta,{\mathcal{C}_{b} \left(X,Z\right)})-$ structure, moreover if $X$ is compact and $Y$ is locally compact then it is compatible; 2. Let $f\in\prod_{x\in\ X}{\mathbf{M}}_{x}$ be such that $\sup_{(x,s)\in X\times K} \|f(x)(s)\|_{B(Z)}<\infty$ for all $K\in Comp(Y)$ then $ (a) \Leftrightarrow (b) \Leftrightarrow (c) \Leftrightarrow (d)$, where 1. $f\in\Gamma^{x_{\infty}}(\pi_{{\mathbf{M}}})$; 2. $(\forall K\in Comp(Y)) (\forall v\in{\mathcal{C}_{b} \left(X,Z\right)})$ $$\lim_{x\to x_{\infty}} \sup_{s\in K} \| f(x)(s)v(x) - f(x_{\infty})(s)v(x) \| =0$$ 3. $f:X\to{\mathcal{C}_{c} \left(Y,B_{s}(Z)\right)}$ continuous at $x_{\infty}$; 4. $(\forall K\in Comp(Y)) (\forall w\in Z)$ $$\lim_{x\to x_{\infty}} \sup_{s\in K} \| f(x)(s)w - f(x_{\infty})(s)w \| =0.$$ By Rmk. \[21022406\] and Lemma \[22312406\] we have that $(5)$ of Def. \[10282712\] follows. $\Gamma(\pi)\simeq{\mathcal{C}_{b} \left(X,Z\right)}$ hence by Rmk. \[21500412b\] the others requests of Def. \[10282712\] follow. Thus the first sentence of statement $(1)$. If $X$ is compact by Lemma \[22312406\] and Rmk. \[17150312\] follows that ${\mathcal{M}}\simeq\Gamma(\pi_{{\mathbf{M}}})$, moreover by Rmk. \[21500412b\] we have ${\mathcal{E}}(\Theta)={\mathcal{E}}$ and finally ${\mathcal{E}}\doteq \Gamma(\pi)\simeq{\mathcal{C}_{b} \left(X,Z\right)} $. Hence the second sentence of statement $(1)$ follows if we show that ${\mathcal{M}}_{t} \bullet {\mathcal{C}_{b} \left(X,Z\right)} \subseteq {\mathcal{C}_{b} \left(X,Z\right)}$. To this end fix $v\in{\mathcal{C}_{b} \left(X,Z\right)}$, $F\in{\mathcal{M}}$, $s\in Y$ and $K_{s}$ a compact neighbourhood of $s$, which there exists by the hypothesis that $Y$ is locally compact. Then we have for all $x,x_{0}\in X$ $$\begin{aligned} \label{16102506} & \| F(x)(s)v(x)-F(x_{\infty})(s)v(x_{0}) \| \leq \\ & C(F,K_{s}) \|v(x)-v(x_{0})\| + \|\left(F(x)(s)- F(x_{0})(s) \right)v(x_{0})\| \end{aligned}$$ By considering that $F\in{\mathcal{C}_{b} \left(X,{\mathcal{C}_{c} \left(Y,B_{s}(Z)\right)}\right)}$ and that $s\in K_{s}$ we have that $\lim_{x\to x_{0}} \|(F(x)(s)-F(x_{0})(s))v(x_{0})\| =0$. Hence by we deduce that $F_{s}\bullet v$ is continuous at $x_{0}$, so continuous on $X$, in particular $X$ being compact it is also $\|\cdot\|_{Z}-$bounded. Thus $F_{s}\bullet v\in{\mathcal{C}_{b} \left(X,Z\right)}$ and the second sentence of the statement follows. Fix $f\in\prod_{x\in\ X}{\mathbf{M}}_{x}$ such that $(\forall K\in Comp(Y)) (\sup_{(x,s)\in X\times K} \|f(x)(s)\|_{B(Z)}<\infty)$. $(a)\Leftrightarrow(b)$ follows by Lemma \[15482712\], the fact that ${\mathcal{M}}\subseteq\Gamma(\pi_{{\mathbf{M}}})$ by Rmk. \[17150312\], and by $(H:X\ni x\mapsto f(x_{\infty})\in {\mathcal{C}_{c} \left(Y,B_{s}(Z)\right)})\in{\mathcal{M}}$, indeed $H$ it is bounded and continuous being constant, moreover $\sup_{(x,s)\in X\times K} \|H(x)(s)\|_{B(Z)} = \sup_{s\in K} \|f(x_{\infty})(s)\|_{B(Z)} <\infty$, for all $K\in Comp(Y)$. $(b)\Rightarrow(d)$ follows by the fact that $(X\ni x\mapsto w\in Z)\in{\mathcal{C}_{b} \left(X,Z\right)}$, and $(c)\Leftrightarrow(d)$ is trivial. For all $K\in Comp(Y)$, $x\in X$ and $s\in K$ $$\begin{aligned} \| (f(x)(s) - f(x_{\infty})(s)) v(x) \| & \leq \\ \| f(x)(s)v(x) - f(x_{\infty})(s)v(x_{\infty}) \| + \| f(x_{\infty})(s)v(x_{\infty}) - f(x_{\infty})(s)v(x) \| & \leq \\ \|f(x)(s)(v(x)-v(x_{\infty}))\| + \|(f(x)(s)-f(x_{\infty})(s))v(x_{\infty})\| + \|f(x_{\infty})(s)(v(x_{\infty})-v(x))\| & \leq \\ \left( \|f(x)(s)\| + \|f(x_{\infty})(s)\| \right) \|v(x_{\infty})-v(x)\| + \|(f(x)(s)-f(x_{\infty})(s))v(x_{\infty})\| & \leq \\ C(f,K)\|v(x_{\infty})-v(x)\| + \|(f(x)(s)-f(x_{\infty})(s))v(x_{\infty})\|, \end{aligned}$$ where $C(f,K) \doteqdot\sup_{(x,s)\in X\times K}$. Hence $(d)$ implies $(b)$. \[21031238\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure, $Y_{0}\subset Y$ and ${\mathcal{V}}\in\prod_{x\in X}{\mathfrak{M}}_{x}$. We say that ${\mathcal{V}}$ is equicontinuous on $Y_{0}$ iff $(\forall j\in J) (\exists a>0) (\exists\,j_{1}\in J) (\forall z\in X) (\forall v_{z}\in{\mathfrak{E}}_{z}) $ $$\label{15422103} \sup_{t\in Y_{0}} \nu_{j}\left({\mathcal{V}}(z)(t)v_{z}\right) \leq a \nu_{j_{1}}(v_{z})$$ While ${\mathcal{V}}$ is *equicontinuous* iff it is equicontinuous on $Y$. Finally ${\mathcal{V}}$ is *pointwise equicontinuous* iff it is equicontinuous on every point of $Y$ and *compactly equicontinuous* iff it is equicontinuous on every compact of $Y$. Note that in case ${\mathfrak{V}}$ is trivial with costant stalk $E$ then ${\mathcal{V}}$ is equicontinuous on $Y_{0}$ if and only if it is equicontinuous in the standard sense the following set of maps $\{ {\mathcal{V}}_{0}(z)(t) \in{\mathcal{L}}(E) \mid (z,t)\in X\times Y_{0} \}$, where ${\mathcal{V}}_{0} \in \left({\mathcal{L}}(E)^{Y} \right)^{X}$ such that ${\mathcal{V}}(z)=(z,{\mathcal{V}}_{0}(z))$ for all $z\in X$. \[20492003\] Let ${\mathfrak{V}}$ be trivial with costant stalk $E$, $A^{0}\in Bounded(E)$, $x_{\infty}\in X$ and [^13] $$\label{16062103} \begin{cases} {\mathcal{E}}_{0} \subseteq {\mathcal{C}_{b} \left(X,E\right)} \\ {\mathcal{E}}_{0} \text{ equicontinuous set at $x_{\infty}$} \\ \{(X\ni x\mapsto a\in E)\mid a\in A^{0}\} \subset {\mathcal{E}}_{0}. \end{cases}$$ Moreover let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure such that for all $x\in X$ $${\mathfrak{M}}_{x} = {\mathcal{C}_{c} \left(Y,{\mathcal{L}}_{S_{x}}(\{x\}\times E)\right)}.$$ and $$\begin{cases} {\mathcal{E}} =\prod_{x\in X} \{x\}\times{\mathcal{E}}_{0} \\ \Theta = \{ B_{A^{0}} \} \end{cases}$$ where $B_{A^{0}}(x) \doteqdot \{x\}\times A^{0}$, then $$\label{23412003} \begin{cases} S_{x} = \{x\}\times A^{0},\forall x\in X \\ {\mathfrak{M}}_{x} \simeq \{x\}\times {\mathcal{C}_{c} \left(Y,{\mathcal{L}}_{A^{0}}(E)\right)}. \\ {\mathfrak{M}} = \bigcup_{x\in X} {\mathfrak{M}}_{x} \simeq \bigcup_{x\in X} \{x\} \times {\mathcal{C}_{c} \left(Y,{\mathcal{L}}_{A^{0}}(E)\right)} \\ \prod_{x\in X} {\mathfrak{M}}_{x} \simeq \prod_{x\in X} \{x\} \times {\mathcal{C}_{c} \left(Y,{\mathcal{L}}_{A^{0}}(E)\right)} \simeq {\mathcal{C}_{c} \left(Y,{\mathcal{L}}_{A^{0}}(E)\right)}^{X}. \end{cases}$$ If ${\mathfrak{W}}$ is full and $$\{ X\ni x\mapsto \tau_{f}(x) =(x,f)\in{\mathfrak{M}}_{x} \mid f \in {\mathcal{C}_{c} \left(Y,{\mathcal{L}}_{A^{0}}(E)\right)} \} \subset \Gamma(\rho),$$ then for all ${\mathcal{V}}\in\prod_{x\in X}^{b}{\mathfrak{M}}_{x}$, $(1)\Rightarrow(2)$ and $(3)\Leftrightarrow(4)$, where 1. ${\mathcal{V}}\in\Gamma^{x_{\infty}}(\rho)$ 2. ${\mathcal{V}}_{0} \in{\mathcal{C}_{} \left(X,{\mathcal{C}_{c} \left(Y,{\mathcal{L}}_{A^{0}}(E)\right)}\right)}$, 3. ${\mathcal{V}}$ is compactly equicontinuous and ${\mathcal{V}}\in\Gamma^{x_{\infty}}(\rho)$ 4. ${\mathcal{V}}$ is compactly equicontinuous and ${\mathcal{V}}_{0} \in{\mathcal{C}_{} \left(X,{\mathcal{C}_{c} \left(Y,{\mathcal{L}}_{A^{0}}(E)\right)}\right)}$. Here in $(2)-(4)$ we consider the isomorphism $\prod_{x\in X} {\mathfrak{M}}_{x} \simeq {\mathcal{C}_{c} \left(Y,{\mathcal{L}}_{A^{0}}(E)\right)}^{X}$, and set ${\mathcal{V}}_{0} \in {\mathcal{C}_{c} \left(Y,{\mathcal{L}}_{A^{0}}(E)\right)}^{X}$ such that ${\mathcal{V}}(x)=(x,{\mathcal{V}}_{0}(x))$ for all $x\in X$. For all $x\in X$ by ${\mathcal{B}}_{B_{A^{0}}}^{x} = \{(x,v_{0}(x)) \mid v_{0}\in{\mathcal{E}}_{0}, v_{0}(X)\subseteq A^{0}\} $ so ${\mathcal{B}}_{B_{A^{0}}}^{x}\subseteq A^{0}$. Moreover by construction $(X\ni x\mapsto a\in E) \in{\mathcal{E}}_{0}$ for all $a\in A^{0}$, thus ${\mathcal{B}}_{B_{A^{0}}}^{x}= A^{0}$. Thus the first equality in follows, the others are trivial. By Proposition \[16572003\] $$\label{15102103} (1) \Leftrightarrow \lim_{z\to x_{\infty}} \sup_{t\in K} \sup_{v_{0}\in {\mathcal{E}}_{0} \cap B_{A^{0}} } \nu_{j}\left(( {\mathcal{V}}_{0}(z)(t)- {\mathcal{V}}_{0}(x_{\infty})(t))v_{0}(z)\right) =0.$$ Moreover by construction we deduce that $\{(X\ni x\mapsto a\in E)\mid a\in A^{0}\} \subset {\mathcal{E}}_{0} \cap B_{A^{0}}$, so $(2)$ follows by $(1)$ and . Let $v_{0}\in{\mathcal{E}}_{0}$ then for all $z\in X$ and $t\in Y$ $$\begin{aligned} {1} \label{15352103} ({\mathcal{V}}(z)(t)- {\mathcal{V}}(x_{\infty})(t)) v_{0}(z) &= {\mathcal{V}}(z)(t)(v_{0}(z)-v_{0}(x_{\infty})) + \notag\\ ({\mathcal{V}}(z)(t)- {\mathcal{V}}(x_{\infty})(t)) v_{0}(x_{\infty}) & + {\mathcal{V}}(x_{\infty})(t) (v_{0}(z)-v_{0}(x_{\infty})).\end{aligned}$$ Moreover by the hypothesis of equicontinuity at $x_{\infty}$ of the set ${\mathcal{E}}_{0}$, for all $j\in J$ $$\label{15542103} \lim_{z\to x_{\infty}} \sup_{v_{0}\in{\mathcal{E}}_{0}} \nu_{j}(v_{0}(z)-v_{0}(x_{\infty})) =0.$$ By and for all $j\in J$ there exists $j_{1}\in J$ and $a>0$ such that for all $z\in X$ $$\begin{aligned} {1} \label{15562103} & \sup_{t\in K} \sup_{v_{0}\in {\mathcal{E}}_{0} \cap B_{A^{0}} } \nu_{j}\left(( {\mathcal{V}}_{0}(z)(t)- {\mathcal{V}}_{0}(x_{\infty})(t))v_{0}(z)\right) \leq \notag\\ & 2a \sup_{v_{0}\in {\mathcal{E}}_{0} \cap B_{A^{0}} } \nu_{j_{1}} \left(v_{0}(z)-v_{0}(x_{\infty})\right) + \notag\\ & \sup_{t\in K} \sup_{v_{0}\in {\mathcal{E}}_{0} \cap B_{A^{0}} } \nu_{j}\left( {\mathcal{V}}(z)(t)- {\mathcal{V}}(x_{\infty})(t)\right) v_{0}(x_{\infty}).\end{aligned}$$ Therefore by , and by $(4)$ follows $$\lim_{z\to x_{\infty}} \sup_{t\in K} \sup_{v_{0}\in {\mathcal{E}}_{0} \cap B_{A^{0}} } \nu_{j}\left(( {\mathcal{V}}_{0}(z)(t)- {\mathcal{V}}_{0}(x_{\infty})(t))v_{0}(z)\right) =0.$$ Hence $(1)$ follows by . Main Claim ---------- The following is the preparatory definition for the first main structure of the paper. \[ $ Graph(\left({\mathfrak{V}}_{1},{\mathfrak{V}}_{2}\right) $ \] \[12432110bis\] Let $ {\mathfrak{V}}_{i} \doteqdot {\left\langle {\left\langle {\mathfrak{E}}_{i},\tau_{i}{\right\rangle}},\pi_{i},X,{\mathfrak{N}}_{i}{\right\rangle}}_{i=1}^{2} $ be a couple of bunldes of $\Omega-$spaces and with the notations used in Remark \[21262110\] let $ \bigoplus_{i=1}^{2} {\mathfrak{V}}_{i} = {\left\langle {\left\langle {\mathfrak{E}}({\mathbf{E}}^{\oplus}),\tau({\mathbf{E}}^{\oplus},{\mathcal{E}}^{\oplus}){\right\rangle}},\pi_{{\mathbf{E}}^{\oplus}},X,{\mathfrak{n}}^{\oplus}{\right\rangle}} $ be the bundle direct sum of the family $ \{ {\mathfrak{V}}_{i} \}_{i=1}^{2} $. Then we say that ${\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}$ is a *graph section* or $ {\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}} \in Graph\left(\left( {\mathfrak{V}}_{1},{\mathfrak{V}}_{2} \right)\right) $ if 1. $ {\mathcal{T}} \in \prod_{x\in X} Graph(\left({\mathfrak{E}}_{1}\right)_{x} \times \left({\mathfrak{E}}_{2}\right)_{x}) $; 2. $x_{\infty}\in X$; 3. $$\boxed{ \Phi \subseteq \Gamma^{x_{\infty}}(\pi_{{\mathbf{E}}^{\oplus}}) }$$ $\Phi$ is a linear space such that Graph inclusion : $ (\forall x\in X) (\forall \phi\in\Phi) (\phi(x)\in{\mathcal{T}}(x)) $ Asymptotic Graph : $$\label{19350512bis} \boxed{ \left\{ \phi(x_{\infty}) \mid \phi \in \Phi \right\} = {\mathcal{T}}(x_{\infty}) }.$$ \[ $ Pregraph(\left({\mathfrak{V}}_{1},{\mathfrak{V}}_{2}\right) $ \] \[16161212bis\] Let $ {\mathfrak{V}}_{i} \doteqdot {\left\langle {\left\langle {\mathfrak{E}}_{i},\tau_{i}{\right\rangle}},\pi_{i},X,{\mathfrak{N}}_{i}{\right\rangle}}_{i=1}^{2} $ be a couple of bunldes of $\Omega-$spaces and $ \bigoplus_{i=1}^{2} {\mathfrak{V}}_{i} = {\left\langle {\left\langle {\mathfrak{E}}({\mathbf{E}}^{\oplus}),\tau({\mathbf{E}}^{\oplus},{\mathcal{E}}^{\oplus}){\right\rangle}},\pi_{{\mathbf{E}}^{\oplus}},X,{\mathfrak{n}}^{\oplus}{\right\rangle}} $ the bundle direct sum of the family $ \{ {\mathfrak{V}}_{i} \}_{i=1}^{2} $. Then we say that ${\left\langle {\mathcal{T}}_{0},x_{\infty},\Phi{\right\rangle}}$ is a *pregraph section* or $ {\left\langle {\mathcal{T}}_{0},x_{\infty},\Phi{\right\rangle}} \in Pregraph\left( {\mathfrak{V}}_{1},{\mathfrak{V}}_{2} \right) $ if 1. $x_{\infty}\in X$; 2. $ {\mathcal{T}}_{0} \in \prod_{x\in X-\{x_{\infty}\}} Graph(\left({\mathfrak{E}}_{1}\right)_{x} \times \left({\mathfrak{E}}_{2}\right)_{x}) $; 3. $ \Phi \subseteq \Gamma^{x_{\infty}}(\pi_{{\mathbf{E}}^{\oplus}}) $ such that $\Phi$ is a linear space and $(\forall x\in X-\{x_{\infty}\}) (\forall \phi\in\Phi) (\phi(x)\in{\mathcal{T}}_{0}(x))$. We shall see in Lemma \[13001512b\] that it is possible to construct by any pregraph section $ {\left\langle {\mathcal{T}}_{0},x_{\infty},\Phi{\right\rangle}} $ with suitable properties, a corresponding graph section ${\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}$ such that ${\mathcal{T}}{\upharpoonright}(X-\{x_{0}\})={\mathcal{T}}_{0}$, while ${\mathcal{T}}(x_{\infty})$ is defined by . To this end it is sufficient to show that ${\mathcal{T}}(x_{\infty}) \in Graph({\mathfrak{E}}_{x_{\infty}} \times{\mathfrak{E}}_{x_{\infty}})$. \[23112110bis\] Notice that the fundamental requirement that any $\phi\in\Phi$ is a selection continuous in $x_{\infty}$ implies that $$\begin{cases} \left\{ \lim_{z\to x_{\infty}} \phi(z) \mid \phi\in\Phi \right\} = {\mathcal{T}}(x_{\infty}) \in Graph(\left({\mathfrak{E}}_{1}\right)_{x_{\infty}} \times \left({\mathfrak{E}}_{2}\right)_{x_{\infty}}) \\ \text{with} \\ \phi(z)\in {\mathcal{T}}(z) \in Graph(\left({\mathfrak{E}}_{1}\right)_{z} \times \left({\mathfrak{E}}_{2}\right)_{z}),\, \forall z\in X-\{x_{\infty}\}, \end{cases}$$ which justify the name of asymptotic graph given to . Moreover by setting $X\ni z\mapsto \phi_{i}(z) \doteqdot \Pr_{i}^{z}(\phi(z))$ we have by Corollary \[17571212\] we have for all $i=1,2$ $$\begin{cases} \label{09382412bis} \left\{ \lim_{z\to x_{\infty}} \phi_{i}(z) \mid \phi\in\Phi \right\} = \Pr_{i}^{x_{\infty}}({\mathcal{T}}(x_{\infty})) \text{with} \\ \phi(z)\in {\mathcal{T}}(z) \in Graph(\left({\mathfrak{E}}_{1}\right)_{z} \times \left({\mathfrak{E}}_{2}\right)_{z}),\, \forall z\in X-\{x_{\infty}\}. \end{cases}$$ Finally for $i=1,2$ by Corollary \[17571212\] and Corollary \[28111707\] we have $(1_{i}) \Leftrightarrow (2_{i})$ $(1_{i})$ : $(\exists\, \sigma\in\Gamma(\pi)) (\sigma(x_{\infty}) =\phi_{i}(x_{\infty}))$ such that $$(\forall j\in J) (\lim_{z\to x_{\infty}} \nu_{j}(\phi_{i}(z)-\sigma(z))=0);$$ $(1_{ii})$ : $(\forall\sigma\in\Gamma(\pi) \mid \sigma(x_{\infty}) =\phi_{i}(x_{\infty}))$ we have $$(\forall j\in J) (\lim_{ z\to x_{\infty}} \nu_{j}(\phi_{i}(z)-\sigma(z))=0).$$ \[ Class $ \Delta{\left\langle {\mathfrak{V}},{\mathfrak{D}},\Theta,{\mathcal{E}}{\right\rangle}} $ \] \[15312011bis\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{D}},X,\{pt\}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure and let us denote ${\mathfrak{D}} \doteqdot {\left\langle {\left\langle {\mathfrak{B}},\gamma{\right\rangle}},\eta,X,{\mathfrak{L}}{\right\rangle}}$. Thus $\Omega\,\in \Delta {\left\langle {\mathfrak{V}},{\mathfrak{D}},\Theta,{\mathcal{E}}{\right\rangle}}$ if 1. $ \Omega \subseteq Graph \left( \left( {\mathfrak{V}},{\mathfrak{V}} \right) \right) $; 2. **Projector Section associated to ${\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}$:** $ \forall {\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}} \in \Omega $ $$\label{18270303} \boxed{ \left(\exists\, {\mathcal{P}} \in \Gamma^{x_{\infty}}(\eta) \cap \prod_{x\in X} \Pr({\mathfrak{E}}_{x}) \right) \left(\forall x\in X\right) \left({\mathcal{P}}(x) T_{x} \subseteq T_{x} {\mathcal{P}}(x)\right). }$$ Here $ T_{x}: D_{x} \subseteq {\mathfrak{E}}_{x} \to {\mathfrak{E}}_{x} $ is the map such that ${\mathcal{T}}(x)=Graph(T_{x})$, for all $x\in X$. \[MAIN\] \[19520412bis\] Under the assumptions in Definition \[15312011bis\], possibly with ${\left\langle {\mathfrak{V}},{\mathfrak{D}},X,\{pt\}{\right\rangle}}$ invariant, find elements in the class $$\Delta{\left\langle {\mathfrak{V}},{\mathfrak{D}},\Theta,{\mathcal{E}}{\right\rangle}}.$$ Induction of Sections of Semigroups ------------------------------------ \[ Induction of Sections of Semigroups \] \[19490412bis\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure such that $\{{\mathfrak{E}}_{x}\}_{x\in X}$ is a family of sequentially complete Hlcs and ${\mathbf{U}}({\mathcal{L}}_{S_{x}}({\mathbf{E}}_{x})) \subset {\mathfrak{M}}_{x}$, where we set $ {\mathfrak{V}} \doteqdot {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}} $ and ${\mathfrak{W}} \doteqdot {\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$. Then $\Omega\in \Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}$ if 1. $ \Omega \subseteq Graph(\left({\mathfrak{E}},{\mathfrak{E}})\right) $; 2. **Semigroup Section associated to ${\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}$:**\ $ \forall {\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}} \in \Omega $ $$\boxed{ \exists\, {\mathcal{U}}_{{\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}} \in \Gamma^{x_{\infty}}(\rho) }$$ such that $\forall x\in X$ 1. $ {\mathcal{U}}_{{\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}}(x) $ is an equicontinuous $(C_{0})-$semigroup on ${\mathfrak{E}}_{x}$; 2. $ (\forall x\in X) ({\mathcal{T}}(x)=Graph(R_{x})) $. Here $R_{x}$ is the infinitesimal generator of the semigroup $ {\mathcal{U}}_{{\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}}(x) \in {\mathcal{C}_{c} \left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}} ({\mathfrak{E}}_{x})\right)} $. \[20090412bis\] Under the assumptions in Definition \[19490412bis\], possibly with ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ compatible, find elements in the class $$\Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}.$$ \[20430412bis\] Notice that $ \forall {\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}} \in \Omega $ there exists only one semigroup section associated to it. Moreover ${\mathcal{U}}_{{\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}}$ is characterized by any of the equivalent conditions in Lemma \[15482712\] with $U=X$ and $Y={\mathbb {R}}^{+}$. Induction of a Section of Semigroups - Projectors {#20110412bis} --------------------------------------------------- \[ Induction of a Section of Semigroups - Projectors \] \[18550612bis\] Let $ {\mathfrak{V}} \doteqdot {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}} $, ${\mathfrak{D}} \doteqdot {\left\langle {\left\langle {\mathfrak{B}},\gamma{\right\rangle}},\eta,X,{\mathfrak{L}}{\right\rangle}}$ and ${\mathfrak{W}} \doteqdot {\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$. Moreover let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure such that $\{{\mathfrak{E}}_{x}\}_{x\in X}$ is a family of sequentially complete Hlcs and ${\left\langle {\mathfrak{V}},{\mathfrak{D}},X,\{pt\}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure. Then $\Psi \in \Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{D}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}} $ if 1. $\Psi\subseteq \bigcup_{x_{\infty}\in X} \Gamma^{x_{\infty}}(\rho) $; 2. $ (\forall{\mathcal{U}}\in\Psi) (\forall x\in X) $ $ ({\mathcal{U}}(x) $ is an equicontinuous $(C_{0})-$semigroup on ${\mathfrak{E}}_{x})$; 3. **Projector Section associated to ${\mathcal{U}}$:**\ $(\forall x_{\infty}\in X) (\forall{\mathcal{U}} \in\Psi\cap\Gamma^{x_{\infty}}(\rho))$ $$\label{16221411bis} \boxed{ \left(\exists\, {\mathcal{P}} \in \Gamma^{x_{\infty}}(\eta) \cap \prod_{x\in X} \Pr({\mathfrak{E}}_{x}) \right) \left(\forall x\in X\right) \left( {\mathcal{P}}(x)H_{x} \subseteq H_{x} {\mathcal{P}}(x) \right). }$$ Here $H_{x}$ is the infinitesimal generator of the semigroup $ {\mathcal{U}}(x) \in {\mathcal{C}_{c} \left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}} ({\mathfrak{E}}_{x})\right)}$ for all $x\in X$. \[S-P\] \[20290412bis\] Under the assumptions in Definition \[19490412bis\], possibly with ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ compatible and ${\left\langle {\mathfrak{V}},{\mathfrak{D}},X,\{pt\}{\right\rangle}}$ invariant, find elements in the class $\Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{D}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}.$ Claims \[20090412bis\] and \[20290412bis\] can be used to solve the Main claim \[19520412bis\] indeed \[20340412bis\] Let $ {\mathfrak{V}} \doteqdot {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}} $, ${\mathfrak{D}} \doteqdot {\left\langle {\left\langle {\mathfrak{B}},\gamma{\right\rangle}},\eta,X,{\mathfrak{L}}{\right\rangle}}$ and ${\mathfrak{W}} \doteqdot {\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$. Moreover let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure such that $\{{\mathfrak{E}}_{x}\}_{x\in X}$ is a family of sequentially complete Hlcs and ${\left\langle {\mathfrak{V}},{\mathfrak{D}},X,\{pt\}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure. Assume that 1. $ \Omega\in \Omega\in \Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}} $; 2. $ \Psi \in \Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{D}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}} $; and $$\label{20500412bis} \boxed{ (\forall {\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}} \in\Omega) ( {\mathcal{U}}_{{\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}} \in \Psi). }$$ Then $$\Omega\,\in\Delta {\left\langle {\mathfrak{V}},{\mathfrak{D}},\Theta,{\mathcal{E}}{\right\rangle}},$$ i.e. $\Omega$ satisfies the Main claim \[19520412bis\]. Moreover $$\left( \forall {\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}} \in \Omega \right) \left( \exists\, {\mathcal{P}} \in \Gamma^{x_{\infty}}(\eta) \right) \left( \exists\, {\mathcal{U}} \in \Gamma^{x_{\infty}}(\rho) \right)$$ 1. $ {\mathcal{U}}(x) $ is an equicontinuous $(C_{0})-$semigroup on ${\mathfrak{E}}_{x}$, for all $x\in X$; 2. $ \left(\forall x\in X\right) \left({\mathcal{P}}(x) \in \Pr({\mathfrak{E}}_{x}) \right) $; 3. $ (\forall x\in X) ({\mathcal{T}}(x)=Graph(R_{x})) $; 4. $ \forall x\in X $ $${\mathcal{P}}(x)R_{x} \subseteq R_{x} {\mathcal{P}}(x).$$ Here $R_{x}$ is the infinitesimal generator of the semigroup $ {\mathcal{U}}(x) \in {\mathcal{C}_{c} \left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})\right)}$, for all $x\in X$. Semigroup Approximation Theorems ================================ General Approximation Theorem I {#11542812} ------------------------------- \[15411512b\] For any $ {\mathfrak{V}} \doteqdot {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}} $ bunlde of $\Omega-$spaces and any ${\left\langle {\mathcal{T}}_{0},x_{\infty},\Phi{\right\rangle}} \in Pregraph \left(\left({\mathfrak{V}},{\mathfrak{V}}\right)\right)$, set $X_{0}\doteqdot X-\{x_{\infty}\}$, and for any $\phi\in\Phi$ $\phi_{i}(x)\doteqdot \Pr_{i}^{x}(\phi(x))$ for all $x\in X$ and $i=1,2$. Moreover let us denote by $T_{x}$ the operator in ${\mathfrak{E}}_{x}$ such that $Graph(T_{x})={\mathcal{T}}_{0}(x)$, for all $x\in X_{0}$, while $ {\mathcal{T}} \in \prod_{x\in X} Graph({\mathfrak{E}}_{x} \times{\mathfrak{E}}_{x}) $ so that $$\begin{cases} {\mathcal{T}}{\upharpoonright}X-\{x_{\infty}\} \doteqdot {\mathcal{T}}_{0} \\ {\mathcal{T}}(x_{\infty}) \doteqdot \{\phi(x_{\infty})\mid\phi\in\Phi\}, \end{cases}$$ in addition set $$D(T_{x_{\infty}}) \doteqdot \Pr_{1}^{x_{\infty}}({\mathcal{T}}(x_{\infty})) = \{\phi_{1}(x_{\infty}) \mid\phi\in\Phi\}.$$ Finally for any map $F:A\to B$ set ${\mathcal{R}}(F)\doteqdot F(A)$ the range of $F$. \[15401512b\] Let $ {\mathfrak{V}} \doteqdot {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}} $ be a bunlde of $\Omega-$spaces and ${\left\langle {\mathcal{T}}_{0},x_{\infty},\Phi{\right\rangle}} \in Pregraph \left(\left({\mathfrak{V}},{\mathfrak{V}}\right)\right)$. By Corollary \[17571212\] $\forall\phi\in\Phi$ $$\label{15211512b} \begin{cases} \phi_{i}\in\Gamma^{x_{\infty}}(\pi), i=1,2\\ (\forall x\in X_{0}) (\phi_{2}(x)=T_{x}\phi_{1}(x)). \end{cases}$$ \[13001512b\] Let $ {\mathfrak{V}} \doteqdot {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}} $ be a bundle of $\Omega-$spaces, where ${\mathfrak{N}}\doteqdot\{\nu_{j}\mid j\in J\}$. Moreover ${\left\langle {\mathcal{T}}_{0},x_{\infty},\Phi{\right\rangle}} \in Pregraph \left(\left({\mathfrak{V}},{\mathfrak{V}}\right)\right)$. If for all $x\in X_{0}$, $v_{x}\in Dom(T_{x})$, $\lambda>0$ and $j\in J$ we have $\nu_{j}((\lambda- T_{x})v_{x}) \geq \lambda\nu_{j}(v_{x})$ and $D(T_{x_{\infty}})$ is dense in ${\mathfrak{E}}_{x_{\infty}}$, then $${\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}} \in Graph \left(\left({\mathfrak{V}},{\mathfrak{V}}\right)\right).$$ Moreover the following $$\label{172212312b} T_{x_{\infty}}: D(T_{x_{\infty}}) \ni \phi_{1}(x_{\infty}) \mapsto \phi_{2}(x_{\infty})$$ is a well-defined linear operator in ${\mathfrak{E}}_{x_{\infty}}$ such that $ Graph(T_{x_{\infty}}) = {\mathcal{T}}(x_{\infty}) $ and $\forall v_{x_{\infty}}\in Dom(T_{x_{\infty}})$, $\forall\lambda>0$ and $\forall j\in J$ we have $$\nu_{j}((\lambda- T_{x_{\infty}}) v_{x_{\infty}}) \geq \lambda\nu_{j}(v_{x_{\infty}}).$$ Clearly $ {\mathcal{T}}(x_{\infty}) \in Graph({\mathfrak{E}}_{x_{\infty}} \times{\mathfrak{E}}_{x_{\infty}}) $ if and only if $\phi_{1}(x_{\infty}) ={\mathbf{0}}_{x_{\infty}} $ implies $\phi_{2}(x_{\infty})={\mathbf{0}}_{x_{\infty}}$, $ \forall\phi\in\Phi $, moreover denoting by $T_{x_{\infty}}$ the corresponding operator we have that $T_{x_{\infty}}: D(T_{x_{\infty}})\to{\mathfrak{E}}_{x_{\infty}}$ is a linear operator. Any real map $F$ defined on a topological space is l.s.c. at a point iff $-F$ is u.s.c. at the same point, see [@BourGT $\S 6.2.$ Ch.$4$], thus by [@BourGT Prop. $3$ $\S 6.2.$ Ch.$4$] and [@BourGT (13),$\S 5.6.$ Ch.$4$] $F:X\to{\mathbb {R}}$ is u.s.c. in $a\in X$ iff $\varlimsup_{x\to a}F(x)=F(a)$. Moreover by [@BourGT $\S 6.2.$ Ch.$4$] we know that $F:X\to{\overline}{{\mathbb {R}}}$ is l.s.c. at $a$ iff $F$ is continuous at $a$ providing ${\overline}{{\mathbb {R}}}$ with the following topology $\{\emptyset,[-\infty,\infty],]a,\infty[ \mid a\in{\mathbb {R}}\}$. Thus for any map $\sigma:Y\to X$ continuous at $b$ such that $\sigma(b)=a$ we have that $F\circ\sigma$ is l.s.c. at $a$. Hence because $(-F)\circ\sigma=-(F\circ\sigma)$ we can state that if $F:X\to{\overline}{{\mathbb {R}}}$ is u.s.c. at $a$ then for any map $\sigma:Y\to X$ continuous at $b$ such that $\sigma(b)=a$ we have that $F\circ\sigma$ is u.s.c. at $a$. Therefore by using [@gie $1.6.(ii)$] we have $\forall\sigma\in\Gamma^{x_{\infty}}(\pi)$ and $\forall j\in J$ $$\label{16051512b} \nu_{j}(\sigma(x_{\infty})) = \varlimsup_{x\to x_{\infty}} \nu_{j}(\sigma(x)).$$ Let $\psi\in\Phi$ such that $\psi_{1}(x_{\infty})={\mathbf{0}}_{x_{\infty}}$ thus $\forall\phi\in\Phi$, $\forall\lambda>0$, $\forall x\in X_{0}$ and $\forall j\in J$ we have by and $$\begin{aligned} {2} \label{16401512b} \nu_{j}\left( \lambda\phi_{1}(x_{\infty})- \phi_{2}(x_{\infty})- \lambda\psi_{2}(x_{\infty}) \right) & = \notag \\ \varlimsup_{x\to x_{\infty}} \nu_{j}\left((\lambda-T_{x}) (\phi_{1}(x)+\lambda\psi_{1}(x))\right) & \geq \notag \\ \varlimsup_{x\to x_{\infty}} \lambda \nu_{j}\left(\phi_{1}(x)+ \lambda\psi_{1}(x))\right) & = \lambda \nu_{j}(\phi_{1}(x_{\infty})),\end{aligned}$$ where, the inequality comes by [@BourGT Prop. $11$ $\S 5.6.$ Ch.$4$]) and by the hypothesis $ \nu_{j}\left((\lambda-T_{x}) (\phi_{1}(x)+\lambda\psi_{1}(x))\right) \geq \lambda \nu_{j}\left((\phi_{1}(x)+ \lambda\psi_{1}(x))\right) $ for all $x\in X_{0}$. Now $ \lim_{\lambda\to\infty}v/\lambda = {\mathbf{0}}_{x_{\infty}} $ for any $v\in{\mathfrak{E}}_{x_{\infty}}$, hence by the fact that $\nu_{j}^{x_{\infty}} \doteqdot \nu_{j}{\upharpoonright}{\mathfrak{E}}_{x_{\infty}}$ is a continuous seminorm and by $ (\forall j\in J) (\forall\phi\in\Phi) $ $$\label{17011512b} \nu_{j}\left( \phi_{1}(x_{\infty})-\psi_{2}(x_{\infty}) \right) = \lim_{\lambda\to\infty} \frac{\nu_{j}\left( \lambda\phi_{1}(x_{\infty})- \phi_{2}(x_{\infty})- \lambda\psi_{2}(x_{\infty}) \right)}{\lambda} \geq \nu_{j}(\phi_{1}(x_{\infty})).$$ By hypothesis $ D(T_{x_{\infty}}) = \{\phi_{1}(x_{\infty}) \mid\phi\in{\mathcal{T}}(x_{\infty})\} $ is dense in ${\mathfrak{E}}_{x_{\infty}}$ thus $\nu_{j}(\psi_{2}(x_{\infty}))=0$ for all $j\in J$. Indeed let $j\in J$ and $v\in{\mathfrak{E}}_{x_{\infty}}$ thus $\exists\,\{\phi^{\alpha}\}_{\alpha\in D}$ net in $\Phi$ such that $ \lim_{\alpha\in D} \phi_{1}^{\alpha}(x_{\infty}) = v $ in ${\mathfrak{E}}_{x_{\infty}}$. So by the continuity of $\nu_{j}^{x_{\infty}}$ and by we have $\forall v\in{\mathfrak{E}}_{x_{\infty}}$ $$\nu_{j}\left( v -\psi_{2}(x_{\infty}) \right) = \lim_{\alpha\in D} \nu_{j}\left( \phi_{1}^{\alpha}(x_{\infty}) -\psi_{2}(x_{\infty}) \right) \geq \lim_{\alpha\in D} \nu_{j}\left( \phi_{1}^{\alpha}(x_{\infty})) \right) = \nu_{j}(v).$$ True in particular for $v=3\psi_{2}(x_{\infty})$, which implies $\nu_{j}\left(\psi_{2}(x_{\infty})\right) =0$. Hence $\psi_{2}(x_{\infty})={\mathbf{0}}_{x_{\infty}}$ because of ${\mathfrak{E}}_{x_{\infty}}$ is a Hausdorff lcs for which $\{\nu_{j}^{x_{\infty}}\}_{j\in J}$ is a generating set of seminorms of its topology. Thus $T_{x_{\infty}}$ is a well-defined (necessarly linear) operator in ${\mathfrak{E}}_{x_{\infty}}$ and consequently ${\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}} \in Graph \left(\left({\mathfrak{V}},{\mathfrak{V}}\right)\right)$. Finally $(\forall j\in J) (\forall\phi\in\Phi) (\forall\lambda>0)$ $$\begin{aligned} {1} \nu_{j} ((\lambda-T_{x_{\infty}})\phi_{1}(x_{\infty})) &= \\ \nu_{j} (\lambda\phi_{1}(x_{\infty})- \phi_{2}(x_{\infty})) &= \qquad \text{ by } \eqref{15211512b}, \eqref{16051512b} \\ \varlimsup_{x\to x_{\infty}} \nu_{j} (\lambda\phi_{1}(x)- \phi_{2}(x)) &= \qquad \text{ by } \eqref{15211512b} \\ \varlimsup_{x\to x_{\infty}} \nu_{j} ((\lambda-T_{x})\phi_{1}(x)) &\geq \qquad \text{ by hypoth. and \cite[Prop. $11$ $\S 5.6.$ Ch.$4$]{BourGT}) } \\ \varlimsup_{x\to x_{\infty}} \nu_{j} (\lambda\phi_{1}(x)) &= \nu_{j} (\lambda\phi_{1}(x_{\infty})).\end{aligned}$$ \[161172901\] In addition to the hypotheses and notations of Lemma \[13001512b\] assume that $(\forall x\in X_{0}) (\forall\lambda\in{\mathbb {R}}) (\forall j\in J) (\forall v_{x}\in Dom(T_{x})) $ $$\label{16292901} \nu_{j} (({\mathbf{1}}-\lambda T_{x})v_{x}) \geq \nu_{j}(v_{x}).$$ Thus $(\forall\lambda\in{\mathbb {R}}) (\forall j\in J) (\forall v_{x_{\infty}}\in Dom(T_{x_{\infty}}))$ $$\label{20022901} \nu_{j}(({\mathbf{1}}-\lambda T_{x_{\infty}}) v_{x_{\infty}}) \geq \nu_{j}(v_{x_{\infty}}).$$ Moreover $\forall\lambda\in{\mathbb {R}}$ $$\label{20102901} \begin{cases} \exists\, ({\mathbf{1}}-\lambda T_{x_{\infty}})^{-1} \in{\mathcal{L}} ({\mathcal{R}}({\mathbf{1}}-\lambda T_{x_{\infty}}), {\mathfrak{E}}_{x_{\infty}}), \\ (\forall w\in {\mathcal{R}}({\mathbf{1}}-\lambda T_{x_{\infty}})) (\forall j\in J) \nu_{j}(({\mathbf{1}}-\lambda T_{x_{\infty}})^{-1}w) \leq\nu_{j}(w). \end{cases}$$ Finally $$\label{20342901} {\mathcal{R}}({\mathbf{1}}-\lambda T_{x_{\infty}}) \text{ is closed in ${\mathfrak{E}}_{x_{\infty}}$.}$$ $(\forall j\in J) (\forall\phi\in\Phi) (\forall\lambda\in{\mathbb {R}})$ $$\begin{aligned} {1} \nu_{j} (({\mathbf{1}}- \lambda T_{x_{\infty}})\phi_{1}(x_{\infty})) &= \\ \nu_{j} (\phi_{1}(x_{\infty})- \lambda \phi_{2}(x_{\infty})) &= \qquad \text{ by } \eqref{15211512b}, \eqref{16051512b} \\ \varlimsup_{x\to x_{\infty}} \nu_{j} (\phi_{1}(x)- \lambda\phi_{2}(x)) &= \qquad \text{ by } \eqref{15211512b} \\ \varlimsup_{x\to x_{\infty}} \nu_{j} (({\mathbf{1}}-\lambda T_{x})\phi_{1}(x)) &\geq \qquad \text{ by \eqref{16292901} and \cite[Prop. $11$ $\S 5.6.$ Ch.$4$]{BourGT}) } \\ \varlimsup_{x\to x_{\infty}} \nu_{j} (\phi_{1}(x)) &= \nu_{j} (\phi_{1}(x_{\infty})).\end{aligned}$$ thus follows. Let $\lambda\in{\mathbb {R}}$, by we obtain , indeed $\forall f,g\in Dom(T_{x_{\infty}})$ such that $({\mathbf{1}}-\lambda T_{x_{\infty}})f = ({\mathbf{1}}-\lambda T_{x_{\infty}})g$ we have $\forall j\in J$ $$0 = \nu_{j}(({\mathbf{1}}-\lambda T_{x_{\infty}}) (f-g)) \geq \nu_{j}(f-g),$$ so $f=g$ because of by construction ${\mathfrak{E}}_{x_{\infty}}$ is Hausdorff. Thus the following is a well-set map $$({\mathbf{1}}-\lambda T_{x_{\infty}})^{-1}: {\mathcal{R}}({\mathbf{1}}-\lambda T_{x_{\infty}}) \ni ({\mathbf{1}}-\lambda T_{x_{\infty}})f \mapsto f \in {\mathfrak{E}}_{x_{\infty}},$$ moreover by we obtain the second sentence of , hence the first one follows by the fact that the inverse map of any linear operator is linear. By , [@BourGT Prop. $3$ $\S 3.1.$ Ch.$3$] and [@BourGT Prop. $11$ $\S 3.6.$ Ch.$2$] we deduce that $$\label{20182901} (\exists\,!\, B\in{\mathcal{L}}\left( {\overline}{{\mathcal{R}}({\mathbf{1}}-\lambda T_{x_{\infty}})}, {\mathfrak{E}}_{x_{\infty}}\right)) (B{\upharpoonright}{\mathcal{R}}({\mathbf{1}}-\lambda T_{x_{\infty}}) = ({\mathbf{1}}-\lambda T_{x_{\infty}})^{-1}).$$ Let $w\in {\overline}{{\mathcal{R}}({\mathbf{1}}-\lambda T_{x_{\infty}})}$ thus $\exists\,\{f_{\alpha}\}_{\alpha\in D}$ net in $Dom(T_{x_{\infty}})$ such that $$\label{19252901} w=\lim_{\alpha\in D} ({\mathbf{1}}-\lambda T_{x_{\infty}}) f_{\alpha},$$ therefore by $$\label{19512901a} Bw= \lim_{\alpha\in D} f_{\alpha},$$ while by and $$\begin{aligned} {2} w-Bw &= \lim_{\alpha\in D} \left( (f_{\alpha}-\lambda T_{x_{\infty}}f_{\alpha}) -f_{\alpha}\right) \\ &= \lim_{\alpha\in D} -\lambda T_{x_{\infty}}f_{\alpha}.\end{aligned}$$ So $$\label{19512901b} Bw-w = \lim_{\alpha\in D} \lambda T_{x_{\infty}}f_{\alpha}.$$ By , and the fact that $\lambda T_{x_{\infty}}$ is closed, we obtain $$\begin{cases} Bw\in Dom(T_{x_{\infty}}), \\ \lambda T_{x_{\infty}}(Bw) = Bw-w, \end{cases}$$ which means $w=({\mathbf{1}}-\lambda T_{x_{\infty}}) Bw$, so $w\in {\mathcal{R}}({\mathbf{1}}-\lambda T_{x_{\infty}})$ and follows. \[14253001\] Let us assume the hypotheses of Lemma \[161172901\], moreover let $\lambda\in {\mathbb {R}}-\{0\}$, $\{\lambda_{n}\}_{n\in{\mathbb{N}}}\subset{\mathbb {R}}-\{0\}$ such that $\lim_{{n\in{\mathbb{N}}}}\lambda_{n}=\lambda$. Thus $$\bigcap_{n\in{\mathbb{N}}} {\mathcal{R}}({\mathbf{1}}-\lambda_{n}T_{x_{\infty}}) \subseteq {\mathcal{R}}({\mathbf{1}}-\lambda T_{x_{\infty}}).$$ Set only in this proof $T\doteqdot T_{x_{\infty}}$. Let $n\in{\mathbb{N}}$, by $\exists\,({\mathbf{1}}-\lambda_{n}T)^{-1}: {\mathcal{R}}({\mathbf{1}}-\lambda_{n}T_{x_{\infty}}) \to Dom(t)$ moreover $$\begin{cases} {\mathbf{1}}-\lambda T = \lambda(\lambda^{-1}-T), \\ ({\mathbf{1}}-\lambda_{n}T)^{-1} = \lambda_{n}^{-1} (\lambda_{n}^{-1}-T)^{-1}. \end{cases}$$ Let $g\in\bigcap_{n\in{\mathbb{N}}} {\mathcal{R}}({\mathbf{1}}-\lambda_{n}T_{x_{\infty}})$ thus $$\begin{aligned} {2} ({\mathbf{1}}-\lambda T) ({\mathbf{1}}-\lambda_{n} T)^{-1}g - g &= \frac{\lambda}{\lambda_{n}} (\lambda^{-1}-T) (\lambda_{n}^{-1}-T)^{-1}g - g \\ &= \frac{\lambda}{\lambda_{n}} \left( \lambda^{-1} (\lambda_{n}^{-1}-T)^{-1}g - \lambda_{n}^{-1} (\lambda_{n}^{-1}-T)^{-1}g \right) \\ &= \frac{\lambda}{\lambda_{n}} (\lambda^{-1}- \lambda_{n}^{-1}) (\lambda_{n}^{-1}-T)^{-1}g,\end{aligned}$$ where in the second equality we considered that $-T (\lambda_{n}^{-1}-T)^{-1}g-g = -\lambda_{n}^{-1} (\lambda_{n}^{-1}-T)^{-1} g$ obtained by $(\lambda_{n}^{-1}-T) (\lambda_{n}^{-1}-T)^{-1}g =g$. Thus $\forall j\in J$ by $$\nu_{j} \left( ({\mathbf{1}}-\lambda T) ({\mathbf{1}}-\lambda_{n} T)^{-1}g - g\right) \leq \left| \frac{\lambda}{\lambda_{n}} \right| |\lambda^{-1}-\lambda_{n}^{-1}| \nu_{j}(g).$$ But $\lim_{n\in{\mathbb{N}}} |\lambda^{-1}-\lambda_{n}^{-1}| =1$ and $\lim_{n\in{\mathbb{N}}} |\lambda^{-1}-\lambda_{n}^{-1}| =0$ so $\nu_{j} \left( ({\mathbf{1}}-\lambda T) ({\mathbf{1}}-\lambda_{n} T)^{-1}g - g \right)=0$, for all $j\in J$. Therefore $$\lim_{n\in{\mathbb{N}}} ({\mathbf{1}}-\lambda T) ({\mathbf{1}}-\lambda_{n} T)^{-1}g = g,$$ and the statement follows by . \[17552901\] Under the hypotheses and notations of Lemma \[13001512b\] we have that ${\mathbf{1}}-\lambda T_{x_{\infty}}$ is a closed operator. Let $(a,b)\in {\overline}{Graph({\mathbf{1}}-\lambda T_{x_{\infty}})}$ closure in the space ${\mathfrak{E}}_{x_{\infty}} \times {\mathfrak{E}}_{x_{\infty}}$ with the product topology. Thus $(\forall{\varepsilon}>0) (\forall j\in J) (\exists\,v_{({\varepsilon},j)}\in Dom(T_{x_{\infty}}))$ $$\begin{cases} \nu_{j} (a-v_{({\varepsilon},j)}) <\frac{{\varepsilon}}{2}, \\ \nu_{j} (b-({\mathbf{1}}-\lambda T_{x_{\infty}})v_{({\varepsilon},j)}) <\frac{{\varepsilon}}{2}, \end{cases}$$ so $$\nu_{j} ((b-a)+\lambda T_{x_{\infty}}v_{({\varepsilon},j)}) \leq \nu_{j}(b-({\mathbf{1}}-\lambda T_{x_{\infty}}) v_{({\varepsilon},j)}) + \nu_{j}(a-v_{({\varepsilon},j)}) \leq {\varepsilon}.$$ Therefore $(\forall{\varepsilon}>0) (\forall j\in J) (\exists\,v_{({\varepsilon},j)}\in Dom(T_{x_{\infty}}))$ $$\begin{cases} \nu_{j} (a-v_{({\varepsilon},j)}) <{\varepsilon}, \\ \nu_{j} \left( (b-a)-(-\lambda T_{x_{\infty}}v_{({\varepsilon},j)}) \right), \end{cases}$$ which means $(a,(b-a)) \in{\overline}{Graph(-\lambda T_{x_{\infty}})}$. Moreover $-\lambda T_{x_{\infty}}$ is a closed operator thus $b-a=-\lambda T_{x_{\infty}}a$ or equivalently $(a,b)\in Graph({\mathbf{1}}-\lambda T_{x_{\infty}})$. \[18581512b\] By we have $\forall\phi\in\Phi$ that $ \phi_{1}(x_{\infty}) = \lim_{z\to x_{\infty}} \phi_{1}(z) $ and $ \phi_{2}(x_{\infty}) = \lim_{z\to x_{\infty}} \phi_{2}(z) = \lim_{z\to x_{\infty}} T_{x}\phi_{1}(z) $, hence $$\begin{cases} \phi_{1}(x_{\infty}) = \lim_{z\to x_{\infty}} \phi_{1}(z) \\ T_{x_{\infty}} \phi_{1}(x_{\infty}) = \lim_{z\to x_{\infty}} T_{z} \phi_{1}(z). \end{cases}$$ \[15062301\] Let $\lambda\in{\mathbb {R}}^{+}$ set $$\mu_{\lambda}: {\mathcal{C}_{cs} \left({\mathbb {R}}^{+},{\mathbb {R}}\right)} \ni f \mapsto \int_{{\mathbb {R}}^{+}} e^{-s\lambda} f(s)\,ds,$$ where the integral is with respect to the Lebesgue measure on ${\mathbb {R}}^{+}$. \[16401812b\] Let ${\mathfrak{W}} \doteqdot {\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$ and ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure such that for all $x\in X$ $$\label{18470109} {\mathfrak{M}}_{x} \subseteq \bigcap_{\lambda>0} {\mathfrak{L}}_{1}({\mathbb {R}}^{+}, {\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x});\mu_{\lambda}),$$ about $S_{x}$ and ${\mathfrak{E}}_{x}$ see Definition \[10282712\]. Let $x\in X$, ${\mathcal{O}}\subseteq\Gamma(\rho)$. and ${\mathcal{D}}\subseteq\Gamma(\pi)$. By recalling Def. \[15012602\], we say that ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ has the *Laplace duality property on ${\mathcal{O}}$ and ${\mathcal{D}}$ at $x$*, shortly ${\mathbf{LD}}_{x}({\mathcal{O}},{\mathcal{D}})$ if $$(\forall\lambda>0) ({\mathfrak{L}}(\Gamma_{{\mathcal{O}}}^{x}(\rho))_{\lambda} \bullet \Gamma_{{\mathcal{D}}}^{x}(\pi) \subseteq \Gamma^{x}(\pi)).$$ Moreover we say that ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ has the *full Laplace duality property on ${\mathcal{O}}$ and ${\mathcal{D}}$*, shortly ${\mathbf{LD}}({\mathcal{O}},{\mathcal{D}})$ if $$(\forall\lambda>0) ({\mathfrak{L}}({\mathcal{O}})_{\lambda} \bullet {\mathcal{D}} \subseteq \Gamma(\pi)).$$ Finally ${\mathbf{LD}}$ is for ${\mathbf{LD}}(\Gamma(\rho),\Gamma(\pi))$. Here ${\mathfrak{L}}: \prod_{x\in X} {\mathfrak{M}}_{x} \to \prod_{x\in X} {\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})^{{\mathbb {R}}^{+}}$ such that $(\forall x\in X) (\forall\lambda\in{\mathbb {R}}^{+})$ $${\mathfrak{L}}(F)(x)(\lambda) \doteqdot \int_{0}^{\infty} e^{-\lambda s} F(x)(s)\,ds,$$ where we recall that the integration is with respect to the Lebesgue measure on ${\mathbb {R}}^{+}$ and with respect to the lct on ${\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})$. Finally we used the notations in Def \[15012602\]. \[10412505\] Under the notations and assumptions of Def. \[16401812b\] we have $$\begin{cases} {\mathfrak{L}}(\Gamma_{{\mathcal{O}}}^{x}(\rho)) \subseteq \Gamma_{{\mathcal{O}}}^{x}(\rho) \\ (\forall t>0) (\Gamma_{{\mathcal{O}}}^{x}(\rho)_{t} \bullet \Gamma_{{\mathcal{D}}}^{x}(\pi) \subseteq \Gamma^{x}(\pi)) \end{cases} \Rightarrow {\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}} \text{has the } {\mathbf{LD}}_{x}({\mathcal{O}},{\mathcal{D}}).$$ Similarly $$\begin{cases} {\mathfrak{L}}({\mathcal{O}}) \subseteq {\mathcal{O}} \\ (\forall t>0) ({\mathcal{O}}_{t} \bullet {\mathcal{D}} \subseteq \Gamma(\pi)) \end{cases} \Rightarrow {\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}} \text{has the } {\mathbf{LD}}({\mathcal{O}},{\mathcal{D}}).$$ A useful property is the following one \[21120901\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure satisfying , $x_{\infty}\in X$. Set $S_{z}=\{B_{l}^{z}\mid l\in L\}$, then $\forall z\in X$, $\forall G\in{\mathfrak{L}}_{1}({\mathbb {R}}^{+}, {\mathcal{L}}_{S_{z}}({\mathfrak{E}}_{z});\mu_{\lambda})$ and $\forall w_{z}\in \bigcup_{l\in L}B_{l}^{z}$ $$\label{14481001} \left(\int_{0}^{\infty} e^{-\lambda s} G(s) \,ds\right) w_{z} = \int_{0}^{\infty} e^{-\lambda s} G(s)w_{z}\,ds.$$ Here in the second member the integration is with respect to the lct on ${\mathfrak{E}}_{z}$, while in the first member the integration is with respect to the lct on ${\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})$. Let $z\in X$ and $v\in\bigcup_{l\in L}B_{l}^{z}={\mathfrak{E}}_{z}$ then map ${\mathcal{L}}_{S_{z}}({\mathfrak{E}}_{z})\ni A \mapsto Av\in{\mathfrak{E}}_{z}$ is linear and continuous. Indeed let $l(v)\in L$ such that $v\in B_{l(v)}^{z}$, thus we have $\nu_{j}^{z}(Av) \leq \sup_{w\in B_{l(v)}^{z}} \nu_{j}^{z}(Aw) \doteq p_{j,l(v)}^{z} (A)$. Hence by a well-known result in vector valued integration we have . \[21500412b\] Let ${\mathfrak{V}} \doteqdot {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}} $ be a bundle of $\Omega-$spaces and ${\mathcal{E}}\subseteq\prod_{x\in X}{\mathfrak{E}}_{x}$. Set for all $v\in\prod_{x\in X}{\mathfrak{E}}_{x}$ $$\label{11121419b} \begin{cases} B_{v}:X\ni x \mapsto \{v(x)\}, \\ \Theta \doteqdot \left\{ B_{w} \mid w\in{\mathcal{E}} \right\} \end{cases}$$ Thus $\Theta\subset \prod_{x\in X} Bounded({\mathfrak{E}}_{x})$ and $\forall v\in{\mathcal{E}}$ $$\label{01592912} {\mathcal{E}} \cap B_{v} = \{v\}.$$ Therefore for all $v\in{\mathcal{E}}$, and for all $x\in X$ with the notations of Def. \[10282712\] $$\begin{cases} {\mathbf{D}}(B_{v},{\mathcal{E}}) = \{v\}, \\ {\mathcal{B}}_{B_{v}}^{x} =\{v(x)\},\, \\ S_{x}=\{\{w(x)\} \mid w\in{\mathcal{E}}\}, \\ {\mathcal{E}}(\Theta) = {\mathcal{E}}. \end{cases}$$ \[15210503\] Let ${\mathfrak{V}} \doteqdot {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,\|\cdot\|{\right\rangle}}$ be a Banach bundle. Let $x_{\infty}\in X$ and $ {\mathcal{U}}_{0} \in\prod_{x\in X_{0}} {\mathcal{C}_{} \left({\mathbb {R}}^{+},B_{s}({\mathfrak{E}}_{x})\right)}$ be such that ${\mathcal{U}}_{0}(x)$ is a $(C_{0})-$semigroup of contractions (respectively of isometries) on ${\mathfrak{E}}_{x}$ for all $x\in X_{0}$. Moreover let us denote by $T_{x}$ the infinitesimal generator of the semigroup ${\mathcal{U}}_{0}(x)$ for any $x\in X_{0}$ and set $$\label{15482601} \begin{cases} {\mathcal{T}}_{0}(x) \doteqdot Graph(T_{x}),\, x\in X_{0} \\ \Phi \doteqdot \{ \phi\in\Gamma^{x_{\infty}} (\pi_{{\mathbf{E}}^{\oplus}}) \mid (\forall x\in X_{0}) (\phi(x)\in{\mathcal{T}}_{0}(x)) \} \\ {\mathcal{E}} \doteqdot \{ v \in \Gamma(\pi) \mid (\exists\,\phi\in\Phi) (v(x_{\infty})=\phi_{1}(x_{\infty})) \} \\ \Theta \doteqdot \left\{ B_{w} \mid w\in{\mathcal{E}} \right\}, \end{cases}$$ where ${\left\langle {\left\langle {\mathfrak{E}}({\mathbf{E}}^{\oplus}),\tau({\mathbf{E}}^{\oplus},{\mathcal{E}}^{\oplus}){\right\rangle}},\pi_{{\mathbf{E}}^{\oplus}},X,{\mathfrak{n}}^{\oplus}{\right\rangle}}$ is the bundle direct sum of the family $\{{\mathfrak{V}}, {\mathfrak{V}}\}$. The following is a direct generalization of the definition given in [@kurtz Lm. $2.11$] \[14531903\] Let $ {\mathfrak{V}} \doteqdot {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}} $ be a bundle of $\Omega-$spaces, where ${\mathfrak{N}}\doteqdot\{\nu_{j}\mid j\in J\}$. Moreover let $Y$ be a topological space, $s_{0}\in Y$, $f\in \prod_{x\in X}{\mathfrak{E}}_{x}^{Y}$ and $\{z_{n}\}_{n\in{\mathbb{N}}}\subset X$. Then we say that $\{f(z_{n})\}_{n\in{\mathbb{N}}}$ is *bounded* if $\sup_{(n,s)\in{\mathbb{N}}\times Y} \nu_{j}(f(z_{n})(s))<\infty$ for all $j\in J$. $\{f(z_{n})\}_{n\in{\mathbb{N}}}$ is *equicontinuous at $s_{0}$* if for all $j\in J$ and for all ${\varepsilon}>0$ there exists a neighbourhood $U$ of $s_{0}$ such that for all $s\in U$ we have $\sup_{n\in{\mathbb{N}}}\nu_{j}(f(z_{n})(s) -f(z_{n})(s_{0}))\leq{\varepsilon}$. Finally $\{f(z_{n})\}_{n\in{\mathbb{N}}}$ is *equicontinuous* if $\{f(z_{n})\}_{n\in{\mathbb{N}}}$ is equicontinuous at $s$ for every $s\in Y$. \[19492307\] Let us assume the notations of Def. \[15210503\] and that ${\mathfrak{V}}$ is full. Thus $\{v(x_{\infty})\mid v\in{\mathfrak{E}}\} = \{\phi_{1}(x_{\infty})\mid\phi\in\Phi\}$. By definition follows the inclusion $\subseteq$. ${\mathfrak{V}}$ being full we have $(\forall\phi\in\Phi) (\exists\,v\in\Gamma(\pi)) (v(x_{\infty})=\phi_{1}(x_{\infty}))$. Thus $(\forall\phi\in\Phi) (\exists\,v\in{\mathcal{E}}) (v(x_{\infty})=\phi_{1}(x_{\infty}))$ hence the inclusion $\supseteq$. \[17301812b\] Let ${\mathfrak{V}} \doteqdot {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,\|\cdot\|{\right\rangle}}$ be a Banach bundle. Let $x_{\infty}\in X$ and $ {\mathcal{U}}_{0} \in\prod_{x\in X_{0}} {\mathcal{C}_{} \left({\mathbb {R}}^{+},B_{s}({\mathfrak{E}}_{x})\right)}$ be such that ${\mathcal{U}}_{0}(x)$ is a $(C_{0})-$semigroup of contractions (respectively of isometries) on ${\mathfrak{E}}_{x}$ for all $x\in X_{0}$. $D(T_{x_{\infty}})$ is dense in ${\mathfrak{E}}_{x_{\infty}}$ and $\exists\lambda_{0}>0$ (respectively $\exists\lambda_{0}>0, \lambda_{1}<0$) such that the range ${\mathcal{R}}(\lambda_{0}-T_{x_{\infty}})$ is dense in ${\mathfrak{E}}_{x_{\infty}}$, (respectively the ranges ${\mathcal{R}}(\lambda_{0}-T_{x_{\infty}})$ and ${\mathcal{R}}(\lambda_{1}-T_{x_{\infty}})$ are dense in ${\mathfrak{E}}_{x_{\infty}}$), $${\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}} \in Graph\left({\mathfrak{V}},{\mathfrak{V}}\right),$$ and $T_{x_{\infty}}$ in is the generator of a $C_{0}-$semigroup of contractions (respectively of isometries) on ${\mathfrak{E}}_{x_{\infty}}$. Moreover that $\{v(x)\mid v\in{\mathcal{E}}\}$ is dense in ${\mathfrak{E}}_{x}$ for all $x\in X_{0}$, by taking the notations in , let ${\mathfrak{W}} \doteqdot {\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$ and ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure [^14] such that holds. Assume $ {\mathbf{U}}_{\|\cdot\|_{B({\mathfrak{E}}_{z})}} ({\mathcal{L}}_{S_{z}}({\mathfrak{E}}_{z})) \subseteq {\mathfrak{M}}_{z}$ (respectively ${\mathbf{U}}_{is} ({\mathcal{L}}_{S_{z}}({\mathfrak{E}}_{z})) \subseteq {\mathfrak{M}}_{z}$) for all $z\in X$ [^15] and that there exists $F\in\Gamma(\rho)$ such that $F(x_{\infty})={\mathcal{U}}(x_{\infty})$ and i : ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ has the ${\mathbf{LD}}_{x_{\infty}}(\{F\},{\mathcal{E}})$; **or** it has the ${\mathbf{LD}}(\{F\},{\mathcal{E}})$; ii : $(\forall v\in{\mathcal{E}}) (\exists\,\phi\in\Phi)$ s.t. $\phi_{1}(x_{\infty}) = v(x_{\infty})$ and $(\forall\{z_{n}\}_{n\in{\mathbb{N}}}\subset X \mid \lim_{n\in{\mathbb{N}}}z_{n}=x_{\infty})$ we have that $\{ {\mathcal{U}}(z_{n})(\cdot)\phi_{1}(z_{n}) - F(z_{n})(\cdot)v(z_{n}) \}_{n\in{\mathbb{N}}}$ is a bounded equicontinuous sequence; iii : $X$ is metrizable. $(\forall v\in{\mathcal{E}}) (\forall K\in Compact({\mathbb {R}}^{+}))$ $$\label{02502912} \boxed{ \lim_{z\to x_{\infty}} \sup_{s\in K} \left\| {\mathcal{U}}(z)(s)v(z) - F(z)(s)v(z) \right\|=0,}$$ and $$\label{02512912p} \boxed{ {\mathcal{U}} \in\Gamma^{x_{\infty}}(\rho) .}$$ In particular $$\label{02512912} \{{\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}\} \in \Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}.$$ Here ${\mathcal{T}}$ and $D(T_{x_{\infty}})$ are defined as in Notations \[15411512b\] with ${\mathcal{T}}_{0}$ and $\Phi$ given in , while ${\mathcal{U}} \in\prod_{x\in X} {\mathfrak{M}}_{x}$ such that ${\mathcal{U}}{\upharpoonright}X_{0}\doteqdot {\mathcal{U}}_{0}$ and ${\mathcal{U}}(x_{\infty})$ is the semigroup on ${\mathfrak{E}}_{x_{\infty}}$ generated by $T_{x_{\infty}}$. By Lemma \[13001512b\], [@kurtz Lms.$(2.8)-(2.9)$], and the Hille-Yosida theorem, see [@kurtz Th.$(1.2)$], we have the first sentence of the statement for the case of semigroup of contractions. By [@bra Corollary $3.1.19.$] applied to $T_{x}$, for any $x\in X_{0}$, and by we have $(\forall\lambda\in{\mathbb {R}}) (\forall v_{x_{\infty}}\in Dom(T_{x_{\infty}}))$ $$\label{15363001} \|({\mathbf{1}}-\lambda T_{x_{\infty}}) v_{x_{\infty}})\|_{x_{\infty}} \geq \|v_{x_{\infty}}\|_{x_{\infty}}.$$ Hence by [@bra Corollary $3.1.19.$], $T_{x_{\infty}}$ will be a generator of a strongly continuous semigroup of isometries if we show that $\forall\lambda\in{\mathbb {R}}-\{0\}$ $$\label{15443001} {\mathcal{R}}({\mathbf{1}}-\lambda T_{x_{\infty}}) = {\mathfrak{E}}_{x_{\infty}}.$$ Let us set $$\rho_{0}(T_{x_{\infty}}) \doteqdot \{ \lambda\in {\mathbb {R}}-\{0\} \mid {\mathcal{R}}({\mathbf{1}}-\lambda T) = {\mathfrak{E}}_{x}\}.$$ By $\rho_{0}(T_{x_{\infty}}) = \rho(T_{x_{\infty}}) \cap ({\mathbb {R}}-\{0\})$, where $\rho(T_{x_{\infty}})$ is the resolvent set of $T_{x}$. By [@ds Lemma $7.3.2$] $\rho(T_{x_{\infty}})$ is open in ${\mathbb {C}}$ so $\rho_{0}(T_{x_{\infty}})$ is open in ${\mathbb {R}}-\{0\}$ with respect to the topology on ${\mathbb {R}}-\{0\}$ induced by that on ${\mathbb {C}}$. By Lemma \[14253001\] we deduce that $\rho_{0}(T_{x_{\infty}})$ is also closed in ${\mathbb {R}}-\{0\}$, therefore $\rho_{0}(T_{x_{\infty}})={\mathbb {R}}-\{0\}$ and follows as well that $T_{x_{\infty}}$ is a generator of a strongly continuous semigroup of isometries. Now we shall apply Lemma \[15482712\] in order to show the remaining part of the statement. By the Dupre’ Thm., see for example [@kurtz Cor. $2.10$], and the fact that a metrizable space is completely regular, we deduce by hyp. $(iii)$ that ${\mathfrak{V}}$ is full. Let $v\in{\mathcal{E}}$ be fixed then by , $(\exists\,\phi\in\Phi) (v(x_{\infty})=\phi_{1}(x_{\infty}))$ thus by and Corollary \[28111707\] $$\label{20352312b} \lim_{z\to x_{\infty}} \|v(z)-\phi_{1}(z)\| =0.$$ Now let $F\in\Gamma(\rho)$ of which in hypothesis so in particular $$\label{10182312b} F(x_{\infty})={\mathcal{U}}(x_{\infty}),$$ moreover $\forall s\in{\mathbb {R}}^{+}$ and $z\in X$ $$\begin{aligned} {2} \label{20492312b} \| {\mathcal{U}}(z)(s)v(z) - F(z)(s)v(z) \| & \leq \notag \\ \| {\mathcal{U}}(z)(s)v(z) - {\mathcal{U}}(z)(s)\phi_{1}(z) \| + \|{\mathcal{U}}(z)(s)\phi_{1}(z)-F(z)(s)v(z)\| & \leq \notag \\ \|v(z)-\phi_{1}(z)\| + \|{\mathcal{U}}(z)(s)\phi_{1}(z)-F(z)(s)v(z)\|.\end{aligned}$$ For any $\lambda>0$ let us set $$g_{\infty}^{\lambda} \doteqdot (\lambda-T_{x_{\infty}})^{-1} \phi_{1}(x_{\infty})$$ thus $g_{\infty}^{\lambda} \in Dom(T_{x_{\infty}})$ hence by Remark \[18581512b\] and the construction of $T_{x_{\infty}}$ $ \exists\, \psi^{\lambda} \in \Phi $ such that $$\label{09252412b} \begin{cases} g_{\infty}^{\lambda} = \psi_{1}^{\lambda}(x_{\infty}) = \lim_{z\in x_{\infty}} \psi_{1}^{\lambda}(z) \\ T_{x_{\infty}} g_{\infty}^{\lambda} = \lim_{z\to x_{\infty}} T_{z} \psi_{1}^{\lambda}(z). \end{cases}$$ By and for all $z\in X$ and for all $w_{z}\in \bigcup_{v\in{\mathcal{E}}}v(z)$ $$\label{15261001pre} \left(\int_{0}^{\infty} e^{-\lambda s} F(z)(s) \,ds\right) w_{z} = \int_{0}^{\infty} e^{-\lambda s} F(z)(s)w_{z}\,ds.$$ Moreover by the fact that ${\mathfrak{V}}$ is full we have that for all $\phi\in\Phi$ there exists a $v\in\Gamma(\pi)$ such that $v(x_{\infty})=\phi_{1}(x_{\infty})$, thus by construction of ${\mathcal{E}}$ $$\label{18390103} (\forall\phi\in\Phi) (\exists\,v\in{\mathcal{E}}) (v(x_{\infty})=\phi_{1}(x_{\infty})).$$ Hence by , and for all $\phi\in\Phi$ $$\label{15261001} \left(\int_{0}^{\infty} e^{-\lambda s} F(x_{\infty})(s) \,ds\right) \phi_{1}(x_{\infty}) = \int_{0}^{\infty} e^{-\lambda s} {\mathcal{U}}(x_{\infty})(s) \phi_{1}(x_{\infty}) \,ds.$$ Now set $$\xi \doteqdot {\mathfrak{L}}(F),$$ thus by hypothesis $(i)$ we have for all $\lambda>0$ $$\label{11462412b} \xi(\cdot)(\lambda)v(\cdot) \in \Gamma^{x_{\infty}}(\pi).$$ Moreover $$\begin{aligned} {2} \label{10262412b} \xi(x_{\infty})(\lambda) v(x_{\infty}) & = \xi(x_{\infty})(\lambda) \phi_{1}(x_{\infty}) \notag \\ & = \int_{0}^{\infty} e^{-\lambda s} {\mathcal{U}}(x_{\infty})(s) \phi_{1}(x_{\infty}) \,ds\, \text{ by } \eqref{15261001} \notag \\ & = (\lambda-T_{x_{\infty}})^{-1} \phi_{1}(x_{\infty})\, \text{ by \cite[$(1.3)$]{kurtz}} \notag \\ & \doteq g_{\infty}^{\lambda} = \psi_{1}^{\lambda}(x_{\infty}) \, \text{ by } \eqref{09252412b}.\end{aligned}$$ By the fact that ${\mathfrak{V}}$ is full, by , the fact that $\psi_{1}^{\lambda} \in\Gamma^{x_{\infty}}(\pi)$ by , by and by Corollary \[21492812\] we have $\forall\lambda>0$ $$\label{10352412b} \lim_{z\to x_{\infty}} \|\psi_{1}^{\lambda}(z)- \xi(z)(\lambda)v(z))\| =0.$$ Now $(\forall\lambda>0)(\forall z\in X)$ set $$w^{\lambda}(z) \doteqdot (\lambda{\mathbf{1}}-T_{z}) \psi_{1}^{\lambda}(z),$$ thus $$\begin{aligned} {2} \label{12042412b} \left\| \int_{0}^{\infty} e^{-\lambda s} \left( {\mathcal{U}}(z)(s)\phi_{1}(z)-F(z)(s)v(z) \right) \,ds \right\| & \leq \notag \\ \left\| \int_{0}^{\infty} e^{-\lambda s} {\mathcal{U}}(z)(s)(\phi_{1}(z)-w^{\lambda}(z)) \,ds \right\| + \left\| \int_{0}^{\infty} e^{-\lambda s} \left( {\mathcal{U}}(z)(s)w^{\lambda}(z)-F(z)(s)v(z) \right) \,ds \right\| & \leq \notag \\ \frac{1}{\lambda} \|\phi_{1}(z)-w^{\lambda}(z)\| + \|\psi_{1}^{\lambda}(z)- \xi(z)(\lambda)v(z)) &\|.\end{aligned}$$ Here we consider that by hypothesis and by the first part of the statemet $\|{\mathcal{U}}(z)\|\leq 1$ for all $z\in X$, moreover we applied the Hille-Yosida formula [@kurtz $(1.3)$]. Now $$\begin{aligned} {1} \label{11272412b} \|\phi_{1}(z)-w^{\lambda}(z)\| & = \\ \| \phi_{1}(z)- (\lambda{\mathbf{1}}-T_{z}) \psi_{1}^{\lambda}(z) \| & \leq \notag \\ \| \phi_{1}(z)-v(z) \| + \|v(z)-\lambda\xi(z)(\lambda)v(z) + \lambda\xi(z)(\lambda)v(z) -(\lambda{\mathbf{1}}-T_{z}) \psi_{1}^{\lambda}(z) \| & \leq \notag \\ \|\phi_{1}(z)-v(z)\| + \lambda \|\xi(z)(\lambda)v(z)-\psi_{1}^{\lambda}(z)\| + \| T_{z}\psi_{1}^{\lambda}(z) -(\lambda\xi(z)(\lambda)v(z)-v(z)) & \|. \notag\end{aligned}$$ By $ T_{x_{\infty}} \psi_{1}^{\lambda}(x_{\infty}) = T_{x_{\infty}} g_{\infty}^{\lambda} $ moreover $$\begin{aligned} {1} \label{11402412b} T_{x_{\infty}} g_{\infty}^{\lambda} & = -(\lambda-T_{x_{\infty}}) g_{\infty}^{\lambda} +\lambda g_{\infty}^{\lambda} \notag \\ & = -(\lambda-T_{x_{\infty}}) (\lambda-T_{x_{\infty}})^{-1} \phi_{1}(x_{\infty}) +\lambda g_{\infty}^{\lambda} \notag \\ & = \lambda g_{\infty}^{\lambda} - \phi_{1}(x_{\infty}) = \lambda\xi(x_{\infty})(\lambda) v(x_{\infty}) - v(x_{\infty}),\end{aligned}$$ where in the last equality we used and the construction of $\phi$. By we have that $(X\ni z \mapsto T_{z} \psi_{1}^{\lambda}(z)) \in \Gamma^{x_{\infty}}(\pi) $, hence by , the fact that $\lambda\xi(\cdot)(\lambda)v(\cdot) - v\in\Gamma^{x_{\infty}}(\pi) $ by , we deduce by the fact that ${\mathfrak{V}}$ is full and by Corollary \[21492812\] that $\forall\lambda>0$ $$\label{11542412b} \lim_{z\to x_{\infty}} \|T_{z}\psi_{1}^{\lambda}(z) -(\lambda\xi(z)(\lambda)v(z)-v(z))\|=0.$$ Therefore by , , and $$\lim_{z\to x_{\infty}} \|\phi_{1}(z)-w^{\lambda}(z)\| =0.$$ By this one along with we can state by using that $\forall\lambda>0$ $$\lim_{z\to x_{\infty}} \left\| \int_{0}^{\infty} e^{-\lambda s} \left( {\mathcal{U}}(z)(s)\phi_{1}(z)-F(z)(s)v(z) \right) \,ds \right\| =0.$$ Therefore $\forall\lambda>0$ and $(\forall\{z_{n}\}_{n\in{\mathbb{N}}}\subset X \mid \lim_{n\in{\mathbb{N}}}z_{n}=x_{\infty})$ $$\label{12052412b} \lim_{n\in{\mathbb{N}}} \left\| \int_{0}^{\infty} e^{-\lambda s} \left( {\mathcal{U}}(z_{n})(s)\phi_{1}(z_{n})- F(z_{n})(s)v(z_{n}) \right) \,ds \right\| =0.$$ By , hypothesis $(ii)$ and [@kurtz Lemma $(2.11)$] we have $(\forall\{z_{n}\}_{n\in{\mathbb{N}}}\subset X \mid \lim_{n\in{\mathbb{N}}}z_{n}=x_{\infty})$ and $\forall K\in Compact({\mathbb {R}}^{+})$ $$\lim_{n\in{\mathbb{N}}} \sup_{s\in K} \left\| {\mathcal{U}}(z_{n})(s)\phi_{1}(z_{n}) - F(z)(s)v(z_{n}) \right\|=0.$$ Therefore by hypothesis $(iii)$ and [@BourGT Prop.$10$, $\S2.6$, Ch. $9$], $\forall K\in Compact({\mathbb {R}}^{+})$ $$\label{12102412b} \lim_{z\to x_{\infty}} \sup_{s\in K} \left\| {\mathcal{U}}(z)(s)\phi_{1}(z) - F(z)(s)v(z) \right\|=0.$$ In conclusion by , and we obtain $\forall K\in Compact({\mathbb {R}}^{+})$ $$\label{02012912} \lim_{z\to x_{\infty}} \sup_{s\in K} \left\| {\mathcal{U}}(z)(s)v(z) - F(z)(s)v(z) \right\|=0,$$ hence . By and we obtain . Thus and follow by Lemma \[15482712\], by and by the following one $\forall K\in Compact({\mathbb {R}}^{+})$ and $\forall v\in{\mathcal{E}}$ $$\sup_{z\in X} \sup_{s\in K} \left\| {\mathcal{U}}(z)(s)v(z) \right\| \leq \sup_{z\in X} \|v(z)\| <\infty.$$ where we considered that by construction $\left\| {\mathcal{U}}(z)(s) \right\|\leq 1$, for all $s\in{\mathbb {R}}^{+}$ and $z\in X$ and that $v\in\Gamma(\pi)$. \[17341602\] If ${\mathfrak{W}}$ is full $(\exists\,F\in\Gamma(\rho)) (F(x_{\infty})={\mathcal{U}}(x_{\infty}))$, so hypotheses reduce. Corollary I.Constructions of Equicontinuous sequence ---------------------------------------------------- As the first corollary we give conditions in order to satisfy the bounded equicontinuity of which in hypothesis $(ii)$. \[21343012\] Let us assume the hypotheses of Theorem \[17301812b\] except $(ii)$ replaced by the following one $$(\exists\,G\in\prod_{z\in X} {\mathfrak{L}}_{1}\left({\mathbb {R}}^{+}, {\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})\right) (\exists\,H\in \prod_{z\in X}^{b}{\mathcal{L}}({\mathfrak{E}}_{z})) (\exists\,F\in\Gamma(\rho))$$ such that $F(x_{\infty})={\mathcal{U}}(x_{\infty})$ and $\forall s>0$ $$\label{22053012} \begin{cases} \sup_{x\in X}\sup_{s>0}\|F(x)(s)\|<\infty \\ (\forall s_{1}>0) (\exists\, a>0) (\sup_{u\in[s_{1},s]} \sup_{z\in X} \|G(z)(u)\|\leq a |s-s_{1}|) \\ (\forall z\in X) (F(z)(s) = H(z)+ \int_{0}^{s} G(z)(u)\,du), \end{cases}$$ where the integration is with respect to the Lebesgue measure on $[0,s]$ and with respect to the $lct$ on ${\mathcal{L}}_{S_{z}}({\mathfrak{E}}_{z})$. Then holds the statement of Theorem \[17301812b\]. Let $v\in{\mathcal{E}}$ thus $(\exists\,\phi\in\Phi) (v(x_{\infty})=\phi_{1}(x_{\infty}))$ so $(\forall\{z_{n}\}_{n\in{\mathbb{N}}}\subset X \mid \lim_{n\in{\mathbb{N}}}z_{n}=x_{\infty})$ we have $$\sup_{n\in{\mathbb{N}}} \sup_{s>0} \|{\mathcal{U}}(z_{n})(s) \phi_{1}(z_{n}) - F(z_{n})(s)v(z_{n}) \| \leq \sup_{n\in{\mathbb{N}}} \|\phi_{1}(z_{n})\| + M \sup_{n\in{\mathbb{N}}} \|v(z_{n})\| <\infty.$$ Here in the first inequality we used $\|{\mathcal{U}}(z)(s)\|\leq 1$ for all $z\in X$ and $s>0$ by construction, and $M\doteqdot \sup_{z\in X}\sup_{s>0}\|F(z)(s)\|<\infty$ by hypothesis, while in the second inequality we used the fact that $v\in\prod_{x\in X}^{b}{\mathfrak{E}}_{x}$, by construction and that $\sup_{n\in{\mathbb{N}}} \|\phi_{1}(z_{n})\|<\infty$ because of $\exists\, \varlimsup_{n\in{\mathbb{N}}}\|\phi_{1}(z_{n})\| \in{\mathbb {R}}$ by Remark \[18581512b\] and by construction $\|\cdot\|$ is $u.s.c.$ Moreover by [@kurtz $(1.4)$], and $S_{x}= \{\{w(x)\}\mid w\in{\mathcal{E}}\}$ for all $x\in X$ we have $$\begin{aligned} {1} {\mathcal{U}}(z_{n})(s) \phi_{1}(z_{n}) - F(z_{n})(s)v(z_{n}) & = \int_{0}^{s} \left( {\mathcal{U}}(z_{n})(u) T_{z_{n}} \phi_{1}(z_{n}) - G(z_{n})(u)v(z_{n}) \right) \,du + \\ & + \phi_{1}(z_{n}) - H(z_{n})v(z_{n}).\end{aligned}$$ Thus for any $s_{1},s_{2}\in{\mathbb {R}}^{+}$ $$\begin{aligned} {1} \sup_{n\in{\mathbb{N}}} \| ({\mathcal{U}}(z_{n})(s_{1}) \phi_{1}(z_{n}) - F(z_{n})(s_{1})v(z_{n})) - ({\mathcal{U}}(z_{n})(s_{2}) \phi_{1}(z_{n}) - F(z_{n})(s_{2})v(z_{n})) \| & \leq \\ |s_{1}-s_{2}| \sup_{n\in{\mathbb{N}}} \sup_{u\in[s_{1},s_{2}]} \|{\mathcal{U}}(z_{n})(u) T_{z_{n}} \phi_{1}(z_{n}) - G(z_{n})(u)v(z_{n}) \| & \leq \\ |s_{1}-s_{2}| \sup_{n\in{\mathbb{N}}} (\|T_{z_{n}}\phi_{1}(z_{n})\| -a\|v(z_{n})\|) & \leq J|s_{1}-s_{2}|.\end{aligned}$$ Here in the second inequality we used $\|{\mathcal{U}}(z)(u)\|\leq 1$ by construction and the hypothesis, in the third one the fact that $\sup_{n\in{\mathbb{N}}} \|T_{z_{n}}\phi_{1}(z_{n})\| <\infty$ as well $\sup_{n\in{\mathbb{N}}}\|v(z_{n})\|<\infty$, because of $\exists\, \varlimsup_{n\in{\mathbb{N}}} \|T_{z_{n}}\phi_{1}(z_{n})\| \in{\mathbb {R}}$ and $ \exists\, \varlimsup_{n\in{\mathbb{N}}}\|v(z_{n})\| \in{\mathbb {R}}$ due to the fact that $\|\cdot\|$ is $u.s.c.$ by construction and Remark \[18581512b\] for the first limit and the continuity of $v$ for the second one. Therefore hypothesis $(ii)$ of Theorem \[17301812b\] is satisfied, hence the statement follows by Theorem \[17301812b\]. Corollaries II. Construction of ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ with the ${\mathbf{LD}}$ {#13320803} ------------------------------------------------------------------------------------------------------------------------------------------- Let us start with the following simple result about the relation among full and Laplace duality property. \[19030106\] Let ${\mathfrak{W}} \doteqdot {\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$ and ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure such that ${\mathfrak{V}}$ is a Banach bundle and $x_{\infty}\in X$. Assume that 1. ${\mathfrak{V}}$ and ${\mathfrak{W}}$ are full; 2. ${\mathcal{E}}=\Gamma(\pi)$ and $\Theta$ is given in ; 3. $(\forall F\in\Gamma^{x_{\infty}}(\rho)) (M(F)\doteqdot \sup_{x\in X} \sup_{s\in{\mathbb {R}}^{+}} \|F(x)(s)\|<\infty)$; 4. $(\forall\sigma\in\Gamma(\rho)) (\sup_{x\in X} \sup_{s\in{\mathbb {R}}^{+}} \|\sigma(x)(s)\|<\infty)$; 5. $X$ is metrizable. If ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ has the ${\mathbf{LD}}$ then it has the ${\mathbf{LD}}_{x_{\infty}}$. Let $F\in\Gamma^{x_{\infty}}(\rho)$ and $w\in\Gamma^{x_{\infty}}(\pi)$ thus by hypothesis $(2)$ and Corollary \[28111707\] there exist $\sigma\in\Gamma(\rho)$ and $v\in\Gamma(\pi)$ such that $\sigma(x_{\infty})=F(x_{\infty})$, $v(x_{\infty})=w(x_{\infty})$, and $\forall K\in Comp({\mathbb {R}}^{+})$, $\forall v\in{\mathcal{E}}$ $$\label{19330106} \begin{cases} \lim_{z\to x_{\infty}} \|w(z)-v(z)\|=0 \\ \lim_{z\to x_{\infty}} \sup_{s\in K} \|(F(x)(s)-\sigma(x)(s))v(x)\| &=0. \end{cases}$$ Moreover $\forall\lambda>0$ $$\begin{aligned} {1} \label{20090106} \left\| \int_{0}^{\infty} e^{\lambda s} F(z)(s)w(z)\,ds - \int_{0}^{\infty} e^{\lambda s} \sigma(z)(s)v(z)\,ds \right\| &\leq \notag \\ \left\| \int_{0}^{\infty} e^{\lambda s} F(z)(s)(w(z)-v(z))\,ds \right\| + \left\| \int_{0}^{\infty} e^{\lambda s} (F(z)(s)-\sigma(z)(s))v(z)\,ds \right\| &\leq \notag \\ \frac{1}{\lambda}M(F)\|v(z)-w(z)\| + \int_{0}^{\infty} e^{\lambda s} \left\| (F(z)(s)-\sigma(z)(s))v(z) \right\|\,ds.\end{aligned}$$ By the hypotheses $(3-4)$ $\sup_{z\in X} \sup_{s\in{\mathbb {R}}^{+}} \left\|(F(z)(s)-\sigma(z)(s))v(z)\right\| <\infty$ hence $\forall\{z_{n}\}_{n\in{\mathbb{N}}}\subset X$ such that $\lim_{n\in{\mathbb{N}}}z_{n}=x_{\infty}$ we have by , and a well-known theorem on convergence of sequences of integrals that $\forall\lambda>0$ $$\lim_{n\in{\mathbb{N}}} \left\| \int_{0}^{\infty} e^{\lambda s} F(z_{n})(s)w(z_{n})\,ds - \int_{0}^{\infty} e^{\lambda s} \sigma(z_{n})(s)v(z_{n})\,ds \right\| =0.$$ Thus $\forall\lambda>0$ by hypothesis $(5)$ $$\lim_{z\to x_{\infty}} \left\| \int_{0}^{\infty} e^{\lambda s} F(z)(s)w(z)\,ds - \int_{0}^{\infty} e^{\lambda s} \sigma(z)(s)v(z)\,ds \right\| =0,$$ hence the statement by Corollary \[28111707\]. Now we shall see that in the case of a bundle of normed space we can choose for all $x$ a simple space ${\mathfrak{M}}_{x}$ satisfying . \[18390901\] Let ${\mathfrak{W}} \doteqdot {\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$ and ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure such that for all $x\in X$, ${\mathfrak{E}}_{x}$ is a reflexive Banach space, $S_{x}\subseteq{\mathcal{P}}_{\omega}({\mathfrak{E}}_{x})$ and $${\mathfrak{M}}_{x} \subseteq \left\{ F\in {\mathcal{C}_{c} \left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})\right)} \mid (\forall\lambda>0) \left(\int_{{\mathbb {R}}^{+}}^{*} e^{-\lambda s} \|F(s)\|_{B({\mathfrak{E}}_{x})} \,ds <\infty\right) \right\}.$$ Thus $$\label{14441001} {\mathfrak{M}}_{x} \subset \bigcap_{\lambda>0} {\mathfrak{L}}_{1} ({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x}); \mu_{\lambda}).$$ In particular , and ${\mathbf{U}}_{\|\cdot\|_{B({\mathfrak{E}}_{x})}} ({\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})) \subseteq {\mathfrak{M}}_{x}$ hold if for any $x\in X$ $${\mathfrak{M}}_{x} = \left\{ F\in {\mathcal{C}_{c} \left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})\right)} \mid \sup_{s\in{\mathbb {R}}^{+}} \|F(s)\|_{B({\mathfrak{E}}_{x})} <\infty \right\}.$$ The first sentence follows by [@SilInt Corollary $2.6.$], while the second sentence comes by the first one. ### ${\mathbf{U}}-$Spaces \[12042101\] First of all recall that for any $W,Z$ topological vector spaces over ${\mathbb{K}}\in\{{\mathbb {R}},{\mathbb {C}}\}$ we denote by ${\mathcal{L}}(W,Z)$ the ${\mathbb{K}}-$linear space of all continuous linear map on $W$ and with values in $Z$ and set ${\mathcal{L}}(Z) \doteqdot {\mathcal{L}}(Z,Z)$ and $Z^{*} \doteqdot {\mathcal{L}}(Z,{\mathbb{K}})$. In this section we assume fixed the following data: 1. a set $X$, a locally compact space $Y$ and a Radon measure $\mu$ on $Y$; 2. a family $\{{\mathfrak{E}}_{x}\}_{x\in X}$ of $Hlcs$; 3. a family $\{\tau_{x}\}_{x\in X}$ such that ${\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}} \in Hlcs$, $\forall x\in X$; 4. a family $\{{\mathfrak{N}}_{x}\}_{x\in X}$ such that ${\mathfrak{N}}_{x}\doteqdot\{\nu_{j_{x}}^{x} \mid j_{x}\in J_{x}\}$ is a fundamental set of seminorms on ${\mathfrak{E}}_{x}$, $\forall x\in X$; 5. a family $\{Q_{x}\}_{x\in X}$ such that $Q_{x}\doteqdot\{q_{\alpha_{x}}^{x} \mid \alpha_{x}\in A_{x}\}$ is a fundamental set of seminorms on ${\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}}$, $\forall x\in X$; 6. ${\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}}\in Hlcs$ such that - ${\mathcal{H}} \subseteq \prod_{x\in X}{\mathfrak{E}}_{x}$ as linear spaces; - $\imath_{x}({\mathfrak{E}}_{x}) \subset{\mathcal{H}}$, for all $x\in X$; - $\Pr_{x}\in {\mathcal{L}}\left( {\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}}, {\mathfrak{E}}_{x} \right)$ and $\imath_{x}\in {\mathcal{L}}\left({\mathfrak{E}}_{x}, {\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}}\right)$, for all $x\in X$; - $\exists\,{\mathcal{A}}\subseteq \prod_{x\in X}{\mathcal{L}}({\mathfrak{E}}_{x})$ linear space such that 1. $\theta({\mathcal{A}}){\upharpoonright}{\mathcal{H}} \subseteq {\mathcal{L}}({\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}})$, 2. $\imath_{x} ({\mathcal{L}}({\mathfrak{E}}_{x})) \subseteq{\mathcal{A}}$ for all $x\in X$. Here $\theta$ is defined in Def. \[13021401\]. For any $Z\in Hlcs$ we denote by ${\mathfrak{L}}_{1}(Y,Z,\mu)$ the linear space of all maps on $Y$ and with values in $Z$ which are essentially $\mu-$integrable in the sense described in [@IntBourb Ch. $6$]. While ${\mathcal{C}_{cs} \left(Y,Z\right)}$ denotes the linear space of all continuous maps $f:Y\to Z$ with compact support. Moreover for any family $\{Z_{x}\}_{x\in X}$ of linear spaces and for all $x\in X$ set $\Pr_{x}: \prod_{y\in X}Z_{y} \ni f\mapsto f(x) \in Z_{x}$ and $\imath_{x}: Z_{x} \to \prod_{y\in X}Z_{y} $ such that for all $x\neq y$ and $z_{x}\in Z_{x}$ $\Pr_{y}\circ\imath_{x}(z_{x})={\mathbf{0}}_{y}$, while $\Pr_{x}\circ\imath_{x}=Id_{x}$. Finally we set $${\left\langle \cdot,\cdot{\right\rangle}}: End({\mathcal{H}}) \times {\mathcal{H}} \ni (A,v) \mapsto A(v) \in {\mathcal{H}},$$ and for all $x\in X$ $${\left\langle \cdot,\cdot{\right\rangle}}_{x}: End\left({\mathfrak{E}}_{x}\right) \times {\mathfrak{E}}_{x} \ni (A,v) \mapsto A(v)\in {\mathfrak{E}}_{x}.$$ \[12182503\] Let ${\mathfrak{W}} \doteqdot {\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$ be a bundle of $\Omega-$spaces such that for all $x\in X$ $${\mathfrak{M}}_{x} \subseteq {\mathfrak{L}}_{1}(Y, {\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}};\mu).$$ Set $$\label{14162503} \begin{cases} \blacksquare_{\mu}: \prod_{x\in X} {\mathfrak{L}}_{1}(Y, {\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}};\mu) \times \prod_{x\in X} {\mathfrak{E}}_{x} \to \prod_{x\in X} {\mathfrak{E}}_{x} \\ \blacksquare_{\mu} (H,v)(x) \doteqdot {\left\langle \int_{{\mathbb {R}}^{+}} H(x)(s)\, d\mu(s),v(x){\right\rangle}}_{x} \in {\mathfrak{E}}_{x}. \end{cases}$$ \[14092503\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure satisfying and ${\mathcal{O}}\subseteq\Gamma(\rho)$, ${\mathcal{D}}\subseteq\Gamma(\pi)$. Then $$\label{LD} {\mathbf{LD}}({\mathcal{O}},{\mathcal{D}}) \Leftrightarrow (\forall\lambda>0) \left( \blacksquare_{\mu_{\lambda}} \left({\mathcal{O}},{\mathcal{D}}\right) \subseteq \Gamma(\pi) \right).$$ Similarly for all $x\in X$ $$\label{LDx} {\mathbf{LD}}_{x}({\mathcal{O}},{\mathcal{D}}) \Leftrightarrow (\forall\lambda>0) \left( \blacksquare_{\mu_{\lambda}} \left(\Gamma_{{\mathcal{O}}}^{x}(\rho) ,\Gamma_{{\mathcal{D}}}^{x}(\pi) \right) \subseteq \Gamma^{x}(\pi) \right).$$ \[ ${\mathbf{U}}-$Spaces \] \[14302503\] ${\mathfrak{G}}$ is a ${\mathbf{U}}-$space with respect to $\{{\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}}\}_{x\in X}$, ${\mathfrak{T}}$ and $D$ iff 1. ${\mathfrak{G}}\in Hlcs$; 2. ${\mathfrak{G}} \subset {\mathcal{L}}\left({\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}}\right)$ as linear spaces; 3. $D\subseteq{\mathcal{H}}$; 4. $(\forall T\in lcp) \left(\exists\, \Psi_{T}\in End\left[ End({\mathcal{H}})^{T}, \prod_{x\in X} End({\mathfrak{E}}_{x})^{Y} \right]\right) (\forall\nu\in Radon(T))$ $$\Psi_{T}: {\mathfrak{L}}_{1}(T,{\mathfrak{G}},\nu) \to \prod_{x\in X} {\mathfrak{L}}_{1}\left(T, {\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}};\nu\right) \footnote{ Of course $\Psi_{T}$ here has to be understood as $\Psi_{T}{\upharpoonright}{\mathfrak{L}}_{1}(T,{\mathfrak{G}},\nu)$.} $$ and $\forall{\overline}{F} \in {\mathfrak{L}}_{1}(T,{\mathfrak{G}},\nu) $, $\forall v\in D$. $\forall x\in X$ $$\label{17432203} \boxed{ {\left\langle \int \Psi_{T}({\overline}{F})(x)(s) \,d\nu(s),v(x){\right\rangle}}_{x} = {\left\langle \int{\overline}{F}(s)\,d\nu(s),v{\right\rangle}}(x) }$$ The reason of introducing the concept of ${\mathbf{U}}-$spaces will be clarified by the following \[14492503\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure satisfying , and let ${\mathfrak{G}}$ be a ${\mathbf{U}}-$space with respect to $\{{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})\}_{x\in X}$, ${\mathfrak{T}}$ and ${\mathcal{D}}$. Then $\forall\lambda>0$, ${\overline}{F} \in {\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$, $v\in{\mathcal{D}}$ $$\label{15102503} \blacksquare_{\mu_{\lambda}} (\Psi_{{\mathbb {R}}^{+}}({\overline}{F}),v) = {\left\langle \int{\overline}{F}(s)\,d\mu_{\lambda}(s),v{\right\rangle}}.$$ Moreover if $\exists\,{\mathcal{F}}\subset \bigcap_{\lambda>0} {\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$ such that $\Psi_{{\mathbb {R}}^{+}}({\mathcal{F}})={\mathcal{O}}$ then $$\label{21002103} \boxed{ {\mathbf{LD}}({\mathcal{O}},{\mathcal{D}}) \Leftrightarrow (\forall\lambda>0) ({\left\langle {\mathcal{B}}_{\lambda},{\mathcal{D}}{\right\rangle}} \subseteq\Gamma(\pi)).}$$ Here $${\mathcal{B}}_{\lambda} \doteqdot \left\{ \int{\overline}{F}(s)\,d\mu_{\lambda}(s) \mid{\overline}{F}\in{\mathcal{F}} \right\}.$$ \[13390104\] In particular if $\exists\,{\mathcal{F}}\subset \bigcap_{\lambda>0} {\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$ such that $\Psi_{{\mathbb {R}}^{+}}({\mathcal{F}})={\mathcal{O}}$ then $${\left\langle {\mathfrak{G}},{\mathcal{D}}{\right\rangle}} \subseteq \Gamma(\pi) \Rightarrow {\mathbf{LD}}({\mathcal{O}},{\mathcal{D}}).$$ More in general if $\exists\,{\mathfrak{G}}_{0}$ complete subspace of ${\mathfrak{G}}$ and $\exists\,{\mathcal{F}}\subset \left\{ {\overline}{F}\in \bigcap_{\lambda>0} {\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda}) \mid {\overline}{F}({\mathbb {R}}^{+})\subseteq{\mathfrak{G}}_{0} \right\}$ such that $\Psi_{{\mathbb {R}}^{+}}({\mathcal{F}})={\mathcal{O}}$ then $${\left\langle {\mathfrak{G}}_{0},{\mathcal{D}}{\right\rangle}} \subseteq \Gamma(\pi) \Rightarrow {\mathbf{LD}}({\mathcal{O}},{\mathcal{D}}).$$ follows by , while follows by and Remark \[14092503\]. Thus the ${\mathbf{U}}$ property expressed by is an important tool for ensuring the satisfaction of the ${\mathbf{LD}}$. For this reason the remaining of the present section will be dedicated to the construction of a space ${\mathfrak{G}}$, Def. , which is a ${\mathbf{U}}-$space, see Theorem \[15332203\] and Corollary \[18491004\] for the ${\mathbf{LD}}({\mathcal{O}},{\mathcal{D}})$. \[13021401\] Set $$\begin{cases} \chi_{{\mathcal{H}}}: End({\mathcal{H}}) \to \prod_{x\in X}End({\mathfrak{E}}_{x}), \\ (\forall x\in X) (\forall w\in End({\mathcal{H}})) ((\Pr_{x}\circ\chi_{{\mathcal{H}}})(w) =\Pr_{x}\circ w\circ\imath_{x}), \\ \chi\doteqdot \chi_{\prod_{x\in X}{\mathfrak{E}}_{x}}. \end{cases}$$ Well defined indeed by construction $\imath_{x}({\mathfrak{E}}_{x}) \subset{\mathcal{H}}$, for all $x\in X$. Finally set $$\begin{cases} \theta: \prod_{x\in X}End({\mathfrak{E}}_{x}) \to End\left(\prod_{x\in X}{\mathfrak{E}}_{x}\right), \\ (\forall x\in X) (\forall u\in \prod_{x\in X}End({\mathfrak{E}}_{x})) (\Pr_{x}\circ\theta(u)=\Pr_{x}(u)\circ\Pr_{x}), \\ \theta_{{\mathcal{H}}}: Im(\chi_{{\mathcal{H}}}) \ni u\mapsto \theta(u){\upharpoonright}{\mathcal{H}}. \end{cases}$$ Well-posed by applying [@BourA1 Prop. $4$, $n^{\circ}5$, $\S 1$,Ch. $2$]. \[10211801\] $(\forall x\in X) (\forall u\in \prod_{x\in X}End({\mathfrak{E}}_{x}))$ we have $(\Pr_{x}\circ\theta(u)\circ\imath_{x} = \Pr_{x}(u))$. \[14152703\] The space $\prod_{x\in X}{\mathfrak{E}}_{x}$ with the product topology satisfies the request for the space ${\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}}$ in Notations \[12042101\] with the choice ${\mathcal{A}}=\prod_{x\in X}{\mathcal{L}}({\mathfrak{E}}_{x})$. $\Pr_{x}\in{\mathcal{L}}\left( \prod_{y\in X}{\mathfrak{E}}_{y}, {\mathfrak{E}}_{x} \right)$ by definition of the product topology, moreover $\imath_{x}\in{\mathcal{L}}\left( {\mathfrak{E}}_{x}, \prod_{y\in X}{\mathfrak{E}}_{y} \right)$. Indeed $\imath_{x}$ is clearly linear and by considering that for any net $\{f^{\alpha}\}_{\alpha\in D}$ and any $f$ in $\prod_{y\in X}{\mathfrak{E}}_{y}$, $\lim_{\alpha\in D}f^{\alpha}=f$ if and only if $\lim_{\alpha\in D}f^{\alpha}(y)=f(y)$ for all $y\in X$, we deduce that for any net $\{f_{x}^{\alpha}\}_{\alpha\in D}$ and any $f_{x}$ in ${\mathfrak{E}}_{x}$ such that $\lim_{\alpha\in D}f_{x}^{\alpha}=f_{x}$ we have $\lim_{\alpha\in D} \imath_{x}(f_{x}^{\alpha}) =\imath_{x}(f_{x})$, so $\imath_{x}$ is continuous. Let $x\in X$ and $u\in\prod_{x\in X}{\mathcal{L}}({\mathfrak{E}}_{x})$ so $ \Pr_{x}(u)\circ\Pr_{x} \in{\mathcal{L}}\left( \prod_{y\in X}{\mathfrak{E}}_{y}, {\mathfrak{E}}_{x} \right)$, so $(6a)$ follows by the definition of $\theta$ and [@BourGT Prp. $4$, $No 3$, $\S 2$]. Finally $(6b)$ is trivial. The following is the main structure of the present section. For the definition and properties of locally convex final topologies see [@BourTVS $No 4$, $\S 4$]. \[10221801\] Set for all $x\in X$ $$\begin{cases} G\doteqdot \theta({\mathcal{A}}) {\upharpoonright}{\mathcal{H}}, \\ g_{x}: {\mathcal{L}}({\mathfrak{E}}_{x}) \ni f_{x}\mapsto \imath_{x}\circ f_{x}\circ\Pr_{x} \in End\left( \prod_{y\in X}{\mathfrak{E}}_{y} \right) \\ h_{x}: {\mathcal{L}}({\mathfrak{E}}_{x}) \ni f_{x}\mapsto g_{x}(f_{x}){\upharpoonright}{\mathcal{H}}. \end{cases}$$ We shall denote by **${\mathfrak{G}}$ the lcs $G$ provided with the locally convex final topology of the family of topologies $\{\tau_{x}\}_{x\in X}$ of the $\{{\mathcal{L}}({\mathfrak{E}}_{x})\}_{x\in X}$, for the family of linear mappings $\{h_{x}\}_{x\in X}$**. \[11441302\] Set in $ \prod_{x\in X}End({\mathfrak{E}}_{x}) $ the following binary operation $\circ$. For all $x\in X$ we set $\Pr_{x}(f\circ h)\doteqdot f(x)\circ h(x)$. It is easy to verify that ${\left\langle \prod_{x\in X}End({\mathfrak{E}}_{x}),+,\circ{\right\rangle}}$ is an algebra over ${\mathbb{K}}$ as well as ${\left\langle \prod_{x\in X} {\mathcal{L}}({\mathfrak{E}}_{x}),+,\circ{\right\rangle}}$. \[17141401\] $G\subset {\mathcal{L}}\left({\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}}\right)$, moreover $\theta$ is a morphism of algebras. Finally if ${\mathcal{A}}$ is a subalgebra of $\prod_{x\in X}{\mathcal{L}}({\mathfrak{E}}_{x})$ then $G$ is a subalgebra of ${\mathcal{L}}\left({\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}}\right)$. The first sentence is immediate by $(6a)$ in Notations \[12042101\]. Let $u,v\in\prod_{x\in X}{\mathcal{L}}({\mathfrak{E}}_{x})$ thus for all $x\in X$ $$\begin{aligned} {1} \Pr_{x}\circ\theta(u\circ v) &= (u(x)\circ v(x))\circ\Pr_{x} \\ &= u(x)\circ\Pr_{x}\circ\theta(v) \\ &= \Pr_{x}\circ\theta(u)\circ\theta(v),\end{aligned}$$ so $\theta(u\circ v)=\theta(u)\circ\theta(v)$, similarly we can show that $\theta$ is linear by the linearity of $\Pr_{x}$ for all $x\in X$. Thus $\theta$ is a morphism of algebras, so the last sentence of the statement follows by the first one and the fact that ${\mathcal{A}}$ is an algebra. \[14271401\] $\theta_{{\mathcal{H}}} \circ \chi_{{\mathcal{H}}} (w) = w\circ\imath_{x}\circ\Pr_{x} {\upharpoonright}{\mathcal{H}}$ for all $w\in End({\mathcal{H}})$, Moreover $\theta_{{\mathcal{H}}} (Im(\chi_{{\mathcal{H}}})) \subset Dom(\chi_{{\mathcal{H}}})$ and $\chi_{{\mathcal{H}}}\circ\theta_{{\mathcal{H}}} =Id{\upharpoonright}Im(\chi_{{\mathcal{H}}})$. Let $w\in End({\mathcal{H}})$ thus for all $x\in X$ we have $(\Pr_{x}\circ\theta_{{\mathcal{H}}} \circ\chi_{{\mathcal{H}}})(w) = \Pr_{x}(\chi_{{\mathcal{H}}}(w))\circ\Pr_{x} {\upharpoonright}{\mathcal{H}} = \Pr_{x}\circ w\circ \imath_{x}\circ\Pr_{x}{\upharpoonright}{\mathcal{H}}$ and the first sentence of the statement follows. By the first sentence and the assumption that $\imath_{x}({\mathfrak{E}}_{x})\subset{\mathcal{H}}$ we have $\theta(Im(\chi_{{\mathcal{H}}})){\upharpoonright}{\mathcal{H}} \subset End({\mathcal{H}})$ so $\chi_{{\mathcal{H}}}\circ\theta_{{\mathcal{H}}}$ is well set. Moreover for all $x\in X$ and $u\in Im(\chi_{{\mathcal{H}}})$ we have $\Pr_{x}\left( \chi_{{\mathcal{H}}}\left(\theta(u){\upharpoonright}{\mathcal{H}}\right) \right) = \Pr_{x}\circ\theta(u)\circ\imath_{x} = \Pr_{x}(u) \circ\Pr_{x}\circ\imath_{x} = \Pr_{x}(u)$. \[10231801\] Let $x\in X$, then 1. $g_{x}=\theta\circ\imath_{x}$ so $Im(h_{x})\subseteq G$; 2. $h_{x}\in End({\mathcal{L}}({\mathfrak{E}}_{x}),G)$; 3. $\exists\, h_{x}^{-1}: Im(h_{x})\to{\mathcal{L}}({\mathfrak{E}}_{x})$ and $$\begin{cases} h_{x}^{-1}=\Pr_{x}\circ\chi_{{\mathcal{H}}} {\upharpoonright}Im(h_{x}), \\ Im(h_{x})=\{\theta(\imath_{x}(f_{x})) {\upharpoonright}{\mathcal{H}} \mid f_{x}\in {\mathcal{L}}({\mathfrak{E}}_{x})\}. \end{cases}$$ $\forall y\in X$ we have $$Pr_{y}\circ\theta(\imath_{x}(f_{x})) = \Pr_{y}(\imath_{x}(f_{x}))\circ\Pr_{y} = \begin{cases} {\mathbf{0}}_{y},x\ne y \\ f_{x}\circ\Pr_{x},x=y. \end{cases}$$ Moreover $$Pr_{y}\circ g_{x}(f_{x}) = \Pr_{y}\circ\imath_{x} \circ f_{x} \circ\Pr_{x} = \begin{cases} {\mathbf{0}}_{y},x\ne y \\ f_{x}\circ\Pr_{x},x=y. \end{cases}$$ So the first sentence of statement $(1)$ follows. Thus $h_{x}\left({\mathcal{L}}({\mathfrak{E}}_{x})\right) = g_{x}\left({\mathcal{L}}({\mathfrak{E}}_{x})\right) {\upharpoonright}{\mathcal{H}} = \theta\left(\imath_{x}\left( {\mathcal{L}}({\mathfrak{E}}_{x})\right) \right) {\upharpoonright}{\mathcal{H}}$ so by $(6b)$ of Notations \[12042101\] the second sentence of statement $(1)$ follows. Statement $(2)$ follows by the trivial linearity of $g_{x}$ and by the second sentence of statement $(1)$. Let $f_{x}\in{\mathcal{L}}({\mathfrak{E}}_{x})$ and $w=\imath_{x}\circ f_{x}\circ\Pr_{x}{\upharpoonright}{\mathcal{H}}$. Then by the assumption $(6)$ we have that $w\in End({\mathcal{H}})$, and $\chi_{{\mathcal{H}}}(w)=\imath_{x}(f_{x})$, indeed $\Pr_{x}(\chi_{{\mathcal{H}}}(w)) = \Pr_{x}\circ\imath_{x} \circ f_{x} \circ \Pr_{x}\circ\imath_{x} = f_{x} = \Pr_{x}(\imath(f_{x}))$. Thus $\imath_{x}(f_{x})\in Im(\chi_{{\mathcal{H}}})$ so by Pr. \[14271401\] $\theta(\imath_{x}(f_{x})){\upharpoonright}{\mathcal{H}} \in Dom(\chi_{{\mathcal{H}}})$ and $h_{x}^{-1}$ is well set. Moreover $$\begin{aligned} {2} (\Pr_{x}\circ\chi_{{\mathcal{H}}})\circ h_{x}(f_{x}) & = \Pr_{x}\circ\chi_{{\mathcal{H}}} \circ\theta_{{\mathcal{H}}}(\imath_{x}(f_{x})) \\ &= \Pr_{x}(\imath_{x}(f_{x})) =f_{x},\end{aligned}$$ where the first equality comes by stat. $(1)$ and by $\imath_{x}(f_{x})\in Im(\chi_{{\mathcal{H}}})$, while the second by Prop. \[14271401\]. Finally $$\begin{aligned} {2} g_{x}\circ\Pr_{x}\circ \chi_{{\mathcal{H}}}(\theta(\imath_{x}(f_{x}))) & = g_{x}\circ\Pr_{x}(\imath_{x}(f_{x})) \\ &= g_{x}(f_{x}) = \theta(\imath_{x}(f_{x})).\end{aligned}$$ Thus stat. $(3)$ follows. \[16141302\] If ${\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}}$ is a topological algebra for all $x\in X$ and ${\mathcal{A}}$ is an algebra then ${\mathfrak{G}}$ is a topological algebra. Let us set for all $F\in{\mathfrak{G}}$ $L_{F}:{\mathfrak{G}}\ni H\mapsto F\circ H\in{\mathfrak{G}}$, well set ${\mathfrak{G}}$ being an algebra by Lemma \[17141401\]. Thus for all $x\in X$, $f\in{\mathcal{A}}$ and $l_{x}\in{\mathcal{L}}({\mathfrak{E}}_{x})$ $$\begin{aligned} {2} (L_{\theta(f)}\circ h_{x}) l_{x} &= L_{\theta(f)} (\theta(\imath_{x}(l_{x})) {\upharpoonright}{\mathcal{H}}) = \theta(f\circ\imath_{x}(l_{x})) {\upharpoonright}{\mathcal{H}} \\ &= \left[ \theta\circ\imath_{x}(f(x)\circ l_{x}) \right] {\upharpoonright}{\mathcal{H}} = \left[ g_{x}(f(x)\circ l_{x}) \right] {\upharpoonright}{\mathcal{H}} \\ &= h_{x}(f(x)\circ l_{x}) = (h_{x}\circ L_{f(x)})l_{x},\end{aligned}$$ where $L_{f_{x}}:{\mathcal{L}}({\mathfrak{E}}_{x})\ni s_{x}\mapsto f_{x}\circ s_{x} \in{\mathcal{L}}({\mathfrak{E}}_{x})$ for all $f_{x}\in{\mathcal{L}}({\mathfrak{E}}_{x})$. Here the first and fourth equality follow by Prop. \[10231801\], the second one by Lemma \[17141401\]. Moreover by hypothesis $L_{f(x)}$ is continuous, while $h_{x}$ is continuous by , so $L_{\theta(f)}\circ h_{x}$ is linear and continuous. Therefore $L_{\theta(f)}$ is linear and continous by . Similarly $R_{F}$ is linear and continuous, where $R_{F}:{\mathfrak{G}}\ni H\mapsto H\circ F\in{\mathfrak{G}}$, thus the statement. \[14321401\] Set $$\begin{cases} \Psi_{Y}^{{\mathcal{H}}}: End({\mathcal{H}})^{Y} \to \prod_{x\in X} End({\mathfrak{E}}_{x})^{Y}, \\ (\Pr_{x}\circ\Psi_{Y}^{{\mathcal{H}}})({\overline}{F})(s) = (\Pr_{x}\circ\chi_{{\mathcal{H}}})({\overline}{F}(s)). \end{cases}$$ Moreover set $$\begin{cases} \Lambda: \prod_{x\in X} End({\mathfrak{E}}_{x})^{Y} \to \left(End\left(\prod_{x\in X}{\mathfrak{E}}_{x}\right) \right)^{Y}, \\ \Lambda(F)(s) = \theta(F(\cdot)(s)). \end{cases}$$ $\forall{\overline}{F}\in End({\mathcal{H}})^{Y}$, $\forall F\in \prod_{x\in X} End({\mathfrak{E}}_{x})^{Y} $, $\forall x\in X$ and $\forall s\in Y$, where $F(\cdot)(s) \in\prod_{y\in X}End({\mathfrak{E}}_{x})$ such that $\Pr_{x}(F(\cdot)(s)) = F(x)(s)$. Finally set $$\Lambda_{{\mathcal{A}}}^{Y} \doteqdot \Lambda{\upharpoonright}\left\{ F\in \prod_{x\in X} {\mathcal{L}}({\mathfrak{E}}_{x})^{Y} \mid (\forall s\in Y) (F(\cdot)(s)\in{\mathcal{A}}) \right\}.$$ \[15111401\] Let $x\in X$ and $s\in Y$, then for all ${\overline}{F}\in End({\mathcal{H}})^{Y}$ 1. $(\Pr_{x}\circ\Psi_{Y}^{{\mathcal{H}}})({\overline}{F})(s) = \Pr_{x}\circ{\overline}{F}(s)\circ\imath_{x}$; 2. $\Psi_{Y}^{{\mathcal{H}}} \circ\Lambda_{{\mathcal{A}}}^{Y}=Id$; 3. $Im(\Lambda_{{\mathcal{A}}}^{Y}) \subset G^{Y}$. Stats. $(1)$ and $(3)$ are trivial. Let $F\in Dom(\Lambda_{{\mathcal{A}}}^{Y})$ so $$\begin{aligned} {2} (\Pr_{x}\circ \Psi_{Y}^{{\mathcal{H}}} \circ \Lambda_{{\mathcal{A}}}^{Y} )(F)(s) &= (\Pr_{x}\circ\chi_{{\mathcal{H}}})(\Lambda_{{\mathcal{A}}}^{Y}(F)(s)) = \Pr_{x}\circ\Lambda_{{\mathcal{A}}}^{Y}(F)(s)\circ\imath_{x} \\ &= \Pr_{x}\circ \theta(F(\cdot)(s)) \circ\imath_{x} = \Pr_{x}(F(\cdot)(s)) \circ\Pr_{x} \circ\imath_{x} \\ &= F(x)(s) = \Pr_{x}(F)(s),\end{aligned}$$ and stat. $(2)$ follows. \[16141501\] $(\forall x\in X) (\forall s\in Y) (\forall{\overline}{F}\in G^{Y})$ we have $$(\Pr_{x}\circ\Psi_{Y}^{{\mathcal{H}}}) ({\overline}{F})(s) \circ\Pr_{x} = \Pr_{x}\circ ({\overline}{F}(s))$$ Let ${\overline}{F}\in G^{Y}$ thus $\exists\,U\in{\mathcal{A}}^{Y}$ such that ${\overline}{F}(s)=\theta(U(s)) {\upharpoonright}{\mathcal{H}}$, hence for all $x\in X, s\in Y$ $$\begin{aligned} {2} (\Pr_{x}\circ\Psi_{Y}^{{\mathcal{H}}})({\overline}{F})(s) \circ\Pr_{x} &= \Pr_{x}(\Psi_{Y}^{{\mathcal{H}}}({\overline}{F}))(s) \circ \Pr_{x} \\ &= \Pr_{x}\circ{\overline}{F}(s)\circ\imath_{x} \circ \Pr_{x},& \text{ by Prop. \ref{15111401}} \\ &= (\Pr_{x}\circ \theta(U(s))) {\upharpoonright}{\mathcal{H}} \circ\imath_{x} \circ \Pr_{x} \\ &\doteq (\Pr_{x}(U(s)) \circ\Pr_{x}) {\upharpoonright}{\mathcal{H}} \circ\imath_{x} \circ \Pr_{x} \\ &= \Pr_{x}(U(s)) \circ\Pr_{x} {\upharpoonright}{\mathcal{H}} \\ &\doteq \Pr_{x} \circ \theta(U(s)) {\upharpoonright}{\mathcal{H}} \\ &= \Pr_{x} \circ ({\overline}{F}(s)).\end{aligned}$$ \[11221801\] Let $x\in X$ $$\begin{cases} I_{x}:Hom({\mathcal{L}}({\mathfrak{E}}_{x}),{\mathbb{K}}) \to Hom\left( \prod_{y\in X}{\mathcal{L}}({\mathfrak{E}}_{y}),{\mathbb{K}}\right), \\ I_{x}(t_{x})\doteqdot t_{x}\circ\Pr_{x}. \end{cases}$$ \[12001801\] Let $x\in X$ thus 1. $(\forall t_{x}\in Hom({\mathcal{L}}({\mathfrak{E}}_{x}),{\mathbb{K}})) (\forall y\in X)$ we have $$\begin{cases} I_{x}(t_{x})\circ\chi_{{\mathcal{H}}}\circ h_{y} = t_{x}, x=y \\ I_{x}(t_{x})\circ\chi_{{\mathcal{H}}}\circ h_{y} ={\mathbf{0}}, x\neq y; \end{cases}$$ 2. $(\forall t_{x}\in {\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}}^{*}) (I_{x}(t_{x})\circ\chi_{{\mathcal{H}}}{\upharpoonright}{\mathfrak{G}} \in{\mathfrak{G}}^{*})$ Let $x\in X$ and $t_{x}\in Hom({\mathcal{L}}({\mathfrak{E}}_{x}),{\mathbb{K}})$ thus for all $y\in X$ and $f_{y}\in{\mathcal{L}}({\mathfrak{E}}_{y})$ we have $$\begin{aligned} {1} I_{x}(t_{x})\circ\chi_{{\mathcal{H}}}\circ h_{y} (f_{y}) &= t_{x}\circ\Pr_{x}\circ\chi_{{\mathcal{H}}} (\imath_{y}\circ f_{y}\circ\Pr_{y}{\upharpoonright}{\mathcal{H}}) \\ &= t_{x}\circ (\Pr_{x}\circ\imath_{y}\circ f_{y} \circ\Pr_{y}\circ\imath_{x}),\end{aligned}$$ and stat. $(1)$ follows. Stat. $(2)$ follows by stat. $(1)$ and . The following is the first main result of this section. \[15251401\] We have 1. $\Psi_{Y}^{{\mathcal{H}}} \in Hom({\mathfrak{L}}_{1}(Y,{\mathfrak{G}},\mu), \prod_{x\in X} {\mathfrak{L}}_{1}(Y, {\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}},\mu))$; 2. $(\forall x\in X)(\forall s\in Y) (\forall{\overline}{F}\in{\mathfrak{L}}_{1}(Y,{\mathfrak{G}},\mu)) $ $$\int\Pr_{x}(\Psi_{Y}^{{\mathcal{H}}}({\overline}{F}))(s) \,d\mu(s) = \Pr_{x} \circ \left[ \int{\overline}{F}(s)\,d\mu(s) \right] \circ\imath_{x}.$$ Let $x\in X$, set $$\Delta_{x}:G\ni f \mapsto \Pr_{x}\circ f\circ\imath_{x} \in {\mathcal{L}}({\mathfrak{E}}_{x}).$$ $\Delta_{x}$ is well-defined by Lemma \[17141401\]. By applying $\Delta_{x}\in{\mathcal{L}} ({\mathfrak{G}}, {\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}})$ if and only if $(\forall y\in X) (\Delta_{x}\circ h_{y}\in {\mathcal{L}}({\left\langle {\mathcal{L}}({\mathfrak{E}}_{y}),\tau_{y}{\right\rangle}}, {\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}}))$. Moreover $\forall y\in X$ and $\forall f_{y}\in{\mathcal{L}}({\mathfrak{E}}_{y})$ we have $$(\Delta_{x}\circ h_{y})(f_{y}) = \Pr_{x}\circ\imath_{y}\circ f_{y} \circ\Pr_{y}\circ\imath_{x},$$ so $$\begin{cases} \Delta_{x}\circ h_{y}=Id, x=y \\ \Delta_{x}\circ h_{y}={\mathbf{0}}, x\neq y. \end{cases}$$ In any case $\Delta_{x}\circ h_{y}\in {\mathcal{L}}({\left\langle {\mathcal{L}}({\mathfrak{E}}_{y}),\tau_{y}{\right\rangle}}, {\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}})$, thus $$\Delta_{x}\in{\mathcal{L}} ({\mathfrak{G}}, {\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}})$$ hence $$\label{13351801} (\forall t_{x}\in {\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}}^{*}) (t_{x}\circ \Delta_{x} \in{\mathfrak{G}}^{*}).$$ Therefore $$\begin{aligned} {1} t_{x}\left( \Pr_{x}\circ\left[ \int{\overline}{F}(s)\,d\mu(s) \right] \circ\imath_{x} \right) &= (t_{x}\circ\Delta) \left(\int{\overline}{F}(s)\,d\mu(s)\right) \\ &= \int(t_{x}\circ\Delta)({\overline}{F}(s))\,d\mu(s) \\ &= \int t_{x}\left(\Pr_{x}\circ {\overline}{F}(s)\circ\imath_{x}\right)\,d\mu(s) \\ &= \int t_{x}\left( (\Pr_{x}\circ\Psi_{Y}^{{\mathcal{H}}})({\overline}{F})(s) \right) \,d\mu(s),\end{aligned}$$ where the second equality comes by and , while the last one comes by Proposition \[15111401\]. \[11121501\] Let $Z$ be a topological vector space set $$\begin{cases} {\mathbf{ev}}_{Z} \in Hom(Z,Hom({\mathcal{L}}(Z),Z), \\ (\forall v\in Z) (\forall f\in{\mathcal{L}}(Z)) ({\mathbf{ev}}_{Z}(v)(f)) \doteqdot f(v)). \end{cases}$$ Moreover set $\eta \doteqdot {\mathbf{ev}}_{{\mathcal{H}}}$ and $\forall x\in X$ set ${\varepsilon}_{x} \doteqdot {\mathbf{ev}}_{{\mathfrak{E}}_{x}}$. \[11011501\] Let $D\subseteq{\mathcal{H}}$ thus $(A)\Rightarrow(B)$, where (A) : $(\forall x\in X)(\forall v_{x}\in\Pr_{x}(D)) ({\varepsilon}_{x}(v_{x})\in{\mathcal{L}} ({\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}},{\mathfrak{E}}_{x}))$; (B) : $(\forall v\in D) (\eta(v)\in{\mathcal{L}} ({\mathfrak{G}},{\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}}))$. Let $y\in X$ thus for all $v\in{\mathcal{H}}$ $$\eta(v)\circ h_{y} = \imath_{y}\circ{\varepsilon}_{y}(\Pr_{y}(v)).$$ Hence by $(A)$ and the fact that by construction $\imath_{y}$ is continuous with respect to the topology ${\mathfrak{T}}$ we have for all $v\in D$ $$\eta(v)\circ g_{y} \in {\mathcal{L}}\left( {\left\langle {\mathcal{L}}({\mathfrak{E}}_{y}),\tau_{y}{\right\rangle}}, {\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}} \right).$$ Thus $(B)$ follows by the universal property of any locally final topology, cf. [@BourTVS $(ii)$ of Prop. $5$, $N\,4$, $\S 4$ Ch $2$]. The following is the second main result of the section \[16291501\] Let $D\subseteq{\mathcal{H}}$ and assume $(A)$ of Lemma \[11011501\]. Then $(\forall{\overline}{F} \in{\mathfrak{L}}_{1}(Y,{\mathfrak{G}},\mu)) (\forall x\in X) (\forall v\in D)$ $$\label{15281901} \int {\left\langle \Pr_{x}(\Psi_{Y}^{{\mathcal{H}}} ({\overline}{F}))(s),v(x){\right\rangle}}_{x} \,d\mu(s) = {\left\langle \int{\overline}{F}(s)\,d\mu(s),v{\right\rangle}}(x).$$ Here the integral in the left-side is with respect to the $\mu$ and the lct on ${\mathfrak{E}}_{x}$, while the integral in the right-side is with respect to the $\mu$ and the lct on ${\mathfrak{G}}$. $(\forall{\overline}{F} \in{\mathfrak{L}}_{1}(Y,{\mathfrak{G}},\mu)) (\forall x\in X) (\forall v\in D)$ we have $$\begin{aligned} {1} \Pr_{x}\circ \left[ \int{\overline}{F}(s)\,d\mu(s) \right](v) &= (\Pr_{x}\circ\eta(v)) \left(\int{\overline}{F}(s)\,d\mu(s)\right) \\ &= \int (\Pr_{x}\circ\eta(v)) ({\overline}{F}(s)) \,d\mu(s) \\ &= \int (\Pr_{x}\circ{\overline}{F}(s))(v)\, d\mu(s) \\ &= \int \Pr_{x}(\Psi_{Y}^{{\mathcal{H}}}({\overline}{F}))(s) (v(x)) \,d\mu(s).\end{aligned}$$ Here in the second equality we applied and the fact that $\Pr_{x}\circ\eta(v) \in{\mathcal{L}}({\mathfrak{G}},{\mathfrak{E}}_{x})$ because of Lemma \[11011501\] and the linearity and continuity of $\Pr_{x}$ with respect to the topology ${\mathfrak{T}}$. Finally in the last equality we used Prop. \[16141501\]. The following is the main result of this section \[15332203\] Let $D\subseteq{\mathcal{H}}$ and assume $(A)$ of Lemma \[11011501\]. Then $(\forall{\overline}{F} \in{\mathfrak{L}}_{1}(Y,{\mathfrak{G}},\mu)) (\forall x\in X) (\forall v\in D)$ $$\label{14342203} \boxed{ {\left\langle \int \Pr_{x}(\Psi_{Y}^{{\mathcal{H}}}({\overline}{F}))(s) \,d\mu(s),v(x){\right\rangle}}_{x} = {\left\langle \int{\overline}{F}(s)\,d\mu(s),v{\right\rangle}}(x). }$$ Equivalently ${\mathfrak{G}}$ is a ${\mathbf{U}}-$space with respect to $\{{\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}}\}_{x\in X}$, ${\mathfrak{T}}$ and $D$. Here the integral in the left-side is with respect to the $\mu$ and the lct on ${\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}}$. By $(A)$ of Lemma \[11011501\], stat.$(1)$ of Th. \[15251401\] and we have $(\forall{\overline}{F} \in{\mathfrak{L}}_{1}(Y,{\mathfrak{G}},\mu)) (\forall x\in X) (\forall v\in D)$ $$\int {\left\langle \Pr_{x} (\Psi_{Y}^{{\mathcal{H}}}({\overline}{F}))(s),v(x){\right\rangle}}_{x} \,d\mu(s) = {\left\langle \int \Pr_{x}(\Psi_{Y}^{{\mathcal{H}}}({\overline}{F}))(s) \,d\mu(s),v(x){\right\rangle}}_{x},$$ hence the statement follows by Theorem \[16291501\]. \[16082203\] By and stat.$(2)$ of Th. \[15251401\] $(\forall{\overline}{F} \in{\mathfrak{L}}_{1}(Y,{\mathfrak{G}},\mu)) (\forall x\in X) (\forall v\in D) $ $${\left\langle \int{\overline}{F}(s)\,d\mu(s),v{\right\rangle}}(x) = {\left\langle \int{\overline}{F}(s)\,d\mu(s),\imath_{x}(v(x)){\right\rangle}}(x).$$ Thus for all $v,w\in D$ and $x\in X$ $$v(x)=w(x) \Rightarrow {\left\langle \int{\overline}{F}(s)\,d\mu(s),v{\right\rangle}}(x) = {\left\langle \int{\overline}{F}(s)\,d\mu(s),w{\right\rangle}}(x).$$ \[15111901\] Let ${\mathcal{S}}\in\prod_{x\in X} 2^{Bounded({\mathfrak{E}}_{x})}$ and ${\mathcal{D}}$ such that $$\label{15391901a} \begin{cases} N(x) \doteqdot \bigcup_{l_{x}\in L_{x}} B_{l_{x}}^{x} \text{ is total in ${\mathfrak{E}}_{x}$}, \forall x\in X, \\ {\mathcal{D}}\subseteq {\mathcal{H}} \cap \prod_{x\in X}N(x), \end{cases}$$ where ${\mathcal{S}}(x) =\{B_{l_{x}}^{x}\mid l_{x}\in L_{x}\}$. Assume that for all $x\in X$ the topology $\tau_{x}$ is generated by the set of seminorms $\{ p_{(l_{x},j_{x})}^{x} \mid (l_{x},j_{x})\in L_{x}\times J_{x}\}$, where [^16] $$\label{15391901b} p_{(l_{x},j_{x})}^{x}: {\mathcal{L}}({\mathfrak{E}}_{x})\ni f_{x} \mapsto \sup_{w\in B_{l_{x}}^{x}} \nu_{j_{x}}^{x}(f_{x}w) \in{\mathbb {R}}^{+}.$$ Then 1. $(A)$ of Lemma \[11011501\] for $D={\mathcal{D}}$; 2. holds and ${\mathfrak{G}}$ is a ${\mathbf{U}}-$space with respect to $\{{\mathcal{L}}_{{\mathcal{S}}(x)}({\mathfrak{E}}_{x})\}_{x\in X}$, ${\mathfrak{T}}$ and ${\mathcal{D}}$. By request we have that the lcs ${\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}}$ is Hausdorff so the position is well-set. By construction $(\forall x\in X) (\forall v_{x}\in D(x)) (\exists\,{\overline}{l}_{x}\in L_{x}) (v_{x}\in B_{{\overline}{l}_{x}}^{x})$, so $(\forall f_{x}\in{\mathcal{L}}({\mathfrak{E}}_{x})) (\forall j_{x}\in J_{x})$ $$\begin{aligned} {1} \nu_{j_{x}}^{x} ({\varepsilon}_{x}(v_{x})f_{x}) &= \nu_{j_{x}}^{x}(f_{x}(v_{x})) \\ &\leq p_{({\overline}{l}_{x},j_{x})}^{x}(f_{x}),\end{aligned}$$ hence statement $(1)$ by . Statement $(2)$ follows by statement $(1)$, Theorem \[16291501\] and Theorem \[15332203\] respectively. \[${\mathbf{LD}}({\mathcal{O}},{\mathcal{D}})$\] \[18491004\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure satisfying and $ \Gamma(\pi) \cap{\mathcal{H}} \cap \prod_{x\in X}{\mathcal{B}}_{B}^{x} \ne\emptyset $. Set $$\label{21241004} \begin{cases} {\mathcal{O}}\subseteq\Gamma(\rho) \\ {\mathcal{D}}\subseteq \Gamma(\pi)\cap{\mathcal{H}} \cap \prod_{x\in X}{\mathcal{B}}_{B}^{x} \end{cases}$$ If $\exists\, {\mathcal{F}}\subset \bigcap_{\lambda>0} {\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$ such that $\Psi_{{\mathbb {R}}^{+}}^{{\mathcal{H}}}({\mathcal{F}})={\mathcal{O}}$ then holds. In particular if $\exists\,{\mathcal{F}}\subset \bigcap_{\lambda>0} {\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$ such that $\Psi_{{\mathbb {R}}^{+}}^{{\mathcal{H}}}({\mathcal{F}})={\mathcal{O}}$ then $${\left\langle {\mathfrak{G}},{\mathcal{D}}{\right\rangle}} \subseteq \Gamma(\pi) \Rightarrow {\mathbf{LD}}({\mathcal{O}},{\mathcal{D}}).$$ Here ${\mathcal{B}}_{B}^{x}$, for all $x\in X$, is defined in . By statement $(2)$ of Cor. \[15111901\], Pr. \[14492503\] and Rm. \[13390104\]. \[21211004\] Note that if ${\mathcal{E}}\subset\Theta$, as for example for the positions taken in Rm. \[21500412b\], we have ${\mathcal{E}}\subset\prod_{x\in X}{\mathcal{B}}_{B}^{x}$. Hence if ${\mathcal{E}}\subseteq{\mathcal{H}}$ we have ${\mathcal{E}} \subseteq \Gamma(\pi) \cap {\mathcal{H}} \cap \prod_{x\in X}{\mathcal{B}}_{B}^{x}$. By the previous remark, Cor. \[18491004\] and Thm. \[17301812b\] we can state \[12080805\] Let us assume the hypotheses of Thm. \[17301812b\] made exception for the $(i)$ replaced by the following one: ${\mathcal{E}}\subseteq{\mathcal{H}}$ and $\exists\, {\mathcal{F}}\subset \bigcap_{\lambda>0} {\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$ such that $\Psi_{{\mathbb {R}}^{+}}^{{\mathcal{H}}}({\mathcal{F}})=\Gamma(\rho)$ and $${\left\langle {\mathfrak{G}},{\mathcal{E}}{\right\rangle}} \subseteq \Gamma(\pi).$$ Then all the statements of Thm. \[17301812b\] hold ### Uniform Convergence over ${\mathcal{K}}\in Compact({\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}})$. In this subsection we assume given the following data 1. a ${\mathfrak{V}}$ Banach bundle, a $\left(\Theta,{\mathcal{E}}\right)-$structure ${\left\langle {\mathfrak{V}},{\mathfrak{M}},X,Y{\right\rangle}}$ where $\Theta$ is defined in , where we denote ${\mathfrak{W}} \doteqdot {\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$ and ${\mathfrak{V}} \doteqdot {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,\{\|\cdot\|\}{\right\rangle}}$; 2. a Banach space ${\left\langle {\mathcal{H}},\|\cdot\|_{{\mathcal{H}}}{\right\rangle}}$ such that ${\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}}$ satisfies $(6)$ of Notations \[12042101\], where ${\mathfrak{T}}$ is the topology induced by the norm $\|\cdot\|_{{\mathcal{H}}}$ and $\tau_{x}$ is such that ${\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}} = {\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})$ for every $x\in X$; 3. ${\mathcal{A}}$ as in $(6)$ of Notations \[12042101\]; 4. ${\mathfrak{G}}$, $\Psi_{Y}^{{\mathcal{H}}}$ and $\Lambda_{{\mathcal{A}}}^{Y}$ as defined in Def. \[10221801\] and Def. \[14321401\] respectively. The proof of the following Lemma is an adaptation to the present framework of the proof of [@cho Prop. $5.13$]. \[19260304\] Let ${\mathcal{U}}\in\prod_{x\in X}{\mathfrak{M}}_{x}$ and $x_{\infty}\in X$ moreover assume that 1. ${\mathcal{E}}\subseteq{\mathcal{H}} \subseteq\prod_{x\in X}^{b}{\mathfrak{E}}_{x}$ such that $(\exists\,a>0)(\forall f\in{\mathcal{H}}) (\|f\|_{\sup}\leq a\|f\|_{{\mathcal{H}}})$, where $\|f\|_{\sup}\doteqdot \sup_{x\in X}\|f(x)\|_{x}$; 2. $\exists\,F\in\Gamma(\rho)$ such that $F(x_{\infty})={\mathcal{U}}(x_{\infty})$ and $\{F(\cdot)(s)\mid s\in Y\} \subseteq{\mathcal{A}}$ 3. $\{{\mathcal{U}}(\cdot)(s)\mid s\in Y\} \subseteq{\mathcal{A}}$; 4. $\{{\overline}{F}(s)\mid s\in Y\}$ and $\{{\overline}{{\mathcal{U}}}(s)\mid s\in Y\}$ are equicontinuous as subsets of ${\mathcal{L}}({\left\langle {\mathcal{H}},\|\cdot\|_{{\mathcal{H}}}{\right\rangle}})$, where ${\overline}{{\mathcal{U}}}\doteqdot\Lambda_{{\mathcal{A}}}^{Y}({\mathcal{U}})$. and ${\overline}{F}\doteqdot\Lambda_{{\mathcal{A}}}^{Y}(F)$. Then $(A)\Leftrightarrow(B)$ where : ${\mathcal{U}}\in\Gamma^{x_{\infty}}(\rho)$; : For all ${\mathcal{K}}\in Compact({\mathcal{H}})$ such that ${\mathcal{K}}\subseteq{\mathcal{E}}$ and for all $K\in Compact(Y)$ $$\lim_{z\to x_{\infty}} \sup_{s\in K} \sup_{v\in{\mathcal{K}}} \left\| {\mathcal{U}}(z)(s)v(z) - F(z)(s)v(z) \right\| =0.$$ We shall prove only $(A)\Rightarrow(B)$, indeed the other implication follows by $(3)\Rightarrow(4)$ of Lemma \[15482712\]. So assume $(A)$ to be true. In this proof let us set $B({\mathcal{H}})\doteqdot {\mathcal{L}}({\left\langle {\mathcal{H}},\|\cdot\|_{{\mathcal{H}}}{\right\rangle}})$, moreover $\Psi\doteqdot\Psi_{Y}^{{\mathcal{H}}}$ and $\Lambda\doteqdot\Lambda_{{\mathcal{A}}}^{Y}$, moreover set ${\overline}{F}\doteqdot\Lambda_{{\mathcal{A}}}^{Y}(F)$ for every $F\in\Gamma(\rho)$; thus by stat. $(2)$ of Pr. \[15111401\] $\Psi({\overline}{F})=F$ and $\Psi({\overline}{{\mathcal{U}}})={\mathcal{U}}$. Hence by Pr. \[16141501\] for all $v\in{\mathcal{E}}$ $F\in\Gamma(\rho)$, $z\in X$ and $s\in Y$ $$\label{20160304} {\mathcal{U}}(z)(s)v(z) = ({\overline}{{\mathcal{U}}}v)(z),\, F(z)(s)v(z) = ({\overline}{F}v)(z).$$ By $(A)$ and implication $(4)\Rightarrow(3)$ of Lemma \[15482712\] we have for all $K\in Compact(Y)$ and $v\in{\mathcal{E}}$ $$\label{20150304} \lim_{z\to x_{\infty}} \sup_{s\in K} \left\| {\mathcal{U}}(z)(s)v(z) - F(z)(s)v(z) \right\| =0.$$ Fix ${\mathcal{K}}\in Compact({\mathcal{H}})$ such that ${\mathcal{K}}\subseteq{\mathcal{E}}$, $f\in{\mathcal{K}}$ and ${\varepsilon}>0$, thus by and there exists $U$ neighbourhood of $x_{\infty}$ such that $$\label{20170304} \sup_{s\in K} \sup_{z\in U} \left\| \left[({\overline}{{\mathcal{U}}}(s)-{\overline}{F}(s))f\right](z) \right\| \leq{\varepsilon}/2.$$ Define $$\begin{cases} M \doteqdot \max\{ \sup_{s\in Y} \|{\overline}{F}(s)\|_{B({\mathcal{H}})}, \, \sup_{s\in Y} \|{\overline}{{\mathcal{U}}}(s)\|_{B({\mathcal{H}})} \} \\ \eta\doteqdot{\varepsilon}/4aM \\ {\mathfrak{U}}(f) \doteqdot \{g\in{\mathcal{K}}\mid\|f-g\|_{{\mathcal{H}}}<\eta\}. \end{cases}$$ Thus for all $g\in{\mathfrak{U}}(f)$ $$\begin{aligned} {1} \sup_{z\in U} \sup_{s\in K} \left\| {\mathcal{U}}(z)(s)g(z) - F(z)(s)g(z) \right\| & = \\ \sup_{s\in K} \sup_{z\in U} \left\| \left[({\overline}{{\mathcal{U}}}(s)-{\overline}{F}(s))g\right](z) \right\| & \leq \\ \sup_{s\in K} \sup_{z\in U} \left\| \left[({\overline}{{\mathcal{U}}}(s)-{\overline}{F}(s))f\right](z) \right\| + \sup_{s\in K} \sup_{z\in U} \left\| {\overline}{{\mathcal{U}}}(s)(g-f)(z) \right\| + \sup_{s\in K} \sup_{z\in U} \left\| F(s)(g-f)(z) \right\| & \leq \\ {\varepsilon}/2 + a \sup_{s\in K} \left\| {\overline}{{\mathcal{U}}}(s)(g-f) \right\|_{{\mathcal{H}}} + a \sup_{s\in K} \left\| F(s)(g-f) \right\|_{{\mathcal{H}}} & \leq \\ {\varepsilon}/2 + 2aM \left\|g-f\right\|_{{\mathcal{H}}} & < {\varepsilon}.\end{aligned}$$ Therefore $(B)$ follows by considering that $\{{\mathfrak{U}}(f)\mid f\in{\mathcal{K}}\}$ is an open cover of the compact ${\mathcal{K}}$. Indeed let for example $\{{\mathfrak{U}}(f_{i})\mid i=1,...,n\}$ a finite subcover of ${\mathcal{K}}$ thus by setting $W\doteqdot\bigcap_{i=1}^{n}U_{n}$ with obvious meaning of $U_{i}$, we have $$\sup_{z\in W} \sup_{s\in K} \sup_{g\in{\mathcal{K}}} \left\| {\mathcal{U}}(z)(s)g(z) - F(z)(s)g(z) \right\| < {\varepsilon}.$$ We can set ${\mathcal{H}}=\prod_{x\in X}^{b}{\mathfrak{E}}_{x}$ with the usual norm $\|\cdot\|_{\sup}$. \[${\mathcal{K}}-$Uniform Convergence\] \[10581004\] Let ${\mathfrak{V}} \doteqdot {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,\|\cdot\|{\right\rangle}}$ be a Banach bundle. Let $x_{\infty}\in X$ and $ {\mathcal{U}}_{0} \in\prod_{x\in X_{0}} {\mathcal{C}_{} \left({\mathbb {R}}^{+},B_{s}({\mathfrak{E}}_{x})\right)}$ be such that ${\mathcal{U}}_{0}(x)$ is a $(C_{0})-$semigroup of contractions (respectively of isometries) on ${\mathfrak{E}}_{x}$ for all $x\in X_{0}$. Assume that 1. $D(T_{x_{\infty}})$ is dense in ${\mathfrak{E}}_{x_{\infty}}$; 2. ${\mathfrak{V}}$ and ${\mathfrak{W}}$ satisfy ; 3. $\exists\lambda_{0}>0$ (respectively $\exists\lambda_{0}>0, \lambda_{1}<0$) such that the range ${\mathcal{R}}(\lambda_{0}-T_{x_{\infty}})$ is dense in ${\mathfrak{E}}_{x_{\infty}}$, (respectively the ranges ${\mathcal{R}}(\lambda_{0}-T_{x_{\infty}})$ and ${\mathcal{R}}(\lambda_{1}-T_{x_{\infty}})$ are dense in ${\mathfrak{E}}_{x_{\infty}}$); 4. $ {\mathbf{U}}_{\|\cdot\|_{B({\mathfrak{E}}_{z})}} ({\mathcal{L}}_{S_{z}}({\mathfrak{E}}_{z})) \subseteq {\mathfrak{M}}_{z}$ (respectively ${\mathbf{U}}_{is} ({\mathcal{L}}_{S_{z}}({\mathfrak{E}}_{z})) \subseteq {\mathfrak{M}}_{z}$) for all $z\in X$; 5. ${\mathcal{E}}\subseteq{\mathcal{H}} \subseteq\prod_{x\in X}^{b}{\mathfrak{E}}_{x}$ 6. $X$ is metrizable; 7. $\exists\,{\mathcal{F}}\subset \bigcap_{\lambda>0} {\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$ such that $\Psi_{{\mathbb {R}}^{+}}^{{\mathcal{H}}}({\mathcal{F}})=\Gamma(\rho)$; 8. $(\exists\,F\in\Gamma(\rho)) (F(x_{\infty})={\mathcal{U}}(x_{\infty}))$ such that 1. ${\left\langle \int{\overline}{F}(s)\,d\mu_{\lambda}(s),{\mathcal{E}}{\right\rangle}}\subseteq\Gamma(\pi)$, for all $\lambda>0$; 2. $(\forall v\in{\mathcal{E}}) (\exists\,\phi\in\Phi)$ s.t. $\phi_{1}(x_{\infty}) = v(x_{\infty})$ and $(\forall\{z_{n}\}_{n\in{\mathbb{N}}}\subset X \mid \lim_{n\in{\mathbb{N}}}z_{n}=x_{\infty})$ we have that $\{ {\mathcal{U}}(z_{n})(\cdot)\phi_{1}(z_{n}) - F(z_{n})(\cdot)v(z_{n}) \}_{n\in{\mathbb{N}}}$ is a bounded equicontinuous sequence. Then $$\label{19351004} {\mathcal{U}} \in\Gamma^{x_{\infty}}(\rho).$$ Furthermore if 1. $(\exists\,a>0)(\forall f\in{\mathcal{H}}) (\|f\|_{\sup}\leq a\|f\|_{{\mathcal{H}}})$, 2. $\{F(\cdot)(s)\mid s\in{\mathbb {R}}^{+}\} \subseteq{\mathcal{A}}$ and $\{{\mathcal{U}}(\cdot)(s)\mid s\in{\mathbb {R}}^{+}\} \subseteq{\mathcal{A}}$; 3. $\{{\overline}{F}(s)\mid s\in{\mathbb {R}}^{+}\}$ and $\{{\overline}{{\mathcal{U}}}(s)\mid s\in{\mathbb {R}}^{+}\}$ are equicontinuous as subsets of ${\mathcal{L}}({\left\langle {\mathcal{H}},\|\cdot\|_{{\mathcal{H}}}{\right\rangle}})$. Then for all ${\mathcal{K}}\in Compact({\mathcal{H}})$ such that ${\mathcal{K}}\subseteq{\mathcal{E}}$ and for all $K\in Compact({\mathbb {R}}^{+})$ $$\label{19371004} \lim_{z\to x_{\infty}} \sup_{s\in K} \sup_{v\in{\mathcal{K}}} \left\| {\mathcal{U}}(z)(s)v(z) - F(z)(s)v(z) \right\| =0.$$ Here $D(T_{x_{\infty}})$ is defined as in Notations \[15411512b\] with ${\mathcal{T}}_{0}$ and $\Phi$ given in . While ${\mathcal{U}} \in\prod_{x\in X} {\mathfrak{M}}_{x}$ such that ${\mathcal{U}}{\upharpoonright}X_{0}\doteqdot {\mathcal{U}}_{0}$ and ${\mathcal{U}}(x_{\infty})$ is the semigroup on ${\mathfrak{E}}_{x_{\infty}}$ generated by $T_{x_{\infty}}$ operator defined in . Moreover $\|f\|_{\sup}\doteqdot \sup_{x\in X}\|f(x)\|_{x}$, while ${\overline}{{\mathcal{U}}}\doteqdot\Lambda_{{\mathcal{A}}}^{Y}({\mathcal{U}})$ and ${\overline}{F}\doteqdot\Lambda_{{\mathcal{A}}}^{Y}(F)$. By hyp. $(7)$ and st. $(1)$ of Th. \[15251401\], follows. Moreover follows by hyp. $(5)$, and Rm. \[21211004\]. Hence by hyps. $(7-8a)$, and Cor. \[18491004\] follows the ${\mathbf{LD}}(\{F\},{\mathcal{E}})$. Then follows by Th. \[17301812b\]. follows by and Lm. \[19260304\]. \[23461004\] By st.$(2)$ of Pr. \[15111401\] hyp. $(7)$ is equivalent to the following one $\Lambda_{{\mathcal{A}}}^{{\mathbb {R}}^{+}}(\Gamma(\rho)) \subseteq \bigcap_{\lambda>0} {\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathfrak{G}},\mu_{\lambda})$. In any case the form in hyp. $(7)$ has the advantage to be considered as a tool for constructing $\Gamma(\rho)$. Finally note that $${\left\langle {\mathfrak{G}},{\mathcal{E}}{\right\rangle}}\subseteq\Gamma(\pi) \Rightarrow (8a).$$ ### ${\left\langle {\mathcal{H}},{\mathfrak{T}}{\right\rangle}}$ as Direct Integral of a Continuous Field of left-Hilbert and associated left-von Neumann Algebras. #### Ex1 Assume that ${\mathfrak{V}} = {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}$ is a continuous field of left-Hilbert algebras on $X$. Let ${\mathcal{H}}$ be the direct integral of ${\mathfrak{V}}$ with respect to some finite Radon measure on $X$ and ${\mathcal{B}}\subset{\mathcal{H}}$ a linear space, set $${\mathcal{A}}({\mathcal{B}}) \doteqdot \left\{ X\ni x\mapsto L_{a(x)} \mid a\in{\mathcal{B}}\right\},$$ where $L_{a_{x}}\in B({\mathfrak{E}}_{x})$ for any $a_{x}\in{\mathfrak{E}}_{x}$, is the left multiplication on the left-Hilbert algebra ${\mathfrak{E}}_{x}$. Then ${\mathcal{H}}$ and ${\mathcal{A}}({\mathcal{B}})$ satisfies the requirements in Notations \[12042101\], moreover $$\label{14502904} \boxed{ G({\mathcal{B}}) \doteqdot \theta({\mathcal{A}}({\mathcal{B}})){\upharpoonright}{\mathcal{H}} = L_{{\mathcal{B}}},}$$ where $L_{a}\in B({\mathcal{H}})$ for any $a\in{\mathcal{H}}$, is the left multiplication on the left-Hilbert algebra ${\mathcal{H}}$. If every ${\mathfrak{E}}_{x}$ is unital then ${\mathcal{H}}$ is unital, thus $L_{(\cdot)}$ is an injective (isometric) map of ${\mathcal{H}}$ into $B({\mathcal{H}})$. Therefore under this additional requirement we can take the following identification $$G({\mathcal{B}})\simeq{\mathcal{B}}\text{ as linear spaces}.$$ Let ${\mathbf{H}}\doteqdot \{{\mathbf{H}}^{i}\in\prod_{x\in X}{\mathfrak{E}}_{x} \}_{i=0}^{2}$ such that ${\mathbf{H}}_{x}^{0}$ is a left Hilbert subalgebra of ${\mathfrak{E}}_{x}$, while ${\mathbf{H}}_{x}^{k}$ is a linear subspace of ${\mathbf{H}}_{x}^{0}$, for all $k=1,2$ and $x\in X$. Set $$\label{14582904} \begin{cases} \Gamma(\pi,{\mathbf{H}}) \doteqdot \left\{ \sigma\in{\mathcal{H}} \mid (\forall x\in X) (\sigma(x)\in{\mathbf{H}}_{x}^{0}) \right\} \\ {\mathcal{D}}_{{\mathbf{H}}} \doteqdot \left\{ \sigma\in{\mathcal{H}} \mid (\forall x\in X) (\sigma(x)\in{\mathbf{H}}_{x}^{1}) \right\} \\ {\mathcal{B}}_{{\mathbf{H}}} \doteqdot \left\{ \sigma\in{\mathcal{H}} \mid (\forall x\in X) (\sigma(x)\in{\mathbf{H}}_{x}^{2}) \right\}. \end{cases}$$ Thus $\Gamma(\pi,{\mathbf{H}})$ is a left Hilbert subalgebra of ${\mathcal{H}}$ and ${\mathcal{B}}_{{\mathbf{H}}}, {\mathcal{D}}_{{\mathbf{H}}}$ are linear subspaces of $\Gamma(\pi,{\mathbf{H}})$, so $$\label{14512904} L_{{\mathcal{B}}_{{\mathbf{H}}}}({\mathcal{D}}_{{\mathbf{H}}}) \subseteq \Gamma(\pi,{\mathbf{H}}).$$ By and follows that for all $\sigma\in{\mathcal{B}}_{{\mathbf{H}}}$, $\eta\in{\mathcal{D}}_{{\mathbf{H}}}$ and $y\in X$ $$\label{15172904} \boxed{ \begin{cases} {\left\langle G({\mathcal{B}}_{{\mathbf{H}}}),{\mathcal{D}}_{{\mathbf{H}}}{\right\rangle}} \subseteq \Gamma(\pi,{\mathbf{H}}), \\ {\left\langle \theta\left( x\mapsto L_{\sigma(x)}\right),\eta{\right\rangle}}(y) = \sigma(y)\eta(y). \end{cases} }$$ #### Ex2 Let us consider now the continuous field of left-von Neumann algebras associated to the fixed field of Hilbert algebras, and by abusing of language, let us denote it with the same symbol ${\mathfrak{V}} = {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}$, as well as ${\mathcal{H}}$ will denote the associated direct integral with respect to some finite Radon measure on $X$. Let $\Delta_{x}$ be the modular operator associated to the Hilbert algebra ${\mathfrak{E}}_{x}$ and $\sigma_{x}$ the corresponding modular group. Thus we can set $$\begin{cases} {\mathcal{A}}_{\Delta} \doteqdot\{ S_{t}: X\ni x\mapsto \sigma_{x}(t)\in Aut({\mathfrak{E}}_{x}) \mid t\in{\mathbb {R}}\} \\ G_{\Delta}\doteqdot \theta({\mathcal{A}}_{\Delta}){\upharpoonright}{\mathcal{H}} \\ \Sigma_{t} \doteqdot \theta(S_{t}){\upharpoonright}{\mathcal{H}},\, t\in{\mathbb {R}}. \end{cases}$$ Note that for every $t\in{\mathbb {R}}$, $v\in{\mathcal{H}}$ and $x\in X$ $$\Sigma_{t}(v)(x) =\sigma_{x}(t)(v(x)).$$ Now if we set $$\Gamma(\pi) \doteqdot{\mathcal{H}}$$ for any linear subspace ${\mathcal{D}}$ of ${\mathcal{H}}$ we have $$\boxed{ {\left\langle G_{\Delta},{\mathcal{D}}{\right\rangle}} \subseteq \Gamma(\pi). }$$ Finally note that to ${\mathcal{A}}_{\Delta}$ we can associate the following map $${\overline}{\Sigma}:{\mathbb {R}}^{+}\ni t \mapsto \Sigma_{t} \in G_{\Delta},$$ for which we have for all $x\in X$ $$\Psi_{{\mathbb {R}}}^{{\mathcal{H}}}({\overline}{\Sigma}) (x) = \sigma_{x}.$$ #### Ex3 In the previous example we consider the extreme case in which $\Gamma(\pi)={\mathcal{H}}$. In order to have a model where $\Gamma(\pi)\subset{\mathcal{H}}$ we have to get a more detailed structure, namely the **half-side modular inclusion**. So for any $x\in X$ let ${\left\langle {\mathfrak{N}}_{x}\subset{\mathfrak{E}}_{x},\Omega_{x}{\right\rangle}}$ be a $hsmi^{+}$ and $V_{x}$ the Wiesbrock one-parameter semigroup of unitarities associated to it so $V_{x}\in Hstr({\mathfrak{E}}_{x})^{+}$ such that ${\mathfrak{N}}_{x}=Ad(V_{x}(1)){\mathfrak{E}}_{x}$. Therefore what we are interested in is that for all $t\in{\mathbb {R}}^{+}$ $$\label{20290405} \begin{cases} Ad(V_{x}(t))({\mathfrak{E}}_{x}) \subseteq{\mathfrak{E}}_{x}, \\ Ad(V_{x}(t))({\mathfrak{N}}_{x}) \subseteq{\mathfrak{N}}_{x}. \end{cases}$$ By using the first inclusion in we can set $$\begin{cases} {\mathcal{A}}_{V} \doteqdot\{ V_{t}: X\ni x\mapsto Ad(V_{x}(t)) {\upharpoonright}{\mathfrak{E}}_{x} \in Aut({\mathfrak{E}}_{x}) \mid t\in{\mathbb {R}}\} \\ G_{V}\doteqdot \theta({\mathcal{A}}_{V}){\upharpoonright}{\mathcal{H}} \\ {\overline}{{\mathcal{V}}}_{t} \doteqdot \theta(V_{t}){\upharpoonright}{\mathcal{H}},\, t\in{\mathbb {R}}. \end{cases}$$ Hence for all $x\in X$ and $t\in{\mathbb {R}}$ $$\begin{cases} {\overline}{{\mathcal{V}}}_{t}(v)(x) = Ad(V_{x}(t))v(x) \\ \Psi_{{\mathbb {R}}}^{{\mathcal{H}}}({\overline}{{\mathcal{V}}}) (x)(t) = Ad(V_{x}(t)) \end{cases}$$ Therefore if we set ${\mathcal{D}}$ and $\Gamma(\pi)$ such that $${\mathcal{D}} \subseteq \Gamma(\pi) \doteqdot \int^{\oplus} {\mathfrak{N}}_{x}\, d\mu(x) \subset {\mathcal{H}}$$ then by using the second inclusion in we have $$\boxed{ {\left\langle G_{V},{\mathcal{D}}{\right\rangle}} \subseteq \Gamma(\pi). }$$ #### Inner property of the Tomita-Takesaki modular group For any semi-finite von Neumann algebra ${\mathfrak{N}}$ and any $\phi\in{\mathbf{N}}_{{\mathfrak{N}}}$ faithful we have that the **Tomita-Takesaki modular group $\sigma_{{\mathfrak{N}}}^{\phi}$ is inner** (see [@tak2 Thm. $3.14$ Ch. $VIII$]) i.e. it is implemented by a strongly continuous group morphism $V:{\mathbb {R}}\to U({\mathfrak{N}})$, where $U({\mathfrak{N}})\doteqdot\{U^{-1}=U^{*}\mid U\in{\mathfrak{N}}\}$, so in particular $$\label{23452904a} V({\mathbb {R}})\subset{\mathfrak{N}}.$$ Now let ${\left\langle H_{\phi},\pi_{\phi},\Omega_{\phi}{\right\rangle}}$ be a cyclic representation associated to $\phi$ and ${\mathfrak{N}}_{\phi}\doteqdot\pi({\mathfrak{N}}_{\phi})$ which is a von Neumann algebra $\phi$ being normal, then by immediatedly we have $$\label{23452904b} \boxed{ \pi_{\phi}(V({\mathbb {R}}))\subset{\mathfrak{N}}_{\phi}. }$$ By the invariance $\phi=\phi\circ\sigma_{{\mathfrak{N}}}^{\phi}$, and the cited unitary implementation we obtain that there exists $W_{\phi}$ unitary action on $H_{\phi}$ such that $$\label{23372904} \boxed{ \begin{cases} Ad(W_{\phi}(t))\circ\pi_{\phi} = Ad(\pi_{\phi}(V(t)))\circ\pi_{\phi}, \\ W_{\phi}(t)=\Delta_{\phi}^{it}, \end{cases} }$$ where the second sentence comes by [@tak2 Thm, $1.2$ Ch. $VIII$], with $\Delta_{\phi}$ the modular operator associated to ${\left\langle {\mathfrak{N}}_{\phi},\Omega_{\phi}{\right\rangle}}$. Section of Projections ======================== In sections \[15491702A\] and \[15491702B\], except when explicitedly stated, we shall maintain Notations \[12042101\]. ${\left\langle \nu,\eta,E,Z,T{\right\rangle}}$ invariant set with respect to ${\mathcal{F}}$ {#15491702A} -------------------------------------------------------------------------------------------- \[19560202\] Let $Z,T$ be two locally compact spaces, $E\in Hlcs$, $\nu\in Rad(Z)$ and $\eta\in Rad(T)^{Z}$. Set $${\mathfrak{L}}_{(1,1)} (T,E,\eta,\nu) \doteqdot \left\{ {\overline}{F}\in\bigcap_{\lambda\in Z} {\mathfrak{L}}_{1}(T,E,\eta_{\lambda}) \mid \left( Z\ni\lambda \mapsto \int{\overline}{F}(s) d\eta_{\lambda}(s) \in E \right) \in {\mathfrak{L}}_{1}(Z,E,\nu) \right\}$$ \[19510202\] Let $Z$ be a locally compact space, $\nu\in Rad(Z)$ and $\eta\in Rad(Y)^{Z}$ finally let $D\in\prod_{x\in X}2^{{\mathfrak{E}}_{x}}$ and assume $(A)$ of Lemma \[11011501\]. Thus $(\forall{\overline}{F}\in {\mathfrak{L}}_{(1,1)} (Y,{\mathfrak{G}},\eta,\nu)) (\forall x\in X) (\forall v\in \prod_{y\in X}D(y))$ $$\Pr_{x} \circ \left[ \int \left( \int{\overline}{F}(s)\, d\eta_{\lambda}(s) \right)\, d\nu(\lambda) \right] (v) = \left[ \int \left( \int \Pr_{x}(\Psi({\overline}{F}))(s) v(x)\, d\eta_{\lambda}(s) \right)\, d\nu(\lambda) \right].$$ Let ${\overline}{F}\in {\mathfrak{L}}_{(1,1)} (Y,{\mathfrak{G}},\eta,\nu)$, $x\in X$ and $v\in\prod_{y\in X}D(y)$. By Theorem \[16291501\] $$\Pr_{x} \circ \left[ \int \left( \int{\overline}{F}(s)\, d\eta_{\lambda}(s) \right)\, d\nu(\lambda) \right] (v) = \int \Pr_{x}\circ\Psi \left( \int{\overline}{F}(s)\,d\eta_{(\cdot)}(s) \right) (\lambda) (v(x)) \,d\nu(\lambda).$$ Moreover $\forall\lambda\in Z$ $$\begin{aligned} {2} \Pr_{x}\circ\Psi \left( \int{\overline}{F}(s)\,d\eta_{(\cdot)}(s) \right) (\lambda) (v(x)) &= \Pr_{x} \circ \left( \int {\overline}{F}(s)\, d\eta_{\lambda}(s) \right) \circ \imath_{x} (v(x)) \\ &= \int \Pr_{x}(\Psi({\overline}{F}))(s) \circ \Pr_{x}\circ\imath_{x} (v(x)) \,d\eta_{\lambda}(s) \\ &= \int \Pr_{x}(\Psi({\overline}{F}))(s) v(x) \,d\eta_{\lambda}(s),\end{aligned}$$ where in the first equality we used Prop. \[15111401\], while in the second one Theorem \[16291501\]. Then the statement follows. \[20030202\] $V$ is a *${\left\langle \nu,\eta,E,Z,T{\right\rangle}}$ invariant set with respect to ${\mathcal{F}}$* if 1. $T,Z$ are two locally compact spaces; 2. $E\in Hlcs$ and $V\subseteq E$; 3. $\nu\in Rad(Z)$ and $\eta\in Rad(T)^{Z}$; 4. ${\mathcal{F}} \subseteq {\mathfrak{L}}_{(1,1)} (T,E,\eta,\nu)$ 5. $\forall{\overline}{F}\in{\mathcal{F}}$ $$\left[ \int \left( \int{\overline}{F}(s)\, d\eta_{\lambda}(s) \right)\, d\nu(\lambda) \right] V \subseteq V.$$ \[18310302\] Let us assume the hypotheses of Corollary \[19510202\] and $V$ be a ${\left\langle \nu,\eta,{\mathfrak{G}},Z,Y{\right\rangle}}$ invariant set with respect to ${\mathcal{F}}$ such that $V\cap\prod_{y\in X}D(y)\neq\emptyset$. Then $\forall v\in V\cap\prod_{y\in X}D(y)$ and $\forall{\overline}{F}\in{\mathcal{F}}$ $$\left( X\ni x \mapsto \left[ \int \left( \int \Pr_{x}(\Psi({\overline}{F}))(s) v(x)\, d\eta_{\lambda}(s) \right)\, d\nu(\lambda) \right] \in{\mathfrak{E}}_{x} \right) \in V.$$ By Corollary \[19510202\]. Construction of classes $ \Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{D}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}} $ through invariant sets {#15491702B} --------------------------------------------------------------------------------------------------------------------------- In this Section we shall assume that ${\mathbf{E}}\doteqdot \{{\mathfrak{E}}_{x}\}_{x\in X}$ is a class of complex Banach spaces. Moreover $ Cld({\mathfrak{E}}_{x}) $ denotes, for all $x\in X$, the class of all closed densely defined linear operators $T_{x}: Dom(T_{x}) \subseteq {\mathfrak{E}}_{x} \to {\mathfrak{E}}_{x}$. \[Ch. $9$, $\S1$, $n^{\circ}4$ of [@kato]\] \[13411311biss\] Let $M>1$, $\beta\in{\mathbb {R}}$. Set ${\mathcal{G}}(M,\beta,{\mathbf{E}})$ the class of all $T\in\prod_{x\in X}Cld({\mathfrak{E}}_{x})$ such that $]\beta,\infty[\subseteq P(-T(x))$ [^17] and $(\forall\xi>\beta) (\forall k\in{\mathbb{N}}) (\forall x\in X)$ $$\|(T(x)+\xi)^{-k}\|_{B({\mathfrak{E}}_{x})} \leq M (\xi-\beta)^{-k}.$$ Moreover let us denote by $ \{e^{-tT(x)}\}_{t\in{\mathbb {R}}^{+}} $ the strongly continuous semigroup generated by $-T(x)$. \[ Separation of the Spectrum \] \[13361311biss\] See [@kato $n^{\circ}4$,$\S 6$, Ch. $3$]. Let $M>1$, $\beta\in{\mathbb {R}}$. We say that $T\in {\mathcal{G}}(M,\beta,{\mathbf{E}})$ satisfies the property of separation of the spectrum if $(\exists\,\Gamma) (\forall x\in X) (\exists\,\Sigma_{T(x)}' \subseteq\Sigma(T(x))) (\exists\,A_{T(x)}\in Op({\mathbb {C}}))$ such that $\Gamma$ is a closed curve in ${\mathbb {C}}$, $\Sigma_{T(x)}'$ is bounded and $$\Sigma_{T(x)}' \subset A_{T(x)} \subset {\mathcal{O}}_{i}(\Gamma), \, \Sigma_{T(x)}'' \subset {\mathcal{O}}_{e}(\Gamma).$$ Here ${\mathcal{O}}_{i}(\Gamma)$ is the interior of $\Gamma$, namely the compact set of ${\mathbb {C}}$ whose frontier is $\Gamma$, ${\mathcal{O}}_{e}(\Gamma) \doteqdot \complement {\mathcal{O}}_{i}(\Gamma)$ is the exterior of $\Gamma$, $\Sigma(T(x))$ is the spectrum of $T(x)$, finally $\Sigma_{T(x)}'' \doteqdot \Sigma(T(x)) \cap \complement \Sigma_{T(x)}'$. Let $T\in{\mathcal{G}}(M,\beta,{\mathbf{E}})$ satisfy the property of separation of the spectrum, then $\forall x\in X$ we set $$\label{16171102} P(x) \doteqdot - \frac{1}{2\pi i} \int_{\Gamma} R(T(x);\zeta)\, d\zeta \in B({\mathfrak{E}}_{x}).$$ Moreover set $R_{T}^{\rho}\in\prod_{x\in X} {\mathcal{L}}({\mathfrak{E}}_{x})^{{\mathbb {R}}^{+}}$ such that $R_{T}^{\rho} (x)(s)\doteqdot R(T(x);\rho(s))$, for all $x\in X$ and $s\in K(\Gamma)$, while $R_{T}^{\rho}(x)(s)\doteqdot{\mathbf{0}}$, if $s\in{\mathbb {R}}^{+}-K(\Gamma)$. Here $R(T(x);\cdot): P(T(x)) \ni \zeta \mapsto (T(x)-\zeta)^{-1} \in B({\mathfrak{E}}_{x}) $ is the resolvent map of $T(x)$ and $P(T(x))$ is its resolvent set, while the integration is with respect to the norm topology on $B({\mathfrak{E}}_{x})$. \[15221311biss\] Let $M>1$, $\beta\in{\mathbb {R}}$ and $T\in {\mathcal{G}}(M,\beta,{\mathbf{E}})$ satisfy the property of separation of the spectrum. Then for all $x\in X$ by [@kato Th. $6.17.$, Ch. $3$], $P(x)\in\Pr({\mathfrak{E}}_{x})$ and ${\mathfrak{E}}_{x}= M_{x}' \oplus M_{x}''$ direct sum of two closed subspaces of ${\mathfrak{E}}_{x}$, where $M_{x}'=P(x){\mathfrak{E}}_{x}$ and $M_{x}'' = ({\mathbf{1}}_{x}-P(x)) {\mathfrak{E}}_{x}$. Moreover $T(x)$ decomposes according the previous decomposition, namely $T(x){\upharpoonright}M_{x}' \in B(M_{x}')$ such that $\Sigma(T_{x}{\upharpoonright}M_{x}') =\Sigma_{T_{x}}'$ and $T_{x}{\upharpoonright}M_{x}''$ is a closed operator in $M_{x}''$ such that $\Sigma(T_{x}{\upharpoonright}M_{x}'')=\Sigma_{T_{x}}''$. \[16420402\] Let $K(\Gamma) \subset{\mathbb {R}}^{+}$ a compact set, $A$ an open neighbourhood of $K(\Gamma)$ and $\rho:A\to{\mathbb {C}}$ be such that $\rho\in C_{1}(A,{\mathbb {R}}^{2})$ [^18] and $ \rho(K(\Gamma)) = \Gamma $. Set $\forall s\in K(\Gamma)$, $\eta_{s}\in Radon({\mathbb {R}}^{+})$ such that $$\eta_{s}: {\mathcal{C}_{cs} \left({\mathbb {R}}^{+}\right)}\ni f \mapsto \int_{{\mathbb {R}}^{+}}e^{\rho(s)t}f(t)\,dt.$$ Moreover let $\nu\in Radon({\mathbb {R}}^{+})$ be the ${\mathbf{0}}-$extension of $\nu_{0}\in Radon(K(\Gamma))$ such that $$\nu_{0}:{\mathcal{C}_{cs} \left(K(\Gamma)\right)} \ni g \mapsto \int_{K(\Gamma)} \frac{- g(s)}{2\pi i} \frac{d\rho}{ds}(s)\, ds.$$ Finally let $M>1$, $\beta\in{\mathbb {R}}$ and $T\in{\mathcal{G}}(M,\beta,{\mathbf{E}})$, then we set ${\mathcal{W}}_{T} \in \prod_{x\in X}{\mathbf{U}}(B_{s}({\mathfrak{E}}_{x}))$ such that $(\forall x\in X)(\forall t\in{\mathbb {R}}^{+})$ $$\begin{cases} {\mathcal{W}}_{T}(x)(t) \doteqdot e^{-T(x)t},\\ {\overline}{F}_{T}\doteqdot\Lambda({\mathcal{W}}_{T}), \end{cases}$$ where $\Lambda$ has been defined in Def. \[14321401\]. \[15251311biss\] Let $M>1$, $\beta\in{\mathbb {R}}$ and $T\in{\mathcal{G}}(M,\beta,{\mathbf{E}})$ satisfy the property of separation of the spectrum. Assume that there exists a closed curve $\Gamma$ of which in Definition \[13361311biss\] such that $$\label{19281311biss} Re(\Gamma) \subseteq {\mathbb {R}}^{-}.$$ Then $\forall x\in X$ and $\forall v_{x}\in{\mathfrak{E}}_{x}$ $$\label{17551311biss} P(x)v_{x} = - \frac{1}{2\pi i} \int_{K(\Gamma)} \frac{d\rho}{ds}(s) R(T(x);\rho(s))v_{x} \,ds,$$ and $\forall s\in K(\Gamma)$, $$\label{18150402} R(T(x),\rho(s))v_{x} = \int_{0}^{\infty} e^{\rho(s) t} e^{-tT(x)} v_{x}\,dt = \int_{{\mathbb {R}}^{+}} {\mathcal{W}}_{T}(x)(t)v_{x}\, d\eta_{s}(t).$$ Here the integration is with respect to the norm topology on ${\mathfrak{E}}_{x}$. Moreover let ${\mathfrak{G}}$ be that in Def. \[10221801\] relative to ${\mathbf{E}}$. If ${\overline}{F}_{T}\in {\mathfrak{L}}_{(1,1)} ({\mathbb {R}}^{+},{\mathfrak{G}},\eta,\nu)$ and $V$ is a ${\left\langle \nu,\eta,{\mathfrak{G}},K(\Gamma),{\mathbb {R}}^{+}{\right\rangle}}$ invariant set with respect to $\{{\overline}{F}_{T}\}$, then [^19] $$\label{15351311biss} P\bullet V\subseteq V $$ By , [@IntBourb $IV.35$ Th. $1$], and by the norm continuity of the map $ B({\mathfrak{E}}_{x})\ni A\mapsto Aw\in {\mathfrak{E}}_{x}$ for any $w\in{\mathfrak{E}}_{x}$, we have . Moreover by we can apply [@kato eq. $1.28$, $n^{\circ}3$, $\S 1$, Ch. $9$] and follows by Def. \[16420402\]. Fix $v\in V$ so $\forall x\in X$ $$\begin{aligned} {1} \label{18590402} P(x)v(x) &= - \frac{1}{2\pi i} \int_{K(\Gamma)} \frac{d\rho}{ds}(s) R(T(x);\rho(s))v(x) \,ds \notag \\ &= - \frac{1}{2\pi i} \int_{K(\Gamma)} \frac{d\rho}{ds}(s) \left( \int_{{\mathbb {R}}} {\mathcal{W}}_{T}(x)(t)v(x)\, d\eta_{s}(t) \right) \,ds \notag \\ &=\int_{K(\Gamma)} \left( \int_{{\mathbb {R}}} \Pr_{x} \left( \Psi({\overline}{F}_{T}) \right)(t) v(x)\, d\eta_{s}(t)\right)\, d\nu(s).\end{aligned}$$ Here the first equality comes by , the second one by and the third one by Prop. \[15111401\] and Def. \[16420402\]. Finally with the notations in Corollary \[15111901\] we have that for the strong operator topology we can choose $(\forall x\in X) (S_{x}={\mathcal{P}}_{\omega}({\mathfrak{E}}_{x}))$, thus $D(x)={\mathfrak{E}}_{x}$, for all $x\in X$, and by Corollary \[15111901\] holds $(A)$ of Lemma \[11011501\]. Therefore the statement follows by and Proposition \[18310302\]. \[13511102\] Let us assume the hypotheses and notations of the Main Theorem \[17301812b\], and the hypotheses of Lemma \[15251311biss\] where $T$ is such that $-T(x)$ is the infinitesimal generator of ${\mathcal{U}}(x)$, for all $x\in X$. Moreover let $V\subset\prod_{x\in X}{\mathfrak{E}}_{x}$. Finally let ${\mathfrak{D}} \doteqdot {\left\langle {\left\langle {\mathfrak{B}},\gamma{\right\rangle}},\eta,X,{\mathfrak{L}}{\right\rangle}}$ and ${\left\langle {\mathfrak{V}},{\mathfrak{D}},X,\{pt\}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure such that $$\label{15181102} \Gamma(\eta) \subseteq \left\{ F\in\prod_{x\in X}{\mathcal{L}}({\mathfrak{E}}_{x}) \mid F\bullet V \subseteq V \right\},$$ ${\overline}{F}_{T}\in {\mathfrak{L}}_{(1,1)} ({\mathbb {R}}^{+},{\mathfrak{G}},\eta,\nu)$ and $V$ is a ${\left\langle \nu,\eta,{\mathfrak{G}},K(\Gamma),{\mathbb {R}}^{+}{\right\rangle}}$ invariant set with respect to $\{{\overline}{F}_{T}\}$, $\{{\mathcal{U}}\} \in \Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{D}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}$. Notice that ${\mathcal{E}}(\Theta)={\mathcal{E}}$ so if ${\mathcal{E}} \subseteq V \subseteq \Gamma(\pi)$ then implies that the $\left(\Theta,{\mathcal{E}}\right)-$structure ${\left\langle {\mathfrak{V}},{\mathfrak{D}},X,\{pt\}{\right\rangle}}$ is compatible. By and by and . \[18421202\] Our aim now is to see when $V$ is a ${\left\langle \nu,\eta,{\mathfrak{G}},K(\Gamma),{\mathbb {R}}^{+}{\right\rangle}}$ invariant set with respect to $\{{\overline}{F}_{T}\}$, by maintaining the positions ${\mathcal{E}} \subseteq V \subseteq \Gamma(\pi)$ and , ensuring as remarked, that ${\left\langle {\mathfrak{V}},{\mathfrak{D}},X,\{pt\}{\right\rangle}}$ is compatible. By using Corollary \[28111707\], we can try to show that if $V$ is a ${\left\langle \nu,\eta,{\mathfrak{G}},K(\Gamma),{\mathbb {R}}^{+}{\right\rangle}}$ invariant set with respect to $\Lambda(\Gamma(\rho))$ then $V$ is a ${\left\langle \nu,\eta,{\mathfrak{G}},K(\Gamma),{\mathbb {R}}^{+}{\right\rangle}}$ invariant set with respect to $\Lambda(\Gamma^{x_{\infty}}(\rho))$, hence by , with respect to $\{{\overline}{F}_{T}\}$. Might be at first glance not a weakening, but it is indeed the most powerfull way for constructing bundles of $\Omega-$spaces, namely that described in Def. \[17471910Ba\], allows us to choose, when $X$ is compact, the set $\Gamma(\rho)$, see Remark \[17150312\]. Assume the notations in Def. \[20030202\]. Another way, maybe better than the previous one, is the following. Let $v\in\prod_{x\in X}{\mathfrak{E}}_{x}$ and $A$ be a subalgebra of ${\mathfrak{G}}$, thus if we set $$V\doteqdot {\overline}{\textrm{span}}(Av),$$ closure in $\prod_{x\in X}{\mathfrak{E}}_{x}$, then by Lemma \[17141401\] we obtain $$A V\subseteq V.$$ Thus if ${\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}}$ is a topological algebra for all $x\in X$, it is sufficient to take for example the closure in ${\mathfrak{G}}$ of the algebra $A_{0}$ [^20] generated by any subset of ${\mathfrak{G}}$ which contains the set $$\left\{ \int \left( \int{\overline}{F}(s)\, d\eta_{\lambda}(s) \right)\, d\nu(\lambda) \mid {\overline}{F}\in{\mathcal{F}} \right\}.$$ Notice that by the fact that $P(x)\in\Pr({\mathfrak{E}}_{x})$ for all $x\in X$, we deduce by and Corollary \[19510202\] that $$\label{20081102} a_{T} \doteqdot \int \left( \int{\overline}{F}_{T}(s)\, d\eta_{\lambda}(s) \right)\, d\nu(\lambda) \in\Pr \left(\prod_{x\in X}{\mathfrak{E}}_{x}\right).$$ Thus the algebra $A_{0}$ should be generated by any subset of ${\mathfrak{G}}$ which contains not only the operator $a_{T}$, otherwise $A_{0}={\mathbb{K}}\cdot a_{T}$ and then $A={\mathbb{K}}\cdot a_{T}$ which is not interesting. The last equality comes by the fact that ${\mathbb{K}}\cdot v$ is a closed set for any $v\in Z-\{{\mathbf{0}}\}$ where $Z$ is a topological vector space such that that $Z^{*}$ separates the points of $Z$, for example any Hlcs. Indeed let $v,w\in Z$ such that $v\neq{\mathbf{0}}$ and $\{\lambda_{\alpha}\}_{\alpha\in D}$ a net in $Z$ such that $\lim_{\alpha\in D}\lambda_{\alpha}v=w$. Thus there exists $\phi\in Z^{*}$ such that $\phi(v)\neq 0$ and $\{\lambda_{\alpha}\phi(v)\}_{\alpha\in D}$ is a Cauchy net in ${\mathbb{K}}$. But $\phi(v)\neq 0$ so also $\{\lambda_{\alpha}\}_{\alpha\in D}$ is a Cauchy net in ${\mathbb{K}}$, let $\mu=\lim_{\alpha\in D}\lambda_{\alpha}$. Thus $w=\lim_{\alpha\in D}\lambda_{\alpha}v =\mu v$, which show that ${\mathbb{K}}v$ is closed in $Z$. Construction of classes $ \Delta_{\Theta}{\left\langle {\mathfrak{V}},{\mathfrak{D}},{\mathfrak{W}},{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}} $ through the generalized Lebesgue Theorem {#15511702} --------------------------------------------------------------------------------------------------------------------------- In this section $X$ is a topological space, $Y$ is a locally compact space $\mu$ is a Radon measure on $Y$, and ${\mathfrak{V}} = {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}}$ is a bundle of $\Omega-$spaces, we indicate with $ {\mathfrak{N}}\doteqdot \{ \nu_{j}\mid j\in J \} $ the directed set of seminorms on ${\mathfrak{E}}$. \[22132802\] Let $Z\in Hlcs$ and $\{\psi_{i}\mid i\in I\}$ a fundamental set of seminorms on $Z$. We denote by $$\left(Z^{Y}\right)_{s}$$ the $Hlcs$ whose linear space support is $Z^{Y}$ and whose $Hlct$ is that generated by the following set of seminorms $$\label{20531902} \begin{cases} \{q_{s}^{i}\mid s\in Y, i\in I\}, \\ q_{s}^{i}: Z^{Y} \ni f \mapsto \psi_{i}(f(s)). \end{cases}$$ Moreover for any $B\subseteq Z^{Y}$ we shall denote by $B_{s}$ the $Hlc$ subspace of $\left(Z^{Y}\right)_{s}$. Notice that this definition is well-set being indipendent by the choice of the fundamental set of seminorms, indeed the topology is that of uniform convergence over the finite subsets of $Y$. \[15361702\] Set $$\begin{cases} \overset{\mu}{\blacklozenge}: \prod_{x\in X} {\mathfrak{L}}_{1}(Y,{\mathfrak{E}}_{x},\mu) \to \prod_{x\in X} {\mathfrak{E}}_{x}, \\ \overset{\mu}{\blacklozenge}(H)(x) \doteqdot \int H(x)(s)\, d\mu(s) \in {\mathfrak{E}}_{x}, \end{cases}$$ for all $H\in \prod_{x\in X} {\mathfrak{L}}_{1}(Y,{\mathfrak{E}}_{x},\mu)$ and for all $x\in X$ \[12121902\] Set $$\begin{cases} \bigstar: \prod_{x\in X} {\mathcal{L}}({\mathfrak{E}}_{x})^{Y} \times \prod_{x\in X} {\mathfrak{E}}_{x} \to \prod_{x\in X}{\mathfrak{E}}_{x}^{Y}, \\ (\forall x\in X)(\forall s\in Y) (F\bigstar v)(x)(s) \doteqdot F(x)(s)(v(x)). \end{cases}$$ \[15492502\] *${\left\langle {\mathfrak{V}},{\mathfrak{Z}}{\right\rangle}}$ are $\mu-$related* if 1. ${\mathfrak{Z}} \doteqdot {\left\langle {\left\langle {\mathfrak{T}},\gamma{\right\rangle}},\zeta,X,{\mathfrak{K}}{\right\rangle}}$ be a bundle of $\Omega-$spaces; 2. for all $x\in X$ [^21] $$\label{20242906} \begin{cases} {\mathfrak{T}}_{x} \subseteq Meas(Y,{\mathfrak{E}}_{x},\mu) \bigcap {\mathfrak{L}}_{1}(Y,{\mathfrak{E}}_{x},\mu)_{s}, \\ {\mathfrak{K}}_{x} = \left\{ \sup_{(s,j)\in O} q_{(s,j)}^{x} \mid O\in{\mathcal{P}}_{\omega}(Y\times J) \right\}, \\ q_{(s,j)}^{x}: {\mathfrak{T}}_{x} \ni f_{x}\mapsto \nu_{j}(f_{x}(s)), \forall s\in Y, j\in J; \end{cases}$$ 3. $\Gamma(\zeta) \subset \left[ \prod_{x\in X}^{b} {\mathfrak{T}}_{x} \right]_{ui}$; 4. $\overset{\mu}{\blacklozenge}(\Gamma(\zeta)) \subseteq \Gamma(\pi)$. Here we set for all ${\mathcal{A}}\subseteq \prod_{x\in X}^{b} {\mathfrak{T}}_{x}$ $$\label{17180207} \left[ {\mathcal{A}}\right]_{ui} \doteqdot \left\{ H\in{\mathcal{A}}\mid (\forall j\in J) \left( \int_{Y}^{\bullet} \sup_{x\in X} \nu_{j}(H(x)(s)) \,d|\mu|(s) <\infty \right) \right\}$$ Finally *${\left\langle {\mathfrak{V}},{\mathfrak{Z}},{\mathbf{H}}{\right\rangle}}$ are $\mu-$related* if 1. ${\mathbf{H}}=\{{\mathbf{H}}_{x}\}_{x\in X}$ such that ${\mathbf{H}}_{x}\subseteq {\mathcal{L}}({\mathfrak{E}}_{x})^{Y}$ for all $x\in X$, 2. ${\left\langle {\mathfrak{V}},{\mathfrak{Z}}{\right\rangle}}$ are $\mu-$related 3. $$\label{19090907} \left( \prod_{x\in X}{\mathbf{H}}_{x}\right) \bigstar \left( \prod_{x\in X}{\mathfrak{E}}_{x}\right) \subseteq \prod_{x\in X}{\mathfrak{T}}_{x}.$$ \[**GLT**\] \[15101701\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{Z}}{\right\rangle}}$ be $\mu-$related. Then for all $x\in X$ $$\overset{\mu}{\blacklozenge} \left( \left[\Gamma_{\diamond}^{x}(\zeta) \right]_{ui} \right) \subseteq \Gamma_{\diamond}^{x}(\pi).$$ Let $x\in X$ and $F\in\left[\Gamma_{\diamond}^{x} (\zeta)\right]_{ui}$ thus by Corollary \[28111707\] there exists $\eta\in\Gamma(\zeta)$ such that for all $j\in J,s\in Y$ $$\label{12221702} \begin{cases} F(x)=\eta(x) \\ \lim_{z\to x} \nu_{j}(F(z)(s)-\eta(z)(s)) =0. \end{cases}$$ Fix $j\in J$ thus by [@IntBourb Prop.$6$, $No 2$, $\S 1$, Ch. $6$] for all $z\in X$ $$\label{14550907} \nu_{j} \left( \int (F(z)(s)-\eta(z)(s)) \,d\mu(s) \right) \leq \int^{\bullet} \nu_{j}(F(z)(s)-\eta(z)(s))\, d|\mu|(s)$$ Moreover $\nu_{j}^{z}$ is continuous by definition of bundles of $\Omega-$spaces, while $F(z)$ and $\eta(z)$ are by construction $\mu-$measurable, hence by [@IntBourb Thm. $1$; Cor. $3$, $n^{\circ}3$, $\S\,5$,Ch. $4$] the map $Y\ni s\mapsto \nu_{j}(F(z)(s)-\eta(z)(s))$ is $\mu-$measurable thus $|\mu|-$measurable. Moreover by the hypothesis on $F$ and by $(3)$ of Def. \[15492502\] $$\label{15370907} \int^{\bullet} \nu_{j}(F(z)(s)-\eta(z)(s))\, d|\mu|(s) \leq \int^{\bullet} \left( \sup_{x\in X} \nu_{j}(F(x)(s)) + \sup_{x\in X} \nu_{j}(\eta(x)(s)) \right) d|\mu|(s) <\infty.$$ Therefore by [@IntBourb Prp..$9$, $No 3$, $\S 1$, Ch. $5$] the map $Y\ni s\mapsto \nu_{j}(F(z)(s)-\eta(z)(s))$ is $|\mu|-$ essentially integrable hence by the fact that $\int_{Y}^{\bullet}f\,d|\mu| =\int_{Y}f\,d|\mu|$ for all $|\mu|-$essentially integrable map $f$, we have by $$\label{12541702} \nu_{j} \left( \int (F(z)(s)-\eta(z)(s)) \,d\mu(s) \right) \leq \int \nu_{j}(F(z)(s)-\eta(z)(s)) \, d|\mu|(s).$$ Let $\{z_{n}\}_{n}\subset X$ be such that $\lim_{n\in{\mathbb{N}}}z_{n}=x$ thus by $$\label{12491702A} \lim_{n\in{\mathbb{N}}} \nu_{j}(F(z_{n})(s)-\eta(z_{n})(s)) =0,$$ For all $s\in Y$ $$\nu_{j}(F(z)(s)-\eta(z)(s))\, \leq \sup_{x\in X} \nu_{j}(F(x)(s)) + \sup_{x\in X} \nu_{j}(\eta(x)(s))$$ thus by the hypothesis on $F$, by $(3)$ of Def. \[15492502\], by the fact that $\int_{Y}^{\bullet}\leq\int_{Y}^{*}$, by , and by the Lebesgue Theorem [@IntBourb Th.$6$, $No 7$, $\S 3$, Ch. $4$] we have $$\label{12531702} \lim_{n\in{\mathbb{N}}} \int \nu_{j}(F(z_{n})(s)-\eta(z_{n})(s))\, d|\mu|(s) =0.$$ Finally by , and the fact that $X$ is metrizable we obtain $$\lim_{z\to x} \nu_{j} \left( \int F(z)(s) \, d\mu(s) - \int \eta(z)(s) \, d \mu(s) \right) =0,$$ thus the statement follows by hypothesis $(4)$ and Corollary \[28111707\]. \[ **$\left(\Theta,{\mathcal{E}},\mu\right)-$structures**\] \[17161902\] We say that ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,Y{\right\rangle}}$ is a *$\left(\Theta,{\mathcal{E}},\mu\right)-$ structure* if 1. $ {\mathfrak{V}} \doteqdot {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,{\mathfrak{N}}{\right\rangle}} $ is a bundle of $\Omega-$spaces; 2. ${\mathcal{E}}\subseteq\Gamma(\pi)$; 3. $\Theta\subseteq \prod_{x\in X} Bounded({\mathfrak{E}}_{x})$; 4. $\forall B\in\Theta$ 1. ${\mathbf{D}}(B,{\mathcal{E}}) \ne\emptyset$; 2. $\bigcup_{B\in\Theta}{\mathcal{B}}_{B}^{x}$ is total in ${\mathfrak{E}}_{x}$ for all $x\in X$; 5. ${\mathfrak{Q}} \doteqdot {\left\langle {\left\langle {\mathfrak{H}},\gamma{\right\rangle}},\xi,X,{\mathfrak{Y}}{\right\rangle}}$ is a bundle of $\Omega-$spaces such that for all $x\in X$ $$\label{22512906} \boxed{ \begin{cases} {\mathfrak{H}}_{x} \subseteq {\mathfrak{L}}_{1} \left(Y,{\mathcal{L}}_{S_{x}} ({\mathfrak{E}}_{x}),\mu\right)_{s}, \\ {\mathfrak{Y}}_{x} = \left\{ \sup_{(t,j,B)\in O} P_{(t,j,B)}^{x} \mid O\in{\mathcal{P}}_{\omega} \left(Y\times J\times\Theta\right) \right\} \\ P_{(t,j,B)}^{x}: {\mathfrak{H}}_{x} \ni F\mapsto \sup_{v\in{\mathbf{D}}(B,{\mathcal{E}})} \nu_{j}(F(t)v(x)),\, \forall t\in Y, B\in\Theta, j\in J. \end{cases} }$$ Here $S_{x}$, ${\mathcal{B}}_{B}^{x}$ and ${\mathbf{D}}(B,{\mathcal{E}})$ are defined in . Moreover ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,Y{\right\rangle}}$ is an *invariant $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure* if it is a $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure such that $$\label{18112502} \left\{ F\in\prod_{z\in X}^{b} {\mathfrak{H}}_{z} \mid (\forall t\in Y) (F_{t} \bullet {\mathcal{E}}(\Theta) \subseteq \Gamma(\pi)) \right\} = \Gamma(\xi).$$ \[14350303\] Let $\mu_{\lambda}$ for all $\lambda>0$ be defined as in Def. \[15062301\], let ${\mathfrak{Q}} = {\left\langle {\left\langle {\mathfrak{H}},\gamma{\right\rangle}},\xi,X,{\mathfrak{S}}{\right\rangle}}$, ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,{\mathbb {R}}^{+}{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure, $x\in X$, ${\mathcal{O}}\subseteq\Gamma(\xi)$. and ${\mathcal{D}}\subseteq\Gamma(\pi)$. Set $${\mathbf{Lap}}({\mathfrak{V}})(x) \doteqdot \bigcap_{\lambda>0} {\mathfrak{L}}_{1}({\mathbb {R}}^{+}, {\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x});\mu_{\lambda}).$$ Assume that $$\label{12480707} \Gamma_{{\mathcal{O}}}^{x}(\xi) \bigcap {\mathbf{Lap}}({\mathfrak{V}})(x) \neq \emptyset$$ We say that ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,{\mathbb {R}}^{+}{\right\rangle}}$ has the *weak-Laplace duality property on ${\mathcal{O}}$ and ${\mathcal{D}}$ at $x$*, shortly $w-{\mathbf{LD}}_{x}({\mathcal{O}},{\mathcal{D}})$ if $\forall\lambda>0$ $$\blacksquare_{\mu_{\lambda}} \left( \Gamma_{{\mathcal{O}}}^{x}(\xi) \bigcap {\mathbf{Lap}}({\mathfrak{V}})(x), \Gamma_{{\mathcal{D}}}^{x}(\pi) \right) \subseteq \Gamma^{x}(\pi).$$ \[18072802\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure and denote ${\mathfrak{W}} \doteqdot {\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$. Assume that for all $x\in X$ $$\label{22372906} {\mathfrak{M}}_{x} \subseteq {\mathfrak{L}}_{1} \left(Y,{\mathcal{L}}_{S_{x}} ({\mathfrak{E}}_{x}),\mu\right).$$ Set ${\mathbf{M}}^{\mu} \doteqdot \{{\left\langle {\mathbf{M}}_{x}^{\mu},{\mathfrak{Y}}_{x}{\right\rangle}}\}_{x\in X}$ where for all $x\in X$ $$\begin{cases} {\mathbf{M}}_{x}^{\mu} \doteqdot {\overline}{ \left\{ \sigma(x) \mid \sigma\in \Gamma(\rho) \right\} }, \\ {\mathfrak{Y}}_{x} \doteqdot \{ \sup_{(t,j,B)\in O} P_{(t,j,B)}^{x} \mid O\in{\mathcal{P}}_{\omega}(Y\times J \times\Theta) \}, \end{cases}$$ where the closure is in the space $${\mathfrak{L}}_{1} \left(Y,{\mathcal{L}}_{S_{x}} ({\mathfrak{E}}_{x}),\mu\right)_{s},$$ and ${\mathbf{M}}_{x}^{\mu}$ has to be considered as Hlc subspace of it, finally here $P_{(t,j,B)}^{x}$ is defined on ${\mathbf{M}}_{x}^{\mu}$ as in . Notice that ${\mathbf{M}}^{\mu}$ is a nice family of Hlcs, and that $\Gamma(\rho)$ satisfies by construction $FM(3)$ with respect to ${\mathbf{M}}^{\mu}$. Moreover by and the fact that $\{t\}\in Comp(Y)$ for all $t\in Y$ $$\label{22582906} P_{(t,j,B)}^{x} =q_{(\{t\},j,B)}^{x}.$$ By [@gie Cor.$1.6.(iii)$] we deduce that $\Gamma(\rho)$ satisfies $FM(4)$ with respect to $\{{\left\langle {\mathfrak{M}}_{x},{\mathfrak{R}}_{x}{\right\rangle}}\}_{x\in X}$. Therefore we obtain by and that for all $t\in Y$, $j\in J$, $B\in\Theta$ and for all $\sigma\in\Gamma(\rho)$ $$X\ni x\mapsto P_{(t,j,B)}^{x}(\sigma(x)) \text{ is $u.s.c.$}.$$ Moreover the upper envelope of a finite set of $u.s.c.$ maps is an $u.s.c.$ map, see [@BourGT Thm.$4$,$\S 6.2.$,Ch.$4$], therefore for all $O\in{\mathcal{P}}_{\omega}(Y\times J \times\Theta)$ $$\label{16392906} X\ni x\mapsto \sup_{(t,j,B)\in O} P_{(t,j,B)}^{x}(\sigma(x)) \text{ is $u.s.c.$}.$$ Hence $\Gamma(\rho)$ satisfies $FM(4)$ with respect to ${\mathbf{M}}^{\mu}$. Finally by the boundedness of $\Gamma(\rho)$ by definition and by we have also that for all $\sigma\in\Gamma(\rho)$ and $O\in{\mathcal{P}}_{\omega}(Compact(Y)\times J \times\Theta)$ $$\sup_{x\in X} \sup_{(t,j,B)\in O} P_{(t,j,B)}^{x}(\sigma(x)) <\infty.$$ Therefore we can construct the bundle generated by the couple ${\left\langle {\mathbf{M}}^{\mu},\Gamma(\rho){\right\rangle}}$, see Def. \[17471910Ba\] $${\mathfrak{V}}({\mathbf{M}}^{\mu},\Gamma(\rho))$$ We shall call ${\left\langle {\mathfrak{V}},{\mathfrak{V}}({\mathbf{M}}^{\mu}, \Gamma(\rho)),X,Y{\right\rangle}}$ the *$\left(\Theta,{\mathcal{E}},\mu\right)-$ structure underlying ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$*. \[19421902\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,Y{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure and $A\subset\prod_{x\in X}{\mathfrak{H}}_{x}$. Define $A_{peq}$ as the set of all pointwise equicontinuous elements in $A$, and $A_{ceq}$ as the set of all compactly equicontinuous elements in $A$, see Def. \[21031238\]. \[23222906\] Lemma \[15482712\] holds by replacing a $\left(\Theta,{\mathcal{E}}\right)-$structure ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ with a $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,Y{\right\rangle}}$ and $K\in Comp(Y)$ with $t\in Y$. In what follows when referring to Lemma \[15482712\] for a $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure we shall mean the corresponding result with the replacements described here. \[12151902\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,Y{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure and ${\left\langle {\mathfrak{V}},{\mathfrak{Z}},{\mathbf{H}}{\right\rangle}}$ be $\mu-$related, where ${\mathfrak{Q}} \doteqdot {\left\langle {\left\langle {\mathfrak{H}},\gamma{\right\rangle}},\xi,X,{\mathfrak{Y}}{\right\rangle}}$ and ${\mathbf{H}}_{x}\doteqdot{\mathfrak{H}}_{x}$ for all $x\in X$. Thus $$\Gamma(\xi) \bigstar {\mathcal{E}}(\Theta) \subseteq \Gamma(\zeta) \Rightarrow (\forall x\in X) \left( \Gamma_{\diamond}^{x}(\xi)_{peq} \bigstar \Gamma_{{\mathcal{E}}(\Theta)}^{x}(\pi) \subseteq \Gamma_{\diamond}^{x}(\zeta)\right).$$ Let $j\in J$, $x\in X$ and $w\in\Gamma_{{\mathcal{E}}(\Theta)}^{x}(\pi)$, so there exists $v\in{\mathcal{E}}(\Theta)$ such that $v(x)=w(x)$ then by Cor. \[28111707\] $$\label{14311902} \lim_{z\to x} \nu_{j}(w(z)-v(z)) =0.$$ Moreover let $F\in\Gamma_{\diamond}^{x}(\xi)$, so by Lemma \[15482712\] $\exists\,\sigma\in\Gamma(\xi)$ such that $F(x)=\sigma(x)$ and for all $t\in Y$ $$\label{14301902} \lim_{z\to x} \nu_{j}\left(F(z)(t)v(z)-\sigma(z)(t)v(z) \right)=0.$$ Moreover $(\forall t\in Y) (\exists\,M_{(t,j)}>0) (\exists\,j_{1}\in J) (\forall z\in X)$ $$\begin{aligned} {1} \nu_{j}\left((F\bigstar w)(z)(t)- (\sigma\bigstar v)(z)(t)\right) &= \nu_{j}\left(F(z)(t)w(z)- \sigma(z)(t)v(z)\right) \\ &\leq \nu_{j}\left(F(z)(t)(w(z)-v(z))\right) + \nu_{j}\left(F(z)(t)v(z)-\sigma(z)(t)v(z) \right) \\ &\leq M_{(t,j)} \nu_{j_{1}}(w(z)-v(z)) + \nu_{j}\left(F(z)(t)v(z)-\sigma(z)(t)v(z) \right).\end{aligned}$$ Therefore by and for all $t\in Y$ $$\lim_{z\to x} \nu_{j}\left((F\bigstar w)(z)(t)- (\sigma\bigstar v)(z)(t)\right) =0.$$ Moreover $(\forall t\in Y) (\exists\,M_{(t,j)}>0) (\exists\,j_{1}\in J)$ $$\label{18590907} \sup_{z\in X} \nu_{j}((F\bigstar w)(z)(t)) \leq M_{(t,j)} \sup_{z\in X} \nu_{j_{1}}(w(z)) <\infty.$$ By the antecedent of the implication of the statement we deduce that $\sigma\bigstar v\in\Gamma(\zeta)$ hence the statement follows by Cor. \[28111707\], , by the fact that by $F\bigstar w\in\prod_{x\in X}{\mathfrak{T}}_{x}$ and by . \[17441202bis\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ be a compatible $\left(\Theta,{\mathcal{E}}\right)-$structure. Then for all $x\in X$ $$(\Gamma_{\diamond}^{x}(\rho)_{peq})_{t} \bullet \Gamma_{{\mathcal{E}}(\Theta)}^{x}(\pi) \subseteq \Gamma_{\diamond}^{x}(\pi)$$ Notice that $(F\bigstar v)(t)=F(\cdot)(t)\bullet v$ with obvious meaning of the symbols. Thus if we set $Y=\{pt\}$ the statement follows by Lm. \[12151902\]. \[15262502\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,Y{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure and ${\left\langle {\mathfrak{V}},{\mathfrak{Z}},{\mathbf{H}}{\right\rangle}}$ be $\mu-$related, Thus for all $x\in X$ $$\Gamma(\xi) \bigstar {\mathcal{E}}(\Theta) \subseteq \Gamma(\zeta) \Rightarrow \overset{\mu}{\blacklozenge} \left( \left[ \Gamma_{\diamond}^{x}(\xi)_{peq} \bigstar \Gamma_{{\mathcal{E}}(\Theta)}^{x}(\pi) \right]_{ui} \right) \subseteq \Gamma_{\diamond}^{x}(\pi).$$ Here ${\mathfrak{Q}} \doteqdot {\left\langle {\left\langle {\mathfrak{H}},\gamma{\right\rangle}},\xi,X,{\mathfrak{Y}}{\right\rangle}}$, ${\mathfrak{Z}} \doteqdot {\left\langle {\left\langle {\mathfrak{T}},\delta{\right\rangle}},\zeta,X,{\mathfrak{K}}{\right\rangle}}$ and ${\mathbf{H}}_{x}\doteqdot{\mathfrak{H}}_{x}$ for all $x\in X$. By Theorem \[15101701\] and Lemma \[12151902\]. \[14452602\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,Y{\right\rangle}}$ be an invariant $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure where $\Theta$ defined in . Then for all $x\in X$ $$\label{15232602} \left\{ H\in \left[ \left( \prod_{z\in X}^{b} {\mathfrak{H}}_{z} \right)_{\diamond}^{x} \right]_{peq} \mid (\forall t\in Y) (H(\cdot)(t) \bullet {\mathcal{E}}(\Theta) \subseteq \Gamma^{x}(\pi)) \right\} \subseteq \Gamma_{\diamond}^{x}(\xi).$$ Let $v\in{\mathcal{E}}(\Theta)$, $t\in Y$ and $H$ belong to the set in the left side of . Thus by $\exists\,F\in\Gamma(\xi)$ such that $F_{t} \bullet v \in\Gamma(\pi)$, $F(x)=H(x)$ and $H(\cdot)(t) \bullet v \in \Gamma^{x}(\pi)$ by construction. Then by Corollary \[28111707\] we obtain for all $j\in J$ $$\lim_{z\to x} \nu_{j}(H(z)(t)v(z)-F(z)(t)v(z)).$$ Therefore the statement follows by Lemma \[15482712\] and . \[12012802\] Notice that Lemma \[14452602\] holds if we replace invariant $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure with invariant $\left(\Theta,{\mathcal{E}}\right)-$ structure, see Def. \[10282712\], and assume that $Comp(Y)=\{\{t\}\mid t\in Y\}$. \[21152602\] Assume that ${\mathfrak{V}}$ is a Banach bundle and set ${\mathbf{E}}\doteqdot\{{\mathfrak{E}}_{x}\}_{x\in X}$. By using the notations of Definitions \[16420402\] and \[13361311biss\] assume that $M>1$, $\beta\in{\mathbb {R}}$ and $T\in{\mathcal{G}}(M,\beta,{\mathbf{E}})$ satisfy the property of separation of the spectrum, moreover that there exists a closed curve $\Gamma$ of which in Definition \[13361311biss\] such that $$\label{19281311} \begin{cases} Re(\Gamma) \subseteq {\mathbb {R}}^{-}, \\ \beta\geq 0 \Rightarrow -\beta \notin Re(\Gamma). \end{cases}$$ Moreover assume that for all $\mu\in\{\nu,\eta_{s}\mid s\in K(\Gamma)\}$ 1. ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,{\mathbb {R}}^{+}{\right\rangle}}$ is an invariant $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure [^22]; 2. ${\left\langle {\mathfrak{V}},{\mathfrak{Z}},{\mathbf{H}}{\right\rangle}}$ is $\mu-$related; 3. ${\mathfrak{Q}} $ is full; 4. for all $z\in X$ $$\label{11130307} {\mathcal{C}_{cs} \left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{z}}({\mathfrak{E}}_{z})\right)} \subseteq {\mathfrak{H}}_{z};$$ 5. $\Gamma(\xi) \bigstar {\mathcal{E}}(\Theta) \subseteq \Gamma(\zeta)$. Here $\Theta$ is defined in , ${\mathfrak{Q}} \doteqdot {\left\langle {\left\langle {\mathfrak{H}},\gamma{\right\rangle}},\xi,X,{\mathfrak{R}}{\right\rangle}}$, ${\mathfrak{Z}} \doteqdot {\left\langle {\left\langle {\mathfrak{T}},\gamma{\right\rangle}},\zeta,X,{\mathfrak{K}}{\right\rangle}}$ and ${\mathbf{H}}_{x}\doteqdot{\mathfrak{H}}_{x}$, for all $x\in X$. Then for all $x\in X$ $$\label{12042802} {\mathcal{W}}_{T}\in \Gamma^{x}(\xi) \Rightarrow P\bullet \Gamma_{{\mathcal{E}}(\Theta)}^{x}(\pi) \subseteq \Gamma^{x}(\pi).$$ Moreover if ${\left\langle {\mathfrak{V}},{\mathfrak{D}},X,\{pt\}{\right\rangle}}$ is an invariant $\left(\Theta,{\mathcal{E}}\right)-$structure such that ${\mathfrak{B}}_{x}={\mathcal{L}}\left({\mathfrak{E}}_{x}\right)$ for all $x$ and ${\mathfrak{D}}$ is full then for all $x\in X$ $$\label{12102802} {\mathcal{W}}_{T} \in \Gamma^{x}(\xi) \Rightarrow P \in\Gamma^{x}(\eta),$$ where ${\mathfrak{D}} \doteqdot {\left\langle {\left\langle {\mathfrak{B}},\gamma{\right\rangle}},\eta,X,{\mathfrak{L}}{\right\rangle}}$. In this proof we denote $R_{T}^{\rho}$ simply by $R^{\rho}$. $R^{\rho}$ is $K(\Gamma)-$supported by construction, moreover by the analyticity of the resolvent map $R^{\rho}(x)$ is $\|\cdot\|_{B({\mathfrak{E}}_{x})}-$ continuous hence continuous as a map valued in ${\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})$. [^23] So $$R^{\rho} \in\prod_{z\in X} {\mathcal{C}_{cs} \left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{z}}({\mathfrak{E}}_{z})\right)},$$ hence by follows $$\label{17500107} R^{\rho} \in \prod_{z\in X} {\mathfrak{H}}_{z}.$$ By for all $s\in K(\Gamma)$, $x\in X$ and $\forall v_{x}\in {\mathfrak{E}}_{x}$ $$\label{17051311} \begin{aligned} \| R(T(x),\rho(s))v_{x} \| & \leq \int_{{\mathbb {R}}^{+}}^{*} e^{-|Re(\rho(s))|t} \|e^{-tT(x)}v_{x}\| dt \\ & \leq M\|v_{x}\| \int_{{\mathbb {R}}^{+}}^{*} e^{\left(\beta-|Re(\rho(s))|\right) t} dt = \frac{M\|v_{x}\|}{\beta-|Re(\rho(s))|}, \end{aligned}$$ where $\int_{{\mathbb {R}}^{+}}^{*}$ is the upper integral on ${\mathbb {R}}^{+}$ with respect to the Lebesgue measure. We considered in the first inequality [@IntBourb Prop. $6$, $n^{\circ}$, $\S 1$, Ch. $6$], in the second one the inequality , finally in the equality the Laplace transform of the map $\exp(\beta t)$. Therefore by and $$R^{\rho} \in\left( \prod_{z\in X} {\mathfrak{H}}_{z} \right)_{peq}.$$ Thus by , and $$\label{10312802} R^{\rho} \in\left( \prod_{z\in X}^{b} {\mathfrak{H}}_{z} \right)_{peq}.$$ By and we have that $R^{\rho}\bigstar v \in \prod_{z\in X} {\mathfrak{T}}_{z}$ for all $v\in\prod_{x\in X}^{b}{\mathfrak{E}}_{x}$. By hypothesis we deduce that $\frac{1} {\beta-|Re(\rho(s))|}$ is defined on $K(\Gamma)$, hence continuous and integrable in it, thus by $$\label{17590207} R^{\rho}\bigstar v \in \left[ \prod_{z\in X} {\mathfrak{T}}_{z} \right]_{ui}.$$ By the continuity of $\frac{1} {\beta-|Re(\rho(s))|}$ on $K(\Gamma)$ we deduce that the map $ \frac{|\frac{d\rho}{ds}(s)|} {\beta-|Re(\rho(s))|} $ is integrable in $K(\Gamma)$. Hence by and $$\label{17483006} \sup_{x\in X} \|P(x)\|_{B({\mathfrak{E}}_{x})} \leq D \doteqdot \frac{1}{2\pi i} \int_{K(\Gamma)} \frac{M\left|\frac{d\rho}{ds}(s)\right|} {\beta-|Re(\rho(s))|} \,ds.$$ Therefore for all $v\in{\mathcal{E}}$ by considering that ${\mathcal{E}}\subset\prod_{x\in X}^{b}{\mathfrak{E}}_{x}$ $$\sup_{x\in X} \|P(x)v(x)\|_{B({\mathfrak{E}}_{x})} \leq D\sup_{x\in X} \|v(x)\|_{{\mathfrak{E}}_{x}} <\infty$$ Thus $$\label{15181303} P\in\prod_{x\in X}^{b}{\mathfrak{B}}_{x}.$$ Let $x\in X$ and $v\in\Gamma_{{\mathcal{E}}(\Theta)}^{x}(\pi)$. By for all $s\in K(\Gamma)$ $$\label{17542502} (R^{\rho}\bigstar v)(x)(s) = {\overset{\eta_{s}}{\blacklozenge}} ({\mathcal{W}}_{T}\bigstar v)(x).$$ Moreover by $$\label{17562502} P\bullet v = {\overset{\nu}{\blacklozenge}} (R^{\rho}\bigstar v).$$ Notice that $(R^{\rho}\bigstar v)(x)(s) = \left(R^{\rho}(\cdot)(s)\bullet v \right)(x) $ so by for all $s\in K(\Gamma)$ $$\label{18022502} R^{\rho}(\cdot)(s)\bullet v = {\overset{\eta_{s}}{\blacklozenge}} ({\mathcal{W}}_{T}\bigstar v).$$ If ${\mathcal{W}}_{T}\in\Gamma^{x}(\xi)$ then by [@kato Ch. $9$, $\S1$, $n^{\circ}4$, $(1.37)$] and the hyp. that ${\mathfrak{Q}}$ is full follows that ${\mathcal{W}}_{T} \in \Gamma_{\diamond}^{x}(\xi)_{peq}$. Therefore by Corollary \[15262502\] and for all $s\in K(\Gamma)$ $$\label{13540303} R^{\rho}(\cdot)(s) \bullet v \in\Gamma^{x}(\pi).$$ By construction ${\mathcal{E}}(\Theta) \subseteq\Gamma(\pi)$ so ${\mathcal{E}}(\Theta) \subseteq \Gamma_{{\mathcal{E}}(\Theta)}^{x}(\pi)$, hence $$\label{18042502} R^{\rho}(\cdot)(s) \bullet {\mathcal{E}}(\Theta) \subseteq \Gamma^{x}(\pi).$$ Moreover by the hypothesis that ${\mathfrak{Q}}$ is full and by follows that $$R^{\rho} \in \left[ \left( \prod_{z\in X}^{b} {\mathfrak{H}}_{z} \right)_{\diamond}^{x} \right]_{peq}.$$ Hence by Lemma \[14452602\] and $$\label{18132502} R^{\rho} \in\Gamma_{\diamond}^{x}(\xi)_{peq}.$$ Finally follows by , , and Corollary \[15262502\]. In conclusion by , and the hyp. that ${\mathfrak{D}}$ is full $$P\in \left[ \left( \prod_{z\in X}^{b} {\mathfrak{B}}_{z} \right)_{\diamond}^{x} \right]_{peq}.$$ Thus follows by , Remark \[12012802\] and by ${\mathcal{E}}(\Theta) \subseteq \Gamma_{{\mathcal{E}}(\Theta)}^{x}(\pi)$. \[12500303\] The statements of Corollary \[21152602\] hold by replacing “for all $x\in X$” by “$x_{\infty}\in X$”, and the hypothesis that ${\left\langle {\mathfrak{V}},{\mathfrak{Z}},{\mathbf{H}}{\right\rangle}}$ is $\mu-$related and ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,{\mathbb {R}}^{+}{\right\rangle}}$ is an invariant $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure for all $\mu\in\{\nu,\eta_{s}\mid s\in K(\Gamma)\}$, with the following two 1. ${\left\langle {\mathfrak{V}},{\mathfrak{Z}},{\mathbf{H}}{\right\rangle}}$ is $\nu-$related and ${\left\langle {\mathfrak{V}},{\mathfrak{Q}},X,{\mathbb {R}}^{+}{\right\rangle}}$ is an invariant $\left(\Theta,{\mathcal{E}},\nu\right)-$ structure; 2. $R^{\rho}(\cdot)(s) \bullet {\mathcal{E}} \subseteq \Gamma^{x_{\infty}}(\pi)$ for all $s\in K(\Gamma)$. Indeed follows for $x=x_{\infty}$ and the proof runs as that of Cor. \[21152602\]. The following result shall permit to apply Corollary \[21152602\] to the Main Theorem \[17301812b\]. \[23332802\] Let ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$ be a $\left(\Theta,{\mathcal{E}}\right)-$structure and denote ${\mathfrak{W}} \doteqdot {\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$. Assume that holds for all $x\in X$ and let ${\left\langle {\mathfrak{V}},{\mathfrak{V}} ({\mathbf{M}}^{\mu},\Gamma(\rho)),X,Y{\right\rangle}}$ be the $\left(\Theta,{\mathcal{E}},\mu\right)-$ structure underlying ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,Y{\right\rangle}}$. Then for all $x\in X$ $$\label{17350303} \Gamma_{\diamond}^{x}(\rho) \subseteq \Gamma_{\Gamma(\rho)}^{x} (\pi_{{\mathbf{M}}^{\mu}})$$ where the right-side of the implication has to be considered modulo the canonical isomorphism. By Remark \[17150312\] $\Gamma(\rho) \subseteq \Gamma(\pi_{{\mathbf{M}}^{\mu}})$, modulo the canonical isomorphism. Thus the statement by Lemma \[15482712\] and . \[14460503\] Let ${\mathfrak{V}} \doteqdot {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,\|\cdot\|{\right\rangle}}$ be a Banach bundle. Let $x_{\infty}\in X$ and $ {\mathcal{U}}_{0} \in\prod_{x\in X_{0}} {\mathcal{C}_{} \left({\mathbb {R}}^{+},B_{s}({\mathfrak{E}}_{x})\right)}$ be such that ${\mathcal{U}}_{0}(x)$ is a $(C_{0})-$semigroup of contractions (respectively of isometries) on ${\mathfrak{E}}_{x}$ for all $x\in X_{0}$. Moreover let $T_{x}$ be the infinitesimal generator of the semigroup ${\mathcal{U}}_{0}(x)$ for any $x\in X_{0}$. Assume $D(T_{x_{\infty}})$, as defined in Notations \[15411512b\], to be dense in ${\mathfrak{E}}_{x_{\infty}}$, that $\{v(x)\mid v\in{\mathcal{E}}\}$ is dense in ${\mathfrak{E}}_{x}$ for all $x\in X_{0}$ and $\exists\lambda_{0}>0$ (respectively $\exists\lambda_{0}>0, \lambda_{1}<0$) such that the range ${\mathcal{R}}(\lambda_{0}-T_{x_{\infty}})$ is dense in ${\mathfrak{E}}_{x_{\infty}}$, (respectively the ranges ${\mathcal{R}}(\lambda_{0}-T_{x_{\infty}})$ and ${\mathcal{R}}(\lambda_{1}-T_{x_{\infty}})$ are dense in ${\mathfrak{E}}_{x_{\infty}}$). Set $$\label{16290303} \begin{cases} {\mathcal{T}}_{0}(x) \doteqdot Graph(T_{x}),\, x\in X_{0} \\ \Phi \doteqdot \{ \phi\in\Gamma^{x_{\infty}} (\pi_{{\mathbf{E}}^{\oplus}}) \mid (\forall x\in X_{0}) (\phi(x)\in{\mathcal{T}}_{0}(x)) \} \\ {\mathcal{E}} \doteqdot \{ v \in \Gamma(\pi) \mid (\exists\,\phi\in\Phi) (v(x_{\infty})=\phi_{1}(x_{\infty})) \} \\ \Theta \doteqdot \left\{ \prod_{x\in X}\{v(x)\} \mid v\in{\mathcal{E}} \right\} \\ T_{x_{\infty}}: D(T_{x_{\infty}}) \ni \phi_{1}(x_{\infty}) \mapsto \phi_{2}(x_{\infty}), \end{cases}$$ where ${\left\langle {\left\langle {\mathfrak{E}}({\mathbf{E}}^{\oplus}),\tau({\mathbf{E}}^{\oplus},{\mathcal{E}}^{\oplus}){\right\rangle}},\pi_{{\mathbf{E}}^{\oplus}},X,{\mathfrak{n}}^{\oplus}{\right\rangle}}$ is the bundle direct sum of the family $\{{\mathfrak{V}}, {\mathfrak{V}}\}$. Set $${\mathcal{U}}\in\prod_{x\in X} {\mathbf{U}}_{\|\cdot\|_{B({\mathfrak{E}}_{x})}} ({\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x}))$$ (respectively ${\mathcal{U}}\in\prod_{x\in X} {\mathbf{U}}_{is} ({\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x}))$) such that ${\mathcal{U}}(x)$ is the the $C_{0}-$semigroup of contractions (respectively of isometries) on ${\mathfrak{E}}_{x}$ whose infinitesimal generator is $T_{x}$ for all $x\in X$. The definitions of $T_{x_{\infty}}$ and ${\mathcal{U}}(x_{\infty})$ are well-posed by Theorem \[17301812b\]. Finally let ${\mathcal{T}}$ be defined as in Notations \[15411512b\], with ${\mathcal{T}}_{0}$ and $\Phi$ given in . Recall that by Theorem \[17301812b\] $$\{{\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}\} \in Graph\left({\mathfrak{V}},{\mathfrak{V}}\right).$$ \[**MAIN**${\mathbf{2}}$\] \[13020103\] Let $X$ be compact and ${\mathfrak{V}} \doteqdot {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,\|\cdot\|{\right\rangle}}$ be a Banach bundle. Let $x_{\infty}\in X$ and $ {\mathcal{U}}_{0} \in\prod_{x\in X_{0}} {\mathcal{C}_{} \left({\mathbb {R}}^{+},B_{s}({\mathfrak{E}}_{x})\right)}$ be such that ${\mathcal{U}}_{0}(x)$ is a $(C_{0})-$semigroup of contractions (respectively of isometries) on ${\mathfrak{E}}_{x}$ for all $x\in X_{0}\doteqdot X-\{x_{\infty}\}$. Moreover let $T_{x}$ be the infinitesimal generator of the semigroup ${\mathcal{U}}_{0}(x)$ for any $x\in X_{0}$. Assume that $\{v(x)\mid v\in{\mathcal{E}}\}$ is dense in ${\mathfrak{E}}_{x}$ for all $x\in X$ and $\exists\lambda_{0}>0$ (respectively $\exists\lambda_{0}>0, \lambda_{1}<0$) such that the range ${\mathcal{R}}(\lambda_{0}-T_{x_{\infty}})$ is dense in ${\mathfrak{E}}_{x_{\infty}}$, (respectively the ranges ${\mathcal{R}}(\lambda_{0}-T_{x_{\infty}})$ and ${\mathcal{R}}(\lambda_{1}-T_{x_{\infty}})$ are dense in ${\mathfrak{E}}_{x_{\infty}}$). Let us assume the notations of Definition \[14460503\] moreover let ${\mathfrak{Z}} = {\left\langle {\left\langle {\mathfrak{T}},\delta{\right\rangle}},\zeta,X,{\mathfrak{K}}{\right\rangle}}$ and ${\mathfrak{W}} \doteqdot {\left\langle {\left\langle {\mathfrak{M}},\gamma{\right\rangle}},\rho,X,{\mathfrak{R}}{\right\rangle}}$ be such that 1. ${\mathfrak{W}}$ is full and ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ is a $\left(\Theta,{\mathcal{E}}\right)-$structure; 2. for all $x\in X$ $${\mathcal{C}_{cs} \left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})\right)} \subseteq {\mathfrak{M}}_{x} \subseteq {\mathfrak{L}}_{1}({\mathbb {R}}^{+}, {\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x});\nu) \bigcap \bigcap_{\lambda>0} {\mathfrak{L}}_{1}({\mathbb {R}}^{+}, {\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x});\mu_{\lambda});$$ 3. ${\left\langle {\mathfrak{V}},{\mathfrak{V}}({\mathbf{M}}^{\nu}, \Gamma(\rho)),X,Y{\right\rangle}}$ is invariant and ${\left\langle {\mathfrak{V}},{\mathfrak{Z}},{\mathbf{M}}^{\nu}{\right\rangle}}$ is $\nu-$related; 4. $\Gamma(\rho) \bigstar {\mathcal{E}}(\Theta) \subseteq \Gamma(\zeta)$; 5. ${\mathbf{U}}_{\|\cdot\|_{B({\mathfrak{E}}_{x})}} ({\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})) \subseteq {\mathfrak{M}}_{x}$ (respectively ${\mathbf{U}}_{is} ({\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})) \subseteq {\mathfrak{M}}_{x}$), for all $x\in X$; 6. $\exists\,F\in\Gamma(\rho)$ such that $F(x_{\infty})={\mathcal{U}}(x_{\infty})$ and i : ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ has the ${\mathbf{LD}}_{x_{\infty}}(\{F\},{\mathcal{E}})$; ii : $(\forall v\in{\mathcal{E}}) (\exists\,\phi\in\Phi)$ s.t. $\phi_{1}(x_{\infty}) = v(x_{\infty})$ and $(\forall\{z_{n}\}_{n\in{\mathbb{N}}}\subset X \mid \lim_{n\in{\mathbb{N}}}z_{n}=x_{\infty})$ we have that $\{ {\mathcal{U}}(z_{n})(\cdot)\phi_{1}(z_{n}) - F(z_{n})(\cdot)v(z_{n}) \}_{n\in{\mathbb{N}}}$ is a bounded equicontinuous sequence; iii : $X$ is metrizable. Finally assume that $$X\ni x \mapsto -T_{x}\in Cld({\mathfrak{E}}_{x}),$$ satisfies the property of separation of the spectrum, moreover that there exists a closed curve $\Gamma$ of which in Definition \[13361311biss\], for the position $T(x)=-T_{x}$ for all $x\in X$, and such that $$\begin{cases} Re(\Gamma) \subseteq {\mathbb {R}}^{-}, \\ -1 \notin Re(\Gamma). \end{cases}$$ 1. ${\mathcal{P}} \bullet \Gamma_{{\mathcal{E}}(\Theta)}^{x_{\infty}}(\pi) \subseteq \Gamma^{x_{\infty}}(\pi)$; 2. If ${\mathfrak{D}} \doteqdot {\left\langle {\left\langle {\mathfrak{B}},\tau{\right\rangle}},\eta,X,{\mathfrak{L}}{\right\rangle}}$ is full and ${\left\langle {\mathfrak{V}},{\mathfrak{D}},X,\{pt\}{\right\rangle}}$ is an invariant $\left(\Theta,{\mathcal{E}}\right)-$structure such that $ \Pr\left({\mathfrak{E}}_{x}\right) \subset {\mathfrak{B}}_{x}$ for all $x$ then $$\label{19000107} {\mathcal{P}}\in\Gamma^{x_{\infty}}(\eta),$$ and $$\label{12102802bis} \boxed{ \{{\left\langle {\mathcal{T}},x_{\infty},\Phi{\right\rangle}}\} \in \Delta {\left\langle {\mathfrak{V}},{\mathfrak{D}},\Theta,{\mathcal{E}}{\right\rangle}}, }$$ moreover ${\mathcal{P}}$ satisfies . Here ${\left\langle {\mathfrak{V}},{\mathfrak{V}}({\mathbf{M}}^{\nu}, \Gamma(\rho)),X,{\mathbb {R}}^{+}{\right\rangle}}$ is the $\left(\Theta,{\mathcal{E}},\nu\right)-$ structure underlying ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$. While for all $x\in X$ we set $${\mathcal{P}}(x) \doteqdot - \frac{1}{2\pi i} \int_{\Gamma} R(-T_{x};\zeta)\, d\zeta \in B({\mathfrak{E}}_{x}),$$ with $R(-T_{x};\cdot): P(-T_{x}) \ni \zeta \mapsto (-T_{x}-\zeta)^{-1} \in B({\mathfrak{E}}_{x})$ the resolvent map of $-T_{x}$ and $P(-T_{x})$ is its resolvent set, while the integration is with respect to the norm topology on $B({\mathfrak{E}}_{x})$. By the Dupre’ Thm., see for example [@kurtz Cor. $2.10$], and the fact that a metrizable space is completely regular, we deduce by hyp. $(6.iii)$ that ${\mathfrak{V}}$ is full. So by Prop. \[19492307\] and the density assumption follows that $Dom(T_{x_{\infty}})$ is dense in ${\mathfrak{E}}_{x_{\infty}}$. Thus we can use the notations in Def. \[14460503\] and by Theorem \[17301812b\] we have that ${\mathcal{U}}\in \Gamma^{x_{\infty}} (\rho)$, where ${\mathcal{U}}(x)$ is the $C_{0}-$semigroup generated by $T_{x}$, for all $x\in X$. Thus by Proposition \[23332802\] and that $F(x_{\infty})={\mathcal{U}}(x_{\infty})$ (by hyp. $(6)$) we have $$\label{20180207} {\mathcal{U}} \in \Gamma_{\diamond}^{x_{\infty}} (\pi_{{\mathbf{M}}^{\nu}})$$ Moreover ${\mathfrak{W}}$ being full implies ${\mathfrak{M}}_{x}\subset {\mathbf{M}}_{x}^{\nu}$ so for all $x\in X$ $$\label{20273006} {\mathcal{C}_{cs} \left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})\right)} \subset {\mathbf{M}}_{x}^{\nu}.$$ Therefore statement $(1)$ follows by Remark \[12500303\] applied to the $\left(\Theta,{\mathcal{E}},\nu\right)-$ structure underlying ${\left\langle {\mathfrak{V}},{\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$, by hyp. $(6.i)$ and by the fact that $F\in\Gamma(\pi_{{\mathbf{M}}^{\nu}})$ indeed $\Gamma(\rho)= \Gamma(\pi_{{\mathbf{M}}^{\nu}})$ modulo the canonical isomorphism, see Rmk. \[17150312\]. Statement $(2)$ follows by , , and ${\mathcal{U}} = {\mathcal{W}}_{T}$ for the position $T(x)\doteqdot -T_{x}$ for all $x\in X$. By follows that is equivalent to $(\exists\, H\in\Gamma(\eta)) (\forall v\in{\mathcal{E}})$ $$\label{18550107} \lim_{z\to x_{\infty}} \left\|\left({\mathcal{P}}(z)-H(z)\right)v(z)\right\| =0.$$ \[17020107\] Let $E$ be a Banach space and $S\subseteq Bounded(E)$ such that $\bigcup_{B\in S}B$ is total in $E$. Set $ \mu_{f}: {\mathcal{C}_{cs} \left({\mathbb {R}}^{+}\right)} \ni g\mapsto \int_{{\mathbb {R}}^{+}}f(s)g(s)\,ds$ for all $f\in{\mathfrak{L}}_{\infty}({\mathbb {R}}^{+})$. Then $$\label{19470103} {\mathfrak{L}}_{1}({\mathbb {R}}^{+},B(E)) \subseteq \bigcap_{f\in{\mathfrak{L}}_{\infty}({\mathbb {R}}^{+})} {\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathcal{L}}_{S}(E),\mu_{f}).$$ In particular $$\label{19480103} {\mathfrak{L}}_{1}({\mathbb {R}}^{+},B(E)) \subseteq \bigcap_{\lambda>0} {\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathcal{L}}_{S}(E), \eta_{\lambda})) \bigcap {\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathcal{L}}_{S}(E),\nu)$$ By we deduce that ${\mathfrak{L}}_{\infty}({\mathbb {R}}^{+}) \cdot {\mathfrak{L}}_{1}({\mathbb {R}}^{+},B(E)) \subseteq {\mathfrak{L}}_{1}({\mathbb {R}}^{+},B(E))$, hence for all $f\in{\mathcal{L}}_{\infty}({\mathbb {R}}^{+})$. $${\mathfrak{L}}_{1}({\mathbb {R}}^{+},B(E)) \subset {\mathfrak{L}}_{1}({\mathbb {R}}^{+},B(E),\mu_{f}).$$ Moreover by the fact that the norm topology on $B(E)$ is stronger than the lct on ${\mathcal{L}}_{S}(E)$ we have for all $\mu\in Radon({\mathbb {R}}^{+})$ that $${\mathfrak{L}}_{1}({\mathbb {R}}^{+},B(E),\mu) \subseteq {\mathfrak{L}}_{1}({\mathbb {R}}^{+},{\mathcal{L}}_{S}(E),\mu).$$ Hence follows. Finally follows by . Let ${\mathfrak{V}} \doteqdot {\left\langle {\left\langle {\mathfrak{E}},\tau{\right\rangle}},\pi,X,\|\cdot\|{\right\rangle}}$ be a Banach bundle and $S_{x}\subseteq Bounded({\mathfrak{E}}_{x})$ such that $\bigcup_{B_{x}\in S_{x}}B_{x}$ is total in ${\mathfrak{E}}_{x}$ for all $x\in X$. Set $$\label{20160107a} {\mathbf{M}}_{x} \doteqdot {\mathfrak{L}}_{1}({\mathbb {R}}^{+},B({\mathfrak{E}}_{x})) \cap {\mathcal{C}_{} \left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x})\right)}.$$ Thus $$\label{20160107b} {\mathbf{U}}_{\|\cdot\|_{B({\mathfrak{E}}_{x})}} (B({\mathfrak{E}}_{x})) \subset {\mathbf{M}}_{x}.$$ Therefore by in Thm. \[13020103\] we can replace hyp. $(2)$ with , while by we can replace “ ${\mathcal{U}}_{0} \in\prod_{x\in X_{0}} {\mathcal{C}_{} \left({\mathbb {R}}^{+},B_{s}({\mathfrak{E}}_{x})\right)}$ be such that ${\mathcal{U}}_{0}(x)$ is a $(C_{0})-$semigroup of contractions” with “ ${\mathcal{U}}_{0} \in\prod_{x\in X_{0}} {\mathcal{C}_{} \left({\mathbb {R}}^{+},B({\mathfrak{E}}_{x})\right)}$ be such that ${\mathcal{U}}_{0}(x)$ is a $\|\cdot\|-$continuous semigroup of contractions” and delete hyp. $(4)$. Similar replacement can be performed in Thm. \[17301812b\]. Finally notice that in general do not hold if we replace ${\mathbf{U}}_{\|\cdot\|_{B({\mathfrak{E}}_{x})}} (B({\mathfrak{E}}_{x}))$ with ${\mathbf{U}}_{\|\cdot\|_{B({\mathfrak{E}}_{x})}} ({\mathcal{L}}_{S_{x}}({\mathfrak{E}}_{x}))$, so we cannot take the choice if we want to have results about $C_{0}-$semigroups. Constructions ============= Kurtz Bundle Construction {#17572301} --------------------------- In this section we shall construct construct a special bundle ${\mathfrak{E}}$ of Banach space such that for it the Main Theorem \[17301812b\] reduces to the [@kurtz Th. $2.1.$] showing in this way that (a particular case) of the construction of Kurtz falls into the general setting of bundle of $\Omega-$spaces. In this section we shall use the notations of [@kurtz] with the additional specification of denoting with $\|\cdot\|_{n}$ the norm in the Banach space $L_{n}$. Moreover we denote by $X$ the Alexandrov (one-point) compactification of the locally compact space ${\mathbb{N}}$ with the discrete topology. Here we recall some definitions given in [@kurtz]. ${\left\langle L,\|\cdot\|{\right\rangle}}$ is a Banach space and $\{{\left\langle L_{n},\|\cdot\|_{n}{\right\rangle}}\}_{n\in{\mathbb{N}}}$ is a sequence of Banach spaces, moreover $\{P_{n}\in B(L,L_{n})\}_{n\in{\mathbb{N}}}$ is a sequence of bounded linear operators such that $\forall f\in L$ $$\label{10572601} \lim_{n\to\infty} \|P_{n}f\|_{n}=\|f\|.$$ Given an element $f\in L$ and a sequence $\{f_{n}\}_{n\in{\mathbb{N}}}$ such that $f_{n}\in L_{n}$ for all $n\in{\mathbb{N}}$ we set $$\label{11012601} \lim_{n\to\infty}f_{n} {\overset{K}{=}}f\overset{def}{\Leftrightarrow} \lim_{n\to\infty} \|f_{n}-P_{n}f\|_{n}=0.$$ In addition to the requirements of [@kurtz] we assume also that $$\label{11212601} (\forall n\in{\mathbb{N}}) ({\overline}{P_{n}(L)}=L_{n})$$ We shall set here $L_{\infty}\doteqdot L$, $\|\cdot\| \doteqdot \|\cdot\|_{\infty}$, where $\|\cdot\|$ is the norm on $L$. Finally for all $Z$ we recall that $B_{s}(Z)$ is the locally convex space of all linear bounded operators on $Z$ with the strong operator topology. \[unique\] Let $f,g\in L$ and $\{f_{n}\}_{n\in{\mathbb{N}}}$ such that $f_{n}\in L_{n}$ for all $n\in{\mathbb{N}}$. Then $ (\lim_{n\to\infty}f_{n} {\overset{K}{=}}f) \wedge (\lim_{n\to\infty}f_{n} {\overset{K}{=}}g) \Rightarrow f=g $ Let $(\lim_{n\to\infty}f_{n} {\overset{K}{=}}f)$ and $(\lim_{n\to\infty}f_{n} {\overset{K}{=}}g)$ thus $$\lim_{n\in{\mathbb{N}}} \| P_{n}(f-g) \| \leq \lim_{n\in{\mathbb{N}}} \|P_{n}f-f_{n}\| + \lim_{n\in{\mathbb{N}}} \|P_{n}g-f_{n}\| =0,$$ so the statement follows by . \[10452601\] Set $$\begin{cases} {\mathbf{L}}\doteqdot \{ {\left\langle L_{x},\|\cdot\|_{x}{\right\rangle}} \}_{x\in X}, \\ {\mathcal{E}}(L)\doteqdot \{\sigma^{f}\mid f\in L\}, \end{cases}$$ where $\sigma^{f}\in\prod_{x\in X}L_{x}$ such that $\sigma^{f}(n)\doteqdot P_{n}f$ for all $n\in{\mathbb{N}}$ and $\sigma^{f}(\infty)\doteqdot f$. \[11322601\] By the sequence $\{\|P_{n}f\|_{n}\}_{n\in{\mathbb{N}}}$ is bounded for all $f\in L$ so $\sigma^{f}\in\prod_{x\in X}^{b}L_{x}$. Moreover by ${\mathcal{E}}(L)$ satisfies $FM(4)$, finally by the request it satisfies also $FM(3)$. Therefore we can define the *Kurtz bundle* the following bundle $${\mathfrak{V}}({\mathbf{L}},{\mathcal{E}}(L))$$ generated by the couple ${\left\langle {\mathbf{L}},{\mathcal{E}}(L){\right\rangle}}$, see in Def. \[17471910Ba\]. \[16022601\] By Remark \[17150312\] and the compacteness of $X$ by construction, we have that $$\label{16022601b} {\mathcal{E}}(L) \simeq \Gamma(\pi_{{\mathbf{L}}}).$$ Finally by applying the principle of uniform boundedness, [@kato Th. $1.29$, $No 3$, Ch.$3$], we deduce that the sequence $\{\|P_{n}\|_{B(L,L_{n})}\}_{n\in{\mathbb{N}}}$ is bounded. \[18151602\] Fix $ {\mathcal{U}}_{0} \in\prod_{n\in{\mathbb{N}}} {\mathcal{C}_{} \left({\mathbb {R}}^{+},B_{s}(L_{n})\right)} $ such that ${\mathcal{U}}_{0}(x)$ is a $(C_{0})-$semigroup of isometries on $L_{n}$ for all $n\in{\mathbb{N}}$. Denote by $T_{n}$ the infinitesimal generator of the semigroup ${\mathcal{U}}_{0}(n)$ for any $n\in{\mathbb{N}}$. Let us take the positions , where ${\left\langle {\left\langle {\mathfrak{E}}({\mathbf{E}}^{\oplus}),\tau({\mathbf{E}}^{\oplus},{\mathcal{E}}^{\oplus}){\right\rangle}},\pi_{{\mathbf{E}}^{\oplus}},X,{\mathfrak{n}}^{\oplus}{\right\rangle}}$ is the bundle direct sum of the family $\{{\mathfrak{V}}({\mathbf{L}},{\mathcal{E}}(L)), {\mathfrak{V}}({\mathbf{L}},{\mathcal{E}}(L))\}$. In addition we maintain the Notations \[15411512b\] where ${\mathfrak{V}}$ has to be considered the Kurtz bundle and $x_{\infty}\doteqdot\infty$, thus $ {\mathcal{T}} \in \prod_{x\in X} Graph(L_{x} \times L_{x}) $ so that $ {\mathcal{T}}{\upharpoonright}X-\{\infty\} \doteqdot {\mathcal{T}}_{0} $ and $${\mathcal{T}}(\infty) \doteqdot \{\phi(\infty)\mid\phi\in\Phi\},$$ and $ D(T_{\infty}) \doteqdot \Pr_{1}^{\infty}({\mathcal{T}}(\infty)) = \{\phi_{1}(\infty) \mid\phi\in\Phi\}. $ Finally ${\mathcal{S}} \doteqdot \{S_{x}\}_{x\in X}$ where $(\forall B\in\Theta)(\forall x\in X)$ $$\label{11232712bis} \begin{cases} {\mathbf{D}}(B,{\mathcal{E}}) \doteqdot {\mathcal{E}} \cap \left(\prod_{x\in X}B_{x}\right) \\ {\mathcal{B}}_{B}^{x} \doteqdot \{v(x)\mid v\in{\mathbf{D}}(B,{\mathcal{E}})\} \} \\ S_{x} \doteqdot \{{\mathcal{B}}_{B}^{x}\mid B\in\Theta\}. \end{cases}$$ \[11422601\] Let ${\overline}{f}\in\prod_{x\in X}L_{x}$ Thus $$\lim_{n\to\infty}{\overline}{f}(n){\overset{K}{=}}{\overline}{f}(\infty) \Leftrightarrow {\overline}{f}\in\Gamma^{\infty}(\pi_{{\mathbf{L}}}).$$ By and implication $(3)\Rightarrow(1)$ of Corollary \[28111707\] we have that $\lim_{n\to\infty}{\overline}{f}(n){\overset{K}{=}}{\overline}{f}(\infty)$ implies that $${\overline}{f} \text{ is continuous at $\infty$,}$$ indeed $\sigma^{{\overline}{f}(\infty)}\in \Gamma(\pi_{{\mathbf{L}}})$ modulo isomorphism. By the upper semicontinuity of $\|\cdot\|:{\mathfrak{E}}\to{\mathbb {R}}^{+}$, due to the construction of the bundle ${\mathfrak{V}}({\mathbf{L}},{\mathcal{E}}(L))$ and to [@kurtz $1.6.(ii)$], and by the fact that the composition of any u.s.c. map with any continuous one at a point is an u.s.c. map in the same point, we deduce that $\|\cdot\|\circ{\overline}{f}$ is u.s.c. at $\infty$. Thus $\sup_{x\in X}\|{\overline}{f}(x)\|_{x}<\infty$, indeed we applied to the u.s.c. map $\|\cdot\|\circ{\overline}{f}$ the fact that $X$ is compact (so quasi compact), $-\|\cdot\|\circ{\overline}{f}$ is l.s.c, the [@BourGT Th. $3$, $\S 6.2.$, Ch. $4$] and [@BourGT form.$(2)$, $\S 5.4.$, Ch. $4$]. Therefore $${\overline}{f}\in\prod_{x\in X}^{b}L_{x}.$$ Then ${\overline}{f}\in\Gamma^{\infty}(\pi_{{\mathbf{L}}})$. The remaining implication follows by Corollary \[28111707\] and by the fact that ${\mathfrak{V}}({\mathbf{L}},{\mathcal{E}}(L))$ is full, $X$ being locally compact and by the fact that any Banach bundle over a locally compact space is full, see [@fell Appendix C]. \[15512601\] We have $$\Gamma^{\infty}(\pi_{\mathbf{L}^{\oplus}}) = \left\{ \sigma_{1}\oplus\sigma_{2} \mid \sigma_{i}\in\prod_{x\in X} L_{x}, \lim_{n\to\infty}\sigma_{i}(n) {\overset{K}{=}}\sigma(\infty), i=1,2 \right\}.$$ Here, we used the Convention \[16392601\] and set $(\sigma_{1}\oplus\sigma_{2})(x) \doteqdot \sigma_{1}(x)\oplus\sigma_{2}(x)$. By Convention \[16392601\] and Corollary \[17571212\] $\sigma_{1}\oplus\sigma_{2}$ is continuous at $\infty$ if and only if $\sigma_{i}$ is continuous at $\infty$ for all $i=1,2$. Thus the statement by Proposition \[11422601\]. \[18542601\] Let $ {\mathcal{U}}_{0} \in\prod_{n\in{\mathbb{N}}} {\mathcal{C}_{} \left({\mathbb {R}}^{+},B_{s}(L_{n})\right)} $ be such that ${\mathcal{U}}_{0}(x)$ is a $(C_{0})-$semigroup of contractions on $L_{n}$ for all $n\in{\mathbb{N}}$. Moreover let us denote by $T_{n}$ the infinitesimal generator of the semigroup ${\mathcal{U}}_{0}(n)$ for any $n\in{\mathbb{N}}$. Thus with the positions where ${\mathfrak{V}}$ is the Kurtz bundle we have $$\label{18452801} \begin{cases} \Phi = \left\{ \sigma_{1}\oplus\sigma_{2} \mid (\forall i\in\{1,2\}) (\sigma_{i}\in\prod_{x\in X} L_{x})(1-2) \right\} \\ (1) \lim_{n\to\infty}\sigma_{i}(n) {\overset{K}{=}}\sigma_{i}(\infty) \\ (2) (\forall n\in{\mathbb{N}}) (\sigma_{1}(n),\sigma_{2}(n)) \in Graph(T_{n}), \end{cases}$$ and $$\label{17472801} \begin{cases} {\mathcal{E}} = \left\{ \sigma^{\sigma_{1}(\infty)} \mid \sigma_{1}\in\prod_{x\in X} L_{x} (1-2-3) \right\} \\ (1) \lim_{n\to\infty}\sigma_{1}(n) {\overset{K}{=}}\sigma_{1}(\infty) \\ (2) (\forall n\in{\mathbb{N}}) (\sigma_{1}(n) \in Dom(T_{n})) \\ (3) (\exists\,f\in L_{\infty}) (\lim_{n\to\infty}T_{n} \sigma_{1}(n) {\overset{K}{=}}f). \end{cases}$$ Moreover $\exists\,!\,f$ satisfying $(3)$ in and $(\forall\sigma_{1}\in{\mathcal{E}}) ((\sigma_{1},\sigma_{2})\in\Phi)$, where $\sigma_{2}\in\prod_{x\in X}L_{x}$ such that $(\forall n\in{\mathbb{N}}) (\sigma_{2}(n)\doteqdot T_{n}\sigma_{1}(n))$ and $\sigma_{2}(\infty) \doteqdot f$. The first sentence follows by Proposition \[15512601\], while the second comes by the first one and Lemma \[unique\]. We assume $\exists\, \{I_{n}\in B(L_{n},L)\}_{n\in{\mathbb{N}}}$ such that $$\label{15052601} \begin{cases} \sup_{n\in{\mathbb{N}}}\|I_{n}\|_{B(L_{n},L)} <\infty, \\ (\forall f\in L) (\forall n\in{\mathbb{N}}) (I_{n}\circ P_{n}=Id). \end{cases}$$ Moreover we assume that $$\label{11442701} \varlimsup_{n\to\infty} \|P_{n}\|\leq 1.$$ In addition we assume that $ (\forall g\in L) (\exists\, \sigma_{1}\in\prod_{x\in X} L_{x}) $ such that $$\label{17202701} \begin{cases} (1) \lim_{n\to\infty} \sigma_{1}(n) {\overset{K}{=}}\sigma_{1}(\infty) \\ (2) (\forall n\in{\mathbb{N}}) (\sigma_{1}(n) \in Dom(T_{n})) \\ (3) (\exists\,f\in L_{\infty}) (\lim_{n\to\infty}T_{n} \sigma_{1}(n) {\overset{K}{=}}f) \\ (4) g=\sigma_{1}(\infty). \end{cases}$$ Set $$\label{16582801} {\mathfrak{U}} \doteqdot \left\{ F\in {\mathcal{C}_{} \left({\mathbb {R}}^{+},B_{s}(L)\right)} \mid (\forall s\in{\mathbb {R}}^{+}) (\forall v\in L) (\|F(s)v\|=\|v\|) \right\}.$$ In the following definition we shall give the data for constructing a bundle ${\mathfrak{W}}$ such that ${\left\langle {\mathfrak{V}}({\mathbf{L}},{\mathcal{E}}(L)), {\mathfrak{W}},X,{\mathbb {R}}^{+}{\right\rangle}}$ would be a $\left(\Theta,{\mathcal{E}}\right)-$structure. \[13352701\] Set $P_{\infty}\doteqdot I_{\infty} \doteqdot Id:L\to L$, moreover $\forall U\in{\mathfrak{U}}$ set $F_{U}\in\prod_{x\in X} {\mathcal{C}_{c} \left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}(L_{x})\right)}$ such that $\forall x\in X$ $$\begin{cases} F_{U}(x) \doteqdot P_{x}\circ U(\cdot) \circ I_{x}, \\ P_{x}\circ U(\cdot) \circ I_{x}: {\mathbb {R}}^{+}\ni s \mapsto P_{x}\circ U(s) \circ I_{x} \in B(L_{x}). \end{cases}$$ Now we can define $\forall x\in X$ $${\mathbf{M}}_{x} \doteqdot \textrm{span} \left \{ F_{U}(x) \mid U\in{\mathfrak{U}} \right \}.$$ ${\mathbf{M}}_{x}$ has to be considered as Hlcs with the topology induced by that on ${\mathcal{C}_{c} \left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}(L_{x})\right)}$. [^24] Moreover set $${\mathcal{M}} \doteqdot \textrm{span} \left \{ F_{U} \mid U\in{\mathfrak{U}} \right \}.$$ \[14282701\] ${\mathbf{M}}_{x}$ as Hlcs is well-defined for any $x\in X$, moreover ${\mathcal{M}}\subset \prod_{x\in X}^{b} {\mathbf{M}}_{x}$ and ${\mathbf{M}}_{x}= \{F(x)\mid F\in{\mathcal{M}}\}$. Finally ${\mathcal{M}}$ satisfies $FM(3)-FM(4)$ with respect to ${\mathbf{M}}$. By Remark \[21500412b\] we have that ${\mathcal{C}_{c} \left({\mathbb {R}}^{+},B_{s}(L_{x})\right)} \subset {\mathcal{C}_{c} \left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}(L_{x})\right)}$ hence for the first sentence of the statement it is sufficient to show that $ P_{x} \circ U(\cdot) \circ I_{x} \in {\mathcal{C}_{c} \left({\mathbb {R}}^{+},B_{s}(L_{x})\right)}$ for any $U\in{\mathfrak{U}}$. For $x=\infty$ is trivial so let $n\in{\mathbb{N}}$ and $f_{n}\in L_{n}$ thus for all $s\in{\mathbb {R}}^{+}$ and all net $\{s_{\alpha}\}_{\alpha\in D}$ in ${\mathbb {R}}^{+}$ converging at $s$ we have $$\lim_{\alpha\in D} \| P_{n} \circ U(s_{\alpha}) \circ I_{n} (f_{n}) - P_{n} \circ U(s) \circ I_{n} (f_{n}) \|_{n} = \lim_{\alpha\in D} \| P_{n} (U(s_{\alpha})-U(s)) I_{n}f_{n} \|_{n} =0,$$ where we used the fact that $U$ is strongly continuous and $P_{n}$ is norm continuous by construction. Thus the first sentence of the statement follows. Let $v\in{\mathcal{E}}$ and $U\in{\mathfrak{U}}$ thus $\forall K\in Compact({\mathbb {R}}^{+})$ $$\begin{aligned} {1} \sup_{n\in{\mathbb{N}}} \sup_{s\in K} \| P_{n} U(s) I_{n} v(n) \|_{n} &\leq M \sup_{n\in{\mathbb{N}}} \sup_{s\in K} \| U(s) I_{n} v(n) \|_{\infty} \\ &= M \sup_{n\in{\mathbb{N}}} \| I_{n} v(n) \|_{\infty} \\ &\leq M \sup_{n\in{\mathbb{N}}} \| I_{n} \| \sup_{n\in{\mathbb{N}}} \| v(n) \|_{\infty}<\infty.\end{aligned}$$ Here $M\doteqdot\sup_{n\in{\mathbb{N}}}\|P_{n}\|$, in the second one the hypothesis that $U(s)$ is an isometry for all $s\in{\mathbb {R}}^{+}$, in the final inequality we considered , ${\mathcal{E}}\subset\prod_{x\in X}^{b}L_{x}$ and that $M<\infty$ by Remark \[16022601\]. Therefore by Remark \[21500412b\] ${\mathcal{M}}\subset \prod_{x\in X}^{b} {\mathbf{M}}_{x}$. The equality ${\mathbf{M}}_{x}= \{F(x)\mid F\in{\mathcal{M}}\}$ comes by construction, in particular ${\mathcal{M}}$ satisfies the $FM(3)$ with respect to the ${\mathbf{M}}$. $\forall v\in{\mathcal{E}}$ $$\begin{aligned} {2} \varlimsup_{n\to\infty} \sup_{s\in K} \| P_{n}U(s)I_{n} v(n) \|_{n} &\leq \varlimsup_{n\to\infty} \left(\| P_{n} \| \sup_{s\in K} \| U(s)I_{n} v(n) \|_{n} \right), & \text{\cite[Prop. $11$, $\S 5.6.$ Ch. $4$]{BourGT}} \\ &\leq \varlimsup_{n\to\infty} \|P_{n}\| \varlimsup_{n\to\infty} \sup_{s\in K} \|U(s)I_{n}v(n)\|_{n}, & \text{\cite[Prop. $13$, $\S 5.7.$ Ch. $4$]{BourGT}} \\ &\leq \varlimsup_{n\to\infty} \|I_{n}v(n)\|_{\infty}, & \text{ \eqref{11442701}, \eqref{16582801}} \\ &= \varlimsup_{n\to\infty} \| I_{n}P_{n}f \|_{\infty}, & \text{ $v\in{\mathcal{E}}\subset\Gamma(\pi) \simeq{\mathcal{E}}(L)$ } \\ &= \|f\|_{\infty}, & \text{ \eqref{15052601} } \\ &= \|v(\infty)\|_{\infty}.\end{aligned}$$ Thus by considering that $U$ is a map of isometries we have $$\varlimsup_{n\to\infty} \sup_{s\in K} \| P_{n}U(s)I_{n} v(n) \|_{n} \leq \sup_{s\in K} \| P_{\infty} U(s)I_{\infty} v(\infty) \|_{\infty}.$$ Hence by [@BourGT Prop. $3$, $\S 7.1.$ Ch. $4$] and [@BourGT $(13)$, $\S 5.6.$ Ch. $4$] we deduce that $$X\ni x\mapsto \sup_{s\in K} \| P_{x}U(s)I_{x} v(x) \|_{x} \text{ is $u.s.c.$ at $\infty$},$$ therefore it is $u.s.c.$ on $X$ because of it is continuous in each point in ${\mathbb{N}}$ due to the fact that the topology induced on ${\mathbb{N}}$ by that on $X$ is the discrete topology. So ${\mathcal{M}}$ satisfies the $FM(4)$ with respect to the ${\mathbf{M}}$. \[16232701\] Theorem \[14282701\] allows us to construct a bundle of $\Omega-$space namely the bundle ${\mathfrak{V}}({\mathbf{M}},{\mathcal{M}})$ generated by the couple ${\left\langle {\mathbf{M}},{\mathcal{M}}{\right\rangle}}$, see Def \[17471910Ba\]. \[16282701\] By Remark \[17150312\] and the compactness of $X$ we have $$\label{14222801} {\mathcal{M}} \simeq \Gamma(\pi_{{\mathbf{M}}}).$$ Hence by ${\mathbf{M}}_{x}= \{F(x)\mid F\in{\mathcal{M}}\}$ we have that ${\mathfrak{V}}({\mathbf{M}},{\mathcal{M}})$ is full. \[16332701\] We have that ${\overline}{\bigcup_{B\in\Theta}{\mathcal{B}}_{B}^{x}} =L_{x}$ for all $x\in X$ moreover ${\left\langle {\mathfrak{V}}({\mathbf{L}},{\mathcal{E}}(L)), {\mathfrak{V}}({\mathbf{M}},{\mathcal{M}}),X,{\mathbb {R}}^{+}{\right\rangle}}$ is a $\left(\Theta,{\mathcal{E}}\right)-$structure. By assumptions , , Proposition \[18542601\] and Remark \[21500412b\] we obtain that ${\overline}{\bigcup_{B\in\Theta}{\mathcal{B}}_{B}^{x}} = L_{x}$ for all $x\in X$. The remaining requests for the second sentence of the statement come by the construction of ${\mathcal{M}}$ and ${\mathbf{M}}$. \[13560202\] If $D(T_{x_{\infty}})$ is dense in ${\mathfrak{E}}_{x_{\infty}}$, and $\exists\lambda_{0}>0, \lambda_{1}<0$ such that the ranges ${\mathcal{R}}(\lambda_{0}-T_{x_{\infty}})$ and ${\mathcal{R}}(\lambda_{1}-T_{x_{\infty}})$ are dense in ${\mathfrak{E}}_{x_{\infty}}$), then ${\left\langle {\mathcal{T}},\infty,\Phi{\right\rangle}} \in Graph\left( {\mathfrak{V}}({\mathbf{L}},{\mathcal{E}}(L)), {\mathfrak{V}}({\mathbf{L}},{\mathcal{E}}(L)) \right)$ and the following $$T_{\infty}:D(T_{\infty}) \ni\phi_{1}(\infty) \mapsto \phi_{2}(\infty)$$ is a well-defined operator which is the generator of a $C_{0}-$semigroup of isometries on ${\mathfrak{E}}_{\infty}$. By Propositions \[16332701\] and \[19272701\] we have that the first part of hypotheses of Theorem \[17301812b\] is satisfied so the statement by the first sentence of the statement of Theorem \[17301812b\]. \[16002801\] Let us denote by ${\mathcal{U}}_{\infty}$ the $C_{0}-$semigroup of isometries on $L_{\infty}$. Moreover set ${\mathcal{U}}\in \prod_{x\in X}{\mathbf{U}}_{is}(B_{s}(L_{x}))$ such that ${\mathcal{U}}{\upharpoonright}{\mathbb{N}}={\mathcal{U}}_{0}$ and ${\mathcal{U}}(\infty) = {\mathcal{U}}_{\infty}$. \[17332801\] $(\exists\,F\in\Gamma(\pi_{{\mathbf{M}}})) (F(\infty)={\mathcal{U}}(\infty))$ such that $(\forall v\in{\mathcal{E}}) (\exists\,\phi\in\Phi)$ s.t. $\phi_{1}(x_{\infty}) = v(x_{\infty})$ and $(\forall\{z_{n}\}_{n\in{\mathbb{N}}}\subset X \mid \lim_{n\in{\mathbb{N}}}z_{n}=x_{\infty})$ we have that $\{ {\mathcal{U}}(z_{n})(\cdot)\phi_{1}(z_{n}) - F(z_{n})(\cdot)v(z_{n}) \}_{n\in{\mathbb{N}}}$ is a bounded equicontinuous sequence. Moreover we can choose $F$ such that $F= F_{{\mathcal{U}}_{\infty}}$. By Prop. \[18542601\] and the statement is equivalent to show that $ \forall \sigma_{1}\in\prod_{x\in X} L_{x}$ satisfying $(1-2-3)$ of and $(\forall\{z_{n}\}_{n\in{\mathbb{N}}}\subset X \mid \lim_{n\in{\mathbb{N}}}z_{n}=\infty)$ we have that $$\label{17592801} \{ {\mathcal{U}}(z_{n})(\cdot)\sigma_{1}(z_{n}) - F_{{\mathcal{U}}_{\infty}} (z_{n})(\cdot) \sigma^{\sigma_{1}(\infty)}(z_{n}) \}_{n\in{\mathbb{N}}}$$ is a bounded equicontinuous sequence. Moreover by the second assumption and $$\label{18022801} \{ {\mathcal{U}}(z_{n})(\cdot) \sigma_{1}(z_{n}) - P_{z_{n}} {\mathcal{U}}_{\infty} (z_{n})(\cdot) \sigma_{1}(\infty) \}_{n\in{\mathbb{N}}}$$ is a bounded equicontinuous sequence. Set $\sigma_{2}\in\prod_{x\in X}L_{x}$ such that $\sigma_{2}(x)\doteqdot T_{x}\sigma_{1}(x)$, for all $x\in X$, thus $$\sigma_{i}\in\Gamma^{\infty}(\pi_{{\mathbf{L}}}),$$ for all $i=1,2$, indeed for $i=1$ follows by $(1)$ of and Prop. \[11422601\], while for $i=2$ follows by construction of of $T_{\infty}$, the second sentence of Prop. \[18542601\], the fact that by construction $\Phi\subseteq\Gamma(\pi_{{\mathbf{E}}^{\oplus}})$, see , and finally by Corollary \[17571212\]. Therefore in particular $\sigma_{i}$ is continuous at $\infty$. Thus by considering that $\sigma^{\sigma_{i}(\infty)} \in\Gamma(\pi_{{\mathbf{L}}})$ modulo isomorphism by , that ${\mathfrak{V}}({\mathbf{L}},{\mathcal{E}}(L))$ is full being a Banach bundle over a locally compact space, we deduce by Prop. \[28111555\] $$\lim_{n\in{\mathbb{N}}} \| \sigma_{i}(z_{n}) - \sigma^{\sigma_{i}(\infty)} (\pi\circ\sigma_{i}(z_{n})) \|_{\pi\circ\sigma_{i}(z_{n})} =0.$$ Then by considering that $\pi\circ\sigma_{i}=Id$ because of $\sigma_{i}$ is a selection, we have $$\label{19172801} \lim_{n\in{\mathbb{N}}} \| \sigma_{i}(z_{n}) - P_{z_{n}}\sigma_{i}(\infty) \|_{z_{n}} =0.$$ The statement now follows by , and by using the same argumentation used in proof of [@kurtz Th. $1.2$] for proving a similar result. \[19272701\] With the notations of Def. \[15062301\] we have that $${\mathbf{M}}_{x} \subset \bigcap_{\lambda>0} {\mathfrak{L}}_{1} ({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}(L_{x}); \mu_{\lambda}),$$ and holds. By Proposition \[18390901\]. \[13512801\] $ {\left\langle {\mathfrak{V}}({\mathbf{L}},{\mathcal{E}}(L)), {\mathfrak{V}}({\mathbf{M}},{\mathcal{M}}),X,{\mathbb {R}}^{+}{\right\rangle}}$ has the full Laplace duality property, moreover $\forall U\in{\mathbf{U}}_{1,\|\cdot\|}(B_{s}(L))$, $\forall\lambda>0$ and $\forall f\in L$ we have that $${\mathfrak{L}}(F_{U})(\cdot)(\lambda) \sigma^{f}(\cdot) = \sigma^{(\lambda-T_{U})^{-1}f}.$$ Here $T_{U}$ is the infinitesimal generator of the semigroup $U$. Let $f\in L$ and $U\in{\mathfrak{U}}$ thus for all $x\in X$ and $\lambda>0$ we have $$\begin{aligned} {2} \label{14242801} \int_{0}^{\infty} e^{-\lambda s} P_{x}U(s)I_{x}\sigma^{f}(x)\, ds &= \int_{0}^{\infty} e^{-\lambda s} P_{x}U(s)f\,ds \notag \\ &= P_{x} \int_{0}^{\infty} e^{-\lambda s} U(s)f\,ds,\end{aligned}$$ where the first equality follows by the second assumption \[15052601\], while the second one by the linearity and continuity of $P_{x}$ and by . Thus the first sentence of the statement by and . The second sentence of the statement folllows by the and by Hille-Yosida Theorem, see [@kurtz Th. $1.2.$]. \[12560202\] Let us assume the hypotheses of Corollary \[13560202\]. Then $(\forall g\in L) (\forall K\in Compact({\mathbb {R}}^{+}))$ $$\label{16180202} \lim_{z\to\infty} \sup_{s\in K} \left\| \left({\mathcal{U}}(z)(s)\circ P_{z} - P_{z}\circ{\mathcal{U}}_{\infty}(s)\right) g \right\|=0.$$ Moreover $$\label{02512912pbis} {\mathcal{U}} \in\Gamma^{\infty}(\rho).$$ In particular $$\label{02512912bis} \{{\left\langle {\mathcal{T}},\infty,\Phi{\right\rangle}}\} \in \Delta_{\Theta}{\left\langle {\mathfrak{V}}({\mathbf{L}},{\mathcal{E}}), {\mathfrak{V}}({\mathbf{M}},{\mathcal{M}}),{\mathcal{E}},X,{\mathbb {R}}^{+}{\right\rangle}}.$$ By Proposition \[19272701\] follows , hypothesis $(i)$ of Theorem \[17301812b\] follows by Theorem \[13512801\], $(ii)$ by Theorem , finally $(iii)$ follows by and by the fact that $\{\{n\}\mid n\in{\mathbb{N}}\}$ is a base for the topology on ${\mathbb{N}}$. Thus by Theorem \[17301812b\] we obtain , and $(\forall v\in{\mathcal{E}}) (\forall K\in Compact({\mathbb {R}}^{+}))$ $$\label{16170202} \lim_{z\to\infty} \sup_{s\in K} \left\| {\mathcal{U}}(z)(s)v(z) - F(z)(s)v(z) \right\|=0,$$ where $F$ is any map of which in Theorem \[17332801\]. Now by Theorem \[17332801\] we can take in the previous equation $F=F_{{\mathcal{U}}_{\infty}}$, moreover by and assumption we have $${\mathcal{E}}=\{\sigma^{g}\mid g\in L\},$$ therefore by $\forall s\in{\mathbb {R}}^{+}$, $\forall z\in X$ and $\forall g\in L$ $$F_{{\mathcal{U}}_{\infty}}(z) \sigma^{g}(z) = (P_{z}\circ{\mathcal{U}}_{\infty}(s)) g.$$ Hence by follows . [99.]{} Bourbaki,N. *Algebre 1*, Diffusion C.C.L.S, 1970. Bourbaki,N. *General topology Part 1, Part 2.* Springer-Verlag, 1989. Bourbaki,N. *Topological Vector Spaces.* Springer-Verlag, 1989. 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Chojnacki, Wojciech *Multiplier algebras, Banach bundles, and one-parameter semigroups.* Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), no. 2, 287–322. Dunford,N; Schwartz, J.T. *Linear operators Vol 1,2,3.* Wiley Interscience, 1988. Fell, J. M.; Doran, R. S. *Representations of $\sp *$-algebras, locally compact groups, and Banach $\sp *$-algebraic bundles. Vol. 1-2.* Pure and Applied Mathematics, 126. Academic Press, Inc., Boston, MA, 1988. Gierz, G. *Bundles of topological vector spaces and their duality.* Lecture Notes in Mathematics, 955. Springer-Verlag, 1982. Kato, T. *Perturbation theory for linear operators.* Springer-Verlag, 1980. Kurtz, Thomas G. *Extensions of Trotter’s operator semigroup approximation theorems.* J. Functional Analysis 3 1969 354–375. Lanza de Cristoforis, Massimo *Singular perturbation problems in potential theory and applications.* Complex analysis and potential theory, 131–139, World Sci. Publ., Hackensack, NJ, 2007. Lanza de Cristoforis, Massimo *Asymptotic behavior of the solutions of the Dirichlet problem for the Laplace operator in a domain with a small hole.* A functional analytic approach. Analysis (Munich) 28 (2008), no. 1, 63–93. Lamberti, Pier Domenico; Lanza de Cristoforis, Massimo *A real analyticity result for symmetric functions of the eigenvalues of a domain-dependent Neumann problem for the Laplace operator.* Mediterr. J. Math. 4 (2007), no. 4, 435–449. Li, Hanfeng *Order-unit quantum Gromov-Hausdorff distance.* J. Funct. Anal. 231 (2006), no. 2, 312–360. Jarchow, H. *Locally Convex Spaces.* B.G. Teubner, 1981. Panchapagesan, T. V. *The Bartle-Dunford-Schwartz integral. Integration with respect to a sigma-additive vector measure.* Birkhauser Verlag, 2008 Schwartz, L. *Analyse, I-IV.* Hermann, 1993. Silvestri,B. *Integral equalities for functions of unbounded spectral operators in Banach spaces*. [ http://arxiv.org/abs/0804.3069v2 ]{} Takesaki,M. *Theory of Operator Algebras I.* Springer-Verlag, 2001. Takesaki,M. *Theory of Operator Algebras II.* Springer-Verlag, 2003. Yosida, K. *Functional Analysis.* Springer-Verlag, 1980. [^1]: Later we shall see that the topology on the bundle space of ${\mathfrak{V}}\oplus{\mathfrak{V}}$ will be constructed in order to ensure that the limit in is equivalent to say that $v= \lim_{x\to x_{\infty}} \phi_{1}(x)$ and $T_{x_{\infty}}v= \lim_{x\to x_{\infty}} \phi_{2}(x)$, both limits with respect to the topology on the bundle space ${\mathfrak{E}}$. [^2]: Indeed it is sufficient to take $Y=\{pt\}$ i.e. one point. [^3]: Notice the strong similarity of with . [^4]: Indeed if we set assume that there exists for every $n\in{\mathbb{N}}$ $S_{n}\in B(L_{n},L)$ such that $S_{n} Q_{n}= Id$ then would become $$\label{17520603} (\forall t\in{\mathbb {R}}^{+}) (\forall f\in L) (\lim_{n\to\infty}\sup_{0\leq s\leq t} \| (U_{n}(s) - Q_{n}U(s)S_{n}) Q_{n}f \|_{n}=0).$$ Moreover let ${\left\langle {\mathfrak{M}},\rho,X{\right\rangle}}$ and ${\left\langle {\mathfrak{E}},\pi,X{\right\rangle}}$ be set as in the beginning and assume that $\{\nu_{(K,v)}^{z} \mid(K,v)\in Comp(Y),v\in{\mathcal{E}} \}$ is a fundamental set of seminorms on ${\mathfrak{M}}_{z}$ for every $z\in X$, where ${\mathcal{E}}\subseteq\Gamma(\pi)$. Finally assume that for all $K\in Comp(Y)$, $v\in{\mathcal{E}}$ and for all $z\in X$ and $f^{z}\in{\mathfrak{M}}_{z}$ $$\label{21310603} \nu_{(K,v)}^{z}(f^{z}) \doteqdot \sup_{s\in K} \| f^{z}(s) v(z) \|_{z}.$$ Thus would read: if there exists $\sigma\in\Gamma(\rho)$ such that $\sigma(x_{\infty})=F(x_{\infty})$ then $$\label{17510603} F\in\Gamma^{x_{\infty}}(\rho) \Leftrightarrow (\forall K\in Comp(Y)) (\forall v\in{\mathcal{E}}) (\lim_{z\to x_{\infty}} \sup_{s\in K} \|(F(z)-\sigma(z))v(z)\|_{z} =0).$$ Therefore by setting $X$ the Alexandroff compactification of ${\mathbb{N}}$, $x_{\infty}=\infty$ and for all $n\in{\mathbb{N}}$ $$\label{22270603} \begin{cases} {\mathfrak{E}}_{n} \doteqdot L_{n},\, {\mathfrak{E}}_{\infty}\doteqdot L \\ {\mathfrak{M}}_{n}\doteqdot {\mathcal{C}_{c} \left({\mathbb {R}}^{+},B_{s}(L_{n})\right)} \\ {\mathfrak{M}}_{\infty}\doteqdot {\mathcal{C}_{c} \left({\mathbb {R}}^{+},B_{s}(L)\right)} \\ {\mathcal{E}} \doteqdot \left\{ Qf \mid f\in L \right\}, \end{cases}$$ if there exist conditions under which we can obtain that $$\label{21420603} \begin{cases} \left\{ Qf \mid f\in L \right\} \subseteq \Gamma(\pi) \\ \left\{ QVS \mid V\in{\mathbf{U}}(L) \right\} \subseteq \Gamma(\rho), \end{cases}$$ where $ (Qf)(n) \doteqdot Q_{n}f $, $(Qf)(\infty) \doteqdot f$, while $(QVS)(n) \doteqdot Q_{n}V S_{n}$, $(QVS)(\infty) \doteqdot V$, for all $n\in{\mathbb{N}}$ and ${\mathbf{U}}(L)$, is the class of all $C_{0}-$semigroup on $L$, then by and follows that $${\mathcal{U}} \in \Gamma^{\infty}(\rho),$$ where ${\mathcal{U}}(n)\doteqdot U_{n}$ and ${\mathcal{U}}(\infty)\doteqdot U$. [^5]: Standard in the following sense $ (B,v) \mapsto B(v)$. [^6]: Notice the similarity of notation with Def. \[15012602\]. In any case it will be always clear which definition has to be considered. [^7]: I.e. $ (\forall j_{1},j_{2}\in J) (\exists\,j\in J) (\mu_{j_{1}}^{x},\mu_{j_{1}}^{x} \leq \mu_{j}^{x}) $ with the standard order relation of pointwise order on $ {\mathbb {R}}^{V_{x}} $. [^8]: By applying $5.3.$ of [@gie] and Ch.1. of [@BourGT] we know that such as topology there exists. [^9]: I.e. $\sigma\leftrightarrow f$ iff $ \sigma(x) =(x, f(x)) $ [^10]: An example is when ${\mathfrak{V}}$ is the trivial bundle. [^11]: See Remark \[22550110\]. [^12]: which ensures that the locally convex topology on $ \left({\mathfrak{E}}_{i}\right)_{x} $ generated by the set of seminorms ${\mathfrak{N}}_{i}^{x}$ is exactly the topology induced on it by the topology $\tau_{i}$ on ${\mathfrak{E}}_{i}$, for all $i$ and $x\in X$. [^13]: See [@BourGT Def $1$, $\S 2.1$, Ch. $10$] for the definition of equicontinuous sets. [^14]: Well set indeed by Prop. \[19492307\], the density assumptions and Rem. \[21500412b\] we have that $S_{x}$ is dense in ${\mathfrak{E}}_{x}$ for all $x\in X$. [^15]: See Proposition \[18390901\] for models of ${\mathfrak{M}}$ satisfying and ${\mathbf{U}}_{\|\cdot\|_{B({\mathfrak{E}}_{z})}} ({\mathcal{L}}_{S_{z}}({\mathfrak{E}}_{z})) \subseteq {\mathfrak{M}}_{z}$. [^16]: In others words ${\left\langle {\mathcal{L}}({\mathfrak{E}}_{x}),\tau_{x}{\right\rangle}} = {\mathcal{L}}_{{\mathcal{S}}_{x}}({\mathfrak{E}}_{x})$, see Notations \[notat\] and Def. \[17471910A\]. [^17]: Equivalently $-]\beta,\infty[ \subseteq P(T(x))$, where $P(T(x))$ is the resolvent set of any closed operator $T_{x}$. [^18]: By identifying ${\mathbb {C}}$ with ${\mathbb {R}}^{2}$, so $\rho$ is derivable with contiuous derivative [^19]: See Def. \[15531102\] for $(\bullet)$. [^20]: which is again an algebra by Lemma \[16141302\] [^21]: In case ${\mathfrak{E}}_{x}$ is a Banach space $Meas(Y,{\mathfrak{E}}_{x},\mu) \bigcap {\mathfrak{L}}_{1}(Y,{\mathfrak{E}}_{x},\mu)_{s} = {\mathfrak{L}}_{1}(Y,{\mathfrak{E}}_{x},\mu)_{s}$. [^22]: See Prop. \[18390901\] or more in general [@SilInt Corollary $2.6.$]. [^23]: Indeed for all $E$ normed space the topology on $B(E)$ is stronger of that on ${\mathcal{L}}_{S}(E)$. For that it is sufficient to show $B(E)={\mathcal{L}}_{Bounded(E)}(E)$. To this end note that $\sup_{v\in E_{r}} \|Av\| \leq r\sup_{v\in E_{1}} \|Av\|$ for all $r>0$ and $A\in{\mathcal{L}}(E)$, where $E_{r}=\{v\in E\mid\|v\|\leq r\}$. [^24]: ${\mathcal{C}_{c} \left({\mathbb {R}}^{+},{\mathcal{L}}_{S_{x}}(L_{x})\right)}$ is Hausdorff for all $x\in X$ by the fact that $\bigcup_{B\in\Theta}{\mathcal{B}}_{B}^{x}=L_{x}$, see later Prop. \[16332701\].

--- abstract: | We introduce ring theoretic constructions that are similar to the construction of wreath product of groups. In particular, for a given graph $\Gamma=(V,E)$ and an associate algebra $A,$ we construct an algebra $B=A\, wr\, L(\Gamma)$ with the following property: $B$ has an ideal $I$,which consists of (possibly infinite) matrices over $A$, $B/I\cong L(\Gamma)$, the Leavitt path algebra of the graph $\Gamma$. Let $W\subset V$ be a hereditary saturated subset of the set of vertices \[1\], $\Gamma(W)=(W,E(W,W))$ is the restriction of the graph $\Gamma$ to $W$, $\Gamma/W$ is the quotient graph \[1\]. Then $L(\Gamma)\cong L(W) wr L(\Gamma/W)$. As an application we use wreath products to construct new examples of (i) affine algebras with non-nil Jacobson radicals, (ii) affine algebras with non-nilpotent locally nilpotent radicals. address: - 'Department of Mathematics, King Abdulaziz University, P.O.Box 80203, Jeddah, 21589, Saudi Arabia' - 'Department of Mathematics, King Abdulaziz University, P.O.Box 80203, Jeddah, 21589, Saudi Arabia' author: - Adel Alahmadi - Hamed Alsulami title: Wreath products by a Leavitt path algebra and affinizations --- Actions by Semigroups =====================   Let $S$ be a semigroup with zero, that is, there exists an element $s_{0}$ such that $s_{0}S=\{s_{0}\}= Ss_{0}$. Suppose that the semigroup $S$ acts on a set $X$ both on the left and on the right, that is, there are mappings $S\times X\longrightarrow X$, $X \times S \longrightarrow X$ such that $s_{1}(s_{2}x)=(s_{1}s_{2})x$,$(xs_{1})s_{2}=x(s_{1}s_{2})$ for arbitrary elements $s_{1},s_{2}\in S$;$x \in X$. We assume that $X$ is a set with zero, that is, there exists an element $x_{0}$ such that $s x_{0}=x_{0},x_{0}s=x_{0},x s_{0}=s_{0}x=x_{0}$ for arbitrary elements $s \in S,x \in X$. Suppose further that the left and right actions of the semigroup $S$ on $X$ have the following properties. For arbitrary elements $s\in S,$ $x\in X$: 1. if $s(xs)=x_0$ then $xs=x_0.$ If $s(xs)\neq x_0$ then $s(xs)=x;$ 2. if $(sx)s=x_0$ then $sx=x_0.$ If $(sx)s\neq x_0$ then $(sx)s=x.$ For a field $F$ let $F_{0}[S]$ denote the reduced semigroup algebra, $F_{0}[S]=F[S]/F s_{0}$. Let $A$ be an $F$-algebra. Let $M_{X\times X} (A)$ denote the algebra of possibly infinite $X\times X$ - matrices over $A$ with only finitely many nonzero entries. For elements $s \in S;x,y \in X;a \in A$ let $a_{x,y}$ denote the matrix having $a$ in the $x$-th row and $y$-th column and zeros in all other entries. We will define an algebra structure on $F_{0}[S]+ M_{X\times X}(A)$. For arbitrary elements $s \in S;x,y \in X;a \in A$ we define $$sa_{x,y} = \left\{ \begin{array}{l l} 0, & \quad \text{if $sx=x_{0}$ }\\ a_{sx,y}, & \quad \text{if $sx \neq x_{0}$ } \end{array} \right.$$ $$a_{x,y}s = \left\{ \begin{array}{l l} 0, & \quad \text{if $ys=x_{0}$ }\\ a_{x,ys}, & \quad \text{if $ys \neq x_{0}.$ } \end{array} \right.$$ In particular, $F_{0}[S]a_{x_{0},x}=a_{x,x_{0}}F_{0}[S]=(0)$. \[lem1\] The algebra $F_{0}[S]+ M_{X\times X}(A)$ is associative. The only nontrivial case that we need to check is $(a_{x,y} s)b_{z,t}=a_{x,y} (sb_{z,t}),$ where $x,y,z,t\in X,$ $s\in S.$ If the left hand side is not equal to zero then $ys=z\neq x_0.$ By the property $(1)$ $sz=s(ys)=y,$ which implies associativity. If the right hand side is not equal to zero, then $y=sz\neq x_0.$ As above by $(2)$ $ys=(sz)s=z,$ which again implies associativity. This proves the Lemma. Wreath Product of Algebras ========================== Now let $\Gamma =(V,E)$ be a row finite directed graph with the set of vertices $V$ and the set of edges $E$. For an edge $e\in E$, let $s(e)$ and $r(e)\in V$ denote its source and range respectively. A vertex $v$ for which $s^{-1}(v)$ is empty is called a sink. A path $p=e_{1}...e_{n}$ in a graph $\Gamma $ is a sequence of edges $e_{1}...e_{n}$ such that $r(e_{i})=s(e_{i+1}),$ $ i=1,2,...,n-1$. In this case we say that the path $p$ starts at the vertex $s(e_{1})$ and ends at the vertex $r(e_{n})$. We refer to $n$ as the length of the path $p$. Vertices are viewed as paths of length $0$. The Cohn algebra $C(\Gamma)$ is presented by generators $V\bigcup\limits^{.} E\bigcup\limits^{.}E^{*}$ and relations: $v^{2}=v,\ v \in V;\ vw=wv=0;\ v,\ w \in V,\ v \neq w;$ $s(e)e=e r(e)=e,\ e \in E;\ e^{*}=e^{*}s(e)=r(e)e^{*},\ e \in E;$ $e^{*} f=0;\ e,f \in E,\ e\neq f;\ e^{*}e=r(e),\ e \in E$. Clearly, the set $S=\{pq^{*}| p,q \text{ are paths on } \Gamma\}\cup \{0\}$ is a semigroup with zero and $C(\Gamma)$ is a reduced semigroup algebra. If $X,Y$ are nonempty subsets of the set $V$ then we let $E(X,Y)$ denote the set $\{e\in E \mid\, s(e)\in X, r(e)\in Y\}.$ Let $\mathcal{E}$ be a family of pairwise orthogonal idempotents in $A$. We introduce a set $E(V,\mathcal{E})$ of edges connecting $V$ to idempotents from $\mathcal{E}$ such that for every nonsink vertex $v \in V$ the set of edges set $e \in E (v,\mathcal{E}),s(e)=v$ is finite (possibly empty). If $v$ is a sink in $\Gamma$, then we assume that $E(v,\mathcal{E})=\emptyset$. Now we extend the graph $\Gamma$ to a graph $\widetilde{\Gamma}(\widetilde V,\widetilde E)$, where $\widetilde V=V \cup \mathcal{E}, \widetilde E = E\cup E (V,\mathcal{E}).$ Let $\mathcal{P}$ be the subset of the extended Cohn algebra $C(\widetilde{\Gamma})$, which consists of paths, that start in $\Gamma$ and end in $\mathcal{E}$, and zero, so $\mathcal{P}=\left( \bigcup\limits_{\text{ p is a path} \atop { \text{ on $\Gamma$}}} pE(r(p),\mathcal{E})\right)\cup \{0\}$. The Cohn algebra $C(\Gamma)$ is a subalgebra of the Cohn algebra $C(\widetilde{\Gamma})$. \[lem2\] $C(\Gamma)\mathcal{P}\subseteq \mathcal{P}$. We have $C(\Gamma)\mathcal{P}= \text{span} C(\Gamma)pe$, where $p$ is a path in $\Gamma$ and $e\in E(V,\mathcal{E})$ with $r(p)=s(e)$. Since $C(\Gamma)p\subseteq C(\Gamma)$, it is sufficient to show that $pq^*e\in\mathcal{P}$ for arbitrary paths $p, q$ in $\Gamma$, $r(p)=r(q)$, $s(q)=s(e)$. Furthermore, it is sufficient to prove that $q^*e \in \mathcal{P}$. If length of $q \geq 1$, then $q^*e=0$. If $q$ is a vertex, then $q^*e=e$. This proves the Lemma. By Lemma \[lem2\], $\mathcal{P}$ can be viewed as a left $S$-module. We will define also a structure of a right $S$-action on $\mathcal{P}$ via $p.s=s^{*}p \in P$. \[lem3\] The left and right actions of the semigroup $S$ on $\mathcal{P}$ satisfy $(1),$ $(2).$ Let us check property $(1).$ If $s=0$ or $x=0$ then clearly $xs=0.$ Suppose that $s=pq^*\neq 0, x=p_1\neq0.$ Then $s(xs)=ss^*p_1=pq^*qp^*p_1=pp^*p_1.$ The equality $pp^*p_1=0$ means that the path $p$ is not a beginning of the path $p_1,$ in which case $xs=qp^*p_1=0.$ If $pp^*p_1\neq 0$ then $p$ is a beginning of the path $p_1,$ $p_1=pp_2.$ Now, $s(xs)=pp^*p_1=pp^*pp_2=pp_2=p_1=x.$ Property $(2)$ is checked similarly. This proves the Lemma. Consider the algebra $C(\Gamma)+M_{\mathcal{P}\times\mathcal{ P}}(A)$ that we have defined in Section I. We extend the range function $r$ by $r(0)=1$. Now consider the subalgebra $C(\Gamma)+I$, where $I$ consists of matrices having all $(p,q)$-entries lie in the $r(p)Ar(q).$ Clearly, $I$ is an ideal of the algebra $C(\Gamma)+I$. For a nonsink vertex $v\in V(\Gamma),$ consider the element $CK(v)=CK(v)^{'}-CK(v)^{''},$ where $$CK(v)^{'}=v-\sum\limits_{f\in E(\Gamma)\atop{s(f)=v}}ff^{*},CK(v)^{''}=\sum\limits_{e \in E(v,\mathcal{E})\atop{s(e)=v}}(r(e))_{e,e}.$$ \[lem4\] $I\ CK(v)=CK(v)\ I=(0)$, for any nonsink vertex $v \in V(\Gamma)$. Let $p,q \in\mathcal{ P}$ and $a \in r(p) A r(q),$ where $v \in V(\Gamma)$ is not a sink. We will show that $a_{p,q}CK(v)=0$. If $q$ is the zero or $s(q)\neq v$, then $a_{p,q}v=0$ as $vq=0; a_{p,q}f=0$ as $f^{*}q=0$ and $a_{p,q}r(e)_{e,e}=0$ as $q\neq e$ (the edge $e$ starts at $v$). Now suppose that $q$ be a nonzero path, $s(q)=v$. Suppose at first that length $(q)=1,$ that is, $q=e$ is an edge connecting $v$ with an idempotent $r(e) \in\mathcal{E}$. Then $a_{p,q}v=a_{p,q}; a_{p,q}f=0$ because $f^*q=0$; $a_{p,q}r(e)_{e,e}=(ar(e))_{p,e}=a_{p,q}, a_{p,q}r(e^{'})_{e^{'},e^{'}}=0$ for an edge $e^{'}\in E(v,\mathcal{E}), e^{'}\neq e$. Hence $a_{p,q}CK(v)=0$.\ Now suppose that length $(q)\geq 2$. Then $q=fq^{'},f \in E (v,V(\Gamma))$. In this case $a_{p,q}v=a_{p,q};\ a_{p,q}ff^{*}=a_{p,q};\ a_{p,q}f^{'}f^{'*}=0$, for an edge $f^{'} \in E(v,V(\Gamma)), f^{'}\neq f$. Now $a_{p,q}r(e)_{e,e}=0,$ because $q\neq e$ and again $a_{p,q}CK(v)=0$. We proved that $CK(v)\ I=(0)$. Similarly, $I\ CK(v)=(0)$. This proves the Lemma. \[lem5\] Let $v_{1},....,v_{m}$ be distinct vertices in $V(\Gamma)$. Let $p_{ik},q_{ik},p^{'}_{is},q^{'}_{it}$ be the paths of length $\geq 1$ in $\Gamma$, $r(p_{ik})=r(q_{ik})=r(p^{'}_{is})=r(q^{'}_{it})=v_{i}$. Assume that for each $i$ all paths $p^{'}_{is}$ are distinct; all paths $q^{'}_{it}$ are distinct and all pairs $(p_{ik},q_{ik})$ are distinct. Then the elements $p_{ik}CK(v_{i})^{'}q^{*}_{ik},\ p'_{is}CK(v_{i})^{'},\ CK(v_{i})^{'}q^{'*}_{it},\ CK(v_{i})^{'}$ in $C(\Gamma)$ are linearly independent. Suppose that $\alpha_{ik},\beta_{is},\gamma_{it},\xi_{i} \in F$ and $$\sum_{i,k}\alpha_{ik}p_{ik}CK(v_{i})^{'}q^{*}_{ik}+\sum_{i,s} \beta_{is}p^{'}_{is}CK(v_{i})^{'}+\sum_{i,t}\gamma_{it}CK(v_{i})^{*'}q_{it}+\sum\xi_{i}CK(v_{i})^{'}=0.$$ We take $S_1=\sum\limits_{i,k}\alpha_{ik}p_{ik}CK(v_{i})^{'}q^{*}_{ik},$ $S_2=\sum\limits_{i,s} \beta_{is}p^{'}_{is}CK(v_{i})^{'},$ $S_3=\sum\limits_{i,t}\gamma_{it}CK(v_{i})^{*'}q_{it},$ and $S_4=\sum\limits_{i}\xi_{i}CK(v_{i})^{'}.$ Since semigroup elements involved in different summands $S_{i}, S_j,\ i\neq j$ are distinct, it follows that $S_1=S_2=S_3=S_4 =0.$ Suppose that not all coefficients $\alpha_{ik}$ are equal to zero. Let $d=\max\{\textrm{length}(p_{ik})+\textrm{length}(q_{ik})\ | \alpha_{ik}\neq 0\}$. Suppose that this maximum is achieved at $(i_{0},k_{0})$. Let $f \in E (v_{i},V(\Gamma))$. Then the summand $\alpha_{i_0k_0} p_{i_0k_0}ff^*q^*_{i_0k_0}$ won’t cancel in $S_1.$ Hence all $\alpha_{ik}=0$. Equalities $\beta_{is}=\gamma_{it}=\xi_{i}=0$ are proved similarly. This proves the Lemma. Let $J$ be the ideal of $C(\Gamma)+I$ generated by all elements $CK(v)$, where $v$ runs over all nonsink vertices from $V(\Gamma)$. \[lem6\] $J\cap I=(0).$ It is easy to see that for any edge $g \in E(\Gamma)$ we have $g^{*}CK(v)=CK(v)g=0$. Hence an arbitrary element from the ideal $J$ can be represented as $$x=\sum_{i,k}\alpha_{ik}p_{ik}CK(v_{i})q^{*}_{ik}+\sum_{i,s}\beta_{is}p^{'}_{is}CK(v_{i})+\sum_{i,t}\gamma_{it}CK(v_{i})q^{'*}_{it}+\sum_{i}\xi_{i}CK(v_{i}),$$ where $\alpha_{ik},\beta_{is},\gamma_{it},\xi_{i} \in F;p_{ik},q_{ik},p^{'}_{is},q^{'}_{it}$ are paths on $\Gamma$ of length $\geq 1$. If this element lies in $I$ then $$\sum_{i,k}\alpha_{ik}p_{ik}CK(v_{i})^{'} q^{*}_{ik}+\sum_{i,s}\beta_{is}p^{'}_{is}CK(v_{i})^{'}+\sum_{i,t}\gamma_{it}CK(v_{i})^{'}q^{'*}_{it}+\sum\xi_{i}CK(v_{i})^{'}=0.$$ By Lemma \[lem5\] $x=0$. This proves the Lemma. Following G. Abrams and Z. Mesyan \[2\], we may view the Leavitt path algebra $L(\Gamma)$ as the quotient algebra $C(\Gamma)/N$, where $N$ is the ideal of $C(\Gamma)$ generated by $CK(v)'$ for all nonsink $v\in V$. Now let $B=(C(\Gamma)+I)/J$. Clearly the algebra $B$ has an ideal $(I+J)/J\cong \sum\limits_{p,q \in \mathcal{P}}(r(p)A r(q))_{p,q}$ and the quotient of $B$ modulo this ideal is isomorphic to the Leavitt path algebra $L(\Gamma)$. We will denote the algebra $B$ as $A\,wr\, L(\Gamma)$ and call it the wreath product of the algebra $A$ and the Leavitt path algebra $L(\Gamma)$. Remark that the construction $A\,wr\, L(\Gamma)$ depends on the set of idempotents $\mathcal{E}$ and the set of edges $E(V,\mathcal{E})$. \[prop1\] If $\Gamma$ is a finite graph and the algebra $A$ is finitely generated then $A\,wr\, L(\Gamma)$ is finitely generated. Indeed, if $A=\langle a_1, a_2, \cdots, a_m \rangle$, then the algebra $C(\Gamma)+I$ is generated by $V, E, E^*$ and matrices $(a_i)_{00}, (r(e))_{e,0},(r(e))_{0,e},\, e\in E(V,\mathcal{E}).$ Let us discuss some applications of the wreath product construction to the theory of Leavitt path algebras. A subset $W\subseteq V$ is said to be hereditary if $v \in W$ implies $r(s^{-1}(v))\subseteq W$ \[1\]. The subset $W$ is said to be saturated if $r(s^{-1}(v))\subseteq W$ implies that $v\in W,$ for every non-sink vertex $v\in V$ \[1\]. The Leavitt path algebra $L(\Gamma)$ has a natural $\mathbb{Z}$-gradation: $deg(v)=0, deg(e)=1, deg(e^*)=-1$. A Leavitt path algebra $L(\Gamma)$ is graded simple if and only if $\Gamma$ does not contains proper hereditary and saturated subsets (see \[7\]). Let $W$ be a hereditary and saturated subset of $V$. The graph $\Gamma(W)=(W, E(W,W))$ is the restriction of the graph $\Gamma$ . Consider the graph $\Gamma / W$ with the set of vertices $V\setminus W$ and the set of edges $E\setminus E(V,W)$. The set $\mathcal{E}= V\setminus W$ in $L(\Gamma/ W)$ is the set of pairwise orthogonal idempotents. Vertices from $W$ are connected to idempotents $\mathcal{E}$ via the edges from $E(W,V\setminus W)$. \[prop2\] $L(\Gamma)\cong L(\Gamma(W)) \,wr\, L(\Gamma/W).$ Let $\mathcal{P}$ be the set of pathes on the graph $\Gamma$ such that $s(p)\in V\setminus W,$ $r(p)\in W,$ $r(p)$ is the first vertex on the path $p$ that lies in $W.$ An arbitrary element $a$ from $L(\Gamma)$ can be uniquely represented as $$a=a'+\sum pa_{pq}q^*+\sum pb_{p}+\sum c_q q^*+d$$ where $a'$ is a linear combination of elements $p_1p^*_2$ such that no vertex on $p_1,\, p_2$ lies in $W ;$ $p,\, q\in P\,;$ $a_{pq}, b_p, c_q, d$ lie in the subalgebra of $L(\Gamma)$ generated by $W,\, E(W,W).$ Let $A=L(\Gamma(W)),\, \Gamma(W)=(W, E(W,W)),$ $\mathcal{E}=W,$ $E(V, \mathcal{E})=E(V,W).$ Straightforward verification show that the mapping $L(\Gamma)\rightarrow A\, wr\, L(\Gamma/W),$ $$a'+\sum pa_{pq}q^*+\sum pb_{p}+\sum c_q q^*+d\mapsto a'+\sum_{p,q} (a_{pq})_{p,q}+\sum_{p}(b_{p})_{p,0}+\sum_{q} (c_q)_{0,q}+(d)_{00}$$ is an isomorphism of algebras. From \[5,6\], it follows that the Leavitt path algebra of a finite graph has polynomial growth if and only if it is an iterated wreath product of disjoint unions of cycles and trees. Following \[3\], we call a vertex $v$ in a connected graph $\Gamma(V,E)$ a *balloon* over a nonempty subset $W$ of $V$ if (1) $v\notin W,$ (2) there is a loop $C\in E(v,v),$ (3) $E(v,W)\neq\emptyset,$ (4) $E(v,V)=\{C\}\cup E(v,W),$ and (5) $E(V,v)=\{C\}.$ If $V$ contains a vertex $v$ which is a balloon over $V\setminus \{v\}$, then we say the graph $\Gamma$ is a balloon extension. Now let $\Gamma$ be a graph and $\Gamma'$ be a balloon extension. Then $L(\Gamma')\cong L(\Gamma) \,wr\, L(C)$, where $C$ is a loop. Affinizations of countable dimensional algebras ================================================ By an affinization we mean an embedding of a countable dimensional algebra in an affine ( that is, finitely generated ) algebra with preservation of certain properties. In $1981$ K. Beidar \[8\] constructed an affine algebra with a non-nil Jacobson radical, answering an old question of S. Amitsur. Beidar’s construction was modified and generalized by L. Small ( see \[10\] ). The next step was done by J. Bell \[9\] who constructed affinizations of small Gelfand-Kirillov dimensions. Answering a question of K. Zhevlakov, E. Zelmanov \[14\] constructed an affine algebra with a non-nilpotent locally nilpotent radical. In this section we show how to construct different affinizations using wreath products. ![[]{data-label="1"}](lwp2 "fig:"){width=".1\textwidth"}\ Let $\Gamma(\{v\},\{c\})$ be a loop. The Leavitt path algebra $L(\Gamma)$ is $$\sum_{i\geq 0} Fc^i+\sum_{i\geq 0} F(c^*)^i\cong F[t^{-1},t].$$ Let $A$ be an associative algebra with $1.$ Let $\mathcal{E}=\{1\},$ the only vertex $v$ of the graph $\Gamma$ is connected to $1$ by the edge $e,$ see figure 1 . Then the set $\mathcal{P}$ from the construction of the wreath product $A\,wr\,L(\Gamma)$ is $\mathcal{P}=\{c^i e, \, i\geq 0\}\cup \{0\},$ $A\,wr\,L(\Gamma)=F[c,c^*]+M_{\mathcal{P}\times\mathcal{P}}(A).$ We will identify the set $\mathcal{P}$ with the set of nonnegative integers $\mathbb{N},$ $0\leftrightarrow 0,$ $c^i e\leftrightarrow i+1,\, i\geq 0.$ Then $A\,wr\,L(\Gamma)$ can be identified with $F[t^{-1},t]+M_{\mathbb{N}\times \mathbb{N}}(A),$ where $F[t^{-1},t]$ is an isomorphic copy of the vector space of the algebra of Laurent polynomials ( it is not a subalgebra in $A\,wr\,L(\Gamma)$, see \[6\]); $t^{-1}t=v,\, tt^{-1}=v-(1)_{1,1}; ta_{i,j}=a_{i+1,j}, t^{-1}a_{i,j}=a_{i-1,j}$ for $i\geq 1,$ $t^{-1}a_{0,j}=0;$ $a_{i,j}t=a_{i,j-1}$ for $j\geq 1,$ $a_{i,0}t=0,$ $a_{i,j}t^{-1}=a_{i,j+1}.$ Let $\widetilde{M_{\mathbb{N}\times \mathbb{N}}(A)}$ denote the algebra of infinite $\mathbb{N}\times \mathbb{N}$ matrices over $A$ having only finitely many nonzero entries in each row and in each column. Clearly, $M_{\mathbb{N}\times \mathbb{N}}(A)\triangleleft \widetilde{M_{\mathbb{N}\times \mathbb{N}}(A)}.$ The algebra $L(\Gamma)+\widetilde{M_{\mathbb{N}\times \mathbb{N}}(A)}$ is defined in the same way as the algebra $A\, wr\, L(\Gamma)=L(\Gamma)+M_{\mathbb{N}\times \mathbb{N}}(A).$ Moreover, $M_{\mathbb{N}\times \mathbb{N}}(A)\triangleleft L(\Gamma)+\widetilde{M_{\mathbb{N}\times \mathbb{N}}(A)}.$ Suppose that the algebra $A$ is generated by a countable set $a_0,a_1,\cdots.$ Let $a=\sum\limits_{i=0}^\infty (a_i)_{i,i}\in \widetilde{M_{\mathbb{N}\times \mathbb{N}}(A)}.$ Let $B$ denote the affine algebra generated by $t,t^{-1},$ $a,$ $(1)_{0,0}.$ The following assertion is straightforward. \[prop3\] $F[t^{-1},t]+M_{\mathbb{N}\times \mathbb{N}}(A)\subset B\subset F[t^{-1},t]+\widetilde{M_{\mathbb{N}\times \mathbb{N}}(A)}.$ Now let $\text{ Rad }(A)$ be the Jacobson or the locally nilpotent radical of the algebra $A$ (see \[11\] ). Then $M_{\mathbb{N}\times \mathbb{N}}(\text{ Rad }(A))=\text{ Rad }(M_{\mathbb{N}\times \mathbb{N}}(A))\subseteq \text{ Rad }(B).$ If $F$ is a countable field then there exists a countably generated commutative domain $A_0,$ which is equal to its Jacobson radical. For example, in the field of rational functions $F(t)$ consider the subalgebra $A_0=\left\{ \frac{f(t)}{g(t)}\mid f(0)=0, g(0)=1\right\}.$ It is easy to see that the algebra $A_0$ is countably dimensional and equal to its Jacobson radical. Let $A=F.1+A_0.$ The algebra $B$ of Proposition \[prop3\] is affine. The Jacobson radical of $B$ contains $M_{\mathbb{N}\times \mathbb{N}}(A)$ and therefore is not nil. Ju. M. Rjabuhin \[13\] constructed a prime countably generated locally nilpotent algebra $A_0.$ Let $A= F.1+A_0.$ Then the locally nilpotent radical of the affine algebra $B\subset F[t^{-1},t]+\widetilde{M_{\mathbb{N}\times \mathbb{N}}(A)}$ contains $M_{\mathbb{N}\times \mathbb{N}}(A_0)$ and therefore is not nilpotent. Acknowledgement {#acknowledgement .unnumbered} =============== The authors would like to express their appreciation to professor S. K. Jain for carefully reading the manuscript and for offering his comments. The authors are grateful to the referee for numerous helpful comments. This paper was funded by King Abdulaziz University, under grant No. (57-130-35-HiCi). The authors, therefore, acknowledge technical and financial support of KAU. [9]{} Abrams G., Aranda Pino G. *The Leavitt path algebra of a graph.* J. Algebra 293(2): 319-334 (2005). Abrams G., Mesyan Z., *Simple Lie algebra arising from Leavitt path algebra.* Journal of pure and applied algebra, 216: 2303-2313 (2012). Alahmadi A., Alsulami H., *Simplicity of Lie algebra of skew elements of Leavitt path algebra.* submitted Alahmadi A., Alsulami H., *On the simplicity of Lie algebra of Leavitt path algebra.* submitted Alahmedi A., Alsulami H., Jain S. K., Zelmanov E. *Leavitt path algebras of finite Gelfand-Kirillov dimension.* J. Algebra Appl. 11(6) (2012). Alahmedi A., Alsulami H., Jain S. K., Zelmanov E. *Structure of Leavitt path algebras of polynomial growth.* PNAS 110(38): 15222-15224 (2013). Ara P., Moreno M., Padro E. *Nonstable K-theory for graph algebras .* J. Algebra Represent Theory. (10)157-178 (2007). Beidar K.I., *Radicals of nitely generated algebras.* Uspekhi Mat. Nauk 36(1981), no. 6 (222), 203 - 204. Bell, J. *Examples in finite Gelfand-Kirillov dimension.* J. Algebra 263 (2003), no. 1, 159 - 175. Bell, J.; Small, L. *A question of Kaplansky. Special issue in celebration of Claudio Procesi’s 60th birthday.* J. Algebra 258 (2002), no. 1, 386 - 388. I.N. Herstein, *Noncommutative rings.* Reprint of the 1968 original. With an afterword by L.W. Small. Carus Mathematical Monographs, 15. Mathematical Association of America, Washington, DC, 1994 Kargapolov M, Merzlyakov Y, *Fundamentals of the theory of groups.* New York, Springer-Verlag (1979). Rjabuhin, Ju. M., *A certain class of locally nilpotent rings.* (Russian) Algebra i Logika 7 1968 no. 5, 100–108. Zelmanov, E.I., *An example of a finitely generated prime ring.* Sibirsk Mat. Z. 20 (1979), 423 (in Russian); English trans. Siberian Math. J. 20 (1979), 303–304.

--- abstract: 'We report the results from our analysis of the *Fermi* Large Area Telescope data of the *Fermi* unassociated source 2FGL J1906.5$+$0720, which is a high-ranked candidate pulsar. In order to better study our target, we first update the ephemeris for PSR J1907$+$0602, which is used to help remove any possible contamination due to strong emission from this nearby pulsar. From our analysis, 2FGL J1906.5$+$0720 is confirmed to have a significant low energy cutoff at $\sim$ 1 GeV in its emission (14$\sigma$–18$\sigma$ significance), consistent with those seen in young pulsars. We search for pulsations but no spin frequency signals are found in a frequency range of 0.5–32 Hz. No single model can fully describe the source’s overall [*Fermi*]{} $\gamma$-ray spectrum, and the reason for this is the excess emission detected at energies of $\geq$4 GeV. The high-energy component possibly indicates emission from a pulsar wind nebula, when considering 2FGL J1906.5$+$0720 as a young pulsar. We conclude that 2FGL J1906.5$+$0720 is likely a pulsar based on the emission properties we have obtained, and observations at other energies are needed in order to confirm its pulsar nature.' author: - 'Yi <span style="font-variant:small-caps;">Xing</span> and Zhongxiang <span style="font-variant:small-caps;">Wang</span>' title: 'Searching for $\gamma$-Ray Pulsars among *Fermi* Unassociated Sources: 2FGL J1906.5+0720' --- Introduction ============ Since the *Fermi* Gamma-ray Space Telescope was launched in June 2008, the main instrument on-board—the Large Area Telescope (LAT) has been continuously scanning the whole sky every three hours in the energy range from 20 MeV to 300 GeV, discovering and monitoring $\gamma$-ray sources with much improved spatial resolution and sensitivity comparing to former $\gamma$-ray telescopes [@atw2009]. In 2012 resulting from *Fermi*/LAT data of the first two-year survey, a catalog of 1873 $\gamma$-ray sources was released by @nol2012 as the *Fermi*/LAT second source catalog. Among the $\gamma$-ray sources, approximately 800 and 250 were found to be respectively associated with blazars and active galaxies of uncertain types, and more than 100 were associated with pulsars in our Galaxy. The three types thus account for the majority of the $\gamma$-ray sources detected by *Fermi*. In addition, 575 sources in the catalog have not been associated with any known astrophysical objects [@nol2012]. For the purpose of identifying the nature of these unassociated sources, many follow-up studies, such as classifying their $\gamma$-ray characteristics [@ack2012], searching for radio pulsars [@ray2012], and observing at multi-wavelengths [@tak2012; @ace2013], have been carried out. Because of the relative lack of sources at low Galactic latitudes in many extragalactic source catalogs and the emission contamination by the Galaxy, the Galactic distribution of the *Fermi* unassociated sources were found to concentrate towards the Galactic plane [@nol2012]. More than half of the unassociated sources are located at low latitudes with $|b|<$ 10 [@nol2012], possibly suggesting Galactic origins for most of them. Taking under consideration the types of identified and associated Galactic $\gamma$-ray sources in the catalog, these low-latitude unassociated sources are most likely pulsars, pulsar wind nebulae, supernova remnants, globular clusters, or high-mass binaries. Additionally since identified and associated AGNs or blazars have a nearly isotropic distribution, AGN/blazar origins for these sources can not be excluded. In any case, the low-latitude *Fermi* unassociated sources are the best young pulsar candidates on the basis of currently known Galactic $\gamma$-ray populations, as $\sim$50% of the identified or associated Galactic [*Fermi*]{} sources are pulsars [@nol2012] and the *Fermi*-detected millisecond pulsars are nearly isotropic (see Figure 2 in [@lat2013]). With high rotational energy loss rates (so-called spin-down luminosities), young pulsars are clustered close to the Galactic plane and can be detected to large distances. Aiming to search for new pulsars among the unassociated sources, we selected the pulsar candidates from the *Fermi* second source catalog by requiring $|b|<$ 10 and variability indices (Variability\_Index parameter in the catalog) lower than 41. The variability indices were reported to measure the variability levels of sources, and a value larger than 41.64 indicates $<1$% chance of being a steady source [@nol2012]. We further ranked the candidates by their Signif\_Curve parameters reported in the catalog, which represent the significance of the fit improvement between curved spectra and power-law spectra, as $\gamma$-ray pulsars typically have curved spectra with a form of exponentially cutoff power law. The first ten sources from our selection are listed in Table \[tab:candi\]. The first source listed is 2FGL J1704.9$-$4618, which has the highest Signif\_Curve value of $\sim$9.97$\sigma$ but the lowest detection significance value ($\sim$9$\sigma$; Signif\_Avg parameter in the catalog). For a comparison, the second source in our list 2FGL J1906.5$+$0720 has both high Signif\_Curve ($\sim$9.85$\sigma$) and Signif\_Avg values ($\sim$24$\sigma$), and is ranked the first among candidate pulsars by @lee2012, who applied a Gaussian-mixture model for the ranking. Among the bright $\gamma$-ray sources ($>$20$\sigma$ detection significance), this source is clearly located in the pulsar region in the plane of the curvature significance versus variability index [@rom2012]. We thus carried out detailed study of 2FGL J1906.5$+$0720 by analyzing *Fermi*/LAT data of the source region, and report our results in this paper. In addition, 2FGL J1906.5$+$0720 is located close to a very bright $\gamma$-ray pulsar J1907$+$0602 (Signif\_Avg $\sim$ 55$\sigma$; [@lat2013]). The angular distance between them is approximately 1.3 degrees (see Figure \[fig:ts\]). The pulsar was discovered in the first $\sim$4 month LAT data, revealed with a spin frequency of $\sim$9.378 Hz and a spin-down luminosity of $\sim$2.8$\times$10$^{36}$ erg s$^{-1}$ [@abd2009]. The pulsar is radio faint, making very difficult to study its timing behavior at radio frequencies [@abd2010]. In order to better study our targeted [*Fermi*]{} source by removing possible contamination from PSR J1907$+$0602, we performed timing analysis to the LAT data of the pulsar and include our timing results in this paper. Observations {#sec:obs} ============ LAT is the main instrument on-board the *Fermi* Gamma-ray Space Telescope. It is a $\gamma$-ray imaging instrument which carries out an all-sky survey in the energy range from 20 MeV to 300 GeV [@atw2009]. In our analysis we selected LAT events inside a 20$\times$ 20 region centered at the position of 2FGL J1906.5$+$0720 during a nearly five-year time period from 2008-08-04 15:43:36 to 2013-07-23 20:53:17 (UTC) from the *Fermi* Pass 7 database. Following recommendations of the LAT team, events included were required to have event zenith angles fewer than 100 deg, preventing contamination from the Earth’s limb, and to be during good time intervals when the quality of the data was not affected by the spacecraft events. Analysis and Results {#sec:ana} ==================== Timing Analysis of PSR J1907$+$0602 {#subsec:timing} ----------------------------------- After the [*Fermi*]{} discovery of PSR J1907$+$0602 [@abd2009], its timing solution was updated by @abd2010 and @ray2011 using the LAT data during MJD 54647–55074 and MJD 54682–55211, respectively. In 2013 the *Fermi*/LAT team released the second *Fermi* catalog of $\gamma$-ray pulsars [@lat2013], in which the timing solution for PSR J1907$+$0602 was updated again using the data during MJD 54691–55817. A glitch at MJD 55422 was detected with $\Delta\nu/\nu$ of $\sim$ 4.6 $\times$ 10$^{-6}$ and $\Delta\dot{\nu}/\dot{\nu}$ of $\sim$ 1 $\times$ 10$^{-2}$. In order to study 2FGL J1906.5$+$0720 by being able to remove photons from the nearby pulsar, we performed phase-connected timing analysis to the LAT data of J1907$+$0602 during the nearly five-year time period of MJD 54683–56497. We selected LAT events within 07 centered at the pulsar’s position given in the catalog in the energy range from 50 MeV to 300 GeV, which was suggested by @ray2011. Pulse phases for photons before MJD 55400 were assigned according to the known ephemeris using the *Fermi* plugin of TEMPO2 [@edw2006; @hob2006]. We extracted an ‘empirical Fourier’ template profile, with which we generated the time-of-arrivals (TOAs) of 128 evenly divided observations of the time period. Both the template and TOAs were obtained using the maximum likelihood method described in @ray2011. From the pre-fit residuals we found that the timing model given in the second *Fermi* catalog of $\gamma$-ray pulsars could not fully describe the TOAs after MJD $\sim$55800, suggesting the requirement of an updated timing model. We then iteratively fitted the TOAs to the timing model using TEMPO2. For the glitch because of its relative large amplitude and long interval between the last pre-glitch and the first post-glitch observations, we could not obtain a unique solution to accurately determine its epoch by requiring continuous pulse phase. Instead, we adopted one of the solutions according to the ephemeris we obtained as the glitch epoch, which is closest to that reported in the second *Fermi* catalog of $\gamma$-ray pulsars. The updated ephemeris is given in Table \[tab:timing\], the post-fit timing residuals are shown in Figure \[fig:rms\], and the folded pulse profile and the two-dimensional phaseogram of this pulsar are plotted in Figure \[fig:ftp\]. We defined phase 0.1–0.7 as the onpulse phase interval and phase 0.7–1.1 as the offpulse phase interval (Figure \[fig:ftp\]), using the definition given in @lat2013. Maximum Likelihood Analysis {#subsec:likeli} --------------------------- ### Full data We selected LAT events in an energy range from 100 MeV to 300 GeV for the likelihood analysis, and included all sources within 15 degrees centered at the position of 2FGL J1906.5$+$0720 in the *Fermi* 2-year catalog to make the source model. The spectral function forms of the sources are given in the catalog. The spectral normalization parameters for the sources within 4 degrees from 2FGL J1906.5$+$0720 were left free, and all the other parameters were fixed to their catalog values. In addition we included the spectrum model gal\_2yearp7v6\_v0.fits and the spectrum file iso\_p7v6source.txt in the source model to consider the Galactic and the extragalactic diffuse emission, respectively. The normalizations of the diffuse components were left free. In the *Fermi* 2-year catalog, the $\gamma$-ray emission from 2FGL J1906.5$+$0720 is modeled by a log parabola expressed by $dN/dE=N_{0}(E/E_{b})^{-(\alpha+\beta \ln(E/E_{b}))}$ [@nol2012]. We fixed the break energy to the catalog value of $\sim$1 GeV, and let the indices $\alpha$ and $\beta$ free. We also tested two other models for the source: an exponentially cutoff power law expressed by $dN/dE=N_{0}E^{-\Gamma}exp[-(E/E_{cut})^{b}]$, where $\Gamma$ is the spectral index, $E_{cut}$ is the cutoff energy, and $b$ represents the sharpness of the cutoff, and a simple power law expressed by $dN/dE=N_{0}E^{-\Gamma}$. For the exponentially cutoff power law we note that all pulsars in the second [*Fermi*]{}  $\gamma$-ray pulsar catalog with $b$ values different from 1 (usually smaller than 1 and indicating a sub-exponential cutoff) have $E_{cut}$ higher than 2 GeV [@lat2013]. Considering the $E_{cut}$ values we obtained for this source are lower than 2 GeV, especially when the possible contamination from nearby sources is excluded (see Section 3.2.2, Section 3.4, and Table \[tab:likeli\]), we only used the simple exponentially cutoff shape with $b= 1$ in our analysis. We performed standard binned likelihood analysis with the LAT science tool software package v9r31p1. The obtained spectral results and Test Statistic (TS) values are given in Table \[tab:likeli\], and the TS map of a $\mathrm{5^{o}\times5^{o}}$ region around 2FGL J1906.5$+$0720 is displayed in the left panel of Figure \[fig:ts\]. PSR J1907$+$0602 is kept in the figure to show the proximity of the two sources. From the analysis, we found that the log parabola and the exponentially cutoff power law better fit the LAT data of 2FGL J1906.5$+$0720 than the simple power law, indicating a significant cutoff in the $\gamma$-ray spectrum of 2FGL J1906.5$+$0720 at the low energy of $\sim$1 GeV (Table \[tab:likeli\]). The significance of the break (approximately described by $\sqrt{TS_{break}}\sigma$ $=$ $\sqrt{TS_{LP} - TS_{PL}}\sigma$) of the log parabola is $\sim$16$\sigma$, and the significance of the cutoff (approximately described by $\sqrt{TS_{cutoff}}\sigma$ $=$ $\sqrt{TS_{PL+cutoff} - TS_{PL}}\sigma$) of the exponentially cutoff power law is $\sim$14$\sigma$. ### Offpulse phase intervals of PSR J1907$+$0602 Considering no offpulse $\gamma$-ray emission from PSR J1907$+$0602 was detected by *Fermi* [@ack2011], we repeated binned likelihood analysis described above by including LAT events only during the offpulse phase intervals to prevent possible contamination from the pulsar. The phase intervals are defined in Section \[subsec:timing\]. Since the emission from the pulsar was removed, we excluded this source from the source model. The likelihood fitting results for the different $\gamma$-ray spectral models for 2FGL J1906.5$+$0720 are given in Table \[tab:likeli\], and the TS map of a $\mathrm{5^{o}\times5^{o}}$ region around 2FGL J1906.5$+$0720 is shown in the right panel of Figure \[fig:ts\]. The TS values are significantly increased comparing to those when the full data were used, having doubled the detection significance of 2FGL J1906.5$+$0720. In addition, a low-energy break or cutoff at $\sim$1 GeV in the source’s emission is similarly favored as that in the analysis of the full data. Spectral Analysis {#subsec:sa} ----------------- To obtain a spectrum for 2FGL J1906.5$+$0720, we evenly divided 20 energy ranges in logarithm from 100 MeV to 300 GeV, and used a simple power law to model the emission in each divided energy range. The index of the power law was fixed to the value we obtained before (Table \[tab:likeli\]). This method is less model-dependent and provides a good description for the $\gamma$-ray emission of a source. The spectra from both the full data and the offpulse phase interval data were obtained, which are displayed in Figure \[fig:spec\]. Only spectral points with TS greater than 4 (corresponding to the detection significance of 2$\sigma$) were kept. We plotted the obtained exponentially cutoff power-law fits and log-parabolic fits from the above likelihood analysis in Figure \[fig:spec\]. As can be seen, the first model does not provide a good fit to the LAT spectrum. At energies of greater than several GeV, the fit deviates from the spectrum for both the full data and the offpulse phase interval data of PSR J1907$+$0602. The log parabola better describes the spectra, which is also indicated by the larger TS values obtained with it (Table \[tab:likeli\]), although a small degree of deviations from the spectra can still be seen. These may suggest an additional spectral component at the high energy range. We fit the spectral data points below 2 GeV with exponentially cutoff power laws and obtained $\Gamma$ of 1.4$\pm$0.2 and $E_{cutoff}$ of 1.0$\pm$0.2 GeV from the full data, and $\Gamma$ of 1.6$\pm$0.1 and $E_{cutoff}$ of 0.7$\pm$0.1 GeV from the offpulse phase interval data. The cutoff energy values are within the range of young $\gamma$-ray pulsars (0.4 $< E_{cutoff} <$ 5.9; see Table 9 in [@lat2013]) but lower than that of millisecond $\gamma$-ray pulsars (1.1 $< E_{cutoff} <$ 5.4; see Table 9 in [@lat2013]). The fitting again shows that an additional spectral component is needed. Spatial Distribution Analysis ----------------------------- In the residual TS maps both from the full data and the offpulse phase interval data after removing all sources (Figure \[fig:ts\]), two $\gamma$-ray emission excesses exist. They are located at R.A.=285326 and Decl.= 5855 (equinox J2000.0), with 1$\sigma$ error circle of 007, and R.A.=285293 and Decl.= 7030 (equinox J2000.0), with 1$\sigma$ error circle of 01 (marked by circles in Figure \[fig:ts\]), which were obtained from running ‘gtfindsrc’ in LAT science tools software package. In addition, there is also a tail-like structure in the southeast direction of 2FGL J1906.5$+$0720, which can be clearly seen in the TS map during offpulse intervals. In order to determine whether this tail structure is associated with 2FGL J1906.5$+$0720 or caused by the two nearby sources, we further performed maximum likelihood analysis by including the two sources in the source model. The emission of the two putative sources were modeled by a simple power law. We found that the tail structure was completely removed (see the left panel of Figure \[fig:notail\]), indicating that it is likely caused by the two nearby sources. A $\gamma$-ray spectrum of 2FGL J1906.5$+$0720 was obtained again for the offpulse phase interval data, with the two nearby sources considered. The three spectral models given in Section \[subsec:likeli\] were used. The results are given in Table \[tab:likeli\]. The spectral parameter values are similar to those obtained above. We also fit the obtained spectral data points below 2 GeV with an exponentially cutoff power law (cf. Section \[subsec:sa\]). Nearly the same results were obtained (Table \[tab:likeli\]). These analyses confirm the existence of a high-energy component in the emission of 2FGL J1906.5$+$0720, which is likely not to be caused by contamination from the nearby sources. By constructing TS maps with photons greater than 2 or 5 GeV, we searched for extended emission (e.g., a pulsar wind nebula) at the position of 2FGL J1906.5$+$0720. However, the source profile was always consistent with being a point source. There was no indication for the presence of an additional source responsible for the high-energy component. Timing analysis of 2FGL J1906.5$+$0720 -------------------------------------- Timing analysis was performed to the LAT data of 2FGL J1906.5$+$0720 to search for $\gamma$-ray pulsation signals. We included events in the energy range from 50 MeV to 300 GeV within 1 degree centered at the position of 2FGL J1906.5$+$0720, which is R.A.= 286647, Decl.= 734256, equinox J2000.0 (the catalog position; [@nol2012]). The time period for the event selection was 300-day from 2012-09-26 20:53:17 to 2013-07-23 20:53:17 (UTC). The time-differencing blind search technique described in @atw2006 was applied. The range of frequency derivative $\dot{\nu}$ over frequency $\nu$ we considered was $|\dot{\nu}/\nu |=0$–$1.3\times 10^{-11}$ $s^{-1}$, which is characteristic of pulsars such as the Crab pulsar. A step of $2.332\times 10^{-15}$ $s^{-1}$ was used in the search. The frequency range we considered was from 0.5 Hz to 32 Hz with a Fourier resolution of $1.90735\times 10^{-6}$ Hz. We did not include the parameter ranges characteristic of millisecond pulsars. The source 2FGL J1906.5$+$0720 is located in the Galactic plane and would be possibly a young pulsar such as PSR J1907$+$0602. No significant $\gamma$-ray pulsations from 2FGL J1906.5$+$0720 were detected. We also applied the blind search to the *Fermi*/LAT data of 2FGL J1906.5$+$0720 only during the offpulse phase intervals of PSR J1907$+$0602. No $\gamma$-ray pulsations except the spin frequency signal of PSR J1907$+$0602 were found. In addition, we also searched for any long-period modulations from the source, the detection of which would be indicative of a binary system (see discussion in Section \[sec:dis\]). We constructed power spectra during offpulse phase intervals of PSR J1907$+$0602 in the three energy bands of 0.2–1 GeV, 1–300 GeV, and 5–300 GeV. Light curves of nearly five-year length in the three energy bands were extracted from performing *Fermi*/LAT aperture photometry analysis. The aperture radius was 1 degree, and the time resolution of the light curves was 1000 seconds. The exposures were calculated assuming power law spectra with photon indices obtained by maximum likelihood analysis (Table \[tab:likeli\]), which were used to determine the flux in each time bin. No long-period modulations in the energy bands were found. Discussion {#sec:dis} ========== By carrying out phase-connected timing analysis of the nearly 5-year [*Fermi*]{} $\gamma$-ray data of PSR J1907$+$0602, we have obtained the timing parameters and updated the $\gamma$-ray ephemeris for this pulsar. The obtained timing parameters are similar to those given in the second [*Fermi*]{}  catalog of $\gamma$-ray pulsars [@lat2013]. However the glitch decay time constant is $\sim$99 days, larger than $\sim$33 days given in the catalog. This difference is likely due to the longer time span of the data we analyzed (2-year data was analyzed in the second [*Fermi*]{}  catalog of $\gamma$-ray pulsars; [@lat2013]) and the unstable timing parameters caused by the timing noise. PSR J1907$+$0602 is quite young with a characteristic age of $\sim$19.5 kyr [@abd2010]. The post-fit rms timing residual was 2.1 ms, resulting from our timing analysis (Table \[tab:timing\]). We performed different analyses of the [*Fermi*]{}/LAT data for the unassociated source 2FGL J1906.5$+$0720. Through likelihood analysis with different spectral models, we confirmed that a curved spectrum with a low-energy break or cutoff at $\sim$1 GeV is clearly preferred to a simple power law. The significances of the curvature ($\sim\sqrt{TS}\sigma$) are approximately 14–16 $\sigma$ and 16–18 $\sigma$ for the full data and the offpulse phase interval data, respectively. This feature is characteristic of $\gamma$-ray pulsars detected by *Fermi*. On the basis of the [*Fermi*]{} second pulsar catalog, young $\gamma$-ray pulsars have 0.6 $<\Gamma<$2 and 0.4 GeV $< E_{cutoff} <$ 5.9 GeV, and millisecond $\gamma$-ray pulsars have 0.4 $<\Gamma <$ 2 and 1.1 GeV $< E_{cutoff} <$ 5.4 GeV ([@lat2013]). If 2FGL J1906.5$+$0720 is a pulsar, its Galactic location and spectral feature suggest that it is probably a young pulsar (see, e.g., [@lat2013]). It should be noted that a log parabola, which better fits the spectra of 2FGL J1906.5$+$0720, is usually used to model the spectra of $\gamma$-ray binaries [@nol2012]. However, considering the non-detection of any long-period modulations and the low variability of 2FGL J1906.5$+$0720, a $\gamma$-ray binary is not likely the case for the source. From our spectral analysis, a high-energy component was found to exist at $\geq$4 GeV in the emission of 2FGL J1906.5$+$0720. Considering it as a young pulsar, the component likely originates from its pulsar wind nebula (PWN; e.g., [@gs06]). A pulsar wind generates a termination shock by the interaction of high-energy particles contained in it with the ambient medium, at which particles are re-distributed and can radiate ultra-relativistic emission. In the $\gamma$-ray energy range, the [*Fermi*]{} second source catalog used 69 known PWNe for the automatic source association, and found that nearly all of them (except three) are associated with young pulsars [@nol2012]. However since $\gamma$-ray emission from a pulsar often dominates over that from its PWN, the number of PWNe that have been confirmedly detected by [*Fermi*]{} is limited [@ack2011]. For 2FGL J1906.5$+$0720, our spatial distribution analysis has confirmed the existence of the high-energy component in its spectrum, but the putative PWN would be too small or too faint to be resolved by [*Fermi*]{}. Further X-ray imaging of the source field is needed in order to detect the PWN and thus help verify the pulsar nature for 2FGL J1906.5$+$0720. We have not been able to find any pulsed emission signals from the [*Fermi*]{} data of 2FGL J1906.5$+$0720, which is required to verify the source’s pulsar nature. We note that the LAT blind search sensitivity depends on many parameters, such as the accurate position of the source, the source region used for pulsation search, contamination from background diffuse emission and from nearby sources (given that our target is located at the Galactic plane with several identifiable sources nearby). Using the sensitivity estimation method for the blind searches provided by @dor2011, the pulsed fraction of 2FGL J1906.5$+$0720 should be $\gtrsim$0.57 for a detection probability of $>$68% (the 1-year detection significance is $\sim$20$\sigma$ for the source). @dor2011 also extracted an all-sky detectability flux map to describe the minimum 0.3 – 20 GeV photon flux required for the detection of pulsars with pulsed fractions. In the inner Galactic plane the detectability flux should be higher than $\sim$10$^{-7}$ ph cm$^{-2}$ s$^{-1}$. The 0.3 – 20 GeV photon flux we obtained for 2FGL J1906.5$+$0720 is $<$10$^{-7.1}$ ph cm$^{-2}$ s$^{-1}$ (derived from spectral parameters listed in Table \[tab:likeli\]), suggesting the difficulty of detecting pulsed emission from the source through blind searches. Considering the radio pulsations from the source have been searched several times but with no detection [@ray2012], in order to verify its pulsar nature, X-ray observations are needed. We thank the referee for valuable suggestions. This research was supported by Shanghai Natural Science Foundation for Youth (13ZR1464400), National Natural Science Foundation of China (11373055), and the Strategic Priority Research Program “The Emergence of Cosmological Structures" of the Chinese Academy of Sciences (Grant No. XDB09000000). ZW is a Research Fellow of the One-Hundred-Talents project of Chinese Academy of Sciences. Source Signif\_Curve ($\sigma$) $Gb$ () Variability\_Index Signif\_Avg ($\sigma$) ---------------------- -------------------------- ---------- -------------------- ------------------------ -- -- -- 2FGL J1704.9$-$4618 10.0 $-$3.111 21.3 9.3 2FGL J1906.5$+$0720 9.8 $-$0.002 30.9 24.0 2FGL J1819.3$-$1523 9.2 $-$0.072 30.0 19.3 2FGL J1847.2$-$0236 8.5 $-$0.257 31.3 13.8 2FGL J1856.2$+$0450c 8.4 1.139 18.8 12.3 2FGL J1619.0$-$4650 8.3 2.457 22.2 10.6 2FGL J2033.6$+$3927 8.3 $-$0.382 33.8 13.0 2FGL J1045.0$-$5941 8.3 $-$0.639 21.5 36.1 2FGL J0858.3$-$4333 8.1 1.428 16.5 14.1 2FGL J1739.6$-$2726 8.1 1.906 27.6 15.2 : The first 10 candidate pulsars ranked by Signif\_Curve[]{data-label="tab:candi"} Parameter Value ----------------------------------------------------- ----------------------------------- R.A., $\alpha$ (J2000.0) 19:07:54.7343205 Decl., $\delta$ (J2000.0) 06:02:16.97850 Pulse frequency (s$^{-1}$) 9.3776609432(10) Frequency first derivative (s$^{-2}$) $-$7.62737(7) $\times$ 10$^{-12}$ Frequency second derivative (s$^{-3}$) 1.95(2) $\times$ 10$^{-22}$ Epoch of frequency (MJD) 55422.275976 Dispersion measure (cm$^{-3}$ pc) 82.1 1st glitch epoch (MJD) 55422.155 1st glitch permanent frequency increment (s$^{-1}$) 4.3466(3) $\times$ 10$^{-5}$ 1st glitch frequency deriv increment (s$^{-2}$) $-$7.72(2) $\times$ 10$^{-14}$ 1st glitch frequency increment (s$^{-1}$) 2.10(5) $\times$ 10$^{-7}$ 1st glitch decay time (Days) 99(4) rms timing residual (ms) 2.1 Time system TDB E$_{min}$ 50 MeV Valid range (MJD) 54683–56497 : $\gamma$-ray ephemeris for PSR J1907$+$0602.[]{data-label="tab:timing"} Parameters with no uncertainty reported are fixed to the values given in the second *Fermi* catalog of $\gamma$-ray pulsar [@lat2013] except the glitch epoch. --------------------- -------------------------------------------------- ----------------- ------------------------------ ------------------------------ -- -- -- Spectral model Parameters Full data Offpulse phase interval data Offpulse phase interval data (‘tail’ removed) PowerLaw $\Gamma$ 2.31 $\pm$ 0.02 2.42 $\pm$ 0.02 2.31 $\pm$ 0.02 $G_{\gamma}$ (10$^{-11}$ erg cm$^{-2}$ s$^{-1}$) 15 $\pm$ 0.5 13 $\pm$ 0.4 11 $\pm$ 0.4 TS$_{PL}$ 1101 2437 1595 LogParabola $\alpha$ 2.52 $\pm$ 0.05 2.82 $\pm$ 0.05 2.73 $\pm$ 0.07 $\beta$ 0.35 $\pm$ 0.03 0.37 $\pm$ 0.03 0.51 $\pm$ 0.04 $E_{b}$ (GeV) 1 1 1 $G_{\gamma}$ (10$^{-11}$ erg cm$^{-2}$ s$^{-1}$) 13 $\pm$ 0.5 12 $\pm$ 0.4 9 $\pm$ 0.3 TS$_{LP}$ 1388 2795 1966 PLSuperExpCutoff $\Gamma$ 1.7 $\pm$ 0.2 1.7 $\pm$ 0.1 1.2 $\pm$ 0.1 $E_{cut}$ (GeV) 1.7 $\pm$ 0.8 1.2 $\pm$ 0.2 0.8 $\pm$ 0.1 $G_{\gamma}$ (10$^{-11}$ erg cm$^{-2}$ s$^{-1}$) 13 $\pm$ 5 12 $\pm$ 2 9 $\pm$ 1 TS$_{PL+cutoff}$ 1297 2694 1853 PLSuperExpCutoff $\Gamma$ 1.4 $\pm$ 0.3 1.4 $\pm$ 0.1 1.4 $\pm$ 0.2 obtained by fitting $E_{cut}$ (GeV) 1.0 $\pm$ 0.3 0.5 $\pm$ 0.1 0.6 $\pm$ 0.1 --------------------- -------------------------------------------------- ----------------- ------------------------------ ------------------------------ -- -- -- : Maximum binned likelihood results for 2FGL J1906.5$+$0720[]{data-label="tab:likeli"} The break energies are fixed at 1 GeV. ![TS maps (0.1–300 GeV) of $\mathrm{5^{o}\times5^{o}}$ regions centered at R.A.= 286.647$\mathrm{^{o}}$, Decl.= 6.6$\mathrm{^{o}}$ (equinox J2000.0) extracted from the full data ([*left*]{} panel) and offpulse phase interval data ([*right*]{} panel) of PSR J1907$+$0602. The image scales of the maps are 01 pixel$^{-1}$. Two putative nearby sources are marked by circles.[]{data-label="fig:ts"}](source-1.eps "fig:"){width="3in"} ![TS maps (0.1–300 GeV) of $\mathrm{5^{o}\times5^{o}}$ regions centered at R.A.= 286.647$\mathrm{^{o}}$, Decl.= 6.6$\mathrm{^{o}}$ (equinox J2000.0) extracted from the full data ([*left*]{} panel) and offpulse phase interval data ([*right*]{} panel) of PSR J1907$+$0602. The image scales of the maps are 01 pixel$^{-1}$. Two putative nearby sources are marked by circles.[]{data-label="fig:ts"}](offpulse-source-1.eps "fig:"){width="3in"} ![Post-fit timing residuals for PSR J1907$+$0602.[]{data-label="fig:rms"}](postfit_rms.eps){width="5in"} ![Folded pulse profile and two-dimensional phaseogram in 32 phase bins obtained for PSR J1907$+$0602. For clarity, two rotations are shown on $X$-axis. The gray scale represents the number of photons in each bin.[]{data-label="fig:ftp"}](profile.ps){width="5in"} ![$\gamma$-ray spectra of 2FGL J1906.5$+$0720 extracted from the full data ([*left*]{} panel) and offpulse phase interval data ([*right*]{} panel). The exponentially cutoff power laws and the log parabolas obtained from maximum likelihood analysis (see Table \[tab:likeli\]) are displayed as dashed and dotted curves. The solid curves are the exponentially cutoff power laws obtained by fitting the data points below 2 GeV.[]{data-label="fig:spec"}](n0-1906-20.eps "fig:"){width="3in"} ![$\gamma$-ray spectra of 2FGL J1906.5$+$0720 extracted from the full data ([*left*]{} panel) and offpulse phase interval data ([*right*]{} panel). The exponentially cutoff power laws and the log parabolas obtained from maximum likelihood analysis (see Table \[tab:likeli\]) are displayed as dashed and dotted curves. The solid curves are the exponentially cutoff power laws obtained by fitting the data points below 2 GeV.[]{data-label="fig:spec"}](n0-1906-offpulse-20.eps "fig:"){width="3in"} ![[*Left panel*]{}: TS map (0.1–300 GeV) of $\mathrm{5^{o}\times5^{o}}$ region centered at R.A.= 286.647$\mathrm{^{o}}$, Decl.= 6.6$\mathrm{^{o}}$ (equinox J2000.0) extracted from the offpulse phase interval data of PSR J1907$+$0602 when the two nearby sources (the positions are marked by circles) were removed. The image scale of the map is 01 pixel$^{-1}$. [*Right panel*]{}: $\gamma$-ray spectrum of 2FGL J1906.5$+$0720 extracted from the offpulse phase interval data when the two nearby sources were removed. The dashed and dotted curves represent the exponentially cutoff power law and the log parabola, respectively, obtained from maximum likelihood analysis (see Table \[tab:likeli\]). The solid curve represents the exponentially cutoff power law obtained by fitting the data points below 2 GeV.[]{data-label="fig:notail"}](offpulse-1906-removesource-1.eps "fig:"){width="3in"} ![[*Left panel*]{}: TS map (0.1–300 GeV) of $\mathrm{5^{o}\times5^{o}}$ region centered at R.A.= 286.647$\mathrm{^{o}}$, Decl.= 6.6$\mathrm{^{o}}$ (equinox J2000.0) extracted from the offpulse phase interval data of PSR J1907$+$0602 when the two nearby sources (the positions are marked by circles) were removed. The image scale of the map is 01 pixel$^{-1}$. [*Right panel*]{}: $\gamma$-ray spectrum of 2FGL J1906.5$+$0720 extracted from the offpulse phase interval data when the two nearby sources were removed. The dashed and dotted curves represent the exponentially cutoff power law and the log parabola, respectively, obtained from maximum likelihood analysis (see Table \[tab:likeli\]). 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--- abstract: 'This paper provides a sample of a LaTeX document which conforms, somewhat loosely, to the formatting guidelines for ACM SIG Proceedings.[^1]' author: - Ben Trovato - 'G.K.M. Tobin' - 'Lars Th[ø]{}rv[ä]{}ld' - 'Lawrence P. Leipuner' - Sean Fogarty - Charles Palmer - John Smith - 'Julius P. Kumquat' bibliography: - 'sample-bibliography.bib' subtitle: Extended Abstract title: SIG Proceedings Paper in LaTeX Format --- &lt;ccs2012&gt; &lt;concept&gt; &lt;concept\_id&gt;10010520.10010553.10010562&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computer systems organization Embedded systems&lt;/concept\_desc&gt; &lt;concept\_significance&gt;500&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10010520.10010575.10010755&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computer systems organization Redundancy&lt;/concept\_desc&gt; &lt;concept\_significance&gt;300&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10010520.10010553.10010554&lt;/concept\_id&gt; &lt;concept\_desc&gt;Computer systems organization Robotics&lt;/concept\_desc&gt; &lt;concept\_significance&gt;100&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;concept&gt; &lt;concept\_id&gt;10003033.10003083.10003095&lt;/concept\_id&gt; &lt;concept\_desc&gt;Networks Network reliability&lt;/concept\_desc&gt; &lt;concept\_significance&gt;100&lt;/concept\_significance&gt; &lt;/concept&gt; &lt;/ccs2012&gt; [^1]: This is an abstract footnote

--- abstract: 'An approach to the quantum-classical mechanics of phase space dependent operators, which has been proposed recently, is remodeled as a formalism for wave fields. Such wave fields obey a system of coupled non-linear equations that can be written by means of a suitable non-Hamiltonian bracket. As an example, the theory is applied to the relaxation dynamics of the spin-boson model. In the adiabatic limit, a good agreement with calculations performed by the operator approach is obtained. Moreover, the theory proposed in this paper can take nonadiabatic effects into account without resorting to surface-hopping approximations. Hence, the results obtained follow qualitatively those of previous surface-hopping calculations and increase by a factor of (at least) two the time length over which nonadiabatic dynamics can be propagated with small statistical errors. Moreover, it is worth to note that the dynamics of quantum-classical wave fields here proposed is a straightforward non-Hamiltonian generalization of the formalism for non-linear quantum mechanics that Weinberg introduced recently.' author: - 'Alessandro Sergi [^1]' title: 'Quantum-Classical Dynamics of Wave Fields' --- Introduction ============ There are many instances where a quantum-classical description can be a useful approximation to full quantum dynamics. Typically, a quantum-classical picture often allows one to implement calculable algorithms on computers whenever charge transfer is considered within complex environments, such as those provided by proteins or nano-systems in general [@ksreview]. With respect to this, an algebraic approach has been recently proposed [@qc-bracket; @kcmqc] in order to formulate the dynamics and the statistical mechanics [@qc-stat] of quantum-classical systems. General questions regarding the quantum-classical correspondence have also been addressed within a similar framework [@brumer]. The approach of Refs. [@qc-bracket; @kcmqc] represents quantum-classical dynamics by means of suitable brackets of phase space dependent operators and describes consistently the back-reaction between quantum and classical degrees of freedom. Notably, a particular implementation of this formalism has been used to calculate nonadiabatic rate constants in systems modeling chemical reactions in the condensed phase [@kapral]. However, such schemes have only permitted the simulation of short-time nonadiabatic dynamics because of the time-growing statistical error of the algorithm. Nevertheless, the algebraic approach [@qc-bracket; @kcmqc], underlying the algorithms of Refs. [@kapral], has some very nice features, such as the (above mentioned) proper description of the back-reaction between degrees of freedom, that one should not give up when addressing quantum-classical statistical mechanics. Moreover, quantum-classical brackets define a non-Hamiltonian algebra [@b3] so that their matrix structure allows one to introduce quantum-classical Nosé-Hoover dynamics [@b3] and to define the statistical mechanics of quantum-classical systems with holonomic constraints [@bsilurante]. All of the above features of the formalism are highly desirable when studying complex systems in condensed phases. Therefore, it is worth to search for a reformulation of the theory of Refs. [@kapral; @b3; @bsilurante] that, while maintaining such features, could be used to integrate reliably long-time nonadiabatic dynamics. To this end, one can note that, within standard quantum mechanics, some problems that are formidable to solve by means of the dynamics of operators become much simpler to handle when, instead, the time evolution of wave functions is considered [@ballentine]. Hence, for analogy, it might also happen that, within quantum-classical mechanics, the correspondence between operators and quantum-classical wave functions could open new possibilities for useful approximations in order to carry long-time calculations efficiently. Indeed, finding and applying the correspondence between operator and wave scheme of motion in quantum-classical mechanics is the scope of the present paper. A wave picture for quantum-classical dynamics can be found by direct algebraic manipulation of the equation of motion for the density matrix. In practice, the single equation obeyed by the quantum-classical density matrix is mapped onto two coupled non-linear equations for quantum-classical wave fields. Despite its non-linear character, such a quantum-classical dynamics of phase space dependent wave fields corresponds exactly to the dynamics of phase space dependent operators discussed in Refs. [@qc-bracket; @kcmqc; @kapral; @b3; @bsilurante] and can be used to devise novel algorithms and approximation schemes. The abstract algebraic equations here presented are readily expressed in the adiabatic basis and applied, in order to provide an illustrative example, to the spin-boson model and its relaxation dynamics both in the adiabatic and nonadiabatic limit. By making a suitable equilibrium approximation to the non-linear wave equations, it is found that nonadiabatic dynamics can be propagated, within the wave picture, for time intervals that are a factor of two-three longer than those which have been spanned in Ref. [@qc-sb] by means of the operator theory [@qc-bracket; @kcmqc; @kapral; @b3; @bsilurante]. Such a result is very encouraging for pursuing the long-time integration of the nonadiabatic dynamics of complex systems in condensed phases. Following a line of research that investigates the relations between classical and quantum theories [@qgen], it is worth to note that the wave picture of quantum-classical mechanics, which is introduced in this paper, generalizes within a non-Hamiltonian framework the elegant formalism that Weinberg [@weinberg] proposed for describing possible non-linear effects in quantum mechanics [@nonlinear]. This paper is organized as follows. In Section \[sec:bracket\] the non-Hamiltonian algebra of phase space dependent operators is briefly summarized. In Section \[sec:qcwd\] the quantum-classical dynamics of operators is transformed into a theory for phase space dependent wave fields evolving in time. Such a theory for wave fields is also expressed by means of suitable non-Hamiltonian brackets: in this way a link is found with the generalization of Weinberg’s non-linear formalism given in Appendix \[app:weinberg\]. More specifically, in Appendix \[app:weinberg\], Weinberg’s formalism is briefly reviewed and its symplectic structure is unveiled. Then, this structure is generalized by means of non-Hamiltonian brackets. Therefore, one can appreciate how the generalized Weinberg’s formalism establishes a more comprehensive mathematical framework for non-linear equations of motion, comprising phase space dependent wave fields as a special case. In Section \[sec:qcwdab\] the abstract non-linear equations of motion for quantum-classical fields are represented in the adiabatic basis and some considerations, which pertain to the numerical implementation, are made. By making an equilibrium *ansatz*, in Section \[sec:sb\] the non-linear equations of motion are put into a linear form and the theory is applied to the spin-boson model. Section \[sec:conclusions\] is devoted to conclusions and perspectives. Non-Hamiltonian Mechanics of Quantum-Classical Operators {#sec:bracket} ======================================================== A quantum-classical system is composed of both quantum $\hat{\chi}$ and classical $X$ degrees of freedom, where $X=(R,P)$ is the phase space point, with $R$ and $P$ coordinates and momenta, respectively. Within the operator formalism of Refs. [@qc-bracket; @kcmqc; @b3; @bsilurante], the quantum variables depend from the classical point, $X$, of phase space. The energy of the system is defined in terms of a Hamiltonian operator $\hat{H}=\hat{H}(X)$, which couples quantum and classical variables, by $E={\rm Tr}'\int dX \hat{H}(X)$. The dynamical evolution of a quantum-classical operator $\hat{\chi}(X)$ is given by [@qc-bracket; @kcmqc] $$\begin{aligned} \frac{d}{dt} \hat{\chi}(X,t)&=& \frac{i}{\hbar} \left[\hat{H},\hat{\chi}(X,t)\right]_{\mbox{\tiny\boldmath$\cal B$}} -\frac{1}{2}\left\{\hat{H},\hat{\chi}(X,t)\right\}_{\mbox{\tiny\boldmath$\cal B$}} \nonumber\\ &+&\frac{1}{2}\left\{\hat{\chi}(X,t),\hat{H}\right\}_{\mbox{\tiny\boldmath$\cal B$}} %\nonumber \\ =\left(\hat{H},\hat{\chi}(X,t)\right)\;, \label{eq:qcbracket}\end{aligned}$$ where $$\begin{aligned} \left[\hat{H} , \hat{\chi}\right]_{\mbox{\tiny\boldmath$\cal B$}} &=& \left[\begin{array}{cc} \hat{H} & \hat{\chi}\end{array}\right] \cdot\mbox{\boldmath$\cal B$}\cdot \left[\begin{array}{c} \hat{H} \\ \hat{\chi} \end{array} \right] \label{eq:qlm}\end{aligned}$$ is the commutator and $$\begin{aligned} \{\hat{H},\hat{\chi}\}_{\mbox{\tiny\boldmath$\cal B$}} &=& \sum_{i,j=1}^{2N} \frac{\partial \hat{H}}{\partial X_i}{\cal B}_{i j} \frac{\partial \hat{\chi}}{\partial X_j} \label{Lambda}\end{aligned}$$ is the Poisson bracket [@goldstein]. Both the commutator and the Poisson bracket are defined in terms of the antisymmetric matrix $$\mbox{\boldmath$\cal B$}=\left[\begin{array}{cc}0 & 1\\ -1 & 0\end{array}\right]\;. \label{B}$$ The last equality in Eq. (\[eq:qcbracket\]) defines the quantum-classical bracket. Following Refs. [@b3; @bsilurante; @sergi], the quantum-classical law of motion can be easily casted in matrix form as $$\begin{aligned} \frac{d}{dt} \hat{\chi}&=&\frac{i}{\hbar} \left[\begin{array}{cc} \hat{H} & \hat{\chi} \end{array}\right] \cdot\mbox{\boldmath$\cal D$}\cdot \left[\begin{array}{c} \hat{H} \\ \hat{\chi} \end{array}\right] \nonumber\\ &=&\frac{i}{\hbar}[\hat{H},\hat{\chi}]_{\mbox{\tiny\boldmath$\cal D$}}\;, \label{qclm}\end{aligned}$$ where $$\mbox{\boldmath$\cal D$}=\left[\begin{array}{cc} 0& 1-\frac{\hbar}{2i} \left\{\ldots,\ldots\right\}_{\mbox{\tiny\boldmath$\cal B$}} \\ -1+\frac{\hbar}{2i} \left\{\ldots,\ldots\right\}_{\mbox{\tiny\boldmath$\cal B$}} & 0\end{array}\right]\;. \label{D}$$ The structure of Eq. (\[qclm\]) is that of a non-Hamiltonian commutator, which will be defined below in Eq. (\[eq:gen-qlm\]), and as such generalizes the standard quantum law of motion [@b3]. The antisymmetric super-operator $\mbox{\boldmath$\cal D$}$ in Eq. (\[D\]) introduces a novel mathematical structure that characterizes the time evolution of quantum-classical systems. The Jacobi relation in quantum-classical dynamics is $${\cal J}=\left[\hat{\chi}, \left[\hat{\xi},\hat{\eta}\right]_{\mbox{\tiny\boldmath$\cal D$}} \right]_{\mbox{\tiny\boldmath$\cal D$}} +\left[\hat{\eta},\left[\hat{\chi},\hat{\xi} \right]_{\mbox{\tiny\boldmath$\cal D$}} \right]_{\mbox{\tiny\boldmath$\cal D$}} +\left[\hat{\xi},\left[\hat{\eta},\hat{\chi} \right]_{\mbox{\tiny\boldmath$\cal D$}} \right]_{\mbox{\tiny\boldmath$\cal D$}}. \label{qc-jacobi}$$ The explicit expression of $\cal J$ has been given in Ref. [@b3] where it was shown that it may be different from zero at least in some point $X$ of phase space: for this reason the quantum-classical theory of Refs. [@qc-bracket; @kcmqc; @b3; @bsilurante] can be classified as a non-Hamiltonian theory. It is worth to note that the quantum-classical law of motion in Eq. (\[qclm\]) is a particular example of a more general form of quantum mechanics where time evolution is defined by means of non-Hamiltonian commutators. The non-Hamiltonian commutator between two arbitrary operators $\hat{\chi}$ and $\hat{\xi}$ is defined by $$[\hat{\chi},\hat{\xi}]_{\mbox{\tiny\boldmath$\Omega$}}= \left[\begin{array}{cc}\hat{\chi} & \hat{\xi}\end{array}\right] \cdot\mbox{\boldmath$\Omega$}\cdot \left[\begin{array}{c}\hat{\chi} \\ \hat{\xi}\end{array}\right]\;, \label{eq:gen-quantum-algebra}$$ where $\mbox{\boldmath$\Omega$}$ is an antisymmetric matrix operator of the form $$\mbox{\boldmath$\Omega$}= \left[\begin{array}{cc}0 & f[\hat{\eta}]\\ -f[\hat{\eta}] & 0\end{array}\right]\;,$$ where $f[\hat{\eta}]$ can be another arbitrary operator or functional of operators. Then, generalized equations of motion can be defined as $$\begin{aligned} \frac{d\hat{\chi}}{dt}&=&\frac{i}{\hbar} \left[\begin{array}{cc} \hat{H} & \hat{\chi} \end{array}\right] \cdot\mbox{\boldmath$\Omega$}\cdot \left[\begin{array}{c} \hat{H} \\ \hat{\chi}\end{array}\right] \nonumber\\ &=&\frac{i}{\hbar} [\hat{H} , \hat{\chi}]_{\mbox{\tiny\boldmath$\Omega$}} \;. \label{eq:gen-qlm}\end{aligned}$$ The non-Hamiltonian commutator of Eq. (\[eq:gen-quantum-algebra\]) defines a generalized form of quantum mechanics where, nevertheless, the Hamiltonian operator $\hat{H}$ is still a constant of motion because of the antisymmetry of $\mbox{\boldmath$\Omega$}$. Quantum-classical wave dynamics {#sec:qcwd} =============================== In Refs. [@qc-bracket; @kcmqc], quantum-classical evolution has been formulated in terms of phase space dependent operators. In this scheme of motion operators evolve according to $$\begin{aligned} \hat{\chi}(X,t)&=&\exp\left\{t \left[\hat{H},\ldots\right]_{\mbox{\tiny\boldmath$\cal D$}}\right\} \hat{\chi}(X)\nonumber\\ &=&\exp\left\{it{\mathcal L}\right\}\hat{\chi}(X)\;, \label{eq:qc-heisenberg}\end{aligned}$$ where the last equality defines the quantum-classical Liouville propagator. Quantum-classical averages are calculated as $$\begin{aligned} \langle\hat{\chi}\rangle(t) &=&{\rm Tr}'\int dX\hat{\rho}(X)\hat{\chi}(X,t) \nonumber\\ &=&{\rm Tr}'\int dX\hat{\rho}(X,t)\hat{\chi}(X)\;, \label{eq:qc-average}\end{aligned}$$ where $\hat{\rho}(X)$ is the quantum-classical density matrix and $\hat{\rho}(X,t)=\exp\left\{-it{\mathcal L}\right\}\hat{\rho}(X)$. Either evolving the dynamical variables or the density matrix, one is still dealing with phase space dependent operators: *viz.*, one deals with a form of generalized quantum-classical matrix mechanics. As it has been discussed in the Introduction, this theory has interesting formal features and a certain number of numerical schemes have been proposed to integrate the dynamics and calculate correlation functions [@kapral; @qc-sb; @num-qc]. However, these algorithms have been applied with success only to short-time dynamics because of statistical uncertainties that grow with time beyond numerical tolerance. With this in mind, it is interesting to see which features are found when the quantum-classical theory of Refs. [@qc-bracket; @kcmqc] is mapped onto a scheme of motion where phase space dependent wave fields, instead of operators, are used to represent the dynamics. As it is well known [@ballentine], in standard quantum mechanics, the correspondence between dynamics in the Heisenberg and in the Schrödinger picture rests ultimately on the following operator identity: $$e^{\hat{Y}}\hat{X}e^{-\hat{Y}}=e^{[\hat{Y},\ldots]}\hat{X}\;, \label{eq:expid}$$ where $[\hat{Y},\ldots]\hat{X}\equiv [\hat{Y},\hat{X}]$. Thus, in quantum-classical theory, one would like to derive an operator identity analogous to that in Eq. (\[eq:expid\]). However, as already shown in Ref. [@qc-stat], because of the non associativity of the quantum-classical bracket in Eq. (\[qclm\]), the identity that can be derived is $$\begin{aligned} e^{\frac{it}{\hbar}~\left[\hat{H}, \ldots\right]_{\mbox{\tiny\boldmath$\cal D$}}} \hat{\chi}&=& {\cal S}\left( e^{\frac{it}{\hbar}\overrightarrow{\mathcal H}}\hat{\chi} e^{-\frac{it}{\hbar}\overleftarrow{\mathcal H}}\right)\;, \label{eq:qc-ope-ide}\end{aligned}$$ where the two operators $$\begin{aligned} \overrightarrow{\mathcal H}&=&\hat{H} -\frac{\hbar}{2i} \left\{\hat{H},\ldots\right\}_{\mbox{\tiny\boldmath$\cal B$}} \label{eq:hright}\\ \overleftarrow{\mathcal H}&=&\hat{H} -\frac{\hbar}{2i} \left\{\ldots,\hat{H}\right\}_{\mbox{\tiny\boldmath$\cal B$}} \label{eq:hleft}\end{aligned}$$ have been introduced and $\cal S$ is an ordering operator which is chosen so that the left and the right hand side of Eq. (\[eq:qc-ope-ide\]) coincide by construction [@qc-stat], when the exponential operators are substituted with their series expansion. The existence of such an ordering problem, and of the ordering operator $\cal S$, in Eq. (\[eq:qc-ope-ide\]) is caused by the Poisson bracket parts of the operators in Eqs. (\[eq:hright\]) and (\[eq:hleft\]). Hence, one can imagine that the solution to this problem can be found by dealing properly with these parts of the brackets. To this end, one can consider the quantum-classical equation of motion for the density matrix $$\begin{aligned} \frac{\partial\hat{\rho}}{\partial t}&=& -\frac{i}{\hbar} \left[\begin{array}{cc}\hat{H} & \hat{\rho}\end{array}\right] \nonumber\\ &\cdot& \left[\begin{array}{cc} 0 & 1-\frac{\hbar}{2i} \left\{\ldots,\ldots\right\}_{\mbox{\tiny\boldmath$\cal B$}} % \\ -1+\frac{\hbar}{2i} \left\{\ldots,\ldots\right\}_{\mbox{\tiny\boldmath$\cal B$}} & 0\end{array}\right] \cdot \left[\begin{array}{c}\hat{H}\\\hat{\rho}\end{array}\right]\;. \nonumber \\ \label{eq:rhoW}\end{aligned}$$ As above discussed, in Eq. (\[eq:rhoW\]) the ordering problem arises from the terms in the right hand side containing the Poisson bracket operator $\left\{\ldots,\ldots\right\}_{\mbox{\tiny\boldmath$\cal B$}}$. Then, considering the identity $1=\hat{\rho}\cdot\hat{\rho}^{-1}=\hat{\rho}^{-1}\cdot\hat{\rho}$, Eq. (\[eq:rhoW\]) can be rewritten as $$\begin{aligned} \partial_t \hat{\rho} &=& -\frac{i}{\hbar} \left[\begin{array}{cc}\hat{H} & \hat{1}\end{array}\right] \nonumber\\ &\cdot& \left[\begin{array}{cc} 0 & 1-\frac{\hbar}{2i} \left\{\ldots,\hat{\rho}\right\}_{\mbox{\tiny\boldmath$\cal B$}} \\ -1+\frac{\hbar}{2i} \left\{\hat{\rho},\ldots\right\}_{\mbox{\tiny\boldmath$\cal B$}} & 0\end{array}\right] \cdot \left[\begin{array}{c}\hat{H}\\\hat{1}\end{array}\right] \nonumber \\ &=& -\frac{i}{\hbar} \left[\begin{array}{cc}\hat{H} & \hat{\rho}\hat{\rho}^{-1}\end{array}\right] \nonumber\\ &\cdot& \left[\begin{array}{cc} 0 & 1-\frac{\hbar}{2i} \left\{\ldots,\hat{\rho}\right\}_{\mbox{\tiny\boldmath$\cal B$}} \\ -1+\frac{\hbar}{2i} \left\{\hat{\rho},\ldots\right\}_{\mbox{\tiny\boldmath$\cal B$}} & 0\end{array}\right] \cdot \left[\begin{array}{c}\hat{H}\\\hat{\rho}^{-1}\hat{\rho}\end{array}\right] \nonumber \\ %%%% &=& -\frac{i}{\hbar} \left[\begin{array}{cc}\hat{H} & \hat{\rho}\end{array}\right] \cdot \mbox{\boldmath$\cal D$}_{\mbox{\tiny\boldmath$\cal B$},[\hat{\rho}]} \cdot \left[\begin{array}{c}\hat{H}\\ \hat{\rho}\end{array}\right]\;, \label{eq:rho2}\end{aligned}$$ where $$\begin{aligned} \begin{array}{l} \mbox{\boldmath$\cal D$}_{\mbox{\tiny\boldmath$\cal B$},[\hat{\rho}]} =\\ \left[\begin{array}{cc} 0 & 1-\frac{\hbar}{2i} \left\{\ldots,\ln(\hat{\rho})\right\}_{\mbox{\tiny\boldmath$\cal B$}} \\ -1+\frac{\hbar}{2i} \left\{\ln(\hat{\rho}),\ldots\right\}_{\mbox{\tiny\boldmath$\cal B$}} & 0\end{array}\right] \end{array}\nonumber\\ \label{eq:Drho}\end{aligned}$$ The operator $\mbox{\boldmath$\cal D$}_{\mbox{\tiny\boldmath$\cal B$},[\hat{\rho}]}$ in Eq. (\[eq:Drho\]) depends from the quantum-classical density matrix, $\hat{\rho}$, itself. However, if one momentarily disregards this non-linear dependence, Eq. (\[eq:rho2\]) can be manipulated algebraically in order to develop a wave picture of quantum-classical mechanics. To this end, one can introduce quantum-classical wave fields, $|\psi(X)\rangle$ and $\langle\psi(X)|$, and make the following [*ansatz*]{} for the density matrix $$\begin{aligned} \hat{\rho}(X)&=&\sum_{\iota}w_{\iota}|\psi^{\iota}(X)\rangle\langle\psi^{\iota}(X)| \;,\label{eq:rho-ansatz}\end{aligned}$$ where one has assumed that, because of thermal disorder, there can be many microscopic states $|\psi^{\iota}(X)\rangle$ $({\iota}=1,\ldots,l)$ which correspond to the same value of the macroscopic relevant observables [@balescu]. In terms of the quantum-classical wave fields, $|\psi^{\iota}(X)\rangle$ and $\langle\psi^{\iota}(X)|$, and considering the single state labeled by $\iota$, Eq. (\[eq:rho2\]) becomes $$\begin{aligned} |\dot{\psi}^{\iota}(X)\rangle\langle\psi^{\iota}(X)| &+&|\psi^{\iota}(X)\rangle\langle\dot{\psi}^{\iota}(X)| =\nonumber\\ &-&\frac{i}{\hbar}\left(\hat{H}|\psi^{\iota}(X)\rangle\langle\psi^{\iota}(X)| \right.\nonumber\\ &+& \left.|\psi^{\iota}(X)\rangle\langle\psi^{\iota}(X)|\hat{H}\right) \nonumber\\ &+&\frac{1}{2} \left( \left\{\hat{H},\ln(\hat{\rho})\right\}_{\mbox{\tiny\boldmath$\cal B$}} |\psi^{\iota}(X)\rangle\langle\psi^{\iota}(X)|\right. \nonumber\\ &-&\left. |\psi^{\iota}(X)\rangle\langle\psi^{\iota}(X)| \left\{\ln(\hat{\rho}),\hat{H}\right\}_{\mbox{\tiny\boldmath$\cal B$}} \right)\;.\nonumber\\ \label{eq:wavematrix}\end{aligned}$$ Equation (\[eq:wavematrix\]) can be written as a system of two coupled equations for the wave fields [@fckrk]: $$\begin{aligned} i\hbar\frac{d}{dt}\vert\psi^{\iota}_{(X,t)}\rangle & =& \left(\hat{H}-\frac{\hbar}{2i} \left\{\hat{H},\ln(\hat{\rho}_{(X,t)})\right\}_{\mbox{\tiny\boldmath$\cal B$}} \right)\vert\psi^{\iota}_{(X,t)}\rangle\nonumber\\ %%%%%%%% -i\hbar\langle\psi^{\iota}_{(X,t)}\vert\overleftarrow{\frac{d}{dt}} &=& \langle\psi^{\iota}_{(X,t)}\vert\left(\hat{H} -\frac{\hbar}{2i} \left\{ \ln(\hat{\rho}_{(X,t)}),\hat{H}\right\}_{\mbox{\tiny\boldmath$\cal B$}} \right) \;.\nonumber\\ \label{eq:fckrk}\end{aligned}$$ Equations (\[eq:fckrk\]), which are obeyed by the wave fields, are non-linear since their solution depends self-consistently from the density matrix defined in Eq. (\[eq:rho-ansatz\]). These equations are also non-Hermitian since the operators $\left\{\hat{H},\ln(\hat{\rho})\right\}_{\mbox{\tiny\boldmath$\cal B$}}$ and $\left\{\ln(\hat{\rho}),\hat{H}\right\}_{\mbox{\tiny\boldmath$\cal B$}}$ are not Hermitian. However, this does not cause problems for the conservation of probability. The wave fields $\vert\psi^{\iota}\rangle$ and $\langle\psi^{\iota}\vert$ evolve according to the different propagators $$\begin{aligned} \overrightarrow{\cal U}_{{\mbox{\tiny\boldmath$\cal B$}},[\hat{\rho}]}(t) &=& \exp\left[-\frac{it}{\hbar}\left(\hat{H} -\frac{\hbar}{2i} \left\{\hat{H},\ln(\hat{\rho})\right\}_{\mbox{\tiny\boldmath$\cal B$}} \right)\right]\;,\nonumber\\ &&\\ \overleftarrow{\cal U}_{{\mbox{\tiny\boldmath$\cal B$}},[\hat{\rho}]}(t) &=& \exp\left[-\frac{it}{\hbar} \left(\hat{H} -\frac{\hbar}{2i} \left\{\ln(\hat{\rho}),\hat{H}\right\}_{\mbox{\tiny\boldmath$\cal B$}} \right)\right]\;,\nonumber\\\end{aligned}$$ so that time-propagating wave fields are defined by $$\begin{aligned} \vert\psi^{\iota}(X,t)\rangle&=& \overrightarrow{\cal U}_{{\mbox{\tiny\boldmath$\cal B$}},[\hat{\rho}]}(t) \vert\psi^{\iota}(X)\rangle \\ \langle\psi^{\iota}(X,t)\vert&=& \langle\psi^{\iota}(X,)\vert \overleftarrow{\cal U}_{{\mbox{\tiny\boldmath$\cal B$}},[\hat{\rho}]}(t) \;.\end{aligned}$$ Quantum classical averages can be written as $$\begin{aligned} \langle\hat{\chi}\rangle(t)&=&\int dX\sum_{\iota}w_{\iota} \langle\psi^{\iota}(X,t)| \hat{\chi} |\psi^{\iota}(X,t)\rangle \;. \label{eq:wave-ave}\end{aligned}$$ One can always transform back to the operator picture to show that the probability is conserved. Non-linear wave dynamics by means of non-Hamiltonian brackets ------------------------------------------------------------- The wave equations in (\[eq:fckrk\]) were derived starting from the non-Hamiltonian commutator expressing the dynamics of phase space dependent operators [@b3]. It is interesting to recast quantum-classical wave dynamics itself by means of non-Hamiltonian brackets. It turns out that this form of the wave equations generalizes the mathematical formalism first proposed by Weinberg [@weinberg] in order to study possible non-linear effects in quantum mechanics (see Appendix \[app:weinberg\]). Consider a case in which a single state is present, *i.e.* $\iota=1$. Then, consider the wave fields $\vert\psi\rangle$ and $\langle\psi\vert$ as coordinates of an abstract space, and denote the point of such a space as $$\mbox{\boldmath$\zeta$}=\left[\begin{array}{c}|\psi\rangle\\ \langle\psi|\end{array}\right]\;.$$ Introduce the function $${\cal H}=\langle\psi\vert\hat{H}\vert\psi\rangle\;,$$ and the antisymmetric matrix operator $$\mbox{\boldmath$\Omega$} = \left[\begin{array}{cc}0 & 1-\frac{\hbar}{2i}\frac{\left\{\hat{H},\ln(\hat{\rho})\right\}_{ \mbox{\tiny\boldmath$\cal B$}}\vert\psi\rangle}{\hat{H}\vert\psi\rangle} \\ -1+\frac{\hbar}{2i}\frac{\left\{\ln(\hat{\rho}),\hat{H}\right\}_{ \mbox{\tiny\boldmath$\cal B$}}\vert\psi\rangle}{\langle\psi\vert\hat{H}} & 0 \end{array}\right]$$ Equations (\[eq:fckrk\]) can be written in compact form as $$\begin{aligned} \frac{\partial\mbox{\boldmath$\zeta$}}{\partial t} &=&-\frac{i}{\hbar} \left[ \begin{array}{cc}\frac{\partial{\cal H}}{\partial\vert\psi\rangle} &\frac{\partial{\cal H}}{\partial\langle\psi\vert} \end{array} \right] \cdot\mbox{\boldmath$\Omega$}\cdot \left[\begin{array}{c}\frac{\partial\mbox{\boldmath$\zeta$}}{\partial\vert\psi\rangle} \\\frac{\partial\mbox{\boldmath$\zeta$}}{\partial\langle\psi\vert}\end{array}\right] \nonumber\\ &=&-\frac{i}{\hbar} \left\{{\cal H},\mbox{\boldmath$\zeta$}\right\}_{\mbox{\tiny\boldmath$\Omega$};\zeta}\;. \label{eq:wein-like}\end{aligned}$$ Equations (\[eq:fckrk\]), or their compact “Weinberg-like” form in Eq. (\[eq:wein-like\]), express the wave picture for the quantum-classical dynamics of phase space dependent quantum degrees of freedom [@qc-bracket; @kcmqc]. Such a wave picture makes one recognize the intrinsic non-linearity of quantum-classical dynamics. This specific features will be discussed, among other issues, in the next section. Adiabatic basis representation and surface-hopping schemes {#sec:qcwdab} ========================================================== Equations (\[eq:fckrk\]) are written in an abstract form. In order to devise a numerical algorithm to solve them, one has to obtain a representation in some basis. Of course, any basis can be used but, since one would like to find a comparison with surface-hopping schemes, the adiabatic basis is a good choice. To this end, consider the following form of the quantum-classical Hamiltonian operator: $$\hat{H}=\frac{P^2}{2M}+\hat{h}(R)\;,$$ where the first term provides the kinetic energy of the classical degrees of freedom with mass $M$, while $\hat{h}(R)$ describes the quantum sub-system and its coupling with the classical coordinates $R$. The adiabatic basis is then defined by the following eigenvalue equation: $$\hat{h}\vert\alpha;R\rangle=E_{\alpha}(R)\vert\alpha;R\rangle\;.$$ Since the non-linear wave equations in (\[eq:fckrk\]) have been derived from the bracket equation for the quantum-classical density matrix (\[eq:rhoW\]), by dealing in a suitable manner with the Poisson bracket terms, the most simple way to find the representation of the wave equations (\[eq:fckrk\]) in the adiabatic basis is to first represent Eq. (\[eq:rhoW\]) in such a basis and then deal with the terms arising from the Poisson brackets. The adiabatic representation of Eq. (\[eq:rhoW\]) is [@kcmqc] $$\begin{aligned} \partial_t \rho_{\alpha\alpha^{\prime}}(X,t) &=&-\sum_{\beta\beta^{\prime}} i{\cal L}_{\alpha\alpha^{\prime},\beta\beta^{\prime}} \rho_{\beta\beta^{\prime}}(X,t) \;,\end{aligned}$$ where $$\begin{aligned} i{\cal L}_{\alpha\alpha^{\prime},\beta\beta^{\prime}} &=& i{\cal L}_{\alpha\alpha^{\prime},\beta\beta^{\prime}}^{(0)} \delta_{\alpha\beta}\delta_{\alpha^{\prime}\beta^{\prime}} -J_{\alpha\alpha^{\prime},\beta\beta^{\prime}}\nonumber\\ &=&\left(i\omega_{\alpha\alpha^{\prime}} +iL_{\alpha\alpha^{\prime}}\right) \delta_{\alpha\beta}\delta_{\alpha^{\prime}\beta^{\prime}} -J_{\alpha\alpha^{\prime},\beta\beta^{\prime}}\;. \label{eq:qc-l}\end{aligned}$$ Here, $\omega_{\alpha\alpha^{\prime}}= \left(E_{\alpha}(R)-E_{\alpha^{\prime}}(R)\right)/\hbar \equiv E_{\alpha\alpha^{\prime}}/\hbar$ and $$iL_{\alpha\alpha^{\prime}} =\frac{P}{M}\cdot\frac{\partial}{\partial R} +\frac{1}{2}\left(F_{\alpha}+F_{\alpha^{\prime}}\right) \frac{\partial}{\partial P}\;, \label{eq:ilad}$$ where $$F_{\alpha}= -\langle\alpha;R\vert\frac{\partial\hat{h}(R)}{\partial R} \vert\alpha;R\rangle$$ is the Hellmann-Feynman force for state $\alpha$. The operator $J$ that describes nonadiabatic effects is $$\begin{aligned} J_{\alpha\alpha^{\prime},\beta\beta^{\prime}} &=&-\frac{P}{M}\cdot d_{\alpha\beta} \left(1+\frac{1}{2}S_{\alpha\beta}\cdot \frac{\partial}{\partial P}\right)\delta_{\alpha^{\prime}\beta^{\prime}} \nonumber\\ &&-\frac{P}{M}\cdot d_{\alpha^{\prime}\beta^{\prime}}^* \left(1+\frac{1}{2}S_{\alpha^{\prime}\beta^{\prime}}^*\cdot \frac{\partial}{\partial P}\right)\delta_{\alpha\beta} \;,\nonumber\\ \label{eq:jad}\end{aligned}$$ where $d_{\alpha\beta}=\langle\alpha;R\vert(\partial/\partial R) \vert\beta;R\rangle$ is the nonadiabatic coupling vector and $$S_{\alpha\beta}=E_{\alpha\beta}d_{\alpha\beta} \left(\frac{P}{M}\cdot d_{\alpha\beta}\right)^{-1}\;.$$ Using Eqs. (\[eq:ilad\]) and  (\[eq:jad\]), the equation of motion for the density matrix in the adiabatic basis can be written explicitly as $$\begin{aligned} \partial_t\rho_{\alpha\alpha^{\prime}} &=& -i\omega_{\alpha\alpha^{\prime}}\rho_{\alpha\alpha^{\prime}} -\frac{P}{M}\cdot\frac{\partial}{\partial R}\rho_{\alpha\alpha^{\prime}} \nonumber\\ && -\frac{1}{2}\left(F_{\alpha}+F_{\alpha^{\prime}}\right)\cdot \frac{\partial}{\partial P}\rho_{\alpha\alpha^{\prime}} \nonumber\\ &&-\sum_{\beta}\frac{P}{M}\cdot d_{\alpha\beta} \left(1+\frac{1}{2}S_{\alpha\beta}\cdot\frac{\partial}{\partial P}\right) \rho_{\beta\alpha^{\prime}} \nonumber\\ &&-\sum_{\beta^{\prime}}\frac{P}{M}\cdot d_{\alpha^{\prime}\beta^{\prime}}^* \left(1+\frac{1}{2}S_{\alpha^{\prime}\beta^{\prime}}^* \cdot\frac{\partial}{\partial P}\right) \rho_{\alpha\beta^{\prime}} \;.\nonumber\\ \label{eq:rho-eq-ad}\end{aligned}$$ The wave fields $\vert\psi^{\iota}(X)\rangle$ and $\langle\psi^{\iota}(X)\vert$ can be expanded in the adiabatic basis as $$\begin{aligned} \vert\psi^{\iota}(X)\rangle&=&\sum_{\alpha}\vert\alpha;R\rangle \langle\alpha;R\vert\psi^{\iota}(X)\rangle =\sum_{\alpha}C_{\alpha}^{\iota}\vert\alpha;R\rangle\nonumber\\ \langle\psi^{\iota}(X)\vert&=&\sum_{\alpha}\langle\psi^{\iota}\vert\alpha;R\rangle \langle\alpha;R\vert =\sum_{\alpha}\langle\alpha;R\vert C_{\alpha}^{\iota *}(X) \;,\nonumber\\\end{aligned}$$ and the density matrix in Eq. (\[eq:rho-ansatz\]) becomes $$\begin{aligned} \rho_{\alpha\alpha^{\prime}}(X,t) &=&\sum_{\iota}w_{\iota}C_{\alpha}^{\iota}(X,t)C_{\alpha^{\prime}}^{\iota *}(X,t) \;. \label{eq:rho-ansatz-ad}\end{aligned}$$ In order to find two separate equations for $C_{\alpha}^{\iota}$ and $C_{\alpha^{\prime}}^{\iota *}$, one cannot insert Eq. (\[eq:rho-ansatz-ad\]) directly into Eq. (\[eq:rho-eq-ad\]) because of the presence of the derivatives with respect to the phase space coordinates $R$ ad $P$. One must set Eq. (\[eq:rho-eq-ad\]) into the form of a multiplicative operator acting on $\rho_{\alpha\alpha^{\prime}}$. To this end, for example, consider $$\begin{aligned} \frac{\partial}{\partial P}\rho_{\beta\alpha^{\prime}} &=&\sum_{\gamma}\left(\frac{\partial}{\partial P}\rho_{\beta\gamma} \right)\delta_{\gamma\alpha^{\prime}} =\sum_{\gamma\mu}\left(\frac{\partial}{\partial P}\rho_{\beta\gamma}\right) \rho_{\gamma\mu}^{-1}\rho_{\mu\alpha^{\prime}} \nonumber\\ &=&\sum_{\mu}\frac{\partial(\ln\hat{\rho})_{\beta\mu}}{\partial P} \rho_{\mu\alpha^{\prime}}\;. \label{eq:der-mani}\end{aligned}$$ Equation (\[eq:der-mani\]) shows how to transform formally a derivative operator acting on $\hat{\rho}$ into a multiplicative operator which, however, depends on $\hat{\rho}$ itself. Therefore, Eq. (\[eq:rho-eq-ad\]) becomes $$\begin{aligned} \partial_t\rho_{\alpha\alpha^{\prime}} &=& -\frac{i}{\hbar}E_{\alpha}\rho_{\alpha\alpha^{\prime}} +\frac{i}{\hbar}E_{\alpha^{\prime}}\rho_{\alpha\alpha^{\prime}} \nonumber\\ &-&\sum_{\beta}\frac{P}{M}\cdot d_{\alpha\beta}\rho_{\beta\alpha^{\prime}} -\sum_{\beta^{\prime}}\frac{P}{M}\cdot d_{\alpha^{\prime}\beta^{\prime}}^* \rho_{\alpha\beta^{\prime}} \nonumber\\ &-&\frac{1}{2}\sum_{\mu}\frac{P}{M}\cdot \frac{\partial(\ln\rho)_{\alpha\mu}}{\partial R} \rho_{\mu\alpha^{\prime}} \nonumber\\ &-&\frac{1}{2}\sum_{\mu}\frac{P}{M}\cdot \frac{\partial(\ln\rho)_{\mu\alpha^{\prime}} }{\partial R} \rho_{\alpha\mu} \nonumber\\ &-&\frac{1}{2}\sum_{\mu}F_{\alpha} \frac{\partial(\ln\rho)_{\alpha\mu}}{\partial P}\rho_{\mu\alpha^{\prime}} \nonumber\\ &-&\frac{1}{2}\sum_{\mu}F_{\alpha^{\prime}}\cdot \frac{\partial(\ln\rho)_{\mu\alpha^{\prime}}}{\partial P}\rho_{\alpha\mu} \nonumber\\ &-&\frac{1}{2}\sum_{\beta,\mu}\frac{P}{M}\cdot d_{\alpha\beta} S_{\alpha\beta}\cdot\frac{\partial(\ln\hat{\rho})_{\beta\mu}}{\partial P} \rho_{\mu\alpha^{\prime}} \nonumber\\ &-&\frac{1}{2}\sum_{\beta^{\prime},\mu}\frac{P}{M}\cdot d_{\alpha^{\prime}\beta^{\prime}}^* S_{\alpha^{\prime}\beta^{\prime}}^* \cdot\frac{\partial(\ln\hat{\rho})_{\mu\beta^{\prime}} }{\partial P} \rho_{\alpha\mu} \;.\nonumber\\ \label{eq:rho-mani-ad}\end{aligned}$$ Inserting the adiabatic expression for the density matrix, given in Eq. (\[eq:rho-ansatz-ad\]), into Eq. (\[eq:rho-mani-ad\]), one obtains, for each quantum state $\iota$, the following two coupled equations $$\begin{aligned} \dot{C}_{\alpha}^{\iota}(X,t) &=& -\frac{i}{\hbar}E_{\alpha}C_{\alpha}^{\iota}(X,t) -\sum_{\beta}\frac{P}{M}\cdot d_{\alpha\beta}C_{\beta}^{\iota}(X,t) \nonumber\\ &-&\frac{1}{2}\sum_{\beta,\mu}\frac{P}{M}\cdot d_{\alpha\beta} S_{\alpha\beta}\cdot\frac{\partial(\ln\hat{\rho})_{\beta\mu}}{\partial P} C_{\mu}^{\iota}(X,t) \nonumber\\ &-&\frac{1}{2}\sum_{\mu}\frac{P}{M}\cdot \frac{\partial(\ln\rho)_{\alpha\mu}}{\partial R}C_{\mu}^{\iota}(X,t) \nonumber\\ &-&\frac{1}{2}\sum_{\mu}F_{\alpha} \frac{\partial(\ln\rho)_{\alpha\mu}}{\partial P}C_{\mu}^{\iota}(X,t) \label{eq:c}\\ %neweq \dot{C}_{\alpha^{\prime}}^{\iota *}(X,t) &=& +\frac{i}{\hbar}E_{\alpha^{\prime}}C_{\alpha^{\prime}}^{\iota *}(X,t) -\sum_{\beta^{\prime}}\frac{P}{M}\cdot d_{\alpha^{\prime}\beta^{\prime}}^* C_{\beta^{\prime}}^{\iota *}(X,t) \nonumber\\ &-&\frac{1}{2}\sum_{\beta^{\prime},\mu}\frac{P}{M}\cdot d_{\alpha^{\prime}\beta^{\prime}}^* S_{\alpha^{\prime}\beta^{\prime}}^* \cdot\frac{\partial(\ln\hat{\rho})_{\mu\beta^{\prime}} }{\partial P} C_{\mu}^{\iota *}(X,t) \nonumber\\ &-& \frac{1}{2}\sum_{\mu}\frac{P}{M}\cdot \frac{\partial(\ln\rho)_{\mu\alpha^{\prime}} }{\partial R} C_{\mu}^{\iota *}(X,t) \nonumber\\ &-& \frac{1}{2}\sum_{\mu}F_{\alpha^{\prime}}\cdot \frac{\partial(\ln\rho)_{\mu\alpha^{\prime}}}{\partial P}C_{\mu}^{\iota *}(X,t) \;.\label{eq:cstar}\end{aligned}$$ Quantum-classical averages of arbitrary observables can be calculated in the adiabatic as $$\langle\hat{\chi}\rangle(t) = \sum_{\iota}w_{\iota}\sum_{\alpha\alpha^{\prime}} \int dXC_{\alpha}^{\iota}(X,t)C_{\alpha^{\prime}}^{\iota *}(X,t) \chi_{\alpha^{\prime}\alpha}(X)\;, \label{eq:qc-ave-ad}$$ where the coefficients $C_{\alpha}^{\iota}(X,t)$ and $C_{\alpha^{\prime}}^{\iota *}(X,t)$ are evolved according to Eqs. (\[eq:c\]) and (\[eq:cstar\]), respectively. Equations (\[eq:c\]) and (\[eq:cstar\]) are non-linear equations which couple all the adiabatic states used to analyze the system. At this stage, a general discussion about such a non-linear character is required. With a wide consensus, quantum mechanics is considered a linear theory. This leads, for example, to the visualization of quantum transitions as instantaneous *quantum jumps*. The linearity of the theory also determines the need of considering infinite perturbative series which must be re-summed in some way in order to extract meaningful predictions. Density Functional Theory is an example of a non-linear theory [@dft] but it is usually considered just as a computational tool. However, there are other approaches to quantum theory that represent interactions by an intrinsic non-linear scheme [@mead]. It is not difficult to see how this is possible. Matter is represented by waves, these very same waves enter into the definition of the fields defining their interaction [@tomonaga]. This point of view has been pursued by Jaynes [@jaynes] and Barut [@barut], among others. These non-linear approaches depict quantum transitions as abrupt but continuous events [@mead] in which, to go from state $\vert 1\rangle$ to state $\vert 2\rangle$, the system is first brought by the interaction in a superposition $\alpha\vert 1\rangle+\beta\vert 2\rangle$, and then, as the interaction ends, it finally goes to state $\vert 2\rangle$. It is understood that this is made possible by the non-linearity of such theories because, instead, a linear theory would preserve the superposition indefinitely. Incidentally, the picture of the transition process just depicted also emerges from the numerical implementation [@kapral] of the nonadiabatic quantum-classical dynamics of phase space dependent operators [@qc-bracket; @kcmqc]: The action of the operator $J$ in Eq. (\[eq:qc-l\]) can build and destroy coherence in the system by creating and destroying superposition of states. As explained above, this is a feature of a non-linear theory. Such a non-linear character is simply hidden in the operator version of quantum-classical dynamics and clearly manifested by the wave picture of the quantum-classical evolution, which has been introduced in this paper. Since Eqs. (\[eq:c\]) and (\[eq:cstar\]) are non-linear, their numerical integration requires either to adopt an iterative self-consistent procedure (according to which one makes a first guess of $\rho_{\alpha\alpha^{\prime}}$, as dictated by Eq. (\[eq:rho-ansatz-ad\]), calculates the evolved $C_{\alpha}^{\iota}(X,t)$ and $C_{\alpha^{\prime}}^{\iota *}(X,t)$, and then goes into a recursive procedure until numerical convergence is obtained) or to choose a definite form for $\rho_{\alpha\alpha^{\prime}}^G$, following physical intuition, and then calculating the time evolution, according to the form of Eqs. (\[eq:c\]) and (\[eq:cstar\]) which is obtained by using $\rho_{\alpha\alpha^{\prime}}^G$. This last method is already known within the Wigner formulation of quantum mechanics [@lee] as the method of *Wigner trajectories* [@wignertraj]. It is also important to find some importance sampling scheme for the phase space integral in Eq. (\[eq:qc-ave-ad\]). Such sampling scheme may depend on the specific form $\chi_{\alpha\alpha^{\prime}}$ of the observable. It is interesting to note that Eqs. (\[eq:c\]), (\[eq:cstar\]), and (\[eq:qc-ave-ad\]) can be used to address both equilibrium and non-equilibrium problems on the same footing. However, the dynamical picture provided by Eqs. (\[eq:c\]) and (\[eq:cstar\]) is very different both from that of the usual surface-hopping schemes [@tully] and from that of the nonadiabatic evolution of quantum-classical operators [@kapral]. In order to appreciate this, for simplicity, one can consider a situation in which there is no thermal disorder in the quantum degrees of freedom so that $\iota=1$: *viz.*, the density matrix becomes that of a pure state $\rho_{\alpha\alpha^{\prime}}(X,t)\to C_{\alpha}(X,t)C_{\alpha^{\prime}}^{*}(X,t)$. Then, equations (\[eq:c\]) and (\[eq:cstar\]) remain unaltered and one has just to remove the index $\iota$ from the coefficients. Therefore, it can be realized that no classical trajectory propagation, and no state switching are involved by Eqs. (\[eq:c\]) and (\[eq:cstar\]). Instead, in order to calculate averages according to Eq. (\[eq:qc-ave-ad\]), one has to sample phase space points and integrate the matrix equations. In the next section, an equilibrium approximation of Eqs. (\[eq:c\]) and (\[eq:cstar\]), along the lines followed by the method of *Wigner trajectories* [@wignertraj], is given and applied, with good numerical results, to the adiabatic and nonadiabatic dynamics of the spin-boson model. Wave dynamics of the spin-boson model {#sec:sb} ===================================== The theory developed in the previous sections can be applied to simulate the relaxation dynamics of the spin-boson system [@sb]. This system has already been studied within the framework of quantum-classical dynamics of operators in Ref. [@qc-sb] and “exact” numerical results were obtained at short-time by means of an iterative path integral procedure developed by Nancy Makri and co-worker [@makri]. The short-time results of Ref. [@sb] numerically coincide with those obtained by the path integral calculation of Ref. [@makri]. However, as it is shown later by Fig. \[fig:fig2\], the quantum-classical results of Ref. [@sb] have some limitations concerning the numerical stability of the algorithm beyong a certain time length. Using the dimensionless variables of Ref. [@qc-sb], the quantum-classical Hamiltonian operator of the spin-boson system reads $$\begin{aligned} \hat{H}(X)&=&-\Omega\hat{\sigma}_x +\sum_{j=1}^N\left(\frac{P_j^2}{2}+ \frac{1}{2}\omega_j^2R_j^2-c_j\hat{\sigma}_zR_j\right) \nonumber\\ &=& \hat{h}_s+H_b+\hat{V}_c(R)\;,\end{aligned}$$ where $\hat{h}_s=-\Omega\hat{\sigma}_x$ is the subsystem Hamiltonian, $H_b=\sum_{j=1}^NP_j^2/2+1/2\omega_j^2R_j^2= \sum_{j=1}^NP_j^2/2+V_b(R)$ is the Hamiltonian of a classical bath of $N$ harmonic oscillators, and $\hat{V}_c(R)=-\sum_{j=1}^Nc_j\hat{\sigma}_zR_j=\gamma(R)\hat{\sigma}_z$ is the interaction between the subsystem and the bath. An Ohmic spectral density is assumed for the bath. Hence, denoting the Kondo parameter as $\xi_K$ and the cut-off frequency as $\omega_{\rm max}$, the frequencies of the oscillators are defined by $\omega_j=-\ln(1-j\omega_0)$, where $\omega_0=N^{-1}(1-\exp(-\omega_{\rm max}))$, and the constants entering the coupling by and $c_j=\sqrt{\xi_K\omega_0}~\omega_j$. The adiabatic eigenvalues and eigenvectors, respectively, are $$E_{1,2}=V_b\mp\sqrt{\Omega^2+\gamma^2(R)}\;,$$ $$\begin{aligned} \vert 1;R\rangle&=& \frac{1}{\sqrt{2(1+G^2}}\left(\begin{array}{c} 1+G\\ 1-G\end{array} \right) \nonumber\\ \vert 2;R\rangle&=& \frac{1}{\sqrt{2(1+G^2}}\left(\begin{array}{c}-1+G\\ 1+G\end{array} \right)\;,\end{aligned}$$ where $$G(R)=\gamma^{-1}(R)\left[-\Omega+\sqrt{\Omega^2+\gamma^2(R)}\right]\;.$$ The coupling vector $d_{\alpha\alpha'}=\langle\alpha;R\vert\overrightarrow{\partial} /\partial R\vert\alpha';R\rangle$ is $$d_{12}=-d_{21}=(1+G^2)^{-1}\partial G/\partial R\;.$$ Assuming an initially uncorrelated density matrix, where the bath is in thermal equilibrium and the subsystem is in state $\vert\uparrow\rangle$, the initial quantum-classical density matrix in the adiabatic basis takes the form $$\mbox{\boldmath$\rho$}(0) =\mbox{\boldmath$\rho$}_s(0)\rho_b(X)\;,$$ where $$\mbox{\boldmath$\rho$}_s(0)=\frac{1}{2(1+G^2)} \left(\begin{array}{cc}(1+G)^2 & 1-G^2 \\ 1-G^2 & (1-G)^2\end{array}\right)\;,$$ and $$\begin{aligned} \rho_b(X)&=&\prod_{I=1}^N\frac{\tanh(\beta\omega_i/2)}{\omega_i} \nonumber\\ &\times& \exp\left[-\frac{2\tanh(\beta\omega_i/2)}{\omega_i} \left(\frac{P_i^2}{2}+\frac{\omega_i^2R_i^2}{2}\right)\right]\;. \nonumber\\\end{aligned}$$ The process of relaxation from the initial state can be followed by monitoring the subsystem observables $\hat{\sigma_z}$, which in the adiabatic basis reads $$\mbox{\boldmath$\sigma$}_z=\frac{1}{1+G^2}\left( \begin{array}{cc} 2G & 1-G^2 \\ 1-G^2 & -2G\end{array}\right)\;.$$ The adiabatic basis is real so that the initial density matrix of the system can be written as $$\rho_{\alpha\alpha'}(X,0)= \sum_{\alpha=1}^2 \psi_{\alpha}(X,0) \phi_{\alpha'}(X,0) \;,$$ where $$\begin{aligned} \psi_{1}(X,0)= \phi_{1}(X,0) &=&\sqrt{\rho_b(X)}\frac{1+G}{\sqrt{2(1+G^2)}}\;, \nonumber\\ \\ \psi_{2}(X,0)= \phi_{2}(X,0) &=&\sqrt{\rho_b(X)}\frac{1-G}{\sqrt{2(1+G^2)}}\;. \nonumber\\\end{aligned}$$ Such coefficients enter into the calculation of the observable $$\langle\mbox{\boldmath$\sigma$}_z(t)\rangle = \sum_{\alpha\alpha'}\int dX \phi_{\alpha'}(X,t)\sigma_{z}^{\alpha'\alpha}(X) \psi_{\alpha}(X,t)\;. \label{eq:sigma-sb}$$ The coefficients evolve in time according to Eqs. (\[eq:c\]) and (\[eq:cstar\]), where one must set $C_{\alpha}^{\iota}\equiv \psi_{\alpha}$ and $C_{\alpha'}^{\iota *}\equiv \phi_{\alpha'}$. In order to devise an effective computational scheme for such equations, one could assume that the density matrix entering Eqs. (\[eq:c\]) and (\[eq:cstar\]) is taken to be that at $t=\infty$, when the total system (subsystem plus bath) has reached thermal equilibrium. The equilibrium quantum-classical density matrix is known as a series expansion in $\hbar$ [@qc-stat]. If one makes the additional assumption of complete decoherence at $t=\infty$, only the ${\cal O}(\hbar^0)$ term can be taken $$\rho_{e}^{(0)\alpha\alpha'}(X) =Z_0^{-1}e^{-\beta(\sum_jP_j^2/2+E_{\alpha}(R))}\delta_{\alpha\alpha'} \;,$$ where $Z_0=\sum_{\alpha\alpha'}\int dX\rho_{e}^{(0)\alpha\alpha'}(X)$. Then $$\begin{aligned} \frac{\partial\ln \rho_{e}^{(0)\alpha\alpha'} }{\partial R} &=&-\beta\frac{\partial E_{\alpha}}{\partial R}\delta_{\alpha\alpha'} \equiv \beta F_{\alpha}(R)\delta_{\alpha\alpha'}\;, \\ \frac{\partial\ln \rho_{e}^{(0)\alpha\alpha'} }{\partial P} &=&-\beta P\delta_{\alpha\alpha'}\;.\end{aligned}$$ Equations (\[eq:c\]) and (\[eq:cstar\]) become $$\begin{aligned} \frac{d}{dt}{\psi}_{\alpha}(X,t) &=& -iE_{\alpha}\psi_{\alpha}(X,t)\nonumber\\ &-&\sum_{\beta}P\cdot d_{\alpha\beta} \left(1-\frac{\beta}{2}E_{\alpha\beta}\right) \psi_{\beta}(X,t) \label{eq:c2}\\ %neweq \frac{d}{dt}{\phi}_{\alpha^{\prime}}(X,t) &=& iE_{\alpha^{\prime}}\phi_{\alpha^{\prime}}(X,t) \nonumber\\ &-&\sum_{\beta^{\prime}}P\cdot d_{\alpha^{\prime}\beta^{\prime}} \left(1-\frac{\beta}{2}E_{\alpha^{\prime}\beta^{\prime}}\right) \phi_{\beta^{\prime}}(X,t) \;. \nonumber\\ \label{eq:cstar2}\end{aligned}$$ In Eqs. (\[eq:cstar\]) and (\[eq:cstar2\]) the terms $\pm(\beta/2)P\cdot F_{\alpha} \psi_{\alpha}$ (and the analogous terms with $\xi_{\alpha'}$) cancel each other. In the adiabatic basis $d_{11}(R)=d_{22}(R)=0$. Hence, defining the matrix $$\begin{aligned} \mbox{\boldmath$\Sigma$} &=& \left[\begin{array}{cc} -iE_1 & -P\cdot d_{12}\left(1-\frac{\beta}{2}E_{12}\right)\\ P\cdot d_{12}\left(1+\frac{\beta}{2}E_{12}\right) & -iE_2 \end{array}\right]\;, \nonumber\\\end{aligned}$$ Equations (\[eq:c2\]) and (\[eq:cstar2\]) can be written as $$\begin{aligned} \frac{d}{dt}\left[\begin{array}{c}{\psi}_1 \\ {\psi}_2\end{array}\right] &=& \mbox{\boldmath$\Sigma$}\cdot \left[\begin{array}{c}\psi_1 \\ \psi_2\end{array}\right] \;, \quad \frac{d}{dt}\left[\begin{array}{c}{\phi}_1 \\ {\phi}_2\end{array}\right] = \mbox{\boldmath$\Sigma$}^*\cdot \left[\begin{array}{c}\phi_1 \\ \phi_2\end{array}\right] \;,\nonumber\\ \label{eq:matrixSigma}\end{aligned}$$ which can be integrated by means of the simple algorithm $\mbox{\boldmath$\Psi$}(X,d\tau)=\mbox{\boldmath$\Psi$}(X,0)+ d\tau\mbox{\boldmath$\Theta$}(X,0)\cdot \mbox{\boldmath$\eta$}(X,0)$, where $\mbox{\boldmath$\Psi$}= (\mbox{\boldmath$\psi$},\mbox{\boldmath$\phi$})$ and $\mbox{\boldmath$\Theta$}=(\mbox{\boldmath$\Sigma$}, \mbox{\boldmath$\Sigma$}^*)$. The phase space part of the initial values of $\mbox{\boldmath$\psi$}$ and $\mbox{\boldmath$\phi$}$ can be used as the weight for sampling the coordinates $X$ entering the classical integral in Eq. (\[eq:sigma-sb\]). Then, for each initial value $X$, Eqs. (\[eq:matrixSigma\]) must be integrated in time so that averages can be calculated. It is worth to note that in such a wave scheme the Eulerian point of view of quantum-classical dynamics [@kapral; @qc-sb] is preserved. This is different from what happens in the original operator approach [@kapral; @qc-sb], where in order to devise an effective time integration scheme by means of the Dyson expansion, one is forced to change from the Eulerian point of view (according to which the phase space point is fixed and the quantum degrees of freedom evolve in time *at* this fixed phase space point) to the Lagrangian point of view, where phase space trajectories are generated. Moreover, it must be noted that the numerical integration of Eqs. (\[eq:matrixSigma\]) provides directly the nonadiabatic dynamics without the need to introduce surface-hopping approximations. In order to be able of comparing the results with those presented in Ref. [@qc-sb], the numerical values of the parameters specifying the spin-boson system have been chosen to be $\beta=0.3$, $\Omega=1/3$, $\omega_{\rm max}=3$, $\xi_K=0.007$, and $N=200$. Figure \[fig:fig1\] shows the results in the adiabatic case, obtained by setting $d_{12}=0$ in Eqs. (\[eq:matrixSigma\]). One can see that, in spite of the simple approximation of the form of the density matrix made in the equations of motion, the wave theory provides results which are in good agreement with those obtained with the operator approach of Ref. [@qc-sb]. Instead, Fig. \[fig:fig2\] shows the results of the nonadiabatic calculation. This is to be compared with the results of the operator theory [@qc-sb] (which are identical with the exact” ones of Ref. [@makri]). Of course, since different ways of dealing with the nonadiabatic effects are used in the two approaches the results do not need to be the same. However, the results of the wave theory follow qualitatively those of Ref. [@qc-sb] while improving substantially the statistical convergence and increasing the length of the time interval spanned by a factor of $2-3$. Such results are particularly encorauging and suggest the possible application of the wave theory here proposed, for example, to the calculation of nonadiabatic rate constants of complex systems in the condensed phase [@ksreview]. Conclusions {#sec:conclusions} =========== In this paper the approach to the quantum-classical mechanics of phase space dependent operators has been remodeled as a non-linear formalism for wave fields. It has been shown that two coupled non-linear equations for phase space dependent wave fields correspond to the single equation for the quantum-classical density matrix in the operator scheme of motion. The equations of motion for the wave fields have been re-expressed by means of a suitable bracket and it has been shown that the emerging formalism generalizes within a non-Hamiltonian framework the non-linear quantum mechanical formalism that has been proposed recently by Weinberg. Finally, the non-linear wave equations have been represented into the adiabatic basis and have been applied, after a suitable equilibrium approximation, to the numerical study of the adiabatic and nonadiabatic dynamics of the spin-boson model. Good results have been obtained. In particular, the time interval that can be spanned by the nonadiabatic calculation within the wave scheme of motion turns out to be a factor of two-three longer than that accessible within the operator scheme of motion. This encourages one to pursue the application of the wave scheme of motion to the calculation of correlation functions for systems in the condensed phase. Future works will be specifically devoted to such an issue. [**Acknowledgment**]{} I acknowledge Professor Kapral for suggesting the possibility of mapping the quantum-classical dynamics of operators into a wave scheme of motion. I am also very grateful to Professor P. V. Giaquinta for continuous encouragement and suggestions. Finally, discussions with Dr Giuseppe Pellicane during the final stage of this work are gratefully acknowledged. Weinberg’s formalism {#app:weinberg} ==================== Consider a quantum system in a state described by the wave fields $|\Psi\rangle$ and $\langle\Psi|$, where Dirac’s bra-ket notation is used to denote $\Psi(r)\equiv \langle r|\Psi\rangle$ and $\Psi^*(r)\equiv \langle \Psi| r\rangle$. Observables are defined by functions of the type $$a=\langle\Psi|\hat{A}|\Psi\rangle\;,$$ where the operators are Hermitian, $\hat{A}=\hat{A}^{\dag}$. Weinberg’s formalism can be introduced by defining Poisson brackets in terms of the wave fields $|\Psi\rangle$ and $\langle\Psi|$. To this end, one considers the wave fields as “phase space” coordinates $\mbox{\boldmath$\zeta$}\equiv(|\Psi\rangle , \langle\Psi|)$, so that $\zeta_1=|\Psi\rangle$ and $\zeta_2=\langle\Psi|$, and then introduce brackets of observables as $$\begin{aligned} \{a,b\}_{\mbox{\tiny\boldmath$\cal B$}}&=&\sum_{\alpha=1}^2\frac{\partial a}{\partial \zeta_{\alpha}} {\mathcal B}_{\alpha\beta}\frac{\partial b}{\partial \zeta_{\beta}}\;. \label{eq:poissonbracket}\end{aligned}$$ The bracket in Eq. (\[eq:poissonbracket\]) defines a Lie algebra and a Hamiltonian systems. Typically, the Jacobi relation is satisfied, *i.e.* $ {\cal J}=\left\{a,\left\{b,c\right\}_{\mbox{\tiny\boldmath$\cal B$}} \right\}_{\mbox{\tiny\boldmath$\cal B$}} +\left\{c\left\{a,b\right\}_{\mbox{\tiny\boldmath$\cal B$}} \right\}_{\mbox{\tiny\boldmath$\cal B$}} +\left\{b,\left\{c,a\right\}_{\mbox{\tiny\boldmath$\cal B$}} \right\}_{\mbox{\tiny\boldmath$\cal B$}}=0$. In order to obtain the usual quantum formalism, one can introduce the Hamiltonian functional in the form $${\cal H}[|\psi\rangle , \langle\psi|]\equiv {\cal H}[\mbox{\boldmath$\zeta$}] = \langle\psi|\hat{H}|\psi\rangle \;, \label{eq:h_qm}$$ where $\hat{H}$ is the Hamiltonian operator of the system. Equations of motion for the wave fields can be written in compact form as $$\frac{\partial\mbox{\boldmath$\zeta$}}{\partial t}=\frac{i}{\hbar} \{ {\cal H}[\mbox{\boldmath$\zeta$}] , \mbox{\boldmath$\zeta$} \}_{\mbox{\tiny\boldmath$\cal B$}} \;. \label{eq:wein_eqofm}$$ The compact form of Eq. (\[eq:wein\_eqofm\]) can be set into an explicit form as $$\begin{aligned} \frac{\partial}{\partial t}|\Psi\rangle&=&\frac{i}{\hbar} \frac{\partial{\cal H}}{\partial\langle\Psi|} {\mathcal B}_{21} \label{eq:wein_eqofm1} \\ \frac{\partial}{\partial t} \langle\Psi| &=&\frac{i}{\hbar} \frac{\partial{\cal H}}{\partial\vert\Psi\rangle} {\mathcal B}_{12} \label{eq:wein_eqofm2} \;.\end{aligned}$$ It is easy to see that, when the Hamiltonian function is chosen as in Eq. (\[eq:h\_qm\]), Eq. (\[eq:wein\_eqofm\]), or its explicit form (\[eq:wein\_eqofm1\]-\[eq:wein\_eqofm2\]), gives the usual formalism of quantum mechanics. It is worth to remark that in order not to alter gauge invariance, the Hamiltonian and the other observables must obey the homogeneity condition: $${\cal H}=\langle\Psi|(\partial{\cal H}/\partial\zeta_2)\rangle =\langle(\partial{\cal H}/\partial\zeta_1)|\Psi\rangle \;.\label{eq:homogeneity}$$ Weinberg showed how the formalism above sketched can be generalized in order to describe non-linear effects in quantum mechanics [@weinberg]. To this end, one must maintain the homogeneity condition, Eq. (\[eq:homogeneity\]), on the Hamiltonian but relax the constraint which assumes that the Hamiltonian must be a bilinear function of the wave fields. Thus, the Hamiltonian can be a general function given by $$\tilde{\cal H}=\sum_{i=1}^n\rho^{-i}{\cal H}_i\;,$$ where $n$ is arbitrary integer that fixes the order of the correction, ${\cal H}_0=h$, and $$\begin{aligned} {\cal H}_1&=&\rho^{-1}\int dr dr'dr''dr'''\Psi^*(r)\Psi^*(r') \nonumber\\ &\times& G(r,r',r'',r''')\Psi(r'')\Psi(r''')\;,\end{aligned}$$ with analogous expressions for higher order terms. Applications and thorough discussions of the above formalism can be found in Ref. [@weinberg]. Once Weinberg’s formalism is expressed by means of the symplectic form in Eq. (\[eq:wein\_eqofm\]), it can be generalized very easily in order to obtain a non-Hamiltonian quantum algebra. To this end, one can substitute the antisymmetric matrix $\mbox{\boldmath$\cal B$}$ with another antisymmetric matrix $\mbox{\boldmath$\Omega$}=\mbox{\boldmath$\Omega$}[\mbox{\boldmath$\zeta$}]$, whose elements might be functionals of $\mbox{\boldmath$\zeta$}\equiv(\vert\Psi\rangle,\langle\Psi\vert)$ obeying the homogeneity condition in Eq. (\[eq:homogeneity\]). By means of $\mbox{\boldmath$\Omega$}$ a non-Hamiltonian bracket $\left\{ \ldots ,\ldots \right\}_{\mbox{\tiny\boldmath$\Omega$}}$ can be defined as $$\begin{aligned} \left\{ a,b\right\}_{\mbox{\tiny\boldmath$\Omega$}}&=& \sum_{\alpha=1}^2\frac{\partial a}{\partial\zeta{\alpha}} {\Omega}_{\alpha\beta}[\zeta]\frac{\partial b}{\partial\zeta{\beta}}\;. \label{eq:nhbracket}\end{aligned}$$ In general, the bracket in Eq. (\[eq:nhbracket\]) does no longer satisfy the Jacobi relation $${\cal J}= \left\{ a,\left\{ b,c\right\}_{\mbox{\tiny\boldmath$\Omega$}} \right\}_{\mbox{\tiny\boldmath$\Omega$}} +\left\{ c\left\{ a,b\right\}_{\mbox{\tiny\boldmath$\Omega$}} \right\}_{\mbox{\tiny\boldmath$\Omega$}} +\left\{ b,\left\{ c,a\right\}_{\mbox{\tiny\boldmath$\Omega$}} \right\}_{\mbox{\tiny\boldmath$\Omega$}} \neq 0\;.\label{eq:njacobi}$$ Thus, non-Hamiltonian equations of motion can be written as $$\frac{\partial\mbox{\boldmath$\zeta$}}{\partial t}=\frac{i}{\hbar} \left\{ {\cal H},\mbox{\boldmath$\zeta$}\right\}_{\mbox{\tiny\boldmath$\Omega$}} \;. \label{eq:wein_nheqofm}$$ In principle, the non-Hamiltonian theory, specified by Eqs. (\[eq:nhbracket\]), (\[eq:njacobi\]), and (\[eq:wein\_nheqofm\]), can be used to address the problem of non-linear correction to quantum mechanics, as it was done in Refs. [@weinberg]. 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--- author: - 'Xián O. Camanho' title: Phase transitions in general gravity theories --- Introduction {#intro} ============ Higher-curvature corrections to the Einstein-Hilbert (EH) action appear in any sensible theory of quantum gravity as next-to-leading orders in the effective action and some, [*e.g.*]{} the Lanczos-Gauss-Bonnet (LGB) action [@Lanczos], also appear in realizations of string theory [@GBstrings1]. This quadratic combination is particularly important as any quadratic term can be brought to the LGB form, $\mathcal{R}^2=R_{\mu\nu\alpha\beta}R^{\mu\nu\alpha\beta}-R_{\mu\nu}R^{\mu\nu}+R^2$, via field redefinitions. Due to the non-linearity of the equations of motion, these theories generally admit more than one maximally symmetric solution, $R_{\mu \nu \alpha\beta}=\Lambda_i(g_{\mu\alpha}g_{\nu\beta}-g_{\mu\beta}g_{\nu\alpha})$; (A)dS vacua with effective cosmological constants $\Lambda_{i}$, whose values are determined by a polynomial equation [@BoulwareDeser], -2mm $$\Upsilon [\Lambda] \equiv \sum_{k=0}^{K}c_{k}\,\Lambda^{k} = c_{K}\prod_{i=1}^{K}\left( \Lambda -\Lambda _{i}\right) =0 ~. \label{cc-algebraic}$$ $K$ being the highest power of curvature (without derivatives) in the field equations. $c_0=1/L^2$ and $c_1=1$ give canonically normalized cosmological and EH terms, $c_{k\geq 2}$ are the LGB and higher order couplings (see [@JDEere] for details). Any vacua is [*a priori*]{} suitable in order to define boundary conditions for the gravity theory we are interested in; [*i.e.*]{} we can define sectors of the theory as classes of solutions that asymptote to a given vacuum [@CE]. In that way, each branch has associated static solutions, representing either black holes or naked singularities, -2mm $$ds^{2}=-f(r)\,dt^{2}+\frac{dr^{2}}{g(r)}+r^{2}\ d\Omega_{d-2}^{2} ~, \qquad \qquad f,g \xrightarrow{r\rightarrow \infty} -\Lambda_i r^2 ~, \label{bhansatz}$$ and other solutions with the same asymptotics. The main motivation of the present work is that of studying transitions between different branches of solutions. This is important in order to investigate whether a new type of instability involving non-perturbative solutions occurs in the theory. This new kind of phase transitions have been recently investigated in the context of LGB [@Camanho2012] and Lovelock gravities [@comingsoon]. Higher order free particle ========================== The existence of branch transitions in higher curvature gravity theories is a concrete expression of the multivaluedness problem of these theories. In general the canonical momenta, $\pi_{ij}$, are not invertible functions of the velocities, $\dot{g}^{ij}$ [@Teitelboim1987]. An analogous situation may be illustrated by means of a simple one-dimensional example [@Henneaux1987b]. Consider a free particle lagrangian containing higher powers of velocities, -1mm $$L(\dot{x})=\frac{1}{2}\dot{x}^2-\frac13\dot{x}^3+\frac1{17}\dot{x}^4 \label{paction}$$ In the hamiltonian formulation the equation of motion just implies the constancy of the conjugate momentum, $\frac{d}{dt}p=0$. However, being this multivalued (also the hamiltonian), the solution is not unique. Fixing boundary conditions $x(t_{1,2})=x_{1,2}$, an obvious solution would be constant speed $ \dot{x}=(x_2-x_1)/(t_2-t_1)\equiv v $ but we may also have jumping solutions with constant momentum and the same mean velocity. ![Lagrangian and momentum for the action (\[paction\]). For the same mean velocity $v$, the action is lower for jumps between $v_\pm$ (big dot) than for constant speed, the minimum action corresponding to the value on the dashed line ([*effective*]{} Lagrangian).[]{data-label="fig:1"}](L-v2.eps "fig:") ![Lagrangian and momentum for the action (\[paction\]). For the same mean velocity $v$, the action is lower for jumps between $v_\pm$ (big dot) than for constant speed, the minimum action corresponding to the value on the dashed line ([*effective*]{} Lagrangian).[]{data-label="fig:1"}](p-v2.eps "fig:") In our example, for mean velocities corresponding to multivalued momentum (see figure \[fig:1\]) solutions are infinitely degenerate as the jumps may occur at any time and unboundedly in number as long as the mean velocity is the same. Nevertheless, this degeneracy is lifted once the value of the action is taken into account. The minimal action path is the naive one for mean velocities outside the range covered by the dashed line whereas in that interval it corresponds to arbitrary jumps between the velocities of the two extrema. The [*effective*]{} Lagrangian (dashed line) is a convex function of the velocities and the effective momentum dependence corresponds to the analogous of the Maxwell construction from thermodynamics (see [@comingsoon] for a detailed explanation of this one-dimensional example). Generalized Hawking-Page transitions ==================================== In the context of General Relativity in asymptotically AdS spacetimes, the Hawking-Page phase transition [@HawkingPage] is the realization that above certain temperature the dominant saddle in the gravitational partition function comes from a black hole, whereas for lower temperatures it corresponds to the thermal vacuum. The [*classical*]{} solution is the one with least Euclidean action among those with a smooth Euclidean section. When one deals with higher curvature gravity there is a crucial difference that has been overlooked in the literature. In addition to the usual continuous and differentiable metrics (\[bhansatz\]), one may construct distributional metrics by gluing two solutions corresponding to different branches across a spherical shell or [*bubble*]{} [@wormholes; @wormholes2]. The resulting solution will be continuous at the bubble –with discontinuous derivatives, even in absence of matter. The higher curvature terms can be thought of as a sort of matter source for the Einstein tensor. The existence of such [*jump*]{} metrics, as for the one-dimensional example, is due to the multivaluedness of momenta in the theory. In the gravitational context, continuity of momenta is equivalent to the junction conditions that need to be imposed on the bubble. In the EH case, Israel junction conditions [@Israel1967], being linear in velocities, also imply the continuity of derivatives of the metric. The generalization of these conditions for higher curvature gravity contain higher powers of velocities, thus allowing for more general situations. Static bubble configurations, when they exist, have a smooth Euclidean continuation. It is then possible to calculate the value of the action and compare it to all other solutions with the same asymptotics and temperature. This analysis has been performed for the LGB action [@Camanho2012] for unstable boundary conditions [@BoulwareDeser]. The result suggests a possible resolution of the instability through bubble nucleation. In the case of LGB gravity there are just two possible static spacetimes to be considered in the analysis for the chosen boundary conditions; the thermal vacuum and the static bubble metric, the usual spherically symmetric solution (\[bhansatz\]) displaying a naked singularity. For low temperatures the thermal vacuum is the preferred solution whereas at high temperatures the bubble will form, as indicated by the change of sign on the relative free energy. The bubble pops out in an unstable position and may expand reaching the boundary in a finite time thus changing the asymptotics and charges of the solution, from the initial to the inner ones. Still, if the free energy is positive the system is metastable. It decays by nucleating bubbles with a probability given, in the semiclassical approximation, by $e^{-F/T}$. Therefore, after enough time, the system will always end up in the stable horizonful branch of solutions, the only one usually considered as relevant. This is then a natural mechanism that selects the general relativistic vacuum among all the possible ones, the stable branch being the endpoint of the initial instability. Discussion ========== The phenomenon described here is quite general. It occurs also for general Lovelock gravities [@comingsoon] and presumably for more general classes of theories. In the generic case, however, the possible situations one may encounter are much more diverse. We may have for instance stable bubble configurations as opposed to the unstable ones discussed above or even bubbles that being unstable cannot reach the boundary of the spacetime. Other generalizations may include transitions between positive and negative values of $\Lambda_i$ and even non-static bubble configurations. Another situation one may think of is that of having different gravity theories on different sides of the bubble. This has a straightforward physical interpretation if we consider the higher order terms as sourced by other fields that vary accross the bubble. For masses above $m^2>\|\Lambda_{\pm}\|$ a bubble made of these fields will be well approximated by a thin wall and we may integrate out the fields for the purpose of discussing the thermodynamics. If those fields have several possible vacuum expectation values leading to different theories we may construct interpolating solutions in essentially the same way discussed above. In this case the energy carried by the bubble can be interpreted as the energy of the fields we have integrated out. The author thanks A. Gomberoff for most interesting discussions, and the Front of pro-Galician Scientists for encouragement. He is supported by a spanish FPU fellowship. This work is supported in part by MICINN and FEDER (grant FPA2011-22594), by Xunta de Galicia (Consellería de Educación and grant PGIDIT10PXIB206075PR), and by the Spanish Consolider-Ingenio 2010 Programme CPAN (CSD2007-00042). [99]{} C. Lanczos, Annals Math.  [**39**]{}, 842 (1938). B. Zwiebach, Phys. Lett. B [**156**]{}, 315 (1985). D. Boulware and S. Deser, Phys. Rev. Lett.  [**55**]{}, 2656 (1985). J. D. Edelstein, these proceedings (and references therein). X. O. Camanho and J. D. Edelstein, Class. Quantum Grav. [**30**]{}, 035009 (2013). X. O. Camanho, J. D. Edelstein, G. Giribet, A. Gomberoff, Phys. Rev. D [**86**]{}, 124048 (2012) X. O. Camanho [*et al*]{}., to appear. C. Teitelboim and J. Zanelli, Class. Quantum Grav., [**4**]{}, 125 (1987). M. Henneaux, C. Teitelboim, and J. Zanelli, Phys. Rev. A, [**36**]{}, 4417 (1987). S. W. Hawking and D. N. Page, Commun. Math. Phys.  [**87**]{}, 577 (1983). E. Gravanis and S. Willison, Phys. Rev. D [**75**]{}, 084025 (2007). C. Garraffo, G. Giribet, E. Gravanis and S. Willison, J. Math. Phys.  [**49**]{}, 042502 (2008). W. Israel, Nuovo Cimento B [**48**]{}, 463 (1967).

--- abstract: 'Here we show that there exist internal gravity waves that are inherently unstable, that is, they cannot exist in nature for a long time. The instability mechanism is a one-way (irreversible) harmonic-generation resonance that permanently transfers the energy of an internal wave to its higher harmonics. We show that, in fact, there are countably infinite number of such unstable waves. For the harmonic-generation resonance to take place, nonlinear terms in the free surface boundary condition play a pivotal role, and the instability does not obtain for a linearly-stratified fluid if a simplified boundary condition such as rigid lid or linear form is employed. Harmonic-generation resonance presented here also provides a mechanism for the transfer of the energy of the internal waves to the higher-frequency part of the spectrum where internal waves are more prone to breaking, hence losing energy to turbulence and heat and contributing to oceanic mixing.' author: - 'Y. Liang, Ahmad Zareei, M.-Reza Alam[^1]' title: Inherently Unstable Internal Gravity Waves due to Resonant Harmonic Generation --- Introduction ============ Internal gravity waves, outcome of perpetually agitated density-stratified oceans, are known to play a critical role in the dynamics of our planet’s energy balance: they absorb energy to form, carry energy over long distances as they propagate, and release energy where they break [@Staquet2002]. The latter phenomenon usually gives rise to considerable mixing [c.f. @Ferrari2008] whereby nutrients also get distributed, which is vital for a wide range of marine life [@Boyd2007; @Harris2012]. More than a century long research has shed a lot of light on various features of internal gravity waves. Nevertheless, many aspects of their inception and fate is yet not well understood [e.g. @Alford2015]. Specifically, the precise mechanism that transfers energy from longer waves to the high-frequency part of the spectrum, where internal waves are more prone to breaking, is yet a matter of dispute. Aside from linear processes such as interaction of internal waves with the seabed topography and sloped continental shelves , several nonlinear instability mechanisms have also been put forward. For instance, we now know that internal waves may undergo instability due to triad resonance . All discovered destabilizing mechanisms for an internal wave (few named above), however, have one thing in common that they require some type of perturbations in order to get initiated. These perturbations can come from, for instance, seabed corrugations or presence of other waves forming resonance triads. Here, we show that there are internal gravity waves in the ocean that are *inherently* unstable, that is, they simply cannot sustain their form. Through the mechanism studied here, specific internal waves *naturally* (without requiring any perturbation) give up their energy *permanently* to their higher harmonics through a one-way irreversible harmonic-generation resonance mechanism. Governing Equations and the Dispersion Relation =============================================== Consider the propagation of internal waves in an inviscid, incompressible, adiabatic and stably stratified fluid of density $\rho(x,y,z,t)$, bounded by a free surface on the top and a rigid seafloor at the depth $h$. Let’s define a Cartesian coordinate system with $x,y$-axes on the mean free surface and $z$-axis positive upward. Newton’s second law, conservation of mass, and conservation of energy provide five equations for the evolution of the components of the velocity vector $\b{u} = \{u, v, w\}$, density $\rho$, and the pressure $p$. These governing equations together with three boundary conditions (two kinematic boundary conditions on the free surface and the seabed, and one dynamic boundary condition on the free surface) uniquely determine the five unknowns and the surface elevation $\eta(x,y,t)$ [e.g. @Thorpe1966]. We assume internal waves are small perturbations to a stable background state at equilibrium. Therefore, density can be written as $\rho(x,y,z,t) = \bar{\rho}(z)+ \rho' (x, y, z,t)$ where $\bar{\rho}(z)$ is the background (unperturbed) density. Similarly, we define a pressure perturbation $p'$ via $p=\bar p(z)+p'(x,y,z,t)$ such that $d\bar p(z)/dz=-\bar{\rho}(z) g$. With some standard manipulation, the governing equations can be written in terms of either of the five variables involved in this problem. We choose to write the equation, as is customary, in terms of the vertical component of the velocity, $w$. These equations then read [see e.g. @Thorpe1966 or Appendix] &\^2w+N\^2\^2\_Hw=(,’),&-h&lt;z&lt;,\[101\]\ &-g\^2\_H w = (,p’,), & z=0,\[102\]\ & w=0, &z=-h.\[103\] where $\nabla^2_H=\p^2/\p x^2+\p^2/\p y^2$ is the horizontal Laplacian, $N^2=-{g/\rho_0~ \d \bar{\rho}(z)}/{ \d z}$ is the Brunt-V[ä]{}is[ä]{}l[ä]{} frequency in which $\rho_0=\bar{\rho}(z=0)$ is the density on the free surface, and $\E,\F$ are nonlinear functions of their arguments. To perform a perturbation analysis, we assume that the solution to can be expressed in terms of a convergent series, i.e. $$\begin{aligned} \label{110} w(\x,t)=\ep w^{(1)}(\x,t)+\ep^2 w^{(2)}(\x,t)+\O(\ep^3),\end{aligned}$$ where $\ep\ll1$ is a measure of steepness of the waves involved and $w^{(i)}\sim \O(1)$. Similar expressions hold for $u,v,\rho'$ and $p'$. Substituting into and collecting terms of the same magnitude, then at the leading order $\O(\ep)$ the linearized equations are obtained. We focus our attention here on the two-dimensional problem with a linear mean density profile, i.e. $\bar{\rho}(z)=\rho_0 (1-a z)$ which gives a constant Brunt-V[ä]{}is[ä]{}l[ä]{} frequency $N=\sqrt{ga}$ [c.f. e.g. @Martin1972]. Looking for a progressive wave solution of the leading order (linearized) equation in the form $w^{(1)}=W(z)\sin({\bf k}\cdot {\bf x} -\omega t)$ the following dispersion relations result: $$\begin{aligned} \label{2000} \hspace{-.70cm} \D(k,\omega)= \begin{cases} \omega^2-\f{g k}{\sqrt{1-{N^2}/{\omega^2}}}\tanh\lp kh \sqrt{1-{N^2}/{\omega^2}}\rp=0,& \omega>N\\ \omega^2-\f{g k}{\sqrt{N^2/\omega^2-1}}\tan\lp kh \sqrt{N^2/\omega^2-1} \rp=0, & \omega<N \end{cases}\end{aligned}$$ ![Plot of the dimensionless frequency $\omega/N$ as a function of dimensionless wavenumber $kh$ of free internal waves (i.e. $\D(k,\omega)=0$) in a fluid of linearly stratified density $\rho(z) =\rho_0(1 - az)$, with $ah$ = 0.05. Associated with each wavenumber there is one surface wave and an infinite number of internal wave modes (blue solid-line branches). Frequency of internal waves cannot exceed the Brunt-V[ä]{}is[ä]{}l[ä]{} frequency $N$, and all branches of the dispersion relation curve are capped at $\omega/N$=1. We also plot contours of $\D(2k,2\omega)$=0 (red dash-dotted lines) whose intersections with blue lines (shown by red circles) mark waves whose second harmonics are also solutions to the dispersion relation. These second harmonics are at the intersections of contours of $\D(k/2,\omega/2)$=0 (green dashed lines) and $\D(k,\omega)$=0 and are marked by black squares. The second harmonic of the wave at “***a***" (mode 2), is the wave “***A***" (mode 1) and so on. Note that second harmonic waves are at least one mode lower than the original waves.[]{data-label="fig1"}](disp94.eps){width="8cm"} Solutions to the above dispersion relation identify permissible frequency and wavenumber of free propagating waves. Contours of $\D(k,\omega)=0$ are shown in figure \[fig1\] in which we plot the dimensionless frequency $\omega/N$ as a function of dimensionless wavenumber $kh$ (blue solid curves). For $\omega>N$ only one solution exists in the first quadrant (with its mirrors in the other quadrants). This solution corresponds to a wave whose associated fluid particle motion is maximum near the free surface and decreases as the depth increases. Therefore this is basically a classical *surface* wave which is a little perturbed because of stratification. For $\omega<N$ there is an infinite number of solutions to . The first member of this set, is the continuation of the surface wave branch (the left-most branch in figure \[fig1\]), but the rest identify waves with associated fluid particle activities that are minimum near the free surface and the seabed, but gain one (or more) maximum/maxima somewhere inside the fluid domain. Therefore these branches show *internal* waves. The number of maxima in the amplitude of velocity along the vertical water line determines the mode number of the branch (the first three are marked on the right-side of figure \[fig1\] with arrows). A cut-off frequency $\omega/N$=1 sets an upper frequency limit for internal waves. It is to be noted that the dispersion relation , although has a different form, is in fact graphically very close to the one under rigid lid assumption $\omega=N/\sqrt{1+(n\pi/kh)^2}$. But clearly the inclusion of the effect of the free surface in the former has changed its form. Harmonic Generation =================== With the linear solution to and its properties at hand, we move to the second order equation by collecting $\O(\ep^2)$ terms obtained from the substitution of into . The second order equation has the exact same form as of the leading order equation on its left-hand side, but with nonlinear terms, arising from nonlinear functions $\E,\F$, on its right-hand side. These nonlinear terms are multiplication of the linear solution (and its derivatives) and, it turns out that, they constitute forcing terms with wavenumber and frequency $(2k,2\om)$. Now if $\D(2k,2\om)$=0, then it means that these forcing terms have a harmonic which is the same as the natural harmonic of the linear system. This is a resonance scenario through which a new second order solution may emerge and may grow large enough to the extent that it becomes comparable to the leading order solution (beyond which the naive expansion is not valid anymore). We would like to note a subtle point here that at the second order the right-side of is identically zero [see e.g. @Tabaei2005]. Therefore a potential harmonic-generation resonance is only possible through the nonlinear terms in the free surface boundary condition . If a rigid-lid assumption or a linearized form of the free surface boundary condition is employed (which is usually the case in the investigation of internal waves) the resonance harmonic generation will simply not obtain unless a non-uniform stratification or Boussinesq terms are considered. To see whether it is possible to satisfy the resonance condition required for the second harmonic of an internal wave to exist as a free propagating wave, we also plot in figure \[fig1\] the contours of $\D(2k,2\om)$=0 (red dash-dotted curves) for first four internal wave modes. Intersections of these contours with the contours of $\D(k,\om)$=0 (solid blue curves) identify waves whose second harmonics are also solution to the dispersion relation. Some of these intersections are marked in figure \[fig1\] by red circles and are identified by lowercase characters. It is easy to find the corresponding second harmonic by multiplying a designated (red circle) frequency and wavenumber by a factor of two, or alternatively by finding intersections of contours of $\D(k/2,\om/2)$=0 (green dashed curves) and $\D(k,\omega)$ (denoted by black squares and uppercase characters). Specifically, “***A***" is the second harmonic of $``\textit{\textbf{a}}"$, $``\textit{\textbf{B}}"$ is the second harmonic of $``\textit{\textbf{b}}"$ and so on. It is to be noted that the second harmonic of an internal wave always belongs to a lower mode than the mode of the original wave. For example, second harmonic of wave “***a***" (mode 2) is the wave “***A***" (mode 1), and second harmonic of wave “***c***" (mode 3) is the wave “***C***" (mode 1). A comprehensive collection of all internal waves (for $kh<$12) whose second harmonics are also free propagating waves is shown in figure \[fig2\]. We are basically plotting in this figure all intersections points of contours of $\D(k,\om)=0$ and contours of $\D(2k,2\om)=0$. Clearly there will be an infinite (but countable) number of such solutions. Through the approach described above, a similar exercise can be performed for waves with their third harmonic being free waves. These solutions are marked by blue circles in figure \[fig2\], and waves with their fourth harmonics lie on the dispersion relation curves are shown by green triangles, and this search can continue indefinitely. ![There are countably infinite number of internal waves that are unstable to their second harmonic. Physically this means that specific incident internal waves of wavenumber and frequency ($k,\omega$) will give up their energy permanently (in a one-way irreversible process, c.f. equation ) to their second harmonic ($2k,2\om$). The necessary condition for this to happen is $\D(k,\om)=\D(2k,2\om)$=0. These waves, for the parameters of figure \[fig1\], are shown here by red squares. Similar story holds for another set of waves that are unstable to their third harmonic (blue circles, necessary codition $\D(k,\om)=\D(3k,3\om)$=0), and fourth harmonic (green triangles, $\D(k,\om)=\D(4k,4\om)$=0) and so on. Instability to higher harmonics are clearly much weaker when compared with the instability to the second harmonic.[]{data-label="fig2"}](2nd3rd4th_v3.eps){width="8cm"} Results and Discussions ======================= To determine the strength of the resonance (i.e. the rate of growth of the resonant wave), and the dynamics of the energy interplay between a wave and its second harmonic, here we perform a multiple scale perturbation analysis. The basic assumption is that the amplitude of the original wave and its second harmonic are both functions of spatial variables and time (i.e. $\x,t$), *as well as* a slow spatial variable in the direction of propagation $x_1=\ep x$. Physically speaking, we allow the amplitude of both waves to *slowly* vary as waves propagate. Mathematically speaking this is written as $$\begin{aligned} \label{210} w(x,x_1,z,t)=\ep w_1(x,x_1,z,t)+\ep^2 w_2(x,x_1,z,t)+\O(\ep^3)\end{aligned}$$ where $w_i\sim\O(1)$. Expressions of the same form are assumed to hold for other variables $u,\rho',p'$ and $\eta$. Similar to regular perturbation methodology, described earlier in the paper to gain insight into the problem, by substituting into the governing equation and collecting terms of the same order in $\ep$. We assume the original wave with wavenumber and frequency $k,\omega$ has the amplitude $\A_1(x_1)$ and the amplitude of resonant second harmonic wave with wavenumber and frequency $2k,2\omega$ is $\B_2(x_1)$. At the second order, applying a compatibility condition (to avoid unbounded solutions) the following two equations governing spatial evolution of $\A_1(x_1)$ and $\B_2(x_1)$ emerge (see Appendix for details of derivation) &=\_1\^2(x\_1),\[201\]\ &=\_1(x\_1)\_2(x\_1),\[202\] where $$\begin{aligned} &\alpha=-\f{6m_1^2\om\cos m_2h}{ g k(2m_2h+\sin 2 m_2 h) },\nn\\ &\beta=-\f{k[\sin m_2 h(4m_1\cos^2 m_1h -3m_1) -2m_2\cos m_2h\sin 2m_1h]}{2\omega(2m_1h+\sin 2m_1h)}. \nn\end{aligned}$$ in which $m_1=k\sqrt{N^2-\omega^2}/\omega$ and $m_2=k\sqrt{N^2-4\omega^2}/\omega$. Spatial evolution of the normalized amplitude of the original wave $\A_1/\A_{10}$ (where $\A_{10}=\A_1(x=0)$) and its second harmonic $\B_2/\A_{10}$ as a function of spatial distance of propagation $x/\lambda_i$ (where $\lambda_i=2\pi/k$= wavelength of the original wave) is shown in figure \[fig3\] respectively by blue-dashed line and solid-red line. In accordance with the case presented in figure \[fig1\], we choose $ah$=0.05 and if we consider that waves are propagating in a water of depth $h$=1 km, then $N$=0.02 rad/s. For this case, figure \[fig3\]a corresponds to the point “***a***" ($kh$=3.144772,$\omega/N$=0.4472136) with its second harmonic at “***A***" in figure \[fig1\], and figure \[fig3\]b corresponds to point “***c***" ($kh$=5.132426,$\omega/N$=0.4780914) with its second harmonic at “***C***". In the former case the interaction is (spatially) faster by a factor of about two, while in the latter case the relative amplitude of the resonant wave is higher. The most striking aspect of the solution is that the interaction is one-way. This can be seen from in which, because $\A_1^2$ is always positive, then $\d \B_2/d x_1$ can never change sign. As a result the magnitude of $\B_2$ can only increase (the sign of $\alpha$ only contributes to 0 or $\pi$ radian phase shift to the wave), and that’s why the direction of energy can never change. This is in contrast to typical triad resonance interactions [e.g. @Alam2010; @Alam2012c] and harmonic generation in shallow water waves [e.g. @Alam2007] where energy initially flows from original waves to resonant waves, but then when the amplitude of resonant waves is large enough the flow of energy reverses. Here, energy only goes from the original wave to the second harmonic and stays there permanently. The original wave will be gone forever. Our multiple scales results conserve energy, as expected. In figure \[fig3\]c (which corresponds to the case presented in figure \[fig3\]b) we plot the energy flux ($E_f=E\times C_g$ where $E$ is the energy per unit area and $C_g$ is the group velocity) normalized by the energy flux of the original wave at the beginning ($E_{f,\A_{10}}$). Plotted are the normalized energy flux of the original wave $E_{f,\A_1}$ (blue-dashed line), the second harmonic $E_{f,\B_2}$, and the summation of the fluxes $E_{f,total}=E_{f,\A_1}+E_{f,\B_2}$. As expected from the conservation of energy the latter is constant and equal to unity. ![Spatial evolution of the amplitude of the original wave (blue dashed line) and its resonant second harmonic (red solid line) correspond to the point “***a***" in figure \[fig1\] ($kh$=3.144772, $\omega/N$=0.4472136, $aA_{10}/N$=0.0023, fig. a), and the point “***c***" in figure \[fig1\] ($kh$=5.132426, $\omega/N$=0.4780914, $aA_{10}/N$=0.0023, fig. b). In figure (a) energy goes from mode 2 to mode 1, whereas in figure (b) energy goes from mode 3 to mode 1. Figure (c) shows the energy flux of each wave as well as the sum of the energy fluxes (green dash-dotted line). As expected from energy conservation, the overall energy flux is unchanged in the domain of interaction. []{data-label="fig3"}](amp_hg05_h1000_pointa_v7.eps "fig:"){width="7cm"} (-205,82)[(a)]{}\ ![Spatial evolution of the amplitude of the original wave (blue dashed line) and its resonant second harmonic (red solid line) correspond to the point “***a***" in figure \[fig1\] ($kh$=3.144772, $\omega/N$=0.4472136, $aA_{10}/N$=0.0023, fig. a), and the point “***c***" in figure \[fig1\] ($kh$=5.132426, $\omega/N$=0.4780914, $aA_{10}/N$=0.0023, fig. b). In figure (a) energy goes from mode 2 to mode 1, whereas in figure (b) energy goes from mode 3 to mode 1. Figure (c) shows the energy flux of each wave as well as the sum of the energy fluxes (green dash-dotted line). As expected from energy conservation, the overall energy flux is unchanged in the domain of interaction. []{data-label="fig3"}](amp_hg05_h1000_pointc_v7.eps "fig:"){width="7cm"} (-205,82)[(b)]{}\ ![Spatial evolution of the amplitude of the original wave (blue dashed line) and its resonant second harmonic (red solid line) correspond to the point “***a***" in figure \[fig1\] ($kh$=3.144772, $\omega/N$=0.4472136, $aA_{10}/N$=0.0023, fig. a), and the point “***c***" in figure \[fig1\] ($kh$=5.132426, $\omega/N$=0.4780914, $aA_{10}/N$=0.0023, fig. b). In figure (a) energy goes from mode 2 to mode 1, whereas in figure (b) energy goes from mode 3 to mode 1. Figure (c) shows the energy flux of each wave as well as the sum of the energy fluxes (green dash-dotted line). As expected from energy conservation, the overall energy flux is unchanged in the domain of interaction. []{data-label="fig3"}](energy_hg05_h1000_pointc_v7.eps "fig:"){width="7cm"} (-200,73)[(c)]{} To cross validate our results with direct simulation, we use the adaptive Navier-Stokes solver code SUNTANS (Stanford Unstructured Nonhydrostatic Terrain-following Adaptive Navier-Stokes Simulator [@Fringer2006]). As a nonhydrostatic parallel ocean model with the capability to implement nonlinear free surface, SUNTANS has been validated and widely used in studying internal waves (and associated turbulence and mixing) in stratified waters [@Zhang2011; @Kang2010; @Wang2011; @Kang2012a; @Walter2012; @Zhang2011; @Wang2011]. Here we consider propagation of waves in a stratified water of density gradient $a=1\times10^{-3}$ m$^{-1}$, depth 100m and in a domain of horizontal extent 20km. We choose $5\times10^3$ grid points in the $x$ direction and 100 layers in the $z$ direction. We specify the velocity of fluid particle at the left boundary to match that of desired incoming wave. Therefore the left boundary that serves as the incoming wave boundary (wave maker). We set the right-side vertical boundary of the domain as the no-penetration and slip-free, and therefore it will act as a rigid vertical wall. We consider the case of incident internal wave of mode 2 ($kh$=3.147946, $\omega/N$=0.4472136, vertical velocity amplitude $w$=0.1m/s) with its second harmonic being a mode 1 wave. The time step $\delta t/T$=1/35000 where $T$ is the period of the incident wave. We run the simulation until the incident wave arrives at the wall, at which point the amplitude of both waves (incident and its second harmonics) have reached a steady state in the domain of interest $0<x/\lambda_i<20$. As is seen from figure \[fig4\], the curve of $\B_2/\A_{10}$ vs $x/\lambda_1$ gives an initial slope of 0.0107 which is the same as our theoretical prediction . ![Comparison of analytical results with the direct simulation. The physical parameters used are: density gradient $a$=${1\e-3}$ $m^{-1}$, water depth h=100m, dimensionless wave number $kh$=3.147946, dimensionless frequency $\omega/N$=0.4472136, the amplitude of vertical velocity of the incident wave $A_{10}$=0.1 m/s. []{data-label="fig4"}](suntans_v4){width="6.5cm"} To see how fast this instability evolves temporally, we present in figure \[fig42\] results of the temporal evolution of the parent wave and its second harmonic, with the same physical parameters as in the case in figure \[fig3\]a. The governing equation takes a similar form to except the derivatives that are with respect to the time $t$ and clearly expressions for $\alpha$ and $\beta$ are different. The qualitative trend is as the spatial case, and the figure suggests that we need time $\sim\O(1000)$ times the period of the initial wave to see the majority of the energy transmitted to the second harmonic. With the chosen Brunt-V[ä]{}is[ä]{}l[ä]{} frequency of $N=0.022$s$^{-1}$, the time scale is about 10 days, which is of the same order as that of parametric subharmonic instability of the M2 internal tide ($\sim$ 2-5 days) [c.f. e.g. @Gerkema2006c; @MacKinnon2005a]. This is somewhat expected as both mechanisms are *second-order* resonance instabilities. We would like to comment here that Parametric Subharmonic Instability is dependent upon perturbation waves in the domain in order to get started [@Davis1967; @muller1986]. In direct simulations, this is achieved by adding random noise to the simulation domain. In experimental studies the required noise already exists in the domain due to unavoidable imperfections. Hence, usually in experiments Parametric Subharmonic Instability is automatically obtained (similar to inevitable Benjamin-Feir instability) and destabilizes “single" internal waves to a number of other subharmonic waves [e.g. @Martin1972]. We would like to emphasize that the underlying mechanism of resonant harmonic generation studied here is different than that of Parametric Subharmonic Instability in that the former mechanism, among other characteristics, is an “inherent" instability that does not require ambient perturbations to get started. Three snapshots of the overall vertical velocity $w$ are shown in figure \[fig42\]b that highlights the horizontal and vertical structures of the wave field at different times. Energy goes from the original internal wave (b1) slowly to its higher harmonics, first leading to modulation of the original wave (b2), but eventually the entire energy is at the second harmonic (b3) whose frequency and horizontal wavenumber are double the horizontal frequency and wavenumber of the initial wave. The interaction is only between the initial wave and its second harmonic, and no cascading to different waves is ensued. ![Time evolution of internal gravity waves that satisfy the conditions of harmonic generation. The physical parameters used here correspond to figure \[fig3\]a. (a) Time evolution of the amplitudes of the two waves. $T_0$ is the period of the parent wave. (b) Snapshots of the wave field at $t=0$ (b1), $t=500T_0$ (b2), and $1500T_0$(b3). The presented mechanism initially results in an modulation of the primary wave, and eventually all the energy goes to the second harmonic.[]{data-label="fig42"}](HT.eps "fig:"){width="8cm"} (-245,82)[(a)]{}\ ![Time evolution of internal gravity waves that satisfy the conditions of harmonic generation. The physical parameters used here correspond to figure \[fig3\]a. (a) Time evolution of the amplitudes of the two waves. $T_0$ is the period of the parent wave. (b) Snapshots of the wave field at $t=0$ (b1), $t=500T_0$ (b2), and $1500T_0$(b3). The presented mechanism initially results in an modulation of the primary wave, and eventually all the energy goes to the second harmonic.[]{data-label="fig42"}](ss.eps "fig:"){width="8cm"} (-245,82)[(b)]{}\ We would like to note that the harmonic generation mechanism proposed here occurs for non-uniformly stratified ocean as well. As pointed out previously, a necessary condition for this to occur is that for a given wave of frequency $\omega$ and wavenumber $k$, the two conditions $\D(k,\omega)=0$ and $\D(2k,2\omega)=0$ are satisfied. We present two examples in figure \[fig5\] for two different density profiles: parabolic $\rho(z)=\rho_0 (1+a z^2)$, where $a=1\times10^{-5} \text{m}^{-2}$ (figure \[fig5\]a), and exponential $\rho(z)=\rho_0 [1+\delta-\delta \exp(az)]$, where $a=0.1 \text{m}^{-1}, \delta=0.1$ (figure \[fig5\]b). The Brunt-V[ä]{}is[ä]{}l[ä]{} frequencies increase with depth in one case and decrease in the other. It can been seen from figure \[fig5\] that there are, as well, infinitely countable intersections (i.e., solutions) between the two sets of curves. In these cases, contrary to the constant $N$ case, the linear wave solution does not satisfy the nonlinear Boussinesq-Euler equation and hence inclusion of the nonlinear free surface condition is not necessary for the harmonic generation to appear. The analysis for these two cases can be carried out by following the same logic presented in the appendix, although mathematically more involved and closed-form explicit solutions will be tedious, if not impossible, to obtain. As discussed above, when density profile is not uniform and hence $N$ is not constant, two sources contribute to the harmonic generation: 1- nonlinearities in the free surface boundary condition, i.e. right-hand side of equation , and 2- nonlinear terms in the momentum equation, i.e. right-hand side of equation . As presented before, for a constant $N$ case the latter source is absent. For nonuniform stratifications, it is of interest to evaluate the relative importance of these two contributors. The motivation is to see whether the classical rigid-lid assumption would be a good approximation in estimating energy exchange due to harmonic generation when the density profile is non-uniform (we already presented this is not the case for a uniform stratification). We comment on this question briefly by considering the exponential density profile as it is representative of actual pycnoclines. For such a density profile, the vertical structure of an internal wave $W(z)$ are obtained as W(z)=- where, ${\mathit J}$ and ${\mathit Y}$ are Bessel function of the first and second kinds respectively, and =,    =,   p=. By substituting this equation into the right-hand side of equations and the relative importance of the right-hand side of the two equations are obtained. We consider a wave with the wave number and frequency $kh=14.14$ and $\omega/N_\text{max}=0.30$, taken from one solution in figure \[fig5\](b). It turns out that, in this example and for instance (at $z=-20$m), the relative magnitude of nonlinear terms on the right-hand side of the Boussinesq-Euler equation to those of free surface boundary condition (i.e. to ) is about 30%. This means that the contribution of free surface boundary condition to the harmonic generation may be multiple times larger than the nonlinear terms in the momentum equation. We would like to emphasize that the resonant harmonic generation studied here is different from *non-resonant* harmonic generation that obtains in non-uniform stratifications [e.g. @Sutherland2016]. The former occurs for specific internal waves, transfers the entire energy to the second harmonic and is a one-way process, whereas the latter obtains for all internal waves but since it is non-resonant only transfers a portion of energy from the initial wave to the superharmonics. ![Plots of $\D(k,\omega)=0$, $\D(2k,2\omega)=0$ and $\D(k/2,\omega/2)=0$ for parabolic and exponential density profiles. (a) The density is $\rho(z)=\rho_0 (1+a z^2)$, where $a=1\times10^{-5} \text{m}^{-2}$. Water depth is $h=100$m. The Brunt-V[ä]{}is[ä]{}l[ä]{} frequency $N=\sqrt{-2gaz}$. $N_\text{max}=\sqrt{2gah}$. (b) The density is $\rho(z)=\rho_0 [1+\delta-\delta \exp(az)]$, where $a=0.1 \text{m}^{-1}, \delta=0.1$. Water depth is $h=100$m. The Brunt-V[ä]{}is[ä]{}l[ä]{} frequency $N=\sqrt{ga\delta\exp(az)}$. $N_\text{max}=\sqrt{ga\delta}$.[]{data-label="fig5"}](1.eps "fig:"){width="8cm"} (-235,82)[(a)]{}\ ![Plots of $\D(k,\omega)=0$, $\D(2k,2\omega)=0$ and $\D(k/2,\omega/2)=0$ for parabolic and exponential density profiles. (a) The density is $\rho(z)=\rho_0 (1+a z^2)$, where $a=1\times10^{-5} \text{m}^{-2}$. Water depth is $h=100$m. The Brunt-V[ä]{}is[ä]{}l[ä]{} frequency $N=\sqrt{-2gaz}$. $N_\text{max}=\sqrt{2gah}$. (b) The density is $\rho(z)=\rho_0 [1+\delta-\delta \exp(az)]$, where $a=0.1 \text{m}^{-1}, \delta=0.1$. Water depth is $h=100$m. The Brunt-V[ä]{}is[ä]{}l[ä]{} frequency $N=\sqrt{ga\delta\exp(az)}$. $N_\text{max}=\sqrt{ga\delta}$.[]{data-label="fig5"}](2.eps "fig:"){width="8cm"} (-235,82)[(b)]{}\ Similarly, since the linear wave solution is not the exact solution to the fully nonlinear problem if taking into the non-Boussinesq terms in the Euler equation, the instability mechanism can also be initiated without a nonlinear free surface. We also would like to note that non-Boussinesq effects (and corresponding terms in the governing equation) can also lead to the generation of resonant super harmonics. In other words, harmonic generation can be obtained in uniform stratification with the rigid lid, if non-Boussinesq terms are not neglected. However, as will be shown shortly, the relative importance of such routing of energy is orders of magnitude less than that of nonlinear free surface avenue: Following the same procedure for getting , except not making Boussinesq approximation, we obtain, &w+\_H(w)- (u)- (v)+g\_H\ &+2+ +2(w)+(w)-()\ &-\[()\]-()=0. Comparing the Boussinesq terms in the above equation and the quadratic terms on the free surface boundary condition, i.e., , we find that their ratio is order of $a/k$. For the two cases presented above, the ratios are less than 1.6%.\ The harmonic generation through a nonlinear free surface involves two energy transfer processes simultaneously, in one process $(k,\omega)$ wave loses energy to $(2k,2\omega)$ wave and generates the second harmonic; in the other the $(2k,2\omega)$ wave forms a triad resonance with $(k,\omega)$ wave and sends energy back in an opposite way. The triad resonance condition is clearly satisfied since $2\omega-\omega=\omega$ and $2k-k=k$. If at the initial moment we only have $(k,\omega)$ wave (i.e. no second harmonic in the domain) then harmonic generation triumphs over triad resonance in transferring energy, resulting in the combined effect that energy only goes in one direction. But it can be shown that there are initial conditions combinations under which the energy at first goes from $(2k,2\omega)$ wave to the $(k,\omega)$ wave, but even in that case eventually the entire energy is transferred to the $(2k,2\omega)$ wave. This can be shown rigorously through Lyapunov stability theorem. Basically, the objective is to prove that $\A_1=0$ and $\B_2$ reaching its maximum is the asymptotically stable solution of the dynamical system described by equation . To prove this, we first note that can be put in the following form \_2\^2 - \_1\^2 = 0. Therefore, since $\f{\alpha}{\beta}$ can be shown to be always negative(see Appendix B), we have \[3457\] \_2\^2 - \_1\^2 = \^2, where $\G$ is a constant that is determined by the initial conditions. Equation shows that $(\A_1,\B_2)$ are always moving on an ellipse. If we define a Lyapunov function $V(\A_1) = \G - \alpha/|\alpha|[ \int_{0}^{x}\alpha \A_1(x)^2 dx+\B_2(0)]$, we find that $V(\A_1)\geqslant 0$ and $d V/d x<0$. According to Lyapunov asymptotic stability theorem, the solution of the system converges to $\A_1=0$ from any starting points as $x$ goes to $\infty$. Conclusion ========== Here we reported that certain internal gravity waves are inherently unstable and are not able to sustain their form. This new instability mechanism, a result of resonance harmonic-generation, draws the energy of an internal wave and hands it over to its second (or generally higher) harmonic. This resonance is distinguished from the classical triad resonance (and associated subharmonic instability) in that 1- a single wave may undergo the instability without requiring any external perturbation, and 2- the transfer of energy is irreversible and the original wave permanently loses its energy to its second harmonic. Extension of the results presented here to the third and higher harmonic generation is straightforward, but the strength of energy exchange in higher harmonics is much weaker. Derivation of the Interaction Equation ====================================== Consider the propagation of waves in an inviscid, incompressible, adiabatic and stably stratified fluid of density $\rho(x,y,z,t)$, bounded by a free surface on the top and a rigid seafloor with a depth $h$ at the bottom. We consider a Cartesian coordinate system with $x,y$-axes on the mean free surface and $z$-axis positive upward. Equations governing the evolution of the velocity vector $\b{u} = \{u, v, w\}$, density $\rho$, pressure $p$ and surface elevation $\eta$ under Boussinesq approximation read &\_0=-p-g z,  -h&lt;z&lt;\[g101\]\ &=0,  -h&lt;z&lt;\[g102\]\ &=0,   -h&lt;z&lt;\[g103\]\ &\_[t]{}=(z-),  z=\[g104\]\ &=0,  z=\[g105\]\ &w=0,  z=-h,\[g106\] where $g$ is the gravitational acceleration, and $\rho_0=\bar{\rho}(z=0)$ is the mean density on the free surface with $\bar{\rho}(z)$ the background (unperturbed) density such that $\rho = \bar{\rho}(z)+ \rho' (x, y, z,t)$. Equation is the momentum equation (Euler’s equations), comes from conservation of salt, and is continuity equation that together form five equations for five unknown variables of the problem (three components of velocity ${\bf u}$, pressure $p$ and density $ \rho$). Equations - are boundary conditions on the free surface and the bottom. Similar to the density perturbation $\rho'$, we define a pressure perturbation $p'$ via $p=\bar p(z)+p'(x,y,z,t)$ such that $d\bar p(z)/dz=-\bar{\rho}(z) g$. For the linear terms of the governing equation to be only in terms of $w$ we calculate $\p/\p z[\g \cdot (\text{\ref{g101}})]-\lap (\text{\ref{g101}})_3$, where (\[g101\])$_3$ denotes the $z$ component of (\[g101\])(likewise, 1,2 for $x,y$ will be used later). We obtain \[2011\] w-u-v+\_H\^2(w)+\_H\^2  ’ =0, where $\lap_{H}=\p^2/\p x^2+\p^2/\p y^2$ is the horizontal Laplacian. Taking the time derivative of and substituting $\rho'$ from the expansion of , i.e., \[g202\] +’+w=0. and denoting $N^2=-{g/\rho_0 \d \bar{\rho}(z)}/{ \d z}$ as the Brunt-V[ä]{}is[ä]{}l[ä]{} frequency, we obtain \[401\] w+N\^2 \_[H]{}w= u+v-\_H\^2(w)+ \_[H]{}(’). Expanding the surface dynamic boundary condition and keeping terms up to the second order we obtain \[302\] =w\_0 g-p’-+\_0 g+ gw ,     [at]{}  z=0. where the last term can be equivalently written as $-N^2\bar\rho_0\eta w$. Here $p'$ can be substituted from an expression obtained from taking the $x$ derivative of (\[g101\])$_1$ added to the $y$ derivative of (\[g101\])$_2$: \[301\] -+\_[H]{}+u+v=0. Substituting $\p p'/\p t$ from in we obtain \[402\] -g\_[H]{}w=& u+v\ &+\_[H]{}g+N\^2w-\_[H]{}p’+. Therefore the governing equation and boundary conditions correct to $\O(\ep^2)$ reduce to , and . For the ease of referring, we rewrite these equations here: w+N\^2 \_[H]{}w=& u+v-w\ &-w+ \_[H]{}’    -h&lt;z&lt;0,\[3001\]\ -g\_[H]{}w=& u+v+\_[H]{}g+N\^2 w\ & - \_[H]{}p’+     z=0,\[3002\]\ w=&0,  z=-h.\[3003\] Equations and are identical to equations (A2) and (A7) of [@Thorpe1966][^2]. To perform a weakly nonlinear analysis, we assume that internal waves in the system described above are small perturbations from the mean state of water at rest, i.e. $\bf u,\rho',p'\sim {\mathcal O}(\epsilon)$, where $\epsilon$ is a measure of the wave steepness. Considering the two-dimensional problem and assuming that the solution to this problem can be expressed in terms of a convergent series we define w(x,x\_1,z,t)=w\_1(x,x\_1,z,t)+\^2 w\_2(x,x\_1,z,t)+Ø(\^3) where $x_1=\epsilon x$ is the slow spatial variable and $\ep\ll1$ is a measure of steepness of the waves and $w_i\sim\O(1)$. Similar expressions exist for other variables, i.e. $u=\ep u_1+\ep^2u_2+\O(\ep^3)$, $\rho'=\ep \rho'_1+\ep^2\rho'_2+\O(\ep^3)$, $p'=\ep p'_1+\ep^2p'_2+\O(\ep^3)$ and $\eta=\ep \eta_1+\ep^2\eta_2+\O(\ep^3)$ with $u_i,\rho'_i,p'_i\sim\O(1)$ being functions of $x,x_1,z,t$, and $\eta_i\sim\O(1)$ being functions of $x,x_1$ and $t$. Upon substitution into the governing equation, at the leading order $\O(\ep)$ we obtain &w\_1+N\^2 \_[H]{}w\_1=0 &-h&lt;z&lt;0,\ &-g\_[H]{}w\_1=0& z=0,\ &w\_1=0, &z=-h. At the second order $\O(\ep^2)$ we have w\_2+N\^2 \_[H]{}w\_2=&-2+N\^2 w\_1+\_1u\_1\ & +\_1v\_1-\_1w\_1-\_1w\_1\ &+ \_[H]{}(\_1’\_1), -h z 0.\[121\]\ -g\_[H]{}w\_2=&2 g + \_1u\_1+\_1v\_1\ &+\_[H]{}g\_1+N\^2 w\_1 \_1\[122\]\ &- \_[H]{}\_1p’\_1 +\_1, z=0.\ w\_2=&0, z=-h. We now assume that waves with wavenumber and frequency $(k,\omega)$ and $(2k,2\omega)$ satisfy the internal waves dispersion relation (which is obtained from the linear equation ) (k,)=\^2-kh , i.e. $\D(k,\omega)=0$ and $\D(2k,2\omega)=0$, and that they both exist in our domain of interest, though potentially with different amplitudes. The propagating wave solution to the linear equation then obtains as w\_1(x,x\_1,z,t)=&A\_1(x\_1)m\_1(z+h)+B\_1(x\_1)m\_1(z+h)\ +&A\_2(x\_1)m\_2(z+h)+B\_2(x\_1)m\_2(z+h) in which $A_1,A_2,B_1,B_2$ are amplitudes of each wave, $m_1^2=k^2(N^2-\omega^2)/\omega^2$ and $m_2^2=k^2(N^2-4\omega^2)/\omega^2$. Other variables $u,\rho',p'$ and $\eta$ can be found respectively via continuity equation , energy equation , kinematic surface boundary condition and the dynamic free surface boundary condition . The left-hand side of the second order equation is identical in the form to the first order equation , but the right hand side of is clearly non-zero and is a nonlinear function of the leading order solution $\u_1,\rho_1',p'_1,\eta_1$. It turns out, after substitution, that the right hand side contains terms with harmonics which are the same as the harmonics of the leading order equation (secular terms). A compatibility condition then must be enforced to make sure that the solution does not go unbounded, which is clearly unphysical. This compatibility condition determines the spatial behavior of the coefficients $A_i,B_i$. While the formulation presented here is general, our primary interest is when an initial wave with wavenumber and frequency $(k,\omega)$ *resonates* its second harmonic $(2k,2\omega)$ whose initial amplitude is zero. Therefore in the following we use the adjectives *original* and *resonant* waves to refer to $(k,\omega)$ and $(2k,2\omega)$ waves respectively. We would like to emphasize that the formulation is general and works for any initial condition of the two waves. We will also comment that the presented approach can be easily extended for third and higher-harmonic generation. Without loss of generality we assume that $B_1=0$, which only has to do with our choice of coordinate system. But we keep $A_2$ and $B_2$, since they determine the phase of the resonant wave $(2k,2\omega)$ with respect to the original wave $(k,\omega)$. The general solution to the second order problem takes the form w\_2(x,x\_1,z,t)=&C\_[11]{}(x\_1,z)   +C\_[12]{}(x\_1,z)\ +&C\_[21]{}(x\_1,z)+C\_[22]{}(x\_1,z) where $C_{ij}(x_1,z)$’s are to be determined from . Substituting and into , and collecting same sine and cosine terms we obtain four ordinary differential equations for $C_{ij}$ ($i,j$=1,2): -\^2&C\_[1i,zz]{}-m\_1\^2\^2C\_[1i]{}=E\_[1i]{}, & -h&lt;z&lt;0,\ -\^2&C\_[1i,z]{}+gk\^2C\_[1i]{}=F\_[1i]{} & z=0,\ &C\_[1i]{}=0, & z=-h, -4\^2&C\_[2i,zz]{}-4m\_2\^2\^2C\_[2i]{}=E\_[2i]{}, & -h&lt;z&lt;0,\[161\]\ -4\^2&C\_[2i,z]{}+4gk\^2C\_[2i]{}=F\_[2i]{}, & z=0,\[162\]\ &C\_[2i]{}=0, & z=-h,\[163\] where $E_{i1},F_{i1}$ are the coefficients of $\sin(ikx-i\om t)$, and $E_{i2},F_{i2}$ are the coefficients of $\cos(ikx-i\om t)$ in the right-hand side of and respectively. Let’s first consider the equation for $C_{22}$, we obtain E\_[22]{}=-m\_2(z+h) ,     F\_[22]{}=4gkm\_2h for which the solution to that satisfies the boundary condition is C\_[22]{}(x\_1,z)=-(z+h)m\_2(z+h) . Upon substitution into we obtain gk =0 therefore, since the coefficient is nonzero then $\d A_2/\d x_1\equiv$0. Physically speaking, this expression says that the amplitude $A_2$ does not change as waves propagate, or in other words, $A_2$ does not take part in the energy exchange. We, therefore, set $A_2$ equal to zero for the rest of the derivation. We use the same approach as above for $C_{21}$. We have &E\_[21]{}=m\_2(z+h) ,\ &F\_[21]{}=-4gkm\_2h -23m\^2+\^2 m\_1hA\_1\^2(x\_1) for which C\_[21]{}(x\_1,z)=(z+h)m\_2(z+h) , and upon substitution into we obtain =A\_1\^2(x\_1), where =-. For $C_{11}(x_1,z)$, we obtain $E_{11}=F_{11}=0$ and therefore the equation for $C_{11}(x_1,z)$ does not provide any extra information on $A_1(x_1)$ and $B_2(x_1)$. For $C_{12}$ we obtain E\_[21]{}=&m\_1(z+h)\ &+ I\^[+]{}(m\_1+m\_2)(z+h)+I\^[-]{}(m\_1-m\_2)(z+h)A\_1(x\_1) B\_2(x\_1)\ F\_[21]{}=&2gkm\_1(z+h) +J A\_1(x\_1)B\_2(x\_1) where I\^[+]{}&=(m\_2+2m\_1)(m\_2\^2+3k\^2-m\_1\^2)\ I\^[-]{}&=(2m\_1-m\_2)(m\_1\^2-3k\^2-m\_2\^2)\ J&=. We obtain C\_[12]{}(x\_1,z)=&-(z+h)m\_1(z+h)\ &++A\_1(x\_1) B\_2(x\_1). Substituting into , we obtain =A\_1(x\_1)B\_2(x\_1), where =-. If we define the actual amplitudes $\A_1=\ep A_1$ and $\B_2=\ep B_2$ (note that $w=\ep w_1+\O(\ep^2)$) then &=\_1\^2\ &=\_1\_2, with the same $\alpha,\beta$ repeated here: &=-,\ &=-. Proof of the Sign of $\alpha/\beta$ =================================== For the sign of $\alpha/\beta$, we have $$\text{sign}(\frac{\alpha}{\beta})=\frac{\cos m_2 h}{\sin m_2 h (4 m_1 \cos^2 m_1 h-3 m_1)-2 m_2 \cos m_2 h \sin 2 m_1 h}.$$ It is equvalent to the sign of $$\tan m_2 h (4 m_1 \cos^2 m_1 h-3 m_1)-2 m_2 \sin 2 m_1 h.$$ Rearranging terms of the above expression gives, $$\label{10001} \cos^2 m_1 h (4 m_1\tan m_2 h -4 m_2\tan m_1 h )-3 m_1\tan m_2 h,$$ After substitution of the disperion relation, $$\omega^2=\frac{gk^2}{m_1}\tan m_1h$$ and $$\omega^2=\frac{gk^2}{m_2}\tan m_2h,$$ can be simplified to $$-3 m_1\tan m_2 h=-3 m_1 m_2\frac{\omega^2}{gk^2}$$ which is less than zero always. [10]{} C. Staquet and J. Sommeria. . , 34:559–593, 2002. Raffaele Ferrari and Carl Wunsch. Ocean circulation kinetic energy: Reservoirs, sources, and sinks. , 41(1):253, 2008. Philip W Boyd. Biogeochemistry: iron findings. , pages 10–11, 2007. Graham Harris. . 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--- address: | Department of Physics, University of California\ Riverside, CA 92521, USA author: - ERNEST MA title: | MODELS OF NEUTRINO MASS AND INTERACTIONS\ FOR NEUTRINO OSCILLATIONS --- \#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{} Neutrino Masses =============== In the minimal standard model, under the gauge group $SU(3)_C \times SU(2)_L \times U(1)_Y$, the leptons transform as: $$\left[ \begin{array} {c} \nu_e \\ e \end{array} \right]_L, \left[ \begin{array} {c} \nu_\mu \\ \mu \end{array} \right]_L, \left[ \begin{array} {c} \nu_\tau \\ \tau \end{array} \right]_L \sim (1, 2, -1/2); ~~~ e_R, ~ \mu_R, ~ \tau_R \sim (1, 1, -1).$$ There is also the Higgs scalar doublet $(\phi^+, \phi^0) \sim (1, 2, 1/2)$ whose nonzero vacuum expectation value $\langle \phi^0 \rangle = v$ breaks $SU(2)_L \times U(1)_Y$ to $U(1)_Q$. Whereas charged leptons acquire masses proportional to $v$, the absence of $\nu_R$ implies that $m_{\nu_i} = 0$. If nonzero neutrino masses are desired (which are of course necessary for neutrino oscillations), then we must ask “What is the nature of this mass?" and “What new physics goes with it?" If $\nu_R$ does not exist, one way to have $m_\nu \neq 0$ is to add a Higgs triplet $(\xi^{++}, \xi^+, \xi^0)$. Each $\nu_L$ then gets a Majorana mass. However, $\langle \xi^0 \rangle$ must be very small, and if the lepton number being carried by $\xi$ is spontaneously violated [@1], the decay of $Z$ to the associated massless Goldstone boson (the triplet Majoron) and its partner would count as two extra neutrinos. Since the effective number of light neutrinos in $Z$ decay is now measured [@2] to be $2.989 \pm 0.012$, the triplet Majoron model is clearly ruled out. If one $\nu_R \sim (1, 1, 0)$ exists for each $\nu_L$, the most general $2 \times 2$ neutrino mass matrix linking $(\bar \nu_L, \bar \nu_R^c)$ to $(\nu_L^c, \nu_R)$ is given by $${\cal M} = \left[ \begin{array} {c@{\quad}c} m_L & m_D \\ m_D & m_R \end{array} \right].$$ If $m_L = 0$ and $m_D << m_R$, we get the famous seesaw mechanism [@3] $$m_\nu \sim {m_D^2 \over m_R}.$$ Here, $\nu_L - \nu_R^c$ mixing is $m_D/m_R$ and $m_R$ is the scale of new physics. In this minimal scenario, new physics enters only through $m_R$, hence there is no other observable effect except for a nonzero $m_\nu$. Actually, $m_D/m_R$ is in principle observable but it is in practice far too small. In general, the mass matrix of Eq. (2) yields two nondegenerate interacting Majorana neutrinos (unless $m_L = m_R = 0$ is maintained exactly). If both eigenvalues are small, the effective number of neutrinos counted in Big Bang Nucleosynthesis may be as high as six, instead of the usual three, depending on the mass splitting and mixing in each case [@4]. The smallness of neutrino masses may be indicative of their radiative origin. Many papers have been written on the subject. For a brief review, see Ref. 5. There are three one-loop mechanisms: the exchange of two scalar bosons with one fermion mass insertion; the exchange of one scalar boson with three fermion mass insertions; and the coupling to a scalar boson which gets a radiative vacuum expectation value through a fermion loop with five mass insertions. A prime example of the first mechanism is the Zee model [@6]. Here the minimal standard model is extended to include a charged scalar singlet $\chi^+$ and a second scalar doublet $(\eta^+, \eta^0)$. We then have the coupling $$f_{ij} \chi^+ (\nu_i l_j - l_i \nu_j),$$ which by itself would require $\chi^+$ to have lepton number $-2$. However, this model also allows the cubic scalar coupling $$\chi^- (\phi^+ \eta^0 - \phi^0 \eta^+),$$ hence lepton number is broken explicitly. A radiative Majorana mass matrix is thus obtained through the exchange and mixing of $\chi^+$ and the physical linear combination of $\phi^+$ and $\eta^+$. Since $f_{ij}$ of Eq. (4) is zero for $i = j$ and $\phi^+$ couples $\nu_i$ to $l_i$ with strength proportional to $m_{l_i}$ which is also the one fermion mass insertion required, the $3 \times 3$ neutrino mass matrix for $\nu_e$, $\nu_\mu$ and $\nu_\tau$ is of the form $${\cal M}_\nu \propto \left[ \begin{array} {c@{\quad}c@{\quad}c} 0 & 0 & f_{e \tau} m_\tau^2 \\ 0 & 0 & f_{\mu \tau} m_\tau^2 \\ f_{e \tau} m_\tau^2 & f_{\mu \tau} m_\tau^2 & 0 \end{array} \right] + {\cal O} (m_\mu^2).$$ This means that $\nu_\tau$ is almost degenerate with a linear combination of $\nu_\mu$ and $\nu_e$ in this model. This may have a practical application in present neutrino-oscillation phenomenology [@7]. There are also three two-loop mechanisms: the exchange of three scalar bosons which are tied together by a cubic coupling; the exchange of two $W$ bosons; and the exchange of $W_L$ and $W_R$ which mix at the one-loop level. The second mechanism [@8] is unique in that it requires only one additional $\nu_R$ beyond the standard model. In this specific case, one $\nu_L$ gets a seesaw mass and the other two get two-loop masses proportional to this mass and as functions of the charged-lepton masses with double GIM suppression [@9]. A detailed analytical and numerical study of this mechanism has been made [@10]. Finally let me return to the triplet-Higgs mechanism. If lepton number is violated explicitly by the coupling of $\xi$ to the scalar doublet $\phi$, then one may let $\xi$ be very heavy and integrate it out to obtain the following effective nonrenormalizable interaction: $${1 \over M} [\phi^0 \phi^0 \nu_i \nu_j + \phi^+ \phi^0 (\nu_i l_j + l_i \nu_j) + \phi^+ \phi^+ l_i l_j] + h.c.$$ For $M \sim 10^{13}$ GeV, one gets $m_\nu \sim$ few eV. This is the most economical solution and could also be a realistic model of leptogenesis [@11] in the early universe which gets converted at the electroweak phase transition into the present observed baryon asymmetry. Neutrino Oscillations ===================== Present experimental evidence for neutrino oscillations [@12] includes the solar $\nu_e$ deficit which requires $\Delta m^2$ of around $10^{-5}$ eV$^2$ for the MSW explanation or $10^{-10}$ eV$^2$ for the vacuum-oscillation solution, the atmospheric neutrino deficit in the ratio $\nu_\mu + \bar \nu_\mu / \nu_e + \bar \nu_e$ which implies a $\Delta m^2$ of around $10^{-2}$ eV$^2$, and the LSND experiment which indicates a $\Delta m^2$ of around 1 eV$^2$. Three different $\Delta m^2$ necessitate four neutrinos, but the invisible width of the $Z$ boson as well as Big Bang Nucleosynthesis allow only three. If all of the above-mentioned experiments are interpreted correctly as due to neutrino oscillations, we are faced with a theoretical challenge in trying to understand how three can equal four. I will focus on addressing this issue rather than trying to review the many theoretical models for the three known neutrinos. Three Neutrinos and One Light Singlet ===================================== One possibility is that there is a light singlet neutrino $\nu_S$ in addition to the three known doublet neutrinos $\nu_e$, $\nu_\mu$, and $\nu_\tau$. This is necessary so that it is not counted in the effective number of neutrinos in $Z$ decay [@2]. On the other hand, it has to mix with the doublet neutrinos for it to be relevant to oscillation experiments. Hence it is also contrained [@4] by Big Bang Nucleosynthesis. Using all available data, a model-independent analysis [@13] shows that the $4 \times 4$ neutrino mass matrix must separate approximately into two blocks: one for $\nu_e - \nu_S$ and the other for $\nu_\mu - \nu_\tau$, the latter with large mixing. An example of a specific model of this kind already exists [@14]. The neutrino interaction eigenstates are related to the mass eigenstates by $$\left[ \begin{array} {c} \nu_S \\ \nu_e \\ \nu_\mu \\ \nu_\tau \end{array} \right] = \left[ \begin{array} {c@{\quad}c@{\quad}c@{\quad}c} -s & c & s''/\sqrt 2 & s''/\sqrt 2 \\ c & s & -s'/\sqrt 2 & s'/\sqrt 2 \\ -s' & 0 & -1/\sqrt 2 & 1/\sqrt 2 \\ 0 & -s'' & 1/\sqrt 2 & 1/\sqrt 2 \end{array} \right] \left[ \begin{array} {c} \nu_1 \\ \nu_2 \\ \nu_3 \\ \nu_4 \end{array} \right],$$ where $m_1 = 0$, $m_2 \sim 2.5 \times 10^{-3}$ eV, $m_3 \sim m_4 \sim 2.5$ eV, with $\Delta m_{34}^2 \sim 1.8 \times 10^{-2}$ eV$^2$; $s \sim s' \sim 0.04$, but $s''$ is undetermined. Note that $m_{\nu_e} < m_{\nu_S}$ is necessary for the MSW solution [@15] of the solar neutrino deficit. Note also that $\nu_\mu$ and $\nu_\tau$ are pseudo-Dirac partners, hence the mixing angle for atmospheric neutrino oscillations is 45 degrees. What is the nature of this light singlet? and how does it mix with the usual neutrinos? There have been some discussions on these questions in the past two or three years. One idea [@16] is that it is the fermion partner of the massless Goldstone boson of a sponatneously broken global symmetry, such as lepton number (hence a Majorino) in supersymmetry. Another [@17] is that it is the fermion partner of a scalar field corresponding to a flat direction (hence a modulino) in the supersymmetric Higgs potential. If the standard model is extended to include a mirror $[SU(2) \times U(1)]'$ sector, then $\nu_S$ may be identified as a mirror neutrino, either in a theory where the mirror symmetry is broken [@18] or one where it is exact [@19]. In the latter case, maximal mixing between the three known neutrinos and their mirror counterparts would occur and Big Bang Nucleosynthesis would count six neutrinos under normal conditions. Both questions can be answered naturally in a model [@20] based on $E_6$ inspired by superstring theory. In the fundamental [**27**]{} representation of $E_6$, outside the 15 fields belonging to the minimal standard model, there are 2 neutral singlets. One ($N$) is identifiable with the right-handed neutrino because it is a member of the [**16**]{} representation of $SO(10)$; the other ($S$) is a singlet also under $SO(10)$. In the reduction of $E_6$ to $SU(3)_C \times SU(2)_L \times U(1)_Y$, an extra U(1) gauge factor may survive down to the TeV energy scale. It could be chosen such that $N$ is trivial under it, but $S$ is not. This means that $N$ is allowed to have a large Majorana mass so that the usual seesaw mechanism works for the three doublet neutrinos. At the same time, $S$ is protected from having a mass by the extra U(1) gauge symmetry, which I call $U(1)_N$. However, it does acquire a small mass from an analog of the usual seesaw mechanism because it can couple to doublet neutral fermions which are present in the [**27**]{} of $E_6$ outside the [**16**]{} of $SO(10)$. Renaming $S$ as $\nu_S$, the $3 \times 3$ mass matrix spanning $\nu_S$, $\nu_E$, and $N_E^c$ is given by $${\cal M} = \left[ \begin{array} {c@{\quad}c@{\quad}c} 0 & m_1 & m_2 \\ m_1 & 0 & m_E \\ m_2 & m_E & 0 \end{array} \right].$$ Hence $m_{\nu_S} \sim 2 m_1 m_2/m_E$, which is a singlet-doublet seesaw rather than the usual doublet-singlet seesaw. Furthermore, the mixing of $\nu_S$ with the doublet neutrinos is also possible through these extra doublet neutral fermions. The spontaneous breaking of $U(1)_N$ is not possible without also breaking the supersymmetry, hence both are assumed to occur at the TeV energy scale, resulting in a rich $Z'$ and Higgs phenomenology [@21]. Three Neutrinos and One Anomalous Interaction ============================================= If one insists on keeping only the usual three neutrinos and yet try to accommodate all present data, how far can one go? It has been pointed out by many authors [@22] that both solar and LSND data can be explained, as well as most of the atmospheric data except for the zenith-angle dependence. It is thus worthwhile to consider the following scenario [@23] whereby a possible anomalously large $\nu_\tau$-quark interaction may mimic the observed zenith-angle dependence of the atmospheric data. Consider first the following approximate mass eigenstates: $$\begin{aligned} \nu_1 &\sim& \nu_e ~~~ {\rm with} ~ m_1 \sim 0, \\ \nu_2 &\sim& c_0 \nu_\mu + s_0 \nu_\tau ~~~ {\rm with} ~ m_2 \sim 10^{-2} ~{\rm eV}, \\ \nu_3 &\sim& -s_0 \nu_\mu + c_0 \nu_\tau ~~~ {\rm with} ~ m_3 \sim 0.5 ~{\rm eV},\end{aligned}$$ where $c_0 \equiv \cos \theta_0$, $s_0 \equiv \sin \theta_0$, and $\theta_0$ is not small. Allow $\nu_1$ to mix with $\nu_3$ with a small angle $\theta '$ and the new $\nu_1$ to mix with $\nu_2$ with a small angle $\theta$, then the LSND data can be explained with $\Delta m^2 \sim 0.25$ eV$^2$ and $2 s_0 s' c' \sim 0.19$ and the solar data can be understood as follows. Consider the basis $\nu_e$ and $\nu_\alpha \equiv c_0 \nu_\mu + s_0 \nu_\tau$. Then $$-i {d \over {dt}} |\nu \rangle_{e,\alpha} = \left( p + {{\cal M}^2 \over {2p}} \right) |\nu \rangle_{e,\alpha},$$ where $${\cal M}^2 = {\cal U} \left[ \begin{array} {c@{\quad}c} 0 & 0 \\ 0 & m_2^2 \end{array} \right] {\cal U}^\dagger + \left[ \begin{array} {c@{\quad}c} A+B & 0 \\ 0 & B+C \end{array} \right].$$ In the above, $A$ comes from the charged-current interaction of $\nu_e$ with $e$, $B$ from the neutral-cuurent interaction of $\nu_e$ and $\nu_\alpha$ with the quarks and electrons, and $C$ from the assumed anomalous $\nu_\tau$-quark interaction. Let $${\cal U} = \left( \begin{array} {c@{\quad}c} c & s \\ -s & c \end{array} \right),$$ then the resonance condition is $$m_2^2 \cos 2 \theta - A + C = 0,$$ where [@24] $$A - C = 2 \sqrt 2 G_F (N_e - s_0^2 \epsilon'_q N_q) p.$$ In order to have a large $\epsilon'_q$ and yet satisfy the resonance condition for solar-neutrino flavor conversion, $m_2$ should be larger than its canonical value of $2.5 \times 10^{-3}$ eV, and $\epsilon'_q$ should be negative. \[If $\epsilon'_q$ comes from $R$-parity violating squark exchange, then it must be positive, in which case an inverted mass hierarchy, [*i.e.*]{} $m_2 < m_1$ would be needed. If it comes from vector exchange, it may be of either sign.\] Assuming as a crude approximation that $N_q \simeq 4 N_e$ in the sun, the usual MSW solution with $\Delta m^2 = 6 \times 10^{-6}$ eV$^2$ is reproduced here with $$s_0^2 \epsilon'_q \simeq -3.92 = -4.17 (m_2^2/10^{-4}{\rm eV}^2) + 0.25.$$ The seemingly arbitrary choice of $\Delta m_{21}^2 \sim 10^{-4}$ eV$^2$ is now sen as a reasonable value so that $\epsilon'_q$ can be large enough to be relevant for the following discussion on the atmospheric neutrino data. Atmospheric neutrino oscillations occur between $\nu_\mu$ and $\nu_\tau$ in this model with $\Delta m^2_{32} \sim 0.25$ eV$^2$, the same as for the LSND data, but now it is large relative to the $E/L$ ratio of the experiment, hence the factor $\cos \Delta m^2 (L/2E)$ washes out and $$P_0 (\nu_\mu \rightarrow \nu_\mu) = 1 - {1 \over 2} \sin^2 2 \theta_0 \simeq 0.66 ~~ {\rm for} ~~ s_0 \simeq 0.47.$$ In the standard model, this would hold for all zenith angles. Hence it cannot explain the present experimental evidence that the depletion is more severe for neutrinos coming upward to the detector through the earth than for neutrinos coming downward through only the atmosphere. This zenith-angle dependence appears also mostly in the multi-GeV data and not in the sub-GeV data. It is this trend which determines $\Delta m^2$ to be around $10^{-2}$ eV$^2$ in this case. As shown below, the hypothesis that $\nu_\tau$ has anomalously large interactions with quarks will mimic this zenith-angle dependence even though $\Delta m^2$ is chosen to be much larger, [*i.e.*]{} 0.25 eV$^2$. Consider the basis $\nu_\mu$ and $\nu_\tau$. Then the analog of Eq. (13) holds with Eq. (14) replaced by $${\cal M}^2 = {\cal U}_0 \left[ \begin{array} {c@{\quad}c} 0 & 0 \\ 0 & m_3^2 \end{array} \right] {\cal U}_0^\dagger + \left[ \begin{array} {c@{\quad}c} B & 0 \\ 0 & B + C \end{array} \right].$$ The resonance condition is then $$m_3^2 \cos 2 \theta_0 + C = 0,$$ where $N_q$ in $C$ now refers to the quark number density inside the earth and the factor $s_0^2$ in Eq. (17) is not there. If $C$ is large enough, the probability $P_0$ would not be the same as the one in matter. Using the estimate $N_q \sim 9 \times 10^{30}$ m$^{-3}$ and defining $$X \equiv \epsilon'_q E_\nu/(10 ~{\rm GeV}),$$ the effective mixing angles in matter are given by $$\begin{aligned} \tan 2 \theta_m^E &=& {{\sin 2 \theta_0} \over {\cos 2 \theta_0 + 0.091 X}} ~~ {\rm for} ~ \nu, \\ \tan 2 \bar \theta_m^E &=& {{\sin 2 \theta_0} \over {\cos 2 \theta_0 - 0.091 X}} ~~ {\rm for} ~ \bar \nu.\end{aligned}$$ For sub-GeV neutrinos, $X$ is small so matter effects are not very important. For multi-GeV neutrinos, $X$ may be large enough to satisfy the resonance condition of Eq. (21). Assuming adiabaticity, the neutrino and antineutrino survival probabilities are given by $$\begin{aligned} P(\nu_\mu \rightarrow \nu_\mu) &=& {1 \over 2} (1 + \cos 2 \theta_0 \cos 2 \theta_m^E), \\ \bar P(\bar \nu_\mu \rightarrow \bar \nu_\mu) &=& {1 \over 2} (1 + \cos 2 \theta_0 \cos 2 \bar \theta_m^E).\end{aligned}$$ Since $\sigma_\nu \simeq 3 \sigma_{\bar \nu}$, the observed ratio of $\nu + \bar \nu$ events is then $$P_m \simeq {{3 r P + \bar P} \over {3 r + 1}},$$ where $r$ is the ratio of the $\nu_\mu$ to $\bar \nu_\mu$ flux in the upper atmosphere. The atmospheric data are then interpreted as follows. For neutrinos coming down through only the atmosphere, $P_0 = 0.66$ applies. For neutrinos coming up through the earth, $P_m \simeq P_0 \simeq 0.66$ as well for the sub-GeV data. However, for the multi-GeV data, if $X = -15$, then $P = 0.31$ and $\bar P = 0.76$, hence $P_m$ is lowered to 0.39 if $r = 1.5$ or 0.42 if $r = 1.0$. The apparent zenith-angle dependence of the data may be explained. Conclusion and Outlook ====================== If all present experimental indications of neutrino oscillations turn out to be correct, then either there is at least one sterile neutrino beyond the usual $\nu_e$, $\nu_\mu$, and $\nu_\tau$, or there is an anomalously large $\nu_\tau$-quark interaction. The latter can be tested at the forthcoming Sudbury Neutrino Observatory (SNO) which has the capability of neutral-current detection. The predicted $\Delta m^2$ of 0.25 eV$^2$ in $\nu_\mu$ to $\nu_e$ and $\nu_\tau$ oscillations will also be tested at the long-baseline neutrino experiments such as Fermilab to Soudan 2 (MINOS), KEK to Super-Kamiokande (K2K), and CERN to Gran Sasso. More immediately, the new data from Super-Kamiokande, Soudan 2, and MACRO on $\nu_\mu + \bar \nu_\mu$ events through the earth should be analyzed for such an effect. For a zenith angle near zero, the $\Delta m^2 \sim 10^{-2}$ eV$^2$ oscillation scenario should have $R \sim 1$, whereas the $\Delta m^2 \sim 0.25$ eV$^2$ oscillation scenario (with anomalous interaction) would have $R = P_0 \sim 0.66$. Furthermore, if $\nu$ and $\bar \nu$ can be distinguished (as proposed in the HANUL experiment), then to the extent that $CP$ is conserved, matter effects can be isolated. Neutrino physics is on the verge of major breakthroughs. New experiments in the next several years will be decisive in leading us forward in its theoretical understanding, and may even discover radically new physics beyond the standard model. Acknowledgments {#acknowledgments .unnumbered} =============== I thank Biswarup Mukhopadhyaya and all the other organizers for their great hospitality and a stimulating workshop. This work was supported in part by the U. S. Department of Energy under Grant No. DE-FG03-94ER40837. References {#references .unnumbered} ========== [99]{} G. B. 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--- abstract: | We present the results of a systematic study of the evolution of low- and intermediate-mass X-ray binaries (LMXBs and IMXBs). Using a standard Henyey-type stellar-evolution code and a standard model for binary interactions, we have calculated 100 binary evolution sequences containing a neutron star and a normal-type companion star, where the initial mass of the secondary ranges from 0.6 to 7[[$\,M_\odot$]{}]{} and the initial orbital period from $\sim 4\,$hr to $\sim 100\,$d. This grid of models samples the entire range of parameters one is likely to encounter for LMXBs and IMXBs. The sequences show an enormous variety of evolutionary histories and outcomes, where different mass-transfer mechanisms dominate in different phases. Very few sequences resemble the classical evolution of cataclysmic variables, where the evolution is driven by magnetic braking and gravitational radiation alone. Many systems experience a phase of mass transfer on a thermal timescale and may briefly become detached immediately after this phase (for the more massive secondaries). In agreement with previous results (Tauris & Savonije 1999), we find that all sequences with (sub-)giant donors up to $\sim 2{{\mbox}{$\,M_\odot$}}$ are stable against dynamical mass transfer. Sequences where the secondary has a radiative envelope are stable against dynamical mass transfer for initial masses up to $\sim 4{{\mbox}{$\,M_\odot$}}$. For higher initial masses, they experience a delayed dynamical instability after a stable phase of mass transfer lasting up to $\sim 10^6\,$yr. Systems where the initial orbital period is just below the bifurcation period of $\sim 18\,$hr evolve towards extremely short orbital periods (as short as $\sim 10\,$min). For a 1[[$\,M_\odot$]{}]{} secondary, the initial period range that leads to the formation of ultracompact systems (with minimum periods less than $\sim 40\,$min) is 13 to 18hr. Since systems that start mass transfer in this period range are naturally produced as a result of tidal capture, this may explain the large fraction of ultracompact LMXBs observed in globular clusters. The implications of this study for our understanding of the population of X-ray binaries and the formation of millisecond pulsars are also discussed. author: - 'Ph. Podsiadlowski' - 'S. Rappaport and E. Pfahl' title: 'Evolutionary Binary Sequences for Low- and Intermediate-Mass X-Ray Binaries' --- Introduction ============ Low-mass X-ray binaries (LMXBs) were discovered nearly 40 years ago, and there are now $\sim 100$ known in the Galaxy. Based on their short orbital periods of $\la 10\,$d and the absence of luminous companion stars, it is generally inferred that the donor stars in these systems are typically low-mass stars (i.e., $\la 1 M_{\odot}$). However, to-date Cyg X-2 provides the only case in which a low mass for the donor star has actually been confirmed dynamically (Casares, Charles, & Kuulkers 1998; Orosz & Kuulkers 1998). Nonetheless, a fairly compelling picture of LMXBs has emerged over the years, wherein a low-mass donor star, of varying evolutionary states, transfers mass through the inner Lagrange point to a neutron star (Lewin, van Paradijs, & van den Heuvel 1995). Only relatively recently, however, has attention been focused on the possibility that many, or perhaps most, of the current LMXBs descended from systems with intermediate-mass donor stars (hereafter IMXBs). It has long been conventional wisdom that, if the donor star in an X-ray binary is significantly higher in mass than the accreting neutron star, mass transfer would be unstable on a dynamical timescale, and therefore such systems would not survive. The first systematic study which indicated that such a view was too simplistic was carried out by Pylyser & Savonije (1988, 1989), who considered compact binaries with initial donor masses up to $2\,M_{\odot}$ and initial orbital periods of $\la 2\,$d. Tauris & Savonije (1999) extended this work to show that, even if the donor star is a giant, dynamical mass transfer is avoided provided that the initial donor mass is $\la 2\,M_{\odot}$. More recent theoretical work in trying to understand the origin of the “LMXB” Cyg X-2 and, in particular, the high intrinsic luminosity of the donor star indicates that the mass of the donor must originally have been substantially larger ($\sim 3.5\,M_{\odot}$) than the current value of $\sim 0.6{{\mbox}{$\,M_\odot$}}$ (King & Ritter 1999; Podsiadlowski & Rappaport 2000). The case of Cyg X-2 is particularly important since it provides direct observational evidence that, even when the mass-transfer rate exceeds the Eddington rate by several orders of magnitude, such intermediate-mass systems can survive this phase of high mass-transfer by ejecting most of the transferred mass and subsequently mimick LMXBs. Independently, Davies & Hansen (1998) have suggested that IMXBs may be the progenitors of recycled pulsars in globular clusters. All of these recent developments have led to a resurgence in interest in IMXBs (also see Kolb et al. 2000; Tauris, van den Heuvel, & Savonije 2000). In order to approach this problem in a more systematic way, we have carried out binary stellar evolution calculations which cover a broad grid of starting binary parameters, specifically the mass of the donor star, $M_2$, and the orbital period at the start of the mass-transfer phase, $P_{\rm orb}$. At fixed $M_2$, the value of initial orbital period effectively determines the evolutionary state of the donor star. This library of models comprises 17 different donor-star masses between $0.6\, M_{\odot}$ and $7\,M_{\odot}$, and up to 8 different evolutionary states (or, alternatively, values of $P_{\rm orb}$). The initial orbital periods span the range from $\sim 4$ hours to 100 days. The starting parameter values associated with this library of models are summarized in Figure 1. In this figure we show the initial binary parameters in a Hertzsprung-Russell (H-R) diagram for the companion star. Evolutionary tracks for stars of the same mass, but which evolve as single stars are superposed for reference. Contours of constant initial orbital period for the case of Roche lobe overflow onto a neutron star of $1.4\,M_{\odot}$ are also included. In our binary evolution models, once mass transfer has commenced, it is sustained by either (i) systemic angular-momentum losses (e.g., magnetic braking or gravitational radiation), or (ii) expansion of the donor star due to nuclear and/or thermal evolution. The mass transfer may proceed on any of the timescales implicit in the mechanisms listed above, or may in fact proceed on a dynamical timescale under certain conditions. All of these are explored in detail in this study. During the mass transfer phases, these objects will generally appear as X-ray sources (possibly LMXBs, IMXBs). These sources could be steady or transient, depending on the size and temperature of the accretion disk and on the mass transfer rate through the disk. At the end of the mass-transfer phase, many of these systems will become binary radio pulsars, wherein the neutron star has been spun up to high rotation rates by the accretion of matter. One of the main objectives of this study is to provide a library of models that covers the whole range of parameters for LMXBs and IMXBs using a self-consistent set of binary calculations and to discuss the various physical phenomena encountered in the process. In a subsequent study (Pfahl, Podsiadlowski, & Rappaport 2001), we will use this library to study the population of LMXBs and IMXBs as a whole by integrating them into a binary population synthesis code and by comparing the results with the observed population. In §2 of this paper we describe in detail the stellar evolution code and the binary model used in this study. In §3 we discuss the various types of binary sequences encountered and compare them to previous studies. In §4 we consider the end products of this evolution and present a new case study for the formation of ultracompact X-ray binaries. Finally in §5 and 6, we discuss the implications of these results for the population of X-ray binaries and the formation of binary millisecond pulsars. Binary Calculations =================== The Stellar-Evolution Code -------------------------- All calculations were carried out with an up-to-date, standard Henyey-type stellar evolution code (Kippenhahn, Weigert, & Hofmeister 1967), which uses OPAL opacities (Rogers & Iglesias 1992) complemented with those from Alexander & Ferguson (1994) at low temperatures[^1]. We use solar metallicity ($Z=0.02$), a mixing-length parameter $\alpha=2$ and assume 0.25 pressure scale heights of convective overshooting from the core, following the recent calibration of this parameter by Schröder, Pols, & Eggleton (1997) and Pols et al. (1997). To include the effects of pressure ionization in the equation of state, which is important for low-mass stars, we adopted the thermodynamically self-consistent formalism of Eggleton, Faulkner, & Flannery (1973) and calibrated the continuum depression term so that our models for single stars compare well with the detailed models of Baraffe et al. (1998). Our models agree with these models typically within a few per cent in radius (for masses as low as $0.1\,M_{\odot}$), although their luminosities may differ by as much as $\sim 20$ per cent. This may be due, in part, to the fact that we required a helium abundance of 0.295 for $\alpha=2$ to produce a good solar model at the present age of the Sun, as compared to their value of 0.282 for $\alpha=1.9$. Our single-star models become fully convective at a mass of $0.351\,M_{\odot}$. The Binary-Evolution Code ------------------------- Each of the binaries initially consists of a neutron-star primary with an initial mass $M_1 = 1.4\,M_{\odot}$ and a normal-type secondary of mass $M_2$. The effective radius of the Roche lobe, $R_{\rm L}$, is calculated with the formula of Eggleton (1983), $$R_{\rm L} = a\, {0.49\,q^{-2/3}\over 0.6\,q^{-2/3} + \ln (1+q^{-1/3})},$$ where $a$ is the orbital separation and $q=M_1/M_2$ the mass ratio of the binary components. To calculate the mass-transfer rate, $\dot{M}$, we adopted the prescription of Ritter (1988), $$\dot{M} = \dot{M}_0\,e^{R-R_{\rm L}\over H_{\rm p}},$$ where $R$ is the radius of the secondary and $H_{\rm p}$ the pressure scale height at its surface. The constant $\dot{M}_0$ is calculated according to the model of Ritter (1988). The solution of this equation requires an iteration in the stellar models. We follow the method described by Braun (1997), which uses a combined secant/bisection method (the Brent method; see, e.g., Press et al. 1992). Angular-momentum loss due to gravitational radiation is calculated according to the standard formula (Landau & Lifshitz 1959; Faulkner 1971), $${d\,\ln J_{\rm GR}\over dt} = -{32\over 5}\,{G^3\,c^5}{M_1 M_2 (M_1+M_2)\over a^4},$$ where $G$ and $c$ are the gravitational constant and vacuum speed of light, respectively. To calculate the angular-momentum loss due to magnetic braking, we use the prescription of Rappaport, Verbunt, & Joss (1993) (their eq. 36 with $\gamma =4$), which is based on the magnetic-braking law of Verbunt & Zwaan (1981), $${d\,J_{MB}\over dt} = -3.8\times 10^{-30}\,M_2\,R^4\,\omega^3 \mbox{\ \ \rm dyn\,\,cm}. $$ In this equation $\omega$ is the angular rotation frequency of the secondary, assumed to be synchronized with the orbit. We only include full magnetic braking if the secondary has a sizable convective envelope, taken to be at least $2\,\%$ in mass (see also Pylyser & Savonije 1988). For secondaries with convective envelopes smaller than $2\,\%$, we reduce the efficiency of magnetic braking by an [*ad hoc*]{} factor $\exp\{-0.02/q_{\rm conv} + 1\}$, where $q_{\rm conv}$ is the fractional mass of the convective envelope. We also assume that magnetic braking stops when the secondaries become fully convective (Rappaport et al. 1983; Spruit & Ritter 1983). We do not follow the tidal evolution before the onset of mass transfer (see, e.g., Witte & Savonije 2001), but start our binary sequences assuming that the systems have already circularized when the secondaries are close to filling their Roche lobes; to be precise we start our calculations when the mass transfer rate as given by equation (2) is $\sim 10^{-14}\,M_{\odot}\,$yr$^{-1}$. For each sequence, we need to specify what fraction, $\beta$, of the mass lost by the donor is accreted by the neutron star and the specific angular momentum of any matter that is lost from the system. We scale the latter with the specific orbital angular momentum of the neutron star, i.e., assume that the angular momentum loss due to mass loss from the system is given by $${d\,J_{ML}\over dt} = - \alpha\,(1-\beta)\,a_1^2\,\omega\,\dot{M},$$ where $\alpha$ is an adjustable parameter and $a_1$ the orbital radius of the neutron star. The change of the orbital separation due to the systemic mass loss alone can be calculated analytically according to $${a^1\over a^0} = \left({M_2^1\over M_2^0}\right)^{C_1}\, \left({M_1^1\over M_1^0}\right)^{C_2}\,\left({M_1^1+M_2^1\over M_1^0 + M_2^0}\right)^{C_3},$$ where superscripts 0 indicate initial values and superscripts 1 final values and where the exponents are given by $$\begin{aligned} C_1&=&-2\nonumber\\ C_2&=&-2-2\alpha\,(1-\beta)/\beta\\ C_3&=&1-2\alpha\nonumber.\end{aligned}$$ When $\beta=0$, equation (6) has to be replaced by $$\begin{aligned} {a^1\over a^0} &=& \left({M_2^1\over M_2^0}\right)^{C_1}\, \,\left({M_1+M_2^1\over M_1 + M_2^0}\right)^{C_3}\,\nonumber\\ \noalign{\vspace{5pt}} &&\hspace{1cm}\times \exp\left\{2\alpha\,\left({M_2^1-M_2^0\over M_1}\right)\right\}.\end{aligned}$$ In all of our sequences, we take $\alpha$ to be 1, which implicitly assumes that all the mass lost from the system is lost from the neighborhood of the neutron star (or its accretion disk), and set $\beta$, somewhat arbitrarily, equal to 0.5. In addition, we limit the maximum accretion rate onto the neutron star to the Eddington accretion rate, taken to be $\dot{M} = 2\times 10^{-8}{{\mbox}{$\,M_\odot\,$yr$^{-1}$}}$ and kept constant throughout each run. In our calculations with relatively massive secondaries, the mass-transfer rate can exceed the Eddington accretion rate by many orders of magnitude. Most of this excess mass must be lost from the system, as the case of Cyg X-2 has demonstrated. This mass loss may, for example, occur in the form of a relativistic jet from the accreting neutron star or a radiation-pressure driven wind from the outer parts of the accretion disk (see, e.g., King & Begelman 1999). Evidence for both of these processes is seen in the X-ray binary SS 433 (Blundell et al. 2001), the only system presently known to be in an extreme super-Eddington mass-transfer phase. Since the pressure scale height at the surface of the donor is generally a small fraction of the stellar radius (often as low as $\sim 10^{-4}\,R$), the calculation of the mass-transfer rate according to equation (2) requires that the radius of the star be calculated to very high precision. To avoid discontinuous changes in radius and hence $\dot{M}$, it is important that the chemical profile of the initial star is well resolved and that abrupt changes in the surface abundances (for example, as a result of dredge-up) are avoided. To calculate mass loss efficiently, we introduced a moving mesh in the outermost 5% of the mass of the star. We also assumed that the outermost $10^{-4}$ of the envelope mass of the donor star was in thermal equilibrium. This is necessary since in each time step we typically take off a much larger fraction of the mass of the star and since the structure variables often change by a large factor in this outermost layer. It is also justified since the thermal timescale of this layer is much shorter than any mass-loss timescale encountered in this study. (We have extensively tested that our results are not sensitive to these assumptions, at least for the mass-loss rates obtained, where we generally limited the maximum mass-loss rate to $10^{-4}\,M_{\odot}\,$yr$^{-1}$.) Despite of these precautions, our calculated mass-loss rates are occasionally subject to numerical oscillations. These tend to be almost negligible for stars with radiative envelopes (typically less than a few percent), but can be several 10’s of per cent for stars with convective envelopes and occasionally much larger for evolved giants (in particular during dredge-up phases). We note that, in all of the plots of $\dot{M}$ presented in this paper, these oscillations, which do not affect the secular evolution of the systems, have been averaged out. Tests and Comparisons --------------------- To test our binary evolution code, we chose to calculate the standard evolution of a cataclysmic variable (CV), initially consisting of a white dwarf of $0.6\,M_{\odot}$ and a secondary of $0.8\,M_{\odot}$ (here we assumed that all of the mass transferred from the secondary was lost from the system). In this calculation, the system experienced a period gap between 2.4 and 3.1hr, somewhat smaller than the observed gap (Ritter & Kolb 1998), but consistent with previous results for our adopted magnetic-braking law (Rappaport, Verbunt, & Joss 1983). The minimum period in this calculation was 75min, somewhat longer than the minimum period found in the most detailed studies of CVs (see, e.g., Kolb & Baraffe 1999; Howell, Nelson, & Rappaport 2001). We also compared our calculations to other recent similar binary calculations by a number of different authors, in particular the calculations by Pylyser & Savonije (1988, 1989); Han, Tout, & Eggleton (2000); Langer et al. (2000); Kolb et al. (2000); Tauris et al. (2000). For comparable models, we generally find excellent agreement between our calculations and the calculations of these authors. The only significant discrepancy to note is the early case B calculation for Cyg X-2 by Kolb et al.(2000), where the secondary has an initial mass of 3.5[[$\,M_\odot$]{}]{} and has just evolved off the main sequence at the beginning of mass transfer. While our early-case B model (see Podsiadlowski & Rappaport 2000) is in excellent agreement with a similar calculation by Tauris et al.(2000), in the Kolb et al. model, the early super-Eddington phase is much longer, and as a consequence the mass-transfer rate in the subsequent slower phase about an order of magnitude lower than in our model. We do not know the reason for this discrepancy, whether it has to do with the treatment of mass loss at these very high rates ($\sim 10^{-5}\,M_{\odot}\,$yr$^{-1}$) or whether it is caused by differences in the structure of the initial models (U. Kolb 2000, private communication). For example Kolb et al. (2000) do not include convective overshooting in their calculations; this produces a different chemical profile just outside the hydrogen-exhausted core, which may affect the evolution of the secondary (the evolutionary track of their secondary in the H-R diagram is indeed quite different). Until this discrepancy is resolved, we note that there is some uncertainty in the modeling of these systems with extreme mass-transfer rates. \[fig:2a\] Results of Binary Calculations ============================== Altogether we carried out 100 binary stellar evolution calculations with initial secondary masses ranging from $0.6\,M_{\odot}$ to $7\,M_{\odot}$ and covering, in a fairly uniform manner, all evolutionary stages likely to be encountered for LMXBs/IMXBs, with orbital periods from 4hr to 100d (see Fig. 1). In Figure 2 we present the evolutionary tracks of these calculations both in a secondary mass – orbital period diagram ($\log M_2\,$–$\,\log P_{\rm orb}$; Fig. 2a) and in a traditional Hertzsprung-Russell (H-R) diagram (Fig. 2b), where the color coding indicates how much time a system spends in a particular region of the diagrams. What these figures do not show very well, however, is the actual variety in these sequences. Some 70 of the 100 sequences are qualitatively different with respect to the importance and the order of different mass-transfer driving mechanisms, the occurrence of detached phases, the final end products, etc. Indeed there are very few sequences that resemble the classical CV evolution where mass-transfer is driven solely by gravitational radiation and magnetic braking. Instead of presenting all of these sequences in detail, we will discuss the various physical phenomena encountered and illustrate them with particular evolutionary sequences. In the appendix we present the key characteristics of each sequence in tabular form. As Figure 2 shows, the sequences can be broadly divided into three classes: (1) and (2) systems evolving to long periods and short periods, respectively, and (3) more massive systems experiencing dynamical mass transfer and spiral-in (the short yellow tracks). The systems evolving towards short and long orbital periods are separated by the well-known bifurcation period that has been studied by several authors in the past (see, in particular, Tutukov et al. 1985; Pylyser & Savonije 1988; Ergma 1996; Ergma & Sarna 1996). For our binary model, the bifurcation period occurs around 18hr for a 1[[$\,M_\odot$]{}]{} model (see §4.2 for a systematic case study). This implies that all 1[[$\,M_\odot$]{}]{} models that start mass transfer on or just off the main sequence evolve towards short periods, while for the more massive secondaries only relatively unevolved secondaries do so (see Fig. 1), in agreement with the findings of Pylyser & Savonije (1988). However, the value of the bifurcation period and the behavior of the evolutionary tracks near the bifurcation period is very sensitive to the model assumptions, in particular the magnetic-braking law (Pylyser & Savonije 1988 and § 4.2) and the assumptions about mass loss from the system (Ergma 1996; Ergma & Sarna 1996). Because of the strong divergence of tracks below and about the bifurcation period, one would expect very few systems with final orbital periods near the bifurcation period (Pylyser & Savonije 1988) unless a system started its evolution very close to it initially (Ergma 1996). Figure 2 also shows that, for the more massive systems, the initial evolution is very rapid. As a direct consequence, very few systems should be observable in this early rapid phase, and X-ray binaries are most likely to have a relatively low-mass secondary when they are observed at the present epoch, even if they had a much more massive companion initially. Low-Mass Models and the Role of Magnetic Braking ------------------------------------------------ If the secondary is a relatively unevolved low-mass star initially (with mass $\la 1{{\mbox}{$\,M_\odot$}}$), the only important mechanisms driving mass transfer are systemic angular-momentum losses due to magnetic braking and gravitational radiation. This type of evolution is similar to the classical evolution of CVs. The systems evolve towards shorter periods, may experience a period gap when magnetic braking stops being effective (when the secondary becomes fully convective) and ultimately reach a minimum period just before hydrogen burning is extinguished (Paczyński & Sienkiewicz 1981; Rappaport, Joss, & Webbink 1982). Beyond the period minimum (which depends on the evolutionary stage of the initial model), the secondaries follow the mass-radius relation for degenerate stars and the systems will expand, driven by gravitational radiation alone. This classical CV-like evolution is illustrated in Figures 3 and 4 for three binary sequences with initial secondaries of 1[[$\,M_\odot$]{}]{} and different evolutionary stages (at the beginning, the middle and the end of the main sequence). Figure 3 shows the evolution of orbital period and mass-transfer rate as a function of time since the beginning of mass transfer (the evolutionary tracks of the secondaries in the H-R diagram are shown in Fig. 4). These calculations serve to illustrate several points, already found in previous studies (see, in particular, Pylyser & Savonije 1989). The maximum mass-transfer rate is of order a few $10^{-9}{{\mbox}{$\,M_\odot\,$yr$^{-1}$}}$, where the more evolved secondaries experience the lower rates. Indeed, this behavior is also often found for more massive secondaries, where the somewhat evolved stars generally tend to be more stable than the unevolved ones. The period gap for the calculation with the initially unevolved secondary is substantially smaller (2.8 to 3.1hr) than the period gap for a similar CV calculation where the secondary is a white dwarf of 0.6[[$\,M_\odot$]{}]{}(2.4 to 3.1hr). While the donor stars become fully convective at more-or-less the same orbital period, they have different masses, 0.336[[$\,M_\odot$]{}]{}and 0.273[[$\,M_\odot$]{}]{}, respectively, since the donor in the LMXB case is not as much out of thermal equilibrium as in the CV case (the magnetic-braking timescale is a factor of $\sim 2.2$ longer in the LMXB case, while the gravitational-radiation timescale is a factor of $\sim 2.6$ shorter; cf eqs. 3 and 4). The location and the extent of the period gap decreases for the more evolved systems and completely disappears for the most evolved one. The reason is that the more evolved secondaries become fully convective at a lower mass, which implies a shorter orbital period for the system; but at shorter orbital periods, the timescales for angular-momentum loss due to gravitational radiation and magnetic braking become more comparable, hence producing a smaller gap. Since this type of evolution is similar to the classical CV evolution, it has the obvious implication, as emphasized by Pylyser & Savonije (1989), that the vast majority of secondaries in CVs have to be essentially unevolved initially to prevent the appearance of too many systems in the observed period gap (Ritter & Kolb 1998). The minimum periods decrease for the more evolved systems, again consistent with previous studies (see §4.2 for further discussion). The system that started mass transfer \[fig:fig3\] when the secondary had just completed hydrogen burning in the center attains a minimum period of 48min (note the spike in $\dot{M}$ near the minimum period in this case). \[fig:fig4\] Thermal timescale mass transfer ------------------------------- If the donor is initially more massive than the accretor, the Roche lobe radius generally shrinks. If this radius is less than the thermal equilibrium radius of a star of the same mass, the secondary can no longer stay in thermal equilibrium and mass transfer will proceed on a thermal timescale or, in more extreme cases, on a dynamical timescale (see § 3.3; for general reviews of thermal timescale mass transfer see, e.g., Paczyński 1970; Ritter 1996, and for other recent discussions DiStefano et al. 1997; Langer et al. 2000; King et al. 2001). It is customary to analyze the stability of mass transfer in terms of mass–radius exponents where $$\xi_{\rm eq} = \left({d\ln R\over d\ln M}\right)_{\rm eq},\,\,\, \xi_{\rm RL} = \left({d\ln R\over d\ln M}\right)_{\rm RL}\nonumber$$ $$\mbox{and\,\,\,}\xi_{\rm ad} = \left({d\ln R\over d\ln M}\right)_{\rm ad}$$ define, respectively, the mass–radius exponents for stars in thermal equilibrium, for the Roche-lobe, and for stars losing mass adiabatically. If the Roche-lobe radius shrinks more rapidly than the adiabatic radius (i.e., if $\xi_{\rm RL}>\xi_{\rm ad}$), then there is no hydrostatic solution for which the secondary can fill its Roche lobe (as defined by eq. 2) and mass transfer will proceed on a dynamical timescale (see § 3.3). The case where mass transfer is dynamically stable, but occurs on a thermal timescale is given by the inequalities $\xi_{\rm ad} > \xi_{\rm RL} > \xi_{\rm eq}$. For stars with radiative envelopes, $\xi_{\rm ad}$ is generally very large initially, and in most situations much larger than $\xi_{\rm RL}$ ($\xi_{\rm RL}$ generally depends on the mass ratio and any changes in orbital separation due to the transfer of mass and systemic mass and angular momentum losses; see, e.g., Rappaport et al. 1983). To illustrate this, we show approximate adiabatic mass–radius exponents, $\xi_{\rm ad}$, in Figure 5 for stars with initial masses from 1.2–2.2[[$\,M_\odot$]{}]{}and the corresponding mass–radius relations (these were obtained by taking mass off these stars at a high constant rate of $10^{-5}{{\mbox}{$\,M_\odot\,$yr$^{-1}$}}$). The large initial values for $\xi_{\rm ad}$ imply that the star has to lose very little mass to shrink significantly. This simply reflects the fact that, in radiative stars, a large fraction of the envelope (in radius) contains very little mass (for example, in an unevolved 2.1[[$\,M_\odot$]{}]{} star, the outer 40% of the radius contains just 1% of the mass). Once this low-density, high-entropy layer is lost, $\xi_{\rm ad}$ drops dramatically to a value of order 1 and ultimately becomes negative when the convective, flat-entropy core is exposed. From this stage on, the radius of the star increases with further mass loss. Since the whole star expands dramatically in this phase, it will be very underluminous for its mass (since most of the internal luminosity drives the expansion) and nuclear burning will ultimately be turned off because of a dramatic decrease in the central temperature. As long as $\xi_{\rm ad}$ remains larger than $\xi_{\rm RL}$, mass transfer remains dynamically stable. The star, which is undersized for its mass, will expand and try to relax to its equilibrium radius. It is this relaxation of the star on a thermal timescale that gives this mode of mass transfer its name. A characteristic mass-transfer rate for this phase is often defined by an expression of the form (see, e.g., Rappaport, DiStefano, & Smith 1994; Langer et al. 2000) $$\dot{M}_{\rm th} = {(M_2^i - M_1^i)\over t_{\rm KH}},$$ where $M_2^i$ and $M_1^i$ are the initial masses of the secondary (the mass donor) and the primary, respectively, and $t_{\rm KH}$ is the Kelvin-Helmholtz timescale of the secondary (i.e., the thermal timescale of the whole star), $$t_{\rm KH} \simeq {G M_2^2\over 2 R L},$$ \[fig:fig5\] where $R$ and $L$ are the radius and the nuclear luminosity of the secondary. As shown by Langer et al. (2000), equation (11) tends to overestimate the actual mass-transfer rate by up to an order of magnitude for solar-metallicity stars. In Figure 6 we present the binary sequence for a 2.1[[$\,M_\odot$]{}]{} star that starts to fill its Roche lobe near the end of the main sequence (when its central hydrogen abundance was $X_c = 0.096$). Indeed, the maximum mass-transfer rate of $\sim 2\times 10^{-7}{{\mbox}{$\,M_\odot\,$yr$^{-1}$}}$ is about an order of magnitude lower than what equation (11) would predict. Since the mass-loss timescale is much longer than the thermal timescale, the secondary is only moderately out of thermal equilibrium throughout the high $\dot{M}$ phase. Figure 7 shows a more extreme example of thermal timescale mass transfer where the secondary has an initial mass of 4.0[[$\,M_\odot$]{}]{} and is in a similar evolutionary phase as the secondary in Figure 6. This system is, in fact, on the brink of experiencing a delayed dynamical instability (see § 3.3). In this case, equation (11) provides a good estimate for the [*average*]{} mass-transfer rate in the thermal mass-transfer phase of $\sim 4\times 10^{-6}{{\mbox}{$\,M_\odot\,$yr$^{-1}$}}$, but, as discussed by Langer et al. (2000), it does not describe the detailed behavior of this phase very well. In the turn-on phase, which lasts of order a Kelvin-Helmholtz time, $\dot{M}$ is significantly less than $\dot{M}_{\rm th}$, simply because there is so little mass in the outer layers of the donor star and very little mass needs to be transferred for the secondary to adjust its radius to the shrinking Roche-lobe radius (which again is reflected in the large adiabatic mass–radius exponent). Equation (11) also does not provide a good estimate for the maximum $\dot{M}$ of $1.7\times 10^{-4}{{\mbox}{$\,M_\odot\,$yr$^{-1}$}}$. The reason is that, at this high mass-transfer rate, only the outer layers will be able to adjust thermally and drive the expansion and that, in this case, $t_{\rm KH}$ in equation (11) should be replaced by the shorter thermal timescale of this layer. What fraction of the envelope can adjust thermally also depends to a large degree on how the Roche-lobe radius changes (through the Roche-lobe filling constraint, i.e., eq. 2), which in turn depends on external factors such as the change of the mass ratio and systemic mass and angular momentum loss and not on the internal properties of the secondary. In the somewhat extreme example shown in Figure 7, the secondary evolves essentially adiabatically near the peak in $\dot{M}$. The secondary will only be able to re-establish thermal equilibrium once the Roche-lobe radius starts to expand (generally after the mass ratio has been reversed). At this stage, the secondary will be significantly undersized and underluminous for its mass. However, this equilibration phase itself will take a full Kelvin-Helmholtz time and a significant amount of mass ($\sim 0.4{{\mbox}{$\,M_\odot$}}$ in the sequence shown in Fig. 7) will still be transferred before the secondary has re-established thermal equilibrium. To illustrate this thermal relaxation phase further, we calculated a separate mass-loss sequence for an unevolved 2.1[[$\,M_\odot$]{}]{} star losing mass at a constant rate of $10^{-6}{{\mbox}{$\,M_\odot\,$yr$^{-1}$}}$ ($\sim 5 \times \dot{M}_{\rm th}$) until its mass had been reduced to 1[[$\,M_\odot$]{}]{} and then let it relax until it reestablished thermal equilibrium. Figure 8 shows the evolution of the radius (solid curve) and Figure 9 the evolution of the entropy profile in this calculation. First note in Figure 9 that only the outer layers of the secondary (in mass) are able to thermally adjust significantly (and only at early times). During the mass-loss phase, the radius of the star is always substantially smaller than the equilibrium radius of a star of the same mass (shown as a dashed curve in Fig. 8). In the subsequent relaxation phase, however, the radius overshoots the equilibrium radius by about 15%. The reason is that a star does not relax in a uniform, homologous way, but different parts of the star adjust on their local thermal timescales which vary throughout the star. This mismatch of timescales drives a thermal wave through the star (associated with a luminosity wave) which causes the overshooting in radius (and luminosity). This is also the reason why the system in Figure 7 becomes detached immediately after the thermal mass-transfer phase[^2]. The binary sequence shown in Figure 6 may, at early times, represent the evolution for a system like HZ Her/Her X-1, which has an orbital period of 41hr and contains a slightly evolved secondary of $\sim 2.35{{\mbox}{$\,M_\odot$}}$ (e.g., Joss & Rappaport 1984). At late times, the sequence may be appropriate for a system like the LMXB X-ray pulsar GRO J1744-28 with an orbital period of 11.8d (Finger et al. 1996) and a donor mass that is likely in the range $0.2\,$–$\,0.4\,M_\odot$ (Rappaport & Joss 1997). After the initial high-$\dot{M}$ phase, mass transfer is driven by the nuclear evolution of the star and starts to rise towards the end of the main-sequence phase. The system becomes briefly detached at the point of hydrogen exhaustion (associated with a brief shrinkage in radius). The subsequent peak in $\dot{M}$ occurs when magnetic braking has become most active. As the system expands, magnetic braking becomes less effective and consequently $\dot{M}$ starts to decrease. At some point nuclear evolution on the sub-giant branch becomes the dominant mass-transfer driving mechanism, and $\dot{M}$ rises as the nuclear timescale becomes shorter. Eventually, the system becomes detached when the secondary has a mass of 0.322[[$\,M_\odot$]{}]{}. Despite this low mass, the secondary still ignites helium in its core and ultimately ends its evolution as a HeCO white dwarf (see §4.1). Figure 10 shows a sequence similar to Figure 6 for a secondary with an initial mass of 2.1[[$\,M_\odot$]{}]{}, except that it is less evolved initially (its initial, central hydrogen abundance was $X_c = 0.489$). This sequence may provide a model for the X-ray binary Sco X-1 with an orbital period of 18.9hr. In this case, the high observed luminosity of Sco X-1 would be the result of thermal timescale mass transfer. In this particular model, the secondary of Sco X-1 is predicted to have a mass of $\sim 2{{\mbox}{$\,M_\odot$}}$ and resemble an A or F star (absent any X-ray heating effects; see the corresponding evolutionary track in Fig. 4). After the thermal timescale phase, mass transfer is driven by the nuclear evolution of the core. As the star develops a convective envelope, magnetic braking takes over as the dominant mass-transfer driving mechanism, causing a spike in the mass-transfer rate. As the system expands, magnetic braking becomes less effective and the system becomes briefly detached as it evolves up the giant branch. Eventually, after the secondary has lost most of its hydrogen-rich envelope, it evolves away from the giant branch and ends its evolution as a He white dwarf with a mass of 0.231[[$\,M_\odot$]{}]{}. The binary sequence in Figure 7 represents a slightly more massive version of the evolution that may explain the evolutionary history of Cyg X-2 (see Podsiadlowski & Rappaport 2000). After the thermal timescale phase, the system becomes detached and stays detached for the next $\sim 4\times 10^7\,$yr. In this phase, the secondary has the appearance of a $\sim 2.5 {{\mbox}{$\,M_\odot$}}$ main-sequence star in the H-R diagram (see Fig. 4), except that it is significantly undermassive with a mass of only $\sim 1{{\mbox}{$\,M_\odot$}}$ and has a low surface hydrogen abundance of 0.34 (by mass). The companion may appear as a slightly spun-up radio pulsar, having accreted $\sim 0.007{{\mbox}{$\,M_\odot$}}$ of material. The secondary starts to transfer mass again shortly after exhausting hydrogen in its core. In this second mass-transfer phase, the evolution is driven by nuclear shell burning (similar to the case AB model of Podsiadlowski et al. 2000). The final mass of the HeCO white dwarf is 0.466[[$\,M_\odot$]{}]{}. Dynamically unstable mass transfer ---------------------------------- A dynamical mass-transfer instability occurs when the Roche-lobe radius shrinks more rapidly (or expands less slowly) than the star can adjust either thermally or adiabatically, i.e., when $\xi_{\rm RL} > \xi_{\rm ad}$ (see § 3.2). This will then most likely lead to a common-envelope and a spiral-in phase (Paczyński 1976). For a binary initially consisting of a (sub-)giant and a neutron star, the system will either merge completely to form a rapidly rotating single object (a Thorne-Żytkow object? Thorne & Żytkow 1977) or become a short-period binary with a white-dwarf companion if the envelope is ejected. For a fully convective star (approximated by an $n=1.5$ polytrope), the condition $\xi_{\rm RL} = \xi_{\rm ad}$ defines a critical mass ratio $q_{\rm crit}\simeq 3/2$ (where $q=M_1/M_2$; see, e.g., Faulkner 1971; Paczyński & Sienkiewicz 1972; Rappaport et al. 1982). If the accreting star is a neutron star of 1.4[[$\,M_\odot$]{}]{}, this criterion applied literally would imply that mass transfer would be dynamically unstable if the secondary is a giant larger than $\sim 0.9{{\mbox}{$\,M_\odot$}}$. However, (sub-)giants are generally not well represented by fully convective polytropes. For example, Hjellming & Webbink (1987) (also see Soberman, Phinney, & van den Heuvel 1997) showed that the fact that (sub-)giants have degenerate cores of finite mass can increase $q_{\rm crit}$ significantly. This criterion also does not take into account any time delay between the onset of mass transfer and the appearance of the dynamical instability, during which a substantial amount of mass may already be transferred in a stable manner. In our calculations, dynamical instability manifests itself by the fact that we can no longer satisfy the Roche-lobe filling constraint in equation (2). The secondary will subsequently overfill its Roche lobe by an ever increasing amount. Since we generally limit the maximum mass-transfer rate to $10^{-4}{{\mbox}{$\,M_\odot\,$yr$^{-1}$}}$, some stars near the brink of dynamical instability will not be able to satisfy equation (2) in our calculations, but would do so without the constraint of a maximum mass-transfer rate. We therefore assume that all systems where the secondary overfills its Roche lobe by at most a few per cent for a short amount of time are stable against dynamical mass transfer[^3]. In all other cases, we continued the calculations until the overflow factor ($f_{\rm over}\equiv R/R_{\rm L}$) exceeded a value of 1.5. Tauris & Savonije (1999) have recently examined the dynamical stability for X-ray binaries with giant donors in detail, using realistic binary stellar evolution calculations, and found that all systems with (sub-)giant donor masses $\la 2{{\mbox}{$\,M_\odot$}}$ were dynamically stable (they assumed an initial neutron-star mass of 1.3[[$\,M_\odot$]{}]{}). In our calculations we also find that all sequences with donor stars up to 1.8[[$\,M_\odot$]{}]{} are dynamically stable, irrespective of evolutionary phase. Indeed, even for more massive secondaries, we often find that mass transfer is either dynamically stable (if the secondaries start mass transfer at the beginning of their ascent of the giant branch) or that the secondaries overfill their Roche lobes by only a relatively moderate amount (the most evolved secondaries in our sequences with initial masses of 2.1, 2.4, and 2.7[[$\,M_\odot$]{}]{} overfill their Roche lobes by at most 9, 12, and 13%, respectively; see Table A1). While the latter is likely to lead to the formation of a common envelope, it is not obvious that it necessarily leads to a spiral-in phase, since there is no friction between the immersed binary and the envelope, as long as the envelope can remain tidally locked to the orbiting binary (see, e.g., Sawada et al. 1984). In Figure 11 we present the binary sequence for the most evolved 1.8[[$\,M_\odot$]{}]{} secondary we calculated. The initial peak in the mass-transfer rate is very high ($\sim 9\times 10^{-6}{{\mbox}{$\,M_\odot$}}$), but mass-transfer remains stable. Even after the mass ratio has been reversed, $\dot{M}$ is significantly super-Eddington. The system becomes temporarily detached when the H-burning shell starts to move into the region with a gradient in hydrogen abundance, established during the hydrogen core burning phase, and the giant shrinks significantly (Thomas 1967). We find these temporarily detached phases in most of our sequences where the secondary evolves up the giant branch. (These detached phases on the giant branch have also been found in a number of other recent studies; Tauris & Savonije 1999; Han et al. 2000; N. Langer 1999 \[private communication\].) Many of the systems in which the initial secondary mass is $4\,M_{\odot}$ and probably all systems more massive than $\sim 4.5\,M_{\odot}$ experience dynamical mass transfer (see Fig. 1 and Table A1), resulting in the spiral-in of the neutron star inside the secondary. However, in all systems where the \[fig:fig12\] secondary is still on the main sequence when mass transfer starts, this dynamical instability is delayed (see Hjellming & Webbink 1987) since the secondaries initially have radiative envelopes with large adiabatic mass–radius exponents (as discussed in § 3.2), which stabilizes them against dynamical mass transfer. Dynamical instability occurs once the radiative part of the envelope with a steeply rising entropy profile (the entropy spike near the surface in Fig. 9) has been lost and the core with a relatively flat entropy profile starts to determine the reaction of the star to mass loss. This delay may last for up to $\sim 10^6\,$yr; during this time the system should still be detectable as an X-ray binary, with a very high mass-transfer rate and quite possibly some unusual properties (such as SS433?) in the last $10^4\,$–$\,10^5\,$yr before the onset of the dynamical instability. Figure 12 illustrates the case of a delayed dynamical instability for an initially unevolved 4.5[[$\,M_\odot$]{}]{} secondary. The early mass-transfer phase can be divided into two separate phases: (1) a phase of atmospheric Roche-lobe overflow where $\dot{M}$ increases exponentially (according to eq. 2) because the radius of the star approaches the Roche-lobe radius; and (2) a phase (labelled ‘radiative’ in Fig. 12) where the high-entropy material in the low-density envelope of the secondary is lost. The binary parameters remain essentially unchanged in the first phase, lasting $\sim 1.2\times 10^6\,$yr in this example, but the secondary loses $\sim 0.5\,{{\mbox}{$\,M_\odot$}}$ in the second much shorter phase, lasting only $\sim 10^5\,$yr, and both the radius and the orbital period shrink drastically. At the onset of the dynamical instability (which we here take as the point when $\dot{M}$ exceeds $10^{-4}{{\mbox}{$\,M_\odot\,$yr$^{-1}$}}$), the secondary is extremely underluminous and has the appearance of a $\sim 1.3{{\mbox}{$\,M_\odot$}}$ main-sequence star in the H-R diagram (see Fig. 4). \[fig:fig13\] End Products ============ Pulsars with He, HeCO White Dwarfs ---------------------------------- In Figure 13 we show the final distribution of the calculated systems in the secondary mass – orbital period plane (for sequences that avoided dynamical instability). Here the size of the symbols indicates how much mass a neutron star has accreted. In systems with large symbols, the neutron star has accreted at least 0.2[[$\,M_\odot$]{}]{} (for our accretion prescription) and may be reasonably expected to appear as a millisecond pulsar. A circle indicates that the secondary ends its evolution as a He white dwarf. Note that the He white dwarfs form a sequence that quite closely follows the relation between white-dwarf mass and orbital period for wide binary radio pulsars as calculated by Rappaport at al. (1995; solid and dashed curves). The new sequence may be somewhat steeper at low masses (also see Ergma 1996; Tauris & Savonije 1999) and lies systematically below the average sequence of Rappaport et al.(1995). The latter can be easily understood since, when the secondaries become detached from their Roche lobes, they have already evolved somewhat away from a Hayashi track and are hotter (and hence smaller) than a giant of the same core mass (see Fig. 4), an effect that could not be easily included without detailed binary evolution calculations. This suggests that one should rescale the average relation of Rappaport et al. (1995) by a factor $\sim 0.65$. In systems with triangle symbols, the secondaries ignite helium in the core and generally burn helium in a hot OB subdwarf phase (after mass transfer has been completed). Note that the lowest mass of a helium star for which helium can be ignited is $\sim 0.3{{\mbox}{$\,M_\odot$}}$, the minimum mass for helium ignition in non-degenerate cores (see, e.g., Kippenhahn & Weigert 1990). While the more massive helium stars convert most of their mass into carbon and oxygen (typically having a helium-rich envelope of at most a few per cent), the lower-mass helium stars only burn helium completely in the core and end their evolution with large helium envelopes (this was found first in calculations by Iben & Tutukov \[1985\] and more recently by Han et al. \[2000\]). It is not clear at the present time whether the fact that these low-mass HeCO white dwarfs have large CO cores has detectable, observational consequences. The most interesting aspect of the systems with HeCO white dwarfs is, of course, that most of them lie well below the white-dwarf – orbital period relation without having experienced a common-envelope phase \[fig:fig14\] (also see Podsiadlowski & Rappaport 2000; Tauris et al. 2000). Finally, it is worth noting that most of the He white dwarfs with masses $< 0.4\,M_{\odot}$ and even some of the more massive HeCO white dwarfs experience several dramatic hydrogen shell flashes (typically 2 to 4) before settling on the white-dwarf cooling sequence. These flashes have been extensively discussed in the literature (e.g., Kippenhahn, Thomas, & Weigert 1968; Iben & Tutukov 1986). More recently, Sarna, Ergma, & Gerškevitš-Antipova (2000) published a detailed study of hydrogen shell flashes for low-mass He white dwarfs and their implications for the calculations of cooling ages in companions of binary millisecond pulsars. During these flashes, the luminosity typically rises by a factor of 1000 and the radius increases by a factor of 10 or more on timescales of a few decades. Indeed, during these flashes the secondaries tend to fill their Roche lobes again, leading to several short mass-transfer phases with mass-transfer rates that are often much higher than the rates achieved in earlier phases (typically with $\dot{M}\sim \mbox{several}\times 10^{-6}{{\mbox}{$\,M_\odot\,$yr$^{-1}$}}$). In Figure 14 we present an example of a He white dwarf of 0.199[[$\,M_\odot$]{}]{} which experiences 3 such flashes (the secondary originally had a mass of 1.4[[$\,M_\odot$]{}]{} and filled its Roche lobe near the end of the main sequence). Ultracompact X-ray Binaries --------------------------- As Figures 2a and 13 show, systems with initial orbital periods below the bifurcation period ($\sim 18\,$hr) become ultracompact binaries with minimum orbital periods in the range of 11$\,$–$\,83\,$min. The shortest period is similar to the $11\,$min period in the X-ray binary 4U 1820-30 in the globular cluster NGC 6624 (Stella et al. 1987). Unlike the two better-known models for the formation of this system, this evolutionary channel involves neither a direct collision (Verbunt 1987) nor a common-envelope phase (Bailyn & Grindlay 1987; Rasio, Pfahl, & Rappaport 2000) and therefore constitutes an attractive alternative scenario for 4U 1820-30. This alternative evolutionary path for the origin of 4U1820-30 was originally suggested by Tutukov et al. (1987)[^4]. Fedorova & Ergma (1989) made the first detailed case study of this scenario and showed that, if mass transfer starts near or just after the point of central hydrogen exhaustion, orbital periods as short as 8 min could be attained and that a system like 4U 1820-30 can pass through an orbital of 11 min twice, while approaching the period minimum and after having passed through it. In related studies, Nelson, Rappaport & Joss (1986) and Pylyser & Savonije (1988) found minimum periods as short as 34 and 38 min, respectively. These authors were not specifically trying to explain 4U 1820-30 but their Galactic counterparts 4U 1626-67 and 4U 1915-05 with orbital periods of 41min (Middleditch et al. 1981; Chakrabarty 1998) and 50min (Chou, Grindlay, & Bloser 2001), respectively. \[fig:fig17\] To determine the shortest orbital period that can be attained through this channel, we performed a separate series of binary calculations for a $1\,M_{\odot}$ secondary with parameters appropriate for 4U 1820-30 in the metal-rich globular cluster NGC 6624 (i.e., with Z=0.01, Y=0.27). Our results confirm the earlier results of Fedorova & Ergma (1989) that, if the secondaries start mass transfer near the end of core hydrogen burning (or, in fact, just beyond), the secondaries transform themselves into degenerate helium stars and that orbital periods as short as $\sim 5\,$min can be attained without the spiral-in of the neutron star inside a common envelope. The top panel in Figure 15 shows the relation between initial orbital period and the minimum period, while the other panels give the secondary mass, $M_2$, mass-transfer rate, $\dot{M}$, and surface hydrogen abundance, $X_{\rm s}$, at the minimum period. There is a fairly large range of initial orbital periods ($13\,$–$\,17.7\,$hr) which leads to ultra-compact LMXBs with a minimum orbital period of less than $30\,$min. The drop in $P_{\rm min}$ at $13\,$hr occurs for a model where the secondary has just exhausted hydrogen in the center at the beginning of mass transfer. The shortest minimum period is attained for systems just below the bifurcation period (in this case $\sim 18\,$hr). The mass-transfer rate at the minimum period increases significantly as the minimum period decreases, simply because the time scale for gravitational radiation, which drives the evolution at this stage, becomes so short. Figure 16 shows the details of four representative sequences: sequence (a; black) has an initially unevolved secondary, in sequence (b; green) the secondary has just completed hydrogen burning in the center, while in sequences (c; red) and (d; blue) the secondaries have pure helium cores of 0.024[[$\,M_\odot$]{}]{} and 0.028[[$\,M_\odot$]{}]{}, respectively, at the beginning of mass transfer. Open circles show when the sequences reach the period minimum, and other symbols indicate when they pass through the periods of the three known ultracompact X-ray binaries in globular clusters (see Table 1). The shortest minimum periods are obtained for systems that exhaust all of the hydrogen left in their surface layers just near the point where they become degenerate and hence manage to transform themselves into essentially pure helium white dwarfs. In the phase where the hydrogen shell is being extinguished, the luminosity of the secondary briefly increases in the sequences with the shortest minimum periods (top panel), before the secondaries descend on the cooling sequence for He white dwarfs. In sequence (d; blue), the secondary becomes detached at an orbital period of 4.3hr. Gravitational radiation then causes the system to shrink, and the secondary starts to fills its Roche lobe again at an orbital period of 35min. Note also that after the minimum period, $\dot{M}$ generally drops dramatically. In Table 1 we list all X-ray binaries in globular clusters with known orbital periods. Strikingly, three of the six systems have ultra-short periods. In this table, we also list the timescales for the orbital period changes for the three compact systems based on the sequences shown in Figure 16 as well as on a simple model where the secondary is a fully degenerate He white dwarf and the system is driven by gravitational radiation alone (labelled ‘GR’). The 11min system (4U 1820-30 = X1820-303) is particularly interesting since its orbital period is observed to be decreasing rather than increasing (Tan et al. 1991), as would be expected for a fully degenerate secondary. While it had been argued that this apparent orbital period decrease could be caused by gravitational acceleration within the globular cluster (Tan et al. 1991), van der Klis et al. (1993) subsequently concluded that, using a more realistic mass model for the globular cluster NGC 6624, it was unlikely that the negative $\dot{P}$ could be fully explained by cluster acceleration. On the other hand, as Table 1 shows, the theoretical $P/\dot{P}$ in sequence (c; red) is in excellent agreement with the observed value, and the value in sequence (d; blue) is in reasonable agreement (Fedorova & Ergma \[1989\] obtained similar values). This provides additional support for this evolutionary channel for these ultracompact globular-cluster systems. One potentially testable prediction is that some of these systems still contain hydrogen in their envelopes (up to 40% by mass in the sequences shown). After the period minimum, the secondaries in sequences (c; red) and (d; blue) become pure He white dwarfs, and their evolution is identical to that of systems with He white dwarfs, driven by gravitational radiation alone. Since the value of the bifurcation period is sensitive to the adopted magnetic-braking law, the range of initial orbital periods which leads to ultracompact systems will also depend on it. To examine this dependence, we carried out two additional series of calculations where we increased and decreased the efficiency of magnetic braking by a factor of 5 with respect to our standard model, respectively. As expected, the bifurcation period increased to $\sim 20\,$hr for the more efficient magnetic-braking law and decreased to $\sim 15\,$hr for the less efficient one. In both cases, we obtained systems with minimum periods as short as 9 and 16min, respectively (note, however, that this exploration was not as comprehensive as for the standard case). It is quite remarkable that one of our binary sequences (sequence c; red), with an initial orbital period around $17\,$hr, appears to be a suitable sequence to explain all six LMXBs in globular clusters whose orbital periods are presently known, from the system with the longest period (AC211/X2127+119 in M15; $P_{\rm orb}=17.1\,$hr; Ilovaisky et al. 1993) to the 11-min binary. Furthermore, systems with an initial period in the range of $13\,$–$\,18\,$hr are quite naturally produced as a result of the tidal capture of a neutron star by a main-sequence star (Fabian, Pringle, & Rees 1975; DiStefano & Rappaport 1992) and are not the generally expected outcome of a 3- or 4-body exchange interaction (see, e.g., Rasio et al. 2000). To illustrate this, we calculated the orbital period at which mass transfer commences, the contact period, for systems that form as a result of the tidal capture of a 1[[$\,M_\odot$]{}]{} normal star by a 1.4[[$\,M_\odot$]{}]{}neutron star. Figure 17 shows the contact period as a function of the relative capture distance (i.e., the ratio of initial periastron distance to the radius of the star) for different radii (i.e., evolutionary stages on the main sequence) of the secondary at the epoch of capture. In these calculations we assumed that the systems formed by tidal capture circularized quickly (on a timescale short compared to the magnetic-braking timescale and the evolutionary timescale of the secondary) and that the system was brought into contact by the combined effects of magnetic braking (causing the orbit to shrink) and the evolution of the secondary. It has been estimated that for the tidal capture of a main-sequence star by a neutron star, the initial periastron distance has to be $\la 3$ stellar radii (Fabian et al. 1975; Press & Teukolsky 1977; McMillan, McDermott, & Taam 1987). It also has to be larger than $\sim 1.5$ stellar radii, so that in the circularized system (which has a separation twice the initial periastron distance) the secondary underfills its Roche lobe. As Figure 17 shows, this range of initial capture distances produces systems which start mass transfer between 9 and 21hr, depending on the radius of the secondary at the time of capture. The range of radii chosen corresponds to the change in radius of a 1[[$\,M_\odot$]{}]{} star on the main sequence (i.e., as it evolves from the zero-age main sequence to the terminal-age main sequence). Since all stars on the main sequence have roughly equal probability for a dynamical encounter with a neutron star, this predicts a fairly uniform distribution of contact periods in this range. In fact, stars with a larger radius are somewhat more likely to be captured since the capture cross section increases linearly with radius (Fabian et al. 1975; DiStefano & Rappaport 1992). This suggests that at least half of the X-ray binaries formed by tidal capture may start mass transfer with orbital periods in the range of 13 to 18hr, the range which produces ultracompact systems. This may explain the surprisingly large fraction of ultracompact systems (3 out of 6 systems with known orbital periods) in globular clusters. One well-recognized problem with the tidal-capture scenario for the formation of LMXBs in globular clusters is the fact that the total energy that needs to be tidally dissipated during the capture and subsequent circularization process is of order the binding energy of the secondary. Since all of this energy is deposited in the secondary, it may lead to its destruction either by dynamical effects (Rasio & Shapiro 1991) or due to the thermally driven expansion of the secondary that is tidally heated (McMillan et al. 1987; Ray, Kembhavi, & Antia 1987; Podsiadlowski 1996). As a consequence, the initial capture distance for which a 1[[$\,M_\odot$]{}]{} star can be captured and survive the process may be much more limited than the range used above. Indeed, this is the reason why it has been popular in recent years to dismiss this formation channel altogether. However, we would like to emphasize (1) that the details of the tidal capture process, in particular the response of the secondary to tidal heating (see Podsiadlowski 1996), are still rather uncertain; (2) that, as shown here, tidal capture naturally produces the range of orbital periods actually observed in globular-cluster LMXBs; and (3) that alternative scenarios, 3- or 4-body interactions, do not generally lead to systems in the observed range. This suggests to us that it is not only premature to rule out tidal capture as a formation scenario for LMXBs, but that the LMXBs in globular clusters with well-determined orbital periods actually provide observational evidence in its favor. An unbiased re-examination of the whole process is therefore clearly warranted. \[fig:fig18\] Application to the Population of X-Ray Binaries =============================================== In the previous sections we have presented the results of our 100 binary evolution models for LMXBs and IMXBs which cover a systematic grid of binary parameters at the onset of mass transfer. We have shown specifically that a number of these binary sequences pass through states which would closely resemble many of the well known and individually studied LMXBs and IMXBs. This includes X-ray binaries with long and short orbital periods, very low- to moderate-mass secondaries, a wide range of X-ray luminosities, and systems in and out of globular clusters. All but a few of our 100 binary evolution sequences started with donor stars of mass $> 1 M_{\odot}$, and the majority had donors $> 2 M_{\odot}$. One of the more striking results of these calculations is that most of the evolution time these systems spend as an X-ray binary occurs [*after*]{} the mass of the donor star has been reduced to $\la 1 M_\odot$ (see Fig. 2a). Thus, a large fraction of the systems which we commonly refer to as “LMXBs” may actually have started their lives as “IMXBs.” This has important implications for both (i) the retention of neutron stars in binaries at the time of the supernova explosion which gives birth to the neutron star (i.e., it is easier to keep the neutron star bound with a 2 or 3 $M_{\odot}$ companion than with a truly low-mass companion), and (ii) the evolutionary state of the companion stars in LMXBs that we observe today (i.e., they are probably not nearly as [*un*]{}evolved as was previously assumed). One observational consequence is that many of these systems should be hydrogen deficient and helium enriched, and that the surface composition of many secondaries should show evidence for CNO processing (i.e., be enhanced in N and depleted in C and O). Our calculations predict that at a particular orbital period ($\ga 80\,$min), the surface hydrogen abundance can vary typically between $\sim 1/3$ of the solar value and solar (at shorter orbital periods, all systems should be hydrogen-deficient). While the hydrogen (or helium) abundance can usually not be measured directly, an increased helium abundance affects the behavior of accretion, e.g., by increasing the Eddington accretion rate. This could provide some indirect evidence for helium enrichment in some of these systems. While it is, of course, rather gratifying to be able to “explain” possible evolutionary paths leading to some of the best known X-ray binaries, there remain several outstanding issues: (1) how unique are the evolutionary paths we have found; (2) are types of systems suggested by other reasonably long-lived phases of our binary evolutions represented in the observed binary X-ray source population; and (3) is our complete ensemble of binary evolution models consistent with the overall population of observed LMXBs and IMXBs? In order to properly investigate these questions one needs to carry out a full binary population synthesis (BPS) study, starting from primordial binaries and utilizing a library of binary evolution models of the type we have generated. Such a BPS study is beyond the scope of the present paper, but has been initiated (Pfahl et al. 2001). One objective of a BPS study will be to produce probability distributions, at the current epoch, for finding LMXBs and IMXBs with various values of $M$, $\dot M$, $P_{\rm orb}$, as well as in different evolutionary states, locations in the Galaxy, space velocities, and so forth. Such a study will involve weighting each of the binary evolutions in our library by the probability that each of the initial binary parameters would be realized in nature. For the present study we utilize our evolution tracks to produce a simplified estimate of the likelihood of finding LMXBs and IMXBs in various locations in the $\dot M-P_{\rm orb}$ plane. The choice of these two parameters – $\dot M$ and $P_{\rm orb}$ – is motivated by the fact that these are the easiest to determine for LMXBs. In fact, observationally, very few LMXBs and IMXBs (HZ Her/Her X-1, Cyg X-2) have well determined constituent masses or have much known about the state of the donor star. On the other hand, if an estimate of the distance is known, then $\dot M$ can be inferred from the X-ray luminosity (at least in the case of conservative mass transfer), and the orbital period can be inferred from X-ray or optical photometry, rather than requiring Doppler measurements of either the companion or the neutron star. To relate our binary models to the Galactic population of LMXBs and IMXBs on a statistical basis, we have constructed a plot which estimates the probability of finding an LMXB or IMXB in a particular region of the $\dot M-P_{\rm orb}$ plane. We do not attempt to weight each of the binary evolution runs (in the library). We do, however, take into account the amount of time spent in a particular part of the evolution, as each of our model binaries traverses the $\dot M-P_{\rm orb}$ plane. We make the implicit assumption of a steady-state production of LMXBs and IMXBs which then proceed through their entire evolution, well within the lifetime of the Galaxy. This, of course, will become less valid for systems with evolutionary phases comparable to the age of the Galaxy. To construct our probability distribution in the $\dot M-P_{\rm orb}$ plane, we proceeded as follows. First, we divided up the $\dot M-P_{\rm orb}$ plane into a finely spaced, discrete, two-dimensional array. Each of the 100 binary evolution tracks was then placed into this array, weighted by the evolution time spent in each element of the array. The probability of finding an LMXB/IMXB in any particular array element is then proportional to the combined evolution time of all the tracks passing through that array element. However, since there are only 100 evolution tracks, the entire $\dot M-P_{\rm orb}$ plane is not completely sampled (for an analogous sampling effect in the $M-P_{\rm orb}$ plane see Fig. 2a). In order to circumvent this problem somewhat, we computed, for each value of $P_{\rm orb}$, a cumulative probability distribution in $\dot M$. We then utilized these to compute contours of constant probability which are plotted in Figure 18. The central heavy curve is the median value of $\dot M$, while the contours on either side are in increments of $10\%$ in probability, except for the top and bottom curves which represent $1\%$ and $99\%$ of the systems. The shaded region represents $50\%$ of all systems around the median. As one can see from a perusal of Figure 18, there should theoretically be a general positive correlation between orbital period (for $P_{\rm orb} \ga 1$ hr) and $\dot M$, with the value of $\dot M$ a few hundred times larger at periods of $\sim$100 days as compared with 1 hr. At a given $P_{\rm orb}$, typically half of the systems are contained within a range of about a factor of $\sim 6$ in $\dot M$, centered on the median value. The next step is to compare the model results shown in Figure 18 with the positions of known LMXBs and IMXBs in this diagram. We excluded all obviously transient LMXBs and IMXBs since, in most cases, it is unclear how to estimate the long-term average X-ray luminosity[^5]. We then selected 16 LMXBs and IMXBs (i) whose orbital periods are known, (ii) whose X-ray luminosities do not vary wildly, and (iii) where a distance to the source could be estimated. These are shown overplotted on Figure 18; they include 2 “Z sources” (triangles), 8 “atoll sources”(squares), 3 X-ray pulsars (stars), and 3 “accretion disk corona sources” (circles). (For references see, e.g., van Paradijs 1995; Christian & Swank 1997.) For the latter group of sources, the observed X-ray flux is thought to be severely affected by an accretion disk corona, and therefore the inferred value of $\dot M$ is shown only as a lower limit. We used a simple factor of $10^{-8} M_{\odot}\, {\rm yr}^{-1} \equiv 10^{38}$ ergs$^{-1}$ in converting X-ray luminosity to mass-accretion rate. The first obvious fact in comparing the theoretical probability distribution to the locations of known LMXBs and IMXBs in the $\dot M-P_{\rm orb}$ plane (Fig. 18) is that only a relative handful lie plausibly in or near the shaded region. In fact, 10 of the 16 sources lie at or outside the $1\%$ upper and lower probability contours. The largest discrepancies come from luminous LMXBs with shorter orbital periods (i.e., $\la$ 1 day). There are several important caveats to note before viewing the comparison made in Figure 18 as being grossly discrepant. First, as mentioned above, the evolution tracks that went into the production of the probability contours in Figure 18 are not weighted by the relative probabilities of achieving their initial binary configurations in nature. Second, no transient X-ray sources have been included in the figure. Many of these sources probably have mean values of $\dot{M}$ of $\la 3 \times 10^{-10} M_{\odot}\, {\rm yr}^{-1}$ (and possibly even higher for the larger values of $P_{\rm orb}$; van Paradijs 1996; King, Kolb, & Sienkiewicz 1997) which cover a substantial portion of the shaded (high probability) region. Third, there are serious observational selection effects to consider, in that it is generally true that the most luminous X-ray sources are studied in detail, yielding higher probabilities of optical identifications which, in turn, can lead to orbital period determinations. At least the first two of these shortcomings of Figure 18 will be addressed in our binary population synthesis study (Pfahl et al. 2001). One potentially very important effect that has not been included in our binary calculations is the effect of X-ray irradiation on the secondary which could significantly alter the evolution of these systems and increase the mass-accretion rate by either driving a strong wind from the secondary (Ruderman et al.1989) or by causing significant expansion of the secondary (Podsiadlowski 1991; Harpaz & Rappaport 1991). While these irradiation effects are still poorly understood, even a relatively moderate irradiation-driven expansion of the secondary may cause mass-transfer cycles (Hameury et al. 1993) characterized by relatively short phases of enhanced mass transfer and long detached phases. During the X-ray phases these systems would appear to be much more luminous than without the inclusion of X-ray irradiation effects. We also plan to examine this possibility in our BPS study. Application to Binary Millisecond Pulsars ========================================= There are currently about 1400 radio pulsars known (Taylor, Manchester, & Lyne 1993; V. Kaspi 2001, private communication). Of these, $\sim 100$ have at least one of the following properties (V. Kaspi 2001, private communication): (i) a very short pulse period ($\sim 77$ with $P \lesssim 12\,$ms); (ii) a relatively weak magnetic field ($\sim 46$ with $B \lesssim 10^{10}\,$G); (iii) membership in a binary system ($\sim 66$); and/or (iv) location in a globular cluster ($\sim 45$). These systems are widely believed to be “recycled" pulsars, i.e., NSs whose magnetic field has decayed away and which have been spun up to high rotation rates by the accretion of matter from a companion star (see, e.g., Bhattacharya & van den Heuvel 1991). In the Galactic plane, there are several distinct classes of binary radio pulsars. One major class involves systems with low-mass companions ($0.10\,$–$\, 0.4\, M_\odot$) and nearly circular orbits. These range in $P_{\rm orb}$ from a fraction of a day to 1000 days. There is a dearth of these pulsars in the period range of 12 and 68 days. The masses of most of the companions to these pulsars are known only approximately from the measured mass functions. Based on the scenario for their formation, which involves stable mass transfer from a low-mass giant, there is a theoretically predicted relation between the orbital period of these systems and the mass of the remnant companion white dwarf (see, e.g., Rappaport et al. 1995). In fact, the locus of points in Figure 12 tracing the maximum value of $P_{\rm orb}$ at any given white dwarf mass matches the theoretical relation rather closely (see also § 4.1). Most of the model systems helping to define this relation, however, have orbital periods between $\sim$12 and 120 days – at least the first half of which fall in the period “gap” found observationally. There is also another cluster of model systems with $P_{\rm orb}$ between $\sim$11 and 85 minutes; these are of shorter periods than any of the binary pulsars discovered thus far in the Galactic disk. Again we note the caveat discussed in § 5 that our library of binary models has not been weighted according to the probability of achieving their initial binary parameters at the onset of mass transfer, e.g., in the context of a full binary population synthesis calculation. Another class of binary radio pulsars are the ones with substantially more massive white dwarf companions which distinctly do not fit the scenario described above (with a low-mass giant donor) and do not lie in the $P_{\rm orb}\,$–$\,M_{\rm wd}$ plane near the associated theoretical relationship. These systems also have nearly circular orbits and $P_{\rm orb}$ in the range of $\sim 1\,$–10 days. It has been proposed for some time now that these systems result from donor stars which are more massive than the neutron star, thereby leading to unstable mass transfer and a common envelope phase (see, e.g., Taam & van den Heuvel 1986). Our models with donors that are initially of intermediate mass naturally lead to this type of system [*without*]{} a common envelope phase (see also § 4 and Tauris et al. 2000). Such model systems are found in abundance in Figure 13 (triangles situated well below and to the right of the theoretical curve). Yet another class of binary radio pulsars are systems that contain planetary mass companions (i.e., $M \la 0.02 M_\odot$) which are in the process of being ablated by the radiation from the pulsar (e.g., 1957+20; Fruchter, Stinebring, & Taylor 1988). In this regard, we note that in our binary evolution calculations, the mass transfer is allowed to continue until either the donor star becomes detached from its Roche lobe or its non-degenerate envelope has been completely stripped. Of course, as the neutron star accretes matter from the companion it will be spun up by accretion torques – the maximum spin period being determined by a combination of, $\dot M$, the total mass accreted, and the strength of the neutron star’s magnetic field. For weak surface magnetic fields (i.e., $\la 10^9\,$G), the accretion-induced spin period is given approximately by $P = 3.5\times (\Delta M/0.01 M_\odot)^{-1/2}$ ms, where $\Delta M$ is the accreted mass. For higher magnetic fields, the minimum accretion induced rotation period scales as $1.9 B_{9}^{6/7}$ ms, for an Eddington-limited luminosity, where $B_9$ is the surface dipole field strength in units of $10^9\,$G (see, e.g., Bhattacharya & van den Heuvel 1991). Thus, at some point in the evolution, prior to the exhaustion of the donor’s envelope, a spun-up neutron star may turn on as a radio pulsar. This could have two important consequences for the subsequent evolution of the binary. First, the pulsar radiation (in the form of both electromagnetic waves and a relativistic wind) may exert sufficient pressure on the incoming accretion flow that all further accretion is halted. Second, the pulsar radiation may ablate material from the donor star, and in some cases possibly evaporate it altogether (see, e.g., Ruderman, Shaham, & Tavani 1989; van den Heuvel & van Paradijs 1988, Bhattacharya & van den Heuvel 1991). In the present binary evolution calculations, we do not take either a possible pulsar turnon into account or the subsequent effects of the pulsar radiation on the donor star. In our BPS study we will attempt to use simplified prescriptions to handle both of these processes, i.e., pulsar turnon and ablation of the donor, although there are obviously still many uncertainties concerning both of these. The final class of binary pulsars we comment on consists of a pair of neutron stars. While these systems are important for exploring binary evolution, acting as laboratories for general relativity, and yielding potentially detectable gravity wave signals when they merge, our study does not shed any new light on their formation. This results from the fact that our highest mass donor stars are $7~M_\odot$ which is too low to form a second neutron star. Many of the same classes of binary radio pulsars that are found in the plane have also been discovered in globular clusters. In particular, there are 22 radio pulsars known in 47 Tuc (Camilo et al. 2000), 8 in M15 (Anderson 1992), and 2 each in M5, M13, Ter 5, and NGC 6624. At least 10 of the radio pulsars in 47 Tuc are in binary systems with periods ranging from 1.5 hr to 2 days. This abundance of recycled pulsars in globular clusters is widely attributed to the dense stellar environment which can lead to 2-, 3-, and 4-body stellar encounters at interestingly high rates. Thus, through a combination of processes such as 2-body tidal capture (e.g., Fabian, Pringle, & Rees 1975; Di Stefano & Rappaport 1992) and exchange interactions where a field neutron star replaces a normal star in a binary, numerous neutron star binaries should be formed (see, e.g., Rasio, Pfahl, & Rappaport 2000; Rappaport et al. 2001). We have already discussed in § 4.2 how the globular-cluster X-ray sources fit our evolution scenarios. However, we need to check if the binary millisecond pulsars also fit naturally into the same evolutionary scenarios. In 47 Tuc, there is a group (“A”) of five pulsars which have masses of $0.02-0.03~M_\odot$ and periods of 1.5–5.5hr, while a second group (“B”) has masses approximately 10 times higher and periods of 0.1–2d. The evolutionary scenarios for producing the ultracompact X-ray binaries discussed in §4.2 may also provide a possible path to the formation of binary radio pulsars of the type found in the 47 Tuc group A pulsars. In the process of evolving to very short periods (11–83 min)., these mass transfer binaries will naturally pass through orbital periods of 1.5–5 hr. However, at these periods the donor masses are substantially larger than the typical values of $\sim 0.025 M_\odot$ found for the companions in the group A systems. On the other hand, if at some point in the binary evolution, the neutron star has been spun up to msec periods, pulsar radiation may turn on and both shut off further mass transfer and ablate the donor star until it has been reduced to planetary mass. It is also true that after the minimum period is reached in the ultracompact systems (see §4.2), the orbit will, in principle, expand back into the range of about an hour or so within a Hubble time. In order for such systems to return all the way back to periods of 1.5–5 hours, some other effect, such as sustained tidal heating, would be required (as is also invoked for the case of the post-common envelope scenario proposed by Rasio et al. 2000). A characteristic of our “ultracompact” evolutionary scenario is that the correct initial orbital periods between $\sim13-18$ hr arise naturally from tidal capture in globular clusters. By contrast, the common envelope scenario proposed by Rasio et al. (2000) follows more naturally from the wider orbits left by 3-body encounters. The orbital period range of $\sim 0.1-2$ days for the group B pulsars is traversed in many of our binary evolutions (see, e.g., Fig. 2a). However, in all but 2 cases, the masses of the donor stars in this period range are much higher than the group B pulsar companions. In addition, the donor stars at this phase of the evolution are still quite H-rich and would therefore not resemble the inferred He white dwarf companions of these pulsars. Again, it is possible that these evolutions are interrupted by the turn-on of a strong radio pulsar at just the right values of $P_{\rm orb}$ to match the group B systems. At present, we do not have a good explanation for how this would happen, nor do we know the reason for the existence of two rather distinct groups of msec pulsars. A long-standing problem in our understanding of millisecond pulsars, known as the birthrate problem, is that, in the standard model of LMXBs, the birthrate of LMXBs appears to be a factor of 10 to 100 lower than the birthrate of millisecond pulsars, which are believed to be their direct descendants. This problem exists for millisecond pulsars both in the Galactic disk (Kulkarni & Narayan 1988; Johnston & Bailes 1991) and in globular clusters ( Fruchter & Goss 1990; Kulkarni, Narayan, & Romani 1990). This discrepancy may be the result of an overestimate of the LMXB lifetime, typically taken to be $\sim 5\times 10^9\,$yr in these estimates. If a large fraction of X-ray binaries are IMXBs, one might expect that the time these systems spend as X-ray emitters could be significantly reduced, which would then alleviate the problem. However, our calculations show that even IMXBs spend most most of their X-ray active lifetime as low-mass systems and that consequently the duration of the X-ray active lifetime is generally not much lower than for true LMXBs (see the column $\Delta t_{\dot{M}}$ in Table A1). Thus the inclusion of IMXBs does not immediately solve the birthrate problem. This problem may also be related to the problem of the low median X-ray luminosities found in our calculations (as discussed in § 5) and may have a similar resolution: if irradiation-driven mass-transfer cycles operate for low-mass systems (see § 5), these would not only increase the mass-transfer rates during the X-ray active portion of the cycles but also reduce the duration of the X-ray active lifetime of these systems by a proportionate amount. This could provide a simultaneous solution to both of these problems, the X-ray luminosity and the birthrate problem. As we have shown, our binary evolution models pertain directly to the binary radio pulsars found in both the Galactic plane and in globular clusters. However, before we can draw definitive conclusions about the relative and absolute populations of the different classes of these objects, in both the plane and in clusters, we must await the results of our binary population synthesis study (Pfahl et al.2001). Summary and Outlook =================== As this study has shown, the evolution of low- and intermediate-mass X-ray binaries is much more complex than previously believed, and the standard model for these systems where mass transfer is driven only by magnetic braking, gravitational radiation, and occasionally nuclear evolution is the exception rather than the rule. Our evolutionary sequences show an enormous variety of evolutionary channels which may explain the large diversity in observed systems. Indeed many of the best-studied systems in the Galactic disk (e.g., Her X-1, Cyg X-2, Sco X-1, GRO J1744-28, 4U 1626-67 and 4U 1915-05) as well as all globular clusters sources with known orbital periods can be identified with particular sequences in our library of models. This demonstrates the importance of these results for our understanding of X-ray binaries and millisecond pulsars, believed to be their descendants. However, our results also show that there are a number of problems still remaining, in particular the low median X-ray luminosities as compared to the luminosity of well observed systems, the orbital-period distribution of millisecond pulsars, and the millisecond pulsar birthrate problem. To shed more light on the significance of these discrepancies, we have initiated a systemic binary population synthesis study (Pfahl et al. 2001) where we implement this library of models in a population synthesis code. This will not only allow us to quantify these discrepancies more precisely, but also to examine possible solutions (e.g., irradiation-driven cycles, pulsar turnon, pulsar evaporation, etc.) and should ultimately help us to improve our understanding of the evolution X-ray binaries and the formation of millisecond pulsars. Acknowledgments {#acknowledgments .unnumbered} =============== We would like to thank Ron Remillard, Mike Muno, and Phil Charles for providing important observational information on several LMXBs and many useful discussions. This work was in part supported by the National Aeronautics and Space Administration under ATP grant NAG5-8368. In this appendix, we present in tabular form (Table A1) some of the main characteristics of the 100 binary sequences in this study, where the selected parameters depend on the type of evolution encountered. In all sequences, the initial mass of the compact object, assumed to be a neutron star, was taken to be 1.4[[$\,M_\odot$]{}]{}. The initial mass of the secondary, $M_2$, ranges from 0.6 to 7[[$\,M_\odot$]{}]{}. For these masses, the table first lists (in the first row) the initial parameters for each sequence: the initial central hydrogen mass fraction, $X_c^i$; the initial fractional mass of the H-exhausted core, $M_c^i/M$; the age of the secondary at the beginning of mass transfer, $t^i\,$(yr). The next parameters give the key binary parameters at the end of each calculation and the type of end product, where $M_2^f$ and $M_1^f$ are the final masses of the secondary and primary (in [[$\,M_\odot$]{}]{}), respectively, $P^f$ is the final orbital period (in d), $\Delta t$ gives the total time since the beginning of mass transfer (in yr) and ‘Type’ indicates the type of end product (‘short’: compact system; ‘He’: wide system with a He white dwarf secondary; ‘HeCO’: wide system with a HeCO white dwarf secondary; ‘del dyn’: delayed dynamical instability; ‘dyn’: dynamical mass transfer; ‘?’ indicates that the system may be dynamically unstable). Note that the calculations were terminated at different points for the different types of evolution. For the ultracompact systems, the calculations were generally terminated just after the period minimum when the secondaries have become fully degenerate. In systems where the secondary becomes a He or a HeCO white dwarf, the calculations are continued either up to the point where the secondary has settled on the cooling sequence for degenerate stars or until the beginning of the first hydrogen shell flash (although in many cases, we continued the calculations through all flashes). In systems that experience a dynamical instability, the calculations were generally terminated when the secondaries overfilled their Roche lobes by a factor of 1.5. In cases, where the maximum overflow factor was less than 1.5, we continued the calculations (pretending that the systems did not experience a spiral-in phase) just as for the systems that were dynamically stable (these are the systems marked with ‘?’ in the ‘Type’ column). The next three parameters in the first row give the average and the maximum mass-transfer rate, $<\dot{M}>$ and $\dot{M}_{\rm max}$ (in [[$\,M_\odot\,$yr$^{-1}$]{}]{}), respectively, and the total duration of the mass-transfer phase for each sequence, $\Delta t_{\dot{M}}$ (in yr), where we considered only phases where $\dot{M}$ exceeded a rate of $10^{-12}{{\mbox}{$\,M_\odot\,$yr$^{-1}$}}$. The last column in the first row indicates whether the secondary experienced hydrogen shell flashes before settling onto the sequence for degenerate stars. 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However, such a model requires a complex triple-star interaction in order to form a non-degenerate helium star at the present epoch (see the discussion in van der Klis et al.  1993). [^5]: We note that the issue of determining the long-term luminosity of an X-ray source is quite difficult – even for so called “steady sources”. The entire history of X-ray astronomy is shorter than 40 years; while the typical timesteps in our binary evolution code may range from $10^2 - 10^7$ years. Therefore, even X-ray sources which appear steady over the entire history of X-ray astronomy may, in fact, be transient over the longer term, e.g., comparable to the time steps in our evolution code.

--- abstract: 'A coupling-constant definition is given based on the compositeness property of some particle states with respect to the elementary states of other particles. It is applied in the context of the vector-spin-1/2-particle interaction vertices of a field theory, and the standard model. The definition reproduces Weinberg’s angle in a grand-unified theory. One obtains coupling values close to the experimental ones for appropriate configurations of the standard-model vector particles, at the unification scale within grand-unified models, and at the electroweak breaking scale.' author: - 'J. Besprosvany' date: 'Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, México 01000, D. F., México ' title: 'Standard-model coupling constants from compositeness' --- =msbm10 =-.3in =8.5in '\#1[[19\#1i i\#1]{}]{} v\#1[[\#1]{}]{} ‘@=11 @ @.2326ex = 1.5ex plus 1pt 22 pt The coupling constants are the dimensionless numbers that measure the strength of nature’s interactions. Their values are fixed by experiment in the standard model (SM) of elementary particles, and depend on the energy scale. Clues to the origin of their values are suggested from the relations among the quantum numbers of the SM particles. In general, the realization of unity among physical variables, originally thought as disconnected, has led to a new understanding and connections among additional ones. For example, by linking electric and magnetic phenomena, Maxwell’s theory showed that light is a phenomenon of the kind, and predicted its velocity in terms of likewise parameters. Indeed, recently proposed SM extensions including a unifying principle are able to provide information on the coupling constant values. Thus, grand unification$\cite{unification}$ assumes that the gauge groups describing the interactions originate in a common group, and it predicts a single unified coupling, to which distinct couplings indeed appear to converge at high energy. It is also able to predict the coupling-constant ratios. In addition, compactification configurations of additional dimensions associated to interactions[@Weinbergcoup], and the dilaton-field ground state in string theory[@Green] predict their values, but, as yet, not uniquely. Information on the coupling constants may be also derived from extended-spin models[@Jaime]. Even if the underlying dynamics is not obvious, these connections may become manifest through symmetry arguments, which give additional information. Composite models are another class of unifying theories that address the SM particle-multiplicity problem. Utilizing the connections among the quantum numbers of the 27 or so SM particles, these particles are constructed in terms of fewer elementary fields[@haplon]. The SM Poincaré symmetry and gauge-invariant interactions provide the link. In general, these symmetries dictate the few quantum numbers that describe a particle state. These are the configuration or momentum coordinates, the spin, the gauge-group representation, and the flavor for quarks and leptons. Flavor characterizes only fermions. In the SM, fermions belong in the spin-1/2 Lorentz representation, and the gauge bosons are vectors. Similarly, fermions belong in the fundamental representation of the gauge group, while the vector bosons belong in the adjoint. This means that the gauge and spin quantum numbers of the latter can be constructed in terms of the former. In the case of composite models, this facilitates their modelling in terms of simpler fields. However, it is difficult then to reproduce the SM dynamics without introducing additional fields and interactions, which, in turn, reduces the models’ predictability. Also, no additional substructure of the SM particles has been found. Another appealing idea is to assume that the vector bosons are composed of the SM fermions. A quantum electrodynamics model was proposed in which the photon is constructed from an electron and a positron[@Bjorken]. This model requires an unobservable space asymmetry, and its renormalizability rules are unclear. In this paper, we use the experimentally derived compositeness property of the SM particles to get information on the SM coupling constants. We focus on those vector quantum numbers that can be constructed in terms of those of the fermions. This is a remarkable SM property; fermions could otherwise belong to other representations transforming according to the Lorentz and gauge groups, without satisfying this property. As with grand unification, which assumes a connection among the quantum numbers of the vector bosons, this paper assumes a connection among those of the spin-1/2 particles and vector bosons. The associated symmetry provides the coupling information. In particular, the application of quantum mechanical rules leads to normalization constants, and Clebsch-Gordan coefficients that relate both representations, and ultimately relate to the coupling constants. We will also find that the grand-unified coupling ratio prescription is reproduced. In addition, we show that this assumption is consistent with the SM. Indeed, we apply an equivalent field-theory formulation that makes this kind of compositeness explicit, keeping the SM assumption that the fields are fundamental, unlike the composite-model case; all the SM predictions are therefore maintained. Thus, while composite models require additional fields in terms of which SM or new particles are constructed, this assumption is model independent. Hence, the putative problems associated with substructure compositeness are not encountered. We first give a general coupling-constant definition based on the normalization and the compositeness property of some particle states with respect to other particle elementary states. Using the Wigner spinor classification of Lorentz representations, one may express SM fields in terms of their spinor components. It follows that the SM Lagrangian and its fields can be rewritten and reinterpreted in this way. Finally, we classify the configurations of the vector particles in relation to their SM and grand-unified theory content, calculate corresponding coupling values at the electroweak breaking and unification scales, and present final comments. Quantum numbers characterize particles, and the normalized state $|w_i \rangle$ represents a particle with eigenvalue $w_i$ of the appropriate operator. The numbers $a_{ij}$ in the composite state $$\begin{aligned} \label {composite} | W \rangle =\frac{1}{\sqrt{N}}\sum_{i,j} a_{ij} |w_i \rangle| w_j \rangle ,\end{aligned}$$ normalized with $$\begin{aligned} \label {normalisation} N=\sum_{i,j} a_{ij}^* a_{ij},\end{aligned}$$ fix $\langle w_i w_j| W \rangle$. The same amplitude is reproduced by the corresponding operator $\hat W=\frac{1}{\sqrt{N}}\sum_{i,j} a_{ij} |w_i \rangle\langle w_j | $, satisfying $tr \hat W^\dagger \hat W=1$, through $\langle w_i |\hat W |w_j\rangle$. Thus, both structures keep the same information, and the same normalization prescription may be applied. $\hat W$ is also the most general operator acting on the $|w_i \rangle$ states. Symmetry can determine the coefficients $a_{ij}^\lambda$, up to a constant, where $\lambda$ labels the representation components of such symmetry. For example, the only (non-axial) vector operator that can be constructed out of spin-1/2 particle states is the Dirac matrix $\gamma_0\gamma^\mu$[@Dirac]; $\partial^\mu$ stems from configuration space, and, when coupled to a vector field, it is not relevant in the SM vector-spin-1/2 interaction Lagrangian because it is neither renormalizable nor gauge invariant. For each $\mu$ (no sum) $tr \gamma_0\gamma^\mu \gamma_0\gamma^\mu=4$ normalizes covariantly the operator, and fully determines it by providing the remaining constant; so is the case for the corresponding composite state $| W \rangle$. Hence, the matrix element between the spin states $|i \rangle$ and $|j \rangle$ $$\begin{aligned} \label{matrixelement} \langle i |\hat W^\mu|j \rangle\end{aligned}$$ is determined with $\hat W^\mu=\frac{1}{2}\gamma_0\gamma^\mu$. The four-entry $\hat W^\mu$ acts on the space spanned essentially by the spin-1/2 particle, its antiparticle, and their two spin polarizations. This procedure can be generalized to the case of greater number of degrees of freedom, using the rules for the direct product of vector spaces and the generalized operator that acts on such a space. The normalization for $M$ such operators, $\hat W^T= \hat W_1... \hat W_M$, is the product of the traces of each operator $\hat W_i$ in its space. The vertex interaction Lagrangian $\int {\mathcal L}_{f}$ with density $ {\mathcal L}_{f}=-\frac{1}{2}gA^a_\mu{\Psi^\alpha}^\dagger\gamma_0 \gamma^\mu G^a\Psi^\alpha$ is determined from Poincaré and gauge invariance. In general, the latter determines the interactions of the vector bosons with the other particles, and among themselves, up to the coupling constant $g$. In particular, $ {\mathcal L}_{f}$ is the only boson-spin-1/2 vertex. In the SM the fermions belong in the fundamental representation. The vertex can be consistently viewed as the expectation value of the tensor-product operator $\hat W^{\mu a}=g \gamma_0\gamma^\mu G^a1_x 1_\alpha, $ with vector components $A^a_\mu(x), $ acting upon the spin-1/2 particles $\Psi^\alpha(x)$; $\mu$ is the spin-1 index, $G^a$ the gauge-group representation matrix of the fermions, $a$ the group-representation index, $x$ the spacetime coordinate with the diagonal[^1] $1_x=|x\rangle \langle x |$, and $1_\alpha$ the unit matrix over the flavor $\alpha$. A composite state $A^a_\mu(x)|x\rangle |\mu \rangle | a\rangle$, with $|\mu \rangle$, $| a\rangle$ elements as in Eq. \[composite\], underlies the operator association leading to $\hat W^{\mu a}$: $|\mu \rangle\rightarrow (\gamma_0\gamma^\mu)_{\sigma\eta},$ $|a \rangle\rightarrow G^a_{bc}$, $|x \rangle\rightarrow|x\rangle \langle x| $; the fermion state is $\Psi_{\eta c}^\alpha(x)$. All are written explicitly in terms of $\sigma$, $\eta$ spin-1/2 indices, $b,$ $c,$ gauge-group representation indices, and the flavor. $A_\mu^a\hat W^{\mu a}$ is also the expression for the vector field in spin space, treated, e. g., in Ref. [@Wald] (the same generalization is applied to the gauge degrees of freedom). In that reference, a spinor description of the Lorentz representations is given. At each spacetime point, tensor spinorial objects are defined. In particular, a real basis of (bi)spinorial objects is constructed that spans the Lorentz vector representation. The component elements of such a basis are essentially constructed out of the unit and the Pauli matrices. A map is defined between these bispinor objects and vectors. Their identification follows from the fact that they have the same transformation properties. In fact, Maxwell’s equations can be equivalently formulated in terms of such objects, as two Dirac equations[@Bargmann]. The other Lagrangian terms can also be reinterpreted and formulated in terms of spin-projected fields. Canonical quantization in quantum field theory normalizes $A^a_\mu$; the compositeness assumption further imposes such condition on the $\hat W_i$ operators, which fully normalizes $A^a_\mu\hat W^{\mu a} $. In general, $A^a_\mu $ can be understood as an element in a polarization or group basis $A^a_\mu =tr n_{\mu }^a A^b_\nu n^{\nu b}$, where in our case $n^{\nu b }=\hat W^{\nu b}$, and it is assumed to be normalized. Indeed, we recognize in the vertex $$\begin{aligned} \label {vertex}{\mathcal L}_{f}=-{\Psi_{\sigma b}^\alpha(x)}^\dagger A^a_\mu(x)\Psi_{\eta c}^\alpha(x) \langle \sigma |\gamma_0\gamma^\mu | \eta \rangle \frac{1}{2}g \langle b | G^a | c \rangle\end{aligned}$$ the matrix elements in Eq. \[matrixelement\], and the gauge-group ones. Within the compositeness assumption, we equate each matrix element in Eq. \[vertex\] with that of the composite vector in Eq. \[matrixelement\], and similarly for the group-representation matrices, all of which contain operators acting upon the spin-1/2 particles, which leads to the identification $$\begin{aligned} \label {identi} g\rightarrow 2\sqrt{ \frac{1}{ N}}.\end{aligned}$$ The normalization $N$ is calculated as in Eq. \[normalisation\], with the convention for the $\gamma$-matrices $$\begin{aligned} \label {groupnorgam} tr \gamma_\mu\gamma_\nu=4 g_{\mu\nu} ,\end{aligned}$$ and irreducible representations $$\begin{aligned} \label {groupnor} trG_i G_j=2 \delta_{ij}. \end{aligned}$$ Essentially, we are setting normalization constants for the matrix elements in Eq. \[vertex\], which connect representations, and can be viewed as Clebsch-Gordan coefficients. ${\mathcal L}_{f}$ contains sums over matrix elements for each $\mu$ and $a$, which determine the coupling constant; only two polarizations $\mu$ have a physical-state interpretation, while gauge and Lorentz invariance demand a unique value. Quantum field theory admits arbitrary coupling constants for a vertex, which are obtained experimentally. The theoretical assignment of $g$ complements this theory. In comparing the fermion states with the vector ones, we find that the latter are composite only in the Lorentz and the gauge groups, whereas the configuration variable $x$ is elementary for both types of field. In general, an additional fermion index $\beta$ independent of $A^a_\mu$ corresponds to $\hat W_F=\sum |\beta \rangle\langle \beta| =1_F$, a unit operator present in the vertex, not contributing to the coupling constant. This is the flavor’s case. However, there are two consistent coupling definitions when such a kind of operator acts in a fermion subspace. Thus, e.g., $ SU(2)_L$ generators in a grand-unified theory such as $SU(5)$ are constructed with their lepton ($l$) and baryon ($b$) components as $G_{SU(2)_Ll}+(G_{SU(2)_Lb}\times 1_{SU(3)})$, with $1_{SU(3)}$ a projection operator in color space (leptons are color singlets); $1_{SU(3)}$ does not commute with some $SU(5)$ generators, and the associated vector-field components interact with the other unified-group ones. Physically, this [*full*]{} case corresponds to active degrees of freedom. In a lower energy regime, the symmetry is broken, and the interactions are truncated to the weak $SU(2)_L$ and the other SM interactions, while $1_{SU(3)}$ commutes with these generators. Then, in this [*reduced*]{} case, $1_{SU(3)}$ drops out of the calculation. Grand-unified theory predicts coupling-constant ratios under the condition that the SM generators belong to the same unified-group representation, which determines Weinberg’s angle at the unification energy scale[@unification], and the running of the coupling of each interaction gives values at lower energies. Similarly, the configuration of the fields’ group representations $G_i$ gives a clue to the energy scale. To obtain unified and SM couplings we specify the normalized vector-field polarizations and gauge-group generators. The couplings are calculated using the fermion quantum numbers, which are the generators’ eigenvalues, and make the generators themselves (the Cartan subset). A generation of SM left-handed \[quarks; leptons\] is classified by $[Q,u^c,d^c;$ $L,e^c]$, with $L=(e,\nu)$, $Q=(u,d)$ $SU(2)_L$ doublets, and $u^c$, $d^c$, $e^c$, charge-conjugate singlets, according to their color-weak-hypercharge $SU(3)\times SU(2)_L\times U(1)_Y$ groups; the latter can be viewed as subgroups of the $SU(5)$ grand-unified theory. The multiplets are $ [( 3,2,1/3), (\bar 3,1,-4/3),(\bar 3,1,2/3);$ $(1,2,-1), (1,1,2)]$. The fermions fit neatly into the $\bf 5$ and $\bf 10$ representations of this group. The hypercharge $Y$ and the weak interaction have different $\sigma_{\mu\pm}=\frac{1}{2}(1\pm \gamma_5)\gamma_0\gamma_\mu$ components, with the pseudoscalar $\gamma_5=-i\gamma_0\gamma_1\gamma_2\gamma_3$, which uses ${\mathcal L}_{f}$ with possibly different $\hat W^a_{\mu\pm}=\sigma_{\mu\pm}G^a_\pm $ components, in an obvious notation. One gets for $Y$ in the [*full*]{} configuration, with the above quantum numbers, the rules in Eqs. \[normalisation\] and \[identi\], and conventions in Eqs. \[groupnorgam\] and \[groupnor\], $g^\prime=2/[{2 (2 + 2^2 + 6(\frac{1}{3})^2+ 3 (\frac{2}{3})^2 +3 (\frac{4}{3})^2)]^{1/2}}$ $=\frac{1}{2}\sqrt{\frac{3}{5}}$, where the 2 in the denominator normalizes each chiral component $\sigma_{\mu\pm}$, to which corresponds one massless fermion polarization. The first two terms in the parenthesis $2+2^2=1^2+1^2+2^2$ are the lepton hypercharges and, the last three are the quark hypercharges; their multiplicity is taken into account. $Y$ may be also viewed as a generator of the $SU(5)$ interaction. Two coupling definitions apply for the weak $SU(2)_L$ interaction, one of whose generators has diagonal components $I_{(l,b)} =(1,-1)$. For the [*full*]{} configuration, $g^{uni}=$ $2/{[2(1+3)(1^2+1^2)]^{1/2}}$ $ =\frac{1}{2}$, where the second factor in the denominator counts the lepton and quark doublets, which in turn give the third factor. Non-supersymmetric unified models[@DGLee] give experimentally consistent unification couplings of $g_{ex}^{uni}\sim.52 -.56$, at $10^{14}-10^{16}$ GeV. From the SM[@Glashow]-[@Salam], $tan(\theta_W)=g^\prime/g^{uni}$, and one reproduces the $SU(5)$ unification result for Weinberg’s angle$\cite{quinn}$ $sin^2(\theta_W^{uni})=3/8$. In general, the coupling definition in Eq. \[identi\] is consistent with the grand-unified prescription for such a coupling ratio. The [*reduced*]{} configuration of the normalized weak vector implies that the color components drop from the calculation. It gives the same weight to quarks as to leptons, as is necessary if one omits unification-group information. We should get information on the electroweak-breaking scale to the extent that these weak and hypercharge configurations describe on-shell Z and W vector bosons. We find $g^{le} =2 /[{2 (2)(1^2+1^2) )]^{1/2}}=\frac{1}{\sqrt{2}}\approx .707$, while at the $M_Z$ scale$\cite{tables}$, $g_{ex}=.649519(20)$, where one standard-deviation uncertainty for the last digits is given in parenthesis. Each isospin doublet component corresponds to a different hypercharge isospin singlet; this suggests, extending the rule to color components, that only the [*full*]{} configuration need be considered for $Y.$ Thus, $g'\approx .387$ is between $g_{ex}^\prime(M_Z)=.35603(6)$ and the unified hypercharge values $\sqrt{\frac{3}{5}}g_{ex}^{uni}\sim .40-.43$; the relatively narrow range provides a test of the prediction. From $tan(\theta_W)=g^\prime/g^{le}$, we find $sin^2(\theta_W)=3/13\approx .23078,$ while at $M_Z$ $sin^2(\theta_{Wex})= .23113(15).$ One may also interpret the [*reduced*]{} weak configuration within the minimal supersymmetric model, with a unified[@Amaldi] $g^{Suni}_{ex}=.69(4)$ at $10^{15.8\pm .4}$ GeV. $g^{le}$ reproduces a value also within a narrow low and high energy range. For the gluons’ coupling in the $1_{SU(2)_L}$-[*reduced*]{} case we use the $\lambda_3$ Pauli-matrix fundamental component of the $SU(3)$ (any other generator would also do) with the convention of Eq. \[groupnor\] $g_s =2 /[{2 (2)(1^2+1^2) )]^{1/2}}=1/\sqrt{2}\approx .707,$ or $\alpha_s=\frac{g_s^2}{4 \pi}\approx .040$, while $\alpha_{s(ex)}(M_Z)=.1172(20)$. Then $g_s$ provides a lower limit around the unification scale. While only the fermion-vector vertex has been examined, the results are valid for a more general Lagrangian. The coupling constants in the other Lagrangian terms get a unique value, because gauge invariance demands it for each gauge group. All along, flavor is assumed to belong to the [*reduced*]{} configuration, for it does not influence interactions. In a grand-unified theory and in the SM, the electroweak-field components at the unification scale, and at the symmetry-breaking scale, are determined, respectively, through the ratios of the electroweak couplings, namely, Weinberg’s angle. The SM fermions and bosons, and their simple interactions conform to a compositeness assumption. Under this assumption, the allowed vertices and the fields’ normalized polarization generate the coupling constants. Specifically, these are obtained by associating composite-field configurations both to the unification scale and the W and Z particle regime. Already at tree level, Weinberg’s angle is reproduced for the $SU(5)$ unified theory, and a value close to the experimental one is obtained at the $M_Z$ electroweak-breaking scale, which validates the ascribed configuration in each regime. This set of two coupling constant or Weinberg angle values provides a connection between the two energy scales through, e.g., the renormalization group equations, which have to be supplemented with boundary conditions. Although the low-energy ratio $tan (\theta_{W}) $ does not contain couplings at precisely the same energy, it contains information on the group-generator structure, stemming from the compositeness assumption; this is not in contradiction with the coupling running that should be applied, and whose corrections cancel among the two couplings. The couplings are also interpreted consistently within the minimal supersymmetric model. The calculation of $\theta_{W}$ can be viewed as complement to, or as alternative to, that of $\theta_{W}^{uni}$. In the first approach, the coupling constants relate energies in the unified and symmetry-breaking scales. In the second approach, one obtains information on the $M_Z$ scale, understood as fundamental[@Dim]. The paper’s approach may also be applied in other extensions, which require only the consideration of reducible representations. The compositeness hypothesis is supported with coupling constants obtained among a limited number of allowed configurations, and that reproduce experimental values, which are within a narrow range at different energy scales. [99]{} H. Georgi, and S. L. Glashow, Phys. Rev. Lett. [**32**]{}, 438 (1974). S. Weinberg, Phys. Lett. B [**125**]{}, 265 (1983). M. B. Green, J. H. Schwarz, and E. Witten, [*Superstring theory*]{}, Vols. 1 & 2, (Cambridge University Press, Cambridge, 1987). J. Besprosvany, Gauge and space-time symmetry unification, Int. 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<span style="font-variant:small-caps;">A generalization of</span> <span style="font-variant:small-caps;">random self-decomposability</span> NEELOLPALAM, S. N. Park Road Thrissur-680 004, India. e-mail: *ssatheesh1963@yahoo.co.in* Department of Statistics, Prajyoti Niketan College Thrissur-680 301, India. e-mail: *esandhya@hotmail.com* [**Abstract.**]{} The notion of random self-decomposability is generalized here. Its relation to self-decomposability, Harris infinite divisibility and its connection with a stationary first order generalized autoregressive model are presented. The notion is then extended to $\mathbf{Z_+}$-valued distributions. [**Mathematics Subject Classification.**]{} 60E05, 60E07, 62E10, 62M10. [**Keywords.**]{} self-decomposability, random self-decomposability, geometric infinite divisibility, Harris infinite divisibility, geometric distribution, Harris distribution, characteristic function, probability generating function. Introduction {#sec1} ============ The role of self-decomposable (SD) distributions in first order autoregressive (AR(1)) models of the form $$X_n = cX_{n-1}+\epsilon_n,$$ described by random variables (*r.v.*s) $\{X_n, n\in Z\}$, innovations (*i.i.d. r.v.*s) $\{\epsilon_n\}$ and $c\in (0,1)$ such that for each $n$, $\epsilon_n$ is independent of $X_{n-1}$, has been discussed by many authors, *see e.g.* Bouzar and Satheesh (2008) and the references therein. Recently Kozubowski and Podgórski (2010) has introduced the notion of random self-decomposability of distributions on the reals motivated by stationary solutions to the AR(1) model $$X_n = \begin{cases} \epsilon_n, \text{ with probability $p$}, \\ cX_{n-1}+\epsilon_n, \text{ with probability $(1-p)$,} \end{cases}$$ described by *r.v.*s $\{X_n, n\in Z\}$, innovations $\{\epsilon_n\}$ and $c\in [0,1]$ such that for each $n$, $\epsilon_n$ is independent of $X_{n-1}$. **Definition 1.1** A charcteristic function (CF), of a probability distribution, $\psi(t)$ is randomly self-decomposable (RSD) if for each $p,c\in [0,1]$ there exists a CF $\psi_{c,p}(t)$ such that $$\psi(t)= \psi_{c,p}(t)\{p+(1-p)\psi(ct)\}.$$ Kozubowski and Podgórski (2010) then discusses the relation of RSD laws to SD laws and geometrically infinitely divisible (GID) laws. In proposition 2.3 they prove, in an elegent manner, that the class of RSD laws equals the intersection of the classes of GID laws and SD laws. They also discuss a variety of examples. We need the following in our discussion. **Definition 1.2** Harris$(a,k)$ distribution on $\{1, 1+k, 1+2k, ....\}$ is described by its probability generating function (PGF) $$P(s)= \frac{s}{\{a-(a-1)s^k\}^{1/k}}, \text {$k>0$ integer and $a>1$.}$$ **Definition 1.3** A CF $\psi(t)$ is Harris-ID (HID) if for each $p\in(0,1)$ there exists a CF $\psi_p(t)$ such that $$\psi(t)=\frac{\psi_p(t)}{\{a-(a-1)\psi_{p}^{k}(t)\}^{1/k}}, p=\frac{1}{a}.$$ **Theorem 1.1** (Satheesh *et al.* (2008)) A CF $\psi(t)$ is HID *iff* $$\psi(t)=\frac{1}{(1-\log h(t))^{1/k}}$$ where $k>0$ integer and $h(t)$ is some CF that is ID. When $k=1$ Harris distribution becomes the geometric(*p*) distribution on $\{1, 2, ...\}$ with $p=\frac{1}{a}$. For more on this distribution see Sandhya *et al.* (2008). Certain aspects of HID laws and generalized AR(1) models have been discussed in Satheesh *et al.* (2008). In section 2, the notion of RSD is generalized, its relation to SD laws and HID laws are presented and its connection to a stationary generalized AR(1) model is given. The notion is then extended to $\mathbf{Z_+}$-valued distributions in section 3. We closely follow the development in Kozubowski and Podgórski (2010). Generalizing RSD distributions {#sec2} ============================== **Remark 2.1** In the paragraph after their Proposition 3.1 Kozubowski and Podgórski (2010) state that AR(1) processes described by (1.2) cannot be constructed with either (general) gamma or Gaussian distributions for $X_n$ as neither of them are GID although both are SD. However, it should be noted that gamma$(\alpha,\lambda)$ distributions (*equation* (2.10)) are GID if $\alpha \leq 1$, *see e.g.* Yannaros (1988) or Sandhya (1991). **Definition 2.1** A CF $\psi(t)$ is Harris-RSD (HRSD) if for each $c\in (0,1]$ and each $p\in[0,1) $ there exists a distribution with CF $\psi_{c,p}(t)$ such that $$\psi(t)= \psi_{c,p}(t)\{p+(1-p)\psi^{k}(ct)\}^{1/k}.$$ **Remark 2.2** With the above nomenclature the RSD defined by Kozubowski and Podgórski (2010) is geometric RSD (GRSD) because it bridges the notions of SD and GID where as our definition bridges the notions of SD and HID. When $p=0$ equation (2.1) reduces to $$\psi(t)=\psi(ct) \psi_{c}(t)$$ where $\psi_{c}(t)= \psi_{c,0}(t)$, that is $\psi(t)$ is SD. On the other hand when $c=1$ equation (2.1) becomes $$\psi(t)= \psi_{p}(t)\{p+(1-p)\psi^{k}(t)\}^{1/k}.$$ where $\psi_{p}(t)=\psi_{1,p}(t)$. Solving for $\psi(t)$ we get $$\psi(t)= \frac{\psi_p(t)}{\{a-(a-1)\psi_p^k(t)\}^{1/k}} ; a=\frac1p.$$ That is $\psi(t)$ is HID. Denoting the classes of HRSD, SD and HID distributions by $\mathcal{C}_{HRSD}$, $\mathcal{C}_{SD}$ and $\mathcal{C}_{HID}$ the above discussion shows that $\mathcal{C}_{HRSD}\subset \mathcal{C}_{SD} \cap \mathcal{C}_{HID}$. In the next Proposition we show that we have equality here. **Proposition 2.1** We have $\mathcal{C}_{HRSD} = \mathcal{C}_{SD} \cap \mathcal{C}_{HID}$. Further, whenever the CF $\psi(t) \in \mathcal{C}_{HRSD}$, the CF $\psi_{c,p}(t)$ in (2.1) can be written as $$\psi_{c,p}(t)=\psi_{c}(t). \psi_{p}(ct)$$ where $\psi_{c}(t)$ and $\psi_{p}(t)$ are given by $$\psi_{c}(t)= \frac {\psi(t)}{\psi(ct)}$$ $$\psi_{p}(t)= \frac {\psi(t)} {\{p+(1-p)\psi^{k}(t)\}^{1/k}}$$ *Proof.* If the CF $\psi(t)$ is SD then for each $c\in(0,1]$ the function $\psi_{c}(t)$ in (2.6) is a genuine CF and similarly if $\psi(t)$ is HID then for each $p\in[0,1)$ the function $\psi_{p}(t)$ in (2.7) also is a genuine CF. Consequently (2.5) is a well defined CF and hence (2.1) holds, proving the assertion. Now let us consider a generalization of the AR(1) sequence (1.2). Here $\{X_{n}\}$ is composed of $k$ independent AR(1) sequences $\{Y_{n,i}\},i=1,2, \dots\ k$ and where for each *n*, $\{Y_{n,i}\}$ are independent. That is, for each $n$, $X_{n}=\sum_{i=1}^k Y_{n,i}$ and $\epsilon_{n}=\sum_{i=1}^k \epsilon_{n,i}$ where $\{Y_{n,i}\}$ is an *i.i.d* sequence and similarly $\{\epsilon_{n,i}\}$ is also an *i.i.d* sequence, $k$ being a fixed positive integer. Further, it is also assumed that for each $n$, $\epsilon_{n,i}$ is independent of $Y_{n-1,i}$ for all $i=1,2, \dots\ k$. Situations where such a model can be useful have been discussed in Satheesh *et al.* (2008). $$\sum_{i=1}^k Y_{n,i}= \begin{cases} \sum_{i=1}^k \epsilon_{n,i}, \text{ with probability }p,\\ \sum_{i=1}^k cY_{n-1,i} + \sum_{i=1}^k \epsilon_{n,i}, \text{ with probability}(1-p). \end{cases}$$ In terms of CFs and assuming stationarity we get $$\psi_Y(t) = \psi_\epsilon (t) \{p+(1-p)\psi_Y^k (ct)\}^{1/k}.$$ The following Proposition is now clear. **Proposition 2.2*a*** If $\{Y_{n,i}\}$ describes the AR(1) model (2.8) that is stationary for each $c\in (0,1]$ and $p\in[0,1)$ then the distribution of $\{Y_{n,i}\}$ is HRSD. Now proceeding as in the proof of Theorem 2.2 in Bouzar and Satheesh (2008) we have the following converse to Proposition 2.2*a*. **Proposition 2.2*b*** If $\psi_{Y}(t)$ is a CF that is HRSD with $\psi_{c,p}(t)=\psi_\epsilon(t)$ for each $c\in (0,1]$ and $p\in[0,1)$ then there exists a stationary AR(1) model described by (2.8) with $\psi_{Y}(t)$ the CF of $\{Y_{n,i}\}$ and $\psi_\epsilon(t)$ that of the innovations $\{\epsilon_{n,i}\}$. **Example 2.1** Gamma$(\frac{1}{k},\lambda)$ distributions has CF $$\psi(t)=\frac{1}{(1-i \lambda t)^{1/k}}, \text {$k>0$ integer, $\lambda >0$}$$ is HID. Further since it is also SD, this distribution is HRSD. **Example 2.2** Let the CF $\psi(t)$ be Harris-sum-stable for every $c\in(0,1)$. Then $$\psi(t)=\psi(ct).\frac{1}{\{a-(a-1)\psi^k(ct)\}^{1/k}}.$$ The second factor on the RHS is also a genuine CF being a Harris-sum of $\psi(t)$ where the support of this Harris distribution is $\mathbf{Z_+}$. Thus $\psi(t)$ is SD. Hence if we take $h(t)$ as a stable CF in Theorem 1.1 then $\psi(t)$ in (1.6) is Harris-sum-stable for every $c\in(0,1)$ and we have a general procedure to construct CFs that are HRSD. With $k=1$ above, we have the corresponding geometric-sum-stable laws and a procedure to construct the examples in Kozubowski and Podgórski (2010). Discrete analogue of HRSD distributions {#sec3} ======================================= Steutel and van Harn (1979) had developed discrete SD (DSD) distributions. We now introduce RSD and HRSD for $\mathbf{Z_+}$-valued distributions. Some aspects of discrete HID (DHID) laws and generalized AR(1) models on $\mathbf{Z_+}$ have been discussed in Satheesh *et al.* (2010*b*). **Definition 3.1** A PGF $P(s)$ is DHID if for each $p\in(0,1)$ there exists a PGF $P_p(s)$ such that $$P(s)=\frac{P_p(s)}{\{a-(a-1)P_{p}^{k}(s)\}^{1/k}}, p=\frac{1}{a}.$$ **Theorem 3.1** (Satheesh *et al.* (2010*a*)) A PGF $P(s)$ is DHID *iff* $$P(s)=\frac{1}{(1-\log R(s))^{1/k}},$$ where $k>0$ integer and $R(s)$ is a PGF that is DID. **Definition 3.2** A PGF $P(s)$ is discrete HRSD (DHRSD) if for each $c\in (0,1]$ and $p\in[0,1) $ there exists a PGF $P_{c,p}(s)$ such that $$P(s)= P_{c,p}(s)\{p+(1-p)P^{k}(1-c+cs)\}^{1/k}.$$ Denoting the classes of DHRSD, DSD and DHID distributions by $\mathcal{C}_{DHRSD}$, $\mathcal{C}_{DSD}$ and $\mathcal{C}_{DHID}$ we can proceed as in Section 2 to arrive at **Proposition 3.1** We have $\mathcal{C}_{DHRSD} = \mathcal{C}_{DSD} \cap \mathcal{C}_{DHID}$. Further, whenever $P(s) \in \mathcal{C}_{DHRSD}$, the PGF $P_{c,p}(s)$ in (3.3) can be written as $$P_{c,p}(s)=P_{c}(s). P_{p}(1-c+cs)$$ where $P_{p}(s)$ and $P_{c}(s)$ are given by $$P_{c}(s)= \frac {P(s)}{P(1-c+cs)}$$ $$P_{p}(s)= \frac {P(s)} {\{p+(1-p)P^{k}(s)\}^{1/k}}$$ Again, considering the $\mathbf{Z_+}$-valued analogue of the generalized AR(1) scheme (2.8) with $\odot$, the binomial thinning operator in Steutel and van Harn (1979) we have the INAR(1) model $$\sum_{i=1}^k Y_{n,i}= \begin{cases} \sum_{i=1}^k \epsilon_{n,i}, \text{ with probability }p,\\ \sum_{i=1}^k c \odot Y_{n-1,i} + \sum_{i=1}^k \epsilon_{n,i}, \text{ with probability}(1-p). \end{cases}$$ Assuming stationarity we have the following Propositions as in Section 2. **Proposition 3.2*a*** If $\{Y_{n,i}\}$ describes the stationary INAR(1) model (3.7) for each $c\in (0,1]$ and $p\in[0,1)$ then the distribution of $\{Y_{n,i}\}$ is DHRSD. Conversely, **Proposition 3.3*b*** If $P_{Y}(s)$ is a PGF that is DHRSD with $P_{c,p}(s)= P_\epsilon(s)$ for each $c\in (0,1]$ and $p\in[0,1)$ then there exists a stationary INAR(1) model described by (3.7) with $P_{Y}(s)$ the PGF of $\{Y_{n,i}\}$ and $P_\epsilon(s)$ that of $\{\epsilon_{n,i}\}$. **Example 3.1** Negative binomial$(\frac{1}{k},\lambda)$ distributions with PGF $$P(s)=\frac{1}{(1+\lambda(1-s))^{1/k}}, \text {$k>0$ integer, $\lambda >0$}$$ is DHID. Further since it is also DSD, this distribution is HRSD. **Example 3.2** We may also proceed in a general frame work as done in Example 2.2 to construct PGFs that are HRSD. Satheesh and Sandhya (2010) has proposed a further generalization of HRSD distributions based on the notion of $\mathcal{N}ID$ distributions of Gnedenko and Korolev (1996). [99]{} Bouzar, N and Satheesh, S (2008), Comments on $\alpha$-decomposability, *Metron*, **LXVI**, 243–252. Gnedenko, B V and Korolev, V Yu. (1996), Random Summation, limit Theorems and Applications, CRC Press, Boca Raton. Kozubowski, T J and Podgórski, K (2010), Random self-decomposability and autoregressive processes, *Statis. Probab. Lett.*, doi:10.1016/j.spl.2010.06.014. Sandhya, E (1991), *Geometric Infinite Divisibility and Applications*, Unpublished Ph.D. thesis submitted to the Department of Statistics, University of Kerala, India. Sandhya, E; Sherly, S and Raju, N (2008), Harris family of discrete distributions, in *Some recent innovations in Statistics*, A special volume in honour of Professor T S K Moothathu, published by the Department of Statistics, University of Kerala, India, 57–72. Satheesh, S; Sandhya, E and Rajasekharan, K E (2008), A generalization and extension of an autoregressive model, *Statis. Probab. Lett.*, **78**, 1369–1374. Satheesh, S; Sandhya, E and Lovely, A T (2010*a*), Limit distributions of random sums of $\mathbf{Z_+}$-valued random variables, *Commu. Statist.-Theor. Meth.*, **39**, 1979–1984. Satheesh, S; Sandhya, E and Lovely, A T (2010*b*), Random infinite divisibility on $\mathbf{Z_+}$ and generalized INAR models, *ProbStat Forum*, **3**, 108–117. Satheesh, S and Sandhya, E (2010), A further generalization of random self-decomposability, *submitted*. Steutel, F W and van Harn, K (1979), Discrete analogues of selfdecomposability and stability, *Ann. Probab.*, **7**, 893–899. Yannaros, N (1988), On Cox processes and gamma renewal processes, *J. appl. Probab.*, **25**, 423–427.

--- abstract: 'We show that the Lambert series $f(x)=\sum d(n) x^n$ is irrational at $x=1/b$ for negative integers $b < -1$ using an elementary proof that finishes an incomplete proof of Erdős.' author: - 'J. Vandehey' title: On an incomplete argument of Erdős on the irrationality of Lambert series --- Introduction ============ Chowla [@chowla] conjectured that the functions $$f(x) = \sum_{n=1}^\infty \frac{x^n}{1-x^n} \qquad \text{and} \qquad g(x) = \sum_{n=1}^\infty \frac{x^n}{1-x^n}(-1)^{n+1}$$ are irrational at all rational values of $x$ satisfying $|x|<1$. For such $x$ the above functions may be rewritten as $$f(x) = \sum_{n=1}^\infty d(n) x^n \qquad \text{and} \qquad g(x) = \frac{1}{4}\sum_{n=1}^\infty r(n)x^n,$$ where $d(n)$ is the number of divisors of $n$ and $r(n)$ is the number of representations of $n$ as a sum of two squares. Erdős [@erdos1] proved that for any integer $b>1$, the value $f(1/b)$ is irrational. He did so by showing that $f(1/b)$ written in base $b$ contains arbitrary long strings of $0$’s without terminating on $0$’s completely. If we take $b<-1$ to be a negative integer, then Erdős’ methods show by the same method that $f(1/b)$ in base $|b|$ contains arbitrary long strings of $0$’s; however, Erdős claims without proof that showing it will not terminate on $0$’s can be done using similar methods. It is not clear what method Erdős intended, and in later papers (including his review of similar irrationality results [@erdos2]) Erdős only refers to proving the case of positive $b$. Since then, several proofs have been offered for the irrationality of the $b<-1$ case and far more general theorems besides. Much credit is often given to Bezivin [@bezivin] and Borwein [@borwein] for proving the first major generalizations of these results; and other results can be often be found in the literature under the term of the $q$-analogue of the logarithm or, simply, the $q$-logarithm. However, these results are proved using entirely different techniques than what Erdős uses and leaves open the question of whether his method could have finished the proof. Erdős’ method can be extended to the following stronger result with a virtually identical proof. \[thm:erdos\] Let $b>1$ be a positive integer and $\mathcal{A}$ be any finite set of non-negative integers. Then for any sequence $\{a_n\}_{n=1}^\infty$ taking values in $\mathcal{A}$ such that the sequence does not end on repeated $0$’s, we have that $$\sum_{n=1}^\infty d(n) \frac{a_n }{b^n}$$ is irrational. Theorem \[thm:erdos\] has the following curious corollary. Let $a_n(x)$ be the $n$th base $b$ digit of a number $x$ in $(0,1)$. (If $x$ has two base $b$ expansions, then we chose the one which does not end on repeated $0$’s.) Then the map $$x=\sum_{n=1}^\infty \frac{a_n(x)}{b^n} \longmapsto \sum_{n=1}^\infty d(n) \frac{a_n(x)}{b^n}$$ has its image in $\mathbb{R}\setminus \mathbb{Q}$ and is also continuous at all $x$ that do not have a representation as a finite base $b$ expansion. We could replace the condition that $a_n$ be in the finite set $\mathcal{A}$ with a restriction that $0\le a_n \le \phi(n)$ for some sufficiently slowly growing integer-valued function $\phi$. It would be interesting to know what the fastest growing $\phi$ for which the Theorem \[thm:erdos\] holds would be. In this paper, we will prove the following extension of Theorem \[thm:erdos\]. \[thm:main\] Let $b>1$ be a positive integer and $\mathcal{A}$ be any finite set of integers *that does not contain* $0$. Then for any sequence of $\{a_n\}_{n=1}^\infty$ taking values in $\mathcal{A}$, we have that $$\sum_{n=1}^\infty d(n)\frac{a_n}{b^n}$$ is irrational. The new ingredient to extend Erdős’ method is finding arbitrarily long strings of zeros *that are known to be preceded by a non-zero number*, and to find these strings arbitrarily far into the base $|b|$ expansion. In particular, by taking $a_n=(-1)^n$, this proves that $f(1/b)$ is irrational for negative integers $b<1$ as well, completing Erdős’ proof. Proof of Theorem \[thm:main\] ============================= We will require a result mentioned by Alford, Granville, and Pomerance [@agp p. 705]. The function $\pi(N;d,a)$ equals the number of primes up to $N$ that are congruent to $a$ modulo $d$. \[prop:agb\] Let $0<\delta<5/12$. Then there exist positive integers $N_0$ and $\overline{\mathcal{D}}$ dependent only on $\delta$, such that the bound $$\pi(N;d,a) \ge \frac{N}{2\varphi(d) \log N}$$ holds for all $N>N_0$; all moduli $d$ with $1 \le d \le N^\delta$, except, possibly for those $d$ that are multiples of some element in $\mathcal{D}(N)$, a set of at most $\overline{\mathcal{D}}$ different integers that all exceed $\log N$; and all a relatively prime to $d$. We begin our proof much as Erdős did his. Let $b \ge 2$ be a fixed positive integer, let $\mathcal{A}$ be a finite set of integers that does not contain $0$, and let $N$ be a large positive integer that is allowed to vary. Define $k$ in terms of $N$ by $$k=k(N):=\lfloor \left( \log{N} \right)^{1/10} \rfloor.$$ Let $j_0$ be a fixed integer, independent of $N$, so that $2 \max_{a\in \mathcal{A}} |a| / b^{j_0} <1$. Let $0<\delta<5/12$ be some sufficiently small fixed constant, and let $N_0=N_0(\delta)$ and $\overline{\mathcal{D}}=\overline{\mathcal{D}}(\delta)$ be the corresponding constants from Proposition \[prop:agb\]. Let $N_1 >N_0$ be large enough so that for any $N> N_1$, the interval $((\log N)^2, 2 (\log N)^2)$ cotains at least $u+\overline{\mathcal{D}}$ primes, where $u=u(N)=k(k-1)/2$. In addition, for such $N>N_1$, let $\mathcal{D}(N)$ be the set of exceptional moduli from Proposition \[prop:agb\]. Since we assume that $\delta$ is constant, $|\mathcal{D}(N)|\le \overline{\mathcal{D}}$ is bounded. For each $D$ in $\mathcal{D}(N)$, let $\tilde{p}_D$ denote the smallest prime strictly greater than $(\log N)^2$ that divides $D$, if such a prime exists, and then let $p_1< p_2< \dots <p_u$ be the smallest $u$ primes strictly greater than $(\log N)^2$ that are not equal to $\tilde{p}_D$ for any $D\in \mathcal{D}(N)$; by assumption on $N$, we have that each such $p_i$ is less than $2(\log N)^2$. Finally, let $$A:= \prod_{i=1}^{ j_0(j_0-1)/2}p_i^b \prod_{i=j_0(j_0+1)/2+1 }^{ u} p_i^b,$$ so that, in particular, $A$ is not a multiple of any $D$ in $\mathcal{D}(N)$; moreover, provided $N$ is sufficiently large, we have $$A<(2(\log N)^2)^{bk(k-1)/2} \le N^\delta.$$ If we set $$A= \prod_{1 \le i \le k^2} p_i^b$$ then since $p_i < 2 \left( \log{N} \right)^2$ for $i \le k^2$ and sufficiently large $N$, we have that—again for sufficiently large $N$— $$A<e^{(\log{N})^{1/4}}.$$ By the Chinese remainder theorem, there exists an integer $r$, with $0 \le r \le A-1$, such that $$r+j\equiv \prod_{i=j(j-1)/2+1}^{j(j+1)/2} p_i^{b-1} \pmod{\prod_{i=j(j-1)/2+1}^{j(j+1)/2} p_i^{b}}, \qquad 0 \le j \le k-1, j\neq j_0.$$ (The exception $j\neq j_0$ marks the key difference between this proof and Erdős’.) Since all the $p_i$’s are bounded below by $(\log N)^2$, we have that $r$ necessarily tends to infinity as $N$ does, although possibly much slower. With this value of $r$, for any integer of the form $$r+mA, \qquad 0 \le m <\lfloor N/A \rfloor,$$ we have that $$\label{eq:congruence} d(r+mA+j) \equiv 0 \pmod{b^{j+1}}, \qquad 0 \le j <k,\ j \neq j_0$$ by the multiplicity of $d(\cdot)$. Moreover, $r+j_0$ is relatively prime to $A$, since each $p$ dividing $A$ also divides some $r+j$, with $0 \le j <k,$ $j \neq j_0$; the largest $j$ can be is $k\le (\log N)^{1/10}$, but all primes dividing $A$ are at least $(\log N)^2$. We can also apply Proposition \[prop:agb\] to see that $$\label{eq:agbcor} \pi (N,A,r+j_0)\ge \frac{N}{2\varphi(A)\log N}.$$ Erdős in [@edros1] also proved the following result, which we give here without reproof. (While our construction of $A$ and $r$ are different from Erdős’, they satisfy all the requirements for Erdős’ proof technique to still hold.) \[lem:erdos\] With $A$, $r$, $b$, and $k$ all as above, the number of $m < \lfloor N/A \rfloor$ such that $$\sum_{n> r+k+mA} d(n)\frac{1}{b^n} > \frac{1}{b^{r+k/2+mA}}$$ is less than $$\frac{10cN(\log{N})^2}{A2^{k/4}}$$ for some constant $c$ independent of all variables. Regardless of how large $c$ is, we have, for sufficently large $N$, that $$\frac{N}{2\varphi(A)\log N} \ge \frac{10cN(\log{N})^2}{A2^{k/4}}.$$ Therefore, by combining Lemma \[lem:erdos\] with and , we see that for sufficiently large $N$ there exists some $m_0 < \lfloor N/A\rfloor$ such that $$\label{eq:upper} b^{j+1} | d(r+m_0 A+j) , \qquad 0 \le j <k, j \neq j_0,$$ $$\label{eq:middle} r+m_0 A + j_0 \text{ is prime},$$ and $$\label{eq:lower} \sum_{n> r+k+m_0 A} \left| d(n) \frac{a_n }{b^n} \right| \le \frac{\max_{a\in \mathcal{A}} |a|}{b^{r+k/2+mA}}.$$ Now consider a particular sequence $(a_n)$ with each $a_n \in \mathcal{A}$ together with the sum $$\sum_{n=1}^\infty d(n) \frac{a_n }{b^n} .$$ By , the partial sum $$\sum_{\substack{n \le r+k+m_0 A \\ n \neq r+j_0+m_0 A}} d(n) \frac{a_n }{b^n},$$ when written in base $b$, has its last non-zero digit in the $(r-1+m_0A)$th place or earlier.[^1] In addition, by , the partial sum $$\sum_{n > r+k+m_0 A} d(n) \frac{a_n }{b^n}$$ when written in base $b$ has its first non-zero digit in the $$(r+k/2+m_0 A-\lceil \log_b \max_{a\in \mathcal{A}} |a| \rceil)\text{th}$$ place or later. The number $$d(r+j_0+m_0A) \frac{a_{r+j_0+m_0A} }{b^{r+j_0+m_0A}}=\frac{2a_{r+j_0+m_0A} }{b^{r+j_0+m_0A}}$$ when written in base $b$ has its non-zero digits only between the $(r+m_0 A)$th and $(r+j_0+m_0 A)$th place, and it has at least one such non-zero digit. Thus the full sum has a string of at least $k/2+O(1)$ zeroes immediately preceded by a non-zero digit starting somewhere between the $(r+m_0 A)$th and $(r+j_0+m_0 A)$th place. So as $N$ increases to infinity, we can find arbitrarily long strings of $0$’s (which corresponds to $k$ increasing to infinity) immediately preceded by a non-zero digit, and we find these strings arbitrarily far out in the expansion (since $r$ also tends to infinity). The base $b$ digits cannot therefore be periodic and hence the sum is irrational. This completes the proof. $$\begin{aligned} n &\equiv q_1^{b-1} \pmod{q_1^b}\notag \\ n +2 &\equiv (q_2 q_3 q_4)^{b-1} \pmod{(q_2 q_3 q_4)^b} \notag \\ n +3 &\equiv (q_5 q_6 q_7 q_8)^{b-1} \pmod{(q_5 q_6 q_7 q_8)^b} \label{eq:congruence} \\ &\cdots \notag \\ n+k-1 &\equiv (q_u q_{u+1} \ldots q_{u+k-1})^{b-1} \pmod{(q_u q_{u+1} \ldots q_{u+k-1})^b}. \notag\end{aligned}$$ The absence of a congruence for $n+1$ is intentional and marks a key difference between the proof here and Erdős’ proof. Let $D=D(N)$ be the exceptional modulus from Proposition \[prop:agb\]. If $D$ and $A$ share a prime factor $p_j$, let $B=A/p_j^t$; otherwise, let $B=A/p_1^t$. Then, we have that $D$ does not divide $B$. Let $q_1, q_2, \ldots$ be the various distinct prime factors of $B$ in no particular order. Then, by the Chinese remainder theorem, there exists an $x$ modulo $B$ which satisfies the following congruences: $$\begin{aligned} x &\equiv q_1^{t-1} \pmod{q_1^t}\\ x +2 &\equiv (q_2 q_3 q_4)^{t-1} \pmod{(q_2 q_3 q_4)^t}\\ x +3 &\equiv (q_5 q_6 q_7 q_8)^{t-1} \pmod{(q_5 q_6 q_7 q_8)^t}\\ &\cdots\\ x+k-1 &\equiv (q_u q_{u+1} \ldots q_{u+k-1})^{t-1} \pmod{(q_u q_{u+1} \ldots q_{u+k-1})^t}\end{aligned}$$ where $u = k(k-1)/2 -1$. The absence of a congrence for $x+1$ is intentional. With this choice of $x$ modulo $B$, we have that any integer of the form $$x+yB, \qquad 0 \le y < \lfloor N/B \rfloor$$ we have by the multiplicity of $d(n)$ that $$d(x+j+yA) \equiv 0 \pmod{t^{j+1}}$$ for all $0 \le j < k$, $j \neq 1$. We have that $x+1$ is relatively prime to $B$, since each $q_i$ divides a number of the form $x+j$ with $0\le j <k$, $j\neq 1$, and since each $q_i$ is larger than $k$. By construction, $D$ does not divide $B$ so we can apply Proposition \[prop:agb\] to see that $$\pi(N,B,x+1) \gg \frac{N}{\phi(B) \log{N}}$$ We furthermore can apply the above theorem to see how often $x+1+yB$ is prime, since $(x+1,B)=1$ (there exists $0 \le j < k$, $j \neq 1$ for which a given prime dividing $B$ also divides $x+j$ and since $k< \log{n}$ and each prime dividing $B$ is greater than $\log{n}^2$ the result follows) and $D_0 \nmid B$. Then, since $B < n^{\delta}$ for any $\delta$ provided $n$ is sufficiently large, we then have that $$\pi (n,B,x+1) > \frac{n}{2 \phi(B) \log{n}}.$$ Combining this with the above work, we can see that for any large enough $n$ there exists some $y_0>1$ such that $$\sum_{r> x+k+y_0 B} \left| \frac{d(r)}{t^r} \right| < \frac{1}{t^{x+k/2+y_0 B}}$$, $x+1+y_0 B$ is prime, and for all $0 \le j < k$, $j \neq 1$, we have that $t^{j+1} | d(x+j+y_0 B)$. Erdős himself proved that for any large enough $n$ there exists some $B$ and $y_0>1$ such that $$\sum_{r> x+k+y_0 B} \left| \frac{d(r)}{t^r} \right| < \frac{1}{t^{x+k/2+y_0 B}}$$, and for all $0 \le j < k$, we have that $t^{j+1} | d(x+j+y_0 B)$. To distinguish these two, we call the previous statement fact 1, and this statement fact 2. This allows us to prove the following: For any integer $t>1$, the function $$f(x)=\sum_{r=1}^{\infty} \frac{x^r}{1-x^r} = \sum_{r=1}^{\infty} d(r) x^r$$ is irrational for $x=-1/t$. Consider the number $a=\sum_{r=1}^{\infty} d(r) (-1/t)^r$. If $a$ is rational, then the fractional parts of $ta,t^2 a, t^3a,\ldots$ should eventually be zero or cyclic depending on whether or not the denominator of $a$ is a power of $t$ or not. So given a large $n$, consider fact 2 above and multiply $a$ by $t^{x+y_0 B-1}$. Clearly the first $x+y_0 B-1$ terms become integers so are not considered in the fractional part. And the terms $r=x+j+y_0 B$ for $0 \le j < k$ will also be integers since the power of $t$ that divides the numerator will cancel the remaining part of the denominator after multiplying by $t^{x+y_0 B-1}$. Thus the fractional part of $t^{x+y_0 B-1} a$ equals $$t^{x+y_0 B-1} \sum_{r> x+k+y_0 B} \frac{d(r)}{(-t)^r} < t^{-1-k/2},$$ which can be made arbitrarily small by letting $n$ be large. Thus we must fall into the case where $ta,t^2 a, t^3a,\ldots$ is eventually zero. However, by fact 1, we can find another even larger $n'$ so that its corresponding $x'+y'_0 B'$ is larger than the $n$ from the previous paragraph. So when we multiply $a$ by $t^{x'+y'_0 B' - 1}$, the fractional part equals $$t^{x+y_0 B-1} \left( \frac{d(x'+y'_0 B'+1)}{(-t)^{x'+y'_0 B'+1}} + \sum_{r> x+k+y_0 B} \frac{d(r)}{(-t)^r} \right) \pmod{1}$$ but this first term is just $\pm 2/t^2$ while the latter sum tends to zero. Thus $ta,t^2 a, t^3a,\ldots$ is not eventually zero, so $a$ cannot be rational. Now consider an arbitrary number in $(0,1]$ and its base $t$ expansion $\sum_{r=1}^{\infty} a_r / t^r$, $a_r \in {0,1,\ldots,b-1}$. If a number has two different expansions (ending on infinite $0$’s or $(b-1)$’s) we will chose the infinite $(b-1)$’s as the proper expansion, so all numbers have infinite sums. So now we ask, when will $\sum_{r=1}^{\infty} d(r) a_r / t^r$ be irrational? The answer here is always. We simply apply fact 2 again with arbitrary large $n$ to see that $\left\{ t^{x+y_0 B-1} \sum_{r=1}^{\infty} d(r) a_r / t^r \right\} = t^{x+y_0 B-1} \sum_{r> x+k+y_0 B} d(r)a_r/ t^r$ and this last value is strictly between $0$ and $(b-1)/t^{x+k/2+y_0 B}$. Thus the $t$-power multiples of the sum $\sum_{r=1}^{\infty} d(r) a_r / t^r$ have fractional parts that can be arbitrarily close to $0$ without being $0$, thus $\sum_{r=1}^{\infty} d(r) a_r / t^r$ must be irrational. If we instead ask when $\sum_{r=1}^{\infty} d(r)(-1)^r a_r / t^r$ is irrational, the answer is not quite as satisfying. In order to make certain that the tail of the series does not vanish completely, in our proof above, we wanted to have some large prime $p$ and consider $d(p)(-1)^p / t^p$. If the corresponding digit $a_p$ is $0$ then this method of attack fails. So, we know $\sum_{r=1}^{\infty} d(r)(-1)^r a_r / t^r$ will be irrational if for all sufficiently large primes, $a_p \neq 0$. We will now show that $d(r)$ has a similar irrationality-creating property in certain generalized Luroth series expansions. Our definitions of the GLS expansion come predominantly from the work of Dajani and Kraaikamp. Consider a partition $\mathcal{I}=\left\{ (l_n,r_n]: n \in \mathbb{D}\right\}$ of $(0,1]$ where $\mathbb{D} \subset \mathbb{N}$ and $\sum_{n \in \mathbb{D}} (r_n -l_n) = 1$. For each $n \in \mathbb{D}$, let $I_n = (l_n,r_n]$. Every real number in $(0,1]$ corresponds to a unique expansion of the form $$\sum_{r=1}^{\infty} h_r \prod_{i=1}^{r-1} s_i$$ where $h_r,s_r$ are the left endpoint and interval length of some $I_n$ respectively. This is known as a GLS($\mathcal{I}$) expansion. We call a partition $\mathcal{I}$, as defined above, regular if each $l_n,r_n$ is rational, and when put in lowest terms the denominator of $l_n$ divides the denominator of $r_n-l_n$. Further, call $\mathcal{I}$ $t$-regular if in addition to being regular, the denominator of $r_n-l_n$ (in lowest terms) is divisible by $t$ for all $n$. If $\mathcal{I}$ is a $t$-regular partition where the numerator of each $r_n-l_n$ is $1$, then for all GLS($\mathcal{I}$) expansions $$\sum_{r=1}^{\infty} h_r \prod_{i=1}^{r-1} s_i$$ we have that $$\sum_{r=1}^{\infty} d(r) h_r \prod_{i=1}^{r-1} s_i$$ is irrational. If $\mathcal{I}$ is a $t$-regular partition where the numerator of each $r_n-l_n$ is relatively prime to $t$ and for each prime $p$ dividing a numerator, we have $$\sum_{p^k || (r_n-l_n)} k*(r_n-l_n) <0$$ then for lebesgue-almost all GLS($\mathcal{I}$) expansions $$\sum_{r=1}^{\infty} h_r \prod_{i=1}^{r-1} s_i$$ we have that $$\sum_{r=1}^{\infty} d(r) h_r \prod_{i=1}^{r-1} s_i$$ is irrational. Provided no $l_n=0$ the above results also hold if $d(r)$ is replaced with $d(r)(-1)^r$. In the first case, note that since $s_r$ always takes the form of $1$ divided by some positive multiple of $t$ that we can write $h_r \prod_{i=1}^{r-1} s_i$ as $a_r/b_r$ with integers $a_r,b_r$, where $a_r$ is the numerator of $h_r$, $b_r|b_{r+1}$ and $t^{r-1}|b_r$. For the second case, it suffices to consider those numbers normal in the GLS($\mathcal{I}$) expansion, that is, those expansions $\sum_{r=1}^{\infty} h_r \prod_{i=1}^{r-1} s_i$ where $$\lim_{r \to \infty} \frac{1}{r} \#\{ h_r = l_n\}$$ is well-defined and equals $r_n-l_n$. Our condition $\sum_{p^k || (r_n-l_n)} k*(r_n-l_n) <0$ implies that for all sufficiently large $r$ we can write $h_r \prod_{i=1}^{r-1} s_i$ as $a_r/b_r$ with integers $a_r,b_r$, where $a_r$ is the numerator of $h_r$, $b_r|b_{r+1}$ and $t^{r-1}|b_r$. Acknowledgements ================ The author acknowledges support from National Science Foundation grant DMS 08-38434 “EMSW21-MCTP: Research Experience for Graduate Students.” The author would also like to thank Paul Pollack and Paul Spiegelhalter for their assistance. [10]{} W. Alford, A. Granville, and C. Pomerance. There are infinitely many Carmichael numbers *Annals of Mathematics*, 140 (1994), 703–722 J.-P. Bézivin. Indépendance linéaire des valeurs des solutions transcendantes de certaines équations fonctionnelles *Manuscripta Math*, 61 (1988), no. 1, 103–129. P. Borwein. On the irrationality of $ \sum (1/(q^n+r))$ *J. Number Theory*, 37 (1991), no. 3, 253–259. S. Chowla. On series of the Lambert type which assume irrational values for rational values of the argument. , 13 (1947). P. Erdős. On arithmetical properties of Lambert series *J. Indian Math. Soc.*, 12 (1948), 63–66. P. Erdős. On the irrationality of certain series: problems and results *New Advances in Transcendence Theory*, 102–109, Cambridge Univ. Press, Cambridge, 1988. [^1]: Here we switch back to the convention that finite expansions are assumed to end on an infinite string of zeros.

--- abstract: 'Using THz spectroscopy in external magnetic fields we investigate the low-temperature charge dynamics of strained HgTe, a three dimensional topological insulator. From the Faraday rotation angle and ellipticity a complete characterization of the charge carriers is obtained, including the 2D density, the scattering rate and the Fermi velocity. The obtained value of the Fermi velocity provides further evidence for the Dirac character of the carriers in the sample. In resonator experiments, we observe quantum Hall oscillations at THz frequencies. The 2D density estimated from the period of these oscillations agrees well with direct transport experiments on the topological surface state. Our findings open new avenues for the studies of the finite-frequency quantum Hall effect in topological insulators.' author: - 'A. M. Shuvaev' - 'G. V. Astakhov' - 'G. Tkachov' - 'C. Brüne' - 'H. Buhmann' - 'L. W. Molenkamp' - 'A. Pimenov' bibliography: - 'lit\_HgTe.bib' title: Terahertz Quantum Hall Effect in a Topological Insulator --- Three dimensional topological insulators [@hasan_rmp_2010; @qi_prb_2008] have attracted much interest recently, as they exhibit a number of unusual and non-trivial properties, such as protected conducting states on the surfaces of the sample. Unusual electrodynamics, such as a universal Faraday effect and an anomalous Kerr rotation have been predicted [@tse_prl_2010; @tse_prb_2011; @maciejko_prl_2010; @tkachov_prb_2011] for these surface states, their observation is still outstanding. We showed recently that strained HgTe, where the strain lifts the light-hole–heavy-hole degeneracy that normally is present in bulk HgTe, is a very promising 3D topological insulator [@brune_prl_2011]. This is because at low temperatures parasitic effects due to bulk carriers are practically absent. In static transport experiments a strained 70 nm thick HgTe layer [@brune_prl_2011] exhibits a quantum Hall effect (QHE), yielding direct evidence that the charge carriers in these layers are confined to the topological two dimensional (2D) surface states of the material. These findings are further corroborated by recent Faraday rotation data [@hancock_prl_2011] in a similar layer, which have been obtained using a terahertz time-domain technique. In this work, we present the results of low temperature terahertz Faraday cw transmission experiments on another strained HgTe film. The carrier density, Fermi velocity and the scattering rate can be reliably determined from these data. In particular, we obtain the Fermi velocity $v_F = 0.52 \cdot 10^6$ m/s, which is in excellent agreement with the Faraday rotation experiments [@hancock_prl_2011] and the dc Shubnikov-de Haas measurements [@brune_prl_2011] on 70-nm-thick strained HgTe films as well as with band-structure calculations for the surface states in 3D topological insulators (see e.g. Ref. [@liu_prb_2010]). In the same sample we observe quantum Hall-induced oscillations at terahertz frequencies, providing further evidence for the 2D character of the conductivity. In the case of topological insulators, no finite frequency QHE has been reported up to now. The sample studied in this work is a coherently strained 52-nm-thick nominally undoped HgTe layer, grown by molecular beam epitaxy on an insulating CdTe substrate [@becker_pss_2007]. Transmittance experiments at terahertz frequencies (100 GHz $< \nu <$ 800 GHz) have been carried out in a Mach-Zehnder interferometer arrangement [@volkov_infrared_1985; @pimenov_prb_2005] which allows measurement of the amplitude and phase shift of the electromagnetic radiation in a geometry with controlled polarization. Using wire grid polarizers, the complex transmission coefficient can be measured both in parallel and crossed polarizers geometry. Static magnetic fields, up to 8 Tesla, have been applied to the sample using a split-coil superconducting magnet. To interpret the experimental data we use the ac conductivity tensor $\hat{\sigma} (\omega)$ obtained in the classical (Drude) limit from the Kubo conductivity of topological surface states (see e.g. Ref. [@tse_prb_2011]). The diagonal, $\sigma_{xx} (\omega)$, and Hall, $\sigma_{xy} (\omega)$, components of the conductivity tensor as functions of THz frequency $\omega$ can be written as: $$\begin{aligned} && \sigma_{xx} (\omega)=\sigma_{yy} (\omega) = \frac{1-i \omega \tau}{(1-i \omega \tau)^2 +(\Omega_c \tau)^2} \sigma_0 \,, \label{sxx}\\ && \sigma_{xy} (\omega)=-\sigma_{yx} (\omega)= \frac{\Omega_c \tau}{(1-i \omega \tau)^2 +(\Omega_c \tau)^2} \sigma_0 \,. \label{sxy}\end{aligned}$$ Here, $\Omega_c = eBv_F/\hbar k_F$ is the cyclotron frequency, $\sigma_0$ is the dc conductivity, $B$ is the magnetic field, $v_F$, $k_F$, $e$, and $\tau$ are the Fermi velocity, Fermi wave-number, charge, and scattering time of the carriers, respectively. For the Dirac spin-helical surface states the Fermi wave-number depends on the 2D carrier density, $n_{2D}$, through relation $k_F=\sqrt{4\pi n_{2D}}$, with no spin degeneracy. The transmission spectra can then be calculated using a transfer matrix formalism [@berreman_josa_1972; @shuvaev_epjb_2011; @shuvaev_prl_2011] which takes multiple reflection within the substrate into account. The electrodynamic properties of the CdTe substrate have been obtained in a separate experiment on a bare substrate. Further details of the fitting procedure can be found in the Supplementary information to Ref. [@shuvaev_prl_2011]. Neglecting any substrate effects, the complex transmission coefficients in parallel ($t_p$) and crossed ($t_c$) polarizers geometry can be written as: $$\begin{aligned} && t_p =\frac{4+2\Sigma_{xx}} {4+4\Sigma_{xx}+\Sigma_{xx}^2+\Sigma_{xy}^2} \,, \label{tp}\\ && t_c =\frac{2\Sigma_{xy}} {4+4\Sigma_{xx}+\Sigma_{xx}^2+\Sigma_{xy}^2} \,. \label{tc}\end{aligned}$$ Here $\Sigma_{xx}$ and $\Sigma_{xy}$ are effective dimensionless 2D conductivities, defined as: $\Sigma_{xx}=\sigma_{xx}dZ_0$ and $\Sigma_{xy}=\sigma_{xy}dZ_0$ with the HgTe film thickness $d=52$nm and the vacuum impedance $Z_0 \approx 377\,\Omega$. In order to self-consistently obtain the parameters of the quasiparticles, the field-dependent complex transmission $t_p(B)$ and $t_c(B)$ for $\nu =$0.17 THz, 0.35 THz and 0.75 THz and the zero-field transmittance spectra $|t_c(\omega)|^2$ have been fitted simultaneously. ![*Magnetic field dependence of the transmission in strained HgTe.* (a-c) Transmission amplitude in parallel polarizers ($t_p$) geometry, showing cyclotron resonance at the positions indicated by the arrows. The frequency of the experiments is indicated in the panels. The inset shows the frequency dependent transmittance in zero external magnetic field, $|t_p(B=0)|^2$. Symbols: experiment, solid lines: simultaneous fit of all data with the Drude model as described in the text.[]{data-label="ftran"}](ftran.eps){width="0.6\linewidth"} The inset in Fig. \[ftran\] shows the transmittance spectrum of the HgTe film at zero magnetic field. The characteristic oscillations in the spectrum, with a period of about 58 GHz, are due to Fabry-Pérot type interferences within the CdTe substrate. The absolute transmittance in the interference maxima is close to 95%, which reflects the low effective conductance of our HgTe film, $\Sigma_{xx} \ll 1$. At low frequencies, the maximum transmittance decreases and approaches $|t_p|^2 \simeq 0.7$ in the zero frequency limit. Such a behavior is typical for Drude carriers with a scattering rate in the frequency region of the experiment. Indeed, the solid line in the transmission spectra represents a Drude fit with the parameters given in the first row of Tab. \[tab\]. From the fits we obtain the Fermi velocity $v_F = 0.52 \cdot 10^6$ m/s. This value is very close both to $v_F = (0.51 \div 0.58) \cdot 10^6$ m/s as determined in the Faraday rotation experiments on a 70-nm-thick strained HgTe film [@hancock_prl_2011] and to $v_F = 0.42 \cdot 10^6$ m/s as extracted from dc Shubnikov-de Haas measurements on a patterned 70-nm-thick strained HgTe layer [@brune_prl_2011]. The obtained value of the Fermi velocity is also in very good agreement with the band-structure-theory result $v_F = 0.51 \cdot 10^6$ m/s for the linear (Dirac) part of the surface-state spectrum in topological insulators (see e.g. Ref. [@liu_prb_2010]). As an additional check of the 2D surface carrier dynamics in our sample, we have analyzed the terahertz transmission data of Ref. [@shuvaev_prl_2011] for a 70-nm-thick strained HgTe film at high temperature $T=200$ K and for a bulk (1000-nm-thick) unstrained HgTe sample. In both cases, the electrodynamics is governed by massive bulk carriers, for which the values of $v_F$ turn out to be much larger than the Dirac surface-state velocity, i.e., $v_F \approx 0.5 \cdot 10^6$ m/s (Tab. \[tab\]). ----------------------------------------------------------------------------------------------------------------------------------------------------- Sample $T$(K) $n_{2D}$(cm$^{-2}$) $v_F$(ms$^{-1}$) $1/2\pi $G_{2D}=\sigma_0 \cdot d\ (\Omega^{-1})$ \tau$ (GHz) ------------------------------------------ -------- --------------------- ------------------ ------------- ------------------------------------------ 52 nm (strained) \[this work\] 2 $1.08\cdot10^{11}$ $0.52\cdot 10^6$ 250 $7.6\cdot 10^{-4}$ 70 nm (strained) [@shuvaev_prl_2011] 4 $4.8\cdot10^{10}$ $0.38\cdot 10^6$ 210 $4.3\cdot 10^{-4}$ 200 $1.5\cdot10^{12}$ $1.63\cdot 10^6$ 360 $5.3\cdot 10^{-3}$ 1000 nm (unstrained) [@shuvaev_prl_2011] 3 $4.2\cdot10^{11}$ $0.99\cdot 10^6$ 240 $2.8\cdot 10^{-3}$ 200 $4.9\cdot10^{13}$ $9.36\cdot 10^6$ 360 $1.9\cdot 10^{-1}$ ----------------------------------------------------------------------------------------------------------------------------------------------------- ![*Complex Faraday angle $\theta + i \eta$ in HgTe.* Bottom panels: Faraday rotation, top panels: ellipticity for the same frequencies as in Fig. \[ftran\]. The inset sketches the definitions of the Faraday rotation $\theta$ and ellipticity $\eta$. Symbols: experiment, solid lines: simultaneous fit of all data with the Drude model as described in the text. Angular units are radians.[]{data-label="fang"}](fang.eps){width="0.9\linewidth"} Figure \[ftran\] shows the magnetic field dependent transmittance of the HgTe film in Faraday geometry and for parallel orientation of polarizer and analyzer. According to Eq. (\[tp\]), the transmittance in parallel polarizers ($t_p$) depends mainly on $\Sigma_{xx}$. For all three frequencies two clear minima in the transmitted signal are observed in the range below $\pm 1$T. The minima in $|t_p|$ roughly correspond to the cyclotron resonance energy and scale with magnetic field. This may be understood taking into account that in our case $\Sigma \ll 1$ and Eqs. (\[tp\],\[tc\]) simplify to: $$\label{trsimple} t_p \simeq 1- \Sigma_{xx}/2; \quad t_c \simeq \Sigma_{xy}/2 \ .$$ In the limit $\omega\tau \gg 1$, Eq. (\[sxx\]) may be approximated by $$\label{sxx1} \sigma_{xx} \simeq \frac{1-i \omega \tau}{(\Omega_c ^2-\omega ^2)\tau^2} \sigma_0 \ ,$$ which leads to a resonance like feature for $\Omega_c=\omega$. Thus, the positions and widths of the minima in Fig. \[ftran\] are directly connected with the parameter $v_F/k_F$ and the scattering rate $\tau^{-1}$ of the charge carriers. Figure \[fang\] shows the complex Faraday angle $\theta + i \eta$ as obtained at the same frequencies as in Fig. \[ftran\]. The polarization rotation $\theta$ and the ellipticity $\eta$ are obtained from the transmission data using: $$\begin{aligned} && \tan(2\theta)=2\Re(\chi)/(1-|\chi|^2)\ , \\ && \sin(2\eta)=2\Im(\chi)/(1+|\chi|^2)\ .\end{aligned}$$ Here $\chi=t_c/t_p$ and the definitions of $\theta + i \eta$ are shown graphically in the inset to Fig. \[fang\]. A direct interpretation of the complex Faraday angle is in general not possible because of the interplay of $\sigma_{xx}$ and $\sigma_{xy}$ in the data. In the low frequency limit, $\omega\tau \ll 1$ Eq. (\[sxy\]) simplifies to the static result $\sigma_{xy} = \Omega_c \tau\sigma_0 /(1 +(\Omega_c \tau)^2) $. The last expression has a maximum at $\Omega_c (B) = \tau^{-1} $, which leads to maxima in $t_c$ and $\theta$ at about the same field value. Therefore, the Faraday angle provides a direct and an independent way of obtaining the scattering rate $1/\tau$. The solid line in Fig. \[fang\] are the fits which have been done simultaneously for all results presented above. In total, the parameters of the charge carriers have been obtained by simultaneously fitting ten data sets. The quite reasonable fit of all results proves that a single type of charge carriers dominates the electrodynamics in the range of frequencies and magnetic fields used in these experiments. ![*Faraday rotation in HgTe within resonator geometry.* (a) - Ellipticity, (b) - Faraday angle. Symbols - experiment, lines - fits according to Eqs. (1-4). Upper inset shows the experimental geometry within a Copper meshes resonator. Lower inset shows a magnified view of the Faraday angle demonstrating QHE oscillations.[]{data-label="fres"}](fres.eps){width="0.7\linewidth"} Very solid evidence for the two dimensional character of the carriers probed in the Faraday rotation experiments would be the observation of quantum Hall plateaus, similar to the observation of the QHE in [@brune_prl_2011]. However, the accuracy of the experiments shown above does not allow to observe the QHE. In order to solve this problem, we have performed further Faraday transmission experiments on the same sample, now using a resonator geometry as shown in the inset of Fig. \[fres\]. ![*Terahertz quantum Hall effect in HgTe.* (a) Two dimensional conductance: $G_{xx}$, (b) derivative of $G_{xy}$ ($dG_{xy}/dB^{-1}$). The data have been obtained within a resonator geometry and are plotted as a function of inverse magnetic field. The experimental data are shown as solid lines for frequencies as indicated. Dashes in the bottom panel marks the minima for negative magnetic fields. (c) Numbered positions of the minima in $G_{xx}$ and in the derivative of $G_{xy}$ for 0.14 THz and 0.19 THz. Straight lines yield interpolation to the origin. []{data-label="fqhe"}](fqhe.eps){width="0.95\linewidth"} In these experiments, the sample is placed in the middle of a Fabry-Pérot resonator defined by metallic meshes. We have utilized Cu meshes with a 200 $\mu$m period. The distance between adjacent maxima of the resonator is $\simeq 51$GHz. In the frequency range between 100 and 200 GHz the quality factor of the loaded resonator is about $Q \sim 10$. This indicates that, effectively, the radiation passes about ten times through the sample before reaching the detector, which effectively increases the sensitivity to fine details by roughly the same value. As shown in Fig. \[fres\], in the resonator experiments the field dependence of the Faraday rotation and the ellipticity appears qualitatively similar to that in Fig. \[fang\]. An exact calculation of the complex transmission coefficients within a resonator is complicated because of the increased number of parameters. Therefore, in this case we utilize the simple equations Eqs. (\[sxx\])-(\[tc\]) which neglect the effect of the substrate and the resonator completely. Nevertheless, as clearly seen in Fig. \[fres\], the fits based on the simplified expressions reproduce the experimental results reasonably well. Fitting of the signals for parallel and crossed polarizers yields within experimental accuracy the same parameters as in the experiments without a resonator. The only parameter which differs from the results without a resonator is the absolute value of the conductivity. This is of course expected, and results from multiple transmission in the resonator and the influence of the substrate. The main advantage of the resonator experiments is a higher sensitivity to the details of the field-dependent transmission. In addition to an overall field dependence similar to that in Figs. \[ftran\] and \[fang\], a tiny modulation of the signal can now be observed. To convert this modulation to a conventional presentation, we have inverted the transmittance curves into the 2D conductivity, using Eqs. (\[tp\] and \[tc\]). Because the absolute transmittance is not well-defined in the resonator experiments, we have scaled the absolute 2D conductance to agree with the data without a resonator. The final results expressed in form of the effective 2D conductance $G_{xx,xy}=\Sigma_{xx,xy}/Z_0$ are shown in Fig. \[fqhe\]. Fig. \[fqhe\]a shows the real part of the two dimensional conductance $G_{xx}$ as a function of inverse magnetic field. Clear oscillations in the conductance can be observed in this presentation. In general, the phenomenology of the QHE at terahertz frequencies is not well understood [@hols_prl_2002; @ikebe_prl_2010]. Existing experiments are generally limited to frequencies below 100 GHz and they are analyzed using scaling exponents [@sondi_rmp_1997; @hols_prl_2002]. In the resonator experiments, the field dependent oscillations can be observed both with parallel and crossed polarizers. Contrary to $G_{xx}$, the off-diagonal conductance $G_{xy}$ shows a substantial field dependence even in high magnetic fields. Therefore, no clear QHE signal can be directly detected in $G_{xy}$. In order to extract the QHE information from these data, we have plotted the derivative of the $G_{xy}$ as a function of an inverse magnetic field ($dG_{xy}/dB^{-1}$) in Fig. \[fqhe\]b. The derivative has the advantage of being insensitive to any residual slowly varying signals, and, importantly, the expected steps in $G_{xy}$ are transformed into the minima of the derivative. Finally, in order to analyze the quantum Hall effect, both the minima in $G_{xx}$ and in the derivative of $G_{xy}$ have been taken into account. In Fig. \[fqhe\]a,b the results at finite frequencies are compared with dc QHE on the same sample. The periodicity of the oscillations in the dc experiments is slightly different because of different carrier concentration at the sample surface, induced by exposure to photoresist and the presence of ohmic contacts. The main results of the QHE experiments are represented in Fig. \[fqhe\]c demonstrating an approximate equidistant positioning of all minima (labeled by number $N$) in inverse magnetic fields $B^{-1}$ with the period of $\Delta B^{-1} = 0.18$ T$^{-1}$. This periodicity reflects the dependence of the number of the occupied Landau levels on $B^{-1}$. In a total, we detect the oscillations up to index number $\pm 10$; also the first oscillations with the Landau level index $\pm 1$ are clearly observed in the data. From the periodicity of these oscillations the effective 2D carrier density can be estimated according to the free electron expression $n_{2D}=e/(h\Delta B^{-1}) \simeq 1.4 \cdot 10^{11}$cm$^{-2}$. This value agrees reasonably well with the density $n_{2D}=1.08 \cdot 10^{11}$cm$^{-2}$ obtained directly from fitting the transmittance and the Faraday rotation on the basis of the Drude model (Tab. \[tab\]). Therefore, we may conclude that charge carriers which are responsible for the terahertz electrodynamics at low temperatures reveal 2D behavior. To further characterize the electron system in our sample we extrapolated the dependence $N(B^{-1})$ to the origin (see straight lines in Fig. \[fqhe\]c), which corresponds to the limit of very strong magnetic fields. At the origin we find a finite value $N \approx 1/2$ instead of $N=0$ as would be the case for the conventional QHE. Previously, similar extrapolated values were reported for graphene (see e.g. Ref. [@novoselov_nature_2005]), zero-gap HgTe quantum wells [@buttner_nphys_2011] and strained 70 nm-thick HgTe films [@brune_prl_2011], i.e. for materilas with 2D Dirac-like charge carriers encoding a nonzero Berry phase. We therefore believe that our teraherz QHE also indicates the 2D Dirac-like behavior. In conclusion, we have analyzed the terahertz Faraday rotation in a strained HgTe film. From these data all relevant parameters of the charge carriers can be obtained. In addition, terahertz quantum Hall effect oscillations have been observed within the same experiment, which proved the two-dimensional character of the conductivity. We thank E. M. Hankiewicz for valuable discussion. This work was supported by the by the German Research Foundation DFG (SPP 1285, FOR 1162) the joint DFG-JST Forschergruppe on ’Topological Electronics’, the ERC-AG project ’3-TOP’, and the Austrian Science Funds (I815-N16).

--- abstract: | The bulk viscosity, $\zeta$ and its ratio with the shear viscosity, $\zeta/\eta$ have been studied in an anisotropically expanding pure glue plasma in the presence of turbulent color fields. It has been shown that the anisotropy in the momentum distribution function of gluons, which has been determined from a linearized transport equation eventually leads to the bulk viscosity. For the isotropic (equilibrium) state, a recently proposed quasi-particle model of pure $SU(3)$ lattice QCD equation of state has been employed where the interactions are encoded in the effective fugacity. It has been argued that the interactions present in the equation of state, significantly contribute to the bulk viscosity. Its ratio with the shear viscosity is significant even at $1.5 T_c$. Thus, one needs to take in account the effects of the bulk viscosity while studying the hydrodynamic expansion of QGP in RHIC and LHC. [**Keywords**]{}: Bulk viscosity; Shear viscosity; Quark-gluon plasma; Quasi-particle; Chromo-Weibel instability. author: - Vinod Chandra title: On the bulk viscosity of anisotropically expanding hot QCD plasma --- TIFR-TH/11-29 . Introduction ============= It is by now well established that Quark-gluon plasma (QGP) has been created in RHIC experiments, and is a strongly coupled fluid [@expt].There have been first few reports of QGP in Pb-Pb collisions $@ 2.76$ Tev in LHC[@lhc], which reconfirm the formation of strongly coupled fluid. QGP at RHIC has shown a robust collective phenomenon, [*viz.*]{}, the elliptic flow[@flow_rhic]. In the heavy-ion collisions at LHC, there are other interesting flows, [*viz.*]{}, the dipolar, and the triangular flow which are sensitive to the initial collision geometry [@flow_lhc]. In this concern, we refer the reader to the very recent interesting studies [@bhalerao; @alice], where these new kind of flows at LHC have been investigated. The shear and bulk viscosities ($\eta$ and $\zeta$) characterize dissipative processes in the hydrodynamic evolution of a fluid. The former accounts for the entropy production due to the transformation of the shape of hydrodynamic system at a constant volume. On the other hand, latter accounts for the entropy production at the constant rate of change of the volume of the system (in the context of RHIC the system stands for the fireball). These transport parameters serve as the inputs from the hydrodynamic evolution of the fluid. Their determination has to be done separately from a microscopic theory (either from a transport equation with appropriate force, collision and source terms or from the field theoretic approach using Green-Kubo formula). It has been found that QGP possess a very tiny value of the shear viscosity to entropy density ratio, $\eta/s$ [@shrvis]. On the other hand, bulk viscosity has achieved considerable attention in the context of QGP in RHIC after the interesting reports on its rising value close to the QCD transition temperature [@khz1; @khz2]. In the recent investigations, these transport coefficients are found to be sensitive to the interactions [@chandra_eta1; @chandra_eta2], and nature of the phase transition in QCD [@moore]. The computation of transport coefficients in lattice QCD is a very non-trivial exercise, due to several uncertainties and inadequacy in their determination. Despite, there are a few first results computed from lattice QCD for bulk and shear viscosities [@meyer; @nakamura] which have observed a small value of $\eta/s$, and a large value for $\zeta/s$ at RHIC. While determining the behavior of the spectral function in [@meyer], a contribution coming from a $\delta$-function has not been taken in to account. This issue has been discussed extensively in [@tmr]. The spectral density has been modified by incorporating the contributions from the $\delta$-function by Meyer in [@meyer1]. However, a more refined lattice studies on $\eta$ and $\zeta$ are awaited in the near future with less dependence on the lattice artifacts and uncertainties. Subsequently, the possible impact of the large bulk viscosity of QGP in RHIC have been studied by several authors; Song and Heinz [@heinz] have studied, in detail, the interplay of shear and bulk viscosities in the context of collective flow in heavy ion collisions. Their study revealed that one can not simply ignore the bulk viscosity while modeling QGP in heavy ion collisions. In this context, there are other interesting studies reported in the literature [@den; @raj1; @hirano; @raj; @efaaf; @pion; @fries]. The role of bulk viscosity in freeze out phenomenon has been reported in [@torri; @hirano]. Effects of bulk viscosity in hadronic phase, and in the hadron emission have been reported in [@boz]. There has been a wealth of recent literature on the computations of bulk viscosity in the context of cosmology [@cosmo], strange quark matter [@sm], and neutron stars [@ns]. The noteworthy point is that most of works devoted to study the hydrodynamic evolution of QGP, employ constant value of $\eta/s$ [@shhydro] and $\zeta/s$ [@bulkhydro]. This may not be desirable, in the light of experimental and phenomenological observation for QGP at RHIC. The work presented in this paper is an attempt to achieve, (i) temperature dependence of transport coefficients, in particular, $\zeta$, (ii) to understand the large bulk viscosity of QGP. In this study, we shall take inputs from the computations of bulk viscosity in quasi-particle models [@sakai; @quasi1], and combine the understanding with a transport theory determination of $\zeta$ in the presence of Chromo-Weibel instabilities [@bmuller; @chromw]. In this context the shear viscosity of QGP has already been addressed [@bmuller; @bmuller1; @chandra_eta1; @chandra_eta2], and we find very interesting results. As it is well emphasized by Pratt [@pratt] that there may be a variety of physical phenomena which can lead to viscous effects in QGP. Among them, in this paper, we are particularly interested in the viscous effects which get contributions from the classical chromo-fields. The idea adopted here is based on the mechanism, earlier proposed to explain the small viscosity of a weakly coupled, but expanding hot QCD plasma [@bmuller; @bmuller1]. This mechanism is based on the particle transport theory in turbulent plasmas [@dupree] which are characterized by strongly excited random field modes in the certain regimes of instability, which coherently scatter the charged particles and thus reduce the rate of momentum transport.This eventually leads to the suppression of the transport coefficients in plasmas. This phenomenon in electro-magnetic (EM) plasmas has been studied in [@niu], and generalized by Asakawa, Bass and Müller [@bmuller] to the Non-Abelian plasma (QCD), and further employed for the realistic QGP EOS in [@chandra_eta1; @chandra_eta2]. As it is emphasized in [@bmuller2], the sufficient condition for the spontaneous formation of turbulent, partially coherent fields is the presence of instabilities in the gauge fields due to the presence of charged particles. This condition is met in both EM plasmas with an anisotropic momentum distribution [@weibel] of charged particles and in QGP with an anisotropic distribution of thermal partons [@sma]. Here, we shall argue that the similar mechanism can lead to a large bulk viscosity for the hot QCD plasma for the temperatures relevant at RHIC and heavy ion collisions at LHC. The paper is organized as follows. In Sec. II, we present the general formalism to determine the transport parameters from a transport equation with a Vlasov term. We have neglected the collision and source term, while obtaining bulk viscosity. In Sec. III, we discuss the temperature dependence of bulk viscosity and its comparison with the shear viscosity. Finally, in Sec. IV, we present the conclusions and outlook. Transport parameters within a quasi-particle model ================================================== The determination of transport coefficients requires modeling beyond the equilibrium properties, in terms of the collision terms and other transport parameters, and also the nature of perturbation to the equilibrium distribution. In particular, their determination within linearized transport theory needs knowledge of EOS and the equilibrium momentum distribution functions of particles, which constitute the plasma. We shall first discuss the modeling of the EOS within a quasi-particle model. The EOS chosen here is the pure $SU(3)$ gauge theory EOS [@kar]. We subsequently discuss the setting up of the transport equation and the determination of $\zeta$. The quasi-particle model ------------------------ Lattice QCD is the best, and most powerful technique to extract non-perturbative information on the equation of state for QGP [@lat_eos; @lat_eos1]. Recently, we have proposed a quasi-particle model to describe the lattice data on pure $SU(3)$ gauge theory pressure (LEOS), and studied the bulk and transport properties of QGP [@chandra_eta2], which is utilized in obtaining the temperature dependence of bulk viscosity here. In this description, quasi-gluon distribution function extracted from LEOS possess the following form, $$\label{eq1} f_{eq}= \frac{z_g \exp(-\beta p)}{\bigg(1-z_g\exp(-\beta p)\bigg)}.$$ It has further been argued[@chandra_eta2] that the model is in the spirit of Landau theory of Fermi liquids. The connection with the Landau’s theory is apparent from the single quasi-gluon energy, which gets non-trivial contributions from the quasi-particle excitations. The dispersion relation (single particle energy) came out to be, $$\label{eq2} E_p=p+T^2\partial_T \ln(z_g).$$ The main feature of the description is the mapping of strongly interacting LEOS in to a system of non-interacting/weakly interacting quasi-gluons (free up to the temperature dependent fugacity, $z_g$ which encodes all the interactions, and the dispersion relation in Eq.(\[eq2\])). This enables us to tackle highly non-trivial strong interaction in QCD in a very simplified manner while studying the properties of QGP. Interestingly, Eq.(\[eq2\]), which is obtained from the thermodynamic definition of the energy-density in terms of Grand-canonical QCD partition functions, ensures the thermodynamic consistency in hot QCD, and reproduces the lattice results on the trace anomaly correctly. This is also true for the recently proposed quasi-particle model which describes the $(2+1)$-flavor lattice QCD [@vinod]. This quasi-particle understanding of hot QCD has been quite successful in describing the realistic QGP equations of state, and in investigating the bulk and transport properties of QGP [@vinod_quasi; @vinod_quasi1; @chandra_eta1; @chandra_eta2]. We shall utilize Eqs.(\[eq1\]) and (\[eq2\]) to determine the bulk and shear viscosities within transport theory framework here. Note that there are other quasi-particle approaches to describe lattice QCD EOS based on effective thermal masses for quasi-partons [@kamp; @pesh; @bannur; @rebhan; @thaler], approaches based on Polyakov loop [@polyakov], and quasi-particle models with gluon condensate [@ella; @casto]. Recently, transport coefficients for QGP within the effective mass models in the relaxation time approximation have been reported in [@bluhm; @aks]. As argued in [@vinod_quasi1], our model is distinct from all these approaches, but equally successful in describing the thermodynamics of QGP. Determination of the transport coefficients ------------------------------------------- We now consider the important physical quantities, the bulk viscosity, $\zeta$, its ratio with entropy density, $\zeta/s$. For the entropy density, we again utilize the lattice results quoted in [@chandra_eta2]. These quantities are very crucial to understand the QGP in RHIC. Their determination requires knowledge of the collisional properties of the medium when it is perturbed away from equilibrium. To determine these quantities, we adopt approach of  [@bmuller; @bmuller1; @chandra_eta2]. The shear viscosity had been determined in [@chandra_eta2], which we shall utilize to study the ratio $\zeta/\eta$ in the later part. Here, we consider $\zeta$ and determine it from a transport equation. The determination of bulk viscosity has been done in a multi-fold way. Firstly, we need an appropriate modeling of distribution function for the equilibrium state. Secondly, we need to set up an appropriate transport equation to determine the form of the perturbation to the distribution function. These two steps eventually determine the bulk viscosity. For the former step, we employ the quasi-particle model for LEOS discussed earlier. We shall leave the analysis in the case of full QCD for future investigations. The bulk viscosity has two contributions same as the shear viscosity in [@bmuller], (i) from the Vlasov term which captures the long range component of the interactions, and (ii) the collision term, which models the short range component of the interaction. Here, we shall only concentrate on the former case. The determination of shear and bulk viscosities from an appropriate collision term will be a matter of future investigations. Importantly, the analysis adopted here is based on weak coupling limit in QCD, therefore, the results are shown beyond $1.3 T_c$ assuming the validity of weak coupling results for QGP there. ### Formalism Let us first briefly outline the standard procedure of determining transport coefficients in transport theory [@landau; @bmuller]. The bulk and shear viscosities, $\zeta$ and $\eta$ of QGP in terms of equilibrium parton distribution functions are obtained by comparing the microscopic definition of the stress tensor with the macroscopic definition of the viscous stress tensor. The microscopic definition of the stress tensor is, $$\label{eq8} T_{i k}=\int \frac{d^3p}{(2\pi)^3 E_p} p_i p_k f(\vec{p},\vec{r}).$$ On the other hand, macroscopic expression for the viscous stress tensor is given by, $$\label{eq9} T_{i k}=P\delta_{i k}+\epsilon u_i u_k-2\eta(\nabla u)_{i k} -\zeta \delta_{i k} \nabla\cdot\vec{u},$$ where $(\nabla u)_{i k}$ is the traceless, symmetrized velocity gradient, and $\nabla\cdot\vec{u}$ is the divergence of the fluid velocity field. $E_p$ accounts for the dispersion relation. To determine $\zeta$ an $\eta$, one writes the gluon distribution function as $$\label{eqn9c} f(\vec{p},\vec{r})=\frac{1}{{z_{g}}^{-1}\exp(\beta u\cdot p-f_1(\vec{p},\vec{r}))- 1}.$$ Assuming that $f_1(\vec{p},\vec{r})$ is a small perturbation to the equilibrium distribution, we expand $f(\vec{p},\vec{r})$ and keep the linear order term in $f_1$; this leads to, $$\begin{aligned} f(\vec{p},\vec{r})&=&f_0({\bf p})+\delta f(\vec{p},\vec{r})\nonumber\\ &=&f_0({\bf p})\bigg(1+f_1(\vec {p},\vec{r})(1+ f_0({\bf p}))\bigg),\end{aligned}$$ where $f_0({\bf p})$ is the isotropic distribution function, as we shall see that this will be same as the equilibrium thermal distribution function of the quasi-gluons, in the rest frame of the fluid. As discussed in [@chandra_eta2], $\zeta$ and $\eta$ are determined by taking the following form of the perturbation $f_1$, $$\label{eq11} f_1(\vec{p},\vec{r})=-\frac{1}{E_p T^2} p_i p_j \bigg({\Delta_1}(p)\nabla u)_{i j}+{\Delta_2}(\vec{p})(\nabla.u)\delta_{ij}\bigg),$$ where the dimensionless functions $\Delta_{1}(p), \Delta_{2}(\vec{p})$ measure the deviation from the equilibrium configuration. $\Delta_1(p)$, $\Delta_2(\vec{p})$, lead to $\eta$ and $\zeta$ respectively. Note that $\Delta_1(p)$ is an isotropic function of the momentum in contrast to $\Delta_2(\vec{p})$, which is an anisotropic in momentum $\vec{p}$. Since $\zeta$ and $\eta$ are Lorentz scalars; they may be evaluated conveniently in the local rest frame. In the local rest frame of the fluid $f_0\equiv f_{eq}$. Considering the a boost invariant longitudinal flow, $\nabla\cdot u=\frac{1}{\tau}$ and, $(\nabla u)_{i j} = \frac{1}{3\tau} diag (-1, -1,2)$, in the local rest frame, we find that $f_1(p)$ takes the form, $$\label{eq9a} f_1(\vec{p})=-\frac{{\Delta_1}(p)}{E_p T^2\tau}\bigg(p_z^2-\frac{p^2}{3}\bigg)-\frac{{\Delta_2}(\vec{p})}{E_p T^2\tau} p^2,$$ where $\tau$ is the proper time($\tau=\sqrt{t^2-z^2}$). The shear and bulk viscosities are obtained in terms of entirely unknown function $\Delta_1(p)$ and $\Delta_2(\vec{p})$ as, $$\label{eq10} \eta=\frac{\nu_g}{15 T^2}\int \frac{d^3 p}{8\pi^3} \frac{p^4}{E_p^2} \Delta_1(p)f_{eq}(1+f_{eq}),$$ $$\label{xi} \zeta=\frac{\nu_g}{3 T^2} \int \frac{d^3 p}{8\pi^3} \frac{p^2}{E_p^2} (p^2-3 c^2_s E_p^2)\Delta_2(\vec{p}) f_{eq}(1+f_{eq}).$$ In these expressions, $\nu_g\equiv 2(N_c^2-1)$ is the degrees of freedom. Notice that while obtaining the expression for the bulk viscosity, we have exploited the Landau-Lifshitz condition for the stress energy tensor. The factor $-(3 c^2_s E_p^2)$ in the [*rhs*]{} of Eq.(\[xi\]) is coming only because of that. For details, we refer the reader to [@quasi1]. The determinations of $\Delta_1(p)$, and $\eta$ have already been done in [@chandra_eta1; @chandra_eta2]. We shall utilize these results to fix the temperature dependence of $\zeta$ in the later part of the analysis. Now, we shall focus on the determination of the unknown function $\Delta_2(\vec{p})$ and $\zeta$. ### Determination of $\Delta_2(\vec{p})$ For simplicity, we consider the purely chromo-magnetic plasma for our analysis. The modeling of transport equation for full chromo-electromagnetic plasma is straight forward[@bmuller; @chandra_eta1; @chandra_eta2] and differs by simple factors. Here, we only quote the mathematical form of the drift term and the Vlasov term (For details see [@bmuller; @chandra_eta1]). The drift term in the transport equation for the full chromo-electromagnetic plasma for LEOS is obtained as, $$\begin{aligned} \label{eqd} (v\cdot\partial)f_{eq}(p)&=&f_{eq}(1+f_{eq})\bigg[\frac{p_i p_j}{E_p T} (\nabla u)_{ij}\nonumber\\&-&\frac{m^2_D <E^2> \tau^{el}_m E_p}{3T^2 {\partial {\mathcal E}}/{\partial T}}+(\frac{p^2}{3 E^2_p}-c^2_s)\frac{E_p}{T}(\nabla\cdot\vec{u})\bigg],\nonumber\\\end{aligned}$$ where $c^2_s$ is the speed of sound. The other notations are kept same as in [@chandra_eta2]. Note that $<E^2>$ stands for the chromo-electric field, $\tau_{el}$ relaxation time associated with the instability[@bmuller]. In Eq.(\[eqd\]), first term contributes to the shear viscosity, second term contributes to the thermal conductivity, and the third term contributes to the bulk viscosity. Since, we are considering the purely chromo-magnetic plasma, so the second term will not be present. On the other hand the force term (Vlasov term) which we denote as ${\bf V}_A$, is obtained as[@bmuller; @chandra_eta2] follows, $${\bf V}_A=\frac{g^2 C_2}{2(N_c^2-1) E_p^2}<B^2> \tau_m {\bf L^2},$$ where $C_2=N_c$, $<B^2>$ denotes chromo-magnetic field, $\tau_m$ is the times scale associated with instability in the field, and the operator ${\bf L^{2}}$ is $$\begin{aligned} {\bf L^2}&=&-(\vec{p}\times\partial_{\vec p})^2+(\vec{p}\times\partial_{\vec p})\vert_z^2\nonumber\\ &&\equiv -(L^{p})^2+({L^{p}}_{z})^2.\nonumber\\\end{aligned}$$ Since ${\bf L^2}$ contains angular momentum operator $L^{p}$, therefore it gives non-vanishing contribution while operating on an anisotropic function of $\vec{p}$. It will always lead to the vanishing contribution while operating on an isotropic function of $\vec{p}$. Therefore, ${\bf V}_A f_{eq}\equiv 0$. Now, we write the transport equation containing only those terms which contribute to bulk viscosity $\zeta$ as, $$\begin{aligned} (\frac{p^2}{3 E^2_p}&-&c^2_s)\frac{E_p}{T}(\nabla\cdot\vec{u})f_{eq}(1+f_{eq}) \nonumber\\=&&\frac{g^2 C_2}{3(N_c^2-1) E_p^2}<B^2> \tau_m {\bf L}^2 \ f_1(\vec{p},\vec{r})f_{eq}(1+f_{eq}).\nonumber\\\end{aligned}$$ Substituting for $f_1$ in term of the unknown function $\Delta_2(\vec{p})$ and rearranging above equation, we obtain a differential equation for $\Delta_2(\vec{p})$ as, $${\bf L^2} \Delta_2(\vec{p})=\frac{2(N_c^2-1)T E_p^2}{N_c g^2 <B^2> \tau_m p^2 }(\frac{p^2}{3}-c^2_s\ E_p^2)\nonumber\\$$ Now, using the fact that ${\bf L^2}$ only operates on the anisotropic function of $\vec{p}$, we can write, $$\label{delp} \Delta_2(\vec{p})=\frac{2(N_c^2-1)T E_p^2}{N_c g^2 <B^2> \tau_m p^2 }(\frac{p^2}{3}-c^2_s\ E_p^2)\times g(\vec{p}),$$ where $g(\vec{p})$ can be determined from the following condition, $${\bf L^2}\ g(\vec{p})= 1,$$ which leads to, $$\label{gp} g(\vec{p})=\frac{1}{2}\ln(\frac{p_x^2+p_y^2}{p^2_0})\equiv \ln(\frac{p_T}{p_0}).$$ Since, at high temperature average value of the energy is $3\ T$. Employing equipartition theorem for relativistic massless gas, we obtain $p^2_0=6 \ T^2$. Substituting Eq.(\[gp\]) in Eq.(\[delp\]), we obtain, $$\begin{aligned} \label{1} \Delta_2(\vec{p})=\frac{2(N_c^2-1)T E_p^2}{N_c g^2 <B^2> \tau_m p^2 }(\frac{p^2}{3}-c^2_s\ E_p^2)\ \ln(\frac{p_T}{p_0}) \end{aligned}$$ The determination of bulk viscosity is incomplete unless we know not only the temperature dependence of the speed of sound square, $c_s^2$, and the the collective contributions of quasi-particle to the single particle energy, $T^2\partial_{T}\ln(z_g)$ but also the quantity $g^2 <B^2> \tau_m$. We determine first two quantities using the quasi-particle model. As from Ref.[@chandra_eta2], the trace anomaly in terms of effective quasi-particle number density and effective gluon fugacity reads, $$\frac{(\epsilon-3P)}{T^4}=\frac{{\cal N}_g}{T^3} \lbrace T\partial_T \ln(z_g)\rbrace.$$ The thermodynamic quantities can be obtained using the well known thermodynamic relations. In particular, the energy density and the entropy density was shown to be in almost perfect agreement with the lattice data [@chandra_eta2]. We determine, $c^2_s$ by employing a method reported in [@gupta]. The temperature dependence is shown in Fig. 1. To relate the denominator of Eq. (\[1\]) to the gluon quenching parameter, $\hat{q}$ we go the light cone frame. In this frame, Eq.(\[1\]) can be rewritten as, $$\label{eqz} \Delta_2(\vec{p})=\frac{4(N_c^2-1)T E_p^2}{N_c g^2 <E^2+B^2> \tau_m p^2 }(\frac{p^2}{3}-c^2_s\ E_p^2)\ \ln(\frac{p_T}{p_0}).$$ The gluon quenching parameter, $\hat{q}$ is related with the denominator of [*rhs*]{} of the above equation as [@bmuller1], $$\hat{q}=\frac{2 g^2 N_c}{3(N_c^2-1)} <E^2+B^2> \tau_m .$$ Now,employing Eq.(\[eqz\]) in Eq.(8), we obtain the $\zeta$ as, $$\begin{aligned} \label{eqxi} \zeta &=& \frac{(N_c^2-1)}{3 T \pi^2 \hat{q}} \int_0^\infty \int_{-\infty}^{\infty}\ p_{T} dp_{T} dp_{z} (\frac{p^2}{3}-c^2_s\ E_p^2)^2\times\nonumber\\ && \ln(\frac{p_T}{p_0})\times f_{eq}(1+f_{eq}).\end{aligned}$$ On the other hand, if we employ the results of [@chandra_eta2] for $\Delta_1(p)$ in Eq.(10) for $\eta$, we obtain, $$\label{eqet} \eta= \frac{T^6}{\hat{q}} \frac{64(N_c^2-1)}{ 3\pi^2} PolyLog[6,z_g],$$ where $N_c=3$ and $PolyLog[6,z_g]=\sum_{k=1}^\infty \frac{{z_g}^k}{k^6}$. ![\[bulks\] (Color online) The ratio of bulk viscosity, $\zeta$ to the shear viscosity, $\eta$ as a function of temperature. The leading order (LO) result of $\zeta/\eta$ has been obtained from the data taken from Refs.[@chen_bulk; @chen_shear], and shown as dashed line. For the sake of comparison, we have multiplied the leading order $\zeta/\eta$ by a factor of hundred.](bulktoshear_mod.eps) Now scaling, all the quantities in the integrand in Eq.(\[eqxi\]) by $T$, and rewriting Eq.(\[eqet\]) in the form given below, we obtain, $$\label{eqhat} \zeta=\frac{T^6}{\hat{q}} I_{1}(T/T_c);\ \eta=\frac{T^6}{\hat{q}} I_{2}(T/T_c),$$ where $I_{1}(T/T_c)$, is evaluated by integrating the [*rhs*]{} of Eqs.(\[eqxi\]) numerically, and $I_{2}(T/T_c)\equiv\frac{8^3}{3\pi^3} PolyLog[6,z_g]$. The $T/T_c$ scaling of these quantities is coming from the temperature dependence of the effective gluon fugacity, $z_g$. Here, $T_c$ is taken to be $0.27\ GeV$ [@zantow]. Clearly, the quantity which can be determined unambiguously in our approach is the ratio $\zeta/\eta\equiv {I_{1}(T/T_c)}/{I_{2}(T/T_c)}$. In the recent past, Chen [*et. al*]{} [@chen_bulk; @chen_shear] have computed the leading order shear and bulk viscosities for purely gluonic plasma. This is nothing but the collisional contribution to these transport parameters for a gluonic plasma. It is to be instructive to compare the results on $\zeta/\eta$ obtained in the present work with those reported in [@chen_bulk]. This has been shown in Fig. 1, where both the results on $\zeta/\eta$ are plotted as a function of $T/T_c$. Note that while obtaining the temperature dependence of the ratio $\zeta/\eta$, we have employed the two-loop expression for the running coupling constant at finite temperature quoted in [@chen_bulk]. Quantitatively the ratio $\zeta/\eta$ is much smaller than what we have obtained from the diffusive Vlasov term. If we compare the two curves on the ratio $\zeta/\eta$ shown in Fig. 1, we find that in contrast to our prediction on $\zeta/\eta$, the leading order result suggests the near conformal picture of hot QCD even at lower temperatures. Next, we discuss the interplay of the two contributions to the bulk viscosity, [*viz.*]{}, the anomalous, and the leading order (collisional). As it is emphasized in [@bmuller], these two contributions for $\eta$ are inverse additive. Their inverse additivity has been argued from the additivity of various rates in the hot QCD medium. In the case of weak coupling, the former is predominant. It seems that a similar additivity of the inverse of two contributions to $\zeta$,[*viz.*]{} (denoted as $\zeta_a$ and $\zeta_c$ respectively) may perhaps be valid. This could be understood as follows: since $\zeta_a$ is inversely proportional to the $\hat{q}$ (transport rate), on the other hand collisional $\zeta_c$ will be inversely proportional to the collision rate. Following the argument previously mentioned, one may write, $\zeta_T^{-1}=\zeta_{a}^{-1}+\zeta_c^{-1}$, where $\zeta_T$ denotes the total bulk viscosity. This inverse additivity of $\zeta_a$, and $\zeta_c$ at weak coupling, suggests that the collisional bulk viscosity (leading order) will dominate over the anomalous one, since the former is quantitatively much smaller than the latter. However, one has be very cautious while comparing these two contributions for the temperature ranges relevant for QGP at RHIC. This is because of the strongly coupled fluid like picture of QGP. At this moment, we do not know whether the inverse additivity of $\zeta$ will be followed at the temperatures which are closer to $T_c$ or not. This is a very crucial issue, and will require much deeper investigations, which is beyond the scope of the present work. Henceforth, we denote the anomalous bulk viscosity as $\zeta$ dropping the subscript, [*a*]{}. We now proceed to discuss the temperature dependence of $\zeta/\eta$ and $\zeta/s$. ![\[bulkpert\] (Color online) Comparison of the ratio $\zeta/\eta$ with the perturbative QCD, and strongly coupled theories. The quantities $R_{pert}$, and $R_{str}$ are defined in terms of the ratios $\frac{\zeta}{\eta (c_s^2-\frac{1}{3})^2}$, and $\frac{\zeta}{\eta (c_s^2-\frac{1}{3})}$ respectively. Here, [*pert*]{} stands for perturbative QCD, and [*str*]{} stands for the strongly coupled near conformal theories.](bulk_ps.eps) ![\[bulk\] (Color online) $\zeta/s$ and $\eta/s$ as a function of temperature. The dashed (green) line denotes $\zeta/s$ and solid(red) line denotes the $\eta/s$.](zets_etas.eps) Temperature dependence of $\zeta/\eta$ and $\zeta/s$ ---------------------------------------------------- In our analysis, determination of the temperature dependence of $\zeta$ and $\eta$ is incomplete, without the knowledge of the temperature dependence of $\hat{q}$ in QGP. This issue was addressed by fixing the temperature dependence of $\hat{q}$ by calculating the soft part of the energy density and the relaxation time associated with the instability of chromo-fields [@chandra_eta2]. To do that we take inputs from the phenomenological values of $\hat{q}$, which is known at a particular temperature [@hatq]. Here, we have utilized the same transport equation and quasi-particle model developed for pure $SU(3)$ lattice QCD EOS, as in [@chandra_eta2]. Therefore, we employ the temperature dependence of $\eta/s$ to obtain the temperature dependence of $\zeta/s$. This is quite easier to do, since the ratio, $\zeta/\eta$ can easily be obtained from Eq. (\[eqhat\]). The temperature dependence of $\zeta/\eta$ is shown in Fig. 1, $\zeta/\eta$ relative to perturbative QCD prediction [@arnold], and strongly coupled near conformal gauge theories [@confo] is shown in Fig. 2. On the other hand, $\zeta/s$ and $\eta/s$ are shown together in Fig. 3. Let us discuss their behavior one by one. From Fig. 1, it is clear that $\zeta/\eta$ is equally significant while studying the hydrodynamic evolution of hot QCD matter until we reach $T= 2 T_c$. As we go to the higher temperatures the ratio further decreases and eventually vanishes when $c_s^2=\frac{1}{3}$ and the dispersion relation $E_p=p$. Quantitatively, $\zeta/\eta \sim 2.3$ at $1.3 T_c$; $1.0$ at $1.5 T_c$; $0.2$ at $2.0 T_c$. Therefore, for $T\geq 2.5 T_c$, one can ignore $\zeta$ over $\eta$. In other words, the hot QCD becomes almost conformal there. The ratio, $\zeta/\eta$ decreases as we increase the temperature. The decrease is quite steeper until we reach $T=2.0 T_c$. For higher values of $T$ it is much slower. It is hard to make clear cut statement in regard to the behavior of $\zeta/\eta$ with temperature, since by looking at Eqs. (23) and (24), it is clear that the behavior of $\zeta/\eta$ as a function of temperature is mainly governed by the temperature dependence of trace anomaly (through quasi-gluon dispersion relation), speed of sound, $c_s^2$ and temperature dependence of $z_g$ and gluon quenching parameter, $\hat{q}$. To compare the perturbative QCD prediction of the ratio $\zeta/\eta$, we consider $R_{pert}\equiv \frac{\zeta}{\eta (c_s^2-\frac{1}{3})^2}$, where $(c_s^2-\frac{1}{3})$ can be thought of as the measure of conformal symmetry, which we call conformal measure. For scalar field theories, $\zeta/\eta= 15 (c_s^2-\frac{1}{3})^2$ [@scalar], and this has been found to be true for a photon gas coupled with hot matter by Weinberg [@weinberg]. The pre-factor $15$ is not fixed for perturbative QCD but the scaling $\zeta/\eta\sim (c_s^2-\frac{1}{3})^2$is valid[@arnold]). Note that in certain strongly coupled near conformal theories with gravity dual the ratio $\zeta/\eta$ shows linear dependence on the conformal measure [@confo]. To compare with the latter, we consider the ratio $R_{str}\equiv\frac{\zeta}{\eta (c_s^2-\frac{1}{3})}$. We have shown the behavior of $R_{pert}$ and $R_{str}$ as a function of temperature in Fig. 2. Clearly, none of these two scaling are respected by the ratio $\zeta/\eta$ in Fig. 2 even at $2.5\ T_c$. It is safer to say that $\zeta/\eta$ for LEOS which is obtained from transport equation with Vlasov-Dupree term [@bmuller; @chandra_eta2] neither shows linear nor the quadratic dependence with the conformal measure, $(c_s^2-\frac{1}{3})$. However, one can realize the quadratic scaling of $\zeta/\eta$ with the conformal measure in a certain limiting case. It is easy to say from Eqs.(24) and (25) that for $E_p=p$ ($p<<T^2\partial_T (ln(z_g))$), if the thermal distribution of quasi-gluons shows near ideal behavior, and with constant value of $\hat{q}/T^3$, the quadratic scaling can be achieved. Moreover, this may perhaps be realized at higher temperatures which are not relevant for QGP in RHIC and LHC. If we compare qualitatively our prediction of $R_{pert}$, and $R_{str}$ with the leading order result of [@chen_bulk] (see Fig. 4 of this Ref.) , we find opposite trend of these quantities at very high temperature. The former decreases, although slowly, in contrast to the latter, as a function of temperature. This could perhaps be because of their origin from the distinct physical processes in hot QCD medium. The slow decreases of the former at higher temperatures, could be understood as the effect of thermal distribution function of quasi-gluons (through $z_g$, since $z_g$ increases very slowly as a function of temperature, and will asymptotically approach to unity). Finally, in Fig. 3, we have shown the temperature dependence of $\zeta/s$ and $\eta/s$. The $\zeta/s$ decreases as with increasing temperature for $T\geq 1.5$, in contrast to $\eta/s$. As mentioned earlier, $\zeta/s$ and $\eta/s$ becomes equal around $1.5 T_c$ (below which $\zeta/s$ is higher, and lower for higher temperatures.). Again the behavior is predominantly controlled by the behavior of $c_s^2$, and the trace anomaly through the modified dispersion relation with temperature. Conclusions and future prospects ================================ In conclusion, we have estimated the temperature dependence of bulk viscosity to entropy density ratio ($\zeta/s$), and bulk viscosity relative to shear viscosity, $\zeta/\eta$ within a quasi-particle model for pure glue QCD at high temperature by employing transport theory. We have determined $\zeta/\eta$, exactly and unambiguously. In our analysis, these quantities get contributions from the instabilities in the chromo-electromagnetic fields due to the anisotropic thermal distribution of the partons in QGP. The mechanism has succeeded in explaining the small $\eta/s$ and large value of the ratio $\zeta/\eta$. In fact, $\zeta/\eta$ is around $2.3$ at $1.3\ T_c$, of the order of unity at $1.5\ T_c$, and 0.2 at $2\ T_c$. This tells us that the breaking of conformal symmetry in hot QCD plays crucial role even at $2\ T_c$. In consequence, shear and bulk viscosities are equally important while studying the hydrodynamic evolution of QGP at RHIC and LHC. One cannot simply ignore bulk viscosity even at $2.0 T_c$ while modeling the heavy ion collisions. Moreover, $\eta/s$ increases as a function of temperature, in contrast to $\zeta/s$ beyond $1.5 T_c$. As expected $\zeta/s$ and $\zeta/\eta$ are vanishingly small beyond $2.5 T_c$. This may be due to the fact that conformal measure is very small there, and the speed of sound is closer to $1/3$. We have compared our predictions on $\zeta/\eta$ to the leading order result on the same quantity obtained by [@chen_bulk]. Interestingly, in the perturbative region (temperatures beyond $1.5 T_c$), our study also agree with the near conformal picture of hot QCD similar to leading order results of Chen [*et. al*]{} [@chen_bulk]. On the other hand the predictions are in contrast at lower temperatures. However, this may not be thought of as the complete story, an adequate analysis on the interplay of our predictions on $\zeta$, and leading order prediction is very much desired, and will be a matter of future investigations. We have addressed the temperature dependence of the bulk and shear viscosities of pure glue sector of hot QCD only. An extension to full QCD including collision term, employing the understanding of [@vinod], will be a matter of future investigations. We strongly believe that a similar analysis will also be valid in the case of full QCD. The most interesting study would be to include the temperature dependence of $\eta/s$ and $\zeta/s$ in the existing hydro codes to model QGP, and see how various observables get modifications. Moreover, future directions may include exploration on the effects of $\eta$ and $\zeta$ on the quarkonia suppression in heavy ion collisions along the lines of [@chandra_iitr; @dumitru]. 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--- abstract: 'A hybrid map representation, which consists of a modified generalized Voronoi Diagram (GVD)-based topological map and a grid-based metric map, is proposed to facilitate a new frontier-driven exploration strategy. Exploration frontiers are the regions on the boundary between open space and unexplored space. A mobile robot is able to construct its map by adding new space and moving to unvisited frontiers until the entire environment has been explored. The existing exploration methods suffer from low exploration efficiency in complex environments due to the lack of a systematical way to determine and assign optimal exploration command. Leveraging on the abstracted information from the GVD map (global) and the detected frontier in the local sliding window, a global-local exploration strategy is proposed to handle the exploration task in a hierarchical manner. The new exploration algorithm is able to create a modified tree structure to represent the environment while consolidating global frontier information during the self-exploration. The proposed method is verified in simulated environments, and then tested in real-world office environments as well.' author: - 'Wenchao Gao, Matthew Booker, Jiadong Wang [^1]' title: 'Self-Exploration in Complex Unknown Environments using Hybrid Map Representation' --- =1 Introduction ============ Traditional robotic navigation requires a known or predefined map before navigation goals can be determined and executed by the motion planner [@Bruemmer2009]. As the robotics industry grows rapidly, the ability to investigate and operate independently in an unknown environment becomes essential for an advanced robot to be considered fully autonomous. According to [@frontier1997], self-exploration and mapping can be defined as the action of autonomously moving through an unknown environment while building a map that can be used for subsequent navigation. In literature, solutions for self-exploration in unknown environments have been reported and divided into two categories: randomized-based searches [@random_walk; @greedy_mapping; @Oriolo2004; @Umari2017] and frontier-driven strategies [@frontier1997; @Banos2002; @Keidar2012; @Senarathne2013]. In the first category, straightforward approaches employ randomized selection mechanisms [@random_walk] or greedy based searches [@greedy_mapping] to explore the environment. Although simple and fast, such strategies yield locally optimal solutions but do not guarantee global optimization in many cases. To address the issue, Sensor-based Random Tree (SRT) method [@Oriolo2004], which can be considered as a goal-oriented exploration strategy, bias the randomized generation of configurations towards unexplored areas. However, these approaches suffer from the problem of revisiting explored places. Recently, a new exploration strategy leveraging on Rapidly-exploring Random Trees (RRT) utilizes the randomized tree expansion to detect and prioritize unknown spaces [@Umari2017]. RRT techniques ensure complete search coverage and can be extended to higher dimensions, but result in a lower exploration efficiency when searching in complex spaces, such as office areas with narrow corridors [@tencon2017]. More efficient approaches make use of the concept of map frontier. The key idea of this branch is to determine the next desired goal based on frontiers, i.e. boundaries between the known and unknown cells in an occupancy grid map. In the pioneer work of [@frontier1997], frontier edges are required to be segmented from a dynamical occupancy grid map in order to determine potential targets. The selected target will be assigned as a temporary destination point. To improve the frontier detection efficiency, in [@Banos2002] and [@Keidar2012], a series of target points in the grid map reveal the quality of the candidate points around a frontier which will be evaluated according to some criteria. Senarathne et al. develop an efficient approach to segment frontiers by only detecting intermediate changes to cells in the current exploration map and only the updated grid cells are considered for the frontier segmentation [@Senarathne2013]. To produce accurate maps, metric or grid-based SLAM techniques are frequently incorporated with frontier-driven exploration [@metric1; @metric2]. However, these approaches usually have to process the entire map to detect the desired frontier. If the map is dynamically updated and becomes larger, more computational resources and memories are required [@Keidar2012], which often prohibits the exploration efficiency in large-scale environments. Another issue with the current frontier-driven methods is that they usually have poor capability to efficiently select and assign frontier in a systematic way when the search area is large and cluttered, resulting in back and forth exploration over visited places in a complex searching space. Some other approaches that leverage the use of a topological map [@topological1; @topological2] have been proposed to represent the unknown environment in a qualitative manner. A local and global decision-making mechanism for self-exploration is proposed in [@topological1], where a bubble searching mechanism based on local geometric features is used to determine robot orientation and a topological map is built to move the robot between different topological nodes globally. The problem of high computational cost in large-scale environments can be alleviated, and yet a place recognition algorithm is a prerequisite for this method, making it vulnerable in places that are difficult to be recognized. The topology of the environment is encoded in a Generalized Voronoi Diagram (GVD) in [@topological2]. The GVD containing the key geometric information can be interpreted as an efficient topological representation of an indoor/outdoor environment. However, in return, the topological methods lose the metric property and may encounter the problem of ambiguous spatial reasoning between topological classes. To further improve the exploration efficiency, recent works [@hybrid1; @hybrid2] propose a hybrid map representation using metric and topological information. In [@hybrid1], an ear-based exploration strategy makes use of GVD-based topological graph and extended Kalman filter (EKF) to track the pose of the robot. The ear-based strategy is reported to facilitate loop closure in SLAM process, assuming that several small obstacles exist in the search space. An incrementally constructed GVD for frontier-based exploration is introduced to completely solve the pose-SLAM problem in [@hybrid2]. However, the proposed diagram carries redundancies resulting in chaotic exploration decisions which degrade exploration efficiency. Motivation and Overall Strategy {#motivation} =============================== The goal of work is to develop an efficient self-exploration navigator which maximizes the mapping coverage as quickly as possible in an unknown environment. To gather the local metric information efficiently, a modified frontier-based method is proposed to realize a stem-first exploration. The frontier method is employed due to its efficiency in local unexplored space searching and ease of integration with the grid-based SLAM techniques such as [@Slam2005]. Considering the limitation of the current frontier methods, the concept of topology is introduced to consolidate all frontier information from a global vision and systematically determine optimal unexplored places for the mapping agent. Different from the existing methods, the working space is categorized into two parts: “Stem" and “Branches". The region of “Stem" can be considered as the main road in a metric map or the backbone in a topological way, while “Branches" are rest scattered areas. By taking the robot heading information and map topology into account, the proposed strategy prefers to navigate along the “stem" to explore the main structure of the space first. And then prioritize the unexplored spaces (Branches) based on a global decision making. The global call will be activated to choose an optimal area to explore when the “stem" has been fully explored or the robot change its orientation rapidly in a cross-road or dead-end. Borrowing the idea of hybrid map representation [@hybrid1; @hybrid2], an innovative hierarchical exploration algorithm is proposed in this paper. The hierarchical strategy has been designed in a global-local-cooperative fashion. More specifically, in a lower level control, the local desired frontiers pushing the robot to stay on the main road are determined and assigned to the navigator within a sliding local window. Globally, a GVD-based topological planner taking the role of an upper level decision maker is developed to abstract the metric information of all global frontiers through a modified tree structure named as multi-root tree. It is noteworthy that the hierarchical strategy is proposed to achieve a systematic way of exploring complex unknown environments by combining the benefits of both metric and topological map information. Preliminary Terminology {#terminology} ======================= In this section, we provide the definition of functions and symbols related to the proposed approach. **Occupancy Grid**: The representation of a map that divides the space into grid cells. **Search Space** $\mathbb{R}^2$: The set of the whole search space. This set in $2D$ consists of free $\mathbb{R}_f$, occupied $\mathbb{R}_o$, and unknown space $\mathbb{R}_u$, i.e, $\mathbb{R}=\mathbb{R}_f\cup\mathbb{R}_o\cup\mathbb{R}_u$ **Frontiers** $\mathcal {F}$: A list $\mathcal {F}=\{f_0,...,f_j \}$ that stores all nearby frontier nodes. The desired frontier $f^* \in \mathcal {F}$ will be assigned as the exploration goal. **Utility Cost** $C$: This cost function is defined to determine the most desirable frontier $f^*$ to be explored from list $\mathcal {F}$. **Topological Node** $N$: A set of nodes $N=\{\nu_0,...,\nu_k\}$ denoting the location of a GVD vertex. Nodes along the main path are called “stem nodes", the others located in the branches of the GVD graph are named “branch nodes". **Edge** $E$: An edge linking two topological nodes. Edges are divided into two categories: edges between two stem nodes $\eta\in E$ and edges connecting to the branch nodes $\epsilon\in E$. **Topological Map** $G$: A graph-based map constructed by edges and topological nodes, i.e. $G=(N,E)$. A graphic example showing the hybrid map representation is illustrated in Fig. \[graphic\_example\], containing the detected frontiers and the topological map in an office area. The frontiers in $\mathcal {F}$ are highlighted by blue boundary lines for all unexplored areas. Stem nodes are denoted by red (linked to frontiers) and green (not directly linked to frontiers) dots. Branch nodes are highlighted in black. The map $G$ is connected by red edges ($\eta$) along the main path and green edges ($\epsilon$) at the branches. Methodology =========== Assuming that the environment can be represented in a topological form, the main idea of the proposed strategy is to take a backbone traversal by examining the topological stem and then explore the remaining areas at the branches. A cycle of two stages of decision making is designed to implement the idea, i.e. local frontier detector and GVD topological planner. The two stages cooperate with each other in a hierarchical way. The lower level stage is to obtain new information by moving the robot to the boundaries between open space ($\mathbb{R}_f$) and unknown space ($\mathbb{R}_u$) inside a sliding window. Metric information measured by laser is used to build up environmental structures and detect local frontiers. The upper level stage, leveraging on topology, deals with the global exploration planning when local information is not reliable. Hierarchical Exploration Strategy --------------------------------- Fig. \[framework\] shows the framework of the hierarchical exploration strategy. As can be seen, a pose SLAM method named “Karto" [@karto] is deployed to map multiple unexplored areas. Taking odom data and sensor information as inputs, it produces a metric map and the robots location for the task handler to generate exploration goals. Each goal generated by the task handler will be assigned to the path planner, resulting in a series of velocity commands to drive the robot into new territory. \[framework\] 1. **while** $\emph{HierarchicalPlanner}=1$ **do** 2. run *LocalFrontierDetector* 3. **if** ($\mathcal {F}=\emptyset$) **or** ($\emph{angleChanged}>=150 \degree $)\ **then** $\rhd task \ filter$ 4. run *GVDTopologicalPlanner* 5. **if** ($v^*\neq \emptyset$) **then** 6. $goal\leftarrow v^*$ 7. run *SendGoal(goal)* 8. **continue** 9. **else if** ($v^* =\emptyset$) **and** ($\mathcal {F}=\emptyset$) 10. $\emph{HierarchicalPlanner}=0$ $\rhd end \ task$ 11. **else** 12. $goal\leftarrow f^*$ 13. run *SendGoal(goal)* \[hierarchial\] Our task handler is summarized in **Algorithm \[hierarchial\]**. Two planning stages (‘*LocalFrontierDetector’* and ‘*GVDTopologicalPlanner*’) are designed in a hierarchical way that the robot prioritizes local frontier searching in a sliding window (*line 2*), and makes upper-level decision by requesting the GVD topological map (*line 4*) based on the conditions specified by the task filter (*line 3*). The task filter will trigger ‘*GVDTopologicalPlanner*’ and deactivate ‘*LocalFrontierDetector*’ when either of the two conditions are satisfied: 1) no local frontier is detected; 2) the robot changes its orientation significantly in a short period of time (2 seconds). ‘*LocalFrontierDetector*’ will be activated again once the GVD exploration goal is reached. Only one exploration goal from either stage will be activated for each iteration to prevent sending multiple commands to the robot. The filtered exploration goals are assigned to the path planner to obtain velocity commands to actuate the robot (*line 7 and 12*). The exploration task terminates when both planning stages return *NULL* (*line 9*). It is proven by simulation and experimental results that the hierarchical task handler is able to effectively combine the two planning stages by taking advantage of both metric and topological information. The way to determine the desired frontier $f^*$ and the GVD exploration goal $v^*$ will be discussed in the following two subsections. Oriented Local Frontier-driven Exploration ------------------------------------------ Within the occupancy grid any unknown cells adjacent to free cells are grouped together into regions. The centroid of each region (above a certain minimum size) can be considered as a frontier node $f_j$. The frontier list $\mathcal {F}_t$ contains all the valid frontier nodes at time $t$. The most widely used frontier-driven approaches [@frontier1997; @Umari2017; @exploration2007] determine the desired frontier $f^*$ by taking into consideration the frontier size and distance, which also has been referred as the greedy frontier-driven exploration. More specifically, in these greedy approaches $f^*$ is selected by minimizing the following utility function: $$\label{nsingsta} C(f_j) = (\omega_d \times {f^D_j} - \omega_s \times {f^S_j}) \nonumber$$ $$\label{old_cost} f^* = \emph{Arg}\operatorname*{min}_{f_j \in \mathcal {F}} \Bigg(C(f_j)\bigg)$$ where $f^D_j$ is the Euclidean distance from the robot to the frontier node and $f^S_j$ is the grid size of the frontier area. $\omega_d, \omega_s$ are weighting parameters associated with the two terms. By minimizing the utility function, the robot takes the shortest path from its current location to the boundary containing the most unknown information. To be noted that the optimal frontier $f^*$ is selected among all the detected frontiers in the global map every iteration. It has been reported in [@frontier1997] that by constantly moving to new frontiers, the robot is able to extend its map into new space until the entire environment has been explored. However, Eq. 1 can be inefficient when the unknown environment is complex and dynamic (e.g, office area with narrow corridors, secluded cubicles, and possible moving pedestrians) as it only takes into account Euclidean distance as opposed to actual travel distance. In cases where a frontier is behind a large object, such as a wall, the Euclidean distance is lower than the travel distance. The cluttered environment thus causes the robot to travel back-and-forth repeatedly over explored locations reducing the exploration efficiency. To avoid this, an efficient way of measuring the travel distance from the robot position to the candidate frontiers must be developed. Standard path planning techniques such as A\* searching and RRT can solve the problem yet at a higher computational cost such as [@Umari2017]. The following frontier detection mechanism combining with the topological tree representation has the ability solve the back-and-forth trap efficiently. A new utility cost $\widetilde{C}$ is designed to incorporate the frontier orientation information in ‘*LocalFrontierDetector*’. As a result, the modified frontier detector ensures exploring in a certain direction as far as possible before turning or backtracking. The utility cost $\widetilde{C}(\widehat{f}_j)$ can be written as: $$\label{new_cost} \widetilde{C}(\widehat{f}_j) = (\|\widehat{f}^D_j\| - \|\widehat{f}^S_j\| + \|\widehat{f}^R_j\| )$$ where, $\widehat{f}^D_j, \widehat{f}^S_j$ are defined the same as Eq. (\[old\_cost\]). $\widehat{f}^R_j$ denotes the steering angle to face each frontier node. And the candidate for desired frontier has been narrowed down to those frontiers $\widehat{f}_j$ within a certain distance around the robot (local window) to speed up the searching process (global information will be handled by the global decision maker). The three cost components are normalized into the range of $[0,1]$ to balance the overall utility cost and omit the process of parameter selection. The scaled values $ \|\widehat{f}^D_j\|$, $\|\widehat{f}^S_j\|$, $\|\widehat{f}^R_j\|$ are computed as below: $$\label{normalization} \|\widehat{f}^D_j\|=\frac{\widehat{f}^{D}_j - \widehat{f}^{D}_{min}}{\widehat{f}^{D}_{max} - \widehat{f}^{D}_{min}} \nonumber$$ $$\label{normalization} \|\widehat{f}^S_j\|=\frac{\widehat{f}^S_j - \widehat{f}^S_{min}}{\widehat{f}^S_{max} - \widehat{f}^S_{min}},\ \|\widehat{f}^R_j\|=\frac{\widehat{f}^R_j - \widehat{f}^R_{min}}{\widehat{f}^R_{max} - \widehat{f}^R_{min}}$$ The “min" and “max" sign indicate the minimal and maximum value of each cost component. The local desirable frontier node $\widehat{f}^*$, thus, can be determined when $\widetilde{C}(\widehat{f}_j)$ is minimized as: $$\label{nsingsta} \widehat{f}^* = \emph{Arg}\operatorname*{min}_{\widehat{f}_j \in \mathcal {F}} \Bigg(\widetilde{C}(\widehat{f}_j)\bigg)$$ Considering a 2D navigation scenario, the pose associated with the desired frontier $\widehat{f}^*(x,y,\theta)$ is the output of exploration stage ‘*LocalFrontierDetector*’. GVD-based Topological Planner ----------------------------- The GVD graph representation is used to obtain the topological structure of the grid map [@gvd]. An common approach is to create a GVD-Matrix $M$ that can indicate whether a cell belongs to GVD. To get the topological map $G$, image processing techniques are applied to recognize the intersection points in $M$. These points are stored as nodes in the topological node set $N$. Moreover, points of GVD which cross frontiers will be attributed to $N$ as well. Once all the topological nodes are detected, the edges $E$ are generated by gathering the GVD’s cells between nodes using a point queue. One novelty of this paper is that a modified version of the tree data structure called multi-root tree is introduced to represent the graph. Within set $N$, each stem node can be regarded as a root of a binary tree, the other nodes are treated as branch nodes. Branch nodes connecting to frontiers are defined as leaf nodes. They can be traced back to the corresponding root in the same tree. Therefore, the graph can be considered as a combination of multiple tree structures. By consolidating all the roots, the topological planner is able to access the abstracted frontier information of the whole graph and make exploration decisions at the global level. Based on the concept of a multi-root tree, the planner can be divided into four steps: 1) Determining the stem and branch nodes of the GVD graph; 2) Transforming the GVD graph into a multi-root tree and tracing all frontier information to the roots; 3) Finding the nearest root $v^{key}$ to the current robot pose; 4) Determining the best stem node as the exploration goal $v^*$, based on a specific score function. 1. **$MainPathSearch(v^c)$** 2. **Input**: a node $v^c$. 3. **Output**: $L^c$ the longest path length from $v^c$, $N^c$ the corresponding set of nodes. 4. **Initialization**: boolean type vector $Visited$ that indicates whether a node has been visited. 5. $Visited[v^c]$ $\xleftarrow{}$ 1. 6. $L^c$ $\xleftarrow{}$ 0. 7. $N^c$ $\xleftarrow{}$ empty vector. 8. **if** $Neighbour^c = \emptyset$ **then** 9. **return** $(L^c, N^c)$ 10. **for** $v^n: Neighbour^c$ 11. **if** $Visited[v^c] \neq 1$ 12. $(L^n, N^n)$ $\xleftarrow{}$ $MainPath Search(v^n)$ 13. $TempL^c$ $\xleftarrow{}$ $L^n + L^{nc}$ 14. **if** $TempL^c > L^c$ 15. **then** $(L^c, N^c)$ $\xleftarrow{}$ $(TempL^c, N^n)$ 16. **else if** $TempL^c > threshold^l $ 17. **then** **add** $N^n$ into $subN$ 18. **add** $v^c$ into $N^c$ 19. **return** $(L^c, N^c)$ \[gvd\_dfs\] Seperating GVD nodes into stem and branch nodes is a vital step. A depth-first-search (DFS) algorithm is developed in a recursive way as shown in **Algorithm \[gvd\_dfs\]**. $Neighbour^c$ is the set of nodes that directly connect to the current node. $v^n\in Neighbour^c$ is one of the neighbouring nodes to the current node $v^c$. $L^{nc}$ is the length of the edge connecting $v^c$ and $v^n$. The recursive mechanism (*line 6 to line 13*) enables the algorithm to search and store the longest path of nodes ($N^c$) which is considered the main path ($L^c$) in the graph. The results can be seen in Fig.\[graphic\_example\], where the stem and branch nodes are properly clustered into two groups. In order to abstract the frontier information from leaf nodes, Fig.\[refine\] illustrates how the multi-root tree is constructed. The original topological graph is shown in (a), where $V^s_1$ and $V^s_2$ are stem nodes and also the roots of two tree structures. All the leaf nodes labelled as $V^b$ need to be backtracked to $V^s$. From (b) to (c), the blue nodes are fused into their parents nodes and transmit the frontier information up to the root level layer by layer. When no more blue nodes can be fused, the process is finished as shown in (d). By performing step 2), all information at the branches is transmitted to the root so that the processed stem nodes are able to represent the whole unexplored space. In the proposed multi-root tree structure all stem nodes are at the same root level with no parent node. This structure tackles the ordering problem seen in other tree structures that occurs when the stem nodes create a cycle. \[refine\] \[scorediagram\] Next, in order to make a global decision, the robot pose is taken as a reference to search for the nearest stem node $v^{key}$, highlighted in blue in Fig. \[scorediagram\]. For instance, there are three possible paths, $P_1$, $P_2$ and $P_3$. The starting nodes of these three paths are $v_1^{1}$, $v_2^{1}$ and $v_3^{1}$, respectively. Similarly the most distant stem node of each path is denoted as $v_1^{end}$, $v_2^{end}$ and $v_3^{end}$. All branch nodes $v^b$ are fused into the root nodes through the previous steps. For the sake of efficiency in global planning, all stem nodes along each path are taken into consideration when determining the exploration goal. Thanks to steps 1) and 2), the processed stem nodes containing frontier information can be used to design a score function that evaluates individual node scores. By summing the node scores in one path we can obtain the total score for that path. The $NodeScore$ and $PathScore$ functions are expressed as: $$\label{scorefunction} Node Score(v^j) = \sum_{i=1}^k S_{i} \cdot e^{-d^j}$$ $$\label{scorefunction} Path Score = \frac{\sum_{j=1}^n Node Score(v^j)}{log(l+1)}$$ where, $k$ is the number of frontiers associated with $v^j$. $S_i$ is the size of the $ith$ frontier associated with $v^j$. $d^j$ is the real world travel distance between $v^{key}$ and $v^j$, obtained by GVD. $n$ represents the number of nodes within the current path. While $l$ can be considered as the approximated real world travel distance to the last stem node in the path ($v^{end}$). As described before, all GVD nodes $v\in N$ are equidistant to nearby obstacles [@gvd]. By taking advantage of this property, the travel path between two nodes has a great ability to avoid obstacles and its distance is shortest. Hence, we can approximate travel distances by counting the number of GVD points between two nodes and multiplying it by the resolution of the corresponding grid map (measuring the length of Edge $E$ between topological nodes as shown in Fig. \[graphic\_example\]), instead of applying an extra path planning algorithm. By incorporating $l$ and $d^j$, the global planner avoids the back-and-forth trap by selecting stem nodes based on their approximate travel distances. The optimal path $P^*$ is the one with the highest $PathScore$ value, which takes all the global frontier information into account. Simultaneously, the first node of path $P^*$ is chosen as the exploration goal $v^*$, which is also the output of the global planning stage. To be noted, the topological planner, though takes extra computation, is activated only in certain conditions (**Algorithm \[hierarchial\]**). The testing results show that the overall processing speed will not be significantly affected. Experiments {#experiment} =========== Simulation Results ------------------ To evaluate the proposed algorithm, two sketched map models shown in Fig. \[gazebo\] are generated in Robot-Operating-System (ROS) Gazebo simulator. The size of two maps are listed in Table \[comparison\]. The simulator can generate realistic robot movement and sensor data combined with noise. The proposed exploration algorithm is implemented in ROS environment, and compared against the open-source greedy frontier-driven exploration method [@online] and the RRT detector in [@Umari2017]. Off-the-Shelf ROS packages are used in the implementation of SLAM (Karto) and motion planning (ROS Navigation Stack). The simulation results using the proposed approach are presented in Fig. \[simulation\], where (a)-(d) show the exploration process for map model ‘$Sketched_1$’; and (e)-(h) for map model ‘$Sketched_2$’. The GVD-based topological nodes and edges are denoted by different coloured dots and lines. As mentioned, the backbone extracted by **Algorithm \[gvd\_dfs\]** is shown as the red line. The stem nodes are denoted as red and green dots. Red indicates nodes that are associated with frontier information, through the multi-root tree transformation, while green indicates nodes that have no frontier information. At the early stage, the number of red nodes is quite high, but they quickly turn to green nodes as the robot clears frontiers. The current robot pose is represented by a green marker, which is at the center of the yellow box. The yellow box visualizes the local sliding window, while the blue boundaries indicate the list of $\mathcal {F}$, however only the frontier inside the local window will be considered during the ‘*LocalFrointerDetector*’. \[gazebo\] \[simulation\] To further evaluate the advantage of the proposed approach, a graph revealing exploration coverage against time is plotted in Fig. \[timing\]. The data is calculated by taking the average value of 20 simulation runs. To evaluate the robustness of different methods, the exploration performance under four different initial poses (shown by the red arrows in Fig. \[gazebo\]) has been tested. Obviously, with equivalent time spending, more unknown space can be detected by using the hierarchical strategy than the other two methods, especially in the early stages of exploration. In other words, the exploration coverage grows the fastest by using the proposed approach. The RRT exploration [@Umari2017], consuming high computational resources, shows a poor performance for the first map model due to low sampling efficiency when using the randomized-tree in a complex environment with multiple tight corridors. In Table \[comparison\], the time taken to cover $60\%$, $80\%$ and $90\%$ is shown along with the average processing rate of each approach (default rate is $20 hz$). During the simulation, the last $10\%$ coverage for each approach consisted of small frontiers that the robot had difficulty in navigating to. The greedy frontier approach [@online], always explores the closest and biggest frontier but lacks exploration plan with a global vision. As a result it consumes slightly less computational resources, but results in a much longer exploration time for the large-scale map of $Sketched_2$. \[timing\] Experimental Results -------------------- The real robot platform is built based on a differential-drive mobile platform (Pioneer). A Hokuyo UTM-30-LX laser scanner is mounted about $30cm$ from the ground, which allows the environment around the robot to be scanned. Different exploration algorithms are implemented on the mentioned platform and tested in our office (9th and 13th floor of Fusionopolis, Singapore). As both maps consist of long and narrow corridors with shielded cubicles/rooms on both sides, they would be considered challenging for self-exploration. The estimated size of each floor is given in Table \[comparison\] as well. \[exploration\_9th\] \[exploration\_13th\] \[comparison\] The experimental results using the proposed approach are illustrated in Fig. \[exploration\_9th\] and Fig. \[exploration\_13th\] respectively. The exploration process, including map building and topological graph construction, is gradually demonstrated from the starting position in (a) until most of the space has been explored in (d). Subfigures (a)-(d) are snapshots of the visualization tool ‘rviz’ during self-exploration. The corresponding topological graph $G$ shown in subfigures (e)-(h) updates with respect to the robot position, which is indicated by the green marker. Although graph $G$ is shown in each step, the ‘*GVDTopologicalPlanner*’ may not be activated each time. Frontiers $\mathcal {F}$ within the local window (yellow box) denoted by blue boundaries are used to determine $f^*$. By running the ‘*LocalFrontierDetector*’ the navigational coordinates for the desired frontier is selected and is displayed as a yellow dot. Whenever ‘*GVDTopologicalPlanner*’ is activated, the exploration goal $v^*$ can be determined via the four steps mentioned in Section \[methodology\].C. As a result, the global planner was activated and the exploration goal was marked by a red dot in both figures when the robot reached an dead-end in Fig. \[exploration\_9th\](b) or when no available frontier existed in the local window in Fig. \[exploration\_13th\](c). The performance of real robot exploration is summarized in Table \[comparison\]. In both cases, the time used to cover $60\%$ and $80\%$ of the whole map is significantly less when using the hierarchial approach. For the greedy approach, a similar exploration speed can be achieved to cover $90\%$ of the map in a much simpler environment of $Office_{13th}$, where the space is relatively open and regular. However, the same approach failed to explore the more complex environment $Office_{9th}$ up to $90\%$ as it was travelling back and forth repeatedly over explored location. A example of exploration trajectory performed by the greedy frontier is shown in Fig. \[standard\_perform\](a), where quite a few back and forth movements can be observed inside the highlighted region, while no such behaviour can be observed under the proposed framework in Fig. \[standard\_perform\](b). Note that the extra CPU usage by taking the hierarchial architecture is within an acceptable range. Therefore, in terms of time expenditure and map coverage, the experimental results validate the superior exploration efficiency of the hierarchical exploration strategy compared to the greedy method in complex office environments. In fact, an additional advantage of using the hierarchial approach can be inferred that the stem-preferred strategy could also increase the probability of loop-closure during the SLAM process. Meanwhile, according to Table \[comparison\], the RRT exploration method [@Umari2017] showing a poor search coverage and low processing rate performed less inefficiently than the proposed one as well. \[standard\_perform\] Conclusion and Discussion {#conclusion} ========================= In this paper, by combining the benefits of metric and topological map, an exploration strategy using hybrid map representation is proposed to solve a challenging problem of self-exploration and mapping in a complex environment. Two planning stages are designed to collaborate with each other hierarchically. The lower-level stage prioritizing the local frontier along with the robot motion direction will force the robot to follow the main road (if it is available). The upper-level planner, leveraging on the technique of GVD, is able to make global decisions based on a modified tree data structure called a multi-root tree. By consolidating the frontier information from all leaf nodes, a systematic way of determining the optimal exploration goal can be achieved. The exploration goal determined by either planning stage is assigned to the robot base to work together with the process of SLAM. The proposed approach is evaluated in both simulation and experimental environments by comparing against two other methods. According to the results, the proposed approach achieves the greatest exploration efficiency when exploring in a typical office area. 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--- abstract: 'A computable ring is a ring equipped with mechanical procedure to add and multiply elements. In most natural computable integral domains, there is a computational procedure to determine if a given element is prime/irreducible. However, there do exist computable UFDs (in fact, polynomial rings over computable fields) where the set of prime/irreducible elements is not computable. Outside of the class of UFDs, the notions of irreducible and prime may not coincide. We demonstrate how different these concepts can be by constructing computable integral domains where the set of irreducible elements is computable while the set of prime elements is not, and vice versa. Along the way, we will generalize Kronecker’s method for computing irreducibles and factorizations in $\mathbb{Z}[x]$.' address: - | Department of Mathematics and Statistics\ Grinnell College\ Grinnell, Iowa 50112 U.S.A. - | Department of Mathematics and Statistics\ Grinnell College\ Grinnell, Iowa 50112 U.S.A. - | Department of Mathematics and Statistics\ Grinnell College\ Grinnell, Iowa 50112 U.S.A. author: - Leigh Evron - 'Joseph R. Mileti' - 'Ethan Ratliff-Crain' title: Irreducibles and Primes in Computable Integral Domains --- [^1] Introduction ============ In an integral domain, there are two natural definitions of basic “atomic" elements: irreducibles and primes. We recall these standard algebraic definitions. Let $A$ be an integral domain, i.e. a commutative ring with $1 \neq 0$ and with no zero divisors (so $ab = 0$ implies either $a = 0$ or $b=0$). Recall the following definitions. 1. An element $u \in A$ is a [*unit*]{} if there exists $w \in A$ with $uw = 1$. We denote the set of units by $U(A)$. Notice that $U(A)$ is a multiplicative group. 2. Given $a,b \in A$, we say that $a$ and $b$ are [*associates*]{} if there exists $u \in U(A)$ with $au = b$. 3. An element $p \in A$ is [*irreducible*]{} if it nonzero, not a unit, and has the property that whenever $p = ab$, either $a$ is a unit or $b$ is a unit. An equivalent definition is that $p \in A$ is irreducible if it is nonzero, not a unit, and its divisors are precisely the units and the associates of $p$. 4. An element $p \in A$ is [*prime*]{} if it nonzero, not a unit, and has the property that whenever $p \mid ab$, either $p \mid a$ or $p \mid b$. 5. $A$ is a [*unique factorization domain*]{}, or [*UFD*]{}, if it has the following two properties: - For each $a \in A$ such that $a$ is nonzero and not a unit, there exist irreducible elements $r_1,r_2,\dots,r_n \in A$ with $a = r_1r_2 \cdots r_n$. - If $r_1,r_2,\dots,r_n,q_1,q_2,\dots,q_m \in A$ are all irreducible and $r_1r_2 \cdots r_n = q_1q_2 \cdots q_m$, then $n = m$ and there exists a permutation $\sigma$ of $\{1,2,\dots,n\}$ such that $r_i$ and $q_{\sigma(i)}$ are associates for all $i$. It is a simple fact that if $A$ is an integral domain, then every prime element of $A$ is irreducible. Although the converse is true in any UFD, it does fail for general integral domains. For example, in the integral domain $\mathbb{Z}[\sqrt{-5}]$, there are two different factorizations of $6$ into irreducibles: $$2 \cdot 3 = 6 = (1 + \sqrt{-5})(1 - \sqrt{-5}).$$ Since $U(\mathbb{Z}(\sqrt{-5})) = \{1,-1\}$, these two factorizations are indeed distinct. This example also shows that $2$ is an irreducible element that is not prime because $2 \mid (1 + \sqrt{-5})(1 - \sqrt{-5})$ but $2 \nmid 1 + \sqrt{-5}$ and $2 \nmid 1 - \sqrt{-5}$. In fact, all four of the above irreducible factors are not prime. For another example that will be particularly relevant for our purposes, let $A$ be the subring of $\mathbb{Q}[x]$ consisting of those polynomials whose constant term and coefficient of $x$ are both integers, i.e. $$A = \{a_0 + a_1x + a_2x^2 + \dots + a_nx^n \in \mathbb{Q}[x] : a_0 \in \mathbb{Z} \text{ and } a_1 \in \mathbb{Z}\}.$$ In this integral domain, all of the normal integer primes are still irreducible (by a simple degree argument), but none of them are prime in $A$ because given any integer prime $p \in \mathbb{Z}$, we have that $p \mid x^2$ since $\frac{x^2}{p} \in A$, but $p \nmid x$ as $\frac{x}{p} \notin A$. We are interested in the extent to which the irreducible and prime elements can differ in an integral domain. As just discussed, the set of prime elements is always a subset of the set of irreducible elements, but it may be a proper subset. Can one of these sets be significantly more complicated than the other? We approach this question from the point of view of computability theory. We begin with the following fundamental definition. A [*computable ring*]{} is a ring whose underlying set is a computable set $A \subseteq \mathbb{N}$, with the property that $+$ and $\cdot$ are computable functions from $A \times A$ to $A$. For a general overview of results about computable rings and fields, see [@SHTucker]. Computable fields together with computable factorizations in polynomial rings over those fields have received a great deal of attention ([@FrohlichShep], [@MetakidesNerode], [@Rabin]), and [@MillerNotices] provides an excellent overview of work in this area. In particular, there exists a computable field $F$ such that the set of primes in $F[x]$ is not computable (see [@MillerNotices Lemma 3.4] or [@SHTucker Section 3.2] for an example). Moreover, there is a computable UFD such that the set of primes is as complicated as possible in the arithmetical hierarchy (see [@JoeDamir]). For our purposes, we will only need the first level of this hierarchy (see [@Soare Chapter 4] for more information). Let $Z \subseteq \mathbb{N}$. - We say that $Z$ is a $\Sigma_1^0$ set, or [*computably enumerable*]{}, if there exists a computable $R \subseteq \mathbb{N}^2$ such that $$i \in Z \Longleftrightarrow (\exists x) R(x,i).$$ - We say that $Z$ is a $\Pi_1^0$ set if there exists a computable $R \subseteq \mathbb{N}^2$ such that $$i \in Z \Longleftrightarrow (\forall x) R(x,i).$$ Notice that the complement of $\Sigma_1^0$ set is a $\Pi_1^0$ set, and the complement of $\Pi_1^0$ set is a $\Sigma_1^0$ set. Although every computable set is both a $\Sigma_1^0$ set and $\Pi_1^0$ set, there exists a $\Sigma_1^0$ set that is not computable, such as the set of natural numbers coding programs that halt. The complement of a noncomputable $\Sigma_1^0$ set is a noncomputable $\Pi_1^0$ set. We will use the following standard fact (see [@Soare Section II.1]) \[p:Sigma1IffRangeComputable\] An infinite set $Z \subseteq \mathbb{N}$ is $\Sigma_1^0$ if and only if there exists a computable injective function $\alpha \colon \mathbb{N} \to \mathbb{N}$ such that $\text{range}(\alpha) = Z$. We will prove that there exists a computable integral domain where the set of irreducible elements is computable while the set of prime elements is not, and also there exists a computable integral domain where the set of prime elements is computable while the set of irreducible elements is not. Thus, these two notions can be wildly different. Our approach will be to code an arbitrary $\Pi_1^0$ set into the set of irreducible (resp. prime) elements while maintaining control over the set of prime (reps. irreducible) elements. Moreover, our integral domains will extend $\mathbb{Z}$ and we will perform our noncomputable coding into the normal integer primes as in [@JoeDamir]. Strongly Computable Finite Factorization Domains ================================================ In Section 3, we will build a computable integral domain $A$ such that the set of irreducible elements of $A$ is computable but the set of prime elements of $A$ is not computable. The idea is that we will turn off the primeness of a normal integer prime $p_i$ in response to a $\Sigma_1^0$ event (such as program $i$ halting) by introducing a new element $x$ with $p_i \mid x^2$ but $p_i \nmid x$. In doing this, we will expand $A$ and we will want to ensure that we can compute the irreducible elements in the resulting integral domain. Since we are adding a new element, this construction will be analogous to expanding our original $A$ to the polynomial ring $A[x]$. However, there is a potential problem here in that even if the irreducible elements of an integral domain $A$ are computable, it need not be the case the the irreducible elements of $A[x]$ are computable. In fact, as mentioned in the introduction, there are computable fields $F$ (where the irreducibles are trivially computable because no element is irreducible) such that the irreducibles of $F[x]$ are not computable. To remedy this situation, we will ensure that the integral domains in our construction have a stronger property. As motivation, we first summarize Kronecker’s method for finding the divisors of an element $\mathbb{Z}[x]$, and hence for determining whether an element is irreducible. Let $f(x) \in \mathbb{Z}[x]$ be nonzero, and let $n = \deg(f(x))$. We try to restrict the set of possible divisors to a finite set that we need to check. Since the degree function is additive, notice that any divisor of $f(x)$ has degree at most $n$. Now perform the following: - Notice that if $g(x) \in \mathbb{Z}[x]$ and $g(x) \mid f(x)$ in $\mathbb{Z}[x]$, then $g(a) \mid f(a)$ for all $a \in \mathbb{Z}$. - Find $n+1$ many points $a \in \mathbb{Z}$ with $f(a) \neq 0$ (which exist because $f(x)$ has at most $n$ roots). Notice that each such $f(a)$ has only finitely many divisors in $\mathbb{Z}$. - For each of the possible choices of the divisors of these values in $\mathbb{Z}$, find the unique interpolating polynomial in $\mathbb{Q}[x]$ of degree at most $n$. - Check if any of these polynomials are in $\mathbb{Z}[x]$, and if so, check if they divide $f(x)$ in $\mathbb{Z}[x]$. - Compile the resulting list of divisors. Therefore, we can compute the finite set of divisors of any element of $\mathbb{Z}[x]$. Since we know the units of $\mathbb{Z}[x]$, it follows that we can computably determine if an element of $\mathbb{Z}[x]$ is irreducible. The key algebraic fact that makes Kronecker’s method work is that every nonzero element of $\mathbb{Z}$ has only finitely many divisors. Integral domains with this property were defined and studied in [@AAZ-Factor1; @AAZ-Factor2; @AndersonMullins]. Let $A$ be an integral domain. - $A$ is a [*finite factorization domain*]{}, or FFD, if every nonzero element has only finitely many divisors up to associates. - $A$ is a [*strong finite factorization domain*]{} if every nonzero element has only finitely many divisors. We now define an effective analogue of strong finite factorization domains. In addition to wanting our ring to be computable, we also want the stronger property that we can compute the finite set of divisors of any nonzero element. Instead of using the word “strong" twice, we adopt the following definition. A [*strongly computable finite factorization domain*]{}, or SCFFD, is a computable integral domain $A$ equipped with a computable function $D$ such that for all $a \in A \backslash \{0\}$, we have that $D(a)$ is (a canonical index for) the finite set of divisors of $a$ in $A$. Let $A$ be an SCFFD equipped with divisor function $D$. 1. The set $U(A)$ is a finite set that can be computed from $A$. 2. The set of irreducible elements of $A$ is computable. For the first claim, simply notice that $U(A) = D(1)$. For the second, given any $a \in A$, we have that $a$ is irreducible if and only it nonzero, not a unit, and its only divisors are units and associates. Suppose then that we are given an arbitrary $a \in A$. We can check whether $a$ is zero or a unit (by part 1), and if either is true, then $a$ is not irreducible. Otherwise, then since $a \neq 0$, we can compute the finite set $D(a)$ of divisors of $a$. Since we can also compute the finite set $U(A)$, we can examine each $b \in D(a)$ in turn to determine whether $b \in U(A)$ or whether there exists $u \in U(A)$ with $b = au$. If this is true for all $b \in D(a)$, then $a$ is irreducible in $A$, and otherwise it is not. If we include an additional assumption that $A$ is a UFD, then we have a converse to the previous result. Let $A$ be a computable integral domain with the following properties: - $A$ is a UFD. - $U(A)$ is finite. - The set of irreducible elements of $A$ is computable. We can then equip $A$ with a computable function $D$ so that $A$ becomes an SCFFD. We first argue that we can computably factor elements of $A$ into irreducibles. Let $a \in A$ be nonzero and not a unit. Since the set of irreducibles of $A$ is computable, we can check whether $a$ is irreducible. If not, we search until we find two nonzero nonunit elements of $A$ whose product is $a$. We can now check if these factors are irreducible, and if not we can repeat to factor them. Notice that this process must eventually produce finitely many irreducibles whose product is $a$ by König’s Lemma together with the fact that there are no infinite descending chains of strict divisibilities in a UFD. We now define our function $D$. Let $a \in A \backslash \{0\}$ be arbitrary. Check if $a \in U(A)$ (which is possible because $U(A)$ is finite and computable from $A$), and if so, define $D(a)$ to equal $U(A)$. If $a \notin U(A)$, then we we can computably factor it into irreducibles $q_i$ so that $a = q_1q_2 \dots q_n$. Since $U(A)$ is finite, we can now computably check if any of the $q_i$ are associates of each other, and if so we can find witnessing units. Thus, we can write $a = up_1^{k_1} \cdots p_m^{k_m}$ where $w \in U(A)$, each $p_i$ is irreducible, each $k_i \in \mathbb{N}^+$, and $p_i$ and $p_j$ are not associates whenever $i \neq j$. Since $A$ is a UFD, we then have that the set of divisors of $a$ equals the set of elements of the form $wp_1^{\ell_1} \cdots p_m^{\ell_m}$ where $w \in U(A)$ and $0 \leq \ell_i \leq k_i$ for all $i$. Thus, we can define $D(a)$ to be this finite set. In contrast, there are SCFFDs that are not UFDs, such as $\mathbb{Z}[\sqrt{-5}]$. More generally, the ring of integers in any imaginary quadratic number field is an SCFFD. To see this, Let $K$ be an imaginary quadratic number field, and fix an integral basis of $\mathcal{O}_K$. Using this integral basis, we can view $\mathcal{O}_K$ as a computable integral domain in such a way that the norm function and divisibility relation are both computable on $\mathcal{O}_K$ (see [@JoeDamir Proposition 1.4]). Given any $n \in \mathbb{N}$, there are only finitely many elements of norm $n$, and moreover we can compute the finite set of such elements. Now given any nonzero $a \in A$, we can compute $N(a)$, examine all elements of norm dividing $N(a)$, and check which of them divide $a$ (since the divisibility relation is computable) to compute the set of divisors of $a$. Let $A$ be a computable integral domain and let $F$ be the field of fractions of $A$. Recall that elements of $F$ are equivalence classes of pairs of elements of $A$. If we were to allow multiple representations of elements, we can of course work with pairs of elements of $A$ and define addition and multiplication on these elements computably. Nonetheless, a computable ring is defined in a way that forbids such multiple representations, so it is not immediately obvious that we can view $F$ as a computable field. However, since a computable integral domain is coded as a subset of $\mathbb{N}$, we can view pairs of elements $(a,b) \in A^2$ with $b \neq 0$ as being coded by elements of $\mathbb{N}^2$, which in turn can be coded by elements of $\mathbb{N}$. Thus, we can view the field of fractions $F$ as a computable field by working only with pairs $(a,b)$ such that there is no strictly smaller pair $(c,d)$ in the usual ordering of $\mathbb{N}$ with $ad = bc$. In this way, we can still define addition and multiplication computably be searching back for the smallest equivalent representative. In general, for a computable integral domain $A$, it may not be possible to build the field of fractions as a computable extension of $A$, because it may not be possible to determine when an element $\frac{a}{b} \in F$ is actually an element of $A$. The issue is that we may not be able to determine if $b \mid a$ because the divisibility relation may not be computable. However, we have the following. If $A$ is an SCFFD, then the field of fractions of $A$ is a computable field, and we can computably build it as an extension of $A$. Notice that in the field of fractions of $A$, we have that $\frac{a}{b} \in A$ if and only if $b \mid a$, which is if and only if $b \in D(a)$. Now since $A$ is a computable integral domain, it is coded as a subset of $\mathbb{N}$. We can now add on minimal pairs $(a,b)$ such that $b \nmid a$. With this, we can define addition and multiplication In fact, we can computably “reduce" fractions over an SCFFD to lowest terms, as we now show. \[p:ReduceFractionsOverSCFFD\] Let $A$ be an SCFFD and let $F$ be the field of fractions of $A$. Given an arbitrary pair of elements $a,b \in R$ with $b \neq 0$, we can computably find a pair of elements $c,d \in R$ with $d \neq 0$, with $\frac{c}{d} = \frac{a}{b}$ in $F$, and such that the only common divisors of $c$ and $d$ are the units of $A$. First notice that if $a = 0$, then we may take $c = 0$ and $d = 1$. Suppose then that $a \neq 0$. Since we also have that $b \neq 0$, we can now computably determine the finite set of divisors of each of $a$ and $b$, and thus can computably build the finite set $S$ of common divisors of $a$ and $b$, i.e. $S = D(a) \cap D(b)$. For each $r \in S$, we can computably determine the number $|\{s \in S : s \mid r\}| = |D(r) \cap S|$. Fix an $r \in S$ such that $|\{s \in S : s \mid r\}|$ is as large as possible. Since $r$ is a common divisor of $a$ and $b$, we can computably search for $c,d \in A$ such that $rc = a$ and $rd = b$. Notice that $d \neq 0$ (because $b \neq 0$) and $\frac{a}{b}= \frac{c}{d}$. Suppose now that $t$ is a common divisor of $c$ and $d$. We then have that $rt$ is a common divisor of $a$ and $b$, so $rt \in S$. By definition of $R$, this implies that $|\{s \in S : s \mid rt\}| \leq |\{s \in S : s \mid r\}|$. Since $\{s \in S : s \mid r\} \subseteq \{s \in S : s \mid rt\}$, it follows that $\{s \in S : s \mid r\} \subseteq \{s \in S : s \mid rt\}$. Thus ,$|\{s \in S : s \mid rt\}| = |\{s \in S : s \mid r\}|$. In particular, we must have $rt \mid r$, so $t \in U(A)$. Notice this reduction need not be unique, even up to units. In the SCFFD $\mathbb{Z}[\sqrt{-5}]$ we have that $$\frac{2}{1+\sqrt{-5}} = \frac{1-\sqrt{-5}}{3}$$ where there are no nonunit common factors for the numerator and denominator of either side. By [@AAZ-Factor1 Proposition 5.3] and [@AndersonMullins Theorem 5], if $A$ is a (strong) finite factorization domain, then so is $A[x]$. We now prove an effective analogue of this result. Notice first that if $A$ is a finite integral domain, then $A$ is a finite field, and $A[x]$ is trivially an SCFFD because given $f(x) \in A[x] \backslash \{0\}$, every divisor $g(x)$ of $f(x)$ must satisfy $\deg(g(x)) \leq \deg(f(x))$, and so we need only check each of the finitely many possibilities (which is possible because we can computably search for quotients and remainders). We now handle the infinite case. \[t:PolynomialRingOverSCFFDisSCFFD\] If $A$ is an infinite SCFFD, then so is $A[x]$. Moreover, given an index for a function $D$ witnessing that $A$ is an SCFFD, we can computably obtain an index for a function $D'$ extending $D$ to witness the fact that $A[x]$ is an SCFFD. Before jumping into the proof, we give two lemmas. \[l:InterpoteAndCheckIfInA\] Let $A$ be an SCFFD, let $n \in \mathbb{N}^+$, let $a_0, a_1, \dots, a_n \in A$ be distinct and let $b_0, b_1, \dots, b_n \in R$. Let $F$ be the field of fractions of $A$. There is exactly one polynomial $p(x) \in F[x]$ of degree at most $n$ with $p(a_i) = b_i$ for all $i$. Furthermore, we can computably construct $p(x)$ in $F[x]$, and can computably determine if $p(x) \in A[x]$.\ Uniqueness follows from that fact that if two polynomials over a field having degree at most $n$ agree at $n+1$ points, then they must be the same polynomial. For existence, using Lagrange’s method of interpolation for $n+1$ distinct points of the form $(a_i, b_i)$ will result in a polynomial of the following form: $$p(x) = \sum_{i=0}^{n} b_i \cdot \frac{(x-a_0) \cdots (x-a_{i-1}) (x-a_{i+1}) \cdots (x-a_n)}{(a_i-a_0) \cdots (a_i-a_{i-1}) (a_i-a_{i+1})\cdots (a_i-a_n)}$$ Notice that the denominator is nonzero because $A$ is an integral domain and $a_i \neq a_j$ whenever $i \neq j$. We can computably expand $p(x)$ to write it as $p(x) = \sum_{i=0}^n \frac{c_i}{d_i} x^i$. We then have that $p(x) \in A[x]$ if and only if $d_i \mid c_i$ for all $i$, which we can verify by checking if $d_i \in D(c_i)$ for all $i$. \[l:DivisibilityRelationOnPolyRingIsComputable\] Suppose that $A$ is an SCFFD. The divisibility relation on $A[x]$ is computable, i.e. given $f(x),g(x) \in A[x]$, we can computably determine if $f(x) \mid g(x)$ in $A[x]$. Let $f(x),g(x) \in A[x]$ be arbitrary. If $g(x) = 0$, then trivially we have $f(x) \mid g(x)$. Suppose then that both $g(x)$ is nonzero. Perform polynomial long division (or search) to find $q(x),r(x) \in F[x]$ with $f(x) = q(x)g(x) + r(x)$ and either $r(x) = 0$ or $\deg(r(x)) < \deg(g(x))$. Since quotients and remainders are unique in $F[x]$, we have that $g(x) \mid f(x)$ in $A[x]$ if and only if $q(x) \in A[x]$ and $r(x) = 0$. Since we can computably determine if an element of $F[x]$ is in $A[x]$ as in Lemma \[l:InterpoteAndCheckIfInA\], this completes the proof. Let $f(x) \in A[x]$ be arbitrary, and let $n = \deg(f(x))$. Suppose that $g(x) \in A[x]$ is such that $g(x) \mid f(x)$. First notice that $\deg(g(x)) \leq n$ because the degree function is additive (as $A$ is an integral domain). Now if we fix $h(x) \in A[x]$ with $g(x)h(x) = f(x)$, we then have $g(a)h(a) = f(a)$ for all $a \in A$, so since $f(a),g(a),h(a) \in A$ for all $a \in A$, we have that $g(a) \mid f(a)$ for all $a \in A$. Search until we find $n+1$ many distinct elements $a_0,a_1,\dots,a_n \in A$ such that $f(a_i) \neq 0$ for all $i$ (such $a_i$ exist because $A$ is infinite and $f(x)$ has at most $n$ roots in $A$). Since $A$ is an SCFFD, we have that $f(a_i)$ has only finitely many divisors for each $i$, and we can compute the finite sets $D(f(a_i))$. Suppose that we pick elements $b_i \in D(f(a_i))$ for each $i$. From Lemma \[l:InterpoteAndCheckIfInA\], there is a unique element $p(x) \in F[x]$ with $\deg(p(x)) \leq n$ and $p(a_i) = b_i$ for all $i$, and we can compute this polynomial $p(x)$ and determine if $p(x) \in A[x]$. As we do this for each choice of the $b_i$, we obtain a finite subset of $A[x]$ of all possible divisors of $f(x)$. Now using Lemma \[l:DivisibilityRelationOnPolyRingIsComputable\], we can thin out this set to form the actual finite set of divisors of $f(x)$. Irreducibles Computable and Primes Noncomputable ================================================ Let $A$ be an integral domain that is an SCFFD and suppose that $q$ is a prime of $A$. Suppose that we want to destroy the primeness of $q$ while maintaining its irreducibility (say in response to a $\Sigma_1^0$ event such as the halting of a program). The idea is to introduce a new element $x$ so that $q \mid x^2$ but $q \nmid x$. If we let $F$ be the field of fractions of $A$, then we can accomplish this by working in $F[x]$, and extending $A$ to the subring $A[\frac{x^2}{q}]$ of $F[x]$. More explicitly, $A[\frac{x^2}{q}]$ is the set of all polynomials of the form $$a_0 + a_1x + \frac{a_2}{q} \cdot x^2 + \frac{a_3}{q} \cdot x^3 + \frac{a_4}{q^2} \cdot x^4 + \frac{a_5}{q^2} \cdot x^5 + \dots + \frac{a_n}{q^{\lfloor n/2 \rfloor}} \cdot x^n$$ where each $a_i \in A$. Although this works, we will find it more convenient notationally to work with subring $B$ of $F[x]$ consisting of those polynomials of the form $$a_0 + a_1x + \frac{a_2}{q^2} \cdot x^2 + \frac{a_3}{q^3} \cdot x^3 + \frac{a_4}{q^4} \cdot x^4 + \frac{a_5}{q^5} \cdot x^5 + \dots + \frac{a_n}{q^n} \cdot x^n$$ where each $a_i \in A$. \[t:PropoertiesAfterDestroyingOnePrime\] Let $A$ be an SCFFD and let $q \in A$ be prime. Let $F$ be the field of fractions of $A$ and let $B$ be the subring of $F[x]$ consisting of those polynomials of the form $$a_0 + a_1x + \frac{a_2}{q^2} \cdot x^2 + \frac{a_3}{q^3} \cdot x^3 + \frac{a_4}{q^4} \cdot x^4 + \frac{a_5}{q^5} \cdot x^5 + \dots + \frac{a_n}{q^n} \cdot x^n$$ where each $a_i \in A$. We then have the following. 1. \[e:DivisorsPreservedInPrimeDestruction\] For any $a \in A$, the set of divisors of $a$ in $A$ equals the set of divisors of $a$ in $B$. 2. $B$ is an SCFFD. Moreover, given $A$, $q$, and an index for a function $D$ witnessing that $A$ is an SCFFD, we can computably build $B$ as an extension of $A$ and obtain an index for a function $D'$ witnessing that $B$ is an SCFFD with the property that $D'(a) = D(a)$ for all $a \in A$. 3. \[e:UnitsPreservedInPrimeDestruction\] $U(B) = U(A)$. 4. If $p$ is irreducible in $A$, then $p$ is irreducible in $B$. 5. If $p_1,p_2 \in A$ are irreducibles that are not associates in $A$, then they are not associates in $B$. 6. $q$ is not prime in $B$. 7. If $p$ is a prime of $A$ that is not an associate of $q$, then $p$ is prime in $B$. <!-- --> 1. Let $a \in A$. Clearly, if an element of $A$ divides $a$ in $A$, then it divides $a$ in $B$. For the converse, since the degree function is additive on $F[x]$, if $f(x),g(x) \in B$ are such that $a = f(x)g(x)$, then we must have $\deg(f(x)) = 0 = \deg(g(x))$, and hence $f(x),g(x) \in A$. 2. Notice first that we computably build $B$ as an extension of $A$ trivially, because if $\frac{a}{q^k} = \frac{b}{q^k}$, then $a = b$ (so there is no issue of distinct representations). The proof that $B$ is an SCFFD is analogous to the proof of Theorem \[t:PolynomialRingOverSCFFDisSCFFD\], with a few straightforward modifications. Given $f(x) \in B$ with $\deg(f(x)) = n$, to determine the divisors of $f(x)$ in $B$, we note the following: - Notice that if $f(x) \in B$ and $a \in A$, then in general it need not be the case that $f(a) \in A$. However, we will only plug in values $q^i$ for $i \geq n$ to avoid this issue. Suppose then that $g(x) \in B$ with $g(x) \mid f(x)$, and fix $h(x) \in B$ with $g(x)h(x) = f(x)$. We then have that $\deg(g(x)) \leq n$ and $\deg(h(x)) \leq n$. Thus, for any $i \geq n$, we have $f(q^i), g(q^i), h(q^i) \in A$, and so $g(q^i) \mid f(q^i)$ in $A$. Since there are infinitely many $i \geq n$, and these $q^i$ provide an infinite supply of distinct elements (because $A$ is an integral domain), we can plug in $n+1$ many such values with $f(q^i) \neq 0$ to form the basis for our Lagrange interpolations. - We can computably determine if an element $p(x) \in F[x]$ is actually an element of $B$. The key question is given $a,b \in A$ with $b \neq 0$ and a $k \geq 2$, can we determine if we can write an element $\frac{a}{b}$ of $F$ in the form $\frac{c}{q^k}$. Notice that this is possible if and only if there exists $c \in A$ with $aq^k = bc$, which is if and only if $b \mid aq^k$. Since $A$ is an SCFFD, we can computably determine if $b \in D(aq^k)$, and furthermore in this case we can computably find $c$ with $bc = aq^k$ and hence $\frac{a}{b}= \frac{c}{q^k}$, Thus, we can determine if an element of $F[x]$ is an element of $B$, and if so write it in the above form. - The divisibility relation is computable on $B$ as in Lemma \[l:DivisibilityRelationOnPolyRingIsComputable\], because we can computably determine if an element of $F[x]$ is an element of $B$ as just mentioned. This shows that $B$ is an SCFFD and allows us to compute $D'$ uniformly from $A$ and $D$. Finally, notice that $D'$ extends $D$ by \[e:DivisorsPreservedInPrimeDestruction\]. 3. Immediate from \[e:DivisorsPreservedInPrimeDestruction\] and the fact that $U(B) = D(1)$. 4. This follow from \[e:DivisorsPreservedInPrimeDestruction\] and \[e:UnitsPreservedInPrimeDestruction\]. 5. Immediate from \[e:UnitsPreservedInPrimeDestruction\]. 6. Notice that $q$ is nonzero and not a unit by \[e:UnitsPreservedInPrimeDestruction\]. We have that $q \mid x^2$ in $B$ because $\frac{1}{q} \cdot x^2 = \frac{q}{q^2} \cdot x^2 \in B$, but $q \nmid x$ because $\frac{1}{q} \cdot x \notin B$ as $q$ is not a unit (and this is the only possible witness for divisibility because $F[x]$ is an integral domain). Therefore, $q$ is not prime in $B$. 7. Let $p$ be a prime of $A$ that is not an associate of $q$. Notice that $p$ is nonzero and not a unit of $B$ by \[e:UnitsPreservedInPrimeDestruction\]. Let $f(x),g(x) \in B$, and suppose that $p \mid f(x)g(x)$ in $B$. Write out $$\begin{aligned} f(x) & = a_0 + a_1x + \frac{a_2}{q^2} \cdot x^2 + \frac{a_3}{q^3} \cdot x^3 + \dots + \frac{a_n}{q^n} \cdot x^n \\ g(x) & = b_0 + b_1x + \frac{b_2}{q^2} \cdot x^2 + \frac{b_3}{q^3} \cdot x^3 + \dots + \frac{b_n}{q^n} \cdot x^n \\ f(x)g(x) & = c_0 + c_1x + \frac{c_2}{q^2} \cdot x^2 + \frac{c_3}{q^3} \cdot x^3 + \dots + \frac{c_n}{q^n} \cdot x^n\end{aligned}$$ Since $p \mid f(x)g(x)$ in $B$, we have that $p \mid c_i$ in $A$ for all $i$. Assume that $p \nmid f(x)$ and $p \nmid g(x)$ in $B$. Then there must exist $i$ and $j$ such that $p \nmid a_i$ in $A$ and $p \nmid b_j$ in $A$. Let $k$ and $\ell$ be largest possible such that $p \nmid a_k$ in $A$ and $p \nmid b_{\ell}$ in $A$. Now element $c_{k+\ell}$ will be a sum of terms, one of which will be $a_kb_{\ell}q^j$ for some $j \in \{0,1\}$, while other terms will be divisible by $p$ in $A$. Since $p$ divides $c_{k+\ell}$, it follows that $p \mid a_kb_{\ell}q^j$ in $A$. However, this is a contradiction because $p$ is prime in $A$ but divides none of $a_k$, $b_{\ell}$, or $q$ (the last because $p$ is not an associate of $q$ in $A$). We now show that we can code an arbitrary $\Pi_1^0$ set into the primes of an integral domain $A$ while maintaining the computability of the irreducible elements. In fact, we perform our coding within the normal integer primes and can make the resulting integral domain an SCFFD. \[t:Pi1ControlOfPrimes\] Let $S$ be a $\Sigma_1^0$ set, and let $p_0,p_1,p_2,\dots$ list the usual primes from $\mathbb{N}$ in increasing order. There exists an SCFFD $A$ such that: - $\mathbb{Z}$ is a subring of $A$. - $U(A) = \{1,-1\}$. - Every $p_i$ is irreducible in $A$. - $p_i$ is prime in $A$ if and only if $i \notin S$. If $S = \emptyset$, this is trivial by letting $A = \mathbb{Z}$. Assume then that $S \neq \emptyset$. If $S$ is finite, say $|S| = n$, then we can trivially fix a computable injective function $\alpha \colon \{1,2,\dots,n\} \to \mathbb{N}$ with $\text{range}(\alpha) = S$. If $S$ is infinite, then we can fix a computable injective function $\alpha \colon \mathbb{N} \to \mathbb{N}$ with $\text{range}(\alpha) = S$ by Proposition \[p:Sigma1IffRangeComputable\]. We build our computable SCFFD $A$ in stages, starting by letting $A_0 = \mathbb{Z}$ and letting $D_0(a)$ be the finite set of divisors of $a$ for all $a \in \mathbb{Z} \backslash \{0\}$. Suppose that we are at a stage $k$ and have constructed an SCFFD $A_k$ together with witnessing function $D_k$. We now extend $A_k$ to $A_{k+1}$ by destroying the primality of $p_{\alpha(k)}$ as in the construction of Theorem \[t:PropoertiesAfterDestroyingOnePrime\] using a new indeterminate $x_k$. In other words, letting $F_k$ be the field of fractions of $A_k$, we let $A_{k+1}$ be the subring of $F_k[x]$ consisting of those polynomials of the form $$a_0 + a_1x + \frac{a_2}{p_{\alpha(k)}^2} \cdot x_k^2 + \frac{a_3}{p_{\alpha(k)}^3} \cdot x_k^3 + \frac{a_4}{p_{\alpha(k)}^4} \cdot x_k^4 + \dots + \frac{a_n}{p_{\alpha(k)}^n} \cdot x_k^n$$ where each $a_i \in A_k$. We continue this process through the construction of $A_n$ if $|S| = n$, and infinitely often if $S$ is infinite. Using Theorem \[t:PropoertiesAfterDestroyingOnePrime\], the following properties hold by induction on $k$: - $A_k$ is an SCFFD with witnessing function $D_k$ extending $D_i$ for all $i < k$. - $U(A_k) = \{1,-1\}$. - Every $p_i$ is irreducible in $A_k$. - $p_i$ is prime in $A_k$ if and only if $i \notin \{\alpha(1),\alpha(2),\dots,\alpha(k)\}$. Now if $S$ is finite, say $|S| = n$, then it follows that the integral domain $A_n$ has the required properties. Suppose then that $S$ is infinite, and let $A = A_{\infty} = \bigcup_{k=0}^{\infty} A_k$. Also, let $D = \bigcup_{k=1}^{\infty} D_k$, which makes sense because the $D_i$ extend each other as functions. Notice that $D$ is a computable function and that for any $a \in A_k$, we have that the set of divisors of $a$ in $A$ equals the set of divisors of $a$ in $A_k$, so $D(a) = D_k(a)$ is the finite set of divisors of $a$ in $A$. Therefore, $A$ is an SCFFD as witnessed by $D$. Since $U(A_k) = \{1,-1\}$ for all $k \in \mathbb{N}$, it follows that $U(A) = \{1,-1\}$. Since we maintain the units and divisibility at each stage, it also follows that every $p_i$ is irreducible in $A$. We now show that $p_i$ is prime in $A$ if and only if $i \notin S$. First notice that each $p_i$ is nonzero and not a unit of $A$. - Suppose first that $i \notin S$. We then have that $i \notin \text{range}(\alpha)$, so $p_i$ is prime in every $A_k$ by the last property above. Let $a,b \in A$, and suppose that $p_i \mid ab$ in $A$. Fix $c \in A$ with $p_ic = ab$. Go to a point $k$ where each of $p_i,a,b,c$ exist. We then have that $p_i \mid ab$ in $A_k$, so as $p_i$ is prime in $A_k$, either $p_i \mid a$ in $A_k$ or $p_i \mid b$ in $A_k$. Therefore, either $p_i \mid a$ in $A$ or $p_i \mid b$ in $A$. It follows that $p_i$ is prime in $A_k$. - Suppose now that $i \in S$. Thus, we can fix $k \in \mathbb{N}$ with $\alpha(k) = i$. We then have that $p_i$ is not prime in $A_{k+1}$ by the last property above. Fix $a,b \in A_{k+1}$ such that $p_i \mid ab$ in $A_{k+1}$ but $p_i \nmid a$ in $A_{k+1}$ and $p_i \nmid b$ in $A_{k+1}$. Since the $D_i$ extend each other as functions, and $A$ is an SCFFD as witnessed by $D$, it follow that $p_i \mid ab$ in $A$ but $p_i \nmid a$ in $A$ and $p_i \nmid b$ in $A$. Therefore, $p_i$ is not prime in $A$. There exists a computable integral domain $A$ such that the set of irreducible elements of $A$ is computable but the set of prime elements of $A$ is not computable. Fix a noncomputable $\Sigma_1^0$ set $S$, and let $A$ be the SCFFD given by Theorem \[t:Pi1ControlOfPrimes\]. Since $A$ is an SCFFD, it is a computable integral domain and the set of irreducible elements of $A$ is computable. However, the set of prime elements of $A$ is not computable, because if we could compute it, then we could compute $S$, which is a contradiction. Primes Computable and Irreducibles Noncomputable ================================================ Consider the subring $A = \mathbb{Z} + x\mathbb{Z} + x^2\mathbb{Q}[x]$ of $\mathbb{Q}[x]$. In other words, $A$ is the set of polynomials of the form $q_0 + q_1x + q_2x^2 + \dots + q_nx^n$ where $q_0 \in \mathbb{Z}$ and $q_1 \in \mathbb{Z}$. As mentioned in the introduction, each normal integer prime is irreducible in $A$ but is not prime in $A$. It is also a standard fact for $p(x) \in A$, we have that $p(x)$ is prime in $A$ if and only if $p(x)$ is irreducible in $\mathbb{Q}[x]$ and $p(0) \in \{1,-1\}$. We will generalize this construction by replacing $\mathbb{Z}$ with an arbitrary integral domain. Suppose that $R$ is an integral domain, and let $F$ be its field of fractions. Consider the subring $A = R + xR + x^2F[x]$ of $F[x]$, i.e. $A$ is the set of polynomials of the form $q_0 + q_1x + q_2x^2 + \dots + q_nx^n$ where $q_0 \in R$ and $q_1 \in R$. Such an integral domain $A$ is particularly nice from our perspective because the irreducibles in $R$ will remain irreducible in $A$ (so all of the complexity of irreducibles remain), but no element of $R$ is prime in $A$ (so any complexity of primes is “erased"). Moreover, we can reduce the complexity of primality of elements of $A$ to that of irreducibles in the polynomial ring over a field, about which a great deal is understood. \[l:PrimesInAAreNonconstantAndIrreducible\] Let $R$ be an integral domain with field of fractions $F$. Consider the subring $A = R + xR + x^2F[x]$ of $F[x]$. Let $p(x) \in A$. If $p(x)$ is prime in $A$, then $p(x)$ is non-constant and irreducible in $F[x]$. We prove the contrapositive, i.e. if $p(x) \in A$ is either constant or not irreducible, then $p(x)$ is not prime in $A$. Suppose first that $p(x)$ is a constant, and fix $k \in R$ with $p(x) = k$. If $k \in \{0\} \cup U(R)$, then $k$ is either zero or a unit, so $k$ is not prime in $A$ by definition. Suppose then that $k \notin \{0\} \cup U(R)$. Notice that $k \mid x^2$ in $A$ because $\frac{1}{k} \cdot x^2 \in A$, but $k \nmid x$ in $A$ because $\frac{1}{k} \cdot x \notin A$. Therefore, $p(x) = k$ is not prime in $A$. Suppose now that $p(x) \in A$ is non-constant and not irreducible in $F[x]$. Since $p(x)$ is non-constant, it is not a unit in $F[x]$. Fix $g(x),h(x) \in F[x]$ with $p(x) = g(x)h(x)$ and such that $0 < \deg(g(x)) < \deg(p(x))$ and $0 < \deg(h(x)) < \deg(p(x))$. Now since $g(x),h(x) \in F[x]$, the constant terms and coefficients of $x$ in these polynomials need not be in $R$. Let $b$ be the product of the denominators of these coefficients in $g(x)$, and let $c$ be the product of the denominators of these coefficients in $h(x)$. We then have that $p(x) \cdot bc = (b \cdot g(x)) \cdot (c \cdot h(x))$ where both $b \cdot g(x) \in A$ and $c \cdot h(x) \in A$. Since $bc \in R \subseteq A$, we have that $p(x) \mid (b \cdot g(x)) \cdot (c \cdot h(x))$ in $A$. However, notice that $p(x) \nmid b \cdot g(x)$ in $A$ because $\deg(b \cdot g(x)) < \deg(p(x))$ and $p(x) \nmid c \cdot h(x)$ because $\deg(c \cdot h(x)) < \deg(p(x))$. Therefore, $p(x)$ is not prime in $A$. \[l:CharacterizePrimesInA\] Let $R$ be an integral domain with field of fractions $F$. Consider the subring $A = R + xR + x^2F[x]$ of $F[x]$. Let $p(x) \in A$ and suppose that $p(x)$ is irreducible in $F[x]$. The following are equivalent. 1. $p(x)$ is prime in $A$. 2. For all $f(x) \in F[x]$, if $p(x)f(x) \in A$, then $f(x) \in A$. 3. For all $g(x) \in A$ such that $p(x) \mid g(x)$ in $F[x]$, we have that $p(x) \mid g(x)$ in $A$. $(1) \rightarrow (2)$: Suppose first that $p(x)$ is prime in $A$. We know that no constants are prime in $A$ from above, so $p(x)$ is non-constant. Let $f(x) \in F[x]$ be such that $p(x)f(x) \in A$. We prove that $f(x) \in A$. Write $f(x) = q_0 + q_1x + \dots + q_nx^n$ where each $q_i \in F$. Let $d$ be the product of the denominators of $q_0$ and $q_1$. Now $d \in R \subseteq A$ and $d \cdot f(x) \in A$, hence $p(x) \mid p(x) \cdot d \cdot f(x)$ in $A$, i.e. $p(x) \mid d \cdot (p(x)f(x))$ in $A$. Since $p(x)$ is prime in $A$, either $p(x) \mid d$ in $A$ or $p(x) \mid p(x)f(x)$ in $A$. The former is impossible because $p(x)$ is non-constant, so we must have that $p(x) \mid p(x)f(x)$ in $A$. Fix $h(x) \in A$ with $p(x)h(x) = p(x)f(x)$. Since $F[x]$ is an integral domain, we conclude that $f(x) = h(x) \in A$. $(2) \rightarrow (3)$: Immediate. $(3) \rightarrow (1)$: Let $g(x),h(x) \in A$ and suppose that $p(x) \mid g(x)h(x)$ in $A$. Since $A$ is a subring of $F[x]$, we then have that $p(x) \mid g(x)h(x)$ in $F[x]$. Now $p(x)$ is irreducible in $F[x]$, so since $F[x]$ is a UFD, we know that $p(x)$ is prime in $F[x]$. Thus, either $p(x) \mid g(x)$ in $F[x]$ or $f(x) \mid h(x)$ in $F[x]$. Using $(3)$, we conclude that either $p(x) \mid g(x)$ in $A$ or $p(x) \mid h(x)$ in $A$. Therefore, $p(x)$ is prime in $A$. \[p:ClassifyPrimesInSubringOfFX\] Let $R$ be an integral domain that is not a field, and let $F$ be its field of fractions. Consider the subring $A = R + xR + x^2F[x]$ of $F[x]$. An element $p(x) \in A$ is prime in $A$ if and only if $p(x)$ is irreducible in $F[x]$ and $p(0) \in U(R)$. We first prove that if $p(x) \in A$ does not satisfy $p(0) \notin U(R)$, then $p(x)$ is not prime in $A$. If $p(0) = 0$, then fixing any nonzero nonunit $b \in R$ (which exists because $R$ is not a field), we have $p(x) \cdot \frac{x}{b} \in A$ but $\frac{x}{b} \notin A$, so $p(x)$ is not prime in $A$ by Lemma \[l:CharacterizePrimesInA\]. Suppose then that $p(0) \notin \{0\} \cup U(R)$. Write $p(x) = q_nx^n + \dots + q_2x^2 + ax + b$ where $a,b \in R$ and $b \notin \{0\} \cup U(R)$. We have $$\begin{aligned} p(x) \cdot \left(\frac{1}{b} \cdot x\right) & = (q_nx^n + \dots + q_2x^2 + ax + b) \cdot \left(\frac{1}{b} \cdot x\right) \\ & = \left(\frac{q_n}{b}\right) \cdot x^{n+1} + \dots + \left(\frac{q_2}{b}\right) \cdot x^3 + \left(\frac{a}{b}\right) \cdot x^2+x\end{aligned}$$ Thus, $f(x) \cdot \frac{1}{b} \cdot x \in A$ but $\frac{1}{b} \cdot x \notin A$, so $f(x)$ is not prime in $A$ by Lemma \[l:CharacterizePrimesInA\]. We have just shown that $p(x) \in A$ is prime in $A$, then $p(0) \in U(R)$. We also know that if $p(x) \in A$ is prime in $A$, then $p(x)$ is irreducible in $F[x]$ by Lemma \[l:PrimesInAAreNonconstantAndIrreducible\]. Suppose conversely that $p(x)$ is irreducible in $F[x]$ and that $p(0) \in U(R)$. Using Lemma \[l:CharacterizePrimesInA\], to show that $p(x)$ is prime in $A$ it suffices to show that whenever $f(x) \in F[x]$ is such that $p(x)f(x) \in A$, then we must have $f(x) \in A$. Suppose then that $f(x) \in F[x]$ and $p(x)f(x) \in A$. Write $$\begin{aligned} f(x) & = q_0 + q_1x + q_2x^2 + \dots + q_nx^n \\ p(x) & = a_0 + a_1x + r_2x^2 + \dots + r_nx^n\end{aligned}$$ where $a_0 \in U(R)$, $a_1 \in R$, each $q_i \in F$, and each $r_i \in F$. We then have that $p(x)f(x) \in F[x]$ with $$p(x)f(x) = q_0a_0 + (q_0a_1 + a_0q_1) x + \dots$$ As $p(x)f(x) \in A$, we know that $q_0a_0 \in R$ and $q_0a_1 + a_0q_1 \in R$. Since $q_0a_0 \in R$ and $a_0 \in U(R)$, it follows that $q_0 \in R$. Using this together with the facts that $a_1 \in R$ and $q_0a_1 + a_0q_1 \in R$, it follows that $a_0q_1 \in R$. Applying again the fact that $a_0 \in U(R)$, we conclude that $q_1 \in R$. Since $q_0,q_1 \in R$, it follows that $p(x) \in A$. With these results in hand, we now proceed to construct an integral domain $R$ with a complicated set of irreducible elements. We will want our $R$ to have a “nice" field of fractions $F$ in the sense that the irreducibles of $F[y]$ will be computable. \[l:DestorySigma1PrimesLemma\] Let $S$ be a $\Sigma_1^0$ set, and let $p_0,p_1,p_2,\dots$ list the usual primes from $\mathbb{N}$ in increasing order. There exists a computable UFD $R$ such that: - $\mathbb{Z}$ is a subring of $R$, and in fact $$\mathbb{Z}[x_1,x_2,\dots] \subseteq R \subseteq \mathbb{Q}(x_1,x_2,\dots),$$ where there are infinitely many indeterminates if $S$ is infinite, and exactly $n$ of them if $|S| = n$. - $U(R) = \{1,-1\}$. - $p_i$ is irreducible in $R$ if and only if $i \notin S$. If $S = \emptyset$, this is trivial by letting $A = \mathbb{Z}$. Assume then that $S \neq \emptyset$. If $S$ is finite, say $|S| = n$, then we can trivially fix a computable injective function $\alpha \colon \{1,,2\dots,n\} \to \mathbb{N}$ with $\text{range}(\alpha) = S$. If $S$ is infinite, then we can fix a computable injective function $\alpha \colon \mathbb{N} \to \mathbb{N}$ with $\text{range}(\alpha) = S$ by Proposition \[p:Sigma1IffRangeComputable\]. We build our computable UFD $R$ in stages, starting by letting $R_0 = \mathbb{Z}$. Suppose that we are at a stage $k$ and have constructed through the integral domain $R_k$. We now destroy the irreducibility of $p_{\alpha(k)}$ by letting $R_{k+1} = R_k[x_k,\frac{p_{\alpha(k)}}{x_k}]$ as in [@JoeDamir Section 3]. We continue this process through the construction of $R_{n+1}$ if $|S| = n$, and infinitely often if $S$ is infinite. Using [@JoeDamir Proposition 3.3 and Theorem 3.10], the following properties hold by induction on $k$: - $R_k$ is a Noetherian UFD. - $\mathbb{Z}[x_1,x_2,\dots,x_k] \subseteq R_k \subseteq \mathbb{Q}(x_1,x_2,\dots,x_k)$. - $U(R_k) = \{1,-1\}$. - $p_i$ is irreducible in $R_k$ if and only if $i \notin \{\alpha(1),\alpha(2),\dots,\alpha(k)\}$. Now if $S$ is finite, say $|S| = n$, then it follows that the integral domain $R_n$ has the required properties. Suppose then that $S$ is infinite, and let $R = R_{\infty} = \bigcup_{k=0}^{\infty} R_k$. We then have that $R$ has the required properties by the proofs in [@JoeDamir Section 4] (although they are significantly easier in this case because we never change the units). \[t:Pi1ControlOfIrreducibles\] Let $S$ be a $\Sigma_1^0$ set, and let $p_0,p_1,p_2,\dots$ list the usual primes from $\mathbb{N}$ in increasing order. There exists a computable integral domain $A$ such that: - $\mathbb{Z}$ is a subring of $A$. - $U(A) = \{1,-1\}$. - No $p_i$ is prime in $A$. - The set of prime elements of $A$ is computable. - $p_i$ is irreducible in $A$ if and only if $i \notin S$. Let $R$ be the integral domain given by Lemma \[l:DestorySigma1PrimesLemma\]. Let $F$ be the field of fractions of $R$. Since $$\mathbb{Z}[x_1,x_2,\dots] \subseteq R \subseteq \mathbb{Q}(x_1,x_2,\dots)$$ (where there are infinitely many indeterminates if $S$ is infinite, and exactly $n$ of them if $|S| = n$) and the field of fractions of $\mathbb{Z}[x_1,x_2,\dots]$ is $\mathbb{Q}(x_1,x_2,\dots)$, it follows that $F = \mathbb{Q}(x_1,x_2,\dots)$. Let $A$ be the subring $R + yR + y^2F[y]$ of $F[y]$. Now we clearly have that $\mathbb{Z}$ is a subring of $A$ and $U(A) = \{1,-1\}$. Also, each $p_i$ is a constant polynomial in $A$, so is not prime in $A$ by Lemma \[l:PrimesInAAreNonconstantAndIrreducible\]. By [@FrohlichShep Theorem 4.5], the set of irreducible elements of $F[y]$ is computable, so since $U(R) = \{1,-1\}$, we may use Proposition \[p:ClassifyPrimesInSubringOfFX\] to conclude that the set of prime elements of $A$ is computable. Finally, by Lemma \[l:DestorySigma1PrimesLemma\], we have that $p_i$ is irreducible in $R$ if and only if $i \notin S$. 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--- abstract: 'In this paper we first demonstrate continuous noisy speech recognition using electroencephalography (EEG) signals on English vocabulary using different types of state of the art end-to-end automatic speech recognition (ASR) models, we further provide results obtained using EEG data recorded under different experimental conditions. We finally demonstrate decoding of speech spectrum from EEG signals using a long short term memory (LSTM) based regression model and Generative Adversarial Network (GAN) based model. Our results demonstrate the feasibility of using EEG signals for continuous noisy speech recognition under different experimental conditions and we provide preliminary results for synthesis of speech from EEG features.' author: - | Gautam Krishna[^1]\ Brain Machine Interface Lab\ The University of Texas at Austin\ ``\ Yan Han[^2]\ Brain Machine Interface Lab\ The University of Texas at Austin\ Co Tran[^3]\ Brain Machine Interface Lab\ The University of Texas at Austin\ Mason Carnahan\ Brain Machine Interface Lab\ The University of Texas at Austin\ Ahmed H Tewfik\ Brain Machine Interface Lab\ The University of Texas at Austin\ bibliography: - 'neurips\_2019.bib' title: 'State-of-the-art Speech Recognition using EEG and Towards Decoding of Speech Spectrum From EEG' --- Introduction ============ Electroencephalography (EEG) is a non-invasive way of measuring electrical activity of human brain. In [@krishna2019speech] authors demonstrated deep learning based automatic speech recognition (ASR) using EEG signals for a limited English vocabulary of four words and five vowels. In [@krishna20] authors demonstrated continuous ASR using the same set of EEG features used in [@krishna2019speech] for larger English vocabulary. The work presented in this paper is different from work presented in reference [@krishna20] as this paper introduces two new sets of EEG features, provide EEG based speech recognition results for additional conditions like listen, listen and spoken. In [@krishna20] authors provided results only for spoken condition. In addition, in this paper we provide speech recognition results using a new end-to-end model called RNN transducer model and also demonstrate preliminary results for speech synthesis using EEG signals. Finally in this paper we provide speech recognition results evaluated on data sets consisting of more number of subjects than the ones used in reference [@krishna20]. Recently in [@anumanchipalli2019speech] researchers demonstrated synthesizing speech from electrocorticography (ECoG) signals recorded for spoken English sentences. ECoG is an invasive technique for measuring electrical activity of human brain. In [@ramsey2017decoding] authors demonstrated speech recognition using ECoG signals. In [@zhao2015classifying] the authors used classification approach for identifying phonological categories in imagined and silent speech. In this paper we demonstrate continuous noisy speech recognition using EEG signals recorded in parallel with speech for spoken English sentences, EEG signals recorded in parallel while the subjects were listening to utterances of the same English sentences and finally we demonstrate speech recognition by concatenating both this sets of EEG features. Inspired from the unique robustness to environmental artifacts exhibited by the human auditory cortex [@yang1991auditory; @mesgarani2011speech] we used EEG data recorded in presence of background noise for this work and demonstrated lower word error rate (WER) for smaller corpus using EEG features. We first conducted speech recognition experiments using the EEG features used by authors in [@krishna2019speech; @krishna20] and we further conducted experiments using two more different feature sets which are more commonly used by neuroscientists studying EEG signals. In this paper we provide comparison of the speech recognition performance results obtained using these different feature sets. EEG has the big advantage of being a non invasive technique compared to ECoG which is an invasive technique, making EEG based brain computer interface (BCI) technology easily deployable and it can be used by subjects without the need of undergoing a neurosurgery to implant ECoG electrodes. We believe speech recognition using EEG will help people with speaking disabilities to use voice activated technologies with better user experience, help with speech restoration and also potentially introduce a new form of thought based communication. Inspired from the results presented in [@anumanchipalli2019speech] we used long short memory (LSTM) [@hochreiter1997long] based regression model, generative adversarial network (GAN) [@goodfellow2014generative], wasserstein generative adversarial networks (WGAN) [@arjovsky2017wasserstein] to decode the Mel-frequency cepstral coefficients (MFCC) features of the audio that the subjects were listening from the EEG signals which were recorded in parallel while they were listening to the audio as well as we decode MFCC features of the sound that the subjects spoke out from the EEG signals which were recorded in parallel with their speech. Automatic Speech Recognition System Models ========================================== In this section we briefly describe the ASR models that were used in this work. We used end to end ASR models which directly maps the EEG features to text. We did experiments using three different types of end to end ASR models, namely: Connectionist Temporal Classification (CTC) model [@graves2006connectionist; @graves2014towards], Attention based RNN encoder decoder model [@cho2014learning; @chorowski2015attention; @bahdanau2014neural] and RNN transducer model [@graves2012sequence; @graves2013speech]. For all the models the number of time steps of the encoder was equal to the product of sampling frequency of EEG features and sequence length. Since different subjects spoke with different rate and listening utterances were of different length, there was no fixed value for the encoder time steps, so we used Tensorflow’s dynamic RNN cell for the encoder. Connectionist Temporal Classification (CTC) ------------------------------------------- In our work we used a single layer gated recurrent unit (GRU) [@chung2014empirical] with 128 hidden units as encoder for the CTC network. The decoder consists of a combination of a dense layer and a softmax activation. The output at every time step of the GRU layer is fed into the decoder network. We used CTC loss function with adam optimizer [@kingma2014adam] and during inference time we used CTC beam search decoder. The mathematical details of CTC loss function computation is covered in [@graves2014towards; @krishna20]. A dynamic algorithm is used to compute the CTC loss. In our work we used character based CTC ASR model and the model was trained for 800 epochs to observe loss convergence. RNN Encoder-Decoder or Attention model -------------------------------------- RNN encoder - decoder ASR model consists of a RNN encoder and a RNN decoder with attention mechanism. We used a single layer GRU with 512 hidden units for both encoder and decoder. A dense layer followed by softmax activation is used after the decoder GRU to get the prediction probabilities. We used cross entropy as loss function with adam as the optimizer. We used teacher forcing algorithm [@williams1989learning] to train the model. The model was trained for 150 epochs to observe loss convergence. During inference time we used beam search decoder. The labels are augmented using two special tokens namely the start token and end token which indicates beginning and end of a sentence. During inference time the label prediction process stops when the end token label is predicted. The mathematical details of the attention mechanism used in our attention model are covered in references [@krishna20; @bahdanau2014neural; @chorowski2015attention]. More specifically we used the exact attention mechanism used by authors in [@krishna20]. RNN Transducer model -------------------- The RNN transducer model consists of an encoder model working in parallel with a prediction network over the output tokens. We used LSTM with 128 hidden units for both our encoder and prediction network. The encoder and prediction network outputs are passed to a joint network which uses tanh activation to compute logits, which are passed to softmax layer to get the prediction probabilities. During inference time, beam search decoder was used. The RNN transducer model was trained for 200 epochs using stochastic gradient descent optimizer to optimize RNN T loss[@graves2012sequence]. We used character based RNN transducer model for this work. More details of RNN transducer model are covered in [@graves2012sequence; @graves2013speech]. Design of Experiments for building the database =============================================== We built three databases for this work. All the subjects who took part in the experiments were healthy UT Austin undergraduate, graduate students in their early twenties for all the databases. For the first database A, 20 subjects took part in the experiment. Out of the 20 subjects, 8 were females and rest were males. Only five out of the 20 subjects were native English speakers. Each one of them was asked to speak the first 9 English sentences from USC-TIMIT database[@narayanan2014real] three times and their simultaneous speech and EEG signals were recorded. The sentences were shown to them on a computer screen. This data was recorded in presence of background noise of 65dB. Music played from our lab computer was used as the source of generating background noise. For the second database B, 15 subjects took part in the experiment. Out of the 15 subjects, three were females and rest were males. Only two out of the 15 subjects were native English speakers. Each one of them was asked to listen to the utterances of the first 9 English sentences from USC-TIMIT database[@narayanan2014real] and then they were asked to speak the utterances that they listened to. Their EEG was recorded in parallel while they were listening to the utterances and also their simultaneous speech and EEG signals were recorded while they were speaking out the utterances that they listened to. This data was recorded in presence of background noise of 50dB. The utterances that the subjects listened to were also recorded. Then the 15 subjects were asked to repeat the same experiment two more times. For the third database C, five female and five male subjects took part in the experiment. Each one of them was asked to read out the first 30 sentences from USC-TIMIT database [@narayanan2014real] and their simultaneous speech and EEG signals were recorded. This data was recorded in absence of external background noise. Then the 10 subjects were asked to repeat the same experiment two more times. Throughout this paper we will refer to the acoustic features for spoken speech as spoken MFCC, acoustic features for the listening utterances that were recorded as listen MFCC, EEG features recorded in parallel with spoken speech as spoken EEG and EEG features recorded in parallel while the subjects were listening to the utterances as listen EEG. We used Brain Vision EEG recording hardware. Our EEG cap had 32 wet EEG electrodes including one electrode as ground. We used EEGLab [@delorme2004eeglab] to obtain the EEG sensor location mapping. It is based on standard 10-20 EEG sensor placement method for 32 electrodes. EEG and Speech feature extraction details ========================================= For preprocessing of EEG we followed the same method as described by the authors in [@krishna2019speech; @krishna20]. EEG signals were sampled at 1000Hz and a fourth order IIR band pass filter with cut off frequencies 0.1Hz and 70Hz was applied. A notch filter with cut off frequency 60 Hz was used to remove the power line noise. EEGlab’s [@delorme2004eeglab] Independent component analysis (ICA) toolbox was used to remove other biological signal artifacts like electrocardiography (ECG), electromyography (EMG), electrooculography (EOG) etc from the EEG signals. We extracted three different EEG feature sets for this work. All EEG features were extracted at a sampling frequency of 100 Hz. The first EEG feature set was same as the ones used by the authors in [@krishna2019speech; @krishna20] namely root mean square, zero crossing rate, moving window average, kurtosis and power spectral entropy with frequency bands value equal to none (power per band was same as the power spectral density). For this set, EEG feature dimension was 31(channels) $\times$ 5 or 155. The second set of features were the magnitudes of short time Fourier Transform of the EEG signals, discrete time wavelet based spectral entropy using approximation and detailed coefficients. We used db4 wavelet. Frequency bands value was kept as none. For every window we extracted only level one coefficients. For this set, EEG feature dimension was 31(channels) $\times$ 3 or 93. The third set of features were power spectral entropy based on delta, theta, alpha and beta EEG frequency bands {0.5,4,7,12,30}Hz, hurst exponent and petrosian fractal dimension. This features are more commonly used by neuroscientists studying EEG signals. For this set, EEG feature dimension was 31(channels) $\times$ 3 or 93. For both listening speech and spoken speech we extracted MFCC 13 features and then we computed first and second order differentials (delta and delta-delta) thus having total MFCC 39 features. The MFCC features were also sampled at 100Hz frequency. For spectral entropy calculations, we used the python neurokit library. For hurst exponent and petrosian fractal dimension calculation we used python pyeeg library. EEG Feature Dimension Reduction Algorithm Details ================================================= We used non linear dimension reduction methods to denoise the EEG feature space. The tool we used for this purpose was Kernel Principle Component Analysis (KPCA) [@mika1999kernel]. We plotted cumulative explained variance versus number of components to identify the right feature dimension for each feature set. We used KPCA with polynomial kernel of degree 3 [@krishna2019speech; @krishna20]. For the first feature set, 155 dimension was reduced to 30. For the second feature set, 93 dimension was reduced to 50. For the third feature set when we plotted explained variance versus number of components we observed that it was best to keep the original dimension. We used python scikit library for performing KPCA. The cumulative explained variance plot is not supported by the library for KPCA as KPCA projects features to different feature space, hence for getting explained variance plot we used normal PCA but after identifying the right dimension we used KPCA to perform dimension reductions. We further computed delta, delta and delta features, thus the final feature dimension of EEG feature set 1 was 90 (30 $\times$ 3), for feature set 2 final dimension was 150 (50 $\times$ 3) and for feature set 3 final dimension was 279 (93 $\times$ 3). The same preprocessing and dimension reduction methodology was followed for both spoken EEG and listen EEG data. Models to predict Listen MFCC from Listen EEG ============================================= In this section we briefly describe the architectures of the deep learning models that we used to predict listen MFCC features from listen EEG features. For this decoding problem we considered feature set 1, feature set 2 and feature set 3 for listen EEG and delta, delta-delta features were not considered, hence the listen MFCC dimension was 13. We tried two different approaches to solve this problem. 1) using a LSTM based regression model and 2) using generative model. For both the approaches, during test time, we used three evaluation metrics: RMSE, Normalized RMSE and Mel cepstral distortion (MCD)between the model output and listen MFCC features from test set. The RMSE values were normalized by dividing the RMSE values with the absolute difference between the maximum and minimum value in the test set observation vector. LSTM based regression model --------------------------- Our LSTM based regression model consists of two layers of LSTM with 128 hidden units in each layer. The final LSTM layer is connected to a time distributed dense layer with 13 hidden units. Root mean squared error was used as the loss function and the model was trained for 200 epochs to observe loss convergence and adam optimizer was used [@kingma2014adam]. 90 % of the data was used to train the model and remaining 10 % was used as test set. Generative Model ---------------- Generative Adversarial Network (GAN) consists of two networks namely the generator model and the discriminator model which are trained simultaneously. The generator model learns to generate data from a latent space and the discriminator model evaluates whether the data generated by the generator is fake or is from true data distribution. The training objective of the generator is to fool the discriminator. Our generator model consists of two layers of LSTM with 128 hidden units in each layer followed by a time distributed dense layer with 13 hidden units. During training, real listen EEG features say with dimension 30 (when used with EEG feature set 1) from training set are fed into the generator model and the generator outputs a vector of dimension 13, which can be considered as fake listen MFCC. This is explained in figure 1. In the figure 1, Fmfcc denotes fake MFCC. The discriminator model consists of two single layered LSTM’s with 128 hidden units connected in parallel. At each training step a pair of inputs are fed into the discriminator. The discriminator takes (real listen EEG features, fake listen MFCC) and (real listen EEG features, real listen MFCC) pairs. The outputs of the two parallel LSTM’s are concatenated and then fed to another LSTM with 128 hidden units. The last time step of the final LSTM is fed into the dense layer with sigmoid activation function. This is explained in figure 2. In order to define the loss functions for both our generator and discriminator model let us first define few terms. Let $P_{e_f}$ be the sigmoid output of the discriminator for (real listen EEG features, fake listen MFCC) input pair and let $P_{e_s}$ be the sigmoid output of the discriminator for (real listen EEG features, real listen MFCC) input pair. Then we can define the loss function of generator as $-\log (P_{e_f})$ and loss function of discriminator as $-\log (P_{e_s}) - \log(1-P_{e_f})$. In order to get better stabilized training, we also tried implementing the same idea using WGAN [@arjovsky2017wasserstein] where the loss function is earth mover’s distance or wasserstein 1 distance instead of the log loss. Both GAN and WGAN models were trained for 500 epochs and adam optimizer was used [@kingma2014adam]. During test time listen EEG features from the test set is fed into the generator and generator generates listen MFCC features. 90 % of the data was used as the training set and remaining as test set. Predicting Spoken MFCC from Spoken EEG ====================================== For predicting spoken MFCC from spoken EEG we used database C, same models used for predicting listen MFCC from listen EEG but we considered only feature set 1 for spoken EEG and delta, delta-delta features were not considered, hence the spoken MFCC dimension was 13 and spoken EEG dimension was 30 for this particular decoding problem. Results ======= The attention model was predicting a word at every time step while the other two types of ASR models were predicting a character at every time step. For CTC and RNN transducer, word based model training was not stable, hence we used only character based model for both CTC and RNN transducer. The performance metric for attention model was word error rate (WER) and character error rate (CER) was the performance metric for both CTC and RNN transducer model. For both attention model and CTC model for both data sets A and B, 80 % of the data was used as training set, 10 % for validation set and remaining 10 % for test set. For RNN transducer model, for data set A data from the first 18 subjects was used as training set and remaining two subjects data as validation and test set respectively. For data set B, data from the first 12 subjects was used as training set, next two subjects data was used as validation set and last subject data was used as test set. We observed that this way of data splitting reduced the over fitting phenomena for all the models. We used Nvidia Quadro GV100 GPU with 32 GB video memory for training all the models. Table 1 shows the results obtained during test time for CTC and attention model when the models were trained using only data set A with feature set 1 EEG features. Table 2 shows the test time results for RNN transducer model for the same data set. Tables 3, 4, 5, 6 and 7 shows the results obtained during test time for CTC and attention model when the models were trained using data set B for different experiments. In general we observed that for all the models that were used to perform speech recognition using EEG features, the error rate during test time went up as the corpus size increase. We believe as the corpus size increase the deep learning models need to be trained with more number of examples to achieve better performance during test time. We also observed that the models gave comparable performance when trained with the EEG feature set 1 and feature set 3. As the corpus increase, training with feature set 2 gave higher error rates compared to other two feature sets, hence we didn’t perform more experiments with feature set 2. Table 5 shows the test time result for feature set 2 training with different models for one type of experiment. We didn’t perform more experiments with the RNN transducer model as it demonstrated poor performance even when tested on smaller corpus for the different feature sets. In [@krishna2019speech] authors demonstrated that EEG sensors T7 and T8 contributed most to the ASR test time accuracy, hence we tried performing ASR experiments using EEG data from only T7, T8 electrodes without performing dimension reduction using data set B with EEG feature set 1 and we observed error rates of 70 % WER for attention model for spoken EEG, 73.3 % WER for attention model for listen EEG, **68** % CER for CTC model for spoken EEG and **66** % CER for CTC model for listen EEG during test time for predicting 9 sentences ( the complete test time corpus). We noticed that the CTC model error rate for listen EEG using T7, T8 sensors data was slightly lower than the error rate obtained when the model was trained using the data from all 31 sensors followed by dimension reduction as seen from table 3. The other error rates obtained after T7, T8 training were comparable (but were not lower) to the results obtained when the models were trained using the data from all 31 sensors followed by dimension reduction for other ASR experiments. For predicting listen MFCC features from listen EEG features, the LSTM regression model demonstrated lowest average normalized RMSE, RMSE and MCD during test time compared to GAN and WGAN models as shown in tables 8,9 and 10. WGAN model demonstrated better test time performance than GAN model when trained using EEG feature set 1 and 2. Also the WGAN model demonstrated better training loss convergence for both generator and discriminator models compared to the GAN’s generator and discriminator models for all the EEG feature sets. For predicting spoken MFCC features from spoken EEG features, the LSTM regression model again demonstrated lowest average normalized RMSE, RMSE and MCD during test time compared to GAN and WGAN models as shown in table 11. We computed WER values for CTC and RNN transducer model for some experiments, in our opinion since CTC and RNN-T models were predicting characters at every time step, CER is a better performance metric than WER for those models. For data set B, for listen condition the CTC model gave WER values 52.6 %, 87.09 %, 88.88 % and 94.9 % for number of sentences = {3,5,7,9} respectively using EEG feature set 1 and for spoken condition the same model gave WER values 73.6 %, 83.8 %, 91.1% and 91.5 % respectively using the same EEG feature set 1. For data set B, for listen condition the RNN-T model gave WER values 92.98 %, 69.89 %, 70.37 % and 92.66 % for number of sentences = {3,5,7,9} respectively using EEG feature set 1 and for spoken condition the same model gave WER values 64.91 %, 82.8 %, 79.26% and 93.79 % respectively using EEG feature set 2. --- --------- ---- ------ ------ 3 [ 19]{} 19 32.5 0 5 [ 20]{} 29 54 54.8 7 22 42 67.5 64.4 9 [ 23]{} 55 69 60 --- --------- ---- ------ ------ : Results for data set A with feature set 1 --- --------- ------- 3 [ 19]{} 41.09 5 [ 20]{} 60.34 7 22 66.22 9 [ 23]{} 64.26 --- --------- ------- : Results for RNN Transducer model on data set A with feature set 1 --- --------- ---- ---- ------ 3 [ 19]{} 19 59 0 5 [ 20]{} 29 65 48 7 22 42 70 64.4 9 [ 23]{} 55 73 68.3 --- --------- ---- ---- ------ : Results for data set B with feature set 1 --- --------- ---- ---- ------ 3 [ 19]{} 19 36 0 5 [ 20]{} 29 65 54 7 22 42 68 66.6 9 [ 23]{} 55 68 61.6 --- --------- ---- ---- ------ : Results for data set B with feature set 1 --- --------- ---- ------ ---- 3 [ 19]{} 19 27.5 0 5 [ 20]{} 29 57.3 45 7 22 42 67 57 9 [ 23]{} 55 72 70 --- --------- ---- ------ ---- : Results for data set B with feature set 2 --- --------- ---- ------ ---- 3 [ 19]{} 19 48.3 0 5 [ 20]{} 29 63.1 41 7 22 42 62 51 9 [ 23]{} 55 61 66 --- --------- ---- ------ ---- : Results for data set B with feature set 3 (Listen) --- --------- ---- ---- ---- 3 [ 19]{} 19 40 0 5 [ 20]{} 29 54 32 7 22 42 62 62 9 [ 23]{} 55 63 68 --- --------- ---- ---- ---- : Results for data set B with feature set 3 (Spoken) [llll]{} **Model** & & &\ GAN & 0.216 & 63.25 & 10.926\ WGAN & 0.201 & 60.45 & 10.356\ ------------ LSTM Regression ------------ : Results for predicting listen MFCC from listen EEG feature set 1 & 0.0291 & 6.45 & 1.413\ [llll]{} **Model** & & &\ GAN & 0.209 & 64.018 & 10.818\ WGAN & 0.2 & 61.21 & 10.336\ ------------ LSTM Regression ------------ : Results for predicting listen MFCC from listen EEG feature set 2 & 0.0266 & 7.954 & 1.34\ [llll]{} **Model** & & &\ GAN & 0.208 & 63.12 & 10.748\ WGAN & 0.209 & 63.27 & 10.578\ ------------ LSTM Regression ------------ : Results for predicting listen MFCC from listen EEG feature set 3 & 0.0289 & 8.526 & 1.444\ [llll]{} **Model** & & &\ GAN & 0.193 & 73.77 & 13.249\ WGAN & 0.188 & 72.149 & 12.953\ ------------ LSTM Regression ------------ : Results for predicting spoken MFCC from spoken EEG & 0.126 & 48.449 & 5.737\ Discussion ========== From neuroscience perspective, listen EEG can be considered as the brain activity of a subject while hearing and processing intend to speak. Intend to speak includes the brain activity of the subjects responsible for recognizing phonemes present, words present, meaning, start and end of the utterances that they were hearing. Spoken EEG can be considered as the brain activity of a subject while articulating the speech. Concatenation of listen and spoken EEG can be considered as mixture of brain activity involving hearing, intend to speak and articulation. We believe speech recognition using listen EEG will help with speech restoration for people who can not speak at all and speech recognition using spoken EEG will help with improving quality of speech for people with speaking difficulties like broken speech. Speech recognition using concatenation of listen and spoken EEG is an interesting area which need further exploration. We believe it will help with speech restoration for people suffering from mild to severe speaking disabilities. Our work provides only preliminary results for speech recognition using EEG. Conclusion and Future work ========================== In this paper we demonstrated continuous noisy speech recognition using different EEG feature sets and we demonstrated LSTM based regression method, GAN model to predict listen acoustic features from listen EEG features, to predict spoken acoustic features from spoken EEG features with very low normalized RMSE during test time. We observed that for speech recognition using EEG, we were able to achieve low error rates during test time for smaller corpus size and error rates went up as we increased the corpus size. We further observed that attention model and CTC model demonstrated better performance than RNN transducer model for speech recognition using EEG. We observed that for predicting MFCC features from EEG features, LSTM based regression model demonstrated better test time performance than GAN and WGAN model. We believe the speech recognition results can be improved by training the models with more number of examples. Another possible reason for low speech recognition accuracy might be the nature of the data set used. Our data set had a mix of non native and native English speakers with majority of the subjects being non native English speakers for both the data bases. For future work we would like to conduct experiments with equal number of training examples from both native and non native speakers and investigate whether it will help in improving ASR test time performance. For the speech synthesis problem, griffin lim reconstruction [@griffin1984signal] algorithm can be used to convert the predicted listen mfcc, spoken mfcc features to interpretable audio. As seen from our results, we observed high MCD values, we believe to reduce the test time MCD values, the models need to be trained with much larger data set. Also we observed that both GAN and WGAN models were difficult to train compared to the simple LSTM regression model and the training loss showed lot of fluctuations in convergence rate for both GAN and WGAN models. Our initial hypothesis was that both GAN and WGAN should demonstrate better results than LSTM regression as the GAN model learns the loss function compared to a fixed loss function in the case of LSTM regression but we observed LSTM regression model demonstrating better test time results than both GAN and WGAN models. We believe GAN results can be improved by pre training the generator, adding regularization terms to the GAN loss function etc. This will be considered for our future work. In this work we provide only preliminary results for speech synthesis using EEG. For future work we would also like to build a much larger speech EEG corpus and train the model with more examples, include implicit language model during training, include an external language model during inference time and see if our results can be improved. We would also like to see if the results can be improved by concatenating all the three EEG feature sets. For future work, we would also like to perform EEG based speech recognition and speech synthesis experiments per each subject, in that case we would need to collect lot of speech EEG data per each subject. We plan to publish the speech EEG databases used in this work to help advancement of research in this area. ### Acknowledgments {#acknowledgments .unnumbered} We would like to thank Kerry Loader and Rezwanul Kabir from Dell, Austin, TX for donating us the GPU to train the models used in this work. The first author would like to thank Satyanarayana Vusirikala from CS department, UT Austin for his helpful discussion on generative adversarial networks. ### A Additional Figures and Tables {#a-additional-figures-and-tables .unnumbered} Tables 12, 13 and 14 shows the test time results for RNN transducer model for data set B for different experimental conditions ( Spoken, Listen, concatenation of spoken and listen). Table 15 shows the test time result for CTC and attention model for data set B with EEG feature set 2. --- --------- ------- 3 [ 19]{} 58.91 5 [ 20]{} 62.03 7 22 64.43 9 23 69.30 --- --------- ------- : Results for RNN Transducer model on data set B with feature set 1 (Listen) --- --------- ------- 3 [ 19]{} 65.08 5 [ 20]{} 56.05 7 22 73.21 9 23 71.03 --- --------- ------- : Results for RNN Transducer model on data set B with feature set 1 (Spoken) --- --------- ------- 3 [ 19]{} 65.08 5 [ 20]{} 65.82 7 22 75.45 9 23 75.83 --- --------- ------- : Results for RNN Transducer model on data set B with feature set 1 (Spoken + Listen) --- --------- ---- ------ ------ 3 [ 19]{} 19 60.5 31.5 5 [ 20]{} 29 63.2 41.9 7 22 42 71.9 57.7 9 [ 23]{} 55 72.3 63.3 --- --------- ---- ------ ------ : Result for Data set B with feature set 2 (Listen) EEG sensor placement ==================== Figure 3 shows the EEG sensor location mapping obtained using EEG lab tool box. Figures 4 and 5 shows the three dimensional view of the EEG sensor locations. Training loss convergence plots =============================== Figure 6 shows the training loss convergence for attention model when trained on data set A using EEG feature set 1 for number of sentences equal to three. Figure 7 shows the training loss convergence for CTC model when trained on data set A using EEG feature set 1 for number of sentences equal to three. We observed that the training loss converged for all the models for all the experiments. Cumulative explained variance plots =================================== Figure 8 shows the cumulative explained variance plot for EEG feature set 1. From the plot it is clear that the optimal feature dimension required to represent the complete feature set is 30. Figure 9 shows the cumulative explained variance plot for EEG feature set 3. From the plot it is clear that the explained variance is not converging to an optimal value, hence we didn’t perform dimension reduction for this feature set. So the EEG feature dimension for feature set 3 was 93. Figure 10 shows the cumulative explained variance plot for EEG feature set 2. In this case it is very difficult to figure out the optimal feature dimension required to represent the complete feature set as the convergence point is not clearly observable. So we used a development set to figure out the optimal feature dimension as 50 for EEG feature set 2. Additional results for predicting listen MFCC from listen EEG ============================================================= Figures 11 and 12 shows the training loss plot for WGAN and Figures 13 and 14 shows the training loss plot for GAN for feature set 1. As seen from the figures, WGAN generator and discriminator models showed better training loss convergence than GAN. Figure 15 shows the training loss convergence for the LSTM regression model for feature set 1. Figures 16, 17 and 18 shows the test time performance of WGAN, GAN and LSTM regression models respectively for predicting listen MFCC features from listen EEG features using feature set 1. The plots shows the normalized RMSE values per each test set sample. Figures 19, 20 and 21 shows the test time performance of WGAN, GAN and LSTM regression models respectively for predicting listen MFCC features from listen EEG features using feature set 2. Figures 22, 23 and 24 shows the test time performance of WGAN, GAN and LSTM regression models respectively for predicting listen MFCC features from listen EEG features using feature set 3. Figures 25, 26 and 27 shows the test time performance of WGAN, GAN and LSTM regression models respectively for predicting spoken MFCC from spoken EEG. [^1]: Equal author contribution [^2]: Equal author contribution [^3]: Equal author contribution

--- abstract: 'The existence of the five-quark Fock states for the intrinsic charm quark in the nucleons was suggested some time ago, but conclusive evidence is still lacking. We generalize the previous theoretical approach to the light-quark sector and study possible experimental signatures for such five-quark states. In particular, we compare the $\bar d - \bar u$ and $\bar u + \bar d - s -\bar s$ data with the calculations based on the five-quark Fock states. The qualitative agreement between the data and the calculations is interpreted as evidence for the existence of the intrinsic light-quark sea in the nucleons. The probabilities for the $|uudu\bar{u}\rangle$ and $|uudd\bar{d}\rangle$ Fock states are also extracted.' author: - 'Wen-Chen Chang' - 'Jen-Chieh Peng' title: ' Flavor Asymmetry of the Nucleon Sea and the Five-Quark Components of the Nucleons' --- The possible existence of a significant $u u d c \bar c$ five-quark Fock component in the proton was proposed some time ago by Brodsky, Hoyer, Peterson, and Sakai (BHPS) [@brodsky80] to explain the unexpectedly large production rates of charmed hadrons at large forward $x_F$ region. In the light-cone Fock space framework, the probability distribution of the momentum fraction (Bjorken-$x$) for this nonperturbative “intrinsic" charm (IC) component was obtained [@brodsky80]. The intrinsic charm originating from the five-quark Fock state is to be distinguished from the “extrinsic" charm produced in the splitting of gluons into $c \bar c$ pairs, which is well described by QCD. The extrinsic charm has a “sea-like" characteristics with large magnitude only at the small $x$ region. In contrast, the intrinsic charm is “valence-like" with a distribution peaking at larger $x$. The presence of the intrinsic charm component can lead to a sizable charm production at the forward rapidity ($x_F$) region. The $x$ distribution of the intrinsic charm in the BHPS model was derived with some simplifying assumptions. Recently, Pumplin [@pumplin06] showed that a variety of light-cone models in which these assumptions are removed would still predict the $x$ distributions of the intrinsic charm similar to that of the BHPS model. The CTEQ collaboration [@pumplin06] has also examined all relevant hard-scattering data sensitive to the presence of the IC component, and concluded that the existing data are consistent with a wide range of the IC magnitude, from null to 2-3 times larger than the estimate by the BHPS model. This result shows that the experimental data are not yet sufficiently accurate to determine the magnitude or the $x$ distribution of the IC. In an attempt to further study the role of five-quark Fock states for intrinsic quark distributions in the nucleons, we have extended the BHPS model to the light quark sector and compared the predictions with the experimental data. The BHPS model predicts the probability for the $u u d Q \bar Q$ five-quark Fock state to be approximately proportional to $1/m_Q^2$, where $m_Q$ is the mass of the quark $Q$ [@brodsky80]. Therefore, the light five-quark states $u u d u \bar u$ and $u u d d \bar d$ are expected to have significantly larger probabilities than the $u u d c \bar c$ state. This suggests that the light quark sector could potentially provide more clear evidence for the roles of the five-quark Fock states, allowing the specific predictions of the BHPS model, such as the shape of the quark $x$ distributions originating from the five-quark configuration, to be tested. To compare the experimental data with the prediction based on the intrinsic five-quark Fock state, it is essential to separate the contributions of the intrinsic quark and the extrinsic one. Fortunately, there exist some experimental observables which are free from the contributions of the extrinsic quarks. As discussed later, the $\bar d - \bar u$ and the $\bar u + \bar d - s - \bar s$ are examples of quantities independent of the contributions from extrinsic quarks. The $x$ distribution of $\bar d - \bar u$ has been measured in a Drell-Yan experiment [@e866]. A recent measurement of $s + \bar s$ in a semi-inclusive deep-inelastic scattering (DIS) experiment [@hermes] also allowed the determination of the $x$ distribution of $\bar u + \bar d - s - \bar s$. In this paper, we compare these data with the calculations based on the intrinsic five-quark Fock states. The qualitative agreement between the data and the calculations provides evidence for the existence of the intrinsic light-quark sea in the nucleons. For a $|u u d Q \bar Q\rangle$ proton Fock state, the probability for quark $i$ to carry a momentum fraction $x_i$ is given in the BHPS model [@brodsky80] as $$P(x_1, ...,x_5)=N_5\delta(1-\sum_{i=1}^5x_i)[m_p^2-\sum_{i=1}^5\frac{m_i^2}{x_i}]^{-2}, \label{eq:prob5q_a}$$ where the delta function ensures momentum conservation. $N_5$ is the normalization factor for five-quark Fock state, and $m_i$ is the mass of quark $i$. In the limit of $m_{4,5} >> m_p, m_{1,2,3}$, where $m_p$ is the proton mass, Eq. \[eq:prob5q\_a\] becomes $$P(x_1, ...,x_5)=\tilde{N}_5\frac{x_4^2x_5^2}{(x_4+x_5)^2} \delta(1-\sum_{i=1}^5 x_i), \label{eq:prob5q_b}$$ where $\tilde{N}_5 = N_5/m_{4,5}^4$. Eq. \[eq:prob5q\_b\] can be readily integrated over $x_1$, $x_2$, $x_3$ and $x_4$, and the heavy-quark $x$ distribution [@brodsky80; @pumplin06] is: $$\begin{aligned} P(x_5)=\frac{1}{2} \tilde{N}_5 x_5^2[\frac{1}{3} (1-x_5) (1+10x_5+x_5^2) \nonumber \\ -2x_5(1+x_5)\ln (1/x_5)]. \label{eq:prob5q_d}\end{aligned}$$ One can integrate Eq. \[eq:prob5q\_d\] over $x_5$ and obtain the result ${\cal P}^{c \bar c}_5 = \tilde{N}_5/3600$, where ${\cal P}^{c \bar c}_5$ is the probability for the $|u u d c \bar c\rangle$ five-quark Fock state. An estimate of the magnitude of ${\cal P}^{c \bar c}_5$ was given by Brodsky et al. [@brodsky80] as $\approx 0.01$, based on diffractive production of $\Lambda_c$. This value is consistent with a bag-model estimate [@donoghue77]. ![The $x$ distributions of the intrinsic $\bar Q$ in the $u u d Q \bar Q$ configuration of the proton from the BHPS model [@brodsky80]. The solid curve is plotted using the expression in Eq. \[eq:prob5q\_d\] for $\bar c$. The other three curves, corresponding to $\bar c$, $\bar s$, and $\bar d$ in the five-quark configurations, are obtained by solving Eq. \[eq:prob5q\_a\] numerically. The same probability ${\cal P}^{Q \bar Q}_5$ (${\cal P}^{Q \bar Q}_5= 0.01$) is used for the three different five-quark states.[]{data-label="fig_5q_c_s_d"}](fig_5q){width="50.00000%"} The solid curve in Fig. \[fig\_5q\_c\_s\_d\] shows the $x$ distribution for the charm quark ($P(x_5)$) using Eq. \[eq:prob5q\_d\], assuming ${\cal P}^{c \bar c}_5 = 0.01$. Since this analytical expression was obtained for the limiting case of infinite charm-quark mass, it is of interest to compare this result with calculations without such an assumption. To this end, we have developed the algorithm to calculate the quark distributions using Eq. \[eq:prob5q\_a\] with Monte-Carlo techniques. The five-quark configuration of $\{x_1,...,x_5\}$ satisfying the constraint of Eq. \[eq:prob5q\_a\] is randomly sampled. The probability distribution $P(x_i)$ can be obtained numerically with an accumulation of sufficient statistics. We first verified that the Monte-Carlo calculations in the limit of very heavy charm quarks reproduce the analytical result for $P(x_5)$ in Eq. \[eq:prob5q\_d\]. We then calculated $P(x_5)$ using $m_u = m_d = 0.3$ GeV/$c^2$, $m_c = 1.5$ GeV/$c^2$, and $m_p = 0.938$ GeV/$c^2$, and the result is shown as the dashed curve in Fig. \[fig\_5q\_c\_s\_d\]. The similarity between the solid and dashed curves shows that the assumption adopted for deriving Eq. \[eq:prob5q\_d\] is adequate. It is important to note that the Monte-Carlo technique allows us to calculate the quark $x$ distributions for other five-quark configurations when $Q$ is the lighter $u$, $d$, or $s$ quark, for which one could no longer assume a large mass. As mentioned above, the insufficient accuracy of existing data as well as the inherently small probability for intrinsic charm due to the large charm-quark mass make it difficult to confirm the existence of the intrinsic charm component in the proton. On the other hand the five-quark states involving only lighter quarks, such as $|u u d u \bar u\rangle$, $|u u d d \bar d\rangle$, and $|u u d s \bar s\rangle$, might be more easily observed experimentally. We have calculated the $x$ distributions of the $\bar s$ and $\bar d$ quarks in the BHPS model for the $|u u d s \bar s\rangle$ and $|u u d d \bar d\rangle$ configurations, respectively, using Eq. \[eq:prob5q\_a\]. The mass of the strange quark is chosen as 0.5 GeV/c$^2$. In Fig. \[fig\_5q\_c\_s\_d\], we show the $x$ distributions of $\bar s$ and $\bar d$, together with that of $\bar c$. In order to focus on the different shapes of the $x$ distributions, the same value of ${\cal P}^{Q \bar Q}_5$ is assumed for these different five-quark states. Figure \[fig\_5q\_c\_s\_d\] shows that the $x$ distributions of the intrinsic $\bar Q$ shift progressively to lower $x$ region as the mass of the quark $Q$ decreases. The $x$ distributions of $\bar Q$ originating from the gluon splitting into quark-antiquark pair ($g \to Q \bar Q$) QCD processes are localized at the low-$x$ region. Figure \[fig\_5q\_c\_s\_d\] illustrates an important advantage for identifying the IC component, namely, the intrinsic charm component is better separated from the extrinsic charm component as a result of their different $x$ distributions. Nevertheless, the probability for intrinsic lighter quarks are expected to be significantly larger than for the heavier charm quark. The challenge is to identify proper experimental observables which allow a clear separation of the intrinsic light quark component from the extrinsic QCD component. As we discuss next, the quantities $\bar d(x) - \bar u(x)$ and $\bar u(x) + \bar d(x) - s(x) - \bar s(x)$ are suitable for studying the intrinsic light-quark components of the proton. ![Comparison of the $\bar d(x) - \bar u(x)$ data with the calculations based on the BHPS model. The dashed curve corresponds to the calculation using Eq. \[eq:prob5q\_a\] and Eq. \[eq:intdbarubar2\], and the solid and dotted curves are obtained by evolving the BHPS result to $Q^2 = 54.0$ GeV$^2$ using $\mu = 0.5$ GeV and $\mu = 0.3$ GeV, respectively.[]{data-label="fig_dbar-ubar"}](fig_dbarubar){width="50.00000%"} The first evidence for an asymmetric $\bar u$ and $\bar d$ distribution came from the observation [@amaudruz91] that the Gottfried Sum Rule [@gottfried67] was violated. The striking difference between the $\bar d$ and $\bar u$ distributions was clearly observed subsequently in the proton-induced Drell-Yan [@na51; @e866] and semi-inclusive DIS experiments [@hermes_sidis]. This large flavor asymmetry was in qualitative agreement with the meson cloud model which incorporates chiral symmetry [@MCM1]. Reviews on this subject can be found in Refs. [@tony; @kumano; @garvey]. The $\bar d(x) - \bar u(x)$ data from the Fermilab E866 Drell-Yan experiment at the $Q^2$ scale of 54 GeV$^2$ [@e866] is shown in Fig. \[fig\_dbar-ubar\]. The $\bar d(x) - \bar u(x)$ distribution is of particular interest for testing the intrinsic light-quark contents in the proton, since the perturbative $g \to Q \bar Q$ processes are expected to generate $u \bar u$ and $d \bar d$ pairs with equal probabilities and thus have no contribution to this quantity. In the BHPS model, the $\bar u$ and $\bar d$ are predicted to have the same $x$ dependence if $m_u = m_d$. It is important to note that the probabilities of the $|u u d d \bar d\rangle$ and $|u u d u \bar u\rangle$ configurations, ${\cal P}^{u \bar u}_5$ and ${\cal P}^{d \bar d}_5$, are not known from the BHPS model, and remain to be determined from the experiments. Non-perturbative effects such as Pauli-blocking [@feynman] could lead to different probabilities for the $|u u d d \bar d\rangle$ and $|u u d u \bar u\rangle$ configurations. Nevertheless the shape of the $\bar d(x) - \bar u(x)$ distribution shall be identical to those of $\bar d(x)$ and $\bar u(x)$ in the BHPS model. Moreover, the normalization of $\bar d(x) - \bar u(x)$ is already known from the Fermilab E866 Drell-Yan experiment as $$\int^{1}_{0} (\bar d(x) - \bar u(x)) dx = 0.118 \pm 0.012 . \label{eq:intdbarubar1}$$ This allows us to compare the $\bar d(x) - \bar u(x)$ data with the calculations from the BHPS model, since the above integral is simply equal to ${\cal P}^{d \bar d}_5 - {\cal P}^{u \bar u}_5$, i.e. $$\int^{1}_{0} (\bar d(x) - \bar u(x)) dx = {\cal P}^{d \bar d}_5 - {\cal P}^{u \bar u}_5 = 0.118 \pm 0.012 . \label{eq:intdbarubar2}$$ Figure \[fig\_dbar-ubar\] shows the calculation of the $\bar d(x) - \bar u(x)$ distribution (dashed curve) from the BHPS model, together with the data. The $x$-dependence of the $\bar d(x) - \bar u(x)$ data is not in good agreement with the calculation. It is important to note that the $\bar d(x) - \bar u(x)$ data in Fig. \[fig\_dbar-ubar\] were obtained at a rather large $Q^2$ of 54 GeV$^2$ [@e866]. In contrast, the relevant scale, $\mu^2$, for the five-quark Fock states is expected to be much lower, around the confinement scale. This suggests that the apparent discrepancy between the data and the BHPS model calculation in Fig. \[fig\_dbar-ubar\] could be partially due to the scale dependence of $\bar d(x) - \bar u(x)$. We adopt the value of $\mu = 0.5$ GeV, which was chosen by Glück, Reya, and Vogt [@grv] in their attempt to generate gluon and quark distributions in the so-called “dynamical approach" starting with only valence-like distributions at the initial $\mu^2$ scale and relying on evolution to generate the distributions at higher $Q^2$. We have evolved the predicted $\bar d(x) - \bar u(x)$ distribution from $Q_0^2 = \mu^2 =0.25$ GeV$^2$ to $Q^2 = 54$ GeV$^2$. Since $\bar d(x) - \bar u(x)$ is a flavor non-singlet parton distribution, its evolution from $Q_0$ to $Q$ only depends on the values of $\bar d(x) - \bar u(x)$ at $Q_0$, and is independent of any other parton distributions. The solid curve in Fig. \[fig\_dbar-ubar\] corresponds to $\bar d(x) - \bar u(x)$ from the BHPS model evolved to $Q^2=$ 54 GeV$^2$. Significantly improved agreement with the data is now obtained. This shows that the $x$-dependence of $\bar d(x) -\bar u(x)$ is quite well described by the five-quark Fock states in the BHPS model provided that the $Q^2$-evolution is taken into consideration. It is interesting to note that an excellent fit to the data can be obtained if $\mu = 0.3$ GeV is chosen (dotted curve in Fig. \[fig\_dbar-ubar\]) rather than the more conventional value of $\mu = 0.5$ GeV. We have also found good agreement between the HERMES $\bar d(x) - \bar u(x)$ data at $Q^2 = 2.3 GeV^2$ [@hermes_sidis] with calculation using the BHPS model. ![Comparison of the $x(\bar d(x) + \bar u(x) - s(x) - \bar s(x))$ data with the calculations based on the BHPS model. The dashed curve corresponds to the calculation using Eq. \[eq:prob5q\_a\], and the solid and dotted curves are obtained by evolving the BHPS result to $Q^2 = 2.5$ GeV$^2$ using $\mu = 0.5$ GeV and $\mu = 0.3$ GeV, respectively.[]{data-label="fig_sbar-dubar"}](fig_deltas){width="50.00000%"} We now consider the quantity $\bar u(x) + \bar d(x) - s(x) - \bar s(x)$. New measurements of charged kaon production in semi-inclusive DIS by the HERMES collaboration [@hermes] allow the extraction of $x(s(x) + \bar s(x))$ at $Q^2 = 2.5$ GeV$^2$. Combining this result with the $x(\bar d(x) + \bar u(x))$ distributions determined by the CTEQ group (CTEQ6.6) [@cteq], the quantity $x(\bar u(x) + \bar d(x) - s(x) - \bar s(x))$ can be obtained and is shown in Fig. \[fig\_sbar-dubar\]. This approach for determining $x(\bar u(x) + \bar d(x) - s(x) - \bar s(x))$ is identical to that used by Chen, Cao, and Signal in their recent study [@signal] of strange quark sea in the meson-cloud model [@MCM2]. An interesting property of $\bar u + \bar d - s - \bar s$ is that the contribution from the extrinsic sea vanishes, just like the case for $\bar d - \bar u$. Therefore, this quantity is only sensitive to the intrinsic sea and can be compared with the calculation of the intrinsic sea in the BHPS model. We have $$\begin{aligned} \bar u(x) + \bar d(x) - s(x) - \bar s(x) = \nonumber \\ P^{u \bar u}(x_{\bar u}) + P^{d \bar d}(x_{\bar d}) - 2 P^{s \bar s}(x_{\bar s}), \label{eq:udssbar_p5}\end{aligned}$$ where $P^{Q \bar Q}(x_{\bar Q})$ is the $x$-distribution for $\bar Q$ in the $|u u d Q \bar Q\rangle$ Fock state. Although the shapes of the intrinsic $\bar u, \bar d, s, \bar s$ distributions can be readily calculated from the BHPS model, the relative magnitude of the intrinsic strange sea versus intrinsic non-strange sea is unknown. We have adopted the assumption that the probability of the intrinsic sea is proportional to $1/m_Q^2$, as stated earlier. This implies that ${\cal P}^{s \bar s}_5/(\frac{1}{2}({\cal P}^{u \bar u}_5 + {\cal P}^{d \bar d}_5)) = {m_{\bar u}^2}/{m_{\bar s}^2} \approx 0.36$ for $m_{\bar u} = 0.3$ GeV/c$^2$ and $m_{\bar s} = 0.5$ GeV/c$^2$. With this assumption, we can now compare the $x(\bar u(x) + \bar d(x) - s(x) - \bar s(x))$ data with the calculation using the BHPS model, shown as the dashed curve in Fig. \[fig\_sbar-dubar\]. The prediction of the BHPS model is found to be shifted to larger $x$ relative to the data. This apparent discrepancy could again partially reflect the different scales of the theory and the data. Since $\bar u + \bar d - s - \bar s$ is a flavor non-singlet quantity, we can readily evolve the BHPS prediction to $Q^2 =2.5$ GeV$^2$ using $Q_0 = \mu = 0.5$ GeV and the result is shown as the solid curve in Fig. \[fig\_sbar-dubar\]. Better agreement between the data and the calculation is achieved after the scale dependence is taken into account. It is interesting to note that a better fit to the data can again be obtained with $\mu = 0.3$ GeV, shown as the dotted curve in Fig. \[fig\_sbar-dubar\]. From the comparison between the data and the BHPS calculation using $\mu = 0.5$ GeV in Fig. \[fig\_sbar-dubar\], one can determine the sum of the probabilities for the $|u u d u \bar u\rangle$ and $|u u d d \bar d\rangle$ configurations, $\Sigma {\cal P}^{\bar d \bar u}_5$ ($= {\cal P}^{d \bar d}_5 + {\cal P}^{u \bar u}_5)$. We found that $\Sigma {\cal P}^{\bar d \bar u}_5 = 0.471$. Together with Eq. \[eq:intdbarubar2\], we have $${\cal P}^{u \bar u}_5 = 0.176;~~~~~{\cal P}^{d \bar d}_5 = 0.294. \label{eq:ud_value}$$ It is remarkable that the $\bar d(x) - \bar u(x)$ and the $\bar d(x) + \bar u(x) - s(x) - \bar s(x)$ data not only allow us to check the predicted $x$-dependence of the five-quark $|u u d u \bar u\rangle$ and $|u u d d \bar d\rangle$ Fock states, but also provide a determination of the probabilities for these two states. As expected, the extracted values for the five-quark Fock states probabilities in Eq. \[eq:ud\_value\] depends on the assumption for the probability of the $|uuds \bar s\rangle$. For the limiting case of ${\cal P}^{s \bar s}_5=0$, we obtain ${\cal P}^{u \bar u}_5 = 0.097$ and ${\cal P}^{d \bar d}_5 = 0.215$, which reflect the range of uncertainty of the extracted values. It is interesting to note that values obtained in Eq. \[eq:ud\_value\] are consistent with the $1/m_Q^2$ assumption for the probability of the $|u u d Q \bar Q\rangle$ Fock state. If one uses the bag model estimate of ${\cal P}^{c \bar c}_5\sim 0.01$ [@donoghue77], the $1/m_Q^2$ dependence would then imply that ${\cal P}^{d \bar d}_5$ to be $\sim 0.01 (m_c^2/m_d^2) \sim 0.25$, consistent with the results of Eq. \[eq:ud\_value\]. In conclusion, we have generalized the existing BHPS model to the light-quark sector and compared the calculation with the $\bar d - \bar u$ and $\bar u + \bar d - s - \bar s$ data. 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--- abstract: 'We use models of thermal evolution and XUV-driven mass loss to explore the composition and history of low-mass low-density transiting planets. We investigate the Kepler-11 system in detail and provide estimates of both the current and past planetary compositions. We find that a H/He envelope on Kepler-11b is highly vulnerable to mass loss. By comparing to formation models, we show that in situ formation of the system is extremely difficult. Instead we propose that it is a water-rich system of sub-Neptunes that migrated from beyond the snow line. For the broader population of observed planets, we show that there is a threshold in bulk planet density and incident flux above which no low-mass transiting planets have been observed. We suggest that this threshold is due to the instability of H/He envelopes to XUV-driven mass loss. Importantly, we find that this mass loss threshold is well reproduced by our thermal evolution/contraction models that incorporate a standard mass loss prescription. Treating the planets’ contraction history is essential because the planets have significantly larger radii during the early era of high XUV fluxes. Over time low mass planets with H/He envelopes can be transformed into water-dominated worlds with steam envelopes or rocky super-Earths. Finally, we use this threshold to provide likely minimum masses and radial velocity amplitudes for the general population of *Kepler* candidates. Likewise, we use this threshold to provide constraints on the maximum radii of low-mass planets found by radial velocity surveys.' author: - 'Eric D. Lopez' - 'Jonathan J. Fortney$^1$' - Neil Miller bibliography: - 'myreferences.bib' title: 'How Thermal Evolution and Mass Loss Sculpt Populations of Super-Earths and Sub-Neptunes: Application to the Kepler-11 System and Beyond' --- Introduction ============ In recent years, the frontier of the search for extrasolar planets has pushed towards ever smaller and more Earth-like worlds. We now know of dozens of Neptune mass planets and have even found the first definitively rocky extrasolar planets [Batalha2011,Leger2009]{}. In between, transit searches have begun finding a population of low-mass low-density “super-Earths”. Beginning with the discovery of GJ1214b [Charbonneau2009]{}, these planets represent a new class of exoplanets that do not have any analog in our Solar System. Basic questions about their composition, structure, and formation are still unknown. Are these, in fact, scaled up versions of the Earth that simply have thick hydrogen/helium envelopes atop of rock/iron cores? Or are they instead scaled down versions of Neptune that are rich in water and other volatile ices? The distinction between water-poor super-Earths or water-rich sub-Neptunes has fundamental implications for how these planets formed. So far these low-mass low-density (hereafter LMLD) planets have only been found well inside the snow-line. If these planets only contain rock, iron, and hydrogen/helium, then it is possible they formed close to their current orbits [Hansen2011]{}. However, if a significant fraction of their mass is in water, then they must have formed beyond the snow-line and migrated in to their current locations [Alibert2011,Ida2010,Rogers2011]{}. The Kepler-11 system [Lissauer2011a]{} is an extremely powerful tool for exploring the features of LMLD planets. With six transiting planets orbiting a close solar analog, it is the richest extrasolar system currently known. Moreover, five of the planets have masses from Transit Timing Variations (TTVs), and all five of these fall into the low-mass low-density regime in between Earth and Neptune. These five planets are all interior to Mercury’s orbit, with periods from 10 to 47 days. This provides a unique laboratory to test the possible composition, formation, and evolution of LMLD planets and how these vary as a function of both period and planet mass. Transiting planets with measured masses, like those in Kepler-11, are particularly valuable because we can determine their mean density. All the planets in Kepler-11 have densities too low for pure rock, and therefore must have some sort of thick envelope of volatiles. Likewise, all the planets except Kepler-11b are less dense than pure water and so must have at least some hydrogen/helium. Unfortunately, mass and radius alone cannot uniquely determine a planet’s composition. In general, there is a large degeneracy between the relative amounts of rock, iron, water, and hydrogen/helium [Rogers2010a]{}. This problem is particularly acute for planets with radii $\approx 2-4 \: R_{\mathrm{\oplus}}$, since in this range any of these four constituents can be important. Indeed these sorts of degeneracies have long been a focus of studies of Uranus and Neptune [Hubbard1991,Fortney2011a]{}. One possible solution to the composition problem is to obtain multi-wavelength transmission spectra, as has been done for GJ1214b [Bean2011, Desert2011, Croll2011]{}. Since hydrogen-rich atmospheres have much larger scale heights at a given temperature, near infrared water and methane absorption features will be much more prominent for planets with hydrogen/helium envelopes [Kempton2009,Kempton2010]{}. Unfortunately, these observations are extremely time intensive and even then the possible presence of clouds can make their interpretation difficult. Even worse, nearly all the systems found by *Kepler* are too faint for these observations with current telescopes. An alternative is to develop models of the formation and evolution of low-mass planets to try and predict what compositions can form and how those compositions change as a planet evolves. In particular, hydrodynamic mass loss from extreme ultra-violet (XUV) heating can remove large amounts of hydrogen/helium from highly irradiated LMLD planets. Models of XUV driven mass loss were first developed to study water loss from early Venus [Hunten1982, Kasting1983]{}, and hydrogen loss from the early Earth [Sekiya1980, Watson1981]{}. These kinds of models have since been developed to study mass loss from hot Jupiters \[e.g.,\]\[\][Lammer2003, Yelle2004, Murray-Clay2009, Ehrenreich2011, Owen2012]{}, where there is strong evidence that atmospheric escape is an important physical process [Vidal-Madjar2004, Davis2009, Lecavelier2010, Lecavelier2012]{}. In Sections \[masslosssec\], \[watersec\], and \[formationsec\] we show that energy-limited hydrodynamic mass loss models, coupled with models of thermal evolution and contraction, can distinguish between water-poor super-Earth and water-rich sub-Neptune scenarios in Kepler-11. Moreover, these models make powerful predictions for the density distribution of the entire population of LMLD transiting planets. In particular, observations show that there is threshold in the bulk density - incident flux distribution above which there are no LMLD planets. In Section \[fdsec\] we examine this threshold and show how it can by reproduced using our thermal evolution models coupled with standard hydrodynamic mass loss prescriptions. Finally, in Section \[constraintsec\] we explore how this threshold can be used to obtain important constraints on planets without measured densities: We constrain the maximum radii of non-transiting radial velocity planets, and the minimum masses of *Kepler* candidates. Our Model ========= Planet Structure {#intsec} ---------------- We have built on previous work in [Fortney2007]{} and [Nettelmann2011]{} to develop models of the thermal evolution of LMLD planets. To simplify what is undoubtedly a complex interior structure for real planets, we construct model planets with well-defined layers. Low-mass planets are likely to have a significant fraction of their mass in iron and silicate rocks. For simplicity, we assume that these materials are contained in a isothermal rocky core with Earth-like proportions of 2/3 silicate rock and 1/3 iron. For the rock, we use the ANEOS [Thompson1990]{} olivine equation of state (EOS); while for the iron, we use the SESAME 2140 Fe EOS [Lyon1992]{}. On top of this rock/iron core we then attach an interior adiabat. The composition of this adiabat depends on the planet model being considered. For this work, we consider three classes of LMLD planets: rocky super-Earths with H/He envelopes, water-worlds that have pure water envelopes, and sub-Neptunes with a water layer in between the core and the upper H/He layer. For the water-rich sub-Neptune models we assume that this intermediate water-layer has the same mass as the rock/iron core. We choose this value because it is comparable to the water to rock ratio need to fit Kepler-11b as a water-world. This allows us to explore the proposition that all five Kepler-11 planets started out with similar compositions, but that mass loss has subsequently distinguished them. For hydrogen/helium we use the [Saumon1995]{} EOS. Meanwhile for water we use the ab-initio H2O-REOS EOS developed by [Nettelmann2008]{} and [French2009]{}, which was recently confirmed up to 7 Mbar in laboratory experiments [Knudson2012]{}. In the Kepler-11 system, our models predict that water will be in the vapor, molecular fluid, and the ionic fluid phases. The interiors are too hot for high pressure ice phases. Finally, we model the radiative upper atmosphere by assuming that the planet becomes isothermal at pressures where the adiabat is cooler than the planet’s equilibrium temperature, assuming 20% Bond albedo and uniform re-radiation. We then calculate the radius at 10 mbar which we take to be the transiting radius. We connect the different layers of our models models by requiring that pressure and temperature are continuous across boundaries. We then solve for the interior structure assuming hydrostatic equilibrium. A given model is defined by its mass, composition (i.e., the relative proportions in H/He, water, and the rock/iron core) and the entropy of its interior H/He adiabat. By tracking changes in composition and entropy we can then connect these models in time and study the thermal and structural evolution of a given planet. Thermal Evolution {#tesec} ----------------- In order to obtain precise constraints on composition, it is important to fully model how a planet cools and contracts due to thermal evolution. Models that only compute an instantaneous structure [Rogers2010b]{} by necessity must vary the intrinsic luminosity of the planet over several orders of magnitude, which can introduce large uncertainties in the current composition. Obtaining precise constraints from thermal evolution is essential when considering mass loss, since mass loss histories are highly sensitive to uncertainties in the current composition. Moreover, since mass loss depends strongly on planetary radius (to the third power), the mass loss and thermal histories are inextricably linked. Modeling this contraction requires a detailed understanding of a planet’s energy budget. By tracking the net luminosity of a planet, we know how the specific entropy $S$, (i.e., the entropy per unit mass) of the interior adiabat changes with time. For a given mass and composition, this adiabat then defines the planet’s structure and so we can track the planet’s total radius as the model cools and contracts with time. Equation (\[thermaleq\]) shows the energy budget for our models and how this relates to the change in entropy $dS/dt$. $$\label{thermaleq} \int_{M_{\mathrm{core}}}^{M_{\mathrm{p}}} dm \frac{T dS}{dt} = - L_{\mathrm{int}} + L_{\mathrm{radio}} - c_{\mathrm{v}} M_{\mathrm{core}} \frac{dT_{\mathrm{core}}}{dt}$$ The left hand side shows the rate of change of the thermal energy of the interior adiabat. Positive terms on the right hand side represent energy sources that heat and inflate a planet, while negative terms represent energy losses that allow a planet to cool and contract. The term $L_{\mathrm{int}} = L_{\mathrm{eff}}-L_{\mathrm{eq}}$ describes the intrinsic luminosity due to radiation from the planet, where $L_{\mathrm{eq}}$ is the planet’s luminosity due only to absorbed stellar radiation. The $L_{\mathrm{radio}}$ term describes heating due to radioactive decay. The important isotopes are $^{235}$U, $^{40}$K, $^{238}$U, and $^{232}$Th. These have half lives of 0.704, 1.27, 4.47, and 14.1 Gyr, respectively. We assume meteoritic abundances given by [Anders1989]{}. We do not consider the early decay of $^{26}Al$, since we only consider models that are at least 10 Myr after planet formation. The $L_{\mathrm{radio}}$ term has only a minor effect on our models since it is typically an order of magnitude smaller than the other terms in equation (\[thermaleq\]). Lastly, there is the $dT_{\mathrm{core}}/dt$ term, which represents the delay in cooling due to the thermal inertia of the rocky core. As the interior adiabat cools, the core isotherm must also cool, as $T_{\mathrm{core}}$ equals the temperature at the bottom of the adiabat. When the core makes up a large fraction of the planet’s mass, this can significantly slow down the planet’s rate of contraction. We assume a core heat capacity of $c_v = 0.5-1.0$ $\mathrm{J \, K^{-1} \, g^{-1}}$ [Alfe2002, Guillot1995, Valencia2010]{} as in [Nettelmann2011]{}. This range covers values appropriate for both the cores of the Earth and Jupiter. For our three layer sub-Neptune models we still us the mass in rock and iron for $M_{\mathrm{core}}$, since the water layer is generally too hot for ice phases and so it is assumed to be fully convective. For a given interior structure, we determine the intrinsic flux from the interior, at given $S$ of the adiabat, via interpolation in a grid of model atmospheres. The values of $T_{\mathrm{int}}$ (a parametrization of the interior flux), $T_{\mathrm{eq}}$, and $T_{\mathrm{eff}}$ are tabulated on a grid of surface gravity, interior specific entropy, and incident flux for 50$\times$ solar metallicity H/He atmospheres (similar to Neptune). This corresponds to a metal mass fraction of $Z \approx 0.35$ and a mean molecular weight of $\mu \approx 3.5$ $\mathrm{g \, mol^{-1} }$. The grid is the same as that described in [Nettelmann2011]{} for LMLD planet GJ 1214b, where a more detailed description can be found. Here we do expand on that grid to now include a range of incident fluxes, as was done for giant planets in [Fortney2007]{}. In choosing the initial entropy for our evolution model, we assume a “hot start” for model; i.e., we start the models out with a large initial entropy. When then allow the models to cool and contract until either 10 Myr or 100 Myr which is when we begin the coupled thermal and mass loss evolution. This is a common but important assumption. However, in general our thermal evolution models are insensitive to the initial entropy choice by $\sim100$ Myr as in [Marley2007]{}. As a result, we present results at both 10 and 100 Myr. Moreover, to gain confidence in our 10 Myr models we examined the effect of starting those models with a lower initial entropy. Specifically, we ran models in which we started the 10 Myr with the entropies found at 100 Myr. This allowed us to separate the effect of the stellar XUV evolution, from any “hot start” vs. “cold start” uncertainties. Future progress in modeling the formation of water-rich sub-Neptune planets \[e.g.,\]\[\][Rogers2011]{} may allow for an assessment of the most realistic initial specific entropies. XUV-Driven Mass Loss {#masslosssec} -------------------- Close-in planets like those in Kepler-11 are highly irradiated by extreme ultraviolet (EUV) and x-ray photons. These photons photoionize atomic hydrogen high in a planet’s atmosphere, which in turn produces significant heating [Hunten1982]{}. If this heating is large enough, it can generate a hydrodynamic wind that is capable of removing significant mass, potentially including heavier elements as well [Kasting1983]{}. We couple this XUV driven mass loss to our thermal evolution models following the approach of [Jackson2010]{} and [Valencia2010]{}, which explored possible mass loss histories for CoRoT-7b [Leger2009,Queloz2009]{}. Similar approaches have also been used to study the coupled evolution of hot Jupiters \[e.g.,\]\[\][Baraffe2004, Baraffe2005, Hubbard2007a, Hubbard2007b]{} and hot Neptunes [Baraffe2006]{}. A common approach to estimate the mass loss rate is to assume that some fixed fraction of the XUV energy incident on a planet is converted into heat that does work on the atmosphere to remove mass. This is known as the energy-limited approximation [Watson1981]{} and allows a relatively simple analytic description of mass loss rates. $$\label{masslosseq} \dot{M}_{\mathrm{e-lim}} \approx \frac{\epsilon \pi F_{\mathrm{XUV}} R_{\mathrm{XUV}}^3}{G M_{\mathrm{p}} K_{\mathrm{tide}}}$$ $$K_{\mathrm{tide}} = (1 - \frac{3}{2 \xi} + \frac{1}{2 \xi^3})$$ $$\xi = \frac{R_{\mathrm{Hill}}}{R_{\mathrm{XUV}}}$$ Equation (\[masslosseq\]) describes our estimate of the mass loss rate based on the formulation from [Erkaev2007]{}. $F_{\mathrm{XUV}}$ is the total flux between $1-1200$ Å, which is given by Ribas(2005) for Sun-like stars. For stars older than 100 Myr, Ribas found that at 1 AU $F_{\mathrm{XUV}} = 29.7 \tau^{-1.23} \: \mathrm{erg \, s^{-1} \, cm^{-2}}$, where $\tau$ is the age of the star in Gyr. Using this power law, we scale the XUV flux to the appropriate age and semi-major axis for each planet in our models. Although Ribas only targeted Sun-like stars, [Sanz-Forcada2010]{} found similar results for a wide range of stellar types from M3 to F7. Hereafter, we will simply refer to the entire $1-1200$ Å$;$ spectrum as XUV. $R_{\mathrm{XUV}}$ is the planetary radius at which the atmosphere becomes optically thick to XUV photons, which [Murray-Clay2009]{} find occurs at pressures around a nanobar, in the hot Jupiter context. For our work, we assume that the atmosphere is isothermal between the optical and XUV photospheres. This neglects heating from photo-disassociation, which should occur around a $\mathrm{\mu bar}$ [Kempton2012a]{}. However, this effect should be relatively small and if anything will lead to slight underestimate of the mass loss rate. We vary pressure of the XUV photosphere from 0.1 nbar to 10 nbar to include the uncertainty in the structure of the XUV photosphere. For H/He atmospheres on LMLD planets, the nbar radius is typically 10-20% larger than the optical photosphere. $K_{\mathrm{tide}}$ is a correction factor that accounts for the fact that mass only needs to reach the Hill radius to escape [Erkaev2007]{}. For planets like Kepler-11b today this correction factor increases the mass loss rate by $\sim$ 10%, however at early times it can increase the rate by as much as a factor of 2. Finally, $\epsilon$ is an efficiency factor that parametrizes the fraction of the incident XUV flux that is converted into usable work. This efficiency is set by radiative cooling, especially via Lyman $\alpha$, and can depend on the level of incident flux [Murray-Clay2009]{}. Kepler-11 is a $8\pm2$ Gyr old Sun-like star. Using the power law from Ribas et al., this implies that current XUV flux at Kepler-11f is $\approx 37 \: \mathrm{erg \, s^{-1} \, cm^{-2}}$. Similarly, when Kepler-11 was 100 Myr old, the flux at Kepler-11b was $\approx 6\times10^4 \: \mathrm{erg \, s^{-1} \, cm^{-2}}$. [Murray-Clay2009]{} found that at XUV fluxes over $10^5 \: \mathrm{erg \, s^{-1} \, cm^{-2}}$, relevant for many hot Jupiters, mass loss becomes radiation/recombination-limited and highly inefficient. However, at the lower XUV fluxes relevant for the Kepler-11 system mass loss is roughly linear with $F_{\mathrm{XUV}}$ and has efficiencies $\sim 0.1-0.3$. For this work, we assume a default efficiency of $\epsilon=0.1\pm^{0.1}_{0.05}$, although we do examine the effects of lower efficiencies. While we predominantly investigate the loss of H/He envelopes, in some limited cases for Kepler-11b, we also assume this holds for steam envelope loss. [ccccccccc]{}\[h!\] Kepler-11b & $4.3\pm^{2.2}_{2.0}$ & $0.3\pm^{1.1}_{0.25}\%$ & $40\pm^{41}_{29}\%$ & n/a\ \ Kepler-11c & $13.5\pm^{4.8}_{6.1}$ & $4.6\pm^{2.7}_{2.3}\%$ & n/a & $0.3\pm^{1.0}_{0.1}\%$\ \ Kepler-11d & $6.1\pm^{3.1}_{1.7}$ & $8.2\pm^{2.7}_{2.4}\%$ & n/a & $1.3\pm^{0.9}_{0.8}\%$\ \ Kepler-11e & $8.4\pm^{2.5}_{1.9}$ & $17.2\pm^{4.1}_{4.2}\%$ & n/a & $5.5\pm^{2.3}_{3.0}\%$\ \ Kepler-11f & $2.3\pm^{2.2}_{1.2}$ & $4.1\pm^{1.8}_{1.5}\%$ & n/a & $0.4\pm^{0.6}_{0.2}\%$\ \ \[currenttab\] One important implication from equation (\[masslosseq\]) is that mass loss rates are much higher when planets are young. This is due to two reasons. Planetary radii are considerably larger due to residual heat from formation. Moreover, at 100 Myr $F_{\mathrm{XUV}}$ was $\approx 500$ times higher than it is currently [Ribas2005]{}. As a result, most of the mass loss happens in a planet’s first Gyr. Thus although a planet’s envelope may be stable today, its composition may have changed significantly since formation. Likewise, a considerable amount of mass will be lost between the end of planet formation at $\sim10$ Myr [Calvet2002]{} and 100 Myr. Following the x-ray observations of [Jackson2012]{}, we assume that at ages younger than 100 Myr the stellar XUV flux saturates and is constant at the 100 Myr value. Unfortunately, the observations for 10-100 Myr do not cover the EUV (100-1200 Å) part of the spectrum, so there is some uncertainty as to whether this saturation age is uniform across the entire XUV spectrum. Nonetheless, our constraints on the formation of Kepler-11 in sections \[sesec\] and \[formationsec\] come from the lower limits we are able to place on the initial compositions. Assuming that the EUV saturates along with the x-rays is conservative assumption in terms of the amount of mass that is lost. In general, models of LMLD planets that assume H/He envelopes today will predict much larger mass loss histories then models that assume steam envelopes. Partly, this is because the lower mean molecular weight of hydrogen. Mostly, however, it is because when we integrate the compositions back in time from the present, the addition of a small amount of H/He has much larger impact on a planet’s radius than a small amount of water. A larger radius in the past in turn means a higher mass loss rate; and so the integrated mass loss history becomes much more substantial for H/He envelopes. Application to Kepler-11 ======================== Current Compositions from Thermal Evolution ------------------------------------------- The first step in trying to understand the formation and history of a planetary system is to identify the possible current compositions for each of the planets in the absence of any mass loss. This then gives us estimates for the current masses of each planet’s core, which we then use as the starting point for all of our calculations with mass loss. Figure \[mrfig\] shows the Kepler-11 planets in a mass-radius diagram along with curves for different possible compositions. For all planets, we color-code by the incident bolometric flux they receive. The Kepler-11 planets are shown by filled circles with identifying letters next to each one. The other known transiting exoplanets in this mass and radius range are shown by the open squares. In order of increasing radius, these are Kepler-10b [Batalha2011]{}, Kepler-36b [Carter2012]{}, CoRoT-7b [Leger2009,Queloz2009,Hatzes2011]{}, Kepler-20b [Fressin2011,Gautier2011]{}, Kepler-18b [Cochran2011]{}, 55 Cancri e [Winn2011,Demory2011]{}, GJ 1214b [Charbonneau2009]{}, Kepler-36c [Carter2012]{}, Kepler-30b [Fabrycky2012, Sanchis-Ojeda2012]{}, and GJ 3470b [Bonfils2012]{}. Lastly, the open triangles show the four planets in our own solar system that fall in this range: Venus, Earth, Uranus, and Neptune. The curves show various possible compositions. The solid black curve shows a standard Earth-like composition with 2/3 rock and 1/3 iron as described in Section \[intsec\]. The other curves show compositions with thick water or H/He envelopes atop an Earth-like core. These curves include thermal evolution without mass loss to 8 Gyr, the age of Kepler-11. The blue dashed curves show the results for 50% and 100% water-worlds computed at $T_{\mathrm{eq}}=700$ K, approximately the average temperature of the five inaner planets. Likewise, the dotted orange curves show the results for H/He envelopes; however, here each curve is tailored to match a specific Kepler-11 planet and is computed at the flux of that planet. These fits are listed in greater detail in table \[currenttab\]. Here we list the mass of each planet taken from [Lissauer2011a]{}; the H/He fractions needed to match each planet’s current radius for a water-poor super-Earth model; the water fraction needed to match Kepler-11b as a water-world; and the H/He fractions needed to fit Kepler-11c, d, e, and f as sub-Neptunes with an intermediate water layer, as described in Section \[intsec\]. As described in section \[sesec\], we varied the planetary albedo, the heat capacity of the rocky core, and the observed mass, radius, current age, and incident flux. Figure \[mrfig\] and table \[currenttab\] clearly show the degeneracy between various compositions that we are attempting to untangle. There are now four planets including Kepler-11b that can easily be fit either as water-worlds or as water-rich sub-Neptunes with $<2\%$ of their mass in H/He. However, it is worth looking closer at Kepler-11b in particular. It is this the only planet in the system which does not require any hydrogen or helium to match its current radius, although it must have some sort of volatile envelope. Moreover, it is also the most irradiated and it is fairly low gravity. As a result, adding a small amount of hydrogen to its current composition has a large impact on the bulk density, which in turn makes the planet more vulnerable to mass loss, as seen in Eq. (\[masslosseq\]). A clearer picture for this planet emerges when including XUV driven mass-loss and relatively strong constraints from formation models discussed in Section \[formationsec\]. Thus, if there is hope of using mass loss to constrain the composition and formation of the system, it likely lies with Kepler-11b. Mass Loss for a Super-Earth Scenario {#sesec} ------------------------------------ [cccccccccc]{}\[h!\] Kepler-11b & $34.6\pm^{6.5}_{28.2}$ & $87.6\pm^{6.6}_{85.4}\%$ & $44.8\pm^{9.7}_{10.1}$ & $90.4\pm^{5.1}_{8.2}\%$\ \ Kepler-11c & $13.7\pm^{4.7}_{5.8}$ & $6.0\pm^{5.0}_{3.2}\%$ & $14.2\pm^{4.3}_{3.1}$ & $9.1\pm^{28}_{7.3}\%$\ \ Kepler-11d & $6.7\pm^{2.8}_{0.6}$ & $16.5\pm^{22}_{8.5}\%$ & $7.8\pm^{12.8}_{0.8}$ & $28\pm^{56}_{17}\%$\ \ Kepler-11e & $8.8\pm^{2.3}_{1.6}$ & $21.2\pm^{6.0}_{3.2}\%$ & $9.7\pm^{2.5}_{1.9}$ & $28.1\pm^{10.5}_{7.7}\%$\ \ Kepler-11f & $3.1\pm^{5.2}_{0.2}$ & $29\pm^{58}_{24}\%$ & $3.4\pm^{6.6}_{0.4}$ & $35\pm^{57}_{25}\%$\ \ \[masslosstab\] Now that we have estimates for the present day compositions, we will begin considering the effects of mass loss. We will compute mass loss histories that when evolved to the present day, match the current mass and composition. This then tells us what the mass would have to be in the past to result in the current mass and composition. As discussed in Section \[masslosssec\], there is uncertainty in stellar XUV fluxes ages younger than 100 Myr; as a result, we will present results both at 10 Myr and 100 Myr after planet formation. First we will consider water-poor super-Earth models for each planet, which have H/He envelopes atop Earth-like rocky cores. As discussed in Section \[masslosssec\], H/He envelopes are particularly susceptible to mass loss. As an example, Figure \[k11bfig\] shows four possible cooling histories for Kepler-11b. The solid lines show thermal evolution without any mass loss while the dashed lines include mass loss. The orange curves are for water-poor super-Earth models, while the blue curves show water-world models. The red cross shows the current radius and age of Kepler-11b. These curves illustrate the impacts of both thermal evolution and mass loss on the radius of a low-mass planet. The water-world models require that 40% of the current mass must be in water to match the current radius. Assuming our standard efficiency $\epsilon=0.1$, implies an initial composition of 43% water at 10 Myr. This illustrates the relative stability of water envelopes. On the other hand, the dashed orange curve shows the vulnerability of H/He layers. Here we have assumed a efficiency 5$\times$ lower $\epsilon=0.02$ and yet more mass is lost than in the water-world scenario. Even at this low efficiency, Kepler-11b would have to initially be 11% H/He and 4.8 $M_{\mathrm{\oplus}}$ to retain the 0.3% needed to match the current radius. This also shows the large increase in radius that can result from even a relatively modest increase in the H/He mass. Table \[masslosstab\] summarizes the results for Kepler-11 b-f for the water-poor super-Earth scenario. We list the masses predicted by our models when the planets were 10 and 100 Myr old. In addition, we list the fraction of the planets’ masses in the H/He envelope at each age. These results are further illustrated in Figure \[lossfig\]a. Here we have plotted the mass and H/He fraction for each planet at 10 Myr, 100 Myr, and today. Each color corresponds to a particular planet with the squares indicating the current masses and compositions, the circles the results at 100 Myr, and the triangles the results at 10 Myr. In order to calculate the uncertainty on these results, we varied the mass loss efficiency $\epsilon$ from 5-20% and varied the XUV photosphere from 0.1-10 nbar. Likewise, we varied the planetary Bond albedo from 0-0.80 and varied the heat capacity of the rocky core from 0.5-1.0 $\mathrm{J \, g^{-1} \, K^{-1}}$. Also, as discussed in section \[tesec\], we varied the initial entropy for the 10 Myr models, to account for undertainties in “hot-start” vs. “cold-start”. Finally, we factored in the observed uncertainties in mass, radius, and incident flux. Clearly, Kepler-11b is vulnerable to extreme mass loss if it has a H/He envelope atop a rock/iron core. Although less than 1% H/He today, if it is a water-poor super-Earth it could have been have over $\sim90\%$ H/He in the past. At 10 Myr, its mass would have been $45\pm10$ $M_{\mathrm{\oplus}}$, an order of magnitude higher than the current value. Kepler-11b is able to undergo such extreme mass loss because its high XUV flux and the low mass of its rocky core put it in a regime where it is possible to enter a type of runaway mass loss. This happens when the mass loss timescale is significantly shorter then the cooling timescale. After the planet initially loses mass it has an interior adiabat and rocky core that are significantly hotter than would otherwise be expected for a planet of its mass and age. This is because the interior still remembers when the planet was more massive and has not had sufficient time to cool. As a result, the planet will stay inflated for some time and the density stays roughly constant and can actually decrease. A similar effect was seen by [Baraffe2004]{} when they studied coupled thermal evolution and mass loss models for core-less hot Jupiters. We find that this process generally shuts off once the composition drops below $\sim$20% H/He. At that point the presence of the core forces the total radius to shrink even if the planet is unable to cool efficiently. Figure \[runawayfig\] shows this process as Kepler-11b loses mass for three different values of its current mass and therefore its core mass. The curves correspond to the best fit mass from transit-timing as well as the 1$\sigma$ error bars. This shows that the timing of this runaway loss event depends strongly on the mass of the rock/iron core. Super-Earth models of Kepler-11b are unusual in that they are subject to tremendous mass loss and yet they retain a small amount of H/He today. Typically models that start out $\sim90\%$ H/He either experience runaway mass loss and lose their H/He envelopes completely, or they never enter the runaway regime and remain over 50% H/He. The uncertainty in the initial composition of Kepler-11b is due to uncertainty in its TTV mass. At a given current mass, the range of Kepler-11b models that will retain an envelope that is $<1\%$ H/He is extremely narrow. In this sense, the current composition of Kepler-11b requires a rare set of initial conditions if it is a water-poor super-Earth. [cccccccccc]{}\[h!\] Kepler-11b & $4.4\pm^{2.2}_{2.0}$ & $41\pm^{39}_{28}\%$ & $4.5\pm^{2.1}_{1.8}$ & $43\pm^{38}_{29}\%$\ \ Kepler-11c & $13.7\pm^{4.8}_{4.6}$ & $1.8\pm^{18}_{1.4}\%$ & $15.2\pm^{25}_{1.2}$ & $12\pm^{70}_{10}\%$\ \ Kepler-11d & $6.8\pm^{2.8}_{1.2}$ & $11.6\pm^{17}_{8.7}\%$ & $7.6\pm^{7.6}_{0.9}$ & $21\pm^{49}_{10}\%$\ \ Kepler-11e & $9.1\pm^{2.1}_{2.0}$ & $12.8\pm^{4.6}_{6.7}\%$ & $9.7\pm^{2.6}_{2.0}$ & $18\pm^{12}_{10}\%$\ \ Kepler-11f & $2.9\pm^{3.8}_{0.5}$ & $21\pm^{62}_{17}\%$ & $4.0\pm^{4.9}_{1.5}$ & $43\pm^{44}_{36}\%$\ \ \[watertab\] Counterintuitively, if Kepler-11b is more massive today then its implied mass in the past is actually lower. This is because a higher mass today would imply a more massive core, which would increase the planet’s density and decrease its mass loss rate. As a result, a more massive model for Kepler-11b today is less vulnerable to mass loss and so less H/He is needed in the past in order to retain 0.3% today. At 100 Myr, there is a very large uncertainty in the composition due to the uncertainty in the core mass. However, even if we assume the 1$\sigma$ error bar 6.5 $M_{\mathrm{\oplus}}$, Kepler-11b would still be at least 37 $M_{\mathrm{\oplus}}$ and at least 83% H/He at 10 Myr. In section \[formationsec\], we will compare this to models of in situ formation and show that such a scenario is unlikely. On the other hand, Kepler-11c is not particularly vulnerable to mass loss, at least using the best fit mass from transit timing, despite having the second highest flux in system. This is because of the relatively large mass of its rocky core; the high gravity means additional H/He has a more modest effect of the planet’s radius and therefore on the mass loss rate. In fact, along with the incident XUV flux the mass of the rocky core is the single largest factor that determines whether a given planet will be vulnerable to mass loss. As a result, the dominant sources of uncertainty in our mass loss models are the uncertainties in the masses from TTV. These dominate over all the theoretical uncertainties in the thermal evolution and mass loss models. The uncertainty in planet mass from transit timing is particular large for Kepler-11c. If its mass is close to the 1$\sigma$ low value, then it is possible Kepler-11c has undergone more substantial mass loss similar to Kepler-11d-f. Fortunately, as more quarters of data are processed the mass estimates from TTV will become more precise [Agol2005,Holman2005]{}. Finally, Kepler-11d, e, and f are modestly vulnerable to mass loss and are consistent with having originated with $\sim 20\%$ H/He at 100 Myr and $\sim 30\%$ H/He at 10 Myr. In Section \[hillsec\] we will discuss these results in terms of orbital stability. The Water-Rich Scenario {#watersec} ----------------------- Next we consider a water-rich scenario where the entire system formed beyond the snow line. We assume that Kepler-11c-f are water-rich sub-Neptunes as described in Section \[intsec\], while Kepler-11b is currently a water-world. Otherwise the thermal mass loss histories are calculated in the same manner as the water-poor super-Earth scenario. For Kepler-11c-f we calculate the planet mass H/He fraction at 10 and 100 Myr, assuming that only H/He is lost. For Kepler-11b, we examine the vulnerability of both H/He and steam envelopes atop water-rich interiors. The results are summarized in Table \[watertab\] which list the water fraction for a water-world model of Kepler-11b and the H/He fraction for water-rich sub-Neptune models of Kepler-11c-f. Likewise, the results for c-f are shown in Figure \[lossfig\]b. In general, these three layer models are slightly more vulnerable to mass loss than the water-poor super-Earth models presented in section \[sesec\]. Mostly this is because models with a water layer have hotter interiors that cool more slowly. Since models For example, for Kepler-11c without mass loss the models presented in Table \[currenttab\], at 8 Gyr the final entropy in the H/He layer is 6.6 $\mathrm{k_b}$/baryon for the water-rich sub-Neptune model versus 5.8 for the water-poor super-Earth model. The second reason is that counter-intuitively the water-rich sub-Neptune models are slightly more vulnerable to mass loss precisely because they have less of the planet’s mass in H/He today. For a planet that has less H/He today, adding a small amount of H/He at the margin has a larger impact on the planet’s radius and therefore on the mass loss rate. For Kepler-11c-f the results are broadly similar to the those for the water-poor super-Earth scenario. Kepler-11c is again the least vulnerable to mass loss; while Kepler-11d is again the most vulnerable of the four planets that we model as water-rich sub-Neptunes. However, all four of these planets are consistent with having been $\sim10-20\%$ H/He at 100 Myr and $\sim20-30\%$ H/He at 10 Myr. If Kepler-11b was always a water-world, then mass loss was never important for it. Between 10 Myr and the present, it only drops from 43% to 40% water. Moreover, if Kepler-11b was initially a water-rich sub-Neptune similar to the other planets in the system, it could have easily stripped its H/He outer envelope. If we start Kepler-11b at 100 Myr as a water-rich sub-Neptune similar to the other planets with $30\%$ H/He atop 4.3 $M_{\mathrm{\oplus}}$ of rock and water, then assuming $\epsilon=0.1$ the entire H/He envelope will be stripped by 300 Myr. We can set upper limits on the initial mass and H/He fraction of 70 $M_{\mathrm{\oplus}}$ and 94% if Kepler-11b was originally a water-rich sub-Neptune. These are however strictly upper limits, a H/He layer could have been lost at any time between formation and now. Therefore, all five planets are consistent with a scenario is which they formed as water-rich sub-Neptunes with $\sim10\%$ H/He at 100 Myr and $\sim20\%$ H/He at 10 Myr. A Mass Loss Threshold for Low-Mass Low-Density Planets {#fdsec} ====================================================== Although Kepler-11 provides a unique case-study, it is essential to explore how mass loss impacts the larger population of LMLD transiting planets. Figure \[shorelinefig\] shows the bolometric flux these planets receive at the top of their atmospheres vs. their bulk densities. As in Figure \[mrfig\], filled circles show the Kepler-11 planets with the letters indicating each planet. Likewise, the open squares show the other transiting exoplanets that are less than 15 Earth masses. For reference, we have also plotted all other transiting planets between 15 and 100 $M_{\mathrm{\oplus}}$ as gray crosses [Wright2011]{}. The colors indicate possible compositions. All planets with a best-fit mass and radius that lies below a pure rock curve are colored red. These include Kepler-10b, Kepler-36b, CoRoT-7b, and just barely Kepler-20b. Planets that are less dense than pure rock but more dense than pure water, indicating that the could potentially be water-worlds, are colored blue. These include Kepler-11b, Kepler-18b, and 55 Cancri e. Meanwhile those planets that must have a H/He envelope to match their radius are colored orange. These include Kepler-11c, d, e, and f, Kepler-30b, Kepler-36c, GJ 1214b, and GJ 3470b. The dashed black lines show curves of constant mass loss rate according to equation (\[masslosseq\]), assuming $\epsilon=0.1$ and $K_{\mathrm{tide}}=1$. These curves are linear in this plot since the instantaneous mass loss rate goes as the flux over the density. Although Figure \[shorelinefig\] plots the bolometric flux today, we can relate this to an XUV flux at a given time using the [Ribas2005]{} power law for sun-like stars described in Section \[masslosssec\]. The curves show the flux today required to lose mass at 1 $M_{\mathrm{\oplus}} \, \mathrm{Gyr^{-1}}$ when the planets were 1 Gyr old and 100 Myr old, along with another curve showing 0.1 $M_{\mathrm{\oplus}} \, \mathrm{Gyr^{-1}}$ at 100 Myr. Since most of the mass loss happens in the first few hundred Myrs, the bottom two curves can roughly be considered as the respective thresholds for mass loss being important and being unimportant for LMLD planets. One possible explanation of this mass loss threshold is that it caused by XUV driven mass loss from H/He envelopes on low-mass planets. LMLD planets that form above the 100 Myr 1 $M_{\mathrm{\oplus}} \, \mathrm{Gyr^{-1}}$ curve lose mass, increase in density and move to the right until they lie below this threshold. The planets that are left above this line are mostly rocky or at the very least probably do not have H/He envelopes. Planets more massive than $\sim 15$ $M_{\mathrm{\oplus}}$ are not affected since they have a larger reservoir of mass and the loss of a few earth masses of volatiles isn’t sufficient to significantly change their bulk density. To illustrate this, we have plotted our predictions for the bulk densities of each of the Kepler-11 planets at 100 Myr, including the effects of both mass loss and thermal evolution. These are indicated by the shadowed letters at the left of Figure \[shorelinefig\]. The situation becomes even clearer if we instead we plot flux against mass times density as in Figure \[improvedfig\]. The timescale for XUV mass loss goes like $\rho M_{\mathrm{p}}/F_{\mathrm{XUV}}$, so lines in this diagram are constant mass loss timescales. Now the threshold is much clearer and applies to all planets up to all planets with H/He envelopes. This also removes any effects from the somewhat arbitrary 15 $M_{\mathrm{\oplus}}$ cut.The sparsity of planets at low flux and high density is almost certainly a selection effect, since these are likely to be planets with long periods and small radii. However, the interesting result is that there is appears to be a critical mass loss timescale above which we do not find any planets with H/He envelopes. In particular, all five of the inner Kepler-11 planets lie nicely along this threshold. Moreover, of the three planets that lie above the critical mass loss timescale, two are likely rocky. $$\label{tmleq} t_{\mathrm{loss}} = \frac{M_{\mathrm{p}}}{\dot{M}} = \frac{G M_{\mathrm{p}}^2}{\pi \epsilon R_{\mathrm{p}}^3 F_{\mathrm{XUV,E100}}} \frac{F_{\mathrm{\oplus}}}{F_{\mathrm{p}}}$$ The dashed black line in Figure \[improvedfig\] shows our best fit for this critical mass loss timescale. Equation \[tmleq\] defines this mass loss timescale. Here $\epsilon=0.1$ is the mass loss efficiency, $F_{\mathrm{XUV,E100}} = 504$ $\mathrm{erg \, s^{-1} \, cm^{-2}}$ is the XUV flux at the Earth when it was 100 Myr old, and $F_{\mathrm{p}}$ is the current incident bolometric flux at a planet. We find a best fit with $t_{\mathrm{loss,crit}}\approx 12$ Gyr. However, while equation \[tmleq\] accounts for the higher XUV fluxes at earlier times, it does not include the effects of larger radii at formation. The will reduce $t_{\mathrm{loss}}$ by at least another order of magnitude. A similar mass loss threshold was proposed by [Lecavelier2007]{}. Unfortunately, at that time there were relatively few transiting planets and no known transiting super-Earths. As a result, the authors we mostly limited to hot Jupiters from radial velocity surveys and were forced to use a scaling law to estimate radii. Here we are able to confirm the existence of a mass loss threshold and extend it all the way down to $\sim 2$ $M_{\mathrm{\oplus}}$. This mass loss threshold could also help explain features in occurrence rate of planets found by *Kepler*. [Howard2011a]{} found that the frequency of 2-4 $R_{\mathrm{\oplus}}$ *Kepler* planet candidates dropped off exponentially for periods within 7 days. This 7 day cutoff corresponds to an incident bolometric flux of 200 $F_{\mathrm{\oplus}}$. There are five planets with measured densities in figure \[shorelinefig\] that lie above 200 $F_{\mathrm{\oplus}}$. Of these five, three planets are consistent with being rocky and two with being water-worlds; none of the five requires a H/He atmosphere to match its observed mass and radius. If all low mass planets orbiting within 7 days lose their H/He atmospheres, then their radii will shrink from 2-4 $R_{\mathrm{\oplus}}$ to $<$2 $R_{\mathrm{\oplus}}$. This could naturally explain the drop off in 2-4 $R_{\mathrm{\oplus}}$ candidates at short periods. Reproducing the Mass Loss Threshold {#reproducesec} ----------------------------------- In order to fully examine whether the mass loss threshold in Figure \[shorelinefig\] can be explained by atmospheric mass loss, we performed a small parameter study with $\sim$800 mass loss models across a wide range of initial masses, compositions, and incident fluxes. For each model we ran thermal evolution and mass loss starting at 10 Myr around a Sun-like star. We ran models with initial masses of 2, 4, 8, 16, 32, and 64 $M_{\mathrm{\oplus}}$. We assumed water-poor super-Earth compositions, meaning H/He envelopes on top Earth-like cores, with initial compositions of 1, 2, 5, 10, 20, and 40% H/He. Finally we varied the incident bolometric flux from 10 to 1000 $F_{\mathrm{\oplus}}$, in order to cover the range of observed planets in Figures \[shorelinefig\] and \[improvedfig\]. We then recorded the resulting masses, densities, and compositions at various ages. The results are shown in Figure \[predictfig\]. As in Figure \[improvedfig\], each panel plots the total incident flux at the top of the atmosphere vs. the planet mass times density assuming different mass loss histories for our full suite of models. The size of each point indicates the mass of the planet, while the color indicates the fraction of its mass in the H/He envelope. The top left panel shows the initial distribution at 10 Myr before we start any mass loss. The other two top panels show the results at 100 Myr and 10 Gyr for our standard mass loss efficiency $\epsilon=0.1$. Meanwhile, the bottom panels show the results at 1 Gyr for a range of different efficiencies. These range from highly inefficient mass loss $\epsilon=0.01$, to our standard efficiency $\epsilon = 0.1$, and finally extremely efficient mass loss $\epsilon=1$. In each panel, as planets cool and lose mass the points move to the right, shrink, and become bluer (less H/He). For reference, we have re-plotted our critical mass loss timescale from Figure \[improvedfig\] in each of the result panels. As we can see, models with mass loss do in general result in a threshold roughly corresponding to a critical mass loss timescale. Moreover, the mass loss threshold observed in Figure \[shorelinefig\] is well reproduced by mass loss models with $\epsilon \approx 0.1$. This is similar to the efficiencies found by detailed models of mass loss from hot Jupiters in the energy-limited regime [Murray-Clay2009]{}. This suggests that our assumption of comparable mass loss efficiencies for LMLD planets is reasonable. It is also apparent that the threshold already in place by 100 Myr, and subsequent evolution has a relatively minor effect. We also examined the effect of beginning our parameter study at 100 rather than 10 Myr; however, this did not significantly affect the location of the threshold. Previous mass loss evolution models \[e.g.,\]\[\][Hubbard2007a, Hubbard2007b, Jackson2012, Owen2012]{} have also predicted mass loss thresholds. However, our models are the first to fully include the effects of coupled mass loss and thermal evolution for LMLD planets. We are able confirm and explain the observed threshold seen in Figure \[improvedfig\] in a region of parameter space where most of the [*Kepler*]{} planets are being found. Constraints On Mass and Radius for the General Population {#constraintsec} --------------------------------------------------------- If we use the critical mass loss timescale curve from Figure \[improvedfig\] as a approximation for the observed mass loss threshold, then we can write down a simple expression for the threshold. This is shown in equation (\[fdeq\]), which is valid for planets around Sun-like stars with $F_{\mathrm{p}}< 500$ $F_{\mathrm{\oplus}}$. The 500 $F_{\mathrm{\oplus}}$ cut excludes highly irradiated rocky planets like Kepler-10b and CoRoT-7b. These planets may have once had volatiles in the past, but they are likely rocky today and so H/He mass loss is no longer relevant. This cut also excludes the region where energy-limited escape breaks down and mass loss becomes radiation and recombination limited [Murray-Clay2009]{} $$\label{fdeq} \rho M_{\mathrm{p}} \ge \frac{ 3\epsilon F_{\mathrm{XUV,E100}} } { 4 G } \frac{ F_{\mathrm{p}} } { F_{\mathrm{\oplus}} } t_{\mathrm{loss,crit}}$$ The exciting implication of equation (\[fdeq\]) is that we can use it to obtain lower limits on mass for the much larger population of *Kepler* super-Earths and sub-Neptunes for which we do not have measured densities. This will help identify promising targets for follow-up work with radial velocity observations. This is shown in equation (\[masslimeq\]). $$\label{masslimeq} M_{\mathrm{p}} \ge \sqrt{ \frac{\pi\epsilon F_{\mathrm{XUV,E100}}}{G} \frac{F_{\mathrm{p}}}{F_{\mathrm{\oplus}}} t_{\mathrm{loss,crit}}} \, R_{\mathrm{p}}^{3/2}$$ Table \[masslimtab\] applies equation (\[masslimeq\]) to a list of *Kepler* candidates smaller than 4 $R_{\mathrm{\oplus}}$ that are well suited to radial-velocity follow-up. We excluded any planets with $F_{\mathrm{p}}> 500$ $F_{\mathrm{\oplus}}$, since equation (\[masslimeq\]) is not valid in that regime. Also, we limited the sample to only those planets with minimum radial velocity semi-amplitudes $K_{\mathrm{min}}>1.0 \: \mathrm{m \, s^{-1}}$ around stars with Kepler magnitude brighter than 13, since these will be the most promising for RV follow-up. In the end, this leaves us with a list of 38 likely detectable targets, eight of which (KOIs 104.01, 107.01, 123.01, 246.01, 262.02, 288.01, 984.01, and 1241.02) have $K_{\mathrm{min}}>2.0 \: \mathrm{m \, s^{-1}}$. Finally, we can also use the mass loss threshold to find an upper limit on the radii of non-transiting planets from radial velocity surveys with $F_{\mathrm{p}} < 500$ $F_{\mathrm{\oplus}}$. This is done in equation (\[radiuslimeq\]). $$\label{radiuslimeq} R_{\mathrm{p}} \le ( \frac{G}{\pi\epsilon F_{\mathrm{XUV,E100}} t_{\mathrm{loss,crit}}}\frac{F_{\mathrm{\oplus}}}{F_{\mathrm{p}}} )^{1/3} M_{\mathrm{p}}^{2/3}$$ Discussion ========== Kepler-11: Comparison to Formation Models, Implications for Migration {#formationsec} --------------------------------------------------------------------- By itself, the constraints from mass loss do not tell us whether Kepler-11 is a system of water-poor super-Earths or water-rich sub-Neptunes. Instead we need to compare our estimates of the initial compositions to models of planet formation. By doing so we can examine whether our estimates of the original compositions for a water-poor super-Earth scenario are consistent with the maximum H/He fraction that can be accreted during in situ formation. [Ikoma2012]{} examine the accretion of H/He atmospheres onto the rocky cores of hot water-poor super-Earths. In particular, they examine the in situ formation of the Kepler-11 system. In addition to a planet’s core mass and temperature, the amount of H/He accreted will depend strongly on the lifetime and dust grain opacity of the accretion disk. As with thermal evolution, the need to cool the rocky core can slow the contraction of the accreting atmosphere and limit the final H/He fraction. They are able to set hard upper limits on the initial compositions for in situ formation by assuming a grain-free, long-lived ($\sim 1$ Myr in the inner 0.2 AU) accretion disk and ignoring the delay in accretion due to cooling the core. In particular, [Ikoma2012]{} find that Kepler-11b could not have accreted more than 10% of its mass in H/He if it formed in situ. Moreover, using a more typical disk lifetime of $10^5$ yr [Gorti2009]{} and including the effect of cooling the core implies that Kepler-11b was $<1\%$ H/He at formation. On the other hand, in Table \[masslosstab\] we showed that thermal evolution and mass loss models predict that if Kepler-11b is a water-poor super-Earth then it was $87\pm^{7}_{85}\%$ and at least $82\%$ at 10 Myr. Combined with the results of [Ikoma2012]{}, this disfavors in situ formation of Kepler-11b. This result appears robust to any uncertainties in thermal evolution or mass loss models. Even if we only look after the period of run-away mass loss, at 3 Gyr Kepler-11b was still 10$\%$ H/He, the maximum allowed by [Ikoma2012]{}. Likewise, we find that Kepler-11f was at least 10$\%$ H/He at 10 Myr, even though the [Ikoma2012]{} models predict that it cannot have accreted its current composition of 4$\%$ H/He if it formed in situ. Furthermore, the co-planar, tightly packed, circular orbits in the system strongly suggest that it could have undergone type 1 migration [Ida2010]{}. As a result, we disfavor in situ formation of the system. If the Kepler-11 system did not form at its current location, then one possibility is that it formed at or beyond the snow-line and then Type 1 migrated to it is current location [Rogers2011]{}. If this is the case, then it is likely a system of water-rich sub-Neptunes and water-worlds as discussed in Section \[watersec\]. As we showed in Section \[watersec\], Kepler-11b is very stable to mass-loss if it is a water-world. If it was initially a water-rich sub-Neptune, it could have easily lost its H/He layer in the first few 100 Myr. Likewise, Kepler-11c-f are all consistent having formed as water-rich sub-Neptunes with $\sim20\%$ of their mass in H/He. The other possibility is that Kepler-11 is a system of water-poor super-Earths that has nonetheless undergone significant migration. For a grain-free accretion disk that lasts $10^6$ yr at 550 K, the critical mass for run-away accretion drops to 5 $M_{\mathrm{\oplus}}$ [Ikoma2012]{}. This implies that Kepler-11b could possibly have formed as a water-poor super-Earth at or beyond the current orbit of 11f. Nonetheless, this assumes a completely grain-free long-lived disk, which may not be realistic. Furthermore, this scenario still requires that Kepler-11b was $\sim$90% H/He when it formed, while all the other planets in the system are consistent with more modest initial compositions. As a result, we favor the water-rich sub-Neptune scenario. Kepler 11: Mass Loss and Orbital Stability {#hillsec} ------------------------------------------ One possible result of significant mass loss is that it could impact the orbital stability of closely packed multi-planet systems like Kepler-11. Although this system is stable in its current configuration, it might not be with the initial masses determined by our models. One relatively simple stability check is to calculate the separation between pairs of planets in terms of their mutual Hill spheres. Figure \[hillfig\] plots the separation in mutual Hill spheres ($\Delta$) between adjacent pairs of planets at both 10 Myr and the present, assuming a water-poor super-Earth composition. [Smith2009]{} found that systems with five or more planets tended to de-stabilize when $\Delta<9$. This threshold is shown as dashed gray lines in figure \[hillfig\]. Although Kepler-11b-c currently lies well below this threshold, [Lissauer2011a]{} showed that the system is nevertheless stable today because planets b and c are dynamically decoupled from the other four planets and so act more like a two planet system. For two planet systems the absolute minimum stable separation is $\Delta=2\sqrt{3}=3.46$ [Gladman1993]{}. This second stability threshold is shown by the dotted lines in figure \[hillfig\]. The Hill radius goes as $M_p^{1/3}$, as a result the change in $\Delta$ from mass loss is relatively modest; nonetheless, the stability of the system is in danger. At 10 Myr, Planets d-e do lie below the approximate $\Delta>9$ stability threshold; however, both pairs on either side of d-e are still relatively stable which may help stabilize the system. More importantly, the separation of planets b-c at 10 Myr skirts dangerously close, $\Delta=3.8\pm^{0.5}_{0.4}$, to the critical $\Delta>2\sqrt{3}$ stability threshold. More detailed modeling needs to be done to assess the impact of mass loss on orbital stability; nonetheless, the $\Delta>2\sqrt{3}$ stability threshold provides another strong reason to be skeptical of a water-poor super-Earth scenario for Kepler-11b. The major caveat to this stability analysis is that we assume that all of the orbits are stationary even as the planets lose mass. This is motivated by [Adams2011]{}, which showed that in the presence of a modest planetary magnetic field XUV driven mass loss from hot Jupiters tends to come out along the magnetic poles. Assuming that the magnetic field is sufficiently strong, dipolar, and perpendicular to the plane of the orbit, then mass loss won’t have any impact on the orbit. In general however, the directionality of mass loss will be an extremely complicated problem determined by the interaction of the ionized hydrodynamic wind, the planetary magnetic field, and the stellar wind. [Boue2012]{} showed that if the mass loss is directed in the plane of the orbit, then it can have a significant impact on both semi-major axis and eccentricity. Future Work ----------- Further confirmation of the mass loss threshold will depend on getting reliable mass estimates for more LMLD planets. Fortunately, there is a large population of super-Earth sized planets in multi-planet systems found by *Kepler* [Lissauer2011b]{}. For some of these systems TTV can be used to determine masses [Agol2005]{}. Moreover, this will become possible for more systems as more quarters of data are collected. Likewise, as more quarters of transit data are analyzed, previous mass constraints from TTV will become more precise. For Kepler-11 this will allow tighter constraints on both the current and past compositions. In order to better understand mass loss, there is also a strong need to acquire more XUV observations of young ($\sim10$ Myr) stars. Currently the best estimates of EUV fluxes are for G and K stars older than 100 [Ribas2005,Sanz-Forcada2010]{}. Meanwhile, planet formation ends and mass loss becomes important after a few Myr [Calvet2002, Alexander2006]{}. A large amount of planetary mass will be lost in the first 100 Myr and this will depend strongly on the stellar XUV flux. [Jackson2012]{} recently found that x-ray fluxes saturate for stars younger than 100 Myr; however, this needs to be investigated at other wavelengths. Likewise, more UV observations of transiting exospheres are needed. Currently, we only have observations for a handful of hot Jupiters [Vidal-Madjar2004, Lecavelier2010, Lecavelier2012]{}. More observations are needed, especially for Neptune and super-Earth sized planets. Likewise, it is also important get more observations of XUV fluxes and flares from M dwarfs, which are known to be highly active [Reiners2012]{}. This is particularly important for super-Earths in the habitable zones of late M dwarfs where the stability of habitable atmospheres could depend on XUV driven mass loss. The three planets orbiting M-stars in figure \[improvedfig\] all lie an order of magnitude below the threshold for Sun-like stars. This could be due to mass loss from X-ray flares, but without a larger sample size it is impossible to say. The Recently [France2012]{} obtained the first FUV spectrum for GJ876; however, more data are needed. Likewise, non-equilibrium mass loss models need to be developed to understand the impact of flares. In the area of modeling mass loss, we have utilized a relatively simple model that appears to work well for hot Jupiters, although mass loss rates have only been constrained for HD 209458b [Vidal-Madjar2004]{} and HD 189733b [Lecavelier2010, Lecavelier2012]{}. On a broader scale, there may be evidence for a lost population of hot Jupiter planets at very high levels of XUV irradiation \[e.g.,\]\[\][Davis2009]{}. Since we have suggested that H/He mass loss may be more important for more modestly irradiated LMLD planets than for average hot Jupiters (since the smaller planets have much smaller H/He masses, mass loss for LMLD planets can actually *transform* the very structure of the planets), we encourage further detailed models to quantitatively access mass loss from these atmospheres. Conclusions =========== In order to better understand the structure, history, and formation of low-mass planets, we constructed coupled thermal evolution and mass loss models of water-poor super-Earths, water-worlds, and water-rich sub-Neptunes. The Kepler-11 system represents a new class of low-mass low-density planets that offers a unique test-bed for such models and gives us powerful insights on planet formation and evolution. Applying this understanding more broadly, we find a relation between a planet’s mass, density, and its incident flux that matches the observed population. Moreover, this threshold can help constrain the properties of hundreds of planets. Our primary conclusions are: - XUV-driven hydrogen mass loss coupled with planetary thermal evolution is a powerful tool in understanding the composition and formation of low-mass low-density planets. - A coupled model is essential for this work, due to the much larger planetary radii in the past, when XUV fluxes were significantly higher. - In situ formation of the Kepler-11 system is disfavored, instead it could be a system of water-rich sub-Neptunes that formed beyond the snow line. - If Kepler-11b is a water-poor super-Earth then it likely formed with $\sim 90\%$ H/He beyond 0.25 AU. We believe this is unlikely and instead show that Kepler-11 b-f all could have originated as water-rich sub-Neptunes with $\sim 20\%$ H/He initially. If this is the case, Kepler-11b could have lost its H/He envelope and become a water-world today for a wide range of initial masses and compositions. - There is a sharp observed threshold in incident flux vs. planet density times mass above which we do not find planets with H/He envelopes. To date, low-density planets have not been found above this threshold. - This mass loss threshold is well reproduced by our coupled thermal evolution and mass loss models. - This threshold can be used to provide limits on planet mass or radius for the large population of low-mass low-density planets without measured densities. - In particular, we have identified promising *Kepler* targets for RV follow-up. [ccccccccc]{}\[h!\] $70.01$ & $12.50$ & $10.85$ & $80.3$ & $3.09$ & $0.73$ & $3.9$ & $1.21$\ \ $70.02$ & $12.50$ & $3.69$ & $343.5$ & $1.92$ & $3.0$ & $3.9$ & $1.75$\ \ $85.01$ & $11.02$ & $5.85$ & $403.4$ & $2.35$ & $2.4$ & $5.8$ & $1.83$\ \ $94.02$ & $12.21$ & $10.42$ & $209.9$ & $3.43$ & $1.0$ & $7.4$ & $1.92$\ \ $104.01$ & $12.90$ & $2.50$ & $233.4$ & $3.36$ & $1.1$ & $7.5$ & $4.08$\ \ $105.01$ & $12.87$ & $8.98$ & $130.3$ & $3.35$ & $0.82$ & $5.6$ & $1.94$\ \ $107.01$ & $12.70$ & $7.25$ & $301.7$ & $3.09$ & $1.4$ & $7.6$ & $2.29$\ \ $110.01$ & $12.66$ & $9.94$ & $220.5$ & $2.92$ & $1.3$ & $5.9$ & $1.62$\ \ $115.02$ & $12.79$ & $7.12$ & $409.1$ & $1.88$ & $3.4$ & $4.2$ & $1.29$\ \ $117.02$ & $12.49$ & $4.90$ & $436.9$ & $1.70$ & $4.1$ & $3.7$ & $1.25$\ \ $122.01$ & $12.35$ & $11.52$ & $108.8$ & $2.78$ & $1.0$ & $3.9$ & $1.06$\ \ $123.01$ & $12.37$ & $6.48$ & $461.4$ & $2.64$ & $2.2$ & $7.4$ & $2.46$\ \ $124.01$ & $12.94$ & $12.69$ & $227.8$ & $3.00$ & $1.2$ & $6.3$ & $1.65$\ \ $246.01$ & $10.00$ & $5.39$ & $404.8$ & $2.53$ & $2.2$ & $6.5$ & $2.27$\ \ $257.01$ & $10.87$ & $6.88$ & $308.6$ & $2.61$ & $1.8$ & $5.9$ & $1.80$\ \ $262.02$ & $10.42$ & $9.37$ & $491.8$ & $2.79$ & $2.1$ & $8.3$ & $2.20$\ \ $277.01$ & $11.87$ & $16.23$ & $177.4$ & $3.82$ & $0.79$ & $8.0$ & $1.91$\ \ $280.01$ & $11.07$ & $11.87$ & $154.9$ & $2.52$ & $1.3$ & $4.0$ & $1.16$\ \ $281.01$ & $11.95$ & $19.55$ & $192.3$ & $3.46$ & $0.95$ & $7.2$ & $1.99$\ \ $285.01$ & $11.57$ & $13.74$ & $180.5$ & $3.38$ & $0.96$ & $6.7$ & $1.61$\ \ $288.01$ & $11.02$ & $10.27$ & $433.9$ & $3.11$ & $1.6$ & $9.2$ & $2.14$\ \ $291.02$ & $12.85$ & $8.12$ & $247.8$ & $2.14$ & $2.2$ & $3.9$ & $1.25$\ \ $295.01$ & $12.32$ & $5.31$ & $339.7$ & $1.77$ & $3.4$ & $3.5$ & $1.22$\ \ $297.01$ & $12.18$ & $5.65$ & $482.1$ & $1.65$ & $4.6$ & $3.7$ & $1.25$\ \ $301.01$ & $12.73$ & $6.00$ & $399.2$ & $1.75$ & $3.8$ & $3.7$ & $1.17$\ \ $323.01$ & $12.47$ & $5.83$ & $166.2$ & $2.17$ & $1.7$ & $3.3$ & $1.21$\ \ $984.01$ & $11.63$ & $4.28$ & $259.7$ & $3.19$ & $1.2$ & $7.4$ & $2.91$\ \ $987.01$ & $12.55$ & $3.17$ & $404.8$ & $1.28$ & $6.1$ & $2.3$ & $1.03$\ \ $1117.01$ & $12.81$ & $11.08$ & $327.5$ & $2.20$ & $2.4$ & $4.7$ & $1.13$\ \ $1220.01$ & $12.99$ & $6.40$ & $441.4$ & $1.95$ & $3.4$ & $4.6$ & $1.52$\ \ $1241.02$ & $12.44$ & $10.50$ & $485.3$ & $3.84$ & $1.3$ & $13.3$ & $3.17$\ \ $1597.01$ & $12.68$ & $7.79$ & $423.6$ & $2.67$ & $2.0$ & $7.2$ & $1.86$\ \ $1692.01$ & $12.56$ & $5.96$ & $175.9$ & $2.65$ & $1.3$ & $4.6$ & $1.61$\ \ $1781.01$ & $12.23$ & $7.83$ & $61.3$ & $3.29$ & $0.58$ & $3.7$ & $1.38$\ \ $1781.02$ & $12.23$ & $3.00$ & $219.6$ & $1.94$ & $2.4$ & $3.2$ & $1.63$\ \ $1921.01$ & $12.82$ & $16.00$ & $172.9$ & $3.09$ & $1.0$ & $5.7$ & $1.28$\ \ $1929.01$ & $12.73$ & $9.69$ & $251.7$ & $2.00$ & $2.4$ & $3.6$ & $1.11$\ \ $2067.01$ & $12.58$ & $13.24$ & $347.2$ & $2.97$ & $1.6$ & $7.6$ & $1.69$\ \ \[masslimtab\] {width="6.0in" height="4.3in"} {width="5.in" height="3.57in"} {width="5.35in" height="3.57in"} {width="5.in" height="6.43in"} {width="6.in" height="4.3in"} {width="6.in" height="4.3in"} {width="6.in" height="3.4in"} {width="6.in" height="4.3in"}

--- abstract: 'We show the first lattice QCD results on the axial charge $g_A^{N^*N^*}$ of $N^*(1535)$ and $N^*(1650)$. The measurements are performed with two flavors of dynamical quarks employing the renormalization-group improved gauge action at $\beta$=1.95 and the mean-field improved clover quark action with the hopping parameters, $\kappa$=0.1375, 0.1390 and 0.1400. In order to properly separate signals of $N^*(1535)$ and $N^*(1650)$, we construct 2$\times$2 correlation matrices and diagonalize them. Wraparound contributions in the correlator, which can be another source of signal contaminations, are eliminated by imposing the Dirichlet boundary condition in the temporal direction. We find that the axial charge of $N^*(1535)$ takes small values as $g_A^{N^*N^*}\sim {\mathcal O}(0.1)$, whereas that of $N^*(1650)$ is about 0.5, which is found independent of quark masses and consistent with the predictions by the naive nonrelativistic quark model.' author: - 'Toru T. Takahashi and Teiji Kunihiro' title: 'Axial charges of N(1535) and N(1650) in lattice QCD with two flavors of dynamical quarks' --- [**Introduction.**]{} Chiral symmetry is an approximate global symmetry in QCD, the fundamental theory of the strong interaction; this symmetry together with its spontaneous breaking has been one of the key ingredients in the low-energy hadron or nuclear physics. Due to its spontaneous breaking, up and down quarks, whose current masses are of the order of a few MeV, acquire the large constituent masses of a few hundreds MeV, and are consequently responsible for about 99% of mass of the nucleon and hence that of our world. Thus one would say that chiral condensate $\langle \bar \psi \psi \rangle$, the order parameter of the chiral phase transition, plays an essential role in the hadron-mass genesis in the light quark sector. On the other hand, chiral symmetry gets restored in systems where hard external energy scales such as high-momentum transfer, temperature($T$), baryon density and so on exist, owing to the asymptotic freedom of QCD. Then, are all hadronic modes massless in such systems? Can hadrons be massive even without non-vanishing chiral condensate? An interesting possibility was suggested some years ago by DeTar and Kunihiro [@DeTar:1988kn], who showed that nucleons can be [*massive even without the help of chiral condensate*]{} due to the possible [*chirally invariant mass terms*]{}, which give [*degenerated*]{} finite masses to the members in the chiral multiplet (a nucleon and its parity partner) even when chiral condensate is set to zero: To show this for a finite-$T$ case, they introduced a linear sigma model which offers a nontrivial chiral structure in the baryon sector and a mass-generation mechanism completely and essentially different from that by the spontaneous chiral symmetry breaking. Interestingly enough, their chiral doublet model has recently become a source of debate as a possible scenario of [*observed parity doubling in excited baryons* ]{} [@Jaffe:2005sq; @Jaffe:2006jy; @Glozman:2007jt; @Jido:1999hd; @Jido:2001nt; @Lee:1972], although their original work [@DeTar:1988kn] was supposed to be applied to finite-$T$ systems. It is thus an intriguing problem to reveal the chiral structure of excited baryons in the light quark sector beyond model considerations. One of the key observables which are sensitive to the chiral structure of the baryon sector is axial charges [@DeTar:1988kn]. The axial charge of a nucleon $N$ is encoded in the three-point function $$\langle N| A_\mu^a |N\rangle = \bar u \frac{\tau^a}{2} [ \gamma_\mu \gamma_5 g_A(q^2) + q_\mu \gamma_5 h_A(q^2) ] u.$$ Here, $A_\mu^a \equiv \bar Q \gamma_\mu \gamma_5 \frac{\tau^a}{2} Q$ is the isovector axial current. The axial charge $g_A$ is defined by $g_A(q^2)$ with the vanishing transferred momentum $q^2=0$. It is a celebrated fact that the axial charge $g_A^{NN}$ of $N(940)$ is 1.26. Though the axial charges in the chiral broken phase can be freely adjusted with higher-dimensional possible terms and cannot be the crucial clues for the chiral structure [@Jaffe:2005sq; @Jaffe:2006jy], they would surely reflect the internal structure of baryons and would play an important role in the clarification of the low-energy hadron dynamics. In this paper, we show the first unquenched lattice QCD study [@Takahashi:2007ti] of the axial charge $g_A^{N^*N^*}$ of $N^*(1535)$ and $N^*(1650)$. We employ $16^3\times 32$ lattice with two flavors of dynamical quarks, generated by CP-PACS collaboration [@AliKhan:2001tx] with the renormalization-group improved gauge action and the mean-field improved clover quark action. We choose the gauge configurations at $\beta=1.95$ with the clover coefficient $c_{\rm SW}=1.530$, whose lattice spacing $a$ is determined as 0.1555(17) fm. We perform measurements with 590, 680, and 680 gauge configurations with three different hopping parameters for sea and valence quarks, $\kappa_{\rm sea},\kappa_{\rm val}=0.1375,0.1390$ and $0.1400$, which correspond to quark masses of $\sim$ 150, 100, 65 MeV and the related $\pi$-$\rho$ mass ratios are $m_{\rm PS}/m_{\rm V}=0.804(1)$, $0.752(1)$ and $0.690(1)$, respectively. Statistical errors are estimated by the jackknife method with the bin size of 10 configurations. Our main concern is the axial charges of the negative-parity nucleon resonances $N^*(1535)$ and $N^*(1650)$ in $\frac12^-$channel. We then have to construct an optimal operator which dominantly couples to $N^*(1535)$ or $N^*(1650)$. We employ the following two independent nucleon fields, $ N_1(x)\equiv \varepsilon_{\rm abc}u^a(x)(u^b(x)C\gamma_5 d^c(x)) $ and $ N_2(x)\equiv \varepsilon_{\rm abc}\gamma_5 u^a(x)(u^b(x)C d^c(x)), $ in order to construct correlation matrices and to separate signals of $N^*(1535)$ and $N^*(1650)$. (Here, $u(x)$ and $d(x)$ are Dirac spinor for u- and d- quark, respectively, and $a,b,c$ denote the color indices.) Even after the successful signal separations, there still remain several signal contaminations mainly because lattices employed in actual calculations are finite systems: Signal contaminations ([*a*]{}) [*by scattering states*]{}, ([*b*]{}) [*by wraparound effects*]{}. [Comment to ([*a*]{}) :]{} Since our gauge configurations are unquenched ones, the negative parity nucleon states could decay to $\pi$ and N, and their scattering states could come into the spectrum. The sum of the pion mass $M_\pi$ and the nucleon mass $M_N$ is however in our setups heavier than the masses of the lowest two states (would-be $N^*(1535)$ and $N^*(1650)$) in the negative parity channel. We then do not suffer from any scattering-state signals. [Comment to ([*b*]{}) :]{} The other possible contamination is wraparound effects [@Takahashi:2005uk]. Let us consider a two-point baryonic correlator $\langle N^*(t_{\rm snk}) \bar N^*(t_{\rm src})\rangle$ in a Euclidean space-time. Here, the operators $N^*(t)$ and $\bar N^*(t)$ have nonzero matrix elements, $\langle 0|N^*(t)|N^*\rangle$ and $\langle N^*|\bar N^*(t)|0\rangle$, and couple to the state $|N^*\rangle$. Since we perform unquenched calculations, the excited nucleon $N^*$ can decay into $N$ and $\pi$, and even when we have no scattering state $|N+\pi\rangle$, we could have another “scattering states”. The correlator $\langle N^*(t_{\rm snk}) \bar N^*(t_{\rm src})\rangle$ can still accommodate, for example, the following term. $$\begin{aligned} &&\langle \pi|N^*(t_{\rm snk})|N\rangle \langle N| \bar N^*(t_{\rm src})|\pi\rangle \nonumber \\ &\times&e^{-E_N(t_{\rm snk}-t_{\rm src})}\times e^{-E_\pi (N_t-t_{\rm snk}+t_{\rm src})}.\end{aligned}$$ Here, $N_t$ denotes the temporal extent of a lattice. Such a term is quite problematic and mimic a fake plateau at $E_N-E_\pi$ in the effective mass plot because it behaves as $\sim e^{-(E_N-E_\pi)(t_{\rm snk}-t_{\rm src})}$. Although these contaminations disappear when one employ enough large-$N_t$ lattice, our lattices do not have so large $N_t$. In order to eliminate such contributions, we impose the Dirichlet condition on the temporal boundary for valence quarks, which prevents valence quarks from going over the boundary. Though the boundary is still transparent for the states with the same quantum numbers as vacuum, [*e.g.*]{} glueballs, such contributions will be suppressed by the factor of $e^{-E_GN_t}$ and we neglect them in this paper. (Wraparound effects can be found even in quenched calculations [@Takahashi:2005uk].) [**Formulation.**]{} We here give a brief introduction to our formulation [@Takahashi:2005uk; @Burch:2006cc]. Let us assume that we have a set of $N$ independent operators, $O_{\rm snk}^I$ for sinks and $O_{\rm src}^{I\dagger}$ for sources. We can then construct an $N\times N$ correlation matrix ${\cal C}^{IJ}(T) \equiv \langle O_{\rm snk}^I(T)O_{\rm src}^{J\dagger}(0)\rangle =C_{\rm snk}^\dagger\Lambda(T)C_{\rm src} $. Here, $ (C^\dagger_{\rm snk})_{Ii}\equiv \langle 0 | O_{\rm snk}^I | i \rangle $ and $ (C_{\rm src})_{jI}\equiv \langle j | O_{\rm src}^{J \dagger} | 0 \rangle $ are general matrices, and $\Lambda(T)_{ij}$ is a diagonal matrix given by $ \Lambda(T)_{ij}\equiv \delta_{ij} e^{- E_iT}. $ The optimal source and sink operators, ${\cal O}_{\rm src}^{i \dagger}$ and ${\cal O}_{\rm snk}^{i}$, which couple dominantly (solely in the ideal case) to $i$-th lowest state, are obtained as $ {\cal O}_{\rm src}^{i \dagger}=\sum_J O^{J\dagger}_{\rm src} (C_{\rm src})^{-1}_{Ji} $ and $ {\cal O}_{\rm snk}^{i}=\sum_J (C^\dagger_{\rm snk})^{-1}_{iJ} O^J_{\rm snk}, $ since $(C^\dagger_{\rm snk})^{-1} {\cal C}(T)(C_{\rm src})^{-1} =\Lambda(T)$ is diagonal. Besides overall constants, $(C_{\rm src})^{-1}$ and $(C^\dagger_{\rm snk})^{-1}$ are obtained as the right and left eigenvectors of ${\cal C}^{-1}(T+1){\cal C}(T)$ and ${\cal C}(T){\cal C}(T+1)^{-1}$, respectively. The zero-momentum-projected point-type operators, $$N_1(t)\equiv \sum_{\bf x}\varepsilon_{\rm abc} u^a({\bf x},t)(u^b({\bf x},t)C\gamma_5 d^c({\bf x},t))$$ and $$N_2(t)\equiv \sum_{\bf x}\varepsilon_{\rm abc}\gamma_5 u^a({\bf x},t)(u^b({\bf x},t)C d^c({\bf x},t)),$$ are chosen for the sinks. For the sources, we employ the following wall-type operators in the Coulomb gauge, $$\overline{N_1}(t)\equiv \sum_{{\bf x_1},{\bf x_2},{\bf x_3}} \varepsilon_{\rm abc}\bar u^a({\bf x_1},t) (\bar u^b({\bf x_2},t)C\gamma_5 \bar d^c({\bf x_3},t))$$ and $$\overline{N_2}(t)\equiv \sum_{{\bf x_1},{\bf x_2},{\bf x_3}} \varepsilon_{\rm abc}\gamma_5 \bar u^a({\bf x_1},t) (\bar u^b({\bf x_2},t)C \bar d^c({\bf x_3},t)).$$ The parity is flipped by multiplying the operator by $\gamma_5$; $N^+_i(t)\equiv N_i(t)$ and $N^-_i(t)\equiv \gamma_5 N_i(t)$. The optimized sink (source) operators ${\cal N}^\pm_i$ ($\overline{{\cal N}^\pm_i}$), which couple dominantly to the $i$-th lowest state are constructed as $$\begin{aligned} {\cal N}_i^\pm(t)&=& N^\pm_1(t) +\left[ {(C^{\pm \dagger}_{\rm snk})^{-1}_{i2}} / {(C^{\pm \dagger}_{\rm snk})^{-1}_{i1}} \right] N^\pm_2(t) \\ &\equiv& N^\pm_1(t) +L_i^\pm N^\pm_2(t),\end{aligned}$$ and $$\begin{aligned} \overline{{\cal N}^\pm_i}(t)&=& \overline{N_1^\pm}(t) +\left[ {(C^\pm_{\rm src})^{-1}_{2i}} / {(C^\pm_{\rm src})^{-1}_{1i}} \right] \overline{N_2^\pm}(t) \\ &\equiv& \overline{N_1^\pm}(t) +R_i^\pm \overline{N_2^\pm}(t).\end{aligned}$$ Now that we have constructed optimized operators, we can easily compute the (non-renormalized) vector and axial charges $g_{V,A}^{\pm{\rm [lat]}}$ for the positive- and negative-parity nucleons via three-point functions with the so-called sequential-source method [@Sasaki:2003jh]. In practice, we evaluate $g_{V,A}^{\pm{\rm [lat]}}(t)$ defined as $$g_{V,A}^{\pm{\rm [lat]}}(t) = \frac { {\rm Tr}\ \Gamma_{V,A} \langle B(t_{\rm snk}) J_\mu^{V,A}(t) \overline B(t_{\rm src}) \rangle } { {\rm Tr}\ \Gamma_{V,A} \langle B(t_{\rm snk}) \overline B(t_{\rm src}) \rangle },$$ and extract $g_{V,A}^{\pm{\rm [lat]}}$ by the fit $g_{V,A}^{\pm{\rm [lat]}}=g_{V,A}^{\pm{\rm [lat]}}(t)$ in the plateau region. $B(t)$ denotes the (optimized) interpolating field for nucleons, and $\Gamma_{V,A}$ are $\gamma_\mu \frac{1+\gamma_4}{2}$ and $\gamma_\mu \gamma_5 \frac{1+\gamma_4}{2}$, respectively. $J_\mu^{V,A}(t)$ are the vector and the axial vector currents inserted at $t$. We show in Fig. \[3pointfunc\] $g_{A}^{-0{\rm [lat]}}(t)$ for $N^*(1535)$ as a function of the current insertion time $t$. They are rather stable around $t_{\rm src}<t<t_{\rm snk}$. ![\[3pointfunc\] The non-renormalized axial charge of $N^*(1535)$, $g_{A}^{-0{\rm [lat]}}(t)$, as a function of the current insertion time $t$. ](ga1535.eps) We finally reach the renormalized charges $g_{A,V}^\pm=\widetilde Z_{A,V}g^{\pm{\rm [lat]}}_{A,V}$ with the prefactors $\widetilde Z_{A,V} \equiv 2\kappa u_0 Z_{V,A} \left( 1+b_{V,A}\frac{m}{u_0} \right) $, which are estimated with the values listed in Ref. [@AliKhan:2001tx]. [1.0]{} [@cccccccccccccc]{} $\kappa$ & $L_1^+$ & $R_1^+$ & $L_1^-$ & $R_1^-$ & $L_2^+$ & $R_2^+$ & $L_2^-$ & $R_2^-$ & $M_\pi$ & $E_1^+$ & $E_1^-$ & $E_2^+$ & $E_2^-$\ 0.1375 & $-$0.4341 & $-$0.4573 & 0.0355 & 0.0126 & $-$1353 & $-$314.1 & $-$1.432 & $-$1.302 & 0.8985(5) & 1.696(1) & 2.137(10) & 2.524(53) & 2.141(14)\ 0.1390 & $-$0.4526 & $-$0.4552 & 0.1115 & $-$0.2036 & $-$845.9 & $-$228.1 & $-$2.729 & $-$1.084 & 0.7351(5) & 1.459(1) & 1.854(13) & 2.162(44) & 1.908(17)\ 0.1400 & $-$0.1605 & $-$0.3552 & 0.0990 & $-$0.0151 & $-$408.9 & $-$143.6 & $-$1.510 & $-$1.038 & 0.6024(6) & 1.270(2) & 1.665(15) & 2.046(67) & 1.733(25)\ C.L. & - & - & - & - & - & - & - & - & - & 0.936(3) & 1.277(25) & 1.570(109)& 1.411(38)\ [1.0]{} [@ccccccccccc]{} $\kappa$ & ${g^{0+{\rm [lat]}}_V}$ & ${g^{0-{\rm [lat]}}_V}$ & ${g^{0+(u){\rm [lat]}}_A}$ & ${g^{0+(d){\rm [lat]}}_A}$ & ${g^{0+{\rm [lat]}}_A}$ & ${g^{0+{\rm}}_V}$ & ${g^{0-{\rm}}_V}$ & ${g^{0+{\rm}}_A}$ & $\widetilde Z_V$ & $\widetilde Z_A$\ 0.1375 & 4.208( 8) & 3.844( 76) & 3.852( 42) & $-$1.073(49) & 4.925( 24) & 1.045(1) & 0.989( 19) & 1.247(8) & 0.2530 & 0.2576\ 0.1390 & 4.492(10) & 4.152(160) & 3.978( 94) & $-$1.244(44) & 5.222(126) & 1.089(1) & 1.036( 60) & 1.261(7) & 0.2446 & 0.2491\ 0.1400 & 4.663( 9) & 4.380(206) & 3.952(136) & $-$1.150(55) & 5.102(145) & 1.115(2) & 1.048(111) & 1.261(8) & 0.2390 & 0.2434\ [1.0]{} [@ccccccccc]{} $\kappa$ & ${g^{0-(u){\rm [lat]}}_A}$ & ${g^{0-(d){\rm [lat]}}_A}$ & ${g^{0-{\rm [lat]}}_A}$ & ${g^{1-(u){\rm [lat]}}_A}$ & ${g^{1-(d){\rm [lat]}}_A}$ & ${g^{1-{\rm [lat]}}_A}$ & ${g^{0-{\rm}}_A}$ & ${g^{1-{\rm}}_A}$\ 0.1375 & 0.336(194) & $-$0.257(118) & 0.592(226) & 3.308(234) & 1.189(209) & 2.119(359) & 0.152(58) & 0.546(093)\ 0.1390 & $-$0.710(251) & 0.081(119) & $-$0.791(272) & 3.423(495) & 1.243(420) & 2.180(730) & $-$0.197(68) & 0.543(182)\ 0.1400 & 0.189(257) & $-$0.129(178) & 0.318(297) & 3.530(516) & 1.339(405) & 2.190(676) & 0.077(72) & 0.533(165)\ We show the fitted values of $L_{1,2}^\pm$ and $R_{1,2}^\pm$ in Table \[fittedval\]. $L(T)$ and $R(T)$ are rather stable and show a plateau from relatively small value of $T$ ($T\sim 2$), which is the same tendency as that found in Ref. [@Burch:2006cc]. We plot in Fig. \[optparam\] $L_i^\pm(T)$ and $R_i^\pm(T)$ obtained at $\kappa$=0.1390, for the purpose of reference. ![\[optparam\] As typical examples, $L_1^-$ and $R_1^-$ obtained at $\kappa$=0.1390 are plotted. ](L01_neg_1390.eps "fig:") ![\[optparam\] As typical examples, $L_1^-$ and $R_1^-$ obtained at $\kappa$=0.1390 are plotted. ](R01_neg_1390.eps "fig:") The energies $E_{1,2}^\pm$ are extracted from two-point correlation functions by the exponential fit as $\langle {\cal N}^\pm_i(t_{\rm src}+T) \overline{{\cal N}^\pm_i}(t_{\rm src}) \rangle = C \exp(-E_i^\pm T)$ in the large-$T$ region. The value at each hopping parameter is found to coincide with that in the original paper by the CP-PACS collaboration [@AliKhan:2001tx], with deviations of 0.1% to 1%. We here perform simple linear chiral extrapolations for $E_{1,2}^\pm$. The chirally extrapolated values as well as those at each hopping parameter for $E_{1,2}^\pm$ in lattice unit are listed in Table \[fittedval\]. Although the mass $E_1^+$ of the ground-state positive-parity nucleon at the chiral limit is overestimated in our analysis ($a^{-1}=1.267$ GeV), this failure comes from our simple linear fit. ![\[AxialVectorC\] The renormalized vector and axial charges of the positive- and the negative-parity nucleons are plotted as a function of the squared pion mass $m_\pi^2$. [**upper panel**]{}: The results of the vector charges. The solid line is drawn at ${g}_V=1$ for reference. [**lower panel**]{}: The results of the axial charges. The solid line is drawn at ${g}_A=1.26$ and the dashed line is drawn at ${g}_A=0$. ](vector.eps "fig:") ![\[AxialVectorC\] The renormalized vector and axial charges of the positive- and the negative-parity nucleons are plotted as a function of the squared pion mass $m_\pi^2$. [**upper panel**]{}: The results of the vector charges. The solid line is drawn at ${g}_V=1$ for reference. [**lower panel**]{}: The results of the axial charges. The solid line is drawn at ${g}_A=1.26$ and the dashed line is drawn at ${g}_A=0$. ](axial.eps "fig:") [**Results.**]{} We first take a stock of the vector charges $g_V^{0\pm}$ of the ground-state positive- and negative-parity nucleons as well as the axial charge $g_A^{0+}$ of the ground-state positive-parity nucleon, which are well known and can be the references. We show $g_V^{0\pm}$, the vector charges of the positive- and the negative-parity nucleons obtained with three hopping parameters $\kappa$=0.1375, 0.1390 and 0.1400, in the upper panel in Fig. \[AxialVectorC\], where the vertical axis denotes $g_V^{0\pm}$ and the horizontal one the squared pion masses. (These values are also listed in Table \[fittedval\].) The vector charges should be unity if the charge conservation is exact, whereas we can actually find about 10% deviations in Table \[fittedval\] or in the upper panel in Fig. \[AxialVectorC\]. Such unwanted deviations are considered to arise due to the discretization errors: The present lattice spacing is about 0.15 fm, which is far from the continuum limit. In fact, the decay constants obtained with the same setup as ours deviate from the continuum values by ${\cal O}(10)$%. We should then count at least 10% ambiguities in our results. The axial charge $g_A^{0+}$ of the positive parity nucleon is also shown in the lower panel in Fig. \[AxialVectorC\]. As found in the previous lattice studies, the axial charge of the positive parity nucleon shows little quark-mass dependence, and they lie around the experimental value 1.26. We finally show the axial charges of the negative-parity nucleon resonances in the lower panel in Fig. \[AxialVectorC\]. One finds at a glance that the axial charge $g_A^{0-}$ of $N^*(1535)$ takes quite small value, as $g_A^{0-}\sim {\mathcal O}(0.1)$ and that even the sign is quark-mass dependent. While the wavy behavior might come from the sensitiveness of $g_A^{0-}$ to quark masses, this behavior may indicate that $g_A^{0-}$ is rather consistent with zero. These small values are not the consequence of the cancellation between u- and d-quark contributions. The u- and d-quark contributions to $g_A^{0-}$ are in fact individually small, which one can find in the columns named as $g_A^{0-(u)[{\rm lat}]}$ and $g_A^{0-(d)[{\rm lat}]}$ in Table \[fittedval\]. We additionally make some trials with lighter u- and d-quark masses at $\kappa$=0.1410. Since we have less gauge configurations and the statistical fluctuation is larger at this kappa, we fail to find a clear plateau in the effective mass plots of the two-point correlators and the extracted mass $E_1^-$ of the negative-parity state cannot be reliable. Leaving aside these failures, we try to extract $g_A^{0-}$. The result is added in the lower panel in Fig. \[AxialVectorC\] as a faint-colored symbol, which is consistent with those obtained at other $\kappa$’s. On the other hand, the axial charge $g_A^{1-}$ of $N^*(1650)$ is found to be about 0.55, which has almost no quark-mass dependence. The striking feature is that these axial charges, $g_A^{0-}\sim 0$ and $g_A^{1-}\sim 0.55$, are consistent with the naive nonrelativistic quark model calculations [@Nacher:1999vg; @Glozman:2008vg], $g_A^{0-}= -\frac19$ and $g_A^{1-}= \frac59$. Such values are obtained if we assume that the wave functions of $N^*(1535)$ and $N^*(1650)$ are $|l=1, S=\frac12\rangle$ and $|l=1, S=\frac32\rangle$ neglecting the possible state mixing. (Here, $l$ denotes the orbital angular momentum and $S$ the total spin.) In the chiral doublet model [@DeTar:1988kn; @Glozman:2007jt], the small $g_A^{N^*N^*}$ is realized when the system is decoupled from the chiral condensate $\langle \bar \psi \psi \rangle$. The small $g_A^{0-}$ of $N^*(1535)$ then does not contradict with the possible and attempting scenario, the [*chiral restoration scenario in excited hadrons*]{} [@Glozman:2007jt]. If this scenario is the case, the origin of mass of $N^*(1535)$ (or excited nucleons) is essentially different from that of the positive-parity ground-state nucleon $N(940)$, which mainly arises from the spontaneous chiral symmetry breaking. However, the non-vanishing axial charge of $N^*(1650)$ unfortunately gives rise to doubts about the scenario. In order to reveal the realistic chiral structure, studies with much lighter u,d quarks will be indispensable. A study of the axial charge of Roper, as well as the inclusion of strange sea quarks could also cast light on the low-energy chiral structure of baryons and the origin of mass. [**Conclusions.**]{} In conclusion, we have performed the first lattice QCD study of the axial charge $g_A^{N^*N^*}$ of $N^*(1535)$ and $N^*(1650)$, with two flavors of dynamical quarks employing the renormalization-group improved gauge action at $\beta$=1.95 and the mean-field improved clover quark action with the hopping parameters, $\kappa$=0.1375, 0.1390 and 0.1400. We have found the small axial charge $g_A^{0-}$ of $N^*(1535)$, whose absolute value seems less than 0.2 and which is almost independent of quark mass, whereas the axial charge $g_A^{1-}$ of $N^*(1650)$ is found to be about 0.55. These values are consistent with the naive nonrelativistic quark model predictions, and could not be the favorable evidences for the chiral restoration scenario in (low-lying) excited hadrons. Further investigations on the axial charges of $N^*(1535)$ or other excited baryons will cast light on the chiral structure of the low-energy hadron dynamics and on where hadronic masses come from. All the numerical calculations were performed on NEC SX-8R at RCNP and CMC, Osaka University, on SX-8 at YITP, Kyoto University, and on BlueGene at KEK. The unquenched gauge configurations employed in our analysis were all generated by CP-PACS collaboration [@AliKhan:2001tx]. We thank L. Glozman, D. Jido, S. Sasaki, and H. Suganuma for useful comments and discussions. This work is supported by a Grant-in-Aid for Scientific research by Monbu-Kagakusho (No. 17540250), the 21st Century COE “Center for Diversity and University in Physics”, Kyoto University and Yukawa International Program for Quark-Hadron Sciences (YIPQS). [99]{} C. DeTar and T. Kunihiro, Phys. Rev.  D [**39**]{}, 2805 (1989). L. Y. Glozman, Phys. Rev. Lett.  [**99**]{}, 191602 (2007) \[arXiv:0706.3288 \[hep-ph\]\]. R. L. Jaffe, D. Pirjol and A. Scardicchio, Phys. Rev. Lett.  [**96**]{}, 121601 (2006) \[arXiv:hep-ph/0511081\]. R. L. Jaffe, D. Pirjol and A. Scardicchio, Phys. Rept.  [**435**]{}, 157 (2006) D. Jido, T. Hatsuda and T. Kunihiro, Phys. Rev. Lett.  [**84**]{}, 3252 (2000) D. Jido, M. Oka and A. Hosaka, Prog. Theor. Phys.  [**106**]{}, 873 (2001) \[arXiv:hep-ph/0110005\]. B. W. Lee, Chiral Dynamics, Gordon and Breach, New York, 1972 T. T. Takahashi and T. Kunihiro, arXiv:0711.1961 \[hep-lat\]. A. Ali Khan [*et al.*]{} \[CP-PACS Collaboration\], Phys. Rev.  D [**65**]{}, 054505 (2002) \[Erratum-ibid.  D [**67**]{}, 059901 (2003)\] T. T. Takahashi, T. Umeda, T. Onogi and T. Kunihiro, Phys. Rev.  D [**71**]{}, 114509 (2005) \[arXiv:hep-lat/0503019\]. T. Burch, C. Gattringer, L. Y. Glozman, C. Hagen, D. Hierl, C. B. Lang and A. Schafer, Phys. Rev.  D [**74**]{}, 014504 (2006) \[arXiv:hep-lat/0604019\]. S. Sasaki, K. Orginos, S. Ohta and T. Blum \[the RIKEN-BNL-Columbia-KEK Collaboration\], Phys. Rev.  D [**68**]{}, 054509 (2003) \[arXiv:hep-lat/0306007\]. J. C. Nacher, A. Parreno, E. Oset, A. Ramos, A. Hosaka and M. Oka, Nucl. Phys.  A [**678**]{}, 187 (2000) \[arXiv:nucl-th/9906018\]. L. Y. Glozman and A. V. Nefediev, arXiv:0801.4343 \[hep-ph\].

--- bibliography: - 'cosmoBell.bib' --- =15.5pt [ ]{} [Daniel Green$^1$ and Rafael A. Porto$^2$]{} *$^1$ Department of Physics, University of California, San Diego, La Jolla, CA 92093, USA* *$^2$ Deutsches Elektronen-Synchrotron DESY, Notkestrasse 85, 22607 Hamburg, Germany* [**Abstract**]{} Structure in the Universe is widely believed to have originated from [*quantum*]{} fluctuations during an early epoch of accelerated expansion. Yet, the patterns we observe today do not distinguish between quantum or classical primordial fluctuations; current cosmological data is consistent with either possibility. We argue here that a detection of primordial non-Gaussianity can resolve the present situation, and provide a litmus-test for the quantum origin of cosmic structure. Unlike in quantum mechanics, vacuum fluctuations cannot arise in classical theories and therefore long-range classical correlations must result from (real) particles in the initial state. Similarly to flat-space scattering processes, we show how basic principles require these particles to manifest themselves as poles in the $n$-point functions, in the so-called folded configurations. Following this observation, and assuming fluctuations are [*(i)*]{} correlated over large scales, and [*(ii)*]{} generated by local evolution during an inflationary phase, we demonstrate that: [*the absence of a pole in the folded limit of non-Gaussian correlators uniquely identifies the quantum vacuum as the initial state*]{}. In the same spirit as Bell’s inequalities, we discuss how this can be circumvented if locality is abandoned. We also briefly discuss the implications for simulations of a non-Gaussian universe. Introduction ============ Cosmological observations strongly suggest that structure in the universe originated from minute fluctuations present in the very early universe, prior to the hot big bang [@Hu:1996yt; @Spergel:1997vq; @Dodelson:2003ip]. A compelling possibility is that these density perturbations were produced through quantum mechanical zero-point fluctuations in the vacuum [@Mukhanov:1981xt; @Hawking:1982cz; @Guth:1982ec; @Starobinsky:1982ee; @Bardeen:1983qw], and then were stretched over long distances by rapid accelerated expansion (inflation). In one brush, this idea unveils a beautiful connection between the largest structures in the cosmos and the fundamental laws of physics at the smallest scales. Yet, current data [@Akrami:2018odb; @Akrami:2019izv] could equally be explained if inflation had stretched classical statistical fluctuations instead. In the same fashion as Bell’s program back in the 1960’s put quantum mechanics to the test [@Bell:1964kc], our goal here is to bring the quantum origin of the density fluctuations, realized in a majority of models, into a well-defined statement that can be confronted with future observations. Unfortunately, one cannot simply perform experiments with the entire universe. We only get to observe the one we inhabit, and only have access to an effectively classical probability distribution of fluctuations [@Grishchuk:1990bj]. Classic tests of quantum mechanics, such as Bell’s inequalities [@Bell:1964kc], cannot be directly applied in this case. As a result, despite a long history (e.g. [@Starobinsky:1986fx; @Grishchuk:1990bj; @Campo:2005sv; @Lim:2014uea; @Martin:2015qta; @Goldstein:2015mha; @Nelson:2016kjm; @Choudhury:2016cso; @Martin:2017zxs; @Shandera:2017qkg; @dePutter:2019xxv]), there has been limited progress identifying observational connections between the quantum initial state and the classical universe we observe today. Recently, a step towards a potential signature was suggested by Maldacena [@Maldacena:2015bha]. For a judiciously chosen model, the dynamics during inflation effectively performs a Bell-type measurement, storing the result in the final probability distribution. The proposal does not suggest a generic observational test; yet, although baroque, Maldacena’s model is a proof of principle that the primordial fluctuations can [*remember*]{} their quantum origin. In this letter we will pursue these ideas further, and provide a testable prediction of the quantum nature of the initial state. We will argue that non-linear local evolution of the density fluctuations can indeed store its quantum origin in the correlations observed at late times. Concretely, we will show how only quantum mechanics can produce the type of long-range correlations typical of the vacuum state, while classical fluctuations are necessarily produced by (highly-excited) states with their own characteristic features. The basic picture, illustrated in Fig. \[fig1\], is the following: Non-Gaussian correlations in the quantum-vacuum are associated with ‘particle-creation’. In contrast, and due to locality, causal classical evolution must also include the decay of particles in the initial state. Hence, even though both vacuum and classical effects produce correlations on large scales at late times, the latter necessarily encode its distinctive physical origin, yielding distinguishable signatures from the case of quantum-vacuum fluctuations. In particular — in analogy with flat-space [*polology*]{} [@Weinberg:1995mt] — an associated pole must be present for classical $n$-point functions (beyond the power spectrum). Moreover, a [*width*]{} will also be generated, through dissipation [@Berera:1995ie; @Berera:1998px; @Green:2009ds; @LopezNacir:2011kk; @LopezNacir:2012rm; @Turiaci:2013dka], which effectively smooths these poles to produce a [*bump*]{} at physical momenta, as in particle colliders. The existence of poles by itself may not be sufficient to show that classical physics is the culprit. For instance, quantum excited states can also develop the same pole structure [@Flauger:2013hra]. Yet, we will demonstrate that [*the absence of this signature — in otherwise observable long-range non-Gaussian correlations — can only be explained by quantum zero-point effects.*]{} In other words, in a classical framework consistent with locality, tampering with the analytic structure of the correlators in an attempt to remove the poles, will unavoidably alter the structure at large scales, as expected from our intuition in flat space. On the other hand, long-range correlations — as those featured in the vacuum state — may be produced without the associated poles if locality is violated. We will illustrate the role of local causal evolution in an illuminating example. Our analysis is also motivated by the practical issue of simulating a universe with non-Gaussian initial conditions. Typically, generating initial conditions with non-local correlations from a Gaussian map requires high-dimensional integration [@Smith:2006ud; @Schmidt:2010gw; @Scoccimarro:2011pz]. If these initial conditions were generated by local classical evolution instead, one could simply produce them via a Gaussian map evolved in time, and potentially speed up the simulations.[^1] However, as we show here, such a procedure — or any local evolution for that matter — will not accurately reproduce the non-Gaussian probability distribution obtained from quantum fluctuations. This result may also have some deeper relevance in quantum versus classical computing. ![Late-time observations measure correlations of the adiabatic density fluctuation, $\zeta(\x,\tau)$, produced from non-linear time evolution in the early universe. The particle’s propagation is illustrated by the solid lines, while the dashed line represents the absence of the corresponding mode at late times. [*Left:*]{} Quantum-vacuum fluctuations arise as the correlated production of three particles due to non-linear effects. This process would violate energy conservation in flat space, and thus produces no poles at physical momenta [@Flauger:2013hra]. [*Right:*]{} Classical fluctuations only arise in a state containing physical particles, as local variations in the particle density, e.g. [@Berera:1995ie; @Berera:1998px; @Green:2009ds; @LopezNacir:2011kk; @LopezNacir:2012rm; @Turiaci:2013dka]. The non-linear evolution that leads to net particle creation also allows for decays (or annihilation). These processes appear as poles at physical momenta.[]{data-label="fig1"}](figure_v2.pdf){width="\textwidth"} Cosmic Quantumness ================== We are interested in comparing the predictions of quantum and classical physics for the statistics of the initial density perturbations, assuming that the non-Gaussianity is produced from local non-linear evolution, namely it is not present in the initial state. Gaussian Fluctuations {#gaussian-fluctuations .unnumbered} --------------------- For concreteness, we will assume that the adiabatic density fluctuations, $\zeta(\x,\tau)$, arise from an effectively massless field during inflation propagating in a de Sitter background,[^2] ds\^2 = -dt\^2 + a(t)\^2 d\^2 = a()\^2 (-d\^2 + d \^2) = (-d\^2 + d \^2) , with $a$ the scale factor in physical $(t)$ and conformal $(\tau)$ time, respectively. Recall the (constant) Hubble expansion parameter is given by $H \equiv \dot a(t)/a(t)$, where we use the notation (throughout this paper) $\dot f \equiv \partial_t f = a^{-1}\partial_\tau f$, for derivatives w.r.t. the physical time. The modes of the density perturbations obey (,) = e\^[i ]{} \[ a\^\_(1-i k) e\^[i k ]{} +a\_[-]{} (1+i k) e\^[-i k ]{} \]  , \[eq:norm\] where the normalization, $\Delta_\zeta$, is so chosen to coincide with the observed amplitude of adiabatic fluctuations. Since the field is real, we have $(a^\dagger_\k)^\dagger= a_{-\k}$. The statistical differences arise once we compare quantum versus classical correlation functions. - [**Quantum**]{}. The $a^\dagger_\k$ are creation [*operators*]{} in a Hilbert space, satisfying: \[a\^\_, a\_[’]{}\] = (- ’) a\_|0 = 0, which readily imply 0| a\_[’]{} a\^\_|0 = (-’), 0| a\^\_a\_[’]{}|0 = 0 , in the vacuum state. In what follows, we will define $\langle 0| [\ldots]|0\rangle \to \langle [\ldots]\rangle_q$ for convenience. - [**Classical**]{}. The $a^\dagger_\k$ are stochastic [*parameters*]{}, which obey the following statistical properties: a\^\_a\_[’]{}\_c = (-’) = a\_[’]{} a\^\_\_c, as an ensemble average. Notice that the second equality is only valid for classical fluctuations, since it implies that the stochastic parameters commute. Although produced by different mechanisms, both the classical and quantum-vacuum fluctuations are normalized to give rise to the same equal-time correlation function, \_() \_[’]{}() = (1+k\^2 \^2) (2 )\^3 (+’) , in the absence of interactions. Therefore, measurements of the power spectrum alone are not sufficient to distinguish between them. On the other hand, for unequal times the quantum and classical two-point functions do not agree, reflecting the non-zero commutator, $[\zeta(\x,\tau),\dot \zeta(\x,\tau)]\neq 0$, in the quantum theory. This distinction plays a key role when interactions are present. We will illustrate the main difference explicitly in the next section. However, we can also understand the root of the discrepancy as follows. Let us introduce the real and imaginary parts of $\zeta$, such that \[eq:sines\] (,) = e\^[i ]{}  , with $a_\k = \frac{1}{2} [a_{R,\k} + i a_{I,\k}]$. In these variables, the classical statistics obey a\_[R, ]{} a\_[R,’]{} \_c = (+’) = a\_[I, ]{} a\_[I,’]{} \_c , which implies that the independent fluctuations are sines and cosines. On the other hand, in relativistic quantum mechanics — due to locality/causality — we speak instead of positive [*and*]{} negative frequency (energy) modes. The former are [*annihilated*]{} (by definition) in the vacuum state, while virtual particles can be produced. This textbook observation is the key that allows us to demonstrate how non-linear interactions can discern between classical and quantum correlations. Non-Gaussianity {#non-gaussianity .unnumbered} --------------- In order to gain intuition, we will consider an illustrative example with the interaction Hamiltonian $H_{\rm int} = -\frac{\lambda}{3!} \dot \zeta^3$. This choice will allow us to perform explicit computations without losing generality.[^3] As we shall see, our conclusions will be rooted in well-established principles, and therefore do not depend on the type of interaction as long as it respects locality (see Appendix \[app:comm\]). [**Quantum**]{}. The standard (in-in) calculation [@Weinberg:2005vy] in the vacuum state yields (with $|\k_i|=k_i$) \[vacuum\] \_[\_1]{} \_[\_2]{} \_[\_3]{} ’\_q &=& id’\ &=& 2 [Im]{} \_[-]{}\^0 d’ ’\^2 e\^[i (k\_1+k\_2+k\_3)’]{}\ &=&  , up to the momentum conserving $\delta$-function, which is denoted by the primed brackets $\langle \rangle'$. Notice, for $k_i\neq 0$, we have a pole in the total energy: $k_t \equiv k_1+k_2+k_3$. This is due to the fact that, for cosmological (in-in) correlators, the [*would-be*]{} energy-conserving $\delta$-function becomes a factor of $1/k_t^n$, for non-negative integer $n$. Via analytic continuation, as $k_t \to 0$, the residue of this pole is intimately connected to the flat-space $S$-matrix, with the order of the pole ($n=3$ in this case) related to the number of derivatives at the local interaction. In the quantum vacuum, the correlation is produced by the creation of three (virtual) particles ($0 \to 3$), which are subsequently [*measured*]{} at later times (see Fig. \[fig1\]). The uncertainty principle in an expanding universe permits a — minimal amount of — violation of the conservation of energy, $\Delta t \sim H^{-1}$, which is forbidden classically. As expected, since there are no real particles to scatter in the vacuum, there are no other processes allowed nor poles at physical momenta. [**Classical**]{}. We determine the bispectrum by solving the classical equations of motion perturbatively. Using the Green’s function, obeying \[eq:Green2\] \_[’]{}G\_(, ’) = 2 \_\^2 , we find at first order in $\lambda$ \[eq:classicalevol\] \^[(2)]{}\_() = (-H’)\^[-1]{} ( \_[’]{} G\_(,’) ) \_[’]{}\^[(1)]{}\_(’) \_[’]{}\^[(1)]{}\_[-]{} (’)  , where $\zeta_\k^{(1)}$ represents the Gaussian field. Hence, using $\zeta_\k \approx \zeta_\k^{(1)}+\zeta_\k^{(2)}$, the leading contribution to the bispectrum at $\tau = 0$ becomes \_[\_1]{} \_[\_2]{} \_[\_3]{} ’\_c =\ = \[classic\] . As anticipated, there are poles at physical momenta in addition to the one at $k_t=0$. These poles are due to classical fluctuations of physical (real) particles in the initial state, which (non-linearly) interact to produce long-range non-Gaussian correlations (see Fig. \[fig1\]). For instance, physical particles can decay (annihilate) via on-shell $1 \to 2$ ($2\to1$) processes, and therefore are associated with the poles in the so-called [*folded limit*]{} [@Babich:2004gb], where $k_1 \to k_2 +k_3$ and permutations thereof. [**Signatures of Quantum Origin:**]{}. The above example illustrates a general property of (in-in) inflationary correlation functions: poles at physical momenta arise from the scattering of real particles present in the initial state. For quantum-vacuum fluctuations there are no real particles, only virtual, yet the poles are still present (by analytic continuation) at negative energies. This is more than just an isolated result mimicking our flat-space intuition. In fact, notice that the overall coefficients of the poles, either in the quantum (\[vacuum\]) or classical correlation (\[classic\]) are related, and ultimately linked to the scattering amplitude in the flat-space limit [@Maldacena:2011nz; @Raju:2012zr; @Arkani-Hamed:2015bza; @Arkani-Hamed:2018kmz; @Arkani-Hamed:2018bjr; @Benincasa:2018ssx]. Hence, following basic principles, causality guarantees that any process that creates (real) particles at local events, is necessarily accompanied by physical poles in the correlation functions [@Lehmann:1954rq]. The specific form of the interaction controls the resulting polynomial in momentum and/or time dependence, and hence only affects the residue of the poles.[^4] Let us emphasize that this is an unavoidable conclusion, which does not depend on the form of the (local) interaction. As a consequence, since there are no vacuum fluctuations in classical mechanics, quantum mechanics is the only way we can guarantee a non-Gaussian signal without violations of locality/causality, while avoiding the existence of poles at physical momenta. As usual [@Weinberg:1995mt], decay processes will introduce a finite width which softens the behavior in the folded limit. However, unlike the drift towards the complex plane found in flat space, for (in-in) correlators in an expanding universe the poles move away from the ‘mass-shell’ but remain real. While the existence of a width usually happens at higher orders in perturbation theory, models with strong dissipation will exhibit this softening already at tree-level [@LopezNacir:2011kk; @LopezNacir:2012rm], see Appendix \[app:diss\]. Classical Non-localities ======================== A crucial aspect of Bell’s inequalities is that they may be circumvented by non-local theories with hidden-variables at the classical level [@Bell:1964kc; @Bohm:1951xw; @Bohm:1951xx]. Similarly, locality plays a key role in inferring the quantum nature of the cosmological signal. For our purposes here, it will be sufficient to find an example of a theory which reproduces the same correlators as in the quantum vacuum, but violates locality. At the same time, we will show that enforcing local causal evolution, while attempting to remove the poles, also alters the type of long-range correlations that are expected in the vacuum state. Hidden-variables {#hidden-variables .unnumbered} ---------------- For illustrative purposes, we consider a [*complex*]{} scalar field, which may be decomposed as \_() = \[ a\^\_ (1-i k ) e\^[i k ]{}+ b\_[-]{} (1+i k ) e\^[-i k ]{}\], obeying classical Gaussian statistics, \[eq:complex\] a\^\_a\_[’]{} \_c = a\_[’]{} a\^\_\_c = (-’), b\_b\^\_[’]{} \_c = 0. Let us assume now the existence of a Lagrangian such that the following modified Green’s function \_[’]{}G\_(0, ’) G\^[eff]{}\_(0,’) = e\^[- i k ’]{} ,\[green2\] applies in the $\tau \to 0$ limit. Notice that it includes only positive frequency modes. We will not specify the nature of the interaction leading to the above properties, and therefore we treat it as a hidden-variable theory. Using with $\zeta \to \phi$ and $\lambda \to \lambda_\phi$, we find \[eq:nonlocal\] \_(0) = \_ d’ e\^[- i k ’]{} \^\*\_(’) \^\*\_[-]{}(’). From this expression we can calculate the bispectrum as usual, yielding \[eq:nonlocal\_bi\] \_[\_1]{}\_[\_2]{} \_[\_3]{} ’ &=& i \_H\^[-1]{} \_[-]{}\^0 d’ ’\^2 e\^[-i (k\_1+k\_2+k\_3)’]{}\ &=& . Since $\phi$ is real at late times, it can be converted into density fluctuations after inflation, e.g. $\phi(\tau \to 0 )\approx \zeta$. Up to an overall constant, the result reproduces the same statistical map for the quantum vacuum in (up to higher order effects which are not relevant here). This theory, however, is non-local as we can see directly from . In particular, locality demands that the Green’s function in coordinate space must vanish outside of the light-cone. Yet, we have $G^{\rm eff}_{\k}(\tau' \to 0) \simeq k^{-1}$, resembling the Coulomb potential, which is non-zero everywhere in space. As a consequence, this theory propagates information instantaneously everywhere in space-time. Causality {#causality .unnumbered} --------- The failure in the above example is rooted in basic principles. Causality in a relativistic theory demands the presence of a negative frequency mode (‘anti-particle’) [@feynman_feynman_weinberg_1987],[^5] which is precisely what gives rise to the poles at physical momenta.[^6] Local causal evolution requires that the Green’s function (effectively) must take the form \[eq:local\_Green\] G\^[eff,causal]{}\_(0, ’) (k ’), as seen in (\[eq:Green2\]), including the negative frequency modes. Unlike three anti-particles annihilating into nothing, as in the non-local example, a causal theory — with a non-zero number of particles in the initial state — must also produce poles at physical momentum (in the folded limit), from particle-creation and annihilation at local events. In principle, the reader may object that causality could allow for a Green’s function which is analytic everywhere in $\k$, even at finite $\tau'$, e.g. G\^[eff,causal]{}\_(0, ’) = ’\^ k\^[2]{},\[template2\]where $\beta$ is a non-negative integer. By construction, causality is preserved without positive and negative frequency modes. However, because of analyticity, the Green’s function is localized in real space. Hence, while it always vanishes outside the light-cone, it also does not generate long-range correlations. Such a Green’s function can only arise in the limit in which the particle’s velocities are small. This is, of course, consistent with the absence of anti-particles in non-relativistic theories. However, this theory must be ‘UV-completed’ into a (local) relativistic one, which will then re-introduce positive and negative frequency modes. More generally, Green’s functions with non-analytic behavior in $\k$, necessarily require cancellations between positive and negative frequency modes to remain causal in the $\tau'\to 0$ limit. This is precisely the case in our example, see e.g. (\[eq:local\_Green\]). (Amusingly, using templates of the form in , the associated bispectra takes a form similar to the ones used in computationally efficient simulations [@Scoccimarro:2011pz].) The analytic properties of the correlators can also be modified due to the presence of a finite width. In that case, the positive and negative frequency modes will appear in the Green’s function, but the would-be divergence is regulated in the folded limit. Hence, the poles are moved slightly away from their physical ‘on-shell’ values. Similarly to the non-relativistic case, a finite width can also alter the type of long-range correlations in otherwise local/causal theories. We elaborate on this point in Appendix \[app:diss\]. Conclusions and Outlook ======================= The origin of structure as a result of vacuum fluctuations is a purely quantum mechanical phenomenon, for classical effects can only arise when states contain (many) physical particles. Moreover, due to causality, non-linear interactions that allow for the creation of particles must be accompanied by processes in which particles are also able to decay. While the creation of (virtual) particles is allowed, decays are forbidden in vacuum, which gives rise to a dramatic difference in the types of non-Gaussian correlations arising in classical versus quantum-vacuum fluctuations. The distinction between the two results, as well as the role of locality, is also manifest in the manipulations involved in the derivation of the three-point function. In general, one can show that the difference between the quantum-vacuum and classical computation may be written as: $$\begin{aligned} &\qquad \qquad \qquad \qquad\qquad\langle \zeta(\x_1,\tau)\zeta(\x_2,\tau) \zeta(\x_3,\tau)\rangle_q -\langle \zeta(\x_1,\tau)\zeta(\x_2,\tau) \zeta(\x_3,\tau)\rangle_c = \\ & \frac{i \lambda}{24} \sum_\sigma \int_{-\infty}^{\tau} d^3 \x' d\tau' a^4(\tau') [\zeta(\x_1, \tau),\hat D_{\sigma(1)} \zeta(\x', \tau')][\zeta(\x_2, \tau),\hat D_{\sigma(2)} \zeta(\x', \tau')][\zeta(\x_3, \tau),\hat D_{\sigma(3)} \zeta(\x', \tau')]\,, \nonumber\end{aligned}$$ where $\hat D_{\ell=1,2,3}$ are local differential operators that characterize the type of interaction(s), and $\sigma$ is a permutation (see Appendix \[app:comm\]). The above expression neatly illustrates the link between late time measurements in a quantum state and Bell-type correlations at an earlier time, which are encoded in the (non-vanishing) commutators. Moreover, because of causality, the commutators vanish at space-like separation. Therefore, the above difference is built up from interactions in the overlap between the past light-cones of the points $\x_1,\x_2$ and $\x_3$. This implies that the information encoded in the correlators cannot be modified by local operations at late times. For quantum-vacuum fluctuations, the absence of a pole in the [*folded*]{} limit of the bispectrum thus becomes a unique signature of local causal evolution. Limits of various $n$-point functions have been known to encode important physical information, e.g. [@Maldacena2; @Creminelli; @Assassi:2012zq; @Flauger:2013hra; @Goldberger:2013rsa; @Arkani-Hamed:2015bza; @Holman:2007na; @LopezNacir:2011kk]. An enhanced ‘soft limit’ (with soft internal or external momenta) is due to additional (light) fields, while the folded limit is enhanced for excited states. Yet, as we have demonstrated here, the [*absence*]{} of an enhancement in folded configurations cannot occur with classical fluctuations, which would provide — barring violations of locality — striking evidence for the quantum origin of structure in the Universe. Although current observations are consistent with a Gaussian spectrum, surveys of increasing volume and sensitivity will continue the search for non-Gaussianity [@Meerburg:2019qqi]. A true pole in the folded limit of the $n$-point functions would have diverging signal-to-noise [@Babich:2004gb]. While the non-zero width of physical particles will make the signal-to-noise finite, it is detectable nonetheless. However, enhanced dissipation may increase the width, thus reducing the signal-to-noise in the folded limit. At the same time, dissipation also increases the amplitude of the overall non-Gaussianty [@LopezNacir:2011kk; @Turiaci:2013dka; @Porto:2014sea], which is presently constrained by data [@Akrami:2019izv]. Hence, as a matter of principle, it is possible (but potentially challenging) to distinguish between the spectrum of classical and quantum fluctuations. Moreover, as we have emphasized, the analytic structure of their respective shapes is clearly distinct, which suggests an analysis in position-space might provide a more stringent test, perhaps along the line discussed in [@Munchmeyer:2019wlh]. Even though we have restricted ourselves here to the case of density fluctuations, our results are also relevant for the quantum origin of primordial gravitational waves [@Senatore:2011sp; @Porto:2014sea; @Shandera:2019ufi]. A detection would both be a measure of the energy scale of inflation and provide a putative signal of quantum gravity. However, such a signal could also be the consequence of classical production mechanisms [@Senatore:2011sp; @Porto:2014sea]. Yet, as emphasized in [@Porto:2014sea; @Mirbabayi:2014jqa], classical production of gravitational waves also introduces a measurable non-Gaussian signature in the density perturbations. As we have shown, the [*shape*]{} of the associated non-Gaussianity will ultimately reveal the origin of these fluctuations. Finally, the question we addressed here is also of potential interest beyond cosmological applications. We have shown that a local classical algorithm cannot reproduce the non-Gaussian correlations found from quantum evolution. This is a limitation of classical computing. It is unclear, on the other hand, if there exists a quantum algorithm that offers a significant computational advantage over classical approaches which include non-local (acausal) evolution. Nevertheless, it suggests an interesting connection between (quantum) cosmology and (quantum) computing, which deserves further study. [**Acknowledgements**]{} We are grateful to Daniel Baumann, Roland de Putter, Raphael Flauger, John McGreevy, Alec Ridgway, Uroš Seljak, Benjamin Wallisch and Matias Zaldarriaga for helpful discussions. We would like to thank also the participants of the ‘Amplitudes meet Cosmology’ workshop[^7] for useful conversations. D.G. is supported by the US Department of Energy under grant no. DE-SC0019035. R.A.P. acknowledges financial support from the ERC Consolidator Grant “Precision Gravity: From the LHC to LISA" provided by the European Research Council (ERC) under the European Union’s H2020 research and innovation programme (grant agreement No. 817791), as well as from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy (EXC 2121) ‘Quantum Universe’ (390833306). Bell-type Correlations {#app:comm} ====================== For our example in the main text, with $H_{\rm int} = -\frac{\lambda}{3!} \dot \zeta^3$, the three-point function in the in-in formalism may be written as: && (\_1,)(\_2,) (\_3,)\_q = -i\_[-]{}\^ d’ d\^3 ’ a\^4(’)\ && = \_[-]{}\^[t]{} dt’ d\^3 x’ a\^3(t’) , where in the second equality we restored the physical time, with $\dot G(t,t') =\partial_{t'} G(t,t')$, for notational convenience. As it turns out, the only difference from the classical calculation is the operator ordering. For Gaussian correlators, the quantum and classical power spectra are in fact related by (, ) (’,’) \_c= , where, for the case of a de Sitter background, we have $\dot \zeta(\tau) =-H \tau \partial_{\tau}\zeta(\tau)$. As a result, when computing correlation functions in perturbation theory we can substitute (, ) (’,’) \_q &=& (x, ) (’,’) \_c+\ &=&(, ) (’,’) \_c + H’ \_[’]{}G(,’;\_1-’). Hence, the difference between the quantum and classical three-point functions becomes &(\_1,)(\_2,) (\_3,)\_q - (\_1,)(\_2,) (\_3,)\_c =\ &+ \_[-]{}\^ d’ d\^3 x’ a\^4(’) \[(\_1, ),(’, ’)\]\[(\_2, ),(’, ’)\]\[(\_3, ),(’, ’)\] , where the commutators are $c$-numbers in this case. The above derivation can be extended to generic cubic interactions of the form $H_{\rm int} = -\frac{\lambda}{3!}\Pi_\ell(\hat D_\ell \zeta )$, with $\hat D_{\ell=1,2,3}$ some local differential operators. In such cases, we find $$\begin{aligned} &\qquad \qquad \qquad\qquad\quad \zeta(\x_1,\tau)\zeta(\x_2,\tau) \zeta(\x_3,\tau)\rangle_q - \langle \zeta(\x_1,\tau)\zeta(\x_2,\tau) \zeta(\x_3,\tau)\rangle_c = \\ & \frac{i \lambda}{24} \sum_\sigma \int_{-\infty}^{\tau} d\tau' d^3 \x' a^4(\tau') [\zeta(\x_1, \tau),\hat D_{\sigma(1)} \zeta(\x', \tau')][\zeta(\x_2, \tau),\hat D_{\sigma(2)} \zeta(\x', \tau')][\zeta(\x_3, \tau),\hat D_{\sigma(3)} \zeta(\x', \tau')]\,, \nonumber\end{aligned}$$ where $\sigma$ is a permutation of $(1,2,3)$. This result can also be generalized to higher order interactions, with $\ell>3$. Although somewhat expected, this formula explicitly reveals how the quantum answer differs from the classical computation, by an integral over the product of three commutators. As we demonstrated here, this fact has deep consequences for the analytic properties of the correlation functions. Decay Width {#app:diss} =========== When particles are unstable, the Green’s function develops new analytic structure. This can be illustrated simply for the harmonic oscillator. In the presence of a dissipative term we have + +\_0\^2 = [O]{}(t), where ${\cal O}$ is the associated [*noise*]{}, expected from the fluctuation-dissipation theorem. Assuming strong dissipation, $\gamma \dot \zeta \gg \ddot \zeta$, the Green’s function becomes, G\_() G\_(t-t’) e\^[-(t-t’)]{}(t-t’), with $\Gamma=\omega_0^2/\gamma$. The pole has moved to the complex plane, signaling an exponential decay in time. If the dissipation is strong enough, the power spectrum will be then dominated by the noise. This happens in an expanding universe, provided $\gamma \gg H$ [@LopezNacir:2011kk]. However, this is not the only change. The exact Green’s function becomes more elaborate, see e.g. [@LopezNacir:2011kk]. In the limit $\tau \to 0$ and $\tau' \to \infty$, it takes the form G\_(z, z\^) ( z’),\[greenb\] where $\nu = 3/2+ \gamma/(2H)$, $z = - k c_s \tau$ and $z' = -k c_s \tau'$, with $c_s$ the sound-speed. Notice the exponential decay in physical time has become polynomial decay in conformal time. In the strong dissipative regime, the non-Gaussianity is also dominated by the noise. Assuming the latter to be Gaussian, the former is produced by non-linear effects. As shown in [@LopezNacir:2011kk], the dominate term scales as $\gamma \k^2 \zeta^2$, generated by non-linear corrections to the response. The derivation is somewhat involved [@LopezNacir:2011kk], yet the basic features in the bispectrum are captured by the template B(k\_i,k\_c) &,\[template\] plus permutations, with $k_c \sim k_\star \log(\gamma/H)$ for some reference scale $k_\star$. The functions $F(k_i,k_c)$ and $G(k_i,k_c)$ are analytic in the momenta, and made out of products of the sort e.g. $k_1^2 k_2^2$ and $k_1^2k_2^3$, respectively, and $f_{\rm NL}^{\rm eq}$ is the standard non-Gaussianity parameter in the equilateral configuration, e.g. [@Babich:2004gb]. (A similar template was introduced in [@Turiaci:2013dka] with $k_c/k_\star\simeq 3/4$ in the regime studied there.) The presence of the (shifted) poles is thus manifest.[^8] There are, however, two main differences with respect to our flat-space intuition. First of all, the poles remain on the real axis, and secondly the width depends [*logarithmically*]{} on $\gamma/H$. This is due to the properties of the Green’s function in . At the end of the day, the width smooths the would-be pole in the folded limit, producing instead an enhanced signal which is correlated to the size of the non-Gaussianity [@LopezNacir:2011kk; @Porto:2014sea]. In principle, for sufficiently large $\gamma/H$, the pole may become ‘too wide’ to be observed. While still classical, in practice this could be read as having no poles, thus misinterpreted as zero-point fluctuations. Yet, from the scaling with momenta in , we see that with the poles removed the bispectrum does not reproduce the long-range correlations found in the vacuum. This can be seen already with the first term, which yields e.g. $1/(k_2^3k_3^3)$, that is clearly more [*localized*]{} in space than vacuum fluctuations. [^1]: We thank Uroš Seljak for first raising this point. [^2]: For illustrative purposes, we will ignore slow-roll corrections, e.g. the spectral tilt. [^3]: We assume cubic interactions are not forbidden by symmetries. (Notice, though feeble, gravity unavoidably sets a floor for the bispectrum [@Maldacena2; @Cabass:2016cgp].) It is straightforward to show the same result applies to all the $n$-point functions. [^4]: The overall cancellation in vacuum of the (intermediate) poles arising at physical momenta can be also seen as the appearance of negative residues, due to virtual processes. In fact, the structure of the poles is a direct consequence of the presence of both positive [*and*]{} negative frequency modes in the classical computation, whereas the vacuum annihilates the former. For classical (highly-excited) states, there is a mismatch between positive and negative contributions, due to real particles in the initial state, leading to the remaining poles seen in . [^5]: <https://www.youtube.com/watch?v=MDZaM-Bi-kI> [^6]: We can see this connection more explicitly as follows: The Gaussian theory preserves an effective $U(1)$ charge and (\[eq:complex\]) corresponds to an excitation of particle anti-particle pairs. The non-linear evolution in (\[eq:nonlocal\]) breaks this charge conservation and allows three anti-particles to annihilate into nothing (energy need not be conserved in the cosmological background). The correlation we observe in (\[eq:nonlocal\_bi\]) is the net gain of 3 particles due to this annihilation. [^7]: [ https://www.simonsfoundation.org/event/amplitudes-meet-cosmology-2019/]( https://www.simonsfoundation.org/event/amplitudes-meet-cosmology-2019/) [^8]: We can also see the smoothing of the poles explicitly in the derivation of the three-point function. For example, consider the simple case of non-Gaussian noise with a non-zero three-point function. Then, in the folded limit, $B(k_1,k_2,k_3) \sim \int_{-\infty} d\tilde\tau \tilde\tau^{4-3\left(1+\gamma/(2H)\right)}$, which is finite provided $\gamma/H>0$.

--- abstract: | Counterexamples to some old-standing optimization problems in the smooth convex coercive setting are provided. We show that block-coordinate, steepest descent with exact search or Bregman descent methods do not generally converge. Other failures of various desirable features are established: directional convergence of Cauchy’s gradient curves, convergence of Newton’s flow, finite length of Tikhonov path, convergence of central paths, or smooth Kurdyka-Łojasiewicz inequality. All examples are planar.\ These examples are based on general smooth convex interpolation results. Given a decreasing sequence of positively curved $C^k$ convex compact sets in the plane, we provide a level set interpolation of a $C^k$ smooth convex function where $k\geq2$ is arbitrary. If the intersection is reduced to one point our interpolant has positive definite Hessian, otherwise it is positive definite out of the solution set. Furthermore, given a sequence of decreasing polygons we provide an interpolant agreeing with the vertices and whose gradients coincide with prescribed normals. author: - 'Jérôme Bolte[^1] and Edouard Pauwels[^2]' date: Draft of title: Curiosities and counterexamples in smooth convex optimization --- Introduction ============ Questions and method -------------------- One of the goals of convex optimization is to provide a solution to a problem with stable and fast algorithms. The quality of a method is generally assessed by the convergence of sequences, rate estimates, complexity bounds, finite length of relevant quantities and other quantitative or qualitative ways. Positive results in this direction are numerous and have been the object of intense research since decades. To name but a few: gradient methods e.g., [@newmirovsky1983problem; @Nesterov; @Boyd], proximal methods e.g., [@PLC], alternating methods e.g., [@Beck; @wright2015coordinate], path following methods e.g., [@Aus99; @NN], Tikhonov regularization e.g. [@Golub], semi-algebraic optimization e.g., [@jon; @BNPS], decomposition methods e.g., [@PLC; @Beck], augmented Lagrangian methods e.g., [@Bertsek] and many others. Despite this vast literature, some simple questions remain unanswered or just partly tackled, even for [*smooth convex coercive*]{} functions. Does the alternating minimization method, aka Gauss-Seidel method, converge? Does the steepest descent method with exact line search converge? Do mirror descent or Bregman methods converge? Does Newton’s flow converge? Do central paths converge? Is the gradient flow directionally stable? Do smooth convex functions have the Kurdyka-Łojasiewicz property? In this article we provide a negative answer to all these questions. Our work draws inspiration from early works of de Finetti [@definetti1949stratificazioni], Fenchel [@fenchel51], on convex interpolation, but also from Torralba’s PhD thesis [@torralba96] and the more recent [@bolte2010characterization], where some counterexamples on the Tikhonov path and Łojasiewicz inequalities are provided. The convex interpolation problem, see [@definetti1949stratificazioni], is as follows: given a monotone sequence of convex sets[^3] may we find a convex function interpolating each of these sets, i.e., having these sets as sublevel sets? Answers to these questions for [*continuous*]{} convex functions were provided by de Finetti, and improved by Fenchel [@fenchel51], Kannai [@kannai77], and then used in [@torralba96; @bolte2010characterization] for building counterexamples. We improve this work by providing, for $k\geq 2$ arbitrary, a general $C^k$ interpolation theorem for positively curved convex sets, imposing at the same time the positive definiteness of its Hessian out of the solution set. An abridged version could be as follows. Let $\left( T_i \right)_{i\in {\mathbb{Z}}}$ be a sequence of compact convex subsets of ${\mathbb{R}}^2$, with positively curved $C^k$ boundary, such that $T_i\subset{\mathrm{int}\,}T_{i+1}$ for all $i$ in ${\mathbb{Z}}$. Then there exists a $C^k$ convex function $f$ having the $T_i$ as sublevel sets with positive definite Hessian outside of the set: $$\operatorname*{argmin}f=\bigcap_{i\in{\mathbb{Z}}} T_i.$$ We provide several additional tools (derivatives computations) and variants (status of the solution set, Legendre functions, Lipschitz continuity). Whether our result is generalizable to general smooth convex sequences, i.e., with vanishing curvature, seems to be a very delicate question whose answer might well be negative. Our central theoretical result is complemented by a discrete approximate interpolation result “of order one" which is particularly well adapted for building counterexamples. Given a nested collection of polygons, one can indeed build a smooth convex function having level sets interpolating its vertices and whose gradients agree with prescribed normals. Our results are obtained by blending parametrization techniques, Minkovski summation, Bernstein approximations and convex analysis. As sketched below, our results offer the possibility of building counterexamples in convex optimization by restricting oneself to the construction of countable collections of nested convex sets satisfying some desirable properties. In all cases failures of good properties are caused by some curvature oscillations. A digest of counterexamples --------------------------- Counterexamples provided in this article can be classified along three axes: structural counterexamples[^4], counterexamples for convex optimization algorithms and ordinary differential equations. In the following, the term “nonconverging” sequence or trajectory means, a sequence or a trajectory with at least two distinct accumulation points. Unless otherwise stated, convex functions have full domain. [*The following results are proved for $C^k$ convex functions on the plane with $k\geq 2$.*]{} #### Structural counterexamples - **Kurdyka-Łojasiewicz:** There exists a $C^k$ convex function whose Hessian is positive definite outside its solution set and which does not satisfy the Kurdyka-Łojasiewicz inequality. This is an improvement on [@bolte2010characterization]. - **Tikhonov regularization path:** There exists a $C^k$ convex function $f$ such that the regularization path $$\begin{aligned} x(r)= \operatorname*{argmin}\left\{ f(y) + r \|y\|^2:y\in {\mathbb{R}}^2\right\}, \,\,r\in(0,1) \end{aligned}$$ has infinite length. This strengthens a theorem by Torralba [@torralba96]. - **Central path:** There exists a continuous Legendre function $h \colon [-1,1]^2 \mapsto {\mathbb{R}}$, $C^k$ on the interior, and $c$ in ${\mathbb{R}}^2$ such that the central path $$\begin{aligned} x(r) = \operatorname*{argmin}\left\{ \left\langle c, y \right\rangle + r h(y):y\in D\right\} \end{aligned}$$ does not have a limit as $r \to 0$. #### Algorithmic counterexamples: - **Gauss-Seidel method (block coordinate descent):** There exists a $C^k$ convex function with positive definite Hessian outside its solution set and an initialization $ (u_0,v_0)$ in ${\mathbb{R}}^2$, such that the alternating minimization algorithm $$\begin{aligned} u_{i+1} &= \operatorname*{argmin}_{u \in {\mathbb{R}}} f(u, v_i) \\ v_{i+1} &= \operatorname*{argmin}_{v \in {\mathbb{R}}}f(u_{i+1}, v) \end{aligned}$$ produces a bounded nonconverging sequence $((u_i,v_i))_{i\in {\mathbb{N}}}$. - **Gradient descent with exact line search:** There exists a $C^k$ convex function $f$ with positive definite Hessian outside its solution set and an initialization $x_0$ in ${\mathbb{R}}^2$, such that the gradient descent algorithm with exact line search $$\begin{aligned} x_{i+1} &= \operatorname*{argmin}_{t \in {\mathbb{R}}} f(x_i + t \nabla f(x_i)) \end{aligned}$$ produces a bounded nonconvergent sequence. - **Bregman or mirror descent method:** There exists a continuous Legendre function $h \colon [-1,1]^2 \mapsto {\mathbb{R}}$, $C^k$ on the interior, a vector $c$ in ${\mathbb{R}}^2$ and an initialization $x_0$ in $(-1,1)^2$ such that the Bregman recursion $$\begin{aligned} x_{i+1} = \nabla h^*(\nabla h(x_i) - c) \end{aligned}$$ produces a nonconverging sequence. The couple $(h,\langle c,\cdot\rangle$) satisfies the smoothness properties introduced in [@bauschke2016descent]. #### Continuous time ODE counterexamples: - **Continuous time Newton method:** There exists a $C^k$ convex function with positive definite Hessian outside its solution set, and an initialization $x_0$ in ${\mathbb{R}}^2$ such that the continuous time Newton’s system $$\begin{aligned} \dot{x}(t) &= - \left[\nabla^2 f(x(t))\right]^{-1} \nabla f(x(t)), \,\,t\geq 0,\\ x(0) &= x_0 \end{aligned}$$ has a solution approaching $\operatorname*{argmin}f$ which does not converge. - **Directional convergence for gradient curves:** There exists a $C^k$ convex function with $0$ as a unique minimizer and a positive definite Hessian on ${\mathbb{R}}^2\setminus\{0\}$, such that for any non stationary solution to the gradient system $$\begin{aligned} \dot{x}(t) &= - \nabla f(x(t)) \end{aligned}$$ the direction $x(t) / \|x(t)\|$ does not converge. - **Hessian Riemannian gradient dynamics:** There exists a continuous Legendre function $h \colon [-1,1]^2 \mapsto {\mathbb{R}}$, $C^k$ on the interior, a linear function $f$ and a nonconvergent solution to the following system $$\begin{aligned} \dot{x}(t) = - \nabla_H f(x(t)), \end{aligned}$$ where $H = \nabla^2 h$ is the Hessian of $h$ and $\nabla_H f = H^{-1} \nabla f$ is the gradient of $f$ in the Riemannian metric induced by $H$ on $(-1,1)^2$. #### Pathological sequences and curves Our counterexamples lead to sequences or paths in ${\mathbb{R}}^2$ which are related to a function $f$ by a certain property (see examples above) and have a certain type of pathology. For illustration purposes, we provide sketches of the pathological behaviors we met in Figure \[fig:pathologicalbehaviors\]. Preliminaries ============= Let us set ${\mathbb{N}^*}={\mathbb{N}}\setminus\{0\}$. For $p$ in ${\mathbb{N}}^*$, the Euclidean scalar product in ${\mathbb{R}}^p$ is denoted by $\langle\cdot,\cdot\rangle$, otherwise stated the norm is denoted by $\|\cdot\|$. Given subsets $S,T$ in ${\mathbb{R}}^p$, and $x$ in ${\mathbb{R}}^p$, we define $${\mathrm{dist}}(x,S):=\inf \left\{\|x-y\|:y\in S\right\},$$ and the Hausdorff distance between $S$ and $T$, $${\mathrm{dist}}(S,T)=\max\left(\sup_{x\in S}{\mathrm{dist}}(x,T),\sup_{x\in T}{\mathrm{dist}}(x,S)\right).$$ Throughout this note, the assertion “$g$ is $C^k$ on $D$” where $D$ is not an open set is to be understood as “$g$ is $C^k$ on an open neighborhood of $D$”. Given a map $G \colon X \mapsto A \times B$ for some space $X$, $[G]_1 \colon X \mapsto A$ denotes the first component of $G$. Continuous convex interpolations -------------------------------- We consider a sequence of compact convex subsets of ${\mathbb{R}}^p$, $\left(T_i \right)_{i \in {\mathbb{Z}}}$ such that $T_{i+1} \subset \mathrm{int}\ T_i$. Finding a continuous convex interpolation of $\left(T_i \right)_{i \in {\mathbb{Z}}}$ is finding a convex continuous function which makes the $T_i$ a sequence of level sets. We call this process [*continuous convex interpolation*]{}. This questioning was present in Fenchel [@fenchel51] and dates back to de Finetti [@definetti1949stratificazioni], let us also mention the work of Crouzeix [@crouzeix80] revolving around this issue. Such constructions have been shown to be realizable by Torralba [@torralba96], Kannai [@kannai77], using ideas based on Minkowski sum. The validity of this construction can be proved easily using the result of Crouzeix [@crouzeix80] which was already present under different and weaker forms in the works of de Finetti and Fenchel. Let $f \colon {\mathbb{R}}^p \to {\mathbb{R}}$ be a quasiconvex function. The functions $$\begin{aligned} F_x \colon \lambda \mapsto \sup\left\{ \langle z,x\rangle: f(z)\leq \lambda\right\} \end{aligned}$$ are concave for all $x$ in ${\mathbb{R}}^p$, if and only $f$ is convex. \[th:crouzeix\] Our goal is to build smooth convex interpolation for sequences of smooth convex sets. To make such a construction we shall use [*nonlinear*]{} [*Minkowski*]{} [*interpolation*]{} between level sets. We shall also rely on Bernstein approximation which we now describe. Bernstein approximation ----------------------- We refer to the monograph [@lorentz1954bernstein] by G.G. Lorentz. The main properties of Bernstein polynomials to be used in this paper are the following: - Bernstein approximation is linear in its functional argument $f$ and “monotone” which allows to construct an approximation using only positive combination of a finite number of function values. - There are precise formulas for derivatives of Bernstein approximation. They involve repeated finite differences. So approximating piecewise affine function with high enough degree leads to an approximation for which corner values of derivatives are controlled while the remaining derivatives are vanishing (up to a given order). - Bernstein approximation is shape preserving, which means in particular that approximating a concave function preserves concavity. The main idea to produce a smooth interpolation which preserves level sets is depicted in Figure \[fig:ideaBernstein\] where we use Bernstein approximation to interpolate smoothly between three points and controlling the successive derivatives at the end points of the interpolation. Let us now be specific. Given $f$ defined on the interval $[0,1]$, the [*Bernstein polynomial*]{} of order $d \in {\mathbb{N}^*}$ associated to $f$ is given by $$\begin{aligned} B_d(x) = B_{d,f}(x) = \sum_{k=0}^d f\left( \frac{k}{d} \right) {d \choose k} x^k(1-x)^{d-k}, \mbox{ for $x\in [0,1].$} \label{eq:bernsteinPolynomial}\end{aligned}$$ #### Derivatives and shape preservation: For any $h $ in $ (0,1)$ and $x $ in $ [0,1-h]$, we set $\Delta_h^1 f(x) = f(x+h) - f(x)$ and recursively for all $k $ in $ {\mathbb{N}^*}$, $\Delta_h^k f(x) = \Delta\left( \Delta_h^{k-1} f(x) \right)$. We fix $d \neq0$ in ${\mathbb{N}}$ and for $h = \frac{1}{d}$ write $\Delta_h^k = \Delta^k$. Then for any $m \leq d$, we have $$\begin{aligned} B_d(x)^{(m)} &= d(d-1)\ldots(d-m+1) \sum_{k=0}^{d - m} \Delta^m f\left( \frac{k}{d} \right) {{d-m} \choose k} x^k(1-x)^{d-k-m}, \label{eq:bernsteinDerivative}\end{aligned}$$ for any $x$ in $[0,1]$. If $f$ is increasing (resp. strictly increasing), then $\Delta^1f(x) \geq 0$ (resp. $\Delta^1f(x) > 0$) for all $x$ and $B'_d$ is positive (resp. strictly positive) and $B_d$ is increasing (resp. strictly increasing). Similarly, if $f$ is concave, then $\Delta^2 f(x) \leq 0$ for all $x$ so that $B^{(2)}_d \leq 0$ and $B_d$ is concave. From , we infer $$\label{e:born} \left|B_d(x)^{(m)}\right|\leq d(d-1)\ldots(d-m+1)\sup_{k\in\{0,\ldots,d-m\}} \left|\Delta^m f\left( \frac{k}{d} \right)\right|$$ for $x$ in $[0,1]$. #### Approximation of piecewise affine functions: The following lemma will be extensively used throughout the proofs. \[lem:convexInterpol\] Let $q_0,q_1 \in {\mathbb{R}}^p$, $\lambda_-<\lambda_0< \lambda_1 < \lambda_+$ and $0<e_1,e_0<1$. Set $\Theta = \left( q_0,q_1,,\lambda_-,\lambda_0,\lambda_1,\lambda_+, e_0,e_1 \right)$ and define $\gamma_\Theta \colon [0,1] \mapsto {\mathbb{R}}^p$ through $$\begin{aligned} \small \gamma_\Theta(t)= \begin{cases} q_0 \left( 1 + \frac{e_0}{\lambda_0 - \lambda_-} \left( t (\lambda_+- \lambda_-) \right) \right) & \text{ if } 0\leq t \leq \frac{\lambda_0 - \lambda_-}{\lambda_+ - \lambda_-} \\ q_0(1 + e_0) \left( \frac{\lambda_1 - \lambda_- - t (\lambda_+ - \lambda_-)}{\lambda_1 - \lambda_0} \right) + q_1(1 - e_1) \left( \frac{ \lambda_- - \lambda_0 + t (\lambda_+ - \lambda_-)}{\lambda_1 - \lambda_0} \right)& \text{ if } \frac{\lambda_0 - \lambda_-}{\lambda_+ - \lambda_-} \leq t \leq \frac{\lambda_1 - \lambda_-}{\lambda_+ - \lambda_-} \\ q_1\left( 1 + \frac{(t-1)(\lambda_+-\lambda_-)e_1}{\lambda_+ - \lambda_1} \right) & \text{ if } \frac{\lambda_1 - \lambda_-}{\lambda_+ - \lambda_-} \leq t \leq 1. \end{cases} \end{aligned}$$ The curve $\gamma_\Theta$ in ${\mathbb{R}}^{p+1}$ is the affine interpolant between the points $q_0$, $ (1 + e_0) q_0$, $(1 - e_1) q_1$ and $ q_1$. For any $m $ in $ {\mathbb{N}}$, we choose $d$ in ${\mathbb{N}}^*$ such that $$\begin{aligned} \label{e:deg} \frac{m}{d} \leq \min\left\{ \frac{\lambda_0 - \lambda_-}{\lambda_+ - \lambda_-}, 1 - \frac{\lambda_1 - \lambda_-}{\lambda_+ - \lambda_-} \right\}. \end{aligned}$$ We consider a Bernstein-like reparametrization of $\tilde\gamma_\Theta $ given by $$\begin{aligned} \tilde\gamma_\Theta \colon [\lambda_-,\lambda_+] &\mapsto {\mathbb{R}}^p\\ \lambda & \mapsto \sum_{k=0}^d \tilde\gamma_\Theta \left( \frac{k}{d} \right){d \choose k} \left(\frac{\lambda - \lambda_-}{\lambda_+ - \lambda_-} \right)^k\left(1 - \frac{\lambda - \lambda_-}{\lambda_+ - \lambda_-} \right)^{d-k}. \end{aligned}$$ Then the following holds, for any $2 \leq l \leq m$, $\tilde\gamma_\Theta $ is $C^m$ and $$\begin{aligned} \tilde\gamma_\Theta (\lambda_-)&=q_0& \tilde\gamma_\Theta (\lambda_+)&=q_1\\ \tilde\gamma_\Theta '(\lambda_-)&=\frac{e_0}{\lambda_0- \lambda-} q_0& \tilde\gamma_\Theta '(\lambda_+)&=\frac{e_1}{\lambda_+ - \lambda_1} q_1\\ \tilde\gamma_\Theta ^{(l)}(\lambda_-) &=0 &\tilde\gamma_\Theta ^{(l)}(\lambda_+) &= 0. \end{aligned}$$ Furthermore, if $\gamma_\Theta $ has monotone coordinates (resp. strictly monotone, resp. concave, resp. convex), then so has $\tilde\gamma_\Theta $. Note that the dependence of $\tilde\gamma_\Theta $ in $(q_0,q_1)$ is linear so that the dependence of $\tilde\gamma_\Theta $ in $(q_0,q_1)$ is also linear. Hence $\tilde\gamma_\Theta $ is of the form $\lambda \mapsto a(\lambda) q_0 + b(\lambda)q_1$. We can restrict ourselves to $p = 1$ since the general case follows from the univariate case applied coordinatewise. If $p=1$, then $\tilde\gamma_\Theta = B_{f_{\Theta,d}} \circ A$, where $A \colon \lambda \mapsto \frac{\lambda - \lambda_-}{\lambda_+-\lambda_-}$. We have $\tilde\gamma_\Theta (0) = q_0$, $\tilde\gamma_\Theta (1) = q_1$ and $\Delta^1 f_{\Theta}(0) = \frac{\lambda_+ - \lambda_-}{\lambda_0 - \lambda_-}\frac{e_0}{d} q_0$, $\Delta^1 f_{\Theta}\left( 1 - \frac{1}{d} \right) = \frac{\lambda_+ - \lambda_-}{\lambda_+ - \lambda_1}\frac{e_1}{d} q_1$ and $\Delta^{(l)} f_{\Theta}(0) = \Delta^{(l)} f_{\Theta}\left( 1 - \frac{l}{d} \right) =0$. The results follow from the expressions in and and the chain rule for $\tilde\gamma_\Theta = B_{f_{\Theta,d}} \circ A$. The last property of $\tilde\gamma_\Theta $ is due to the shape preserving property of Bernstein approximation and the fact that $\tilde\gamma_\Theta = B_{f_{\Theta,d}} \circ L$. \[rem:alignedInterpolation\] [(a) [**\[Affine image\]**]{} Using the notation of Lemma \[lem:convexInterpol\], if $(\lambda_-,q_0)$, $(\lambda_0, (1 + e_0) q_0)$, $(\lambda_1, (1 - e_1) q_1)$ and $(\lambda_+, q_1)$ are aligned, then the interpolation is actually an affine function.\ (b) [**\[Degree of the interpolants\]**]{} Observe that the degree of the Bernstein interpolant is connected to the slopes of the piecewise path $\lambda$ by .]{} Smooth convex interpolation =========================== Being given a subset $S$ of ${\mathbb{R}}^p$, we denote by $\mathrm{int}(S)$ its interior, $\bar S$ its closure and ${\mathrm{bd}\,}S=\bar S\setminus \mathrm{int}(S)$ its boundary. Let us recall that the [*support function*]{} of $S$ is defined through $$\sigma_S(x)=\sup\left\{\langle y,x\rangle :y\in S\right\}\in {\mathbb{R}}\cup\{+\infty\}.$$ Smooth parametrization of convex rings -------------------------------------- A [*convex ring*]{} is a set of the form $C_1\setminus C_2$ where $C_1\subset C_2$ are convex sets. Providing adequate parameterizations for such objects is key for interpolating $C_1$ and $C_2$ by some (regular) convex function. The following assertion plays a fundamental role. Let $T_-,T_+ \subset {\mathbb{R}}^2$ be convex, compact with $C^k$ boundary ($k \geq 2$) and positive curvature. Assume that, $T_- \subset \mathrm{int} (T_+)$ and $0 \in \mathrm{int}(T_-)$. \[ass:curvature\] The positive curvature assumption ensures that the boundaries can be parametrized by their normal, that is, for $i=-,+$, there exists $$\begin{aligned} c_i \colon {\mathbb{R}}/ 2 \pi {\mathbb{Z}}\mapsto {\mathrm{bd}\,}(T_i)\end{aligned}$$ such that the normal to $T_i$ at $c_i(\theta)$ is the vector $n(\theta) = (\cos(\theta),\sin(\theta))^T$ and $\dot{c}_i(\theta) = \rho_{i}(\theta) \tau(\theta)$ where $\rho_i > 0$ and $\tau(\theta) = (-\sin(\theta),\cos(\theta))$. In this setting, it holds that $c_i(\theta) = \mbox{argmax}_{y \in T_i} \left\langle n(\theta), y\right\rangle$. The map $c_i$ is the inverse of the Gauss map and is $C^{k-1}$ (see [@schneider1993convex] Section 2.5). \[l:curv\] Let $T_-,T_+$ be as in Assumption \[ass:curvature\] with normal parametrizations as above. For $a,b\geq0$ with $a+b>0$ set $T=aT_-+bT_+$. Then $T$ has positive curvature and its boundary is given by $${\mathrm{bd}\,}T=\{a\,c_-(\theta)+b\,c_+(\theta): \theta \in {\mathbb{R}}/ 2 \pi {\mathbb{Z}}\},$$ with the natural parametrization ${{\mathbb{R}}/2\pi{\mathbb{Z}}\,}\ni \theta \to a\,c_-(\theta)+b\,c_+(\theta).$ We may assume $ab>0$ otherwise the result is obvious. Let $x$ be in ${\mathrm{bd}\,}T$ and denote by $n(\theta)$ the normal vector at $x$ for a well chosen $\theta$, so that $x={\mathrm{argmax}\,}\left\{\langle y,n(\theta)\rangle\right\}: y \in T\}$. Observe that the definition of the Minkowski sum yields $$\begin{aligned} \max_{y\in T}\langle y,n(\theta)\rangle & = & \max_{(v,w)\in T_-\times T_+}\langle a v+bw,n(\theta)\rangle \\ &=& a \max_{v\in T_-} \langle v,n(\theta)\rangle+b\max_{w\in T_+} \langle w,n(\theta)\rangle \end{aligned}$$ so that $$\begin{aligned} \langle x,n(\theta)\rangle & =& a \langle c_-(\theta),n(\theta)\rangle+b\langle c_+(\theta),n(\theta)\rangle \\ & = & \langle a c_-(\theta)+b c_+(\theta),n(\theta)\rangle \end{aligned}$$ which implies by extremality of $x$ that $x= a c_-(\theta)+b c_+(\theta)$. Conversely, for any such $x$, $n(\theta)$ defines a supporting hyperplane to $T$ and $x$ must be on the boundary of $T$. The other results follow immediately. In the following fundamental proposition, we provide a smooth parametrization of the convex ring $T_+\setminus {\mathrm{int}\,}T_-$. The major difficulty is to control tightly the derivatives at the boundary so that the parametrizations can be glued afterward to build smooth interpolants. Let $T_-,T_+$ be as in Assumption \[ass:curvature\] with their normal parametrization as above. Fix $k\geq 2$, $\lambda_-< \lambda_0 < \lambda_1 < \lambda_+$ and $0 < e_0,e_1<1$. Choose $d $ in $ {\mathbb{N}}^*$, such that $$\begin{aligned} \frac{k}{d} \leq \min\left\{ \frac{\lambda_0 - \lambda_-}{\lambda_+ - \lambda_-}, 1 - \frac{\lambda_1 - \lambda_-}{\lambda_+ - \lambda_-} \right\}. \end{aligned}$$ Consider the map $$\begin{aligned} G \colon [\lambda_-,\lambda_+] \times {\mathbb{R}}/2\pi{\mathbb{Z}}&\mapsto {\mathbb{R}}^2 \nonumber\\ (\lambda, \theta) \mapsto \tilde\gamma_\Theta(\lambda) \label{eq:mapG} \end{aligned}$$ with $\Theta = (c_-(\theta), c_+(\theta), \lambda_-, \lambda_0, \lambda_1, \lambda_+, e_0,e_1)$ and $\tilde\gamma$ as given by Lemma \[lem:convexInterpol\]. Assume further that: ${(\mathcal{M})}$ For any $\theta \in {\mathbb{R}}/ 2\pi{\mathbb{Z}}$, $\lambda \mapsto \left\langle G(\lambda, \theta), n(\theta)\right\rangle$ has strictly positive derivative on $[\lambda_-,\lambda_+]$. Then the image of $G$ is ${\mathcal{R}}:=T_+ \setminus \mathrm{int}(T_-)$, $G$ is $C^k$ and satisfies, for any $2 \leq l \leq k$ and any $m $ in $ {\mathbb{N}}^*$, $$\begin{aligned} \frac{\partial^m G}{\partial \theta^m}(\lambda_-,\theta) &= c_-^{(m)}(\theta)& \frac{\partial^m G}{\partial \theta^m}(\lambda_+,\theta) &= c_+^{(m)}(\theta)\\ \frac{\partial^{m+1} G}{\partial \lambda\partial \theta^m} (\lambda_-,\theta) &= c_-^{(m)}(\theta) \frac{e_0}{\lambda_0 - \lambda_-}& \frac{\partial^{m+1} G}{\partial \lambda\partial \theta^m} (\lambda_+,\theta) &= c_+^{(m)}(\theta) \frac{e_1}{\lambda_+-\lambda_1}\\ \frac{\partial^{l+m} G}{\partial \lambda^l \partial \theta^m }(\lambda_-,\theta) &= 0& \frac{\partial^{l+m} G}{\partial \lambda^l \partial \theta^m }(\lambda_+,\theta) &= 0. \end{aligned}$$ Besides $G$ is a diffeomorphism from its domain on to its image. Set ${\mathcal{R}}\ni x \mapsto (f(x), \theta(x))$ to be the inverse of $G$. Then $f$ is $C^k$ and in addition, for all $x$ in $ {\mathbb{R}}$, $$\begin{aligned} \nabla f(x) =\quad& \frac{1}{\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle} \, n(\theta(x))\nonumber\\ \nabla \theta(x) =\quad& \frac{1}{\left\langle\frac{\partial G}{\partial \theta}(f(x),\theta(x)),\tau(\theta(x))\right\rangle} \tau(\theta(x)) \nonumber\\ &- \frac{\left\langle \frac{\partial G}{\partial \lambda} (f(x),\theta(x)), \tau(\theta(x))\right\rangle}{\left\langle \frac{\partial G}{\partial \lambda} (f(x),\theta(x)), n(\theta(x))\right\rangle \left\langle\frac{\partial G}{\partial \theta}(f(x),\theta(x)),\tau(\theta(x))\right\rangle } n(\theta(x)) \nonumber\\ \nabla^2 f(x)=\quad&\frac{\left\langle\frac{\partial G}{\partial \theta}(f(x),\theta(x)),\tau(\theta(x))\right\rangle}{\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle} \nabla \theta(x) \nabla \theta (x)^T \nonumber\\ &- \frac{\left\langle \frac{\partial^2 G}{\partial \lambda^2}(f(x),\theta(x)), n(\theta(x))\right\rangle}{\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle} \nabla f(x) \nabla f(x)^T, \label{eq:gradient} \end{aligned}$$ where all denominators are positive. \[prop:diffeoMorphism\] [ Note that $G$ is actually well defined and smooth on an open set containing its domain. As we shall see it is also a diffeomorphism from an open set containing its domain onto its image.]{} Note that by construction, we have $G(\lambda,\theta) = a(\lambda) c_-(\theta) + b(\lambda)c_+(\theta)$ for some polynomials $a$ and $b$ which are nonnegative on $[\lambda_-,\lambda_+]$. The formulas for the derivatives follow easily from this remark, the form of $a$ and $b$ and Lemma \[lem:convexInterpol\]. Set, for any $\lambda $ in $ [\lambda_-,\lambda_+]$, $T_\lambda = a(\lambda) T_- + b(\lambda) T_+$. The resulting set $T_\lambda$ is convex and has a positive curvature by Lemma \[l:curv\], and for $\lambda$ fixed $G(\lambda,\cdot)$ is the inverse of the Gauss map of $T_\lambda$, which constitutes a parametrization by normals of the boundary. Assume that $\lambda < \lambda'$, using the monotonicity assumption ${(\mathcal{M})}$, we have for any $\theta,\theta'$, $$\begin{aligned} \left\langle n(\theta'), G(\lambda,\theta) \right\rangle& \leq \sup_{y \in T_{\lambda}} \left\langle n(\theta'),y \right\rangle \\ & = \left\langle n(\theta'), G(\lambda,\theta') \right\rangle\\ &< \left\langle n(\theta'), G(\lambda',\theta') \right\rangle \end{aligned}$$ so that $G(\lambda,\theta) \neq G(\lambda',\theta')$. Furthermore, we have by definition of $G(\lambda',\theta')$ $$\begin{aligned} G(\lambda,\theta) \in \bigcap_{\theta' \in {\mathbb{R}}/2\pi{\mathbb{Z}}} \left\{ y,\; \left\langle y, n(\theta')\right\rangle \leq \left\langle n(\theta'), G(\lambda',\theta') \right\rangle \right\} = T_{\lambda'}, \end{aligned}$$ where the equality follows from the convexity of $T_{\lambda'}$. By convexity and compactness, this entails that $T_\lambda = \mathrm{conv}({\mathrm{bd}\,}(T_\lambda)) \subset {\mathrm{int}\,}T_{\lambda'}$. Let us show that the map $G$ is bijective, first consider proving surjectivity. Let $f$ be defined on $T_+ \setminus \mathrm{int}(T_-)$ through $$\begin{aligned} f \colon x \mapsto \inf_{}\left\{ \lambda : \lambda\geq \lambda_-, \; x \in T_\lambda = a(\lambda) T_- + b(\lambda) T_+\right\}. \label{eq:defIntropol} \end{aligned}$$ Since $a(\lambda_+)=0, b(\lambda_+)=1$ this function is well defined and by compactness and continuity the infimum is achieved. It must hold that $x $ belongs to $ {\mathrm{bd}\,}(T_{f(x)})$, indeed, if $f(x) = \lambda_-$, then $x $ belongs to $ {\mathrm{bd}\,}(T_-)$ and otherwise, if $x $ is in $ \mathrm{int}(T_{\lambda'})$ for $\lambda' > \lambda_-$, then $f(x) < \lambda'$. We deduce that $x$ is of the form $G(f(x),\theta)$ for a certain value of $\theta$, so that $G$ is surjective. As for injectivity, we have already seen a first case, the monotonicity assumption ${(\mathcal{M})}$ ensures that $\lambda \neq \lambda'$ implies $G(\lambda,\theta) \neq G(\lambda',\theta')$ for any $\theta,\theta'$. Furthermore, we have the second case, for any $\lambda $ in $ [\lambda_-,\lambda_+]$ and any $\theta$, $G(\lambda,\theta) = \arg\max_{y\in T_\lambda} \left\langle y,n(\theta) \right\rangle$ so that $\theta \neq \theta'$ implies $G(\lambda,\theta) \neq G(\lambda,\theta')$. So in all cases, $(\lambda,\theta) \neq (\lambda',\theta')$ implies that $G(\lambda,\theta) \neq G(\lambda',\theta')$ and $G$ is injective. Let us now show that the map $G$ is a local diffeomorphism by estimating its Jacobian map. Since $0 \in \mathrm{int}(T_-)$, we have for any $\lambda,\theta$, $$\begin{aligned} 0 &< \sup_{y \in T_-} \left\langle y, n(\theta) \right\rangle \\ &= \left\langle G(\lambda_-,\theta),n(\theta) \right\rangle \\ & \leq a(\lambda) \left\langle c_-(\theta), n(\theta)\right\rangle + b(\lambda) \left\langle c_+(\theta), n(\theta) \right\rangle, \end{aligned}$$ and both scalar products are positive so that $a(\lambda) + b(\lambda) > 0$. Hence, for any $\theta $ in $ {\mathbb{R}}/ 2\pi {\mathbb{Z}}$, $$\begin{aligned} \label{eq:derivativeGtheta} \frac{\partial G}{\partial \theta}(\lambda,\theta) = (a(\lambda) \rho_-(\theta) + b(\lambda) \rho_+ (\theta)) \tau(\theta), \end{aligned}$$ with $a(\lambda) \rho_-(\theta) + b(\lambda) \rho_+ (\theta) > 0$. Furthermore by assumption $\lambda\to \max_{y \in T_\lambda} \left\langle y, n(\theta) \right\rangle=\left\langle G(\lambda,\theta),n(\theta) \right\rangle$ has strictly positive derivative in $\lambda$, whence $$\begin{aligned} \left\langle \frac{\partial G}{\partial \lambda} (\lambda,\theta), n(\theta)\right\rangle > 0. \end{aligned}$$ We deduce that for any fixed $\theta$, in the basis $(n(\theta), \tau(\theta))$, the Jacobian of $G$, denoted $J_G$, is triangular with positive diagonal entries. More precisely fix $\lambda, \theta$ and set $x = G(\lambda, \theta)$ such that $\lambda = f(x)$, $\theta = \theta(x)$. In the basis $(n(\theta), \tau(\theta))$, we deduce from that the Jacobian of $G$ is of the form $$\begin{aligned} J_G(\lambda, \theta) = \begin{pmatrix} \alpha &0\\ \gamma&\beta \end{pmatrix}, \end{aligned}$$ where $$\begin{aligned} \alpha &= \left\langle \frac{\partial G}{\partial \lambda} (\lambda,\theta), n(\theta)\right\rangle>0 \\ \beta &= \left\langle \frac{\partial G}{\partial \theta} (\lambda,\theta), \tau(\theta)\right\rangle = a(\lambda) \rho_-(\theta) + b(\lambda) \rho_+ (\theta) >0\\ \gamma &= \left\langle \frac{\partial G}{\partial \lambda} (\lambda,\theta), \tau(\theta)\right\rangle. \end{aligned}$$ It is thus invertible and we have a local diffeomorphism. We deduce that $$\begin{aligned} J_G(\lambda, \theta)^{-1} = \begin{pmatrix} \alpha^{-1} &0\\ \frac{-\gamma}{\alpha\beta}&\beta^{-1} \end{pmatrix}. \end{aligned}$$ We have $J_G(\lambda,\theta)^{-1} = J_{G^{-1}}(x)$ so that the first line is $\nabla f(x)$ and second line is $\nabla \theta(x)$, which proves the claimed expressions for gradients. We also have $d n(\theta) / d\theta = \tau(\theta)$ so that $$\begin{aligned} J_{n \circ \theta}(x) = \tau(\theta(x)) \nabla \theta(x)^T. \end{aligned}$$ Differentiating the gradient expression, we obtain ($\nabla$ denotes gradient with respect to $x$): $$\begin{aligned} &\nabla^2 f(x)\\ =\quad& \frac{\partial}{\partial x} \left(\frac{1}{\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle} \, n(\theta(x))\right)\\ =\quad& \frac{1}{\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle} \, J_{n\circ \theta}(x) \\ &+ n(\theta(x)) \nabla \left(\frac{1}{\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle }\right)^T \\ =\quad& \frac{1}{\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle} \, J_{n\circ \theta}(x) \\ &- n(\theta(x)) \nabla \left(\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle \right)^T \frac{1}{\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle^2} \\ =\quad & \frac{1}{\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle} \left( \tau(\theta) \nabla \theta(x)^T - \nabla f(x) \nabla \left(\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle \right)^T\right). \end{aligned}$$ We have $$\begin{aligned} =\quad&\nabla \left(\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle \right)^T \\ =\quad & \frac{\partial G}{\partial \lambda} (f(x), \theta(x))^T J_{n\circ\theta}(x) + n(\theta(x))^T J_{\frac{\partial G}{\partial \lambda}}(f(x),\theta(x)) J_{G^{-1}} (x) \\ =\quad & \left\langle \frac{\partial G}{\partial \lambda}(f(x), \theta(x)), \tau(\theta(x)) \right\rangle \nabla \theta(x)^T + \left\langle n (\theta(x)), \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta)\right\rangle \nabla f(x)^T, \end{aligned}$$ where, for the last identity, we have used the fact that $$\begin{aligned} n(\theta(x))^T J_{\frac{\partial G}{\partial \lambda}}(f(x),\theta(x)) &= \left\langle n (\theta(x)), \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta)\right\rangle n(\theta(x))^T + \left\langle n (\theta(x)), \frac{\partial^2 G}{\partial \lambda \partial \theta}(\lambda,\theta)\right\rangle \tau(\theta(x))^T\\ &= \left\langle n (\theta(x)), \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta)\right\rangle n(\theta(x))^T\\ n(\theta(x))^T J_{G^{-1}}(x) &= n(\theta(x))^T J_G(f(x),\theta(x))^{-1}\\ &= \frac{1}{\left\langle \frac{\partial G}{\partial \lambda} (f(x),\theta(x)), n(\theta(x))\right\rangle} n(\theta(x))^T = \nabla f(x)^T. \end{aligned}$$ We deduce that $$\begin{aligned} &\nabla^2 f(x)\\ =\quad & \frac{1}{\left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle}\\ &\Bigg( \tau(\theta(x)) \nabla \theta(x)^T - \left\langle \frac{\partial G}{\partial \lambda}(f(x), \theta(x)), \tau(\theta(x)) \right\rangle \nabla f(x) \nabla \theta(x)^T \\ &- \nabla f(x) \nabla f(x)^T \left\langle n (\theta(x)), \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta)\right\rangle \Bigg). \end{aligned}$$ We have $$\begin{aligned} & \tau(\theta(x)) \nabla \theta(x)^T - \left\langle \frac{\partial G}{\partial \lambda}(f(x), \theta(x)), \tau(\theta(x)) \right\rangle \nabla f(x) \nabla \theta(x)^T \\ =\quad& \left(\tau(\theta(x)) - \frac{\left\langle \frac{\partial G}{\partial \lambda}(f(x), \theta(x)), \tau(\theta(x)) \right\rangle}{\left\langle \frac{\partial G}{\partial \lambda}(f(x), \theta(x)), n(\theta(x)) \right\rangle} n(\theta(x)) \right) \nabla \theta(x)^T\\ =\quad& (a(\lambda) \rho_-(\theta) + b(\lambda) \rho_+ (\theta)) \nabla \theta(x) \nabla \theta (x)^T \end{aligned}$$ So that we actually get $$\begin{aligned} &\nabla^2 f(x) \left\langle \frac{\partial G}{\partial \lambda} (f(x), \theta(x)), n(\theta(x)) \right\rangle\\ =\quad & (a(\lambda) \rho_-(\theta) + b(\lambda) \rho_+ (\theta)) \nabla \theta(x) \nabla \theta (x)^T - \left\langle n (\theta(x)), \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta)\right\rangle \nabla f(x) \nabla f(x)^T\\ =\quad &\left\langle\frac{\partial G}{\partial \theta}(\lambda,\theta),\tau(\theta(x))\right\rangle \nabla \theta(x) \nabla \theta (x)^T - \left\langle n (\theta(x)), \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta)\right\rangle \nabla f(x) \nabla f(x)^T. \end{aligned}$$ This concludes the proof. Smooth convex interpolation of smooth positively curved convex sequences ------------------------------------------------------------------------ In this section, we consider an indexing set $I$ with either $I = {\mathbb{N}}$ or $I = {\mathbb{Z}}$, and an increasing sequence of compact convex sets $\left( T_i \right)_{i\in I}$ such that for any $i $ in $ I$, the couple $T_+ := T_{i+1}$, $T_-: = T_i$ satisfies Assumption \[ass:curvature\]. In particular, for each $i $ in $ I$, $T_i$ is compact, convex with $C^k$ boundary and positive curvature. We denote by $c_i$ the corresponding parametrization by the normal, $T_i \subset \mathrm{int}\, T_{i+1}$. With no loss of generality we assume $\displaystyle 0 \in \cap_{i \in I} T_i$. This is our main theoretical result. \[th:smoothinterp\] Let $I = {\mathbb{N}}$ or $I = {\mathbb{Z}}$ and $\left( T_i \right)_{i\in I}$ such that for any $i \in I$, $T_i \subset {\mathbb{R}}^2$ and the couple $T_+ := T_{i+1}$, $T_-: = T_i$ satisfies Assumption \[ass:curvature\]. Then there exists a $C^k$ convex function $$\begin{aligned} f \colon {\mathcal{T}}:= \mathrm{int}\left(\bigcup_{i\in I} T_i \right) \mapsto {\mathbb{R}}\end{aligned}$$ such that \(i) $T_i$ is a sublevel set of $f$ for all $i $ in $ I$. \(ii) We have $$\begin{aligned} \operatorname*{argmin}f= \begin{cases} \bigcap_I T_i &\quad \text{ if } I = {\mathbb{Z}}\\ \{0\}&\quad \text{ if } I = {\mathbb{N}}. \end{cases} \end{aligned}$$ \(iii) $\nabla^2 f $ is positive definite on ${\mathcal{T}}\setminus \operatorname*{argmin}f$, and if $I = {\mathbb{N}}$, it is positive definite throughout ${\mathcal{T}}$. $\:$\ [*Preconditionning.*]{} We have that $0 \in \cap_{i \in I } \mathrm{int}(T_i)$. Hence for any $i $ in $ I$ and $0 \leq \alpha <1$, $\alpha T_i \in \mathrm{int}(T_i)$. Furthermore, for $\alpha>0$, small enough, $(1 + \alpha)T_{i} \subset (1-\alpha) \mathrm{int}(T_{i+1})$. Set $\alpha_0$ such that $(1 + \alpha_0)T_0 \subset (1-\alpha_0) \mathrm{int} (T_1)$. By forward (and backward if $I = {\mathbb{Z}}$) induction, for all $i $ in $ I$, we obtain $\alpha_i>0$ such that $$\begin{aligned} (1 + \alpha_i) T_i &\subset (1-\alpha_{i}) \mathrm{int} (T_{i+1}). \end{aligned}$$ Setting for all $i $ in $ I$, $\epsilon_{i+1} = \min\{\alpha_i,\alpha_{i+1}\}$ ($\epsilon_0 = \alpha_0$ if $I = {\mathbb{N}}$), we have for all $i $ in $ I$ $$\begin{aligned} (1 + \epsilon_i) T_i \subset (1 + \alpha_i) T_i &\subset (1-\alpha_{i}) \mathrm{int} (T_{i+1}) \subset(1-\epsilon_{i+1}) \mathrm{int} (T_{i+1}). \end{aligned}$$ For all $i $ in $ I$, we introduce $$\begin{aligned} S_{3i} &= T_i,\\ S_{3i + 1} &= (1 + \epsilon_i) T_i\\ S_{3i + 2} &= (1 - \epsilon_{i+1}) T_{i+1}.\end{aligned}$$ We have a new sequence of strictly increasing compact convex sets $\left( S_i\right)_{i \in I }$. [*Value assignation.*]{} For each $i $ in $ I $, we set $$\begin{aligned} K_i = \max_{\|x\| = 1} \frac{\sigma_{S_{i+1}}(x) - \sigma_{S_i}(x)}{ \sigma_{S_{i}}(x) - \sigma_{S_{i-1}}(x)} \in \left( 0, + \infty \right).\end{aligned}$$ Note that for all $i $ in $ I$, $K_{3i} = 1$. We choose $\lambda_{1} = 2$, $\lambda_0 = 1$ and for all $i $ in $ I $, $$\label{value} \lambda_{i+1} = \lambda_i + K_i(\lambda_i - \lambda_{i-1}).$$ By construction, we have for all $i $ in $ I $ and all $\theta \in {\mathbb{R}}/ 2 \pi {\mathbb{Z}}$, $$\begin{aligned} \frac{\sigma_{S_{i+1}}(n(\theta)) - \sigma_{S_{i}}(n(\theta))}{\lambda_{i+1} - \lambda_{i}} \leq \frac{\sigma_{S_{i}}(n(\theta)) - \sigma_{S_{i-1}}(n(\theta)) }{\lambda_{i} - \lambda_{i-1}}.\end{aligned}$$ If $I = {\mathbb{Z}}$, this entails $$\begin{aligned} 0 < \lambda_i - \lambda_{i - 1} \leq \frac{\lambda_{1}- \lambda_0}{\sigma_{S_{1}}(n(\theta)) - \sigma_{S_{0}}(n(\theta))} (\sigma_{S_{i}}(n(\theta)) - \sigma_{S_{i-1}}(n(\theta))),\end{aligned}$$ and the right-hand side is summable over negative indices $i\leq 0$, so that $\lambda_{i} \to \underline{\lambda} \in {\mathbb{R}}$ as $i \to -\infty$. In all cases $(\lambda_i)_{i\in I}$ is an increasing sequence bounded from below. [*Local interpolation.*]{} We fix $i $ in $ I $ and consider the function $G_i$ described in Proposition \[prop:diffeoMorphism\] with $T_+ = S_{3i + 3} = T_{i+1}$, $T_- = S_{3i} = T_i$, $\lambda_+ = \lambda_{3i+3}$, $\lambda_{1} = \lambda_{3i+2}$, $\lambda_0 = \lambda_{3i+1}$, $\lambda_- = \lambda_{3i}$, $e_0 = \epsilon_i$, $e_1 = \epsilon_{i+1}$. By linearity, we have for any $(\lambda,\theta) \in [\lambda_-,\lambda_+] \times {\mathbb{R}}/ 2\pi {\mathbb{Z}}$, $$\begin{aligned} \left\langle G_i(\lambda,\theta),n(\theta)\right\rangle = \tilde \gamma_{\Theta}(\lambda)\end{aligned}$$ where $ \tilde \gamma_{\Theta}$ is as in Lemma \[lem:convexInterpol\] with input data $q_0 = \left\langle c_{3i}(\theta), n(\theta)\right\rangle = \sigma_{S_{3i}}(n(\theta))$, $q_1 = \left\langle c_{3i+3}(\theta), n(\theta)\right\rangle = \sigma_{S_{3i+3}}(n(\theta))$, and $\lambda_-,\lambda_0,\lambda_1,\lambda_+,e_1,e_0$ as already described. This corresponds to the Bernstein approximation of the piecewise affine interpolation between the points $$\begin{aligned} &\left( \lambda_{3i}, \sigma_{S_{3i}}(n(\theta))\right)\nonumber\\ &\left( \lambda_{3i+1}, \sigma_{S_{3i+1}}(n(\theta))\right),\nonumber \\ &\left( \lambda_{3i+2}, \sigma_{S_{3i+2}}(n(\theta))\right),\nonumber \\ &(\lambda_{3i+3}, \sigma_{3i+3}(n(\theta))), \label{eq:piecewiseAffineInterpolExplicit}\end{aligned}$$ By construction of $\left( K_i \right)_{i \in I }$, we have for all $\theta$, $$\begin{aligned} 0 < \frac{\sigma_{S_{3i+3}}(n(\theta)) - \sigma_{S_{3i+2}}(n(\theta)) }{\lambda_{3i+3} - \lambda_{3i + 2}} \leq \frac{\sigma_{S_{3i+2}}(n(\theta)) - \sigma_{S_{3i+1}}(n(\theta)) }{\lambda_{3i+2} - \lambda_{3i + 1}} \leq \frac{\sigma_{S_{3i+1}}(n(\theta)) - \sigma_{S_{3i}}(n(\theta)) }{\lambda_{3i+1} - \lambda_{3i}}.\end{aligned}$$ Whence the affine interpolant between points in is strictly increasing and concave, and by using the shape preserving properties of Bernstein polynomials, $\left\langle G_i(\lambda,\theta),n(\theta)\right\rangle$ has strictly positive derivative. As a consequence $G_i$ is a diffeomorphism and its derivatives are as in Proposition \[prop:diffeoMorphism\]. Furthermore $$\begin{aligned} \lambda \mapsto \left\langle G_i(\lambda,\theta),n(\theta)\right\rangle \end{aligned}$$ is a $C^k$ concave function of $\lambda$. [*Global interpolation.*]{} Recall that $\underline{\lambda} = \inf_{i \in I} \lambda_i>-\infty$ and set $\bar{\lambda} = \sup_{i \in I} \lambda_i\in (-\infty,+\infty]$. For any $\lambda \in (\underline{\lambda},\bar{\lambda})$, there exists a unique $i_\lambda \in I$ such that $\lambda \in [\lambda_{3i_\lambda}, \lambda_{3i_\lambda+3})$. Define $$\begin{aligned} G \colon (\underline{\lambda}, \bar{\lambda}) \times {\mathbb{R}}/ 2\pi {\mathbb{Z}}&\mapsto {\mathbb{R}}^2 \\ (\lambda,\theta) &\mapsto G_{i_\lambda} (\lambda, \theta).\end{aligned}$$ Fix $i$ in $I$. The boundary of $T_{i+1}$ is given by $G_{i+1}(\lambda_{3i+3}, {\mathbb{R}}/ 2 \pi {\mathbb{Z}}) = G_{i}(\lambda_{3i+3}, {\mathbb{R}}/ 2 \pi {\mathbb{Z}})$ with actually $$\label{coinc} G_{i+1}(\lambda_{3i+3}, \theta) = G_{i}(\lambda_{3i+3}, \theta ), \mbox{ for all }\theta \mbox{ in }{\mathbb{R}}/ 2 \pi {\mathbb{Z}}.$$ Since $K_{3i} = 1$, we have $$\begin{aligned} \lambda_{3i+1} - \lambda_{3i} = \lambda_{3i} - \lambda_{3i-1}.\end{aligned}$$ The expressions of the derivatives in Proposition \[prop:diffeoMorphism\] and ensure that the derivatives of $G_{i+1}$ and $G_{i}$ agree on ${\lambda_{3i+3}} \times {\mathbb{R}}/ 2 \pi {\mathbb{Z}}$ up to order $k$. Hence $G$ is a local diffeomorphism. Bijectivity of each $G_i$ ensure that $G$ is also bijective and thus $G$ is a diffeomorphism. Furthermore $$\begin{aligned} \lambda \mapsto \left\langle G(\lambda,\theta),n(\theta)\right\rangle \end{aligned}$$ is $C^k$ piecewise concave and thus concave. [*Extending $G$.*]{} If $I = {\mathbb{N}}$, we may assume without loss of generality that $S_0 = B$ the Euclidean ball and $S_1 = 5/3 S_0$, which corresponds to $\epsilon_0 = 2/3$, eventually after adding a set in the list and rescaling. Let $\phi$ denote the function described in Lemma \[lem:interpolationAroundZero\] and $G_{-1}$ be described as in Lemma \[lem:diffGauge2\]. This allows to extend $G$ for $\lambda \in [0,1]$, $G$ is then $C^k$ on $(0,\bar{\lambda}) \times{\mathbb{R}}/ 2\pi {\mathbb{Z}}$ by using Lemma \[lem:diffGauge2\] and Proposition \[prop:diffeoMorphism\]. This does not affect the differentiability, monotonicity and concavity properties of $G$. [*Defining the interpolant $f$.*]{} We assume without loss of generality that $\underline{\lambda} = 0$. We set $f$ to be the first component of the inverse of $G$ so that it is defined on $G^{-1}\left( (0,\bar{\lambda}) \times {\mathbb{R}}/ 2\pi {\mathbb{Z}}\right)$. We extend $f$ as follows: - $f(0) = 0$ if $I = {\mathbb{N}}$, - $f = 0$ on $\cap_{i \in I } T_i$ if $I = {\mathbb{Z}}$. Since $G$ is $C^k$ and non-singular on $ (0,\bar{\lambda}) \times {\mathbb{R}}/ 2\pi {\mathbb{Z}}$, the inverse mapping theorem ensures that $f$ is $C^k$ on $\mathrm{int}({\mathcal{T}}) \setminus \arg\min_{{\mathcal{T}}} f$. [*Convexity of $f$.*]{} For any $\theta $ in $ {\mathbb{R}}/ 2\pi {\mathbb{Z}}$, $$\begin{aligned} (0, \bar{\lambda}) &\mapsto {\mathbb{R}}_+\\ \lambda &\mapsto \sup_{z\in[f \leq \lambda]} n(\theta)^Tz \end{aligned}$$ is equal to $\left\langle G(\lambda,\theta),n(\theta)\right\rangle$ which is concave. It can be extended at $\lambda = 0$ by continuity. This preserves concavity hence, using Theorem \[th:crouzeix\], we have proved that $f$ is convex and $C^k$ on ${\mathcal{T}}\setminus \operatorname*{argmin}_{{\mathcal{T}}}f$. [*Smoothness around the argmin and Hessian positivity.*]{} If $I = {\mathbb{N}}$, then the interpolant defined in Lemma \[lem:diffGauge2\] ensures that $f$ is proportional to the norm squared around $0$. Hence it is $C^k$ around $0$ with positive definite Hessian. We may compose $f$ with the function $g \colon t \mapsto \sqrt{t^2 + 1} + t$ which is increasing and has positive second derivative. This ensures that the resulting Hessian is positive definite outside $\operatorname*{argmin}f$ and thus everywhere since $$\begin{aligned} \nabla^2 g \circ f = g' \nabla^2 f + g'' \nabla f \nabla f^T\end{aligned}$$ is positive definite thanks to the expressions for the Hessian of $f$ in Proposition \[prop:diffeoMorphism\] If $I = {\mathbb{Z}}$, we let all the derivatives of $f$ vanish around the solution set. The smoothing Lemma \[lem:CkSmoothing\] applies and provides a function $\phi$ with positive derivative on $(0,+\infty)$, such that $\phi \circ f$ is convex, $C^k$ with prescribed sublevel sets. Furthermore, we remark that $$\begin{aligned} \nabla^2 \phi \circ f = \phi' \nabla^2 f + \phi'' \nabla f \nabla f^T.\end{aligned}$$ We may compose $\phi \circ f$ with the function $g \colon t \mapsto \sqrt{t^2 + 1} + t$ which is increasing with positive second derivative, the expressions for the Hessian of $f$ in Proposition \[prop:diffeoMorphism\] ensure that the resulting Hessian is positive definite out of $\operatorname*{argmin}f$. \[rem:alignedLevelSets\][In view of Remark \[rem:alignedInterpolation\], if we have $T_{i+1} = \alpha T_i$ for some $0<\alpha < 1$ and $i $ in $ {\mathbb{Z}}$, then the interpolated level sets between $T_i$ and $T_{i+1}$ are all of the form $s T_i$ for $\alpha \leq s \leq 1$.]{} \[rem:strictConvexity\][ Recall that strict convexity of a differentiable function amounts to the injectivity of its gradient. In Theorem \[th:smoothinterp\] if there is a unique minimizer, then the invertibility of the Hessian outside argmin $f$ ensures that our interpolant is strictly convex (note that this is automatically the case if $I = {\mathbb{N}}$).]{} Considerations on Legendre functions {#s:legendre} ------------------------------------ The following proposition provides some interpolant with additional properties as global Lipschitz continuity and finiteness properties for the dual function. At this stage these results appear as merely technical but they happen to be decisive in the construction of counterexamples involving Legendre functions. The properties of Legendre functions can be found in [@rockafellar1970convex Chapter 6]. We simply recall here that, given a convex body $C$ of ${\mathbb{R}}^p$, a convex function $h:C\to {\mathbb{R}}$ is [*Legendre*]{} if it is differentiable on ${\mathrm{int}\,}C$ and if $\nabla h$ defines a bijection from ${\mathrm{int}\,}C$ to $\nabla h({\mathrm{int}\,}C)$ with in addition $$\lim_{\begin{array}{l} x\in {\mathrm{int}\,}C\\ x\to z\end{array}} \|\nabla h(x)\|=+\infty,$$ for all $z$ in ${\mathrm{bd}\,}C$. We also assume that $\mbox{epi}\,f:=\{(x,\lambda):f(x)\leq \lambda\}$ is closed in ${\mathbb{R}}^{p+1}$. The [*Legendre conjugate*]{} or [*dual function*]{} of $h$ is defined through $$h^*(z)=\sup\left\{\langle z,x\rangle -h(x):x \in C\right\},$$ for $z$ in ${\mathbb{R}}^p$, and its domain is $D:=\left\{z\in{\mathbb{R}}^p:h(z)<+\infty\right\}.$ The function $h^*$ is differentiable on the interior of $D$, and the inverse of $\nabla h:{\mathrm{int}\,}C \to {\mathrm{int}\,}D$ is $\nabla h^*:{\mathrm{int}\,}D \to {\mathrm{int}\,}C$. We start with a simple technical lemma on the compactness of the domain of a Legendre function. Let $h \colon {\mathbb{R}}^2 \mapsto {\mathbb{R}}$ be a globally Lipschitz continuous Legendre function, and set $D = \mathrm{int}(\mathrm{dom}(h^*))$ where $h^* \colon {\mathbb{R}}^2 \mapsto {\mathbb{R}}$ is the conjugate of $h$. For each $\lambda \geq \min_{{\mathbb{R}}^2}h$ let $\sigma_\lambda$ be the support function associated to the set $\left\{ z \in {\mathbb{R}}^2, \, h(z) \leq \lambda \right\}$. The following are equivalent - $h^*(x)\leq 0$ for all $x \in D$. - For all $y \in {\mathbb{R}}^2$, $\sigma_{h(y)}(\nabla h(y)) \leq h(y)$. In both cases $h^*$ has compact domain. \[lem:legendreBounded\] Let us establish beforehand the following formula $$\begin{aligned} \label{e:h} h^*(z) &= \sigma_{h(y)}(\nabla h(y))- h(y), \end{aligned}$$ with $y = \nabla h^* (z)$ and $y\in{\mathbb{R}}^2$. Since $h^*$ is Legendre, we have for all $y$ in $D$, $$\begin{aligned} h^*(z) &= \sup_{y \in {\mathbb{R}}^2} \left\langle z, y \right\rangle - h(y) = \left\langle z, \nabla h^*(z) \right\rangle - h(\nabla h^*(z)). \end{aligned}$$ We have, setting $y = \nabla h^*(z)$ $$\begin{aligned} \left\langle z, \nabla h^*(z) \right\rangle = \left\langle \nabla h(y),y \right\rangle = \sigma_{h(y)}(\nabla h(y)) \end{aligned}$$ because $\nabla h(y)$ is normal to the sublevel set of $h$ which contains $y$ in its boundary. Hence we have $h^*(z) = \sigma_{h(y)}(\nabla h(y))- h(y)$ with $y = \nabla h^* (z)$, that is holds. Since $\nabla h^* \colon D \mapsto {\mathbb{R}}^2$ is a bijection, the equivalence follows. In this case the domain of $h^*$ is closed because $h^*$ is bounded and lower semicontinuous. The domain is also bounded by the Lipschitz continuity of $h$, whence compact. \[th:globallyLipshitz\] Let $\left( S_i \right)_{i\in {\mathbb{N}}}$ be such that for any $i $ in $ I$, $T_- = S_{i}$, $T_+ = S_{i+1}$ satisfy Assumption \[ass:curvature\] and there exists a sequence $\left( \epsilon_i \right)_{i \in {\mathbb{N}}}$ in $(0,1)$ such that for all $i\geq 1$, $(1 - \epsilon_i)^{-1} S_{3i-1} = (1 +\epsilon_i)^{-1} S_{3i+1}=S_{3i}$.\ Assume in addition that, $$\begin{aligned} \inf_{\|x\| = 1}& \sigma_{S_i}(x) - \sigma_{S_{i-1}}(x) =1 + O\left(\frac{1}{i^3}\right)&\mbox{ (non degeneracy)}\label{nd}, \\ \sup_{\|x\|=1}& \left|\frac{\sigma_{S_{i+1}}(x) - \sigma_{S_i}(x)}{\sigma_{S_{i}}(x) - \sigma_{S_{i-1}}(x)} - 1\right| = O\left( \frac{1}{i^3} \right)& \mbox{ (moderate growth)}\label{mg}. \end{aligned}$$ Then there exists a convex $C^k$ function $h \colon {\mathbb{R}}^2 \mapsto {\mathbb{R}}$, such that - For all $i $ in $ {\mathbb{N}}$, $S_{3i}$ is a sublevel set of $h$, - $h$ has positive definite Hessian, - $h$ is globally Lipschitz continuous, - $h^*$ has a compact domain $D$ and is $C^k$ and strictly convex on $\mathrm{int}(D)$. The construction of $h$ follows the exact same principle as that of Theorem \[th:smoothinterp\]. This ensures that the first two points are valid. Note that equation implies that the sets sequence grows by at least a fixed amount in each direction as $i$ grows. Hence we have ${\mathcal{T}}= {\mathbb{R}}^2$.\ *Global Lipschitz continuity of $h$:* The values of $h$ are defined through $$\begin{aligned} \lambda_{i+1}-\lambda_{i}&=K_i(\lambda_{i}-\lambda_{i-1}),\;\forall i\in {\mathbb{N}}^*\\ K_i &= \max_{\|x\| = 1} \frac{\sigma_{S_{i+1}}(x) - \sigma_{S_i}(x)}{ \sigma_{S_{i}}(x) - \sigma_{S_{i-1}}(x)} \in \left( 0, + \infty \right), \end{aligned}$$ so that $K_i = 1 + O(1/i^3)$ thanks to equation . Note that the moderate growth assumption entails $$\begin{aligned} \sup_{i \in {\mathbb{N}}} \sigma_{S_{i+1}}(x) - \sigma_{S_{i}}(x) = O\left( \prod_{i \in {\mathbb{N}}^*} K_i\right) = O(1). \label{eq:suportDiffBounded} \end{aligned}$$ For $i\geq 1$, one has $$\label{for} \lambda_{i+1}-\lambda_{i}=\prod_{1 \leq j \leq i} K_j(\lambda_1-\lambda_0).$$ On the other hand using the bounds , and the identity , there exists a constant $\kappa>0$ such that for all $i \geq 1$, all $\theta $ in $ {\mathbb{R}}/ 2 \pi {\mathbb{Z}}$, $$\begin{aligned} \frac{\sigma_{S_{i+1}}(n(\theta)) - \sigma_{S_{i}}(n(\theta)) }{\lambda_{i+1} - \lambda_{i}} = \frac{\left(\sigma_{S_{i+1}}(n(\theta)) - \sigma_{S_{i}}(n(\theta))\right) }{\prod_{j=1}^i K_j (\lambda_{1} - \lambda_{0})} \geq \kappa>0. \label{eq:tempDerivLambda}\end{aligned}$$ By the interpolation properties described in Lemma \[lem:convexInterpol\] the function $\left\langle G(\lambda,\theta),n(\theta)\right\rangle$ constructed in Theorem \[th:smoothinterp\] has derivative with respect to $\lambda$ greater than $\kappa$. Recalling the expression of the gradient as given in Proposition \[prop:diffeoMorphism\] (and the concavity of $G$ with respect to $\lambda$), this shows that $$\|\nabla h(x)\|\leq \frac{1}{\kappa}$$ for all $x $ in $ {\mathbb{R}}^2$, and by the mean value theorem, $h$ is globally Lipschitz continuous on ${\mathbb{R}}^2$. *Properties of the dual function:* $h$ is Legendre, its conjugate $h^*$ is therefore Legendre. From the definiteness of $\nabla^2 h$ and the fact that $\nabla h \colon {\mathbb{R}}^2 \mapsto \mathrm{int}(D)$ is a bijection, we deduce that $h^*$ is $C^k$ by the inverse mapping theorem. So the only property which we need to establish is that $h^*$ has a compact domain, in other words, using Lemma \[lem:legendreBounded\], it is sufficient to show that $\sup_{x \in \mathrm{int} D} h^*(x) \leq 0$. Using the notation of the proof of Theorem \[th:smoothinterp\], we will show that it is possible to verify that, for all $\lambda,\theta$ in the domain of $G$ $$\begin{aligned} \label{eq:toBeCheckedForBoundedness} \frac{\left\langle n(\theta), G(\lambda,\theta)\right\rangle}{\left\langle \frac{\partial G}{\partial \lambda}(\lambda,\theta), n(\theta) \right\rangle} \leq \lambda.\end{aligned}$$ Equation is indeed the coordinate form of the characterization given in Lemma \[lem:legendreBounded\]. Let us observe that $$\begin{aligned} \label{e:int}\frac{\partial}{\partial \lambda} \left( \left\langle n(\theta), G(\lambda,\theta)\right\rangle - \lambda \left\langle \frac{\partial G}{\partial \lambda}(\lambda,\theta), n(\theta) \right\rangle \right) = -\lambda \left\langle \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta), n(\theta) \right\rangle,\end{aligned}$$ and since $G$ is concave, the right hand side is positive. Assume that we have proved that, $$\label{e:lim} \lambda \mapsto \sup_\theta -\lambda \left\langle \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta), n(\theta) \right\rangle$$ has finite integral as $\lambda \to \infty$. Since the function $$\label{e:funTheta} \theta \mapsto \lambda \left\langle \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta), n(\theta) \right\rangle$$ is continuous on ${\mathbb{R}}/2\pi{\mathbb{Z}}$ for any $\lambda$, Lebesgue dominated convergence theorem would ensure that $$\begin{aligned} \theta \mapsto \int_{\lambda \geq \lambda_0} -\lambda \left\langle \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta), n(\theta) \right\rangle d \lambda\end{aligned}$$ is continuous in $\theta$, so that: $$\begin{aligned} \sup_{\theta} \left[\lim_{\lambda \to \infty} \left\langle n(\theta), G(\lambda,\theta)\right\rangle - \lambda \left\langle \frac{\partial G}{\partial \lambda}(\lambda,\theta), n(\theta) \right\rangle\right]<+\infty.\end{aligned}$$ Shifting values if necessary, we could assume that this upper bound is equal to zero to obtain equation . The latter being the condition required in Lemma \[lem:legendreBounded\], we would have reached a conclusion. Let us therefore establish that is integrable over ${\mathbb{R}}_+$. Recall that $G$ is constructed using the Bernstein interpolation given in Lemma \[lem:convexInterpol\] between successive values of $\lambda$. As a result, for a fixed $\theta$, the function $\left\langle n(\theta), G(\lambda,\theta)\right\rangle $ is the interpolation of the piecewise affine function interpolating $$\begin{aligned} &\left( \lambda_{3i}, \sigma_{S_{3i}}(n(\theta))\right)\nonumber\\ &\left( \lambda_{3i+1}, \sigma_{S_{3i+1}}(n(\theta))\right),\nonumber \\ &\left( \lambda_{3i+2}, \sigma_{S_{3i+2}}(n(\theta))\right),\nonumber \\ &(\lambda_{3i+3}, \sigma_{3i+3}(n(\theta))), \label{eq:piecewiseAffineInterpolExplicit2}\end{aligned}$$ as in equation . This interpolation is concave and increasing. Assumption ensures that $K_j = 1 + O(1/j^3)$. Then $$\begin{aligned} \prod_{j=1}^m K_j = \bar{K} + O(1 / j^{2})\end{aligned}$$ where $\bar{K}$ is the finite, positive limit of the product (we can for example perform integral series comparison after taking the logarithm). The recursion on the values writes for all $i\geq1$ $$\begin{aligned} \lambda_{i+1} = \lambda_i + K_i(\lambda_i - \lambda_{i-1}),\end{aligned}$$ so that $$\begin{aligned} \lambda_{i+1} - \lambda_i= (\lambda_1 - \lambda_0) \prod_{j=1}^i K_i = (\lambda_1 - \lambda_0) \bar{K} + O(1/i^{2}).\end{aligned}$$ This means that the gap between consecutive values tends to be constant. Thus by in Lemma \[lem:convexInterpol\], see also Remark \[rem:alignedInterpolation\], the degree of the Bernstein interpolants is bounded. Using this bound together with inequality , providing bounds for the derivatives of Bernstein’s polynomial, ensure that, for all $\lambda $ in $ [\lambda_{3i}, \lambda_{3i+3})$: $$\begin{aligned} &\left|\left\langle \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta), n(\theta) \right\rangle\right| \\ =\,& O\left( \max _{j = 3i+2, 3i+1}\left| \frac{\sigma_{S_{j+1}}(n(\theta)) - \sigma_{S_{j}}(n(\theta))}{\lambda_{j+1} - \lambda_{j}} - \frac{\sigma_{S_{j}}(n(\theta)) - \sigma_{S_{j-1}}(n(\theta)) }{\lambda_{j} - \lambda_{j-1}} \right| \right).\end{aligned}$$ Now for any $j = 3i+2, 3i+1$, $$\begin{aligned} &\left| \frac{\sigma_{S_{j+1}}(n(\theta)) - \sigma_{S_{j}}(n(\theta))}{\lambda_{j+1} - \lambda_{j}} - \frac{\sigma_{S_{j}}(n(\theta)) - \sigma_{S_{j-1}}(n(\theta)) }{\lambda_{j} - \lambda_{j-1}} \right| \\ =\,& \frac{1}{\lambda_{j+1} - \lambda_{j}} \left| (\sigma_{S_{j+1}}(n(\theta)) - \sigma_{S_{j}}(n(\theta))) - \frac{\lambda_{j+1} - \lambda_{j}}{\lambda_{j} - \lambda_{j-1}} (\sigma_{S_{j}}(n(\theta)) - \sigma_{S_{j-1}(n(\theta)) }) \right|\\ =\,& \left(1 / ((\lambda_1 - \lambda_0) \bar{K}) + O(1/i^2)\right)\\ &\times \left| (\sigma_{S_{j+1}}(n(\theta)) - \sigma_{S_{j}}(n(\theta))) - (1 + O(1/i^2)) (\sigma_{S_{j}}(n(\theta)) - \sigma_{S_{j-1}(n(\theta)) }) \right|\\ =\,& \left(1 / ((\lambda_1 - \lambda_0) \bar{K}) + O(1/i^2)\right)\\ &\times \left| (\sigma_{S_{j+1}}(n(\theta)) - \sigma_{S_{j}}(n(\theta))) - (\sigma_{S_{j}}(n(\theta)) - \sigma_{S_{j-1}(n(\theta)) }) \right|\end{aligned}$$ where the last identity follows from the triangle inequality because using $\sigma_{S_{j}}(n(\theta)) - \sigma_{S_{j-1}}(n(\theta)) = O(1)$ in . Hence $$\begin{aligned} &\left|\left\langle \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta), n(\theta) \right\rangle\right| \\ =\,& \left(1 / ((\lambda_1 - \lambda_0) \bar{K}) + O(1/i^2)\right)\\ &\times O\left( \max _{j = 3i+2, 3i+1}\left| \sigma_{S_{j+1}}(n(\theta)) - \sigma_{S_{j}}(n(\theta)) - (\sigma_{S_{j}}(n(\theta)) - \sigma_{S_{j-1}}(n(\theta)) ) \right| \right)\\ =\,& \left(1 / ((\lambda_1 - \lambda_0) \bar{K}) + O(1/i^2)\right)\\ &\times O\left( \max _{j = 3i+2, 3i+1}\left|\sigma_{S_{j}}(n(\theta)) - \sigma_{S_{j-1}}(n(\theta)) \right| \times \left| \frac{\sigma_{S_{j+1}}(n(\theta)) - \sigma_{S_{j}}(n(\theta))}{\sigma_{S_{j}}(n(\theta)) - \sigma_{S_{j-1}}(n(\theta)) } - 1 \right| \right)\\ =\,& O(1/i^3),\end{aligned}$$ where the last inequality follows from and . Now as $i \to \infty$, $\lambda_{3i} \sim \lambda_{3i + 3} \sim i c$ for some constant $c>0$ and $$\begin{aligned} \sup_{\lambda \in [\lambda_{3i}, \lambda_{3i+3}], \theta \in [0, 2 \pi]} - \lambda \left\langle \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta), n(\theta) \right\rangle = O(1 / i^2)\end{aligned}$$ and $$\begin{aligned} \sup_{ \theta \in [0, 2 \pi]} - \lambda \left\langle \frac{\partial^2 G}{\partial \lambda^2}(\lambda,\theta), n(\theta) \right\rangle\end{aligned}$$ has finite integral as $\lambda \to \infty$. This implies and it concludes the proofs. Smooth convex interpolation for sequences of polygons ===================================================== Given a sequence of points $A_1,\ldots,A_n$, we denote by $A_1\ldots A_n$ the polygon obtained by joining successive points ending the loop with the segment $[A_n,A_1]$. In the sequel we consider mainly convex polygons, so that the vertices $A_1,\ldots,A_n$ are also the extreme points. The purpose of this section is first to show that polygons can be approximated by smooth convex sets with prescribed normals under weak assumptions. Figure \[fig:illustrPolySmooth\] illustrates the result we would like to establish: given a target polygon with prescribed normals at its vertices, we wish to construct a smooth convex set interpolating the vertices with the desired normals and whose distance to the polygon is small. Then given a sequence of nested polygons, we provide a smooth convex function which interpolates the polygons in the sense described just above. Given a closed nonempty convex subset $S$ of ${\mathbb{R}}^p$ and $x$ in $S$, we recall that the [*normal cone to $S$ at $x$*]{} is $$N_S(x)=\left\{z\in {\mathbb{R}}^p:\langle z,y-x\rangle \leq 0,\forall y\in S\right\}.$$ Such vectors will often simply called normals (to $S$) at $x$. Smooth approximations of polygons --------------------------------- \[lem:approxSegment\] For any $r_-,r_+ > 0$, $t_- > 0$, $t_+<0$ and $\epsilon > 0$, $m \in {\mathbb{N}}$, $m \geq 3$, there exists a strictly concave polynomial function $p \colon [0,1] \mapsto [0,\epsilon]$ such that $$\begin{aligned} p(0) &= 0 &p(1) &= 0\\ p'(0) &= t_-& p'(1) &= t_+\\ p''(0) &= -r_- &p''(1) &= -r_+.\\ p^{(q)}(0) & = 0 \quad q \in\{3, \ldots, m\}. \end{aligned}$$ ![Arrows designate the prescribed normals. We construct a strictly convex set with smooth boundary entirely contained in the auxiliary (blue) polygon. This set interpolates the normals and the distance to the original (red) polygon can be chosen arbitrarily small. The degree of smoothness of the boundary can be chosen arbitrarily high.[]{data-label="fig:illustrPolySmooth"}](illustrPolySmooth){width=".7\textwidth"} Let us begin with preliminary remarks. Consider for any $a,b $ in $ {\mathbb{R}}$, the function $$\begin{aligned} f \colon t \mapsto a(t-b)^2. \end{aligned}$$ For any $t $ in $ {\mathbb{R}}$, $q $ in $ {\mathbb{N}}$, $q > 2$, and any $c > 0$, we have $$\begin{aligned} f(t + c) - f(t) &= \Delta^1_c f(t) = a c ( 2(t-b) + c)\nonumber\\ f(t + 2 c) - 2 f(t+c) + f(t) &= \Delta^2_c f(t) = a c ( 2 (t + c - b) + c - 2(t-b) - c) = 2ac^2\nonumber\\ \Delta_c^q f(t) &= 0 \label{eq:approxSegment00} \end{aligned}$$ [*Choosing the degree $d$ and constructing the polynomial.* ]{} For any $d $ in $ {\mathbb{N}}$, $d \geq 2m + 1$, we set $$\begin{aligned} a_-(d) &= \frac{-dr_-}{2(d - 1)}<0&b_-(d) &= \frac{1}{2d} \left( 1 + 2 t_-\frac{d-1}{r_-} \right)> 0\nonumber\\ a_+(d) &= \frac{-dr_+}{2(d - 1)}<0&b_+(d) &= 1 + \frac{1}{2d} \left( -1 + 2t_+\frac{d-1}{r_+} \right)< 1, \label{eq:approxSegment0} \end{aligned}$$ and define the functions $$\begin{aligned} f_d \colon s &\mapsto \begin{cases} a_-(d) (( s - b_-(d))^2 - b_-(d)^2) & \text{ if } s\leq b_-(d)\\ - a_-(d) b_-(d)^2& \text{ if } s\geq b_-(d) \end{cases}\\ g_d \colon t &\mapsto \begin{cases} a_+(d) (( s - b_+(d))^2 - (1-b_+(d))^2) & \text{ if } s\geq b_+(d)\\ - a_+(d) (1-b_+(d))^2& \text{ if } s\leq b_+(d). \end{cases} \end{aligned}$$ Furthermore, we set $$\begin{aligned} f \colon t &\mapsto \begin{cases} \frac{r_-}{2} \left( \left( \frac{t_-}{r_-} \right)^2- \left(s - \frac{t_-}{r_-} \right)^2 \right) & \text{ if } s\leq \frac{t_-}{r_-}\\ \frac{r_-}{2} \left( \frac{t_-}{r_-} \right)^2 & \text{ if } s\geq \frac{t_-}{r_-} \end{cases}\\ g \colon t &\mapsto \begin{cases} \frac{r_+}{2} \left(\left( \frac{t_+}{r_+} \right)^2 - \left(s -1 - \frac{t_+}{r_+} \right)^2 \right) & \text{ if } s\geq 1 + \frac{t_+}{r_+}\\ \frac{r_+}{2} \left( \frac{t_+}{r_+} \right)^2 & \text{ if } s\leq 1 + \frac{t_+}{r_+}. \end{cases} \end{aligned}$$ Note that $b_-(d) \to t_- / r_-$, $b_+(d) \to 1 + t_+ / r_+$, $a_-(d) \to - r_-/2$ and $a_+(d) \to - r_+/2$ as $d \to \infty$ so that $f_d \to f$ and $g_d \to g$ uniformly on $[0,1]$. For any $d$, $f_d$ is concave increasing and $g_d$ is concave decreasing and all of them are Lipschitz continuous on $[0,1]$ with constants that do not depend on $d$. Note also that $f(0) = 0 < g(0)$ and $g(1) = 0 < f(1)$. We choose $d \geq 2m + 1$ such that $$\begin{aligned} f_d\left( \frac{m}{d} \right) &\leq \min\left( \epsilon, g_d\left( \frac{m}{d} \right) \right)\nonumber\\ g_d\left(1 - \frac{m}{d} \right) &\leq \min\left( \epsilon, f_d\left(1 - \frac{m}{d} \right) \right). \label{eq:approxSegment1} \end{aligned}$$ Such a $d$ always exists because in both cases, the left hand side converges to $0$ and the right hand side converges to a strictly positive term as $d$ tends to $\infty$. For such a $d$, we set $h \colon s \mapsto \min\left\{ f_d(s), g_d(s),\epsilon \right\}$. By construction, $h$ is concave, agrees with $f_d$ on $\left[ 0,\frac{m}{d} \right] \subset [0,1/2]$ and with $g_d$ on $\left[ 1- \frac{m}{d},1 \right] \subset [1/2,1]$. Using equation with $c = 1/d$, we deduce that $$\begin{aligned} d (d-1) \Delta^2 h(0) &= d (d-1) \Delta^2 f_d(0) = d(d-1)\frac{2 a_-(d)}{d^2} = -r_-\\ d \Delta h(0) &= d \Delta f_d(0) = a_-(d)\left( \frac{1}{d} -2b_-(d)\right) = t_-\\ \Delta^{q}h(0) &= 0 = \Delta^{q}f_d(0) \quad \forall m \geq q \geq 3\\ d (d-1) \Delta^2 h\left(1 - \frac{2}{d}\right) &= d (d-1) \Delta^2 g_d\left(1 - \frac{2}{d}\right)= d(d-1)\frac{2 a_+(d)}{d^2} = -r_+\\ d \Delta h\left( 1 - \frac{1}{d} \right) &= d \Delta g_d\left( 1 - \frac{1}{d} \right) = a_+(d) \left( \frac{-1}{d} + 2 (1 - b_+(d))\right) = t_+\\ \Delta^{q}h\left( 1 - \frac{m}{d} \right) &= \Delta^{q}g_d\left( 1 - \frac{m}{d} \right) = 0 \quad \forall m \geq q \geq 3. \end{aligned}$$ From the concavity of $h$ and the derivative formula , we deduce that the polynomial $B_{h,d}$ satisfies the desired properties. ![Illustration of the approximation result of Lemma \[lem:approxSegment\] with $\epsilon = 0.1$, $m = 3$, $t_- =0.7$, $t_+ = -2.2$, $r_- = 2$ and $r_+ = 0.2$. The resulting polynomial is of degree 66. Numerical estimations of the first and second order derivatives at 0 and 1 match the required values up to 3 precision digits. The polynomial is strongly concave, however, this is barely visible because the strong concavity constant is extremely small. []{data-label="fig:illustrApproxSeg"}](illustrApproxSeg){width="\textwidth"} We deduce the following result Let $a > 0$, $r > 0$, $\epsilon > 0$, and an integer $m \geq 3$. Consider two unit vectors: $v_-$ with strictly positive entries and $v_+$ with first entry strictly positive and second entry strictly negative. Then there exists a $C^m$ curve $\gamma \colon [0,M] \mapsto {\mathbb{R}}^2$, such that 1. $\|\gamma'\| = 1$. 2. $\gamma(0) = (-a, 0) := A$ and $\gamma(1) = (0,a) : = B$. 3. $\gamma'(0) = v_-$ and $\gamma'(1) = v_+$. 4. $\|\gamma''(0)\| = \|\gamma''(-1)\| = r$. 5. $\mathrm{det}(\gamma',\gamma'') < 0$ along the curve. 6. $\gamma^{(q)}(0) = \gamma^{(q)}(1) = 0$ for any $3 \leq q \leq m$. 7. ${\mathrm{dist}}(\gamma([0,M]), [A,B]) \leq \epsilon$. \[lem:existenceCurve\] Consider the graph of a polynomial as given in Lemma \[lem:approxSegment\] with $t_- = v_-[2] / v_-[1]> 0$, $t_+ = v_+[2] / v_+[1]< 0$ and $r_- = \frac{r}{2a} (1 + t_-^2)^{\frac{3}{2}}$, $r_+ = \frac{r}{2a} (1 + t_+^2)^{\frac{3}{2}}$ and $\epsilon/2a$ as an approximation parameter. This graph is parametrized by $t$. It is possible to reparametrize it by arclength to obtain a $C^m$ curve $\gamma_0$ whose tangents at $0$ is $T_-$, at $1$ is $T_+$, and whose curvature at $0$ and $1$ is $- \frac{r}{2a}$. Furthermore, $\gamma_0$ has strictly negative curvature whence item 5. Consider the affine transform: $x \mapsto 2a(x - 1/2)$, $y \mapsto 2a y$. This results in a $C^m$ curve $\gamma$, parametrized by arclength which satisfies the desired assumptions. Let $S=A_1...A_n$ be a convex polygon. For each $i$, let $V_i$ be in $ N_S(A_i)$ such that the angle between $V_i$ and each of the two neighboring faces is within $\left( \frac{\pi}{2},\pi \right)$. Then for any $\epsilon > 0$ and any $m \geq 2$, there exists a compact convex set $C \subset {\mathbb{R}}^2$ such that - the boundary of $C$ is $C^m$ with non vanishing curvature, - $S \subset C$, - for any $i = 1,\ldots, n$, $A_i $ in $ {\mathrm{bd}\,}(C)$ and the normal cone to $C$ at $A_i$ is given by $V_i$, - $\max_{y \in C} {\mathrm{dist}}(y,S) \leq \epsilon$. \[lem:polyGon\] We assume without loss of generality that $A_1,\ldots,A_n$ are ordered clockwise. For $i = 1, \ldots, n-1$, and each segment $[A_i,A_{i+1}]$, we may perform a rotation and a translation to obtain $A_i = -(a,0)$ and $A_{i+1} = (a,0)$. Working in this coordinate system, using the angle condition on $V_i$, we may choose $v^-_i$, $v^+_{i+1}$ satisfying the hypotheses of Lemma \[lem:existenceCurve\] respectively orthogonal to $V_i$ and $V_{i+1}$. Choosing $r = 1$, we obtain $\gamma_i \colon [0,M_i] \mapsto {\mathbb{R}}^2$ as given by Lemma \[lem:existenceCurve\]. Rotation and translations affect only the direction of the derivatives of curves, not their length. Hence, it is possible to concatenate curves $\left( \gamma_i \right)_{i=1}^{n-1}$ and to preserve the $C^m$ properties of the resulting curve. At end-points, tangents and second order derivatives coincide while higher derivatives vanish. Furthermore the curvature has constant sign and does not vanish. We obtain a closed $C^m$ curve which defines a convex set which satisfies all the requirements of the lemma. [Given any polygon, choosing normal vectors as given by the direction of the bissector of each angles ensure that the above assumptions are satisfied. Hence all our approximation results hold given polygon without specifying the choice of outer normals.]{} \[rem:noNormal\] Smooth convex interpolants of polygonal sequences ------------------------------------------------- [For $n\geq3$, let $A_1\ldots A_n $ be a convex polygon $S$ and $V_i $ be in $ N_S(V_i)$ for $i=1,\ldots,n$. We say that $\left( A_i,V_i \right)_{i=1}^n $ is [*interpolable*]{} if for each $i = 1,\ldots,n$, the angle between $V_i$ and each of the two neighboring faces of the polygon is in $\left( \frac{\pi}{2},\pi \right)$. The collection $\left( A_i,V_i \right)_{i=1}^n $ is called [*a polygon-normal pair*]{}.]{} \[def:interpolable\] Let $I = {\mathbb{Z}}$ or $I = {\mathbb{N}}$. Let $\left( PN_i \right)_{i \in I}$ be a sequence of interpolable polygon-normal pairs. Setting for $i $ in $ I$, $PN_i = \left\{ \left( A_{j,i} \right)_{j=1}^{n_i}, \left( V_{j,i} \right)_{j=1}^{n_i} \right\}$ where $n_j $ is in $ {\mathbb{N}}$ and denoting by $T_i$ the polygon $A_{1,i}\ldots A_{n_i,i}$, we say that the sequence $\left( PN_i \right)_{i \in I}$ is strictly increasing if for all $i $ in $ I$, $T_i \subset \mathrm{int}(T_{i+1})$. Let $\left( PN_i \right)_{i \in I}$ be a strictly increasing sequence of interpolable polygon-normal pairs. A sequence $(\epsilon_i)_{i \in I}$ in $(0,1)$ is said to be [*admissible*]{} if $0 \in \mathrm{int}(T_i)$ for each $i $ in $ I$ and $$\gamma T_i\subset{\mathrm{int}\,}T_{i+1}$$ for all $\gamma \in [1-\epsilon_i,1+\epsilon_i].$ We have the following corollary of Theorem \[th:smoothinterp\]. Let $I = {\mathbb{Z}}$ or $I = {\mathbb{N}}$. Let $\left( PN_i \right)_{i \in I}$ be a strictly increasing sequence of interpolable polygon-normal pairs and $(\epsilon_i)_{i \in I}$ be admissible. Set $\displaystyle\mathcal{T}:=\mathrm{int}\left(\cup_{i\in I}T_i \right).$ Then for any $k $ in $ {\mathbb{N}}$, $k \geq 2$ there exists a $C^k$ convex function $f \colon \mathcal{T} \mapsto {\mathbb{R}}$, and an increasing sequence $(\lambda_i)_{i \in I}$, with $\inf_{i \in I} \lambda_i > -\infty$, such that for each $i $ in $ I$ - $T_i \subset \left\{ x,\,f(x) \leq \lambda_i \right\}$. - ${\mathrm{dist}}(T_i, \left\{ x,\,f(x) \leq \lambda_i \right\}) \leq \epsilon_i$. - For each $i $ in $ I$, $j$ in $\{1,\ldots,n_i\}$, we have $f(A_{i,j}) = \lambda_i$ and $\nabla f(x)$ is colinear to $V_{i,j}$. - $\nabla^2 f$ is positive definite outside $\operatorname*{argmin}f$. When there is a unique minimizer then $\nabla^2f$ is positive definite throughout ${\mathcal{T}}$ [(]{}this is the case when $I = {\mathbb{N}}$ or when $I={\mathbb{Z}}$ and $\cap_{i \in I} T_i$ is a singleton[)]{}. \[cor:polygonDecrease\] We add two remarks which will be useful for directional convergence issues and the construction of Legendre functions: - If two consecutive elements of the sequence of interpolable polygon-normal pairs are homothetic with center $0$ in the interior of both polytopes, then the restriction of the resulting convex function to this convex ring can be constructed such that all the sublevel sets within this ring are homothetic with the same center. - If further conditions are imposed on the elements of a strictly increasing interpolable polygon-normal pair, then the resulting function can be constructed to be Legendre and globally Lipschitz continuous (that is, its Legendre conjugate has bounded support). This is a consequence of Proposition \[th:globallyLipshitz\] and will be detailled in the next section. More on Legendre functions and a pathological function with polyhedral domain ----------------------------------------------------------------------------- Using intensively polygonal interpolation, we build below a finite continuous Legendre function $h$ on an $\ell^\infty$ square with oscillating “mirror lines": $t\to \nabla h^*(\nabla h(x_0)+tc)$. We start with the following preparation proposition related to the Legendre interpolation of Proposition \[th:globallyLipshitz\]. Let $\left( PN_i \right)_{i \in {\mathbb{N}}^*}$ be a strictly increasing sequence of interpolable polygon-normal pairs. Setting for $i$ in ${\mathbb{N}}^*$, $PN_i = \left\{ \left( A_{j,i} \right)_{j=1}^{n_i}, \left( V_{j,i} \right)_{j=1}^{n_i} \right\}$ where $n_j$ is in ${\mathbb{N}}^*$ and denoting by $T_i$ the polygon $A_{1,i}\ldots A_{n_i,i}$, we assume that $$T_i=3i P,\,\forall i \in {\mathbb{N}}^*,$$ where $P$ is a fixed polygon which contains the unit Euclidean disk.\ Then for any $l$ in ${\mathbb{N}}$, $l \geq 2$, there exists a strictly increasing sequence of sets $\left( S_i \right)_{i \in {\mathbb{N}}, \,i\geq 2}$, such that for $j \geq 1$, - $S_{3j}$ interpolates the normals of $PN_j$ in the sense of Lemma \[lem:polyGon\] with $\mathrm{dist}(S_{3j}, T_j) \leq 1 / (4(3j+2)^l)$ - $S_{3j - 1} = \frac{3j - 1}{3_j} S_{3j}$ - $S_{3j + 1} = \frac{3j + 1}{3_j} S_{3j}$ This sequence has the following properties - there exists $c>0$, such that for all $j $ in ${\mathbb{N}}$, $j \geq 3$ and for all unit vector $x$, $$\begin{aligned} c\geq\sigma_{S_{j+1}}(x) - \sigma_{S_j}(x) \geq 1 - \frac{1}{(j+1)^l}. \label{eq:interpolGauge1} \end{aligned}$$ - for all unit vector $x$, $$\begin{aligned} \left| \frac{\sigma_{S_{j+1}}(x) - \sigma_{S_j}(x)}{ \sigma_{S_{j}}(x) - \sigma_{S_{j-1}}(x)} -1\right| \leq \frac{1}{j^l},\:\forall j\geq 3. \label{eq:interpolGauge2} \end{aligned}$$ \[lem:interpolGauge\] Set for all $j$ in ${\mathbb{N}}^*$, $\delta_j = \frac{1}{4(3j+2)^l}$ and let $S_{3j}$ be the $\delta_j$ interpolant of $T_j=3jP$ as given by Lemma \[lem:polyGon\] so that ${\mathrm{dist}}(S_{3j}, 3jP) \leq \delta_j$. Since $P$ contains the unit ball, $$\label{inc} 3j P \subset S_{3j} \subset (3j + \delta_j) P.$$ Now set $$\begin{aligned} S_{3j - 1} &= \frac{3j - 1}{3_j} S_{3j}\\ S_{3j + 1} &= \frac{3j + 1}{3_j} S_{3j}. \end{aligned}$$ For any $j$ in ${\mathbb{N}}^*$, it is clear that $S_{3j-1} \subset \mathrm{int}(S_{3j})$ and $S_{3j} \subset \mathrm{int}(S_{3j+1})$. Furthermore, by , we have $$\begin{aligned} S_{3j+1} \subset \frac{3j+1}{3j} ( 3j + \delta_j) P \subset ((3j+1) + 2\delta_j) P \subset \mathrm{int}((3j+2) P) \subset \mathrm{int} (S_{3j+2}) \end{aligned}$$ so that we indeed have a strictly increasing sequence of sets. We obtain from the construction, for any $j$ in ${\mathbb{N}}^*$, and any unit vector $x$, $$\begin{aligned} \sigma_{S_{3j}}(x) - \sigma_{S_{3j-1}}(x) &=\sigma_{S_{3j+1}}(x) - \sigma_{S_{3j}}(x)\notag \\ &= \frac{1}{3j} \sigma_{S_{3j}}(x) \in \left[ \sigma_P(x), \left( 1 + \frac{\delta_j}{3j} \right) \sigma_P(x) \right] \subset \left[ 1, \left(1+\frac{\delta_j} {3j} \right) \sigma_P(x) \right]\notag \\ \sigma_{S_{3j+2}}(x) - \sigma_{S_{3j+1}}(x) &\leq \sigma_P(x) (3j +2) \left( 1 + \frac{\delta_{j+1}}{3j+3} \right) - \sigma_P(x)(3j+1)\notag\\ \sigma_{S_{3j+2}}(x) - \sigma_{S_{3j+1}}(x) &\geq \sigma_P(x) (3j +2) - \sigma_P(x)(3j+1)\left( 1 + \frac{\delta_j}{3j} \right) \notag\\ & = \sigma_P(x) \left( 1 - \delta_j \frac{3j+1}{3j} \right)\notag\\ &\geq 1 - \delta_j \frac{4}{3} \geq 1 - \frac{1}{(3j+2)^l}\label{intermed} \end{aligned}$$ This proves . We deduce that for all $j$ in ${\mathbb{N}}^*,$ $$\begin{aligned} \frac{\sigma_{S_{3j+1}}(x) - \sigma_{S_{3j}}(x)}{ \sigma_{S_{3j}}(x) - \sigma_{S_{3j-1}}(x)}& = 1, \mbox{ for all nonzero vector $x$,} \\ \max_{\|x\| = 1} \frac{\sigma_{S_{3j+2}}(x) - \sigma_{S_{3j+1}}(x)}{ \sigma_{S_{3j+1}}(x) - \sigma_{S_{3j}}(x)} &\leq (3j +2) \left( 1 + \frac{\delta_{j+1}}{3j+3} \right) - (3j+1)\\ &= 1 + \frac{3j+2}{3j+3} \delta_{j+1} \leq 1 + \frac{1}{(3j +1)^l},\\ \min_{\|x\| = 1} \frac{\sigma_{S_{3j+2}}(x) - \sigma_{S_{3j+1}}(x)}{ \sigma_{S_{3j+1}}(x) - \sigma_{S_{3j}}(x)} &\underset{\eqref{intermed}}{\geq} \frac{1 - \delta_j \frac{3j+1}{3j} }{\left( 1 + \frac{\delta_j}{3j} \right) }\\ &\geq \left( 1 - \delta_j \frac{3j+1}{3j} \right) \left( 1 - \frac{\delta_j}{3j} \right) \\ & \geq 1 - \delta_j\left( \frac{3j+1}{3j} + \frac{1}{3j} \right) \geq 1 - \delta_j \frac{5}{3} \geq 1 - \frac{1}{(3j+1)^l}. \end{aligned}$$ Furthermore, using the fact that $t \mapsto \frac{1+t}{1-t}$ is increasing on $(-\infty,1)$ and the fact that $\delta_{j+1} \leq \delta_j$, $$\begin{aligned} \max_{\|x\| = 1} \frac{\sigma_{S_{3j+3}}(x) - \sigma_{S_{3j+2}}(x)}{ \sigma_{S_{3j+2}}(x) - \sigma_{S_{3j+1}}(x)} &\underset{\eqref{intermed}}{\leq} \frac{1 + \frac{\delta_{j+1}}{3j + 3}}{1 - (3j+1) \frac{\delta_j}{3j}} \notag\\ &\leq\frac{1 + (3j+1)\frac{\delta_{j}}{3j}}{1 - (3j+1) \frac{\delta_j}{3j}} \notag\\ &\leq\frac{1 + \delta_j\frac{4}{3}}{1 - \delta_j \frac{4}{3}}\label{borntobewild}. \end{aligned}$$ Setting $s(t)= (1+t)/(1-t)$, we have, for all $t \leq 1/2$ $$\begin{aligned} s'(t) &= \frac{2}{(1-t)^2}, \, s(0)=1\\ s''(t) &= \frac{4}{(1-t)^3} \leq 24,\, s'(0)=2. \end{aligned}$$ Thus $s(t) \leq 1 + 2 t + 12 t^2$ on $(-\infty,1/2]$. Since $\frac{4}{3} \delta_j \leq \frac{4}{75} \leq \frac{1}{2}$, we deduce from the previous remark and above: $$\begin{aligned} \max_{\|x\| = 1} \frac{\sigma_{S_{3j+3}}(x) - \sigma_{S_{3j+2}}(x)}{ \sigma_{S_{3j+2}}(x) - \sigma_{S_{3j+1}}(x)} &\leq 1 + \frac{8}{3} \delta_j + \frac{64}{3} \delta_j^2 = 1 + \delta_j\left( \frac{8}{3} + \frac{64}{3} \delta_j \right)\\ &\leq 1 + \delta_j\left(3 + \frac{64}{3 \times 25}\right) \leq 1 + 4 \delta_j = 1 + \frac{1}{(3j+2)^l}. \end{aligned}$$ Finally using again, $$\begin{aligned} \min_{\|x\| = 1} \frac{\sigma_{S_{3j+3}}(x) - \sigma_{S_{3j+2}}(x)}{ \sigma_{S_{3j+2}}(x) - \sigma_{S_{3j+1}}(x)} &\geq \frac{1 }{(3j +2) \left( 1 + \frac{\delta_{j+1}}{3j+3} \right) - (3j+1) } \\ &= \frac{1 }{1 + \delta_{j+1}\frac{3j+2}{3j+3} } \\ &\geq 1 - \delta_{j+1}\frac{3j+2}{3j+3} \geq 1 - \delta_{j+1} \geq 1 - \frac{1}{(3j+2)^l}. \end{aligned}$$ This proves the desired result. Combining Lemma \[lem:interpolGauge\] and Proposition \[th:globallyLipshitz\], we obtain the following result. Let $\left( PN_i \right)_{i \in {\mathbb{N}}^*}$ be a strictly increasing sequence of interpolable polygon-normal pairs. Set for $i$ in ${\mathbb{N}}^*$, $PN_i = \left\{ \left( A_{j,i} \right)_{j=1}^{n_i}, \left( V_{j,i} \right)_{j=1}^{n_i} \right\}$ where $n_i$ is in ${\mathbb{N}}^*$, denote by $T_i$ the polygon $A_{1,i}\ldots A_{n_i,i}$, and assume that for all $i $ in $ {\mathbb{N}}^*$, $T_i=3i P$ where $P$ is a fixed polygon which contains the unit disk in its interior.\ Then for any $k $ in $ {\mathbb{N}}$, $k \geq 2$ and all $l \geq 3$, there exists a $C^k$ globally Lipschitz continuous Legendre function, $h \colon {\mathbb{R}}^2 \mapsto {\mathbb{R}}$, and an increasing sequence $(\lambda_i)_{i \in {\mathbb{N}}}$, such that for each $i $ in $ {\mathbb{N}}$: - $T_i \subset \left\{ x,\,h(x) \leq \lambda_i \right\}$, - ${\mathrm{dist}}(T_i, \left\{ x,\,h(x) \leq \lambda_i \right\}) \leq \frac{1}{4(3i+2)^l}$, - For any $x$ with $h(x) = \lambda_i$ and $\nabla h(x)$ is colinear to $V_i$ for each vertex $x$ of $T_i$, - $h$ has positive definite Hessian and is globally Lipschitz continuous, - $h^*$ has compact domain and is $C^k$ on the interior of its domain. \[th:Legendre\] The function $h^*$ constructed in Theorem \[th:Legendre\] has compact polygonal domain and is continuous on this domain. \[cor:legendreContinuity\] Since $P$ is a polygon and contains the unit Euclidean disk, the gauge function of $3P$ is polyhedral with full domain ${\mathbb{R}}^2$, call it $\omega$. Denote by $P^{\circ}$ the polar of $P$. This is a polytope and since $\omega$ is the gauge of $P$, we actually have $\omega = \sigma_{P^{\circ}}$, the support function of the polar of $P$ [@schneider1993convex Theorem 1.7.6]. Hence the the convex conjugate of $\omega$ is the indicator of the polytope $P^\circ$ [@rockafellar1970convex Theorem 13.2]. It can be easily seen from the proof of Proposition \[th:globallyLipshitz\] that $\lambda_i = \alpha i + r_i$ with $r(i) = O(1)$ as $i \to \infty$. Without loss of generality, we may suppose that $\alpha = 1$ (this is a simple rescaling) so that there is a positive constant $c$ such that $|\lambda_i - i| \leq c$ for all $i$. Let $h$ be given as in Theorem \[th:Legendre\], fix $i \geq 1$ and $x \in {\mathbb{R}}^2$ such that $\lambda_{i-1} \leq h(x) \leq \lambda_i$. We have in ${\mathbb{R}}^2$ $$\begin{aligned} \{y:\, \omega(y) \leq i-1 \} \subset \{y:\, h(y) \leq \lambda_{i-1} \} \subset \{y:\, h(y) \leq \lambda_{i} \} \subset \{y:\, \omega(y) \leq i+1 \} \end{aligned}$$ and hence $$\begin{aligned} i - 1 \leq \omega(x) \leq i+1 \end{aligned}$$ and we deduce that $$\begin{aligned} \omega(x) - 2 - c \leq i - 1 -c \leq \lambda_{i-1} \leq h(x) \leq \lambda_i \leq i+c \leq \omega(x) + c + 1. \end{aligned}$$ Since $i$ was arbitrary, this shows that there exists a constant $C>0$ such that $|h(x) - \omega(x)|\leq C$ for all $x \in {\mathbb{R}}^2$. Recall that $z \mapsto \sup_{y \in {\mathbb{R}}^2} \left\langle y,z\right\rangle - \omega(y)$ is the indicator function of $P^\circ$, hence, $$\begin{aligned} z \in P^{\circ} & \Rightarrow \sup_{y \in {\mathbb{R}}^2} \left\langle y,z\right\rangle - \omega(y) = 0 \Rightarrow \sup_{y \in {\mathbb{R}}^2} \left\langle y,z\right\rangle - h(y) \leq C < +\infty \\ z \not\in P^{\circ} & \Rightarrow \sup_{y \in {\mathbb{R}}^2} \left\langle y,z\right\rangle - \omega(y) = +\infty \Rightarrow \sup_{y \in {\mathbb{R}}^2} \left\langle y,z\right\rangle - h(y) = +\infty \end{aligned}$$ which shows that the domain of $h^*$ is actually $P^{\circ}$ which is a polytope. Now, $h^*$ is convex and lower semicontinuous on $P^{\circ}$, invoking the results of [@gale68convex], it is also upper semicontinuous on $B^*$ and finally it is continuous on $B^*$. For any $\theta \in \left( \frac{-\pi}{4},\frac{\pi}{4} \right)$ there exists a Legendre function $h\colon {\mathbb{R}}^2 \mapsto {\mathbb{R}}$ whose domain is a closed square, continuous on this domain and $C^k$ on its interior, such that for all $i \in {\mathbb{N}}^*$, $\nabla h^*(i, 0)$ is proportional to $(\cos(\theta), (-1)^i \sin(\theta))$. \[cor:legendreOscillating\] For $x=(u,v)$, set $\|x\|_1=|u|+|v|$, and let $P = \left\{ x \in {\mathbb{R}}^2,\, \|x\|_1 \leq 2 \right\}$. Let us construct a strictly increasing sequence of interpolable polygon-normal pairs $\left( PN_i \right)_{i \in {\mathbb{N}}^*}$ as follows, we fix $\theta \in \left( \frac{-\pi}{4},\frac{\pi}{4} \right)$ and set for all $i\in {\mathbb{N}}^*$ : - $T_i = 3i P$, the polygon associated to the $i$-th term $PN_i$ of the sequence, - except at the rightmost corner, consider the normals given by the canonical basis vectors and their opposite, - at the rightmost corner, $(6i,0)$, one chooses the normal given by the vector $$(\cos(\theta), (-1)^i \sin(\theta)).$$ We now invoke Theorem \[th:Legendre\] to obtain a Lipschitz continuous Legendre function, denoted $h^*$, with full domain having all the $T_i$ as sublevel sets and satisfying the hypotheses of the corollary. Rescaling by a factor $6$ and setting $h = h^{**}$ gives the result. Counterexamples in continuous optimization ========================================== We are now in position to apply our interpolation results to build counterexamples to classical problems in convex optimization. We worked on situations ranging from structural questions to qualitative behavior of algorithms and ODEs. Through 9 counterexamples we tried to cover a large spectrum but there are many more possibilities that are left for future research. Some example are constructed from decreasing sequences of convex sets, they can be interpolated using Theorem \[th:smoothinterp\] with $I = {\mathbb{Z}}$, indexing the sequence with negative indices and adding artificially additional sets for positive indices. Nonetheless we sometimes depart from the notations of the first sections and index these sequences by ${\mathbb{N}}$ even though they are decreasing for simplification purposes. Kurdyka-Łojasiewicz inequality may not hold ------------------------------------------- The following result is proved in [@bolte2010characterization], it was crucial to construct a $C^2$ convex function which does not satisfy Kurdyka-Łojasiewicz (KL) inequality. [[@bolte2010characterization Lemma 35]]{} There exists a decreasing sequence of compact convex sets $\left( T_i \right)_{i\in {\mathbb{N}}}$ such that for any $i $ in $ {\mathbb{N}}$, $T_- = T_{i+1}$ and $T_+ = T_i$ satisfy Assumption \[ass:curvature\] and $$\begin{aligned} \sum_{i=0}^{+\infty} {\mathrm{dist}}(T_i, T_{i+1}) = +\infty \end{aligned}$$ \[lem:tamsLoja\] As a corollary, we improve the counterexample in [@bolte2010characterization] and provide a $C^k$ convex counterexamples for any $k\geq 2 $ in $ {\mathbb{N}}$. There exists a $C^k$ convex function $f \colon {\mathbb{R}}^2 \mapsto {\mathbb{R}}$ which does not satisfy KL inequality. More precisely, for any $r > \inf f$ and $\varphi \colon [\inf f, r] \mapsto {\mathbb{R}}$ continuous and differentiable on $(\inf f, r)$ with $\varphi' > 0$ and $\varphi(\inf f) = 0$, we have $$\begin{aligned} \inf \{\| \nabla (\varphi \circ f)(x) \|: \: x\in {\mathbb{R}}^2, \,\inf f < f(x) < r \} = 0. \end{aligned}$$ Using [@schneider1993convex Theorem 1.8.13], each $T_i$ can be approximated up to arbitrary precision by a polygon. Hence we may assume that all $T_i$ are polygonal while preserving the property of Lemma \[lem:tamsLoja\] as well as Assumption \[ass:curvature\]. Furthermore, using Lemma \[lem:polyGon\] and Remark \[rem:noNormal\] each $T_i$ can in turn be approximated with arbitrary precision by a convex set with $C^k$ boundary and positive curvature. Hence we may also assume that all $T_i$ satisfy both the result of Lemma \[lem:tamsLoja\] and have $C^k$ boundary with nonvanishing curvature. Reversing the order of the sets and adding additional sets artificially, we are in the conditions of application of Theorem \[th:smoothinterp\] with $I = {\mathbb{Z}}$ and the resulting $f$ follows from the same argument as in [@bolte2010characterization Theorem 36]. Block coordinate descent may not converge ----------------------------------------- ![Illustration of the alternating minimization (resp. exact line search) example: on the left, the sublevel sets in gray and the corresponding alternating minimization (resp. exact line search) sequence in dashed lines. On the right the interpolating polygons together with their normal vectors as in Lemma \[lem:polyGon\].[]{data-label="fig:illustrAltMin"}](illustrAltMin1 "fig:"){width=".45\textwidth"}![Illustration of the alternating minimization (resp. exact line search) example: on the left, the sublevel sets in gray and the corresponding alternating minimization (resp. exact line search) sequence in dashed lines. On the right the interpolating polygons together with their normal vectors as in Lemma \[lem:polyGon\].[]{data-label="fig:illustrAltMin"}](illustrAltMin2 "fig:"){width=".4\textwidth"} The following polygonal construction is illustrated in Figure \[fig:illustrAltMin\]. For any $n\geq2$ in ${\mathbb{N}}$, we set $$\begin{aligned} A_n &= \left(\frac{1}{4} + \frac{1}{n}, \frac{1}{4} + \frac{1}{n}\right)\\ B_n &= \left( \frac{1}{4} + \frac{1}{2( n-1)} + \frac{1}{2n}, \frac{1}{4} + \frac{1}{2n} + \frac{1}{2(n+1)} \right)\\ C_n &= \left(\frac{1}{4} + \frac{1}{n}, - \frac{1}{4} - \frac{1}{n}\right)\\ D_n &= \left(-\frac{1}{4} - \frac{1}{n}, - \frac{1}{4} - \frac{1}{n}\right)\\ E_n &= \left(-\frac{1}{4} - \frac{1}{n}, + \frac{1}{4} + \frac{1}{n}\right).\end{aligned}$$ This defines a convex polygon. We may choose the normals at $A_n, C_n, D_n, E_n$ to be bisectors of the corresponding corners and the normal at $B_n$ to be horizontal (see Figure \[fig:illustrAltMin\]). Rotating by an angle of $-\frac{n\pi}{2}$ and repeating the process indefinitely, we obtain the sequence of polygons depicted in Figure \[fig:illustrAltMin\]. It can be checked that the polygons form a strictly decreasing sequence of sets, as for $n>1$, the polygon $A_nB_nC_nD_nE_n$ is contained in the interior of the square $A_{n-1}C_{n-1}D_{n-1}E_{n-1}$. This fulfills the requirement of Corollary \[cor:polygonDecrease\]. There exists a $C^k$ convex function $f \colon {\mathbb{R}}^2 \mapsto {\mathbb{R}}$ and an initialization $x_0=(u_0,v_0)$ such that the recursion, for $i \geq 1$ $$\begin{aligned} u_{i+1} &\in \operatorname*{argmin}_{u} f(u,v_i) \\ v_{i+1} &\in \operatorname*{argmin}_{v} f(u_{i+1},v) \end{aligned}$$ produces a non converging sequence $\displaystyle (x_i)_{i\in{\mathbb{N}}}=\left((u_i,v_i)\right)_{i\in{\mathbb{N}}}$. \[cor:altMin\] We apply Corollary \[cor:polygonDecrease\] to the proposed decreasing sequence and by choosing $(u_0,v_0) = B_2$ for example. This requires to shift indices (start with $i = 2$) and use Theorem \[th:smoothinterp\] with $I = {\mathbb{Z}}$. Note that the optimality condition for partial minimization and the fact that level sets have nonvanishing curvature ensure that the partial minima are unique. In the nonsmooth convex case cyclic minimization is known to fail to provide the infimum value, see e.g., [@auslender p. 94]. Smoothness is sufficient for establishing value convergence (see e.g. [@beck2013convergence; @wright2015coordinate] and references therein), whether it is enough or not for obtaining convergence was an open question. Our counterexample closes this question and shows that cyclic minimization does not yield converging sequences even for $C^k$ convex functions. This result also closes the question for the more general nonconvex case for which we are not aware of a nontrivial counterexample for convergence of alternating minimization. Let us mention however Powell’s example [@powell1973search] which shows that cyclic minimization with three blocks does not converge for smooth functions. It would also be interesting to understand how our result may impact dual methods and counterexamples in that field, as for instance the recent three blocks counterexample in [@chen]. Gradient descent with exact line search may not converge -------------------------------------------------------- Gradient descent with exact line search is governed by the recursion: $$x^+\in \operatorname*{argmin}\left\{f(y):y=x-t\nabla f(x),\,t\in{\mathbb{R}}\right\},$$ where $x$ is a point in the plane. Observe that the step coincides with partial minimization when the gradient $\nabla f(x)$ is colinear to one of the axis of the canonical basis. From the previous section, we thus deduce the following. There exists a $C^k$ convex function $f \colon {\mathbb{R}}^2 \mapsto {\mathbb{R}}$ and an initialization $z_0$ in the plane such that the recursion, for $i \geq 1$ $$\begin{aligned} x_{i+1} \in \operatorname*{argmin}\left\{f(y): y=x_i-t\nabla f(x_i),\, t\in{\mathbb{R}}\right\} \end{aligned}$$ produces a well defined non converging sequence $\left(x_i \right)_{i \in {\mathbb{N}}}$. \[cor:exactLineSearch\] Convergence failure for gradient descent with exact line search is new up to our knowledge. Let us mention that despite non convergence, the constructed sequence satisfy sublinear convergence rates in function values [@beck2013convergence]. Tikhonov regularization path may have infinite length ----------------------------------------------------- ![Illustration of the Tikhonov regularization example, on the left in gray, polygons used to build the sublevel sets of the constructed $f$ and the corresponding solutions to for some values of $r$ (solutions are joined by dotted lines). On the right the normal to be chosen to apply Lemma \[lem:polyGon\] (for $n = 1$, see main text for details). The point $P$ represents $\bar x$, it sits on the $x$-axis and is constantly contained in the normal cone at $B_n$ for any $n \geq 1$.[]{data-label="fig:regTorralba"}](regTorralba1 "fig:"){width=".38\textwidth"}![Illustration of the Tikhonov regularization example, on the left in gray, polygons used to build the sublevel sets of the constructed $f$ and the corresponding solutions to for some values of $r$ (solutions are joined by dotted lines). On the right the normal to be chosen to apply Lemma \[lem:polyGon\] (for $n = 1$, see main text for details). The point $P$ represents $\bar x$, it sits on the $x$-axis and is constantly contained in the normal cone at $B_n$ for any $n \geq 1$.[]{data-label="fig:regTorralba"}](regTorralba2 "fig:"){width=".55\textwidth"} Following [@torralba96], we consider for any $r > 0$ $$\begin{aligned} x(r) = \operatorname*{argmin}\left\{ f(x) + r \|x - \bar{x}\|_2^2:x\in {\mathbb{R}}^2\right\} \label{eq:tikhonovRegularization}\end{aligned}$$ where $f$ is $C^k$ convex and where $\bar x$ is any anchor point. We would like to show that the curve $r \mapsto x(r)$ may have infinite length. Torralba provided a counterexample in his PhD Thesis for [*continuous*]{} convex functions, see [@torralba96]. This work extends his result to smooth $C^k$ convex functions in ${\mathbb{R}}^2$. For any $n $ in $ {\mathbb{N}}^*$, we set $$\begin{aligned} A_n &= \left(\frac{2}{n}, \frac{2}{n}\right)\\ B_n &= \left(\frac{2}{n} + \frac{1}{n^2}, -\frac{1}{n} \right)\\ C_n &= \left(\frac{2}{n}, - \frac{2}{n}\right)\\ D_n &= \left(- \frac{2}{n}, - \frac{2}{n}\right)\\ E_n &= \left(- \frac{2}{n}, \frac{2}{n}\right).\end{aligned}$$ This is depicted in Figure \[fig:regTorralba\]. For all $n \geq 1$, denote by $M_n$ the point on the $x$ axis above $B_n$ and $N_n$, the intersection of the normal cone at $B_n$ and the $x$ axis. We have $$\begin{aligned} \frac{M_nN_n}{M_nB_n} = n \times M_nN_n= \frac{A'_nB_n}{A_nA'_n} = \frac{3/n}{1/n^2} = 3 n,\end{aligned}$$ so that for all $n \geq 1$, $M_nN_n = 3$ and $N_n = (3 + 2/n + 1 / n^2, 0)$. Choosing $P = (7,0)$, since for $n \geq 1$, $3 + 2/n + 1 / n^2 \leq 6 < 7$ , this shows that $P$ constantly belongs to the interior of the normal cone at $B_n$ for all $n \geq 1$. The sequence of level sets is constructed as in Figure \[fig:regTorralba\] by considering alternating symmetries with respect to the $x$-axis of the sequence of polygons above. It can be checked that the polygons form a strictly decreasing sequence of sets, as for $n>1$, the polygon $A_nB_nC_nD_nE_n$ is contained in the interior of the square $A_{n-1}C_{n-1}D_{n-1}E_{n-1}$. We choose the normal at $A_n, C_n, D_n, E_n$ to belong to the bisector at the corner and the normal at $B_n$ to be proportional to the vector $B_n P$. Applying Corollary \[cor:polygonDecrease\], we construct $f$ and choose $\bar{x} = P$ in to obtain the following: There exists a $C^k$ strictly convex function $f \colon {\mathbb{R}}^2 \mapsto {\mathbb{R}}$ and $\bar{x} \in {\mathbb{R}}^2$ such that the curve $x((0,1))$ given by has infinite length. \[cor:homotopy\] We apply Corollary \[cor:polygonDecrease\] with $I = {\mathbb{Z}}$ and revert the indices set to match the sequence that we have described. For any $n \geq 1$ there exists a value of $\lambda_n$ such that $f(B_n)= \lambda_n$ and $\nabla f(B_n)$ is colinear to the vector $B_n P$. Set $$\begin{aligned} r = \frac{\|\nabla f(B_n)\|}{2 B_nP} \end{aligned}$$ we have $\nabla f(B_n) + 2r(B_n - P) = 0$ which is the optimality condition in with $\bar{x} = P$. Hence we have shown that there exists a value of $r$ such that $B_n$ is the solution to . Since $n$ was arbitrary this is true for all $n$ and the curve $r \mapsto x(r)$ has to go through a sequence of points whose second coordinate is of the form $(-1)^n/n$ for all $n \geq 1$. Since this sequence is not absolutely summable, the curve has infinite length. This result is in contrast with the definable case for which we have finite length by the monotonicity lemma, since the whole trajectory is definable and bounded. Secants of gradient curves at infinity may not converge ------------------------------------------------------- #### Thom’s gradient conjecture and Kurdyka-Mostowski-Parusinski’s theorem A theorem of Łojasiewicz [@lojasiewicz1984trajectoires] asserts that bounded solutions to the gradient system $$\begin{aligned} \dot{x}(t) = - \nabla f(x(t))\end{aligned}$$ converge when $f$ is a real analytic potential. Thom conjectured in [@thom1989problemes] that this convergence should occur in a stronger form: trajectories converging to a given $\bar{x}$ should admit a tangent at infinity, that is $$\begin{aligned} \label{secants} \frac{x(t) - \bar{x}}{\|x(t) - \bar{x}\|}\end{aligned}$$ should have a limit as $t \to \infty$. Lines passing through $\bar x$ and having as a slope are called [*secants of $x$ at $\bar x$*]{}. This conjecture was proved to be true in [@kurdyka2000proof]. In the convex world, it is well known that solutions to the gradient system converge for general potentials (this is a Féjer monotonicity argument due to Bruck); see also the original approaches by Manselli and Pucci [@Manselli] and Daniilidis et al. [@dan]. It is then natural to wonder whether this convergence satisfies higher order rigidity properties as in the analytic case. The answer turns out to be negative in general yielding a quite mysterious phase portrait. #### Absence of tangential convergence for convex potentials ![$A = (-5,0)$, $B=\left( - \frac{5}{3} - \frac{25}{16}, \frac{10}{3} + \frac{5}{4} \right)$, $C=\left( \frac{-5}{2},5 \right)$, $D=\left( 0,5 \right)$, $E = \left( 5,0 \right)$, $F = \left( 0, -5 \right)$. All normals are chosen to be bisectors except $w$ which is parallel to the line $(DE)$. The vector $v$ is orthogonal to the segment $[BC]$. The point $C'$ is obtained by considering the intersection between the line $(Bw)$ (starting from $B$ with direction $w$), and the segment $[OC]$. The points $A'$, $B'$, $D'$, $E'$, $F'$ are obtained by performing a scaling of $A,B,D,E,F$ of a factor $\frac{OC'}{OC}$. The polygon $A''B''C''D''E''F''$ is $ABCDEF$ scaled by a factor $\frac{OC' + OC}{2 OC}$.[]{data-label="fig:thom"}](thom){width=".6\textwidth"} The construction given in this paragraph is more complex than the previous ones, we start with a technical lemma which will be the basic building block for our counterexample. Let $S$ be a convex set with $C^k$ boundary interpolating $ABCDEF$ in Figure \[fig:thom\] and let $g$ be the gauge function associated to $S$. The function $g$ is differentiable outside the origin. Consider any initialization $x_0 $ in $ [BC]$ with corresponding trajectory solution to the equation $$\begin{aligned} \dot{x}(t) &= - \nabla g(x(t)), \, t\geq 0,\\ x(0) &= x_0. \end{aligned}$$ Set $\bar{t} = \sup_{x(t) \in OBC} t$, we have $\bar{t} < + \infty$ and $x(\bar{t}) $ in $ [CC']$. \[lem:techThom\] The fact that $g$ is differentiable comes from the fact that its subgradient is uniquely determined by the normal cone to $S$ which has dimension one because of the smoothness of the boundary of $S$. Since $S$ is interpolating the polygon, we have $g(B) = g(C) = 1$. Furthermore, we have for all $t$, $\frac{d}{dt} g(x(t)) = -\|\nabla g(x(t))\|^2= -1$, thence $\bar{t} \leq 1 - g(C')$. By homogeneity, for any $x \neq 0$ and $s > 0$, $\nabla g (sx) = \nabla g(x)$. For any $x $ in $ [BC]$, by convexity $$\begin{aligned} 0 \leq \left\langle C-x,\nabla g(C) - \nabla g(x) \right\rangle = \left\langle C-B,\nabla g(C) - \nabla g(x) \right\rangle \frac{\|C-x\|}{\|C-B\|}, \end{aligned}$$ and therefore $$\begin{aligned} -\left\langle C - B, \nabla g(x) \right\rangle \geq -\left\langle C - B, \nabla g(C) \right\rangle \label{eq:thomGradient1} \end{aligned}$$ By homogeneity of $g$, is true for any $x$ in the triangle $OCB$ (different from $0$) and thus in the triangle $C'CB$ . Denote by $y$ the solution to the equation $$\begin{aligned} \dot{y} &= - \nabla g(C)\\ y(0) &= B, \end{aligned}$$ which integrates to $y(t) = B - t w$ for all $t$. Equation ensures that for any $0\leq t \leq \bar{t}$ $$\begin{aligned} \frac{d}{dt} (\left\langle C - B, x(t) \right\rangle) &\geq \frac{d}{dt} (\left\langle C - B, y(t) \right\rangle) \end{aligned}$$ Hence, we have for any $0\leq t \leq \bar{t}$, integrating on $[0,t]$ $$\begin{aligned} \left\langle C - B, x(t) \right\rangle &\geq \left\langle C - B, y(t) \right\rangle + \left\langle C - B, x_0 - B \right\rangle \nonumber\\ &\geq \left\langle C - B, y(t) \right\rangle. \label{eq:thomGradient2} \end{aligned}$$ Furthermore, for all $x $ in $ [BC]$, we have $$\begin{aligned} 1 = \|\nabla g(x)\|^2 = \frac{1}{\|C - B\|^2}\left\langle C - B, \nabla g(x) \right\rangle^2 + \left\langle v, \nabla g(x) \right\rangle^2, \end{aligned}$$ because $v$ is orthogonal to $C-B$. The first term is maximal for $x = C$ and thus the second term is minimal for $x = C$, we have thus for all $x $ in $ [BC]$ $$\begin{aligned} 0 < \left\langle \nabla g(C), v \right\rangle = \left\langle - \nabla g(C), -v \right\rangle \leq \left\langle \nabla g(x), v \right\rangle = \left\langle -\nabla g(x), -v \right\rangle \leq 1. \label{eq:thomGradient3} \end{aligned}$$ Equation holds for all $x$ in $OCB$ different from $O$ by homogeneity. We deduce that for all $0 \leq t \leq \bar{t}$, we have $$\begin{aligned} \frac{d}{dt} (\left\langle -v, x(t) \right\rangle) &\geq \frac{d}{dt} (\left\langle -v, y(t) \right\rangle) \end{aligned}$$ and by integration $$\begin{aligned} \left\langle -v, x(t) \right\rangle &\geq \left\langle -v, y(t) \right\rangle + \left\langle -v, x_0 - B \right\rangle \nonumber\\ &= \left\langle -v, y(t) \right\rangle. \label{eq:thomGradient4} \end{aligned}$$ Hence, in the coordinate system $(C - B, -v)$, which is orthogonal, for all $t $ in $ [0,\bar{t}]$, $x(t)$ has larger coordinates than $y(t)$. The trajectory $y(t)$, of equation $t \mapsto B - t w$ is the line going from $B$ to $C'$. From equations and , we may write for all $t$ in $ [0,\bar{t}]$, $x(t) = y(t) + \alpha (t) (C-B) + \beta(t) (-v)$ where $\alpha$ and $\beta$ are positive functions. Since $y(t)$ belongs to the line $(BC')$, this shows that $x(t)$ has to be above this line for all $t \geq 0$, $t\leq \bar{t}$ and actually, $x(\bar{t}) $ in $ BCC'$. Hence at time $\bar{t}$, we have $x(\bar{t}) $ in $ [CC']$. This holds true because $x(\bar{t})$ is on the boundary of $OCB$ and on the boundary of $BCC'$. Hence either $x(\bar{t}) $ in $ [CC']$, either $x(\bar{t}) $ in $ [BC]$. Equation ensures that if $x(\bar{t}) $ in $ [BC]$ then $x(\bar{t}) = C$ which concludes the proof. There exists a $C^k$ strictly convex function on ${\mathbb{R}}^2$ with a unique minimizer $\bar{x}$, such that any nonconstant solution to the gradient flow equation $$\begin{aligned} \dot{x}(t) = -\nabla f(x(t)) \end{aligned}$$ is such that $$\begin{aligned} \frac{x(t) - \bar{x}}{\|x(t) - \bar{x}\|} \end{aligned}$$ does not have a limit as $t \to \infty$.\ The function $f$ has a positive definite Hessian everywhere except at $0$. \[cor:Thom\] We assume without loss of generality that $\bar{x} = O$ is the origin. Writting $x(t) = (r(t),\theta(t))$ in polar coordinate, we will construct a function $f$ such that each solution to the ODE produces nonconverging trajectories $\theta(t)$. We start with an interpolating set $S_0 = ABCDE$ as in Lemma \[lem:techThom\] and let $S_1 = A'B'C'D'E'$ be its scaled version as described in Figure \[fig:thom\]. Let $\alpha$ be the value of the angle $\widehat{BOC}$ and $m = \left\lceil \frac{2 \pi}{\alpha} \right\rceil + 1$. We have $$\begin{aligned} \frac{2\pi}{m} < \alpha. \end{aligned}$$ To obtain $S_{2}$, we rotate $S_0$ by an angle $2\pi / m$, we denote $S'_0$ the resulting set. We rescale $S'_0$ by a factor $\beta $ in $ (0,1)$ so that $\beta S'_0$ lies in the interior of $S_1$. Call the resulting set $S_2$ and $S_3$ is obtained from $S_2$ exactly the same way as $S_1$ is obtained from $S_0$. We repeat the same process indefinitely to obtain a strictly decreasing sequence of $C^k$ sets. Note that for any $k $ in $ {\mathbb{N}}$, $S_{2km}$ and $S_{2km + 1}$ are homothetic to $S_0$. We now invoke Corollary \[cor:polygonDecrease\] (with $I = {\mathbb{Z}}$ and revert the indices) to obtain a $C^k$ function $f$ with those prescribed level sets. Using Remark \[rem:alignedLevelSets\] it turns out that the level sets of $f$ between $S_0$ and $S_1$ are simple scalings of $S_0$. Hence the gradient curves of $f$ and those of the gauge function of $S$ are the same between $S_0$ and $S_1$, up to time reparametrization. Using Lemma \[lem:techThom\] any trajectory crossing $[BC]$ in Figure \[fig:thom\], must also be crossing $[CC']$ and leave the triangle $BOC$ in finite time. The same statement holds after scaling the level sets and since for all $k $ in $ {\mathbb{N}}$, $S_{2km}$ and $S_{2km+1}$ are homothetic to $S_0$, this shows that no solution stays indefinitely in the triangle $BOC$. Lemma \[lem:techThom\] still holds after rotations and by our construction, for any triangle $T$ obtained by rotating $BOC$ by a multiple of $2\pi/m$, no trajectory stays indefinitely within $T$. Since $2\pi / m < \alpha$, the union of these triangles $U$ contains $O$ in its interior. Note first that any gradient curve converges to $\bar x$. Let us argue by contradiction and assume that there exists a continuous gradient curve $t \mapsto z(t)$ distinct from the stationary solution $\bar{x}$, such that $$\begin{aligned} \frac{z(t) - \bar{x}}{\|z(t) - \bar{x}\|} \end{aligned}$$ converges. This exactly means that the angle $\theta(t)$ of the curve has a limit in $[0,2\pi)$ as $t$ goes to infinity. There is a rotation of $BOC$ by a multiple of $2\pi/m$ whose interior intersects the half line given by the direction $\theta$, call this triangle $T$. The directional convergence entails that there exists $t_0 \geq 0$ such that $z(t)$ belongs to $T$ for all $t \geq t_0$. Hence $z$ can not be a gradient curve. To complete the proof, we may add disks of increasing size to the list of sets to obtain a full domain function and invoke Theorem \[th:smoothinterp\] with $I = {\mathbb{Z}}$. Newton’s flow may not converge ------------------------------ Given a twice differentiable convex function $f$, we define the open set $\Omega:=\{x\in{\mathbb{R}}^2:\nabla^2f \mbox{ is invertible}\}$ and we consider maximal solutions to the differential equation $$\begin{aligned} \dot{x}(t) = - \nabla^2 f(x(t))^{-1} \nabla f(x(t)), \label{eq:newton}\end{aligned}$$ on $\Omega$. This is the continuous counterpart of Newton’s method, it has been studied in [@aubin1984differential] and [@alvarez1998dynamical]. Let $x_0$ be in $\Omega$, there exists a unique maximal nontrivial interval $I$ containing $0$ and a unique solution $x$ to on $I$ with $x(0) = x_0$. Equation may be rewritten as $$\begin{aligned} \frac{d}{dt} \nabla f(x(t)) = - \nabla f(x(t))\end{aligned}$$ and thus for all $t $ in $ I$, we have $$\begin{aligned} \nabla f(x(t)) = e^{-t} \nabla f(x_0). \label{eq:newtonIntegrated}\end{aligned}$$ If we could ensure that $I = {\mathbb{R}}$ and $f$ has oscillating gradients close to its minimum, then entails that the direction of the gradient is constant along the solution, which requires oscillations in space to compensate for gradient oscillations. For any $k\geq2$, there exists a $C^k$ convex coercive function $f \colon {\mathbb{R}}^2 \mapsto {\mathbb{R}}$ and an initial condition $x_0 $ in $ {\mathbb{R}}^2$ such that the solution to is bounded, defined on ${\mathbb{R}}$ and has at least two distinct accumulation points.\[cor:newton\] The counterexample is sketched in Figure \[fig:illustrNewton\], the construction is the same as for Corollary \[cor:altMin\] but instead of doing quarter rotations, we use symmetry with respect to the first axis. We can then call for Corollary \[cor:polygonDecrease\] to construct the function $f$ and equation ensures that the solution interval is unbounded. ![Illustration of the continuous time Newton’s dynamics. On the left, the “skeletons" of the sublevel sets in gray and a sketch of the corresponding curve. On the right, the normals to be chosen in order to apply Lemma \[lem:polyGon\].[]{data-label="fig:illustrNewton"}](illustrNewton "fig:"){width=".45\textwidth"}![Illustration of the continuous time Newton’s dynamics. On the left, the “skeletons" of the sublevel sets in gray and a sketch of the corresponding curve. On the right, the normals to be chosen in order to apply Lemma \[lem:polyGon\].[]{data-label="fig:illustrNewton"}](illustrAltMin2 "fig:"){width=".4\textwidth"} Bregman descent (mirror descent) may not converge ------------------------------------------------- The mirror descent algorithm was introduced in [@newmirovsky1983problem] as an efficient method to solve constrained convex problems. In [@beck2003mirror], this method is shown to be equivalent to a projected subgradient method, using non-Euclidean projections. It plays an important role for some categories of constrained optimization problem; see e.g., [@bauschke2016descent] for recent developments and [@Dragomir19] for a surprising example. Let us recall beforehand some definitions. Given a Legendre function $h$ with domain ${\mathrm{dom}\,}h$, define the [*Bregman distance*]{}[^5] associated to $h$ as $D_h(u,v)=h(u)-h(v)-\langle \nabla h(v),u-v\rangle$ where $u$ is in $ {\mathrm{dom}\,}h$ and $v$ is in the interior of ${\mathrm{dom}\,}h$. Given a smooth convex function $f$ that we wish to minimize on $\overline{ {\mathrm{dom}\,}h}$, we consider the Bregman method $$x_{i+1}=\operatorname*{argmin}\left\{\langle \nabla f(x_i),u-x_i\rangle+\lambda D_h(u,x_i):u\in {\mathbb{R}}^2\right\},$$ where $x_0$ is in ${\mathrm{int}\,}{\mathrm{dom}\,}h$ and $\lambda>0$ is a step size. When the above iteration is well defined, e.g. when ${\mathrm{dom}\,}h$ is bounded, it writes: $$x_{i+1}=\nabla h^*\left(\nabla h(x_i)-\lambda \nabla f(x_i)\right).$$ In [@bauschke2016descent] the authors identified a generalized smoothness condition which confers good minimizing properties to the above method: $$\begin{aligned} \label{hsmooth}Lh-f \mbox{ convex},\\ \lambda\in(0,L).\label{step}\end{aligned}$$ The corollary below shows that such an algorithm may not converge, even though we assume the cost to satisfy , the step to satisfy , and the Legendre function to have a compact domain. There exists a Legendre function $h\colon D \mapsto {\mathbb{R}}$, defined on a closed square $D$, continuous on $D$, a vector $c$ in $ {\mathbb{R}}^2$, and $x_0$ in ${\mathbb{R}}^2$ such that the Bregman descent recursion $$\begin{aligned} x_{i+1} = \nabla h^*\left( \nabla h(x_i) - c \right), \end{aligned}$$ produces a bounded sequence $(x_i)_{i \in {\mathbb{N}}}$ which has at least two distinct accumulation points. \[cor:mirrorDescent\] We fix $\theta \in \left( \frac{-\pi}{4},\frac{\pi}{4} \right)$, $\theta \neq 0$, and consider $h$ constructed in Corollary \[cor:legendreOscillating\] and choose $c = (-1,0)$. In this case the Bregman descent recursion writes for all $i$ in ${\mathbb{N}}$, $$\begin{aligned} \nabla h (x_{i+1}) - \nabla h(x_i) = -c \end{aligned}$$ so that we actually have $\nabla h (x_{i}) - \nabla h(x_{0}) = \nabla h (x_{i}) = - i c$ and thus $$x_i = \nabla h^*(-ic) = \nabla h^*(i,0).$$ By Corollary  \[cor:legendreOscillating\], we have for all $i \in {\mathbb{N}}$ that $\nabla h^*(i,0)$ proportional to $(\cos(\theta), (-1)^i \sin(\theta))$. Since the norm of the gradient of $h^*$ cannot vanish at infinitiy (no flat direction) and is bounded, this proves that the sequence $(x_i)_{i \in {\mathbb{N}}}$ has at least two accumulation points which is the desired result. Central paths of Legendre barriers may not converge --------------------------------------------------- Consider the problem $$\begin{aligned} \min_{x \in D} \left\langle c, x\right\rangle \label{eq:LP}\end{aligned}$$ where $D$ is a subset of ${\mathbb{R}}^2$ and $c$. Given a Legendre function $h$ on $D$, we introduce the $h$ central path through $$\begin{aligned} x(r) = \operatorname*{argmin}\left\{ \left\langle c, x\right\rangle + r h(x):x\in{\mathbb{R}}^2\right\} \label{eq:interiorLegendre}\end{aligned}$$ where $r>0$ is meant to tend to $0$. Central paths are one of the essential tools behind interior point methods, see e.g., [@NN; @Aus99] and references therein. Note that the accumulation points of $x(r)$ as $r \to 0$, have to be in the the solution set of . It is even tempting to think that the convergence of the path to some specific minimizer could occur, as it is the case for many barriers, see e.g. [@Aus99]. We have however: There exists a Legendre function $h\colon D \mapsto {\mathbb{R}}$, defined on a closed square $D$, continuous on $D$, a vector $c$ in $ {\mathbb{R}}^2$, such that the $h$ central path $r \mapsto x(r)$ has two distinct accumulation points. \[cor:interiorPoint\] The optimality condition which characterizes $x(r)$ for any $r > 0$ writes, $$\begin{aligned} x(r)= \nabla h^*\left( \frac{c}{r} \right), \end{aligned}$$ and the construction is the same as in Corollary \[cor:mirrorDescent\]. Hessian Riemannian gradient dynamics may not converge ----------------------------------------------------- The construction of this paragraph is similar to the two previous paragraphs. Consider a $C^k$ ($k\geq 2$) Legendre function $h \colon D \mapsto {\mathbb{R}}$ and the continuous time dynamics $$\begin{aligned} \dot{x}(t) = - \nabla_H f(x(t)), \, t\geq 0, \label{eq:hessianRiemannian}\end{aligned}$$ where $H = \nabla^2 h$ is the Hessian of $h$ and $\nabla_H f = H^{-1} \nabla f$ is the gradient of some differentiable function $f$ in the Riemannian metric induced by $H$ on ${\mathrm{int}\,}D$. Such dynamics were considered in [@BT03; @alvarez2004hessian]. We have the following result: There exists a Legendre function $h\colon D \mapsto {\mathbb{R}}$, defined on a closed square $D$, continuous on $D$, a vector $c$ in $ {\mathbb{R}}^2$, and $x_0$ in ${\mathbb{R}}^2$ such that the solution to with $f=\langle c,\cdot\rangle$ has two distinct accumulation points. \[cor:hessianRiemannian\] Equation may be rewritten $$\begin{aligned} \frac{d}{dt} \nabla h(x(t))= - \nabla f(x(t)), \end{aligned}$$ so choosing $c = (-1,0)$, we have for all $t \in {\mathbb{R}}$, $\nabla h(x(t)) = \nabla h(x(0)) + (t,0) = (t,0)$ and the construction is the same as in Corollary \[cor:mirrorDescent\]. Appendix ======== There exists a $C^\infty$ strictly increasing concave function $\phi\colon [0,1] \mapsto [0,1]$ such that $$\begin{aligned} \phi(t) &= \sqrt{2t/3} \quad \forall t \leq 1/6\\ \phi(1) &= 1 \\ \phi'(1) &= 2/3\\ \phi^{(m)}(1) &= 0, \quad \forall m \geq 2 \end{aligned}$$ \[lem:interpolationAroundZero\] Consider a $C^\infty$ function $g_0 \colon {\mathbb{R}}\mapsto [0,1]$ such that $g_0 = 1$ on $(-\infty,-1)$, $g_0 = 0$ on $(1, +\infty)$ (for example convoluting the step function with a smooth bump function). Set $g(t) = \frac{1}{2}\left( g_0(t) + 1 - g_0(-t) \right)$ we have that $g$ is $C^\infty$, $g = 1$ on $(-\infty,-1)$, $g = 0$ on $(1, +\infty)$ and $g(t) + g(-t) = 1$ for all $t$. We have $$\begin{aligned} \int_{-1}^1 g(s) ds = 1\\ \int_{-1}^1 \left( \int_{-1}^t g(s)ds \right) dt = 1 \end{aligned}$$ Set $\phi_0 \colon [-3,3] \mapsto {\mathbb{R}}$, such that $$\begin{aligned} \phi_0(t) = \int_{-3}^t \left( \int_{-3}^r g(s) ds\right)dr. \end{aligned}$$ For all $r $ in $ [-3,3]$, we have $$\begin{aligned} \int_{-3}^r g(s) ds = \begin{cases} r+3 & \text{ if } r \leq -1\\ 2 + \int_{-1}^r g(s)ds & \text{ if } -1 \leq r \leq 1\\ 3 & \text{ if } r \geq 1 \end{cases} \end{aligned}$$ and thus $$\begin{aligned} \phi_0(t) = \begin{cases} \frac{t^2}{2} - 9/2 + 3(t+3) & \text{ if } t \leq -1\\ 2 + 2(t + 1) + \int_{-1}^t \left( \int_{-1}^r g(s)ds \right)dr& \text{ if } -1 \leq t \leq 1\\ 6 + 3(t-1) & \text{ if } 1 \geq t \end{cases} \end{aligned}$$ and in particular $\phi_0(3) = 12$ and $\phi_0'(3) = 3$. Set $\phi_1(s) = \phi_0(6 s -3)/12$. $$\begin{aligned} \phi_1(0) &= 0\\ \phi_1(t) &= \left(\frac{(6t-3)^2}{2} - 9/2 + 2(3t )\right)/12 = 3 t^2 / 2 = \text{ if } t \leq 1/3\\ \phi_1(1) &= 1\\ \phi_1'(1) &= 3/2. \end{aligned}$$ $\phi_1$ is stricly increasing, let $\phi \colon [0,1] \mapsto [0,1]$ denote the inverse of $\phi_1$, we have $$\begin{aligned} \phi(1) &= 1\\ \phi'(1) & = 2 / 3\\ \phi(t) &= \sqrt{2t/3} \text{ if } t\leq 1/6. \end{aligned}$$ Consider any strictly increasing $C^k$ function $\phi \colon (0,2) \mapsto {\mathbb{R}}$ such that $\phi(1) = 1$ and $\phi^{(m)}(1) = 0$, $m = 2,\ldots k$. Then the function $$\begin{aligned} G \colon (0,2) \times {\mathbb{R}}/ 2 \pi {\mathbb{Z}}& \mapsto {\mathbb{R}}^2 \\ (s,\theta) &\mapsto \phi(s) n(\theta) \end{aligned}$$ is diffeomorphism which satisfies for any $m=1 \ldots,k$ and $l =2,\ldots, k$, $$\begin{aligned} &\frac{\partial^m G}{\partial \theta^m}(1,\theta) = n^{(m)}(\theta)\\ &\frac{\partial^{m+1} G}{\partial \lambda\partial \theta^m} (1,\theta) = \phi'(1)n^{(m)}(\theta) \\ &\frac{\partial^{l+m} G}{\partial \lambda^l \partial \theta^m }(\lambda_-,\theta) = 0. \end{aligned}$$ \[lem:diffGauge2\] [**: Combinatorial Arbogast-Faà di Bruno Formula (from [@ma2009higher]).**]{} Let $g \colon {\mathbb{R}}\mapsto {\mathbb{R}}$ and $f \colon {\mathbb{R}}^p \mapsto [0, +\infty)$ be $C^k$ functions. Then we have for any $m \leq k$ and any indices $i_1,\ldots,i_m \in \left\{ 1,\ldots, p \right\}$. $$\begin{aligned} \frac{\partial^m}{\prod_{l=1}^{m}\partial x_{i_l}} g \circ f(x) = \sum_{\pi \in {\mathcal{P}}} g^{(|\pi|)}(f(x)) \prod_{B \in \pi} \frac{\partial^{|B|} f}{\prod_{l \in B}\partial x_{i_l}}(x), \end{aligned}$$ where ${\mathcal{P}}$ denotes all partitions of $\left\{ 1,\ldots, m \right\}$, the product is over subsets of $\left\{ 1,\ldots,m \right\}$ given by the partition $\pi$ and $|\cdot|$ denotes the number of elements of a set. We rewrite this as follows $$\begin{aligned} \frac{\partial^m}{\prod_{l=1}^{m}\partial x_{i_l}} g \circ f(x) = \sum_{k = 1}^m\sum_{\pi \in {\mathcal{P}}_k} g^{(k)}(f(x)) \prod_{B \in \pi} \frac{\partial^{|B|} f}{\prod_{l=1}^{m}\partial x_{i_l}}(x), \end{aligned}$$ where ${\mathcal{P}}_k$ denotes all partitions of size $k$ of $\left\{ 1,\ldots, m \right\}$. \[lem:faaDiBruno\] [From [@bolte2010characterization Lemma 45]]{} Let $h $ in $ C^0\left( (0,r_0],{\mathbb{R}}_+^* \right)$ be an increasing function. Then there exists a function $\psi $ in $ C^\infty({\mathbb{R}},{\mathbb{R}}_+)$ such that $\psi = 0$ on, ${\mathbb{R}}_-$ and $0 < \psi(s) \leq h(s)$ for any $s$ in $(0,r_0]$ and $\psi$ is increasing on ${\mathbb{R}}$ \[lem:CinfiniteLowerBound\] Let $D \subset {\mathbb{R}}^p$ be a nonempty compact convex set and $f \colon D \mapsto {\mathbb{R}}$ convex, continuous on $D$ and $C^k$ on $D \setminus \operatorname*{argmin}_{D} f$. Assume further that $\operatorname*{argmin}_D f \subset \mathrm{int}(D)$, $k \geq 1$, with $\min_D f = 0$. Then there exists $\phi \colon {\mathbb{R}}\mapsto {\mathbb{R}}_+$, $C^k$, convex and increasing with positive derivative on $(0,+\infty)$, such that $\phi\circ f$ is convex and $C^k$ on $D$. \[lem:CkSmoothing\] By a simple translation, we may assume that $\min_D f = 0$ and $\max_D f = 1$. Any convex function is locally Lipschitz continuous on the interior of its domain so that $f$ is globally Lipschitz continuous on $D$ and its gradient is bounded. Hence, $f^2$ is $C^1$ and convex on $D$. We now proceed by recursion. For any $m =1,\ldots, k$, we let $Q_m$ denote the $m$-order tensor of partial derivatives of order $m$. Fix $m$ in $\{1,\ldots,k\}$. Assume that $f$ is $C^m$ throughout $D$ while it is $C^{m+1}$ on $D \setminus \arg\min_D f$. Note that all the derivatives up to order $m$ are bounded. We wish to prove that $f$ is globally $C^{m+1}$. Consider the increasing function $$\begin{aligned} h \colon (0,1] &\mapsto {\mathbb{R}}_+^*\\ s &\mapsto \frac{s}{1 + \sup_{s \leq f(x) \leq 1}\|Q_{m+1}(x)\|_{\infty}} \end{aligned}$$ and set $\psi$ as in Lemma \[lem:CinfiniteLowerBound\]. Recall that $\psi$ is $C^\infty$ and all its derivative vanish at $0$ and $\psi \leq h$ on $(0,1]$. Let $\phi$ denote the anti-derivative of $\psi$ such that $\phi(0) = 0$. $\phi$ is $C^\infty$ and convex increasing on ${\mathbb{R}}$ and, since its derivatives at $0$ vanish as well, one has, for any $q $ in $ {\mathbb{N}}$, $\phi^{(q)}(z) = o(z)$. Consider the function $\phi \circ f$. It is $C^m$ on $D$ and it has bounded derivatives up to order $m$. Furthermore, it is $C^{m+1}$ on $D \setminus \operatorname*{argmin}_D f$. Let $\bar{y} $ in $ \operatorname*{argmin}_D f$. If $\bar{y} $ in $ \mathrm{int}(\operatorname*{argmin}_D f)$, then $f$ and $\phi \,\circ f$ have derivatives of all order vanishing at $\bar{y}$. Assuming that $\bar{y} $ in $ \operatorname*{argmin}_D f\setminus \mathrm{int}(\operatorname*{argmin}_D f)$. By the induction assumption and Lemma \[lem:faaDiBruno\], we have for any indices $i_1,\ldots,i_m \in \left\{ 1,\ldots, p \right\}$ and any $h $ in $ {\mathbb{R}}^p$: $$\begin{aligned} &\frac{\partial^m}{\prod_{l=1}^{m}\partial x_{i_l}} (\phi \circ f)(\bar{y} + z) - \frac{\partial^m}{\prod_{l=1}^{m}\partial x_{i_l}} (\phi \circ f)(\bar{y}) \\ =\;& \frac{\partial^m}{\prod_{l=1}^{m}\partial x_{i_l}}( \phi \circ f)(\bar{y} + z) \\ =\;&\sum_{q = 1}^{m}\sum_{\pi \in {\mathcal{P}}_q} \phi^{(q)}(f(\bar{y} + z)) \prod_{B \in \pi} \frac{\partial^{|B|} f}{\prod_{l=1}^{m}\partial x_{i_l}}(\bar{y} + z). \end{aligned}$$ All the derivatives of $f$ are of order less or equal to $m$ and thus remain bounded as $z \to 0$. Further more $f$ is Lipschitz continuous on $D$ so that $f(\bar{y} + z) = O(\|z\|)$ near $0$, and, for any $q $ in $ {\mathbb{N}}$, $\phi^{(q)}(f(\bar{y} + z)) = o(\|z\|)$. Hence $\phi \circ f$ has derivative of order $m+1$ at $\bar{y}$ and it is $0$. Since $\operatorname*{argmin}_D f \subset \mathrm{int}(D)$, we may consider any sequence of point $(y_{j})_{j \in {\mathbb{N}}}$ in $D \setminus \operatorname*{argmin}_D f$ converging to $\bar{y}$. By Lemma \[lem:faaDiBruno\], we have for any indices $i_1,\ldots,i_{m+1} \in \left\{ 1,\ldots, p \right\}$, and any $j $ in $ {\mathbb{N}}$, $$\begin{aligned} \frac{\partial^{(m+1)}}{\prod_{l=1}^{m+1}\partial x_{i_l}} (\phi \circ f)(y_j) &= \phi'(f(y_j)) \frac{\partial^{(m+1)} f}{\prod_{l=1}^{m}\partial x_{i_l}}(y_j) + \sum_{q = 2}^{m+1}\sum_{\pi \in \Pi_q} \phi^{(q)}(f(y_j)) \prod_{B \in \pi} \frac{\partial^{|B|} f}{\prod_{l=1}^{m}\partial x_{i_l}}(x)\\ &\leq h(f(y_j))\frac{\partial^{(m+1)} f}{\prod_{l=1}^{m}\partial x_{i_l}}(y_j) + \sum_{q = 2}^{m+1}\sum_{\pi \in \Pi_q} \phi^{(q)}(f(y_j)) \prod_{B \in \pi} \frac{\partial^{|B|} f}{\prod_{l=1}^{m}\partial x_{i_l}}(x)\\ &= f(y_j) \frac{\frac{\partial^{(m+1)} f}{\prod_{l=1}^{m}\partial x_{i_l}}(y_j)}{1 + \sup_{f(y_j) \leq f(x) \leq 1}\|Q_{m+1}(x)\|_{\infty}} + O(f(y_j))\\ &= O(f(y_j)), \end{aligned}$$ where the inequality follows from the construction of $\phi$. The third step follows using the definition of $h$ and the fact that, for any $q \geq 2$, 1. Each partition of $\left\{ 1,\ldots,m+1 \right\}$ of size $q$ contains subsets of size at most $m$. Thus in the product, the terms $\partial^{|B|} f$ correspond to bounded derivatives of $f$ by the induction hypothesis. 2. $\phi^{(q)}(a) = o(a)$ as $a \to 0$. The last step stems from the fact that the ratio has asbolute value less than $1$. This shows that the derivatives of order $m+1$ of $\phi \circ f$ are decreasing to $0$ as $j \to \infty$ and $\phi \circ f$ is actually $C^{m+1}$ and convex on $D$. The result follows by induction up to $m = k$ and by the fact that a composition of increasing convex functions is increasing and convex. Let $p \colon {\mathbb{R}}_+ \mapsto {\mathbb{R}}_+$ be concave increasing and $C^1$ with $p' \geq c$ for some $c > 0$. Assume that there exists $ A > 0$ such that for all $x $ in $ {\mathbb{R}}_+$ $$\begin{aligned} p(x) - x p'(x) \leq A. \end{aligned}$$ Then setting $a= A/c$, we have for all $x \geq a$, $$\begin{aligned} p(x-a) - x p'(x-a) \leq 0 \end{aligned}$$ \[lem:techConcaveAsymptotic\] For all $x \geq a$, we have $$\begin{aligned} f(x-a) - (x-a)f'(x-a) \leq A, \end{aligned}$$ hence $$\begin{aligned} f(x-a) - xf'(x-a) \leq A - af'(x-a) \leq A - ac = 0. \end{aligned}$$ [**Acknowledgements.**]{} The authors acknowledge the support of AI Interdisciplinary Institute ANITI funding, through the French “Investing for the Future – PIA3” program under the Grant agreement n°ANR-19-PI3A-0004, Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant numbers FA9550-19-1-7026, FA9550-18-1-0226, and ANR MasDol. J. Bolte acknowledges the support of ANR Chess, grant ANR-17-EURE-0010 and ANR OMS. [apalike]{} Alvarez, F., Bolte, J. and Brahic, O. (2004). 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--- abstract: | The decoherence of a two-state tunneling molecule, such as a chiral molecule or ammonia, due to collisions with a buffer gas is analyzed in terms of a succession of quantum states of the molecule satisfying the conditions for a consistent family of histories. With $\hbar\omega$ the separation in energy of the levels in the isolated molecule and $\gamma$ a decoherence rate proportional to the rate of collisions, we find for $\gamma \gg \omega$ (strong decoherence) a consistent family in which the molecule flips randomly back and forth between the left- and right-handed chiral states in a stationary Markov process. For $\gamma < \omega$ there is a family in which the molecule oscillates continuously between the different chiral states, but with occasional random changes of phase, at a frequency that goes to zero at a phase transition $\gamma = \omega $. This transition is similar to the behavior of the inversion frequency of ammonia with increasing pressure, but will be difficult to observe in chiral molecules such as D$_2$S$_2$. There are additional consistent families both for $\gamma > \omega $ and for $\gamma < \omega $. In addition we relate the speed with which chiral information is transferred to the environment to the rate of decrease of complementary types of information (e.g., parity information) remaining in the molecule itself. author: - 'Patrick J. Coles' - Vlad Gheorghiu - 'Robert B. Griffiths' title: Consistent histories for tunneling molecules subject to collisional decoherence --- Introduction\[sct1\] ==================== The decoherence produced by the interaction of a quantum system with its environment is ubiquitous in nature and plays an important role in current quantum theory in at least two ways. First, it is widely believed that decoherence helps understand how the classical physics of macroscopic objects emerges as an approximation to underlying quantum mechanical laws. Second, decoherence is the great enemy of quantum computation, quantum cryptography, and other schemes seeking to utilize specifically quantum effects for particular processes. For both reasons it is important to study specific microscopic models from which one can hope to obtain general principles for decoherence. The present paper is the study of a simple two-level system which can be thought of as a crude microscopic model of chiral molecules or ammonia in which the lowest quantum energy levels correspond to the nearly degenerate eigenstates of a double-well potential, with decoherence occuring through collisions with particles in the environment. Microscopic studies of decoherence are often framed in terms of a master equation for the density operator of the decohering system. Such descriptions are perfectly valid, but because they represent the average of a large ensemble of nominally identical systems, each with a different specific time development, they provide less information and less physical insight than the actual history of a single system. For example, in the phenomenon of intermittent fluorescence a single ion in a trap shows intermittent light and dark periods when it does or does not scatter resonance radiation [@PlKn98]. This behavior is not directly reflected in the density operator, even though from the latter one can deduce parameters which govern the statistical behavior of the individual ion. Another way to understand the limitations of the density operator description is to consider its classical analog for a Brownian particle confined to a small but macroscopic volume of a fluid by rigid walls. The probability distribution $\rho({\bm{r}},t)$ of the particle position ${\bm{r}}$ will eventually tend to a constant over the region accessible to the particle, whereas the particle itself will continue to exhibit a sort of random walk. More details of what is going on in this steady-state situation is provided by the joint probability distribution of the sequence of successive positions ${\bm{r}}_1,\,{\bm{r}}_2,\ldots$ of the particle at a sequence of times $t_1,\,t_2,\ldots$, that is, its *history*. Averaging over a large number of histories will yield $\rho({\bm{r}},t)$, but in the process the information needed for a more detailed temporal description of the particle is lost. In the quantum case unravelings of the master equation provide a more detailed description of the microscopic time development, but these are often viewed as mathematical artifacts having no necessary connection with what is really going on in the quantum system. There are many possible unravelings; which, if any are correct? Standing in the way of answering this question is the infamous *measurement problem* of quantum foundations: textbook quantum mechanics introduces probabilities by means of measurements, but cannot say what it actually is that is being measured. However, the *consistent histories* or *decoherent histories*—hereafter simply referred to as *histories*—formulation of quantum mechanics, has no measurement problem, and provides the tools needed to identify trajectories or sequences of events that actually correspond to physical processes. Or, putting it another way, it allows one to identify certain classes of microscopic stochastic processes which can be consistently described in a fully quantum mechanical terms. The histories approach has previously been applied to quantum optical systems by Brun [@Brn02; @PhysRevLett.78.1833], though we believe the material presented here is the first application to the case of tunneling molecules, including chiral molecules. Early in the development of quantum mechanics the question arose as to why chiral molecules are observed in left- and right-handed versions even though the quantum ground state should be a symmetrical combination of the two forms. Hund [@Hund1927] provided the first step in addressing this paradox when he pointed out that the two enantiomers correspond to the two wells of a symmetrical potential with two minima, and that the time required to tunnel from one well to another for a typical chiral molecule is extremely long. A second step was provided by Simonius [@PhysRevLett.40.980] who observed that interaction with the environment of a suitable sort (i.e., decoherence, though when he wrote that term was not yet current) can stabilize the chiral states for periods substantially longer than the tunneling time. At present it seems widely accepted that such decoherence is an important aspect of the stability of chiral molecules, though there have been dissenting voices, e.g. [@PhysLettA.147.411]. The time dependence of the two-state model introduced in Sec. \[sct2\], when analyzed in terms of consistent histories using the principles discussed in Sec. \[sct3\], and applied to specific consistent families in Sec. \[sct4\], yields some insight into this stability problem. In particular, we find that if the rate of decoherence due to collisions ${\gamma }$ (a parameter in our model) is much larger than the tunneling rate ${\omega }$ in an isolated molecule, there is a consistent family in which the molecule spends a long but random period of time in each of the chiral states before flipping to the one of opposite chirality, in a two-state Markov process. As ${\gamma }$ decreases the flips become more rapid and the “dressed” quantum states between which the flips occur become less and less chiral, with this type of family finally disappearing at a phase transition ${\gamma }= {\omega }$. For ${\gamma }< {\omega }$ there is a different consistent family with a rapid but continuous oscillation of the molecule back and forth between its chiral states, interrupted at random times by a change in phase. There are a variety of other consistent families, and these are discussed, along with their physical interpretation, in Sec. \[sct4\]. Most chiral molecules in most circumstances will be in the strong decoherence regime. We give some approximate numerical values in Sec. \[sbct6.1\] for D$_2$S$_2$ in a buffer gas of helium, as it has been the subject of some careful calculations in [@PhysRevLett.103.023202]. On the other hand the ammonia molecule, which though not itself a chiral molecule can behave like one in certain rotational states, has an inversion (tunneling) transition with a frequency that goes to zero with increasing pressure. This is probably an example of, or at least very similar to, the ${\gamma }= {\omega }$ phase transition, for reasons discussed in Sec. \[sbct6.2\]. An intuitive way of characterizing the stochastic time development in the histories formalism is to quantify the information dynamics, e.g., the loss over time of information about the system’s initial state due to random hopping. Such information loss is ultimately connected to decoherence [@Jsao03], which can alternatively be viewed as flow of information about the system to the environment [@Zrk03], and in Sec. \[sct5\] we illustrate the quantitative connection between these two views of decoherence for our model. In our model decoherence corresponds to a flow of chiral information—i.e., is the molecule left or right handed?—to the environment. In Sec. \[sct5\] we analyze this using quantitative measures defined in Sec. \[sbct3.3\], and compare the flow of chiral information to the environment with the decrease of complementary types of information (e.g., parity information) about the earlier state of the molecule that remain in the molecule itself at later times. Our conclusions are summarized in Sec. \[sct7\], which also indicates some ways in which the results reported here could be usefully extended. A few mathematical derivations and details are placed in appendices. Microscopic model and master equation\[sct2\] ============================================= Double-well potential and collisions\[sbct2.1\] ----------------------------------------------- We consider a quantum system, the molecule, with a double-well potential in which the two lowest energy eigenstates, ${|0\rangle}$ (even parity) and ${|1\rangle}$ (odd parity), are sufficiently well separated in energy from all the higher levels that the latter can be ignored. The Hamiltonian is of the form $$H = (1/2)\hbar{\omega }Z, \label{eqn1}$$ where $Z = {{|0\rangle}{\langle 0|}} - {{|1\rangle}{\langle 1|}}$ is the Pauli operator ${\sigma }_3={\sigma }_z$, so the energy splitting between the levels is $\hbar{\omega }$. The linear combinations $${|R\rangle}=\bigl({|0\rangle}+{|1\rangle}\bigr)/ \sqrt{2},\quad {|L\rangle}=\bigl({|0\rangle}-{|1\rangle}\bigr)/ \sqrt{2},\quad \label{eqn2}$$ represent the right- and left-handed chiral forms of the molecule, or in ammonia the nitrogen on one or the other side of the plane formed by the hydrogens. In real molecules there are, of course, additional degrees of freedom—rotations, vibrations, etc. We are assuming that for our purposes these can be ignored, i.e., the Hilbert space can be approximated as a tensor product of these other degrees of freedom with the two levels representing the tunneling, with negligible coupling between them. Hence the isolated molecule can be thought of as oscillating or tunneling between the ${|R\rangle}$ and ${|L\rangle}$ states at an angular frequency ${\omega }$. In the Bloch sphere picture the kets ${|R\rangle}$ and ${|L\rangle}$ correspond to the points on the positive and negative $x$ axis, and the sphere rotates about the $z$ axis as time increases. Next we assume the molecule collides randomly with other particles (atoms or molecules), and the duration of each collision is short compared to the times we are interested in. Successive collisions need not be independent of each other, but we assume that correlations die away rapidly after some *correlation time* $\tau_c$, which could be shorter than the average time between collisions in a dilute gas, but might be significantly longer in a dense gas or liquid. We will consider properties of the molecule at a succession of times $t_0$, $t_1$, $t_2$…, where the $m$’th time interval, ${\Delta }t_m=t_{m+1}-t_m$ is always greater than $\tau_c$, and ideally should be significantly greater than $\tau_c$. That is, we are using a description which is coarse-grained in time; the importance of this will appear later. During the $m$’th time interval there may be zero or one or more collisions of other particles with the molecule, and different collisions can have different effects. We shall assume that the probability distribution for these collisions in a particular interval, both for the times at which they occur and the effects which they have on the molecule, are *statistically independent* of what happens in other intervals. Obviously this cannot be exactly correct, but on physical grounds it seems reasonable provided ${\Delta }t_m$ is not too short, which is why we assume that it is larger than $\tau_c$. In addition we assume, as is appropriate for a steady state situation, that this probability distribution depends only on the length ${\Delta }t_m$ of the interval, and not otherwise on $t_m$. The next assumption is that at the beginning of a time interval of length ${\Delta }t$ the molecule and the environment can be adequately described, for the purposes of what happens next, as a tensor product of a molecule state and some density operator for the environment.[^1] The latter is the quantum analog of a probability distribution for incoming particles which might collide with the molecule during this time interval. This density operator can be “purified” by regarding it as arising from an entangled pure state between the environment and an auxiliary reference system, which we also take to be part of the environment. The overall time development of the molecule and the environment during the interval ${\Delta }t$ is then given by a unitary time development operator, corresponding to an appropriate Hamiltonian, acting on the system and environment, resulting in an isometry mapping the Hilbert space ${\mathcal{H}}_M$ of the molecule onto ${\mathcal{H}}_M{\otimes }{\mathcal{H}}_E$, where ${\mathcal{H}}_E$ is the Hilbert space of the environment. If one traces out the environment the result is a quantum operation or channel from ${\mathcal{H}}_M$ to itself: the channel input is the molecule at the beginning of the time interval ${\Delta }t$ and its output is the molecule at the end of this interval. It is represented by a completely-positive trace-preserving (CPTP) superoperator ${\mathcal{T}}({\Delta }t)$ from the space $\hat{\mathcal{H}}_M$ of linear operators on ${\mathcal{H}}_M$ to itself. Tracing out the molecule instead of environment at the end of the time interval results in a corresponding CPTP map ${\mathcal{T}}^c({\Delta }t)$ from $\hat{\mathcal{H}}_M$ to $\hat{\mathcal{H}}_E$, representing the *complementary* channel. See, for example, [@PhysRevA.83.062338] for further details on how the direct and complementary channel are related to the isometry. The assumption of statistical independence of successive time intervals, and that the environment is in a steady state, allows us to treat the interval from $t_1$ to $t_2$ in the same way as the interval from $t_0$ to $t_1$. Thus a succession of time intervals can be thought of, so far as the molecule is concerned, as a set of channels in series, with dynamics corresponding to an appropriate composition of the superoperators ${\mathcal{T}}({\Delta }t_m)$ for the corresponding intervals. The use of a superoperator ${\mathcal{T}}({\Delta }t)$ that depends only on the length ${\Delta }t$ of the time interval may give rise to the misleading impression that we are assuming exactly the same number and type of collisions for any interval of length ${\Delta }t$. But this is not so. To understand why, consider a classical stochastic process for which the independence of successive time intervals justifies using a Markov model, and for simplicity assume that all the time intervals are of equal length. Let $j$ be a discrete index labeling molecule states at a single time, $j_m$ its value at the beginning of the $m$’th time interval, and $M^{(n)}(j',j)$ the Markov matrix for a transition $j{\rightarrow }j'$ if precisely $n$ collisions occur in one time interval. The probability distribution for a collection of histories that all begin in the state $j_1$ conditioned on a specified set $n_1,n_2,\ldots n_f$ of numbers of collisions in the different intervals is $$\begin{aligned} & \Pr(j_1,j_2,\ldots j_{f+1} {\,|\,}n_1,n_2,\ldots n_f)= \notag\\ &M^{(n_f)}(j_{f+1},j_f)\cdots M^{(n_2)}(j_3,j_2)M^{(n_1)}(j_2,j_1) \label{eqn3}\end{aligned}$$ On the other hand, if the sequence of collision numbers is *not* known, the probability *not* conditioned on this information is given by $$\begin{aligned} & \Pr(j_1,j_2,\ldots j_{f+1}) = \notag\\ &M^{(av)}(j_{f+1},j_f)\cdots M^{(av)}(j_3,j_2)M^{(av)}(j_2,j_1) \label{eqn4}\end{aligned}$$ where $M^{(av)}_{j'j}$ is the averaged Markov matrix, $$M^{(av)}(j',j) = \sum_n \Pr(n) M^{(n)}(j',j), \label{eqn5}$$ and $\Pr(n)$ the probability of $n$ collisions during a single time interval. In the quantum case the single superoperator ${\mathcal{T}}({\Delta }t)$ is the analog of an averaged Markov matrix, and it will allow us to correctly compute the probability of a sequence of histories as long as we do not try and condition it on more detailed information about the initial state of the environment at the beginning of the time interval. Explicit form for the superoperator ${\mathcal{T}}$ {#sbct2.2} --------------------------------------------------- As noted above, ${\mathcal{T}}({\Delta }t)$ only makes physical sense for ${\Delta }t$ greater than some correlation time $\tau_c$. Keeping this in mind, it is nonetheless very convenient to think of the argument of ${\mathcal{T}}({\Delta }t)$ as a continuous variable, which we shall hereafter denote by $t$, thus ${\mathcal{T}}({t})$. This superoperator can be written in various ways, e.g., using Kraus operators or as a matrix using some basis of the operator space of a qubit. A convenient basis is provided by the Pauli operators: ${\sigma }_0={I}$, ${\sigma }_1 = X$, ${\sigma }_2 = Y$, ${\sigma }_3 = Z$, in terms of which we write $${\mathcal{T}}({t}){\sigma }_j = \sum_k {\mathbf{T}}_{kj}({t}) {\sigma }_k, \label{eqn6}$$ using a matrix ${\mathbf{T}}$ of real coefficients whose first row (because ${\mathcal{T}}$ is trace preserving) is $(1,0,0,0)$. The remaining rows constitute a collection of 12 (real) parameters which are only constrained by inequalities that ensure that ${\mathcal{T}}$ is completely positive. Applying ${\mathcal{T}}({t})$ to a density operator $\rho = \sum_j {\boldsymbol{\rho}}_j {\sigma }_j$ at $t=0$, with the coefficients $\{{\boldsymbol{\rho}}_j\}$ regarded as a column vector ${\boldsymbol{\rho}}$, results in a density operator $\bar\rho = \sum_j \bar{{\boldsymbol{\rho}}}_j {\sigma }_j$ at time $t$, where $ \bar {{\boldsymbol{\rho}}} = {\mathbf{T}}\cdot {\boldsymbol{\rho}} $. Rather than explore the entire parameter space, we have assumed that ${\mathbf{T}}$ has the particularly simple form $${\mathbf{T}}({t}) = e^{{t}{\mathbf{S}}},\quad {\mathbf{S}}= \begin{pmatrix} 0 & 0 & 0 & 0\\ 0 & 0 & -{\omega }& 0\\ 0 & {\omega }& -2\gamma & 0\\ 0 & 0 & 0 & -2\gamma \end{pmatrix}, \label{eqn7}$$ where ${\omega }$ is the precession frequency for ${|R\rangle}$ to ${|L\rangle}$ and back again for the isolated molecule—the energy difference between ${|0\rangle}$ and ${|1\rangle}$ is $\hbar{\omega }$—and ${\gamma }\geq 0$ is the *rate of decoherence*. Justification based on scattering theory for this form of ${\mathbf{T}}$ has been discussed in [@HornbergerEPL2007; @PhysRevLett.103.023202]. To see the motivation behind , first consider the case of the isolated molecule with no decoherence, ${\gamma }=0$. Then $$\label{eqn8} {\mathbf{T}}({t})= {\mathbf{R}}({t})=\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & \cos {\omega }{t}& -\sin {\omega }{t}& 0\\ 0 & \sin {\omega }{t}& \cos {\omega }{t}& 0\\ 0 & 0 & 0 & 1 \end{pmatrix}$$ corresponds to precession about the $z$ axis in a Bloch sphere picture. Next, suppose that ${\omega }=0$, so that only decoherence is present. Then $${\mathbf{T}}({t}) \approx {\mathbf{D}}({t}) =\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 &0 & 0\\ 0 & 0 & 1-2{\gamma }{t}& 0\\ 0 & 0 & 0 &1-2{\gamma }{t}\end{pmatrix} \label{eqn9}$$ when ${t}$ is small, and $$\label{eqn10} {\mathbf{T}}({t})={\mathbf{D}}({t})\cdot {\mathbf{R}}({t})+{\mathbf{O}}({t}^2) ={\mathbf{R}}({t})\cdot {\mathbf{D}}({t})+{\mathbf{O}}({t}^2).$$ with ${\mathbf{O}}({t}^2)$ a second order correction. Thus combines the competing effects of decoherence and the molecule’s internal dynamics. The motivation behind is a simple physical picture in which if the environment is initially in the state ${|E\rangle}$ its interaction with the molecule during a collision corresponds to the unitary transformation $$\begin{aligned} {|L\rangle}{\otimes }{|E\rangle} &\rightarrow {|L\rangle}{\otimes }\bigl( \sqrt{1-2{p}}\, {|E\rangle} + \sqrt{2{p}}\, {|E'\rangle}\bigr), \nonumber\\ {|R\rangle}{\otimes }{|E\rangle} &\rightarrow {|R\rangle} {\otimes }\bigl(\sqrt{1-2{p}}\, {|E\rangle} + \sqrt{2{p}}\, {|E''\rangle}\bigr), \label{eqn11} \end{aligned}$$ where the environment states ${|E\rangle}$, ${|E'\rangle}$ and ${|E''\rangle}$ are orthonormal. The intuitive idea is that the distinction between ${|L\rangle}$ and ${|R\rangle}$ is carried off to the distinct environmental states ${|E'\rangle}$ and ${|E''\rangle}$ with an amplitude that increases with ${p}$, a quantity lying between 0 and 1/2 which is a measure of the effectiveness of the decoherence. The decohering effect is unchanged if ${|E\rangle}$ on the right side of is replaced with any other state ${|\bar E\rangle}$ as long as it is orthogonal to ${|E'\rangle}$ and ${|E''\rangle}$, i.e., if the alteration does not depend upon the difference between ${|L\rangle}$ and ${|R\rangle}$. One can represent the channel corresponding to by three Kraus operators associated with ${|E\rangle}$, ${|E'\rangle}$ and ${|E''\rangle}$, but an equally good form uses just two Kraus operators $\sqrt{1-{p}}\;{I}$ and $\sqrt{{p}}\;X$ corresponding to a “bit flip” channel in [@NielsenChuang:QuantumComputation] p. 376. When ${p}=0$ there is no decoherence (a perfect channel) whereas for ${p}=1/2$ the collision “collapses” the molecule into either ${|R\rangle}$ or ${|L\rangle}$. If one sets $p={\gamma }{t}$ the superoperator corresponding to the process is given by , and this makes sense for ${t}$ of the order of the time between collisions. However, as noted above in Sec. \[sbct2.1\], the superoperator ${\mathcal{T}}({t})$ can appropriately represent a situation in which the number of collisions in the interval ${t}$ is a random quantity. The matrix ${\mathbf{T}}(t)$ and the density operator ${\boldsymbol{\rho}}(t)$ thought of as a column vector satisfy the simple linear differential equations: $$\frac{d{\mathbf{T}}}{dt} = {\mathbf{S}}\cdot{\mathbf{T}},\quad \frac{d{\boldsymbol{\rho}}}{dt} = {\mathbf{S}}\cdot{\boldsymbol{\rho}}. \label{eqn12}$$ The second is equivalent to a master equation in Lindblad form $$\label{eqn13} \frac{d\rho}{dt} = -{\mathrm{i}}[H ,\rho]/\hbar +\gamma (X\rho X-\rho),$$ with $H /\hbar={\omega }Z/2$ as in . Solutions to and can of course be expressed as linear combinations of exponentials of the form $e^{{\lambda }_j t}$, where $${\lambda }_1 = 0,\; {\lambda }_2 = -{\gamma }+ \xi,\; {\lambda }_3 = -{\gamma }-\xi,\; {\lambda }_4 = -2{\gamma }\label{eqn14}$$ $$\xi= \sqrt{{\gamma }^2-{{\omega }}^2} \label{eqn15}$$ are the eigenvalues of the matrix ${\mathbf{S}}$. Note that ${\lambda }_2$ and ${\lambda }_3$ occur in solutions of the form $e^{{\lambda }t}$ to the damped oscillator equation $d^2x/dt^2 + 2{\gamma }dx/dt + {{\omega }}^2x=0$. Thus for ${\gamma }< {{\omega }}$ they are complex conjugates of each other lying on a circle of radius ${{\omega }}$ in the complex plane, corresponding to oscillatory solutions, while for ${\gamma }> {{\omega }}$ both are real and negative, corresponding to damped motion without oscillation. Critical damping ${\gamma }={{\omega }}$ corresponds to a phase transition in the sense of a changeover between two qualitatively different types of behavior. The explicit form of ${\mathbf{T}}(t)$ is given in Appendix \[apdxA\]. General aspects of consistent histories and information\[sct3\] =============================================================== Introduction to histories {#sbct3.1} ------------------------- In the (consistent or decoherent) histories formalism a *history* is a sequence of quantum properties, identified by projectors onto appropriate subspaces of the quantum Hilbert space, at a succession of times $t_1 < t_2 <\cdots <t_f$; see Ch. 8 of . In the situation at hand we use a sample space of mutually exclusive histories formed by assuming that at time $t_m$ the properties of interest to us correspond to a collection $\{P_m^{{\alpha }_m}\}$ of projectors which form a decomposition of the identity: $$\sum_{\alpha_m}P_m^{\alpha_m}={I}, \quad ({P_m^{\alpha_m}})^\dagger={P_m^{\alpha_m}}=({P_m^{\alpha_m}})^2. \label{eqn16}$$ Here the subscript $m$ labels the time, while the superscript ${\alpha }_m$ is not an exponent but instead a label to differentiate the projectors at this time. Choosing at each time a property from the corresponding decomposition of the identity yields a history represented by a projector $$Y^{{\bm{{\alpha }}}} = P_1^{{\alpha }_1}{\odot }P_2^{{\alpha }_2} {\odot }\cdots P_f^{{\alpha }_f},\quad {\bm{{\alpha }}} = ({\alpha }_1,{\alpha }_2,\ldots {\alpha }_f) \label{eqn17}$$ on the history Hilbert space $\breve {\mathcal{H}}={\mathcal{H}}^{{\odot }f} = {\mathcal{H}}{\odot }{\mathcal{H}}{\odot }\cdots {\mathcal{H}}$ formed by the tensor product of the Hilbert space with itself $f$ times. Here ${\odot }$ is a tensor product symbol with the same significance as ${\otimes }$, but employed to distinguish different times. The physical significance of $Y^{{\bm{{\alpha }}}}$ can be seen by reading as “property $P_1^{{\alpha }_1}$ at time $t_1$ followed by property $ P_2^{{\alpha }_2}$ at time $t_2$ followed by….” For a closed system in which the unitary (Schrödinger) time development from $t_m$ to $t_{m+1}$ is described by the operator $U_{m+1,m}$, probabilities (probabilistic weights) can be assigned using the *decoherence functional* [@RBGriffiths:ConsistentQuantumTheory] $$\begin{aligned} {\mathbb{D}}(Y^{{\bm{\alpha}}},Y^{{\bm{\beta}}}) = {{\rm Tr}}[& P^{{\alpha }_f}_{f} U_{f,{f-1}}\dotsm P^{{\alpha }_2}_{2}U_{2,1}P^{{\alpha }_1}_{1} \Psi_0 \notag\\ & P^{{\beta }_1}_{1}U_{1,2} P^{{\beta }_2}_{2}\dotsm U_{{f-1},f}P^{{\beta }_f}_{f} ], \label{eqn18}\end{aligned}$$ where $\Psi_0$ is some initial state, provided the *consistency conditions* $${\mathbb{D}}(Y^{{\bm{\alpha}}},Y^{{\bm{\beta}}}) = 0 \text{ whenever } {\bm{{\alpha }}}\neq {\bm{{\beta }}} \label{eqn19}$$ are satisfied. Here ${\bm{{\alpha }}}\neq{\bm{{\beta }}}$ means that for at least one time $t_m$ it is the case that ${\alpha }_m\neq {\beta }_m$. When holds one assigns the positive *weight* $W({\bm{{\alpha }}}) = {\mathbb{D}}(Y^{{\bm{{\alpha }}}},Y^{{\bm{{\alpha }}}})$ to the history $Y^{{\bm{{\alpha }}}}$. The probability of each history is its weight divided by the total weight of all the histories; if $\Psi_0$ is a normalized density operator this total weight is 1 and the probability of history ${\bm{{\alpha }}}$ is $W({\bm{{\alpha }}})$. The Hilbert space for the present discussion is ${\mathcal{H}}= {\mathcal{H}}_M{\otimes }{\mathcal{H}}_E$, where ${\mathcal{H}}_M$ is the Hilbert space of the molecule and ${\mathcal{H}}_E$ that of the environment. However, the histories of interest to us refer to properties of the molecule, not the environment, and we employ the usual convention that $P_m^{{\alpha }_m}$ representing one of these properties can denote both a projector on ${\mathcal{H}}_M$ or its counterpart $P_m^{{\alpha }_m}{\otimes }I_E$ on ${\mathcal{H}}$. For the initial state we let $\Psi_0 = I_M {\otimes }{{|\Phi_E\rangle}{\langle \Phi_E|}}$, where $${|\Phi_E\rangle} = {|E_1\rangle}{\otimes }{|E_2\rangle}{\otimes }\cdots{|E_{f-1}\rangle} \label{eqn20}$$ is a “giant” tensor product state on the environment chosen in such a way that during the time interval between $t_m$ and $t_{m+1}$ the molecule will interact only with the piece ${|E_m\rangle}$ in this tensor product in a manner determined by $U_{m+1,m}$; after that this part of the environment can be ignored so far as the molecule is concerned. In particular, if we take a partial trace over the environment of the middle portion on the right side of at time $t_2$, the interaction of the molecule with ${|E_1\rangle}$ is chosen so that $${{\rm Tr}}_E\left[ U_{2,1}P^{{\alpha }_1}_{1} \Psi_0 P^{{\beta }_1}_{1}U_{1,2} \right] = {\mathcal{T}}_{2,1}(P^{{\alpha }_1}_{1} P^{{\beta }_1}_{1}), \label{eqn21}$$ where ${\mathcal{T}}_{2,1} = {\mathcal{T}}(t_2-t_1)$ is the superoperator that maps the state of the molecule at the beginning of this time interval to its state at the end. In the same way, if the partial trace over the environment is carried out at time $t_3$ the result will be $${\mathcal{T}}_{3,2}(P_2^{{\alpha }_2} {\mathcal{T}}_{2,1}(P^{{\alpha }_1}_{1} P^{{\beta }_1}_{1}) P_2^{{\beta }_2}), \label{eqn22}$$ with $ {\mathcal{T}}_{3,2} = {\mathcal{T}}(t_3-t_2)$, and similarly for later times. Consequently, for our model the decoherence functional is given by $$\begin{aligned} {\mathbb{D}}(Y^{{\bm{\alpha}}},Y^{{\bm{\beta}}}) = {{\rm Tr}}_M [ &P^{{\alpha }_f}_{f} {\mathcal{T}}_{f,f-1}(\dotsm P^{{\alpha }_2}_{2}{\mathcal{T}}_{2,1}(P^{{\alpha }_1}_{1}\notag \\ & P^{{\beta }_1}_{1})P^{{\beta }_2}_{2}\dotsm )P^{{\beta }_f}_{f} ], \label{eqn23}\end{aligned}$$ an expression which no longer makes any (direct) reference to the environment. See Sec. III of [@PhysRevD.48.2728] for a more detailed argument. If all the projectors in the decomposition $\{P_m^{{\alpha }_m}\}$ are rank 1, which is to say they project onto pure states of the molecule, and the consistency conditions are satisfied, then the probabilities (corresponding to the diagonal elements of the decoherence functional ) are those of a memoryless hopping process - a Markov process. If the time steps ${t}_m$ are identical and the same decomposition is used at every time, this process is stationary (homogeneous, i.e. same Markov matrix at each timestep), but in general it is nonstationary (inhomogeneous). Both cases are of interest for our model, as discussed below in Sec. \[sct4\]. Forwards and backwards conditions {#sbct3.2} --------------------------------- Finding collections of histories such that the consistency condition is satisfied is made somewhat easier by the following observation. Suppose it is the case that for every $m$ between 1 and $f-1$, if $Q$ is a linear combination of the projectors in the set $\{P_m^{{\alpha }_m}\}$, then ${\mathcal{T}}_{m+1,m}(Q)$ is a linear combination of the projectors in the set $\{P_{m+1}^{{\alpha }_{m+1}}\}$. When this *forward condition* is satisfied, the family of histories will be consistent, as can be seen in the following way. The functional ${\mathbb{D}}(Y^{{\bm{\alpha}}},Y^{{\bm{\beta}}})$ in will vanish if ${\alpha }_1\neq{\beta }_1$, since $P^{{\alpha }_1}_{1} P^{{\beta }_1}_{1}=0$. If the forward condition is satisfied, ${\mathcal{T}}_{2,1}(P^{{\alpha }_1}_1)$ will be a linear combination of projectors in the collection $\{P_2^{{\alpha }_2}\}$, and will therefore commute with any projector in this collection. Consequently, $P_2^{{\alpha }_2}{\mathcal{T}}_{2,1}(P_1^{{\alpha }_1})P_2^{{\beta }_2} = P_2^{{\alpha }_2}P_2^{{\beta }_2}{\mathcal{T}}_{2,1}(P_1^{{\alpha }_1})$ will vanish whenever ${\alpha }_2\neq{\beta }_2$ since $P_2^{{\alpha }_2}P_2^{{\beta }_2}=0$, and if it does not vanish it will be some linear combination of the $\{P_2^{{\alpha }_2}\}$. Proceeding in the same way for larger $m$ one sees that ${\mathbb{D}}(Y^{{\bm{\alpha}}},Y^{{\bm{\beta}}})$ will vanish if, for any $m$, ${\alpha }_m\neq{\beta }_m$. Note that the forward condition is a sufficient but not a necessary condition for consistency. The same is true of the *backward condition*: for every $m$ between $f$ and $2$ it is the case that if $Q$ is a linear combination of the projectors in $\{P_m^{{\alpha }_m}\}$, then ${\mathcal{T}}_{m,m-1}^\dagger(Q)$ is a linear combination of the projectors in $\{P_{m-1}^{{\alpha }_{m-1}}\}$. Here ${\mathcal{T}}_{m,m-1}^\dagger$ denotes the adjoint of the superoperator with respect to the Frobenius inner product: ${\langle {\mathcal{T}}^\dagger(A),B\rangle } = {\langle A,{\mathcal{T}}(B)\rangle }$ where ${\langle A,B\rangle } = {{\rm Tr}}(A{^\dagger }B)$. The proof of consistency when the backward condition is satisfied proceeds in the same way as for the forwards condition, but in reverse. Start with and rewrite the argument inside the trace by first cycling $P_f^{{\beta }_f}$ to become the first term, and then replacing ${\mathcal{T}}_{f,f-1}$ with ${\mathcal{T}}_{f,f-1}^\dagger$ acting on $P_f^{{\beta }_f} P_f^{{\alpha }_f}$, and continue this cycling process to convert all ${\mathcal{T}}$ to ${\mathcal{T}}^\dagger$. \[Note that since ${\mathcal{T}}$ is a (completely) positive superoperator, $({\mathcal{T}}{^\dagger }(A)){^\dagger }= {\mathcal{T}}{^\dagger }(A{^\dagger })$.\] In the case of qubits, the situation of primary interest for the present paper, one can show that consistent families of histories of the type must satisfy either the forward or the backward condition. Measuring information {#sbct3.3} --------------------- We will want to discuss and quantify the information about the initial state of the molecule as time goes on. We can do this within the context of the histories formalism. Alternatively, we can do this in the context of the quantum channel formalism, i.e., quantifying the distinguishability of density operators at the output of a quantum channel, and as we will see there is some connection between the two approaches. Let us first consider information from the histories perspective. Suppose some consistent family of histories uses the projective decompositions $P_1=\{P_1^j\}$ at time $t_1$ and $P_m=\{P_m^k\}$ at time $t_m$, with $t_1< t_m$ \[for simplicity here we replaced the indices ${\alpha }_1$ and ${\alpha }_m$ in and with $j$ and $k$\]. As in [@PhysRevA.76.062320], we will equate the notion of a projective decomposition, like $P_1$, with a type of information about the system, in our case the molecule. A convenient measure of how much of the $P_1$ type of information about the molecule remains at time $t_m$ is the Shannon mutual information $$\label{eqn24} H(P_1 {\,\hbox{:}\,}P_m)= H(P_1)+H(P_m)- H(P_1,P_m),$$ where $H(P_1)$ is the familiar Shannon entropy. In particular if $P_1 $, $P_m$, and $P_{m'}$ are projective decompositions associated with a consistent family at three successive times, and if the probabilities correspond to a Markov process, then (see, e.g., p. 510 of [@NielsenChuang:QuantumComputation]) $H(P_1 {\,\hbox{:}\,}P_{m'})$ cannot be greater than $H(P_1 {\,\hbox{:}\,}P_m)$: the information about the initial situation can only decrease with time. For simplicity, in what follows we will set $\Pr(P_1^j) = 1/d_1$ for all $j$, where $d_1$ is the number of projectors in the decomposition $P_1$. Then $H(P_1{\,\hbox{:}\,}P_1)=H(P_1)=\log d_1$, and hence the information decays from its initial value of $\log d_1$ as time goes on. Now, alternatively, consider the quantum channel perspective, where we will quantify how much of the $P_1$ type of information remains at time $t_m$ by measuring the *distinguishability* of the conditional density operators at the output of the relevant quantum channel. (This approach was taken in [@PhysRevA.83.062338].) To measure distinguishability of density operators, in particular if these density operators do not commute, we need a measure that is inherently quantum-mechanical, which is provided by the Holevo function $$\label{eqn25} \chi(\{p_j,\rho_j\}):=S(\sum_j p_j\rho_j)-\sum_j p_jS(\rho_j)$$ defined for an ensemble $\{p_j,\rho_j\}$, where $p_j$ is the probability assigned to the density operator $\rho_j$, and $S(\rho):=-{{\rm Tr}}(\rho\log \rho)$ is the von Neumann entropy. Applying this measure to the ensemble $\{1/d_1, {\mathcal{T}}_{m,1} (P_1^j)\}$, where ${\mathcal{T}}_{m,1} $ is the quantum channel that governs the molecule’s evolution from $t_1$ to $t_m$, gives a quantitative measure of how much $P_1$ information remains at time $t_m$, and we write this as $$\hat\chi(P_1,{\mathcal{T}}(t))= \hat\chi(P_1,{\mathcal{T}}_{m,1}) : = \chi( \{ \frac{1}{d_1} ,\frac{{\mathcal{T}}_{m,1}(P_1^j)}{{{\rm Tr}}(P_1^j)} \}), \label{eqn26}$$ where $t=t_m-t_1$. Equations  and give two alternative ways to measure the loss of information from the system over time. Equation  has the advantage of a clear conceptual interpretation, whereas Equation  has the advantage of being easy to compute since one does not need to go through the histories analysis to compute it. Fortunately, there is a connection between these approaches. It turns out, see the argument in Appendix \[apdxC\], that for a family satisfying the forward consistency condition $$\label{eqn27} H(P_1 {\,\hbox{:}\,}P_m)= \hat\chi(P_1,{\mathcal{T}}_{m,1}).$$ A similar sort of connection holds for families satisfying the backward consistency condition (but involving the adjoint channel ${\mathcal{T}}_{m,1}{^\dagger }$), but for simplicity we will focus on families satisfying the forward condition to illustrate information flows in Sect. \[sct5\]. One can also quantify information flow from the molecule to the environment with the quantum channel approach by using the complementary channel with superoperator ${\mathcal{T}}^c$, introduced in Sec. \[sbct2.1\]. In fact, ${\mathcal{T}}^c$ is completely determined by ${\mathcal{T}}$ up to an isometry on its output (the environment), which does not affect distinguishability measures like $\chi$. Hence, the following information measure is well-defined: $$\hat\chi(P_1,{\mathcal{T}}^c(t))= \hat\chi(P_1,{\mathcal{T}}^c_{m,1}) : = \chi( \{ \frac{1}{d_1} ,\frac{{\mathcal{T}}^c_{m,1}(P_1^j)}{{{\rm Tr}}(P_1^j)} \}), \label{eqn28}$$ where $t=t_m-t_1$. It quantifies the amount of the $P_1$ type of information about the molecule (at time $t_1$) that is present in the environment at time $t_m$. Though one cannot in general equate this $\hat\chi(P_1,{\mathcal{T}}^c_{m,1}) $ with a Shannon mutual information between the molecule and the environment, the former provides, as is well-known (e.g., p. 531 of [@NielsenChuang:QuantumComputation]), an upper bound on the latter. We note that there can be a tradeoff in sending information to the environment and preserving it in the molecule, which is most dramatic for complementary or *mutually-unbiased* bases $P_1$ and $P'_1$: $$\begin{aligned} \label{eqn29} \hat\chi(P_1,{\mathcal{T}}_{m,1})+ \hat\chi(P'_1,{\mathcal{T}}^c_{m,1})\leq \log d_1.\end{aligned}$$ This inequality is from Corollary 6 of [@PhysRevA.83.062338]. Consistent families for our model {#sct4} ================================= Differential equations {#sbct4.1} ---------------------- Our model has only two states, and therefore any (nontrivial) decomposition of the identity involves only projectors of rank 1 onto pure states. Thus a consistent history family corresponds to a two-state Markov process (sometimes called a “telegraph process”); in general this process is nonstationary: the transition rates depend upon the time. While such a process can be discussed using discrete times separated by finite intervals, the results are simpler and the mathematical expressions more transparent if one adopts a continuous time approximation with differential equations in place of difference equations. It should, of course, be kept in mind that the processes here described are not truly continuous, since time intervals shorter than the correlation time $\tau_c$ introduced in Sec. \[sct2\] lack physical significance. The continuous time approach should be satisfactory as long as both ${\omega }\tau_c$ and ${\gamma }\tau_c$ are small compared to 1. Note that this condition can still be true even when ${\gamma }$ is large, as long as $\tau_c$ decreases as $1/{\gamma }$, which seems physically plausible. For families satisfying the forward consistency condition the relevant differential equations can be obtained in the following way. At a particular time the decomposition of the identity will correspond to two projectors, call them $\rho_0$ and $\rho_1$, represented by end points or antipodes of a diameter of the Bloch sphere. Let the direction of this diameter be denoted by the usual polar and azimuthal angles ${\theta }$ and $\phi$: the $z$ axis at ${\theta }=0$ and the $x$ axis at ${\theta }=\pi/2$, $\phi=0$. Which end of the diameter corresponds to these angles does not matter for the following discussion. The locations of these end points after a short time interval is determined by the master equation . One can show that because of the form of ${\mathbf{S}}$ in they are still located on a diameter of the Bloch sphere, but are now a bit closer to its center. The rate of change of the diameter’s direction is represented by the differential equations $$\label{eqn30} \frac{d\phi}{dt}={\omega }-{\gamma }\sin 2\phi,\quad \frac{d{\theta }}{dt}={\gamma }\sin 2{\theta }\cos^2 \phi,$$ whereas the shift towards the center can be used to calculate the instantaneous transition rate $$\kappa = {\gamma }(1- \sin^2 {\theta }\cos^2 \phi), \label{eqn31}$$ which enters the rate equations $$dp_0/dt = {\kappa }(-p_0 + p_1),\quad dp_1/dt = {\kappa }(p_0 - p_1) \label{eqn32}$$ for the probabilities associated with these two states. The backwards consistency condition can be analyzed in a similar way, and leads to the differential equations $$\frac{d\phi}{dt}={\omega }+{\gamma }\sin 2\phi,\quad \frac{d{\theta }}{dt}=-{\gamma }\sin 2{\theta }\cos^2 \phi. \label{eqn33}$$ governing the direction of the diameter, and to exactly the same expression for the transition rate. For a more detailed derivation of these formulas see Appendix \[apdxBB\]. Stationary families {#sbct4.2} ------------------- If the angles ${\theta }$ and $\phi$ which determine the diameter for the projectors forming a consistent family do not change with time the Markov process is stationary or homogeneous, in the sense that the states and the transition probabilities do not change with time; of course the actual state of the molecule is varying randomly as it hops back and forth between the two states. The simplest case is what we call the $z$ family, in which ${\theta }=0$ (or $\pi$), thus $d{\theta }/dt=0$ in or and $d\phi/dt$ is irrelevant. The two projectors $({I}+{\sigma }_z)/2$ and $({I}-{\sigma }_z)/2$ correspond, respectively to the even parity (ground) and odd parity (excited) states of the isolated molecule. Thus we have a two-state stationary Markov process in which the molecule spends a certain amount of time in the even parity state before flipping instantaneously (in our continuous time approximation) to the odd parity state where it remains for a random time interval before flipping back. The time $\tau$ between flips is a random variable with an exponential distribution $e^{-{\gamma }\tau}$, since setting ${\theta }=0$ in gives $${\kappa }_z = {\gamma }\label{eqn34}$$ for the transition rate. On average the molecule spends an equal amount of time in both states, which means that in our model the environment has an effective temperature $T\gg \hbar{\omega }/k_B$. ![(a) The steady-state solutions for $\phi$ correspond to the intersections of $\sin 2\phi $ (solid curve) with $\pm {\omega }/{\gamma }$ (dashed lines), shown here for ${\gamma }> {\omega }> 0$ (“strong decoherence” regime). (b) These steady-state solutions are plotted schematically on the Bloch sphere, as if the $z$-axis is going into the page. \[fgr1\]](fig1.eps) In addition to the $z$ family just discussed there are stationary families in which the projectors correspond to points in the $x$-$y$ or equatorial plane of the Bloch sphere, so ${\theta }=\pi/2$ with $d\phi/dt=0$ in and , and thus $$\sin 2\phi = \pm ({\omega }/{\gamma }). \label{eqn35}$$ For $0<{\omega }/{\gamma }<1$ there are four solutions as shown in Fig. \[fgr1\], which coalesce into two for ${\omega }/{\gamma }=1$. For ${\omega }/{\gamma }> 1$ these families disappear, leaving the $z$ family as the only stationary family. In the limit of strong decoherence, small ${\omega }/{\gamma }$, two of the families approach the $x$ axis and two the $y$ axis of the Bloch sphere, so we shall refer to them as the dressed $x$- and dressed $y$-families. The associated transition rates ${\kappa }_x$ and ${\kappa }_y$ are given by $-1/2$ times the corresponding eigenvalues of ${\mathbf{S}}$, see : $${\kappa }_x = -{\lambda }_2/2 = ({\gamma }-\xi)/2,\quad {\kappa }_y = -{\lambda }_3/2 = ({\gamma }+\xi)/2. \label{eqn36}$$ As ${\gamma }/{\omega }$ becomes very large the dressed $x$-families approach the $x$ or chirality basis ${|R\rangle}$ and ${|L\rangle}$ of , and the transition rate ${\kappa }_x\approx{\omega }^2/4{\gamma }$ becomes very slow. Thus these families represent long-lived (almost) chiral states when decoherence is rapid compared with the the tunneling rate. (But see the further discussion in Sec. \[sbct4.5\].) Nonstationary families {#sbct4.3} ---------------------- The equations and can be integrated in closed form to obtain the bases corresponding to nonstationary consistent families, Appendix \[apdxB\]. However, the solutions are fairly complicated expressions. The time evolution for some cases in which ${\omega }=1$ and ${\gamma }$ is either less than or larger than $1$ is shown in Fig. \[fgr2\]. For ${\gamma }< {\omega }$ the diameter rotates continuously about the $z$ axis (the discontinuities in $\phi$ are of course artifacts of the plot) with an angular frequency of $$\eta = \sqrt{{\omega }^2 - {\gamma }^2}, \label{eqn37}$$ while the polar angle ${\theta }$ tends either to $\pi/2$ for the forward or to $0$ (equivalently, $\pi$) for the backward consistency condition. In the limit in which ${\gamma }$ tends to 0, no decoherence, one has a simple rotation of the diameter of the consistent family about the $z$ axis at a rate ${\omega }$ with ${\theta }$ fixed. This same tendency is seen in the dependence of ${\theta }$ on time when ${\gamma }> {\omega }$, whereas $\phi$ more or less rapidly approaches one of the values corresponding to a stationary family. Note that along with the continuous change of basis there is a random flipping from one of the basis states to the other at a rate given by , so one is dealing with a nonstationary Markov process. Thus in the Bloch sphere picture, for ${\gamma }< {\omega }$, the families “corkscrew” about the $z$ axis (going away from or towards this axis for the forward or backward families, respectively), with $\phi$ periodically coming back to the same value at time intervals that are integer multiples of $\pi / \eta$. If ${\theta }= \pi /2$ it remains constant, so the same basis reoccurs after an interval of $\pi / \eta$. If only these discrete times are considered, the result is what one might call a *stroboscopic* family which can be thought of as a discrete time stationary Markov process.  Phase transition {#sbct4.4} ---------------- As the parameters ${\gamma }$ and ${\omega }$ vary there is a phase transition, a qualitative change of behavior, when they are equal. This manifests itself in a variety of related ways. For ${\gamma }< {\omega }$ the eigenvalues of ${\mathbf{S}}$ include a complex-conjugate pair ${\lambda }_2$ and ${\lambda }_3$, , which coalesce into a single degenerate eigenvalue at the transition, and thereafter, for ${\gamma }> {\omega }$, become a pair of distinct real eigenvalues. This is, of course, precisely the behavior one finds in a classical one-dimensional oscillator when the damping passes through the critical value. For ${\gamma }> {\omega }$ these eigenvalues are the decay rates for the dressed-$x$ and dressed-$y$ continuous stationary Markov processes discussed in Sec. \[sbct4.2\]. On the other hand, as ${\gamma }$ decreases towards ${\omega }$ from above, the four stationary families shown in Fig. \[fgr1\](b) coalesce into two, corresponding to diameters of the Bloch sphere midway between the $x$ and $y$ axes, and for ${\gamma }< {\omega }$ they no longer exist: the only remaining continuous stationary family is the $z$ family. As noted above in Sec. \[sbct4.3\], for ${\gamma }< {\omega }$ there is a new class of “stroboscopic” families defined using a periodic time interval. As ${\gamma }$ approaches ${\omega }$ from below this period becomes infinitely long. The behavior of nonstationary continuous families is also different for ${\gamma }< {\omega }$ and ${\gamma }> {\omega }$. For the former $\phi$ increases indefinitely and monotonically with time, although this motion, which is simply linear when ${\gamma }=0$, becomes more and more “jerky” as ${\gamma }$ increases towards ${\omega }$. See the example for ${\gamma }/{\omega }= 1/2$ in Fig. \[fgr2\]. For ${\gamma }> {\omega }$, $\phi$ approaches a fixed value with increasing time, and no longer “winds.” (Again, the damped harmonic oscillator provides a helpful analogy.) One might say that the nonstationary continuous families transition from a damped oscillatory character to a purely damped character as ${\gamma }/{\omega }$ increases, passing through the critical value of 1. In terms of its mathematical structure as represented in the master equation this is a dynamical quantum phase transition of the sort discussed in quantum optics for two level systems in Ch. 11 of [@WlMl94] and in [@GrLs10], and in a more general context in [@Pstw07; @Rttr09; @Rttr10]. It appears that the vanishing of the inversion transition in ammonia is of this type; see the discussion in Sec. \[sbct6.2\] below. We believe that ours is the first attempt to explore dynamical properties near such a transition using the histories approach. Physical interpretation {#sbct4.5} ----------------------- Each consistent family contains a collection of histories, and each history a particular succession of micrscopic properties (subspaces of ${\mathcal{H}}_M$). One and only one history from this collection will describe the behavior of a particular molecule during a particular interval of time. There is no need to make any reference to measurements, though it is in principle possible (i.e., does not violate the laws of quantum mechanics) to use a succession of suitably idealized measurements to determine which of these histories is actually realized. But because one is dealing with a system exhibiting “quantum” behavior, i.e., in a regime in which a classical description is not adequate, it is important to keep in mind certain respects in which quantum descriptions differ from their classical counterparts. In particular, two consistent families of histories may be mutually incompatible with each other in such a way that they cannot be combined into a single description that makes sense. A well-known example is a spin-half particle where incompatibility arises from the fact that the operators for angular momentum in different directions do not commute with each other, and hence have no common eigenvectors. It makes good (quantum) sense to say, for example, that $S_x=+1/2$ (in units of $\hbar$), or that $S_z=+1/2$, but there is no quantum property, no subspace in the Hilbert space, that corresponds to $S_x=+1/2$ *and* $S_z=+1/2$. So one cannot ascribe simultaneous existence to $S_z$ and $S_x$. In the consistent histories approach this inability to combine incompatible descriptions is codified as the single framework rule, and the consistency conditions discussed above in Sec. \[sct3\] serve to extend this rule from a single time to a sequence of times. In addition, just as two incompatible consistent families or descriptions cannot be combined into a single description, they also cannot be compared: it makes no sense to ask which of two incompatible families is the “correct” one, or to look for some law of nature that single out one against another. Each consistent family provides its own quantum description in a way roughly analogous to looking at a mountain from different locations. For a detailed discussion of these points we refer the reader to [@Grff12]. With reference to a tunneling molecule, consider the situation in which ${\gamma }$ is much larger than ${\omega }$, strong decoherence. There is a stationary family, Sec. \[sbct4.2\], the “chiral” family, in which the molecule hops back and forth at a comparatively slow rate between the (dressed) left-handed and right-handed chiral states. (There are actually two of these dressed-$x$ families, but when decoherence is strong there is very little difference between them.) This is the family to use if one is interested in understanding why a specific chirality, the left or right-handed form of the molecule, can persist for a very long time in a situation of strong decoherence. It provides a description in terms of a stochastic two-state Markov process in which the rate of hopping from left to right-handed or vice versa is a well-defined function, ${\kappa }_x$ in , of the parameters that enter the model. (Each hop is instantaneous on the time scale used for our description, in which intervals less than the correlation time $\tau_c$ do not enter; see Sec. \[sct2\].) As the rate of decoherence decreases, the hopping time becomes shorter and the amount of “dressing” required to produce a consistent family increases, which means that even though this family continues to provide a correct quantum description, it no longer corresponds to a simple physical picture of a definite left- or right-handed molecule when ${\gamma }$ becomes comparable to ${\omega }$. In addition to this chiral family there is a $z$ or “parity” family in which the molecule hops back and forth at random time intervals between parity eigenstates (energy eigenstates of the isolated molecule), at a rate given by the decoherence rate ${\gamma }$, see . The parity family is incompatible with the chiral family discussed above, and they cannot be combined. One should not try and imagine them as going on simultaneously; to do so would be to make the same mistake as supposing that $S_x$ and $S_z$ for a spin-half particle can simultaneously possess values. On the other hand, just as it is possible to measure either $S_x$ or $S_z$, but not both simultaneously, it is also possible in principle (without violating the laws of quantum mechanics) to determine by measurements the succession of events that occur in a parity family, or by a different set of measurements those occurring in a chiral family. Thus a relatively rapid but random flipping back and forth between parity eigenstates is a valid physical picture of the succession of microscopic states of the molecule, one which can be used both when the decoherence is strong and when it is weak. There is in addition a third stationary family for ${\gamma }> {\omega }$, the dressed $y$ family, which has a relatively rapid hopping rate in the strong decoherence regime, and eventually merges with the chiral family as the decoherence rate decreases. We do not have a simple name or physical interpretation for this family. In the regime where decoherence is weak, ${\gamma }<{\omega }$, there are no truly stationary families, apart from the parity family discussed above. A relatively simple nonstationary family is the one that employs an “equatorial” basis in the $x$-$y$ plane of the Block sphere, ${\theta }=\pi/2$, rotating at an average angular speed $\eta$, see . Let us call this the “tunneling” family, since it corresponds in physical terms to the molecule oscillating back and forth between the two potential wells. As ${\gamma }$ increases the rate of tunneling decreases and eventually goes to zero at the phase transition ${\gamma }={\omega }$. In addition to the tunneling, the phase $\phi$ undergoes random changes by $\pi$, instantaneous on the time scale we are using, at a rate, , proportional to ${\gamma }$, but also depending on the value of $\phi$. Thus we have a nonstationary Markov process. The random flipping rate increases with ${\gamma }$ at the same time as the tunneling rate is decreasing, so the simple physical picture of the molecule tunneling from one potential well to the other breaks down upon approaching the phase transition ${\gamma }={\omega }$. For larger values of ${\gamma }$ this consistent family no longer exists. Information Flows\[sct5\] ========================= In the previous section we found various consistent frameworks for discussing the stochastic trajectory (for our model) of a tunneling molecule. We now wish to study the dynamics of information, e.g., the loss of information about the molecule’s original state as time progresses. Section \[sbct3.3\] discussed how the Shannon mutual information between the original state and the state at some later time, for the forward or backward consistent family, is equivalent to a particular Holevo $\chi$ quantity. Here, for simplicity, we will focus on families satisfying the forward condition, for which the information remaining in the molecule is given by , and that flowing to the environment by . These quantities are shown in Fig. \[fgr3\] for $P_1 = Z$ (parity basis) and $P_1=X$ (chirality basis), both for information remaining in the molecule ${\mathcal{T}}(t)$ and that flowing to the environment ${\mathcal{T}}^c(t)$. Figure \[fgr3\](a) shows a case of strong decoherence, ${\gamma }/{\omega }=2.5$, where the curves as a function of time are quite smooth, consistent with the fact, Fig. \[fgr2\](a), that the consistent family is rapidly approaching a stationary family. For weak decoherence, Fig. \[fgr3\](b) with ${\gamma }/{\omega }=0.05$, the consistent family is not stationary and the alternating rises and plateaus reflect this fact. The top curves in both (a) and (b) represent the sums, see below, for one type of information remaining in the molecule and a mutually unbiased type flowing to the environment. ![Information flows in terms of the $\hat \chi$ information measure (see text for definition), (a) in the strong decoherence regime with ${\gamma }= 2$ and ${\omega }= 0.8$, and (b) in the weak decoherence regime with ${\gamma }= 1$ and ${\omega }= 20$. The measure $\hat \chi$ is in units of bits, and the time $t$ is in units of $0.8 / {\omega }$ in (a) and $20/{\omega }$ in (b).[]{data-label="fgr3"}](fig3a.eps "fig:") ![Information flows in terms of the $\hat \chi$ information measure (see text for definition), (a) in the strong decoherence regime with ${\gamma }= 2$ and ${\omega }= 0.8$, and (b) in the weak decoherence regime with ${\gamma }= 1$ and ${\omega }= 20$. The measure $\hat \chi$ is in units of bits, and the time $t$ is in units of $0.8 / {\omega }$ in (a) and $20/{\omega }$ in (b).[]{data-label="fgr3"}](fig3b.eps "fig:") For short times the individual information measures can be computed, using the expressions for ${\mathcal{T}}$ and ${\mathcal{T}}^c$ in Appendix \[apdxA\] and noting that $S({\mathcal{T}}^c({{|\psi\rangle}{\langle \psi|}}))=S({\mathcal{T}}({{|\psi\rangle}{\langle \psi|}}))$ for any pure state ${|\psi\rangle}$, to obtain $$\begin{aligned} \label{eqn38} \hat{\chi}(X,{\mathcal{T}}({t}))&=1 - O[{t}^2\log (1/t)],\notag \\ \hat{\chi}(Z,{\mathcal{T}}({t}))&=1 - {\gamma }t\log(1/{\gamma }t)-{\gamma }t + O[{t}^2\log (1/t)],\notag \\ \hat{\chi}(X,{\mathcal{T}}^c({t}))&= {\gamma }t\log(1/{\gamma }t)+{\gamma }t + O[{t}^2\log (1/t)],\notag \\ \hat{\chi}(Z,{\mathcal{T}}^c({t}))&= O[{t}^2\log (1/t)],\end{aligned}$$ Here $O[\;]$ means that the correction term is of this or possibly some higher order. These expressions are consistent with $\hat{\chi}(X,{\mathcal{T}}({t}))$ and $\hat{\chi}(Z,{\mathcal{T}}^c({t}))$ having zero slope at $t=0$, and $\hat{\chi}(Z,{\mathcal{T}}({t}))$ and $\hat{\chi}(X,{\mathcal{T}}^c({t}))$ having infinite slope at $t=0$, as depicted in Fig. \[fgr3\]. That $\hat{\chi}(Z,{\mathcal{T}}^c({t}))$ in has no term linear in $t$ seems plausible in that the decoherence mechanism in our model has been chosen specifically to carry $X$ information into the environment. The uppermost curves in Figs. \[fgr3\](a) and \[fgr3\](b) represent $$\begin{aligned} \label{eqn39} \hat{\chi}(Z,{\mathcal{T}}({t}))+\hat{\chi}(X,{\mathcal{T}}^c({t}))&= \notag\\ \hat{\chi}(Z,{\mathcal{T}}^c({t}))+\hat{\chi}(X,{\mathcal{T}}({t}))& =1 - O[{t}^2\log (1/t)],\end{aligned}$$ where the first equality comes from Theorem 3 of [@PhysRevA.83.062338], and the second from ; the correction must be negative in view of the bound in . Another consequence of is that for small ${t}$ the flow of chiral information to the environment is compensated by a decrease of parity information remaining in the molecule: $$\frac{\mathrm{d}}{\mathrm{d}{t}}\hat{\chi}(X,{\mathcal{T}}^c({t})) =-\frac{\mathrm{d}}{\mathrm{d}{t}}\hat{\chi}(Z,{\mathcal{T}}({t})) + O[{t}\log (1/t)]. \label{eqn40}$$ However, as noted above, both sides of diverge logarithmically as ${t}\to 0$. Of course, these expressions lack physical meaning for times shorter than $\tau_c$, and thus the divergence is a mathematical artifact. Nonetheless, this makes it difficult to define rates of flow of information in a mathematically clean way using the $\hat{\chi}$ measure. An alternative which avoids the divergence is to replace the von Neumann entropy $S$ in the definition with the quadratic entropy $$\label{eqn41} S_Q(\rho)=1-{{\rm Tr}}(\rho^2).$$ In the case of a qubit channel with $W$ an orthonormal basis, projectors $W^1+W^2=I$, and Pauli operator ${\sigma }_W := W^1-W^2$, the measure defined in becomes $$\label{eqn42} \hat{\chi}_Q\left(W,{\mathcal{T}}({t})\right)= \frac{1}{2}{{\rm Tr}}\left[\Bigl({\mathcal{T}}({\sigma }_W)\Bigr)^2 \right],$$ with a similar expression for the complementary channel if ${\mathcal{T}}$ is replaced by ${\mathcal{T}}^c$. One can use the usual Pauli representation to write $$\label{eqn43} {\sigma }_W = {\bm{n}}\cdot{{\bm{\sigma}}},$$ where the $x$, $y$, and $z$ components of the ${\sigma }_W$ Pauli operator are given by the unit vector $${\bm{n}}=\{n_x,n_y,n_z\} =\{\sin \theta \cos \phi,\sin \theta \sin \phi, \cos \theta\}. \label{eqn44}$$ Using the short-time expressions for ${\mathcal{T}}$ and ${\mathcal{T}}^c$ given in Appendix \[apdxA\], one finds that for the direct and complementary channels $$\begin{aligned} \hat{\chi}_Q\left(W,{\mathcal{T}}({t})\right) &=1-4\gamma{t}\left[1-n_x^2 \right] + O({t}^2), \label{eqn45}\\ \hat{\chi}_Q\left(W,{\mathcal{T}}^c({t})\right) &=4\gamma{t}n_x^2 + O({t}^2). \label{eqn46}\end{aligned}$$ Setting $W=X$ in and $W=V$ in , where $V$ is some basis in the $y$-$z$ plane and thus mutually unbiased relative to $X$, the analog of with $\hat{\chi}$ replaced by $\hat{\chi}_Q$ is $$\frac{d}{d{t}} \hat{\chi}_Q(X,{\mathcal{T}}^c({t})) = -\frac{d}{d{t}}\hat{\chi}_Q(V,{\mathcal{T}}({t})) =4\gamma \label{eqn47}$$ at $t=0$, so the derivatives are now finite. Thus when one uses the $\hat{\chi}_Q$ measure the rate of flow of $X$ information to the environment equals the rate of decrease within the molecule of any type of information associated with a basis in the $y$-$z$ plane. D$_2$S$_2$ and NH$_3$ {#sct6} ===================== D$_2$S$_2$ {#sbct6.1} ---------- An order-of-magnitude calculation of the decoherence rate ${\gamma }$ for a D$_2$S$_2$ molecule immersed in a gas of helium atoms shows that this system is in the strong decoherence regime under typical conditions. The flux $q$ of helium atoms (atoms per unit area per unit time) is their concentration times their average velocity, given by $q= P\cdot\sqrt{8/(\pi m_{\text{He}} k_b T)}$, assuming an ideal gas with pressure $P$ and temperature $T$, with $m_{\text{He}}$ and $k_b$ the mass of a helium atom and Boltzmann’s constant [@KauzmannKineticTheory]. At room temperature $T=300$ K and a pressure $P$ of 1 atmosphere this gives $q\approx 3 \times 10^{28}$ atoms s$^{-1}$ m$^{-2}$. Multiplying $q$ by approximate cross sections $\sigma_{\text{col}}\approx 1000 a_0^2$ and $\sigma_{\text{dec}}\approx 100 a_0^2$ for collisions and decoherence, taken from [@PhysRevLett.103.023202], with $a_0$ the Bohr radius, leads to a collision rate of approximately $9 \times 10^{10} s^{-1}$ and a decoherence rate of $$\gamma\approx 9 \times 10^{9} s^{-1}. \label{eqn48}$$ The estimated tunneling rate is ${\omega }\approx 176$ rad/s after correcting[^2] the value published in [@PhysRevLett.103.023202] by a factor of $2\pi$. Thus $$\label{eqn49} \gamma/{\omega }\approx 5 \times 10^7,$$ which means strong decoherence, for which our results in Sec. \[sct4\] indicate that chirality is both a consistent description as well as a stable property, to a very good approximation. Probing the regime ${\gamma }\leq {\omega }$ for this molecule would seem quite difficult as it would involve very low pressures. Replacing deuterium with hydrogen and/or sulfur with oxygen leads to chiral molecules, e.g. H$_2$O$_2$, that have significantly higher tunneling frequencies [@Quack2001] and hence may be candidates for probing the ${\gamma }\leq {\omega }$ regime. NH$_3$ {#sbct6.2} ------ In the electronic ground state of ammonia NH$_3$ the nitrogen lies to one side of the plane defined by the three hydrogens, but there is a relatively low potential barrier separating it from the mirror image state on the other side, and tunneling in this double-well potential exhibits itself in the well-known inversion transition at a frequency of about 24 GHz. Although the molecule is not chiral, when it is rotating about an axis passing through the nitrogen and the midpoint between the hydrogen atoms the symmetry operation of parity (inversion) moves the nitrogen to the other side of the plane, changing the sign of the electric dipole, while leaving the angular momentum unchanged. Consequently, the energy levels with a nonzero quantum number $K$ for this component of angular momentum are split into two parity eigenstates by the inversion transition in a way similar to that in a chiral molecule. The tunneling transition has been observed directly by microwave absorption, which at low pressure exhibits a set of closely-spaced lines associated with the different rotational states [@Twns46]. As the pressure increases the lines broaden and merge, and the center of the merged line shifts towards lower frequencies, reaching zero frequency at a pressure of about 2 atmospheres [@BlLb50]. It has been suggested, e.g. [@Wght95], that at pressures above this transition the ammonia molecule adopts a “pyramidal” shape with the nitrogen on one side of the hydrogen plane, analogous to the shape of a chiral molecule with a definite handedness. Deuterated ammonia ND$_3$ shows similar behavior, except that the low pressure tunneling frequency is now at 1.6 GHz, and the center of the broadened line tends to zero frequency at a pressure of 0.12 atmospheres [@BrMr53]. The shift towards zero frequency has been analyzed theoretically using two different approaches. The first, exemplified by [@JLPT02], and with similar ideas in [@GrSc04; @GnBr11] among others, starts with the observation that since the ammonia molecule possesses a significant electric dipole moment when the nitrogen is on one side of the hydrogen plane there will be a strong dipole-dipole interaction between nearby molecules. It is then proposed that this produces a sort of mean-field effect in which the polarization of one molecule influences its neighbors in such a way that eventually as the pressure increases and the molecules come closer together, the double well potential for a single molecule is changed into one with a single minimum on one side of the hydrogen plane, resulting in molecules of pyramidal shape. An alternative approach found in [@BRvn65; @BRvn66; @HrDn07; @BhBs11] focuses instead on the decohering effects of collisions between gas molecules. It is argued that these collisions in addition to broadening the lines can also lower the tunneling frequency as the pressure, and thus the collision rate, increases. From this perspective the electric dipole-dipole interaction, while significant in determining the collision cross section and the effects of collisions, is not the fundamental source of the line shift to lower frequencies. The latter ought still to be present if ammonia is a dilute component in a nonpolar buffer gas. Of particular significance for this second point of view is the work of Ben-Reuven [@BRvn65; @BRvn66], who argued on theoretical grounds that when proper account is taken of the effects of collisions the line shape, absorption as a function of frequency, is not adequately represented by the Van Vleck and Weisskopf formula [@VVWs45] used earlier in [@BlLb50] to analyze the experimental data. He proposed an alternative line shape function with three parameters, ${\gamma }$, ${\zeta }$ and ${\delta }$, proportional to the collision rate, and thus the pressure, to fit the experimental microwave absorption data for NH$_3$ and (with a different choice of parameters) ND$_3$ over a range of frequencies and pressures sufficient to include that at which the tunneling frequency goes to zero. It is noteworth that this fit was achieved for all pressures and frequencies using just these three parameters, whereas the earlier analysis of Bleaney and Loubser [@BlLb50] was carried out by adjusting two parameters separately for each pressure. The validity of Ben-Reuven’s analysis is supported by the fact that more recent data on microwave absorption by ammonia in mixtures of hydrogen and helium (of interest in studies of the atmospheres of Jupiter and the other giant planets) has been fitted using his line shape formula for the tunneling transition [@DvSK11] with, of course, different choices of parameters for the different species scattering from the ammonia molecule. Since neither hydrogen nor helium has an electric dipole moment, this tends to support the idea that collisions, rather than dipole-dipole interactions as such, are what drive the transtion frequency to zero in pure ammonia gas as the pressure rises. The numbers given in [@DvSK11] would suggest a phase transition at about 20 atmospheres for ammonia in a buffer gas of hydrogen at room temperature. Replacing NH$_3$ with ND$_3$ should bring the transition pressure down by a factor of 15, and replacing hydrogen with a gas of some other nonpolar molecule might be advantageous. Thus a direct experimental test of whether dipole-dipole interactions are or are not essential for understanding the vanishing of the tunneling frequency seems feasible. Our very simple decoherence model corresponds to setting ${\gamma }={\zeta }$ and ${\delta }=0$ in Ben-Reuven’s theory as it applies to a two-level system. In fact, he achieved a good fit to the experimental data with ${\delta }=0$, but with ${\gamma }$ larger than ${\zeta }$ by a factor of around 1.3, see p. 21 of [@BRvn66]. To have ${\gamma }$ larger than ${\zeta }$ in our model would require our adding another source of decoherence. The phase transition present in our model is also clearly present in Ben-Reuven’s work; see the discussion of the spectrum of the perturbed Liouville matrix in Sec. 4C of [@BRvn66], where the eigenvalues change character when ${\zeta }$ passes through the value ${\omega }_0+{\delta }$; this is the counterpart of our ${\gamma }={\omega }$. Hence it seems that the vanishing of the inversion frequency in ammonia with increasing pressure is an instance of the sort of phase transition that occurs in our model. A more detailed comparison, which we have not attempted, would require our including an additional mechanism for decoherence to make ${\gamma }$ larger than ${\zeta }$, and dealing with complications caused by the presence in ammonia of a number of different rotational states. Nonetheless, we think our considerations provide some insight into the sense in which the ammonia molecule can be said to be “pyramidal” in the gas at high pressure and lack this feature at low pressures. Namely, when collisions are sufficiently frequent there is a consistent family of histories, the chiral or $x$ family discussed in Sec. \[sbct4.5\], in which in quantum mechanical terms the molecule spends a time much longer than the tunneling time in a pyramidal shape (or a “dressed” state close to it) which is “chiral” in the sense that electric dipole moment has a definite orientation relative to the angular momentum, with occasional random hops between the two pyramidal possibilities. As the pressure decreases towards the transition pressure the pyramidal picture begins to break down: the hops become more frequent between dressed states, which are starting to lose their pyramidal character. At still lower pressures it is better to think of the molecule as continuously tunneling back and forth, rather than possessing a fixed pyramidal form, with a period that diverges as the pressure rises to its value at the transition.   Conclusion \[sct7\] ===================== We have shown how the decoherence of a two-state tunneling molecule, a chiral molecule or ammonia, in the presence of a buffer gas can be described in terms of a succession of quantum states of the molecule itself that form a consistent family of histories, on a sufficiently coarse time scale so that intervals are always longer than a correlation time. Our model is described by just two parameters, a tunneling rate $\omega$ and a decoherence rate $\gamma$, and its essential properties depend upon the ratio ${\gamma }/{\omega }$. In addition we have studied the flow of information to the environment, along with its retention by the molecule itself, during the process of decoherence. We found a large variety of consistent families, some of which are stationary (in the sense of Markov processes) and some of which are not. In the regime ${\gamma }/{\omega }\gg 1$ of strong decoherence there is a stationary family (actually two closely related families) in which the molecule spends a relatively long time in one of its chiral states before flipping to the other chirality, and eventually flipping back again, in a stationary Markov (“telegraph”) process, with a transition rate which is approximately ${\omega }^2/4{\gamma }$ for ${\gamma }\gg {\omega }$, and hence quite slow compared to the tunneling rate ${\omega }$. Thus this “chiral” family explains the persistence of chirality for a long period of time when there is strong decoherence. However, as ${\gamma }/{\omega }$ decreases, the transitions between chiral states become more frequent and the states themselves (the “dressed $x$” states of Sec. \[sbct4.2\]) lose their chiral character, until finally this family disappears entirely at a phase transition ${\gamma }/{\omega }=1$. We have found two other stationary consistent families for ${\gamma }/{\omega }> 1$. One of them is the “parity” family in which the molecule is at each of the times considered in one of the two states of definite parity (the energy eigenstates of the isolated molecule), but with a transition rate of ${\gamma }$ between them, thus a rapid flipping compared to transitions between chiral states when the decoherence is large. This family, present at all values of ${\gamma }/{\omega }$, is incompatible, in the quantum mechanical sense, with the chiral family: while both provide valid quantum descriptions, they cannot be employed simultaneously; see the discussion in Sec. \[sbct4.5\]. The other stationary family for ${\gamma }/{\omega }> 1$ (again there are actually two families) is the “dressed $y$” family of Sec. \[sbct4.2\]. It involves a relatively rapid flipping between two orthogonal quantum states for which we do not have a simple physical interpretation. Like the chiral family this one only exists for ${\gamma }> {\omega }$. For ${\gamma }< {\omega }$ there is a nonstationary “tunneling” family in which the molecule oscillates back and forth between the two chiral states at a rate that goes to zero as ${\gamma }/{\omega }$ increases to 1. On top of this relatively smooth oscillation there are random changes in phase which constitute a nonstationary Markov process, with a rate that increases with ${\gamma }$. This tunneling family disappears at the phase transition ${\gamma }={\omega }$. The only truly stationary family in the regime ${\gamma }< {\omega }$ is the parity family. In addition, both for ${\gamma }< {\omega }$ and for ${\gamma }> {\omega }$ there are a variety of nonstationary consistent families in which the orthogonal basis used to describe the quantum system tends with time towards one of the stationary families or, for ${\gamma }< {\omega }$, the tunneling family. It seems likely that most chiral molecules under most conditions will be in a regime of strong decoherence ${\gamma }\gg {\omega }$, see the remarks about D$_2$S$_2$ in Sec. \[sbct6.1\]. Whereas it can only be thought of as “chiral” when in an appropriate rotational state, ammonia, including its deuterated form ND$_3$, is an example of a tunneling molecule in which the transition at ${\gamma }/{\omega }=1$ can be readily observed in the laboratory. Indeed, it appears that it has already been observed; see the discussion in Sec. \[sbct6.2\]. One wonders if additional experiments, perhaps using technques other than, or in addition to, microwave absorption might be helpful in elucidating its behavior near the phase transtion. In addition to consistent quantum families of histories we have studied, within the scope of our simple model, the flow of information from a tunneling molecule to its environment, along with the loss of information in the molecule itself. In Sec. \[sct5\] we used a perspective in which at a later time the information about the quantum state of the molecule at an earlier time is thought of as a quantum channel, while similar information present at this later time in the environment constitutes a complementary channel. What happens in both cases depends strongly on the type of information considered. Given that our model of decoherence, Sec. \[sbct2.2\], is based on the flow of chiral ($X$) information—is the molecule left or right handed?—to the environment, we were not surprised to find this exhibited in our quantitative measures, together with a rapid decrease of “complementary” types of information, corresponding to bases mutually unbiased with respect to $X$, retained within the molecule itself. Indeed, there is a direct quantitative relationship for short times if one uses a Holevo type of information measure, and an exact equality in the instantaneous rates, , if in the Holevo measure von Neumann entropy is replaced with quadratic entropy in order to render the rates finite. There are a number of ways in which lines of investigation initiated in this paper could be further extended. Parity-violation effects could be modeled as a small energy splitting between chiral states [@QuackReview2002]. Also, our model contains only one mechanism for decoherence, Sec. \[sct2\]; adding a second would allow a serious comparison with Ben-Reuven’s formula for ammonia as discussed in Sec. \[sbct6.2\]. Obtaining the correct physical interpretation might prove difficult given the complexity of the rotational states, even for ammonia present as a dilute component in a buffer gas. Fluorescence from a two-level atom, where decoherence arises from spontaneous decay, could be a simpler system for studying the dynamical phase transition. This transition has been studied in terms of correlations among scattered photons in [@GrLs10], and it would be of interest to supplement this with a description of how the atom itself behaves as a function of time. Indeed, even the decay of an isolated atom initially in an excited state has not, so far as we know, been examined using consistent families, and studying them might yield valuable physical insights. Acknowledgments {#acknowledgments .unnumbered} =============== The research described here was supported by the Office of Naval Research. V. Gheorghiu acknowledges additional support from the Natural Sciences and Engineering Research Council of Canada (NSERC) and from a Pacific Institute for Mathematical Sciences (PIMS) postdoctoral fellowship. Explicit expressions for ${\mathbf{T}}$ and its eigenvectors\[apdxA\] ===================================================================== The general expression for ${\mathbf{T}}(t)=e^{t{\mathbf{S}}}$ in is $${\mathbf{T}}(t)=\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & e^{-{\gamma }t}a(t) & -e^{-{\gamma }t}b(t) & 0 \\ 0 & e^{-{\gamma }t}b(t) & -e^{-{\gamma }t}c(t) & 0 \\ 0 & 0 & 0 & e^{-2{\gamma }t} \\ \end{pmatrix} \label{eqn50}$$ where, for ${\gamma }> {\omega }$ $$\begin{aligned} a(t) &= \cosh \xi t +({\gamma }/\xi)\sinh \xi t \notag\\ b(t) &= ({\omega }/\xi)\sinh \xi t \notag\\ c(t) &= \cosh \xi t -({\gamma }/\xi)\sinh \xi t \notag\\ \xi &:= \sqrt{{\gamma }^2-{{\omega }}^2}, \label{eqn51}\end{aligned}$$ whereas for ${\gamma }< {\omega }$ $$\begin{aligned} a(t) &= \cos \eta t +({\gamma }/\eta)\sin \eta t \notag\\ b(t) &= ({\omega }/\eta)\sin \eta t \notag\\ c(t) &= \cos \eta t -({\gamma }/\eta)\sin \eta t \notag\\ \eta &:= \sqrt{{{\omega }}^2-{\gamma }^2}. \label{eqn52}\end{aligned}$$ For ${\gamma }= {\omega }$ one has $$a(t) = 1+{\gamma }t,\quad b(t) = {\gamma }t,\quad c(t) = 1-{\gamma }t, \label{eqn53}$$ where of course ${\gamma }$ could be replaced by ${\omega }$. For the eigenvalues ${\lambda }_1$ and ${\lambda }_4$ in the left and right eigenvectors of ${\mathbf{T}}$ are, trivially, $(1,0,0,0)$ and $(0,0,0,1)$, respectively. For ${\lambda }_2$ and ${\lambda }_3$, the unnormalized left ${\bm{v}}$ and (transposed) right ${\bm{w}}$ eigenvectors are: $$\begin{aligned} {2} {\bm{v}}_2 &= (0,\xi - {\gamma },{\omega },0),\quad & {\bm{w}}_2 &= (0,{\gamma }-\xi,{\omega },0), \notag\\ {\bm{v}}_3 &= (0,-{\gamma }-\xi,{\omega },0), \quad & {\bm{w}}_3 &= (0,{\gamma }+\xi,{\omega },0), \label{eqn54}\end{aligned}$$ where for ${\gamma }< {\omega }$ replace $\xi$ with $i\eta$. For short times $t \ll1$ one has $$\label{eqn55} {\mathbf{T}}(t)=\begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & -{\omega }t & 0\\ 0 & {\omega }t & 1-2\gamma t & 0\\ 0 & 0 & 0 & 1-2\gamma t \end{pmatrix}+{\mathbf{O}}( t^2).$$ for the direct channel and $$\begin{aligned} \label{eqn56} {\mathbf{T}}^c(t)=& \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 2\sqrt{\gamma}\sqrt{ t} -\gamma^{3/2} t^{3/2} & {\omega }\sqrt{\gamma} t^{3/2} & 0\\ 0 & 0 & 0 & 0\\ 1-2\gamma t & 0 & 0 & 0 \end{pmatrix}\notag\\ &+{\mathbf{O}}( t^2).\end{aligned}$$ for the complementary channel. During this short time interval, one can represent the direct channel ${\mathcal{T}}$ using using only 2 Kraus operators, instead of 4 required by the most general qubit channel. The physical intuition behind this is that during this time interval the system of interest interacts only with a qubit environment, being effectively “decoupled” from the other environmental qubit (we remind the reader that the most general qubit evolution requires an interaction with an environment that is represented by at least 2 qubits, see e.g. [@NielsenChuang:QuantumComputation]). Differential equations for consistent families\[apdxBB\] ======================================================== As discussed in Sec. \[sbct4.1\], the forwards and backwards consistency conditions specify how a diameter of the Bloch sphere rotates. To determine this for the forwards condition, consider the density operator which at the initial time is at one end of the diameter, and write it in the form $$\rho = {\textstyle\frac{1}{2} }(I + {\bm{r}}\cdot{\bm{{\sigma }}}),\quad {\bm{r}} = r{\bm{n}}, \label{eqn57}$$ using the notation of and . Because ${\mathcal{T}}$ is unital the master equation for ${\boldsymbol{\rho}}$ is equivalent to $$d{\bm{r}}/dt = {\mathbf{\bar S}}\cdot{\bm{r}}, \label{eqn58}$$ or $$\Bigl(\frac{dr}{dt}\Bigr){\bm{n}} + \Bigl[ r\frac{d{\bm{n}}}{dt}\Bigr] =\Bigl({\bm{n}}\cdot{\mathbf{\bar S}}\cdot{\bm{n}}\Bigr){\bm{n}} + \Bigl[{\mathbf{\bar S}}\cdot{\bm{r}} - \left({\bm{n}}\cdot{\mathbf{\bar S}}\cdot{\bm{n}}\right){\bm{n}}\Bigr], \label{eqn59}$$ where ${\mathbf{\bar S}}$ is the lower right $3\times 3$ block of ${\mathbf{S}}$ in . Set $r=1$, thus ${\bm{r}} ={\bm{n}}$, and take the dot product of both sides of with ${\bm{n}}$, noting that ${\bm{n}}$ and $d{\bm{n}}/dt$ are necessarily orthogonal to each other, to obtain: $$\begin{aligned} dr/dt &= {\bm{n}}\cdot{\mathbf{\bar S}}\cdot{\bm{n}}, \label{eqn60} \\ d{\bm{n}}/dt &= {\mathbf{\bar S}}\cdot{\bm{n}} - ({\bm{n}}\cdot{\mathbf{\bar S}}\cdot{\bm{n}}){\bm{n}}. \label{eqn61}\end{aligned}$$ Note that ${\bm{n}}\cdot{\mathbf{\bar S}}\cdot{\bm{n}}$ depends only on the symmetrical part of ${\mathbf{\bar S}}$, which is to say the dissipative term, proportional to ${\gamma }$, in the master equation . The differential equations in are equivalent to when ${\bm{n}}$ is written in polar coordinates. To obtain the differential equations for the backwards consistency condition, replace ${\mathbf{\bar S}}$ in with ${\mathbf{\bar S}}{^\dagger }$, corresponding to the adjoint superoperator ${\mathcal{T}}{^\dagger }$, and $d/dt$ with $-d/dt$. The resulting differential equations are equivalent to with ${\gamma }$ replaced with $-{\gamma }$, thus . The consistency conditions are related to the motion in the Bloch sphere of the diameter that corresponds to the (instaneous) orthonormal basis. However, the instantaneous hopping rate ${\kappa }$ for the Markov process can be calculated using the Born rule for a very short time interval during which one can assume that the diameter remains fixed, as its motion (the change in basis) only contributes to higher order. When $r=1$, ${\kappa }$ as defined in is equal to $(-1/2)dr/dt$. Thus, using , $${\kappa }=(-1/2) ({\bm{n}}\cdot{\mathbf{\bar S}}\cdot{\bm{n}}) ={\gamma }(1-n_x^2), \label{eqn62}$$ which, transformed to polar coordinates, is . Time-integrated solutions for consistent families\[apdxB\] ========================================================== It is helpful to define $\mu (t):=\tan \phi(t)$ and $\nu(t):=\tan {\theta }(t)$. Then and can be integrated to give the following explicit solutions for the non-stationary consistent families: $$\begin{aligned} \mu(t)&=\mu(0)+({\omega }\mp 2{\gamma }\mu(0)+ {\omega }\mu(0)^2)\frac{\sinh \xi t}{\xi \cosh \xi t +(\pm {\gamma }-{\omega }\mu(0))\sinh \xi t}\notag\\ \nu(t)&=\nu(0)e^{\pm {\gamma }t}\sqrt{1\pm ({\gamma }/\xi) \frac{1-\mu(0)^2}{1+\mu(0)^2} \sinh 2 \xi t +(2{\gamma }/\xi^2) \Bigl({\gamma }\mp \frac{2{\omega }\mu(0)}{1+\mu(0)^2} \Bigr) \sinh^2 \xi t } \label{eqn63}\end{aligned}$$ where the top (or bottom) symbol in $\pm$ or $\mp$ is the solution to (or ) for the family satisfying the forward (or backward) condition. In the case that ${\omega }>{\gamma }$, one can replace every occurrence of $\xi=\sqrt{{\gamma }^2-{\omega }^2}$ in with $\eta=\sqrt{{\omega }^2-{\gamma }^2}$, provided that $\sinh$ and $\cosh$ are replaced by $\sin$ and $\cos$. Equality of mutual information and $\chi$ measure when forwards consistency conditions satisfied \[apdxC\] ========================================================================================================== The key observation is that, when the forward condition is satisfied, ${\mathcal{T}}_{m,1}(P^j_1)=\sum_k q_{kj} P^k_m$ for each $j$ and hence the ensemble of density operators at the channel output commute with each other, so quantum (von Neumann) entropies of these density operators become classical (Shannon) entropies in the basis that diagonalizes these density operators. Denoting $r^j_1:= {{\rm Tr}}(P_1^j)$ and $r^k_m:= {{\rm Tr}}(P_m^k)$, we have $$\begin{aligned} \hat\chi(P_1,{\mathcal{T}}_{m,1})=& S(\sum_j p_j {\mathcal{T}}_{m,1}(P^j_1)/r^j_1)-\sum_j p_j S({\mathcal{T}}_{m,1}(P^j_1)/r^j_1)\notag\\ =&S(\sum_{j,k} p_j q_{kj} P^k_m /r^j_1)-\sum_j p_j S(\sum_k q_{kj} P^k_m/r^j_1) \notag\\ =&H(\{ \sum_j p_jq_{kj} r^k_m/ r^j_1 \}_k)+ \sum_{j,k} (p_j q_{kj} r^k_m /r^j_1) S(P^k_m /r^k_m) \notag\\ &-\sum_j p_j H(\{ q_{kj} r^k_m /r^j_1 \}_k ) - \sum_{j,k} (p_j q_{kj} r^k_m /r^j_1) S(P^k_m /r^k_m) \notag\\ =&H(\{ \sum_j p_jq_{kj} r^k_m/ r^j_1 \}_k)-\sum_j p_j H(\{ q_{kj} r^k_m /r^j_1 \}_k ) \notag\\ =&H(P_m)-H(P_m |P_1 )=H(P_1{\,\hbox{:}\,}P_m), \label{eqn64}\end{aligned}$$ where the $k$ subscript in $\{\cdot \}_k$ indicates that the set is generated by allowing $k$ to vary. In this derivation, we used a property of the von Neumann entropy, for orthogonal density operators, given on page 513 of [@NielsenChuang:QuantumComputation]. [10]{} M. B. 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Tunneling properties of non-planar molecules in a gas medium. , 84:062115, 2011. arXiv:1110.3285v1 \[cond-mat.stat-mech\]. J. H. Van Vleck and V. F. Weisskopf. On the shape of collision-broadened lines. , 17, 227-236, 1945. Kiruthika Devaraj, Paul G. Steffes, and Bryan M. Karpowicz. Reconciling the centimeter- and millimeter-wavelength ammonia absorption spectra under jovian conditions: [E]{}xtensive millimeter-wavelength measurements and a consistent model. , 212:224–235, 2011. Martin Quack. How important is parity violation for molecular and biomolecular chirality? , 41:4618–460, 2002. [^1]: Working out the general connection of decoherence with thermodynamic irreversibility and the properties of steady state is, at a fundamental level, an unsolved problem. Our hope is that its eventual resolution will justify present practice by the experts in both the quantum information and decoherence communities, whose example we are following here. We note that some detailed quantum mechanical treatments of decoherence for situations similar to ours can be found in, e.g., [@HornbergerEPL2007; @PhysRevB.82.245417] and references therein. [^2]: Private communication from K. Hornberger

--- address: | Physique Théorique et Mathématique,\ Université Libre de Bruxelles, C.P. 231, 1050 Brussels, Belgium.\ International Solvay Institutes, Brussels, Belgium.\ Department of Fundamental Physics,\ Chalmers University of Technology, 412 96 Göteborg, Sweden.\ $\mathrm{Email: christoffer.petersson@ulb.ac.be,~christoffer.petersson@chalmers.se}$ author: - CHRISTOFFER PETERSSON title: MULTILEPTON AND MULTIPHOTON SIGNATURES OF SUPERSYMMETRY AT THE LHC --- Introduction ============ The discovery of a scalar boson, with properties in agreement with the predictions of the Standard Model (SM), reinforces the desire to understand the origin of the electroweak scale. The quadratic sensitivity of the scalar mass to ultraviolet physics suggests the presence of new states beyond the SM (BSM) at or below the TeV scale. But, in spite of the impressive range of searches performed by the LHC collaborations during Run I, no such new states have been observed so far. While these null results may be taken as an indication of the absence of new states in this energy range, they may also be taken as a motivation to push forward in new directions in the exploration of TeV scale physics. Supersymmetric (SUSY) extensions of the SM have the potential to both stabilize the electroweak scale and explain why it is hierarchically smaller than the Planck scale. The current bounds on superpartner masses are discomforting, but one should perhaps keep in mind that many searches are designed to probe particular SUSY extensions which are [*minimal*]{} in terms of their particle content and the underlying assumptions. However, given the non-minimality of, for example, the particle content of the SM, with three generations of quarks and leptons with a hierarchical mass spectrum, it could be that minimality is not a good guiding principle. By going beyond minimality in terms of model building, non-standard phenomenology can easily arise, with new search channels opening up and/or with standard search channels closing down. In the fist part of this note, by allowing for spectra beyond those of minimal models, we discuss an example of a BSM model that both fits a slight excess in the data and that predicts non-standard signatures which are currently not being targeted at the LHC. In the second part, we discuss a scenario where a deviation from the minimal model building assumptions opens up new search channels, while evading constraints coming from the standard ones. We consider simplified models based on the framework of gauge mediated SUSY breaking (GMSB), with R-partity conservation. In Section \[multileptons\], based on the paper [@multileptons], we do the exercise of explaining a slight excess in terms of multilepton events observed by the CMS collaboration [@CMSmultileptons]. We provide a simple model that can explain the excess, without being excluded by any other data, and we discuss how to best probe this model. In Section \[multiphotons\], based on the paper [@multiphotons], we discuss how the standard phenomenology of GMSB is modified if SUSY is broken in more than one hidden sector.[^1] In such multiple hidden sector models, the final state spectrum is typically softer than in standard GMSB, which implies that existing LHC searches are not very sensitive to these kind of models. The upshot is that these models typically give rise to additional (soft) photons in the final state, and we propose new searches designed to probe them. Multilepton signatures {#multileptons} ====================== Let us start by discussing the small excess observed by the CMS collaboration in a search for events with three or more leptons with 19.5 fb${}^{-1}$ of data at $\sqrt{s}=8$ TeV [@CMSmultileptons]. This small excess was seen in the final state category of events with three electrons or muons,[^2] one hadronically decaying tau lepton ($\tau_h$), low hadronic activity[^3] and no tagged b-jets. In this category, CMS observed (expected) in the three bins of missing transverse energy $\MET{<}50$GeV, 50${<}\MET{<}100$GeV and $\MET{>}100$GeV, respectively. The probability to observe 22 events in the combined $\MET$-range, when $10.1\pm2.4$ events were expected, is about 1%. However, when taking into account the fact that they search in 64 independent categories, the probability for this fluctuation in the combined $\MET$-range is about 50%, while the joint probability to observe such an excess in all the three $\MET$-bins is about 5% [@CMSmultileptons]. The most likely explanation for this slight excess is that it is due to a statistical fluctuation and that it will go away with more data. Nevertheless, we take the opportunity to perform the exercise of trying to fit this excess with some BSM physics. We consider two simplified models of GMSB, denoted by [**M.I**]{} and [**M.II**]{}, with spectra given in Figure \[fig:models\]. These models were studied in [@CMSmultileptons]. Here we extend that study by taking into account the exclusion bounds arising from pair production of the next-to-lightest SUSY particle (NLSP), determining the best fit model, considering other relevant searches and discussing prospects and possible new searches designed to probe the best fit model. Concerning the particle content of the models in Figure \[fig:models\], as always in GMSB, the lightest SUSY particle (LSP) is the nearly massless gravitino $\widetilde{G}$. In model [**M.I**]{}/[**M.II**]{} we take the NLSP to be the right-handed stau/sleptons, $\tilde{\tau}_R/\tilde{\ell}_R$, where “slepton" refers to either a selectron or a smuon, $\tilde{\ell}_R=\tilde{e}_R,\tilde{\mu}_R$. The next-to-NLSP (NNLSP) is the $\tilde{\ell}_R/\tilde{\tau}_R$, while the Bino $\widetilde{B}$ is taken to be heavier. All remaining superpartners are assumed to be sufficiently heavy and effectively decoupled. While such a decoupling is typically not possible in minimal GMSB models, where the relations among the soft masses are completely determined in terms of the gauge quantum numbers, it is possible to realize such spectra within the framework of See [@selectronNLSP] for a complete characterization of models realizing the non-standard GMSB spectrum of the simplified model [**M.II**]{} in Figure \[fig:models\]. Concerning the decay channels, since we assume R-parity, the NLSP only has one decay mode, i.e. to its SM partner and the gravitino. In contrast, the NNLSP has two possible decay channels, either the two-body decay to its SM partner and the gravitino, or the three-body decay, via an off-shell Bino, to the NLSP. For the parameter space region we are interested in, where the gravitino mass is in the range 0.1eV${<} \,m_{3/2} {<} \,10$eV, the NNLSP coupling to the gravitino is strongly suppressed compared to the gauge couplings entering the three-body decay, and the dominant NNLSP decay mode is the three-body decay. At the LHC, the models [**M.I**]{} and [**M.II**]{} give rise to the processes shown in Figure \[fig:processes\], and the final states $4\tau+2\ell+\MET$ and $2\tau+4\ell+\MET$, respectively. Hence, NNLSP pair production gives rise to multilepton events which could be relevant for the CMS search [@CMSmultileptons]. In order to see if we can fit the excess in [@CMSmultileptons], we simulate the two processes in Figure \[fig:processes\] at the LHC and analyze of data at $\sqrt{s}=8$ TeV, with kinematic and geometric cuts applied in accordance with the CMS search. ![Spectra for the simplified models [**M.I**]{} and [**M.II**]{}.[]{data-label="fig:models"}](Petersson_models.pdf){width="0.55\linewidth"} ![\[fig:processes\] The NNLSP pair production processes for models [**M.I**]{} (left) and [**M.II**]{} (right).](Petersson_slep.pdf "fig:"){width=".37\textwidth"} ![\[fig:processes\] The NNLSP pair production processes for models [**M.I**]{} (left) and [**M.II**]{} (right).](Petersson_stau.pdf "fig:"){width=".37\textwidth"} In Figure \[fig:results\], we show the number of signal events the processes in Figure \[fig:processes\] give rise to in the stau/slepton mass plane, where model [**M.I**]{}/[**M.II**]{} corresponds to the lower/upper triangular half plane. Figure \[fig:results\] (left) corresponds to the final state category where CMS observed the excess, and we see that both model [**M.I**]{} and [**M.II**]{} contain regions in the mass plane where the number of signal events fill the gap between the observed and expected number of events. So far we have only discussed the multilepton final states arising from the pair production of the NNLSP. Of course, in these models, there will also be pair production of the NLSP. In model [**M.II**]{}, the pair produced NLSP sleptons decay to the final state $\ell^+\ell^- {+}\MET$. The current bound on such right-handed sleptons is $m_{\tilde{\ell}_R}>245$ GeV [@ATLASsleptons], which actually excludes the entire parameter space region of [**M.II**]{} relevant for explaining the CMS excess. In model [**M.I**]{}, the pair produced NLSP staus decay to the final state $\tau^+\tau^-{+}\MET$. The current strongest bound on such right-handed staus is still the one set by LEP at As can be seen from Figure \[fig:results\], even if this stau mass bound imposes a non-trivial constraint on [**M.I**]{}, there still remains a parameter space region of [**M.I**]{} that can explain the CMS excess. The best fit model, obtained by considering the three $\MET$-bins individually, has $m_{\tilde{\ell}_R}=145$GeV and $m_{\tilde{\tau}_R}=90$GeV. ![\[fig:results\] The number of signal events in the $m_{\tilde{\ell}_R}/m_{\tilde{\tau}_R}$ mass plane, where we have combined the three $\MET$-bins. The left plot corresponds to the category discussed in the text, where CMS observed an excess. The middle plot corresponds to a similar category, but where the invariant mass of the OSSF lepton pair is instead *within* a window of $\pm15$GeV around the $Z$ boson mass. The right plot corresponds to the category where the final state consists of four electrons or muons, but no hadronic tau. In all three plots, the number of expected and observed events are indicated, as well as the LHC and LEP exclusion bounds arising from NLSP pair production.](Petersson_1tau-OSSF1-offZ.pdf "fig:"){width=".32\textwidth"} ![\[fig:results\] The number of signal events in the $m_{\tilde{\ell}_R}/m_{\tilde{\tau}_R}$ mass plane, where we have combined the three $\MET$-bins. The left plot corresponds to the category discussed in the text, where CMS observed an excess. The middle plot corresponds to a similar category, but where the invariant mass of the OSSF lepton pair is instead *within* a window of $\pm15$GeV around the $Z$ boson mass. The right plot corresponds to the category where the final state consists of four electrons or muons, but no hadronic tau. In all three plots, the number of expected and observed events are indicated, as well as the LHC and LEP exclusion bounds arising from NLSP pair production.](Petersson_1tau-OSSF1-onZ.pdf "fig:"){width=".32\textwidth"} ![\[fig:results\] The number of signal events in the $m_{\tilde{\ell}_R}/m_{\tilde{\tau}_R}$ mass plane, where we have combined the three $\MET$-bins. The left plot corresponds to the category discussed in the text, where CMS observed an excess. The middle plot corresponds to a similar category, but where the invariant mass of the OSSF lepton pair is instead *within* a window of $\pm15$GeV around the $Z$ boson mass. The right plot corresponds to the category where the final state consists of four electrons or muons, but no hadronic tau. In all three plots, the number of expected and observed events are indicated, as well as the LHC and LEP exclusion bounds arising from NLSP pair production.](Petersson_0tau-OSSF1-offZ.pdf "fig:"){width=".32\textwidth"} In Figure \[fig:results\] (middle) and (right), we show the number of signal events we obtain for two of the other final state categories in [@CMSmultileptons]. We see that the best fit model gives rise to a number of signal events that is within the 1$\sigma$ variation from the SM prediction in the middle plot. Also in the right plot there is no conflict with the data. We refer the reader to the paper [@multileptons] for discussions about data from other categories and searches. Concerning the possibility to probe this best fit model, it is important to realize that, in the last decay step of the process in Figure \[fig:processes\] (left), each of the 90 GeV staus decays to two approximately massless particles, a gravitino and a tau, which therefore roughly share the energy. Since the taus dominantly decay hadronically, many of these signal processes give rise to at least two hadronic taus which are hard enough to allow for reconstruction. We find that the most promising search channel for the best fit model would be in the final state involving two hadronic taus, two or three electrons or muons and $\MET$. In paper [@multileptons] we estimate the number of signal events the best fit model would contribute with to these final states, and we find that, already with the existing data set, such a search could have very good sensitivity. Multiphoton signatures {#multiphotons} ====================== If SUSY is realized in Nature it must be in a broken phase at low energies. A model-independent consequence of (global) SUSY breaking is the presence of a spin 1/2 Goldstone mode, the goldstino. Upon coupling to gravity, the goldstino is eaten by the spin 3/2 gravitino, becoming its longitudinal components, and the gravitino becomes massive. This is in analogy with electroweak symmetry breaking in the SM, where the Goldstone bosons are eaten by the $Z$ and $W$ bosons, which become massive. When the mass of the gravitino is small compared to the energy scale under consideration, in analogy with the equivalence theorem in the SM, the gravitino can be replaced by its longitudinal goldstino components. We consider this case, where the gravitino mass is small and where the communication of SUSY breaking to the visible sector, which we take to be the MSSM, is done via gauge interactions (i.e. within the framework of GMSB). In this section we investigate how the usual phenomenology of GMSB models is modified if we make the non-minimal assumption that SUSY is broken in more than one hidden sector.[^4] If SUSY is broken in $n$ hidden sectors, there will be $n$ neutral spin 1/2 goldstino-like fermions in the spectrum. However, there is only one particular linear combination that corresponds to the true goldstino mode, i.e. the one that is eaten by the gravitino. The remaining $n{-}1$ linear combinations correspond to so called pseudo-goldstini which, in contrast to the true goldstino, are not protected by the Goldstone shift symmetry and therefore they acquire masses, both at the tree and the radiative level. In comparison to standard GMSB models, where it is assumed that SUSY is broken in only one hidden sector and that there is only the nearly massless gravitino below the SM superpartners, multiple hidden sector models will, in addition, contain a tower of these massive pseudo-goldstini. In the case where the lightest SM superpartner is a Bino-like neutralino, its dominant decay channel is generically to a photon and the heaviest of the pseudo-goldstini. The reason is that the strength of the neutralino couplings are related to the mass of the pseudo-goldstini and the larger the mass, the stronger the coupling. In paper [@multiphotons] we also show that, in models with more than two hidden sectors, i.e. with more than one pseudo-goldstino, the heaviest pseudo-goldstino can decay promptly to a photon and a lighter pseudo-goldstino.[^5] Hence, in comparison to standard GMSB, the final states will contain additional photons. In order to illustrate the characteristic features of GMSB models with multiple hidden sectors and with a Bino-like neutralino being the lightest SM superpartner, let us for concreteness take the number of hidden sectors to be three, and thereby the number of massive pseudo-goldstini to be two. Above the Bino-like neutralino, depending on the ultraviolet model one has in mind, one could consider different superpartners which could be produced at the LHC. Here, as an example, we include the right-handed sleptons in the simplified model we consider. For simplicity, we take all three families of sleptons to be mass-degenerate, and with a change of notation with respect to the Section \[multileptons\], “slepton" here refers to a selectron, smuon or stau, $\tilde{\ell}_R=\tilde{e}_R,\tilde{\mu}_R,\tilde{\tau}_R$. ![\[Spec+proc\] The spectrum of the simplified model we consider, and the process it gives rise to.](Petersson_modelsPGLD2.pdf "fig:"){width=".35\textwidth"} ![\[Spec+proc\] The spectrum of the simplified model we consider, and the process it gives rise to.](Petersson_sleptonPGLD2.pdf "fig:"){width=".45\textwidth"} In the simplified model depicted in Figure \[Spec+proc\] (left), the main production mode is slepton pair production and the relevant process is shown in Figure \[Spec+proc\] (right). In this model, the heaviest pseudo-goldstino, to which the neutralino decays via a photon, is denoted by $\tilde{G}''$. This pseudo-goldstino subsequently decays promptly to the lighter pseudo-goldstino $\tilde{G}'$ which, in contrast to $\tilde{G}''$, is stable on collider time scales. Eventually $\tilde{G}'$ will decay to the gravitino $\tilde{G}$ but since that coupling is very small, this decay takes place outside the detector. Thus, from the point of view of collider physics, the gravitino plays no role and therefore it is not included in Figure \[Spec+proc\] (left). The process in Figure \[Spec+proc\] (right) gives rise to the final state $\ell^+\ell^- + 4\gamma+\MET$. Note that, if we would consider a different production mechanism by replacing the sleptons by other SM superpartners, then the OSSF lepton pair in Figure \[Spec+proc\] (right) would be replaced by the corresponding SM partners. In order to be as model-independent as possible in terms of the production mode, let us focus on the last two decay steps in the process, from which the photons and $\MET$ originate. In comparison to standard GMSB, where each of the two neutralinos decays to a photon and a nearly massless gravitino, since the pseudo-goldstini are massive, the emitted photons will here be softer. Moreover, since some of the $\MET$ that would be carried away by $\tilde{G}''$ is transformed into soft photon energy in the last decay step, also the amount of $\MET$ will here be smaller. In paper [@multiphotons] we show that, due to the hard cuts on the transverse momentum of the photons ($p_T^\gamma$) and the $\MET$, the existing LHC searches for GMSB are not very sensitive to multiple hidden sector models. Instead, what one should do in order to probe these models is to relax the $p_T^\gamma$ and $\MET$ cuts, but require additional soft photons in the final state. In [@multiphotons] we find that a search for four photons, each with $p_T^\gamma>20$GeV, and $\MET>$50GeV could easily lead to a discovery (or very strong constraints) already with the existing amount data, i.e.  Conclusions =========== In the first part of this note we provided a possible explanation of a slight excess, observed by the CMS collaboration, in terms of a simplified model of GMSB, with a gravitino LSP, a stau NLSP at around 90GeV and mass-degenerate selectron/smuon at around 145GeV. Since the stau mass is close to the LEP bound it will be interesting to see if the LHC at some point will be able to set a stronger bound on the stau mass. The search we propose that would best probe this model is in the final state $2\tau_h+(2/3)\ell+\MET$. In the second part, we discussed GMSB models in which SUSY is broken in more than one hidden sector. The two key features of such multiple hidden sector models were that the final state spectrum was generically softer than in standard GMSB, making the existing LHC searches poorly sensitive, and the presence of additional decay steps, where soft photons are emitted, opening up new search channels. The search we propose, that would be model-independent in terms of the production mode, is an inclusive search in the final state $(3/4)\gamma+\MET$, with minimal $p_T$-requirements on the photons. We focused on the example of slepton pair production which, due to the presence of a lepton pair in the final state, could be probed by a search in the final state $2\ell +2\gamma+\MET$, which would capture also the case where one or two photons are too soft to allow for reconstruction. Acknowledgments {#acknowledgments .unnumbered} =============== I would like to thank J.D’Hondt, K.De Causmaecker, G.Ferretti, B.Fuks, A.Mariotti, K.Mawatari and D.Redigolo for the collaborations on the projects I presented here. This work is supported by the Swedish Research Council (VR) under the contract 637-2013-475, by IISN-Belgium (conventions 4.4511.06, 4.4505.86 and 4.4514.08) and by the “Communauté Française de Belgique" through the ARC program and by a “Mandat d’Impulsion Scientifique" of the F.R.S.-FNRS. Finally I would like to thank the organizers of the conference “Rencontres de Moriond 2014, Electroweak Session" for their effort in organizing a very nice and fruitful meeting. References {#references .unnumbered} ========== [99]{} D.Redigolo, [*Phys.Lett.*]{}B 731, 7 (2014), \[arXiv:1310.0018 \[hep-ph\]\]. CMS Collaboration, arXiv:1404.5801 \[hep-ex\]. G.Ferretti, A.Mariotti, K.Mawatari and C.Petersson, [*JHEP*]{}, 1404:126 (2014), \[arXiv:1312.1698 \[hep-ph\]\]. 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--- abstract: | For fixed compact connected Lie groups H $\subseteq$ G, we provide a polynomial time algorithm to compute the multiplicity of a given irreducible representation of H in the restriction of an irreducible representation of G. Our algorithm is based on a finite difference formula which makes the multiplicities amenable to Barvinok’s algorithm for counting integral points in polytopes. The Kronecker coefficients of the symmetric group, which can be seen to be a special case of such multiplicities, play an important role in the geometric complexity theory approach to the P vs. NP problem. Whereas their computation is known to be \#P-hard for Young diagrams with an arbitrary number of rows, our algorithm computes them in polynomial time if the number of rows is bounded. We complement our work by showing that information on the asymptotic growth rates of multiplicities in the coordinate rings of orbit closures does not directly lead to new complexity-theoretic obstructions beyond what can be obtained from the moment polytopes of the orbit closures. Non-asymptotic information on the multiplicities, such as provided by our algorithm, may therefore be essential in order to find obstructions in geometric complexity theory. author: - - - bibliography: - 'multiplicities.bib' title: | Computing Multiplicities\ of Lie Group Representations --- Introduction {#section:introduction} ============ The decomposition of Lie group representations into irreducible sub-representations is a fundamental problem of mathematics with a variety of applications to the sciences. In atomic and molecular physics (Clebsch–Gordan series), as well as in high-energy physics, this problem has been studied extensively [@weyl50; @wigner59; @wigner73], perhaps most famously in Ne’eman and Gell-Mann’s eight-fold way of elementary particles [@neeman; @gellmann2; @gellmann]. In pure mathematics, the combinatorial resolution of the problem of decomposing tensor products of irreducible representations of the unitary group by Knutson and Tao has been a recent highlight with a long history of research [@fulton00; @knutsontao99]. More recently, the theories of quantum information [@keylwerner01; @christandlmitchison06; @klyachko06], computation and complexity [@baconchuangharrow07], as well as the geometric complexity theory approach to the ${\mathbf P}$ vs. ${\mathbf{NP}}$ problem [@mulmuleysohoni01; @mulmuleysohoni08; @burgisserlandsbergmaniveletal11] have brought the representation theory of Lie groups to the attention of the computer science community. In this paper, we study the problem of computing multiplicities of Lie group representations: \[main problem\] Let $f \colon H \rightarrow G$ be a homomorphism between compact connected Lie groups $H$ and $G$. The *subgroup restriction problem for $f$* is to determine the multiplicity $m^\lambda_\mu$ of the irreducible $H$-representation $V_{H,\mu}$ in the irreducible $G$-representation $V_{G,\lambda}$ when given as input the highest weights $\mu$ and $\lambda$ (specified as bitstrings containing their coordinates with respect to fixed bases of fundamental weights, see ). The name *subgroup restriction problem* comes from the archetypical case where the map $f$ is induced by the inclusion of a subgroup $H \subseteq G$. is also known as the *branching problem*. The main result of this paper is a polynomial-time algorithm for : \[A\] For any homomorphism $f \colon H \rightarrow G$ between compact connected Lie groups $H$ and $G$, there is a polynomial-time algorithm for the subgroup restriction problem for $f$. Indeed, we describe a concrete algorithm (). In particular, for any fixed $\lambda$ and $\mu$ the stretching function $k \mapsto m^{k \lambda}_{k \mu}$ can be evaluated in polynomial time. \[Aprime\] For any homomorphism $f \colon H \rightarrow G$ between compact connected Lie groups $H$ and $G$, positivity of the coefficients $m^\lambda_\mu$ can be decided in polynomial time. Mulmuley conjectures that deciding positivity of the multiplicities $m^\lambda_\mu$ is possible in polynomial time if the group homomorphism $f$ is also part of the input [@mulmuley07]. can be regarded as supporting evidence that this conjecture might in fact be true for general $f$ (note that for specific families of homomorphisms, such as those corresponding to the Littlewood–Richardson coefficients, positivity can be decided in polynomial time [@knutsontao99; @mulmuleysohoni05]). However, any approach to deciding positivity that proceeds by computing the actual multiplicities is of course expected to fail, since the latter problem is well-known to be ${\#\mathbf P}$-hard [@narayanan06; @burgisserikenmeyer08]. We establish by deriving a novel formula for the multiplicities $m^\lambda_\mu$ (), which is obtained in three steps: First, we restrict from the group $G$ to its maximal torus $T_G$; the corresponding weight multiplicities can be computed efficiently by using the classical Kostant multiplicity formula [@kostant59; @cochet05] or in fact by evaluating a single vector partition function [@billeyguilleminrassart04; @bliem08; @bliem10] (). Second, we restrict all weights to a maximal torus $T_H$ of $H$. Third, we recover the multiplicity of an irreducible $H$-representation by using a finite-difference formula (). By carefully combining the first two steps, can be reduced to counting integral points in certain rational convex polytopes of bounded dimension, which can be done efficiently by using Barvinok’s algorithm [@barvinok94; @dyerkannan97; @barvinokpommersheim99] (see also [@dyer91; @cook92; @welledabaldonibeckcochetetal06]). The multiplicity formula itself has intrinsic interest beyond its application to algorithmics. One insight that is immediate from our result is the piecewise quasi-polynomial nature of the multiplicities $m^\lambda_\mu$ (). Let us now turn to the computation of the *Kronecker coefficients* $g_{\lambda,\mu,\nu}$, which arise in the decomposition of tensor products of irreducible representations of the symmetric group $S_k$ [@fulton97]: $$[\lambda] \otimes [\mu] = \bigoplus_\nu g_{\lambda,\mu,\nu} \, [\nu],$$ where we denote by $[\lambda]$ the irreducible representation of $S_k$ labeled by the Young diagram $\lambda$ with $k$ boxes (). Kronecker coefficients are notoriously difficult to study, and finding an appropriately strong combinatorial interpretation is one of the outstanding problems of classical representation theory. They appear naturally in geometric complexity theory, where their efficient computation has been subject to various conjectures [@mulmuley07], as well as in quantum information theory in the context of the marginal problem and coding theory [@christandlmitchison06; @daftuarhayden04; @klyachko04; @klyachko06; @christandlharrowmitchison07; @harrow05]. Using Schur–Weyl duality, the Kronecker coefficients for Young diagrams with a bounded number of rows can be equivalently characterized in terms of a single subgroup restriction problem for compact connected Lie groups (). Therefore, by they can also be computed efficiently: \[B\] For any fixed $d \in {\mathbb Z}_{> 0}$, there exists a polynomial-time algorithm for computing the Kronecker coefficient $g_{\lambda,\mu,\nu}$ given as input Young diagrams $\lambda$, $\mu$ and $\nu$ with at most $d$ rows. That is, the algorithm runs in $O(\operatorname{poly}(\log k))$ where $k$ is the number of boxes of the Young diagrams. Positivity of Kronecker coefficients for Young diagrams with a bounded number of rows can be decided in polynomial time. By specializing our technique, we get a clean closed-form expression for the Kronecker coefficients (), which not only nicely illustrates its effectiveness, but also implies piecewise quasi-polynomiality for bounded height (a feature that has only been noticed in a special case [@briandorellanarosas09]). Moreover, it is immediate from our formula that the problem of computing Kronecker coefficients with unbounded height is in ${\mathbf{GapP}}$, as first proved in [@burgisserikenmeyer08]. Similar conclusions can be drawn for the plethysm coefficients, which can also be formulated in terms of subgroup restriction problems [@fultonharris91]. Like the Kronecker coefficients, they play a fundamental role in geometric complexity theory [@burgisserlandsbergmaniveletal11; @burgisserchristandlikenmeyer11b] and quantum information theory [@klyachko06; @christandlschuchwinter10]. In practice, our algorithms appear to be rather fast as long as the rank of the Lie group $G$ is not too large. In the case of Kronecker coefficients for Young diagrams with two rows, we can easily go up to $k=10^8$ boxes using commodity hardware. In contrast, all other software packages known to the authors cannot go beyond only a moderate number of boxes ($k=10^2$ on the same hardware as used above). Moreover, by distributing the computation of weight multiplicities onto several processors, we have been able to compute Kronecker coefficients for Young diagrams with three rows and $k=10^5$ boxes in a couple of minutes.[^1] We hope that our algorithm will provide a useful tool in experimental mathematics, theoretical physics, and geometric complexity theory. Our final result concerns the asymptotics of multiplicities in the general algebro-geometric setup of the geometric complexity theory approach to proving the $\mathbf{VP} \neq \mathbf{VNP}$ conjecture, an algebraic version of the ${\mathbf P}\neq {\mathbf{NP}}$ conjecture. Recall that, in a nutshell, this approach amounts to showing that for certain pairs of projective subvarieties $X$ and $Y$ one is not contained in the other; this would then imply complexity-theoretic lower bounds. Both the permanent vs. determinant problem, which is equivalent to the $\mathbf{VP}$ vs. $\mathbf{VNP}$ problem [@valiant79], as well as the complexity of matrix multiplication [@strassen69] can be formulated in this framework [@mulmuleysohoni01; @mulmuleysohoni08; @burgisserikenmeyer11; @burgisserlandsbergmaniveletal11]. More concretely, let us denote by $m_{H,X,k}(\mu)$ the multiplicity of the dual of an irreducible $H$-representation $V_{H,\mu}$ in the $k$-th graded part of the coordinate ring of $X$, and similarly for $Y$ (cf.  for precise definitions). Then, $$\label{mult crit} X \subseteq Y \;\Rightarrow\; m_{H,X,k}(\mu) \leq m_{H,Y,k}(\mu)$$ for all $\mu$ and $k \geq 0$. Therefore, the existence of $\mu$ and $k$ such that $m_{H,X,k}(\mu) > m_{H,Y,k}(\mu)$ proves that $X \not\subseteq Y$; such a pair $(\mu,k)$ is called an *obstruction* [@mulmuleysohoni08]. One can relax this implication further and instead compare the support of the multiplicity functions, $$X \subseteq Y \;\Rightarrow\; \left( m_{H,X,k}(\mu) \neq 0 \Rightarrow m_{H,Y,k}(\mu) \neq 0 \right).$$ Since computing multiplicities in general coordinate rings is a difficult problem, it is natural to instead study their asymptotic behavior. Following an idea of Strassen [@strassen], it has been proposed in [@burgisserikenmeyer11] to consider the *moment polytope*, $$\Delta_X := \overline {\bigcup_{k=1}^\infty \left\{ \frac \mu k : m_{H,X,k}(\mu) \neq 0 \right\}},$$ which is a compact convex polytope that represents the asymptotic support of the stretching function. Moment polytopes do have a geometric interpretation, which should facilitate their computation [@brion87]. Clearly, $$\label{mo po crit} X \subseteq Y \;\Rightarrow\; \Delta_X \subseteq \Delta_Y.$$ However, preliminary results suggest that the right-hand side moment polytope $\Delta_Y$ might be trivially large in the cases of interest [@burgisserchristandlikenmeyer11; @burgisserikenmeyer11; @kumar11; @burgisserlandsbergmaniveletal11], and therefore insufficient for finding complexity-theoretic obstructions. It has therefore recently been suggested to study the *asymptotic growth* of multiplicities (e.g., [@grochowrusek12 §2.2]). The natural object is the *Duistermaat–Heckman measure*, which is defined as the weak limit $$\label{definition duistermaat-heckman} \operatorname{DH}_X := \lim_{k \rightarrow \infty} \frac 1 {k^{d_X}} \sum_{\mu \in \Lambda^*_{H,+}} m_{H,X,k}(\mu) \, \delta_{\mu/k},$$ where $d_X \in {\mathbb Z}_{\geq 0}$ is the appropriate exponent such that $\operatorname{DH}_X$ is a non-zero finite measure [@okounkov96]. The Duistermaat–Heckman measure has a continuous density function $f_X$ with respect to Lebesgue measure on the moment polytope; it is supported on the entire moment polytope (both statements follow from the main result of [@okounkov96]). For well-behaved varieties, Duistermaat–Heckman measures have a geometric interpretation [@heckman82; @guilleminsternberg82b; @sjamaar95; @meinrenken96; @meinrenkensjamaar99; @vergne98; @teleman00], which makes their computation potentially much more tractable [@guilleminlermansternberg88; @guilleminlermansternberg96; @christandldorankousidiswalter12] (this connection is however less clear in the singular cases relevant to geometric complexity theory). In this context, our main technical result is the following (see for the proof): \[C\] The exponent $d_X$ is equal to $\dim X - R_X$, where $R_X$ is the number of positive roots of $H$ that are not orthogonal to all points of the moment polytope $\Delta_X$. The significance of is that the order of growth of the “smoothed” multiplicities, as captured by the Duistermaat–Heckman measures, does only depend on the dimension of the orbit closures and on their moment polytopes. Now suppose that we are in the situation that $X$ and $Y$ cannot be separated by using moment polytopes, i.e., $\Delta_X \subseteq \Delta_Y$. For the orbit closures $X$ and $Y$ that one tries to separate in geometric complexity theory, one can show that $\dim X < \dim Y$ [@burgisserlandsbergmaniveletal11; @burgisserikenmeyer11]. Then, $X \subseteq Y$ would imply that $d_X < d_Y$ (). But this means that we *cannot* deduce from and a criterion of the form $$X \subseteq Y \;\Rightarrow\; f_X(\mu) \leq f_Y(\mu) \qquad (\forall \mu),$$ since in order to take the weak limit we need to divide by different powers of $k$. Therefore, Duistermaat–Heckman measures do not directly give rise to new obstructions, indicating that a more refined understanding of the behavior of multiplicities in coordinate rings might be required. Preliminaries {#section:preliminaries} ============= In this paper we will use basic notions of the theory of compact Lie groups [@fultonharris91; @cartersegalmacdonald95; @kirillov08; @knapp02]. Let $G$ be a compact connected Lie group with Lie algebra $\mathfrak g$. We fix a maximal torus $T_G \subseteq G$ and denote by $\mathfrak t_G$ its Lie algebra, the corresponding Cartan subalgebra. We write $\Lambda_G = \ker {\left.\exp\vphantom{\big|}\right|_{\mathfrak t_G}}$ for the integral lattice and $\Lambda^*_G$ for the weight lattice, which we can consider as a subset of $\mathfrak t^*_G$. The Weyl group $W_G$ acts on $\mathfrak t_G^*$ by reflections through the hyperplanes orthogonal to the roots. Let us choose a set of positive roots $R_{G,+} \subseteq \Lambda^*_G$. This determines a positive Weyl chamber $\mathfrak t^*_{G,+}$, as well as a basis of fundamental weights $\{\omega^G_1,\ldots,\omega^G_{r_G}\}$, where $r_G = \dim T_G$ is the rank of the Lie group, and the Weyl vector $\rho = \frac 1 2 \sum_{\alpha \in R_{G,+}} \alpha$. The set of dominant weights $\Lambda^*_{G,+}$ is by definition the intersection of the weight lattice and the positive Weyl chamber. The fundamental theorem of the representation theory of compact connected Lie groups is the fact that the irreducible (complex) representations of $G$ can be labeled by their *highest weight* $\lambda \in \Lambda^*_{G,+}$ [@knapp02]; for every element $\lambda \in \Lambda^*_{G,+}$ there exists a unique irreducible representation $V_{G,\lambda}$ with this highest weight. Given an arbitrary finite-dimensional (complex) $G$-representation $V$, we can always decompose it into irreducible sub-representations $ V \cong \bigoplus_{\lambda \in \Lambda^*_{G,+}} m_{G,V}(\lambda) \, V_{G,\lambda} $. We shall call the function $m_{G,V}$ thus defined the *highest weight multiplicity function*. If we restrict the representation to the maximal torus, we can similarly decompose into irreducible representations. Since $T_G$ is a compact Abelian group, we can always jointly diagonalize its action, and it follows that the irreducible representations are one-dimensional. The joint eigenvalues can be encoded as a weight $\beta \in \Lambda^*_G$, and we will denote the corresponding irreducible representation of $T_G$ by ${\mathbb C}_\beta$. The decomposition $ V \cong \bigoplus_{\beta \in \Lambda^*_G} m_{T_G,V}(\beta) \, {\mathbb C}_\beta $ then defines the *weight multiplicity function* $m_{T_G,V}$. We also set $[k] = \{ 1,\ldots,k \}$, and write $f \sim g$ for the asymptotic equivalence $\lim_{k \rightarrow \infty} f(k)/g(k) = 1$. An equivalent way of encoding weight multiplicities is in terms of the (formal) *character*, $$\operatorname{ch}V = \sum_\beta m_{T_G,V}(\beta) \, e^\beta,$$ which can be understood as the generating function of $m_{T_G,V}$. Formally, $\operatorname{ch}V$ is an element of the group ring ${\mathbb Z}[\Lambda^*_G]$, which consists of (finite) linear combinations of basis elements $e^\beta$ subject to the relation $e^\beta e^{\beta'} = e^{\beta + \beta'}$. The character of an irreducible representation $V_{G,\lambda}$ is given by the *Weyl character formula* [@knapp02 p. 319], $$\label{weyl character formula} \operatorname{ch}V_{G,\lambda} = \frac {\sum_{w \in W_G} \det(w) \, e^{w(\lambda + \rho)}} {e^\rho \prod_{\alpha \in R_{G,+}} \left( 1 - e^{-\alpha} \right)}.$$ Observe that we have $$\label{derivation kostant partition function} \begin{aligned} \frac 1 {\prod_{\alpha \in R_{G,+}} \left( 1 - e^{-\alpha} \right)} &= \prod_{\mathclap{\alpha \in R_{G,+}}} \left( 1 + e^{-\alpha} + e^{-2\alpha} + \ldots \right) \\ &= \sum_{\mathclap{\beta \in \Lambda^*_G}} \phi_{R_{G,+}}(\beta) e^{-\beta}, \end{aligned}$$ where $\phi_{R_{G,+}}$ is the *Kostant partition function* given by the formula $$\label{kostant partition function} \phi_{R_{G,+}}(\beta) = \# \{ (x_j) \in {\mathbb Z}^{{\lvertR_{G,+}\rvert}}_{\geq 0} : \sum_j x_j \alpha_j = \beta \}.$$ That is, $\phi_{R_{G,+}}$ counts the number of ways that a weight can be written as a sum of positive roots (this number is always finite since the positive roots span a proper cone). It follows directly from and and that $$\begin{aligned} \operatorname{ch}V_{G,\lambda} = &\sum_{w \in W_G} \det(w) \sum_{\beta \in \Lambda^*_G} \phi_{R_{G,+}}(\beta) e^{w(\lambda + \rho) - \rho - \beta} \\ = &\sum_{\beta \in \Lambda^*_G} \sum_{w \in W_G} \det(w) \, \phi_{R_{G,+}}(w(\lambda + \rho) - \rho - \beta) e^\beta.\end{aligned}$$ In other words, the multiplicity of a weight $\beta$ in an irreducible representation $V_{G,\lambda}$ is given by the well-known *Kostant multiplicity formula* [@kostant59], $$\label{kostant multiplicity formula} m_{T_G,V_{G,\lambda}}(\beta) = \sum_{w \in W_G} \det(w) \, \phi_{R_{G,+}}(w(\lambda + \rho) - \rho - \beta).$$ For any fixed group $G$, the Kostant partition function can be evaluated efficiently by using Barvinok’s algorithm [@barvinok94], since it amounts to counting points in a convex polytope in an ambient space of fixed dimension. Therefore, weight multiplicities for fixed groups $G$ can be computed efficiently. This idea has been implemented by Cochet [@cochet05] to compute weight multiplicities for the classical Lie algebras (using the method presented in [@welledabaldonibeckcochetetal06] instead of Barvinok’s algorithm). We remark that the problem of computing weight multiplicities is of course the special case of where $H$ is the maximal torus $T_G \subseteq G$. Weight Multiplicities as a Single Partition Function {#weight-multiplicities-as-a-single-partition-function .unnumbered} ---------------------------------------------------- If $G$ is semisimple, we can find $s, t \in {\mathbb Z}_{\geq 0}$ and group homomorphisms $A \colon {\mathbb Z}^s \rightarrow {\mathbb Z}^t$ and $B \colon \Lambda^*_G \oplus \Lambda^*_G \rightarrow {\mathbb Z}^t$ such that $$\label{bliem multiplicity formula} m_{T_G,V_{G,\lambda}}(\beta) = \phi_A \left( B {\left(\begin{smallmatrix}\lambda \\ \beta\end{smallmatrix}\right)} \right) \>\> (\forall \lambda \in \Lambda^*_{G,+}, \beta \in \Lambda^*_G),$$ where $\phi_A$ is the *vector partition function* defined by $$\phi_A(y) = \# \{ x \in {\mathbb Z}^s_{\geq 0} : A x = y \}.$$ Note that this improves over the Kostant multiplicity formula , where weight multiplicities are expressed as an alternating sum over vector partition functions. In particular, is an evidently positive formula. It has been established by Billey, Guillemin, and Rassart for the Lie algebra ${\mathfrak{su}}(d)$ [@billeyguilleminrassart04], and was later extended to the general case by Bliem [@bliem08] by considering Littelmann patterns [@littelmann98] instead of Gelfand–Tsetlin patterns [@gelfandtsetlin88]. The assumption of semisimplicity for is not a restriction. Indeed, if $G$ is a general compact connected Lie group then its Lie algebra can always decomposed as $$\label{compact decomposition} \mathfrak g = [\mathfrak g,\mathfrak g] \oplus \mathfrak z,$$ where the commutator $[\mathfrak g, \mathfrak g]$ is the Lie algebra of a compact connected semisimple Lie group $G_{\operatorname{ss}}$, and where $\mathfrak z$ the Lie algebra of the center $Z(G)$ of $G$ [@knapp02 Corollary 4.25]. Let us choose a maximal torus $T_{G_{\operatorname{ss}}}$ of $G_{\operatorname{ss}}$ that is contained in $T_G$. Consider now an irreducible $G$-representation $V_{G,\lambda}$ with highest weight $\lambda$. By Schur’s lemma, each element in $Z(G)$ acts by a scalar. Therefore, all weights $\beta$ that appear in the weight-space decomposition have the same restriction to $\mathfrak z$. It follows that $$\label{semisimple restriction} m_{T_G,V_{G,\lambda}}(\beta) = \begin{cases} m_{T_{G_{\operatorname{ss}}},V_{G_{\operatorname{ss}},\lambda_{\operatorname{ss}}}}(\beta_{\operatorname{ss}}) & \text{if $\lambda_z = \beta_z$},\\ 0 & \text{otherwise}, \end{cases}$$ where we write $\mu_{\operatorname{ss}}$ and $\mu_z$ for the restriction of a weight $\mu$ to the Cartan subalgebra of $[\mathfrak g,\mathfrak g]$ and to $\mathfrak z$, respectively. These multiplicities can therefore be evaluated by using . The Finite Difference Formula {#section:finite difference formula} ============================= Let $V$ be an arbitrary finite-dimensional representation of the compact, connected Lie group $G$. Clearly, we can compute the weight multiplicity function $m_{T_G,V}$ from the highest weight multiplicity function $m_{G,V}$ by using any of the classical formulas and , or by evaluating the vector partition function described in . By “inverting” the Weyl character formula, the converse can also be achieved: \[steinberg lemma\] The highest weight and weight multiplicity function of a finite-dimensional $G$-representation $V$ are related by $$m_{G,V} = {\left.\left(\prod_{\alpha \in R_{G,+}} - D_\alpha \right) m_{T_G,V}\vphantom{\big|}\right|_{\Lambda^*_{G,+}}},$$ where $(D_\alpha m)(\lambda) = m(\lambda + \alpha) - m(\lambda)$ is the finite-difference operator in direction $\alpha$. Note that any two of the operators $D_\alpha$ commute, so that their product is independent of the order of multiplication. By linearity, it suffices to establish the lemma for a single irreducible representation $V = V_{G,\lambda}$ of highest weight $\lambda$. The Weyl character formula can be rewritten in the form $$\label{weyl for steinberg} \prod_{\alpha > 0} \left( 1 - e^{-\alpha} \right) \operatorname{ch}V_{G,\lambda} = \sum_{w \in W_G} \det(w) \, e^{w(\lambda + \rho) - \rho}.$$ If we identify elements in ${\mathbb Z}[\Lambda^*_G]$ with functions on the weight lattice, applying finite-difference operators $D_\alpha$ corresponds to multiplication by $\left(e^{-\alpha} - 1 \right)$. Therefore, the left-hand side of is identified with $\left( \prod_{\alpha \in R_{G,+}} - D_\alpha \right) m_{T_G,V_{G,\lambda}}$. Now consider the right-hand side of . Since $\lambda + \rho$ is a strictly dominant weight, it is sent by any Weyl group element $w \neq 1$ to the interior of another Weyl chamber. That is, there exists a positive root $\alpha \in R_{G,+}$ such that $\langle \alpha, w(\lambda + \rho) \rangle < 0$. In particular, $w(\lambda + \rho) - \rho$ is never dominant unless $w = 1$. It follows that the restriction of $\left( \prod_{\alpha \in R_{G,+}} - D_\alpha \right) m_{T_G,V_{G,\lambda}}$ to $\Lambda^*_{G,+}$ is equal to the indicator function of $\{\lambda\}$, i.e., equal to the highest weight multiplicity function of $V_{G,\lambda}$. The idea of using for determining multiplicities of irreducible representations goes back at least to Steinberg [@steinberg61], who proved a formula for the multiplicity $c_{\lambda,\mu}^\nu$ of an irreducible representation $V_{G,\nu}$ in the tensor product $V_{G,\lambda} \otimes V_{G,\mu}$. These multiplicities $c_{\lambda,\mu}^\nu$ are called the *Littlewood–Richardson coefficients* for $G$. Steinberg’s formula involves an alternating sum over the Kostant partition function ; it can be evaluated efficiently as described by Cochet [@cochet05]. De Loera and McAllister give another method for computing Littlewood–Richardson coefficients [@deloeramcallister06], which applies Barvinok’s algorithm to results by Berenstein and Zelevinsky [@berensteinzelevinsky01]. Since the tensor products of irreducible $G$-representations are just the irreducible representations of $G \times G$, the problem of computing Littlewood–Richardson coefficients is again a special case of . The following consequence of the proof of will be convenient in the sequel: \[steinberg corollary\] Write $\prod_{\alpha \in R_{G,+}} \left( 1 - e^{-\alpha} \right) = \sum_{\gamma \in \Gamma_G} c_\gamma e^{-\gamma}$ with $\Gamma_G \subseteq \Lambda^*_G$ finite and all $c_\gamma \neq 0$. Then, $$m_{G,V}(\lambda) = \sum_{\gamma \in \Gamma_G} c_\gamma \, m_{T_G,V}(\lambda + \gamma).$$ In particular, it is evident from that, for any fixed group $G$, the multiplicity of an irreducible representation in some representation $V$ can be computed efficiently from the weight multiplicities of $V$ by computing a finite linear combination. Multiplicities for the Subgroup Restriction Problem {#section:finite difference formula for subgroup restrictions} =================================================== Every $G$-representation $V$ can be considered as (“restricts to”) a representation of $H$ by setting $$\label{restriction} h \cdot v := f(h) \cdot v \qquad (\forall h \in H),$$ and the subgroup restriction problem for $f$, as defined in , amounts to determining the multiplicity $m^\lambda_\mu$ of a given irreducible representation of $H$ in the restriction of a given irreducible representation of $G$. In this section we will derive a formula for these multiplicities (), which will be the main ingredient of the algorithm presented in below. It will also follow from this formula that the $m^\lambda_\mu$ are given by a piecewise quasi-polynomial function[^2] in $\lambda$ and $\mu$ (). Let us choose the maximal torus $T_H \subseteq H$ in such a way that $f(T_H) \subseteq T_G$, and denote the corresponding Cartan subalgebra by $\mathfrak t_H$. Of course, this implies that the induced Lie algebra homomorphism $\operatorname{Lie}(f)$ sends the Cartan subalgebra of $H$ in the one of $G$. Since $f$ is a group homomorphism, $\operatorname{Lie}(f)$ restricts to a homomorphism between the integral lattices, $F \colon \Lambda_H \rightarrow \Lambda_G, ~ X \mapsto \operatorname{Lie}(f) X$. The dual map between the weight lattices is given by $$\label{definition dual map} F^* \colon \Lambda_G^* \rightarrow \Lambda_H^*, \quad \beta \mapsto \beta \circ F = {\left.\beta \circ \operatorname{Lie}(f)\vphantom{\big|}\right|_{\Lambda_H}}.$$ The following is well-known and easily follows from the definitions: \[weight restriction\] Let $V$ be a representation of $G$ and $v \in V$ a weight vector of weight $\beta \in \Lambda^*_G$. If we restrict the action to $H$ via then $v$ is a weight vector of weight $F^*(\beta) \in \Lambda^*_H$. Let us also fix systems of positive roots $R_{H,+}$ for $H$. This in turn determines the set of dominant weights $\Lambda^*_{H,+}$ as well as a basis of fundamental weights $(\omega^H_j)$ as described in . Let us also set $r_H = \dim T_H$. Our strategy for solving the subgroup restriction problem for $f$ then is the following: Given an irreducible representation $V_{G,\lambda}$ of $G$, we can determine its weight multiplicities with respect to the maximal torus $T_G$ by using any of the formulas presented in . We then obtain weight multiplicities for $T_H$ by restricting according to . Finally, we reconstruct the multiplicity of an irreducible representation $V_{H,\mu}$ by using the finite-difference formula (/). If this procedure was translated directly into an algorithm, the runtime would be polynomial in the coefficients of $\lambda$ (with respect to the basis of fundamental weights), i.e., exponential in their bitlength, since the number of weights is of the order of the dimension of the irreducible representation $V_{G,\lambda}$, which according to the *Weyl dimension formula* is given by the polynomial $\prod_{\alpha \in R_{G,+}} {\langle \alpha, \lambda + \rho \rangle} / {\langle \alpha, \rho \rangle}$ (cf. the formula by Straumann [@straumann65]). We will now show that it is possible to combine the weight multiplicity formula with the restriction map $F^*$ in a way that will later give rise to an algorithm that runs in polynomial time in the bitlength of the input: \[main theorem\] Let $f \colon H \rightarrow G$ be a homomorphism of compact connected Lie groups. Then we can find $s, s', u \in {\mathbb Z}_{\geq 0}$ and group homomorphisms $\mathcal A \colon {\mathbb Z}^{s+s'} \rightarrow {\mathbb Z}^u$ and $\mathcal B \colon \Lambda^*_G \oplus \Lambda^*_H \rightarrow {\mathbb Z}^u$ with the following property: For every irreducible representation $V_{G,\lambda}$ of $G$ and $V_{H,\mu}$ of $H$, the multiplicity $m^\lambda_\mu$ of the latter in the former is given by $$m^\lambda_\mu = \sum_{\gamma \in \Gamma_H} c_\gamma \, \# \{ x \in {\mathbb Z}^s_{\geq 0} \oplus {\mathbb Z}^{s'} : \mathcal A x = \mathcal B {\begin{pmatrix}\lambda \\ \mu + \gamma\end{pmatrix}} \},$$ where the (finite) set $\Gamma_H$ and the coefficients $(c_\gamma)$ are defined by $\prod_{\alpha \in R_{H,+}} \left( 1 - e^{-\alpha} \right) = \sum_{\gamma \in \Gamma_H} c_\gamma e^{-\gamma}$ and $c_\gamma \neq 0$. In fact, we can choose $s = O(r_G^2)$, $s' \leq r_G$ and $u = O(r_G^2) + r_H$. By definition and , we have $m^\lambda_\mu = m_{H,V_{G,\lambda}}(\mu) = \sum_{\gamma \in \Gamma_H} c_\gamma \, m_{T_H,V_{G,\lambda}}(\mu + \gamma)$. In view of , the multiplicity of a $T_H$-weight $\delta \in \Lambda^*_H$ in the irreducible $G$-representation $V_{G,\lambda}$ is given by $$m_{T_H,V_{G,\lambda}}(\delta) ~=~ \sum_{\mathclap{\substack{\beta \in \Lambda^*_G\\ F^*(\beta) = \delta}}} m_{T_G,V_{G,\lambda}}(\beta).$$ As in , let us now decompose the Lie-algebra $\mathfrak g = [\mathfrak g, \mathfrak g] \oplus \mathfrak z$. Denote the Lie group corresponding to $[\mathfrak g,\mathfrak g]$ by $G_{\operatorname{ss}}$ and choose a maximal torus $T_{G_{\operatorname{ss}}}$ which is contained in $T$. Using , $$\begin{aligned} \sum_{\mathclap{\substack{\beta \in \Lambda^*_G\\ F^*(\beta) = \delta}}} m_{T_G,V_{G,\lambda}}(\beta) ~~=~~ \sum_{\mathclap{\substack{\beta_{\operatorname{ss}} \in \Lambda^*_{G_{\operatorname{ss}}}\\ C_{\operatorname{ss}} \beta_{\operatorname{ss}} + C_z \lambda_z = \delta}}} m_{T_{G_{\operatorname{ss}}},V_{G_{\operatorname{ss}},\lambda_{\operatorname{ss}}}}(\beta_{\operatorname{ss}}), \end{aligned}$$ where we have decomposed $F^*$ as a sum of two homomorphisms $C_{\operatorname{ss}} \colon \Lambda^*_{G_{\operatorname{ss}}} \rightarrow \Lambda^*_H$ and $C_z \colon \Lambda^*_{Z(G)} \rightarrow \Lambda^*_H$. Let us now choose group homomorphisms $A \colon {\mathbb Z}^s \rightarrow {\mathbb Z}^t$ and $B = B_1 \oplus B_2 \colon \Lambda^*_{G_{\operatorname{ss}}} \oplus \Lambda^*_{G_{\operatorname{ss}}} \rightarrow {\mathbb Z}^t$ such that holds for the weight multiplicities for $G_{\operatorname{ss}}$. For this, $s$ and $t$ can be taken of order $O(r_G^2)$ [@bliem08 Proposition 19]. Then, $$\begin{aligned} \label{constructive} &\sum_{\mathclap{\substack{\beta_{\operatorname{ss}} \in \Lambda^*_{G_{\operatorname{ss}}}\\ C_{\operatorname{ss}} \beta_{\operatorname{ss}} + C_z \lambda_z = \delta}}} m_{T_{G_{\operatorname{ss}}},V_{G_{\operatorname{ss}},\lambda_{\operatorname{ss}}}}(\beta_{\operatorname{ss}}) \\ =~&\sum_{\mathclap{\substack{\beta_{\operatorname{ss}} \in \Lambda^*_{G_{\operatorname{ss}}}\\ C_{\operatorname{ss}} \beta_{\operatorname{ss}} + C_z \lambda_z = \delta}}} \# \{ x \in {\mathbb Z}^s_{\geq 0} : A x = B {\left(\begin{smallmatrix}\lambda_{\operatorname{ss}} \\ \beta_{\operatorname{ss}}\end{smallmatrix}\right)} \} \\ =~&\# \{ (x,\beta_{\operatorname{ss}}) : {\left(\begin{smallmatrix} A & - B_2\\ 0 & C_{\operatorname{ss}} \end{smallmatrix}\right)} {\left(\begin{smallmatrix}x \\ \beta_{\operatorname{ss}}\end{smallmatrix}\right)} = {\left(\begin{smallmatrix}B_1 \lambda_{\operatorname{ss}} \\ - C_z \lambda_z + \delta\end{smallmatrix}\right)} \}\\ =~&\# \{ (x,\beta_{\operatorname{ss}}) : \underbrace{{\left(\begin{smallmatrix} A & - B_2\\ 0 & C_{\operatorname{ss}} \end{smallmatrix}\right)}}_{=: \mathcal A} {\left(\begin{smallmatrix}x \\ \beta_{\operatorname{ss}}\end{smallmatrix}\right)} = \underbrace{{\left(\begin{smallmatrix} B_1 & 0 & 0\\ 0 & -C_z & {\mathbf 1}\end{smallmatrix}\right)}}_{=: \mathcal B} {\left(\begin{smallmatrix}\lambda_{\operatorname{ss}} \\ \lambda_z \\ \delta\end{smallmatrix}\right)} \}. \end{aligned}$$ After choosing a basis of the lattice $\Lambda^*_{G_{\operatorname{ss}}}$ we arrive at the asserted formula (with $s' = \dim T_{G_{\operatorname{ss}}}$ and $u = t + r_H$). We stress that the proof of is constructive: The maps $\mathcal A$ and $\mathcal B$, whose existence is asserted by the theorem, are defined in in terms of $A$ and $B$, whose construction is described explicitly in [@billeyguilleminrassart04 Proof of Theorem 2.1] (for the case of $\mathfrak g = {\mathfrak{su}}(d)$) and in [@bliem08 §4] (for the general case). See for an illustration in the context of the Kronecker coefficients. If one uses the Kostant multiplicity formula instead of in the proof of then one arrives at a similar formula for the multiplicities $m^\lambda_\mu$ involving an additional alternating sum over the Weyl group of $G$. After completion of this work, we have learned of [@heckman82 Lemma 3.1] which is derived in this spirit. Piecewise Quasi-Polynomiality {#piecewise-quasi-polynomiality .unnumbered} ----------------------------- Let us use the fundamental weight bases fixed above to identify $\Lambda^*_G \cong {\mathbb Z}^{r_G}$ and $\Lambda^*_H \cong {\mathbb Z}^{r_H}$. The group homomorphisms $\mathcal A$ and $\mathcal B$ correspond to matrices with integer entries, which we shall denote by the same symbols. Observe that the formula in in essence amounts to counting the number $n(y) := \# \left( \Delta_{\mathcal A, \mathcal B}(y) \cap {\mathbb Z}^{s+s'} \right)$ of integral points in certain rational convex polytopes of the form $$\label{weight restriction polytope} \Delta_{\mathcal A,\mathcal B}(y) := \{ x \in {\mathbb R}^{s+s'} : x_1, \ldots, x_s \geq 0, \mathcal A x = \mathcal B y \},$$ parametrized by $y \in {\mathbb Z}^{r_G+r_H}$. Explicitly, $$\label{concrete alternating sum} m^\lambda_\mu = \sum_{\gamma \in \Gamma_H} c_\gamma n(\lambda, \mu + \gamma).$$ It is well-known that $n(y)$ is a *piecewise quasi-polynomial* function in $y$ [@claussloechner98]. \[pageref:quasi-poly\] That is, there exists a decomposition of ${\mathbb Z}^{r_G+r_H}$ into polyhedral chambers such that on each chamber $C$ the function $n(y)$ is given by a single quasi-polynomial, i.e., there exists a sublattice $L \subseteq {\mathbb Z}^{r_G+r_H}$ of finite index and polynomials $(p_z)$ with rational coefficients, labeled by the finitely many points $z \in {\mathbb Z}^{r_G+r_H} / L$, such that $n(y) = p_{[y]}(y)$ for all $y \in {\mathbb Z}^{r_G+r_H}$ (cf. [@verdoolaegeseghirbeylsetal07 §2.2]). We record the following immediate consequence: \[abstract cor\] For any fixed group homomorphism $f \colon H \rightarrow G$, the multiplicities $m^\lambda_\mu$ are given by a piecewise quasi-polynomial function in $\lambda$ and $\mu$. In particular, this implies that the *stretching function* $k \mapsto m^{k\lambda}_{k\mu}$ is a quasi-polynomial function for large $k$. This is in fact true for all $k$, as has been observed in [@mulmuley07] (cf. [@meinrenkensjamaar99] for more general quasi-polynomiality results on convex cones, and also [@baldonivergne10] for further discussion). Polynomial-Time Algorithm for the Subgroup Restriction Problem {#section:algorithm} ============================================================== In this section we will formulate our algorithm for the subgroup restriction problem, . Recall that, by , the computation of the multiplicities $m^\lambda_\mu$ effectively reduces to counting the number of integral points in certain rational convex polytopes of the form . We shall suppose that the highest weights $\lambda$ and $\mu$, which are the *input* to our algorithm, are given in terms of their coordinates with respect to the fundamental weight bases fixed in . Clearly, for each of the finitely many $\gamma \in \Gamma_H$, the description of the polytope $\Delta_{\mathcal A,\mathcal B}(\lambda,\mu+\gamma)$ (say, in terms of linear inequalities) is of polynomial size in the bitlength of the input. It follows that *Barvinok’s algorithm* can be used to compute the number of integral points in each of these polytopes in polynomial time [@barvinok94] (see also [@dyerkannan97; @barvinokpommersheim99]). This gives rise to the following polynomial-time algorithm for , thereby establishing : \[main algorithm\] Let $f \colon H \rightarrow G$ be a homomorphism of compact connected Lie groups. Given as input two highest weights $\lambda \in \Lambda^*_G \cong {\mathbb Z}^{r_G}$ and $\mu \in \Lambda^*_H \cong {\mathbb Z}^{r_H}$, encoded as bitstrings containing their coordinates with respect to the fundamental weight bases fixed above, the following algorithm computes the multiplicity $m^\lambda_\mu$ in polynomial time in the bitlength of the input: $m \gets 0$ $n \gets \# \left( \Delta_{\mathcal A,\mathcal B}(\lambda, \mu + \gamma) \cap {\mathbb Z}^{s+s'} \right)$ as computed by Barvinok’s algorithm (see discussion above) $m \gets m + c_\gamma n$ **return** $m$ Here, $\Delta_{\mathcal A,\mathcal B}(y)$ denotes the rational convex polytope defined in , and the finite index set $\Gamma_H \subseteq \Lambda^*_H$ as well as the coefficients $(c_\gamma)$ are defined in the statement of . There are at least two software packages which have implemented Barvinok’s algorithm, namely <span style="font-variant:small-caps;">LattE</span> [@deloeradutrakoppeetal11] and <span style="font-variant:small-caps;">barvinok</span> [@verdoolaegeseghirbeylsetal07; @verdoolaegebruynooghe08]. In we have reported on the performance of our implementation of for computing Kronecker coefficients using the latter package. The existence of a polynomial-time algorithm for in fact already follows abstractly from , since in order to compute $m^\lambda_\mu$ we merely have to evaluate a *fixed* piecewise quasi-polynomial function. This piecewise quasi-polynomial can be computed algorithmically by using a variant of Barvinok’s algorithm which is also implemented in the <span style="font-variant:small-caps;">barvinok</span> package; see [@verdoolaegeseghirbeylsetal07 Proposition 2] and also [@barvinokpommersheim99 (5.3.1)]. Kronecker Coefficients {#section:kronecker} ====================== As explained in the introduction, the Kronecker coefficients play an important role in geometric complexity theory and quantum information theory. In this section, we will describe precisely how they can be computed using our methods. Let us recall the language of Young diagrams which is commonly used in this context [@fulton97]. A *Young diagram* with $r$ rows and $k$ boxes is given by an ordered list of integers $\lambda_1 \geq \ldots \geq \lambda_r > 0$ with $\sum_i \lambda_i = k$. It can be visualized as an arrangement of $k$ boxes in $r$ rows with $\lambda_j$ boxes in the $j$-th row. We set $\lambda_j = 0$ for all $j > r$. We will now consider the *unitary group* $\operatorname{U}(d)$, which consists of the unitary $d \times d$-matrices. Let us fix a system of positive roots and denote the corresponding basis of fundamental weights by $(\omega_j)$. To each Young diagram $\lambda$ with at most $d$ rows we associate the irreducible representation of $\operatorname{U}(d)$ with highest weight equal to $\sum_{j=1}^d \left( \lambda_j - \lambda_{j+1} \right) \omega_j$. Every polynomial irreducible representation of $\operatorname{U}(d)$ arises in this way. By a slight abuse of notation, we identify Young diagrams with the corresponding highest weights. More generally, we can associate to every integer vector $\beta \in {\mathbb Z}^d$ the weight $\sum_{j=1}^d \left( \beta_j - \beta_{j+1} \right) \omega_j$, where we set $\beta_{d+1} = 0$. This defines a bijection between ${\mathbb Z}^d$ and the weight lattice $\Lambda^*_{\operatorname{U}(d)}$ of $\operatorname{U}(d)$. In particular, the positive roots fixed above correspond to the integer vectors of the form $(\ldots,0,1,0,\ldots,0,-1,0,\ldots)$. The *Kronecker coefficient* $g_{\lambda,\mu,\nu}$ associated with triples of Young diagrams $\lambda$, $\mu$ and $\nu$ with $k$ boxes each and at most $a$, $b$ and $c$ rows, respectively, can then be defined in terms of the following subgroup restriction problem of compact, connected Lie groups: Let $H = \operatorname{U}(a) \times \operatorname{U}(b) \times \operatorname{U}(c)$ and $G = \operatorname{U}(abc)$ and consider the homomorphism $f \colon H \rightarrow G$ given by sending a triple of unitaries $(U,V,W)$ to their tensor product $U \otimes V \otimes W$. The Kronecker coefficient $g_{\lambda,\mu,\nu}$ is then given by the multiplicity of the irreducible $H$-representation $V_{H,(\lambda,\mu,\nu)} = V_{\operatorname{U}(a),\lambda} \otimes V_{\operatorname{U}(b),\mu} \otimes V_{\operatorname{U}(c),\nu}$ in the restriction of the symmetric power $\operatorname{Sym}^k({\mathbb C}^{abc})$, which is the irreducible $G$-representation labeled by the Young diagram $(k)$ consisting of a single row with $k$ boxes. That is, $$\label{kronecker definition} g_{\lambda,\mu,\nu} = m^{(k)}_{\lambda,\mu,\nu}(f)$$ This definition in fact does not depend on the concrete values chosen for $a$, $b$ and $c$, as can be seen by rephrasing it in terms of the representation theory of the symmetric group $S_k$ [@burgisserlandsbergmaniveletal11 §8] (but of course $a$, $b$ and $c$ have to be chosen at least as large as the number of rows of the Young diagrams). Moreover, it is evident that the Kronecker coefficients are symmetric in the variables $\lambda$, $\mu$, and $\nu$. It follows that, for any fixed choice of $a$, $b$ and $c$, can be used to compute the Kronecker coefficient given Young diagrams with at most $a$, $b$ and $c$ rows, respectively, in polynomial time in the input size, or equivalently in time $O(\operatorname{poly}(\log k))$, where $k$ is the number of boxes of the Young diagrams. This establishes . Let us again stress that the problem of computing Kronecker coefficients is known to be ${\#\mathbf P}$-hard in general [@burgisserikenmeyer08]; hence we do not expect that there exists a polynomial-time algorithm without any assumption on the number of rows of the Young diagrams. When computing Kronecker coefficients using the above method, we are only interested in the representation $V_{\operatorname{U}(abc),(k)} = \operatorname{Sym}^k({\mathbb C}^{abc})$, not in arbitrary irreducible representations of $\operatorname{U}(abc)$. By specializing the construction described in to this one-parameter family of representations, we obtain the following result: \[optimized kronecker\] The multiplicity of a weight $\delta = (\delta^{A},\delta^B,\delta^C) \in {\mathbb Z}^a \oplus {\mathbb Z}^b \oplus {\mathbb Z}^c \cong \Lambda^*_H$ (we use the identifications fixed at the beginning of ) in the irreducible $G$-representation $\operatorname{Sym}^k({\mathbb C}^{a b c})$ is equal to the number of integral points in the rational convex polytope $$\begin{aligned} &\Delta(k, \delta) = \Big\{ (x_{l,m,n}) \in {\mathbb R}^{a b c}_{\geq 0} \;:\; \sum_{l,m,n} x_{l,m,n} = k, \\ &\quad \sum_{m,n} x_{l,m,n} = \delta^{A}_l, \sum_{l,n} x_{l,m,n} = \delta^{B}_m, \sum_{l,m} x_{l,m,n} = \delta^{C}_n \Big\}. \end{aligned}$$ It follows that the Kronecker coefficient for Young diagrams $\lambda, \mu, \nu$ with $k$ boxes and at most $a$, $b$ and $c$ rows, respectively, is given by the formula $$g_{\lambda,\mu,\nu} = \sum_{\gamma \in \Gamma_H} c_\gamma \, \# \left( \Delta(k, (\lambda,\mu,\nu)+\gamma) \cap {\mathbb Z}^{abc} \right),$$ where $\Gamma_H$ and $(c_\gamma)$ are defined as in the statement of . It is well-known that the weight spaces for the action of $\operatorname{U}(d)$ on $\operatorname{Sym}^k({\mathbb C}^d)$ are all one-dimensional and that the set of weights corresponds to the integer vectors in the standard simplex rescaled by $k$ [@fulton97]. In our case, $d = a b c$, so that the weights are just the integral points of the polytope $$\Big\{ x = (x_{l,m,n})_{l \in [a], m \in [b], n \in [c]} \in {\mathbb R}^{a b c}_{\geq 0} \;:\; \sum_{l,m,n} x_{l,m,n} = k \Big\}.$$ Moreover, the dual map $F^* \colon \Lambda^*_{\operatorname{U}(abc)} \rightarrow \Lambda^*_{\operatorname{U}(a) \times \operatorname{U}(b) \times \operatorname{U}(c)}$ as defined in is given by $$\begin{cases} {\mathbb Z}^{abc} &\rightarrow {\mathbb Z}^a \oplus {\mathbb Z}^b \oplus {\mathbb Z}^c \\ (x_{l,m,n}) &\mapsto \Big( \sum_{m,n} x_{l,m,n}, \sum_{l,n} x_{l,m,n}, \sum_{l,m} x_{l,m,n} \Big). \end{cases}$$ We conclude that the multiplicity of a weight $\delta = (\delta^A, \delta^B, \delta^C)$ for $\operatorname{U}(a) \times \operatorname{U}(b) \times \operatorname{U}(c)$ is given by the number of integral points in the polytope $\Delta(k,\delta)$ described above. Just as for our main algorithm, gives rise to a polynomial-time algorithm for computing Kronecker coefficients with a bounded number of rows. This second algorithm runs faster than the generic one presented earlier, since the ambient space ${\mathbb R}^{abc}$ has a smaller dimension than what we would get from the construction described in the proof of . We remark that the time complexity for unbounded $a$, $b$ and $c$ can be deduced from [@barvinokpommersheim99]. Asymptotics {#section:asymptotics} =========== In this section we will prove our result on the generic order of growth of multiplicities in the coordinate ring of a projective variety (). We will work in the following general setup: Let $V$ be a finite-dimensional rational representation of $H$, and suppose that $X$ is an $H$-stable closed subvariety of the associated projective space ${\mathbb P}(V)$. The homogeneous coordinate ring ${\mathbb C}[X]$ is graded, and we can decompose each part into its irreducible components, $$\label{alggeo setup} {\mathbb C}[X] = \bigoplus_{k=0}^\infty {\mathbb C}[X]_k = \bigoplus_{k=0}^\infty \bigoplus_{\mu} m_{H,X,k}(\mu) \, V_{H,\mu}^*,$$ where, following the usual conventions, we have decomposed with respect to the dual representations $V_{H,\mu}^*$. The *stretching function* is then by definition $k \mapsto m_{H,X,k}(k\mu)$. We stress that in contrast to [@mulmuley07], where it was assumed that $X$ has at most rational singularities, we do not even require that $X$ is a normal variety [@hartshorne77]. This is highly relevant for geometric complexity theory, since it was recently shown in [@kumar10b] and [@burgisserikenmeyer11] that the studied varieties (the orbit closures of the determinant and permanent on the one hand, and of the matrix multiplication tensor and the unit tensor on the other hand) are in fact never normal except in trivial situations. The subgroup restriction problem for a rational group homomorphism $f \colon H \rightarrow G$ can be realized in the above setup: Indeed, for any highest weight $\lambda \in \Lambda^*_{G,+}$ consider $X = \mathcal O_{G,\lambda}$, the coadjoint orbit through $\lambda$, with the induced action of $H$. This variety can be canonically embedded into projective space as the orbit of the highest weight vector in ${\mathbb P}(V_{G,\lambda})$, and it is a consequence of the Borel–Weil theorem that ${\mathbb C}[\mathcal O_{G,\lambda}] = \bigoplus_{k=0}^\infty V_{G,k\lambda}^*$. By comparing with it follows that $m_{H,\mathcal O_{G,\lambda},k}(\mu) = m^{k \lambda}_{\mu}$. In particular, the above definition of the stretching function, $k \mapsto m_{H,\mathcal O_{G,\lambda},k}(k\mu)$, coincides with our previous usage, $k \mapsto m^{k\lambda}_{k\mu}$. By the Hilbert–Serre theorem, the function $k \mapsto \dim {\mathbb C}[X]_k$ is a polynomial of degree $\dim X$ for large $k$ [@hartshorne77 Theorem I.7.5]. Hence there exists a constant $A > 0$ such that $$\begin{aligned} &A \, k^{\dim X} \sim \dim {\mathbb C}[X]_k \\ ~=~ &\sum_{\mathclap{\mu \in \Lambda^*_{H,+}}} m_{H,X,k}(\mu) \, \dim V_\mu ~=~ \sum_{\mathclap{\mu \in \Delta_X \cap \frac 1 k \Lambda^*_{H,+}}} m_{H,X,k}(k \mu) \, \dim V_{k \mu}, \end{aligned}$$ where for the last equality we have used the definition of the moment polytope $\Delta_X$. By the Weyl dimension formula, we have $$\begin{aligned} &\dim V_{k\mu} ~=~ \prod_{\mathclap{\alpha \in R_{H,+}}} \frac {\langle \alpha, k \mu + \rho \rangle} {\langle \alpha, \rho \rangle} \\ ~=~ &\left( ~~~ \prod_{\mathclap{\substack{\alpha \in R_{H,+} \\ \alpha \not\perp \Delta_X}}} \frac {\langle \alpha, \mu \rangle} {\langle \alpha, \rho \rangle} \right) k^{R_X} + O(k^{R_X-1}) \end{aligned}$$ for the representations that occur in ${\mathbb C}[X]$. The coefficient $P(\mu) = \prod_{\alpha \in R_{H,+}, \alpha \not\perp \Delta_X} {\langle \alpha, \mu \rangle} / {\langle \alpha, \rho \rangle}$ is a polynomial function in $\mu$. Since $\Delta_X$ is compact, we can therefore find a constant $C > 0$ such that $$\dim V_{k \mu} \leq C \, k^{R_X} \qquad (\forall k, \mu \in \Delta_X \cap \frac 1 k \Lambda^*_{H,+}).$$ It follows that $$\begin{aligned} &\sum_{\mathclap{\mu \in \Delta_X \cap \frac 1 k \Lambda^*_{H,+}}} m_{H,X,k}(k \mu) \, \dim V_{k \mu} \\ \leq~&C \, k^{R_X} \sum_{\mathclap{\mu \in \Delta_X \cap \frac 1 k \Lambda^*_{H,+}}} m_{H,X,k}(k \mu) \sim C \, k^{R_X + d_X} \int d\operatorname{DH}_X, \end{aligned}$$ so that $\dim X \leq R_X + d_X$. On the other hand, since $\operatorname{DH}_X$ is Lebesgue-absolutely continuous, the boundary of the moment polytope does not carry any measure. We can therefore find a compact set $K$ contained in the (relative) interior of the moment polytope which has positive measure with respect to $\operatorname{DH}_X$. Note that $P(\mu)$ is positive for all $\mu$ contained in the interior of the moment polytope (indeed, for all positive roots $\alpha$ with $\alpha \not\perp \Delta_X$ there exists $\nu \in \Delta_X$ such that $\langle \alpha, \nu \rangle > 0$; since we can always write $\mu$ as a proper convex combination of $\nu$ and some other point $\nu' \in \Delta_X$, it follows that $\langle \alpha, \mu \rangle > 0$). This implies that on the compact set $K$ we can bound $P(\mu)$ from below by a positive constant. Thus there exists a constant $D > 0$ (depending on $K$) such that $$\dim V_{k \mu} \geq D \, k^{R_X} \qquad (\forall \mu \in K \cap \frac 1 k \Lambda^*_{H,+}).$$ Consequently, $$\begin{aligned} &\sum_{\mathclap{\mu \in \Delta_X \cap \frac 1 k \Lambda^*_{H,+}}} m_{H,X,k}(k \mu) \, \dim V_{k \mu} \geq~\sum_{\mathclap{\mu \in K \cap \frac 1 k \Lambda^*_{H,+}}} m_{H,X,k}(k \mu) \, \dim V_{k \mu} \\ \geq~&D \, k^{R_X} \sum_{\mathclap{\mu \in K \cap \frac 1 k \Lambda^*_{H,+}}} m_{H,X,k}(k \mu) \sim D \, k^{R_X + d_X} \int_K d\operatorname{DH}_X. \end{aligned}$$ We conclude that also $\dim X \geq R_X + d_X$, hence we have equality. Let us now elaborate on the argument presented at the end of the introduction, where we showed that Duistermaat–Heckman measures do not directly give rise to new complexity-theoretic obstructions. For this, we consider a pair of projective subvarieties $X$ and $Y$ with $\dim X < \dim Y$, as is the case for the orbit closures of relevance to GCT. Let us assume that $\Delta_X \subseteq \Delta_Y$, so that the moment polytopes alone do not already give rise to an obstruction. Clearly, this implies that $R_X \leq R_Y$. \[polytope smaller lemma\] Let $\Delta_X \subseteq \Delta_Y$ and $R_X < R_Y$. Then, $\dim \Delta_X < \dim \Delta_Y$. Note that we have $$\dim \Delta_X = \dim \operatorname{aff}\Delta_X \leq \dim \operatorname{aff}\Delta_Y = \dim \Delta_Y,$$ with equality if and only if the two affine hulls $\operatorname{aff}\Delta_X \subseteq \operatorname{aff}\Delta_Y$ are equal. Now by assumption there exists a positive root $\alpha \in R_{H,+}$ that is orthogonal to all points in $\Delta_X$ (i.e., for all $p \in \Delta_X$, $\alpha \perp p$), but not to all points in $\Delta_Y$. It follows that $\alpha$ is also orthogonal to all points in the affine hull of $\Delta_X$, but not to all points in the affine hull of $\Delta_Y$. Therefore, we have $\operatorname{aff}\Delta_X \subsetneq \operatorname{aff}\Delta_Y$. \[different lebesgue lemma\] Let $\dim \Delta_X < \dim \Delta_Y$. Then, $X \subseteq Y$ implies $d_X < d_Y$. If $X \subseteq Y$ then it is immediate from and that $d_X \leq d_Y$. Let us suppose for a moment that in fact $d_X = d_Y$. Then it follows from that $$\int_{\Delta_X} d\operatorname{DH}_X(\mu) \, g(\mu) \leq \int_{\Delta_Y} d\operatorname{DH}_Y(\mu') \, g(\mu')$$ for any test function $g$. In particular, this inequality would hold for $g$ the indicator function of $\Delta_X$. But this is clearly impossible, since $\operatorname{DH}_Y$ is absolutely continuous with respect to Lebesgue measure on $\Delta_Y$, for which $\Delta_X$ is a set of measure zero. \[dimension lemma\] Let $\dim X < \dim Y$. Then, $X \subseteq Y$ implies $d_X < d_Y$. Clearly, $X \subseteq Y$ implies that $\Delta_X \subseteq \Delta_Y$ and $R_X \leq R_Y$. If $R_X = R_Y$ then the assertion follows directly from , since $$d_X = \dim X - R_X < \dim Y - R_Y = d_Y.$$ Otherwise, if $R_X < R_Y$, it follows from combining and . As described in the introduction, the upshot of the above is that we cannot directly deduce from a new criterion for obstructions based on the Duistermaat–Heckman measure that goes beyond what is provided by the moment polytope. Acknowledgements {#acknowledgements .unnumbered} ================ We would like to thank Aravind Asok, Emmanuel Briand, Peter Bürgisser, David Gross, Christian Ikenmeyer, Stavros Kousidis, Graeme Mitchison, Mercedes Rosas, Volkher Scholz, and Michèle Vergne for helpful discussions. This work is supported by the Swiss National Science Foundation (grant PP00P2–128455 and 200021\_138071), the German Science Foundation (grants CH 843/1–1 and CH 843/2–1), and the National Center of Competence in Research ‘Quantum Science and Technology’. [^1]: A preliminary implementation of the algorithm is available upon request from the authors. [^2]: In the context of this paper, a quasi-polynomial function is a polynomial function with periodic coefficients; see p.  for the precise definition. It should not to be confused with the notion of quasi-polynomial time complexity.

--- title: 'Modeling of Transport through Submicron Semiconductor Structures: A Direct Solution of the Coupled Poisson-Boltzmann Equations' --- We report on a computational approach based on the self-consistent solution of the steady-state Boltzmann transport equation coupled with the Poisson equation for the study of inhomogeneous transport in deep submicron semiconductor structures. The nonlinear, coupled Poisson-Boltzmann system is solved numerically using finite difference and relaxation methods. We demonstrate our method by calculating the high-temperature transport characteristics of an inhomogeneously doped submicron GaAs structure where the large and inhomogeneous built-in fields produce an interesting fine structure in the high-energy tail of the electron velocity distribution, which in general is very far from a drifted-Maxwellian picture. [2]{} The carrier dynamics in submicron structures is far from thermal equilibrium due to strong and rapidly varying external and built-in electric fields. Hot electron and ballistic effects dominate the transport characteristics and the electron velocity distribution function in such systems is far from a drifted-Maxwellian description. In order to fully take into account the nonequilibrium nature of the transport, a full solution of the semiclassical Boltzmann transport equation (BTE) is required. Although the Monte Carlo method has been very popular for the solution of the BTE in semiconductor device simulation [@jacoboni], several works [@barangerPRB87]-[@majoranaCOMPEL04] have recently solved the BTE by direct methods, thus allowing noise-free spatial and temporal resolution of the electron distribution function, which in the Monte Carlo method may be difficult to obtain due to the statistical nature of the approach. In this paper, we present a straight-forward approach to calculate the electron distribution function, $f(x,v)$, for submicron inhomogeneous semiconductor structures by solving the steady-state BTE self-consistently with the Poisson equation. We solve the strictly two-dimensional (2D) BTE (one dimension corresponding to position and one to velocity) and treat scattering within the relaxation time approximation (RTA) where each individual scattering mechanism is represented by a characteristic scattering rate that can be derived from quantum mechanical scattering theory. We demonstrate our approach for submicron, inhomogeneously doped structures and discuss the general nonequilibrium transport characteristics. Basic equations =============== The Boltzmann equation describes the dynamics of the semiclassical distribution function, $f({\bf r}, {\bf v}, t)$, under the influence of electric and magnetic fields, as well as different scattering processes. In the absence of a magnetic field, the 2D phase-space, steady-state BTE in the RTA is written according to: $$-\frac{eE(x)}{m^{\ast}}\frac{\partial f(x,v)}{\partial v}+ v\frac{\partial f(x,v)}{\partial x}=-\frac{f(x,v)-f_{LE}(x,v)}{\tau(\varepsilon)}~, \label{bte}$$ where $m^{\ast}$ is the electron effective mass in the parabolic band approximation, and $f_{LE}(x,v)$ is a local equilibrium distribution function appropriate to a local density, applied field and equilibrium lattice temperature, $T_{0}$, to which the distribution function $f(x,v)$ relaxes at a relaxation rate $\tau(\varepsilon)^{-1}$. As the local equilibrium function, we choose in the following a Maxwell-Boltzmann (MB) distribution at $T_{0}$, normalized to the local density $n(x)$ $$f_{LE}(x,v)=n(x)\left [ \frac{m^{\ast}}{2\pi kT_{0}} \right ]^{1/2} e ^{-\frac{m^{\ast} v^{2}}{2k_{B}T_{0}}}~. \label{mb}$$ The inhomogeneous electric field, $E(x)$, in the BTE, originating from the spatially dependent electron and doping densities, $n(x)$ and $N_{D}(x)$, is obtained from the Poisson equation $$\frac{d ^{2} \phi}{d x^{2}} = -\frac{dE}{dx}= -e \frac{N_{D}(x) - n(x)}{\epsilon \epsilon_{0}} = -\rho(x), \label{poisson}$$ where $\epsilon$ is the static dielectric constant. Since the electron density is related to the distribution function by $$n(x)=\int^{\infty}_{-\infty} f(x,v)dv~, \label{density}$$ the Poisson and Boltzmann equations constitute a coupled, nonlinear set of equations, and thus, Eqs. (\[bte\]-\[density\]) have to be solved self-consistently. Numerical procedure =================== The numerical procedure consists, in short, of initializing the system parameters, discretizing Eqs. (\[bte\]-\[density\]) on a 2D grid in phase-space, performing the self-consistent Poisson-Boltzmann loop and, upon convergence, calculate and output the electron distribution function, electric field and the desired moments of the BTE. In the calculations, after initialization, we rescale the system parameters and the equations according to $$x^{\prime}=x/L_{D},~v^{\prime}=v\tau/L_{D}, \label{scaling}$$ where $L_{D}=\sqrt{\epsilon \epsilon_{0}k_{B}T_{0}/e^{2}N}$ is the Debye length, $N=\max[N_{D}(x)]$ and $\tau$ is a characteristic scattering time. The choice of grid size and resolution depends to a large extent on the system parameters and the electrostatics present in the device. In order to reproduce details due to strong and rapidly varying electric fields, we choose the spatial grid step size to be smaller than the Debye length, $L_{D}$, defined above. In velocity space, on the other hand, the discrete grid step size needs to be small enough to resolve fine structure in the distribution function, as well as give accurate results for the moments of the BTE. In addition, the grid needs to be large enough, in velocity, in order to capture the full information in the high-energy tail of the distribution function, and in position, in order to damp out the effects of the contact boundaries. The Poisson and Boltzmann equations are solved by finite difference and iterative relaxation methods [@numrec]. For the Poisson equation (\[poisson\]), we use forward and backward Euler differences according to $$L^{+}_{x}L^{-}_{x}\phi_{j}=\frac{\phi_{j+1}-2\phi_{j}+ \phi_{j-1}}{(\delta x)^{2}}= -\rho_{j}~, \label{poissondifference}$$ where $L^{+}_{x}\phi(x)=(\phi_{j+1}-\phi_{j})/\delta x$ and $L^{-}_{x}\phi(x)=(\phi_{j}-\phi_{j-1})/\delta x$ denote forward and backward Euler steps, respectively. The resulting matrix equation is solved iteratively using successive overrelaxation (SOR) [@numrec]. For the solution of the BTE, we adopt an upwind finite difference scheme [@fatemiJCP93] which amounts to the following discretization of the partial derivatives in Eq. (\[bte\]): $$\begin{aligned} \frac{\partial f}{\partial v} & = & L_{v}^{+[-]}f(x,v)~~E(x)>0~[E(x)\leq 0] \\ \frac{\partial f}{\partial x} & = & L_{x}^{+[-]}f(x,v)~~v<0~[v\geq 0]~. \label{btedifference}\end{aligned}$$ As for the Poisson equation, we use SOR to solve the matrix equation resulting from the discretization of Eq. (\[bte\]). For the boundary conditions of the Poisson-Boltzmann equations we adopt the following: For the potential, the values at the system boundaries, denoted (l)eft and (r)ight are fixed to $\phi(x_{l})=U_{0}$ and $\phi(x_{r})=0$, respectively, corresponding to an externally applied voltage $U_{0}$. The electron density is allowed to fluctuate freely around the boundaries, subject to the condition of global charge neutrality, which is enforced between each successive iteration in the self-consistent Poisson-Boltzmann loop. We choose the size of the highly-doped contacts to be large enough such that the electron density and the electric field deep inside the contacts is constant. For the electron distribution function four boundary conditions can be defined in the 2D phase-space. At the velocity cut-off in phase-space, we choose $f(x,v_{max})=f(x,-v_{max})= f_{LE}(x,v)$ which is reasonable since we assume $v_{max}\geq 30k_{B}T_{0}$ in the calculations. At such high velocities, the electron population is negligible and of the same order as the local equilibrium distribution $f_{LE}(x,v)$. At the contact boundaries, we assume that the electric field is low and constant (as verified in the calculations), and thus, the homogeneous solution to the BTE in the linear response regime of transport applies. Hence, $$f(x_{i},v)=f_{LE}(x_{i},v)[1-vE(x_{i})\tau(\varepsilon)/k_{B}T_{0}],$$ where $i=l,r$. The iterative Poisson-Boltzmann loop consists of an updating procedure for the electric field, electron distribution function and electron density using Eqs. (\[poisson\], \[bte\], \[density\]), until convergence. The convergence criterion is determined and checked in terms of the evolution of the $L_{2}$ norm of the potential and density variations between subsequent iterations. Typically, the results are converged when the $L_{2}$ norms for the potential and density are on the order of $10^{-3}$ of the original values. Between subsequent iterations, we employ linear mixing in the electron density, according to $$n^{\prime}(x)=(1-\alpha) n^{old}(x)+\alpha n^{new}(x)~,$$ where $n^{old}(x)$ is the input density to the Poisson solver, $n^{new}(x)$ is the new density obtained from the solution of the BTE using the new electric field obtained from the Poisson solver, and $n^{\prime}(x)$ is the final density that is used as an input to the next iteration in the Poisson-Boltzmann loop. The convergence and stability of the self-consistent loop are strongly dependent on the system parameters and the nonequilibrium nature of the electronic system. If the system is strongly out of equilibrium, displaying large variations and strengths of the electric field, the mixing parameters $\alpha$ may have to be chosen as small as a few percent, thus affecting the overall runtime. Furthermore, for highly doped structures, the convergence is slower, partly due to the required small grid size in position due to the small Debye length, but also due to the slow convergence in the SOR procedure in the BTE, where the stability of the numerical scheme is given in terms of a Courant-Friedrich-Levy type condition [@numrec]. Still, the computational demands for the calculations reported in this paper are modest. Numerical results ================= In the following, we demonstrate our numerical approach with calculations of the transport characteristics of a model GaAs $n^{+}-n^{-}-n^{+}-n^{-}-n^{+}$ structure with the doping densities $n^{+}$=10$^{23}$ m$^{-3}$ and $n^{-}$=10$^{19}$ m$^{-3}$. In order to highlight the effects of inhomogeneities and scattering while keeping the nature of the scattering structureless, we use a constant scattering time $\tau=2.5\cdot 10^{-13}$ s, which corresponds to realistic mobilities of GaAs at room temperature for which the calculations have been performed. The central $n^{-}-n^{+}-n^{-}$ region has the dimensions 200/200/200 nm, whereas the contacts are 1 $\mu$m long. {width="70.00000%"} In Fig. \[fig1\] we show the electric field and potential energy around the central region of the system described above, subject to an applied bias voltage $V_{b}=-0.5$ V. Due to the charge imbalance, electrons diffuse towards the lightly doped regions, where potential barriers are formed and, correspondingly, a large and inhomogeneous electric field on the order of 10 kV/cm is formed, even in the absence of an external applied voltage. As a finite voltage is applied to the device, the majority of the potential drop occurs over the submicron central region, giving rise to a strongly inhomogeneous field distribution, in contrast to the $n^{+}$ contact regions, where the field in comparison is very low and constant. The electron velocity distribution in the central region of the structure is shown in Fig. \[fig2\](a), for five specific spatial points as depicted in Fig. \[fig1\]. Figure \[fig2\](b) shows a contour plot of the full spatial dependence of $f(x,v)$ in that region. It is clear that the inhomogeneous electric field gives rise to a strong spatial dependence of the velocity distribution function along the direction of transport, and that the distribution function in the central region is very far from thermal equilibrium. In the outermost highly doped $n^{+}$ regions, where the field is low and constant the distribution is simply a shifted Maxwellian. In the lightly doped $n^{-}$ regions on the other hand, the velocity distribution is highly asymmetric and develops a narrow peak that rapidly shifts toward higher velocity along the direction of transport. This peak contains quasi-ballistic electrons which are accelerated by the strong electric field in the central region, and thus, have a considerably larger average velocity compared to the electrons in the contacts. Close to the potential barrier the distribution function is suppressed at low velocities due to the skimming of the distribution of incoming electrons, as well as the restriction of drain induced electron flow with $v<0$ due to the potential barrier. However, the low-velocity contribution to the distribution function gradually increases away from the barrier, as thermionically injected electrons gradually are thermalized and the lower effective barrier height allows electrons from the $n^{+}$ regions to penetrate the lightly-doped region. Thus, the total distribution function consists of a quasi-ballistic, high-velocity and a diffusive, low-velocity contributions, which gives the total distribution function a highly non-Maxwellian broad and asymmetric shape. Furthermore, the presence of the two barriers creates an additional quasi-ballistic structure in the high-energy tail of the distribution function in the second $n^{-}$ region, as electrons that traverse the intermediate $n^{+}$ region ballistically get an additional acceleration toward higher velocities by the electric field in the second $n^{-}$ region, thus creating two high-velocity electron beams. These features emphasize the highly nonequilibrium nature of the electron transport in these type of systems and demonstrate that our method is capable of taking them fully into account. {width="70.00000%"} Conclusions =========== We have presented a numerical method for the solution of the steady-state, coupled Poisson-Boltzmann equations for the study of inhomogeneous, submicron semiconductor structures and demonstrated our approach on a submicron GaAs structure with strong built-in electric fields. We have shown that our method is capable of taking into account the strong nonequilibrium transport properties that arise in such systems due to the presence of very large and inhomogeneous electric fields, and that interesting structure is present in the high-energy tail of the distribution function, caused by quasi-ballistic electrons.  \ [**Acknowledgments**]{}\  \ This work was supported by the Indiana 21st Century Research and Technology Fund. [99]{} C. Jacoboni, and P. Lugli, [*The Monte Carlo Method for Semiconductor Device Simulation*]{} (Springer-Verlag, Wien, 1989). H. U. Baranger, and J. W. Wilkins, “Ballistic structure in the electron distribution function of small semiconducting structures: General features and specific trends,” [*Physical Review B*]{} [ **36**]{}, 1487 (1987). E. Fatemi, and F. Odeh, “Upwind finite difference solution of Boltzmann Equation applied to electron transport in semiconductor devices”, [*Journal of Computational Physics*]{} [**108**]{}, 209 (1993). A. Majorana, and R. M. Pitadella, “A finite difference scheme solving the Boltzmann-Poisson system for semiconductor devices”, [*Journal of Computational Physics*]{} [ **174**]{}, 649 (2001). J.-H. Rhew [*et al.*]{}, “A numerical study of ballistic transport in nanoscale MOSFET”, [*Solid State Electronics*]{} [ **46**]{}, 1899 (2002). J. A. Carrillo [*et al.*]{}, “A WENO-solver for the transients of Boltzmann-Poisson system for semiconductor devices: performance and comparisons with Monte Carlo methods”, [*Journal of Computational Physics*]{} [ **184**]{}, 498 (2003). A. Majorana [*et al.*]{}, “Charge transport in 1D silicon devices via Monte Carlo simulation and Boltzmann-Poisson solver”, [*COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering*]{} [ **23**]{}, 410 (2004). W. H. Press [*et al.*]{}, [*Numerical Recipes in C*]{} (Cambridge University Press, Cambridge, 1992).

--- abstract: 'A robust classification method is developed on the basis of sparse subspace decomposition. This method tries to decompose a mixture of subspaces of unlabeled data (queries) into class subspaces as few as possible. Each query is classified into the class whose subspace significantly contributes to the decomposed subspace. Multiple queries from different classes can be simultaneously classified into their respective classes. A practical greedy algorithm of the sparse subspace decomposition is designed for the classification. The present method achieves high recognition rate and robust performance exploiting joint sparsity.' author: - | Tomoya Sakai\ Institute of Media and Information Technology, Chiba University\ 1-33 Yayoi, Inage, Chiba, Japan\ [tsakai@faculty.chiba-u.jp]{} bibliography: - 'mybib.bib' title: Multiple Pattern Classification by Sparse Subspace Decomposition --- Introduction ============ Classification is a task of assigning one or more class labels to unlabeled data (query data). A collection of labeled data (training data) is available for the classification. The patterns or signals to be classified are usually groups of measurement data expressed as high-dimensional vectors. Depending on purposes, we need pattern classifiers that can answer - a label to each of queries, - a label to a set of queries, - a few labels to each of queries, - a label “invalid” to an unclassifiable query. We develop a framework of using subspaces for all these functionalities. We regard the unlabeled data as a mixture of subspaces. The key idea is to decompose it into the subspaces of classes as few as possible. Only the classes explaining concisely the mixture are relevant to the unlabeled data. In the classification, the unlabeled data are usually supposed to belong to a few (typically one) classes. Therefore, the classification process can be interpreted as sparse decomposition of the subspace mixture. This work is inspired by the recently developing field of compressed sensing [@Donoho06; @Candes06a; @Candes06b; @Candes08RIP; @Candes08intro] and its innovative applications to robust face recognition [@Wright08], action recognition [@Yang09], computer vision and image processing [@Wright09]. The essential idea of these works is to exploit the prior knowledge that a signal is sparse and compressible. The theory of compressed sensing is very helpful and informative for us to answer questions such as “How many measurements are enough for the pattern recognition?” and “What is the role of feature extraction?” It is worthy to explore the potential of sparse decomposition for substantial improvement of the subspace methods. The rest of this paper is organized as follows. Section \[sec:preliminaries\] provides preliminary details and definitions of subspace representation for sparse decomposition. In Section \[sec:classification\], we propose a classification method named [*sparse subspace method*]{}, which exploits the sparseness property for the classification tasks described above. A practical algorithm of the sparse subspace decomposition is presented in Section \[sec:decomposition\]. We show some tentative evaluation results of the sparse subspace method using a face database in Section \[sec:experiments\] before concluding in Section \[sec:conclusion\]. Preliminaries {#sec:preliminaries} ============= Let $\mtr S_k\in\mathbb{R}^{d\times n_k}$ be a matrix of training dataset of $k$-th class ($k=1,\dots,C$), in which $n_k$ labeled patterns are represented as the $d$-dimensional column feature vectors. We describe as follows the linear subspaces, their union, block sparsity, and sparse linear representation of a subspace. We also define a classification space where the sparsity should be encouraged. #### Linear subspaces of training datasets The class subspace is defined as a vector subspace whose elements are the feature vectors of labeled data. We describe the subspace as a vector subspace in the normed space: $$\mathcal{S}_k\defas\span\mtr S_k\subset(\mathbb{R}^d,l^2).$$ $S_k$ approximates the $k$-th class subspace. We denote the dimensionality of $\mathcal{S}_k$ by $\dim\mathcal{S}_k=\rank\mtr S_k$. #### Union of subspaces The union of subspaces is the subspace obtained by combining the feature vectors of each class. $$\mathcal{S}\defas\cup_{k=1}^C\mathcal{S}_k=\span\mtr S \subseteq(\mathbb{R}^d,l^2)$$ Here, $\mtr S$ is the concatenation of $\mtr S_k$ as $$\mtr S\defas[\mtr S_1,\dots,\mtr S_C]\in\mathbb{R}^{d\times N} \label{eq:concatenated training datasets}$$ and $N\defas\sum_{k=1}^C n_k$. The dimensionality of $\mathcal{S}$ is denoted by $\dim\mathcal{S}=\rank\mtr S$. We say that the subspaces $\mathcal{S}_k$ ($k=1,\dots,C$) are independent if and only if any subspace $\mathcal{S}_k$ is not a subset of the union of the other subspaces, , $\mathcal{S}_k\not\subset\cup_{i\neq k}^C\mathcal{S}_i$ for $\forall k$. #### Linear representation of vector(s) Given sufficient training dataset, a $d$-dimensional vector $\vec q$ of unlabeled data (hereafter “query” vector) will be approximately represented as a linear combination of vectors from class subspaces. $$\vec q =\sum_{k=1}^C\mtr S_k\vecg\alpha_k =\mtr S\vecg\alpha \label{eq:linear representation}$$ Here, $\vecg\alpha_k\in(\mathbb{R}^{n_k},l^2)$ is a vector of coefficients corresponding to the $k$-th class, and $$\vecg\alpha\defas\bmatrix{c}{\vecg\alpha_1\\ \vdots\\ \vecg\alpha_C}\in(\mathbb{R}^N,l^2) \label{eq:concatenated coefficients}$$ is the concatenation of $\vecg\alpha_k$. If a set of queries is given as a matrix $$\mtr Q\defas[\vec q^{(1)},\dots,\vec q^{(n)}]\in\mathbb{R}^{d\times n}, \label{eq:query matrix}$$ then we will solve $$\mtr Q=\mtr S\mtr A. \label{eq:linear representation of vectors}$$ Here, $$\mtr A\defas[\vecg\alpha^{(1)},\dots,\vecg\alpha^{(n)}] \in\mathbb{R}^{N\times n}$$ is the matrix of unknown coefficients, and $$\vecg\alpha^{(j)}\defas\bmatrix{c}{\vecg\alpha_1^{(j)}\\ \vdots\\ \vecg\alpha_C^{(j)}}\in\mathbb{R}^N$$ is the concatenated vector of coefficients for the $j$-th query. The matrix $\mtr A$ can also be described as $$\mtr A=\bmatrix{c}{\mtr A_1\\ \vdots\\ \mtr A_C} \label{eq:row stacked form}$$ where $$\mtr A_k\defas [\vecg\alpha_k^{(1)},\dots,\vecg\alpha_k^{(n)}] \in\mathbb{R}^{n_k\times n}.$$ The systems of linear equations as (\[eq:linear representation of vectors\]) is called the problem for multiple measurement vectors (MMV), while the case of a single measurement $n=1$ as (\[eq:linear representation\]) is referred to as SMV [@Chen05; @Cotter05; @Eldar08]. The query vectors correspond to the measurements in this context. #### Uniqueness The solution $\vecg\alpha$ to (\[eq:linear representation\]) or $\mtr A$ to (\[eq:linear representation of vectors\]) exists if and only if $$\vec q^{(j)}\in\mathcal{S} \;\;\forall j, \label{eq:existence condition}$$ , the queries lie on the union of class subspaces. For $\dim\mathcal{S}<d$, the solution does not always exist. The solution may be dense even if it exists. Most components are nonzero despite the fact that at most $n$ class subspaces are relevant to $n$ queries. This problem is due to invalid situation where training datasets are insufficient to identify the class, uniquely. The actual problem we should cope with is the underdetermined case $d=\dim\mathcal{S}<N$, , the dimensionality of the union of subspaces is less than the total number $N$ of training samples. Unless the training data matrices $\mtr S_k$ are rank-degenerated so that $\dim\mathcal{S}<d$, the $C$ subspaces of training data cannot be independent in the $d$-dimensional space. There is an infinite number of ways to express the query vector by the linear combination of the subspace bases. The underdetermined problem requires regularization to select a unique solution. A sparse solution indicating relevant classes would be preferable. #### Block sparsity A vector $\vecg\xi\in(\mathbb{R}^N,l^0)$ is called $m$-sparse if $||\vecg\xi||_0\leq m$. Here, $||\cdot||_0$ denotes the $l^0$ norm, which counts the nonzero vector components. As the support of a function is the subset of its domain where it is nonzero, the support of a vector $\vecg\xi$ is defined as $\mathcal{T}=\{i|\xi_i\neq 0\}$. The $l^0$ norm is the cardinality of the support. We define a block-wise sparsity level in a similar manner to [@Eldar08]. Let $f_{\mathcal{N}}$ be a map from $\forall\mtr X\in(\mathbb{R}^{N\times n},l^F)$ to $\vecg\gamma\in(\mathbb{R}_{+}^C,l^0)$ according to a list $\mathcal{N}\defas\{n_1,\dots,n_C\}$ such that $$f_{\mathcal{N}} : \bmatrix{c}{\mtr X_1\\ \vdots\\ \mtr X_C} \rightarrow \bmatrix{c}{ ||\mtr X_1||_F\\ \vdots \\ ||\mtr X_C||_F }\defas\vecg\gamma. \label{eq:map to l0 classification space}$$ Here, $\mtr X_k\in(\mathbb{R}^{n_k\times n},l^F)$ is the $k$-th row block of $\mtr X$ with respect to $\mathcal{N}$, and $||\cdot||_F$ denotes the Frobenius norm $l^F$. Clearly, $$f_{\mathcal{N}} : \bmatrix{c}{\vec x_1\\ \vdots\\ \vec x_C} \rightarrow \bmatrix{c}{ ||\vec x_1||_2\\ \vdots \\ ||\vec x_C||_2 }$$ for $n=1$. A vector $\vec x\in(\mathbb{R}^N,l^2)$ is called block $M$-sparse over $\mathcal{N}$ if $\vec x_k\neq \vec 0$ for at most $M$ indices $k$. The block sparsity is measured as $$||\vec x||_{0,\mathcal{N}} \defas ||f_{\mathcal{N}}(\vec x)||_0. \label{eq:block sparsity}$$ That is, $||\cdot||_{0,\mathcal{N}}$ counts the number of nonzero blocks. We measure the row block sparsity of a matrix $\mtr X\in(\mathbb{R}^{N\times n},l^F)$ over $\mathcal{N}$ as $$||\mtr X||_{0,\mathcal{N}} \defas ||f_{\mathcal{N}}(\mtr X)||_0. \label{eq:row block sparsity}$$ A matrix $\mtr X$ is row block $M$-sparse if $||\mtr X||_{0,\mathcal{N}}\leq M$. We remark that the row block $M$-sparse matrix $\mtr X$ can be converted into a block $M$-sparse vector $\vectorize(\mtr X^\top)$. Here, the operator $\vectorize$ transforms a matrix into a column vector by stacking all the columns of the matrix. For $\mathcal{N}\defas\{n_1,\dots,n_C\}$ and $\mathcal{N}'\defas\{nn_1,\dots,nn_C\}$, the block sparsity of $\mtr X\in\mathbb{R}^{N\times n}$ over $\mathcal{N}$ is preserved as $$||\mtr X||_{0,\mathcal{N}}=||\vectorize(\mtr X^\top)||_{0,\mathcal{N}'}.$$ #### Sparse representation of subspace In the underdetermined case, the columns of matrix $\mtr S\in\mathbb{R}^{d\times N}$ represent an overcomplete basis of $\mathbb{R}^d$ for $d<N$. Equation (\[eq:linear representation\]) and (\[eq:linear representation of vectors\]) can be consistent with infinitely many solutions $\vecg\alpha$ and $\mtr A$, respectively. We denote the subspace of query vector(s) by $\mathcal{Q}=\span\vec q$ or $\span\mtr Q$. If a possible solution $\vecg\alpha$ or $\mtr A$ is block sparse over $\mathcal{N}=\{n_1,\dots,n_C\}$, the query subspace $\mathcal{Q}$ consists of a small minority of class subspaces corresponding to nonzero $\vecg\alpha_k$ or $\mtr A_k$. In other words, the query subspace is sparsely represented by the class subspaces. The sparsity of the subspace representation can be quantified as $||\vecg\alpha||_{0,\mathcal{N}}$ or $||\mtr A||_{0,\mathcal{N}}$. #### Classification space By definition, the block sparsity $||\vecg\alpha||_{0,\mathcal{N}}$ or $||\mtr A||_{0,\mathcal{N}}$ is measured by the $l^0$ norm of the $C$-dimensional vector $\vecg\gamma\defas f_{\mathcal{N}}(\vecg\alpha)$ or $f_{\mathcal{N}}(\mtr A)$. The components of $\vecg\gamma$ imply the degrees of class membership. The sparser $\vecg\gamma$ is, the more certainly the class label of each query is identified. The sparsity is properly measured by the $l^0$ norm. Therefore, we refer to the normed space $\mathcal{C}=(\mathbb{R}_{+}^C,l^0)$, where $\vecg\gamma$ resides, as the classification space. Classification based on sparse subspace representation {#sec:classification} ====================================================== From the viewpoint of classification, each query vector is supposed to be composed only of vectors from the subspace of a class to which the query is classified. The subspace spanned by the query vectors should be represented as sparsely as possible by the class subspaces concerned with the queries. In our notation, the $C$-dimensional vector in the classification space, $\vecg\gamma\defas f_{\mathcal{N}}(\vecg\alpha)$ or $f_{\mathcal{N}}(\mtr A)$, is intended to be sparsest. The sparsity is properly measured by the $l^0$ norm of $\vecg\gamma$. Therefore, we incorporate minimization of the $l^0$ norm in the classification framework. Formulation {#subsec:formulation} ----------- Let $\mtr S\in\mathbb{R}^{d\times N}$ be the concatenation of $\mtr S_k\in\mathbb{R}^{d\times n_k}$ ($k=1,\dots,C$, $d=\rank\mtr S<N=\sum_{k=1}^Cn_k$), , the matrices of training datasets. Given the matrix $\mtr Q\in\mathbb{R}^{d\times n}$ of $n$ query vectors, we solve the $l^0$-minimization problem: $$\min_{\mtr A} ||\mtr A||_{0,\mathcal{N}} \quad\mbox{subject to}\quad \mtr Q=\mtr S\mtr A. \label{eq:sparse decomposition by minimization}$$ Here, $\mathcal{N}$ specifies the sizes of row blocks for sparsification. Typically, $\mathcal{N}=\{n_1,\dots,n_C\}$. The matrix $\mtr A$ is released from being row-block sparse if $\mathcal{N}=\mathcal{N}_1\defas\{\forall n_i\!=\!1,i\!=\!1,\dots,N\}=\{1,\dots,1\}$. One can rewrite the problem (\[eq:sparse decomposition by minimization\]) as $$\begin{aligned} && \min_{\mtr A} ||\vectorize(\mtr A^\top)||_{0,\mathcal{N}'} \quad\mbox{subject to}\quad\nonumber\\ &&\qquad\qquad\vectorize(\mtr Q^\top)=(\mtr S\otimes\mtr I_n)\vectorize(\mtr A^\top) \label{eq:sparse decomposition by minimization vectorized}\end{aligned}$$ where $\otimes$ denotes the Kronecker product, and $\mtr I_n$ is the identity matrix of size $n$. The list $\mathcal{N}'$ defines the block sizes of the $nN$-dimensional vector $\vectorize(\mtr A^\top)$. The $l^0$-minimization problem (\[eq:sparse decomposition by minimization vectorized\]) is well investigated in the literature [@Eldar08]. The uniqueness of the solution is guaranteed under the condition called block restricted isometry property (block RIP). Assuming $\vec q^{(j)}\in\mathcal{S}$, the RIP condition for our problem can be described as $$\begin{aligned} && (1-\delta_{M|\mathcal{N}'})||\vec v||_2^2 \nonumber\\ &&\qquad\leq ||(\mtr S\otimes\mtr I_n)\,\vec v||_2^2 \nonumber\\ &&\qquad\qquad\leq (1+\delta_{M|\mathcal{N}'})||\vec v||_2^2\quad \forall\vec v\in{\mathbb{R}^{nN}}. \label{eq:block RIP condition}\end{aligned}$$ where $\delta_{M|\mathcal{N}'}$ is called the block-RIP constant dependent on the block sparsity $M$ over $\mathcal{N}'$. In practice, we normalize the blocks $\mtr S_k$ in order for the matrix $\mtr S\otimes\mtr I_n$ to satisfy the condition. The block RIP condition is less stringent than the standard RIP condition, which is widely used in the field of compressed sensing [@Donoho06; @Candes06a; @Candes06b; @Candes08RIP; @Candes08intro]. Dimensionality reduction ------------------------ In (\[eq:sparse decomposition by minimization\]), we assume the linear system $\mtr Q=\mtr S\mtr A$ to be underdetermined as $d=\rank\mtr S<N$, and regularize it by the $l^0$ minimization. Actually, we do not have to deal with the queries and training data in a space of dimension $d\geq N$. The recent works in the emerging area of compressed sensing show that a small number of projections of a sparse vector can contain its salient information enough to recover the vector with regularization that promotes sparsity [@Donoho06; @Candes06b; @Candes06c]. The statements in [@Candes05; @Rudelson05] guaranteeing the recovery are described as follows. Let $\vec x\defas\mtr\Psi^\top\vec s$ be a $d$-dimensional vector represented by a $m$-sparse vector $\vec s\in\mathbb{R}^d$ using a basis $\mtr\Psi^\top\in\mathbb{R}^{d\times d}$. Then, $\vec s$ can be reconstructed from a $\hat d$-dimensional vector $\hat{\vec x}\defas\mtr\Phi\vec x$ with probability $1-e^{-\mathcal{O}(\hat d)}$. Here, $\mtr\Phi\in\mathbb{R}^{\hat d\times d}$ is a random matrix and $\hat d\geq\hat d_0\defas\mathcal{O}(m\log(d/m))$. \[th:measurement for reconstruction\] Specially, $\hat d\geq 2m\log(d/\hat d)$ holds if $m\ll d$ [@Candes06; @Donoho09]. It is also possible to recover the sparse vector $\vec s$ from a small number of projections, $\hat{\vec x}$, with overwhelming probability in more general case where $\mtr\Phi$ and $\mtr\Psi$ are incoherent [@Candes06; @Candes07; @Candes08intro]. The reconstructability in Theorem \[th:measurement for reconstruction\] suggests that one can obtain the $d$-dimensional $m$-sparse solution from a much lower $\hat d$-dimensional vector after linear transformation. Wright [@Wright08] showed, in their framework of face recognition based on sparse representation, that the computational cost is reduced without significant loss of recognition rate by linear transformations into lower dimensional feature spaces, such as Eigenfaces, Fisherfaces, Laplacianfaces, downsampling, and random projection. These transformations act as dimensionality reduction that preserves information for the recognition. Especially, random projection is a data-independent dimensionality reduction technique, and one can exactly recover the original $d$-dimensional vector. For this reason, we employ the dimensionality reduction if $d$ is too high for computation. Classifiers ----------- #### $n$-to-one classifier Since the minimizer $\mtr A$ for (\[eq:sparse decomposition by minimization\]) is a row block $M$-sparse matrix, the $M$ blocks indicate the $M_C$ $(M_C\leq M$) classes concerned with the query subspaces. For the task of classifying all $n$ queries into one class ($M_C=1$), we calculate the residuals $r_k$ of the representations by the class subspaces. $$r_k(\mtr Q;\mtr A)\defas||\mtr Q-\mtr S_k\mtr A_k||_F. \label{eq:residual}$$ The residuals quantify the dissimilarities between the query subspace and the class subspaces. Note that most of the residuals are $||\mtr Q||_F$ because of the sparsity. If the query subspace $\mathcal{Q}$ can be approximately represented by one of the class subspaces, the class label is identified as $$\arg\min_k r_k(\mtr Q;\mtr A). \label{eq:n-to-one classifier}$$ This classification method achieves the same task as the mutual subspace methods [@Maeda85; @Yamaguchi98; @Fukui03] in a fundamentally different strategy. The mutual subspace methods are robust owing to the multiple queries. The robustness is further enhanced by the block sparsification in our scheme. The $l^0$ minimization in (\[eq:sparse decomposition by minimization\]) encourages the vector of class membership degrees, $f_{\mathcal{N}}(\mtr A)$, to be as sparse as possible in the classification space. For the underdetermined problem with a sparse solution, the recent works in the emerging area of compressed sensing [@Donoho06; @Candes06a; @Candes06b; @Candes08RIP] prove the exact recovery under the $l^0$ or $l^1$ regularization. Since the $l^0$ / $l^1$ minimizer is very insensitive to outliers, the sparse representation is robust compared to the conventional representations by $l^2$-based regularization PCA. We also remark that if $n=1$ and $\mathcal{N}=\mathcal{N}_1$, our $n$-to-one classification is exactly the same as the sparse representation-based classification (SRC) proposed in [@Wright08]. Our classification based on sparse subspace representation is therefore an extension of the SRC for multiple queries. #### $n$-to-ones classifier It is also possible to classify $n$ queries into their respective classes. We calculate $C\times n$ residual matrix whose $kj$-th entry measures the dissimilarity between the $j$-th query and its reconstruction in the $k$-th subspace: $$r_k^{(j)}(\mtr Q;\mtr A)\defas ||\vec q^{(j)}-\mtr S_k\vecg\alpha_k^{(j)}||_2. \label{eq:residual vector}$$ Note that most of the residual entries are $||\vec q^{(j)}||_2$ because of the sparsity. If the query subspace $\mathcal{Q}$ can be approximately represented by union of a small number of class subspaces, the class label for the $j$-th query is identified as $$\arg\min_k r_k^{(j)}(\mtr Q;\mtr A). \label{eq:multi-classifier}$$ Again, our method is expected to be robust owing to the multiple queries. Furthermore, the classes irrelevant to the queries are strongly excluded by the $l^0$ minimization. Therefore, the classifier (\[eq:multi-classifier\]) can detect the respective class for each query without giving the number of relevant classes. #### $n$-to-$M$ classifier Let us mention the potential of the sparse subspace representation for finding $n$-to-$M$ relations, although we do not go into the detail of this type of multiple classification in this paper. If a query simultaneously belongs to multiple classes, the query vector is represented as a linear combination of vectors from the subspaces of the relevant classes. The residuals $r_k^{(j)}$ for such query cannot be zero, but the relevant classes are found by thresholding $r_k^{(j)}$. Thus, each of $n$ queries is assigned to some of $M$ classes. #### Classification validity A classifier should answer “invalid” if the given query belongs to an unknown class. As suggested in [@Wright08], such an unclassifiable query is perceived to be so by measuring how the nonzero components of $\mtr A$ concentrate on a single class. Wright defined the sparsity concentration index (SCI), which quantifies the validity of the classification [@Wright08]. One may compute the SCI for each column of $\mtr A$ to validate the corresponding query. Sparse subspace method ---------------------- Our classification method based on the sparse subspace representation is summarized in Algorithm \[alg:SSM\]. $\mtr Q\in\mathbb{R}^{d\times n}$: matrix of $n$ queries as (\[eq:query matrix\]),$\mtr S\in\mathbb{R}^{d\times N}$: concatenated matrix of training datasets as (\[eq:concatenated training datasets\]),$\mathcal{N}$: list of row block sizes; $\mathcal{L}$: set of class labels; perform dimensionality reduction of $\mtr Q$ and $\mtr S$ if $d$ is intractably high;\[step:dimensionality reduction\] normalize the columns of $\mtr S$ to have unit $l^2$ norm; decompose $\mtr Q$ with respect to $\mtr S$ to obtain the sparse subspace representation. \[step:sparse decomposition\] find the class label $\mathcal{L}=\{\arg\min_k r_k(\mtr Q;\mtr A) \}$ or\ $\mathcal{L}=\{\arg\min_k r_k^{(1)}(\mtr Q;\mtr A),\dots,\arg\min_k r_k^{(n)}(\mtr Q;\mtr A)\}$. The major concern is the sparse subspace decomposition of $\mathcal{Q}$ at Step \[step:sparse decomposition\]. In the next section, we present a practical algorithm of the decomposition, SSD-ROMP, which efficiently and stably provides approximate solution to (\[eq:sparse decomposition by minimization\]). Sparse subspace decomposition {#sec:decomposition} ============================= The sparse decomposition of $\mtr Q$ in (\[eq:sparse decomposition by minimization\]) is considered as a MMV problem whose solution is row-block sparse. The solution has two important characteristics: the column vectors $\vecg\alpha^{(j)}$ of $\mtr A$ share nonzero blocks as their support, and the block partitions are fixed by $\mathcal{N}$ in advance. Prior work on MMV ----------------- Configuration of the nonzero entries in the solution $\mtr A$ is called the joint sparsity model (JSM) [@Baron05; @Marco05a]. There are some prior works on the MMV problems with several JSMs [@Chen05; @Cotter05; @Baron05; @Marco05a; @Tropp06; @Eldar08; @Duarte09]. Most of them [@Chen05; @Cotter05; @Baron05; @Marco05a; @Chen06; @Tropp06] focus on a JSM in which the column vectors $\vecg\alpha^{(j)}$ simply share their support $\mathcal{T}$. This JSM is the special case of our row-block sparsity model with $\mathcal{N}=\mathcal{N}_1$ described in Section \[subsec:formulation\]. Efficient algorithms for the MMV problem with this JSM have been designed as the extensions of greedy algorithms such as matching pursuit (MP) and orthogonal matching pursuit (OMP) [@Pati93; @Davis97; @Tropp04; @Donoho06robust; @Tropp07]. OMP is an efficient algorithm that can recover a $m$-sparse vector from a $\mathcal{O}(m\log N)$-dimensional vector [@Tropp07]. It iteratively selects the basis (column of $\mtr A$) with the largest contribution to the current residual to reduce greedily the representation error at each iteration. The existing MP- and OMP-based algorithms for the MMV problem can be directly used for our problem with the row-block sparsity model only when $\mathcal{N}=\mathcal{N}_1$. Eldar and Mishali [@Eldar08] introduced the block sparsity model and block RIP condition applicable to MMV problems including ours. The uniqueness was guaranteed in \[subsec:formulation\]. By $l^1$ convex relaxation, we can cast the vectorized version in (\[eq:sparse decomposition by minimization vectorized\]) as $$\begin{aligned} && \min_{\mtr A} ||\vectorize(\mtr A^\top)||_{1,\mathcal{N}'} \quad\mbox{subject to}\quad\nonumber\\ &&\qquad\qquad\vectorize(\mtr Q^\top)=(\mtr S\otimes\mtr I_n)\vectorize(\mtr A^\top). \label{eq:sparse decomposition by l1 minimization vectorized}\end{aligned}$$ Here, we redefine $f_{\mathcal{N}}$ as a map from $(\mathbb{R}^{N\times n},l^F)$ to $(\mathbb{R}_{+}^C,l^1)$ in the same form as (\[eq:map to l0 classification space\]), and define [^1] $$||\mtr A||_{1,\mathcal{N}}\defas||f_{\mathcal{N}}(\mtr X)||_1.$$ According to [@Eldar08], this $l^1$ minimization problem is a second order cone problem (SOCP). Sub-optimal algorithm --------------------- $\mtr Q\in\mathbb{R}^{d\times n}$: matrix of $n$ queries as (\[eq:query matrix\]),$\mtr S\in\mathbb{R}^{d\times N}$: concatenated matrix of training datasets as (\[eq:concatenated training datasets\]),$\mathcal{N}$: list of row block sizes,$M_0$: sparsity level; $\mtr A$: row-block sparse matrix as (\[eq:row stacked form\]), $\mathcal{I}$: set of indices of nonzero blocks; let the index set $\mathcal{I}\defas\emptyset$ and residual $\mtr R\defas\mtr Q$; $\mtr U\defas\mtr S^\top\mtr R$;\[step:u\] $\vecg\gamma\defas f_{\mathcal{N}}(\mtr U)$; let $\mathcal{J}$ be a set of indices of the $M_0$ biggest components of $\vecg\gamma$, or all of its nonzero components, whichever set is smaller; sort $\mathcal{J}$ in descending order of the components $\vecg\gamma$; among all subsets $\mathcal{J}_0\subset\mathcal{J}$ such that $\gamma_i\leq 2\gamma_j$ for all $i<j\in\mathcal{J}_0$, choose $\mathcal{J}_0$ with the maximal energy $||\gamma|_{\mathcal{J}_0}||_2^2\defas \displaystyle\sum_{k\in\mathcal{J}_0}\gamma_k^2$; $\mathcal{I}\defas\mathcal{I}\cup\mathcal{J}_0$; $\vecg\alpha^{(j)}\defas\displaystyle\arg\min_{\vecg\alpha}||\vec q^{(j)}-\sum_{k\in\mathcal{I}}\mtr S_k\vecg\alpha||_2$;\[step:lsq\] $\mtr R\defas\mtr Q-\displaystyle\sum_{k\in\mathcal{I}}\mtr S_k\mtr A_k$; $||\mtr R||_F=0$ or $\card\mathcal{I}\geq 2M_0$. (16,10.7) (-0.8,7.7)[{height="3cm"}]{} (-0.8,4.3)[{height="3cm"}]{} (-0.8,0.9)[{height="3cm"}]{} (2,7.7)[{height="3cm"}]{} (9,7.7)[{height="3cm"}]{} (2,4.3)[{height="3cm"}]{} (9,4.3)[{height="3cm"}]{} (2,0.9)[{height="3cm"}]{} (9,0.9)[{height="3cm"}]{} (0.8,7.7)[(a)]{} (0.8,4.3)[(b)]{} (0.8,0.9)[(c)]{} (5.5,0.2)[SSM]{} (12.5,0.2)[SRC]{} We present a practical greedy algorithm of the block sparse decomposition. Although there are optimization packages that solve the SOCP in polynomial time, we prefer a simple and efficient algorithm of the sparse recovery like the MP and OMP. As compared with the signal recovery in compressed sensing, approximate solutions may be enough for the classification purpose. Since the sparsity level is at most $\mathcal{O}(n)$ for $n$ queries, we want the decomposition algorithm to work efficiently in the case of extreme sparseness. We adopt the regularized OMP (ROMP) [@Needell09] because it can stably provide approximate solution from noisy queries. We modify the ROMP to seek for the nonzero row blocks of the solution as shown in Algorithm \[alg:SSD-ROMP\]. This algorithm selects multiple row-blocks of $\mtr S^\top\mtr S\mtr A$ that have comparable magnitudes measured by $f_\mathcal{N}$ at each iteration. Note that the algorithm requires the additional parameter $M_0=\mathcal{O}(M)$ although the solution is insensitive to this parameter. Intensive computations are the matrix multiplication at Step \[step:u\] and the least squares problem at Step \[step:lsq\], which cost $\mathcal{O}(nNd)$ and $\mathcal{O}(nM_0^2d)$ time, respectively. The cost of least squares problem can be reduced to $\mathcal{O}(nM_0d)$ by the conjugate gradient (CG) method as suggested in [@Needell09]. The total running time of Algorithm \[alg:SSD-ROMP\] is $\mathcal{O}(nM_0^2Nd)$ or $\mathcal{O}(nM_0Nd)$ using CG. Experiments {#sec:experiments} =========== We demonstrate our sparse subspace method (SSM) described in Algorithm \[alg:SSM\]. We perform face recognition experiments using a cropped version of the Extended Yale Face Database B [@GeBeKr01; @KCLee05]. The database consists of 2,414 face images of 38 individuals. We randomly select half of the images of each subject for the training dataset ($n_k\approx 32$, $k=1,\dots,38$), and the other half for queries. Each image is expressed as a $d=192\times 168=32,\!256$ dimensional vector storing the grayscale values. #### One-to-one classification Figure \[fig:YaleB redisuals salt papper\] shows examples of one-to-one classification. The SSM tries to answer a class label for a single query. We reduced the dimensionality to $\hat d=1,\!024$ by the Gaussian random projection at Step \[step:dimensionality reduction\] in Algorithm \[alg:SSM\]. We set the block sizes $\mathcal{N}=\{n_1,\dots,n_{38}\}$ and the sparsity level $M_0=4$ in Algorithm \[alg:SSD-ROMP\]. Since SSM behaves as the SRC [@Wright08] when $\mathcal{N}=\mathcal{N}_1$, we also executed the SRC implemented with ROMP. The SSM and SRC, including the random projection, run in less than 0.2 seconds on a moderate workstation. For the valid query image of subject \#16 as Fig. \[fig:YaleB redisuals salt papper\](a), we see that only the residual $r_{16}$ is significantly small. The SSM and SRC stably detect $r_{16}$ as the smallest even if the query is contaminated with noise as shown in Fig \[fig:YaleB redisuals salt papper\](b) before the dimensionality reduction. We also observe in Fig \[fig:YaleB redisuals salt papper\](c) that none of the residuals can be significantly small for the invalid query (taken from the UMIST face database [@UMIST]). In all cases, the residuals tend to be left undisturbed in SSM although the classification results are the same as SRC. This indicates that irrelevant class subspaces are ruled out by the block sparse model. #### $n$-to-one classification For different numbers $n$ of queries, we evaluated the recognition rate of $n$-to-one classifier with respect to reduced feature dimension $\hat d$ by Gaussian random projection. For $\hat d>120$, the recognition rate increases with $n$ and $\hat d$ as shown in Fig. \[fig:YaleB recognition rate vs dim\]. The rate is enhanced to more than 99% at $\hat d>350$ with $n\geq 4$ queries. The perfect classification is achieved at $\hat d>400$ with $n\geq 8$ queries. The $n$-to-one classifier provides better performance than the one-to-one classifier applied to each query, because the $n$-to-one classifier takes advantage of the joint sparsity. However, the SSM did not improve the recognition rate at low dimensions $\hat d<120$. We should cope with this matter in the future work. #### $n$-to-ones classification We also performed the $n$-to-ones classification. Figure \[fig:YaleB multiple face classification\] shows an example using the Extended Yale Face Database B. We gave the classifier five query images, three of which are taken from subject \# 5 and two from \# 29. These five queries are classified into their respective classes indicated by the significantly small residuals. (8,6) (0,0)[![Recognition rates of $n$-to-one classifier on Extended Yale B database, with respect to feature dimension.[]{data-label="fig:YaleB recognition rate vs dim"}](YaleB_smm_rate_dim_rp.eps "fig:"){height="6cm"}]{} (7.8,8.8) (-0.6,0)[![An example of $n$-to-ones classification. Residuals $r_k^{(1)},\dots,r_k^{(5)}$ are shown from top to bottom. Each of $n=5$ queries is classified into one of two classes $k=5$ and $29$.[]{data-label="fig:YaleB multiple face classification"}](SMM_M4_RP1024_residuals_YaleB_id262_266_282_1782_1795_cid5_5_29_29.eps "fig:"){height="9.5cm"}]{} Concluding remarks {#sec:conclusion} ================== We have developed the sparse subspace method (SSM), which enables us to classify multiple queries into their respective classes, simultaneously. The SSM is based on the sparse decomposition of the query subspace. The query subspace is represented only by the relevant class subspaces. Since this sparse decomposition can be cast as the MMV problem with a row-block joint sparsity model, the uniqueness, robustness and recovery of the solution are guaranteed under the block RIP condition. We realized the block sparse decomposition by modifying the greedy algorithm ROMP. We experimentally showed that the classification of multiple queries improves the recognition rate on a face database. The joint sparsity model and the decomposition algorithm should be improved further. More detailed performance evaluation also remains in the future work. [^1]: The norm $||\cdot||_{2,\mathcal{I}}$ defined in [@Eldar08] is the same as our $||\cdot||_{1,\mathcal{N}}$, and it is actually the $l^1$ norm through $f_{\mathcal{N}}$ as we defined.

--- abstract: | The large-scale energy spectrum in two-dimensional turbulence governed by the surface quasi-geostrophic (SQG) equation $$\partial_t(-\Delta)^{1/2}\psi+J(\psi,(-\Delta)^{1/2}\psi) =\mu\Delta\psi+f$$ is studied. The nonlinear transfer of this system conserves the two quadratic quantities $\Psi_1=\langle[(-\Delta)^{1/4}\psi]^2\rangle/2$ and $\Psi_2=\langle[(-\Delta)^{1/2}\psi]^2\rangle/2$ (kinetic energy), where $\langle\cdot\rangle$ denotes a spatial average. The energy density $\Psi_2$ is bounded and its spectrum $\Psi_2(k)$ is shallower than $k^{-1}$ in the inverse-transfer range. For bounded turbulence, $\Psi_2(k)$ in the low-wavenumber region can be bounded by $Ck$ where $C$ is a constant independent of $k$ but dependent on the domain size. Results from numerical simulations confirming the theoretical predictions are presented. author: - 'CHUONGV. TRAN[^1]' - 'JOH NC.BOWMAN' date: 12 May 2004 and in revised form 05 November 2004 title: 'Large-scale energy spectra in surface quasi-geostrophic turbulence' --- Introduction ============ The dynamics of a three-dimensional stratified rapidly rotating fluid is characterized by the geostrophic balance between the Coriolis force and pressure gradient. The nonlinear dynamics governed by the first-order departure from this linear balance is known as quasi-geostrophic dynamics and is inherently three-dimensional. The theory of quasi-geostrophy is interesting and the research performed on this subject constitutes a rich literature (see, for example, Charney 1948, 1971; Rhines 1979; Pedlosky 1987). This theory renders a variety of two-dimensional models that are appealing for their relative simplicity and yet sufficiently sophisticated to capture the underlying dynamics of geophysical fluids. One such model, the so-called surface quasi-geostrophic (SQG) equation, is the subject of the present study. Quasi-geostrophic flows can be described in terms of the geostrophic streamfunction $\psi(\x,t)$. The vertical dimension $z$ is usually taken to be semi-infinite and the horizontal extent may be either bounded or unbounded. Normally, decay conditions are imposed as $z\rightarrow\infty$. At the flat surface boundary $z=0$, the vertical gradient of $\psi(\x,t)$ matches the temperature field $T(\x,t)$, i.e. $T(\x,t)|_{z=0}=\partial_z\psi(\x,t)|_{z=0}$. For flows with zero potential vorticity, this surface temperature field can be identified with $(-\Delta)^{1/2}\psi$, where $\Delta$ is the (horizontal) two-dimensional Laplacian. Here, the operator $(-\Delta)^{1/2}$ is defined by $(-\Delta)^{1/2}\widehat\psi(\k)=k\widehat\psi(\k)$, where $k=|\k|$ is the wavenumber and $\widehat\psi(\k)$ is the Fourier transform of $\psi(\x)$. The conservation equation governing the advection of the temperature $(-\Delta)^{1/2}\psi$ by the surface flow is (Blumen 1978; Pedlosky 1987; Pierrehumbert, Held & Swanson 1994; Held 1995) $$\begin{aligned} \label{Tadvection} \partial_t(-\Delta)^{1/2}\psi+J(\psi,(-\Delta)^{1/2}\psi)&=&0,\end{aligned}$$ where $J(\varphi,\phi)=\partial_x\varphi\partial_y\phi -\partial_y\varphi\partial_x\phi$. This equation is known as the SQG equation. In this paper a forced-dissipative version of (\[Tadvection\]) is studied. A dissipative term of the form $\mu\Delta\psi$, where $\mu>0$, which results from Ekman pumping at the surface, is considered (Constantin 2002; Tran 2004). Since $(-\Delta)^{1/2}\psi$ is the advected quantity, this physical dissipation mechanism corresponds to the (hypoviscous) dissipation operator $\mu(-\Delta)^{1/2}$. The dissipation coefficient $\mu$ has the dimension of velocity and is not vanishingly small in the atmospheric context (Constantin 2002). The system is assumed to be driven by a forcing $f$, for which the spectral support is confined to wavenumbers $k\ge s>0$ (in bounded turbulence, wavenumber zero is replaced by the minimum wavenumber). Thus, the forced-dissipative SQG equation can be written as $$\begin{aligned} \label{governing} \partial_t(-\Delta)^{1/2}\psi+J(\psi,(-\Delta)^{1/2}\psi) &=&\mu\Delta\psi+f.\end{aligned}$$ It is customary in the classical theory of turbulence to consider a doubly periodic domain of size $L$; the unbounded case is obtained [*via*]{} the limit $L\rightarrow\infty$. The Jacobian operator $J(\cdot,\cdot)$ admits the identities $$\begin{aligned} \label{id} \langle\chi J(\varphi,\phi)\rangle=-\langle\varphi J(\chi,\phi)\rangle =-\langle\phi J(\varphi,\chi)\rangle,\end{aligned}$$ where $\langle\cdot\rangle$ denotes the spatial average. As a consequence, the nonlinear term in (\[governing\]) obeys the conservation laws $$\begin{aligned} \label{conservation} \langle\psi J(\psi,(-\Delta)^{1/2}\psi)\rangle= \langle(-\Delta)^{1/2}\psi J(\psi,(-\Delta)^{1/2}\psi)\rangle=0.\end{aligned}$$ It follows that the two quadratic quantities $\Psi_\theta=\langle |(-\Delta)^{\theta/4}\psi|^2\rangle/2=\int\Psi_\theta(k)\,\dk$, where $\theta=1,2$, are conserved by nonlinear transfer. Here, $\Psi_\theta(k)$ is defined by $\Psi_\theta(k)=k^\theta\Psi(k)$, $\Psi(k)$ is the power density of $\psi$ associated with wavenumber $k$ and $\theta$ is a real number. Note that $\Psi_2(k)$ is the kinetic energy spectrum and $\Psi_2$ is the kinetic energy density. The simultaneous conservation of two quadratic quantities by advective nonlinearities is a common feature in incompressible fluid systems in two dimensions. Some familiar systems in this category are the Charney–Hasegawa–Mima equation (Hasegawa & Mima 1978; Hasegawa, Maclennan & Kodama 1979) and the class of $\alpha$ turbulence equations (Pierrehumbert 1994), which includes both the Navier–Stokes and the SQG equations. These conservation laws, together with the scale-selectivity of the dissipation and unboundedness of the domain, are the building block of the classical dual-cascade theory (Fj[ø]{}rtoft 1953; Kraichnan 1967, 1971; Leith 1968; Batchelor 1969). This theory, when applied to the present case, implies that $\Psi_1$ cascades to low wavenumbers (inverse cascade) and $\Psi_2$ cascades to high wavenumbers (direct cascade). For some recent discussion on the possibility of a dual cascade in various two-dimensional systems, including the Navier–Stokes and SQG equations, see [@TS02], Tran & Bowman (2003b,2004) and [@T04]. The inverse cascade toward wavenumber $k=0$ would eventually evade viscous dissipation altogether because the spectral dissipation rate vanishes as $k\rightarrow0$. Hence, according to the classical picture, $\Psi_1$ necessarily grows unbounded, by a steady growth rate $\d\Psi_1/\dt>0$, as $t\rightarrow\infty$. Strictly speaking, one may have to address the possibility of a dissipated inverse cascade, i.e. one for which the dissipation of $\Psi_1$ occurs at scales much larger than the forcing scale and for which $\d\Psi_1/\dt$ has a zero time mean. Such a cascade is not a plausible scenario (and is not the traditional undissipated inverse cascade) in fluid systems, dissipated by a single viscous operator, where the viscous dissipation rate diminishes toward the large scales. A discussion of this issue can be found in [@T04]. In this study, upper bounds are derived for the time averages of the kinetic energy density $\Psi_2$ and of the large-scale spectrum $\Psi_2(k)$. These bounds are derived from the governing equation, involving simple but rigorous estimates. The bound on $\Psi_2$ is valid in both unbounded and bounded cases, and a straightforward consequence of this bound is a bound on the energy spectrum, which also applies to both unbounded and bounded turbulence. Another bound on the large-scale energy spectrum is derived by estimating the nonlinear triple-product term representing the inverse transfer of $\Psi_1$. This result applies to bounded turbulence since upper bounds for the triple-product term are inherently domain-size dependent. The difficulties of extending this result to the unbounded case are discussed. Some numerical results confirming the theoretical predictions are presented. Bounded dynamical quantities ============================ A notable feature of unbounded incompressible fluid turbulence in two dimensions is the appearance of infinite quadratic quantities (per unit area): namely, the kinetic energy density $\Psi_2$ for Navier–Stokes turbulence and $\Psi_1$ for the SQG case. According to the classical theory (applied to the SQG case), a (steady) injection of $\Psi_1$, applied around some finite wavenumber $s$, cascades to ever-larger scales, leading to an unbounded growth of $\Psi_1$ (this is presumably the case for the general quadratic invariant $\Psi_\alpha$ in the so-called $\alpha$ turbulence; [*cf.*]{} Tran 2004). In other words, if the classical inverse cascade is realizable, unbounded incompressible fluid turbulence in two dimensions constitutes an ill-posed problem, in the sense that a key quadratic invariant becomes infinite. Of course, there still exist finite quadratic quantities, in particular the dissipation agent for each quadratic invariant. This section is concerned with these quantities. On multiplying (\[governing\]) by $\psi$ and $(-\Delta)^{1/2}\psi$ and taking the spatial average of the resulting equations, noting from the conservation laws that the nonlinear terms identically vanish, one obtains evolution equations for $\Psi_1$ and $\Psi_2$, $$\begin{aligned} \label{Psi1evolution} \frac{\d}{\dt}\Psi_1&=&-2\mu\Psi_2+\langle f\psi\rangle,\\ \label{Psi2evolution} \frac{\d}{\dt}\Psi_2&=&-2\mu\Psi_3 +\langle f(-\Delta)^{1/2}\psi\rangle.\end{aligned}$$ Using the Cauchy–Schwarz and geometric–arithmetic mean inequalities, one obtains upper bounds on the injection terms in (\[Psi1evolution\]) and (\[Psi2evolution\]): $$\begin{aligned} \label{forcebounds} \langle f\psi\rangle&\le&\langle|(-\Delta)^{1/2}\psi|^2\rangle^{1/2} \langle|(-\Delta)^{-1/2}f|^2\rangle^{1/2} \le\mu\Psi_2+\mu^{-1}F_{-2},\nonumber\\ \langle f(-\Delta)^{1/2}\psi\rangle &\le& \langle|(-\Delta)^{3/4}\psi|^2\rangle^{1/2} \langle|(-\Delta)^{-1/4}f|^2\rangle^{1/2} \le\mu\Psi_3+\mu^{-1}F_{-1},\end{aligned}$$ where the ‘integration by parts’ rule $\langle(-\Delta)^\theta\phi \chi\rangle=\langle(-\Delta)^{\theta'}\phi(-\Delta)^{\theta''}\chi\rangle$, for $\theta=\theta'+\theta''$, has been used and $F_\theta=\langle |(-\Delta)^{\theta/4}f|^2\rangle/2$. Substituting (\[forcebounds\]) in (\[Psi1evolution\]) and (\[Psi2evolution\]) yields $$\begin{aligned} \label{evolbound1} \frac{\d}{\dt}\Psi_1&\le&-\mu\Psi_2+\mu^{-1}F_{-2},\\ \label{evolbound2} \frac{\d}{\dt}\Psi_2&\le&-\mu\Psi_3+\mu^{-1}F_{-1}.\end{aligned}$$ To avoid unnecessary complications, zero initial conditions are assumed, so that for $T>0$ the time means $\langle\d\Psi_1/\dt\rangle_t= \Psi_1(T)/T$ and $\langle\d\Psi_2/\dt\rangle_t$ are non-negative. One can then deduce upper bounds on the time means $\langle\Psi_2\rangle_t$ and $\langle\Psi_3\rangle_t$, which are valid regardless of whether or not $\Psi_1$ remains finite in the limit $t\rightarrow\infty$: $$\begin{aligned} \label{averagebound1} \langle\Psi_2\rangle_t &\le& \mu^{-2}\langle F_{-2}\rangle_t,\\ \label{averagebound2} \langle\Psi_3\rangle_t &\le& \mu^{-2}\langle F_{-1}\rangle_t.\end{aligned}$$ For $\theta\in(2,3)$, $\langle\Psi_\theta\rangle_t$ is also bounded. Indeed, from the Hölder inequalities $\Psi_\theta\le\Psi_2^{3-\theta}\Psi_3^{\theta-2}$ ([*cf.*]{} Tran 2004) and $\langle\Psi_2^{3-\theta}\Psi_3^{\theta-2}\rangle_t\le \langle\Psi_2\rangle_t^{3-\theta}\langle\Psi_3\rangle_t^{\theta-2}$, one can deduce from (\[averagebound1\]) and (\[averagebound2\]) that $$\begin{aligned} \langle\Psi_\theta\rangle_t &\le& \langle\Psi_2\rangle_t^{3-\theta}\langle\Psi_3\rangle_t^{\theta-2} \le \mu^{-2} \langle F_{-2}\rangle_t^{3-\theta}\langle F_{-1}\rangle_t^{\theta-2}.\end{aligned}$$ This result implies that for $\theta\in(2,3)$, $\langle\Psi_\theta\rangle_t$ is bounded, provided that both $\langle F_{-1}\rangle_t$ and $\langle F_{-2}\rangle_t$ are bounded. This condition is assured if $s>0$ and $F_0$ is bounded, a condition normally required of the forcing, because $F_{-2}\le F_{-1}/s\le F_0/s^2$. One may even consider a class of forcing for which $F_0=\infty$ and $F_{-2}\le F_{-1}/s<\infty$. Upper bounds of the above type on dynamical quantities are rather trivial for bounded turbulence. However, they are important in the unbounded case, for two reasons. First, the scale-selective viscous dissipation allows for the possibility of unbounded growth of certain quadratic quantities toward the low wavenumbers. Hence, rigorous bounds on dynamical quantities are not as abundant as in the bounded case. Second, analytic studies of the nonlinear triple-product transfer function are difficult in unbounded domains. In the absence of pointwise estimates for the spectrum, these bounds are particularly useful for qualitative estimates of the large-scale distribution of energy. For example, [@T04] uses inequality (\[averagebound1\]) to argue that the energy spectrum $\Psi_2(k)$ should be shallower than $k^{-1}$, as $k\rightarrow0$. Large-scale energy spectrum =========================== In this section, it is shown that the physical laws of SQG dynamics admit only large-scale energy spectra shallower than $k^{-1}$. This result is due in part to the fact that the simultaneous conservation of $\Psi_1$ and $\Psi_2$ allows virtually no kinetic energy to get transferred toward the low wavenumbers, so that only large-scale kinetic energy spectra shallower than $k^{-1}$ are possible. Shell-averaged energy spectrum ------------------------------ For a given wavenumber $r$, let us denote by $S=S(r)$ the wavenumber shell between $k=r/2$ and $k=3r/2$, i.e. $S(r)=\{\k : r/2 \le k \le 3r/2\}$. The shell-averaged energy spectrum $\overline\Psi_2(r)$ over $S(r)$ is defined by $$\begin{aligned} \label{spectrum} \overline\Psi_2(r)&=&\frac{1}{r}\int_{r/2}^{3r/2}\Psi_2(k)\,\dk.\end{aligned}$$ In the present case of a doubly periodic domain of size $L$, the Fourier representation of the stream function is $\psi(\x)=\sum_{\k}\exp\{i\k\cdot\x\}\widehat\psi(\k)$, where $\k=2\pi L^{-1}(k_x,k_y)$ with $k_x$ and $k_y$ being integers not simultaneously zero. Let $\psi(S)$ denote the component of $\psi$ spectrally supported by $S$, i.e. $\psi(S)=\sum_{\k\in S}\exp \{i\k\cdot\x\}\widehat\psi(\k)$. One has $$\begin{aligned} \label{ineq} \sup_{\x}|\nabla\psi(S)| &\le& \sum_{\k\in S}k|\widehat\psi(\k)| \le \left(\sum_{\k\in S}1\sum_{\k\in S}k^2|\widehat\psi(\k)|^2\right)^{1/2} \le cLr\Psi_2^{1/2}(S),\end{aligned}$$ where the Cauchy–Schwarz inequality is used, the sum $\sum_{\k\in S}1=(cLr)^2$ is the number of wavevectors in $S$, $c$ is an absolute constant of order unity and $\Psi_2(S)$ is the contribution to the kinetic energy from $S$. Upper bounds for the energy spectrum ------------------------------------ A simple upper bound for $\overline\Psi_2(k)$, which is applicable to both the unbounded and bounded cases, can be derived from (\[averagebound1\]). In fact, it follows from (\[averagebound1\]) and (\[spectrum\]) that $$\begin{aligned} \label{spectbound1} \langle\overline\Psi_2(k)\rangle_t&=&\frac{1}{k}\int_{k/2}^{3k/2} \langle\Psi_2(\kappa)\rangle_t\,\d\kappa \le \mu^{-2}\langle F_{-2}\rangle_tk^{-1}.\end{aligned}$$ This bound is supposed to apply to $k$ in the inverse-transfer region. For $k$ in the direct-transfer region, (\[averagebound2\]) yields $$\begin{aligned} \label{spectbound2} \langle\overline\Psi_2(k)\rangle_t&=&\frac{1}{k}\int_{k/2}^{3k/2} \langle\Psi_2(\kappa)\rangle_t\,\d\kappa \le \frac{2}{k^2}\int_{k/2}^{3k/2} \langle\Psi_3(\kappa)\rangle_t\,\d\kappa \le 2\mu^{-2}\langle F_{-1}\rangle_tk^{-2}.\end{aligned}$$ The upper bound (\[spectbound1\]) suggests that dimensional analysis arguments, which predict a large-scale $k^{-1}$ energy spectrum, are not well justified. If a persistent inverse cascade of $\Psi_1$ exists ($\d \Psi_1/\dt > 0$), then the energy $\Psi_2$ ought to acquire a value such that $\Psi_2<\mu^{-2}F_{-2}$. In the unbounded case, the large-scale energy spectrum then needs to be strictly shallower than $k^{-1}$, to ensure that the dissipation of $\Psi_1$ does not increase without bound as the inverse cascade proceeds toward $k=0$. On the other hand, if no inverse cascade of $\Psi_1$ exists, then a $k^{-1}$ energy spectrum with limited extent is possible. If viscous dissipation mechanisms with degrees higher than that of the natural dissipation are considered, then the upper bounds derived above are not valid. Nevertheless, diminishing energy transfer towards the lowest wavenumbers appears to be consistent only with spectra shallower than $k^{-1}$ (for low-wavenumber convergence of the energy integral). The numerical results reported in §4 are well suited to this expectation. An upper bound for the large-scale energy spectrum, based on the nonlinear transfer term, can be derived for the bounded case. This analysis employs elementary but rigorous estimates of the triple-product term. For $3k/2<s$, the evolution of $\Psi_1(S(k))$ is governed by $$\begin{aligned} \frac{\d}{\dt}\Psi_1(S)&=&-\langle\psi(S) J(\psi,(-\Delta)^{1/2}\psi)\rangle-2\mu\Psi_2(S) \nonumber\\ &=&\langle(-\Delta)^{1/2}\psi J(\psi,\psi(S))\rangle-2\mu\Psi_2(S)\nonumber\\ &\le&\langle|(-\Delta)^{1/2}\psi||\nabla\psi||\nabla\psi(S)|\rangle -2\mu\Psi_2(S)\nonumber\\ &\le&\sup_{\x}|\nabla\psi(S)|\langle|(-\Delta)^{1/2}\psi||\nabla\psi|\rangle -2\mu\Psi_2(S)\nonumber\\ &\le&2cLk\Psi_2^{1/2}(S)\Psi_2-2\mu\Psi_2(S)\nonumber\\ &\le&c^2\mu^{-1}L^2k^2\Psi_2^2-\mu\Psi_2(S)\nonumber\\ &=&c^2\mu^{-1}L^2k^2\Psi_2^2-\mu k\overline\Psi_2(k),\end{aligned}$$ where the second equality is a consequence of (\[id\]) and the second last and last inequalities follow from (\[ineq\]) and the geometric–arithmetic mean inequality, respectively. It follows that $$\begin{aligned} \label{spectbound3} \langle\overline\Psi_2(k)\rangle_t&\le&c^2\mu^{-2}L^2k\langle\Psi_2^2\rangle_t.\end{aligned}$$ A notable feature of (\[spectbound3\]) is its dependence on the fluid domain size. The presence of $L$ in this upper bound is natural: the upper bound $\sup_{\x}|\nabla\psi(S)|$, which is associated with the fluid velocity at scales $\approx k^{-1}$, is inherently domain-size dependent. There are no known analytic estimates that allow one to derive an upper bound on the nonlinear transfer function $\langle\psi(S)J(\psi,(-\Delta)^{1/2}\psi)\rangle$ in terms of ‘intensive quantities’ only. This difficulty arises not only in the present estimate but also in other analytic estimates of the transfer function. In other words, the nonlinear triple-product term is intrinsically domain-size dependent. This problem considerably limits our ability to assess the nonlinear transfer in unbounded systems. Finally, it is worth mentioning that although the upper bound (\[spectbound3\]) has a linear dependence on $k$, it may be more excessive than the bound $\mu^{-2}\langle F_{-2}\rangle_t k^{-1}$ derived earlier (even for very low wavenumbers). The reason is that $L^2k\ge k^{-1}$ and the prefactor $c^2\langle\Psi_2^2\rangle_t$ may not be as optimal as $\langle F_{-2}\rangle_t$. Numerical results ================= This section reports results from numerical simulations that illustrate the realization of large-scale spectra shallower than $k^{-1}$. Equation (\[governing\]) is simulated in a doubly periodic square of side $2\pi$, where the forcing $\widehat{f}(\bm k)$ is nonzero only for those wavevectors $\bm k$ having magnitudes lying in the interval $K=[59,61]$: $$\begin{aligned} \label{forcing} \widehat{f}(\bm k)&=&\frac{\epsilon}{N}\frac{\widehat{\psi}(\bm{k})} {2\Psi_1(k)}.\end{aligned}$$ Here $\epsilon=1$ is the constant energy injection rate and $N$ is the number of distinct wavenumbers in $K$. The (constant) injection rate of $\Psi_1$ is $\epsilon/s\approx1/60$, where $1/s\approx1/60$ is the mean of $k^{-1}$ over $K$. This type of forcing was used by [@Shepherd87], [@T04] and [@TB04] in numerical simulations of a large-scale zonal jet on the so-called beta-plane and of Navier–Stokes turbulence. The attractive aspect of (\[forcing\]), as noted in [@Shepherd87], is that it is steady. Dealiased $683^2$ and $1365^2$ pseudospectral simulations ($1024^2$ and $2048^2$ total modes) were performed. Three dissipative forms were considered: $2.5\times10^{-2}\Delta\psi$, $-4\times10^{-4}(-\Delta)^{3/2}\psi$, and $-6\times10^{-6}\Delta^2\psi+\mu\Delta\psi$ for several values of $\mu$. The first case represents the natural dissipation of the SQG dynamics due to Ekman pumping, as mentioned earlier. The second case represents thermal (molecular) diffusion since $(-\Delta)^{1/2}\psi$ is equivalent to the fluid temperature. The third case—the mixed hyperviscous/Ekman dissipation form—is considered in order to demonstrate that even slight amounts of Ekman damping will inhibit the formation of an inverse cascade. Unlike [@Smith02], the case of mechanical friction \[$\propto(-\Delta)^{1/2}\psi$\] was not considered. The higher resolution was used for the first (natural dissipation) case and the lower resolution was used for the second and third cases. All simulations were initialized with the spectrum $\Psi_2(k)=10^{-5}\pi k/(60^2+k^2)$. Figure \[sqg2\] shows the time-averaged steady-state kinetic energy spectrum for the case of the natural dissipation term $2.5\times10^{-2}\Delta\psi$. The dissipation agents of $\Psi_1$ and $\Psi_2$ are, respectively, $\Psi_2$ (energy) and $\Psi_3$. The value of the energy, $0.3333$, implies that the dissipation of $\Psi_1$, averaged in the same period, is $0.01666$. This amounts to virtually all of the injection rate $1/60$. Hence, there exists no inverse cascade of $\Psi_1$ to the large scales and both $\Psi_1$ and $\Psi_2$ are steady. The small-scale energy spectrum scales as $k^{-3.5}$, so that the spectrum $\Psi_3(k)$ scales as $k^{-2.5}$. This scaling means that the energy dissipation occurs mainly around the forcing region and is consistent with the bound (\[spectbound2\]). Unlike Navier–Stokes turbulence, for which the inverse energy cascade is robust and can be simulated at relatively low resolution, it was noticed that no choice for the value of $\mu$ at the present resolution could be used to simulate an inverse cascade of $\Psi_1$. It is not known whether an inverse cascade of $\Psi_1$ is realizable at higher resolutions, using a smaller value of $\mu$. Nevertheless, this observation suggests that $\Psi_1$ is ‘reluctant’ to cascade to the large scales, as compared with the more robust inverse energy cascade in Navier–Stokes turbulence. ![The time-averaged steady-state energy spectrum $\Psi_2(k)$ [*vs.*]{}  $k$ for the dissipation term $2.5\times10^{-2}\Delta\psi$.[]{data-label="sqg2"}](sqg2) Figure \[sqgviscous1\] shows the kinetic energy spectrum averaged between $t=37.3$ and $t=38.7$, for a lower viscous degree. The dissipation agents of $\Psi_1$ and $\Psi_2$ are, respectively, $\Psi_3$ and $\Psi_4$ (enstrophy). The value of $\Psi_3$ is $20$, implying that the dissipation of $\Psi_1$ is $1.6\times10^{-2}$. This amounts to about $96\%$ of the injection rate $1/60$. The inverse cascade then carries only a few percent of the injection of $\Psi_1$ to the large scales. The small-scale energy spectrum scales as $k^{-4.5}$, so that the enstrophy spectrum $\Psi_4(k)$ scales as $k^{-2.5}$. Most of the energy dissipation occurs around the forcing region, consistent with a ‘weak’ inverse cascade (one that does not carry virtually all of the injection of $\Psi_1$ toward $k=0$; [*cf.*]{} Tran and Bowman 2004, Tran 2004). No direct cascade is possible for bounded turbulence in equilibrium or for unbounded turbulence in the presence of a weak inverse cascade. ![The quasisteady energy spectrum $\Psi_2(k)$ [*vs.*]{} $k$ averaged between $t=37.3$ and $t=38.7$ for the dissipation term $-4\times10^{-4}(-\Delta)^{3/2}\psi$.[]{data-label="sqgviscous1"}](sqgviscous1) Similarly, Figure \[sqgviscous2mixed\] shows the kinetic energy spectrum averaged between $t=15.7$ and $t=16.5$ for the mixed dissipation $-6\times10^{-6}\Delta^2\psi+\mu\Delta\psi$, using three different values of $\mu$. When $\mu=0$, the dissipation agents of $\Psi_1$ and $\Psi_2$ are, respectively, $\Psi_4$ (enstrophy) and $\Psi_5$. The value of the enstrophy, $1208$, implies that the dissipation of $\Psi_1$ is $1.45\times10^{-2}$, amounting to about $87\%$ of the injection rate $1/60$. The small-scale energy spectrum scales as $k^{-5}$, so that $\Psi_5(k)$ scales as $k^{-2}$. Again, this scaling means that most of the energy dissipation occurs around the forcing region and that the inverse cascade is weak. We note that as $\mu$ is increased, the inverse cascade becomes increasingly weak. We emphasize this behaviour by plotting in Fig. \[invstrengthvnuL\] the inverse cascade [*strength*]{} $r=1-2s(\mu\Psi_2+6\times10^{-6}\Psi_4)/\epsilon$ for six different values of $\mu$. ![The quasisteady energy spectrum $\Psi_2(k)$ [*vs.*]{} $k$ averaged between $t=15.67$ and $t=16.52$ for the dissipation term $-6\times10^{-6}\Delta^2\psi+\mu\Delta\psi$, using three different values of $\mu$. []{data-label="sqgviscous2mixed"}](sqgviscous2mixed) ![The decay of the inverse cascade strength $r$ for the dissipation term $-6\times10^{-6}\Delta^2\psi+\mu\Delta\psi$ as $\mu$ is increased.[]{data-label="invstrengthvnuL"}](invstrengthvnuL) Unlike Navier–Stokes turbulence, for which the enstrophy acquires its near-equilibrium value once a discernible inverse-transfer range has formed, the energy in SQG turbulence can remain significantly less than its equilibrium value until a very wide inverse-transfer range has developed. For example, for a one-decade Navier–Stokes inverse-transfer range (achievable in numerical simulations), the enstrophy acquires 95% of its projected equilibrium value (calculated with a $k^{-5/3}$ energy spectrum extrapolated to $k=0$). On the other hand, for a one-decade SQG inverse-transfer range, the energy acquires only 66% of its projected equilibrium value (calculated with a $k^{-0.7}$ energy spectrum extrapolated to $k=0$, as realized in the present simulations; [*cf.*]{} the $\mu=0$ case of Figure \[sqgviscous2mixed\]). This means that one needs a considerably wider inverse-transfer region for SQG turbulence than for Navier–Stokes turbulence, in order to approach a quasi-steady state. This problem is in addition to the resolution limitations at the small scales for both cases. Due to the steep spectrum in the inverse-transfer region, the energy in the $\mu=0$ case of Figure \[sqgviscous2mixed\] has not acquired a value considerably close to its equilibrium value. This means that the system is still well within the transient phase, However, the dissipation of $\Psi_1$ (proportional to the enstrophy) cannot grow considerably (without significant change to the existing spectrum), because of the high degree of viscosity, which makes the dissipation of $\Psi_1$ relatively insensitive to growth of the large-scale energy. Conclusion and discussion ========================= In this paper, the kinetic energy density of SQG turbulence and its large-scale spectrum have been studied. For the unbounded case, upper bounds are derived for the time means of the kinetic energy density and of the large-scale energy spectrum, averaged over a narrow window of wavenumbers. Another result is an upper bound on the the time mean of the large-scale energy spectrum, which is derived for the bounded case. Numerical results confirming the predicted slopes of the large-scale energy spectrum are presented and discussed. An important feature in SQG turbulence that gives rise to the rigorous upper bound on the time mean of the kinetic energy density in the unbounded case is that the kinetic energy is the dissipation agent of the inverse-cascading candidate $\Psi_1$. This fact is due to the hypoviscous nature of the dissipation operator $(-\Delta)^{1/2}$, a natural physical dissipation mechanism of SQG dynamics (Ohkitani 1997; Constantin 2002; Tran 2004). If $(-\Delta)^{1/2}$ is replaced by an operator of the form $(-\Delta)^{\eta}$, where $\eta>1/2$, then the simple analysis of Section 2 fails to show that the time mean of the energy density $\langle\Psi_2\rangle_t$ is bounded, although it may remain so for low degrees of viscosity $\eta$. The reason is that the amount of energy getting transferred to wavenumbers lower than a given wavenumber $k$ decreases at least as rapidly as $k$, so that the spectral dissipation rate $\propto k^{2\eta}$, a consequence of the dissipation operator $(-\Delta)^{\eta}$, may be sufficiently strong to balance the diminishing inverse energy transfer and keep the energy from growing unbounded. Numerical simulations of SQG turbulence were performed, using the natural dissipation operator $(-\Delta)^{1/2}$ and two viscous operators $\Delta$ and $(-\Delta)^{3/2}$. The results show large-scale energy spectra shallower than $k^{-1}$, consistent with the theoretical prediction. There have been attempts to explain, within the context of SQG turbulence (Constantin 2002; Tung & Orlando 2003), the kinetic energy spectra observed in the laboratory experiment of [@Baroud02] and in the atmosphere. In the former case, the turbulence in a rotating tank is driven at a sufficiently small scale to allow for a wide inverse-transferring range. A $k^{-2}$ spectrum extending over nearly two wavenumber decades lower than the forcing wavenumber is observed. In the latter case, a $k^{-5/3}$ spectrum is observed in the mesoscales (see Frisch 1995 and Tung & Orlando 2003 and references therein), which correspond to wavenumbers higher (lower) than the forcing wavenumber if the energy released from baroclinic instability (thunderstorms) is considered to be the driving force. The $-2$ power-law scaling observed in [@Baroud02] for the wavenumber range lower than the forcing wavenumber is excessively steeper than the permissible scalings derived in this work. 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--- abstract: 'Several theorems are demonstrated that determine the sufficient conditions for the existence of synchronized states (periodical and chaotic) and also of travelling waves in a CML. Also are analytically proven the existence of period-doubling cascades for the mentioned patterns. The temporal state of any oscillators are completely characterized. The given results are valid for a number of arbitrary oscillators whose individual dynamics is ruled by an arbitrary $C^{2}$ function.' author: - 'Mª Dolores Sotelo Herrera${}^{a}$ & Jesús San Martín${}^{a,b}$' date:   title: 'An Analytical Study in Coupled Map Lattices of Syncronized States and Travelling Waves, and of their Period-Doubling Cascades' --- ${}^{a}$ Departamento de Matemática Aplicada, E.U.I.T.I., Universidad Politécnica de Madrid. Ronda de Valencia 3, 28012 Madrid Spain\ ${}^{b}$ Departamento de Física Matemática y de Fluidos, U.N.E.D. Senda del Rey 9, 28040 Madrid Spain\ Corresponding author: jsm@dfmf.uned.es Systems showing patterns as a consequence of the interaction among their diverse components are really frequent, in any field that one can imagine: neuronal activity within the brain, or the function of organs as a whole within the body, drivers on a motorway, birds flying in a group, a network of computers, coupled lasers, crystal growth, etc. The result of the interaction of the individual elements generates structures that manifest in the system as a whole. In these processes, one should consider two things: the behavior of any individual and the interaction among them. If we consider the traffic example, it is clear that the behavior of an individual driver, that is his decision to drive in a particular way or another, is certainly different when there are few cars on a motorway or when there is a traffic jam (in which case he will be guided by traffic patterns). Broadly speaking, all of these systems consist of a group of elements coupled by some kind of process, and at the same time, every element of the group is ruled by its own local dynamics. The understanding of such systems is extraordinarily complicated, since there are no particular mathematical tools developed to study them. One way to confront this problem is to discretize spatial and temporal variables as well as to fix inter-individual interactions as well as the individual dynamics. The result is a Coupled Map Lattice (CML) [@libro; @kaneko]: a chain of coupled elements (called oscillators), each situated on a discrete point of the lattice, whose individual dynamics is ruled by a discrete map. Despite the spatial and temporal variables are discretized, state variables remain continuous. In the last few years, CML have been extensively studied since the work of Kaneko and colaborators [@Kaneko89; @Kaneko90a; @Kaneko90b; @kaneko91a; @kaneko91b], and from the beginning, they have shown themselves to be exceptional modelling spatially extended systems. The use of this study has been extended into diverse scientific branches with an extraordinary variety of applications in physics, biology, chemistry, social sciences, and engineering modeling. [@PhysicaD; @Chaos]   A typical evolution equation for a CML [@libro; @kaneko] is given by  $$X_{i}(n+1)=(1-\alpha)f(X_{i}(n))+\frac{\alpha}{m}\sum_{j=1}^{m}f(X_{j}(n))\label{eq:uno}$$ $$i=1,...,m$$ where $X_{i}(n)$ represents the state of the oscilator located at node “i” of the lattice, in the instant “n”. The parameter $\alpha$ weights the coupling among oscilators. Periodic conditions are assumed in the boundaries, given as $$X_{i}(n)=X_{i+m}(n)\;\;\forall i$$ Depending on the value of $\alpha$ , the system behavior changes from the independent evolution of each oscilator (for $\alpha=0$) up to a mean field approach (for $\alpha=1$). For intermediate values $0<\alpha<1$ the system is ruled by both local and global mechanisms. The general form of the coupling term is given by$$\frac{\alpha}{m}\sum_{j=1}^{m}w_{ij}f(X_{j}(n))$$ where the $w_{ij}$ measure the weights between the $j$-th oscilator and the $i$-th one. To achieve a symmetrical and spatially invariant coupling, it is usually taken $w_{ij}=\bar{w}_{\vert i-j\vert}$. Sometimes, the coupling term will be written as $$\frac{\alpha}{m}\sum_{j=1}^{m}f(X_{j}(n))$$ (mean field), or $$\frac{1}{2}\left[f(X_{j-1}(n))+f(X_{j+1}(n))\right]$$ (nearest-neighbor coupling). However, this last description is not adequate when we are dealing with a supercritical bifurcation threshold, because the coherence lengths are usually quite large [@Chate1988]. Given that, in this paper, we want to study bifurcations in CML, we will use the mean field approach. Another important point, that must be considered, is the updating of oscillators; they can be synchronous (all oscillators are updated simultaneously) or asynchronous (oscillators are updated one at a time) [@Atmans1; @Mehta]. Choosing one or the other depends on whether oscillators communicate among them much quicker that the updating time of the system as a whole, which is ruled by the evolution equation . In this paper we will refer to synchronous systems. In the scientific literature, the majority of the results, referring to CML, are numerical results, as we will see later. The awesome richness of numerical results is restricted by a fixed and finite set of parameter values, and a finite number of oscillators in the CML, which supposes a limitation for adequate understanding of certain phenomena. In particular, the transition to chaos by period duplication needs the period to tend to infinity. It is also necessary the number of oscillators to be infinite, in a finite region, for the understanding of the onset of turbulence in fluids and plasmas; otherwise, there would be a cutoff in the wave numbers that could be studied because the lattice would have a finite spatial resolution. Mathematical proofs would be desirable to characterize syncronized states, traveller wave bifurcations and other behaviours. Fortunately, numerical results point out us what to look for and where. In this paper, analytical proofs, in CML, of the existence of syncronized states and travelling waves will be given. It will be proved that both patterns will go under a period doubling cascade as $f$ , in , does. These behaviours will be completely characterized, giving analytical expressions of the temporal evolution of every oscillator. The fixed points of CML, generated in period-doubling cascades, will be essentially the fixed points of $f^{m2^{k}}$($m$ number of oscillators in CML). As $f$ determines the individual dynamics (see (\[eq:uno\])), what is shown is the emergence of global properties from the local dynamics of a single oscilator. We have tried to keep the widest generality in the results; therefore, theorems have been proved using an arbtirary $C^{2}$ function $f(x;r)$, instead of working with the logistic equation (or any toplogically conjugated functions) as usual. Perturbative methods will be used to obtain analytical solutions. The inversion of functional matrices of arbitrary size is fundamental in the proofs of the theorems; given that whenever the inverse matrix exists, it is unique, it will not be necessary to explain the calculation leading to it: it will be enough to check that the proposed matrix (in the corresponding theorem) is the inverse matrix one was looking for. The matrices appearing during the demostration process will not be circulant; therefore, usual analytical inversion processes of circulant matrix inversion will not be valid. This paper is organized as follows. First, synchronized states will be considered, this solution being quite straightforward, it will indicate how to face up to the more complicated travelling waves in the next section. Both results will be used to study the period-doubling cascades of the patterns. The paper concludes with a section indicating connections of this work with other researchs. Regular and chaotic synchronization =================================== In this section straightforward analytical results will be presented for synchronization in CML, that is, for all the oscilators having the same value at anytime. This is a striking behaviour, in particular when chaotic syncronization is produced, where chaotic systems are very sensitive to perturbations and it is supposed that any slight modification generated by the coupling of the oscillators of CML would destroy the synchronization. The mathematical approach to this problem is far from being unique [@Anteneodo].   Let $$X_{i}(n+1)=(1-\alpha)f(X_{i}(n))+\frac{\alpha}{m}\sum_{i=1}^{m}f(X_{i}(n))\:\;\quad i=1,\,\dots,\, m\label{eq:tres}$$ be the CML, with $m$ oscillators, being $\alpha$ the coupling parameter and $f(x)$ a function depending on a parameter $r$, in function of which the system $y_{n+1}=f(y_{n};r)$ shows fixed points for some arbitrary period $p$. Fixed points of the system. Stationary synchronized state --------------------------------------------------------- It is straightforward to get the fixed points of the system. If the function $f(x)$ has a fixed point in $x^{*}$ then $(x^{*},x^{*},\dots^{m)},x^{*})$ will be a fixed point of the system given by (\[eq:tres\]), since if$$X_{i}(n)=x^{*}\quad i=1,\,\dots,\, m$$ it turns out that$$f(X_{i}(n))=f(x^{*})=x^{*}=X_{i}(n)\quad i=1,\,\dots,\, m$$ and$$\begin{array}{c} X_{i}(n+1)=(1-\alpha)f(x^{*})+{\displaystyle \frac{\alpha}{m}\sum_{i=1}^{m}}f(x^{*})=f(x^{*})=x^{*}\\ i=1,...,m\end{array}$$ as it was wanted to prove. It is then deduced that:$$X(n)=(x^{*},x^{*},...^{m)},x^{*})\label{eq:dos}$$ is a stationary sychronized state of the system. It is observed that if chosen $x^{*}\;\mbox{and}\: r$ for $f(x^{*};r)$ to determine a periodical or chaotic evolution of $x^{*}$, then the result would be that CML would have correspondingly periodical or chaotic synchronization, being this a proof of the existence of synchronized states both periodical and chaotic. In contrast from the CML with nearest neighbour coupling, there is not an upper limit in the number of oscillators “$m$” such that stable synchronous chaotic state exists [@Bohr1993]. See figures \[fig:SSS1\]-\[fig:SSS4\].   Let us now study the linear stability of fixed points, where the eigenvalues of jacobian matrix will be calculated. The jacobian matrix is given by: $$\left(\frac{\partial X_{i}(n+1)}{\partial X_{j}(n)}\right)_{X^{*}}=\left(\begin{array}{cccc} (1-\frac{m-1}{m}\alpha) & \frac{\alpha}{m} & \cdots & \frac{\alpha}{m}\\ \frac{\alpha}{m} & (1-\frac{m-1}{m}\alpha) & \cdots & \frac{\alpha}{m}\\ \vdots & \vdots & \ddots & \vdots\\ \frac{\alpha}{m} & \frac{\alpha}{m} & \cdots & (1-\frac{m-1}{m}\alpha)\end{array}\right)f^{\prime}(x^{*})$$ where $X^{*}=\left(x^{*},x^{*},...,x^{*}\right)$ and its eigenvalues are $$\left\{ \begin{array}{cc} \lambda=f^{\prime}(x^{*}) & \mbox{single}\\ \lambda=(1-\alpha)f^{\prime}(x^{*}) & \mbox{ multiplicity }(m-1)\end{array}\right.$$ therefore, the fixed point given by (\[eq:dos\]), or what would be the same, the stationary schronized state, would be stable whenever $x^{*}$ is a stable fixed point of $f(x)$. Periodical synchronized states ------------------------------ The existence of periodical syncronized states is reflected in theorem 1, shown below. The way to proceed with this proof is similar to the one used to obtain the stationary synchronized state. [Theorem 1.]{} : Let $\left\{ x_{1}^{*},x_{2}^{*},...,x_{p}^{*}\right\} $ be a $p$-periodic orbit of $C^{1}$ function $f$. Then the CML given by $$X_{i}(n+1)=(1-\alpha)f(X_{i}(n))+\frac{\alpha}{m}\sum_{i=1}^{m}f(X_{i}(n))\qquad i=1,\dots,\, m$$ 1. shows a synchronized state of the same period as the function $f$. The sychronized states are as follows $$(x_{j}^{*},x_{j}^{*},\ldots^{m)},x_{j}^{*})_{j=1,\,\dots,\, p}$$ 2. the sychronized states $(x_{j}^{*},x_{j}^{*},\ldots^{m)},x_{j}^{*})_{j=1,\,\dots,\, p}$ have the same stability as the fixed point $x_{j}^{*}$ of $f^{p}$, $j=1,\,\dots,\, p$. [Proof]{} :   1. Taking, for a given time $n$,$$X_{i}(n)=x_{1}^{*}\quad i=1,\,\dots,\, m$$ [ ]{}results in$$f^{p}(X_{i}(n))=X_{i}(n)\quad i=1,\,\dots,\, m$$ with the first iteration of CML being: $$X_{i}(n+1)=(1-\alpha)f(x_{1}^{*})+\frac{\alpha}{m}\sum_{j=1}^{m}f(x_{1}^{*})=f(x_{1}^{*})\qquad i=1,\,\dots,\, m$$ and the $p$-th iteration being:$$\begin{array}{rl} X_{i}(n+p)= & (1-\alpha)f(X_{i}(n+p-1))+{\displaystyle \frac{\alpha}{m}}{\displaystyle \sum_{j=1}^{m}}f(X_{j}(n+p-1))\\ = & f^{p}(x_{1}^{*})=x_{1}^{*}\qquad i=1,\,...,\, m\end{array}$$ As a result, the CML shows $p$ fixed points $(x_{j}^{*},x_{j}^{*},\cdots x_{j}^{*})_{j=1,\cdots,p}$ of period $p$, which constitute one sychronized state of the same period. These orbits of period $p$ constitute patterns of the CML. 2. To study the stability of the fixed points $(x_{j}^{*},x_{j}^{*},\cdots x_{j}^{*})_{j=1,\cdots,p}$ it is enough to perform it in $X_{1}^{*}=(x_{1}^{*},x_{1}^{*},\cdots x_{1}^{*})$, because $f^{p^{\prime}}(x_{j}^{*})$ has the same value for every fixed point $x_{j}^{*}$ of the $p$-periodic orbit. Let us calculate the eigenvalues of the jacobian matrix of the $p$-th iterate in that point. To calculate this Jacobian matrix, one must observe the following, applying the chain rule: $$\begin{array}{l} \left(\frac{\partial X_{i}(n+p)}{\partial X_{j}(n)}\right)_{X_{1}^{*}}=\\ \\\qquad\left(\begin{array}{cccc} (1-\frac{m-1}{m}\alpha) & \frac{\alpha}{m} & \cdots & \frac{\alpha}{m}\\ \frac{\alpha}{m} & (1-\frac{m-1}{m}\alpha) & \cdots & \frac{\alpha}{m}\\ \vdots & \vdots & \ddots & \vdots\\ \frac{\alpha}{m} & \frac{\alpha}{m} & \cdots & (1-\frac{m-1}{m}\alpha)\end{array}\right)f^{\prime}(x_{p}^{*})\left(\frac{\partial X_{i}(n+p-1)}{\partial X_{j}(n)}\right)_{X_{1}^{*}}\end{array}$$ finally as a result:$$\begin{array}{l} \left(\frac{\partial X_{i}(n+p)}{\partial X_{j}(n)}\right)_{X_{1}^{*}}=\\ \\\:\left(\begin{array}{cccc} \frac{1}{m}+\frac{m-1}{m}(1-\alpha)^{p} & \frac{1}{m}-\frac{1}{m}(1-\alpha)^{p} & \cdots & \frac{1}{m}-\frac{1}{m}(1-\alpha)^{p}\\ \frac{1}{m}-\frac{1}{m}(1-\alpha)^{p} & \frac{1}{m}+\frac{m-1}{m}(1-\alpha)^{p} & \cdots & \frac{1}{m}-\frac{1}{m}(1-\alpha)^{p}\\ \vdots & \vdots & \ddots & \vdots\\ \frac{1}{m}-\frac{1}{m}(1-\alpha)^{p} & \frac{1}{m}-\frac{1}{m}(1-\alpha)^{p} & \cdots & \frac{1}{m}+\frac{m-1}{m}(1-\alpha)^{p}\end{array}\right)\prod_{i=1}^{p}f^{\prime}(x_{i}^{*})\end{array}$$ $$$$ with eigenvalues $$\left\{ \begin{array}{cc} \lambda={\displaystyle \prod_{i=1}^{p}}f^{\prime}(x_{i}^{*}))=f^{p\prime}(x_{1}^{*}) & \mbox{single}\\ \lambda=(1-\alpha)^{p}{\displaystyle \prod_{i=1}^{p}}f^{\prime}(x_{i}^{*})=(1-\alpha)^{p}f^{p\prime}(x_{1}^{*}) & \mbox{ multiplicity }(m-1)\end{array}\right.\label{eq:cuatro}$$ So, $(x_{1}^{*},\cdots,x_{1}^{*})$ is a stable point of the CML, and therefore, it is in a stable sychronized state of period $p$, whenever $x_{1}^{*}$ is the stable fixed point $f^{p}$. Furthermore, as $f^{p\prime}(x_{1}^{*})=f^{p\prime}(x_{j}^{*})\quad j=1,\dots,\, m$ all points have the same stability. Keep in mind that if in **Theorem 1** $p=1$, then the stationary sychronized state previously studied is recovered; and because of this, it will undergo the period-doubling process that will be described in what follows. Period doubling cascade of periodic synchronized states ------------------------------------------------------- One would expect that if the function $f^{p}$, from **Theorem 1**, undergoes a period doubling cascade, then the CML given by (\[eq:tres\]) shows a duplication cascade in the sychronized states of period $p$ derived in **Theorem 1**. [Theorem 2.]{} : The synchronized states of period $p$ given by Theorem 1 undergo a period-doubling cascade as does $f^{p}$. [Proof]{} :   The proof is straightforward using Theorem 1, simply using successive substitution of $p$ by $p\cdot2,\, p\cdot2^{2},\,\dots,\, p\cdot2^{n},\,\dots$ every time that the $p$-periodic orbit of $f$ undergoes a period doubling bifurcation according to that theorem. [Note:]{} : See figures \[fig:p4\] and \[fig:p8\]. A nonexistence theorem ---------------------- It has been proven, in Theorem 1, the existences of $p$-period synchronized states in the CML, formed by the points of $p$-periodic orbit of $f$. One may ask whether the $n$-tuple of the form $$\begin{array}{rl} (X_{1}(n),X_{2}(n),\cdots,X_{p}(n))= & (x_{j}^{*},f(x_{j}^{*}),...,f^{p-1}(x_{j}^{*}))\\ f^{p}(x_{j}^{*})= & x_{j}^{*}\quad j=1,\cdots p\end{array}$$ that indicates that every oscillator is positioned in the successive points of the $p$-periodic orbit, generates a pattern of period $p$ in the CML; that is to say, a travelling wave. Nevertheless, this presumption is false, as shown below. [Theorem 3.]{} : Let $\left\{ x_{1}^{*},x_{2}^{*},...,x_{p}^{*}\right\} $ be a $p$-periodic orbit of the function $f$, then the CML given by $$X_{i}(n+1)=(1-\alpha)f(X_{i}(n-1))+\frac{\alpha}{p}\sum_{i=1}^{p}f(X_{i}(n-1))\qquad i=1,\,\dots,\, p$$ does not have a $p$-periodic orbit of the form $$(X_{1}(n),X_{2}(n),\cdots,X_{p}(n))=(x_{j}^{*},f(x_{j}^{*}),...,f^{p-1}(x_{j}^{*}))$$ $x_{j}^{*}$ being any of the $p$ points of the $p$-periodic orbit. [Proof]{} :   The following initial conditions are taken$$(X_{1}(n),X_{2}(n),\cdots,X_{p}(n))=(x_{1}^{*},x_{2}^{*},...,x_{p}^{*})$$ After the first iteration, it will become$$\left\{ \begin{array}{c} X_{1}(n+1)=x_{2}^{*}=(1-\alpha)f(x_{1}^{*})+\frac{\alpha}{p}(f(x_{1}^{*})+...+f(x_{p}^{*}))\\ X_{2}(n+1)=x_{3}^{*}=(1-\alpha)f(x_{2}^{*})+\frac{\alpha}{p}(f(x_{1}^{*})+...+f(x_{p}^{*}))\\ \cdots\\ X_{p}(n+1)=x_{1}^{*}=(1-\alpha)f(x_{p}^{*})+\frac{\alpha}{p}(f(x_{1}^{*})+...+f(x_{p}^{*}))\end{array}\right.$$ therefore:$$\left\{ \begin{array}{c} x_{2}^{*}=(1-\alpha)x_{2}^{*}+\frac{\alpha}{p}(x_{1}^{*}+...+x_{p}^{*})\\ x_{3}^{*}=(1-\alpha)x_{3}^{*}+\frac{\alpha}{p}(x_{1}^{*}+...+x_{p}^{*})\\ \cdots\\ x_{1}^{*}=(1-\alpha)x_{1}^{*}+\frac{\alpha}{p}(x_{1}^{*}+...+x_{p}^{*})\end{array}\right.$$ operating it results in:$$\left\{ \begin{array}{c} \alpha x_{2}^{*}=\frac{\alpha}{p}(x_{1}^{*}+...+x_{p}^{*})\\ \alpha x_{3}^{*}=\frac{\alpha}{p}(x_{1}^{*}+...+x_{p}^{*})\\ \cdots\\ \alpha x_{1}^{*}=\frac{\alpha}{p}(x_{1}^{*}+...+x_{p}^{*})\end{array}\right.$$ from which it is deduced that$$x_{1}^{*}=x_{2}^{*}=...=x_{p}^{*}$$ in contradiction with $x_{1}^{*}\neq x_{2}^{*}\neq...\neq x_{p}^{*}$. This negative result, about $p$-periodic waves, brings us to question the conditions under which they are produced. This study is conducted in the following section.   Analytical study of patterns in weakly coupled CML ================================================== Travelling waves ---------------- It is has been proven in Theorem 3 that the $p$-periodic orbit of the function $f(x;r)$ is not inherited by the system, but it is easily observed that if $\alpha=0$ then a wave of this period exists in the CML. Given that for $\alpha=0$ the wave exists, one would wonder if for a small coupling $\alpha\ll1$, the CML admits a pertubative solution. For this study, we will substitute $\alpha$ with $\varepsilon\alpha$, having $\varepsilon\ll1$ and assuming the new $\alpha$ is $O(1)$, in the CML given by (\[eq:uno\]). [Theorem 4.]{} : Let $\left\{ x_{1}^{*},x_{2}^{*},...,x_{p}^{*}\right\} $ be a $p$-periodic orbit of a $C^{2}$ function $f$, such that $f^{p\prime}(x_{i}^{*})\neq1$, $i=1,\,\dots,\, p$, then the CML given by $$\begin{array}{c} X_{i}(n+1)=(1-\varepsilon\alpha)f(X_{i}(n))+{\displaystyle \frac{\alpha\varepsilon}{p}\sum_{j=1}^{p}}f(X_{j}(n))\\ i=1,\,\dots,\, p\quad\varepsilon\ll1\end{array}\label{eq:cinco}$$ shows a $p$-periodic solution given by$$\begin{array}{c} X_{i}(n+j)=x_{i+j}^{*}+\varepsilon A_{i+j}\\ \begin{array}{c} i=1,\,\dots,\, p\\ j=0,\,\dots,\, p-1\end{array}\end{array}$$ where$$A_{k}=\frac{\alpha}{(1-(f^{p}(x_{1}))^{\prime})}\sum_{j=1}^{p}\left[f^{p-j+k-1}(x_{j+1}^{*})\right]^{\prime}\left(\left(x_{j+1}^{*}\right)+\frac{1}{p}\sum_{l=1}^{p}x_{l}^{*}\right)\qquad k=1,\dots,p$$ with periodic conditions$$\begin{array}{c} A_{i+p}=A_{i}\\ x_{i+p}^{*}=x_{i}^{*}\end{array}\quad i=1,\,\dots,\, p$$ [Proof]{} :   The periodic orbit given by $$\begin{array}{c} X_{i}(n+j)=x_{i+j}^{*}+\varepsilon A_{i+j}\\ \begin{array}{c} i=1,\dots,p\\ j=0,\dots,p-1\end{array}\end{array}$$ will exist when the following system$$\left\{ \begin{array}{c} X_{i}(n)=x_{i}^{*}+\epsilon A_{i}\\ X_{i}(n+1)=x_{i+1}^{*}+\epsilon A_{i+1}\\ \vdots\\ X_{i}(n+p-1)=x_{i+p-1}^{*}+\epsilon A_{i+p-1}=x_{i-1}^{*}+\epsilon A_{i-1}\\ X_{i}(n+p)=x_{i}^{*}+\epsilon A_{i}\end{array}\right.\, i=1,...,p\label{eq:seis}$$ is compatible and determined. As$$X_{i}(n+1)=(1-\varepsilon\alpha)f(X_{i}(n))+\frac{\epsilon\alpha}{p}\sum_{j=1}^{p}f(X_{j}(n))$$ from (\[eq:seis\]), it reults in$$x_{i+1}^{*}+\varepsilon A_{i+1}=(1-\varepsilon\alpha)f(x_{i}^{*}+\varepsilon A_{i})+\frac{\varepsilon\alpha}{p}\sum_{j=1}^{p}f(x_{j}^{*}+\varepsilon A_{j})\label{siete}$$ Performing the expansion$$f(x_{i}^{*}+\varepsilon A_{i})=f(x_{i}^{*})+\varepsilon A_{i}f^{\prime}(x_{i}^{*})+O(\varepsilon^{2})$$ and replacing in (\[siete\]), the system results in$$\begin{array}{c} x_{i+1}^{*}+\varepsilon A_{i+1}=x_{i+1}^{*}+\varepsilon A_{i}f^{\prime}(x_{i}^{*})-\varepsilon\alpha x_{i+1}^{*}+{\displaystyle \frac{\varepsilon\alpha}{p}\sum_{j=1}^{p}}(x_{j+1}^{*})+O(\varepsilon^{2})\\ i=1,...,p\end{array}$$ Solving the system to order $\varepsilon$ it is obtained:$$-A_{i}f^{\prime}(x_{i}^{*})+A_{i+1}=\alpha x_{i+1}^{*}+\frac{\alpha}{p}\sum_{j=1}^{p}x_{j+1}^{*}\qquad i=1,\dots,p$$ whose matricial expression is:[$$\left(\begin{array}{cccccc} -f^{\prime}(x_{1}^{*}) & 1 & 0 & 0 & \cdots & 0\\ 0 & -f^{\prime}(x_{2}^{*}) & 1 & 0 & \cdots & 0\\ 0 & 0 & -f^{\prime}(x_{3}^{*}) & 1 & \cdots & 0\\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots\\ 1 & 0 & 0 & 0 & \cdots & -f^{\prime}(x_{p}^{*})\end{array}\right)\left(\begin{array}{c} A_{1}\\ A_{2}\\ A_{3}\\ \vdots\\ A_{p}\end{array}\right)=\alpha\left(\begin{array}{c} -x_{2}^{*}+\frac{1}{p}\Sigma_{j=1}^{p}x_{j}^{*}\\ -x_{3}^{*}+\frac{1}{p}\Sigma_{j=1}^{p}x_{j}^{*}\\ -x_{4}^{*}+\frac{1}{p}\Sigma_{j=1}^{p}x_{j}^{*}\\ \vdots\\ -x_{1}^{*}+\frac{1}{p}\Sigma_{j=1}^{p}x_{j}^{*}\end{array}\right)\label{nueve}$$ ]{} This is a system of $p$ equations and $p$ unknowns whose coefficient matrix has determinant $$(-1)^{p}\prod_{i=1}^{p}f^{\prime}(x_{i}^{*})+(-1)^{p+1}$$ Given that $\prod_{i=1}^{p}f^{\prime}(x_{i}^{*})=f^{p\prime}(x_{i}^{*})\neq1$ (by hypothesis) the system is compatible and determined for every $\alpha\neq0$. Moreover, the solution of the system is different from the trivial one, since the independent term column is not null $x_{1}^{*}\neq x_{2}^{*}\neq\cdots\neq x_{p}^{*}$, that is$$\left(\begin{array}{c} -x_{2}^{*}+{\displaystyle \frac{1}{p}\sum_{j=1}^{p}}x_{j}^{*}\\ -x_{3}^{*}+{\displaystyle \frac{1}{p}\sum_{j=1}^{p}}x_{j}^{*}\\ -x_{4}^{*}+{\displaystyle \frac{1}{p}\sum_{j=1}^{p}}x_{j}^{*}\\ \vdots\\ -x_{1}^{*}+{\displaystyle \frac{1}{p}\sum_{j=1}^{p}}x_{j}^{*}\end{array}\right)\neq\left(\begin{array}{c} 0\\ 0\\ 0\\ \vdots\\ 0\end{array}\right)$$   It does not matter which oscillator is considered for study of the evolution of the system, as the algebraic system obtained is always the same. The solution of the system in (\[nueve\]) can be obtained directly by inversion and results in: $$\left(\begin{array}{c} A_{1}\\ A_{2}\\ \vdots\\ A_{p}\end{array}\right)=\alpha\left(\begin{array}{ccccc} -f^{\prime}(x_{1}^{*}) & 1 & 0 & \cdots & 0\\ 0 & -f^{\prime}(x_{2}^{*}) & 1 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 1 & 0 & 0 & \cdots & -f^{\prime}(x_{p}^{*})\end{array}\right)^{-1}\left(\begin{array}{c} -x_{2}^{*}+{\displaystyle \frac{1}{p}\sum_{j=1}^{p}}x_{j}^{*}\\ -x_{3}^{*}+{\displaystyle \frac{1}{p}\sum_{j=1}^{p}}x_{j}^{*}\\ \vdots\\ -x_{1}^{*}+{\displaystyle \frac{1}{p}\sum_{j=1}^{p}}x_{j}^{*}\end{array}\right)$$ The inversion of the matrix (which is not a circulant one) results in the following$$\left(\begin{array}{c} A_{1}\\ A_{2}\\ A_{3}\\ \vdots\\ A_{p}\end{array}\right)=\alpha\frac{1}{(-1)^{p+1}(1-(f^{p}(x_{1}^{*}))^{\prime})}MN\label{eq:diez}$$ where the matrix $M$ is given by [ ]{} [$$M=\left(\begin{array}{ccccc} f^{\prime}(x_{2}^{*})\cdots f^{\prime}(x_{p}^{*}) & f^{\prime}(x_{3}^{*})\cdots f^{\prime}(x_{p}^{*}) & f^{\prime}(x_{4}^{*})\cdots f^{\prime}(x_{p}^{*}) & \cdots & 1\\ 1 & f^{\prime}(x_{3}^{*})\cdots f^{\prime}(x_{p}^{*})f^{\prime}(x_{1}^{*}) & f^{\prime}(x_{4}^{*})\cdots f^{\prime}(x_{p}^{*})f^{\prime}(x_{1}^{*}) & \cdots & f^{\prime}(x_{1}^{*})\\ f^{\prime}(x_{2}) & 1 & f^{\prime}(x_{4}^{*})\cdots f^{\prime}(x_{p}^{*})f^{\prime}(x_{1}^{*})f^{\prime}(x_{2}^{*}) & \cdots & f^{\prime}(x_{1}^{*})f^{\prime}(x_{2}^{*})\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ f^{\prime}(x_{2}^{*})\cdots f^{\prime}(x_{p-1}^{*}) & f^{\prime}(x_{3}^{*})\cdots f^{\prime}(x_{p-1}^{*}) & f^{\prime}(x_{4}^{*})\cdots f^{\prime}(x_{p-1}^{*}) & \cdots & f^{\prime}(x_{1}^{*})f^{\prime}(x_{2}^{*})\cdots f^{\prime}(x_{p-1}^{*})\end{array}\right)\label{eq:matrizM}$$ ]{}and $N$ by $$N=\left(\begin{array}{c} -x_{2}^{*}+{\displaystyle \frac{1}{p}\sum_{j=1}^{p}}x_{j}^{*}\\ -x_{3}^{*}+{\displaystyle \frac{1}{p}\sum_{j=1}^{p}}x_{j}^{*}\\ -x_{4}^{*}+{\displaystyle \frac{1}{p}\sum_{j=1}^{p}}x_{j}^{*}\\ \vdots\\ -x_{1}^{*}+{\displaystyle \frac{1}{p}\sum_{j=1}^{p}}x_{j}^{*}\end{array}\right)$$   After operating in (\[eq:diez\]) it results in: $$\begin{array}{c} A_{k}=\frac{\alpha}{(-1)^{p+1}(1-(f^{p}(x_{1}))^{\prime})}{\displaystyle \sum_{j=1}^{p}}\left[f^{p-j+k-1}(x_{j+1}^{*})\right]^{\prime}\left(\left(x_{1+j}^{*}\right)+{\displaystyle \frac{1}{p}\sum_{l=1}^{p}}x_{l}^{*}\right)\\ k=1,...,p\end{array}\label{soluc}$$ Every $A_{k}\neq0$ because the solution is known to be different from the trivial one. The solution obtained is valid at order $O(\varepsilon^{2})$ while $\varepsilon\ll\frac{1}{1-f^{p\prime}(x_{1}^{*})}$. Period doubling cascade for travelling waves in a CML ----------------------------------------------------- Period-doubling transitions to chaos have already been observed a long time ago, in CML with nearest neighbour coupling, using the Mandelbrot map [@Selection1992]. The existence of this phenomenon is not relegated only to the quadratic functions, and its existence can be proved for any function (as we will demonstrate) undergoing a period-doubling cascade; therefore, this phenomenon must be very frequent. [Theorem 5.]{} : Let $f:I\rightarrow I$ be a $C^{2}$ funtion depending on some parameter, in function of which the $2^{p}$-periodic orbit of the map $x_{n+1}=f(x_{n})$ undergoes a period-doubling cascade. Let $\left\{ x_{i,2^{p+q}}^{*}\right\} _{i=1}^{2^{p+q}}$ be the $2^{p+q}$-period orbit of the cascade, $q\in\mathbb{N}$, where it is noted that $f^{k}(x_{i,2^{p+q}}^{*})=x_{i+k,2^{p+q}}^{*}$. The CML given by $$\begin{array}{rl} X_{i}(n+1) & =(1-\varepsilon\alpha)f(X_{i}(n))+{\displaystyle \frac{\alpha\varepsilon}{2^{p}}\sum_{j=1}^{2^{p}}}f(X_{j}(n))\\ i & =1,\cdots,2^{p}\quad\varepsilon\ll1\end{array}\label{eq:ONCE}$$ has a $2^{p+q}$-periodic solution given by$$\begin{array}{rl} X_{i}(n+j)= & x_{2^{q}(i-1)+1+j,2^{p+q}}^{*}+\varepsilon A_{2^{q}(i-1)+1+j}\\ i= & 1,\dots,2^{p}\quad j=0,\dots,2^{p+q}\end{array}$$ where $$\begin{array}{c} A_{k}=\frac{\alpha}{(-1+(f^{2^{p+q}}(x_{1}^{*}))^{\prime})}{\displaystyle \sum_{j=1}^{2^{p+q}}}\left[f^{2^{p+q}-j+k-1}(x_{j+1,2^{p+q}}^{*})\right]^{\prime}\left(\left(x_{1+j,2^{p+q}}^{*}\right)+\frac{1}{p}S_{j}\right)\\ k=1,\dots,2^{p+q}\end{array}$$ with$$S_{j}=\left\{ \begin{array}{lcl} {\displaystyle \sum_{i=1}^{2^{p}}}x_{2^{q}(i-1)+2,2^{p+q}}^{*} & \quad & \mbox{if}\quad j=[2^{q}]\\ {\displaystyle \sum_{i=1}^{2^{p}}}x_{2^{q}(i-1)+3,2^{p+q}}^{*} & \quad & \mbox{if}\quad j=[2^{q}]+1\\ {\displaystyle \sum_{i=1}^{2^{p}}}x_{2^{q}(i-1)+4,2^{p+q}}^{*} & \quad & \mbox{if}\quad j=[2^{q}]+2\\ {\displaystyle \sum_{i=1}^{2^{p}}}x_{2^{q}(i-1)+5,2^{p+q}}^{*} & \quad & \mbox{if}\quad j=[2^{q}]+3\\ \qquad\qquad\vdots & \vdots & \qquad\quad\vdots\\ {\displaystyle \sum_{i=1}^{2^{p}}}x_{2^{q}(i-1)+2^{q}+1,2^{p+q}}^{*} & \quad & \mbox{if}\quad j=[2^{q}]+2^{q}-1\end{array}\right.$$ where $[2^{q}]$ represents a multiple of $2^{q}$. This $2^{p+q}$-periodic solution fulfills the periodicity condition$$\begin{array}{c} A_{i+2^{p+q}}=A_{i}\\ x_{i+2^{p+q},2^{p+q}}^{*}=x_{i,2^{p+q}}^{*}\end{array}$$ [Proof]{} :   **Note:** Notice that although there are just $2^{p}$ oscillators the periodic wave will have period $2^{p+q}$, therefore, the system that is dealt with in the proof will have $2^{p+q}$ equations. Let $2^{p}$ oscillators be with the initial conditions given by $$X_{i}(n)=x_{2^{q}(i-1)+1,2^{p+q}}^{*}+\varepsilon A_{2^{q}(i-1)+1}\quad i=1,\cdots,2^{p}\label{eq:17}$$ Initial conditions are fixed points of $f^{2^{p+q}}$ (taken one every $2^{q}$) plus a perturbation that must be calculated. A $2^{p+q}$-periodic orbit will exist whenever the system[$$\left\{ \begin{array}{rl} X_{i}(n)= & x_{2^{q}(i-1)+1,2^{p+q}}^{*}+\varepsilon A_{2^{q}(i-1)+1}\\ X_{i}(n+1)= & x_{2^{q}(i-1)+2,2^{p+q}}^{*}+\varepsilon A_{2^{q}(i-1)+2}\\ & \vdots\\ X_{i}(n+2^{p+q})= & x_{2^{q}(i-1)+1+2^{p+q},2^{p+q}}^{*}+\varepsilon A_{2^{q}(i-1)+2^{p+q}}\\ = & x_{2^{q}(i-1)+1,2^{p+q}}^{*}+\varepsilon A_{2^{q}(i-1)+1}=X_{i}(n)\end{array}\right.\, i=1,...,2^{p}\label{eq:dieciocho}$$ ]{}is compatible and determined. From equation (\[eq:ONCE\]) and subtituting the second equality in (\[eq:dieciocho\]), we have $$\begin{array}{c} x_{2^{q}(i-1)+2,2^{p+q}}^{*}+\varepsilon A_{2^{q}(i-1)+2}=(1-\varepsilon\alpha)f(x_{2^{q}(i-1)+1,2^{p+q}}^{*}+\varepsilon A_{2^{q}(i-1)+1})+\\ +{\displaystyle \frac{\alpha\varepsilon}{2^{p}}\sum_{j=1}^{2^{p}}}f(x_{2^{q}(j-1)+1,2^{p+q}}^{*}+\varepsilon A_{2^{q}(j-1)+1})\end{array}\label{eq:diecinueve}$$ Performing a Taylor expansion of $f$ to order $O(\varepsilon^{2})$ and substituting in (\[eq:diecinueve\]) the following is obtained$$\begin{array}{c} x_{2^{q}(i-1)+2,2^{p+q}}^{*}+\varepsilon A_{2^{q}(i-1)+2}=x_{2^{q}(i-1)+2,2^{p+q}}^{*}+\\ +\varepsilon A_{2^{q}(i-1)+1}f^{\prime}(x_{2^{q}(i-1)+1,2^{p+q}}^{*})-\varepsilon\alpha x_{2^{q}(i-1)+2,2^{p+q}}^{*}+\frac{\varepsilon\alpha}{2^{p}}\sum_{i=1}^{2^{p}}x_{2^{q}(i-1)+2,2^{p+q}}^{*}+O(\varepsilon^{2})\end{array}$$ Doing exactly the same with the next equality in (\[eq:dieciocho\]) the following is obtained:$$\begin{array}{c} x_{2^{q}(i-1)+3,2^{p+q}}^{*}+\varepsilon A_{2^{q}(i-1)+3}=x_{2^{q}(i-1)+3,2^{p+q}}^{*}+\\ +\varepsilon A_{2^{q}(i-1)+2}f^{\prime}(x_{2^{q}(i-1)+2,2^{p+q}}^{*})-\varepsilon\alpha x_{2^{q}(i-1)+3,2^{p+q}}^{*}+{\displaystyle \frac{\varepsilon\alpha}{2^{p}}\sum_{i=1}^{2^{p}}}x_{2^{q}(i-1)+3,2^{p+q}}^{*}+O(\varepsilon^{2})\end{array}$$ and with the last equality in (\[eq:dieciocho\]), we get the following equation:$$\begin{array}{c} x_{2^{q}(i-1)+1,2^{p+q}}^{*}+\varepsilon A_{2^{q}(i-1)+1}=x_{2^{q}(i-1)+1,2^{p+q}}^{*}+\\ +\varepsilon A_{2^{q}(i-1)}f^{\prime}(x_{2^{q}(i-1),2^{p+q}}^{*})-\varepsilon\alpha x_{2^{q}(i-1)+1,2^{p+q}}^{*}+{\displaystyle \frac{\varepsilon\alpha}{2^{p}}\sum_{i=1}^{2^{p}}}x_{2^{q}(i-1)+1,2^{p+q}}^{*}+O(\varepsilon^{2})\end{array}$$ The former $2^{p+q}$-equations, for the oscillator i, represent a linear system, whose matricial expresion is:[$$\begin{array}{c} \left(\begin{array}{ccccc} -f^{\prime}(x_{2^{q}(i-1)+1,2^{p+q}}^{*}) & 1 & 0 & \cdots & 0\\ 0 & -f^{\prime}(x_{2^{q}(i-1)+2,2^{p+q}}^{*}) & 1 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 1 & 0 & 0 & \cdots & -f^{\prime}(x_{2^{q}(i-1)+2^{p+1},2^{p+q}}^{*})\end{array}\right).\left(\begin{array}{c} A_{2^{q}(i-1)+1}\\ A_{2^{q}(i-1)+2}\\ \vdots\\ A_{2^{q}(i-1)+2^{p+q}}\end{array}\right)=\\ \\=\alpha\left(\begin{array}{c} -x_{2^{q}(i-1)+2,2^{p+q}}^{*}+{\displaystyle \frac{1}{2^{p}}\sum_{i=1}^{2^{p+1}}}x_{2^{q}(i-1)+2,2^{p+q}}^{*}\\ -x_{2^{q}(i-1)+3,2^{p+q}}^{*}+{\displaystyle \frac{1}{2^{p}}\sum_{i=1}^{2^{p+1}}}x_{2^{q}(i-1)+3,2^{p+q}}^{*}\\ \vdots\\ -x_{2^{q}(i-1)+1,2^{p+q}}^{*}+{\displaystyle \frac{1}{2^{p}}\sum_{i=1}^{2^{p+1}}}x_{2^{q}(i-1)+1,2^{p+q}}^{*}\end{array}\right)\end{array}\label{quince}$$ ]{} The previous matricial expresion represents a system of $2^{p+q}$ equations with $2^{p+q}$ unknowns, and being the determinant of coefficient matrix$$(-1)^{2^{p+q}}\prod_{i=1}^{2^{p+q}}f^{\prime}(x_{i}^{*})-(-1)^{2^{p+q}}=\left[f^{2^{p+q}}(x_{i}^{*})\right]^{\prime}-1\neq0$$ (period doubling bifurcations take place when $\left[f^{2^{p+q}}(x_{i}^{*})\right]^{\prime}=-1$). Thus the system is compatible and determined for every $\alpha$ and, as in the previous cases, its solution is different from the trivial one for $\alpha\neq0$. The solution is obtained directly from (\[quince\]) by inversion:$$\begin{array}{c} \left(\begin{array}{c} A_{2^{q}(i-1)+1}\\ A_{2^{q}(i-1)+2}\\ \vdots\\ A_{2^{q}(i-1)+2^{p+q}}\end{array}\right)=\frac{\alpha}{(-1+[f^{2^{p+q}}(x_{1}^{*})]^{\prime})}MN\end{array}\label{eq:veinte}$$ where$$M=\left(\begin{array}{ccccc} -f^{\prime}(x_{2^{q}(i-1)+1,2^{p+q}}^{*}) & 1 & 0 & \cdots & 0\\ 0 & -f^{\prime}(x_{2^{q}(i-1)+2,2^{p+q}}^{*}) & 1 & \cdots & 0\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 1 & 0 & 0 & \cdots & -f^{\prime}(x_{2^{q}(i-1)+2^{p+q},2^{p+q}}^{*})\end{array}\right)^{-1}$$ has already been calculated (see \[eq:matrizM\]) and$$N=\left(\begin{array}{c} -x_{2^{q}(i-1)+2,2^{p+q}}^{*}+{\displaystyle \frac{1}{2^{p}}\sum_{i=1}^{2^{p+1}}}x_{2^{q}(i-1)+2,2^{p+q}}^{*}\\ -x_{2^{q}(i-1)+3,2^{p+q}}^{*}+{\displaystyle \frac{1}{2^{p}}\sum_{i=1}^{2^{p+1}}}x_{2^{q}(i-1)+3,2^{p+q}}^{*}\\ \vdots\\ -x_{2^{q}(i-1)+1,2^{p+q}}^{*}+{\displaystyle \frac{1}{2^{p}}\sum_{i=1}^{2^{p+1}}}x_{2^{q}(i-1)+1,2^{p+1}}^{*}\end{array}\right)$$ After operating in (\[eq:veinte\]) it results in: $$A_{k}=\frac{\alpha}{(-1)^{2^{p+q}}(1-[f^{2^{p+q}}(x_{1}^{*})]^{\prime})}\sum_{j=1}^{2^{p+q}}\left[f^{2^{p+q}-j+k-1}(x_{j+1,2^{p+q}}^{*})\right]^{\prime}\left(\left(x_{1+j,2^{p+q}}^{*}\right)+\frac{1}{p}S_{j}\right)$$ $$k=1,\,...,\,2^{p+q}$$ with$$S_{j}=\left\{ \begin{array}{lcl} {\displaystyle \sum_{i=1}^{2^{p}}}x_{2^{q}(i-1)+2,2^{p+q}}^{*} & \quad & \mbox{if}\quad j=[2^{q}]\\ {\displaystyle \sum_{i=1}^{2^{p}}}x_{2^{q}(i-1)+3,2^{p+q}}^{*} & \quad & \mbox{if}\quad j=[2^{q}]+1\\ {\displaystyle \sum_{i=1}^{2^{p}}}x_{2^{q}(i-1)+4,2^{p+q}}^{*} & \quad & \mbox{if}\quad j=[2^{q}]+2\\ {\displaystyle \sum_{i=1}^{2^{p}}}x_{2^{q}(i-1)+5,2^{p+q}}^{*} & \quad & \mbox{if}\quad j=[2^{q}]+3\\ \qquad\qquad\vdots & \vdots & \qquad\quad\vdots\\ {\displaystyle \sum_{i=1}^{2^{p}}}x_{2^{q}(i-1)+2^{q}+1,2^{p+q}}^{*} & \quad & \mbox{if}\quad j=[2^{q}]+2^{q}-1\end{array}\right.$$ where $[2^{q}]$ represents a multiple of $2^{q}$, with the periodicity conditions $$\begin{array}{c} A_{i+2^{p+q}}=A_{i}\\ x_{i+2^{p+q},2^{p+q}}^{*}=x_{i,2^{p+q}}^{*}\end{array}$$ due to the cyclic character of the $2^{p+q}$-periodic orbit. The solution obtained is valid at order $O(\varepsilon^{2})$ while $\varepsilon\ll\frac{1}{1-[f^{2^{p+q}}(x_{1}^{*})]^{\prime}}$. The saddle-node orbit ($[f^{2^{p+q}}(x_{1}^{*})]^{\prime}\neq1$) has been avoided by the condition $q\in\mathbb{N}$ ($0\not\in\mathbb{N}$), and therefore there has been, at least, one period-doubling bifurcation. ### Remarks: {#remarks .unnumbered} 1. Notice that this theorem indicates that a CML, with $2^{p}$ oscillators, has originally a $2^{p}$-period travelling wave. As the $2^{p}$-periodic orbit of $f$ duplicates ($q$ increases) to a $2^{p+q}$-periodic orbit so does the travelling wave of the CML. 2. The theorem 5 does not impose any restriction to the $2^{p}$-periodic orbit of $f$. In the case of $f$ presenting $2^{p}$-periodic windows, this $2^{p}$-periodic orbit could belong to a period-doubling cascade in the canonical window, or originate from a $2^{p}$-periodic saddle-node orbit. In the former case, $f$ undergoes a period-doubling cascade, in the latter it is $f^{2^{p}}$ who goes through a period-doubling cascade (this period-doubling cascade would be located inside a $2^{p}$-periodic window). The conclusion is straightforward: the CML will not have just one $2^{p}$-periodic wave, undergoing a period-doubling cascade; there will be as many as $2^{p_{1}}$-periodic windows of $f$, with $p_{1}\leq p$ ($p_{1}=0$ would be the canonical window). 3. Similar arguments can be done for synchronized state cascades, subject to the condition that the duplicating orbit does not have prime period: it can be a $p\cdot q$-periodic orbit in the canonical window, or a $q$-periodic orbit in the $p$-periodic window, or a $p$-periodic orbit in a $q$-periodic window, that afterwards will undergo period bifurcation cascade. 4. There is a fact that could be not observed at a first reading of theorem 5: the points used to construct the perturbative solution both can be stable and unstable. 5. Since it has been deduced that the CML undergoes a period doubling cascade and that this cascade has its origin in the period doubling cascade of the $f$, it is concluded that the CML inherits the dynamics of $f$. Discussion and conclusions ========================== Several theorems have been proved that show the sufficiency conditions of existence of synchronized states (periodic and chaotic) and travelling waves in CML. Also it has been analytically determined the value that describes the state of each oscillator at any moment. The results of the theorems are as general as possible. This is due to two facts. Firstly, the CML, with which we have worked, has a number of arbitrary oscillators. Second, the function $f$, that rules the dynamics of every oscillator, is also arbitrary with the condition that it undergoes a period-doubling cascade. The results have the following consequences and link with other research: 1. The emergence of the global properties from the local ones has been proved. The global dynamics inherits the dynamics of every oscillator: fixed points of the system come essencially from the fixed points of the map (that governs the dynamics of every oscillator) compounded with itself $m\cdot2^{k}$ times (being $m$ number of oscillators of the CML). In particular, this result has been observed recently in numerical computations [@Palaniyandi]. Our results are an explicit analytical expresion of the results of Lemaitré y Chaté [@Lemaitre], who proved, in CML, the traslation of the local properties to a spatiotemporal level. 2. The dynamics of a CML has been studied, where the individual dynamics of every oscillator is ruled by an arbitrary function $f$; being few the analytical results on the matter, one normally only works with quadratic functions or piece-wise linear functions [@Atmans1; @Anteneodo; @Liyorke]. The one presented here is an interesting generalization that permits the calculation of properties associated with the states and their evolution. 3. Two limitations that are present when numerical techniques are applied have been overcome: 1. Spurious results, due to finite precision in computer simulations [@Grebogi; @Czhou2000; @Czhou2002], have been avoided. 2. Limits tending to infinity can be used for analytical solutions, both with the number of oscillators in CML and the number of bifurcations in the period doubling cascade. On the one hand, the numerical simulation with a large number of oscillators becomes unaffordable due to the computation time that would be necessary as the number of oscillators grows. On the other hand, as it has been indicated in the introduction, the study of the onset of the turbulence in a fluid to be properly understood would need many oscillators, the more the better. As a direct consequence, of taking the limit in the period doubling cascade, it is deduced the existence of waves of arbitrary period, tending to infinity as the parameter bifurcation gets closer to the Myrberg-Feigenbaum point. This is a response to the established question of Gade and Amritkar in their work [@Pmgade] where they found the wavelength-doubling bifurcation. Another question raised by Gade and Amritkar in that same paper was: is there more than one value such that, if the parameter value tends to it, then the period of the travelling wave tends to infinity?. The response again in the afirmative; in fact, there are infinite values that are the correspondent Myrberg-Feigenbaum points of the windows inside the canonical window. The position of these values is determined by the Saddle-Node Bifurcation Cascades [@Sm1] and the relation between period doubling cascade and Saddle-Node Bifurcation Cascades is also known [@Sm2]. [10]{} Kaneko K. Theory and applications of coupled map lattices. New York: Wiley; 1993. Kaneko K. Chaotic but Regular Posi-nega Switch among Coded Attractors by Cluster Size Variation. Phys. Rev. Lett. 1989; 63: 219-224. Kaneko K. Clustering, Coding, Switching, Hierarchical Ordering, and Control lin Network of Chaotic Elements. Physica D 1990; 41: 131-172. Kaneko K. Globally Coupled Chaos Violates Law of Large Numbers. Phys. Rev. Lett. 1990; 65: 1391-1394. Kaneko K. Partition Complexity in Network of Chaotic Elements. J. Phys. A 1991; 24: 2107-2119. Kaneko K. Globally Coupled Circle Maps. Physica D 1991; 54: 5-19 Special issue. Physica D 1997:103\] Special issue: Chaos 1992;2(3)\] Chaté H, Manneville P. Spatio-temporal intermittency in coupled map lattices. Physica D 1988; 32:409-422. Atmanspacher H, Scheingraber H. Inherent global stabilization of unstable local behavior in coupled map lattices. Int J Bifurcat Chaos 2005; 5(15):16651676. Mehta M, Sinha S. Asynchronous updating of coupled maps leads to synchronization. Chaos 2000; 10:350-358. Anteneodo C, de S. Pinto SE, Batista AM, Viana RL. Analytical results for coupled-map lattices with long-range interactions. Phys. Rev. E 2003; 68:045202(R) [\[]{}Erratum Phys. Rev. E 2004; 69:029904\] Bohr T, Christensen OB. Size dependence, coherence and scaling in turbulent coupled map lattices. Phys. Rev. Lett. 1989; 63:2161-2164 Willeboordse FH. Selection of Windows, Attractors and Self-similar Patterns in a Coupled Map Lattice. Chaos, Solitons and Fractals 1992; 2:609-634. Lemaître A, Chaté H. Nonperturbative Renormalization Group for Chaotic Coupled Map Lattices. Phys. Rev. Lett. 1998; 80(25):5528-5531. Palaniyandi P, Muruganandam P, Laksmanan M. Coexistence of synchronized and desynchronized patterns in coupled chaotic dynamical systems. Chaos, Solitons and Fractals 2006;doi:10. 1016/j.chaos.2006.08.004 Li P, Li Z, Halang WA, Chen G. Li-Yorke chaos in a spatiotemporal chaotic system. Chaos, Solitons and Fractals 2007; 33: 335-341. Grebogi C, Hammel SM, Yorke JA, Sauer T. Shadowing of physical trajectories in chaotic dynamics: containment and refinement. Phys. Rev. Lett. 1990; 65:1527-1530. Zhou C, Lai C-H. Analysis of spurious synchronization with positive conditional Lyapunov exponents in computer simulations. Physica D 2000; 135:1-23. Zhou C, Kurths J. Noise-induced phase synchronization and synchronization transitions in chaotic oscillators. Phys. Rev. Lett. 2002; 88:230602. Gade PM, Amritkar RE. Wavelength-doubling bifurcations in one-dimensional coupled logisitc map. Phys. Rev. E 1994; 49(4):2617-2622. San Martín J. Intermittency cascades. Chaos Solitons and Fractals 2007; 32:816-831. San Martín J, Rodriguez-Perez D. Conjugation of cascades. Chaos Solitons and Fractals. (In Press doi:10.1016/J.Chaos.2007.01.073)

--- abstract: | We consider the problem of estimating a spatially varying density function, motivated by problems that arise in large-scale radiological survey and anomaly detection. In this context, the density functions to be estimated are the background gamma-ray energy spectra at sites spread across a large geographical area, such as nuclear production and waste-storage sites, military bases, medical facilities, university campuses, or the downtown of a city. Several challenges combine to make this a difficult problem. First, the spectral density at any given spatial location may have both smooth and non-smooth features. Second, the spatial correlation in these density functions is neither stationary nor locally isotropic. Finally, at some spatial locations, there is very little data. We present a method called multiscale spatial density smoothing that successfully addresses these challenges. The method is based on recursive dyadic partition of the sample space, and therefore shares much in common with other multiscale methods, such as wavelets and Pólya-tree priors. We describe an efficient algorithm for finding a maximum a posteriori (MAP) estimate that leverages recent advances in convex optimization for non-smooth functions. We apply multiscale spatial density smoothing to real data collected on the background gamma-ray spectra at locations across a large university campus. The method exhibits state-of-the-art performance for spatial smoothing in density estimation, and it leads to substantial improvements in power when used in conjunction with existing methods for detecting the kinds of radiological anomalies that may have important consequences for public health and safety. Key words: radiological survey, density estimation, spatial statistics, Bayesian nonparametrics, total-variation denoising, fused lasso author: - 'Wesley Tansey[^1]' - 'Alex Athey[^2]' - 'Alex Reinhart[^3]' - 'James G. Scott[^4]' bibliography: - 'spatial\_density\_multiscale.bib' date: 'This version: ' title: 'Multiscale spatial density smoothing: an application to large-scale radiological survey and anomaly detection' --- Detecting radiation anomalies {#sec:introduction} ============================= Overview of approach {#sec:preliminaries} ==================== Spatial smoothing via graph-based denoising {#sec:spatial_smoothing} =========================================== Simulations {#sec:simulations} =========== Radiological survey and anomaly detection at UT-Austin {#sec:anomaly_example} ====================================================== Conclusions {#sec:conclusions} =========== #### Acknowledgements. The authors thank Patrick Vetter of the UT Applied Research Laboratories for his assistance with the pilot studies described here; the University of Texas Police Department for their ongoing collaboration with data collection; and Ryan Tibshirani of CMU for sharing his expertise on algorithms for the graph-fused lasso. Pre-processing {#app:remarks} ============== Protocol for estimating empirical spectra {#app:protocol} ========================================= Further details of Bayesian method {#app:bayes} ================================== [^1]: Department of Computer Science, University of Texas at Austin. [^2]: Applied Research Laboratories, University of Texas at Austin. [^3]: Department of Statistics, Carnegie Mellon University, <areinhar@stat.cmu.edu> (corresponding author). [^4]: Department of Information, Risk, and Operations Management; Department of Statistics and Data Sciences, University of Texas at Austin.

--- abstract: 'Proton acceleration by using a 620-TW, $18$-J laser pulse of peak intensity of $5\times 10^{21}$ W/cm$^{2}$ irradiating a disk target is examined using three-dimensional particle-in-cell simulations. It is shown that protons are accelerated efficiently to high energy for a “light” material in the first layer of a double-layer target, because a strongly inhomogeneous expansion of the first layer occurs by a Coulomb explosion within such a material. Moreover, a large movement of the first layer for the accelerated protons is produced by radiation-pressure-dominant acceleration. A time-varying electric potential produced by this expanding and moving ion cloud accelerates protons effectively. In addition, using the best material for the target, one can generate a proton beam with an energy of $200$ MeV and an energy spread of 2$\%$.' author: - Toshimasa Morita title: Laser ion acceleration by using the dynamic motion of a target --- INTRODUCTION ============ Recently, there has been great progress in compact laser systems, with dramatic improvements in both laser power and peak intensity. Ion acceleration by laser pulses has proved to be very useful in applications using compact laser systems. Laser-driven fast ions are expected to be useful in many applications such as hadron therapy, [@SBK; @MVL] fast ignition for thermonuclear fusion, [@ROT; @BRM; @ATH] laser-driven heavy ion colliders, [@ESI1; @BEE] and other applications that use the high-energy ions. Although the achieved proton energy at present is not high enough for some applications such as hadron therapy, which requires 200-MeV protons, other methods can be considered for generating higher energy protons. One simple way is by using a higher power laser. However, current power capabilities of compact lasers are insufficient; moreover, laser power enhancement will result in a cost increase of the accelerator. Therefore, it is important to study conditions for generating higher energy protons with lower laser power and energy by using some special techniques. [@BWP; @FVM; @Toncian; @HAC; @YAH; @PRK; @PPM; @HSM] In this paper, I show a way to obtain $200$-MeV protons by using a laser pulse whose intensity is $I_0 \approx 10^{21}$ W/cm$^{2}$, energy is $\mathcal{E}_{las} \leq 20$ J, and power is $P \approx 500$ TW. I use three-dimensional (3D) particle-in-cell (PIC) simulations to investigate how high-energy, high-quality protons can be generated by a several-hundred-terawatt laser. I study the proton acceleration during the interaction of the laser pulse with a double-layer target composed of a high-$Z$ atom layer coated with a hydrogen layer (see Fig. \[fig:fig01\]). As suggested in Refs. and , a quasimonoenergetic ion beam can be obtained using targets of this type. Our aim is to obtain a high-energy ($\mathcal{E} \approx 200$ MeV) and high-quality ($\Delta \mathcal{E}/\mathcal{E}\leq 2\%$) proton beam using a relatively moderate power laser. In the following sections, I show the dependence of the proton energy on the material of the first layer and that the high-energy protons can be generated by optimally combining a couple of ion acceleration schemes. ION ACCELERATION ================= I consider ion acceleration by a charged disk. The charged disk is produced by a laser pulse with sufficiently high intensity irradiating a thin foil. Many electrons are driven from the foil by the laser pulse, although the ions of the foils almost stay at their initial positions because they are much heavier than the electrons. Therefore, the thin foil will have a charge, which induces an electrostatic field. Ions located on the foil surface are accelerated by this electric field. The $x$ component of the electric field of a positively charged thin disk is $$E_x(x)=\frac{\rho l}{2\epsilon_0} \left(1-\frac{x}{\sqrt{x^{2}+R^{2}}} \right), \label{exx}$$ where $\rho$ is the charge density, $l$ is the disk thickness, $\epsilon_0$ is the vacuum permittivity, and $R$ is the charged disk radius. I assume that the $x$ axis is normal to the disk surface placed at the disk center. The solid curve in Fig. \[fig:fig01\] shows this electric field. The ions, i.e. protons, are accelerated in this electric field, although it rapidly decreases as a function of distance from the target surface. The electric field decreases to $10\%$ at $x =2R$, which is the distance equal to the diameter of the target and can be considered to be the spot size of a laser pulse. Therefore, generating higher energy protons requires producing a higher surface charge density, $\rho l$, or increasing $R$. The former requires a higher intensity laser and the latter requires a higher power laser. The rapidly decreasing accelerating field and its narrow width lead to inefficient proton acceleration by the charged disk. In this paper, I present ways to improve these inefficiences and to generate high-energy protons effectively. Here, let us define the some terms. In laser ion acceleration, the ions are accelerated in some electric field, $E$. We assume that for an ion of mass of $m$ and charge $q$, the force on it from the electric field is $qE$. The equation of motion is $qE=\frac{d}{dt}(mv)$, where $v$ is the ion velocity. This equation can be written as $$E=\frac{d}{dt}(\tilde{m}v), \label{emv}$$ where $\tilde{m}=m/q$. $\tilde{m}$ is the resistance to movement of an ion in a certain electric field, $E$; therefore we call $\tilde{m}$ “mass” in this paper. This expression shows that the smaller $\tilde{m}$ ions can experience greater acceleration in a certain electric field, $E$. Therefore, small-“mass” ions will be called “light,” and big-“mass” ions will be called “heavy.” Ions of the same “mass” undergo the same movement in a certain electric field. Note that $\tilde{m}$ is equal to the inverse of the well-known parameter $q/m$, the charge-to-mass ratio. I use $\tilde{m}$ in this paper because it makes it very simple and easy to image the movement of charged particles in an electric field. ![ Configuration of a double-layer target. The $x$ component of the electric field, $\tilde{E}_x(x)$, is normalized by its maximum, $\rho l/2\epsilon_0$, of an electrically charged disk on the $x$ axis (solid curve). Protons are accelerated in this electric field. []{data-label="fig:fig01"}](fig-01.pdf){width="10.0cm"} Figure \[fig:fig01\] shows that the accelerating protons exit the electric field in a short time. This means that the electric field produced is not used enough for proton acceleration. Therefore, we should create a situation in which the protons experience this electric field longer for efficient acceleration. If the electric potential moves in the direction of the moving protons, the protons will experience the electric field longer. In other words, the charged first layer keeps pushing the moving protons. I present two ways to create this situation. ![ The first layer using “light” materials produces a strongly inhomogeneous expansion due to the Coulomb explosion (light pattern). The expanding first layer moves at average velocity $V$ in the direction of laser propagation by RPDA. The electric potential moves in the $x$ direction as a result of these effects. []{data-label="fig:fig02"}](fig-02.pdf){width="10.0cm"} One way this situation can be created by the use of a Coulomb explosion of the first layer. Figure \[fig:fig02\] shows that the first layer disk undergoes a strongly inhomogeneous expansion owing to the Coulomb explosion. This expansion raises the moving electric potential for the accelerating protons. In other words, many ions in the first layer are distributed close to the accelerating protons keeping a comparatively high density and move in the proton direction. The acceleration rate is higher when the expansion velocity of the first layer ions is higher. This means that the strong Coulomb expansion operates effectively for proton acceleration. The Coulomb explosion level is determined by the “mass” of the ions composing the first layer. Equation (\[emv\]) shows that “light” ions have a high expansion velocity. That is, “light” ions undergo a stronger a Coulomb explosion and should be generating higher energy protons. Another way to induce movement of the first layer is by radiation-pressure-dominant acceleration (RPDA). Figure \[fig:fig02\] shows that the first layer, which expands by a Coulomb explosion (with an ellipsoidal light pattern), is moving with velocity $V$ in the laser propagation direction (proton direction) by RPDA. This movement leads the moving electric potential. Higher $V$ values generate higher energy protons, since the protons experience the accelerating electric field over a longer time by following the electric potential. A portion of the energy and momentum transferred from the laser pulse to the electrons is imparted to the ions via a charge separation field. That is, the ions get accelerated by this field, and the “light” ions have higher velocity (Eq. (\[emv\])). Thus the “light” ions experience a higher first layer velocity and should be generating higher energy protons. One can obtain higher energy protons by using a “light” material in the first layer, as is corroborated by the simulations described below. The simulations were performed with a 3D massively parallel electromagnetic code, based on the PIC method. [@CBL] SIMULATION OF FIRST LAYER MATERIALS {#sim-a} =================================== In this section, I study the dependence of the proton energy on the first layer materials of the double-layer target by using simulations. Simulation parameters --------------------- Here, I show the parameters used in the simulations. The spatial coordinates are normalized by the laser wavelength $\lambda=0.8$ $\mu$m and time is measured in terms of the laser period, $2\pi/\omega$. I use an idealized model, in which a Gaussian linearly polarized laser pulse is incident on a double-layer target represented by a collisionless plasma. The laser pulse with dimensionless amplitude $a=q_eE_{0}/m_{e}\omega c=50$, which corresponds to a laser peak intensity of $5\times 10^{21}$ W/cm$^{2}$, is $10\lambda $ long in the propagation direction, $27$ fs in duration, and focused to a spot with size $4\lambda $ (FWHM), which corresponds to a laser peak power of $620$ TW and a laser energy of $18$ J. Here, the laser peak power is calculated by using $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} I(y,z)dydz$ and the laser energy is calculated by using $\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}I(y,z,t)dydzdt$, where $I$ is the laser intensity. The laser pulse is normally incident on the target. The electric field is oriented in the $y$ direction. I use this laser pulse in all simulations in this paper. Both layers of the double-layer target are shaped as disks. The first layer has a diameter of $8\lambda $ and a thickness of $0.5\lambda $. The second, hydrogen, layer is narrower and thinner; its diameter is $4\lambda $ and its thickness is $0.03\lambda $. The electron density inside the first layer is $n_{e}=3\times 10^{22}$ cm$^{-3}$ and inside the hydrogen layer it is $n_{e}=9\times 10^{20}$ cm$^{-3}$. The total number of quasiparticles is $8\times 10^{7}$. The number of grid cells is equal to $3300\times1024\times 1024$ along the $X$, $Y$, and $Z$ axes, respectively. Correspondingly, the simulation box size is $120\lambda \times 36.5\lambda \times 36.5\lambda$. The boundary conditions for the particles and for the fields are periodic in the transverse ($Y$,$Z$) directions and absorbing at the boundaries of the computation box along the $X$ axis. $xyz$ coordinates are used in the text and figures; the origin of the coordinate system is located at the center of the rear surface of the initial first layer, and the directions of the $x$, $y$, and $z$ axes are the same as those of the $X$, $Y$, and $Z$ axes, respectively. That is, the $x$ axis denotes the direction perpendicular to the target surface and the $y$ and $z$ axes lie in the plane of the target surface. Although the first layer material can be varied, the number of ions is the same in all cases and the ionization state of each ion is assumed to be $Z_{i}=+6$, [@HSB; @HKM; @JHK] since the laser parameters and the target geometry are fixed. Proton energy as a function of first layer materials ---------------------------------------------------- To examine the dependence of the proton energy, $\mathcal{E}$, on the first layer material, I performed simulations using different materials at normal incidence to the laser pulse. First, I show two simulation results, one in which the first layer consists of carbon (a “light” material) and the other is in which it consists of gold (a “heavy” material). The results are shown in Figs. \[fig:fig03\]–\[fig:fig09\]. ![ Particle distribution and electric field magnitude (isosurface for value $a=2$) for carbon (a) and gold (b). Half of the electric field box has been removed to reveal the internal structure. Shown are the initial shape of the target and the laser pulse ($t=0$), the interaction of the target and laser pulse ($t=25,50\times 2\pi/\omega$), and the first layer shape and the accelerated protons (color scale) ($t=75,100\times 2\pi/\omega$). []{data-label="fig:fig03"}](fig-03.pdf){width="10.0cm"} Figure \[fig:fig03\] shows the particle distribution and electric field magnitude of both cases at each time. We see that the carbon ions are distributed over a much wider area by its Coulomb explosion than are the gold ions. However, the deformation of the laser pulse is similar in both cases at all times. The Coulomb explosion process of the first layer is much slower than the laser pulse progress. The first layer is almost undeformed at $t=25\times 2\pi/\omega$ when the laser pulse is just around the target and it has the strong interactions with the target. A big deformation of the first layer appears at $t>25\times 2\pi/\omega$ when the laser pulse passes through or reflects from the target. Therefore, the explosion of the first layer is an almost simple Coulomb explosion without other effects. The proton energy obtained in the carbon case is much higher than that in the gold case. The average proton energy at $t=100\times 2\pi/\omega$, $\mathcal{E}_\mathrm{ave}$, is $63$ MeV for the carbon case and $38$ MeV for the gold case (a factor of 1.7 higher). ![ Distribution of the ions of the first layer and protons (color scale) in the carbon (a) and gold (b) cases at $t=100\times 2\pi/\omega$; a two-dimensional projection is shown looking along the $z$ axis. In the carbon case, the distribution area of the carbon ion cloud is very wide and it moves in the $x$ direction. []{data-label="fig:fig04"}](fig-04.pdf){width="8.0cm"} The reason why higher energy protons are obtained in the carbon case is that the first layer deformation effectively contributes to the proton acceleration. Let us examine the deformation of the first layer. Figure \[fig:fig04\] shows a cross section of the ion density distribution near the $(x,y,z=0)$ plane at $t=100\times 2\pi/\omega$, as seen by looking along the $z$ axis. In the carbon case, the strong Coulomb explosion distributes the carbon ions ellipsoidally, and the distribution area is much wider than in the gold case. The expansion of the cloud of carbon ions is strongly inhomogeneous and it appears to be substantially elongated in the longitudinal direction. The surface of the carbon ion cloud is close to the acceleration protons, which means that the protons keep getting pushed by a comparatively strong force. Moreover, the center point of the ion cloud is moving in the $x$ direction, at a coordinate of $3.1\lambda$, so that the electrostatic potential is moving in the direction of the protons. In contrast, in the gold case, the ion distribution is very compact and the distance between the ion cloud surface and the protons is much greater than in the carbon case. In addition, it almost does not move, with the coordinate of the center point of gold ions being $0.0\lambda$. Moreover, the proton positions in the carbon case have travelled further along the $x$ axis than in the gold case, since the protons in the carbon case have higher energy (higher velocity) than in the gold case. ![ (a) Average proton energy $\mathcal{E}$ versus time, as obtained in the simulation shown in Figs. \[fig:fig03\]. The proton energy of the carbon case is higher than in the gold case at all times. (b) Proton energy spectrum obtained in the simulation at $t=100\times 2\pi/\omega$, for the cases of gold and carbon, normalized by the maximum in the former case. []{data-label="fig:fig05"}](fig-05.pdf){width="8.0cm"} Here, let us consider the variation of the proton energy in time for both cases. Figure \[fig:fig05\](a) shows the average proton energy versus time. The proton energy in the carbon case is always higher than in the gold case and difference in energy grows in time. The protons are accelerated relatively quickly until a time of about $t=50\times2\pi/\omega$ in both cases. In the gold case, the acceleration almost saturates at $t>50\times2\pi/\omega$, although in the carbon case it is still increasing compared with the gold case. This is because the electric field seen by protons in the carbon case is greater and continues longer than in the gold case, owing to the expansion and movement of first layer (see Fig. \[fig:fig04\]). Correspondingly, their energy gain is also greater. In the carbon case the proton energy is 1.7 times higher than in the gold case and the energy spread is almost the same in the both cases, as seen in Fig. \[fig:fig05\](b). The energy spread, $\Delta\mathcal{E}/\mathcal{E}_\mathrm{ave}$, is 6%and 8$\%$ for the cases of carbon and gold, respectively. Higher energy protons can be obtained in the carbon first layer, as a result of the stronger electric field and movement of the first layer. I consider the differences in the electric field and the first layer movement in the two cases. ![ The $x$ component of the electric field, $E_x(x)$ (solid lines) and the proton position (circles) at each time. In the gold case, in the inset, the electric field simply decreases with distance from the target surface. In the carbon case, it does not decrease much and has a peak that moves to the right side. []{data-label="fig:fig07"}](fig-06.pdf){width="8.0cm"} First, I show the differences in the electric field between the two cases. Figure \[fig:fig07\] shows the $x$ component of the electric field along the $x$ coordinate (solid lines), $E_x(x)$, and the proton position (circular dots) at each time. In the gold case (see Fig. \[fig:fig07\](b)), the electric field simply decreases with the distance from the surface of the initial target at all times. Therefore, the electric field that protons experience (see the circular dots) simply decreases too. In contrast, in the carbon case (Fig. \[fig:fig07\](a)), the electric field peaks at points near the protons for each time. The electric field moves in the direction of the accelerating protons keeping the same shape (see Fig. \[fig:fig07\](a): $t=75$ and $t=100\times2\pi/\omega$). The electric fields at the proton positions slowly decrease, and those are higher than in the gold case at all times. At the proton position at $t=50\times2\pi/\omega$, the value of $E_x$ is $2.4$ and $1.4$ TV/m for the cases of carbon and gold ions, respectively. ![ Spatial distribution of particles and the electric field magnitude at early simulation times for the carbon case. A two-dimensional projection is shown looking along the $z$ axis. The solid bar is a projection of the first layer ($t=0$). The gas pattern denotes electrons that are pushed out from the target by the laser pulse ($t=10, 20\times 2\pi/\omega$) and distributed to the rear area (propagation direction) of the target. This produces the movement of the first layer ions. []{data-label="fig:fig08"}](fig-07.pdf){width="13.0cm"} Next, I consider the movement of the first layer. Figure \[fig:fig08\] shows the laser pulse and the target at an early time, $t<20\times2\pi/\omega$, for the carbon case, as seen by looking along the $z$ axis. The laser pulse moves from the left-hand size to the right-hand side, and the color shows the electric field magnitude. We can see that many electrons (gas pattern) are pushed out from the target by the ponderomotive force from the laser pulse. Those pushed-out electrons are distributed to the rear area (propagation direction) of the target and they move forward in the laser propagation direction, although the carbon ions stay at their initial position. This charge separation produces the strong electric field. Then, the carbon ions are moved in the laser propagation direction by this electric field (which is RPDA). ![ (a) Velocity of the first layer of the target in the $x$ direction normalized by the speed of light, $V_x/c$, as a function of time. The velocity of the carbon target is much higher than that for the gold case. (b) Movements of the first layer in the $x$ direction normalized by the wavelength, $x/\lambda$, as a function of time. []{data-label="fig:fig09"}](fig-08.pdf){width="8.0cm"} Figure \[fig:fig09\] shows the first layer velocity, $V_x$, normalized by the speed of light, and the first layer position, normalized by the wavelength, for the $x$ direction as a function of time. These are averaged values for all ions of the first layer. The first layer velocity rises rapidly at the initial time, $t \sim 20\times2\pi/\omega,$ when the laser pulse is still around the target (see Fig. \[fig:fig08\]) and the velocity is constant at time $t>25\times2\pi/\omega,$ after the laser pulse passes through or reflects off the target. The increase in the fist layer velocity stops at $t \approx 20\times 2\pi/\omega$, since by this time there is no clearly one-sided distribution of the electrons like that at time $t \approx 10\times2\pi/\omega$ (see Fig. \[fig:fig08\]). The first layer velocity in the carbon case is $14$ times that in the gold case at $t>25\times 2\pi/\omega$. This means strong RPDA occurs in the carbon case. The movement in the carbon case is much greater than in the gold case too, and the difference in distance between the carbon case and the gold case grows with time. This indicates that the moving electric potential operates efficiently in the carbon case. Incidentally, the velocity for the $y$ direction, $V_y$, of the first layer is relatively very small. It is about $1/1000$ of $V_x$ in both the carbon and gold cases. Movement for the $y$ direction is very small too, because they are normal to the incidence of the laser pulse. ![ Average proton energy, normalized by energy in the gold case, shown with respect to the ratio between the charge state, $Z_{\rm i}$, and the atomic number, $A$, of ions making up the first layer. []{data-label="fig:fig10"}](fig-09.pdf){width="7.0cm"} In above considerations, I showed that the higher energy protons can be obtained for “lighter” material in the first layer by comparing carbon and gold. Here, I investigate the effect using additional materials. Figure \[fig:fig10\] shows the average proton energy for different materials comprising the first layer. The proton energy is normalized by the energy in the case of gold ions. We see that the average proton energy almost linearly depends on the ratio of $Z_{\rm i}/A$, where $Z_{\rm i}$ is the charge state and $A$ is the atomic mass number of the ion. Higher energy protons can be obtained by using a larger ratio of $Z_{\rm i}/A$, i.e., “light,” material for the first layer in the double-layer target. In experiments on laser-driven ion acceleration, a CH polymer target exhibited higher energy protons [@SNAV] than a metallic target. [@CLAR] COMPONENTS OF THE PROTON ENERGY {#sim-b} =============================== In the previous section, I showed that higher energy protons can be obtained by using “light” material in the first layer, because a strong Coulomb explosion and a greater first layer movement toward the accelerating protons occur in such materials. These effects augment the acceleration by the electric field of the charged first layer disk. In this section, I show the amount of each effect of the acceleration on the total proton energy in the carbon case. ![ (a) 3D view and (b) the projection onto the $(x,y)$ plane for a laser pulse irradiating onto the other side, the proton layer side, of a double-layer target. Shown are the particle distribution and electric field magnitude (isosurface for value $a=2$); half of the electric field box has been removed. The protons are accelerated in the $+x$ direction, and the C ions move in the $-x$ direction by RPDA. []{data-label="fig:fig11"}](fig-10.pdf){width="10.0cm"} To examine the effect of the first layer velocity, $V$, I performed simulations with a laser pulse by reversing the irradiation direction of the laser (reverse irradiating) in the previous carbon case. Thus the hydrogen layer was put on the front side of the target (see Fig. \[fig:fig11\]). ![ Proton energy spectrum obtained in the simulation at $t=100\times 2\pi/\omega$, for the reverse irradiation case, $C^-$, and the positive irradiation (previous carbon) case, $C^+$, normalized by the maximum in the former case. []{data-label="fig:fig12"}](fig-11.pdf){width="7.0cm"} The results are shown in Figs. \[fig:fig11\] and \[fig:fig12\]. The protons are accelerating in the $+x$ direction even when the laser pulse irradiates the target in the reversed way. The average proton energy in this case, $\mathcal{E}^{-}$, is $32$ MeV. Since the proton layer is very thin and small, it has less effect on the first layer velocity. Therefore, the first layer velocity, $V^-$, has the same absolute value as in positive irradiation, the case of the previous section, but the sign is opposite, $V^-(t)=-V(t)$, where $V(t)$ is the first layer velocity in time of the positive case. Therefore, the first layer movement has a negative effect on the proton energy in the reverse irradiating case. Using this result, we can estimate the amount of proton energy attributable to the first layer velocity, $\mathcal{E}_{V}$. The proton energy in the positive irradiating case, $\mathcal{E}^{+}$, and in the reverse irradiation case, $\mathcal{E}^{-}$, are written as $$\mathcal{E}^{+}=\mathcal{E}_a+\mathcal{E}_V, \label{evp}$$ $$\mathcal{E}^{-}=\mathcal{E}_a-\mathcal{E}_V, \label{evm}$$ where $\mathcal{E}_a$ is the proton energy without the work done by the first layer velocity (i.e., attributable to the acceleration by the electric field of the charged disk of the first layer and the Coulomb explosion of the first layer ions). Hence we obtain the work done by the first layer velocity, $\mathcal{E}_{V}=(\mathcal{E}^{+}-\mathcal{E}^{-})/2$, which is $16$ MeV. Therefore, the ratio of the effect of the first layer velocity, RPDA, is $25\%$ of the total proton energy in the carbon case. In our simulations, the laser intensity and energy of $I_0=5\times 10^{21}$W/cm$^{2}$ and $\mathcal{E}_{las}=18$ J, respectively, are not enough for the RPDA regime in full scale, but this shows that the RPDA regime has a strong effect even at this laser power level. The proton energy without the work done by the first layer velocity, $\mathcal{E}_a$, is $48$ MeV. Next, I examine the work done by the Coulomb explosion of the first layer. To do so, I performed a simulation with a laser pulse reverse irradiating in the gold case. In this case, the average proton energy, $\mathcal{E}^{'-}$, is $32$ MeV. In the positive irradiation case, the gold case of the previous section, the average proton energy, $\mathcal{E}^{'+}$, was $38$ MeV. The difference between these values is very small, because the first layer velocity, $V$, is very small (see Figs. \[fig:fig04\] and \[fig:fig09\]). This means that the work done by the first layer velocity is negligibly small in the gold case. The Coulomb explosion effect is negligible, as shown in previous section, too. Therefore, we estimate that the work without the first layer velocity and Coulomb explosion in the carbon case, $\mathcal{E}_0 \approx (\mathcal{E}^{'+}+\mathcal{E}^{'-})/2$, is $35$ MeV. This is almost the energy from acceleration by only the electric field of the charged disk of the first layer. The ratio of this effect in the total energy of the carbon case is $55\%$. Then we obtain the work done by the Coulomb explosion as $\mathcal{E}_C=\mathcal{E}_a-\mathcal{E}_0=13$ MeV. The ratio of the Coulomb explosion effect, $\mathcal{E}_{C}/\mathcal{E}^{+}$, is $20\%$ in the carbon case (see Fig. \[fig:fig13\]). ![ Proton energy of each acceleration mechanism in the carbon case. The proton energy produced by the electric field of the nonmoving first layer is almost half the total proton energy, and RPDA and the Coulomb explosion each amount to almost a quarter of the total proton energy. []{data-label="fig:fig13"}](fig-12.pdf){width="10.0cm"} Another effect to consider is that of the protons being dragged by the electrons that are pushed out of the target by the laser pulse. I estimate this effect by using the gold case results. The electrons are mainly pushed out in the laser propagation direction. Therefore, in the reverse irradiation case, the dragging effect by electrons acts to reduce the proton energy. The proton energy in the positive irradiating case, $\mathcal{E}^{'+}$, and in the reverse irradiation case, $\mathcal{E}^{'-}$, are written as $\mathcal{E}^{'+}=\mathcal{E}^{'}_a+\mathcal{E}^{'}_V+\mathcal{E}^{'}_d$ and $\mathcal{E}^{'-}=\mathcal{E}^{'}_a-\mathcal{E}^{'}_V-\mathcal{E}^{'}_d$, where $\mathcal{E}^{'}_V$ is the work done by the first layer velocity, $\mathcal{E}^{'}_d$ is the work done by the dragging by the electrons, and $\mathcal{E}^{'}_a$ is other work. The effect of the first layer velocity plus the dragging by electrons, $\mathcal{E}^{'}_V+\mathcal{E}{'}_d=(\mathcal{E}^{'+}-\mathcal{E}^{'-})/2$, is $3$ MeV. Since $\mathcal{E}^{'}_V>0$, the effect of the dragging by the electrons is $\mathcal{E}^{'}_d<3$ MeV, and the ratio is less than $5\%$ of the total energy of the carbon case. Therefore, the effect whereby the protons are dragged by the electrons is very small in our simulations. This is because the electrons move very fast compared with the ions and protons, the distance between electrons and protons quickly becomes very large compared with that between the first layer ions and protons, and, moreover, the electrons are distributed over a very wide area. The electric force decreases with distance by second order. Therefore, the force the electrons exert on the protons is very small compared with the force exerted by the ions. Next, I consider the theory for the work done by the first layer velocity. I assume that the proton velocity $v > V$ and $v^2 \ll c^2$. The theoretical formula [@MBEKK] is written as $\mathcal{E}_V=mV^2(1+\sqrt{2\mathcal{E}_0/mV^2+1})$, where $m$ is the proton mass and $\mathcal{E}_0$ is the proton energy in the case of a nonmoving first layer. Denoting the proton velocity in the case of a nonmoving first layer by $v_0$ and assuming $v_0^2 \ll c^2$ we obtain $$\tilde{\mathcal{E}}_V=2\tilde{V}(\tilde{V}+\sqrt{1+\tilde{V}^2}), \label{eve0}$$ where $\tilde{\mathcal{E}_V}$ is the normalized proton energy of the work done by the first layer velocity, $\mathcal{E}_V/\mathcal{E}_0$, and $\tilde{V}$ is the normalized velocity of the first layer, $V/v_0$. ![ Proton energy produced by RPDA, $\tilde{\mathcal{E}}_V$, as a function of first layer velocity, $\tilde{V}$. The proton energy, $\mathcal{E}_V$, is normalized by the proton energy of the nonmoving first layer case, $\mathcal{E}_V/\mathcal{E}_0$, and the first layer velocity, $V$, is normalized by the proton velocity of the nonmoving first layer case, $V/v_0$. The theoretical result (solid line) is given by equation (\[eve0\]). The simulation results for the gold and carbon cases are plotted with a square and a circle, respectively. []{data-label="fig:fig14"}](fig-13.pdf){width="8.0cm"} In Fig. \[fig:fig14\], I present this theoretical dependence of $\tilde{\mathcal{E}}_V$ on $\tilde{V}$, by using formula (\[eve0\]). The proton energy by RPDA, $\mathcal{E}_V$, grows rapidly with increasing first layer velocity $V$. The simulation results (circle and square dot) are plotted on the figure. The theoretical calculations and the simulation results agree well. Since the first layer velocity in the carbon case is not so high, $V/v_0=0.13$, Fig. \[fig:fig14\] shows that if we produce a high velocity in the first layer, protons with a few times higher energy can be obtained. In addition, assuming $\tilde{V}^2 \ll 1$ we obtain the simpler formula $\tilde{\mathcal{E}}_V=2\tilde{V}$. Generating A 200-MeV proton beam ================================ I show the way to obtain a $200$-MeV proton beam, using the same laser pulse as in the previous section with the same normal incidence. In the previous section, I showed that higher energy protons can be obtained by using “light” material in the first layer. The “lightest” material is hydrogen. Therefore, we could get higher energy protons by using hydrogen for the first layer. I evaluate this contribution by simulation. In this case, even if we put the second layer of thin hydrogen on the hydrogen first layer, the second layer has no meaning. Therefore, I use a simple hydrogen disk, without a second layer. The hydrogen disk target has the same shape and size as that of the first layer of the double-layer target used previously. The electron density inside the target is $n_{e}=9\times 10^{22}$ cm$^{-3}$. Because the hydrogen cloud generated by the target is distributed over a wider area than in the previous case, I define a wider simulation box for $Y$ and $Z$ directions. The number of grid cells is equal to $5000\times 3000\times 3000$ along the $X$, $Y$, and $Z$ axes, respectively. Correspondingly, the simulation box size is $113\lambda \times 67.5\lambda \times 67.5\lambda$. The total number of quasiparticles is $4\times 10^{8}$. The other simulation parameters are the same as those used in previous sections. ![ (a) Particle distribution and electric field magnitude (isosurface for value $a=2$), showing the initial shape of the target and the laser pulse ($t=0$) and the interaction of the target and laser pulse ($t=25,50\times 2\pi/\omega$). Half of the electric field box has been removed to reveal the internal structure. For protons, the color corresponds to energy. (b) Two-dimensional projection of the particle distribution, shown looking along the $z$ axis. Half of the proton cloud has been removed to reveal the internal structure. []{data-label="fig:fig-h1"}](fig-h1.pdf){width="10.0cm"} Figure \[fig:fig-h1\](a) shows the particle distribution and the electric field magnitude in time. At $t=25\times 2\pi/\omega$, the laser pulse is just around the target and it has the strong interactions with a target. The target maintains its initial disk shape at this time. After $t=25\times 2\pi/\omega$, the laser pulse passes through or reflects off of the target, and the proton cloud produced by Coulomb explosion is growing in time. Figure \[fig:fig-h1\](b) shows a cross section of the ion cloud at each time. Hydrogen ions (protons) are classified by color in terms of energy. We see that the exploded hydrogen disk (hydrogen ions) is distributed over a very wide area. The expansion of the cloud of hydrogen ions is inhomogeneous and the cloud is elongated in the longitudinal direction. ![ Proton energy spectrum obtained in the simulation at $t=100\times 2\pi/\omega$. []{data-label="fig:fig-h2"}](fig-h2.pdf){width="8.0cm"} Figure \[fig:fig-h2\] shows the proton energy spectrum at $t=100\times 2\pi/\omega$. The maximum energy is $\mathcal{E}_\mathrm{max}=200$ MeV and the average energy is $\mathcal{E}_\mathrm{ave}=25$ MeV. The vertical axis is given in units of number of protons per $1$ MeV. We can estimate the number of obtained protons by using the required energy and energy width from Fig. \[fig:fig-h2\]. We see that we can obtain five times higher proton energy than in the gold case, even by using the same laser pulse, by using an optimum target material. ![ (a) Velocity of the hydrogen target in the $x$ direction, normalized by the speed of light, $V_x/c,$ as a function of time. (b) Movement of the hydrogen target in the $x$ direction, normalized by the wavelength, $x/\lambda$, as a function of time. []{data-label="fig:fig-h3"}](fig-h3.pdf){width="8.0cm"} Figure \[fig:fig-h3\] shows the target average velocity, normalized by the speed of light, and the target average position, normalized by the wavelength, in the $x$ direction as a function of time. The target velocity rises rapidly at the initial time, $t \sim 20\times2\pi/\omega,$ when the laser pulse is still around the target and the velocity is constant at time $t>25\times2\pi/\omega,$ after the laser pulse passes through or reflects off of the target. This is similar to the carbon case (Fig. \[fig:fig09\]), although the target velocity of this case is $2.5$ times that in the carbon case at time $t>25\times 2\pi/\omega$. That means strong RPDA appears in this case, because the target velocity is attributable to RPDA. The movement of the target is much greater than in the carbon case too. In the hydrogen disk target, the protons are accelerated more efficiently than in the carbon case, because hydrogen is much “lighter” than carbon. Because I used the same laser pulse in all simulations, the numbers of electrons pushed out from the target are estimated to be almost the same in carbon and hydrogen disk cases at the initial simulation time. Therefore, the target surface charge is almost the same in the two cases, and the proton energy by the charged disk electric field can be estimated using the same considerations as in Section \[sim-b\], yielding $35$ MeV. Therefore, we can estimate that the proton energy by RPDA and Coulomb explosion is $\approx$165 MeV. We can say that the RPDA and Coulomb explosion effect is much stronger in the hydrogen disk case compared with the carbon case. We see that protons are distributed in different areas based on each energy level (see Fig. \[fig:fig-h1\]). Moreover, high-energy protons are distributed on the $+x$ side edge of the proton cloud and are moving in the $+x$ direction. Therefore, we could select only high-energy protons by using a pinhole with a shutter. The shutter must have high enough accuracy and the ability to shield unwanted particles and radiation. The shutter speed must be very fast near the target. The accuracy becomes coarser if the shutter position moves to a position farther from the target, because the distance between the high-energy protons and the low-energy protons grows in time. It may be difficult to construct a shutter that satisfies both the timing accuracy and shielding ability, however, we could get a similar result by using a magnet and a slit. The path of a proton is changed for each energy level by a magnetic field, and high-energy protons can be taken out by passing through a slit. I show the results using the previous method. ![ Distributions of protons at $t=100\times 2\pi/\omega$ and the selected area of the proton bunch. []{data-label="fig:fig-h4"}](fig-h4.pdf){width="7.0cm"} ![ Proton energy spectrum obtained by cutting off the proton bunch at $t=100\times 2\pi/\omega$, normalized by the maximum. []{data-label="fig:fig-h5"}](fig-h5.pdf){width="8.0cm"} The cutoff position is shown in Fig. \[fig:fig-h4\]. Figure \[fig:fig-h5\] shows energy spectrum of cutoff protons at $t=100\times 2\pi/\omega$. We obtain a proton beam with a maximum energy of $\mathcal{E}_\mathrm{max}=200$ MeV and an average energy of $\mathcal{E}_\mathrm{ave}=193$ MeV with an energy spread of $\Delta\mathcal{E}/\mathcal{E}_\mathrm{ave}=2.3\%$ and a particle number of $2.6\times10^7$. This proton beam has high enough energy and quality for some applications (e.g., in medical applications). CONCLUSIONS =========== Proton acceleration driven by a laser pulse irradiating a disk target is investigated with the help of 3D PIC simulations. I have found higher energy protons are obtained by using “light” materials for the target. As seen in simulations, for these materials a strongly inhomogeneous expansion of the disk target occurs owing to the Coulomb explosion, which plays an important role, and RPDA has a strong effect. The time-varying electric potential of the inhomogeneous expanding ion cloud and the movement of the ion cloud for the protons efficiently accelerate protons. The proton beam energy can be substantially increased by using a “light” material for the target. In our simulations, the laser intensity and energy, $I_0=5\times 10^{21}$ W/cm$^{2}$ and $\mathcal{E}_{las}=18$ J, are not enough to reach the RPDA regime in full scale, but the RPDA regime has a big effect even at this laser power level. Although I show simulation results by using a simple hydrogen disk, it may be difficult to fabricate such a hydrogen disk; a CH$_n$ foil target with a high $n$ value should be a good substitute. The laser parameters used in this paper—intensity, power, energy, and spot size—are ones already existing in current laser systems. Therefore, we should be able to generate $200$-MeV protons now. ACKNOWLEDGMENTS {#acknowledgments .unnumbered} =============== I thank P. Bolton, S. V. Bulanov, T. 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--- abstract: 'In the core-degenerate (CD) scenario for the formation of Type Ia supernovae (SNe) the Chandrasekhar or super-Chandrasekhar mass white dwarf (WD) is formed at the termination of the common envelope phase or during the planetary nebula phase, from a merger of a WD companion with the hot core of a massive asymptotic giant branch (AGB) star. The WD is destructed and accreted onto the more massive core. In the CD scenario the rapidly rotating WD is formed shortly after the stellar formation episode, and the delay from stellar formation to explosion is basically determined by the spin-down time of the rapidly rotating merger remnant. The spin-down is due to the magneto-dipole radiation torque. Several properties of the CD scenario make it attractive compared with the double-degenerate (DD) scenario. (1) Off-center ignition of carbon during the merger process is not likely to occur. (2) No large envelope is formed. Hence avoiding too much mass loss that might bring the merger remnant below the critical mass. (3) This model explains the finding that more luminous SNe Ia occur preferentially in star forming galaxies.' --- The Core Degenerate (CD) Scenario ================================= Observations and theoretical studies cannot teach us yet whether both the double-degenerate (DD) and single degenerate (SD) scenarios for SNe Ia can work, only one of them, or none (e.g., [@Livio2001], [@Maoz2010], [@Howell2011]). I suggest to pay more attention to the *core-degenerate* (CD) scenario that overcomes some difficulties in the DD and SD scenarios ([@Ilkov2011; @KashiSoker2011], where more details can be found). The merger of a WD with the core of an AGB star was studied in the past ([@Sparks1974], [@Livio2003], [@Tout2008]). Livio & Riess (2003) suggested that the merger of the WD with the AGB core leads to a SN Ia that occurs at the end of the CE phase or shortly after, and can explain the presence of hydrogen lines. In the CD scenario the possibility of a very long time delay (up to $10^{10}$ yr) is considered as well. Because of its rapid rotation the super-Chandrasekhar WD does not explode ([@Yoon2005]). The CD scenario is summarized schematically in Figure \[fig:fig1\]. ![A schematic summary of the core-degenerate (CD) scenario for SNe Ia (from Ilkov & Soker 2011). []{data-label="fig:fig1"}](sokerfig1.eps) Contrary to the view presented by Mario Livio in his review talk, I think the CD scenario is not a branch of the DD scenario, but rather a distinguish scenario. Both the CD and DD scenarios require the merger of the remnants of AGB stars (the core or the descendant WD) to form a degenerate WD above the critical mass. However, there are three key ingredients in the CD scenario that distinguish it from the DD scenario. (1) The hot core is more massive than the companion cold WD. (2) The merger should occur while the core is still large, hence hot. This limits the merger to occur within $\sim 10^5$ yr after the common envelope phase. Kashi & Soker (2011) showed that this condition can be met when the AGB star is massive. (3) In the CD scenario most of the delay between the binary formation time and the explosion is due to the spinning-down time of the merger product. The spinning-down is due to the magneto-dipole radiation torque (and not gravitational waves; see [@Ilkov2011]). In the DD scenario most of the delay time is the spiraling-in time of the two WDs (caused by gravitational radiation). The strong points of the CD scenario ==================================== They most important factor is that the hot core is larger than its final radius when it becomes a cold WD. At $\sim 10^5$ yr after it left the AGB the radius of a $M_{\rm core } \sim 0.7-0.8 M_\odot$ remnant is $\xi \simeq 1.2$ times its final radius as a cold WD ([@Bloecker1995]). This more or less limits the time period over which merger must occur. Most likely the merger will occur much earlier, while the core is still large $\xi > 1.2$. Since in the CD scenario the core is more massive than the WD companion, the WD companion will be destructed. I now raise some strong points of the CD scenario, and compare it with the DD scenario. **Carbon ignition off-center.** The main problem for the DD scenario is that in many cases an off-center carbon ignition occurs (e.g., [@SaioNomoto2004]) leading to accretion induced collapse (AIC) rather than a SNe Ia. Yoon et al. (2007) raised the possibility that in a merger process where the more massive WD is hot, off-axis ignition of carbon is less likely to occur. The reason is that a hot WD is larger, such that its potential well is shallower and the peak temperature of the destructed WD (the lighter WD) accreted material is lower. Hence, in such a case the supercritical-mass remnant is more likely to ignite carbon in the center at a later time, leading to a SN Ia. Namely, the merger remnant becomes a rapidly rotating massive WD, that can collapse only after it loses sufficient angular momentum. **Mass loss of the merger product.** Consider two merging cold WDs in the DD scenario. The less massive WD is destructed, and its mass is accreted onto the more massive WD. The gravitational well of the more massive WD is much deeper than that of the destructed WD (e.g., [@Dan2011]), and a large amount of energy is liberated $\sim 10^{50}$ erg. If the remnant radiates the extra energy during a very short time $t_r$, we would expect for a very bright event with a peak luminosity of $L_{\rm merg} \sim 10^8 (t_r/10~{\rm yr})^{-1} L_\odot$. This by itself will be at an almost SN luminosity. Do we observe such objects? If the energy release time is longer, the material of the destructed WD has time to expand and form a giant-like structure ([@Shen2011]). According to Shen et al. (2011) the giant-like phase lasts for $\sim 10^4$ years and its luminosity is half the Eddington limit. Such giants with a solar composition lose mass at a rate of few$\times 10^{-5} M_\odot~{\rm yr}^{-1}$ ([@Willson2007]). When the carbon rich atmosphere of the merger remnant is considered the mass loss rate will be higher even. Therefore, over the giant-like structure phase that lasts for $\sim 10^4 {\rm yr}$, the remnant might lose about half a solar mass and decrease below the critical mass for explosion. In the CD scenario the more massive WD is hot, and the potential well is much lower. Assume a WD with a radius of $R_{\rm WD} \propto M_{\rm WD}^{-1/3}$ and a core with a radius of $R_{\rm core} \propto \xi M_{\rm core}^{-1/3}$. Then the ratio of the potentials is $$\frac {\Psi_{\rm core}} {\Psi_{\rm WD}} \simeq \frac{1}{\xi} \left( \frac {M_{\rm core}}{M_{\rm WD}} \right)^{4/3} = 1 \left( \frac{\xi}{1.5} \right)^{-1} \left( \frac {M_{\rm core}/0.8M_\odot}{M_{\rm WD}/0.6M_\odot} \right)^{4/3} . \label{eq:ed}$$ The crude equality of potentials implies that the destruction of the less massive WD and the accretion of its mass onto the core will not release large amount of energy, and no formation of a giant-like structure will take place. The merger remnant will not have a large radius, and no substantial mass loss will take place. The merger remnant will continue to evolve as a massive central star of a planetary nebulae. **More luminous SNe Ia in star forming galaxies** The strong magnetic fields required in the present model for the spin-down mechanism most likely will enforce a rigid rotation within a short time scale due the WD being a perfect conductor. 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--- abstract: 'We examine the results of Chiral Effective Field Theory ([$\chi$EFT]{}) for the scalar- and spin-dipole polarisabilities of the proton and neutron, both for the physical pion mass and as a function of ${\ensuremath{m_\pi}}$. This provides chiral extrapolations for lattice-QCD polarisability computations. We include both the leading and sub-leading effects of the nucleon’s pion cloud, as well as the leading ones of the $\Delta(1232)$ resonance and its pion cloud. The analytic results are complete at [N${}^{2}$LO]{} in the $\delta$-counting for pion masses close to the physical value, and at leading order for pion masses similar to the Delta-nucleon mass splitting. In order to quantify the truncation error of our predictions and fits as $68$% degree-of-belief intervals, we use a Bayesian procedure recently adapted to EFT expansions. At the physical point, our predictions for the spin polarisabilities are, within respective errors, in good agreement with alternative extractions using experiments and dispersion-relation theory. At larger pion masses we find that the chiral expansion of all polarisabilities becomes intrinsically unreliable as ${\ensuremath{m_\pi}}$ approaches about $300\;{\ensuremath{\mathrm{MeV}}}$—as has already been seen in other observables. [$\chi$EFT]{}also predicts a substantial isospin splitting above the physical point for both the electric and magnetic scalar polarisabilities; and we speculate on the impact this has on the stability of nucleons. Our results agree very well with emerging lattice computations in the realm where [$\chi$EFT]{}converges. Curiously, for the central values of some of our predictions, this agreement persists to much higher pion masses. We speculate on whether this might be more than a fortuitous coincidence.' --- 6th November 2015\ Final version 18 May 2016, accepted by Eur. Phys. J. **A**. [**Harald W. Grie[ß]{}hammer$^{a}$**]{}[^1], [**Judith A. McGovern$^{b}$**]{}[^2] *and* [**Daniel R. Phillips$^{c}$**]{}[^3] [*$^a$ Institute for Nuclear Studies, Department of Physics,\ The George Washington University, Washington DC 20052, USA*]{}\ [*$^b$ School of Physics and Astronomy, The University of Manchester,\ Manchester M13 9PL, UK*]{}\ [*$^c$ Department of Physics and Astronomy and Institute of Nuclear and Particle Physics, Ohio University, Athens, Ohio 45701, USA*]{} --------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Suggested Keywords: Effective Field Theory, lattice QCD, chiral extrapolation, proton, neutron and nucleon polarisabilities, spin polarisabilities, Chiral Perturbation Theory, $\Delta(1232)$ resonance, Bayesian statistics, uncertainty/error estimates. --------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Introduction ============ \[sec:introduction\] The polarisabilities of a composite system are among its most basic properties; see e.g. [@Griesshammer:2012we] for a recent review. At a classical level, they reflect how much freedom charged constituents have to rearrange under the application of external electromagnetic fields, while in quantum mechanics they indicate how easily electromagnetic interactions induce transitions to low-lying excited states. They therefore encode information about the symmetries and strengths of constituents’ interactions with each other and with the photon. As well as the usual electric (${\ensuremath{\alpha_{E1}}}$) and magnetic (${\ensuremath{\beta_{M1}}}$) polarisabilities, a spin-half object like the nucleon has four “spin-polarisabilities" ($\gamma_i$). These are less obvious in their effects but encode the spin-dependent response and can, for instance, be related to effects analogous to birefringence and Faraday rotation for long-wavelength electromagnetic radiation. In the nucleon, the lightest relevant excitation involves the creation of a virtual charged pion. This mechanism is expected to dominate the electric polarisability and contribute significantly to others, too. The exploration of nucleon polarisabilities was therefore a natural early application of Chiral Perturbation Theory in the baryonic sector [@Jenkins:1991ne; @Bernard:1991rq; @Bernard:1995dp] which predicts the behaviour of each polarisability as it diverges in the chiral limit ${\ensuremath{m_\pi}}\to0$ [@Bernard:1991rq]. On the other hand, in the real world the excitation energy of the $\Delta(1232)$ resonance, ${\ensuremath{\Delta_{\scriptscriptstyle M}}}\equiv {\ensuremath{M_\Delta}}-{\ensuremath{M_\mathrm{N}}}$, is about $300\;{\ensuremath{\mathrm{MeV}}}$, and thus not very much larger than the physical pion mass. Furthermore, the strong magnetic N$\Delta$ dipole transition should give a large paramagnetic contribution to the magnetic polarisability. The inclusion of the Delta as an explicit degree of freedom in Chiral Effective Field Theory [@Jenkins:1991ne; @Butler:1992ci; @Hemmert:1996xg; @Hemmert:1997ye] enables quantitative predictions to be made for Compton scattering [@Pascalutsa:2002pi; @Hildebrandt:2003fm]. This EFT has recently been used in the most accurate extant determinations of the electric and magnetic polarisabilities of the proton and neutron from Compton scattering data [@Griesshammer:2012we; @McGovern:2012ew; @Myers:2014ace]. This progress in the theory of polarisabilities is coupled to an upsurge of interest in new experiments that are devoted to obtaining or refining our knowledge of all the polarisabilities, electric, magnetic and spin, of both the proton and neutron [@Weller:2009zz; @HIGSPAC; @Downie:2011mm; @Huber:2015uza], with results from MAXlab [@Myers:2014ace; @Myers:2015aba] and MAMI [@Martel:2014pba] published within the last year. The calculation of nucleon polarisabilities directly from the QCD action is also an aim of lattice QCD. The need to incorporate electromagnetic fields in the computation creates challenges, which means that this is a fairly new endeavour, but several groups now have published results [@Chang:2015qxa; @Lujan:2014qga; @Detmold:2010ts; @Primer:2013pva; @Hall:2013dva; @Engelhardt:2011qq; @Engelhardt:2007ub; @Engelhardt:2010tm; @Engelhardt:2015; @Freeman:2014kka]. Since all are at pion masses substantially above the physical pion mass, the question of how to extrapolate to the real world is of pressing interest, and can be addressed within [$\chi$EFT]{}. Our analysis provides a bridge between data and lattice QCD, where a direct computation of Compton scattering would be highly nontrivial. Polarisabilities are therefore fundamental characteristics of hadrons, and benchmarks for our understanding of hadronic structure; a summary of their importance and best ways to access them was also provided by a number of theorists in Ref. [@Griesshammer:2014xla]. Furthermore, their values have other implications, some examples of which we now discuss. First, the Cottingham Sum rule relates the doubly-virtual forward Compton scattering amplitude, and hence the proton-neutron difference in ${\ensuremath{\beta_{M1}}}$, to the proton-neutron electromagnetic mass difference [@WalkerLoud:2012bg; @WalkerLoud:2012en; @Erben:2014hza; @Thomas:2014dxa; @Gasser:2015dwa]. The relation between the mass difference and the polarisabilities proceeds via a low-energy theorem for the subtraction function in the Cottingham formula at vanishing momentum, which is related to ${\ensuremath{\beta_{M1}}}^{(\text{p-n})}$ [@WalkerLoud:2012bg; @Gasser:2015dwa]. When one uses present knowledge on ${\ensuremath{\beta_{M1}}}^{(\text{p-n})}$ as input and models the subtraction function along the lines suggested in Refs. [@WalkerLoud:2012bg; @WalkerLoud:2012en; @Erben:2014hza], the uncertainty in the polarisability contributes sizeably to the uncertainty in the mass difference. Conversely, assuming knowledge about the electromagnetic part of the mass difference provides a constraint on the polarisabilities [@Thomas:2014dxa]. Either scenario tests our understanding of the subtle interplay between electromagnetic and strong interactions in a fundamental observable. Second, the magnetic polarisability, ${\ensuremath{\beta_{M1}}}$, is also crucial for the two-photon-exchange contribution to the Lamb shift in muonic hydrogen [@Pachucki; @Carlson:2011dz; @Pohl:2013yb], the least-known ingredient of the “proton-radius puzzle”. The aim of this paper is thus two-fold. Firstly, we will present the analytic expressions and numerical results for all static dipole polarisabilities as they enter in the Compton amplitudes used in the recent proton and neutron analyses [@McGovern:2012ew; @Myers:2014ace]. There is considerable evidence that the extraction of ${\ensuremath{\alpha_{E1}^{(\mathrm{p})}}}$ and ${\ensuremath{\beta_{M1}^{(\mathrm{p})}}}$ from unpolarised Compton scattering is robust against variations in the spin polarisabilities [@Griesshammer:2012we; @McGovern:2012ew; @Lensky:2014efa]. This means that polarisation observables are the best place to determine these latter quantities [@observables], and programs at MAMI and HI$\gamma$S are engaged in that pursuit [@Weller:2009zz; @HIGSPAC; @Downie:2011mm; @Huber:2015uza]. Secondly, we use our expressions to predict the running of the polarisabilities with the pion mass, the better to compare with lattice computations at numerically less costly, heavier, pion masses. In both these contexts, we pay particular attention to the uncertainties of our predictions and extractions, which are of two types. The impact of statistical errors on data on [$\chi$EFT]{}parameters can ultimately be reduced by experimental and non-EFT-related efforts. However, there is also a “truncation error” which is intrinsic to an EFT, and it is that we focus on in this paper. Because [$\chi$EFT]{}gives a perturbative series for all polarisabilities, this truncation error accounts for the fact that we only have computed up to a finite order in the EFT expansion [@Cacciari:2011ze; @Furnstahl:2015rha]. Without its proper appraisal—and that of any other uncertainties entering the [$\chi$EFT]{}result—the significance of any agreement or discrepancy between theory and experiment cannot be assessed [@Furnstahl:2014xsa]. On a technical note we mention here that although polarisabilities are frequency-dependent functions (see e.g. [@Hildebrandt:2003fm; @Babusci:1998ww; @Griesshammer:2012we]), this paper is concerned with the static values, that is the limit as $\omega\to0$. We will report these in the canonical units of $10^{-4}~{\rm fm}^3$ for the scalar polarisabilities, and $10^{-4}~{\rm fm}^4$ for the spin ones. Preliminary findings were reported in Ref. [@talkMAMI] and provided for inclusion in Refs. [@Martel:2014pba; @Lujan:2014qga]. The presentation is organised as follows. In Sect. \[sec:formalism\], we define the chiral power counting in the regimes relevant for lattice computations and summarise the analytic results for the scalar and spin polarisabilities of the proton and neutron. After presenting their values and uncertainties for physical pion masses in Sect. \[sec:values\], we detail our procedure to assign Bayesian degree-of-belief intervals (Sects. \[sec:theoryerrors\] and \[sec:Finding-Theory-Uncertainties\]) and discuss convergence checks. Section \[sec:extrapolations\] extends this procedure to pion masses above the physical point, provides predictions with error bars in Fig. \[fig:allpols\], and concludes with speculations on the relationship of our findings to the proton-neutron mass splitting and anthropic arguments. We then compare with available lattice computations in Sect. \[sec:lattice\], and add Conclusions and an Appendix. Chiral EFT with Dynamical Delta(1232) for Polarisabilities ========================================================== \[sec:formalism\] Chiral Regimes and Power Counting {#sec:regimes} --------------------------------- Compton scattering on nucleons in [$\chi$EFT]{}has been reviewed in Refs. [@Griesshammer:2012we; @McGovern:2012ew], to which we refer the reader for notation and the relevant parts of the chiral Lagrangian. Here, we briefly discuss the power counting, which is crucial for our considerations, and sketch the results. Recall that Compton scattering exhibits three typical low-energy scales in [$\chi$EFT]{}with a dynamical Delta: the pion mass ${\ensuremath{m_\pi}}$ as the typical chiral scale; the Delta-nucleon mass splitting ${\ensuremath{\Delta_{\scriptscriptstyle M}}}\approx 300\;{\ensuremath{\mathrm{MeV}}}$; and the photon energy $\omega$. Each provides a small, dimensionless expansion parameter when measured in units of a natural “high” scale $\Lambda_\chi\gg{\ensuremath{\Delta_{\scriptscriptstyle M}}},{\ensuremath{m_\pi}},\omega$ at which the theory is to be expected to break down because new degrees of freedom enter. For static scalar polarisabilities, one considers the part of the amplitude which is quadratic in $\omega$ as $\omega\to0$, and for spin polarisabilities the one cubic in $\omega$. That leaves two parameters: $$P({\ensuremath{m_\pi}}) \equiv \frac{{\ensuremath{m_\pi}}}{\Lambda_\chi} \qquad \epsilon \equiv \frac{{\ensuremath{M_\Delta}}-{\ensuremath{M_\mathrm{N}}}}{\Lambda_\chi}\approx0.4\;\;, \label{eq:expparams}$$ where for simplicity we take one common breakdown scale $\Lambda_\chi\approx 650\;{\ensuremath{\mathrm{MeV}}}$ for both expansions, consistent with the masses of the $\omega$ and $\rho$ as the next-lightest exchange mesons; we also count ${\ensuremath{M_\mathrm{N}}}\sim\Lambda_\chi$. This scale is only weakly dependent on ${\ensuremath{m_\pi}}$, and we will treat it as constant. The fact that these two expansion parameters have a very different functional dependence on ${\ensuremath{m_\pi}}$ has important consequences for chiral extrapolations. The Delta-nucleon mass splitting depends only weakly on the pion mass, and hence $\epsilon$ is independent of ${\ensuremath{m_\pi}}$ at the order to which we work. By definition, though, the chiral parameter $P({\ensuremath{m_\pi}})$ does change significantly with ${\ensuremath{m_\pi}}$. We therefore identify three regimes relevant in contemporary lattice computations, based on the relative size of $P$ and $\epsilon$. We stress that regimes are not clearly separated, but transition from one regime to the next is gradual. In **regime (i)**, around the physical pion mass, ${\ensuremath{m_\pi}}\approx{\ensuremath{m_\pi^\text{phys}}}$, we follow Pascalutsa and Phillips [@Pascalutsa:2002pi] and exploit a numerical coincidence to define a single expansion parameter $\delta$: $$\label{eq:deltacountingi} \mbox{regime (i): } \delta\approx\epsilon\approx\sqrt{P({\ensuremath{m_\pi^\text{phys}}})}\approx0.4\;\;.$$ This is, of course, the regime relevant for the analysis of real-world Compton scattering data, and hence this power counting determines the contributions which were included in Refs. [@Griesshammer:2012we; @McGovern:2012ew] which should be consulted for more details. As the pion mass increases, we move into **regime (ii)**, ${\ensuremath{m_\pi}}\approx{\ensuremath{\Delta_{\scriptscriptstyle M}}}\approx 300\;{\ensuremath{\mathrm{MeV}}}$. The two expansion parameters are now numerically of comparable size, $P({\ensuremath{m_\pi}})\approx\epsilon$, but their ${\ensuremath{m_\pi}}$ dependence is still different. It is then appropriate to identify $$\label{eq:deltacountingii} \mbox{regime (ii): } \epsilon\approx P({\ensuremath{m_\pi}}\approx{\ensuremath{\Delta_{\scriptscriptstyle M}}})\approx 0.4$$ as the sole expansion parameter [@Jenkins:1991ne; @Butler:1992ci; @Hemmert:1996xg; @Hemmert:1997ye]. Finally in **regime (iii)**, ${\ensuremath{m_\pi}}\to\Lambda_\chi$, [$\chi$EFT]{}becomes inapplicable because the chiral expansion does not converge. A chiral extrapolation of any observable can be expected to hold qualitatively at best. In Sect. \[sec:lattice\], we will see that [$\chi$EFT]{}’s polarisabilities agree in the main with extant lattice results at such pion masses, but why this should be so is unclear. The corresponding uncertainties are certainly impossible to quantify with present techniques. Note that we will not discuss the power counting for very small pion masses, ${\ensuremath{m_\pi}}\ll {\ensuremath{M_\Delta}}- {\ensuremath{M_\mathrm{N}}}$. This regime near the chiral limit is fascinating, but it will be some time before lattice computations explore it. Dipole Polarisabilities in Regime (i) {#sec:pols} ------------------------------------- Here we bring together the relevant expressions for the various contributions to the polarisabilities: $\pi$N loops (calculated to subleading order), $\pi\Delta$ loops, Delta pole diagrams, and low-energy constants from the fourth-order $\pi$N Lagrangian. The expressions for the dipole polarisabilities are mostly published, but they are scattered in the literature. The numerical values of all variables are listed in Refs. [@Griesshammer:2012we; @McGovern:2012ew][^4]. When we cite the expressions for all polarisabilities, we are excluding any “non-structure" effects which would persist for a point-like nucleon with an anomalous magnetic moment. For the spin polarisabilities we also exclude the contribution of the $\pi^0$ pole [@Griesshammer:2012we]. These definitions are standard in the literature, except that the backward spin polarisability ${\ensuremath{\gamma_{\pi}}}$ is often given without subtraction of the pion pole contribution of $\mp45.9$ for the proton/neutron. In terms of the multipole polarisabilities, the forward and backward spin polarisabilities are defined as $$\label{eq:fbspin} {\ensuremath{\gamma_{0}}}:=-{\ensuremath{\gamma_{E1E1}}}-{\ensuremath{\gamma_{M1M1}}}-{\ensuremath{\gamma_{E1M2}}}-{\ensuremath{\gamma_{M1E2}}}\;\;,\;\; {\ensuremath{\gamma_{\pi}}}:=-{\ensuremath{\gamma_{E1E1}}}+{\ensuremath{\gamma_{M1M1}}}-{\ensuremath{\gamma_{E1M2}}}+{\ensuremath{\gamma_{M1E2}}}\;\;.$$ We work in a heavy-baryon framework, except for the Delta pole, whose case is explained shortly. In $\delta$ counting, in regime (i), the leading contribution to the Compton scattering amplitudes is the Thomson term which is ${\mathcal{O}}(e^2)$, followed by leading $\pi$N loops which are ${\mathcal{O}}(e^2\delta^2)$, then diagrams with a single Delta propagator which are ${\mathcal{O}}(e^2\delta^3)$, and finally subleading $\pi$N loops and LECs (counter terms) at ${\mathcal{O}}(e^2\delta^4)$. In two cases, our expressions also contain terms which are higher-order in $\delta$ counting. First, we do not expand the $\pi\Delta$ loop expressions in powers of ${\ensuremath{m_\pi}}/{\ensuremath{\Delta_{\scriptscriptstyle M}}}\approx\delta$; this has the advantage that the expression remains valid as we move towards regime (ii) ${\ensuremath{m_\pi}}\sim {\ensuremath{\Delta_{\scriptscriptstyle M}}}$. However, we do omit higher-order graphs that are suppressed by ${\ensuremath{\Delta_{\scriptscriptstyle M}}}/\Lambda_\chi$ relative to the leading ones. The other exception is that we use a covariant Delta propagator for its pole graphs. Since no loops or renormalisation are involved, this is simply a convenient way of accounting for some kinematic higher-order effects, including the electric $\gamma$N$\Delta$ coupling, which are relevant in the regime $\omega\sim{\ensuremath{\Delta_{\scriptscriptstyle M}}}$. For our present purposes these are not necessary, but as they are small at the physical point and do not affect the running with ${\ensuremath{m_\pi}}$, we retain them for consistency with our previous work [@McGovern:2012ew]. Before presenting the formulae, we note a subtlety which arises when one counts polarisabilities rather than amplitudes. The electric and magnetic polarisabilities ${\ensuremath{\alpha_{E1}}}$ and ${\ensuremath{\beta_{M1}}}$ follow the power-counting outlined above; since in the amplitudes they multiply two powers of $\omega\sim{\ensuremath{m_\pi}}\sim\delta^2$, the contributions are ${\mathcal{O}}(e^2\delta^{-2})\sim {\ensuremath{m_\pi}}^{-1}$ from the leading $\pi$N pieces, ${\mathcal{O}}(e^2\delta^{-1})\sim {\ensuremath{\Delta_{\scriptscriptstyle M}}}^{-1}$ from Delta contributions and ${\mathcal{O}}(e^2\delta^0)$ from subleading $\pi$N pieces, generating $\ln{\ensuremath{m_\pi}}$ as well as ${\ensuremath{m_\pi}}$-independent contributions. On the other hand, the spin polarisabilities $\gamma_i$ multiply three powers of $\omega\sim{\ensuremath{m_\pi}}$ and hence start at ${\mathcal{O}}(e^2\delta^{-4})\sim {\ensuremath{m_\pi}}^{-2}$. However, there are no contributions at ${\mathcal{O}}(e^2\delta^{-3})\sim ({\ensuremath{m_\pi}}{\ensuremath{\Delta_{\scriptscriptstyle M}}})^{-1}$ since the Delta-nucleon mass difference acts as an infrared cut-off, forbidding Delta contributions to diverge in the chiral limit. Instead, the Delta contributions start at ${\mathcal{O}}(e^2\delta^{-2})\sim {\ensuremath{\Delta_{\scriptscriptstyle M}}}^{-2}$, and these, together with the subleading pion loops which are ${\mathcal{O}}(e^2\delta^{-2})\sim {\ensuremath{m_\pi}}^{-1}$, form the next non-zero contribution to the (isoscalar) spin polarisabilities. This has consequences for our error estimates, which are more reliable for ${\ensuremath{\alpha_{E1}}}$ and ${\ensuremath{\beta_{M1}}}$ where we have three nonzero terms in the $\delta$ expansion series, than they are for the $\gamma_i$ where there are only two. In either case, the last contribution calculated is of order $\delta^2$ relative to leading. ### piN Loops {#sec:Npiloops} The leading-order (LO) contributions from the pion cloud around the nucleon, Fig. \[fig:LOpiN\], were first calculated by Bernard, Kaiser and Mei[ß]{}ner [@Bernard:1991rq; @Bernard:1995dp]: $$\begin{aligned} &{\ensuremath{\alpha_{E1}}}^{\pi\text{N, LO}} = 10 {\ensuremath{\beta_{M1}}}^{\pi\text{N, LO}} = \frac{5{\alpha_{\scriptscriptstyle\text{EM}}}{g_{\scriptscriptstyle A}}^2}{96\pi{\ensuremath{f_\pi}}^2{\ensuremath{m_\pi}}}\label{eq:abpiN-LO}\\ &{\ensuremath{\gamma_{E1E1}}}^{\pi\text{N, LO}}=5{\ensuremath{\gamma_{M1M1}}}^{\pi\text{N, LO}}= -5{\ensuremath{\gamma_{M1E2}}}^{\pi\text{N, LO}}=-5{\ensuremath{\gamma_{E1M2}}}^{\pi\text{N, LO}}= -\frac{5{\alpha_{\scriptscriptstyle\text{EM}}}{g_{\scriptscriptstyle A}}^2}{96\pi^2{\ensuremath{f_\pi}}^2{\ensuremath{m_\pi}}^2}\;\;. \label{eq:gpiN-LO}\end{aligned}$$ As motivated above, they diverge in the chiral limit and are indeed ${\mathcal{O}}(e^2{\ensuremath{m_\pi}}^{-1}\sim e^2\delta^{-2})$ for the scalar polarisabilities, and ${\mathcal{O}}(e^2{\ensuremath{m_\pi}}^{-2}\sim e^2\delta^{-4})$ for the spin ones. For future reference we note that the $\pi^0$-pole contributes $${\ensuremath{\gamma_{E1E1}}}^{\pi^0}=-{\ensuremath{\gamma_{M1M1}}}^{\pi^0}= {\ensuremath{\gamma_{E1M2}}}^{\pi^0}=-{\ensuremath{\gamma_{M1E2}}}^{\pi^0}=\tau_3\;\frac{e^2{g_{\pi{\scriptscriptstyle\text{NN}}}}}{16\pi^3{\ensuremath{f_\pi}}{\ensuremath{M_\mathrm{N}}}m_{\pi^0}^2}\;, \label{eq:pi-pole}$$ where $\tau_3$ is the third Pauli matrix in isospin space. This has the numerical value of $11.5$ for the proton at the physical pion mass (with ${g_{\pi{\scriptscriptstyle\text{NN}}}}^2/(4\pi)=13.64$ [@Rentmeester:1999vw]). ![(Colour online) Leading contributions to the polarisabilities from the pion cloud around a nucleon in [$\chi$EFT]{}. Interactions without symbol from ${\mathcal{L}}_{\pi\mathrm{N}}^{(1)}$ [@Bernard:1995dp]. Permuted and crossed diagrams not displayed.[]{data-label="fig:LOpiN"}](graphs-piN-LO.pdf){width="72.00000%"} The first chiral corrections, shown in Fig. \[fig:corrpiN\], were found by Bernard et al. [@bksm93] for the scalar polarisabilities (logarithmic in ${\ensuremath{m_\pi}}$) and by Kumar, McGovern and Birse [@VijayaKumar:2000pv] for the spin ones (one inverse power of ${\ensuremath{m_\pi}}$): $$\begin{aligned} {\ensuremath{\alpha_{E1}}}^{\pi\text{N, corr}} = & \frac{{\alpha_{\scriptscriptstyle\text{EM}}}}{24\pi^2{\ensuremath{f_\pi}}^2}\left[ \left(\frac{2(3+\tau_3){g_{\scriptscriptstyle A}}^2}{{\ensuremath{M_\mathrm{N}}}}-c_2\right)\ln\frac{{\ensuremath{m_\pi}}}{{\ensuremath{m_\pi^\text{phys}}}}+ \left(\frac{(27+8\tau_3){g_{\scriptscriptstyle A}}^2}{4{\ensuremath{M_\mathrm{N}}}} -(2c_1+\frac{c_2}{2}-c_3)\right) \right]{\nonumber}\\ {\ensuremath{\beta_{M1}}}^{\pi\text{N, corr}} = & \frac{{\alpha_{\scriptscriptstyle\text{EM}}}}{24\pi^2{\ensuremath{f_\pi}}^2}\left[ \left(\frac{3(2+(1+{\ensuremath{\kappa^{(\mathrm{s})}}})\tau_3){g_{\scriptscriptstyle A}}^2}{{\ensuremath{M_\mathrm{N}}}}-c_2\right) \ln\frac{{\ensuremath{m_\pi}}}{{\ensuremath{m_\pi^\text{phys}}}}\right. \label{eq:abpiN-corr}\\ &\left.\hspace{10ex}+\left(\frac{(13+6(1+{\ensuremath{\kappa^{(\mathrm{s})}}})\tau_3){g_{\scriptscriptstyle A}}^2}{4{\ensuremath{M_\mathrm{N}}}}+ (2c_1-\frac{c_2}{2}-c_3)\right)\right]{\nonumber}\\ \label{eq:gpiN-corr} \begin{split} {\ensuremath{\gamma_{E1E1}}}^{\pi\text{N, corr}} = & \frac{{\alpha_{\scriptscriptstyle\text{EM}}}{g_{\scriptscriptstyle A}}^2}{384\pi{\ensuremath{f_\pi}}^2{\ensuremath{M_\mathrm{N}}}{\ensuremath{m_\pi}}} 11(2+\tau_3) \\ {\ensuremath{\gamma_{M1M1}}}^{\pi\text{N, corr}} = & \frac{{\alpha_{\scriptscriptstyle\text{EM}}}{g_{\scriptscriptstyle A}}^2}{384\pi{\ensuremath{f_\pi}}^2{\ensuremath{M_\mathrm{N}}}{\ensuremath{m_\pi}}} \left(15+4{\ensuremath{\kappa^{(\mathrm{v})}}}+4(1+{\ensuremath{\kappa^{(\mathrm{s})}}})\tau_3\right)\\ {\ensuremath{\gamma_{E1M2}}}^{\pi\text{N, corr}} = & \frac{{\alpha_{\scriptscriptstyle\text{EM}}}{g_{\scriptscriptstyle A}}^2}{384\pi{\ensuremath{f_\pi}}^2{\ensuremath{M_\mathrm{N}}}{\ensuremath{m_\pi}}} (-6-\tau_3)\\ {\ensuremath{\gamma_{M1E2}}}^{\pi\text{N, corr}} = & \frac{{\alpha_{\scriptscriptstyle\text{EM}}}{g_{\scriptscriptstyle A}}^2}{384\pi{\ensuremath{f_\pi}}^2{\ensuremath{M_\mathrm{N}}}{\ensuremath{m_\pi}}} \left(-1+2{\ensuremath{\kappa^{(\mathrm{v})}}}-2(1+{\ensuremath{\kappa^{(\mathrm{s})}}})\tau_3\right)\;\;, \end{split}\end{aligned}$$ where ${\ensuremath{\kappa^{(\mathrm{s})}}}={\ensuremath{\kappa^{(\mathrm{p})}}}+{\ensuremath{\kappa^{(\mathrm{n})}}}=-0.12$ and ${\ensuremath{\kappa^{(\mathrm{v})}}}={\ensuremath{\kappa^{(\mathrm{p})}}}-{\ensuremath{\kappa^{(\mathrm{n})}}}=3.71$ are the anomalous magnetic moments of the nucleon, and $c_{1,2,3}$ low-energy constants from the next-to-leading order (NLO) $\pi$N Lagrangian, determined, e.g. from $\pi$N scattering. We set the renormalisation scale in the chiral logarithms to be ${\ensuremath{m_\pi^\text{phys}}}$. ![(Colour online) Subleading contributions to the polarisabilities from the pion cloud around a nucleon in [$\chi$EFT]{}. Notation as in Fig. \[fig:LOpiN\]; square: vertex from ${\mathcal{L}}_{\pi\mathrm{N}}^{(2)}$ [@Bernard:1995dp]. Permuted and crossed diagrams not displayed.[]{data-label="fig:corrpiN"}](graphs-piN-NLO.pdf){width="82.00000%"} To the order we work, these are the *only* ${\ensuremath{m_\pi}}$-dependent contributions which contain an isovector component and hence differentiate between proton and neutron polarisabilities. For ${\ensuremath{\alpha_{E1}}}$ and ${\ensuremath{\beta_{M1}}}$, the chiral logarithm provides a parameter-free and rather strong ${\ensuremath{m_\pi}}$-dependence in the difference—besides an ${\ensuremath{m_\pi}}$-independent offset. The pion-mass dependence of the proton-neutron split in the spin polarisabilities is stronger, scaling with ${\ensuremath{m_\pi}}^{-1}$, but will turn out to be considerably smaller than the theoretical uncertainties of our predictions. See discussions in Sects. \[sec:results\], \[sec:isovector\] and \[sec:lattice\]. ### Low-Energy Coefficients {#sec:CTs} In addition to the loops, there are contributions to polarisabilities directly from the ${\ensuremath{m_\pi}}$- and ${\ensuremath{\Delta_{\scriptscriptstyle M}}}$-independent low-energy coefficients (LECs) multiplying operators in the Lagrangian. Their finite parts subsume physics which is unrelated to the pion cloud or to the Delta, generically: $$\label{eq:scalarLEC} \xi^{\text{LEC}}=\xi^{\text{(s)LEC}}+\tau_3\, \xi^{\text{(v)LEC}}$$ for the isoscalar and isovector LECs of any polarisability $\xi\in\{{\ensuremath{\alpha_{E1}}},{\ensuremath{\beta_{M1}}},\gamma_i\}$. Of these, only ${\ensuremath{\alpha_{E1}}}^{\text{LEC}}$ and ${\ensuremath{\beta_{M1}}}^{\text{LEC}}$ need to be included as counter terms at the same order as the $\pi$N corrections to ${\ensuremath{\alpha_{E1}}}$ and ${\ensuremath{\beta_{M1}}}$ of eqs. ; they absorb the renormalisation-point dependence of the chiral logarithms induced by the divergent loops of Fig. \[fig:corrpiN\]. In Ref. [@McGovern:2012ew], we determined the proton values by fitting to a statistically consistent proton Compton database detailed in Ref. [@Griesshammer:2012we]. The results, given in eq. , are constrained by the Baldin sum rule ${\ensuremath{\alpha_{E1}^{(\mathrm{p})}}}+{\ensuremath{\beta_{M1}^{(\mathrm{p})}}}=13.8\pm0.4$ [@Olmos:2001], though fitting ${\ensuremath{\alpha_{E1}^{(\mathrm{p})}}}$ and ${\ensuremath{\beta_{M1}^{(\mathrm{p})}}}$ separately produces results which are consistent within the uncertainties. In the same work, it was found that a good fit to (unpolarised) proton Compton scattering data at ${\mathcal{O}}(e^2\delta^4)$ in the amplitudes could only be achieved if one also determines ${\ensuremath{\gamma_{M1M1}}}^{\text{(p)LEC}}$. This makes ${\ensuremath{\gamma_{M1M1}^{(\mathrm{p})}}}$ also a fitted quantity. The neutron scalar polarisabilities are obtained from deuteron targets. In Ref. [@Myers:2014ace], we extracted scalar polarisabilities for the neutron from the statistically-consistent world data, updated with the recent high-quality data from MAX-lab [@Myers:2015aba], with the neutron’s Baldin sum rule ${\ensuremath{\alpha_{E1}^{(\mathrm{n})}}}+{\ensuremath{\beta_{M1}^{(\mathrm{n})}}}=15.2\pm0.4$ as a constraint [@Levchuk:1999zy]. This extraction was carried out at one order lower, ${\mathcal{O}}(e^2\delta^3)$, which means that the theoretical uncertainties are larger, but there was no need to fit ${\ensuremath{\gamma_{M1M1}^{(\mathrm{n})}}}$. For the sake of the present study, we take a minimalist approach and assume that the ${\ensuremath{\gamma_{M1M1}}}$ LEC we promoted by one order is purely isoscalar, while all other short-distance contributions to the spin polarisabilities enter at higher order. In Sect. \[sec:results\], we will show that the dependence of ${\ensuremath{\gamma_{M1M1}}}$ on the pion mass provides supporting evidence for this. ### Delta(1232) Pole Contribution Since ${\ensuremath{\Delta_{\scriptscriptstyle M}}}$ is about $30\%$ of ${\ensuremath{M_\mathrm{N}}}$, recoil effects in this part of the amplitude are expected to be sizeable. We thus choose to include purely kinematic, relativistic effects in the Delta pole contribution (see leftmost graph of Fig. \[fig:Deltapole\]). The results in Lorentz-covariant kinematics were implicit in Ref. [@McGovern:2012ew], but are stated here for the first time. Related results, but with several errors, were given in Ref. [@Pascalutsa:2003zk]. $$\begin{aligned} \label{eq:abDelta} \begin{split} {\ensuremath{\alpha_{E1}}}^{\Delta} &= -\frac{2{\alpha_{\scriptscriptstyle\text{EM}}}b_2^2}{9{\ensuremath{M_\mathrm{N}}}^2(2{\ensuremath{M_\mathrm{N}}}+{\ensuremath{\Delta_{\scriptscriptstyle M}}})} \\ {\ensuremath{\beta_{M1}}}^{\Delta} &= \frac{2{\alpha_{\scriptscriptstyle\text{EM}}}b_1^2}{9{\ensuremath{M_\mathrm{N}}}^2{\ensuremath{\Delta_{\scriptscriptstyle M}}}} \end{split}\\ \begin{split} \label{eq:gDelta} {\ensuremath{\gamma_{E1E1}}}^{\Delta}&= \frac{{\alpha_{\scriptscriptstyle\text{EM}}}}{18{\ensuremath{M_\mathrm{N}}}^3}\left[ -\frac{b_1^2}{{\ensuremath{\Delta_{\scriptscriptstyle M}}}}+\frac{b_1b_2}{2{\ensuremath{M_\mathrm{N}}}+{\ensuremath{\Delta_{\scriptscriptstyle M}}}}- \frac{2b_2^2({\ensuremath{M_\mathrm{N}}}+{\ensuremath{\Delta_{\scriptscriptstyle M}}})}{(2{\ensuremath{M_\mathrm{N}}}+{\ensuremath{\Delta_{\scriptscriptstyle M}}})^2}\right]\\ {\ensuremath{\gamma_{M1M1}}}^{\Delta}&= \frac{{\alpha_{\scriptscriptstyle\text{EM}}}}{18{\ensuremath{M_\mathrm{N}}}^3}\left[ \frac{2b_1^2({\ensuremath{M_\mathrm{N}}}+{\ensuremath{\Delta_{\scriptscriptstyle M}}})}{{\ensuremath{\Delta_{\scriptscriptstyle M}}}^2}-\frac{b_1b_2}{{\ensuremath{\Delta_{\scriptscriptstyle M}}}}+ \frac{b_2^2}{2{\ensuremath{M_\mathrm{N}}}+{\ensuremath{\Delta_{\scriptscriptstyle M}}}}\right]\\ {\ensuremath{\gamma_{E1M2}}}^{\Delta}&= \frac{{\alpha_{\scriptscriptstyle\text{EM}}}}{18{\ensuremath{M_\mathrm{N}}}^3}\left[ -\frac{b_1^2}{{\ensuremath{\Delta_{\scriptscriptstyle M}}}}+\frac{3b_1b_2}{2{\ensuremath{M_\mathrm{N}}}+{\ensuremath{\Delta_{\scriptscriptstyle M}}}}\right]\\ {\ensuremath{\gamma_{M1E2}}}^{\Delta}&= \frac{{\alpha_{\scriptscriptstyle\text{EM}}}}{18{\ensuremath{M_\mathrm{N}}}^3}\left[ -\frac{3b_1b_2}{{\ensuremath{\Delta_{\scriptscriptstyle M}}}}+\frac{b_2^2}{2{\ensuremath{M_\mathrm{N}}}+{\ensuremath{\Delta_{\scriptscriptstyle M}}}}\right] \end{split}\end{aligned}$$ These contributions reduce at leading order in the heavy-baryon limit, ${\ensuremath{M_\mathrm{N}}}\to\infty$ and $b_2\to0$, to the results by Hemmert et al. [@hhk97; @hhkk98], namely ${\ensuremath{\gamma_{M1M1}}}^{\Delta}={\ensuremath{\beta_{M1}}}^{\Delta}/(2\Delta_M)$. All other polarisabilities are zero in this limit. Here, $b_1$ and $b_2$ are the magnetic and electric $\gamma$N$\Delta$ couplings respectively. Their leading chiral loop corrections were derived in Ref. [@McGovern:2012ew] but enter an order of ${\ensuremath{m_\pi}}/{\Lambda_\chi}\sim \delta^2$ higher, and are thus omitted from the results above. ### pi-Delta Loops The leading contributions from the pion cloud around the Delta, Fig. \[fig:piDelta\], were calculated in Refs. [@hhk97; @hhkk98] using the heavy-baryon approximation: $$\begin{aligned} \label{eq:abpiDelta} \begin{split} {\ensuremath{\alpha_{E1}}}^{\pi\Delta} &= \frac{{\alpha_{\scriptscriptstyle\text{EM}}}{g_{\pi{\scriptscriptstyle\text{N}\Delta}}}^2}{54\pi^2{\ensuremath{f_\pi}}^2} \left[\frac{9{\ensuremath{\Delta_{\scriptscriptstyle M}}}}{{\ensuremath{\Delta_{\scriptscriptstyle M}}}^2-{\ensuremath{m_\pi}}^2}+ \frac{{\ensuremath{\Delta_{\scriptscriptstyle M}}}^2-10{\ensuremath{m_\pi}}^2}{({\ensuremath{\Delta_{\scriptscriptstyle M}}}^2-{\ensuremath{m_\pi}}^2)^\frac{3}{2}}{\;F\!\left(\frac{{\ensuremath{\Delta_{\scriptscriptstyle M}}}}{{\ensuremath{m_\pi}}}\right)}\right]\\ {\ensuremath{\beta_{M1}}}^{\pi\Delta} &= \frac{{\alpha_{\scriptscriptstyle\text{EM}}}{g_{\pi{\scriptscriptstyle\text{N}\Delta}}}^2}{54\pi^2{\ensuremath{f_\pi}}^2} \frac{1}{({\ensuremath{\Delta_{\scriptscriptstyle M}}}^2-{\ensuremath{m_\pi}}^2)^\frac{1}{2}}{\;F\!\left(\frac{{\ensuremath{\Delta_{\scriptscriptstyle M}}}}{{\ensuremath{m_\pi}}}\right)}\\ \end{split}&\\ \label{eq:gpiDelta} \begin{split} {\ensuremath{\gamma_{E1E1}}}^{\pi\Delta} &= \frac{{\alpha_{\scriptscriptstyle\text{EM}}}{g_{\pi{\scriptscriptstyle\text{N}\Delta}}}^2}{108\pi^2{\ensuremath{f_\pi}}^2} \left[\frac{{\ensuremath{\Delta_{\scriptscriptstyle M}}}^2+5{\ensuremath{m_\pi}}^2}{({\ensuremath{\Delta_{\scriptscriptstyle M}}}^2-{\ensuremath{m_\pi}}^2)^2}+ \frac{{\ensuremath{\Delta_{\scriptscriptstyle M}}}({\ensuremath{\Delta_{\scriptscriptstyle M}}}^2-7{\ensuremath{m_\pi}}^2)}{({\ensuremath{\Delta_{\scriptscriptstyle M}}}^2-{\ensuremath{m_\pi}}^2)^\frac{5}{2}} {\;F\!\left(\frac{{\ensuremath{\Delta_{\scriptscriptstyle M}}}}{{\ensuremath{m_\pi}}}\right)}\right]\\ {\ensuremath{\gamma_{M1M1}}}^{\pi\Delta} &= -{\ensuremath{\gamma_{E1M2}}}^{\pi\Delta}= -{\ensuremath{\gamma_{M1E2}}}^{\pi\Delta} = - \frac{{\alpha_{\scriptscriptstyle\text{EM}}}{g_{\pi{\scriptscriptstyle\text{N}\Delta}}}^2}{108\pi^2{\ensuremath{f_\pi}}^2}\! \left[\frac{1}{{\ensuremath{\Delta_{\scriptscriptstyle M}}}^2-{\ensuremath{m_\pi}}^2}- \frac{{\ensuremath{\Delta_{\scriptscriptstyle M}}}}{({\ensuremath{\Delta_{\scriptscriptstyle M}}}^2-{\ensuremath{m_\pi}}^2)^\frac{3}{2}}\!{\;F\!\left(\frac{{\ensuremath{\Delta_{\scriptscriptstyle M}}}}{{\ensuremath{m_\pi}}}\right)}\right]\!, \end{split}&\end{aligned}$$ with $F(x)=\mathrm{arcsinh}(\sqrt{x^2-1})$ and ${g_{\pi{\scriptscriptstyle\text{N}\Delta}}}$ the $\pi$N$\Delta$ coupling constant. These results are real, continuous and non-singular for all ratios ${\ensuremath{m_\pi}}/{\ensuremath{\Delta_{\scriptscriptstyle M}}}>0$. ![(Colour online) Leading contributions to the polarisabilities from the $\Delta(1232)$ (leftmost) and its pion cloud. Notation as in Fig. \[fig:LOpiN\]; covariant $\gamma$N$\Delta$ and heavy-baryon $\pi$N$\Delta$ vertices from Ref. [@McGovern:2012ew]. Permuted and crossed diagrams not displayed.[]{data-label="fig:Deltapole"}](graphs-DeltapiDelta.pdf){width="90.00000%"} \[fig:piDelta\] Polarisabilities and Uncertainties at the Physical Point ======================================================== \[sec:physpoint\] Summary {#sec:values} ------- The formulae of the previous section with the LECs for ${\ensuremath{\alpha_{E1}^{(\mathrm{p})}}}$, ${\ensuremath{\beta_{M1}^{(\mathrm{p})}}}$, ${\ensuremath{\alpha_{E1}^{(\mathrm{n})}}}$, ${\ensuremath{\beta_{M1}^{(\mathrm{n})}}}$ and ${\ensuremath{\gamma_{M1M1}}}^\text{(s)}$ fitted to unpolarised Compton scattering data give the values of the scalar polarisabilities of the nucleons in units of $10^{-4}~{\rm fm}^3$ as follows [@McGovern:2012ew; @Myers:2014ace]: $$\begin{aligned} \label{eq:scalar-pol-values} \begin{split} {\ensuremath{\alpha_{E1}^{(\mathrm{p})}}}=&10.65\pm0.35(\text{stat})\pm0.2(\text{Baldin}) \pm0.3(\text{theory})\\ {\ensuremath{\beta_{M1}^{(\mathrm{p})}}}=&\phantom{0}3.15\mp0.35(\text{stat})\pm0.2(\text{Baldin}) \mp0.3(\text{theory})\\ {\ensuremath{\alpha_{E1}^{(\mathrm{n})}}}=&11.55\pm1.25(\text{stat})\pm0.2(\text{Baldin})\pm0.8(\text{theory})\\ {\ensuremath{\beta_{M1}^{(\mathrm{n})}}}=&\phantom{0}3.65\mp1.25(\text{stat})\pm0.2(\text{Baldin}) \mp0.8(\text{theory})\;\;, \end{split}\end{aligned}$$ with $\chi^2=113.2$ for $135$ degrees of freedom for the proton, and $45.2$ for $44$ for the neutron. Notice that due to the imposition of the Baldin sum rule for each nucleon, both the statistical and theory errors are anticorrelated between ${\ensuremath{\alpha_{E1}}}$ and ${\ensuremath{\beta_{M1}}}$; see also Sect. \[sec:Finding-Theory-Uncertainties\]. For the spin polarisabilities, in units of $10^{-4}~{\rm fm}^4$: $$\begin{aligned} \label{eq:spin-pol-values} {\ensuremath{\gamma_{E1E1}^{(\mathrm{p})}}}&=-1.1\pm1.9(\text{theory}){\hspace{1em}}{\hspace{1em}}&{\hspace{1em}}{\hspace{1em}}{\ensuremath{\gamma_{E1E1}^{(\mathrm{n})}}}&=-4.0\pm1.9(\text{theory})\nonumber\\ {\ensuremath{\gamma_{M1M1}^{(\mathrm{p})}}}&= {\phantom{-}}2.2\pm0.5(\text{stat})\pm0.6(\text{theory})& {\ensuremath{\gamma_{M1M1}^{(\mathrm{n})}}}&={\phantom{-}}1.3\pm0.5(\text{stat})\pm0.6(\text{theory})\nonumber\\ {\ensuremath{\gamma_{E1M2}^{(\mathrm{p})}}}&=-0.4\pm0.6(\text{theory})& {\ensuremath{\gamma_{E1M2}^{(\mathrm{n})}}}&=-0.1\pm0.6(\text{theory})\nonumber\\ {\ensuremath{\gamma_{M1E2}^{(\mathrm{p})}}}&={\phantom{-}}1.9\pm0.5(\text{theory})& {\ensuremath{\gamma_{M1E2}^{(\mathrm{n})}}}&={\phantom{-}}2.4\pm0.5(\text{theory})\end{aligned}$$ The central values for the proton were given in Ref. [@McGovern:2012ew] and cited, with an estimation of uncertainties supplied by the current authors, in Refs. [@talkMAMI; @Martel:2014pba][^5]. A justification of the theoretical uncertainties in eqs.  and , which are derived from order-by-order convergence of the results, will be the subject of the next two subsections. Here, we only remark that they encompass $68$% intervals but that the corresponding probability is not distributed in a Gau[ß]{}ian manner. Note also that the statistical error from fitting ${\ensuremath{\gamma_{M1M1}^{(\mathrm{p})}}}$ along with ${\ensuremath{\alpha_{E1}^{(\mathrm{p})}}}$ and ${\ensuremath{\beta_{M1}^{(\mathrm{p})}}}$ in Ref. [@McGovern:2012ew] is inherited by related quantities, including ${\ensuremath{\gamma_{M1M1}^{(\mathrm{n})}}}$; cf. Sect. \[sec:CTs\]. Figure \[fig:abvalues\] illustrates the pattern of convergence and the $1\sigma$ ellipses for the scalar polarisabilities. At present, there is only a weak signal that proton and neutron polarisabilities differ, and then only if the Baldin sum rule is used. The ellipses are obtained by adding all uncertainties (statistical, Baldin, and truncation) in quadrature. The truncation error plays a minor role, so although, strictly speaking, its non-Gau[ß]{}ian distribution (see Sect. \[sec:theoryerrors\] below) mandates a more sophisticated treatment of error combination, that would not lead to ellipses which are appreciably different from those shown here. [|l|l|llll|]{} ------------------------------------------------------------------------ & this work: ${\mathcal{O}}(e^2\delta^{-2})$&NLO B$\chi$PT & MAMI & DR(I) & DR(II)\ ------------------------------------------------------------------------ ${\ensuremath{\gamma_{E1E1}^{(\mathrm{p})}}}$&$-1.1\pm1.9$& $-3.3\pm 0.8$&$-3.5\pm1.2$ & $-3.4$&$-4.3$\ ------------------------------------------------------------------------ ${\ensuremath{\gamma_{M1M1}^{(\mathrm{p})}}}$& ${\phantom{-}}2.2\pm0.5(\text{stat})\pm0.6$&${\phantom{-}}2.9\pm 1.5$&${\phantom{-}}3.2\pm0.9$& ${\phantom{-}}2.7$&${\phantom{-}}2.9$\ ------------------------------------------------------------------------ ${\ensuremath{\gamma_{E1M2}^{(\mathrm{p})}}}$&$-0.4\pm0.6$&${\phantom{-}}0.2\pm 0.2 $&$-0.7\pm1.2$& ${\phantom{-}}0.3 $0.15 0.15$ $&${\phantom{-}}0.0$\ ------------------------------------------------------------------------ ${\ensuremath{\gamma_{M1E2}^{(\mathrm{p})}}}$&${\phantom{-}}1.9\pm0.5$&${\phantom{-}}1.1\pm0.3$&${\phantom{-}}2.0\pm0.3$& ${\phantom{-}}1.9$&${\phantom{-}}2.2$\ ------------------------------------------------------------------------ ${\ensuremath{\gamma_{0}^{(\mathrm{p})}}}$&$-2.6\pm0.5(\text{stat})\pm1.8$&$-0.9\pm1.4$&$-1.0\pm0.1\pm0.1$& $-1.5$&$-0.8$\ ------------------------------------------------------------------------ ${\ensuremath{\gamma_{\pi}^{(\mathrm{p})}}}$&${\phantom{-}}5.5\pm0.5(\text{stat})\pm1.8$&${\phantom{-}}7.2\pm1.7$&${\phantom{-}}8.0\pm1.8$& ${\phantom{-}}7.8$&${\phantom{-}}9.4$\ Table \[tab:spinpols\] gives the comparison with the recent MAMI extraction of the proton spin polarisabilities from the first double-polarised Compton measurements [@Martel:2014pba]. These extractions were constrained to reproduce the listed values of two linear combinations, namely the forward and backward spin polarisabilities $\gamma_0$ [@Ahrens:2001qt; @Dutz; @Pasquini:2010zr] and $\gamma_\pi$ [@Camen]. (Note, however, that the value for ${\ensuremath{\gamma_{0}}}$ is much more model-independent than that for ${\ensuremath{\gamma_{\pi}}}$.) Within their stated uncertainties, they all overlap our $68$% confidence intervals. This agreement, and specifically the agreement for ${\ensuremath{\gamma_{M1M1}^{(\mathrm{p})}}}$, supports our previously-developed strategy of promoting and fitting the isoscalar LEC of ${\ensuremath{\gamma_{M1M1}}}$ to unpolarised data [@McGovern:2012ew]. Future experiments on double-polarisation observables on the proton and light nuclei that are running or approved at MAMI and [HI$\gamma$S]{}will provide high-accuracy data for more conclusive comparisons [@Weller:2009zz; @HIGSPAC; @Downie:2011mm; @Huber:2015uza]. A [$\chi$EFT]{}analysis of the polarised scattering data for the proton is forthcoming [@observables]. At this order, the only differences between neutron and proton spin polarisabilities come from the pion-cloud corrections of eq. . Little is known about the neutron spin polarisabilities. Our prediction ${\ensuremath{\gamma_{0}^{(\mathrm{n})}}}=0.5\pm0.5(\mathrm{stat}) \pm1.8$ is certainly compatible with the expectation that this quantity should be “about zero” [@Schumacher:2005an]. For ${\ensuremath{\gamma_{\pi}^{(\mathrm{n})}}}$, our value is $7.7\pm0.5(\mathrm{stat})\pm1.8$. This places us on the low end of the range $12.7\pm4.0$ extracted in a DR framework from the reaction $\gamma\text{d}\to\gamma {\rm p} {\rm n}$ [@Kossert:2002ws], but uncertainties in that theory may be underestimated [@Griesshammer:2012we]. A Theory Of Theoretical Uncertainties {#sec:theoryerrors} ------------------------------------- The intrinsic uncertainty of any EFT calculation comes from the truncation of the EFT series at a finite order $k$. It is clear that definitive results for terms in the series that have not been computed are impossible to obtain. Nevertheless, estimates of the EFT truncation error can be made using the same strategy as in other perturbative quantum field theories, i.e., by combining knowledge of the perturbative parameter with reasonable assumptions about the behavior of higher-order coefficients. Ref. [@Cacciari:2011ze] used Bayesian methods to implement this strategy for perturbative QCD, and Ref. [@Furnstahl:2015rha] adapted the approach to EFTs. Other methods for estimating the truncation error are certainly possible, but this Bayesian framework has the advantage that it allows clear specification of premises: it facilitates a rigorous derivation of the theoretical uncertainties from a particular assumption about the behaviour of higher-order coefficients in the EFT series. Note that we work in the leading-omitted-term approximation, i.e. we assume the error associated with that term dominates the theoretical uncertainty. As long as [$\chi$EFT]{}is convergent, this is a reasonable assumption (cf. discussion of its accuracy below). We also point out that Bayesian methods can aid in the extraction and uncertainty estimation of [$\chi$EFT]{}LECs that appear in the nucleon polarisabilities [@Schindler:2008fh; @Wesolowski:2015fqa], but we do not pursue that avenue here, instead applying them only to truncation errors. Suppose we have computed $k$ non-trivial orders of a generic polarisability $\xi$, with the first (leading) order being $c_0$: $$\centralvaluexi= \sum_{n=0}^{k-1} c_n \delta^n\;\;. \label{eq:xiexp}$$ The canonical EFT estimate of the truncation error, , is then determined by the largest magnitude of the coefficients $c_n$ as follows: $$\scalexi=\max_n \{|c_n|: n=0, \ldots, k-1\}\times\delta^k \;\;. \label{eq:errorestimate}$$ Equation  was the basis of the determination of the truncation error of scalar polarisabilities in Refs. [@Griesshammer:2012we; @McGovern:2012ew]. An entirely equivalent error estimation was recently articulated and advocated for two- and three-nucleon-system observables in Refs. [@Epelbaum:2014; @Epelbaum:2014sza; @Binder:2015mbz]. We want to understand how to interpret such a truncation error. To do that, we compute the probability distribution function (pdf), denoted ${{\rm pr}}(\errorxi|I)$, namely the degree of belief that a polarisability will take a specific value which differs by an amount $\error$ from the calculated central value $\centralvaluexi$ of the [$\chi$EFT]{}prediction at order $k$. This belief is based upon the available information $I$ which includes the order $k$ of the calculation, the behaviour of the [$\chi$EFT]{}series, our expectations regarding naturalness, and—in the case of fitted polarisabilities—data at the physical pion mass. There is no reason to assume this pdf will be Gau[ß]{}ian; in general, it is not. Nevertheless, it can still be integrated to compute degree-of-belief intervals (DoB intervals), such as the Bayesian analogue of the usual 68% (1$\sigma$) and 95% (2$\sigma$) confidence intervals. Since this is a Bayesian approach, a choice of prior for the coefficients $c_j$ associated with the higher-order terms is mandatory. We implement a prior which assigns uniform probability to any value of the omitted coefficients, up to some unspecified maximum, as for prior $A_\epsilon^{(1)}$ of Ref. [@Furnstahl:2015rha]. These assumptions lead to analytic expressions for the probability distribution: $$\text{pr}(\errorxi | \scalexi,k)=\frac{k}{k+1}\; \frac{1}{2\scalexi}\times\left\{\begin{array}{ll} 1&\mbox{ for } |\errorxi | \le \scalexi\\ \left({\displaystyle}\frac{\scalexi}{ |\errorxi|}\right)^{k+1} &\mbox{ for } |\errorxi | > \scalexi\end{array}\right. \label{eq:theoryerror}$$ For $k=1,2,3$, these pdfs are illustrated for in Fig. \[fig:combinedpdfs\]. By integration of the pdf, we define $\sigma_\xi$ such that $[\centralvaluexi-\sigma_\xi;\centralvaluexi+\sigma_\xi]$ is the 68% DoB interval. This is also how we define the theoretical or truncation error in eqs.  and , and throughout this work. For comparison, the standard EFT error estimate, $[\centralvaluexi-\scalexi;\centralvaluexi+ \scalexi]$, is a $\frac{k}{k+1} \times 100$% DoB interval. For two of the cases we are concerned with, namely $k=2$ and $k=3$, these are $67\%$ and $75\%$ intervals, respectively, so there is little difference between $\scalexi$ and $\sigma_\xi$. We close with a few comments. First, the truncation uncertainty is not distributed in a Gau[ß]{}ian way. This can be understood as follows. Each time a new order is calculated, more information is gleaned about the largest-possible coefficient in the series. That the probability is equidistributed for any value inside the standard EFT interval between zero and $R_\xi$ is inherited from our choice of prior. On the other hand, the probability of finding a coefficient larger than the maximum of those obtained thus far becomes smaller the more orders are known—a fact represented by the steeper power-law falloff above the maximum as the EFT calculation’s order increases. Indeed, the $95\%$ DoB interval of the pdf in eq.  is not twice as large as the $68$% interval $\sigma_\xi$, as would be expected for a Gau[ß]{}ian pdf; see also the examples in Fig. \[fig:combinedpdfs\] below. Instead, it lies at about $7\sigma$ for $k=1$, $2.6\sigma$ for $k=2$, and $1.9\sigma$ for $k=3$. Second, in cases where we know that one of the $c_n$ is [*a priori*]{} zero, e.g. by symmetry arguments, since the power-law falloff of the pdf outside the “standard” EFT interval is determined by the number of non-trivial orders computed, the value of $k$ to be used in eq.  must then be reduced by one. Different interpretations of the naturalness of EFT coefficients are encoded in different priors, and those in turn can produce somewhat different 68% DoB intervals. In Ref. [@Furnstahl:2015rha], several possibilities for the naturalness prior were considered and it was demonstrated that—once three orders in the EFT series are known—those different interpretations of naturalness led to a 10–15% variation in the truncation error. However, the variation is larger if fewer coefficients in the series have been computed, as then the specific form of the prior plays more of a role in determining the final error. Furthermore, our invocation of the first-omitted-term approximation means that our error bands have a fractional uncertainty that could be as large as ${\mathcal O}(\delta)$, although the actual impact of terms beyond the (first omitted) $\delta^k$ term of Eq. (\[eq:errorestimate\]) depends on whether coefficients at order $k$ and beyond are correlated, uncorrelated, or anti-correlated [@Furnstahl:2015rha]. Regardless, this suggests that the truncation error computed here should be understood to itself have an accuracy of $\pm 20$%. The errors quoted below should be read with this in mind. For uniformity of presentation, we quote all errors to one decimal place, but in certain instances this may constitute spurious precision. Assigning Theoretical Uncertainties to Spin Polarisabilities {#sec:Finding-Theory-Uncertainties} ------------------------------------------------------------ The values of the spin polarisabilities at ${\mathcal{O}}(e^2\delta^{-4})$ and ${\mathcal{O}}(e^2\delta^{-2})$ are given in Table \[tab:convergence\]. The first order is isoscalar, while the next one contains both isoscalar and isovector components. The series for the isoscalar spin polarisabilities is therefore computed to an accuracy of $k=2$ nonzero orders; the contribution of order $\delta^1$ relative to LO is zero, and so we know only $c_0$ and $c_2$ in eq. . We hence obtain the results for the isoscalar remainder $\scalexi$ from eq.  shown in Table \[tab:convergence\]. In practice $c_2$ is the larger coefficient in each case, and so $\scale_{\gamma_i}$ is $\delta$ times the $e^2\delta^{-2}$ contribution. Comparing with eq.  for $k=2$, we find that this remainder can be interpreted as a $67$% DoB interval. For the isovector spin polarisabilities we only have one order, and so $\scale_{\gamma_i}$ is $\delta$ times the total and the corresponding DoB interval is only $50$%. [|l|llll|]{} ------------------------------------------------------------------------ &${\ensuremath{\gamma_{E1E1}}}$&${\ensuremath{\gamma_{M1M1}}}$&${\ensuremath{\gamma_{E1M2}}}$&${\ensuremath{\gamma_{M1E2}}}$\ isoscalar $e^2\delta^{-4}$& $-5.7$&$-1.1$&$\phantom{+}1.1$&$\phantom{+}1.1$\ isoscalar $e^2\delta^{-2}$&$-2.6\pm1.3$&$\phantom{+}1.8 \pm0.5$&$-0.3\pm0.6$ &$\phantom{+}2.2\pm0.4$\ isovector $e^2\delta^{-2}$&$\phantom{+}1.5\pm0.6$& $\phantom{+}0.5\pm0.2$& $-0.1\pm0.1$& $-0.2\pm0.1$\ To get the pdf of the truncation error for an individual nucleon, we convolute the two pdfs: $$\text{pr}_\xi(\errorxi)\equiv\text{pr}(\errorxi | \scale{}^\text{(s)}_\xi,k^\text{(s)}, \scale{}^\text{(v)}_\xi,k^\text{(v)}) =\int\limits_{-\infty}^\infty{\mathrm{d}}y\, \text{pr}(y| \scale{}^\text{(s)}_\xi,k^\text{(s)}) \text{pr}(\errorxi-y|\ \scale{}^\text{(v)}_\xi,k^\text{(v)})\;\;, \label{eq:combining}$$ with obvious notation for isovector and isoscalar pieces. As Fig. \[fig:combinedpdfs\] shows, this smears out the individual pdfs considerably so that they look somewhat more Gau[ß]{}ian. Results are the same for the proton and neutron. The additional convolution makes the relation between $95$% and $68$% DoBs depend on both $\errorxi^\text{(s)}$ and $\errorxi^\text{(v)}$. ![(Colour online) Examples of pdfs. Left: The pdf of ${\ensuremath{\alpha_{E1}}}-{\ensuremath{\beta_{M1}}}$ at the physical point for the proton (red solid) and neutron (blue dashed) as resulting from the fits underlying eq. , at [N${}^{2}$LO]{} for the proton ($k=3$, $\scale_{{\ensuremath{\alpha_{E1}}}-{\ensuremath{\beta_{M1}}}}=0.6$ in eq. ) and NLO for the neutron ($k=2$, $\scale_{{\ensuremath{\alpha_{E1}}}-{\ensuremath{\beta_{M1}}}}=1.6$). Right: The pdf of the truncation error in ${\ensuremath{\gamma_{E1E1}}}$ at the physical point. The green dotted line shows the isovector combination’s truncation error ($k^{(\text{v})}=1$, $ \scale{}^\text{(v)}_{{\ensuremath{\gamma_{E1E1}}}}=0.6$), the blue dashed line that of the isoscalar ($k^{(\text{s})}=2$, $ \scale{}^\text{(s)}_{{\ensuremath{\gamma_{E1E1}}}}=1.3$), and the red line is the pdf that results from the integration in eq. . The solid (dashed) grey line denotes the 68% (95%) DoB interval. Pdfs for all polarisabilities are available in the Appendix.[]{data-label="fig:combinedpdfs"}](posterioralphaminusbetapn+gammaee.pdf){width="80.00000%"} In all cases, simply adding isoscalar and isovector errors linearly yields ranges that are near-identical to the 68% DoB intervals. One could also choose to study proton and neutron convergence patterns separately, instead of those of the isoscalar and isovector quantities. In general this produces somewhat smaller errors; it never gives larger ones. For example, the proton uncertainty of ${\ensuremath{\gamma_{E1E1}}}$ would be about equally large, while that for the neutron would be quite a bit smaller ($\pm0.7$). Whether one uses the proton-neutron or the isospin basis is a question of choice. We take the latter because its error assessments are more conservative and because [$\chi$EFT]{}is most naturally formulated in the isospin basis. We now turn to ${\ensuremath{\gamma_{M1M1}}}$ since its expansion is more complicated. As mentioned above, it is, strictly speaking, not a free parameter at the order to which we work, but in practice its proton value was obtained from a fit to unpolarised proton data. This requires the promotion of a LEC from the fifth-order $\pi$N Lagrangian. A further complication is that its Delta-pole contribution of $+2.8$, which is nominally suppressed by $\delta^2$, is more than twice the LO contribution. The contribution of $\pi\Delta$ loops is tiny. The origin of this, of course, is the large size of the magnetic $\gamma$N$\Delta$ coupling $b_1$, whose square enters the leading pole contribution for ${\ensuremath{\gamma_{M1M1}}}$ but not the other spin polarisabilities (see eq. ). As a result, the pole contribution is about $6$ times as large as if $b_1$ were of natural size. There is precedent for promoting terms involving coefficients which are unnaturally large, see e.g. [@Griesshammer:2000mi], and hence for ${\ensuremath{\gamma_{M1M1}}}$ we treat the $b_1^2$ contribution to the Delta pole as suppressed by only one power of $\delta$, giving $k=3$ nontrivial orders in our expansion. The calculation is still complete to [N${}^{2}$LO]{} as any other graphs involving $b_1^2$ will be suppressed by $\delta^4$; thus only the error estimate is affected by the reordering. With neither of these adjustments, we would predict the proton value to be $6.4 \pm 3.0$, which is in fact still compatible with our quoted result of $2.2\pm0.5(\text{stat})\pm0.6(\text{theory})$. However several lines of evidence, though not conclusive, suggest the latter is a more appropriate central value and uncertainty; these include the close similarity of the values in the third and fourth-order extractions of the scalar polarisabilities discussed in Ref. [@McGovern:2012ew], as well as the DR and MAMI values quoted in Table \[tab:spinpols\]. Finally, the forward and backward spin polarisabilities of eq.  are not independent of the multipole spin polarisabilities. Their errors could thus be assessed naively as $\pm2.1$ by adding the individual multipole errors in quadrature, if the individual errors were Gau[ß]{}ian distributed. Since both also inherit the large Delta-pole contribution as well as the fitted LEC from ${\ensuremath{\gamma_{M1M1}}}$, we instead assess their theoretical uncertainties by the same prescription as for ${\ensuremath{\gamma_{M1M1}}}$. As both are linear combinations of quantities which are sensitive to different multipolarities, and hence different physical mechanisms, [*a priori*]{} we expect the two errors to be similar. Surprisingly, we find an error of ${\ensuremath{\gamma_{\pi}}}$ that is half that of ${\ensuremath{\gamma_{0}}}$. We see no physical reason for this. Furthermore, when in Sect. \[sec:errorregimei\], we look at values of ${\ensuremath{m_\pi}}$ greater than the physical value, the difference between the ${\ensuremath{\gamma_{0}}}$ and ${\ensuremath{\gamma_{\pi}}}$ errors disappears for ${\ensuremath{m_\pi}}\gtrsim170\;{\ensuremath{\mathrm{MeV}}}$. This suggests that the rather small uncertainty for ${\ensuremath{\gamma_{\pi}}}$ found via the order-by-order prescription at ${\ensuremath{m_\pi^\text{phys}}}$ is accidental. Therefore, for all ${\ensuremath{m_\pi}}$, we will use as the error of ${\ensuremath{\gamma_{\pi}}}$ that derived by Bayesian criteria for ${\ensuremath{\gamma_{0}}}$. The resulting ${\ensuremath{\gamma_{\pi}^{(\mathrm{p})}}}$ uncertainty of $\pm1.8$ at ${\ensuremath{m_\pi^\text{phys}}}$ is then close to the added-in-quadrature result. We close with remarks which pertain to scalar polarisabilities; further discussion can be found in Sect. \[sec:errorregimei\]. Our previously-published uncertainties of eq.  were also obtained by considering order-by-order convergence, though without the rigorous framework provided here. For the proton, there are three non-vanishing orders ($k=3$) and the uncertainties should be interpreted as the $75\%$ DoB interval for the pdf, rather than $68$% DoB intervals. For the neutron, though, $k=2$ for the extracted scalar polarisabilities, so the DoB is $67$%, but statistical uncertainties there far outweigh the theoretical accuracy. For either nucleon, our previous remainder estimates $R_\xi$ and the new $68$%-DoBs $\sigma_\xi$ are identical after rounding. A determination for the neutron at ${\mathcal{O}}(e^2\delta^4)$ in the amplitudes is forthcoming [@dcompton-delta4]. Figure \[fig:combinedpdfs\] shows the plots of the pdfs for ${\ensuremath{\alpha_{E1}}}-{\ensuremath{\beta_{M1}}}$, the one free variable after the Baldin sum rule is used as a constraint. Convergence Tests: Selected Higher-Order Terms {#sec:someho} ---------------------------------------------- To check our error estimate at ${\ensuremath{m_\pi^\text{phys}}}$, we look at the effect of including selected, renormalisation-group-invariant, higher-order corrections. This does not lead to increased accuracy; in general that is only achieved when all contributions at a given order are known. It can, however, add credence to our procedure if contributions fall within the derived error estimates. Since Refs. [@Griesshammer:2012we; @McGovern:2012ew; @Myers:2014ace] have already discussed various such effects for the scalar polarisabilities, we here consider only the spin polarisabilities. As mentioned in Sect. \[sec:pols\], some contributions are known in variants which differ by contributions at ${\mathcal{O}}(\delta^{3})$ relative to LO, i.e. at the first order not calculated. We will also use these variants to discuss the reliability of our uncertainty estimate of the ${\ensuremath{m_\pi}}$-dependence in Sect. \[sec:tests\]. 1. Vertex corrections to $b_1$ and $b_2$ are reported in Ref. [@McGovern:2012ew]. They add $0.1$ units to ${\ensuremath{\gamma_{E1E1}}}$ and ${\ensuremath{\gamma_{E1M2}}}$, $-0.37$ to ${\ensuremath{\gamma_{M1M1}}}$, and $-0.2$ to ${\ensuremath{\gamma_{M1E2}}}$, well within our proposed theoretical uncertainties. 2. The $\pi\Delta$ contributions in the heavy-baryon version of eqs.  and differ by terms of ${\mathcal{O}}(e^2\delta^{-1})$ from a covariant version [@Lensky:2015awa]. The latter add $-0.2$ to ${\ensuremath{\gamma_{E1E1}}}$, $-0.4$ to ${\ensuremath{\gamma_{M1M1}}}$, $0.3$ to ${\ensuremath{\gamma_{E1M2}}}$ and $0.03$ to ${\ensuremath{\gamma_{M1E2}}}$, again in accord with our error bars. 3. The difference between the covariant Delta-pole result of eqs.  and and the heavy-baryon one enters at ${\mathcal{O}}(e^2\delta^{-1})$ as well. The latter sets the $E2$-coupling $b_2$ and recoil effects to zero and uses a different value of $b_1^\text{HB}\approx4.8$ [@Griesshammer:2012we; @McGovern:2012ew]. We find the heavy-baryon version adds $0.35$ units to ${\ensuremath{\gamma_{E1E1}}}$, $0.8$ to ${\ensuremath{\gamma_{M1M1}}}$, $0.4$ to ${\ensuremath{\gamma_{E1M2}}}$ and $-0.3$ to ${\ensuremath{\gamma_{M1E2}}}$. These are within the quoted uncertainties. In each case, the changes quoted for ${\ensuremath{\gamma_{M1M1}}}$ do not take into account the fact that it needs to be refitted once such effects are added; we expect this to decrease the magnitude of higher-order effects. Chiral Extrapolations ===================== \[sec:extrapolations\] A chiral extrapolation of the central values for the polarisabilities simply uses the pion-mass dependence of the [$\chi$EFT]{}results in Sect. \[sec:pols\] to generate the functions $\xi({\ensuremath{m_\pi}})$. The final results are shown as solid red (proton) and blue (neutron) lines in all the plots of Sect. \[sec:results\]. Just as at the physical point, the chiral predictions have truncation errors which are assessed as before, but now are functions of ${\ensuremath{m_\pi}}$; we will make a few technical remarks on their evaluation in Sect. \[sec:errorregimei\]. The truncation uncertainties are represented in the plots by shaded bands for as far as we trust them; see Sect. \[sec:errorregimeii\]. These error bands at unphysical pion masses correspond only to the “theory” error. The full error is then obtained by combining this with the (${\ensuremath{m_\pi}}$-independent) statistical and (for scalar polarisabilities) Baldin-sum-rule errors. Therefore, in the plots, we mark the total error only at the physical point, adding all errors linearly. In cases where the physical-point error is larger than the width of the band, the whole band can be moved up or down within this difference; see details in Sect. \[sec:results\]. This presentation has been chosen to highlight the running of the error with the pion mass, and because the other sources of error depend on the current state of the data, and thus may change in future. The different sources of errors could of course be combined in a Bayesian formalism, but that is outside the scope of this paper. Finally, we do not consider any implicit ${\ensuremath{m_\pi}}$-dependence of $g_A$, ${\ensuremath{f_\pi}}$, ${\ensuremath{M_\mathrm{N}}}$, ${\ensuremath{\Delta_{\scriptscriptstyle M}}}$, $c_i$, etc. All such effects are of higher order in the chiral counting. For example, the correction to $g_A({\ensuremath{m_\pi}})=g_A^0+{\mathcal{O}}({\ensuremath{m_\pi}}^2)$ [@Procura:2006gq] is suppressed by two orders in ${\ensuremath{m_\pi}}/\Lambda_\chi$ and hence beyond the accuracy of our results. Theoretical Uncertainties near the Physical Point {#sec:errorregimei} ------------------------------------------------- We first consider the uncertainties in regime (i), ${\ensuremath{m_\pi}}\approx{\ensuremath{m_\pi^\text{phys}}}$. There, we have included those contributions to the polarisabilities that are required in the physical low-energy Compton amplitudes up to ${\mathcal{O}}(e^2\delta^4)$. Omitted terms are therefore ${\mathcal{O}}(\delta^3)$ relative to leading. Ignoring a few details to be discussed shortly, we estimate the remainder $R_\xi({\ensuremath{m_\pi}})$ and $68$% DoBs $\sigma_\xi({\ensuremath{m_\pi}})$ at any given ${\ensuremath{m_\pi}}$ in regime (i) just as we did at ${\ensuremath{m_\pi^\text{phys}}}$ in Sect. \[sec:Finding-Theory-Uncertainties\]. The only difference is in the expansion parameter $\delta$, which was defined in Sect. \[sec:formalism\] and enters in eqs.  and . So long as we remain well below ${\ensuremath{m_\pi}}\approx{\ensuremath{\Delta_{\scriptscriptstyle M}}}$, it could be taken to be constant, $\delta\approx0.4$. But, in practice, some chiral corrections to pion loops grow linearly with ${\ensuremath{m_\pi}}$, and once ${\ensuremath{m_\pi}}\approx{\ensuremath{\Delta_{\scriptscriptstyle M}}}$ the whole counting changes: Delta graphs become as important as nucleon ones; see Sect. \[sec:regimes\] and the next subsection. Hence, we choose a value of $\delta$ which increases with ${\ensuremath{m_\pi}}$, $$\delta({\ensuremath{m_\pi}})=0.4 \frac{{\ensuremath{m_\pi}}}{{\ensuremath{m_\pi^\text{phys}}}}\;\;.$$ For the three spin polarisabilities which are pure predictions, ${\ensuremath{\gamma_{E1E1}}}$, ${\ensuremath{\gamma_{E1M2}}}$ and ${\ensuremath{\gamma_{M1E2}}}$, there is little more be said. As at the physical point, at any given ${\ensuremath{m_\pi}}$ we derive the isoscalar and isovector uncertainties from the order-by-order convergence of their respective series, and obtain the total uncertainty by convoluting the corresponding pdfs. The overall scale of their isoscalar uncertainty $R_{\gamma_i}^{(\text{s})}$ turns out to be dominated by $\delta$ times the ${\mathcal{O}}(e^2 \delta^{-2})$ contribution at all pion masses we consider. In each case, the absolute error increases rather modestly as a function of ${\ensuremath{m_\pi}}$, mainly because an important part of the ${\mathcal{O}}(e^2 \delta^{-2})$ contribution is actually falling as $1/{\ensuremath{m_\pi}}$ while the expansion parameter $\delta({\ensuremath{m_\pi}})$ grows linearly. The ordering of contributions to ${\ensuremath{\gamma_{M1M1}}}$, ${\ensuremath{\gamma_{0}}}$ and ${\ensuremath{\gamma_{\pi}}}$ for ${\ensuremath{m_\pi}}\neq{\ensuremath{m_\pi^\text{phys}}}$ also mirrors exactly that at the physical point as discussed in Sect. \[sec:Finding-Theory-Uncertainties\]: terms quadratic in the rather large coupling $b_1^2$ are promoted by one order. The scalar polarisabilities require a little more discussion, since we did not derive in detail their errors at the physical point, these having been determined through fits to data [@McGovern:2012ew; @Myers:2014ace]. First, we recall that the fit of ${\ensuremath{\alpha_{E1}^{(\mathrm{n})}}}$ and ${\ensuremath{\beta_{M1}^{(\mathrm{n})}}}$ to deuteron data used amplitudes only at NLO, while that for the proton used those at one order higher. This leads to substantially larger truncation errors at the physical point than that for the proton: $\pm 0.8$ as opposed to $\pm 0.3$. However, their running with ${\ensuremath{m_\pi}}$ is predicted in [$\chi$EFT]{}to [N${}^{2}$LO]{} for both. Thus, when generating the scalar-polarisability error bands in the neutron case, we use the same uncertainties as for the proton, while including the larger truncation uncertainty in the error bar at the physical point. Once again, any change in the central value of ${\ensuremath{\alpha_{E1}^{(\mathrm{n})}}}$ and ${\ensuremath{\beta_{M1}^{(\mathrm{n})}}}$ resulting from a new fit to deuteron data at order $\delta^2$ rather than $\delta^1$ relative to LO only serves to move the curve up and down. A construction of isoscalar and isovector uncertainties as for the spin-polarisabilities, with reasonable assumptions as to how much ${\ensuremath{\alpha_{E1}^{(\mathrm{n})}}}-{\ensuremath{\beta_{M1}^{(\mathrm{n})}}}$ may shift in an future [N${}^{2}$LO]{} fit, produces near-identical bands. Furthermore, in the proton fits of Ref. [@McGovern:2012ew], the actual fit parameter was ${\ensuremath{\alpha_{E1}^{(\mathrm{p})}}}-{\ensuremath{\beta_{M1}^{(\mathrm{p})}}}$, with ${\ensuremath{\alpha_{E1}^{(\mathrm{p})}}}+{\ensuremath{\beta_{M1}^{(\mathrm{p})}}}$ fixed by the Baldin sum rule. The scale $\scale$ of the uncertainty was obtained by considering ${\ensuremath{\alpha_{E1}^{(\mathrm{p})}}}-{\ensuremath{\beta_{M1}^{(\mathrm{p})}}}$ as *predicted* at LO—${\mathcal{O}}(e^2\delta^{-2})$—and *fit* at the next two orders. By fitting the amplitudes at NLO, the LECs ${\ensuremath{\alpha_{E1}}}^{\text{(p)LEC}}$ and ${\ensuremath{\beta_{M1}}}^{\text{(p)LEC}}$ are promoted by one order. In practice, the proton fits at orders $\delta^1$ and $\delta^2$ relative to LO gave almost identical values for ${\ensuremath{\alpha_{E1}}}^\text{(p)LEC} - {\ensuremath{\beta_{M1}}}^\text{(p)LEC}$. To reflect this, we treat the LECs as part of the ${\mathcal{O}}(e^2\delta^{-1})$ contribution to ${\ensuremath{\alpha_{E1}^{(\mathrm{p})}}}$ and ${\ensuremath{\beta_{M1}^{(\mathrm{p})}}}$. The additional contribution at ${\mathcal{O}}(e^2\delta^{0})$ is then subtracted to give zero at the physical point, and so affects ${\ensuremath{\alpha_{E1}^{(\mathrm{p})}}}-{\ensuremath{\beta_{M1}^{(\mathrm{p})}}}$ only for ${\ensuremath{m_\pi}}\neq {\ensuremath{m_\pi^\text{phys}}}$. Beyond the physical point, we continue to treat ${\ensuremath{\alpha_{E1}}}+{\ensuremath{\beta_{M1}}}$ and ${\ensuremath{\alpha_{E1}}}-{\ensuremath{\beta_{M1}}}$ as independent. However, the Baldin sum rule for ${\ensuremath{\alpha_{E1}}}+{\ensuremath{\beta_{M1}}}$ is special: at the physical point it provides extrinsic information in our NLO and [N${}^{2}$LO]{} fits and should be reproduced exactly. In fact, higher-order effects in the scalar-polarisability sum for the proton at the physical point are tiny compared to the LO prediction of $13.9$ from eq. . We therefore base our error estimates for ${\ensuremath{\alpha_{E1}^{(\mathrm{p})}}}+{\ensuremath{\beta_{M1}^{(\mathrm{p})}}}$ beyond the physical point only on the shifts from the LO term, i.e. we drop $c_0$ in the construction of $\scalexi$ in eq. ; we then have $k=2$ for ${\ensuremath{\alpha_{E1}^{(\mathrm{p})}}}+{\ensuremath{\beta_{M1}^{(\mathrm{p})}}}$. We considered several variants of this procedure, but they do not substantially alter the outcome: while the [$\chi$EFT]{}uncertainty at ${\ensuremath{m_\pi^\text{phys}}}$ is indeed zero, it grows very rapidly with ${\ensuremath{m_\pi}}$, see Fig. \[fig:constraints\]. A several-hundred-per-cent error at ${\ensuremath{m_\pi}}=350\;{\ensuremath{\mathrm{MeV}}}$, while technically correct for a theory defined as an expansion around the chiral limit, seems overly pessimistic. Indeed, it defies the quite conservative expectation that ${\ensuremath{\alpha_{E1}}}+{\ensuremath{\beta_{M1}}}$ tends asymptotically to a constant as ${\ensuremath{m_\pi}}$ gets bigger, since the nucleon size, and hence the typical size over which its charged constituents can fluctuate, will not increase with ${\ensuremath{m_\pi}}$. The [$\chi$EFT]{}error evolution for ${\ensuremath{\alpha_{E1}^{(\mathrm{p})}}}- {\ensuremath{\beta_{M1}^{(\mathrm{p})}}}$ is more modest, and is more representative of the truncation error in [$\chi$EFT]{}predictions for polarisabilities. Well away from the physical point and the constraint provided by the Baldin sum rule, there is no particular logic to using ${\ensuremath{\alpha_{E1}}}\pm {\ensuremath{\beta_{M1}}}$, rather than ${\ensuremath{\alpha_{E1}}}$ and ${\ensuremath{\beta_{M1}}}$ separately, as the quantities via which we assess convergence of the [$\chi$EFT]{}series. Once ${\ensuremath{m_\pi}}$ reaches $250\;{\ensuremath{\mathrm{MeV}}}$, ${\ensuremath{\alpha_{E1}}}$ and ${\ensuremath{\beta_{M1}}}$ separately have much better convergence properties than does ${\ensuremath{\alpha_{E1}}}+ {\ensuremath{\beta_{M1}}}$. For pion masses in that vicinity, taking the error only from ${\ensuremath{\alpha_{E1}}}-{\ensuremath{\beta_{M1}}}$ gives errors consistent with those obtained from analysing ${\ensuremath{\alpha_{E1}}}$ and ${\ensuremath{\beta_{M1}}}$ separately. So, in what follows, we simply, if somewhat arbitrarily, assign zero truncation uncertainty for the evolution of ${\ensuremath{\alpha_{E1}}}+{\ensuremath{\beta_{M1}}}$ in order to generate our final results for ${\ensuremath{\alpha_{E1}}}$ and ${\ensuremath{\beta_{M1}}}$. We then use different colours for the bands in the plot of ${\ensuremath{\alpha_{E1}}}+ {\ensuremath{\beta_{M1}}}$ in Fig. \[fig:constraints\] to indicate that they are not used to derive corridors for ${\ensuremath{\alpha_{E1}}}$ or ${\ensuremath{\beta_{M1}}}$. The widths of the corridors of the scalar polarisabilities are set by half the uncertainty in ${\ensuremath{\alpha_{E1}}}-{\ensuremath{\beta_{M1}}}$, and their corridors are anti-correlated. The Baldin-related error is indicated, along with the statistical one, only at the physical point. \[sec:tests\] Figure \[fig:higherorders\] shows the evolution of the error bands for all six polarisabilities, but *not* the evolution of central values with ${\ensuremath{m_\pi}}$, the more clearly to display the behaviour of the uncertainties. As corroborating evidence for this error assessment, we proceed as in Sect. \[sec:someho\] to take advantage of the fact that three classes of (isoscalar) pion-mass-dependent higher-order corrections are known. However, only the first two display ${\ensuremath{m_\pi}}$-dependence: vertex corrections (blue dashed line) [@McGovern:2012ew], and the inclusion of (some) $1/{\ensuremath{M_\mathrm{N}}}$ corrections in $\pi \Delta$ loops via a covariant calculation (red dot-dashed line) [@Lensky:2015awa]. Both effects are much smaller than our $68\%$ uncertainty bands for all $6$ polarisabilities, even in regime (ii), providing a potential hint that the errors may be over-estimated. We can safely conclude that our analysis of uncertainties passes reasonable checks both at the physical point (Sect. \[sec:values\]) and in regard to the ${\ensuremath{m_\pi}}$ dependence. ![(Colour online) Error bands on theoretical results for ${\ensuremath{m_\pi}}$-dependence of polarisabilities. For greater clarity, the ${\ensuremath{m_\pi}}$-evolution of the central value is not shown. The lines demonstrate the added effect of selected next-order corrections to the ${\ensuremath{m_\pi}}$-dependence of the polarisabilities, normalised to zero at ${\ensuremath{m_\pi^\text{phys}}}$. Blue dashed: $\gamma$N$\Delta$ vertex corrections [@McGovern:2012ew]; red dot-dashed: a set of “relativistic" corrections to $\pi\Delta$ amplitudes [@Lensky:2015awa]. []{data-label="fig:higherorders"}](higherorder-corrections.pdf){width="80.00000%"} Theoretical Uncertainties Well Beyond the Physical Pion Mass {#sec:errorregimeii} ------------------------------------------------------------ For lattice calculations with ${\ensuremath{m_\pi}}$ markedly larger than ${\ensuremath{m_\pi^\text{phys}}}$ the regime (i) power counting employed in Sect. \[sec:errorregimei\] is no longer appropriate, as discussed in Sect. \[sec:regimes\]. For ${\ensuremath{m_\pi}}\sim{\ensuremath{\Delta_{\scriptscriptstyle M}}}$ the scales $P({\ensuremath{m_\pi}})$ and $\epsilon$ of eq.  are of the same order, so there is no suppression of graphs according to the numbers of Delta propagators in them. Leading $\pi$N and $\pi\Delta$ loops and Delta-pole graphs are all on the same footing, contributing to the scalar polarisabilities at ${\mathcal{O}}(e^2 \epsilon^{-1})$ and to the spin polarisabilities at ${\mathcal{O}}(e^2 \epsilon^{-2})$. The subleading $\pi$N loops included above are suppressed by one power of $\epsilon$ but do not constitute a complete set of contributions at this order. There are, for instance, Delta-pole graphs with $\pi$N and $\pi\Delta$ loop corrections to the magnetic and electric $\gamma$N$\Delta$ vertices, and graphs like those of Fig. \[fig:corrpiN\] with intermediate Delta propagators replacing one or more nucleon propagators. Some, but not all, of the latter are included in a covariant calculation [@Lensky:2015awa]. Thus, in regime (ii) our calculation is only complete at leading order; there are omitted effects already at relative order $\epsilon$. Though we have not shown the results here, we did carry out a regime-(ii) error analysis. We found that for ${\ensuremath{m_\pi}}\sim {\ensuremath{\Delta_{\scriptscriptstyle M}}}$ the bands we generated were in broad agreement with the regime-(i) bands continued into this region. This is unsurprising in view of the fact that regime (i) and regime (ii) are not clearly separated; transition from one to the other is gradual. In any case, it seems that [$\chi$EFT]{}does not adequately describe the behaviour of many observables seen in lattice computations for larger values of ${\ensuremath{m_\pi}}$ [@McGovern:2006fm; @Djukanovic:2006xc; @Schindler:2007dr]. For example, computations show the nucleon mass rising linearly with ${\ensuremath{m_\pi}}$ beyond $300\;{\ensuremath{\mathrm{MeV}}}$, not displaying the more complex dependence involving higher powers predicted by [$\chi$EFT]{} [@WalkerLoud:2008bp; @Walker-Loud:2013yua]. Similar difficulties for [$\chi$EFT]{}are seen in other observables, too [@Bernard:2006te; @Beane:2015]. Given these issues, we will show predictions for [$\chi$EFT]{}in regime (ii), but allow the regime-(i) error bars to fade away beyond ${\ensuremath{m_\pi}}\approx 250\;{\ensuremath{\mathrm{MeV}}}$, i.e. as they become unreliable, and disappear altogether beyond about $350\;{\ensuremath{\mathrm{MeV}}}$. At such pion masses, linear extrapolations in ${\ensuremath{m_\pi}}$—as used in Refs. [@Walker-Loud:2013yua; @Beane:2015] and earlier studies—may describe lattice QCD results, but they are not justified within [$\chi$EFT]{}, and are more-or-less uncontrolled. We thus refer to results in this regime not as “chiral predictions” but as “chiral curves” and speculate as to why such extrapolations may be successful at the end of Sect. \[sec:lattice\]. Results: Pion-Mass Dependence of Polarisabilities {#sec:results} ------------------------------------------------- ![(Colour online) Predicted ${\ensuremath{m_\pi}}$-dependence of the dipole polarisabilities in [$\chi$EFT]{}. The full result is represented by solid lines and labelled for the respective nucleon: proton results are coloured red with red corridors of [$\chi$EFT]{}uncertainties and symbol $\protect\textcolor{red}{\blacksquare}$ at ${\ensuremath{m_\pi^\text{phys}}}$ (slightly offset to smaller ${\ensuremath{m_\pi}}$ for better visibility); neutron results are blue with blue corridors and at ${\ensuremath{m_\pi^\text{phys}}}$ (slightly offset to larger ${\ensuremath{m_\pi}}$). Error bars at the physical point add statistical, theory and Baldin-sum-rule errors linearly, as applicable. Green lines are isoscalar, with dotted: leading $\pi$N contributions; dashed (for ${\ensuremath{\alpha_{E1}}}$, ${\ensuremath{\beta_{M1}}}$ and ${\ensuremath{\gamma_{M1M1}}}$ only): Delta pole and leading $\pi\Delta$ pieces added, plus isoscalar LECs for ${\ensuremath{\alpha_{E1}}}$, ${\ensuremath{\beta_{M1}}}$. The vertical lines show the position of the physical pion mass and the Delta-nucleon mass difference. See text for additional details.[]{data-label="fig:allpols"}](alphabeta-combined-errors-500.pdf){width="100.00000%"} ![(Colour online) Predicted ${\ensuremath{m_\pi}}$-dependence of the dipole polarisabilities in [$\chi$EFT]{}. The full result is represented by solid lines and labelled for the respective nucleon: proton results are coloured red with red corridors of [$\chi$EFT]{}uncertainties and symbol $\protect\textcolor{red}{\blacksquare}$ at ${\ensuremath{m_\pi^\text{phys}}}$ (slightly offset to smaller ${\ensuremath{m_\pi}}$ for better visibility); neutron results are blue with blue corridors and at ${\ensuremath{m_\pi^\text{phys}}}$ (slightly offset to larger ${\ensuremath{m_\pi}}$). Error bars at the physical point add statistical, theory and Baldin-sum-rule errors linearly, as applicable. Green lines are isoscalar, with dotted: leading $\pi$N contributions; dashed (for ${\ensuremath{\alpha_{E1}}}$, ${\ensuremath{\beta_{M1}}}$ and ${\ensuremath{\gamma_{M1M1}}}$ only): Delta pole and leading $\pi\Delta$ pieces added, plus isoscalar LECs for ${\ensuremath{\alpha_{E1}}}$, ${\ensuremath{\beta_{M1}}}$. The vertical lines show the position of the physical pion mass and the Delta-nucleon mass difference. See text for additional details.[]{data-label="fig:allpols"}](gammas-combined-errors-500.pdf){width="100.00000%"} The solid lines in Figs. \[fig:allpols\] and \[fig:constraints\] are [$\chi$EFT]{}predictions for the pion-mass dependence of the dipole polarisabilities, together with their theoretical uncertainties, computed as specified in Sects. \[sec:errorregimei\] and \[sec:errorregimeii\]. These are complete at order $\delta^2$ relative to LO for all polarisabilities, and include three nonzero orders for ${\ensuremath{\alpha_{E1}}}$, ${\ensuremath{\beta_{M1}}}$, ${\ensuremath{\gamma_{M1M1}}}$, ${\ensuremath{\gamma_{0}}}$ and ${\ensuremath{\gamma_{\pi}}}$, and two nonzero orders for the other polarisabilities. Proton results are colour-coded in red, neutron ones in blue, and isoscalar ones in green. The leading $\pi$N loops (green dotted) constitute LO in regime (i). The green dashed curves represent the complete isoscalar result at order $\delta^1$ relative to LO in regime (i), as detailed in Sect. \[sec:Finding-Theory-Uncertainties\] and eq. . No new contribution arises at this order for ${\ensuremath{\gamma_{E1E1}}}$, ${\ensuremath{\gamma_{E1M2}}}$ and ${\ensuremath{\gamma_{M1E2}}}$, and so this curve is absent there. In regime (ii), the solid and dashed curves are both only complete to LO, while the dotted curve is not even the full LO isovector result. ![(Colour online) Pion-mass dependence of the Baldin sum rule (top left), ${\ensuremath{\alpha_{E1}}}-{\ensuremath{\beta_{M1}}}$ (top right), ${\ensuremath{\gamma_{0}}}$ (bottom left) and ${\ensuremath{\gamma_{\pi}}}$ (bottom right). For ${\ensuremath{\alpha_{E1}}}+{\ensuremath{\beta_{M1}}}$, error bars at ${\ensuremath{m_\pi^\text{phys}}}$ only reflect the uncertainties of the Baldin sum rules [@Olmos:2001; @Levchuk:1999zy]; in [$\chi$EFT]{}, such errors are input and are separate from the intrinsic [$\chi$EFT]{}errors. The [$\chi$EFT]{}uncertainty corridors of ${\ensuremath{\alpha_{E1}}}+{\ensuremath{\beta_{M1}}}$ are coloured differently to indicate that they are not used to derive corridors for ${\ensuremath{\alpha_{E1}}}$ or ${\ensuremath{\beta_{M1}}}$. The experimental value of ${\ensuremath{\gamma_{0}^{(\mathrm{p})}}}$ is indicated by the symbol $\protect\textcolor{red}{\bullet}$ [@Ahrens:2001qt; @Dutz]; this result’s uncertainty is smaller than the size of the symbol. Colour coding as in Fig. \[fig:allpols\]. See text for further details.[]{data-label="fig:constraints"}](Baldin+alphaminusbeta-combined-errors.pdf "fig:"){width="\linewidth"}\ ![(Colour online) Pion-mass dependence of the Baldin sum rule (top left), ${\ensuremath{\alpha_{E1}}}-{\ensuremath{\beta_{M1}}}$ (top right), ${\ensuremath{\gamma_{0}}}$ (bottom left) and ${\ensuremath{\gamma_{\pi}}}$ (bottom right). For ${\ensuremath{\alpha_{E1}}}+{\ensuremath{\beta_{M1}}}$, error bars at ${\ensuremath{m_\pi^\text{phys}}}$ only reflect the uncertainties of the Baldin sum rules [@Olmos:2001; @Levchuk:1999zy]; in [$\chi$EFT]{}, such errors are input and are separate from the intrinsic [$\chi$EFT]{}errors. The [$\chi$EFT]{}uncertainty corridors of ${\ensuremath{\alpha_{E1}}}+{\ensuremath{\beta_{M1}}}$ are coloured differently to indicate that they are not used to derive corridors for ${\ensuremath{\alpha_{E1}}}$ or ${\ensuremath{\beta_{M1}}}$. The experimental value of ${\ensuremath{\gamma_{0}^{(\mathrm{p})}}}$ is indicated by the symbol $\protect\textcolor{red}{\bullet}$ [@Ahrens:2001qt; @Dutz]; this result’s uncertainty is smaller than the size of the symbol. Colour coding as in Fig. \[fig:allpols\]. See text for further details.[]{data-label="fig:constraints"}](gamma0+gammapi-combined-errors.pdf "fig:"){width="\linewidth"} The symbols at the physical point are the [$\chi$EFT]{}predictions or, for ${\ensuremath{\alpha_{E1}}}$, ${\ensuremath{\beta_{M1}}}$ and ${\ensuremath{\gamma_{M1M1}}}$, fits. Their error bars are found by adding the theory and—where applicable—statistical plus Baldin-sum-rule errors linearly, as discussed at the beginning of Sect. \[sec:extrapolations\]. When this total error exceeds the width of the band at the physical point, the entire band can be floated up or down within that difference. We see that the higher-order graphs have only a modest effect on the running of ${\ensuremath{\alpha_{E1}}}-{\ensuremath{\beta_{M1}}}$, ${\ensuremath{\gamma_{E1E1}}}$, ${\ensuremath{\gamma_{E1M2}}}$ and ${\ensuremath{\gamma_{M1E2}}}$ with ${\ensuremath{m_\pi}}$, but a major effect in the case of ${\ensuremath{\gamma_{M1M1}}}$, ${\ensuremath{\beta_{M1}}}$, ${\ensuremath{\gamma_{0}}}$, ${\ensuremath{\alpha_{E1}}}+{\ensuremath{\beta_{M1}}}$, and, to some degree, ${\ensuremath{\gamma_{\pi}}}$. In addition, ${\ensuremath{\alpha_{E1}}}-{\ensuremath{\beta_{M1}}}$ is almost purely isoscalar. At the physical point, this is a simple consequence of the fact that the fitted proton and neutron values are very similar, cf. eq. . Beyond that, its isovector component grows only logarithmically with ${\ensuremath{m_\pi}}$ and with a small pre-factor; see eq. . As already noted, we find it impossible to assign credible errors to ${\ensuremath{\alpha_{E1}}}+{\ensuremath{\beta_{M1}}}$, because of its poor convergence. The error bands on ${\ensuremath{\alpha_{E1}}}$ and ${\ensuremath{\beta_{M1}}}$ in Fig. \[fig:allpols\] are therefore simply half those of ${\ensuremath{\alpha_{E1}}}-{\ensuremath{\beta_{M1}}}$, and are anti-correlated. The uncertainties in the scalar polarisabilities then appear quite small relative to their magnitudes in both regimes (i) and (ii), while the uncertainties of the spin polarisabilities are comparable to their sizes. ![(Colour online) Pion-mass dependence of ${\ensuremath{\beta_{M1}}}$ and ${\ensuremath{\gamma_{M1M1}}}$ in regime (i). Legend as in Fig. \[fig:allpols\]. []{data-label="fig:zoomin"}](betagammaMM-zoomin-combined-errors.pdf){width="100.00000%"} As can be seen in Fig \[fig:allpols\], in most cases the sub-leading $\pi$N loops do not have a major effect on the trend of the polarisabilities. However for ${\ensuremath{\beta_{M1}}}$ and ${\ensuremath{\gamma_{M1M1}}}$, shown in the chirally relevant ${\ensuremath{m_\pi}}$-range in Fig. \[fig:zoomin\], they change functions which are monotonically decreasing (resp. increasing) with ${\ensuremath{m_\pi}}$ at low orders into ones that are increasing (resp. decreasing) at ${\ensuremath{m_\pi^\text{phys}}}$. Indeed, both these polarisabilities could be regarded as somewhat fine-tuned at the physical pion mass, at least compared to their value at an arbitrary ${\ensuremath{m_\pi}}< {\ensuremath{m_\pi^\text{phys}}}$. This is interesting in light of the well-known puzzle that the magnetic polarisability has a physical value which is much smaller than predicted by the strong paramagnetic effects of the Delta. That higher-order terms in the ${\ensuremath{m_\pi}}/{\ensuremath{M_\mathrm{N}}}$ expansion could change the lower-order trend in ${\ensuremath{\beta_{M1}}}$ was pointed out in Ref. [@Lensky:2009uv]. At ${\ensuremath{m_\pi^\text{phys}}}$, pion-loop effects cancel against Delta-pole excitations to render the magnetic polarisability much smaller than either, and produce ${\ensuremath{\beta_{M1}^{(\mathrm{p})}}}\approx {\ensuremath{\beta_{M1}^{(\mathrm{n})}}}$. Below the physical point, the isovector $N\pi$ corrections of eqs.  and gain increasing statistical significance, destroying this cancellation for both proton and neutron; ${\ensuremath{\beta_{M1}^{(\mathrm{n})}}}$ quickly approaches its “natural” size in this low-${\ensuremath{m_\pi}}$ region, but flattens out once ${\ensuremath{m_\pi}}>{\ensuremath{m_\pi^\text{phys}}}$, while ${\ensuremath{\beta_{M1}^{(\mathrm{p})}}}$ changes from a decreasing to an increasing function around ${\ensuremath{m_\pi}}=50$ MeV, and after that has a trend that is dictated by the sub-leading (isovector) $\pi$N loops. A similar pattern is observed in ${\ensuremath{\gamma_{M1M1}}}$, whose leading dependence with $1/{\ensuremath{m_\pi}}^2$ dictates a much more prominent “turn-over” for ${\ensuremath{m_\pi}}< {\ensuremath{m_\pi^\text{phys}}}$. In this case the isovector component is smaller than for ${\ensuremath{\beta_{M1}}}$, but it is still statistically significant. The fact that ${\ensuremath{\gamma_{M1M1}^{(\mathrm{p})}}}$ is somewhat smaller at the physical point than its generic size for ${\ensuremath{m_\pi}}< {\ensuremath{m_\pi^\text{phys}}}$ may be related to the necessity to use ${\ensuremath{\gamma_{M1M1}}}$ as a parameter in our recent extraction of the proton’s scalar polarisabilities in [$\chi$EFT]{}, cf. Sect. \[sec:CTs\], even though the corresponding LEC only enters one order higher [@McGovern:2012ew]. These issues merit further study. For the shape and physical-point value of these quantities to be markedly affected by contributions of still higher order, our error corridors would have to underestimate those effects. Our examination of select higher-order effects in Sect. \[sec:errorregimei\] showed no such problems. Lattice computations closer to the chiral limit would provide excellent tests, but are numerically quite challenging. Isovector Magnetic Polarisability and the Anthropic Principle {#sec:isovector} ------------------------------------------------------------- Isovector contributions of sub-leading $\pi$N loops enter at order $\delta^2$ relative to LO for all polarisabilities, and the LECs of the scalar polarisabilities at the same order have isovector parts as well; see eq. . One thus expects that ${\ensuremath{\alpha_{E1}^{(\mathrm{v})}}}$ and ${\ensuremath{\beta_{M1}^{(\mathrm{v})}}}$ are about $20\%$ of ${\ensuremath{\alpha_{E1}^{(\mathrm{s})}}}$ and ${\ensuremath{\beta_{M1}^{(\mathrm{s})}}}$, respectively. (Recall that we define $\xi^\text{(s,v)}={\frac{1}{2}}(\xi^\text{(p)}\pm\xi^\text {(n)})$.) However, eq.  implies that these isovector combinations are zero within present uncertainties at the physical pion mass. This may signal another instance of fine-tuning between loops and short-distance physics at the physical point. This can be quantified via the variation of the isovector polarisabilities with ${\ensuremath{m_\pi}}$ (or equivalently $m_\mathrm{q}$). At this order, the fitted LECs are ${\ensuremath{m_\pi}}$-independent, so the relevant rate of change is determined completely by long-distance physics associated with the subleading $\pi$N loops of eq. : $$\label{eq:lnderivative} \left.\frac{{\mathrm{d}}{\ensuremath{\beta_{M1}^{(\mathrm{v})}}}}{{\mathrm{d}}\ln m_\mathrm{q}}\right|_{{\ensuremath{m_\pi^\text{phys}}}} =0.65\pm0.4\;\;,\qquad \left.\frac{{\mathrm{d}}{\ensuremath{\alpha_{E1}^{(\mathrm{v})}}}}{{\mathrm{d}}\ln m_\mathrm{q}}\right|_{{\ensuremath{m_\pi^\text{phys}}}}=0.7 \pm 0.4\;\;,$$ in $10^{-4}\;{\ensuremath{\mathrm{fm}}}^3$, and with a Bayesian estimate of the truncation error. Therefore, both ${\ensuremath{\alpha_{E1}^{(\mathrm{v})}}}$ and ${\ensuremath{\beta_{M1}^{(\mathrm{v})}}}$ vary strongly away from ${\ensuremath{m_\pi^\text{phys}}}$. Indeed, we have already seen in Figs. \[fig:allpols\] and \[fig:zoomin\] that the similarity of proton and neutron polarisabilities disappears for ${\ensuremath{m_\pi}}\ne{\ensuremath{m_\pi^\text{phys}}}$. While the isovector component of ${\ensuremath{\alpha_{E1}}}- {\ensuremath{\beta_{M1}}}$ remains small at all ${\ensuremath{m_\pi}}$ (see Fig. \[fig:constraints\]), the degeneracy of ${\ensuremath{\beta_{M1}^{(\mathrm{p})}}}$ and ${\ensuremath{\beta_{M1}^{(\mathrm{n})}}}$ at the physical point does seem to be something of an accident. Lattice results at ${\ensuremath{m_\pi}}=806\;{\ensuremath{\mathrm{MeV}}}$ corroborate this for ${\ensuremath{\beta_{M1}}}$ [@Chang:2015qxa]; see Sect. \[sec:lattice\] for further discussion. The Cottingham Sum rule relates the Compton scattering amplitude to the electromagnetic part of the proton-neutron self-energy difference, $\delta{\ensuremath{M_\mathrm{N}}}^\mathrm{em} \equiv M_\mathrm{p}^\mathrm{em}-M_\mathrm{n}^\mathrm{em}$ [@WalkerLoud:2012bg; @WalkerLoud:2012en; @Erben:2014hza; @Thomas:2014dxa; @Gasser:2015dwa], leading us to speculate about a potential rationale for this apparent coincidence. Though the topic is not without controversy and there is some scale-dependence in assigning strong and electromagnetic self-energy differences, there is broad agreement that $\delta{\ensuremath{M_\mathrm{N}}}^\mathrm{em}$ and${\ensuremath{\beta_{M1}^{(\mathrm{v})}}}$ are connected, since the latter is related to the value of a component of the integrand in the Cottingham sum rule at $q^2=0$ [@WalkerLoud:2012bg; @Erben:2014hza; @Gasser:2015dwa]. The strength of this connection depends, though, on assumptions about the integrand, and is hotly debated at present. According to the analysis of Ref. [@WalkerLoud:2012bg], at the physical point the principal contributions to $\delta{\ensuremath{M_\mathrm{N}}}^\mathrm{em}$ are an elastic piece of $[0.77\pm0.03]\;{\ensuremath{\mathrm{MeV}}}$, and an inelastic piece which is dominated by a term proportional to ${\ensuremath{\beta_{M1}^{(\mathrm{v})}}}$: $$\label{eq:linkage} \delta{\ensuremath{M_\mathrm{N}}}^{\beta}({\ensuremath{m_\pi}})= -A \,{\ensuremath{\beta_{M1}^{(\mathrm{v})}}}({\ensuremath{m_\pi}})\;\;.$$ The size of $A$ can be obtained from Ref. [@WalkerLoud:2012bg]’s value of $\delta{\ensuremath{M_\mathrm{N}}}^\beta\approx0.5\;{\ensuremath{\mathrm{MeV}}}$ for ${\ensuremath{\beta_{M1}^{(\mathrm{v})}}}=-0.5$. Both the elastic and inelastic part of $\delta {\ensuremath{M_\mathrm{N}}}^\mathrm{em}$ involve integrals over form factors, well-known for the elastic contribution and estimated for the inelastic one. Assuming the pertinent scale in these form factors is associated with non-chiral physics, the variation of $\delta {\ensuremath{M_\mathrm{N}}}^\mathrm{em}$ with quark mass will come mainly from the magnetic moment in the elastic term, and from ${\ensuremath{\beta_{M1}}}$ in the inelastic term. Lattice QCD shows, though, that nucleon magnetic moments are rather insensitive to the quark mass [@Beane:2014ora], so we are left with $\delta{\ensuremath{M_\mathrm{N}}}^{\beta}$ as the dominant source of variation of $\delta{\ensuremath{M_\mathrm{N}}}^\mathrm{em}$ with quark mass. If we assume $A$ of eq.  is $m_\pi$ independent, we can estimate the variation of $\delta{\ensuremath{M_\mathrm{N}}}^\beta$ as $$\left.\frac{{\mathrm{d}}\delta{\ensuremath{M_\mathrm{N}}}^{\beta}({\ensuremath{m_\pi}})}{{\mathrm{d}}\ln m_\mathrm{q}}\right|_{{\ensuremath{m_\pi^\text{phys}}}}=-0.65\;{\ensuremath{\mathrm{MeV}}}\;\;. \label{eq:estimate}$$ Here, we do not give uncertainties, since we cannot quantify them on some of the assumptions being made. This value is not negligible relative to the quark-mass variation in Ref. [@Bedaque:2010hr]: $\left.\frac{{\mathrm{d}}\delta{\ensuremath{M_\mathrm{N}}}^\text{strong}}{{\mathrm{d}}\ln m_\mathrm{q}}\right|_{{\ensuremath{m_\pi^\text{phys}}}}\approx-2.1\;{\ensuremath{\mathrm{MeV}}}$ (obtained under the assumption that $m_\mathrm{u}/m_\mathrm{d}$ remains constant). Of course, the slope of $\delta{\ensuremath{M_\mathrm{N}}}^{\beta}$ could be smaller than our estimate, or somehow cancelled by other effects in $\delta{\ensuremath{M_\mathrm{N}}}^{\rm em}$. But the estimate makes it plausible that—contrary to what was assumed heretofore in many works, such as Ref. [@Bedaque:2010hr]—the variation of $\delta{\ensuremath{M_\mathrm{N}}}^\mathrm{em}$ is not negligible in comparison to that of the strong part. If $\delta{\ensuremath{M_\mathrm{N}}}^\beta$ does indeed produce the largest variation of $\delta {\ensuremath{M_\mathrm{N}}}^\mathrm{em}$ with $m_\mathrm{q}$, our estimate suggests that the quark-mass dependence of the proton-neutron mass splitting may be significantly enhanced—or indeed reduced for ${\ensuremath{\beta_{M1}^{(\mathrm{p})}}}>{\ensuremath{\beta_{M1}^{(\mathrm{n})}}}$, which is within today’s allowed range. As a consequence, the neutron life-time would then either be substantially shortened as ${\ensuremath{m_\pi}}$ increases (if ${\ensuremath{\beta_{M1}^{(\mathrm{p})}}}<{\ensuremath{\beta_{M1}^{(\mathrm{n})}}}$), or as ${\ensuremath{m_\pi}}$ decreases (if ${\ensuremath{\beta_{M1}^{(\mathrm{p})}}}>{\ensuremath{\beta_{M1}^{(\mathrm{n})}}}$). A neutron that is too short-lived to allow Big-Bang Nucleosynthesis to proceed to ${}^4$He presumably makes carbon-based life impossible. This putative connection between a small ${\ensuremath{\beta_{M1}^{(\mathrm{v})}}}$ and the Anthropic Principle deserves additional investigation. On a more prosaic level, the connection to $\delta {\ensuremath{M_\mathrm{N}}}^\beta$ was used by Thomas et al. to extract values for ${\ensuremath{\beta_{M1}^{(\mathrm{v})}}}$ from the RBC lattice results for the electromagnetic self-energy of the nucleon [@Blum:2010ym]. They deduced small negative values (between $-0.5$ and $0$) at four pion masses between $279$ and $683\;{\ensuremath{\mathrm{MeV}}}$ [@Thomas:2014dxa]. Neither our results for ${\ensuremath{\beta_{M1}^{(\mathrm{v})}}}$ nor the lattice computations by NPLQCD support this finding [@Chang:2015qxa]; see Sect. \[sec:lattice\]. We point out here that the analysis of Ref. [@Thomas:2014dxa] assumes that other contributions to $\delta {\ensuremath{M_\mathrm{N}}}^{\rm em}$ are completely negligible. Comparisons of direct measurements of polarisabilities in lattice QCD with our analysis are more satisfying, and it is to these that we now turn. Comparison with Lattice Computations ==================================== \[sec:lattice\] ![(Colour online) Comparison to lattice-QCD computations. Lattice computations of ${\ensuremath{\alpha_{E1}}}$ (top left): $\protect\textcolor{blue}{\blacktriangle}$ (neutron) Lujan et al. [@Lujan:2014qga]; $\protect\textcolor{red}{\times}$ (proton) and $\protect\textcolor{blue}{+}$ (neutron) Detmold et al. [@Detmold:2010ts] (slight ${\ensuremath{m_\pi}}$-offset for better visibility); $\protect\textcolor{blue}{\blacktriangledown}$ (neutron) Engelhardt/LHPC [@Engelhardt:2007ub; @Engelhardt:2010tm]. Lattice computations of ${\ensuremath{\beta_{M1}}}$ (top right): $\protect\textcolor{blue}{\bullet}$ (neutron) Hall et al. [@Primer:2013pva; @Hall:2013dva]; (proton) and $\protect\textcolor{blue}{\circ}$ (neutron) NPLQCD [@Chang:2015qxa]. Gray “ghost points” found by shifting all lattice results by $+3\times10^{-4}\;{\ensuremath{\mathrm{fm}}}^3$. Lattice computation of ${\ensuremath{\gamma_{E1E1}^{(\mathrm{n})}}}$ (bottom): $\protect\textcolor{blue}{\blacktriangledown}$ (neutron) Engelhardt et al. [@Engelhardt:2011qq; @Engelhardt:2015], see text for qualifier. For Refs. [@Primer:2013pva; @Hall:2013dva; @Engelhardt:2011qq; @Engelhardt:2015], the reported lattice errors are smaller than our symbol sizes. Further notation as in Fig. \[fig:allpols\], including $\protect\textcolor{red}{\blacksquare}$ for proton values and for neutron ones at ${\ensuremath{m_\pi^\text{phys}}}$.[]{data-label="fig:lattice"}](lattice-alpha+beta-n+p.pdf "fig:"){width="\textwidth"}\ ![(Colour online) Comparison to lattice-QCD computations. Lattice computations of ${\ensuremath{\alpha_{E1}}}$ (top left): $\protect\textcolor{blue}{\blacktriangle}$ (neutron) Lujan et al. [@Lujan:2014qga]; $\protect\textcolor{red}{\times}$ (proton) and $\protect\textcolor{blue}{+}$ (neutron) Detmold et al. [@Detmold:2010ts] (slight ${\ensuremath{m_\pi}}$-offset for better visibility); $\protect\textcolor{blue}{\blacktriangledown}$ (neutron) Engelhardt/LHPC [@Engelhardt:2007ub; @Engelhardt:2010tm]. Lattice computations of ${\ensuremath{\beta_{M1}}}$ (top right): $\protect\textcolor{blue}{\bullet}$ (neutron) Hall et al. [@Primer:2013pva; @Hall:2013dva]; (proton) and $\protect\textcolor{blue}{\circ}$ (neutron) NPLQCD [@Chang:2015qxa]. Gray “ghost points” found by shifting all lattice results by $+3\times10^{-4}\;{\ensuremath{\mathrm{fm}}}^3$. Lattice computation of ${\ensuremath{\gamma_{E1E1}^{(\mathrm{n})}}}$ (bottom): $\protect\textcolor{blue}{\blacktriangledown}$ (neutron) Engelhardt et al. [@Engelhardt:2011qq; @Engelhardt:2015], see text for qualifier. For Refs. [@Primer:2013pva; @Hall:2013dva; @Engelhardt:2011qq; @Engelhardt:2015], the reported lattice errors are smaller than our symbol sizes. Further notation as in Fig. \[fig:allpols\], including $\protect\textcolor{red}{\blacksquare}$ for proton values and for neutron ones at ${\ensuremath{m_\pi^\text{phys}}}$.[]{data-label="fig:lattice"}](lattice-gammaE1E1n.pdf "fig:"){width="47.50000%"} In Fig. \[fig:lattice\], we compare our findings to emerging lattice-QCD computations of dipole polarisabilities. We do not report calculations without sea quarks—i.e. those that set the fermion determinant to one—and we have selected only references for pion masses up to about $850\;{\ensuremath{\mathrm{MeV}}}$, with values that were either extrapolated to infinite volume and infinitesimal lattice spacing, or for which the authors estimated such effects to be irrelevant at present accuracies; cf. Refs. [@Detmold:2006vu; @Tiburzi:2014zva]. To our knowledge, the work by Engelhardt/LHPC [@Engelhardt:2007ub; @Engelhardt:2010tm] on ${\ensuremath{\alpha_{E1}^{(\mathrm{n})}}}$ is the only extant calculation which meets these criteria and also accounts for the charges of the sea quarks themselves. At present, all other computations use uncharged sea quarks whose mass is identical to that of the valence quarks. Several efforts to include charged sea-quark effects are ongoing [@Freeman:2014kka; @Engelhardt:2007ub; @Engelhardt:2010tm]. We report lattice uncertainties as stated in the sources; a thorough appraisal of the lattice computations is not our goal. Note that an analysis of the consistency of lattice computations with our ${\ensuremath{m_\pi}}$-dependent predictions for polarisabilities cannot proceed by simple “standard-deviation counting", because the uncertainties in the shape of $\xi({\ensuremath{m_\pi}})$, and thus also the theory errors at different pion masses, are highly correlated. Ref. [@Stump:2001gu] derives a modified $\chi^2$, whose use would be one way to account for such systematic errors. We find that the [$\chi$EFT]{}prediction for ${\ensuremath{\alpha_{E1}^{(\mathrm{n})}}}$ agrees well with the available computations, of Lujan et al. [@Lujan:2014qga], Detmold et al. [@Detmold:2010ts], and Engelhardt/LHPC [@Engelhardt:2007ub; @Engelhardt:2010tm]. For the proton, only the result of Detmold et al. for ${\ensuremath{\alpha_{E1}}}$ at ${\ensuremath{m_\pi}}\approx400\;{\ensuremath{\mathrm{MeV}}}$ meets our selection criteria; it is quite compatible with the chiral curve. In regime (i) and (ii) these are statistically rigorous statements, since our error corridors provide estimates of higher-order effects—albeit with decreasing reliability as ${\ensuremath{m_\pi}}$ increases. The agreement for electric polarisabilities is so good that we feel we must stress that our results are not fits to lattice computations; the physical value is determined by Compton-scattering data on the proton and deuteron, and the pion-mass dependence exclusively by chiral dynamics. For the magnetic polarisability of the neutron, ${\ensuremath{\beta_{M1}^{(\mathrm{n})}}}$, we cannot reproduce the magnitude reported by the Adelaide group [@Primer:2013pva; @Hall:2013dva]. NPLQCD [@Chang:2015qxa] also reports results for ${\ensuremath{\beta_{M1}^{(\mathrm{n})}}}$ and ${\ensuremath{\beta_{M1}^{(\mathrm{p})}}}$ at $806\;{\ensuremath{\mathrm{MeV}}}$; the former agrees well with the Adelaide results. The authors of this paper point out that, since this pion mass corresponds to the flavour SU(3) limit, their result for the isovector combination ${\ensuremath{\beta_{M1}^{(\mathrm{p})}}}-{\ensuremath{\beta_{M1}^{(\mathrm{n})}}}$ is unaffected by their neglect of sea-quark charges, in contradistinction to the isoscalar one. Interestingly, their isovector result agrees nearly exactly with the chiral curve. This is illustrated by the grey “ghost points” in the top right panel of Fig. \[fig:lattice\], to which we have added constant isoscalar contribution of $3\times10^{-4}\;{\ensuremath{\mathrm{fm}}}^3$. This uncannily good agreement—at a pion mass of $800\;{\ensuremath{\mathrm{MeV}}}$, which is certainly outside the radius of convergence of [$\chi$EFT]{}—may of course be pure coincidence. Nevertheless, the lattice result of a sizeable isovector splitting at ${\ensuremath{m_\pi}}=806\;{\ensuremath{\mathrm{MeV}}}$ seems to support our finding in Sect. \[sec:isovector\] that ${\ensuremath{\beta_{M1}^{(\mathrm{p})}}}\approx {\ensuremath{\beta_{M1}^{(\mathrm{n})}}}$ at the physical point is something of a coincidence. Lattice QCD calculations at intermediate pion masses will either strengthen or rebut this conclusion. They can also check if the [$\chi$EFT]{}prediction of a significant ${\ensuremath{\alpha_{E1}^{(\mathrm{v})}}}$ for ${\ensuremath{m_\pi}}> {\ensuremath{m_\pi^\text{phys}}}$ is realised in QCD. Engelhardt et al. have also reported the first lattice study of a spin polarisability [@Engelhardt:2011qq; @Engelhardt:2015]. The neutron’s ${\ensuremath{\gamma_{E1E1}}}$ shows a minuscule but nonzero signal within the reported uncertainties. These values are without the subtraction of the pion-pole contribution in eq. . However, a calculation with uncharged sea quarks, like this one, has a number of pathologies. For example, in the two-flavour variant considered in Ref. [@Detmold:2006vu], the isovector “pion-pole” contribution of the physical particle $\propto{g_{\scriptscriptstyle A}}\tau_3$ must be supplemented by a degenerate isoscalar-scalar ghost which couples with an unknown strength $g_1$ to the nucleon. If $g_1$ has a similar magnitude to ${g_{\scriptscriptstyle A}}$, then even the sign of the total “pion-pole" contribution is unknown. The total lattice values of ${\ensuremath{\gamma_{E1E1}^{(\mathrm{n})}}}$ are extremely small, so in the absence of a very fine-tuned cancellation between “pion-pole” and structure contributions, it is likely that both are small. We therefore feel justified in placing the lattice points—which include both the physical and pathological pion-pole contributions—on the same plot as our pion-pole-subtracted curves. In most cases, lattice groups account for the differences between identifying polarisabilities as the terms quadratic in the electromagnetic fields, and the canonical definition via non-pole parts of the Compton amplitudes; see Refs. [@Lvov:1993fp; @Bawin:1996nz; @Schumacher:2005an; @Lee:2014iha] for further discussion. We follow Ref. [@engelprivcomm] in adding the Dirac-Foldy contribution of $\alpha_{\mathrm{EM}}({\ensuremath{\kappa^{(\mathrm{n})}}})^2/(4{\ensuremath{M_\mathrm{N}}}^3)\approx 0.7$ to ${\ensuremath{\alpha_{E1}^{(\mathrm{n})}}}$ in Ref. [@Engelhardt:2010tm]. Similarly, $\alpha_{\mathrm{EM}}/(4{\ensuremath{M_\mathrm{N}}}^3)\approx0.2$ should be subtracted from ${\ensuremath{\beta_{M1}^{(\mathrm{p})}}}$ at ${\ensuremath{m_\pi}}=806\;{\ensuremath{\mathrm{MeV}}}$ in Ref. [@Chang:2015qxa], but the effect is well within the lattice uncertainties [@tiburziprivcomm]. The surprising agreement between some of our chiral curves and lattice results at very large pion masses may be accidental. This agreement occurs far outside the [$\chi$EFT]{}radius of convergence, and could just be a coincidence. But it is striking, and so we close this section with a testable speculation as to why it could be more than an accident. First we note that, at the order to which we work, the chiral expansion does not produce positive powers of ${\ensuremath{m_\pi}}$; our [$\chi$EFT]{}result includes at most a logarithmic divergence as ${\ensuremath{m_\pi}}\to\infty$. Were we to go further in the expansion, we would encounter the usual problem of contributions that grow with more and more powers of ${\ensuremath{m_\pi}}$. The very mild dependence on pion mass seen in several lattice observables (see e.g. [@WalkerLoud:2008bp; @Walker-Loud:2013yua]) can then only be reconciled with [$\chi$EFT]{}through ever-increasing fine-tuning between terms of different, higher orders. In the case of the lattice polarisabilities for ${\ensuremath{m_\pi}}\gtrsim 400\;{\ensuremath{\mathrm{MeV}}}$, such pion-mass-independence is more akin to that expected in a heavy-constituent-quark model, or in the classical Lorentz model which considers heavy, charged particles in a harmonic-oscillator potential. Indeed, this smooth ${\ensuremath{m_\pi}}$-evolution might be considered generic. Of course, QCD must provide a smooth interpolation between the chiral and heavy-quark regimes; cf. Ref. [@Leinweber:1999ig] for the case of baryon masses. It is plausible that results like ours, with weak ${\ensuremath{m_\pi}}$-dependence at low chiral orders and large pion masses, provide reasonable extrapolations into the regime in which [$\chi$EFT]{}is *a priori* inapplicable. Such a “principle of chiral persistence” suggests that our chiral extrapolations for the spin polarisabilities may turn out to match future lattice computations at high pion masses, too. If such “chiral persistence" is not a feature of QCD, then those computations will reveal the agreement seen thus far to indeed be accidental. Summary and Conclusions ======================= \[sec:conclusions\] In this paper, we have presented the static scalar and spin dipole polarisabilities of both the proton and neutron in [$\chi$EFT]{}as a function of the pion mass. We have included the leading and sub-leading effects of the nucleon’s pion cloud, together with the leading contributions of the $\Delta(1232)$ and its pion cloud. We have differentiated between two pion-mass regimes. Close to the physical ${\ensuremath{m_\pi}}$, our results are complete at second order in the small expansion parameter $\delta\approx\sqrt{{\ensuremath{m_\pi}}/\Lambda_\chi}\approx {\ensuremath{\Delta_{\scriptscriptstyle M}}}/\Lambda_\chi$. This corresponds to three non-vanishing orders for the scalar polarisabilities and two for the spin polarisabilities. For ${\ensuremath{m_\pi}}\sim{\ensuremath{\Delta_{\scriptscriptstyle M}}}$, however, the results are complete only at leading order since contributions are reordered: leading $\pi$N, $\Delta(1232)$ and $\pi\Delta$ effects are all of similar size. A central goal of the paper is to provide reproducible estimates of uncertainties from within the [$\chi$EFT]{}framework which are as objective as feasible. At each pion mass, we have used a recently developed statistical interpretation of standard order-by-order EFT convergence estimates to derive $68\%$ degree-of-belief intervals. The resulting probability distributions are non-Gau[ß]{}ian. They are based on several assumptions: the error associated with the first omitted term in the [$\chi$EFT]{}series dominates the uncertainty; the corresponding EFT coefficient is “natural” in units of the breakdown scale; and the size of this first omitted term grows linearly with ${\ensuremath{m_\pi}}$. The inclusion of select higher-order effects indicates that for pion masses below about $250\;{\ensuremath{\mathrm{MeV}}}$ our uncertainties are, if anything, overestimated. In fact, basic physical arguments imply that our truncation-error for ${\ensuremath{\alpha_{E1}}}+{\ensuremath{\beta_{M1}}}$ is markedly too large for ${\ensuremath{m_\pi}}> {\ensuremath{m_\pi^\text{phys}}}$, so we somewhat arbitrarily assign zero truncation error to it. Truncation errors must be combined with uncertainties in the input parameters, like the error on ${\ensuremath{\alpha_{E1}}}+ {\ensuremath{\beta_{M1}}}$ from the Baldin sum rule. A framework that combines all these errors in a statistically consistent way is under development [@Furnstahl:2014xsa; @Wesolowski:2015fqa]. At the physical pion mass, the truncation errors of the spin polarisabilities augment our previously published prediction of the central values, and our recent fits of the scalar polarisabilities. In all cases, the Bayesian method provides a rigorous theory error. Our spin polarisabilities agree well, within their errors, with available extractions from data, and with the predictions of both dispersion relations and a formulation of [$\chi$EFT]{}with relativistic baryons. For the neutron electric polarisability and ${\ensuremath{\gamma_{E1E1}^{(\mathrm{n})}}}$, agreement between our [$\chi$EFT]{}predictions and extant lattice computations at ${\ensuremath{m_\pi}}\lsim 350$ MeV is remarkably good. Beyond this pion mass, there are doubts about the convergence of [$\chi$EFT]{}, and the error bars we derived are certainly not trustworthy. Nevertheless, if we extrapolate the central value of our [$\chi$EFT]{}curves into this regime, agreement persists for lattice results on both ${\ensuremath{\alpha_{E1}^{(\mathrm{n})}}}$ and ${\ensuremath{\alpha_{E1}^{(\mathrm{p})}}}$ at ${\ensuremath{m_\pi}}\approx 400$ MeV—within the uncertainties on the lattice numbers. Taking such an extrapolation for the isovector magnetic polarisability out to ${\ensuremath{m_\pi}}\approx 800$ MeV yields striking agreement with the recent results of Ref. [@Chang:2015qxa]. This is surprising, given that [$\chi$EFT]{}is certainly not convergent at such a large pion mass. We speculate that such an extrapolation of the [$\chi$EFT]{}curve does better than we have any right to expect, because—at the order to which we work—the pion-mass dependence is tame enough to permit smooth evolution into the functional dependence of a constituent-quark model. While most of the present lattice results are at too high a pion mass to be reliably extrapolated to the physical point, they still corroborate important aspects of our findings. A cancellation between $\pi$N loops and short-distance mechanisms encoded in LECs makes the magnetic scalar and spin polarisabilities small at ${\ensuremath{m_\pi^\text{phys}}}$, but this cancellation does not persist away from the physical point. Similar fine-tuning leads to a physical-world proton-neutron difference that, for both the scalar electric and magnetic polarisabilities, is consistent with zero within present uncertainties. Both of these proton-neutron degeneracies are lifted away from the physical point. In the near future, lattice calculations could examine the onset of these cancellations as ${\ensuremath{m_\pi}}$ is lowered towards, and even below, $250\;{\ensuremath{\mathrm{MeV}}}$. We pointed out that ${\ensuremath{\beta_{M1}^{(\mathrm{v})}}}$ may have a previously-neglected impact on the variation of the proton-neutron mass difference with the quark mass. The critical role of the neutron lifetime in Big-Bang Nucleosynthesis then suggests an anthropic argument may explain what otherwise appears to be a coincidentally small value of ${\ensuremath{\beta_{M1}^{(\mathrm{v})}}}$. Finally, we look forward to a next-order calculation of the polarisabilities in [$\chi$EFT]{}. This includes subleading effects of the pion cloud around the Delta. The associated LECs in the [$\chi$EFT]{}with an explicit $\Delta(1232)$ are, in principle, constrained by data from other processes, e.g. $\pi$N scattering and pion photoproduction. But in practice, their values have non-negligible uncertainties [@Gellas:1998wx; @Pascalutsa:2006up; @Yao:2016vbz]. Still, such an ${\cal O}(e^2 \delta^5)$ calculation could yield more accurate predictions at the physical point, in particular for spin polarisabilities and isovector parts. It may also allow a better assessment of the convergence of the chiral series for ${\ensuremath{m_\pi}}\sim{\ensuremath{\Delta_{\scriptscriptstyle M}}}$, where only leading-order accuracy is available at present. Acknowledgements {#acknowledgements .unnumbered} ================ This project was prompted by a request from A. Alexandru. We are indebted to him and to M. Lujan for discussions and encouragement; to M. Savage for well-timed advice, crucial inspiration, and sharing findings of the NPLQCD collaboration prior to publication; and to S. Beane for discussions on the behaviour of low-energy quantities outside the chiral regime. We also gratefully acknowledge correspondence with A. Alexandru, M. Engelhardt and B. Tiburzi concerning details of their respective lattice computations, and with M. Hoferichter and H. Leutwyler on the impact of ${\ensuremath{\beta_{M1}^{(\mathrm{v})}}}$ on the proton-neutron mass difference. Finally, we are grateful to the organisers and participants of the workshops <span style="font-variant:small-caps;">Compton Scattering off Protons and Light Nuclei: Pinning Down the Nucleon Polarizabilities</span> (2013) and <span style="font-variant:small-caps;">Lattice Nuclei, Nuclear Physics and QCD—Bridging the Gap</span> (2015), both at the ECT\*, Trento (Italy), and <span style="font-variant:small-caps;">Bound States and Resonances in Effective Field Theories and Lattice QCD Calculations</span>, Benasque (Spain, 2014) for financial support, and stimulating presentations and atmosphere. 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Shapes and Profiles of Corridors of Uncertainties ================================================= \[app:appendix\] We show in Fig. \[fig:spinposterior\] the pdfs for all polarisabilities at the physical pion mass, and in Fig. \[fig:higherorders-supplement\] the theoretical error corridors of the results for the $m_\pi$-dependence of $\alpha_{E1}\pm\beta_{M1}$, $\gamma_0$ and $\gamma_\pi$, as detailed in Sect. \[sec:Finding-Theory-Uncertainties\]. [^1]: Email: hgrie@gwu.edu [^2]: Email: judith.mcgovern@manchester.ac.uk [^3]: Email: phillips@phy.ohiou.edu [^4]: These contain a merely typographical error for ${\ensuremath{\kappa^{(\mathrm{s})}}}=-0.12$. [^5]: The errors of $\{\pm1.8;\pm0.7;\pm0.4;\pm0.4\}$ cited in Refs. [@talkMAMI; @Martel:2014pba], though supplied by us, differ slightly from these. That is because eq.  reflects the difference in power-counting for amplitudes and polarisabilities discussed in Sect. \[sec:pols\]; considers isoscalar and isovector convergence separately; and uses the Bayesian framework described below to calculate 68% intervals from the EFT truncation error. With the possible exception of ${\ensuremath{\gamma_{E1M2}}}$, we do not consider the change in uncertainties significant—cf. the discussion below of how well uncertainties can be estimated.

--- abstract: 'We propose a universal gate set acting on a qubit formed by the degenerate ground states of a Coulomb-blockaded time-reversal invariant topological superconductor island with spatially separated Majorana Kramers pairs: the “Majorana Kramers Qubit". All gate operations are implemented by coupling the Majorana Kramers pairs to conventional superconducting leads. Interestingly, in such an all-superconducting device, the energy gap of the leads provides another layer of protection from quasiparticle poisoning independent of the island charging energy. Moreover, the absence of strong magnetic fields – which typically reduce the superconducting gap size of the island – suggests a unique robustness of our qubit to quasiparticle poisoning due to thermal excitations. Consequently, the Majorana Kramers Qubit should benefit from prolonged coherence times and may provide an alternative route to a Majorana-based quantum computer.' author: - Constantin Schrade and Liang Fu title: Quantum Computing with Majorana Kramers Pairs --- @twocolumnfalse 1.5truecm In recent years an increasing number of platforms have been proposed for realizing time-reversal invariant topological superconductors (TRI TSCs) [@bib:Schnyder2008]. Among the most notable platforms are nanowires and topological insulators in contact to unconventional superconductors (SCs) [@bib:Wong2012; @bib:Nagaosa2013; @bib:Zhang2013; @bib:Dumitrescu2014] and conventional SCs [@bib:Klinovaja2014; @bib:Gaidamauskas2014; @bib:Schrade2017; @bib:Klinovaja20142; @bib:Yan2018; @bib:Hsu2018], proximity-induced Josephson $\pi$-junctions in nanowires and topological insulators [@bib:Keselman2013; @bib:Haim2014; @bib:Schrade2015; @bib:Borla2017] as well as TSCs with an emergent time-reversal symmetry (TRS) [@bib:Huang2017; @bib:Reeg2017; @bib:Hu2017; @bib:Maisberger2017]. A common feature of TRI TSCs is that they host spatially separated Majorana Kramers pairs (MKPs) which form robust, zero energy modes protected by TRS. In spite of much fundamental interest in the properties of MKPs [@bib:Chung2013; @bib:Li2016; @bib:Pikulin2016; @bib:Kim2016; @bib:Camjayi2017; @bib:Bao2017; @bib:Schrade2018], a yet unsolved question is if MKPs can be employed for applications in quantum computation. Here, we answer this question in the affirmative. The purpose of this work is to introduce a qubit formed by the degenerate ground states of a Coulomb-blockaded TRI TSC island with spatially separated MKPs: the “Majorana Kramers Qubit" (MKQ). We depict the minimal experimental setup for a single MKQ in Fig. \[fig:1\]. It comprises two SC leads which separately couple to two distinct MKPs on a U-shaped TRI TSC island. The two SC leads are weakly coupled among themselves by spin-flip and normal tunnelling barriers. Within this setup, we will implement single-qubit Clifford gates by making use of a measurement-based approach to quantum computing [@bib:Bonderson2008; @bib:Litinski2017]. Moreover, to achieve universal quantum computation we will implement a $\pi/8$-gate as well as a two-MKQ entangling gate by pulsing of tunnel couplings. ![(Color online) Setup consisting of a U-shaped, mesoscopic TRI TSC island (gray) realizing a MKQ. Tunable tunnel couplings (white, dashed) connect SC leads $\ell=\text{L,R}$ (red) to the MKPs $\gamma_{\ell,s}$ (yellow) with $s={\uparrow},{\downarrow}$. The SC leads themselves are also connected by a spin-flip and a normal tunnelling barrier with lengths $d, d'$. To facilitate Cooper pair splitting between these two tunnelling barriers and the TRI TSC island we require that the separation of the tunnelling contacts is smaller than the coherence length $\xi_{\text{SC}}$ of the SC leads. Moreover, to avoid couplings of the MKPs to fermionic corner modes [@bib:Loss2015], the length of the vertical segments of the TRI TSC islands are much longer than the MKP localization length $\xi_{\text{MKP}}$. Lastly, a gate voltage $V$ tunes the charge on the TRI TSC island via a capacitor with capacitance $C$. []{data-label="fig:1"}](Fig1){width="0.75\linewidth"} The main conceptual lesson we will learn is that Majorana-based quantum computing is possible without invoking the need for magnetic fields. Besides that, there two interesting, yet more practical, features of our setup which are noteworthy: (1) Within the single-MKQ setup of Fig. \[fig:1\], single-electron tunnelling from the SC leads does not only require overcoming the charging energy of the TRI TSC island but also the breaking of a Cooper pair in the leads. Consequently, the SC gap of the leads provides an additional layer of protection against quasiparticle poisoning, independent of the island charging energy. (2) Quasiparticle poisoning due to thermal excitations within the TRI TSC island is strongly suppressed the SC gap of the island itself. Critically, the energy gap of a TRI TSC island is conceivably larger than the energy gap of TRS-breaking Majorana islands [@bib:Fu2010; @bib:Vijay2015; @bib:Vijay2016; @bib:Landau2016; @bib:Plugge2016; @bib:Vijay2016_2; @bib:Aasen2016; @bib:Karzig2016; @bib:Plugge2017; @bib:Schrade2018_2; @bib:Gau2018] since there is no magnetic field that would reduce the SC gap size. As a consequence, the MKQ should benefit from improved coherence times and may be a viable route towards a robust quantum computer. [*Setup.*]{} As shown in Fig. \[fig:1\], our setup comprises a U-shaped TRI TSC islands hosting MKPs $\gamma_{\ell,s}$ with $s={\uparrow},{\downarrow}$ at spatially well separated boundaries $\ell=\text{L,R}$. The two members of a MKP are related by TRS, $$\mathcal{T}\gamma_{\ell,{\uparrow}}\mathcal{T}^{-1}=\gamma_{\ell,{\downarrow}}, \; \mathcal{T}\gamma_{\ell,{\downarrow}}\mathcal{T}^{-1}=-\gamma_{\ell,{\uparrow}}.$$ We assume that the dimensions of the horizontal island segments exceed the localization lengths $\xi_{\text{MKP}}$ of the MKPs. This avoids couplings of the MKPs to fermionic modes that are potentially localized at the island corners [@bib:Loss2015] and, thereby, ensures that the MKPs are, in fact, robust zero-energy states protected by TRS. Since the TRI TSC island is of mesoscopic size, it acquires a charging energy given by $$U_{C} = \left(ne-Q\right)^{2}/ 2C.$$ Here, $Q$ is the island gate charges that is continuously tunable with a voltage across a capacitor with capacitance $C$. We assume that the gate charge $Q/e$ is tuned close to an even or odd integer for both islands. A sufficiently large charging energy $e^{2}/2C$ then fixes the joint parity of the MKPs on the TRI TSC island to [@bib:Fu2010; @bib:Xu2010] $$\label{TotalParity} \gamma_{\text{L},{\uparrow}}\gamma_{\text{R},{\uparrow}}\gamma_{\text{L},{\downarrow}}\gamma_{\text{R},{\downarrow}} = (-1)^{n_0}.$$ This constraint reduces the four-fold degeneracy of the ground state at zero charging energy, to a two-fold degenerate ground state which forms the MKQ. The Pauli operators acting on each of the two MKQs can be written as bilinears in the Majorana operators, $$\begin{split} \hat{x}&=i\gamma_{\text{R},{\uparrow}}\gamma_{\text{L},{\downarrow}}, \quad \hat{y}=i\gamma_{\text{R},{\uparrow}}\gamma_{\text{R},{\downarrow}} , \quad \hat{z}=i\gamma_{\text{R},{\downarrow}}\gamma_{\text{L},{\downarrow}}. \end{split}$$ Under TRS, the Pauli operators transform as $\mathcal{T}\hat{x}\mathcal{T}^{-1}=(-1)^{n_0}\hat{x}$, $\mathcal{T}\hat{y}\mathcal{T}^{-1}=-\hat{y}$ and $\mathcal{T}\hat{z}\mathcal{T}^{-1}=(-1)^{n_0}\hat{z}$. In our setup, we choose to address the MKQ by weakly coupling each MKP to a separate $s$-wave SC lead. The Hamiltonian for the two SC leads reads $$H_{SC}=\sum_{\ell=\text{L,R}}\sum_{{{{\bf{k}}}}} \Psi_{\ell,{{{\bf{k}}}}}^\dagger \left( \xi_{{{{\bf{k}}}}}\eta_{z}+\Delta_{\ell}\eta_{x}e^{i\varphi_{\ell}\eta_{z}} \right)\Psi_{\ell,{{{\bf{k}}}}},$$ where $\Psi_{\ell,{{{\bf{k}}}}}=(c_{\ell,{{{\bf{k}}}}{\uparrow}},c^{\dag}_{\ell,-{{{\bf{k}}}}{\downarrow}})^{T}$ is a Nambu spinor with $c_{\ell,{{{\bf{k}}}}s}$ the electron annihilation operator at momentum ${{{\bf{k}}}}$ and spin $s$ in lead $\ell$. The Pauli matrices $\eta_{x,y,z}$ are acting in Nambu-space. Furthermore, $\xi_{{{{\bf{k}}}}}$ is the normal state dispersion and $\Delta_{\ell},\varphi\equiv\varphi_{\text{L}}-\varphi_{\text{R}}$ denote the magnitude and the relative phase difference of the SC order parameters. We assume sufficiently low temperatures such that no quasiparticle states in the SC leads are occupied with notable probability and can couple to the MKPs. The most general tunneling Hamiltonian between the MKPs on the islands and the fermions on the $\ell$-SC lead is given by, $$\begin{aligned} \label{Eq4} H_{T} &= \sum_{\ell=\text{L,R}} \sum_{{{{\bf{k}}}},s} \lambda_{\ell} c^{\dag}_{\ell,{{{\bf{k}}}}s} \gamma_{\ell,s} e^{-i\phi/2} + \text{H.c.}, \end{aligned}$$ where we have diagonalized the tunnelling Hamiltonian in spin-space by an appropriate rotation of the lead fermions [@bib:Schrade2018]. In particular, this rotation also constraints the point-like tunnelling amplitudes $\lambda_{\ell}$ to be real numbers. Furthermore, we remark that the operators $e^{\pm i\phi/2}$ raise/lower the total island charges by one unit, $[n,e^{\pm i\phi/2}]=\pm e^{\pm i\phi/2}$, while the MBSs operators $\gamma_{\ell,s}$ flip the respective electron number parities. As evident from Fig. \[fig:1\], there are two types of couplings between the SC leads: The first type is an *indirect coupling* via the TRI TSC islands which is induced by the tunnelling Hamiltonian of Eq. . The second type is a *direct coupling* via two additional tunnelling barriers. The first tunnelling barrier only allows for normal tunnelling described by the tunnelling Hamiltonian, $$H_{N} = t_{N} \sum_{{{{\bf{k}}}}} c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}}c_{\text{L},{{{\bf{k}}}}{\uparrow}} + c^{\dag}_{\text{L},{{{\bf{k}}}}{\downarrow}} c_{\text{R},{{{\bf{k}}}}{\downarrow}} + \text{H.c.},$$ with $t_{N}$ a complex, point-like tunnelling amplitude and $|t_{N}|\ll\lambda_{\ell}$. The second barrier only allows for spin-flip tunnelling described by a tunnelling Hamiltonian. $$H_{S} = t_{S} \sum_{{{{\bf{k}}}}} c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}}c_{\text{L},{{{\bf{k}}}}{\downarrow}} - c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} c_{\text{R},{{{\bf{k}}}}{\downarrow}} + \text{H.c.},$$ with $t_{S}\ll\lambda_{\ell}$ a complex, point-like tunnelling amplitude and $|t_{S}|\ll\lambda_{\ell}$. We propose two ways to engineer such tunnellling barriers: (1) We consider barriers with a finite *intrinsic* spin-orbit coupling with spin-orbit length $\lambda_{\text{SO}}$ as well as different barrier lengths $d,d'$. Tuning $\lambda_{\text{SO}}/d'$ ($\lambda_{\text{SO}}/d$) to a positive integer (positive half integer) realizes a barrier with pure normal (spin-flip) tunnelling [@bib:Bercioux2015]. (2) We consider barriers without intrinsic spin-orbit coupling but with an *engineered* spin-orbit coupling due to a local, rotating magnetic field induced by a series of nanomagnets [@bib:Karmakar2011; @bib:Klinovaja2013]. By adjusting either the period of the rotating field through the separation of the nanomagnets or again the length of the barriers, we can realize barriers with pure normal or spin-flip tunnelling. In summary, we conclude that the full Hamiltonian of our setup is given by $H=U_{C}+H_{SC}+H_{T}+H_{N}+H_{S}$. ![(Color online) (a) To third order in the tunnelling amplitudes, a Cooper pair moves between the two SC lead by splitting up between the normal tunnelling barrier and the TRI TSC island. The spin-flip tunnelling barrier is fully depleted by a local gate. The resulting terms in the effective Hamiltonian are $\propto\hat{z}$. (b) Same as (a) but this time the Cooper pair splits up between the spin-flip tunnelling barrier and the TRI TSC island. The normal tunnelling barrier is fully depleted by a local gate. The resulting terms in the effective Hamiltonian are now $\propto\hat{x}$. []{data-label="fig:2"}](Fig2){width="\linewidth"} [*Single-qubit Clifford gates.*]{} In this section, we will implement single-qubit Clifford gates by “Majorana tracking" [@bib:Litinski2017]. This means that for a given circuit of single-qubit Clifford gates we record all Pauli operator redefinitions on a classical computer and use the quantum hardware only to perform suitable measurements of the $\hat{x}, \hat{y}, \hat{z}$-Pauli operators at the end of the computation. As a starting point, for measuring the $\hat{z}$-Pauli operator, we consider the situation when a local gate depletes the spin-flip tunnelling barrier between the two SC leads such that $\text{Im}\, t_{S}=0$ for $n_0$ even and $\text{Re}\, t_{S}=0$ for $n_{0}$ odd. In this case, to second order in the tunnelling amplitudes $t_{N}$, Cooper pairs tunnel between the SC leads only via the normal tunnelling barrier inducing a finite Josephson coupling $J_{N}\sim |t_{N}|^{2}/\Delta$. In particular, a Josephson coupling due to Cooper tunnelling between each SC lead and the TRI TSC is highly unfavorable as a result of the substantial island charging energy [@bib:Schrade2018_2]. We, hence, recognize that the island charging energy plays two significant roles in our setup: First, it suppresses quasiparticle poisoning due to single-electron tunnelling from the environment. Second, it also suppresses local mixing terms $\propto\hat{y}$ due to Cooper pair tunnelling between each SC lead and the TRI TSC island. Such local mixing terms are – as noticed in the previous literature [@bib:Wolms2014] – of importance for TRI TSCs with zero charging energy and, as we will see, can be utilized for measuring the $\hat{y}$-Pauli operator. As a next step, we note that to third order in the tunnelling amplitudes $t_{N},\lambda_{\text{L}},\lambda_{\text{R}}$ Cooper pair splitting sequences between the TRI TSC island and the normal tunnelling barrier induce additional Josephson couplings, $J_{z}$ for $n_{0}$ even and $J'_{z}$ for $n_{0}$ odd. We depict an example of such a Cooper pair splitting sequence in Fig. \[fig:2\](a). In a first process, a Cooper pair on the left SC lead breaks up and one of the electrons tunnels via the normal tunnelling barrier to the right SC lead. This leaves the left SC lead in an excited state with one quasiparticle above the SC gap. In a second process, the quasiparticle on the left SC tunnels to the TRI TSC island and increments its charge by one unit. While the left SC returns to its ground state in this way, the TRI TSC island is now in an excited state with one excess charge. It, therefore, requires a third process to remove the extra charge from the TRI TSC by recombining it to a Cooper pair on the right SC lead. Critically, the tunnelling events via both the normal tunnelling barrier and the TRI TSC island conserve the electron spin. For that reason, the just described third-order sequences contribute terms $\propto\hat{z}=i\gamma_{\text{R},{\downarrow}}\gamma_{\text{L},{\downarrow}}=(-1)^{n_{0}}i\gamma_{\text{L},{\uparrow}}\gamma_{\text{R},{\uparrow}}$. As the last step, we point out that to fourth order in the tunnelling amplitudes, Cooper pairs tunnel between the two SC leads only via the TRI TSC island yielding a Josephson coupling $J$ with a sign determined by the joint parity of all four MBSs on the TRI TSC island [@bib:Schrade2018]. In the limit of weak tunnel couplings, $\pi\nu_{\ell}\lambda_{\ell}^{2}\ll\Delta,e^{2}/2C$ with $\nu_{\ell}$ the normal-state density of states per spin of the $\ell$-SC lead at the Fermi energy, we compute the amplitudes of all above-mentioned sequences perturbatively. Up to fourth order in the tunnelling amplitudes, we then summarize our results in an effective Hamiltonians acting on the ground states of the SC leads and the TRI TSC island. For $n_{0}$ even and $n_{0}$ odd, we find that, $$\begin{split} &H_{z,\text{even}}=-(J_{N}-J+\hat{z}\, J_{z})\cos\varphi, \\ &H_{z,\text{odd}}=-(J_{N}+J)\cos\varphi + \hat{z} \, J'_{z} \sin\varphi, \label{Hz} \end{split}$$ where the detailed microscopic forms of the Josephson couplings $J$ and $J_{z}, J'_{z}$ are given in [@bib:Schrade2018] and [@bib:supplemental], respectively. Here, it suffices to remark that $J_{z}\neq0$ $(J'_{z}\neq0)$ provided $\text{Im}\, t_{N}\neq0$ ($\text{Re} \, t_{N}\neq0$). Moreover, we point out that the both effective Hamiltonian exhibit TRS: For $H_{z,\text{even}}$ both $\hat{z}$ and $\cos\varphi$ are time-reversal even, while for $H_{z,\text{odd}}$ both $\hat{z}$ and $\sin\varphi$ are time-reversal odd. To measure the $z$-eigenvalue of the $\hat{z}$-Pauli operator, we adopt a two-step protocol: (1) First, we separately measure the Josephson current through the normal tunnelling barrier and through the TRI TSC island to determine $J_{N}$ and $J$. (2) Second, we measure the Josephson current through the entire device. For $n_{0}$ even, the latter is given by $I=I_{c}\sin\varphi$ with the critical current $I_c =2e(J_{N}-J+z\, J_{z})/\hbar$ fixing the $z$-eigenvalue. For $n_0$ odd, the current phase relation is of the form $I=I_{c}\sin(\varphi+\varphi_0)$. This time it is not the critical current $I_{c}=2e\,\text{sgn}(J_{N}+J)\sqrt{(J_{N}+J)^{2}+(J'_z)^{2}}/\hbar$ but the anomalous phase shift $\varphi_{0}=z\,\arctan[J'_{z}/(J_{N}+J)]$ which fixes the $z$-eigenvalue. We note that the finite anomalous phase shift results from the $\hat{z}$-eigenstates breaking TRS when $n_{0}$ odd. For the measurement of the $\hat{x}$-Pauli operator, we now shift our attention to a fully depleted normal tunnelling barrier such that $\text{Im}\, t_{N}=0$ for $n_0$ even and $\text{Re}\, t_{N}=0$ for $n_{0}$ odd. Similar to our earlier discussions, second order co-tunnelling events induce a Josephson coupling $J_{S}\sim|t_{S}|^{2}/\Delta$ as a result of Cooper pair tunnelling via the spin-flip tunnelling barrier whereas fourth order co-tunnelling events mediate a Josephson coupling $J$ via the TRI TSC island. However, a qualitative difference to the preceding considerations arises for the third order Cooper pair splitting sequences, see Fig. \[fig:2\](b). These sequences now demand two spin-flips, one for an electron to move through the spin-flip tunnelling barrier and one for an electron to move through the TRI TSC island. As a consequence, the third-order sequences now contribute terms $\propto\hat{x}=i\gamma_{\text{R},{\uparrow}}\gamma_{\text{L},{\downarrow}}=(-1)^{n_0}i\gamma_{\text{R},{\downarrow}}\gamma_{\text{L},{\uparrow}}$. More explicitly, up to fourth order in the weak tunnel couplings, the effective Hamiltonians for $n_0$ even and $n_0$ odd are given by, $$\begin{split} &H_{x,\text{even}}=-(J_{N}-J+\hat{x}\, J_{x})\cos\varphi, \\ &H_{x,\text{odd}}=-(J_{N}+J)\cos\varphi + \hat{x} \, J'_{z} \sin\varphi. \label{Hx} \end{split}$$ Here, $J_{x}, J'_{x}$ denote the Josephson couplings due to the Cooper pair splitting processes. For their microscopic form, see [@bib:supplemental]. Here, we only remark that $J_{x}\neq 0$ ($J'_{x}\neq 0$) granted that $\text{Im}\,t_{S}\neq0$ ($\text{Re}\,t_{S}\neq0$). We further point out that both effective Hamiltonian exhibit TRS: For $H_{x,\text{even}}$ both $\hat{x}$ and $\cos\varphi$ are time-reversal even, while for $H_{x,\text{odd}}$ both $\hat{x}$ and $\sin\varphi$ are time-reversal odd. To measure the $\hat{x}$-Pauli operator, we readily see that the effective Hamiltonians are of the same form as those in Eq. . Consequently, our previously introduced measurement protocol for the $\hat{z}$-Pauli operator immediately carries over to the measurement of the $\hat{x}$-Pauli operator. At this point, it is worth mentioning that a potential error source for the $\hat{x}$, $\hat{z}$-measurements occurs when both $\text{Im}\, t_{N}\neq0$, $\text{Im}\, t_{S}\neq0$ for $n_0$ even or $\text{Re}\, t_{N}\neq0$, $\text{Re}\, t_{S}\neq0$ for $n_{0}$ odd. In practice, this happens either when one of the tunnelling barriers is not fully depleted, or the barrier lengths $d,d'$ are not appropriately adjusted to the spin-orbit length $\lambda_{\text{SO}}$. Fortunately, this constitutes a static hardware error which can be addressed prior to all experiments. In particular, the error can be made controllably small with a careful design of a *conventional* Josephson junction. In the last part of this section, we address $\hat{y}$-measurements. These require the tuning the charging energy of the TRI TSC to zero which is attainable – on demand – by coupling the TRI TSC island to a bulk SC through a gate-tunable valve [@bib:Aasen2016]. Critically, even at zero charging energy the value of the joint fermion parity in Eq.  remains protected as a result of the lead SC gap. However, unlike in the case of a substantial charging energy, Cooper pairs can now tunnel in a second order process between each SC lead and the TRI TSC island inducing a Josephson coupling $\propto\hat{y}$ [@bib:Chung2013]. Consequently, the resulting Josephson current provides a means for measuring the $\hat{y}$ eigenvalue. The details of this measurement scheme are discussed in [@bib:Chung2013]. [*Universal quantum computation.*]{} So far, we have discussed the implementation of single-qubit Clifford gates. However, for universal quantum computation, we need to supplement the single-qubit Clifford gates by a single-qubit $\pi/8$-gate and a two-qubit entangling gate [@bib:Brylinski2001]. Clearly, by pulsing the Josephson couplings in the effective Hamiltonians of Eq.  and we can perform arbitrary rotations on the MKQ Bloch sphere and, therefore, in particular, a $\pi/8$-gate. For this procedure, phase-independent contributions – which were irrelevant for the Josephson current – should now be included in the effective Hamiltonians, see [@bib:supplemental]. For a two-qubit entangling gate, we consider the setup of Fig. \[fig:3\] which comprises two SC leads addressing two MKQs $a,b$. Here, a local gate fully depletes both the normal and the spin-flip tunnelling barrier, $t_{N}=t_{S}=0$. Provided that the width of the SC leads is much smaller than the SC coherence length $\xi_{\text{SC}}$, a Cooper pair can now split up between the two TRI TSC islands and generate entanglement between the MKQs. For symmetric couplings and a ground state charge $n_0$ for both TRI TSC islands, we have computed the amplitudes of these processes in the weak coupling limit. An effective anisotropic Heisenberg interaction summarizes the results, $$\begin{split} \label{HEff_Two_Qubit} H_{ab} &=J_{y}\hat{y}_{a}\hat{y}_{b}\\ &+[J_{xz}+(-1)^{n_{0}+1}J'_{xz}\cos\varphi] (\hat{x}_{a}\hat{x}_{b}+\hat{z}_{a}\hat{z}_{b}). \end{split}$$ For the microscopic form of the coupling constants $J_{xz},J'_{xz},J_{y}$, see [@bib:supplemental]. We note that Heisenberg interaction can be made isotropic by choosing the SC phase difference such that $\tilde{J}\equiv J_{y}=J_{xz}+(-1)^{n_{0}+1}J'_{xz}\cos\varphi$. Pulsing the couplings for a duration $\tau$ defined by $ \int^{\tau}_{0} \tilde{J}(t')\ \mathrm{d}t'= \pi/2 \ (\text{mod}\ \pi) $ then implements a $\sqrt{\text{SWAP}}$-gate via the unitary time evolution operator. The latter, in combination with single-qubit gates, is sufficient for universal quantum computing [@bib:Loss1998]. ![(Color online) Setup of two MKQs $a,b$ coupled to two SC leads. The width of the leads is much smaller than their SC coherence length $\xi_{\text{SC}}$, thereby, permitting Cooper pair splitting between the two MKQs. The resulting anisotropic Heisenberg interaction between the two MKQs is used to construct a two-qubit entangling gate. []{data-label="fig:3"}](Fig3){width="0.8\linewidth"} [*Conclusions.*]{} In this work, we have introduced a qubit formed by the degenerate ground states of a Coulomb-blockaded TRI TSC island with spatially well separated MKPs: the “Majorana Kramers Qubit". 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Constantin Schrade and Liang Fu\ [*Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139*]{} In the Supplemental Material, we provide more details on the derivation of the effective Hamiltonians required for the implementation of the single- and two-qubit quantum gates. Effective Hamiltonian for the single-qubit gates ================================================ In this first section of the Supplemental Material, we derive the effective Hamiltonians for the single-qubit gates as given in Eq. (9) and (10) of the main text. For simplicity, we adopt two assumptions: First, we assume that the lead SC gaps are of equal magnitude, $\Delta\equiv\Delta_{\ell}$. Second, we assume that the gate charge of the TRI TSC island $Q/e$ is tuned to an integer value $n_{0}$. In this, so-called, Coulomb-valley regime the ground state of the TRI TSC island consists of $n_{0}$ units of charge. At the same time, adding/removing a single unit of charge from the TRI TSC island comes at an equal energy cost of $U\equiv e^{2}/2C$.\ Initially, we derive the effective Hamiltonian given in Eq. (9) of the main text that is used for the measurement of the $\hat{z}$-Pauli operator. Our focus are the contributions to the effective Hamiltonian which are most important for the measurement protocol and which occur at third-order in the tunnelling amplitudes. They are given by, $$\begin{split} H^{(3)}_{z}&= - P_{n_0} H_{T,z} \left(\left[H_{0}+U_{C}\right]^{-1}\left[1-P_{n_0}\right]H_{T,z}\right)^{2}P_{n_0}. \label{H3z} \end{split}$$ Here, $H_{0}=U_{C}+H_{\text{SC}}$ denotes the Hamiltonian of the uncoupled system and $H_{T,z}=H_{T}+H_{N}$ denotes the total tunnelling Hamiltonian. For the moment, we have set $t_{S}=0$. However, after having derived both effective Hamiltonians given in Eqs. (9) and (10) of the main text, we will see that it is sufficient to require $\text{Im}(t_{S})=0$ for $n_0$ even and $\text{Re}(t_{S})=0$ for $n_{0}$ odd. Finally, we note that $P_{n_0}=\Pi_{n_0}\Pi_{\text{BCS}}$ where $\Pi_{n_0}$ is a projector on the ground state of the TRI TSC island with $n_0$ units of charge and $\Pi_{\text{BCS}}$ is a projector on the BCS (Bardeen-Cooper-Schrieffer) ground states of the SC leads. As the first step in our derivation, we interpret Eq.  as the weighted sum of all three-step sequences of intermediate states that map the ground state manifold of $H_{0}$ onto itself. For the moment, we will only discuss the three-step sequences which comprise the transport of a Cooper pair from the left to the right SC lead or vice versa, see Fig. \[fig:1\_SM\]. Two example sequences of the type shown in Fig. \[fig:1\_SM\](a) are, $$\begin{split} &\quad\ P_{n_{0}} ( \lambda c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}} \gamma_{\text{R},{\uparrow}} e^{-i\phi/2} ) ( \lambda \gamma_{\text{L},{\uparrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} e^{i\phi/2} ) ( t^{*}_{N}c^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}}c_{\text{L},-{{{\bf{k}}}}{\downarrow}} ) P_{n_{0}} \\ &= t^{*}_{N}\lambda^{2} P_{n_{0}} ( c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}} \gamma_{\text{R},{\uparrow}} \gamma_{\text{L},{\uparrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} c^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}}c_{\text{L},-{{{\bf{k}}}}{\downarrow}} ) P_{n_{0}} \\ &= (-1)^{n_{0}+1} t^{*}_{N}\lambda^{2} ( \Pi_{n_0} \gamma_{\text{R},{\downarrow}} \gamma_{\text{L},{\downarrow}} \Pi_{n_0} ) ( \Pi_{\text{BCS}} c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} c^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}}c_{\text{L},-{{{\bf{k}}}}{\downarrow}} \Pi_{\text{BCS}}) \\ &= i(-1)^{n_{0}} t^{*}_{N}\lambda^{2} ( \Pi_{n_0} \hat{z} \Pi_{n_0} ) ( \Pi_{\text{BCS}} c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} c^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}}c_{\text{L},-{{{\bf{k}}}}{\downarrow}} \Pi_{\text{BCS}}) \\ &= i(-1)^{n_{0}+1} e^{i(\varphi_{\text{L}}-\varphi_{\text{R}})} t^{*}_{N}\lambda^{2} (u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2} ( \Pi_{n_0} \hat{z} \Pi_{n_0} ) ( \Pi_{\text{BCS}} \gamma_{\text{R},-{{{\bf{k}}}}{\downarrow}} \gamma_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}} \gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \Pi_{\text{BCS}}) \\ &= i (-1)^{n_{0}} e^{i(\varphi_{\text{L}}-\varphi_{\text{R}})} t^{*}_{N}\lambda^{2} (u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2} ( \Pi_{n_0} \hat{z} \Pi_{n_0} ) \Pi_{\text{BCS}} \\ \\ &\quad\ P_{n_{0}} ( \lambda c^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}} \gamma_{\text{R},{\downarrow}} e^{-i\phi/2} ) ( \lambda \gamma_{\text{L},{\downarrow}} c_{\text{L},-{{{\bf{k}}}}{\downarrow}} e^{i\phi/2} ) ( t_{N}c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}}c_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{n_{0}} \\ &= t_{N}\lambda^{2} P_{n_{0}} ( c^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}} \gamma_{\text{R},{\downarrow}} \gamma_{\text{L},{\downarrow}} c_{\text{L},-{{{\bf{k}}}}{\downarrow}} c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}}c_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{n_{0}} \\ &= t_{N}\lambda^{2} ( \Pi_{n_0} \gamma_{\text{R},{\downarrow}} \gamma_{\text{L},{\downarrow}} \Pi_{n_0} ) ( \Pi_{\text{BCS}} c^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}} c_{\text{L},-{{{\bf{k}}}}{\downarrow}} c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} \Pi_{\text{BCS}} ) \\ &= -i t_{N}\lambda^{2} ( \Pi_{n_0} \hat{z} \Pi_{n_0} ) ( \Pi_{\text{BCS}} c^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}} c_{\text{L},-{{{\bf{k}}}}{\downarrow}} c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} \Pi_{\text{BCS}} ) \\ &= i t_{N}\lambda^{2} e^{i(\varphi_{\text{L}}-\varphi_{\text{R}})} (u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2} ( \Pi_{n_0} \hat{z} \Pi_{n_0} ) ( \Pi_{\text{BCS}} \gamma_{\text{R},{{{\bf{k}}}}{\uparrow}} \gamma_{\text{L},-{{{\bf{k}}}}{\downarrow}} \gamma^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}} \gamma^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}} \Pi_{\text{BCS}} ) \\ &= -it_{N}\lambda^{2} e^{i(\varphi_{\text{L}}-\varphi_{\text{R}})} (u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2} ( \Pi_{n_0} \hat{z} \Pi_{n_0} ) \Pi_{\text{BCS}} \end{split}$$ We remark that in the fifth line of both calculations, we have rewritten the electron operators of SC leads in terms of Bogoliubov quasiparticles, $c_{\ell,{{{\bf{k}}}}{\uparrow}}=e^{i\varphi_{\ell}/2}(u_{{{{\bf{k}}}}}\gamma_{\ell,{{{\bf{k}}}}{\uparrow}}+v_{{{{\bf{k}}}}}\gamma^{\dag}_{\ell,-{{{\bf{k}}}}{\downarrow}})$ and $c_{\ell,-{{{\bf{k}}}}{\downarrow}}=e^{i\varphi_{\ell}/2}(u_{{{{\bf{k}}}}}\gamma_{\ell,-{{{\bf{k}}}}{\downarrow}}-v_{{{{\bf{k}}}}}\gamma^{\dag}_{\ell,{{{\bf{k}}}}{\uparrow}})$. Adding these two sequences as well as their hermitian-conjugated counterparts, multiplying by the energy denominator $-1/(E_{{{{\bf{k}}}}}+U)(2E_{{{{\bf{k}}}}})$ and carrying out the summation over all momenta, yields the contribution, $$\begin{split} &-\hat{z} J^{\text{(a)}}_{z} \cos\varphi \quad \text{with} \quad J^{\text{(a)}}_{z}=4\lambda^{2}\,\text{Im}(t_{N}) \sum_{{{{\bf{k}}}}}\frac{(u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2}}{(E_{{{{\bf{k}}}}}+U)E_{{{{\bf{k}}}}}} \quad \text{for} \ n_{0} \ \text{even}, \\ &\quad\ \hat{z} J'^{\text{(a)}}_{z} \sin\varphi \quad  \text{with} \quad J'^{\text{(a)}}_{z}=-4\lambda^{2}\,\text{Re}(t_{N}) \sum_{{{{\bf{k}}}}}\frac{(u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2}}{(E_{{{{\bf{k}}}}}+U)E_{{{{\bf{k}}}}}} \quad \text{for} \ n_{0} \ \text{odd}. \end{split}$$ Here, for notational brevity, we have dropped the projectors on the ground state manifold of $H_{0}$. As the second step, we evaluate the Cooper pair transport sequences corresponding to Figs. \[fig:1\_SM\](b) to (f) in a similar way. This gives, $$\begin{split} &-\hat{z} J_{z} \cos\varphi \quad \text{with} \quad J_{z}=J^{\text{(a)}}_{z}+J^{\text{(b)}}_{z}+J^{\text{(c)}}_{z}+J^{\text{(d)}}_{z}+J^{\text{(e)}}_{z}+J^{\text{(f)}}_{z} \quad \text{for} \ n_{0} \ \text{even}, \\ &\hspace{12pt}\hat{z} J'_{z} \sin\varphi \quad \text{with} \quad J'_{z}=J'^{\text{(a)}}_{z}+J'^{\text{(b)}}_{z}+J'^{\text{(c)}}_{z}+J'^{\text{(d)}}_{z}+J'^{\text{(e)}}_{z}+J'^{\text{(f)}}_{z} \quad \text{for} \ n_{0} \ \text{odd}, \end{split}$$ where we have introduced the coupling constants, $$\begin{split} &J^{\text{(a)}}_{z}=J^{\text{(b)}}_{z}=J^{\text{(e)}}_{z}=J^{\text{(f)}}_{z} \quad , \quad J^{\text{(c)}}_{z}=J^{\text{(d)}}_{z}=8\lambda^{2}\,\text{Im}(t_{N}) \sum_{{{{\bf{k}}}}}\left(\frac{u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}}}{E_{{{{\bf{k}}}}}+U}\right)^{2}, \\ &J'^{\text{(a)}}_{z}=J'^{\text{(b)}}_{z}=J'^{\text{(e)}}_{z}=J'^{\text{(f)}}_{z} \quad , \quad J'^{\text{(c)}}_{z}=J'^{\text{(d)}}_{z}=-8\lambda^{2}\,\text{Re}(t_{N}) \sum_{{{{\bf{k}}}}}\left(\frac{u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}}}{E_{{{{\bf{k}}}}}+U}\right)^{2}. \end{split}$$ Adding the second order contribution, which corresponds to a conventional Josephson effect through the normal tunnelling barrier, as well as the fourth order contribution, which corresponds to a parity-controlled $2\pi$-Josephson effect through the TRI TSC island [@bib:Schrade2018_SM], we arrive at the effective Hamiltonian given in Eq. (9) of the main text, $$\begin{split} &H_{z,\text{even}}=-(J_{N}-J+\hat{z}\, J_{z})\cos\varphi \quad, \quad H_{z,\text{odd}}=-(J_{N}+J)\cos\varphi + \hat{z} \, J'_{z} \sin\varphi. \end{split}$$ Up to this point, we have only considered Cooper pair transport sequences which induce terms proportional to the SC phase difference in the effective Hamiltonian. Terms that are independent of the SC phase difference do not modify the Josephson current and, for that reason, have been omitted for the $\hat{z}$-measurement protocol. However, terms which are independent of the SC phase difference are of relevance when pulsing the tunnel couplings to obtain a $\pi/8$-gate. For that reason, we now provide a derivation of these contributions. First, we again examine two example sequence of the type shown in Fig. \[fig:1\_SM\](a), $$\begin{split} &\quad\ P_{n_{0}} ( \lambda \gamma_{\text{R},{\downarrow}} c_{\text{R},-{{{\bf{k}}}}{\downarrow}} e^{i\phi/2} ) ( \lambda c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}} \gamma_{\text{L},{\downarrow}} e^{-i\phi/2} ) ( t^{*}_{N}c^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}}c_{\text{L},-{{{\bf{k}}}}{\downarrow}} ) P_{n_{0}} \\ & = t^{*}_{N}\lambda^{2} P_{n_{0}} ( \gamma_{\text{R},{\downarrow}} c_{\text{R},-{{{\bf{k}}}}{\downarrow}} c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}} \gamma_{\text{L},{\downarrow}} c^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}} c_{\text{L},-{{{\bf{k}}}}{\downarrow}} ) P_{n_{0}} \\ &= t^{*}_{N}\lambda^{2} (\lambda \Pi_{n_0} \gamma_{\text{R},{\downarrow}} \gamma_{\text{L},{\downarrow}} \Pi_{n_0} ) ( \Pi_{\text{BCS}} c_{\text{R},-{{{\bf{k}}}}{\downarrow}} c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}} c^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}} c_{\text{L},-{{{\bf{k}}}}{\downarrow}} \Pi_{\text{BCS}}) \\ &= -i t^{*}_{N}\lambda^{2} ( \Pi_{n_0} \hat{z} \Pi_{n_0} ) ( \Pi_{\text{BCS}} c_{\text{R},-{{{\bf{k}}}}{\downarrow}} c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}} c^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}} c_{\text{L},-{{{\bf{k}}}}{\downarrow}} \Pi_{\text{BCS}}) \\ &= -i t^{*}_{N}\lambda^{2} (u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2} ( \Pi_{n_0} \hat{z} \Pi_{n_0} ) ( \Pi_{\text{BCS}} \gamma_{\text{R},-{{{\bf{k}}}}{\downarrow}} \gamma_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma^{\dag}_{\text{R},-{{{\bf{k}}}}{\downarrow}} \gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \Pi_{\text{BCS}}) \\ &= i t^{*}_{N}\lambda^{2} (u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2} ( \Pi_{n_0} \hat{z} \Pi_{n_0} ) \Pi_{\text{BCS}}, \\ \\ &\quad\ P_{n_{0}} ( \lambda \gamma_{\text{R},{\uparrow}} c_{\text{R},{{{\bf{k}}}}{\uparrow}} e^{i\phi/2} ) ( \lambda c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{\text{L},{\uparrow}} e^{-i\phi/2} ) ( t_{N}c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}}c_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{n_{0}} \\ &= t_{N}\lambda^{2} P_{n_{0}} ( \gamma_{\text{R},{\uparrow}} c_{\text{R},{{{\bf{k}}}}{\uparrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{\text{L},{\uparrow}} c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{n_{0}} \\ &= (-1)^{n_{0}+1} t_{N}\lambda^{2} ( \Pi_{n_0} \gamma_{\text{R},{\downarrow}} \gamma_{\text{L},{\downarrow}} \Pi_{n_0} ) ( \Pi_{\text{BCS}} c_{\text{R},{{{\bf{k}}}}{\uparrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} \Pi_{\text{BCS}} ) \\ &= i(-1)^{n_{0}} t_{N}\lambda^{2} ( \Pi_{n_0} \hat{z} \Pi_{n_0} ) ( \Pi_{\text{BCS}} c_{\text{R},{{{\bf{k}}}}{\uparrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} c^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} \Pi_{\text{BCS}} ) \\ &= i(-1)^{n_{0}} t_{N}\lambda^{2} (u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2} ( \Pi_{n_0} \hat{z} \Pi_{n_0} ) ( \Pi_{\text{BCS}} \gamma_{\text{R},{{{\bf{k}}}}{\uparrow}} \gamma_{\text{L},-{{{\bf{k}}}}{\downarrow}} \gamma^{\dag}_{\text{R},{{{\bf{k}}}}{\uparrow}} \gamma^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}} \Pi_{\text{BCS}} ) \\ &= i(-1)^{n_{0}+1} t_{N}\lambda^{2} (u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2} ( \Pi_{n_0} \hat{z} \Pi_{n_0} ) \Pi_{\text{BCS}} \end{split}$$ ![(Color online) Third-order sequences of intermediate states (up to hermitian-conjugation) relevant for the single-qubit gates. The enumerated blue lines label the tunnel couplings which are turned on for a given intermediate step within a third-order sequence. []{data-label="fig:1_SM"}](Fig1_SM){width="\linewidth"} When combining the two sequences with their hermitian-conjugated counterparts, their contributions cancel each other for $n_{0}$ odd but yield a finite contribution for $n_{0}$ even. Incorporating all possible contributions, as shown in Figs. \[fig:1\_SM\](a) to (f), we find that for $n_{0}$ even the effective Hamiltonian changes to, $$\begin{split} &H_{z,\text{even}}\rightarrow\hat{z}\tilde{J}_{z}-(J_{N}-J+\hat{z}\, J_{z})\cos\varphi \quad \text{with} \quad \tilde{J}_{z}=\tilde{J}_{z,a}+\tilde{J}_{z,b}+\tilde{J}_{z,c}+\tilde{J}_{z,d}+\tilde{J}_{z,e}+\tilde{J}_{z,f}. \end{split}$$ Here, we have defined additional coupling constants, $$\begin{split} &\tilde{J}_{z,a}=\tilde{J}_{z,b}=\tilde{J}_{z,e}=\tilde{J}_{z,f}=-4\lambda^{2}\,\text{Im}(t_{N}) \sum_{{{{\bf{k}}}}}\frac{(u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2}}{(E_{{{{\bf{k}}}}}+U)E_{{{{\bf{k}}}}}} \quad , \quad \tilde{J}_{z,c}=\tilde{J}_{z,d}=8\lambda^{2}\,\text{Im}(t_{N}) \sum_{{{{\bf{k}}}}}\left(\frac{u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}}}{E_{{{{\bf{k}}}}}+U}\right)^{2}. \end{split}$$ We again remark that the effective Hamiltonian for $n_{0}$ odd remains unchanged as a result of the cancellations discussed above.\ We now proceed by deriving the effective Hamiltonian given in Eq. (10) of the main text that is used for measuring the $\hat{x}$-Pauli operator. The all-important third-order contributions to the effective Hamiltonian are given by, $$\begin{split} H^{(3)}_{x}&= - P H_{T,x} \left(\left[H_{0}+U_{C}\right]^{-1}\left[1-P\right]H_{T,x}\right)^{2}P, \label{H3x} \end{split}$$ where $H_{T,x}=H_{T}+H_{S}$ denotes the total tunnelling Hamiltonian. For now, we have again set $t_{N}=0$. However, it will become clear after this derivation that it is sufficient to require $\text{Im}(t_{N})=0$ for $n_0$ even and $\text{Re}(t_{N})=0$ for $n_{0}$ odd. To begin, we consider two example sequences of the type shown in Fig. \[fig:1\_SM\](a), $$\begin{split} &\quad\ P_{n_{0}} ( \lambda c^{\dag}_{\text{R},{{{\bf{k}}}}{\downarrow}} \gamma_{\text{R},{\downarrow}} e^{-i\phi/2} ) ( \lambda \gamma_{\text{L},{\uparrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} e^{-i\phi/2} ) ( t_{S}c^{\dag}_{\text{R},-{{{\bf{k}}}}{\uparrow}}c_{\text{L},-{{{\bf{k}}}}{\downarrow}} ) P_{n_{0}} \\ &= t_{S}\lambda^{2} P_{n_{0}} ( c^{\dag}_{\text{R},{{{\bf{k}}}}{\downarrow}} \gamma_{\text{R},{\downarrow}} \gamma_{\text{L},{\uparrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} c^{\dag}_{\text{R},-{{{\bf{k}}}}{\uparrow}}c_{\text{L},-{{{\bf{k}}}}{\downarrow}} ) P_{n_{0}} \\ &= t_{S}\lambda^{2} ( \Pi_{n_0} \gamma_{\text{R},{\downarrow}} \gamma_{\text{L},{\uparrow}} \Pi_{n_0} ) ( \Pi_{\text{BCS}} c^{\dag}_{\text{R},{{{\bf{k}}}}{\downarrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} c^{\dag}_{\text{R},-{{{\bf{k}}}}{\uparrow}} c_{\text{L},-{{{\bf{k}}}}{\downarrow}} \Pi_{\text{BCS}} ) \\ &= i(-1)^{n_{0}+1}t_{S}\lambda^{2} ( \Pi_{n_0} \hat{x} \Pi_{n_0} ) ( \Pi_{\text{BCS}} c^{\dag}_{\text{R},{{{\bf{k}}}}{\downarrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} c^{\dag}_{\text{R},-{{{\bf{k}}}}{\uparrow}} c_{\text{L},-{{{\bf{k}}}}{\downarrow}} \Pi_{\text{BCS}} ) \\ &= i(-1)^{n_{0}+1} e^{i(\varphi_{\text{L}}-\varphi_{\text{R}})} t_{S}\lambda^{2} (u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2} ( \Pi_{n_0} \hat{x} \Pi_{n_0} ) ( \Pi_{\text{BCS}} \gamma_{\text{R},-{{{\bf{k}}}}{\uparrow}} \gamma_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma^{\dag}_{\text{R},-{{{\bf{k}}}}{\uparrow}} \gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \Pi_{\text{BCS}}) \\ &= i(-1)^{n_{0}} e^{i(\varphi_{\text{L}}-\varphi_{\text{R}})} t_{S}\lambda^{2} (u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2} ( \Pi_{n_0} \hat{x} \Pi_{n_0} ) \Pi_{\text{BCS}}, \\ \\ &\quad\ P_{n_{0}} ( \lambda c^{\dag}_{\text{R},-{{{\bf{k}}}}{\uparrow}} \gamma_{\text{R},{\uparrow}} e^{-i\phi/2} ) ( \lambda \gamma_{\text{L},{\downarrow}} c_{\text{L},-{{{\bf{k}}}}{\downarrow}} e^{i\phi/2} ) ( - t^{*}_{S}c^{\dag}_{\text{R},{{{\bf{k}}}}{\downarrow}}c_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{n_{0}} \\ &= - t^{*}_{S}\lambda^{2} P_{n_{0}} ( c^{\dag}_{\text{R},-{{{\bf{k}}}}{\uparrow}} \gamma_{\text{R},{\uparrow}} \gamma_{\text{L},{\downarrow}} c_{\text{L},-{{{\bf{k}}}}{\downarrow}} c^{\dag}_{\text{R},{{{\bf{k}}}}{\downarrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{n_{0}} \\ &= - t^{*}_{S}\lambda^{2} ( \Pi_{n_0} \gamma_{\text{R},{\uparrow}} \gamma_{\text{L},{\downarrow}} \Pi_{n_0} ) ( \Pi_{\text{BCS}} c^{\dag}_{\text{R},-{{{\bf{k}}}}{\uparrow}} c_{\text{L},-{{{\bf{k}}}}{\downarrow}} c^{\dag}_{\text{R},{{{\bf{k}}}}{\downarrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} \Pi_{\text{BCS}} ) \\ &= i t^{*}_{S}\lambda^{2} ( \Pi_{n_0} \hat{x} \Pi_{n_0} ) ( \Pi_{\text{BCS}} c^{\dag}_{\text{R},-{{{\bf{k}}}}{\uparrow}} c_{\text{L},-{{{\bf{k}}}}{\downarrow}} c^{\dag}_{\text{R},{{{\bf{k}}}}{\downarrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} \Pi_{\text{BCS}} ) \\ &= i t^{*}_{S}\lambda^{2} ( \Pi_{n_0} \hat{x} \Pi_{n_0} ) ( \Pi_{\text{BCS}} c^{\dag}_{\text{R},{{{\bf{k}}}}{\downarrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} c^{\dag}_{\text{R},-{{{\bf{k}}}}{\uparrow}} c_{\text{L},-{{{\bf{k}}}}{\downarrow}} \Pi_{\text{BCS}} ) \\ &= ie^{i(\varphi_{\text{L}}-\varphi_{\text{R}})} t^{*}_{S}\lambda^{2} (u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2} ( \Pi_{n_0} \hat{x} \Pi_{n_0} ) ( \Pi_{\text{BCS}} \gamma_{\text{R},-{{{\bf{k}}}}{\uparrow}} \gamma_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma^{\dag}_{\text{R},-{{{\bf{k}}}}{\uparrow}} \gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \Pi_{\text{BCS}} ) \\ &= - ie^{i(\varphi_{\text{L}}-\varphi_{\text{R}})} t^{*}_{S}\lambda^{2} (u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2} ( \Pi_{n_0} \hat{x} \Pi_{n_0} ) \Pi_{\text{BCS}} \end{split}$$ Here, we have made the inessential assumption that the normal state dispersion of the SC obeys $\xi_{{{{\bf{k}}}}}=\xi_{-{{{\bf{k}}}}}$ such that $u_{{{{\bf{k}}}}}=u_{-{{{\bf{k}}}}}$, $v_{{{{\bf{k}}}}}=v_{-{{{\bf{k}}}}}$ and $E_{{{{\bf{k}}}}}=E_{-{{{\bf{k}}}}}$. Adding the two sequencesas well as their hermitian-conjugated counterparts, multiplying by the energy denominator $-1/(E_{{{{\bf{k}}}}}+U)(2E_{{{{\bf{k}}}}})$ and carrying out the summation over all momenta, gives the contribution, $$\begin{split} &-\hat{x} J^{(\text{a})}_{x} \cos\varphi \quad \text{with} \quad J^{(\text{a})}_{x}=-4\lambda^{2}\,\text{Im}(t_{S}) \sum_{{{{\bf{k}}}}}\frac{(u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2}}{(E_{{{{\bf{k}}}}}+U)E_{{{{\bf{k}}}}}} \quad \text{for} \ n_{0} \ \text{even}, \\ &\quad\ \hat{x} J'^{(\text{a})}_{x} \sin\varphi \quad  \text{with} \quad J'^{(\text{a})}_{x} =-4\lambda^{2}\,\text{Re}(t_{S}) \sum_{{{{\bf{k}}}}}\frac{(u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2}}{(E_{{{{\bf{k}}}}}+U)E_{{{{\bf{k}}}}}} \quad \text{for} \ n_{0} \ \text{odd}. \end{split}$$ For notational brevity, we have again omitted the projectors on the ground state manifold of $H_{0}$. In a similar way, we also evaluate the Cooper pair transport sequences given in Fig. \[fig:1\_SM\](b) to (f). This yields, $$\begin{split} &-\hat{x} J_{x} \cos\varphi \quad \text{with} \quad J_{x}=J^{\text{(a)}}_{x}+J^{\text{(b)}}_{x}+J^{\text{(c)}}_{x}+J^{\text{(d)}}_{x}+J^{\text{(e)}}_{x}+J^{\text{(f)}}_{x} \quad \text{for} \ n_{0} \ \text{even}, \\ &\hspace{12pt}\hat{z} J'_{x} \sin\varphi \quad \text{with} \quad J'_{x}=J'^{\text{(a)}}_{x}+J'^{\text{(b)}}_{x}+J'^{\text{(c)}}_{x}+J'^{\text{(d)}}_{x}+J'^{\text{(e)}}_{x}+J'^{\text{(f)}}_{x} \quad \text{for} \ n_{0} \ \text{odd}. \end{split}$$ Here, we have defined the coupling constants, $$\begin{split} &J^{\text{(a)}}_{x}=J^{\text{(b)}}_{x}=J^{\text{(e)}}_{x}=J^{\text{(f)}}_{x} \quad , \quad J^{\text{(c)}}_{x}=J^{\text{(d)}}_{x}=-8\lambda^{2}\,\text{Im}(t_{S}) \sum_{{{{\bf{k}}}}}\left(\frac{u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}}}{E_{{{{\bf{k}}}}}+U}\right)^{2}, \\ &J'^{\text{(a)}}_{x}=J'^{\text{(b)}}_{x}=J'^{\text{(e)}}_{x}=J'^{\text{(f)}}_{x} \quad , \quad J'^{\text{(c)}}_{x}=J'^{\text{(d)}}_{x}=-8\lambda^{2}\,\text{Re}(t_{S}) \sum_{{{{\bf{k}}}}}\left(\frac{u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}}}{E_{{{{\bf{k}}}}}+U}\right)^{2}. \end{split}$$ Once we add the second order contribution, which corresponds to a conventional Josephson effect via the spin-flip tunnelling barrier, as well as the fourth order contribution, which corresponds to a parity-controlled $2\pi$-Josephson effect via the TRI TSC island [@bib:Schrade2018_SM], we arrive at the effective Hamiltonian of Eq. (10) in the main text, $$\begin{split} &H_{x,\text{even}}=-(J_{S}-J+\hat{x}\, J_{x})\cos\varphi \quad, \quad H_{x,\text{odd}}=-(J_{S}+J)\cos\varphi + \hat{x} \, J'_{x} \sin\varphi. \end{split}$$ So far, we have again only considered Cooper pair transport sequences leading to terms in the effective Hamiltonian that depend on the SC phase difference. Terms which are independent of the SC phase difference do not affect the Josephson current and, for that reason, have been omitted for the $\hat{x}$-measurement protocol. However, those terms are clearly of relevance when pulsing the tunnel couplings to obtain a $\pi/8$-gate. As for the effective Hamiltonian for the $\hat{z}$-measurement protocol, we find that contributions that are independent of the SC phase difference and $\propto\hat{x}$ only occur for $n_{0}$ even. More concretely, the effective Hamiltonian modifies to, $$\begin{split} &H_{x,\text{even}}\rightarrow\hat{x}\tilde{J}_{x}-(J_{S}-J+\hat{x}\, J_{x})\cos\varphi \quad \text{with} \quad \tilde{J}_{x}=\tilde{J}^{\text{(a)}}_{x}+\tilde{J}^{\text{(b)}}_{x}+\tilde{J}^{\text{(c)}}_{x}+\tilde{J}^{\text{(d)}}_{x}+\tilde{J}^{\text{(e)}}_{x}+\tilde{J}^{\text{(f)}}_{x}, \end{split}$$ where the additional coupling constants are given by, $$\begin{split} &\tilde{J}^{\text{(a)}}_{x}=\tilde{J}^{\text{(b)}}_{x}=\tilde{J}^{\text{(e)}}_{x}=\tilde{J}^{\text{(f)}}_{x}=4\lambda^{2}\,\text{Im}(t_{S}) \sum_{{{{\bf{k}}}}}\frac{(u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}})^{2}}{(E_{{{{\bf{k}}}}}+U)E_{{{{\bf{k}}}}}} \quad , \quad \tilde{J}^{\text{(c)}}_{x}=\tilde{J}^{\text{(d)}}_{x}=-8\lambda^{2}\,\text{Im}(t_{S}) \sum_{{{{\bf{k}}}}}\left(\frac{u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}}}{E_{{{{\bf{k}}}}}+U}\right)^{2}. \end{split}$$ Before closing this first section of the Supplemental Material, we note that the third-order contributions to $H_{z,\text{even}}$ and $H_{x,\text{even}}$ only depend on $\text{Im}(t_{N})$ and $\text{Im}(t_{S})$, respectively. In particular, spin-flip tunnelling contribution with $\text{Re}(t_{S})\neq0$ do not alter the form of $H_{z,\text{even}}$. Similarly, normal tunnelling contribution with $\text{Re}(t_{N})\neq0$ also do not alter the form of $H_{x,\text{even}}$. Hence, we are able to relax our initial assumptions from $t_{S}=0$ to $\text{Im}(t_{S})=0$ for $H_{z,\text{even}}$ and from $t_{N}=0$ to $\text{Im}(t_{N})=0$ for $H_{x,\text{even}}$. Similar arguments apply to $H_{z,\text{odd}}$ and $H_{x,\text{odd}}$. Here, our initial assumption $t_{S}=0$ relaxes to $\text{Re}(t_{S})=0$ for $H_{z,\text{odd}}$ and $t_{N}=0$ relaxes to $\text{Re}(t_{N})=0$ for $H_{x,\text{odd}}$. Effective Hamiltonian for the two-qubit gates ============================================= In this second section of the Supplemental Material, we sketch the derivation of the effective Hamiltonian for the two-qubit gates as given in Eq. (11) of the main text. Up to fourth order in the couplings, the general form of the effective Hamiltonian reads, $$\begin{split} H_{ab}&= - P_{ab} H_{T,ab} \left[H_{0,ab}^{-1}(1-P_{ab})H_{T,ab}\right]^{3}P_{ab}. \end{split}$$ ![(Color online) Fourth-order sequences of intermediate states (up to hermitian-conjugation and mirror operations $m_{x}$, $m_{y}$) relevant for the two-qubit gate. []{data-label="fig:2_SM"}](Fig2_SM){width="0.9\linewidth"} Here, we have dropped the second order contribution as it only leads to a constant shift in energy and consequently contributes neither to the entanglement generation between the two TRI TS islands nor the Josephson current between the SC grains. Moreover, $H_{0,ab}$ is the Hamiltonian of the uncoupled system comprised of the two SC leads and the two TRI TSC islands, $H_{T,ab}$ is the tunnelling Hamiltonian between the two SC leads and the TRI TSC islands and $P_{ab}$ is the projector on the reduced Hilbert space consisting of the BCS ground states of the SC leads as well as the charge ground states of both TRI TS islands. For simplicity, we will assume that both the TRI TSC islands as well as all their tunnel couplings to the SC leads are identical. In particular, both TRI TSC islands are tuned to a Coulomb valley with $n_{0}$ units of charge in the ground state and both are coupled to the ground through capacitors of equal capacitance $C$. The tunnelling amplitude between the TRI TSC islands and the SC leads will be denoted by $\lambda$. To evaluate the effective Hamiltonian, we first list the different types of sequences of intermediate states, see Fig. \[fig:2\_SM\]. Then we compute the effective Hamiltonian for each type of sequence separately. A summation over all the different types then produces the final expression for the effective Hamiltonian, $$\begin{split} H_{ab} &= H^{\text{(a)}}_{ab} + H^{\text{(b)}}_{ab} + H^{\text{(c)}}_{ab} + H^{\text{(d)}}_{ab} + H^{\text{(e)}}_{ab} + H^{\text{(f)}}_{ab} \\ &+ H^{\text{(g)}}_{ab} + H^{\text{(h)}}_{ab} + H^{\text{(i)}}_{ab} + H^{\text{(j)}}_{ab} + H^{\text{(k)}}_{ab} + H^{\text{($\ell$)}}_{ab} \end{split}$$ Before going into the details of the derivation, we first display the full result for all types of contributions, $$\begin{split} &H^{\text{(a)}}_{ab} = (-1)^{n_{0}+1}2J_{1}\cos\varphi \quad, \quad H^{\text{(b)}}_{ab} = (-1)^{n_{0}+1}2J_{2}\cos\varphi \quad, \quad H^{\text{(c)}}_{ab} = (-1)^{n_{0}+1}2J_{3}\cos\varphi \\ &H^{\text{(d)}}_{ab} = [(-1)^{n_{0}+1}J_{1}\cos\varphi-J_{4}](\hat{x}_{a}\hat{x}_{b}+\hat{z}_{a}\hat{z}_{b}) \quad, \quad H^{\text{(e)}}_{ab} = [(-1)^{n_{0}+1}J_{2}\cos\varphi-J_{5}](\hat{x}_{a}\hat{x}_{b}+\hat{z}_{a}\hat{z}_{b}) \\ &H^{\text{(f)}}_{ab} = [(-1)^{n_{0}+1}J_{7}\cos\varphi+J_{6}](\hat{x}_{a}\hat{x}_{b}+\hat{z}_{a}\hat{z}_{b}) \quad\ , \quad H^{\text{(g)}}_{ab} = [(-1)^{n_{0}+1}J_{9}\cos\varphi+J_{8}](\hat{x}_{a}\hat{x}_{b}+\hat{z}_{a}\hat{z}_{b}) \\ & H^{\text{(h)}}_{ab}= [(-1)^{n_{0}+1}J_{11}\cos\varphi-J_{10}](\hat{x}_{a}\hat{x}_{b}+\hat{z}_{a}\hat{z}_{b})\hspace{5pt} , \quad H^{\text{(i)}}_{ab} = [(-1)^{n_{0}+1}J_{11}\cos\varphi+J_{12}](\hat{x}_{a}\hat{x}_{b}+\hat{z}_{a}\hat{z}_{b}) \\ &H^{\text{(j)}}_{ab} = -J_{13}\hat{y}_{a}\hat{y}_{b} +J_{14}\hat{y}_{a}\hat{y}_{b} \quad, \quad H^{\text{(k)}}_{ab} = J_{15}\hat{y}_{a}\hat{y}_{b} + J_{16}\hat{y}_{a}\hat{y}_{b} \quad, \quad H^{\text{($\ell$)}}_{ab} = -J_{17}\hat{y}_{a}\hat{y}_{b} -J_{18}\hat{y}_{a}\hat{y}_{b}. \end{split}$$ Here, we have introduced the coupling constants $$\begin{split} &J_{1}=16\lambda^{4} \sum_{{{{\bf{k}}}},{{\bf{q}}}}\frac{u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}}u_{{{\bf{q}}}}v_{{{\bf{q}}}}}{(E_{{{{\bf{k}}}}}+U)^{2}(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}})} \hspace{75pt}, \quad J_{2}=16\lambda^{4} \sum_{{{{\bf{k}}}},{{\bf{q}}}}\frac{u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}}u_{{{\bf{q}}}}v_{{{\bf{q}}}}}{(E_{{{{\bf{k}}}}}+U)(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}})(E_{{{\bf{q}}}}+U)} \\ &J_{3}=16\lambda^{4} \sum_{{{{\bf{k}}}},{{\bf{q}}}}\frac{u_{{{{\bf{k}}}}}v_{{{{\bf{k}}}}}u_{{{\bf{q}}}}v_{{{\bf{q}}}}}{(E_{{{{\bf{k}}}}}+U)(4U)(E_{{{\bf{q}}}}+U)} \hspace{65pt}, \quad J_{4}=8\lambda^{4} \sum_{{{{\bf{k}}}},{{\bf{q}}}} \frac{ (u_{{{\bf{q}}}}v_{{{{\bf{k}}}}})^{2}+(v_{{{\bf{q}}}}u_{{{{\bf{k}}}}})^{2} } { (E_{{{{\bf{k}}}}}+U)^{2}(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}}) } \\ &J_{5}=8\lambda^{4} \sum_{{{{\bf{k}}}},{{\bf{q}}}} \frac{ (u_{{{\bf{q}}}}v_{{{{\bf{k}}}}})^{2}+(v_{{{\bf{q}}}}u_{{{{\bf{k}}}}})^{2} } { (E_{{{\bf{q}}}}+U)(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}})(E_{{{{\bf{k}}}}}+U) } \hspace{44pt}, \quad J_{6}=8\lambda^{4} \sum_{{{{\bf{k}}}},{{\bf{q}}}} \frac{ (u_{{{{\bf{k}}}}}u_{{{\bf{q}}}})^{2}+(v_{{{{\bf{k}}}}}v_{{{\bf{q}}}})^{2} } { (E_{{{{\bf{k}}}}}+U)^{2}(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}}+2U) } \\ &J_{7}=16\lambda^{4} \sum_{{{{\bf{k}}}},{{\bf{q}}}} \frac{ u_{{{{\bf{k}}}}}u_{{{\bf{q}}}}v_{{{{\bf{k}}}}}v_{{{\bf{q}}}} } { (E_{{{{\bf{k}}}}}+U)^{2}(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}}+2U) } \hspace{52pt},\quad J_{8}=8\lambda^{4} \sum_{{{{\bf{k}}}},{{\bf{q}}}} \frac{ (u_{{{{\bf{k}}}}}u_{{{\bf{q}}}})^{2}+(v_{{{{\bf{k}}}}}v_{{{\bf{q}}}})^{2} } { (E_{{{{\bf{k}}}}}+U)(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}}+2U)(E_{{{\bf{q}}}}+U) } \\ &J_{9}=16\lambda^{4} \sum_{{{{\bf{k}}}},{{\bf{q}}}} \frac{ u_{{{{\bf{k}}}}}u_{{{\bf{q}}}}v_{{{{\bf{k}}}}}v_{{{\bf{q}}}} } { (E_{{{{\bf{k}}}}}+U)(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}}+2U)(E_{{{\bf{q}}}}+U) } \hspace{17pt},\quad J_{10}=8\lambda^{4} \sum_{{{{\bf{k}}}},{{\bf{q}}}} \frac{ (u_{{{\bf{q}}}}v_{{{{\bf{k}}}}})^{2}+(v_{{{\bf{q}}}}u_{{{{\bf{k}}}}})^{2} } { (E_{{{{\bf{k}}}}}+U)(2U)(E_{{{\bf{q}}}}+U) } \\ &J_{11}=16\lambda^{4} \sum_{{{{\bf{k}}}},{{\bf{q}}}} \frac{ u_{{{{\bf{k}}}}}u_{{{\bf{q}}}}v_{{{{\bf{k}}}}}v_{{{\bf{q}}}} } { (E_{{{{\bf{k}}}}}+U)(2U)(E_{{{\bf{q}}}}+U) } \hspace{64pt},\quad J_{12}=8\lambda^{4} \sum_{{{{\bf{k}}}},{{\bf{q}}}} \frac{ (u_{{{{\bf{k}}}}}u_{{{\bf{q}}}})^{2}+(v_{{{{\bf{k}}}}}v_{{{\bf{q}}}})^{2} } { (E_{{{{\bf{k}}}}}+U)(2U)(E_{{{\bf{q}}}}+U) } \\ &J_{13}= 8\lambda^{4} \sum_{{{{\bf{k}}}},{{\bf{q}}}} \frac{ (u_{{{\bf{q}}}}v_{{{{\bf{k}}}}}-u_{{{{\bf{k}}}}}v_{{{\bf{q}}}})^{2} } { (E_{{{{\bf{k}}}}}+U)(2U)(E_{{{\bf{q}}}}+U) } \hspace{68pt},\quad J_{14}= 8\lambda^{4} \sum_{{{{\bf{k}}}},{{\bf{q}}}} \frac{ (u_{{{{\bf{k}}}}}u_{{{\bf{q}}}}+v_{{{{\bf{k}}}}}v_{{{\bf{q}}}})^{2} } { (E_{{{{\bf{k}}}}}+U)^{2}(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}}+2U) } \\ &J_{15}= 8\lambda^{4} \sum_{{{{\bf{k}}}},{{\bf{q}}}} \frac{ (u_{{{{\bf{k}}}}}u_{{{\bf{q}}}}+v_{{{{\bf{k}}}}}v_{{{\bf{q}}}})^{2} } { (E_{{{\bf{q}}}}+U)(2U)(E_{{{{\bf{k}}}}}+U) } \hspace{68pt},\quad J_{16}= 8\lambda^{4} \sum_{{{{\bf{k}}}},{{\bf{q}}}} \frac{ (u_{{{{\bf{k}}}}}u_{{{\bf{q}}}}+v_{{{{\bf{k}}}}}v_{{{\bf{q}}}})^{2} } { (E_{{{\bf{q}}}}+U)(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}}+2U)(E_{{{{\bf{k}}}}}+U) } \\ &J_{17}=8\lambda^{4} \sum_{{{{\bf{k}}}},{{\bf{q}}}} \frac{ (u_{{{\bf{q}}}}v_{{{{\bf{k}}}}}-u_{{{{\bf{k}}}}}v_{{{\bf{q}}}})^{2} } { (E_{{{\bf{q}}}}+U)(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}})(E_{{{{\bf{k}}}}}+U) } \hspace{43pt},\quad J_{18}=8\lambda^{4} \sum_{{{{\bf{k}}}},{{\bf{q}}}} \frac{ (u_{{{\bf{q}}}}v_{{{{\bf{k}}}}}-u_{{{{\bf{k}}}}}v_{{{\bf{q}}}})^{2} } { (E_{{{{\bf{k}}}}}+U)^{2} (E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}}) }. \end{split}$$ As a next step, we collect the different contributions. This simplifies the expression of the effective Hamiltonian to $$\begin{split} H_{ab} = (-1)^{n_{0}+1}J_{0}\cos\varphi + [(-1)^{n_{0}+1}J'_{xz}\cos\varphi+J_{xz}](\hat{x}_{a}\hat{x}_{b}+\hat{z}_{a}\hat{z}_{b}) + J_{y}\hat{y}_{a}\hat{y}_{b} \end{split}$$ with the coupling constants $$\begin{split} J_{0}&=J_{1}+J_{2}+J_{3} \\ J'_{xz}&=J_{1}+J_{2}+J_{7}+J_{9}+2J_{11} \\ J_{xz}&=-J_{4}-J_{5}+J_{6}+J_{8}-J_{10}+J_{12} \\ J'_{y}&=J_{13}+J_{14}-J_{15}+J_{16}+J_{17}-J_{18}. \end{split}$$ Now that we have presented the full expression for the effective Hamiltonian we will give an overview of the derivation of the individual contributions. Sequences of intermediate states corresponding to contributions $\boldsymbol{\propto J_{1},J_{2},J_{7},J_{9},J_{11}}$ --------------------------------------------------------------------------------------------------------------------- In this first subsection, we discuss sequences of intermediate states which lead to contributions $\propto J_{1},J_{2},J_{7},J_{9},J_{11}$ in the effective Hamiltonian. Initially, we will present two examples which yield contributions $\propto J_{1}$ in the effective Hamiltonian. Subsequently, we will explain how additional examples for the contributions $\propto J_{2},J_{7},J_{9},J_{11}$ can be obtained from these findings. Our first example is given by $$\begin{split} &\quad\ P_{ab}( \lambda c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}} \gamma_{b,\text{L},{\downarrow}} e^{-i\phi_{2}/2} ) ( \lambda \gamma_{b,\text{R},{\downarrow}} c_{\text{R},-{{\bf{q}}}{\downarrow}} e^{i\phi_{2}/2} ) ( \lambda \gamma_{a,\text{R},{\uparrow}} c_{\text{R},{{\bf{q}}}{\uparrow}} e^{i\phi_{1}/2} ) ( \lambda c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{a,\text{L},{\uparrow}} e^{-i\phi_{1}/2} ) P_{ab} \\ &= P_{ab}\lambda^{4} ( c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}} \gamma_{b,\text{L},{\downarrow}} \gamma_{b,\text{R},{\downarrow}} c_{\text{R},-{{\bf{q}}}{\downarrow}} \gamma_{a,\text{R},{\uparrow}} c_{\text{R},{{\bf{q}}}{\uparrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{a,\text{L},{\uparrow}} ) P_{ab} \\ &= - P_{ab}\lambda^{4} ( \gamma_{a,\text{R},{\uparrow}} \gamma_{a,\text{L},{\uparrow}} \gamma_{b,\text{R},{\downarrow}} \gamma_{b,\text{L},{\downarrow}} ) ( c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}} c_{\text{R},-{{\bf{q}}}{\downarrow}} c_{\text{R},{{\bf{q}}}{\uparrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= (-1)^{n_{0}+1}P_{ab}\lambda^{4} \hat{z}_{a}\hat{z}_{b} ( c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}} c_{\text{R},-{{\bf{q}}}{\downarrow}} c_{\text{R},{{\bf{q}}}{\uparrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= (-1)^{n_{0}} e^{i(\varphi_{\text{R}}-\varphi_{\text{L}})} P_{ab}\lambda^{4} v_{{{{\bf{k}}}}} u_{{{\bf{q}}}} v_{{{\bf{q}}}} u_{{{{\bf{k}}}}} \hat{z}_{a}\hat{z}_{b} ( \gamma_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{\text{R},-{{\bf{q}}}{\downarrow}} \gamma^{\dag}_{\text{R},-{{\bf{q}}}{\downarrow}} \gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= (-1)^{n_{0}} e^{i(\varphi_{\text{R}}-\varphi_{\text{L}})} P_{ab}\lambda^{4} v_{{{{\bf{k}}}}} u_{{{\bf{q}}}} v_{{{\bf{q}}}} u_{{{{\bf{k}}}}} \hat{z}_{a}\hat{z}_{b} P_{ab} \label{c1} \end{split}$$ and thus leads to a term $\propto\hat{z}_{a}\hat{z}_{b}$. In the second example the coupling to the MKPs is different compared to first example, $$\begin{split} &\quad\ P_{ab}( \lambda c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}} \gamma_{b,\text{L},{\downarrow}} e^{-i\phi_{2}/2} ) ( \lambda \gamma_{b,\text{R},{\uparrow}} c_{\text{R},{{\bf{q}}}{\uparrow}} e^{i\phi_{2}/2} ) ( \lambda \gamma_{a,\text{R},{\downarrow}} c_{\text{R},-{{\bf{q}}}{\downarrow}} e^{i\phi_{1}/2} ) ( \lambda c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{a,\text{L},{\uparrow}} e^{-i\phi_{1}/2} ) P_{ab} \\ &= P_{ab}\lambda^{4} ( c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}} \gamma_{b,\text{L},{\downarrow}} \gamma_{b,\text{R},{\uparrow}} c_{\text{R},{{\bf{q}}}{\uparrow}} \gamma_{a,\text{R},{\downarrow}} c_{\text{R},-{{\bf{q}}}{\downarrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{a,\text{L},{\uparrow}} ) P_{ab} \\ &= P_{ab}\lambda^{4} ( \gamma_{b,\text{R},{\uparrow}} \gamma_{b,\text{L},{\downarrow}} \gamma_{a,\text{L},{\uparrow}} \gamma_{a,\text{R},{\downarrow}} ) ( c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}} c_{\text{R},{{\bf{q}}}{\uparrow}} c_{\text{R},-{{\bf{q}}}{\downarrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= (-1)^{n_{0}} P_{ab}\lambda^{4} \hat{x}_{a}\hat{x}_{b} ( c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}} c_{\text{R},{{\bf{q}}}{\uparrow}} c_{\text{R},-{{\bf{q}}}{\downarrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= (-1)^{n_{0}} e^{i(\varphi_{\text{R}}-\varphi_{\text{L}})} P_{ab}\lambda^{4} v_{{{{\bf{k}}}}} u_{{{\bf{q}}}} v_{{{\bf{q}}}} u_{{{{\bf{k}}}}} \hat{x}_{a}\hat{x}_{b} ( \gamma_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{\text{R},{{\bf{q}}}{\uparrow}} \gamma^{\dag}_{\text{R},{{\bf{q}}}{\uparrow}} \gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= (-1)^{n_{0}} e^{i(\varphi_{\text{R}}-\varphi_{\text{L}})} P_{ab}\lambda^{4} v_{{{{\bf{k}}}}} u_{{{\bf{q}}}} v_{{{\bf{q}}}} u_{{{{\bf{k}}}}} \hat{x}_{a}\hat{x}_{b} P_{ab} \label{c2} \end{split}$$ which leads to a term $\propto\hat{x}_{a}\hat{x}_{b}$. The energy denominator for both examples is given by $-1/[(E_{{{{\bf{k}}}}}+U)^{2}(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}})]$. Hence, after combining the above results with the corresponding hermitian-conjugated sequences and summing over all momenta, we indeed find a contribution $\propto J_{1}$. Finally, we point out that examples for contributions $\propto J_{2},J_{7},J_{9},J_{11}$ can be obtained by suitably commuting the terms in the first line of Eq.  and Eq. . The different coupling constants arise because the energy denominators for the resulting processes will be different than the ones we used for the two examples given above. Sequences of intermediate states corresponding to contributions $\boldsymbol{\propto J_{4},J_{5},J_{6},J_{8},J_{10},J_{12}}$ ---------------------------------------------------------------------------------------------------------------------------- In this second subsection, we continue our overview on the sequences of intermediate states which contribute to the effective Hamiltonian. More specifically, we will examine sequences that give contributions $\propto J_{4},J_{5},J_{6},J_{8},J_{10},J_{12}$. As a first step, we introduce two examples which produce contributions $\propto J_{4}$. The first example is given by $$\begin{split} &\quad\ P_{ab}( \lambda \gamma_{b,\text{L},{\uparrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} e^{i\phi_{2}/2} ) ( \lambda c^{\dag}_{\text{R},{{\bf{q}}}{\uparrow}} \gamma_{b,\text{R},{\uparrow}} e^{-i\phi_{2}/2} ) ( \lambda \gamma_{a,\text{R},{\uparrow}} c_{\text{R},{{\bf{q}}}{\uparrow}} e^{i\phi_{1}/2} ) ( \lambda c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{a,\text{L},{\uparrow}} e^{-i\phi_{1}/2} ) P_{ab} \\ &= P_{ab}\lambda^{4} ( \gamma_{b,\text{L},{\uparrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} c^{\dag}_{\text{R},{{\bf{q}}}{\uparrow}} \gamma_{b,\text{R},{\uparrow}} \gamma_{a,\text{R},{\uparrow}} c_{\text{R},{{\bf{q}}}{\uparrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{a,\text{L},{\uparrow}} ) P_{ab} \\ &= - P_{ab}\lambda^{4} ( \gamma_{a,\text{R},{\uparrow}} \gamma_{a,\text{L},{\uparrow}} \gamma_{b,\text{R},{\uparrow}} \gamma_{b,\text{L},{\uparrow}} ) ( c_{\text{L},{{{\bf{k}}}}{\uparrow}} c^{\dag}_{\text{R},{{\bf{q}}}{\uparrow}} c_{\text{R},{{\bf{q}}}{\uparrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= P_{ab}\lambda^{4} \hat{z}_{a}\hat{z}_{b} ( c_{\text{L},{{{\bf{k}}}}{\uparrow}} c^{\dag}_{\text{R},{{\bf{q}}}{\uparrow}} c_{\text{R},{{\bf{q}}}{\uparrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= P_{ab}\lambda^{4} \hat{z}_{a}\hat{z}_{b} ( v_{{{\bf{q}}}} u_{{{{\bf{k}}}}} )^{2} ( \gamma_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{\text{R},-{{\bf{q}}}{\downarrow}} \gamma^{\dag}_{\text{R},-{{\bf{q}}}{\downarrow}} \gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= P_{ab}\lambda^{4} \hat{z}_{a}\hat{z}_{b} ( v_{{{\bf{q}}}} u_{{{{\bf{k}}}}} )^{2} P_{ab} \label{c3} \end{split}$$ and it leads to a term $\propto\hat{z}_{a}\hat{z}_{b}$. The second example is given by $$\begin{split} &\quad\ P_{ab}( \lambda \gamma_{b,\text{L},{\uparrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} e^{i\phi_{2}/2} ) ( \lambda c^{\dag}_{\text{R},{{\bf{q}}}{\downarrow}} \gamma_{b,\text{R},{\downarrow}} e^{-i\phi_{2}/2} ) ( \lambda \gamma_{a,\text{R},{\downarrow}} c_{\text{R},{{\bf{q}}}{\downarrow}} e^{i\phi_{1}/2} ) ( \lambda c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{a,\text{L},{\uparrow}} e^{-i\phi_{1}/2} ) P_{ab} \\ &= P_{ab}\lambda^{4} ( \gamma_{b,\text{L},{\uparrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} c^{\dag}_{\text{R},{{\bf{q}}}{\downarrow}} \gamma_{b,\text{R},{\downarrow}} \gamma_{a,\text{R},{\downarrow}} c_{\text{R},{{\bf{q}}}{\downarrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{a,\text{L},{\uparrow}} ) P_{ab} \\ &= - P_{ab}\lambda^{4} ( \gamma_{a,\text{L},{\uparrow}} \gamma_{a,\text{R},{\downarrow}} \gamma_{b,\text{L},{\uparrow}} \gamma_{b,\text{R},{\downarrow}} ) ( c_{\text{L},{{{\bf{k}}}}{\uparrow}} c^{\dag}_{\text{R},{{\bf{q}}}{\downarrow}} c_{\text{R},{{\bf{q}}}{\downarrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= P_{ab}\lambda^{4} \hat{x}_{a}\hat{x}_{b} ( c_{\text{L},{{{\bf{k}}}}{\uparrow}} c^{\dag}_{\text{R},{{\bf{q}}}{\downarrow}} c_{\text{R},{{\bf{q}}}{\downarrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= P_{ab}\lambda^{4} \hat{x}_{a}\hat{x}_{b} ( u_{{{{\bf{k}}}}} v_{{{\bf{q}}}} )^{2} ( \gamma_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{\text{R},-{{\bf{q}}}{\uparrow}} \gamma^{\dag}_{\text{R},-{{\bf{q}}}{\uparrow}} \gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= P_{ab}\lambda^{4} \hat{x}_{a}\hat{x}_{b} ( u_{{{{\bf{k}}}}} v_{{{\bf{q}}}} )^{2} P_{ab}, \label{c4} \end{split}$$ and gives a term $\propto\hat{x}_{a}\hat{x}_{b}$. The energy denominator for both examples is given by $-1/[(E_{{{{\bf{k}}}}}+U)^{2}(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}})]$. After carrying out the summation over all momenta, we see that both examples indeed contribute to the term $\propto J_{4}$ in the effective Hamiltonian. Moreover, we remark that examples for sequences of intermediate states that give contributions $\propto J_{5}$ and $\propto J_{10}$ can be obtained by appropriately commuting the terms in the round brackets in the first line of Eqs.  and . The required energy denominator for the examples $\propto J_{5}$ is $-1/[(E_{{{\bf{q}}}}+U)(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}})(E_{{{{\bf{k}}}}}+U)]$ and for examples $\propto J_{10}$ it is $-1/[(E_{{{\bf{q}}}}+U)(2U)(E_{{{{\bf{k}}}}}+U)]$. We now proceed by presenting two examples of sequences of intermediate states leading to contributions $\propto J_{6}$. The first example is given by $$\begin{split} &\quad\ P_{ab}( \lambda \gamma_{b,\text{L},{\uparrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} e^{i\phi_{2}/2} ) ( \lambda \gamma_{a,\text{R},{\uparrow}} c_{\text{R},{{\bf{q}}}{\uparrow}} e^{i\phi_{1}/2} ) ( \lambda c^{\dag}_{\text{R},{{\bf{q}}}{\uparrow}} \gamma_{b,\text{R},{\uparrow}} e^{-i\phi_{2}/2} ) ( \lambda c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{a,\text{L},{\uparrow}} e^{-i\phi_{1}/2} ) P_{ab} \\ &= P_{ab}\lambda^{4} ( \gamma_{b,\text{L},{\uparrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{a,\text{R},{\uparrow}} c_{\text{R},{{\bf{q}}}{\uparrow}} c^{\dag}_{\text{R},{{\bf{q}}}{\uparrow}} \gamma_{b,\text{R},{\uparrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{a,\text{L},{\uparrow}} ) P_{ab} \\ &= P_{ab}\lambda^{4} ( \gamma_{a,\text{R},{\uparrow}} \gamma_{a,\text{L},{\uparrow}} \gamma_{b,\text{R},{\uparrow}} \gamma_{b,\text{L},{\uparrow}} ) ( c_{\text{L},{{{\bf{k}}}}{\uparrow}} c_{\text{R},{{\bf{q}}}{\uparrow}} c^{\dag}_{\text{R},{{\bf{q}}}{\uparrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= - P_{ab}\lambda^{4} \hat{z}_{a}\hat{z}_{b} ( c_{\text{L},{{{\bf{k}}}}{\uparrow}} c_{\text{R},{{\bf{q}}}{\uparrow}} c^{\dag}_{\text{R},{{\bf{q}}}{\uparrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= - P_{ab}\lambda^{4} \hat{z}_{a}\hat{z}_{b} ( u_{{{\bf{q}}}} u_{{{{\bf{k}}}}} )^{2} ( \gamma_{\text{L},{{\bf{q}}}{\uparrow}} \gamma_{\text{R},{{\bf{q}}}{\uparrow}} \gamma^{\dag}_{\text{R},{{\bf{q}}}{\uparrow}} \gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= - P_{ab}\lambda^{4} \hat{z}_{a}\hat{z}_{b} ( u_{{{\bf{q}}}} u_{{{{\bf{k}}}}} )^{2} P_{ab} \label{c5} \end{split}$$ leading to a term $\propto\hat{z}_{a}\hat{z}_{b}$ and the second example is given $$\begin{split} &\quad\ P_{ab}( \lambda \gamma_{b,\text{L},{\uparrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} e^{i\phi_{2}/2} ) ( \lambda \gamma_{a,\text{R},{\downarrow}} c_{\text{R},{{\bf{q}}}{\downarrow}} e^{i\phi_{1}/2} ) ( \lambda c^{\dag}_{\text{R},{{\bf{q}}}{\downarrow}} \gamma_{b,\text{R},{\downarrow}} e^{-i\phi_{2}/2} ) ( \lambda c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{a,\text{L},{\uparrow}} e^{-i\phi_{1}/2} ) P_{ab} \\ &= P_{ab}\lambda^{4} ( \gamma_{b,\text{L},{\uparrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{a,\text{R},{\downarrow}} c_{\text{R},{{\bf{q}}}{\downarrow}} c^{\dag}_{\text{R},{{\bf{q}}}{\downarrow}} \gamma_{b,\text{R},{\downarrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{a,\text{L},{\uparrow}} ) P_{ab} \\ &= P_{ab}\lambda^{4} ( \gamma_{a,\text{L},{\uparrow}} \gamma_{a,\text{R},{\downarrow}} \gamma_{b,\text{L},{\uparrow}} \gamma_{b,\text{R},{\downarrow}} ) ( c_{\text{L},{{{\bf{k}}}}{\uparrow}} c_{\text{R},{{\bf{q}}}{\downarrow}} c^{\dag}_{\text{R},{{\bf{q}}}{\downarrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= - P_{ab}\lambda^{4} \hat{x}_{a}\hat{x}_{b} ( c_{\text{L},{{{\bf{k}}}}{\uparrow}} c_{\text{R},{{\bf{q}}}{\downarrow}} c^{\dag}_{\text{R},{{\bf{q}}}{\downarrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= - P_{ab}\lambda^{4} \hat{x}_{a}\hat{x}_{b} ( u_{{{{\bf{k}}}}} u_{{{\bf{q}}}} )^{2} ( \gamma_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{\text{L},{{\bf{q}}}{\downarrow}} \gamma^{\dag}_{\text{L},{{\bf{q}}}{\downarrow}} \gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= - P_{ab}\lambda^{4} \hat{x}_{a}\hat{x}_{b} ( u_{{{{\bf{k}}}}} u_{{{\bf{q}}}} )^{2} P_{ab} \label{c6} \end{split}$$ producing a term $\propto\hat{x}_{a}\hat{x}_{b}$. The energy denominator for both examples is given by $-1/[(E_{{{{\bf{k}}}}}+U)^{2}(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}}+2U)]$. Hence, once we have summed over all momenta, we conclude that both examples give contributions $\propto J_{6}$. Finally, we remark that examples for contributions $\propto J_{8}$ and $\propto J_{12}$ can be obtained by appropriately commuting the terms in the round brackets in the first line of Eqs.  and . The only difference occurs in the energy denominator. The latter is given by $-1/[(E_{{{\bf{q}}}}+U)(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}}+2U)(E_{{{{\bf{k}}}}}+U)]$ for the contributions $\propto J_{8}$ and by $-1/[(E_{{{\bf{q}}}}+U)(2U)(E_{{{{\bf{k}}}}}+U)]$ for the contributions $\propto J_{12}$. Sequences of intermediate states corresponding to contributions $\boldsymbol{\propto J_{13},J_{15}}$ ---------------------------------------------------------------------------------------------------- In this third subsection we discuss sequences of intermediate states that lead to contributions $\propto J_{13}, J_{14}$ in the effective Hamiltonian. An examples that lead to a contribution $\propto J_{13}$ is given by $$\begin{split} &\quad\ P_{ab}( \lambda c^{\dag}_{\text{L},{{\bf{q}}}{\downarrow}} \gamma_{b,\text{L},{\downarrow}} e^{-i\phi_{2}/2} ) ( \lambda \gamma_{a,\text{L},{\downarrow}} c_{\text{L},{{\bf{q}}}{\downarrow}} e^{i\phi_{1}/2} ) ( \lambda \gamma_{b,\text{L},{\uparrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} e^{i\phi_{2}/2} ) ( \lambda c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{a,\text{L},{\uparrow}} e^{-i\phi_{1}/2} ) P_{ab} \\ &= P_{ab}\lambda^{4} ( c^{\dag}_{\text{L},{{\bf{q}}}{\downarrow}} \gamma_{b,\text{L},{\downarrow}} \gamma_{a,\text{L},{\downarrow}} c_{\text{L},{{\bf{q}}}{\downarrow}} \gamma_{b,\text{L},{\uparrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{a,\text{L},{\uparrow}} ) P_{ab} \\ &= - P_{ab}\lambda^{4} ( \gamma_{a,\text{L},{\uparrow}} \gamma_{a,\text{L},{\downarrow}} \gamma_{b,\text{L},{\uparrow}} \gamma_{b,\text{L},{\downarrow}} ) ( c^{\dag}_{\text{L},{{\bf{q}}}{\downarrow}} c_{\text{L},{{\bf{q}}}{\downarrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= P_{ab}\lambda^{4} \hat{y}_{a}\hat{y}_{b} ( c^{\dag}_{\text{L},{{\bf{q}}}{\downarrow}} c_{\text{L},{{\bf{q}}}{\downarrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= P_{ab}\lambda^{4} ( v_{{{\bf{q}}}} u_{{{{\bf{k}}}}} )^{2} \hat{y}_{a}\hat{y}_{b} ( \gamma_{\text{L},-{{\bf{q}}}{\uparrow}} \gamma^{\dag}_{\text{L},-{{\bf{q}}}{\uparrow}} \gamma_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= P_{ab}\lambda^{4} ( v_{{{\bf{q}}}} u_{{{{\bf{k}}}}} )^{2} \hat{y}_{a}\hat{y}_{b} P_{ab}. \label{c9} \end{split}$$ and gives a term $\propto\hat{y}_{a}\hat{y}_{b}$. The energy denominator for thhis sequences is given by $-1/[(E_{{{\bf{q}}}}+U)(2U)(E_{{{{\bf{k}}}}}+U)]$. Hence, after summing over all momenta we verify that both sequences indeed contribute to the term $\propto J_{13}$ in the effective Hamiltonian. Providing examples of sequences that give contributions $\propto J_{15}$ in the effective Hamiltonian requires us to swap the first two terms in the round brackets in the first line of Eq. . However, these terms do not commute. Hence, we need to re-evaluate the modified sequences. We find that, $$\begin{split} &\quad\ P_{ab}( \lambda \gamma_{a,\text{L},{\downarrow}} c_{\text{L},{{\bf{q}}}{\downarrow}} e^{i\phi_{1}/2} ) ( \lambda c^{\dag}_{\text{L},{{\bf{q}}}{\downarrow}} \gamma_{b,\text{L},{\downarrow}} e^{-i\phi_{2}/2} ) ( \lambda \gamma_{b,\text{L},{\uparrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} e^{i\phi_{2}/2} ) ( \lambda c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{a,\text{L},{\uparrow}} e^{-i\phi_{1}/2} ) P_{ab} \\ &= P_{ab}\lambda^{4} ( \gamma_{a,\text{L},{\downarrow}} c_{\text{L},{{\bf{q}}}{\downarrow}} c^{\dag}_{\text{L},{{\bf{q}}}{\downarrow}} \gamma_{b,\text{L},{\downarrow}} \gamma_{b,\text{L},{\uparrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{a,\text{L},{\uparrow}} ) P_{ab} \\ &= P_{ab}\lambda^{4} ( \gamma_{a,\text{L},{\uparrow}} \gamma_{a,\text{L},{\downarrow}} \gamma_{b,\text{L},{\uparrow}} \gamma_{b,\text{L},{\downarrow}} ) ( c_{\text{L},{{\bf{q}}}{\downarrow}} c^{\dag}_{\text{L},{{\bf{q}}}{\downarrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= - P_{ab}\lambda^{4} \hat{y}_{a}\hat{y}_{b} ( c_{\text{L},{{\bf{q}}}{\downarrow}} c^{\dag}_{\text{L},{{\bf{q}}}{\downarrow}} c_{\text{L},{{{\bf{k}}}}{\uparrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= - P_{ab}\lambda^{4} ( u_{{{\bf{q}}}} u_{{{{\bf{k}}}}} )^{2} \hat{y}_{a}\hat{y}_{b} ( \gamma_{\text{L},{{\bf{q}}}{\downarrow}} \gamma^{\dag}_{\text{L},{{\bf{q}}}{\downarrow}} \gamma_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= - P_{ab}\lambda^{4} ( u_{{{\bf{q}}}} u_{{{{\bf{k}}}}} )^{2} \hat{y}_{a}\hat{y}_{b} P_{ab}. \label{c11} \end{split}$$ The energy denominator for this sequence is still given by $-1/[(E_{{{\bf{q}}}}+U)(2U)(E_{{{{\bf{k}}}}}+U)]$. Consequently, after summing over all momenta, we recognize that the example given in Eq.  contributes to the term $\propto J_{15}$ in the effective Hamiltonian. Sequences of intermediate states corresponding to contributions $\boldsymbol{\propto J_{14},J_{16}}$ ---------------------------------------------------------------------------------------------------- In this fourth subsection, we discuss sequences of intermediate states that contribute to the terms $\propto J_{14},J_{16}$ in the effective Hamiltonian. We begin by presenting an examples fo a sequence of intermediate states that yield contribution $\propto J_{14}$ in the effective Hamiltonian, $$\begin{split} &\quad\ P_{ab}( \lambda c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}} \gamma_{b,\text{L},{\downarrow}} e^{-i\phi_{2}/2} ) ( \lambda \gamma_{a,\text{L},{\downarrow}} c_{\text{L},-{{\bf{q}}}{\downarrow}} e^{i\phi_{1}/2} ) ( \lambda \gamma_{b,\text{L},{\uparrow}} c_{\text{L},{{\bf{q}}}{\uparrow}} e^{i\phi_{2}/2} ) ( \lambda c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{a,\text{L},{\uparrow}} e^{-i\phi_{1}/2} ) P_{ab} \\ &= P_{ab}\lambda^{4} ( c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}} \gamma_{b,\text{L},{\downarrow}} \gamma_{a,\text{L},{\downarrow}} c_{\text{L},-{{\bf{q}}}{\downarrow}} \gamma_{b,\text{L},{\uparrow}} c_{\text{L},{{\bf{q}}}{\uparrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{a,\text{L},{\uparrow}} ) P_{ab} \\ &= - P_{ab}\lambda^{4} ( \gamma_{a,\text{L},{\uparrow}} \gamma_{a,\text{L},{\downarrow}} \gamma_{b,\text{L},{\uparrow}} \gamma_{b,\text{L},{\downarrow}} ) ( c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}} c_{\text{L},-{{\bf{q}}}{\downarrow}} c_{\text{L},{{\bf{q}}}{\uparrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= P_{ab}\lambda^{4} \hat{y}_{a}\hat{y}_{b} ( c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}} c_{\text{L},-{{\bf{q}}}{\downarrow}} c_{\text{L},{{\bf{q}}}{\uparrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= - P_{ab}\lambda^{4} v_{{{{\bf{k}}}}} u_{{{\bf{q}}}} v_{{{\bf{q}}}} u_{{{{\bf{k}}}}} \hat{y}_{a}\hat{y}_{b} ( \gamma_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{\text{L},-{{\bf{q}}}{\downarrow}} \gamma^{\dag}_{\text{L},-{{\bf{q}}}{\downarrow}} \gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= - P_{ab}\lambda^{4} v_{{{{\bf{k}}}}} u_{{{\bf{q}}}} v_{{{\bf{q}}}} u_{{{{\bf{k}}}}} \hat{y}_{a}\hat{y}_{b} P_{ab} \label{c14} \end{split}$$ The energy denominator for this example is given by $-1/[(E_{{{{\bf{k}}}}}+U)^{2}(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}}+2U)]$. This means that after summing over all momenta, both sequences indeed contribute to the term $\propto J_{14}$ in the effective Hamiltonian. Finally, we note that examples for sequences of intermediate states leading to contributions $\propto J_{16}$ can be obtained by commuting the first two terms in the round brackets in the first line of Eq.  and adapting the energy denominators accordingly. Sequences of intermediate states corresponding to contributions $\boldsymbol{\propto J_{17},J_{18}}$ ---------------------------------------------------------------------------------------------------- In this final subsection, we give examples on sequences of intermediate states which lead to contributions $\propto J_{17}, J_{18}$ in the effective Hamiltonian. An example leading to a contribution $\propto J_{18}$ is given by $$\begin{split} &\quad\ P_{ab}( \lambda c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}} \gamma_{b,\text{L},{\downarrow}} e^{-i\phi_{2}/2} ) ( \lambda \gamma_{b,\text{L},{\uparrow}} c_{\text{L},{{\bf{q}}}{\uparrow}} e^{i\phi_{2}/2} ) ( \lambda \gamma_{a,\text{L},{\downarrow}} c_{\text{L},-{{\bf{q}}}{\downarrow}} e^{i\phi_{1}/2} ) ( \lambda c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{a,\text{L},{\uparrow}} e^{-i\phi_{1}/2} ) P_{ab} \\ &= P_{ab}\lambda^{4} ( c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}} \gamma_{b,\text{L},{\downarrow}} \gamma_{b,\text{L},{\uparrow}} c_{\text{L},{{\bf{q}}}{\uparrow}} \gamma_{a,\text{L},{\downarrow}} c_{\text{L},-{{\bf{q}}}{\downarrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{a,\text{L},{\uparrow}} ) P_{ab} \\ &= P_{ab}\lambda^{4} ( \gamma_{a,\text{L},{\uparrow}} \gamma_{a,\text{L},{\downarrow}} \gamma_{b,\text{L},{\uparrow}} \gamma_{b,\text{L},{\downarrow}} ) ( c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}} c_{\text{L},{{\bf{q}}}{\uparrow}} c_{\text{L},-{{\bf{q}}}{\downarrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= - P_{ab}\lambda^{4} \hat{y}_{a}\hat{y}_{b} ( c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}} c_{\text{L},{{\bf{q}}}{\uparrow}} c_{\text{L},-{{\bf{q}}}{\downarrow}} c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= - P_{ab}\lambda^{4} v_{{{{\bf{k}}}}} u_{{{\bf{q}}}} v_{{{\bf{q}}}} u_{{{{\bf{k}}}}} \hat{y}_{a}\hat{y}_{b} ( \gamma_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{\text{L},{{\bf{q}}}{\uparrow}} \gamma^{\dag}_{\text{L},{{\bf{q}}}{\uparrow}} \gamma^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} ) P_{ab} \\ &= - P_{ab}\lambda^{4} v_{{{{\bf{k}}}}} u_{{{\bf{q}}}} v_{{{\bf{q}}}} u_{{{{\bf{k}}}}} \hat{y}_{a}\hat{y}_{b} P_{ab}. \label{c16} \end{split}$$ The energy denominator is given by $-1/[(E_{{{{\bf{k}}}}}+U)^{2}(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}})]$. Hence, after summing over all momenta, we find a contribution $\propto J_{18}$. An example for a sequence of intermediate states that yields a contribution $\propto J_{17}$ can be obtained by swapping the first two terms in round brackets in the first line of Eq. . More concretely, $$\begin{split} &\quad\ P_{ab}( \lambda \gamma_{b,\text{L},{\uparrow}} c_{\text{L},{{\bf{q}}}{\uparrow}} e^{i\phi_{2}/2} ) ( \lambda c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}} \gamma_{b,\text{L},{\downarrow}} e^{-i\phi_{2}/2} ) ( \lambda \gamma_{a,\text{L},{\downarrow}} c_{\text{L},-{{\bf{q}}}{\downarrow}} e^{i\phi_{1}/2} ) ( \lambda c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{a,\text{L},{\uparrow}} e^{-i\phi_{1}/2} ) P_{ab} \\ &= P_{ab}( \lambda c^{\dag}_{\text{L},-{{{\bf{k}}}}{\downarrow}} \gamma_{b,\text{L},{\downarrow}} e^{-i\phi_{2}/2} ) ( \lambda \gamma_{b,\text{L},{\uparrow}} c_{\text{L},{{\bf{q}}}{\uparrow}} e^{i\phi_{2}/2} ) ( \lambda \gamma_{a,\text{L},{\downarrow}} c_{\text{L},-{{\bf{q}}}{\downarrow}} e^{i\phi_{1}/2} ) ( \lambda c^{\dag}_{\text{L},{{{\bf{k}}}}{\uparrow}} \gamma_{a,\text{L},{\uparrow}} e^{-i\phi_{1}/2} ) P_{ab} \\ &= - P_{ab}\lambda^{4} v_{{{{\bf{k}}}}} u_{{{\bf{q}}}} v_{{{\bf{q}}}} u_{{{{\bf{k}}}}} \hat{y}_{a}\hat{y}_{b} P_{ab}. \end{split}$$ The energy denominator is now given by $-1/[(E_{{{\bf{q}}}}+U)(E_{{{{\bf{k}}}}}+E_{{{\bf{q}}}})(E_{{{{\bf{k}}}}}+U)]$. Consequently, after summing over all momenta, we find a contribution $\propto J_{17}$ in the effective Hamiltonian.\ \ In summary, in this second section of the Supplemental Material, we have provided an extensive overview of the numerous contributions which make up the effective Hamiltonian for the two-qubit gate as given in Eq. (11) of the main text. [99]{} C. Schrade and L. Fu, Phys. Rev. Lett. **120**, 267002 (2018).

--- abstract: 'We re-examine the physics of supercritical nuclei, specially focusing on the scattering phase $\delta_{\varkappa}$ and its dependence on the energy $\varepsilon$ of the diving electronic level, for which we give both exact and approximate formulas. The Coulomb potential $Z\alpha/r$ is rounded to the constant $Z\alpha/R$ for $r < R$. We confirm the resonant behavior of $\delta_{\varkappa}$ that we investigate in details. In addition to solving the Dirac equation for an electron, we solve it for a positron, in the field of the same nucleus. This clarifies the interpretation of the resonances. Our results are compared with claims made in previous works.' author: - 'S.I. Godunov' - 'B. Machet' - 'M.I. Vysotsky' bibliography: - 'references.bib' title: 'Resonances in positron scattering on a supercritical nucleus and spontaneous production of $e^{+}e^{-}$ pairs' --- Introduction ============ The Coulomb problem for a nucleus with charge $Z>Z_{\rm cr}$ was recently analysed [@Kuleshov] by solving the Dirac equation for an electron in the external field of this nucleus. Because of the specificity of the Dirac equation that accounts simultaneously for electrons and positrons this problem gets connected to the scattering of positrons (holes in the Dirac sea) on the nucleus (see below). The behavior of the scattering amplitude was found to be very peculiar: it contains resonances and their energies, obtained from an analytical formula found in [@Kuleshov], $$\varepsilon = -\xi + \frac{i}{2} \gamma, \;\; \xi > m, \;\; \gamma > 0, \label{eq:1}$$ correspond to poles of the $S$ matrix located above the left cut, on the second (unphysical) sheet of the energy plane. The resonances in positron scattering were discussed in Refs. [@MRG1:1972; @MRG2:1972]. At $Z<Z_{\rm cr}$, the width $\gamma$ vanishes, and this equation describes the usual bound states of electrons in the Coulomb field of the nucleus. When $Z>Z_{\rm cr}$, $\gamma\neq0$ makes these states quasistationary [@Mur:1976wh; @Popov:1976dh]. For electrons, as $Z$ increases, the transition from bound states to resonant states corresponds to the diving of the bound states, which start at $\varepsilon=+m$, downwards into the lower continuum. In the present paper, in order to clarify the situation, we will also study “the Dirac equation for positron”. By this we mean here the standard Dirac equation with the substitution of electron charge $e$ by $-e$. Now, as $Z$ increases, bound states raise up from $\varepsilon=-m$ and become resonant in the upper continuum. For $Z<Z_{\rm cr}$, the interpretation of these bound states (also noted in [@GMR] chapter 4.3) is the following. For obvious reasons they cannot be $\left(e^{+}N^{+}\right)$ bound states, but are just our previous $\left(e^{-}N^{+}\right)$ bound states. There is no more information in there[^1]. For $Z>Z_{\rm cr}$, we find that $\left(e^{+}N^{+}\right)$ resonances occur at the energies $$\varepsilon_{\rm p} = \xi - \frac{i}{2} \gamma, \;\; \xi > m, \;\; \gamma > 0, \label{eq:2}$$ which now correspond to poles of the $S$ matrix below the right cut of the energy plane, also, as it should be, on the second, unphysical, sheet. This result confirms the proposal made in [@Kuleshov] that the sign of the energy in (\[eq:1\]) should be reversed. This change of sign we are accustomed to when dealing with holes in the lower continuum: the absence of an electron with energy $-\varepsilon$ is then interpreted as the presence of a positron with energy $\varepsilon$. It is now to be operated on the empty states of the energy levels that dive into the lower continuum. Our consideration of the Dirac equation for positrons therefore helps to clarify the nature and position of the resonances. No physical interpretation for them was suggested in [@Kuleshov]. It was only claimed that spontaneous $e^{+}e^{-}$ pair production by naked nuclei at $Z>Z_{\rm cr}$, as discussed in [@MRG1:1972; @MRG2:1972; @Voronkov:1961; @Gershtein:1969; @Greiner:1969; @Popov:1970-1; @Popov:1970-2; @Gerstein:1969-lett; @Popov:1970nz; @Popov:1970-ZhETF-2; @Zeldovich:1972; @Zeldovich:1971; @KP:2014; @Gershtein1973; @Okun:1974rza; @GMR; @GMM], does not occur. We, however, do not see any sensible objection to the occurrence of this process: an empty state diving into the lower continuum gets filled by one electron of the Dirac sea; the resulting hole in the sea is the positron that gets ejected by the nucleus the charge of which has become $Z-1$. The characteristic time of this emission process is $1/\gamma$, in agreement with the results obtained in [@MRG1:1972; @MRG2:1972; @Voronkov:1961; @Gershtein:1969; @Greiner:1969; @Popov:1970-1; @Popov:1970-2; @Gerstein:1969-lett; @Popov:1970nz; @Popov:1970-ZhETF-2; @Zeldovich:1972; @Zeldovich:1971; @KP:2014; @Gershtein1973; @Okun:1974rza; @GMR; @GMM]. Furthermore, spontaneous production of $e^+e^-$ pairs was recently observed in the numerical solution of the Dirac equation in the case of heavy ion collisions [@Maltsev:2014qna; @Maltsev2017]. The plan of the paper is as follows. In Section \[sec:lower\], following [@Kuleshov] and using the Dirac equation, we study the scattering of states of the lower continuum on a supercritical nucleus. In addition to reproducing the approximate results obtained in [@Kuleshov] we get explicit results without using an expansion over the parameter $m\times R$, where $R$ is the nucleus radius. Such an expansion being good for electrons does not work for heavy particles, for example, muons [@Mur:1976wh; @Popov:1976dh]. In Section \[sec:upper\], we use instead the Dirac equation for positrons (see above) and study the scattering of states of its upper continuum on a supercritical nucleus. We conclude in Section \[sec:conclusions\]. Lower continuum wave functions and scattering phases in the Coulomb field of a supercritical nucleus {#sec:lower} ==================================================================================================== The radial functions of the Dirac equation $F(r) \equiv rf(r)$ and $G(r) \equiv rg (r)$ are determined by the following differential equations [@Bethe; @Bethe2; @BLP]: $$\left\{ \begin{aligned} &\frac{dF}{dr} + \frac{\varkappa}{r}F - \left(\varepsilon + m - V(r)\right)G = 0,\\ &\frac{dG}{dr} - \frac{\varkappa}{r}G + \left(\varepsilon - m - V(r)\right)F = 0, \end{aligned} \right. \label{eq:3}$$ where $\varkappa = -(j+1/2) = -1, -2,\dots$ for $j = l + 1/2$ and $\varkappa = (j +1/2)= 1,2,3\dots$ for $j = l-1/2$ and the ground state corresponds to $\varkappa = -1$ (let us note that in [@Kuleshov] the Dirac equation with the substitution $F\Rightarrow -F$ is used). In order to deal with the case $Z\alpha >1$ the Coulomb potential should be regularised at $r=0$ [@PomSmo:1945]. To do this we shall approximate the nucleus as a homogeneous charged sphere with radius $R$ (the so-called rectangular cutoff). Thus, the potential in which the Dirac equation should be solved looks like: [V(r)=]{} -, & $r < R$, \[eq:potential\_r&lt;R\]\ -, & $r > R$. \[eq:potential\_r&gt;R\] \[eq:potential\] At small distances $r< R$, substituting expression (\[eq:potential\_r&lt;R\]) into (\[eq:3\]), we obtain the Dirac equation with a constant potential, the solution of which is expressed through Bessel functions. In order to obtain finite $f$ and $g$ at $r=0$ among the two sets of solutions the one with a positive index of the Bessel function should be selected[^2]: $$\left( \begin{array}{l} F \\ G \end{array}\right) = {\rm const}\cdot\sqrt{\beta r}\cdot \left( \begin{array}{l} \mp J_{\mp(1/2 + \varkappa)} (\beta r)\\ J_{\pm(1/2 - \varkappa)} (\beta r) \frac{\beta}{\varepsilon+m+\frac{Z\alpha}{R}} \end{array} \right),\; r < R, \label{5}$$ where $\beta = \sqrt{(\varepsilon + Z\alpha /R)^2 - m^2}$. Upper (lower) signs should be taken for $\varkappa < 0$ ($\varkappa > 0$). For $r>R$, we need the solution of the Dirac equation for the Coulomb potential. We introduce the standard quantity $\lambda$ which, for $-m < \varepsilon < m$, equals $\lambda = \sqrt{(m-\varepsilon)(m+\varepsilon)} \equiv -ik$, where $k$ is the electron momentum. Here we have to make an important remark. Since later we are going to look for resonances in the complex $\varepsilon$ plane, we must carefully define the square roots used here. Each of them, $\sqrt{m-\varepsilon}$ and $\sqrt{m+\varepsilon}$, are defined on two Riemann sheets of the complex $\varepsilon$ plane. To avoid ambiguous expressions let us introduce a uniquely defined function ${\rm sqrt(z)}$ as follows: $$\label{eq:sqrt_general_0} {\rm sqrt}\left(|z|e^{i{\rm Arg}(z)}\right)=\sqrt{|z|}e^{i{\rm Arg}(z)/2},\text{ for } {\rm Arg}(z)\in(-\pi;\pi].$$ For example $$\label{eq:sqrt_0} {\rm sqrt}\left(z\right)= \begin{cases} i& \text{for }z=-1+i\cdot0,\\ i& \text{for }z=-1,\\ -i& \text{for }z=-1-i\cdot0. \end{cases}$$ It is therefore the first branch of the function $\sqrt{z}$ with the cut $(-\infty;0)$. The second branch is given by $-{\rm sqrt}(z)$. This definition is also very convenient because the square root is defined in this way in many numerical tools for computers. Switching branches of both square roots, $\sqrt{m-\varepsilon}$ and $\sqrt{m+\varepsilon}$, leads to the same value of $\lambda$. Therefore, $\lambda$ is defined on the two Riemann sheets according to: $$\label{eq:lambda_sheets} \lambda= \begin{cases} {\rm sqrt}\left(m-\varepsilon\right)\cdot {\rm sqrt}\left(m+\varepsilon\right)& \text{on the physical sheet,}\\ -{\rm sqrt}\left(m-\varepsilon\right)\cdot {\rm sqrt}\left(m+\varepsilon\right)& \text{on the unphysical sheet,} \end{cases}$$ with two cuts originating, respectively, from each of the square roots (see Fig. \[fig:cuts\])[^3]. From general arguments of scattering theory, we know that electron bound states are located at real $\varepsilon$ in the interval $-m < \varepsilon < m$. Unbound electron states are located above the right cut and unbound positron states below the left cut. ![The plane of complex energy $\varepsilon$.[]{data-label="fig:cuts"}](pics/cuts.pdf){width="4.5in"} In what follows we shall use the following conventions for the “$\sqrt{~}$” symbol: $$\begin{aligned} \label{eq:roots} \sqrt{m+\varepsilon}&= \begin{cases} {\rm sqrt}\left(m+\varepsilon\right)&\text{on the physical sheet,}\\ -{\rm sqrt}\left(m+\varepsilon\right)&\text{on the unphysical sheet,} \end{cases}\\ \sqrt{m-\varepsilon}&={\rm sqrt}\left(m-\varepsilon\right)\text{ on both sheets.}\end{aligned}$$ It does not matter which root changes sign when we go to the second sheet since we can always also change the signs of both. We are looking for solution written in the standard form [@BLP][^4]: $$\left( \begin{array}{c} F \\ G \end{array} \right) = \left( \begin{array}{c} \sqrt{m+\varepsilon} \\ -\sqrt{m-\varepsilon} \end{array} \right) {\rm exp} (-\rho/2)\rho^{i\tau} \left( \begin{array}{c} Q_1 + Q_2 \\ Q_1 - Q_2 \end{array} \right), \label{8}$$ where $\tau=\sqrt{\left(Z\alpha\right)^{2}-\varkappa^{2}}$, $\rho = 2\lambda r = -2ikr$, $Q_1$ and $Q_2$ are determined by differential equations, the solutions of which are Kummer confluent hypergeometric functions $_1F_1(\alpha,\beta,z)$ (also sometimes noted $F(\alpha,\gamma,z)$ like in [@BLP]). In textbooks dealing with the case $Z \alpha <1$, $R=0$, only solutions regular at $r=0$ are considered. We must instead here take into account both type of solutions of the equations for $Q_1$ and $Q_2$. The formulas for the $Q_i$ are derived in Appendix \[sec:Q1Q2\]. From (\[A6\]) to (\[A8\]) we get: $$\hspace{-7mm}\left\{ \begin{aligned} Q_1 &= C\cdot\frac{-\frac{iZ\alpha m}{k}+\varkappa} {-i\tau+\frac{iZ\alpha\varepsilon}{k}} \cdot {_1F_{1}}\left(i\tau-\frac{iZ\alpha\varepsilon}{k},2i\tau +1,\rho\right)+\\ &\hspace{40mm}+D\cdot\frac{-\frac{iZ\alpha m}{k} +\varkappa} {i\tau+\frac{iZ\alpha\varepsilon}{k}} \rho^{-2i\tau}{_{1}F_{1}}\left(-i\tau - \frac{iZ\alpha\varepsilon}{k}, -2i\tau+1, \rho\right),\\ Q_2 &= C\cdot{_{1}F_{1}}\left(1+i\tau - \frac{iZ\alpha\varepsilon}{k}, 2i\tau+1, \rho\right) + D\rho^{-2i\tau}{_{1}F_{1}}\left(1-i\tau - \frac{iZ\alpha\varepsilon}{k}, -2i\tau+1, \rho\right), \end{aligned}\right.\label{eq:9}$$ where $C$ and $D$ are arbitrary coefficients[^5]. The scattering phase $\delta_{\varkappa}(\varepsilon,Z)$ is determined by investigating the behavior of the wave function at large $r$. To this purpose, the asymptotic expansion of $_1F_1$ at large $|z|$ $$_{1}F_{1}(\alpha, \gamma, z)\Big\rvert_{|z| \to \infty} = \frac{\Gamma(\gamma)}{\Gamma(\gamma-\alpha)}(-z)^{-\alpha} [1+ O(1/z)] + \frac{\Gamma(\gamma)}{\Gamma(\alpha)} e^z z^{\alpha -\gamma} [1+O(1/z)] \label{11}$$ is very useful. Using the asymptotic expansion (\[11\]) for the Kummer functions occurring in (\[eq:9\]) gives: $$\begin{aligned} \left.\left( \begin{array}{c} F \\ G \end{array} \right)\right|_{r\to\infty} &= A\cdot\left( \begin{array}{c} \sqrt{m+\varepsilon} \\ -\sqrt{m-\varepsilon} \end{array} \right)\times\\ \times\Biggl(C&\left[ e^{-\frac{\rho}{2}}\frac{\Gamma\left(2i\tau+1\right)} {\Gamma\left(1+i\tau+\frac{iZ\alpha\varepsilon}{k}\right)} \frac{\frac{iZ\alpha m}{k}-\varkappa}{i\tau-\frac{iZ\alpha\varepsilon}{k}} \rho^{i\tau}\left(-\rho\right)^{-i\tau} \left(-\rho\right)^{\frac{iZ\alpha\varepsilon}{k}} \pm e^{\frac{\rho}{2}}\frac{\Gamma\left(2i\tau+1\right)} {\Gamma\left(1+i\tau-\frac{iZ\alpha\varepsilon}{k}\right)} \rho^{-\frac{iZ\alpha\varepsilon}{k}} \right] +\\ +D&\left[ e^{-\frac{\rho}{2}}\frac{\Gamma\left(-2i\tau+1\right)} {\Gamma\left(1-i\tau+\frac{iZ\alpha\varepsilon}{k}\right)} \frac{\frac{iZ\alpha m}{k}-\varkappa}{-i\tau-\frac{iZ\alpha\varepsilon}{k}} \rho^{-i\tau}\left(-\rho\right)^{i\tau} \left(-\rho\right)^{\frac{iZ\alpha\varepsilon}{k}} \pm e^{\frac{\rho}{2}}\frac{\Gamma\left(-2i\tau+1\right)} {\Gamma\left(1-i\tau-\frac{iZ\alpha\varepsilon}{k}\right)} \rho^{-\frac{iZ\alpha\varepsilon}{k}} \right]\Biggr), \label{12}\end{aligned}$$ where the upper sign corresponds to $F$ and the lower sign corresponds to $G$. The ratio $$\begin{aligned} \frac{\left(-\rho\right)^{\frac{iZ\alpha\varepsilon}{k}}} {\rho^{-\frac{iZ\alpha\varepsilon}{k}}}\end{aligned}$$ yields the Coulomb logarithm (for real $\varepsilon$ below the left cut it gives $\exp\left[\frac{2iZ\alpha\varepsilon}{k}\ln\left(2kr\right)\right]$). Since the latter does not contribute to the differential scattering cross section at nonzero angle $\theta$, we will omit this term in our further calculations. From the general formula $$\begin{aligned} \left.\left( \begin{array}{c} F \\ G \end{array} \right)\right|_{r\to\infty} \propto\left( \begin{array}{c} \sqrt{m+\varepsilon} \\ -\sqrt{m-\varepsilon} \end{array} \right) \left\{e^{i(kr+\frac{Z\alpha\varepsilon}{k}\ln(2kr))} e^{2i\delta} \pm e^{-i(kr+\frac{Z\alpha\varepsilon}{k}\ln(2kr))}\right\}\end{aligned}$$ it follows that the ratio of the remaining coefficients define the scattering phase $\delta_{\varkappa}$ (on the real axis below the left cut $e^{-\rho/2}\equiv e^{ikr}$ corresponds to the outgoing wave and $e^{\rho/2}\equiv e^{-ikr}$ corresponds to the incoming wave): $$\begin{aligned} \label{eq:phase} e^{2i\delta_{\varkappa}} &=-\frac{1}{\varkappa+\frac{iZ\alpha m}{k}}\cdot \frac {\frac{C}{D}\cdot\frac{\Gamma\left(2i\tau\right)} {\Gamma\left(i\tau+\frac{iZ\alpha\varepsilon}{k}\right)} \rho^{i\tau}\left(-\rho\right)^{-i\tau} - \frac{\Gamma\left(-2i\tau\right)} {\Gamma\left(-i\tau+\frac{iZ\alpha\varepsilon}{k}\right)} \rho^{-i\tau}\left(-\rho\right)^{i\tau}} {\frac{C}{D}\cdot\frac{\Gamma\left(2i\tau\right)} {\Gamma\left(1+i\tau-\frac{iZ\alpha\varepsilon}{k}\right)} - \frac{\Gamma\left(-2i\tau\right)} {\Gamma\left(1-i\tau-\frac{iZ\alpha\varepsilon}{k}\right)}},\end{aligned}$$ where $$\begin{aligned} \rho^{i\tau}\left(-\rho\right)^{-i\tau} &= \exp\left[i\tau\ln\left(\rho\right)-i\tau\ln\left(-\rho\right)\right] =\exp\left[-\tau\left( {\rm Arg}\left[\rho\right]-{\rm Arg}\left[-\rho\right] \right)\right]=\\ &=e^{-\pi\tau\cdot{\rm sign}\left[{\rm Arg}\left[\rho\right]\right]}.\nonumber\end{aligned}$$ The resonance of the scattering amplitude corresponds to the pole of the $S$-matrix element $S\equiv e^{2i\delta}$ and from (\[eq:phase\]) we immediately get an equation for the position of this pole in the $\varepsilon$-plane: $$\begin{aligned} \label{eq:pole} \frac{C}{D}\cdot\frac{\Gamma\left(2i\tau\right)} {\Gamma\left(1+i\tau-\frac{iZ\alpha\varepsilon}{k}\right)} - \frac{\Gamma\left(-2i\tau\right)} {\Gamma\left(1-i\tau-\frac{iZ\alpha\varepsilon}{k}\right)}=0.\end{aligned}$$ In what follows we will match the solutions at $r<R$ and $r>R$ to obtain the ratio $C/D$, such that we can calculate the phase $\delta_{\varkappa}$ and find the poles of the $S$ matrix which correspond to the energy levels. This procedure can be performed both exactly and approximately. Exact results ------------- With the help of the exact formulas (\[5\]), (\[8\]), and (\[eq:9\]) we get the ratio $C/D$ from matching $F/G$ at $r=R+0$ and $r=R-0$: $$\label{eq:CD_exact} \frac{C}{D}= -\rho_{0}^{-2i\tau}\cdot\frac{F_{g}^{-}-MF_{f}^{-}}{F_{g}^{+}-MF_{f}^{+}},$$ where $$\begin{aligned} M&=\pm\frac{\sqrt{m+\varepsilon}}{\sqrt{m-\varepsilon}} \cdot \frac{J_{\pm\left(1/2-\varkappa\right)}\left(\beta R\right)} {J_{\mp\left(1/2+\varkappa\right)}\left(\beta R\right)} \cdot \frac{\beta}{\varepsilon+m+\frac{Z\alpha}{R}},\label{eq:M}\\ \label{eq:Ff} F_{f}^{\pm}&= {_{1}F_{1}}\left(\alpha_{1}^{\pm},\gamma^{\pm},\rho_{0}\right) \frac{\frac{iZ\alpha m}{k}-\varkappa}{\alpha_{1}^{\pm}} +{_{1}F_{1}}\left(\alpha_{2}^{\pm},\gamma^{\pm},\rho_{0}\right),\\ \label{eq:Fg} F_{g}^{\pm}&= {_{1}F_{1}}\left(\alpha_{1}^{\pm},\gamma^{\pm},\rho_{0}\right) \frac{\frac{iZ\alpha m}{k}-\varkappa}{\alpha_{1}^{\pm}} -{_{1}F_{1}}\left(\alpha_{2}^{\pm},\gamma^{\pm},\rho_{0}\right),\end{aligned}$$ and $$\label{eq:args} \alpha_{1}^{\pm}=\pm i\tau-\frac{iZ\alpha\varepsilon}{k}, \alpha_{2}^{\pm}=1\pm i\tau-\frac{iZ\alpha\varepsilon}{k}, \gamma^{\pm}=\pm 2i\tau+1, \rho_{0}=-2ikR.$$ The numerical evaluation of the square roots in (\[eq:M\]) and of $k$ in (\[eq:args\]) for *real* $\varepsilon$ is somewhat tricky since one should carefully choose the side of the cut to use. Due to the definition (\[eq:sqrt\_general\_0\])–(\[eq:sqrt\_0\]) of the ${\rm sqrt}()$ function the expression ${\rm sqrt}\left(m+\varepsilon\right)$ gives, for *real* $\varepsilon$, the values above the cut such that $-{\rm sqrt}\left(m+\varepsilon\right)$ should be used. It corresponds *formally* to calculating the scattering phase on the second (unphysical) sheet. The same holds for $k$. For any *real* $\varepsilon$ it is also possible to use $k={\rm sqrt}\left(\varepsilon^{2}-m^{2}\right)$ which chooses the correct side of the cut; then $\sqrt{m+\varepsilon}/\sqrt{m-\varepsilon}=-ik/\left(m-\varepsilon\right)$. With the help of (\[eq:CD\_exact\]) we can calculate the scattering phase $\delta_{\varkappa}$ defined by (\[eq:phase\]). In the domain $\varepsilon<-m$, $\delta_{\varkappa}(\varepsilon,Z)$ gives the scattering phase of a positron with energy $\varepsilon_{\rm p}=-\varepsilon>m$ on the nucleus (for real $\varepsilon<-m$ we get ${\rm Arg}\left[\rho\right]<0$). Its dependence on $\varepsilon_{\rm p}$ for $\varkappa = -1$ and $Z = 232$ is shown in Fig. \[fig:phase\] (compare with Fig. 3 of [@Kuleshov]). The scattering phase $\delta_{\varkappa}$ exhibits a resonance behavior; it goes through $\pi/2$ at $\varepsilon_{\rm p}/m\approx 5.06$. We obtain the equation for the position of the poles by substituting (\[eq:CD\_exact\]) into (\[eq:pole\]) $$\begin{aligned} \label{eq:pole_detailed} \frac{\Gamma\left(-2i\tau\right)} {\Gamma\left(2i\tau\right)}\cdot \frac{\Gamma\left(1+i\tau-\frac{iZ\alpha\varepsilon}{k}\right)} {\Gamma\left(1-i\tau-\frac{iZ\alpha\varepsilon}{k}\right)}= -\rho_{0}^{-2i\tau}\cdot\frac{F_{g}^{-}-MF_{f}^{-}}{F_{g}^{+}-MF_{f}^{+}}.\end{aligned}$$ The solutions of (\[eq:pole\_detailed\]) can be found by scanning the complex $\varepsilon$ plane. This is the method that we used to find the exact positions[^6] of the $S$-matrix poles (see Fig. \[fig:e\_of\_Z\] and Table \[tab:e\_of\_Z\]). The energies $\varepsilon$ of the quasistationary states are located above the left cut on the second sheet of the complex $\varepsilon$ plane[^7]: $$\varepsilon = -\xi + \frac{i}{2}\gamma,~\xi>m,~\gamma>0. \label{eq:resonances}$$ Approximate results ------------------- In [@Kuleshov] the approximation $1/R\gg\varepsilon,m$ was used. In this section we are going to reproduce their results and compare them to the exact ones. Being interested in the case $Z\alpha \gtrsim 1$ and taking into account the smallness of the nucleus radius in comparison with the electron Compton wavelength $1/m$ we obtain that $\beta \approx Z\alpha/R$ in (\[5\]). The solution of the system (\[eq:3\]) at $r > R$ should match (\[5\]) at $r=R$, in particular the ratio $F/G$ of both solutions at $r=R$ should coincide. Substituting (\[eq:potential\_r&gt;R\]) in (\[eq:3\]) at $r\to 0$ we easily get $$\left.\left( \begin{array}{l} F \\ G \end{array}\right)\right|_{r \to 0} = \eta_\sigma r^\sigma \left( \begin{array}{c} -1 \\ \frac{Z\alpha}{\sigma - \varkappa} \end{array} \right) + \eta_{-\sigma} r^{-\sigma} \left( \begin{array}{c} -1 \\ \frac{Z\alpha}{-\sigma - \varkappa} \end{array} \right), \label{6}$$ where $\sigma = \sqrt{\varkappa^2 - Z^2\alpha^2}$ and $\eta_\sigma$ and $\eta_{-\sigma}$ are arbitrary constants. Matching the ratios $F/G$ from (\[6\]) and (\[5\]) at $r=R$ we obtain $$\frac{\eta_\sigma}{\eta_{-\sigma}} = \frac{\sigma -\varkappa}{\sigma +\varkappa}\cdot \frac{Z\alpha J_{\mp(1/2 +\varkappa)}(Z\alpha) \pm (\sigma + \varkappa)J_{\pm(1/2 -\varkappa)}(Z\alpha)} {Z\alpha J_{\mp(1/2 +\varkappa)}(Z\alpha) \mp (\sigma - \varkappa) J_{\pm(1/2 -\varkappa)}(Z\alpha)} \cdot\frac{R^{-\sigma}}{R^\sigma}=\tan\theta, \label{7}$$ which coincides with Eq. (13) from [@Kuleshov]. In the case $Z\alpha > |\varkappa|$ one should substitute $\sigma$ by $i\tau$ (where, as before, $\tau = \sqrt{Z^2\alpha^2 - \varkappa^2}$): $$\frac{\eta_\tau}{\eta_{-\tau}} = \frac{i\tau -\varkappa}{i\tau +\varkappa}\cdot \frac{Z\alpha J_{\mp(1/2 +\varkappa)}(Z\alpha) \pm (i\tau + \varkappa) J_{\pm(1/2 -\varkappa)}(Z\alpha)} {Z\alpha J_{\mp(1/2 +\varkappa)}(Z\alpha) \mp (i\tau - \varkappa) J_{\pm(1/2 -\varkappa)}(Z\alpha)} \cdot\frac{R^{-i\tau}}{R^{i\tau}}= e^{2i\theta}. \label{77}$$ The modulus of the r.h.s of (\[77\]) can be easily checked to be unity, this is why we can rewrite it as an $\exp{\left(2i\theta\right)}$ with real $\theta$. The expansion of (\[8\]) at small $\rho$ contains terms $\sim\rho^{i\tau}$ and $\rho^{-i\tau}$. Comparing this expansion with (\[6\]) and substituting $\sigma \to i\tau$, $\eta_\sigma \to \eta_\tau$, $\eta_{-\sigma}\to \eta_{-\tau}$ yields: $$\eta_\tau = C\cdot(-2ik)^{i\tau} \frac{i\tau - \varkappa + Z\alpha\sqrt{\frac{m-\varepsilon} {m+\varepsilon}}}{i\tau - \frac{iZ\alpha\varepsilon}{k}}, \; \eta_{-\tau} = D\cdot(-2ik)^{-i\tau} \frac{-i\tau - \varkappa + Z\alpha\sqrt{\frac{m-\varepsilon} {m+\varepsilon}}}{-i\tau - \frac{iZ\alpha \varepsilon}{k}}. \label{10}$$ Getting an equation for $C/D$ needs matching (\[77\]) with $\eta_{\tau}/\eta_{-\tau}$ obtained from (\[10\]): $$\begin{aligned} \label{eq:CD} \frac{C}{D}=e^{2i\theta}\cdot \frac{\left(-2ik\right)^{-i\tau}}{\left(-2ik\right)^{i\tau}}\cdot \frac{Z\alpha\sqrt{m-\varepsilon}+\left(-i\tau-\varkappa\right)\sqrt{m+\varepsilon}} {Z\alpha\sqrt{m-\varepsilon}+\left(i\tau-\varkappa\right)\sqrt{m+\varepsilon}}\cdot \frac{i\tau-\frac{iZ\alpha\varepsilon}{k}}{-i\tau-\frac{iZ\alpha\varepsilon}{k}}.\end{aligned}$$ Two sets of approximations were made in deriving (\[eq:CD\]): i. to get $\eta_\tau/\eta_{-\tau}$ at $r=R-0$ we replaced $\beta R$ with $Z \alpha$ and used (\[6\]) which was itself derived for $Z \alpha /r \gg \varepsilon,m$; ii. to get $\eta_\tau/\eta_{-\tau}$ at $r=R+0$ we expanded (\[8\]) and (\[eq:9\]) at $\rho\ll 1$. For $m\cdot R = 0.031$ one cannot expect an accuracy better than 3% and, with growing $|\varepsilon|$ it can even get worse. The accuracy of the final result is not easy to guess from the start, and the best way is to compare it with the exact solution which was found in the previous subsection. Note that all results in [@Kuleshov] are based on the asymptotic behavior (\[6\]) and are therefore approximate by default. Substituting (\[eq:CD\]) into (\[eq:phase\]) we obtain the approximate expression for the scattering phase $\delta_{\varkappa}$. Its dependence on $\varepsilon_{\rm p}\equiv-\varepsilon$ for $\varkappa = -1$ and $Z = 232$ is shown in Fig. \[fig:phase\] (compare with Fig. 3 of [@Kuleshov]). The scattering phase $\delta_{\varkappa}$ exhibits a resonance behavior; it goes through $\pi/2$ at $\varepsilon_{\rm p}/m\approx 4.88$. Let us note that on the real axis of $\varepsilon$ the expression for the scattering phase $\delta_{\varkappa}$ can be written in the same form as in [@Kuleshov] (see Appendix \[sec:real\_phase\]). ![Dependence on $\varepsilon_{\rm p}$ of the scattering phase $\delta_{-1}(\varepsilon_{\rm p}, 232)$ ($Z=232$ and $\varkappa=-1$) for a nucleus with radius $R=0.031/m$. The blue solid line corresponds to the exact phase, the green dashed line corresponds to the approximate one.[]{data-label="fig:phase"}](pics/phase.pdf){width="6.5in"} The positions of the $S$ matrix poles are defined by the same equality (\[eq:pole\]) with $C/D$ given by (\[eq:CD\]): $$\begin{aligned} e^{2i\theta}= \frac{\left(-2ik\right)^{i\tau}}{\left(-2ik\right)^{-i\tau}}\cdot \frac{\Gamma\left(-2i\tau\right)}{\Gamma\left(2i\tau\right)}\cdot \frac{\Gamma\left(1+i\tau-\frac{iZ\alpha\varepsilon}{k}\right)} {\Gamma\left(1-i\tau-\frac{iZ\alpha\varepsilon}{k}\right)}\cdot \frac{Z\alpha\sqrt{m-\varepsilon}+\left(-i\tau+\varkappa\right)\sqrt{m+\varepsilon}} {Z\alpha\sqrt{m-\varepsilon}+\left(i\tau+\varkappa\right)\sqrt{m+\varepsilon}}, \label{eq:eq_poles}\end{aligned}$$ where the l.h.s. is defined by (\[77\]). The r.h.s. coincides with Eq. (26) from [@Kuleshov]. The exact expression (\[eq:pole\_detailed\]) is, of course, more complicated, but, anyhow, special functions have to be evaluated numerically in both cases. The accuracy of ${\rm Re}[\varepsilon]$ obtained by the approximate procedure is quite reasonable (see Fig. \[fig:e\_of\_Z\] and Table \[tab:e\_of\_Z\]); however it is much worse for ${\rm Im}[\varepsilon]$, for example $\approx15\%$ at $Z=186$. This is why it is worth getting the exact values of the energies $\varepsilon$. The question that we want to address now is the origin of the resonance and how it transforms for $Z < Z_{\rm cr}$. At $Z < Z_{\rm cr}$ the resonances become bound states, the energies of which are determined by the same type of matching at $r=R$ as before (it is convenient to replace now, in (\[eq:eq\_poles\]), $k$ by $i\lambda$, since on the real axis, for $-m<\varepsilon<+m$, $\lambda$ is real positive): $${\rm exp}(2i\theta) = \frac{(2\lambda)^{i\tau}}{(2\lambda)^{-i\tau}}\cdot \frac{\Gamma(-2i\tau)}{\Gamma(2i\tau)}\cdot \frac{\Gamma\left(1+i\tau-\frac{Z\alpha\varepsilon}{\lambda}\right)} {\Gamma\left(1-i\tau-\frac{Z\alpha\varepsilon}{\lambda}\right)}\cdot \frac{Z\alpha\sqrt{m-\varepsilon}+(\varkappa-i\tau)\sqrt{m+\varepsilon}} {Z\alpha\sqrt{m-\varepsilon}+(\varkappa+i\tau)\sqrt{m+\varepsilon}}. \label{171}$$ Last, for $Z\alpha < |\varkappa|$ we must change $i\tau$ into $\sigma = \sqrt{\varkappa^2 - Z^2\alpha^2}$: $$\tan\theta = \frac{(2\lambda)^\sigma}{(2\lambda)^{-\sigma}}\cdot \frac{\Gamma(-2\sigma)}{\Gamma(2\sigma)}\cdot \frac{\Gamma\left(1+\sigma-\frac{Z\alpha\varepsilon}{\lambda}\right)} {\Gamma\left(1-\sigma-\frac{Z\alpha\varepsilon}{\lambda}\right)}\cdot \frac{Z\alpha\sqrt{m-\varepsilon}+(\varkappa-\sigma)\sqrt{m+\varepsilon}} {Z\alpha\sqrt{m-\varepsilon}+(\varkappa +\sigma)\sqrt{m+\varepsilon}}. \label{181}$$ ![The dependence of the ground state energy on $Z$. The square markers are for the exact values of the energy (see (\[eq:pole\_detailed\])) and the round markers are for the approximate ones calculated with the help of (\[eq:eq\_poles\]). The correspondence between color and $Z$ is shown in the legend (the real part of the energy is monotonically decreasing). At $Z=Z_{\rm cr}$ the bound states become resonances with positive ${\rm Im}[\varepsilon]$.[]{data-label="fig:e_of_Z"}](pics/ground_level_210.pdf){width="6.5in"} [|c|c|c|c|c|]{} $Z$ & ${\rm Re}\left(\varepsilon_{\rm appr}\right)$ & ${\rm Im}\left(\varepsilon_{\rm appr}\right)$ & ${\rm Re}\left(\varepsilon\right)$ & ${\rm Im}\left(\varepsilon\right)$\ Let us consider for example $Z\alpha<1$, for which taking a point-like nucleus is reliable. At the limit $R\to0$, the r.h.s. of (\[7\]) becomes infinite. Therefore, the spectrum of the Dirac equation is given by the poles of (\[181\]). They are given by the poles of $\Gamma\left(1+\sigma - \frac{Z\alpha\varepsilon}{\lambda}\right)$: $$\sqrt{\varkappa^2-Z^2\alpha^2} - \frac{Z\alpha\varepsilon}{\sqrt{m^2-\varepsilon^2}} = -1, -2, \dots\equiv -n_r, \label{19}$$ to which must be added, for $\varkappa<0$, the zero of the last term in the denominator of (\[181\])[^8]: $$Z\alpha\sqrt{m-\varepsilon}+(\varkappa +\sigma)\sqrt{m+\varepsilon}=0 \;\Rightarrow\;\sqrt{\varkappa^2-Z^2\alpha^2} - \frac{Z\alpha\varepsilon}{\sqrt{m^2 -\varepsilon^2}} = 0 \equiv n_{r}. \label{20}$$ The electron bound states at $Z < Z_{\rm cr}$ become therefore resonances at $Z > Z_{\rm cr}$; the poles of the $S$ matrix corresponding to the latter describe positron-nucleus scattering. The trajectory of the ground state energy with growing $Z$ is shown in Fig. \[fig:e\_of\_Z\] (see also Table \[tab:e\_of\_Z\]). Let us notice the unusual signs of both real and imaginary parts of the resonance energy. It was suggested in [@Kuleshov] that the sign of the energy should be reversed, under the claim that the corresponding state is a resonance in the positron-nucleus system. Such a sign reversal is usual for holes in the lower continuum of the Dirac equation: the absence of an electron of energy $-\varepsilon$ is equivalent to the presence of a positron with energy $\varepsilon$. Advocating for the same procedure in the case at hands looks a priori suspicious since the resonances that we found originate from electron bound energy levels (however, also empty) that dive from $\varepsilon=+m$ downwards into the lower continuum (and will return upwards to $+m$ if $Z$ decreases). An interpretation of the phenomenon in terms of electrons looks therefore more intuitive. In order to resolve this (apparent) puzzle, we shall solve in the next section the Dirac equation for positrons, which describes the scattering of a positron in the upper continuum on a nucleus. The Dirac equation for positrons: upper continuum wave functions and scattering phases in the Coulomb field of a supercritical nucleus {#sec:upper} ====================================================================================================================================== Changing the sign of $Z\alpha$ in (\[eq:potential\]), we get instead of (\[eq:3\]) $$\left\{ \begin{aligned} &\frac{d\tilde F}{dr} + \frac{\varkappa}{r} \tilde F - (\varepsilon +m -\tilde V(r)) \tilde G = 0,\\ &\frac{d\tilde G}{dr} - \frac{\varkappa}{r} \tilde G + (\varepsilon -m -\tilde V(r)) \tilde F = 0, \end{aligned} \right. \label{21}$$ where [V(r) =]{} , & $r > R$, \[eq:pos\_potential\_r&gt;R\]\ , & $r < R$. Notice that (\[eq:3\]) gives (\[21\]) by the set of transformations $\varkappa\to -\varkappa$, $\varepsilon\to -\varepsilon$, $F\to\tilde G$ and $G\to\tilde F$. The states in the upper continuum ($\varepsilon > m$) describe positron scattering on a nucleus. Since a positron cannot form a bound state with a positively charged nucleus, one could think that no resonance at $Z > Z_{\rm cr}$ will occur, nor the resonant behavior of the scattering phase found in [@Kuleshov] and reproduced in Section \[sec:lower\]. The central issue is therefore to investigate whether bound states and resonances arise or not in the Dirac equation for positrons (\[21\]). Solving (\[21\]) at $r < R$ we obtain $$\left( \begin{array}{l} \tilde F \\ \tilde G \end{array}\right) = {\rm const}\cdot\sqrt{\tilde\beta r}\cdot \left( \begin{array}{l} \pm J_{\mp(1/2 + \varkappa)} (\tilde\beta r) \\ J_{\pm(1/2 - \varkappa)} (\tilde\beta r) \frac{\tilde\beta}{\varepsilon+m-\frac{Z\alpha}{R}} \end{array} \right),\; r < R, \label{23}$$ where $\tilde\beta = \sqrt{(\varepsilon - Z\alpha/R)^2 - m^2}$ and, at small distances, where the solution (\[23\]) will be used, $\tilde\beta \approx \beta \approx Z\alpha/R$. The upper (lower) signs in (\[23\]) should be taken for $\varkappa < 0$ ($\varkappa > 0$). Note that the sign of $\tilde F$ is opposite to that of $F$ in (\[5\]), while the signs of $\tilde G$ and $G$ coincide. Substituting in (\[21\]) the Coulomb potential (\[eq:pos\_potential\_r&gt;R\]) and going to the limit $r \to 0$ we get: $$\left.\left( \begin{array}{l} \tilde F \\ \tilde G \end{array} \right)\right|_{r \to 0} = \tilde\eta_\sigma r^\sigma \left( \begin{array}{c} -1 \\ \frac{-Z\alpha}{\sigma -\varkappa} \end{array} \right) + \tilde\eta_{-\sigma} r^{-\sigma} \left( \begin{array}{c} -1 \\ \frac{-Z\alpha}{-\sigma -\varkappa} \end{array} \right). \label{24}$$ Note that the sign of $\tilde G$ is opposite to that of $G$ in (\[6\]), while the signs of $\tilde F$ and $F$ coincide. Thus, when matching the ratios of $\tilde F/\tilde G$ from (\[23\]) and (\[24\]) at $r=R$ we obtain equations identical to (\[7\]), (\[77\]) with the change $\eta\to\tilde\eta$. Like in (\[8\]), we look for solutions of the form $$\left( \begin{array}{l} \tilde F \\ \tilde G \end{array} \right) = \left( \begin{array}{c} \sqrt{m+\varepsilon} \\ -\sqrt{m-\varepsilon} \end{array} \right) {\rm exp}(-\rho/2) \rho^{i\tau} \left( \begin{array}{c} \tilde Q_1 + \tilde Q_2 \\ \tilde Q_1 - \tilde Q_2 \end{array} \right), \label{25}$$ where as before $\rho = 2\lambda r = -2ikr$. The expressions for $\tilde Q_1$ and $\tilde Q_2$ are given by (\[eq:9\]), where $Z\alpha$ should be substituted by $-Z\alpha$: $$\hspace{-7mm}\left\{ \begin{aligned} \tilde Q_1 &= C\cdot\frac{\frac{iZ\alpha m}{k} + \varkappa} {-i\tau-\frac{iZ\alpha\varepsilon}{k}}\cdot{_{1}F_{1}}\left( i\tau + \frac{iZ \alpha \varepsilon}{k}, 2i\tau + 1, \rho \right) + \\ &\hspace{40mm} + D\cdot\frac{\frac{iZ\alpha m}{k} + \varkappa}{i\tau-\frac{iZ\alpha\varepsilon}{k}}\cdot \rho^{-2i\tau}\cdot {_{1}F_{1}}\left(-i\tau + \frac{iZ\alpha\varepsilon}{k}, -2i\tau+1, \rho\right), \\ \tilde Q_2 &= C\cdot{_{1}F_{1}}\left(1+i\tau + \frac{iZ\alpha\varepsilon}{k}, 2i\tau + 1, \rho \right) + D\rho^{-2i\tau}\cdot{_{1}F_{1}}\left( 1-i\tau + \frac{iZ\alpha\varepsilon}{k}, -2i\tau+1, \rho \right). \end{aligned} \right. \label{26}$$ Since we are interested in resonant states, we demand that only terms $\propto\exp[ikr]$ (outgoing waves) survive at $r\to\infty$. For $\varepsilon<m$, $\exp[ikr]$ becomes $\exp[-\lambda r]$, which describes bound states. In $\tilde Q_1$, the coefficient of the $\exp[-ikr]$ term, being damped by an extra $1/r$, does not contribute, and the condition for the terms proportional to $\exp[-ikr]$ to be absent in $\tilde Q_2$ is $$C\cdot\frac{\Gamma(2i\tau)} {\Gamma\left(1+i\tau+\frac{iZ\alpha\varepsilon}{k}\right)} - D\cdot\frac{\Gamma(-2i\tau)} {\Gamma\left(1-i\tau+\frac{iZ\alpha\varepsilon}{k}\right)} = 0. \label{27}$$ Substituting (\[26\]) into (\[25\]) at the limit $r\to 0$, we reproduce (\[24\]) for $$\begin{aligned} \frac{\tilde\eta_\tau}{\tilde\eta_{-\tau}} & = \frac{(-2ik)^{i\tau}}{(-2ik)^{-i\tau}}\cdot \frac{\Gamma(-2i\tau)}{\Gamma(2i\tau)}\cdot \frac{\Gamma\left(1+i\tau+\frac{iZ\alpha\varepsilon} {k}\right)} {\Gamma\left(1-i\tau+\frac{iZ\alpha \varepsilon} {k}\right)}\cdot \frac{Z\alpha\sqrt{m-\varepsilon} -(-i\tau+\varkappa)\sqrt{m+\varepsilon}} {Z\alpha\sqrt{m-\varepsilon} -(i\tau+\varkappa)\sqrt{m+\varepsilon}}. \end{aligned} \label{28}$$ Matching Eqs. (\[28\]) and (\[77\]) yields an equation for the energies of the resonant states. After the substitution of ($\varkappa, \varepsilon$) by ($-\varkappa, -\varepsilon$), it coincides with the similar equation that we obtained in Section \[sec:lower\]. Thus, resonances also arise as solutions of the Dirac equation for positrons, at energies $\varepsilon=\xi-\frac{i}{2}\gamma,~\xi>m,~\gamma>0$[^9]. After making the same substitutions as in Section \[sec:lower\], we get equations that coincide with (\[171\]), (\[181\])). This clears the mystery concerning the resonances that we have found there. Positrons states of negative energies should be interpreted in terms of electrons. At $Z < Z_{\rm cr}$ we just found electron–nucleus bound states — with growing $Z$, the energy of the bound particle moves from $-m$ (at $Z=0$) to $+m$ (see also footnote \[footnote:pos\_bound\_states\]) and, at $Z > Z_{\rm cr}$ it becomes complex and located on the second sheet below the right cut. Equations for the scattering phase $\delta_{\varkappa}$ analogous to (\[eq:phase\]), (\[77\]), (\[eq:CD\]) in Section \[sec:lower\] can be written. They coincide with these equations after changing $\varkappa\to-\varkappa$ and $\varepsilon\to-\varepsilon$. It is therefore not necessary to solve the Dirac equation for positrons as we did in this section. It is enough to note that, after substitution of $\varepsilon$ by $-\varepsilon$, $\varkappa$ by $-\varkappa$, $F$ by $\tilde G$ and $G$ by $\tilde F$, Eq. (\[eq:3\]) becomes (\[21\]) with $V(r)$ converted to $\tilde V(r)$. In this way, the formulas of Section \[sec:upper\] can be directly deduced from the ones of Section \[sec:lower\]. Conclusions {#sec:conclusions} =========== In Sections \[sec:lower\] and \[sec:upper\], the scattering of positrons on a supercritical nucleus was studied. It has the spectacular resonance behavior discovered in [@MRG1:1972; @MRG2:1972; @Kuleshov]. In the present paper, results with an exact dependence on the parameter $m\times R$ have been obtained on both sheets of the complex energy plane in the form convenient for numerical evaluation. However, one can hardly hope to study this phenomenon experimentally: even if a supercritical nucleus can be produced in heavy ions collisions, its life time will be so short that one cannot scatter a positron on it, not to mention the still bigger challenge of making a target with supercritical nuclei. Let us note that since the elastic scattering matrix was found to be unitary (the scattering phase is real) there are no inelastic processes in the positron scattering on supercritical nucleus. More realistic is the hope to detect the emission of positrons from a short-lived supercritical nucleus eventually produced in heavy ions collisions. Indeed we do not agree with the claim made in the abstract of [@Kuleshov] (and in contradiction with [@Zeldovich:1971] in particular) that the spontaneous production of $e^+e^-$ pairs from a supercritical nucleus does not occur. On the contrary, we believe that the resonance found in [@Kuleshov] in the system positron—supercritical nucleus is precisely the signal for pair production. It occurs when, as $Z$ grows, an empty electron level dives into the lower continuum of the Dirac equation. In the absence of the nucleus, this empty state in the lower continuum would just mean the presence of a positron. The presence of the nucleus makes the energy of this state complex, and its lifetime is precisely $1/\gamma$. In this lapse of time, an electron from the sea with the same energy $-\xi$ located far from the nucleus can penetrate in its vicinity. It partially screens the charge of the nucleus and, at the same time, an empty electron state arises in the Dirac sea. This is the positron which gets repulsed to infinity by the nucleus. Let us suppose that solutions of the Dirac equation we get are approximately valid also when an electron screens nuclear potential, being embedded in the lower continuum. It means our solutions for the resonance energy and width are almost valid. It well can be so, since electric charge of one electron is small and it is situated far from nucleus, $r \approx 1/m$. So, the obtained width (imaginary part of energy) is the lifetime of positron in the vicinity of nucleus, which is already surrounded by diving electron. Therefore this is the lifetime of the system of nucleus, electron and positron with respect to positron emission to infinity, so it is an average time of $e^{+}e^{-}$ pair production (in reality two independent pairs are produced because of electron spin degeneracy). The potential barrier which holds the positron in the vicinity of the nucleus is shown in Fig. 2 of [@Zeldovich:1971]; its penetration time is given by the analytical formulas (4.14, 4.15), and the results of numerical calculations are shown in Fig. 13 of the same review paper. We reproduced the curve shown in Fig. 13 from the dependence $\gamma(Z)$ that we obtained in Section \[sec:lower\] for the energy of the Gamov (quasistationary) state. Let us finally mention that we agree with the description of the stable states of a supercritical nucleus made in Section 6 of [@Kuleshov]: empty states in the upper continuum, empty discrete levels, and occupied states in the lower continuum. The levels of the lower continuum that get occupied by electrons after the diving process form the so-called “charged vacuum”; it has charge $-n$, where $n$ is the number of these levels. The $n$ positrons that get emitted compensate for this negative charge. A supercritical nucleus is no longer naked and its electric charge is partially screened by these electrons. Indirect evidence of such a phenomenon is found in graphene physics [@Wang734; @NaturePhysics]. We thank O.V. Kancheli, V.D. Mur, V.A. Novikov, and M.I. Eides for useful discussions. We are grateful to V.M. Shabaev who provided us with references [@Maltsev:2014qna; @Maltsev2017]. S.G. is supported by RFBR under grants 16-32-60115 and 16-32-00241, by the Grant of President of Russian Federation for the leading scientific Schools of Russian Federation, NSh-9022-2016, and by the “Dynasty Foundation”. M.V. is supported by RFBR under grant 16-02-00342. M.V. is grateful to LPTHE, CNRS and Sorbonne Univesité for hospitality and funding during the first steps (projet IDEX PACHA OTP-53897) and the last steps of this work. Functions $Q_1$ and $Q_2$ {#sec:Q1Q2} ========================= Substituting (\[8\]) into the Dirac equations (\[eq:3\]) we get: $$\begin{aligned} &\rho(Q_1^\prime + Q_2^\prime) + (i\tau +\varkappa)(Q_1 +Q_2)-\rho Q_2 + Z\alpha \sqrt{\frac{m-\varepsilon}{m+\varepsilon}} (Q_1 - Q_2) = 0,\\ &\rho(Q_1^\prime - Q_2^\prime) + (i\tau -\varkappa)(Q_1 - Q_2)+\rho Q_2 -Z\alpha \sqrt{\frac{m+\varepsilon}{m-\varepsilon}} (Q_1 + Q_2) = 0, \end{aligned} \label{A1}$$ where a prime means the derivative with respect to $\rho$. The sum and difference of the two equations (\[A1\]) give (compare with Eq. (36.5) from [@BLP]): $$\begin{aligned} &\rho Q_1^\prime + \left(i\tau - \frac{iZ\alpha\varepsilon}{k}\right)Q_1 + \left(\varkappa - \frac{iZ\alpha m}{k}\right) Q_2 = 0,\\ &\rho Q_2^\prime + \left(i\tau - \rho +\frac{iZ\alpha\varepsilon}{k}\right)Q_2 + \left(\varkappa + \frac{iZ\alpha m}{k}\right) Q_1 = 0. \end{aligned} \label{A2}$$ Eliminating $Q_1$ or $Q_2$ gives $$\begin{aligned} &\rho Q_1^{\prime\prime} + (2 i\tau +1 -\rho)Q_1^\prime + \left(\frac{iZ\alpha \varepsilon}{k} - i\tau\right) Q_1 = 0,\\ &\rho Q_2^{\prime\prime} + (2 i\tau +1 - \rho)Q_2^\prime + \left(\frac{iZ\alpha \varepsilon}{k} - 1 - i\tau\right) Q_2=0. \end{aligned} \label{A3}$$ Unlike in the case of a point-like nucleus, we do not demand here that the solutions of (\[A3\]) be regular at $\rho=0$. We accordingly consider linear superpositions of the two independent solutions of the second order differential equations (\[A3\]) with arbitrary coefficients. First let us recall that the general solution of the equation $$zu^{\prime\prime} + (\gamma - z)u^\prime -\alpha u = 0 \label{A4}$$ is: $$u = C_1\cdot{_{1}F_{1}}(\alpha, \gamma, z) + C_2\cdot z^{1-\gamma}{_{1}F_{1}}(\alpha -\gamma +1, 2-\gamma, z) , \label{A5}$$ where $C_1$ and $C_2$ are arbitrary coefficients while the $_1F_1$ are the Kummer confluent hypergeometric functions. Thus for the solutions of (\[A3\]) we obtain: $$\hspace{-3mm} \begin{aligned} Q_1 & = A\cdot{_{1}F_{1}}\left(i\tau - \frac{iZ\alpha\varepsilon}{k}, 2i\tau + 1, \rho \right)+ B\cdot\rho^{-2i\tau} {_{1}F_{1}}\left( -i\tau -\frac{iZ\alpha\varepsilon}{k}, -2i\tau+1, \rho \right),\\ Q_2 & = C\cdot{_{1}F_{1}}\left(1 +i\tau - \frac{iZ\alpha \varepsilon}{k}, 2i\tau + 1, \rho \right) + D\cdot\rho^{-2i\tau}{_{1}F_{1}}\left(1 -i\tau -\frac{iZ\alpha\varepsilon}{k}, -2i\tau+1, \rho \right), \end{aligned} \label{A6}$$ where $A$, $B$, $C$, and $D$ are arbitrary coefficients. At small $z$, $_1F_1=1+\mathcal{O}(z)$. Substituting the expansions of (\[A6\]) at small $\rho$ into the first equation in (\[A2\]) determines $A$ and $B$, respectively, in terms of $C$ and $D$: $$\begin{aligned} &\left( i\tau - \frac{iZ\alpha\varepsilon}{k}\right) A + \left(\varkappa - \frac{iZ\alpha m}{k}\right) C = 0, \label{A7}\\ &\left( -i\tau - \frac{iZ\alpha\varepsilon}{k}\right) B + \left(\varkappa - \frac{iZ\alpha m}{k}\right) D = 0. \label{A8}\end{aligned}$$ Plugging then $A$ and $B$ obtained from (\[A7\]) and (\[A8\]) into (\[A6\]) yields (\[eq:9\]). The scattering phase according to [@Kuleshov] {#sec:real_phase} ============================================= Considering real $\varepsilon<-m$ below the left cut we can rewrite the expression for the scattering phase in a more compact form. Let us introduce the following notations equivalent to those used in [@Kuleshov]: $$\begin{aligned} \label{eq:kuleshov_notations} \exp(i\varphi) &= e^{2i\theta}\frac{(2k)^{-i\tau}\Gamma(2i\tau)} {(2k)^{i\tau}\Gamma(-2i\tau)},\\ a &= \frac{Z\alpha\sqrt{m-\varepsilon} +(-i\tau+\varkappa)\sqrt{m+\varepsilon}} {\Gamma(1-i\tau - \frac{iZ\alpha\varepsilon}{k})},\\ b &= \frac{Z\alpha\sqrt{m-\varepsilon} -(-i\tau+\varkappa)\sqrt{m+\varepsilon}} {\Gamma(1-i\tau + \frac{iZ\alpha\varepsilon}{k})}.\end{aligned}$$ With these notations the approximate ratio $C/D$ defined by (\[eq:CD\]) can be written in the following way: $$\begin{aligned} \label{eq:CD_Kuleshov_numerator} C/D &=e^{i\varphi-\pi\tau}\cdot\frac{a^{*}}{b}\cdot \frac{\Gamma\left(-2i\tau\right)} {\Gamma\left(2i\tau\right)}\cdot \frac{\Gamma\left(i\tau+\frac{iZ\alpha\varepsilon}{k}\right)} {\Gamma\left(-i\tau+\frac{iZ\alpha\varepsilon}{k}\right)}=\\ \label{eq:CD_Kuleshov_denominator} &=e^{i\varphi-\pi\tau}\cdot\frac{b^{*}}{a}\cdot \frac{\Gamma\left(-2i\tau\right)} {\Gamma\left(2i\tau\right)}\cdot \frac{\Gamma\left(1+i\tau-\frac{iZ\alpha\varepsilon}{k}\right)} {\Gamma\left(1-i\tau-\frac{iZ\alpha\varepsilon}{k}\right)},\end{aligned}$$ where we used $$\frac{\left(-i\right)^{-i\tau}}{\left(-i\right)^{i\tau}}=e^{-\pi\tau},$$ and $$\begin{aligned} \left(-i\tau-\frac{iZ\alpha\varepsilon}{k}\right) & \left(Z\alpha\sqrt{m-\varepsilon}+ \left(i\tau-\varkappa\right)\sqrt{m+\varepsilon}\right)=\nonumber\\ \nonumber &=-i\tau\left(Z\alpha\sqrt{m-\varepsilon} +\left(i\tau-\varkappa\right)\sqrt{m+\varepsilon}\right) -iZ\alpha\varepsilon\left(\frac{Z\alpha}{i\sqrt{m+\varepsilon}} +\frac{i\tau-\varkappa}{i\sqrt{m-\varepsilon}}\right)=\\ \nonumber &=\frac{-i\tau\left(i\tau-\varkappa\right)\left(m+\varepsilon\right) -\left(Z\alpha\right)^{2}\varepsilon}{\sqrt{m+\varepsilon}} +\frac{-i\tau\left(Z\alpha\right)\left(m-\varepsilon\right) -Z\alpha\varepsilon\left(i\tau-\varkappa\right)}{\sqrt{m-\varepsilon}}=\\ \nonumber &=\frac{\left(Z\alpha\right)^{2}m +\varkappa\left(i\tau-\varkappa\right)\left(m+\varepsilon\right)} {\sqrt{m+\varepsilon}} +\frac{-\left(i\tau-\varkappa\right)Z\alpha m-\varkappa Z\alpha\left(m-\varepsilon\right)}{\sqrt{m-\varepsilon}}= \nonumber \\ &=Z\alpha m\left(\frac{Z\alpha}{\sqrt{m+\varepsilon}}+ \frac{-i\tau+\varkappa}{\sqrt{m-\varepsilon}}\right) -\varkappa\left(Z\alpha\sqrt{m-\varepsilon} +\left(-i\tau+\varkappa\right)\sqrt{m+\varepsilon}\right)=\nonumber\\ &=\left(Z\alpha\sqrt{m-\varepsilon}+ \left(-i\tau+\varkappa\right)\sqrt{m+\varepsilon}\right)\left(\frac{iZ\alpha m}{k}-\varkappa\right),\end{aligned}$$ where we used the relation $$\begin{aligned} \left(-i\tau-\varkappa\right)\left(i\tau-\varkappa\right)= \tau^{2}+\varkappa^{2}=\left(Z\alpha\right)^{2}.\end{aligned}$$ Then, the scattering phase $\delta_{\varkappa}$ can be written as follows (by substituting (\[eq:CD\_Kuleshov\_numerator\]) and (\[eq:CD\_Kuleshov\_denominator\]) into the numerator and the denominator of (\[eq:phase\]) respectively): $$e^{2i\delta_{\varkappa}} = -\frac{{\rm exp}\left(\frac{\pi\tau}{2}+\frac{i\varphi}{2}\right)a^* -{\rm exp}\left(-\frac{\pi\tau}{2}-\frac{i\varphi}{2}\right)b} {{\rm exp}\left(\frac{\pi\tau}{2}-\frac{i\varphi}{2}\right)a -{\rm exp}\left(-\frac{\pi\tau}{2} +\frac{i\varphi}{2}\right)b^*}. \label{13}$$ In eq.(22) of [@Kuleshov] the phase $\delta_{\varkappa}$ is expressed through the ratio $f^*/f$, where $f$ is the Jost function. Our result differs from eq.(23) of [@Kuleshov] by the substitution $\varphi\to\varphi/2$ (it seems that there is a misprint in [@Kuleshov]). Qualitative explanation of the resonance phenomena in the $e^+ N^+$ system {#sec:eff_potential} ========================================================================== The effective potential for an electron in the field of a supercritical nucleus is derived in [@Zeldovich:1971] from the Dirac equation, for $\varepsilon\approx-m$. It is attractive at short distances, repulsive at large distances, with a Coulomb barrier in between. We derive below, in a similar way, the effective potential for a positron in the field of a similar nucleus, in the vicinity of $\varepsilon=+m$. As already noticed at the beginning of Section \[sec:upper\], the Dirac equation (\[eq:3\]) for electrons becomes (\[21\]) for positrons after the following substitutions: $$\varkappa\to-\varkappa, \;\; \varepsilon\to-\varepsilon, \;\; F \to \tilde G, \;\; G \to \tilde F, \;\; V(r) \to \tilde V(r) = -V(r). \label{C1}$$ To proceed like in [@Zeldovich:1971], we deduce the second order differential equation satisfied by $\tilde G$ from (\[21\]), which is $$\label{eq:G_shred} \tilde G'' +\frac{\tilde V'}{\varepsilon-m-\tilde V}\left(\tilde G'-\frac{\varkappa}{r}\tilde G\right) +\left(\left(\varepsilon-\tilde V\right)^{2}-m^{2} +\frac{\varkappa\left(1-\varkappa\right)}{r^{2}}\right)\tilde G=0,$$ in which “$~'~$” means here derivation with respect to $r$. In order to transform this equation into a Schrödinger-like equation, the following change of variables must be operated $$\label{eq:G_to_chi} \tilde G=\chi\sqrt{m-\varepsilon+\tilde V}.$$ Thus, we get: $$\label{eq:chi_shred} \chi''+ k^{2}\chi=0,$$ where $k^{2}=2m\left(E-U\right)$, $E=\frac{\varepsilon^{2}-m^{2}}{2m}$. The effective potential is seen to be made of two terms: $U=U_{1}+U_{2}$, where: $$\label{eq:U1} U_{1}=\frac{\varepsilon}{m}\tilde V-\frac{1}{2m}\tilde V^{2} -\frac{\varkappa\left(1-\varkappa\right)}{2mr^{2}},$$ and $$\label{eq:U2} U_{2}=\frac{\tilde V''}{4m\left(\varepsilon-m-\tilde V\right)} +\frac{3}{8m}\frac{\left(\tilde V'\right)^{2}}{\left(\varepsilon-m-\tilde V\right)^{2}} +\frac{\varkappa\tilde V'}{2mr\left(\varepsilon-m-\tilde V\right)}.$$ It coincides with the equation obtained in [@Zeldovich:1971] after the substitution $\varepsilon\to-\varepsilon$, $\varkappa\to-\varkappa$ and $\tilde V\to-V$. We are interested in positrons with $\varepsilon\approx m$. At large distances the first term in $U_1$ dominates, and describes the repulsion of the positron by the nucleus. For the ground state $\varkappa=1$ the centrifugal term in $U_1$ vanishes. Finally, for $\varepsilon=m$ and $\varkappa=1$, we get from (\[eq:U1\]) and (\[eq:U2\]) $$\label{eq:eff_potential} U = \frac{Z\alpha}{r} + \frac{3-4(Z\alpha)^2}{8mr^{2}}.$$ At short distances the terms $\propto 1/r^2$ dominates and, for a supercritical nucleus, they lead to attraction, while the Coulomb term dominates at $r\geq1/m$. This attractive force explains the existence of resonances in the $e^{+}N^{+}$ system while bound state cannot exist due to the narrowness of the well. Let us note that the fall to the center occurs only for $Z\alpha>1$ when the coefficient in front of the term $\propto -1/r^2$ becomes larger than $1/8m$ (see [@LL3], eq. (35.10)). We are grateful to V.A. Novikov who brought our attention to this feature. In the problem under consideration the finite nucleus size prevents the fall to the center. [^1]: The reader can be convinced as regards this interpretation as follows. In the present simple formalism, which only uses the Dirac equation, the energy of a bare positron, which is obtained by simply taking the limit $Z\to0$, is found to be $-m$. Since the “production” of such a particle costs at least the energy $+m$, the result that is obtained can only be interpreted in term of an electron with energy $-(-m)=+m$. This is what we mean by the statement that “there is no more information”. A more satisfying description of positrons can only be achieved in the framework of Quantum Field Theory, where creating (annihilating) an electron and annihilating (creating) a positron both occur in the expansion of the field operator $\psi$ in terms of creation and annihilation operators.\[footnote:pos\_bound\_states\] [^2]: Any solution with a negative index of the Bessel function is not normalizable, so it should be discarded. [^3]: The procedure used in [@Kuleshov] amounts to stating that, below the left cut, $\lambda = -i\sqrt{(m-\varepsilon)(-m-\varepsilon)}$. So doing, $\sqrt{-m-\varepsilon}$ is defined with the same cut $(-\infty;-m)$ as $\sqrt{m+\varepsilon}$, with positive values below the cut. With such a definition, $-i\sqrt{-m-\varepsilon}={\rm sqrt}\left(m+\varepsilon\right)$ everywhere on the physical sheet, not only below the left cut. There is no need to rewrite formulas in this way since, when numerical outputs are needed, we should return to the original definition (\[eq:lambda\_sheets\]). Let us note that on the first sheet formulas (17) and (26) from [@Kuleshov] are exactly the same. [^4]: Let us note that changing the signs of both square roots is still permitted since it leads to changing the sign of the full wave function. [^5]: Unlike in [@Kuleshov] we did not feel necessary to use Tricomi functions. [^6]: Due to the unitarity of the $S$ matrix, there is a zero of $e^{2i\delta_{\varkappa}}$ at $\varepsilon=-\xi-\frac{i}{2}\gamma$ that is symmetric to the pole $\varepsilon=-\xi+\frac{i}{2}\gamma$ with respect to the real axis. It corresponds to incoming waves instead of the outgoing waves that we selected. [^7]: It corresponds to ${\rm Re}[k]>0$, ${\rm Im}[k]=\frac{\displaystyle{\rm Re}[\varepsilon]{\rm Im}[\varepsilon]} {\displaystyle{\rm Re}[k]}<0$. In [@MRG1:1972; @MRG2:1972; @Zeldovich:1971] the resonance in positron scattering on a supercritical nucleus was discussed. [^8]: This term is proportional to the sum $\left(\sqrt{\varkappa^{2}-\left(Z\alpha\right)^{2}}-\frac{\displaystyle Z\alpha\varepsilon} {\displaystyle\sqrt{m^{2}-\varepsilon^{2}}}\right)+\left(\varkappa+ \frac{\displaystyle Z\alpha m}{\displaystyle\sqrt{m^{2}-\varepsilon^{2}}}\right)$. It is easy to check that, when the first term vanishes, so does the second. Their sum increasing monotonically when $\varepsilon$ increases can vanish only once. [^9]: One may wonder how a positron, being repelled from the positively charged nucleus, can form a quasistationary resonance state with it. This unusual phenomena is explained in Appendix \[sec:eff\_potential\].

--- abstract: 'The local structure of  in the incommensurate charge density wave (IC-CDW) state has been obtained using atomic pair distribution function (PDF) analysis of x-ray diffraction data. Local atomic distortions in the Te-nets due to the CDW are larger than observed crystallographically, resulting in distinct short and long Te-Te bonds. Observation of different distortion amplitudes in the local and average structures are explained by the discommensurated nature of the CDW since the PDF is sensitive to the local displacements within the commensurate regions whereas the crystallographic result averages over many discommensurated domains. The result is supported by STM data. This is the first quantitative local structural study within the commensurate domains in an IC-CDW system.' author: - 'H. J. Kim$^1$, C. D. Malliakas$^2$, A. Tomic$^1$, S. H. Tessmer$^1$, M. G. Kanatzidis$^2$ & S. J. L. Billinge$^{1*}$' title: 'Local atomic structure and discommensurations in the charge density wave of [CeTe$_{\boldmath{3}}$]{} ' --- Incommensurate charge density waves (IC-CDWs) are a fundamental property of low-dimensional metals [@grune;b;dwis94] and also underly the novel properties of correlated electron oxides such as cuprates in the pseudo-gap state [@versh;s04; @hoffm;s02; @hanag;n04], and manganites at high doping [@loudo;prl05]. Knowing the nature of local atomic displacements (Peierls distortions) in the IC-CDWs is crucial to understand such factors as electron-lattice coupling [@milwa;n05], yet this information is difficult to obtain quantitatively. Here we solve this problem by taking the novel approach of using a local structural method, the atomic pair distribution function (PDF) technique [@egami;b;utbp03], to determine the local atomic displacements with high precision in the system . IC-CDWs, and the underlying atomic displacements, can be uniform incommensurate modulations or locally commensurate waves separated by narrow domain walls, known as discommensurations [@mcmil;prb76], where the phase of the wave changes rapidly. Here we show that the IC-CDW in  is discommensurated and obtain for the first time the quantitative local atomic displacements within the commensurate domains. In the case of *incommensurate* CDWs, superlattice peaks observed crystallographically yield the average distorted structure. Except in the cases where the domains are periodically arranged, giving rise to satellite peaks [@monct;prb75], it is not possible to determine whether the underlying CDW is truly incommensurate or forms a discommensurated structure with commensurate regions separated by domain walls [@mcmil;prb76]. A number of techniques have been successful at differentiating between the truly incommensurate and discommensurated cases. The earliest verification of a discommensurated phase came from photoemission spectroscopy evidence that the Ta 4$f$ states in 1$T$-TaS$_2$ had the same splitting in the commensurate and nearly-commensurate states [@hughe;cop76]. Photoemission is a local probe and found distinct Ta environments rather than a broad continuum expected from a purely incommensurate state. Similarly, another local probe, nuclear magnetic resonance (NMR), found distinct Knight-shifts for three Se sites in the incommensurate state of 2$H$-TaSe$_2$, similar to the commensurate phase [@suits;prl80; @suits;prb81]. High resolution atomic imaging methods have also contributed to this debate. The strain fields due to the domain walls were observed in dark field transmission electron microscopy (TEM) measurements [@chen;prl81]. Interestingly, atomic resolution images in real-space have difficulty in resolving discommensurated domains [@gibso;prl83; @ishig;prb91; @kuwab;pssa86; @steed;u86]. However, Fourier analysis of scanning tunneling microscopy (STM) images can be a reliable measure, as discussed in detail by Thomson $et~al.$ [@thoms;prb94]. As in the case of the NMR and photoemission studies, the PDF approach described here makes use of the fact that the local structure deviates from the average in the discommensurated case. By comparing atomic displacements determined from the PDF with those determined crystallographically we establish the presence of commensurate domains, but crucially, also obtain quantitatively the atomic structure within these domains. This novel approach is here applied to the incommensurate phase of CeTe$_{3}$. In its undistorted form, CeTe$_{3}$ takes the NdTe$_{3}$ structure type with space group $Cmcm$ [@lin;ic65]. It forms a layered structure with ionic \[CeTe\] layers sandwiched between two Te layers. These sandwich layers stack together with weak van der Waals forces to form the 3-dimensional structure. Te ions in the Te layers form a square-net with 3.1 Å Te-Te bonds. The structure is shown in Fig. \[fig;stm\](a). ![\[fig;stm\] (a) The crystal structure of CeTe$_{3}$ with the square Te net that supports the CDW highlighted. The reduced unit cell on the Te net is indicated by the red dashed box (****). (b) A representative STM image from the square Te net showing the CDW. On the expanded image, the network of Te bonds is superimposed. (c) The Fourier transform of the STM data. To achieve a high signal-to-noise ratio, the transform represents the average of 24 images (each image was $27~{\times}~27$ nm). The square Te net gives rise to four distinct peaks (L), with peaks related to the CDW oriented at $45^\circ$, as indicated by the arrow. The fundamental CDW peak (corresponding to a wavelength of $\approx 15$ Å) and the $\lambda/2$ harmonic are labeled 1 and 3, respectively. Peaks 2 and 4 are in close proximity to 3, implying a characteristic discommensuration length of 38 Å, as described in the text. Peak 5 corresponds to the diagonal of the Te net. This component of the Te net may be enhanced due to the underlying crystal structure; the CDW-lattice interaction may also enhance this peak.](fig1_cete3_PRL2006.ps){width="2.7in"} The electronic bands crossing the Fermi level are Te $p$-bands from the 2D square nets [@dimas;prb95] and the CDW forms in these metallic layers. In the CDW state an incommensurate superlattice is observed [@malli;jacs05], with a wavevector characteristic of a strong Fermi-surface nesting vector in the electronic structure [@gweon;prl98; @broue;prl04; @komod;prb04; @laver;prb05]. This is a surprisingly stable and simple single-$q$ IC-CDW state in an easily cleavable 2D square net making the RETe$_3$ (RE=Rare Earth) systems ideal for studying the IC-CDW state [@dimas;prb95]. The atomic distortions giving rise to the superlattice have been solved crystallographically from single crystal x-ray diffraction data [@malli;jacs05]. The incommensurate wavelength of the distortion is close to $25a/7$, where $a$ is the lattice parameter of the undistorted phase. The distorted structure is in the $Ama2$ spacegroup [@malli;jacs05]. From the crystallography alone it is not possible to determine whether this distorted structure is truly incommensurate or whether discommensurations form between short-range commensurate domains. The X-ray PDF experiment was conducted on a fine powder of prepared as described in Ref. [@malli;jacs05]. CeTe$_{3}$ powder was loosely packed in a flat plate with thickness of 1.0 mm sealed with kapton tape. Care must be taken when grinding this material or turbostratic disorder is introduced, significantly modifying the stacking of the layers. Diffraction data were collected at 300 K using the rapid acquisition pair distribution function (RA-PDF) technique [@chupa;jac03]. Standard corrections [@chupa;jac03; @egami;b;utbp03] were made using the program PDFgetX2 [@qiu;jac04] to obtain the properly normalized total scattering function, $S(Q)$, [@egami;b;utbp03] which was truncated at $Q_{max}$ of 25 Å$^{-1}$ before Fourier transforming to obtain the PDF, $G(r)= \frac{2}{\pi}\int_{0}^{\infty} Q [S(Q)-1] \sin (Qr)\> dQ$. Structural models are fit to the data using the program PDFFIT [@proff;jac99]. The PDF of , measured at room temperature, is shown in Fig. \[fig;first PDF peak\](a). ![(a) The PDF of CeTe$_{3}$ at room temperature. In (b)-(d) the first peak of the experimental PDF of CeTe$_{3}$ (****) is plotted with the calculated PDF (****) from various models: (b) the undistorted crystal structure model ($Cmcm$), (c) the distorted crystallographic model and (d) the local structural model determined by the PDF refinement over the range $2.5 < r < 6.37$ Å. The difference between the experimental and calculated PDFs (****) is plotted below the data in each panel. The shoulder due to the Peierls distortions in the Te-nets is indicated by an arrow. []{data-label="fig;first PDF peak"}](fig2_cete3_PRL2006.ps){width="2.4in"} The PDF gives the probability of finding an atom at a distance-*r* away from another atom. The nearest neighbor peak around 3.1 Å comes from the Te-Te bond in the nets and the Ce-Te bond in the intergrowth layers. This is shown on an expanded scale in Figs. \[fig;first PDF peak\](b)-(d). A shoulder is evident on the low-$r$ side of the peak. This feature is robust; it is much larger than the statistical and systematic errors and is reproduced in measurements of isostructural compounds NdTe and PrTe. Fig. \[fig;first PDF peak\](b) shows the fit to this peak of the undistorted crystal structure model ($Cmcm$), where only symmetry allowed atomic positions and isotropic thermal factors were refined. The result clearly does not explain this shoulder which originates from short Te-Te bonds in the Te-net. Surprisingly however, the PDF calculated from the *distorted* structure determined crystallographically [@malli;jacs05] also does not explain this shoulder well. In this case the atoms were fixed at the crystallographically determined positions and isotropic thermal factors were refined. This resulted in a better fit to the first peak (Fig. \[fig;first PDF peak\](c)); however, the fit is not ideal and required a large value of $U_{iso}$ for the Te atoms in the nets ($U_{iso}=0.0152(2)$ Å$^{2}$). The value was two times larger than $U_{iso}$ of the Ce and Te atoms in the ionic \[CeTe\] layers (0.0077(2) Å$^{2}$ and 0.0080(2) Å$^{2}$ for $U_{iso}$ of Ce and Te atoms, respectively). The large fluctuation in the difference curve in Fig. \[fig;first PDF peak\](b) arises because the real distribution of Te-Te bond-lengths in the data is broader than in the undistorted model. This fluctuation in the difference curve is smaller in Fig. \[fig;first PDF peak\](c) because the distortions of the Te net in the $Ama2$ crystallographic model result in a broader Te-Te bond-length distribution. However, clearly the distorted-crystallographic model still has a Te-Te bond-length distribution that is narrower than in the data. We therefore refined the Te-net distortions directly in the PDF by allowing the atomic positions in the model to vary. The model was constrained to have the $Ama2$ symmetry and the same unit cell was used as in the distorted crystallographic model. The refinement result for $2.5 < r < 6.37$ Å is shown in Fig. \[fig;first PDF peak\](d). As well as giving a significantly better fit to the low-$r$ region of the PDF, this refinement resulted in much smaller and more physical thermal factors on the planar Te ions. The model of the local structure refined from the PDF gives a broader range of Te-Te bond lengths (from 2.83 Å to 3.36 Å) than the crystallographic distorted model (from 2.94 Å to 3.26 Å). It is also interesting to see the shape of the bond-length *distributions* for the Te-Te bonds in the Te-nets from these two models. These are shown in Fig. \[fig;bond distributions of Te-nets\]. ![Bond-length distributions in the Te-nets refined from the PDF over various $r$-ranges (****) (a) $r_{max}$=6.37 [Å]{} (b) $r_{max}$=6.37 [Å]{} (c) $r_{max}$=14.5 [Å]{} (d) $r_{max}$=27.1 [Å]{}. The $r_{min}$ value was fixed to 2.5 [Å]{} for all the cases. The bond distribution from the distorted crystallographic model (****) is plotted in (a), (c), and (d) for comparison. In (b) the bond distribution of the local structural model ($r_{max}$=6.37 [Å]{}) is fit with two Gaussians. The fit is shown as a red line (****) and the two Gaussian sub-components in green (****). []{data-label="fig;bond distributions of Te-nets"}](fig3_cete3_PRL2006.ps){width="2.4in"} The blue solid line shows the bond-length distribution refined from the PDF and the red line is the distribution from the crystallographic model [@malli;jacs05]. For direct comparison the distributions are plotted using the same thermal broadening of 0.007 Å$^{2}$. The distorted-crystallographic model has broad, but symmetric and Gaussian bond-length distribution coming from the continuous distribution of Te-Te bond lengths in the *average* IC-CDW. On the other hand, the local structure refinement ($r_{max}=6.37$ Å) yields a bond-length distribution that is clearly bimodal and is separated into distinct “short" and “long" Te-Te distances. This is emphasized in Fig. \[fig;bond distributions of Te-nets\](b) where we show a fit of two, well separated, Gaussian curves to the PDF-refined bond-length distribution. This behavior is characteristic of oligomerization with Te forming bonded and non-bonded interactions with its neighbors in the net [@patsc;pccp02] that would be expected in a commensurate structure. Since we know that the modulation is incommensurate on average, this is strong evidence that the structure consists of commensurate domains separated by discommensurations. As $r_{max}$ in the PDF-refinements is increased the PDF refined distribution crosses over towards the crystallographic result and by $r_{max}=27.1$ Å, resembles it rather closely (Fig. \[fig;bond distributions of Te-nets\](d)). We have applied STM on the exposed Te net of a cleaved single crystal of , grown according to the method described in Ref. [@malli;jacs05]. Measurements were done at 300 K in the constant current mode of the STM. Data were acquired with a bias voltage of 100 mV and with a tunneling current of 0.6 nA. Fig. \[fig;stm\](b) shows a representative atomic resolution image with the CDW modulation clearly visible oriented at 45$^\circ$ to the net. To investigate the images for discommensurations, we examine the corresponding two-dimensional Fourier transform, shown in Fig. \[fig;stm\](c). As indicated by the labels, in addition to the fundamental CDW peak (1), four more peaks lie along the CDW direction (2-5). Although the transforms of real space images resemble diffraction data, symmetry requirements intrinsic to diffraction data do not apply. As demonstrated by Thomson and co-workers, the Fourier transforms of STM images that exhibit true discommensurations always have extra peaks in proximity to the fundamental CDW peak [@thoms;prb94]. This arises from the fact that a discommensurate CDW can be expressed as the product of a uniformly incommensurate CDW and a modulation envelope [@thoms;prb94]. The wavelengths of the envelope are given by the differences in the wave vectors of closely spaced peaks. The longest such wavelength in our images is 38 Å, corresponding to peaks 2-3 and 3-4 in the Fourier transform, indicating that a discommensuration separation of this length-scale exists. This is consistent with the refined PDF behavior which crosses over from the local to the average behavior for a refinement range of around 27 Å(Fig. \[fig;bond distributions of Te-nets\]), which would be expected to occur at around, or a little above, the *radius* of the local domains. As well as the bond-length distributions, the local and average structure refinements allow us to study the patterns of atomic displacents due to the IC-CDW. ![Te-Te bond length deviation from the average value as refined crystallographically (a) and from the PDF (b). The deviation $r_{bond}-r_0$ is defined such that $r_{i}$ is the Te-Te bond length of the $i^{th}$ bond (bond-index $i$) in the unit cell and $r_0=3.1$ Å. (c) Schematic of the arrangements of “short" bonds within the unit-cell highlighting the formation of oligomers. Short bonds are defined as those lying within $\pm$ $\sigma$ of the center of the first Gaussian of the bimodal distribution in Fig. \[fig;bond distributions of Te-nets\] (b), indicated as a yellow band in the Figure. The Gray markers represent Te atoms. Red (blue) lines and markers are bonds lying in the top (bottom) row of the unit cell. The CDW is out of phase between these two rows of bonds.[]{data-label="fig;spatial bond length distribution"}](fig4_cete3_PRL2006.ps){width="2.4in"} The average structure refinement [@malli;jacs05] results in an almost perfectly sinusoidal pattern of bond-lengths, with the wavelength of the CDW (Fig. \[fig;spatial bond length distribution\](a)), clearly identifying these as Peierls distortions. The local structural model was refined in the same unit cell and space-group, but results in a much more square-wave like distribution, consistent with the distinct short and long Te-Te distances described above (Fig. \[fig;spatial bond length distribution\](b)). Fig. \[fig;spatial bond length distribution\](c) shows the pattern of bonded Te-Te atoms that results when the short distances determined from the PDF data are plotted in the unit cell. In this way, the Peierls distortions due to the IC-CDW can be thought of as forming oligomers in the Te net. In this picture, the discommensurations occur when the pattern of oligomers has defects. This is a common picture in the chemistry literature [@patsc;jacs01; @malli;jacs05], though we note that this picture is not supported by the crystallographic results shown in (Fig. \[fig;spatial bond length distribution\](a)) and needed the application of a local structural method to show that it has a physical reality beyond its heuristic value. The refined parameters of the low-$r_{max}$ PDF refinements yield quantitatively the atomic displacements within the commensurate domains. This is the first demonstration of the use of the PDF to obtain quantitatively the atomic displacements (Peierls distortion) within the commensurate domains of a discommensurated IC-CDW. This opens the way to a quantitative first principles calculations and a better microscopic understanding of the IC-CDW state. We gratefully acknowledge P. M. Duxbury, S. D. Mahanti, and D. I. Bilc for discussions and D. 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--- abstract: 'In modified gravity the propagation of gravitational waves (GWs) is in general different from that in general relativity. As a result, the luminosity distance for GWs can differ from that for electromagnetic signals, and is affected both by the dark energy equation of state $\wde(z)$ and by a function $\delta(z)$ describing modified propagation. We show that the effect of modified propagation in general dominates over the effect of the dark energy equation of state, making it easier to distinguish a modified gravity model from $\Lambda$CDM. We illustrate this using a nonlocal modification of gravity that has been shown to fit remarkably well CMB, SNe, BAO and structure formation data, and we discuss the prospects for distinguishing nonlocal gravity from $\Lambda$CDM with the Einstein Telescope. We find that, depending on the exact sensitivity, a few tens of standard sirens with measured redshift at $z\sim 0.4$, or a few hundreds at $1\,\lsim\, z\, \lsim\, 2$, could suffice.' author: - Enis Belgacem - Yves Dirian - Stefano Foffa - Michele Maggiore bibliography: - 'myrefs\_massive.bib' title: | The gravitational-wave luminosity distance\ in modified gravity theories --- Introduction ============ The observation of the GWs from the neutron star binary coalescence GW170817 [@TheLIGOScientific:2017qsa] and of the associated $\gamma$-ray burst GRB 170817A [@Goldstein:2017mmi; @Savchenko:2017ffs; @Monitor:2017mdv] has marked the opening of the era of multi-messenger astronomy. In the near future more events of this type are expected, while, on a time-scale of 1-2 decades, the space interferometer LISA [@Audley:2017drz] and a third-generation ground-based interferometer such as the Einstein Telescope (ET) [@Sathyaprakash:2012jk] could extend these observations to large redshifts. One of the most interesting targets of third-generation detectors is the measurement of the luminosity distance with standard sirens [@Schutz:1986gp; @Dalal:2006qt; @MacLeod:2007jd; @Nissanke:2009kt; @Cutler:2009qv; @Sathyaprakash:2009xt; @Zhao:2010sz; @DelPozzo:2011yh; @Nishizawa:2011eq; @Taylor:2012db; @Camera:2013xfa; @Tamanini:2016zlh; @Caprini:2016qxs; @Cai:2016sby]. Currently, all the studies on the subject have been performed using the standard expression of the luminosity distance in a theory with a dark energy (DE) density $\rde(z)$, \[dLem\] d\_L(z)=\_0\^z , where \[E(z)\] E(z)= , and, as usual, $\rho_0=3H_0^2/(8\pi G)$ and $\ora$ and $\oma$ are the radiation and matter density fractions, respectively. The evolution of the DE density is determined by its equation of state (EoS) function $\wde(z)$ through the conservation equation \_[DE]{}+3H(1+)=0 . Then, all works on cosmological applications of standard sirens either choose a simple phenomenological parametrization of $\wde(z)$, such as the $(w_0,w_a)$ parametrization $w_{\rm DE}(a)= w_0+(1-a) w_a$ [@Chevallier:2000qy; @Linder:2002et] and provide forecasts on the accuracy to which $(w_0,w_a)$ can be measured, or develop methods for attempting a model-independent reconstruction of the function $\wde(z)$. The most natural motivation for a non-trivial dark energy EoS is the assumption that gravity is modified at cosmological scales. Here we point out, through the study of an explicit model, that in a generic modified gravity theory is not necessarily the correct luminosity distance for GWs (see also [@Deffayet:2007kf; @Saltas:2014dha; @Lombriser:2015sxa; @Nishizawa:2017nef; @Arai:2017hxj; @Amendola:2017ovw]), and we further show that the difference between the GW luminosity distance $d_L^{\,\rm gw}$ and the standard electromagnetic luminosity distance $d_L^{\,\rm em}$ gives an effect that can be significantly larger than that due to a non-trivial dark energy EoS. Tensor perturbations in modified gravity ======================================== Let us first recall that, in GR, the free propagation of tensor perturbations in a Friedmann-Robertson-Walker (FRW) background is described by \[4eqtensorsect\] ”\_A+2[H]{}’\_A+k\^2\_A=0 , where $\tilde{h}_A(\eta, \vk)$ are the Fourier modes of the GW amplitude, $A=+,\times$ labels the two polarizations, $\eta$ denotes conformal time, the prime denotes $\pa_{\eta}$, and ${\cal H}=a'/a$. Introducing a field $\tilde{\chi}_A(\eta, \vk)$ from \[4defhchiproofs\] \_A(, )= \_A(, ) , becomes ”\_A+$k^2-a''/a$ \_A=0 . Both in matter dominance and in the recent DE dominated epoch $a''/a\sim 1/\eta^2$. For sub-horizon modes $k\eta\gg 1$, and therefore $a''/a$ can be neglected compared to $k^2$. For GWs observed at ground- or space-based interferometers this holds to huge accuracy: for instance, for a GW frequency $f\sim 10^2$ Hz, (k)\^[-2]{}\~(500 [km]{}/H\_0\^[-1]{})\^2\~10\^[-41]{} . Then, we can write simply ”\_A+k\^2 \_A=0 . This shows that the dispersion relation of tensor perturbations is $\omega=k$, i.e. GWs propagate at the speed of light (that we have set to one). On the other hand, the factor $1/a$ in tells us how the GW amplitude decreases in the propagation over cosmological distances from the source to the observer and, for inspiraling binaries, leads to the standard dependence of the GW amplitude $\tilde{h}_A(\eta, \vk)\propto 1/d_L(z)$; see e.g. Section 4.1.4 of [@Maggiore:1900zz]. In a generic modified gravity theory both the coefficient of the $k^2$ term and that of the $2{\cal H}$ term in (as well as the source term, that we have not written explicitly) can be different. This has already been observed in various explicit models. In particular, in the DGP model [@Dvali:2000hr] (which, in the self-accelerated branch, is by now ruled out by the presence of instabilities at the level of cosmological perturbations [@Luty:2003vm; @Nicolis:2004qq; @Gorbunov:2005zk; @Charmousis:2006pn]), at cosmological scales gravity leaks into extra dimensions, and this affects the $1/d_L(z)$ behavior of a gravitational signal [@Deffayet:2007kf]. The same effect has been found for Einstein-Aether models and for scalar-tensor theories of the Horndeski class [@Saltas:2014dha; @Lombriser:2015sxa; @Arai:2017hxj; @Amendola:2017ovw]. A modified propagation equation for tensor modes can be included in the general effective field theory approach to dark energy developed in [@Gleyzes:2014rba], and the relevance of this effect for standard sirens has already been pointed out, in a scalar-tensor theory of the Horndeski class, in [@Lombriser:2015sxa].[^1] A change in the coefficient of the $k^2$ term in gives a propagation speed of GWs different from the speed of light. The GW170817/GRB 170817A event now puts a very stringent limit on such a modification, at the level $|c_{\rm gw}-c|/c< O(10^{-15})$ [@Monitor:2017mdv], which rules out a large class of scalar-tensor and vector-tensor modifications of GR [@Creminelli:2017sry; @Sakstein:2017xjx; @Ezquiaga:2017ekz; @Baker:2017hug]. Let us then focus on the effect of modifying the coefficient of the $2{\cal H}$ term, i.e. let us consider a propagation equation of the form \[prophmodgrav\] ”\_A +2 [H]{}\[1-()\] ’\_A+k\^2\_A=0 , with $\delta(\eta)$ some function (we will present in Section \[sect:modpropNL\] an explicit example of a modified gravity model where GW propagation is described by such an equation). In this case we introduce $\tilde{\chi}_A(\eta, \vk)$ from \[4defhchiproofsRR\] \_A(, )= \_A(, ) , where \[deftildea\] =[H]{}\[1-()\] , and we get $\tilde{\chi}''_A+(k^2-\tilde{a}''/\tilde{a}) \tilde{\chi}_A=0$. Once again, inside the horizon the term $\tilde{a}''/\tilde{a}$ is totally negligible, so GWs propagate at the speed of light. However, in the propagation across cosmological distances, $\tilde{h}_A$ now decreases as $1/\tilde{a}$ rather than $1/a$. Then, in such a modified gravity model we must distinguish between an electromagnetic luminosity distance $d_L^{\,\rm em}(z)$ and a GW luminosity distance $d_L^{\,\rm gw}(z)$, and the GW amplitude of a coalescing binary at redshift $z$ will now be proportional to $1/d_L^{\,\rm gw}(z)$, where d\_L\^[gw]{}(z)= d\_L\^[em]{}(z) = d\_L\^[em]{}(z) , and $d_L^{\,\rm em}(z)\equiv d_L(z)$ is the standard luminosity distance (\[dLem\]) for electromagnetic signals. is equivalent to $(\log a/\tilde{a})'=\delta(\eta) {\cal H}(\eta)$, which is easily integrated and gives \[dLgwdLem\] d\_L\^[gw]{}(z)=d\_L\^[em]{}(z){-\_0\^z (z’)} . Modified propagation in nonlocal gravity {#sect:modpropNL} ======================================== To illustrate this effect, and the relative roles of $\wde(z)$ and $\d(z)$ in $d_L^{\,\rm gw}(z)$, we consider an explicit modified gravity model, but, as will be clear, the results that we find are more general. The model that we consider is a nonlocal modification of gravity that has been introduced and much studied in recent years by our group. The underlying physical idea is that, even if the fundamental action of gravity is local, the corresponding quantum effective action, that includes the effect of quantum fluctuations, is nonlocal. These nonlocalities are well understood in the ultraviolet regime, where their computation is by now standard textbook material [@Birrell:1982ix; @Mukhanov:2007zz; @Shapiro:2008sf], but are much less understood in the infrared (IR), which is the regime relevant for cosmology. IR effects in quantum field theory in curved space have been studied particularly in de Sitter space where strong effects, due in particular to the propagator of the conformal mode [@Antoniadis:1986sb], have been found. However, the whole issue of IR corrections in de Sitter space is unsettled, because of the intrinsic difficulty of the problem. Given the difficulty of a pure top-down approach, we have taken an alternative and more phenomenological strategy. In general, strong IR effects manifest themselves through the generation of nonlocal terms, proportional to inverse powers of the d’Alembertian operator, in the quantum effective action. For instance, in QCD the strong IR fluctuations generate a term [@Boucaud:2001st; @Capri:2005dy; @Dudal:2008sp] \[Fmn2\] d\^4x F\_ F\^ , in the quantum effective action, where $\Fmn$ is the non-abelian field strength. This nonlocal term corresponds to giving a mass $m_g$ to the gluons: indeed, choosing the Lorentz gauge and expanding in powers of the gauge field $A^{\mu}$, the above terms gives a gluon mass term $m_g^2 {\rm Tr} (A_{\mu}A^{\mu})$, plus extra nonlocal interactions. Note that the use of a nonlocal operator such as $\iBox$ allows us to write a mass term without violating gauge invariance. However, this only makes sense at the level of quantum effective actions, where nonlocalities are unavoidably generated by quantum loops whenever the theory contains massless or light particles. The fundamental action of a quantum field theory, in contrast, must be local. Thus, nonlocal terms of this form describe dynamical mass generation by quantum fluctuations at the level of the quantum effective action. In a similar spirit, we have studied a model of gravity based on the quantum effective action \[RR\] \_[RR]{}=d\^[4]{}x $$R-\frac{1}{6} m^2R\frac{1}{\Box^2} R$$ , where $\mplr$ is the reduced Planck mass and $m$ is a new mass parameter that replaces the cosmological constant of $\Lambda$CDM. This model was proposed in [@Maggiore:2014sia], following earlier work in [@Maggiore:2013mea], and it can be shown that the nonlocal term in corresponds to a dynamical mass generation for the conformal mode of the metric [@Maggiore:2015rma; @Maggiore:2016fbn]. Recently, some evidence for the nonlocal term in has also been found from non-perturbative studies in lattice gravity [@Knorr:2018kog]. A detailed comparison with cosmological data and Bayesian parameter estimation has been carried out in [@Dirian:2014ara; @Dirian:2014bma; @Dirian:2016puz; @Dirian:2017pwp; @Belgacem:2017cqo], where it has been found that the model fits cosmic microwave background (CMB), supernovae (SNe), baryon acoustic oscillation (BAO), structure formation and local $H_0$ measurements at a level statistically indistinguishable from $\Lambda$CDM (with the same number of parameters, since $m$ replaces $\Lambda$); furthermore, parameter estimation gives a large value of the Hubble parameter, which basically eliminates the tension between the [*Planck*]{} CMB data [@Planck_2015_CP] and the local $H_0$ measurements [@Riess:2016jrr]. The parameter $m$ is also fixed by Bayesian parameter estimation from CMB, SNe and BAO data, and turns out to be of order $H_0$. The model has been reviewed in [@Maggiore:2016gpx] and, more recently, in [@Belgacem:2017cqo], to which we refer the reader for a detailed discussion of conceptual aspects and phenomenological consequences. We will refer to it as the “RR" model. ![The ratio $d_L^{\,\rm gw}(z)/d_L^{\,\rm em}(z)$ in the RR model.[]{data-label="fig:dLgw_over_dLem"}](dLgw_over_dLem.pdf){width="35.00000%"} ![The relative differences $\Delta d_L/d_L$ between the RR model and $\Lambda$CDM for three different cases. Green, dot-dashed curve: the relative difference $(d_L^{\rm RR,em}-d_L^{\Lambda{\rm CDM}})/d_L^{\Lambda{\rm CDM}}$ using the same values of $h_0$ and $\oma$ (taken for definiteness as $h_0=0.7013$ and $\oma=0.2922$). Dashed magenta curve: the same, but using for each model its own mean values of $h_0$ and $\oma$. Blue solid line: the relative difference $\Delta d^{\rm gw}_L/d_L\equiv (d_L^{\rm RR,gw}-d_L^{\Lambda{\rm CDM}})/d_L^{\Lambda{\rm CDM}}$ using again for each model its own mean values of $h_0$ and $\oma$.[]{data-label="fig:Deltad_over_d"}](Deltad_over_d.pdf){width="35.00000%"} The equation of tensor perturbations in the RR model has been derived in [@Dirian:2016puz] and, for the free propagation, has indeed the form (\[prophmodgrav\]), with \[defdeltaperRR\] = , where $\bar{V}$ is the background evolution of an auxiliary field that is introduced to rewrite in local form (see e.g. Section 3 of [@Belgacem:2017cqo] for review), and $\gamma= m^2/(9H_0^2)$. For this form of $\delta(z)$, the integral in can be computed analytically by transforming the integration over $dz$ into an integration over $d\bar{V}$, which gives \[dLgwdLemRR\] d\_L\^[RR,gw]{}(z)=d\_L\^[RR,em]{}(z) , so in the RR model the ratio $d_L^{\,\rm gw}(z)/d_L^{\,\rm em}(z)$ is a local function of $\bar{V}(z)$. We plot this ratio in Fig. \[fig:dLgw\_over\_dLem\]. In the RR model, scalar perturbations obey a modified Poisson equation with an effective Newton constant that, for modes well inside the horizon, is given by [@Dirian:2014ara] \[Geffdiz\] G\_[eff]{}(z)= . Then, can be rewritten as \[dLgwdLemGeff\] d\_L\^[RR,gw]{}(z)=d\_L\^[RR,em]{}(z) , that nicely ties modified GW propagation to the modification in the growth of structures. Quite remarkably, this is exactly the same relation found recently in a subclass of Horndeski models [@Linder:2018jil]. In Fig. \[fig:Deltad\_over\_d\] we show the relative difference $\Delta d_L/d_L$ for three different cases. The upper curve is the relative difference between the electromagnetic luminosity distance in the RR model and the luminosity distance of $\Lambda$CDM, when we use the same fiducial values for $h_0$ and $\oma$. In this case we see that, over a range of redshifts relevant for third-generation interferometers, the relative difference is of order $2\%$. However, this is not the quantity relevant to observations. For each model, the actual predictions are those obtained by using its own best-fit values (or the mean values, or the priors) of the cosmological parameters, which are found by comparing the model with a set of cosmological data and performing Bayesian parameter estimation. For the RR model, as well as for $\Lambda$CDM, this is obtained by computing the cosmological perturbations of the model, inserting them in a Boltzmann code, and constraining the model with observations by using a Markov Chain Monte Carlo. For the RR model this has been done in [@Dirian:2014bma; @Dirian:2016puz; @Dirian:2017pwp; @Belgacem:2017cqo]. Here we will use for definiteness the values in Table 3 of [@Belgacem:2017cqo], where we used as datasets the [*Planck*]{} CMB data, a compilation of BAO data, the SNe data from the JLA dataset, and the local measurement of $H_0$. In this case for $\Lambda$CDM we get the mean values $h_0=0.681(5)$ and $\oma =0.305(7)$, while for the “minimal" RR model (in which a parameter $u_0$ that determines the initial condition of an auxiliary field is set to zero; the limit of large $u_0$ brings the model closer and closer to $\Lambda$CDM) we get $h_0=0.701(7)$ and $\oma =0.292(8)$. The corresponding result for $(d_L^{\rm RR,em}-d_L^{\Lambda{\rm CDM}})/d_L^{\Lambda{\rm CDM}}$ is given by the dashed, magenta curve in Fig. \[fig:Deltad\_over\_d\] and we see that, at redshifts $z\, \gsim\, 1$, is one order of magnitude smaller than the green curve. This is easily understood. Parameter estimation is basically performed by comparing the predictions of each model to a set of fixed distance indicators, such as those given by the peaks of the CMB or by the BAO scale. Thus, the parameters in each model are adjusted so to reproduce these distance measurements at large redshift, and therefore have the tendency to compensate the differences in luminosity distance (or in comoving distance or in angular diameter distance) induced by the different functional forms of $\wde(z)$. As a result, at redshifts $z\, \gsim\, 0.5$, $|\Delta d_L|/d_L$ is reduced by about one order of magnitude, to a value $(0.2-0.4)\%$, which is much more difficult to observe. It is clear, from the above physical explanation, that this effect is quite general in modified gravity models, and we have detected it in the RR model simply because in this case a detailed Bayesian parameter estimation was already available. The two upper curves in Fig. \[fig:Deltad\_over\_d\] give the relative difference of the [*electromagnetic*]{} luminosity distances, which is relevant for standard candles. For standard sirens we rather need to compare the GW luminosity distance of the RR model, $d_L^{\rm RR,gw}$, to the luminosity distance $d_L^{\Lambda{\rm CDM}}$ of $\Lambda$CDM (which, in contrast, is the same for GWs and for electromagnetic signals). The result of this comparison, using again the respective mean values of the parameters for the RR model and for $\Lambda$CDM, is given by the lower curve (blue, solid line) in Fig. \[fig:Deltad\_over\_d\]. We see that the difference, in absolute value, now raises again to values of order $3\%$, and the sign of the difference is opposite. From these results we can draw some interesting conclusions. First, the existence in generic modified gravity theories of a notion of GW luminosity distance, a priori different from the electromagnetic luminosity distance, makes in principle possible a conceptually clean test of modifications of GR. If the luminosity distance derived from a set of standard candles turns out to be different from the result obtained with standard sirens, this will be a “smoking gun" evidence for modified gravity (see also [@Saltas:2014dha; @Lombriser:2015sxa; @Nishizawa:2017nef]). A second point is that, at the redshifts $z\, \gsim \, 1$ relevant for LISA and ET, the deviation from the $\Lambda$CDM prediction induced by $\d(z)$ is much larger than that induced by $\wde(z)$, i.e., in absolute value, in Fig. \[fig:Deltad\_over\_d\] the blue solid curve is larger than the magenta dashed curve.[^2] Note also that, if one measures a deviation from $\Lambda$CDM of the type of the blue solid curve in Fig. \[fig:Deltad\_over\_d\] and tries to interpret it as due to a non-trivial dark-energy equation of state, neglecting the possibility of modified GW propagation, one would conclude that this is a signature of a non-phantom $\wde(z)$ (which results in a negative value for $\Delta d_L/d_L$). However, this interpretation could be totally wrong. In our case, the blue solid curve in Fig. \[fig:Deltad\_over\_d\] is produced in a model, such as the RR model, that has a phantom DE equation of state, and the effect is not due to $\wde(z)$, but is rather dominated by $\d(z)$. Comparison with the Einstein Telescope ====================================== In a generic modified gravity model there will be both differences in the propagation of GWs and in their production mechanism, compared to GR. The two effects are decoupled, the former affecting the luminosity distance, as we have seen, and the latter the phase of the GW signal. The modification to the production mechanism depends on how much the modified gravity theory differs from GR at the distance scale $L$ of the binary system (and on whether it contains extra radiative degrees of freedom). In the RR model there are no extra radiative degrees of freedom, and the static solution of the theory reduces smoothly to that of GR at distances $L\ll m^{-1}\simeq H_0^{-1}$, with corrections of order $(mL)^2$ [@Kehagias:2014sda; @Maggiore:2014sia]. For $L$ of the order of the size of an astrophysical binary this correction is utterly negligible and does not affect GW production. A more subtle point is whether the time dependence (\[Geffdiz\]) of the effective Newton constant, found on cosmological scales, can be extrapolated down to the scale of a coalescing binary (see [@Barreira:2014kra] and Appendix B of [@Dirian:2016puz] for discussion). In any case, the effect on the waveform due to modifications of GW propagation in general dominates over modification of GW production, since the former gives an effect that accumulates over the distance to the source [@Yunes:2016jcc], and here we focus on it. ![The absolute value of $\Delta d^{\rm gw}_L/d_L\equiv (d_L^{\rm RR,gw}-d_L^{\Lambda{\rm CDM}})/d_L^{\Lambda{\rm CDM}}$, where both $d_L^{\rm RR,gw}$ and $d_L^{\Lambda{\rm CDM}}$ are computed using the respective mean values of the parameters (blue solid line), compared with an estimate of the total error on $\Delta d^{\rm gw}_L/d_L$ for ET (magenta, dashed) and the contribution to the error due to lensing (green, dot-dashed).[]{data-label="fig:comparisonErrorET"}](comparisonErrorET.pdf){width="35.00000%"} In Fig. \[fig:comparisonErrorET\] we show | | , and we compare it with an estimate of the total error in ET due to instrumental noise plus lensing  [@Zhao:2010sz], and with the separate contribution to the error due to lensing  [@Sathyaprakash:2009xt]. While the instrumental error is inversely proportional to the signal-to-noise ratio of the source, and can in principle be decreased by improving the detector sensitivity, the error due to lensing is due to intervening matter structures that affect the GW propagation, and provides a lower limit on the error of a third-generation interferometer (unless suitable delensing techniques are applied). Note that at very low redshifts, $z\, \lsim\, 0.05$, the error in the uncertainty in the local Hubble flow (not shown in the figure) will eventually dominate. Given that with $N$ measurements we improve the accuracy by a factor $\sqrt{N}$, from this plot we find that, to reach a sensitivity of the order of the signal, we need about 7 standard sirens (with measured redshift) at $z\simeq 0.4$, or about $44$ standard sirens at $z\simeq 1$, or $130$ at $z\simeq 2$. Thus, a significant signal-to-noise ratio could be obtained with a few tens of standard sirens at $z\sim 0.5$, or a few hundreds at $z\sim 1-2$. Of course, these numbers should only be taken as indicative, since the sensitivity of third-generation interferometers is still quite tentative. Note also that current errors on the estimate of cosmological parameters such as $h_0$ and $\oma$ are of order $2\%$. This induces a corresponding theoretical error in the prediction, that is not negligible compared to the predicted value of $|\Delta d^{\rm gw}_L/d_L|$. However, by the time that third-generation interferometers will operate, further improvement in cosmological parameter estimation is expected from mid-future observations such as the EUCLID mission [@Laureijs:2011gra], DESI [@DESI-1] or SKA [@Bull:2015nra]; otherwise, a larger number of sources will be necessary. In any case, a third-generation interferometer such as ET is expected to detect millions of binary mergers, of which possibly $O(10^3-10^4)$ could have an electromagnetic counterpart. Prospects for dark energy studies using standard sirens therefore look bright. [**Acknowledgments.**]{} The work of the authors is supported by the Fonds National Suisse and by the SwissMap National Center for Competence in Research. [^1]: A general formalism for testing gravity with GW propagation has been recently presented in [@Nishizawa:2017nef]. Ref. [@Yunes:2016jcc] gives a detailed discussion of the constraints obtained from the first two observations of BH-BH coalescences, GW150914 and GW151226, both on modified GW generation and on modified GW propagation due to a non-trivial dispersion relation of the graviton. [^2]: Of course, in a given specific modified gravity model, the function $\delta (z)$ could simply be zero, or anyhow such that $|\delta(z)|\ll |1+\wde(z)|$, in which case the main effect would come from $\wde(z)$. What our argument shows is that, in a generic modified gravity model where the deviation of $\d(z)$ from zero and the deviation of $\wde(z)$ from $-1$ are of the same order, the effect of $\d(z)$ dominates.

--- abstract: | We prove that the bicrossed product of two groups is a quotient of the pushout of two semidirect products. A matched pair of groups $(H, G, \alpha, \beta)$ is deformed using a combinatorial datum $(\sigma, v, r)$ consisting of an automorphism $\sigma$ of $H$, a permutation $v$ of the set $G$ and a transition map $r: G\to H$ in order to obtain a new matched pair $\bigl(H, (G,*), \alpha', \beta' \bigl)$ such that there exist an $\sigma$-invariant isomorphism of groups $H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G \cong H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \,(G,*)$. Moreover, if we fix the group $H$ and the automorphism $\sigma \in {{\rm Aut}\,}(H)$ then any $\sigma$-invariant isomorphism $H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G \cong H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G'$ between two arbitrary bicrossed product of groups is obtained in a unique way by the above deformation method. As applications two Schreier type classification theorems for bicrossed product of groups are given. address: 'Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei 14, RO-010014 Bucharest 1, Romania' author: - 'A. L. Agore' - 'G. Militaru' title: Deformations of a matched pair and Schreier type theorems for bicrossed product of groups --- [^1] Introduction {#introduction .unnumbered} ============ The aim of the paper is to bring back to attention and revitalize one of the most famous open problems of group theory formulated in the first half of the last century([@Douglas], [@Ore], [@Redei]). It can be seen as the dual of the more famous *extension problem* of O. L. Hölder and it is called the *factorization problem*. The statement is very simple and tempting: *Let $H$ and $G$ be two given groups. Describe and classify up to an isomorphism all groups $E$ that factorize through $H$ and $G$: i.e. $E$ contains $H$ and $G$ as subgroups such that $E = H G$ and $ H \cap G = 1$.* Leaving aside the classification part introduced above, the first part of the problem was formulated in 1937 by O. Ore [@Ore] but it roots are much older and descend to E. Maillet’s 1900 paper [@Maillet]. Even though if the statement is very simple, as many famous problems in mathematics are, little progress has been made since then. We dare to say that this one is even more difficult than the more popular extension problem. In the case of two cyclic groups $H$ and $G$, not both finite, the problem was started by L. Rédei in [@Redei] and finished by P.M. Cohn in [@Cohn], without the classification part introduced above. To the best of our knowledge this seems to be the only case where the complete answer is known. If $H$ and $G$ are both finite cyclic groups the problem is more difficult and seems to be still an open question, even though J. Douglas [@Douglas] has devoted four papers and over two dozen theorems to the subject. Recently, in [@ACIM Theorem 2.1] the problem was solved in the case that one of the finite cyclic groups is of prime order. Using a famous theorem of Frobenius a Schur-Zassenhaus type theorem was proven: any group $E$ that factorizes through two finite cyclic groups, one of them being of prime order, is isomorphic to a semidirect product between the two cyclic groups of the same order. More popular in group theory was the converse of the factorization problem: given a group $E$ find all *exact factorizations* of it, that is, all subgroups $H$ and $G$ of $E$ such that $E = H G$ and $ H \cap G = 1$. Starting with the 1980’s various papers dealing with this problem were written (see [@Ba], [@Gi], [@Pr], [@WW] and their list of references). Derived from this problem is the following: describe and characterize the class of (finite simple) groups that do not admit an exact factorization between two proper subgroups. Having in mind the abelian case such a group will be called an *indecomposable group*: the quaternion group $Q$, $\ZZ_{p^{n}}$ for a prime integer $p$ or the alternating group $A_{6}$ are typical examples of indecomposable groups. An important step related to the factorization problem was the construction of the bicrossed product $H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ associated to a matched pair $(H, G, \alpha, \beta)$ given by M. Takeuchi [@Takeuchi]: $\alpha$ is a left action of the group $G$ on the set $H$, $\beta$ is a right action of the group $H$ on the set $G$ satisfying two compatibility conditions. A group $E$ factorizes through two subgroups $H$ and $G$ if and only if there exists a matched pair $(H, G, \alpha, \beta)$ such that $$\theta : H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G \rightarrow E, \qquad \theta (h, g) = hg$$ is an isomorphism of groups. Thus the factorization problem can be restated in a computational manner as follows: *Let $H$ and $G$ be two given groups. Describe all matched pairs $(H, G, \alpha, \beta)$ and classify up to an isomorphism all bicrossed products $H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$.* The motivation for the above problem is triple: first of all, the problem presents an interest in itself in group theory. On the other hand the construction of the bicrossed product provides the easiest way of constructing finite quantum groups [@masu], hence the classification theorems from group level lead us to classification theorems for finite quantum groups. Finally, the bicrossed product construction at the level of groups served as a model for similar constructions in other fields of mathematics like: algebras [@cap], coalgebras [@CIMZ], groupoids [@AA], Hopf algebras [@Takeuchi], locally compact groups [@baaj] or locally compact quantum groups [@VV], Lie Algebras [@Mic] or Lie groups [@Kro]. Thus, the above problem can be easily formulated for each of the above different levels where the bicrossed product construction was made. For instance, at the level of algebras (the bicrossed product of two algebras is also called *twisted tensor product algebra*) the first steps were already made in the last years: the story started with [@CIMZ Examples 2.11] where all bicrossed product between two group algebras of dimension two are completely described and classified. Recently, the classification of all bicrossed product between the algebras $k^2$ and $k^m$ was finished in [@Pena] and the description of some bicrossed products between two polynomial algebras $k[X]$ and $k[Y]$ was started in [@gucci]. On the other hand, in [@Jara] only a sufficient condition for the isomorphism between two bicrossed products of algebras that fix one of the algebra is given under the name of *invariance under twisting* problem. This paper is devoted to the classification part of the factorization problem at the group level. Namely we shall ask the following question: when are two bicrossed products $H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ and $H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G$ isomorphic? The organization of the paper is the following: in [Section \[se:1\]]{} we recall the construction of the bicrossed product of two groups given by M. Takeuchi. It is a generalization of the semidirect product construction for the case when none of the factors is required to be normal. The first natural question arises: how far is a bicrossed product from being a semidirect product? [Proposition \[pr:pushout\]]{} gives the first answer to the question: we prove that the bicrossed product of two groups is a quotient of the pushout of two semidirect products over the direct product of the subgroups of invariants of the actions $\alpha$ and $\beta$. In [Section \[se:2\]]{} we start the classification part of the factorization problem. The main result is [Theorem \[th:deformation\]]{} : for any matched pair of groups $(H, G, \alpha, \beta)$ and any triple $(\sigma, v, r)$, consisting of an automorphism $\sigma$ of $H$, a permutation $v$ on the set $G$ and a transition map $r: G\to H$ satisfying a certain compatibility condition, a new matched pair $\bigl(H, (G,*), \alpha', \beta' \bigl)$ is constructed such that there exists an $\sigma$-invariant isomorphism of groups $H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G \cong H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \,(G,*)$. The importance of the result is given by the converse: if we fix the group $H$ and the automorphism $\sigma \in {{\rm Aut}\,}(H)$, then any $\sigma$-invariant isomorphism $H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G \cong H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G'$ between two arbitrary bicrossed products of groups is obtained in a unique way by the above deformation method. As applications in [Section \[se:3\]]{} two Schreier type classification theorems for bicrossed products of groups are given. They are formulated using the language of category theory. Let $H$ and $G$ be two fixed groups: we define a category $B_{1}(H,G)$ having as object the set of all matched pairs $(H, G, \alpha, \beta)$ and morphisms are defined as morphisms between two bicrossed products that fix one of the groups. [Theorem \[th:sch1\]]{} gives a bijection between the set of objects of the skeleton of the category $B_{1}(H,G)$ and a certain pointed set $K^{2}(H, G)$ that will play for the classification problem of bicrossed products the same role as the second cohomology group does for the classification of the extension problem. Returning to the question of how far a bicrossed product is from being a semidirect product, [Corollary \[co:lake1\]]{} and [Corollary \[co:lake2\]]{} give two necessary and sufficient conditions for a bicrossed product to be isomorphic to a semidirect product of groups in the category $B_{1}(H,G)$. [Theorem \[th:sch22\]]{} is the second Schreier type theorem for bicrossed products: this time we fix two groups $H$, $G$ and $\beta: G \times H \rightarrow G$ a right action of the group $H$ on the set $G$ and the classification theorem is more restrictive than the one given in [Theorem \[th:sch1\]]{}. In the last section we give some examples: we compute and count explicitly the set of all matched pairs $(C_3, C_m, \alpha, \beta)$, where $C_m$ is a cyclic group of order $m$, and the pointed set $K^{2} (C_{3}, C_{6})$ constructed in [Theorem \[th:sch1\]]{} is shown to have three elements. Preliminaries ============= [\[se:1\]]{} Let us fix the notation that will be used throughout the paper. Let $H$ and $G$ be two groups and $\alpha : G \times H \rightarrow H$ and $\beta : G \times H \rightarrow G$ two maps. We use the notation $$\alpha (g, h) = g\triangleright h \quad \text{and} \quad \beta (g, h) = g\triangleleft h$$ for all $g\in G$ and $h\in H$. The map $\alpha$ (resp. $\beta$) is called trivial if $g\triangleright h = h$ (resp. $g\triangleleft h = g$) for all $g\in G$ and $h\in H$. We recall that $\alpha$ is an action as automorphism if it is a left action of the group $G$ on the set $H$ and $g \triangleright (h_1 h_2) = (g \triangleright h_1) (g \triangleright h_2)$, for all $g\in G$, $h_1$, $h_2 \in H$. Similarly, $\beta$ is an action as automorphism if it is a right action of the group $H$ on the set $G$ and $(g_1 g_2) \lhd h = (g_1 \lhd h) (g_2 \lhd h)$ for all $g_1$, $g_2 \in G$ and $h\in H$. ${{\rm Aut}\,}(H)$ is the group of automorphisms of $H$ and $C_n$ is the cyclic group of order $n$. Let $H$ and $G$ be two groups with the multiplications $m_{H}: H \times H\rightarrow H$, $m_{G}:G \times G\rightarrow G$, units $1_H$ and respectively $1_G$ and $R: G \times H \rightarrow H \times G$ a map. We shall define a new multiplication on the set $H \times G$ using $R$ instead of the usual flip $\tau: G\times H \to H\times G$, $\tau (g, h) = (h, g)$ as follows: $$m_{H \times G, R}: H \times G \times H \times G \rightarrow H \times G, \qquad m_{H \times G, R}:= (m_{H} \times m_{G})\circ (I \times R \times I)$$ Let $\alpha := \pi_{1} \circ R : G \times H \rightarrow H$, $\beta :=\pi_{2} \circ R: G \times H \rightarrow G$, where $\pi_{i}$ is the projection on the $i$-component; we shall denote $\alpha (g, h) = g \rhd h$ and $\beta (g, h) = g \lhd h$, for all $g\in G$ and $h\in H$. Then $R (g, h) = (g \rhd h, g \lhd h)$ and the multiplication $m_{H \times G, R}$ on $H \times G$ can be explicitly written as follows: $$(h_{1}, g_{1}) \cdot_{R} (h_{2}, g_{2}) = \bigl( h_{1}(g_{1} \rhd h_{2}), \, (g_{1} \lhd h_{2})g_{2}\bigl)$$ for all $h_{1}$, $h_{2} \in H$ and $g_{1}$, $g_{2} \in G$. It can be easily shown that $(H \times G, m_{H \times G, R})$ is a group with $(1_{H}, 1_{G})$ as a unit if and only if $(H, G, \alpha, \beta)$ is a *matched pair* in the sense of Takeuchi ([@Takeuchi]): i.e. $\alpha$ is a left action of the group $G$ on the set $H$, $\beta$ is a right action of the group $H$ on the set $G$ and the following two compatibility conditions hold: $${\label{eq:2}} g \rhd (h_{1} h_{2}) = (g \rhd h_{1})\bigl((g \lhd h_{1}) \rhd h_{2}\bigl)$$ $${\label{eq:3}} (g_{1} g_{2}) \lhd h = \bigl(g_{1} \lhd (g_{2} \rhd h)\bigl)(g_{2} \lhd h)$$ for all $h$, $h_{1}$, $h_{2} \in H$ and $g$, $g_{1}$, $g_{2} \in G$. It follows from [(\[eq:2\])]{} and [(\[eq:3\])]{} that: $${\label{eq:4}} g \rhd 1_H = 1_H \quad {\rm and}\quad 1_G \lhd h = 1_G$$ for all $h \in H$ and $g \in G$. If $(H, G, \alpha, \beta)$ is a matched pair, the new group obtained on the set $H \times G$ will be denoted by $H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G = H\bowtie \, G$ and will be called the *bicrossed product* ([*knit product*]{} or [*Zappa-Sz' ep product*]{}) of $H$ and $G$. We note that $i_H : H\to H\bowtie \, G$, $i_H (h) = (h, 1)$ and $i_G : G\to H\bowtie \, G$, $i_G (g) = (1, g)$, for all $h\in H$, $g\in G$ are morphisms of groups and hence $H \times \{1\}\cong H$ and $\{1\} \times G \cong G$ are subgroups of $H\bowtie \, G$. Moreover, every element $(h, g)$ of $H\bowtie \, G$ can be written uniquely as a product of an element of $H \times \{1\}$ and of an element of $\{1\} \times G$ as follows: $$(h,g) = (h,1) \cdot (1,g)$$ Conversely, this observation characterizes the bicrossed product. Let $E$ be a group $H$, $G\leq E$ be subgroups such that any element of $E$ can be written uniquely as a product of an element of $E$ and an element of $G$. Then there exists a matched pair $(H, G, \alpha, \beta)$ such that $$\theta : H\bowtie \, G \ \rightarrow E, \qquad \theta (h, g) = hg$$ is a group isomorphism ([@Takeuchi]). The maps $\alpha$ and $\beta$ play a symmetric role: if $(H, G, \alpha, \beta)$ is a matched pair then we can construct a new matched pair $(G, H, \tilde{\alpha}, \tilde{\beta} )$ such that there exists a canonical isomorphism of groups $H\, _{\alpha}\bowtie_{\beta} \, G \cong G\, _{\tilde{\alpha}}\bowtie_{\tilde{\beta}} \, H$ ([@ACIM Proposition 2.5]). [\[re:2.4.90\]]{} Let $H$ and $G$ be two groups and $\beta : G \times H \rightarrow G$ the trivial action. Then $(H, G, \alpha, \beta)$ is a matched pair if and only if $\alpha : G \times H \rightarrow H$ is an action of $G$ on $H$ as group automorphisms. In this case the bicrossed product $H\bowtie \, G$ is exactly the left version of the semidirect product $H {}_{\alpha}\ltimes G$. Assume now that the map $\alpha$ is the trivial action. Then $(H, G, \alpha, \beta)$ is a matched pair if and only if $\beta$ is a right action of $H$ on $G$ as group automorphisms. Is this case the bicrossed product $H\bowtie \, G$ is exactly the right version of the semidirect product $H\rtimes_{\beta} \, G$. It can be easily proved that $H\rtimes_{\beta} \, G \cong G {}_{\varphi}\ltimes H$, where $\varphi = \varphi_{\beta}$ is the action of $H$ on $G$ as group automorphisms given by $$\varphi : H \rightarrow {{\rm Aut}\,}(G), \qquad \varphi (h)(g) = \Bigl( g^{-1} \triangleleft {h^{-1}} \Bigl)^{-1}$$ for all $h\in H$ and $g\in G$ ([@ACIM Remark 2.6]). A matched pair $(H, G, \alpha, \beta)$ is called *proper* if $\alpha$ and $\beta$ are both nontrivial actions. The above Remark shows that the semidirect product is a special case of the bicrossed product construction. It is therefore natural to ask the converse: *Can a bicrossed product be obtained from semidirect products of groups?* In what follows we shall give a first answer to this question: a bicrossed product can be obtained as a quotient of a pushout of two semidirect products of groups. Let $(H, G, \alpha, \beta)$ be a matched pair and let us denote by ${\rm Fix}(H)$ and ${\rm Fix}(G)$ the invariants of the two actions $\alpha$ and $\beta$ : $${\rm Fix} (H) : = \{h \in H \mid g \rhd h = h, \, \forall g \in G\}, \quad {\rm Fix}(G) : = \{g \in G \mid g \lhd h = g, \, \forall h \in H\}$$ Using the compatibility conditions [(\[eq:2\])]{} and [(\[eq:3\])]{} we shall prove that ${\rm Fix}(H)$ is a subgroup of $H$ and ${\rm Fix}(G)$ a subgroup of $G$. Indeed, from [(\[eq:4\])]{} we obtain that $1_H \in {\rm Fix}(H)$ and for $h_{1}$, $h_{2} \in {\rm Fix}(H)$ we have: $$g \rhd (h_{1}h_{2}) \stackrel{{(\ref{eq:2})}} {=} (g \rhd h_{1})\bigl((g \lhd h_{1}) \rhd h_{2}\bigl) = h_{1}h_{2}$$ i.e. $h_{1}h_{2} \in {\rm Fix}(H)$. On the other hand: $$1_H \stackrel{{(\ref{eq:4})}} {=} g \rhd 1_H = g \rhd (h_{1}^{-1}h_{1}) \stackrel{{(\ref{eq:2})}} {=} (g \rhd h_{1}^{-1})\bigl((g \lhd h_{1}^{-1}) \rhd h_{1}\bigl) = (g \rhd h_{1}^{-1})h_{1}$$ Thus $g \rhd h_{1}^{-1} = h_{1}^{-1}$, i.e. $h_{1}^{-1} \in {\rm Fix}(H)$. In a similar way we can show that ${\rm Fix}(G)$ is a subgroup of $G$. Using the compatibility condition [(\[eq:2\])]{} we obtain that the map given by: $$\varphi_{\rhd}: {\rm Fix}(G) \rightarrow {\rm Aut}(H), \quad \varphi_{\rhd}(g)(h) := g \rhd h$$ for all $g \in {\rm Fix}(G)$, $h\in H$ is a morphism of groups. Thus we can construct the left version of the semidirect product associated to the triple $(H, \, {\rm Fix}(G), \, \varphi_{\rhd})$: that is $H {}_{\varphi_{\rhd}}\!\! \ltimes {\rm Fix}(G) : = H\times {\rm Fix}(G)$ with the multiplication: $$(h, g) (h', g') = \bigl(h(g \rhd h'), \, gg'\bigl)$$ for all $h$, $h' \in H$ and $g$, $g' \in {\rm Fix}(G)$. Similarly, using [(\[eq:3\])]{} we obtain that the map given by: $$\psi_{\lhd}: {\rm Fix}(H) \rightarrow {\rm Aut}(G), \quad \psi_{\lhd}(h)(g):= g \lhd h$$ for all $h \in {\rm Fix}(H)$, $g\in G$ is a morphism of groups and we can construct the right version of the semidirect product associated to the triple $(G, \, {\rm Fix}(H), \, \psi_{\lhd})$: i.e. ${\rm Fix}(H) \rtimes_{\psi_{\lhd}} G := {\rm Fix}(H) \times G$ with the multiplication: $$(h, g)(h', g') = \bigl(hh', \, (g \lhd h')g'\bigl)$$ for all $h$, $h' \in {\rm Fix}(H)$ and $g$, $g' \in G$. Moreover, the inclusion maps $$\overline{i}: {\rm Fix}(H) \times {\rm Fix}(G) \hookrightarrow H {}_{\varphi_{\rhd}}\!\! \ltimes {\rm Fix}(G) \quad {\rm and} \quad \overline{j} : {\rm Fix}(H) \times {\rm Fix}(G) \hookrightarrow {\rm Fix}(H) \rtimes_{\psi_{\lhd}} G$$ are morphisms of groups by straightforward verifications. On the other hand we can easily prove that the canonical inclusions $$i: H {}_{\varphi_{\rhd}}\!\! \ltimes {\rm Fix}(G) \hookrightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G, \quad i(h,g) = (h,g)$$ and $$j: {\rm Fix}(H) \rtimes_{\psi_{\lhd}} G \hookrightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G, \quad j(h,g) = (h,g)$$ are morphisms of groups. Indeed for $h$, $h'\in H$ and $g$, $g' \in {\rm Fix}(G)$ we have: $$i(h, g) \cdot i(h', g') = \bigl( h (g \rhd h'), (g \lhd h') g'\bigl)\, \stackrel{g \in {\rm Fix}(G)} {=} \bigl( h(g \rhd h'), gg'\bigl) = i\bigl((h, g) (h',g')\bigl)$$ Thus the two semidirect products constructed above, $H {}_{\varphi_{\rhd}}\!\! \ltimes {\rm Fix}(G)$ and ${\rm Fix}(H) \rtimes_{\psi_{\lhd}} G$, are subgroups of the bicrossed product $H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$. To conclude, we obtained a commutative diagram in the category of groups $${\label{eq:sfibr}} \begin {CD} Fix(H)\times Fix(G) @>\overline{j}>> Fix(H)\rtimes_{\psi} G\\ @VV\overline{i}V @VVjV\\ H {}_\varphi \ltimes Fix(G)@>i>> H {}_\alpha \bowtie_\beta G \end{CD}$$ Using the construction of the pullback in the category of groups it follows that the pair $({\rm Fix}(H) \times {\rm Fix}(G), (\overline{i}, \overline{j}))$ is a pullback of the morphisms $i: H {}_{\varphi_{\rhd}}\!\! \ltimes {\rm Fix}(G) \hookrightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ and $j: {\rm Fix}(H) \rtimes_{\psi_{\lhd}} G \hookrightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$. [\[pr:pushout\]]{} Let $(H, G, \alpha, \beta)$ be a matched pair of groups and $\bigl(X, (\varphi, \psi)\bigl)$ be the pushout in the category of groups of the diagram $$\begin {CD} Fix(H)\times Fix(G) @>\overline{j}>> Fix(H)\rtimes_{\psi} G\\ @VV\overline{i}V @VV\varphi V\\ H {}_\varphi \ltimes Fix(G)@>\psi >> X \end{CD}$$ Then the bicrossed product $H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ is isomorphic to a quotient group of $X$. The diagram [(\[eq:sfibr\])]{} is commutative and $\bigl(X, (\varphi, \psi)\bigl)$ is the pushout of the pair $(\overline{i}, \, \overline{j})$: thus there exists an unique morphism of groups $\theta : X \rightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ such that $\theta \circ \psi = i$ and $\theta \circ \varphi = j$. Let $(h, g) \in H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$: as $(h, 1_G) \in H {}_{\varphi_{\rhd}}\!\! \ltimes Fix(G)$ and $(1_H, g) \in Fix(H) \rtimes_{\psi_{\lhd}} G$ we obtain $$(h, g) = (h, 1_G)(1_H, g) = i(h, 1_G) j(1_H, g) = \theta \bigl( \psi(h, 1_G)\bigl) \theta \bigl(\varphi(1_H, g) \bigl) = \theta \bigl(\psi(h, 1_G) \varphi(1_H, g) \bigl)$$ that is $\theta$ is surjective. Thus $H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ is a quotient group of $X$. We end the section with a problem that can be of interest for a further study: *Let “P” be a property in the category of groups. Give a necessary and sufficient condition such that $H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ has the property “P”.* In the following we give an example in the case that “P” is the property of being abelian or cyclic. Let $(H, G, \alpha, \beta)$ be a matched pair of groups. Then: 1. The center of the bicrossed product $H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ is given by: $$Z\bigl(H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G \bigl) = \{(h, g) \in {\rm Fix}(H)\times {\rm Fix}(G) \mid g \rhd x = h^{-1}xh, \, y \lhd h = gyg^{-1}, \forall x \in H, y \in G \}$$ 2. $H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ is an abelian group if and only if $H$ and $G$ are abelian groups and $\alpha$ and $\beta$ are the trivial actions; 3. $H\,{}_{\alpha}\!\!\bowtie_{\beta} \, G$ is a cyclic group if and only if $\alpha$ and $\beta$ are the trivial actions and $H$, $G$ are finite cyclic groups of coprime orders. An element $(h, g) \in H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ belongs to the center of the group if and only if $(h, g) (x, 1) = (x, 1) (h, g)$ and $(h, g) (1, y) = (1, y) (h, g)$, for all $x \in H$ and $y \in G$. This is equivalent to $h(g \rhd x) = xh$, $g \lhd x = g$, $y \rhd h = h$ and $(y \lhd h)g = gy$, for all $x \in H$, $y \in G$. Hence $h \in {\rm Fix}(H)$, $g \in {\rm Fix}(G)$, $g \rhd x = h^{-1}xh$, $y \lhd h = gyg^{-1}$ for all $x \in H$, $y \in G$. (2) follows from (1) and (3) follows from (2) and the Chinese lemma: a direct product of two groups is a cyclic group if and only if they are finite, cyclic of coprime order. Deformation of a matched pair ============================= [\[se:2\]]{} Let $H$ be a group and $\sigma \in {{\rm Aut}\,}(H)$ an automorphism of $H$. We define the category $\mathcal{C}(H, \sigma)$ as follows: an object of $\mathcal{C}(H, \sigma)$ is a triple $(G, \alpha, \beta)$ such that $(H, G, \alpha, \beta)$ is a matched pair of groups. A morphism $\psi : (G', \alpha', \beta') \rightarrow (G, \alpha, \beta)$ in $\mathcal{C}(H, \sigma)$ is a morphism of groups $\psi: H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G' \rightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ such that the following diagram $${\label{eq:D1}} \begin{CD} H@>i_H>> H\, {}_{\alpha'}\bowtie_{\beta'}G' \\ @VV\sigma V @VV\psi V\\ H@>i_H>> H _{\alpha} \bowtie_{\beta} G \end{CD}$$ is commutative. A (iso)morphism $\psi: H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G' \rightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ in the category $\mathcal{C}(H, \sigma)$ will be called a *$\sigma$-invariant (iso)morphism* between the two bicrossed products. The following key proposition describes explicitly the morphisms of $\mathcal{C}(H, \sigma)$ and gives a necessary and sufficient condition for two bicrossed products $H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G'$ and $H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ to be isomorphic in the category $\mathcal{C}(H, \sigma)$. If $G'$ is a new group we shall denote by “$*$” the multiplication of $G'$ and $\alpha'(g',h) = g' \rhd' h$, $\beta' (g',h) = g' \lhd' h$, for all $g'\in G'$ and $h\in H$. [\[pr:1\]]{} Let $H$ be a group, $\sigma \in {{\rm Aut}\,}(H)$ and $(H, G, \alpha, \beta)$, $(H, G', \alpha', \beta')$ two matched pairs. There exists a one to one correspondence between the set of all morphisms $\psi: H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G' \rightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ in the category $\mathcal{C}(H, \sigma)$ and the set of all pairs $(r,v)$, where $r: G' \rightarrow H$, $v: G' \rightarrow G$ are two maps such that: $$\begin{aligned} \sigma(g' \rhd' h)r(g' \lhd' h) &=& r(g')\bigl(v(g') \rhd \sigma(h)\bigl){\label{eq:p1}} \\ v(g' \lhd' h) &=& v(g') \lhd \sigma(h) {\label{eq:p2}} \\ r(g_{1}' * g_{2}') &=& r(g_{1}') \bigl( v(g_{1}') \rhd r(g_{2}') \bigl) {\label{eq:p3}} \\ v(g_{1}' * g_{2}') &=& \bigl( v(g_{1}') \lhd r(g_{2}')\bigl) v(g_{2}') {\label{eq:p4}}\end{aligned}$$ for all $g', g_{1}', g_{2}' \in G'$, $h \in H$. Through the above bijection $\psi$ is given by $${\label{eq:p5}} \psi(h, \, g') = \bigl(\sigma(h)r(g'), \, v(g')\bigl)$$ for all $h \in H$, $g' \in G'$. Moreover, $\psi : H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G' \rightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ is an isomorphism in $\mathcal{\mathcal{C}}(H,\sigma)$ if and only if the map $v: G' \to G$ is bijective. A morphism of groups $\psi : H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G' \rightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ that makes the diagram [(\[eq:D1\])]{} commutative is uniquely defined by two maps $r = r_{\psi}: G' \rightarrow H$, $v = v_{\psi}:G' \rightarrow G$ such that $\psi(1,g') = \bigl(r(g'),v(g')\bigl)$ for all $g' \in G'$. In this case $\psi$ is given by : $$\psi(h, g') = \psi\bigl((h,1)\cdot (1,g')\bigl) = \bigl(\sigma(h), 1\bigl) \cdot \bigl(r(g'), v(g')\bigl) = \bigl(\sigma(h)r(g'), v(g')\bigl)$$ for all $h \in H$ and $g' \in G'$. As $\psi(1,1)=(1,1)$ we obtain that $r(1)=1$ and $v(1)=1$. We shall prove now that $\psi$ is a morphism of groups if and only if the compatibility conditions [(\[eq:p1\])]{} - [(\[eq:p4\])]{} hold for the pair $(r,v)$. It is enough to check the condition $\psi(xy) = \psi(x) \psi(y)$ only for generators $x$, $y \in \bigl(H\times \{1\}\bigl)\cup \bigl(\{1\}\times G' \bigl)$ of the bicrossed product $H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G'$. Since $\sigma$ is an automorphism of $H$, we have to check only for $x = (1, g')$, $y = (h, 1)$ and $x = (1, g_{1}')$, $y = (1, g_{2}')$. The condition $\psi\bigl((1,g')(h,1)\bigl) = \psi(1,g')\psi(h,1)$ is equivalent to [(\[eq:p1\])]{} - [(\[eq:p2\])]{} and the condition $\psi\bigl((1,g_{1}')(1,g_{2}')\bigl) = \psi(1,g_{1}')\psi(1,g_{2}')$ is equivalent to [(\[eq:p3\])]{} - [(\[eq:p4\])]{}. Note that the normalization conditions $v(1) = 1$ and $r(1) = 1$ where used to obtain [(\[eq:p1\])]{} and [(\[eq:p4\])]{}. Conversely, the normalization conditions follow from [(\[eq:p1\])]{} - [(\[eq:p4\])]{} in the following manner: first, for $g' = 1$ in [(\[eq:p1\])]{} we obtain $\sigma(h)r(1) = r(1)\bigl(v(1) \rhd \sigma(h)\bigl) $ for all $h \in H$. Since $\sigma$ is an automorphism we have : $${\label{eq:th2}} hr(1) = r(1)\bigl(v(1) \rhd h\bigl)$$ for all $h \in H$. Now let $g_{1}'=1$ in [(\[eq:p3\])]{} to obtain: $$\begin{aligned} r(g_{2}') = r(1)\bigl(v(1) \rhd r(g_{2}')\bigl) \stackrel{{(\ref{eq:th2})}} {=} r(g_{2}')r(1)\end{aligned}$$ thus $r(1) = 1$. Finally we let $g_{2}' = 1$ in [(\[eq:p4\])]{} to obtain $v(g_{1}') = v(g_{1}')v(1)$, thus $v(1) = 1$. It remains to be proven that $\psi$ given by [(\[eq:p5\])]{} is an isomorphism if and only if $v: G'\to G$ is a bijective map. Assume first that $\psi$ is an isomorphism. Then $v$ is surjective and for $g_{1}'$, $g_{2}' \in G'$ such that $v(g_{1}') = v(g_{2}')$ we have: $$\psi(1, g_{2}') = \bigl(r(g_{2}'), v(g_{2}')\bigl) = \bigl(r(g_{2}'), v(g_{1}')\bigl) = \psi\bigl(\sigma^{-1}(r(g_{2}'))\sigma^{-1}(r(g_{1}')^{-1}), g_{1}'\bigl)$$ Hence $g_{1}' = g_{2}'$ and $v$ is injective. Conversely, assume that $v$ is bijective. If $\psi(h, g') = (1, 1)$ we obtain that $\sigma(h)r(g') = 1$ and $v(g') = 1 = v(1)$. It follows from here that $g' = 1$ and $\sigma(h) = 1 = \sigma(1)$, i.e. $h = 1$. Hence $\psi$ is injective. Let $(h, g) \in H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ and $g' \in G'$ such that $v(g') = g$. Then $\psi\bigl(\sigma^{-1}(h)\sigma^{-1}(r(g')^{-1}), g'\bigl) = (h, g)$ i.e. $\psi$ is an isomorphism of groups. We shall prove now the main result of this section: **(Deformation of a matched pair)**[\[th:deformation\]]{} Let $(H, G, \alpha, \beta)$ be a matched pair of groups, $(\sigma, v, r)$ be a triple where $\sigma \in {{\rm Aut}\,}(H)$, $v: G \to G$ is a bijective map, $r: G \rightarrow H$ is a map such that $v (1_G) = 1_G$, $r (1_G) = 1_H$ and the following compatibility condition: $${\label{eq:crossed}} r \circ v^{-1}\bigl((v(g_{1}) \lhd r(g_{2}))v(g_{2})\bigl) = r(g_{1})\bigl(v(g_{1}) \rhd r(g_{2})\bigl)$$ holds for all $g_{1}, g_{2} \in G$. On the set $G$ we define a new multiplication $*$ and two new actions $\beta' : G\times H \rightarrow G$, $\alpha' : G\times H \rightarrow H$ given by: $$\begin{aligned} g_{1} * g_{2} &:=& v^{-1}\bigl((v(g_{1}) \lhd r(g_{2}))v(g_{2})\bigl) {\label{eq:def1}}\\ g \lhd' h &:=& v^{-1}\bigl(v(g) \lhd \sigma(h)\bigl) {\label{eq:def2}}\\ g \rhd' h &:=& \sigma^{-1}(r(g))\sigma^{-1}(v(g) \rhd \sigma(h)) \sigma^{-1}\bigl(r \circ v^{-1}(v(g) \lhd \sigma(h))^{-1}\bigl) {\label{eq:def3}}\end{aligned}$$ for all $g_1$, $g_2$, $g \in G$ and $h\in H$. Then: 1. $(G,*)$ is a group structure on the set $G$ with $1_{G}$ as a unit; 2. $\bigl(H, (G,*), \alpha', \beta'\bigl)$ is a matched pair of groups and $$\psi : H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, (G,*) \rightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G, \quad \psi(h,g) := \bigl(\sigma(h)r(g), v(g)\bigl)$$ is a $\sigma$-invariant isomorphism of groups. 3. Any $\sigma$-invariant isomorphism of groups $H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G \cong H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ arises as above. \(1) Let $g \in G$. Then $$g * 1_G = v^{-1}\bigl((v(g) \lhd r(1_G))v(1)\bigl) = v^{-1}\bigl(v(g)\bigl) = g$$ and $$1_G * g = v^{-1}\bigl((v(1_G) \lhd r(g))v(g)\bigl) = v^{-1}\bigl(v(g)\bigl) = g$$ Hence $1_{G}$ is a unit for $*$. Let $g_{1}$, $g_{2}$, $g_{3} \in G$. Then: $$\begin{aligned} v \bigl(\underline{(g_{1} * g_{2})} * g_{3}\bigl) &\stackrel{{(\ref{eq:def1})}} {=}& v\bigl[\underline{ v^{-1}\bigl((v(g_{1}) \lhd r(g_{2}))v(g_{2})\bigl) * g_{3} }\bigl]\\ &\stackrel{{(\ref{eq:def1})}} {=}& \bigl[ \underline{\bigl((v(g_{1}) \lhd r(g_{2}))v(g_{2}) \bigl) \lhd r(g_{3})}\bigl]v(g_{3})\\ &\stackrel{{(\ref{eq:3})}} {=}& \bigl[\bigl(\underline{v(g_{1}) \lhd r(g_{2})\bigl) \lhd \bigl(v(g_{2}) \rhd r(g_{3})\bigl)\bigl] \bigl(v(g_{2}) \lhd r(g_{3}) \bigl)} v(g_{3})\\ &\stackrel{\lhd-{\rm~right~action~}}{=}& \bigl[v(g_{1}) \lhd \underline{\bigl(r(g_{2})(v(g_{2}) \rhd r(g_{3}))\bigl)}\bigl]\bigl(v(g_{2}) \lhd r(g_{3})\bigl)v(g_{3})\\ &\stackrel{{(\ref{eq:def1})}} {=}& \bigl[v(g_{1}) \lhd r \circ v^{-1}\bigl((v(g_{2}) \lhd r(g_{3}))v(g_{3})\bigl)\bigl]\\ &&\bigl(v(g_{2}) \lhd r(g_{3})\bigl)v(g_{3})\\ &{=}& v\bigl[g_{1} * \underline{v^{-1}\bigl((v(g_{2}) \lhd r(g_{3})) v(g_{3})\bigl)} \bigl]\\ &\stackrel{{(\ref{eq:def1})}} {=}& v\bigl(g_{1} * (g_{2} * g_{3})\bigl)\end{aligned}$$ i.e. the multiplication $*$ is associative as $v$ is a bijection. Let $g \in G$ and define $g' := v^{-1}\bigl(v(g)^{-1} \lhd r(g)^{-1}\bigl)$. Then: $$\begin{aligned} v(g' * g) &=& \bigl( v(g') \lhd r(g) \bigl) v(g) = \Bigl( \bigl (v(g)^{-1} \lhd r(g)^{-1} \bigl) \lhd r(g)\Bigl) v(g)\\ &=& v(g)^{-1} v(g) = 1 = v (1)\end{aligned}$$ i.e. $g' * g = 1$ as $v$ is bijective. Thus every element $g \in G$ has a left inverse, i.e. $(G,*)$ is a group. \(2) The proof can be done directly through a long computation but we prefer the following approach: first we remark that the defining relations [(\[eq:def1\])]{}, [(\[eq:def2\])]{}, [(\[eq:def3\])]{} are exactly the compatibility conditions [(\[eq:p4\])]{}, [(\[eq:p2\])]{}, and respectively [(\[eq:p1\])]{} from [Proposition \[pr:1\]]{} and the compatibility condition [(\[eq:crossed\])]{} is exactly [(\[eq:p3\])]{} with the $*$ operations as defined by [(\[eq:def1\])]{}. Moreover the map $$\psi : H \times (G,*) \rightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G, \quad \psi(h,g) = \bigl(\sigma(h)r(g), v(g)\bigl)$$ is a bijection between the set $ H \times (G, *)$ and the group $H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$. With this observation in mind, in order to prove that $\bigl(H, (G,*), \alpha', \beta'\bigl)$ is a matched pair it is enough to show that the group structure obtained by transferring the group structure from the bicrossed product $H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G,$ to the set $ H \times (G, *)$ via the bijective map $\psi$ is exactly the one of a bicrossed product on the set $ H \times (G, *)$ associated to the actions $\alpha'$ and $\beta'$. In other words, we have to prove that $$\bigl( h( g \rhd' h'), \, (g \lhd' h')g'\bigl) = \psi^{-1} \Bigl(\psi(h,g) \cdot \psi(h',g')\Bigl)$$ for all $h$, $h' \in H$, $g$, $g' \in G$ or equivalently, as $\psi$ is bijective $${\label{eq:def4}} \psi\bigl((h,g)\cdot (h',g')\bigl) = \psi(h,g) \cdot \psi(h',g')$$ for all $h$, $h' \in H$, $g$, $g' \in G$. This reduces to proving the following two conditions: $${\label{eq:c11}} \sigma\bigl(h(g \rhd' h')\bigl) r\circ v^{-1}\bigl[\bigl(v(g \lhd' h') \lhd r(g')\bigl)v(g')\bigl] = \sigma(h) r(g)\bigl(v(g) \rhd (\sigma(h')r(g'))\bigl)$$ and $${\label{eq:c2}} \bigl(v(g \lhd' h') \lhd r(g')\bigl)v(g') = \bigl(v(g) \lhd (\sigma(h')r(g'))\bigl)v(g')$$ for any $h$, $h' \in H$, $g$, $g' \in G$. We have: $$\begin{aligned} \sigma(h) r(g)\bigl(v(g) \rhd (\sigma(h')r(g'))\bigl)&\stackrel{{(\ref{eq:2})}} {=}& \sigma(h) r(g)\bigl(v(g) \rhd \sigma(h')\bigl)\bigl((v (g) \lhd \sigma(h')) \rhd r(g')\bigl)\\ &\stackrel{{(\ref{eq:def2})}} {=}& \sigma(h) r(g)\bigl(v(g) \rhd \sigma(h')\bigl)\bigl(v (g \lhd'h') \rhd r(g')\bigl)\\ &{=}& \sigma(h) r(g)\bigl(v(g) \rhd \sigma(h')\bigl)r(g \lhd' h')^{-1}r(g \lhd' h')\\ &&\bigl(v (g \lhd'h') \rhd r(g')\bigl)\\ &\stackrel{{(\ref{eq:crossed})}} {=}& \sigma(h) r(g)\bigl(v(g) \rhd \sigma(h')\bigl) r(g \lhd' h')^{-1}r\bigl((g\lhd' h') * g'\bigl)\\ &\stackrel{{(\ref{eq:def2})},{(\ref{eq:def1})}} {=}& \sigma(h) r(g)\bigl(v(g) \rhd \sigma(h')\bigl) r\circ v^{-1}\bigl(v(g) \lhd \sigma(h)\bigl)^{-1}\\ && r\circ v^{-1}\bigl[\bigl(v (g \lhd'h')\lhd r(g')\bigl)v(g')\bigl]\\ &\stackrel{{(\ref{eq:def3})}} {=}& \sigma(h)\sigma(g \rhd' h') r\circ v^{-1}\bigl[\bigl(v(g \lhd' h')\lhd r(g')\bigl)v(g')\bigl]\\ &{=}& \sigma\bigl(h(g \rhd' h')\bigl) r\circ v^{-1}\bigl[\bigl(v(g \lhd' h') \lhd r(g')\bigl)v(g')\bigl]\end{aligned}$$ hence [(\[eq:c11\])]{} holds. Moreover: $$\bigl(v(g \lhd' h') \lhd r(g')\bigl) \stackrel{{(\ref{eq:def2})}} {=} \bigl[\bigl(v(g) \lhd \sigma(h')\bigl) \lhd r(g')\bigl]v(g') = \bigl(v(g) \lhd (\sigma(h')r(g'))\bigl)v(g')$$ It follows that [(\[eq:c2\])]{} and hence [(\[eq:def4\])]{} holds and we are done. \(3) Follows from (2) and the [Proposition \[pr:1\]]{}. Schreier type theorems for bicrossed products ============================================= [\[se:3\]]{} In this section we shall prove two Schreier type classification theorems for bicrossed products. Let $H$ and $G$ be two fixed groups. Let $MP(H,G):= \{(\alpha,\beta) ~|~ (H, G, \alpha, \beta) {\rm~is~}{\rm~a~}{\rm~matched~}{\rm~pair}\}$. We define $B_{1}(H,G)$ to be the category having as objects the set $MP(H,G)$ and the morphisms defined as follows: $\psi : (\alpha', \beta') \to (\alpha, \beta)$ is a morphism in $B_{1}(H,G)$ if and only if $\psi: H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G \rightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ is a morphism of groups such that $\psi \circ i_H = i_H$, where $i_H (h) = (h, 1)$ is the canonical inclusion. Thus a morphism in the category $B_{1}(H,G)$ is a morphism between two bicrossed products of $H$ and $G$ that fix $H$. Considering $\sigma = Id_{H}$ and $G' = G$ in [Proposition \[pr:1\]]{} we obtain the following: [\[co:cosch\]]{} Let $(H, G, \alpha, \beta)$ and $(H, G, \alpha', \beta')$ two matched pairs. There exists a one to one correspondence between the set of all morphisms $\psi: H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G \rightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ in the category $B_{1}(H,G)$ and the set of all pairs $(r, v)$, where $r: G \rightarrow H$, $v: G \rightarrow G$ are two maps such that: $$\begin{aligned} (g \rhd' h)r(g \lhd' h) &=& r(g)(v(g) \rhd h){\label{eq:s1}} \\ v(g \lhd' h) &=& v(g) \lhd h {\label{eq:s2}} \\ r(g_{1}g_{2}) &=& r(g_{1})\bigl(v(g_{1}) \rhd r(g_{2}) \bigl) {\label{eq:s3}} \\ v(g_{1}g_{2}) &=& \bigl(v(g_{1}) \lhd r(g_{2})\bigl)v(g_{2}) {\label{eq:s4}}\end{aligned}$$ for all $h \in H$, $g$, $g_{1}$, $g_{2} \in G$. Through the above bijection $\psi$ is given by $${\label{eq:d}} \psi(h,g) = \bigl(hr(g),v(g)\bigl)$$ and $\psi : H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G \rightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ is an isomorphism of groups that fixes $H$ if and only if $v: G\to G$ is a bijective map. We are led to the following: Let $H$ and $G$ be two groups. Two pairs $(\alpha, \beta)$, $(\alpha', \beta') \in MP(H,G)$ are called *1-equivalent* and we denote this by $(\alpha, \beta) \approx_{1} (\alpha', \beta')$ if and only if there exists a pair $(r, v)$, where $r: G \rightarrow H$, $v: G \rightarrow G$ are two maps such that $v$ is bijective and the relations [(\[eq:s1\])]{} - [(\[eq:s4\])]{} hold. Thus, using [Corollary \[co:cosch\]]{}, we obtain that $(\alpha, \beta) \approx_{1} (\alpha', \beta')$ if and only if there exists an isomorphism $(\alpha, \beta) \cong (\alpha', \beta')$ in the category $B_{1}(H,G)$. In particular, $\approx_{1}$ is an equivalence relation on the set $MP(H,G)$ and we have proved: **(First Schreier type theorem for bicrossed products)**[\[th:sch1\]]{} Let $H$ and $G$ be two groups. There exists a bijection between the set of objects of the skeleton of the category $B_{1}(H,G)$ and the pointed quotient set $MP(H,G)/\approx_{1}$. We shall use the following notation: $K^{2}(H,G):= M(H,G)/\approx_{1}$. A general problem arises in order to have a classification type theorem for bicrossed products of two given groups $H$ and $G$: *Compute $K^{2}(H,G)$ for two given groups $H$ and $G$.* In the last section we shall compute explicitly the set $K^2 (C_3, C_6)$. $K^{2}(H,G)$ is a pointed set by the equivalence class of the pair $(\alpha_{0}$, $\beta_{0})$, where $\alpha_{0}$, $\beta_{0}$ are the trivial actions. For $\alpha := \alpha_{0}$ and $\beta := \beta_{0}$ we obtain from relations [(\[eq:s1\])]{} - [(\[eq:s4\])]{} that $\lhd'$ is the trivial action, $r: G\to H$ and $v: G\to G$ are morphisms of groups and $g \rhd' h = r(g)hr(g)^{-1}$, for all $g\in G$ and $h\in H$. For every morphism of groups $r: G\to H$ we shall denote by $\rhd_{r}$ the action $g \rhd_{r} h := r(g)hr(g)^{-1}$, for all $g\in G$ and $h\in H$. Hence $\widehat{(\alpha_{0}, \beta_{0})} = \{(\rhd_{r}, \beta_{0}) ~|~ r: G\to H ~~ {\rm ~is~ a~ morphism~ of~ groups}\}$. We restate this observation as follows: let $H$ and $G$ be two groups. Then there exists $(H, G, \alpha', \beta')$ a matched pair such that $H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G \cong H\times G$ (isomorphism of groups that fixes $H$) if and only if the action $\beta'$ is trivial and there exists a morphism of groups $r: G \rightarrow H$ such that the action $\alpha'$ is given by $g \rhd' h = r(g) h r(g)^{-1}$ for all $g \in G$, $h \in H$. More generally, as a first application of [Theorem \[th:sch1\]]{}, we shall prove the following necessary and sufficient condition for a bicrossed product to be isomorphic to a left version of a semidirect product in the category $B_{1}(H,G)$: [\[co:lake1\]]{} Let $H$, $G$ be two groups and $\alpha : G\times H \to H$ be an action as automorphisms of $G$ on $H$. The following statements are equivalent: 1. There exists $(H, G, \alpha', \beta')$ a matched pair of groups such that $H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G \cong H {}_{\alpha}\ltimes G$ an isomorphism of groups that fixes $H$. 2. The action $\beta'$ is trivial and there exists a pair $(r, v)$, where $v\in {{\rm Aut}\,}(G)$ is an automorphism of $G$, $r: G \to H$ is a map such that $${\label{eq:s3'}} r(g_{1}g_{2}) = r(g_{1})\bigl(v(g_{1}) \rhd r(g_{2}) \bigl)$$ for all $g_1$, $g_2 \in G$ and the action $\alpha '$ is given by $${\label{eq:s1'}} g \rhd' h = r(g)(v(g) \rhd h) r(g)^{-1}$$ for all $g\in G$ and $h\in H$. The isomorphism $\psi: H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G \to H {}_{\alpha}\ltimes G$ in $B_{1}(H,G)$ is given by $\psi (h, g) = (h r(g), v(g))$, for all $h\in H$, $g\in G$. We apply [Corollary \[co:cosch\]]{} in the case that $\beta$ is the trivial action. It this context, using the fact that $v$ is bijective, it follows from [(\[eq:s2\])]{} that the action $\beta'$ is trivial and [(\[eq:s4\])]{} reduces to the fact that $v$ is a morphism, hence an automorphism of $G$. Finally, [(\[eq:s1\])]{} and [(\[eq:s3\])]{} are exactly [(\[eq:s1’\])]{} and [(\[eq:s3’\])]{}. We shall give now a necessary and sufficient condition for a bicrossed product to be isomorphic to a right version of a semidirect product in the category $B_{1}(H,G)$: [\[co:lake2\]]{} Let $H$, $G$ be two groups and $\beta : G\times H \to G$ be an action as automorphisms of $H$ on $G$. The following statements are equivalent: 1. There exists $(H, G, \alpha', \beta')$ a matched pair of groups such that $H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G \cong H \rtimes_{\beta} G$ is an isomorphism of groups that fixes $H$. 2. There exists a pair $(r, v)$, where $r: G \to H$ is a morphism of groups, $v: G\to G$ is a bijective map such that $${\label{eq:s4''}} v(g_{1}g_{2}) = \bigl(v(g_{1}) \lhd r(g_{2})\bigl)v(g_{2})$$ for all $g_1$, $g_2 \in G$ and the actions $\alpha '$ and $\beta'$ are given by $$\begin{aligned} g \rhd' h &=& r(g) \, h \, \Bigl ( r\circ v^{-1} \bigl (v(g) \lhd h\bigl) \Bigl)^{-1} {\label{eq:s1''}}\\ g \lhd' h &=& v^{-1} \bigl( v(g) \lhd h \bigl) {\label{eq:s2''}}\end{aligned}$$ for all $g\in G$ and $h\in H$. The isomorphism $\psi: H\, {}_{\alpha'}\!\! \bowtie_{\beta'} \, G \to H \rtimes_{\beta} G$ in $B_{1}(H,G)$ is given by $\psi (h, g) = (h r(g), v(g))$, for all $h\in H$, $g\in G$. We apply [Corollary \[co:cosch\]]{} in the case that $\alpha$ is the trivial action. It this context, it follows from [(\[eq:s3\])]{} that $r$ is a morphism of groups, while [(\[eq:s1\])]{} and [(\[eq:s2\])]{} are exactly [(\[eq:s1”\])]{} and [(\[eq:s2”\])]{}. In what follows we will prove the second Schreier type theorem for bicrossed products: it is the analogue of the theorem regarding group extensions. Let $(H, G, \alpha, \beta)$ be a matched pair. Then the natural projections $\pi_{G}:H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G \rightarrow G$, $\pi_{H}:H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G \rightarrow H$ are not morphisms of groups. We will fix two groups $H$, $G$ and $\beta: G \times H \rightarrow G$ a right action of the group $H$ on the set $G$. We define $${\rm Ker}(\beta):= \{h \in H ~|~ g \lhd h = g, \forall g \in G\}$$ We denote by $MP_{\beta}(H,G):= \{\alpha ~|~ (H, G, \alpha, \beta) {\rm~is~}{\rm~a~}{\rm~matched~}{\rm~pair~}\}$. Let $B_{2}^{\beta}(H,G)$ be the category having $MP_{\beta}(H,G)$ as the set of objects and the morphisms defined as follows: $\psi: \alpha' \rightarrow \alpha$ is a morphism in $B_{2}^{\beta}(H,G)$ if and only if $\psi: H\, {}_{\alpha'}\!\! \bowtie_{\beta} \, G \rightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ is a morphism of groups such that $${\label{eq:diag2}} \psi \circ i_H = i_H ~~~{\rm and}~~~ \pi_G \circ \psi = \pi_G$$ [\[pr:2\]]{} Let $(H, G, \alpha', \beta)$, $(H, G, \alpha, \beta)$ be two matched pairs. There exists a one to one correspondence between the set of all morphisms $\psi: \alpha' \rightarrow \alpha$ in the category $B_{2}^{\beta}(H,G)$ and the set of all maps $r:G \rightarrow {\rm Ker} (\beta)$ such that : $$\begin{aligned} (g \rhd' h)r(g \lhd h) &=& r(g)(g \rhd h){\label{eq:p1'}} \\ r(g_{1} g_{2}) &=& r(g_{1})\bigl(g_{1} \rhd r(g_{2})\bigl) {\label{eq:p3'}}\end{aligned}$$ for all $g, g_{1}, g_{2} \in G$, $h \in H$. Through the above bijection the morphism $\psi$ is given by $${\label{eq:p5'}} \psi(h,g) = \bigl(hr(g), g\bigl)$$ for all $h \in H$, $g \in G$ and $\psi$ is an isomorphism of groups i.e. $B_{2}^{\beta}(H,G)$ is a grupoid [^2]. For any morphism of groups $\psi: H\, {}_{\alpha'}\!\! \bowtie_{\beta} \, G \rightarrow H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ such that [(\[eq:diag2\])]{} hold there exists a unique map $r: G\to H$ such that $\psi (h, g) = (h r(g), g)$, for all $h\in H$ and $g\in G$. Now we are in a position to use [Proposition \[pr:1\]]{} for $G' = G$, $\beta' = \beta$ and $v = Id_G$. We obtain [(\[eq:p1’\])]{} and [(\[eq:p3’\])]{} by considering $v = Id_{G}$ in [(\[eq:p1\])]{}, respectively [(\[eq:p3\])]{}. On the other hand [(\[eq:p2\])]{} is trivially fulfilled and [(\[eq:p4\])]{} becomes $g_{1} = g_{1} \lhd r(g_{2})$ for all $g_{1}$, $g_{2} \in G$, i.e. ${\rm Im}(r) \subseteq {\rm Ker}(\beta)$. If $\beta$ is a faithful action (i.e. ${\rm Ker}(\beta) = 1$) then $B_{2}^{\beta}(H,G)$ is a discret grupoid i.e. there exists a morphism $\psi: \alpha' \rightarrow \alpha$ if and only if $\alpha = \alpha'$. Indeed, in this case $r(g) = 1$ for all $g \in G$, [(\[eq:p3’\])]{} is trivially fulfilled and [(\[eq:p1’\])]{} reduces to $g \rhd h = g \rhd' h$ i.e. $\alpha = \alpha'$. Hence, in this case the skeleton of the category $B_{2}^{\beta}(H,G)$ is the set $MP_{\beta}(H,G)$. Let $H$, $G$ be two groups and $\beta : G \times H \rightarrow G$ be a right action. Two elements $\alpha'$ and $\alpha$ of $MP_{\beta}(H,G)$ are called $\approx_{2}$-equivalent and we denote this by $\alpha' \approx_{2} \alpha$ if there exists a map $r:G \rightarrow {\rm Ker}(\beta)$ such that the relations [(\[eq:p1’\])]{} and [(\[eq:p3’\])]{} hold. From [Proposition \[pr:2\]]{} we obtain that $\alpha' \approx_{2} \alpha$ if and only if there exists an isomorphism $\alpha' \cong \alpha$ in $B_{2}^{\beta}(H,G)$. Hence $\approx_{2}$ is an equivalence relation on $MP_{\beta}(H,G)$ and we obtained the following: **(The second Schreier type theorem for bicrossed products)**[\[th:sch22\]]{} Let $H$, $G$ be two groups and $\beta: G \times H \rightarrow G$ be a right action. There exists a bijection between the set of objects of the skeleton of the category $B_{2}^{\beta}(H,G)$ and the quotient set $MP_{\beta}(H,G)/\approx_{2}$. It is possible that the set $MP_{\beta}(H,G)$ (and hence $MP_{\beta}(H,G)/\approx_{2}$) is the empty set. However, if $\beta : G\times H \to G$ is an action as automorphisms then $MP_{\beta}(H,G)$ is nonempty as it contains the trivial action $\alpha_0$. In this case the quotient set $MP_{\beta}(H,G)/\approx_{2}$ is a pointed set by the equivalence class of the trivial action $\alpha_0$. It follows from [Proposition \[pr:2\]]{} that $$\widehat{\alpha_{0}} = \{\alpha' | \, \alpha' (g, h) = r(g) h r(g \lhd h)^{-1}, {\rm for ~ some ~} r: G\to {\rm Ker}(\beta) ~~ {\rm ~ a~ morphism~ of~ groups}\}$$ We record this observation in the following: Let $H$, $G$ be two groups, $\beta: G \times H \rightarrow G$ an action as automorphisms and $H\rtimes_{\beta} G$ the right version of the semidirect product. The following statements are equivalent: 1. There exists a matched pair $(H, G, \alpha, \beta)$ such that the bicrossed products $H\, {}_{\alpha}\!\! \bowtie_{\beta} \, G$ and $H\rtimes_{\beta} G$ are isomorphic in the category $B_{2}^{\beta}(H,G)$; 2. There exists a morphism of groups $r: G \rightarrow {\rm Ker}(\beta)$ such that the action $\alpha$ is given by $g \rhd h = r(g) h r(g \lhd h)^{-1}$ for all $g \in G$, $h \in H$. Examples ======== [\[se:4\]]{} In this section we describe all matched pairs between $C_{n}$ and $C_{m}$, for $n \in \{2,3\}$ and $m \in \NN^{*}$ arbitrary. First, let us introduce some notation. We denote by $a$ a generator of the cyclic group $C_n$ and $b$ a generator of $C_m$. The set of group morphisms from the group $C_n$ to the group of automorphisms ${{\rm Aut}\,}(C_m)$ will be denoted by $\varsigma (n, m)$. Such a morphism $\vartheta : C_n \rightarrow {{\rm Aut}\,}(C_m)$ is uniquely determined by a positive integer $t\in [m-1] := \{1, 2, \cdots, m-1\} $ such that $m|t^n -1$ and $${\label{eq:2.4.399}} \vartheta : C_n \rightarrow {{\rm Aut}\,}(C_m), \qquad \vartheta (a) (b) = b^t$$ Therefore, one can equivalently think of $\varsigma (n, m)$ as the subgroup of $U(\ZZ_m)$ consisting of all solutions in $\ZZ_{m}$ of the equation $x^n = 1$. Using the fact that if $m= 2^{a_0} p_1^{a_1}\cdots p_k^{a_k}$ with $p_1$, $\cdots$, $p_k$ odd primes, then $${{\rm Aut}\,}(C_m)\cong U( \ZZ_m ) \cong U( \ZZ_{2^{a_0}}) \times U(\ZZ_{p_1^{a_1}}) \times \cdots \times U(\ZZ_{p_k^{a_k}})$$ it is a routine computation to check that $$|\varsigma (n, m)| = \left \{\begin{array}{rcl} \prod_{i=1}^k (n, p_i^{a_i} - p_i^{a_i -1}), \, & \mbox { if }& 4\!\not| m\\ (n,2) (n, 2^{a_0 -2}) \prod_{i=1}^k (n, p_i^{a_i} - p_i^{a_i -1}), \, & \mbox { if }& 4|m \end{array} \right.$$ In particular $$|\varsigma (2, m) | = \left \{\begin{array}{rcl} 2^k, \, & \mbox {\rm if }& \, a_0\leq 1\\ 2^{k+1}, \, & \mbox {\rm if }& \, a_0=2\\ 2^{k+2}, \, & \mbox {\rm if }& \, a_0\geq 3 \end{array} \right.$$ and $$| \varsigma (p, m) | = \left \{\begin{array}{rcl} \prod_{i=1}^k (p, p_i -1), \, & \mbox {\rm if }& \, p^2\!\not| m\\ p\prod_{i=1}^k (p, p_i -1), \, & \mbox {\rm if }& \, p^2|m \end{array} \right.$$ for an odd prime $p$. Let $m$ be a positive integer. Then $(C_2, C_m, \alpha, \beta)$ is a matched pair if and only if the action $\alpha$ is trivial and there exists a positive integer $t\in [m-1]$ such that $m|t^2 -1$ and $\beta = \beta_t : C_m \times C_2 \rightarrow C_m$ is given by $${\label{eq:2.4.6}} \beta (b^i, a) = b^{it}, \quad \beta (b^i, 1) = b^i$$ for any $i= 0, \cdots, m-1$. In particular, there are $|\varsigma (2, m)|$ matched pairs $(C_2, C_m, \alpha, \beta)$. Indeed, as $\alpha$ is an action we get $b\triangleright a \neq 1 = b \triangleright 1$. Thus $b \triangleright a = a$ which implies that $\alpha$ is trivial. Thus $(C_2, C_m, \alpha, \beta)$ is a matched pair if and only if $\beta ' : C_2 \rightarrow {{\rm Aut}\,}(C_m)$, $\beta' (x) (y) := \beta (y, x)$ is a morphism of groups, so by letting $n=2$ in [(\[eq:2.4.399\])]{} we obtain that $(C_2, C_m, \alpha, \beta)$ is a matched pair if and only if there exists $t\in [m-1]$ such that $m|t^2 -1$ and $\beta (b, a) = b^t$. The formula [(\[eq:2.4.6\])]{} follows as $\beta$ is an action. In order to describe all matched pairs $(C_3, C_m, \alpha, \beta)$ we need the following observation. [\[re:knitauto\]]{} Let $(H, G, \alpha, \beta)$ be a matched pair such that $\alpha$ is an action of $G$ on $H$ as group automorphisms. Then the compatibility condition [(\[eq:2\])]{} from the definition of a matched pair is equivalent to $(g\triangleleft {h_1}) \triangleright h_2 = g \triangleright h_2$ that can be written as $${\label{eq:KS4'}} g^{-1}(g\triangleleft {h_1}) \in {\rm Stab}_G (h_2)$$ for any $g\in G$, $h_1$, $h_2\in H$. Thus if $\alpha$ is an action as automorphisms then $(H, G, \alpha, \beta)$ is a matched pair if and only if [(\[eq:3\])]{} and [(\[eq:KS4’\])]{} hold. The condition [(\[eq:KS4’\])]{} gives important information regarding $\beta$: the elements $g^{-1}\beta (g, h)$ act trivially on $H$ for any $g\in G$ and $h\in H$. Now we can describe all matched pairs $(C_3, C_m, \alpha, \beta)$. [\[pr:2.4.45\]]{} Let $m$ be a positive integer, $\alpha : C_m \times C_3 \rightarrow C_3$, $\beta : C_m \times C_3 \rightarrow C_m$ two maps and $t\in [m-1]$ such that $m|t^3 -1$. Then: 1. Let $\alpha$ be the trivial action and $\beta = \beta_t : C_m \times C_3 \rightarrow C_m$ given by $${\label{eq:2.4.90}} \beta (b^i, a) = b^{it}, \quad \beta (b^i, a^2) = b^{it^2}, \quad \beta (b^i, 1) = b^i$$ for any $i= 0, \cdots, m-1$. Then $(C_3, C_m, \alpha, \beta_t)$ is a matched pair. There are no other matched pairs $(C_3, C_m, \alpha, \beta)$ if $m$ is odd. 2. Assume that $m$ is even. Let $\beta$ be the trivial action and $\alpha : C_m \times C_3 \rightarrow C_3$ given by $\alpha (b^j, 1) = 1$ and $${\label{eq:2.4.740}} \alpha (b^j, a) = \left \{\begin{array}{rcl} a, \, & \mbox {\rm if $j$ is even }\\ a^2, \, & \mbox {\rm if $j$ is odd } \end{array} \right.$$ $${\label{eq:2.4.750}} \alpha (b^j, a^2) = \left \{\begin{array}{rcl} a^2, \, & \mbox {\rm if $j$ is even }\\ a, \, & \mbox {\rm if $j$ is odd } \end{array} \right.$$ for all $j = 1, \cdots, m-1$. Then $(C_3, C_m, \alpha, \beta)$ is a matched pair. 3. Assume that $m = 6 u$ for some positive integer $u$ and that $\alpha$ is described by [(\[eq:2.4.740\])]{} and [(\[eq:2.4.750\])]{}. Then there exist two matched pairs $(C_3, C_m, \alpha, \beta)$, $(C_3, C_m, \alpha, \beta')$, where $\beta$ and $\beta'$ are given by $${\label{eq:24.11}} \beta (b^{2k+1}, a) = b^{2u +2k +1}, \qquad \beta (b^{2k+1}, a^2) = b^{4u+ 2k +1}$$ $${\label{eq:24.11'}} \beta (b^{2k}, a) = \beta (b^{2k}, a^2) = b^{2k}$$ and $${\label{eq:24.12}} \beta' (b^{2k+1}, a) = b^{4u +2k +1}, \qquad \beta' (b^{2k+1}, a^2) = b^{2u+ 2k + 1}$$ $${\label{eq:24.12'}} \beta'(b^{2k}, a) = \beta'(b^{2k}, a^2) = b^{2k}$$ for all nonnegative integers $k$. In this case there are $2+ |\varsigma (3, m)|$ matched pairs between $C_3$ and $C_m$. 4. There are no other matched pairs on $(C_3, C_m, \alpha, \beta)$ other than the ones described above. We assume first that $\alpha$ is the trivial action. It follows from [Remark \[re:2.4.90\]]{} that $(C_3, C_m, \alpha, \beta)$ is a matched pair if and only if $\beta ' : C_3 \rightarrow {{\rm Aut}\,}(C_m)$, $\beta' (x) (y) := \beta (y, x)$ is a morphism of groups; setting $n=3$ in [(\[eq:2.4.399\])]{} we get that $(C_3, C_m, \alpha, \beta)$ is a matched pair if and only if there exists $t\in [m-1]$ such that $m|t^3 -1$ and $\beta (b, a) = b^t$. The formula [(\[eq:2.4.90\])]{} follows as $\beta$ is an action. Assume now that $m$ is odd. It follows from [(\[eq:4\])]{} that $b^i \triangleright 1 = 1$ for all $i= 0, \cdots, m-1$. If $b \triangleright a = a$ we obtain that $\alpha$ is trivial. Assume that $b \triangleright a = a^2$. Then $b \triangleright a^2 = a$ and $\alpha$ is given by [(\[eq:2.4.740\])]{}, [(\[eq:2.4.750\])]{}. If $m$ is odd we obtain: $a = 1\triangleright a = b^m \triangleright a = a^2$, contradiction. Hence, for an odd $m$ the action $\alpha$ must be trivial and (i) is proved. Assume now that $m$ is even and $\beta$ is the trivial action. Then $(C_3, C_m, \alpha, \beta)$ is a matched pair if and only if $\alpha$ is an action of $C_m$ on $C_3$ as group automorphisms. The map $\alpha$ given by [(\[eq:2.4.740\])]{}, [(\[eq:2.4.750\])]{} is such an action corresponding to $$C_m \rightarrow {{\rm Aut}\,}(C_3), \quad b \mapsto (a \mapsto a^2)$$ Therefore (ii) is proved. We shall prove now (iii) and (iv). Let $(C_3, C_m, \alpha, \beta)$ be a matched pair. We have proved that $\alpha : C_m \times C_3 \rightarrow C_3$ is either the trivial action or it is given by [(\[eq:2.4.740\])]{}, [(\[eq:2.4.750\])]{} if $m$ is even. We assume now that $m$ is even and that $\alpha$ is given by [(\[eq:2.4.740\])]{}, [(\[eq:2.4.750\])]{}. Then $\alpha$ is an action of $C_m$ on $C_3$ as automorphisms and $${\rm Stab}_{C_m} (a) = {\rm Stab}_{C_m} (a^2) = <b^2>$$ Using [Remark \[re:knitauto\]]{} we get that $b^{-1}\beta (b, a) \in \, <b^2>$ and $b^{-1}\beta (b, a^2) \in \, <b^2>$ and [(\[eq:2\])]{} holds automatically. Let $l$, $t \in \{0, 1, \cdots, m/2 -1 \}$ such that $${\label{eq:4.444}} \beta (b, a) = b^{2l+1}, \qquad \beta (b, a^2) = b^{2t+1}$$ We shall extend $\beta$ for each element of $C_m \times C_3$ using [(\[eq:3\])]{} as defining relations and the fact that $\beta$ is an action. First we define $\beta$ for each pair $(b^i, a)$ such that [(\[eq:3\])]{} holds. We have $$\beta (b^2, a ) = (bb)\triangleleft a \stackrel{{(\ref{eq:3})} } = (b\triangleleft {(b\triangleright a)}) (b\triangleleft a) \stackrel{{(\ref{eq:2.4.740})} } = (b\triangleleft {a^2})(b \triangleleft a) = b ^{2(t+l +1)}$$ and $$\beta (b^3, a ) = (bb^2)\triangleleft a \stackrel{{(\ref{eq:3})} } = (b\triangleleft {(b^2\triangleright a)}) (b^2\triangleleft a) \stackrel{{(\ref{eq:2.4.740})} } = (b\triangleleft {a}) (b^2\triangleleft a) = b ^{4l + 2t +3}$$ Using the induction we can prove $${\label{eq:4.445}} \beta (b^{2k}, a) = b^{2k(l+t+1)} , \qquad \beta (b^{2k+1}, a) = b^{(2k+2)l+2kt+2k+1}$$ for any $k = 0, 1, \cdots$. We note that $$b^{2t+1} = \beta (b, a^2) = b\triangleleft {a^2} = (b\triangleleft a)\triangleleft a = \beta (b^{2l+1}, a)\stackrel{{(\ref{eq:4.445})} } = b^{(2l+2)l + 2lt + 2l + 1}$$ As the order of $b$ is $m$ we get a first compatibility condition for $l$ and $t$: $${\label{eq:c1}} m | 2(l^2 + 2l + lt -t)$$ Now we define $\beta$ for each pair $(b^i, a^2)$ using [(\[eq:3\])]{} repeatedly. We have: $$\beta (b^2, a^2) = (bb)\triangleleft {a^2} \stackrel{{(\ref{eq:3})} }= (b\triangleleft {(b\triangleright a^2)}) (b\triangleleft {a^2}) \stackrel{ {(\ref{eq:2.4.750})} } =( b\triangleleft a)( b\triangleleft {a^2}) = b^{2(l + t + 1 )}$$ $$\beta (b^3, a^2) = (bb^2)\triangleleft {a^2} \stackrel{{(\ref{eq:3})} }= (b\triangleleft {(b^2\triangleright a^2)}) (b^2 \triangleleft {a^2}) \stackrel{ {(\ref{eq:2.4.750})} } = (b\triangleleft {a^2}) (b^2\triangleleft {a^2}) = b^{2l + 4t + 3}$$ Using the induction we can easily prove that $${\label{eq:4.445'}} \beta (b^{2k}, a^2) = b^{2k(l+t+1)} , \qquad \beta (b^{2k+1}, a^2) = b^{ 2kl + (2k+2)t + 2k + 1}$$ for any $k = 0, 1, \cdots$. Moreover, keeping in mind [(\[eq:4.445\])]{} we find that $$\beta (b^{2k}, a) = \beta (b^{2k}, a^2) = b^{2k(l+t+1)}$$ On the other hand $\beta$ is a right action and $a^3 = 1$. Hence: $$b^{2k} = \beta (b^{2k}, 1) = (b^{2k}\triangleleft {a^2})\triangleleft a = (b^{2k}\triangleleft {a})\triangleleft a = b^{2k}\triangleleft {a^2} = b^{2k(l+t+1)}$$ As the order of $b$ is $m$ we obtain a second compatibility condition between $l$ and $t$: $m | 2k(l+t)$ for any $k = 0, 1, \cdots$ which is equivalent to: $${\label{eq:c5}} m | 2(l+t)$$ From this condition and [(\[eq:c1\])]{} we obtain $${\label{eq:c1'}} m | 2(2l - t)$$ Let now $m = 2r$. We have to find $l$, $t \in \{1, 2, \cdots, r-1 \}$ such that $$m | 2(l+t) \quad {\rm and} \quad m | 2(2l -t)$$ Equivalently, we have to solve in $\ZZ_r$ the system of equations $${\label{eq:4.600}} \left \{\begin{array}{rcl} \hat{l} + \hat{t} = \hat{0} \\ \hat{2}\hat{l} - \hat{t} = \hat{0} \end{array} \right.$$ The equation $\hat{3} \hat{l} = \hat{0}$ has $(3, r)$ solutions in $\ZZ_r$. If $3$ does not divide $m$ then the unique solution of the system is $\hat{l} = \hat{t} = \hat{0}$ and therefore $\beta$ is the trivial action. If $3$ divides $r$ let $u$ be such that $r = 3u$. Then the system [(\[eq:4.600\])]{} has three solutions $$\hat{l_1} = \hat{t}_1 = \hat{0}; \qquad \hat{l}_2 = \hat{u}, \, \hat{t}_2 = \hat{2}\hat{u}, \qquad \hat{l_3} = \hat{2}\hat{u}, \, \hat{t_3} = \hat{4}\hat{u}$$ The first solution gives that the action $\beta$ is trivial and the last two solutions give exactly the two actions $\beta$ described in [(\[eq:24.11\])]{} and [(\[eq:24.12\])]{}. We showed that the smallest example of a proper matched pair (i.e. one in which both actions are nontrivial) between two finite cyclic groups is the one between the groups $C_3$ and $C_6$. According to [Proposition \[pr:2.4.45\]]{} there exist exactly four matched pairs $(C_3, C_6, \alpha, \beta)$ namely: 1. $\alpha_{0}$ and $\beta_{0}$ are the trivial actions; 2. $\beta_{0}$ is the trivial action and $\alpha_{1}$ is defined by: $$\begin{aligned} b^j\rhd_{1} a = \left \{\begin{array}{rcl} a, \, & \mbox {\rm if $j$ is even }\\ a^2, \, & \mbox {\rm if $j$ is odd } \end{array} \right. , \quad b^j \rhd_{1} a^2 = \left \{\begin{array}{rcl} a^2, \, & \mbox {\rm if $j$ is even }\\ a, \, & \mbox {\rm if $j$ is odd } \end{array} \right.\end{aligned}$$ for all $j = 1, \cdots, 5$. 3. $\alpha_{2}$ and $\beta_{2}$ are defined by : $$\begin{aligned} b^j\rhd_{2} a = \left \{\begin{array}{rcl} a, \, & \mbox {\rm if $j$ is even }\\ a^2, \, & \mbox {\rm if $j$ is odd } \end{array} \right. , \quad b^j \rhd_{2} a^2 = \left \{\begin{array}{rcl} a^2, \, & \mbox {\rm if $j$ is even }\\ a, \, & \mbox {\rm if $j$ is odd } \end{array} \right.\end{aligned}$$ and $$\begin{aligned} b^j\lhd_{2} a = \left \{\begin{array}{rcl} b^{j}, \, & \mbox {\rm if $j$ is even }\\ b^{j+2}, \, & \mbox {\rm if $j$ is odd } \end{array} \right. , \quad b^j \lhd_{2} a^2 = \left \{\begin{array}{rcl} b^{j}, \, & \mbox {\rm if $j$ is even }\\ b^{j+4}, \, & \mbox {\rm if $j$ is odd } \end{array} \right.\end{aligned}$$ for all $j = 1, \cdots, 5$. 4. $\alpha_{2}$ and $\beta_{3}$ are defined by: $$\begin{aligned} b^j\lhd_{3} a = \left \{\begin{array}{rcl} b^{j}, \, & \mbox {\rm if $j$ is even }\\ b^{j+4}, \, & \mbox {\rm if $j$ is odd } \end{array} \right. , \quad b^j \lhd_{3} a^2 = \left \{\begin{array}{rcl} b^{j}, \, & \mbox {\rm if $j$ is even }\\ b^{j+2}, \, & \mbox {\rm if $j$ is odd } \end{array} \right.\end{aligned}$$ for all $j = 1, \cdots, 5$. We shall now classify all bicrossed products $C_3 \bowtie C_6$ that fixes the group $C_3$, i.e. we shall determine the pointed set $K^{2}(C_{3},C_{6})$ from [Theorem \[th:sch1\]]{}. $K^{2}(C_{3},C_{6})$ is a pointed set with three elements. In particular, any bicrossed product $C_3 \bowtie C_6$ that fixes the group $C_3$ is isomorphic to one of the following three groups: $$C_3 \times C_6, \quad < a, b \, | \, a^3 = 1, \, b^6 = 1, \, ba = a^2 b > , \quad < a, b \,|\, a^3 = 1, \, b^6 = 1, \, ba = a^2b^3 >$$ Let $(\alpha', \beta')$ be a matched pair such that $(\alpha_{0}, \beta_{0}) \approx_{1} (\alpha', \beta')$. The relations [(\[eq:s1\])]{} - [(\[eq:s2\])]{} collapse into: $$\begin{aligned} (g \rhd' h) r(g \lhd ' h) &=& r(g)h {\label{eq:ex1}}\\ v(g \lhd ' h) &=& v(g) {\label{eq:ex2}}\end{aligned}$$ Since $v$ is a bijective map, it follows from [(\[eq:ex2\])]{} that $\beta'$ is the trivial action. Furthermore, from [(\[eq:ex1\])]{} we obtain that $\alpha'$ is also the trivial action, that is, the equivalence class of $(\alpha_{0}, \beta_{0})$ is trivial. By similar arguments it follows that the equivalence class of $(\alpha_{1}, \beta_{0})$ is also trivial. Consider now $r: C_{6} \rightarrow C_{3}$ be the trivial morphism of groups and $v: C_{6} \rightarrow C_{6}$ the automorphism given by $v(b) := b^{5}$. By a straightforward computation it follows that: $$\begin{aligned} g \rhd_{2} h = v(g)\rhd_{2} h, \quad v(g \lhd_{3} h) = v(g) \lhd_{2} h\end{aligned}$$ hence $(\alpha_{2}, \beta_{2}) \approx_{1} (\alpha_{2}, \beta_{3})$. Thus, $K^{2}(C_{3},C_{6})$ is a set with three elements. It is easy to see that $B_{2}^{\beta}(C_{3},C_{6})$ is a singleton or a set with two elements for any right action $\beta$. Indeed, it is obvious that $B_{2}^{\beta_{2}}(C_{3},C_{6}) = \{(\alpha_{2}, \beta_{2})\}$ and $B_{2}^{\beta_{3}}(C_{3},C_{6}) = \{(\alpha_{2}, \beta_{3})\}$. Now suppose that $(\alpha_{0}, \beta_{0}) \approx_{2} (\alpha_{1}, \beta_{0})$. From [(\[eq:p1’\])]{} we obtain that $g \rhd_{0} h = g \rhd_{1} h$ for all $g \in C_{3}$, $h \in C_{6}$ which is a contradiction. Thus $B_{2}^{\beta_{0}}(C_{3},C_{6}) = \{(\alpha_{0}, \beta_{0}), (\alpha_{1}, \beta_{0})\}$. We end the section by showing the difficulty of the problem of finding all matched pairs $(C_n, C_m, \alpha, \beta)$ between arbitrary finite cyclic groups. For any positive integer $k$ we denote by $\ZZ_k$ the ring of residue classes modulo $k$ and by $S(\ZZ_k)$ the set of bijective functions from $\ZZ_k$ to itself. Let $\alpha : C_m \times C_n \rightarrow C_n$, and $\beta: C_m \times C_n \rightarrow C_m$ be two actions. They are completely determined by two maps $$\theta : \ZZ_n \rightarrow \ZZ_n \quad \text{and} \quad \phi: \ZZ_m \rightarrow \ZZ_m$$ such that $$\alpha (b, a^x ) = a ^{\theta (x)}, \qquad \beta (b^y, a) = b^{\phi(y)}$$ for any $x \in \ZZ_n$, $y \in \ZZ_m$. As $\alpha$ and $\beta$ are actions we obtain that $\theta$ and $\phi$ are bijections (and hence they can be regarded as elements of $S(\ZZ_n)$ and respectively $S(\ZZ_m)$) and $${\label{eq:douglas1}} \alpha (b^i, a^x) = a ^{\theta^i(x)}, \qquad \beta (b^y, a^j) = b^{\phi^j (y)}$$ for all positive integers $i$, and $j$. In particular, $${\label{eq:douglas2}} \theta^m (x) = x \quad\text{and}\quad \phi^n (y) = y$$ for all $x \in \ZZ_n$ and $y\in \ZZ_m$. Hence, as an element of $S(\ZZ_n)$ the order of $\theta$ is a divisor of $m$ and as an element of $S(\ZZ_m)$ the order of $\phi$ is a divisor of $n$. Using [(\[eq:douglas1\])]{} we obtain immediately that the compatibility conditions [(\[eq:2\])]{}, [(\[eq:3\])]{} defining a matched pair are equivalent to $$\theta ^y (x + z) - \theta^y (z) = \theta^{\phi^z (y)}(x)$$ $$\phi ^y (x + z) - \phi^y (z) = \phi ^{\theta^z (y)}(x)$$ for all $x$, $y$, $z$ and all operations are in the relevant rings. Finally, [(\[eq:4\])]{} is equivalent to $\theta (0) = 0$ and $\phi (0) = 0$. To conclude, we have the following [\[pr:douglasconditions\]]{} Let $m$ and $n$ be two positive integers. There is a one to one correspondence between the set of all matched pairs $(C_n, C_m, \alpha, \beta)$ and the set of pairs $(\theta, \phi) \in S(\ZZ_n) \times S(\ZZ_m)$ such that $${\label{eq:douglas3}} \theta (0) = 0, \qquad \phi (0) = 0$$ $${\label{eq:douglas4}} ord(\theta) | m, \qquad ord(\phi) | n$$ $${\label{eq:douglas5}} \theta ^y (x + z) - \theta^y (z) = \theta ^{\phi^z (y)}(x)$$ $${\label{eq:douglas6}} \phi ^y (x + z) - \phi^y (z) = \phi ^{\theta^z (y)}(x)$$ for all $x$, $y$, $z$ (all the operations are taking place in the relevant rings). This is [@Douglas Theorem I] and the pairs $(\theta, \phi) \in S(\ZZ_n) \times S(\ZZ_m)$ satisfying the above conditions are called in [@Douglas] [*conjugate special substitutions*]{}. 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--- abstract: 'While the analysis of airborne laser scanning (ALS) data often provides reliable estimates for certain forest stand attributes – such as total volume or basal area – there is still room for improvement, especially in estimating species-specific attributes. Moreover, while information on the estimate uncertainty would be useful in various economic and environmental analyses on forests, a computationally feasible framework for uncertainty quantifying in ALS is still missing. In this article, the species-specific stand attribute estimation and uncertainty quantification (UQ) is approached using Gaussian process regression (GPR), which is a nonlinear and nonparametric machine learning method. Multiple species-specific stand attributes are estimated simultaneously: tree height, stem diameter, stem number, basal area, and stem volume. The cross-validation results show that GPR yields on average an improvement of 4.6% in estimate RMSE over a state-of-the-art k-nearest neighbors (kNN) implementation, negligible bias and well performing UQ (credible intervals), while being computationally fast. The performance advantage over kNN and the feasibility of credible intervals persists even when smaller training sets are used.' author: - 'Petri Varvia, Timo Lähivaara, Matti Maltamo, Petteri Packalen, Aku Seppänen[^1][^2][^3][^4]' bibliography: - 'IEEEabrv.bib' - 'bibliography.bib' title: Gaussian process regression for forest attribute estimation from airborne laser scanning data --- forest inventory, LiDAR, area based approach, machine learning, Gaussian process 1000 Copyright notice {#copyright-notice .unnumbered} ================ P. Varvia, T. Lähivaara, M. Maltamo, P. Packalen and A. Seppänen, “Gaussian Process Regression for Forest Attribute Estimation From Airborne Laser Scanning Data,” in IEEE Transactions on Geoscience and Remote Sensing. doi: 10.1109/TGRS.2018.2883495\ 2018 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. Introduction ============ Forest inventories based on airborne laser scanning (ALS) are becoming increasingly popular. Therefore, it is more and more important to have well performing methods for the estimation/prediction of stand attributes, such as basal area and tree height. Coupled with the prediction procedures, efficient methods for the quantification of prediction uncertainty are also urgently needed for forestry planning and assessment purposes [@kangas2018]. Operational forest inventories employing ALS data are most often implemented with the area based approach (ABA) [@naesset2002]. In ABA, metrics used as predictor variables are calculated from the ALS returns within a plot or grid cell. Using training plots with field-measured stand attributes, a model is formulated between the stand attributes and ABA metrics. This statistical model is then used to predict the stand attributes for each grid cell [@reutebuch2005; @maltamobook] and the predictions are finally aggregated to the desired area, e.g. to a stand. Although tree species is among the most important attributes of forest inventory, the ALS research does not particularly reflect this. One reason for this is that in many biomes the number of tree species is so high that it is practically impossible to separate them by remote sensing. In the Nordic countries, however, the majority of the growing stock comes from three economically valuable tree species. The species-specific prediction is approached two ways in Nordic countries: in Norway, stands are stratified according to tree species by visual interpretation of aerial images before the actual ALS inventory [@naesset2004], whereas in Finland, stand attributes are predicted by tree species using a combined set of metrics from ALS data and aerial images [@packalen2007]. In both approaches, aerial images are used to improve the discrimination of tree species. Uncertainty estimation is a key component in strategic inventories that cover large areas [@mandallaz2007]. ALS can be used in that context too. For example, ALS metrics can be used as auxiliary variables in model-based (e.g. [@staahl2010]) or model assisted (e.g. [@gregoire2010]) estimation of some forest parameter. Typically, sample mean and sample variance are estimated to the area of interest (e.g. 1000000 ha) using a certain number (e.g. 500) of sample plots and auxiliary variables covering all population elements. In the stand level forest management inventories, the situation is different: the point estimate and its confidence intervals are needed for each stand and there may not be any sample plots in most stands. Today, most ALS inventories can be considered as stand level management inventories. Commonly in ABA, when using prediction methods such as linear regression or , only point estimates without accompanying uncertainty metrics are computed. Plot or cell level prediction uncertainty has garnered some research interest in recent years and several methods of predicting plot/cell level variance have been proposed [@junttila2008a; @finley2013; @magnussen2016]. Recently, a Bayesian inference approach to quantify uncertainty within the framework of the ABA was proposed by Varvia *et al.* [@varvia]. The main shortcoming in the method proposed in [@varvia] is that it is computationally costly: wall-to-wall uncertainty quantification of a large forest area would require considerable computer resources. Gaussian process regression (GPR) [@rasmussenbook] is a machine learning method that provides an attractive alternative; compared with the more widely used machine learning methods, such as artificial neural networks [@niska2010neural; @alsdeeplearning], GPR also produces an uncertainty estimate for the prediction. Univariate GPR was tested for estimation of several total stand attributes by Zhao *et al.* [@alsgpr], where it was found to significantly outperform (log)linear regression. In this paper, we propose a multivariate GPR for simultaneous estimation of species-specific stand attributes within ABA. The estimation accuracy of GPR is compared with kNN and the uncertainty quantification performance with the Bayesian inference method of [@varvia]. Furthermore, the effect of training set size on its performance is evaluated. Materials ========= The same test data as in [@varvia] is used in this study. In this section, the data set is briefly summarized, for detailed description, see e.g. [@packalen2009; @Packalen2012]. The test area is a managed boreal forest located in Juuka, Finland. The area is dominated by Scots pine (*Pinus sylvestris* L.) and Norway spruce (*Picea abies* (L.) Karst.), with a minority of deciduous trees, mostly downy birch (*Betula pubescens* Ehrh.) and silver birch (*Betula pendula* Roth.). The deciduous trees are considered as a single group. The field measurements were done during the summers of 2005 and 2006. Total of 493 circular sample plots of radius 9 m are used in this study. The diameter at breast height (DBH), tree and storey class, and tree species were recorded for each tree with DBH larger than 5 cm and the height of one sample tree of each species in each storey class was measured. The heights of other trees on the plot were predicted using a fitted Näslund’s height model [@naslund]. The species-specific stand attributes were then calculated using the measured DBH and the predicted heights. The stand attributes considered in this study are tree height ($H_{\mathrm{gm}}$), diameter at breast height ($D_{\mathrm{gm}}$), stem number ($N$), basal area ($\mathit{BA}$), and stem volume ($V$). The ALS data and aerial images were captured in 13 July 2005 and 1 September 2005, respectively. The ALS data has a nominal sampling density of 0.6 returns per square meter, with a footprint of about 60 cm at ground level. The orthorectified aerial images contain four channels (red, green, blue, and near infrared). A total of $n_x=77$ metrics were computed from the ALS point cloud and aerial images and used in ABA. The metrics include canopy height percentiles, the corresponding proportional canopy densities, the mean and standard deviation of the ALS height distribution, the fraction of above ground returns (i.e. returns with $z>2$ m), and metrics computed from the LiDAR intensity. From the aerial images, the mean values of each channel were used along with two spectral vegetation indices [@packalen2009]. Methods {#sec:methods} ======= Let us denote a vector consisting of the stand attributes by $\mathbf{y}\in\mathbb{R}^{15}$; the vector $\mathbf{y}$ contains the species-specific (pine, spruce, deciduous) $H_{\mathrm{gm}}$, $D_{\mathrm{gm}}$, $N$, $\mathit{BA}$, and $V$, resulting in a total of $n_y=15$ variables. The vector of predictors (ALS and aerial image metrics) is denoted by $\mathbf{x}\in\mathbb{R}^{n_x}$. The general objective is to learn a nonlinear regression model $$\label{thefunc} \mathbf{y}=f(\mathbf{x})+\mathbf{e},$$ where $\mathbf{e}$ is an error term, from a set of $n_t$ training data $(\mathbf{Y}_t,\mathbf{X}_t)$. Let $\mathbf{Y}$ be a finite collection of points $\mathbf{y}^{(i)}=f(\mathbf{x}^{(i)})+\mathbf{e}^{(i)}$ concatenated in a long vector. In Gaussian process regression [@rasmussenbook], the joint probability distribution of these points $\mathbf{Y}$ is modeled as a multivariate normal distribution, with mean $\boldsymbol{\mu}_{\mathbf{y}}$ and covariance $\boldsymbol{\Gamma}_{\mathbf{y}}$ written as functions of $\mathbf{x}$: $$\begin{aligned} &\boldsymbol{\mu}_{\mathbf{y}} = \mathbf{m}(\mathbf{x}) = \mathbb{E}\{f(\mathbf{x})\},\\ &\boldsymbol{\Gamma}_{\mathbf{y}}=\mathbf{K}(\mathbf{x},\mathbf{x}^{\prime}) = \mathbb{E}\{(f(\mathbf{x})-\mathbf{m}(\mathbf{x}))(f(\mathbf{x}^{\prime})-\mathbf{m}(\mathbf{x}^{\prime}))^T\},\end{aligned}$$ where $(\:\cdot\:)^T$ is the matrix transpose. Let now $\mathbf{Y} = \begin{bmatrix} \mathbf{Y}_t & \mathbf{y}_*\end{bmatrix}^T$, that is, $\mathbf{Y}$ a vector consisting of the training data $\mathbf{Y}_t$, and a new point $\mathbf{y}_*$ which we want to estimate, using the corresponding measurement $\mathbf{x}_*$. For simplification, we set $\mathbf{m}(\mathbf{x})=0$. The mean term mostly affects the behavior when extrapolating far away from the space covered by the training data. The joint distribution of $\mathbf{Y}$ is then $$\label{joint} \begin{bmatrix} \mathbf{Y}_t \\ \mathbf{y}_* \end{bmatrix} \sim\mathcal{N}\left(0, \begin{bmatrix} \mathbf{K}(\mathbf{X}_t,\mathbf{X}_t)+\mathbf{E} & \mathbf{K}(\mathbf{x}_*,\mathbf{X}_t)^T \\ \mathbf{K}(\mathbf{x}_*,\mathbf{X}_t) & \mathbf{K}(\mathbf{x}_*,\mathbf{x}_*)+\mathbf{E}_* \end{bmatrix}\right),$$ where $\mathbf{E}$ and $\mathbf{E}_*$ describe the covariance of the error $\mathbf{e}$, i.e. uncertainty of $\mathbf{y}$. In this work, we use $\mathbf{E}*=0.1\mathbf{D}$, where $\mathbf{D}$ is a diagonal matrix that contains the sample variances of the training data $\mathbf{Y}_t$ on the main diagonal. The error matrix $\mathbf{E}= 0.1\mathbf{D}\otimes \mathbf{I}$, where $\otimes$ is the Kronecker product and $\mathbf{I}\in\mathbb{R}^{n_t\times n_t}$ is an identity matrix. For brevity, following shorthand notations are introduced: $$\begin{aligned} &\mathbf{K} = \mathbf{K}(\mathbf{X}_t,\mathbf{X}_t)\in\mathbb{R}^{n_yn_t\times n_yn_t} \\ &\mathbf{K}_* = \mathbf{K}(\mathbf{x}_*,\mathbf{X}_t)\in\mathbb{R}^{n_y\times n_yn_t}.\end{aligned}$$ In GPR, the kernel matrices $\mathbf{K}$ and $\mathbf{K}_* $ are constructed based on a covariance function. In this study we use stationary Matérn covariance function with $\nu=3/2$, fixed length scale $l=10$, and $\sigma=1$: $$\label{matern} k(\mathbf{x},\mathbf{x^\prime}) = \left(1+\frac{\sqrt{3}d(\mathbf{x},\mathbf{x^\prime})}{10}\right)\exp\left(-\frac{\sqrt{3}d(\mathbf{x},\mathbf{x^\prime})}{10}\right),$$ where the distance metric $d(\mathbf{x},\mathbf{x^\prime})$ is the Euclidean distance. The covariance function $k(\mathbf{x},\mathbf{x^\prime})$ describes the covariance between the vectors $\mathbf{x}$ and $\mathbf{x^\prime}$ based on the distance between the vectors. The covariance function is the core component of GPR that specifies properties such as smoothness of the regressor. The covariance function is used to construct univariate kernel matrices $$\begin{aligned} &K(i,j) = k(\mathbf{x}_t^{(i)},\mathbf{x}_t^{(j)})\in\mathbb{R}^{n_t\times n_t} \\ &K_*(1,j) = k(\mathbf{x}^*,\mathbf{x}_t^{(j)})\in\mathbb{R}^{1\times n_t}.\end{aligned}$$ To get from the univariate kernels to multivariate kernels used in , the so-called separable kernel [@bonilla2008] is used: $$\begin{aligned} &\mathbf{K} = \boldsymbol{\Gamma}_{\mathbf{y}}\otimes K \\ &\mathbf{K}_* = \boldsymbol{\Gamma}_{\mathbf{y}}\otimes K_*,\end{aligned}$$ where $\boldsymbol{\Gamma}_{\mathbf{y}}\in\mathbb{R}^{n_y\times n_y}$ is a (prior) covariance for $\mathbf{y}$. In this work, $\boldsymbol{\Gamma}_{\mathbf{y}}$ is approximated by the sample covariance of $\mathbf{Y}_t$. It should be noted, that $\sigma=1$ is chosen in the kernel function , because the (prior) variances of $\mathbf{y}$ are added to the covariance kernel in this step through $\boldsymbol{\Gamma}_{\mathbf{y}}$. From the joint density , the conditional density of $\mathbf{y}_*$ given the training data and the measurement $\mathbf{x}_*$ is $$\mathbf{y}_*\:|\:\mathbf{Y}_t,\mathbf{X}_t,\mathbf{x}_*\sim\mathcal{N}(\mathbf{m}(\mathbf{y}_*),\boldsymbol{\Gamma}_{\mathbf{y}_*}),$$ where $$\begin{aligned} \label{predmean} &\boldsymbol{\mu}_{\mathbf{y}_*} = \mathbf{K}_*(\mathbf{K}+\mathbf{E})^{-1}\mathbf{y}_t \\ \label{predcov} &\boldsymbol{\Gamma}_{\mathbf{y}_*}= \boldsymbol{\Gamma}_{\mathbf{y}} + \mathbf{E}_* - \mathbf{K}_*(\mathbf{K}+\mathbf{E})^{-1}\mathbf{K}_*^T.\end{aligned}$$ The predictive mean $\boldsymbol{\mu}_{\mathbf{y}_*} $ is now the point estimate for the unknown vector of stand attributes $\mathbf{y}_*$ and $\boldsymbol{\Gamma}_{\mathbf{y}_*}$ provides the estimate covariance. As can be seen from the equation , the final prediction is a linear combination of the training data values $\mathbf{y}_t$. This mathematical connection to linear models is expected, because general linear model can be written as a special case of GPR [@rasmussenbook]. Correcting for negative predictions ----------------------------------- Unlike kNN and certain other machine learning methods, GPR extrapolates outside the training data. As an unwanted side effect of this extrapolation behavior, GPR can produce unrealistic negative predictions for the stand attributes. Several statistically rigorous methods for constraining the GPR predictions have been proposed [@DaVeiga2012; @jidling2017], but these methods increase the computational cost significantly and are nontrivial to implement. Here we adopt a simpler correction. For the point prediction, we compute a maximum a posteriori estimate by solving $$\hat{\mathbf{y}}_*=\mathrm{arg}\:\underset{\hat{\mathbf{y}}}{\mathrm{min}}\left\{(\hat{\mathbf{y}}-\boldsymbol{\mu}_{\mathbf{y}_*})^T\boldsymbol{\Gamma}_{\mathbf{y}_*}^{-1}(\hat{\mathbf{y}}-\boldsymbol{\mu}_{\mathbf{y}_*})\right\},\;\hat{\mathbf{y}}\geq 0.$$ The prediction $\hat{\mathbf{y}}_*$ is the mode of the truncated GPR predictive density. If the original GPR predictive mean is non-negative, $\hat{\mathbf{y}}_*$ is simply $\boldsymbol{\mu}_{\mathbf{y}_*}$. Due to the complicated structure of the marginal densities of a truncated multivariate Gaussian distribution [@Horrace2005], correcting the predictive intervals exactly is not computationally practical. Instead, the univariate Gaussian marginals of the GPR predictive density are truncated at zero. If the original 95% predictive interval for a stand attribute is $[a,b]$ and $a<0$, set $\hat{a}=0$ and calculate the new corrected upper bound $\hat{b}$ using the cumulative distribution of univariate truncated Gaussian by solving $$\Phi(\hat{b},\mu_{y_{*}},\sigma_{y_{*}}) = 0.95+0.05\Phi(0,\mu_{y_{*}},\sigma_{y_{*}}),$$ where $\Phi(\:\cdot\:,\mu_{y_{*}},\sigma_{y_{*}})$ is the cumulative distribution function of the univariate Gaussian distribution with the mean and standard deviation from the GPR predictive distribution. If $a\geq0$, the interval $[a,b]$ does not change. The corrected interval $[\hat{a},\hat{b}]$ is not a proper predictive interval of the truncated predictive distribution, unless $\boldsymbol{\Gamma}_{\mathbf{y}_*}$ is strictly diagonal. Reference methods ----------------- The GPR point estimates are compared with a state-of-the-art kNN algorithm. We select ten predictors from the (transformed) data using a simulated annealing -based optimization approach of [@Packalen2012] and use the most similar neighbor (MSN) method for selecting the neighbors. The number of neighbors is chosen to be $k=5$, as in [@packalen2009; @Packalen2012]. The predictor selection is done using the whole data set and leave-one-out cross-validation. The prediction credible intervals provided by GPR are compared with the Bayesian inference approach [@varvia]. In the Bayesian approach the posterior predictive density: $$\label{alsextposterior} \pi(\mathbf{y}_*|\mathbf{x}) \propto \begin{cases} \mathcal{N}(\mathbf{x}|\hat{\mathbf{A}}\boldsymbol{\phi}(\mathbf{y}_*)+\hat{\boldsymbol{\mu}}_{\mathbf{e}|\mathbf{y}},\hat{\boldsymbol{\Gamma}}_{\mathbf{e}|\mathbf{y}}) &\\ \qquad\qquad\quad\;\cdot\:\mathcal{N}(\mathbf{y}_*|\hat{\boldsymbol{\mu}}_{\boldsymbol{\theta}},\hat{\boldsymbol{\Gamma}}_{\mathbf{y}}), & \mathbf{y}_*\geq 0 \\ 0, & \mathbf{y}_*< 0, \end{cases}$$ is constructed based on the training data and the new measurement. The model matrix $\hat{\mathbf{A}}$, conditional (residual) error statistics $\hat{\boldsymbol{\mu}}_{\mathbf{e}|\mathbf{y}}$ and $\hat{\boldsymbol{\Gamma}}_{\mathbf{e}|\mathbf{y}}$, and the prior statistics $\hat{\boldsymbol{\mu}}_{\boldsymbol{\theta}}$ and $\hat{\boldsymbol{\Gamma}}_{\mathbf{y}}$ are learned from the training data. The density is then sampled using a Markov chain Monte Carlo method. The point estimate and 95% credible intervals are then calculated from the samples. {width="\textwidth"} Performance assessment ---------------------- The proposed GPR method is first evaluated using leave-one-out cross-validation (i.e. $n_t=492$). From the results, relative root mean square error (RMSE%), relative bias (bias%), and credible interval coverage (CI%) are calculated. Credible interval coverage is the percentage of the test plots where the field measured value of a stand attribute lies inside the computed 95% prediction interval; CI% thus has the ideal value of 95%. In addition to conducting a leave-one-out cross-validation, the effect of the number of training plots is evaluated. Training set sizes from $n_t=20$ to $n_t=400$ are tested with a stepping of 20. The cross-validation is performed by first randomly sampling $n_t$ plots to be used as a training set and then randomly selecting a single test plot from the remaining $493-n_t$ plots. This procedure is repeated 2000 times for each $n_t$ value. This way the number samples for each tested $n_t$ stays constant. The effect of training set size is only evaluated for GPR and kNN, due to the high computational cost of the reference Bayesian inference approach. Results and discussion {#sec.results} ====================== Species-specific attributes --------------------------- The RMSE% comparison between the GPR predictions, kNN and Bayesian inference is shown in the Figure \[fig:rmse\] for all estimated stand attributes. The numerical RMSE% value is shown above each bar. For pine, which is the dominant species in the study area, the GPR and kNN estimates have fairly equivalent performance: GPR is slightly better for all the stand attributes except height and basal area. The Bayesian linear inference estimates are notably worse. In the minority species (spruce and deciduous), the GPR estimates have consistently better RMSE% than kNN or Bayesian linear. On average, the relative improvement over kNN is 6.5% for the minority species and 4.6% for all species. Figure \[fig:bias\] shows a similar comparison of relative bias between the evaluated methods for all the estimated stand attributes. The numerical bias% value is printed for each bar. GPR estimates show smaller than 2% absolute bias for all the stand attributes, except the spruce basal area and volume. kNN shows small bias in the spruce attributes, but has a large bias in deciduous basal area and volume. The Bayesian linear results show notable bias in $N$, $\mathit{BA}$, and $V$. {width="\textwidth"} The CI coverages of GPR and the reference Bayesian inference method are compared in Figure \[fig:ci\]. The numerical CI% value is shown above each bar; the ideal value is here 95%. The CI% for the Bayesian linear estimates fall short of the 95% target, that is, the prediction intervals that are too narrow. The GPR prediction intervals perform well on basal area and stem volume, with good coverage also on stem number. The GPR CI% for these stand attributes is consistently better than the Bayesian linear. The GPR prediction intervals for height and diameter are overconfident, especially in the deciduous variables, and the performance is roughly similar to the Bayesian linear estimates. {width="140mm"} Total attributes ---------------- Point estimates and credible intervals for the total stem number, basal area, and stem volume were calculated from the species-specific results. The point estimates were computed by summing up the corresponding species-specific estimates. The GPR prediction interval for the total attributes is acquired from the prediction covariance $\boldsymbol{\Gamma}_{\mathbf{y}_*}$, because summation is a linear transformation. The results for the total attributes are shown in Figure \[fig:totvars\]. In RMSE%, GPR estimates show the best performance. Bayesian linear estimates have lower RMSE% than kNN in the total basal area and volume, while kNN is better in the stem number. In the relative bias, GPR has fairly low bias and performs worst in the total stem volume. kNN has consistent slight bias, while the Bayesian linear estimates show large bias in the stem number. In credible interval coverage, GPR produces too wide intervals for basal area and stem volume (CI% between 98-99%). The Bayesian linear intervals are, on the other hand, with CI% around roughly 80%. {width="140mm"} Effect of training set size --------------------------- RMSE% versus training set size is shown in Figure \[fig:rmsetrain\]. The dashed minimum line corresponds to the species-specific stand attribute with the lowest RMSE%, maximum to the stand attribute with the highest RMSE%, and mean is the average over the stand attributes. As expected, the RMSE% increases for both methods when the training set size decreases. GPR keeps the slight performance advantage over kNN even when using smaller training sets. The improvement in performance for training sets larger than c. 200 plots is fairly small. ![Lowest, average, and the highest relative RMSE as a function of training set size for GPR and kNN estimates.[]{data-label="fig:rmsetrain"}](rmse_nt.png){width="70mm"} Figure \[fig:biastrain\] shows the relative bias as a function of the training set size. When the training set size decreases, the estimated bias increases in both positive and negative directions, but with a general negative tendency. Smaller training sets are less likely to cover the full range of variation of the stand attributes in the population, which results in underestimation of large values: this would explain the observed tendency in bias. The largest negative biases produced by kNN are consistently larger than in GPR. ![Lowest, average, and the highest relative bias as a function of training set size for GPR and kNN estimates.[]{data-label="fig:biastrain"}](bias_nt.png){width="70mm"} Figure \[fig:citrain\] shows the CI% of the GPR estimates versus the training set size. The average CI% increases slightly as training set size is decreased, the lowest CI% increases considerably, while the highest CI% drops somewhat. The generally too narrow credible intervals signify overconfidence in the predictions, which implies that either the GP model is not optimal in its current formulation, or the stand attributes have not sufficiently explained the variation in the predictors. The latter explanation might cause that there are usually contradicting training data (i.e. training points that are close in the stand attribute space, but distant in the predictor space) in large training sets, which might partly explain the slight improvement of CI% when training set size decreases. With the Bayesian inference approach, on the contrary, a substantial drop in CI% in smaller training set sizes would be expected based on the results in [@varvia]. ![Lowest, average, and the highest CI% as a function of training set size for GPR estimates.[]{data-label="fig:citrain"}](ci_nt.png){width="70mm"} Discussion {#sec:disc} ---------- Conceptually, GPR is a non-parametric machine learning method that has similarities with kNN. Thus, many approaches proposed for improvement of kNN estimates within ABA could be also utilized to further improve GPR estimates. GPR seems to be insensitive to multicollinearity and quite large numbers of predictors can been used simultaneously [@alsgpr]. In this paper, fairly traditional ABA metrics were used, adding additional predictors, such as $\alpha$-shape [@alphashape] or composite metrics [@zhao2009], could potentially improve prediction performance. Additionally, dimension reduction, for example by using principal component analysis (PCA) [@junttila2015], would probably improve performance when using small training sets. Besides PCA, the deep belief network pretraining proposed in [@hinton2008] could be beneficial. The prediction step of GPR is not computationally much more costly than using kNN. The most computationally expensive part is the GPR model training, which requires computing the matrix inverse of a large matrix (see the equations and ). However, the matrix inverse can be precomputed for a given set of training data. After this, computing the prediction and the prediction interval only requires calculating matrix products. In the LOO case ($n_t=492$), computing the GPR prediction and intervals for a plot/cell took on average 345 ms in Matlab on a AMD Ryzen 1700X (3.4 GHz) processor, this is more expensive than kNN (1.5 ms), but still feasible for practice. For comparison, computing the Bayesian linear estimate took on average 18.5 s per plot. The training of GPR took 12.7 seconds. The present work used fixed length scale, covariance function, and error magnitude, because finding the optimal values for these (hyper)parameters automatically is generally a nonconvex and computationally difficult optimization problem. The values, $l=10$ for the correlation length, and 10% variance for the error $e$, were found by manual testing. The sub-optimal choice of these parameters might explain some of the tendency to produce too narrow prediction intervals for some stand attributes. Additionally, several commonly used covariance functions were tested; Matérn $3/2$ covariance function was found to be the best performing. More advanced covariance functions, such as nonstationary covariance functions [@paciorek2004] or spectral mixture covariance functions [@wilson2013; @wilson2016], could potentially improve prediction accuracy. Further research is still needed on finding the optimal model formulation. Due to extrapolation, GPR can produce unrealistic negative predictions. In this study, the negative predictions were corrected in a post-processing step. In the LOO cross-validation, total of 628 negative predictions occurred on 215 plots. Of these, 386 (61.5%) occurred in cases where the corresponding field-measured value was zero (i.e. a missing tree species). Furthermore, 93% of the negative predictions happened in cases where the corresponding field measurement was less than half of the average value of the stand attribute in the data set. The negative predictions thus occurred most commonly when predicting small stand attribute values. Additionally, 1039 cases where the field-measured stand attribute was zero were predicted to have a positive, non-zero value. Due to the more probable occurrence when predicting small values and the relatively symmetric distribution of the prediction error, the behavior of the negative predictions seems to be in line with the Gaussianity assumption in the GPR. In this study, GPR showed better reliability in all considered stand attribute predictions except the mean height of pine (the RMSE of basal area predictions of pine were practically the same for GPR and kNN) and the relative improvement of GPR predictions over the state-of-the-art method kNN were rather large being on average 4.6%. This is contrary to the earlier studies where the reliability of ALS based forest inventory system has been examined by comparing different estimation methods. For example, Maltamo *et al.* [@Maltamo2015] used visual pre-classification of aerial images to divide the study data into strata according to the main tree species and stand development stages. The aim was to improve species-specific estimates by applying more homogeneous reference data in kNN but the results were contradictory. The pre-classification did improve the accuracy of some species-specific stand attributes compared to the kNN estimates which applied whole study data as reference, but for some species-specific estimates the accuracy decreased. It is also notable that usually the accuracy of minor tree species did not improve, whereas in the present study the improvement was substantial especially for the minor species. Similar contradictory results have been obtained when comparing different statistical methods, such as neural networks or Bayesian approach [@niska2010neural; @varvia]. For example, Niska *et al.* [@niska2010neural] obtained more accurate species-specific volume estimates using neural networks at plot level than kNN but on the other hand kNN was more accurate on the stand level. R[ä]{}ty *et al.* [@raty2018] compared kNN estimates in which the species-specific estimates were obtained either by simultaneous imputation for all the species (as in this study) or by separate imputation for each species. The results concerning separate imputations were promising, but again, the results were contradictory. Conclusions =========== In this article, the feasibility of Gaussian process regression for the estimation of species-specific stand attributes within the area based approach was evaluated. In addition to testing the prediction performance, the prediction credible intervals were also evaluated. GPR estimates were compared with a state-of-the-art kNN-based algorithm and a linear Bayesian inference based method. The effect of training set size on the performance was also examined. The GPR estimates showed on average a 4.6% relative improvement in RMSE over the reference kNN method in the leave-one-out cross-validation, generally smaller bias, and credible interval performance on par with the linear Bayesian inference. The GPR estimates kept the advantage even when tested using smaller training set sizes. Especially the credible interval performance proved robust with respect to the training set size. The promising performance of GPR, the feasible computational cost, and that it provides prediction intervals make GPR an attractive method to use in forestry applications. Especially the plot level prediction uncertainty information provides many potential improvements in forest planning. [Petri Varvia]{} was born in Karttula, Finland, in 1988. He received M.Sc. and Ph.D. degrees in Applied Physics from the University of Eastern Finland in 2013 and 2018, respectively. He is currently a postdoctoral researcher at the Laboratory of Mathematics in the Tampere University of Technology. His scholarly interests include statistical inverse problems, Bayesian statistics and remote sensing. [Timo Lähivaara]{} received the M.Sc. and Ph.D. degrees from the University of Kuopio, Finland, and University of Eastern Finland in 2006 and 2010, respectively. Currently, he is a senior researcher at the Department of Applied Physics in the University of Eastern Finland. His research interests are in computational wave problems and remote sensing. [Matti Maltamo]{} was born in Jyväskylä, Finland in 1965. He received the M.Sc., Lic.Sc., and D.Sc. degrees (with honors) in forestry from the University of Joensuu, Joensuu, Finland, in 1988, 1992, and 1998, respectively. He is currently the Professor of Forest Mensuration Science with the Faculty of Science and Forestry, University of Eastern Finland. He has also worked as a visiting professor at the research group of professor Erik Naesset at the Norwegian University of Life Sciences. He was together with Naesset and Jari Vauhkonen the editor of the textbook “Forestry Applications of Airborne Laser Scanning – concepts and case studies” published in 2014. He has published about 165 scientifically refereed papers. His specific research topic is Forestry Applications of ALS. He is an Associate Editor of the journal Canadian Journal of Forest Research. Prof. Maltamo won together with professor Juha Hyyppä the First Innovation prize of the Finnish Society of Forest Science in 2010 about “Bringing airborne laser scanning to Finland. Maltamo also obtained bronze A.K. Cajander medal of the Finnish Society of Forest Science, 2012 [Petteri Packalen]{} was born in Rauma, Finland, in 1973. He received the M.Sc, Lic.Sc., and D.Sc. degrees in Forestry from the University of Joensuu, Joensuu, Finland, in 2002, 2007, and 2009, respectively. Currently, he is an Associate Professor in optimization of multi-functional forest management (Tenure Track) with the School of Forest Sciences, Faculty of Science and Forestry, University of Eastern Finland, Joensuu, Finland. Previously, he has been an Assistant, Senior Assistant, and Professor with the Faculty of Forestry, University of Joensuu. From August 2011 to July 2012, he was a Visiting Research Scientist at the Oregon State University, Corvallis, OR, USA. He has authored over 80 peer-reviewed research articles. Recently, his focus has been on time series, nearest neighbor imputation, combined use of ALS and spectral data in forest inventory, and the use of ALS in wildlife management. Since 2007, he has also been a Consultant for remote sensing-based forest inventory. His research interests include both practical and theoretical aspects of utilizing remote-sensing data in the monitoring and assessment of the forest environment. [Aku Seppänen]{} is an Associate Professor in the Department of Applied Physics at University of Eastern Finland, Kuopio. He received the M.Sc. and Ph.D. degrees from the University of Kuopio, Finland, in 2000 and 2006, respectively, and has authored 50 journal articles, 36 conference papers and 3 book chapters. His research interests are in statistical and computational inverse problems. He is one of the PIs in the Centre of Excellence in Inverse Modelling and Imaging (2018-2025) appointed by the Academy of Finland. The applications of his research include, e.g., industrial process imaging, non-destructive material testing and remote sensing of forest. [^1]: This work was supported by the Finnish Cultural Foundation, North Savo Regional fund, the Academy of Finland (Project numbers 270174, 295341, 295489, and 303801, and Finnish Centre of Excellence of Inverse Modelling and Imaging 2018-2025), and the FORBIO project (The Strategic Research Council, Grant No. 293380). [^2]: P. Varvia was with the Department of Applied Physics, University of Eastern Finland, FI-70211 Kuopio, Finland. He is now with the Laboratory of Mathematics, Tampere University of Technology, FI-33101 Tampere, Finland (e-mail: petri.varvia@gmail.com). [^3]: T. Lähivaara and A. Seppänen are with the Department of Applied Physics, University of Eastern Finland, FI-70211 Kuopio, Finland. [^4]: M. Maltamo and P. Packalen are with the School of Forest Sciences, University of Eastern Finland, FI-80101 Joensuu, Finland.

--- abstract: 'The lack of our knowledge on how angular momentum is transported in accretion disks around black holes has prevented us from fully understanding their high energy emissions. We briefly highlight some theoretical models, emphasizing the energy flow and electron energization processes. More questions and uncertainties are raised from a plasma physics point of view.' author: - 'H. Li, S. A. Colgate, M. Kusunose and R.V.E. Lovelace' title: On the Particle Heating and Acceleration in Black Hole Accretion Systems --- Introduction ============ Figure \[fig-1\] shows three (roughly) contemporaneous broad band high energy emission spectra from three galactic black hole candidates (GBHCs; Grove et al. 1998). Although it is conventional to interpret the soft black-body-like component below $\sim 10$ keV as coming from an optically thick Shakura-Sunyaev (SS) disk, the origin of the hard X-ray continuum (and its extension into soft X-rays during the low-hard state) is a constant source of debate. Extracting a physically sensible model through a maze of high quality spectral and timing data on these systems remains a great challenge. Recently, there seems to be a renewed interest in understanding particle heating/acceleration in accretion disks. We attribute this to the observations of: possible $> 0.5$ MeV emissions from Cyg X-1 and GRO J0422; the powerlaw component of GRO J1655 extending to at least 800 keV without a cutoff (Tomsick et al. 1998); and relativistic radio jets from sources like GRO J1655 and GRS 1915. Furthermore, the clearly laid-out physical requirements of ADAF models (which have enjoyed much success, see Narayan et al. 1998 for a review) also prompted further discussions on particle heating. In this review we will mostly discuss a few models for the so-called low-hard state where the spectrum ($\nu F_\nu$) is peaking around 100-200 keV. We apologize for not able to cover all the models (see Liang 1998 for a recent extensive review). The powerlaw tail that seems to extend beyond $500$ keV during the soft-high state also begs explanation, though the total energy contained in this tail is perhaps $< 10\%$ of the total emission, so we will place less emphasis on them. We will focus on the electron energization processes of these theoretical models. We will not discuss any detailed spectral and temporal analyses (see other articles in this volume). Even so, we quickly realized that writing on this topic is a very difficult task because we find many questions and confusions with no clear and definite answers. Some Models for the Origin of Hard X-rays and Gamma-rays ======================================================== In all the models discussed here, the physics of angular momentum transport (or “$\alpha$” viscosity) during accretion is not well understood. As a direct consequence, unfortunately, modeling energy dissipation in accretion disks has many [*ad hoc*]{} elements. Quite generally, the matter (surface density $\Sigma$) in accretion disk is evolved as (taken from Papaloizou & Lin 1995) $${\partial \Sigma \over \partial t} - \frac{1}{r} \frac{\partial}{\partial r}\left[F_1 + F_2 + F_3\right] -S_{\Sigma} = 0$$ where $F_1 \propto \partial (\langle \nu \rangle \Sigma r^{1/2})/\partial r$ is the local viscous transport with viscosity $\langle \nu \rangle$ (i.e., the standard $\alpha-$disk viscosity or from MHD turbulence by Balbus & Hawley 1991, 1998); $F_2 \propto S_{\Sigma}J$ is the advective loss with $J$ being the angular momentum carried by the source/sink ($S_{\Sigma}$) material (i.e., magnetic flux and/or winds, Blandford & Payne 1982); $F_3 \propto \Lambda$ is the external perturbation (i.e., tidal interactions). Three models (or their variants) are usually employed for explaining the high energy emissions, namely, the SS model, the SLE model (Shapiro et al. 1976), and the ADAF model. All of them use the local viscous transport prescription (the $F_1$ term) and the energy is also dissipated locally at the disk. In SS model disk is optically thick and geometrically thin, and the plasma is also highly collisional. The heat deposited from transporting angular momentum is successfully radiated away so that disk remains thin ($H \ll R$). In SLE and ADAF models, however, an inner, hot ($T_e \sim 100$ keV), optically thin ($\tau \leq 1$) and two-temperature ($T_i \gg T_e$) region is postulated. This region is then cooled via various radiation processes, such as thermal Compton scattering and Synchrotron. The arguments for the existence of this hot, optically thin region might be summarized as follows: if local viscous energy dissipation [*only*]{} heats protons, and if there is only Coulomb coupling between electrons and protons, then when the energy input rate is high enough, the system will become unstable if the cooling via radiation is not quick enough, so the plasma has to expand and become optically thin. Here, we want to emphasize that the accreting plasma, during this transition from an optically thick, thin disk to an optically thin, quasi-spherical state, has also changed from [*highly collisional*]{} to essentially [*collisionless*]{}. This brings up several immediate questions which are related to the above “if”. Open Questions ============== Will local viscous energy dissipation only heat protons? -------------------------------------------------------- Bisnovatyi-Kogan & Lovelace (1997) first discussed this issue and argued that dissipation in such a magnetized collisionless plasma predominantly heats the electrons owing to reconnection of the random magnetic field. On the other hand, Quataert (1998) and Gruzinov (1998) have argued that conditions for ADAF could be true in the high $\beta = P_{\rm plasma}/P_{\rm magnetic} \geq 5$ limit by calculating the linear damping rates of short wavelength modes in a hot (but nonrelativistic) plasma. in an (implicit) almost uniform magnetic field. Note that even though MHD turbulence phenomenology was used in both papers, the damping rates are valid in the linear regime for plasma waves only (see below for further discussion). But these calculations perhaps are not answering the question of how to form the optically thin region in the first place because they are damping rates in the [*collisionless*]{} limit. Instead, one perhaps might first evaluate the energy dissipation processes (with an understanding of $\alpha$ viscosity) in the [*collisional*]{} limit which is the physical state initially. These collisions ensure thermal electron and proton distributions and efficient energy exchange between them, especially at the so-called transition radius in ADAF ($10^3-10^4 r_s$). If one uses Balbus-Hawley instability (see also Velikov 1959 and Chandrasekhar 1981) as the origin of the viscosity in the disk, then the gravitational energy is mostly released in large scale (longest wavelength of the magnetic field changes) and this energy will amplify the field first (instead of going into heating the particles). Once the nonlinear saturation is reached (say with magnetic energy density being $10\%$ of the kinetic energy density of the shear flow), we are actually faced with two possibilities, namely, whether the magnetic fields will be expelled (or escape) from the disk, or they will have to dissipate locally in the disk. Bisnovatyi-Kogan & Lovelace (1997) argued for the second possibility (but see Blackman 1998). Since we know that both the fluid and magnetic Reynolds numbers are exceedingly large in these flows, any “classical” viscous and ohmic dissipations will happen on timescales longer than the age of the universe, thus efficient magnetic reconnection has been sought as the primary candidate for energy dissipation in the disk. They further argued that current-driven instabilities in this turbulent plasma will give rise to large local $E_{\parallel}$ which mostly accelerate electrons. Thus, up to half of the magnetic energy input goes directly to electrons and is subsequently radiated away, and the disk will always stay thin and optically thick. The uncertainties in these arguments are nevertheless quite large since we don’t fully understand MHD turbulence, let alone its dissipation via kinetic effects. For example, it is unclear whether such reconnection sites are populated throughout the plasma so that most fluid elements encounter such regions. There has been some detailed numerical simulations with magnetic Reynolds number up to 1000 (Ambrosiano et al. 1988) in which test particles are observed to get accelerated by the induced small scale electric fields associated with reconnection sites in turbulent MHD flows. If indeed the magnetic energy dissipation is through accelerating particles by the induced electric fields (this is a big if), since electrons are the current carriers, it is hard to imagine that protons receive most of the energy. Is there any collective process that could ensure efficient energy exchange between protons and electrons besides Coulomb? -------------------------------------------------------------------------------------------------------------------------- Putting aside the uncertainties discussed above, if there is indeed an optically thin, hot, two-temperature plasma region, a pertinent question is how much energy electrons can get. This question is, unfortunately, ill-fated again because we do not know how to formulate the problem. Another way to look at it is how to identify the free energy, since most plasma instabilities require a good knowledge of the free energy as determined by the system configuration. For example, is there a relative drift between protons and electrons and can fast electrons be regarded as a beam to an Maxwellian proton core distribution? Is there temperature anisotropy parallel and perpendicular to background magnetic fields, etc.? Begelman & Chiueh (1988) have studied some plasma instabilities in detail and found plausible ways of transferring energy from ions to electrons, under the conditions that a substantial level of MHD turbulence will give a large enough proton density gradient (or curvature drifts) so that proton drift velocity can be large enough to drive certain modes unstable. The fluctuating electric field parallel to the magnetic field will then accelerate electrons. The applicability of this instability is again hampered by our lack of knowledge of the presumed MHD turbulence. Narayan & Yi (1995) argued that this mechanism does not work well in ADAF. A conceptual difficulty is that the typical modes excited by protons (having most of the energy) are below the proton gyrofrequency $\Omega_{ci}$. This makes resonance with the electrons difficult. But a possible avenue is to have protons excite (almost) perpendicular modes (i.e., high $k_\perp$ and very small $k_\parallel$). Then the resonant conditions for electrons to resonate with these waves are easier to satisfy. More work is needed to explore these possibilities. Could accretion disk have a magnetically dominated, hot corona like our Sun? ---------------------------------------------------------------------------- The formation of a “structured corona” was first proposed by Galeev et al. (1979). In this model a radial quadrupole field is wound up by differential rotation into an enhanced toroidal field. Then the helicity of the presumed convective “turbulence” converts a fraction of the toroidal flux back into poloidal field and hence produces an exponentiating dynamo that saturates by back reaction. This is the classical $\alpha - \Omega$ dynamo although not identified as same in the paper. Furthermore the saturation or back reaction limit of this disk dynamo is assumed to be the random loops of flux characteristic of the solar surface. One important step in the above model is the requirement of vertical (thermal) convection in the $\{R,z\}$ plane. The convective motion may be driven by heat released at or near the mid-plane. Lin et al. (1993) have shown that under specific conditions of opacity and equation of state that convective instability should occur both linearly and nonlinearly, thus leading to large amplitude cells. However, the convective cells are highly constrained radially. The problems of restrictive initial conditions and the restrictive cell geometry leads one to conclude that this is not the universal mechanism needed to explain accretion disks. Colgate & Petschek (1986) showed that to drive convective cells whose displacement radially is of the order of the disk height $h$, (unrealistic) efficiency of the energy flow (a Carnot cycle of $\sim 100\%$ efficiency) is necessary to drive these convective eddies, and the cells created are also highly restrictive, tall but narrow radially (i.e., similar as Lin et al.). Thus the existence of strong convective turbulence is doubtful. The result of this lack of convective turbulence with rising plumes is to negate the origin of the helicity invoked in the structured corona model. Electron Energization ===================== Besides the possible role of magnetic reconnection in accelerating electrons which is observed in the solar corona (Tsuneta 1996), there are more standard processes which involve wave-particle interactions (see Kuijpers and Melrose 1996). Shock acceleration is not considered here. We give a quick review of the electron energization by plasma waves and turbulence. Particle heating/acceleration – linear and quasilinear theory ------------------------------------------------------------- Linear Vlasov equation is usually used to describe the collisionless plasma, which is a good approximation of astrophysical plasmas. Linearization of the Vlasov equation yields various dispersion relations $\omega = \omega({\bf k})$ which describe how the system will respond to [*small*]{} electrostatic and electromagnetic perturbations. Since the field energy of low frequency fluctuations (i.e., $\omega < \Omega_{\rm cp}$) is predominantly magnetic, particles generally experience strong pitch-angle scattering before they can be energized. Of fundamental importance is the wave-particle resonance, that is, given an electromagnetic fluctuation of frequency $\omega$ and wavevector ${\bf k}$, a charged particle ($q, m$) is considered to be in resonance with this fluctuation when $$\omega - k_{\parallel} v_{\parallel} - \ell \Omega_0 / \gamma = 0, ~~~~~~~~~~\ell = 0, \pm 1, \pm 2,~ \dots \eqno(1)$$ where the nonrelativistic gyrofrequency $\Omega_0 = |q|B_0/mc$, and $v_{\parallel}$ and $\gamma$ are the particle’s parallel velocity and Lorentz factor, respectively. When the harmonic number $\ell = 0$, the resonance is referred to as the Landau or Cherenkov resonance, and implies that the particle speed along the magnetic field matches the speed of the parallel wave electric or magnetic field. If $|\ell| > 0$, the process is called gyroresonance, and there is a matching between the wave transverse electric field and the cyclotron motion of the particle. The sign of $\ell$ depends upon the transverse polarization of the wave and the sign of $q$: if the transverse wave electric field and the particle rotate in the same sense about $B_0$ in the plasma frame, then $\ell$ is positive. In most settings, only $\ell = \pm 1$ is of importance. The key quantities are the plasma $\beta=n(T_i+T_e)/(B^2/8\pi)$ factor and the temperature ratio $T_e/T_i$. Furthermore, we have: $\bullet$ Linear theory. The linear theory of plasma waves and instabilities is often reduced to a linear dispersion equation with a complex $\omega$, whose imaginary part gives the growth or damping of certain modes. Recent studies by Gruzinov (1998) and Quataert (1998) belong to this case. The usual candidates for wave-particle resonances are: for $\ell=0$, the transit time damping (TTD) for particle with oblique fast magnetosonic waves and Landau damping (LD) with kinetic Alfven waves; for $|\ell| \geq 1$, gyroresonances between the proton/Alfven wave and the electron/whistler wave. $\bullet$ Quasilinear theory. A detailed physical understanding of pitch-angle scattering and stochastic acceleration is beautifully presented in Karimabadi et al. (1992), using nonlinear orbit theory with the Hamiltonian formalism. In the presence of a continuum of plasma waves, the number of resonances between the particle and waves is greatly increased to a point that the trapping width associated with one particular resonance can overlap with neighboring resonances, thus allowing particles “jump” from one resonance to another. As particles sample different resonances, they gain energy in a “ladder-climbing” fashion. Hence the description stochastic acceleration. This approach has been adopted in several studies on electron acceleration by fast-mode waves and whistler waves in accretion disk (Li et al. 1996, Li & Miller 1997). We typically find that the electron distribution is hybrid with a nonthermal tail, which is responsible for the production of $> 500$ keV emissions in several GBHCs. We have built a computer code which solves 3 coupled, time-dependent kinetic equations for particles, photons and waves, respectively. Namely, $$\frac{\partial N_e}{\partial t} = -\frac{\partial}{\partial E} \left\{ \left[\Big\langle \frac{dE}{dt}\Big\rangle + \left(\frac{dE}{dt}\right)_{\rm loss} \right] N_e \right\} + \frac{1}{2} \, \frac{\partial^2}{\partial E^2} \left[(D + D_c) N_e \right]$$ $$\frac{\partial W_{\rm T}}{\partial t} = \frac{\partial}{\partial k} \left[ k^2 D \, \frac{\partial}{\partial k} \, \left( k^{-2} W_{\rm T} \right) \right] -\gamma W_{\rm T} + Q_{\rm W}\delta(k-k_0)$$ $$\frac{\partial n_{\rm ph}(\varepsilon)}{\partial t} = - n_{\rm ph}(\varepsilon) \, \int dE \, N_e(E) \, R(\varepsilon, E) +$$ $$\int\int d\varepsilon^{\prime} \, dE \, P(\varepsilon;\varepsilon^{\prime},E) \, n_{\rm ph}(\varepsilon^{\prime}) N_e(E) \mbox{} + \dot{n}_{\rm ext}(\varepsilon) + \dot{n}_{\rm emis}(\varepsilon) - \dot{n}_{\rm abs}(\varepsilon) - \frac{n_{\rm ph}(\epsilon)}{t_{\rm esc}} \, .$$ The particle distribution can be arbitrary. This allows us to determine from all the interactions whether the distribution is thermal or nonthermal. Pair production is not included so far. The Coulomb terms are also implemented for arbitrary particle distributions. Accurate Compton scattering is treated as a scattering matrix (Coppi 1992) with the full cross section. The Cyclo-Syn. process is calculated according to Robinson & Melrose (1984) which enters both as a cooling and heating term (Ghisellini et al. 1988). Syn-self absorption is also included. The radiation part of the kinetic code is tested against Monte Carlo simulations (Kusunose, Li, & Coppi 1998) and is found to be very good for $\tau \leq 3$ and for both thermal and nonthermal electron distributions. Figure \[fig-2\] shows the time evolution of particles (upper panels), waves (middle) and photons (lower panels) from the start of continuous wave injection until the steady state is reached. There are 20 curves in each plot which are from $t=0 - 10 \tau_{dyn}$, where $\tau_{dyn} = R / c \sim 1.5\times 10^{-3}$ sec. These runs are made with application to optically thin environment in mind. The plasma density $n$ is varied from $\tau = 0.1 - 1$. At early times, the particle distribution softens first as shown in upper panels, due to that waves have not fully cascaded (i.e., small $\langle k \rangle$ as shown in middle panel), and losses dominate at high energies. As waves cascade over the inertial range, $\langle k \rangle$ quickly grows to a level that acceleration overcomes all losses, electrons are then energized out of the thermal background and the nonthermal hard tail forms. The photon spectra indicates that gamma-rays can be produced when $\tau < 0.5$. Furthermore, note that the nonthermal tails start to develop at $E/m_e c^2 \sim 0.13$ (corresponding to $v_{\rm A}/c = 0.46$), this nicely confirms the fact that only particles with $v > v_{\rm A}$ can be accelerated. MHD turbulence, are they an ensemble of waves? ---------------------------------------------- The above described calculations, both linear and quasilinear, can be broadly regarded as “dissipation” in a general MHD turbulence theory. Finding a dynamical model that might adequately describe the evolution of magnetic fluctuations (such as equation (3) above) is at best phenomenological. In the dissipation range, the physics of the couplings that connect fluid and kinetic scales is not understood at all. A critical assumption that is employed in all the kinetic calculations is that the magnetic fluctuations that cascade from large scales to small scales could be regarded as an ensemble of kinetic waves with a well-defined dispersion relation to describe them. This view is by no means proven, though it allows us to get an estimate of the particle heating rate since the kinetic theory is significantly more advanced (see an application of such an approach to the interplanetary magnetic field dissipation range, Leamon et al. 1998). On the other hand, the dynamics of MHD turbulence has been studied using statistical theories and simulations (e.g., Kraichnan & Montgomery 1980; Shebalin et al. 1983; Matthaeus & Lamkin 1986), and has never been convincingly presented or developed within a normal-mode, perturbation-type of framework. A further complication is that most (MHD) turbulence theory is based on the incompressible fluid model, how it will “carry-over” to compressible astrophysical flow is still an open question. MHD turbulence Truncation ------------------------- Recently, the assumption of a cascade to smaller scales of MHD turbulence is criticized in dynamo theory. It has been argued that both the more rapid folding of magnetic flux as well as the smaller energy density at small scale ensures rapid saturation or back reaction by the field stress, immobilizing the small scale fluid motions expected from the Kolmogorov spectrum. This will truncate the turbulent spectrum at the back reaction scale, initially the smallest and progressively reaching the largest. Since the energy input to the turbulence is assumed to be primarily at the largest scale, this leaves one with negligible power at the small scale (Cattaneo & Vainshtein 1991; Kulsrud & Anderson 1992; Gruzinov & Diamond 1994; Cattaneo 1994). The remaining largest scale is that of the disk itself. In general all particle instabilities presumably leading to particle heating require large local gradients in some aspect of their phase space, i.e. temperature, density, and velocities, etc. Furthermore, all the free gravitational energy must flow through these gradients. This requirement, however, will not be met if the small scale turbulent motions are strongly damped by the back reaction of the field itself. We therefore look for a solution to this paradox in large scale magnetic structures. A Sketch View ============= Here, we outline some plausible physical pictures about what might be happening in an accretion flow. Most these are ideas that have not been thoroughly investigated. It is also clear that there are obvious gaps which need to be filled with rigorous calculations. Hydrodynamic transport and high-soft state ------------------------------------------ Many investigations have sought a linear instability deriving energy from the Keplerian flow to produce a growing mode leading, in the non-linear limit, to turbulence. The Papaloizou & Pringle instability (1984) seems to be the most studied instability but its relevance to Keplerian accretion disks has been questioned (Balbus & Hawley 1998). Recently we have identified a linear instability in Keplerian disk leading to Rossby waves and presumably Rossby vortices in the nonlinear limit (Lovelace et al. 1998). This instability grows most effectively from a large radial gradient in entropy. It has the advantage that the nonlinear limit consists of co-planar, co-rotating vortices (Nelson et al. in preparation) that require only a radial gradient, not vertical gradient of entropy. The radial gradient, we believe, is astrophysically reasonable because all disks are presumably fed by matter at some outer radius by, for example, Roche-lobe overflow in low-mass X-ray binaries. If there is no angular momentum removal mechanism, the matter will accumulate until it builds up enough to trap heat, and variations in entropy would then render the onset of the above instability. We do not, however, expect this instability to lead to turbulence in the usual sense of convective turbulence. An ensemble of co-planar vortices does not lead to significant vertical flow as compared to the usual picture of convective turbulence where buoyant plumes would convect heat released at the the mid-plain to the disk surface. We expect the angular momentum transport is done via nonlinear interactions of these vortices with the background flow, but this has to be addressed by extensive hydro simulations. The heat flow derived from the “viscosity” of the ensemble of Rossby vortices must be removed by radiation flow. We expect this not to be a problem because the radiation thickness of the disk, $\tau$, is small enough such that the effective diffusion velocity, $v_{\rm diff} \simeq c/(3 \tau) >> v_{\phi}$. Under these conditions, the disk solution will be essentially the same as the SS disk. Thus, this picture might be applied to the high-soft state of GBHCs. Relatively speaking, magnetic fields do not play a major role during this state but some nonthermal processes (such as a weak magnetic outflow) might be responsible for the powerlaw component from $20$ keV - 1 MeV. Role of large scale magnetic fields and low-hard state ------------------------------------------------------ As pointed out by Blandford & Payne (1982), large scale magnetic fields can also be very effective in removing the disk angular momentum. These large scale magnetic outflows could be a hydromagnetic wind (Blandford & Payne 1982), or it might be a nearly force-free helix (Poynting flux) with very little matter as discussed by Lovelace et al. (1987, 1997). The accreting plasma from, say a companion star, is likely to be magnetized. In the advection of this flux with the mass flow, there will necessarily be a convergence and strengthening of the field. In the region where an $\alpha$-viscosity prevails and the field acts as a passive marker of the flow, there will be both advection and diffusion. The diffusion radially outwards depends upon probably the same diffusion coefficient which allows the diffusion of angular momentum. Hence there will be a unique relationship between advection inwards and diffusion outwards leading to the relationship, $B_{z} \propto r^{-3/2}$ (see also Bisnovatyi-Kogan & Lovelace 1997 in which they argued $B_r \propto r^{-2}$). If the initial field strength advected with the mass flow at the outer disk radius is large enough, then the field energy density could become comparable to the Keplerian stress at a certain radius. Bisnovatyi-Kogan & Lovelace (1997) argued that magnetic flux then has to be destroyed at the disk via reconnection. Alternatively, instead of destroying the flux, magnetic fields (presumably tied to the companion star) could be twisted such as they will remove the angular momentum of the flow and take away the released gravitational energy. So the energy dissipation (into radiation) might not be at the disk at all. The reconnection dissipation of the current supporting the torsion of the magnetic field will perhaps lead to the non-thermal emission of GBHCs. In fact, there is ample evidence in AGNs that perhaps most of the energy release is in the outflow/jet. Such a picture could also apply to GBHCs with the hard X-ray to gamma-ray emissions produced via nonthermal processes (such as Syn. or SSC) in the magnetized outflow away from the disk. If the initial field strength advected with the mass flow at the outer disk radius is weaker, then the amplified magnetic field (by $r^{-3/2}$) may never be greater than the Keplerian stress, thus is not the dominant channel of angular momentum transport. 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--- abstract: 'We recently demonstrated a 1$^{\textrm{st}}$-order axial gradiometer SQUID system which is operated in a liquid He dewar with negligible noise contribution. The achieved close to SQUID limited measured coupled energy sensitivity $\varepsilon_{c}$ of $\sim 30\,h$ corresponds to a white field noise below 180 aT Hz${^{-1/2}}$. In order to further improve the SQUID noise performance, the junction capacitance was reduced by decreasing its lateral size to below $1~\mu$m. This was realized by extending the fabrication process for sub-micrometer-sized Josephson Junctions based on the HfTi self-shunted junction technology to an SIS process with AlO$_{\textrm{x}}$ as the insulating layer. We achieved energy sensitivities of 4.7$\,h$ and 20$\,h$ at 4.2 K for uncoupled and coupled SQUIDs, respectively. We also investigated the temperature dependence of the noise of the uncoupled SQUIDs and reached an energy sensitivity of 0.65 $h$ in the white noise regime at 400 mK.' author: - 'Jan-Hendrik Storm, Oliver Kieler, Rainer Körber[^1]' bibliography: - 'TAS\_2019\_0303\_arxiv.bib' title: 'Towards ultra-sensitive SQUIDs based on sub-micrometer-sized Josephson Junctions' --- [Shell : Sub-micrometer-sized Josephson Junctions for ultra-sensitive SQUIDs]{} sub-$\mu$m Josephson Junctions, SQUID, Noise Introduction ============ The SQUID is the most sensitive detector for magnetic flux and state-of-the-art PTB current sensor SQUIDs reach a coupled energy sensitivity of about 30 $h$ when operated at 4.2 K. For the detection of weak magnetic signals from room temperature samples, e.g. encountered in biomagnetism, SQUIDs are typically coupled to a superconducting pick-up coil and operated in a glass fiber liquid He dewar. In our latest system, the thermal noise from the superinsulation and thermal shields could be avoided enabling a close to SQUID-limited white magnetic field noise $S_{B}^{1/2}$ below 180 aT Hz${^{-1/2}}$ for a 45 mm diameter gradiometric pick-up coil [@Storm2017; @Storm2019]. Hence, improvements in SQUID performance would be beneficial for biomagnetism, but also for other applications where SQUID noise is the limiting factor. For an uncoupled dc SQUID of inductance $L_{\textrm{SQ}}$, critical current $I_{\textrm{c}}$, shunt resistance $ R_{\textrm{N}}$ and junction capacitance $C$, the design parameters ${\beta_{c}=2\pi I_{\textrm{c}} R_{\textrm{N}}^{2}C/\Phi_{0}}$ and $\beta_{L}=2 L_{\textrm{SQ}}I_{\textrm{c}}/\Phi_{0}$ are chosen close to 1 for optimal noise performance. In this case, numerical simulations yield for the energy sensitivity per unit bandwidth $\varepsilon\approx 16 k_{\textrm{B}}T(L_{\textrm{SQ}}C)^{1/2}$ where $k_{\textrm{B}}$ is the Boltzmann constant and $T$ the temperature [@Clarke2004]. The energy sensitivity of the uncoupled SQUID is determined experimentally by ${\varepsilon=S_{\Phi}/(2L_{\textrm{SQ}})}$ where $S_{\Phi}$ is the measured flux noise power density. For a SQUID with integrated input coil (current sensor SQUID), the coupled energy sensitivity ${\varepsilon_{c}=\varepsilon/k^{2}}$ is referred to the input coil (with inductance $L_{\textrm{i}}$) via the coupling coefficient $k=M/(L_{\textrm{SQ}}L_{\textrm{i}})^{1/2}$. Here, $M$ is the mutual inductance between the input coil and the SQUID loop. The equivalent field noise can be obtained from $S_{B}^{1/2}=S_{\Phi}^{1/2}L_{\textrm{tot}}/(M_{\textrm{i}}A_{\textrm{p}})=(2\varepsilon_{c}/L_{\textrm{i}})^{1/2}L_{\textrm{tot}}/A_{\textrm{p}}$ where $L_{\textrm{tot}}$ is the total inductance of the input circuit and $A_{\textrm{p}}$ the field sensitive area of the pick-up loop. The aforementioned simulations show that an improvement in the energy resolution $\varepsilon$ is possible by: 1. lowering the SQUID inductance $L_{\textrm{SQ}}$ by decreasing the size of the SQUID loop 2. reducing the Josephson junction (JJ) capacitance $C$ by decreasing the junction area 3. cooling down the SQUID device to reduce thermal noise in the shunt resistors While approach 1) has been implemented in nano-SQUIDs, it is impractical for current sensor SQUIDs with high inductance pick-up coils (${\sim\mu}$H) as coupling to the small SQUID loop becomes excessively difficult. Consequently, in this work, we present our development of dc SQUIDs based on sub-micrometer-sized Josephson Junctions to reduce the JJ capacitance and thereby increase sensitivity. We also present the noise performance for temperatures as low as 400 mK. This is important for applications where the SQUIDs are cooled to below 4.2 K. Sub-micrometer-sized Josephson Junctions ======================================== Junction technology ------------------- ![Technology for sub-micrometer-sized Josephson Junctions.[]{data-label="fig:figure1"}](figure1.pdf){width=".90\columnwidth"} SQUIDs based on cross-type submicron-sized JJs have been reported in the literature [@Schmelz2017; @Luomahaara2018]. We chose a fabrication process for the sub-micrometer-sized JJs based on the established HfTi self-shunted junction technology developed at PTB for JJs arrays [@Hagedorn2006] and nano-SQUIDs [@Bechstein2017]. The technology has been extended to a superconductor-insulator-superconductor (SIS) process utilizing conventional AlO$_{\textrm{x}}$ as the insulating layer with a nominal critical current density of 1 kA cm$^{-2}$. The fabrication is indicated in Fig. \[fig:figure1\] and uses electron-beam lithography and a chemical mechanical planarization (CMP). The trilayer JJ is patterned using inductively coupled plasma reactive ion etching (ICP-RIE) for the Nb counter electrode (200 nm) and ion beam etching for the insulating junction layer ([20 nm Al + x nm AlO$_{\textrm{x}}$]{}). This is followed by an anodization to electrically isolate the JJ edges and an SEM image of a $(0.7\times 0.7)~\mu$m$^{2}$ junction after this step is shown in Fig. \[fig:figure2\]a). To pattern the Nb base electrode (160 nm), ICP-RIE is used once more. After depositing the SiO$_{2}$ insulation between the base and the Nb wiring, superfluous SiO$_{2}$ is removed via CMP to reveal the junction contacts and for planarization of the wafer surface. For the realization of superconducting connections between the base and wiring electrodes, vias in the insulation are opened by ICP-RIE. Subsequently, patterning of the resistance layer AuPd (75 nm) for the shunt resistors is done with a lift-off process and the final Nb wiring (560 nm) is structured using again ICP-RIE. An SEM image of a $(0.7\times 0.7)~\mu$m$^{2}$ AlO$_{\textrm{x}}$ junction with the wiring electrode is shown in Fig. \[fig:figure2\]b). ![SEM image of a $(0.7\times 0.7)~\mu$m$^{2}$ AlO$_{\textrm{x}}$ junction. a) after anodization b) with wiring electrode.[]{data-label="fig:figure2"}](figure2.pdf){width="0.78\columnwidth"} Junction characterization ------------------------- ![$I$-$V$ curve of the $(0.8\times 0.8)~\mu$m$^{2}$ junction array consisting of 10 JJs at 4.2 K. Division of the abscissa by 10 gives $V_{g}$ for a single junction.[]{data-label="fig:figure3"}](figure3.pdf){width="0.90\columnwidth"} The $I$-$V$ curves of various series junction arrays were measured at 4.2 K. Exemplary data for the $(0.8\times 0.8)~\mu$m$^{2}$ JJ array are shown in Fig. \[fig:figure3\] and the extracted parameters are given in Table \[tab:IVcurve\]. The critical current $I_{\textrm{c,IV}}$ is significantly reduced compared to the nominal values and the results from the shunted JJs in the miniature SQUIDs (sec. III) due to finite temperature and rf interference. The gap voltage $V_{g}$ is determined at $I_{\textrm{c,IV}}$ and the subgap resistance $R_{S}$ at 2 mV. ----------- ------------------ --------------------- --------- ------------------ ------------------ --------------- -- JJ length $I_{\textrm{c}}$ $I_{\textrm{c,IV}}$ $V_{g}$ $R_{\textrm{N}}$ $R_{\textrm{S}}$ $R_{S}/R_{N}$ ($\mu$m) ($\mu$A) ($\mu$A) (mV) ($\Omega$) (k$\Omega$) 0.8 6.40$^{a}$ 3.36 2.63 234 4.79 20.5 0.7 4.90$^{a}$ 2.34 2.63 299 6.01 20.0 0.6 3.60$^{a}$ 1.64 2.63 397 9.98 25.1 ----------- ------------------ --------------------- --------- ------------------ ------------------ --------------- -- : Parameters of the sub-micron junctions at 4.2 K.[]{data-label="tab:IVcurve"} $^{a}$ nominal values SQUIDs based on sub-$\mu$m-sized JJs ==================================== SQUID parameters ---------------- For the evaluation of the Nb-AlO$_{\textrm{x}}$-Nb sub-micron-junction process, miniature SQUID magnetometers with $L_{\textrm{SQ}}=70~\textrm{pH}$ were fabricated. These have square junctions with side lengths of $0.8\,\mu$m and differ in their $R_{\textrm{N}}$. The design parameters for ${T=4.2}$ K are given in Table \[tab:Design\_par\] together with the values for a test SQUID with conventional JJs of 2.5 $\mu$m size. Current sensor SQUIDs with $L_{\textrm{SQ}}=80~\textrm{pH}$ and ${(0.7\times 0.7)}~\mu$m$^{2}$ JJs were also fabricated and characterized. All sub-micrometer-sized JJ devices were fabricated on a set of two wafers with nominally identical fabrication parameters. ------ ----------- ------------------ ------------------- ------------------ ------------- ---------------- ---------------------------- ----------------------------- -- -- \# JJ length $I_{\textrm{c}}$ $C$ $R_{\textrm{N}}$ $\beta_{c}$ $ \beta_{L}$ $\varepsilon_{\textrm{w}}$ $\varepsilon_{\textrm{th}}$ ($\mu$m) ($\mu$A) ($\mathrm{f\/F}$) ($\Omega$) ($h$) ($h$) SQ-1 0.8 6.4 40$^{a}$ 47 1.7 0.43 4.1$^{b}$ 1.7 SQ-2 0.8 6.4 40$^{a}$ 34 0.90 0.43 5.3 2.4 2.5$^{c}$ 6.55 400 10 0.80 0.44 27 8.0 ------ ----------- ------------------ ------------------- ------------------ ------------- ---------------- ---------------------------- ----------------------------- -- -- : Parameters of miniature SQUIDs with $L_{\textrm{SQ}}=70~\textrm{pH}$ at 4.2 K.[]{data-label="tab:Design_par"} $^{a}$ estimated from 400 mK data\ $^{b}$ mean value\ $^{c}$ equivalent square length of octagonal shaped JJ The $V$-$\Phi$ curve at 4.2 K of miniature SQUID SQ-2 is shown in the inset of Fig. \[fig:figure4\]. The indentation appearing at ${\approx90\,\mu\textrm{V}}$ corresponding to a Josephson frequency of ${\approx44~\textrm{GHz}}$ is caused by a $\lambda/4$ resonance of the interconnection stripline between SQUID and wire bonds. $I_{c}$ was calculated from the bias current needed for maximum voltage and we obtained a mean (eight devices) of $6.2\,\mu$A with a standard deviation of $0.5\,\mu$A for a single JJ. The capacitance of the ${(0.8\times 0.8)}~\mu$m$^{2}$-sized JJ was estimated from the $V$-$\Phi$ curve behavior of SQ-2 at 400 mK. Here, the measured $I_{\textrm{c}}$ increased from 6.6 to 7.2 $\mu$A in good agreement with the theoretical expectation [@Ambegaokar1963a] resulting in a noise parameter ${\Gamma=2\pi k_{\textrm{B}}T/(I_{\textrm{c}}\Phi_{0})=2.3\times 10^{-3}}$. In this case, suppression of hysteresis by thermal noise is reduced and $\beta_{c}$ needs to be close to one. We estimate ${C\approx 40~\mathrm{f\/F}}$ leading to ${\beta_{c}=0.99}$ for SQ-2. This is also consistent with ${C\propto A}$ where $A$ is the JJ area. In contrast, SQ-1 ${(\beta_{c}=1.7)}$ showed hysteresis in the $V$-$\Phi$ curve at 400 mK. Performance ----------- ![Energy sensitivity of 4 SQUIDs with $(0.8\times 0.8)~\mu$m$^{2}$ (SQ-1: $R_{\textrm{N}}=47\,\Omega$, SQ-2: $R_{\textrm{N}}=34\,\Omega$) and one SQUID with $(2.5\times 2.5)$ $\mu$m$^{2}$ JJs. The dashed lines give the white noise values $\varepsilon_{\textrm{w}}$. Inset shows $V$-$\Phi$ curves of SQ-2 with ${(0.8\times 0.8)}~\mu$m$^{2}$ JJs at 4.2 K for bias currents from 3 to 12 $\mu$A.[]{data-label="fig:figure4"}](figure4.pdf){width="0.90\columnwidth"} We first discuss the performance determined at 4.2 K. The measurements were carried out in a liquid He bath using a direct read out scheme and a single-stage configuration. The equivalent flux noise due to voltage and current noise of the XXF read-out electronics [@magnicon] was subtracted to obtain the intrinsic SQUID flux noise $S_{\Phi}$. At 4.2 K our devices fulfill $\Gamma\beta_{L}< 0.2$ and the theoretical limit $\varepsilon_{\textrm{th}}$ can be estimated by: $$\varepsilon_{\textrm{th}}\approx 2(1+\beta_{L})\Phi_{0}k_{\textrm{B}}T/I_{\textrm{c}} R_{\textrm{N}} \label{eq:epsilon}$$ which is valid for an arbitrary value of $\beta_{L}$ [@Clarke2004]. As shown in Fig. \[fig:figure4\], the intrinsic white energy sensitivity $\varepsilon_{\textrm{w}}$ was about 4.1 and 5.3$\,h$ for SQ-1 and SQ-2, respectively. Those values are a factor of more than 5 better compared to our conventional technology with junction sizes of ${(2.5\times 2.5)}$ $\mu$m$^{2}$ which shows ${\varepsilon_{\textrm{w}}=27~h}$. The slightly larger experimental $\varepsilon$ for SQ-2 is also expected theoretically using (\[eq:epsilon\]), however, the absolute values are about a factor of 2 larger. A possible origin might be suboptimal biasing or insufficient subtraction of the electronics contribution $S_{\Phi,\textrm{amp}}$ at 4.2 K as $V_{\Phi}$ is reduced in this case. A two-stage read-out scheme should be employed in future studies. The energy sensitivity increases at lower frequencies with a typical value 37$\,h$ at 1 Hz corresponding to 900 n$\Phi_{0}$ Hz$^{-1/2}$ which was reproducible for the different devices. Interestingly, the low frequency noise follows a $1/f^{\alpha}$ behavior with two distinct $\alpha$-values. For the $2.5~\mu$m-sized JJ SQUID and SQ-1 (chip T3) we find ${\alpha\sim1}$ consistent with critical current fluctuations as the origin. In contrast, the remaining SQUIDs show ${\alpha\sim0.6}$ at 4.2 K as has also been observed previously [@Drung2011; @Schmelz2017]. Temperature dependence ---------------------- ![Energy sensitivity of SQ-2 with ($800\times 800$) nm$^{2}$ for different temperatures. The dashed lines give the white noise values $\varepsilon_{\textrm{w}}$.[]{data-label="fig:figure5"}](figure5.pdf){width="0.90\columnwidth"} Measurements below 4.2 K were carried out on a pulse-tube cooler equipped with a $^{3}$He sorption unit with the samples mounted in the vacuum space inside an aluminum and lead shield. No special provisions were made to thermally anchor the device to the cold stage of the cryostat. Consequently, the thermal link is mainly provided by the copper wiring used for biasing the SQUID. Fig. \[fig:figure5\] shows the energy sensitivity for temperatures down to 400 mK together with the fitted white noise values $\varepsilon_{\textrm{w}}$ (dashed lines) for SQ-2. In agreement with our previous work [@Drung2011], we observe a reduction in the white noise level $\varepsilon_{\textrm{w}}$, an increase of the low frequency noise and a shift of the $1/f$-noise corner to higher frequencies with decreasing temperatures. ![Energy sensitivity of magnetometer SQUIDs with $(0.8\times 0.8)~\mu$m$^{2}$ for different temperatures together with the theoretical curves.[]{data-label="fig:figure6"}](figure6.pdf){width="0.90\columnwidth"} In discussing $\varepsilon_{\textrm{w}}(T)$, we refer to Fig. \[fig:figure6\] which also contains the 4.2 K-values for all SQ-1 chips. Data for lower temperatures are not included due to the aforementioned hysteresis. At 400 mK we determine $\varepsilon_{\textrm{w}}=0.65$ $h$ for SQ-2. In addition, the theoretical curve obtained by combining (\[eq:epsilon\]) and the temperature dependence $I_{\textrm{c}}(T)$ as given in [@Ambegaokar1963a] is shown. For ${T\leq 4.2}$ K one finds ${I_{\textrm{c}}(T)/I_{\textrm{c}}(0)\approx 1}$ so that $\varepsilon_{\textrm{w}}(T)$ is dominated by the linear $k_{\textrm{B}}T$-term in (\[eq:epsilon\]). For the lower temperatures there is reasonable agreement, whereas at 4.2 K the measured $\varepsilon_{\textrm{w}}$ are significantly larger as already mentioned above. It is worth discussing the temperature dependence of the low frequency behavior in more detail. The unusual increase with decreasing temperature has been observed before [@Wellstood1987] and for 4.2 and 0.4 K the excess noise follows roughly $1/f^{\alpha}$ with ${\alpha\sim0.6}$. However, for the intermediate temperatures of 1 K and 2 K, a low frequency plateau emerges which is particularly clear for the 1 K data. Whether this behavior is intrinsic to the fabrication technology remains unclear and requires further studies on a larger number of samples. Fully integrated current sensor SQUIDs -------------------------------------- ![Coupled energy sensitivity of fully integrated current sensor SQUID with $(0.7\times 0.7)~\mu$m$^{2}$ JJs. Inset shows the set of $V$-$\Phi$ curves for bias currents ranging from 2, 4 to 14 $\mu$A.[]{data-label="fig:figure7"}](figure7.pdf){width="0.90\columnwidth"} Based on these results, integrated current sensors were designed and fabricated having a SQUID inductance of 80 pH and $(0.7\times 0.7)~\mu$m$^{2}$ JJs. For the input circuit consisting of a 400 nH input coil, a proven double-transformer design was used with a coupling coefficient of ${k=0.75}$ [@Drung2007]. The expected white coupled energy sensitivity of this design is approximately 10$\,h$ at 4.2 K. In Fig. \[fig:figure7\] the results show that we achieve an $\epsilon_{c,\textrm{w}}$ of 20$\,h$. The mismatch is most likely due to a parasitic resistance between SQUID and input coil of 75 $\Omega$. The low frequency noise follows approximately $1/f$ and the set of $V$-$\Phi$-curves, given in the inset of Fig. \[fig:figure7\], does not show any irregularities, e.g. due to resonances, for SQUID voltages up to $\sim 180~\mu$V. This could be avoided by the implementation of on-chip rf-filters compared to the miniature magnetometers. Due to a limited number of fabrication runs the technology has not reached maturity, but the insufficient isolation observed in the current sensor device has been eliminated. Initially, as shown here, a good homogeneity of the JJs critical current could be achieved. However, after extensive alterations in the production technology, the restoration of this condition turns out to be difficult and is an ongoing task. Conclusions =========== In this work we presented a proof of concept of a new generation of SQUID sensors based on sub-micrometer-sized JJs. The fabrication technology was adapted from an established process for SNS junction and is based on electron-beam lithography and CMP. Small test magnetometer SQUIDs with ${(0.8\times 0.8)}~\mu$m$^{2}$ JJs validated the approach for which we obtained energy sensitivities of about $5~h$ at 4.2 K. When the devices were cooled to 400 mK we achieved below $1~h$ in reasonable agreement theoretical expectations. A current sensor SQUID utilizing a double transformer scheme suffered from an insufficient isolation and reached $\varepsilon_{\textrm{w}}$ of 20 $h$. For a reliable fabrication technology we expect a performance of $10~h$ at 4.2 K and about $1~h$ when cooled to 400 mK. In order to minimize electronics noise contributions, a 2-stage read-out scheme will be employed. Recent improvements in liquid He glass-fiber dewar construction circumvented the limiting thermal noise contributions and one can expect a significant impact of such novel devices in biomagnetic measurements. In addition, they will be very useful in fundamental science experiments where SQUIDs are often operated in a superconducting shield environment and cooled down for increased sensitivity. Acknoledgement {#acknoledgement .unnumbered} ============== The authors thank M. Petrich, J. Felgner, K. Störr and Th. Weimann for fabrication. [^1]: This work was supported in part by the European Union’s Horizon 2020 research and innovation program under Grant 686865, by the DFG under Grant KO 5321/1-1 and Grant KI 698/3-2. *(Corresponding author: Rainer Körber.)* J.-H. Storm was with Physikalisch-Technische Bundesanstalt, 10587 Berlin, Germany. O. Kieler is with Physikalisch-Technische Bundesanstalt, 38116 Braunschweig, Germany. R. Körber is with Physikalisch-Technische Bundesanstalt, 10587 Berlin, Germany. (e-mail: rainer.koerber@ptb.de) Color versions of one or more of the figures in this paper are available online at http://ieeexplore.org

--- abstract: | The asymptotic behaviour of the solutions of Poincaré’s functional equation $f(\lambda z)=p(f(z))$ ($\lambda>1$) for $p$ a real polynomial of degree $\geq2$ is studied in angular regions $W$ of the complex plain. It is known [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions; @Derfel_Grabner_Vogl2008:zeta_function_laplacian] that $f(z)\sim\exp(z^\rho F(\log_\lambda z))$, if $f(z)\to\infty$ for $z\to\infty$ and $z\in W$, where $F$ denotes a periodic function of period $1$ and $\rho=\log_\lambda\deg(p)$. In the present paper we refine this result and derive a full asymptotic expansion. The constancy of the periodic function $F$ is characterised in terms of geometric properties of the Julia set of $p$. For real Julia sets we give inequalities for multipliers of Pommerenke-Levin-Yoccoz type. The distribution of zeros of $f$ is related to the harmonic measure on the Julia set of $p$. author: - | GREGORY DERFEL\ Department of Mathematics and Computer Science,\ Ben Gurion University of the Negev, Beer Sheva 84105, Israel\ e-mail PETER J. GRABNER[^1]\ Institut für Analysis und Computational Number Theory (Math A),\ Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria\ e-mail\ - | FRITZ VOGL\ Institut für Analysis und Scientific Computing, Technische Universität Wien,\ Wiedner Hauptstraße 8–10, 1040 Wien, Austria\ e-mail title: Complex asymptotics of Poincaré functions and properties of Julia sets --- *Dedicated to Robert F. Tichy on the occasion of his ^th^ birthday.* Introduction {#sec:introduction} ============ Historical remarks {#sec:historical-remarks} ------------------ In his seminal papers [@Poincare1886:une_classe_etendue; @Poincare1890:une_classe_nouvelle] H. Poincaré has studied the equation $$\label{Eq 1} f(\lambda z)= R(f(z)),\quad z \in {\mathbb{C}},$$ where $R(z)$ is a rational function and $\lambda\in{\mathbb{C}}$. He proved that, if $R(0)=0$, $R'(0)=\lambda$, and $|\lambda|>1$, then there exists a meromorphic or entire solution of (\[Eq 1\]). After Poincaré, (\[Eq 1\]) is called [*the Poincaré equation*]{} and solutions of (\[Eq 1\]) are called [*the Poincaré functions* ]{}. The next important step was made by G. Valiron [@Valiron1923:lectures_on_general; @Valiron1954:fonctions_analytiques], who investigated the case, where $R(z)=p(z)$ is a polynomial, i.e. $$\label{eq:poincare} f(\lambda z)=p(f(z)),\quad z \in {\mathbb{C}},$$ and obtained conditions for the existence of an entire solution $f(z)$. Furthermore, he derived the following asymptotic formula for $M(r)=\max_{|z|\leq r}|f(z)|$: $$\label{Eq 3} \log M(r)\sim r^{\rho}F\left(\frac{\log r}{\log |\lambda|}\right), \quad r\rightarrow \infty.$$ Here $F(z)$ is a $1$-periodic function bounded between two positive constants, $\rho=\frac{\log d}{\log |\lambda|}$ and $d=\deg p(z)$. Different aspects of the Poincaré functions have been studied in the papers [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions; @Derfel_Grabner_Vogl2008:zeta_function_laplacian; @Eremenko_Levin1989:periodic_points_polynomials; @Eremenko_Sodin1990:iterations_rational_functions; @Ishizaki_Yanagihara2005:borel_and_julia; @Romanenko_Sharkovsky2000:long_time_properties]. In particular in [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions], in addition to (\[Eq 3\]), asymptotics of entire solutions $f(z)$ on various rays $re^{i \vartheta}$ of the complex plane have been found. It turns out that this asymptotic behaviour heavily depends on the arithmetic nature of $\lambda$. For instance, if $\operatorname{\mathrm{arg}}\lambda=2\pi\beta$, and $\beta$ is irrational, then $f(z)$ is unbounded along any ray $\operatorname{\mathrm{arg}}z={\vartheta}$ (cf. [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions]). Assumptions {#sec:assumptions} ----------- In the present paper we concentrate on the simplest, but maybe most important case for applications, namely, when $\lambda$ is real and $p(z)$ is a real polynomial (i. e. all coefficients of $p(z)$ are real). It is known from [@Valiron1954:fonctions_analytiques] and [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions] that, if $f(z)$ is an entire solution of , then the only admissible values for $f_0=f(0)$ are the fixed points of $p(z)$ (i. e. $p(f_0)=f_0$). Moreover, entire solutions exist, if and only if there exists an $n_0\in{\mathbb{N}}$ such that $$\lambda^{n_0}=p'(f_0).$$ It was proved in [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions Propositions 2.1–2.3] that the general case may be reduced to the simplest case $$f(0)=p(0)=0\text{ and }p'(0)=\lambda>1$$ by a change of variables. In the same vein, we can assume without loss of generality that $f'(0)=1$ and the polynomial $p$ is monic (i. e. the leading coefficient is $1$) $$\label{eq:poly} p(z)=z^d+p_{d-1}z^{d-1}+\cdots+p_1z.$$ Poincaré and Schröder equations {#sec:poinc-schr-equat} ------------------------------- The functional equation with the additional (natural) conditions $f(0)=0$ and $f'(0)=1$ is closely related to Schröder’s functional equation (cf. [@Schroeder1871:uber_iterierte_funktionen]) $$\label{eq:schroeder} g(p(z))=\lambda g(z),\quad g(0)=0\text{ and }g'(0)=1$$ which was used by G. Koenigs [@Koenigs1884:recherches_sur_integrales; @Koenigs1885:nouvelles_recherches_sur] to study the behaviour of $p$ under iteration around the repelling fixed point $z=0$. By definition, $g$ is the local inverse of $f$ around $z=0$. Both functions together provide a linearisation of $p$ around its repelling fixed point $z=0$ $$g(p(f(z)))=\lambda z\text{ and }g(p^{ (n)}(f(z)))=\lambda^n z,$$ where $p^{(n)}(z)$ denotes the $n$-th iterate of $p$ given by $p^{(0)}(z)=z$ and $p^{(n+1)}(z)=p(p^{(n)}(z))$. We note here that and are also called Schröder equation by some authors. For instance, the value distribution of solutions of the Poincaré (alias Schröder) equation has been investigated in [@Ishizaki_Yanagihara2005:borel_and_julia]. Branching processes and diffusion on fractals {#sec:branch-proc-diff} --------------------------------------------- Iterative functional equations occur in the context of branching processes (cf. [@Harris1963:theory_branching_processes]). Here a probability generating function $$q(z)=\sum_{n=0}^\infty p_nz^n$$ encodes the offspring distribution, where with $p_n\geq0$ is the probability that an individual has $n$ offspring in the next generation (note that $q(1)=1$). The growth rate $\lambda=q'(1)$ decides whether the population is increasing ($\lambda>1$) or dying out $\lambda\leq1$. In the first case the branching process is called *super-critical*. The probability generating function $q^{(n)}(z)$ ($n$-th iterate of $q$) encodes the distribution of the size $X_n$ of the $n$-th generation under the offspring distribution $q$. In the case of a super-critical branching process it is known that the random variables $\lambda^{-n}X_n$ tend to a limiting random variable $X_\infty$. The moment generating function of this random variable $$f(z)=\mathbb{E}e^{-zX_\infty}$$ satisfies the functional equation (cf. [@Harris1963:theory_branching_processes]) $$f(\lambda z)=q(f(z)),$$ which is , if $q$ is a polynomial. Furthermore, this equation can be transformed into , if $q$ is conjugate to a polynomial by a Möbius transformation, especially $q(z)=\frac1{p(1/z)}$, where $p$ is a polynomial. Branching processes have been used in [@Barlow1998:diffusions_on_fractals; @Barlow_Perkins1988:brownian_motion_sierpinski; @Lindstroem1990:brownian_motion_nested] to model time for the Brownian motion on certain types of self-similar structures such as the Sierpiński gasket. In this context the zeros of the solution of are the eigenvalues of the infinitesimal generator of the diffusion (“Laplacian”), if the generating function of the offspring distribution is conjugate to a polynomial (cf. [@Derfel_Grabner_Vogl2008:zeta_function_laplacian; @Grabner1997:functional_iterations_stopping; @Malozemov_Teplyaev2003:self_similarity_operators; @Teplyaev2004:spectral_zeta_function; @Teplyaev2007:spectral_zeta_functions]). In this case the zeros of $f$ have to be real, since they are eigenvalues of a self-adjoint operator. This motivates the investigation of real Julia sets in Section \[sec:real-julia-set\]. Contents {#sec:contents} -------- The paper is organised as follows. In Section \[sec:asympt-infin-fatou\] we study the asymptotic behaviour of $f(z)$ in those sectors $W$ of the complex plane, where $$\label{eq:infty} f(z)\to\infty \text{ for } z\to\infty,\quad z\in W.$$ It was proved in [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions; @Derfel_Grabner_Vogl2008:zeta_function_laplacian] that implies $$f(z)\sim\exp\left(z^\rho F\left(\frac{\log z}{\log\lambda}\right)\right) \text{ for }z\to\infty,\quad z\in W,$$ where $F(z)$ is a periodic function of period $1$. In Section \[sec:asympt-infin-fatou\] we will refine this result to a full asymptotic expansion of $f(z)$, which takes the form $$\label{eq:f-asymp-1} f(z)=\exp\left(z^\rho F\left(\log_\lambda z\right)\right)+ \sum_{n=0}^\infty c_n\exp\left(-nz^\rho F\left(\log_\lambda z\right)\right),$$ where $F$ is a periodic function of period $1$ holomorphic in some strip depending on $W$ and $\rho=\log_\lambda d$. The proof is based on an application of the Böttcher function at $\infty$ of $p(z)$. We note here that E. Romanenko and A. Sharkovsky [@Romanenko_Sharkovsky2000:long_time_properties] have studied equation on ${\mathbb{R}}$ (rather than ${\mathbb{C}}$) and obtained a full asymptotic expansion of this type by Sharkovsky’s method of “first integrals” or “invariant curves”. Further analysis of the periodic function $F$ occurring in is presented in Section \[sec:furth-analys-peri\], where the Fourier coefficients of $F$ are related to the Böttcher function at $\infty$ of $p(z)$ and the harmonic measure on the Julia set of $p$. In Section \[sec:asympt-finite-fatou\] the asymptotic behaviour of $f(z)$ is studied in sectors that are related to basins of attraction of finite attracting fixed points. In Section \[sec:zeros-poinc-funct\] we relate geometric properties of the Julia set to the location of the zeros of $f$. Section \[sec:real-julia-set\] is devoted to the special case of real Julia sets ${\mathcal{J}}(p)$. Here we prove, in particular, the following inequalities of Pommerenke-Levin-Yoccoz type for multipliers of fixed points $\xi$: $$\label{eq:pommerenke} p(\xi)=\xi\Rightarrow \begin{cases} |p'(\xi)|\geq d&\text{ for }\min{\mathcal{J}}(p)<\xi<\max{\mathcal{J}}(p)\\ |p'(\xi)|\geq d^2&\text{ for }\xi=\min{\mathcal{J}}(p)\text{ or }\xi=\max{\mathcal{J}}(p). \end{cases}$$ Furthermore, equality can hold only, if $p$ is linearly conjugate to a Chebyshev polynomial of the first kind. In Section \[sec:zeta-funct-poinc\] we continue the study of Dirichlet generating functions of zeros of Poincaré functions that we started in [@Derfel_Grabner_Vogl2008:zeta_function_laplacian] in the context of spectral zeta functions on certain fractals. We relate the poles and residues of the zeta function of $f$ to the Mellin transform of the harmonic measure $\mu$ on the Julia set of $p$. Furthermore, we show a connection between the zero counting function of $f$ and the harmonic measure $\mu$ of circles around the origin. Relation of complex asymptotics and the Fatou set {#sec:relat-compl-asympt} ================================================= Throughout the rest of the paper we will use the following notations and assumptions. Let $p$ be a real polynomial of degree $d$ as in . We always assume that $p(0)=0$ and $p'(0)=a_1=\lambda$ with $|\lambda|>1$. We refer to [@Beardon1991:iteration_rational_functions; @Milnor2006:dynamics_complex] as general references for complex dynamics. We denote the Riemann sphere by ${\mathbb{C}}_\infty$ and consider $p$ as a map on ${\mathbb{C}}_\infty$. We recall that the Fatou set ${\mathcal{F}}(p)$ is the set of all $z\in{\mathbb{C}}_\infty$ which have an open neighbourhood $U$ such that the sequence $(p^{(n)})_{n\in{\mathbb{N}}}$ is equicontinuous on $U$ in the chordal metric on ${\mathbb{C}}_\infty$. By definition ${\mathcal{F}}(p)$ is open. We will especially need the component of $\infty$ of ${\mathcal{F}}(p)$ given by $$\label{eq:Fatou-infty} {\mathcal{F}}_\infty(p)=\left\{z\in{\mathbb{C}}\mid \lim_{n\to\infty}p^{(n)}(z)=\infty\right\},$$ as well as the basins of attraction of a finite attracting fixed point $w_0$ ($p(w_0)=w_0$, $|p'(w_0)|<1$) $$\label{eq:Fatou-w0} {\mathcal{F}}_{w_0}(p)=\left\{z\in{\mathbb{C}}\mid \lim_{n\to\infty}p^{(n)}(z)=w_0\right\}.$$ The complement of the Fatou set is the Julia set ${\mathcal{J}}(p)={\mathbb{C}}_\infty\setminus{\mathcal{F}}(p)$. The filled Julia set is given by $$\label{eq:filled-Julia} {\mathcal{K}}(p)=\left\{z\in{\mathbb{C}}\mid (p^{(n)}(z))_{n\in{\mathbb{N}}}\text{ is bounded}\right\}= {\mathbb{C}}\setminus{\mathcal{F}}_\infty(p).$$ Furthermore, it is known that (cf. [@Falconer2003:fractal_geometry]) $$\label{eq:boundary} \partial{\mathcal{K}}(p)=\partial{\mathcal{F}}_\infty(p)={\mathcal{J}}(p).$$ In the case of polynomials this can be used as an equivalent definition of the Julia set. We will also use the notations $$\label{eq:W_alpha_beta} W_{\alpha,\beta}=\left\{z\in{\mathbb{C}}\setminus\{0\}\mid \alpha<\arg z<\beta\right\}$$ and $$B(z,r)=\left\{w\in{\mathbb{C}}\mid |z-w|<r\right\}.$$ Asymptotics in the infinite Fatou component {#sec:asympt-infin-fatou} ------------------------------------------- In [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions; @Derfel_Grabner_Vogl2008:zeta_function_laplacian] the asymptotics of the solution of the Poincaré equation was given. We want to present a different approach here, which gives a full asymptotic expansion. \[thm:poincare-asymp\] Let $f$ be the entire solution of the Poincaré equation for a real polynomial $p$ with $\lambda=p'(0)>1$. Assume further that the Fatou component of $\infty$, ${\mathcal{F}}_\infty(p)$ contains an angular region $W_{\alpha,\beta}$. A : Then the following asymptotic expansion for $f$ is valid for all $z\in W_{\alpha,\beta}$ large enough $$\label{eq:f-asymp} f(z)=\exp\left(z^\rho F\left(\log_\lambda z\right)\right)+ \sum_{n=0}^\infty c_n\exp\left(-nz^\rho F\left(\log_\lambda z\right)\right),$$ where $F$ is a periodic function of period $1$ holomorphic in the strip $$\left\{z\in{\mathbb{C}}\mid \frac{\alpha}{\log\lambda}<\Im z<\frac{\beta}{\log\lambda} \right\}$$ and $\rho=\log_\lambda d$. Furthermore, $$\label{eq:Re>0} \forall z\in W_{\alpha,\beta}:\Re z^\rho F(\log_\lambda z)>0$$ holds. B : Let $g$ denote the Böttcher function associated with $p$, *i. e.* $$\label{eq:boettcher} (g(z))^d=g(p(z))$$ in some neighbourhood of $\infty$. Its inverse function is given by the Laurent series around $\infty$ $$\label{eq:Boettcher-inverse-cn} g^{(-1)}\left(w\right)=w+\sum_{n=0}^\infty\frac{c_n}{w^n}.$$ Then we have $$f(z)=g^{(-1)}\left(\exp\left(z^\rho F\left(\log_\lambda z\right)\right)\right)$$ and $c_n$ can be determined from the coefficients of $p$. We recall that $p$ has a super-attracting fixed point of order $d=\deg p$ at infinity. We consider the Böttcher function $g$ associated with this fixed point (cf. [@Beardon1991:iteration_rational_functions; @Blanchard1984:complex_analytic_dynamics; @Boettcher1905:beitraege_zur_theorie; @Kuczma_Choczewski_Ger1990:iterative_functional_equations]), which satisfies the functional equation in some neighbourhood of infinity. The Böttcher function has a Laurent expansion around infinity given by $$\label{eq:Boettcher-Laurent} g(z)=z+\sum_{n=0}^\infty\frac{b_n}{z^n},$$ which converges for $|z|>R$ for some $R>0$. The coefficients $(b_n)_{n\in{\mathbb{N}}_0}$ can be determined uniquely from the coefficients of the polynomial $p$. Using the Böttcher function we can rewrite the Poincaré equation assuming that $|f(z)|>R$ $$\label{eq:Poincare-Boettcher} (g(f(z)))^d=g(p(f(z)))=g(f(\lambda z)).$$ From this we derive that $h(z)=g(f(z))$ satisfies the much simpler functional equation $$(h(z))^d=h(\lambda z),$$ which only holds for those values $z$ for which $|f(z)|>R$. This equation has solutions $$\label{eq:h(z)} h(z)=\exp\left(z^\rho F\left(\log_\lambda z\right)\right)$$ with $\rho=\log_\lambda d$ and $F$ a periodic function of period $1$ holomorphic in some strip parallel to the real axis. Since $|h(z)|>1$ for all $z$ with $|f(z)|>R$ by the properties of the function $g$, we have . By $g$ is invertible in some neighbourhood of $\infty$ and we can write where the coefficients $c_n$ depend only on the coefficients of the polynomial $p$. This function satisfies the functional equation $$\label{eq:Boettcher-inverse} g^{(-1)}(w^d)=p(g^{(-1)}(w))$$ for $w$ in some neighbourhood of $\infty$. Inserting into yields giving an exact and asymptotic expression for $f(z)$. \[rem8\] E. Romanenko and A. Sharkovsky have studied equation on ${\mathbb{R}}$ (rather than on ${\mathbb{C}}$) in [@Romanenko_Sharkovsky2000:long_time_properties]. Applying Sharkovsky’s method of “first integrals” (“invariant graphs”) they obtained a full asymptotic formula of type for all solutions $f(x)$, such that $f(x)\to\infty$ for $x\to\infty$. Böttcher functions, Green functions, and constancy of the periodic function $F$ {#sec:bottch-funct-green} ------------------------------------------------------------------------------- We will make frequent use of the integral representation of the Böttcher function $$\label{eq:Boettcher-int} g(z)=\exp\left(\int_{{\mathcal{J}}(p)}\log(z-x)\,d\mu(x)\right),$$ where $\mu$ denotes the harmonic measure on the Julia set ${\mathcal{J}}(p)$ (cf. [@Bessis_Geronimo_Moussa1984:mellin_transforms_associated; @Brolin1965:invariant_sets_under; @Ransford1995:potential_theory_complex_plane]). This shows that $g$ is holomorphic on any simply connected subset of ${\mathcal{F}}_\infty(p)$. The measure $\mu$ can be given as the weak limit of the measures $$\label{eq:mu_n} \mu_n=\frac1{d^n}\sum_{p^{(n)}(x)=\xi}\delta_x,$$ where $\xi$ can be chosen arbitrarily (not exceptional) and $\delta_x$ denotes the unit point mass at $x$ (cf. [@Brolin1965:invariant_sets_under; @Ransford1995:potential_theory_complex_plane]). The function $g(z)$ can be continued to any simply connected subset $U$ of ${\mathbb{C}}_\infty\setminus{\mathcal{K}}(p)$ (this follows for instance from the integral representation ). Furthermore, it follows from [@Beardon1991:iteration_rational_functions Lemma 9.5.5] and that $$g(U)\subset\{z\in{\mathbb{C}}_\infty\mid |z|>1\}.$$ The function $\log|g(z)|$ is the Green function for the logarithmic potential on ${\mathcal{F}}_\infty(p)$ (cf. [@Beardon1991:iteration_rational_functions Section 9]). Combining classical potential theory with polynomial iteration theory we get $$\label{eq:Julia-condition} \lim_{\substack{z\to z_0\\ z\in {\mathcal{F}}_\infty(p)}}|g(z)|=1\Leftrightarrow z_0\in{\mathcal{J}}(p),$$ where the implication $\Leftarrow$ is [@Beardon1991:iteration_rational_functions Lemma 9.5.5]. The opposite implication is a general property of the Green function (cf. [@Garnett_Marshall2005:harmonic_measure Chapter III], and [@Ransford1995:potential_theory_complex_plane Section 6.5]) combined with the fact that $\partial{\mathcal{F}}_\infty(p)={\mathcal{J}}(p)$ for polynomial $p$. \[thm:constant\] The periodic function $F$ occurring in the asymptotic expression for $f$ is constant, if and only if the polynomial $p$ is either linearly conjugate to $z^d$ or to the Chebyshev polynomial of the first kind $T_d(z)$. The periodic function $F$ is constant, if and only if the function $h(z)=g(f(z))$ introduced above satisfies $$\label{eq:h-exact} h(z)=\exp\left(Cz^\rho\right)$$ for some constant $C\neq0$. This implies that for any $w_0\in{\mathcal{J}}(p)\setminus\{0\}$ the function $g$ has an analytic continuation to some open neighbourhood of $w_0$. Thus can be replaced by $$|g(w_0)|=1 \Leftrightarrow w_0\in{\mathcal{J}}(p)$$ in our case. By this is equivalent to $w_0=f(z_0)$ for $Cz_0^\rho\in i{\mathbb{R}}$. Since $Cz^\rho\in i{\mathbb{R}}$ describes an analytic curve (with a possible cusp at $z=0$), the Julia set of $p$ is the image of this curve under the entire function $f$, thus itself an analytic arc. By [@Hamilton1995:length_julia_curves Theorem 1] ${\mathcal{J}}(p)$ can only be an analytic arc, if the Julia set of $p$ is either a line segment or a circle. The Julia set is a line segment, if and only if $p$ is linearly conjugate to the Chebyshev polynomial $T_d$ (cf. [@Beardon1991:iteration_rational_functions Theorem 1.4.1]); the Julia set is a circle, if and only if $p$ is linearly conjugate to $z^d$ (cf. [@Beardon1991:iteration_rational_functions Theorem 1.3.1]). Suppose that the periodic function $F$ is constant. If $p$ is linearly conjugate to a monomial, then the Böttcher function $g$ and therefore its inverse are linear functions. In this case $\rho=1$. (We recall that we generally assume that $f'(0)=1$.) If $p$ is linearly conjugate to a Chebyshev polynomial, $g^{(-1)}$ is linearly conjugate to the Joukowski function $z+\frac1z$. In this case $\rho=1$, if $0$ is an inner point of the line segment ${\mathcal{J}}(p)$, and $\rho=\frac12$, if $0$ is an end point of the line segment ${\mathcal{J}}(p)$ (cf. Sections \[sec:negative-julia-set\] and \[sec:julia-set-has\]). Furthermore, the asymptotic series is finite, if the periodic function $F$ is constant. Further analysis of the periodic function {#sec:furth-analys-peri} ----------------------------------------- In this section we relate the periodic function $F$ occurring in to the local behaviour of the Böttcher function at the fixed point $f(0)=0$. This will allow to express the Fourier coefficients of $F$ in terms of residues of the Mellin transform (cf. [@Doetsch1971:handbuch_der_laplace; @Oberhettinger1974:tables_mellin_transforms]) of the harmonic measure $\mu$ given by . This Mellin transform was introduced and studied in [@Bessis_Geronimo_Moussa1984:mellin_transforms_associated]. A similar relation was also used in [@Grabner1997:functional_iterations_stopping] to derive an asymptotic expression for $f$ in a special case. We will use the relation $$\label{eq:G(w)} G(w)=\log g(w)=\int_{{\mathcal{J}}(p)}\log(w-x)\,d\mu(x)$$ between the (complex) “Green function” $G$ and the Böttcher function $g$. Assume that the Fatou component ${\mathcal{F}}_\infty(p)$ contains an angular region centred at the fixed point $0$. Furthermore, assume that $\lim_{w\to 0}g(w)=1$. Then holds in this angular region. This fact can be used to analyse the local behaviour of $\log g(w)$ around $w=0$: $$\label{eq:logg} \log g(w)=\left(f^{(-1)}(w)\right)^\rho F\left(\log_\lambda f^{(-1)}(w)\right)= w^\rho F\left(\log_\lambda w\right)+{\mathcal{O}}(w^{\rho+1}).$$ Thus the behaviour of the Green function $G$ at the point $0$ exhibits the same periodic function $F$ as the asymptotic expansion of $\log f$ around $\infty$. ![Paths of integration.[]{data-label="fig:paths"}](path.eps){width="0.8\hsize"} We now relate the Green function $G(w)$ to the Mellin transform of $\mu$ $$\label{eq:Mellin} M_\mu(s)=\int_{{\mathcal{J}}(p)}(-x)^s\,d\mu(x),$$ where the branch cut for the function $(-x)^s$ is chosen to connect $0$ with $\infty$ without any further intersection with ${\mathcal{J}}(p)$. Following the computations in [@Bessis_Geronimo_Moussa1984:mellin_transforms_associated Section 5] we obtain $$M_\mu(s)=\frac1{2\pi i}\oint_\Gamma(-z)^s\,dG(z)= \frac1{2\pi i}\oint_{\Gamma_R}(-z)^s\,dG(z).$$ For $\Re s<0$ we have for the circle of radius $R$ $$\left|\frac1{2\pi i}\int_{|z|=R}(-z)^s\,dG(z)\right|\ll R^{\Re s},$$ which allows to let $R\to\infty$ in this case. This gives $$\begin{gathered} M_\mu(s)=\frac1{2\pi i}\left(\int_{\Lambda_+}(-z)^s\,dG(z)- \int_{\Lambda_-}(-z)^s\,dG(z)\right)\\ =\frac{e^{-i\pi s}-e^{i\pi s}}{2\pi i}\int_0^\infty x^sG'(x)\,dx= s\frac{\sin\pi s}\pi\int_0^\infty x^{s-1}G(x)\,dx,\end{gathered}$$ which relates the Mellin transform of the measure $\mu$ to the Mellin transform of the function $G(z)$ $$\label{eq:Mellin-G} {\mathcal{M}}G(s)=\int_0^\infty x^{s-1}G(x)\,dx=\frac\pi{s\sin\pi s}M_\mu(s)\text{ for } -\rho<\Re s<0.$$ The function $M_\mu(s)$ (and therefore ${\mathcal{M}}G(s)$ by ) has an analytic continuation by the following observation $$\label{eq:continuation} M_\mu(s)=\frac1d\sum_{k=1}^d\int_{{\mathcal{J}}(p)}(-p_k^{(-1)}(x))^s\,d\mu(x),$$ where $p_k^{(-1)}$ ($k=1,\ldots,d$) denote the $d$ branches of the inverse function of $p$; we choose the numbering so that $p_1^{(-1)}(0)=0$. The summands for $k=2,\ldots,d$ are clearly entire functions in $s$, since the integrand is bounded away from $0$ and $\infty$. For the summand with $k=1$ we observe that $$\label{eq:approx} p_1^{(-1)}(x)=\frac1\lambda x+{\mathcal{O}}(x^2)\text{ for }x\to0.$$ Inserting this into gives $$\begin{gathered} M_\mu(s)=\frac1d\lambda^{-s}\int_{{\mathcal{J}}(p)}(-x)^s\,d\mu(x)+ \frac1d\lambda^{-s}\int_{{\mathcal{J}}(p)}(-x)^s{\mathcal{O}}(x)\,d\mu(x)\\+ \frac1d\sum_{k=2}^d\int_{{\mathcal{J}}(p)}(-p_k^{(-1)}(x))^s\,d\mu(x),\end{gathered}$$ where the second term on the right-hand-side originates from inserting the holomorphic function ${\mathcal{O}}(x^2)$ from into the integrand, which gives a function holomorphic in a larger domain. Thus we obtain $$\label{eq:Mellin-continuation} M_\mu(s)=\frac1{d\lambda^s-1}H(s)$$ for some function $H(s)$ holomorphic for $\Re s>-\rho-1$ ($\rho=\log_\lambda d$). The numerator $d\lambda^s-1$ has zeros at $s=-\rho+\frac{2k\pi i}{\log\lambda}$ ($k\in{\mathbb{Z}}$), which give possible poles for the function $M_\mu(s)$. Using the full Taylor expansion of $p_1^{(-1)}(x)$ instead of the ${\mathcal{O}}$-term in would yield the existence of a meromorphic continuation of $M_\mu(s)$ to the whole complex plane. Taking and together gives the analytic continuation of ${\mathcal{M}}G(s)$ to $-\rho-1<\Re s<0$. Then the Mellin inversion formula (cf. [@Doetsch1971:handbuch_der_laplace]) gives (for $-\rho<c<0$) $$\begin{gathered} \label{eq:Mellin-inv-G} G(x)=\frac1{2\pi i}\int\limits_{c-i\infty}^{c+i\infty}{\mathcal{M}}G(s)x^{-s}\,ds= \frac1{2\pi i}\int\limits_{c-i\infty}^{c+i\infty}\frac\pi{s\sin\pi s} \frac1{d\lambda^s-1}H(s)x^{-s}\,ds\\ =\frac1{2\pi i}\int\limits_{-\rho-\frac12-i\infty}^{-\rho-\frac12+i\infty} \frac\pi{s\sin\pi s} \frac1{d\lambda^s-1}H(s)x^{-s}\,ds+ \sum_{k\in{\mathbb{Z}}}\operatorname*{\mathrm{Res}}_{s=-\rho+\frac{2k\pi i}{\log\lambda}} {\mathcal{M}}G(s)x^{-s}.\end{gathered}$$ The integral in the second line is ${\mathcal{O}}(x^{\rho+\frac12})$, the sum of residues can be evaluated further to give the Fourier expansion of the periodic function $F$ $$\label{eq:Fourier-F} \sum_{k\in{\mathbb{Z}}}\operatorname*{\mathrm{Res}}_{s=-\rho+\frac{2k\pi i}{\log\lambda}} {\mathcal{M}}G(s)x^{-s}=x^\rho\sum_{k\in{\mathbb{Z}}}f_k e^{2k\pi i\log_\lambda x}= x^\rho F(\log_\lambda x).$$ The Fourier coefficients $f_k$ are given by $$\begin{gathered} \label{eq:fk} f_k=\operatorname*{\mathrm{Res}}_{s=-\rho-\frac{2k\pi i}{\log\lambda}}{\mathcal{M}}G(s)= \frac\pi{\left(-\rho-\frac{2k\pi i}{\log\lambda}\right) \sin\pi\left(-\rho-\frac{2k\pi i}{\log\lambda}\right)} \operatorname*{\mathrm{Res}}_{s=-\rho-\frac{2k\pi i}{\log\lambda}}M_\mu(s)\\ =\frac\pi{\left(-\log d-2k\pi i\right) \sin\pi\left(-\rho-\frac{2k\pi i}{\log\lambda}\right)} H\left(-\rho-\frac{2k\pi i}{\log\lambda}\right).\end{gathered}$$ Asymptotics in a finite Fatou component – analysis of asymptotic values {#sec:asympt-finite-fatou} ----------------------------------------------------------------------- It is clear from the functional equation for $f$ that any asymptotic value of $f$ has to be an attracting fixed point of the polynomial $p$ (including $\infty$). Thus the analysis in Section \[sec:asympt-infin-fatou\] can be interpreted as the behaviour of $f$ when approaching the asymptotic value $\infty$. In the present section we extend this analysis to all asymptotic values. First we study the case of a finite attracting, but not super-attracting fixed point. Let $w_0$ be an attracting fixed point of $p$ and denote $\eta=p'(w_0)\neq0$ ($|\eta|<1$). Then there exists a solution $\Psi$ of the Schröder equation $$\label{eq:schroeder-Psi} \eta\Psi(z)=\Psi(p(z)),\quad \Psi(w_0)=0,\text{ and }\Psi'(w_0)=1,$$ which is holomorphic in ${\mathcal{F}}_{w_0}(p)$ (for instance, the sequence $(\eta^{-n}(p^{(n)}(z)-w_0))_{n\in{\mathbb{N}}}$ converges to $\Psi$ on any compact subset of ${\mathcal{F}}_{w_0}(p)$). Assume now that ${\mathcal{F}}_{w_0}(p)$ contains an angular region $W_{\alpha,\beta}\cap B(0,r)$ for some $r>0$. Then by conformity of $f$ some angular region at the origin is mapped into $W_{\alpha,\beta}\cap B(0,r)$. We consider the function $$j(z)=\Psi(f(z)),$$ which satisfies the functional equation $$\label{eq:Psif} j(\lambda z)=\Psi(f(\lambda z))=\Psi(p(f(z)))=\eta\Psi(f(z))=\eta j(z).$$ This equation has the solution $$\label{eq:jz} j(z)=z^{\log_\lambda\eta}H(\log_\lambda z)$$ with some periodic function of period $1$, holomorphic in some strip. This periodic function can never be constant, since otherwise $j(z)$ would have an analytic continuation to the slit complex plane. From this it would follow that $f$ is bounded in the slit complex plane, a contradiction. The function $\Psi$ has a holomorphic inverse around $0$ $$\Psi^{(-1)}(z)=w_0+z+\sum_{n=2}^\infty \psi_nz^n$$ which allows us to write $$\label{eq:fw0} f(z)=\Psi^{(-1)}\left(z^{\log_\lambda\eta}H(\log_\lambda z)\right)= w_0+z^{\log_\lambda\eta}H(\log_\lambda z)+ \sum_{n=2}^\infty \psi_nz^{n\log_\lambda\eta}(H(\log_\lambda z))^n,$$ which is valid in the angular region $W_{\alpha,\beta}$ for $z$ large enough. This gives an exact and asymptotic expression for $f$ in an angular region. In the case of a super-attracting fixed point $w_0$ we have $p'(w_0)=0$. Assume that the first $k-1$ derivatives of $p$ vanish in $w_0$, but the $k$-th derivative is non-zero. Then $p(z)=(z-w_0)^kP(z)$ with $P(w_0)=A\neq0$. We use the solution $g$ of the corresponding Böttcher equation $$\label{eq:boettcher-w0} g(p(z))=A(g(z))^k\quad g(w_0)=0,\quad g'(w_0)=1$$ to linearise $$g(f(\lambda z))=g(p(f(z)))=A(g(f(z)))^k.$$ Thus the function $h(z)=g(f(z))$ satisfies $$h(\lambda z)=A(h(z))^k.$$ This equation has solutions $$h(z)=A^{-\frac1{k-1}}\exp\left(z^{\log_\lambda k} L\left(\log_\lambda z\right)\right)$$ for a periodic function $L$ of period $1$ and a suitable choice of the $(k-1)$-th root. Furthermore, by the fact that $\lim_{z\to\infty}h(z)=0$ we have $$\Re\left(z^{\log_\lambda k} L\left(\log_\lambda z\right)\right)<0\text{ for } f(z)\in{\mathcal{F}}_{w_0}(p).$$ using the local inverse of $g$ around $0$ we get $$\begin{gathered} \label{eq:f-asymp-w0} f(z)=g^{(-1)}\left(A^{-\frac1{k-1}}\exp\left(z^{\log_\lambda k} L\left(\log_\lambda z\right)\right)\right)\\ =w_0+A^{-\frac1{k-1}}\exp\left(z^{\log_\lambda k} L\left(\log_\lambda z\right)\right)(1+o(1)).\end{gathered}$$ Summing up, we have proved \[thm10\] Let $w_0$ be an attracting fixed point of $p$ such that the Fatou component ${\mathcal{F}}_{w_0}(p)$ contains an angular region $W_{\alpha,\beta}\cap B(0,r)$ for some $r>0$. Then the asymptotic behaviour of $f$ for $z\to\infty$ and $z\in W_{\alpha,\beta}$ is given by , if $\eta=p'(w_0)\neq0$, and by , if $p(z)-w_0$ has a zero of order $k$ in $w_0$. The periodic function $H$ in cannot be constant, because otherwise $f(z)$ would be bounded. The periodic function $L$ in can only be constant, if $p$ is linearly conjugate to $z^k$, by the same arguments as in the proof of Theorem \[thm:constant\] (the case of Chebyshev polynomials does not occur, because they only have repelling finite fixed points). As a consequence of Ahlfors’ theorem on asymptotic values (cf. [@Goluzin1969:geometric_theory_functions]) and Valiron’s theorem on the growth of $f$ (cf. [@Valiron1923:lectures_on_general; @Valiron1954:fonctions_analytiques]) we get an upper bound for the number of attracting fixed points of a polynomial. Let $p$ be a real polynomial of degree $d>1$ and let $$\gamma=\max\left\{|p'(z)|\mid p(z)=z\right\}.$$ Then the number of (finite) attracting fixed points of $p$ is bounded by $2\log_\gamma d$, i.e. $$\label{eq:ahlfors} \#\left\{z\in{\mathbb{C}}\mid p(z)=z\wedge |p'(z)|<1\right\}\leq 2\log_\gamma d.$$ Zeros of the Poincaré function and Julia sets {#sec:zeros-poinc-funct} ============================================= In this section we relate the distribution of zeros of the Poincaré function in angular regions to geometric properties of the Julia set ${\mathcal{J}}(p)$ of the polynomial $p$. \[thm8\] Let $p$ be a real polynomial with $p(0)=0$ and $p'(0)=\lambda>1$. Then the following are equivalent 1. \[thm8.1\] $\displaystyle{\forall r>0: W_{\alpha,\beta}\cap {\mathcal{J}}(p)\cap B(0,r)\neq\emptyset}$ 2. \[thm8.3\] $W_{\alpha,\beta}$ contains a zero of $f$. 3. \[thm8.2\] $W_{\alpha,\beta}$ contains infinitely many zeros of $f$. We first remark that \[thm8.3\] and \[thm8.2\] are trivially equivalent, since $f(z_0)=0$ implies that $f(\lambda^n z_0)=0$. For the proof of “\[thm8.1\]$\Rightarrow$ \[thm8.3\]” we take $0<{\varepsilon}<\frac{\beta-\alpha}2$ so small that $$\forall r>0: W_{\alpha+{\varepsilon},\beta-{\varepsilon}}\cap {\mathcal{J}}(p)\cap B(0,r)\neq\emptyset.$$ Then we take $r>0$ so small that $$\label{eq:angle} W_{\alpha+{\varepsilon},\beta-{\varepsilon}}\cap B(0,r)\subset f\left(W_{\alpha,\beta}\right),$$ which is possible by conformity of $f$ and $f'(0)=1$. Since the preimages of $0$ are dense in ${\mathcal{J}}(p)$, there exists $\eta\in W_{\alpha+{\varepsilon},\beta-{\varepsilon}}\cap B(0,r)$ and $n\in{\mathbb{N}}$ such that $p^{(n)}(\eta)=0$. By there exists $\xi\in W_{\alpha,\beta}$ such that $f(\xi)=\eta$, from which we obtain $$f(\lambda^n\xi)=p^{(n)}(f(\xi))=p^{(n)}(\eta)=0.$$ For the proof of “\[thm8.2\]$\Rightarrow$ \[thm8.1\]” we take $z_0\in W_{\alpha,\beta}$ with $f(z_0)=0$. Then $$\forall n\in{\mathbb{N}}: f(\lambda^{-n}z_0)\in{\mathcal{J}}(p).$$ For any $r>0$ and $n$ large enough $f(\lambda^{-n}z_0)\in W_{\alpha,\beta}\cap B(0,r)$, which gives \[thm8.1\]. Similar arguments show \[thm:zeros-on-line\] Let $p$ be a real polynomial with $p(0)=0$ and $p'(0)=\lambda>1$. Then $$\label{eq:negative-zeros} {\mathcal{J}}(p)\subset{\mathbb{R}}^-\cup\{0\}\Leftrightarrow \text{ all zeros of }f \text{ are non-positive real}$$ and $$\label{eq:real-zeros} {\mathcal{J}}(p)\subset{\mathbb{R}}\Leftrightarrow \text{ all zeros of }f\text{ are real.}$$ Real Julia set {#sec:real-julia-set} ============== \[lem1\] Let $p$ be a real polynomial of degree $d>1$. Then the Julia-set ${\mathcal{J}}(p)$ is real, if and only if there exists an interval $[a,b]$ such that $$\label{eq:inv-int} p^{(-1)}\left([a,b]\right)\subseteq[a,b].$$ Assume first that ${\mathcal{J}}(p)\subset{\mathbb{R}}$ and take the interval $[a,b]=[\min{\mathcal{J}}(p),\max{\mathcal{J}}(p)]$. Let ${\varepsilon}>0$. Since ${\mathcal{J}}(p)$ is perfect, there exist $\xi,\eta\in{\mathcal{J}}(p)$ with $a<\xi<a+{\varepsilon}<b-{\varepsilon}<\eta<b$. All preimages of $\xi$ and $\eta$ are in ${\mathcal{J}}(p)$ by the invariance of ${\mathcal{J}}(p)$. Furthermore, all these preimages are distinct. Therefore, every value $x\in[\xi,\eta]$ has exactly $d$ distinct preimages in $[a,b]$ by continuity of $p$. Since ${\varepsilon}$ was arbitrary and the two points $a,b$ also have all their preimages in ${\mathcal{J}}(p)\subset[a,b]$, we have proved . Assume on the other hand that $[a,b]$ satisfies . Since the map $p$ has only finitely many critical values, there exists $x\in[a,b]$ such that the backward iterates of $x$ are dense in the Julia set. By all these backward iterates are real; therefore ${\mathcal{J}}(p)$ is real. \[rem1\] By the above proof we can always assume $[a,b]=[\min {\mathcal{J}}(p),\max {\mathcal{J}}(p)]$. Furthermore, we have $$p\left(\{\min {\mathcal{J}}(p),\max {\mathcal{J}}(p)\}\right)\subseteq\{\min {\mathcal{J}}(p),\max {\mathcal{J}}(p)\},$$ which implies that at least one of the two end points of this interval is either a fixed point, or they form a cycle of length $2$. \[thm1\] Let $p$ be a polynomial of degree $d>1$ with real Julia set ${\mathcal{J}}(p)$. Then for any fixed point $\xi$ of $p$ with $\min {\mathcal{J}}(p)<\xi<\max {\mathcal{J}}(p)$ we have $|p'(\xi)|\geq d$. Furthermore, $|p'(\min{\mathcal{J}}(p))|\geq d^2$ and $|p'(\max {\mathcal{J}}(p))|\geq d^2$. Equality in one of these inequalities implies that $p$ is linearly conjugate to the Chebyshev polynomial $T_d$ of degree $d$. This theorem can be compared to [@Buff2003:bieberbach_conjecture_dynamics Theorem 2] and [@Levin1991:pommerenke's_inequality; @Pommerenke1986:conformal_mapping_iteration], where estimates for the derivative of $p$ for connected Julia sets are derived. Furthermore, in [@Eremenko_Levin1992:estimation_characteristic_exponents] estimates for $\frac1n\log|(p^{(n)})'(z)|$ for periodic points of period $n$ are given. Before we give a proof of the theorem, we present a lemma, which is of some interest on its own. A similar result is given in [@Levin1980:distribution_zeros_entire Chapter V, Section 2, Lemma 3]. \[lem2\] Let $f$ be holomorphic in the angular region $W_{\alpha,\beta}$ If there exists a positive constant $M$ such that $$\forall z\in W_{\alpha,\beta}:|f(z)|\geq M,$$ then $$\forall {\varepsilon}>0\,\,\, \exists A,B>0\,\,\, \forall z\in W_{\alpha+{\varepsilon},\beta-{\varepsilon}}: |f(z)|\leq B\exp(A|z|^\kappa)$$ with $\kappa=\frac\pi{\beta-\alpha}$. Without loss of generality we can assume that $M=1$, $\alpha=-\frac\pi2$, and $\beta=\frac\pi2$. In this case $\kappa=1$. The function $$v(z)=\log|f(z)|$$ is a positive harmonic function in the right half-plane. Thus it can be represented by the Nevanlinna formula (cf. [@Levin1996:lectures_entire_functions p.100]) $$\label{eq:nevanlinna} v(x+iy)=\frac x\pi\int_{-\infty}^\infty\frac{d\nu(t)}{|z-it|^2}+\sigma x,$$ where $\nu$ denotes a measure satisfying $$\int_{-\infty}^\infty\frac{d\nu(t)}{1+t^2}<\infty$$ and $\sigma\geq0$. In the region given by $|\arg z|\leq\frac\pi2-{\varepsilon}$ and $|z|>1$ we have $$|z-it|\geq\max(|t|\sin{\varepsilon},|z|\sin{\varepsilon})\geq\max(1,|t|)\sin{\varepsilon}.$$ From this it follows that $$|z-it|^2\geq\frac12(1+t^2)\sin^2{\varepsilon},$$ which gives $$\int_{-\infty}^\infty\frac{d\nu(t)}{|z-it|^2}\leq \frac2{\sin^2{\varepsilon}}\int_{-\infty}^\infty\frac{d\nu(t)}{1+t^2}\leq B_{\varepsilon}$$ for $|z|\geq1$ and some $B_{\varepsilon}>0$. Setting $A=\frac1\pi B_{\varepsilon}+\sigma$ and observing that $x\leq|z|$ completes the proof. Without loss of generality we may assume that the fixed point $\xi=0$. Then we consider the solution $f$ of the Poincaré equation $$f(\lambda z)=p(f(z))$$ with $\lambda=p'(0)$. We assume first that $\lambda>0$. First we consider the case $\min {\mathcal{J}}(p)<\xi<\max {\mathcal{J}}(p)$. In this case the function $f(z)/z$ tends to infinity uniformly for $z\to\infty$ in the region ${\varepsilon}\leq\arg z\leq\pi-{\varepsilon}$ for any ${\varepsilon}>0$ by Theorem \[thm:poincare-asymp\]. Furthermore, we know that $$|f(z)|\geq C\exp(A|z|^{\log_\lambda d})$$ in this region for some positive constants $A$ and $C$. Since $f(z)/z$ does not vanish at $z=0$, this function satisfies the hypothesis of Lemma \[lem2\], from which we derive that $$\log_\lambda d\leq\frac\pi{\pi-2{\varepsilon}}$$ holds for any ${\varepsilon}>0$, which implies $\lambda=p'(0)\geq d$. The proof in the case $\xi=\max {\mathcal{J}}(p)$ runs along the same lines. The function $f(z)/z$ tends to infinity uniformly in any region $|\arg z|\leq\pi-{\varepsilon}$ in this case, which by Lemma \[lem2\] implies $$\log_\lambda d\leq\frac\pi{2\pi-2{\varepsilon}}$$ for all ${\varepsilon}>0$, and consequently $\lambda=p'(0)\geq d^2$. For negative $\lambda=p'(0)$ we apply the same arguments to $p^{(2)}$. For the proof of the second assertion of the theorem, we first assume that the fixed point $\xi=0$ satisfies $a=\min {\mathcal{J}}(p)<0<\max {\mathcal{J}}(p)=b$ and that $p'(0)=d$. We know that for a suitable linear conjugate $q$ of the Chebyshev polynomial $T_d$ we have $q'(0)=d$ and ${\mathcal{J}}(q)=[a,b]$ with $0\in(a,b)$. Let us assume now that $p'(0)=d$ and ${\mathcal{J}}(p)$ is a Cantor subset of the real line, or after a rotation that ${\mathcal{J}}(p)$ is a Cantor subset of the imaginary axis (this makes notation slightly simpler). By arguments, similar to those in the beginning of Section \[sec:furth-analys-peri\] we can write $$\label{eq:harmonic} H(z)=\Re\log g(f(z))=\int_{{\mathcal{J}}(p)}\log|f(z)-x|\,d\mu(x).$$ Since $\Re\log g(.)$ is the Green function of ${\mathcal{J}}(p)$ with pole at $\infty$ (cf. [@Beardon1991:iteration_rational_functions Lemma 9.5.5] or [@Ransford1995:potential_theory_complex_plane]), we know that $H(z)\geq0$ for all $z\in{\mathbb{C}}$ and $H(z)=0$, if and only if $f(z)\in{\mathcal{J}}(p)$ (since ${\mathcal{K}}(p)={\mathcal{J}}(p)$ in the present case). By Theorem \[thm:poincare-asymp\] we have $$\label{eq:periodic} H(z)=\Re \left(zF(\log_d z)\right)=x\Re(F(\log_d z))-y\Im(F(\log_d z)) \text{ for }z=x+iy,$$ and by Theorem \[thm:constant\] the function $F$ is not constant in the present case. The periodic function $\Im F(t+i{\varphi})$ has zero mean, since the mean of $F$ is real. Thus $\Im F(t+i{\varphi})$ attains positive and negative values for any ${\varphi}$. We now take $z=iy\in i{\mathbb{R}}^+$ to obtain $$H(iy)=-y\Im(F(\log_d y+i\frac\pi{2\log d})).$$ Since $\Im F$ attains positive values by the above argument, we get a contradiction to $H(z)\geq0$ for all $z$. A similar argument shows that for $0=\max {\mathcal{J}}(p)$ and $p'(0)=d^2$ the assumption that the Julia set is not an interval leads to the same contradiction. \[rem10\] Lemma 6.4 in [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions] proves Theorem \[thm1\] for the special case of quadratic polynomials. The proof given in [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions] is purely geometrical. \[rem11\] We have a purely real analytic proof for $|p'(\max{\mathcal{J}}(p))|\geq d^2$, which is motivated by the proof of the extremality of the Chebyshev polynomials of the first kind given in [@Rivlin1974:chebyshev_polynomials]. However, we could not find a similar proof for the other assertions of the theorem. The Julia set is a subset of the negative reals {#sec:negative-julia-set} ----------------------------------------------- As a consequence of Lemma \[lem2\] we get that any solution of the Poincaré equation for a polynomial with Julia set contained in the negative real axis has order $\leq\frac12$. The only solutions of a Poincaré equation with order $\frac12$ in this situation are the functions $$f(z)=\frac1a\left(\cosh\sqrt{2az}-1\right)$$ for $$p(z)=(T_d(az+1)-1)/a,$$ where $a\in{\mathbb{R}}^+$ and $T_d$ denotes the Chebyshev polynomial of the first kind of degree $d$. This is also the only case where the periodic function $F$ in is constant in this situation. \[cor15\] Assume that $p$ is a real polynomial such that ${\mathcal{J}}(p)$ is real and all coefficients $p_i$ ($i\geq2$) of $p$ are non-negative. Then ${\mathcal{J}}(p)\subset{\mathbb{R}}^-\cup\{0\}$ and therefore $$\label{eq:simple-asymp} f(z)\sim\exp\left(z^\rho F\left(\frac{\log z}{\log\lambda}\right)\right)$$ for $z\to\infty$ and $|\operatorname{\mathrm{arg}}z|<\pi$. Here $F$ is a periodic function of period $1$ holomorphic in the strip given by $|\Im w|<\frac\pi{\log\lambda}$. Furthermore, for every ${\varepsilon}>0$ $\Re e^{i\rho\operatorname{\mathrm{arg}}z}F(\frac{\log z}{\log\lambda})$ is bounded between two positive constants for $|\operatorname{\mathrm{arg}}z|\leq\pi-{\varepsilon}$. From [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions Lemmas 6.4 and 6.5] it follows that $f(z)$ has only non-positive real zeros. Then by Theorem \[thm:zeros-on-line\] ${\mathcal{J}}(p)\subset{\mathbb{R}}^-\cup\{0\}$. Finally, the assertion follows by applying [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions Theorem 7.5]. In order to illustrate the above results, we shall turn to the equation $$f(5z)=4f(z)^2-3f(z),$$ which arises in the description of Brownian motion on the Sierpiński gasket [@Derfel_Grabner_Vogl2008:zeta_function_laplacian; @Kroen2002:green_functions_self; @Kroen_Teufl2004:asymptotics_transition_probabilities; @Teplyaev2004:spectral_zeta_function]. Here $p(z)=4z^2-3z$, and the fixed point of interest is $f(0)=1$. This fits into the assumptions of Section \[sec:assumptions\] only after substituting $g(z)=4(f(z)-1)$, where $g$ satisfies $$g(5z)=g(z)^2+5g(z).$$ Now Corollary \[cor15\] may be applied to this equation (the preimages of $0$ are real by [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions Lemma 6.7]) to give . Note also that $p'(0)=5>4=2^2$ in accordance with Theorem \[thm1\]. The Julia set has positive and negative elements {#sec:julia-set-has} ------------------------------------------------ Again as a consequence of Theorem \[thm1\] the solution of the Poincaré equation for a polynomial with real Julia set with positive and negative elements has order $\leq1$. The only solution of a Poincaré equation of order $1$ in this situation are the functions $$f(z)=\frac1a\left(\cos\left(a\frac{z-\frac{2k\pi}{d-1}} {\sin\frac{k\pi}{d-1}}\right)-\xi_k\right)$$ for $$p(z)=\frac1a\left(T_d(a(z+\xi_k))-\xi_k\right),$$ where $a\in{\mathbb{R}}^+$ and $\xi_k=\cos\frac{k\pi}{d-1}$ for $1\leq k<\frac{d-1}2$. This is again the only case where the periodic function $F$ in is constant in this situation. The Zeta function of the Poincaré function {#sec:zeta-funct-poinc} ========================================== In [@Derfel_Grabner_Vogl2008:zeta_function_laplacian] the zeta function of a fractal Laplace operator was related to the zeta function of certain Poincaré functions. Asymptotic expansions for the Poincaré functions were then used to give a meromorphic continuation of these zeta functions as well as information on the location of their poles and values of residues. In this section we give a generalisation of these results to polynomials whose Fatou set contains an angular region $W_{-\alpha,\alpha}$ around the positive real axis. In this case the solution $f$ of has no zeros in an angular region $W_{-\alpha,\alpha}$. Furthermore, from the Hadamard factorisation theorem we get $$\label{eq:hadamard} f(z)=z\exp\left(\sum_{\ell=1}^k(-1)^{\ell-1}\frac{e_\ell z^\ell}\ell\right) \prod_{\substack{f(-\xi)=0\\\xi\neq0}}\!\!\left(1+\frac z\xi\right) \exp\left(-\frac z\xi+\frac {z^2}{2\xi^2}+\cdots+ (-1)^{k-1}\frac{z^k}{k\xi^k}\right),$$ where $k=\lfloor\log_\lambda d\rfloor$. By the discussion in [@Derfel_Grabner_Vogl2008:zeta_function_laplacian Section 5] the values $e_1,\ldots,e_k$ are given by the first $k$ terms of the Taylor series of $\log\frac{f(z)}z$ $$\log\frac{f(z)}z=\sum_{\ell=1}^k(-1)^{\ell-1}\frac{e_\ell z^\ell}\ell +{\mathcal{O}}(z^{k+1}).$$ The zeta function of $f$ is now defined as $$\label{eq:zeta_f} \zeta_f(s)=\sum_{\substack{f(-\xi)=0\\\xi\neq0}}\xi^{-s},$$ where $\xi^{-s}$ is defined using the principal value of the logarithm, which is sensible, since $\xi$ is never negative real by our assumption on ${\mathcal{F}}_\infty(p)$. The function $\zeta_f(s)$ is holomorphic in the half plane $\Re s>\rho$. In [@Derfel_Grabner_Vogl2008:zeta_function_laplacian] we used the equation $$\label{eq:zeta-weierstrass} \int_0^\infty\left(\log f(x)-\log x- \sum_{\ell=1}^k(-1)^{\ell-1}\frac{e_\ell x^\ell}\ell\right) x^{-s-1}\,dx=\zeta_f(s)\frac\pi{s\sin\pi s},$$ which holds for $\rho<\Re s<k+1$, to derive the existence of a meromorphic continuation of $\zeta_f$ to the whole complex plane. There ([@Derfel_Grabner_Vogl2008:zeta_function_laplacian Theorem 8]) we obtained $$\operatorname*{\mathrm{Res}}_{s=\rho+\frac{2k\pi i}{\log\lambda}}\zeta_f(s)= -\frac{f_k}\pi\left(\rho+\frac{2\pi ik}{\log\lambda}\right) \sin\pi\left(\rho+\frac{2\pi ik}{\log\lambda}\right),$$ where $f_k$ is given by . From this we get $$\label{eq:Res-zeta_f} \operatorname*{\mathrm{Res}}_{s=\rho+\frac{2k\pi i}{\log\lambda}}\zeta_f(s)= -\operatorname*{\mathrm{Res}}_{s=-\rho-\frac{2k\pi i}{\log\lambda}}M_\mu(s).$$ This shows that the function $$\label{eq:zeta_f-M_mu} \zeta_f(s)-M_\mu(-s)$$ is holomorphic in $\rho-1<\Re s<\rho+1$, since the single poles on the line $\Re s=\rho$ cancel. This fact was used in [@Grabner1997:functional_iterations_stopping] to derive an analytic continuation for $\zeta_f(s)$. \[thm9\] Let $f$ be the entire solution of and assume that $p$ is neither linearly conjugate to a Chebyshev polynomial nor to a monomial and that $W_{-\alpha,\alpha}\subset{\mathcal{F}}_\infty(p)$ for some $\alpha>0$. Then the following assertions hold 1. \[enum2\] the limit $\lim_{t\to\infty}t^{-\rho}\log f(t)$ does not exist. 2. \[enum1\] $\zeta_f(s)$ has at least two non-real poles in the set $\rho+2\pi i\sigma{\mathbb{Z}}$ $\sigma=\frac1{\log\lambda}$. 3. \[enum5\] the limit $\lim_{x\to0}x^{-\rho}G(x)$ with $G$ given by does not exist. Equation in Theorem \[thm:poincare-asymp\] (see also [@Derfel_Grabner_Vogl2007:asymptotics_poincare_functions]) implies that $$z^{-\rho}\log f(z)=F(\log_\lambda z)+o(1)\text{ for }z\to\infty\text{ and } z\in W_{-\alpha,\alpha}$$ with a periodic function $F$ of period $1$. Theorem \[thm:constant\] implies that $F$ is a non-constant . Thus the limit in \[enum2\] does not exist. Since the periodic function $F$ is non-constant, there exists a $k_0\neq0$ such that the Fourier-coefficients $f_{\pm k_0}$ do not vanish. By we have $$\log f(z)=z^\rho\sum_{k\in{\mathbb{Z}}}f_k z^{\frac{2k\pi i}{\log\lambda}}+ {\mathcal{O}}(z^{-M})$$ for any $M>0$. By properties of the Mellin transform (cf. [@Paris_Kaminski2001:asymptotics_mellin_barnes]), every term $Az^{\rho+i\tau}$ in the asymptotic expansion of $\log f(z)$ corresponds to a first order pole of the Mellin transform of $\log f(z)$ with residue $A$ at $s=\rho+i\tau$. Since $f_{k_0}\neq0$, from we have simple poles of $\zeta_f(s)$ at $s=\rho\pm\frac{2k_0\pi i}{\log\lambda}$. Assertion \[enum5\] follows from \[enum2\] by . In the following we consider the zero counting function of $f$ $$\label{eq:N(x)} N_f(x)=\sum_{\substack{|\xi|<x\\f(\xi)=0}}1.$$ \[thm11\] Let $f$ be the entire solution of . Then the following are equivalent 1. \[enum3\] the limit $\lim_{x\to\infty}x^{-\rho}N_f(x)$ does not exist. 2. \[enum4\] the limit $\lim_{t\to0}t^{-\rho}\mu(B(0,t))$ does not exist. For the proof of the equivalence of \[enum3\] and \[enum4\] we observe that by the fact that $f'(0)=1$, there is an $r_0>0$ such that $f:B(0,r_0)\to{\mathbb{C}}$ is invertible. For the following we choose $n=\lfloor\log_\lambda(x/r_0)\rfloor+k$ and let the integer $k>0$ be fixed for the moment. Then we use the functional equation for $f$ to get $$N_f(x)=\#\left\{\xi\mid f(\lambda^n\xi)=p^{(n)}(f(\xi))=0 \wedge|\xi|<x\lambda^{-n}\right\} \!=\!\#\!\left(p^{(-n)}(0)\cap f(B(0,x\lambda^{-n}))\right).$$ This last expression can now be written in terms of the discrete measure $\mu_n$ given in $$N_f(x)=d^n\mu_n\left(f(B(0,x\lambda^{-n}))\right).$$ By the weak convergence of the measures $\mu_n$ (cf. [@Brolin1965:invariant_sets_under]) we get for $x\to\infty$ (equivalently $n\to\infty$) $$N_f(x)=d^n\mu(f(B(0,x\lambda^{-n})))+o(d^n)= x^\rho(x\lambda^{-n})^{-\rho}\mu(f(B(0,x\lambda^{-n})))+o(x^\rho).$$ By our choice of $n$ we have $r_0\lambda^{-k-1}\leq x\lambda^{-n}\leq r_0\lambda^{-k-1}$, which makes the first term dominant. From this it is clear that the existence of the limit $$\lim_{x\to\infty}x^{-\rho}N_f(x)=C$$ is equivalent to $$\mu(f(B(0,t)))=Ct^\rho\text{ for }r_0\lambda^{-k}\leq t< r_0\lambda^{-(k-1)}.$$ Since $k$ was arbitrary this implies $$\label{eq:mu(B)} \mu(f(B(0,t)))=Ct^\rho\text{ for }0< t< r_0.$$ It follows from $f'(0)=1$ that $$\label{eq:mu(f(B))-mu(B))} \forall{\varepsilon}>0:\exists\delta>0:\forall t<\delta: B(0,(1-{\varepsilon})t)\subset f(B(0,t))\subset B(0,(1+{\varepsilon})t).$$ Thus the existence of the limit in assertion \[enum4\] is equivalent to $$\lim_{t\to0}t^{-\rho}\mu(f(B(0,t)))=C.$$ Thus \[enum3\] and \[enum4\] are equivalent. If ${\mathcal{J}}(p)$ is real and disconnected then the limits in Theorem \[thm11\] do not exist. Furthermore, it is known that the limit $$\lim_{t\to0}t^{-\rho}\mu(f(B(w,t)))=C$$ does not exist for $\mu$-almost all $w\in{\mathcal{J}}(p)$ (cf. [@Mattila1995:geometry_sets_measures Theorem 14.10]), if $\rho$ is not an integer. 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Soc.* **359** (2007) 4339–4358 (electronic). . *Lectures on the [G]{}eneral [T]{}heory of [I]{}ntegral [F]{}unctions* (Private, Toulouse, 1923). . *Fonctions [A]{}nalytiques* (Presses Universitaires de France, Paris, 1954). [^1]: This author is supported by the Austrian Science Foundation FWF, project S9605, part of the Austrian National Research Network “Analytic Combinatorics and Probabilistic Number Theory”.

--- abstract: 'A system of functional equations relating the Euler characteristics of moduli spaces of stable representations of quivers and the Euler characteristics of (Hilbert scheme-type) framed versions of quiver moduli is derived. This is applied to wall-crossing formulas for the Donaldson-Thomas type invariants of M. Kontsevich and Y. Soibelman, in particular confirming their integrality.' author: - | Markus Reineke\ Fachbereich C - Mathematik\ Bergische Universität Wuppertal\ D - 42097 Wuppertal, Germany\ e-mail: reineke@math.uni-wuppertal.de title: '**Cohomology of quiver moduli, functional equations, and integrality of Donaldson-Thomas type invariants**' --- Introduction ============ In [@KS], a framework for the definition of Donaldson-Thomas type invariants for Calabi-Yau categories endowed with a stability structure is developed. One of the key features of this setup is a wall-crossing formula for these invariants, describing their behaviour under a change of stability structure in terms of a factorization formula for automorphisms of certain Poisson algebras defined using the Euler form of the category.\ In [@RWC], such factorization formulas are interpreted using quiver representations, their moduli spaces, and Hall algebras. The main result of [@RWC] interprets the factorization formula in terms of generating series of the Euler characteristic of the smooth models of [@SM], which can be viewed as Hilbert schemes in the setup of quiver moduli:\ In the general framework of [@KR; @LBNCC], series of moduli spaces of stable representations of quivers are viewed as the commutative ‘approximations’ to a fictitious noncommutative geometry of (the path algebras of) quivers. In this framework, the smooth models can be viewed as Hilbert schemes of points of this noncommutative geometry (for example, in the case of moduli spaces of semisimple representations of quivers, the smooth models parametrize finite codimensional left ideals in the path algebra of the quiver, in the same way as the Hilbert schemes of points of an affine variety parametrize finite codimensional ideals in the coordinate ring of the variety; see [@SM Section 6]). Since path algebras of quivers are of global dimension $1$, this setup thus describes aspects of a one-dimensional noncommutative geometry.\ The first aim of this paper (after reviewing some facts on quiver moduli in Section \[recoll\]) is to develop a (one-dimensional, noncommutative) analog of the result [@Ch] calculating the generating series of Euler characteristics of Hilbert schemes of points of a threefold $X$ as the $\chi(X)$-th power of the MacMahon series (see [@BF Theorem 4.12], [@MNOP Conjecture 1] for the corresponding statement for Donaldson-Thomas invariants). Namely, we relate the (generating series of) Euler characteristics of moduli spaces of stable quiver representations and Euler characteristics of their smooth models by a coupled system of functional equations, see Theorem \[t42\], Corollary \[corsd\]. This is achieved using a detailed analysis of a Hilbert-Chow type morphism from a smooth model to a moduli space of semistable representations, whose fibres are non-commutative Hilbert schemes (see Section \[fe1\]). The explicit cell decompositions for the latter, constructed in [@SM], yield functional equations for the Euler characteristic; see Section \[section4\].\ The second aim is to prove the integrality conjecture [@KS Conjecture 1] for the Donaldson-Thomas type invariants appearing in the wall-crossing formula of [@KS]; see Section \[app\]. These numbers arise by a factorization of the generating series of Euler characteristics as an Euler product (this process can thus be interpreted as fitting a genuinely noncommutative (one-dimensional) object into a commutative (three-dimensional) framework). Using the functional equations mentioned above, we can interprete this process as passing to the compositional inverse of an Euler product, and elementary number-theoretic considerations in Section \[number\] yield the desired integrality property (it should be noted that a similar process appears in [@Sti] in relating modular forms and instanton expansions). We also confirm a conjectural formula of [@KS] for diagonal Donaldson-Thomas type invariants using recent results of [@Weist].\ [**Acknowledgments:**]{} I would like to thank T. Bridgeland, V. Jovovic, S. Mozgovoy, Y. Soibelman, H. Thomas, V. Toledano-Laredo and T. Weist for interesting discussions concerning this work. Recollections on quiver moduli {#recoll} ============================== In this section, we fix some notation and collect information on moduli spaces of stable representations of quivers and some of their variants, like Hilbert schemes of path algebras and the smooth models of [@SM]. See [@Rmoduli] for an overview over these moduli spaces and the techniques used to prove some of the results cited below.\ Let $Q$ be a finite quiver, with set of vertices $I$, and arrows written as $\alpha:i\rightarrow j$ for $i,j\in I$. Denote by $r_{i,j}$ the number of arrows from $i\in I$ to $j\in I$ in $Q$. Define $\Lambda={\bf Z}I$, with elements written in the form $d=\sum_{i\in I}d_ii$, and define $\Lambda^+={\bf N}I\subset \Lambda$. We will sometimes use locally finite quiver, for which the set of vertices is possibly infinite, but with only finitely many arrows starting or ending in each single vertex. Dimension vectors for locally finite quivers are assumed to be supported on a finite subquiver.\ Introduce a non-symmetric bilinear form $\langle\_,\_\rangle$ (the Euler form) on $\Lambda$ by $$\langle d,e\rangle=\sum_{i\in I}d_ie_i-\sum_{\alpha:i\rightarrow j}d_ie_j$$ for $d,e\in\Lambda$; we thus have $\langle i,j\rangle=\delta_{i,j}-r_{i,j}$. For a functional $\Theta\in\Lambda^*={\rm Hom}_{\bf Z}(\Lambda,{\bf Z})$ (called a stability), define the slope of $d\in\Lambda^+\setminus 0$ as $\mu(d)=\Theta(d)/\dim d$, where $\dim d=\sum_{i\in I}d_i$. For $\mu\in{\bf Q}$, define $$\Lambda^+_\mu=\{d\in\Lambda^+\setminus 0\, ,\, \mu(d)=\mu\}\cup\{0\}$$ (a subsemigroup of $\Lambda^+$), and ${{}^\prime\!\Lambda}_\mu^+=\Lambda^+_\mu\setminus 0$.\ We consider complex finite dimensional representations $M$ of $Q$, consisting of a tuple of complex vector spaces $M_i$ for $i\in I$ and a tuple of ${\bf C}$-linear maps $M_\alpha:M_i\rightarrow M_j$ indexed by the arrows $\alpha:i\rightarrow j$ of $Q$. The dimension vector ${\underline{\dim}}M\in\Lambda^+$ is defined by $({\underline{\dim}}M)_i=\dim_{\bf C}M_i$. The abelian ${\bf C}$-linear category of all such representations is denoted by ${\rm mod}_{\bf C}Q$.\ Define the slope of a non-zero representation $M$ of $Q$ as the slope of its dimension vector, thus $\mu(M)=\mu({\underline{\dim}}M)$. Call $M$ semistable (for the choice of stability $\Theta$) if $\mu(U)\leq \mu(M)$ for all non-zero subrepresentations $U$ of $M$, and call $M$ stable if $\mu(U)<\mu(M)$ for all proper non-zero subrepresentations $U$ of $M$. Finally, call $M$ polystable if it isomorphic to a direct sum of stable representations of the same slope. The full subcategory ${\rm mod}_{\bf C}^\mu Q$ of all semistable representations of slope $\mu\in{\bf Q}$ is an abelian subcategory, that is, it is closed under extensions, kernels and cokernels. Its simple (resp. semisimple) objects are precisely the stable (resp. polystable) representations of $Q$ of slope $\mu$.\ Note that in the case $\Theta=0$, all representations are semistable, and the stable (resp. polystable) ones are just the simples (resp. semisimples).\ By [@King], for every $d\in\Lambda^+$, there exists a (typically singular) complex variety $M_d^{\rm sst}(Q)$ whose points parametrize the isomorphism classes of polystable representations of $Q$ of dimension vector $d$. In case $\Theta=0$, the variety $M_d^{sst}(Q)$ is affine, parametrizing isomorphism classes of semisimple representations of $Q$ of dimension vector $d$; it will be denoted by $M_d^{ssimp}(Q)$. This variety always contains a special point $0$ corresponding to the semisimple representations $\bigoplus_{i\in I}S_i^{d_i}$, where $S_i$ denotes the one-dimensional representation of $Q$ concentrated at a vertex $i\in I$, and with all arrows represented by zero maps. Note that all $M_d^{ssimp}(Q)$ reduce to the single point $0$ if $Q$ has no oriented cycles. There exists a projective morphism from $M_d^{sst}(Q)$ to $M_d^{ssimp}(Q)$.\ The variety $M_d^{sst}(Q)$ admits the following Luna type stratification (that is, a finite decomposition into locally closed subsets) induced by the decomposition types of polystable representations: let $\xi=((d^1,\ldots,d^s),(m_1,\ldots,m_s))$ be a pair consisting of a tuple of dimension vectors of the same slope as $d$ and a tuple of non-negative integers, such that $d=\sum_{i=1}^sm_id^i$. We call such $\xi$ a polystable type for $d$. Analogously to [@LBP] in the case of trivial stability, the set of all polystable representations $M$ such that $M=\bigoplus_{i=1}^sU_i^{m_i}$ for pairwise non-isomorphic stable representations $U_i$ of dimension vector $d^i$ forms a locally closed subset of $M_d^{sst}(Q)$, denoted by $S_\xi$.\ Let $n\in\Lambda^+$ be another dimension vector, and fix complex vector spaces $V_i$ of dimension $n_i$ for $i\in I$. A pair $(M,f)$ consisting of a semistable representation $M$ of $Q$ of dimension vector $d$ and a tuple $f=(f_i:V_i\rightarrow M_i)$ of ${\bf C}$-linear maps is called stable in [@SM] if the following condition holds: if $U$ is a proper subrepresentation of $M$ containing the image of $f$ (in the sense that $f_i(V_i)\subset U_i$ for all $i\in I$), then $\mu(U)<\mu(M)$. Two such pairs $(M,f)$, $(M',f')$ are called equivalent if there exists an isomorphism $\varphi:M\rightarrow M'$ interwining the additional maps, that is, such that $f_i'=\varphi_i\circ f_i$ for all $i\in I$.\ By [@SM], there exists a smooth complex variety $M_{d,n}^\Theta(Q)$, called a smooth model for $M_d^{sst}(Q)$, whose points parametrize equivalence classes of stable pairs as above. It admits a projective morphism $\pi_d:M_{d,n}^\Theta(Q)\rightarrow M_d^{sst}(Q)$.\ In the case of trivial stability, the smooth model (a Hilbert scheme for the path algebra of $Q$) ${\rm Hilb}_{d,n}(Q):=M_{d,n}^0(Q)$ parametrizes arbitrary representations $M$ of $Q$ of dimension vector $d$, together with maps $f_i:V_i\rightarrow M_i$ whose images generate the representation $M$. There exists a projective morphism $\pi:{\rm Hilb}_{d,n}(Q)\rightarrow M_d^{ssimp}(Q)$. We denote by ${\rm Hilb}_{d,n}^{nilp}(Q)$ the inverse image under $\pi$ of the special point $0\in M_d^{ssimp}(Q)$; it parametrizes pairs $(M,f)$ as above, with $M$ being a nilpotent representation, in the sense that all maps $M_{\alpha_n}\circ\ldots\circ M_{\alpha_1}$ representing oriented cycles $i_1\stackrel{\alpha_1}{\rightarrow}i_2\stackrel{\alpha_2}{\rightarrow}\ldots\stackrel{\alpha_n}{\rightarrow}i_1$ in $Q$ are nilpotent.\ Following [@ALB], for any polystable type $\xi$ for $d$ as above, introduce new (called local) quiver data $Q_\xi$, $d_\xi$, $n_\xi$ as follows: the quiver $Q_\xi$ has vertices $1,\ldots,s$ with $\delta_{i,j}-\langle d^i,d^j\rangle$ arrows from $i$ to $j$ for $i,j=1,\ldots,s$. The dimension vector $d_\xi$ is defined by $(d_\xi)_i=m_i$ for $i=1,\ldots,s$, and the dimension vector $n_\xi$ is defined by $(n_\xi)_i=n\cdot d^i$. With this notation, we have the following result (see [@SM]): \[strat\] The variety $M_{d,n}^\Theta(Q)$ admits a stratification (in the sense defined above) by the locally closed subsets $M_{d,n}^\Theta(Q)_\xi=\pi_d^{-1}S_\xi$. Each $M_{d,n}^\Theta(Q)_\xi$ admits a fibration (that is, an étale locally trivial surjection) over the corresponding Luna stratum $S_\xi$, whose fibre is isomorphic to ${\rm Hilb}_{d_\xi,n_\xi}^{nilp}(Q_\xi)$. By a cell decomposition of a variety $X$ we mean a filtration $\emptyset=X_0\subset X_1\subset\ldots\subset X_s=X$ by closed subvarieties, such that the complements $X_s\setminus X_{s-1}$ are isomorphic to affine spaces.\ For every vertex $i\in I$, we construct a (locally finite) tree quiver $Q_i$ as follows: the vertices $\omega$ of $Q_i$ are indexed by the paths in $Q$ starting in $i$ (including the empty path from $i$ to $i$ of length $0$); there is an arrow $\omega\rightarrow\alpha\omega$ for every path $\omega$ from $i$ to $j$ and every arrow $\alpha:j\rightarrow k$. Note that $Q_i$ has a unique source, corresponding to the empty path. By a subtree $T$ of $Q_i$ we mean a full subquiver which is closed under taking predecessors. The dimension vector ${\underline{\dim}}T$ is defined by setting $({\underline{\dim}}T)_j$ as the number of paths $\omega\in T$ which end in $j$. By an $n$-forest we mean a tuple $T_*=(T_{i,k})_{i\in I,\, k=1,\ldots,n_i}$ of subtrees $T_{i,k}$ of $Q_i$; its dimension vector is defined as ${\underline{\dim}}T_*=\sum_{i\in I}\sum_{k=1}^{n_i}{\underline{\dim}}T_{i,k}$. It is proved in [@SM] that \[celldec\] For all $d$ and $n$, the Hilbert scheme ${\rm Hilb}_{d,n}(Q)$ admits a cell decomposition, whose cells are parametrized by the $n$-forests of dimension vector $d$. Functional equation for $\chi({\rm Hilb}_{d,n}(Q))$ and the big local quiver {#fe1} ============================================================================ It follows immediately from Theorem \[celldec\] that the Euler characteristic of the Hilbert scheme ${\rm Hilb}_{d,n}(Q)$ can be computed as the number of $n$-forests of dimension vector $d$. This allows us to characterize the generating function of these Euler characteristics by a functional equation. For all $n\in \Lambda^+$, we write $$F^n(t)=\sum_{d\in\Lambda^+}\chi({\rm Hilb}_{d,n}(Q))t^d\in{\bf Q}[[\Lambda]].$$ \[grafting\] The series $F^n(t)$ are the uniquely determined elements of ${\bf Q}[[\Lambda]]$ satisfying the following functional equations: 1. For all $n\in\Lambda^+$, we have $F^n(t)=\prod_{i\in I}F^i(t)^{n_i}$, 2. for all $i\in I$, we have $F^i(t)=1+t_i\prod_{j\in I}F^j(t)^{r_{i,j}}.$ [[**Proof:** ]{}]{}Comparing coefficients of $t^d$ in both sides of the first identity, we see that the first claim reduces to the definition of $n$-forests. With the same method, the second identity reduces to the existence of a bijection between subtrees of $Q_i$ of dimension vector $d$ and tuples $(T_{j,k})_{j\in I,\, ,k=1,\ldots,r_{i,j}}$ of subtrees $T_{j,k}$ of $Q_j$ such that $\sum_{j\in I}\sum_{k=1}^{r_{i,j}}{\underline{\dim}}T_{j,k}=d-i$. Such a bijection is provided, by definition of the trees $Q_i$, by grafting the subtrees $T_{j,k}$ to a common root $i$ to obtain any subtree of $Q_i$ exactly once. [$\Box$]{} [[**Remark:** ]{}]{}In the special case of a quiver with a single vertex and a number of loops, this functional equation is derived in [@RNCHilb]. For all $d,n\in\Lambda^+$, we have $\chi({\rm Hilb}_{d,n}^{nilp}(Q))=\chi({\rm Hilb}_{d,n}(Q))$. [[**Proof:** ]{}]{}We adopt an argument used in [@CBVdB]. There is a natural ${\bf C}^*$-action on representations of $Q$ by rescaling the maps representing the arrows by a common factor. This action induces actions on ${\rm Hilb}_{d,n}(Q)$ and $M_d^{ssimp}(Q)$, for which the map $\pi_d: {\rm Hilb}_{d,n}(Q)\rightarrow M_d^{simp}(Q)$ is equivariant. Moreover, there exists a unique fixed point for the action of ${\bf C}^*$ on $M_d^{ssimp}(Q)$, namely the point $0$, to which all points of $M_d^{ssimp}(Q)$ attract, in the sense that $\lim_{t\rightarrow 0}t\cdot M=0$ for all $M\in M_d^{ssimp}(Q)$. Therefore, all points of ${\rm Hilb}_{d,n}(Q)$ admit a well-defined limit in the projective variety $\pi^{-1}(0)={\rm Hilb}_{d,n}^{nilp}(Q)$. For each connected component $C$ of ${\rm Hilb}_{d,n}^{nilp}(Q)$, we have its attractor $A_C$ consisting of all points of ${\rm Hilb}_{d,n}(Q)$ whose limit belongs to $C$. By the Bialynicki-Birula theorem [@BB], the attractors $A_C$ are affine fibrations over the components $C$. Consequently, the Euler characteristics of ${\rm Hilb}_{d,n}(Q)$ and of ${\rm Hilb}_{d,n}^{nilp}(Q)$ coincide.[$\Box$]{} Now we fix data $Q,\Theta,\mu,n$ as before, and associate to it a locally finite quiver (called the big local quiver) $\widetilde{Q}$ as follows: the vertices of $\widetilde{Q}$ are indexed by pairs $(d,i)$ in ${{}^\prime\!\Lambda}^+_\mu\times{\bf N}$. The number of arrows from vertex $(d,i)$ to $(d',i')$ is given as $\delta_{d,d'}\cdot\delta_{i,i'}-\langle d,d'\rangle$. For a function $l:{{}^\prime\!\Lambda}^+_\mu\rightarrow{\bf N}$, we define $\widetilde{Q}_l$ as the full subquiver of $\widetilde{Q}$ supported on the set of vertices $(d,i)$ for $d\in{{}^\prime\!\Lambda}^+_\mu$ and $1\leq i\leq l(d)$.\ We define dimension vectors $\widetilde{n}$ for the various quivers $\widetilde{Q}_l$ by $\widetilde{n}_{(d,i)}=n\cdot d$. The product $S(\widetilde{Q})=\prod_{d\in{{}^\prime\!\Lambda}^+_\mu}S_\infty$ of infinite symmetric groups acts on the vertices of $\widetilde{Q}$ by permutation $(\sigma_e)_{e\in\Lambda^+_\mu}(d,i)=(d,\sigma_d(i))$; this restricts to an action of $\prod_{d\in{{}^\prime\!\Lambda}^+_\mu}S_{l(d)}$ on $\widetilde{Q}_l$.\ For a polystable type $\xi=((d^1,\ldots,d^s),(m_1,\ldots,m_s))$ as above, we can view the local quiver $Q_\xi$ as the quiver $\widetilde{Q}_{l_\xi}$ just defined, where the function $l_\xi$ is given by defining $l_\xi(d)$ as the number of indices $1\leq j\leq s$ such that $d=d^j$. The dimension vector $d_\xi$ for $Q_\xi$ can then be viewed as a dimension vector $\widetilde{d}_\xi$ for $Q_l$. This dimension vector can be made unique by assuming that its entries $(\widetilde{d}_xi)_{d,i}$, for fixed $d\in{{}^\prime\!\Lambda}_\mu^+$, form a partition, that is, $(\widetilde{d}_\xi)_{(d,1)}\geq\ldots\geq(\widetilde{d}_xi)_{(d,l_\xi(d))}$. Therefore, we call dimension vectors $\widetilde{d}$ of $\widetilde{Q}_l$ partitive if $\widetilde{d}_{(d,1)}\geq\ldots\geq\widetilde{d}_{(d,l(d))}$ for all $d\in\Lambda^+_\mu$; the set of all partitive dimension vectors for $\widetilde{Q}$ (resp. $\widetilde{Q}_l$) is denoted by $\Lambda(\widetilde{Q})^\geq$ (resp. $\Lambda(\widetilde{Q}_l)^\geq$). We have a natural specialization map $\nu:\Lambda(\widetilde{Q}_l)^+\rightarrow\Lambda^+_\mu$ given by $\nu(d,i)=d$.\ We consider the generating function $$R^n_l(t)=\sum_{\widetilde{d}\in\Lambda^+(\widetilde{Q}_l)}\chi({\rm Hilb}_{\widetilde{d},\widetilde{n}}(\widetilde{Q}_l)t^{\nu(\widetilde{d})}\in{\bf Z}[[\Lambda^+_\mu]],$$ the specialization of the generating function $F^{\widetilde{n}}$ for the quiver $\widetilde{Q}_l$ with respect to the map $\nu$. By the natural $\prod_{d\in{{}^\prime\!\Lambda}^+_\mu}S_{l(d)}$-symmetry of $\widetilde{Q}_l$, we have $R^{(d,i)}_l(t)=R^{(d,j)}_l(t)$ for all $d\in\Lambda^+_\mu\setminus 0$ and all $1\leq i,j\leq l(d)$. We denote this series by $R^{(d)}_l(t)$. Applying Proposition \[grafting\] and the definition of $\widetilde{Q}_l$, we get $$R^n_l(t)=\prod_{d\in{{}^\prime\!\Lambda}^+_\mu}R^{(d)}_l(t)^{l(d)\cdot(n\cdot d)}$$ and $$R_l^{(d)}(t)=1+t^d\cdot R_l^{(d)}(t)\cdot\prod_{e\in{{}^\prime\!\Lambda}^+_\mu}R^{(e)}_l(t)^{-\langle d,e\rangle\cdot l(e)}.$$ Call a dimension vector for $\widetilde{Q}_l$ faithful if all its entries are non-zero, and denote by $\Lambda(\widetilde{Q}_l)^{++}$ the set of all such dimension vectors. Define $${{}^\prime\!R}^n_l(t)=\sum_{\widetilde{d}\in\Lambda^+(\widetilde{Q}_l)^{++}}\chi( {\rm Hilb}_{\widetilde{d},\widetilde{n}}(\widetilde{Q}_l)t^d\in{\bf Z}[[\Lambda^+_\mu]].$$ Using again the symmetry of $\widetilde{Q}_l$, we see that $$R^n_l(t)=\sum_{l':{{}^\prime\!\Lambda}^+_\mu\rightarrow{\bf N}}\prod_{d\in{{}^\prime\!\Lambda}^+_\mu}\binom{l(d)}{l'(d)}\cdot{{}^\prime\!R}^n_{l'}(t).$$ Let $\chi:\Lambda^+_\mu\setminus 0\rightarrow{\bf Z}$ be a function with arbitrary integer values (in contrast to the function $l$ considered so far), and define a formal series by $$R^n_\chi(t)=\sum_{l':{{}^\prime\!\Lambda}^+_\mu\rightarrow{\bf N}}\prod_{d\in{{}^\prime\!\Lambda}^+_\mu}{\binom{\chi(d)}{l'(d)}}\cdot{{}^\prime\!R}^n_{l'}(t).$$ Simililarly to the above, we have series ${{}^\prime\!R}_l^{(d)}(t)$ and $R_\chi^{(d)}(t)$ for $d\in\Lambda^+_\mu$ as special cases of the series ${{}^\prime\!R}_l^n(t)$ and $R_\chi^n(t)$, respectively. \[trick\] The series $R^n_\chi(t)$ are given by the functional equations $$R^n_\chi(t)=\prod_{d\in{{}^\prime\!\Lambda}^+_\mu}R^{(d)}_\chi(t)^{\chi(d)\cdot(n\cdot d)}$$ and $$R_\chi^{(d)}(t)=1+t^d\cdot R_\chi^{(d)}\cdot\prod_{e\in{{}^\prime\!\Lambda}^+_\mu}R^{(e)}_\chi(t)^{-\langle d,e\rangle\cdot \chi(e)}.$$ [[**Proof:** ]{}]{}It is easy to see that there exist unique series $S_\chi^d(t)$ for all functions $\chi$ as above and all $d\in{{}^\prime\!\Lambda}^+_\mu$ fulfilling the equations $$S_\chi^d(t)=1+t^d\cdot S_\chi^d(t)\cdot\prod_{e\in{{}^\prime\!\Lambda}^+_\mu}S^e_\chi(t)^{-\langle d,e\rangle\cdot \chi(e)},$$ since these functional equations define recursions determining the coefficients of the series. These coefficients depend polynomially on the values $\chi(d)$. The same holds for the coefficients of the series $R_\chi^{(d)}(t)$ by definition. Now the equality $S_\chi^d(t)=R_\chi^{(d)}(t)$ holds for all functions $\chi$ with values in ${\bf N}$, thus it has to hold for arbitrary $\chi$.[$\Box$]{} Functional equation for $\chi(M_{d,n}^\Theta(Q))$ {#section4} ================================================= We start with a calculation of Euler characteristics of strata of symmetric products of a variety, which should be well-known. Denote by $\mathcal{P}$ the set of all partitions. For $\lambda$ in $\mathcal{P}$, denote by $m_i(\lambda)$ the multiplicity of $i$ in $\lambda$, that is, the number of indices $j$ such that $\lambda_j=i$. For a variety $X$, we denote by $S^nX$ its $n$-th symmetric power, that is, the quotient of $X^n$ by the natural action of the symmetric group $S_n$. The product variety $X^n$ admits a stratification by strata $X^n_I$, where $I=(I_1,\ldots,I_k)$ is a decomposition of $\{1,\ldots,n\}$ into pairwise disjoint subsets. Namely, $X^n_I$ is defined as the set of ordered tuples $(x_1,\ldots,x_n)$ such that $x_i=x_j$ if and only if $i,j$ belong to the same subset $I_l$. Obviously, $X^n_I$ is isomorphic to $X^k_{(1,\ldots,1)}$, the set of unordered $k$-tuples of pairwise different points in $X$. Any $I$ as above induces a partition $\lambda(I)$ of $n$, with parts being the cardinalities of the subsets $I_k$ forming $I$. The image of $X^n_I$ under the quotient map $\pi:X^n\rightarrow S^nX$ depends only on the partition $\lambda=\lambda(I)$ and is denoted by $S^n_\lambda X$. The inverse image under $\pi$ of $S^n_\lambda X$ is precisely the union of the strata $X^n_I$ such that $\lambda(I)=\lambda$. Moreover, the fibre of $\pi$ over a point in $S^n_\lambda X$ is finite of cardinality $\frac{n!}{\lambda_1!\ldots\lambda_k!}$. The number of decompositions $I$ such that $\lambda(I)=\lambda$ equals $$\frac{n!}{\lambda_1!\cdot\ldots\cdot\lambda_k!}\cdot\frac{1}{\prod_i(m_i(\lambda)!)}.$$ An easy induction shows that the Euler characteristic in cohomology with compact support $\chi$ of $X^n_{(1,\ldots,1)}$ equals $$\chi(X)(\chi(X)-1)\ldots(\chi(X)-n+1)={n!}{\binom{\chi(X)}{n}}.$$ We have thus proved: \[lemmasymm\] For all partitions $\lambda$ of $n$, we have $$\chi(S^n_\lambda X)=\frac{1}{\prod_im_i(\lambda)!}\chi(X)(\chi(X)-1)\ldots(\chi(X)-k+1)=\frac{1}{\prod_im_i(\lambda)!}k!{\binom{\chi(X)}{k}}.$$ We can now consider the generating function of the Euler characteristics of arbitrary smooth models, using the big local quiver notation of the previous section.\ In particular, to a polystable type $\xi$, we have associated a partitive dimension vector $p$ for $\widetilde{Q}$ (resp. a large enough $\widetilde{Q}_l$); we denote the stratum $S_\xi$ by $S_p$. With the above notation, we have $$S_p\simeq\prod_{d\in{{}^\prime\!\Lambda}^+_\mu}S_{p(d)}^{|p(d)|}M_d^{st}(Q)$$ by definition of $S_\xi$. Theorem \[strat\] can now be rephrased as stating that $M_{d,n}^\Theta(Q)$ admits a stratification indexed by partitive dimension vectors $p\in\Lambda(\widetilde{Q})^+$ such that $\nu(p)=d$. Each stratum is a locally trivial fibration over $S_p$, with fibre isomorphic to ${\rm Hilb}_{p,\widetilde{n}}^{nilp}(\widetilde{Q})$. We thus have, using Lemma \[lemmasymm\] for the second equality: $$\begin{aligned} \chi(M_{d,n}^\Theta(Q))&=&\sum_p\chi(S_p)\cdot\chi({\rm Hilb}_{p,\widetilde{n}}^{nilp}(\widetilde{Q}))\\ &=&\sum_p\prod_{d\in{{}^\prime\!\Lambda}_\mu^+}\frac{1}{\prod_im_i(p(d))!}l(p(d))!{\binom{\chi(M_d^{st}(Q))}{l(p(d))}}\cdot\chi({\rm Hilb}_{p,\widetilde{n}}^{nilp}(\widetilde{Q})),\end{aligned}$$ the sum running over all partitive dimension vectors $p$ for $\widetilde{Q}$ such that $\nu(p)=d$.\ Considering the generating function, we thus have $$\sum_{d\in\Lambda^+_\mu}\chi(M_{d,n}^\Theta(Q))t^d$$ $$=\sum_{p\in\Lambda(\widetilde{Q})^\geq}\prod_{d\in{{}^\prime\!\Lambda}^+_\mu}\left(\frac{1}{\prod_im_i(p(d))!}l(p(d))!{\binom{\chi(M_d^{st}(Q))}{l(p(d))}}\right)\cdot\chi({\rm Hilb}_{p,\widetilde{n}}^{nilp}(\widetilde{Q}))t^{\nu(p)}.$$ Sorting by lengths of the partitions, this can be rewritten as $$\sum_{l:{{}^\prime\!\Lambda}^+_\mu\rightarrow{\bf N}}\sum_{p\in\Lambda(\widetilde{Q}_l)^\geq}\prod_{d\in{{}^\prime\!\Lambda}^+_\mu}\left(\frac{1}{\prod_im_i(p(d))!}l(d))!{\binom{\chi(M_d^{st}(Q))}{l(d))}}\right)\cdot\chi({\rm Hilb}_{p,\widetilde{n}}^{nilp}(\widetilde{Q}))t^{\nu(p)}.$$ We want to extend the range of summation in the inner sum to arbitrary dimension vectors for each $\widetilde{Q}_l$ without changing the sum. By the symmetry property of $\widetilde{Q}$ (resp. $\widetilde{Q}_l$), we can do this by incorporating a factor which counts the number of derangements of a given partitive dimension vector $p$ into arbitrary dimension vectors. This number is precisely $$\prod_{d\in{{}^\prime\!\Lambda}^+_\mu}\frac{l(p(d))!}{\prod_im_i(p(d))!},$$ this factor being already present. Thus, the above sum equals $$\sum_{l:{{}^\prime\!\Lambda}^+_\mu\rightarrow{\bf N}}\sum_{\widetilde{d}\in\Lambda(\widetilde{Q}_l)^{++}}\prod_{d\in{{}^\prime\!\Lambda}^+_\mu}{\binom{\chi(M_d^{st}(Q)}{l(d)}}\cdot\chi({\rm Hilb}_{\widetilde{d},\widetilde{n}}^{nilp}(\widetilde{Q}))t^{\nu(\widetilde{d})},$$ the inner sum now running over all faithful dimension vectors for $\widetilde{Q}_l$. Using the previous notation, this equals $$\sum_{l:{{}^\prime\!\Lambda}^+_\mu\rightarrow {\bf N}}\prod_{d\in{{}^\prime\!\Lambda}_\mu^+}{\binom{\chi(M_d^{st}(Q))}{l(d)}}{{}^\prime\!R}_l^n(t)=R^n_\chi(t)$$ for the function $\chi$ defined by $\chi(d)=\chi(M_d^{st}(Q))$. By Lemma \[trick\], we arrive at the following result: \[t42\] The generating function of Euler characteristics of smooth models is defined by the functional equations $$\sum_{d\in\Lambda^+_\mu}\chi(M_{d,n}^\Theta(Q))t^d=\prod_{d\in{{}^\prime\!\Lambda}^+_\mu}R^{d}(t)^{\chi(M_d^{st}(Q))\cdot(n\cdot d)}$$ and $$R^{d}(t)=1+t^d\cdot R^d(t)\cdot\prod_{e\in{{}^\prime\!\Lambda}^+_\mu}R^{e}(t)^{-\langle d,e\rangle\cdot \chi(M_e^{st}(Q))}.$$ To make the nature of these functional equations more transparent, we will define a slight variant of the generating functions. Writing $$Q^n_\mu(t)=\sum_{d\in\Lambda_\mu^+}\chi(M_{d,n}^\Theta(Q))t^d,$$ we have $Q^n_\mu(t)=\prod_{i\in I}Q^{i}_\mu(t)^{n_i}$ by the previous theorem. This suggests the definition $Q^\eta_\mu(t)=\prod_{i\in I}Q^{i}_\mu(t)^{\eta(i)}$ for an arbitrary linear functional $\eta\in\Lambda^*$, so that $Q^{n\cdot}_\mu(t)=Q^n_\mu(t)$ for all $n\in\Lambda^+$. In particular, we consider $S^d_\mu(t)=Q^{\langle d,\_\rangle}_\mu(t)$ for $d\in{{}^\prime\!\Lambda}^+_\mu$. \[corsd\] The series $S^d_\mu(t)$ for $d\in{{}^\prime\!\Lambda}^+_\mu$ are given by the functional equations $$S^d_\mu(t)=\prod_{e\in{{}^\prime\!\Lambda}_\mu^+}(1-t^eS^e_\mu(t))^{-\langle d,e\rangle\cdot\chi(M_e^{st}(Q))}.$$ [[**Proof:** ]{}]{}By the definitions and Theorem \[t42\], we have $$S^d_\mu(t)=\prod_{e\in{{}^\prime\!\Lambda}_\mu^+}R^e(t)^{\langle d,e\rangle\cdot\chi(M_e^{st}(Q))}.$$ The last line of Theorem \[t42\] can be restated as $$R^d(t)=(1-t^d\prod_{e\in{{}^\prime\!\Lambda}_\mu^+}R^e(t)^{-\langle d,e\rangle\cdot\chi(M_e^{st}(Q))})^{-1},$$ thus $$R^d(t)=(1-t^dS^d_\mu(t))^{-1}.$$ Substituting this in the factorization of $S^d_\mu(t)$ yields the desired equation.[$\Box$]{} Duality for Euler products {#number} ========================== Let $F(t)\in{\bf Q}[[t]]$ be a formal power series with constant term $F(0)=1$. Then we can write $F(t)$ as an Euler product $$\label{eulerproduct} F(t)=\prod_{i\geq 1}(1-(-t)^i)^{-ia_i}$$ for $a_i\in{\bf Q}$ (note the sign convention, which is essential in the following; see the example at the end of this section). We can also characterize $F(t)$ as the unique solution of a functional equation of the form $$\label{functionalequation} F(t)=\prod_{i\geq 1}(1-(tF(t))^i)^{ib_i}$$ for $b_i\in{\bf Q}$; see the remark below for the proof.\ The main result of this section is: \[duality\] In the above notation, we have $b_i\in{\bf Z}$ for all $i\geq 1$ if and only if $a_i\in{\bf Z}$ for all $i\geq 1$. [[**Remark:** ]{}]{}Writing $H(t)=-tF(t)$, we have, by a straightforward calculation, $$H(t)=-t\prod_{i\geq 1}(1-(-t)^i)^{-ia_i}$$ and $$t=-H(t)\prod_{i\geq 1}(1-(-H(t))^i)^{-ib_i}.$$ This means that $H(t)$ is the compositional inverse of the series $$-t\prod_{i\geq 1}(1-(-t)^i)^{-ib_i}.$$ This shows that the series $F(t)$ can be characterized by a functional equation of the form (\[functionalequation\]) for unique $b_i$, and it shows the symmetry of the statement in the theorem. Thus, we only have to prove integrality of the $a_i$ given integrality of the $b_i$.\ As the first step towards the proof of the theorem, we will derive an explicit formula for the $a_i$ in terms of the $b_i$ by applying Lagrange inversion to the functional equation (\[functionalequation\]. We use the following version of Lagrange inversion: \[lagrangeinversion\] Suppose that power series $F(t)$, $G(t)\in{\bf Q}[[t]]$ with $G(0)\not=0$ are related by $F(t)=G(tF(t))$. Then, for all $k,d\in{\bf Z}$, we have $$(k+d)[t^d]F(t)^k=k[t^d]G(t)^{k+d},$$ where $[t^d]F(t)$ denotes the $t^d$-coefficient of the series $F(t)$. [[**Proof:** ]{}]{}Apply [@St Theorem 5.4.2] using the notation $f(t)=tF(t)$ and $d=n-k$.[$\Box$]{} \[coeffpartition\] For all $d\in{\bf N}$ and all $c_i\in{\bf Z}$ for $i\geq 1$, we have $$\prod_{i\geq 1}(1-t^i)^{-c_i}=\sum_{\lambda\vdash d}\prod_{i\geq 1}\binom{c_i+\lambda_i-\lambda_{i+1}-1}{\lambda_i-\lambda_{i+1}},$$ the sum ranging over all partitions $\lambda$ of $d$. [[**Proof:** ]{}]{}We have $$(1-t)^{-c}=\sum_{k\geq 0}\binom{c+k-1}{k}t^k,$$ and therefore $$\begin{aligned} \prod_{i\geq 1}(1-t^i)^{-c_i}&=&[t^d]\prod_{i\geq 1}\sum_{k_i\geq 0}\binom{c_i+k_i-1}{k_i}t^k_i=\\ &&[t^d]\sum_{k_1,k_2,\ldots\geq 0}\prod_{i\geq 1}\binom{c_i+k_i-1}{k_i}t^{\sum_ik_i}=\\ &&[t^d]\sum_{\lambda}\prod_{i\geq 1}\binom{c_i+\lambda_i-\lambda_{i+1}-1}{\lambda_i-\lambda_{i+1}}t^{|\lambda},\end{aligned}$$ where the last sum ranges over all partitions $\lambda$, which are related to sequences $k_1,k_2,\ldots\geq 0$ via $\lambda_i=\sum_{j\geq i}k_j$.[$\Box$]{} [[**Remark:** ]{}]{}Here and in the following, we make frequent use of binomial coefficients $\binom{a}{b}$ for $a\in{\bf Z}$ using $$\label{negativebinom}\binom{-a+b-1}{b}=(-1)^b\binom{a}{b}$$ Using these preparations, we can state the desired formula relating the coefficients $a_i$ and $b_i$: \[moebiusinversion\] With the above notation, we have, for all $d\geq 1$: $$\label{mif}d^2a_d=\sum_{e|d}\mu(d/e)(-1)^e\sum_{\lambda\vdash e}(-1)^{\lambda_1}\prod_{i\geq 1}\binom{ib_ie}{\lambda_i-\lambda_{i+1}},$$ where the first sum ranges over all divisors of $d$, and $\mu$ denotes the number-theoretic Moebius function. [[**Proof:** ]{}]{}We apply Lemma \[lagrangeinversion\] to the functional equation (\[functionalequation\]) using $$G(t)=\prod_{i\geq 1}(1-t^i)^{ib_i}$$ and get $$\label{lifn}(k+d)[t^d]\prod_{i\geq 1}(1-(-t)^i)^{-ia_ik}=k[t^d]\prod_{i\geq 1}(1-t^i)^{ib_i(k+d)}.$$ Lemma \[coeffpartition\] allows us to write the left hand side of (\[lifn\]) as $$(k+d)(-1)^d\sum_{\lambda\vdash d}\prod_{i\geq 1}\binom{ia_ik+\lambda_i-\lambda_{i+1}-1}{\lambda_i-\lambda_{i+1}},$$ and the right hand side of (\[lifn\]) as $$k\sum_{\lambda\vdash d}\prod_{i\geq 1}\binom{-ib_i(k+d)+\lambda_i-\lambda_{i+1}-1}{\lambda_i-\lambda_{i+1}}.$$ We use (\[negativebinom\]) and substitute $k$ by $X$ to rewrite (\[lifn\]) as $$\label{rewrite}X\sum_{\lambda\vdash d}(-1)^{\lambda_1}\prod_{i\geq 1}\binom{ib_i(X+d)}{\lambda_i-\lambda_{i+1}}=(-1)^d(X+d)\sum_{\lambda\vdash d}\prod_{i\geq 1}\binom{ia_iX+\lambda_i-\lambda_{i+1}-1}{\lambda_i-\lambda_{i+1}}.$$ Both sides behaving polynomially in $X$, equality for all $X\in{\bf Z}$ thus implies equality of the polynomials. We want to compare the linear $X$-terms (the constant terms being $0$) of both sides. Note the following property:\ The polynomial $\binom{aX+b+c-1}{c}$ has constant $X$-coefficient $\binom{b+c-1}{c}$, and the polynomial $\binom{aX+c-1}{c}$ has linear $X$-coefficient $a/c$.\ Applying this, we see that the left hand side of (\[rewrite\]) has linear $X$-coefficient $$\sum_{\lambda\vdash d}(-1)^{\lambda_1}\prod_{i\geq 1}\binom{ib_id}{\lambda_i-\lambda_{i+1}}.$$ To analyze the linear $X$-coefficient of the right hand side of (\[rewrite\]), note first that the constant $X$-coefficient of each product $$\label{product}\prod_{i\geq 1}\binom{ia_iX+\lambda_i-\lambda_{i+1}-1}{\lambda_i-\lambda_{i+1}}$$ equals zero. Its linear $X$-term is non-zero only if exactly one factor appears, that is, if there is only one non-zero difference $\lambda_i-\lambda_{i+1}$. In this case, the partition $\lambda$ of $d$ equals $$\lambda=\underbrace{(d/i,\ldots,d/i)}_{i\mbox{-times}}$$ for a divisor $i$ of $d$. Thus, the product (\[product\]) reduces to $$\binom{ia_iX+d/i-1}{d/i},$$ having linear $X$-coefficient $(ia_i)/(d/i)=i^2a_i/d$ by the above. We conclude that the linear $X$-coefficient of the right hand side of (\[rewrite\]) equals $$(-1)^d\sum_{i|d}i^2a_i.$$ Comparison of both linear $X$-coefficients thus yields $$\sum_{i|d}i^2a_i=(-1)^d\sum_{\lambda\vdash d}(-1)^{\lambda_1}\prod_{i\geq 1}\binom{ib_id}{\lambda_i-\lambda_{i+1}}.$$ After Moebius inversion, we arrive at the claimed formula (\[mif\]).[$\Box$]{} To prove integrality of the $a_d$ given integrality of all $b_i$, we thus have to prove that the right hand side of (\[mif\]) is divisible by $d^2$. This can be tested on the prime divisors of $d$. Denoting by $$m(d)=m_p(d)=\max\{m\,:\, p^m|d\}$$ the multiplicity of a prime $p$ as a divisor of $d$, we thus have to prove divisibility by $p^{2m_p(d)}$ of the right hand side of (\[mif\]) for all primes $p$. We prepare this proof by stating certain divisibility/congruence properties of binomial coefficients. \[kummer\] Let $p$ be a prime. For $a,b\in{\bf Z}$ and $b\geq 0$, we have $$p^{\max(m_p(a)-m_p(b),0)}|\binom{a}{b}.$$ [[**Proof:** ]{}]{}By a result of Kummer (see, for example, [@Gr]), the exact power of $p$ diving $\binom{a}{b}$ equals the number of ‘carries’ when subtracting $b$ from $a$ in base $p$, at least when $a\geq 0$. This can be generalized to $a\in{\bf Z}$ using $$\label{negbinom}\binom{-a}{b}=(-1)^b\frac{a}{a+b}\binom{a+b}{b}.$$ The lemma follows.[$\Box$]{} \[gessel\] Let $p$ be a prime, and define $\mu_p=0,1,2$ provided $p=2$, $p=3$, $p\geq 5$, respectively. Assume $p|a,b$ for integers $a$, $b$ with $b\geq 0$. Define $\eta$ as $-1$ if $p=2$ and $b\equiv 2\equiv a-b\bmod 4$, and as $1$ otherwise. Then $$\binom{a}{b}\equiv\eta\binom{a/p}{b/p}\bmod p^r,$$ for $$r\leq m_p(a)+m_p(b)+m_p(a-b)+m_p(\binom{a/p}{b/p})-\mu_p.$$ In case $p=2$, we also have $$\binom{a}{b}\equiv\binom{a/2}{b/2}\bmod 4.$$ [[**Proof:** ]{}]{}The general statement (usually [@Ge; @Gr] attributed to Jacobsthal [@Jac]) is proved in [@Ge Theorem 2.2], with the assumption $a\geq 0$ there removed by (\[negbinom\]). For the congruence modulo $4$, we calculate as in the proof of [@Ge Theorem 2.2]: $$\binom{a}{b}=\binom{a/2}{b/2}\prod_{\stackrel{i=1}{2\nmid i}}^b(1+2(a-b)/i)\equiv\binom{a/2}{b/2}(1+2(a-b)\sum_{\stackrel{i=1}{2\nmid i}}^b 1/i)\equiv$$ $$\equiv\binom{a/2}{b/2}(1+(a-b)(b/2)^2)\bmod 4.$$ The term $(a-b)(b/2)^2$ is congruent to $1 \bmod 4$ except when $b/2$ is odd and $a/2$ is even, in which case it is congruent to $-1 \mod 4$. But in this case, $\binom{a/2}{b/2}$ is even by Lemma \[kummer\].[$\Box$]{} From the previous two lemmas, we derive divisibility/congruence properties of the product of binomial coefficients appearing in (\[mif\]). \[claim2\] Let $p$ be a prime dividing $e\geq 0$. If $\lambda$ is a partition of $e$ which is not divisible by $p$ (that is, some coefficient $\lambda_i$ is not divisible by $p$), we have $$p^{2m(e)}|\prod_{i\geq 1}\binom{ib_ie}{\lambda_i-\lambda_{i+1}}.$$ [[**Proof:** ]{}]{}To shorten notation, we write $m=m_p(e)$ and $c_i=\lambda_i-\lambda_{i+1}$ for $i\geq 1$, thus $e=\sum_{i\geq 1}ic_i$. Lemma \[kummer\] yields $$p^{\max(m+m(i)-m(c_i),0)}|\binom{ib_ie}{c_i};$$ we thus have to prove $$\label{ineq}\sum_{i:c_i\not=0}\max(m+m(i)-m(c_i),0)\geq 2m$$ provided some $c_i\not=0$ is not divisible by $p$. Let $i_0$ be an index such that $m(c_{i_0})=0$.\ Let $m_0$ be the minimum over all $m(i)+m(c_i)$. Since $e=\sum_iic_i$, we can distinguish two cases: either $m_0=m$ (case 1), or $m_0<m$ and the minimum is obtained at least twice (case 2). For case 1 we have, in particular, $m(i_0)\geq m$, thus $$\max(m+m(i_0)-m(c_{i_0}),0)\geq 2m,$$ and (\[ineq\]) follows.\ For case 2, let $i_1,i_2$ be two different indices where the minimum $m_0$ is obtained. For $s=1,2$, we have $m+m(i_s)-m(c_{i_s})\geq 0$, since otherwise, $$m>m_0=m(c_{i_s})+m(i_s)\geq m(c_{i_s})>m+m(i_s),$$ a contradiction. Again we distinguish two cases: first, assume that $i_0$ coincides with, say, $i_1$. Then we can estimate $$\begin{aligned} &&\sum_{i:c_i\not=0}\max(m+m(i)-m(c_i),0)\\ &\geq&\max(m+m(i_0)-m(c_{i_0}),0)+\max(m+m(i_2)-m(c_{i_2}),0)\\ &=&2m+m_0+m(i_2)-m(c_{i_2})=2m+2m(i_2)\geq 2m,\end{aligned}$$ and (\[ineq\]) follows. Second, assume that $i_0$ differs from $i_1$, $i_2$. Like in the previous case, we can estimate $$\begin{aligned} &&\sum_{i:c_i\not=0}\max(m+m(i)-m(c_i),0)\\ &\geq&3m+m(i_0)+m(i_1)+m(i_2)-m(c_{i_1})-m(c_{i_2})\\ &\geq&2m+m-m_0+2m(i_1)+2m(i_2)\geq 2m,\end{aligned}$$ and (\[ineq\]) follows again.[$\Box$]{} \[claim1\] Let $p$ be a prime dividing $e\geq 0$. If $\lambda=p\mu$ is a partition of $e$ divisible by $p$, then $$\label{cong}\prod_{i\geq 1}\binom{ib_ie}{\lambda_i-\lambda_{i+1}}\equiv (-1)^{(p-1)(e/p+\mu_1)}\prod_{i\geq 1}\binom{ib_ie/p}{\mu_i-\mu_{i+1}}\bmod p^{2m_p(e)}.$$ [[**Proof:** ]{}]{}So assume that $\lambda=p\mu$, and denote again $m=m(e)$ and $c_i=\lambda_i-\lambda_{i+1}$. Applying the general congruence of Lemma \[gessel\] to a non-trivial (that is, $c_i\not=0$) factor of the left hand side of (\[cong\]), we get $$\binom{ib_ie}{c_i}\equiv\eta_i\binom{ib_ie/p}{c_i/p}\bmod p^{r_i},$$ where the sign $\eta_i$ is $-1$ only in case $p=2$, $c_i/2$ odd, $ib_ie/2-c_i/2$ odd, and $$\begin{aligned} \label{est}\nonumber r_i&=&m(ib_ie)+m(c_i)+m(ib_ie-c_i)+m(\binom{ib_ie/p}{c_i/p}-\mu_p\\ \nonumber &\geq&m(e)+m(i)+m(c_i)+\min(m(e)+m(i),m(c_i))+\\ \nonumber &&\max(m(e)+m(i)-m(c_i),0)-\mu_p\\ &=&2m+2m(i)+m(c_i)-\mu_p.\end{aligned}$$ Suppose first that $p\geq 3$. Then $r_i\geq 2m$ using $m(c_i)\geq 1$ and $\mu_p\leq 1$. The sign in (\[cong\]) vanishes due to the even factor $p-1$, and $\eta_i=1$. The congruence (\[cong\]) follows.\ Next, assume that $p=2$ and $m\geq 2$. Then the estimate (\[est\]) only assures congruence of the binomial coefficients $\bmod 2^{2m-1}$ in case $i$ and $c_i/2$ are odd, thus $$\binom{ib_ie}{c_i}\equiv\eta_i\binom{ib_ie/2}{c_i/2}+\varepsilon_i\bmod 2^{2m},$$ where $\varepsilon_i\in\{0,2^{2m-1}\}$, non-triviality only being possible if $i$ and $c_i/2$ are odd. Then $$\begin{aligned} \label{complex}\nonumber \prod_{i\geq 1}\binom{ib_ie}{c_i}&\equiv&\prod_{i\geq 1}(\eta_i\binom{ib_ie}{c_i/2}+\varepsilon_i)\\ &\equiv&\prod_{i\geq 1}\eta_i\binom{ib_ie/2}{c_i/2}+\sum_{i\geq 1}\varepsilon_i\prod_{j\not=i}\eta_j\binom{jb_je}{c_j/2}\bmod 2^{2m},\end{aligned}$$ since all multiple products of the $\varepsilon_i$ vanish $\bmod 2^{2m}$. For the same reason, we only have to consider summands in (\[complex\]) for which $\varepsilon_i\not=0$ and each factor $$\binom{jb_je/2}{c_j/2}$$ is odd. Since $m\geq 2$, this can only happen (using Lemma \[kummer\]) in the case that $m(c_j)\geq m(j)+m$ for all $j\not=i$ such that $c_j\not=0$. But then $$2^m|e-\sum_{j\not=i:c_j\not=0}jc_j=ic_i,$$ a contradiction to the assumptions $m(ic_i)=1$ (by $\varepsilon_i\not=0$) and $m\geq 2$. Thus, we have proved that $$\prod_{i\geq 1}\binom{ib_ie}{c_i}\equiv\prod_{i\geq 1}\eta_i\cdot\prod_{i\geq 1}\binom{ib_ie/2}{c_i/2}\bmod 2^{2m},$$ and we have to compare the sign $\prod_i\eta_i=(-1)^u$ to the sign of (\[cong\]). Using $m\geq 2$, we have $$\begin{aligned} u&=&{|\{i\geq 1\, :\, c_i/2\mbox{ odd}, ib_ie/2-c_i/2\mbox{ odd}\}|}\\ &=&{|\{i\geq 1\, :\, c_i/2\mbox{ odd}\}|}.\end{aligned}$$ The sign in (\[cong\]) equals $$(-1)^{e/2+\sum_ic_i/2},$$ and we are done.\ Finally, consider the case $p=2$ and $m=1$. Then the statement on congruences $\bmod 4$ of Lemma \[gessel\] yields $$\prod_{i\geq 1}\binom{ib_ie}{c_i}\equiv\prod_{i\geq 1}\binom{ib_ie/2}{c_i/2}\bmod 4,$$ and again we only have to consider the sign. The sign in (\[cong\]) equals $$(-1)^{1+\sum_i c_i/2}.$$ We have $e/2=\sum_iic_i/2$, thus the sum $\sum_{{2}{\nmid}\,{i}}c_i/2$ is odd. Suppose $\sum_ic_i/2$ is even (the only case in which the sign of (\[cong\]) potentially differs from $1$). Then $\sum_{2|i}c_i/2$ is odd. Thus, there exists an even index $i$ with $c_i/2$ odd. In this case, the binomial coefficient $$\binom{ib_ie/2}{c_i/2}$$ is even, and the sign is irrelevant $\bmod 4$.[$\Box$]{} With these preparations, we can finish the\ [**Proof of Theorem \[duality\]:**]{} Assume that $p$ is a prime such that $m=m(d)=m_p(d)\geq 1$. The divisors $e$ of $d$ for which $\mu(d/e)$ is non-zero fulfill $m(e)=m(d)$ or $m(e)=m(d)-1$, that is, they are of the form $e$ or $e/p$ for a divisor $e$ of $d$ such that $m(e)=m(d)$. We can thus split the right hand side of (\[mif\]) into the following difference: $$\begin{aligned} \label{diff}\nonumber &&\sum_{e|d\, :\, m(e)=m(d)}\mu(d/e)(-1)^{e}\sum_{\lambda\vdash e}(-1)^{\lambda_1}\prod_{i\geq 1}\binom{ib_ie}{\lambda_i-\lambda_{i+1}}\\ &-&\sum_{e|d\, :\, m(e)=m(d)}\mu(d/e)(-1)^{e/p}\sum_{\mu\vdash e/p}(-1)^{\mu_1}\prod_{i\geq 1}\binom{ib_ie/p}{\mu_i-\mu_{i+1}}.\end{aligned}$$ Now consider a summand of the first sum of (\[diff\]) corresponding to a partition $\lambda$ of $e$. If $\lambda$ is not divisible by $p$, then Lemma \[claim2\] shows that the summand is divisible by $p^{2m(e)}=p^{2m(d)}$. If $\lambda=p\mu$ is divisible by $p$, then Lemma \[claim1\] shows that the summand is congruent $\bmod p^{2m(d)}$ to the summand of the second sum of (\[diff\]) corresponding to the partition $\mu$. In other words, the difference of the two sums in (\[diff\]) vanishes $\bmod p^{2m(d)}$, proving the theorem.[$\Box$]{} For the application to the integrality of certain Donaldson-Thomas type invariants in the following section, we need a slight generalization of Theorem \[duality\]. We treat this case separately, although a second inspection of the proofs leading to Theorem \[duality\] is neccessary, to avoid additional complications in the notation used so far. \[genduality\] Let $F(t)\in{\bf Q}[[t]]$ be a power series with $F(0)=1$. For $N\in{\bf Z}$, write $$F(t)=\prod_{i\geq 1}(1-((-1)^Nt)^i)^{-ia_i}$$ for $a_i\in{\bf Q}$. We can characterize $F(t)$ as the solution to a functional equation of the form $$F(t)=\prod_{i\geq 1}(1-(tF(t)^N)^i)^{ib_i}$$ for unique $b_i\in{\bf Q}$. Under these assumptions, we have $b_i\in{\bf Z}$ for all $i\geq 1$ if and only if $a_i\in{\bf Z}$ for all $i\geq 1$. [[**Proof:** ]{}]{}The argument used in the remark following Theorem \[duality\], using the power series $H(t)=t(-F(t))^N$, shows existence and uniqueness of the $b_i$, as well as the symmetry of the statement of Theorem \[genduality\]. Applying Proposition \[moebiusinversion\] to $G(t)=F(t)^N$ yields the following explicit formula for all $d\geq 1$: $$\label{gmif}d^2a_d=\frac{1}{N}\sum_{e|d}\mu(d/e)(-1)^{Ne}\sum_{\lambda\vdash e}(-1)^{\lambda_1}\prod_{i\geq 1}\binom{Nib_ie}{\lambda_i-\lambda_{i+1}}.$$ Now any summand of (\[gmif\]) is divisible by $N$, thus the denominator $N$ in (\[gmif\]) cancels. Next, note that none of our arguments (Lemma \[claim2\], \[claim1\]) for the proof of Theorem \[duality\] uses any divisibility properties of the $b_i$, thus these arguments are valid when replacing $b_i$ by $Nb_i$, yielding an additional divisibility by $N$.\ The only additional difficulty is the sign in the statement of Lemma \[claim1\], which now reads $$(-1)^{(p-1)(Ne/p+\mu_1)}.$$ Repeating the sign considerations in the proof of Lemma \[claim1\], we see that we can concentrate on the case $p=2$ and $m(e)=1$, where the sign now reads $$(-1)^{N+\sum_i c_i/2}.$$ The argument of the proof of Lemma \[claim1\] is still valid in case $N$ is odd. On the other hand, if $N$ is even, we can choose an index $i$ such that $c_i/2$ is odd, and Lemma \[kummer\] shows that the binomial coefficient $$\binom{Nib_ie/2}{c_i/2}$$ is even, the sign thus being again irrelevant $\bmod 4$. [$\Box$]{} [[**Example:** ]{}]{}We consider the example $b_i=0$ for all $i\geq 2$ and denote $b=b_1$. Then $F(t)$ is the solution to the functional equation $$F(t)=(1-tF(t)^N)^b,$$ and we want to factor $F(t)$ as $$F(t)=\prod_{i\geq 1}(1-((-1)^Nt)^i)^{-ia_i}.$$ The formula (\[gmif\]) gives $$a_d=\frac{1}{Nd^2}\sum_{e|d}\mu(d/e)(-1)^{(N+1)e}\binom{Nbe}{e}.$$ In particular, we have $a_1=(-1)^{N+1}b$ and $$a_2=\frac{b(2Nb-(1+(-1)^{N+1})}{4},$$ and we see that the choice of signs is essential for the integrality of the $a_d$ given by Theorem \[genduality\].\ The particular case $N=1$, $b=-1$ gives a factorization (\[eulerproduct\]) for the generating function $$F(t)=\frac{1-\sqrt{1-4t}}{2t}$$ of Catalan numbers with $$a_d=\frac{1}{d^2}\sum_{e|d}(-1)^e\mu(d/e)\binom{2e-1}{e},$$ which is (up to signs) sequence A131868 in [@OEIS]. Application to Donaldson-Thomas invariants and wall-crossing formulas {#app} ===================================================================== In this section, we apply the results of the previous sections to the setup of [@RWC]. We assume that $Q$ is a quiver without oriented cycles, thus we can order the vertices as $I=\{i_1,\ldots,i_r\}$ in such a way that $k>l$ provided there exists an arrow $i_k\rightarrow i_l$. Denote by $\{\_,\_\}$ the skew-symmetrization of $\langle\_,\_\rangle$, thus $\{d,e\}=\langle d,e\rangle-\langle e,d\rangle$. Define $b_{ij}=\{i,j\}$ for $i,j\in I$.\ We consider the formal power series ring $B={\bf Q}[[\Lambda^+]]={\bf Q}[[x_i: i\in I]]$ with topological basis $x^d=\prod_{i\in I}x_i^{d_i}$ for $d\in\Lambda^+$. The algebra $B$ becomes a Poisson algebra via the Poisson bracket $$\{x_i,x_j\}=b_{ij}x_ix_j\mbox{ for }i,j\in I.$$ Define Poisson automorphisms $T_i$ of $B$ by $$T_i(x_j)=x_j\cdot(1+x_i)^{\{i,j\}}$$ for all $i,j\in I$.\ We study a factorization property in the group ${\rm Aut}(B)$ of Poisson automorphisms of $B$ involving a descending product $\prod^\leftarrow_{\mu\in{\bf Q}}$ indexed by rational numbers, which is indeed well-defined. The main result of [@RWC] states (in the notation of the previous section): \[mainth\] In the group ${\rm Aut}(B)$, we have a factorization $$T_{i_1}\circ\ldots\circ T_{i_r}=\prod^\leftarrow_{{\mu\in{\bf Q}}}T_\mu,$$ where $$T_\mu(x^d)=x^d\cdot Q_\mu^{\{\_,d\}}(x).$$ Here $Q_\mu^\eta(x)$ denotes the specialization of the series $Q_\mu^\eta(t)$ of Section \[section4\] from the variables $t_i$ to the variables $x_i$. Let $\Phi\in{\rm Aut}(\Lambda)$ be the map induced on dimension vectors by the inverse Auslander-Reiten translation; $\Phi$ is a Coxeter element of the corresponding Weyl group determined by the property $$\langle \Phi(d),e\rangle=-\langle e,d\rangle.$$ Then we have $$\{\_,d\}=\langle-({\rm id}+\Phi)d,\_\rangle$$ and thus using Corollary \[corsd\]: The automorphisms $T_\mu$ of Theorem \[mainth\] can be written as $$T_\mu(x_d)=x^d\cdot S_\mu^{-({\rm id}+\Phi)d}(x).$$ We specialize to the generalized Kronecker quiver $K_m$ with set of vertices $I=\{i,j\}$ and $m$ arrows from $j$ to $i$. Choose the generators $x=-x_i$ and $y=-x_j$ of $B$; then $B_m={\bf Q}[[x,y]]$ with Poisson bracket $\{x,y\}=mxy$. For $a,b\in{\bf Z}$ with $a,b\geq 0$ and $a+b\geq 1$, we define a Poisson automorphism $T_{a,b}^{(m)}$ of $B$ by $$T_{a,b}^{(m)}:\left\{\begin{array}{lll}x&\mapsto&x(1-(-1)^{mab}x^ay^b)^{-mb},\\ y&\mapsto&y(1-(-1)^{mab}x^ay^b)^{ma}\end{array}\right.$$ as in [@KS 1.4]. More generally, for an arbitrary series $F(t)\in{\bf Z}[[t]]$ with $F(0)=1$, we define as in [@GPS 0.1]: $$T_{a,b,F(t)}^{(m)}:\left\{\begin{array}{lll}x&\mapsto&xF(x^ay^b)^{-mb},\\ y&\mapsto&yF(x^ay^b)^{ma}.\end{array}\right.$$ Note that the automorphisms $T_{a,b}^{(m)}$ for fixed slope $a/b$ commute, thus $$\label{commute} \prod_{i\geq 1}(T_{ia,ib}^{(m)})^{d_i}=T_{a,b,F(t)}^{(m)}$$ for $$F(t)=\prod_{i\geq 1}(1-((-1)^{mab}t)^i)^{id_i}.$$ We can now use our main results Theorem \[genduality\], Theorem \[mainth\] to confirm [@KS Conjecture 1]: \[conj1\] Writing $$T_{1,0}^{(m)}T_{0,1}^{(m)}=\prod^{\leftarrow}_{b/a\mbox{ decreasing}}(T_{a,b}^{(m)})^{d(a,b,m)},$$ we have $d(a,b,m)\in{\bf Z}$ for all $a,b,m$. [[**Proof:** ]{}]{}We choose the stability $\Theta=j^*$ (in fact, the only non-trivial stability, see [@Rmoduli 5.1]). By Theorem \[mainth\], we have a factorization $$\label{factkron}T_{1,0}^{(m)}T_{0,1}^{(m)}=T_iT_j=\prod^\leftarrow_{{\mu\in{\bf Q}}}T_\mu,$$ where $$T_\mu(x^d)=x^d\cdot Q_\mu^{\{\_,d\}}(x).$$ Given $\mu\in{\bf Q}$, we write $\mu=b/(a+b)$ for coprime nonnegative $a,b\in{\bf Z}$ and choose integers $c$ and $d$ such that $ac+bd=1$. We have $\Lambda_\mu^+={\bf N}t^{(a,b)}$. Defining $$G_\mu(t)=Q_\mu^i(t)^cQ_\mu^j(t)^d\in{\bf Z}[[\Lambda_\mu^+]],$$ the proof of [@RWC Theorem 6.1] shows that $$G_\mu(t)^a=Q_\mu^i(t)\mbox{ and }G_\mu(t)^b=Q_\mu^j(t).$$ Similarly to Corollary \[corsd\], we can find a functional equation for $G_\mu(t)$. We denote $\chi_\mu(k)=\chi(M_{(ka,kb)}^{st}(K_m))$ for $k\geq 1$ and $N=-\langle (a,b),(a,b)\rangle=mab-a^2-b^2$ and apply the first formula of Theorem \[t42\]: $$\begin{aligned} G_\mu(t)&=&Q_\mu^i(t)^cQ_\mu^j(t)^d=Q_\mu^{(c,d)}(t)\\ &=&\prod_{k\geq 1}R^{(ka,kb)}(t)^{\chi_\mu(k)\cdot k\cdot(ac+bd)}\\ &=&\prod_{k\geq 1}R^{(ka,kb)}(t)^{k\chi_\mu(k)}.\end{aligned}$$ Applying the second formula of Theorem \[t42\], this yields $$\begin{aligned} G_\mu(t)&=&\prod_{k\geq 1}(1-t^{(ka,kb)}\prod_{l\geq 1}R^{(la,lb)}(t)^{klN\chi\mu(l)})^{-l\chi_\mu(l)}\\ &=&\prod_{l\geq 1}(1-t^{(ka,kb)}G_\mu(t)^{kN})^{-k\chi_\mu(k)}.\end{aligned}$$ Thus, the series $G_\mu(t)$ fulfills the functional equation $$\label{777}G_\mu(t)=\prod_{k\geq 1}(1-(t^{(a,b)}G_\mu(t)^N)^k)^{-k\chi_\mu(k)}.$$ By Theorem \[genduality\], $G_\mu(t)$ admits a factorization $$\label{888}G_\mu(t)=\prod_{k\geq 1}(1-((-1)^Nt^{(a,b)})^k)^{kd_\mu(k)}$$ for $d_\mu(k)\in{\bf Z}$ for all $k\geq 1$.\ Defining $F_\mu(t)\in{\bf Z}[[t]]$ by $F_\mu((-1)^{a+b}t^{(a,b)})=G_\mu(t)$, we have $$\label{fact16}T_\mu=T_{a,b,F_\mu(t)}^{(m)}$$ (the sign appearing due to the convention $x=-x_i$, $y=-x_j$) and $$\begin{aligned} F_\mu(t)&=&\prod_{k\geq 1}(1-((-1)^{N+a+b}t)^k)^{kd_\mu(k)}\\ &=&\prod_{k\geq 1}(1-((-1)^{mab}t)^k)^{kd_\mu(k)}.\end{aligned}$$ By (\[commute\]) and (\[fact16\]), this yields $$T_\mu=\prod_{k\geq 1}(T_{ka,kb}^{(m)})^{d_\mu(k)}.$$ Together with the factorization (\[factkron\]), this yields the factorization claimed in the theorem, with $d(ka,kb,m)=d_\mu(k)$.[$\Box$]{} Using a result result of T. Weist, we can also confirm a conjecture in [@KS 1.4] concerning the diagonal term of the factorization in Theorem \[conj1\]: For all $k\geq 1$, we have $$d(k,k,m)=\frac{1}{(m-2)k^2}\sum_{i|k}\mu(k/i)(-1)^{mi+1}\binom{(m-1)^2i-1}{i}.$$ [[**Proof:** ]{}]{}By [@Weist 6.2], we have $\chi(M_{d,d}^{st}(K_m))=0$ for $d\geq 2$, whereas $M_{1,1}^{st}(K_m)\simeq{\bf P}^{m-1}$. In the notation of (\[777\]), (\[888\]) above, we can apply the example at the end of the previous section with $b=-m$ and $N=m-2$ and arrive at the claimed formula.[$\Box$]{} [99]{} J. Adriaenssens, L. Le Bruyn: [*Local quivers and stable representations.*]{} Comm. Algebra 31 (2003), no. 4, 1777–-1797. K. Behrend, B. Fantechi: [*Symmetric obstruction theories and Hilbert schemes of points on threefolds.*]{} Algebra Number Theory 2 (2008), no. 3, 313–345. A. Bialynicki-Birula: [*Some theorems on actions of algebraic groups.*]{} Ann. Math. (2) 98 (1973), 480-–497. V. Brun, J. O. Stubban, J. E. Fjeldstad, R. Tambs Lyche, K. 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--- author: - 'Felix Kahlhoefer,' - 'Kai Schmidt-Hoberg,' - Thomas Schwetz - and Stefan Vogl title: Implications of unitarity and gauge invariance for simplified dark matter models --- Introduction ============ After the successful discovery of a Higgs Boson consistent with the predictions of the Standard Model (SM), the focus of the current and upcoming runs of the Large Hadron Collider (LHC) at 13 TeV will be to discover evidence for physics beyond the SM. Among the prime targets of this search is dark matter (DM), which has so far only been observed via its gravitational interactions at astrophysical and cosmological scales. Since no particle within the SM has the required properties to explain these observations, DM searches at the LHC are necessarily searches for new particles. In fact, LHC DM searches are also likely to be searches for new interactions. Given the severe experimental constraints on the interactions between DM and SM particles, it is a plausible and intriguing possibility that the DM particle is part of a (potentially rich) hidden sector, which does not couple directly to SM particles or participate in the known gauge interactions. In this setup, the visible sector interacts with the hidden sector only via one or several new mediators, which have couplings to both sectors. In the simplest case the mass of these mediators is large enough that they can be integrated out and interactions between DM particles and the SM can be described by higher-dimensional contact interactions [@Beltran:2008xg; @Beltran:2010ww]. This effective field theory (EFT) approach has been very popular for the analysis and interpretation of DM searches at the LHC [@Goodman:2010ku; @Fox:2011pm; @Rajaraman:2011wf]. Nevertheless, as any effective theory it suffers from the problem that unitarity breaks down if the relevant energy scales become comparable to the cut-off scale of the theory [@Shoemaker:2011vi; @Fox:2012ee; @Busoni:2013lha; @Busoni:2014sya; @Xiang:2015lfa] (for other examples of applying unitarity arguments in the DM context see refs. [@Griest:1989wd; @Walker:2013hka; @Endo:2014mja; @Hedri:2014mua]). The easiest way to avoid this problem appears to be to explicitly retain the (lightest) mediator in the theory. The resulting models are referred to as simplified DM models, in which couplings are only specified after electroweak symmetry breaking (EWSB) and no ultraviolet (UV) completion is provided [@Abdallah:2015ter]. Compared to the EFT approach, simplified models have a richer phenomenology [@Busoni:2013lha; @Buchmueller:2013dya; @Buchmueller:2014yoa; @Harris:2014hga; @Garny:2014waa; @Buckley:2014fba; @Jacques:2015zha; @Alves:2015dya; @Choudhury:2015lha], including explicit searches for the mediator itself [@Frandsen:2012rk; @Fairbairn:2014aqa; @Chala:2015ama]. Moreover, it is possible to achieve the DM relic abundance in large regions of parameter space [@Busoni:2014gta; @Chala:2015ama; @Blennow:2015gta]. Constraining the parameter space of simplified DM models is therefore a central objective of experimental collaborations [@Khachatryan:2014rra; @Aad:2015zva; @Abercrombie:2015wmb]. In the present work we focus on the case of a spin-1 $s$-channel mediator [@Dudas:2009uq; @Fox:2011qd; @Frandsen:2012rk; @Alves:2013tqa; @Arcadi:2013qia; @Jackson:2013pjq; @Jackson:2013rqp; @Duerr:2013lka; @Duerr:2014wra; @Lebedev:2014bba; @Hooper:2014fda; @Martin-Lozano:2015vva; @Alves:2015pea; @Alves:2015mua; @Blennow:2015gta; @Duerr:2015wfa; @Heisig:2015ira]. Our central observation is that the simplified model approach is not generally sufficient to avoid the problem of unitarity violation at high energies and that further amendments are required if the model is to be both simple and realistic. In particular, a spin-1 mediator with axial couplings violates perturbative unitarity at large energies, pointing towards the presence of additional new physics to restore unitarity. Indeed, the simplest way to restore unitarity is to assume that the spin-1 mediator is the gauge boson of an additional $U(1)'$ gauge symmetry [@Holdom:1985ag; @Babu:1997st] and that its mass as well as the DM mass are generated by a new Higgs field in the hidden sector. The famous Lee-Quigg-Thacker bound [@Lee:1977eg] implies that the additional Higgs boson cannot be arbitrarily heavy and may therefore play an important role for LHC and DM phenomenology. In particular, it can mix with the SM-like Higgs boson and mediate interactions between DM particles and quarks. Furthermore, we require for a consistent simplified DM model that the coupling structure respects gauge invariance of the full SM gauge group before EWSB (see [@Bell:2015sza] for a similar discussion in the EFT context). If the mediator has axial couplings to quarks, this requirement implies that the new mediator will also have couplings to leptons and mixing with the SM $Z$ boson, both of which are tightly constrained by experiments. Much weaker constraints are obtained for the simplified DM model containing a spin-1 mediator with vectorial couplings to quarks. Constraints from direct detection can be evaded if the mediator has only axial couplings to DM, which naturally arises in the case that the DM particle is a Majorana fermion. We discuss the importance of loop-induced mixing effects in this context, which can play a crucial role for both direct detection experiments and LHC phenomenology. The outline of the paper is as follows. Starting from a simplified model for a mediator, we explore in section \[sec:unitarity\] the implications of perturbative unitarity, deriving a number of constraints on the model parameters and in particular an upper bound on the scale of additional new physics. In section \[sec:higgs\] we then consider the case where this additional new physics is a Higgs field in the hidden sector and derive an upper bound on the mass of the extra Higgs boson. We then discuss additional constraints on the SM couplings implied by gauge invariance. Section \[sec:axial\] focuses on the case of non-zero axial couplings between SM fermions and the mediator, whereas in section \[sec:vector\] we assume that the SM couplings of the mediator are purely vectorial. Finally, we discuss the experimental implications of a possible mixing between the SM Higgs and the hidden sector Higgs in section \[sec:higgsmixing\]. A discussion of our results and our conclusions are presented in section \[sec:discussion\]. Unitarity constraints on simplified models {#sec:unitarity} ========================================== Brief review of $S$ matrix unitarity constraints ------------------------------------------------ Consider the scattering matrix element $\mathcal{M}_{if}(s, \cos \theta)$ between 2-particle initial and final states ($i,f$), with $\sqrt{s}$ and $\theta$ being the centre of mass energy and scattering angle, respectively. We define the helicity matrix element for the $J$th partial wave by $$\label{eq:Jexpansion} \mathcal{M}_{if}^J(s) = \frac{1}{32\pi} \beta_{if} \int_{-1}^1 \mathrm{d}\cos \theta \, d^J_{\mu \mu'}(\theta) \, \mathcal{M}_{if}(s, \cos \theta) \,,$$ where $d^J_{\mu \mu'}$ is the $J$th Wigner d-function, $\mu$ and $\mu'$ denote the total spin of the initial and the final state (see e.g. [@Chanowitz:1978mv]), and $\beta_{if}$ is a kinematical factor. In the high-energy limit $s \to \infty$, which we are going to consider below, $\beta_{if} \to 1$. The right-hand side of eq.  is to be multiplied with a factor of $1/\sqrt{2}$ each if the initial or final state particles are identical [@Schuessler:2007av]. Unitarity of the $S$ matrix implies $$\begin{aligned} {\rm Im}(\mathcal{M}_{ii}^J) & = \sum_f | \mathcal{M}_{if}^J|^2 \nonumber\\ &= | \mathcal{M}_{ii}^J|^2 + \sum_{f \neq i} | \mathcal{M}_{if}^J|^2 \ge | \mathcal{M}_{ii}^J|^2 \label{eq:unity}\end{aligned}$$ for all $J$ and all $s$. The sum over $f$ in the first line runs over all possible final states. Restricting these to be all possible 2-particle states leads to a conservative bound. If the relation is strongly violated for matrix elements calculated at leading order in perturbation theory one can conclude that either higher-order terms in perturbation theory restore unitarity (i.e. break-down of perturbativity) or that the theory is not complete and additional contributions to the matrix element are needed. From eq.  one obtains the necessary conditions $$\label{eq:unitary-bound} 0 \le {\rm Im}({\mathcal{M}}_{ii}^J) \le 1\,, \quad \left| \text{Re}({\mathcal{M}}_{ii}^J) \right| \le \frac{1}{2} \,.$$ In the following we will apply these inequalities to leading-order matrix elements in order to identify regions in parameter space where perturbative unitarity is violated. Since these matrix elements are always real in the present context, only the second constraint will be relevant. If the matrix $\mathcal{M}_{if}^J$ is diagonalized the inequality in eq.  becomes an equality. Hence, stronger constraints can be obtained by considering the full transition matrix connecting all possible 2-particle states with each other (or some submatrix thereof) and calculating the eigenvalues of that matrix. Then the bounds from eq.  have to hold for each of the eigenvalues [@Schuessler:2007av]. We note that $d^J_{00}(\theta) = P_J(\cos \theta)$, where $P_J$ are the Legendre polynomials. If initial and final state both have zero total spin, eq.  therefore becomes identical to the familiar partial wave expansion of the matrix element. In the following we will focus on the $J=0$ partial wave, which typically provides the strongest constraint. Since $d^0_{\mu\mu'}$ is non-zero only for $\mu = \mu' = 0$, we then obtain from eq.  $$\mathcal{M}_{if}^0(s) = \frac{1}{64\pi} \beta_{if} \, \delta_{\mu0} \delta_{\mu'0} \int_{-1}^1 \mathrm{d}\cos \theta \, \mathcal{M}_{if}(s, \cos \theta) \,.$$ Application to a simplified model with a $Z'$ mediator ------------------------------------------------------ Let us consider a simplified model for a spin-1 mediator $Z'^\mu$ with mass $m_{Z'}$ and a Dirac DM particle $\psi$ with mass $m_\text{DM}$.[^1] The most general coupling structure is captured by the following Lagrangian: $$\label{eq:L_VA} \mathcal{L} = - \sum_{f = q,l,\nu} Z'^\mu \, \bar{f} \left[ g_{f}^V \gamma_\mu + g_f^A \gamma_\mu \gamma^5 \right] f - Z'^\mu \, \bar{\psi} \left[ g_\text{DM}^V \gamma_\mu + g_\text{DM}^A \gamma_\mu \gamma^5 \right] \psi \; .\\$$ Although these interactions appear renormalisable, the presence of a massive vector boson implies that perturbative unitarity may be violated at large energies. In the following, we will study this issue in detail and derive constraints on the parameter space of the model. Let us first consider diagrams between 2-fermion states with the $Z'$ as mediator. The appropriate propagator for the mediator is $$\langle Z'^\mu(k)Z'^\nu(-k)\rangle = \frac{1}{k^2 - m_{Z'}^2} \left(g^{\mu\nu} - \frac{k^\mu k^\nu}{m_{Z'}^2}\right) \;,$$ where $k^\mu$ is the momentum of the mediator. For the case of a gauge boson this corresponds to unitary gauge in which the Goldstone boson has been absorbed. Since we are interested in the high-energy behaviour of the theory we concentrate on the second term, which does not vanish in the limit $k \rightarrow \infty$. This corresponds to restricting to the longitudinal component of the mediator, $Z'_L$, which dominates at high energy [@Chanowitz:1978mv].[^2] For instance, considering DM annihilations, we can contract the longitudinal part of the propagator with the DM current. Making use of $k = p_1 + p_2$, where $p_1$ and $p_2$ are the momenta of the two DM particles in the initial state, leads to a factor $$\begin{aligned} k^\mu \bar{v}(p_2) \left( g^V_\text{DM} \gamma_\mu + g^A_\text{DM} \gamma_\mu \gamma^5 \right) u(p_1) & = \bar{v}(p_2) \left[ g^V_\text{DM} (\slashed{p}_2 + \slashed{p}_1) + g^A_\text{DM} (\slashed{p}_2 \gamma^5 - \gamma^5 \slashed{p}_1) \right] u(p_1) \nonumber \\ & = - 2 \, g^A_\text{DM} \, m_\text{DM} \, \bar{v}(p_2) \gamma^5 u(p_1) \;.\end{aligned}$$ Hence, the second term in the propagator behaves exactly like a pseudoscalar with mass $m_{Z'}$ and couplings to DM equal to $2 \, g^A_\text{DM} \, m_\text{DM} / m_{Z'}$, just like the Goldstone boson present in Feynman gauge. Note that the term is independent of the vector couplings. The same argument holds for the quark couplings, which are found to be given by $2 \, g^A_f \, m_f / m_{Z'}$. This consideration suggests that perturbative unitarity will not only lead to bounds on $g^{V,A}$, but also on the combination $g^A_f \, m_f / m_{Z'}$. We can make this statement more precise by applying the methods outlined in the previous subsection to the self-scattering of two DM particles or two SM fermions. We obtain for any fermion $f$ with axial couplings $g^A_f \neq 0$ that the fermion mass must satisfy the bound $$m_f \lesssim \sqrt{\frac{\pi}{2}} \frac{m_{Z'}}{g^A_f} \; . \label{eq:DMmass}$$ Here $f$ can be any fermion, including SM fermions and the DM particle. As suggested by the above discussion we do not obtain any bound on the masses of fermions with purely vectorial couplings, nor on the scale of new physics. Let us now turn to the discussion of processes involving $Z'$ in the external state, in particular $Z'$ with longitudinal polarisation. For concreteness, we study the process $\psi \bar{\psi} \rightarrow Z'_L Z'_L$.[^3] At large momenta, $k^2 \gg m_{Z'}^2$, the polarisation vectors of the gauge bosons can be replaced by $\epsilon_L^\mu(k) = k^\mu / m_{Z'}$. One might therefore expect the matrix element for this process to grow proportional to $s / m_{Z'}^2$. However, such a term is absent due to a cancellation between the $t$- and $u$-channel diagram. To obtain a non-zero contribution, one needs to include a mass insertion along the fermion line [@Shu:2007wg]. It turns out that the contribution proportional to $g^V_\text{DM}$ still cancels in this case and that the leading contribution at high energies becomes proportional to $(g^A_\text{DM})^2 \sqrt{s} \, m_\text{DM} / m_{Z'}^2$. As a result, perturbative unitarity is violated unless [@Hosch:1996wu; @Shu:2007wg; @Babu:2011sd][^4] $$\label{eq:s} \sqrt{s} < \frac{\pi \, m_{Z'}^2}{(g^A_\text{DM})^2 \, m_\text{DM}} \; .$$ For larger energies new physics must appear to restore unitarity. This can be accomplished by including an additional diagram with an $s$-channel Higgs boson, since both contributions have the same high-energy behaviour. The consideration above implies an upper bound on the mass of the Higgs that breaks the $U(1)'$ and gives mass to the $Z'$: $$m_s < \frac{\pi \, m_{Z'}^2}{(g^A_\text{DM})^2 \, m_\text{DM}} \; . \label{eq:higgsmass}$$ We will discus the consequences of such an extension of the minimal model in section \[sec:higgs\]. ![Parameter space forbidden by the requirement of perturbative unitarity in the $\sqrt{s}-m_{Z'}$ plane for $m_\text{DM}=500\:\text{GeV}$. The constraint resulting from DM scattering is shown in grey (solid), the constraint resulting from DM annihilation into $Z'$s is shown in blue (dashed). Thick (thin) lines correspond to $g^A_\text{DM}=1$ ($g^A_\text{DM}=0.1$). In these cases, the $Z'$ can never be lighter than about $400\:\text{GeV}$ ($40\:\text{GeV}$) irrespective of the UV completion.[]{data-label="fig:ubound"}](uni.pdf){height="0.3\textheight"} In summary we have found that there are two different types of constraints on the parameters of this simplified model, even for perturbative couplings. For non-vanishing axial couplings there is an energy scale for which the theory violates perturbative unitarity and needs to be UV completed, see eq. . In addition, imposing that the coupling between the longitudinal component of the vector mediator and the DM particle remain perturbative, we find that the vector mediator cannot be much lighter than the DM, see eq. . This constraint is not related to missing degrees of freedom and is therefore completely independent of the UV completion. We illustrate both constraints in figure \[fig:ubound\] for different axial couplings and a DM mass $m_\text{DM} = 500\:\text{GeV}$. To conclude this section, we emphasise that for pure vector couplings of the $Z'$ ($g^A_\text{DM} = g^A_f = 0$) the simplified model considered in this section is well-behaved in the UV in the sense that there is no problem with perturbative unitarity.[^5] Indeed in this specific case a bare mass term for the dark matter is allowed such that it is sufficient to generate the vector boson mass via a Stueckelberg mechanism without the need for additional degrees of freedom [@Stueckelberg:1900zz; @Kors:2005uz]. However, this specific coupling configuration is highly constrained, since it is very difficult to evade bounds from direct detection experiments and still reproduce the observed DM relic abundance. This is illustrated in figure \[fig:vector\] where we show the parameter region excluded by the bound on the spin-independent DM-nucleon scattering cross section from LUX [@Akerib:2013tjd] and the parameter region where the DM annihilation cross section becomes so small that DM is overproduced in the early Universe. One can clearly see that only a finely-tuned region of parameter space close to the resonance $m_\text{DM} = m_{Z'}/2$ is still allowed. For the rest of the paper, we will therefore not consider this case further and always assume that at least one of the vector couplings vanishes such that direct detection constraints can be weakened. ![Vector(SM)–Vector(DM): Parameter space excluded by the bound on the spin-independent DM-nucleon scattering cross section from LUX (green, dashed) and the parameter region where the DM annihilation cross section becomes so small that DM is overproduced in the early Universe (red, solid).[]{data-label="fig:vector"}](vectorvector.pdf){height="0.3\textheight"} Including an additional Higgs field {#sec:higgs} =================================== As we have seen in the previous section, for non-zero axial couplings the simplified model violates perturbative unitarity at high energies, implying that additional new physics must appear below these scales. This observation motivates a detailed discussion of how to generate the vector boson mass from an additional Higgs mechanism. To restore unitarity let us therefore now consider the case that the $Z'$ is the gauge boson of a new $U(1)'$ gauge group. To break this gauge group and give a mass to the $Z'$, we introduce a dark Higgs singlet $S$, which needs to be complex in order to allow for a $U(1)'$ charge. We then obtain the following Lagrangian $$\mathcal{L} = \mathcal{L}_\text{SM} + \mathcal{L}_\text{DM} + \mathcal{L}'_\text{SM} + \mathcal{L}_\text{S} \; ,$$ where the first term is the usual SM Lagrangian and the second term describes the interactions of DM. The third term contains the interactions between SM states and the new $Z'$ gauge boson while the fourth term contains the extended Higgs sector. Implications for the dark sector -------------------------------- As mentioned above, it is well-motivated from a phenomenological perspective to consider the case that vector couplings to the $Z'$ mediator vanish in at least one of the two sectors, so that direct detection is suppressed. On the DM side this is naturally achieved for a Majorana fermion, which we will focus on from now. We therefore write $$\psi = \left( \begin{array}{c} \chi \\ \epsilon \chi^\ast \end{array} \right) \; ,$$ where $\chi$ is a Weyl spinor. We assume that $\chi$ carries a charge $q_\text{DM}$ under the new $U(1)'$ gauge group, such that under a gauge transformation $$\psi \rightarrow \exp\left[i \, g' q_\text{DM} \, \alpha(x) \, \gamma^5\right] \psi \; ,$$ where $g'$ is the gauge coupling of the new $U(1)'$. The kinetic term for $\psi$ can hence be written as $$\mathcal{L}_\text{kin} = \frac{1}{2} \bar{\psi} (i \slashed{\partial} - g' \, q_\text{DM} \, \gamma^5 \slashed{Z}') \psi = \frac{i}{2} \bar{\psi} \slashed{\partial} \psi - \frac{1}{2} g_\text{DM}^A Z'^\mu \bar{\psi} \gamma^5 \gamma_\mu \psi \; ,$$ with $g_\text{DM}^A \equiv g' q_\text{DM}$. The $U(1)'$ charge forbids a Majorana mass term. Nevertheless, if the Higgs field $S$ carries charge $q_S = - 2 q_\text{DM}$, we can write down the gauge-invariant combination $$\mathcal{L}_\text{mass} = -\frac{1}{2} y_\text{DM} \bar{\psi} (P_L S + P_R S^\ast) \psi \; .$$ Including the kinetic and potential terms for the Higgs singlet, the full dark Lagrangian therefore reads $$\begin{aligned} \mathcal{L}_\text{DM} = & \frac{i}{2} \bar{\psi} \slashed{\partial} \psi - \frac{1}{2} g_\text{DM}^A Z'^\mu \bar{\psi} \gamma^5 \gamma_\mu \psi - \frac{1}{2} y_\text{DM} \bar{\psi} (P_L S + P_R S^\ast) \psi \,, \nonumber \\ \mathcal{L}_S = & \left[ (\partial^\mu + i \, g_S \, Z'^\mu) S \right]^\dagger \left[ (\partial_\mu + i \, g_S \, Z'_\mu) S \right] + \mu_s^2 \, S^\dagger S - \lambda_s \left(S^\dagger S \right)^2 \,.\end{aligned}$$ Once the Higgs singlet aquires a vacuum expectation value (vev), it will spontaneously break the $U(1)'$ symmetry, thus giving mass to the $Z'$ gauge boson and the DM particle. After symmetry breaking, we obtain the following Lagrangian (defining $S = 1/\sqrt{2} (s + w)$ and using $g_S \equiv g'q_S = -2g_\text{DM}^A$) $$\begin{aligned} \mathcal{L} = & \frac{i}{2} \bar{\psi} \slashed{\partial} \psi - \frac{1}{4} F'^{\mu\nu}F'_{\mu\nu} - \frac{1}{2} g_\text{DM}^A Z'^\mu \bar{\psi} \gamma^5 \gamma_\mu \psi - \frac{m_\text{DM}}{2} \bar{\psi} \psi - \frac{y_\text{DM}}{2\sqrt{2}} s \bar{\psi} \psi \nonumber \\ & + \frac{1}{2} m_{Z'}^2 \, Z'^\mu Z'_\mu + \frac{1}{2} \partial^\mu s \partial_\mu s + 2 (g_\text{DM}^A)^2 \, Z'^\mu Z'_\mu (s^2 + 2\,s\,w) + \frac{\mu_s^2}{2} (s+w)^2 - \frac{\lambda_s}{4} (s+w)^4 \; ,\end{aligned}$$ with $F'^{\mu\nu} = \partial^\mu Z'^\nu - \partial^\nu Z'^\mu$ and $$\label{eq:masses} m_\text{DM} = \frac{1}{\sqrt{2}} \, y_\text{DM} \, w\,,\quad m_{Z'} \approx 2 g_\text{DM}^A \, w \,.$$ If the SM Higgs is charged under the $U(1)'$ the $Z'$ mass will receive an additional contribution from the SM Higgs vev, see eq.  below. Electroweak precisison data requires that this contribution is small, and therefore we neglect this term in eq.  and for the rest of this subsection. Note that without loss of generality we can choose $w$ and $y_\text{DM}$ to be real (ensuring real masses) by absorbing complex phases in the field definitions for $S$ and $\psi$.[^6] As discussed above, the mass of the additional Higgs particle must satisfy $$m_s < \frac{\pi \, m_{Z'}^2}{(g^A_\text{DM})^2 \, m_\text{DM}}$$ in order for perturbative unitarity to be satisfied, which when substituting the masses of the $Z'$ and DM becomes $$m_s < \frac{4\sqrt{2} \pi w}{y_\text{DM}} \;.$$ Once we include such a new particle coupling to the $Z'$, however, there are additional scattering processes such as $s s \rightarrow s s$ that need to be taken into account when checking perturbative unitarity [@Basso:2011na]. Here we consider the scattering of the states $ss/\sqrt{2}$ and $Z'_L Z'_L/\sqrt{2}$. In the limit $\sqrt{s} \gg m_s \gg m_{Z'}$, the $J=0$ partial wave of the scattering matrix takes the form [@Lee:1977eg] $$\lim_{\sqrt{s} \rightarrow \infty} \mathcal{M}^0_{if} = - \frac{(g^A_\text{DM})^2 m_s^2}{8 \pi m_{Z'}^2} \begin{pmatrix} 3 & 1\\ 1 & 3 \end{pmatrix} \; .$$ Partial wave unitarity requires the real part of the largest eigenvalue, which corresponds to the eigenvector $(ss + Z'_L Z'_L)/2$, to be smaller than $1/2$. We hence obtain the inequality $$m_s \leq \frac{\sqrt{\pi} \, m_{Z'}}{g_\text{DM}^A} = \sqrt{4 \pi} w \; . \label{eq:perturb}$$ This inequality together with eq. (\[eq:DMmass\]) gives a stronger bound on the Higgs mass than the one obtained in eq. (\[eq:higgsmass\]). In other words, the bound in (\[eq:higgsmass\]) can never actually be saturated in this UV completion. We note that eqs.  and can be unified to $$\label{eq:bound_w} \sqrt{\pi} \, \frac{m_{Z'}}{g_\text{DM}^A} \ge \text{max}\left[ m_s , \sqrt{2} m_\text{DM}\right] \,.$$ Implications for the visible sector ----------------------------------- For the discussion above we only needed to consider the DM part of the Lagrangian. Let us now also look at the coupling to the SM, see e.g. [@Carena:2004xs]. The interactions between SM states and the new $Z'$ gauge boson can be written as $$\begin{aligned} \mathcal{L}'_\text{SM} = & \left[ (D^\mu H)^\dagger (-i \, g' \, q_H \, Z'_\mu \, H) + \text{h.c.} \right] + g'^2 \, q_H^2 \, Z'^\mu Z'_\mu \, H^\dagger H \nonumber \\ & - \sum_{f = q,\ell,\nu} g' \, Z'^\mu \, \left[ q_{f_L} \, \bar{f}_L \gamma_\mu f_L + q_{f_R} \, \bar{f}_R \gamma_\mu f_R \right] \; ,\end{aligned}$$ where $D^\mu$ denotes the SM covariant derivative. We can now immediately write down a list of relations between the different charges $q$ required by gauge invariance of the SM Yukawa terms:[^7] $$\begin{aligned} \label{eq:charges} q_H = q_{q_L} - q_{u_R} = q_{d_R} - q_{q_L} = q_{e_R} - q_{\ell_L} \; .\end{aligned}$$ After electroweak symmetry breaking, we obtain $$\begin{aligned} \mathcal{L}'_\text{SM} & = \frac{1}{2} \frac{e \, g' \, q_{H}}{ s_\mathrm{W} \, c_\mathrm{W}} (h+v)^2 \, Z^\mu Z'_\mu + \frac{1}{2} g'^2 \, q_{H}^2 \, (h+v)^2 \, Z'^\mu Z'_\mu \nonumber \\ & \quad - \sum_{f = q,l,\nu} \frac{1}{2} g' Z'^\mu \, \bar{f} \left[ (q_{f_R} + q_{f_L}) \gamma_\mu + (q_{f_R} - q_{f_L}) \gamma_\mu \gamma^5 \right] f \; . \label{eq:LprSM}\end{aligned}$$ Comparing the second line of eq.  with eq.  we can read off the vector and axial vector couplings of the fermions: $$\label{eq:g_f_VA} g_f^V = \frac{1}{2}g'(q_{f_R} + q_{f_L}) \,,\quad g_f^A = \frac{1}{2}g'(q_{f_R} - q_{f_L}) \,.$$ It is well known that a $U(1)'$ under which only SM fields are charged is in general anomalous, unless the SM fields have very specific charges (e.g. $U(1)_{B-L}$ is anomaly free). The relevant anomaly coefficients can e.g. be found in [@Carena:2004xs]. The presence of these anomalies implies that the theory has to include new fermions to cancel the anomalies. While these fermions can be vectorlike with respect to the SM, they will then need to be chiral with respect to the $U(1)'$. The mass of the additional fermions is therefore constrained by the breaking scale of the $U(1)'$. In particular, the bound from eq.  applies to these fermions as well and therefore they cannot be decoupled from the low-energy theory. It is however interesting to note that the anomaly involving two gluons and a $Z'$ is proportional to $$A_{ggZ'} = 3 \left( 2q_{q_L} - q_{u_R} - q_{d_R} \right) \; ,$$ which always vanishes if we restrict the charges based on gauge invariance of the Yukawa couplings (see eq. ). This implies that no new coloured states are needed to cancel the anomalies, greatly reducing the sensitivity of colliders to these new states.[^8] In any case, there are many different possibilities for cancelling the anomalies via new fermions. While the existence of additional fermions will lead to new signatures, a detailed investigation of these is beyond the scope of this work. If the SM Higgs is charged under $U(1)'$ ($q_H \neq 0$) the mass of the $Z'$ receives a contribution from both Higgses: $$\label{eq:mZpr} m_{Z'}^2 = (g'q_H v)^2 + 4(g_\text{DM}^A w)^2 \,,$$ and we obtain a mass mixing term of the form $\delta m^2 \, Z^\mu Z'_\mu$ with $$\begin{aligned} \delta m^2 = \frac{1}{2} \frac{e \, g' \, q_{H}}{ s_\mathrm{W} \, c_\mathrm{W}} v^2 \; ,\end{aligned}$$ where $s_\mathrm{W} \,(c_\mathrm{W})$ is the sine (cosine) of the Weinberg angle. As we are going to discuss below, electroweak precision data requires $|\delta m^2| \ll |m_Z-m_{Z'}|$ (see also App. \[app:Z\]). Using $m_Z = e v / (2 s_\mathrm{W} c_\mathrm{W})$, $g'q_s = -2g_\text{DM}^A$, and neglecting order one factors this requirement implies either $g'q_H \ll e$ or $q_s w \gg v$. In the parameter regions of interest it follows from those conditions that the first term in eq.  is small and hence the mass of the $Z'$ is dominated by the vev of the dark Higgs. Taking into account eqs.  and , the condition $|\delta m^2| \ll |m_Z - m_{Z'}|$ then implies either small axial couplings ($g_f^A \ll 1$) or $m_{Z'} \gg m_Z$. We are going to present more quantitative results in the next section and discuss a number of interesting experimental signatures resulting from the new interactions due to eq. . To conclude this section, it should be noted, that the Lagrangian introduced above is UV-complete (up to anomalies) and gauge invariant but does not correspond to the most general realization of this model. In particular the term $$\mathcal{L} \supset - \lambda_{hs} (S^*S)(H^\dagger H) \,,$$ which will lead to mixing between the SM Higgs $h$ and the dark Higgs $s$ can be expected to be present at tree level. Furthermore, the term $$\label{eq:kin-mix} \mathcal{L} \supset -\frac{1}{2} \sin \epsilon F'^{\mu\nu} B_{\mu\nu} \; ,$$ which generates kinetic mixing between the $Z'$ and the $Z$-boson, respects all symmetries of the Lagrangian. It can be argued that $\epsilon$ might vanish at high scales in certain UV-completions, but even in this case kinetic mixing is necessarily generated at the one-loop level and can have a substantial impact on EWPT. We will return to these issues and the resulting phenomenology of the model in sections \[sec:vector\] and \[sec:higgsmixing\]. For the moment, however, we are going to neglect these additional effects and focus on the impact of $\delta m \neq 0$, which necessarily leads to mass mixing between the neutral gauge bosons in the case of non-vanishing axial couplings. Non-zero axial couplings to SM fermions {#sec:axial} ======================================= Let us start with the case that axial couplings on the SM side are non-vanishing. An immediate consequence is that the SM Higgs is charged under the $U(1)'$, which follows from eqs.  and for $g_f^A \neq 0$. Note that these equations also imply that it is inconsistent to set the vectorial couplings for all quarks equal to zero. For example, if we impose that the vectorial couplings of up quarks vanish, i.e. $g_u^V = 0$, eq.  implies $q_{u_R} = - q_{q_L}$, which using eq.  leads to $g_d^V = 2 g' q_{q_L}$. In the following, whenever $g^A_f \neq 0$, we always fix $g^V_f = g^A_f$, which corresponds to setting $q_{q_L} = q_{\ell_L} = 0$. Furthermore, eq.  requires that $Z'$ couplings are flavour universal and leptons couple with the same strength to the $Z'$ as quarks. This conclusion could potentially be modified by considering an extended Higgs sector, e.g. a two-Higgs-doublet model. Here we focus on the simplest case where a single Higgs doublet generates all SM fermion masses. This implies that the leading search channel at the LHC will be dilepton resonances, which give severe constraints. In principle also electron-positron colliders can constrain this scenario efficiently. Limits on a $Z'$ lighter than 209 GeV derived from LEP data imply $g \lesssim 10^{-2}$ [@Agashe:2014kda] (see also [@Appelquist:2002mw; @LEP:2003aa]). We do not include LEP constraints here since other constraints will turn out to be at least equally strong. For general couplings, the partial decay width of the mediator into SM fermions is given by $$\Gamma(Z'\rightarrow f\bar{f}) = \frac{m_{Z'} N_c}{12\pi} \sqrt{1-\frac{4 m_f^2}{m_{Z'}^2}} \, \left[(g^{V}_{f})^2+(g^{A}_{f})^2 + \frac{m_f^2}{m_{Z'}^2}\left(2 (g^{V}_{f})^2 - 4 (g^{A}_{f})^2\right) \right] \; ,$$ where $N_c = 3 \;(1)$ for quarks (leptons). The decay width into DM pairs is $$\Gamma(Z'\rightarrow \psi\psi) = \frac{m_{Z'}}{24\pi} (g^{A}_\text{DM})^2 \left(1-\frac{4 m_\text{DM}^2}{m_{Z'}^2}\right)^{3/2} \; .$$ Consequently, for $m_\text{DM} \ll m_{Z'}$ and $g^A_{\ell} = g^A_{q} \ll g^A_\text{DM}$ the branching ratio into $\ell = e,\,\mu$ is given by $\text{BR}(R\rightarrow \ell \ell) \approx 8 (g^A_{\ell})^2 / (g^A_\text{DM})^2$. For $m_\text{DM} > m_{Z'} / 2$, on the other hand, the branching ratio is given by $\text{BR}(R\rightarrow \ell \ell) \approx 0.08\text{--}0.10$ depending on the ratio $m_{Z'} / m_t$. We implement the latest ATLAS dilepton search [@Aad:2014cka], complemented by a Tevatron dilepton search [@Jaffre:2009dg] for the low mass region, and show the resulting bounds in figure \[fig:axialaxial\]. One can see that the bounds strongly depend on the assumed branching ratio of the $Z'$. As a conservative limiting case we show $g_\text{DM}^A=1$ and $m_\text{DM}=100$ GeV, which leads to a rather large branching fraction into DM and hence suppressed bounds. The second benchmark, $m_\text{DM}=500$ GeV, allows for $Z'$ decays to DM only for rather heavy $Z'$s, leading to correspondingly more restrictive dilepton constraints. Overall the bounds turn out to be very stringent and the $Z'$ coupling to leptons and quarks needs to be significantly smaller than unity for $100 \; \text{GeV} \lesssim m_{Z'} \lesssim 4 \; \text{TeV}$, so that dijet constraints are basically irrelevant in this case given that $g_q=g_l$. ![Axial+Vector(SM)–Axial(DM): Parameter space forbidden by constraints from dilepton resonance searches from ATLAS (light green, dashed) and Tevatron (dark green, dashed), electroweak precision observables (blue, dotted) and DM overproduction (red, solid) in the $m_{Z'}-g_{q,l}^A$ parameter plane for two exemplary DM masses 100 GeV (left) and 500 GeV (right). In the shaded region to the left of the vertical grey line the $Z'$-mass violates the bound from perturbative unitarity from eq. .[]{data-label="fig:axialaxial"}](axialaxial_100_label.pdf "fig:"){height="0.3\textheight"}![Axial+Vector(SM)–Axial(DM): Parameter space forbidden by constraints from dilepton resonance searches from ATLAS (light green, dashed) and Tevatron (dark green, dashed), electroweak precision observables (blue, dotted) and DM overproduction (red, solid) in the $m_{Z'}-g_{q,l}^A$ parameter plane for two exemplary DM masses 100 GeV (left) and 500 GeV (right). In the shaded region to the left of the vertical grey line the $Z'$-mass violates the bound from perturbative unitarity from eq. .[]{data-label="fig:axialaxial"}](axialaxial_500.pdf "fig:"){height="0.3\textheight"} The fact that the SM Higgs is charged also implies potentially large corrections to electroweak precision observables. In particular we obtain the non-diagonal mass term $\delta m^2 \, Z^\mu Z'_\mu$ leading to mass mixing between the SM $Z$ and the new $Z'$. The diagonalisation required to obtain mass eigenstates is discussed in the appendix. In the absence of kinetic mixing between the $U(1)'$ and the SM $U(1)$ gauge bosons ($\epsilon = 0$), the resulting effects can be expressed in terms of the mixing parameter $\xi = \delta m^2 /(m_Z^2-m_{Z'}^2)$ (see eq.  in the appendix with $\epsilon=0$). In particular, we can calculate the constraints from electroweak precision measurements, which are encoded in the $S$ and $T$ parameters. To quadratic order in $\xi$ we find [@Frandsen:2011cg] $$\begin{aligned} \alpha S = & - 4 c_\mathrm{W}^2 s_\mathrm{W}^2 \xi^2 \; , \nonumber\\ \alpha T = & \xi^2\left(\frac{m_{Z'}^2}{m_{Z}^2}-2\right) \; , \label{eq:ST}\end{aligned}$$ where $\alpha=e^2/4\pi$. The resulting bounds are shown in figure \[fig:axialaxial\]. To infer our bounds we use the $90\%$ CL limit on the $S$ and $T$ parameters as given in [@Agashe:2014kda]. Note that the bound from electroweak precision data is completely independent of the $Z'$ couplings to the DM as well as the DM mass. Hence, the same bound would apply also in the case of Dirac DM with vector couplings to the $Z'$. Note however that since vectorial couplings to quarks are necessarily non-zero if $g_q^A \neq 0$ (see eqs. , ), there will be very stringent bounds from direct detection experiments on any model with $g^V_\text{DM} \neq 0$ due to unsuppressed spin-independent scattering. For Majorana DM, the vectorial coupling always vanishes and the constraints from direct detection are much weaker. ![Axial+Vector(SM)–Axial(DM): Parameter space forbidden by constraints from dilepton resonance searches (green, dashed) and electroweak precision observables (blue, dotted) in the $m_\text{DM}-m_{Z'}$ plane for four different sets of couplings. We also show the regions excluded by DM overproduction (red), direct detection bounds (purple, dot-dashed) and the parameter space where perturbative unitarity is violated (grey). For the relic density calculation we have assumed that the mass of the hidden sector Higgs saturates the unitarity bound.[]{data-label="fig:axialaxial2"}](aamm1_monojet.pdf "fig:"){height="0.3\textheight"}![Axial+Vector(SM)–Axial(DM): Parameter space forbidden by constraints from dilepton resonance searches (green, dashed) and electroweak precision observables (blue, dotted) in the $m_\text{DM}-m_{Z'}$ plane for four different sets of couplings. We also show the regions excluded by DM overproduction (red), direct detection bounds (purple, dot-dashed) and the parameter space where perturbative unitarity is violated (grey). For the relic density calculation we have assumed that the mass of the hidden sector Higgs saturates the unitarity bound.[]{data-label="fig:axialaxial2"}](aamm025_label_monojet.pdf "fig:"){height="0.3\textheight"} ![Axial+Vector(SM)–Axial(DM): Parameter space forbidden by constraints from dilepton resonance searches (green, dashed) and electroweak precision observables (blue, dotted) in the $m_\text{DM}-m_{Z'}$ plane for four different sets of couplings. We also show the regions excluded by DM overproduction (red), direct detection bounds (purple, dot-dashed) and the parameter space where perturbative unitarity is violated (grey). For the relic density calculation we have assumed that the mass of the hidden sector Higgs saturates the unitarity bound.[]{data-label="fig:axialaxial2"}](aamm01_monojet.pdf "fig:"){height="0.3\textheight"}![Axial+Vector(SM)–Axial(DM): Parameter space forbidden by constraints from dilepton resonance searches (green, dashed) and electroweak precision observables (blue, dotted) in the $m_\text{DM}-m_{Z'}$ plane for four different sets of couplings. We also show the regions excluded by DM overproduction (red), direct detection bounds (purple, dot-dashed) and the parameter space where perturbative unitarity is violated (grey). For the relic density calculation we have assumed that the mass of the hidden sector Higgs saturates the unitarity bound.[]{data-label="fig:axialaxial2"}](aamm0025_monojet.pdf "fig:"){height="0.3\textheight"} In figure \[fig:axialaxial2\] we show the constraints from electroweak precision data as well as LHC dilepton searches in the $m_\text{DM}-m_{Z'}$ plane for different values of the axial vector coupling to fermions. In the lower right corner of the plots (grey area) the perturbative unitarity condition from eq.  is violated. We also show the region excluded by direct detection searches (dark region in the lower left corners). For the axial-axial couplings DM-nucleus scattering proceeds through spin-dependent interactions, with a scattering cross section given by $$\sigma^\text{SD}_N = \frac{3 \, a_N^2 \, (g^A_\text{DM})^2 \, (g^A_q)^2}{\pi} \frac{\mu^2}{m_{Z'}^4} \; ,$$ where $\mu$ is the DM-nucleon reduced mass, $N = p,\,n$ and $a_p = -a_n = 1.18$ is the effective nucleon coupling [@Agashe:2014kda]. This is the dominant contribution in this case as the vector-axial coupling combination is even further suppressed. In the plots we show the bound on the spin-dependent scattering cross section that can be calculated from the published LUX results [@Akerib:2013tjd], following the method described in [@Feldstein:2014ufa]. We observe that in this case direct detection is never competitive with other constraints. The red solid curves in Figs. \[fig:axialaxial\] and \[fig:axialaxial2\] show the parameter values that lead to the correct relic abundance. In order to calculate the relic abundance we have implemented the model in micrOMEGAs\_v4 [@Belanger:2014vza], assuming that the mass of the Higgs singlet saturates the unitarity bound and setting the mixing with the SM Higgs to zero.[^9] In the regions shaded in red (to the right/above the solid curve) there is overproduction of DM. In this region additional annihilation channels are required to avoid overclosure of the Universe, since the interactions provided by the $Z'$ are insufficient to keep DM in thermal equilibrium long enough. Such additional interactions could be obtained for instance from the scalar mixing discussed in section \[sec:higgsmixing\]. Conversely, to the left/below the red solid curve the model does not provide all of the DM matter in the Universe, since the annihilation rate is too high. Let us briefly discuss the various features that can be observed in the relic abundance curve. First there is a significant decrease of the predicted abundance as the DM mass crosses the top-quark threshold, $m_\chi > m_t$, resulting from the fact that the $s$-wave contribution to the annihilation cross section is helicity suppressed and hence annihilation into top-quarks becomes the dominant annihilation channel as soon as it is kinematically allowed. The second feature occurs at $m_\chi \sim m_{Z'}$ and reflects the resonant enhancement of the annihilation process $\chi \chi \rightarrow q \bar{q}$ as the mediator can be produced on-shell. A third visible feature is a very narrow resonance at $2 \, m_\chi \sim m_s = \sqrt{\pi} \, m_{Z'} / g^A_\text{DM}$ due to a resonant enhancement of the process $\chi \chi \rightarrow s \rightarrow Z' Z$. The position and magnitude of this effect depends on the mass of the dark Higgs, which has been (arbitrarily) fixed to saturate the unitarity bound. However, even for this extreme choice, it turns out to give a non-negligible contribution to the relic abundance. For $m_\chi > m_{Z'}$ direct annihilation into two mediators becomes possible, leading to a significant decrease of the predicted relic abundance. Finally, the fact that the relic abundance curve in figure \[fig:axialaxial2\] touches the unitarity bound for high DM masses reflects the well-known unitarity bound on the mass of a thermally produced DM particle [@Griest:1989wd]. All in all we find the case with non-vanishing axial couplings on the SM side to be strongly constrained by dilepton searches as well as electroweak precision observables, implying that in a UV complete model this is where a signal should first be seen. For comparison, we show recent bounds from LUX as well as from the CMS monojet search [@Khachatryan:2014rra].[^10] We find that these searches, as well as searches for dijet resonances, are not competitive. Note that in figure \[fig:axialaxial2\] we assume $g_\text{DM}^A = 1$. We comment on smaller couplings on the DM side later in the context of figure \[fig:smallgA\]. Let us now look at the case where axial couplings to quarks are taken to be zero, which will turn out to be somewhat less constrained. Purely vectorial couplings to SM fermions {#sec:vector} ========================================= Let us now consider the case with purely vectorial couplings on the SM side, i.e. $g^A_q = g^A_\ell = g^A_\nu = 0$. In this case the SM Higgs does not carry a $U(1)'$ charge and therefore the charges of quarks and leptons are independent. In particular, it is conceivable that $g^V_q \gg g^V_\ell$, so that constraints from dilepton resonance searches can be evaded. Also there can in principle be a flavour dependence of the $Z'$ couplings to quarks. Nevertheless, to avoid large flavour-changing neutral currents, we will always assume the same coupling for all quark families in what follows [@Abdallah:2015ter]. Finally, in contrast to the case discussed above, tree-level $Z-Z'$ mass mixing is absent. It therefore seems plausible that the $Z'$ is the only state coupling to both the visible and the dark sector. Nevertheless, as mentioned above, potentially important effects in this scenario can be kinetic mixing of the $U(1)$ gauge bosons as well as effects induced by the dark Higgs, which we are going to discuss below. Let us just mention that all these effects will also be present in the scenario discussed in the previous section. They are, however, typically less important than the effects of tree-level $Z$-$Z'$ mixing. We first consider the effects of kinetic mixing between the $Z'$ and the SM hypercharge gauge boson $B$: $$\mathcal{L} \supset -\frac{1}{2} \sin \epsilon \, F'^{\mu\nu} B_{\mu\nu} \; ,$$ where $F'^{\mu\nu} = \partial^\mu Z'^\nu - \partial^\nu Z'^\mu$ and $B^{\mu\nu} = \partial^\mu B^\nu - \partial^\nu B^\mu$. A non-zero value of $\epsilon$ leads to mixing between the $Z'$ and the neutral gauge bosons of the SM (see App. \[app:Z\]). As in the case of mass mixing discussed above, there are strong constraints on kinetic mixing from searches for dilepton resonances and electroweak precision observables. ![Vector(SM)–Axial(DM): Parameter space forbidden by constraints from ATLAS and Tevatron dileptons (green, dashed), electroweak precision observables (blue, dotted) and relic DM overproduction (red, solid) in the $m_{Z'}$-$\epsilon$ parameter plane (left) and the $m_\text{DM}$-$m_{Z'}$ parameter plane (right). In both panels we show the parameter space where perturbative unitarity is violated (grey). For the relic density calculation we have assumed that the mass of the hidden sector Higgs saturates the unitarity bound.[]{data-label="fig:va"}](vectoraxial_100.pdf "fig:"){height="0.3\textheight"}![Vector(SM)–Axial(DM): Parameter space forbidden by constraints from ATLAS and Tevatron dileptons (green, dashed), electroweak precision observables (blue, dotted) and relic DM overproduction (red, solid) in the $m_{Z'}$-$\epsilon$ parameter plane (left) and the $m_\text{DM}$-$m_{Z'}$ parameter plane (right). In both panels we show the parameter space where perturbative unitarity is violated (grey). For the relic density calculation we have assumed that the mass of the hidden sector Higgs saturates the unitarity bound.[]{data-label="fig:va"}](vectoraxialmm.pdf "fig:"){height="0.3\textheight"} The dilepton couplings induced via the kinetic mixing parameter $\epsilon$ can be inferred from the mixing matrices and are given in the appendix, cf. eq. . The $S$ and $T$ parameters are given by $$\begin{aligned} \alpha S = & 4 c_\mathrm{W}^2 s_\mathrm{W} \xi (\epsilon - s_\mathrm{W} \xi) \; , \nonumber\\ \alpha T = & \xi^2\left(\frac{m_{Z'}^2}{m_{Z}^2}-2\right)+2 s_\mathrm{W} \xi \epsilon \; , \label{eq:ST2}\end{aligned}$$ where for $\delta m^2 = 0$ the mixing parameter $\xi$ is given by $\xi = m_{Z}^2 s_\mathrm{W} \epsilon / (m_{Z}^2 - m_{Z'}^2)$ at leading order. If $\epsilon$ is sizeable, i.e. if mixing is present at tree level, the resulting bounds can be quite strong. This expectation is confirmed in figure \[fig:va\]. Note that the relic density curves shown in figure \[fig:va\] are basically independent of $\epsilon$, because freeze-out is dominated by direct $Z'$ exchange for the adopted choice of couplings. While tree-level mixing is tightly constrained, it is reasonable to expect that $\epsilon$ vanishes at high scales, for example if both $U(1)$s originate from the same underlying non-Abelian gauge group, as in Grand Unified Theories. Since quarks carry charge under both $U(1)'$ and $U(1)_Y$, quark loops will still induce kinetic mixing at lower scales [@Holdom:1985ag], but the magnitude of $\epsilon$ can be much smaller than what we considered above. The precise magnitude of the kinetic mixing depends on the underlying theory, but if we assume that $\epsilon(\Lambda) = 0$ at some scale $\Lambda \gg 1\:\text{TeV}$, the kinetic mixing at a lower scale $\mu > m_t$ will be given by [@Carone:1995pu] $$\epsilon(\mu) = \frac{e \, g^V_q}{2 \pi^2 \, \cos \theta_\text{W}} \log \frac{\Lambda}{\mu} \simeq 0.02 \, g^V_q \, \log \frac{\Lambda}{\mu} \; .$$ We can use this equation (setting $\mu = m_{Z'}$) to translate the bounds from figure \[fig:va\] into constraints on $g^V_q$. The results of such an analysis are shown in figure \[fig:va\_loop\] assuming $\Lambda = 10$ TeV. As can be seen in figure \[fig:va\_loop\] (left), searches for dilepton resonances give again stringent constraints, implying $g_q^V < 0.1$ for $m_{Z'} = 200\:\text{GeV}$ and $g_q^V < 1$ for $m_{Z'} = 1\:\text{TeV}$. ![Vector(SM)–Axial(DM): Parameter space forbidden by constraints from ATLAS and Tevatron dileptons (green, dashed) and electroweak precision observables (blue, dotted) in the $m_{Z'}$-$g^V_q$ parameter plane (left) and the $m_\text{DM}$-$m_{Z'}$ parameter plane (right), assuming that $\epsilon=0$ at $\Lambda = 10\:\text{TeV}$ so that kinetic mixing is only induced at the one-loop level. In the right panel we show also the region excluded by LHC monojet (orange, dashed) and dijet (violet, dot-dashed) searches due to tree-level $Z'$ exchange for the adopted coupling choice. In both panels we show the parameter space where perturbative unitarity is violated (grey). For the relic density calculation we have assumed that the mass of the hidden sector Higgs saturates the unitarity bound. []{data-label="fig:va_loop"}](vectoraxial_loop_100.pdf "fig:"){height="0.3\textheight"}![Vector(SM)–Axial(DM): Parameter space forbidden by constraints from ATLAS and Tevatron dileptons (green, dashed) and electroweak precision observables (blue, dotted) in the $m_{Z'}$-$g^V_q$ parameter plane (left) and the $m_\text{DM}$-$m_{Z'}$ parameter plane (right), assuming that $\epsilon=0$ at $\Lambda = 10\:\text{TeV}$ so that kinetic mixing is only induced at the one-loop level. In the right panel we show also the region excluded by LHC monojet (orange, dashed) and dijet (violet, dot-dashed) searches due to tree-level $Z'$ exchange for the adopted coupling choice. In both panels we show the parameter space where perturbative unitarity is violated (grey). For the relic density calculation we have assumed that the mass of the hidden sector Higgs saturates the unitarity bound. []{data-label="fig:va_loop"}](vectoraxialmm_loop.pdf "fig:"){height="0.3\textheight"} In the right panel of figure \[fig:va\_loop\] we also show the constraints coming from LHC searches for monojets (i.e. jets in association with large amounts of missing transverse energy) and for dijet resonances, adopted from ref. [@Chala:2015ama].[^11] These limits are independent of the kinetic mixing $\epsilon$ since they originate from the tree-level $Z'$ exchange and probe larger values of $m_{Z'}$ and smaller values of $m_\text{DM}$. Nevertheless, dilepton resonance searches and EWPT give relevant constraints for small $m_{Z'}$ and large $m_\text{DM}$, which are difficult to probe with monojet and dijet searches. We conclude from figure \[fig:va\_loop\] that the combination of constraints due to loop-induced kinetic mixing and bounds from LHC DM searches leave only a small region in parameter space (a small strip close to the $Z'$ resonance), where DM overabundance is avoided. While this result depends somewhat on our choice $\Lambda = 10$ TeV and $\mu = m_{Z'}$, it is only logarithmically sensitive to these choices. It is worth emphasising that the unitarity constraints shown in figures \[fig:axialaxial2\]– \[fig:va\_loop\] depend sensitively on the choice of $g^A_\text{DM}$ (cf. figure \[fig:ubound\]). We therefore show in figure \[fig:smallgA\] how these constraints change if we take $g^A_\text{DM} = 0.1$ rather than $g^A_\text{DM} = 1$. In this case both $m_\text{DM}$ and $m_s$ can be much larger than $m_{Z'}$. At the same time, however, relic density constraints become significantly more severe, excluding almost the entire parameter space with $m_\text{DM} < m_{Z'}$ (apart from the resonance region $m_\text{DM} \sim m_{Z'}$). Even for $m_\text{DM} > m_{Z'}$ is it difficult to reproduce the observed relic abundance, because the annihilation channel $\chi \chi \rightarrow Z' Z'$ is significantly suppressed due to the smallness of $g^A_\text{DM}$. It only becomes relevant close to the unitarity bound, where also the process $\chi \chi \rightarrow Z Z'$ mediated by the dark Higgs gives a sizeable contribution. While in the case with axial couplings on the SM side the region compatible with thermal freeze-out becomes fully excluded by dilepton resonance searches, the case with vector couplings on the SM side is very difficult to probe at colliders and direct detection, leading to a small allowed parameter region close to the bound from perturbative unitarity. ![Constraints for small DM couplings ($g^A_\text{DM} = 0.1$). The left panel considers the case Axial+Vector(SM)–Axial(DM) and should be compared to figure \[fig:axialaxial2\]. The right panel considers the case Vector(SM)–Axial(DM) with loop-induced kinetic mixing, assuming that $\epsilon=0$ at $\Lambda = 10\:\text{TeV}$ (cf. figure \[fig:va\_loop\]).[]{data-label="fig:smallgA"}](aamm_gAx01_gAq01_label.pdf "fig:"){height="0.3\textheight"}![Constraints for small DM couplings ($g^A_\text{DM} = 0.1$). The left panel considers the case Axial+Vector(SM)–Axial(DM) and should be compared to figure \[fig:axialaxial2\]. The right panel considers the case Vector(SM)–Axial(DM) with loop-induced kinetic mixing, assuming that $\epsilon=0$ at $\Lambda = 10\:\text{TeV}$ (cf. figure \[fig:va\_loop\]).[]{data-label="fig:smallgA"}](vectoraxialmm_loop_gax01_gvq01.pdf "fig:"){height="0.3\textheight"} In addition to the effects of kinetic mixing, we have shown above that for $g^A_\text{DM} \neq 0$ the dark sector necessarily contains a new Higgs particle. The presence of this additional Higgs can change the phenomenology of the model in two important ways. First, loop-induced couplings of the dark Higgs to SM states may give an important contribution to direct detection signals. And second, there may be mixing between the SM Higgs and the dark Higgs, leading to pertinent modifications of the properties of the SM Higgs as well as opening another portal for DM-SM interactions. We will discuss loop-induced couplings in this section and then return to a detailed study of the Higgs potential in the next section. For $g^A_q = g^V_\text{DM} = 0$, scattering in direct detection experiments is momentum-suppressed in the non-relativistic limit and the corresponding event rates are very small. This conclusion may change if loop corrections induce unsuppressed scattering [@Haisch:2013uaa]. Indeed, at the one-loop level the dark Higgs can couple to quarks and can therefore mediate unsuppressed spin-independent interactions. The resulting interaction can be written as $\mathcal{L} \propto \sum_q \, m_q \, s \, \bar{q} q$. After integrating out heavy-quark loops as well as the dark Higgs this interaction leads to an effective coupling between DM and nucleons of the form $\mathcal{L} \propto f_N \, m_N \, m_\text{DM} \, \bar{N}N \, \bar{\psi}\psi$, where $m_N$ is the nucleon mass, $N = p, n$ and $f_N \approx 0.3$ is the effective nucleon coupling. In the non-relativistic limit, the diagram in the left of figure \[fig:SI\] induces the effective interaction $$\mathcal{L}_\text{eff} \supset \frac{(g^A_\text{DM})^2 \, (g^V_q)^2}{\pi^2} \frac{1}{m_s^2 \, m_{Z'}^2} \times m_\text{DM} \, f_N \, m_N \, \bar{N} N \, \bar{\psi} \psi \; .$$ The corresponding spin-independent scattering cross section is given by $$\sigma_N^\text{SI} = \frac{m_\text{DM}^2 \, f_N^2 \, m_N^2 \, \mu^2}{\pi} \frac{(g^A_\text{DM})^4 \, (g^V_q)^4}{\pi^4} \frac{1}{m_s^4 \, m_{Z'}^4} \; ,$$ where $\mu$ is the DM-nucleon reduced mass. For masses of order $300\:\text{GeV}$ and couplings of order unity this expression yields $\sigma_N^\text{SI} \sim 10^{-46} \:\text{cm}^2$, which is below the current bounds from LUX but well within the potential sensitivity of XENON1T. ![Contributions to spin-independent scattering.[]{data-label="fig:SI"}](diagrams.pdf){width="75.00000%"} We note that there are two additional diagrams (shown in the second and third panel of figure \[fig:SI\]) that also lead to unsuppressed spin-independent scattering of DM particles [@Haisch:2013uaa]. For $m_\text{DM} \gg m_N$, the resulting contribution is given by $$\mathcal{L}_\text{eff} \supset \frac{(g^A_\text{DM})^2 \, (g^V_q)^2}{4 \pi^2} \frac{m_\text{DM}^2 - m_{Z'}^2 + m_{Z'}^2 \log (m_{Z'}^2 / m_\text{DM}^2)}{m_{Z'}^2 (m_{Z'}^2 - m_\text{DM}^2)^2} \times m_\text{DM} \, f_N \, m_N \, \bar{N} N \, \bar{\psi} \psi \; .$$ If there is no large hierarchy between $m_\text{DM}$, $m_{Z'}$ and $m_s$, this contribution is of comparable magnitude to the one from dark Higgs exchange and interference effects can be important. Moreover, there may be a relevant contribution from loop-induced spin-dependent scattering. We leave a detailed study of these effects to future work. Mixing between the two Higgs bosons {#sec:higgsmixing} =================================== In addition to the loop-induced couplings of the dark Higgs to SM fermions discussed in the previous section, such couplings can also arise at tree-level from mixing. In fact, an important implication of the presence of a second Higgs field is that the two Higgs fields will in general mix, thus modifying the properties of the mostly SM-like Higgs. Furthermore, the mixing opens up the so-called Higgs portal between the DM and SM particles, leading to a much richer DM phenomenology than in the case of DM-SM interactions only via the vector mediator. The mixing between the scalars is due to an additional term in the scalar potential: $$V(S, H) \supset \lambda_{hs} (S^*S)(H^\dagger H) \,.$$ The coupling $\lambda_{hs}$ is a free parameter, independent of the vector mediator. For non-zero $\lambda_{hs}$, the scalar mass eigenstates $H_{1,2}$ are given by $$\begin{aligned} H_1 = s \sin \theta + h \cos \theta \nonumber \\ H_2 = s \cos \theta - h \sin \theta \end{aligned}$$ where, as shown in App. \[app:scalar\], $$\theta \approx - \frac{\lambda_{hs} \, v \, w}{m_s^2 - m_h^2} + \mathcal{O}(\lambda_{hs}^3) \; .$$ We emphasise that perturbative unitarity implies that $m_s$ cannot be arbitrarily large (for given $m_{Z'}$ and $g^A_\text{DM}$) and hence it is impossible to completely decouple the dark Higgs. The resulting Higgs mixing leads to three important consequences. First, the (mostly) dark Higgs obtains couplings to SM particles, enabling us to produce it at hadron colliders and to search for its decay products (or monojet signals). Second, the properties of the (mostly) SM-like Higgs, in particular its total production cross section and potentially also its branching ratios, are modified. And finally, both Higgs particles can mediate interactions between DM and nuclei, leading to potentially observable signals at direct detection experiments. Higgs portal DM has been extensively studied, see for instance [@Djouadi:2011aa; @Lebedev:2012zw; @LopezHonorez:2012kv; @Baek:2012uj; @Walker:2013hka; @Esch:2013rta; @Freitas:2015hsa] for an incomplete selection of references. A full analysis of Higgs mixing effects is beyond the scope of the present paper. Nevertheless, to illustrate the magnitude of potential effects, let us consider the induced coupling of the SM-like Higgs $H_1 \approx h$ to DM particles $$\mathcal{L} \supset - \frac{m_\text{DM} \, \sin \theta}{2 \, w} h \, \bar{\psi} \psi \simeq \frac{m_\text{DM} \, \lambda_{hs} \, v}{2 (m_s^2 - m_h^2)} h \, \bar{\psi} \psi \; .$$ For small $\lambda_{hs}$, the resulting direct detection cross section is given by [@Djouadi:2011aa] $$\sigma_N^\text{SI} \simeq \frac{\mu^2}{\pi \, m_h^4} \frac{f_N^2 \, m_N^2 \, m_\text{DM}^2\,\lambda_{hs}^2}{(m_s^2 - m_h^2)^2} \; ,$$ where we can neglect an additional contribution from the exchange of a dark Higgs provided $m_s^4 \gg m_h^4$. The parameter regions excluded by the LUX results [@Akerib:2013tjd] are shown in figure \[fig:higgsmixing\] (green regions). ![Constraints on $m_s$ and $\lambda_{hs}$ from bounds on the Higgs invisible branching ratio (blue, dotted) and from bounds on the spin-independent DM-nucleon scattering cross section (green, dashed). In the grey parameter region unitarity constraints are in conflict with the stability of the potential.[]{data-label="fig:higgsmixing"}](HiggsPlot1.pdf "fig:"){height="0.3\textheight"}![Constraints on $m_s$ and $\lambda_{hs}$ from bounds on the Higgs invisible branching ratio (blue, dotted) and from bounds on the spin-independent DM-nucleon scattering cross section (green, dashed). In the grey parameter region unitarity constraints are in conflict with the stability of the potential.[]{data-label="fig:higgsmixing"}](HiggsPlot2.pdf "fig:"){height="0.3\textheight"} We note that (in the linear approximation) the direct detection cross section is independent of $w$ and does therefore not depend on $m_{Z'}$ or $g^A_\text{DM}$. Nevertheless $w$ is not arbitrary, because unitarity gives a lower bound $\sqrt{4\pi} w > \text{max}\left[\sqrt{2} m_\text{DM}, m_s \right]$. At the same time, stability of the Higgs potential requires $4 \lambda_s \, \lambda_h > \lambda_{hs}$. These two inequalities can only be satisfied at the same time if $$m_\text{DM} < \frac{\sqrt{2 \pi} \, m_s \, m_h}{v} \; .$$ In figure \[fig:higgsmixing\], we show the parameter region where unitarity and stability are in conflict in grey. Finally, if the DM mass is sufficiently small, the SM-like Higgs can decay into pairs of DM particles, with a partial width given by [@Djouadi:2011aa] $$\Gamma^\text{inv} = \frac{1}{8 \pi} \frac{m_\text{DM}^2 \, \lambda_{hs}^2}{(m_s^2 - m_h^2)^2} v^2 m_h \left(1-\frac{4 \, m_\text{DM}^2}{m_h^2}\right)^{3/2} \; .$$ The invisible branching fraction is tightly constrained by LHC measurements: $\text{BR}(h\rightarrow \text{inv}) < 0.27$ [@Khachatryan:2014jba]. Furthermore, a combined fit from ATLAS and CMS yields $\mu = 1.09^{+0.11}_{-0.10}$ for the total Higgs signal strength [@ATLAS-CONF-2015-044], which can be used to deduce $\text{BR}(h\rightarrow \text{inv}) < 0.11$ at 95% CL. The resulting constraints, compared to the ones on $\sigma_N^\text{SI}$ from LUX, are shown in figure \[fig:higgsmixing\] (blue regions). The crucial observation is that the necessary presence of a dark Higgs will in general induce additional signatures and therefore lead to new ways to constrain models with a $Z'$ mediator using both direct detection experiments and Higgs measurements. However, since $\lambda_{hs}$ and $m_s$ are effectively free parameters, it is difficult to directly compare the constraints shown in figure \[fig:higgsmixing\] to the ones obtained from monojet and dijet searches at the LHC. Nevertheless, we can conservatively estimate the relevance of these effects by fixing the dark Higgs mass $m_s$ to the largest value consistent with perturbative unitarity. ![Constraints in the $m_{Z'}$-$m_\text{DM}$ plane for different values of $\lambda_{hs}$, taking the mass of the hidden sector Higgs to saturate the unitarity bound. The blue (dotted) region is excluded by bounds on the Higgs invisible branching ratio and the green (dashed) region is in conflict with bounds on the spin-independent DM-nucleon scattering cross section. The orange (dashed) region shows constraints from the CMS monojet search, the purple (dot-dashed) region is excluded by a combination of dijet searches from the LHC, Tevatron and UA2 (adopted from ref. [@Chala:2015ama]). In the grey parameter region unitarity constraints are in conflict with the stability of the potential, the red region corresponds to DM overproduction. Note the change of scale in these figures.[]{data-label="fig:higgsmixing2"}](HiggsPlot_01_label.pdf "fig:"){height="0.3\textheight"}![Constraints in the $m_{Z'}$-$m_\text{DM}$ plane for different values of $\lambda_{hs}$, taking the mass of the hidden sector Higgs to saturate the unitarity bound. The blue (dotted) region is excluded by bounds on the Higgs invisible branching ratio and the green (dashed) region is in conflict with bounds on the spin-independent DM-nucleon scattering cross section. The orange (dashed) region shows constraints from the CMS monojet search, the purple (dot-dashed) region is excluded by a combination of dijet searches from the LHC, Tevatron and UA2 (adopted from ref. [@Chala:2015ama]). In the grey parameter region unitarity constraints are in conflict with the stability of the potential, the red region corresponds to DM overproduction. Note the change of scale in these figures.[]{data-label="fig:higgsmixing2"}](HiggsPlot_02.pdf "fig:"){height="0.3\textheight"} The resulting constraints in the conventional $m_{Z'}$-$m_\text{DM}$ parameter plane with fixed couplings are shown in figure \[fig:higgsmixing2\]. For comparison we show the constraints from the CMS monojet search [@Khachatryan:2014rra] and a combination of dijet searches from the LHC, Tevatron and UA2 (adopted from ref. [@Chala:2015ama]). We find that the additional constraints due to Higgs mixing provide valuable complementary information in the parameter region with small $m_{Z'}$ and large $m_\text{DM}$, which is difficult to probe with monojet or dijet searches. Note that for $m_{Z'} > 2\:\text{TeV}$ (not shown in figure \[fig:higgsmixing2\]) there is still an allowed parameter region if either $m_\text{DM} \approx m_{Z'} / 2$ or $m_\text{DM} > m_{Z'}$ (cf. figure 6). Furthermore, it is worth emphasising that for smaller values of $m_s$ significantly stronger constraints are expected from Higgs mixing. Moreover, these constraints are independent of the SM couplings of the $Z'$ and will therefore become increasingly important in the case of small $g_q$. Discussion and Outlook {#sec:discussion} ====================== In this paper we have studied the so-called simplified model approach to DM used to parametrise the interactions of a DM particle with the SM via one or several new mediators. It should be clear that simplified models are considered merely as an effective description, used as a tool to combine different DM search strategies. Nevertheless, it is important that such models fulfil basic requirements, such as gauge invariance and that perturbative unitarity is guaranteed in the regions of the parameter space where the model is used to describe data. To ensure gauge invariance, one needs to impose certain relations between the different couplings, whereas it is necessary to introduce additional states in order to restore perturbative unitarity. We have illustrated these issues by considering a simplified model consisting of a fermionic DM particle and a vector mediator, which may for example be the $Z'$ gauge boson of a new $U(1)'$ gauge symmetry in the hidden sector. The phenomenology of this model depends decisively on whether the couplings of the mediator are purely vectorial or whether there are non-zero axial couplings (implying that left- and right-handed fields are charged differently under the new $U(1)'$). Since the coupling structure on the SM side may be different from the one of the DM side, there are four different cases of interest: purely vectorial couplings on both sides, non-zero axial couplings on either the SM side or the DM side, and non-zero axial couplings in both sectors. Our results can be summarized as follows: 1. \[it:VV\] [**Vector(SM)–Vector(DM):**]{} In this case no additional new physics is needed to guarantee perturbative unitarity and the mass of the $Z'$ can be generated via the Stueckelberg mechanism. This model is however highly constrained phenomenologically and a thermal DM is excluded for large parts of the parameter space due to strong limits on the spin-independent DM-nucleon scattering cross section. Generally, if at least one of the axial couplings is non-zero one needs new physics to unitarize the longitudinal component of the $Z'$. As a simple example we consider a SM-singlet Higgs breaking the dark $U(1)'$. Unitarity then requires the mass of the new Higgs to be comparable to the $Z'$ mass. Models with non-zero axial couplings are therefore expected to have a rich phenomenology with promising experimental signatures in DM direct detection experiments and invisible Higgs decays as well as additional DM annihilation channels. 2. \[it:AA\] [**Axial(SM)–Axial(DM):**]{} The crucial observation in this case is that gauge invariance of the SM Yukawa terms requires that the SM Higgs has to be charged under the $U(1)'$. This requirement has important implications for phenomenology: - Electroweak symmetry breaking leads to mass mixing between the $Z'$ and the SM $Z$-boson, which is strongly constrained by EWPT. - The axial couplings of SM fermions to the $Z'$ are necessarily flavour universal and equal for quarks and leptons. Hence, it is not possible to couple the DM particle to quarks without also inducing couplings of the $Z'$ to leptons. Since the LHC is very sensitive to dilepton resonances, the resulting bounds severely constrain the model (dominating over constraints from monojet and dijet searches). 3. \[it:AV\] [**Axial(SM)–Vector(DM):**]{} The constraints from EWPT and dilepton resonance searches are largely independent of the coupling between the $Z'$ and DM and therefore also apply in the case of purely vectorial couplings on the DM side. However, gauge invariance of the Yukawa couplings implies that it is impossible for the $Z'$ to have purely axial couplings to quarks. Consequently, as soon as there is a vectorial coupling on the DM side, one necessarily obtains a vector-vector component inducing unsuppressed spin-independent DM-nucleus scattering, which is strongly constrained by direct detection (see item \[it:VV\] above). 4. \[it:VA\] [**Vector(SM)–Axial(DM):**]{} In contrast to the couplings between the $Z'$ and quarks it is possible for the DM-$Z'$ coupling to be purely axial. Indeed, this situation arises naturally in the case that the DM particle is a Majorana fermion such that the vector current vanishes. If the couplings on the SM side are purely vectorial (i.e. left- and right-handed SM fields have the same charge), the SM Higgs is uncharged under the $U(1)'$ and consequently the constraints discussed in item \[it:AA\] do not apply. Furthermore, the tree-level direct detection cross section is velocity suppressed, leading to much weaker constraints on this particular scenario. Nevertheless, sizeable spin-independent DM-nucleus scattering can be induced at loop level. In addition, kinetic mixing between the $Z'$ and SM gauge bosons (at tree level or loop-induced) can be potentially important for EWPT and dilepton signatures. Assuming $\epsilon = 0$ at $\Lambda=10$ TeV, we find that bounds from searches for dilepton resonances due to loop-induced kinetic mixing can still be relevant and give constraints that are complementary to the ones obtained from monojet and dijet searches. All in all we find that imposing gauge invariance and conservation of perturbative unitarity has important implications for the phenomenology of DM interacting via a vector mediator and that relevant experimental signatures are not captured by considering only the interactions of the vector mediator with DM and quarks. This observation is relevant for the interpretation of various recent analyses of $Z'$-based simplified models, e.g. [@Lebedev:2014bba; @Hooper:2014fda; @Chala:2015ama; @Alves:2015dya; @Blennow:2015gta; @Heisig:2015ira]. Indeed, the $Z'$ model considered here is severely constrained by EWPT and dilepton resonance searches, either due to tree-level effects or loop-induced kinetic mixing. Moreover, the general expectation is that the mixing between the dark Higgs and the SM Higgs is sizeable and that as a result Higgs portal interactions are present in addition to the interactions mediated by the $Z'$. The weakest constraints are obtained in the case of purely vectorial couplings on the SM side and purely axial couplings on the DM side. Indeed, this is the only case where LHC monojet and dijet searches are potentially competitive with other kinds of constraints. In all cases that we have considered we find the hypothesis of thermal DM production to be under significant pressure. In large regions of the parameter space which is still allowed by experiments additional annihilation channels (beyond the $Z'$-mediated interactions) are necessary to avoid DM overabundance. A more systematic parameter scan of the model will be performed in a forthcoming publication [@upcoming]. Two final comments are in order. First, in this work we have not taken into account gauge anomalies. In general new fermions are needed to cancel the anomalous triangle diagrams, potentially leading to additional signatures and further constraints on the model. However, due to the constraints implied by gauge invariance of the SM Yukawa terms, the gluon-gluon-$Z'$ anomaly vanishes automatically, so that the new fermions need not be charged under colour, making them difficult to probe at the LHC. Second, the requirement of universality of all axial fermion charges (including leptons) follows from the gauge invariance of the SM Yukawa term. It relies on the fact that in the SM all fermion masses are generated by the same Higgs doublet. If the Higgs sector is more complicated, for example in a two-Higgs-doublet model, this condition is relaxed and it is possible to have different axial couplings to up- and down-type quarks or different axial couplings to quarks and to leptons. In any case, such extensions of the SM go significantly beyond the simplified model approach and would most likely have a number of implications for Higgs physics, EWPT and other searches for new physics. In conclusion we would like to emphasise that one of the most intriguing implications of our study is that a model with a vector mediator should generically also contain a scalar mediator, corresponding to the dark Higgs that generates the vector mass. In the limit that the mass of the vector mediator is much larger than the mass of the scalar, our $Z'$ model can also be used to study simplified models with a scalar mediator, where gauge invariance and perturbative unitarity can be similarly problematic. Indeed, our findings suggest that a strict distinction between simplified models with scalar and vector mediators is unnatural and many of the issues with these models may be best addressed in a more realistic set-up combining the two. Future direct detection experiments together with the upcoming runs of the LHC will be able to thoroughly explore the parameter space of such a realistic simplified model and test the hypothesis of DM as a thermal relic. Acknowledgements {#acknowledgements .unnumbered} ---------------- We thank Sonia El Hedri, Juan Herrero-Garcia, Matthew McCullough and Oscar Stål for helpful discussions, and Michael Duerr and Jure Zupan for valuable comments on the manuscript. FK would like to thank the Oskar Klein Center and Stockholm University for hospitality. This work is supported by the German Science Foundation (DFG) under the Collaborative Research Center (SFB) 676 Particles, Strings and the Early Universe as well as the ERC Starting Grant ‘NewAve’ (638528). Coupling structure from mixing {#ap:mixing} ============================== Gauge boson mixing {#app:Z} ------------------ In this appendix we discuss the mixing of a gauge boson $\hat{Z}'$ of a new $U(1)'$ gauge group with the SM $U(1)_Y$ gauge field $\hat B$ and the neutral component $\hat W^3$ of the $SU(2)_\mathrm{L}$ weak fields, where we use hats to denote the interaction eigenstates in the original basis. The mixing will then lead to the mass eigenstates $Z'$, $Z$ and $A$. Following the discussion in [@Frandsen:2011cg], we consider an effective Lagrangian including both kinetic mixing and mass mixing (see also [@Babu:1997st]) $$\begin{aligned} {\cal L} =& \; {\cal L}_{SM} -\frac{1}{4}\hat{X}^{\mu\nu}\hat{X}_{\mu\nu} + {\frac{1}{2}} m_{\hat Z'}^2 \hat{Z'}_\mu \hat{Z'}^\mu - {\frac{1}{2}} \sin \epsilon\, \hat{B}_{\mu\nu} \hat{X}^{\mu\nu} +\delta m^2 \hat{Z}_\mu \hat{Z'}^\mu \;, \label{eq:Lappendix}\end{aligned}$$ where $\hat{X}^{\mu \nu} \equiv \partial^\mu \hat{Z'}^\nu - \partial^\nu \hat{Z'}^\mu$. Furthermore, we have defined $\hat{Z}\equiv \hat{c}_\mathrm{W} \hat{W}^3- \hat{s}_\mathrm{W} \hat{B}$, where $\hat{s}_\mathrm{W} \, (\hat{c}_\mathrm{W})$ is the sine (cosine) of the Weinberg angle and $\hat g', \, \hat g$ are the corresponding gauge couplings. The field strengths are diagonalised and canonically normalised via the following two consecutive transformations [@Babu:1997st; @Chun:2010ve; @Frandsen:2011cg] $$\begin{aligned} \label{eq:Zpmixing} \left(\begin{array}{c} \hat B_\mu \\ \hat W_\mu^3 \\ \hat Z'_\mu \end{array}\right) & = \left(\begin{array}{ccc} 1 & 0 & -t_\epsilon \\ 0 & 1 & 0 \\ 0 & 0 & 1/c_\epsilon \end{array}\right) \left(\begin{array}{c} B_\mu \\ W_\mu^3 \\ Z'_\mu \end{array}\right) \ , \\ \left(\begin{array}{c} B_\mu \\ W_\mu^3 \\ Z'_\mu \end{array}\right) & = \left(\begin{array}{ccc} \hat c_\mathrm{W} & -\hat s_\mathrm{W} c_\xi & \hat s_\mathrm{W} s_\xi \\ \hat s_\mathrm{W} & \hat c_\mathrm{W} c_\xi & - \hat c_\mathrm{W} s_\xi \\ 0 & s_\xi & c_\xi \end{array} \right) \left(\begin{array}{c} A_\mu \\ Z_\mu \\ R_\mu \end{array}\right) \; ,\end{aligned}$$ where $$\begin{aligned} t_{2\xi}=\frac{-2c_\epsilon(\delta m^2+m_{\hat Z}^2 \hat s_\mathrm{W} s_\epsilon)} {m_{\hat Z'}^2-m_{\hat Z}^2 c_\epsilon^2 +m_{\hat Z}^2\hat s_\mathrm{W}^2 s_\epsilon^2 +2\,\delta m^2\,\hat s_\mathrm{W} s_\epsilon} \; . \label{eq:xi}\end{aligned}$$ For $\epsilon \ll 1$ and $\delta m^2 \ll m_{\hat Z}^2, m_{\hat Z'}^2$, this equation can be approximated by $$\label{eq:xi_def} \xi = \frac{\delta m^2 + m_{\hat Z}^2 \hat s_\mathrm{W} \epsilon}{m_{\hat Z}^2 - m_{\hat Z'}^2} \; .$$ The mass eigenvalues $m_Z$ and $m_{Z'}$ are given by $$\begin{aligned} m_Z^2 & = m_{\hat Z}^2 (1+{\hat s}_\mathrm{W} \, t_\xi \, t_\epsilon)+\frac{\delta m^2 \, t_\xi}{c_\epsilon} \nonumber \\ & \approx m_{\hat Z}^2 + (m_{\hat Z}^2 - m_{\hat Z'}^2) \xi^2 \; , \\ m_{Z'}^2 & = \frac{m_{\hat Z'}^2 + \delta m^2 ({\hat s}_\mathrm{W} \, s_\epsilon-c_\epsilon \, t_\xi)}{c_\epsilon^2 \, (1+{\hat s}_\mathrm{W} \, t_\xi \, t_\epsilon)} \nonumber \\ & \approx m_{\hat Z'}^2 + m_{\hat Z'}^2 \xi (\xi - {\hat s_\mathrm{W}} \epsilon) - m_{\hat Z}^2 (\xi - {\hat s_\mathrm{W}} \epsilon)^2\; .\end{aligned}$$ We define the ‘physical’ weak angle via $$\begin{aligned} s_\mathrm{W}^2 \, c_\mathrm{W}^2=\frac{\pi \, \alpha(m_{Z})}{\sqrt{2} \, G_\mathrm{F} \, m_{Z}^2} \; , \label{eq:swcw}\end{aligned}$$ where $\alpha = e^2 / (4\pi)$. Eq. (\[eq:swcw\]) also holds with the replacements $s_\mathrm{W}\to\hat s_\mathrm{W}$, $c_\mathrm{W}\to\hat c_\mathrm{W}$ and $m_{Z}\to m_{\hat Z}$, leading to the identity $s_\mathrm{W} \, c_\mathrm{W} \, m_{Z}=\hat s_\mathrm{W} \, \hat c_\mathrm{W} \, m_{\hat Z}$. This equation implies $$s_\mathrm{W}^2 = \hat s_\mathrm{W}^2 - \frac{\hat s_\mathrm{W}^2 \, \hat c_\mathrm{W}^2}{\hat c_\mathrm{W}^2 - \hat s_\mathrm{W}^2} \left(1 - \frac{m_{\hat Z'}^2}{m_{\hat Z}^2}\right) \xi^2 \; .$$ These equations allow us to fix $\hat s_\mathrm{W}$ and $m_{\hat Z}$ in such a way that we reproduce the experimentally well-measured quantities $s_\mathrm{W}$ and $m_{Z}$. The couplings of the $Z'$ to SM fermions induced via mixing can e.g. be found in [@Frandsen:2012rk]. Of particular interest to our current analysis are the couplings to leptons which are strongly constrained. In terms of the mixing parameters they can be written as $$\begin{aligned} g_{\ell}^\mathrm{V} &= \frac{1}{4}(3 {\hat g'} (\hat s_\mathrm{W} s_\xi-c_\xi t_\epsilon)- {\hat g}\hat c_\mathrm{W} s_\xi) \ , & g_{\ell}^\mathrm{A} &= -\frac{1}{4} ({\hat g'} (\hat s_\mathrm{W} s_\xi-c_\xi t_\epsilon) + {\hat g} \hat c_\mathrm{W} s_\xi) \; , \label{eq:dilepton}\end{aligned}$$ with $\hat{g}$ and $\hat{g}'$ the fundamental gauge couplings of $SU(2)_\text{L}$ and $U(1)_Y$. Scalar mixing {#app:scalar} ------------- Considering the SM Higgs $h$ plus the dark Higgs $s$, the most general scalar potential after electroweak and dark symmetry breaking can be written as $$V(s, h) = - \frac{\mu_s^2}{2} (s+w)^2 - \frac{\mu_h^2}{2} (h+v)^2 + \frac{\lambda_h}{4} (h+v)^4 + \frac{\lambda_s}{4} (s+w)^4 + \frac{\lambda_{hs}}{4} (h+v)^2(s+w)^2 \; .$$ For $\lambda_{hs} = 0$, we obtain the usual formulas $$\begin{aligned} v^2 & = \frac{\mu_h^2}{\lambda_h} \; , \quad m_h^2 = 2 \, \lambda_h \, v^2 \; ,\\ w^2 & = \frac{\mu_s^2}{\lambda_s} \; , \quad m_s^2 = 2 \, \lambda_s \, w^2 \; .\end{aligned}$$ In this case, there is no mixing between the two Higgs fields even at one-loop level. Nevertheless, there is no reason why $\lambda_{hs}$ should be negligible and therefore the two fields will in general mix. One then obtains for the minimum (assuming $4 \, \lambda_h \, \lambda_s > \lambda_{hs}^2$) $$\begin{aligned} v^2 & = 2 \frac{2 \, \lambda_s \, \mu_h^2 - \lambda_{hs} \, \mu_s^2}{4 \, \lambda_s \, \lambda_h - \lambda_{hs}^2} \; ,\\ w^2 & = 2 \frac{2 \, \lambda_h \, \mu_s^2 - \lambda_{hs} \, \mu_h^2}{4 \, \lambda_s \, \lambda_h - \lambda_{hs}^2} \; ,\;\end{aligned}$$ and for the mass squared eigenvalues $$m_{1,2}^2 = \lambda_h \, v^2 + \lambda_s w^2 \mp \sqrt{(\lambda_s w^2 - \lambda_h v^2)^2 + \lambda_{hs}^2 w^2 v^2} \; .$$ The corresponding mass eigenstates are $$\begin{aligned} H_1 = s \sin \theta + h \cos \theta \nonumber \\ H_2 = s \cos \theta - h \sin \theta \end{aligned}$$ with $$\tan 2\theta = \frac{\lambda_{hs} \, v \, w}{\lambda_h \, v^2 - \lambda_s \, w^2} \; .$$ For small $\lambda_{hs}$ we find $m_1^2 \approx 2 \, \lambda_h \, v^2 \equiv m_h^2$ and $m_2^2 \approx 2 \, \lambda_s \, w^2 \equiv m_s^2$. 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[^2]: It turns out that for certain processes the transversal part of the propagator leads to a logarithmic divergence for $m_{Z'}^2 \ll s$. This divergence is not related to the UV completeness of the theory, but signals breakdown of perturbativity in the IR, see also [@Hedri:2014mua]. By restricting to the longitudinal components of the $Z'$ [@Chanowitz:1978mv] we can avoid the occurence of those IR divergences. [^3]: Note that this process corresponds to an off-diagonal element of $\mathcal{M}_{if}$, with $i \neq f$, whereas the bounds from eq.  apply for diagonal elements. In order to apply the unitarity constraint we consider the $2\times 2$ submatrix of $\mathcal{M}_{if}$ spanned by the states $\psi \bar{\psi}$ and $Z'_L Z'_L$. For $s\to \infty$ only the off-diagonal element survives, and hence the eigenvalues of the matrix become equal to the off-diagonal element, and we can apply eq. . [^4]: Our result differs from the one in [@Shu:2007wg] by a factor $1/\sqrt{2}$. [^5]: As discussed below there can be anomalies which require additional fermions. [^6]: This will no longer be true if we allow for an explicit mass term for $\psi$. In this case the relative phase between $y_\text{DM}$ and the mass term is physical (see e.g. [@LopezHonorez:2012kv]). Here we do not allow for an explicit mass term and we assume that the vev of the singlet is the only source of $U(1)'$ symmetry breaking. [^7]: If right-handed neutrinos exist their charge $q_{\nu_R}$ would be constrained by $q_H = q_{\ell_L} - q_{\nu_R}$ to allow for a Yukawa term with the lepton doublet. In the following we assume that if right-handed neutrinos exist they are heavy enough to decouple from all relevant phenomenology. [^8]: This conclusion is in disagreement with the observations made in [@Hooper:2014fda]. [^9]: Note that since $\xi$ can be large in some regions of parameter space, it is not a good approximation to expand the annihilation cross section in $\xi$. We therefore use the exact expression for the mixing between the neutral gauge bosons in terms of $\epsilon$ and $\delta m^2$ as derived in the appendix. [^10]: To interpret the CMS results in the context of our model, we implement our model in `Feynrules v2` [@Alloul:2013bka] and simulate the monojet signal with `MadGraph v5` [@Alwall:2014hca] and `Pythia v6` [@Sjostrand:2006za]. Imposing a cut on the missing transverse energy of $\slashed{E}_T > 450\:\text{GeV}$, we exclude all parameter points that predict a contribution to the monojet cross section larger than $7.8\:\text{fb}$. We find good agreement between this procedure and an analogous implementation using `CalcHEP v3` [@Belyaev:2012qa] and `DELPHES v3` [@deFavereau:2013fsa]. [^11]: Note that as long as the mediator is produced on-shell, the production cross section is proportional to $(g^V_q)^2 + (g^A_q)^2$ and hence it is a good approximation to apply the bounds obtained for $g^V_q = 0$ and $g^A_q \neq 0$ also to the case $g^A_q = 0$ and $g^V_q \neq 0$. However, ref. [@Chala:2015ama] assumes a Dirac DM particle, while we focus on Majorana DM. As a result, the invisible branching fraction will be somewhat smaller in our case and bounds from dijet resonance searches will be strengthened. The dijet bounds we show are therefore conservative.

--- bibliography: - 'Planck\_bib.bib' - 'Zodi.bib' - 'SSO4ken/SSO\_extra.bib' title: ' 2013 results. XIV. Zodiacal emission' --- ,[–]{} Introduction {#sec:introduction} ============ This paper, one of a set associated with the 2013 release of data from the [^1] mission [@planck2011-1.1], describes the measurement of Zodiacal emission using  data. Zodiacal light, or the reflection of sunlight from small dust particles in our Solar System, can be seen by eye at dawn or dusk in dark locations, and contributes significantly to the diffuse sky brightness at optical and near-infrared wavelengths. The study of Zodiacal emission, or the thermal re-emission of absorbed energy from these interplanetary dust (IPD) particles has been enabled by the advent of infrared astronomical techniques, and it is now known to dominate the diffuse sky brightness over most of the sky between 10 and 50$\micron$ [see, for example, @Leinert1997]. Full-sky, infrared satellite surveys, in particular, have allowed us to begin to determine the structure of the density of the IPD [@Hauser84; @kelsall1998; @fixsendwek2002; @Pyo2010]. One of the full-sky models of Zodiacal emission most easily adapted for  and most commonly used at longer wavelengths is based on work from the Cosmic Background Explorer Diffuse Infrared Brightness Experiment (*COBE*/DIRBE) team [@kelsall1998 hereafter K98]. Others are presented in [@Good1986], [@Rowan-Robinson1990; @Rowan-Robinson1991], [@Jones1993], [@Vrtilek1995], [@Wright1998], and [@Rowan-Robinson2012]. The K98 model comprises the well-known diffuse cloud, three sets of dust bands first discovered by *IRAS* [@Low1984], and a circumsolar ring and Earth-trailing feature, hinted at in *IRAS* and confirmed in DIRBE [called a ‘blob’ in K98. See @Reach1995 and references therein]. @fixsendwek2002 have used data from the Far Infrared Absolute Spectrophotometer (FIRAS) to extend measurements of the diffuse cloud to longer wavelengths, but given its modest angular resolution and large uncertainties in the submillimetre region, could not say more about the smaller angular-scale Zodiacal features. ’s sensitivity allows it to detect and measure the emissivity of the diffuse Zodiacal cloud at long wavelengths, and its angular resolution also allows it to characterize the smaller-scale components of the Zodiacal. This paper continues as follows: in Sect. \[sec:planck:mission\] we describe the relevant aspects of the  mission for this analysis, along with it’s observation strategy and data processing; in Sect. \[sec:detection\] we describe how  detects Zodiacal emission; and in Sect. \[sec:model\] we briefly describe the *COBE* Zodiacal emission model. The fit to the /HFI data is described in Sect. \[sec:fit\], and the results of the fit are discussed in Sect. \[sec:discussion\]. We conclude in Sect. \[sec:conclusion\]. The  Mission {#sec:planck:mission} ============ , comprised of the High Frequency Instrument, or HFI, and the Low Frequency Instrument, or LFI, was launched in May of 2009. The mission as a whole is described in [@planck2013-p01]. This work uses only data at frequencies of 100GHz and higher. At these frequencies,  observed the entire sky in six broad frequency bands from 100 to 857GHz, with corresponding angular resolutions from roughly 97 to 46 [@planck2013-p03c]. Orbit, Scanning Strategy and Dates of Observation ------------------------------------------------- While ’s orbit and scanning strategy are described in depth in [@planck2011-1.1] and [@planck2013-p01], we give a synopsis of the elements relevant to our analysis here.  orbits around the second Sun-Earth Lagrange point, and is thus always close to the Ecliptic plane and about 1.01 AU from the Sun and 0.01 AU from the Earth. Its focal plane scans the sky once per minute, with each detector always observing on a circle approximately $85^\circ$ from its spin axis. A simplified video showing this scanning strategy can be found at the ESA website[^2]. In addition, and not visible in the video noted above, the spin axis traces the Sun-Earth vector, but with an additional “cycloid” component, so that in the Sun-Earth frame the spin axis is always $7.5^\circ$ degrees from the Sun-Earth vector and circles around it twice per year. This cycloid component results in differing total amounts of IPD in ’s line of sight for different observations of the same point on the distant celestial sphere. This is shown schematically in Fig. \[fig:Scan\]. ![ Schematic representation of the geometry of ’s measurements, which shows that it can view different amounts of Zodiacal emission while looking at the same point on the distant sky. The plane of the ecliptic is in the plane of the diagram. The Sun is in the center of the circles. The solid black line represents the orbit of the Earth and . The dashed line at the outer edge of the shaded ring represents the orbit of Jupiter, beyond which we assume there is no contribution to the Zodiacal emission from IPD. Panel (a) shows a case where the phase of the scan cycloid and the location of the observed point on the sky yield two measurements for which the lines of sight through the IPD is roughly equal, and the same Zodiacal signal is seen. Panel (b) shows a case where the phase of the scan cycloid and the location of the observed point on the sky yield different total columns of IPD along the lines of sight, and thus a different Zodiacal signal is seen in each of the two measurements. *Note that this figure is highly stylized and not to scale.* []{data-label="fig:Scan"}](Scan.pdf){width="88mm"} As nearly the entire sky is seen twice each year, the Planck team divides the observations into “surveys” of approximately six-month duration. The exact definition of the beginning and end of each survey was agreed upon within the  team. The basic characteristics are that each survey lasts about six months and covers a maximum of sky, with a minimum of overlap between the beginnings and ends of the survey. During any single one of these surveys, some pixels near the ecliptic poles are observed multiple times, as are the pixels near the ecliptic plane which are seen both at the beginning and at the end of the survey. The bulk of the sky, however, is observed only during well-defined periods, usually less than a single week. In Fig. \[fig:jd\] we show the Julian dates of observations of those pixels on the sky for which the observation times during survey 1 spanned one week or less. The analogous plot for survey 2 is similar in nature, and the corresponding maps for surveys 3 and 4 are quite similar to those of surveys 1 and 2, as the scanning strategy for surveys 3 and 4 were almost identical to those of surveys 1 and 2, respectively. ![The Julian date of observation of pixels on the sky during survey 1, for a single detector, in Galactic coordinates. There are only very small differences between maps for different detectors. The grid lines show ecliptic coordinates, with the darker lines representing the ecliptic plane and the line of zero ecliptic longitude. Undefined pixels, which were either not observed at all, or which were observed multiple times over a period that spanned more than one week and are thus not used in this analysis, are shown as the uniform gray band.[]{data-label="fig:jd"}](jd_h857-1_s1_d07_v53.jpg){width="88mm"} Data Processing --------------- The overall HFI data processing is described in [@planck2011-1.7] and [@planck2013-p03]. Given the time-dependent nature of the Zodiacal signal seen with the  scanning strategy, this analysis is done using the individual survey 1-4 maps. This allows us to exclude from the analysis regions of the sky and periods of time where the column of IPD viewed by  is not constant. The HFI instrument has a number of horns at each measurement frequency [@planck2013-p03c Fig. 9]. Working with individual horn maps, rather than the co-added frequency maps, allows us to adjust the response of each detector so that they are uniform for a source with a Zodiacal spectrum, rather than that used for a CMB spectrum, as is done in the standard processing [@planck2013-p03d]. At 100, 143, 217, and 353GHz, some of these horns contain two polarization sensitive bolometers [PSBs; @Jones2003]. As we are not addressing polarisation here, for those horns that have PSBs, we combine the maps from each of the two detectors within the given horn with a simple average. As the evaluation of the model to be presented in Sect. \[sec:model\] involves calculating emission from a number of points along each line of sight and summing them, the computations are time consuming. To mitigate this to some extent, we use $13\parcm7\times13\parcm7$ pixels, rather than the original $1\parcm7\times1\parcm7$ HFI pixels, reducing the number of map pixels from 50 million to a bit less than 800 thousand [that is, we use HEALPix pixels with $N_\mathrm{side}$ of 256 rather than 2048; @gorski2005]. While this does reduce our sensitivity to finer scale structures, it is not a serious hindrance, since we will be making comparisons with DIRBE, which had a still larger beam. Smaller pixels will be used in future work as more detail is teased out of the data. Pre-launch estimates of ’s ability to detect Zodiacal emission and an estimate of the possible level of contamination at the highest  frequencies were presented in [@maris2006]. More recent predictions have addressed the possibility of Zodiacal contamination at lower frequencies [@diego2010] and speculated that emission from dust in the outer Solar System might contribute to large-scale anomalies which have been reported in data from the Wilkinson Microwave Anisotropy Probe (*WMAP*) at large angular scales [@maris2011; @Hansen2012]. Detection {#sec:detection} ========= The existence of the Zodiacal emission in the  maps is straightforward to demonstrate by exploiting the fact, noted above, that different  surveys often sample different columns of IPD while observing the same location on the distant celestial sphere. Fig. \[fig:surveymaps\] shows the first (top) and second (middle) survey maps for the 857-1 detector, along with the difference of these two maps (bottom). Similar differences for all HFI frequencies, using data from all horns at each frequency, are shown in the left-hand panel of Fig. \[fig:beforeAndAfterJackknives\]. There are at least three features that stand out in the map difference: (1) The scale has been reduced immensely, but the Galactic plane is still visible in this difference map; (2) the “arcs” at the top and the bottom of the difference map are the images of the Galactic centre as seen through the instrument’s far sidelobes (FSLs); and (3) the Zodiacal emission can be seen as the variations following the ecliptic plane. We begin with the Zodiacal emission. ![Single-survey maps in Galactic coordinates for the 857-1 detector. *Top:* Survey 1 map. *Middle:* Survey 2 map. *Bottom:* Survey 2 minus survey 1 difference map. This bottom image shows the Zodiacal emission and the residual Galactic emission effects discussed in the text. The units are MJy/sr, assuming a spectrum inversely proportional to frequency. Undefined pixels are shown in gray. These occur in pixels which either have not been observed during the survey, were observed during the passage of a planet, or a small number of other events. In the top two plots, pixels which were observed over periods longer than a week were not masked, and thus the masked regions are smaller in the top two images than that in the difference (bottom) panel.[]{data-label="fig:surveymaps"}](857-1_W_TauDeconv_survey1_I.jpg "fig:")\ ![Single-survey maps in Galactic coordinates for the 857-1 detector. *Top:* Survey 1 map. *Middle:* Survey 2 map. *Bottom:* Survey 2 minus survey 1 difference map. This bottom image shows the Zodiacal emission and the residual Galactic emission effects discussed in the text. The units are MJy/sr, assuming a spectrum inversely proportional to frequency. Undefined pixels are shown in gray. These occur in pixels which either have not been observed during the survey, were observed during the passage of a planet, or a small number of other events. In the top two plots, pixels which were observed over periods longer than a week were not masked, and thus the masked regions are smaller in the top two images than that in the difference (bottom) panel.[]{data-label="fig:surveymaps"}](857-1_W_TauDeconv_survey2_I.jpg "fig:")\ ![Single-survey maps in Galactic coordinates for the 857-1 detector. *Top:* Survey 1 map. *Middle:* Survey 2 map. *Bottom:* Survey 2 minus survey 1 difference map. This bottom image shows the Zodiacal emission and the residual Galactic emission effects discussed in the text. The units are MJy/sr, assuming a spectrum inversely proportional to frequency. Undefined pixels are shown in gray. These occur in pixels which either have not been observed during the survey, were observed during the passage of a planet, or a small number of other events. In the top two plots, pixels which were observed over periods longer than a week were not masked, and thus the masked regions are smaller in the top two images than that in the difference (bottom) panel.[]{data-label="fig:surveymaps"}](survey_jack_857-1_year1.jpg "fig:") Since these are difference maps, the Zodiacal emission is seen in the ecliptic plane as both positive and as negative, depending upon the relative geometry of the IPD and the  satellite when a given location was observed in the two surveys. While the amplitude of the Zodiacal emission signal is reduced in this differential process [this will be described in Sect. \[sec:model\], and in particular Fig. \[fig:zodiMaps\]; see also @maris2006], it has the advantage that the bulk of the Galactic and extra-galactic signal, the main contaminants in the analysis at high frequencies, is efficiently removed with little effort. What remains of the Galactic signal arises from effects such as beam asymmetries and imperfections in the transfer function removal [@planck2013-p03c]. To mitigate these residual Galactic effects, we generally do not use data within 10 degrees of the Galactic plane in this analysis. We have confirmed that changing this cut to 5 or to 20 degrees does not change the conclusions of this analysis, and emphasize that contrary to other analyses, the fact that we are analysing difference maps makes this work less sensitive to Galactic contamination. It should be noted that the Zodiacal emission is much dimmer than many other background components in the  data. Whereas the Zodiacal emission dominates the sky in some *IRAS* and *COBE* bands, this is never the case for , with the cosmic infrared background and Galactic dust dominating at high frequencies, and the CMB itself dominating at lower frequencies. This makes our differencing scheme appealing, but also restricts the analysis in some ways. It is, for example, difficult to look at individual scans, or slices of the sky, as has been done successfully with *IRAS*. We are obligated to use almost the entire sky, and use a model of the Zodiacal emission to interpret variations, rather than being able to directly interpret the total Zodiacal emission on the sky. Model {#sec:model} ===== The goal of this section is to describe the Zodiacal and far sidelobe templates we will create to fit to the map at the bottom of Fig. \[fig:surveymaps\]. Zodiacal Components ------------------- The *COBE*/DIRBE Zodiacal emission model is described in depth in K98, but we review the salient parts here. ### Diffuse cloud {#sec:cloudmath} The density of the diffuse IPD cloud, having both radial and vertical dependence, is taken to be of the form $$n_c\left(\mathbf{R}\right) = n_0 R_c^{-\alpha} \left\{ \begin{array}{cl} e^{-\beta\left(\zeta^2/2\mu\right)^\gamma} & \mathrm{if}\ \zeta < \mu \\ e^{-\beta\left(\zeta -\mu/2\right)^\gamma} & \mathrm{if}\ \zeta\geq\mu \\ \end{array} \right.,$$ where $$R_c = \sqrt{\left(x-x_0\right)^2 + \left(y-y_0\right)^2 + \left(z-z_0\right)^2},$$ $$\zeta = \left|Z_c\right|/R_c,$$ $$\begin{aligned} Z_c & = & \left(x-x_0\right) \cdot\sin\left(\Omega_R\right) \cdot\sin\left(i_R\right) \nonumber \\ & - & \left(y-y_0\right) \cdot\cos\left(\Omega_R\right) \cdot\sin\left(i_R\right) + \left(z-z_0\right)\cdot\cos(i_R), \end{aligned}$$ and $\alpha$, $\beta$, $\gamma$, $\mu$, $x_0$, $y_0$, $z_0$, $\Omega_R$, and $i_R$ are parameters describing the location and shape of the cloud. This form (and others used elsewhere which are similar) is based on an approximation of a model of particles orbiting the Sun, accounting for drag from the Poynting-Robertson effect, and with a modified fan distribution used to describe the changes in density above and below the plane of the ecliptic. See K98 for more details and references. The numerical values for the parameters can be found in K98, or from the LAMBDA website[^3]. This is shown, for both survey 1 and survey 2, as well as their difference, assuming an emissivity of one, in the bottom row of Fig. \[fig:zodiMaps\]. $\begin{array}{cc|c} \includegraphics[width=60mm]{Z857JackSim_survey1_band3.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_band3.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_band3.jpg} \\ \includegraphics[width=60mm]{Z857JackSim_survey1_band2.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_band2.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_band2.jpg} \\ \includegraphics[width=60mm]{Z857JackSim_survey1_band1.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_band1.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_band1.jpg} \\ \includegraphics[width=60mm]{Z857JackSim_survey1_cring.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_cring.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_cring.jpg} \\ \includegraphics[width=60mm]{Z857JackSim_survey1_tblob.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_tblob.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_tblob.jpg} \\ \hline \includegraphics[width=60mm]{Z857JackSim_survey1_cloud.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_cloud.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_cloud.jpg} \end{array}$ ### Dust bands {#sec:bandmath} The Zodiacal dust bands were first seen by *IRAS* [@Low1984], and appear as pairs of bright, parallel bands equally spaced above and below the ecliptic plane. They were quickly associated with asteroid families, and then understood to be the relics of asteroid collisions or collapses [@Dermott1984; @sykes86]. [@Reach1997] study them in detail. The K98 model contains three bands[^4] called, reasonably enough, bands 1, 2 and 3. They appear at ecliptic latitudes around $\pm 1\pdeg4$, $\pm 10\deg$, and $\pm 15\deg$. *IRAS*, having higher angular resolution than DIRBE, was able find more bands, dubbed $\alpha$, $\beta$, $\gamma$, E/F, G/H, J/K, and M/N [@Sykes1990], though some are more firmly detected than others. The K98 band 1, or *IRAS* band $\gamma$, was originally associated with the Eos family of asteroids [@Dermott1984], but [@Grogan2001] called this into question and @Nesvorny2003 [hereafter N03] found better correspondence with the Veritas family of asteroids. K98 note that their band 2 is a blend of *IRAS* bands $\alpha$ and $\beta$ [@Sykes1990]. [@sykes86] tentatively associated the $\alpha$ band with the Themis family of asteroids, and @Nesvorny2008 [hereafter N08] has narrowed this association to a cluster within this family associated with the Beagle asteroid. The $\beta$ band was associated by [@sykes86] with the Koronis family of asteroids, and N03 narrowed this to the Karin cluster within the Koronis family. Anticipating the discussion in Sect. \[subsec:Bands\], we note also that the $\beta$ band appears brighter than the $\alpha$ band [@Sykes1988; @Reach1997; @Nesvorny2008]. K98 states that their band 3, that furthest from the ecliptic plane, has been associated with both the Io and Maria families of asteroids [@Sykes1988; @Reach1997], corresponding to *IRAS* bands J/K and M/N. N03 have more recently noted, however, that 4652 Iannini and/or 845 Naema may be better asteroid associations for the J/K band-pair and that 1521 Seinajoki may work better for the M/N pair. We summarize these associations in table \[tab:bandInfo\]. K98 $\delta_\zeta$ *IRAS* $i_p$ $i_p$ Type Age a ----- ---------------- ---------- ------- ------ ----------- ------- ------------- -------------- ------- 1 8.78 $\gamma$ 9.35 490 Veritas 9.26 C / Ch/... 8.3 3.169 $\alpha$ 1.34 656 Beagle 1.34 .../.../ C $\lesssim$10 3.157 $\beta$ 2.11 832 Karin 2.11 .../.../ S 5.8 2.866 845 Naema 11.96 .../ C /... $\gtrsim$20 ... 4562 Iannini 12.17 .../.../ S $\lesssim$5 2.644 M/N 15.0 1521 Seinajoki 15.02 .../.../... ... 2.852 For each of the three dust bands in the *COBE* model, the density is given by $$n_{B}\left(\mathbf{R}\right) = \frac{3N_0}{R} e^{-\left(\zeta/\delta_\zeta\right)^6} \left(1 + \frac{(\zeta/\delta_\zeta)^{p}}{v_B}\right) (1-e^{-(R/\delta_R)^{20}}), \label{eq:bandDensity}$$ where we have used a simplified notation based on that of K98, where one can also find the numerical values for the parameters. Note that Eqn. \[eq:bandDensity\] matches the code used for the Zodiacal model (which can be found on the LAMBDA website), but that there is a factor of $1/v_B$ difference between Eqn. \[eq:bandDensity\] and Eqn. 8 of K98 ($v_B$ is a shape parameter, not a frequency). Also, K98 assumed that the emissivities of the three sets of bands were all equal. We relax this assumption below and allow the emissivities of each of the sets of bands to be different. The bands are shown, assuming unit emissivity, as the first, second, and third rows in Fig. \[fig:zodiMaps\]. ### Circumsolar ring and Earth-trailing feature {#sec:ringmath} The functional form for the density of the circumsolar ring is taken to be $$n_{B}\left(\mathbf{R}\right) = n_{SR}\cdot e^{-\left(R-R_{SR}\right)^2/\sigma_{rSR}^2-\left|Z_R\right|/\sigma_{zSR}}.$$ Similar to the treatment of the bands, K98 assumed that the emissivity of the circumsolar ring was the same as that of the Earth-trailing feature, below. We also relax this assumption, and allow them to be different. The shape of the expected signal from the circumsolar ring is shown in the fourth row of Fig. \[fig:zodiMaps\]. The density of the Earth-trailing feature is given by $$n_{B}\left(\mathbf{R}\right) = n_{TB}\cdot e^{-\left(R-R_{TB}\right)^2/\sigma_{rTB}^2-\left|Z_R\right|/\sigma_{zTB} -\left(\theta-\theta_{TB}\right)^2/\sigma_{\theta TB}^2}.$$ We note that for both the circumsolar ring and the Earth-trailing feature, there is a typo in the text of K98 – a factor of 2 in the denominator of the first and third terms in the exponential has been added in the text, compared to what is in the code. We follow the code. The expected signal from the Earth-trailing feature is shown in the fifth row of Fig. \[fig:zodiMaps\]. ### Integrated Emission The total Zodiacal emission is calculated as $$I_x\left(\nu\right) = \epsilon_x\int d\mathbf{R} \cdot n_x\left(\mathbf{R}\right) \cdot B\left(\nu, T\left(\mathbf{R}\right)\right),$$ where $x$ is the Zodiacal component, $\nu$ is the frequency, $\mathbf{R}$ is a location in the Solar System, and $B\left(\nu, T\left(\mathbf{R}\right)\right)$ is the Planck function for the given frequency and temperature at the given location, given by $T_0/R^\delta$, with $T_0$ and $\delta$ being parameters. $\epsilon$ is the emissivity for the given component, which we will be finding with our fit. $n_x$ is the density for the given component, described above. The integral is done along the line of sight, from the location of the satellite to 5.2AU. Galactic Emission Seen Through Sidelobes {#subsec:galFSL} ---------------------------------------- Like all real telescopes,  has some small sensitivity to radiation arriving at the telescope from off-axis, often called spillover or ‘far sidelobes’. Note that here we are referring to light coming from more than five degrees from the instrument’s nominal pointing direction, which should not be confused with ‘near sidelobes’, discussed in the HFI transfer function and beams papers [@planck2013-p03c see also [@tauber2010b]]. Figure \[fig:sidelobeDiagram\] shows the main stray light routes. ![Origin of far sidelobes. The ‘SR Spillover’ (for ‘Secondary Reflector Spillover’; the lowest set of rays on the left of the figure) arrives at the focal plane from outside the secondary mirror, directly from the sky. The ‘PR Spillover’ (for ‘Primary Reflector Spillover’), arrives at the focal plane from above the primary mirror and reflects off of the secondary to arrive at the focal plane. The set of rays between these two contributions represents the main beam. The ‘baffle’ contribution, light reaching the focal plane after reflecting from the inner sides of the baffles, is not shown here. It is often included as part of the SR Spillover. Adopted from [@tauber2010b]. []{data-label="fig:sidelobeDiagram"}](Rays-sketch-2.pdf){width="88mm"} The secondary reflector (SR) spillover arises from radiation that reaches the focal plane without reflecting off of the primary reflector. As such, a major component is radiation from the general direction of the telescope boresight, though well outside the main beam. The ‘baffle’ contribution to the SR spillover results from radiation which reflects off of the baffles to arrive at the focal plane. The primary reflector (PR) spillover arises from radiation that comes to the satellite from just above the primary mirror, reflects off the secondary mirror and arrives at the detectors. At the highest  frequencies, the Galactic centre is bright enough to be seen through these far sidelobes, even if faintly. Since the orientations of these sidelobes change as the instrument scans, the survey differences done to detect the Zodiacal emission are also sensitive tests of the FSLs. Though the Galactic emission mechanism and amplitude is different, analogous effects are discussed for the LFI in [@planck2013-p02d]. To this point, resource constraints have limited this study to a single far sidelobe calculation for all detectors. We use a GRASP calculation of the far sidelobes for the 353-1 horn [see Fig. 9 of @planck2013-p03c] and do not attempt to correct for differences in frequency or location for other horns. While this is not optimal, and will be improved in later releases, we note that the primary, large-scale features of the far sidelobes are defined by the telescope, rather that the horns or their placement, so first-order effects should be captured. Some of the limitations imposed by this are discussed in Sect. \[sec:farSidelobeResults\]. To make templates of what we might see from the Galactic centre through the far sidelobes, we use the  simulation software described in [@reinecke2006], but use GRASP calculations of the the various far sidelobe components instead of main beams. As inputs to the simulations, we use the actual  maps at the appropriate frequency. This allows us to account for differences in the brightness of the Galaxy as a function of frequency with minimal effort – the Galactic templates should already be calibrated in the correct units at the given frequency. The far sidelobe templates are made at the timeline level and run through the relevant parts of the pipeline software. In particular, the offset removal, or ‘destriping’ must be done on the templates before fitting in order to get reasonable fit results. The resulting templates made using these FSL calculations with a  857GHz sky as input are shown in the bottom three rows of Fig. \[fig:FSLs\]. $\begin{array}{cc|c} \includegraphics[width=60mm]{Z857JackSim_survey1_Dipolar_direct.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_Dipolar_direct.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_Dipolar_direct.jpg} \\ \includegraphics[width=60mm]{Z857JackSim_survey1_Dipolar_secondary.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_Dipolar_secondary.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_Dipolar_secondary.jpg} \\ \includegraphics[width=60mm]{Z857JackSim_survey1_Dipolar_baffle.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_Dipolar_baffle.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_Dipolar_baffle.jpg} \\ \hline \includegraphics[width=60mm]{Z857JackSim_survey1_Galactic_direct.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_Galactic_direct.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_Galactic_direct.jpg} \\ \includegraphics[width=60mm]{Z857JackSim_survey1_Galactic_secondary.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_Galactic_secondary.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_Galactic_secondary.jpg} \\ \includegraphics[width=60mm]{Z857JackSim_survey1_Galactic_baffle.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_Galactic_baffle.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_Galactic_baffle.jpg} \end{array}$ One factor for which we do not account with these templates is the difference in spillover between the different frequencies [@Lamarre2010; @tauber2010b]. Since we illuminate more of the telescope at lower frequencies than at higher frequencies, if our templates simply scale with the spillover, we will expect different fit values for our templates at different frequencies. To illustrate the relative contributions of these various templates, in Fig. \[fig:evolution\] we show a series of maps. Maps in each row are similar to those in the previous row, except that one more template or group of templates has been added to form the new row. For all rows, the first column corresponds to data from the first   survey, the second column corresponds to data from the second  survey, and the third column is the difference of the second column minus the first. The first row shows the sum of all the far sidelobes; the second row shows the result when we add dust band 1 to the far sidelobes – note that the scales change from the first to the second row; the third row shows the sum of the far sidelobes and the first two bands; the fourth row shows the sum of the far sidelobes and all the dust bands; the fifth row shows what’s in the fourth row, plus the circumsolar ring and Earth-trailing feature; finally, the bottom row shows the sum of the far sidelobes and all Zodiacal templates (and the scales have again changed). As this is simply illustrative, we have assumed unit emissivities for the Zodiacal components, and multiplied the far sidelobe components by a factor of 15 (which we will see in Sect. \[sec:fit\] is representative of the most extreme, multi-moded case). With these caveats in mind, the survey difference of the sums of all the components, the lower right image, can be compared to the bottom image in Fig. \[fig:surveymaps\]. $\begin{array}{cc|c} \includegraphics[width=60mm]{Z857JackSim_survey1_5.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_5.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_5.jpg} \\ \hline \includegraphics[width=60mm]{Z857JackSim_survey1_6.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_6.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_6.jpg} \\ \includegraphics[width=60mm]{Z857JackSim_survey1_7.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_7.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_7.jpg} \\ \includegraphics[width=60mm]{Z857JackSim_survey1_8.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_8.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_8.jpg} \\ \includegraphics[width=60mm]{Z857JackSim_survey1_10.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_10.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_10.jpg} \\ \hline \includegraphics[width=60mm]{Z857JackSim_survey1_11.jpg} & \includegraphics[width=60mm]{Z857JackSim_survey2_11.jpg} & \includegraphics[width=60mm]{Z857JackSim_surveyD_11.jpg} \end{array}$ Spectrum {#sec:fit} ======== We fit the data shown in the bottom panel of Fig. \[fig:surveymaps\], as well as the analogous data at other frequencies and for the second year of observations, to a constant plus combinations of the templates shown in Sect. \[sec:model\], Figs. \[fig:zodiMaps\] and \[fig:FSLs\]. For each fit, we assume that the survey difference map at sky pixel $p$, called $D_{p}$, can be modelled as $$D_{p} = \sum_t \left( \epsilon_{2,t}\cdot T_{2,t,p}-\epsilon_{1,t}\cdot T_{1,t,p} \right) + \mathrm{constant},$$ where $\epsilon_{\left[1|2\right],t}$ is the emissivity fit for template $t$ during survey 1 or 2 at the given frequency and $T_{\left[1|2\right],t,p}$ is the value of the $t^\mathrm{th}$ template at pixel $p$ for survey 1 or 2, calculated as described in Sect. \[sec:model\]. For example, for our ‘basic’ fit, we will have 19 templates – one each for the diffuse cloud, circumsolar ring, Earth-trailing feature and each of the three dust bands, as well as one each for the Galactic far sidelobes. All of these are repeated twice; once for each survey in a yearly difference map. Finally we also fit to an overall constant, to which  is not sensitive. We then minimize $$\chi^2 = \sum_p \left(\Delta_{p} - D_{p}\right)^2.$$ Separating the templates into surveys has the disadvantage of increasing the number of parameters in our fits. While we do not expect either the emissivity of the Zodiacal emission or the far sidelobe calculations to change from survey-to-survey, we separate them in this way for two reasons. The first is simply as a basic reality check – if we see significant differences between fits to two different surveys, we should be sceptical. Beam asymmetries or transfer function effects, for example, might cause differences from survey-to-survey, as might imperfections in the model itself. The second reason is to calculate the error bars. As just noted, we are often as concerned by systemic effects as much as by “random” noise. By separating the data by survey, we may calculate error bars using the standard error of the successive measurements as a proxy for the uncertainties, rather than propagating white noise estimates. This should allow us a more conservative estimate of our uncertainties, which accounts for model deficiencies or low levels of systematics which change by survey (the aforementioned beam asymmetries and transfer functions, for example). A note about weighting: the bulk of the Zodiacal emission is, of course, in the ecliptic plane. , on the other hand, has more statistical weight, on a pixel-by-pixel basis, at the ecliptic poles. We therefore use uniform weights over pixels, rather than statistical weights, since this would down-weight specifically the regions with our signal. As mentioned above, we fit each of the two years’ survey difference maps to a cloud, circumsolar ring, Earth-trailing feature, three bands and three far sidelobe templates, plus a constant. The results for the four 857 GHz horns are shown in Fig. \[fig:fit857\]. Averaging over horns and surveys at all six HFI frequencies yields Fig. \[fig:noDipoleSidelobesFit\]. Numerical values are given in tables \[tab:fullFit\], \[tab:fullFitBands\] and \[tab:galacticSidelobeFit\]. {width="180mm"}  $\nu$ (GHz) ------------- -------------------- -------------------- -------------------- 857 0.301 $\pm$ 0.008 0.578 $\pm$ 0.359 0.423 $\pm$ 0.114 545 0.223 $\pm$ 0.007 0.591 $\pm$ 0.203 -0.182 $\pm$ 0.061 353 0.168 $\pm$ 0.005 -0.211 $\pm$ 0.085 0.676 $\pm$ 0.149 217 0.031 $\pm$ 0.004 -0.185 $\pm$ 0.143 0.243 $\pm$ 0.139 143 -0.014 $\pm$ 0.010 -0.252 $\pm$ 0.314 -0.002 $\pm$ 0.180 100 0.003 $\pm$ 0.022 0.163 $\pm$ 0.784 0.252 $\pm$ 0.455 : Emissivities of the diffuse cloud, circumsolar ring, and Earth-trailing feature from fit result averages.[]{data-label="tab:fullFit"} $\nu$ (GHz) Band 1 Band 2 Band 3 ------------- ------------------- ------------------- ------------------- 857 1.777 $\pm$ 0.066 0.716 $\pm$ 0.049 2.870 $\pm$ 0.137 545 2.235 $\pm$ 0.059 0.718 $\pm$ 0.041 3.193 $\pm$ 0.097 353 2.035 $\pm$ 0.053 0.436 $\pm$ 0.041 2.400 $\pm$ 0.100 217 2.024 $\pm$ 0.072 0.338 $\pm$ 0.047 2.507 $\pm$ 0.109 143 1.463 $\pm$ 0.103 0.530 $\pm$ 0.073 1.794 $\pm$ 0.184 100 1.129 $\pm$ 0.154 0.674 $\pm$ 0.197 1.106 $\pm$ 0.413 : Emissivities of the three dust bands from fit result averages.[]{data-label="tab:fullFitBands"} ------- ----------------- ----- ----------------- ---------------- ------- $\nu$ (GHz) Direct Baffle 100 -25.8 $\pm$ 5.7 7 26.3 $\pm$ 15.7 56.7 $\pm$ 5.7 10 143 -9.1 $\pm$ 4.1 6 13.0 $\pm$ 4.8 23.8 $\pm$ 5.4 10 217 0.6 $\pm$ 1.1 5 6.3 $\pm$ 2.1 6.5 $\pm$ 1.3 6 353 -1.2 $\pm$ 0.5 1 -4.3 $\pm$ 2.1 3.6 $\pm$ 0.7 1 545 7.7 $\pm$ 1.7 15 8.8 $\pm$ 3.1 7.9 $\pm$ 1.0 1 857 17.1 $\pm$ 3.4 1.5 23.9 $\pm$ 4.2 16.7 $\pm$ 3.1 0.005 ------- ----------------- ----- ----------------- ---------------- ------- : Fit coefficients for the Galaxy seen through the far sidelobes.[]{data-label="tab:galacticSidelobeFit"} $\begin{array}{cc|c} \includegraphics[width=60mm]{Z857SurveyJackknifeWiZodi.jpg} & \includegraphics[width=60mm]{Z857SurveyJackknifeNoZodi.jpg} & \includegraphics[width=60mm]{Z857Correction.jpg} \\ \includegraphics[width=60mm]{Z545SurveyJackknifeWiZodi.jpg} & \includegraphics[width=60mm]{Z545SurveyJackknifeNoZodi.jpg} & \includegraphics[width=60mm]{Z545Correction.jpg} \\ \hline \includegraphics[width=60mm]{Z353SurveyJackknifeWiZodi.jpg} & \includegraphics[width=60mm]{Z353SurveyJackknifeNoZodi.jpg} & \includegraphics[width=60mm]{Z353Correction.jpg} \\ \includegraphics[width=60mm]{Z217SurveyJackknifeWiZodi.jpg} & \includegraphics[width=60mm]{Z217SurveyJackknifeNoZodi.jpg} & \includegraphics[width=60mm]{Z217Correction.jpg} \\ \includegraphics[width=60mm]{Z143SurveyJackknifeWiZodi.jpg} & \includegraphics[width=60mm]{Z143SurveyJackknifeNoZodi.jpg} & \includegraphics[width=60mm]{Z143Correction.jpg} \\ \includegraphics[width=60mm]{Z100SurveyJackknifeWiZodi.jpg} & \includegraphics[width=60mm]{Z100SurveyJackknifeNoZodi.jpg} & \includegraphics[width=60mm]{Z100Correction.jpg} \end{array}$ Discussion {#sec:discussion} ========== As discussed in the HFI processing paper [@planck2013-p03] and in the  explanatory supplement [@planck2013-p28], these fits are used to create the implied Zodiacal and sidelobe emission in each HFI observation, which can then be removed and the maps are recreated. The survey 2 minus survey 1 difference maps for the 857-1 horn both before and after Zodiacal and far sidelobe removal, are shown in Fig. \[fig:beforeAndAfterJackknives\]. In this section we discuss the fit implications further. Far Sidelobes {#sec:farSidelobeResults} ------------- In addition to fit values obtained for the Galaxy seen in each component of the far sidelobes, the Cols. labelled ‘Pred.’ in table \[tab:galacticSidelobeFit\] show the expected values of the spillover, normalized to that of the 353GHz channel, from [@tauber2010b]. These are the ratios of the expected spillover in each frequency, compared to that at 353GHz, the frequency for which the sidelobe calculations were done. Since the fit values account for the changes in Galactic emission with frequency, if our predictions and data were perfect, the fit values would match those of the predictions. The FSL signature is clearly visible at 857GHz in the bottom panel of Fig. \[fig:surveymaps\], and this is reinforced in table \[tab:galacticSidelobeFit\]. As the 857 and 545GHz channels are multi-moded, the differences are not that surprising; it is difficult to do the calculations necessary for the prediction. In addition, the specifications for the horn fabrication were quite demanding, and small variations, though still within the mechanical tolerances, could give large variations in the amount of spillover. For the lower frequency, single-moded channels, however, the situation is different. There is no clear detection of PR spillover. While the significant negative values may indicate some low-level, large-scale systematic, there seems to be nothing with the distinctive signature of primary spillover at frequencies between 100 and 353GHz. For the direct contribution of the secondary SR spillover, the situation is similar at 353GHz, but at 217 and 143GHz we are finding a 3$\sigma$ detection at about the level expected, while at 100GHz the value is about 2.5 times higher than expected, though the signal-to-noise on the detection is less than 2$\sigma$. The baffle contribution to the SR spillover seems to be in accord with expectations at 353 and 217GHz, and higher than what is predicted at 100GHz. The values for the PR spillover, which is the most distinctive of the far sidelobe patterns, and presumably the easiest to disentangle from other effects, would indicate that the PR spillover values in table 2 of [@tauber2010b] may be slightly overestimated. The values for the direct contribution of the SR spillover roughly confirm the far sidelobe calculations. The baffle contribution to the SR spillover seems a bit high. We take the ensemble of these numbers as rough confirmation that our beam calculations are not drastically incorrect, but do not use the specific numbers in either [@planck2013-p03c] or [@planck2013-p03b]. Similar conclusions are drawn for the LFI in [@planck2013-p02d]. We have also included a template of the Dipole as seen through the far sidelobes in the fits in some fits to check to see if they are detected. As expected, they are not. The results quoted above are from fits which do not include these Dipole templates. Diffuse cloud {#diffuse-cloud} ------------- Figure \[fig:noDipoleSidelobesFit\] shows the emissivity of the diffuse cloud falling off with increasing wavelength, as would be expected for particles with characteristic sizes of order 30. The dashed line in this figure shows a flat emissivity to 150, with values proportional to the frequency-squared at longer wavelengths, for comparison with Fig. 2 of [@fixsendwek2002], who used FIRAS data to investigate the far-infrared/sub-millimetre behaviour of the Zodiacal cloud. These results are consistent with their conclusions[^5]. As the diffuse cloud is so much brighter in the mid-infrared than in the CMB, its relatively low level at  wavelengths has been exploited in [@planck2013-p03d] to set in-flight limits on any possible out-of-band leaks in the instrument’s spectral transmission. Circumsolar ring and Earth-trailing feature {#circumsolar-ring-and-earth-trailing-feature} ------------------------------------------- We draw no conclusions about the circumsolar ring or the Earth-trailing feature. The fit values obtained for their emissivities are inconsistent from frequency-to-frequency, their values often being negative. This remains true for  data when the fits are done requiring that the two components have the same emissivities, as was done in K98. Inspection of the panels labelled ‘Ring’ and ’Feat.’ (i.e., the middle- and lower-left panels) of Fig. \[fig:fit857\] show systematic differences between results from even- and odd-numbered surveys (that is, the circles and squares seem to be systematically different, regardless of whether they are blue or red). ’s observing pattern was different for odd and even numbered surveys, but very similar for either even numbered surveys alone, or for odd numbered surveys alone. Random noise is not an issue, since the measurements with similar observations are repeatable, so this is either an indication that the circumsolar ring and Earth-trailing model templates themselves need improvement at  wavelengths, or that systematic errors are affecting these specific components. We note that the conclusions presented elsewhere in this work remain essentially the same whether or not we include the circumsolar ring and Earth-trailing feature. Bands {#subsec:Bands} ----- An interesting feature of Fig. \[fig:noDipoleSidelobesFit\], and the primary result of this work, is the difference in the emissivities of the bands compared to that of the diffuse cloud, pointing towards differences in the particles in these different components. While there may be hints of this in the longest wavelength DIRBE data, the effect becomes clear at  wavelengths. This is not necessarily unexpected. The composition of the diffuse cloud is still disputed, but is often claimed to be both asteroidal and cometary [see, for example, @Kortenkamp1998; @Nesvorny2010; @Tsumura2010]. Since the bands, on the other hand, are understood to be asteroidal debris only [@sykes86], the difference may simply be a reflection of these different origins. The fact that the fitted emissivities of bands 1 and 3 rise above unity is perplexing. At first glance, one might imagine some new, cold component in the cloud which would cause an excess that might be interpreted as an excess in emissivity in some other component. However, to peak around 545GHz, this component would have to have a temperature of the order of 10K, and therefore be much more distant than most of the dust usually associated with the Zodiacal cloud. While enticing, it is difficult to understand how such a component could survive the differencing process used in this analysis, which reduces signals from distant sources more than those nearby, or how such a component could mimic an excess in two dust bands above from the ecliptic plane, but not do the same in the other Zodiacal cloud components. One might worry that covariance between between the various components might be causing problems in the fitting procedure. To check this, we have repeated the fit including and omitting various combinations of the the circumsolar ring and Earth-trailing feature, or both, and assuming their emissivities were independent or equal. In no case did the difference between the diffuse cloud and the dust bands disappear. The excess may ultimately be explained by degeneracies in the model for the density of the bands. As presented in Sect. \[sec:bandmath\], the normalisation of the density of particles is completely degenerate with the emissivity for each band. In addition, the emission is also roughly proportional to the temperature normalisation, because we are observing in the Rayleigh-Jeans tail of the Zodiacal emission. While any overall change in the temperature of the IPD particles would scale all components of the Zodiacal emission, because temperature is nearly inversely proportional to distance from the Sun, the location of the bands are important. While the excess over 1 is too large to be explained by errors in distance and thus temperatures alone, one might appeal to a change in a combination of distance, particle density normalisation and emissivity of these bands in particular to arrive at mutually consistent results for both  and DIRBE. As this will involve simultaneous work with both  and DIRBE data, it is beyond the scope of this paper. Bands 1 and 3 also seem to show different behaviour than band 2. Since bands 1 and 3 are both at high ecliptic latitude, while band 2 is not, one might again worry that one of the other templates to which we are fitting might have significant overlap with a subset of the bands, which in turn causes an apparent difference in emissivities. To check this, we have repeated the fits with and without various combinations of the cloud, circumsolar ring and Earth-trailing feature, as well as the far sidelobes. In all cases, bands 1 and 3 are always significantly different than dust band 2. When the diffuse cloud itself is omitted from the fit, the emissivity of dust band 2 goes up, but is still distinctively different from that of dust bands 1 and 3. As dust band 2 is a combination of the *IRAS* $\alpha$ and $\beta$ bands, one may also worry that one of these two is more important for the shorter *IRAS* and *COBE* wavelengths, but that the other might be more important for the longer  wavelengths. We note that its emission is dominated by contributions from the Karin/Koronis family [see @Nesvorny2008 Fig. 1], but have therefore confirmed specifically that varying the $\delta_\zeta$ parameter of the second band between values appropriate for either $\alpha$ or $\beta$ does not remove this difference (see table \[tab:bandInfo\]). If the age of band 2 were significantly different from those of bands 1 and 3, we might argue that Poynting-Robertson drag had depleted some of the bands of more smaller particles than the others [@Wyatt2011]. N03 and N08, however, have estimated the ages of most of the asteroid families which might be associated with the bands (reproduced in table \[tab:bandInfo\]), and the age of any of the associations with band 2 is between those of any of the possible associations with bands 1 or 3. These same figures tend to rule out modifications of the material properties due to photo-processing or solar wind exposure for differing periods. Band 2 also seems to be roughly the same distance from the Sun as the other two bands, so it is difficult to appeal to differences in environment as the cause. We speculate on the following to explain any differences: Veritas, the asteroid family proposed to be associated with dust band 1, is classified as carbonaceous [@Bus2002]. As noted above, the *IRAS* $\beta$ band, associated with the Karin family of asteroids, seems to dominate the emission from dust band 2. Karin and its larger sibling, Koronis, are classified as siliceous, or stony, objects [@Bus2002; @Carvano2010]. While the dust band 3 has a number of asteroid families which may be contributing to it (see table \[tab:bandInfo\]), we propose that the emission is dominated by carbonaceous-based asteroid families (three quarters of the asteroids in the Solar System are carbonaceous), and that the difference in emissivities between dust band 2 and dust bands 1 and 3 arises from this difference in composition. The differing emissivities may be either due to this intrinsic composition difference, or the size-frequency distribution of particles that results from different kinds of asteroids colliding [@Grogan2001 for example]. This explanation would not be valid if it were to turn out that dust band 3 was dominated by dust associated with the Iannini asteroid, for example, as it is siliceous. Implications for the CMB ------------------------ The right-hand column of Fig. \[fig:beforeAndAfterJackknives\] shows the Zodiacal emission implied by the fits, created by subtracting the maps made applying the Zodiacal emission correction from those that were made without it applied. One can see here the difference in the relative amplitudes of the bands versus the diffuse cloud, the bands being relatively more important at low frequencies than at high. Figure \[fig:Cls\] shows the power spectra of the Zodiacal correction maps, all in units of $\left(\mu\mathrm{K}_\mathrm{CMB}\right)^2$. Here, the cloud is seen at multipoles of less than about 10, while the bands and other structures are see in higher multipoles. ![Power spectra of the Zodiacal correction maps shown in the right-hand column of Fig. \[fig:beforeAndAfterJackknives\]. Black plus signs, grey crosses, cyan triangles, orange diamonds, blue circles, and red squares represent, respectively, 857, 545, 353, 217, 143, and 100GHz. For the CMB channels, even multipoles are shown with filled symbols, while odd multipoles are shown with empty symbols. This “even-odd” pattern is a reflection of the symmetry around the ecliptic plane – odd multipoles are almost absent, as they would indicate structure in the maps that were anti-symmetric about the ecliptic plane. A standard $\Lambda$CDM CMB temperature spectrum is shown as the grey line roughly half-way down the plot, orders of magnitude above the Zodiacal spectrum in the  CMB channels.[]{data-label="fig:Cls"}](ZCorrectionCls.pdf){width="88mm"} At 143GHz, The signal reaches a few $\mu\mathrm{K}_\mathrm{CMB}$ in the map, while the power spectrum has values of the order of one$\left(\mu\mathrm{K}_\mathrm{CMB}\right)^2$. The absence of power in the odd multipoles is a reflection of the north-south symmetry of the signal. These can be compared with the CMB temperature spectrum, which is shown as the black line about halfway between the top and the bottom of the plot[^6]. The Zodiacal emission correction spectra are orders of magnitude smaller than the CMB spectrum. The Zodiacal emission does not compromise ’s cosmological results. Conclusion {#sec:conclusion} ========== Zodiacal emission has long been an important foreground for searches for the extra-galactic background at infrared wavelengths. With the ever-increasing sensitivity of CMB experiments, it will soon become important to account for in sub-millimetre and CMB analyses as well. The K98 model does fairly well in modelling the diffuse Zodiacal cloud emission at  wavelengths, as long as appropriate emissivities are assumed. It does less well, however, in modelling the other features. Because they appear to be more emissive at these frequencies than the cloud, the bands contribute more to the Zodiacal emission relative to the diffuse cloud at CMB frequencies (i.e., near 143GHz, or 2.1mm). The 2013  release includes both maps which have had Zodiacal emission removed, and maps which have not had Zodiacal emission removed. We note that [@planck2013-p06] found better results using maps that had not had the Zodiacal emission removed to make estimations of the CMB and other astrophysical components in the HFI maps. The component separation methods used there naturally correct for a large amount of Zodiacal emission, as it is spectrally similar to Galactic dust emission in the  CMB channels. In addition, one can see in Fig. \[fig:noDipoleSidelobesFit\] and table \[tab:fullFit\] that the diffuse cloud is not actually detected at 143GHz, the frequency which carries the most weight in  CMB analyses. The Zodiacal emission removal, of course, depends delicately on how this non-detection is treated. This will be addressed in future  releases. Improvements in modelling of the circumsolar ring, Earth-trailing feature, and the dust bands, as well as inclusion of fainter and partial bands [e.g., @Espy2009] should be done to make truly “clean” CMB maps. As these bands are believed to be the products of asteroid collisions, further study of these bands at these wavelengths may also inform us on the nature of the intermediate-sized particles created during the destruction of the associated asteroids. We may hope to learn not only more about the size distribution, but also the differences between, for example, the results of a collisions involving siliceous and carbonaceous asteroids. Just as material from asteroid collisions contributes to Zodiacal emission, material shed from comets must also contribute. As part of the HFI data reduction process, we mask out solar system objects which would cause “noise” in the final sky maps. We searched for comets as part of this process, but found only one (Christensen). For completeness, appendix \[sec:Ast\] presents this information, as it is not possible to extract from the data in the 2013  map release. We have not yet detected extended tails of comets. One of the primary goals of the next stage of analyses, once this “nearby” IPD has been completely removed, will be to search for dust associated with the Kuiper Belt, or set limits upon it. The full-mission  data release will include polarisation information. While it is not expected, limits will be put on possible contamination of the polarisation of the CMB by polarised Zodiacal emission, as has been done here for temperature. Work is now underway to address all these points for the next  data release. While the signal is quite small – at CMB wavelengths the signal we are discussing is orders of magnitude smaller than the primary CMB anisotropies, it is detectable and should be subtracted from the data. There will be improvements in dust modelling, improvements in satellite modelling, and additions to address polarisation. The ultimate goal will be simultaneous analyses with *IRAS*, *COBE*, and Akari and other to understand the large-scale Zodiacal emission from the near-infrared to the microwave. This paper benefited from exchanges with Dale Fixsen, Tom Kelsall and Janet Weiland. We acknowledge the IN2P3 Computer Center (<http://cc.in2p3.fr>) for providing a significant amount of the computing resources and services needed for this work. We acknowledge the use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA). Support for LAMBDA is provided by the NASA Office of Space Science. The development of  has been supported by: ESA; CNES and CNRS/INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MICINN, JA and RES (Spain); Tekes, AoF and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland); RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); and PRACE (EU). A description of the  Collaboration and a list of its members, including the technical or scientific activities in which they have been involved, can be found at <http://www.sciops.esa.int/index.php?project=planck&page=Planck_Collaboration>. Basic Behaviour --------------- Figure \[fig:Asts\] demonstrates some of the basics of the asteroid detections. For each measurement, we denote the measurement as $f$, the distance between  and the object as $d$, and the distance between the Sun and the object as $s$. These quantities for the first measurement of each asteroid will be denoted $f_0$, $d_0$, and $s_0$. In the top panel, we show asteroids that were detected in multiple surveys. Assuming the temperature of the object at any time is inversely proportional to $\sqrt{s}$, we would expect $$f\sqrt{s} \propto \frac{1}{d^2},$$ which is roughly seen in the data.  In the bottom panel, we show the 545-to-857GHz spectral indices, as well as the 545-to-353GHz spectral indices for Ceres and Vesta. [^1]:  (http://www.esa.int/Planck) is a project of the European Space Agency (ESA) with instruments provided by two scientific consortia funded by ESA member states (in particular the lead countries France and Italy), with contributions from NASA (USA) and telescope reflectors provided by a collaboration between ESA and a scientific consortium led and funded by Denmark. [^2]: <http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=47345>. [^3]: Legacy Archive for Microwave Background Data Analysis – NASA (<http://lambda.gsfc.nasa.gov/>). [^4]: While three bands are described in the text of K98, there are actually four in the code. There is, however, no ambiguity, as the density of the fourth band is set to zero in the code at the LAMBDA site. [^5]: Care must be taken with direct comparisons between the emissivities quoted in [@fixsendwek2002], who quote emissivities relative to a 245K cloud, and those in this work, which follow K98 and assume a cloud of temperature 286K at 1AU from the Sun. [^6]: Made with CAMB – <http://camb.info>

--- abstract: 'We describe our efforts to understand large-scale (10’s–100’s kpc) relativistic jet systems through observations of the highest-redshift quasars. Results from a VLA survey search for radio jets in $\sim$30 z$>$3.4 quasars are described along with new $Chandra$ observations of 4 selected targets.' author: - 'C. C. Cheung, Ł. Stawarz' - 'A. Siemiginowska, D. E Harris, D. A. Schwartz' - 'J. F. C. Wardle, D. Gobeille' - 'N. P. Lee' title: The Highest Redshift Relativistic Jets --- Why High-redshift Jets? ======================= It is now well established that X-ray emission is a common feature of kiloparsec-scale radio jets [see @har06 for a recent review and the associated website, http://hea-www.harvard.edu/XJET/]. The spectral energy distributions (SEDs) of the powerful quasar jets are predominantly characterized as “optically faint”, with the spectra rising between the optical and X-ray bands. Current models for this ‘excess’ X-ray emission posit either inverse Compton (IC) scattering off CMB photons in a (still) relativistic kpc-scale jet or an additional high-energy synchrotron emitting component. In the simplest scenario, such models have diverging predictions at high redshift. Specifically, we expect a strong redshift dependence in the monochromatic flux ratio, $f_{X}/f_{r}~\propto~U_{\rm CMB}~\propto~(1+z)^{4}$ for IC/CMB, whereas in synchrotron models, we expect no such dependence, $f_{X}/f_{r}~\propto~(1+z)^{0}$. As a first order test of this simple idea, our approach is to study the highest-redshift relativistic jets. Such jets probe the physics of the earliest (first $\sim$1 Gyr of the Universe in the quasars studied) actively accreting supermassive black hole systems and are interesting for other reasons. For instance, the ambient medium in these high-redshift galaxies is probably different [e.g., @dey06] and this may manifest in jets with different morphologies, increased dissipation, and slower than their lower-redshift counterparts. Most $Chandra$ studies of quasar jets have so far targeted known arcsecond-scale radio jets [e.g., @sam04; @mar05], as most known examples are at $z$ $\stackrel{<}{{}_\sim}$2 [@liu02]. There are currently only two high-$z$ quasars with well-established kpc-scale X-ray jet detections: GB 1508+5714 at $z$=4.3 [@sie03; @yua03; @che04] and 1745+624 at $z$=3.9 [@che06]. They are observed to have large $f_{X}/f_{r}$ values as expected in the IC/CMB model [@sch02; @che04], although the small number of high-$z$ detections preclude any definitive statements [@kat05; @che06]. We have therefore carried out a VLA survey in search of new radio jets in a sample of high-$z$ quasars (§ \[sec-vla\]) and new $Chandra$ observations of a small subset (§ \[sec-cxo\]). This contribution presents some results from these observations. For the redshifts considered, $z$=3.4 to 4.7, 1$''$ corresponds to 7.4 to 6.5 kpc ($H_{0}=70~$km s$^{-1}$ Mpc$^{-1}$, $\Omega_{\rm M}=0.3$ and $\Omega_{\rm \Lambda}=0.7$). Observations of a High-Redshift Quasar Sample ============================================= VLA Imaging Survey\[sec-vla\] ----------------------------- Using NED, we assembled a sample of z$>$3.4 flat-spectrum radio quasars for imaging with the VLA. We did not aim for our sample to be a complete one as current samples of lower-z X-ray jets are inhomogenous also. With archival [@lee05] and new VLA observations, we find that radio jets in this redshift range are common with a $\sim$50$\%$ detection rate [@che05 and in preparation]. Examples of new radio jets detected from our observations are shown in Figure \[fig-1\]. $Chandra$ Observations\[sec-cxo\] --------------------------------- A small percentage of the radio jets from our radio study (§ \[sec-vla\]) are extended enough ($>$2.5” long) to study with $Chandra$. We observed four of them with short snapshot $Chandra$ observations (Figure \[fig-2\]). We detected bright X-ray counterparts to the jets in the quasars J1421–0643 [$z$=3.689; @ell01] and GB 1428+4217 [$z$=4.72; @hoo98]; the latter detection is currently the highest-redshift kpc-scale radio and X-ray jet known. We did not detect the X-ray counterparts to the radio jets in 1239+376 [z=3.819; @ver96] and J1754+6737 [$z$=3.6; @vil99]. The 2/4 X-ray jet detection rate of our high-$z$ sample is comparable to that of lower-$z$ samples [@sam04; @mar05]. Discussion and Summary ====================== Previous $Chandra$ imaging studies of a number of z$>$4 radio loud quasars do not reveal significant extended X-ray emission [@bas04; @lop06]. However, in these studies, there were no pre-existing information on possible radio structures in the target objects and any definitive statements regarding the nature of the X-ray emission mechanism in jets at high-redshifts may be premature. In fact, in one case where there was evidence of an extended X-ray structure [J2219–2719; @lop06], our VLA observation revealed a radio counterpart (Figure \[fig-1\]). In our approach, we began with a VLA survey of a sample of z$>$3.4 quasars and found radio jets to be relatively common ($\sim$50$\%$ detection rate). These jets are quite luminous; with a confident detection of a 1 mJy knot at 1.4 GHz, this corresponds to luminosities of 1.5 $\times$10$^{42}$ erg s$^{-1}$ ($z$=3.4) to 3.1 $\times$10$^{42}$ erg s$^{-1}$ ($z$=4.7). With the radio survey results, we found only a few radio jets to have sufficient angular extent to be imaged with $Chandra$. The detection rate of X-ray counterparts of the high-z radio jets (2/4) is similar to that of lower-z radio jet samples [@sam04; @mar05]. The implications of these observations for models of X-ray emission from large-scale jets will be described in forthcoming publications. The National Radio Astronomy Observatory is operated by Associated Universities, Inc. under a cooperative agreement with the National Science Foundation. This research was funded in part by NASA through contract NAS8-39073 (A. S., D. E. H., D. A. S.) and $Chandra$ Award Numbers GO7-8114 (C. C. C., Ł. S., J. F. C. W., D .G.) issued by the $Chandra$ X-Ray Observatory Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of NASA under contract NAS8-39073. Radio astronomy at Brandeis University is supported by the NSF and NASA. Ł. S. is supported by MEiN through research project 1-P03D-003-29 from 2005-2008. Bassett, L. C., et al. 2004, , 128, 523 Cheung, C.C. 2004, , 600, L23 Cheung, C.C., Wardle, J.F.C., & Lee, N.P. 2005, in 22nd Texas Symposium on Relativistic Astrophysics, Eds. P. Chen et al., (Palo Alto: SLAC) 1613 Cheung, C.C., Stawarz, Ł., & Siemiginowska, A. 2006, , 650, 679 De Young, D. 2006, Astron. Nachr., 327, 231 Ellison, S. L., Yan, L., Hook, I. M., Pettini, M., Wall, J. V., & Shaver, P. 2001, , 379, 393 Harris, D.E., Krawczynski, H. 2006, ARAA, 44, 463 Hook, I.M., & McMahon, R.G. 1998, MNRAS, 294, L7 Hook, I.M., McMahon, R.G., Shaver, P.A., & Snellen, I.A.G. 2002, A&A, 391, 509 Kataoka, J., & Stawarz, [Ł]{}. 2005, ApJ, 622, 797 Lee, N. P. 2005, Undergraduate Honors Thesis, Brandeis University Liu, F. K., & Zhang, Y. H. 2002, , 381, 757 Lopez, L. A., Brandt, W. N., Vignali, C., Schneider, D. P., Chartas, G., & Garmire, G. P. 2006, , 131, 1914 Marshall, H. L., et al. 2005, ApJS, 156, 13 Sambruna, R. M., Gambill, J. K., Maraschi, L., Tavecchio, F., Cerutti, R., Cheung, C. C., Urry, C. M., & Chartas, G. 2004, , 608, 698 Schwartz, D.A. 2002, , 569, L23 Siemiginowska, A., Smith, R.K., Aldcroft, T.L., Schwartz, D.A., Paerels, F., & Petric, A.O. 2003, L, 598, L15 Vermeulen, R. C., Taylor, G. B., Readhead, A. C. S., & Browne, I. W. A. 1996, , 111, 1013 Villani, D., & di Serego Alighieri, S. 1999, , 135, 299 Xu, W., Readhead, A.C.S., Pearson, T.J., Polatidis, A.G., & Wilkinson, P.N. 1995, ApJS, 99, 297 Yuan, W., Fabian, A.C., Celotti, A., & Jonker, P.G. 2003, , 346, L7

--- abstract: 'We investigate the notion of symplectic divisorial compactification for symplectic 4-manifolds with either convex or concave type boundary. This is motivated by the notion of compactifying divisors for open algebraic surfaces. We give a sufficient and necessary criterion, which is simple and also works in higher dimensions, to determine whether an arbitrarily small concave/convex neighborhood exist for an $\omega$-orthogonal symplectic divisor (a symplectic plumbing). If deformation of symplectic form is allowed, we show that a symplectic divisor has either a concave or convex neighborhood whenever the symplectic form is exact on the boundary of its plumbing. As an application, we classify symplectic compactifying divisors having finite boundary fundamental group. We also obtain a finiteness result of fillings when the boundary can be capped by a symplectic divisor with finite boundary fundamental group.' author: - 'Tian-Jun Li and Cheuk Yu Mak[^1]' title: Symplectic Divisorial Capping in Dimension 4 --- Introduction ============ In this paper, a [*symplectic divisor*]{} refers to a connected configuration of finitely many closed embedded symplectic surfaces $D=C_1 \cup \dots \cup C_k$ in a symplectic 4 dimensional manifold (possibly with boundary or non-compact) $(W, \omega)$. $D$ is further required to have the following properties: $D$ has empty intersection with $ \partial W$, no three $C_i$ intersect at a point, and any intersection between two surfaces is transversal and positive. The orientation of each $C_i$ is chosen to be positive with respect to $\omega$. Since we are interested in the germ of a symplectic divisor, $W$ is sometimes omitted in the writing and $(D,\omega)$, or simply $D$, is used to denote a symplectic divisor. A closed regular neighborhood of $D$ is called a plumbing of $D$. The plumbings are well defined up to orientation preserving diffeomorphism, so we can introduce topological invariants of $D$ using any of its plumbings. In particular, $b_2^{\pm}(D)$ is defined as $b_2^{\pm}$ of a plumbing. Similarly, we define the [*boundary*]{} of the divisor $D$, and we call the fundamental group of the boundary [*boundary fundamental group*]{} of $D$. In the same vein, when $\omega$ is exact on the boundary of a plumbing, we say that $\omega$ is exact on the boundary of $D$. A plumbing $P(D)$ of $D$ is called a [*concave (resp. convex) neigborhood*]{} if $P(D)$ is a strong concave (resp. convex) filling of its boundary. A symplectic divisor $D$ is called [*concave*]{} (resp. [*convex*]{}) if for any neighborhood $N$ of $D$, there is a concave (resp. convex) neighborhood $P(D) \subset N$ for the divisor. Through out this paper, all concave (resp. convex) fillings are symplectic strong concave (resp. strong convex) fillings and we simply call it cappings or concave fillings (resp. fillings or convex fillings). Suppose that $D$ is a concave (resp. convex) divisor. If a symplectic gluing ([@Et98]) can be performed for a concave (resp. convex) neighborhood of $D$ and a symplectic manifold $Y$ with convex (resp. concave) boundary to obtain a closed symplecitc manifold, then we call $D$ a [**capping**]{} (resp. [**filling**]{}) divisor. In both cases, we call $D$ a [**compactifying**]{} divisor of $Y$. Motivation ---------- We provide some motivation from two typical families of examples in algebraic geometry together with some general symplectic compactification phenomena. Suppose $Y$ is a smooth affine algebraic variety over $\mathbb{C}$. Then $Y$ can be compactified by a divisor $D$ to a projective variety $X$. By Hironaka’s resolution of singularities theorem, we could assume that $X$ is smooth and $D$ is a simple normal crossing divisor. In this case, $Y$ is a Stein manifold and $D$ has a concave neighborhood induced by a plurisubharmonic function on $Y$ ([@ElGr91]). Moreover, $Y$ is symplectomorphic to the completion of a suitably chosen Stein domain $\overline{Y}\subset Y$ (See e.g. [@McL12]). Therefore, compactifying $Y$ by $D$ in the algebro geometric situation is analogous to gluing $\overline{Y}$ with a concave neighborhood of $D$ along their contact boundaries [@Et98]. On the other hand, suppose we have a compact complex surface with an isolated normal singularity. We can resolve the isolated normal singularity and obtain a pair $(W,D)$, where $W$ is a smooth compact complex surface and $D$ is a simple normal crossing resolution divisor. In this case, we can define a Kähler form near $D$ such that $D$ has a convex neighborhood $P(D)$. If the Kähler form can be extended to $W$, then the Kähler compactification of $W-D$ by $D$ is analogous to gluing the symplectic manifold $W-Int(P(D))$ with $P(D)$ along their contact boundaries. From the symplectic point of view, there are both flexibility and constraints for capping a symplectic 4 manifold $Y$ with convex boundary. For flexibility, there are infinitely many ways to embed $Y$ in closed symplectic 4-manifolds (Theorem 1.3 of [@EtHo02]). This still holds even when $Y$ has only weak convex boundary (See [@El04] and [@Et04]). For constraints, it is well-known that (e.g. [@Hu13]) $Y$ does not have any exact capping. From these perspectives, divisor cappings might provide a suitable capping model to study (See also [@Ga03] and [@Ga03c]). On the other hand, divisor fillings have been studied by several authors. For instance, it is known that they are the maximal fillings for the canonical contact structures on Lens spaces (See [@Li08] and [@BhOz14]). In this setting, the following questions are natural: Suppose $D$ is a symplectic divisor. \(i) When is $D$ also a compactifying divisor? \(ii) What symplectic manifolds can be compactified by $D$? A Flowchart {#A Flowchart} ----------- Regarding the first question, observe that a divisor is a capping (resp. filling) divisor if it is concave (resp. convex), and embeddable in the following sense: If a symplectic divisor $D$ admits a symplectic embedding into a closed symplectic manifold $W$, then we call $D$ an [**embeddable**]{} divisor. We recall some results from the literature for the filling side. It is proved in [@GaSt09] that when the graph of a symplectic divisor is negative definite, it can always be perturbed to be a convex divisor. Moreover, a convex divisor is always embeddable, by [@EtHo02], hence a filling divisor. However, a concave divisor is not necessarily embeddable. An obstruction is provided by [@Mc90] (See Theorem \[McDuff\]). Our first main result: \[MAIN\] Let $D \subset (W,\omega_0)$ be a symplectic divisor. If the intersection form of $D$ is not negative definite and $\omega_0$ restricted to the boundary of $D$ is exact, then $\omega_0$ can be deformed through a family of symplectic forms $\omega_t$ on $W$ keeping $D$ symplectic and such that $(D,\omega_1)$ is a concave divisor. In particular, if $D$ is also an embeddable divisor, then it is a capping divisor after a deformation. It is convenient to associate an augmented graph $(\Gamma,a)$ to a symplectic divisor $(D,\omega)$, where $\Gamma$ is the graph of $D$ and $a$ is the area vector for the embedded symplectic surfaces (See Section \[Preliminary\] for details). The intersection form of $\Gamma$ is denoted by $Q_{\Gamma}$. Suppose $(\Gamma,a)$ is an augmented graph with $k$ vertices. Then, we say that $(\Gamma,a)$ satisfies the positive (resp. negative) [**GS criterion**]{} if there exists $z \in (0,\infty)^k$ (resp $(-\infty,0]^k$) such that $Q_{\Gamma}z=a$. A symplectic divisor is said to satisfy the positive (resp. negative) GS criterion if its associated augmented graph does. One important ingredient for the proof of Theorem \[MAIN\] is the following result. \[MAIN2\] Let $(D,\omega)$ be a symplectic divisor with $\omega$-orthogonal intersections. Then, $(D,\omega)$ has a concave (resp. convex) neighborhood inside any regular neighborhood of $D$ if $(D,\omega)$ satisfies the positive (resp. negative) GS criterion. The construction is essentially due to Gay and Stipsicz in [@GaSt09], which we call the GS construction. We remark that GS criteria can be verified easily. They are conditions on wrapping numbers in disguise. Therefore, by a recent result of Mark McLean [@McL14], Proposition \[MAIN2\] can be generalized to higher dimensions with GS criteria being replaced accordingly. Moreover, using techniques in [@McL14], we establish the necessity of the GS criterion and answer the uniqueness question in [@GaSt09]. \[obstruction-GS\] Let $D \subset (W,\omega)$ be an $\omega$-orthogonal symplectic divisor. If $(D,\omega)$ does not satisfy the positive (resp. negative) GS criterion. Then, there is a neighborhood $N$ of $D$ such that any plumbing $P(D) \subset N$ of $D$ is not a concave (resp. convex) neighborhood. \[uniqueness-GS\] Let $(D,\omega_i)$ be $\omega_i$-orthogonal symplectic divisors for $i=0,1$ such that both satisfy the positive (resp. negative) GS criterion. Then the concave (resp. convex) structures on the boundary of $(D,\omega_0)$ and $(D,\omega_1)$ via the GS construction are contactomorphic. In particular, when $\omega_0=\omega_1$, the contact structure constructed via GS construction is independent of choices, up to contactomorphism. Summarizing Theorem \[MAIN\] and Proposition \[MAIN2\], we have \[QHS\] Let $(D,\omega)$ be a symplectic divisor with $\omega$ exact on the boundary of $D$. Then $D$ is either a concave divisor or a convex divisor, possibly after a symplectic deformation. More precise information is illustrated by the following schematic flowchart. (exact) \[startstop\] [$\omega|_{\partial P(D)}$ exact?]{}; (not exact) \[startstop2, below of=exact\] [No concave nor convex neighborhood]{}; (definite) \[startstop, right of=exact\] [$Q_D$ negative definite?]{}; (convex) \[startstop2, below of=definite\] [Admits a convex neighborhood]{}; (GS criterion) \[startstop, right of=definite\] [$(D,\omega)$ satisfies positive GS criterion?]{}; (concave) \[startstop2, right of=GS criterion\] [Admits a concave neighborhood]{}; (deformation) \[startstop3, below of=GS criterion\] [No small concave neighborhood, but admits one after a deformation]{}; (exact) – node\[right\][no]{}(not exact); (exact) – node\[above\][yes]{}(definite); (definite) – node\[right\][yes]{}(convex); (definite) – node\[above\][no]{}(GS criterion); (GS criterion) – node\[above\][yes]{}(concave); (GS criterion) – node\[right\][no]{}(deformation); For a general divisor $(D,\omega)$, which is not necessarily $\omega$-orthogonal, the corresponding results for Proposition \[MAIN2\] and Theorem \[uniqueness-GS\] are still valid (See Proposition \[McLean0\] and Proposition \[McLean\]), by McLean’s construction. The generalization of Theorem \[obstruction-GS\] is a bit subtle. For an embeddable divisor $(D,\omega)$, we obtain in Theorem \[obstruction-closed case\] that if it does not satisfy the positive GS criterion then there is a neighborhood $N$ of $D$ such that any plumbing $P(D) \subset N$ is not a concave neighborhood. Divisors with Finite Boundary $\pi_1$ ------------------------------------- Using Theorem \[MAIN\], Theorem \[obstruction-closed case\], Proposition \[McLean0\] and Proposition \[McLean\], we classify, in our second main results Theorem \[main classification theorem\] and Theorem \[complete classification\], capping divisors (not necessarily $\omega$-orthogonal) with finite boundary fundamental group. This allows us to answer the second question if $D$ is a capping divisor with finite boundary fundamental group. As a consequence, we show that only finitely many minimal symplectic manifolds can be compactified by $D$, up to diffeomorphism. More details are described in Section \[Finiteness\]. Moreover, we also investigate special kinds of symplectic filling. In Section \[Non-Conjugate Phenomena\], we study pairs of symplectic divisors that compactify each other. \[Conjugate Definition\_Divisor\] For symplectic divisors $D_1$ and $D_2$, we say that they are [**conjugate**]{} to each other if there exists plumbings $P(D_1)$ and $P(D_2)$ for $D_1$ and $D_2$, respectively, such that $D_1$ is a capping divisor of $P(D_2)$ and $D_2$ is a filling divisor of $P(D_1)$. On the other hand, it is also interesting to investigate the category of symplectic manifolds having symplectic divisorial compactifications. Affine surfaces are certainly in this category. In this regard, symplectic cohomology could play an important role. Growth rate of symplectic cohomology has been used in [@Se08] and [@McL12] to distinguish a family of cotangent bundles from affine varieties. The proof actually applies to any Liouville domain which admits a divisor cap, so certain boundedness on the growth rate is necessary for a Liouville domain to be in this category. Finally, for symplectic manifolds in this category we would like to define invariants in terms of the divisorial compactifications (See [@LiZh11] for a related invariant). The remaining of this article is organized as follows. In Section \[Preliminary\], we give the proof of Proposition \[MAIN2\], Theorem \[obstruction-GS\] and Theorem \[uniqueness-GS\]. Section \[Operation on Divisors\] is mainly devoted to the proof of Theorem \[MAIN\]. We give the statement and proof of the classification of compactifying divisors with finite boundary fundamental group in Section \[Classification of Symplectic Divisors Having Finite Boundary Fundamental Group\]. Acknowledgements {#acknowledgements .unnumbered} ---------------- The authors would like to thank Mark McLean for many helpful discussions, in particular for explaining the canonical contact structure in [@McL14]. They would also like to thank David Gay, Ko Honda, Burak Ozbagci, Andras Stipsicz, Weiwei Wu and Weiyi Zhang for their interest in this work. They are also grateful to Laura Starkston for stimulating discussions. Contact Structures on the Boundary {#Preliminary} ================================== Essential topological information of a symplectic divisor can be encoded by its [*graph*]{}. The graph is a weighted finite graph with vertices representing the surfaces and each edge joining two vertices representing an intersection between the two surfaces corresponding to the two vertices. Moreover, each vertex is weighted by its genus (a non-negative integer) and its self-intersection number (an integer). If each vertex is also weighted by its symplectic area (a positive real number), then we call it an [*augmented graph*]{}. Sometimes, the genera (and the symplectic area) are not explicitly stated. For simplicity, we would like to assume the symplectic divisors are connected. In what follow, we call a finite graph weighted by its self-intersection number and its genus (resp. and its area) with no edge coming from a vertex back to itself a graph (resp. an augmented graph). For a graph (resp. an augmented graph) $\Gamma$ (resp. $(\Gamma,a)$), we use $Q_{\Gamma}$ to denote the intersection matrix for $\Gamma$ (resp. and $a$ to denote the area weights for $\Gamma$). We denote the determinant of $Q_{\Gamma}$ as $\delta_{\Gamma}$. Moreover, $v_1,\dots,v_k$ are used to denote the vertices of $\Gamma$ and $s_i$, $g_i$ and $a_i$ are self-intersection, genus and area of $v_i$, respectively. Notice that, $\omega$ being exact on the boundary of a plumbing is equivalent to $[\omega]$ being able to be lifted to a relative cohomological class. Using Lefschetz duality, this is in turn equivalent to $[\omega]$ being able to be expressed as a linear combination $\sum\limits_{i=1}^k z_i[C_i]$, where $z_i \in \mathbb{R}$ and $D=C_1 \cup \dots \cup C_k$. As a result, $\omega$ is exact on the boundary of a plumbing if and only if there exist a solution $z$ for the equation $Q_{\Gamma}z=a$ (See Subsection \[wrapping numbers\] for a more detailed discussion). We also remark that the germ of a symplectic divisor $(D,\omega)$ with $\omega$-orthogonal intersections is uniquely determined by its augmented graph $(\Gamma,a)$ (See [@McR05] and Theorem 3.1 of [@GaSt09]) and a symplectic divisor can always be made $\omega$-orthogonal after a perturbation (See [@Go95]). \[2-1\] The graph $$\xymatrix{ \bullet^{2}_{v_1} \ar@{-}[r] & \bullet^{1}_{v_2}\\ }$$ where both vertices are of genus zero, represents a symplectic divisor of two spheres with self-intersection $2$ and $1$ and intersecting positively transversally at a point. \[realizable definition\] A graph $\Gamma$ is called [**realizable**]{} (resp. [**strongly realizable**]{}) if there is an embeddable (resp. compactifying) symplectic divisor $D$ such that its graph is the same as $\Gamma$. In this case, $D$ is called a realization (resp. strongly realization) of $\Gamma$. Similar to Definition \[realizable definition\], we can define realizability and strongly realizability for an augmented graph. If the area weights attached to $\Gamma$ is too arbitrary, it is possible that $(\Gamma,a)$ is not strongly realizable but $\Gamma$ is strongly realizable. Existence --------- In this subsection, Proposition \[MAIN2\] is given via two different approaches, namely, GS construction and McLean’s construction. ### Existence via the GS construction [@GaSt09] $(X,\omega,D,f,V)$ is said to be an [**orthogonal neighborhood 5-tuple**]{} if $(X,\omega)$ is a symplectic 4-manifold with $D$ being a collection of closed symplectic surfaces in $X$ intersecting $\omega$-orthogonally such that $f:X \to [0,\infty)$ is a smooth function with no critical value in $(0,\infty)$ and with $f^{-1}(0)=D$, and $V$ is a Liouville vector field on $X-D$. Moreover, if $df(V)>0$ (resp $<0$), then $(X,\omega,D,f,V)$ is called a convex (resp concave) neighborhood 5-tuple. \[fig GS construction\] (0,4.5) node (yaxis) \[above\] [$y$]{} |- (6,0) node (xaxis) \[right\] [$x$]{}; (1.5,0.9) coordinate (a\_1) – (4.5,0.9) coordinate (a\_2); (1.5,0.9) coordinate (b\_1) – (1.5,3.9) coordinate (b\_2); (2.4,1.8) coordinate (d\_1)– (5.4,1.8) coordinate (d\_2); (2.4,1.8) coordinate (e\_1)– (2.4,3) coordinate (e\_2); (2.5,3.3) node \[left\] [$R_{e_{\alpha \beta},v_{\alpha}} $]{}; (2.4,3) coordinate (j\_1)– (2.4,4.3) coordinate (j\_2); (1.5,3.9) coordinate (f\_1)– (2.4,4.3) coordinate (f\_2); (1.5,2.4) coordinate (g\_1)– (2.4,2.8) coordinate (g\_2); (3,0.9) coordinate (h\_1)– (3.9,1.8) coordinate (h\_2); (3.8,1.3) node \[right\] [$R_{e_{\alpha \beta},v_{\beta}} $]{}; (4.5,0.9) coordinate (i\_1)– (5.4,1.8) coordinate (i\_2); (c) at (intersection of a\_1–a\_2 and b\_1–b\_2); (yaxis |- c) node\[left\] [$z_{\beta}'$]{} -| (xaxis -| c) node\[below\] [$z_{\alpha}'$]{}; (1.5,2.4) – (0,2.4) node\[left\] [$z_{\beta}'+\epsilon$]{}; (1.5,3.9) – (0,3.9) node\[left\] [$z_{\beta}'+2\epsilon$]{}; (3,0.9) – (3,0) node\[below\] [$z_{\alpha}'+\epsilon$]{}; (4.5,0.9) – (4.5,0) node\[below\] [$z_{\alpha}'+2\epsilon$]{}; Figure \[fig GS construction\] In [@GaSt09], Gay and Stipsicz constructed a convex orthogonal neighborhood 5-tuple $(X,\omega,D,f,V)$ when the augmented graph $(\Gamma,a)$ of $D$ satisfies the negative GS criterion. We first review their construction and an immediate consequence will be Proposition \[MAIN2\]. Let $z$ be a vector solving $Q_{\Gamma}z=a$ with $z \in (-\infty,0]^k$. Then $z'=(z_1',\dots,z_n')^T=\frac{-1}{2\pi}z$ has all entries being non-negative. We remark that the $z'$ we use corresponds to the $z$ in [@GaSt09]. For each vertex $v$ and each edge $e$ meeting the chosen $v$, we set $s_{v,e}$ to be an integer. These integers $s_{v,e}$ are chosen such that $\sum\limits_{\text{e meeting v}} s_{v,e}=s_v$ for all $v$, where $s_v$ is the self-intersection number of the vertex $v$. Also, set $x_{v,e}=s_{v,e}z_v'+z_{v'}'$, where $v'$ is the other vertex of the edge $e$. For each edge $e_{\alpha \beta}$ of $\Gamma$ joining vertices $v_\alpha$ and $v_\beta$, we construct a local model $N_{e_{\alpha \beta}}$ as follows. Let $\mu:\mathbb{S}^2 \times \mathbb{S}^2 \to [z_{\alpha}',z_{\alpha}'+1] \times [z_{\beta}',z_{\beta}'+1]$ be the moment map of $\mathbb{S}^2 \times \mathbb{S}^2$ onto its image. We use $p_1$ for coordinate in $[z_{\alpha}',z_{\alpha}'+1]$, $p_2$ for coordinate in $[z_{\beta}',z_{\beta}'+1]$ and $q_i \in \mathbb{R}/2\pi$ be the corresponding fibre coordinates so $\theta=p_1dq_1+p_2dq_2$ gives a primitive of the symplectic form $dp_1 \wedge dq_1 + dp_2 \wedge dq_2$ on the preimage of the interior of the moment image. Fix a small $\epsilon >0$ and let $D_1=\mu^{-1}(\{ z_{\alpha}'\} \times [z_{\beta}', z_{\beta}' + 2\epsilon])$ be a symplectic disc. Let also $D_2=\mu^{-1}([z_{\alpha}', z_{\alpha}' + 2\epsilon] \times \{ z_{\beta}' \})$ be another symplectic disc meeting $D_1$ $\omega$-orthogonal at the point $\mu^{-1}(\{z_{\alpha}'\} \times \{z_{\beta}' \})$. Our local model $N_{e_{\alpha \beta}}$ is going to be the preimage under $\mu$ of a region containing $\{ z_{\alpha}'\} \times [z_{\beta}', z_{\beta}' + 2\epsilon] \cup [z_{\alpha}', z_{\alpha}' + 2\epsilon] \times \{ z_{\beta}' \}$. A sufficiently small $\delta$ will be chosen. For this $\delta$, let $R_{e_{\alpha \beta},v_{\alpha}}$ be the closed parallelogram with vertices $(z_{\alpha}',z_{\beta}'+\epsilon), (z_{\alpha}',z_{\beta}'+2\epsilon), (z_{\alpha}'+\delta,z_{\beta}'+2\epsilon-s_{v_{\alpha},e_{\alpha \beta}} \delta), (z_{\alpha}'+\delta,z_{\beta}'+\epsilon-s_{v_{\alpha},e_{\alpha \beta}} \delta)$. Also, $R_{e_{\alpha \beta},v_{\beta}}$ is defined similarly as the closed parallelogram with vertices $(z_{\alpha}'+\epsilon,z_{\beta}'), (z_{\alpha}'+2 \epsilon,z_{\beta}'), (z_{\alpha}'+2\epsilon-s_{v_{\beta},e_{\alpha \beta}} \delta,z_{\beta}'+\delta), (z_{\alpha}'+\epsilon-s_{v_{\beta},e_{\alpha \beta}} \delta,z_{\beta}'+\delta)$. We extend the right vertical edge of $R_{e_{\alpha \beta},v_{\alpha}}$ downward and extend the top horizontal edge of $R_{e_{\alpha \beta},v_{\beta}}$ to the left until they meet at the point $(z_{\alpha}'+\delta,z_{\beta}'+\delta)$. Then, the top edge of $R_{e_{\alpha \beta},v_{\alpha}}$, the right edge of $R_{e_{\alpha \beta},v_{\beta}}$, the extension of right edge of $R_{e_{\alpha \beta},v_{\alpha}}$, the extension of top edge of $R_{e_{\alpha \beta},v_{\beta}}$, $\{ z_{\alpha}'\} \times [z_{\beta}', z_{\beta}' + 2\epsilon]$ and $[z_{\alpha}', z_{\alpha}' + 2\epsilon] \times \{ z_{\beta}' \}$ enclose a region. After rounding the corner symmetrically at $(z_{\alpha}'+\delta,z_{\beta}'+\delta)$, we call this closed region $R$. Now, we set $N_{e_{\alpha \beta}}$ to be the preimage of $R$ under $\mu$. See Figure \[fig GS construction\]. On the other hand, for each vertex $v_{\alpha}$, we also need to construct a local model $N_{v_{\alpha}}$. Let $g_{\alpha}$ be the genus of $v_{\alpha}$. We can form a genus $g_{\alpha}$ compact Riemann surface $\Sigma_{v_{\alpha}}$ such that the boundary components one to one correspond to the edges meeting $v_{\alpha}$. We denote the boundary component corresponding to $e_{\alpha \beta}$ by $\partial_{e_{\alpha \beta}} \Sigma_{v_{\alpha}}$. There exists a symplectic form $\bar{\omega}_{v_{\alpha}}$ and a Liouville vector field $\bar{X}_{v_{\alpha}}$ on $\Sigma_{v_{\alpha}}$ such that when we give the local coordinates $(t,\vartheta_1) \in (x_{v_{\alpha}, e_{\alpha \beta}}-2\epsilon,x_{v_{\alpha}, e_{\alpha \beta}}-\epsilon] \times \mathbb{R}/2\pi \mathbb{Z}$ to the neighborhood of the boundary component $\partial_{e_{\alpha \beta}} \Sigma_{v_{\alpha}}$, we have that $\bar{\omega}_{v_{\alpha}}=dt \wedge d\vartheta_1$ and $\bar{X}_{v_{\alpha}}=t \partial_t$. Now, we form the local model $N_{v_{\alpha}}=\Sigma_{v_{\alpha}} \times \mathbb{D}^2_{\sqrt{2\delta}}$ with product symplectic form $\omega_{v_{\alpha}}=\bar{\omega}_{v_{\alpha}}+rdr \wedge d\vartheta_2$ and Liouville vector field $X_{v_{\alpha}}=\bar{X}_{v_{\alpha}}+(\frac{r}{2}+\frac{z_{v_{\alpha}}'}{r})\partial_r$, where $(r,\vartheta_2)$ is the standard polar coordinates on $\mathbb{D}^2_{\sqrt{2\delta}}$. Finally, the GS construction is done by gluing these local models appropriately. To be more precise, the preimage of $R_{e_{\alpha \beta},v_{\alpha}}$ of $N_{e_{\alpha \beta}}$ is glued via a symplectomorphism preserving the Liouville vector field to $[x_{v_{\alpha}, e_{\alpha \beta}}-2\epsilon,x_{v_{\alpha}, e_{\alpha \beta}}-\epsilon] \times \mathbb{R}/2\pi \mathbb{Z} \times \mathbb{D}^2_{\sqrt{2\delta}}$ of $N_{v_{\alpha}}$ and other matching pieces are glued similarly. When $\delta >0$ is chosen sufficiently small, this glued manifold give our desired convex orthogonal neighborhood 5-tuple with the symplectic divisor having graph $\Gamma$. We remark the whole construction works exactly the same if all entries of $z'$ are negative. In this case, all entries of $z$ are positive and we get the desired concave orthogonal neighborhood 5-tuple if $(\Gamma,a)$ satisfies the positive GS criterion. Now, if we have an $\omega'$-orthogonal divisor $(D',\omega')$ with augmented graph $(\Gamma,a)$, which is the same as that of the concave orthogonal neighborhood 5-tuple $(X,\omega,D,f,V)$, then there exist neighborhood $N'$ of $D'$ symplectomorphic to a neighborhood of $D$ and sending $D'$ to $D$ (See [@McR05] and [@GaSt09]). Therefore, a concave neighborhood of $D$ in $N$ give rise to a concave neighborhood of $D'$ in $N'$. This finishes the proof of Proposition \[MAIN2\]. ### Existence in Higher Dimensions via Wrapping Numbers {#wrapping numbers} To understand the geometrical meaning of the GS criteria, we recall wrapping numbers from [@McL14] and [@McL12]. Then, another construction for Proposition \[MAIN2\] is given. Let $(D,P(D),\omega)$ be a plumbing of a symplectic divisor. If $\omega$ is not exact on the boundary of $D$, then there is no Liouville flow $X$ near $\partial P(D)$ such that $\alpha=i_X \omega$ and $d\alpha=\omega$. Therefore, $D$ does not have concave nor convex neighborhood. When $\omega$ is exact on the boundary, let $\alpha$ be a $1$-form on $P(D)-D$ such that $d \alpha=\omega$. Let $\alpha_c$ be a $1$-form on $P(D)$ such that it is $0$ near $D$ and it equals $\alpha$ near $\partial P(D)$. Note that $[\omega-d\alpha_c] \in H^2(P(D), \partial P(D);\mathbb{R})$. Let its Lefschetz dual be $-\sum\limits_{i=1}^k \lambda_i [C_i] \in H_2(P(D); \mathbb{R})$. We call $\lambda_i$ the wrapping number of $\alpha$ around $C_i$. Also, there is another equivalent interpretation of wrapping numbers. If we symplectically embed a small disc to $P(D)$ meeting $C_i$ positively transversally at the origin of the disc, then the pull-back of $\alpha$ equals $ \frac{r^2}{2}d\vartheta + \frac{\lambda_i}{2\pi} d\vartheta +df$, where $(r,\vartheta)$ is the polar coordinates of the disc and $f$ is some function defined on the punctured disc. (See the paragraph before Lemma 5.17 of [@McL12]). From this point of view, we can see that the $z_i$’s in the GS criteria are minus of the wrapping numbers $-\lambda_i$’s for a lift of the symplectic class $[\omega] \in H^2(P(D);\mathbb{R})$ to $H^2(P(D),\partial P(D);\mathbb{R})$. In particular, $Q$ being non-degenerate is equivalent to lifting of symplectic class being unique, which is in turn equivalent to the connecting homomorphism $H^1(\partial P(D); \mathbb{R}) \to H^2(P(D),\partial P(D);\mathbb{R})$ is zero. When $Q$ is degenerate and for a fixed $\omega$, the equation $Qz=a$ having no solution for $z$ is equivalent to $\omega|_{\partial P(D)}$ being not exact. Similarly, when $Qz=a$ has a solution for $z$, then the solution is unique up to the kernel of $Q$, which corresponds to the unique lift of $\omega$ up to the image of the connecting homomorphism $H^1(\partial P(D); \mathbb{R}) \to H^2(P(D),\partial P(D);\mathbb{R})$. To summarize, we have Let $(D,\omega)$ be a symplectic divisor. Then, lifts $[\omega-d\alpha_c] \in H^2(P(D),\partial P(D);\mathbb{R})$ of the symplectic class $[\omega]$ are in one-to-one correspondence to the solution $z$ of $Q_Dz=a$ via the minus of Lefschetz dual $PD([\omega-d\alpha_c])= -\sum\limits_{i=1}^k \lambda_i [C_i]$ and $z_i=-\lambda_i$. Proposition \[MAIN2\] can be generalized to arbitrary dimension if we apply the constructions in the recent paper of McLean [@McL14]. We first recall an appropriate definition of a symplectic divisor in higher dimension (See [@McL14] or [@McL12]). Let $(W^{2n},\omega)$ be a symplectic manifold with or without boundary. Let $C_1, \dots, C_k$ be real codimension $2$ symplectic submanifolds of $W$ that intersect $\partial W$ trivially (if any). Assume all intersections among $C_i$ are transversal and positive, where positive is defined in the following sense. \(i) For each $I \subset \{1, \dots,k \}$, $C_I=\cap_{i \in I}C_i$ is a symplectic submanifold. \(ii) For each $I, J \subset \{1, \dots, k\}$ with $C_{I \cup J} \neq \emptyset$, we let $N_1$ be the symplectic normal bundle of $C_{I \cup J}$ in $C_I$ and $N_2$ be the symplectic normal bundle of $C_{I \cup J}$ in $C_J$. Then, it is required that the orientation of $N_1 \oplus N_2 \oplus TC_{I \cup J}$ is compatible with the orientation of $TW|_{C_{I \cup J}}$. We remark that the condition (ii) above guarantees that no three distinct $C_i$ intersect at a common point when $W$ is four dimensions. Therefore, this higher dimension definition coincides with the one we use in four dimension. To make our paper more consistent, in higher dimension, we call $D= C_1 \cup \dots \cup C_k$ a symplectic divisor if $D$ is moreover connected and the orientation of each $C_i$ is induced from $\omega^{n-1}|_{C_i}$. Now, for each $i$, let $N_i$ be a neighborhood of $C_i$ such that we have a smooth projection $p_i: N_i \to C_i$ with a connection rotating the disc fibers. Hence, for each $i$, we have a well-defined radial coordinate $r_i$ with respect to the fibration $p_i$ such that $C_i$ corresponds to $r_i=0$. Let $\bar{\rho}: [0,\delta) \to [0,1]$ be a smooth function such that $\bar{\rho}(x)=x^2$ near $x=0$ and $\bar{\rho}(x)=1$ when $x$ is close to $\delta$. Moreover, we require $\bar{\rho}'(x) \ge 0$. A smooth function $f: W-D \to \mathbb{R}$ is called compatible with $D$ if $f=\sum\limits_{i=1}^k \log(\bar{\rho}(r_i))+\bar{\tau}$ for some smooth $\bar{\tau}: W \to \mathbb{R}$ and choice of $\bar{\rho}(r_i)$ as above. Here is the analogue of Proposition \[MAIN2\] in arbitrary even dimension. \[McLean0\] Suppose $f: W^{2n}-D \to \mathbb{R}$ is compatible with $D$ and $D$ is a symplectic divisor with respect to $\omega$. Suppose $\theta \in \Omega^1(W^{2n}-D)$ is a primitive of $\omega$ on $W^{2n}-D$ such that it has positive (resp. negative) wrapping numbers for all $i=1, \dots, k$. Then, there exist $g: W^{2n}-D \to \mathbb{R}$ such that $df(X_{\theta+dg}) > 0$ (resp. $df(-X_{\theta+dg}) > 0$) near $D$, where $X_{\theta+dg}$ is the dual of $\theta+dg$ with respect to $\omega$. In particular, $D$ is a convex (resp. concave) divisor. This is essentially contained in Propositon 4.1 of [@McL14]–the only new statement is the last sentence. And Proposition 4.1 in [@McL14] is stated only for the case in which wrapping numbers are all positive, however, the proof there goes through without additional difficulty for the other case. We give here the most technical lemma adapted to the case of negative wrapping numbers and ambient manifold being dimension four for the sake of completeness. We remark that the $\omega$-orthogonal intersection condition is not required in his construction. Given $D=D_1 \cup D_2 \subset (U,\omega)$, where $D_1$ and $D_2$ are symplectic $2$-discs intersecting each other positively and transversally at a point $p$. Suppose $\theta \in \Omega^1(U-D)$ is a primitive of $\omega$ on $U-D$ such that it has negative wrapping numbers with respect to both $D_1$ and $D_2$. Then there exists $g$ such that for all smooth functions $f:U-D \to \mathbb{R}$ compatible wtih $D$, we have that $df(-X_{\theta+dg}) > c_f \|\theta+dg\| \| df\|$ near $D$, where $c_f>0$ is a constant depending on $f$. Also $c_1 \| db\| < \| \theta+dg \| < c_2 \| db\|$ near $D$ for some smooth function $b$ compatible with $D$, where $c_1$ and $c_2$ are some constants. By possibly shrinking $U$, we give a symplectic coordinate system at the intersection point $p$ such that $D_1=\{x_1=y_1=0\}$ and $0$ corresponds to $p$. Let $\pi_1$ be the projection to the $x_2,y_2$ coordinates. Write $x_1=r \cos \vartheta$ and $y_1=r \sin \vartheta$ and let $\tau=\frac{r^2}{2}$. Let $U_1=U-D_1$ and $\tilde{U_1}'$ be the universal cover of $U_1$ with covering map $\alpha$. Give $\tilde{U_1}'$ the coordinates $(\tilde{x_1},\tilde{y_1},\tilde{x_2},\tilde{y_2})$ coming from pulling back the coordinates of $(\tau,\vartheta,x_2,y_2)$ by the covering map. Then, the pulled back symplectic form on $\tilde{U_1}'$ is given by $d\tilde{x_1} \wedge d\tilde{y_1} + d\tilde{x_2} \wedge d\tilde{y_2}$. Hence, we can enlarge $\tilde{U_1}'$ across $\{ \alpha^*\tau=\tilde{x_1}=0 \}$ to $\tilde{U_1}$ by identifying $\tilde{U_1}'$ as an open subset of $\mathbb{R}^4$ with standard symplectic form. Let $L_{\vartheta_0}=\{(\tau,\vartheta_0,x_2,y_2) \in \tilde{U_1}| \tau,x_2,y_2 \in \mathbb{R}\}$, which is a $3$-manifold depending on the choice of $\vartheta_0$. Let $T$ be the tangent space of $D_2$ at $0$ and identify it as a $2$ dimensional linear subspace in $(x_1,y_1,x_2,y_2)$ coordinates. Then, $l_{\vartheta_0}=\alpha(L_{\vartheta_0} \cap \tilde{U_1}') \cap T$ is an open ray starting from $0$ in $U$ because $D_1$ and $D_2$ are assumed to be transversal. If we pull back the tangent space of $l_{\vartheta_0}$ to the $(\tilde{x_1},\tilde{y_1},\tilde{x_2},\tilde{y_2})$ coordinates in $\tilde{U_1}'$, it is spanned by a vector of the form $(1,0,a_{\vartheta_0},b_{\vartheta_0})$ for some $a_{\vartheta_0},b_{\vartheta_0}$. We identify this vector as a vector at $(0,\vartheta_0,0,0)$ and call it $v_{\vartheta_0}$. Notice that the $\omega$-dual of $v_{\vartheta_0}$ is $d\tilde{y_1}-b_{\vartheta_0}d\tilde{x_2}+a_{\vartheta_0}d\tilde{y_2}$, for all $\vartheta_0 \in [0,2\pi]$. Let $X_1$ be a vector field on $\tilde{U_1}$ such that $X_1=\frac{\lambda_1}{2\pi}v_{\vartheta_0}$ at $(\tilde{x_1},\tilde{y_1},\tilde{x_2},\tilde{y_2})=(0,\vartheta_0,0,0)$ for all $\vartheta_0 \in [0,2\pi]$, where $\lambda_1$ is the wrapping number of $\theta$ with respect to $D_1$. We also require the $\omega$-dual of $X_1$ to be a closed form on $\tilde{U_1}$. This can be done because the $\omega$-dual of $X_1$ restricted to $\{\tilde{x_1}=\tilde{x_2}=\tilde{y_2}=0\}$ is closed. Furthermore, we can also assume $X_1$ is invariant under the $2\pi \mathbb{Z}$ action on $\tilde{y_1}$ coordinate. Note that $d\tilde{x_1}(X_1)=\frac{\lambda_1}{2\pi} < 0$ at $(0,\vartheta_0,0,0)$ for all $\vartheta_0$ so we have $d\tilde{x_1}(X_1) < 0$ near $\{\tilde{x_1}=\tilde{x_2}=\tilde{y_2}=0\}$. Let the $\omega$ dual of $X_1$ be $\tilde{q_1}$, which is exact as it is closed in $\tilde{U_1}$. Now, $\tilde{q_1}$ can be descended to a closed 1-form $q_1$ in $U_1$ under $\alpha$ with wrapping numbers $\lambda_1$ and $0$ with respect to $D_1$ and $D_2$, respectively. We can construct another closed 1-form $q_2$ in $U_2$ in the same way as $q_1$ with $D_1$ and $D_2$ swapped around. Notice that $q_1+q_2$ is a well-defined closed 1-form in $U-D$ with same wrapping numbers as that of $\theta$. Let $\theta'=\theta_1+q_1+q_2$ be such that $d(\theta')=\omega$ and $\theta_1$ has bounded norm. Since $\theta'$ has the same wrapping numbers as that of $\theta$, we can find a function $g: U-D \to \mathbb{R}$ such that $\theta'=\theta+dg=\theta_1+q_1+q_2$. We want to show that $df(-X_{\theta+dg}) > c_f \|\theta+dg\| \| df\|$ near $D$. It suffices to show that $df(-X_{q_1+q_2}) > c_f \|q_1+q_2\| \| df\|$ near $D$ as $\| \theta_1 \|$ is bounded. Since $f=\sum\limits_{i=1}^n \log(\rho(r_i))+\bar{\tau}$ for some smooth $\bar{\tau}: M \to \mathbb{R}$, it suffices to show that $\sum\limits_{i=1}^2 (d\log(x_i'^2+y_i'^2)) (-X_{q_1+q_2}) > c_f \|q_1+q_2\| \| \sum_{i=1}^2 (d\log(x_i'^2+y_i'^2))\|$, where $(x_1',y_1',x_2',y_2')$ are smooth coordinates adapted to the fibrations used to define compatibility. To do this, we pick a sequence of points $p_k \in U-D$ converging to $0$. Then $\frac{X_{q_1}}{\| q_1 \|}$ at $p_k$ converges (after passing to a subsequence) to a vector transversal to $D_1$ but tangential to $D_2$. The analogous statement is true for $\frac{X_{q_2}}{\| q_2 \|}$. Hence we have $\sum\limits_{i=1}^2 (d\log(x_i'^2+y_i'^2)) (-X_{q_1+q_2}) > c_f \sum\limits_{i=1}^2 \|q_i\| \| (d\log(x_i'^2+y_i'^2))\|$ and thus get the desired estimate (See [@McL14] for details). On the other hand, $c_1 \| db\| < \| \theta+dg \| < c_2 \| db\|$ near $D$ for some smooth function $b$ compatible with $D$ is easy to achieve by taking $b=C\sum\limits_{i=1}^2 (d\log(x_i'^2+y_i'^2))$ near $D$. Careful readers will find that when constructing a convex neighborhood, the GS construction works when wrapping numbers are all non-negative while McLean’s constructions work only when wrapping numbers are all positive. We end this subsection with a lemma saying that the GS construction is not really more powerful than McLean’s construction in dimension four. \[non-negative wrapping numbers\] Let $(D^{2n-2},\omega)$ be a symplectic divisor with $n>1$. Suppose $\omega$ is exact on the boundary with $\alpha$ being a primitive on $P(D)-D$. If the wrapping numbers of $\alpha$ are all non-negative, then all are positive. Suppose the wrapping numbers $\lambda_i$ of $\alpha$ are all non-negative and $\lambda_1=0$. Then, $\alpha$ can be extend over $C_1-\cup_{1 \in I, |I| \ge 2}C_I$, where we recall $C_I$ with $1 \in I$ are the symplectic submanifold of $C_1$ induced from intersection with other $C_i$. Therefore, $$\int_{C_1} \omega^{n-1} = \int_{P(\cup_{1 \in I, |I| \ge 2} C_I)} \omega^{n-1} - \int_{\partial P(\cup_{1 \in I, |I| \ge 2} C_I)} \alpha \wedge \omega^{n-2},$$ where $P(\cup_{1 \in I, |I| \ge 2} C_I)$ is a regular neighborhood of $\cup_{1 \in I, |I| \ge 2} C_I$ in $C_1$. We claim that $\int_{P(\cup_{1 \in I, |I| \ge 2} C_I)} \omega^{n-1} - \int_{\partial P(\cup_{1 \in I, |I| \ge 2} C_I)} \alpha \wedge \omega^{n-2} \le 0$ so we will arrive at a contradiction. We first assume that if $1 \in I$, then $C_I = \emptyset$ except $C_1$ and $C_{ \{1,2 \}}$. As a submanifold of $C_1$, $P(\cup_{1 \in I, |I| \ge 2} C_I)=P(C_{\{1,2 \}})$ can be symplectically identified with a closed $2$-disc bundle over $C_{ \{1,2 \}}$. For each fibre, $\alpha|_{\text{fibre}}= \frac{r^2}{2}d\vartheta + \frac{\lambda_2}{2\pi} d\vartheta +df$, where $(r,\vartheta)$ is the polar coordinates of the disc and $f$ is a smooth function defined on the punctured disc. Without loss of generality, we can assume $P(C_{\{1,2\}})$ is taken such that symplectic connection rotates the fibre and we have a well defined one form $\frac{\lambda_2}{2\pi} d\vartheta$ on $P(C_{\{1,2\}})-C_{\{1,2\}}$. Then, $\alpha- \frac{\lambda_2}{2\pi} d\vartheta -df$ can be defined over $P(C_{\{1,2\}})$ for some $f$ defined on $P(C_{\{1,2\}})-C_{\{1,2\}}$ and $$\begin{aligned} \int_{\partial P(C_{\{1,2\}})} (\alpha -\frac{\lambda_2}{2\pi} d\vartheta) \wedge \omega^{n-2} &=&\int_{\partial P(C_{\{1,2\}})} (\alpha -\frac{\lambda_2}{2\pi} d\vartheta-df) \wedge \omega^{n-2}\\ &=&\int_{P(C_{\{1,2\}})} rdr \wedge d\vartheta \wedge \omega^{n-2}\\ &=&\int_{P(C_{\{1,2\}})} \omega^{n-1}.\end{aligned}$$ Therefore, $$\begin{aligned} \int_{P(C_{\{1,2\}})} \omega^{n-1}- \int_{\partial P(C_{\{1,2\}})} \alpha \wedge \omega^{n-2} &=& -\int_{\partial P(C_{\{1,2\}})} \frac{\lambda_2}{2\pi} d\vartheta \wedge \omega^{n-2} \\ &=& -\lambda_2 \int_{C_{\{1,2\}}} \omega^{n-2} \le 0\end{aligned}$$ It is not hard to see that this argument can be generalized to more than two $C_I$ being non-empty, where $1 \in I$. This completes the proof. Obstruction ----------- In this subsection we prove Theorem \[obstruction-GS\] and Theorem \[obstruction-closed case\]. We first prove Theorem \[obstruction-GS\], in which $(D,\omega)$ is assumed to be $\omega$-orthogonal. Then the proof for Theorem \[obstruction-closed case\], which is similar, is sketched. ### Energy Lower Bound Given an $\omega$-orthogonal symplectic divisor $D =C_1 \cup \dots \cup C_k$ in a 4-manifold $(W,\omega)$, for each $i$, let $N_i$ be a neighborhood of $C_i$ together with a symplectic open disk fibration $p_i: N_i \to C_i$ such that the symplectic connection induced by $\omega$-orthogonal subspace of the fibers rotates the symplectic disc fibres. Hence, for each $i$, we have a well-defined radial coordinate $r_i$ with respect to the fibration $p_i$ such that $C_i$ corresponds to $r_i=0$. Also, $N_i$ are chosen such that the disk fibers are symplectomorphic to the standard open symplectic disk with radius $\epsilon_i$. We also assume $\min_{i=1}^k {r_i}=r_1$ (or simply $r_1=r_2=\dots=r_k$). Moreover, we require $p_{ij}: N_i \cap N_j \to C_{ij}$ to be a symplectic $\mathbb{D}^2 \times \mathbb{D}^2$ fibration such that $p_i|_{N_i \cap N_j}$ is the projection to the first factor and $p_j|_{N_i \cap N_j}$ is the projection to the second factor. Such choice of $p_i$ and $N_i$ exist (See Lemma 5.14 of [@McL12]). Let $(D=C_1 \cup \dots \cup C_k ,\omega)$ be a symplectic divisor with $p_i$ and $N_i$ as above. There exist an $\omega$-compatible almost complex structure $J_N$ on $N=\cup_{i=1}^k N_i$ such that $C_i$ are $J_N$-holomorphic, the projections $p_i$ are $J_N$-holomorphic and the fibers are $J_N$-holomorphic. Using $p_{ij}$, we can define a product complex structure on $\mathbb{D}^2 \times \mathbb{D}^2 = N_i \cap N_j$. Since $p_{ij}$ are compatible with $p_i$ and $p_j$, we can extend this almost complex structure such that $J|_{C_i}$ and $J|_{C_j}$ are complex structures, $(p_l)_*J=J|_{C_l}$ and $J(r_l\partial_{r_l})=\partial_{\vartheta_l}$ for $l=i,j$, where $(r_i,\vartheta_i)$ and $(r_j,\vartheta_j)$ are polar coordinates of the disk fiber for $p_i$ and $p_j$, respectively. Although $\vartheta_i$ and $\vartheta_j$ are not well-defined if the disk bundle has non-trivial Euler class, $\partial_{\vartheta_l}$ are well-defined for $k=i,j$. Since the almost complex structure $J$ is ‘product-like’, $J$ is compatible with the symplectic form $\omega$. We call this desired almost complex structure $J_N$. Now, we consider a partial compactification of $N=\cup_{i=1}^k N_i$ in the following sense. Consider a local symplectic trivialization of the symplectic disk bundle induced by $p_1$, $B_1 \times \mathbb{D}^2$, where $B_1 \subset C_1$ is symplectomorphic to the standard symplectic closed disk with radius $\tau$. We assume that $4\epsilon_1 < \tau$. We recall that $\mathbb{D}^2$ is equipped with a standard symplectic form with radius $\epsilon_1$. Choose a symplectic embedding of $\mathbb{D}^2_{\epsilon_1}$ to $S^2_{\epsilon}$ with $\epsilon$ slightly large than $\epsilon_1$, where $S^2_{\epsilon}$ is a symplectic sphere of area $\pi \epsilon^2$. We glue $\cup_{i=1}^k N_i$ with $B_1 \times S^2_{\epsilon}$ along $B_1 \times \mathbb{D}^2_{\epsilon_1}$ with the identification above. This glued manifold is called $\bar{N}$ and the compatible almost complex structure constructed above can be extended to $\bar{N}$, which we denote as $J_{\bar{N}}$. We further require that $\{ q \} \times S^2_{\epsilon}$ is $J_{\bar{N}}$-holomorphic for every $q \in B_1$. We want to get an energy uniform lower bound for $J$-holomorphic curves representing certain fixed homology class, for those $J$ that are equal to $J_{\bar{N}}$ away from a neighborhood of the divisor $D$. Let $N^{\delta} = \cup_{i=1}^k \{ r_i \le \delta \} \subset \bar{N}$, where $r_i$ are the radial coordinates for the disk fibration $p_i$. \[energy lower bound\] Let $\delta_{\min} >0$ be small and $\delta_{\max}>0$ be slightly less than $\epsilon_1$. Let $q_{\infty} \in B_1 \times S^2_{\epsilon}$ be a point in $\bar{N}-N$ and the first coordinate of which is the center of $B_1$. Let $J$ be an $\omega$-compatible almost complex structure such that $J=J_{\bar{N}}$ on $\bar{N}-N^{\delta_{\min}}$. If $u:\mathbb{C}P^1 \to \bar{N}$ is a non-constant $J$ holomorphic curve passing through $q_{\infty}$, then either $u^*\omega([\mathbb{C}P^1]) > 1.9\pi (\delta_{\max}^2 - \delta_{\max}\delta_{\min}) $ or the image of $u$ stays inside $N^{\delta_{\max}} \cup B_1^{\frac{\tau}{2}} \times S^2_{\epsilon}$, where $B_1^{\frac{\tau}{2}}$ is a closed sub-disk of $B_1$ with the same center but radius $\frac{\tau}{2}$. Let us assume $u^*\omega([\mathbb{C}P^1]) \le 1.9\pi (\delta_{\max}^2 - \delta_{\max}\delta_{\min}) $. Otherwise, we have nothing to prove. Also, we can assume $u$ intersect $\partial N^{\delta_{\min}}$ and $\partial N^{\delta_{\max}}$ transversally, by slightly adjusting $\delta_{\min}$ and $\delta_{\max}$. Passing to the underlying curve if necessary, we can also assume $u$ is somewhere injective. Consider the portion of $u$ inside $Int(B_1^{\frac{\tau}{2}}) \times S^2_{\epsilon}-N^{\delta_{\min}}$. Let $\bar{p_1}$ be the projection to the first factor. We have $\bar{p_1} \circ u|_{u^{-1}(Int(B_1^{\frac{\tau}{2}}) \times S^2_{\epsilon}-N^{\delta_{\min}})}$ is a holomorphic map because $J=J_{\bar{N}}$ in $Int(B_1^{\frac{\tau}{2}}) \times S^2_{\epsilon}-N^{\delta_{\min}}$ and $J_{\bar{N}}$ splits as a product. This map is also proper. Therefore, the map is either a surjection or a constant map. If it is a surjection, then $u^*\omega([\mathbb{C}P^1]) > \pi (\frac{\tau}{2})^2 >1.9\pi (\delta_{\max}^2 - \delta_{\max}\delta_{\min})$. Contradiction. If it is a constant map, then the image of $u|_{u^{-1}(Int(B_1^{\frac{\tau}{2}}) \times S^2_{\epsilon}-N^{\delta_{\min}})}$ is the fiber. Hence, we get $u^*\omega([\mathbb{C}P^1]) \ge \pi (\epsilon^2 - \delta_{\min}^2) > \pi (\delta_{\max}^2 - \delta_{\min}^2)$. If the image of $u$ does not stay inside $N^{\delta_{\max}} \cup B_1^{\frac{\tau}{2}} \times S^2_{\epsilon}$, then there is a point $q_*$ outside this region, lying inside the image of $u$ and $N^{\epsilon_1}-N^{\delta_{\max}}$. We can assume $q_*$ is an injectivity point of $u$. In particular, it also means that $u^{-1}(\bar{N}-N^{\delta_{\min}})$ is disconnected. Consider the connected component $\Sigma$ of $u^{-1}(\bar{N}-N^{\delta_{\min}})$, which contains the preimage of $q_*$ under $u$. Using one of the projections $p_i$, depending on the position of $q_*$, we can identify a neighborhood of $q_*$ as $Int(\mathbb{D}^2_{\delta_{\max}-\delta_{\min}}) \times (Int(\mathbb{D}^2_{\epsilon_1}) -\mathbb{D}^2_{\delta_{\min}})$, where $\mathbb{D}^2_{\delta_{\min}}$ has the same center as $\mathbb{D}^2_{\epsilon_1}$ and they are closed disks with radii $\delta_{\min}$ and $\epsilon_1$, respectively. We call this neighborhood $N_{q_*}$. Also, we still have $J=J_{\bar{N}}$ and $J_{\bar{N}}$ splits as a product in $N_{q_*}$. Similar as before, by projection to the factors, we see that $\int_{\Sigma \cap u^{-1}(N_{q_*})} u^*\omega \ge \min \{ \pi(\delta_{\max}-\delta_{\min})^2, \pi \epsilon_1^2-\pi \delta_{\min}^2 \}$. Therefore, we have $$u^*\omega([\mathbb{C}P^1]) \ge \int_{u^{-1}(Int(B_1^{\frac{\tau}{2}}) \times S^2_{\epsilon}-N^{\delta_{\min}})} u^*\omega+\int_{\Sigma \cap u^{-1}(N_{q_*})} u^*\omega > 1.9\pi (\delta_{\max}^2 - \delta_{\max}\delta_{\min})$$ Contradiction. ### Theorem \[obstruction-GS\] and Theorem \[obstruction-closed case\] We recall the terminology [*GW triple*]{} used in [@McL14]. For a symplectic manifold $(W,\omega)$ (possibly non-compact), a homology class $[A] \in H_2(W;\mathbb{Z})$ and a family of compatible almost complex structures $\mathcal{J}$ such that \(1) $\mathcal{J}$ is non-empty and path connected. \(2) there is a relative compact open subset $U$ of $W$ such that for any $J \in \mathcal{J}$, any compact genus $0$ nodal $J$-holomorphic curve representing the class $[A]$ lies inside $U$. \(3) $c_1(TW)([A])+n-3=0$. $GW_0(W,[A],\mathcal{J})$ is called a GW triple. The key property of a GW triple is the following. \[GW\] Suppose $GW_0(W,[A],\mathcal{J})$ is a GW triple. Then, the GW invariants $GW_0(W,[A],J_0)$ and $GW_0(W,[A],J_1)$ are the same for any $J_0,J_1 \in \mathcal{J}$. In particular, if $GW_0(W,[A],J_0) \neq 0$, then for any $J \in \mathcal{J}$, there is a nodal closed genus $0$ $J$-holomorphic curve representing the class $[A]$. One more technique that we need to use is usually called neck-stretching (See [@BEHWZ03] and the references there-in). Given a contact hypersurface $Y \subset W$ separating $W$ with Liouville flow $X$ defined near $Y$. We call the two components of $W-Y$ as $W^-$ and $W^+$, where $W^-$ is the one containing $D$. Then, $Y$ has a tubular neighborhood of the form $(-\delta,\delta)\times Y$ induced by $X$, which can be identified as part of the symplectization of $Y$. By this identification, we can talk about what it means for an almost complex structure to be translation invariant and cylindrical in this neighborhood. If one choose a sequence of almost complex structures $J_i$ that “stretch the neck” along $Y$ and a sequence of closed $J_i$-holomorphic curve $u_i$ with the same domain such that there is a uniform energy bound, then $u_i$ will have a subseguence ’converge’ to a $J_{\infty}$-holomorphic building. The fact that we need to use is the following. \[SFT\] Suppose we have a sequence of $\omega$-compatible almost complex structure $J_i$ and a sequence of nodal closed genus $0$ $J_i$-holomorphic maps $u_i$ to $W$ representing the same homology class in $W$ such that the image of $u_i$ stays inside a fixed relative compact open subset of $W$. Assume $J_i$ stretch the neck along a separating contact hypersurface $Y \subset W$ with respect to a Liouville flow $X$ defined near $Y$. Assume that the image of $u_i$ has non-empty intersection with $W^-$ and $W^+$, respectively, for all $i$. Then, there are proper genus $0$ $J_{\infty}$-holomorphic maps (domains are not compact) $u_{\infty}^-:\Sigma^- \to W^-$ and $u_{\infty}^+:\Sigma^+ \to W^+$ such that $u_{\infty}^-$ and $u_{\infty}^+$ are asymptotic to Reeb orbits on $Y$ with respect to the contact form $\iota_X \omega$. In our notations, $u_{\infty}^-$ and $u_{\infty}^+$ are certain irreducible components of the top/bottom buildings but not necessary the whole top/bottom buildings. Also, $u_{\infty}^-$ does not necessarily refer to the bottom building because we do not declare the direction of the Liouville flow near $Y$. We are finally ready to prove Theorem \[obstruction-GS\]. The following which we are going to prove implies Theorem \[obstruction-GS\]. \[obstruction\] Let $D \subset (W,\omega)$ be an $\omega$-orthogonal symplectic divisor with area vector $a=(\omega[C_1], \dots, \omega[C_k])$. Let $z=(z_1,\dots,z_k)$ be a solution of $Q_Dz=a$. If one of the $z_i$ is non-positive (resp. positive), there is a small neighborhood $N^{\delta_{\min}} \subset W$ of $D$ such that there is no plumbing $(P(D),\omega|_{P(D)}) \subset (N^{\delta_{\min}},\omega|_{N^{\delta_{\min}}})$ of $D$ being a capping (resp. filling) of its boundary $(\partial P(D),\alpha)$ with $\alpha$ being the contact form, where $\alpha$ is any primitive of $\omega$ defined near $\partial P(D)$ with wrapping numbers $-z$. \[fig Liouville flow\] (0,2) node (yaxis) \[above\] [$y$]{} |- (2,0) node (xaxis) \[right\] [$x$]{}; (0,-2) |- (-2,0); (1,0) .. controls (2,1) and (-0.3,0.2) .. (-1.3,-0.1); (-1.3,-0.1) .. controls (-1.5,-0.3) and (-0.5,-2) .. (1,0); (-0.5,-1.2) node [$(u_{\infty}^-)^{-1}(\partial_{\eta} P(D))$]{}; (0.5,0.5) – (0.4,0.3); (-1,0.2) – (-0.8,-0.2); (0.5,-0.8) – (0.4,-0.5); (0.05,0.05) – (0.15,0.15); (0.05,-0.05)– (0.15,-0.15); (-0.05,-0.05) – (-0.15,-0.15); (-0.05,0.05) – (-0.15,0.15); Figure \[fig Liouville flow\] The direction of the arrows indicates the direction of the Liouville flow $X_{\Sigma^-}$ and $q_0$ is identified with the origin. This is a schematic picture and we do not claim that $(u_{\infty}^-)^{-1}(\partial_{\eta} P(D))$ is connected. We first prove the case that one of the $z_i$ is non-positive. Without loss of generality, assume $z_1 \le 0$. We use the notation in Lemma \[energy lower bound\]. In particular, we have symplectic disk fibration $p_i:N_i \to C_i$, the partial compactification $\bar{N}$ and its $\omega$-compatible almost complex structure $J_{\bar{N}}$. We also have $N^{\delta} = \cup_{i=1}^k \{ r_i \le \delta \} \subset \bar{N}$ and so on. We want to prove the statement with $N^{\delta_{\min}}$ being a small neighborhood of $D$, where $\delta_{\min}$ is so small such that $1.9\pi (\delta_{\max}^2 - \delta_{\max}\delta_{\min}) > \pi \epsilon^2$. We recall that we have $\epsilon$ slightly larger than $\epsilon_1$ and $\epsilon_1$ is slightly larger than $\delta_{\max}$. We also recall that we have $B_1 \times S^2_{\epsilon} \subset \bar{N}$ and $\{q\} \times S^2_{\epsilon}$ has symplecitc area $\pi \epsilon^2$ for any $q \in B_1$. Suppose the contrary, assume $(P(D),\omega|_{P(D)}) \subset (N^{\delta_{\min}},\omega|_{N^{\delta_{\min}}})$ caps its boundary $(\partial P(D),\alpha)$. Let $X$ be the corresponding Liouville flow near $\partial P(D)$. We do a small symplectic blow-up centered at $q_{\infty}$ and this blow-up is so small that it is done in $\bar{N}-N$. We call this blown-up manifold $(\bar{N}',\omega_{\bar{N}'})$ and pick a $\omega_{\bar{N}'}$-compatible almost complex structure $J_{\bar{N}'}$ such that the blow-down map is $(J_{\bar{N}'},J_{\bar{N}})$-holomorphic, the exceptional divisor is $J_{\bar{N}'}$-holomorphic and $J_{\bar{N}'}=J_{\bar{N}}$ in $N$. Let the exceptional divisor be $E$ and the proper transform of the sphere fiber in $B_1 \times S^2_{\epsilon}$ containing $q_{\infty}$ be $A$. We have $GW_0(\bar{N}',[A],J_{\bar{N}'})=1$ by automatic transversality or argue as in the end of Step $4$ of the proof of Theorem 6.1 in [@McL14] for higher dimensions. Since blow-up decreases the area, $\omega_{\bar{N}'}([A])<\omega_{\bar{N}}(Bl_*[A])=\pi \epsilon^2$, where $Bl$ is the blow-down map. This gives us the energy upper bound. By the same argument as in Lemma \[energy lower bound\], we have that for any $\omega_{\bar{N}'}$-compatible almost complex structure $J$ such that $J=J_{\bar{N}'}$ on $\bar{N}'-N^{\delta_{\min}}$, any (nodal) $J$-holomorphic curve representing the class $[A]$ stays inside a fixed relative compact open subset of $\bar{N}'$. As a result, we have a GW triple $GW_0(\bar{N}',[A],\mathcal{J})$, where $\mathcal{J}$ is the family of compatible almost complex structure that equals $J_{\bar{N}'}$ on $\bar{N}'-N^{\delta_{\min}}$. By Proposition \[GW\], we have a nodal closed genus $0$ $J$-holomorphic curve for any $J \in \mathcal{J}$. Now, since $P(D) \subset N^{\delta_{\min}}$, we can choose a sequence $J_i$ in $\mathcal{J}$ such that it stretches the neck along $(\partial P(D),\alpha)$ and $J_i=J_{\bar{N}'}$ very close to $D$. We have a corresponding sequence of nodal closed genus $0$ $J_i$-holomorphic curve $u_i$ to $\bar{N}'$. By Proposition \[SFT\], we have a proper genus $0$ $J_{\infty}$-holomorphic maps $u_{\infty}^-:\Sigma^- \to Int(P(D))$ such that $u_{\infty}^-$ is asymptotic to Reeb orbits on $\partial P(D)$ with respect to the contact form $\alpha$. By the direction of the flow, $u_{\infty}^-$ corresponds to the top building. In general, the top building can be reducible. In our case, since $[A]\cdot [C_1]=1$ and $ [A] \cdot [C_i]=0$ for $i=2,\dots,k$, if the top building is reducible, there is some irreducible component lying inside $Int(P(D))-D$, by positivity of intersection and $D$ being $J_{\infty}$-holomorphic. Since $\omega_{\bar{N}'}$ is exact on $Int(P(D))-D$, any irreducible component lying inside $Int(P(D))-D$ most have non-compact domain and converge asymptotically to Reeb orbits on $Y$. By the direction of the Reeb flow, we get a contradiction by Stoke’s theorem. (cf. Proposition 8.1 of [@McL14] or Step 3 of proof of Theorem 6.1 in [@McL14] or Lemma 7.2 of [@AbSe10]) Therefore, we conclude that there is only one irreducible component which is exactly $u_{\infty}^-$ and the image of $u_{\infty}^-$ intersect $C_1$ transversally once. Let $q_0 \in \Sigma^-$ be the point that maps to the intersection. Let also $\mathbb{D}^2_{q_0}$ be a Darboux disk around $q_0$ and $i=u_{\infty}^-|_{\mathbb{D}^2_{q_0}}$. Now, we want to draw contradiction using the existence of $u_{\infty}^-$. Since $\omega$ is exact on $\partial P(D)$, it is exact in $N^{\delta_{\min}}-D$. Extend $\alpha$ to be a primitive of $\omega$ in $N^{\delta_{\min}}-D$ and we still denote it as $\alpha$. By assumption, $[\omega-d \alpha_c]$ is Lefschetz dual to $\sum\limits_{i=1}^k z_i[C_i]$. In particular, $i^* \alpha$ has wrapping number $-z_1$ around $q_0$ on $\mathbb{D}^2_{q_0}-q_0$. In other words, $[i^*\alpha-\frac{r^2}{2}d\theta]$ is cohomologous to $\frac{-z_1}{2\pi} d\theta$ in $H^1(\mathbb{D}^2_{q_0}-q_0,\mathbb{R})$, where $(r,\theta)$ are the polar coordinates. We have $$i^*\alpha=\frac{r^2}{2}d\theta+\frac{-z_1}{2\pi} d\theta+df$$ for some function $f$ on $\mathbb{D}^2_{q_0}-q_0$. When we choose the extension of $\alpha$ to $N^{\delta_{\min}}-D$, we can choose in a way that $i^*\alpha=\frac{r^2}{2}d\theta+\frac{-z_1}{2\pi} d\theta$ because the image of $i$ is away from $\partial P(D)$. Notice that the $(u_{\infty}^-)^*\omega$ dual of $(u_{\infty}^-)^*\alpha$ defines a Liouville vector field $X_{\Sigma^-}$ on $\Sigma^- -q_0$ away from critical points of $u_{\infty}^-$ (if any). This Liouville flow equals to the component of $X$ in $T(u_{\infty}^-(\Sigma^-))$ near $\partial P(D)$, when we write down the decomposition of $X$ into $T(u_{\infty}^-(\Sigma^-))$-component and its $\omega_{\bar{N}'}$-orthogonal complement component. Here, $T(u_{\infty}^-(\Sigma^-))$ denotes the tangent bundle of the image of $u_{\infty}^-$, which is well-defined near $\partial P(D)$. Since $u_{\infty}^-$ is $J_{\infty}$-holomorphic and it asymptotic converges to Reeb orbits of $(\partial P(D),\alpha)$, $X=-J_{\infty}R_{\alpha}$ has non-zero $T(u_{\infty}^-(\Sigma^-))$-component near $\partial P(D)$. Moreover, since $X$ points inward with respect to $P(D)$, so is $X_{\Sigma^-}$ on $\Sigma^-$ near infinity. In particular, if we take $\partial_{\eta}P(D)$ to be the flow of $\partial P(D)$ with respect to $X$ for a sufficiently small time, then $X_{\Sigma^-}$ is pointing inward along $(u_{\infty}^-)^{-1}(\partial_{\eta} P(D))$. On the other hand, $i^*\alpha=\frac{r^2}{2}d\theta+\frac{-z_1}{2\pi} d\theta$. Therefore, the Liouville vector field $X_{\Sigma^-}$ near $q_0$ equals $(\frac{r}{2}+\frac{-z_1}{2\pi r}) \partial_r$ and hence points outward with respect to $\mathbb{D}^2_{q_0}$ (This is where we use $z_1 \le 0$). As a result, we get a compact codimension $0$ submanifold $\Sigma^-_0$ with boundary in $\Sigma^--q_0$ that has Liouville flow pointing inward along the boundaries and $(u_{\infty}^-|_{\Sigma^-_0})^*\omega$ has a globally defined primitive $(u_{\infty}^-|_{\Sigma^-_0})^*\alpha$. It gives a contradiction by Stoks’s theorem and $\int_{\Sigma^-_0} (u_{\infty}^-)^*\omega \ge 0$. See Figure \[fig Liouville flow\]. For the other case, we assume $z_1$ is positive. In this case, the argument is basically the same but we need to use $u_{\infty}^+$ instead of $u_{\infty}^-$. We have $u_{\infty}^+$ intersecting the exceptional divisor $E$ transversally exactly once. By blowing down, we get a corresponding pseudo-holomorphic map $Bl \circ u_{\infty}^+ $. Let the point on $\Sigma^+$ that maps to the exceptional divisor be $q_0$ as before. Then, the $(u_{\infty}^+)^*\omega$ dual of $(u_{\infty}^+)^*\alpha$ defines a Liouville vector field on $\Sigma^+ -q_0$ away from critical points of $u_{\infty}^+$ (if any). Similar as before, we get a contradiction by Stoke’s theorem and the fact that $u_{\infty}^+$ restricted to any subdomian has non-negative energy. This completes the proof. We remark that the same proof can be generalized to higher dimensional $\omega$-orthogonal divisors. The only thing that need to be changed is to use Monotonicity lemma to get the energy lower bound in Lemma \[energy lower bound\] instead of using surjectivity. We leave it to interested readers. On the other hand, this is not obvious to the authors that how one can remove the $\omega$-orthogonal assumption in Theorem \[obstruction\]. Therefore, Theorem \[obstruction\] is not enough for our application. To deal with this issue, we have the following Theorem \[obstruction-closed case\]. \[obstruction-closed case\] Let $(D,\omega)$ be a symplectic divisor in a closed symplectic manifold $(W,\omega)$. There exists a neighborhood $N$ of $D$ such that there is no concave neighborhood $P(D)$ inside $N$ with the Liouville form $\alpha$ on $\partial P(D)$ having a non-negative wrapping number among its wrapping numbers. The proof is in the same vein of that of Theorem \[obstruction\]. However, we need $W$ to be closed to help us to run the argument this time. \[fig obstruction closed\] (-3.5,0) – (3.5,0) node (xaxis) \[right\] [$C_1$]{}; (-2,0) node \[below\] [$4\tau$]{} – (-2,2); (2,0) node \[below\] [$4\tau$]{} – (2,2); (-2,0.2) – (-2,2); (2,0.2) – (2,2); (-2,2) – (2,2); (-2,0.2) – (2,0.2); (-1.5,1.3) node \[left\] [$N_1$]{}; (-1.5,0.8) – (-1.5,1.7); (1.5,0.8) – (1.5,1.7); (-1.5,1.7) – (1.5,1.7); (-1.5,0.8) – (-0.25,0.8); (0.25,0.8) – (1.5,0.8); (-0.25,0.8) – (-0.25,0.8); (-1,1) – (-1,1.5); (1,1) – (1,1.5); (-1,1.5) – (1,1.5); (-1,1) – (1,1); (-1,1) – (-0.5,1); (0.5,1) – (1,1); (-1,1.3) node \[left\] [$N_2$]{}; (-0.5,1) – (-0.5,1.2); (0.5,1) – (0.5,1.2); (-0.5,1.2) – (0.5,1.2); (-0.25,0.8) – (-0.25,1.2); (0.25,0.8) – (0.25,1.2); (-0.5,1.2) – (-0.25,1.2); (0.25,1.2) – (0.5,1.2); (0,1.35) node [$q_{\infty}$]{}; (0,1.2) node [$\ast$]{}; (-1.5,0.8) –(-1.5,0) node \[below\] [$3\tau$]{}; (1.5,0.8) –(1.5,0) node \[below\] [$3\tau$]{}; (-1,1) –(-1,0) node \[below\] [$2\tau$]{}; (1,1) –(1,0) node \[below\] [$2\tau$]{}; (-0.5,1) –(-0.5,0) node \[below\] [$\tau$]{}; (0.5,1) –(0.5,0) node \[below\] [$\tau$]{}; (-0.25,0.8) –(-0.25,0) node \[below\] [$\frac{\tau}{2}$]{}; (0.25,0.8) –(0.25,0) node \[below\] [$\frac{\tau}{2}$]{}; (2,0.2) –(2.5,0.2) node \[right\] [$\epsilon_5$]{}; (1.5,0.8) –(2.5,0.8) node \[right\] [$\epsilon_4$]{}; (1,1) –(2.5,1) node \[right\] [$\epsilon_3$]{}; (0.5,1.2) –(2.5,1.2) node \[right\] [$\epsilon_2$]{}; (1,1.5) –(2.5,1.5) node \[right\] [$\epsilon_1$]{}; (1,1.7) –(2.5,1.7) node \[right\] [$\epsilon_0$]{}; (2,2) –(2.5,2) node \[right\] [$\epsilon$]{}; Figure \[fig obstruction closed\]: The bottom line represents $C_1$. The region enclosed by thickest solid lines represents $N_2$. The region enclosed by less thick solid lines represents $N_1$ For any $J=J_{\overline{W}}$ on $N_1$, every closed genus $0$ $J$-holomorphic curve $u$ passing through $q_{\infty}$ stays away from $N_2$ if energy of $u$ is sufficiently small. Same as before, we start with an energy estimate. Suppose $\alpha$ is a primitive of $\omega$ defined near $D$ but not defined on $D$. Let the wrapping numbers of $\alpha$ be $-z_i$. Suppose $z_1$ is non-positive. Identify a neighborhood $M_1$ of $C_1$ with a symplectic disk bundle over $C_1$ with symplectic connection rotating the fibers. Assume the fibers are symplecticomorphic to standard symplectic disk of radius $\epsilon$. Pick a point $p$ on $C_1$ such that there is a Darboux disk of radius $4\tau$ and $\tau > c\epsilon$ for some $c>0$ to be determined (we can achieve this by choosing a small $\epsilon$ in advance). Identify the neighborhood of $p$ as a product of closed disks $\mathbb{D}^2_{4\tau} \times \mathbb{D}^2_{\epsilon}$. Choose $\epsilon_i$ for $i=0,1,\dots,5$ to be determined such that $\epsilon > \epsilon_0 > \epsilon_1 > \dots > \epsilon_5 > 0$. Now, cut out the closed region $\mathbb{D}^2_{2\tau} \times (\mathbb{D}^2_{\epsilon_1}-Int(\mathbb{D}^2_{\epsilon_3}))$ from $W$ and call it $W_0$. We partially compactify $W_0$ to be $\overline{W}$ by gluing $W_0$ and $Int(\mathbb{D}^2_{\tau}) \times S^2_{\epsilon_2}$ along $Int(\mathbb{D}^2_{\tau}) \times Int(\mathbb{D}^2_{\epsilon_3})$ by identifying $Int(\mathbb{D}^2_{\epsilon_3})$ with a choose of symplectic embedding to $S^2_{\epsilon_2}$, where $S^2_{\epsilon_2}$ is a symplectic sphere of symplectic area $\pi \epsilon_2^2$. We define $N_3 \subset N_2 \subset N_1 \subset N \subset \overline{W}$ to be the following subset of $\overline{W}$. Notice that $N$, $N_1$ and $N_2$ are closed (but not compact) and $N_3$ is open. $$N=(\mathbb{D}^2_{4\tau} \times \mathbb{D}^2_{\epsilon}-\mathbb{D}^2_{2\tau} \times (\mathbb{D}^2_{\epsilon_1}-Int(\mathbb{D}^2_{\epsilon_3})))\cup Int(\mathbb{D}^2_{\tau}) \times S^2_{\epsilon_2}$$ $$N_1=(\mathbb{D}^2_{4\tau} \times (\mathbb{D}^2_{\epsilon}-Int(\mathbb{D}^2_{\epsilon_5}))-\mathbb{D}^2_{2\tau} \times (\mathbb{D}^2_{\epsilon_1}-Int(\mathbb{D}^2_{\epsilon_3})))\cup Int(\mathbb{D}^2_{\tau}) \times S^2_{\epsilon_2}$$ $$\begin{aligned} N_2&=&((\mathbb{D}^2_{3\tau} \times (\mathbb{D}^2_{\epsilon_0}-Int(\mathbb{D}^2_{\epsilon_4}))-\mathbb{D}^2_{2\tau} \times (\mathbb{D}^2_{\epsilon_1}-Int(\mathbb{D}^2_{\epsilon_3}))-Int(\mathbb{D}^2_{\frac{\tau}{2}}) \times Int(\mathbb{D}^2_{\epsilon_3})) \\ &&\cup (Int(\mathbb{D}^2_{\tau})-Int(\mathbb{D}^2_{\frac{\tau}{2}})) \times S^2_{\epsilon_2}\end{aligned}$$ $$N_3=Int(\mathbb{D}^2_{\tau}) \times S^2_{\epsilon_2}-Int(\mathbb{D}^2_{\tau}) \times \mathbb{D}^2_{\epsilon_3}$$ Then, $\overline{W}$ is our desired manifold to run the argument above (See Figure \[fig obstruction closed\]). Let $J_{\overline{W}}$ be an $\omega$-compatible almost complex structure on $\overline{W}-N_3$ such that $J_{\overline{W}}$ is split as a product in $N-N_3$. Then, extend $J_{\overline{W}}$ naturally over $N_3$ such that it is ’product-like’ making the $S^2_{\epsilon_2}$ sphere fibers $J_{\overline{W}}$-holomorphic. We still call this $J_{\overline{W}}$. Let $q_{\infty} \in N_3$ be a point in $N$ such that it lies in the $S^2_{\epsilon_2}$ sphere fiber at $p$. Similar as before, for any $\omega$-compatible almost complex structure $J$ such that $J=J_{\overline{W}}$ on $N_1$ and any closed genus $0$ $J$-holomorphic curve $u$ to $\overline{W}$ passing through $q_{\infty}$, we must have the image of $u$ stays inside $\overline{W}-N_2$ or the energy of $u$, $\int_{\mathbb{C}P^1}u^*\omega$, greater than a lower bound depending on $\epsilon_5$ (once $c$ and $\epsilon_i$ for $i=0,\dots,4$ are determined). It should be convincing that one can choose a choice of $c$ and $\epsilon_i$ such that any $J$-holomorphic curve representing the class $[S^2_{\epsilon_2}]$, the spherical fiber class at $p$, and passing through $q_{\infty}$ has to stay inside $\overline{W}-N_2$. Since $W$ is closed, $\overline{W}-N_2$ is a relative compact open subset which we can use to define the GW triple below. We claim that $D$ does not have a concave neighborhood $P(D)$ lying inside $W-N_1$ with the Liouville contact form $\alpha'$ defined near $\partial P(D)$ having the same wrapping numbers as that of $\alpha$. Suppose on the contrary, there were such a $P(D)$. Then, by a $C^0$ perturbation near the intersection points of $C_i$ in $D$, we can assume that $D$ is $\omega$-orthogonal and it still lies inside $P(D)$. We do a sufficiently small blow-up at $q_{\infty}$ as before and let the proper transform of the $S^2_{\epsilon_2}$ fiber containing $q_{\infty}$ be $A$. We have a GW triple $GW_0(\overline{W},[A],\mathcal{J})$, where $\mathcal{J}$ is the family of $\omega$-compatible almost complex structure $J$ such that $J=J_{\overline{W}}$ on $N_1$. Since $D$ is now $\omega$-orthogonal, we can find $J\in \mathcal{J}$ such that $D$ is $J$-holomorphic (notice that, there exists symplectic divisor with no almost complex structure making all irreducible components pseudo-holomorphic simultaneously). Then, we find a sequence $J_i \in \mathcal{J}$ making $D$ $J_i$-holomorphic for all $i$ and stretch the neck along $\partial P(D)$ as before to draw contradiction. Uniqueness ---------- In this subsection we show that any contact structure obtained from the GS construction is contactomorphic to one from the McLean’s construction. Then Theorem \[uniqueness-GS\] follows from the uniqueness of McLean’s construction. In fact, we are going to prove the following more precise version of Theorem \[uniqueness-GS\]. \[canonical contact structure\_proof\] Suppose $D=\cup_{i=1}^k C_i$ is a symplectic divisor with each intersection point being $\omega$-orthogonal such that the augmented graph $(\Gamma,a)$ satisfies the positive (resp. negative) GS criterion. Then, the contact structures induced by the positive (resp. negative) GS criterion are contactomorphic, independent of choices made in the construction and independent of $a$ as long as $(\Gamma,a)$ satisfies positive GS criterion. Moreover, if $D$ arises from resolving an isolated normal surface singularity, then the contact structure induced by the negative GS criterion is contactomorphic to the contact structure induced by the complex structure. On the other hand, if $D$ is the support of an effective ample line bundle, then the contact structure induced by the positive GS criterion is contactomorphic to that induced by a positive hermitian metric on the ample line bundle. ### Uniqueness of McLean’s construction We recall the uniqueness part of McLean’s construction, which can be regarded as a more complete version of Proposition \[McLean0\]. \[McLean\]\[cf. Corollary 4.3 and Lemma 4.12 of [@McL14]\] Suppose $f_0,f_1: W-D \to \mathbb{R}$ are compatible with $D$ and $D$ is a symplectic divisor with respect to both $\omega_0$ and $\omega_1$ having positive transversal intersections. Suppose $\theta_j \in \Omega^1(W-D)$ is a primitive of $\omega_j$ on $W-D$ such that it has positive (resp. negative) wrapping numbers for all $i=1, \dots, k$ and for both $j=0,1$. Suppose, for both $j=0,1$, there exist $g_j: W-D \to \mathbb{R}$ such that $df(X^j_{\theta_j+dg_j}) > 0$ (resp. $df(-X^j_{\theta_j+dg_j}) > 0$) near $D$, where $X^j_{\theta_j+dg_j}$ is the dual of $\theta_j+dg_j$ with respect to $\omega_j$. Then, for sufficiently negative $l$, we have that $(f_0^{-1}(l),\theta_0+dg_0|_{f_0^{-1}(l)})$ is contactomorphic to $(f_1^{-1}(l),\theta_1+dg_1|_{f_1^{-1}(l)})$. Moreover, when $(W,D,\omega)$ arises from resolving a normal isolated surface singularity, then the link with contact structure induced from complex line of tangency is contactomorphic to this canonical contact structure. The first thing to note is that the choice of $g_j$ for $j=0,1$ always exist (cf. Proposition \[McLean0\] above, Proposition 4.1 and Proposition 4.2 in [@McL14]). Moreover, by the definition of compatible function, it also always exist. In other words, Proposition \[McLean\] implies that in dimension four, for any symplectic form $\omega_0$ and $\omega_1$ making $D$ a divisor such that they have primitives $\theta_0$ and $\theta_1$ on $W-D$ with positive (resp. negative) wrapping numbers, the contact structures constructed by McLean’s construction with respect to $\theta_0$ and $\theta_1$ are contactomorphic. Proposition \[McLean\] is literally not exactly the same as Corollary 4.3 and Lemma 4.12 in [@McL14] so we want to make clear why it is still valid after we have made the changes. We remark that if $\theta_0 $ and $\theta_1$ have positive wrapping numbers, then $\theta_t=(1-t)\theta_0+t\theta_1$ has positive wrapping numbers for all $t$ and $f_t=(1-t)f_0+tf_1$ is compatible with $D$ for all $t$. As a symplectic divisor, we always assume $C_i$ have positive orientations with respect to the symplectic form for all $i$. In other words, both $\omega_0|_{C_i}$ and $\omega_1|_{C_i}$ are positive and hence $D$ is a symplectic divisor with respect to $d\theta_t$ for all $t$. Therefore, we get a deformation of $\omega_t$ and the first half of Proposition \[McLean\] with $\theta_0 $ and $\theta_1$ having positive wrapping numbers follows from Corollary 4.3 of [@McL14]. The analogous statement for the first half of Proposition \[McLean\] with $\theta_0 $ and $\theta_1$ having negative wrapping numbers follows similarly as in the case where $\theta_0 $ and $\theta_1$ have positive wrapping numbers. On the other hand, Lemma 4.12 of [@McL14] requires that the resolution is obtained from blowing up. Although there exist a resolution such that it is not obtained from blowing up in complex dimension three or higher, every resolution for an isolated normal surface singularity can be obtained by blowing up the unique minimal model, where the minimal model is obtained from blowing up the singularity. Therefore, the second half of Proposition \[McLean\] follows. ### Proof of Theorem \[uniqueness-GS\] To prove Proposition \[canonical contact structure\_proof\] using Proposition \[McLean\], the remaining task is to construct an appropriate disc fibration having a connection rotating fibers for the local models in the GS-construction. Then, the constructions of $\theta$, $f$ and $g$ will be automatic. We give the fibration in the following Lemma. \[canonical contact structure\_tech\] Let $z_1'$ and $z_2'$ be two positive numbers. Let $\mu: \mathbb{S}^2 \times \mathbb{S}^2 \to [z_1',z_1'+1] \times [z_2',z_2'+1]$ be the moment map of $\mathbb{S}^2 \times \mathbb{S}^2$ onto its image. Fix a small $\epsilon >0$ and let $D_1=\mu^{-1}(\{ z_1'\} \times [z_2', z_2' + 2\epsilon])$ be a symplectic disc. Fix a number $s \in \mathbb{R}$ first and then let $\delta>0$ be sufficiently small. Let $Q$ be the closed polygon with vertices $(z_1',z_2'), (z_1'+\delta,z_2'), (z_1'+\delta,z_2'+2\epsilon-s\delta), (z_1',z_2'+2\epsilon)$. Using the $(p_1,q_1,p_2,q_2)$ coordinates described in the GS-construction above, we define a map $\pi: \mu^{-1}(Q) \to D_1$ by sending $(p_1,q_1,p_2,q_2)$ to $(z_1',*,p_2+\frac{(2\epsilon-t(p_1,p_2)-\rho(t(p_1,p_2)))p_1}{\delta}, q_2)$, where $\rho: [0, 2\epsilon] \to [0,2\epsilon-s\delta]$ is a smooth strictly monotonic decreasing function with $\rho(0)=2\epsilon-s\delta$ and $\rho(2\epsilon)=0$ such that $\rho'(t)=-1$ for $t\in [0, \epsilon]$ and near $t=2\epsilon$. This can be done as $\delta$ is sufficiently small. Moreover, $t(p_1,p_2)$ is the unique $t$ solving $p_2-(z_2'+2\epsilon-t)=\frac{(\rho(t)-(2\epsilon-t))(p_1-z_1')}{\delta}$ and $*$ means that there is no $q_1$ coordinate above $(z_1',x)$ for any $x$ so $q_1$ coordinate is not relevant. Then, we have that $\pi$ gives a symplectic fibration with each fibre symplectomorphic to $(\mathbb{D}^2_{\sqrt{2\delta}},\omega_{std})$ and the symplectic connection of $\pi$ has structural group lies inside $U(1)$. Moroever, fibres are symplectic orthogonal to the base. \[fig disk fibration\] (0,4.5) node (yaxis) \[above\] [$y$]{} |- (6,0) node (xaxis) \[right\] [$x$]{}; (1.5,0.9) coordinate (a\_1) – (4.5,0.9) coordinate (a\_2); (1.5,0.9) coordinate (b\_1) – (1.5,3.9) coordinate (b\_2); (2.4,1.8) coordinate (d\_1)– (5.4,1.8) coordinate (d\_2); (2.4,0.9) coordinate (e\_1)– (2.4,3) coordinate (e\_2); (2.2,2.5) node \[left\] [$Q$]{}; (2.4,3) coordinate (j\_1)– (2.4,4.3) coordinate (j\_2); (1.5,3.9) coordinate (f\_1)– (2.4,4.3) coordinate (f\_2); (1.5,2.4) coordinate (g\_1)– (2.4,2.8) coordinate (g\_2); (1.5,3.6) – (2.4,4); (1.5,3.4) – (2.4,3.8); (1.5,3.2) – (2.4,3.6); (1.5,3.0) – (2.4,3.4); (1.5,2.8) – (2.4,3.2); (1.5,2.6) – (2.4,3); (1.5,1.1) – (2.4,1.1); (1.5,1.3) – (2.4,1.3); (1.5,1.5) – (2.4,1.6); (1.5,1.7) – (2.4,1.9); (1.5,1.9) – (2.4,2.2); (1.5,2.1) – (2.4,2.5); (3,0.9) coordinate (h\_1)– (3.9,1.8) coordinate (h\_2); (4.5,0.9) coordinate (i\_1)– (5.4,1.8) coordinate (i\_2); (c) at (intersection of a\_1–a\_2 and b\_1–b\_2); (yaxis |- c) node\[left\] [$z_{2}'$]{} -| (xaxis -| c) node\[below\] [$z_{1}'$]{}; (1.5,2.4) – (0,2.4) node\[left\] [$z_{2}'+\epsilon$]{}; (1.5,3.9) – (0,3.9) node\[left\] [$z_{2}'+2\epsilon$]{}; (3,0.9) – (3,0) node\[below\] [$z_{1}'+\epsilon$]{}; (4.5,0.9) – (4.5,0) node\[below\] [$z_{1}'+2\epsilon$]{}; Figure \[fig disk fibration\]. The arrows give the schematic picture for the projection $\pi$. First, we want to explain what $t(p_1,p_2)$ means geometrically. $\rho(2 \epsilon -t)$ is an oriented diffeomorphism from $[0, 2\epsilon] \to [0,2\epsilon-s\delta]$ so it can be viewed as a diffeomorphism from the left edge of $Q$ to the right edge of $Q$. $p_2-(z_2'+2\epsilon-t)=\frac{(\rho(t)-(2\epsilon-t))(p_1-z_1')}{\delta}$, which we call $L_t$, is the equation of line joining the point $(z_1',z_2'+2\epsilon-t)$ and $(z_1'+\delta,z_2'+\rho(t))$. Therefore, for a point $(p_1,p_2)$, $t(p_1,p_2)$ is such that $L_{t(p_1,p_2)}$ contains the point $(p_1,p_2)$. Moreover, $p_2+\frac{(2\epsilon-t(p_1,p_2)-\rho(t(p_1,p_2)))p_1}{\delta}$ is the $p_2$-coordinate of the intersection between line $L_{t(p_1,p_2)}$ and the left edge of $Q$, $\{ p_1=z_1' \}$. See Figure \[fig disk fibration\]. To prove the Lemma, we pick $\kappa$ close to $2 \epsilon$ from below such that $\rho'(t)=-1$ for all $t \in [\kappa, 2\epsilon]$. Let $\Delta$ be $\pi^{-1}(\mu^{-1}(\{ z_1' \} \times [z_2'+2\epsilon-\kappa,z_2'+2\epsilon]))$ We give a smooth trivialization of $\pi|_{\Delta}$ as follows. Let $\Phi: [0,\kappa] \times \mathbb{R}/2\pi \mathbb{Z} \times \mathbb{D}^2_{\sqrt{2\delta}} \to \Delta$ be given by sending $(t,\vartheta_1,\tau,\vartheta_2)$ to $(p_1,q_1,p_2,q_2)=(z_1'+\tau,-s \vartheta_1+\vartheta_2,(z_2'+2\epsilon-t)+\frac{(\rho(t)-(2\epsilon-t))\tau}{\delta},-\vartheta_1)$, where $t,\vartheta_1$ are the coordinates of $[0,\kappa]$ and $\mathbb{R}/2\pi$, respectively, and $(\tau=\frac{r^2}{2},\vartheta_2)$ is such that $(r,\vartheta_2)$ is the standard polar coordinates of $\mathbb{D}^2_{\sqrt{2\delta}}$. In particular, $\tau \in [0,\delta]$. Note that, $\Phi$ is well-defined and it is a diffeomorphism. Let $\pi_{\Phi}:[0,\kappa] \times \mathbb{R}/2\pi \mathbb{Z} \times \mathbb{D}^2_{\sqrt{2\delta}} \to [0,\kappa] \times \mathbb{R}/2\pi \mathbb{Z} \times \mathbb{D}^2_{\sqrt{2\delta}}$ be the projection to the first two factors. Then, we have $\pi \circ \Phi= \Phi \circ \pi_{\Phi}$. Notice that, when $\tau=0$, the $\vartheta_2$-coordinate degenerates and it corresponds to $p_1=z_1'$ and the $q_1$-coordinate degenerates. To investigate this fibration under the trivialization, we have $$\begin{aligned} \Phi^{*}\omega & = \Phi^{*}(dp_1 \wedge dq_1+dp_2 \wedge dq_2) \\ & = d\tau \wedge (-sd\vartheta_1+d\vartheta_2) \\ & \quad + (-dt+\frac{\tau}{\delta}dt+\frac{\rho'(t)\tau}{\delta}dt+ \frac{\rho(t)-(2\epsilon-t)}{\delta}d\tau) \wedge (-d\vartheta_1)\\ & = (1-\frac{\tau}{\delta}-\frac{\rho'(t)\tau}{\delta})dt \wedge d\vartheta_1 + d\tau \wedge d\vartheta_2 + (\frac{2\epsilon-t-\rho(t)}{\delta}-s) d\tau \wedge d\vartheta_1\end{aligned}$$ For a fibre, we have $t$ and $\vartheta_1$ being contant so the the symplectic form restricted on the fibre is $ d\tau \wedge d\vartheta_2$, which is the standard one. Hence, each fibre is symplectomorphic to $(\mathbb{D}^2_{\sqrt{2\delta}},\omega_{std})$. When $\tau=0$, the symplectic form equals $dt \wedge d\vartheta_1$ so the base is symplectic and fibres are symplectic orthogonal to the base. Moreover, the vector space that is symplectic orthogonal to the fibre at a point is spanned by $\partial_{t}$ and $\partial_{\vartheta_1}-(\frac{2\epsilon-t-\rho(t)}{\delta}-s)\partial_{\vartheta_2}$ so the symplectic connection has structural group inside $U(1)$. Finally, we remark that $\rho(0)=2\epsilon-s\delta$ and $\rho'(t)=-1$ when $t$ is close to $0$, hence $\Phi^{*}\omega=dt \wedge d\vartheta_1 + d\tau \wedge d\vartheta_2$. Therefore, when $t$ is close to $0$, the trivialization $\Phi$ actually coincide with the gluing symplectomorphism in the GS construction from preimage of $R_{e_{\alpha \beta},v_{\alpha}}$ to $[x_{v_{\alpha}, e_{\alpha \beta}}-2\epsilon,x_{v_{\alpha}, e_{\alpha \beta}}-\epsilon] \times \mathbb{R}/2\pi \mathbb{Z} \times \mathbb{D}^2_{\sqrt{2\delta}}$ , up to a translation in $t$-coordinate. On the other hand, $\pi|_{\mu^{-1}(Q)-\Delta}$ is clearly a symplectic fibration with all the desired properties described in the Lemma as it corresponds to the trivial projection by sending $(p_1,q_1,p_2,q_2)$ to $(z_1', *, p_2,q_2)$. This finishes the proof of this Lemma. We remark that the disc fibration above gives a fibration on the local models $N_{e_{\alpha \beta}}$ and it is compatible with the trivial fibration on the local models $N_{v_{\alpha}}$ so they give a well-defined fibration after gluing all the local models $N_{v_{\alpha}}$ and $N_{e_{\alpha \beta}}$. Now, we are ready to prove Proposition \[canonical contact structure\_proof\]. Let $(D=\cup_{i=1}^k C_i,W, \omega)$ be a symplectic plumbing. First, we assume the intersection form of $D$ is negative definite (or equivalently, the augmented graph satisfies the negative GS criterion). By [@GaSt09], $D$ satisfies the negative GS criterion. Therfore, by possibly shrinking $W$, we can assume $W$ is a symplectic plumbing constructed from the negative GS criterion. A byproduct of the construction is the existence of a primitive of $\omega$ on $W-D$, $\theta$, given by contracting $\omega$ by the Liouville vector field. From the construction, in the $N_{v_{\alpha}}$ local model, we have $\theta=\iota_{\bar{X}_{v_{\alpha}}+(\frac{r}{2}+\frac{z_{v_{\alpha}}'}{r})\partial_r} (\bar{\omega}_{v_{\alpha}}+rdr \wedge d\vartheta_2)=\iota_{\bar{X}_{v_{\alpha}}}\bar{\omega}_{v_{\alpha}}+(\frac{r^2}{2}+z_{v_{\alpha}}')d\vartheta_2$. When we restrict it to a fibre, we can see that the wrapping numbers of $\theta$ with respect to $C_{v_{\alpha}}$ is $2 \pi z_{v_{\alpha}}'$, which is positive. Here, $C_{v_{\alpha}}$ is the smooth symplectic submanifold corresponding to the vertex $v_{\alpha}$. Note that we have $\lambda=-z=2\pi z'$ in our convention above. By tracing back the negative GS construction, we see that Lemma \[canonical contact structure\_tech\] provides a desired symplectic fibrations we needed to apply Proposition \[McLean\]. In particular, this symplectic fibrations give us well-defined $r_i$-coordinates near the divisor. As a result, one can set $f=\sum\limits_{i=1}^k \log(\rho(r_i))$ and $g=0$ and get that $df(X_\theta) > 0$ near $D$ and $(f^{-1}(l),\theta|_{f^{-1}(l)})$ is precisely the contact manifold obtained from the negative GS criterion. In particular, $(f^{-1}(l),\theta|_{f^{-1}(l)})$ is the canonical one with respect to $(W,D)$. If we have made another set of choices in the construction, we get that $(\bar{f}^{-1}(l),\theta|_{\bar{f}^{-1}(l)})$ is the canonical one with respect to $(\bar{W},\bar{D})$. Then, since $(W,D)$ is diffeomorphic to $(\bar{W},\bar{D})$, we can pull back the compatible function and the $1$-form on $(\bar{W},\bar{D})$ to $(W,D)$. By Proposition \[McLean\], the two contact manifolds are contactomorphic. Same argument works to show that this contact structure is independent of symplectic area $a$ as long as $(\Gamma,a)$ still satisfies GS criteria. Also, when $D$ is arising from isolated normal surface singularity, contact structure of its link is contactomorphic that induced by GS-criterion, by Proposition \[McLean\], again. This finishes the case when $D$ is negative definite. Now, we assume that $(D,\omega)$ satisfies the positive GS criterion and $D$ is $\omega$-orthogonal. By the same reasoning as bove, the contact structure induced by the positive GS criterion is independent of choices. Suppose $D$ is also the support of an effective ample line bundle. Pick a hermitian metric $\| . \|$ and a section $s$ with zero being $\sum\limits_{i=1}^k z_i C_i$, where $z_i > 0$. Let $f=-\log \| s\|$, $\theta= -d^cf$ and $\omega= d\theta$, where $d^cf=df \circ J_{std}$. Then, $\theta$ induces a contact structure on the boundary of plumbing of $(D,\omega)$ with negative wrapping numbers (See Lemma 5.19 of [@McL12]). Moreover, $f$ is compatible with $D$ and $df(-X_{\theta}) > 0$ near $D$ (See Lemma 2.1 of [@McL12] or Lemma 4.12 of [@McL14]). Hence the contact structure induced by $\theta$ is contactomorphic to the canonical one by Proposition \[McLean\], which is contactomorphic to the one induced by the positive GS criterion. Examples of Concave Divisors {#Comments on the Induced Contact Structure} ---------------------------- In this subsection, we are going to see five illuminating examples. The first one is the simplest kind of symplectic divisor. The second one illustrates Theorem \[obstruction-GS\] is no longer valid if the plumbing chosen is not close to the divisor. In particular, there is a concave divisor which admits a convex neighborhood but it is not a convex divisor. The third one is a frequently used example when studying Lefschetz fibration. The forth one is a concave divisor with non-fillable contact structure on the boundary. The last one shows that the constructed contact structure on the boundary is not necessarily contactomorphic to the standard one that one might expect if the divisor is concave. \[single vertex\] A symplectic surface with self-intersection $n$ admits a concave (resp convex) boundary when $n>0$ (resp $n<0$). When $n=0$, a symplectic form cannot make both the surface symplectic and the restriction to boundary be exact so it has no convex or concave neighborhood. In fact, more is true, by a result of Eliasberg [@El90], $\mathbb{S}^1 \times \mathbb{S}^2$ cannot be a convex boundary of any symplecyic form on $\mathbb{D}^2 \times \mathbb{S}^2$. In contrast, althought a symplectic torus with self-intersection zero has no concave nor convex neighborhood, a Lagrangian torus has self-intersection zero and has a convex neigborhood. ([@Mc91]) In [@Mc91], McDuff constructed a symplectic form on $(S\Sigma_g \times [0,1],\omega)$ such that it has disconnected convex boundary, where $S\Sigma_g$ is a circle boundle of a genus $g$ surface and $g>1$. The contact structure near $S\Sigma_g \times \{0\}$ is contactomorphic to the concave boudary near a self-intersection $2g-2$ symplectic genus g surface. The contact structure near $S\Sigma_g \times \{1\}$ is contactomorphic to the convex boundary near a Lagrangian genus g surface. If one glues a symplectic closed disc bundle $P(D)$ over a symplectic genus g surface $D$ with $(S\Sigma_g \times [0,1],\omega)$ along $S\Sigma_g \times \{0\}$. One gets a plumbing of the surface with convex boundary. This suggests that a symplectic genus $g$ ($g > 1$) surface can have both concave and convex neighborhood, depending on the symplectic form and the neighborhood. Notice that $D$ is trivially $\omega$-orthogonal. It illustrates that the assumption on $P(D)$ being sufficiently close to $D$ in Theorem \[obstruction-GS\] cannot be dropped. Moreover, by Theorem \[obstruction-GS\], $D$ is a concave divisor but not a convex divisor although it admits convex neighborhood. \[section fiber\] Suppose there is a symplectic Lefschetz fibration $(X,\omega)$ over $\mathbb{CP}^1$ with generic fibre $F$ and a symplectic section $S$ of self-intersection $-n$ ($n \ge 0$). Let $D=F \cup S$, then the augmented graph of $D$ always satisfies the positive GS criterion regardless the area weights of the surfaces. Suppose also that $S$ is perturbed to be $\omega$-orthogonal to $F$. Then, Proposition \[MAIN2\] (or, Proposition \[McLean0\] if one does not want to perturb) shows that $D$ is a concave divisor. In other words, the complement of a concave neighborhood of $D$ is a convex filling of its boundary. This fits well to the well-known fact that the complement of a regular neighborhood of $D$ is a Stein domain. Moreover, this construction has been successfully used to find exotic Stein fillings [@AkEtMaSm08]. \[b\_2\^+\] Let $(\Gamma,a)$ be an augmented graph satisfying the positive GS criterion and $D$ be a realization. Suppose there are two genera zero vertices with self-intersection $s_1,s_2$ such that either \(i) they are adjacent to each other and $s_1 >s_2\ge 1$, or \(ii) they are not adjacent to each other with $s_1 \ge 1$ and $s_2 \ge 0$. Then, $D$ is a concave divisor but not a capping divisor. Suppose on the contrary, the boundary has a convex fillings $Y$. Then, we can glue $D$ with $Y$ to obtain a closed symplectic 4 manifold $W$. By McDuff’s theorem [@Mc90] (see also Theorem \[McDuff\] below), $W$ is rational or ruled and hence have $b_2^+ =1$. For (i), the two spheres generates a positive two dimensional subspace of $H_2(W)$ with respect to the intersection form. Thus, we get a contradiction. For (ii), it suffices to consider tha case $s_1=1$ and $s_2=0$. By the Theorem in [@Mc90], one can assume the sphere with self-intersection $1$ represent the hyperplane class $H$, with respect to an orthonormal basis $\{H,E_1,\dots,E_n \}$ for $H_2(W)$. The two spheres being disjoint implies the one with self-intersection $0$ has homology class being a linear combination of exceptional classes. Since the sphere is symplectic, the linear combination is non-trivial. Thus, we get a contradiction. Let $\Gamma$ be the graph in Example \[2-1\]. $$Q_{\Gamma}= \left( \begin{array}{cc} 2 & 1 \\ 1 & 1 \end{array} \right),$$ Then the boundary fundamental group of $\Gamma$ is the free group generated by $e_1$ and $e_2$ modulo the relations $e_1e_2^1=e_2^1e_1$, $1=e_1^2e_2$ and $1=e_1e_2$ (See Lemma \[representation\] below). Therefore, the boundary of the plumbing according to $\Gamma$ has trivial fundamental group and hence diffeomorphic to a sphere. It is easily see that the corresponding augmented graph $(\Gamma,a)$ satisfies the positive GS criterion if and only if the area weights satisfy $a_1 < a_2 < 2a_1$, where $a_i$ is the area weight of $v_i$. In other words, if $a_1 < a_2 < 2a_1$, by Proposition \[MAIN2\] and Lemma \[b\_2\^+\], we get an overtwisted contact structure on $S^3$ ($S^3$ has only one tight contact structure which is fillable). \[non-standard example\] There is a capping divisor $D$ with graph as in the following Figure, by Theorem \[main classification theorem\]. However, $D$ is not conjugate to any other divisor (See Definition \[Conjugate Definition\_Divisor\]), by Lemma \[first chern class\]. $\partial P(D)$ is diffeomorphic to the boundary of the plumbing of the resolution of a tetrahedral singularity and the latter one which is equipped with a standard contact structure has a conjugate. Therefore, the fillable contact strucutre on $\partial P(D)$ is not the standard one. Applying the method in [@Li08],[@BhOn12] and [@St13], one can obtain a finiteness result on the number of minimal symplectic manifolds that can be compactified by $D$, up to diffeomorphism (See Proposition \[bounds\]). $$\xymatrix{ \bullet^{-3} \ar@{-}[r]& \bullet^{-2} \ar@{-}[r] \ar@{-}[d]& \bullet^{-2} \ar@{-}[r] & \bullet^{1} \\ &\bullet^{-2} \\ }$$ Operations on Divisors {#Operation on Divisors} ====================== In this section we first apply the inflation operation to establish Theorem \[MAIN\]. Then, we explain in details the resulting flowchart and illustrate how to reduce the classification problem into a problem of graphs. Finally, we introduce blow up and the [**dual**]{} blow up operations, which are essential to the next section. Theorem \[MAIN\] ---------------- The proof of Theorem \[MAIN\] involes two inputs. The first one is a linear algebraic lemma, which is simple but important. The second one which is called inflation lemma allows us to deform the symplectic form to our desired one so as to apply Propositon \[MAIN2\]. ### A key lemma The following linear algebraic Lemma related to the positive GS criterion will be crucial. \[trichotomy\] Let $Q$ be a k by k symmetric matrix with off-diagonal entries being all non-negative. Assume that there exist $a \in (0,\infty)^k$ such that there exist $z \in \mathbb{R}^k$ with $Qz=a$. Suppose also that $Q$ is not negative definite. Then, there exists $z \in (0,\infty)^k$ such that $Qz \in (0,\infty)^k$. When $k=1$, it is trivial. Suppose the statement is true for (k-1) by (k-1) matrix and now we consider a k by k matrix $Q$. Let $q_{i,j}$ be the $(i,j)^{th}$-entry of $Q$. First observe that if $q_{i,i} \ge 0$, for all $i=1,\dots,k$, then the statement is true. The reason is that if each row has a positive entry, then $z=(1,\dots,1)$ works. If there exist a row with all $0$, then there is no $a \in (0,\infty)^k$ such that there exist $z \in \mathbb{R}^k$ with $Qz=a$. Therefore, we might assume $q_{k,k}<0$. Let $l_j=-\frac{q_{k,j}}{q_{k,k}} \ge 0$, for $j <k$, and let $B$ be the lower triangular matrix given by $$b_{i,j} = \left \{ \begin{array}{l l} \delta_{i,j} & \quad \text{if $i \neq k$ or $(i,j)=(k,k)$}\\ l_j & \quad \text{if $i = k$ and $(i,j) \neq (k,k)$} \end{array} \right.$$ Let $M=B^TQB$. Then, $$m_{i,j} = \left \{ \begin{array}{l l} q_{i,j}-\frac{q_{i,k}q_{k,j}}{q_{k,k}} & \quad \text{if $(i,j) \neq (k,k)$}\\ q_{k,k} & \quad \text{if $(i,j) = (k,k)$} \end{array} \right.$$ In particular, $m_{i,k}=m_{k,j}=0$, for all $i$ and $j$ less than $k$. We can write $M$ as a direct sum of a k-1 by k-1 matrix $M'$ with the 1 by 1 matrix $q_{k,k}$ in the obvious way. Notice that the off diagonal entries of $M'$ are all non-negative. Let $a=(a_1, \dots,a_k)^T$ and $z=(z_1,\dots,z_k)^T$ such that $Qz=a$. Let also $\overline{z}=(\overline{z}_1,\dots,\overline{z}_k)^T=B^{-1}z$ and $\overline{a}=(\overline{a}_1, \dots,\overline{a}_k)^T=B^Ta$. Then, $Qz=a$ is equivalent to $M\overline{z}=\overline{a}$. Here, $\overline{z}_i=z_i$, for $i <k$, and $\overline{z}_k=z_k-\sum\limits_{i=1}^{k-1}l_iz_i$. On the other hand, $\overline{a}_i=a_i+l_ia_k$, for all $i < k$, and $\overline{a}_k=a_k$. By assumption, there exist $a \in (0,\infty)^k$ such that there exist $z \in \mathbb{R}^k$ with $Qz=a$. So we have $(\overline{a}_1,\dots, \overline{a}_{k-1})^T \in (0,\infty)^k$ and $M' (z_1,\dots,z_{k-1})^T=(\overline{a}_1,\dots, \overline{a}_{k-1})^T$. Apply induction hypothesis, we can find $y \in (0,\infty)^{k-1}$ such that $M'y \in (0,\infty)^{k-1}$. Pick $y_k >0$ such that $q_{k,k}(y_k-\sum\limits_{i=1}^{k-1}l_iy_i) >0$ but sufficient close to zero. Then, let $\overline{z}=(y_1,\dots,y_{k-1},y_k-\sum\limits_{i=1}^{k-1}l_iy_i)^T$ and tracing it back. We have $Q(y_1,\dots,y_k)^T \in (0,\infty)^k$. Regarding the negative GS criterion, we remark that one can show the following. (It is mentioned in [@GaMa11] with additional assumption but the additional assumption can be removed.) Suppose $Q$ is a symmetric matrix with off-diagonal entries being non-negative. Then, the following statements are equivalent. \(a) For any $a \in (0,\infty)^n$, there exist $z \in (-\infty,0)^n$ satisfying $Qz=a$. (a2) For any $a \in (0,\infty)^n$, there exist $z \in (-\infty,0]^n$ satisfying $Qz=a$. \(b) There exist $a \in (0,\infty)^n$ such that there exist $z \in (-\infty,0)^n$ satisfying $Qz=a$. (b2) There exist $a \in (0,\infty)^n$ such that there exist $z \in (-\infty,0]^n$ satisfying $Qz=a$. \(c) $Q$ is negative definite. The implication from (a) to (b), (a2) to (b2), (a) to (a2), (b) to (b2) are trivial. (c) implying (a2) is Lemma 3.3 of [@GaSt09] and a moment thought will justify (c)+(a2) implying (a), which is hiddenly used in [@GaSt09]. (b) implying (c) is similar to the proof of Lemma \[trichotomy\]. To be more precise, one again use induction on the size of $Q$ and change the basis using $B$. Therefore, an augmented graph $(\Gamma,a)$ satisfies the negative GS criterion if and only if $Q_{\Gamma}$ is negative definite. In particular, when a graph $\Gamma$ is negative definite, the negative GS criterion is always satisfied, independent of the area weights. ### Inflation Now, it comes the second input. \[inflation\](Inflation, See [@LaMc96] and [@LiUs06]) Let $C$ be a smooth symplectic surface inside $(W,\omega)$. If $[C]^2 \ge 0$, then there exists a family of symplectic form $\omega_t$ on $W$ such that $[\omega_t]=[\omega]+tPD(C)$ for all $t \ge 0$. If $[C]^2 < 0$, then there exists a family of symplectic form $\omega_t$ on $W$ such that $[\omega_t]=[\omega]+tPD(C)$ for all $0 \le t < -\frac{\omega[C]}{[C]^2}$. Also, $C$ is symplectic with respect to $\omega_t$ for all $t$ in the range above. Moreover, if there is another smooth symplectic surface $C'$ intersect $C$ positively and $\omega$-orthogonally, then $C'$ is also symplectic with respect to $\omega_t$ for all $t$ in the range above. Here, $PD(C)$ denotes the Poincare dual of $C$. When $[C]^2 < 0$, one can see that $([\omega]+tPD(C))[C] > 0$ if and only if $t < -\frac{\omega[C]}{[C]^2}$. Therefore, the upper bound of $t$ in this case comes directly from $\omega_t[C] > 0$. We remarked that one can actually do inflation for a larger $t$ but one cannot hope for $C$ being symplectic anymore when $t$ goes beyond $-\frac{\omega[C]}{[C]^2}$. ### Proof \[Proof of Theorem \[MAIN\]\] First of all, we can isotope $D$ to $D'$ such that every intersection of $D'$ is $\omega_0$-orthogonal, using Theorem 2.3 of [@Go95]. Since every intersection of $D$ is transversal and no three of $C_i$ intersect at a common point, such an isotopy can be extended to an ambient isotopy. Now, instead of isotoping $D$, we can deform $\omega_0$ through the pull back of $\omega_0$ along the isotopy. As a result, we can assume $D$ is $\omega_0$-orthogonal. Now, we want to construct a family of realizations $D_t$ of $\Gamma$, by deforming the symplectic form, such that the augmented graph of $D_1$ satisfies the positive GS criterion. Let $D=D_0=C_1 \cup \dots \cup C_k$ and let also the area weights of $D_0$ with respect to $\omega_0$ be $a$. Since $\omega$ is exact on $\partial P(D)$, there exists $z$ such that $Q_{\Gamma}z=a$. Also, by assumption and Lemma \[trichotomy\], there exists $\overline{z} \in (0, \infty)^k$ such that $Q_{\Gamma}\overline{z}=\overline{a} \in (0, \infty)^k$. Let $z^t=z+t(\overline{z}-z)$ and $a^t=a+t(\overline{a}-a)=Q_{\Gamma}z^t \in (0, \infty)^k$. We want to construct a realization $D_1$ of $\Gamma$ with area weights $a^1$. If this can be done, then the augmented graph of $D_1$ will satisfy the positive GS criterion. Observe that, it suffices to find a family of symplectic forms $\omega_t$ such that $[\omega_t]=[\omega_0]+t \sum\limits_i (\overline{z_i}-z_i)PD([C_i])$ and a corresponding family of $\omega_t$-symplectic divisor $D_t=C_1 \cup \dots \cup C_k$. The reason is that $C_i$ has symplectic area equal the $i^{th}$ entry of $a^t$ under the symplectic form $[\omega_t]=[\omega_0]+t \sum\limits_i (\overline{z_i}-z_i)PD([C_i])$. However, we need to modify this natural choice of family a little bit. Without loss of generality, we can assume $\overline{z_i}>z_i$ for all $1 \le i \le k$. We can choose a piecewise linear path $p^t$ arbitrarily close to $z^t$ such that each piece is parallel to a coordinate axis and moving in the positive axis direction. Since satisfying the positive GS criterion is an open condition, we can choose $p^t$ such that $Q_{\Gamma}p^t \in (0, \infty)^k$. The fact that $p^t$ is chosen such that $Q_{\Gamma}p^t$ is entrywise greater then zero allows us to do inflation along $p^t$ to get out desired family of $\omega_t$ and $D_t$, by Lemma \[inflation\]. Therefore, we arrive at a symplectic form $\omega_1$ such that the augmented graph of $(D,\omega_1)$, denoted by $(\Gamma,a)$, satisfies the positive GS criterion. We finish the proof by applying Propositon \[MAIN2\]. \[symplecitc cone\] The proof of Theorem \[MAIN\] implies that for any $a \in (0,\infty)^k \cap Q_D (0,\infty)^k$, there is a symplectic deformation making the augmented graph of $(D,\omega_1)$ to be $(\Gamma,a)$. First suppose $D$ is not negative definite. By Theorem \[MAIN\], $\omega$ being exact on the boundary implies $D$ is a concave divisor after a symplectic deformation. If $D$ is negative definite, then $\omega$ is necessarily exact on the boundary with unique lift of $[\omega]$ to a relative second cohomology class. Moroever, the discussion after the proof of Lemma \[trichotomy\] implies that $D$ satisifes negative GS criterion and hence $D$ is a convex divisor. Flowchart and Reducing Classification Problem to Graph ------------------------------------------------------ We offer a detailed explanation of the flowchart. Given a divisor $(D,\omega)$ (not necessarily $\omega$-orthogonal, see Proposition \[McLean0\]), the first obstruction of whether $D$ admits a concave or convex neighborhood comes from $\omega$ being not exact on the boundary of $D$. In this case, $[\omega]$ cannot be lifted to a relative second cohomology class and $Q_{D}z=a$ has no solution for $z$. If $\omega$ is exact on the boundary, we look at the solutions $z$ for the equation $Q_{D}z=a$. When $Q_D$ is negative definite (in this case $\omega$ is necessarily exact on the boundary), there is a unique solution for $z$ and all the entries for this solution is negative. Therefore, $(D,\omega)$ satisfies the negative GS criterion and $D$ is convex (Proposition \[MAIN2\] or Proposition \[McLean0\]). If $\omega$ is exact on the boundary but $Q_D$ is not negative definite, the situation becomes a bit more complicated. There might be more than one solution for $z$ (when $Q_D$ is degenerate). If we are lucky that there is one solution $z$ with all entries being positive, then $D$ is concave (Proposition \[MAIN2\] or Proposition \[McLean0\]). However, it is possible that all the solutions $z$ have at least one entry being non-positive. In this case, if $D$ is $\omega$-orthogonal or $D$ lies inside a closed symplectic manifold, there is a small neigborhood $N$ of $D$ such that $D$ has no convex nor conave neighborhood inside $N$ (Theorem \[obstruction-GS\] and Theorem \[obstruction-closed case\]). However, we can choose an area vector $\bar{a}$ such that there is a solution $\bar{z}$ for $Q_D\bar{z}=\bar{a}$ with all entries of $\bar{z}$ being positive (Lemma \[trichotomy\]). Geometrically, we can do inflation (Lemma \[inflation\]) to deform the symplectic form such that $(D,\bar{\omega})$ has area vector $\bar{a}$. Then, $(D,\bar{\omega})$ is concave (Proposition \[MAIN2\] or Proposition \[McLean0\]). This is exactly the proof of Theorem \[MAIN\]. (exact) \[startstop\] [$\omega|_{\partial P(D)}$ exact?]{}; (not exact) \[startstop2, below of=exact\] [No concave nor convex neighborhood]{}; (definite) \[startstop, right of=exact\] [$Q_D$ negative definite?]{}; (convex) \[startstop2, below of=definite\] [Admits a convex neighborhood]{}; (GS criterion) \[startstop, right of=definite\] [$(D,\omega)$ satisfies positive GS criterion?]{}; (concave) \[startstop2, right of=GS criterion\] [Admits a concave neighborhood]{}; (deformation) \[startstop3, below of=GS criterion\] [No small concave neighborhood, but admits one after a deformation]{}; (exact) – node\[right\][no]{}(not exact); (exact) – node\[above\][yes]{}(definite); (definite) – node\[right\][yes]{}(convex); (definite) – node\[above\][no]{}(GS criterion); (GS criterion) – node\[above\][yes]{}(concave); (GS criterion) – node\[right\][no]{}(deformation); In Section \[Classification of Symplectic Divisors Having Finite Boundary Fundamental Group\], we investigate capping (i.e. embeddable and concave) divisors with boudary fundamental group. Before doing this, we want to see how we use Theorem \[MAIN\], Proposition \[McLean0\], Theorem \[obstruction-closed case\] and Proposition \[McLean\] to reduce the problem to the realizability of its graph. \[reduce to graph\] Suppose $\Gamma$ is realizable. If the graph of $(D,\omega)$ is $\Gamma$, then $(D,\omega)$ is a capping divisor if and only if $(D,\omega)$ satisfies the positive GS criterion. Let $(\overline{D},\overline{\omega}) \subset \overline{W}$ be a realization of $\Gamma$ (See Definition \[realizable definition\]). In other words, $\overline{W}$ is a closed symplectic manifold and $\Gamma$ is the graph of $\overline{D}$. We first assume $(D,\omega)$ satisfies the positive GS criterion. By Theorem \[MAIN\] (See Remark \[symplecitc cone\]), we can find an $\overline{\omega}'$-orthogonal capping divisor $(\overline{D},\overline{\omega}')$ by doing symplectic deformation in $\overline{W}$ as long as its augmented graph $(\Gamma,\overline{a})$ satisfies the positive GS criterion (this is equivalent to $\overline{a} \in (0,\infty)^k \cap Q_{\Gamma} (0,\infty)^k$). Therefore, we can choose $\overline{a}$ such that $\overline{a}=a$, where $a$ is the area vector of $(D,\omega)$. Hence the augmented graphs of $(D,\omega)$ and $(\overline{D},\overline{\omega}')$ are the same and satisfy the positive GS criterion. By Proposition \[McLean0\], $(D,\omega)$ is a concave divisor. Moreover, Proposition \[McLean\] implies that the contact structures constructed on $P(D)$ and $P(\overline{D})$ are contactomorphic. Therefore, we can cut $Int(P(\overline{D}))$ from $\overline{W}$ and glue it with $P(D)$ along the boundary to get a closed symplectic manifold $W$. As a result, $(D,\omega)$ is a capping divisor. For the other direction, $(D,\omega)$ is a capping divisor. In particular, $(D,\omega)$ can be embedded into a closed symplectic manifold. By Theorem \[obstruction-closed case\], there is a neighborhood $N$ of $D$ such that there is no concave neighborhood $P(D)$ of $D$ lying inside $N$ if $(D,\omega)$ does not satisfies positive GS criterion. Therefore, $(D,\omega)$ is not a concave divisor. Contradiction. Having this, we are going to focus our study on graphs and Section \[Classification of Symplectic Divisors Having Finite Boundary Fundamental Group\] is solely the classification of realizable graphs with finite boundary fundamental group. Blow Up and Dual Blow Up ------------------------ The symplectic blow up and blow down operations have obvious analogues in the category of graphs and augmented graphs. We will describe these operations for augmented graphs. Both of these operations will play an important role in the classification of capping divisors with finite boundary $\pi$ in the next section. Let $(\Gamma,a)$ be an augmented graph. In the following figure, $(\tilde{\Gamma},\tilde{a})$ is obtained by blowing up $(\Gamma,a)$ at $v_1$ and $(\tilde{\tilde{\Gamma}},\tilde{\tilde{a}})$ is obtained by blowing up $(\tilde{\Gamma},\tilde{a})$ at an edge between $v_1$ and $v_0$. If the weight for the first blow up is $a_0$, then the area of $v_1$ and $v_0$ in $(\tilde{\Gamma},\tilde{a})$ is $a_1-a_0$ and $a_0$, respectively. If the weight for the second blow up is $a_{-1}$, then the area of $v_1$, $v_{-1}$ and $v_0$ in $(\tilde{\tilde{\Gamma}},\tilde{\tilde{a}})$ is $a_1-a_0-a_{-1}$, $a_{-1}$ and $a_0-a_{-1}$, respectively. \[blow up figure\] $$\xymatrix@R=1pc @C=1pc{ \dots \ar@{-}[r] & \bullet_{v_1}^{-y} \ar@{-}[r] & \dots &\dots \ar@{-}[r] & \bullet_{v_1}^{-1-y} \ar@{-}[r] \ar@{-}[d] & \dots & \dots \ar@{-}[r] & \bullet_{v_1}^{-2-y} \ar@{-}[r] \ar@{-}[d] & \dots \\ & & & & \bullet_{v_0}^{-1} & & & \bullet_{v_{-1}}^{-1} \ar@{-}[d] & \\ (a)~(\Gamma, a) & & & (b)~(\tilde{\Gamma},\tilde{a}) & & & (c)~(\tilde{\tilde{\Gamma}},\tilde{\tilde{a}}) & \bullet_{v_0}^{-2} & }$$ If one graph $\Gamma$ can be obtained from another graph $\Gamma'$ through blow ups and blow downs, then we call $\Gamma$ and $\Gamma'$ [*equivalent*]{}. A graph is called minimal if no blow down can be performed. \[0-0&gt;1\] $\xymatrix{ \bullet^{0} \ar@{-}[r] & \bullet^{0}\\ } $ is equivalent to $\xymatrix{ \bullet^{1}} $: $\xymatrix@R=1pc @C=2.5pc{ \bullet^{0} \ar@{-}[r] & \bullet^{0} & \ar[r] & & \bullet^{-1} \ar@{-}[r] & \bullet^{-1} \ar@{-}[r] & \bullet^{-1}\\ \ar[r] & & \bullet^{-1} \ar@{-}[r] & \bullet^{0} & \ar[r] & & \bullet^{1} \\} $ We remark that both realizablility and strongly realizability (See Definition \[realizable definition\]) are stable under blow ups and blow downs for graphs. However, there is no obvious reason for (strong) realizability to be stable under blow ups for augmented graphs (See Lemma \[stability of criterion\_blow up\] below). \[stability of criterion\_blow up\] Suppose $(\Gamma,a)$ is an augmented graph and $(\tilde{\Gamma},\tilde{a})$ be obtained from a single blow up of $(\Gamma,a)$ with weight $a_0$. If $(\Gamma,a)$ satisfies the negative GS criterion, then so does $(\tilde{\Gamma},\tilde{a})$. If $(\Gamma,a)$ satisfies the positive GS criterion, then so does $(\tilde{\Gamma},\tilde{a})$ for $a_0$ being sufficiently small. Suppose that $Q_{\Gamma}z=a$ for a vector $z=(z_1, \dots, z_k)^{T}$. We need to consider two cases. First, if $\tilde{\Gamma}$ is obtained from blowing up at the vertex of $v_1$ of $\Gamma$, then $\tilde{z}=(\tilde{z_0},\tilde{z_1},\dots,\tilde{z_k})^T=(z_1-a_0,z_1,z_2, \dots, z_k)^{T}$ satisfies $Q_{\tilde{\Gamma}}\tilde{z}=\tilde{a}$. Secondly, if $(\tilde{\Gamma},\tilde{a})$ is obtained from blowing up an edge between $v_1$ and $v_2$, then $\tilde{z}=(\tilde{z_0},\tilde{z_1},\dots,\tilde{z_k})^T=(z_1+z_2-a_0,z_1,z_2, \dots, z_k)^{T}$ satisfies $Q_{\tilde{\Gamma}}\tilde{z}=\tilde{a}$. In any of the above two cases, the Lemma follows. This illustrates the difference between convex and concave boundary. After blowing up a concave neighborhood 5-tuple that is obtained from the positive GS criterion, we might no longer be able to apply the GS criterion to get a concave neighborhood 5-tuple if the weight of blow up is too large. However, if we blow up a convex neighborhood 5-tuple that is obtained from the negative GS criterion, we can still get a convex neighborhood 5-tuple by the criterion again. For a graph $\Gamma$ and a vertex $v_1$ of $\Gamma$, we use $\Gamma^{(v_1)}$ to denote the graph that is obtained by adding two genera zero and self-intersection number zero vertices to the vertex $v_1$ of $\Gamma$ as illustrated in the following Figure. It is clear that $\Gamma^{(v_1)}$ is equivalent to attaching a single vertex of genus $0$ and self-intersection $1$ to $v_1$ and adding the self-intersection of $v_1$ by $1$ as in the following Figure (See Example \[0-0&gt;1\]). We denote it by $\overline{\Gamma^{(v_1)}}$ and call it the [**dual**]{} blow up of $\Gamma$ at $v_1$. \[dual blow up example\] $$\xymatrix @R=1pc @C=1pc { \dots \ar@{-}[r] & \bullet_{v_1}^{-y} \ar@{-}[r] & \dots &\dots \ar@{-}[r] & \bullet_{v_1}^{-y} \ar@{-}[r] \ar@{-}[d] & \dots & \dots \ar@{-}[r] & \bullet_{v_1}^{1-y} \ar@{-}[r] \ar@{-}[d] & \dots \\ & & & & \bullet_{v_0}^{0} \ar@{-}[d] & & & \bullet_{v_0}^{1} \\ (a)~\Gamma & & & (b)~\Gamma^{(v_1)} & \bullet_{v_{-1}}^{0} & & (c)~\overline{\Gamma^{(v_1)}} & }$$ By comparing $\overline{\Gamma^{(v_1)}}$ and the blown-up graph of $\Gamma$ at $v_1$ (See Figure \[blow up figure\](b)), we can regard the dual blow up as a dual operation of blow up. We remark that in [@Ne81] the dual blow up operation is also called blow up, and it is mentioned in [@Ne81] that the blow up and dual blow up operations do not change the oriented diffeomorphism type of the boundary of the plumbing. Capping Divisors with Finite Boundary $\pi_1$ {#Classification of Symplectic Divisors Having Finite Boundary Fundamental Group} ============================================= In this section, we classify capping divisors with finite boundary fundamental group. For completion and illustration, the classification of filling divisors with finite boundary fundamental group is given in subsection \[Filling classification\]. Different from the proof of filling divisors, the study for capping divisors requires essential symplectic input. Then, we illustrate the (strong) realizability of the graphs in type (P) and thus finish the proof of Theorem \[main classification theorem\]. Moreover, we sketch the proof of finiteness of fillings result (Proposition \[bounds\]) and study a conjugate phenomenon in subsection \[Fillings\]. Statement of Classification --------------------------- We use $<n,\lambda>$ to denote the following linear graph, where $\lambda$ and $n$ are both positive integers and $\lambda < n$, $$\xymatrix{ \bullet^{-d_1} \ar@{-}[r] & \bullet^{-d_2} \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{-d_k}\\ }$$ where each vertex has genus zero and $d_i \ge 2$ are the minus of the self-intersection numbers such that $$\frac{n}{\lambda}=d_1-\frac{1}{d_2-\frac{1}{\dots-\frac{1}{d_k}}}.$$ In what follows, we use $[d_1,\dots,d_k]$ to denotes the continuous fraction so the condition above is just $\frac{n}{\lambda}=[d_1,\dots,d_k]$. Moreover, we use $<y;n_1,\lambda_1;n_2,\lambda_2;n_3,\lambda_3>$ to denote the following graph with exactly one branching point. $$\xymatrix{ \bullet^{-d_k} \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{-d_1} \ar@{-}[r]& \bullet^{-y} \ar@{-}[r] \ar@{-}[d]& \bullet^{-b_1} \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{-b_l}\\ & & &\bullet^{-c_1} \ar@{-}[d] \\ & & &\vdots \\ & & &\bullet^{-c_m}\\ }$$ where all vertices have genera zero and we require the self intersection numbers satisfies $\frac{n_1}{\lambda_1}=[d_1,\dots,d_k] $, $\frac{n_2}{\lambda_2}=[b_1,\dots,b_l] $ and $\frac{n_3}{\lambda_3}=[c_1,\dots,c_m]$. We call the vertex with self-intersection $-y$ to be the central vertex. \[Five Types\] We define eight special types of graphs as follows. Type(N1): empty graph, Type(N2): linear graph $<n,\lambda>$, for $0 < \lambda < n$, $(n,\lambda)=1$, Type(N3): one branching point graph $<y;2,1;n_2,\lambda_2;n_3,\lambda_3>$, where $(n_2, n_3)$ is one of the pairs $(3, 3)$, $(3, 4)$, $(3, 5)$, or $(2, n)$, for some $n \ge 2$ and $0 < \lambda_i < n_i$, $(n_i,\lambda_i)=1$, and $y \ge 2$, Type(P1): linear graph $\xymatrix{ \bullet^{0} \ar@{-}[r] & \bullet^{0}\\ } $, Type(P2): (linear) dual blown up graph $\overline{\Gamma^{(v)}}$ where $\Gamma=<n,n-\lambda>$ is of type (N2) and $v$ is the left-end vertex, Type(P3): one branching point graph $<3-y;2,1;n_2,n_2-\lambda_2;n_3,n_3-\lambda_3>$, where $<y;2,1;n_2,\lambda_2;n_3,\lambda_3>$ is of type (N3), Type(P4): (one branch point) dual blown up graph $\overline{\Gamma^{(v)}}$ where $\Gamma$ is of type (N2) and $v$ is not an end vertex, Type(P5): (one or two branch points) dual blown up graphs $\overline{\Gamma^{(v)}}$ where $\Gamma$ is of type (N3) and $v$ is any vertex in $T$. Type (N) graphs have $b_2^+=0$ and type (P) graphs have $b_2^+=1$. These graphs are going to be our focus for the remaining of the paper. We remark that the set of type (N2) graphs is the same as the set of linear graphs with all self-intersection less than $-1$. Therefore, the set of type (P2) graphs is the same as the set of dual blow up at the right-end vertex of (N2) graphs, by symmetry. \[main classification theorem\] Let $\Gamma$ be a graph with finite boundary fundamental group. If $Q_{\Gamma}$ is not negative definite, then $\Gamma$ is realizable if and only if $\Gamma$ satisfies one of the following conditions. \(A) $\Gamma$ is equivalent to a graph in types (P1), (P2), (P3), (P4), or \(B) $\Gamma$ is equivalent to a graph $T^{(v)}$ in type (P5) such that (B)(i) $y \neq 2$, or (B)(ii) $y=2$ and $v$ is a vertex labeled by a subscript $Y$ in a graph from Figure \[Tetrahedral\] to Figure \[Dihedral\], where $-y$ is the self-intersection of the central vertex of $T$ for (B)(i) and (B)(ii). In particular, if $\Gamma$ is realizable, then we have $b_2^+(Q_{\Gamma})=1$ , $\delta_{\Gamma} \neq 0 $ and $\Gamma$ is strongly realizable. \[main classification theorem\_convex\] Suppose $\Gamma$ is a graph with finite boundary fundamental group. If $Q_{\Gamma}$ is negative definite, which means that $b_2^+(Q_{\Gamma})=0$ here, then $\Gamma$ is equivalent to a graph in type (N). Moreover, any type (N) graph can be realized as a resolution graph of an isolated quotient singularity. Notice that a graph with finite boundary fundamental group must have non-degenerate intersection form (See Lemma \[order\]) except the empty graph. Therefore, $b_2^+(Q_{\Gamma})=0$ is equivalent to $Q_{\Gamma}$ being negative definite. To be consistent, the empty graph is considered to be negative definite in this paper. Using Theorem \[main classification theorem\] and Proposition \[main classification theorem\_convex\], we give the classification of filling divisors and capping divisors with finite boundary fundamental group. \[complete classification\] Let $(D,\omega)$ be a divisor (not necessarily $\omega$-orthogonal) with finite boundary fundamental group. Then $(D,\omega)$ is a capping divisor if and only if it satisifies the positive GS criterion and its graph is equivalent to a realizable graph in type (P). On the other hand, $(D,\omega)$ is a filling divisor if and only if its graph is equivalent to a graph in type (N). The statement for capping divisor follows directly from Proposition \[reduce to graph\] and Theorem \[main classification theorem\]. On the other hand, if $(D,\omega)$ is of type $N$, it is negative definite and hence satisfies the negative GS criterion. By Proposition \[McLean\], $(D,\omega)$ is convex and we can close it up (by [@EtHo02]), thus is a filling divisor. If $(D,\omega)$ is a filling divisor but not in type (N), then $(D,\omega)$ is in type (P) and has $b_2^+(D)=1$, by Theorem \[main classification theorem\] and Proposition \[main classification theorem\_convex\]. We can close it up to a closed symplectic manifold $(W,\omega)$, which must be rational since it contains the divisor $D$ (for the reason why $W$ is rational if the graph of $D$ is equivalent to one in type (P3), see [@BhOn12], for the other, see Theorem \[McDuff\]). Therefore, $b_2^+(W)=1$. However, $\omega|_{W-P(D)}$ descends to a relative class in the cap, $W-P(D)$, and thus has positive square (i.e $[\omega|_{W-P(D)} -d\alpha_c]^2 >0$ for any choice of primitive $\alpha$ of $\omega$ defined near $P(D)$). Therefore, $b_2^+(W-P(D)) \ge 1$. However, $b_2^+(W)=b_2^+(P(D))+b_2^+(W-P(D))$ as $\partial P(D)$ is a rational homology sphere. Contradiction. Filling Divisors with Finite Boundary $\pi_1$ {#Filling classification} --------------------------------------------- We prove Proposition \[main classification theorem\_convex\] in this subsection. ### Topological Restrictions We first recall some topological constraints for a configuration to have finite boundary fundamental group. \[graph theoretic definition\] Suppose we have a graph $\Gamma$. The boundary fundamental group of $\Gamma$, denoted by $\pi_1(\Gamma)$, is the fundamental group of the boundary of the plumbing of the configuration represented by $\Gamma$. We call $\Gamma$ spherical, cyclic, finite cyclic if $\pi_1(\Gamma)$ is trivial, cyclic, finite cyclic. A branch point (or branch vertex) of a graph is a vertex with at least three branches. A branch at a vertex $v$ also refers to the sub-graph $\Gamma$ obtained by deleting $v$ and all other branches linking to $v$. A simple branch $\gamma$ is a branch that is linear. An extremal branch point is a branch point with only one non-simple branch. Finally, for a connected sub-graph $\gamma$, $\delta_{\gamma}$ denotes the determinant of the intersection form of $\gamma$. \[tree\] Let $T$ be a graph with finite $\pi_1(T)$. Then, all of its vertices have genera zero and $T$ is a finite tree. Therefore, from now on, all vertices are assumed to have genera zero and the number above a vertex is the self-intersection number of the vertex. Here, we give the concrete representation for boundary fundamental group. \[representation\]([@Hi66]) Let $T$ be a finite tree such that the genera of all vertices are zero. Label the vertices as $v_i$ for $i=1, \dots, n$ and let $q_{ij}=[v_i][v_j] \in \mathbb{Z}$ be the $(i,j)^{th}$-entry of the intersection form of $T$. Then, $\pi_1(T)$ is isomorphic to the free group generated by $e_1,\dots,e_n$ modulo the relations $$e_ie_j^{q_{ij}}=e_j^{q_{ij}}e_i,\quad \hbox{for any } i, j$$ and $$1=\prod_{1 \le j\le n}e_j^{q_{ij}}, \quad \hbox{for any } i.$$ \[order\] ([@Hi66]) Let $T$ be a finite tree such that the genera of all vertices are zero. Then, the order of the abelianization of $\pi_1(T)$ is finite if and only if $\delta_T \neq 0$. In this case, $\delta_T$ equals the order of the abelianization of $\pi_1(T)$. \[non-spherical\] For a type (N2) linear graph $T$, $\delta_T \neq 1$ and hence $\pi_1(T)$ is nontrivial by Lemma \[order\]. \[key\] Let $T$ be a minimal tree and $v$ a vertex in $T$. \(i) If $\pi_1(T)$ is cyclic, then there are at most two non-spherical branches at $v$. \(ii) If $\pi_1(T)$ is finite, then there are at most three non-spherical branches at $v$. Moreover, if there are three non-spherical branches, then they are all finite cyclic. The proof is based on the representation in Lemma \[representation\] and the group theoretical result in the following lemma. See Lemma 3.1 and 3.2 of [@Sh85] \[quotient\] Let $G_1, \dots, G_n$ be non-trivial groups and let $t_i \in G_i$ be an arbitrary element. Then, \(i) for $n \ge 4$, $G_1 * \dots *G_n/(\prod_{i=1}^{n}t_i=id)$ is infinite. \(ii) for $n \ge 3$, $G_1 * \dots *G_n/(\prod_{i=1}^{n}t_i=id)$ is non-trivial and non-cyclic. \(iii) $G_1*G_2*G_3/(\prod_{i=1}^{3}t_i=id)$ is finite if and only if $G_i$ are all cyclic groups generated by $t_i$ with $(G_1,G_2,G_3)$ isomorphic to one of the following unordered triples $(\mathbb{Z}/2\mathbb{Z},\mathbb{Z}/2\mathbb{Z},\mathbb{Z}/k\mathbb{Z})$, ($k \ge 2$), $(\mathbb{Z}/2\mathbb{Z},\mathbb{Z}/3\mathbb{Z},\mathbb{Z}/3\mathbb{Z}) $, $(\mathbb{Z}/2\mathbb{Z},\mathbb{Z}/3\mathbb{Z},\mathbb{Z}/4\mathbb{Z})$, or $(\mathbb{Z}/2\mathbb{Z},\mathbb{Z}/3\mathbb{Z},\mathbb{Z}/5\mathbb{Z})$. \[lens space analysis\] Suppose $T$ is of the form $\xymatrix@R=1pc @C=1pc{ T_1 \ar@{.}[r] & \bullet_{v_0} \ar@{-}[r] \ar@{.}[d]& \dots \ar@{-}[r] & \bullet_{v_r} \ar@{.}[d] \ar@{.}[r] & T_3 \\ & T_2 & & T_4 } $ with $r \ge 1$, where $T_i$ are non-spherical branches (not necessarily simple) such that $\delta_{T_1}\delta_{T_2} \neq 0$ and $\delta_{T_3}\delta_{T_4} \neq 0$. Suppose also that the boundary of the plumbing of $\xymatrix@R=1pc @C=1pc{ T_1 \ar@{.}[r] & \bullet_{v_0} \ar@{.}[r] & T_2 \\} $ and $\xymatrix@R=1pc @C=1pc{ T_3 \ar@{.}[r] & \bullet_{v_r} \ar@{.}[r] & T_4 \\} $ are diffeomorphic to a lens space or $\mathbb{S}^2 \times \mathbb{S}^1$. Then, $\pi_1(T)$ contains $\mathbb{Z} \oplus \mathbb{Z} $ as a subgroup. See Lemma 3.3 and 3.4 of [@Sh85] It is time to mention the following [**Fact**]{}: type (N) graphs have finite boundary fundamental group. To be more precise, (N1) graph is spherical, (N2) graphs are finite cyclic, and (N3) graphs are finite and non-cyclic. This is well known in algebraic geometry: graphs in type (N2) correspond to resolution graphs of cyclic quotient singularities and the graphs in type (N3) correspond to resolution graphs of dihedral, tetrahedral, octahedral and icosahedral singularities (cf. [@Br68] Satz $2.11$). One can also prove this fact directly by Lemma \[representation\] and Lemma \[quotient\]. It is easy for (N2) graphs. And for an (N3) graph $T$, $\pi_1(T)$ is finite as it can be realized as a finite extension of a finite group (basically by Lemma \[quotient\]), and it is non-cyclic because it has a non-cyclic quotient. ### Proof of Proposition \[main classification theorem\_convex\] {#classification} In this subsection we are going to make use of the constraints above to prove Proposition \[main classification theorem\_convex\]. \[simple branch\] Let $T$ be a negative definite, minimal tree with no branch point. Then $T$ is of type (N1) or (N2). In particular, $T$ is finite cyclic. A connected genus zero tree has no branch point, so it is linear. Linearity and minimality ensure no $-1$ vertices, while being negative definite ensures that each vertex has self-intersection less than 0. So $T$ is the empty graph (N1), or a linear graph with all vertices having self-intersection less than $-1$, which is a type (N2) graph. \[T shape\] Let $T$ be a minimal tree with exactly one branch point $v$. Suppose all the self-intersection of vertices in the branches are negative (satisfied if $T$ is negative definite). Then, $\pi_1(T)$ is finite if and only if $T$ is a (N3) or (P3) graph. In particular, $\pi_1(T)$ is not cyclic if $\pi_1(T)$ is finite. See Theorem 4.3 of [@Sh85]. The proof is purely topological. Now, we deal with the case that there are more than $1$ branch points. \[Tech\] Let $T$ be a negative definite, minimal tree with $k \ge 2$ branch points. Then, $\pi_1(T)$ is non-cyclic and infinite. The proof follows [@Sh85] closely. It is convenient to make two observations first. [**Observation 1**]{}: For any branch point, by the minimality assumption, all self-intersections of vertices in simple branches are less than $-1$. Therefore every simple branch of $T$ is not spherical from Example \[non-spherical\]. [**Observation 2**]{}: \[observation\] Let $T$ be a negative definite, minimal tree with $k \ge 2$ branch points. Suppose $\pi_1(T)$ is finite or cyclic. Let $v$ be a branch point and $\Gamma$ a branch at $v$. Suppose there are at least two non-spherical branches at $v$ other than $\Gamma$ (it is satisfied if $v$ is an extremal branch point and $\Gamma$ is the non-simple branch), then $\Gamma$ is finite cyclic and is either (a)a negative definite minimal tree with $k-1$ branch points, OR (b)a negative definite minimal tree with $k-2$ branch points, OR (c)not minimal. In case (c), there exists a branch point of $T$, $v_2$, which is linked to $v$, with exactly three branches and the self-intersection of $v_2$ is $-1$. Suppose first that $\pi_1(T)$ is cyclic. Since there are at least two non-spherical branches at $v$ other than $\Gamma$, $\Gamma$ is spherical by Lemma \[key\](i) applied to $v$. If $\pi_1(T)$ is finite, then $\Gamma$ is finite cyclic by Lemma \[key\](ii) applied to $v$. Therefore in either case $\pi_1(T)$ is finite cyclic. Moreover, if $\Gamma$ is minimal, it is either in (a) or (b). If $\Gamma$ is not minimal, it is in (c). In this case, the only possible $-1$ vertex that can be blown down is the vertex in $\Gamma$ linked to $v$, which we call it $v_2$. Since $T$ is minimal, $v_2$ has self-intersection $-1$ means that it is a branch point of $T$ but it can be blown down in $\Gamma$ means that it is not a branch point of $\Gamma$. Therefore, the result follows. An extremal branch point satisfies the assumption because there are at least two simple branches at $v$, which are non-spherical by Observation 1. We are going to first establish the claim of Lemma \[Tech\] for the cases $k=2$ and $k=3$, then prove by contradiction using induction on $k$. Label the vertices of $T$ as $v_1, \dots, v_m$ with the corresponding self-intersection $s_1, \dots, s_m$. First suppose $k=2$ and $v_1$, $v_2$ are the two branch points of $T$ with $\pi_1(T)$ cyclic or finite. If one of $v_1$, $v_2$ has three simple branches, say $v_2$, we denote the non-simple branch at $v_1$ by $\gamma$. Apply Lemma \[observation\] to $v_1$, $\gamma$ is in (a). However, negative definite minimal tree with exactly one branch point is not cyclic (See Lemma \[T shape\]). Contradiction. Thus, both $v_1$ and $v_2$ have only two simple branches. Let the two simple branches at $v_1$ be $T_1$ and $T_2$ and those at $v_2$ be $T_3$ and $T_4$. Then, the assumptions of Lemma \[lens space analysis\] for $v_0=v_1$ and $v_r=v_2$ are satisfied. Thus, $\pi_1(T)$ contains $\mathbb{Z} \oplus \mathbb{Z}$, contradiction. For $k=3$, let $v_2$, $v_1$ and $v_3$ be the three branch points of $T$ and suppose $T$ is finite or cyclic. We have two of the three branch points are extremal, say $v_2$ and $v_3$. Let $\Gamma$ be the non-simple branch at $v_2$ and $\Gamma'$ be the non-simple branch at $v_3$. Apply Lemma \[observation\] at $v_2$, we have $\Gamma$ is not minimal because we have shown that negative definite minimal trees with exactly one or two branch points are not finite cyclic. Thus, we must have $v_1$ is linked to $v_2$ and $v_1$ has only one simple branch in $T$, which we denote by $T_0$. Moreover, we have $s_1 = -1$ and by symmetry, $v_1$ is also linked to $v_3$. Observe that, we must have all vertices in $T_0$ having self-intersection $-2$, otherwise $\pi_1(\Gamma)$ is still not cyclic. Since $T$ is negative definite, both $s_2$ and $s_3$ are not $-1$. If $v_2$ or $v_3$ has three or more simple branches, then we can apply Lemma \[key\](ii) at $v_1$ and Lemma \[T shape\] to the branch at $v_1$ which is minimal and having exactly one branch point to get a contradiction. In other words, $v_2$ or $v_3$ has exactly two simple branches. Therefore, we have $T$ is of the following form with all $T_i$ being simple branches. $\xymatrix{ T_1 \ar@{.}[r] & \bullet_{v_2} \ar@{-}[r] \ar@{.}[d]& \bullet^{-1}_{v_1} \ar@{-}[d] \ar@{-}[r] & \bullet_{v_3} \ar@{.}[d] \ar@{.}[r] & T_3 \\ & T_2 & T_0 & T_4 } $ Notice that, both $\Gamma$ and $\Gamma'$ are equivalent to a linear graph because all vertices in $T_0$ have self-intersection $-2$. Let $\gamma=\xymatrix {T_1 \ar@{-}[r] & \bullet_{v_2} \ar@{-}[r] & T_2}$. Since $s_2 \neq -1$, we have $\delta_{\gamma} \neq 0$ and $\pi_1(\gamma)$ nontrivial. Hence, we can apply Lemma \[lens space analysis\] for $v_0=v_1$ and $v_r=v_3$. This is because the $T_i$ in Lemma \[lens space analysis\] are not assumed to be simple branches and $\Gamma'$ is equivalent to a linear graph. Hence, we get a contradiction. This finishes the study of $k=3$. In general, we assume the statement is true for $n <k$ and we deal with the case with $T$ having $k \ge 4$ branch points. Let $v_1$ be an extremal branch point and $\Gamma$ is the non-simple branch. The induction hypothesis and Lemma \[observation\] imply that $\Gamma$ is not minimal. In particular, there is a branch point of $T$, say $v_2$, is linked to $v_1$ with $s_2=-1$. If none of the three branches of $v_2$ in $T$ is simple, no matter how we blow down $\Gamma$, its minimal model has at least two branch points, contradicting to the above $k=2$ case or induction hypothesis. Hence, $v_2$ has a simple branch. Since $T$ is negative definite and $s_2=-1$, we must have $s_1 \le -2$. Thus, the branch at $v_2$ containing $v_1$ is not spherical. Let $\Gamma'$ be the non-simple branch at $v_2$ not containing $v_1$. Applying Lemma \[observation\] at $v_2$, we get that $\Gamma'$ is not minimal and hence there exist $v_3$, which is linked to $v_2$ with $s_3=-1$. The existence of two adjacent vertices, $v_2$ and $v_3$, having self-intersections $-1$ contradicts to $T$ being negative definite. Finally, we can complete the proof of Proposition \[main classification theorem\_convex\] by Lemma \[simple branch\], Lemma \[T shape\], Lemma \[Tech\] and the classification of isolated quotient surface singularities in [@Br68]. In [@Li08] and [@BhOz14], they study the filling of the lens spaces with the canonical contact structure. These correspond to the graphs in (N2). It is proved that the divisor filling is the maximal one among all the fillings and all other fillings can be obtained by rational blow downs of the divisor filling [@BhOz14]. Therefore, divisor filling is interesting to investigate. More Topological Restrictions ----------------------------- \[0-0\] Let $T$ be a tree and $v$ a vertex of $T$. Then, $\pi_1(T)=\pi_1(T^{(v)})$. It is a direct computation using Lemma \[representation\]. Label the vertices of $T$ as $v_1,\dots,v_n$ and let $v=v_1$. Label the two additional self-intersection $0$ vertices in $T^{(v)}$ as $v_{-1}$ and $v_0$, where $v_0$ is the one linked to $v_1$. We compare $\pi_1(T^{(v)})$ and $\pi_1(T)$. In terms of generators, $\pi_1(T^{(v)})$ has two additional generators, namely $e_{-1}$ and $e_0$. In terms of relations, there are two new relation and one of the relation in $\pi_1(T)$ is changed. The two new relations are given by $1=e_0$ and $1=e_{-1}e_1$. The relation $1=\prod_{1 \le j\le n}e_j^{q_{1j}}$ is changed to $1=e_0\prod_{1 \le j\le n}e_j^{q_{1j}}$. However, we have $1=e_0$, which means the changed relation is actually unchanged. Moreover, adding the generator $e_{-1}$ with the relation $1=e_{-1}e_1$ is doing nothing to the group so we arrive the conclusion. It is time to state the following fact. \[P graph\] Type P graphs have finite boundary fundamental groups. More precisely, (P1) graph is spherical, (P2) and (P4) graphs are finite cyclic, (P3) and (P5) graphs are finite and non-cyclic. Clear for the (P1) graph. Since (N2) graph is finite cyclic, so are (P2) and (P4) graphs by by Lemma \[0-0\]. (P3) and (P5) graphs and finite and non-cyclic by Lemmas \[T shape\] and \[0-0\]. In [@Ne81], it is mentioned that the dual blow up (which is called blow up there) does not change the oriented diffeomorphism type of the boundary of the plumbing. \[00\] Let $T$ be a minimal tree and $v$ a vertex in $T$. Suppose $\gamma$ is a simple branch at $v$ with $\gamma$ not equivalent to $\xymatrix{\bullet^0 \\} $ and some of the vertices having non-negative self-intersection. Then, $T$ is equivalent to a minimal tree $T'$ obtained by replacing the branch $\gamma$ by an equivalent branch $\xymatrix@R=1pc @C=1pc{ \bullet_{v'} \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{0} \ar@{-}[r] & \bullet^{0}_{x} \\ } $. where $v'$ is the vertex linked to $v$, $x$ is an end vertex and the self-intersection of $v$ may possibly be changed. We first make the following observations. If $T$ has a sub-tree of the form $\xymatrix@R=1pc @C=1pc{ \dots \ar@{-}[r] & \bullet^{q} \ar@{-}[r] & \bullet^{0} \ar@{-}[r] & \bullet^{b} \ar@{-}[r]& \dots \\ } $, then $T$ is equivalent to $T'$ where $T'$ is obtained by changing the sub-tree to $\xymatrix@R=1pc @C=1pc{ \dots \ar@{-}[r] & \bullet^{q+1} \ar@{-}[r] & \bullet^{0} \ar@{-}[r] & \bullet^{b-1} \ar@{-}[r]& \dots \\ } $. Also, if $T$ has a sub-tree of the form $\xymatrix@R=1pc @C=1pc{ \dots \ar@{-}[r] & \bullet^{q} \ar@{-}[r] & \bullet^{0} \\ } $, then $T$ is equivalent to $T'$ where $T'$ is obtained by changing the sub-tree to $\xymatrix@R=1pc @C=1pc{ \dots \ar@{-}[r] & \bullet^{q+1} \ar@{-}[r] & \bullet^{0} \\ } $. Let $u$ be the vertex in $\gamma$ having self-intersection non-negative, By possibly blowing up successively at edges linked to $u$, we assume that $u$ has self-intersection $0$. By the first observation, we can make the end vertex $w$ of $\gamma$ to have self-intersection $0$. Since $\gamma$ is not equivalent to $\xymatrix{ \bullet_{0} \\ } $, there is a vertex $w'$ which is different from $v$ and is linked to the end vertex $w$. Now, we blow down all $-1$ vertices in $\gamma-\{w,w'\}$. By the second observation and the fact that $w$ has self-intersection $0$, we obtain $T'$ as we want. For details, see Lemma 4.1 of [@Sh85]. \[negative definite spherical\] A spherical negative definite tree is not minimal. In other words, it is equivalent to an empty graph. See Section $3$ of [@Hi66] Symplectic Restrictions {#symplectic} ----------------------- We are going to provide symplectic input to give more constraints on the trees that we are interested in. In this subsection, $L_i$ appearing as a superscript of a vertex represents the homology class of the corresponding sphere. First, we recall a theorem of McDuff. \[McDuff\]([@Mc90]) Let $M$ be a closed symplectic 4-manifold. Suppose there exist an embedded symplectic sphere $C$ with positive (resp. zero) self-intersection. Then, $M$ is symplectic rational (resp. ruled). Moreover, if $(M,C)$ is relatively minimal and $[C]^2=0$, then there exists a symplectic deformation equivalent from $(M,C)$ to $(N,F)$, where $N$ is a symplectic sphere bundle over a closed symplectic surface and $F$ is a fibre. \[E\] Let $M$ be a symplectic 4-manifold. Suppose there exist an embedded symplectic sphere $C$ with \(i) positive self-intersection, or \(ii) zero self-intersection and there exists another embedded symplectic sphere $C'$ that intersect $C$ transversally once. Then $H_1(M)=0$, $M$ is rational and $b_2^+=1$. \[R\] Let $D$ be a symplectic divisor in a closed symplectic manifold $W$. Then, the graph of $D$ does not have a sub-tree of the form, $$\xymatrix{ \bullet^{L_1} \ar@{-}[r] & \bullet^{L_3} \ar@{-}[d] \ar@{-}[r] & \bullet^{L_4} \ar@{-}[r] & \bullet^{L_5}_x\\ & \bullet^{L_2} \\ }$$ with $[L_3]^2=-1$ and $[L_5]^2 \ge 0$, and all vertices having genera zero. By Corollary \[E\], we get $H_1(W)=0$ and $W$ is rational. Without loss of generality, we may assume $[L_5]^2=0$ (by possibly blowing up at regular points for the sphere representing $x$). Since $[L_3]^2=-1$, $W$ is not minimal and thus symplectomorphic deformation equivalent to blown-up of a Hizerburch surface, and $[L_5]$ is the fibre class of the Hizerburch surface by Theorem \[McDuff\]. Let $\{f,s,e_1,\dots,e_N\}$ be a basis for $H_2(W)$ such that $f$, $s$ and $e_i$ correspond to the fibre class, section class and the exceptional classes, respectively. Suppose $s^2=n$. Then, we recall the first chern class of the Hirzeburch surface is $(2-n)f+2s$, thus the first chern class of $W$ is $c_1(W)=(2-n)f+2s-e_1-\dots-e_N$. Moreover, $e_i$ can be a prior chosen so that $[L_3]=e_1$. Suppose there is an embedded symplectic sphere in $W$ with class $[S]=\alpha f+\beta s +a_1e_1+\dots+a_Ne_N$. Then, adjunction formula gives $$(2-n)\beta+2\alpha +2n\beta+a_1+\dots+a_N=2\alpha \beta+\beta^2n-a_1^2-\dots-a_N^2+2.$$ Suppose $[S]f=1$. Then $\beta=1$ and the formula reduces to $$a_1^2+\dots+a_N^2+a_1+\dots+a_N=0$$ and hence $a_i=0$ or $-1$ for all $i$. Suppose on the contrary $[S]f=0$. Then $\beta=0$ and the formula reduces to $$2\alpha+a_1^2+\dots+a_N^2+a_1+\dots+a_N=2.$$ Now, we want to study the homology of $L_i$ and draw contradiction. We recall that $[L_5]=f$ and $[L_3]=e_1$. Since $1=[L_4][L_5]=[L_4]f$, we apply the adjunction formula derived above and get $[L_4]=\alpha f+s+\epsilon_1 e_1+\dots+\epsilon_N e_N$ for some $\alpha$, where $\epsilon_i$ equals $0$ or $-1$ for all $i$. Since $1=[L_4][L_3]=[L_4]e_1$, $[L_4]$ is of the form $\alpha f+s-e_1+\epsilon_2 e_2+\dots+\epsilon_N e_N$. If we write $[L_1]=\overline{\alpha} f+\beta s +a_1e_1+\dots+a_Ne_N$, then $[L_1]f=0$, $[L_1]e_1=1$ and adjunction imply $[L_1]=\overline{\alpha} f-e_1+a_2e_2+\dots+a_Ne_N$ and $$\begin{aligned} \label{1} 2\overline{\alpha}+a_2^2+\dots+a_N^2+a_2+\dots+a_N=2\end{aligned}$$ Moreover, by $[L_1][L_4]=0$, we have $$\begin{aligned} \label{2} \overline{\alpha}-1-\epsilon_2a_2-\dots-\epsilon_N a_N=0\end{aligned}$$ By equations \[1\] and \[2\], we have $$\begin{aligned} &&a_2^2+\dots+a_N^2+(1+2\epsilon_2)a_2+\dots+(1+2\epsilon_N)a_N \\ &=& a_2(a_2+(-1)^{\epsilon_2})+\dots+a_N(a_N+(-1)^{\epsilon_N}) \\ &=&0\end{aligned}$$ Therefore, $a_i=0,-1$ if $\epsilon_i=0$ and $a_i=0,1$ if $\epsilon_i=-1$, for all $i$. Similarly, $[L_2]=\overline{\beta}f-e_1+b_2e_2+\dots+b_Ne_N$ with $b_i=0,-1$ if $\epsilon_i=0$ and $b_i=0,1$ if $\epsilon_i=-1$, for all $i$. By $[L_1][L_2]=0$, we have $-1-a_2b_2-\dots-a_Nb_N=0$. When $\epsilon_i=0$, we have both $a_i$ and $b_i$ equals $0$ or $-1$, thus $a_ib_i \ge 0$. Similarly, if $\epsilon_i=-1$, we still have $a_ib_i \ge 0$. Therefore, $-1-a_2b_2-\dots-a_Nb_N=0$ gives a contradiction. By the previous two Lemmas, we can now state one more basic consequence. \[0\] Let $T$ be a minimal graph of an embeddable divisor with finite $\pi_1$. Then, at any branch point $v$, no branch can be a single vertex $u$ with self-intersection zero. Suppose there is such a vertex $u$. Since $\pi_1(u)$ is infinite and $v$ has at least three branches, by Lemma \[key\](ii), there is a spherical branch at $v$, which we call $\gamma$. By Theorem \[McDuff\] again, the homology class $[u]$ represents a fiber class. Suppose $b_2^+(\gamma) \neq 0$, then there exists a class $[P]$ which is a linear combination of homology classes of vertices of $\gamma$ such that $[P]^2 >0$. Moreover, $[P][u]=0$. A basis for blown-up Hizerburch surface is given by $\{[u],s,e_1,\dots,e_n\}$, where $s$ is a section class and $e_i$ are the exceptional classes. Therefore, $[P][u]=0$ implies $[P]$ is a linear combination of $[u]$ and $e_i$. As a result, we have $[P]^2 \le 0$, which is a contradiction. Therefore, $b_2^+(\gamma)=0$. Since $\delta_{\gamma} \neq 0$, $b_2^+(\gamma)=0$ implies $\gamma$ is negative definite (Lemma \[order\]). Hence, by Lemma \[negative definite spherical\], $\gamma$ is not minimal. Therefore, $T$ has a sub-tree of the form, $$\xymatrix{ \bullet^{L_1} \ar@{-}[r] & \bullet^{-1}_w \ar@{-}[d] \ar@{-}[r] & \bullet^{L_4}_v \ar@{-}[r] & \bullet^{0}_u\\ & \bullet^{L_2} \\ }$$ Contradiction to Lemma \[R\]. Suppose $T$ is a realizable minimal tree with $\pi_1(T)$ being finite. Suppose also that there is a non-negative self-intersection vertex in a simple branch of $T$. Then, $T$ is equivalent to $T'^{(v)}$ for another minimal tree $T'$. We can apply Lemma \[0\] to ensure that the proof of Lemma \[00\] goes through and hence the result follows. By Corollary \[E\], $T$ has $b_2^+=1$ and hence $T'$ is negative definite. Moreover, by Lemma \[representation\], we can see that $\pi_1(T)=\pi_1(T')$ and hence finite. Therefore, what we need to do next is the classification minimal tree $T$ with $\pi_1(T)$ being finite and all self-intersection in simple branches being negative. By Lemma \[T shape\], we just need to consider the case that there are more than $1$ branch point. \[Tech2\] Let $T$ be a minimal realizable graph. Suppose $k \ge 2$ be the number of branch points of $T$. Suppose also all self-intersection of vertices in simple branches are less than $-1$. Then, $\pi_1(T)$ is non-cyclic and infinite. In particular, if $T$ is minimal, non-negative definite and $\pi_1(T)$ is finite, then $T$ is equivalent to a type (P) graph. The essence of the proof is the same as in Lemma \[Tech\] (See [@Sh85]). Since we do not assume our tree $T$ satisfies $b_2^+=1$, we cannot apply the result in [@Sh85] directly. Instead, we need to use Corollary \[E\] to guarantee $b_2^+=1$ whenever it is needed in the proof. This is required when we study the case that $T$ has $k=3$ branch points. We remark that in that case, there are two $-1$ vertices linked to each other so that blowing down one of them gives us an embedded symplectic sphere with self-intersection $0$. Therefore, we can apply Corollary \[E\] in that case to finish the proof of the first assertion. Moreover, by Lemma \[T shape\], the second assertion also follows. Proof of Theorem \[main classification theorem\] ------------------------------------------------ Having Lemma \[Tech2\], we can now focus on the study of type (P) graphs. Type (P1), (P2), (P3) are relatively easy to study and we are going to first go through it. Then, complete classification of realizability of type (P4) and (P5) graphs are given, which in turn completes the proof of Theorem \[main classification theorem\]. Finally, we are going to show that many graphs in type (P5) do not have their conjugate. ### Type (P1), (P2), (P3) {#Type (P1) to (P3)} We start with type (P1). By Example \[0-0&gt;1\], we have $\xymatrix{ \bullet^{0} \ar@{-}[r] & \bullet^{0}\\ } $ is equivalent to $\xymatrix{ \bullet^{1}} $. Then, by Example \[single vertex\], it is strongly realizable. Moreover, it corresponds to a capping divisor of the empty graph. Instead of answering the realizability of graphs in type (P2) and (P3) directly, we observe that graphs in type (2) and type (3) are all considered in [@BhOn12]. The (P2) graphs correspond to compactifying divisors for cyclic quotient singularities and the (P3) graphs correspond to compactifying divisors for the dihedral, tetrahedral, octahedral and icosahedral singularities. In particular, all (P2) and (P3) graphs are strongly realizable. The only less obvious correspondence between (P2), (P3) graphs and the graphs considered in [@BhOn12] are (P3) graphs of the form $<y;2,1;n_2,\lambda_2;n_3,\lambda_3>$ with $y \le 1$ and $(n_2, n_3)=(2, n)$. We denote the following graph in [@BhOn12] as $(c,c_1, \dots, c_k)$, where $[c,c_1,\dots,c_k]=\frac{n}{n-q}>1$ and $c,c_i \ge 2$ for all $i$. These are the graphs of compactifying divisors of dihedral singularities used in [@BhOn12]. $$\xymatrix{ \bullet^{-2} \ar@{-}[r]& \bullet^{-1} \ar@{-}[r] \ar@{-}[d]& \bullet^{-c+1} \ar@{-}[r] & \bullet^{-c_1} \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{-c_k}\\ &\bullet^{-2} \\ }$$ Observe that $(c,c_1, \dots, c_k)$ is the same as $<1;2,1;2,1;q,n-q>$ if one extends the definition to the case that $q < n-q$. Every graph $(c,c_1, \dots, c_k)$ in [@BhOn12] with $c,c_i \ge 2$ is equivalent to a (P3) graph $<y;2,1;2,1;n,\lambda>$ with $y \le 1$ and $0 < \lambda < n$ and vice versa. Observe that $<y;2,1;2,1;n,\lambda>$ is equivalent to $$\xymatrix@R=1pc @C=1pc{ \bullet^{-2} \ar@{-}[r]& \bullet^{-1} \ar@{-}[r] \ar@{-}[d]& \bullet^{-1}_{v} \ar@{-}[r] & \bullet^{-2} \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{-2} \ar@{-}[r] & \bullet^{-d_1-1}_{w} \ar@{-}[r] & \bullet^{-d_2} \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{-d_k}\\ &\bullet^{-2} \\ }$$ where there are $-y$ many self-intersection $-2$ spheres between the spheres named $v$ and $w$ as subscript and $\frac{n}{\lambda}=[d_1,\dots,d_k]$. Hence, it is of the form $(c,c_1, \dots, c_{-a+k-1})$ with $$[c,c_1,\dots,c_{-d},c_{-d+1},c_{-d+2}, \dots,c_{-d+k-1}] =[2,2,\dots,2,d_1+1,d_2,\dots,d_k].$$ This defines a map from the set of $<y;2,1;2,1;n,\lambda>$ with $y \le 1$ and $0 < \lambda < n$ to the set of $(c,c_1, \dots, c_k)$ and $c,c_i \ge 2$. Moreover, the inverse exists. Knowing that the graphs in type (P1) to type (P3) are realizable, as remarked before, we can determine whether a divisor with its graph being in type (P1) to type (P3) is a capping divisor or not, by Proposition \[reduce to graph\]. We remarked that for any graph $T$ in (P2), there is a unique (N2) graph $T'$ such that the dual blow up of $T'$ at the left-end vertex is $T$. In fact, by symmetry, there is also a unique (N2) graph $T"$ such that the dual blow up of the right-end vertex of $T"$ is $T$. To be more precise, $T"$ is obtained from $T'$ by rewriting the self-intersections from left to right to from right to left. ### Type (P4) and (P5) {#Type (P4) and (P5)} Suppose a graph $\overline{T^{(v)}}$ in type (P4) or (P5) admits a realization $D$ in a closed symplectic manifold $W$. By Theorem \[McDuff\] again, the existence of the self-intersection $1$ sphere implies that $W$ is rational. After preparation, now we are ready to study the realizability of type (P4) and (P5) graphs. We recall that for a type (P4) graph, $T^{(v)}$, $v$ is not an end vertex of $T$. We show that all (P4) graphs are realizable but we some graphs in type (P5) are not realizable. \[standard realization\] Suppose $T=<n,\lambda>= \xymatrix{ \bullet^{-d_1} \ar@{-}[r] & \bullet^{-d_2} \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{-d_k}\\ }$ and $v_1$ is the vertex with self-intersection $-d_j$ and $j \neq 1,k$. Then, $\overline{T^{(v_1)}}$ and hence the (P4) graph $T^{(v_1)}$ is realizable by some symplectic divisor $D$. On the other hand, suppose $T_2=<y;2,1;n_2,\lambda_2;n_3,\lambda_3>$ is a graph in type (N3). Then, $T_2^{(v)}$ is realizable if $y \neq 2$. We start with two algebraic lines in $\mathbb{CP}^2$ and call it $C_0$ and $C_1$. We blow up $\mathbb{CP}^2$ at $d_j$ distinct regular points at $C_1$ away from the intersection point of $C_0$ and $C_1$. Label the exceptional spheres as $E_1, E_{j+1}, E_1^{2}, \dots, E_1^{d_j-1}$. Call the proper transform of $C_0$ and $C_1$ as $C_0$ and $C_1$ again. Moreover, we call $E_1$ and $E_{j+1}$ to be $C_2$ and $C_{j+1}$, respectively. Then, we blow up $d_{j-1}-1$ many distinct regular points on $C_2$ that are away from the intersection points. Denote the exceptional spheres as $E_2, E_2^{2}, E_2^{3}, \dots, E_2^{d_{j-1}-1}$. Call the proper transform of $C_i$’s as $C_i$’s again and we call $E_2$ as $C_3$. We keep blowing up at regular points inductively and similarly on $C_3$ up to $C_{j-1}$ and denotes $E_{j-1}$ as $C_j$. Now,we blow up $C_j$ at $d_1-1$ many distinct regular points and call the exceptional spheres as $E_j, E_j^{2}, E_j^{3}, \dots, E_j^{d_{1}-1}$. This time, we do not let $C_{j+1}$ to be $E_j$ (we actually defined $C_{j+1}=E_{j+1}$). We get the second branch $C_2 \cup \dots \cup C_j$ of $C_1$ ($C_0$ is the first branch of $C_1$). Now, we blow up similarly for $E_{j+1}=C_{j+1}$ and we can get the last branch $C_{j+1} \cup \dots \cup C_k$ of $C_1$. This gives an embeddable symplectic divisor $D=C_0 \cup \dots \cup C_k$ that realize $\overline{T^{(v_1)}}$. For the type (P5) graph, we also consider $\overline{T_2^{(v)}}$ instead of $T_2^{(v)}$. We assume that $v$ is a vertex with two branches. The case when $v$ is the vertex with three branches is similar. We start with $\mathbb{CP}^2$ with $D$ being union of two distinct $\mathbb{CP}^1$, denoted by $C_1$ and $C_2$. Without loss of generality, we can assume $\overline{T_2^{(v)}}$ is of the form $$\xymatrix@R=1pc @C=1pc{ \bullet^{-d_k} \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{-d_1} \ar@{-}[r]& \bullet^{-y}_c \ar@{-}[r] \ar@{-}[d]& \bullet^{-b_1} \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{1-b_j}_v \ar@{-}[r] \ar@{-}[d]& \dots \ar@{-}[r] & \bullet^{-b_l}\\ & & &\bullet^{-c_1} \ar@{-}[d] & & & \bullet^{1}_p &\\ & & &\vdots \\ & & &\bullet^{-c_m}\\ }$$ Let we denote the sphere with self-intersection $-d_s$, $-b_s$ ($s \neq j$) and $-c_s$ as $C^\alpha_s$, $C^\beta_s$ ($s \neq j$) and $C^\gamma_s$, respectively. Also, denote the sphere with self-intersection $-y$, $1-b_j$ and $1$ as $C^c$ $C^v$ and $C^p$, respectively. It suffices to consider the case that $d_s=b_s=c_s=2$ for all $s$ because we can obtain the other cases by extra blow-ups. It is possible to obtain $\overline{T_2^{(v)}}$ with homology of the spheres indicated below by iterative blow-ups starting from $D$ similar to that in Lemma \[standard realization\]. Here $h$ is the hyperplane class and $e_i$’s are the exceptional classes resulting from single blow-ups. $[C^\alpha_s]=e_{i^{\alpha_{s-1}}_2}-e_{i^{\alpha_{s}}_2}$ for $2 \le s \le k$; $[C^\alpha_1]=e_{i^{\alpha_1}_1}-e_{i^{\alpha_1}_2}$; $[C^\gamma_s]=e_{i^{\gamma_{s-1}}_2}-e_{i^{\gamma_{s}}_2}$ for $2 \le s \le m$; $[C^\gamma_1]=e_{i^{\gamma_1}_1}-e_{i^{\gamma_1}_2}$; $[C^c]=e_{i^{\beta_1}_2}-e_{i^{\alpha_1}_1}-e_{i^{\gamma_1}_1}$; $[C^\beta_s]=e_{i^{\beta_{s+1}}_2}-e_{i^{\beta_{s}}_2}$ for $1 \le s \le j-2$; $[C^\beta_{j-1}]=e_{i^{\beta_{j-1}}_1}-e_{i^{\beta_{j-1}}_2}$; $[C^v]=h-e_{i^{\beta_{j-1}}_1}-e_{i^{\beta_{j+1}}_1}$; $[C^\beta_{j+1}]=e_{i^{\beta_{j+1}}_1}-e_{i^{\beta_{j+1}}_2}$; $[C^\beta_s]=e_{i^{\beta_{s-1}}_2}-e_{i^{\beta_{s}}_2}$ for $j+2 \le s \le l$, and $[C^p]=h$. This shows the existence of a realization for the type (P5) graph. To aid the non-realizablility study of some type (P5) graphs, we recall a combinatorical argument given by Lisca (See Proposition 4.4 of [@Li08]). \[combinaotric Lisca\] Suppose $W=\mathbb{CP}^2 \# N \overline{\mathbb{CP}^2}$ equipped with a symplectic form $\omega$ coming from blown-up of the Fubini-Study form $\omega_{FS}$. Let $D=C_0 \cup C_1 \cup \dots \cup C_k$ be a symplectic divisor with linear graph and $C_0$ corresponds to one of the two end vertices. Suppose the self-intersection of $C_i$ is $-b_i$ for $2 \le i \le k$, $[C_0]^2=1$ and $[C_1]^2=1-b_1$, where $b_i \ge 1$ for all $i$. Suppose $\{h,e_1,\dots,e_N\} \subset H_2(\mathbb{CP}^2\#N\overline{\mathbb{CP}^2};\mathbb{Z})$ forms an orthogonal basis with $h$ being the line class and $e_i^2=-1$. Assume also $[C_0]=h$. Then, $[C_1]=h-e_{i^1_1}-e_{i^1_2}-\dots-e_{i^1_{b_1}}$ and $[C_j]=e_{i^j_1}-e_{i^j_2}-\dots-e_{i^j_{b_j}}$ for $2 \le j\le k$, where, for any $\alpha$, $e_{i^\alpha_m} \neq e_{i^\alpha_n}$ for $m \neq n$. From now on, when we write the homology of a sphere, say $C$, we might simply write $h-e.-e.-\dots-e.$ and $e.-e.-\dots-e.$ to represent the homology class of $C$. In this case, the different $e.$’s in $[C]$ are understood to be distinct exceptional classes as in the conclusion of the Proposition \[combinaotric Lisca\]. \[realizable\] Suppose $T=<y;2,1;n_2,\lambda_2;n_3,\lambda_3>$ is a graph in type (N3). If $v$ is the vertex with three branches, then $T^{(v)}$ is realizable if and only if $y \neq 2$. The realizability part is already covered by Lemma \[standard realization\] so we are going to show the other direction. Suppose $y=2$ and, on the contrary, there were a realization of $T^{(v)}$ in a closed symplectic manifold. Then, we have $\overline{T^{(v)}}$ is also realizable and we have the following graph. $$\xymatrix{ & & & \bullet^{1}_p \ar@{-}[d] \\ \bullet^{-d_k} \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{-d_1}_{v_1} \ar@{-}[r]& \bullet^{-1}_v \ar@{-}[r] \ar@{-}[d]& \bullet^{-b_1}_{v_2} \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{-b_l}\\ & & &\bullet^{-c_1}_{v_3} \ar@{-}[d] \\ & & &\vdots \\ & & &\bullet^{-c_m}\\ }$$ By Theorem \[McDuff\] we can assume the positive sphere (the one with subscript $p$) has homology class $h$. Then, the only vertex with 4 branches ($v$) has to have homology class of the form $h-e_1-e_2$, by Proposition \[combinaotric Lisca\]. Here, as usual, $e_1$ and $e_2$ are exceptional classes formed by blowups. We recall that Proposition \[combinaotric Lisca\] ensure that the vertices $v_i$ has homology of the form $e_{j_1}-e_{j_2}-\dots-e_{j_t}$ for some distinct $e_{j_s}$ $1 \le s \le t$. To give the positive one contribution of the intersection of vertex $v_i$ with $v$, modulo symmetry, two of three vertices, $v_1$, $v_2$ and $v_3$ has homology class of the form $e_1-e.-\dots-e.$, where $e.$ are distinct exceptional classes not equal to $e_1$. However, it contradict to the zero intersection of any pair of $v_i$, $i=1,2,3$. Using the same line of reasoning, one can determine completely which graph is realizable and which is not and we put the results in the following. Therefore, the proof of Theorem \[main classification theorem\] is completed. A vertex with subscribe $Y$ indicates that if it is $v$, then the corresponding $T^{(v)}$ is realizable. Otherwise, the subscribe is $X$. \[Tetrahedral\] Tetrahedral $$\xymatrix{ \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \\ & &\bullet^{-2}_X \\ }$$ $$\xymatrix{ \bullet^{-2}_Y \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-3}_Y \\ & &\bullet^{-2}_Y \\ }$$ $$\xymatrix{ \bullet^{-3}_Y \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-3}_Y \\ &\bullet^{-2}_Y \\ }$$ \[Octahedral\] Octahedral $$\xymatrix{ \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \\ & & \bullet^{-2}_X \\ }$$ $$\xymatrix{ \bullet^{-3}_Y \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \\ &\bullet^{-2}_Y \\ }$$ $$\xymatrix{ \bullet^{-2}_Y \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-4}_Y \\ & &\bullet^{-2}_Y \\ }$$ $$\xymatrix{ \bullet^{-3}_Y \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-4}_Y \\ &\bullet^{-2}_Y \\ }$$ \[Icosahedral\] Icosahedral $$\xymatrix{ \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \\ & &\bullet^{-2}_X \\ }$$ $$\xymatrix{ \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-2}_X \ar@{-}[r] & \bullet^{-3}_X \\ & &\bullet^{-2}_X \\ }$$ $$\xymatrix{ \bullet^{-3}_Y \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \\ &\bullet^{-2}_Y \\ }$$ $$\xymatrix{ \bullet^{-2}_X \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-3}_Y \ar@{-}[r] & \bullet^{-2}_Y \\ & &\bullet^{-2}_Y \\ }$$ $$\xymatrix{ \bullet^{-3}_Y \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-2}_X \ar@{-}[r] & \bullet^{-3}_Y \\ & \bullet^{-2}_Y \\ }$$ $$\xymatrix{ \bullet^{-2}_Y \ar@{-}[r] & \bullet^{-2}_X \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-5}_Y\\ & &\bullet^{-2}_Y \\ }$$ $$\xymatrix{ \bullet^{-3}_Y \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-3}_Y \ar@{-}[r] & \bullet^{-2}_Y \\ &\bullet^{-2}_Y \\ }$$ $$\xymatrix{ \bullet^{-3}_Y \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-5}_Y \\ &\bullet^{-2}_Y \\ }$$ \[Dihedral\] Dihedral $$\xymatrix{ \bullet^{-2}_Y \ar@{-}[r]& \bullet^{-2}_X \ar@{-}[r] \ar@{-}[d]& \bullet^{-d_1}_N \ar@{-}[r] & \dots \ar@{-}[r] & \bullet^{-d_k}_N \\ & \bullet^{-2}_Y \\ }$$ where $N=Y$ if $d_1 \ge 3$ and $N=X$ if $d_1=2$. This completes the classification of realizable graph with finite boundary fundamental group. Remarks on Fillings {#Fillings} ------------------- In this subsection, we study the fillings for a given capping divisor $D$. First, we sketch the proof of the finiteness of fillings. Then, we study the conjugate phenomena. Finally, Liouville domain as a filling is considered. ### Finiteness {#Finiteness} \[bounds\] Suppose $D$ is a capping divisor with finite boundary fundamental group. Then, up to diffeomorphism, only finitely many minimal symplectic manifolds can be compactified by $D$. \[Stetch of proof\] We follow the strategy in [@Li08],[@BhOn12] and [@St13]. We remark that this question is answered in [@BhOn12] for graphs in type (P1), (P2), (P3). Therefore, it suffices to consider the case that the graph of $D$ is a graph in type (P4) or (P5), which are all dual blown up graphs. Suppose $D$ is a capping divisor for a symplectic manifold $Y$, and let $W$ be the resulting closed manifold. By Theorem \[McDuff\], $W$ is rational since there is a positive sphere $Q$ in $D$ due to the dual blow up. We can pick an orthonormal basis for $\{h,e_1, \dots, e_n \} $ for $H_2(W)$ such that $h^2=1$, $e_i^2=-1$ and $\omega(e_i) > 0$ for all $1 \le i \le N$. Moreover, we can assume the positive sphere $Q$ is of class $h$ by Theorem \[McDuff\]. Let $C_j$ be the sphere corresponding to the vertex $v$ and suppose that its self-intersection is $1-d_j$. By Proposition \[combinaotric Lisca\], the homology of $C_j$ is $[C_j]=h-e_{i^j_1}-\dots-e_{i^j_{d_j}}$ for some $i^j_1, \dots, i^j_{d_j}$ distinct. Moreover, we know that for the other spheres, the homology is of the form $e.-e.-\dots-e.$. If we do iterative symplectic blow-downs away from the positive sphere $Q$, we will end up with $(\mathbb{CP}^2,\omega_0)$, and the image of $D$ under the blow-down maps can be made to be union of exactly $2$ $J$-holomorphic spheres for some $\omega_0$-tamed $J$ if the blow-down maps are carefully chosen. By keeping track of the homological effects of the blow-downs, one can classify all possible $Y$ using the same reasoning as in [@Li08], [@OhOn05] and [@BhOn12]. In particular, one can obtain finiteness. Fixing a capping divisor $D$, we would like to investigate whether there are bounds for topological complexity among all minimal symplectic manifold that can be compactified by $D$. The answer is no in general. By Donaldson’s celebrated construction of Lefschetz pencil, any closed symplectic 4 manifold can be decomposed into a disc bundle over a closed symplectic surface $\Sigma$ glued with a Stein domain. In this case, we can view the symplectic surface $\Sigma$ as a symplectic capping divisor for the Stein domain. One of the interesting problems in this case is to bound the topological complexity for a given genus $g$. Some finiteness results of the topological complexity are obtained in [@Sm01] when the Lefschetz pencil has small genus. However, it is proved in [@BaMo12] that there is no bound of the Euler number of the filling when the genus is greater than $10$. In our setting, we allow ’reducible’ symplectic capping divisor so one should hope for obtaining some finiteness results when the symplectic capping divisor has small geometric genus. By Proposition \[bounds\], bounds for diffeomorphic invariants are obtained when $D$ has finite boundary fundamental group, which is a special case of geometric genus being zero. ### Non-Conjugate Phenomena {#Non-Conjugate Phenomena} Graphs in type (N) are resolution graphs of distinct quotient singularities. In particular, if $T_1$ and $T_2$ are graphs in (N), then they have different boundary fundamental groups except both $T_1$ and $T_2$ are resolution graphs of cyclic singularities (See e.g. [@Br68] Satz $2.11$, fourth column of the table). When two graphs $T$ and $T^c$ admit strong realizations $D$ and $D^c$ respectively such that $D$ and $D^c$ are conjugate to each other, we say that $T$ is conjugate to $T^c$. In Section \[Type (P1) to (P3)\], we mentioned that each graph $T^N$ in type (N) has a conjugate graph $T^P$ in type (P1), (P2) or (P3), and vice versa. We are going to show that many type (P5) graphs do not share this phenomena. There should be many ways to do it and we would like to use the first Chern class. When $T$ admits a realization $D=C_1 \cup \dots \cup C_k$ inside a closed symplectic manifold $W$, the first Chern class of $W$ descends to the first Chern class $c_1^D$ for $P(D)$, where $P(D)$ is a plumbing of $D$. Since $\partial P(D)$ is a rational homology sphere, $c_1^D$ lifts to a class uniquely in $H^2(P(D),\partial P(D), \mathbb{Q})$ by the Mayer-Vietoris sequence, which we still denote as $c_1^D$. Then, by the Lefschetz-Poincare duality, we can identify it with a class in $H_2(P(D), \mathbb{Q})$, which is generated by $[C_1],\dots,[C_k]$. Keeping the notations as in the previous paragraph, we call $(c_1^T)^2+k$ the characterizing number of $T$ and denote it by $n^T$. For $Y=W-P(D)$, we define the characterizing number of $Y$ to be $n^Y=(c_1^Y)^2+b_2(Y)$, where $c_1^Y$ and $b_2(Y)$ are the first Chern class and second Betti number of $Y$, respectively. \[first chern class\] Suppose $T$ is a graph in special types that admits a realization $D=C_1 \cup \dots \cup C_k$ in $W$. Let $b=(s_1+2,\dots,s_k+2)^T$ and write $c_1^T=\sum\limits_{i=1}^k w_i[C_k]$. Then, \(i) $w=(w_1,\dots,w_k)^T$ satisfies $Q_Tw=b$, \(ii) if $T$ is of type (P) and $Y=W-P(D)$, then we have $n^T+n^Y=10$. \(iii) if $T^{(v)}$ is of type (P5) and $n^{T}+n^{T^{(v)}} \neq 10$, then there is no graph conjugate to $T^{(v)}$. \[non-standard contact structure\] Suppose $T$ is a type (N3) graph and $T^{v}$ is a dual blow up of $T$. If $n^{T}+n^{T^{(v)}} \neq 10$ and $T^{(v)}$ is realizable, then it follows from (iii) that, on the diffeomorphic boundaries of plumbings, the contact structure $\xi^{T^{v}}$ induced by the positive GS criterion on $T^{(v)}$ is not contactomorphic to the canonical contact structure $\xi^T$, which is induced by the negative GS criterion on $T$. In this case, we can actually use the capping divisor $T^{(v)}$ to classify the symplectic fillings of the non-standard contact structure $\xi^{T^{v}}$ on the boundary of plumbing of $T^{(v)}$ (See also subsection \[Finiteness\]). Since the first Chern class is induced by a symplectic form, adjunction formula works in $P(D)$. Therefore, we have $2+s_i=c_1^T[C_i]$, where $s_i$ is the self-intersection of $C_i$. Hence, (i) follows immediately. For (ii), since $W$ is rational, we have $(c_1^W)^2+b_2(W)=10$. By the Mayer-Vietoris sequence, we have $H_2(P(D),\mathbb{Q}) \bigoplus H_2(Y,\mathbb{Q})= H_2(W, \mathbb{Q})$. Thus, $b_2(W)=b_2(P(D))+b_2(Y)$ and $(c_1^W)^2=(c_1^T)^2+(c_1^Y)^2$, which proves (ii). Finally, if the complement of plumbing of $T^{(v)}$ is a plumbing of a symplectic divisor, say $D'$, then $D'$ must be negative definite. By Lemma \[0-0\], we know that $\pi_1(D')=\pi_1(T^{(v)})=\pi_1(T)$. Among the type (N3) graphs, the boundary fundamental group uniquely characterize the graph (See [@Br68] Satz $2.11$, fourth column of the table). Therefore, by the classification Theorem \[main classification theorem\], the graph of $D'$ must be $T$ and hence, (ii) implies (iii). Consider the resolution graph of $E_8$ singularities, which is given by $$\xymatrix{ \bullet^{-2} \ar@{-}[r]& \bullet^{-2} \ar@{-}[r]& \bullet^{-2} \ar@{-}[r] \ar@{-}[d]& \bullet^{-2} \ar@{-}[r] & \bullet^{-2} \ar@{-}[r] &\bullet^{-2} \ar@{-}[r] &\bullet^{-2}\\ & &\bullet^{-2} \\ }$$ By Lemma \[first chern class\](i), the first Chern class is $c_1^{E_8}=0$ The following graph is a symplectic capping divisor of a plumbing of $E_8$, which we call $E_8^c$. $$\xymatrix{ \bullet^{-2}_{v_2} \ar@{-}[r]& \bullet^{-1}_{v_1} \ar@{-}[r] \ar@{-}[d]& \bullet^{-3}_{v_3} \\ &\bullet^{-5}_{v_4} \\ }$$ Then, by Lemma \[first chern class\](i), we have $c_1^{E_8^c}=2[C^{v_1}]+[C^{v_2}]+[C^{v_3}]+[C^{v_4}]$, where $C^{v_i}$ is the sphere corresponding to $v_i$. Direct calculation gives $(c_1^{E_8})^2=-2$ which is also predicted by Lemma \[first chern class\](ii). By a direct computation using mathematica, if $T$ in (N3) does not correspond to dihedral singularity, then there are only seven different (P5) graphs $T^{(v)}$ that satisfy $n^{T}+n^{T^{(v)}}=10$. Moreover, by Theorem \[main classification theorem\], only four of them are realizable and they are given by the followings. Therefore, these four graphs are the only exception that Lemma \[first chern class\](iii) cannot conclude anything among all graphs in (P5) not arising from dihedral resolution graph. For the following four type (N3) graphs $T$, the corresponding four type (P5) graphs $T^{(v)}$ satisfy $n^{T}+n^{T^{(v)}}=10$. $$\xymatrix{ \bullet^{-2} \ar@{-}[r] & \bullet^{-2} \ar@{-}[r]& \bullet^{-7} \ar@{-}[r] \ar@{-}[d]& \bullet^{-2} \ar@{-}[r] & \bullet^{-2} \ar@{-}[r] & \bullet^{-2} \\ & & \bullet^{-2}_v \\ }$$ $$\xymatrix{ \bullet^{-3} \ar@{-}[r]& \bullet^{-4} \ar@{-}[r] \ar@{-}[d]& \bullet^{-2}_v \ar@{-}[r] & \bullet^{-2} \ar@{-}[r] & \bullet^{-2} \\ &\bullet^{-2} \\ }$$ $$\xymatrix{ \bullet^{-2}_v \ar@{-}[r] & \bullet^{-2} \ar@{-}[r]& \bullet^{-8} \ar@{-}[r] \ar@{-}[d]& \bullet^{-2} \ar@{-}[r] & \bullet^{-2} \ar@{-}[r] & \bullet^{-2} \ar@{-}[r] & \bullet^{-2} \\ & &\bullet^{-2} \\ }$$ $$\xymatrix{ \bullet^{-3} \ar@{-}[r]& \bullet^{-3} \ar@{-}[r] \ar@{-}[d]& \bullet^{-3}_v \ar@{-}[r] & \bullet^{-2} \\ &\bullet^{-2} \\ }$$ ### Liouville Domain It is also interesting to know when a symplectic manifold, in particular, Liouville domain, can be compactified by a symplectic capping divisor. Affine varieties are this kind of Liouville domains. Thus, in some sense we can regard such Liouville domains as symplectic analogues of affine varieties. For an affine surface $X$, log Kodaira dimension can be defined for the pair $(V,D)$, where $V$ is the completion of $X$ and $V-D=X$. Moreover, this holomorphic invariant is independent of the compactification. When the affine surface $X$ is a homology plane (also called affine acyclic), McLean actually showed in [@McL12-2] that the log Kodaira dimension is also a symplectic invariant. Therefore, among all the Louville domains, (rational) homology planes are particularly interesting. It is a classical question in algebraic geometry to classify all (rational) homology planes (See the last Section of [@Mi00] and [@Za98]). A common feature for such an affine variety is that its completion is a rational surface. As we have seen, symplectic $4-$manifolds that can be compactified by a capping divisor with finite boundary fundamental group also share this phenomena. In particular, it would be interesting to know what symplectic capping divisors can compactify a Liouville domain that is a rational homology disk but the completion is not a rational surface. Another classical question is to determine all singularities that admits a rational homology disks smoothing. If the resolution graph for one such singularity is $\Gamma$, then in particular, the plumbing of $\Gamma$ can be symplectically filled by the rational homology disk. We remark that this question is completely answered using techniques ranging from smooth topology, symplectic topology and algebraic geometry (See [@StSzWa08], [@BhSt11] and [@PaShSt14]). Using the same reasoning as in Lemma \[first chern class\], we have the following Lemma. \[rational homology disks filling\] Suppose $T$ is a realizable type (P) graph. If $T$ can symplectic divisorial compactify a rational homology disk, then we have $n^T=10$. By mathematica, we find that there are only four type (P5) graphs $T^{(v)}$ that are not arising from dihedral resolution graph and satisfy $n^{T^{(v)}}=10$. Among these four, only three of them are realizable and they are listed in the following. In particular, it means that apart from these three, all other strongly realizable graphs in type (P5) not arising from dihedral resolution graph cannot sympelctic divisorial compactify a rational homology disk. For the following three (N3) graphs $T$, the corresponding three (P5) graphs $T^{(v)}$ satisfy $n^{T^{(v)}}=10$. $$\xymatrix{ \bullet^{-2}_v \ar@{-}[r] & \bullet^{-2} \ar@{-}[r]& \bullet^{-2} \ar@{-}[r] \ar@{-}[d]& \bullet^{-3} \\ & &\bullet^{-2} \\ }$$ $$\xymatrix{ \bullet^{-2} \ar@{-}[r] & \bullet^{-2} \ar@{-}[r]& \bullet^{-2} \ar@{-}[r] \ar@{-}[d]& \bullet^{-3}_v \\ & &\bullet^{-2} \\ }$$ $$\xymatrix{ \bullet^{-2} \ar@{-}[r] & \bullet^{-2} \ar@{-}[r]& \bullet^{-6} \ar@{-}[r] \ar@{-}[d]& \bullet^{-2} \ar@{-}[r] & \bullet^{-2} \ar@{-}[r] & \bullet^{-2} \ar@{-}[r] & \bullet^{-2}_v \\ & &\bullet^{-2} \\ }$$ Note that since most of the graphs in type (P5) do not have conjugate, the contact structure on the boundary is non-standard. Therefore, the consideration above is not covered by [@StSzWa08], [@BhSt11] and [@PaShSt14] (See Remark \[non-standard contact structure\]). [99]{} M. Abouzaid and P. Seidel. 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--- abstract: 'Non-Gaussianity generated in inflation can be contributed by two parts. The first part, denoted by $f_{NL}^{\delta N}$, is the contribution from four-point correlation of inflaton field which can be calculated using $\delta N$ formalism, and the second part, denoted by $f_{NL}^{int}$, is the contribution from the three-point correlation function of the inflaton field. We consider the two contributions to the non-Gaussianity in noncanonical warm inflation throughout (noncanonical warm inflation is a new inflationary model which is proposed in [@Zhang2014]). We find the two contributions are complementary to each other. The four-point correlation contribution to the non-Gaussianity is overwhelmed by the three-point one in strong noncanonical limit, while the conclusion is opposite in the canonical case. We also discuss the influence of the field redefinition, thermal dissipative effect and noncanonical effect to the non-Gaussianity in noncanonical warm inflation.' author: - 'Xiao-Min Zhang' - 'Hong-Yang Ma' - 'Peng-Cheng Chu' - 'Jian-Yang Zhu' title: 'Primordial non-Gaussianity in noncanonical warm inflation: three- and four-point correlations' --- [^1] \[sec1\]Introduction ==================== Inflation, as an necessary supplement to the standard model of the Universe, is an important branch of cosmology which can successfully solve the problems such as horizon, flatness and monopole [@Guth1981; @Linde1982; @Albrecht1982]. Another charming feature of inflation scenario is that it can give a natural mechanism to clarify the observed anisotropy of the cosmological microwave background (CMB) and the large scale structure exactly [@Weinberg; @LiddleLyth; @Dodelson]. Generally speaking, there are two kinds of inflationary theory till now: standard inflation, or sometimes called cold inflation, and warm inflation. Warm inflation was first proposed by A. Berera in 1995 [@BereraFang; @Lisa2004; @Berera2000], and then has been developed a lot in the past twenty years, especially in the fields of perturbation theory [@Berera2000; @BereraIanRamos; @Lisa2004; @MossXiong; @Chris2009], the micro-mechanism realization and dissipative issue of warm inflation [@MossXiong2006; @Berera1999; @BereraIanRamos], and the consistency issue of warm inflation [@Ian2008; @Campo2010; @Zhang2015; @Zhang2014; @ZhangTachyon; @ZhangZhu]. Standard and warm inflation share the advantages of solving horizon, flatness and monopole problems and generating nearly scale-invariant power spectrum. And warm inflation has its own advantages and improvements, such as curing the “$\eta$" problem [@etaproblem] and the problem of overlarge amplitude of the inflaton suffered in some standard inflationary models [@Berera2005; @BereraIanRamos], and relaxing the strict slow roll conditions in standard inflation greatly. A most distinct difference between standard and warm inflation is the origin of density fluctuations. The cosmological perturbations can naturally arise from vacuum quantum fluctuations in standard inflation [@Weinberg; @LiddleLyth; @Dodelson; @Bassett2006] while thermal fluctuations in warm inflation [@BereraFang; @Lisa2004; @Berera2000]. Warm inflation contains rich information about particle physics and can broad the scope of inflationary theory greatly. Some models that are already ruled out by new Planck observations [@PLANCKI2015] in standard inflation can again be in very good agreement with the Planck results in warm inflationary theory. When studying inflation, one typically calculates the power spectrum of scalar perturbations and the amplitude of gravitational waves. These perturbation quantities, although very important, contain only two-point correlation statistics information. Two-point correlation information in perturbations is too limited to discriminate among a large range of inflationary models. There exists a so-called ‘degeneracy problem’ (i.e. a single set of observables maps to a range of different inflation models) [@Eassona2013] in inflation theory. Even a precise measurement of the spectral index, the running of spectral index, and the detection of gravitational wave will not allow us efficiently discriminate among them. So we need the important information contained in primordial non-Gaussianity of inflation. The three-point function of curvature perturbation $\zeta$, or its Fourier transform, the bispectrum represents the lowest order statistics able to distinguish non-Gaussian from Gaussian perturbations [@Heavens1998; @Ferreira1998]. In this paper we will concentrate on the lowest order non-Gaussianity. Non-Gaussianity contains useful message of inflation, which can help to distinguish different inflationary models. Two-point correlation perturbations, i.e. power spectrums of scalar and tensor modes, generated in canonical standard inflation are already clear issues [@Weinberg; @LiddleLyth; @Dodelson; @Bassett2006]. Many works also has been concentrated on the perturbations of noncanonical standard inflation. The research of scalar power spectrum, spectral index, the amplitude of gravitational wave and consistency relation shows that the sound speed, which is an character quantity describing noncanonical effect in noncanonical inflation, plays an important role in the two-point perturbation quantities [@Mukhanov19991; @Mukhanov19992]. Non-Gaussianity, especially the three-point correlation in noncanonical standard inflation was researched in [@Creminelli2003; @Tong2004; @ChenHuang2007], and these works found that a low sound speed can much enhance the level of non-Gaussianity. Many works calculate non-Gaussianity generated by multi-field inflation and reach the conclusion that multi-field inflation has more enhanced non-Gaussianity than single field inflation [@Vernizzi2006; @Battefeld2007; @Tower2010]. Non-Gaussianity in warm inflation was analysed specially from different opinion in some works [@MossXiong; @Zhang2015; @Zhang2016; @MarGil2014; @Gupta2002; @Gupta2006]. In related works such as [@MossXiong; @Gupta2006; @Gupta2002; @MarGil2014], non-Gaussianity generated in canonical warm inflation was performed. Papers [@Gupta2002; @Gupta2006] concentrated on the temperature independent warm inflationary case and [@MarGil2014; @IanMoss2011] focused on the more complicated temperature dependent case. Thermal dissipation effect can increase non-Gaussianity to some extent. Canonical field was often used as inflaton in the research of warm inflation. Noncanonical warm inflation was first proposed in [@Zhang2014] and broaden the scope of inflationary picture. Non-Gaussianity in noncanonical warm inflation was first considered in our previous work [@Zhang2015], and we get the result that small sound speed and large dissipation strength can both enhance the magnitude of non-Gaussianity. The works above all considered non-Gaussianity generated by inflaton fields in linear large-scale evolution of perturbations. More than ten years ago, $\delta N$ formalism, a gauge-invariant description of nonlinear curvature perturbation on large scales, was proposed to calculate the issue of non-Gaussianity [@Lyth2005; @Zaballa2005; @Vernizzi2006; @Battefeld2007; @Tower2010]. Nonlinear parameter $f_{NL}$ is often introduced to parameterize the magnitude of non-Gaussianity. Nonlinear parameter obtained by $\delta N$ formalism, i.e. $f_{NL}^{\delta N}$, is nearly scale independent, while nonlinear parameter generated by the intrinsic non-Gaussianities of inflaton fields in linear cosmological perturbation theory, i.e. $f_{NL}^{int}$ is often scale dependent. If the inflaton fields are Gaussian to sufficient accuracy, such as in canonical multi-field inflation, intrinsic result of non-Gaussianity $f_{NL}^{int}$ is overwhelmed by $\delta N$ result $f_{NL}^{\delta N}$ [@Sasaki2016; @Vernizzi2006]. The two effects are complementary to each under field redefinition in standard inflation [@Sasaki2016]. Non-Gaussianity in canonical warm inflation was calculated from the $\delta N$ view in the work [@Zhang2016], which is allowed by recent observations [@PLANCKNG2015]. That $f_{NL}^{\delta N}$ is less than one in large scale in canonical warm inflation is due to the overdamped thermal term, which can make the slow roll more easily to be satisfied. In this paper we will analyse non-Gaussianity throughout in noncanonical warm inflation both from $\delta N$ view and intrinsic view. Since the intrinsic non-Gaussianity of inflaton field in noncanonical warm inflation is more prominent than in canonical inflation and the calculation of non-Gaussianity from $\delta N$ view is still absent, we’ll calculate the $\delta N$ part non-Gaussianity, discuss the contributions to non-Gaussianity from both view and make comparisons between them. We also try to find how noncanonical effect and thermal effect influence the non-Gaussianity in noncanonical warm inflation. The paper is organized as follows: In Sec. \[sec2\], we introduce noncanonical warm inflationary scenario briefly and review the basic equations and important parameters of the new picture. In Sec. \[sec3\], we introduce non-Gaussian perturbation, $\delta N$ formalism and the evolution equations of inflaton perturbations in noncanonical warm inflation. Then we calculate the nonlinear parameter $f_{NL}$ from both $\delta N$ view and intrinsic view in noncanonical warm inflation concretely and give discussions of the non-Gaussian results respectively in Sec. \[sec4\]. Finally, we draw the conclusions in Sec. \[sec5\]. \[sec2\]The framework of noncanonical warm inflation ==================================================== Different from standard inflation, the scalar inflaton field is not isolated, but has interactions with other sub-dominated fields in warm inflation. Thanks to the interactions, a significant amount of radiation was produced constantly during the inflationary epoch, so the Universe is hot with a non-zero temperature $T$. There’s a strong possibility that a warm Universe can happen [@Chris2008; @Berera2016]. The total matter action of the multi-component Universe in noncanonical warm inflation (noncanonical warm inflation is a kind of new inflationary model where noncanonical field behaves as inflaton [@Zhang2014]) is $$\label{action} S=\int d^4x \sqrt{-g} \left[ \mathcal{L}(\phi,X)+\mathcal{L}_R+\mathcal{L}_{int}\right],$$ where $\mathcal{L}(\phi,X)$ is the Lagrangian density of the noncanonical inflaton field, $\mathcal{L}_R$ is the Lagrangian density of radiation fields and $\mathcal{L}_{int}$ denotes the interaction between the scalar fields. In the Friedmann-Robertson-Walker (FRW) Universe, the mean inflaton field is homogeneous, i.e. $\phi=\phi(t)$. Under some assumptions and calculations, we can get the evolution equation of the inflaton field by varying the action with respect to the inflaton field [@BereraFang; @Berera1999; @Zhang2014]: $$\mathcal{L}_{X}c_{s}^{-2}\ddot{\phi}+(3H\mathcal{L}_{X}+\Gamma)\dot{\phi}+V_{eff,\phi}(\phi,T)=0, \label{EOMphi}$$ where $H$ is the Hubble parameter which satisfies the Friedmann equation: $$\label{Friedmann} 3H^2=8\pi G\rho.$$ In Eq. (\[EOMphi\]), $c_{s}^{2}=P_{X}/\rho_{X}=\left(1+2X\mathcal{L}_{XX}/\mathcal{L}_{X}\right)^{-1}$ is the sound speed which describes the traveling speed of scalar perturbations, $\Gamma$ is the dissipation coefficient and $V_{eff,\phi}(\phi,T)$ is the effective potential acquired thermal corrections. The subscripts $\phi$ and $X$ denote a derivative in our paper. The effective potential $V_{eff}(\phi,T)$ is different from the zero-temperature potential $V(\phi)$ in cold inflation. The thermal correction to the potential is constrained to be small enough by the slow roll conditions in warm inflation [@Ian2008; @Campo2010; @Zhang2014]. For simplicity we’ll write $V_{eff}$ as $V$ hereinafter. The term $\Gamma \dot{\phi}$ in the evolution equation describes the dissipation effect of $\phi$ to radiations [@BereraFang; @Berera2005; @Berera2000; @BereraIanRamos], which is a thermal damping term. In some papers, $\Gamma $ is often set to be a constant for simplicity to analyse [@Herrera2010; @Xiao2011; @Taylor2000]. Considering different microphysical basis of the interactions between inflaton and other fields, different form of $\Gamma$ can been obtained [@MossXiong2006; @MarGil2013; @BereraIanRamos]. Generally speaking, $\Gamma$ can be a function of inflaton field and even Universe temperature. An important parameter in warm inflationa is the dissipation strength which is defined as: $$\label{r} r=\frac{\Gamma}{3H}.$$ This parameter describes the effectiveness of warm inflation, where $r\gg1$ refers to strong regime of warm inflation and $r\ll1$ refers to weak regime of warm inflation. Thermal dissipative effect of warm inflation is accompanied by the production of entropy. The expression for entropy density from thermodynamics is $s=-\partial f/\partial T$, and we have $s\simeq -V_T$ for that the free energy $f=\rho -Ts$ is dominated by potential during inflation. The total energy density of the multi component Universe is $$\rho =\frac 12\dot{\phi}^2+V(\phi ,T)+Ts. \label{rho}$$ and the total pressure is given by $$p=\frac 12\dot{\phi}^2-V(\phi ,T). \label{p}$$ Combining the energy-momentum conservation $$\dot{\rho}+3H(\rho +p)=0, \label{conservation}$$ with Eq. (\[EOMphi\]), we can get the entropy production equation: $$T\dot{s}+3HTs=\Gamma \dot{\phi}^2. \label{entropy}$$ The equation above is equivalent to the radiation energy density producing equation $\dot{\rho}_r+4H\rho _r=\Gamma \dot{\phi}^2$, when the thermal correction to the effective potential is small enough in slow roll inflation. Inflation is often associated with slow-roll approximation to drop the highest derivative terms in the equations of motion, thus we can get the slow roll equations of noncanonical warm inflation: $$\dot{\phi}=-\frac{V_\phi}{3H(\mathcal{L}_{X}+r)}, \label{SRdotphi}$$ $$Ts=r\dot{\phi}^2, \label{SRTs}$$ $$H^2=\frac{8\pi G}3 V, \label{SRH}$$ $$4H\rho _r=\Gamma \dot{\phi}^2. \label{SRrho}$$ The validity of the slow roll approximation depends on the slow roll conditions given by systemic stability analysis [@Ian2008; @Campo2010; @Lisa2004; @Zhang2014]. The slow roll conditions are associated with some important slow roll parameters defined as $$\epsilon =\frac{M_p^2}{2}\left(\frac{V_{\phi}}{V}\right) ^2, \eta =M_p^2\frac {V_{\phi \phi}}{V}, \beta =M_p^2\frac{V_{\phi}\Gamma_{\phi}}{V\Gamma},$$ When dealing with warm inflation, we’ll need two additional slow roll parameters: $$b=\frac {TV_{\phi T}}{V_{\phi}}$$ and $$c=\frac{T\Gamma_T}{\Gamma}$$ to describe the temperature dependence of effective potential and dissipation coefficient in warm inflation [@Ian2008; @Campo2010]. These slow roll parameters are potential slow roll (PSR) parameters, which have relations with inflation potential and are different from Hubble slow roll (HSR) parameters. HSR parameters are invariant under field redefinition while PSR parameters are not. The slow-roll approximations can be guaranteed when $$\begin{aligned} \epsilon\ll\frac{\mathcal{L}_{X}+r}{c^2_s},\beta\ll\frac{\mathcal{L}_{X}+r}{c^2_s},\eta\ll\frac{\mathcal{L}_{X}}{c^2_s}, \nonumber \\ b\ll\frac{min\{\mathcal{L}_{X},r\}}{(\mathcal{L}_{X}+r)c^2_s},~~~~|c|<4~~~ , \label{SRcondition}\end{aligned}$$ in noncanonical warm inflationary scenario [@Zhang2014]. The additional parameter $c$ is not necessarily small, but a stability analysis of warm inflation shows that $|c|<4$ for a consistent model [@Ian2008; @ZhangZhu; @Zhang2014]. These slow roll conditions are more easy to be satisfied than in canonical warm inflation, let alone standard inflation. The number of e-folds in warm inflation is given by $$N=\int H dt=\int\frac{H}{\dot{\phi}}d\phi\simeq-\frac{1}{M_p^2}\int_{\phi_{\ast}} ^{\phi_{end}}\frac{V(\mathcal{L}_X+r)}{V_{\phi}}d\phi, \label{efold}$$ where $M_p^2=\frac 1{8\pi G}$. \[sec3\]Non-Gaussian perturbations of inflation =============================================== \[sec31\]$\delta N$ formalism and non-Gaussianity ------------------------------------------------- Now we’ll give a brief introduction of $\delta N$ formalism and primordial non-Gaussianity of inflation. $\delta N$ formalism was proposed in [@Lyth2005; @David2005; @Starobinsky; @Sasaki1996; @Sasaki1998] and then often used in calculating the non-Gaussianity of double and multi-field inflationary models [@Vernizzi2006; @Battefeld2007; @Tower2010]. The primordial curvature perturbation on uniform density hypersurfaces of the Universe, denoted by $\zeta$, is already present a few Hubble times before cosmological scales start to enter the horizon. And observations suggest the perturbation $\zeta$ was Gaussian term dominated with a nearly scale-invariant spectrum. Considering small perturbations in the background of the flat FRW Universe with scale factor $a(t)$, the spatial metric is given by $$\label{gij} g_{ij}=a^2(t)e^{2\zeta(t,\mathbf{x})}\gamma_{ij}(t,\mathbf{x})=\tilde{a}^2(t,\mathbf{x})\gamma_{ij}(t,\mathbf{x}),$$ where $\gamma_{ij}(t,\mathbf{x})$ has unit determinant and accounts for the tensor perturbation. We can find that according to this definition, $\zeta$ is the perturbation in $\ln \tilde{a}$. According to $\delta N$ formalism [@Lyth2005; @David2005; @LythMalik; @Starobinsky; @Sasaki1996; @Sasaki1998], $\zeta$, evaluated at some time $t$, is equivalent to the perturbation of the number of e-foldings $N(t,\mathbf{x})$ from an initial flat hypersurface at $t=t_{in}$, to a finial uniform density or, equivalently, comoving hypersurface at the time of $t$. Thus we have $$\label{zeta} \zeta(t,\mathbf{x})=\delta N \equiv N(t,\mathbf{x})-N_0(t),$$ where $N(t,\mathbf{x})\equiv\ln[\frac{\tilde{a}(t)}{a(t_{in})}]$ and $N_0(t)\equiv\ln[\frac{a(t)}{a(t_{in})}]$. The evolution of the Universe is supposed to be determined mainly by one or more inflaton fields during inflationary epoch. Choosing the convenient flat slicing gauge and considering perturbations, we can expand each scalar field in the form $\Phi_i(t,\mathbf{x})=\phi_i(t)+\delta\phi_i(t,\mathbf{x})$. As mentioned above, the curvature perturbation $\zeta$ is almost Gaussian, so we can expand $\zeta$ up to second order for good accuracy: $$\label{zeta2} \zeta(t,\mathbf{x})=\delta N\simeq \sum_i N_{,i}(t)\delta\phi_i+\frac12\sum_{ij} N_{,ij}(t)\delta\phi_i\delta\phi_j,$$ where $N_{,i}\equiv\frac{\partial N}{\partial\phi_i}$ and $N_{,ij}\equiv\frac{\partial^2 N}{\partial\phi_i\partial\phi_j}$. They may be entirely responsible for any observed non-Gaussianity if the field perturbations are pure Gaussian, which are the contributions of four-point correlations. However, the inflaton field perturbation in noncanonical warm inflation deviates from pure Gaussian distribution to some extent that larger than in canonical inflation. Thus we also need to compute non-Gaussianity generated by intrinsic non-Gaussianity of inflaton field, i.e. the three-point correlations of field. The power spectrum of the curvature perturbation $\zeta$, denoted by $\mathcal{P}_{\zeta}$, is defined as $$\label{spectrum} \langle\zeta_{\mathbf{k}_1}\zeta_{\mathbf{k}_2}\rangle\equiv(2\pi)^3\delta^3(\mathbf{k}_1+\mathbf{k}_2) \frac{2\pi^2}{k_1^3}\mathcal{P}_{\zeta}(k_1),$$ and $\mathcal{P}_{\zeta}(k)\equiv\frac{k^3}{2\pi^2}P_{\zeta}(k)$. The lowest order non-Gaussianity is three-point function of curvature perturbation, or its Fourier transform, the bispectrum, which is defined through $$\label{bispectrum} \langle\zeta_{\mathbf{k}_1}\zeta_{\mathbf{k}_2}\zeta_{\mathbf{k}_3}\rangle\equiv(2\pi)^3\delta^3 (\mathbf{k}_1+\mathbf{k}_2+\mathbf{k}_3)B_{\zeta}(k_1,k_2,k_3).$$ Its normalization is specified by the nonlinear parameter $f_{NL}$ through $$\label{fnl} B_{\zeta}(k_1,k_2,k_3)\equiv -\frac65f_{NL}(k_1,k_2,k_3)\left[P_{\zeta}(k_1)P_{\zeta}(k_2)+cyclic \right].$$ Observational limits are usually put on the nonlinear parameter and it is often used to describe the level of non-Gaussianity effectively. We can compute the bispectrum through Eq. (\[zeta2\]) and the calculation can yield $$\begin{aligned} \label{threepoint} \langle\zeta_{\mathbf{k}_1}\zeta_{\mathbf{k}_2}\zeta_{\mathbf{k}_3}\rangle =\sum_{ijk}N_{,i}N_{,j}N_{,k} \langle\delta\phi^{i}_{\mathbf{k}1}\delta\phi^{j}_{\mathbf{k}2}\delta\phi^{k}_{\mathbf{k}3}\rangle ~~~~~~~~~~~\nonumber\\ +\frac12\sum_{ijkl}N_{,i}N_{,j}N_{,kl}\langle\delta\phi^{i}_{\mathbf{k}1}\delta\phi^{j}_{\mathbf{k}2} (\delta\phi^{k}\star\delta\phi^{l})_{\mathbf{k}3}\rangle+perms,\end{aligned}$$ where a star denotes the convolution and the correlation functions higher than four-point are neglected. The first line in the equation above, a three-point correlation, is the contribution from the intrinsic non-Gaussianity of the inflaton fields, which can be scale dependent; while the second line, a four-point correlation, is scale independent and can be calculated conveniently by using $\delta N$ formalism. Based on the $\delta N$ formalism, we can get the part of non-Gaussianity generated by four-point correlation. The expression of $\delta N$ part nonlinear parameter is given by [@Lyth2005; @Boubekeur]: $$\label{fNL} -\frac35 f_{NL}^{\delta N}=\frac{\sum_{ij}N_{,i}N_{,j}N_{,ij}}{2\left[\sum_i N^2_{,i}\right]^2}.$$ We can see that the $f_{NL}^{\delta N}$ term is scale independent. The total non-Gaussianity should be described by $f_{NL}=f_{NL}^{\delta N}+f^{int}_{NL}$. \[sec32\]thermal fluctuations of inflaton field ----------------------------------------------- In noncanonical warm inflation, there is only one scalar field acting as inflaton and we can expand the full inflaton as $\Phi(\mathbf{x},t)=\phi(t)+\delta\phi(\mathbf{x},t)$, where $\delta\phi$ is the small perturbation around the homogenous background field $\phi(t)$ as usual. The evolution of inflaton is in overdamped regime in noncanonical warm inflation due to the enhanced Hubble damping term and thermal damping term. The evolution of the inflaton perturbations is very slow in slow roll regime as indicated in [@Lisa2004], so the evolution equation of full inflaton in slow roll noncanonical warm inflation can be given by [@Berera2000; @Taylor2000; @Zhang2014]: $$\label{EOMphik} \frac{d\Phi(\mathbf{k},t)}{dt}=\frac{1}{3H\mathcal{L}_X+\Gamma}\left[-k^2\mathcal{L}_X\delta\phi(\mathbf{k},t)-V_{\phi} (\Phi(\mathbf{k},t))+\xi(\mathbf{k},t)\right],$$ where $\xi$ is the thermal stochastic noise in thermal system with zero mean $\langle\xi\rangle=0$. In the high temperature limit $T\rightarrow\infty$, the noise source is Markovian: $\langle\xi(\mathbf{k},t)\xi(\mathbf{k'},t')\rangle=2\Gamma T(2\pi)^3\delta^3(\mathbf{k}-\mathbf{k'})\delta(t-t')$ [@Lisa2004; @Gleiser1994]. Thermal noise term in warm inflation is a kind of Gaussian distributed white noise [@Berera2000]. Since the leading order inflaton perturbation is linear response to the thermal noise, it is also Gaussian distributed. So if we want to calculate the predicted bispectrum of inflaton perturbation from Eq. (\[EOMphik\]), we should expand the inflaton fluctuations to second order at least: $\delta\phi(\mathbf{x},t)=\delta\phi_1(\mathbf{x},t)+\delta\phi_2(\mathbf{x},t)$, where $\delta\phi_1=\mathcal{O}(\delta\phi)$ and $\delta\phi_2=\mathcal{O}(\delta\phi^2)$. Then the equations of motion for the first and second order fluctuations in Fourier space can be obtained from Eq. (\[EOMphik\]): $$\begin{aligned} \label{deltaphi1} \frac{d}{dt}\delta\phi_1(\mathbf{k},t)&=&\frac{1}{3H\mathcal{L}_{X}+\Gamma}\left[-\mathcal{L}_Xk^2\delta\phi_1(\mathbf{k},t)\right.\nonumber\\ &-&\left.V_{\phi\phi}(\phi(t))\delta\phi_1(\mathbf{k},t)+\xi(\mathbf{k},t)\right],\end{aligned}$$ $$\begin{aligned} \label{deltaphi2} \frac{d}{dt}\delta\phi_2(\mathbf{k},t)&=&\frac{1}{3H\mathcal{L}_{X}+\Gamma}\left[-\mathcal{L}_Xk^2\delta\phi_2(\mathbf{k},t)-V_{\phi\phi} (\phi(t))\delta\phi_2(\mathbf{k},t)\right.\nonumber\\ &-&\left.\frac12V_{\phi\phi\phi}(\phi(t))\int\frac{dp^3}{(2\pi)^3}\delta\phi_1(\mathbf{p},t)\delta\phi_1(\mathbf{k} -\mathbf{p},t)\right. \nonumber\\ &-& \left. k^2\mathcal{L}_{XX}\int\frac{dp^3}{(2\pi)^3}\delta\phi_1(\mathbf{p},t)\delta X_1(\mathbf{k} -\mathbf{p},t)\right],\end{aligned}$$ The equation of motion for the fluctuations is obtained through perturbing the evolution equation of the full inflaton to second order. The analytic solutions of first and second order fluctuations, $\delta\phi_1$ and $\delta\phi_2$, can be obtained by solving the two equations above. And then the non-Gaussianity generated by intrinsic non-Gaussian distributions of inflaton can be obtained [@Zhang2015], which we’ll analyse concretely in next section. \[sec4\]Non-Gaussianity in noncanonical warm inflation ====================================================== There’s only one inflaton field in noncanonical warm inflation, so only one $\delta\phi_i$ is relevant, then Eq. (\[zeta2\]) reduces to $$\label{zeta3} \zeta(t,\mathbf{x})=N_{,i}\delta\phi_i+\frac12 N_{,ii}\left(\delta\phi_i\right)^2,$$ so we can get $$\label{fNL1} -\frac35f_{NL}^{\delta N}=\frac12\frac{N_{,ii}}{N^2_{,i}}.$$ Since there is only one $\delta\phi_i$, without ambiguity, we can rewrite $N_{,i}$ as $N_{\phi}$ and $N_{,ii}$ as $N_{\phi\phi}$ below. Through Eq. (\[efold\]), we can get $$\label{Nphi} N_{\phi}=-\frac{1}{M_p^2}\frac{V(\mathcal{L}_{X}+r)}{V_{\phi}},$$ Observational limits of primordial non-Gaussianity generated by inflation are usually put on the nonlinear parameter. And it’s estimated on the time of horizon crossing, which is well inside the slow roll inflationary regime. So we’ll first calculate the $\delta N$ part nonlinear parameter $f_{NL}^{\delta N}$ in slow roll approximation. Now we will consider a general dissipative coefficient case with $\Gamma=\Gamma(\phi,T)$ in slow roll noncanonical warm inflation. From Eq. (\[Nphi\]), we can get $$\label{Nphiphi1} N_{\phi\phi}=-\frac{1}{M_p^2}\left[(\mathcal{L}_{X}+r)-\frac{(\mathcal{L}_{X}+r)\eta}{2\epsilon}-\frac r2+\frac{\beta r}{2\epsilon}\right].$$ Then we can obtain the $\delta N$ part nonlinear parameter in noncanonical warm inflation with a general dissipation coefficient from Eqs (\[fNL1\]), (\[Nphi\]) and (\[Nphiphi1\]): $$\label{fNLdeltaN} f_{NL}^{\delta N}=\frac{5\epsilon}{3(\mathcal{L}_{X}+r)}-\frac{5\eta}{6(\mathcal{L}_{X}+r)}-\frac{5r\epsilon} {6(\mathcal{L}_{X}+r)^2}+\frac{5r\beta}{6(\mathcal{L}_{X}+r)^2}.$$ As $\delta N$ formalism indicated, the $\delta N$ part nonlinear parameter $f_{NL}^{\delta N}$ is scale independent, for it can be decided only by nonperturbative background equations. Considering the slow roll conditions Eq. (\[SRcondition\]) in noncanonical warm inflation, we can find from the equation above that $|f_{NL}^{\delta N}|\sim \mathcal{O}\left(\epsilon/(\mathcal{L}_{X}+r)\right)\lesssim1$, which is a first order small quantity in slow roll approximation. As the slow roll conditions suggest, during the inflationary epoch, the amplitude of non-Gaussianity is quite small and can grow slightly along with the inflation of Universe. Thus the level of non-Gaussiaity generated by four-point correlation in noncanonical warm inflation, characterized by parameter $f_{NL}^{\delta N}$, is not significant as in canonical warm inflation. Since the $\delta N$ form non-Gaussianity is not large enough, it’s unsafe to use this part to represent the whole primordial non-Gaussianity generated by inflation as some papers performed [@Zhang2016; @David2005]. So the calculation of non-Gaussianity generated by three-point correlation functions of inflaton field is also necessary. Non-Gaussianity generated by intrinsic non-Gaussianity of inflaton, characterized by $f_{NL}^{int}$, is considered in some papers [@Gupta2002; @Gupta2006; @MossXiong; @Zhang2015; @Zhang2016; @MarGil2014]. Paper [@Zhang2015] researched non-Gaussianity in noncanonical warm inflation with a temperature independent dissipative coefficient preliminarily and yields: $$\label{fNLint} f_{NL}^{int}=-\frac56 \ln\sqrt{\frac{3(\mathcal{L}_X+r)}{\mathcal{L}_X}}\left[\frac{\epsilon\varepsilon}{(\mathcal{L}_X+r)^2} +\left(\frac{1}{c_s^2}-1\right)\right].$$ The intrinsic part nonlinear parameter $f_{NL}^{int}$ is estimated for the three wavenumber $\mathbf{k}_1$, $\mathbf{k}_2$, $\mathbf{k}_3$ all within a few e-folds of exiting the horizon, i.e. there is a mild hierarchy among them. The intrinsic nonlinear parameter $f_{NL}^{int}$ is weakly dependent on the time and different wavenumbers, so it has a good scale independent approximation. We can see that $f_{NL}^{int}$ can be much greater than one, and thereby much greater than $f_{NL}^{\delta N}$ when the sound speed $c_s$ is low. A low sound speed can much enhance the magnitude of primordial non-Gaussianity in noncanonical warm inflation, thus $f_{NL}$ should be dominated by $f_{NL}^{int}$ term. The first term in Eq. (\[fNLint\]) is a second order small quantity, while $|f_{NL}^{\delta N}|$ is a first order small quantity, so the first term in $f_{NL}^{int}$ is absolutely overwhelmed by the second term and can be neglected. Then $f_{NL}^{int}\cong -\frac56\left(\frac{1}{c_s^2}-1\right)\ln\sqrt{\frac{3(\mathcal{L}_X+r)}{\mathcal{L}_X}}~\sim \mathcal{O}\left(\frac{1}{c_s^2}-1\right)$, which suggests both strong noncanonical effect (characterized by low sound speed $c_s$) and strong dissipative strength (characterized by large $r$) contribute to large magnitude of non-Gaussianity, but obviously the contribution of low sound speed is more significant. The whole non-Gaussianity should be described by the nonlinear parameter $f_{NL}=f_{NL}^{\delta N}+f_{NL}^{int}$. The two part are complementary to each other and both are not invariant under field redefinition. We can find that the non-Gaussianity in noncanonical warm inflation is dominated by intrinsic non-Gaussianity of inflaton field from the discussions above. The term $f_{NL}^{int}$ cannot be overlooked, instead it plays important role in non-Gaussian problems of noncanonical warm inflation. We know that PSR parameters are all related to inflaton field and thus they are not invariant quantities under field redefinition, while Hubble parameter $H$, HSR parameters, sound speed $c_s$ are invariant quantities. When a Lagrangian density $\mathcal{L}(X,\phi)$ is given, the parameters $X$ and thus $\mathcal{L}_{X}$ are both variant under field redefinition. The important characteristic parameters in warm inflation, $\Gamma$, and thus $r$ are also variant under field redefinition. Through Eqs. (\[fNLdeltaN\]) and (\[fNLint\]), we can see that both $f_{NL}^{\delta N}$ and $f_{NL}^{int}$ are not invariant under field redefinition. While the total $f_{NL}$ should be an invariant quantity under field redefinition, so the two parts $f_{NL}^{\delta N}$ and $f_{NL}^{int}$ are complementary to each other. Since the calculation of $f_{NL}^{int}$ part is more complicated especially in noncanonical warm inflation, we can try to choose an appropriate field gauge to simplify the calculation of total non-Gaussianity to some extent. The non-Gaussian result can be well inside the region allowed by Planck observations [@PLANCKNG2015] when $c_s$ is not small enough. The non-Gaussian results in noncanonical warm inflation can reduce to canonical case when $c_s\rightarrow1$: $$\label{fnlncanonical} f_{NL}^{\delta N}=\frac{5\epsilon}{3(1+r)}-\frac{5\eta}{6(1+r)}-\frac{5r\epsilon}{6(1+r)^2}+\frac{5r\beta}{6(1+r)^2},$$ and $$\label{fnlintcanonical} f_{NL}^{int}=-\frac56\ln\sqrt{3(1+r)}\frac{\epsilon\varepsilon}{(1+r)^2},$$ where $\varepsilon=2M_p^2\frac{V_{\phi\phi\phi}}{V_{\phi}}$ can be seen as a first order slow-roll small quantity which has the same magnitude as the slow-roll parameter $\eta$ in the monomial potential case. In canonical warm inflation, the situation is quite different from that in noncanonical warm inflation. In canonical warm inflation, $f_{NL}=f_{NL}^{\delta N}+f^{int}_{NL}$ is dominated by the $f_{NL}^{\delta N}$ term, since the term $f^{\delta N}_{NL}$ is a first order slow roll small quantity while $f_{NL}^{int}$ is a second order small quantity. From Eqs. (\[fnlncanonical\]) and (\[fnlintcanonical\]), we can see that primordial non-Gaussianity in canonical warm inflation is not significant, which is quite different from noncanonical case. The intrinsic non-Gaussian results represented by Eqs. (\[fNLint\]) and (\[fnlintcanonical\]) are obtained in the warm inflationary case with a temperature independent dissipative coefficient $\Gamma=\Gamma(\phi)$. The non-Gaussianity generated in canonical warm inflation with a more complicated temperature dependent dissipative coefficient $\Gamma=\Gamma(\phi,T)$ is considered in [@IanMoss2011], where the authors found that the non-Gaussianity can be significant to some extent. In temperature dependent case, the inflaton and radiation fluctuations are coupled to each other, the analytic result for power spectrum is too hard to be obtained, and we can only get a numerical result [@Chris2009]. The coupling between inflaton fluctuations and radiation fluctuations can make warm inflation stronger in most cases and also enhance the magnitude of non-Gaussianity [@IanMoss2011]. Some papers proposed a general form of the dissipative coefficient in warm inflation like $\Gamma=C_{\phi}\frac{T^m}{\phi^{m-1}}$ [@Zhangyi2009] and $\Gamma=\Gamma_0(\frac{\phi}{\phi_0})^n(\frac{T}{\tau_0})^m$ [@Lisa2004; @Campo2010], where $n$ and $m$ are integers. In these general forms, the characteristic warm inflationary parameter $c=m$, and the warm inflationary case can reduce to temperature independent case when $c=0$. The non-Gaussianity in canonical temperature dependent warm inflation can depend on the parameter $c$ by a function $f(c)$ [@IanMoss2011], where the function $f(c)$ is greater than 1 in most cases (i.e. the cases with $c>0$) while reduce to 1 when $c=0$. We can see that the coupling between inflaton fluctuations and radiation fluctuations can enhance the magnitude of non-Gaussianity in canonical warm inflation, the result should still hold qualitatively (i.e. the magnitude of non-Gaussianity in temperature dependent noncanonical warm inflation is enhanced by a factor $f(c)>1$ compared to temperature independent case) in noncanonical warm inflation with a different and more complicated form of $f(c)$. The quantitative analysis is complicated to some extent and we’ll concentrate completely on this problem in our next work. \[sec5\]conclusions and discussions =================================== In this paper, we investigate the whole primordial non-Gaussianity generated in noncanonical warm inflation. We give a brief introduction of noncanonical warm inflationary theory. Non-Gaussianity generated by inflation is often described by nonlinear parameter $f_{NL}$ and it can be divided into two parts: $f_{NL}^{\delta N}$ and $f_{NL}^{int}$. The first part describes the contribution of the four-point correlation of inflaton perturbation and the second part is due to the three-point correlation, i.e. the intrinsic non-Gaussianity of inflaton field. The two parts are complementary to each and they together can describe the primordial non-Gaussianity in inflation entirely. In addition, the two parts are both variant under field redefinition, while the whole $f_{NL}$ should be invariant. $\delta N$ formalism is convenient to use and so is often used in calculating the non-Gaussianities in multi-field inflation theories. We introduce $\delta N$ formalism and the evolution equation of the perturbation of inflaton field in noncanonical warm inflation briefly. Noncanonical warm inflation is dominated by one inflaton field, so we use the $\delta N$ formalism that reduces to single field case to calculate the parameter $f_{NL}^{\delta N}$. The $\delta N$ part nonlinear parameter $f_{NL}^{\delta N}$ in noncanonical warm inflation is scale-independent. Using $\delta N$ formalism, we obtain the expression of $f_{NL}^{\delta N}$ in noncanonical warm inflation with a general coefficient $\Gamma=\Gamma(\phi,T)$. We reach the conclusion that $f_{NL}^{\delta N}$ can be expressed as a linear combination of the PSR parameters, so it’s a first order small quantity in slow roll approximation with the order $|f_{NL}^{\delta N}|\sim \mathcal{O}\left(\frac{\epsilon}{\mathcal{L}_{X}+r}\right)$. That indicates the $\delta N$ part non-Gaussianity generated by noncanonical warm inflation is insignificant as in canonical single-field inflation. Since the magnitude of non-Gaussianity represented by $f_{NL}^{\delta N}$ is not large enough, it’s unsafe to ignore the non-Gaussianity comes from the self-interaction of the inflaton field. 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--- abstract: 'For a metric space $X$, let $\mathsf FX$ be the space of all nonempty finite subsets of $X$ endowed with the largest metric $d^1_{\mathsf FX}$ such that for every $n\in{\mathbb N}$ the map $X^n\to\mathsf FX$, $(x_1,\dots,x_n)\mapsto \{x_1,\dots,x_n\}$, is non-expanding with respect to the $\ell^1$-metric on $X^n$. We study the completion of the metric space $\mathsf F^1\!X=(\mathsf FX,d^1_{\mathsf FX})$ and prove that it coincides with the space $\mathsf Z^1\!X$ of nonempty compact subsets of $X$ that have zero length (defined with the help of graphs). We prove that each subset of zero length in a metric space has 1-dimensional Hausdorff measure zero. A subset $A$ of the real line has zero length if and only if its closure is compact and has Lebesgue measure zero. On the other hand, for every $n\ge 2$ the Euclidean space ${\mathbb R}^n$ contains a compact subset of 1-dimensional Hausdorff measure zero that fails to have zero length.' address: - '$^{1}$Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Naukova 3b, Ukraine' - '$^{2}$Ivan Franko National University of Lviv, Ukraine' - '$^{3}$Institute of Mathematics, Jan Kochanowski University, Kielce, Poland' author: - 'Iryna Banakh$^1$, Taras Banakh$^{2,3}$ and Joanna Garbulińska-Wȩgrzyn$^{3}$' title: 'The completion of the hyperspace of finite subsets, endowed with the $\ell^1$-metric' --- Introduction ============ Given a metric space $X$ with metric $d_X$, denote by $\mathsf KX$ the space of all nonempty compact subsets of $X$, endowed with the Hausdorff metric $d_{\mathsf KX}$ defined by the formula $$d_{\mathsf KX}(A,B)=\max\{\max_{a\in A}\min_{b\in B}d_X(a,b),\max_{b\in B}\min_{a\in A}d_X(b,a)\}.$$ The metric space $\mathsf KX$, called the [*hyperspace*]{} of $X$, plays an important role in General Topology [@Beer §3.2], [@Eng 4.5.23] and Theory of Fractals [@Edgar §2.5], [@Fal §9.1]. It is well-known [@Eng 4.5.23] that for any complete (and compact) metric space $X$ its hyperspace $\mathsf KX$ is complete (and compact). The hyperspace $\mathsf KX$ contains an important dense subspace $\mathsf FX$ consisting of nonempty finite subsets of $X$. The density of $\mathsf FX$ in $\mathsf KX$ implies that for a complete metric space $X$, the hyperspace $\mathsf KX$ is a completion of the hyperspace $\mathsf FX$. In [@BBKZ §30] it was shown that the Hausdorff metric $d_{\mathsf FX}$ on $\mathsf FX$ coincides with the largest metric on $\mathsf FX$ such that for every $n\in{\mathbb N}$ the map $X^n\to \mathsf FX$, $x\mapsto x[n]:=\{x(i):i\in n\}$, is non-expanding, where $X^n$ is endowed with the $\ell^\infty$-metric $$d^\infty_{X^n}(x,y)=\max_{i\in n}d_X(x(i),y(i)).$$ Here we identify the natural number $n$ with the set $\{0,\dots,n-1\}$ and think of the elements of $X^n$ as functions $x:n\to X$. Let us recall that a function $f:Y\to Z$ between metric spaces $(Y,d_Y)$ and $(Z,d_Z)$ is [*non-expanding*]{} if $d_Z(f(y),f(y'))\le d_Y(y,y')$ for any $y,y'\in Y$. It is well-known that the $\ell^\infty$-metric $d^\infty_{X^n}$ on $X^n$ is the limit at $p\to\infty$ of the $\ell^p$-metrics $d^p_{X^n}$ on $X^n$, defined by the formula: $$d^p_{X^n}(x,y)=\Big(\sum_{i=1}^nd_X(x(i),y(i))^p\Big)^{\frac1p}\mbox{ \ for \ $x,y\in X^n$.}$$ Given any metric space $(X,d)$ and any number $p\in[1,\infty]$, let $d^p_{\mathsf FX}$ be the largest metric $d^p_{\mathsf FX}$ on the set $\mathsf FX$ such that for every $n\in{\mathbb N}$ the map $X^n\to \mathsf FX$, $x\mapsto x[n]$, is non-expanding with respect to the $\ell^p$-metric $d^p_{X^n}$ on $X^n$. The metric $d^p_{\mathsf FX}$ was introduced in [@BBKZ], where it was shown that $d^p_{\mathsf FX}$ is a well-defined metric on $\mathsf FX$ such that $$d_{\mathsf FX}=d^\infty_{\mathsf FX}\le d^p_{\mathsf FX}\le d^1_{\mathsf FX},$$ where $d_{\mathsf FX}$ stands for the Hausdorff metric on $\mathsf FX$. By $\mathsf F^p\!X$ we will denote the metric space $(\mathsf FX,d^p_{\mathsf FX})$. So, $\mathsf F^\infty\! X$ coincides with the hyperspace $\mathsf FX$ endowed with the Hausdorff metric. As we already know, for any complete metric space $X$, the completion $\hat {\mathsf F}^\infty\!X$ of the metric space $\mathsf F^\infty\!X$ can be identified with the hyperspace $\mathsf KX$ endowed with the Hausdorff metric. In this paper we study the completion $\hat {\mathsf F}^1\!X$ of the metric space $\mathsf F^1\!X=(\mathsf FX,d^1_{\mathsf FX})$ and show that it can be identified with the space $\mathsf Z^1\!X$ of nonempty compact subsets of zero length in $X$. Sets of zero length are defined with the help of graphs. By a [*graph*]{} we understand a pair $\Gamma=(V,E)$ consisting of a set $V$ of vertices and a set $E$ of edges. Each edge $e\in E$ is a nonempty subset of $V$ of cardinality $|e|\le 2$. A graph $(V,E)$ is [*finite*]{} if its set of vertices $V$ is finite. In this case the set of edges $E$ is finite, too. For a graph $\Gamma=(V,E)$, a subset $C\subseteq V$ is [*connected*]{} if for any vertices $x,y\in C$ there exists a sequence of vertices $c_0,\dots,c_n\in C$ such that $c_0=x$, $c_n=y$ and $\{c_{i-1},c_i\}\in E$ for every $i\in\{1,\dots,n\}$. The maximal connected subsets of $V$ are called the [*connected components*]{} of the graph $\Gamma$. It is easy to see that two connected components of $\Gamma$ either coincide or are disjoint. For a vertex $x\in V$ by $\Gamma(x)$ we shall denote the unique connected component of the graph $\Gamma$ that contains the point $x$. By a [*graph in a metric space*]{} $(X,d_X)$ we understand any graph $\Gamma=(V,E)$ with $V\subseteq X$. In this case we can define the [*total length*]{} $\ell(\Gamma)$ of $\Gamma$ by the formula $$\ell(\Gamma)=\sum_{\{x,y\}\in E}d_X(x,y).$$ If $E$ is infinite, then by $\sum\limits_{\{x,y\}\in E}d_X(x,y)$ we understand the (finite or infinite) number $$\sup\limits_{E'\in \mathsf FE}\sum\limits_{\{x,y\}\in E'}d_X(x,y).$$ For a subset $C\subseteq X$ by $\overline C$ we denote the closure of $C$ in the metric space $(X,d_X)$. Given a subset $A$ of a metric space $X$, denote by $\mathbf \Gamma_{\!X\!}(A)$ the family of graphs $\Gamma=(V,E)$ with finitely many connected components such that $V\subseteq X$ and $A\subseteq\overline V$. Observe that the family $\mathbf \Gamma_{\!X\!}(A)$ contains the complete graph on the set $A$ and hence $\mathbf \Gamma_{\!X\!}(A)$ is not empty. The set $A$ is defined to have [*zero length in*]{} $X$ if for any ${\varepsilon}>0$ there exists a graph $\Gamma\in\mathbf \Gamma_{\!X\!}(A)$ of total length $\ell(A)<{\varepsilon}$. In Proposition \[p:zero\] we shall prove that each set $A$ of zero length in a metric space $X$ is totally bounded and has 1-dimensional Hausdorff measure equal to zero. For a metric space $X$, denote by $\mathsf ZX$ the family of nonempty compact subsets of zero length in $X$. It is clear that each finite subset of $X$ has zero length, so $\mathsf FX\subseteq \mathsf ZX\subseteq \mathsf KX$. Now we define the metric $d^1_{\mathsf ZX}$ on the set $\mathsf ZX$. Given two compact sets $A,B\in \mathsf ZX$, let ${\mathbf\Gamma}_{\!X\!}(A,B)$ be the family of graphs $\Gamma=(V,E)$ in $X$ such that - $A\cup B\subseteq\overline{V}$; - $\Gamma$ has finitely many connected components; - for every connected component $C$ of $\Gamma$ we have $A\cap\overline{C}\ne\emptyset\ne B\cap\overline C$. The conditions (i),(ii) imply that $A\cup B\subseteq\overline{V}=\bigcup_{x\in V}\overline{\Gamma(x)}$. Observe that the family $\mathbf \Gamma_{\!X\!}(A,B)$ contains the complete graph on the set $A\cup B$ and hence is not empty. For two compact subsets $A,B\in \mathsf ZX$, let $$d^1_{\mathsf ZX}(A,B):=\inf_{\Gamma\in\mathbf \Gamma_{\!X\!}(A,B)}\ell(\Gamma).$$ By a [*completion*]{} of a metric space $X$ we understand any complete metric space containing $X$ as a dense subspace. The following theorem is the main result of this paper. \[t:main\] Let $X$ be a metric space and $d_X$ be its metric. 1. The function $d^1_{\mathsf ZX}$ is a well-defined metric on $\mathsf ZX$. 2. $d_{\mathsf KX}(A,B)\le d^1_{\mathsf ZX}(A,B)$ for any $A,B\in \mathsf ZX$. 3. $d^1_{\mathsf ZX}(A,B)=d^1_{\mathsf FX}(A,B)$ for any finite sets $A,B\in \mathsf FX$. 4. $\mathsf FX$ is a dense subset in the metric space $\mathsf Z^1\!X:=(\mathsf ZX,d^1_{\mathsf ZX})$. 5. If the metric space $X$ is complete, then so is the metric space $\mathsf Z^1\!X=(\mathsf ZX,d^1_{\mathsf ZX})$. 6. If $Y$ is a dense subspace in $X$, then $d^1_{\mathsf ZY}(A,B)=d^1_{\mathsf ZX}(A,B)$ for any $A,B\in \mathsf ZY$. 7. If $\bar X$ is a completion of the metric space $X$, then $\mathsf Z^1\!\bar X$ is a completion of the metric space $\mathsf F^1\!X$. The proof of Theorem \[t:main\] is divided into seven lemmas. \[l:H\] $d_{\mathsf KX}(A,B)\le d^1_{\mathsf ZX}(A,B)$ for any $A,B\in \mathsf ZX$. To derive a contradiction, assume that $d_{\mathsf KX}(A,B)>d^1_{\mathsf ZX}(A,B)$ for some compact sets $A,B\in \mathsf ZX$. By the definition of $d^1_{\mathsf ZX}$, there exists a graph $\Gamma\in\mathbf \Gamma_{\!X\!}(A,B)$ such that $\ell(\Gamma)<d_{\mathsf KX}(A,B)$. Choose a positive real number ${\varepsilon}$ such that $\ell(\Gamma)+2{\varepsilon}<d_{\mathsf KX}(A,B)$. Since $\Gamma$ has finitely many connected components and $A\cup B\subseteq\overline V$, for any point $a\in A$ there exists a connected component $C$ of the graph $\Gamma$ such that $a\in\overline C$ . By the definition of the family $\mathbf \Gamma_{\!X\!}(A,B)$, the intersection $\overline C\cap B$ contains some point $b'\in B$. Since $a,b'\in\overline C$, there are points $c,c'\in C$ such that $d_X(a,c)<{\varepsilon}$ and $d_X(b',c')<{\varepsilon}$. Since the set $C$ is connected in the graph $\Gamma=(V,E)$, there exists a sequence $c_0,\dots,c_n\in C$ of pairwise distinct points such that $c_0=c$, $c_n=c'$, and $\{c_{i-1},c_i\}\in E$ for all $i\in\{1,\dots,n\}$. Since the points $c_0,\dots,c_n$ are pairwise distinct, the edges $\{c_{0},c_1\},\{c_1,c_2\},\dots,\{c_{n-1},c_n\}$ of the graph $\Gamma$ are pairwise distinct and then $$d_X(a,b')\le d_X(a,c_0)+\sum_{i=1}^nd_X(c_{i-1},c_i)+d_X(c_n,c')<{\varepsilon}+\ell(\Gamma)+{\varepsilon}.$$ Then $\min_{b\in B}d_X(a,b)\le d_X(a,b')< 2{\varepsilon}+\ell(\Gamma)$ and $\max_{a\in A}\min_{b\in B}<2{\varepsilon}+\ell(\Gamma)$. By analogy we can prove that $\max_{b\in B}\min_{a\in A}d_X(b,a)<2{\varepsilon}+\ell(\Gamma)$. Then $$d_{\mathsf KX}(A,B)=\max\{\max_{a\in A}\min_{b\in B}d(a,b),\max_{b\in B}\min_{a\in A}d(b,a)\}<2{\varepsilon}+\ell(\Gamma)<d_{\mathsf KX}(A,B),$$ which is a desired contradiction completing the proof of the lemma. \[l:metric\] $d^1_{\mathsf ZX}$ is a well-defined metric on $\mathsf ZX$. Given any sets $A,B,C\in\mathsf ZX$, we need to verify the three axioms of metric: 1. $0\le d^1_{\mathsf ZX}(A,B)<\infty$ and $d^1_{\mathsf ZX}(A,B)=0$ iff $A=B$, 2. $d^1_{\mathsf ZX}(A,B)=d_{\mathsf ZX}^1(B,A)$, 3. $d^1_{\mathsf ZX}(A,B)\le d^1_{\mathsf ZX}(A,C)+d^1_{\mathsf ZX}(C,B)$. 1\. First we show that $d^1_{\mathsf ZX}(A,A)=0$ for any $A\in \mathsf ZX$. Since the set $A$ has zero length, for any ${\varepsilon}>0$ there exists a graph $\Gamma=(V,E)$ in $X$ with finitely many connected components such that $A\subseteq\overline V$ and $\ell(\Gamma)<{\varepsilon}$. Replacing $\Gamma$ by a suitable subgraph, we can assume that the closure of each connected component of $\Gamma$ intersects the set $A$. Then $A\in\mathbf \Gamma_{\!X\!}(A,A)$ and hence $$d_{\mathsf ZX}^1(A,A)\le\ell(\Gamma)<{\varepsilon}.$$ Since ${\varepsilon}>0$ was arbitrary, $d_{\mathsf ZX}^1(A,A)=0$. If sets $A,B\in \mathsf ZX$ are distinct, then by Lemma \[l:H\], $d_{\mathsf ZX}^1(A,B)\ge d_{\mathsf KX}(A,B)>0$ (as the Hausdorff metric $d_{\mathsf KX}$ is a metric). The proof of the first axiom of metric will be complete as soon as we check that $d^1_{\mathsf ZX}(A,B)<\infty$ for any $A,B\in \mathsf ZX$. Since the sets $A,B$ have zero length, there exist graphs $\Gamma_A=(V_A,E_A)$ and $\Gamma_B=(V_B,E_B)$ with finitely many connected components such that $A\subseteq\overline V_{\!A}$, $B\subseteq\overline V_{\!B}$ and $\ell(\Gamma_A)+\ell(\Gamma_B)<1$. Let $D$ be a finite subset of $V_A\cup V_B$ intersecting every connected component of the graphs $\Gamma_A$ and $\Gamma_B$. Consider the graph $\Gamma=(V,E)$ where $V=V_A\cup V_B$ and $E=E_A\cup E_B\cup E_D$ where $E_D:=\{e\subseteq D:|e|=2\}$. It is easy to see that the graph $\Gamma$ is connected and belongs to the family $\mathbf \Gamma_{\!X\!}(A,B)$. Then $$d^1_{\mathsf ZX}(A,B)\le \ell(\Gamma)\le \ell(\Gamma_A)+\ell(\Gamma_B)+\sum_{\{x,y\}\in E_D}d_X(x,y)<\infty.$$ 2\. The definition of the distance $d^1_{\mathsf ZX}$ implies that $d^1_{\mathsf ZX}(A,B)=d^1_{\mathsf ZX}(B,A)$ for any $A,B\in \mathsf ZX$. 3\. Finally we check the triangle inequality for $d^1_{\mathsf ZX}$. Given any $A,B,C\in \mathsf ZX$ and ${\varepsilon}>0$, it suffices to show that $$d^1_{\mathsf ZX}(A,C)\le d^1_{\mathsf ZX}(A,B)+d^1_{\mathsf ZX}(B,C)+4{\varepsilon}.$$ By the definition of the distances $d^1_{\mathsf ZX}(A,B)$ and $d^1_{\mathsf ZX}(B,C)$, there exist graphs $\Gamma\in\mathbf \Gamma_{\!X\!}(A,B)$ and $\Gamma'\in\mathbf \Gamma_{\!X\!}(B,C)$ such that $\ell(\Gamma)<d^1_{\mathsf ZX}(A,B)+{\varepsilon}$ and $\ell(\Gamma')<d^1_{\mathsf ZX}(B,C)+{\varepsilon}$. By the definition of the families $\mathbf \Gamma_{\!X\!}(A,B)$ and $\mathbf\Gamma_{\!X\!}(B,C)$, the graphs $\Gamma=(V,E)$ and $\Gamma'=(V',E')$ have finitely many connected components and their closures meet the sets $A,B$ and $B,C$, respectively. Fix a finite set $D\subseteq V$ intersecting all connected components of the graph $\Gamma$ and a finite set $D'\subseteq V'$ intersecting all connected components of the graph $\Gamma'$. Fix a function $f:D\to B$ assigning to each point $x\in D$ a point $f(x)\in B\cap\overline{\Gamma(x)}$. Since $B\subseteq\overline V=\bigcup_{x\in V}\overline{\Gamma(x)}$, for every $b\in B$ there exists a point $g(b)\in V$ such that $b\in \overline{\Gamma(g(b))}$. Since $b\in \overline{\Gamma(g(b))}$ we can replace $g(b)$ by a suitable point in the connected component $\Gamma(g(b))$ and additionally assume that $d(b,g(b))<{\varepsilon}/{|D|}$. Next, do the same for the graph $\Gamma'$: choose a function $f':D'\to B$ such that $f(x)\in B\cap\overline{\Gamma'(x)}$ for every $x\in D'$, and a function $g':B\to V'$ such that $b\in \overline{\Gamma'(g'(b))}$ and $d(b,g'(b))<{\varepsilon}/|D'|$ for every $b\in B$. Consider the graph $\Gamma''=(V'',E'')$ where $V''=V\cup V'$ and $$E''=E\cup E'\cup\big\{\{f(x),g'(f(x))\}:x\in D\big\}\cup\big\{\{f'(x),g(f'(x))\}:x\in D'\big\}.$$ It can be shown that $\Gamma''\in\mathbf \Gamma_{\!X\!}(A,C)$ and hence $$\begin{gathered} d^1_{\mathsf ZX}(A,C)\le\ell(\Gamma'')\le\ell(\Gamma)+\ell(\Gamma')+\sum_{x\in D}d\big(f(x),g'(f(x))\big)+\sum_{x\in D'}d\big(f'(x),g(f'(x))\big)<\\ \big(d^1_{\mathsf ZX}(A,B)+{\varepsilon}\big)+\big(d^1_{\mathsf ZX}(B,C)+{\varepsilon}\big)+|D|\cdot\frac{{\varepsilon}}{|D|}+|D'|\cdot\frac{{\varepsilon}}{|D'|}=d^1_{\mathsf ZX}(A,B)+d^1_{\mathsf ZX}(B,C)+4{\varepsilon}.\end{gathered}$$ Given any finite sets, $A,B\in \mathsf FX$, let $\mathbf \Gamma^{\mathsf f}_{\!X\!}(A,B)$ be the subfamily of finite graphs in $\mathbf \Gamma_{\!X\!}(A,B)$. \[l:BBKZ\] $d^1_{\mathsf ZX}(A,B)=d^1_{\mathsf FX}(A,B)=\inf\limits_{\Gamma\in\mathbf \Gamma^{\mathsf f}_{\!X\!}(A,B)}\ell(\Gamma)$ for all $A,B\in \mathsf FX$. Fix any finite sets $A,B\in \mathsf FX$ and put $I=\inf\limits_{\Gamma\in\mathbf \Gamma_{\!X\!}(A,B)}\ell(\Gamma)$ and $I_{\mathsf f}=\inf\limits_{\Gamma\in\mathbf \Gamma^{\mathsf f}_{\!X\!}(A,B)}\ell(\Gamma)$. The equality $d^1_{\mathsf FX}(A,B)=I_{\mathsf f}$ was proved in Theorem 30.4 in [@BBKZ]. So, it suffices to show that $I=I_{\mathsf f}$. The inequality $I\le I_{\mathsf f}$ is trivial and follows from the inclusion $\mathbf \Gamma^{\mathsf f}_{\!X\!}(A,B)\subseteq \mathbf \Gamma_{\!X\!}(A,B)$. The inequality $I_{\mathsf f}\le I$ will follow as soon as we show that $I_{\mathsf f}\le I+5{\varepsilon}$ for any ${\varepsilon}>0$. Given any ${\varepsilon}>0$, find a graph $\Gamma\in\mathbf\Gamma_{\!X\!}(A,B)$ such that $\ell(\Gamma)<I+{\varepsilon}$. By the definition of the family $\mathbf\Gamma_{\!X\!}(A,B)$, for every $a\in A$ we can find a point $v(a)\in V$ such that $a\in \overline{\Gamma(v(a))}$ and $B\cap\overline{\Gamma(v(a))}$ contains some point $\beta(a)$. Since $\beta(a)\in\overline{\Gamma(v(a))}$, there exists a point $u(a)\in \Gamma(v(a))$ such that $d_X(u(a),\beta(a))<{\varepsilon}/|A|$. Since $a\in \overline{\Gamma(f(x))}$, we can replace $v(a)$ by a suitable point in the connected component $\Gamma(v(a))$ and additionally assume that $d_X(a,v(a))<{\varepsilon}/|A|$. Since the points $v(a),u(a)$ belong to the same connected component of the graph $\Gamma$, there exist a number $n_a\in{\mathbb N}$ and a sequence $v_0(a),\dots,v_{n_a}(a)\in V$ such that $v_0(a)=v(a)$, $v_{n_a}(a)=u(a)$ and $\{v_{i-1}(a),v_i(a)\}\in E$ for every $i\in\{1,\dots,n_a\}$. Now do the same with the set $B$: for every point $b\in B$ choose points $\alpha(b)\in A$ and $v'(b),u'(b)\in V$ such that $b\in \overline{\Gamma(v'(b))}$, $\alpha(b)\in A\cap\overline{\Gamma(v'(b))}$, $d_X(b,v'(b))<{\varepsilon}/|B|$, $u'(b)\in\Gamma(v'(b))$, and $d_X(\alpha(b),u'(b))<{\varepsilon}/|B|$. Since the points $v'(b),u'(b)$ belong to the same connected component of the graph $\Gamma$, there exist $m_a\in{\mathbb N}$ and a sequence $v_0'(b),\dots,v'_{m_b}(b)\in V$ such that $v'_0(b)=v'(b)$, $v'_{m_b}(b)=u'(b)$ and $\{v'_{i-1}(b),v'_i(b)\}\in E$ for every $i\in\{1,\dots,m_a\}$. Now consider the finite graph $\Gamma'=(V',E')$ with the set of vertices $$V'=A\cup B\cup\bigcup_{a\in A}\{v_i(a):1\le i\le n_a\}\cup \bigcup_{b\in B}\{v'_i(b):1\le i\le m_a\}$$ and the set of edges $$\begin{gathered} E'=\Big(\bigcup_{a\in A}\big\{\{a,v(a)\},\{u(a),\beta(a)\},\{v_{i-1}(a),v_i(a)\}:1\le i\le n_a\big\}\Big)\cup\\ \Big(\bigcup_{b\in B}\big\{\{b,v'(b)\},\{u'(b),\alpha(b)\},\{v'_{i-1}(b),v'_i(b)\}:1\le i\le m_a\big\}\Big).\end{gathered}$$ It is easy to see that $\Gamma'\in\mathbf\Gamma^{\mathsf f}_{\!X\!}(A,B)$ and hence $$\begin{gathered} I_{\mathsf f}\le \ell(\Gamma')\le\\ \ell(\Gamma)+\sum_{a\in A}\big(d_X(a,v(a))+d_X(u(a),\beta(a))\big)+\sum_{b\in B}\big(d_X(b,v'(b))+d_X(\alpha(b),u'(b))\big)<\\ I+{\varepsilon}+2{\varepsilon}+2{\varepsilon}=I+5{\varepsilon}.\end{gathered}$$ \[l:dense\] For any dense subset $Y\subseteq X$, the set $\mathsf FY$ is dense in the metric space $\mathsf Z^1\!X=(\mathsf ZX,d^1_{\mathsf ZX})$. Given any $A\in \mathsf ZX$ and ${\varepsilon}>0$, it suffices to find a set $B\in\mathsf FY$ such that $d^1_{\mathsf ZX}(A,B)<2{\varepsilon}$. Since $\ell(A)=0$, there exists a graph $\Gamma=(V,E)$ in $X$ such that $\Gamma$ has finitely many connected components, $A\subseteq \overline{V}$ and $\ell(A)<{\varepsilon}$. Choose a finite set $B'\subseteq V$ that meets each connected component of the graph $\Gamma$ and consider the subset $B''=\{b\in B':\overline{\Gamma(b)}\cap A\ne\emptyset\}$. It is easy to see that $\Gamma\in\mathbf \Gamma_{\!X\!}(A,B'')$ and hence $d^1_{\mathsf ZX}(A,B'')\le\ell(\Gamma)<{\varepsilon}$. Using the density of the set $Y$ in $X$, choose a finite set $B\subseteq Y$ and a surjective function $f:B''\to B$ such that $d_X(x,f(x))<{\varepsilon}/|B''|$ for all $x\in B''$. Consider the graph $\Gamma'=(V',E')$ with the set of vertices $V'=B''\cup f(B'')$ and the set of edges $E'=\{\{x,f(x)\}:x\in B''\}$. Observe that $\Gamma'\in\mathbf \Gamma_{\!X\!}(B'',B)$ and hence $d^1_{\mathsf ZX}(B,B'')\le\ell(\Gamma')<\sum_{x\in B''}d_X(x,f(x))<{\varepsilon}$. Then $$d^1_{\mathsf ZX}(A,B)\le d^1_{\mathsf ZX}(A,B'')+d^1_{\mathsf ZX}(B'',B)<{\varepsilon}+{\varepsilon}=2{\varepsilon}.$$ \[l:complete\] If the metric space $X$ is complete, then so is the metric space $\mathsf Z^1\!X$. We need to prove that each Cauchy sequence in the space $\mathsf Z^1\!X$ is convergent. Since the space $\mathsf F^1\!X$ is dense in $\mathsf Z^1\!X$ (see Lemmas \[l:BBKZ\], \[l:dense\]), it suffices to prove that each Cauchy sequence in $\mathsf F^1\!X$ converges to some set $A\in \mathsf ZX$. So, fix a Cauchy sequence $\{A_n\}_{n\in{\omega}}\subseteq\mathsf F^1\!X$. Since $d_{\mathsf FX}=d^\infty_{\mathsf FX}\le d^1_{\mathsf FX}$, the sequence $(A_n)_{n\in{\omega}}$ remains Cauchy in the Hausdorff metric $d_{\mathsf FX}$. By the completeness of the hyperspace $\mathsf KX$, the sequence $(A_n)_{\in{\omega}}$ converges (in the Hausdorff metric $d_{\mathsf KX}$) to some nonempty compact set $A\in \mathsf KX$. It remains to show that $A\in \mathsf ZX$ and the sequence $(A_n)_{n\in{\omega}}$ converges to $A$ in the metric space $\mathsf Z^1\!X$. Given any ${\varepsilon}>0$, use the Cauchy property of the sequence $(A_n)_{n\in{\omega}}$ and find an increasing number sequence $(n_k)_{k\in{\omega}}$ such that $$d^1_{\mathsf FX}(A_{n_k},A_i)<\frac{\varepsilon}{2^{k+1}}$$for any $k\in{\omega}$ and $i\ge n_k$. By Lemma \[l:BBKZ\], for every $k\in{\omega}$ there exists a graph $\Gamma_k\in\mathbf \Gamma^{\mathsf f}_{\!X\!}(A_{n_k},A_{n_{k+1}})$ such that $\ell(\Gamma_k)<\frac{\varepsilon}{2^{k+1}}$. Now consider the graph $\Gamma=(V,E)$ with $V=\bigcup_{k\in{\omega}}V_k$ and $E=\bigcup_{k\in{\omega}}E_k$ and observe that each connected component of the graph $\Gamma$ meets the finite set $A_{n_0}$, which implies that $\Gamma$ has finitely many connected components. Taking into account that $A$ is the limit of the sequence $(A_{n_k})_{k\in{\omega}}$ in the Hausdorff metric, we conclude that $A\subseteq\overline{\bigcup_{k\in{\omega}}A_{n_k}}\subseteq\overline V$ and the closure of each connected component of $\Gamma$ meets the set $A$. Then $\Gamma\in\mathbf \Gamma_{\!X\!}(A)$ and $$\ell(A)\le\ell(\Gamma)\le\sum_{k\in{\omega}}\ell(\Gamma_k)<\sum_{k\in{\omega}}\frac{\varepsilon}{2^{k+1}}={\varepsilon}.$$ This shows that $\ell(A)=0$ and $A\in \mathsf ZX$. It remains to show that the sequence $(A_n)_{n\in{\omega}}$ converges to $A$ in the metric space $\mathsf Z^1\!X$. Since this sequence is Cauchy, it suffices to show that the subsequence $(A_{n_k})_{k\in{\omega}}$ converges to $A$. For every $k\in{\omega}$, consider the graph $\widetilde\Gamma_k=(\widetilde V_k,\widetilde E_k)$ with the set of vertices $\widetilde V_k=\bigcup_{i=k}^\infty V_k$ and the set of edges $\widetilde E_k=\bigcup_{i=k}^\infty E_k$. It can be shown that $\widetilde\Gamma_k\in\mathbf \Gamma_{\!X\!}(A,A_{n_k})$ and hence $$d^1_{\mathsf ZX}(A,A_{n_k})\le\ell(\widetilde\Gamma_k)\le\sum_{i=k}^\infty\ell(\Gamma_i)<\sum_{i=k}^\infty\frac{\varepsilon}{2^{i+1}}=\frac{\varepsilon}{2^k}\underset{k\to\infty}{\;\longrightarrow\;} 0,$$which means that the sequence $(A_{n_k})_{k\in{\omega}}$ converges to $A$ in the metric space $\mathsf Z^1\!X$. \[l:equal\] If $Y$ is a dense subspace of $X$, then $d^1_{\mathsf ZX}(A,B)=d^1_{\mathsf ZY}(A,B)$ for every $A,B\in \mathsf ZY$. The inequality $d^1_{\mathsf ZX}(A,B)\le d^1_{\mathsf ZY}(A,B)$ is trivial and follows from the inclusion$\mathbf \Gamma_{\!Y\!}(A,B)\subseteq \mathbf \Gamma_{\!X\!}(A,B)$. Assuming that $d^1_{\mathsf ZX}(A,B)<d^1_{\mathsf ZY}(A,B)$, find ${\varepsilon}>0$ such that $d^1_{\mathsf ZX}(A,B)+7{\varepsilon}<d^1_{\mathsf ZY}(A,B)$. Using Lemma \[l:dense\], choose finite sets $A',B'\in \mathsf FY$ such that $d^1_{\mathsf ZY}(A,A')<{\varepsilon}$ and $d^1_{\mathsf ZY}(B,B')<{\varepsilon}$. Then also $d^1_{\mathsf ZX}(A,A')\le d^1_{\mathsf ZY}(A,A')<{\varepsilon}$ and $d^1_{\mathsf ZX}(B,B')\le d^1_{\mathsf ZY}(B,B')<{\varepsilon}$. Applying the triangle inequality, we obtain $$\begin{gathered} d^1_{\mathsf ZX}(A',B')<d^1_{\mathsf ZX}(A',A)+d^1_{\mathsf ZX}(A,B)+d^1_{\mathsf ZX}(B,B')\le 2{\varepsilon}+d^1_{\mathsf ZX}(A,B)<\\ 2{\varepsilon}+d^1_{\mathsf ZY}(A,B)-7{\varepsilon}\le d^1_{\mathsf ZY}(A,A')+d^1_{\mathsf ZY}(A',B')+d^1_{\mathsf ZY}(B',B)-5{\varepsilon}<\\ {\varepsilon}+d^1_{\mathsf ZY}(A',B')+{\varepsilon}-5{\varepsilon}=d^1_{\mathsf ZY}(A',B')-3{\varepsilon}.\end{gathered}$$ By Lemma \[l:BBKZ\], there exists a finite graph $\Gamma=(V,E)\in\mathbf \Gamma^{\mathsf f}_{\!X\!}(A',B')$ such that $$\ell(\Gamma)<d^1_{\mathsf ZX}(A',B')+{\varepsilon}.$$ Since $Y$ is dense in $X$, we can find a function $f:V\to Y$ such that $f(x)=x$ if $x\in Y$ and $d_X(f(x),x)<{\varepsilon}/|E|$ if $x\in V\setminus Y$. Consider the graph $\Gamma'=(V',E')$ with the set of vertices $V'=f(V)$ and the set of edges $E'=\{\{f(x),f(y)\}:\{x,y\}\in E\}$. Observe that the graph $\Gamma'$ belongs to the family $\mathbf \Gamma_{\!Y\!}^{\mathsf f}(A',B')$ and hence $$\begin{gathered} d^1_{\mathsf ZY}(A',B')\le\ell(\Gamma')=\sum_{\{x',y'\}\in E'}d_X(x',y')\le\sum_{\{x,y\}\in E}d_X(f(x),f(y))\le \\ \sum_{\{x,y\}\in E}(d_X(f(x),x)+d_X(x,y)+d_X(y,f(y))< \sum_{\{x,y\}\in E}(\tfrac{\varepsilon}{|E|}+d_X(x,y)+\tfrac{\varepsilon}{|E|})<\\ 2{\varepsilon}+\sum_{\{x,y\}\in E}d_X(x,y)=2{\varepsilon}+\ell(\Gamma)<2{\varepsilon}+d^1_{\mathsf ZX}(A',B')+{\varepsilon}<d^1_{\mathsf ZY}(A',B'),\end{gathered}$$ which is a desired contradiction showing that $d^1_{\mathsf ZX}(A,B)=d^1_{\mathsf ZY}(A,B)$. If $\bar X$ is a completion of $X$, then the complete metric space $\mathsf Z^1\!\bar X$ is a completion of the metric space $\mathsf F^1\!X$. By Lemma \[l:complete\], the metric space $\mathsf Z^1\!\bar X$ is complete. By Lemmas \[l:BBKZ\] and \[l:equal\], for any $A,B\in \mathsf FX$ we have $$d^1_{\mathsf FX}(A,B)=d^1_{\mathsf ZX}(A,B)=d^1_{\mathsf Z\bar X}(A,B),$$ so the metric space $\mathsf F^1\!X$ is a subspace of the complete metric space $\mathsf Z^1\!\bar X$. By Lemma \[l:dense\], the space $\mathsf FX$ is dense in $\mathsf Z^1\!\bar X$. This means that $\mathsf Z^1\!\bar X$ is a completion on $\mathsf F^1\!X$. Now we discuss the interplay between zero length and 1-dimensional Hausdorff measure. A subset $A$ of a metric space $X$ is defined to have [*$1$-dimensional Hausdorff measure zero*]{} if for any ${\varepsilon}>0$ there exists a countable set $C\subseteq X$ and a function $\epsilon:C\to(0,1]$ such that $\sum_{c\in C}\epsilon(c)<{\varepsilon}$ and $A\subseteq \bigcup_{c\in C}B(c,\epsilon(c))$. Here and further on by $$B(x,\delta)=\{y\in X:d_X(x,y)<\delta\}\mbox{ \ and \ }B[x,\delta]=\{y\in X:d_X(x,y)\le\delta\}$$ we denote respectively the open and closed balls of radius $\delta$ around a point $x$ in the metric space $(X,d_X)$. \[p:zero\] If a subset $A$ of a metric space $(X,d_X)$ has zero length, then it is totally bounded, its closure has zero length and also $\bar A$ has 1-dimensional Hausdorff measure zero. If $A$ has zero length, then for every ${\varepsilon}>0$ there exists a graph $\Gamma=(V,E)$ in $X$ that has finitely many connected components such that $\ell(\Gamma)<{\varepsilon}$ and $A\subseteq\overline V$. Then also $\bar A\subseteq\overline V$, which means that $\bar A$ has zero length. To see that $\bar A$ has 1-dimensional Hausdorff measure zero, choose a finite set $D\subseteq V$ that meets each connected component of $V$ in a single point. Then $\{\Gamma(x)\}_{x\in D}$ is a finite disjoint cover of $V$. For every $x\in D$ let $\epsilon(x):=\sup_{y\in \Gamma(x)}d_X(x,y)$ and observe that $V\subseteq\bigcup_{x\in D}B(x,\epsilon(x))$. The connectedness of $\Gamma(x)$ implies that $\epsilon(x)\le\ell(\Gamma(x))$ and $\sum_{x\in D}\epsilon(x)\le\ell(\Gamma)<{\varepsilon}$. Choose any $\delta>0$ such that $|D|\cdot\delta+\sum_{x\in D}\epsilon(x)<{\varepsilon}$ and observe that $$\bar A\subseteq\overline V\subseteq \bigcup_{x\in D}B[x,\epsilon(x)]\subseteq \bigcup_{x\in D}B(x,\epsilon(x)+\delta).$$ Since $\sum_{x\in D}(\epsilon(x)+\delta)=|D|\cdot\delta+\sum_{x\in D}\epsilon(x)<{\varepsilon}$, and ${\varepsilon}$ is arbitrary, the set $\bar A$ has 1-dimensional Hausdorff measure zero. For subsets of the real line we have the following characterization. \[p:zero2\] For a subset $A$ of the real line the following conditions are equivalent: 1. $A$ has zero length; 2. the closure $\bar A$ is compact and has zero length; 3. the closure $\bar A$ is compact and has $1$-dimensional Hausdorff measure zero; 4. the closure $\bar A$ is compact and has Lebesgue measure zero. The implications $(1){\Rightarrow}(2){\Rightarrow}(3)$ were proved in Proposition \[p:zero\]. The implication $(3){\Rightarrow}(4)$ follows from the definition of the Lebesgue measure (as the 1-dimensional Hausdorff measure) on the real line. To prove that $(4){\Rightarrow}(1)$, assume that the closure $\bar A$ is compact and has Lebesgue measure zero. Take any ${\varepsilon}>0$. Using the compactness of the set $\bar A$ and the regularity of the Lebesgue measure, construct inductively a decreasing sequence $(U_k)_{k\in{\omega}}$ of bounded open neighborhoods of $\bar A$ such that for every $k\in{\omega}$ the following conditions are satisfied: - $\overline U_{\!k+1}\subset U_k$; - the set $U_k$ has Lebesgue measure $\lambda(U_k)<{\varepsilon}/2^{k}$; - $U_k=\bigcup_{i=1}^{n_k}(a_{i,k},b_{i,k})$ for some $n_k\in{\mathbb N}$ and real numbers $a_{1,k}<b_{1,k}\le\!\cdots\!\le a_{n_k,k}<b_{n_k,k}$ such that $A\cap (a_{i,k},b_{i,k})\ne\emptyset$ for every $i\in\{1,\dots,n_k\}$. For every $k\in{\omega}$ let $$a'_{i,k}:=\min\{a_{j,k+1}:j\in\{1,\dots,n_{k+1}\},\;a_{i,k}<a_{j,k+1}\}$$ and observe that $a'_{i,k}\le\min \big(\bar A\cap(a_{i,k},b_{i,k})\big)$ and hence $|a_{i,k}-a'_{i,k}|\le|a_{i,k}-b_{i,k}|$. For every $k\in{\mathbb N}$, let $$\Omega_k=\big\{i\in\{1,\dots,n_k-1\}:\exists j\in\{1,\dots,n_{k-1}\}\;\;\;(b_{i,k},a_{i+1,k})\subseteq (a_{j,k-1},b_{j,k-1})\big\}.$$ Consider the graph $\Gamma=(V,E)$ with the set of vertices $$V=\bigcup_{k\in{\omega}}\{a_{i,k},b_{i,k}:1\le i\le n_k\}$$ and the set of edges $$E=\big\{\{a_{i,k},b_{i,k}\},\{a_{i,k},a'_{i,k}\}:k\in{\omega},\;i\in\{1,\dots,n_k\}\big\}\cup\big\{\{b_{i,k},a_{i+1,k}\}:k\in{\mathbb N},\;i\in\Omega_k\big\}.$$ It is easy to see that $A\subseteq\bar A\subseteq \overline V$ and each connected component of the graph $\Gamma$ intersects the set $\{a_{i,0}:1\le i\le n_0\}$. Therefore, $\Gamma$ has finitely many connected components. Also $$\begin{gathered} \ell(\Gamma)\le \sum_{k=0}^\infty \sum_{i=1}^{n_k}(|b_{i,k}-a_{i,k}|+|a'_{i,k}-a_{i,k}|)+\sum_{k=1}^\infty \sum_{i\in\Omega_k}|a_{i+1,k}-b_{i,k}|<\\ 2\cdot\sum_{k=0}^\infty \sum_{i=1}^{n_k}|b_{i,k}-a_{i,k}|+\sum_{k=1}^\infty\sum_{j=1}^{n_{k-1}}|b_{i,k-1}-a_{i,k-1}|= 3\cdot\sum_{k=0}^\infty \sum_{i=1}^{n_k}|b_{i,k}-a_{i,k}|\le\\ 3\cdot\sum_{k=0}^\infty\lambda(U_k)<3\sum_{k=0}^\infty\frac{\varepsilon}{2^{k}}=3{\varepsilon}, \end{gathered}$$ which implies that the set $A$ has zero length. \[p:R\] For the real line $X={\mathbb R}$, the identity inclusion $\mathsf Z^1\!X\to \mathsf KX$ is a topological embedding. Because of Lemma \[l:H\], it suffices to prove that for every $A\in\mathsf ZX$ and ${\varepsilon}>0$ there exists $\delta>0$ such that for any $B\in\mathsf ZX$ the inequality $d_{\mathsf KX}(A,B)<\delta$ implies $d^1_{\mathsf ZX}(A,B)<{\varepsilon}$. By Proposition \[p:zero2\], the set $\bar A$ is compact and has Lebesgue measure zero. By the regularity of the Lebesgue measure on the real line, there exists an open neighborhood $U$ of $\bar A$ in ${\mathbb R}$ such that $U=\bigcup_{i=1}^n(a_i,b_i)$ for some sequence $a_1<b_1<a_2<b_2<\dots<a_n<b_n$ such that $\sum_{i=1}^n|b_i-a_i|<\tfrac19{\varepsilon}$. By the proof of Proposition \[p:zero2\], there exists a graph $\Gamma_{\!A}=(V_{\!A},E_A)$ such that $\bar A\subseteq \overline V_{\!\!A}$, $\ell(\Gamma_{\!A})<3\cdot\tfrac19{\varepsilon}=\tfrac13{\varepsilon}$, and each connected component of $\Gamma_{\!A}$ intersects the set $\{a_i\}_{i=1}^n$. Find $\delta>0$ such that every set $B\in\mathsf KX$ with $d_{\mathsf KX}(A,B)<\delta$ is contained in $U$. Take any set $B\in\mathsf ZX$ with $d_{\mathsf KX}(A,B)<\delta$. Then $B\subseteq U$ and by the proof of Proposition \[p:zero2\], there exists a graph $\Gamma_{\!B}=(V_{\!B},E_{\!B})$ with finitely many components such that $\overline B\subseteq\overline V_{\!B}\subset U$ and $\ell(\Gamma_{\!B})<3\cdot\frac19 {\varepsilon}=\frac13{\varepsilon}$. Let $D\subseteq V$ be a finite set intersecting each connected component of the graph $\Gamma_{\!B}$. For every $i\in\{1,\dots,n\}$, write the set $\{a_i\}\cup \big(D\cap (a_i,b_i)\big)$ as $\{a_{i,0},\dots,a_{i,m_i}\}$ for some points $a_{i,0}<\dots<a_{i,m_i}$. It follows that $a_{i,1}=a_i$ and $a_{i,m_i}\le b_i$, which implies $\sum_{j=1}^{m_i}|a_{i,j}-a_{i,j-1}|\le|b_i-a_i|$. Consider the graph $\Gamma=(V,E)$ with the set of vertices $V=V_A\cup V_B$ and the set of edges $$E=E_A\cup E_B\cup\bigcup_{i=1}^n\big\{\{a_{i,j-1},a_{i,j}\}:j\in\{1,\dots,m_i\}\big\}.$$ It can be shown that $\Gamma\in\mathbf\Gamma_{\!X\!}(A,B)$ and hence $$d^1_{\mathsf ZX}(A,B)\le\ell(\Gamma)\le\ell(\Gamma_A)+\ell(\Gamma_B)+\sum_{i=1}^n\sum_{j=1}^{m_i}|a_{i,j}-a_{i,j-1}|<\tfrac13{\varepsilon}+\tfrac13{\varepsilon}+\sum_{i=1}^n|b_i-a_i|<\tfrac23{\varepsilon}+\tfrac19{\varepsilon}<{\varepsilon}.$$ Proposition \[p:R\] is specific for the real line and does not hold for higher-dimensional Euclidean spaces. To prove this fact, let us recall the definition of the upper box-counting dimension $\overline{\dim}_B(X)$ of a metric space $X$. Given any ${\varepsilon}>0$, denote by $N_{\varepsilon}(X)$ by the smallest cardinality of a cover of $X$ by subsets of diameter $\le {\varepsilon}$. Observe that the metric space $X$ is totally bounded iff $N_{\varepsilon}(X)$ is finite for every ${\varepsilon}>0$. If $X$ is not totally bounded, then put $\overline{\dim}_B(X)=\infty$. If $X$ is totally bounded, then let $$\overline{\dim}_B(X):=\limsup_{{\varepsilon}\to+0}\frac{\ln N_{\varepsilon}(X)}{\ln(1/{\varepsilon})}\in[0,\infty].$$ By [@Fal §3.2], for every $n\in{\mathbb N}$, every bounded set $X\subseteq{\mathbb R}^n$ with nonempty interior has $\overline{\dim}_B(X)=n$. In the following proposition we endow the hyperspace $\mathsf FX$ with the Hausdorff metric. \[p:dim\] Let $X$ be a metric space and $Y\subseteq X$ be a subspace of $X$ such that $\overline{\dim}_B(Y)>1$. Then for any $l\in{\mathbb N}$ there exists a nonempty finite subset $A\subseteq Y$ such that $d^1_{\mathsf FX}(A,\{x\})\ge l$ for any singleton $\{x\}\subseteq X$. To derive a contradiction, assume that there exists $l\in{\mathbb N}$ such that for any finite set $A\subseteq Y$ there exists $x\in X$ such that $d^1_{\mathsf FX}(A,\{x\})< l$. We are going to show that $N_{2{\varepsilon}}(Y)\le (2l+1)/{\varepsilon}$ for every ${\varepsilon}\in(0,1]$. Given any ${\varepsilon}\in(0,1]$, use the Kuratowski-Zorn Lemma and find a maximal subset $M$ in $Y$, which is $2{\varepsilon}$-separated in the sense that $d_X(y,z)\ge 2{\varepsilon}$ for any distinct points $y,z\in M$. The maximality of the set $M$ implies that $Y\subseteq \bigcup_{y\in M}B(y,2{\varepsilon})$. We claim that $|M|\le (1+2l)/{\varepsilon}$. To derive a contradiction, assume that $|M|>(1+2l)/{\varepsilon}$. In this case we can find a finite subset $A\subseteq M$ such that $|A|>(1+2l)/{\varepsilon}$. The choice of the number $l$ ensures that $d^1_{ZX}(A,\{x\})< l$ for some $x\in X$. By Lemma \[l:BBKZ\], there exists a finite graph $\Gamma\in\mathbf\Gamma_{\!X\!}(\{x\},A)$ such that $\ell(\Gamma)<l$. Since each connected component of the graph $\Gamma$ meets the singleton $\{x\}$, the graph $\Gamma=(V,E)$ is connected. Replacing $\Gamma$ by a minimal connected subgraph, we can assume that $\Gamma$ is a tree. By Lemma \[l:tree\] (proved below), there exists a sequence $v_0,\dots,v_n\in V$ such that - $V=\{v_0,\dots,v_n\}$; - $\big\{\{v_{i-1},v_i\}:1\le i\le n\big\}\subseteq E$; - for every $e\in E$ the set $\big\{i\in\{1,\dots,n\}:\{v_{i-1},v_i\}=e\big\}$ contains at most two elements. Choose a sequence of real numbers $t_0,\dots,t_n$ such that $t_0=0$ and $t_i-t_{i-1}=d_X(v_i,v_{i-1})$ for every $i\in\{1,\dots,n\}$. The condition (iii) implies that $t_n\le 2\ell(\Gamma)<2l$. Then the set $T=\{t_0,\dots,t_n\}$ has $$N_{\varepsilon}(T)<1+\frac{t_n}{\varepsilon}<1+\frac{2l}{{\varepsilon}}\le \frac{1+2l}{{\varepsilon}}.$$ Taking into account that the map $T\to V$, $t_i\mapsto v_i$, is non-expanding, we conclude that $N_{\varepsilon}(A)\le N_{\varepsilon}(V)\le N_{\varepsilon}(T)< (1+2l)/{\varepsilon}$. Since the set $A$ is $2{\varepsilon}$-separated, it has cardinality $|A|=N_{\varepsilon}(A)<(1+2l)/{\varepsilon}$, which contradicts the choice of $A$. This contradiction shows that $|M|\le (1+2l)/{\varepsilon}$ and then $N_{2{\varepsilon}}(Y)\le |M|\le (1+2l)/{\varepsilon}$ for any ${\varepsilon}>0$. Taking the upper limit at ${\varepsilon}\to+0$, we obtain the upper bound $$\overline{\dim}_B(Y)=\limsup_{{\varepsilon}\to+0}\frac{\ln N_{\varepsilon}(Y)}{\ln(1/{\varepsilon})}=\limsup_{{\varepsilon}\to+0}\frac{\ln N_{2{\varepsilon}}(Y)}{-\ln(1/(2{\varepsilon}))}\le\limsup_{{\varepsilon}\to+0}\frac{\ln((1+2l)/{\varepsilon})}{\ln(1/(2{\varepsilon}))}=1,$$ which contradicts our assumption. \[l:tree\] For any finite tree $\Gamma=(V,E)$, there exists a sequence $v_0,\dots,v_n\in V$ such that - $V=\{v_0,\dots,v_n\}$, - $\big\{\{v_{i-1},v_i\}:1\le i\le n\big\}=E$, and - for every edge $e\in E$ the set $\{i\in\{1,\dots,n\}:\{v_{i-1},v_i\}=e\}$ contains at most two elements. This lemma will be proved by induction on the cardinality $|V|$ of the tree $V$. If $|V|=1$, then let $v_0$ be the unique vertex of $X$ and observe that the sequence $v_0$ has the properties (i)–(iii). Assume that for some $k\ge 2$ the lemma has been proved for all trees on $<k$ vertices. Let $\Gamma=(V,E)$ be any tree with $|V|=k$. By [@Diestel 1.5.1], the tree $\Gamma$ has exactly $k-1$ edges. Consequently, there exists a vertex $v\in V$ having a unique neighbor $u\in V\setminus\{v\}$ in the tree $(V,E)$. Put $V'=V\setminus\{v\}$, $E'=E\setminus\big\{\{u,v\}\big\}$ and observe that $(V',E')$ is a tree on $k-1$ vertices. By the inductive assumption, there exists a sequence $v'_1,\dots,v'_n\in V'$ such that $V'=\{v'_1, \dots, v'_n\}$, $\big\{\{v'_{i-1},v'_i\}:i\in \{1,\dots,n\}\big\}=E'$, and for every $e\in E'$ the set $\{i\in\{1,\dots,n\}:\{v'_{i-1},v'_i\}=e\}$ contains at most two elements. Find an index $j\in\{1,\dots,n\}$ such that $v'_j=u$ and consider the sequence $v_0,\dots,v_{n+1}$, where $v_i=v'_i$ for $i\le j$, $v_{j+1}=v$, and $v_{i}=v'_{i-2}$ for $i\in \{j+1,\dots,n+2\}$. It is easy to see that the sequence $v_0,\dots,v_{n+2}$ has the properties (i)–(iii). Proposition \[p:dim\] implies the following corollary, in which by $\mathsf FX$ we denote the hyperspace of nonempty finite subsets of $X$, endowed with the Hausdorff metric. Let $X$ be a metric space. If for some point $x\in X$ the identity map $\mathsf FX\to\mathsf F^1\!X$ is continuous at $\{x\}$, then the point $x$ has a neighborhood $O_x\subseteq X$ with box-counting dimension $\overline{\dim}_B(O_x)\le 1$. Assuming that the identity map $\mathsf FX\to\mathsf Z^1\!X$ is continuous at $\{x\}$, we can find $\delta>0$ such that for any set $A\in\mathsf FX$ with $d_{\mathsf FX}(A,\{x\})<\delta$ we have $d_{\mathsf FX}^1(A,\{x\})<1$. Let $O_x:=B(x,\delta)$. Assuming that $\overline{\dim}_B(O_x)>1$, we can apply Proposition \[p:dim\] and find a finite set $A\subseteq O_x$ such that $d^1_{\mathsf FX}(A,\{x\})>1$. On the other hand, the inclusion $A\subseteq O_x=B(x,\delta)$ implies that $d_{\mathsf FX}(A,x)<\delta$ and hence $d_{\mathsf FX}^1(A,\{x\})<1$ by the choice of $\delta$. This contradiction shows that $\overline{\dim}_B(O_x)\le 1$. Finally, we present an example showing that the equivalence $(2)\Leftrightarrow(3)$ in Proposition \[p:zero2\] does not hold for higher-dimensional Euclidean spaces. Assume that $X$ is a complete metric space such that every nonempty open set $U\subseteq X$ has box-counting dimension $\overline{\dim}_B(U)>1$. Then every nonempty open set $U$ contains a compact subset $A\subseteq U$ such that $A$ has $1$-dimensional Hausdorff measure zero but fails to have zero length. Choose any point $x_0\in U$ and a positive number ${\varepsilon}_0$ such that $B[x_0,{\varepsilon}_0]\subseteq U$. Put $A_0=\{x_0\}$. For every $n\in{\mathbb N}$ we shall inductively choose a finite subset $A_n\subseteq X$, a positive real number ${\varepsilon}_n$, and a map $r_n:A_n\to A_{n-1}$, satisfying the following conditions: - $A_{n-1}\subseteq A_n$; - ${\varepsilon}_n\le \frac1{2^n|A_n|}$; - $B[x,{\varepsilon}_n]\cap B[y,{\varepsilon}_n]=\emptyset$ for any distinct points $x,y\in A_n$; - $r_n(x)=x$ for any $x\in A_{n-1}$; - $B[x,{\varepsilon}_n]\subseteq B(r_n(x),{\varepsilon}_{n-1})$ for any $x\in A_{n-1}$; - $d^1_{\mathsf FX}(\{x\},r_n^{-1}(x))>n$ for every $x\in A_{n-1}$. Assume that for some $n\in{\mathbb N}$ we have constructed a set $A_{n-1}$ and a number ${\varepsilon}_{n-1}>0$ satisfying the condition (iii). By our assumption, for every $y\in A_{n-1}$ the ball $B(y,{\varepsilon}_{n-1})$ has $\overline{\dim}_BB(y,{\varepsilon}_{n-1})>1$. By Proposition \[p:dim\], the ball $B(y,{\varepsilon}_{n-1})$ contains a finite subset $A'_y$ such that $d^1_{\mathsf FX}(A'_y,\{y\})>n$. The definition of the metric $d^1_{\mathsf FX}$ implies that $d^1_{\mathsf FX}(A'_y\cup\{y\},\{y\})=d^1_{\mathsf FX}(A'_y,\{y\})>n$. Let $A_n=\bigcup_{y\in A_{n-1}}(\{y\}\cup A'_y)$ and $r_n:A_n\to A_{n-1}$ be the map assigning to each point $x\in A_n$ the unique point $y\in A_{n-1}$ such that $x\in A'_y\cup\{y\}$. It is clear that the $A_n$ satisfies the inductive condition (i) and the function $r_n$ satisfies the conditions (iv), (vi). Now choose any number ${\varepsilon}_n$ satisfying the conditions (ii), (iii) and (v). This completes the inductive step. After completing the inductive construction, consider the compact set $$A=\bigcap_{n\in{\omega}}\bigcup_{x\in A_n}B[x,{\varepsilon}_n]\subseteq U$$ in $X$. We claim that the set $A$ has 1-dimensional Hausdorff measure zero. Given any ${\varepsilon}>0$, find $n\in{\omega}$ such that $\frac2{2^n}<{\varepsilon}$ and observe that $A\subseteq\bigcup_{x\in A_n}B(x,2{\varepsilon}_n)$ and $$\sum_{x\in A_n}2{\varepsilon}_n<\sum_{x\in A_n}\frac2{2^n|A_n|}=\frac2{2^n}<{\varepsilon},$$ witnessing that the 1-dimensional Hausdorff measure of $A$ is zero. Assuming that $A$ has zero length, we calculate the distance $d^1_{\mathsf ZX}(A,A_0)<\infty$ and find a graph $\Gamma\in\mathbf \Gamma_{\!X\!}(A,A_0)$ such that $\ell(\Gamma)<\infty$. Since each component of $\Gamma$ intersects the singleton $A_0=\{x_0\}$, the graph $\Gamma$ is connected. Take any integer number $n>\ell(\Gamma)$ and conclude that for every $x\in A_{n-1}$ we have $\{x\}\cup r_n^{-1}(x)\subseteq A\subseteq\overline V$ and hence $\Gamma\in\mathbf\Gamma_{\!X\!}(\{x\},r_n^{-1}(x))$. By Lemma \[l:BBKZ\], $$d^1_{\mathsf FX}(\{x\},r_n(x))=d^1_{\mathsf ZX}(\{x\},r_n(x))\le\ell(\Gamma)<n,$$ which contradicts the inductive condition (vi). This contradiction shows that the set $A$ fails to have zero length. There are interesting algorithmic problems related to efficient calculating the distance $d^1_{\mathsf FX}(A,B)$ between nonempty finite subsets $A,B$ of a metric space. For a nonempty finite subset $A$ of the Euclidean plane ${\mathbb R}^2$ and a singleton $B=\{x\}\subset{\mathbb R}^2$, the problem of calculating the distance $d^1_{\mathsf FX}(A,B)$ reduces to the classical Steiner’s problem [@Steiner] of finding a tree of the smallest length that contains the set $A\cup B$. This problem is known [@Holby] to be computationally very difficult. On the other hand, for nonempty finite subsets of the real line, there exists an efficient algorithm [@MO] of complexity $O(n\ln n)$ calculating the distance $d^1_{\mathsf F{\mathbb R}}(A,B)$ between two sets $A,B\in\mathsf F\mathbb R$ of cardinality $|A|+|B|\le n$. Also there exists an algorithm of the same complexity $O(n\ln n)$ calculating the Hausdorff distance $d_{\mathsf F{\mathbb R}}(A,B)$ between the sets $A,B$. Finally, let us remark that the evident brute force algorithm for calculating the Hausdorff distance $d_{\mathsf FX}(A,B)$ between nonempty finite subsets of an arbitrary metric space $(X,d_X)$ has complexity $O(|A|{\cdot}|B|)$. Here we assume that calculating the distance between points requires a constant amount of time. T. Banakh, [*A quick algorithm for calculating the $\ell^1$-distance between two finite sets on the real line*]{}, (https://mathoverflow.net/a/277928/61536), 04.08.2017. T. Banakh, V. Brydun, L. Karchevska, M. Zarichnyi, [*The $\ell^p$-metrization of functors with finite supports*]{}, preprint. G. Beer, [*Topologies on closed and closed convex sets*]{}, Kluwer Academic Publishers Group, Dordrecht, 1993. M. Brazil, R. Graham, D.A. Thomas, M. Zachariasen, [*On the history of the Euclidean Steiner tree problem*]{}, Arch. Hist. Exact Sci. [**68**]{}:3 (2014), 327–354. R. Diestel, [*Graph Theory*]{}, GTM [**173**]{}. Springer-Verlag, Berlin, 2005. G. Edgar, [*Measure, Topology, and Fractal Geometry*]{}, Springer, New York, 2008. R. Engelking, [*General Topology*]{}, Heldermann Verlag, Berlin, 1989. K. Falconer, [*Fractal Geometry. Mathematical foundations and applications*]{}, John Wiley & Sons, Inc., Hoboken, NJ, 2003. M.W. Bern, R. Graham,[*The shortest-network problem*]{}, Scientific American. [**260**]{}:1 (1989) 84–89.

--- abstract: 'We consider the problem of online learning and its application to solving minimax games. For the online learning problem, Follow the Perturbed Leader (FTPL) is a widely studied algorithm which enjoys the optimal ${O\left({T^{1/2}}\right)}$ *worst case* regret guarantee for both convex and nonconvex losses. In this work, we show that when the sequence of loss functions is *predictable*, a simple modification of FTPL which incorporates optimism can achieve better regret guarantees, while retaining the optimal worst case regret guarantee for unpredictable sequences. A key challenge in obtaining these tighter regret bounds is the stochasticity and optimism in the algorithm, which requires different analysis techniques than those commonly used in the analysis of FTPL. The key ingredient we utilize in our analysis is the dual view of perturbation as regularization. While our algorithm has several applications, we consider the specific application of minimax games. For solving smooth convex-concave games, our algorithm only requires access to a linear optimization oracle. For Lipschitz and smooth nonconvex-nonconcave games, our algorithm requires access to an optimization oracle which computes the perturbed best response. In both these settings, our algorithm solves the game up to an accuracy of ${O\left({T^{-1/2}}\right)}$ using $T$ calls to the optimization oracle. An important feature of our algorithm is that it is highly parallelizable and requires only $O(T^{1/2})$ iterations, with each iteration making ${O\left({T^{1/2}}\right)}$ parallel calls to the optimization oracle.' author: - | Arun Sai Suggala\ Carnegie Mellon University\ `asuggala@andrew.cmu.edu`\ Praneeth Netrapalli\ Microsoft Research, India\ `praneeth@microsoft.com`\ bibliography: - 'local.bib' title: 'Follow the Perturbed Leader: Optimism and Fast Parallel Algorithms for Smooth Minimax Games' --- Introduction {#sec:intro} ============ In this work, we consider the problem of online learning, where in each iteration, the learner chooses an action and observes a loss function. The goal of the learner is to choose a sequence of actions which minimizes the cumulative loss suffered over the course of learning. The paradigm of online learning has many theoretical and practical applications and has been widely studied in a number of fields, including game theory and machine learning. One of the popular applications of online learning is in solving minimax games arising in various contexts such as boosting [@freund1996game], robust optimization [@chen2017robust], Generative Adversarial Networks [@goodfellow2014generative]. In recent years, a number of efficient algorithms have been developed for regret minimization. These algorithms fall into two broad categories, namely, Follow the Regularized Leader (FTRL) [@mcmahan2017survey] and FTPL [@kalai2005efficient] style algorithms. When the sequence of loss functions encountered by the learner are convex, both these algorithms are known to achieve the optimal ${O\left({T^{1/2}}\right)}$ worst case regret [@cesa2006prediction; @hazan2016introduction]. While these algorithms have similar regret guarantees, they differ in computational aspects. Each iteration of FTRL involves implementation of an expensive projection step. In contrast, each step of FTPL involves solving a linear optimization problem, which can be implemented efficiently for many problems of interest [@garber2013playing; @gidel2016frank; @hazan2020projection]. This crucial difference between FTRL and FTPL makes the latter algorithm more attractive in practice. Even in the more general nonconvex setting, where the loss functions encountered by the learner can potentially be nonconvex, FTPL algorithms are attractive. In this setting, FTPL requires access to an offline optimization oracle which computes the perturbed best response, and achieves ${O\left({T^{1/2}}\right)}$ worst case regret [@suggala2019online]. Furthermore, these optimization oracles can be efficiently implemented for many problems by leveraging the rich body of work on global optimization [@horst2013handbook]. Despite its importance and popularity, FTPL has been mostly studied for the worst case setting, where the loss functions are assumed to be adversarially chosen. In a number of applications of online learning, the loss functions are actually benign and predictable [@rakhlin2012online]. In such scenarios, FTPL can not utilize the predictability of losses to achieve tighter regret bounds. While [@rakhlin2012online; @suggala2019online] study variants of FTPL which can make use of predictability, these works either consider restricted settings or provide sub-optimal regret guarantees (see Section \[sec:bg\] for more details). This is unlike FTRL, where optimistic variants that can utilize the predictability of loss functions have been well understood [@rakhlin2012online; @rakhlin2013optimization] and have been shown to provide faster convergence rates in applications such as minimax games. In this work, we aim to bridge this gap and study a variant of FTPL called Optimistic FTPL (OFTPL), which can achieve better regret bounds, while retaining the optimal worst case regret guarantee for unpredictable sequences. The main challenge in obtaining these tighter regret bounds is handling the stochasticity and optimism in the algorithm, which requires different analysis techniques to those commonly used in the analysis of FTPL. In this work, we rely on the dual view of perturbation as regularization to derive regret bounds of OFTPL. To demonstrate the usefulness of OFTPL, we consider the problem of solving minimax games. A widely used approach for solving such games relies on online learning algorithms [@cesa2006prediction]. In this approach, both the minimization and the maximization players play a repeated game against each other and rely on online learning algorithms to choose their actions in each round of the game. In our algorithm for solving games, we let both the players use OFTPL to choose their actions. For solving smooth convex-concave games, our algorithm only requires access to a linear optimization oracle. For Lipschitz and smooth nonconvex-nonconcave games, our algorithm requires access to an optimization oracle which computes the perturbed best response. In both these settings, our algorithm solves the game up to an accuracy of ${O\left({T^{-1/2}}\right)}$ using $T$ calls to the optimization oracle. While there are prior algorithms that achieve these convergence rates [@he2015semi; @suggala2019online], an important feature of our algorithm is that it is highly parallelizable and requires only $O(T^{1/2})$ iterations, with each iteration making ${O\left({T^{1/2}}\right)}$ parallel calls to the optimization oracle. We note that such parallelizable algorithms are especially useful in large-scale machine learning applications such as training of GANs, adversarial training, which often involve huge datasets such as ImageNet [@russakovsky2015imagenet]. Preliminaries and Background Material {#sec:bg} ===================================== #### Online Learning. The online learning framework can be seen as a repeated game between a learner and an adversary. In this framework, in each round $t$, the learner makes a prediction for some compact set ${\mathcal{X}}$, and the adversary simultaneously chooses a loss function and observe each others actions. The goal of the learner is to choose a sequence of actions $\{{{\ensuremath{\mathbf{x}}}}_t\}_{t=1}^T$ so that the following notion of regret is minimized: When the domain ${\mathcal{X}}$ and loss functions $f_t$ are convex, a number of efficient algorithms for regret minimization have been studied. Some of these include deterministic algorithms such as Online Mirror Descent, Follow the Regularized Leader (FTRL) [@hazan2016introduction; @mcmahan2017survey], and stochastic algorithms such as Follow the Perturbed Leader (FTPL) [@kalai2005efficient]. In FTRL, one predicts ${{\ensuremath{\mathbf{x}}}}_t$ as ${\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}} \sum_{i=1}^{t-1}{\left\langle {\nabla}_i, {{\ensuremath{\mathbf{x}}}}\right\rangle} + R({{\ensuremath{\mathbf{x}}}})$, for some strongly convex regularizer $R$, where ${\nabla}_i = {\nabla}f_i({{\ensuremath{\mathbf{x}}}}_i)$. FTRL is known to achieve the optimal $O(T^{1/2})$ worst case regret in the convex setting [@mcmahan2017survey]. In FTPL, one predicts ${{\ensuremath{\mathbf{x}}}}_t$ as $m^{-1}\sum_{j=1}^m{{\ensuremath{\mathbf{x}}}}_{t,j}$, where ${{\ensuremath{\mathbf{x}}}}_{t,j}$ is a minimizer of the following linear optimization problem: ${\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}} {\left\langle \sum_{i=1}^{t-1}{\nabla}_i - \sigma_{t,j}, {{\ensuremath{\mathbf{x}}}}\right\rangle}.$ Here, $\{\sigma_{t,j}\}_{j=1}^m$ are independent random perturbations drawn from some appropriate probability distribution such as exponential distribution or uniform distribution in a hyper-cube. Various choices of perturbation distribution gives rise to various FTPL algorithms. When the loss functions are linear, @kalai2005efficient show that FTPL achieves ${O\left({T^{1/2}}\right)}$ expected regret, irrespective of the choice of $m$. When the loss functions are convex, @hazan2016introduction showed that the deterministic version of FTPL (*i.e.,* as $m \to \infty$) achieves ${O\left({T^{1/2}}\right)}$ regret. While projection free methods for online convex learning have been studied since the early work of [@hazan2012projection], surprisingly, regret bounds of FTPL for finite $m$ have only been recently studied [@hazan2020projection]. @hazan2020projection show that for Lipschitz and convex functions, FTPL achieves ${O\left({T^{1/2} + m^{-1/2}T }\right)}$ expected regret, and for smooth convex functions, the algorithm achieves ${O\left({T^{1/2} + m^{-1}T}\right)}$ expected regret. When either the domain ${\mathcal{X}}$ or the loss functions $f_t$ are non-convex, no deterministic algorithm can achieve $o(T)$ regret [@cesa2006prediction; @suggala2019online]. In such cases, one has to rely on randomized algorithms to achieve sub-linear regret. In randomized algorithms, in each round $t$, the learner samples the prediction ${{\ensuremath{\mathbf{x}}}}_t$ from a distribution $P_t \in {\mathcal{P}}$, where ${\mathcal{P}}$ is the set of all probability distributions supported on ${\mathcal{X}}$. The goal of the learner is to choose a sequence of distributions $\{P_t\}_{t=1}^T$ to minimize the expected regret $ \sum_{t = 1}^T {\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P_t}\left[f_t({{\ensuremath{\mathbf{x}}}})\right]} - \inf_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}\sum_{t=1}^Tf_t({{\ensuremath{\mathbf{x}}}}). $ A popular technique to minimize the expected regret is to consider a linearized problem in the space of probability distributions with losses $\Tilde{f}_t(P) = {\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P}\left[f_t({{\ensuremath{\mathbf{x}}}})\right]}$ and perform FTRL in this space. In such a technique, $P_t$ is computed as: $ {\mathop{\rm argmin}}_{P \in {\mathcal{P}}} \sum_{i=1}^{t-1} \Tilde{f}_i(P) + R(P), $ for some strongly convex regularizer $R(P).$ When $R(P)$ is the negative entropy of $P$, the algorithm is called entropic mirror descent or continuous exponential weights. This algorithm achieves ${O\left({T^{1/2}}\right)}$ expected regret for bounded loss functions $f_t$. Another technique to minimize expected regret is to rely on FTPL [@gonen2018learning; @suggala2019online]. Here, the learner generates the random prediction ${{\ensuremath{\mathbf{x}}}}_t$ by first sampling a random perturbation $\sigma$ and then computing the perturbed best response, which is defined as ${\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}} \sum_{i=1}^{t-1}f_i({{\ensuremath{\mathbf{x}}}}) - {\left\langle \sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle}$. In a recent work, @suggala2019online show that this algorithm achieves ${O\left({T^{1/2}}\right)}$ expected regret, whenever the sequence of loss functions are Lipschitz. We now briefly discuss the computational aspects of FTRL and FTPL. Each iteration of FTRL (with entropic regularizer) requires sampling from a non-logconcave distribution. In contrast, FTPL requires solving a nonconvex optimization problem to compute the perturbed best response. Of these, computing the perturbed best response seems significantly easier since standard algorithms such as gradient descent seem to be able to find approximate global optima reasonably fast, even for complicated tasks such as training deep neural networks. #### Online Learning with Optimism. When the sequence of loss functions are convex and predictable, @rakhlin2012online [@rakhlin2013optimization] study optimistic variants of FTRL which can exploit the predictability to obtain better regret bounds. Let $g_t$ be our guess of ${\nabla}_t$ at the beginning of round $t$. Given $g_t$, we predict ${{\ensuremath{\mathbf{x}}}}_t$ in Optimistic FTRL (OFTRL) as $ {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}{\left\langle \sum_{i=1}^{t-1}{\nabla}_i + g_t, {{\ensuremath{\mathbf{x}}}}\right\rangle} + R({{\ensuremath{\mathbf{x}}}}). $ Note that when $g_t=0$, OFTRL is equivalent to FTRL. [@rakhlin2012online; @rakhlin2013optimization] show that the regret bounds of OFTRL only depend on $(g_t-{\nabla}_t)$. Moreover, these works show that OFTRL provides faster convergence rates for solving smooth convex-concave games. In contrast to FTRL, the optimistic variants of FTPL have been less well understood. [@rakhlin2012online] studies OFTPL for linear loss functions. But they consider restrictive settings and their algorithms require the knowledge of sizes of deviations $(g_t-\nabla_t)$. [@suggala2019online] studies OFTPL for the more general nonconvex setting. The algorithm predicts ${{\ensuremath{\mathbf{x}}}}_t$ as ${\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}} \sum_{i=1}^{t-1}f_i({{\ensuremath{\mathbf{x}}}}) +g_t({{\ensuremath{\mathbf{x}}}}) - {\left\langle \sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle}$, where $g_t$ is our guess of $f_t$. However, the regret bounds of [@suggala2019online] are sub-optimal and weaker than the bounds we obtain in our work (see Theorem \[thm:oftpl\_noncvx\_regret\]). Moreover, [@suggala2019online] does not provide any consequences of their results to minimax games. We note that their sub-optimal regret bounds translate to sub-optimal rates of convergence for solving smooth minimax games. #### Minimax Games. Consider the following problem, which we refer to as minimax game: $\min_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}\max_{{{\ensuremath{\mathbf{y}}}}\in {\mathcal{Y}}} f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})$. In these games, we are often interested in finding a Nash Equilibrium (NE). A pair $(P,Q)$, where $P$ is a probability distribution over ${\mathcal{X}}$ and $Q$ is a probability distribution over ${\mathcal{Y}}$, is called a NE if: $\sup_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P}\left[f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})\right]}\leq{\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P, {{\ensuremath{\mathbf{y}}}}\sim Q}\left[f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})\right]} \leq \inf_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}{\mathbb{E}_{{{\ensuremath{\mathbf{y}}}}\sim Q}\left[f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})\right]}.$ A standard technique for finding a NE of the game is to rely on no-regret algorithms [@cesa2006prediction; @hazan2016introduction]. Here, both ${{\ensuremath{\mathbf{x}}}}$ and ${{\ensuremath{\mathbf{y}}}}$ players play a repeated game against each other and use online learning algorithms to choose their actions. The average of the iterates generated via this repeated game can be shown to converge to a NE. #### Projection Free Learning. Projection free learning algorithms are attractive as they only involve solving linear optimization problems. Two broad classes of projection free techniques have been considered for online convex learning and minimax games, namely, Frank-Wolfe (FW) methods and FTPL based methods. @garber2013playing consider the problem of online learning when the action space ${\mathcal{X}}$ is a polytope. They provide a FW method which achieves ${O\left({T^{1/2}}\right)}$ regret using $T$ calls to the linear optimization oracle. @hazan2012projection provide a FW technique which achieves ${O\left({T^{3/4}}\right)}$ regret for general online convex learning with Lipschitz losses and uses $T$ calls to the linear optimization oracle. In a recent work, @hazan2020projection show that FTPL achieves ${O\left({T^{2/3}}\right)}$ regret for online convex learning with smooth losses, using $T$ calls to the linear optimization oracle. This translates to ${O\left({T^{-1/3}}\right)}$ rate of convergence for solving smooth convex-concave games. Note that, in contrast, our algorithm achieves ${O\left({T^{-1/2}}\right)}$ convergence rate in the same setting. @gidel2016frank study FW methods for solving convex-concave games. When the constraint sets ${\mathcal{X}},{\mathcal{Y}}$ are *strongly convex*, the authors show geometric convergence of their algorithms. In a recent work, @he2015semi propose a FW technique for solving smooth convex-concave games which converges at a rate of ${O\left({T^{-1/2}}\right)}$ using $T$ calls to the linear optimization oracle. We note that our simple OFTPL based algorithm achieves these rates, with the added advantage of parallelizability. That being said, @he2015semi achieve dimension free convergence rates in the Euclidean setting, where the smoothness is measured w.r.t $\|\cdot\|_2$ norm. In contrast, the rates of convergence of our algorithm depend on the dimension. #### Notation. $\|\cdot\|$ is a norm on some vector space, which is typically $\mathbb{R}^d$ in our work. $\|\cdot\|_{*}$ is the dual norm of $\|\cdot\|$, which is defined as $\|{{\ensuremath{\mathbf{x}}}}\|_{*} = \sup\{{\left\langle {{\ensuremath{\mathbf{u}}}}, {{\ensuremath{\mathbf{x}}}}\right\rangle}: {{\ensuremath{\mathbf{u}}}}\in\mathbb{R}^d, \|{{\ensuremath{\mathbf{u}}}}\|\leq 1\}$. We use ${\Psi_1}, {\Psi_2}$ to denote norm compatibility constants of $\|\cdot\|$, which are defined as ${\Psi_1}= \sup_{{{\ensuremath{\mathbf{x}}}}\neq 0} \|{{\ensuremath{\mathbf{x}}}}\|/\|{{\ensuremath{\mathbf{x}}}}\|_2,\ {\Psi_2}= \sup_{{{\ensuremath{\mathbf{x}}}}\neq 0} \|{{\ensuremath{\mathbf{x}}}}\|_2/\|{{\ensuremath{\mathbf{x}}}}\|.$ We use the notation $f_{1:t}$ to denote $\sum_{i=1}^tf_i$. In some cases, when clear from context, we overload the notation $f_{1:t}$ and use it to denote the set $\{f_1,f_2\dots f_t\}$. For any convex function $f$, $\partial f({{\ensuremath{\mathbf{x}}}})$ is the set of all subgradients of $f$ at ${{\ensuremath{\mathbf{x}}}}$. For any function $f:{\mathcal{X}}\times {\mathcal{Y}}\to \mathbb{R}$, $f(\cdot,{{\ensuremath{\mathbf{y}}}}), f({{\ensuremath{\mathbf{x}}}},\cdot)$ denote the functions ${{\ensuremath{\mathbf{x}}}}\rightarrow f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}), {{\ensuremath{\mathbf{y}}}}\rightarrow f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}).$ For any function $f:{\mathcal{X}}\to\mathbb{R}$ and any probability distribution $P$, we let $f(P)$ denote ${\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P}\left[f({{\ensuremath{\mathbf{x}}}})\right]}.$ Similarly, for any function $f:{\mathcal{X}}\times{\mathcal{Y}}\to\mathbb{R}$ and any two distributions $P,Q$, we let $f(P,Q)$ denote ${\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P,{{\ensuremath{\mathbf{y}}}}\sim Q}\left[f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})\right]}.$ For any set of distributions $\{P_j\}_{j=1}^m$, $\frac{1}{m}\sum_{j=1}^mP_j$ is the mixture distribution which gives equal weights to its components. We use $\text{Exp}(\eta)$ to denote the exponential distribution, whose CDF is given by $P(Z\leq s) =1-\exp(-s/\eta).$ Dual view of Perturbation as Regularization {#sec:duality} =========================================== In this section, we present a key result which shows that when the sequence of loss functions are convex, every FTPL algorithm is an FTRL algorithm. Our analysis of OFTPL relies on this dual view to obtain tight regret bounds. This duality between FTPL and FTRL was originally studied by @hofbauer2002global, where the authors show that any FTPL algorithm, with perturbation distribution admitting a strictly positive density on $\mathbb{R}^d$, is an FTRL algorithm w.r.t some convex regularizer. However, many popular perturbation distributions such as exponential and uniform distributions don’t have a strictly positive density. In a recent work, @abernethy2016perturbation point out that the duality between FTPL and FTRL holds for very general perturbation distributions. However, the authors do not provide a formal theorem showing this result. Here, we provide a proposition formalizing the claim of [@abernethy2016perturbation]. \[prop:ftpl\_ftrl\_connection\] Consider the problem of online convex learning, where the sequence of loss functions $\{f_t\}_{t=1}^T$ encountered by the learner are convex. Consider the deterministic version of FTPL algorithm, where the learner predicts ${{\ensuremath{\mathbf{x}}}}_t$ as ${\mathbb{E}_{\sigma}\left[{\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}{\left\langle {\nabla}_{1:t-1}-\sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle}\right]}$. Suppose the perturbation distribution is absolutely continuous w.r.t the Lebesgue measure. Then there exists a convex regularizer $R:\mathbb{R}^d\to \mathbb{R}\cup\{\infty\}$, with domain ${\text{dom}(R)}\subseteq {\mathcal{X}}$, such that $ {{\ensuremath{\mathbf{x}}}}_t = {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}} {\left\langle {\nabla}_{1:t-1}, {{\ensuremath{\mathbf{x}}}}\right\rangle}+R({{\ensuremath{\mathbf{x}}}}). $ Moreover, and ${{\ensuremath{\mathbf{x}}}}_t = \partial R^{-1}\left(-{\nabla}_{1:t-1}\right),$ where $\partial R^{-1}$ is the inverse of $\partial R$ in the sense of multivalued mappings. Online Learning with OFTPL {#sec:onlinelearning} ========================== Online Convex Learning ---------------------- **Input:** Perturbation Distribution ${P_{\text{PRTB}}},$ number of samples $m,$ number of iterations $T$ Denote ${\nabla}_0 = 0$ Let $g_t$ be the guess for ${\nabla}_t$ Sample $\sigma_{t,j}\sim {P_{\text{PRTB}}}$ ${{\ensuremath{\mathbf{x}}}}_{t,j}\in {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}{\left\langle {\nabla}_{0:t-1}+ g_t-\sigma_{t,j}, {{\ensuremath{\mathbf{x}}}}\right\rangle}$ Play ${{\ensuremath{\mathbf{x}}}}_t=\frac{1}{m}\sum_{j=1}^m {{\ensuremath{\mathbf{x}}}}_{t,j}$ Observe loss function $f_t$ In this section, we present the OFTPL algorithm for online convex learning and derive an upper bound on its regret. The algorithm we consider is similar to the OFTRL algorithm (see Algorithm \[alg:oftpl\_cvx\]). Let $g_t[f_1\dots f_{t-1}]$ be our guess for ${\nabla}_t$ at the beginning of round $t$, with $g_1 = 0$. To simplify the notation, in the sequel, we suppress the dependence of $g_t$ on $\{f_{i}\}_{i=1}^{t-1}$. Given $g_t$, we predict ${{\ensuremath{\mathbf{x}}}}_t$ in OFTPL as follows. We sample independent perturbations $\{\sigma_{t,j}\}_{j=1}^m$ from the perturbation distribution ${P_{\text{PRTB}}}$ and compute ${{\ensuremath{\mathbf{x}}}}_t$ as $m^{-1}\sum_{j=1}^m{{\ensuremath{\mathbf{x}}}}_{t,j}$, where ${{\ensuremath{\mathbf{x}}}}_{t,j}$ is a minimizer of the following linear optimization problem $${{\ensuremath{\mathbf{x}}}}_{t,j} \in {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}} {\left\langle {\nabla}_{1:t-1} + g_t - \sigma_{t,j}, {{\ensuremath{\mathbf{x}}}}\right\rangle}.$$ We now present our main theorem which bounds the regret of OFTPL. A key quantity the regret depends on is the *stability* of predictions of the deterministic version of OFTPL. Intuitively, an algorithm is stable if its predictions in two consecutive iterations differ by a small quantity. To capture this notion, we first define function ${{\nabla}\Phi}:\mathbb{R}^d \to \mathbb{R}^d$ as: $ {{\nabla}\Phi\left(g\right)} = {\mathbb{E}_{\sigma}\left[{\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}\left\langle g-\sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle\right]}. $ Observe that ${{\nabla}\Phi\left(\nabla_{1:t-1} + g_t\right)}$ is the prediction of the deterministic version of OFTPL. We say the predictions of OFTPL are stable, if ${{\nabla}\Phi}$ is a Lipschitz function. The predictions of OFTPL are said to be $\beta$-stable w.r.t some norm $\|\cdot\|$, if $$\forall g_1,g_2\in\mathbb{R}^d \quad \|{{\nabla}\Phi\left(g_1\right)} - {{\nabla}\Phi\left(g_2\right)}\|_{*} \leq \beta \|g_1-g_2\|.$$ \[thm:oftpl\_regret\] Suppose the perturbation distribution ${P_{\text{PRTB}}}$ is absolutely continuous w.r.t Lebesgue measure. Let $D$ be the diameter of ${\mathcal{X}}$ w.r.t $\|\cdot\|$, which is defined as $D= \sup_{{{\ensuremath{\mathbf{x}}}}_1,{{\ensuremath{\mathbf{x}}}}_2\in{\mathcal{X}}} \|{{\ensuremath{\mathbf{x}}}}_1-{{\ensuremath{\mathbf{x}}}}_2\|.$ Let and suppose the predictions of OFTPL are ${C}\eta^{-1}$-stable w.r.t $\|\cdot\|_*$, where ${C}$ is a constant that depends on the set $\mathcal{X}.$ Finally, suppose the sequence of loss functions $\{f_t\}_{t=1}^T$ are Holder smooth and satisfy $$\forall {{\ensuremath{\mathbf{x}}}}_1,{{\ensuremath{\mathbf{x}}}}_2\in {\mathcal{X}}\quad \|{\nabla}f_t({{\ensuremath{\mathbf{x}}}}_1)-{\nabla}f_t({{\ensuremath{\mathbf{x}}}}_2)\|_* \leq L\|{{\ensuremath{\mathbf{x}}}}_1-{{\ensuremath{\mathbf{x}}}}_2\|^{\alpha},$$ for some constant $\alpha \in [0,1]$. Then the expected regret of Algorithm \[alg:oftpl\_cvx\] satisfies $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} {\mathbb{E}\left[\sum_{t=1}^Tf_t({{\ensuremath{\mathbf{x}}}}_t) - f_t({{\ensuremath{\mathbf{x}}}})\right]} &\leq \eta D + \sum_{t=1}^T \frac{{C}}{2\eta}{\mathbb{E}\left[\|{\nabla}_t-g_{t}\|_{*}^2\right]} - \sum_{t=1}^T\frac{\eta}{2{C}} {\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2\right]} \\ &\quad + LT\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{1+\alpha}.\end{aligned}$$ where ${{\ensuremath{\mathbf{x}}}}_t^{\infty} = {\mathbb{E}\left[{{\ensuremath{\mathbf{x}}}}_t|g_t,f_{1:t-1}, {{\ensuremath{\mathbf{x}}}}_{1:t-1}\right]}$ and $\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty} = {\mathbb{E}\left[\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}|f_{1:t-1}, {{\ensuremath{\mathbf{x}}}}_{1:t-1}\right]}$ and $\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}$ denotes the prediction in the $t^{th}$ iteration of Algorithm \[alg:oftpl\_cvx\], if guess $g_{t}=0$ was used. Here, ${\Psi_1}, {\Psi_2}$ denote the norm compatibility constants of $\|\cdot\|.$ Regret bounds that hold with high probability can be found in Appendix \[sec:hp\_bounds\]. The above Theorem shows that the regret of OFTPL only depends on $\|{\nabla}_t-g_{t}\|_{*}$, which quantifies the accuracy of our guess $g_t$. In contrast, the regret of FTPL depends on $\|{\nabla}_t\|_{*}$ [@hazan2016introduction]. This shows that for predictable sequences, with an appropriate choice of $g_t$, OFTPL can achieve better regret guarantees than FTPL. As we demonstrate in Section \[sec:games\], this helps us design faster algorithms for solving minimax games. Note that the above result is very general and holds for any absolutely continuous perturbation distribution. The key challenge in instantiating this result for any particular perturbation distribution is in showing the stability of predictions. Several past works have studied the stability of FTPL for various perturbation distributions such as uniform, exponential, Gumbel distributions [@kalai2005efficient; @hazan2016introduction; @hazan2020projection]. Consequently, the above result can be used to derive tight regret bounds for all these perturbation distributions. As one particular instantiation of Theorem \[thm:oftpl\_regret\], we consider the special case of $g_t = 0$ and derive regret bounds for FTPL, when the perturbation distribution is the uniform distribution over a ball centered at the origin. \[cor:ftpl\_cvx\_gaussian\] Suppose the perturbation distribution is equal to the uniform distribution over $\{{{\ensuremath{\mathbf{x}}}}:\|{{\ensuremath{\mathbf{x}}}}\|_2 \leq (1+d^{-1})\eta\}.$ Let $D$ be the diameter of ${\mathcal{X}}$ w.r.t $\|\cdot\|_2$. Then ${\mathbb{E}_{\sigma}\left[\|\sigma\|_2\right]} = \eta$, and the predictions of OFTPL are $dD\eta^{-1}$-stable w.r.t $\|\cdot\|_2$. Suppose, the sequence of loss functions $\{f_t\}_{t=1}^T$ are $G$-Lipschitz and satisfy $\sup_{{{\ensuremath{\mathbf{x}}}}\in \mathcal{X}} \|{\nabla}f_t({{\ensuremath{\mathbf{x}}}})\|_2 \leq G$. Moreover, suppose $f_t$ satisfies the Holder smooth condition in Theorem \[thm:oftpl\_regret\] w.r.t $\|\cdot\|_2$ norm. Then the expected regret of Algorithm \[alg:oftpl\_cvx\], with guess $g_t = 0$, satisfies $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} {\mathbb{E}\left[\sum_{t=1}^Tf_t({{\ensuremath{\mathbf{x}}}}_t) - f_t({{\ensuremath{\mathbf{x}}}})\right]} &\leq \eta D + \frac{dDG^2 T}{2\eta} + LT\left(\frac{D}{\sqrt{m}}\right)^{1+\alpha}.\end{aligned}$$ This recovers the regret bounds of FTPL for general convex loss functions, derived by [@hazan2020projection]. Online Nonconvex Learning {#sec:online_noncvx} ------------------------- **Input:** Perturbation Distribution ${P_{\text{PRTB}}},$ number of samples $m$, number of iterations $T$ Denote $f_0=0$ Let $g_t$ be the guess for $f_t$ Sample $\sigma_{t,j}\sim {P_{\text{PRTB}}}$ ${{\ensuremath{\mathbf{x}}}}_{t,j} \in {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}f_{0:t-1}({{\ensuremath{\mathbf{x}}}})+g_t({{\ensuremath{\mathbf{x}}}})-\sigma_{t,j}({{\ensuremath{\mathbf{x}}}})$ Let $P_t $ be the empirical distribution over $\{{{\ensuremath{\mathbf{x}}}}_{t,1}, {{\ensuremath{\mathbf{x}}}}_{t,2} \dots {{\ensuremath{\mathbf{x}}}}_{t,m}\}$ Play ${{\ensuremath{\mathbf{x}}}}_t$, a random sample generated from $P_t$ Observe loss function $f_t$ We now study OFTPL in the nonconvex setting. In this setting, we assume the sequence of loss functions belong to some function class ${\mathcal{F}}$ containing real-valued measurable functions on ${\mathcal{X}}$. Some popular choices for ${\mathcal{F}}$ include the set of Lipschitz functions, the set of bounded functions. The OFTPL algorithm in this setting is described in Algorithm \[alg:oftpl\_noncvx\]. Similar to the convex case, we first sample random perturbation functions $\{\sigma_{t,j}\}_{j=1}^m$ from some distribution ${P_{\text{PRTB}}}$. Some examples of perturbation functions that have been considered in the past include $\sigma_{t,j}({{\ensuremath{\mathbf{x}}}}) = {\left\langle \bar{\sigma}_{t,j}, {{\ensuremath{\mathbf{x}}}}\right\rangle},$ for some random vector $\bar{\sigma}_{t,j}$ sampled from exponential or uniform distributions [@gonen2018learning; @suggala2019online]. Another popular choice for $\sigma_{t,j}$ is the Gumbel process, which results in the continuous exponential weights algorithm [@maddison2014sampling]. Letting, $g_t$ be our guess of loss function $f_t$ at the beginning of round $t$, the learner first computes ${{\ensuremath{\mathbf{x}}}}_{t,j}$ as ${\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}\sum_{i = 1}^{t-1}f_i({{\ensuremath{\mathbf{x}}}})+g_t({{\ensuremath{\mathbf{x}}}})-\sigma_{t,j}({{\ensuremath{\mathbf{x}}}}).$ We assume access to an optimization oracle which computes a minimizer of this problem. We often refer to this oracle as the *perturbed best response* oracle. Let $P_t$ denote the empirical distribution of $\{{{\ensuremath{\mathbf{x}}}}_{t,j}\}_{j=1}^m$. The learner then plays an ${{\ensuremath{\mathbf{x}}}}_t$ which is sampled from $P_t$. Algorithm \[alg:oftpl\_noncvx\] describes this procedure. We note that for the online learning problem, $m=1$ suffices, as the expected loss suffered by the learner in each round is independent of $m$; that is ${\mathbb{E}\left[f_t({{\ensuremath{\mathbf{x}}}}_t)\right]} = {\mathbb{E}\left[f_t({{\ensuremath{\mathbf{x}}}}_{t,1})\right]}$. However, the choice of $m$ affects the rate of convergence when Algorithm \[alg:oftpl\_noncvx\] is used for solving nonconvex nonconcave minimax games. Before we present the regret bounds, we introduce the *dual space* associated with ${\mathcal{F}}$. Let $\|\cdot\|_{{\mathcal{F}}}$ be a seminorm associated with ${\mathcal{F}}$. For example, when ${\mathcal{F}}$ is the set of Lipschitz functions, $\|\cdot\|_{{\mathcal{F}}}$ is the Lipschitz seminorm. Various choices of $({\mathcal{F}},\|\cdot\|_{{\mathcal{F}}})$ induce various distance metrics on ${\mathcal{P}}$, the set of all probability distributions on ${\mathcal{X}}$. We let ${\gamma_{{\mathcal{F}}}}$ denote the Integral Probability Metric (IPM) induced by $({\mathcal{F}},\|\cdot\|_{{\mathcal{F}}})$, which is defined as $${\gamma_{{\mathcal{F}}}}(P,Q) = \sup_{f\in{\mathcal{F}},\|f\|_{{\mathcal{F}}} \leq 1}\Big|{\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P}\left[f({{\ensuremath{\mathbf{x}}}})\right]} - {\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim Q}\left[f({{\ensuremath{\mathbf{x}}}})\right]}\Big|.$$ We often refer to $({\mathcal{P}}, {\gamma_{{\mathcal{F}}}})$ as the dual space of $({\mathcal{F}},\|\cdot\|_{{\mathcal{F}}})$. When ${\mathcal{F}}$ is the set of Lipschitz functions and when $\|\cdot\|_{{\mathcal{F}}}$ is the Lipschitz seminorm, ${\gamma_{{\mathcal{F}}}}$ is the Wasserstein distance. Table \[tab:ipm\] in Appendix \[sec:primal\_dual\_spaces\] presents examples of ${\gamma_{{\mathcal{F}}}}$ induced by some popular function spaces. Similar to the convex case, the regret bounds in the nonconvex setting depend on the stability of predictions of OFTPL. Suppose the perturbation function $\sigma({{\ensuremath{\mathbf{x}}}})$ is sampled from ${P_{\text{PRTB}}}$. For any $f\in{\mathcal{F}}$, define random variable ${{\ensuremath{\mathbf{x}}}}_f(\sigma)$ as ${\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}} f({{\ensuremath{\mathbf{x}}}})-\sigma({{\ensuremath{\mathbf{x}}}}).$ Let ${{\nabla}\Phi\left(f\right)}$ denote the distribution of ${{\ensuremath{\mathbf{x}}}}_f(\sigma)$. The predictions of OFTPL are said to be $\beta$-stable w.r.t $\|\cdot\|_{{\mathcal{F}}}$ if $$\forall f,g\in{\mathcal{F}}\quad {\gamma_{{\mathcal{F}}}}({{\nabla}\Phi\left(f\right)}, {{\nabla}\Phi\left(g\right)}) \leq \beta \|f-g\|_{{\mathcal{F}}}.$$ \[thm:oftpl\_noncvx\_regret\] Suppose the sequence of loss functions $\{f_t\}_{t=1}^T$ belong to $({\mathcal{F}}, \|\cdot\|_{{\mathcal{F}}})$. Suppose the perturbation distribution ${P_{\text{PRTB}}}$ is such that ${\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} f({{\ensuremath{\mathbf{x}}}}) - \sigma({{\ensuremath{\mathbf{x}}}})$ has a unique minimizer with probability one, for any $f\in{\mathcal{F}}$. Let ${\mathcal{P}}$ be the set of probability distributions over ${\mathcal{X}}$. Define the diameter of ${\mathcal{P}}$ as $D= \sup_{P_1,P_2\in{\mathcal{P}}} {\gamma_{{\mathcal{F}}}}(P_1,P_2).$ Let $\eta={\mathbb{E}\left[\|\sigma\|_{{\mathcal{F}}}\right]}$. Suppose the predictions of OFTPL are ${C}\eta^{-1}$-stable w.r.t $\|\cdot\|_{{\mathcal{F}}}$, for some constant ${C}$ that depends on ${\mathcal{X}}$. Then the expected regret of Algorithm \[alg:oftpl\_noncvx\] satisfies $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} {\mathbb{E}\left[\sum_{t=1}^Tf_t({{\ensuremath{\mathbf{x}}}}_t)-f_t({{\ensuremath{\mathbf{x}}}})\right]} &\leq \eta D + \sum_{t=1}^T \frac{{C}}{2\eta}{\mathbb{E}\left[\|f_t-g_{t}\|_{{\mathcal{F}}}^2\right]} -\sum_{t=1}^T \frac{\eta}{2{C}}{\mathbb{E}\left[{\gamma_{{\mathcal{F}}}}(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})^2\right]},\end{aligned}$$ where $P_t^{\infty} = {\mathbb{E}\left[P_t|g_t, f_{1:t-1}, P_{1:t-1}\right]}, $ $\Tilde{P}_{t}^{\infty} = {\mathbb{E}\left[\Tilde{P}_{t-1}|f_{1:t-1}, P_{1:t-1}\right]}$ and $\Tilde{P}_{t-1}$ is the empirical distribution computed in the $t^{th}$ iteration of Algorithm \[alg:oftpl\_noncvx\], if guess $g_t = 0$ was used. We note that, unlike the convex case, there are no known analogs of Fenchel duality for infinite dimensional function spaces. As a result, more careful analysis is needed to obtain the above regret bounds. Our analysis mimics the arguments made in the convex case, albeit without explicitly relying on duality theory. As in the convex case, the key challenge in instantiating the above result for any particular perturbation distribution is in showing the stability of predictions. In a recent work, [@suggala2019online] consider linear perturbation functions $\sigma({{\ensuremath{\mathbf{x}}}}) = {\left\langle \bar{\sigma}, {{\ensuremath{\mathbf{x}}}}\right\rangle},$ for $\bar{\sigma}$ sampled from exponential distribution, and show stability of FTPL. We now instantiate the above Theorem for this setting. \[cor:ftpl\_noncvx\_exp\] Consider the setting of Theorem \[thm:oftpl\_noncvx\_regret\]. Let ${\mathcal{F}}$ be the set of Lipschitz functions and $\|\cdot\|_{{\mathcal{F}}}$ be the Lipschitz seminorm, which is defined as $\|f\|_{{\mathcal{F}}}=\sup_{{{\ensuremath{\mathbf{x}}}}\neq {{\ensuremath{\mathbf{y}}}}\text{ in }{\mathcal{X}}} |f({{\ensuremath{\mathbf{x}}}})-f({{\ensuremath{\mathbf{y}}}})|/\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{y}}}}\|_1$. Suppose the perturbation function is such that $\sigma({{\ensuremath{\mathbf{x}}}}) = {\left\langle \bar{\sigma}, {{\ensuremath{\mathbf{x}}}}\right\rangle}$, where $\bar{\sigma} \in \mathbb{R}^d$ is a random vector whose entries are sampled independently from $\text{Exp}(\eta)$. Then ${\mathbb{E}_{\sigma}\left[\|\sigma\|_{{\mathcal{F}}}\right]} = \eta\log{d}$, and the predictions of OFTPL are ${O\left({d^2D\eta^{-1}}\right)}$-stable w.r.t $\|\cdot\|_{{\mathcal{F}}}$. Moreover, the expected regret of Algorithm \[alg:oftpl\_noncvx\] is upper bounded by We note that the above regret bounds are tighter than the regret bounds of [@suggala2019online], where the authors show that the regret of OFTPL is bounded by ${O\left({\eta D\log{d} + \sum_{t=1}^T \frac{d^2D }{\eta}{\mathbb{E}\left[\|f_t-g_{t}\|_{{\mathcal{F}}}^2\right]}}\right)}$. These tigher bounds help us design faster algorithms for solving minimax games in the nonconvex setting. Minimax Games {#sec:games} ============= We now consider the problem of solving minimax games of the following form $$\label{eqn:minimax_game} \min_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}} \max_{{{\ensuremath{\mathbf{y}}}}\in {\mathcal{Y}}} f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}).$$ Nash equilibria of such games can be computed by playing two online learning algorithms against each other [@cesa2006prediction; @hazan2016introduction]. In this work, we study the algorithm where both the players employ OFTPL to decide their actions in each round. For convex-concave games, both the players use the OFTPL algorithm described in Algorithm \[alg:oftpl\_cvx\] (see Algorithm \[alg:oftpl\_cvx\_games\] in Appendix \[sec:cvx-games\]). The following theorem derives the rate of convergence of this algorithm to a Nash equilibirum (NE). \[thm:oftpl\_cvx\_smooth\_games\_uniform\] Consider the minimax game in Equation . Suppose both the domains ${\mathcal{X}},{\mathcal{Y}}$ are compact subsets of $\mathbb{R}^d$, with diameter Suppose $f$ is convex in ${{\ensuremath{\mathbf{x}}}}$, concave in ${{\ensuremath{\mathbf{y}}}}$ and is smooth w.r.t $\|\cdot\|_2$ $$\begin{aligned} \|{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{2}+ \|{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{2} \leq L\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}'\|_2 + L\|{{\ensuremath{\mathbf{y}}}}-{{\ensuremath{\mathbf{y}}}}'\|_2.\end{aligned}$$ Suppose Algorithm \[alg:oftpl\_cvx\_games\] is used to solve the minimax game. Suppose the perturbation distributions used by both the players are the same and equal to the uniform distribution over $\{{{\ensuremath{\mathbf{x}}}}:\|{{\ensuremath{\mathbf{x}}}}\|_2 \leq (1+d^{-1})\eta\}.$ Suppose the guesses used by ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players in the $t^{th}$ iteration are ${\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}), {\nabla}_{{{\ensuremath{\mathbf{y}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})$, where $\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}$ denote the predictions of ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players in the $t^{th}$ iteration, if guess $g_t = 0$ was used. If Algorithm \[alg:oftpl\_cvx\_games\] is run with $\eta = 6dD(L+1), m = T$, then the iterates $\{({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)\}_{t=1}^T$ satisfy $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[f\left(\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}\right) - f\left({{\ensuremath{\mathbf{x}}}},\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{y}}}}_t\right)\right]}= {O\left({\frac{dD^2(L+1)}{T}}\right)}.\end{aligned}$$ Rates of convergence which hold with high probability can be found in Appendix \[sec:hp\_bounds\]. We note that Theorem \[thm:oftpl\_cvx\_smooth\_games\_uniform\] can be extended to more general noise distributions and settings where gradients of $f$ are Holder smooth w.r.t non-Euclidean norms, and ${\mathcal{X}},{\mathcal{Y}}$ lie in spaces of different dimensions (see Theorem \[thm:oftpl\_cvx\_smooth\_games\] in Appendix). The above result shows that for smooth convex-concave games, Algorithm \[alg:oftpl\_cvx\_games\] converges to a NE at ${O\left({T^{-1}}\right)}$ rate using $T^2$ calls to the linear optimization oracle. Moreover, the algorithm runs in ${O\left({T}\right)}$ iterations, with each iteration making ${O\left({T}\right)}$ parallel calls to the optimization oracle. We believe the dimension dependence in the rates can be removed by appropriately choosing the perturbation distributions based on domains ${\mathcal{X}}, {\mathcal{Y}}$ (see Appendix \[sec:pert\_dist\_choice\]). We now consider the more general nonconvex-nonconcave games. In this case, both the players use the nonconvex OFTPL algorithm described in Algorithm \[alg:oftpl\_noncvx\] to choose their actions. Instead of generating a single sample from the empirical distribution $P_t$ computed in $t^{th}$ iteration of Algorithm \[alg:oftpl\_noncvx\], the players now play the entire distribution $P_t$ (see Algorithm \[alg:oftpl\_noncvx\_games\] in Appendix \[sec:ncvx-games\]). Letting $\{P_t\}_{t=1}^T, \{Q_t\}_{t=1}^T$, be the sequence of iterates generated by the ${{\ensuremath{\mathbf{x}}}}$ and ${{\ensuremath{\mathbf{y}}}}$ players, the following theorem shows that $\left(\frac{1}{T}\sum_{t=1}^TP_t,\frac{1}{T}\sum_{t=1}^TQ_t\right)$ converges to a NE. \[thm:oftpl\_noncvx\_smooth\_games\_exp\] Consider the minimax game in Equation . Suppose the domains ${\mathcal{X}},{\mathcal{Y}}$ are compact subsets of $\mathbb{R}^d$ with diameter $D = \max\{\sup_{{{\ensuremath{\mathbf{x}}}}_1,{{\ensuremath{\mathbf{x}}}}_2\in{\mathcal{X}}} \|{{\ensuremath{\mathbf{x}}}}_1-{{\ensuremath{\mathbf{x}}}}_2\|_1, \sup_{{{\ensuremath{\mathbf{y}}}}_1,{{\ensuremath{\mathbf{y}}}}_2\in{\mathcal{Y}}} \|{{\ensuremath{\mathbf{y}}}}_1-{{\ensuremath{\mathbf{y}}}}_2\|_1\}$. Suppose $f$ is Lipschitz w.r.t $\|\cdot\|_1$ and satisfies $$\begin{aligned} \max\left\lbrace\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}, {{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} \|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})\|_{\infty}, \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}\|{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})\|_{\infty}\right\rbrace\leq G.\end{aligned}$$ Moreover, suppose $f$ satisfies the following smoothness property $$\begin{aligned} \|{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{\infty} + \|{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{\infty} \leq L\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}'\|_1 + L\|{{\ensuremath{\mathbf{y}}}}-{{\ensuremath{\mathbf{y}}}}'\|_1.\end{aligned}$$ Suppose both ${{\ensuremath{\mathbf{x}}}}$ and ${{\ensuremath{\mathbf{y}}}}$ players use Algorithm \[alg:oftpl\_noncvx\_games\] to solve the game with linear perturbation functions $\sigma({{\ensuremath{\mathbf{z}}}})={\left\langle \bar{\sigma}, {{\ensuremath{\mathbf{z}}}}\right\rangle}$, where $\bar{\sigma} \in \mathbb{R}^d$ is such that each of its entries is sampled independently from $\text{Exp}(\eta)$. Suppose the guesses used by ${{\ensuremath{\mathbf{x}}}}$ and ${{\ensuremath{\mathbf{y}}}}$ players in the $t^{th}$ iteration are $f(\cdot,\Tilde{Q}_{t-1}), f(\Tilde{P}_{t-1},\cdot)$, where $\Tilde{P}_{t-1},\Tilde{Q}_{t-1}$ denote the predictions of ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players in the $t^{th}$ iteration, if guess $g_t = 0$ was used. If Algorithm \[alg:oftpl\_noncvx\_games\] is run with $\eta = 10d^2D(L+1), m = T$, then the iterates $\{(P_t,Q_t)\}_{t=1}^T$ satisfy $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[f\left(\frac{1}{T}\sum_{t=1}^TP_t,{{\ensuremath{\mathbf{y}}}}\right) - f\left({{\ensuremath{\mathbf{x}}}},\frac{1}{T}\sum_{t=1}^TQ_t\right)\right]}& = {O\left({\frac{d^2D^2(L+1)\log{d}}{T}}\right)}\\ &\quad + {O\left({\min\left\lbrace D^2L, \frac{d^2G^2\log{T}}{LT}\right\rbrace}\right)}.\end{aligned}$$ More general versions of the Theorem, which consider other function classes and general perturbation distributions, can be found in Appendix \[sec:ncvx-games\]. The above result shows that Algorithm \[alg:oftpl\_noncvx\_games\] converges to a NE at ${\Tilde{O}\left({T^{-1}}\right)}$ rate using $T^2$ calls to the perturbed best response oracle. This matches the rates of convergence of FTPL [@suggala2019online]. However, the key advantage of our algorithm is that it is highly parallelizable and runs in ${O\left({T}\right)}$ iterations, in contrast to FTPL, which runs in ${O\left({T^2}\right)}$ iterations. Conclusion {#sec:conclusion} ========== We studied an optimistic variant of FTPL which achieves better regret guarantees when the sequence of loss functions is predictable. As one specific application of our algorithm, we considered the problem of solving minimax games. For solving convex-concave games, our algorithm requires access to a linear optimization oracle and for nonconvex-nonconcave games our algorithm requires access to a more powerful perturbed best response oracle. In both these settings, our algorithm achieves ${O\left({T^{-1/2}}\right)}$ convergence rates using $T$ calls to the oracles. Moreover, our algorithm runs in ${O\left({T^{1/2}}\right)}$ iterations, with each iteration making ${O\left({T^{1/2}}\right)}$ parallel calls to the optimization oracle. We believe our improved algorithms for solving minimax games are useful in a number of modern machine learning applications such as training of GANs, adversarial training, which involve solving nonconvex-nonconcave minimax games and often deal with huge datasets. Dual view of Perturbations as Regularization ============================================ Proof of Theorem \[prop:ftpl\_ftrl\_connection\] ------------------------------------------------ We first define a convex function $\Psi:\mathbb{R}^d \to \mathbb{R}$ as $$\Psi(f)={\mathbb{E}_{\sigma}\left[\sup_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}{\left\langle f+\sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle}\right]} = {\mathbb{E}_{\sigma}\left[\sup_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}{\left\langle f+\sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle}\right]},$$ where perturbation $\sigma$ follows probability distribution ${P_{\text{PRTB}}}$ which is absolutely continuous w.r.t the Lebesgue measure. For our choice of ${P_{\text{PRTB}}}$, we now show that $\Psi$ is differentiable. Consider the function $\psi(g) = \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}{\left\langle g, {{\ensuremath{\mathbf{x}}}}\right\rangle}$. Since $\psi(g)$ is a proper convex function, we know that it is differentiable almost everywhere, except on a set of Lebesgue measure $0$ [see Theorem 25.5 of @rockafellar1970convex]. Moreover, it is easy to verify that ${\mathop{\rm argmax}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}{\left\langle g, {{\ensuremath{\mathbf{x}}}}\right\rangle} \in \partial \psi(g).$ These two observations, together with the fact that ${P_{\text{PRTB}}}$ is absolutely continuous, show that the $\sup$ expression inside the expectation of $\Psi$ has a unique maximizer with probability one. Since the sup expression inside the expectation has a unique maximizer with probability $1$, we can swap the expectation and gradient to obtain [see Proposition 2.2 of @bertsekas1973stochastic] $$\label{eqn:phi_gradient} {\nabla}\Psi(f) = {\mathbb{E}_{\sigma}\left[{\mathop{\rm argmax}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}{\left\langle f+\sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle}\right]}.$$ Note that ${\nabla}\Psi$ is related to the prediction of deterministic version of FTPL. Specifically, ${\nabla}\Psi(-{\nabla}_{1:t-1})$ is the prediction of deterministic FTPL in the $t^{th}$ iteration. We now show that ${\nabla}\Psi(f) = {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} {\left\langle -f, {{\ensuremath{\mathbf{x}}}}\right\rangle} + R({{\ensuremath{\mathbf{x}}}})$, for some convex function $R$. Since all differentiable functions are closed, $\Psi(f)$ is a proper, closed and differentiable convex function over $\mathbb{R}^d$. Let $R({{\ensuremath{\mathbf{x}}}})$ denote the Fenchel conjugate of $\Psi(f)$ $$R({{\ensuremath{\mathbf{x}}}}) = \sup_{f\in {\text{dom}(\Phi)}} {\left\langle {{\ensuremath{\mathbf{x}}}}, f \right\rangle} - \Psi(f),$$ where ${\text{dom}(\Psi)}$ denotes the domain of $\Psi$. Following Theorem \[thm:fenchel\_prop1\] (see Appendix \[sec:fenchel\_conjugate\]), $\Psi(f)$ is the Fenchel conjugate of $R({{\ensuremath{\mathbf{x}}}})$ $$\begin{aligned} \Psi(f) = \sup_{{{\ensuremath{\mathbf{x}}}}\in {\text{dom}(R)}}{\left\langle f, {{\ensuremath{\mathbf{x}}}}\right\rangle} - R({{\ensuremath{\mathbf{x}}}}).\end{aligned}$$ Furthermore, from Theorem \[thm:fenchel\_prop3\] we have $${\nabla}\Psi(f) = {\mathop{\rm argmax}}_{{{\ensuremath{\mathbf{x}}}}\in {\text{dom}(R)}}{\left\langle f, {{\ensuremath{\mathbf{x}}}}\right\rangle} - R({{\ensuremath{\mathbf{x}}}}).$$ We now show that the domain of $R$ is a subset of ${\mathcal{X}}$. This, together with the previous two equations, would then immediately imply $$\begin{aligned} \Psi(f) = \sup_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}{\left\langle f, {{\ensuremath{\mathbf{x}}}}\right\rangle} - R({{\ensuremath{\mathbf{x}}}}),\\ \label{eqn:ftpl_ftrl_connection_proof} {\nabla}\Psi(f) = {\mathop{\rm argmax}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}{\left\langle f, {{\ensuremath{\mathbf{x}}}}\right\rangle} - R({{\ensuremath{\mathbf{x}}}}).\end{aligned}$$ From Theorem \[thm:fenchel\_prop2\], we know that the domain of $R$ satisfies $$\text{ri}({\text{dom}(R)}) \subseteq \text{range} {\nabla}\Psi \subseteq {\text{dom}(R)},$$ where $\text{ri}(A)$ denotes the relative interior of a set $A$. Moreover, from the definition of ${\nabla}\Psi(f)$ in Equation , we have $\text{range} {\nabla}\Psi \subseteq {\mathcal{X}}$. Combining these two properties, we can show that one of the following statements is true $$\begin{aligned} \text{ri}({\text{dom}(R)}) \subseteq \text{range} {\nabla}\Psi \subseteq {\mathcal{X}}\subseteq {\text{dom}(R)},\\ \text{ri}({\text{dom}(R)}) \subseteq \text{range} {\nabla}\Psi \subseteq {\text{dom}(R)} \subseteq {\mathcal{X}}.\end{aligned}$$ Suppose the first statement is true. Since ${\mathcal{X}}$ is a compact set, it is easy to see that If the second statement is true, then ${\text{dom}(R)} \subseteq {\mathcal{X}}$. Together, these two statements imply ${\text{dom}(R)} \subseteq {\mathcal{X}}$. #### Connecting back to FTPL. We now connect the above results to FTPL. From Equation , we know that the prediction at iteration $t$ of deterministic FTPL is equal to ${\nabla}\Psi(-{\nabla}_{1:t-1}).$ From Equation , ${\nabla}\Psi(-{\nabla}_{1:t-1})$ is defined as $${{\ensuremath{\mathbf{x}}}}_t = {\nabla}\Psi(-{\nabla}_{1:t-1}) = {\mathop{\rm argmax}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}{\left\langle -{\nabla}_{1:t-1}, {{\ensuremath{\mathbf{x}}}}\right\rangle} - R({{\ensuremath{\mathbf{x}}}}).$$ This shows that $${{\ensuremath{\mathbf{x}}}}_t = {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}{\left\langle {\nabla}_{1:t-1}, {{\ensuremath{\mathbf{x}}}}\right\rangle} + R({{\ensuremath{\mathbf{x}}}}).$$ So the prediction of FTPL can also be obtained using FTRL for some convex regularizer $R({{\ensuremath{\mathbf{x}}}})$. Finally, to show that $-{\nabla}_{1:t-1} \in \partial R({{\ensuremath{\mathbf{x}}}}_t), {{\ensuremath{\mathbf{x}}}}_t = \partial R^{-1}\left(-{\nabla}_{1:t-1}\right),$ we rely on Theorem \[thm:fenchel\_prop4\]. Since ${{\ensuremath{\mathbf{x}}}}_t = {\nabla}\Psi(-{\nabla}_{1:t-1})$, from Theorem \[thm:fenchel\_prop4\], we have $$-{\nabla}_{1:t-1} \in \partial R({{\ensuremath{\mathbf{x}}}}_t),\quad {{\ensuremath{\mathbf{x}}}}_t = {\nabla}\Psi(-{\nabla}_{1:t-1})= \partial R^{-1}\left(-{\nabla}_{1:t-1}\right),$$ where $\partial R^{-1}$ is the inverse of $\partial R$ in the sense of multivalued mappings. Note that, even though $\partial R$ can be a multivalued mapping, its inverse $\partial R^{-1} = {\nabla}\Psi$ is a singlevalued mapping (this follows form differentiability of $\Psi$). This finishes the proof of the Theorem. Online Convex Learning ====================== Proof of Theorem \[thm:oftpl\_regret\] -------------------------------------- Before presenting the proof of the Theorem, we introduce some notation. ### Notation We define functions $\Phi:\mathbb{R}^d\to \mathbb{R}$, $R:\mathbb{R}^d\to\mathbb{R}$ as follows $$\begin{aligned} &\Phi(f) = {\mathbb{E}_{\sigma}\left[\inf_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}{\left\langle f-\sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle}\right]},\quad R({{\ensuremath{\mathbf{x}}}}) = \sup_{f\in \mathbb{R}^d} {\left\langle f, {{\ensuremath{\mathbf{x}}}}\right\rangle} + \Phi(-f).\end{aligned}$$ Note that $\Phi$ is related to the function $\Psi$ defined in the proof of Proposition \[prop:ftpl\_ftrl\_connection\]. To be precise, $\Psi(f) = -\Phi(-f)$. Moreover, $R({{\ensuremath{\mathbf{x}}}})$ is the Fenchel conjugate of $\Psi$. For our choice of perturbation distribution, $\Psi$ is differentiable (see proof of Proposition \[prop:ftpl\_ftrl\_connection\]). This implies $\Phi$ is also differentiable with gradient ${{\nabla}\Phi}$ defined as $$\begin{aligned} {{\nabla}\Phi\left(f\right)} = {\mathbb{E}_{\sigma}\left[{\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}{\left\langle f-\sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle}\right]}.\end{aligned}$$ Note that ${\nabla}\Phi$ is the prediction of deterministic version of FTPL. In Proposition \[prop:ftpl\_ftrl\_connection\] we showed that $${{\nabla}\Phi\left(f\right)} = {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}{\left\langle f, {{\ensuremath{\mathbf{x}}}}\right\rangle} + R({{\ensuremath{\mathbf{x}}}}).$$ ### Main Argument Since ${{\ensuremath{\mathbf{x}}}}_t^{\infty}$ is the prediction of deterministic version of FTPL, following FTPL-FTRL duality proved in Proposition \[prop:ftpl\_ftrl\_connection\], ${{\ensuremath{\mathbf{x}}}}_t^{\infty}$ can equivalently be written as $${{\ensuremath{\mathbf{x}}}}_t^{\infty} = {{\nabla}\Phi\left({\nabla}_{1:t-1} + g_t\right)} = {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} {\left\langle {\nabla}_{1:t-1} + g_t, {{\ensuremath{\mathbf{x}}}}\right\rangle} + R({{\ensuremath{\mathbf{x}}}}).$$ Similarly, $\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}$ can be written as $$\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}={{\nabla}\Phi\left({\nabla}_{1:t}\right)} = {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} {\left\langle {\nabla}_{1:t}, {{\ensuremath{\mathbf{x}}}}\right\rangle} + R({{\ensuremath{\mathbf{x}}}}).$$ We use the notation ${\nabla}_{1:0}=0$. So $\Tilde{{{\ensuremath{\mathbf{x}}}}}_0^{\infty},{{\ensuremath{\mathbf{x}}}}_1^{\infty}$ are equal to ${\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} R({{\ensuremath{\mathbf{x}}}}).$ From the first order optimality conditions, we have $$-{\nabla}_{1:t-1} - g_t \in \partial R\left({{\ensuremath{\mathbf{x}}}}_t^{\infty}\right),\quad -{\nabla}_{1:t} \in \partial R\left(\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}\right).$$ Define functions $B(\cdot,{{\ensuremath{\mathbf{x}}}}_t^{\infty}), B(\cdot,\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty})$ for any $t\in[T]$ as $$\begin{aligned} B({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{x}}}}_t^{\infty}) &= R({{\ensuremath{\mathbf{x}}}}) - R({{\ensuremath{\mathbf{x}}}}_t^{\infty}) + {\left\langle {\nabla}_{1:t-1} + g_t, {{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}_t^{\infty} \right\rangle},\\ B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}) &= R({{\ensuremath{\mathbf{x}}}}) - R(\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}) + {\left\langle {\nabla}_{1:t}, {{\ensuremath{\mathbf{x}}}}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty} \right\rangle}.\end{aligned}$$ From the stability of predictions of OFTPL we know that: $ \|{{\nabla}\Phi\left(g_1\right)} - {{\nabla}\Phi\left(g_2\right)}\| \leq C\eta^{-1}\|g_1-g_2\|_{*}. $ Following our connection between $\Psi,\Phi$, this implies $ \|{\nabla}\Psi(g_1) - {\nabla}\Psi(g_2)\| \leq C\eta^{-1}\|g_1-g_2\|_{*}. $ This implies the following smoothness condition on $\Psi$ [see Lemma 15 of @shalev2007thesis] $$\Psi(g_2) \leq \Psi(g_1) + {\left\langle {\nabla}\Psi(g_1), g_2-g_1 \right\rangle} + \frac{C\eta^{-1}}{2}\|g_1-g_2\|_{*}^2.$$ Since $\Psi$ is $C\eta^{-1}$-smooth w.r.t $\|\cdot\|_{*}$, following duality between strong convexity and strong smoothness properties (see Theorem \[thm:fenchel\_strong\_convex\_weak\]), we can infer that $R$ is ${C}^{-1}\eta$- strongly convex w.r.t $\|\cdot\|$ norm and satisfies $$B({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{x}}}}_t^{\infty}) \geq \frac{\eta}{2{C}}\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^2, \quad B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}) \geq \frac{\eta}{2{C}}\|{{\ensuremath{\mathbf{x}}}}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}\|^2.$$ We now go ahead and bound the regret of the learner. For any ${{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}$, we have $$\begin{aligned} f_t({{\ensuremath{\mathbf{x}}}}_t) - f_t({{\ensuremath{\mathbf{x}}}}) \stackrel{(a)}{\leq} {\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}, {\nabla}_t \right\rangle} & = {\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t \right\rangle} + {\left\langle {{\ensuremath{\mathbf{x}}}}_t^{\infty} - {{\ensuremath{\mathbf{x}}}}, {\nabla}_t \right\rangle}\\ &= {\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t \right\rangle} + {\left\langle {{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}, {\nabla}_t-g_{t} \right\rangle} + {\left\langle {{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}, g_{t} \right\rangle} \\ &\quad +{\left\langle \Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}-{{\ensuremath{\mathbf{x}}}}, {\nabla}_t \right\rangle}\\ & \leq {\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t \right\rangle} + \|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}\|\|{\nabla}_t-g_{t}\|_{*} + {\left\langle {{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}, g_{t} \right\rangle} \\ &\quad +{\left\langle \Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}-{{\ensuremath{\mathbf{x}}}}, {\nabla}_t \right\rangle},\end{aligned}$$ where $(a)$ follows from convexity of $f$. Next, a simple calculation shows that $$\begin{aligned} {\left\langle {{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}, g_{t} \right\rangle} &= B(\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}) - B(\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty},{{\ensuremath{\mathbf{x}}}}_t^{\infty}) - B({{\ensuremath{\mathbf{x}}}}_t^{\infty},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty})\\ {\left\langle \Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}-{{\ensuremath{\mathbf{x}}}}, {\nabla}_t \right\rangle} &= B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}) - B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty})-B(\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}).\end{aligned}$$ Substituting this in the previous inequality gives us $$\begin{aligned} f_t({{\ensuremath{\mathbf{x}}}}_t) - f_t({{\ensuremath{\mathbf{x}}}}) & \leq {\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t \right\rangle} + \|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}\|\|{\nabla}_t-g_{t}\|_{*} \\ &\quad + B(\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}) - B(\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty},{{\ensuremath{\mathbf{x}}}}_t^{\infty}) - B({{\ensuremath{\mathbf{x}}}}_t^{\infty},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty})\vspace{0.1in}\\ &\quad +B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}) - B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty})-B(\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty})\vspace{0.1in}\\ & = {\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t \right\rangle} + \|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}\|\|{\nabla}_t-g_{t}\|_{*} \\ &\quad + B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}) - B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}) - B(\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty},{{\ensuremath{\mathbf{x}}}}_t^{\infty}) - B({{\ensuremath{\mathbf{x}}}}_t^{\infty},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty})\vspace{0.1in}\\ & \stackrel{(a)}{\leq}{\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t \right\rangle} + \|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}\|\|{\nabla}_t-g_{t}\|_{*} \\ &\quad + B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}) - B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}) - \frac{\eta\|\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^2}{2{C}}-\frac{ \eta\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2}{2{C}},\end{aligned}$$ where $(a)$ follows from strongly convexity of $R$. Summing over $t=1,\dots T$, gives us $$\begin{aligned} \sum_{t=1}^Tf_t({{\ensuremath{\mathbf{x}}}}_t) - f_t({{\ensuremath{\mathbf{x}}}}) &\leq \sum_{t=1}^T{\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t \right\rangle} + \underbrace{B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{0}^{\infty}) - B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{T}^{\infty})}_{S_1} \\ &\quad + \sum_{t=1}^T \|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}\|\|{\nabla}_t-g_{t}\|_{*}\\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T\left(\|\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^2 + \|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2\right).\end{aligned}$$ #### Bounding $S_1$. We now bound $B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{0}^{\infty}) - B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{T}^{\infty})$. From the definition of $B$, we have $$\begin{aligned} B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{0}^{\infty}) - B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{T}^{\infty}) &= R(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{T}^{\infty}) - {\left\langle {\nabla}_{1:T}, {{\ensuremath{\mathbf{x}}}}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_T^{\infty} \right\rangle} -R(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{0}^{\infty}) + {\left\langle {\nabla}_{1:0}, {{\ensuremath{\mathbf{x}}}}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_T^{\infty} \right\rangle}.\end{aligned}$$ Note that ${\nabla}_{1:0} = 0.$ This gives us $$\begin{aligned} B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{0}^{\infty}) - B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{T}^{\infty}) &= R(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{T}^{\infty}) - {\left\langle {\nabla}_{1:T}, {{\ensuremath{\mathbf{x}}}}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_T^{\infty} \right\rangle} -R(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{0}^{\infty}).\end{aligned}$$ We now use duality to convert the RHS of the above equation, which is currently in terms of $R$, into a quantity which depends on $\Phi$. From Proposition \[prop:ftpl\_ftrl\_connection\] we have $$\Phi(g) = -\Psi(-g)=\inf_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}{\left\langle g, {{\ensuremath{\mathbf{x}}}}\right\rangle} + R({{\ensuremath{\mathbf{x}}}}).$$ Since $\Tilde{{{\ensuremath{\mathbf{x}}}}}_{T}^{\infty}$ is the minimizer of ${\left\langle {\nabla}_{1:T}, {{\ensuremath{\mathbf{x}}}}\right\rangle} + R({{\ensuremath{\mathbf{x}}}})$, we have $\Phi({\nabla}_{1:T}) = {\left\langle {\nabla}_{1:T}, \Tilde{{{\ensuremath{\mathbf{x}}}}}_T^{\infty} \right\rangle} + R(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{T}^{\infty})$. Similarly, $\Phi(0) = R(\Tilde{{{\ensuremath{\mathbf{x}}}}}_0^{\infty}).$ Substituting these in the previous equation gives us $$\begin{aligned} B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{0}^{\infty}) - B({{\ensuremath{\mathbf{x}}}},\Tilde{{{\ensuremath{\mathbf{x}}}}}_{T}^{\infty}) &= \Phi({\nabla}_{1:T}) - {\left\langle {\nabla}_{1:T}, {{\ensuremath{\mathbf{x}}}}\right\rangle} -\Phi(0)\\ &={\mathbb{E}_{\sigma}\left[\inf_{{{\ensuremath{\mathbf{x}}}}' \in {\mathcal{X}}}\left\langle {\nabla}_{1:T}-\sigma, {{\ensuremath{\mathbf{x}}}}'\right\rangle\right]} - {\left\langle {\nabla}_{1:T}, {{\ensuremath{\mathbf{x}}}}\right\rangle} - {\mathbb{E}_{\sigma}\left[\inf_{{{\ensuremath{\mathbf{x}}}}' \in {\mathcal{X}}}\left\langle -\sigma, {{\ensuremath{\mathbf{x}}}}'\right\rangle\right]}\\ &\leq {\mathbb{E}_{\sigma}\left[\left\langle {\nabla}_{1:T}-\sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle\right]}- {\left\langle {\nabla}_{1:T}, {{\ensuremath{\mathbf{x}}}}\right\rangle} - {\mathbb{E}_{\sigma}\left[\inf_{{{\ensuremath{\mathbf{x}}}}' \in {\mathcal{X}}}\left\langle -\sigma, {{\ensuremath{\mathbf{x}}}}'\right\rangle\right]}\\ &= {\mathbb{E}_{\sigma}\left[\inf_{{{\ensuremath{\mathbf{x}}}}' \in {\mathcal{X}}}\left\langle \sigma, {{\ensuremath{\mathbf{x}}}}'\right\rangle\right]} - {\mathbb{E}_{\sigma}\left[\left\langle \sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle\right]}\\ &\leq D{\mathbb{E}_{\sigma}\left[\|\sigma\|_{*}\right]} = \eta D\end{aligned}$$ #### Bounding Regret. Substituting this in our regret bound and taking expectation on both sides gives us $$\begin{aligned} {\mathbb{E}\left[\sum_{t=1}^Tf_t({{\ensuremath{\mathbf{x}}}}_t) - f_t({{\ensuremath{\mathbf{x}}}})\right]} &\leq \sum_{t=1}^T{\mathbb{E}\left[{\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t \right\rangle}\right]} + \eta D + \sum_{t=1}^T {\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}\|\|{\nabla}_t-g_{t}\|_{*}\right]} \\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T\left({\mathbb{E}\left[\|\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^2\right]} + {\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2\right]}\right)\\ &\leq \sum_{t=1}^T{\mathbb{E}\left[{\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t \right\rangle}\right]} + \eta D + \sum_{t=1}^T \frac{{C}}{2\eta}{\mathbb{E}\left[\|{\nabla}_t-g_{t}\|_{*}^2\right]} \\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T {\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2\right]}\\\end{aligned}$$ To finish the proof, we make use of the Holder’s smoothness assumption on $f_t$ to bound the first term in the RHS above. From Holder’s smoothness assumption, we have $${\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t - {\nabla}f_t({{\ensuremath{\mathbf{x}}}}_t^{\infty}) \right\rangle} \leq L\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^{1+\alpha}.$$ Using this, we get $$\begin{aligned} {\mathbb{E}\left[{\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t \right\rangle}|g_t, {{\ensuremath{\mathbf{x}}}}_{1:t-1},f_{1:t}\right]} &\leq {\mathbb{E}\left[{\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}f_t({{\ensuremath{\mathbf{x}}}}_t^{\infty}) \right\rangle} + L\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^{1+\alpha}|g_t,{{\ensuremath{\mathbf{x}}}}_{1:t-1},f_{1:t}\right]}\\ & \stackrel{(a)}{=} L{\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^{1+\alpha}|g_t,{{\ensuremath{\mathbf{x}}}}_{1:t-1},f_{1:t}\right]}\\ & \stackrel{(b)}{\leq} {\Psi_1}^{1+\alpha}L{\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|_2^{1+\alpha}|g_t,{{\ensuremath{\mathbf{x}}}}_{1:t-1},f_{1:t}\right]}\\ & \stackrel{(c)}{\leq} {\Psi_1}^{1+\alpha}L{\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|_2^2|g_t,{{\ensuremath{\mathbf{x}}}}_{1:t-1},f_{1:t}\right]}^{(1+\alpha)/2}\\ & \stackrel{(d)}{\leq} L\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{1+\alpha},\end{aligned}$$ where $(a)$ follows from the fact that ${\mathbb{E}\left[{\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}f_t({{\ensuremath{\mathbf{x}}}}_t^{\infty}) \right\rangle}|g_t,{{\ensuremath{\mathbf{x}}}}_{1:t-1},f_{1:t}\right]} = 0$, $(b)$ follows from the definition of norm compatibility constant ${\Psi_1}$, $(c)$ follows from Holders inequality and $(d)$ uses the fact that conditioned on $\{g_t, {{\ensuremath{\mathbf{x}}}}_{1:t-1},f_{1:t}\}$, ${{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}$ is the average of $m$ i.i.d bounded mean $0$ random variables, the variance of which scales as $O(D^2/m)$. Substituting this in the above regret bound gives us the required result. Proof of Corollary \[cor:ftpl\_cvx\_gaussian\] ---------------------------------------------- We first bound ${\mathbb{E}_{\sigma}\left[\|\sigma\|_2\right]}$. Relying on spherical symmetry of the perturbation distribution and the fact that the density of ${P_{\text{PRTB}}}$ on the spherical shell of radius $r$ is proportional to $r^{d-1}$, we get $$\begin{aligned} {\mathbb{E}_{\sigma}\left[\|\sigma\|_2\right]} = \frac{\int_{r=0}^{(1+d^{-1})\eta} r\times r^{d-1} dr}{\int_{r=0}^{(1+d^{-1})\eta} r^{d-1} dr} = \eta.\end{aligned}$$ We now bound the stability of predictions of OFTPL. Our technique for bounding the stability uses similar arguments as  @hazan2020projection (see Lemma 4.2 of [@hazan2020projection]). Recall, to bound stability, we need to show that $\Phi(g) = {\mathbb{E}_{\sigma}\left[\inf_{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}\left\langle g-\sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle\right]}$ is smooth. Let $\phi_0(g) = \inf_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}{\left\langle g, {{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}_{00} \right\rangle}$, where ${{\ensuremath{\mathbf{x}}}}_{00}$ is an arbitrary point in ${\mathcal{X}}$. We can rewrite $\Phi(g)$ as $$\Phi(g) = {\mathbb{E}_{\sigma}\left[\phi_0(g-\sigma)\right]} + {\left\langle g, {{\ensuremath{\mathbf{x}}}}_{00} \right\rangle}.$$ Since the second term in the RHS above is linear in $g$, any upper bound on the smoothness of ${\mathbb{E}_{\sigma}\left[\phi_0(g-\sigma)\right]}$ is also a bound on the smoothness of $\Phi(g)$. So we focus on bounding the smoothness of ${\mathbb{E}_{\sigma}\left[\phi_0(g-\sigma)\right]}$. First note that $\phi_0(g)$ is $D$ Lipschitz and satisfies the following for any $g_1,g_2\in\mathbb{R}^d$ $$\begin{aligned} \phi_0(g_1) - \phi_0(g_2) & = \inf_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}{\left\langle -g_2, {{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}_{00} \right\rangle}-\inf_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}{\left\langle -g_1, {{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}_{00} \right\rangle}\\ & \leq \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}{\left\langle g_1-g_2, {{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}_{00} \right\rangle}\\ &\leq D\|g_1-g_2\|_2.\end{aligned}$$ Letting $\Phi_0(g) = {\mathbb{E}_{\sigma}\left[\phi_0(g-\sigma)\right]}$, Lemma 4.2 of @hazan2020projection shows that $\Phi_0(g)$ is smooth and satisfies $$\|{\nabla}\Phi_0(g_1) - {\nabla}\Phi_0(g_2)\|_2 \leq dD\eta^{-1}\|g_1-g_2\|_2.$$ This shows that the predictions of OFTPL are $dD\eta^{-1}$ stable. The rest of the proof involves substituting $C=dD$ in the regret bound of Theorem \[thm:oftpl\_regret\] and setting $g_t = 0$ and using the fact that $\|{\nabla}_t\|_2\leq G$. Online Nonconvex Learning {#online-nonconvex-learning} ========================= Proof of Theorem \[thm:oftpl\_noncvx\_regret\] ---------------------------------------------- Before we present the proof of the Theorem, we introduce some notation and present some useful intermediate results. We note that unlike the convex case, there are no know Fenchel duality theorems for infinite dimensional setting. So more careful arguments are need to obtain tight regret bounds. Our proof mimics the proof of Theorem \[thm:oftpl\_regret\]. ### Notation Let ${\mathcal{P}}$ be the set of all probability measures on ${\mathcal{X}}$. We define functions $\Phi:{\mathcal{F}}\to \mathbb{R}$, $R:{\mathcal{P}}\to\mathbb{R}$ as follows $$\begin{aligned} &\Phi(f) = {\mathbb{E}_{\sigma}\left[\inf_{P \in {\mathcal{P}}}{\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P}\left[f({{\ensuremath{\mathbf{x}}}})-\sigma({{\ensuremath{\mathbf{x}}}})\right]}\right]},\\ &R(P) = \sup_{f\in {\mathcal{F}}} -{\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P}\left[f({{\ensuremath{\mathbf{x}}}})\right]} + \Phi(f).\end{aligned}$$ Also, note that the function ${{\nabla}\Phi}:{\mathcal{F}}\to{\mathcal{P}}$ defined in Section \[sec:online\_noncvx\] can be written as $$\begin{aligned} {{\nabla}\Phi\left(f\right)} = {\mathbb{E}_{\sigma}\left[{\mathop{\rm argmin}}_{P \in {\mathcal{P}}}{\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P}\left[f({{\ensuremath{\mathbf{x}}}})-\sigma({{\ensuremath{\mathbf{x}}}})\right]}\right]}.\end{aligned}$$ Note that, ${{\nabla}\Phi\left(f\right)}$ is well defined because from our assumption on the perturbation distribution, the minimization problem inside the expectation has a unique minimizer with probability one. To simplify the notation, in the sequel, we use the shorthand notation ${\left\langle P, f \right\rangle}$ to denote ${\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P}\left[f({{\ensuremath{\mathbf{x}}}})\right]}$, for any $P\in{\mathcal{P}}$ and $f\in {\mathcal{F}}$. Similarly, for any $P_1,P_2\in {\mathcal{P}}$ and $f\in {\mathcal{F}}$, we use the notation ${\left\langle P_1-P_2, f \right\rangle}$ to denote ${\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P_1}\left[f({{\ensuremath{\mathbf{x}}}})\right]}-{\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P_2}\left[f({{\ensuremath{\mathbf{x}}}})\right]}$. ### Intermediate Results \[lem:noncvx\_gradient\] For any $g\in {\mathcal{F}}$, $R({{\nabla}\Phi\left(g\right)}) = -{\left\langle {{\nabla}\Phi\left(g\right)}, g \right\rangle} + \Phi(g)$. Define $P_{g,\sigma}$ as $$P_{g,\sigma}={\mathop{\rm argmin}}_{P \in {\mathcal{P}}}{\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P}\left[g({{\ensuremath{\mathbf{x}}}})-\sigma({{\ensuremath{\mathbf{x}}}})\right]}.$$ Note that ${{\nabla}\Phi\left(g\right)} = {\mathbb{E}_{\sigma}\left[P_{g,\sigma}\right]}$. For any $g,h \in {\mathcal{F}}$, we have $$\begin{aligned} \Phi(h) &= {\mathbb{E}_{\sigma}\left[\inf_{P \in {\mathcal{P}}}{\left\langle P, h-\sigma \right\rangle}\right]}\\ &\leq {\mathbb{E}_{\sigma}\left[{\left\langle P_{g,\sigma}, h-\sigma \right\rangle}\right]}\\ &= {\mathbb{E}_{\sigma}\left[{\left\langle P_{g,\sigma}, g-\sigma \right\rangle}\right]} + {\mathbb{E}_{\sigma}\left[{\left\langle P_{g,\sigma}, h-g \right\rangle}\right]}\\ & = \Phi(g) + {\left\langle {{\nabla}\Phi\left(g\right)}, h-g \right\rangle}.\end{aligned}$$ This shows that for any $g,h \in {\mathcal{F}}$ $$\label{eqn:noncvx_phi_convex} \Phi(h) - {\left\langle {{\nabla}\Phi\left(g\right)}, h \right\rangle} \leq \Phi(g) - {\left\langle {{\nabla}\Phi\left(g\right)}, g \right\rangle}.$$ Taking supremum over $h$ of the LHS quantity gives us $$R({{\nabla}\Phi\left(g\right)})=\sup_{h\in {\mathcal{F}}}\Phi(h) - {\left\langle {{\nabla}\Phi\left(g\right)}, h \right\rangle} = \Phi(g) - {\left\langle {{\nabla}\Phi\left(g\right)}, g \right\rangle}.$$ \[lem:noncvx\_phi\_smooth\] The function $-\Phi$ is convex and strongly smooth and satisfies the following inequality for any $g_1,g_2\in {\mathcal{F}}$ $$-\Phi(g_2)\leq -\Phi(g_1) - {\left\langle {{\nabla}\Phi\left(g_1\right)}, g_2-g_1 \right\rangle} + \frac{C}{2\eta}\|g_2-g_1\|_{{\mathcal{F}}}^2.$$ Let $g_1,g_2\in{\mathcal{F}}$ and $\alpha \in [0,1]$. Then $$\begin{aligned} \Phi(\alpha g_1 + (1-\alpha)g_2) &= {\mathbb{E}_{\sigma}\left[\inf_{P \in {\mathcal{P}}}{\left\langle P, \alpha g_1 + (1-\alpha)g_2-\sigma \right\rangle}\right]}\\ & \geq \alpha{\mathbb{E}_{\sigma}\left[\inf_{P \in {\mathcal{P}}}{\left\langle P, g_1-\sigma \right\rangle}\right]} + (1-\alpha){\mathbb{E}_{\sigma}\left[\inf_{P \in {\mathcal{P}}}{\left\langle P, g_2-\sigma \right\rangle}\right]}\\ & = \alpha\Phi(g_1) + (1-\alpha)\Phi(g_2).\end{aligned}$$ This shows that $-\Phi$ is convex. To show smoothness, we rely on the following stability property $$\forall g_1,g_2\in{\mathcal{F}}\quad {\gamma_{{\mathcal{F}}}}({{\nabla}\Phi\left(g_1\right)}, {{\nabla}\Phi\left(g_2\right)}) \leq \frac{C}{\eta} \|g_1-g_2\|_{{\mathcal{F}}}.$$ Let $T$ be an arbitrary positive integer and for $t\in \{0,1,\dots T\}$, define $\alpha_t = t/T$. Let $h = g_2-g_1$. We have $$\begin{aligned} \Phi(g_1)-\Phi(g_2) &= \Phi(g_1 + \alpha_0h)-\Phi(g_1 + \alpha_Th)\\ &=\sum_{t=0}^{T-1}\left(\Phi(g_1 + \alpha_{t}h)-\Phi(g_1 + \alpha_{t+1}h)\right)\end{aligned}$$ Since $-\Phi$ is convex and satisfies Equation , we have $$\begin{aligned} \Phi(g_1)-\Phi(g_2) &=\sum_{t=0}^{T-1}\left(\Phi(g_1 + \alpha_{t}h)-\Phi(g_1 + \alpha_{t+1}h)\right)\\ &\leq -\sum_{t=0}^{T-1} \frac{1}{T}{\left\langle {{\nabla}\Phi\left(g_1+\alpha_{t+1}h\right)}, h \right\rangle}\end{aligned}$$ Using stability, we get $$\begin{aligned} \Phi(g_1)-\Phi(g_2) &\leq -\sum_{t=0}^{T-1} \frac{1}{T}{\left\langle {{\nabla}\Phi\left(g_1+\alpha_{t+1}h\right)}, h \right\rangle}\\ & = \sum_{t=0}^{T-1} \frac{1}{T}\left({\left\langle {{\nabla}\Phi\left(g_1\right)}-{{\nabla}\Phi\left(g_1+\alpha_{t+1}h\right)}, h \right\rangle} - {\left\langle {{\nabla}\Phi\left(g_1\right)}, h \right\rangle}\right)\\ & \stackrel{(a)}{\leq} -{\left\langle {{\nabla}\Phi\left(g_1\right)}, h \right\rangle} + \sum_{t=0}^{T-1} \frac{1}{T}{\gamma_{{\mathcal{F}}}}({{\nabla}\Phi\left(g_1\right)},{{\nabla}\Phi\left(g_1+\alpha_{t+1}h\right)})\|h\|_{{\mathcal{F}}} \\ &\stackrel{(b)}{\leq} -{\left\langle {{\nabla}\Phi\left(g_1\right)}, h \right\rangle} + \sum_{t=0}^{T-1} \frac{C}{T\eta}\|\alpha_{t+1}h\|_{{\mathcal{F}}}\|h\|_{{\mathcal{F}}}\\ &=-{\left\langle {{\nabla}\Phi\left(g_1\right)}, h \right\rangle} + \sum_{t=0}^{T-1} \frac{C\alpha_{t+1}}{T\eta}\|h\|^2_{{\mathcal{F}}}\\ &=-{\left\langle {{\nabla}\Phi\left(g_1\right)}, h \right\rangle} +\frac{C}{\eta}\frac{T+1}{2T}\|h\|^2_{{\mathcal{F}}},\end{aligned}$$ where $(a)$ follows from the definition of ${\gamma_{{\mathcal{F}}}}$ and $(b)$ follows from the stability assumption. Taking $T\to\infty$, we get $$-\Phi(g_2)\leq -\Phi(g_1) - {\left\langle {{\nabla}\Phi\left(g_1\right)}, g_2-g_1 \right\rangle} + \frac{C}{2\eta}\|g_2-g_1\|_{{\mathcal{F}}}^2.$$ \[lem:noncvx\_reg\_strong\_cvx\] For any $P\in {\mathcal{P}}$ and $g \in {\mathcal{F}}$, $R$ satisfies the following inequality $$R(P)\geq R({{\nabla}\Phi\left(g\right)}) +{\left\langle {{\nabla}\Phi\left(g\right)}-P, g \right\rangle} + \frac{\eta}{2{C}}{\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(g\right)})^2.$$ From Lemma \[lem:noncvx\_phi\_smooth\] we know that the following holds for any $g,h\in{\mathcal{F}}$ $$\Phi(g)\geq \underbrace{\Phi(h) + {\left\langle {{\nabla}\Phi\left(h\right)}, g-h \right\rangle} - \frac{C}{2\eta}\|g-h\|_{{\mathcal{F}}}^2}_{\Phi_{\text{lb}, h}(g)}.$$ Define $R_{\text{lb},h}(P)$ as $$R_{\text{lb}}(P) =\sup_{g\in{\mathcal{F}}} -{\left\langle P, g \right\rangle} + \Phi_{\text{lb},h}(g).$$ Since $\Phi(g) \geq \Phi_{\text{lb},h}(g)$ for all $g\in{\mathcal{F}}$, $R(P) \geq R_{\text{lb},h}(P)$ for all $P$. We now derive an expression for $R_{\text{lb},h}(P)$. Note that from Lemma \[lem:noncvx\_gradient\] we have $R({{\nabla}\Phi\left(h\right)}) = -{\left\langle {{\nabla}\Phi\left(h\right)}, h \right\rangle} + \Phi(h)$. Using this, we get $$\begin{aligned} R_{\text{lb},h}(P) &= \sup_{g\in{\mathcal{F}}} -{\left\langle P, g \right\rangle} + \Phi_{\text{lb},h}(g)\\ & \stackrel{(a)}{=} \sup_{g\in{\mathcal{F}}} \left(-{\left\langle P, g \right\rangle} + \Phi(h) + {\left\langle {{\nabla}\Phi\left(h\right)}, g-h \right\rangle} - \frac{C}{2\eta}\|g-h\|_{{\mathcal{F}}}^2\right)\\ &\stackrel{(b)}{=}R({{\nabla}\Phi\left(h\right)}) + \sup_{g\in{\mathcal{F}}} \left({\left\langle {{\nabla}\Phi\left(h\right)}-P, g \right\rangle}-\frac{C}{2\eta}\|g-h\|_{{\mathcal{F}}}^2\right),\end{aligned}$$ where $(a)$ follows from the definition of $\Phi_{\text{lb},h}(g)$ and $(b)$ follows from Lemma \[lem:noncvx\_gradient\]. We now do a change of variables in the supremum of the above expression. Substituting $g' = g - h$, we get $$\begin{aligned} R_{\text{lb},h}(P) & = R({{\nabla}\Phi\left(h\right)}) + {\left\langle {{\nabla}\Phi\left(h\right)}-P, h \right\rangle} + \sup_{g'\in{\mathcal{F}}} \left({\left\langle {{\nabla}\Phi\left(h\right)}-P, g' \right\rangle}-\frac{C}{2\eta}\|g'\|_{{\mathcal{F}}}^2\right).\end{aligned}$$ We now show that $$\sup_{g'\in{\mathcal{F}}} \left({\left\langle {{\nabla}\Phi\left(h\right)}-P, g' \right\rangle}-\frac{C}{2\eta}\|g'\|_{{\mathcal{F}}}^2\right) \geq \frac{\eta}{2C}{\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)})^2.$$ To this end, we choose a $g'' \in {\mathcal{F}}$ such that $$\label{eqn:noncvx_sc_g} \|g''\|_{{\mathcal{F}}} = \frac{\eta}{C}{\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)}),\quad {\left\langle {{\nabla}\Phi\left(h\right)}-P, g'' \right\rangle} = \frac{\eta}{C}{\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)})^2.$$ If such a $g''$ can be found, we have $$\begin{aligned} \sup_{g'\in{\mathcal{F}}} \left({\left\langle {{\nabla}\Phi\left(h\right)}-P, g' \right\rangle}-\frac{C}{2\eta}\|g'\|_{{\mathcal{F}}}^2\right) &\geq {\left\langle {{\nabla}\Phi\left(h\right)}-P, g'' \right\rangle}-\frac{C}{2\eta}\|g''\|_{{\mathcal{F}}}^2\\ & = \frac{\eta}{2C}{\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)})^2.\end{aligned}$$ This would then imply the main claim of the Lemma. $$\begin{aligned} R(P) \geq R_{\text{lb},h}(P) \geq R({{\nabla}\Phi\left(h\right)}) + {\left\langle {{\nabla}\Phi\left(h\right)}-P, h \right\rangle} + \frac{\eta}{2C}{\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)})^2.\end{aligned}$$ #### Finding $g''$. We now construct a $g''$ which satisfies Equation . From the definition of ${\gamma_{{\mathcal{F}}}}$ we know that $${\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)}) = \sup_{\|g'\|_{{\mathcal{F}}}\leq 1} |{\left\langle {{\nabla}\Phi\left(h\right)}-P, g' \right\rangle}|$$ Suppose the supremum is achieved at $g^*$. Define $g''$ as $\frac{\eta s}{C}{\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)})g^*$, where $s = \text{sign}({\left\langle {{\nabla}\Phi\left(h\right)}-P, g^* \right\rangle})$. It can be easily verified that $g''$ satifies Equation . If the supremum is never achieved, the same argument as above can still be made using a sequence of functions $\{g_{n}\}_{n=1}^{\infty}$ such that $$\|g_n\|_{{\mathcal{F}}}\leq 1,\quad \lim_{n\to\infty} |{\left\langle {{\nabla}\Phi\left(h\right)}-P, g_n \right\rangle}| = {\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)}).$$ Define $g''_n$ as $\frac{\eta s_n}{C}{\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)})g_n$, where $s_n = \text{sign}({\left\langle {{\nabla}\Phi\left(h\right)}-P, g_n \right\rangle})$. Since $\lim_{n\to\infty}\|g_n\|_{{\mathcal{F}}} = 1$, we have $\lim_{n \to \infty}\|g''_{n}\|_{{\mathcal{F}}} = \frac{\eta}{C} {\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)})$. Moreover, $$\lim_{n\to\infty} {\left\langle {{\nabla}\Phi\left(h\right)}-P, g''_n \right\rangle} = \lim_{n\to\infty} \frac{\eta}{C}{\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)}) \Big|{\left\langle {{\nabla}\Phi\left(h\right)}-P, g_n \right\rangle}\Big| = \frac{\eta}{C}{\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)})^2.$$ This shows that $$\begin{aligned} \sup_{g'\in{\mathcal{F}}} \left({\left\langle {{\nabla}\Phi\left(h\right)}-P, g' \right\rangle}-\frac{C}{2\eta}\|g'\|_{{\mathcal{F}}}^2\right) &\geq \lim_{n\to\infty}{\left\langle {{\nabla}\Phi\left(h\right)}-P, g''_n \right\rangle}-\frac{C}{2\eta}\|g''_n\|_{{\mathcal{F}}}^2\\ & = \frac{\eta}{2C}{\gamma_{{\mathcal{F}}}}(P,{{\nabla}\Phi\left(h\right)})^2.\end{aligned}$$ This finishes the proof of the Lemma. ### Main Argument We are now ready to prove Theorem \[thm:oftpl\_noncvx\_regret\]. Our proof relies on Lemma \[lem:noncvx\_reg\_strong\_cvx\] and uses similar arguments as used in the proof of Theorem \[thm:oftpl\_regret\]. We first rewrite $P_t, \Tilde{P}_t$ as $$\begin{aligned} P_t &= \frac{1}{m}\sum_{j=1}^m{\mathop{\rm argmin}}_{P \in {\mathcal{P}}}{\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P}\left[\sum_{i = 1}^{t-1}f_i({{\ensuremath{\mathbf{x}}}})+g_t({{\ensuremath{\mathbf{x}}}})-\sigma_{t,j}({{\ensuremath{\mathbf{x}}}})\right]},\\ \Tilde{P}_t &= \frac{1}{m}\sum_{j=1}^m{\mathop{\rm argmin}}_{P \in {\mathcal{P}}}{\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P}\left[\sum_{i = 1}^{t}f_i({{\ensuremath{\mathbf{x}}}})-\sigma'_{t,j}({{\ensuremath{\mathbf{x}}}})\right]}.\end{aligned}$$ Note that $$\begin{aligned} P_t^{\infty} &= {\mathbb{E}\left[P_t|g_t,f_{1:t-1},P_{1:t-1}\right]} = {{\nabla}\Phi\left(f_{1:t-1} + g_t\right)},\\ \Tilde{P}_t^{\infty} &= {\mathbb{E}\left[\Tilde{P}_t|f_{1:t-1},P_{1:t-1}\right]} = {{\nabla}\Phi\left(f_{1:t}\right)},\end{aligned}$$ with $P_1^{\infty} = \Tilde{P}_0^{\infty} ={{\nabla}\Phi\left(0\right)}$. Define functions $B(\cdot,P_t^{\infty}), B(\cdot, \Tilde{P}_t^{\infty})$ as $$\begin{aligned} B(P,P_t^{\infty}) &= R(P) - R(P_t^{\infty}) + {\left\langle P-P_t^{\infty}, f_{1:t-1}+g_t \right\rangle},\\ B(P,\Tilde{P}_t^{\infty}) &= R(P) - R(\Tilde{P}_t^{\infty}) + {\left\langle P-\Tilde{P}_t^{\infty}, f_{1:t} \right\rangle}.\end{aligned}$$ From Lemma \[lem:noncvx\_reg\_strong\_cvx\], we have $$B(P,P_t^{\infty}) \geq \frac{\eta}{2C}{\gamma_{{\mathcal{F}}}}(P, P_t^{\infty})^2,\quad B(P,\Tilde{P}_t^{\infty}) \geq \frac{\eta}{2C}{\gamma_{{\mathcal{F}}}}(P, \Tilde{P}_t^{\infty})^2.$$ For any $P\in {\mathcal{P}}$, we have $$\begin{aligned} {\mathbb{E}\left[f_t({{\ensuremath{\mathbf{x}}}}_t)-f_t(P)\right]} &= {\mathbb{E}\left[f_t(P_t)-f_t(P)\right]} \\ & = {\mathbb{E}\left[{\left\langle P_t-P, f_t \right\rangle}\right]}\\ &={\mathbb{E}\left[{\left\langle P_t-P_t^{\infty}, f_t \right\rangle}\right]} + {\mathbb{E}\left[{\left\langle P_t^{\infty}-P, f_t \right\rangle}\right]}\\ &={\mathbb{E}\left[{\left\langle P_t-P_t^{\infty}, f_t \right\rangle}\right]} + {\mathbb{E}\left[{\left\langle P_t^{\infty}-\Tilde{P}_t^{\infty}, f_t-g_t \right\rangle}\right]} \\ &\quad+ {\mathbb{E}\left[{\left\langle P_t^{\infty}-\Tilde{P}_t^{\infty}, g_t \right\rangle}\right]}+{\mathbb{E}\left[{\left\langle \Tilde{P}_t^{\infty}-P, f_t \right\rangle}\right]}\\ &\stackrel{(a)}{\leq}{\mathbb{E}\left[{\gamma_{{\mathcal{F}}}}(P_t^{\infty}, \Tilde{P}_t^{\infty})\|f_t-g_t\|_{{\mathcal{F}}}\right]}+ {\mathbb{E}\left[{\left\langle P_t^{\infty}-\Tilde{P}_t^{\infty}, g_t \right\rangle}\right]} \\ &\quad+{\mathbb{E}\left[{\left\langle \Tilde{P}_t^{\infty}-P, f_t \right\rangle}\right]},\end{aligned}$$ where $(a)$ follows from the fact that ${\mathbb{E}\left[{\left\langle P_t-P_t^{\infty}, f_t \right\rangle}|g_t, f_{1:t-1}, P_{1:t-1}\right]} = 0$ and as a result ${\mathbb{E}\left[{\left\langle P_t-P_t^{\infty}, f_t \right\rangle}\right]}=0$. Next, a simple calculation shows that $$\begin{aligned} {\left\langle P_t^{\infty}-\Tilde{P}_t^{\infty}, g_{t} \right\rangle} &= B(\Tilde{P}_t^{\infty},\Tilde{P}_{t-1}^{\infty}) - B(\Tilde{P}_t^{\infty},P_t^{\infty}) - B(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})\\ {\left\langle \Tilde{P}_t^{\infty}-P, f_t \right\rangle} &= B(P,\Tilde{P}_{t-1}^{\infty}) - B(P,\Tilde{P}_t^{\infty})-B(\Tilde{P}_t^{\infty},\Tilde{P}_{t-1}^{\infty}).\end{aligned}$$ Substituting this in the previous regret bound gives us $$\begin{aligned} {\mathbb{E}\left[f_t({{\ensuremath{\mathbf{x}}}}_t)-f_t(P)\right]} & \leq {\mathbb{E}\left[{\gamma_{{\mathcal{F}}}}(P_t^{\infty}, \Tilde{P}_t^{\infty})\|f_t-g_t\|_{{\mathcal{F}}}\right]} + {\mathbb{E}\left[B(\Tilde{P}_t^{\infty},\Tilde{P}_{t-1}^{\infty}) - B(\Tilde{P}_t^{\infty},P_t^{\infty}) - B(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})\right]}\\ &\quad +{\mathbb{E}\left[B(P,\Tilde{P}_{t-1}^{\infty}) - B(P,\Tilde{P}_t^{\infty})-B(\Tilde{P}_t^{\infty},\Tilde{P}_{t-1}^{\infty})\right]}\\ & = {\mathbb{E}\left[{\gamma_{{\mathcal{F}}}}(P_t^{\infty}, \Tilde{P}_t^{\infty})\|f_t-g_t\|_{{\mathcal{F}}}\right]} \\ &\quad + {\mathbb{E}\left[B(P,\Tilde{P}_{t-1}^{\infty}) - B(P,\Tilde{P}_t^{\infty}) - B(\Tilde{P}_t^{\infty},P_t^{\infty}) - B(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})\right]}\\ & \stackrel{(a)}{\leq}{\mathbb{E}\left[{\gamma_{{\mathcal{F}}}}(P_t^{\infty}, \Tilde{P}_t^{\infty})\|f_t-g_t\|_{{\mathcal{F}}}\right]} \\ &\quad + {\mathbb{E}\left[B(P,\Tilde{P}_{t-1}^{\infty}) - B(P,\Tilde{P}_t^{\infty})\right]} - {\mathbb{E}\left[\frac{\eta}{2C}{\gamma_{{\mathcal{F}}}}(\Tilde{P}_t^{\infty}, P_t^{\infty})^2 + \frac{\eta}{2C}{\gamma_{{\mathcal{F}}}}(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})^2 \right]}\\ & \stackrel{(b)}{\leq} \frac{C}{2\eta}{\mathbb{E}\left[\|f_t-g_t\|_{{\mathcal{F}}}^2\right]} + {\mathbb{E}\left[B(P,\Tilde{P}_{t-1}^{\infty}) - B(P,\Tilde{P}_t^{\infty})\right]} - {\mathbb{E}\left[ \frac{\eta}{2C}{\gamma_{{\mathcal{F}}}}(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})^2 \right]}\end{aligned}$$ where $(a)$ follows from Lemma \[lem:noncvx\_reg\_strong\_cvx\], and $(b)$ uses the fact that $|xy|\leq \frac{1}{2c}|x|^2 + \frac{c}{2}|y|^2$, for any $x,y$, $c> 0$. Summing over $t=1,\dots T$ gives us $$\begin{aligned} \sum_{t=1}^T {\mathbb{E}\left[f_t({{\ensuremath{\mathbf{x}}}}_t)-f_t(P)\right]} &\leq \underbrace{{\mathbb{E}\left[B(P,\Tilde{P}_{0}^{\infty}) - B(P,\Tilde{P}_T^{\infty})\right]}}_{S_1}+ \sum_{t=1}^T\frac{C}{2\eta}{\mathbb{E}\left[\|f_t-g_t\|_{{\mathcal{F}}}^2\right]}\\ &\quad- \sum_{t=1}^T\frac{\eta}{2C}{\mathbb{E}\left[ {\gamma_{{\mathcal{F}}}}(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})^2 \right]}\end{aligned}$$ To finish the proof of the Theorem, we need to bound $S_1$. #### Bounding $S_1$. From the definition of $B$, we have $$\begin{aligned} B(P,\Tilde{P}_{0}^{\infty}) - B(P,\Tilde{P}_T^{\infty}) &= R(\Tilde{P}_{T}^{\infty}) - {\left\langle P-\Tilde{P}_T^{\infty}, f_{1:T} \right\rangle}-R(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{0}^{\infty}),\end{aligned}$$ where we used the fact that $f_{1:0} = 0$. We now rely on Lemma \[lem:noncvx\_gradient\] to convert the above equation, which is currently in terms of $R$, into a quantity which depends on $\Phi$. Using Lemma \[lem:noncvx\_gradient\], we get $$\begin{aligned} B(P,\Tilde{P}_{0}^{\infty}) - B(P,\Tilde{P}_T^{\infty}) &= \Phi(f_{1:T}) - {\left\langle P, f_{1:T} \right\rangle}-\Phi(0).\end{aligned}$$ From the definition of $\Phi$ we have $$\begin{aligned} B(P,\Tilde{P}_{0}^{\infty}) - B(P,\Tilde{P}_T^{\infty}) &= \Phi(f_{1:T}) - {\left\langle P, f_{1:T} \right\rangle}-\Phi(0)\\ &={\mathbb{E}_{\sigma}\left[\inf_{P' \in {\mathcal{P}}}{\left\langle P', f_{1:T}-\sigma \right\rangle}\right]} - {\left\langle P, f_{1:T} \right\rangle} - {\mathbb{E}_{\sigma}\left[\inf_{P' \in {\mathcal{P}}}{\left\langle P', -\sigma \right\rangle}\right]}\\ &\leq {\mathbb{E}_{\sigma}\left[{\left\langle P, f_{1:T}-\sigma \right\rangle}\right]} - {\left\langle P, f_{1:T} \right\rangle} - {\mathbb{E}_{\sigma}\left[\inf_{P' \in {\mathcal{P}}}{\left\langle P', -\sigma \right\rangle}\right]}\\ &= {\mathbb{E}_{\sigma}\left[\sup_{P' \in {\mathcal{P}}}{\left\langle P', \sigma \right\rangle}\right]} - {\mathbb{E}_{\sigma}\left[\left\langle P,\sigma\right\rangle\right]}\\ &\leq D{\mathbb{E}_{\sigma}\left[\|\sigma\|_{{\mathcal{F}}}\right]} = \eta D,\end{aligned}$$ where the last inequality follows from our bound on the diameter of ${\mathcal{P}}$. Substituting this in the above regret bound gives us the required result. Proof of Corollary \[cor:ftpl\_noncvx\_exp\] -------------------------------------------- To prove the corollary we first show that for our choice of perturbation distribution, ${\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} f({{\ensuremath{\mathbf{x}}}}) - \sigma({{\ensuremath{\mathbf{x}}}})$ has a unique minimizer with probability one, for any $f\in{\mathcal{F}}$. Next, we show that the predictions of OFTPL are stable. ### Intermediate Results Suppose the perturbation function is such that $\sigma({{\ensuremath{\mathbf{x}}}}) = {\left\langle \bar{\sigma}, {{\ensuremath{\mathbf{x}}}}\right\rangle}$, where $\bar{\sigma} \in \mathbb{R}^d$ is a random vector whose entries are sampled independently from $\text{Exp}(\eta)$. Then, for any $f\in{\mathcal{F}}$, ${\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} f({{\ensuremath{\mathbf{x}}}}) - \sigma({{\ensuremath{\mathbf{x}}}})$ has a unique minimizer with probability one. Define ${{\ensuremath{\mathbf{x}}}}_f(\sigma)$ as $${{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}) \in {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} f({{\ensuremath{\mathbf{x}}}}) - {\left\langle \bar{\sigma}, {{\ensuremath{\mathbf{x}}}}\right\rangle}.$$ For any $\bar{\sigma}_1,\bar{\sigma}_2$ we now show that ${{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma})$ satisfies the following monotonicity property $${\left\langle {{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_1)-{{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_2), \bar{\sigma}_1-\bar{\sigma}_2 \right\rangle} \geq 0.$$ From the optimality of ${{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_1),{{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_2)$ we have $$\begin{aligned} f({{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_1)) - {\left\langle \bar{\sigma}_1, {{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_1) \right\rangle} &\leq f({{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_2)) - {\left\langle \bar{\sigma}_1, {{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_2) \right\rangle}\\ & = f({{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_2)) - {\left\langle \bar{\sigma}_2, {{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_2) \right\rangle} + {\left\langle \bar{\sigma}_2-\bar{\sigma}_1, {{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_2) \right\rangle}\\ &\leq f({{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_1)) - {\left\langle \bar{\sigma}_2, {{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_1) \right\rangle}+ {\left\langle \bar{\sigma}_2-\bar{\sigma}_1, {{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_2) \right\rangle}.\end{aligned}$$ This shows that ${\left\langle \bar{\sigma}_2-\bar{\sigma}_1, {{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_2)-{{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}_1) \right\rangle} \geq 0$. To finish the proof of Lemma, we rely on Theorem 1 of @zarantonello1973dense, which shows that the set of points for which a monotone operator is not single-valued has Lebesgue measure zero. Since the distribution of $\bar{\sigma}$ is absolutely continuous w.r.t Lebesgue measure, this shows that ${\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} f({{\ensuremath{\mathbf{x}}}}) - \sigma({{\ensuremath{\mathbf{x}}}})$ has a unique minimizer with probability one. ### Main Argument For our choice of perturbation distribution, ${\mathbb{E}_{\sigma}\left[\|\sigma\|_{{\mathcal{F}}}\right]} = {\mathbb{E}_{\bar{\sigma}}\left[\|\bar{\sigma}\|_{\infty}\right]} = \eta\log{d}$. We now bound the stability of predictions of OFTPL. First note that for our choice of primal space $({\mathcal{F}},\|\cdot\|_{{\mathcal{F}}})$, ${\gamma_{{\mathcal{F}}}}$ is the Wasserstein-1 metric, which is defined as $${\gamma_{{\mathcal{F}}}}(P_1,P_2) = \sup_{f\in{\mathcal{F}}, \|f\|_{{\mathcal{F}}}\leq 1} \Big|{\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P_1}\left[f({{\ensuremath{\mathbf{x}}}})\right]}-{\mathbb{E}_{{{\ensuremath{\mathbf{x}}}}\sim P_2}\left[f({{\ensuremath{\mathbf{x}}}})\right]}\Big| = \inf_{Q\in\Gamma(P_1,P_2)}{\mathbb{E}_{({{\ensuremath{\mathbf{x}}}}_1,{{\ensuremath{\mathbf{x}}}}_2)\sim Q}\left[\|{{\ensuremath{\mathbf{x}}}}_1-{{\ensuremath{\mathbf{x}}}}_2\|_1\right]},$$ where $\Gamma(P_1,P_2)$ is the set of all probability measures on ${\mathcal{X}}\times{\mathcal{X}}$ with marginals $P_1,P_2$ on the first and second factors respectively. Define ${{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma})$ as $${{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma}) \in {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} f({{\ensuremath{\mathbf{x}}}}) - {\left\langle \bar{\sigma}, {{\ensuremath{\mathbf{x}}}}\right\rangle}.$$ Note that ${{\nabla}\Phi\left(f\right)}$ is the distribution of random variable ${{\ensuremath{\mathbf{x}}}}_f(\bar{\sigma})$. @suggala2019online show that for any $f,g\in{\mathcal{F}}$ $${\mathbb{E}_{\bar{\sigma}}\left[\|{{\ensuremath{\mathbf{x}}}}_{f}(\bar{\sigma})-{{\ensuremath{\mathbf{x}}}}_{g}(\bar{\sigma})\|_1\right]} \leq \frac{125d^2D}{\eta}\|f-g\|_{{\mathcal{F}}}.$$ Since ${\gamma_{{\mathcal{F}}}}({{\nabla}\Phi\left(f\right)},{{\nabla}\Phi\left(g\right)}) \leq {\mathbb{E}_{\bar{\sigma}}\left[\|{{\ensuremath{\mathbf{x}}}}_{f}(\bar{\sigma})-{{\ensuremath{\mathbf{x}}}}_{g}(\bar{\sigma})\|_1\right]}$, this shows that OFTPL is ${O\left({d^2D\eta^{-1}}\right)}$ stable w.r.t $\|\cdot\|_{{\mathcal{F}}}$. Substituting the stability bound in the regret bound of Theorem \[thm:oftpl\_noncvx\_regret\] shows that $$\begin{aligned} \sup_{P\in{\mathcal{P}}}{\mathbb{E}\left[\sum_{t=1}^Tf_t({{\ensuremath{\mathbf{x}}}}_t)-f_t(P)\right]} &= \eta D\log{d} \\ &\quad +{O\left({ \sum_{t=1}^T \frac{d^2D }{\eta}{\mathbb{E}\left[\|f_t-g_{t}\|_{{\mathcal{F}}}^2\right]} -\sum_{t=1}^T \frac{\eta}{d^2D }{\mathbb{E}\left[{\gamma_{{\mathcal{F}}}}(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})^2\right]}}\right)}.\end{aligned}$$ Convex-Concave Games {#sec:cvx-games} ==================== Our algorithm for convex-concave games is presented in Algorithm \[alg:oftpl\_cvx\_games\]. Before presenting the proof of Theorem \[thm:oftpl\_cvx\_smooth\_games\_uniform\], we first present a more general result in Section \[sec:cvx\_games\_general\]. Theorem \[thm:oftpl\_cvx\_smooth\_games\_uniform\] immediately follows from our general result by instantiating it for the uniform noise distribution. **Input:** Perturbation Distributions ${P_{\text{PRTB}}}^1,{P_{\text{PRTB}}}^2$ of ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players, number of samples $m,$ iterations $T$ Sample $\{\sigma_{1,j}^1\}_{j=1}^m,$ $\{\sigma_{1,j}^2\}_{j=1}^m$ from ${P_{\text{PRTB}}}^1,{P_{\text{PRTB}}}^2$ ${{\ensuremath{\mathbf{x}}}}_1 = \frac{1}{m}\sum_{j=1}^m\left[{\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} {\left\langle -\sigma_{1,j}^1, {{\ensuremath{\mathbf{x}}}}\right\rangle}\right], {{\ensuremath{\mathbf{y}}}}_1 = \frac{1}{m}\left[\sum_{j=1}^m{\mathop{\rm argmax}}_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} {\left\langle \sigma_{1,j}^2, {{\ensuremath{\mathbf{y}}}}\right\rangle}\right]$ **continue** `//Compute guesses` Sample $\sigma_{t,j}^1\sim {P_{\text{PRTB}}}^1, \sigma_{t,j}^2\sim {P_{\text{PRTB}}}^2$ $\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1,j}= \underset{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}{{\mathop{\rm argmin}}}{\left\langle \sum_{i = 1}^{t-1}{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_i,{{\ensuremath{\mathbf{y}}}}_i) -\sigma_{t,j}^1, {{\ensuremath{\mathbf{x}}}}\right\rangle}$ $\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1,j}= \underset{{{\ensuremath{\mathbf{y}}}}\in {\mathcal{Y}}}{{\mathop{\rm argmax}}}{\left\langle \sum_{i = 1}^{t-1}{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}}_i,{{\ensuremath{\mathbf{y}}}}_i) +\sigma_{t,j}^2, {{\ensuremath{\mathbf{y}}}}\right\rangle}$ $\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1} = \frac{1}{m}\sum_{j=1}^m\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1,j}$, $\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1} = \frac{1}{m}\sum_{j=1}^m\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1,j}$ `//Use the guesses to compute the next action` Sample $\sigma_{t,j}^1\sim {P_{\text{PRTB}}}^1, \sigma_{t,j}^2\sim {P_{\text{PRTB}}}^2$ ${{\ensuremath{\mathbf{x}}}}_{t,j}= \underset{{{\ensuremath{\mathbf{x}}}}\in {\mathcal{X}}}{{\mathop{\rm argmin}}}{\left\langle \sum_{i = 1}^{t-1}{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_i,{{\ensuremath{\mathbf{y}}}}_i)+ {\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})-\sigma_{t,j}^1, {{\ensuremath{\mathbf{x}}}}\right\rangle}$ ${{\ensuremath{\mathbf{y}}}}_{t,j}= \underset{{{\ensuremath{\mathbf{y}}}}\in {\mathcal{Y}}}{{\mathop{\rm argmax}}}{\left\langle \sum_{i = 1}^{t-1}{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}}_i,{{\ensuremath{\mathbf{y}}}}_i)+ {\nabla}_{{{\ensuremath{\mathbf{y}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})+\sigma_{t,j}^2, {{\ensuremath{\mathbf{y}}}}\right\rangle}$ ${{\ensuremath{\mathbf{x}}}}_t=\frac{1}{m}\sum_{j=1}^m {{\ensuremath{\mathbf{x}}}}_{t,j}, {{\ensuremath{\mathbf{y}}}}_t=\frac{1}{m}\sum_{j=1}^m {{\ensuremath{\mathbf{y}}}}_{t,j}$ $\{({{\ensuremath{\mathbf{x}}}}_t, {{\ensuremath{\mathbf{y}}}}_t)\}_{t=1}^T$ General Result {#sec:cvx_games_general} -------------- \[thm:oftpl\_cvx\_smooth\_games\] Consider the minimax game in Equation . Suppose $f$ is convex in ${{\ensuremath{\mathbf{x}}}}$, concave in ${{\ensuremath{\mathbf{y}}}}$ and is Holder smooth w.r.t some norm $\|\cdot\|$ $$\begin{aligned} \|{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{*} \leq L_1\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}'\|^{\alpha} + L_2\|{{\ensuremath{\mathbf{y}}}}-{{\ensuremath{\mathbf{y}}}}'\|^{\alpha},\\ \|{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{*} \leq L_2\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}'\|^{\alpha}+L_1\|{{\ensuremath{\mathbf{y}}}}-{{\ensuremath{\mathbf{y}}}}'\|^{\alpha}.\end{aligned}$$ Define diameter of sets ${\mathcal{X}},{\mathcal{Y}}$ as Let $L=\{L_1,L_2\}$. Suppose both ${{\ensuremath{\mathbf{x}}}}$ and ${{\ensuremath{\mathbf{y}}}}$ players use Algorithm \[alg:oftpl\_cvx\] to solve the minimax game. Suppose the perturbation distributions ${P_{\text{PRTB}}}^1,{P_{\text{PRTB}}}^2,$ used by ${{\ensuremath{\mathbf{x}}}}$, ${{\ensuremath{\mathbf{y}}}}$ players are absolutely continuous and satisfy ${\mathbb{E}_{\sigma\sim {P_{\text{PRTB}}}^1}\left[\|\sigma\|_{*}\right]} = {\mathbb{E}_{\sigma\sim {P_{\text{PRTB}}}^2}\left[\|\sigma\|_{*}\right]}=\eta$. Suppose the predictions of both the players are ${C}\eta^{-1}$-stable w.r.t $\|\cdot\|_*$. Suppose the guesses used by ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players in the $t^{th}$ iteration are ${\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}), {\nabla}_{{{\ensuremath{\mathbf{y}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})$, where $\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}$ denote the predictions of ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players in the $t^{th}$ iteration, if guess $g_t = 0$ was used in that iteration. Then the iterates $\{({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)\}_{t=1}^T$ generated by the OFTPL based algorithm satisfy $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[f\left(\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}\right) - f\left({{\ensuremath{\mathbf{x}}}},\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{y}}}}_t\right)\right]}\leq & 2L_1\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{1+\alpha}+\frac{2\eta D}{T} \\ &+ \frac{20{C}L^2}{\eta} \left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{2\alpha}+10L\left(\frac{5{C}L}{\eta}\right)^{\frac{1+\alpha}{1-\alpha}}\end{aligned}$$ Since both the players are responding to each others actions using OFTPL, using Theorem \[thm:oftpl\_regret\], we get the following regret bounds for the players $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} {\mathbb{E}\left[\sum_{t=1}^Tf({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t) - f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_t)\right]} &\leq L_1T\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{1+\alpha} + \eta D \\ &\quad+ \frac{{C}}{2\eta}\sum_{t=1}^T {\mathbb{E}\left[\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})\|_*^2\right]} \\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T {\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2\right]}.\end{aligned}$$ $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} {\mathbb{E}\left[\sum_{t=1}^T f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}})- f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)\right]} &\leq L_1T\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{1+\alpha} + \eta D \\ &\quad+ \frac{{C}}{2\eta}\sum_{t=1}^T {\mathbb{E}\left[\|{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)-{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})\|_{*}^2\right]} \\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T {\mathbb{E}\left[\|{{\ensuremath{\mathbf{y}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|^2\right]}.\end{aligned}$$ First, consider the regret of the ${{\ensuremath{\mathbf{x}}}}$ player. Since $\|a_1+\dots +a_5\|^2 \leq 5(\|a_1\|^2\dots + \|a_5\|^2)$, we have $$\begin{aligned} \|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})\|_{*}^2 \leq &5\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t)\|_{*}^2\\ &\quad +5\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t)-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})\|_{*}^2\\ &\quad +5\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{*}^2\\ &\quad +5\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})\|_{*}^2\\ &\quad +5\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})\|_{*}^2\\ &\stackrel{(a)}{\leq} 5L_1^2\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^{2\alpha}+5L_1^2\|\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^{2\alpha}\\ &\quad + 5L_2^2\|{{\ensuremath{\mathbf{y}}}}_t-{{\ensuremath{\mathbf{y}}}}_t^{\infty}\|^{2\alpha}+5L_2^2\|\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|^{2\alpha}\\ &\quad + 5\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{*}^2.\end{aligned}$$ where $(a)$ follows from the Holder’s smoothness of $f$. Using a similar technique as in the proof of Theorem \[thm:oftpl\_regret\], relying on Holders inequality, we get $$\begin{aligned} {\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^{2\alpha}|\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}, \Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1},{{\ensuremath{\mathbf{x}}}}_{1:t-1},{{\ensuremath{\mathbf{y}}}}_{1:t-1}\right]} &\leq {\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^{2}|\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}, \Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1},{{\ensuremath{\mathbf{x}}}}_{1:t-1},{{\ensuremath{\mathbf{y}}}}_{1:t-1}\right]}^{\alpha}\\ &\leq {\Psi_1}^{2\alpha}{\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^{2}_2|\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}, \Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1},{{\ensuremath{\mathbf{x}}}}_{1:t-1},{{\ensuremath{\mathbf{y}}}}_{1:t-1}\right]}^{\alpha}\\ &\stackrel{(a)}{\leq} \left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{2\alpha},\end{aligned}$$ where $(a)$ follows from the fact that conditioned on past randomness, ${{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}$ is the average of $m$ i.i.d bounded mean $0$ random variables, the variance of which scales as $O(D^2/m)$. A similar bound holds for the expectation of other quantities appearing in the RHS of the above equation. Using this, the regret of ${{\ensuremath{\mathbf{x}}}}$ player can be upper bounded as $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}{\mathbb{E}\left[\sum_{t=1}^Tf({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t) - f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_t)\right]} &\leq L_1T\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{1+\alpha}+\eta D + \frac{10{C}L^2 T}{\eta}\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{2\alpha} \\ &\quad + \frac{5{C}}{2\eta}\sum_{t=1}^T {\mathbb{E}\left[\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{*}^2\right]} \\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T {\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2\right]}.\end{aligned}$$ Similarly, the regret of ${{\ensuremath{\mathbf{y}}}}$ player can be bounded as $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[\sum_{t=1}^T f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}})- f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)\right]} &\leq L_1T\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{1+\alpha} + \eta D + \frac{10{C}L^2 T}{\eta}\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{2\alpha} \\ &\quad + \frac{5{C}}{2\eta}\sum_{t=1}^T {\mathbb{E}\left[\|{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{*}^2\right]} \\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T {\mathbb{E}\left[\|{{\ensuremath{\mathbf{y}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|^2\right]}.\end{aligned}$$ Summing the above two inequalities, we get $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[\sum_{t=1}^T f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}})- f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_t)\right]} &\leq 2L_1T\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{1+\alpha} + 2\eta D + \frac{20{C}L^2 T}{\eta}\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{2\alpha} \\ &\quad + \frac{5{C}}{2\eta}\sum_{t=1}^T {\mathbb{E}\left[\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{*}^2\right]} \\ &\quad + \frac{5{C}}{2\eta}\sum_{t=1}^T {\mathbb{E}\left[\|{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{*}^2\right]} \\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T \left({\mathbb{E}\left[\|{{\ensuremath{\mathbf{y}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|^2\right]}+{\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2\right]}\right).\end{aligned}$$ From Holder’s smoothness assumption on $f$, we have $$\begin{aligned} {\mathbb{E}\left[\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{*}^2\right]} & \leq 2{\mathbb{E}\left[\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{*}^2\right]}\\ &\quad + 2{\mathbb{E}\left[\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{*}^2\right]}\\ &\stackrel{(a)}{\leq} 2L^2 {\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^{2\alpha}\right]} + 2L^2{\mathbb{E}\left[\|{{\ensuremath{\mathbf{y}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|^{2\alpha}\right]},\end{aligned}$$ Using a similar argument, we get $$\begin{aligned} {\mathbb{E}\left[\|{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{*}^2\right]} \leq 2L^2 {\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^{2\alpha}\right]} + 2L^2{\mathbb{E}\left[\|{{\ensuremath{\mathbf{y}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|^{2\alpha}\right]}.\end{aligned}$$ Plugging this in the previous bound, we get $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[\sum_{t=1}^T f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}})- f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_t)\right]} &\leq 2L_1T\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{1+\alpha} + 2\eta D + \frac{20{C}L^2 T}{\eta}\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{2\alpha}\\ &\quad+\frac{10CL^2}{\eta}\sum_{t=1}^T \left({\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^{2\alpha}\right]} + {\mathbb{E}\left[\|{{\ensuremath{\mathbf{y}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|^{2\alpha}\right]}\right)\\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T \left({\mathbb{E}\left[\|{{\ensuremath{\mathbf{y}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|^2\right]}+{\mathbb{E}\left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2\right]}\right).\end{aligned}$$ #### Case $\alpha=1$. We first consider the case of $\alpha = 1$. In this case, choosing $\eta> \sqrt{20}CL$, we get $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[\sum_{t=1}^T f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}})- f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_t)\right]} &\leq 2L_1T\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{1+\alpha} + 2\eta D + \frac{20{C}L^2 T}{\eta}\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{2\alpha}.\end{aligned}$$ #### General $\alpha$. The more general case relies on AM-GM inequality. Consider the following $$\begin{aligned} \frac{10CL^2}{\eta}\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^{2\alpha} &= \left((2\alpha C)^{\frac{\alpha}{1-\alpha}}\eta^{-\frac{1+\alpha}{1-\alpha}}(10CL^2)^{\frac{1}{1-\alpha}}\right)^{1-\alpha}\left(\frac{\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2}{2\alpha C\eta^{-1}}\right)^{\alpha}\\ &\stackrel{(a)}{\leq} (1-\alpha) \left((2\alpha C)^{\frac{\alpha}{1-\alpha}}\eta^{-\frac{1+\alpha}{1-\alpha}}(10CL^2)^{\frac{1}{1-\alpha}}\right)+ \frac{\eta}{2C}\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2\\ &= \sqrt{20}L \left( \frac{\sqrt{20}CL}{\eta}\right)^{\frac{1+\alpha}{1-\alpha}}+ \frac{\eta}{2C}\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2\end{aligned}$$ where $(a)$ follows from AM-GM inequality. Plugging this in the previous bound, we get $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[\sum_{t=1}^T f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}})- f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_t)\right]}\leq & 2L_1T\left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{1+\alpha}+2\eta D \\ &+ \frac{20{C}L^2T}{\eta} \left(\frac{{\Psi_1}{\Psi_2}D}{\sqrt{m}}\right)^{2\alpha}+4\sqrt{5}LT\left(\frac{\sqrt{20}{C}L}{\eta}\right)^{\frac{1+\alpha}{1-\alpha}}.\end{aligned}$$ The claim of the theorem then follows from the observation that $$\begin{aligned} {\mathbb{E}\left[f\left(\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}\right) - f\left({{\ensuremath{\mathbf{x}}}},\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{y}}}}_t\right)\right]} \leq \frac{1}{T}{\mathbb{E}\left[\sum_{t=1}^T f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}})- f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_t)\right]}.\end{aligned}$$ Proof of Theorem \[thm:oftpl\_cvx\_smooth\_games\_uniform\] ----------------------------------------------------------- To prove the Theorem, we instantiate Theorem \[thm:oftpl\_cvx\_smooth\_games\] for the uniform noise distribution. As shown in Corollary \[cor:ftpl\_cvx\_gaussian\], the predictions of OFTPL are $dD\eta^{-1}$-stable in this case. Plugging this in the bound of Theorem \[thm:oftpl\_cvx\_smooth\_games\] and using the fact that ${\Psi_1}={\Psi_2}=1$ and $\alpha =1$ gives us $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[f\left(\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}\right) - f\left({{\ensuremath{\mathbf{x}}}},\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{y}}}}_t\right)\right]}\leq & 2L\left(\frac{D}{\sqrt{m}}\right)^{2}+\frac{2\eta D}{T} \\ &+ \frac{20dD L^2}{\eta} \left(\frac{D}{\sqrt{m}}\right)^{2}+10L\left(\frac{5dD L}{\eta}\right)^{\infty}.\end{aligned}$$ Plugging in $\eta = 6dD(L+1)$, $m=T$ in the above bound gives us $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[f\left(\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}\right) - f\left({{\ensuremath{\mathbf{x}}}},\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{y}}}}_t\right)\right]}\leq & {O\left({\frac{dD^2(L+1)}{T}}\right)}.\end{aligned}$$ Nonconvex-Nonconcave Games {#sec:ncvx-games} ========================== Our algorithm for nonconvex-nonconcave games is presented in Algorithm \[alg:oftpl\_noncvx\_games\]. Note that in each iteration of this game, both the players play empirical distributions $(P_t,Q_t)$. Before presenting the proof of Theorem \[thm:oftpl\_noncvx\_smooth\_games\_exp\], we first present a more general result in Section \[sec:noncvx\_games\_general\]. Theorem \[thm:oftpl\_noncvx\_smooth\_games\_exp\] immediately follows from our general result by instantiating it for exponential noise distribution. **Input:** Perturbation Distributions ${P_{\text{PRTB}}}^1,{P_{\text{PRTB}}}^2$ of ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players, number of samples $m,$ iterations $T$ Sample $\sigma_{t,j}^1\sim {P_{\text{PRTB}}}^1, \sigma_{t,j}^2\sim {P_{\text{PRTB}}}^2$ ${{\ensuremath{\mathbf{x}}}}_{1,j} = {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} -\sigma_{1,j}^1({{\ensuremath{\mathbf{x}}}})$ ${{\ensuremath{\mathbf{y}}}}_{1,j} = {\mathop{\rm argmax}}_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} \sigma_{1,j}^2({{\ensuremath{\mathbf{y}}}})$ Let $P_1,Q_1 $ be the empirical distributions over $\{{{\ensuremath{\mathbf{x}}}}_{1,j}\}_{j=1}^m, \{{{\ensuremath{\mathbf{y}}}}_{1,j}\}_{j=1}^m$ **continue** `//Compute guesses` Sample $\sigma_{t,j}^1\sim {P_{\text{PRTB}}}^1, \sigma_{t,j}^2\sim {P_{\text{PRTB}}}^2$ $\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1,j}= {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}\sum_{i = 1}^{t-1}f({{\ensuremath{\mathbf{x}}}},Q_i) -\sigma_{t,j}^1({{\ensuremath{\mathbf{x}}}})$ $\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1,j}= {\mathop{\rm argmax}}_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}\sum_{i = 1}^{t-1}f(P_i,{{\ensuremath{\mathbf{y}}}}) +\sigma_{t,j}^2({{\ensuremath{\mathbf{y}}}})$ Let $\Tilde{P}_{t-1} $, $\Tilde{Q}_{t-1} $ be the empirical distributions over $\{\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1,j}\}_{j=1}^m, \{\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1,j}\}_{j=1}^m$ `//Use the guesses to compute the next action` Sample $\sigma_{t,j}^1\sim {P_{\text{PRTB}}}^1, \sigma_{t,j}^2\sim {P_{\text{PRTB}}}^2$ ${{\ensuremath{\mathbf{x}}}}_{t,j}= {\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}\sum_{i = 1}^{t-1}f({{\ensuremath{\mathbf{x}}}},Q_i) + f({{\ensuremath{\mathbf{x}}}},\Tilde{Q}_{t-1}) -\sigma_{t,j}^1({{\ensuremath{\mathbf{x}}}})$ ${{\ensuremath{\mathbf{y}}}}_{t,j}= {\mathop{\rm argmax}}_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}\sum_{i = 1}^{t-1}f(P_i,{{\ensuremath{\mathbf{y}}}}) + f(\Tilde{P}_{t-1},{{\ensuremath{\mathbf{y}}}}) +\sigma_{t,j}^2({{\ensuremath{\mathbf{y}}}})$ Let $P_t, Q_t$ be the empirical distributions over $\{{{\ensuremath{\mathbf{x}}}}_{t,j}\}_{j=1}^m, \{{{\ensuremath{\mathbf{y}}}}_{t,j}\}_{j=1}^m$ $\{(P_t, Q_t)\}_{t=1}^T$ Primal Dual Spaces {#sec:primal_dual_spaces} ------------------ In this section, we present some integral probability metrics induced by popular choices of functions spaces $({\mathcal{F}},\|\cdot\|_{{\mathcal{F}}})$. [||c |c c||]{} ${\gamma_{{\mathcal{F}}}}(P, Q)$ & $\|f\|_{{\mathcal{F}}}$&${\mathcal{F}}$\ \[0.5ex\] Dudley Metric & $\text{Lip}(f)+\|f\|_{\infty}$& $\{f:\text{Lip}(f) + \|f\|_{\infty} < \infty\}$\ ------------------------- Kantorovich Metric (or) Wasserstein-1 Metric ------------------------- : Table showing some popular Integral Probability Metrics. Here $\text{Lip}(f)$ is the Lipschitz constant of $f$ which is defined as $\sup_{{{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{X}}}|f({{\ensuremath{\mathbf{x}}}})-f({{\ensuremath{\mathbf{y}}}})|/\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{y}}}}\|$ and $\|f\|_{\infty}$ is the supremum norm of $f$.[]{data-label="tab:ipm"} & $\text{Lip}(f)$ & $\{f:\text{Lip}(f) < \infty\}$\ Total Variation (TV) Distance & $\|f\|_{\infty}$ & $\{f:\|f\|_{\infty} < \infty\}$\ -------------------------------- Maximum Mean Discrepancy (MMD) for RKHS $\mathcal{H}$ -------------------------------- : Table showing some popular Integral Probability Metrics. Here $\text{Lip}(f)$ is the Lipschitz constant of $f$ which is defined as $\sup_{{{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{X}}}|f({{\ensuremath{\mathbf{x}}}})-f({{\ensuremath{\mathbf{y}}}})|/\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{y}}}}\|$ and $\|f\|_{\infty}$ is the supremum norm of $f$.[]{data-label="tab:ipm"} & $\|f\|_{\mathcal{H}}$ & $\{f:\|f\|_{\mathcal{H}} < \infty\}$\ \[1ex\] General Result {#sec:noncvx_games_general} -------------- \[thm:oftpl\_noncvx\_smooth\_games\] Consider the minimax game in Equation . Suppose the domains ${\mathcal{X}},{\mathcal{Y}}$ are compact subsets of $\mathbb{R}^d$. Let ${\mathcal{F}},{\mathcal{F}}'$ be the set of Lipschitz functions over ${\mathcal{X}},{\mathcal{Y}}$, and $\|g_1\|_{{\mathcal{F}}},\|g_2\|_{{\mathcal{F}}'}$ be the Lipschitz constants of functions $g_2:{\mathcal{Y}}\to\mathbb{R}$ w.r.t some norm $\|\cdot\|$. Suppose $f$ is such that $\max\{\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} \|f(\cdot,{{\ensuremath{\mathbf{y}}}})\|_{{\mathcal{F}}}, \sup_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}\|f({{\ensuremath{\mathbf{x}}}},\cdot)\|_{{\mathcal{F}}'}\}\leq G$ and satisfies the following smoothness property $$\begin{aligned} \|{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{*} \leq L\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}'\| + L\|{{\ensuremath{\mathbf{y}}}}-{{\ensuremath{\mathbf{y}}}}'\|,\\ \|{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{*} \leq L\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}'\|+L\|{{\ensuremath{\mathbf{y}}}}-{{\ensuremath{\mathbf{y}}}}'\|.\end{aligned}$$ Let ${\mathcal{P}},{\mathcal{Q}}$ be the set of probability distributions over ${\mathcal{X}},{\mathcal{Y}}$. Define diameter of ${\mathcal{P}},{\mathcal{Q}}$ as $D = \max\{\sup_{P_1,P_2\in{\mathcal{P}}} {\gamma_{{\mathcal{F}}}}(P_1,P_2), \sup_{Q_1,Q_2\in{\mathcal{Q}}} {\gamma_{{\mathcal{F}}'}}(Q_1,Q_2)\}$. Suppose both ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players use Algorithm \[alg:oftpl\_noncvx\] to solve the game. Suppose the perturbation distributions ${P_{\text{PRTB}}}^1,{P_{\text{PRTB}}}^2,$ used by ${{\ensuremath{\mathbf{x}}}}$, ${{\ensuremath{\mathbf{y}}}}$ players are such that ${\mathop{\rm argmin}}_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}f({{\ensuremath{\mathbf{x}}}})-\sigma({{\ensuremath{\mathbf{x}}}}), {\mathop{\rm argmax}}_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} f({{\ensuremath{\mathbf{y}}}})+\sigma({{\ensuremath{\mathbf{y}}}})$ have unique optimizers with probability one, for any $f$ in ${\mathcal{F}},{\mathcal{F}}'$ respectively. Moreover, suppose ${\mathbb{E}_{\sigma\sim {P_{\text{PRTB}}}^1}\left[\|\sigma\|_{{\mathcal{F}}}\right]} = {\mathbb{E}_{\sigma\sim {P_{\text{PRTB}}}^2}\left[\|\sigma\|_{{\mathcal{F}}'}\right]}=\eta$ and predictions of both the players are ${C}\eta^{-1}$-stable w.r.t norms $\|\cdot\|_{{\mathcal{F}}}, \|\cdot\|_{{\mathcal{F}}'}$. Suppose the guesses used by ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players in the $t^{th}$ iteration are $f(\cdot,\Tilde{Q}_{t-1}), f(\Tilde{P}_{t-1},\cdot)$, where $\Tilde{P}_{t-1},\Tilde{Q}_{t-1}$ denote the predictions of ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players in the $t^{th}$ iteration, if guess $g_t = 0$ was used. Then the iterates $\{(P_t,Q_t)\}_{t=1}^T$ generated by the Algorithm \[alg:oftpl\_cvx\_games\] satisfy the following, for $\eta > \sqrt{3}{C}L$ $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[f\left(\frac{1}{T}\sum_{t=1}^TP_t,{{\ensuremath{\mathbf{y}}}}\right) - f\left({{\ensuremath{\mathbf{x}}}},\frac{1}{T}\sum_{t=1}^TQ_t\right)\right]} &= {O\left({\frac{\eta D}{T} + \frac{{C}D^2L^2}{\eta m}}\right)}\\ &\quad +{O\left({\min\left\lbrace\frac{d{C}{\Psi_1}^2 {\Psi_2}^2G^2 \log(2m)}{\eta m}, \frac{CD^2L^2}{\eta}\right\rbrace}\right)}.\end{aligned}$$ The proof of this Theorem uses similar arguments as Theorem \[thm:oftpl\_cvx\_smooth\_games\]. Since both the players are responding to each others actions using OFTPL, using Theorem \[thm:oftpl\_noncvx\_regret\], we get the following regret bounds for the players $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} {\mathbb{E}\left[\sum_{t=1}^Tf(P_t,Q_t) - f({{\ensuremath{\mathbf{x}}}},Q_t)\right]} &\leq \eta D + \sum_{t=1}^T \frac{{C}}{2\eta}{\mathbb{E}\left[\|f(\cdot,Q_t)-f(\cdot,\Tilde{Q}_{t-1})\|_{{\mathcal{F}}}^2\right]}\\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T {\mathbb{E}\left[{\gamma_{{\mathcal{F}}}}(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})^2\right]},\end{aligned}$$ $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} {\mathbb{E}\left[\sum_{t=1}^T f(P_t,{{\ensuremath{\mathbf{y}}}})- f(P_t,Q_t)\right]} &\leq \eta D + \sum_{t=1}^T \frac{{C}}{2\eta}{\mathbb{E}\left[\|f(P_t,\cdot)-f(\Tilde{P}_{t-1},\cdot)\|_{{\mathcal{F}}'}^2\right]}\\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T {\mathbb{E}\left[{\gamma_{{\mathcal{F}}'}}(Q_t^{\infty},\Tilde{Q}_{t-1}^{\infty})^2\right]},\end{aligned}$$ where $P_t^{\infty},\Tilde{P}_{t-1}^{\infty}, Q_t^{\infty},\Tilde{Q}_{t-1}^{\infty}$ are as defined in Theorem \[thm:oftpl\_noncvx\_regret\]. First, consider the regret of the ${{\ensuremath{\mathbf{x}}}}$ player. We upper bound $\|f(\cdot, Q_t)-f(\cdot,\Tilde{Q}_{t-1})\|^2_{{\mathcal{F}}}$ as $$\begin{aligned} \|f(\cdot, Q_t)-f(\cdot,\Tilde{Q}_{t-1})\|^2_{{\mathcal{F}}} &\leq 3\|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|^2_{{\mathcal{F}}} \\ &\quad + 3\|f(\cdot, Q_t^{\infty})-f(\cdot,\Tilde{Q}_{t-1}^{\infty})\|^2_{{\mathcal{F}}}\\ &\quad + 3\|f(\cdot, \Tilde{Q}_{t-1}^{\infty})-f(\cdot,\Tilde{Q}_{t-1})\|^2_{{\mathcal{F}}}.\end{aligned}$$ We now show that ${\mathbb{E}\left[\|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|^2_{{\mathcal{F}}}|\Tilde{P}_{t-1}, \Tilde{Q}_{t-1},P_{1:t-1},Q_{1:t-1}\right]}$ is $O(1/m)$. To simplify the notation, we let Let ${\mathcal{N}}_{\epsilon}$ be the $\epsilon$-net of ${\mathcal{X}}$ w.r.t $\|\cdot\|$. Then $$\begin{aligned} \|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|_{{\mathcal{F}}} &\stackrel{(a)}{=} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}\|\nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})\|_*\\ & \stackrel{(b)}{\leq} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{N}}_{\epsilon}}\|\nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})\|_* + 2L\epsilon,\end{aligned}$$ where $(a)$ follows from the definition of Lipschitz constant and $(b)$ follows from our smoothness assumption on $f$. Using this, we get $$\begin{aligned} &{\mathbb{E}\left[\|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|^2_{{\mathcal{F}}}|\zeta_t\right]} \leq 2{\mathbb{E}\left[\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{N}}_{\epsilon}}\|\nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})\|_*^2\Big|\zeta_t\right]} + 8L^2\epsilon^2,\end{aligned}$$ Since $f$ is Lipschitz, $\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})\|_*$ is bounded by $G$. So is bounded by $2G$ and is bounded by $2{\Psi_1}G$. Moreover, conditioned on past randomness ($\zeta_t$), $\nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})$ is a sub-Gaussian random vector and satisfies the following bound $$\begin{aligned} {\mathbb{E}\left[{\left\langle {{\ensuremath{\mathbf{u}}}}, \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty}) \right\rangle}|\zeta_t\right]} \leq \exp\left(2{\Psi_1}^2 G^2\|{{\ensuremath{\mathbf{u}}}}\|_2^2/m\right).\end{aligned}$$ From tail bounds of sub-Gaussian random vectors [@hsu2012tail], we have $$\begin{aligned} {\mathbb{P}}\left(\|\nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})\|_2^2 > \frac{4{\Psi_1}^2G^2}{m}(d+2\sqrt{ds} + 2s)\Big|\zeta_t\right) \leq e^{-s},\end{aligned}$$ for any $s>0$. Using union bound, and the fact that $\log|{\mathcal{N}}_{\epsilon}|$ is upper bounded by $d\log\left(1+2D/\epsilon\right)$, we get $$\begin{aligned} {\mathbb{P}}\left(\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{N}}_{\epsilon}}\|\nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})\|_2^2 > \frac{4{\Psi_1}^2G^2}{m}(d+2\sqrt{ds} + 2s)\Big|\zeta_t\right) \leq e^{-s+d\log(1+2D/\epsilon)}.\end{aligned}$$ Let $Z = \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{N}}_{\epsilon}}\|\nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})\|_2^2$. The expectation of $Z$ can be bounded as follows $$\begin{aligned} {\mathbb{E}\left[Z|\zeta_t\right]} &= {\mathbb{P}}(Z \leq a|\zeta_t) {\mathbb{E}\left[Z|\zeta_t, Z \leq a\right]} + {\mathbb{P}}(Z > a|\zeta_t) {\mathbb{E}\left[Z|\zeta_t, Z > a\right]}\\ &\leq a + 4{\Psi_1}^2G^2{\mathbb{P}}(Z > a|\zeta_t).\end{aligned}$$ Choosing $\epsilon=Dm^{-1/2}, s = 3d\log(1+2m^{1/2})$, and $a= \frac{44d{\Psi_1}^2G^2 \log(1+2m^{1/2})}{m}$, we get $${\mathbb{E}\left[Z|\zeta_t\right]} \leq \frac{48d{\Psi_1}^2G^2 \log(1+2m^{1/2})}{m}.$$ This shows that ${\mathbb{E}\left[\|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|^2_{{\mathcal{F}}}|\zeta_t\right]} \leq \frac{96d{\Psi_1}^2{\Psi_2}^2G^2 \log(1+2m^{1/2})}{m} + \frac{8D^2L^2}{m}$. Note that another trivial upper bound for $\|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|_{{\mathcal{F}}}$ is $DL$, which can obtained as follows $$\begin{aligned} \|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|_{{\mathcal{F}}} &=\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} \|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - {\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})\|_*\\ & = \|{\mathbb{E}_{{{\ensuremath{\mathbf{y}}}}_1\sim Q_t,{{\ensuremath{\mathbf{y}}}}_2\sim Q_t^{\infty}}\left[{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_1) - {\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_2)\right]}\|_*\\ &\stackrel{(a)}{\leq} LD,\end{aligned}$$ where $(a)$ follows from the smoothness assumption on $f$ and the fact that the diameter of ${\mathcal{X}}$ is $D$. When $L$ is close to $0$, this bound can be much better than the above bound. So we have $${\mathbb{E}\left[\|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|^2_{{\mathcal{F}}}|\zeta_t\right]} \leq \min\left(\frac{96d{\Psi_1}^2{\Psi_2}^2G^2 \log(1+2m^{1/2})}{m} + \frac{8D^2L^2}{m}, L^2D^2\right).$$ Using this, the regret of the ${{\ensuremath{\mathbf{x}}}}$ player can be bounded as follows $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} {\mathbb{E}\left[\sum_{t=1}^Tf(P_t,Q_t) - f({{\ensuremath{\mathbf{x}}}},Q_t)\right]} &\leq \eta D + \frac{24{C}D^2L^2T}{\eta m}\\ &\quad+\min\left(\frac{288d{C}{\Psi_1}^2{\Psi_2}^2G^2T \log(1+2m^{1/2})}{\eta m}, \frac{3CD^2L^2T}{\eta}\right)\\ &\quad + \sum_{t=1}^T \frac{3{C}}{2\eta}{\mathbb{E}\left[\|f(\cdot,Q_t^{\infty})-f(\cdot,\Tilde{Q}_{t-1}^{\infty})\|_{{\mathcal{F}}}^2\right]}\\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T {\mathbb{E}\left[{\gamma_{{\mathcal{F}}}}(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})^2\right]}.\end{aligned}$$ A similar analysis shows that the regret of ${{\ensuremath{\mathbf{y}}}}$ player can be bounded as $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} {\mathbb{E}\left[\sum_{t=1}^T f(P_t,{{\ensuremath{\mathbf{y}}}})- f(P_t,Q_t)\right]} &\leq \eta D + \frac{24{C}D^2L^2T}{\eta m}\\ &\quad+\min\left(\frac{288d{C}{\Psi_1}^2{\Psi_2}^2G^2T \log(1+2m^{1/2})}{\eta m}, \frac{3CD^2L^2T}{\eta}\right)\\ &\quad + \sum_{t=1}^T \frac{3{C}}{2\eta}{\mathbb{E}\left[\|f(P_t^{\infty},\cdot)-f(\Tilde{P}_{t-1}^{\infty},\cdot)\|_{{\mathcal{F}}'}^2\right]}\\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T {\mathbb{E}\left[{\gamma_{{\mathcal{F}}'}}(Q_t^{\infty},\Tilde{Q}_{t-1}^{\infty})^2\right]},\end{aligned}$$ Summing the above two inequalities, we get $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} {\mathbb{E}\left[\sum_{t=1}^T f(P_t,{{\ensuremath{\mathbf{y}}}})- f(P,Q_t)\right]}&\leq 2\eta D + \frac{48{C}D^2L^2T}{\eta m}\\ &\quad+\min\left(\frac{576d{C}{\Psi_1}^2{\Psi_2}^2G^2T \log(1+2m^{1/2})}{\eta m}, \frac{6CD^2L^2T}{\eta}\right)\\ &\quad + \sum_{t=1}^T \frac{3{C}}{2\eta}{\mathbb{E}\left[\|f(\cdot,Q_t^{\infty})-f(\cdot,\Tilde{Q}_{t-1}^{\infty})\|_{{\mathcal{F}}}^2\right]}\\ &\quad + \sum_{t=1}^T \frac{3{C}}{2\eta}{\mathbb{E}\left[\|f(P_t^{\infty},\cdot)-f(\Tilde{P}_{t-1}^{\infty},\cdot)\|_{{\mathcal{F}}'}^2\right]}\\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T \left({\mathbb{E}\left[{\gamma_{{\mathcal{F}}}}(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})^2\right]} + {\mathbb{E}\left[{\gamma_{{\mathcal{F}}'}}(Q_t^{\infty},\Tilde{Q}_{t-1}^{\infty})^2\right]}\right).\end{aligned}$$ From our assumption on smoothness of $f$, we have $$\|f(\cdot,Q_t^{\infty})-f(\cdot,\Tilde{Q}_{t-1}^{\infty})\|_{{\mathcal{F}}} \leq L {\gamma_{{\mathcal{F}}'}}(Q_t^{\infty},\Tilde{Q}_{t-1}^{\infty}),\quad \|f(P_t^{\infty},\cdot)-f(\Tilde{P}_{t-1}^{\infty},\cdot)\|_{{\mathcal{F}}'} \leq L {\gamma_{{\mathcal{F}}}}(P_t^{\infty},\Tilde{P}_{t-1}^{\infty}).$$ To see this, consider the following $$\begin{aligned} \|f(\cdot,Q_t^{\infty})-f(\cdot,\Tilde{Q}_{t-1}^{\infty})\|_{{\mathcal{F}}}&=\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},\Tilde{Q}_{t-1}^{\infty})\|_{*}\\ &= \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}, \|{{\ensuremath{\mathbf{u}}}}\|\leq 1} {\left\langle {{\ensuremath{\mathbf{u}}}}, {\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},\Tilde{Q}_{t-1}^{\infty}) \right\rangle}\\ &= \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}, \|{{\ensuremath{\mathbf{u}}}}\|\leq 1} {\mathbb{E}_{{{\ensuremath{\mathbf{y}}}}\sim Q_t^{\infty}}\left[{\left\langle {{\ensuremath{\mathbf{u}}}}, {\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}) \right\rangle}\right]}-{\mathbb{E}_{{{\ensuremath{\mathbf{y}}}}\sim \Tilde{Q}_{t-1}^{\infty}}\left[{\left\langle {{\ensuremath{\mathbf{u}}}}, {\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}) \right\rangle}\right]}\\ &\leq {\gamma_{{\mathcal{F}}'}}(Q_t^{\infty}, \Tilde{Q}_{t-1}^{\infty})\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}, \|{{\ensuremath{\mathbf{u}}}}\|\leq 1}\|{\left\langle {{\ensuremath{\mathbf{u}}}}, {\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},\cdot) \right\rangle}\|_{{\mathcal{F}}'}\\ &= {\gamma_{{\mathcal{F}}'}}(Q_t^{\infty}, \Tilde{Q}_{t-1}^{\infty})\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}, \|{{\ensuremath{\mathbf{u}}}}\|\leq 1}\left(\sup_{{{\ensuremath{\mathbf{y}}}}_1\neq {{\ensuremath{\mathbf{y}}}}_2 \in {\mathcal{Y}}}\frac{|{\left\langle {{\ensuremath{\mathbf{u}}}}, {\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_1) \right\rangle}-{\left\langle {{\ensuremath{\mathbf{u}}}}, {\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_2) \right\rangle}|}{\|{{\ensuremath{\mathbf{y}}}}_1-{{\ensuremath{\mathbf{y}}}}_2\|}\right)\\ &\leq {\gamma_{{\mathcal{F}}'}}(Q_t^{\infty}, \Tilde{Q}_{t-1}^{\infty})\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}\left(\sup_{{{\ensuremath{\mathbf{y}}}}_1\neq {{\ensuremath{\mathbf{y}}}}_2 \in {\mathcal{Y}}}\frac{\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_1)-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_2)\|_{*}}{\|{{\ensuremath{\mathbf{y}}}}_1-{{\ensuremath{\mathbf{y}}}}_2\|}\right)\\ &\stackrel{(a)}{\leq} L {\gamma_{{\mathcal{F}}'}}(Q_t^{\infty}, \Tilde{Q}_{t-1}^{\infty}),\end{aligned}$$ where $(a)$ follows from smoothness of $f$. Substituting this in the previous equation, and choosing $\eta > \sqrt{3}{C}L$, we get $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} {\mathbb{E}\left[\sum_{t=1}^T f(P_t,{{\ensuremath{\mathbf{y}}}})- f(P,Q_t)\right]}&\leq 2\eta D + \frac{48{C}D^2L^2T}{\eta m}\\ &\quad+\min\left(\frac{576d{C}{\Psi_1}^2{\Psi_2}^2G^2T \log(1+2m^{1/2})}{\eta m}, \frac{6CD^2L^2T}{\eta}\right)\\\end{aligned}$$ This finishes the proof of the Theorem. We note that a similar result can be obtained for other choice of function classes such as the set of all bounded and Lipschitz functions. The only difference between proving such a result vs. proving Theorem \[thm:oftpl\_noncvx\_smooth\_games\] is in bounding $\|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|_{{\mathcal{F}}}$. Proof of Theorem \[thm:oftpl\_noncvx\_smooth\_games\_exp\] ---------------------------------------------------------- To prove the Theorem, we instantiate Theorem \[thm:oftpl\_noncvx\_smooth\_games\] for exponential noise distribution. Recall, in Corollary \[cor:ftpl\_noncvx\_exp\], we showed that ${\mathbb{E}_{\sigma}\left[\|\sigma\|_{{\mathcal{F}}}\right]} = \eta\log{d}$ and OFTPL is ${O\left({d^2D\eta^{-1}}\right)}$ stable w.r.t $\|\cdot\|_{{\mathcal{F}}}$, for this choice of perturbation distribution (similar results hold for $({\mathcal{F}}',\|\cdot\|_{{\mathcal{F}}'})$). Substituting this in the bounds of Theorem \[thm:oftpl\_noncvx\_smooth\_games\] and using the fact that ${\Psi_1}=\sqrt{d},{\Psi_2}= 1$, we get $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[f\left(\frac{1}{T}\sum_{t=1}^TP_t,{{\ensuremath{\mathbf{y}}}}\right) - f\left({{\ensuremath{\mathbf{x}}}},\frac{1}{T}\sum_{t=1}^TQ_t\right)\right]} &= {O\left({\frac{\eta D\log{d}}{T} + \frac{d^2 D^3L^2}{\eta m}}\right)}\\ &\quad +{O\left({\min\left\lbrace\frac{d^4DG^2 \log(2m)}{\eta m}, \frac{d^2D^3L^2}{\eta}\right\rbrace}\right)}.\end{aligned}$$ Choosing $\eta = 10d^2D(L+1), m=T$, we get $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[f\left(\frac{1}{T}\sum_{t=1}^TP_t,{{\ensuremath{\mathbf{y}}}}\right) - f\left({{\ensuremath{\mathbf{x}}}},\frac{1}{T}\sum_{t=1}^TQ_t\right)\right]} &= {O\left({\frac{d^2 D^2(L+1)\log{d}}{T}}\right)}\\ &\quad +{O\left({\min\left\lbrace\frac{d^2G^2 \log(T)}{ LT}, D^2L\right\rbrace}\right)}.\end{aligned}$$ Choice of Perturbation Distributions {#sec:pert_dist_choice} ==================================== #### Regularization of some Perturbation Distributions. We first study the regularization effect of various perturbation distributions. Table \[tab:reg\_linf\] presents the regularizer $R$ corresponding to some commonly used perturbation distributions, when the action space ${\mathcal{X}}$ is $\ell_{\infty}$ ball of radius $1$ centered at origin. ------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------- Perturbation Distribution ${P_{\text{PRTB}}}$ Regularizer \[0.5ex\] Uniform over $[0,\eta]^d$ $\eta \|{{\ensuremath{\mathbf{x}}}}-1\|_2^2$ Exponential $P(\sigma > t)=\exp(-t/\eta)$ $\displaystyle\sum_i\eta({{\ensuremath{\mathbf{x}}}}_i+1)\left[\log({{\ensuremath{\mathbf{x}}}}_i+1) - (1+\log 2)\right]$ Gaussian $P(\sigma =t)\propto e^{-t^2/2\eta^2}$ $\displaystyle\sum_i \sup_{u\in\mathbb{R}} u\left[{{\ensuremath{\mathbf{x}}}}_i-1+2F(-u/\eta)\right]$ ------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------- : Regularizers corresponding to various perturbation distributions used in FTPL when the action space ${\mathcal{X}}$ is $\ell_{\infty}$ ball of radius $1$ centered at origin. Here, $F$ is the CDF of a standard normal random variable.[]{data-label="tab:reg_linf"} #### Dimension independent rates. Recall, the OFTPL algorithm described in Algorithm \[alg:oftpl\_cvx\_games\] converges at ${O\left({d/T}\right)}$ rate to a Nash equilibrium of smooth convex-concave games (see Theorem \[thm:oftpl\_cvx\_smooth\_games\_uniform\]). We now show that for certain constraint sets ${\mathcal{X}}, {\mathcal{Y}}$, by choosing the perturbation distributions appropriately, the dimension dependence in the rates can *potentially* be removed. Suppose the action set is ${\mathcal{X}}=\{{{\ensuremath{\mathbf{x}}}}:\|{{\ensuremath{\mathbf{x}}}}\|_2 \leq 1\}$. Suppose the perturbation distribution ${P_{\text{PRTB}}}$ is the multivariate Gaussian distribution with mean $0$ and covariance $\eta^{2}I_{d\times d}$, where $I_{d\times d}$ is the identity matrix. We now try to explicitly compute the reguralizer corresponding to this perturbation distribution and action set. Define function $\Psi$ as $$\Psi(f) = {\mathbb{E}_{\sigma}\left[\max_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} {\left\langle f+\sigma, {{\ensuremath{\mathbf{x}}}}\right\rangle}\right]}= {\mathbb{E}_{\sigma}\left[\|f+\sigma\|_2\right]}.$$ As shown in Proposition \[prop:ftpl\_ftrl\_connection\], the regularizer $R$ corresponding to any perturbation distribution is given by the Fenchel conjugate of $\Psi$ $$R({{\ensuremath{\mathbf{x}}}}) = \sup_{f}{\left\langle f, {{\ensuremath{\mathbf{x}}}}\right\rangle} - \Psi(f).$$ Since getting an exact expression for $R$ is a non-trivial task, we only compute an *approximate expression* for $R$. Consider the high dimensional setting (*i.e.,* very large $d$). In this setting, $\|f+\sigma\|_2$, for $\sigma$ drawn from ${\mathcal{N}}(0,\eta^2I_{d\times d})$, can be approximated as follows $$\begin{aligned} \|f+\sigma\|_2 &= \sqrt{\|f\|_2^2 + \|\sigma\|_2^2 + 2{\left\langle f, \sigma \right\rangle}}\\ &\stackrel{(a)}{\approx} \sqrt{\|f\|_2^2 + \eta^2d + 2{\left\langle f, \sigma \right\rangle}}\\ &\stackrel{(b)}{\approx} \sqrt{\|f\|_2^2 + \eta^2d}\end{aligned}$$ where $(a)$ follows from the fact that $\|\sigma\|_2^2$ is highly concentrated around $\eta^2d$ [@hsu2012tail]. To be precise $$\mathbb{P}(\|\sigma\|_2^2 \geq \eta^2(d + 2\sqrt{dt} + 2t)) \leq e^{-t}.$$ A similar bound holds for the lower tail. Approximation $(b)$ follows from the fact that ${\left\langle f, \sigma \right\rangle}$ is a Gaussian random variable with mean $0$ and variance $\eta^2\|f\|_2^2$, and with high probability its magnitude is upper bounded by $\Tilde{O}(\eta\|f\|_2)$. Since $\eta\|f\|_2 \ll \sqrt{d}\eta\|f\|_2 \leq \|f\|_2^2 + \eta^2d$, approximation $(b)$ holds. This shows that $\Psi(f)$ can be approximated as $$\Psi(f) \approx \sqrt{\|f\|_2^2 + \eta^2d}.$$ Using this approximation, we now compute the reguralizer corresponding to the perturbation distribution $$R({{\ensuremath{\mathbf{x}}}}) = \sup_{f}{\left\langle f, {{\ensuremath{\mathbf{x}}}}\right\rangle} - \Psi(f) \approx \sup_{f}{\left\langle f, {{\ensuremath{\mathbf{x}}}}\right\rangle} - \sqrt{\|f\|_2^2 + \eta^2d} = -\eta\sqrt{d}\sqrt{1-\|{{\ensuremath{\mathbf{x}}}}\|_2^2}.$$ This shows that $R$ is $\eta\sqrt{d}$-strongly convex w.r.t $\|\cdot\|_2$ norm. Following duality between strong convexity and strong smoothness, $\Psi(f)$ is $(\eta^2d)^{-1/2}$ strongly smooth w.r.t $\|\cdot\|_2$ norm and satisfies $$\|{\nabla}\Psi(f_1)-{\nabla}\Psi(f_2)\|_2 \leq (\eta^2d)^{-1/2}\|f_1-f_2\|_2.$$ This shows that the predictions of OFTPL are $(\eta^2d)^{-1/2}$ stable w.r.t $\|\cdot\|_2$ norm. We now instantiate Theorem \[thm:oftpl\_cvx\_smooth\_games\] for this perturbation distribution and for constraint sets which are unit balls centered at origin, and use the above stability bound, together with the fact that ${\mathbb{E}_{\sigma}\left[\|\sigma\|_2\right]} \approx \eta\sqrt{d}$. Suppose $f$ is smooth w.r.t $\|\cdot\|_2$ norm and satisfies $$\begin{aligned} \|{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{2}+ \|{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{2} \leq L\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}'\|_2 + L\|{{\ensuremath{\mathbf{y}}}}-{{\ensuremath{\mathbf{y}}}}'\|_2.\end{aligned}$$ Then Theorem \[thm:oftpl\_cvx\_smooth\_games\] gives us the following rates of convergence to a NE $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}{\mathbb{E}\left[f\left(\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}\right) - f\left({{\ensuremath{\mathbf{x}}}},\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{y}}}}_t\right)\right]}\leq & \frac{ 2L_1}{m}+\frac{2\eta\sqrt{d}}{T} \\ &+ \frac{20 L^2}{\eta\sqrt{d}} \left(\frac{ 1}{m}\right)+10L\left(\frac{5 L}{\eta\sqrt{d}}\right)^{\infty}\end{aligned}$$ Choosing $\eta = 6L/\sqrt{d}, m=T$, we get ${O\left({\frac{L}{T}}\right)}$ rate of convergence. Although, these rates are dimension independent, we note that our stability bound is only approximate. More accurate analysis is needed to actually claim that Algorithm \[alg:oftpl\_cvx\_games\] achieves dimension independent rates in this setting. That being said, for general constraints sets, we believe one can get dimension independent rates by choosing the perturbation distribution appropriately. High Probability Bounds {#sec:hp_bounds} ======================= In this section, we provide high probability bounds for Theorems \[thm:oftpl\_regret\], \[thm:oftpl\_cvx\_smooth\_games\_uniform\]. Our results rely on the following concentration inequalities. \[prop:azuma\] Let $X_1,\dots X_K$ be $K$ independent mean $0$ vector-valued random variables such that $\|X_i\|_2\leq B_i$. Then $$\mathbb{P}\left({\|{\sum_{i=1}^KX_i} \|}_2 \geq t\right) \leq 2\exp\left(-c\frac{t^2}{\sum_{i=1}^KB_i^2}\right),$$ where $c>0$ is a universal constant. We also need the following concentration inequality for martingales. \[prop:martingale\_diff\] Let $X_1,\dots X_K \in \mathbb{R}$ be a martingale difference sequence, where ${\mathbb{E}\left[X_i|{\mathcal{F}}_{i-1}\right]} = 0$. Assume that $X_i$ satisfy the following tail condition, for some scalar $B_i>0$ $$\mathbb{P}\left(\Big|\frac{X_i}{B_i}\Big| \geq z\Big| {\mathcal{F}}_{i-1}\right) \leq 2\exp(-z^2).$$ Then $$\mathbb{P}\left(\Big|\sum_{i=1}^K X_i\Big| \geq z\right)\leq 2\exp\left(-c\frac{z^2}{\sum_{i=1}^KB_i^2}\right),$$ where $c>0$ is a universal constant. Online Convex Learning ---------------------- In this section, we present a high probability version of Theorem \[thm:oftpl\_regret\]. \[thm:oftpl\_regret\_hp\] Suppose the perturbation distribution ${P_{\text{PRTB}}}$ is absolutely continuous w.r.t Lebesgue measure. Let $D$ be the diameter of ${\mathcal{X}}$ w.r.t $\|\cdot\|$, which is defined as $D= \sup_{{{\ensuremath{\mathbf{x}}}}_1,{{\ensuremath{\mathbf{x}}}}_2\in{\mathcal{X}}} \|{{\ensuremath{\mathbf{x}}}}_1-{{\ensuremath{\mathbf{x}}}}_2\|.$ Let and suppose the predictions of OFTPL are ${C}\eta^{-1}$-stable w.r.t $\|\cdot\|_*$, where ${C}$ is a constant that depends on the set $\mathcal{X}.$ Suppose, the sequence of loss functions $\{f_t\}_{t=1}^T$ are $G$-Lipschitz w.r.t $\|\cdot\|$ and satisfy $\sup_{{{\ensuremath{\mathbf{x}}}}\in \mathcal{X}} \|{\nabla}f_t({{\ensuremath{\mathbf{x}}}})\|_* \leq G$. Moreover, suppose $\{f_t\}_{t=1}^T$ are Holder smooth and satisfy $$\begin{aligned} \forall {{\ensuremath{\mathbf{x}}}}_1,{{\ensuremath{\mathbf{x}}}}_2\in{\mathcal{X}}\quad \|{\nabla}f_t({{\ensuremath{\mathbf{x}}}}_1)-{\nabla}f_t({{\ensuremath{\mathbf{x}}}}_2)\|_* \leq L\|{{\ensuremath{\mathbf{x}}}}_1-{{\ensuremath{\mathbf{x}}}}_2\|^{\alpha},\end{aligned}$$ for some constant $\alpha \in [0,1]$. Then the regret of Algorithm \[alg:oftpl\_cvx\] satisfies the following with probability at least $1-\delta$ $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}}\sum_{t=1}^Tf_t({{\ensuremath{\mathbf{x}}}}_t) - f_t({{\ensuremath{\mathbf{x}}}}) &\leq \eta D + \sum_{t=1}^T\frac{{C}}{2\eta} \|{\nabla}_t-g_{t}\|_{*}^2-\sum_{t=1}^T \frac{\eta}{2{C}}\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2\\ &\quad+cGD\sqrt{\frac{T\log{2/\delta}}{m}} + cLT\left(\frac{{\Psi_1}^2 {\Psi_2}^2 D^2\log{4T/\delta}}{m}\right)^{\frac{1+\alpha}{2}},\end{aligned}$$ where $c$ is a universal constant, ${{\ensuremath{\mathbf{x}}}}_t^{\infty} = {\mathbb{E}\left[{{\ensuremath{\mathbf{x}}}}_t|g_t,f_{1:t-1}, {{\ensuremath{\mathbf{x}}}}_{1:t-1}\right]}$ and $\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty} = {\mathbb{E}\left[\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}|f_{1:t-1}, {{\ensuremath{\mathbf{x}}}}_{1:t-1}\right]}$ and $\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}$ denotes the prediction in the $t^{th}$ iteration of Algorithm \[alg:oftpl\_cvx\], if guess $g_{t}=0$ was used. Here, ${\Psi_1}, {\Psi_2}$ denote the norm compatibility constants of $\|\cdot\|.$ Our proof uses the same notation and similar arguments as in the proof Theorem \[thm:oftpl\_regret\]. Recall, in Theorem \[thm:oftpl\_regret\] we showed that the regret of OFTPL is upper bounded by $$\begin{aligned} \sum_{t=1}^Tf_t({{\ensuremath{\mathbf{x}}}}_t) - f_t({{\ensuremath{\mathbf{x}}}}) &\leq \sum_{t=1}^T{\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t \right\rangle} + \eta D + \sum_{t=1}^T \|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}\|\|{\nabla}_t-g_{t}\|_{*}\\ &\quad -\frac{\eta}{2{C}}\sum_{t=1}^T\left(\|\Tilde{{{\ensuremath{\mathbf{x}}}}}_t^{\infty}-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^2 + \|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2\right)\\ &\leq \sum_{t=1}^T{\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t \right\rangle} + \eta D + \sum_{t=1}^T\frac{{C}}{2\eta} \|{\nabla}_t-g_{t}\|_{*}^2-\sum_{t=1}^T \frac{\eta}{2{C}}\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2.\end{aligned}$$ From Holder’s smoothness assumption, we have $${\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}_t - {\nabla}f_t({{\ensuremath{\mathbf{x}}}}_t^{\infty}) \right\rangle} \leq L\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^{1+\alpha}.$$ Substituting this in the previous bound gives us $$\begin{aligned} \sum_{t=1}^Tf_t({{\ensuremath{\mathbf{x}}}}_t) - f_t({{\ensuremath{\mathbf{x}}}}) &\leq \underbrace{\sum_{t=1}^T{\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}f_t({{\ensuremath{\mathbf{x}}}}_t^{\infty}) \right\rangle}}_{S_1}+\sum_{t=1}^TL\underbrace{\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^{1+\alpha}}_{S_2} + \eta D \\ &\quad + \sum_{t=1}^T\frac{{C}}{2\eta} \|{\nabla}_t-g_{t}\|_{*}^2-\sum_{t=1}^T \frac{\eta}{2{C}}\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2.\end{aligned}$$ We now provide high probability bounds for $S_1$ and $S_2$. #### Bounding $S_1$. Let $\xi_i = \{g_{i+1}, f_{i+1}, {{\ensuremath{\mathbf{x}}}}_{i}\}$ and let $\xi_{0:t}$ denote the union of sets $\xi_0,\xi_1,\dots, \xi_t$. Let $\zeta_t = {\left\langle {{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}, {\nabla}f_t({{\ensuremath{\mathbf{x}}}}_t^{\infty}) \right\rangle}$ with $\zeta_0=0$. Note that $\{\zeta_t\}_{t=0}^T$ is a martingale difference sequence w.r.t $\xi_{0:T}$. This is because ${\mathbb{E}\left[{{\ensuremath{\mathbf{x}}}}_t|\xi_{0:t-1}\right]} = {{\ensuremath{\mathbf{x}}}}_t^{\infty}$ and ${\nabla}f_t({{\ensuremath{\mathbf{x}}}}_t^{\infty})$ is a deterministic quantity conditioned on $\xi_{0:t-1}$. As a result ${\mathbb{E}\left[\zeta_t|\xi_{0:t-1}\right]}=0$. Moreover, conditioned on $\xi_{0:t-1}$, $\zeta_t$ is the average of $m$ independent mean $0$ random variables, each of which is bounded by $GD$. Using Proposition \[prop:azuma\], we get $$\mathbb{P}\left(|\zeta_t| \geq s\Big| \xi_{0:t-1}\right) \leq 2\exp\left(-\frac{ms^2}{G^2D^2}\right).$$ Using Proposition \[prop:martingale\_diff\] on the martingale difference sequence $\{\zeta_t\}_{t=0}^T$, we get $$\mathbb{P}\left(\Big|\sum_{t=1}^T\zeta_t\Big| \geq s\right)\leq 2\exp\left(-c\frac{ms^2}{G^2D^2T}\right),$$ where $c>0$ is a universal constant. This shows that with probability at least $1-\delta/2$, $S_1$ is upper bounded by $ {O\left({\sqrt{\frac{G^2D^2T\log{\frac{2}{\delta}}}{m}}}\right)}.$ #### Bounding $S_2$. Conditioned on $\{g_t,f_{1:t-1}, {{\ensuremath{\mathbf{x}}}}_{1:t-1}\}$, ${{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}$ is the average of $m$ independent mean $0$ random variables which are bounded by $D$ in $\|\cdot\|$ norm. From our definition of norm compatibility constant ${\Psi_2}$, this implies the random variables are bounded by ${\Psi_2}D$ in $\|\cdot\|_2$. Using Proposition \[prop:azuma\], we get $$\mathbb{P}\left(\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|_2 \geq {\Psi_2}D\sqrt{\frac{c\log{4T/\delta}}{m}}\Bigg| g_t,f_{1:t-1}, {{\ensuremath{\mathbf{x}}}}_{1:t-1}\right) \leq \frac{\delta}{2T}.$$ Since the above bound holds for any set of $\{g_t,f_{1:t}, {{\ensuremath{\mathbf{x}}}}_{1:t-1}\}$, the same tail bound also holds without the conditioning. This shows that $$\mathbb{P}\left(\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|^{1+\alpha} \geq \left(\frac{c{\Psi_1}^2 {\Psi_2}^2 D^2\log{4T/\delta}}{m}\right)^{\frac{1+\alpha}{2}}\right) \leq \frac{\delta}{2T},$$ where we converted back to $\|\cdot\|$ by introducing the norm compatibility constant ${\Psi_1}$. #### Bounding the regret. Plugging the above high probability bounds for $S_1,S_2$ in the previous regret bound and using union bound, we get the following regret bound which holds with probability at least $1-\delta$ $$\begin{aligned} \sum_{t=1}^Tf_t({{\ensuremath{\mathbf{x}}}}_t) - f_t({{\ensuremath{\mathbf{x}}}}) &\leq cGD\sqrt{\frac{T\log{2/\delta}}{m}} + cLT\left(\frac{{\Psi_1}^2 {\Psi_2}^2 D^2\log{4T/\delta}}{m}\right)^{\frac{1+\alpha}{2}} + \eta D \\ &\quad + \sum_{t=1}^T\frac{{C}}{2\eta} \|{\nabla}_t-g_{t}\|_{*}^2-\sum_{t=1}^T \frac{\eta}{2{C}}\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|^2,\end{aligned}$$ where $c>0$ is a universal constant. Convex-Concave Games {#convex-concave-games} -------------------- In this section, we present a high probability version of Theorem \[thm:oftpl\_cvx\_smooth\_games\_uniform\]. \[thm:oftpl\_cvx\_smooth\_games\_uniform\_hp\] Consider the minimax game in Equation . Suppose both the domains ${\mathcal{X}},{\mathcal{Y}}$ are compact subsets of $\mathbb{R}^d$, with diameter Suppose $f$ is convex in ${{\ensuremath{\mathbf{x}}}}$, concave in ${{\ensuremath{\mathbf{y}}}}$ and is Lipschitz w.r.t $\|\cdot\|_2$ and satisfies $$\begin{aligned} \max\left\lbrace\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}, {{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} \|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})\|_{2}, \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}\|{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})\|_{2}\right\rbrace\leq G.\end{aligned}$$ Moreover, suppose $f$ is smooth w.r.t $\|\cdot\|_2$ $$\begin{aligned} \|{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{2}+ \|{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{2} \leq L\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}'\|_2 + L\|{{\ensuremath{\mathbf{y}}}}-{{\ensuremath{\mathbf{y}}}}'\|_2.\end{aligned}$$ Suppose Algorithm \[alg:oftpl\_cvx\_games\] is used to solve the minimax game. Suppose the perturbation distributions used by both the players are the same and equal to the uniform distribution over $\{{{\ensuremath{\mathbf{x}}}}:\|{{\ensuremath{\mathbf{x}}}}\|_2 \leq (1+d^{-1})\eta\}.$ Suppose the guesses used by ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players in the $t^{th}$ iteration are ${\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}), {\nabla}_{{{\ensuremath{\mathbf{y}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})$, where $\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}$ denote the predictions of ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players in the $t^{th}$ iteration, if guess $g_t = 0$ was used. If Algorithm \[alg:oftpl\_cvx\_games\] is run with $\eta = 6dD(L+1), m = T$, then the iterates $\{({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)\}_{t=1}^T$ satisfy the following bound with probability at least $1-\delta$ $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}\left[f\left(\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}\right) - f\left({{\ensuremath{\mathbf{x}}}},\frac{1}{T}\sum_{t=1}^T{{\ensuremath{\mathbf{y}}}}_t\right)\right]= {O\left({\frac{GD\sqrt{\log{\frac{8}{\delta}}}}{T}+ \frac{D^2(L+1)\left(d + \log{\frac{16T}{\delta}}\right)}{T}}\right)}.\end{aligned}$$ We use the same notation and proof technique as Theorems \[thm:oftpl\_cvx\_smooth\_games\], \[thm:oftpl\_cvx\_smooth\_games\_uniform\]. From Theorem \[cor:ftpl\_cvx\_gaussian\] we know that the predictions of OFTPL are $dD\eta^{-1}$ stable w.r.t $\|\cdot\|_2$, for the particular perturbation distribution we consider here. We use this stability bound in our proof. From Theorem \[thm:oftpl\_regret\_hp\], we have the following regret bound for both the players, which holds with probability at least $1-\delta/2$ $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} \left[\sum_{t=1}^Tf({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t) - f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_t)\right] &\leq cGD\sqrt{\frac{T\log{8/\delta}}{m}} + cLT\left(\frac{D^2\log{16T/\delta}}{m}\right) + \eta D \\ &\quad+ \frac{dD }{2\eta}\sum_{t=1}^T \left[\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})\|_2^2\right] \\ &\quad -\frac{\eta}{2dD }\sum_{t=1}^T \left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|_2^2\right].\end{aligned}$$ $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} \left[\sum_{t=1}^T f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}})- f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)\right] &\leq cGD\sqrt{\frac{T\log{8/\delta}}{m}} + cLT\left(\frac{D^2\log{16T/\delta}}{m}\right) + \eta D \\ &\quad+ \frac{dD }{2\eta}\sum_{t=1}^T \left[\|{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)-{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})\|_{2}^2\right] \\ &\quad -\frac{\eta}{2dD }\sum_{t=1}^T \left[\|{{\ensuremath{\mathbf{y}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|_2^2\right].\end{aligned}$$ First, consider the regret of the ${{\ensuremath{\mathbf{x}}}}$ player. From the proof of Theorem \[thm:oftpl\_cvx\_smooth\_games\], we have $$\begin{aligned} \|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1})\|_{2}^2 &\leq 5L^2\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|_2^{2}+5L^2\|\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|_2^{2}\\ &\quad + 5L^2\|{{\ensuremath{\mathbf{y}}}}_t-{{\ensuremath{\mathbf{y}}}}_t^{\infty}\|_2^{2}+5L^2\|\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|_2^{2}\\ &\quad + 5\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{2}^2.\end{aligned}$$ Moreover, from the proof of Theorem \[thm:oftpl\_regret\_hp\], we know that $\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|_2^{2}$ satisfies the following tail bound $$\mathbb{P}\left(\|{{\ensuremath{\mathbf{x}}}}_t-{{\ensuremath{\mathbf{x}}}}_t^{\infty}\|_2^{2} \geq \frac{c D^2\log{16T/\delta}}{m}\right) \leq \frac{\delta}{8T}.$$ Similar bounds hold for the quantities appearing in the regret bound of ${{\ensuremath{\mathbf{y}}}}$ player. Plugging this in the previous regret bounds, we get the following which hold with probability at least $1-\delta$ $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} \left[\sum_{t=1}^Tf({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t) - f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_t)\right] &\leq cGD\sqrt{\frac{T\log{8/\delta}}{m}} + \left(L+\frac{10dDL^2}{\eta}\right)\left(\frac{cD^2\log{16T/\delta}}{m}\right)T \\ &\quad + \eta D+ \frac{5dD }{2\eta}\sum_{t=1}^T \left[\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_2^2\right] \\ &\quad -\frac{\eta}{2dD }\sum_{t=1}^T \left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|_2^2\right].\end{aligned}$$ $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} \left[\sum_{t=1}^T f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}})- f({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}_t)\right] &\leq cGD\sqrt{\frac{T\log{8/\delta}}{m}} + \left(L+\frac{10dDL^2}{\eta}\right)\left(\frac{cD^2\log{16T/\delta}}{m}\right)T \\ &\quad+ \eta D + \frac{5dD }{2\eta}\sum_{t=1}^T \left[\|{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{2}^2\right] \\ &\quad -\frac{\eta}{2dD }\sum_{t=1}^T \left[\|{{\ensuremath{\mathbf{y}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|_2^2\right].\end{aligned}$$ Summing these two regret bounds, we get $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} \left[\sum_{t=1}^Tf({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}) - f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_t)\right] &\leq 2cGD\sqrt{\frac{T\log{8/\delta}}{m}} + \left(L+\frac{10dDL^2}{\eta}\right)\left(\frac{2cD^2\log{16T/\delta}}{m}\right)T + 2\eta D \\ &\quad+ \frac{10dD }{2\eta}\sum_{t=1}^T \left[\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_2^2\right] \\ &\quad+ \frac{10dD }{2\eta}\sum_{t=1}^T \left[\|{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{2}^2\right] \\ &\quad -\frac{\eta}{2dD }\sum_{t=1}^T \left[\|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|_2^2+\|{{\ensuremath{\mathbf{y}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|_2^2\right].\end{aligned}$$ From Holder’s smoothness assumption on $f$, we have $$\begin{aligned} \|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{2}^2 & \leq 2\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{2}^2\\ &\quad + 2\|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{2}^2\\ &\leq 2L^2 \|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|_2^{2} + 2L^2\|{{\ensuremath{\mathbf{y}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|_2^{2},\end{aligned}$$ Using a similar argument, we get $$\begin{aligned} \|{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}}_t^{\infty},{{\ensuremath{\mathbf{y}}}}_t^{\infty})-{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f(\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty},\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty})\|_{2}^2 \leq 2L^2 \|{{\ensuremath{\mathbf{x}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{x}}}}}_{t-1}^{\infty}\|_2^{2} + 2L^2\|{{\ensuremath{\mathbf{y}}}}_t^{\infty}-\Tilde{{{\ensuremath{\mathbf{y}}}}}_{t-1}^{\infty}\|_2^{2}.\end{aligned}$$ Plugging this in the previous bound, and setting $\eta = 6d D(L+1), m=T$, we get the following bound which holds with probability at least $1-\delta$ $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} \left[\sum_{t=1}^Tf({{\ensuremath{\mathbf{x}}}}_t,{{\ensuremath{\mathbf{y}}}}) - f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_t)\right] &\leq {O\left({GD\sqrt{\log{\frac{8}{\delta}}}+ D^2(L+1)\left(d + \log{\frac{16T}{\delta}}\right)}\right)}.\end{aligned}$$ Nonconvex-Nonconcave Games {#nonconvex-nonconcave-games} -------------------------- In this section, we present a high probability version of Theorem \[thm:oftpl\_noncvx\_smooth\_games\_exp\]. \[thm:oftpl\_noncvx\_smooth\_games\_exp\_hp\] Consider the minimax game in Equation . Suppose the domains ${\mathcal{X}},{\mathcal{Y}}$ are compact subsets of $\mathbb{R}^d$ with diameter $D = \max\{\sup_{{{\ensuremath{\mathbf{x}}}}_1,{{\ensuremath{\mathbf{x}}}}_2\in{\mathcal{X}}} \|{{\ensuremath{\mathbf{x}}}}_1-{{\ensuremath{\mathbf{x}}}}_2\|_1, \sup_{{{\ensuremath{\mathbf{y}}}}_1,{{\ensuremath{\mathbf{y}}}}_2\in{\mathcal{Y}}} \|{{\ensuremath{\mathbf{y}}}}_1-{{\ensuremath{\mathbf{y}}}}_2\|_1\}$. Suppose $f$ is Lipschitz w.r.t $\|\cdot\|_1$ and satisfies $$\begin{aligned} \max\left\lbrace\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}, {{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}} \|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})\|_{\infty}, \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}\|{\nabla}_{{{\ensuremath{\mathbf{y}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})\|_{\infty}\right\rbrace\leq G.\end{aligned}$$ Moreover, suppose $f$ satisfies the following smoothness property $$\begin{aligned} \|{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{x}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{\infty} + \|{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}})-{\nabla}_{{\ensuremath{\mathbf{y}}}}f({{\ensuremath{\mathbf{x}}}}',{{\ensuremath{\mathbf{y}}}}')\|_{\infty} \leq L\|{{\ensuremath{\mathbf{x}}}}-{{\ensuremath{\mathbf{x}}}}'\|_1 + L\|{{\ensuremath{\mathbf{y}}}}-{{\ensuremath{\mathbf{y}}}}'\|_1.\end{aligned}$$ Suppose both ${{\ensuremath{\mathbf{x}}}}$ and ${{\ensuremath{\mathbf{y}}}}$ players use Algorithm \[alg:oftpl\_noncvx\_games\] to solve the game with linear perturbation functions $\sigma({{\ensuremath{\mathbf{z}}}})={\left\langle \bar{\sigma}, {{\ensuremath{\mathbf{z}}}}\right\rangle}$, where $\bar{\sigma} \in \mathbb{R}^d$ is such that each of its entries is sampled independently from $\text{Exp}(\eta)$. Suppose the guesses used by ${{\ensuremath{\mathbf{x}}}}$ and ${{\ensuremath{\mathbf{y}}}}$ players in the $t^{th}$ iteration are $f(\cdot,\Tilde{Q}_{t-1}), f(\Tilde{P}_{t-1},\cdot)$, where $\Tilde{P}_{t-1},\Tilde{Q}_{t-1}$ denote the predictions of ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players in the $t^{th}$ iteration, if guess $g_t = 0$ was used. If Algorithm \[alg:oftpl\_noncvx\_games\] is run with $\eta = 10d^2D(L+1), m = T$, then the iterates $\{(P_t,Q_t)\}_{t=1}^T$ satisfy the following with probability at least $1-\delta$ $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}\sum_{t=1}^Tf(P_t,{{\ensuremath{\mathbf{y}}}}) - f({{\ensuremath{\mathbf{x}}}},Q_t)& = {O\left({\frac{d^2D^2(L+1)\log{d}}{T} + \frac{GD}{T}\sqrt{\log{\frac{8}{\delta}}}}\right)}\\ &\quad + {O\left({\min\left\lbrace D^2L, \frac{d^2G^2\log{T}+dG^2\log{\frac{8}{\delta}}}{LT}\right\rbrace}\right)}.\end{aligned}$$ We use the same notation used in the proofs of Theorems \[thm:oftpl\_noncvx\_regret\], \[thm:oftpl\_noncvx\_smooth\_games\]. Let ${\mathcal{F}},{\mathcal{F}}'$ be the set of Lipschitz functions over ${\mathcal{X}},{\mathcal{Y}}$, and $\|g_1\|_{{\mathcal{F}}},\|g_2\|_{{\mathcal{F}}'}$ be the Lipschitz constants of functions w.r.t $\|\cdot\|_1$. Recall, in Corollary \[cor:ftpl\_noncvx\_exp\] we showed that for our choice of perturbation distribution, ${\mathbb{E}_{\sigma}\left[\|\sigma\|_{{\mathcal{F}}}\right]} = \eta \log{d}$ and OFTPL is ${O\left({d^2D\eta^{-1}}\right)}$ stable. We use this in our proof. From Theorem \[thm:oftpl\_noncvx\_regret\], we know that the regret of ${{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}$ players satisfy $$\begin{aligned} \sum_{t=1}^Tf(P_t,Q_t) - f({{\ensuremath{\mathbf{x}}}},Q_t) &\leq \eta D\log{d} + \underbrace{\sum_{t=1}^T{\left\langle P_t-P_t^{\infty}, f(\cdot, Q_t) \right\rangle}}_{S_1} \\ &\quad + \sum_{t=1}^T\frac{cd^2D}{2\eta}\underbrace{\|f(\cdot,Q_t)-f(\cdot,\Tilde{Q}_{t-1})\|_{{\mathcal{F}}}^2}_{S_2}\\ &\quad- \sum_{t=1}^T\frac{\eta}{2cd^2D}{\gamma_{{\mathcal{F}}}}(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})^2\end{aligned}$$ $$\begin{aligned} \sum_{t=1}^T f(P_t,{{\ensuremath{\mathbf{y}}}})- f(P_t,Q_t) &\leq \eta D\log{d}+ \sum_{t=1}^T{\left\langle Q_t-Q_t^{\infty}, f(P_t,\cdot) \right\rangle}\\ &\quad +\sum_{t=1}^T\frac{cd^2D}{2\eta}\|f(P_t,\cdot)-f(\Tilde{P}_{t-1},\cdot)\|_{{\mathcal{F}}'}^2\\ &\quad- \sum_{t=1}^T\frac{\eta}{2cd^2D}{\gamma_{{\mathcal{F}}'}}(Q_t^{\infty},\Tilde{Q}_{t-1}^{\infty})^2,\end{aligned}$$ where $c>0$ is a positive constant. We now provide high probability bounds for $S_1,S_2$. #### Bounding $S_1$. Let $\xi_i = \{\Tilde{P}_{i}, \Tilde{Q}_i, P_{i}, Q_{i+1}\}$ with $\xi_0=\{Q_1\}$ and let $\xi_{0:t}$ denote the union of sets $\xi_0,\dots, \xi_t$. Let $\zeta_t = {\left\langle P_t-P_t^{\infty}, f(\cdot, Q_t) \right\rangle}$ with $\zeta_0 = 0$. Note that $\{\zeta_t\}_{t=0}^T$ is a martingale difference sequence w.r.t $\xi_{0:T}$. This is because ${\mathbb{E}\left[P_t|\xi_{0:t-1}\right]} = P_t^{\infty}$ and $f(\cdot,Q_t)$ is a deterministic quantity conditioned on $\xi_{0:t-1}$. As a result ${\mathbb{E}\left[\zeta_t|\xi_{0:t-1}\right]}=0$. Moreover, conditioned on $\xi_{0:t-1}$, $\zeta_t$ is the average of $m$ independent mean $0$ random variables, each of which is bounded by $2GD$. Using Proposition \[prop:azuma\], we get $$\mathbb{P}\left(|\zeta_t| \geq s\Big| \xi_{0:t-1}\right) \leq 2\exp\left(-\frac{ms^2}{4G^2D^2}\right).$$ Using Proposition \[prop:martingale\_diff\] on the martingale difference sequence $\{\zeta_t\}_{t=0}^T$, we get $$\mathbb{P}\left(\Big|\sum_{t=1}^T\zeta_t\Big| \geq s\right)\leq 2\exp\left(-c\frac{ms^2}{G^2D^2T}\right),$$ where $c>0$ is a universal constant. This shows that with probability at least $1-\delta/8$, $S_1$ is upper bounded by $ {O\left({\sqrt{\frac{G^2D^2T\log{\frac{8}{\delta}}}{m}}}\right)}.$ #### Bounding $S_2$. We upper bound $S_2$ as $$\begin{aligned} \|f(\cdot, Q_t)-f(\cdot,\Tilde{Q}_{t-1})\|^2_{{\mathcal{F}}} &\leq 3\|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|^2_{{\mathcal{F}}} \\ &\quad + 3\|f(\cdot, Q_t^{\infty})-f(\cdot,\Tilde{Q}_{t-1}^{\infty})\|^2_{{\mathcal{F}}}\\ &\quad + 3\|f(\cdot, \Tilde{Q}_{t-1}^{\infty})-f(\cdot,\Tilde{Q}_{t-1})\|^2_{{\mathcal{F}}}.\end{aligned}$$ We first provide a high probability bound for $\|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|^2_{{\mathcal{F}}}$. A trivial bound for this quantity is $L^2D^2$, which can be obtained as follows $$\begin{aligned} \|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|_{{\mathcal{F}}} &=\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}}} \|{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - {\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})\|_{\infty}\\ & = \|{\mathbb{E}_{{{\ensuremath{\mathbf{y}}}}_1\sim Q_t,{{\ensuremath{\mathbf{y}}}}_2\sim Q_t^{\infty}}\left[{\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_1) - {\nabla}_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},{{\ensuremath{\mathbf{y}}}}_2)\right]}\|_{\infty}\\ &\stackrel{(a)}{\leq} LD,\end{aligned}$$ where $(a)$ follows from the smoothness assumption on $f$ and the fact that the diameter of ${\mathcal{X}}$ is $D$. A better bound for this quantity can be obtained as follows. From proof of Theorem \[thm:oftpl\_noncvx\_smooth\_games\], we have $$\begin{aligned} &\|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|^2_{{\mathcal{F}}} \leq 2\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{N}}_{\epsilon}}\|\nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})\|_{\infty}^2 + 8L^2\epsilon^2.\end{aligned}$$ where ${\mathcal{N}}_{\epsilon}$ be the $\epsilon$-net of ${\mathcal{X}}$ w.r.t $\|\cdot\|$. Recall, in the proof of Theorem \[thm:oftpl\_noncvx\_smooth\_games\], we showed the following high probability bound for the RHS quantity $$\begin{aligned} {\mathbb{P}}\left(\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{N}}_{\epsilon}}\|\nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})\|_2^2 > \frac{4dG^2}{m}(d+2\sqrt{ds} + 2s)\right) \leq e^{-s+d\log(1+2D/\epsilon)}.\end{aligned}$$ Choosing $\epsilon=Dm^{-1/2}, s = \log{\frac{8}{\delta}}+d\log(1+2m^{1/2})$, we get the following bound for $\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{N}}_{\epsilon}}\|\nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})\|_2^2$ which holds with probability at least $1-\delta/8$ $$\sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{N}}_{\epsilon}}\|\nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t) - \nabla_{{{\ensuremath{\mathbf{x}}}}}f({{\ensuremath{\mathbf{x}}}},Q_t^{\infty})\|_2^2 \leq \frac{20dG^2}{m}\left(\log{\frac{8}{\delta}}+d\log(1+2m^{1/2})\right).$$ Together with our trivial bound of $D^2L^2$, this gives us the following bound for $\|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|^2_{{\mathcal{F}}} $, which holds with probability at least $1-\delta/8$ $$\|f(\cdot, Q_t)-f(\cdot,Q_{t}^{\infty})\|^2_{{\mathcal{F}}} \leq \min\left(\frac{20dG^2}{m}\left(\log{\frac{8}{\delta}}+d\log(1+2m^{1/2})\right), D^2L^2\right) + \frac{8D^2L^2}{m}.$$ Next, we bound $\|f(\cdot, Q_t^{\infty})-f(\cdot,\Tilde{Q}_{t-1}^{\infty})\|^2_{{\mathcal{F}}}$. From our smoothness assumption on $f$, we have $$\|f(\cdot,Q_t^{\infty})-f(\cdot,\Tilde{Q}_{t-1}^{\infty})\|_{{\mathcal{F}}} \leq L {\gamma_{{\mathcal{F}}'}}(Q_t^{\infty},\Tilde{Q}_{t-1}^{\infty}).$$ Combining the previous two results, we get the following upper bound for $S_2$ which holds with probability at least $1-\delta/8$ $$\begin{aligned} \|f(\cdot, Q_t)-f(\cdot,\Tilde{Q}_{t-1})\|^2_{{\mathcal{F}}} &\leq 3L^2{\gamma_{{\mathcal{F}}'}}(Q_t^{\infty},\Tilde{Q}_{t-1}^{\infty})^2 + \frac{48D^2L^2}{m} \\ &\quad + \min\left(\frac{120dG^2}{m}\left(\log{\frac{8}{\delta}}+d\log(1+2m^{1/2})\right), 6D^2L^2\right).\end{aligned}$$ #### Regret bound. Substituting the above bounds for $S_1,S_2$ in the regret bound for ${{\ensuremath{\mathbf{x}}}}$ player gives us the following bound, which holds with probability at least $1-\delta/2$ $$\begin{aligned} \sum_{t=1}^Tf(P_t,Q_t) - f({{\ensuremath{\mathbf{x}}}},Q_t) &\leq \eta D\log{d} + {O\left({GD\sqrt{\frac{T\log{\frac{8}{\delta}}}{m}}+\frac{d^2D^3L^2T}{\eta m}}\right)} \\ &\quad +{O\left({\min\left(\frac{d^3DG^2T}{\eta m}\left(\log{\frac{8}{\delta}}+d\log(2m)\right), \frac{d^2D^3L^2T}{\eta}\right)}\right)}\\ &\quad+\sum_{t=1}^T\frac{3cd^2DL^2}{2\eta}{\gamma_{{\mathcal{F}}'}}(Q_t^{\infty},\Tilde{Q}_{t-1}^{\infty})^2- \sum_{t=1}^T\frac{\eta}{2cd^2D}{\gamma_{{\mathcal{F}}}}(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})^2\end{aligned}$$ Using a similar analysis, we get the following regret bound for the ${{\ensuremath{\mathbf{y}}}}$ player $$\begin{aligned} \sum_{t=1}^Tf(P_t,Q_t) - f({{\ensuremath{\mathbf{x}}}},Q_t) &\leq \eta D\log{d} + {O\left({GD\sqrt{\frac{T\log{\frac{8}{\delta}}}{m}}+\frac{d^2D^3L^2T}{\eta m}}\right)} \\ &\quad +{O\left({\min\left(\frac{d^3DG^2T}{\eta m}\left(\log{\frac{8}{\delta}}+d\log(2m)\right), \frac{d^2D^3L^2T}{\eta}\right)}\right)}\\ &\quad+\sum_{t=1}^T\frac{3cd^2DL^2}{2\eta}{\gamma_{{\mathcal{F}}}}(P_t^{\infty},\Tilde{P}_{t-1}^{\infty})^2- \sum_{t=1}^T\frac{\eta}{2cd^2D}{\gamma_{{\mathcal{F}}'}}(Q_t^{\infty},\Tilde{Q}_{t-1}^{\infty})^2\end{aligned}$$ Choosing, $\eta = 10d^2D(L+1), m= T$, and adding the above two regret bounds, we get $$\begin{aligned} \sup_{{{\ensuremath{\mathbf{x}}}}\in{\mathcal{X}},{{\ensuremath{\mathbf{y}}}}\in{\mathcal{Y}}}\sum_{t=1}^Tf(P_t,{{\ensuremath{\mathbf{y}}}}) - f({{\ensuremath{\mathbf{x}}}},Q_t)& = {O\left({d^2D^2(L+1)\log{d} + GD\sqrt{\log{\frac{8}{\delta}}}}\right)}\\ &\quad + {O\left({\min\left\lbrace D^2LT, \frac{d^2G^2\log{T}}{L} + \frac{dG^2\log{\frac{8}{\delta}}}{L}\right\rbrace}\right)}.\end{aligned}$$ Background on Convex Analysis ============================= #### Fenchel Conjugate. {#sec:fenchel_conjugate} The Fenchel conjugate of a function $f$ is defined as $$f^*(x^*) = \sup_{x}{\left\langle x, x^* \right\rangle} - f(x).$$ We now state some useful properties of Fenchel conjugates. These properties can be found in @rockafellar1970convex. \[thm:fenchel\_prop1\] Let $f$ be a proper convex function. The conjugate function $f^*$ is then a closed and proper convex function. Moreover, if $f$ is lower semi-continuous then $f^{**} = f$. \[thm:fenchel\_prop3\] For any proper convex function $f$ and any vector $x$, the following conditions on a vector $x^*$ are equivalent to each other - $x^* \in \partial f(x)$ - ${\left\langle z, x^* \right\rangle} - f(z)$ achieves its supremum in $z$ at $z=x$ - $f(x) + f^*(x^*) = {\left\langle x, x^* \right\rangle}$ If $(\text{cl}f)(x) = f(x)$, the following condition can be added to the list - $x\in\partial f^*(x^*)$ \[thm:fenchel\_prop4\] If $f$ is a closed proper convex function, $\partial f^*$ is the inverse of $\partial f$ in the sense of multivalued mappings, *i.e.,* $x \in \partial f^*(x^*)$ iff $x^* \in \partial f(x).$ \[thm:fenchel\_prop2\] Let $f$ be a closed proper convex function. Let $\partial f$ be the subdifferential mapping. The effective domain of $\partial f$, which is the set ${\text{dom}(\partial f)} = \{x|\partial f \neq 0\},$ satisfies $$\text{ri}({\text{dom}( f)}) \subseteq {\text{dom}(\partial f)} \subseteq {\text{dom}( f)}.$$ The range of $\partial f$ is defined as $\text{range} \partial f=\cup\{\partial f(x)|x\in \mathbb{R}^d\}$. The range of $\partial f$ is the effective domain of $\partial f^*$, so $$\text{ri}({\text{dom}( f^*)}) \subseteq \text{range} \partial f \subseteq {\text{dom}( f^*)}.$$ #### Strong Convexity and Smoothness. We now define strong convexity and strong smoothness and show that these two properties are duals of each other. \[def:strong\_convexity\] A function $f:{\mathcal{X}}\to \mathbb{R}\cup\{\infty\}$ is $\beta$-strongly convex w.r.t a norm $\|\cdot\|$ if for all $x,y \in \text{ri}({\text{dom}( f)})$ and $\alpha\in (0,1)$ we have $$f(\alpha x + (1-\alpha)y) \leq \alpha f(x) + (1-\alpha)f(y) - \frac{1}{2}\beta \alpha (1-\alpha) \|x-y\|^2.$$ This definition of strong convexity is equivalent to the following condition on $f$ [see Lemma 13 of @shalev2007thesis] $$f(y) \geq f(x) + {\left\langle g, y-x \right\rangle} + \frac{1}{2}\beta\|y-x\|^2, \quad \text{for any } x,y\in \text{ri}({\text{dom}( f)}), g\in\partial f(x)$$ \[def:strong\_smoothness\] A function $f:{\mathcal{X}}\to \mathbb{R}\cup\{\infty\}$ is $\beta$-strongly smooth w.r.t a norm $\|\cdot\|$ if $f$ is everywhere differentiable and if for all $x,y$ we have $$f(y) \leq f(x) + {\left\langle {\nabla}f(x), y-x \right\rangle} + \frac{1}{2}\beta\|y-x\|^2.$$ \[thm:fenchel\_strong\_convex\_weak\] Assume that $f$ is a proper closed and convex function. Suppose $f$ is $\beta$-strongly smooth w.r.t a norm $\|\cdot\|$. Then its conjugate $f^*$ satisfies the following for all $a,x$ with $u = {\nabla}f(x)$ $$f^*(a+u) \geq f^*(u) + {\left\langle x, a \right\rangle}+\frac{1}{2\beta}\|a\|_*^2.$$ \[thm:fenchel\_strong\_convex\] Assume that $f$ is a closed and convex function. Then $f$ is $\beta$-strongly convex w.r.t a norm $\|\cdot\|$ iff $f^*$ is $\frac{1}{\beta}$-strongly smooth w.r.t the dual norm $\|\cdot\|_{*}$.

--- abstract: 'We propose a generalization of Yang-Mills theory for which the symmetry algebra does not have to be factorized as mutually commuting algebras of a finite-dimensional Lie algebra and the algebra of functions on base space. The algebra of diffeomorphism can be constructed as an example, and a class of gravity theories can be interpreted as generalized Yang-Mills theories. These theories in general include a graviton, a dilaton and a rank-2 antisymmetric field, although Einstein gravity is also included as a special case. We present calculations suggesting that the connection in scattering amplitudes between Yang-Mills theory and gravity via BCJ duality can be made more manifest in this formulation.' --- [UTF8]{}[bsmi]{} **Generalized Yang-Mills Theory and Gravity .6cm** .5in [ Pei-Ming Ho$ $[^1], ]{}\ [ *Department of Physics and Center for Theoretical Sciences\ Center for Advanced Study in Theoretical Sciences\ National Taiwan University, Taipei 106, Taiwan, R.O.C.* ]{}\ Introduction ============ The textbook definition of Yang-Mills (YM) theory is usually based on the choice of a finite dimensional Lie group, and gauge transformations are specified by Lie-group valued functions. However, it is well known that when the base space is noncommutative, the algebra of gauge transformations is a mixture of the finite-dimensional Lie algebra and the algebra of functions on the noncommutative space. As a result, $SU(N)$ gauge symmetry cannot be straightforwardly defined on noncommutative space. In this paper, we consider a minor generalization of the notion of gauge symmetry. We will not only allow the generators of gauge transformations to behave like pseudo-differential operators (as functions on noncommutative space do), but we will also allow them to be not factorized into the part of a finite-dimensional Lie algebra and that of functions on the base space. That is, the gauge symmetry algebra does not have to be defined as the composition of a finite-dimensional Lie algebra and an associative algebra of functions on the base space. With this generalization, it may no longer be possible to view a gauge symmetry as what you get from “gauging” a global symmetry through introducing space-time dependence. A possibility of this generalization was already suggested [@Ho:2001as] for even-dimensional spherical brane configurations in the matrix theory. For example, for the fuzzy-$S^4$ configuration of $n$ D4-branes, the algebra of functions on the 4-dimensional base space is non-associative, but there is an associative algebra for gauge transformations. For large $n$, the gauge symmetry algebra is approximately that of a $U(n)$-bundle (or equivalently a fuzzy-$S^2$ bundle) over $S^4$ [@Ho:2001as]. Another example is the low energy effective theory of a D3-brane in large R-R 2-form field background [@Ho:2013opa]. This theory is S-dual to the noncommutative gauge theory for a D3-brane in large NS-NS $B$-field background. The gauge symmetry to all orders in the dual theory is not given by the noncommutative gauge symmetry, but is characterized by a bracket $\{\cdot, \cdot\}_{**}$ which defines a non-associative algebra on the base space [@Ho:2013opa]. (The gauge symmetry algebra is of course associative.) In this paper, we will show that the gauge symmetry of space-time diffeomorphism is also an example of the generalized gauge symmetry. Accordingly, a class of gravity theories can be interpreted as YM theories. Generically, these theories include a graviton, a dilaton and an anti-symmetric tensor. We will point out that the connection between Yang-Mills theory and gravity (through the color-kinematics duality) is manifest at tree level in 3-point amplitudes. Attempts to interpret gravity as a gauge theory have a long history since the works of Utiyama [@Utiyama], Kibble [@Kibble] and Sciama [@Sciama]. It is well known that General Relativity (GR) can be rewritten as the Chern-Simons theory in 3 dimensions [@Witten:1988hc], and a YM-like theory in 4 dimensions [@MacDowell:1977jt; @Stelle-West], as well as higher dimensions [@Vasiliev:2001wa]. The vielbein and the connection are defined as components of a gauge potential, and the gauge symmetry is $SO(d,2)$, instead of the space-time diffeomorphism. These formulations are based on gauge symmetries in the traditional sense. Our formulation of gravity as a YM theory is different from these formulations. While GR can be formulated as a YM theory, YM theories can also be realized as the low energy effective theories of gravity theories in higher dimensions via suitable compactification. Similar to this scenario of Kaluza-Klein reduction, internal symmetries and external symmetries are treated on equal footing in the generalized YM theories, as we will not distinguish the base space dependence from the internal space dependence in the gauge symmetry algebra. Our formulation of gravity is also reminiscent of teleparallel gravity [@review-tele], which can be interpreted as a gauge theory of the (Abelian) translation group, with the vielbeins playing the role of the gauge potential. In another formulation of gravity [@Cortese:2010ze] in which the vielbeins are identified with the gauge potential, a deformation of the gauge symmetry is considered to achieve the nonlinearity in gravity. In our formulation, on the other hand, the gauge potential is not the vielbein, but the inverse of the vielbein. The plan of this paper is as follows. In Sec.\[GaugeSymm\], we will see how the algebra of diffeomorphism appears as an example of the generalized gauge symmetry. In Sec.\[YM\], the gauge potential for the gauge symmetry of diffeomorphism is essentially the inverse of the vielbein, and the field strength is the torsion of the Weitzenböck connection. We show that the corresponding YM theories with quadratic Lagrangians define a class of gravity theories in Sec.\[YM=GR\]. It will be pointed out in Sec.\[ScattAmp\] that this new formulation of gravity may have significant advantages in its use to compute scattering amplitudes, with relations reminiscent of the double-copy procedure [@BCJ] to derive scattering amplitudes in gravity from YM theories. In Sec.\[HigherGauge\], we comment on extensions of the generalized notion of gauge symmetry to higher form gauge theories. Gauge Symmetry Algebra {#GaugeSymm} ====================== In a naive textbook introduction to non-Abelian gauge symmetry, the gauge transformation parameter $\Lam(x) = \sum_a \Lam^a(x)T_a$ is a sum of products of space-time functions and Lie algebra generators. The Lie algebra of local gauge transformations is spanned by a set of basis elements, say, T\_a(p) e\^[ipx]{} T\_a \[factor\] in the Fourier basis. In this basis, a gauge transformation parameter can be expressed as (x) = \_[a, p]{} \^a(p) T\_a(p), where the sum over $p$ is understood to be the integral $\int d^D p$ for $D$-dimensional space-time. Similarly, the gauge potential can be written as A\_(x) = \_[a, p]{} \^a\_(p) T\_a(p). Normally, for a given finite-dimensional Lie algebra with structure constants $f_{ab}{}^c$, the algebra of gauge transformations has the commutator = \_c f\_[ab]{}\^c T\_c(p+p’), where the structure constants $f_{ab}{}^c$ only involve color indices $a, b, c$. For these cases, the inclusion of functional dependence on the space-time in the generators $T_a(p)$ is trivial, and thus often omitted in discussions. However, for noncommutative gauge symmetries, the structure constants depend not only on the color indices $a, b, c$, but also on the kinematic parameters $p, p'$. For a noncommutative space defined by = i\^, the Lie algebra $U(N)$ gauge symmetry is = \_[p”]{} [**f**]{}\_[ab]{}\^[c]{}(p, p’, p”) T\_c(p”), \[NCUN\] where the structure constants are [^2] \_[ab]{}\^[c]{}(p, p’, p”) = \^[(D)]{}(p+p’-p”), and they involve kinematic parameters $p, p'$ and $p''$. Here $f_{ab}{}^c$ is the structure constant of $U(N)$ and $d_{ab}{}^c$ is defined by {T\_a, T\_b} = d\_[ab]{}\^c T\_c for $T_a$’s in the fundamental representation. In this gauge symmetry algebra, the $U(N)$ Lie algebra and the algebra of functions on the base space are mixed. (This is the obstacle to define noncommutative $SU(N)$ gauge symmetry.) The gauge algebra is non-Abelian even for the Abelian group $U(1)$. To describe the noncommutative $U(N)$ gauge algebra properly, it is a necessity to use the generators (\[factor\]) including functional dependence on the base space. Nevertheless, the noncommutative $U(N)$ gauge symmetry still assumes that the generators can be factorized (\[factor\]), and that $e^{ip\cdot x}$ always commutes with $T_a$. These are unnecessary assumptions for most algebraic calculations in the gauge theory. After all, in field theories, only the coefficients $\tilde{A}^a_{\mu}(p)$ are operators (observables), while the space-time and Lie algebra dependence are to be integrated out (summed over) in the action. It is thus natural to slightly extend the formulation of gauge symmetry (and YM theory) to allow the Lie algebra to be directly defined in terms of the generators $T_a(p)$, without even assuming its factorization into a Lie algebra factor $T_a$ and a function $e^{p\cdot x}$. The integration over space-time and trace of the internal space in the action can be replaced by the Killing form of the Lie algebra of $T_a(p)$. The distinction between internal space and external space is reduced in this description. In short, we propose to study gauge symmetries without assuming its factorization into two associative algebras (an algebra for the functions on the base space and a finite dimensional Lie algebra). Even when it is possible to factorize the generators formally as (\[factor\]), we will not assume that the space-time functions to commute with the algebraic elements $T_a$. (In general, we do not have to use the Fourier basis, and the argument $p$ of $T_a(p)$ can represent labels of any complete basis of functions on the base space.) One of the goals of this paper is to show that this generalization is beneficial, for bringing in new insights into gravity theories. Algebraically, this generalization is very natural. A corresponding geometric notion is however absent at this moment. (It is not clear what [*twisted*]{} bundles would mean.) The notion of bundles on noncommutative space is replaced by projective modules [@Connes], which should be further generalized for our purpose. We postpone the problem of finding a suitable geometric notion for the generalized gauge symmetry to the future. A Generalized Gauge Symmetry Algebra ------------------------------------ For the generalized gauge symmetry, we consider a Lie algebra defined as = \_ f\_[ab]{}\^c(, , ) T\_c(), which may not be decomposed as a product of the algebra of functions on the space-time and a finite-dimensional Lie algebra. Here $\a, \b, \g$ are labels for a complete basis of functions on the base space. For a theory with translational symmetry, it would be natural to use the Fourier basis, and we have = f\_[ab]{}\^c(p, p’) T\_c(p+p’), where the structure constant is actually $f_{ab}{}^c(p, p') \d^{(D)}(p+p'-p'')$ with the Dirac delta function cancelled by the integration over $p''$. The Jacobi identity of this Lie algebra is f\_[ab]{}\^e(p, p’)f\_[ec]{}\^d(p+p’, p”) + f\_[bc]{}\^e(p’, p”)f\_[ea]{}\^d(p’+p”, p) + f\_[ca]{}\^e(p”, p)f\_[eb]{}\^d(p”+p, p’) = 0. Every solution to this equation for $f$ defines a gauge symmetry. It will be interesting to find solutions with non-trivial dependence on momenta, as the case of the noncommutative gauge symmetry. As an example, let us now construct a Lie algebra for generators of the form { T(, p) \^[()]{}T\_[()]{}(p) }, where the basis elements $T_{(\mu)}(p)$ has a space-time index $\mu$ as its internal space index. To construct a concrete example, we will assume that the structure constants are linear in momenta $p$, $p'$, and that it is compatible with Poincare symmetry. We put the index $\mu$ in a parenthesis to remind ourselves that it plays the role of the index of an internal space. The reason why we choose this ansatz for a Lie algebra is that the Lie algebra of space-time diffeomorphisms is of this form. We wish to explore the possibility of rewriting a gravity theory as a generalized YM theory. The translation symmetry implies that the commutators are of the form = \_[\_3]{} f(\_1, p\_1, \_2, p\_2, \_3) T(\_3, p\_1 + p\_2), \[ansatz\] and the Lorentz symmetry implies that the structure constants are Lorentz-invariant functions of the vectors $\teps_1, \teps_2, \teps_3, p_1, p_2$. (The summation over $\tilde{\eps}_3$ is a summation over an orthonormal basis of vectors.) The most general ansatz consistent with all assumptions is thus f(\_1, p\_1, \_2, p\_2, \_3) = (\_1\_3) - (\_2\_3) + (\_1\_2) , where $\a, \b, \g$ are constant parameters. It follows from the ansatz that $$\begin{aligned} &[[T(1), T(2)], T(3)] + [[T(2), T(3)], T(1)] + [[T(3), T(1)], T(2)] = \sum_{\teps_4} \Big\{ \nn \\ &(\teps_1\cdot\teps_4)\left[ - \a\b (\teps_2\cdot\teps_3)(p_1\cdot(p_2-p_3)) + \g [(\teps_2\cdot p_2)(\teps_3\cdot (\a p_1 + \g p_2) - (\teps_3\cdot p_3)(\teps_2\cdot (\a p_1 + \g p_3)] \right] \nn \\ &+ \b (\teps_1\cdot\teps_2)\left[ (\teps_3\cdot(p_1 - p_2)) (\teps_4\cdot[(\a-\b)(p_1 + p_2) - \b(p_1 - p_2)]) - \g (\teps_3\cdot p_3)(\teps_4\cdot(p_1 - p_2)) \right] \nn \\ &+ \mbox{cyclic permutations of $(1, 2, 3)$} \Big\},\end{aligned}$$ where we have used $T(1)$ to represent $T(\teps_1, p_1)$, $T(2)$ to represent $T(\teps_2, p_2)$, etc. In order to satisfy the Jacobi identity, we have to set $\b = \g = 0$. The most general solution is thus equivalent to = \_[\_3]{} i T(3), \[commutator\] by scaling $\a$ to $-i$. More explicitly, it is = \_ iT\_[()]{}(p\_1+p\_2). \[commutator-2\] Incidentally, it is consistent to allow $\a$ to depend on the momenta. For example, Jacobi identity is satisfied for (p\_1, p\_2) = c e\^[p\_1p\_2]{} for arbitrary constant parameters $c$ and $\lam$. It is equivalent to the scaling of the generators by $T(\eps, p) \rightarrow T'(\eps, p) \equiv c\, e^{\lam p^2/2} T(\eps, p)$. Representations --------------- A representation of the algebra constructed above is given by T(, p) (p’) = i (p’) (p’+p), on a linear space with the basis $\{\tilde{\phi}(p)\}$. This expression allows us to interpret $\tilde{\phi}(p)$ as the Fourier modes of a scalar field $\phi(x)$ and $T(\teps, p)$ as the generator of a coordinate transformation with x\^ = \^[()]{} e\^[- i px]{}. That is, T(, p) = e\^[ip\_ x\^]{} \^[()]{}\_. Indices are contracted according to Einstein’s summation convention regardless of whether they are in parentheses or not. In this representation, it is clear that this algebra is that of space-time diffeomorphism. The gauge symmetry of GR arises as the most general gauge symmetry with Lie algebra of the form (\[ansatz\]), assuming that the structure constants are linear in momentum and that they respect Poincare symmetry. A generic element of the Lie algebra is a superposition d\^D p \^[()]{}(p) T\_[()]{}(p) in $D$ dimensions, which can be written as T\_ \^[()]{}(x) \_, \[Teps\] where $\eps^{(\m)}(x)$ is the inverse Fourier transform of $\teps^{(m)}(p)$. In view of this representation (\[Teps\]), it is tempting to interpret the algebra constructed above as merely the result of taking $T_a$’s to be derivatives $\del_a$’s in (\[factor\]) for a traditional gauge symmetry. But if we were really dealing with a traditional gauge symmetry, we would have obtained an Abelian gauge symmetry because $[T_a, T_b] = [\del_a, \del_b] = 0$. The need to generalize the notion of gauge symmetry here is due to the fact that $T_a = \del_a$ does not commute with space-time functions. Incidentally, the traditional interpretation of the torsion in teleparallel gravity is indeed the field strength of an Abelian gauge theory [@Cho:1975dh]. (See eq.(\[torsion-field-strength\]) below.) Matter fields in the gauge theory are classified as representations of the gauge symmetry. Since the gauge symmetry under consideration is the diffeomorphism, we know all about other representations of different spins. Gauge Field of Diffeomorphism {#YM} ============================= Gauge Potential vs Vielbein {#vielbein} --------------------------- We can define a gauge potential for the gauge symmetry algebra (\[commutator\]). In the representation (\[Teps\]), the gauge potential A\_(x) = \_[, p]{} \_\^[()]{}(p)T\_[()]{}(p) = A\_\^[()]{}(x) \_ should transform like A\_(x) = \[D\_, (x)\], \[A-transf\] where D\_ \_ + A\_(x) = (\_\^ + A\_\^[()]{})\_ is the covariant derivative and the gauge transformation parameter is (x) = \^[()]{}(x) \_. More explicitly, the gauge transformation (\[A-transf\]) can be expressed as A\_\^[()]{}(x) = \_ \^[()]{}(x) - \^[()]{}(x) \_ A\_\^[()]{}(x) + A\_\^[()]{}(x) \_ \^[()]{}(x). \[Amn-transf\] Let us recall that the vielbein $e_{\mu}{}^a(x)$ in gravity is defined to transform under general coordinate transformations as e\_\^a(x) = x\^ \_ e\_\^a + e\_\^a \_x\^. The index $a$ on $e_{\mu}{}^a$ labels a local orthonormal Lorentz frame. Under a rotation of the local Lorentz frame, e\_\^a(x) e’\_\^a(x) = \^a\_b(x) e\_\^b(x), \[rotation-0\] where \_[ab]{}(x) = - \_[ba]{}(x) is the parameter for infinitesimal $SO(D-1, 1)$-rotations. The inverse $e_a{}^{\mu}(x)$ of the vielbein is defined by e\_a\^(x) e\_\^b(x) = \_a\^b, e\_\^a(x) e\_a\^(x) = \_\^. The transformation of $e_a{}^{\mu}(x)$ is e\_a\^(x) = x\^ \_ e\_a\^(x) - e\_a\^ \_ x\^. \[e-inv-transf\] Let us now consider a flat background in which e\_a\^(x) = \_a\^ + C\_a\^(x) for a fluctuation denoted by $C_a{}^{\m}$. Here we have chosen a particular frame in which the flat background is given by $e_a{}^{\mu} = \d_a^{\mu}$. The local Lorentz transformation symmetry is not manifest (nonlinearly realized) in terms of the variable $C_a{}^{\mu}$. It follows from (\[e-inv-transf\]) that C\_a\^(x) = - \_[a]{} x\^ + x\^ \_ C\_a\^(x) - C\_a\^ \_ x\^. Comparing this expression with (\[Amn-transf\]), we see that it is tempting to identify $A$ with $C$, and $\Lam$ with $- \d x$. The transformations of $A$ and $C$ are matched by identifying upper (lower) indices with upper (lower) indices. That is, the Lorentz index $\mu$ on $A_{\mu}{}^{(\nu)}$ is to be identified with the local Lorentz frame index $a$ on $C_a{}^{\mu}$, while the internal space index $\nu$ on on $A_{\mu}{}^{(\nu)}$ is to be identified with the space-time coordinate index $\mu$ on $C_a{}^{\mu}$. This may seem peculiar at first sight but it is actually expected. The gauge algebra (\[commutator\]) is defined with the assumption of Poincare symmetry on the base space, so the Lorentz index $\mu$ on the potential $A_{\mu}{}^{(\nu)}$ cannot be identified with the coordinate index $\mu$ on a curved manifold. On the other hand, the internal space index $\nu$ on the potential $A_{\mu}{}^{(\nu)}$ is contracted with the index of a derivative $\del_{\nu}$ of space-time coordinates, hence it is really an index of coordinates. Note that the gauge potential $A$ is still defined as part of the covariant derivative $D = dx^{\mu} D_{\mu}$, and in this sense it is still a 1-form. The one-form index $\mu$ of $A_{\mu}{}^{(\nu)}$ is matched with the frame index, not the 1-form index of the vielbein, only because the geometric interpretation of gravity is changed. The gauge symmetry of gravity is now interpreted as a non-Abelian symmetry on Minkowski space whose transformations involve kinematic vectors. On the other hand, the potential $A_{\mu}{}^{(\nu)}$ is not a pure 1-form as it has a vector-field index $(\nu)$. In the following, we will adopt the conventional notation for vielbeins. Latin letters $a, b, c, \cdots$ are used for indices of local Lorentz frames, and Greek letters $\mu, \nu, \lam, \cdots$ for indices of space-time coordinates. For instance, we will relabel the gauge potential as $A_{a}{}^{(\mu)}$ (without raising or lowering indices), or simply as $A_{a}{}^{\mu}$ without the parenthesis. Despite the fact that $A$ and $C$ transform in exactly the same way under general coordinate transformations, it is not clear yet whether $A$ can be fully identified with $C$. In particular, in pure GR, not only the general coordinate transformation, but also the rotations of local Lorentz frames are gauge symmetries. We also need to check whether there are ghosts or tachyons before we claim that the YM theory of the gauge symmetry of Sec.\[GaugeSymm\] can be interpreted as a gravity theory. This will be the main issue to focus on below. Nevertheless, motivated by this potential identification, we denote the covariant derivative as D\_[a]{} = \_[a]{}\^\_, where we used the notation \_[a]{}\^ \_[a]{}\^ + A\_[a]{}\^. \[def-e-A\] The kinetic term of a scalar field is \^[ab]{}D\_a D\_b = \^ \_ \_ , where the effective metric $\hat{g}_{\m\n}$ naturally arises. It is defined by \_ = \_\^a \_[ab]{} \_\^b, \[metric\] where $\hat{e}_{\m}{}^a$ is by definition the inverse of $\hat{e}_{a}{}^{\m}$. Field Strength vs Torsion ------------------------- The field strength of the non-Abelian gauge symmetry constructed above is F\_[ab]{}(x) = \_[a]{} A\_[b]{}(x) - \_[b]{} A\_[a]{}(x) + \[A\_[a]{}(x), A\_[b]{}(x)\]. In the representation (\[Teps\]), it is F\_[ab]{}(x) = F\_[ab]{}\^[()]{}(x) \_, where F\_[ab]{}\^[()]{}(x) = \_a\^\_\_b\^ - \_b\^\_\_a\^. \[field-strength\] With the analogy between $\hat{e}_{\mu}{}^a$ and the vielbein $e_{\mu}{}^a$, we define \^\_ \_\^ - \_\^. \[torsion\] It is the torsion for the Weitzenböck connection \^\_ \_a\^\_\_\^a \[connection\] used in teleparallel gravity when $\hat{e}_{\mu}{}^a$ is identified with the vielbein $e_{\mu}{}^a$. The field strength and the “torsion” are essentially the same quantity: F\_[ab]{}\^[()]{}(x) = - \_a\^ \_b\^ \^\_, if we think of $\hat{e}_{\mu}{}^a$ and $\hat{e}_a{}^{\m}$ as the quantities used to switch between the two bases $\del_{\m}$ and $D_a$. The “connection” (\[connection\]) satisfies the relation D\_ \_\^a \_ \_\^a - \_\^ \_\^a = 0, and has zero “curvature”: d - = 0. Field-Dependent Killing Form ---------------------------- An interesting feature of the algebra (\[commutator\]) for space-time diffeomorphism is that the Killing form (invariant inner product) has to be field-dependent. For two elements of the Lie algebra $T_f \equiv f^{\m}(x) \del_{\m}$ and $T_{f'} \equiv f'{}^{\n}(x) \del_{\n}$, it is clear that the Killing form should be = d\^D x f\^(x) \_(x) f’\^(x) \[Killing-1\] up to an overall normalization constant factor. Here the measure $\sqrt{\hat{g}} = \det \hat{e}_{\mu}{}^a$ must be present to ensure that the integration is diffeomorphism-invariant. The Killing form can be slightly simplified by a change of basis. Let us use the field-dependent basis $\{D_a\}$. The Killing form for two generators $T_f = f^a D_a$ and $T_{f'} = f'{}^b D_b$ is = d\^D x f\^[a]{}(x) \_[ab]{}(x) f’\^[b]{}(x), \[Killing-2\] where $f^a \equiv f^{\mu} \hat{e}_{\mu}{}^a$ and similarly for $f'{}^b$. (The factor of $\hat{g}_{\m\n}$ in (\[Killing-1\]) is replaced by $\eta_{ab}$.) In this basis, the structure constants are field-dependent: = F\_[ab]{}\^c D\_c, \[DDD\] and the Jacobi identity (the consistency of the Lie algebra) is equivalent to the Bianchi identity of the field strength. YM as Gravity {#YM=GR} ============= YM Action --------- The YM action is given as the norm of the field strength: S\_[YM]{} = d\^D x F\^[abc]{} F\_[abc]{}. \[YM-action\] It is invariant under space-time diffeomorphism. However, it is not invariant under rotations of the local Lorentz frame: \_\^a(x) ’\_\^a(x) = \^a\_b(x) \_\^b(x). \[rotation\] In the absence of the gauge symmetry of local Lorentz frame rotations, the variable $\hat{e}_{\m}{}^a$ contains more degrees of freedom than the genuine vielbein $e_{\m}{}^a$. (This is why we have used a hat to distinguish it from the vielbein.) The YM theory cannot be identified with pure GR. To achieve a YM-like theory equivalent to Einstein’s theory, we should utilize the fact that the internal space index $a$ (for the basis $D_a$) can be contracted with the coordinate index $a$. It allows us to introduce quadratic terms in addition to (\[YM-action\]) in the action. The most general quadratic action is the superposition of three terms: S\_[YM-like]{} = d\^D x . \[general-action-0\] The action remains the same if we simultaneously scale $\kappa^2, \lambda, \alpha, \beta$ by the same factor. Up to this ambiguity, there is a unique choice of the parameters such that this action is invariant under local Lorentz rotations. Teleparallel Gravity -------------------- The action (\[general-action-0\]) is of a form resembling that of the teleparallel gravity, which is equivalent to Einstein’s theory [@review-tele]. The teleparallel gravity has the interpretation as the gauge theory of translational symmetry. The gauge potential is essentially the vielbein: \_\^a e\_\^a - \_\^a, \[A=e-I\] where $\d_{\m}{}^a$ can be replaced by an arbitrary constant matrix. The torsion $T^{\lam}{}_{\m\n}$ of the Weitzenböck connection is essentially the Abelian field strength T\^[a]{}\_ = \_ e\_\^a - \_ e\_\^a = \_ [A]{}\_\^a - \_ [A]{}\_\^a. \[torsion-field-strength\] Despite the fact that the field strength (\[field-strength\]) of the generalized gauge symmetry of diffeomorphism is related to this field strength (\[torsion-field-strength\]) by a mere change of basis in the tangent space, the gauge symmetries are totally different. The gauge symmetry is Abelian in the traditional interpretation of teleparallel gravity, while the diffeomorphism is of course non-Abelian. The action of teleparallel gravity is S\_[TP]{} &=& d\^D x , \[action-TP\] where indices are raised or lowered using the metric $g_{\mu\nu} = e_{\m}{}^a e_{\n}{}^b \eta_{ab}$ and $e = \sqrt{g}$ stands for the determinant of $e_{\mu}^a$. The Lagrangian of this action equals the Hilbert-Einstein Lagrangian up to a total derivative. That is, \_[TP]{} = R\^[(LC)]{} + , where $R^{(LC)}$ is the scalar curvature for the (torsion-free) Levi-Civita connection. Even though the choices of connections are different, the teleparallel gravity action and the Hilbert-Einstein action give exactly the same field equation for the metric and so they are physically equivalent. It is interesting that the inverse of $(I + {\cal A})$ (see (\[A=e-I\])) for the potential ${\cal A}$ of the Abelian group of translations can be identified with $(I + A)$ for the gauge potential $A_{a}{}^{\m}$ of the non-Abelian gauge symmetry of general coordinate transformations. The teleparallel gravity action (\[action-TP\]) is equivalent to the action (\[general-action-0\]) for the choice of parameters $\lam = 1/2, \alpha = 1, \beta = 1$. It is S\_[TP]{} = d\^D x . \[action-TP-2\] The first term is the YM action (\[YM-action\]). The rest of the terms provide the unique combination so that the action is invariant under rotations of local Lorentz frames (\[rotation\]). The field $\hat{e}_{\m}{}^{a}$ can now be identified with the vielbein $e_{\m}{}^a$ in gravity, and the modified YM theory is equivalent to GR. Metric, $B$-field and Dilaton ----------------------------- While the action (\[action-TP-2\]) is equivalent to pure GR, we investigate the most general quadratic action (\[general-action-0\]), which can be equivalently put in the form S = d\^D x , \[general-action\] assuming that the coefficient of the YM term is non-zero. It is invariant under general coordinate transformations for arbitrary constants $\a, \b$. (Compared with (\[general-action-0\]), $\lam = 1 - \a/2$.) The case of teleparallel gravity (\[action-TP\]) corresponds to the choice $\a = \b = 1$. For generic values of $\a, \b$, local rotations of Lorentz frames are no longer gauge symmetries. With fewer gauge symmetries, there are more physical degrees of freedom in the theory. In $D$-dimensional space-time, the fundamental field $\hat{e}_a{}^{\m}$ has $D^2$ components. When the rotation of local Lorentz frames is a gauge symmetry, local Lorentz transformations identify $D(D-1)/2$ components of $\hat{e}_a{}^{\m}$ as gauge artifacts, with the remaining $D(D+1)/2$ components of $\hat{e}_a{}^{\m}$ to be matched with the $D(D+1)/2$ independent components of the metric. Tuning the values of $\a, \b$ slightly away from $1$, we have $D(D-1)/2$ of the components that can no longer be gauged away. The theory with generic values of $\a, \b$ is expected to contain more physical fields in addition to the metric. For coefficients $\a, \b$ with values not too different from $1$, the theory is expected to be a gravity theory including matter fields. After all, the gauge symmetry of general coordinate transformation is always present. In Einstein’s theory of gravity, for a fluctuation of the metric g\_ = \_ + h\_ + , one can choose the vielbein to be symmetric e\_[a]{} = \_[a]{} + h\_[a]{}/2 + as a condition for the local Lorentz frame. Note that in the perturbation theory we are forced to mix the Latin and Greek indices as the space-time is Minkowskian at the lowest order. We will no longer distinguish the indices in the perturbative theory, and use both sets of labells $a, b, c, \cdots$ and $\m, \n, \cdots$ at will. We decompose the field $A_{\m a}$ into the symmetric part and the anti-symmetric part A\_[ab]{} = (h\_[ab]{} + B\_[ab]{})/2, \[A=h+B\] where $h_{ab}$ is symmetric and $B_{ab}$ is anti-symmetric. We identify $h_{ab}$ as the fluctuation of the metric, and only the traceless part of $h_{ab}$ propagates in Einstein’s theory. When the rotation of the local Lorentz frame is not a gauge symmetry, the trace part of $h_{ab}$ and the tensor $B_{ab}$ cannot be gauged away. For the theory to be physically sensible, one has to check that there are no ghosts or tachyons. A necessary condition for linearized field equations of a rank-2 tensor field to be free of ghosts and tachyonic modes is that the anti-symmetric part of the tensor field is decoupled from the symmetric part [@VanNieuwenhuizen:1973fi]. The implication of this criterion to the general quadratic action (\[general-action\]) can be easily derived as follows. First, the cyclic combination F\_[abc]{} + F\_[bca]{} + F\_[cab]{} = (\_a B\_[bc]{} + \_b B\_[ca]{} + \_c B\_[ab]{}) + [O]{}(A\^2) \[FFF\] in the second term of the action (\[general-action\]) involves only the anti-symmetric tensor field $B_{ab}$ at the linearized level. (This was why we chose the peculiar form of the second term in the action (\[general-action\]).) Hence its coefficient $\a$ has no effect on the coupling between $h_{ab}$ and $B_{ab}$. The first and third terms in the action (\[general-action\]) involve both $h_{ab}$ and $B_{ab}$, and the relative magnitude of their coefficients should be fixed to decouple $h_{ab}$ from $B_{ab}$. Since we know that pure GR is free of ghosts and tachyons, the ratio of these coefficients should be identical to that of the teleparallel gravity action (\[action-TP\]). Consequently, the parameter $\b$ should be fixed as [@MuellerHoissen:1983vc] = 1. \[b=1\] It is still necessary to check that the kinetic terms are positive-definite. The parameter $\a$ is constrained by [@MuellerHoissen:1983vc] &lt; 1 \[a&lt;1\] for the kinetic term of the anti-symmetric field $B_{ab}$ to be positive-definite. These two conditions (\[b=1\]) and (\[a&lt;1\]) ensures that the theory is ghost-free and tachyon-free at the free field level. For this class of theories, the propagating modes of the traceless part of $h_{ab}$ should be interpreted as the graviton, and the trace part $h^a{}_a$ as the dilation. There is also a rank-2 anti-symmetric field $B_{ab}$ whose gauge transformation at the lowest order is B\_[ab]{} = \_a \_b - \_b \_a + [O]{}(A). One can define the covariant field strength of $B_{ab}$ as $(F_{abc} + F_{bca} + F_{cab})$ (\[FFF\]). The dilaton can be viewed as a 0-form gauge potential. In the Lorentz gauge \_ \_a\^ = 0, \[Lorentz-gauge\] we have F\^[ab]{}\_b &=& \^[ac]{}\_c \^b\_b - \^[bc]{}\_c \^[a]{}\_b\ &=& \^a + [O]{}(A\^2), \[F-phi\] where h\^b\_b/D, so that $F_{ab}{}^b$ can be interpreted as the field strength of the dilaton. Comparison with Other Formulations of Gravity --------------------------------------------- At first sight, the formulation of gravity outlined above may appear reminiscent to other known formulations of gravity as a gauge theory. We have already commented above how the traditional interpretation of teleparallel gravity (as an Abelian gauge theory) is different from our formulation. There are also non-Abelian gauge theory formulations of gravity, e.g. Chern-Simons gravity in 3D [@Witten:1988hc], MacDowell-Mansouri gravity in 4D [@MacDowell:1977jt] and higher dimensional generalizations [@Vasiliev:2001wa]. In these theories, the gauge potential is of the form A\_ = e\_\^a P\_a + \_\^[ab]{} J\_[ab]{}, \[usual\] where $P_a$ and $J_{ab}$ are generators of the local Poincare algebra $iso(D-1, 1)$ or the conformal algebra $so(D-1, 2)$. The components of the gauge potential are the vielbein $e_{\m}{}^a$ and the spin connection $\om_{\m}{}^{ab}$. In contract, the gauge potential in our theory is (in a certain representation) A\_[a]{} = (\_[a]{}\^-\^\_a) \_, \[ours\] where $\hat{e}_{a}{}^{\m}$ might be identified with the [*inverse*]{} of the vielbein if the action respects the local Lorentz symmetry. Formally, if we identify the translation generator $P_a$ in (\[usual\]) with the derivative $\del_{\m}$ in (\[ours\]), the gauge potential (\[usual\]) resembles (\[ours\]). However, more precisely, there are many important differences. 1. The generators $P_a$ (and $J_{ab}$) in (\[usual\]) commute with functions on the base space, while the derivative in (\[ours\]) does not. 2. The coefficient of $P_a$ is the vielbein, and that of $\del_{\m}$ is the inverse vielbein in the special case of teleparallel gravity. In general, $\hat{e}_{\m}{}^a$ includes more degrees of freedom than the inverse vielbein. 3. Eq. (\[usual\]) is the usual potential associated with the “gauging” of a finite dimensional Lie group, while (\[ours\]) is not the potential for gauging any global symmetry. 4. The field strength for (\[usual\]) is given by the Riemann tensor, and that for (\[ours\]) by the torsion of the Weitzenböck connection. A priori they are not related in any simple way as the Weitzenböck connection is not invariant under local Lorentz transformations while the Riemann tensor is. 5. Despite the fact that the Lagrangians for gravity are quadratic in the field strengths for both potentials (\[usual\]) and (\[ours\]). The former gives the Hilbert-Einstein action. Even in the special case of telelparallel gravity, the latter differs by a total derivative. Supergravity theories are constructed [@SUGRA] based on the YM-like theory for the gauge potential (\[usual\]). It will be interesting to consider the supersymmetrization of the gauge symmetry of diffeomorphism and to derive the supergravity theory as an alternative formulation of supergravity. We leave this project for future publications. Scattering Amplitudes {#ScattAmp} ===================== In recent years, there has been amazing progress in the techniques of calculating scattering amplitudes, as well the understanding of their structures. Among them, a very interesting and mysterious structure is the connection between YM theory and gravity through the so-called double-copy procedure, which utilizes the color-kinematics duality (also known as the BCJ duality) [@BCJ]. According to Ref.[@BCJ], certain gravity theories are double copies of YM theories: the scattering amplitudes of gravity theories can be obtained from those of YM theories with color factors replaced by kinematic factors. In many cases this connection can find its origin in the open-closed string duality (the KLT duality [@KLT]), although there are also other cases in which the string-theory origin is absent at this moment. A complete off-shell field-theoretic explanation of this connection between gravity and YM theories, which applies only to on-shell amplitudes, may not be possible. But it is desirable to understand how much of the on-shell miracle can be understood in an off-shell theory. Earlier efforts in this direction include Refs. [@BjerrumBohr:2012mg; @Monteiro:2014cda]. We propose that the formulation of gravity as a YM theory we constructed above may shed some new light on this problem. Heuristic Explanation {#heuristic} --------------------- Let us first re-examine the double-copy procedure to illustrate our idea. As the simplest example, for the pure YM theory, the color-ordered 3-point amplitude at tree-level is f\_[abc]{} n\^[(3)]{}\_(p, q, -(p+q)) \[fn\] where $f_{abc}$ is the structure constant and $n^{(3)}_{\lam\m\n}(p, q, -(p+q))$ the kinematic factor n\^[(3)]{}\_(p, q, r) (p-q)\_\_ + (q-r)\_\_ + (r-p)\_\_. \[n\] Here $(p, q, r)$ are the momenta of the 3 external legs, and $\lam, \m, \n$ are Lorentz indices labelling the polarizations of the vector fields. The origin of this factor (\[n\]) is the 3-point vertex \_c F\_[(0)]{}\^[c]{} = f\_[abc]{} A\_\^a A\_\^b (\^ A\^[c]{} - \^ A\^[c]{}) \[fAAF\] in the YM Lagrangian, with cyclic permutations of the three factors of $A$ contracted with three external legs, assuming that the basis of the Lie algebra is chosen such that $f_{abc} = f_{bca} = f_{cab}$. Here $F_{(0)}^{\m\n c} \equiv \del^{\m} A^{\n c} - \del^{\n} A^{\m c}$ is the field strength at the lowest order. The double-copy procedure states that the replacement of $f_{abc}$ by $n^{(3)}_{\lam\m\n}(p, q, -(p+q))$ gives the 3-point amplitude of the corresponding gravity theory. In other words, if there is a Lie algebra with indices $a = (\lam, p)$ and structure constants $f_{abc}$ given by $n^{(3)}_{\lam\m\n}(p, q, -(p+q))$, the YM theory would agree with GR at least for 3-point amplitudes. Yet one can check that this choice of structure constants does not satisfy the Jacobi identity. The color-kinematics duality and the double-copy procedure applies to all higher-point amplitudes. For 4-point amplitudes at the tree level, the kinematic factor of color-ordered amplitudes is [^3], skematically, n\_i\^[(4)]{} = n\_i\^[(3)]{}n\_i\^[(3)]{} + m\_i\^[(4)]{} (i = s, t, u), \[nn+m\] where the first term on the right hand side is the contribution from 3-point vertices, and the second term from the 4-point interaction in the YM theory. (The index values $s, t, u$ are labels for the $s$, $t$ and $u$ channels of the Feynman diagrams for tree-level 4-point scattering amplitudes.) The kinematic factors $n_i^{(4)}$ satisfy a linear relation n\_s\^[(4)]{} + n\_t\^[(4)]{} + n\_u\^[(4)]{} = 0 analogous to the Jacobi identity for the color factor (which is quadratic in the structure constants). The relation above would be equivalent to the Jacobi identify for $n^{(3)}$ interpreted as structure constants if the terms $m_i^{(4)}$ were absent. The presence of $m_i^{(4)}$ is the evidence that $n^{(3)}$ cannot be used as structure constants. Often the relation between YM and GR indicated by the double-copy procedure is symbolically represented as (YM)${}^2 = $(GR). This expression is actually misleading, because neither the color factors or the propagators are squared in the gravity theory. Instead, the identification of color factors with kinematic factors (e.g. $f_{abc}$ with $n_{\lam\m\n}(p, q, -(p+q))$ for tree-level 3-point amplitudes) identifies YM directly with GR. It is more appropriate to use (YM)${}^{\prime}$ = GR as the symbolic representation of this connection. The prime on (YM) indicates the modification of YM theory by the replacement of color factors by kinematic factors. Since the color factors are composed of structure constants of the gauge group, we are naturally led to consider the possibility of gauge symmetries with structure constants involving kinematic factors. This was precisely what we did in Sec.\[GaugeSymm\], which led to new formulations of gravity theories as generalized YM theories in Sec.\[YM\] Apparently, the hope for a direct matching between structure constants and the kinematic factors is too naive. First, as the 3-point amplitude is defined on-shell, the structure constant can be different from $n_{\lam\m\n}(p, q, -(p+q))$ when it is off-shell. Secondly, even if the 3-point amplitudes agree with structure constant, it is not clear if higher-point amplitudes will automatically agree with the corresponding color factors, as there will be different on-shell conditions at work. (In fact, eq.(\[nn+m\]) says that the structure constants for 4-point amplitudes are not to be given by the structure constants for 3-point amplitudes due to the extra term $m^{(4)}_i$.) In general, as the BCJ duality only holds on-shell, the correspondence between structure constants and kinematic factors in $n$-point amplitudes is different for different $n$, and it is unclear if there exists an off-shell generalization. It is also highly nontrivial how such on-shell correspondences can be implemented efficiently in a field-theoretic approach. Hence we leave the search for a field-theoretic proof of the validity of the double-copy procedure for future works. Nevertheless, the color-kinematics duality and double-copy procedure motivate us to explore YM theories with Lie algebras involving kinematic factors, and it does lead to a connection between YM theories and gravity theories at the level of Lagrangians as we have shown in Sec.\[YM\]. It will be interesting to see whether the calculation of scattering amplitudes is simplified in this YM-like formulation of gravity, compared with the calculation based on the Hilbert-Einstein action. It will be even more interesting to see if we are getting closer to the simplified on-shell results obtained via the BCJ duality. To gain some intuition about how far or close our theory is to the concise results of the BCJ duality, let us comment on the skematic structure of the 3-point scattering amplitude (\[fn\]). The kinematic factor $n$ comes from the factor $F_{(0)}$, as the only object involving derivatives in the cubic term $f AA F_{(0)}$ (\[fAAF\]) of the YM action. This suggests that, to replace the color factor $f$ by $n$, we should have $f \sim F_{(0)}$ to the lowest order in $A$. But this is precisely what we have: the structure constant (\[DDD\]) in the basis of $D_a$ is the field strength $F$! In the following, we will compute more carefully the 3-point vertices of the YM-like formulation of gravity, and see that the calculation is much simpler than the calculation based on the Hilbert-Einstein action (which involves around 100 terms). We leave higher-point scattering amplitudes for the future. Incidentally, although the Chern-Simons theory in 3D [@Witten:1988hc] and the MacDowell-Mansouri theory in 4D [@MacDowell:1977jt] are also YM-type formulations of gravity, they are first order formulations of GR. One has to first solve the connection in terms of the vielbein before calculating any scattering amplitudes of gravitons. The calculation in those theories is not simpler than a direct computation from the Hilbert-Einstein action. Perturbative Expansion in $A$ ----------------------------- In this section, we focus on the 3-point vertices relevant for the 3-graviton scattering. We consider 3-point vertices of the action (\[general-action\]) for the traceless part of $h_{\m\n}$. First, using (\[FFF\]), one can easily see that the second term in the action (\[general-action\]) is (F\^[abc]{}+F\^[bca]{}+F\^[cab]{})(F\_[abc]{}+F\_[bca]{}+F\_[cab]{}) &=& H\^[(0)abc]{} H\^[(0)]{}\_[abc]{} + [O]{}(H\^[(0)]{} A\^2) + [O]{}(A\^4), where $H^{(0)}_{abc}$ is the field strength of the anti-symmetric tensor field $B_{ab}$ defined at the lowest order: H\^[(0)]{}\_[abc]{} \_a B\_[bc]{} + \_b B\_[ca]{} + \_c B\_[ab]{}. A vertex operator involving only external legs of $h$ appears at ${\cal O}(A^4)$ or higher. The 3-point vertices of $h_{\m\n}$ is thus independent of the parameter $\a$. Secondly, the third term in (\[general-action\]) is the square of F\_[ab]{}\^b &=& \_a A\_b\^b - \_b A\_a\^b + [O]{}(A\^2), where the first term involves the trace of $h_{\m\n}$, and the second term vanishes if we impose the gauge-fixing condition \^b h\_[ab]{} = 0 \[dh=0\] for the graviton field. As a result, in this gauge (\[dh=0\]), the third term of the action (\[general-action\]) is also irrelevant to the 3-point vertex for the traceless part of $h_{\m\n}$. Furthermore, the overall measure of integration is e = 1 + h\_a\^a + [O]{}(A\^2), which is also irrelevant for our consideration. The 3-point vertices for the graviton can thus only come from the YM Lagrangian, and there are only two terms \^[(3)]{} F\^[abc]{}\_[(0)]{}(\[A\_a, A\_b\]\_c - F\^[(0)]{}\_[ab]{}\^[d]{}A\_[dc]{}), \[3-point-1\] where \_c A\_a\^d\_d A\_[bc]{} - A\_b\^d\_d A\_[ac]{} and $F^{(0)}_{abc}$ is the field strength at the zero-th order F\^[(0)]{}\_[abc]{} \_a A\_[bc]{} - \_b A\_[ac]{}. The first term of (\[3-point-1\]) comes from the Lie algebra structure of this generalized YM theory. The second term arises due to the field-dependent inner-product of the Lie algebra. Near the end of Sec.\[heuristic\], we discussed how the formulation of gravity as a generalized YM theory can heuristically explain the double-copy procedure for 3-point amplitudes at tree level. Eq.(\[3-point-1\]) is the exact expression of the heuristic expression $f AAF_{(0)} \sim F_{(0)} A F_{(0)}$ there. It may seem that there is a small discrepancy between $F_{(0)}([A,A]+AF_{(0)})$ (\[3-point-1\]) and $F_{(0)}AF_{(0)}$. But recall that the structure constant $f_{abc}$ is assumed to be cyclic in the double-copy procedure (and in our heuristic discussion in Sec.\[heuristic\]), while $F_{(0)}^{abc}$ is not. The exact expression (\[3-point-1\]) is in fact of the form of $F_{(0)}AF_{(0)}$ but additional terms that have some of the indices permuted. In Sec.\[pert-hatA\] below, we will see a simpler and more direct match with the discussions in Sec.\[heuristic\]. The 3-point vertices for the gravitons are therefore (\^a h\^[bc]{}-\^b h\^[ac]{}) . This is already a very simple expression, especially if we compare it with the expression obtained from the Hilbert-Einstein action. But the expression can be even further simplified as we will shown below. Perturbative Expansion in $\hat{A}$ {#pert-hatA} ----------------------------------- It is more economic, at least at the lowest order, to use the variable $\hat{A}_{\m a}$ defined by \_\^a = \_\^a + \_\^[a]{}, where $\hat{e}_{\m}{}^a$ is the inverse of $\hat{e}_a{}^{\m}$ (\[def-e-A\]). The new variable $\hat{A}_{\m}{}^a$ is merely a field redefinition of $A_a{}^{\m}$. They are related via A\_a\^ = \_a\^ - \_a\^ = - \_a\^ + \_a\^\_\^ + [O]{}(A\^3). Therefore h\_[ab]{} - \_[ab]{} + . Up to sign, the physical (on-shell) amplitudes of $h_{ab}$ and $\hat{h}_{ab}$ should agree. As it is suggested by the notation, we have decomposed $\hat{A}$ as \_[ab]{} = \_[ab]{}/2 + \_[ab]{}, where $\hat{h}_{ab}$ is symmetric and $\hat{B}_{ab}$ is anti-symmetric. The trace part of $\hat{h}_{ab}$ is denoted [^4] \_a\^a. The effective metric (\[metric\]) is \_ = \_\^a\_[ab]{}\_\^b \_ + \_[ab]{} + 2 \_[ab]{} + [O]{}(A\^2). We also have \_\^[a]{}\_a\^ = D (1 + + [O]{}(A\^2)), and thus = 1 + D + [O]{}(A\^2). One can ignore the integration measure $\det\hat{e}$ when the 3-point vertex under consideration does not involve $\hat{\phi}$ as an external leg. Expanding the field strength in powers of $\hat{A}$, we have F\_[abc]{} &=& - \_a\^\_b\^(\_\_[c]{} - \_\_[c]{}), Then, F\_[abc]{}F\^[abc]{} &=& \^[(0)]{}\_[abc]{}\^[(0)abc]{} - 2 \^[(0)abc]{}\_a\^[d]{}\^[(0)]{}\_[dbc]{} + [O]{}(A\^4), \[F2\] where \^[(0)]{}\_[a]{} \_\_[a]{} - \_\_[a]{}. Similar to the perturbative expansion in terms of $A$, the second and third terms in the action (\[general-action\]) do not contribute to 3-point interactions of the traceless part of $h_{ab}$. It is remarkable that for this action (\[general-action\]) there is a single 3-point vertex for graviton interaction (the second term in (\[F2\])) \^[(3)]{} = - 2 \^[(0)abc]{}\_a\^[d]{}\^[(0)]{}\_[dbc]{}. This is precisely of the form $F_{(0)}AF_{(0)}$ needed to explain the double-copy procedure at tree level as we discussed in Sec.\[heuristic\]. This is of course also a significant simplification compared with the usual expression of GR. Higher-Form Gauge Symmetries {#HigherGauge} ============================ In the above we have focused on the gauge symmetries with 1-form potentials and 0-form gauge parameters. We can also apply the same notion of generalization to gauge symmetries with higher-form gauge potentials. The basic ideas of our generalization are the following: 1. \[g1\] The symmetry generators do not have to be factorizable in the form T({f, a}) = \_n f\_n(x) T\_[a\_n]{}, \[factorizable\] where $f_n(x)$’s are functions on the base space and $T_a$’s are elements of a finite-dimensional Lie algebra. 2. \[g2\] Even if the symmetry generators are formally factorable in the form (\[factorizable\]), the objects $T_a$’s do not have to commute with base-space functions. If we take a given principal bundle as a classical manifold and deform its algebra of functions so that it becomes noncommutative, in general, this noncommutative space is not the tensor product of a group and a noncommutative base space. This is a way to construct examples of the comment \[g1\] above. As an example of the comment \[g2\], even though the gauge symmetry constructed in Sec.\[GaugeSymm\] has generators of the form (\[factorizable\]) with $T_a = \del_a$, these $T_a$’s should not be interpreted as generators of a finite-dimensional Lie algebra (otherwise the Lie algebra is Abelian) and they do not commute with space-time functions. For higher-form gauge symmetries, only the Abelian case is well understood. There is no consensus on the definition of non-Abelian higher-form gauge symmetries, [^5] and concrete examples are scarce. Due to this reason, our discussion below cannot be very precise. It is commonly speculated that there is something analogous to the Lie algebra whose elements replace $T_a$ in the factorized formula (\[factorizable\]) of a gauge transformation generator. The analogue of our generalization of gauge symmetry for higher-form gauge symmetries is then referring to a violation of that factorization. The Nambu-Poisson gauge theory [@M5-C; @NP] is one of the few examples of non-Abelian higher-form gauge theories. It was used to describe an M5-brane in a large $C$-field background. It can be viewed as the covariant lift of the Poisson limit of the noncommutative gauge symmetry for a D4-brane in large $B$-field background to a higher dimension. The Nambu-Poisson gauge symmetry has a 2-form potential with 1-form transformation parameters. Its gauge group is the non-Abelian group of volume-preserving-diffeomorphisms. It can be viewed as an example of generalized gauge symmetry for higher form potentials. Incidentally, the fact that the gauge algebra involves kinematic factors is also the reason why it is possible for higher-form gauge symmetries to be non-Abelian. Higher-form global symmetries are always Abelian [@Gaiotto:2014kfa]. Hence an ordinary procedure of “gauging” the global symmetry by introducing space-time dependence to the generators in a way analogous to eq.(\[factorizable\]) can never result in a non-Abelian gauge symmetry (unless the space-time coordinates are noncommutative). Conversely, for a non-Abelian higher-form gauge symmetry, when the transformation parameters are restricted to be constant, all kinematic factors become trivial, and the symmetry algebra becomes Abelian. The Nambu-Poisson gauge symmetry is clearly an example of this fact. The noncommutative $U(1)$ gauge symmetry is the lower-form analogue. Another example of non-Abelian gauge symmetry with a 2-form gauge potential is the low energy effective theory for multiple M5-branes proposed in Ref.[@MM5; @Ho:2012nt]. The M5-branes are compactified on a circle, and the gauge transformation laws distinguish zero-modes from KK modes. The distinct treatment on zero-modes and KK modes can be viewed as a dependence on the kinematic factor (whether the momentum is zero or not), and so it is also an example of the generalized gauge theory for higher forms. There are other examples of non-Abelian gauge symmetry with higher-form gauge potentials [@2form], in addition those mentioned above. It will be interesting to explore further how the idea promoted above on generalized gauge symmetry will help the construction of a mathematical framework for non-Abelian higher-form gauge theories. Acknowledgement {#acknowledgement .unnumbered} =============== The author would like to thank Chong-Sun Chu, Kazuo Hosomichi, Yu-Tin Huang, Takeo Inami for their interest and discussions. 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[^3]: More explicitly [@Zhu:1980sz], n\_s\^[(4)]{} &=& \_1\^\_2\^\_3\^\_4\^ n\^[(3)]{}\_(-p\_4, -p\_3, p\_3+p\_4)n\^[(3)]{}\_\^(p\_1,p\_2,-p\_1-p\_2) + m\_s\^[(4)]{}, where $p_i^{\m}, \eps_i^{\m}$ ($i = 1, 2, 3, 4$) are the momenta and polarization vectors of the external legs, $s = (p_1+p_2)^2$ and m\_s\^[(4)]{} s\[(\_1\_4)(\_2\_3)-(\_1\_3)(\_2\_4)\]. (The choice of $m_s^{(4)}$ is not unique.) The other two kinematic factors $n^{(4)}_t, n^{(4)}_u$ can be obtained by permutations of external legs. [^4]: As we have learned from pure GR, the trace part of the fluctuation of the metric is not a physical propagating mode. Hence we should identify the trace part of $\hat{A}_{ab}$ (and $A_{ab}$) as an independent scalar field. [^5]: The mathematical structure for the symmetry of a 2-form gauge potential is called a non-Abelian gerbe. But there are different versions of its definition.

--- abstract: | We compute the warping of a disc induced by an inclined dipole. We consider a magnetised star surrounded by a thin Keplerian diamagnetic disc with an inner edge that corotates with the star. We suppose the stellar field is a dipole with an axis that is slightly misaligned with the stellar rotation axis. The rotation axes of the disc material orbiting at large distances from the star and that of the star are supposed to coincide. The misalignment of the magnetic and rotation axes results in the magnetic pressure not being the same on the upper and lower surfaces of the disc. The resultant net vertical force produces a warp which appears stationary in a frame corotating with the star. We find that, if viscosity is large enough ($\alpha \sim 0.01$–0.1) to damp bending waves as they propagate away, a smoothly varying warp of the inner region of the disc is produced. The amplitude of the warp can easily be on the order of ten percent of the disc inner radius for reasonably small misalignment angles (less than 30 degrees). Viscous damping also introduces a phase shift between the warp and the forcing torque, which results in the locations of maximum elevation above the disc forming a trailing spiral pattern. We apply these results to recent observations of AA Tau, and show that the variability of its light curve, which occurs with a period comparable to the expected stellar rotation period, could be due to obscuration produced by a warp configuration of the type we obtain. author: - Caroline Terquem - 'John C. B. Papaloizou' date: 'Received / Accepted: 07 June 2000' title: The response of an accretion disc to an inclined dipole with application to AA Tau --- Introduction ============ Objects accreting material through an accretion disc very commonly contain a significant magnetic field. This is the case for accreting white dwarfs in cataclysmic variables, some X–ray binary pulsars and at least some classical T Tauri stars (CCTS). It was first suggested by Bertout et al. ([@Bertout]), as a result of the detection of bright stellar spots, that CTTS may accrete along stellar magnetic field lines. This picture has been further supported by a wide array of observational evidence (see Najita et al. [@Najita] and references therein), including spectroscopic indications of infalling material onto the stellar surface (Edwards et al. [@Edwards2]; Hartmann et al. [@Hartmann]) and the low spin rate of CCTS (Bouvier et al. [@Bouvier1]; Edwards et al. [@Edwards1]). Since T Tauri stars have a large convective envelope, it is likely that at least part of their magnetic field is generated through a dynamo process. However, there may also be a fossil component originating from the molecular cloud out of which the star formed (Tayler [@Tayler]). Recent Zeeman measurements indicate relatively strong field strength at the surface of T Tauri stars, on the order of one kilogauss (Guenther et al. [@Guenther]; Johns-Krull et al. [@Johns]). It is not known what the structure of the field is. At some distance from the star the dipolar component probably dominates, but whether this is the case in the magnetosphere is not clear. However, observations cannot rule out such a coherent field structure (Montmerle et al. [@Montmerle]), and numerical simulations of nonlinear stellar dynamos indicate that a steady dipole mode is the most easily excited one (Brandenburg et al. [@brand1]). Interaction between the stellar magnetic field and the accretion disc has very important consequences for the disc structure, the accretion process (see Ghosh & Lamb [@Ghosh91] and references therein) and the evolution of the stellar rotation (Königl [@Konigl]). In particular, the disc is truncated by the magnetic torque, so that it does not extend down to the stellar surface (Gosh & Lamb [@Ghosh79]). The location of the disc inner radius is determined by the condition that magnetic and viscous torques balance. For CTTS, the radius of the inner cavity is believed to be a few stellar radii (see, e.g., Wang [@Wang]). So far, there are only a few numerical simulations of disc–stellar magnetic field interactions (Hayashi et al. [@Hayashi]; Miller & Stone [@Miller]; Goodson et al. [@Goodson]; Kudoh et al. [@Kudoh]). They all show the disc–magnetosphere interaction to be complex and sensitive to initial and boundary conditions. At this stage, it is not clear what final form a full theoretical model is likely to take. However, analytical or semi–analytical simplified models can still be valuable in pointing out some important processes that may arise in these systems, and the goal of this paper is to describe one of these processes. We note that the magnetic axis and the rotation axis of the disc at large distances from the star may not be aligned, although often, for simplicity, they are assumed to coincide. We assume here that the stellar rotation axis and the disc rotation axis at large distances coincide. Misalignment would then occur if, for instance, the star were to generate a dipole field with magnetic axis misaligned with its spin axis (like in the case of the Earth). It is possible that the interaction with the disc would lead to some evolution of the misalignment angle, but the details are likely to depend on the processes which generate the field. In any case, when such a misalignment is present, the magnetic pressure is not the same on the upper and lower surfaces of the disc. This mismatch generates a net vertical force which excites bending waves and warps inner parts of the disc (Aly [@Aly]). Bending instabilities in a disc subject to a stellar magnetic dipole have been investigated by Agapitou et al. ([@Agapitou], hereafter APT). APT calculated the global bending modes of a disc permeated by both an internally produced poloidal magnetic field and an external dipole field with axis aligned with the disc rotation axis (in this case no warp is induced by the dipole configuration, but free bending modes can be excited by a perturbation which takes the disc out of its equilibrium plane). They found that instability could occur if the magnetic and centrifugal forces were comparable in some region of the disc. They pointed out that such instabilities may result in the periodic variability observed in the light curve of many CTTS. Lai ([@Lai]) studied the warping of a disc induced by an inclined dipole. He calculated the magnetic torque exerted by an inclined dipole on a disc, and studied the stability against vertical displacements of a disc subject to such a torque. In the terms of the APT analysis, he studied the stability of low frequency (as measured in an inertial frame) bending modes corresponding to the modified tilt mode as discussed in APT. We note that, when considering the structure of the disc subject to the inclined dipole, he did not take into account the effects of the distortion of the disc itself on its response, which can have important consequences on the dynamics through wave propagation. But he added the effects of a toroidal field, assumed to be generated by winding up a penetrating vertical field, on the magnetic pressure determining the vertical force on the disc. This contribution is phase shifted with respect to the other contributions and may thus (if not counteracted) cause the modified tilt mode to become unstable, resulting in spontaneous warping. To decide whether this mode can be destabilised requires detailed consideration of the effects of wave propagation and viscosity. We comment that under some conditions warps diffuse away on a timescale much shorter than the viscous timescale (Papaloizou & Pringle [@Pap2]) or propagate away with a velocity on the order of the sound speed (Papaloizou & Lin [@Pap1]) resulting in stabilisation. In this paper, we calculate the structure of a thin Keplerian disc subject to an inclined dipole, taking into account the effects of the distortion of the disc itself on its response, i.e. the full dynamics of the system. For simplicity, we suppose that the disc is diamagnetic, so that it is not permeated by the external stellar field. In principle, the calculations presented here could be extended to more general cases. However, if the disc were not diamagnetic, wrapping of field lines would probably become important, leading to the possible disruption of the magnetosphere (see, e.g., Mikic & Linker [@Mikic]). Also we do not address here the physical processes of accretion or plasma entry into the stellar magnetosphere. We note that because the warp induced in the inner disc appears steady in a frame rotating with the star, any resulting variability would have the same period as that of the star. We comment that the generation of spontaneous warping does not apply to the calculations we present here, since we study a response which is forced by the inclined dipole and has a pattern speed equal to the rotation rate of the star. Thus, in contrast to the considerations of Lai ([@Lai]), it is not a modified tilt mode. This work has been motivated by a recent study of Bouvier et al. ([@Bouvier2]) who report that the light curve of the CTTS AA Tau displays photometric, spectroscopic and polarimetric variations on timescales from a few hours to several weeks. The most striking feature of this light curve is a photometric variability with a period comparable to the expected rotation period of the star. This has been interpreted by Bouvier et al. ([@Bouvier2]) as being due to the occultation of the star by a warp of the inner disc (the system is observed almost edge–on). The authors speculated that the warp could be produced by an inclined dipole. We note that Bouvier et al. ([@Bouvier2]) did not consider AA Tau as being a special case as far as its properties are concerned. They pointed out that only its light curve is unusual, and they interpreted it as being due to the fact that the system is seen almost edge–on. In other words, warping of the inner parts of CTTS discs would not be uncommon, but it could be seen only for particular viewing angles. The plan of the paper is as follows: We begin by considering an equilibrium configuration where the axis of the dipole and the rotation axis of the disc are aligned. This is described in § \[sec:equil\]. We then perturb this equilibrium by slightly inclining the dipole. In § \[sec:pert\] we calculate the resulting perturbed magnetic field and derive the integro–differential equation which has to be solved for the disc vertical displacement. This equation is solved in the WKB approximation, which is valid when the wavelength of the bending waves excited in response to the perturbation is small compared to the disc radius. To allow these waves to damp as they propagate away (and therefore to damp very small wavelength oscillations, which would be unphysical), we include the effects of viscous damping in the integro–differential equation. In § \[sec:results\] we solve this equation numerically and present the results for two different magnetic field equilibria (derived by Aly [@Aly] and Low [@Low]). In both cases we find that, if the viscosity is large enough to damp the waves as they propagate away, a smooth warp configuration of the disc inner parts can exist. The elevation above the disc equilibrium plane (i.e. the stellar equatorial plane) can easily be on the order of ten percent of the disc inner radius for reasonably small misalignment angles (less than 30 degrees). If the viscosity is too small to damp the waves efficiently, the disc inner parts may be disrupted, and it is likely that the evolution is then highly time–dependent. In § \[sec:discussion\] we apply these results to the case of AA Tau, and show that the variability of its light curve, which occurs with a period expected to be the stellar rotation period, can plausibly be explained by a warp configuration of the type we obtain. Disc and aligned dipole {#sec:equil} ======================= We begin by considering a thin disc configuration such that the gas orbits a central rotating star with a dipole field and where the magnetic axis and rotation axes are all aligned. In this situation, the configuration is axisymmetric. For convenience, we work in a frame corotating with the central star with angular velocity $\omega.$ The disc is truncated in its inner parts by the magnetic torque and we will suppose that the inner edge corotates with the star. Magnetic field {#sec:mfield} -------------- The dipole field ${\mathbf B}_{ext}$ due to the central star induces azimuthal currents in the conducting disc which in turn generate an additional poloidal magnetic field ${\mathbf B}_d$. Here, as in APT, we shall assume that the field external to the disc and central star can be approximated as curl free, i.e. assumed to be a vacuum field (the currents external to the disc and star are at infinity). The total axisymmetric poloidal magnetic field, ${\mathbf B}_{ext} + {\mathbf B}_d$, is noted ${\mathbf B}= (B_r,0,B_z)$, where we use cylindrical polar coordinates $(r,\varphi, z)$. The associated Cartesian coordinates are $(x,y,z)$. The flux function $\psi$ is such that: $$B_r= {-1 \over r}{\partial \psi \over \partial z} \; {\rm and}\ B_z = {1\over r}{\partial\psi\over \partial r}. \label{psi}$$ For the stellar dipole field: $$\psi_{ext} = -{\mu_d r^2 \over( r^2 +z^2 )^{3/2 }}, \label{psid}$$ where the magnitude of the stellar magnetic dipole moment is $\mu_d=B_{\ast} R_{\ast}^3$, with $B_{\ast}$ and $R_{\ast}$ being the stellar magnetic field and radius, respectively. For the general axisymmetric poloidal field, the associated current density is ${\mathbf j}=(0,j_{\varphi},0).$ For an infinitesimally thin disc, as we consider here, we define the vertically integrated azimuthal component of the current density, $J,$ such that: $$J=\int^{\infty}_{-\infty} j_{\varphi}{\rm d}z.$$ By integrating the azimuthal component of Ampère’s law through the disc, we obtain: $$B_r^{+}= \mu_0J/2, \label{br}$$ where $B_r^{+}$ denotes the radial component of the magnetic field just outside the upper surface of the disc. Throughout this paper we use MKSA units, and $\mu_0$ denotes the permeability of the vacuum. $B_r$ is antisymmetric with respect to reflection in the disc mid–plane so that its value just outside the lower surface of the disc is $B_r^{-}=-B_r^{+}.$ Force balance {#sec:forceb} ------------- Neglecting pressure forces as in APT, the vertical integration of the radial component of the momentum equation yields the condition for radial equilibrium as: $$\Sigma {\partial \Phi \over \partial r}=\Sigma r \Omega^2 + J B_z, \label{EQ}$$ where $\Sigma$ is the surface density, $\Omega$ is the disc angular velocity measured in an inertial frame, and $\Phi$ is the gravitational potential here taken to be due to a central point mass, $M_{\ast}$, such that: $$\Phi =-{GM_{\ast} \over \sqrt{r^2+z^2}}.$$ If $B_z \ne 0,$ in some regions of the disc a variety of configurations are possible (see APT). These include cases where inner field lines cross the disc and join to the central star and thus may be assumed to corotate with it. Outer field lines may be open in the case of an infinite disc or a finite disc with appropriate boundary conditions. In this paper, for simplicity, we shall perform calculations for the special case where the field is excluded from the disc. This situation arises when the disc is perfectly conducting. Then only surface currents flow in the disc and they screen the stellar dipole field from the disc interior, i.e. they produce an additional field ${\mathbf B}_d$ which cancels ${\mathbf B}_{ext}$ in the disc interior. We note that, since the vertical component of the field is continuous at the disc surface, $B_z$ just outside the disc is zero in this case. The same does not apply for $B_r$, and in general $B^+_r$ and $B^-_r$ are non zero. We remark that, since there is no Lorentz force acting in the disc when $B_z=0$, the angular velocity given by equation (\[EQ\]) is Keplerian. Disc and slightly misaligned dipole {#sec:pert} =================================== We suppose that the system is perturbed from the axisymmetric equilibrium state described above by introducing a slight misalignment between the magnetic axis of the central dipole and the rotation axis of the disc and star (the $z$ axis). This produces a non axisymmetric response in the disc that can be described using linear perturbation theory. The main features of this response is that it takes the form of a warping of the disc, as indicated by Aly ([@Aly]), together with the additional feature of the excitation of bending waves. The response is naturally largest in the inner parts of the disc where the magnetic field is strongest. As we indicate by considering specific examples in § \[sec:results\], it can take the form of a steeply changing inclination of the inner disc orbits which can make it appear to have an inner wall. To calculate the geometry of the disc (that is its elevation above its equilibrium plane), we extend the analysis of APT. To compute the free bending modes of the disc, APT solved an eigenvalue problem where the eigenvalues were the mode frequencies and the eigenfunctions their amplitudes. Here we are interested in a response with frequency equal to that of the forcing term. Since the dipole is anchored in the star, this is the angular velocity of the star. The amplitude of the mode (its spatial dependence) can be found from the mode equation of APT with a specified frequency and the addition of the forcing term. However, a different equilibrium field is adopted, since here, in contrast to APT, there is no internally produced field in the disc. Perturbed magnetic field {#sec:bpert} ------------------------ To calculate the response we suppose the central dipole moment is rotated in the $(x,z)$ plane through a small angle $\delta$ being the inclination to the $z$ axis. The dipole moment is then given by: $$\mbox{\boldmath $\mu$}_d = (\mu_d \delta, 0, \mu_d ). \label{psidm}$$ This contributes to a potential: $$\Phi'_{{\rm M},ext} = \frac{ \mbox{\boldmath $\mu$}_d \cdot {\mathbf r}}{r^3} , \label{phiext1}$$ where ${\mathbf r}$ denotes the position vector, which produces a radial external magnetic field perturbation $B'_{r,ext}=\partial \Phi'_{{\rm M},ext} / \partial r$. Just outside the disc surfaces, this is given by the real part of: $$B'^+_{r, ext} = B'^-_{r, ext} = -{ 2 \mu_d \delta \exp(i\varphi) \over r^3}. \label{bext}$$ Thus the problem reduces to the calculation of the disc response to a field perturbation with azimuthal mode number $m=1.$ This, when acting with the unperturbed azimuthal current, produces a vertical Lorentz force which tends to warp the disc. In other terms, when the dipole is misaligned, the magnetic pressure force is not the same on the upper and lower surfaces of the disc. This mismatch generates a net vertical pressure force which tends to warp the disc. We thus introduce the Lagrangian displacement $\mbox{\boldmath $\xi$}$ which, in a razor–thin disc approximation, has the form: $$\mbox{\boldmath $\xi$} = (0,0,\xi_z).$$ The only non-negligible component is the vertical one (APT), and $\xi_z$ represents the elevation above the disc equilibrium plane. In keeping with the form of the external magnetic field perturbation, we make all the perturbed quantities complex by taking their $\varphi$-dependence to be through a factor $\exp( i \varphi)$, which henceforth will be dropped. The physical perturbations will be recovered by taking the real part of these complex quantities. The Eulerian perturbations of the various quantities are denoted by a prime. The perturbation of the magnetic field interior to the disc, ${\mathbf B}'$, is related to $\mbox{\boldmath $\xi$}$ by the flux freezing condition: $${\mathbf B}' = (B'_r, B'_{\varphi}, B'_z) = \mbox{\boldmath $\nabla$} \mbox{\boldmath $\times$} (\mbox{\boldmath $\xi$} \mbox{\boldmath $\times$} {\mathbf B}).$$ The non-zero components of ${\mathbf B}'$ take the form: $$B'_r = -\xi_z {\partial B_r\over \partial z},\ {\rm and }\ B'_z= {1 \over r}{\partial (rB_r \xi_z)\over \partial r} . \label{bpert}$$ Since $B_r$ is antisymmetric with respect to reflection in the disc mid–plane and $\xi_z$ is independent of $z$ in the thin disc approximation, $B'_z$ is antisymmetric and $B'_r$ is symmetric. As in APT, the vertical component of the perturbed field must be matched to the vertical component of a perturbed vacuum field exterior to the disc. Here this is taken to be a potential field. Therefore we have: $$\frac{\partial \Phi'_{\rm M}}{\partial z} = B'^{+}_z, \label{derphi}$$ where $B'^{+}_z$ is the value of the vertical field perturbation just outside the disc surface and $\Phi'_{\rm M}$ is the magnetic potential associated with the external field perturbation. To find $\Phi'_{\rm M}$, we first subtract out the contribution of the external field perturbation arising from the tilted dipole. At the disc surface, this is (eq. \[\[phiext1\]\]): $$\Phi'_{{\rm M},ext} = { \mu_d \delta \over r^2}, \label{phiext}$$ where as above the factor ${\rm exp}(i \varphi)$ has been dropped. With this contribution removed, the residual potential has no singularity outside the disc. It can be calculated from equation (\[derphi\]) in an analogous manner to finding the gravitational potential due to a disc surface density distribution, with $2 \pi G \Sigma$ (where $G$ is the gravitational constant) being replaced by $B'^+_z$ (see Tagger et al. [@Tagger]; Spruit et al. [@Spruit]; APT). Following this procedure, $\Phi_{\rm M}'$ may be written as the sum of a Poisson integral and $\Phi'_{{\rm M},ext}$: $$\Phi'_{\rm M} = \Phi'_{{\rm M},d} + \Phi'_{{\rm M},ext} , \label{phisum}$$ where $$\Phi'_{{\rm M},d}= - {1 \over 2 \pi} \int^{R_o}_{R_i} \int^{2 \pi}_0 { B'^{+}_z(r') \cos \varphi' \; r' {\rm d}r' {\rm d}\varphi' \over \sqrt{ r'^2 + r^2 - 2rr'\cos(\varphi') + z^2 } }, \label{mpot}$$ where $R_i$ and $R_o$ are the inner and outer radii of the disc, respectively. In the above integral, the $\varphi$-dependence of the perturbed field has been taken into account. Here again, the factor $\exp( i \varphi)$, to which $\Phi'_{{\rm M},d}$ is proportional, has been dropped. Since the disc is perfectly conducting, the vertical component of the field given by equation (\[bpert\]) is continuous at the disc surface. Therefore: $$B'^{+}_z = {1 \over r}{\partial (rB^{+}_r \xi_z)\over \partial r}. \label{bpzp}$$ This expression for $B'^+_z$ can be used in the above integral. The radial component of the magnetic field just outside the surfaces of the disc is given by: $$B'^{+}_r= B'^{-}_r =\left({\partial \Phi'_{\rm M} \over \partial r}\right)_{z=0} = \left({\partial \Phi'_{{\rm M},d} \over \partial r}\right)_{z=0} - {2 \mu_d \delta \over r^3} , \label{pot}$$ where equations (\[phiext\]) and (\[phisum\]) have been used to write the last expression (we have approximated the derivatives at the disc surfaces by their value at $z=0$ because the disc is infinitesimally thin). The disc vertical displacement ------------------------------- The vertical component of the perturbed Lorentz force integrated vertically through the disc is (see APT): $$\int^{\infty}_{-\infty} F'_z {\rm d}z = -{2 B_r^{+} B_r'^{+} \over \mu_0} %- {2 \xi_z B_r^{+} \over \mu_0} {\partial B_z \over \partial r} . \label{force}$$ The term proportional to $\partial B_z / \partial r$, which was present in APT, is zero here since $B_z=0$ in the disc interior and at its surfaces (see § \[sec:forceb\]). We note that this force is simply the perturbed magnetic pressure force $P'_m$ vertically integrated through the disc. The pressure force vertically integrated through the disc is indeed: $$P_m = - \frac{ \left( B_r^{+} \right)^2}{2 \mu_0} + \frac{ \left( B_r^{-} \right)^2}{2 \mu_0} , \label{pm}$$ ($B_z=0$ at the disc surfaces), so that: $$P'_m = - \frac{B^+_r B'^+_r}{\mu_0} + \frac{B^-_r B'^-_r}{\mu_0} = - \frac{2 B^+_r B'^+_r}{\mu_0}, \label{Ppert}$$ where we have used the symmetries of the field. We note that $P'_m$ is non zero because the perturbation induced by the misaligned dipole has an opposite effect on the upper and lower surfaces of the disc: it increases the magnitude of the radial component of the magnetic field on one of the surfaces from $|B^+_r|$ to $|B^+_r|+|B'^+_r|$ while decreasing it on the other surface from $|B^+_r|$ to $|B^+_r|-|B'^+_r|$. The vertically integrated $z$-component of the perturbed equation of motion is: $$\Sigma{{\rm D}^2 \xi_z \over {\rm D}t^2}= -\Sigma \left({\partial^2 \Phi \over \partial z^2}\right)_{z=0} \xi_z + \int^{\infty}_{-\infty} F'_z {\rm d}z \label{mot}$$ where ${\rm D}/{\rm D}t$ denotes the convective derivative. In the problem considered here, the solution is steady in a frame rotating with the central star angular velocity $\omega.$ Then: $$\Sigma{{\rm D}^2 \xi_z \over {\rm D}t^2}= -\Sigma(\omega -\Omega)^2\xi_z. \label{moto}$$ Also, for a point mass potential: $$\left({\partial^2 \Phi \over \partial z^2}\right)_{z=0}={GM \over r^3}=\Omega_{\rm K}^2, \label{gravo}$$ where $\Omega_{\rm K}$ is the Keplerian angular velocity. Using equations (\[pot\]), (\[force\]), (\[moto\]) and (\[gravo\]), equation (\[mot\]) becomes: $$\left[ - (\omega-\Omega)^2 + \Omega_{\rm K}^2 % + {2B_r^{+} \over \mu_0 % \Sigma} {\partial B_z \over \partial r} \right] \xi_z = - {2B_r^+ \over \mu_0 \Sigma} \left[ \left( {\partial \Phi'_{{\rm M},d} \over \partial r} \right)_{z=0} - {2 \mu_d \delta \over r^3} \right] . \label{mod1}$$ The potential $\Phi'_{{\rm M},d}$ can be expressed in term of $\xi_z$ and its derivative with respect to $r$ using eq. (\[mpot\]) and (\[bpzp\]). Therefore equation (\[mod1\]) gives a linear equation for determining the vertical displacement $\xi_z$ induced by the tilted external dipole. The term proportional to $\mu_d$ in this equation acts like a forcing term for the vertical displacement and it causes the disc to become warped and may also excite bending waves. As noted above, since there is no Lorentz force acting in the disc interior, $\Omega$ in equation (\[mod1\]) is the Keplerian angular velocity. We remark that equation (\[mod1\]) is the same as equation (14) of APT with $B_z=0$, but with an additional nonhomogeneous term proportional to $\mu_d$. As noted above, this is expected as here we calculate the response which is forced by the nonaligned dipole. We remark that this nonhomogeneous term is simply the pressure force due to the misaligned component of the dipole vertically integrated through the disc: $$P'_{m,ext} = - \frac{2 B^+_r B'^+_{r,ext}}{\mu_0} = \frac{2 B^+_r}{\mu_0} \frac{2 \mu_d \delta}{r^3},$$ where we have used equation (\[bext\]) (see also eq. \[24\] of Aly [@Aly]). WKB waves {#sec:dispersion} --------- When $\mu_d$ is set to zero in equation (\[mod1\]), this equation has solutions corresponding to free bending waves. In the local limit, these can be found by assuming that $\xi_z \propto \exp(ikr),$ where $k$ is the radial wavenumber, assumed to be very large compared to $1/r$. The integral in equation (\[mpot\]) can be evaluated in the WKB approximation to give, after using (\[bpzp\]) (see APT and references therein): $$\Phi'_{\rm M} = \frac{-B'_z}{|k|} = \frac{ - i k B_r^{+} \xi_z}{|k|} .$$ The local dispersion relation derived from equation (\[mod1\]) is then: $$(\omega - \Omega_{\rm K})^2= \Omega_{\rm K}^2 % + {2B_r^{+}\over \mu_0 \Sigma} {\partial % B_z\over \partial r} + {2(B_r^{+})^2\over \mu_0 \Sigma}|k|, \label{fb}$$ where we have used the fact that the angular velocity is Keplerian. We see from equation (\[fb\]) that bending waves with $m=1$ propagate (with $|k|>0$) exterior to the Lindblad resonances where $(\omega-\Omega_{\rm K}) = \pm \Omega_{\rm K}$. In this paper, we are concerned with the situation where the disc is terminated at an inner boundary $(r=R_i)$ where the local Keplerian rotation rate is close to corotation with the central star, i.e. $\Omega_{\rm K}(R_i) = \omega$. In such a case, only the outer Lindblad resonance (OLR), where $\Omega_{\rm K} = \omega /2$, will exist within the disc, and it occurs at a radius $r= 1.59 R_i.$ Significantly beyond the OLR, the wavenumber associated with the waves is given by: $$|k| r \sim {\mu_0 \omega^2 r \Sigma \over 2 (B_r^{+})^2 } \propto \frac{r^3}{\beta},$$ where we define $\beta$ such that it would be the ratio of magnetic to centrifugal forces if we had a vertical field $B_z \sim B^+_r$ (see § \[sec:results\]). Although $B_z=0$ here, we use $\beta$ as a measure of the strength of $B^+_r$. This is expected to be at most of order unity at the inner edge of the disc and then to decrease rapidly outwards, as the magnetic field decreases (see § \[sec:results\]). If $\beta$ decreases as $r$ increases, the forcing term proportional to $\mu_d$ in equation (\[mod1\]) will excite bending waves in the disc that propagate outwards with increasing wavenumber until they are damped, much as in the case of Saturn’s rings (see Goldreich & Tremaine [@Goldreich]; Shu [@Shu]). In some cases, the wavelength may be so short at the OLR that dissipative processes prevent waves from being properly launched (see below). Viscous dissipation ------------------- In order to allow the waves to damp as they propagate away from the OLR, we add an additional viscous force per unit mass to the right hand side of equation (\[mot\]) of the form $\nu \partial^2 v'_z / \partial r^2$, where $v'_z$ is the Eulerian perturbed vertical velocity and $\nu$ is a kinematic viscosity. Neglecting the variation of $\Omega_K,$ which is reasonable for short wavelength disturbances, this can also be written as $i \nu (\Omega_{\rm K} -\omega) \partial^2 \xi_z / \partial r^2$. For $\nu$ we use a standard ’$\alpha$’ prescription such that $\nu = \alpha H^2 \Omega_{\rm K}$, where $H$ is the disc semi–thickness and $\alpha$ is a constant (Shakura & Sunyaev [@Shakura]). With the incorporation of the above viscous force, equation (\[mod1\]), which gives the forced response of the disc, becomes: $$\left[ - ( \omega - \Omega)^2 + \Omega_{\rm K}^2 % + {2 B_r^{+} \over \mu_0 \Sigma} % { \partial B_z \over \partial r} \right] \xi_z - i \nu ( \Omega_{\rm K} - \omega) {\partial^2 \xi_z \over \partial r^2} = - { 2 B_r^{+} \over \mu_0 \Sigma } \left[ \left( { \partial \Phi'_{{\rm M},d} \over \partial r } \right)_{z=0} - {2 \mu_d \delta \over r^3} \right] . \label{mod}$$ We now comment briefly on the possible origin of this $\alpha$ viscosity. So far, the only process which has been shown to initiate and sustain turbulence in Keplerian discs is the magnetohydrodynamic Balbus–Hawley instability, and it does lend itself to an $\alpha$ formalism (Balbus & Hawley [@Balbus] and references therein). For the disc we consider here, which is not permeated by the external magnetic field, it is difficult to justify the existence of an $\alpha$ viscosity. However, in reality, the disc is probably permeated to at least a small extent by the external field (Ghosh & Lamb [@Ghosh79]). If the ratio of magnetic to thermal pressure is smaller than unity, the disc is then subject to the BH instability, the internal disc field is amplified and bending waves are damped. The values of $\alpha$ produced by the BH instability range roughly from $10^{-3}$ to 0.1 (Hawley et al. [@Hawley1]; Brandenburg et al. [@brand2]), the largest values corresponding to the case where the magnetic field varies on a scale large compared to the disc semithickness (Hawley [@Hawley2]). Numerical response calculations {#sec:results} =============================== Unperturbed magnetic field {#sec:bequil} -------------------------- For the model calculations presented here, we adopt a radial magnetic field corresponding to the situation when the central dipole flux is completely excluded from the disc. We consider both the solutions computed by Aly ([@Aly]) and Low ([@Low]). Aly’s solution corresponds to the case when all the dipole flux goes through the magnetospheric cavity in the middle of the disc, and is singular at the disc inner edge. Low gives a solution that is non singular with some flux escaping to infinity. In these cases we have, for $r$ larger than some radius $r_B$: $$B_r^+ = \mp B_0 \left( \frac{R_o}{r} \right)^3 \left[ \left( \frac{r}{r_B} \right)^2 -1 \right]^{\pm 1/2} , \label{Aly}$$ where we have defined: $$B_0= \frac{4 \mu_d}{\pi R_o^3}. \label{B0}$$ In equation (\[Aly\]), the upper sign corresponds to Low’s solution, whereas the lower sign corresponds to Aly’s solution. Of course, in both cases $B_z=0$ since the disc is diamagnetic. Note that the radial field at the disc surface has opposite sign in the two cases. We comment that $B_r^+$ is negative and positive for Low’s and Aly’s cases, respectively, which is the opposite of the solution obtained by these authors. This is because we have arbitrarily chosen $B_z$ to be positive when the dipole moment $\mu_d$ is positive, in contrast to these authors. In a two dimensional model, $r_B$ is the disc inner radius. However, more realistically it is expected to differ from this by an amount comparable to the vertical thickness of the disc. We adopted the procedure of taking the disc inner radius $R_i$ to be slightly larger than $r_B$, and to check convergence of the results by decreasing $\left( R_i - r_B \right)$. For Aly’s solution, convergence requires that the disc surface density increases rapidly enough towards $R_i$ such as to limit the Lorentz force there. Results ------- We have performed global normal mode calculations for disc models with the unperturbed magnetic field described above. We solve equation (\[mod\]) considered as an integro-differential equation for the response function $\xi_z$ by the method described in APT. We use the dimensionless radius $\varpi=r/R_o$, so that the outer radius is $\varpi_o=1$. In these units, the disc inner radius on the computational grid is taken to be $\varpi_i=R_i/R_o=0.1$, and to corotate with the central star so that $\omega = \Omega_K(R_i)$. The radial interval $\left[\varpi_i, 1 \right]$ is divided into a grid of $n_r$ equally spaced points at positions $\left( \varpi_I \right)_{I=1 \ldots n_r}$ with a spacing ${\Delta \varpi}_I = \varpi_{I+1}-\varpi_{I}$. Equation (\[mod\]) is solved as in APT by discretizing it on the grid so converting it into a matrix inversion problem for $\left( \xi_z \right)_{I=1 \ldots n_r}$. In the calculations presented below, $n_r$ was varied between 500 and 700, but 300 already gave satisfactory results. As already noted in § \[sec:dispersion\], we use the magnetic support $\beta$, which would be the ratio of the Lorentz force to the centrifugal force in the disc if $B_z$ were non zero and on the order of $B^+_r$: $$\beta = \frac{ 2 \left( B^+_r \right)^2 r^2}{\mu_0 \Sigma G M_{\ast}} , \label{beta1}$$ where $G$ is the gravitational constant. Although $B_z=0$ in the disc, $\beta$ will be used as a measure of the strength of the field $B^+_r$. We found it convenient to define the dimensionless quantities $\bar{\Sigma} = \Sigma / \Sigma_i$, where $\Sigma_i$ if the surface density at the disc inner radius, $\bar{B}_r^+ = B_r^+ /B_0$, and $\bar{\beta} = \beta / \beta_0 $ with: $$\beta_0 = \frac{ B_0^2 R_o^2 }{ \mu_0 \Sigma_i G M_{\ast} }. \label{beta0}$$ We then have: $$\bar{\beta} = \frac{2 \varpi^2 \left( \bar{B}_r^+ \right)^2}{\bar{\Sigma}} . \label{beta}$$ For both Low’s and Aly’s magnetic field, we consider models where the magnetic support is large (reaching values of order unity) close to the disc inner edge and decreases rapidly with radius. For Aly’s solution, we take $\beta=1$ at the disc inner edge. For Low’s solution, $\beta$ has to vanish at the same location as $B^+_r$, i.e. at $\varpi_B=r_B/R_o$, otherwise $\xi_z$ is infinite there. We then take $\beta=0$ at $\varpi_B$ and $\beta=1$ at the location where $B^+_r$ has its maximum. From the value of $\beta$ and $\bar{B}_r^+$ at the disc inner radius, we can calculate $\beta_0$. From the profile of $\bar{\beta}$, we calculate $\bar{\Sigma}$ using equation (\[beta\]). We actually take the profile of $\bar{\Sigma}$ which gives the desired $\bar{\beta}$ near the disc inner edge, and we then match it on to a $r^{-3/2}$ profile further away in the disc (since $B^+_r$ decreases rapidly with radius, we do not need to worry about the profile of $\Sigma$ for $\varpi$ larger than a value which turns out to be about 0.2). We note that our results do not depend on the numerical values of $\Sigma_i$, $B_0$, and $R_i$ (or equivalently $R_o$) taken individually, but only on $\beta_0$ and $\delta$. For $R_i$ we could take a few times $R_{\ast}$, and then compute $R_o=10 R_i$. We could then get a value of $B_0$ by using equation (\[B0\]) with $\mu_d = B_{\ast} R_{\ast}^3$. Then $\Sigma_i$ would follow from the expression (\[beta0\]) for $\beta_0$. Depending on the value of $\Sigma_i$, the amount of mass in the annulus between $R_i$ and $R_o$ could be modified by changing the profile of $\bar{\Sigma}$ for $\varpi > 0.2$. An external poloidal magnetic field tends to squeeze the disc. Therefore, we choose a form of the aspect ratio $H/r$ which is zero at the disc inner edge and which increases as the magnetic support decreases. The disc is squeezed by the magnetic field if the magnetic pressure dominates over the thermal pressure. If the sound speed is about a tenth of the Keplerian velocity, then the magnetic and thermal pressures become comparable when the magnetic support is about 0.1. Therefore, we choose a profile of $H/r$ which reaches a constant value (that we take to be 0.1) at the location where $\beta=0.1$. We note that the vertical displacement of the disc at a location $(r, \varphi)$ is: $${\mathcal R}[\xi_z(r)] \cos \varphi - {\mathcal I}[\xi_z(r)] \sin \varphi. \label{physxi}$$ Figures \[fig1\]–\[fig3\] show the magnetic support $\beta$, the disc aspect ratio $H/r$ and the surface density $\Sigma/\Sigma_i$ versus $\varpi$. Also displayed are ${\mathcal R}(\xi_z)/(2\delta R_o)$, i.e. $\xi_z/(2\delta R_o)$ for $\varphi=0$ versus $x/R_o$, and $-{\mathcal I}(\xi_z)/(2\delta R_o)$, i.e. $\xi_z/(2\delta R_o)$ for $\varphi=\pi/2$ versus $y/R_o$, for $\alpha=10^{-3}$ and 0.1. These quantities represent the disc vertical displacement within a factor $2 \delta R_o$ in the $(\varphi=0)$ and $(\varphi=\pi/2)$ half planes, respectively, or, equivalently, in the $(x>0, z)$ and $(y>0, z)$ half planes, respectively (see eq. \[\[physxi\]\]). The different figures correspond to different models. Figure \[fig1\] and \[fig2\] correspond to Aly’s solution and $\varpi_B=0.0975$. In Figure \[fig1\], $1/\beta \propto {\rm exp} \left[ 100 (\varpi-0.14) \right] +1$. The results do not depend significantly on the details of the $H/r$ profile in the disc inner parts (we considered both the case where $H/r$ increases almost linearly and exponentially with radius). We also checked that $\xi_z$ hardly changes when $\varpi_B$ varies between 0.09 and 0.099, providing we keep the same magnetic support (i.e. we change $\Sigma$ accordingly). Figure \[fig2\] corresponds to $\beta \propto B^+_r / \left\{ {\rm exp} \left[ 40 (\varpi-0.14) \right] +1 \right\}$. The results are qualitatively the same as in Figure \[fig1\]. We also ran the case $\beta \propto 1/\varpi^3$, which gave similar results, except that the positive values reached by $\xi_z$ in that case were almost as large as the negative values. Figure \[fig3\] corresponds to Low’s solution and $\varpi_B=0.09999999$. Here again, the results hardly depend on the details of the profile of $H/r$ in the disc inner parts. From equation (\[mod\]), we see that $\xi_z$ vanishes at $\varpi=\varpi_B$ if $\Sigma$ is non zero there. When $\varpi_B$ is very close to $\varpi_i$, we indeed check that $\xi_z$ is almost zero at the disc inner edge. However, since $B^+_r$ varies very rapidly near $\varpi_B$, $\xi_z$ takes some finite value at the inner edge as $\varpi_B$ is moved a little bit away from $\varpi_i$. This is illustrated in Figure \[fig4\], where we plot $\beta$ and $\xi_z$ for different values of $\varpi_B$, ranging from 0.09 to 0.09999999 (all the curves have the same parameters except for $\varpi_B$). The calculations for the low viscosity $\alpha = 10^{-3}$ illustrate the excitation of bending waves. Several oscillations corresponding to bending waves excited at the OLR are visible. These are damped by viscosity before they propagate very far. This damping is enhanced by the increasing wavenumber produced by the decreasing magnetic field. For larger and probably more realistic viscosity $\alpha= 0.1,$ these waves are so heavily damped they are barely visible. The dominant displacement is near the disc inner edge in this case. We note that, when viscous damping is present, the response of the disc is not in phase with the perturbation, and the warp lags behind the dipole. That is, the disc is also twisted, and the twist is trailing. Mathematically, this is illustrated by the fact that the imaginary part of $\xi_z$ is non zero, as seen in the different figures. It may seem surprising that, even when $\alpha$ is as low as $10^{-3}$, the imaginary part of $\xi_z$ is comparable in magnitude to its real part. This is because $\xi_z$ varies rapidly with radius when $\alpha$ is small, and the imaginary part of $\xi_z$ becomes important when $| \partial^2 \xi_z/\partial r^2 |$ is large (see eq. \[\[mod\]\]). We checked however that the imaginary part of $\xi_z$ becomes very small compared to its real part when $\alpha$ is reduced further. Practically, the existence of a twist means that the point of maximum elevation in the disc is not in the plane which contains the dipole axis and the rotation axis of the star, i.e. the $(\varphi=0)$, or equivalently $(x,z)$, plane, but in a plane corresponding to a smaller (negative) $\varphi$. Or, in other words, the elevation does not vanish in the $(\varphi=\pi/2)$, or equivalently $(y,z)$, plane, but in a plane corresponding to a smaller value of $\varphi$. To illustrate this, we show in Fig. \[fig5\] the line of nodes (which joins the points of zero elevation) in the inner parts of the $(x,y)$ plane for both Aly’s and Low’s models and $\alpha=0.1$. We clearly see that the line of nodes is trailing. If there were no twist, it would indeed coincide with the $y$–axis. The curves corresponding to different values of $\varpi_B$ or different models are very similar to those displayed in Fig. \[fig5\]. For $\alpha=10^{-3}$, the line of nodes is more tightly wrapped, because $\xi_z$ oscillates more rapidly. In Fig. \[fig6\] we present a surface plot of the warped disc corresponding to the Low’s model displayed in Fig. \[fig3\]. This clearly shows the presence of an obscuring wall like feature near the disc inner boundary and the trailing twist. Discussion and application to AA Tau {#sec:discussion} ==================================== The light curve of the classical T Tauri star AA Tau displays photometric, spectroscopic and polarimetric variations on timescales from a few hours to several weeks (Bouvier et al. [@Bouvier2]). The most striking feature of this light curve is a photometric variability with a period of 8 to 9 days, comparable to the expected rotation period of the star. This has been interpreted by Bouvier et al. ([@Bouvier2]) as being due to the occultation of the star by a warp of the inner disc (the system is observed almost edge–on). They proposed that this warp be produced by the interaction of the disc with a stellar magnetic dipole tilted with respect to the disc rotation axis. Following Mahdavi & Kenyon ([@Mahdavi]), they assumed that material at the disc inner edge, neglecting warping, would be more likely to accrete along the shorter field line than along the longer one connecting star and disc. In their model, based on this accretion geometry, the disc is elevated above the original equatorial plane at the location where the dipole axis is bent toward the disc. This geometry enables at least part of the stellar hot spots, located at the field line footpoints , to still be on the line of sight when occultation of the star is maximum, in agreement with the observations. In this model, the warp is not calculated self-consistently, and is not actually a real dynamical bending of the disc. The assumption by Mahdavi & Kenyon ([@Mahdavi]) can become incorrect, precisely because it neglects the warping of the disc. If the inner disc is no longer in the original equatorial plane, then material at the inner edge no longer ’sees’ the short and long field lines it would see if it were in the unperturbed disc plane. At the disc inner edge, since $\Omega(R_i) = \Omega_{\rm K}(R_i) = \omega$, we see from equation (\[mod\]) that the sign of $\xi_z$ is the same as that of $P'_m$ (i.e., from eq. \[\[Ppert\]\], the same as that of $- B_r^{+} B'^{+}_r $) for $\varphi=0$, and that determines the direction in which the disc bends. From Figure \[fig3\], we see that for Low’s solution the disc inner edge on the $x$–axis is pushed below the equatorial plane, which is the opposite of what was anticipated by Mahdavi & Kenyon ([@Mahdavi]). For Aly’s solution, the disc is bent above the equatorial plane (see Fig. \[fig1\] and \[fig2\]). It is not surprising that the two cases give different results since $B_r^+$ changes sign from one case to another. To get the elevation of the disc above the equatorial plane, the real and imaginary parts of $\xi_z$ have to be combined through equation (\[physxi\]). In Fig. \[fig7\], we show the maximum elevation $\xi_{z,max}$ (where the $\varphi$ dependence has been taken into account in $\xi_z$), normalized to unity, above the equatorial plane along the $-\varphi$ direction, versus $-\varphi$. This represents the maximum elevation of the disc material located in between the star and an observer looking at the disc almost edge–on and from above along the $-\varphi$ direction. The curves correspond to Aly’s and Low’s models displayed in Fig. \[fig1\] and \[fig3\], respectively, and $\alpha=0.1$. Let us suppose that, to begin with, we look at the system along the $\varphi=0$ direction. Then, as the star rotates, our line of sight moves to smaller, negative values of $\varphi$ (to the right along the $x$–axis of Fig. \[fig7\]). In the case of Aly’s model, this corresponds to the star being occulted at first and until it has rotated by about 90 degrees. Then, as the star rotates further, it becomes less and less obscured, and it can be seen by the observer until it has completed a full rotation. In the case of Low’s model, the star is not obscured at first. Occultation begins only after the star has rotated by about 180 degrees and lasts also for about a quarter of a period. We note that, depending on the direction of the line of sight, the radius at which the elevation is maximum varies (typically between 0.1 $R_o$ and 0.2 $R_o$). It is unlikely that the observer would be able to know at which radius is the part of the disc responsible for the occultation, as the whole structure rotates with the same frequency (that of the star). Indeed, it is important to stress again that a structure located at some radius $r$ does not rotate with the local angular frequency but with that of the star (in other words, the warp appears steady in a frame rotating with the star). With Low’s solution, the disc is pushed below the equatorial plane at the location where the dipole is bent toward the disc. Therefore the stellar hot spots, at the field line footpoints cannot be seen under condition of maximum occultation, unless they are relatively large. In Aly’s case, at least part of the hot spots can still be observed at an occultation maximum because of the phase–lag between the disc response and the perturbation. Indeed, we see in Fig. \[fig7\] that, for the particular Aly’s model represented there, the occultation is maximum when $\varphi \sim -15^{\circ}$, and not when $\varphi=0$. Therefore, it seems the observations favour Aly’s model. To fit the observations, Bouvier et al. ([@Bouvier2]) needed the amplitude of the warp to be about 0.3 times the disc inner radius. We see from Figures \[fig1\]–\[fig3\] that this is easily attained here. There, $\left| \xi_z \right| /(2 \delta R_i)$ ranges from about 0.1 to 1. For $\delta$ up to about $30^{\circ}$ (for which $\sin \delta \simeq \delta$ in radians), this gives $\left| \xi_z \right|/R_i$ from 0.1 to 1. The lower value could even be made larger by considering a profile of $H/r$ with an exponential rather than almost linear increase in Figure \[fig2\]. Therefore only a moderate misalignment angle is required to produce $\left| \xi_z \right| /R_i \sim 0.3$. Figures \[fig1\]–\[fig3\] show that when the viscosity in the disc is small ($\alpha=10^{-3}$ for instance), the vertical displacement of the disc varies rapidly, on a scale smaller than the disc semi–thickness $H$. In that case, we expect the warp to become dispersive (Papaloizou & Lin [@Pap1]), and probably the disc to disrupt. It is likely that different situations arise when a disc interacts with a nonaligned dipole, depending on the detailed physics. In some cases we may get a moderate smoothly varying warp, as described above, in other cases it may be that the disc breaks up and reforms. This may also produce light curve variability. We note that the bending waves which are excited by the tilted dipole transport angular momentum (Papaloizou & Terquem [@Pap3]; Terquem [@Terquem1]), which is deposited in the disc if the waves are damped. Since the dipole rotates faster than the disc in which the waves propagate, the resulting torque opposes accretion onto the star. This effect may produce additional variability, and will be the subject of another paper. Acknowledgements {#acknowledgements .unnumbered} ================ It is a pleasure to thank Jérôme Bouvier for many interesting and stimulating discussions about AA Tau and for useful comments on an earlier draft of this paper, and Claude Bertout for further information about the variability of CCTS. We also thank an anonymous referee and A. P. Goodson whose suggestions helped to improve the quality of this paper. JCBP is grateful to the Laboratoire d’Astrophysique de l’Observatoire de Grenoble for hospitality and visitor support. CT is grateful to the Astronomy Unit at QMW for hospitality and visitor support through PPARC grant PPA/V/O/1997/00261. Agapitou V., Papaloizou J.C.B., Terquem C., 1997, MNRAS 292, 631 Aly J.J., 1980, A&A 86, 192 Balbus S.A., Hawley J.F., 1998, Rev. Mod. 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--- abstract: 'While 3GPP has been developing NB-IoT, the market of Low Power Wide Area Networks has been mastered by cheap and simple Sigfox and LoRa/LoRaWAN technologies. Being positioned as having an open standard, LoRaWAN has attracted also much interest from the research community. Specifically, many papers address the efficiency of its PHY layer. However MAC is still underinvestigated. Existing studies of LoRaWAN do not take into account the acknowledgement and retransmission policy, which may lead to incorrect results. In this paper, we carefully take into account the peculiarities of LoRaWAN transmission retries and show that it is the weakest issue of this technology, which significantly increases failure probability for retries. The main contribution of the paper is a mathematical model which accurately estimates how packet error rate depends on the offered load. In contrast to other papers, which evaluate LoRaWAN capacity just as the maximal throughput, our model can be used to find the maximal load, which allows reliable packet delivery.' author: - bibliography: - 'biblio.bib' title: 'Mathematical Model of LoRaWAN Channel Access [^1]' --- at (current page.south) ; LoRa, LoRaWAN, LPWAN, Channel Access, Performance Evaluation, ALOHA Introduction ============ LoRaWAN is a relatively new protocol designed to provide cheap and reliable wireless connectivity in various Internet of Things scenarios. Being a Low Power Wide Area Network technology operating in the ISM band, it rapidly got popularity in both industry and academic communities. Literature review shows that in spite of numerous studies of its PHY layer [@centenaro2016long; @vangelista2015long; @goursaud2015dedicated], the MAC layer got little attention, even though it has multiple issues [@bankov2016limits; @mikhaylov2016analysis] that limit its performance. However, as LoRaWAN is designed to support networks of thousands of devices, it is crucial not only to consider the performance of this technology in point-to-point scenarios, but also to evaluate its applicability in case of highly-populated networks. To calculate throughput of LoRaWAN networks, in existing studies of the MAC layer (e.g., see [@adelantado2017understanding]), the authors typically use the classical approach for modeling ALOHA networks [@aloha]. The papers (e.g. [@augustin2016study]) also limit the study to unacknowledged mode, which has no control acknowledgements (ACKs). Thus, with no control traffic the throughput increases. However the reliability of transmission decreases. In this paper, we provide a mathematical model for a LoRaWAN network operating in the acknowledged mode. We explain why the usage of classical ALOHA-like approach underestimates the collision probability and develop an accurate mathematical model which takes into account LoRaWAN peculiarities related to retransmission policy. LoRaWAN Channel Access Description ================================== A typical LoRaWAN [@lorawan] network consists of end devices, called *motes*, gateways (GWs), and a server. Motes are connected to the GWs via wireless LoRa links. Gateways gather information from the motes, send it to the server via an IP network, and forward packets from the server to the motes. LoRaWAN devices operate in different ways. Depending on operation, the standard describes three classes of devices. The basic functionality for sporadic uplink data transmission is described as class A operation and is studied in this paper. A LoRaWAN network simultaneously works in several wireless channels. For example, in Europe they can use three main channels and one downlink channel. To transmit a data frame, each mote randomly selects one of the main channels (see Fig. \[fig:channel\_access\]). Having received the frame, the GW sends two ACKs. The first one is sent in the main channel, where the frame was received, $T_1$ after frame reception. The second ACK is sent in the downlink channel after timeout $T_2 = T_1 + \SI{1}{\s}$. If a mote receives no ACK, it makes a retransmission. The standard recommends making a retransmission in a random time drawn from $[1, 1 + W]$ seconds, where $W = 2$. Note that the recommended $W$ is too small and, as we show in the paper leads to the “avalanche effect”. At the PHY layer, LoRaWAN uses Chirp Spread Spectrum modulation. Its main feature is that signals with different spreading factors can be distinguished and received simultaneously, even if they are transmitted in the same time on the same channel. Spreading factor, together with the channel width and the coding rate, determines the data rate. Lower data rates extend transmission range and improve transmission reliability. For the first transmission attempt, the rate is determined by the GW. The standard also recommends decrementing data rate every two consequent transmission failures, limiting the number of retransmissions by $RL = 7$. The first ACK is sent at a data rate that is lower than the data rate for the frame transmission by a configurable offset (it can be zero). The second ACK should always be sent at a fixed data rate, by default the lowest one. (0,2.2) – (11.5,2.2); at (0, 2.9) ; at (0, 1.4) ; (0,0.8) – (11.5,0.8); at (11.5, 1.9) [$t$]{}; at (11.5, 0.5) [$t$]{}; (1, 2.2) rectangle (3.6, 2.8); at (2.3, 2.5) ; (6, 1.6) rectangle (8.4, 2.2); at (7.25, 1.9) ; (8.5, 0.8) rectangle (10.9, 1.4); at (9.7, 1.1) ; (6, 2) – (6, 3.4); (3.6, 0) – (3.6, 3.4); (8.5, 0) – (8.5, 1); (3.6,3.1) – (6,3.1); (3.6,0.2) – (8.5,0.2); at (4.75, 3.3) [$T_1$]{}; at (6, 0.5) [$T_2$]{}; Problem Statement {#sec:scenario} ================= Consider a LoRaWAN network that consists of a GW and $N$ motes and operates in $F$ main channels and one downlink channel. The motes use data rates $0, 1, ..., R$, set by the GW. Let $p_i$ be the probability that a mote uses data rate $i$. We consider that a frame collision occurs when two frames are transmitted in the same channel at the same data rate, and they intersect in time. The motes generate frames according to a Poisson process with total intensity $\lambda$ (the network load). All motes transmit frames with 51-byte Frame Payload which corresponds to the biggest payload that can fit a frame at the lowest data rate. The frames are transmitted in the acknowledged mode, and ACKs carry no frame payload. We consider a situation, when motes have no queue, i.e. if two messages are generated, a mote transmits the most recent one. For the described scenario, it is important not only to know the nominal channel capacity, but also to find the maximal load at which the network can provide reliable communications. In other words, we need *to find the packet error rate (PER) as a function of network load $\lambda$*. Mathematical Model ================== To solve the problem, we develop a mathematical model of the transmission process. As the first transmission attempts are described by the Poisson process, to find the PER in these assumptions, in Section \[first\], we consider the approach used to evaluate ALOHA networks[@aloha] and extended to take into account ACKs. This approach is however inapplicable for retransmissions, because they do not form a Poisson process, so in Section \[retries\] we propose another way to take them into account and thus to improve the accuracy of the model. The First Transmission Attempt {#first} ------------------------------ The first transmission attempt is successful with probability $$\label{eq:success1} P_{S,1} = \sum_{i = 0}^{R} p_{i} P^{Data}_i P^{Ack}_{i},$$ where $P^{Data}_i$ is the probability that the data frame is transmitted without collision at data rate $i$ and $P^{Ack}_i$ is the probability that at least one ACK out of two is received by the mote, provided that the data frame is successful. Since the packets transmitted in different channels and at different rates do not collide, we need to consider separately each combination of channel and data rate. Specifically for rate $i$ and one of $F$ channels, the load equals $r_i = \frac{\lambda p_i}{F}$. A data frame transmission is successful if it intersects with no transmission of another frame or an ACK sent by the GW as a response to previous frame. Let $T^{Data}_{i}$ and $T^{Ack}_i$ be the durations of a data frame and an ACK, respectively, at rate $i$. Intersection with a frame does not occur if no frames are generated in the interval $[-T^{Data}_{i}, T^{Data}_{i}]$, relative to the beginning of the considered frame. For a Poisson process of frame generation, such an event happens with probability $e^{-2 r_i T^{Data}_{i}}$. We consider that the GW cancels ACK transmission if it is receiving a data frame, so a collision can happen only if the ACK is generated in the interval $[-T^{Ack}_{i}, 0]$. The rate of ACK generation is $P^{Data}_i r_i$, so the probability to avoid collision with an ACK is $e^{-r_i P^{Data}_i T^{Ack}_{i}}$. Finally, $P^{Data}_i$ can be found from the following equation: $$P^{Data}_i = e^{-(2 T^{Data}_{i} + P^{Data}_i T^{Ack}_{i}) r_i}.$$ As for ACKs, the probability that at least one ACK arrives is calculated according to the inclusion-exclusion principle: $$P^{Ack}_i = P^{Ack1}_{i} + P^{Ack2}_i - P^{Ack1}_i P^{Ack2}_i,$$ where $P^{Ack1}_{i}$ and $P^{Ack2}_i$ are the probabilities that the first and the second ACK, respectively, is transmitted successfully, provided that data was transmitted at rate $i$. The first ACK is transmitted successfully if no data frame intersects it: $$P^{Ack1}_i = e^{-\left(\min\left(T_1, T^{Data}_{i}\right) + T^{Ack}_{i}\right) r_i}.$$ Here we take the minimum of $T^{Data}_{i}$ and $T_1$, because if a frame exceeds $T_1$, it breaks the acknowledged frame, but such an event is already taken into account by $P^{Data}_i$. The second ACK is transmitted successfully if no data frame is successful in any other channel or at any other data rate, such that its second ACK would intersect the considered one: $$P^{Ack2}_i = e^{-T^{Ack}_{0} \lambda \left(1 - \frac{p_i}{F}\right) \sum_{j = 0}^{R} P^{Data}_j p_j}.$$ Retransmissions {#retries} --------------- Consider a case, when two motes transmit frames with collision, as shown in Fig. \[fig:retransmission\]. Let 0 be the time when the frame of mote A begins, and $x$ be the offset for frame of mote B. Motes choose a channel for retransmission randomly. If they choose different channels, the collision is resolved. Otherwise, with probability $\frac{1}{F}$, they choose the same channel. In this case, let $y$ and $z$ be the times when motes A and B start their retransmission, respectively. The value of $y$ is distributed uniformly in the interval $[\tau, \tau + W]$, where $\tau$ is the frame duration $T$ plus the timeout for the ACK. The value of $z$ is distributed uniformly in the interval $[\tau + x, \tau + x + W]$. The retransmission results in a new collision, if $[z, z + T]$ intersects with $[y, y + T]$, which happens with the probability $$\begin{aligned} P_x &= \frac{\int\limits_{0}^{T} r_i e^{-r_i x} \int\limits_{0}^{W} \int\limits_{x}^{W + x} \frac{\mathbbm{1}\left(y \leq z \leq y + T\right) + \mathbbm{1}\left(z \leq y \leq z + T\right)}{W^2} dz dy dx}{\int\limits_{0}^{T} r_i e^{-r_i x} dx} =\\ &=\frac{T}{W^2} \left(2 W - \frac{3}{2} T - \frac{2}{T r_{i}^2} + \frac{1}{r_i \tanh(\frac{r_i T}{2})}\right),\end{aligned}$$ where $\mathbbm{1}(condition)$ is the indicator function which equals 1 if $condition$ is true and 0 otherwise. Motes have the same probability of being the first and the second one, so the probability that there is no collision equals $$P^{Data}_{i, Re} = 1 - 2 P_x / F.$$ (0,1) – (3.3,1); (4.1,1) – (11,1); at ( 11, 0.6) [$t$]{}; at (0.5, 0.6) [$0$]{}; at (1.3, 0.6) [$x$]{}; at (2.0, 0.6) [$T$]{}; at (3.6, 1) [$...$]{}; at (4.5, 0.6) [$\tau$]{}; at (5.7, 0.6) [$y$]{}; at (7.5, 0.6) [$z$]{}; at (9.5, 0.6) [$\tau + W$]{}; (0.5, 1) rectangle (2.0, 1.5); (1.3, 1) rectangle (2.8, 1.5); (0, 2.2) rectangle (1.5, 2.8); at (3.5, 2.6) [frame of mote A]{}; (6.0, 2.2) rectangle (7.5, 2.8); at (9.5, 2.6) [frame of mote B]{}; (4.5, 0.9) – (4.5, 1.8); (5.7, 1) rectangle (7.2, 1.5); (7.5, 1) rectangle (9.0, 1.5); (9.5, 0.9) – (9.5, 1.8); The average probability of a successful transmission $P_{S}$ is $$P_{S} = P_{1} P_{S, 1} + (1 - P_{1}) P_{S, Re},$$ where $P_{S, Re}$ is the probability of a successful retransmission, calculated as in eq. , using $P^{Data}_{i, Re}$ instead of $P^{Data}_i$, and $P_{1}$ is the probability that the transmission is the first one (not a retry). $P_{1}$ is reverse to the average number of transmission attempts per a frame: $$P_1 = \left(1 + \left(1 - P_{S, 1}\right) \sum\limits_{r = 0}^{RL} \left(1 - P_{S, Re}\right)^r P^{r + 1}_{N}\right)^{-1},$$ where $P_{N} = \sum_{i = 0}^{R} p_i e^{-\frac{\lambda}{N}(T^{Data}_i + T_2 + T^{Ack}_0 + \langle T_{wait} \rangle)}$ is the probability that a new frame does not arrive during the transmission and $\langle T_{wait} \rangle = 1+W/2$ is the average interval that a mote waits before a retransmission. The packet error rate is calculated as $PER = 1 - P_{S}$. The model estimates PER correctly up to such network load, that new frames arrive at the motes as quickly as the motes drop the frames due to inability to resolve collisions after $RL$ retransmission attempts. It means that the load equals $$\lambda^* = F \left(\sum_{i = 0}^{R} p_i \left( T^{Data}_{i} + T_2 + T^{Ack}_{0} + \langle T_{wait} \rangle\right) RL \right)^{-1}.$$ Numerical Results ================= Let us use the developed model to evaluate performance of a LoRaWAN network. As in [@adelantado2017understanding], we consider a scenario, when the motes are distributed uniformly in a circular area with radius of $\SI{1}{\km}$ around the GW, and the path-loss is described by Okumura-Hata model for urban environment. We consider EU 863-880 MHz ISM band. In this case, the data rates are distributed as follows: $p_0 = 0.28, p_1 = 0.2, p_2 = 0.14, p_3 = 0.1, p_4 = 0.08, p_5 = 0.2$. We simulate a network with 1000 motes and compare the average $\mathrm{PER}$ and $\mathrm{PER}_1$ for the first transmission attempt with those obtained with the developed mathematical model. The results are shown in Fig. \[fig:per\]. Because of inefficient retransmission parameters the real $\mathrm{PER}$ is by 50% greater than $\mathrm{PER}_1$. Thus, by taking into account retransmissions, we have significantly improved the accuracy of the model. From Fig. \[fig:per\] we also see that we correctly estimate $\lambda^*$ which is the highest load when we can neglect high-order collisions and the “avalanche effect” inherent to the default retransmission parameters. Non-adaptive and small retransmission window does not allow to resolve collisions with high number of packets, and involving new motes in collisions is faster than packet dropping or collision resolution. This significantly limits the capacity of a LoRaWAN network. While the network can transmit several packets per second, because of a poor retransmission policy the PER rapidly tends to 1, when the load exceeds $10^{-1}$ packets per second. Conclusion {#sec:conclusion} ========== In the paper, we develop the first accurate mathematical model of acknowledged uplink transmissions in LoRaWAN networks with class A devices. We have shown that leaving out of consideration retransmission process significantly overestimates efficiency of a LoRaWAN network. In contrast, our model takes into account peculiarities of the retransmission process and correctly estimates packet error rate when the load is lower than some threshold $\lambda^*$, which is found in the paper. However the area with the higher loads is not interesting from a practical point of view. Indeed, after the load exceeds the described threshold, PER rapidly grows to 1 because retransmissions form an “avalanche”. Thus in this area LoRaWAN cannot provide reliable communications. [^1]: The reported study was partially supported by RFBR, research project No. 15-37-70004 mol\_a\_mos.

--- abstract: 'We study the problem of multiclass classification with an extremely large number of classes ($k$), with the goal of obtaining train and test time complexity logarithmic in the number of classes. We develop top-down tree construction approaches for constructing logarithmic depth trees. On the theoretical front, we formulate a new objective function, which is optimized at each node of the tree and creates dynamic partitions of the data which are both pure (in terms of class labels) and balanced. We demonstrate that under favorable conditions, we can construct logarithmic depth trees that have leaves with low label entropy. However, the objective function at the nodes is challenging to optimize computationally. We address the empirical problem with a new online decision tree construction procedure. Experiments demonstrate that this online algorithm quickly achieves improvement in test error compared to more common logarithmic training time approaches, which makes it a plausible method in computationally constrained large-$k$ applications.' author: - | Anna Choromanska\ Courant Institute of Mathematical Sciences\ New York, NY, USA\ `achoroma@cims.nyu.edu`\ John Langford\ Microsoft Research\ New York, NY, USA\ `jcl@microsoft.com`\ title: Logarithmic Time Online Multiclass prediction --- Introduction ============ The central problem of this paper is computational complexity in a setting where the number of classes $k$ for multiclass prediction is very large. Such problems occur in natural language (Which translation is best?), search (What result is best?), and detection (Who is that?) tasks. Almost all machine learning algorithms (with the exception of decision trees) have running times for multiclass classification which are $\mathcal{O}(k)$ with a canonical example being one-against-all classifiers [@Rifkin2004]. In this setting, the most efficient possible accurate approach is given by information theory [@CnT]. In essence, any multiclass classification algorithm must uniquely specify the bits of all labels that it predicts correctly on. Consequently, Kraft’s inequality ([@CnT] equation 5.6) implies that the expected *computational* complexity of predicting correctly is $\Omega(H(Y))$ per example where $H(Y)$ is the Shannon entropy of the label. For the worst case distribution on $k$ classes, this implies $\Omega(\log(k))$ computation is required. Hence, our goal is achieving $O(\log(k))$ computational time per example[^1] for both training and testing, while effectively using online learning algorithms to minimize passes over the data. The goal of logarithmic (in $k$) complexity naturally motivates approaches that construct a logarithmic depth hierarchy over the labels, with one label per leaf. While this hierarchy is sometimes available through prior knowledge, in many scenarios it needs to be learned as well. This naturally leads to a *partition* problem which arises at each node in the hierarchy. The partition problem is finding a classifier: $c:X \rightarrow \{-1,1\}$ which divides examples into two subsets with a purer set of labels than the original set. Definitions of purity vary, but canonical examples are the number of labels remaining in each subset, or softer notions such as the average Shannon entropy of the class labels. Despite resulting in a classifier, this problem is fundamentally different from standard binary classification. To see this, note that replacing $c(x)$ with $-c(x)$ is very bad for binary classification, but has no impact on the quality of a partition[^2]. The partition problem is fundamentally non-convex for symmetric classes since the average $\frac{c(x) - c(x)}{2}$ of $c(x)$ and $-c(x)$ is a poor partition (the always-$0$ function places all points on the same side). The choice of partition matters in problem dependent ways. For example, consider examples on a line with label $i$ at position $i$ and threshold classifiers. In this case, trying to partition class labels $\{1,3\}$ from class label $2$ results in poor performance. The partition problem is typically solved for decision tree learning via an enumerate-and-test approach amongst a small set of possible classifiers (see e.g. [@ig]). In the multiclass setting, it is desirable to achieve substantial error reduction for each node in the tree which motivates using a richer set of classifiers in the nodes to minimize the number of nodes, and thereby decrease the computational complexity. The main theoretical contribution of this work is to establish a boosting algorithm for learning trees with $O(k)$ nodes and $O(\log k)$ depth, thereby addressing the goal of logarithmic time train and test complexity. Our main theoretical result, presented in Section \[sec:boosting\], generalizes a binary boosting-by-decision-tree theorem [@Kearns95] to multiclass boosting. As in all boosting results, performance is critically dependent on the quality of the *weak learner*, supporting intuition that we need sufficiently rich partitioners at nodes. The approach uses a new objective for decision tree learning, which we optimize at each node of the tree. The objective and its theoretical properties are presented in Section \[sec:framework\]. [l]{}[0.5]{} A complete system with multiple partitions could be constructed top down (as the boosting theorem) or bottom up (as Filter tree [@BeygelzimerLR09]). A bottom up partition process appears impossible with representational constraints as shown in Section \[sec:bottom-up\] in the Supplementary material so we focus on top-down tree creation. Whenever there are representational constraints on partitions (such as linear classifiers), finding a strong partition function requires an efficient search over this set of classifiers. Efficient searches over large function classes are routinely performed via gradient descent techniques for supervised learning, so they seem like a natural candidate. In existing literature, examples for doing this exist when the problem is indeed binary, or when there is a prespecified hierarchy over the labels and we just need to find partitioners aligned with that hierarchy. Neither of these cases applies—we have multiple labels and want to dynamically create the choice of partition, rather than assuming that one was handed to us. Does there exist a purity criterion amenable to a gradient descent approach? The precise objective studied in theory fails this test due to its discrete nature, and even natural approximations are challenging to tractably optimize under computational constraints. As a result, we use the theoretical objective as a motivation and construct a new Logarithmic Online Multiclass Tree (LOMtree) algorithm for empirical evaluation. Creating a tree in an online fashion creates a new class of problems. What if some node is initially created but eventually proves useless because no examples go to it? At best this results in a wasteful solution, while in practice it starves other parts of the tree which need representational complexity. To deal with this, we design an efficient process for recycling orphan nodes into locations where they are needed, and prove that the number of times a node is recycled is at most logarithmic in the number of examples. The algorithm is described in Section \[sec:alg\] and analyzed in Section \[sec:swap-bound\]. And is it effective? Given the inherent non-convexity of the partition problem this is unavoidably an empirical question which we answer on a range of datasets varying from 26 to 105K classes in Section \[sec:experiments\]. We find that under constrained training times, this approach is quite effective compared to all baselines while dominating other $O(\log k)$ train time approaches. What’s new? To the best of our knowledge, the splitting criterion, the boosting statement, the LOMtree algorithm, the swapping guarantee, and the experimental results are all new here. Prior Work ---------- Only a few authors address logarithmic time training. The Filter tree [@BeygelzimerLR09] addresses consistent (and robust) multiclass classification, showing that it is possible in the statistical limit. The Filter tree does not address the partition problem as we do here which as shown in our experimental section is often helpful. The partition finding problem is addressed in the conditional probability tree [@BeygelzimerLLSS09], but that paper addresses conditional probability estimation. Conditional probability estimation can be converted into multiclass prediction [@Bishop:2006:PRM:1162264], but doing so is not a logarithmic time operation. Quite a few authors have addressed logarithmic testing time while allowing training time to be $O(k)$ or worse. While these approaches are intractable on our larger scale problems, we describe them here for context. The partition problem can be addressed by recursively applying spectral clustering on a confusion graph [@BengioWG10] (other clustering approaches include [@journals/informaticaSI/MadzarovGC09]). Empirically, this approach has been found to sometimes lead to badly imbalanced splits [@DengSBL11]. In the context of ranking, another approach uses $k$-means hierarchical clustering to recover the label sets for a given partition [@weston13]. The more recent work [@conf/cvpr/ZhaoX13] on the multiclass classification problem addresses it via sparse output coding by tuning high-cardinality multiclass categorization into a bit-by-bit decoding problem. The authors decouple the learning processes of coding matrix and bit predictors and use probabilistic decoding to decode the optimal class label. The authors however specify a class similarity which is $\mathcal{O}(k^2)$ to compute (see Section $2.1.1$ in [@conf/cvpr/ZhaoX13]), and hence this approach is in a different complexity class than ours (this is also born out experimentally). The variant of the popular error correcting output code scheme for solving multi-label prediction problems with large output spaces under the assumption of output sparsity was also considered in [@DBLP:journals/corr/abs-0902-1284]. Their approach in general requires $O(k)$ running time to decode since, in essence, the fit of each label to the predictions must be checked and there are $\mathcal{O}(k)$ labels. Another approach [@DBLP:journals/corr/AgarwalKKSV13] proposes iterative least-squares-style algorithms for multi-class (and multi-label) prediction with relatively large number of examples and data dimensions, and the work of [@icml2014c2_beijbom14] focusing in particular on the cost-sensitive multiclass classification. Both approaches however have $\mathcal{O}(k)$ training time. Decision trees are naturally structured to allow logarithmic time prediction. Traditional decision trees often have difficulties with a large number of classes because their splitting criteria are not well-suited to the large class setting. However, newer approaches [@Manik; @Prabhu2014] have addressed this effectively at significant scales in the context of multilabel classification (multilabel learning, with missing labels, is also addressed in [@LMLML14]). More specifically, the first work [@Manik] performs brute force optimization of a multilabel variant of the Gini index defined over the set of positive labels in the node and assumes label independence during random forest construction. Their method makes fast predictions, however has high training costs [@Prabhu2014]. The second work [@Prabhu2014] optimizes a rank sensitive loss function (Discounted Cumulative Gain). Additionally, a well-known problem with hierarchical classification is that the performance significantly deteriorates lower in the hierarchy [@LiuLargeScale2005] which some authors solve by biasing the training distribution to reduce error propagation while simultaneously combining bottom-up and top-down approaches during training [@conf/sigir/BennettN09]. The reduction approach we use for optimizing partitions implicitly optimizes a differential objective. A non-reductive approach to this has been tried previously [@MTSWIMC] on other objectives yielding good results in a different context. Framework and theoretical analysis {#sec:framework} ================================== In this section we describe the essential elements of the approach, and outline the theoretical properties of the resulting framework. We begin with high-level ideas. Setting {#sec:outline} ------- We employ a hierarchical approach for learning a multiclass decision tree structure, training this structure in a *top-down* fashion. We assume that we receive examples $x \in {\ensuremath{\mathcal{X}}}\subseteq {\ensuremath{\mathbb{R}}}^d$, with labels $y \in \{1,2,\ldots, k\}$. We also assume access to a hypothesis class ${\ensuremath{\mathcal{H}}}$ where each $h \in {\ensuremath{\mathcal{H}}}$ is a binary classifier, $h~:~{\ensuremath{\mathcal{X}}}\mapsto \{-1, 1\}$. The overall objective is to learn a tree of depth $O(\log k)$, where each node in the tree consists of a classifier from ${\ensuremath{\mathcal{H}}}$. The classifiers are trained in such a way that $h_n(x) = 1$ ($h_n$ denotes the classifier in node $n$ of the tree[^3]) means that the example $x$ is sent to the right subtree of node $n$, while $h_n(x) = -1$ sends $x$ to the left subtree. When we reach a leaf, we predict according to the label with the highest frequency amongst the examples reaching that leaf. In the interest of computational complexity, we want to encourage the number of examples going to the left and right to be *fairly balanced*. For good statistical accuracy, we want to send examples of class $i$ almost exclusively to either the left or the right subtree, thereby refining the *purity* of the class distributions at subsequent levels in the tree. The *purity* of a tree node is therefore a measure of whether the examples of each class reaching the node are then mostly sent to its one child node (pure split) or otherwise to both children (impure split). The formal definitions of *balancedness* and *purity* are introduced in Section \[sec:objective\]. An objective expressing both criteria[^4] and resulting theoretical properties are illustrated in the following sections. A key consideration in picking this objective is that we want to effectively optimize it over hypotheses $h \in {\ensuremath{\mathcal{H}}}$, while streaming over examples in an online fashion[^5]. This seems unsuitable with some of the more standard decision tree objectives such as Shannon or Gini entropy, which leads us to design a new objective. At the same time, we show in Section \[sec:boosting\] that under suitable assumptions, optimizing the objective also leads to effective reduction of the average Shannon entropy over the entire tree. An objective and analysis of resulting partitions {#sec:objective} ------------------------------------------------- We now define a criterion to measure the quality of a hypothesis $h \in {\ensuremath{\mathcal{H}}}$ in creating partitions at a fixed node $n$ in the tree. Let $\pi_i$ denotes the proportion of label $i$ amongst the examples reaching this node. Let $P(h(x) > 0)$ and $P(h(x) > 0 | i)$ denote the fraction of examples reaching $n$ for which $h(x) > 0$, marginally and conditional on class $i$ respectively. Then we define the objective[^6]: $$J(h) = 2\sum_{i=1}^k \pi_i \left| P(h(x) > 0) - P(h(x) > 0 | i) \right|. \label{eqn:objective} \vspace{-0.03in}$$ We aim to *maximize the objective $J(h)$* to obtain high quality partitions. Intuitively, the objective encourages the fraction of examples going to the right from class $i$ to be substantially different from the background fraction for each class $i$. As a concrete simple scenario, if $P(h(x) > 0) = 0.5$ for some hypothesis $h$, then the objective prefers $P(h(x) > 0 | i)$ to be as close to 0 or 1 as possible for each class $i$, leading to pure partitions. We now make these intuitions more formal. The hypothesis $h \in \mathcal{H}$ induces a pure split if $$\alpha := \sum_{i=1}^{k}\pi_i\min(P(h(x) > 0|i), P(h(x)<0|i)) \leq \delta, \vspace{-0.04in}$$ where $\delta \in [0,0.5)$, and $\alpha$ is called the *purity factor*. In particular, a partition is called *maximally pure* if $\alpha = 0$, meaning that each class is sent exclusively to the left or the right. We now define a similar definition for the balancedness of a split. The hypothesis $h \in \mathcal{H}$ induces a balanced split if $$c \leq \underbrace{P(h(x) > 0)}_{=\beta} \leq 1 - c, \vspace{-0.06in}$$ where $c \in (0,0.5]$, and $\beta$ is called the *balancing factor*. A partition is called *maximally balanced* if $\beta = 0.5$, meaning that an equal number of examples are sent to the left and right children of the partition. The balancing factor and the purity factor are related as shown in Lemma \[lemma:obj-to-purity\] (the proofs of Lemma \[lemma:obj-to-purity\] and the following lemma (Lemma \[lemma:maximal\]) are deferred to the Supplementary material). For any hypothesis $h$, and any distribution over examples $(x,y)$, the purity factor $\alpha$ and the balancing factor $\beta$ satisfy $\alpha \leq \min\{(2 - J(h))/(4\beta) - \beta, 0.5\}$. \[lemma:obj-to-purity\] A partition is called *maximally pure and balanced* if it satisfies both $\alpha = 0$ and $\beta = 0.5$. We see that $J(h) = 1$ for a hypothesis $h$ inducing a maximally pure and balanced partition as captured in the next lemma. Of course we do not expect to have hypotheses producing maximally pure and balanced splits in practice. For any hypothesis $h~:~{\ensuremath{\mathcal{X}}}\mapsto \{-1,1\}$, the objective $J(h)$ satisfies $J(h) \in [0,1]$. Furthermore, if $h$ induces a maximally pure and balanced partition then $J(h) = 1$. \[lemma:maximal\] Quality of the entire tree {#sec:boosting} -------------------------- The above section helps us understand the quality of an individual split produced by effectively maximizing $J(h)$. We next reason about the quality of the entire tree as we add more and more nodes. We measure the quality of trees using the average entropy over all the leaves in the tree, and track the decrease of this entropy as a function of the number of nodes. Our analysis extends the theoretical analysis in [@Kearns95], originally developed to show the boosting properties of the decision trees for binary classification problems, to the multiclass classification setting. Given a tree $\mathcal{T}$, we consider the entropy function $G_t$ as the measure of the quality of tree: $$G_t = \sum_{l \in \mathcal{L}}w_l\sum_{i = 1}^k \pi_{l,i}\ln \left( \frac{1}{\pi_{l,i}} \right) \vspace{-0.05in}$$ where $\pi_{l,i}$’s are the probabilities that a randomly chosen data point $x$ drawn from $\mathcal{P}$, where $\mathcal{P}$ is a fixed target distribution over $\mathcal{X}$, has label $i$ given that $x$ reaches node $l$, $\mathcal{L}$ denotes the set of all tree leaves, $t$ denotes the number of internal tree nodes, and $w_l$ is the weight of leaf $l$ defined as the probability a randomly chosen $x$ drawn from $\mathcal{P}$ reaches leaf $l$ (note that $\sum_{l \in \mathcal{L}}w_l = 1$). We next state the main theoretical result of this paper (it is captured in Theorem \[thm:main\]). We adopt the *weak learning* framework. The *weak hypothesis assumption*, captured in Definition \[def:wha\], posits that each node of the tree $\mathcal{T}$ has a hypothesis $h$ in its hypothesis class $\mathcal{H}$ which guarantees simultaneously a “weak” purity and a ”weak” balancedness of the split on any distribution $\mathcal{P}$ over $\mathcal{X}$. Under this assumption, one can use the new decision tree approach to drive the error below any threshold. Let $m$ denote any node of the tree $\mathcal{T}$, and let $\beta_m = P(h_m(x) > 0)$ and $P_{m,i} = P(h_m(x) > 0|i)$. Furthermore, let $\gamma \in \mathbb{R}^{+}$ be such that for all $m$, $\gamma \in (0,\min(\beta_m,1-\beta_m)]$. We say that the *weak hypothesis assumption* is satisfied when for any distribution $\mathcal{P}$ over $\mathcal{X}$ at each node $m$ of the tree $\mathcal{T}$ there exists a hypothesis $h_m \in \mathcal{H}$ such that $J(h_m)/2 = \sum_{i = 1}^k \pi_{m,i}|P_{m,i} - \beta_{m}| \geq \gamma$. \[def:wha\] Under the Weak Hypothesis Assumption, for any $\alpha \in [0,1]$, to obtain $G_t \leq \alpha$ it suffices to make $t \geq (1/\alpha)^\frac{4(1 - \gamma)^2\ln k}{\gamma^2}$ splits. \[thm:main\] We defer the proof of Theorem \[thm:main\] to the Supplementary material and provide its sketch now. The analysis studies a tree construction algorithm where we recursively find the leaf node with the highest weight, and choose to split it into two children. Let $n$ be the heaviest leaf at time $t$. Consider splitting it to two children. The contribution of node $n$ to the tree entropy changes after it splits. This change (entropy reduction) corresponds to a gap in the Jensen’s inequality applied to the concave function, and thus can further be lower-bounded (we use the fact that Shannon entropy is strongly concave with respect to $\ell_1$-norm (see e.g., Example 2.5 in Shalev-Shwartz [@ShaiSS2012])). The obtained lower-bound turns out to depend proportionally on $J(h_n)^2$. This implies that the larger the objective $J(h_n)$ is at time $t$, the larger the entropy reduction ends up being, which further reinforces intuitions to maximize $J$. In general, it might not be possible to find any hypothesis with a large enough objective $J(h_n)$ to guarantee sufficient progress at this point so we appeal to a *weak learning assumption*. This assumption can be used to further lower-bound the entropy reduction and prove Theorem \[thm:main\]. The LOMtree Algorithm {#sec:alg} ===================== **Input**: regression algorithm $R$, max number of tree non-leaf nodes $T$, swap resistance $R_S\:\:\:\:\:\:\:\:\:$ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ Subroutine **SetNode ($v$)** ${\bm m}_v = \emptyset $(${\bm m}_v(y)$ - sum of the scores for class $y$) ${\bm l}_v \:\:= \emptyset $ (${\bm l}_v(y)$ - number of points of class $y$ reaching $v$) ${\bm n}_v \:= \emptyset $ (${\bm n}_v(y)$ - number of points of class $y$ which are used to train regressor in $v$) ${\bm e}_v \:\:\!= \emptyset $ (${\bm e}_v(y)$ - expected score for class $y$) $\:\:\!\!{\bm E}_v \:\!\:\!= 0 $ (expected total score) $C_v \:\:\!\!= 0$(the size of the smallest leafin the subtree with root $v$) Subroutine **UpdateC** ($v$) **While** ($v \neq r$ AND $C_{\textsc{parent}(v)} \neq C_v$) $v = \textsc{parent}(v)$; $C_v = \min(C_{\textsc{left}(v)},C_{\textsc{right}(v)})$ Subroutine **Swap** (v) Find a leaf $s$ for which $(C_s = C_r)$ $s_{\textsc{pa}} \!\!=\!\! \textsc{parent}(s)$; $s_{\textsc{gpa}}$ = <span style="font-variant:small-caps;">grandpa</span>(s); $s_{\textsc{sib}} \!\!=\!\! \textsc{sibling}(s)$ **If** ($s_{\textsc{pa}}$ = <span style="font-variant:small-caps;">left</span>($s_{\textsc{gpa}}$)) <span style="font-variant:small-caps;">left</span>($s_{\textsc{gpa}}$) = $s_{\textsc{sib}}$**Else** <span style="font-variant:small-caps;">right</span>($s_{\textsc{gpa}}$) = $s_{\textsc{sib}}$ **UpdateC** ($s_{\textsc{sib}}$); **SetNode** ($s$); $\textsc{left}(v) = s$; **SetNode** ($s_{\textsc{pa}}$); $\textsc{right}(v) = s_{\textsc{pa}}$ **Create** root $r = 0$: **SetNode** ($r$); $t = 1$ **For each** example $(\bm{x},y)$ **do** Set $j = r$ **Do** **If** ($l_j(y) = \emptyset$) $m_j(y) = 0$; $l_j(y) = 0$;$n_j(y) = 0$;$e_j(y) = 0$ $\bm{l}_j(y)$++ **If**($j$ is a leaf) **If**(${\bm l}_j$ has at least $2$ non-zero entries) **If**($t \!\!<\!\! T$ OR $C_j \!\!-\!\! \max_i{\bm l}_j(i) \!\!>\!\! R_S(C_r\!\!+\!\!1)$) **If** ($t \!\!<\!\! T$) **SetNode** (<span style="font-variant:small-caps;">left</span>($j$));**SetNode** (<span style="font-variant:small-caps;">right</span>($j$));$t$++ **Else** **Swap**(j) $C_{\textsc{left}(j)} \!\!=\!\! \floor*{C_j/2}$;$C_{\textsc{right}(j)} \!\!=\!\! C_j \!\!-\!\! C_{\textsc{left}(j)}$;**UpdateC** (<span style="font-variant:small-caps;">left</span>($j$)) **If**($j$ is not a leaf) **If** $\left(E_j > \bm{e}_j(y)\right)$ $c \!=\! -1$**Else** $c \!=\! 1$ **Train** $h_j$ with example $({\bm x},c)$: $R({\bm x},c)$ $\bm{n}_j(y)+\!\!+$;${\bm m}_j(y)\:+\!\!= h_j(\bm{x})$;$\bm{e}_j(y) =\bm{m}_j(y)/\bm{n}_j(y)$;$E_j = \frac{\sum_{i = 1}^k{{\bm m}_j(i)}}{\sum_{i=1}^k{\bm n}_j(i)}$ **Set** $j$ to the child of $j$ corresponding to $h_j$ **Else** $C_j$++ **break** The objective function of Section \[sec:framework\] has another convenient form which yields a simple online algorithm for tree construction and training. Note that Equation \[eqn:objective\] can be written (details are shown in Section \[sec:explanation\] in the Supplementary material) as $$J(h) = 2\mathbb{E}_i [|\mathbb{E}_x[\mathds{1}(h(x) > 0)] - \mathbb{E}_{x}[\mathds{1}(h(x) > 0|i)]|]. \vspace{-0.05in}$$ Maximizing this objective is a discrete optimization problem that can be relaxed as follows $$J(h) = 2\mathbb{E}_i[|\mathbb{E}_x[h(x)] - \mathbb{E}_{x}[h(x)|i]|], \vspace{-0.05in}$$ where $E_{x}[h(x)|i]$ is the expected score of class i. We next explain our empirical approach for maximizing the relaxed objective. The empirical estimates of the expectations can be easily stored and updated online in every tree node. The decision whether to send an example reaching a node to its left or right child node is based on the sign of the difference between the two expectations: $\mathbb{E}_x[h(x)]$ and $\mathbb{E}_{x}[h(x)|y]$, where $y$ is a label of the data point, i.e. when $\mathbb{E}_x[h(x)] - \mathbb{E}_{x}[h(x)|y] > 0$ the data point is sent to the left, else it is sent to the right. This procedure is conveniently demonstrated on a toy example in Section \[sec:toy\] in the Supplement. During training, the algorithm assigns a unique label to each node of the tree which is currently a leaf. This is the label with the highest frequency amongst the examples reaching that leaf. While testing, a test example is pushed down the tree along the path from the root to the leaf, where in each non-leaf node of the path its regressor directs the example either to the left or right child node. The test example is then labeled with the label assigned to the leaf that this example descended to. The training algorithm is detailed in Algorithm \[alg:OTT\] where each tree node contains a classifier (we use linear classifiers), i.e. $h_j$ is the regressor stored in node $j$ and $h_j(\bf x)$ is the value of the prediction of $h_j$ on example $\bf x$. The stopping criterion for expanding the tree is when the number of non-leaf nodes reaches a threshold $T$. Swapping {#sec:swap-bound} -------- child [node [$\dots$]{} child [node [$j$]{}]{} child [node [$\dots$]{}]{} ]{} child [node [$\dots$]{} child [node […]{}]{} child [node [$s_{\textsc{gpa}}$]{} child [node [$\dots$]{}]{} child [node [$s_{\textsc{pa}}$]{} child [node [$s$]{}]{} child [node [$s_{\textsc{sib}}$]{} child [node [$\dots$]{}]{} child [node [$\dots$]{}]{} ]{} ]{} ]{} ]{}; child [node [$\dots$]{} child [node [$j$]{} child [node [$s$]{}]{} child [node [$s_{\textsc{pa}}$]{}]{} ]{} child [node [$\dots$]{}]{} ]{} child [node [$\dots$]{} child [node […]{}]{} child [node [$s_{\textsc{gpa}}$]{} child [node [$\dots$]{}]{} child [node [$s_{\textsc{sib}}$]{} child [node [$\dots$]{}]{} child [node [$\dots$]{}]{} ]{} ]{} ]{}; \[fig:swap\] Consider a scenario where the current training example descends to leaf $j$. The leaf can split (create two children) if the examples that reached it in the past were coming from at least two different classes. However, if the number of non-leaf nodes of the tree reaches threshold $T$, no more nodes can be expanded and thus $j$ cannot create children. Since the tree construction is done online, some nodes created at early stages of training may end up useless because no examples reach them later on. This prevents potentially useful splits such as at leaf $j$. This problem can be solved by recycling orphan nodes (subroutine **Swap** in Algorithm \[alg:OTT\]). The general idea behind node recycling is to allow nodes to split if a certain condition is met. In particular, node $j$ splits if the following holds: $$C_j - \max_{i \in \{1,2,\dots,k\}}{\bm l}_j(i) > R_S(C_r+1), \label{eq:swap_cond} \vspace{-0.02in}$$ where $r$ denotes the root of the entire tree, $C_j$ is the size of the smallest leaf in the subtree with root $j$, where the smallest leaf is the one with the smallest total number of data points reaching it in the past, ${\bm l}_j$ is a $k$-dimensional vector of non-negative integers where the $i^{\text{th}}$ element is the count of the number of data points with label $i$ reaching leaf $j$ in the past, and finally $R_S$ is a “swap resistance”. The subtraction of $\max_{i \in \{1,2,\dots,k\}}{\bm l}_j(i)$ in Equation \[eq:swap\_cond\] ensures that a pure node will not be recycled. If the condition in Inequality \[eq:swap\_cond\] is satisfied, the swap of the nodes is performed where an orphan leaf $s$, which was reached by the smallest number of examples in the past, and its parent $s_{\textsc{PA}}$ are detached from the tree and become children of node $j$ whereas the old sibling $s_{\textsc{sib}}$ of an orphan node $s$ becomes a direct child of the old grandparent $s_{\textsc{GPA}}$. The swapping procedure is shown in Figure 2. The condition captured in the Inequality \[eq:swap\_cond\] allows us to prove that the number of times any given node is recycled is upper-bounded by the logarithm of the number of examples whenever the swap resistance is $4$ or more (Lemma \[lemma:sb\]). Let the swap resistance $R_S$ be greater or equal to $4$. Then for all sequences of examples, the number of times Algorithm \[alg:OTT\] recycles any given node is upper-bounded by the logarithm (with base $2$) of the sequence length. \[lemma:sb\] Experiments {#sec:experiments} =========== We address several hypotheses experimentally. 1. The LOMtree algorithm achieves true logarithmic time computation in practice. 2. The LOMtree algorithm is competitive with or better than all other logarithmic train/test time algorithms for multiclass classification. 3. The LOMtree algorithm has statistical performance close to more common $O(k)$ approaches. [l]{}[0.6]{} Isolet Sector Aloi ImNet ODP -- -------- -------- ------ ---------- ----------- 52.3 19 17.7 104 3 617 54 128 6144 0.5 7797 9619 108 14.2 1577418 26 105 1000 $\sim$22 $\sim$105 \[tab:dsize\] To address these hypotheses, we conducted experiments on a variety of benchmark multiclass datasets: *Isolet*, *Sector*, *Aloi*, *ImageNet* (*ImNet*) and *ODP*. The details of the datasets are provided in Table \[tab:dsize\]. The datasets were divided into training ($90\%$) and testing ($10\%$). Furthermore, $10\%$ of the training dataset was used as a validation set. The baselines we compared *LOMtree* with are a balanced random tree of logarithmic depth (*Rtree*) and the *Filter tree* [@BeygelzimerLR09]. Where computationally feasible, we also compared with a one-against-all classifier (*OAA*) as a representative $O(k)$ approach. All methods were implemented in the Vowpal Wabbit [@VowpalWabbit] learning system and have similar levels of optimization. The regressors in the tree nodes for *LOMtree*, *Rtree*, and *Filter tree* as well as the *OAA* regressors were trained by online gradient descent for which we explored step sizes chosen from the set $\{0.25,0.5,0.75,1,2,4,8\}$. We used linear regressors. For each method we investigated training with up to $20$ passes through the data and we selected the best setting of the parameters (step size and number of passes) as the one minimizing the validation error. Additionally, for the *LOMtree* we investigated different settings of the stopping criterion for the tree expansion: $T = \{k-1,2k-1,4k-1,8k-1,16k-1,32k-1,64k-1\}$, and swap resistance $R_S = \{4,8,16,32,64,128,256\}$. In Table \[tab:traintime\] and \[tab:peretesttime\] we report respectively train time and per-example test time (the best performer is indicated in bold). Training time (and later reported test error) is not provided for *OAA* on *ImageNet* and *ODP* due to intractability-both are petabyte scale computations. Isolet Sector Aloi -- ------------ ------------ ------------ **16.27s** **12.77s** **51.86s** 19.58s 18.37s 11m2.43s : Per-example test time on all problems. \[tab:traintime\] Isolet Sector Aloi ImNet ODP -- ------------ ------------ ------------ ------------ ------------ **0.14ms** **0.13ms** **0.06ms** **0.52ms** **0.26ms** 0.16 ms 0.24ms 0.33ms 0.21s 1.05s : Per-example test time on all problems. \[tab:peretesttime\] The first hypothesis is consistent with the experimental results. Time-wise *LOMtree* significantly outperforms *OAA* due to building only close-to logarithmic depth trees. The improvement in the training time increases with the number of classes in the classification problem. For instance on *Aloi* training with *LOMtree* is $12.8$ times faster than with $OAA$. The same can be said about the test time, where the per-example test time for *Aloi*, *ImageNet* and *ODP* are respectively $5.5$, $403.8$ and $4038.5$ times faster than *OAA*. The significant advantage of *LOMtree* over *OAA* is also captured in Figure \[fig:impex\]. [l]{}[0.5]{} Next, in Table \[tab:testerr\] (the best logarithmic time performer is indicated in bold) we report test error of logarithmic train/test time algorithms. We also show the binomial symmetrical $95\%$ confidence intervals for our results. Clearly the second hypothesis is also consistent with the experimental results. Since the *Rtree* imposes a random label partition, the resulting error it obtains is generally worse than the error obtained by the competitor methods including *LOMtree* which learns the label partitioning directly from the data. At the same time *LOMtree* beats *Filter tree* on every dataset, though for *ImageNet* and *ODP* (both have a high level of noise) the advantage of *LOMtree* is not as significant. LOMtree Rtree Filter tree OAA -- ----------- ----------- ------------- --------- **6.36** 16.92 15.10 $3.56$ 16.19 **15.77** 17.70 $9.17$ **16.50** 83.74 80.50 $13.78$ **90.17** 96.99 92.12 NA **93.46** 93.85 93.76 NA : Test error ($\%$) and confidence interval on all problems. \[tab:testerr\] The third hypothesis is weakly consistent with the empirical results. The time advantage of *LOMtree* comes with some loss of statistical accuracy with respect to *OAA* where *OAA* is tractable. We conclude that *LOMtree* significantly closes the gap between other logarithmic time methods and *OAA*, making it a plausible approach in computationally constrained large-$k$ applications. Conclusion ========== The LOMtree algorithm reduces the multiclass problem to a set of binary problems organized in a tree structure where the partition in every tree node is done by optimizing a new partition criterion online. The criterion guarantees pure and balanced splits leading to logarithmic training and testing time for the tree classifier. We provide theoretical justification for our approach via a boosting statement and empirically evaluate it on multiple multiclass datasets. Empirically, we find that this is the best available logarithmic time approach for multiclass classification problems. ### Acknowledgments {#acknowledgments .unnumbered} We would like to thank Alekh Agarwal, Dean Foster, Robert Schapire and Matus Telgarsky for valuable discussions. Bottom-up partitions do not work {#sec:bottom-up} ================================ The most natural bottom-up construction for creating partitions is not viable as will be now shown by an example. Bottom-up construction techniques start by pairing labels, either randomly or arbitrarily, and then building a predictor of whether the class label is left or right conditioned on the class label being one of the paired labels. In order to construct a full tree, this operation must compose, pairing trees with size $2$ to create trees of size $4$. Here, we show that the straightforward approach to composition fails. Suppose we have a one dimensional feature space with examples of class label $i$ having feature value $i$ and we work with threshold predictors. Suppose we have 4 classes $1, 2, 3, 4$, and we happen to pair $(1,3)$ and $(2,4)$. It is easy to build a linear predictor for each of these splits. The next step is building a predictor for $(1,3)$ vs $(2,4)$ which is impossible because all thresholds in $(-\infty,1)$, $(2,3)$, and $(4,\infty)$ err on two labels while thresholds on $(1,2)$ and $(3,4)$ err on one label. Proof of Lemma \[lemma:obj-to-purity\] ====================================== We start from deriving an upper-bound on $J(h)$. For the ease of notation let $P_i = P(h(x) > 0 | i)$. Thus $$J(h) = 2\sum_{i=1}^{k}\pi_i\left \lvert P(h(x) > 0 | i) - P(h(x) > 0)\right \rvert = 2\sum_{i=1}^{k}\pi_i\left \lvert P_i - \sum_{j=1}^{k}\pi_jP_j\right \rvert,$$ where $\forall_{i = \{1,2,\dots,k\}}0 \leq P_i \leq 1$. Let $\alpha_i = \min(P_i,1-P_i)$ and recall the purity factor $\alpha = \sum_{i=1}^{k}\pi_i\alpha_i$ and the balancing factor $\beta = P(h(x) > 0)$. Without loss of generality let $\beta \leq \frac{1}{2}$. Furthermore, let $$L_1 = \{i:i \in \{1,2,\dots,k\}, P_i \geq \frac{1}{2}\}, \:\:\:L_2 = \{i:i \in \{1,2,\dots,k\}, P_i \in [\beta,\frac{1}{2})\}$$ $$\text{and} \:\:\:\:\:L_3 = \{i:i \in \{1,2,\dots,k\}, P_i < \beta\}.$$ First notice that $$\beta = \sum_{i=1}^{k}\pi_iP_i = \sum_{i \in L_1}\pi_i(1 - \alpha_i) + \sum_{i \in L_2 \cup L_3}\pi_i\alpha_i = \sum_{i \in L_1}\pi_i - 2\sum_{i \in L_1}\pi_i\alpha_i + \alpha \label{eq:medium}$$ Therefore $$\begin{aligned} \frac{J(h)}{2} &=& \sum_{i=1}^{k}\pi_i\left \lvert P_i - \beta \right \rvert = \sum_{i \in L_1}\pi_i(1 - \alpha_i - \beta) + \sum_{i \in L_2}\pi_i(\alpha_i - \beta) + \sum_{i \in L_3}\pi_i(\beta - \alpha_i)\\ &=& \sum_{i \in L_1}\pi_i(1 - \beta) - \sum_{i \in L_1}\pi_i\alpha_i + \sum_{i \in L_2}\pi_i\alpha_i - \sum_{i \in L_2}\pi_i\beta + \sum_{i \in L_3}\pi_i\beta - \sum_{i \in L_3}\pi_i\alpha_i\end{aligned}$$ Note that $\sum_{i \in L_3}\pi_i = 1 - \sum_{i \in L_1}\pi_i - \sum_{i \in L_2}\pi_i$ and therefore $$\begin{aligned} \frac{J(h)}{2} &=& \sum_{i \in L_1}\pi_i(1 \!-\! \beta) \!-\!\!\! \sum_{i \in L_1}\pi_i\alpha_i \!+\!\!\! \sum_{i \in L_2}\pi_i\alpha_i \!-\!\!\! \sum_{i \in L_2}\pi_i\beta + \beta(1 \!-\!\!\! \sum_{i \in L_1}\pi_i \!-\!\!\! \sum_{i \in L_2}\pi_i) \!-\!\!\! \sum_{i \in L_3}\pi_i\alpha_i\\ &=& \sum_{i \in L_1}\pi_i(1 - 2\beta) - \sum_{i \in L_1}\pi_i\alpha_i + \sum_{i \in L_2}\pi_i\alpha_i + \beta(1 - 2\sum_{i \in L_2}\pi_i) - \sum_{i \in L_3}\pi_i\alpha_i\end{aligned}$$ Furthermore, since $- \sum_{i \in L_1}\pi_i\alpha_i + \sum_{i \in L_2}\pi_i\alpha_i - \sum_{i \in L_3}\pi_i\alpha_i = - \alpha + 2\sum_{i \in L_2}\pi_i\alpha_i$ we further write that $$\begin{aligned} \frac{J(h)}{2} &=& \sum_{i \in L_1}\pi_i(1 - 2\beta)+ \beta(1 - 2\sum_{i \in L_2}\pi_i) - \alpha + 2\sum_{i \in L_2}\pi_i\alpha_i\end{aligned}$$ By Equation \[eq:medium\], it can be further rewritten as $$\begin{aligned} \frac{J(h)}{2} &=& (1 - 2\beta)(\beta + 2\sum_{i \in L_1}\pi_i\alpha_i - \alpha)+ \beta(1 - 2\sum_{i \in L_2}\pi_i) - \alpha + 2\sum_{i \in L_2}\pi_i\alpha_i\\ &=& 2(1 - \beta)(\beta - \alpha) + 2(1 - 2\beta)\sum_{i \in L_1}\pi_i\alpha_i + 2\sum_{i \in L_2}\pi_i(\alpha_i - \beta)\end{aligned}$$ Since $\alpha_i$’s are bounded by $0.5$ we obtain $$\begin{aligned} \frac{J(h)}{2} &\leq& 2(1 - \beta)(\beta - \alpha) + 2(1 - 2\beta)\sum_{i \in L_1}\pi_i\alpha_i + 2\sum_{i \in L_2}\pi_i(\frac{1}{2} - \beta)\\ &\leq& 2(1 - \beta)(\beta - \alpha) + 2(1 - 2\beta)\alpha + 1 - 2\beta\\ &=& 2\beta(1 - \beta) - 2\alpha(1 - \beta) + 2\alpha(1 - 2\beta) + 1 - 2\beta\\ &=& 1 - 2\beta^2 - 2\beta\alpha\end{aligned}$$ Thus: $$\alpha \leq \frac{2 - J(h)}{4\beta} - \beta.$$ Proof of Lemma \[lemma:maximal\] ================================ We first show that $J(h) \in [0,1]$. We start from deriving an upper-bound on $J(h)$, where $h \in \mathcal{H}$ is some hypothesis in the hypothesis class. For the ease of notation let $P_i = P(h(x) > 0 | i)$. Thus $$\begin{aligned} J(h) &=& 2\sum_{i=1}^{k}\pi_i\left \lvert P(h(x) > 0 | i) - P(h(x) > 0)\right \rvert\\ & = &2\sum_{i=1}^{k}\pi_i\left \lvert P_i - \sum_{j=1}^{k}\pi_jP_j\right \rvert, \nonumber \label{eq:objform}\end{aligned}$$ where $\forall_{i = \{1,2,\dots,k\}}0 \leq P_i \leq 1$. The objective $J(h)$ is certainly maximized on the extremes of the $[0,1]$ interval. The upper-bound on $J(h)$ can be thus obtained by setting some of the $P_i$’s to $1$’s and remaining ones to $0$’s. To be more precise, let $$L_1 = \{i:i \in \{1,2,\dots,k\}, P_i = 1\} \text{\:\:\:\:\:\:and\:\:\:\:\:\:} L_2 = \{i:i \in \{1,2,\dots,k\}, P_i = 0\}.$$ Therefore it follows that $$\begin{aligned} J(h) &\leq& 2\left[\sum_{i \in L_1} \pi_i(1 - \sum_{j \in L_1}\pi_j) + \sum_{i \in L_2} \pi_i\sum_{j \in L_1}\pi_j\right]\\ &=& 2\left[\sum_{i \in L_1} \pi_i - ( \sum_{i \in L_1} \pi_i)^2 + (1 - \sum_{i \in L_1} \pi_i) \sum_{i \in L_1} \pi_i\right]\\ &=& 4\left[\sum_{i \in L_1} \pi_i - ( \sum_{i \in L_1} \pi_i)^2\right]\end{aligned}$$ Let $b = \sum_{i \in L_1} \pi_i$ thus $$J(h) \leq 4b(1 - b) = -4b^2 + 4b \label{eq:upper_bound}$$ Since $b \in [0,1]$, it is straightforward that $-4b^2 + 4b \in [0,1]$ and thus $J(h) \in [0,1]$. We now proceed to prove the main statement of Lemma \[lemma:maximal\], if $h$ induces a maximally pure and balanced partition then $J(h) = 1$. Since $h$ is maximally balanced, $P(h(x) > 0) = 0.5$. Simultaneously, since $h$ is maximally pure $\forall_{i = \{1,2,\dots,k\}}(P(h(x) > 0|i) = 0 \:\:\text{or}\:\: P(h(x) > 0|i) = 1)$. Substituting that into Equation \[eq:objform\] yields that $J(h) = 1$. Proof of Theorem \[thm:main\] {#sec:maindetails} ============================= The analysis studies a tree construction algorithm where we recursively find the leaf node with the highest weight, and choose to split it into two children. Consider the tree constructed over $t$ steps where in each step we take one leaf node and split it into two. Let $n$ be the heaviest node at time $t$ and its weight $w_n$ be denoted by $w$ for brevity. Consider splitting this leaf to two children $n_0$ and $n_1$. For the ease of notation let $w_0 = w_{n_0}$ and $w_1 = w_{n_1}$. Also for the ease of notation let $\beta = P(h_n(x) > 0)$ and $P_i = P(h_n(x) > 0|i)$. Let $\pi_i$ be the shorthand for $\pi_{n,i}$ and $h$ be the shorthand for $h_n$. Recall that $\beta = \sum_{i=1}^k\pi_iP_i$ and $\sum_{i=1}^k\pi_i = 1$. Also notice that $w_0 = w(1-\beta)$ and $w_1 = w\beta$. Let ${\bm \pi}$ be the $k$-element vector with $i^{th}$ entry equal to $\pi_i$. Furthermore let $\tilde{G}({\bm \pi}) = \sum_{i = 1}^k \pi_{i}\ln \left( \frac{1}{\pi_{i}} \right)$. Before the split the contribution of node $n$ to $G_t$ was $w\tilde{G}({\bm \pi})$. Let $\pi_{n_0,i} = \frac{\pi_i(1 - P_i)}{1 - \beta}$ and $\pi_{n_1,i} = \frac{\pi_iP_i}{\beta}$ be the probabilities that a randomly chosen $x$ drawn from $\mathcal{P}$ has label $i$ given that $x$ reaches nodes $n_0$ and $n_1$ respectively. For brevity, let $\pi_{n_0,i}$ be denoted by $\pi_{0,i}$ and $\pi_{n_1,i}$ be denoted by $\pi_{1,i}$. Furthermore let ${\bm \pi}_0$ be the $k$-element vector with $i^{th}$ entry equal to $\pi_{0,i}$ and let ${\bm \pi}_1$ be the $k$-element vector with $i^{th}$ entry equal to $\pi_{1,i}$. Notice that ${\bm \pi} = (1 - \beta){\bm \pi}_0 + \beta{\bm \pi}_1$. After the split the contribution of the same, now internal, node $n$ changes to $w((1- \beta)\tilde{G}({\bm \pi}_0) + \beta \tilde{G}({\bm \pi}_1))$. We denote the difference between them as $\Delta_t$ and thus $$\Delta_t := G_t - G_{t+1} = w\left[\tilde{G}({\bm \pi}) - (1- \beta)\tilde{G}({\bm \pi}_0) - \beta \tilde{G}({\bm \pi}_1)\right]. \label{eqn:ent-decrease} \vspace{-0.02in}$$ We aim to lower-bound $\Delta_t$. The entropy reduction of Equation \[eqn:ent-decrease\] [@Kearns95] corresponds to a gap in the Jensen’s inequality applied to the concave function $\tilde{G}(\bm \pi)$. This leads to the lower-bound on $\Delta_t$ given in Lemma \[lem:lower-bound\] (the lemma is proven in Section \[sec:lower-boundproof\] in the Supplementary material). The entropy reduction $\Delta_t$ of Equation \[eqn:ent-decrease\] can be lower-bounded as follows $$\Delta_t \geq \frac{J(h)^2G_t}{8\beta(1-\beta)t\ln k}$$ \[lem:lower-bound\] Lemma \[lem:lower-bound\] implies that the larger the objective $J(h)$ is at time $t$, the larger the entropy reduction ends up being, which further reinforces intuitions to maximize $J$. In general, it might not be possible to find any hypothesis with a large enough objective $J(h)$ to guarantee sufficient progress at this point so we appeal to a *weak learning assumption*. This assumption can be used to further lower-bound $\Delta_t$. The lower-bound can then be used (details are in Section \[sec:maindetails\] in the Supplementary material) to obtain the main theoretical statement of the paper captured in Theorem \[thm:main\]. From the definition of $\gamma$ it follows that $1 - \gamma \geq \beta \geq \gamma$. Also note that the *weak hypothesis assumption* guarantees $J(h) \geq 2\gamma$, which applied to the lower-bound on $\Delta_t$ captured in Lemma \[lem:lower-bound\] yields $$\Delta_t \geq \frac{\gamma^2G_t}{2(1 - \gamma)^2t\ln k}. \vspace{-0.05in}$$ Let $\eta = \sqrt{\frac{8}{(1 - \gamma)^2\ln k}}\gamma$. Then $\Delta_t > \frac{\eta^2G_t}{16t}$. Thus we obtain the recurrence inequality $$G_{t+1} \leq G_t - \Delta_t < G_t - \frac{\eta^2G_t}{16t} = G_t\left[1 - \frac{\eta^2}{16t}\right] \vspace{-0.02in}$$ One can now compute the minimum number of splits required to reduce $G_t$ below $\alpha$, where $\alpha \in [0,1]$. Applying the proof technique from [@Kearns95] (the proof of Theorem 10) gives the final statement of Theorem \[thm:main\]. Proof of Lemma \[lem:lower-bound\] {#sec:lower-boundproof} ================================== Without loss of generality assume that $P_1 \leq P_2 \leq \dots \leq P_k$. As mentioned before, the entropy reduction $\Delta_t$ corresponds to a gap in the Jensen’s inequality applied to the concave function $\tilde{G}(\bm \pi)$. Also recall that Shannon entropy is strongly concave with respect to $\ell_1$-norm (see e.g., Example 2.5 in Shalev-Shwartz [@ShaiSS2012]). As a specific consequence (see e.g. Theorem 2.1.9 in Nesterov [@Nesterov2004]) we obtain $$\Delta_t \geq w\beta(1-\beta)\|{\bm \pi}_0 - {\bm \pi}_1\|_1^2 = \frac{w}{\beta(1-\beta)}\left(\sum_{i = 1}^k\left|\pi_i(P_i - \beta)\right|\right)^2 = \frac{wJ(h)^2}{4\beta(1 - \beta)}, \label{eq:subst}$$ where the last equality results from the definition of $J(h) = 2\sum_{i = 1}^k\pi_i|P_i - \beta|$. Note that the following holds $w \geq \frac{G_t}{2t\ln k}$, where recall that $w$ is the weight of the heaviest leaf in the tree, i.e. the leaf with the highest weight, at round $t$. This leaf is selected to the currently considered split [@Kearns95]. In particular, the lower-bound on $w$ is the consequence of the following $$G_t \!=\! \sum_{l \in \mathcal{L}}\!\!w_l\!\sum_{i = 1}^k\!\! \pi_{l,i}\ln \left( \frac{1}{\pi_{l,i}} \right) \leq \sum_{l \in \mathcal{L}}\!\!w_l\ln k \leq 2tw\ln k,$$ where $w = \max_{l \in \mathcal{L}}w_l$. Thus $w \geq \frac{G_t}{2t\ln k}$ which when substituted to Equation \[eq:subst\] gives the final statement of the lemma. Proof of Lemma \[lemma:sb\] =========================== We bound the number of swaps that any node makes. Consider $R_S = 4$ and let $j$ be the node that is about to split and $s$ be the orphan node that will be recycled (thus $C_r = C_s$). The condition in Equation \[eq:swap\_cond\] implies that the swap is done if $C_j > 4(C_r + 1) = 4(C_s + 1)$. Algorithm \[alg:OTT\] makes $s$ a child of $j$ during the swap and sets its counter to $C_s^{new} = \floor*{C_j/2} \geq 2(C_r + 1) = 2(C_s + 1)$. Then $C_r$ gets updated. Since the value of $C_s^{new}$ at least doubles after a swap and all counters are bounded by the number of examples $n$, the node can be involved in at most $\log_2 n$ swaps. Equivalent forms of the objective function {#sec:explanation} ========================================== Consider the objective function as given in Equation \[eqn:objective\] $$J(h) = 2\sum_{i=1}^k \pi_i \left| P(h(x) > 0) - P(h(x) > 0 | i) \right|.$$ Recall that $\mathcal{X}$ denotes the set of all examples and let $\mathcal{X}_i$ denote the set of examples in class $i$. Also let $|\mathcal{X}|$ denote the cardinality of set $\mathcal{X}$ and let $|\mathcal{X}_i|$ denote the cardinality of set $\mathcal{X}_i$. Then we can re-write the objective as $$\begin{aligned} J(h) &=& 2\sum_{i=1}^k \pi_i \left| \frac{\sum_{x \in \mathcal{X}}\mathds{1}(h(x) > 0)}{|\mathcal{X}|} - \frac{\sum_{x \in \mathcal{X}_i}\mathds{1}(h(x) > 0)}{|\mathcal{X}_i|} \right| \nonumber\\ &=& 2\sum_{i=1}^k \pi_i \left| \mathbb{E}_x[\mathds{1}(h(x) > 0)] - \mathbb{E}_{x}[\mathds{1}(h(x) > 0|i)] \right| \nonumber\\ &=& 2\mathbb{E}_i[\left| \mathbb{E}_x[\mathds{1}(h(x) > 0)] - \mathbb{E}_{x}[\mathds{1}(h(x) > 0|i)] \right|]. \nonumber\end{aligned}$$ Toy example of the behavior of LOMtree algorithm {#sec:toy} ================================================ Figure \[fig:root\_example\] shows the toy example of the behavior of LOMtree algorithm for the first few data points. Without loss of generality we consider the root node (exactly the same actions would be performed in any other tree node). Notice that the algorithm achieves simultaneously balanced and pure split of classes reaching the considered node. $e$ denotes the expectation $\mathbb{E}_x[h(x)]$, and $e1,e2,e3,e4$ denote the expectations $\mathbb{E}_x[h(x)|i = 1]$, $\mathbb{E}_x[h(x)|i = 2]$, $\mathbb{E}_x[h(x)|i = 3]$, and $\mathbb{E}_x[h(x)|i = 4]$. For simplicity we assume score $h(x)$ can only be either $1$ (if the example is sent to the right) or $-1$ (if the example is sent to the left). The figure should be read as follows (we explain how to read first few illustrations): 1. Root is initialized. Expectation $e$ is initialized to $0$. 2. The first example $x1$ comes with label $1$ (we denote it as $(x1,1)$). $e1$ is initialized to $0$. The difference between $e$ and $e1$ is computed: $e - e1 = 0$. The difference is non-positive thus the example is sent to the right child of the root, which is now being created (the left child is created along with the right child as we always create both children of any node simultaneously). 3. Expectations $e$ and $e1$ get updated. It is shown that root and its right child saw an example of class $1$. 4. The second example $x2$ comes with label $2$ (we denote it as $(x2,2)$). $e2$ is initialized to $0$. The difference between $e$ and $e2$ is computed: $e - e2 = 1$. The difference is positive thus the example is sent to the left child of the root. 5. Expectations $e$ and $e2$ get updated. It is shown that root saw examples of class $1$ and $2$, whereas its resp. left and right child saw example of class resp. $2$ and $1$. 6. $\dots$ a\) b) c)\ d) e) f)\ g) h) i)\ j) k) Experiments - dataset details ============================= Below we provide the details of the datasets that we were using for the experiments in Section \[sec:experiments\]: - *Isolet*: downloaded from <http://www.cs.huji.ac.il/~shais/datasets/ClassificationDatasets.html> - *Sector* and *Aloi*: downloaded from <http://www.csie.ntu.edu.tw/~cjlin/libsvmtools/datasets/multiclass.html> - *ImageNet* [@Deng09imagenet:a]: features extracted according to <http://www.di.ens.fr/willow/research/cnn/>, dataset obtained from the authors. - *ODP* [@conf/sigir/BennettN09]: obtained from Paul Bennett. Our version has significantly more classes than reported in the cited paper because we use the entire dataset. [^1]: Throughout the paper by logarithmic time we mean logarithmic time per example. [^2]: The problem bears parallels to clustering in this regard. [^3]: Further in the paper we skip index $n$ whenever it is clear from the context that we consider a fixed tree node. [^4]: We want an objective to achieve its optimum for simultaneously pure and balanced split. The standard entropy-based criteria, such as Shannon or Gini entropy, as well as the criterion we will propose, posed in Equation \[eqn:objective\], satisfy this requirement (for the entropy-based criteria see [@Kearns95], for our criterion see Lemma \[lemma:maximal\]). [^5]: Our algorithm could also be implemented as batch or streaming, where in case of the latter one can for example make one pass through the data per every tree level, however for massive datasets making multiple passes through the data is computationally costly, further justifying the need for an online approach. [^6]: The proposed objective function exhibits some similarities with the so-called Carnap’s measure [@Tentori2007107; @Carnap1962] used in probability and inductive logic.

--- abstract: 'A dominating set of a graph $G$ is a subset $D \subseteq V_G$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is the domination number of $G$. The accurate domination number of $G$, denoted by $\gamma_{\rm a}(G)$, is the cardinality of a smallest set $D$ that is a dominating set of $G$ and no $|D|$-element subset of $V_G \setminus D$ is a dominating set of $G$. We study graphs for which the accurate domination number is equal to the domination number. In particular, all trees $G$ for which $\gamma_{\rm a}(G) = \gamma(G)$ are characterized. Furthermore, we compare the accurate domination number with the domination number of different coronas of a graph.' author: - | $^{1}$Joanna Cyman, $^{2}$Michael A. Henning[^1], and $^{3}$Jerzy Topp\ \ $^1$Faculty of Applied Physics and Mathematics\ Gdańsk University of Technology, 80-233 Gdańsk, Poland\ \ \ $^2$Department of Pure and Applied Mathematics\ University of Johannesburg\ Auckland Park 2006, South Africa\ \ \ $^3$Faculty of Mathematics, Physics and Informatics\ University of Gdańsk, 80-952 Gdańsk, Poland\ title: On Accurate Domination in Graphs --- [**Keywords:** Domination number; Accurate domination number; Tree; Corona.]{}\ Introduction and notation ========================= We generally follow the notation and terminology of [@ChartrandLesniakPing; @Zhang] and [@Haynes...Slater]. Let $G = (V_G,E_G)$ be a graph with vertex set $V_G$ of order $n(G) = |V_G|$ and edge set $E_G$ of size $m(G) = |E_G|$. If $v$ is a vertex of $G$, then the *open neighborhood* of $v$ is the set $N_G(v)=\{u\in V_G\colon uv\in E_G\}$, while the *closed neighborhood* of $v$ is the set $N_G[v]=N_G(v)\cup\{v\}$. For a subset $X$ of $V_G$ and a vertex $x$ in $X$, the set ${{\rm pn}}_G(x,X) = \{v \in V_G \mid N_G[v] \cap X = \{x\}\}$ is called the *$X$-private neighborhood* of the vertex $x$, and it consists of those vertices of $N_G[x]$ which are not adjacent to any vertex in $X \setminus \{x\}$; that is, ${{\rm pn}}_G(x,X) = N_G[x] \setminus N_G[X\setminus\{x\}]$. The *degree* $d_G(v)$ of a vertex $v$ in $G$ is the number of vertices in $N_G(v)$. A vertex of degree one is called a *leaf* and its neighbor is called a *support vertex*. The set of leaves of a graph $G$ is denoted by $L_G$, while the set of support vertices by $S_G$. For a set $S\subseteq V_G$, the subgraph induced by $S$ is denoted by $G[S]$, while the subgraph induced by $V_G \setminus S$ is denoted by $G-S$. Thus the graph $G - S$ is obtained from $G$ by deleting the vertices in $S$ and all edges incident with $S$. Let $\kappa(G)$ denote the number of components of $G$. A *dominating set* of a graph $G$ is a subset $D$ of $V_G$ such that every vertex not in $D$ is adjacent to at least one vertex in $D$, that is, $N_G(x)\cap D\ne \emptyset$ for every $x\in V_G \setminus D$. The *domination number* of $G$, denoted by $\gamma(G)$, is the cardinality of a smallest dominating set of $G$. An *accurate dominating set* of $G$ is a dominating set $D$ of $G$ such that no $|D|$-element subset of $V_G \setminus D$ is a dominating set of $G$. The *accurate domination number* of $G$, denoted by ${\gamma_{\rm a}}(G)$, is the cardinality of a smallest accurate dominating set of $G$. We call a dominating set of $G$ of cardinality $\gamma(G)$ a $\gamma$-*set of $G$*, and an accurate dominating set of $G$ of cardinality ${\gamma_{\rm a}}(G)$ a ${\gamma_{\rm a}}$-*set of $G$*. Since every accurate dominating set of $G$ is a dominating set of $G$, we note that $\gamma(G)\le {\gamma_{\rm a}}(G)$. The accurate domination in graphs was introduced by Kulli and Kattimani [@KulliKattimani], and further studied in a number of papers. A comprehensive survey of concepts and results on domination in graphs can be found in [@Haynes...Slater]. We denote the path and cycle on $n$ vertices by $P_n$ and $C_n$, respectively. We denote by $K_n$ the *complete graph* on $n$ vertices, and by $K_{m,n}$ the *complete bipartite graph* with partite sets of size $m$ and $n$. The accurate domination numbers of some common graphs are given by the following formulas: \[formula\] The following holds.\ 1. For $n \ge 1$, ${\gamma_{\rm a}}(K_n)= \lfloor \frac{n}{2} \rfloor + 1$ and ${\gamma_{\rm a}}(K_{n,n})= n + 1$. 2. For $n > m \ge 1$, ${\gamma_{\rm a}}(K_{m,n}) = m$. 3. For $n \ge 3$, ${\gamma_{\rm a}}(C_n)= \lfloor \frac{n}{3} \rfloor - \lfloor \frac{3}{n} \rfloor+2$. 4. For $n \ge 1$, ${\gamma_{\rm a}}(P_n)= \lceil \frac{n}{3} \rceil$ unless $n \in \{2,4\}$ when ${\gamma_{\rm a}}(P_n)= \lceil \frac{n}{3} \rceil + 1$ [(see Corollary \[wniosek-sciezki\])]{}. In this paper we study graphs for which the accurate domination number is equal to the domination number. In particular, all trees $G$ for which ${\gamma_{\rm a}}(G)= \gamma(G)$ are characterized. Furthermore, we compare the accurate domination number with the domination number of different coronas of a graph. Throughout the paper, we use the symbol ${{\cal A_{\gamma}}}(G)$ (respectively, ${{\cal A_{\gamma_{\rm a}}}}(G)$) to denote the set of all minimum dominating sets (respectively, minimum accurate dominating sets) of $G$. Graphs with ${\gamma_{\rm a}}$ equal to $\gamma$ ================================================ We are interested in determining the structure of graphs for which the accurate domination number is equal to the domination number. The question about such graphs has been stated in [@KulliKattimani]. We begin with the following general property of the graphs $G$ for which ${\gamma_{\rm a}}(G)= \gamma(G)$. \[twierdzenie1\] Let $G$ be a graph. Then ${\gamma_{\rm a}}(G)=\gamma(G)$ if and only if there exists a set $D \in {{\cal A_{\gamma}}}(G)$ such that $D \cap D' \ne \emptyset$ for every set $D' \in {{\cal A_{\gamma}}}(G)$. First assume that ${\gamma_{\rm a}}(G)=\gamma(G)$, and let $D$ be a minimum accurate dominating set of $G$. Since $D$ is a dominating set of $G$ and $|D|={\gamma_{\rm a}}(G)=\gamma(G)$, we note that $D \in {{\cal A_{\gamma}}}(G)$. Now let $D'$ be an arbitrary minimum dominating set of $G$. If $D \cap D' = \emptyset$, then $D' \subseteq V_G \setminus D$, implying that $D'$ would be a $|D|$-element dominating set of $G$, contradicting the fact that $D$ is an accurate dominating set of $G$. Hence, $D \cap D' \ne \emptyset$. Now assume that there exists a set $D \in {{\cal A_{\gamma}}}(G)$ such that $D \cap D' \ne \emptyset$ for every set $D' \in {{\cal A_{\gamma}}}(G)$. Then, $D$ is an accurate dominating set of $G$, implying that ${\gamma_{\rm a}}(G) \le |D| = \gamma(G) \le {\gamma_{\rm a}}(G)$. Consequently, we must have equality throughout this inequality chain, and so ${\gamma_{\rm a}}(G)=\gamma(G)$. It follows from Lemma \[twierdzenie1\] that if $G$ is a disconnected graph, then ${\gamma_{\rm a}}(G)=\gamma(G)$ if and only if ${\gamma_{\rm a}}(H)=\gamma(H)$ for at least one component $H$ of $G$. In particular, if $G$ has an isolated vertex, then ${\gamma_{\rm a}}(G)=\gamma(G)$. It also follows from Lemma \[twierdzenie1\] that for a graph $G$, ${\gamma_{\rm a}}(G)=\gamma(G)$ if $G$ has one of the following properties: (1) $G$ has a unique minimum dominating set (see, for example, [@Fischermann] or [@GuntherHartnellMarkusRall] for some characterizations of such graphs); (2) $G$ has a vertex which belongs to every minimum dominating set of $G$ (see [@Mynhardt]); (3) $G$ has a vertex adjacent to at least two leaves. Consequently, there is no forbidden subgraph characterization for the class of graphs $G$ for which ${\gamma_{\rm a}}(G)=\gamma(G)$, as for any graph $H$, we can add an isolated vertex (or two leaves to one vertex of $H$), and in this way form a graph $H'$ for which ${\gamma_{\rm a}}(H')= \gamma(H')$. The *corona* $F \circ K_1$ of a graph $F$ is the graph formed from $F$ by adding a new vertex $v'$ and edge $vv'$ for each vertex $v \in V(F)$. A graph $G$ is said to be a *corona graph* if $G = F \circ K_1$ for some connected graph $F$. We note that each vertex of a corona graph $G$ is a leaf or it is adjacent to exactly one leaf of $G$. Recall that we denote the set of all leaves in a graph $G$ by $L_G$, and set of support vertices in $G$ by $S_G$. \[twierdzenie-corona\] If $G$ is a corona graph, then ${\gamma_{\rm a}}(G) > \gamma(G)$. Assume that $G$ is a corona graph. If $G = K_1 \circ K_1$, then $G = K_2$ and ${\gamma_{\rm a}}(G) = 2$ and $\gamma(G) = 1$. Hence, we may assume that $G = F \circ K_1$ for some connected graph $F$ of order $n(F) \ge 2$. If $v \in V_G \setminus L_G$, then let ${\overline{v}}$ denote the unique leaf-neighbor of $v$ in $G$. Now let $D$ be an arbitrary minimum dominating set of $G$, and so $D \in {{\cal A_{\gamma}}}(G)$. Then, $|D \cap \{v, {\overline{v}}\}| =1$ for every $v \in V_G \setminus L_G$. Consequently, $D$ and its complement $V_G \setminus D$ are minimum dominating sets of $G$. Thus, $D$ is not an accurate dominating set of $G$. This is true for every minimum dominating set of $G$, implying that ${\gamma_{\rm a}}(G) > \gamma(G)$. \[l:support\] If $T$ is a tree of order at least three, then there exists a set $D \in {{\cal A_{\gamma}}}(T)$ such that the following hold.\ 1. $S_T\subseteq D$. 2. $N_T(v)\subseteq V_T \setminus D$ or $|{{\rm pn}}_T(v,D)|\ge 2$ for every $v \in D \setminus S_T$. Let $T$ be a tree of order $n(T) \ge 3$. Among all minimum dominating sets of $T$, let $D \in {{\cal A_{\gamma}}}(T)$ be chosen that\ (1) $D$ contains as many support vertices as possible.\ (2) Subject to (1), the number of components $\kappa(T[D])$ is as large as possible. If the set $D$ contains a leaf $v$ of $T$, then we can simply replace $v$ in $D$ with the support vertex adjacent to $v$ to produce a new minimum dominating set with more support vertices than $D$, a contradiction. Hence, the set $D$ contains no leaves, implying that $S_T \subseteq D$. Suppose, next, that there exists a vertex $v$ in $D$ that is not a support vertex of $T$ and such that $N_T(v) \not\subseteq V_T \setminus D$. Thus, $v$ has at least one neighbor in $D$; that is, $N_T(v)\cap D \ne \emptyset$. By the minimality of the set $D$, we therefore note that ${{\rm pn}}_T(v,D) \ne \emptyset$. If $|{{\rm pn}}_T(v,D)| = 1$, say ${{\rm pn}}_T(v,D)=\{u\}$, then letting $D' = (D \setminus \{v\}) \cup \{u\}$, the set $D' \in {{\cal A_{\gamma}}}(T)$ and satisfies $S_T \subseteq D \setminus \{v\} \subset D'$ and $\kappa(T[D']) > \kappa(T[D])$, which contradicts the choice of $D$. Hence, if $v \in D$ is not a support vertex of $T$ and $N_T(v) \not\subseteq V_T \setminus D$, then $|{{\rm pn}}_T(v,D)|\ge 2$. We are now in a position to present the following equivalent characterizations of trees for which the accurate domination number is equal to the domination number. \[t:trees\] If $T$ is a tree of order at least two, then the following statements are equivalent:\ 1. $T$ is not a corona graph. 2. There exists a set $D \in {{\cal A_{\gamma}}}(T)$ such that $\kappa(T-D) > |D|$. 3. ${\gamma_{\rm a}}(T)= \gamma(T)$. 4. There exists a set $D \in {{\cal A_{\gamma}}}(T)$ such that $D \cap D' \ne \emptyset$ for every $D' \in {{\cal A_{\gamma}}}(T)$. The statements (3) and (4) are equivalent by Lemma \[twierdzenie1\]. The implication $(3)\Rightarrow (1)$ follows from Lemma \[twierdzenie-corona\]. To prove the implication $(2)\Rightarrow (3)$, let us assume that $D \in {{\cal A_{\gamma}}}(T)$ and $\kappa(T-D) > |D|$. Thus, $\gamma(T-D) \ge \kappa(T-D) > |D|= \gamma(T)$. This proves that $D$ is an accurate dominating set of $T$, and therefore ${\gamma_{\rm a}}(T)= \gamma(T)$. Thus it suffices to prove that (1) implies (2). The proof is by induction on the order of a tree. The implication $(1) \Rightarrow (2)$ is obvious for trees of order two, three, and four. Thus assume that $T$ is a tree of order at least five and $T$ is not a corona graph. Let $D \in {{\cal A_{\gamma}}}(T)$ and assume that $S_T \subseteq D$. Since $T$ is not a corona graph, the tree $T$ has a vertex which is neither a leaf nor adjacent to exactly one leaf. We consider two cases, depending on whether $d_T(v) \ge 3$ for some vertex $v \in S_T$ or $d_T(v)=2$ for every vertex $v \in S_T$. *Case 1. $d_T(v) \ge 3$ for some $v \in S_T$.* Let $v'$ be a leaf of $T$ adjacent to $v$. Let $T'$ be a component of $T-\{v,v'\}$. Now let $T_1$ and $T_2$ be the subtrees of $T$ induced on the vertex sets $V_{T'} \cup \{v,v'\}$ and $V_T \setminus V_{T'}$, respectively. We note that both trees $T_1$ and $T_2$ have order strictly less than $n(T)$. Further, $V(T_1) \cap V(T_2) = \{v,v'\}$, $E(T_1) \cap E(T_2) = \{vv'\}$, and at least one of $T_1$ and $T_2$, say $T_1$, is not a corona graph. Applying the induction hypothesis to $T_1$, there exists a set $D_1 \in {{\cal A_{\gamma}}}(T_1)$ such that $\kappa(T_1 - D_1) > |D_1|$. If $T_2$ is a corona graph, then choosing $D_2$ to be the set of support vertices in $T_2$ we note that $D_2 \in {{\cal A_{\gamma}}}(T_2)$ and $\kappa(T_2 - D_2) = |D_2|$. If $T_2$ is not a corona graph, then applying the induction hypothesis to $T_2$, there exists a set $D_2 \in {{\cal A_{\gamma}}}(T_2)$ such that $\kappa(T_2 - D_2) > |D_2|$. In both cases, there exists a set $D_2 \in {{\cal A_{\gamma}}}(T_2)$ such that $\kappa(T_2 - D_2) \ge |D_2|$. We may assume that all support vertices of $T_1$ and $T_2$ are in $D_1$ and $D_2$, respectively. Thus, $v \in D_1 \cap D_2$, the union $D_1 \cup D_2$ is a $\gamma$-set of $T$, and $\kappa(T-(D_1\cup D_2))= \kappa(T_1-D_1) + \kappa(T_2-D_2) -1> |D_1|+|D_2|-1= |D_1\cup D_2|$. *Case 2. $d_T(v)=2$ for every $v\in S_T$.* We distinguish two subcases, depending on whether $D \setminus S_T \ne \emptyset$ or $D \setminus S_T = \emptyset$. *Case 2.1. $D \setminus S_T \ne \emptyset$.* Let $v$ be an arbitrary vertex belonging to $D \setminus S_T$. It follows from the second part of Lemma \[l:support\] that there are two vertices $v_1$ and $v_2$ belonging to $N_T(v) \setminus D$. Let $R$ be the tree obtained from $T$ by adding a new vertex $v'$ and the edge $vv'$. We note that $D$ is a minimum dominating set of $R$ and $S_R \subseteq D$. Let $R'$ be the component of $R-\{v,v'\}$ containing $v_1$. Now let $R_1$ and $R_2$ be the subtrees of $R$ induced by the vertex sets $V_{R'} \cup \{v,v'\}$ and $V_R \setminus V_{R'}$, respectively. We note that both trees $R_1$ and $R_2$ have order strictly less than $n(T)$. Further, $V(R_1) \cap V(R_2) = \{v,v'\}$, $E(R_1) \cap E(R_2) = \{vv'\}$, and neither $R_1$ nor $R_2$ is a corona graph. By the induction hypothesis, there exists a set $D_1 \in {{\cal A_{\gamma}}}(R_1)$ and a set $D_2 \in {{\cal A_{\gamma}}}(R_2)$ such that $\kappa(R_1 - D_1) > |D_1|$ and $\kappa(R_2 - D_2) > |D_2|$. We may assume that all support vertices of $R_1$ and $R_2$ are in $D_1$ and $D_2$, respectively. Thus, $v \in D_1 \cap D_2$, the union $D_1 \cup D_2$ is a $\gamma$-set of $R$, and $$\begin{array}{lcl} \kappa(T-(D_1\cup D_2)) & = & \kappa(R-(D_1\cup D_2)) - 1 \\ & = & (\kappa(R_1-D_1) + \kappa(R_2-D_2) - 1) - 1 \\ & = & (\kappa(R_1-D_1) - |D_1| + \kappa(R_2-D_2) - |D_2| ) - 2 + |D_1| + |D_2| \\ & \ge & |D_1| + |D_2| \\ & = & |D_1 \cup D_2| + 1 \\ & > & |D_1\cup D_2|. \end{array}$$ *Case 2.2. $D \setminus S_T = \emptyset$.* In this case, we note that $D = S_T$. Let $v$ be an arbitrary vertex belonging to $D$ and assume that $N_T(v) = \{u, w\}$, where $u \in L_T$. If $w \in L_T$, then $T = K_{1,2}$, contradicting the assumption that $n(T) \ge 5$. If $w \in S_T$, then $T = P_4 = K_2 \circ K_1$, contradicting the assumption that $T$ is not a corona graph (and the assumption that $n(T) \ge 5$). Therefore, $w \in V_T \setminus (L_T \cup S_T)$. Thus, $V_T \setminus (L_T\cup S_T)$ is nonempty and $T - D$ has $|D|$ one-element components induced by leaves of $T$ and at least one component induced by $V_T \setminus (L_T \cup S_T)$. Consequently, $\kappa(T-D)\ge |D|+1>|D|$. This completes the proof of Theorem \[t:trees\]. The equivalence of the statements (1) and (3) of Theorem \[t:trees\] shows that the trees $T$ for which ${\gamma_{\rm a}}(T)= \gamma(T)$ are easy to recognize. From Theorem \[t:trees\] and from the well-known fact that $\gamma(P_n) = \lceil n/3\rceil$ for every positive integer $n$, we also immediately have the following corollary which provides a slight improvement on Proposition 3 in [@KulliKattimani]. \[wniosek-sciezki\] For $n \ge 1$, ${\gamma_{\rm a}}(P_n)= \gamma(P_n)= \lceil n/3\rceil$ if and only if $n\in \mathbb{N} \setminus \{2,4\}$. Domination of general coronas of a graph ======================================== Let $G$ be a graph, and let ${{\cal F}}=\{F_v\colon v\in V_G\}$ be a family of nonempty graphs indexed by the vertices of $G$. By $G \circ {{\cal F}}$ we denote the graph with vertex set $$V_{G\circ {{\cal F}}}= (V_G\times \{0\})\cup \bigcup_{v\in V_G}(\{v\}\times V_{F_v})$$ and edge set determined by open neighborhoods defined in such a way that $$N_{G\circ {{\cal F}}}((v,0)) = (N_G(v)\times \{0\})\cup (\{v\}\times V_{F_v})$$ for every $v\in V_G$, and $$N_{G\circ {{\cal F}}}((v,x)) = \{(v,0)\}\cup (\{v\}\times N_{F_v}(x))$$ if $v\in V_G$ and $x\in V_{F_v}$. The graph $G\circ {{\cal F}}$ is said to be the ${{\cal F}}$-*corona* of $G$. Informally, $G \circ {{\cal F}}$ is the graph obtained by taking a disjoint copy of $G$ and all the graphs of ${{\cal F}}$ with additional edges joining each vertex $v$ of $G$ to every vertex in the copy of $F_v$. If all the graphs of the family ${{\cal F}}$ are isomorphic to one and the same graph $F$ (as it was defined by Frucht and Harary [@FruchtHarary]), then we simply write $G\circ F$ instead of $G \circ {{\cal F}}$. Recall that a graph $G$ is said to be a *corona graph* if $G = F \circ K_1$ for some connected graph $F$. The $2$-*subdivided graph* $S_2(G)$ of a graph $G$ is the graph with vertex set $$V_{S_2(G)}= V_G \cup \bigcup_{vu \in E_G}\{(v,vu), (u,vu)\}$$ and the adjacency is defined in such a way that $$N_{S_2(G)}(x)= \{(x,xy)\colon y\in N_G(x)\}$$ if $x\in V_G\subseteq V_{S_2(G)}$, while $$N_{S_2(G)}((x,xy))= \{x\}\cup \{(y,xy)\}$$ if $(x,xy)\in \bigcup_{vu\in E_G}\{(v,vu), (u,vu)\} \subseteq V_{S_2(G)}$. Less formally, $S_2(G)$ is the graph obtained from $G$ by subdividing every edge with two new vertices; that is, by replacing edges $vu$ of $G$ with disjoint paths $(v,(v,vu),(u,vu),u)$. For a graph $G$ and a family ${{\cal P}}= \{ \mathcal{P}(v) \colon v \in V_G\}$, where $\mathcal{P}(v)$ is a partition of the neighborhood $N_G(v)$ of the vertex $v$, by $G \circ \mathcal{P}$ we denote the graph with vertex set $$V_{G\circ \mathcal{P}} =(V_G\times \{1\}) \cup \bigcup_{v\in V_G} (\{v\} \times \mathcal{P}(v))$$ and edge set $$E_{G\circ \mathcal{P}}= \bigcup_{v\in V_G} \{(v,1)(v,A)\colon A \in \mathcal{P}(v)\} \cup \bigcup_{uv\in E_G}\{(v,A)(u,B)\colon (u\in A) \wedge (v\in B) \}.$$ The graph $G\circ \mathcal{P}$ is called the $\mathcal{P}$-*corona* of $G$ and was defined by Dettlaff et al. in [@Dettlaff...Zylinski]. It follows from this definition that if ${{\cal P}}(v)= \{N_G(v)\}$ for every $v\in V_G$, then $G\circ \mathcal{P}$ is isomorphic to the corona $G\circ K_1$. On the other hand, if ${{\cal P}}(v)= \{\{u\}\colon u\in N_G(v)\}$ for every $v\in V_G$, then $G\circ \mathcal{P}$ is isomorphic to the 2-subdivided graph $S_2(G)$ of $G$. Examples of $G\circ K_1$, $G\circ \mathcal{F}$, $G\circ \mathcal{P}$, and $S_2(G)$ are shown in Fig. \[rys1\]. In this case $G$ is the graph $(K_2\cup K_1)+K_1$ with vertex set $V_G=\{v, u, w, z\}$ and edge set $E_G= \{vu, vw, uw, wz\}$, where the family ${{\cal F}}$ consists of the graphs $F_v=F_w=K_1$, $F_z=K_2$, and $F_u= K_2\cup K_1$, while ${{\cal P}}= \{{{\cal P}}(x)\colon x\in V_G\}$ is the family in which ${{\cal P}}(v) =\{\{u, w\}\}$, ${{\cal P}}(u) =\{\{v\}, \{w\}\}$, ${{\cal P}}(w) =\{\{u, v\}, \{z\}\}$, and ${{\cal P}}(z) =\{\{w\}\}$. (9.5,-1.7) (6.5,5) (0,1)[2.2pt]{}[s1]{} (2.5,1.5)[2.2pt]{}[s2]{} (4,1.5)[2.2pt]{}[s3]{} (1,2.5)[2.2pt]{}[s4]{} (0,2.5)[2.2pt]{}[k1]{} (2.5,3)[2.2pt]{}[k2]{} (4,3)[2.2pt]{}[k3]{} (1,4)[2.2pt]{}[k4]{} (0,0.7)[$(v,0)$]{} (2.5,1.2)[$(w,0)$]{} (4,1.2)[$(z,0)$]{} (1.45,2.63)[$(u,0)$]{} (0,2.76)[$(v,1)$]{} (2.5,3.26)[$(w,1)$]{} (4,3.26)[$(z,1)$]{} (1,4.26)[$(u,1)$]{} (-0.5,3.7)[$G\circ K_1$]{} (8.5,5) (0,1)[2.2pt]{}[s1]{} (2.5,1.5)[2.2pt]{}[s2]{} (4,1.5)[2.2pt]{}[s3]{} (1,2.5)[2.2pt]{}[s4]{} (0,2.5)[2.2pt]{}[k1]{} (2.5,3)[2.2pt]{}[k2]{} (3.8,3)[2.2pt]{}[k31]{} (4.2,3)[2.2pt]{}[k32]{} (0.6,4)[2.2pt]{}[k41]{} (1,4)[2.2pt]{}[k42]{} (1.4,4)[2.2pt]{}[k43]{} (0,0.7)[$(v,0)$]{} (2.5,1.2)[$(w,0)$]{} (4,1.2)[$(z,0)$]{} (1.45,2.63)[$(u,0)$]{} (0.2,2.83)[$F_v$]{} (2.7,3.33)[$F_w$]{} (1.5,4.35)[$F_u$]{} (4.3,3.35)[$F_z$]{} (1,4)(0.7,0.25) (4,3)(0.52,0.25) (0,2.5)(0.3,0.2) (2.5,3)(0.3,0.2) (-0.5,3.7)[$G\circ \mathcal{F}$]{} (9.5,-1) (6.5,5) (0,1)[2.2pt]{}[s1]{} (2.25,1.5)[2.2pt]{}[s21]{} (2.7,1.5)[2.2pt]{}[s22]{} (4,1.5)[2.2pt]{}[s3]{} (0.8,2.3)[2.2pt]{}[s41]{} (1.2,2.6)[2.2pt]{}[s42]{} (0,2.5)[2.2pt]{}[k1]{} (2.5,3)[2.2pt]{}[k2]{} (4,3)[2.2pt]{}[k3]{} (1,4)[2.2pt]{}[k4]{} (0,0.7)[$(\!v,\!\{\!u,\!w\!\}\!)$]{} (2.3,1.2)[$(\!w,\!\{\!u,\!v\!\}\!)$]{} (3.144,1.7)[$(\!w,\!\{\!z\!\}\!)$]{} (4,1.2)[$(\!z,\!\{\!w\!\}\!)$]{} (1.109,2.012)[$(\!u,\!\{\!v\!\}\!)$]{} (1.68,2.73)[$(\!u,\!\{\!w\!\}\!)$]{} (0,2.76)[$(v,1)$]{} (2.5,3.26)[$(w,1)$]{} (4,3.26)[$(z,1)$]{} (1,4.26)[$(u,1)$]{} (-0.5,3.7)[$G\circ \mathcal{P}$]{} (8.5,5) (0,1)[2.2pt]{}[s11]{} (0.2,1.3)[2.2pt]{}[s12]{} (2.09,1.77)[2.2pt]{}[s21]{} (2.5,1.3)[2.2pt]{}[s22]{} (2.75,1.5)[2.2pt]{}[s23]{} (4,1.5)[2.2pt]{}[s3]{} (0.8,2.3)[2.2pt]{}[s41]{} (1.2,2.6)[2.2pt]{}[s42]{} (0,2.5)[2.2pt]{}[k1]{} (2.5,3)[2.2pt]{}[k2]{} (4,3)[2.2pt]{}[k3]{} (1,4)[2.2pt]{}[k4]{} (0,0.7)[$(\!v,\!\{\!w\!\}\!)$]{} (0.7,1.33)[$(\!v,\!\{\!u\!\}\!)$]{} (2.55,1)[$(\!w,\!\{\!v\!\}\!)$]{} (3.177,1.7)[$(\!w,\!\{\!z\!\}\!)$]{} (1.7,1.53)[$(\!w,\!\{\!u\!\}\!)$]{} (4,1.2)[$(\!z,\!\{\!w\!\}\!)$]{} (1.109,2.012)[$(\!u,\!\{\!v\!\}\!)$]{} (1.68,2.73)[$(\!u,\!\{\!w\!\}\!)$]{} (0,2.76)[$(v,1)$]{} (2.5,3.26)[$(w,1)$]{} (4,3.26)[$(z,1)$]{} (1,4.26)[$(u,1)$]{} (-0.5,3.7)[$S_2(G)$]{} We now study relations between the domination number and the accurate domination number of different coronas of a graph. Our first theorem specifies when these two numbers are equal for the ${{\cal F}}$-corona $G\circ {{\cal F}}$ of a graph $G$ and a family ${{\cal F}}$ of nonempty graphs indexed by the vertices of $G$. \[twierdzenie-corona-ogolnie\] If $G$ is a graph and ${{\cal F}}= \{F_v \colon v\in V_G\}$ is a family of nonempty graphs indexed by the vertices of $G$, then the following holds.\ 1. $\gamma(G\circ {{\cal F}})= |V_G|$. 2. ${\gamma_{\rm a}}(G\circ {{\cal F}})= \gamma(G\circ {{\cal F}})$ if and only if $\gamma(F_v)>1$ for some vertex $v$ of $G$. 3. $|V_G|\le{\gamma_{\rm a}}(G\circ {{\cal F}})\le |V_G|+\min\{|V_{F_v}|\colon v\in V_G\}$. \(1) It is obvious that $V_G\times \{0\}$ is a minimum dominating set of $G\circ {{\cal F}}$ and therefore $\gamma(G\circ {{\cal F}})= |V_G\times \{0\}|=|V_G|$. \(2) If $\gamma(F_v)>1$ for some vertex $v$ of $G$, then $$\gamma(G\circ {{\cal F}}- (V_G\times \{0\}))= \sum_{v\in V_G}\gamma((G\circ {{\cal F}})[\{v\}\times V_{F_v}])= \sum_{v\in V_G}\gamma(F_v)>|V_G|= |V_G\times \{0\}|$$ and this proves that no subset of $V_{G\circ {{\cal F}}}-(V_G\times \{0\})$ of cardinality $|V_G\times \{0\}|$ is a dominating set of $G\circ {{\cal F}}$. Consequently $V_G\times \{0\}$ is a minimum accurate dominating set of $G\circ {{\cal F}}$ and therefore ${\gamma_{\rm a}}(G\circ {{\cal F}})=\gamma(G\circ {{\cal F}})$. Assume now that $G$ and ${{\cal F}}$ are such that ${\gamma_{\rm a}}(G\circ {{\cal F}})=\gamma(G\circ {{\cal F}})$. We claim that $\gamma(F_v)>1$ for some vertex $v$ of $G$. Suppose, contrary to our claim, that $\gamma(F_v)=1$ for every vertex $v$ of $G$. Then the set $U_v=\{x\in V_{F_v}\colon N_{F_v}[x]=V_{F_v}\}$, the set of universal vertices of $F_v$, is nonempty for every $v\in V_G$. Now, let $D$ be any minimum dominating set of $G\circ {{\cal F}}$. Then, $|D|= \gamma(G\circ {{\cal F}}) = |V_G\times \{0\}|= |V_G|$, $|D\cap (\{(v,0)\} \cup (\{v\}\times U_v))|=1$, and the set $(\{(v,0)\} \cup (\{v\}\times U_v))-D$ is nonempty for every $v\in V_G$. Now, if $\overline{D}$ is a system of representatives of the family $\{(\{(v,0)\} \cup (\{v\}\times U_v))-D\colon v\in V_G\}$, then $\overline{D}$ is a minimum dominating set of $G\circ {{\cal F}}$. Since $\overline{D}$ and $D$ are disjoint, $D$ is not an accurate dominating set of $G\circ {{\cal F}}$. Consequently, no minimum dominating set of $G\circ {{\cal F}}$ is an accurate dominating set and therefore $\gamma(G\circ {{\cal F}}) <{\gamma_{\rm a}}(G\circ {{\cal F}})$, a contradiction. \(3) The lower bound is obvious as $|V_G|=\gamma(G\circ {{\cal F}})\le{\gamma_{\rm a}}(G\circ {{\cal F}})$. Since $(V_G\times \{0\})\cup (\{v\}\times V_{F_v})$ is an accurate dominating set of $G\circ \mathcal{F}$ (for every $v\in V_G$), we also have the inequality ${\gamma_{\rm a}}(G\circ {{\cal F}})\le |V_G|+\min\{|V_{F_v}|\colon v\in V_G\}$. This completes the proof of Theorem \[twierdzenie-corona-ogolnie\]. As a consequence of Theorem \[twierdzenie-corona-ogolnie\], we have the following result. \[twierdzenie-corona2\] If $G$ is a graph, then ${\gamma_{\rm a}}(G\circ K_1)= \gamma(G\circ K_1)+1= |V_G|+1$. Since $\gamma(K_1)=1$, it follows from Theorem \[twierdzenie-corona-ogolnie\] that ${\gamma_{\rm a}}(G\circ K_1)\ge \gamma(G\circ K_1)+1= |V_G|+1$. On the other hand the set $(V_G\times \{0\})\cup \{(v,1)\}$ is an accurate dominating set of $G\circ K_1$ and therefore ${\gamma_{\rm a}}(G\circ K_1)\le |(V_G\times \{0\})\cup \{(v,1)\}| = |V_G|+1$. Consequently, ${\gamma_{\rm a}}(G\circ K_1)= \gamma(G\circ K_1)= |V_G|+1$. From Theorem \[twierdzenie-corona-ogolnie\] we know that ${\gamma_{\rm a}}(G\circ {{\cal F}})= \gamma(G\circ {{\cal F}})=|V_G|$ if and only if the family ${{\cal F}}$ is such that $\gamma(F_v)>1$ for some $F_v\in {{\cal F}}$, but we do not know any general formula for ${\gamma_{\rm a}}(G\circ {{\cal F}})$ if $\gamma(F_v)=1$ for every $F_v\in {{\cal F}}$. The following theorem shows a formula for the domination number and general bounds for the accurate domination number of a ${{\cal P}}$-corona of a graph. \[tw-general-corona\] If $G$ is a graph and ${{\cal P}}=\{\mathcal{P}(v) \colon v\in V_G\}$ is a family of partitions of the vertex neighborhoods of $G$, then the following holds.\ 1. $\gamma(G\circ \mathcal{P})=|V_G|$. 2. ${\gamma_{\rm a}}(G\circ \mathcal{P})\ge |V_G|$. 3. ${\gamma_{\rm a}}(G\circ \mathcal{P})\le |V_G|+\min\{\min\{|{{\cal P}}(v)|\colon v\in V_G\}, 1+\min\{|A|\colon A\!\in \!\bigcup_{v\in V_G}\!{{\cal P}}(v)\}\}$. It follows from the definition of $G\circ \mathcal{P}$ that $V_G\times \{1\}$ is a dominating set of $G\circ \mathcal{P}$, and therefore $\gamma(G \circ \mathcal{P})\le |V_G\times \{1\}|=|V_G|$. On the other hand, let $D \in {{\cal A_{\gamma}}}(G\circ \mathcal{P})$. Then $D \cap N_{G\circ \mathcal{P}}[(v,1)] \ne \emptyset$ for every $v\in V_G$, and, since the sets $N_{G\circ \mathcal{P}}[(v,1)]$ form a partition of $V_{G\circ \mathcal{P}}$, we have $$\gamma(G\circ \mathcal{P})= |D|= |\bigcup_{v\in V_G} \left(D\cap N_{G\circ \mathcal{P}}[(v,1)]\right)|= \sum_{v\in V_G}|D \cap N_{G\circ \mathcal{P}}[(v,1)]| \ge |V_G|.$$ Consequently, we have $|V_G|= \gamma( G\circ {{\cal P}})\le {\gamma_{\rm a}}(G\circ \mathcal{P})$, which proves (1) and (2). From the definition of $G\circ \mathcal{P}$ it also follows that each of the sets $(V_G\times \{1\})\cup N_{G\circ \mathcal{P}}[(v,1)]$ (for every $v\in V_G$) and $(V_G\times \{1\})\cup N_{G\circ \mathcal{P}}[(v,A)]$ (for every $v\in V_G$ and $A\in {{\cal P}}(v)$) is an accurate dominating set of $G\circ \mathcal{P}$. Hence, $$\begin{array}{lcl} |(V_G\times \{1\})\cup N_{G\circ \mathcal{P}}[(v,1)]| & = & |V_G\times \{1\}|+ |N_{G\circ \mathcal{P}}((v,1))| \\ & = & |V_G|+|{{\cal P}}(v)| \\ & \ge & |V_G|+\min\{ |{{\cal P}}(v)|\colon v\in V_G\} \\ & \ge & {\gamma_{\rm a}}(G\circ \mathcal{P}), \end{array}$$ and similarly $$\begin{array}{lcl} |(V_G\times \{1\})\cup N_{G\circ \mathcal{P}}[(v,A)]| & = & |(V_G\times \{1\})\cup \{(v,1)\} \cup N_{G\circ \mathcal{P}}((v,A))| \\ & = & |V_G|+1+ |A| \\ & \ge & |V_G|+1+ \min\{|A|\colon A\in \bigcup_{v\in V_G}{{\cal P}}(v)\}. \end{array}$$ Therefore, $${\gamma_{\rm a}}(G\circ \mathcal{P})\le |V_G|+\min\{\min\{|{{\cal P}}(v)|\colon v\in V_G\}, 1+\min\{|A|\colon A\in \bigcup_{v\in V_G}{{\cal P}}(v)\}\}.$$ This completes the proof of Theorem \[tw-general-corona\]. We do not know all the pairs $(G,{{\cal P}})$ achieving equality in the upper bound for the accurate domination number of a ${{\cal P}}$-corona of a graph, but Theorem \[tw-2-subdivision\] and Corollaries \[wniosek3\] and \[wniosek4\] show that the bounds in Theorem \[tw-general-corona\] are best possible. The next theorem also shows that the domination number and the accurate domination number of a $2$-subdivided graph are easy to compute. \[tw-2-subdivision\] If $G$ is a connected graph, then the following holds.\ 1. $\gamma(S_2(G))=|V_G|$. 2. $|V_G|\le {\gamma_{\rm a}}(S_2(G))\le |V_G|+2$. 3. ${\gamma_{\rm a}}(S_2(G)) = \left\{\begin{array}{rl} |V_G|+2,& \mbox{if \hspace{-0.3ex} $G$ is a cycle,}\\[1ex] |V_G|+1, & \mbox{if \hspace{-0.3ex} $G=K_2$,}\\[1ex] |V_G|,& \mbox{otherwise.}\end{array}\right.$ The statement (1) follows from Theorem \[tw-general-corona\](1). \(2) The inequalities $|V_G|\le {\gamma_{\rm a}}(S_2(G))\le |V_G|+2$ are obvious if $G=K_1$. Thus assume that $G$ is a connected graph of order at least two. Let $u$ and $v$ be adjacent vertices of $G$. Then, $V_G\cup \{(v,vu),(u,vu)\}$ is an accurate dominating set of $S_2(G)$ and we have $|V_G|=\gamma(S_2(G))\le {\gamma_{\rm a}}(S_2(G))\le |V_G\cup \{(v,vu),(u,vu)\}|=|V_G|+2$. \(3) The connectivity of $G$ implies that there are three cases to consider. *Case 1. $|E_G|>|V_G|$.* In this case $S_2(G)-V_G$ has $|E_G|$ components and therefore no $|V_G|$-element subset of $V_{S_2(G)} \setminus V_G$ dominates $S_2(G)$. Hence, $V_G$ is an accurate dominating set of $S_2(G)$ and ${\gamma_{\rm a}}(S_2(G))= |V_G|$. *Case 2. $|E_G|=|V_G|$.* In this case, $G$ is a unicyclic graph. First, if $G$ is a cycle, say $G=C_n$, then $S_2(G)=C_{3n}$ and ${\gamma_{\rm a}}(S_2(G))= {\gamma_{\rm a}}(C_{3n}) =n+2=|V_G|+2$ (see Proposition 3 in [@KulliKattimani]). Thus assume that $G$ is a unicyclic graph which is not a cycle. Then $G$ has a leaf, say $v$. Now, if $u$ is the only neighbor of $v$, then $(V_G \setminus \{v\}) \cup \{(v,vu)\}$ is a minimum dominating set of $S_2(G)$. Since $S_2(G)-((V_G \setminus \{v\})\cup \{(v,vu)\})$ has $|V_G|+1$ components, $(V_G \setminus \{v\})\cup \{(v,vu)\}$ is a minimum accurate dominating set of $S_2(G)$ and ${\gamma_{\rm a}}(S_2(G))= |(V_G \setminus \{v\})\cup \{(v,vu)\}|= |V_G|$. *Case 3. $|E_G|=|V_G|-1$.* In this case, $G$ is a tree. Now, if $G=K_1$, then $S_2(G) =K_1$ and ${\gamma_{\rm a}}(S_2(G))= {\gamma_{\rm a}}(K_1)=1=|V_G|$. If $G=K_2$, then $S_2(G)=P_4$ and ${\gamma_{\rm a}}(S_2(G))= {\gamma_{\rm a}}(P_4)=3=2+1=|V_G|+1$. Finally, if $G$ is a tree of order at least three, then the tree $S_2(G)$ is not a corona graph and by (1) and Theorem \[t:trees\] we have ${\gamma_{\rm a}}(S_2(G))= \gamma(S_2(G))= |V_G|$. As a consequence of Theorem \[tw-2-subdivision\], we have the following results. \[wniosek3\] If $T$ is a tree and ${{\cal P}}=\{\mathcal{P}(v) \colon v\in V_T\}$ is a family of partitions of the vertex neighborhoods of $T$, then $${\gamma_{\rm a}}(T\circ \mathcal{P}) = \left\{ \begin{array}{cl} |V_T|+1 & \mbox{if $|{{\cal P}}(v)|=1$ for every $v \in V_T$} {\vspace{0.1cm}}\\ |V_T| & \mbox{if $|{{\cal P}}(v)|>1$ for some $v \in V_T$.} \end{array} \right.$$ If $|{{\cal P}}(v)|=1$ for every $v \in V_T$, then $T\circ \mathcal{P}= T\circ K_1$ and the result follows from Corollary \[twierdzenie-corona2\]. If $|{{\cal P}}(v)|>1$ for some $v \in V_T$, then the tree $T\circ \mathcal{P}$ is not a corona and the result follows from Theorem \[t:trees\] and Theorem \[tw-general-corona\](1). \[wniosek4\] For $n \ge 3$, if ${{\cal P}}=\{\mathcal{P}(v) \colon v\in V_{C_n}\}$ is a family of partitions of the vertex neighborhoods of $C_n$, then $${\gamma_{\rm a}}(C_n\circ \mathcal{P}) = \left\{ \begin{array}{cl} n+1 & \mbox{if $|\mathcal{P}(v)|=1$ for every $v\in V_{C_n}$} {\vspace{0.1cm}}\\ n+2 & \mbox{if $|\mathcal{P}(v)|=2$ for every $v\in V_{C_n}$} {\vspace{0.1cm}}\\ n & \mbox{otherwise.} \end{array} \right.$$ If $|{{\cal P}}(v)|=1$ for every $v \in V_{C_n}$, then ${C_n}\circ \mathcal{P}= {C_n}\circ K_1$. Thus, by Theorem \[twierdzenie-corona2\], we have ${\gamma_{\rm a}}(C_n \circ \mathcal{P})= {\gamma_{\rm a}}(C_n \circ K_1)= \gamma(C_n \circ K_1)= |V_{C_n}|+1=n+1$. If $|{{\cal P}}(v)|>1$ (and therefore $|{{\cal P}}(v)|=2$) for every $v \in V_{C_n}$, then ${C_n}\circ \mathcal{P}= S_2(C_n)= C_{3n}$. Now, since ${\gamma_{\rm a}}(C_{3n})= n+2$ (as it was observed in [@KulliKattimani]), we have ${\gamma_{\rm a}}({C_n}\circ \mathcal{P})={\gamma_{\rm a}}(C_{3n})= n+2$. Finally assume that there are vertices $u$ and $v$ in $C_n$ such that $|{{\cal P}}(v)|=1$ and $|{{\cal P}}(u)|=2$. Then the sets $$V_{C_n}^1=\{x\in V_{C_n}\colon |{{\cal P}}(x)|=1\} \hspace*{0.5cm} \mbox{and} \hspace*{0.5cm} V_{C_n}^2=\{y\in V_{C_n}\colon |{{\cal P}}(y)|=2\}$$ form a partition of $V_{C_n}$. Without loss of generality we may assume that $x_1,x_2,\ldots,x_k$, $y_1,y_2,\ldots, y_{\ell}, \ldots$, $z_1, z_2,\ldots, z_p, t_1, t_2,\ldots, t_q$ are the consecutive vertices of $C_n$, where $$x_1,x_2,\ldots, x_k \in V_{C_n}^1, y_1,y_2,\ldots, y_{\ell}\in V_{C_n}^2, \ldots, z_1, z_2,\ldots, z_p\in V_{C_n}^1, t_1, t_2,\ldots, t_q\in V_{C_n}^2,$$ and $k+{\ell}+\ldots+p+q=n$. It is easy to observe that $D=\{(x_i,N_{C_n}(x_i))\colon i=1,\ldots,k\}\cup \{(y_j,1)\colon j=1,\ldots,{\ell}\}\cup \cdots \cup \{(z_i,N_{C_n}(z_i))\colon i=1,\ldots, p\}\cup \{(t_j,1)\colon j=1,\ldots,q\}$ is a dominating set of ${C_n}\circ \mathcal{P}$. Since the set $D$ is of cardinality $n=|V_{C_n}|$ and $n= \gamma({C_n}\circ \mathcal{P})$ (by Theorem \[tw-general-corona\](1)), $D$ is a minimum dominating set of ${C_n}\circ \mathcal{P}$. In addition, since ${C_n}\circ \mathcal{P}-D$ has $k+(2+({\ell}-1))+\ldots +p+(2+(q-1))> k+{\ell}+\ldots+p+q=n$ components, that is, since $\kappa({C_n}\circ \mathcal{P}-D)>n$, no $n$-element subset of $V_{{C_n}\circ \mathcal{P}} \setminus D$ is a dominating set of ${C_n}\circ \mathcal{P}$. Thus, $D$ is an accurate dominating set of ${C_n}\circ \mathcal{P}$ and therefore $\gamma({C_n}\circ \mathcal{P})=n$. Closing open problems ===================== We close with the following list of open problems that we have yet to settle. Find a formula for the accurate domination number ${\gamma_{\rm a}}(G\circ {{\cal F}})$ of the ${{\cal F}}$-corona of a graph $G$ depending only on the family ${{\cal F}}=\{F_v\colon v\in V_G\}$ such that $\gamma(F_v)=1$ for every $v\in V_G$. Characterize the graphs $G$ and the families ${{\cal P}}=\{{{\cal P}}(v) \colon v\in V_G\}$ for which ${\gamma_{\rm a}}(G\circ \mathcal{P})= |V_G|+\min\{ \min\{|{{\cal P}}(v)|\colon v\in V_G\}, 1+\min\{|A|\colon A\!\in \!\bigcup_{v\in V_G}\!{{\cal P}}(v)\}\}$. It is a natural question to ask if there exists a nonnegative integer $k$ such that ${\gamma_{\rm a}}(G\circ \mathcal{P})\le |V_G|+k$ for every graph $G$ and every family ${{\cal P}}=\{\mathcal{P}(v) \colon v\in V_G\}$ of partitions of the vertex neighborhoods of $G$. [9]{} G. Chartrand, L. Lesniak, P. Zhang, Graphs and Digraphs. CRC Press, Taylor and Francis Group, Boca Raton, 2016. M. Dettlaff, M. Lemańska, J. Topp, P. Żyliński, Coronas and domination subdivision number of a graph, Bull. Malays. Math. Sci. Soc. (2016). doi:10.1007/s40840-016-0417-0. M. Fischermann, Block graphs with unique minimum dominating sets, Discrete Math. 240 (2001), 247–251. R. Frucht and F. Harary, On the corona of two graphs, Aequ. Math. 4 (1970), 322–324. G. Gunther, B. Hartnell, L.R. Markus, D. Rall, Graphs with unique minimum dominating sets, Congr. Numer. 101 (1994), 55–63. T.W. Haynes, S.T. Hedetniemi, P.J. Slater, Fundamentals of Domination in Graphs. Marcel Dekker Inc., New York, 1998. V.R. Kulli and M.B. Kattimani, Accurate domination in graphs, Advances in Domination Theory I, ed. V.R. Kulli, Vishwa International Publications (2012), 1–8. C.M. Mynhardt, Vertices contained in every minimum dominating set of a tree, J. Graph Theory 31 (1999), 163–177. [^1]: Research supported in part by the South African National Research Foundation and the University of Johannesburg

--- abstract: 'The orbits of the reversible differential system $\dot{x}=-y$, $\dot{y}=x$, $\dot{{z}}=0$, with $x,y \in {\mathbb{R}}$ and ${z}\in {\mathbb{R}}^d$, are periodic with the exception of the equilibrium points $(0,0, {z})$. We compute the maximum number of limit cycles which bifurcate from the periodic orbits of the system $\dot{x}=-y$, $\dot{y}=x$, $\dot{{z}}=0$, using the averaging theory of first order, when this system is perturbed, first inside the class of all polynomial differential systems of degree $n$, and second inside the class of all discontinuous piecewise polynomial differential systems of degree $n$ with two pieces, one in $y> 0$ and the other in $y<0$. In the first case this maximum number is $n^d(n-1)/2$, and in the second is $n^{d+1}$.' address: - '$^1$ Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain.' - '$^{2}$ Departamento de Matemática, Universidade Estadual de Campinas, CP 6065, 13083-859, Campinas, SP, Brazil.' author: - 'Jaume Llibre$^1$, Marco A. Teixeira$^2$ and Iris O. Zeli$^{2}$' title: 'Birth of limit cycles for a class of continuous and discontinuous differential systems in $(d+2)$–dimension' --- Introduction and statements of the main results =============================================== Limit cycles have been used to model the behavior of many real process and different modern devices. In general to prove the existence of limit cycles is a very difficult problem. One way to produce limit cycles is perturbing differential systems that have a linear center. In this case, the limit cycles in a perturbed system bifurcate from the periodic orbits of the unperturbed center. The search for the maximum number of limit cycles that polynomial differential systems of a given degree can have is part of [*$16^{th}$ Hilbert’s Problem*]{} and many contributions have been made in this direction, see for instance [@Hil; @Ily; @Lii] and the references quoted therein. Recently the theory of limit cycles has also been studied in discontinuous piecewise differential systems. The analysis of these systems can be traced from Andronov [*et al.*]{} [@AVK] and still continues to receive attention by researchers. Discontinuous piecewise differential systems is a subject that have been developed very fast due to its strong applications to other branches of science. Currently such systems are one of the connections between mathematics, physics and engineering. These systems model several phenomena in control systems, impact in mechanical systems, nonlinear oscillations and economics see for instance [@Bar; @BSC; @Bro; @Chi; @Ito; @Min]. Recently they have been shown to be also relevant as idealized models for biology [@Kri] and models of cell activity [@Co; @To; @TG]. For more details see Teixeira [@Tei] and all references therein. As we have said it is not simple to determine the existence of limit cycles in a differential system. The simplest case for determining limit cycles is in planar continuous piecewise linear systems when they have only two linear differential systems separated by a straight line. Even in this simple case, only after a delicate analysis it was possible to show the existence of at most of one limit cycle for such systems, see [@FPRT] or an easier proof in [@LMP]. Planar discontinuous piecewise linear differential systems with only two linearity regions separated by a straight line have been studied recently in [@HZ; @HY], among other papers. In [@HZ] some results about the existence of two limit cycles appeared, so that the authors conjectured that the maximum number of limit cycles for this class of piecewise linear differential systems is exactly two. However in [@HY] strong numerical evidence about the existence of three limit cycles was obtained. As far we know the example in [@HY] represents the first discontinuous piecewise linear differential system with two zones with $3$ limit cycles surrounding a unique equilibrium. Recently in [@LP] it is proved that such a system really has three limit cycles. There are several papers studying the limit cycles of the continuous piecewise linear differential systems in ${\mathbb{R}}^3$, see for instance [@CLT; @LP1; @LPR; @LPRT; @LPT]. Our goal is study the periodic solutions of discontinuous piecewise polynomial differential systems in ${\mathbb{R}}^{d+2}$. More precisely the objective of this paper is to study the existence of limit cycles in continuous and discontinuous piecewise polynomial differential systems in ${\mathbb{R}}^{d+2}$, where the discontinuous differential system has two zones of continuity separated by a hyperplane. Without loss of generality we shall assume that the set of discontinuity is the hyperplane $y=0$ in ${\mathbb{R}}^{d+2}$. So we consider the linear differential system in ${\mathbb{R}}^{d+2}$ given by $$\begin{aligned} \label{eq.system.linear} \dot x = & - y \nonumber\\ \dot y = & ~x \\ \dot {z}_\l = & ~0 \nonumber\end{aligned}$$ for $\l=1, \ldots, d$ and $x,y \in {\mathbb{R}}$, ${z}\in {\mathbb{R}}^d$, where the dot denotes derivative with respect to the time $t$, which is reversible with respect to $\phi(x,y,{z})=(x,-y,{z})$ . First we are interested in studying the existence of limit cycles of the continuous polynomial differential system given by $$\begin{aligned} \label{eq.systemX.perturbed} \dot{x} = & -y + \e P_a(x,y,{z}) ,\nonumber\\ \dot{y} = & ~ x + \e P_b(x,y,{z}),\\ \dot{{z}_\l} = & ~\e P_{c_\l}(x,y,{z}), \nonumber \end{aligned}$$ and after we also study the existence of limit cycles of the discontinuous piecewise polynomial differential system formed by two polynomial differential systems separated by the hyperplane $y=0$, namely $$\label{eq.perturbed.discontinuous} \begin{array}{c} \left. \begin{array}{ll} \dot{x} ~ = & -y + \e P_a(x,y,{z}) , \vspace*{0.15 cm}\\ \dot{y} ~ = &~ x + \e P_b(x,y,{z}), \vspace*{0.15 cm}\\ \dot{{z}_\l} = & ~\e P_{c_\l}(x,y,{z}), \end{array} \right\} \quad \hbox{if}\quad y>0 \vspace*{0.25 cm} \\ \left. \begin{array}{ll} \dot{x} ~= & -y + \e Q_a(x,y,{z}) , \vspace*{0.15 cm}\\ \dot{y} ~= &~ x + \e Q_b(x,y,{z}), \vspace*{0.15 cm}\\ \dot{{z}_\l} = & ~\e Q_{c_\l}(x,y,{z}), \end{array} \right\} \quad \hbox{if}\quad y < 0 \end{array}$$ where $\e \neq 0$ is a small parameter and $\l=1, \ldots,d$. In this systems the polynomials $P_a$, $P_b$, $P_{c_\l}$, $Q_\alpha$, $Q_\beta$, $Q_{\gamma_\l}$ are of degree $n$ in the variables $x$, $y$ and ${z}$, more precisely $$\begin{aligned} P_a(x,y,z) = &\sum_{i+j+k=0}^{n} a_{ijk} x^i y^j z^k, \quad P_b(x,y,z)= \sum_{i+j+k=0}^{n} b_{ijk} x^i y^j z^k, \\ P_{c_\l}(x,y,z)= &\sum_{i+j+k=0}^{n} c_{\l ijk} x^i y^j z^k, \quad Q_a(x,y,z) = \sum_{i+j+k=0}^{n} \alpha_{ijk} x^i y^j z^k ,\\ Q_\beta(x,y,z)= & \sum_{i+j+k=0}^{n} \beta_{ijk} x^i y^j z^k, \quad Q_{\gamma_\l}(x,y,z)= \sum_{i+j+k=0}^{n} \gamma_{\l ijk} x^i y^j z^k. \nonumber\end{aligned}$$ In this expressions $k$ is a multi-index and $i+j+k$ denotes $i+j+k_1 + \ldots + k_d$, ${z}^k$ denotes the product $z_1^{k_1} \ldots z_d^{k_d}$ with ${z}=(z_1,\ldots,z_d)\in {\mathbb{R}}^d$, and $a_{ijk}$ denotes the coefficient $a_{ijk_1 \ldots k_d}$ of $x^i y^j z_1^{k_1} \ldots z_d^{k_d}$. It is clear that systems and coincide for $\varepsilon = 0$ and they have linear centers at every plane ${z}=$ constant. In this paper we establish for $\varepsilon \neq 0$ sufficiently small the maximum number of limit cycle of these systems that bifurcate from the periodic orbits of these linear centers using the averaging theory of first order. The following result presents the results for the continuous case. \[th.cycles.systemX\] Using the averaging theory of first order for $|\varepsilon| \neq 0$ sufficiently small the maximum number of limit cycles of the polynomial differential system is at most $n^d(n-1)/2$, and this number is reached. In the next theorem we present results for the discontinuous piecewise polynomial differential system . \[th.cycles.discontinuous\] Using the averaging theory of first order for $|\varepsilon| \neq 0$ sufficiently small the maximum number of limit cycles of the discontinuous piecewise polynomial differential system is at most $n^{d+1}$, and this number is reached. \[cor.1\] Under the assumptions of Theorem \[th.cycles.systemX\] if additionally $a_{00k}=b_{00k}=0$ for all $k$, the limit cycles can be chosen as close to the origin of ${\mathbb{R}}^{d+2}$ as we want. \[cor.2\] Under the assumptions of Theorem \[th.cycles.discontinuous\] if additionally $a_{00k}=b_{00k}=\alpha_{00k}=\beta_{00k}=0$ for all $k$, the number of limit cycles of the discontinuous piecewise polynomial differential system is at most $n^{d}(n-1)$, and this number is reached. Furthermore, the limit cycles can be chosen as close to the origin of ${\mathbb{R}}^{d+2}$ as we want. Corollaries \[cor.1\] and \[cor.2\] provides information of the Hopf bifurcation of systems and . More precisely, Corollaries \[cor.1\] and \[cor.2\] show that at least $n^d(n-1)/2$ and $n^d(n-1)$ limit cycles of systems and can bifurcate from the origin of ${\mathbb{R}}^{d+2}$, respectively. The results of Corollary \[cor.1\] in the particular case $n=2$ coincides with the result obtained in Theorem 1 of [@LZ]. To prove these results we use the classical averaging theory, see for instance [@SVM; @Ve] for a general introduction to this subject. This theory have been used for years to deal with continuous differential systems. The principle of averaging has been extended in many directions and recently in [@LNT] the authors extend the averaging theory for detecting limit cycles of certain discontinuous piecewise differential systems, via the Brouwer degree and the regularization theory. As far as we know this method is one of the best methods for determining limit cycles in discontinuous piecewise differential systems and has already been used by some authors. In [@LLM] the method is used for determining the maximum number of limit cycles that bifurcate from the periodic solutions of some family of isochronous cubic polynomial centers perturbed by discontinuous piecewise cubic polynomial differential systems with two zones separated by a straight line. In [@LM] limit cycles for discontinuous piecewise quadratic differential systems with two zones was studied using the averaging theory. Also in [@No] the averaging theory was applied to provide sufficient conditions for the existence of limit cycles of discontinuous perturbed planar centers when the discontinuity set is a union of regular curves. We have organized this paper as follows. In Section \[section:preliminaries\] we briefly present notation and basic concepts of the averaging theory of first order for continuous differential systems (see Theorem \[Th.continuous.averaging\]) and for discontinuous differential systems (see Theorem \[Th.discontinuous.averaging\]). In Section \[section:proofs\] we present the proof of results of this paper. More precisely, the proof of Theorem \[th.cycles.systemX\] is presented in the Subsection \[section:proof.th.continuous\], and in the Subsection \[section:proof.th.discontinuous\] we prove the Theorem \[th.cycles.discontinuous\]. Finally in the Subsection \[section:proof.cor\] we presented the proofs of the Corollaries \[cor.1\] and \[cor.2\]. Basic results in averaging theory {#section:preliminaries} ================================= In this section we present the basic results from the averaging theory of first order that we shall use for proving the results of this paper. The following theorem provides a method for studying the existence of periodic orbits of a differential system. For more details on the averaging method see for instance [@Ve]. Let $D$ be an open subset of ${\mathbb{R}}^n$. We denote the points of ${\mathbb{R}}\times D$ as $(t,x)$, and we take the variable $t$ as the time. \[Th.continuous.averaging\] Consider the differential system $$\label{eq.system.th.continuous} \dot{x}=\varepsilon F(t,x) + \varepsilon^2 R(t,x,\varepsilon),\quad x(0)=0,$$ where $F: {\mathbb{R}}\times D \to {\mathbb{R}}^n$ and $R: {\mathbb{R}}\times U \times (-\varepsilon,\varepsilon) \to {\mathbb{R}}^n$ are continuous functions and $T-$periodic in the first variable. Define the averaging function $f:D \to {\mathbb{R}}^n$ as $$\label{eq.averaging.function.th} f(x)= \int_0^T F(s,x) {~\mathrm{d}}s,$$ and assume that - the functions $F,~ R, ~D_xF,~ D_x^2F$ and $D_xR$ are defined, continuous and bounded by a constant $M$ (independent of $\varepsilon$) in $[0,\infty) \times D$ and for $\varepsilon \in (0,\varepsilon_0]$, - for $p \in D$ with $f(p)=0$ we have $|J_f(p)| \neq 0$, where $|J_f(p)|$ denotes the determinant of the Jacobian matrix of $f$ evaluated at $p$. Then for $|\varepsilon| > 0$ sufficiently small there exists a $T-$periodic solution $x(t,\varepsilon)$ of the system such that $x(0,\varepsilon) \to p$ as $\varepsilon \to 0$. Now let $h: {\mathbb{R}}\times D \to {\mathbb{R}}$ be a $\mathcal{C}^1$ function with $0\in {\mathbb{R}}$ as a regular value, and $\Sigma = h^{-1}(0)$. Let $X,Y: {\mathbb{R}}\times D \to {\mathbb{R}}^n$ be two continuous vector fields and assume that $h,X$ and $Y$ are $T-$periodic in the variable $t$. We define a *discontinuous piecewise differential system* as $$\label{eq.def.discontinuous.system} \dot{x}=Z(t,x)= \left\{ \begin{array}{c} X(t,x)\quad \text{if} \quad h(t,x) >0,\\ Y(t,x) \quad \text{if} \quad h(t,x) <0. \end{array} \right.$$ We rewrite the discontinuous differential system as follows. Consider the sign function defined in ${\mathbb{R}}\backslash\left\{ 0 \right\}$ as $$\nonumber \text{sign}(u) = \left\{ \begin{array}{c} ~~1 ~\quad \text{if} \quad u>0,\\ -1 \quad \text{if} \quad u<0. \end{array} \right.$$ Then system can be written as $$\nonumber \dot{x}=Z(t,x) = F_1(t,x) + {\operatorname{sign}}\left( h(t,x) \right) F_2(t,x),$$ where $$F_1(t,x)= \dis\frac{1}{2} \left( X(t,x) + Y(t,x) \right) \quad \text{and} \quad F_2(t,x)= \dis\frac{1}{2} \left( X(t,x) - Y(t,x) \right).$$ The following theorem is a version of Theorem \[Th.continuous.averaging\] for studying the periodic solutions of discontinuous differential systems. \[Th.discontinuous.averaging\] Consider the discontinuous differential system $$\label{eq.system1.th.discontinuous.averaging} \dot{x}= \varepsilon F(t,x) + \varepsilon^2 R(t,x,\varepsilon),$$ with $$\begin{aligned} F(t,x)&= F_1(t,x) + {\operatorname{sign}}\left( h(t,x) \right) F_2(t,x),\\ R(t,x,\varepsilon)&= R_1(t,x,\varepsilon) + {\operatorname{sign}}\left( h(t,x) \right) R_2(t,x,\varepsilon),\end{aligned}$$ where $F_1, F_2: {\mathbb{R}}\times D \to {\mathbb{R}}^n$, $R_1,R_2 : {\mathbb{R}}\times D \times (-\varepsilon, \varepsilon) \to {\mathbb{R}}^n$ and $h: {\mathbb{R}}\times D \to {\mathbb{R}}$ are continuous functions, $T-$periodic in the variable $t$. We also suppose that $h$ is a $\mathcal{C}^1$ function with $0$ as a regular value and we denote $\Sigma = h^{-1}(0)$. Define the averaged function $f: D \to {\mathbb{R}}^n$ as $$\label{eq.2.th.averaging.discontinuous} f(x) = \int_0^T F(s,x) {~\mathrm{d}}s,$$ and assume that - the functions $F_1, F_2, R_1, R_2$ and $h$ are locally Lipschitz with respect to $x$; - $\dis\frac{\partial h}{ \partial t}(t,x) \neq 0$ for all $(t,x) \in \Sigma$; - for $p \in C$ with $f(p)=0$, there exist a neighborhood $U \subset C$ of $p$ such that $f(z) \neq 0$ for all $ z \in \overline{U} \backslash \left\{ p \right\}$ and $d_B(f,U,0) \neq 0$ ($d_B$ is the Brouwer degree of $f$ in $p$). Then for $|\varepsilon| >0$ sufficiently small there exists a $T-$periodic solutions $x(t,x)$ of system such that $ x(t, \varepsilon) \to p$ as $\varepsilon \to 0$. For a proof of Theorem \[Th.discontinuous.averaging\] see Theorem $A$ and Proposition $2$ in [@LNT]. Here we emphasize that if $f$ in is $C^1$ then the hypotheses $d_B(f,U,0) \neq 0$ holds if $|J_f(p)| \neq 0$, see for more details [@Ll]. Proof of the main results {#section:proofs} ========================= We devoted this section for the proof of results of this paper. For this we consider the notations introduced in the Introduction. Proof of Theorem \[th.cycles.systemX\] {#section:proof.th.continuous} -------------------------------------- Applying the change of variables $$\label{eq.polar.coordinates} x = r \cos\theta, \quad y =r \sin \theta \quad \text{and} \quad {z}= {z},$$ system becomes $$\nonumber \label{1*} \begin{array}{lll} \dot{\theta}&=& 1+ \displaystyle \frac{\varepsilon}{r}\sum_{i+j+k=0}^{n} r^{i+j} {z}^k \big(a_{ijk} \cos^i\theta \sin^{j+1}\theta + b_{ijk} \cos^{i+1}\theta \sin^j\theta \big),\\ \dot{r}&=&\displaystyle \varepsilon \sum_{i+j+k=0}^{n} r^{i+j} {z}^k \left( a_{ijk} \cos^{i+1}\theta \sin^j\theta + b_{ijk}\cos^{i}\theta \sin^{j+1}\theta\right),\\ \dot{z}_{l}&= &\displaystyle \varepsilon\sum_{i+j+k=0}^{n} c_{lijk}~r^{i+j} {z}^k \cos^i\theta \sin^j\theta, \end{array}$$ Essentially for $\varepsilon\ne 0$ sufficiently small we study the existence of limit cycles bifurcating from periodic orbits of this system when $\varepsilon=0$, and that are contained in the cylindrical annulus $$\label{2*} \tilde{A}= \lbrace (\theta,r,z): r_0 \leq r \leq r_1,~ \theta\in \mathbb S^1, ~ {z}\in {\mathbb{R}}^d \rbrace.$$ So for $\varepsilon$ sufficiently small $\dot{\theta} >0$ for every $(\theta,r,{z}) \in \tilde{A}$. Now taken as new independent variable $\theta$ instead of $t$ so in $\tilde{A}$ can be written as $$\label{eq.systemX.new.coordinates} \begin{array}{lll} r' &=& ~\e F_1(\theta,r, {z}) + \mathcal{O}(\e^2) \vspace*{0.15cm}\\ z'_\l &= & ~ \e F_{\l+1}(\theta,r,z) + \mathcal{O}(\e^2) , \end{array}$$ for $\l=1, \ldots,d$, where the prime denotes derivative with respect to the variable $\theta$, and $$\begin{array}{lll} \label{eq.F1.F2} F_1(\theta,r,z) &= &\dis \sum_{i+j+k=0}^{n} r^{i+j}z^k \big( a_{ijk} \cos^{i+1}\theta \sin^j\theta \vspace*{0.15cm}\\ & & + b_{ijk} \cos^{i}\theta \sin^{j+1}\theta \big), \vspace*{0.15cm}\\ F_{\l+1}(\theta,r,z) &=& \dis \sum_{i+j+k=0}^{n} r^{i+j} z^k~c_{\l ijk} \cos^i\theta \sin^j\theta, \end{array}$$ for $\l=1,2,\ldots,d$. System is $2\pi$–periodic with respect to the independent variable $\theta$, and satisfies the hypotheses of the Theorem \[Th.continuous.averaging\] for $\varepsilon_0$ small and $D$ fixed. Now we compute the averaged function $f=(f_1,f_{2}, \ldots, f_{d+1})$ given in with $T=2\pi$. Now taking $$\mu_{(p,q)} = \displaystyle\int_0^{2\pi}\cos^p\theta\sin^q \theta{~\mathrm{d}}\theta,$$ note that $ \mu_{(p,q)} \neq 0 $ if only if $p$ and $q$ are simultaneously even. So we obtain $$\label{eq.f2.continuous} \begin{array}{lll} f_1(r,{z})&=& \dis \int_0^{2\pi} F_1(\theta,r,{z}){~\mathrm{d}}\theta = \dis \sum_{\footnotesize \begin{matrix} i+j+k=0 \\ i~odd,~j~even \end{matrix}}^{n} r^{i+j} {z}^k~ a_{ijk} ~\mu_{(i+1,j)} \\ & & + \dis \sum_{\footnotesize \begin{matrix} i+j+k=0 \\i~even,~j~odd \end{matrix}}^{n} r^{i+j} {z}^k~ b_{ijk} ~\mu_{(i,j+1)} \vspace*{0.2cm}\\ f_{\l+1}(r,{z})&=& \dis \int_0^{2\pi} F_{\l+1}(\theta,r,{z}){~\mathrm{d}}\theta = \dis \sum_{\footnotesize\begin{matrix} i+j+k=0 \\ i,j~even \end{matrix}}^{n} r^{i+j} {z}^k~c_{\l ijk} ~ \mu_{(i,j)}, \end{array}$$ for $\l=1,2,\ldots,d$. We split the proof in two parts. First assume $n$ is odd, so $$f_1(r,{z})=A_1 r + A_3 r^3 + \ldots + A_n r^n \nonumber$$ where $$\label{eq.Ap.continuous} A_p = \sum_{k=0}^{n-p} {z}^k \left( \sum_{\footnotesize \begin{matrix} i+j=p \\ i~odd,j~even \end{matrix}} a_{ijk} ~\mu_{(i+1,j)} + \sum_{\footnotesize \begin{matrix} i+j=p \\ i~even,j~odd \end{matrix}} b_{ijk} ~\mu_{(i,j+1)} \right),$$ for $p=1,2,\ldots,n$. So we write $f_1=r \bar{f}_1$ with $$\bar{f}_1(r,{z})= A_1 + A_3 r^2 + \ldots + A_n r^{n-1}.$$ Since $r>0$ it is sufficient to solve $(\bar{f}_1,f_{2}, \ldots, f_{d+1})=(0,\ldots,0)$ to determine the number of solutions of $f \equiv 0$. As $\bar{f}_1$ is a polynomial in the variables $r$ and ${z}\in {\mathbb{R}}^d$ of degree $n-1$ and $f_{l+1}$ are polynomials in the variables $r$ and ${z}\in {\mathbb{R}}^d$ of degree $n$ for $l=1,2,\ldots,d$, by Bézout’s theorem (see [@Ful]) $(\bar{f}_1,f_{2}, \ldots, f_{d+1})$ has at most $n^d(n-1)$ solutions. However $\bar{f}_1$ is even on variable $r$ then we consider only solutions with $r>0$, in this case the maximum number of solutions of $f \equiv 0$ is $n^d(n-1)/{2}$. Now we prove that this number is reached. For this, we exhibit a particular case for which this occurs. Let $a_{ij0},b_{ij0} \neq 0$ and we take zero all the other $a_{ijk},b_{ijk}$, then $\bar{f}_1(r,{z})$ is a real polynomial that does not depend of ${z}\in {\mathbb{R}}^d$ of degree $n-1$, that is $$\bar{f}_1= A_1 + A_3 r^2 + \ldots + A_n r^{n-1}$$ where $$\begin{aligned} A_p=\sum_{\footnotesize \begin{matrix} i+j=p \\ i~odd,~j~even \end{matrix}} a_{ij0} ~ \mu_{(i+1,j)} + \sum_{\footnotesize \begin{matrix} i+j=p \\ i~even, ~ j~odd \end{matrix}} b_{ij0} ~ \mu_{(i,j+1)} .\end{aligned}$$ On the other hand, we take $c_{\l ijk}=0$ if $i,j, k_1, k_{l-1},k_{l+1}, \ldots, k_d \neq 0$ for each $\l=1,2,\ldots,d$ so that $$f_{\l+1}(r,z)= \sum_{k_{\l}=0}^{n} {z_{\l}}^{k_{\l}} ~(2 \pi ~c_{{\l} 00 k_{\l}}).$$ In this particular case, we can take $a_{ij0},b_{ij0}$ and $c_{\l 00 k_{\l}}$ in order that all coefficients of $\bar{f}_1$ and $f_{\l+1}$ are linearly independent for all $\l=1,2,\ldots,d$. So we can choose these coefficients in such a way that $\bar{f}_1$ has ${(n-1)}/{2}$ simple positive real roots and $f_{\l+1}$ has $n$ simple real roots for each $\l=1,2,\ldots,d$. Then $(\bar{f}_1,f_{2}, \ldots,f_{d+1})$ has $n^d(n-1)/{2}$ solutions with $r>0$. Now assume $n$ is even. Then $f=r\bar{f}_1$ where $$\bar{f}_1(r,z)= A_1 + A_3r^2 + \ldots + A_{n-1} r^{n-2}.$$ with $A_p$ given in . Note that $\bar{f}_1$ has degree $n-1$ as a polynomial in the variables $r$ and $z$ and $f_{\l+1}$ given in has degree $n$ as polynomials in the variables $r$ and $z$ for all $l=1,2,\ldots,d$, so $(\bar{f}_1,f_{2}, \ldots, f_{d+1})$ have at most $ n^d(n-1)/{2}$ solutions of type $(r,z)$ with $r>0$. To prove that this number is reached we consider $a_{10k_1},b_{01k_1}\neq 0$ and we take zero all the other $a_{ijk},b_{ijk}$. Then $$\bar{f}_1(r,{z})=A_1=\sum_{k_1=0}^{n-1} z_1^{k_1} \left( a_{10k_1} ~\mu_{(2,0)} + b_{01k_1} ~\mu_{(0,2)} \right)$$ is a complete polynomial on variable $z_1$ of degree $n-1$. Now for $\l=2, 3\ldots,d$ we take $c_{1ij0}, c_{\l 00k_{\l}} \neq 0$ and we take zero all the other $c_{\l ijk}$ so that $$\begin{aligned} f_{2}(r,{z}) &= \sum_{\footnotesize \begin{matrix} i+j=0 \\i,j~even \end{matrix}}^{n} r^{i+j} ~c_{1ij0} ~\mu_{(i,j)} ,\\ f_{\l+1}(r,{z}) &=\sum_{k_{\l}=0}^{n} z_\l^{k_\l}~ ~c_{\l 00k_{\l}} 2\pi,\end{aligned}$$ for $\l=2, 3\ldots,d$. Here we take $a_{10k_1},~b_{01k_1}$, $c_{1ij0}$ and $c_{\l 00k_{\l}}$ in such a way for that all coefficients of $\bar{f}_1$ and $f_{\l+1}$ for $\l=1,2,\ldots,d$, are linearly independent. Therefore we can choose these coefficients in order that $\bar{f}_1$ has $n-1$ simple real roots, $f_{2}$ has ${n}/{2}$ simple positive real roots and $f_{\l+1}$ has ${n}$ simple real roots for each $\l=2,3,\ldots,d$ . In this case $(\bar{f}_1,f_{2},\ldots, f_{d+1})=(0,\ldots,0)$ has $n^d(n-1)/2$ solutions with $r>0$. Furthermore by independence of the coefficients these solutions can be taken in a way that the Jacobian of $f$ in all these solutions is nonzero. This completes the proof of Theorem \[th.cycles.systemX\]. Proof of Theorem \[th.cycles.discontinuous\] {#section:proof.th.discontinuous} -------------------------------------------- Analogously to what we did in the previous section we shall study the existence of limit cycles bifurcating of periodic solutions of system when $\varepsilon=0$ which are contained in the cylindrical annulus $\tilde{A}$ defined in . In $\tilde{A}$ we have for $\varepsilon$ sufficiently small $\dot{\theta} >0$ for all $(\theta,r,z) \in \tilde{A}$. Taking $\theta$ as independent variable system in $\tilde{A}$ becomes $$\label{S.new.coord} \begin{array}{c} \left. \begin{array}{lll} r'~ &= & \e F_1(\theta,r, {z}) + \mathcal{O}(\e^2) , \vspace*{0.2 cm}\\ {z}'_{\l} & = & \e F_{\l+1}(\theta,r, {z}) + \mathcal{O}(\e^2) \end{array} \right\} \quad \hbox{if}\quad h(\theta,r,{z}) >0 ,\vspace*{0.3 cm} \\ \left. \begin{array}{lll} r'~ &= & \e G_1(\theta,r, {z}) + \mathcal{O}(\e^2) , \vspace*{0.2 cm}\\ {z}'_{\l} & = & \e G_{\l+1}(\theta,r,{z}) + \mathcal{O}(\e^2) \end{array} \right\} \quad \hbox{if}\quad h(\theta,r, {z}) <0 , \end{array}$$ where $$\label{eq.G1.G2} \begin{array}{lll} G_1(\theta,r,{z})& = &\dis\sum_{i+j+k=0}^{n} r^{i+j} {z}^k \big( \alpha_{ijk} \cos^{i+1}\theta \sin^j\theta \vspace*{0.15cm}\\ & & + \beta_{ijk}\cos^{i}\theta \sin^{j+1} \theta \big), \vspace*{0.15cm} \\ G_{\l+1}(\theta,r,{z})& = &\dis\sum_{i+j+k=0}^{n} r^{i+j} {z}^k \gamma_{\l ijk} \cos^i\theta \sin^j\theta, \end{array}$$ for $\l=1,2,\ldots,d$, $F_1$ and $F_2$ given in , and $h(\theta,r,{z})= \sin \theta$. Then for $(\theta,r,{z}) \in h^{-1}(0)$ we have $\dis\frac{\partial h}{\partial \theta}(\theta, r,z)_{\mid {\theta \in \lbrace 0,\pi \rbrace }}=\cos\theta_{ \mid {\theta \in \lbrace 0,\pi \rbrace}}=\pm 1 \neq 0$. Now, let $$I_{(p,q)}=\displaystyle \int_0^\pi \cos^p\theta\sin^q\theta{~\mathrm{d}}\theta \quad\text{and} \quad J_{(p,q)} = \displaystyle \int_\pi^{2\pi} \cos^p\theta\sin^q \theta {~\mathrm{d}}\theta.$$ Then $(-1)^q J_{(p,q)}= I_{(p,q)}$ and $I_{(p,q)}=J_{(p,q)}=0$ if only if $p$ is odd. Thus we have $$\begin{aligned} \label{f1f2} f_1(r,{z})= &\int_{0}^{\pi}F_1(\theta,r,{z}) {~\mathrm{d}}\theta + \int_{\pi}^{2\pi}G_1(\theta,r,{z}) {~\mathrm{d}}\theta \nonumber\\ = &\sum_{\footnotesize \begin{matrix} i+j+k=0 \\ i~odd \end{matrix}}^{n} r^{i+j} {z}^k \left( a_{ijk} +(-1)^j \alpha_{ijk} \right) I_{(i+1,j)} \nonumber \\ & + \sum_{\footnotesize \begin{matrix} i+j+k=0 \\i~even \end{matrix}}^{n} r^{i+j} {z}^k \left(b_{ijk}-(-1)^j\beta_{ijk} \right) I_{(i,j+1)},\\ f_{\l+1}(r,{z})=& \int_{0}^{\pi}F_{\l+1}(\theta,r,{z}) {~\mathrm{d}}\theta + \int_{\pi}^{2\pi} G_{\l+1}(\theta,r,{z}) {~\mathrm{d}}\theta \nonumber \\ =& \sum_{\footnotesize\begin{matrix} i+j+k=0 \\ i~even \end{matrix}}^{n} r^{i+j} {z}^k~ (c_{\l ijk} +(-1)^j \gamma_{\l ijk}) ~ ~I_{(i,j)}, \nonumber\end{aligned}$$ for $\l=1,2,\ldots,d$. Then $f_{\l+1}$ are polynomials on variables $r$ and $z$ of degree $n$ for each $\l=0,1, \ldots, d$. By Bézout’s theorem $f$ has at most $n^{d+1}$ solutions (with $r >0$). To prove that this number is reached we choose a particular example. So take $a_{ij0} - (-1)^j \alpha_{ij0} , b_{ij0} - (-1)^j \beta_{ij 0} \neq 0$ and we take zero all the other $a_{ij k} - (-1)^j \alpha_{ij k} , b_{ij k} - (-1)^j \beta_{ij k } $. Then ${f}_1$ is a real polynomial on $r$ of degree $n$ that does not depend of ${z}$. Analogously for each $\l=1,2,\ldots,d$ we take $c_{\l 00 k_\l} + \gamma_{\l 00 k_\l} \neq 0$ and we take zero all the other $c_{\l ij k} + \gamma_{\l ij k} $, so that $$f_{\l+1}(r,z) = \sum_{k_\l=0}^{n} z_\l^{k_\l} ~(c_{\l 00 k_\l} + \gamma_{\l 00 k_\l})\pi .$$ Under such conditions all coefficients of ${f}_1$ and $f_{\l+1}$ for $\l=1,2,\ldots,d$ can be taken to be linearly independent from the appropriate choice of $a_{ij0},~ \alpha_{ij0}, ~b_{ij0},~\beta_{ij0}$, $c_{\l 00 k_\l}$ and $\gamma_{\l 00 k_\l}$. Such values can be taken so that ${f}_1$ is a complete real polynomial on variable $r$ of degree $n$, and therefore it can have $n$ simple positive real roots. Furthermore for each $\l=1,2,\ldots,d$ the polynomials $f_{\l+1}$ is a complete real polynomial on variable $z_\l$ with $n$ simple real roots. So for this particular case the number of zeros of $f$ is $ n^{d+1}$. By independence of the coefficients such solutions can be taken so that the Jacobian of $f$ evaluated at them is nonzero. By Theorem \[Th.discontinuous.averaging\] this completes the proof of Theorem \[th.cycles.discontinuous\]. Proofs of Corollaries \[cor.1\] and \[cor.2\] {#section:proof.cor} --------------------------------------------- To prove Corollary \[cor.1\] we follow the steps of the proof presented in subsection \[section:proof.th.continuous\], however we highlight some differences. We consider the change of coordinates given in applied to system taking $a_{00k}=b_{00k}=0$ for all $k$ and instead of we obtain $$\begin{aligned} \dot{\theta}&= 1+ \varepsilon \sum_{i+j+k=0}^{n} r^{i+j} z^k \left(a_{ijk} \cos^i\theta \sin^{j+1}\theta + b_{ijk} \cos^{i+1}\theta \sin^j\theta \right),\\ \dot{r}&= \varepsilon \sum_{i+j+k=0}^{n} r^{i+j}z^k \left( a_{ijk} \cos^{i+1}\theta \sin^j\theta + b_{ijk}\cos^{i}\theta \sin^{j+1}\theta\right),\\ \dot{z}_{\l}&= \varepsilon\sum_{i+j+k=0}^{n} c_{\l ijk}~r^{i+j} z^k \cos^i\theta \sin^j\theta,\end{aligned}$$ for $\l=1,2,\ldots,d$. As $r$ does not appear in the denominator of $\dot{\theta}$, if $\varepsilon$ is sufficiently small $\dot{\theta}>0$ for every $(\theta,r, {z})$ in a ball $B$ of an arbitrary given radius around the origin of ${\mathbb{R}}^{d+2}$. Now in the ball $B$ the variable $r$ can be approximated to the zero as we want, this cannot occur working with the cylindrical annulus $\tilde{A}$ of subsection \[section:proof.th.continuous\]. From here the calculations are done analogously to section \[section:proof.th.continuous\], and we obtain the same maximum number of zeros of the averaging function for the new system with $a_{00k}=b_{00k}=0$. The same above argument can be used for proving Corollary \[cor.2\] and we follow the steps of the proof presented in subsection \[section:proof.th.discontinuous\]. More precisely, we apply the change of coordinates to system considering $a_{00k}=\alpha_{00k}=b_{00k}=\beta_{00k}=0$ for all $k$ in order to obtain an expression for $\dot{\theta}$ in both systems $y>0$ and $y<0$ with denominator that does not depend on $r$. So for $\varepsilon$ sufficiently small we have $\dot{\theta}>0$ for every $(r,\theta, z)$ in the ball $B$ of the proof of Corollary \[cor.1\]. Since the exponent $i+j$ of the variable $r$ on the polynomial $f_1$, given in , is at least one because $i$ is odd, we have $$f_1(r,{z})= A_1 r + A_2 r^2 + \ldots + A_n r^n$$ with $$\label{eq.Ap} \begin{array}{lll} A_p&=& \dis \sum_{k=0}^{n-p} ~ \sum_{\footnotesize \begin{matrix} i+j=p \\ i~odd \end{matrix}} z^k \left( a_{ijk} +(-1)^j \alpha_{ijk} \right) I_{(i+1,j)} \\ & & + \dis \sum_{k=0}^{n-p}~ \sum_{\footnotesize \begin{matrix} i+j=p \\ i~even \end{matrix}} z^k \left(b_{ijk} - (-1)^j \beta_{ijk} \right) I_{(i,j+1)}, \end{array}$$ for $p=1,2,\ldots,n$. Consequently we obtain $f_1=r \bar{f}_1$ where $$\bar{f}_1(r,{z})= A_1 + A_2 r + \ldots + A_n r^{n-1} .$$ As $r >0$, to know the solutions of $(f_1,f_{2}, \ldots, f_{d+1})=(0,\ldots,0)$ is equivalent to solve $(\bar{f}_1,f_{2}, \ldots, f_{d+1})=(0,\ldots, 0)$. But, $\bar{f}_1$ is a polynomial on variables $r$ and $z$ of degree $n-1$ and the functions $f_{\l+1}$ given in are polynomials on variables $r$ and $z$ of degree $n$ for each $\l=1,2, \ldots, d$. By Bézout’s theorem $f$ has at most $n^d(n-1)$ solutions (with $r >0$). To prove that this number is reached we choose a particular example. This step of this proof is done as in subsection \[section:proof.th.discontinuous\] for obtaining the maximum number of zeros of the averaging function for the new system with $a_{00k}=\alpha_{00k}=b_{00k}=\beta_{00k}=0$. Again in the ball $B$ the variable $r$ can be approximated to zero as we want. Acknowledgements {#acknowledgements .unnumbered} ================ The first author is partially supported by a MINECO/FEDER grant MTM2008–03437, an AGAUR grant 2009SGR–0410, an ICREA Academia, FP7–PEOPLE–2012–IRSES–316338 and 318999, and FEDER/UNAB10–4E–378. The second author is partially supported by a FAPESP–BRAZIL grant 2012/18780–0. 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--- abstract: 'Let $X \subset\mathbb{C}^r$ be compact $d$-variety with isolated determinantal singularities and $\omega$ be a $1$-form on $X$ with a finite number of singularities (in the stratified sense). Under some technical conditions on $r$ we use two generalization of Poincaré-Hopf index with the goal of proving a Poincaré-Hopf Type Theorem for $X$.' author: - '[ N. G. Grulha Jr.]{}, [M. S. Pereira ]{} and [H. Santana]{}' title: 'Poincaré-Hopf Theorem for Isolated Determinantal Singularities' --- \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] \[section\] Introduction {#introduction .unnumbered} ============ The Poincaré-Hopf Theorem can be seen as a bridge between combinatorial algebraic topology and differential topology and the Euler characteristic is the main stone in this connection. The Euler characteristic is a very important and well known invariant which appears in mathematics since the first years in primary school and goes up to highlight applications in theoretical physics. To compute the Euler characteristic on the differentiable side of a smooth variety it is necessary to consider the Poincaré Hopf index. However, to adapt this concept on singular varieties, we need to generalize the Poincaré-Hopf index to the singular case. In this context, many generalizations can be considered, such as the different approaches presented in [@KT; @GSV; @BS; @MP]. In [@BSS], the authors present a proof of this type of result in the case where these isolated singularities are complete intersections. In this context, we have the existence and unicity of smoothing, which makes possible to define a generalization for the Poincaré-Hopf index. The next step to continue the research is to use these new indices to find a proof of Poincaré-Hopf Theorem for compact varieties with isolated singularities of determinantal type. In this work, we consider compact varieties with isolated determinantal singularities. To obtain a version of Poincaré-Hopf Theorem in this case, we use techniques similar to the ones used in [@BSS], and some interesting new results about determinantal singularities. Let $X$ be a compact variety with isolated codimension $2$ determinantal singularities. In [@RP], using the unicity of the smoothing, the authors define the Milnor number of $X$ as the middle Betti number of a generic fiber of the smoothing. In a more general setting of determinantal varieties, the results depend on the Euler Characteristic of the stabilization given by the essential smoothing. In that paper, the authors also connect this invariant with the Ebeling and Gusein-Zade index of the $1$-form given by the differential of a generic linear projection defined on the variety. The cases that we consider in this paper is not covered by the ICIS setting. The non-standard behaviour of our setting can be seen because, for instance, we have non-smoothable and smoothable singularities and even in the smoothable case we split in two cases: unicity or not unicity of the smoothing. We consider two different Poincaré-Hopf index generalizations: one, denoted by $Ind_{PH},$ was considered by Ebeling and Gusein-Zade in [@EG] and can be seen as a generalization of the GSV-index [@GSV] and, the other, by $Ind_{PHN}$ defined by Ebeling and Gusein-Zade in [@EG]. In Section 1, we present the basic results about determinantal varieties and indices of $1$-forms and in Section 2, we prove our main result. **Acknowledgements** The authors are grateful to professors Brasselet, Seade and Ruas for their important suggestions about the theme of this paper. We also thank professors Ebeling and Zach for the fruitful conversations about their work, which is essential in this paper, during the Thematic Program on Singularity Theory, at IMPA, Rio de Janeiro, Brazil. The first author was supported by FAPESP, under grant 2019/21181-02, and by CNPq, under grant 303046/2016-3. The second author was partially supported by Proex ICMC/USP in a visit to São Carlos, where part of this work was developed. The third author was supported by FAPESP, grant 2015/25191-9. The authors also thank PROBAL (CAPES-DAAD), grant 88881.198862/2018- 01. Basic Definitions ================= Let $M_{n,p}$ be the set of all $n\times p$ matrices with complex entries, $M^t_{n,p}\subset M_{n,p}$ the subset of matrices that have rank less than $t$, with $1\leq t\leq \min(n,p)$. It is possible to show that $M^t_{n,p}$ is a singular algebraic variety of codimension $(n-t+1)(p-t+1)$ with singular locus $M^{t-1}_{n,p}$ (see[@Bruns]). The set $M^t_{n,p}$ is called *generic determinantal variety*. Let $F=(F_{ij}(x))$ be an $n\times p$ matrix whose entries are complex analytic functions on $U\subset\matC^r$, $0\in U$ and $f$ the function defined by the $t\times t$ minors of $F$. We say that $X$ is a determinantal variety if $X$ is defined by the equation $f=0$ and the codimension of $X$ is $(n-t+1)(p-t+1)$. Using [@EG] and [@Mathias], we present formulas of the Poincaré-Hopf type for compact varieties with isolated determinantal singularities. In order to apply [@Mathias] we need to consider a more general case of essentially isolated determinantal singularities (EIDS) defined by Ebeling e Gusein-Zade. For that, we recall the definition of essentially nonsingular point. A point $x\in X = F^{-1}(M^t_{n,p} )$ is called essentially nonsingular if, at this point, the map $F$ is transversal to the corresponding stratum of the variety $M^t_{n,p}$, where $i = rkF(x) + 1$. Now we present the definition of essentially singular point at the origin. A germ $(X, 0) \subset(\mathbb{C}^r , 0)$ of a determinantal variety has an isolated essentially singular point at the origin if it has only essentially non-singular points in a punctured neighbourhood of the origin in $X$. Let $(X, 0) \subset(\mathbb{C}^r , 0)$ be the germ of an analytic equidimensional variety. It is well known that complete intersections are smoothable and for a determinantal singularity, the existence and uniqueness of the smoothing do not occur in general. Because of that Ebeling and Guzein-Zade introduced the following definition. An essential smoothing $\tilde{X}$ of the EIDS $(X, 0)$ is a subvariety lying in a neighbourhood $U$ of the origin in $\mathbb{C}^r$ and defined by a perturbation $\tilde{F} : U \to M_{n,p}$ of the germ $F$ such that $\tilde{F}$ is transversal to all the strata $M^i_{n,p}\setminus M^{i-1}_{n,p}$ with $i\leq t$. An essential smoothing is not smooth in general, its singular locus is $\tilde{F}^{-1}(M^{t-1}_{n,p})$ and $$\displaystyle{\tilde{X}=\bigcup_{1\leq i\leq t}\tilde{F}^{-1}(M^i_{n,p}\setminus M^{i-1}_{n,p})}.$$ If $X=F^{-1}(M^t_{n,p})$ is an EIDS, $1\leq t\leq min\{n,\,p\}$, an essential smoothing of $X$ is a genuine smoothing if and only if $r <(n - t + 2)(p - t + 2)$ (see [@EG] for more details). In [@RP], the authors obtain the following results that can be seen as a Lê-Greuel type formula for germs of Cohen-Macaulay determinantal variteties of codimension $2$ with isolated singularity at the origin. ([@RP]) Let $(X,0)\subset(\mathbb{C}^4,0)$ be the germ of a determinantal surface with isolated singularity at the origin. Then, $$m_2(X)=\mu(p^{-1}(0)\cap X)+\mu(X),$$ where $m_2(X)$ is the second polar multiplicity of $X$. The $m_{2}(X)$ multiplicity here is a generalization presented by Pereira and Ruas, in the determinantal context, to Gaffney’s multiplicity $m_{d}(X)$ defined in [@Gaffney] for isolated complete intersection singularities. When $\dim(X)=3$, we obtain an expression which reduces to the Lê-Greuel formula when $b_2(X_t)=0$. ([@RP])\[Miriam1.1\] Let $(X,0)\subset(\matC^5,0)$ be the germ of a determinantal variety of codimension $2$ with isolated singularity at the origin. Then, $$m_3(X)=\mu(p^{-1}(0)\cap X)+\mu(X)+b_2(X_t),$$ where $b_2(X_t)$ is the $2$-th Betti number of the generic fiber of $X_t$ and $m_3(X)$ is the polar multiplicity of $X$. In [@Mathias], the author studies the topology of essentially isolated determinantal singularities and obtains a result that describes the homotopy type of the Milnor fiber of an EIDS. Let $L_{n,p}^{k}=M^t_{n,p}\cap H_{k}$ be the $k$-th complex link of $M_{n,p}^t$, where $H_k$ is a plane of codimension $k$ in general position with the generic determinantal variety $M_{n,p}^t$ out of the origin (see [@Tibar]). ( [@Mathias], Corollary 3.5)\[Mathias1.2\] Let $(X, 0)$ be an EIDS given by a holomorphic map germ $F:(\mathbb{C}^r,0)\to (M_{n,p},0)$ such that $X=F^{-1}(M^t_{n,p})$ is smoothable. If $F_u$ is a stabilization of $F$ and $\overline{X}_u=F_u^{-1}(M_{n,p}^t)$ is the determinantal Milnor fiber, then $$\overline{X}_u\cong_{ht} L_{n,p}^{t,np-r}\vee\bigvee_{i=1}^s S^d,$$ where $d=r-(n-t+1)(p-t+1)=\dim(X)$, $L_{n,p}^{t,k}=H_k\cap M_{n,p}^t$ and $H_k$ is a codimensional $k$ hyperplane in general position out of the origin. If $(X,0)\subset (\mathbb{C}^r,0)$ is a smoothable determinantal singularity of codimension $2,$ we have $r<(n-n+2)( (n+1)-n+2)=6$. Moreover using the previous result and Example 3.6 of [@Mathias], we have $$\overline{X}_u\cong_{ht} L_{2,3}^{2,n(n+1)-r}\vee\bigvee_{i=1}^s S^d,$$ and the generic determinantal complex link associated can be calculated by $$L_{2,3}^{2,k}\cong \left\{\begin{array}{lll} S^2 \mbox{ if } k\in\{1,\,2\}\\ \bigvee _{i=1}^{e-1}S^1 \mbox{ if } k=3\\ e\mbox{ points if } k=4\\ \emptyset \mbox{ otherwise} \end{array}\right.$$ with $k=n(n+1)-r$. - If $X$ is a determinantal surface in $\mathbb{C}^4,$ then $$X_u\cong_{ht}L_{2,3}^{2,2}\vee\bigvee_{i=1}^{s} S^2\cong\bigvee_{i=1}^{s+1} S^2.$$ We note that in this case the Milnor number defined in [@RP] is the number of spheres appearing on the previous bouquet. - If $X$ is a determinantal $3$-variety in $\mathbb{C}^5$, then $$X_u\cong_{ht} L_{2,3}^{2,1}\vee\bigvee_{i=1}^s S^3\cong S^2 \vee\bigvee_{i=1}^s S^3.$$ Using Propositions \[Miriam1.1\] and \[Mathias1.2\], since $b_2(X_t)=1$ ([@Mathias]), we obtain the following consequence. Let $(X,0)\subset(\matC^5,0)$ be the germ of a determinantal variety of codimension $2$ with isolated singularity at the origin. Then, $$m_3(X)=\mu(p^{-1}(0)\cap X)+\mu(X)+1.$$ Index of $1$-Forms on Determinantal Varieties ============================================= Let $(X,0)$ be an EIDS represented by a matrix $F=(F_{ij}(x))$, $x\in\matC^r$ and $\tilde{X}$ an essential smoothing of it. In [@EG], Ebeling and Gusein-Zade define indices of $1$-forms on EIDS. If $X$ is a smoothable singularity then the following definition coincides with the definition presented in [@BSS], Section 3.4. Let $(X,0)\subset (\mathbb{C}^r, 0)$ be a EIDS and $\omega$ a $1$-form on $(X,0)$. The Poincaré–Hopf index (PH-index), $Ind_{PH}\, \omega = Ind_{PH}(\omega, X, 0)$, of $\omega$ on $(X,0)$ is the sum of the indices of the zeros of a generic perturbation $\tilde{\omega}$ of the 1-form $\omega$ on the essential smoothing $\tilde{X}$ appearing in the preimage of a neighbourhood of the origin in $(\matC^r,0$). In the case where $\tilde{X}$ is singular we can consider the Poincaré–Hopf–Nash index (PHN-index) that can be defined as follows. Let $\overline{X}$ be the total space of the Nash transform of the variety $\tilde{X}$, $\mathbb{T}$ the Nash bundle over $\tilde{X}$, and $\Pi:\overline{X}\to\tilde{X}$ the associate projection. The $1$-form $\omega$ defines a nonvanishing section $\hat{\omega}$ of the dual bundle $\hat{\mathbb{T}}^*$ over the preimage of the intersection $\tilde{X}\cap S_{\epsilon}$ of the variety $\tilde{X}$ with the sphere $S_{\epsilon}$ centered at the origin. Notice that $\overline{X}$ is a smooth manifold as it was proved in [@EG], p. 06. [@EG] The Poincaré–Hopf–Nash index of the $1$-form $\omega$ on the EIDS $(X,0)$, $ Ind_{PHN}(\omega, X, 0)=Ind_{PHN} \omega $, is the obstruction to extending the nonzero section $\hat{\omega}$ of the dual Nash bundle $\hat{\mathbb{T}}^*$ from the preimage of the boundary $S_{\epsilon} = \partial B_{\epsilon}$ of the ball $B_{\epsilon}$ to the preimage of its interior, i.e., to the manifold $\overline{X}$ or, more precisely, its value (as an element of the cohomology group $H^{2d}(\Pi(\tilde{X}\cap B_{\epsilon}),\Pi(\tilde{X}\cap S_{\epsilon}))$ on the fundamental class of the pair $(\Pi(\tilde{X}\cap B_{\epsilon}),\Pi(\tilde{X}\cap S_{\epsilon}))$. The next proposition is a key ingredient to prove the formulas we present in the following. [@EG] Let $l: M_{n,p}\to\mathbb{C}$ be a generic linear form, and let $L^t_{n,p} = M_{n,p}^t \cap l^{-1}(1)$. Then, for $t\leq n\leq p$, one has $$\overline{\chi}(L^t_{n,p} )=(-1)^t \begin{pmatrix} n-1\\ t-1 \end{pmatrix}.$$ As an immediate consequence, we obtain the following formula. \[Binomial\] If $l:M_{n,p}\to\mathbb{C}^k$ is a linear projection , then $$\overline{\chi}(L^{t,k}_{n,p} )=(-1)^t \begin{pmatrix} n-k\\ t-1 \end{pmatrix},$$ where $L^{t,k}_{n,p}=M_{n,p}^t\cap l^{-1}(\delta)$, $\delta\neq 0$. **Proof.** Let $l:M_{n,p}\to\mathbb{C}^k$ be a linear projection and $l^{-1}(\delta)$ is a plane of codimension $k$ in $M_{n,p}^t$, with $\delta\neq 0$. Then $l^{-1}(\delta)$ is isomorphic to , where $\xi$ is a complex number. Then the result follows by induction applying the previous result on the right space of matrices. $\hfill \blacksquare$ In [@SeadeSuwa; @BSS] the authors proved a Poncaré-Hopf type theorem for compact varieties with isolated complete intersection singularities. Using [@RP] we can extend this result in the case of codimension $2$ determinantal varieties with isolated singularities. We start considering $r < (n-t+2)(p-t+2)$, *i.e.*, smoothable determinantal varieties. In this case, the relation between the PHN-index and the radial index (present in [@EG1]) is given by $$\label{FM} Ind_{PHN}(\omega,X,0)=Ind_{rad}(\omega,X,0)+(-1)^{\dim(X)}\overline{{\chi}}(X_u),$$ with $\overline{{\chi}}(X_u)={\chi}(X_u)-1$. Let $X\subset\mathbb{C}^r$ be a compact surface with isolated singularities $p_1,\ldots,p_l$ such that the germ $X_i=(X,p_i)$, $i=1\ldots,l$, is a germ of isolated singularity and $\omega$ is a $1$-form on $X$ with isolated singularities. Then we use the following formula that is a consequence of the definition of the radial index: $$\label{Radial} Ind_{rad}(\omega,X_i,p_i)=1+\sum_{j=1}^{s_i}Ind_{PHN}(\overline{\omega},X_i,q_j^i),$$ where $\overline{\omega}$ is a $1$-form on $X_i\setminus B(p_i,\epsilon_i')$ which coincides with $\omega$ on $S \cap \partial B(p_i,\epsilon_i)$ and with a radial form on $X\cap \partial B(p_i,\epsilon_i')$ and $q_j^i$ are the singularities of $\overline{\omega}$ on $X_i$ (see [@BSS]). \[Teo1\] Let $S\subset\mathbb{C}^4$ be a compact surface with isolated singularities $p_1,\ldots,p_l$ such that the germ $(S,p_i)$, $i=1\ldots,l$, is a determinantal surface with isolated singularity and $\omega$ a $1$-form on $S$ with isolated singularities. Then $$\sum Ind_{PHN} (\omega, S,p_i)=\chi(S)+\sum \mu(S_i),$$ where $S_i=S\cap B(p_i,\epsilon)$ with $1\leq i\leq l$. **Proof.** Let $0<\epsilon<<1$ such that the representative of the germ of $S$ on $p_i$, $S_i=S\cap B(p_i,\epsilon)$, is a determinantal surface with isolated singularity with $1\leq i\leq l$ ( of the same type, given by $2\times 2$ minors of a $2\times 3$ matrix). Then $\omega|_{S_i}$ has isolated singularities at $p_i$ where $l+1\leq i \leq l+k$. If the value of the $1$-form $\omega$ is not positive on a fixed outward looking normal vector field on the boundary of a tubular neighbourhood of $S$, we can choose $\epsilon_i'<\epsilon_i$ such that $B(p_i,\epsilon_i')$ are the balls coming from the construction of the radial index. If $S'=S\setminus\cup B(p_i,\epsilon_i')$, with $i\in\{1,\ldots, l\}$, then $S'$ is a surface with boundary and $$\sum_{j=l}^{k} Ind_{PHN}(\omega,S',p_{l+j})=\chi(S').$$ In fact, the Euler characteristic of $S$ is equal the sum of Euler characteristic of $S'$ and the number of singularities of $S$ since each $S_i$ is contractible. On the other hand, $$\sum Ind_{PHN}(\omega,S,p_i)=\sum_{i=1}^lInd_{PHN}(\omega,S_i,p_i)+\sum_{j=1}^k Ind_{PHN}(\omega,S',p_{l+j})$$ Using Equations (\[FM\]) and (\[Radial\]), we have $$\begin{aligned} \sum Ind_{PHN}&(\omega,S,p_i)=\\ &=\sum_{i=1}^l\left(Ind_{rad}(\omega,S_i,p_i)+(-1)^2\overline{\chi}(S_i)\right)+\sum_{j=1}^k Ind_{PHN}(\omega,S',p_{l+j})=\\ &=\sum_{i=1}^l\left(1+\sum_{j=1}^{s_i}Ind_{PHN}(\overline{\omega},S_i,q_j^i)+\overline{\chi}(S_i)\right)+\sum_{j=1}^k Ind_{PHN}(\omega,S',p_{l+j}),\\ $$ where $\overline{\omega}$ is a $1$-form on $S_i\setminus B(p_i,\epsilon_i')$ which coincides with $\omega$ on $S \cap \partial B(p_i,\epsilon_i)$ and with a radial form on $S\cap \partial B(p_i,\epsilon_i')$ and $q_j^i$ are the singularities of $\overline{\omega}$ on $S_i$. Hence, $$\begin{aligned} \sum Ind_{PHN}(\omega,S,p_i)&=k+\chi(S')+\sum\overline{\chi}(S_i)=\chi(S)+\sum_{i=1}^l \mu (S_i).\end{aligned}$$ $\hfill \blacksquare$ For simple space curves investigated by Frühbis-Krüger in [@FK] we can prove an analogous result using the Milnor number of an arbitrary reduced curve singularity defined by Buchweitz and Greuel in [@Bg]. \[teo42\] Let $X \subset\mathbb{C}^5$ be a $3$-variety with isolated singularities $p_1,\ldots,p_l$ such that the germ of $X$ at $p_i$, $i=1\ldots,l$, is a determinantal $3$-variety with isolated singularities and $\omega$ a $1$-form on $X$. Then $$\sum Ind_{PHN} (\omega, X,p_i)=\chi(X)-l-\sum_{i=1}^l \mu (X_i)$$ where $X_i=X\cap B(p_i,\epsilon)$, for $0<\epsilon<<1$. **Proof.** Let $0<\epsilon<<1$ such that the representative of the germ $X_i=X\cap B(p_i,\epsilon)$ of $X$ on $p_i$ is a $3$-determinantal variety with isolated singularity and let $q_j$ be the isolated singularities of $\omega$ on $X_i$. Using similar arguments as in the proof of Theorem \[Teo1\], we have $$\begin{aligned} \sum Ind_{PHN}(\omega,X,p_i)&=\chi(X')-\sum\overline{\chi}(X_i)+\#\{\mbox{singular points of } X\}=\\ &=\chi(X)-\sum_{i=1}^l \mu (X_i)-\sum b_2(X_u)=\\ &=\chi(X)-\left(\sum_{i=1}^l (\mu (X_i)+1)\right).\\$$ $\hfill \blacksquare$ In the setting of Theorem \[teo42\], if $X_i$ are simple singularities, then $$\sum Ind_{PHN} (\omega, X,p_i)=\chi(X)-\sum_{i=1}^l\tau(X_i),$$ where $\tau(X_i)$ is the Tjurina number of $X_i$ (see [@RP] ). Let $X \subset\mathbb{C}^r$ be a compact $d$-variety with isolated determinantal singularities $\{p_1,\ldots,p_l\}$, with $r<(n-t+2)(p-t+2)$ and $\omega$ be a $1$-form on $X$ with a finite number of singularities (in the stratified sense) of the same type. Then $$\sum Ind_{PHN}(\omega,X,p_i) =\chi(X)+(-1)^{d}l\left(1+(-1)^{d}+(-1)^{t} \begin{pmatrix}n-1\\ t-1\end{pmatrix} \right)$$ **Proof.** Let $\{U_{i}\}_{i\in I}$ be a finite covering of $X$ such that each $X_{i}=X\cap U_{i}$ can be described as $F_{i}^{-1}(M^t_{n,p})$, where $X_{i}$ has isolated singularity at $p_i$ and $r<(n-t+2)(p-t+2)$. Then $\omega|_{X_i}$ has isolated singularities at $p_i$ where $l+1\leq i \leq k$. If the value of the $1$-form $\omega$ is not positive on a fixed outward looking normal vector field on the boundary of a tubular neighbourhood of $X$, we can choose $\epsilon_i'<\epsilon_i$ such that $B(p_i,\epsilon_i')$ are the balls coming from the construction of the radial index. If $X'=X\setminus\cup B(p_i,\epsilon_i)$, then $X'$ is a determinantal variety with boundary and $\sum_{j=1}^{k} Ind_{PHN}(\omega,X',p_{l+j})=\chi(X')$. In fact, $\chi(X)=\chi(X')+k$ since each $X_i$ is contractible with $i\in\{1,\ldots,l\}$. Nevertheless, $$\sum Ind_{PHN}(\omega,X,p_i)=\sum_{i=1}^lInd_{PHN}(\omega,X_i,p_i)+\sum_{j=1}^k Ind_{PHN}(\omega,X',p_{l+j})$$ Using Equation (\[FM\]), we have $$\begin{aligned} \sum I&nd_{PHN}(\omega,X,p_i)=\\ &=\sum_{i=1}^l\left(Ind_{rad}(\omega,X_i,p_i)+(-1)^{d}\overline{{\chi}}(X_u)\right)+\sum_{j=1}^k Ind_{PHN}(\omega,X',p_{l+j})\\ &=\sum_{i=1}^l\left(1+\sum_{j=1}^{s_i}Ind_{PHN}(\overline{\omega},X_i,q_j^i)+(-1)^{d}\overline{\chi}(X_u)\right)+\sum_{i=1}^k Ind_{PHN}(\omega,X',p_{l+j})\\ &=l+\chi(X')+\sum_{i=1}^l(-1)^{d}\overline{\chi}(X_u)=\chi(X)+\sum_{i=1}^l(-1)^{d}\overline{\chi}(X_u). $$ Moreover, we know that $X_u\cong_{ht} L_{n,p}^{t,np-r}\vee\bigvee_{v=1}^{r_i}S^d$ and $$\overline{\chi}(L^t_{n,p})=(-1)^t \begin{pmatrix} n-1\\ t-1 \end{pmatrix},$$ where $t\leq n\leq p$. Then $$\overline{\chi}(X_u)=(-1)^t \begin{pmatrix} n-1\\ t-1 \end{pmatrix} +\sum_{v=1}^{r_i}((-1)^d+1)$$ Therefore, $$\begin{aligned} \sum Ind_{PHN}(\omega,X,p_i)&=\chi(X)+\sum_{i=1}^l(-1)^{d}\overline{\chi}(X_u)\\ &=\chi(X)+\sum_{i=1}^l\left((-1)^{d+t} \begin{pmatrix}n-1\\ t-1\end{pmatrix} +1+(-1)^{d}\right)\\ &=\chi(X)+(-1)^{d}l\left(1+(-1)^{d}+(-1)^{t} \begin{pmatrix}n-1\\ t-1\end{pmatrix} \right)\\ \end{aligned}$$ $\hfill \blacksquare$ Let us denote by $Eu_{M^t_{n,p}}(0)$ the Euler obstruction of $M^t_{n,p}$ at the origin. Using the previous result and the formula $$Eu_{M^t_{n-1,p}}(0)=\begin{pmatrix}n-1\\ t-1\end{pmatrix}$$ presented in [@GRT], we obtain - If $d$ is odd, then $$\chi(X)=\sum Ind_{PHN}(\omega,X,p_i)+l(-1)^{d+t-1} Eu_{M^t_{n-1,p}}(0).$$ - If $d$ is even, then $$\chi(X)=\sum Ind_{PHN}(\omega,X,p_i)+l(-1)^{d+t-1} Eu_{M^t_{n-1,p}}(0) -2l.$$ That means that the Euler characteristic of $X$ measure, in some sense, the difference between the $PHN$- index on $X$ and the Euler obstruction of $M^t_{n-1,p}$ at the origin. For more details about the Euler obstruction see [@MP] and [@BS]. In the next result, we consider non-smoothable determinantal singularity with isolated singularity, *i.e.*, $r = (n-t+2)(p-t+2)$. In this case, the relation between the PHN- and the radial index present in [@EG1] reduces to $$\begin{aligned} \label{tenth} Ind_{PHN} &(\omega, X, 0)=\nonumber \\ &Ind_{rad} (\omega, X, 0) + (-1)^{\dim(X)} \chi(X_u) + (-1)^{n+p+1}(n - t + 1)\chi((X_u)_{t-1}).\end{aligned}$$ Let $X \subset\mathbb{C}^r$ be a compact $d$-variety with rigid isolated determinantal singularities $\{p_1,\ldots,p_l\}$ and $(X_u^i,0)$ the germ of $X_u$ on $p_i$. Let $\omega=dp$ be a $1$-form on $X$ with a finite number of singularities (in the stratified sense) and $p_i$ the singularities of $\omega|_{X_u^i},$ where $l+1\leq i \leq k$ . Then $$\sum Ind_{PHN}(\omega,X,p_i)=\chi(X)+l(-1)^{d}\overline{\chi}(X_u^i)+(-1)^{n+p+1}(n-t+1)\sum_{i=1}^l\chi((X_u^i)_{t-1}).$$ **Proof.** Let $\{U_{i}\}_{i\in I}$ be a finite covering of $X$ such that each $X_{u}^i=X\cap U_{i}$ can be describe as $$F_i:U_{i}\to M_{n,p}$$ where $F_{i}^{-1}(X^{t}_{n,p})$ has isolated singularity at $p_i$ and $r=(n-t+2)(p-t+2),$ with $\dim(X_u^i)=d$. Using arguments similar to previous ones, we have $$\sum Ind_{PHN}(\omega,X,p_i)=\sum_{i=1}^lInd_{PHN}(\omega,X\cap U_i,p_i)+\sum_{j=1}^k Ind_{PHN}(\omega,X',p_{l+j}) $$ Using relation (\[tenth\]), we have $$\begin{aligned} \sum &Ind_{PHN}(\omega,X,p_i)=\\ &=\sum_{i=1}^l\left(Ind_{rad}(\omega,X_u^i,p_i)+(-1)^{d}\overline{{\chi}}(X_u^i)+(-1)^{n+p+1}(n-t+1)\chi((X_u^i)_{t-1}\right)+\\ &+\sum_{j=1}^k Ind_{PHN}(\omega,X',p_{l+j})=\\ &=\sum_{i=1}^l\left(1+\sum_{j=1}^{s_i}Ind_{PHN}(\overline{\omega},X_u^i,q_j^i)+(-1)^{d}\overline{\chi}(X_u^i)+(-1)^{n+p+1}(n-t+1)\chi((X_u^i)_{t-1}\right)\\ &+\sum_{i=1}^k Ind_{PHN}(\omega,X',p_{l+j})=\\ &=l+\chi(X')+l(-1)^{d}\overline{\chi}(X_u^i)+\sum_{i=1}^l(-1)^{n+p+1}(n-t+1)\chi((X_u^i)_{t-1})=\\ &=\chi(X)+l(-1)^{d}\overline{\chi}(X_u^i)+l(-1)^{n+p+1}(n-t+1)\chi((X_u^i)_{t-1}).\end{aligned}$$ $\hfill \blacksquare$ - In the setting of Corollary \[Binomial\], we can explicitly calculate the sum of indices in terms of Newton binomial. - Notice that if the singularity $X_i$ is rigid for any $i \in \{1,\ldots,l\}$, then $$\sum Ind_{PHN}(\omega,X,p_i) =\chi(X)+l(-1)^{n+p}(n-t+1).$$ Here we conclude this work where it was delivered a generalization of the Poincaré-Hopf theorem for compact determinantal varieties in the both smoothable and non-smoothable cases. 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Introduction ============ One of the goals of the recent nuclear physics is to find the equation of state of nuclear matter. Indeed, the dependence of the pressure on the density of nucleons is a crucial input for a hydrodynamical modeling of heavy ion collisions or of astrophysical events like the big bang, supernova explosions and neutron stars [@SG86]. In the absence of any direct measurement, it is hoped that the equation of state can be deduced from heavy ion collisions via the following scheme. Heavy ion collision data are fitted with the Boltzmann equation (BE) $$\begin{aligned} &&{\partial f_1\over\partial t}+{\partial\varepsilon_1\over\partial k} {\partial f_1\over\partial r}-{\partial\varepsilon_1\over\partial r} {\partial f_1\over\partial k} \nonumber\\ &&=\sum_b\int{dpdq\over(2\pi)^5} \delta\left(\varepsilon_1+\varepsilon_2- \varepsilon_3-\varepsilon_4\right) \nonumber\\ &&\times |T|^2\left(\varepsilon_1+\varepsilon_2,k,p,q,t,r\right) \nonumber\\ &&\times \Bigl[f_3f_4\bigl(1-f_1\bigr)\bigl(1-f_2\bigr)- \bigl(1-f_3\bigr)\bigl(1-f_4\bigr)f_1f_2\Bigr]. \label{1}\end{aligned}$$ Arguments of distributions $f$ and energies $\varepsilon$ are shortened as $f_1\equiv f_a(k,r,t)$, $f_2\equiv f_b(p,r,t)$, $f_3\equiv f_a(k-q,r,t)$, and $f_4\equiv f_b(p+q,r,t)$, with momenta $k,p,q$, coordinate $r$, time $t$, and spin and isospin $a,b$. Once the differential cross sections $|T|^2$ and the functional dependence of energy $\varepsilon$ on the distribution $f$ are fitted, the equation of state is evaluated from the kinetic equation. This scheme has two drawbacks. First, accessible fits of the quasiparticle energy $\varepsilon$ are not sufficiently reliable since two possible fits, momentum-dependent and momentum-independent, result in very contradictory predictions giving hard and soft equations of state, respectively[@BG88]. When this more or less technical problem is resolved in future, one has to face the second drawback: the BE is not thermodynamically consistent with virial corrections to the equation of state. This problem is principal for “how can one infer the equation of state from the BE if the two equations are not consistent?”. A consistency between the kinetic and the thermodynamic theories is a general question for the quantum statistics exceeding the merits of the nuclear matter. Here we approach this question from nonequilibrium Green’s functions. It is shown that the consistency is achieved by a consistent treatment of the quasiclassical limit which results in nonlocal and noninstant corrections to the scattering integral of the BE. The need of nonlocal corrections can be seen on the classical gas of hard spheres. In the scattering integral of (\[1\]), all space arguments of the distributions are identical, i.e., colliding particles $a$ and $b$ are at the same space point $r$. In reality, these particles are displaced by the sum of their radii. This inconsistency has been noticed by Enskog [@CC90] and cured by nonlocal corrections to the scattering integral. The equation of state evaluated from the kinetic equation with the nonlocal scattering integral is of the van der Waals type covering the excluded volume [@CC90; @HCB64]. For nuclear matter, Enskog’s corrections has been first discussed by Malfliet [@M84] and recently implemented by Kortemayer, Daffin and Bauer [@KDB96]. The noninstant corrections are closer to the chemical picture of reacting gases. In the scattering integral of (\[1\]), all time arguments of the distributions are identical what implies that the collision is instant. In reality, the collision has a finite duration which might be quite long when two particles form a resonant state. The resonant two-particle state behaves as an effective short-living molecule. Like in reacting gases [@HCB64], the presence of these molecules reduces the pressure since it reduces the number of freely flying particles. The finite duration of nucleon-nucleon collisions and its thermodynamic consequences has been for the first time discussed only recently by Danielewicz and Pratt [@DP96]. The noninstant scattering integral and its consequencies for the linear response has been also discussed for electrons in semiconductors scattered by resonant levels [@SLM97]. Except for dense Fermi systems, the above intuitively formulated nonlocal and noninstant corrections has been confirmed by systematic approaches. For classical gases, this theory was developed already by Bogoliubov and Green [@B46; @G52]. Obtained gradient contributions to the scattering integral are the lowest order terms of the virial expansion in the kinetic equation [@comdiv]. The first quantum kinetic equation with nonlocal corrections has been derived by Snider [@S60]. Recently, it has been recognized that Snider’s equation is not consistent with the second order virial corrections to equations of state. A consistent quantum mechanical theory of the virial corrections to the BE has been developed from the multiple scattering expansion in terms of Møller operators [@NTL91] and confirmed by Balescu’s formalism [@H90]. Presented treatment extends the nonlocal and noninstant corrections to dense Fermi systems. We follow Baerwinkel [@B69] in starting from nonequilibrium Green’s functions and keeping all gradient contributions to the scattering integral. Baerwinkel’s results are limited to low densities (to avoid medium effects on binary collisions) and not consistent (since he uses the quasiparticle approximation). Here we describe the binary collisions by the Bethe-Goldstone T-matrix which includes the medium effects. Instead of the quasiparticle approximation, the [*extended*]{} quasiparticle approximation is used. This extension is sufficient to gain consistency of the kinetic theory with the virial corrections to thermodynamic quantities. Extended quasiparticle picture ============================== We start our derivation of the kinetic equation from the quasiparticle transport equation first obtained by Kadanoff and Baym [@D84; @SL95] $${\partial f_1\over\partial t}+{\partial\varepsilon_1\over\partial k} {\partial f_1\over\partial r}-{\partial\varepsilon_1\over\partial r} {\partial f_1\over\partial k}= z_1(1-f_1)\Sigma^<_{1,\varepsilon_1}-z_1f_1\Sigma^>_{1,\varepsilon_1}. \label{2}$$ Like in (\[1\]), quasiparticle distribution $f$, quasiparticle energy $\varepsilon$ and wave-function renormalization $z$ are functions of time $t$, coordinate $r$, momentum $k$ and spin and isospin $a$. Self-energy $\Sigma$, taken from nonequilibrium Green’s function in the notation of Kadanoff and Baym [@D84], is moreover a function of energy $\omega$, however, it enters the transport equation only by its value at pole $\omega=\varepsilon_1$. Particular forms of the quasiparticle energy and the scattering integral we derive for a model and an approximation used in nuclear matter for heavy ion collisions in the non-relativistic energy domain. The system is composed of protons and neutrons of equal mass $m$. They interact via an instant potential $V$. We assume no spin-flipping mechanism. As common, the self-energy is constructed from the two-particle T-matrix $T^R$ in the Bethe-Goldstone approximation [@D84; @MR94] as \[$T^R_{\rm sc}\!(1,2,3,4)\!=\!(1\!-\!\delta_{a_1a_2})T^R\!(1,2,3,4)\!+\! {1\over\sqrt{2}}\delta_{a_1a_2}(T^R\!(1,2,3,4)\!-\!T^R\!(1,2,4,3))$\] $$\begin{aligned} \Sigma^<(1,2)&=& T^R_{\rm sc}(1,\bar 3;\bar 5,\bar 6)T^A_{\rm sc}(\bar 7,\bar 8;2,\bar 4) \nonumber\\ &\times &G^>(\bar 4,\bar 3)G^<(\bar 5,\bar 7)G^<(\bar 6,\bar 8), \label{3}\end{aligned}$$ and $\Sigma^>$ is obtained from (\[3\]) by an interchange $>\leftrightarrow <$. Here, $G$’s are single-particle Green’s functions, numbers are cumulative variables, $1\equiv (t_1,r_1,a_1)$, and bars denote internal variables that are integrated over. Before (\[3\]) is plugged in (\[2\]), it has to be transformed into the mixed representation, \[off-diagonal elements in spin and isospin are excluded, $a_1=a_2=a$\] $$\begin{aligned} \Sigma^<(1,2)&=&\int{d\omega\over 2\pi}{dk\over(2\pi)^3} {\rm e}^{ik(r_1-r_2)-i\omega(t_1-t_2)} \nonumber\\ &\times &\Sigma^<_a\left(\omega,k,r,t\right)_ {r={r_1+r_2\over 2},t={t_1+t_2\over 2}}, \label{4}\end{aligned}$$ and all Green’s functions in (\[3\]), too. The self-energy $\Sigma$ is a functional of Green’s functions $G$. This functional $\Sigma[G]$ is converted to the functional of the quasiparticle distribution $\Sigma_\varepsilon[f]$ via the extended quasiparticle approximation [@SL95; @BKKS96] \[$z_1=1+\left. {\partial\over\partial\omega}{\rm Re}\Sigma_{1\omega}\right|_ {\varepsilon_1}$\] $$G^{\begin{array}{c}>\\[-2mm] <\end{array}}_{1,\omega}= \left(\!\begin{array}{c}1\!-\!f_1\\ f_1\end{array}\!\right) 2\pi z_1\delta(\omega-\varepsilon_1)+{\rm Re}{\Sigma^{\begin{array}{c} >\\[-2mm] <\end{array}}_{1,\omega}\over(\omega-\varepsilon_1)^2}, \label{5}$$ where $G_{1,\omega}\equiv G_a(\omega,k,r,t)$ and similarly $\Sigma$. Unlike the plain quasiparticle approximation (without the second term) used by Baerwinkel [@B69], approximation (\[5\]) leads to the consistent theory. The first term brings the on-shell quasiparticle part, the second term is the off-shell contribution. The off-shell part plays four-fold role. First, it justifies the kinetic equation (\[2\]). Equation (\[2\]) has been originally derived from the plain quasiparticle approximation neglecting the off-shell drift.[^1] The off-shell part of $G^<$ in (\[5\]) compensates the off-shell drift so that (\[2\]) is recovered without uncontrollable neglects [@SL95]. Second, in the quasiparticle energy $\varepsilon_1={k^2\over 2m_a}+{\rm Re}\Sigma^R_ {1,\varepsilon_1}$, the off-shell part brings contributions that are essential for the correct binding energy [@KM93]. Third, (\[5\]) provides Wigner’s distribution $\rho=\int{d\omega\over 2\pi}G^<$ as a functional of the quasiparticle distribution $f$ [@SL95; @KM93]. Fourth, in the scattering integral of (\[2\]), the off-shell part results in sequential three-particle processes with the off-shell propagation between the two composing binary processes. Since the three-particle processes are beyond the scope of the present paper, they are excluded from scattering integral. Non-local scattering integral ============================= Now the approximation is specified and we can start to simplify the scattering integral. In contrast to previous treatments of degenerated systems, we keep all terms linear in gradients. The gradient expansion of the self-energy (\[3\]) is a straightforward but tedious task. It results in a one nongradient and nineteen gradient terms that are analogous to those found within the chemical physics [@NTL91; @H90]. All these terms can be recollected into a nonlocal and noninstant scattering integral that has an intuitively appealing structure of the scattering integral in the BE (\[1\]) with Enskog-type shifts of arguments.[^2] In agreement with [@NTL91; @H90], all gradient corrections result proportional to derivatives of the scattering phase shift , $$\begin{array}{lclrcl}\Delta_t&=&{\displaystyle \left.{\partial\phi\over\partial\Omega} \right|_{\varepsilon_1+\varepsilon_2}}&\ \ \Delta_2&=& {\displaystyle\left({\partial\phi\over\partial p}- {\partial\phi\over\partial q}-{\partial\phi\over\partial k} \right)_{\varepsilon_1+\varepsilon_2}}\\ &&&&&\\ \Delta_E&=& {\displaystyle\left.-{1\over 2}{\partial\phi\over\partial t} \right|_{\varepsilon_1+\varepsilon_2}}&\Delta_3&=& {\displaystyle\left.-{\partial\phi\over\partial k} \right|_{\varepsilon_1+\varepsilon_2}}\\ &&&&&\\ \Delta_K&=& {\displaystyle\left.{1\over 2}{\partial\phi\over\partial r} \right|_{\varepsilon_1+\varepsilon_2}}&\Delta_4&=& {\displaystyle-\left({\partial\phi\over\partial k}+ {\partial\phi\over\partial q}\right)_{\varepsilon_1+\varepsilon_2}}. \end{array} \label{8}$$ After derivatives, $\Delta$’s are evaluated at the energy shell $\Omega\to\varepsilon_1+\varepsilon_2$. The corrected BE with the collected gradient terms then reads \[$\Delta_r={1\over 4}(\Delta_2+\Delta_3+\Delta_4)$\] $$\begin{aligned} &&{\partial f_1\over\partial t}+{\partial\varepsilon_1\over\partial k} {\partial f_1\over\partial r}-{\partial\varepsilon_1\over\partial r} {\partial f_1\over\partial k} \nonumber\\ &&=\sum_b\int{dpdq\over(2\pi)^5}\delta\left(\varepsilon_1+\varepsilon_2- \varepsilon_3-\varepsilon_4+2\Delta_E\right)\nonumber\\ &&\times z_1z_2z_3z_4 \Biggl(1-{1\over 2}{\partial\Delta_2\over\partial r} -{\partial\bar\varepsilon_2\over\partial r} {\partial\Delta_2\over\partial\omega}\Biggr) \nonumber\\ &&\times |T_{\rm sc}^R|^2\!\left(\varepsilon_1\!+\!\varepsilon_2\!-\! \Delta_E,k\!-\!{\Delta_K\over 2},p\!-\!{\Delta_K\over 2}, q,r\!-\!\Delta_r,t\!-\!{\Delta_t\over 2}\!\right) \nonumber\\ &&\times\Bigl[f_3f_4\bigl(1-f_1\bigr)\bigl(1-f_2\bigr)- \bigl(1-f_3\bigr)\bigl(1-f_4\bigr)f_1f_2\Bigr]. \label{9}\end{aligned}$$ Unlike in (\[1\]), the subscripts denote shifted arguments: $f_1\equiv f_a(k,r,t)$, $f_2\equiv f_b(p,r\!-\!\Delta_2,t)$, $f_3\equiv f_a(k\!-\!q\!-\!\Delta_K,r\!-\!\Delta_3,t\!-\!\Delta_t)$, and $f_4\equiv f_b(p\!+\!q\!-\!\Delta_K,r\!-\!\Delta_4,t\!-\!\Delta_t)$. The $\Delta$’s are effective shifts and they represent mean values of various nonlocalities of the scattering integral. These shifts enter the scattering integral in the form known from the theory of gases [@CC90; @NTL91; @H90], however, the set of shifts is larger due to the medium effects on the binary collision that are dominated by the Pauli blocking of the internal states of the collision. The physical meaning of the $\Delta$’s is best seen on gradually more complex limiting cases:\ (o) Sending all $\Delta$’s to zero, (\[9\]) reduces to the BE (\[1\]).\ (i) In the classical limit for hard spheres of the diameter $d$, the scattering phase shift $\phi\to\pi-|q|d$ gives $\Delta_4=\Delta_2= {q\over|q|}d$ and all other $\Delta$’s are zero. The Enskog’s nonlocal corrections are thus recovered.\ (ii) For a collision of two isolated particles interacting via the single-channel separable potential, the scattering phase shift does not depend on $r$, $t$, $q$ and $k-p$, while it depends on $k+p$ exclusively via the energy dependency, $\phi\to\phi(\Omega- {1\over 4m}(k+p)^2)$. Then $\Delta_{E,K,2}=0$ and $\Delta_{3,4}= {k+p\over 2m}\Delta_t$. Since ${k+p\over 2m}$ is the center-of-mass velocity, the displacements $\Delta_4=\Delta_3$ represent a distance over which particles fly together as a molecule.\ (iii) For two isolated particles interacting via a general spherical potential, the scattering phase shift reflects the translational and the spherical symmetries. From translation of the center of mass during $\Delta_t$ follows the relation for the molecular flight ${1\over 2}(\Delta_4+\Delta_3-\Delta_2)={k+p\over 2m}\Delta_t$. From the spherical symmetry follows that the sum of relative coordinates of the particles $a$ and $b$ at the end and at the beginning of collision has the direction of the transferred momentum ${1\over 2}(\Delta_4-\Delta_3+\Delta_2)={q\over|q|}d$ (Enskog-type shift), and that the difference has the perpendicular in-plane direction ${1\over 2}(\Delta_4-\Delta_3-\Delta_2)={k-p-q\over|k-p-q|}\alpha$ (rotation of the molecule). The nonlocality of the collision is thus given by three scalars $\Delta_t$, $d$ and $\alpha$.\ (iv) For effectively isolated two-particle collisions (no Pauli blocking but mean-field contributions $U_a(r,t)$ to the energy, i.e., $\phi\to \phi(\Omega\!-\!{1\over 4m}(k\!+\!p)^2\!-\!U_a\!-\!U_b,\ldots)$) all $\Delta$’s become nonzero. The space displacements are the same as in (iii). The energy and the momentum corrections $2\Delta_{E,K}=-\Delta_t{\partial\over\partial t,r}(U_a+U_b)$, represent the energy and the momentum which the effective molecule gains during its short life time $\Delta_t$. These corrections are in fact of three-particle nature, however, only on the mean-field level, thus formally within the binary process. The three scalars, $\Delta_t$, $d$ and $\alpha$, are still sufficient to parameterize all $\Delta$’s. We note that in the energy-conserving $\delta$ function all $\Delta$’s and mean-fields compensate $\delta\left(\varepsilon_1\!+\!\varepsilon_2\!-\!\varepsilon_3\!-\! \varepsilon_4\!-\!2\Delta_E\right)\!=\!\delta\left({k^2\over 2m}\!+\! {p^2\over 2m}\!-\!{(k-q)^2\over 2m}\!-\!{(p+q)^2\over 2m}\right)$. This limit describes the dilute quantum gases and (\[9\]) reduces to the kinetic equation found in [@NTL91; @H90].\ (v) With the medium effects on collisions, all $\Delta$’s become independent. A generally nonparabolic momentum-dependency of the quasiparticle energy does not allow to separate the center-of-mass motion, and an anisotropic, inhomogeneous and time-dependent distribution in the Pauli blocking ruins all symmetries of the collision. The energy conservation does not reduce to the simple conservation of the kinetic energy. The uncompensated residuum of the energy gain contributes to a conversion between the kinetic and the configuration energies of the system. The kinetic equation (\[9\]) is numerically tractable by recent Monte Carlo codes. Kortemayer, Daffin and Bauer [@KDB96] have already studied the kinetic equation with an intuitive extension by Enskog-type displacement, claiming only a little increase in numerical demands compared to the BE. The larger set of $\Delta$’s in (\[9\]) does not require principal changes of the numerical method used in [@KDB96]. The functions $\varepsilon$, $z$, $|T_{\rm sc}^R|^2$, and $\Delta$’s should form a consistent set, i.e., it is preferable to obtain them from Green’s function studies, e.g., like [@ARS94]. Eventual fitting parameters should enter directly the effective nucleon-nucleon interaction. Observables =========== Let us presume that a reasonable set of functions $\varepsilon$, $z$, $|T_{\rm sc}^R|^2$, and $\Delta$’s is known. One can then proceed to evaluate observables. Here we present the density $n_a$ of particles $a$, the density of energy $\cal E$, and the stress tensor ${\cal J}_{ij}$. The observables in question are directly obtained from balance equations which also establish conservation laws. Integrating the kinetic equation (\[2\]) over momentum $k$ with factors $\varepsilon_1,k,1$, one finds that each observable has the standard quasiparticle part following from the drift $$\begin{aligned} {\cal E}^{\rm qp} &=&\sum_a\int{dk\over(2\pi)^3}{k^2\over 2m}f_1 \nonumber\\ &+&{1\over 2}\sum_{a,b}\int{dkdp\over(2\pi)^6} T_{\rm ex}(\varepsilon_1+\varepsilon_2,k,p,0)f_1f_2, \nonumber\\ {\cal J}_{ij}^{\rm qp}&=&\sum_a\int{dk\over(2\pi)^3}\left(k_j {\partial\varepsilon_1\over\partial k_i}+ \delta_{ij}\varepsilon_1\right)f_1- \delta_{ij}{\cal E}^{\rm qp}, \nonumber\\ n_a^{\rm qp}&=&\int{dk\over(2\pi)^3}f_1, \label{10a}\end{aligned}$$ and the $\Delta$-contribution following from the nonlocality of the scattering integral $$\begin{aligned} \Delta {\cal E}&=&{1\over 2}\sum_{a,b}\int{dkdpdq\over(2\pi)^9} P (\varepsilon_1+\varepsilon_2)\Delta_t, \nonumber\\ \Delta {\cal J}_{ij}&=&{1\over 2} \sum_{a,b}\int{dkdpdq\over(2\pi)^9} P \left[(p\!+\!q)\Delta_4+(k\!-\!q)\Delta_3-p\Delta_2\right], \nonumber\\ \Delta n_a&=&\sum_b\int{dkdpdq\over(2\pi)^9} P \Delta_t, \label{10}\end{aligned}$$ where $P=|T_{\rm sc}^R|^22\pi\delta(\varepsilon_1\!+\!\varepsilon_2\!- \!\varepsilon_3\!-\!\varepsilon_4)f_1f_2(1\!-\!f_3\!-\!f_4)$. The arguments denoted by numerical subscripts are identical to those used in (\[1\]), for all $\Delta$’s are explicit. The T-matrix $T_{\rm ex}$ used in the quasiparticle part of energy is the real part of the antisymmetrized Bethe-Goldstone T-matrix, $T^R_{\rm ex} =(1-\delta_{ab})T^R_{\rm sc}+\delta_{ab}\sqrt{2}T^R_{\rm sc}$. The actual observables are sum of the quasiparticle part and the $\Delta$-correction. In the low density limit, the $\Delta$-contributions (\[10\]) become proportional to the square of density. Therefore they turn into the second order virial corrections. In the degenerated system, the density dependence cannot be expressed in the power-law expansion since the T-matrix, and consequently all $\Delta$’s, depend on the density. Nevertheless, we find it instructive to call the $\Delta$-contribution the virial corrections because of their similar structure. The total energy ${\cal E}={\cal E}^{\rm qp}+\Delta{\cal E}$ conserves within kinetic equation (\[2\]). This energy conservation law generalizes the result of Bornath, Kremp, Kraeft and Schlanges [@BKKS96] restricted to non-degenerated systems. At degenerated systems, a new mechanism of energy conversion appears due to the medium effect on binary collisions. This mechanism can be seen writing the energy conservation ${\partial\over\partial t}{\cal E}= {\partial\over\partial t}{\cal E}^{\rm qp}+ {\partial\over\partial t}\Delta{\cal E}=0$ in a form $${\partial{\cal E}^{\rm qp}\over\partial t}= \sum_a\int{dk\over(2\pi)^3}\varepsilon{\partial f_1\over\partial t}- \sum_{a,b}\int{dkdpdq\over(2\pi)^9}P\Delta_E. \label{12}$$ The first term on the right hand side is the drift contribution to the energy balance. It is the only mechanism which appears in the absence of the virial corrections. The second term is the mean energy gain. It provides the conversion of the interaction energy controlled by two-particle correlations into energies of single-particle excitations. By this mechanism, the latent heat hidden in the interaction energy is converted into “thermal” excitations. The non-zero energy conversion, $\Delta_E\not=0$, results from the time-dependency of the scattering phase shift on the quasiparticle distribution via the Pauli blocking of internal states. Accordingly, the energy conversion is a consequence of the in-medium effects. The energy gain has its space counterpart in the momentum gain to the stress forces ${\partial\over\partial r_j}{\cal J}_{ij}$. The energy density contributing to the stress tensor (\[10a\]) has the gradient $${\partial{\cal E}^{\rm qp}\over\partial r_i}= \sum_a\int{dk\over(2\pi)^3}\varepsilon{\partial f_1\over\partial r_i}- \sum_{a,b}\int{dkdpdq\over(2\pi)^9}P\Delta_K. \label{13}$$ In the absence of the virial corrections (\[10\]), the first term of (\[13\]) combines together with the derivative of the first term in (\[10a\]) into the standard quasiparticle contribution. In the presence of the virial corrections, the energy gain is necessary to obtain the correct momentum conservation law. The density of energy ${\cal E}={\cal E}^{\rm qp}+\Delta{\cal E}$ given by (\[10a\]) and (\[10\]) alternatively results from Kadanoff and Baym formula, $${\cal E}=\sum_a\int{dk\over(2\pi)^3}\int{d\omega\over 2\pi} {1\over 2}\left(\omega+{k^2\over 2m}\right)G^<(\omega,k,r,t), \label{11}$$ with $G^<$ in the extended quasiparticle approximation (\[5\]). The particle density $n_a=n_a^{\rm qp}+\Delta n_a$ obtained from (\[5\]) via the definition, $n_a=\int{d\omega\over 2\pi} {dk\over(2\pi)^3}G^<$, also confirms (\[10a\]) and (\[10\]). The equivalence of these two alternative approaches confirms that the extended quasiparticle approximation is thermodynamically consistent with the nonlocal corrections to the scattering integral. For equilibrium distributions, formulas (\[10a\]) and (\[10\]) provide equations of state. Two known cases are worthy of comparison. First, the particle density $n_a=n_a^{\rm qp}+\Delta n_a$ is identical to the quantum Beth-Uhlenbeck equation of state [@BKKS96; @MR94], where $n_a^{\rm qp}$ is called the free density and $\Delta n_a$ the correlated density. Second, the virial correction to the stress tensor has a form of the collision flux contribution known in the theory of moderately dense gases [@CC90; @HCB64]. Summary ======= In this Letter we have derived the kinetic equation (\[9\]) which is consistent with thermodynamic observables (\[10a\]) and (\[10\]) up to the second order virial coefficient. The presented theory extends the theory of quantum gases [@NTL91; @H90] and non-ideal plasma [@BKKS96] to degenerated system. The most important new mechanism is the energy conversion which follows from the medium effect on binary collisions. The proposed corrections can be evaluated from known in-medium T-matrices and incorporated into existing Monte Carlo simulation codes, e.g., with the routine used in [@KDB96]. The nonlocal corrections to the scattering integral and corresponding second order virial corrections to thermodynamic observables enlighten the link between the kinetic equation approach and the hydrodynamical modeling. With expected progress in fits of the single-particle energy and other ingredients of the kinetic equation, the virial corrections can improve our ability to infer the equation of state from the heavy ion collision data. The authors are grateful to P. Danielewicz, D. Kremp and G. Röpke for stimulating discussions. This work was supported from Grant Agency of Czech Republic under contracts Nos. 202960098 and 202960021, the BMBF (Germany) under contract Nr. 06R0884, the Max-Planck-Society and the EC Human Capital and Mobility Programme. H. Stöcker, W. Greiner, Physics Reports [**137**]{}, 277 (1986). G. F. Bertsch, S. Das Gupta, Physics Reports [**160**]{}, 189 (1988). S. Chapman, T. G. Cowling, [*The Mathematical Theory of Non-uniform Gases*]{}, (Cambridge University Press, Third edition 1990), Chap. 16. J. O. Hirschfelder, Ch. F. Curtiss, R. B. Bird, [*Molecular Theory of Gases and Liquids*]{}, Chapts. 6.4a and 9.3, (Wiley, New York 1964). R. Malfliet, Nucl. Phys. [**A 420**]{}, 621 (1984). G. Kortemeyer, F. Daffin, W. Bauer, Phys. Lett. B [**374**]{}, 25 (1996). P. Danielewicz, S. Pratt, Phys. Rev. C [**53**]{}, 249 (1996). V. Špička, P. Lipavský, K. Morawetz, Phys. Rev. B. [**55**]{}, 5084 (1997); 5095 (1997). N. N. Bogoliubov, J. Phys. (USSR) [**10**]{}, 256 (1946); transl. in [*Studies in Statistical Mechanics*]{}, Vol. 1, editors D. de Boer and G. E. Uhlenbeck (North-Holland, Amsterdam 1962). H. S. Green, [*The Molecular Theory of Fluids*]{} (North-Holland, Amsterdam 1952). Later studies of higher-order corrections (for hard spheres) have undermined a trust in any virial expansions of the kinetic equation finding divergencies which result in nonanalytic behavior of the scattering integral with the density, e.g. K. Kawasaki and I. Oppenheim, Phys. Rev. [**139**]{}, A 1763 (1965). The comparison of Enskog’s equation with experiment is, however, a sufficient reason to use this expansion in spite of its questionable theoretical support. R. Snider, J. Chem. Phys. [**32**]{}, 1051 (1960). P. J. Nacher, G. Tastevin, F. Laloë, Annalen der Physik [**48**]{}, 149 (1991); Journal de Physique I [**1**]{}, 181 (1991). M. de Haan, Physica A [**164**]{}, 373 (1990); [**165**]{}, 224 (1990); [**170**]{}, 571 (1191). K. Baerwinkel, Z. Naturforsch. [**24**]{} a, 22 and 38 (1969). P. Danielewicz, Ann. Phys. (N.Y.) [**152**]{}, 239 (1984). V. Špička, P. Lipavský, Phys. Rev. Lett. [**73**]{}, 3439 (1994); Phys. Rev. B [**52**]{}, 14615 (1995). K. Morawetz, G. Röpke, Phys. Rev. E [**51**]{}, 4246 (1995). Th. Bornath, D. Kremp, W. D. Kraeft, M. Schlanges, Phys. Rev. E [**54**]{}, 3274 (1996). H. S. Köhler, R. Malfliet, Phys. Rev. C [**48**]{}, 1034 (1993). T. Alm, G. Röpke, M. Schmidt, Phys. Rev. C [**50**]{}, 31 (1994). [^1]: The well known term $[{\rm Re}G,\Sigma^<]$. [^2]: The basic idea of the recollection can be demonstrated on the following rearrangement of the gradient approximation of a matrix product $C(1,2)=A(1,\bar 3)B(\bar 3,2)$. In the mixed representation $C=AB+{i\over 2}\left( {\partial A\over\partial\omega}{\partial B\over\partial t}- {\partial A\over\partial t}{\partial B\over\partial\omega}- {\partial A\over\partial k}{\partial B\over\partial r}+ {\partial A\over\partial r}{\partial B\over\partial k}\right)$, see [@D84; @SL95]. We denote $\varphi={i\over 2}\ln A$ and rearrange the product as $C=A\left(B+ {\partial \varphi\over\partial\omega}{\partial B\over\partial t}- {\partial \varphi\over\partial t}{\partial B\over\partial\omega}- {\partial \varphi\over\partial k}{\partial B\over\partial r}+ {\partial \varphi\over\partial r}{\partial B\over\partial k}\right)$. The gradient term in brackets can be viewed as a linear expansion of $B$ with all arguments shifted as .

--- author: - | [Nakwoo Kim, Yoonbai Kim${}^{\ast}$ and Kyoungtae Kimm]{}\ [*Department of Physics and Center for Theoretical Physics,*]{}\ [*Seoul National University, Seoul 151-742, Korea*]{}\ [*nakwoo$@$phya.snu.ac.kr, dragon$@$phya.snu.ac.kr*]{}\ [*${}^{\ast}$Department of Physics, Sung Kyun Kwan University, Suwon 440-746, Korea*]{}\ [*yoonbai$@$cosmos.skku.ac.kr*]{} title: | **Charged Black Cosmic String\  ** --- ‘=11 maketitle to 2em **title** 1.5em .5em -------- author -------- 1.5em \#1[preprint[\#1]{}]{} ‘=12 =20.5cm =16.5cm -1.35cm =.285in [**Abstract**]{}\ Global $U(1)$ strings with cylindrical symmetry are studied in anti-de Sitter spacetime. According as the magnitude of negative cosmological constant, they form regular global cosmic strings, extremal black cosmic strings and charged black cosmic strings, but no curvature singularity is involved. The relationship between the topological charge of a neutral global string and the black hole charge is clarified by duality transformation. Physical relevance as straight string is briefly discussed. PACS number(s): 11.27.+d, 04.40-b, 04.70.Bw Keywords: Global vortex; Black hole; Cosmic string Cosmic strings are viable extended objects in cosmology [@VS]. A way to understand basic physical ingredients of cosmic strings is to study a straight string along an axis, which reduces one spatial dimension. Then, the (2+1) dimensional correspondence is the particle-like solitonic excitations so-called vortices in curved spacetime, and the conic space due to massive point source is enough for the description of asymptotic region outside the local vortex core. Recently, black hole solutions have been reported in (2+1) dimensional anti-de Sitter spacetime [@BTZ] in addition to known hyperbolic solutions [@DJ], and these Bañados-Teitelboim-Zanelli (BTZ) black hole solutions have been extensively studied in a variety of models [@Cle]. Here we may raise a question that what are the string-like counterpart of these BTZ black holes in cosmology. Specifically, whether the vortices in anti-de Sitter space can constitute black holes in (2+1)D, or straight black strings in (3+1)D. The objects of our interest are global $U(1)$ vortices [@HS; @Gre]. It has been shown in Ref. [@Gre] that global $U(1)$ strings coupled to Einstein gravity with zero cosmological constant lead to a physical curvature singularity. Then, how does the constant negative vacuum energy affect the global strings? In this paper, we consider the effect of the negative cosmological constant to the global $U(1)$ vortices in (2+1)D and find three types of regular solutions of which base manifolds form (i) smooth hyperbola, (ii) extremal charged black hole and (iii) charged black hole with two horizons. For all these static solutions, the physical singularity can be avoided, which is different from the zero cosmological constant case. Suppose the magnitude of the negative cosmological constant is extremely small like the lower bound of it in the present universe, $|\Lambda| \le 10^{-83}{\rm GeV}^2$. Under this perfect toy environment with no fluctuation the global string may be born as a black string with large horizon size $r_H$ in the early universe, i.e., $r_H \sim 10^6{\rm pc}$ for the grand unification scale and $r_H\sim 10^{-2}{\rm A.U.}$ for the electroweak scale. A cylindrically symmetric metric with boost invariance in the $z$-direction can be written as $$\begin{aligned} \label{cyl} ds^2=e^{2N(r)}B(r)(dt^2-dz^2)-\frac{dr^2}{B(r)}-r^2d\theta^2.\end{aligned}$$ The physics is reduced to (2+1) dimensional one under this metric. Another well-known (2+1)D static metric is written under conformal gauge: $$\begin{aligned} \label{conf} ds^2=\Phi(R)dt^2-b(R)(dR^2+ R^2d\Theta^2). \end{aligned}$$ For a spinless point particle source of mass $m$ at the origin, the general anti-de Sitter solution is $$\begin{aligned} b&=&\frac{4\varepsilon c^2}{ |\Lambda| R^2\Big[ (R/R_0)^{\sqrt{\varepsilon}c} -(R_0/R)^{\sqrt{\varepsilon}c} \Big]^2} \label{beq} \\ \label{phieq} \Phi&=&\sqrt{\varepsilon} \frac{(R/R_0)^{\sqrt{\varepsilon}c} +(R_0/R)^{\sqrt{\varepsilon}c}}{ (R/R_0)^{\sqrt{\varepsilon}c} -(R_0/R)^{\sqrt{\varepsilon}c}},\end{aligned}$$ where $\varepsilon$ is $\pm 1$ for $\Lambda <0$. When $\varepsilon=+1$, a coordinate transformation $$\begin{aligned} r=\frac{2}{|\Lambda|^{1/2}} \frac{1}{|R^{(1-4Gm)}-R^{-(1-4Gm)}|} ~~\mbox{and}~~\theta=(1-4Gm)\Theta ~~(c=1-4Gm)\end{aligned}$$ leads to $$\begin{aligned} ds^{2}=(1+|\Lambda|r^{2})dt^{2} -\frac{dr^2}{1+|\Lambda|r^{2}}-r^{2}d\theta^{2}. \end{aligned}$$ It describes a hyperbola with deficit angle $\delta =8\pi Gm$ where $4Gm<1$ [@DJ]. When $\varepsilon=-1$, another coordinate transformation $$\begin{aligned} r=\frac{c}{|\Lambda|^{1/2}\sin(2 c\ln{R})} ~~{\rm and}~~\theta=\Theta ~~(e^{k\pi/4c} < r <e^{(k+1)\pi/4c}~~{\rm and}~~ c^2=8GM,~k\in {\rm Z})\end{aligned}$$ results in the exterior region of the Schwarzschild type BTZ black hole [@BTZ] with missing information of the point particle mass $m$ in Eqs. (\[beq\]) and (\[phieq\]): $$\begin{aligned} ds^2=(|\Lambda|r^2-8GM)dt^2 -\frac{dr^2}{|\Lambda|r^2-8GM}-r^2d\theta^2.\end{aligned}$$ As expected, the BTZ solution is one of general anti-de Sitter solutions, of which physical meaning was not considered in Ref. [@DJ]. Note that the dimension of $m$ and $M$ has the square of mass in (3+1)D because it represents the mass density per unit length along the string direction. Here we want to solve Einstein equations with both a global string source and constant negative cosmological vacuum energy density. We take a complex scalar field $\phi$ with Lagrange density $$\begin{aligned} {\cal L}= -\frac{1}{16\pi G}(R+2\Lambda) +\frac{1}{2}g^{\mu\nu}\partial_\mu \overline \phi\partial_\nu \phi -\frac{\lambda}{4}(\overline\phi\phi -v^2)^2.\end{aligned}$$ This model admits a string solution of the form $$\begin{aligned} \phi=|\phi|(r)e^{in\theta}.\end{aligned}$$ For the cylindrically symmetric configurations, the Euler-Lagrange equations read under the metric in Eq. (\[cyl\]): $$\begin{aligned} &&\frac{1}{r}\frac{d N}{d r} =8\pi G \Big( \frac{d |\phi|}{dr}\Big)^2 \\ &&\frac{1}{r}\frac{dB}{dr}=2|\Lambda| - 8\pi G\biggl\{ B\Bigl(\frac{d|\phi|}{dr}\Bigl)^{2}+\frac{n^2}{r^2}|\phi|^2 +\frac{\lambda}{2}(|\phi|^2-v^2)^2\biggr\}\\ &&\frac{d^{2}|\phi|}{dr^{2}}+\Bigl(\frac{dN}{dr}+\frac{1}{B}\frac{dB}{dr} +\frac{1}{r}\Bigr)\frac{d|\phi|}{dr}= \frac{1}{B}\Bigl(\frac{n^2|\phi|}{r^2} +\lambda (|\phi|^2-v^2)|\phi|\Bigr).\end{aligned}$$ Though we will consider the spacetime with horizons, we concentrate only on the regular configurations connecting $|\phi|(0)=0$ and $|\phi|(\infty)=v$ smoothly. By asking the reproduction of Minkowski spacetime in the limit of no matter ($T^{\mu}{}_\nu=0$), and zero vacuum energy ($\Lambda\rightarrow 0$), we can choose an appropriate set of boundary conditions, $B(0)=1$ and $N(\infty)=0$. The Einstein equation for the metric function $B(r)$ is then expressed in terms of scalar amplitude $$\begin{aligned} B(r)&=& \exp\Big[-8\pi G \int_r^\infty dr' r' \Big(\frac{d|\phi|}{dr'}\Big)^2\Big] \Bigg\{ 2|\Lambda|\int^{r}_{0} dr' r' \exp\Big[8\pi G \int^\infty_{r'} dr'' r''\Big(\frac{d|\phi|}{dr''}\Big)^2\Big] \\ \nonumber &&\hspace{0mm} -8\pi G\int^r_0 dr'r' \exp\Big[8\pi G \int^\infty_{r'} dr'' r''\Big(\frac{d|\phi|}{dr''}\Big)^2\Big] \bigg(\frac{n^2}{r'^2}|\phi|^2 +\frac{\lambda}{2}(|\phi|^2-v^2)^2\bigg) + e^{N(0)}\Bigg\}. \end{aligned}$$ Under the approximation that $|\phi|=0$ for $r<r_{c}$ and $|\phi|=v$ for $r\ge r_{c}$, we read the possible form of $B(r)$ with the aid of constant $N(r)$ which is a valid approximation for $r\ge r_{c}$: $$\begin{aligned} \label{bform} B(r)\approx |\Lambda|r^2 -8\pi G v^2 n^2 \ln r/r_{c} -4\pi G v^2 n^2 +1.\end{aligned}$$ The core radius $r_{c}=\sqrt{2}n/\sqrt{\lambda} v$ is determined by minimizing the core mass. Another scale, the minimum point of $B(r)$, is $r_m\approx\sqrt{4\pi G v^2/|\Lambda|}n$, and it may coincide with the horizon scale $r_H$ when $2\pi Gv^2 \gg |\Lambda|/\lambda v^2$. Note that the formation of a black hole is favored as the magnitude of the cosmological constant scaled by the square of Higgs mass, $|\Lambda|/\lambda v^2$, becomes small and the symmetry breaking scale $v$ is large. The latter condition is obvious and the former condition can be understood by the balance between the negative energy due to the negative cosmological constant and positive matter contribution. This energy balance also explains the reason why we can have the regular global $U(1)$ vortex solution in singularity-free curved spacetime. Let us recall no-go theorem that this global $U(1)$ scalar model can not support finite-energy static regular vortex configuration in flat spacetime, so the global $U(1)$ vortex contains a logarithmic divergence in its expression of the energy per unit length. This symptom can not be remedied by inclusion of (2+1)D gravity, namely a higher spin (spin 2) field, since (2+1) dimensional Einstein gravity does not have propagating degree. Therefore, the global $U(1)$ vortex coupled to Einstein gravity leads to an unavoidable physical curvature singularity [@Gre]. In the model of our consideration, the negative vacuum energy achieves a kind of energy balance in anti-de Sitter spacetime, and both the scalar field and the curvature are regular everywhere even though we have coordinate singularity at the horizons. From now on we will solve the field equation and Einstein equations, and will show that there exist regular global $U(1)$ vortex solutions in anti-de Sitter space and the base manifolds of these configurations constitute smooth hyperbola, extremal black hole and charged black hole. Near the origin, $|\phi|(r)\sim \phi_0 r^n$ and the leading term of the power series solution of $B(r)$ is given by $$\begin{aligned} B(r)\sim 1+ \Big( \frac{|\Lambda|}{\lambda v^2}-2\pi Gv^2 -\frac{8\pi G \phi_0^2}{\lambda v^2}\delta_{n1}\Big) (\sqrt{\lambda}vr)^2. \label{beq0}\end{aligned}$$ When the cosmological constant rescaled by the symmetry breaking scale is smaller (larger) than the Planck scale, $B(r)$ starts to decrease (increase) near the origin. Despite of the difficulty in systematic series expansion at the asymptotic region, the leading term provides for sufficiently large $r$: $$\begin{aligned} |\phi|(r)&\sim&v-\frac{\phi_{\infty}}{r^{2}}\\ B(r)&\sim&|\Lambda|r^{2}-8\pi Gv^{2}n^{2}\ln{r/r_{c}}- 8G{\cal M}+1 +{\cal O}(1/r^{2}).\label{binf}\end{aligned}$$ Here ${\cal M}$ is the integration constant and will be identified as the core mass of the global vortex. Since the form of Eq. (\[binf\]) is the same as that in Eq. (\[bform\]) and we know that the magnitude of two terms proportional to $G$ can be larger than the sum of other two positive terms for some appropriate large $r$, the existence of horizons for the small magnitude of the negative cosmological constant can easily be confirmed. To elicit the above discussion explicitly, we compute the solitons by use of numerical analysis. Two regular $n=1$ vortex solutions connecting $|\phi|(0)=0$ and $|\phi|(\infty)=v$ are illustrated in Figure 1. When $|\Lambda|/\lambda v^2$ is large enough, i.e., the second term in the right hand side of Eq. (\[beq0\]) is positive for small $r$ ($|\Lambda|/\lambda v^2=1.0$ and $8\pi Gv^2=1.15$), $B(r)$ is monotonically increasing (See $(i-a)$ in Fig. 1). When $|\Lambda|/\lambda v^2$ is intermediate, $B(r)$ has a positive minimum (See $(i-b)$ in Fig. 1). Suppose that there exists a horizon $r_H$ such that $B(r_H)=0$, another boundary condition has to be satisfied, which is obtained from the equation for the scalar field $$\begin{aligned} \frac{d|\phi|}{dr}\bigg|_{r_H}= \frac{|\phi|(r_H)\Big[\frac{n^2}{r_H^2} +\lambda(|\phi|^2(r_H)-v^2)\Big]}{ 8\pi Gr_H\Big[\frac{|\Lambda|}{4\pi G} -\Big( \frac{n^2}{r_H^2}|\phi|^2(r_H) +\frac{\lambda}{2}(|\phi|^2(r_H)-v^2)^2\Big)\Big]} .\end{aligned}$$ Therefore, we solve the equations in one region with two boundaries for a regular solution, in two regions with three boundaries for an extremal black hole and in three regions with four boundaries for a charged black hole. Since the value of $B(r)$ also vanishes at the horizon $r_H$ for the extremal black hole, the position of the horizon and the value of scalar field are obtained in explicit forms: $$\begin{aligned} \label{ehori} r_H =\frac{n}{\sqrt{\lambda}v\left( 1-\sqrt{1-\frac{|\Lambda|}{2\pi G\lambda v^4} }\right)^{1/2}} ~~{\rm and}~~ |\phi|_{H}=v\bigg(1 -\frac{|\Lambda| }{2\pi G \lambda v^4}\bigg)^{1/4}.\end{aligned}$$ When a black hole is formed, an intriguing property should be mentioned: The ratio of horizon scale and the core length ($\sim r_{c}\sim 1/\sqrt{\lambda}v$) tells us that the horizon lies outside the string core. We investigate the extremal black hole configuration for various ($|\Lambda|/\lambda v^2$, $8\pi Gv^2$) values and find one for $|\Lambda|/\lambda v^2=0.1$ and $8\pi Gv^2=1.338$ (See $(ii)$ in Fig. 1). A Reissner-Nordström type charged black hole with two horizons at $r_H^{in}$ and $r_H^{out}$ is also obtained for $|\Lambda|/\lambda v^2=0.1$ and $8\pi Gv^2=1.4$ (See $(iii)$ in Fig. 1). (3600,2160)(0,0) (0,1100)[(0,0)\[c\][$B(r)$]{}]{} (1800,-100)[(0,0)\[l\][$\sqrt\lambda v r$]{}]{} (1541,291)[(0,0)[$\bullet$]{}]{} (1541,391)[(0,0)[$r_H$]{}]{} (1331,291)[(0,0)[$\bullet$]{}]{} (1201,241)[(0,0)[$r_H^{in}$]{}]{} (1821,291)[(0,0)[$\bullet$]{}]{} (1991,241)[(0,0)[$r_H^{out}$]{}]{} (1291,1353)[(0,0)[$(i\!\rm{-}\!{\em a})$]{}]{} (1891,1053)[(0,0)[$(i\!\rm{-}\!{\em b})$]{}]{} (2391,853)[(0,0)[$(ii)$]{}]{} (2691,703)[(0,0)[$(iii)$]{}]{} (3201,50)[(0,0)[6]{}]{} (2728,50)[(0,0)[5]{}]{} (2255,50)[(0,0)[4]{}]{} (1782,50)[(0,0)[3]{}]{} (1309,50)[(0,0)[2]{}]{} (836,50)[(0,0)[1]{}]{} (363,50)[(0,0)[0]{}]{} (313,2060)[(0,0)\[r\][2.5]{}]{} (313,1706)[(0,0)\[r\][2]{}]{} (313,1353)[(0,0)\[r\][1.5]{}]{} (313,999)[(0,0)\[r\][1]{}]{} (313,645)[(0,0)\[r\][0.5]{}]{} (313,291)[(0,0)\[r\][0]{}]{} 1em Although we have a singularity at each horizon, it is not physical singularity but coordinate artifact. This can be checked by reading the (2+1)D Kretschmann scalar: $$\begin{aligned} \lefteqn{R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma} =4G_{\mu\nu}G^{\mu\nu}\label{curv}}&&\\ &=& 4\mbox{Tr} \bigg[\mbox{diag}\Big(-\frac{1}{2r}\frac{dB}{dr},-\frac{1}{2r}\frac{dB}{dr} -\frac{B}{r}\frac{dN}{dr}, -\frac{1}{2}\frac{d^{2}B}{dr^{2}}-\frac{3}{2} \frac{dB}{dr}\frac{dN}{dr}-B\frac{d^{2}N}{dr^{2}}-B\Big(\frac{dN}{dr}\Big)^{2} \Big)^2\bigg].\nonumber \end{aligned}$$ Since both $N(r)$ and $B(r)$ are regular everywhere for all configurations of our consideration, no divergent curvature appears at the horizon, and for the regular scalar configurations that behave $|\phi|(r)\sim r^n$ for small $r$, the Kretschmann scalar is also finite at the origin. Therefore, unlike the global $U(1)$ strings which include unavoidable curvature singularity in the spacetime with zero cosmological constant [@Gre], the string configurations obtained in anti-de Sitter space have no such curvature singularity. Furthermore, since a possible divergent curvature at the core of global vortex is tamed by the regular behavior of scalar amplitude $|\phi|(r)$, no divergence of curvature appears, which is encountered in the charged black hole formed due to infinite electric self energy by a point charge [@BTZ]. Now we look into the planar motions of massive and massless test particles orthogonal to the string direction, which are described by the geodesic equations: $$\begin{aligned} \label{radial} \frac{1}{2}\Big(\frac{dr}{ds}\Big)^2&=& -\frac{1}{2}\bigg( B(r)\Big(m^2 +\frac{L^2}{r^2}\Big) -\frac{\gamma^2}{e^{2N(r)}}\bigg)\end{aligned}$$ with two constants of motion, $\gamma$ and $L$, associated with two Killing vectors $\partial/\partial t$ and $\partial/\partial\theta$ respectively. From Eq. (\[radial\]), the elapsed coordinate time $t$ of a test particle which moves from $r_0$ to $r$ is finite for regular solutions and becomes infinite when it approaches to a horizon ($r\rightarrow r_H$). As we expected, the spacetime with horizons depicts that of a black hole. The radial motion of a massless test particle ($m=0$ in Eq. (\[radial\])) is unbounded for $\gamma\ne 0$. On the other hand, any massive particle ($m=1$ in Eq. (\[radial\])) can never escape the black hole irrespective of values of $\gamma$ and $L$, since the asymptotic space is hyperbolic. The size of the boundary is determined as a function of $\gamma$ and $L$. Obviously, for the radial motion, the boundary $r_{\infty}$ of the massive test particle becomes large as $\gamma$ increases, and then the ratio $r_{\infty}/r_H$ is much larger than one. Details of all possible planar geodesic motions and related physical implication will be provided in Ref. [@KKK]. Under the metric in Eq. (\[cyl\]) the conserved quasilocal mass per unit length measured by the static observer is [@brown]: $$\begin{aligned} 8GM_q=2e^{N(r)}\Big(\sqrt{(|\Lambda|r^2+1)B(r)}-B(r)\Big).\end{aligned}$$ Though the obtained spacetime is not asymptotically flat and thereby $M_{q}$ is not identified by Arnowitt-Deser-Misner mass, we compute it for sufficiently large $r$ $$\begin{aligned} \label{adm} M_q\stackrel{r\rightarrow\infty}{\longrightarrow}\pi n^{2}v^{2} \ln r/r_{c} + {\cal M}.\end{aligned}$$ The first logarithmically divergent term comes from the topological sector of the long range Goldstone degree, and the second finite one is the core mass of the global $U(1)$ vortex, which coincide with those in flat spacetime. If we compare the obtained spacetime of the global vortices with that of the electric field of a point charge [@BTZ], we can easily find a similarity between them except for the core region. The reason why we have this resemblance can be explained by duality transformation [@KL]. The dual transformed theory equivalent to (2+1)D (or (3+1)D) global vortex model of our consideration is written in terms of a dual vector (or second rank antisymmetric tensor) field $A_{\mu}$ (or $A_{\mu\nu}$): $$\begin{aligned} \label{Max} Z&=&\int [g^{\frac{3}{4}}dg_{\mu\nu}][|\phi|^{-2}d|\phi|][dA_{\mu}][d\Theta] \exp\biggl\{i\int d^{3}x\sqrt{g}\Bigl[-\frac{1}{16\pi G}(R+2\Lambda)\nonumber\\ &&\hspace{10mm}+\frac{1}{2}g^{\mu\nu}\partial_{\mu}|\phi|\partial_{\nu}|\phi| -V(|\phi|) -\frac{v^{2}}{4|\phi|^{2}}g^{\mu\nu}g^{\rho\sigma}F_{\mu\rho}F_{\nu\sigma} +\frac{v\epsilon^{\mu\nu\rho}}{2\sqrt{g}}F_{\mu\nu}\partial_{\rho}\Omega\Bigr] \biggr\},\end{aligned}$$ where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$, and $\Omega$ is the topological sector of the Goldstone degree. For the cylindrically symmetric strings with $\Theta=n\theta$, the last term of the Lagrangian in Eq. (\[Max\]) describes the charged point source coupled minimally to a dual gauge field $A_\mu$. The kinetic term of this $A_\mu$ is the Maxwell term for large $r$ ($|\phi|\stackrel{r\rightarrow\infty}{\longrightarrow}v$), but the nonpolynomial interaction ($\sim 1/|\phi|^2 \stackrel{r\rightarrow 0}{\sim}1/r^{2n}$) plays the role of ultraviolet cutoff to remove the possible curvature divergence at the origin. Through this reformulation the role of the topological charge $n$ in the original formulation is transmuted to the electric charge $n$ in the dual transformed theory. Brief comments about physical implication of these black cosmic strings are in order. Let us consider a universe with tiny negative cosmological constant, e.g., our present universe with $|\Lambda| \le 10^{-83}{\rm GeV}^2$. From Eq. (\[ehori\]), the characteristic scale to distinguish regular strings from black strings is about $v\sim 0.3{\rm eV}$. This means that favorite form of survived global strings in anti-de Sitter space is black hole type where the magnitude of the cosmological constant is about the lower bound of the present universe. Estimation of the horizon size in Eq. (\[ehori\]) gives us $r_H\sim 10^{6}{\rm pc}$ for the grand unification scale ($v\sim 10^{15}{\rm GeV}$), $r_H\sim 10^{-2}{\rm A.U.}$ for the electroweak scale ($v\sim 10^2{\rm GeV}$). Though it is estimated in a perfect presumed toy environment, this property that the horizon of GUT scale black cosmic string is larger than the diameter of our galaxy ($\sim 5\times 10^4{\rm pc}$) may imply some incompatibility between the black cosmic string produced in such early universe and the extremely small magnitude of negative cosmological constant. Again, let us emphasize that the scales obtained above are the outcome of three energy scales of big difference, which are the Planck scale (the largest energy scale), the bound of vacuum energy (the smallest measured scale in cosmology) and an intermediate symmetry breaking scale. In this sense, the obtained black string configurations in Fig. 1 seem to be unphysical since $v\sim (10^{18}\sim10^{19}){\rm GeV}$ and $\lambda\sim 10^{-122}$ for $|\Lambda|\sim 10^{-83}{\rm GeV}^2$ and $1/\sqrt{G}\sim 10^{19}{\rm GeV}$. Obtaining black string under the physical situation seems beyond our numerical precision. Even if the global cosmic strings are produced, the lifetime of a typical string loop is very short due to the radiation of gapless Goldstone boson, which is the dominant mechanism for energy loss [@Dav]. The space outside the horizon of black cosmic string is almost flat except for tiny attractive force due to negative cosmological constant as shown in Eq. (\[radial\]) and Eq. (\[binf\]), and then the massless Goldstone bosons can be radiated outside the horizon. However, almost all the energy accumulated inside the horizon remains eternally. Let us remind you that the physical radius of black cosmic string is decided by the horizon which is usually much larger than the size of normal cosmic string (the core radius $ \sim 1/\sqrt{\lambda} v$), but the energy per unit length of both objects is the same as given in Eq. (\[adm\]). Therefore, the very existence of this horizon is expected to change drastically the physics related to the evolution of global $U(1)$ strings, e.g., the intercommuting of two strings or the production of wakes by moving long strings [@VS]. The authors would like to thank A. Hosoya, H. Ikemori, R. Jackiw, Chanju Kim, Hyung Chan Kim and Kimyeong Lee for helpful discussions. This work was supported by the KOSEF(95-0702-04-01-3 and through CTP, SNU). \#1 References {#references .unnumbered} ----------- 40004000 ‘=1000 = [100]{} For a review, see A. Vilenkin and E.P.S. Shellard, [*Cosmic Strings and Other Topological Defects*]{}, (Cambridge, 1994); M.B. Hindmarsh and T.W.B. Kibble, Rept. Prog. Phys. [**58**]{} (1995) 477. M. Bañados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. [**69**]{} (1992) 1849; M. Bañados, M. Henneaux, C. Teitelboim and J. Zanelli, Phys. Rev. D [**48**]{} (1993) 1506. S. Deser and R. Jackiw, Ann. Phys. [**153**]{} (1984) 405 . G. Clement, Phys. Rev. D [**50**]{} 7119 (1994);  Phys. Lett. B [**367**]{} (1996) 70; K.C.K. Chan and R.B. Mann, Phys. Rev. D [**50**]{} (1994) 6385, Erratum-[*ibid*]{} D [**52**]{} 2600; J.S.F. Chan, K.C.K. Chan and R.B. Mann, Phys. Rev. D [**54**]{} (1996) 1535. A. Vilenkin and A.E. Everett, Phys. Rev. Lett. [**48**]{} (1982) 1867; E.P.S. Shellard, Nucl. Phys. B [**283**]{} (1987) 624; D. Harari and P. Sikivie, Phys. Rev. D [**37**]{} (1988) 3438. R. Gregory, Phys. Lett. B [**215**]{} (1988) 663; A.G. Cohen and D.B. Kaplan, Phys. Lett. B [**215**]{} (1988) 67; G.W. Gibbons, M.E. Ortiz and F. Ruiz Ruiz, Phys. Rev. D [**39**]{} (1989) 1546. N. Kim, Y. Kim and K. Kimm, in preparation. J.D. Brown and J.W. York, Phys. Rev. D [**47**]{} (1993) 1407; J.D. Brown, J. Creighton, and R.B. Mann, Phys. Rev. D [**50**]{} (1994) 6394. Y. Kim and K. Lee, Phys. Rev. D [**49**]{} (1994) 2041; K. Lee, Phys. Rev. D [**49**]{} (1994) 4265; C. Kim and Y. Kim, Phys. Rev. D [**50**]{} (1994) 1040. R.L. Davis, Phys. Rev. D [**32**]{} (1985) 3172; A. Vilenkin and T. Vachaspati, Phys. Rev. D [**35**]{} (1987) 1138; For a review, see Ref. [@VS] and the references therein.

--- abstract: | We study $\gamma\gamma$ scattering in noncommutative QED (NCQED) where the gauge field has Yang-Mills type coupling, giving new contributions to the scattering process and making it possible for it to occur at tree level. The process takes place at one loop level in the Standard Model (SM) and could be an important signal for physics beyond SM. But it is found that the Standard Model contribution far exceeds the tree level contribution of the noncommutative case.\ \ [**Keywords**]{}: Noncommutative, gamma-gamma scattering\ \ [**PACS**]{}: 12.60.-i, 13.40.-f author: - | Namit Mahajan[^1]\ [*Department of Physics and Astrophysics,*]{}\ [*University of Delhi, Delhi-110 007, India.*]{} title: 'Noncommutative QED and $\gamma\gamma$ scattering' --- =cmr10 \*[Introduction]{} Noncommutativity of a pair of conjugate variables forms the central theme of quantum mechanics in terms of the Uncertainty Principle. We are quite familiar with the noncommutativity of rotations in ordinary Euclidean space. The idea of noncommutative (NC) space-time can be traced back to the work of Snyder [@snyder]. But more recently, string theory arguments have motivated an extensive study of Quantum Field Theory (QFT) on NC spaces [@douglas]. The noncommutativity of space-time is realised by the coordinate operators, $x_{\mu}$, satisfying $$[x_{\mu},x_{\nu}] = \iota\Theta_{\mu\nu}$$ with $\Theta_{\mu\nu} = \theta \epsilon_{\mu\nu}$. $\theta$ is the noncommutativity parameter with dimensions $(mass)^{-2}$ and $\epsilon _{\mu\nu}$ is a dimensionless antisymmetric matrix with elements ${\mathcal O} (1)$. The field theories formulated on such spaces are non-local and violate Lorentz symmetry. The deviation from the standard theory manifests as violation of Lorentz invariance. We can still expect manifest Lorentz invariance for energies satisfying $E^2\theta << 1$. In the limit $\theta \rightarrow 0$, one expects to recover the standard theory. This is true for the theory at classical level. But at the quantum level, the limit $\theta \rightarrow 0$ does not lead to the commutative theory [@armoni]. The theory of electrons in a strong magnetic field, projected to the lowest Landau level, is a classic example of NC field theory.\ Various attempts, both theoretical and phenomenological, have been made to study QFT on NC spaces. The study of perturbative behaviour and divergence structure [@sheraz], $C$, $P$ and $T$ properties and renormalisability [@shiekh] of such theories has been undertaken. It has been shown that quantum theories with time-like noncommuatativities are not unitary [@gomis]. We shall therefore restrict our discussion to the theories with space-like noncomutativities, although it has been shown that light-like noncommutative theories are also free of pathologies [@gomis1]. To this end, the coordinate commutator simply reads $$[x_i,x_j] = \iota\theta\epsilon_{ij}$$ There have been attempts to write down particle physics models, in particular SM, on such NC spaces [@connes]. From a phenomenological point of view, various scattering processes have been analysed [@pheno; @hewett] along with the attempts to calculate additional contributions to the precisely measured quantities like anomalous magnetic moment [@sh1] and Lamb shift [@sh2] in the noncommutative version of QED. \*[$\gamma\gamma$ scattering in NCQED]{} Consider NCQED i.e. a $U(1)$ noncommutative theory coupled to fermions. The noncommutative version of a theory can be written by replacing the field products by what is called the [*’star product’*]{}. The star ($\ast$) product for any two functions is given by $$f(x)\ast g(x) = f(x)e^{\frac{\iota}{2}\overleftarrow{\partial_{\alpha}} \Theta^{\alpha\beta} \overrightarrow{\partial_{\beta}}}g(x)$$ The NCQED action, using the above line of reasoning, is $$S_{NCQED} = \int d^Dx \Bigg( -\frac{1}{4g^2}F^{\mu\nu}(x)\ast F_{\mu\nu}(x) + \iota\bar{\psi}(x)\gamma^{\mu}\ast D_{\mu}\psi(x) - m\bar{\psi}(x)\ast\psi(x)\Bigg)$$ where $g$ is the coupling and $$F_{\mu\nu} = \partial_{\mu}A_{\nu}(x) - \partial_{\nu}A_{\mu}(x) + \iota g[A_{\mu}(x),A_{\nu}(x)]_{\ast}$$ The covariant derivative is given by $$D_{\mu}\psi(x) = \partial_{\mu}\psi(x) + \iota gA_{\mu}(x)\ast\psi(x)$$ The action is invariant under the noncommutative $U(1)$ transformations obtained by replacing all the products in the standard transformations by the corresponding star products. The noncommutativity is encoded in the star product and from the above expressions it is quite evident that the field strength, even in the case of $U(1)$, theory is nonlinear in gauge field and it is precisely this nonlinearity that gives rise to additional vertices for the gauge field. It is now a straight forward task to derive the Feynman rules from the above action [@sh1], Arfaei et.al [@pheno]. It is found that apart from generating the three and four point vertices for the gauge field self interaction, each interaction vertex picks up a momentum dependent phase factor, whose argument typically has the structure $\frac{\iota}{2} p \wedge k$. The $\wedge$ product, in general, is defined as $$p \wedge k = p_{\mu} \Theta^{\mu\nu} k_{\nu}$$ In the case of theories with only space-like noncommutativities, only the space-space elements contribute and using Eq.(2) it simply reduces to the usual vector cross-product of the two three momenta i.e. $$p \wedge k = \vec{p}\times \vec{k}$$ The process, $\gamma\gamma \longrightarrow \gamma\gamma$ takes place at the one loop level in standard QED as well as SM and thus is quite suppressed. But the presence of Yang-Mills type coupling for the photon field in NCQED enables the process to take place at the tree level. This makes the above process a plausible candidate to look for physics beyond SM at the tree level.\ The diagrams contributing to the scattering process are (155,120)(-5.0,-20) (0,45)(45,0)[3]{}[6]{} (15,50)\[c\][$k_1$]{} (20,40)(30,30) (0,-45)(45,0)[3]{}[6]{} (15,-50)\[c\][$k_2$]{} (20,-40)(30,-30) (45,0)(90,0)[3]{}[5]{} (90,0)(135,45)[3]{}[6]{} (120,50)\[c\][$p_1$]{} (105,30)(115,40) (90,0)(135,-45)[3]{}[6]{} (120,-50)\[c\][$p_2$]{} (105,-30)(115,-40) 1.5cm (155,120)(-5.0,-20) (0,45)(45,0)[3]{}[6]{} (15,50)\[c\][$k_1$]{} (20,40)(30,30) (0,-45)(45,0)[3]{}[6]{} (15,-50)\[c\][$p_1$]{} (30,-30)(20,-40) (45,0)(90,0)[3]{}[5]{} (90,0)(135,45)[3]{}[6]{} (120,50)\[c\][$k_2$]{} (115,40)(105,30) (90,0)(135,-45)[3]{}[6]{} (120,-50)\[c\][$p_2$]{} (105,-30)(115,-40) 1.5cm (155,120)(-5.0,-20) (0,45)(45,0)[3]{}[6]{} (15,50)\[c\][$k_1$]{} (20,40)(30,30) (0,-45)(45,0)[3]{}[6]{} (15,-50)\[c\][$p_2$]{} (30,-30)(20,-40) (45,0)(90,0)[3]{}[5]{} (90,0)(135,45)[3]{}[6]{} (120,50)\[c\][$p_1$]{} (105,30)(115,40) (90,0)(135,-45)[3]{}[6]{} (120,-50)\[c\][$k_2$]{} (115,-40)(105,-30) 1.5cm (155,120)(-5.0,-20) (0,45)(45,0)[3]{}[6]{} (15,50)\[c\][$k_1$]{} (20,40)(30,30) (0,-45)(45,0)[3]{}[6]{} (15,-50)\[c\][$k_2$]{} (20,-40)(30,-30) (45,0)(90,45)[3]{}[6]{} (75,50)\[c\][$p_1$]{} (60,30)(70,40) (45,0)(90,-45)[3]{}[6]{} (75,-50)\[c\][$p_2$]{} (60,-30)(70,-40) Due to the noncommutative nature of the coordinates, the theory is not Lorentz invariant and the results are frame dependent. In writing down the amplitudes corresponding to each of the above diagrams, we assume that $\theta \ll 1$ and make the substitution $\sin(a\theta) \longrightarrow a\theta$, where $a$ is used to generically denote the quantity appearing in the argument of the sine function multiplied to $\theta$.\ Choosing to work in the center of mass frame, we find that the s-channel diagram vanishes. The square of the matrix amplitude reads $$|{\mathcal{M}}_{NC}|^2 = \left(\frac{e\theta}{16}\right)^4~[100s^4 + 96t^4 + 204st^3 + 360s^2t^2 + 250s^3t]$$ and the total (unpolarised) cross section is $$\sigma_{NC} = (1.5 \times 10^{-3}) \alpha_{em}^2s^3\theta^4$$ which for $\sqrt{s}\sim$ TeV and $\theta \leq (10^4~GeV)^{-2}$ as in [@sh2] gives $\sigma_{NC}\sim 10^{-10}$ fb, to be compared with the SM contribution, $\sigma_{SM}\sim$ fb at the same center of mass energy [@jikia]. It is found that at low energies the fermion contribution dominates the SM cross-section while at higher $\sqrt{s}$ ($>$ 100 GeV), it is the W contribution that becomes important. The SM contribution gradually decreases as $\sqrt{s}$ crosses the 500 GeV range. Although, in contrast to SM, the NC cross-section increases monotonously with $\sqrt{s}$, it can never catch up with the SM cross-section for the same energy. \*[Conclusions]{} In this article we have computed the NCQED contribution to the $\gamma\gamma$ scattering and found that even though in this case the process occurs at the tree level as opposed to SM, where it takes place at the one loop level, the SM contribution far exceeds the NC contribution. It is clear that the NCQED contribution will start showing up only when $\theta$ is much larger than the value used here.\ The process has been studied in context of NCQED by Hewett et.al [@hewett] but the authors argue that inspired by recent theories of extra dimensions [@nima], where the effective scale of gravity is ${\mathcal{O}} \sim$ TeV as opposed to the Planck scale, the scale of noncommutativity can, too, be chosen to be around TeV. But a more physical approach would be to use the value of $\theta$, as obtained from studies like Lamb shift [@sh2], to calculate the new contributions. Also the authors have taken into account time-like noncommutativity that may lead to possible non-unitary S-matrix elements. Even for $\theta~\sim~(TeV)^{-2}$ as taken by the authors, the SM contribution is still overwhelmingly large. Thus with the present day and near future experiments, it doesn’t seem possible to get a signal of NCQED from $\gamma\gamma$ scattering. \*[Acknowledgements]{} The author would like to thank University Grants Commission, India for fellowship. [99]{} H. S. Snyder, Phys. Rev. [**71**]{}, 38 (1947). For a recent review see M. R. Douglas and N. A. Nekrasov, hep-th/0106048 and references therein. A. Armoni, Nucl. Phys. [**B 593**]{}, 229 (2001). S. Minwalla, M. V. Raamsdonk and N. Sieberg, hep-th/9912072; T. Krajewski and R. Wulkenhaar, Int. J. Mod. Phys. [**A15**]{}, 1011 (2000); M.Hayakawa, Phys. Lett. [**B 478**]{}, 394 (2000); A. Micu and M. M. Shiekh-Jabbari, hep-th/0008057; C. E. Carlson, C. D. Carone and R. F. Lebed, Phys. Lett. [**B 518**]{}, 201 (2001). C. P. Martin and D. Sanchez-Ruiz, Phys. Rev. Lett. [**83**]{}, 476 (1999); M. M. Shiekh-Jabbari, hep-th/9903107; M. M. Shiekh-Jabbari, Phys. Rev. Lett. [**84**]{}, 5265 (2000). J. Gomis and T. Mehen, hep-th/0005129. O. Aharony, J. Gomis and T. Mehen, hep-th/0006236. A. Connes and J. Lott, Nucl. Phys. Proc. Suppl. [**18B**]{}, 29 (1991); M. Chaichian, P. Presnajder, M. M. Shiekh-Jabbari and A. Tureanu, hep-th/0107055. N. Chair and M. M. Shiekh-Jabbari, hep-th/0009037; P. Mathews, Phys. Rev. [**D 63**]{}, 075007 (2001); S. Baek, D. K. Ghosh, X. G. He and W-Y. P. Hwang, Phys. Rev. [**64**]{}, 056001 (2001); H. Arfaei and M. H. Yavartanoo, hep-th/0010244; S. Godfrey and M. A. Doncheski, hep-ph/0108268. J. L. Hewett, F. J. Petriello and T. G. Rizzo, hep-ph/0010354. I. F. Riad and M. M. Shiekh-Jabbari, hep-th/0008132. N. Chair, M. M. Shiekh-Jabbari and A. Tureanu, Phys. Rev. Lett. [**86**]{}, 2716 (2001). N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. [**B 429**]{}, 263 (1998); I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phy. Lett. [**B 436**]{}, 257 (1998); L. Randall and R. Sundrum, Phys. Rev. Lett [**83**]{}, 3370 (1999); Phys. Rev. Lett. [**83**]{}, 4690 (1999). G. Jikia and A. Tkabladze, Phys. Lett. [**B 323**]{} 453 (1994). [^1]: E–mail : nm@ducos.ernet.in, nmahajan@physics.du.ac.in

--- abstract: 'We propose and investigate an exactly solvable model of non-equilibrium Luttinger liquid on a star graph, modeling a multi-terminal quantum wire junction. The boundary condition at the junction is fixed by an orthogonal matrix $\S$, which describes the splitting of the electric current among the leads. The system is driven away from equilibrium by connecting the leads to heat baths at different temperatures and chemical potentials. The associated non-equilibrium steady state depends on $\S$ and is explicitly constructed. In this context we develop a non-equilibrium bosonization procedure and compute some basic correlation functions. Luttinger liquids with general anyon statistics are considered. The relative momentum distribution away from equilibrium turns out to be the convolution of equilibrium anyon distributions at different temperatures. Both the charge and heat transport are studied. The exact current-current correlation function is derived and the zero-frequency noise power is determined.' --- Ł[L]{} §[S]{} 1[\_1]{} Ø[O]{} [**v**]{} [**Luttinger Liquid in Non-equilibrium\ Steady State**]{}\ [Mihail Mintchev$^1$ and Paul Sorba$^2$]{}\ 1.5 truecm *${}^1$ Istituto Nazionale di Fisica Nucleare and Dipartimento di Fisica dell’Università di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy\ * ${}^2$ Laboratoire de Physique Théorique d’Annecy-le-Vieux, CNRS,\ 9, Chemin de Bellevue, BP 110, F-74941 Annecy-le-Vieux Cedex, France Introduction ============ The universal features of a large class of one-dimensional quantum models, exhibiting gapless excitations with linear spectrum, are successfully described [@Hald; @Haldprl] by the Tomonaga-Luttinger (TL) liquid theory [@T50]-[@ML65]. This theory[^1] applies to various systems, including nanowire junctions and carbon nanotubes, which are available nowadays in experiment [@na1]-[@na3]. For this reason the study of non-equilibrium phenomena in the TL liquid phase attracts recently much attention [@GGM08]-[@Ines]. A typical non-equilibrium setup, considered in the literature, is the junction of two or more semi-infinite leads with electrons at different temperatures and/or chemical potentials. The junction is an interval of finite length $L$, where the electrons injected from the leads interact among themselves. This interaction drives the system away from equilibrium. Differently from the equilibrium TL liquid on the line, the non-equilibrium model defined in this way, is not exactly solvable. Nevertheless, it is extensively studied [@GGM08]-[@Ines] by various methods, including linear response theory, bosonization combined with the non-equilibrium Keldish formalism and perturbation theory. One of the main goals of the present paper is to explore the possibility to construct and analyze an alternative [*exactly solvable*]{} model for a non-equilibrium TL junction. Since the universal features of such a system are expected to manifest themselves in the critical (scale invariant) limit, it is natural to shrink the domain of the non-equilibrium interaction to a point, taking $L\to 0$. For a complete description of the critical regime it is essential to take into account all point-like interactions, which ensure a unitary time evolution of the system. These interactions can be parametrized by a scattering matrix $\S$ localized in the junction point, as shown in the multi-terminal setup displayed in Fig.\[fig1\]. (600,120)(40,250) ![A junction with scattering matrix $\S$ and $n$ semi-infinite leads, connected at infinity to thermal reservoirs with temperatures $\beta_i$ and chemical potentials $\mu_i$.[]{data-label="fig1"}](fig1.pdf "fig:") Each lead contains a TL liquid, which at infinity is in contact with a heat reservoir with (inverse) temperature $\beta_i$ and chemical potential $\mu_i$. Our first step below is to show that there exists a non-equilibrium steady state (NESS), which describes the TL configuration in Fig.\[fig1\]. This state is characterized by non-trivial time independent electric and heat currents, flowing in the leads. The scattering matrix $\S$ is implemented by imposing specific boundary conditions at the junction. It turns out that the boundary conditions, which describe the splitting of the electric steady current in the junction, lead to an exactly solvable problem. In fact, we establish the operator solution in this case and investigate the relative non-equilibrium correlation functions in the NESS representation. The TL theory has been introduced originally [@T50]-[@ML65] for describing fermion systems. It has been understood later on [@Liguori:1999tw]-[@Calabrese:2007ty] that the fermion TL liquid is actually an element of a more general family of [*anyon*]{} TL liquids[^2], which obey Abelian braid statistics. In this paper we explore the general anyon TL liquid, obtaining the conventional fermionic and bosonic ones as a special cases. From the two-point anyon correlation functions we extract the NESS distribution of the TL anyon excitations. In momentum space this non-equilibrium distribution is a [*nested convolution*]{} of equilibrium distributions at different temperatures and chemical potentials. As expected, the convolution depends on the scattering matrix $\S$, which drives the system away from equilibrium. We investigate also the NESS correlators of the electric and energy currents, describing in detail the charge and heat transport in the junction. The zero-frequency noise power is deduced from the two-point current-current correlation function, whose exact expression in terms of hypergeometric functions is established. We prove the breakdown of time reversal invariance as well. The paper has the following structure. In the next section we construct non-equilibrium chiral fields in a NESS on a star graph modeling the junction. We derive here the non-equilibrium Casimir energy and the heat current and compare the latter with the conformal field theory result. In section 3 we develop a non-equilibrium finite temperature operator bosonization procedure. We also establish the operator solution, subject to the current splitting boundary condition at the junction. We show that this condition covers two different physical situations, corresponding to a junction with and without charge dissipation. The non-equilibrium correlation functions are investigated in section 4, where the anyon NESS distributions are derived. The charge and heat transport as well as the noise are also studied there. Section 5 provides a concise outlook of the paper and contains some general observations. The appendix collects some results about the asymptotic properties of the anyon NESS correlators. Non-equilibrium chiral fields on a star graph ============================================= The fundamental building blocks of bosonization away from equilibrium are the free massless scalar field ${\varphi}$ and its dual ${\widetilde{\varphi}}$. The fields ${\varphi}$ and ${\widetilde{\varphi}}$ propagate on a star graph $\Gamma$, which is shown in Fig. \[fig2\] and models the quantum wire junction. (500,70)(-150,20) ![A star graph $\Gamma$ with $n$ edges modelling the junction of $n$ quantum wires.[]{data-label="fig2"}](fig2.pdf "fig:") The edges $E_i$ are half-lines and each point $P$ in the bulk $\Gamma \setminus V$ of $\Gamma$ is uniquely determined by its coordinates $(x,i)$, where $x > 0$ is the distance to the vertex $V$ and $i=1,...,n$ labels the edge. Besides the massless Klein-Gordon equation, the fields ${\varphi}$ and ${\widetilde{\varphi}}$ satisfy the duality relations $$\der_t {\widetilde{\varphi}}(t,x,i) = - \der_x {\varphi}(t,x,i) \, ,\qquad \der_x {\widetilde{\varphi}}(t,x,i) = - \der_t {\varphi}(t,x,i) \, . \label{dual}$$ The initial conditions are fixed by the equal-time canonical commutation relations $$[{\varphi}(t,x,i)\, ,\, {\varphi}(t,y,j)]_{{}_-} = [{\widetilde{\varphi}}(t,x,i)\, ,\, {\widetilde{\varphi}}(t,y,j)]_{{}_-} = 0 \, , \label{ecc1}$$ $$[(\der_t{\varphi}) (t,x,i)\, ,\, {\varphi}(t,y,j)]_{{}_-} = [(\der_t{\widetilde{\varphi}})(t,x,i)\, ,\, {\widetilde{\varphi}}(t,y,j)]_{{}_-} = -\ri \delta_{ij}\delta (x-y) \, . \label{ecc2}$$ In order to determine the dynamics completely, one must impose some boundary conditions at the vertex $x=0$. These conditions are conveniently formulated in terms of the combinations $${\varphi_{i,R}}(t-x) = {\varphi}(t,x,i) + {\widetilde{\varphi}}(t,x,i)\, , \qquad {\varphi_{i,L}}(t+x) = {\varphi}(t,x,i) - {\widetilde{\varphi}}(t,x,i)\, , \label{chi1}$$ which depend on $t-x$ and $t+x$ respectively and define right and left chiral fields ${\varphi_{i,Z}}$ on $\Gamma$. The most general [*scale invariant*]{} boundary conditions, generating a [*unitary time evolution*]{} of ${\varphi}$ and ${\widetilde{\varphi}}$, are parametrized by the orthogonal group $O(n)$ and read [@ks-00]-[@Bellazzini:2008mn] $${\varphi_{i,R}}(\xi ) = \sum_{j=1}^n \S_{ij}\, {\varphi}_{j,L}(\xi ) \, , \qquad \S \in O(n)\, . \label{bc1}$$ These simple conditions capture the universal features of the system and $\S$ has a straightforward physical interpretation: the vertex $V$ of $\Gamma$ represents a [*scale invariant point-like defect*]{}, $\S$ being the associated scattering matrix. The non-equilibrium steady state $\Omega_{\beta,\mu_{{}_b}}$ ------------------------------------------------------------ Our next step is to construct a steady state $\Omega_{\beta,\mu_{{}_b}}$, which captures the evolution of the chiral fields ${\varphi_{i,Z}}$ on $\Gamma$, whose edges are attached at infinity to thermal reservoirs at inverse temperatures $\beta_i$ as shown in Fig. \[fig1\]. In the boson case we take all chemical potentials to be equal[^3], setting $\mu_i = \mu_b$ in all reservoirs. The system is away from equilibrium if $\S$ contains at least one non-trivial transmission coefficient among reservoirs with different temperature. The construction of $\Omega_{\beta,\mu_{{}_b}}$, described below, follows the scheme developed in [@Mintchev:2011mx] and is based on scattering theory. It adapts to the case under consideration some modern ideas [@els-96]-[@st-06] about NESS. The framework is purely algebraic and generalizes the definition [@BR] of equilibrium Gibbs state over the algebra of canonical commutation relations (CCR). We start by observing that the massless Klein-Gordon equation and the relations (\[dual\]) lead to the following representation $$\begin{aligned} &&{\varphi_{i,R}}(\xi) = \int_0^\infty \frac{{{\rm d}}k}{\pi \sqrt{2}}\, \sqrt {\Delta_\lambda (k)} \left [a_i^\ast (k) \e^{\ri k \xi} + a_i(k)\e^{-\ri k \xi}\right ] \, , \label{fil1} \\ && {\varphi_{i,L}}(\xi) = \int_0^\infty \frac{{{\rm d}}k}{\pi \sqrt{2}}\, \sqrt {\Delta_\lambda (k)} \left [a_i^\ast (-k) \e^{\ri k \xi} + a_i(-k) \e^{-\ri k \xi}\right ] \, , \label{fil2}\end{aligned}$$ $\Delta_\lambda $ being some distribution to be fixed below. Using that $\S$ is a real matrix, the boundary condition (\[bc1\]) implies the constraints $$a_i(k) = \sum_{j=1}^n S_{ij}(k)\, a_j (-k) \, , \qquad a^\ast_i (k) = \sum_{j=1}^n S_{ij}(k)\, a^\ast_ j(-k)\, , \label{constr1}$$ where $$S(k) = \theta(-k) \S^t + \theta (k) \S\, , \label{S}$$ $\theta$ is the Heaviside step function and $\S^t$ indicates the transpose of $\S$. From the equal-time commutation relations (\[ecc1\], \[ecc2\]) one infers that the elements $\{a_i(k),\, a^*_i(k)\, :\, k \in {\mbox{${\mathbb R}$}}, \, i=1,...,n\}$ generate the following deformation $\A$ of the algebra of CCR: $$[a_i(k)\, ,\, a_j(p)]_{{}_-} = [a^*_i (k)\, ,\, a^*_j (p)]_{{}_-} = 0 \, , \label{rta1}$$ $$[a_i(k)\, ,\, a^*_j (p)]_{{}_-} = 2\pi [\delta (k-p)\delta_{ij} + S_{ij}(k)\delta(k+p)] \, . \label{rta2}$$ Moreover, (\[ecc1\], \[ecc2\]) imply that $$|k| \Delta_\lambda (k) = 1\, . \label{distr1}$$ There exist a one-parameter family of tempered distributions, which solve this equation in ${\mbox{${\mathbb R}$}}$. A convenient representation of this family is given by [@Liguori:1997vd] $$\Delta_\lambda (k) = \frac{{{\rm d}}}{{{\rm d}}k} \left [\theta (k) \ln \frac {k}{\lambda}\right ]\, , \label{distr2}$$ where $\lambda >0$ is a free parameter with dimension of mass, having well-known infrared origin. The above structure is very general and equations (\[fil1\],\[fil2\]) apply to any representation of the algebra $\A$, which is a simplified version of the so called reflection-transmission (RT) algebra [@Liguori:1996xr]-[@Mintchev:2003ue], describing factorized scattering in integrable models with point-like defects in one dimension. The Fock and the Gibbs state over $\A$ describe equilibrium physics and have been largely explored. We will investigate here the NESS $\Omega_{\beta,\mu_{{}_b}}$, which describes the physical situation shown in Fig. \[fig1\]. For this purpose we first observe that the sub-algebras $\A_\inc$ and $\A_\out$, generated by the elements $\{a_i(k),\, a^*_i(k)\, :\, k<0\}$ and $\{a_i(k),\, a^*_i(k)\, :\, k>0\}$ respectively, parametrize the [*asymptotic incoming*]{} and [*outgoing*]{} fields. Accordingly, both $\A_\inc$ and $\A_\out$ are conventional CCR algebras; in fact the $\delta (k+p)$ term in (\[rta2\]) vanishes if both momenta are negative or positive. It is worth stressing that (\[constr1\]) relate $\A_\inc$ with $\A_\out$ and that the whole RT algebra $\A$ can be generated via (\[constr1\]) either by $\A_\inc$, or by $\A_\out$. The main idea for constructing $\Omega_{\beta , \mu_b}$ is based on this kind of asymptotic completeness property. Starting with an equilibrium state on $\A_\inc$, we will extend it by means of (\[constr1\]) to a non-equilibrium state on the whole algebra $\A$. For this purpose we introduce the edge Hamiltonian and number operators $$h_i = \int_{-\infty}^0 \frac{{{\rm d}}k}{2\pi} |k| a^*_i(k) a_i(k)\, , \qquad n_i = \int_{-\infty}^0 \frac{{{\rm d}}k}{2\pi} a^*_i(k) a_i(k)\, , \label{ac}$$ which describe the [*asymptotic*]{} dynamics at $t=-\infty$ (i.e. before the interaction) in terms of $\A_\inc$. Defining $$K = \sum_{i=1}^n \beta_i (h_i -\mu_b n_i) \, , \qquad \beta_i \geq 0\, , \label{kop}$$ we introduce the equilibrium Gibbs state over $\A_\inc$ in the standard way [@BR]. For any polynomial ${{\cal P}}$ over $\A_\inc$ we set $$\begin{aligned} \left (\Omega_{\beta , \mu_b}\, ,\, {{\cal P}}\bigl (a_i^*(k_i), a_j(p_j)\bigr ) \Omega_{\beta , \mu_b} \right ) \equiv \langle {{\cal P}}\bigl (a_i^*(k_i), a_j(p_j))\bigr ) \rangle_{\beta, \mu_b} = \nonumber \\ \frac{1}{Z} \tr \left [\e^{-K} {{\cal P}}\bigl (a_i^*(k_i), a_j(p_j)\bigr )\right ]\, , \qquad \qquad \qquad \label{def1}\end{aligned}$$ where $k_i<0,\; p_j<0$ and $Z = \tr \left ( \e^{-K}\right )$. All the expectation values (\[def1\]) can be computed [@BR] by purely algebraic manipulations and can be expressed in terms of the two-point functions, which are written in terms of the familiar Bose distribution $$b_i (k) = \frac{\e^{-\beta_i [|k| - \mu_b]}}{1- \e^{-\beta_i [|k| - \mu_b]}} \label{fbd1}$$ in the following way $$\langle a_j^*(p)a_i(k) \rangle_{\beta, \mu_b} = b_i(k) \delta_{ij} 2\pi \delta (k-p)\, , \label{2a}$$ $$\langle a_i(k)a_j^*(p)\rangle_{\beta, \mu_b} = [1+b_i(k)] \delta_{ij} 2\pi \delta (k-p)\, . \label{2b}$$ We stress that (\[2a\],\[2b\]) hold on $\A_\inc$, i.e. only for [*negative*]{} momenta. The common for all reservoirs chemical potential $\mu_b<0$ allows to avoid in (\[2a\],\[2b\]) the infrared singularity at $k=0$. We anticipate that $\mu_b$ has nothing to do with the [*fermion*]{} chemical potentials, appearing in the non-equilibrium bosonization procedure described in the next section, where the limit $\mu_b \to 0^-$ exist and will be performed. The next step is to extend (\[def1\]-\[2b\]) to the whole RT algebra $\A$, namely to [*positive*]{} momenta. Employing (\[constr1\]) one finds $$\begin{aligned} \langle a_j^*(p)a_i(k)\rangle_{\beta, \mu_b} = 2\pi \Bigl\{\Bigl [\theta(-k)b_i(k) \delta_{ij}+ \theta(k)\sum_{l=1}^n \S_{il}\, b_l(k)\, \S^t_{lj}\Bigr ] \delta (k-p) \nonumber \\ + \Bigl [\theta(-k)b_i(k)\, \S^t_{ij} + \theta(k)\S_{ij}\, b_j(k) \Bigr ] \delta (k+p) \Bigr\}\, . \qquad \; \; \, \label{cor1}\end{aligned}$$ The expression for $\langle a_i(k)a_j^*(p)\rangle_{\beta, \mu}$ is obtained from (\[cor1\]) by the substitution $$b_i(k) \longmapsto 1+b_i(k) = \frac{1}{1- \e^{-\beta_i [|k| - \mu_b]}} \, . \label{fbd2}$$ The final step is to compute a generic correlation function. By means of the commutation relations (\[rta1\],\[rta2\]), this problem is reduced to the evaluation of correlators of the form $\langle \prod_{m=1}^M a_{i_m}(k_{i_m}) \prod_{n=1}^N a^\ast_{j_n}(p_{j_n})\rangle_{\beta,\mu_b}$, which can be computed by iteration via $$\begin{aligned} \langle \prod_{m=1}^M a_{i_m}(k_{i_m}) \prod_{n=1}^N a^{\ast j_n}(p_{j_n})\rangle_{\beta,\mu_b} = \qquad \qquad \qquad \nonumber \\ \delta_{MN}\, \sum_{m=1}^M \langle a_{i_1}(k_{i_1})a^{\ast j_m}(p_{j_m})\rangle_{\beta,\mu_b} \, \langle \prod_{m=2}^M a_{i_m}(k_{i_m}) \prod_{n=1\atop {n\not=m} }^N a^{\ast j_n}(p_{j_n}) \rangle_{\beta,\mu_b} \, . \label{gcf2}\end{aligned}$$ We would like to mention in conclusion that the use of the RT algebra $\A$ in the construction of $\Omega_{\beta, \mu_b}$ represents only a convenient choice of coordinates, which has a simple physical interpretation in terms of scattering data and applies to a variety of systems [@Mintchev:2011mx; @Caudrelier:2012xy] with point-like defects. Energy density and energy transport in $\Omega_{\beta,\mu_b}$ ------------------------------------------------------------- In order to illustrate the physical properties of $\Omega_{\beta,\mu_b}$, it is instructive to investigate the non-equilibrium energy density and transport associated with the scalar field ${\varphi}$. The equations of motion imply the conservation $$\der_t \theta_{tt}(t,x,i) - \der_x \theta_{xt}(t,x,i) = 0\, , \label{econs}$$ of the energy-momentum tensor $$\begin{aligned} \theta_{tt}(t,x,i) &=& \frac{1}{2}: \left [ ({\partial}_t {\varphi})({\partial}_t {\varphi}) - {\varphi}({\partial}_x^2 {\varphi}) \right ]:(t,x,i)\, , \label{emtph1}\\ \theta_{xt}(t,x,i) &=& \frac{1}{2} :\left [ ({\partial}_x {\varphi})( {\partial}_t {\varphi}) - {\varphi}({\partial}_x {\partial}_t {\varphi})\right ] :(t,x,i)\, , \label{emtph2}\end{aligned}$$ where $:\cdots :$ denotes the normal product in the algebra $\A$. The boundary condition (\[bc1\]) implies the Kirchhoff rule $$\sum_{i=1}^n \theta_{xt} (t,0,i) = 0\, , \label{K1}$$ which, combined with (\[econs\]), ensures energy conservation. The derivation of $\langle \theta_{tt}(t,x,i)\rangle_{\beta,\mu_b}$ and $\langle \theta_{xt}(t,x,i)\rangle_{\beta,\mu_b}$ is based on the expectation value $$\begin{aligned} \langle : {\varphi}(t_1,x_1,i_1) {\varphi}(t_2,x_2,i_2): \rangle_{\beta,\mu_b} = \qquad \qquad \qquad \nonumber \\ \int_0^\infty \frac{{{\rm d}}k}{2\pi}\Delta_\lambda (k) \Bigl \{\delta_{i_1i_2} b_{i_1}(k) \cos[k(t_{12}+x_{12}] + \S_{i_1i_2} b_{i_2}(k) \cos[k(t_{12}-{\widetilde{x}}_{12}] + \nonumber \\ b_{i_1}(k)\, \S^t_{i_1i_2} \cos[k(t_{12}+{\widetilde{x}}_{12}] +\sum_{j=1}^n \S_{i_1j}\, b_j(k)\, \S^t_{ji_2} \cos[k(t_{12}-x_{12}]\Bigr \}\, , \label{ncf}\end{aligned}$$ where ${\widetilde{x}}_{12} = x_1+x_2$. Plugging (\[ncf\]) in the definitions (\[emtph1\],\[emtph2\]), one gets $$\begin{aligned} \E_i(\beta, \mu_b) \equiv \langle \theta_{tt}(t,x,i)\rangle_{\beta,\mu_b} = \qquad \qquad \qquad \qquad \nonumber \\ \S_{ii} \int_0^\infty \frac{{{\rm d}}k}{\pi}\, k\, \cos(2kx) b_i(k)+ \sum_{j=1}^n \left (\delta_{ij}+\S_{ij}^2 \right ) \frac{1}{2\pi \beta_j^2}\, {\rm Li}_2 \left (\e^{\beta_j \mu_b}\right )\, , \; \; \, \label{tr1}\\ \T_i(\beta, \mu_b) \equiv \langle \theta_{xt}(t,x,i)\rangle_{\beta,\mu_b} = \sum_{j=1}^n \left (\delta_{ij}-\S_{ij}^2\right ) \frac{1}{2\pi \beta_j^2}\, {\rm Li}_2 \left (\e^{\beta_j \mu_b}\right )\, , \; \; \, \label{tr2}\end{aligned}$$ where ${\rm Li}_2$ is the dilogarithm function. Eq. (\[tr1\]) determines the Casimir energy, whereas (\[tr2\]) describes the energy (heat) transport. Both are time-independent, thus confirming that we are dealing with a steady state. Notice also that the energy density is $x$-dependent, which reflects the breaking of translation invariance by the junction and is consistent with the conservation law (\[econs\]). Since $\S$ is an orthogonal matrix, $\T_i(\beta,\mu_b)$ obviously satisfies the Kirchhoff’s rule (\[K1\]). The energy density $\E_i(\beta, \mu_b)$ can be written in the equivalent form $$\E_i(\beta, \mu_b) = \S_{ii} \int_0^\infty \frac{{{\rm d}}k}{\pi}\, k\, \cos(2kx) b_i(k) + \frac{1}{\pi \beta_i^2}\, {\rm Li}_2 \left (\e^{\beta_i \mu_b}\right ) - \sum_{j=1}^n \left (\delta_{ij}-\S_{ij}^2 \right ) \frac{1}{2\pi \beta_j^2}\, {\rm Li}_2 \left (\e^{\beta_j \mu_b}\right )\, , \label{ntr}$$ where the last term vanishes at equilibrium and describes therefore the [*non-equilibrium contribution*]{} to the Casimir energy. For a junction with $n=2$ wires there are two one-parameter families $$\S^+ = \left(\begin{array}{cc}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \\ \end{array} \right)\, , \qquad \S^- = \left(\begin{array}{cc}\cos \theta & \sin \theta \\ \sin \theta & -\cos \theta \\ \end{array} \right)\, , \quad \theta \in [0,2\pi)\, , \label{ex1}$$ of scattering matrices with ${\rm det} (\S^\pm) = \pm 1$. For both families one finds $$\T_1(\beta, \mu_b) = - \T_2(\beta, \mu_b) = \frac{\sin^2 \theta }{2\pi} \left [ \frac{1}{\beta_1^2} {\rm Li}_2 \left (\e^{\beta_1 \mu_b}\right ) - \frac{1}{\beta_2^2} {\rm Li}_2 \left (\e^{\beta_2 \mu_b}\right )\right ]\, , \label{ctr1}$$ which deserves a comment. The heath transport has been investigated recently in the framework of conformal field theory (CFT) for generic central charge $c$ in [@bd-12]. Eq. (\[ctr1\]) confirms the result of [@bd-12] for $c=1$ and extends this result in two directions: imperfect junction with transmission probability $\sin^2(\theta)$ and $\mu_b\not=0$. We observe in this respect that derivation of the dilogarithm terms in (\[ctr1\]) is problematic in a CFT context, because both $\beta_1 \mu_b$ and $\beta_2 \mu_b$ are non-trivial dimensionless parameters. In the limit $\mu_b \to 0^-$ the $x$-dependent integral in (\[tr1\]) can be evaluated explicitly and one finds $$\begin{aligned} \E_i(\beta, 0) &=&\frac{\pi}{12}\sum_{j=1}^n \left (\delta_{ij}+\S^2_{ij} \right )\frac{1}{\beta_j^2} + \S_{ii}\frac{1}{8\pi x^2} - \S_{ii}\frac{\pi}{2\beta_i^2 \left [\sinh \left (2\pi \frac{x}{\beta_i}\right )\right ]^2}\, , \label{tr3} \\ \T_i(\beta, 0) &=& \frac{\pi}{12}\sum_{j=1}^n \left (\delta_{ij} - \S_{ij}^2 \right ) \frac{1}{\beta_j^2}\, . \qquad \label{tr4}\end{aligned}$$ Chiral NESS correlators ----------------------- In the non-equilibrium bosonization procedure, developed below, we will need the correlation functions of the chiral fields (\[fil1\],\[fil2\])) in the NESS $\Omega_{\beta, \mu_b}$. It is easily seen that all of them can be expressed in terms of the distribution $$w (\xi,\beta;\lambda, \mu_b)= \int_0^\infty \frac{{{\rm d}}k}{\pi} \Delta_\lambda (k) \left [\frac{\e^{-\beta [k - \mu_b]}}{1- \e^{-\beta [k - \mu_b]}} \e^{\ri k \xi} + \frac{1}{1- \e^{-\beta [k - \mu_b]}} \e^{-\ri k \xi}\right ]\, . \label{w}$$ The full $\lambda$-dependence and the singularity at $\mu_b=0$ of (\[w\]) are captured by [@Liguori:1999tw] $$w (\xi,\beta;\lambda, \mu_b) = \frac{1}{\pi} \left \{ \frac{2}{\beta |\mu_b|} \ln \frac{|\mu_b|}{\lambda} - \ln \left [2 \ri\, \sinh \left (\frac{\pi}{\beta} \xi - \ri \varepsilon \right ) \right ]\right \} + o(\mu_b)\, , \label{wa}$$ where $o(\mu_b)$ stands for $\lambda$-independent terms, which vanish in the limit $\mu_b \to 0^-$. It is convenient at this point to relate the (up to now free) infrared regularization parameter $\lambda $ to $\mu_b$ by means of $$\lambda = |\mu_b| \, . \label{lambdamu}$$ The limit $\mu_b \to 0^-$ in (\[w\]) now exists and gives the distribution[^4] $$w (\xi,\beta )\equiv \lim_{\mu_b \to 0^-} w (\xi,\beta; \lambda = |\mu_b|, \mu_b) = -\frac{1}{\pi} \ln \left [2 \ri\, \sinh \left (\frac{\pi}{\beta} \xi - \ri \varepsilon \right ) \right ] \, , \label{w1}$$ which is the fundamental block of the NESS chiral correlation functions. In fact, for the two-point correlators one gets $$\begin{aligned} \langle {\varphi}_{i_1,L}(\xi_1) {\varphi}_{i_2,L}(\xi_2) \rangle_\beta &=& \delta_{i_1i_2}\, w (\xi_{12},\beta_{i_1})\, , \label{ll}\\ \langle {\varphi}_{i_1,L}(\xi_1) {\varphi}_{i_2,R}(\xi_2) \rangle_{\beta} &=& w (\xi_{12},\beta_{i_1})\, \S^t_{i_1i_2}\, , \label{lr} \\ \langle {\varphi}_{i_1,R}(\xi_1) {\varphi}_{i_2,L}(\xi_2) \rangle_{\beta} &=& \S_{i_1i_2}\, w (\xi_{12},\beta_{i_2})\, , \label{rl} \\ \langle {\varphi}_{i_1,R}(\xi_1) {\varphi}_{i_2,R}(\xi_2) \rangle_{\beta} &=& \sum_{j=1}^n \S_{i_1j}\, w (\xi_{12},\beta_j)\, \S^t_{ji_2}\, . \label{rr}\end{aligned}$$ where $\xi_{12}\equiv \xi_1-\xi_2$. As expected, the point-like interaction in the vertex of $\Gamma$ induce a non-trivial left-right mixing described by (\[lr\],\[rl\]). Bosonization away from equilibrium ================================== The possibility to express fermions in terms of bosons in 1+1 dimensional space-time has been discovered long ago by Jordan and Wigner [@JW28]. The bosonization technique in the Fock representation of the fields ${\varphi}$ and ${\widetilde{\varphi}}$ has been applied for solving the Tomonaga-Luttinger (TL) model in [@Hald]-[@ML65]. The framework has been extended later [@Liguori:1999tw; @Amaral:2005cqa] to the finite temperature Gibbs representation of ${\varphi}$ and ${\widetilde{\varphi}}$. Both the Fock and Gibbs representations describe equilibrium physics. Our goal in what follows we will to apply the NESS representation, constructed in the previous section, for investigating the non-equilibrium TL liquid in the multi-terminal configuration shown in Fig. \[fig1\]. The Tomonaga-Luttinger model on $\Gamma$ ---------------------------------------- The bulk dynamics is governed by the TL Lagrangian density $${\cal L} = \ri \psi_1^*(\der_t - v_F\der_x)\psi_1 + \ri \psi_2^*(\der_t + v_F\der_x)\psi_2\\ -g_+(\psi_1^* \psi_1+\psi_2^* \psi_2)^2 - g_-(\psi_1^* \psi_1-\psi_2^* \psi_2)^2 \, , \label{lagr}$$ where $\{\psi_\alpha (t,x,i)\,:\, \alpha =1,2\}$ are complex fermion fields, $v_F>0$ is the Fermi velocity and $g_\pm \in {\mbox{${\mathbb R}$}}$ are the coupling constants. The bulk theory has an obvious $U_L(1)\otimes U_R(1)$ symmetry. In fact, the Lagrangian density (\[lagr\]) is left invariant by the two independent phase transformations $$\psi_\a \rightarrow \e^{\ri s_\alpha } \psi_\a \, , \qquad \qquad \; \, \psi^*_\a \rightarrow \e^{-\ri s_\alpha } \psi^*_\a \, , \qquad s_\a \in {\mbox{${\mathbb R}$}}\, ,\quad \a =1,2\, . \label{symm1}$$ implying the current conservation laws $$\der_t \rho_{Z} (t,x,i) -v_F\der_x j_{Z} (t,x,i)= 0\, , \label{conservation}$$ where the charge and current densities are given by $$\rho_{Z} (t,x,i) = \begin{cases} [\psi^*_1\psi_1](t,x,i)\, , & Z=L\, , \\ [\psi^*_2\psi_2](t,x,i)\, , & Z=R\, , \end{cases} \qquad j_{Z} (t,x,i) = \begin{cases} [\psi^*_1\psi_1](t,x,i)\, , & Z=L\, , \\ -[\psi^*_2\psi_2](t,x,i)\, , & Z=R\, . \end{cases} \label{lrcurrents}$$ The currents $j_{Z}(t,x,i)$ have simple physical meaning: $j_{L}(t,x,i)$ and $j_{R}(t,x,i)$ represent the particle excitations moving along the edge $E_i$ towards and away of the vertex $V$ respectively. Interpreting the vertex as a defect, which can be characterized by some scattering matrix, the currents $j_{L}$ and $j_{R}$ describe therefore the [*incoming*]{} and [*outgoing*]{} flows. The current splitting boundary condition ---------------------------------------- It is well known [@Hald]-[@ML65] that the TL model (\[lagr\]) is exactly solvable on the line ${\mbox{${\mathbb R}$}}$. On the graph $\Gamma$ the situation is more involved, because one should take into account the boundary conditions in the vertex $V$. The conditions $$\psi_1(t,0,i) = \sum_{j=1}^n \UU_{ij} \psi_2(t,0,j) \, , \qquad \UU\in U(n)\, , \label{bc2}$$ which work in the free case $g_-=g_+=0$, do not lead [@nfll-99] to exactly solvable problem after switching on the TL interactions. The fact that on ${\mbox{${\mathbb R}$}}$ the quartic bulk interactions in (\[lagr\]) are solved exactly via bosonization suggest to try boundary conditions which, differently from (\[bc2\]), are formulated in terms of real boson fields. In this spirit and according our previous comments on the chiral currents (\[lrcurrents\]), it is quite natural to consider $$j_R(t,0,i) = \sum_{k=1}^n {\mbox{${\mathbb J}$}}_{ik}\, j_L(t,0,k) \, , \qquad {\mbox{${\mathbb J}$}}\in O(n)\, , \label{bc3}$$ which has been proposed and explored first in the two-terminal case in [@SS]. An advantage of (\[bc3\]) is the direct interpretation in terms of [*gauge invariant physical observables*]{}, which represent the basic building blocks of algebraic quantum field theory (see e.g. [@H]). In fact, (\[bc3\]) describes the splitting in the vertex $V$ of the outgoing current $j_R(t,0,i)$ along the edge $E_i$ in incoming currents $j_L(t,0,k)$ along the edges $E_k$. For this reason we refer to ${\mbox{${\mathbb J}$}}$ the as the [*current splitting matrix*]{} and show in the next subsection that ${\mbox{${\mathbb J}$}}$ actually coincides with the boson scattering matrix $\S$. Operator solution of the TL model on $\Gamma$ --------------------------------------------- Referring for the details to [@Bellazzini:2008fu], we recall here the anyon operator solution of the TL model on a star graph $\Gamma$. The solution provides a [*unified description*]{} of all anyon Luttinger liquids and is expressed in terms of the chiral fields (\[fil1\],\[fil2\]) and the parameters $\sigma,\, \tau \in {\mbox{${\mathbb R}$}}$ and the sound velocity $v\in {\mbox{${\mathbb R}$}}$ as follows: $$\begin{aligned} \psi_1(t,x,i) &=& \eta_i :\e^{\ri \sqrt {\pi} \left [\sigma {\varphi_{i,R}}(vt-x) + \tau {\varphi_{i,L}}(vt+x)\right ]}:\, , \label{psi1}\\ \psi_2(t,x,i) &=& \eta_i:\e^{\ri \sqrt {\pi} \left [\tau {\varphi_{i,R}}(vt-x) + \sigma {\varphi_{i,L}}(vt+x)\right ]}:\, . \label{psi2}\end{aligned}$$ Here $: \cdots :$ denotes the normal product in the RT algebra $\A$ and $\eta_i$ are some Klein factors, controlling the statistics of $\psi_\alpha $. In this respect we impose the general anyon exchange relation $$\psi_\alpha^*(t,x_1,i_1) \psi_\alpha (t,x_2,i_2) = \e^{(-1)^{\alpha} \ri \pi \k \eps(x_{12})} \psi_\alpha (t,x_2,i_2)\psi_\alpha^*(t,x_1,i_1)\, , \qquad x_1\not=x_2\, , \label{an1}$$ where $\eps(x)$ is the sign function and $\k > 0$ is the so called [*statistical parameter*]{} which interpolates between bosons ($\k$ - even integer) and fermions ($\k$ - odd integer). A simple realization of the Klein factors is $$\eta_i = \frac{1}{\sqrt {2\pi}}:\e^{\pi\ri (\gamma_i + \gamma^*_i)}: \, , \label{KL1}$$ where $\{\gamma_i,\, \gamma^*_i \, :\, i=1,...,n\}$ generate the auxiliary algebra $$[\gamma_i\, ,\, \gamma_j] = [\gamma^*_i\, ,\, \gamma^*_j] = 0\, , \qquad [\gamma_i\, ,\, \gamma^*_j] = \ri \frac{\k}{2} \epsilon_{ij} \, , \label{KL2}$$ with $\epsilon_{ij}=-1$ for $i<j$, $\epsilon_{ii}=0$ and $\epsilon_{ij}=1$ for $i>j$. In order to fix the solution (\[psi1\],\[psi2\]) completely, one should determine the parameters $\sigma$, $\tau$ and $v$ in terms of coupling constants $g_\pm$ and the statistical parameter $\k$. Using a standard short distance expansion and (\[psi1\],\[psi2\]) one gets the charge and current densities[^5] $$\rho_\pm (t,x,i) \equiv (:\psi^*_1\psi_1: \pm :\psi^*_2\psi_2:)(t,x,i) = \frac{-1}{2\sqrt {\pi }\zeta_\pm } \left [(\der {\varphi_{i,R}})(vt-x) \pm (\der {\varphi_{i,L}})(vt+x)\right ] , \label{rhopm}$$ $$j_\pm (t,x,i) = \frac{v}{2\sqrt {\pi }v_F \zeta_\pm} \left [(\der {\varphi_{i,R}})(vt-x) \mp (\der {\varphi_{i,L}})(vt+x)\right ]\, , \label{jpm}$$ where for convenience the variables $$\zeta_\pm =\tau\pm\sigma \, . \label{zpm}$$ have been introduced. The normalization of (\[rhopm\]) is fixed [@Bellazzini:2008fu] by the Ward identities associated with the electric charge $Q_+$ and the helicity $Q_-$ defined by $$Q_\pm = \sum_{i=1}^n \int_0^\infty {{\rm d}}x\, \rho_\pm(t,x,i)\, . \label{charges}$$ The normalization of (\[jpm\]) in turn is determined by the conservation law $$\der_t \rho_\pm (t,x,i) - v_F\der_x j_\pm (t,x,i) = 0\, . \label{vcons}$$ Plugging (\[psi1\],\[psi2\],\[rhopm\]) in the quantum equations of motion $$\begin{aligned} \ri [\der_t +(-1)^\alpha v_F\der_x] \psi_\alpha (t,x,i)= \nonumber \qquad \qquad \qquad \qquad \\ 2 [g_+:\rho_+(t,x,i)\psi_\alpha : (t,x,i) -(-1)^\alpha g_- :\rho_-(t,x,i)\psi_\alpha :(t,x,i)]\, , \label{TLeqm}\end{aligned}$$ one finds $$\begin{aligned} v \zeta_+^2 &=& v_F \k +\frac{2}{\pi}g_+ \, , \label{sys2}\\ v \zeta_-^2 &=& v_F \k +\frac{2}{\pi}g_- \, . \label{sys3}\end{aligned}$$ Moreover, the exchange relation (\[an1\]) implies $$\zeta_+\, \zeta_- = \kappa\, . \label{sys1}$$ Eqs. (\[sys1\],\[sys2\],\[sys3\]) provide a system for determining $v$ and $\zeta_\pm$ (or equivalently $\sigma$ and $\tau$) in terms of $\k$ and $g_\pm$. The solution is $$\begin{aligned} \zeta_\pm^2 &=& \kappa \left(\frac{\pi \kappa v_F+2g_+}{\pi \kappa v_F+2g_-}\right)^{\pm 1/2}\, , \label{z}\\ v&=&\frac{\sqrt{(\pi \kappa v_F+2g_-)(\pi \kappa v_F+2g_+)}}{\pi \kappa}\, , \label{v}\end{aligned}$$ where the positive roots are taken in the right hand side. The relations (\[z\]) and (\[v\]) represent the anyon generalization [@Bellazzini:2008fu] of the well known result for canonical fermions $\k=1$, where an alternative and frequently used notation [@voit-95] is $$g_2=2(g_+-g_-)\, , \qquad g_4=2(g_+ + g_-)\, , \qquad K=\zeta_-^2=\zeta_+^{-2}\, . \label{altnot}$$ Considering the general anyon solution (\[z\], \[v\]), we assume in what follows that the parameters $\{\k,\, g_\pm\}$ belong to the domain $${\cal D} =\{\k>0,\; 2g_\pm> -\pi \kappa v_F\} \, , \label{physcond}$$ which ensures that $\sigma$, $\tau$ and $v$ are real and finite. Let us discuss finally the current splitting boundary condition (\[bc3\]) and establish the relation between ${\mbox{${\mathbb J}$}}$ and $\S$. Expressing the chiral currents $j_Z$ in terms of the chiral fields ${\varphi_{i,Z}}$, one finds $$\begin{aligned} j_R(t,x,i) &=& \frac{1}{2} (\zeta_-j_- + \zeta_+j_+)(t,x,i) = \frac{v}{2\sqrt {\pi }v_F} \der {\varphi_{i,R}}(vt-x) \, , \label{qcc1} \\ j_L(t,x,i) &=& \frac{1}{2} (\zeta_-j_- - \zeta_+j_+)(t,x,i)= \frac{v}{2\sqrt {\pi }v_F} \der {\varphi_{i,L}}(vt+x) \, , \label{qcc2}\end{aligned}$$ which, according to (\[bc1\]) satisfy the current splitting boundary condition (\[bc3\]), provided that $${\mbox{${\mathbb J}$}}= \S \in O(n) \, . \label{bc4}$$ The symmetry content of the TL junction is strongly influenced by (\[bc4\]). The point is that in the presence of a defect the continuity equation (\[vcons\]) alone is not enough to ensure the electric charge conservation. A direct computation shows indeed that $$\der_t Q_+ = \frac{v}{2\sqrt {\pi} \zeta_+} \sum_{k=1}^n \left (1-\sum_{i=1}^n \S_{ik} \right )(\der \varphi_{k,L})(vt)\, . \label{sym1}$$ The independence of ${\varphi_{i,L}}$ implies that the electric charge $Q_+$ is conserved if and only if $$\sum_{i=1}^n \S_{ik} = 1\, , \qquad \forall\; k=1,...,n\, , \label{K3}$$ which, as expected, is equivalent to the Kirchhoff rule $$\sum_{i=1}^n j_+ (t,0,i) = 0\, . \label{K2}$$ Since $\S \in O(n)$, one infers from (\[K3\]) that $\S^t$ satisfies (\[K3\]) as well. Therefore, the electric charge $Q_+$ is conserved for those $\S$, whose entries along each column (line) sum up to 1. In geometric terms, these scattering matrices belong to the stability subgroup $O_{\bf v}\subset O(n)$ of the $n$-vector ${\bf v} = (1,1,...,1)$. An explicit parametrization of $O_{\bf v}$ in terms of angular variables is given in [@Bellazzini:2009nk]. Summarizing, the condition $\S\in O(n)$ guaranties the energy conservation in the TL-junction. Concerning the electric charge $Q_+$, one must distinguish two different regimes. $Q_+$ is conserved for $\S\in O_{\bf v}$. If instead $\S$ belongs to the complement ${\widetilde{O}}_{\bf v} \equiv O(n)\setminus O_{\bf v}$, there is an external incoming or outgoing charge flow in the junction and $Q_+$ is not conserved. The possibility to describe such imperfect junctions is a remarkable feature of the current splitting boundary condition (\[bc3\]). The physical details about the charge transport in the junction are discussed in section 4.2 below. NESS representation and chemical potentials ------------------------------------------- The crucial property of the operator solution (\[psi1\],\[psi2\],\[z\],\[v\]) is that it is universal, meaning that it applies for any representation of the chiral field algebra generated by ${\varphi_{i,Z}}$. The Fock and Gibbs representations have been largely studied and describe the equilibrium properties of the TL model on $\Gamma$. In order to explore the behavior of the Luttinger liquid away from equilibrium, we investigate below the operator solution in the NESS representation of the RT algebra $\A$, constructed in section 2.2. The first step in this direction is the introduction of the [*fermion*]{} chemical potentials $$\mu_i = k_F -V_i \, , \label{fcp1}$$ where $k_F$ defines the Fermi energy for $\k=1$ and $V_i$ is the external voltage applied to the thermal reservoir in the edge $E_i$ of Fig. \[fig2\]. In what follows we keep $k_F$ fixed and vary eventually the gate voltages $V_i$. As already mentioned, the [*boson*]{} chemical potential $\mu_b<0$ has been introduced for avoiding some infrared singularities at the boson level and has nothing to do with $\mu_i$. In fact, in the chiral correlators (\[ll\]-\[rr\]) we already performed the limit $\mu_b \to 0^-$. In order to recover $\mu_i$, following [@Liguori:1999tw] we introduce the shift $\alpha_\mu$, defined by $${\varphi_{i,L}}(\xi) \longmapsto (\alpha_\mu {\varphi_{i,L}})(\xi ) = {\varphi_{i,L}}(\xi) -\frac{\xi}{\sqrt \pi\, \zeta_+}\, \mu_i \label{fcp2}$$ and, consistently with the boundary condition (\[bc1\]), $${\varphi_{i,R}}(\xi) \longmapsto (\alpha_\mu {\varphi_{i,R}})(\xi ) = {\varphi_{i,R}}(\xi) -\frac{\xi}{\sqrt \pi\, \zeta_+}\sum_{j=1}^n \S_{ij} \mu_j \, . \label{fcp3}$$ The transformations (\[fcp1\],\[fcp2\]) extend to an automorphism $\alpha_{\mu}$ on the whole algebra generated by the chiral fields ${\varphi_{i,Z}}$, which is directly implemented in the operator solution (\[psi1\],\[psi2\],\[rhopm\],\[jpm\]). At this stage the TL correlation functions in the NESS are defined by $$\langle \O_1[{\varphi}_{i_1,Z}] \cdots \O_k[{\varphi}_{i_k,Z}] \rangle_{\beta,\mu} = \langle \O_1[\alpha_\mu {\varphi}_{i_1,Z}] \cdots \O_k[\alpha_\mu {\varphi}_{i_k,Z}]\rangle_{\beta}\, . \label{gcf}$$ In the rest of the paper we focus on the correlation functions (\[gcf\]), which capture the physical properties of the Luttinger liquid with the current splitting boundary condition (\[bc3\]) away from equilibrium. We will show in particular that (\[gcf\]) satisfy the Kubo-Martin-Schwinger (KMS) condition [@BR; @H] at equilibrium, which justifies the introduction of the chemical potentials $\mu_i$ by means of (\[fcp2\],\[fcp3\]). Non-equilibrium TL correlation functions ======================================== Anyon correlators ----------------- We derive here the two-point correlators of $\psi_\alpha (t,x,i)$ defined by (\[psi1\], \[psi2\]) in the NESS and discuss their properties. For this purpose we extend away from equilibrium the finite temperature results of [@Liguori:1999tw]. Using (\[gcf2\]), for $\psi_1$ one finds $$\begin{aligned} \langle \psi_1^*(t_1,x_1,i)\psi_1(t_2,x_2,j)\rangle_{\beta, \mu} = A_{ij}\, B_{ij}(t_{1,2},x_{1,2};\mu) \times \qquad \qquad \qquad \qquad \nonumber \\ \left \{\frac{1}{\frac{\beta_i}{\pi} \sinh \left [\frac{\pi}{\beta_i}(vt_{12}+ {\widetilde{x}}_{12}) - \ri \varepsilon \right ]}\right \}^{\sigma \tau \S_{ij}^t} \left \{\frac{1}{\frac{\beta_j}{\pi} \sinh \left [\frac{\pi}{\beta_j}(vt_{12} - {\widetilde{x}}_{12}) - \ri \varepsilon \right ]}\right \}^{\sigma \tau \S_{ij}} \quad \; \; \nonumber \\ \left \{\frac{1}{\frac{\beta_i}{\pi} \sinh \left [\frac{\pi}{\beta_i}(vt_{12}+ x_{12}) - \ri \varepsilon \right ]}\right \}^{\tau^2 \delta_{ij}} \prod_{k=1}^n \left \{\frac{1}{\frac{\beta_k}{\pi} \sinh \left [\frac{\pi}{\beta_k}(vt_{12} - x_{12}) - \ri \varepsilon \right ]}\right \}^{\sigma^2 \S_{ik}\S_{kj}^t} \, , \label{anc1}\end{aligned}$$ where $$A_{ij} = \frac{\e^{\ri \pi^2 \kappa \epsilon_{ij}/2}}{2\pi} \left (\frac{1}{2\ri}\right )^{(\sigma^2 +\tau^2)\delta_{ij} +\sigma \tau (\S_{ij}+\S^t_{ij})} \, , \label{anc2}$$ $$B_{ij}(t_{1,2},x_{1,2};\mu) = \e^{\ri \{\tau [(vt_1+x_1)\mu_i-(vt_2+x_2)\mu_j]+ \sigma[(vt_1-x_1)\sum_{k=1}^n\S_{ik}\mu_k -(vt_2-x_2)\sum_{k=1}^n\S_{jk}\mu_k]\}/(\sigma+\tau)} \, . \label{anc4}$$ The $\psi_2$-correlator has the analogous form, $$\langle \psi_2^*(t_1,x_1,i)\psi_2(t_2,x_2,j)\rangle_{\beta, \mu} = (\ref{anc1})\quad {\rm with}\quad \sigma \leftrightarrow \tau \, . \label{anc22}$$ The TL junction involves two types of $\psi_1$-$\psi_2$ interactions. First, the Lagrangian (\[lagr\]) contains a $\psi_1$-$\psi_2$ [*bulk coupling*]{} proportional to $(g_+-g_-)$. Second, the current splitting boundary condition (\[bc3\]) provides an additional [*boundary interaction*]{} described by the mixed left-right correlators (\[lr\], \[rl\])). Consequently, the mixed $\psi_1$-$\psi_2$ correlators are non-trivial and have the form $$\begin{aligned} \langle \psi_1^*(t_1,x_1,i)\psi_2(t_2,x_2,j)\rangle_{\beta, \mu} = {\widetilde{A}}_{ij}\, {\widetilde{B}}_{ij}(t_{1,2},x_{1,2};\mu) \times \qquad \qquad \qquad \qquad \nonumber \\ \left \{\frac{1}{\frac{\beta_i}{\pi} \sinh \left [\frac{\pi}{\beta_i}(vt_{12}+ {\widetilde{x}}_{12}) - \ri \varepsilon \right ]}\right \}^{\tau^2 \S_{ij}^t} \left \{\frac{1}{\frac{\beta_j}{\pi} \sinh \left [\frac{\pi}{\beta_j}(vt_{12} - {\widetilde{x}}_{12}) - \ri \varepsilon \right ]}\right \}^{\sigma^2 \S_{ij}} \quad \; \; \nonumber \\ \left \{\frac{1}{\frac{\beta_i}{\pi} \sinh \left [\frac{\pi}{\beta_i}(vt_{12}+ x_{12}) - \ri \varepsilon \right ]}\right \}^{\sigma \tau \delta_{ij}} \prod_{k=1}^n \left \{\frac{1}{\frac{\beta_k}{\pi} \sinh \left [\frac{\pi}{\beta_k}(vt_{12} - x_{12}) - \ri \varepsilon \right ]}\right \}^{\sigma \tau \S_{ik}\S_{kj}^t} \, , \label{manc1}\end{aligned}$$ where $${\widetilde{A}}_{ij} = \frac{\e^{\ri \pi^2 \kappa \epsilon_{ij}/2}}{2\pi} \left (\frac{1}{2\ri}\right )^{\sigma^2 \S_{ij} +\tau^2 \S^t_{ij} +2\sigma \tau \delta_{ij}} \, , \label{manc2}$$ $${\widetilde{B}}_{ij}(t_{1,2},x_{1,2};\mu) = \e^{\ri \{\tau [(vt_1+x_1)\mu_i-(vt_2-x_2)\sum_{k=1}^n\S_{jk}\mu_k]+ \sigma[(vt_1-x_1)\sum_{k=1}^n\S_{ik}\mu_k-(vt_2+x_2)\mu_j ]\}/(\sigma+\tau)} \, . \label{manc4}$$ Finally, $$\langle \psi_2^*(t_1,x_1,i)\psi_1(t_2,x_2,j)\rangle_{\beta, \mu} = (\ref{manc1})\quad {\rm with}\quad \sigma \leftrightarrow \tau \, . \label{manc22}$$ As expected, in the equilibrium limit $\beta_i = \beta$ and $\mu_i=\mu$ for all $i$, the correlators (\[anc1\])–(\[manc22\]) simplify and satisfy the KMS condition, which represents a non-trivial check both on the computation and on the shift (\[fcp2\], \[fcp3\]) introducing the chemical potentials. Let us consider for instance (\[anc1\]), which in this limit takes the form $$\begin{aligned} \langle \psi_1^*(t_1,x_1,i)\psi_1(t_2,x_2,j)\rangle_{\beta, \mu} = A_{ij}\, \e^{\ri [\tau (vt_{12}+x_{12}) + \sigma (vt_{12}-x_{12})]\mu /(\sigma+\tau)}\times \qquad \qquad \nonumber \\ \left \{\frac{1}{\frac{\beta }{\pi} \sinh \left [\frac{\pi}{\beta }(vt_{12}+ {\widetilde{x}}_{12}) - \ri \varepsilon \right ]}\right \}^{\sigma \tau \S_{ij}^t} \left \{\frac{1}{\frac{\beta }{\pi} \sinh \left [\frac{\pi}{\beta }(vt_{12} - {\widetilde{x}}_{12}) - \ri \varepsilon \right ]}\right \}^{\sigma \tau \S_{ij}} \quad \; \; \nonumber \\ \left \{\frac{1}{\frac{\beta }{\pi} \sinh \left [\frac{\pi}{\beta }(vt_{12}+ x_{12}) - \ri \varepsilon \right ]}\right \}^{\tau^2 \delta_{ij}} \left \{\frac{1}{\frac{\beta }{\pi} \sinh \left [\frac{\pi}{\beta }(vt_{12} - x_{12}) - \ri \varepsilon \right ]}\right \}^{\sigma^2 \delta_{ij}} \, . \quad \label{anc5}\end{aligned}$$ Recalling that the KMS automorphism $\varrho_s$ acts on $\psi_\alpha$ as follows, $$\left [\varrho_s \psi_\alpha\right ] (t,x,i) = \e^{\ri s \mu}\, \psi_\alpha (t + s/v,x,i)\, , \label{anc6}$$ one can check that the equilibrium correlator (\[anc5\]) satisfies the KMS condition $$\langle \psi_1^*(t_1,x_1,i)\left [\varrho_{s+\ri \beta}\psi_1\right ](t_2,x_2,j)\rangle_{\beta, \mu} = \langle \left [\varrho_{s}\psi_1\right ](t_2,x_2,j)\psi_1^*(t_1,x_1,i)\rangle_{\beta, \mu} \label{anc7}$$ for all values of the statistical parameter $\k$. The critical scaling dimensions $d_i$ can be extracted from (\[anc1\])–(\[manc22\]) in the limit $\beta_i \to \infty$ and $\mu_i \to 0$. Because of the operator mixing, this is a subtle issue, which has been discussed in full detail in [@Bellazzini:2009nk]. One gets, $$d_i = \frac{1}{2}(\sigma^2 + \tau^2) + \sigma \tau s_i \, , \qquad i=1,...,n \, , \label{dimensions0}$$ where $s_i=\pm 1$ are the eigenvalues of $\S$. As already observed in [@Bellazzini:2006kh], the impact of the vertex interaction is captured by the term $\sigma \tau s_i$, which preserves unitarity in the sense of conformal field theory because $d_i \geq 0$. A remarkable special case is obtained by setting $g_+=g_-\equiv g$. In this case the bulk $\psi_1$-$\psi_2$ coupling vanishes and one is left only with the boundary interaction induced by the current splitting boundary condition (\[bc3\]). From (\[v\]) one gets $$v=v_F + \frac{2g}{\pi \kappa }\, , \label{nv}$$ and (using (\[sys1\]) with $\k > 0$ and $\tau \geq 0$) $$\sigma =0\, , \quad \tau = \sqrt {\kappa }\, . \label{nst}$$ Inserting (\[nst\]) in (\[anc1\])–(\[anc22\]) and localizing the fields in the same edge (i.e. setting $i=j$), one finds that the correlation functions simplify to $$\begin{aligned} C_{11}(vt_{12}+x_{12},i;\beta,\mu) \equiv \langle \psi_1^*(t_1,x_1,i)\psi_1(t_2,x_2,i)\rangle_{\beta, \mu} = \nonumber \\ \frac{1}{2\pi} \left (\frac{1}{2\ri}\right )^\k \e^{\ri (vt_{12}+ x_{12})\mu_i} \left \{\frac{1}{\frac{\beta_i}{\pi} \sinh \left [\frac{\pi}{\beta_i}(vt_{12}+ x_{12}) - \ri \varepsilon \right ]}\right \}^{\k}\, , \label{panc1}\end{aligned}$$ $$\begin{aligned} C_{22}(vt_{12}-x_{12},i;\beta,\mu) \equiv \langle \psi_2^*(t_1,x_1,i)\psi_2(t_2,x_2,i)\rangle_{\beta, \mu} = \qquad \quad \nonumber \\ \frac{1}{2\pi} \left (\frac{1}{2\ri}\right )^\k \prod_{k=1}^n \e^{\ri (vt_{12}- x_{12})\S_{ik}\mu_k} \left \{\frac{1}{\frac{\beta_k}{\pi} \sinh \left [\frac{\pi}{\beta_k}(vt_{12}- x_{12}) - \ri \varepsilon \right ]}\right \}^{\k \S_{ik}^2}. \label{panc2}\end{aligned}$$ The condition $g_+=g_-$ and the left-right asymmetry of the NESS construction in section 2 imply that only left moving (incoming) excitations contribute to $C_{11}$, which therefore coincides with the equilibrium correlator [@Liguori:1999tw]. All the non-equilibrium features are captured by $C_{22}$, which involves only right moving (outgoing) excitations. In fact, in spite of being localized in the edge $E_i$ of the graph, (\[panc2\]) depends on the temperatures and chemical potentials of the all $n$ edges. It is instructive for this reason to derive and compare the Fourier transforms of (\[panc1\], \[panc2\]). We will show first that they can be expressed in terms of the finite temperature TL [*anyon distribution*]{} discovered in [@Liguori:1999tw]. Consider in fact $${\widehat{C}}_{11}(E,p,i;\beta,\mu) \equiv \int_{-\infty}^\infty {{\rm d}}t \int_{-\infty}^\infty {{\rm d}}x\, \e^{-\ri (Evt +px)}\, C_{11}(vt +x,i;\beta,\mu)\, . \label{ad1}$$ Plugging (\[panc1\]) in (\[ad1\]) one gets $${\widehat{C}}_{11}(E,p,i;\beta,\mu) = \frac{\pi^\kappa}{v} \delta(E-p)\, d(p-\mu_i,\beta_i;\k) \, , \label{ad2}$$ where the $\delta$-function fixes the dispersion relation and $d$ is the equilibrium anyon momentum distribution [@Liguori:1999tw] $$\begin{aligned} d(p,\beta;\k) = \frac{\beta^{1-\k} \e^{-\beta p/2}}{2\pi}\, B \left (\frac{\k}{2} - \frac{\ri}{2\pi} \beta p \, , \frac{\k}{2} + \frac{\ri}{2\pi} \beta p\right ) = \qquad \qquad \qquad \nonumber \\ \frac{\beta^{1-\k} \e^{-\beta p/2}}{2\pi \Gamma (\k)}\, \Gamma \left (\frac{\k}{2} - \frac{\ri}{2\pi} \beta p\right ) \Gamma \left (\frac{\k}{2} + \frac{\ri}{2\pi} \beta p\right ) \, , \qquad \k>0\, , \label{ad3}\end{aligned}$$ $B$ and $\Gamma$ being the beta and gamma functions (Euler’s integrals of first and second kind respectively). Notice that for $\k\not=1$ the distribution (\[ad3\]) depends on both $\beta$ and $p$ and not only on the dimensionless combination $\beta p$. Eq. (\[ad3\]) defines a smooth function of $p\in {\mbox{${\mathbb R}$}}$, which satisfies $$\lim_{p\to 0}d(p,\beta;\k) = \frac{\beta^{1-\k} \Gamma^2(\k/2)}{2\pi \Gamma (\k)}\, , \label{as0}$$ and has the following asymptotic behavior: $$\begin{aligned} \lim_{p\to \infty}d(p,\beta;\k) &=& 0\, , \quad \forall\; \k>0\, , \label{as1}\\ \lim_{p\to -\infty}d(p,\beta;\k) &=& \begin{cases} 0\, , \quad &0<\k<1\, ,\\ 1\, , \quad &\k=1\, ,\\ \infty \, , \quad &\k >1\, . \end{cases} \label{as2}\end{aligned}$$ For positive integer $\k$ (i.e. for fermions and bosons) the distribution (\[ad3\]) simplifies to $$d(p,\beta;\k) = \begin{cases} \frac{1}{(1+\e^{\beta p})} \frac{\beta^{-2(n-1)}}{[2(n-1)]!} \prod_{j=1}^{n-1} \bigl |j - \frac{1}{2} - \frac{\ri}{2\pi} p\beta \bigr |^2\, , & \k = 2n-1 \, , \\ \\ \frac{1}{(\e^{\beta p}-1)} \frac{2\pi \beta^{-2n} }{(2n-1)! p} \prod_{j=0}^{n-1} \bigl |j- \frac{\ri}{2\pi} p\beta \bigr |^2\, , & \k = 2n\, , \end{cases} \label{intad}$$ where $n=1,2,...$ and the familiar Fermi and Bose distributions appear as prefactors. The first two fermion and boson distributions are $$\begin{aligned} d(p, \beta;1) = \frac{1}{(1+\e^{\beta p})}\, , \qquad \qquad d(p, \beta; 3) = \frac{1}{(1+\e^{\beta p})} \frac{(\pi^2+p^2\beta^2)}{8\pi^2\beta^2}\, , \label{ad4f} \\ d(p, \beta; 2) = \frac{1}{(\e^{\beta p}-1)} \frac{p}{2\pi}\, , \qquad d(p, \beta; 4) = \frac{1}{(\e^{\beta p}-1)} \frac{p(4\pi^2+p^2\beta^2)}{48\pi^3\beta^2}\, . \label{ad4b}\end{aligned}$$ As expected, in the fermion point $\k=1$ of the TL liquid one gets the familiar Fermi distribution. In spite of the fact that the remaining boson and fermion points ($\k=2,3,...$) have been established in [@Liguori:1999tw; @Ilieva:2000cj] more then a decade ago, to our knowledge their physical meaning and potential applications of (\[intad\]) have not been fully explored. (500,110)(-120,20) ![The distribution $d$ at fixed temperature $\beta=1$ for $\k=1/2$ (dotted blue line) and $\k=3/2$ (dashed black line) compared to the Fermi distribution $\k=1$ (continuous red curve).[]{data-label="fig3"}](fig3.pdf "fig:") (500,110)(-120,20) ![The distribution $d$ at fixed $\k=1/4$ for different temperatures $\beta =0.2$ (continuos red line), $\beta=0.4$ (dashed black line) and $\beta=0.8$ (dotted blue line).[]{data-label="fig4"}](fig4.pdf "fig:") In order to give an idea about the anyon distributions in the interval $0<\k< 1$, we show some of them in Fig.\[fig3\], where the standard Fermi distribution (continuous red curve) is given for comparison. Fig.\[fig4\] displays the behavior of the anyon distribution (\[ad3\]) for fixed $\k=1/4$ and different temperatures. For $0<\k<1$ and with decreasing of the temperature $T\sim 1/\beta$ one observes the formation of a sharp peak at $p=0$ (in agreement with eq. (\[as0\])), which signals a condensation-like phenomenon [@Liguori:1999tw]. Concerning the Fourier transform of (\[panc2\]), it is useful to consider first the case when all the temperatures are equal ($\beta_i = \beta$), the system being driven away from equilibrium only by the voltages $V_i$. In this case $$\begin{aligned} C_{22}(vt_{12}-x_{12},i;\beta,\mu) \equiv \langle \psi_2^*(t_1,x_1,i)\psi_2(t_2,x_2,i)\rangle_{\beta, \mu} = \qquad \quad \nonumber \\ \frac{1}{2\pi} \left (\frac{1}{2\ri}\right )^\k \e^{\ri (vt_{12}- x_{12})\sum_{k=1}^n\S_{ik}\mu_k} \left \{\frac{1}{\frac{\beta}{\pi} \sinh \left [\frac{\pi}{\beta }(vt_{12}- x_{12}) - \ri \varepsilon \right ]}\right \}^{\k}. \label{panc22}\end{aligned}$$ and therefore $${\widehat{C}}_{22}(E,p,i:\beta,\mu) = \frac{\pi^\kappa}{v} \delta(E-p)\, d(p-\sum_{k=1}^n\S_{ik}\mu_k,\beta;\k)\, . \label{ad22}$$ One has still the equilibrium distribution, with the energy shifted by a linear combination of the chemical potentials $\mu_k$, whose coefficients are the $\S$-matrix elements. Finally, in the coordinate space the general expression (\[panc2\]) is a product of $C_{11}$-factors with different temperatures and chemical potentials. One gets therefore in momentum space the nested convolution formula $$\begin{aligned} {\widehat{C}}_{22}(E,p,i:\beta,\mu) = \frac{\pi^\kappa}{v} \delta(E-p)\, \int_{-\infty}^\infty \frac{{{\rm d}}k_1}{2\pi} \int_{-\infty}^\infty \frac{{{\rm d}}k_2}{2\pi} \cdots \int_{-\infty}^\infty \frac{{{\rm d}}k_{n-1}}{2\pi} \times \qquad \nonumber \\ d(k_1-\S_{i1}\mu_1,\beta_1;\k\S^2_{i1})d(k_2-\S_{i2}\mu_2-k_1,\beta_2;\k\S^2_{i2}) \cdots d(p-\S_{in}\mu_n-k_{n-1},\beta_n;\k\S^2_{in})\, . \nonumber \\ \label{ad223}\end{aligned}$$ Being a convolution of distributions, (\[ad223\]) is also a well defined distribution. The NESS $\Omega_{\beta, \mu}$ has therefore a remarkable property: the associated [*non-equilibrium distribution*]{} is simply a [*convolution of equilibrium distributions*]{} with different temperatures and chemical potentials. Since the general form of (\[ad223\]) is quite complicated, it is instructive to consider below the case $n=2$ and $\mu_1=\mu_2=0$, focusing on $$D_2(p;\beta_1,\beta_2;\k,\theta) = \int_{-\infty}^\infty \frac{{{\rm d}}k}{2\pi}\, d(k,\beta_1;\k \cos^2 \theta )d(p-k,\beta_2;\k \sin^2 \theta)\, , \qquad \theta \in [0,\pi)\, . \label{b1}$$ Using the $x$-space representation, at equal temperatures one finds the relation $$D_2(p;\beta,\beta;\k,\theta) = d(k,\beta ;\k)\, , \qquad \forall\; \theta \in [0,\pi)\, . \label{b2}$$ For $\beta_1 \not=\beta_2$ the convolution $D_2$ defines a new distribution. Since we were not able to determine its explicit analytic form, we give some plots which are obtained numerically. The plots in Figs. \[fig5\] and \[fig6\] illustrate the behavior of $D_2$ for different values of $\beta_{1,2}$, $\k$ and $\theta$. We see that even for $\beta_1 \not=\beta_2$ the distribution $D_2$ is similar to $d$ with a kind of “effective" temperature and statistical parameter depending on $\beta_{1,2}$, $\k$ and $\theta$. (500,110)(-120,20) ![The distribution $D_2$ at fixed $\k=1/2$ and $\theta =\pi/4$ for different temperatures $(\beta_1,\beta_2) =(1/2,1)$ (continuos red line), $(\beta_1,\beta_2) =(1,2)$ (dashed black line) and $(\beta_1,\beta_2) =(2,4)$ (dotted blue line).[]{data-label="fig5"}](fig5.pdf "fig:") (500,110)(-120,20) ![The distribution $D_2$ at fixed $\beta_1=1$, $\beta_2=6$ and $\theta=\pi/6$ for statistical parameters $\k =1/3$ (continuos red line), $\k=2/3$ (dashed black line) and $\k=4/3$ (dotted blue line).[]{data-label="fig6"}](fig6.pdf "fig:") Summarizing, we derived above the two-point TL anyon correlation functions away from equilibrium. The results (\[anc1\],\[manc1\]) are expressed as products of $\S$-dependent powers of equilibrium correlators at different temperatures. In agreement with this fact the momentum space anyon NESS distribution is the convolution of equilibrium anyon distributions (\[ad223\]). The equilibrium limit satisfies the KMS conditions. At criticality one is dealing with a $c=1$ conformal field theory, whose anomalous dimensions (\[dimensions0\]) depend not only on the coupling constants $g_\pm$, but also on the scattering matrix $\S$. The above technique allows to compute higher anyon correlation functions as well, but in order to investigate the transport properties of the system, we concentrate below on the electric and energy current correlators away from equilibrium. Charge and heat transport ------------------------- The charge transport in the NESS is described by $$\begin{aligned} \langle \der_t Q_+ \rangle_{\beta, \mu} &=& \frac{v}{2\pi \zeta^2_+} \sum_{i,j=1}^n (\S_{ij} - \delta_{ij}) \mu_j\, , \label{ct00}\\ \langle j_+(t,x,i)\rangle_{\beta, \mu} &=& \frac{v}{2\pi v_F \zeta^2_+} \sum_{j=1}^n (\delta_{ij}-\S_{ij})\mu_j \, , \label{ct01}\end{aligned}$$ which follow by substituting (\[jpm\],\[sym1\]) in (\[gcf\]). Eq. (\[ct00\]) describes the external charge flow in the junction: it is constant in time and is incoming for $\langle \der_t Q_+ \rangle_{\beta, \mu} >0$ and outgoing for $\langle \der_t Q_+ \rangle_{\beta, \mu} <0$. Eq. (\[ct01\]) determines instead the value of the currents along the leads. The charge balance $$\langle \der_t Q_+ \rangle_{\beta, \mu} + v_F \sum_{i=1}^n\langle j_+(t,x,i)\rangle_{\beta, \mu} = 0 \label{ct02}$$ is satisfied and represents an useful check. If $\S\in O_{\bf v}$, the electric charge is conserved $\langle \der_t Q_+ \rangle_{\beta, \mu} =0$ and the $k_F$-dependence in (\[ct01\]) drops out, leading to $$\langle j_+(t,x,i)\rangle_{\beta, \mu} = \frac{v}{2\pi v_F \zeta^2_+} \sum_{j=1}^n (\S_{ij}-\delta_{ij})V_j \, . \label{ct1}$$ The current (\[ct1\]) satisfies the Kirchhoff rule (\[K2\]) and vanishes at equilibrium ($V_i=V$ for all $i$) as it should be. The dependence on the statistical parameter $\k$ is explicit and deserves a comment. In the physical domain $\cal D$, defined by (\[physcond\]), the overall coefficient in front of the sum in (\[ct1\]) is positive, $$G(g_-,\k)\equiv \frac{v}{2\pi v_F\zeta_+^2} = \frac{\pi \kappa v_F + 2g_-}{2 \pi^2 \kappa^2 v_F} >0\, . \label{ct3}$$ For $g_->0$ the coefficient $G$ decreases monotonically with $\k>0$. For $g_-<0$ one has that $\k>-2g_-/\pi v_F$ in the physical domain $\cal D$. In this case the coefficient $G$ increases in the interval $-2g_-/\pi v_F<\k < -4g_-/\pi v_F$, reaching the maximal value $-v_F/16g_-$ and decreases for $\k>-4g_-/\pi v_f$. This behavior is illustrated in Fig. \[fig7\]. (500,110)(-120,20) ![Behavior of $G$ at $g_-=1$ (dashed black line) and $g_-=-1$ (continuous red line).[]{data-label="fig7"}](fig7.pdf "fig:") The current (\[ct1\]) is proportional to the applied external voltages. Non-linear effects are absent in the critical regime under consideration, which implies the conductance tensor $$\G_{ij} =G(g_-,\k)\sum_{j=1}^n (\S_{ij}-\delta_{ij})\, . \label{ct2}$$ We see that the NESS approach, adopted in this paper, confirms the result for $\G$, obtained previously for $\k=1$ by different methods, including renormalization group techniques [@lrs-02; @emabms-05], linear response theory [@Bellazzini:2006kh; @Bellazzini:2008mn] and conformal field theory [@coa-03; @hc-08]. The novelty in (\[ct2\]) is the explicit dependence on the statistical parameter $\k$, shown in Fig. \[fig7\]. A similar computation gives the energy (heat) flow $$\langle \theta_{xt}(t,x,i)\rangle_{\beta,\mu} = \frac{v^2}{8\pi \zeta_+^2}\left [ \mu_i^2 - \left (\sum_{j=1}^n \S_{ij}\mu_j \right )^2\right ] + \frac{\pi v^2}{12}\sum_{j=1}^n \left (\delta_{ij} - \S_{ij}^2 \right )\frac{1}{\beta_j^2}\, , \label{ct04}$$ which satisfies the Kirchhoff rule (\[K1\]) for all $\S \in O(n)$. For $\S\in O_{\bf v}$ the expression (\[ct04\]) takes the form $$\begin{aligned} \langle \theta_{xt}(t,x,i)\rangle_{\beta,\mu} = \qquad \qquad \qquad \qquad \qquad \qquad \nonumber \\ \frac{v^2}{8\pi \zeta_+^2}\left [ V_i^2 - 2k_F \sum_{j=1}^n (\delta_{ij}-\S_{ij})V_j - \left (\sum_{j=1}^n \S_{ij}V_j \right )^2\right ] + \frac{\pi v^2}{12}\sum_{j=1}^n \left (\delta_{ij} - \S_{ij}^2 \right )\frac{1}{\beta_j^2}\, . \label{ct4}\end{aligned}$$ The heat flow depends therefore not only on $k_F$ and the voltages $V_i$, but also on the temperatures $\beta_i$. As before, we consider for illustration the case of $n=2$ wires. Inserting the scattering matrices (\[ex1\]) in (\[ct01\]) one has $$\langle \der_t Q_+ \rangle_{\beta, \mu} = \begin{cases} Gv_F[(\mu_1 +\mu_2)(\cos \theta-1)-(\mu_1-\mu_2)\sin \theta ]\, , & \; \; {\rm det}\, \S =1\, , \\ Gv_F[(\mu_1 +\mu_2)(\sin \theta -1)+(\mu_1-\mu_2)\cos \theta ]\, , & \; \; {\rm det}\, \S =-1\, . \\ \end{cases} \label{ex2}$$ Therefore, $$\langle \der_t Q_+ \rangle_{\beta, \mu} = 0 \Longrightarrow \begin{cases} \theta =0 \quad \Longrightarrow j_+(t,x,1) = j_+(t,x,2) =0 \, , & \; \; {\rm det}\, \S =1\, , \\ \theta =\pi/2 \Longrightarrow j_+(t,x,1) = -j_+(t,x,2)=G(\mu_1-\mu_2)\, , & \; \; {\rm det}\, \S =-1\, , \\ \end{cases} \label{ex3}$$ corresponding respectively to full reflection (disconnected edges) and complete transmission in the junction. Finally, $$\begin{aligned} \langle \theta_{xt}(t,x,1)\rangle_{\beta,\mu} = -\langle \theta_{xt}(t,x,2)\rangle_{\beta,\mu}= \qquad \qquad \qquad \nonumber \\ \frac{v^2}{8\pi \zeta_+^2}[(\mu_1^2-\mu_2^2)\sin^2 \theta -\mu_1\mu_2 \sin 2\theta ] + \frac{\pi v^2\sin^2 \theta}{12}\left (\frac{1}{\beta_1^2}-\frac{1}{\beta_2^2}\right ) \, , \label{ex4}\end{aligned}$$ for both families in (\[ex1\]). Quantum noise ------------- In this section we derive the [*noise power*]{} in the TL junction in Fig. \[fig1\]. For this purpose we need [@bb-00] the two-point [*connected*]{} current-current correlator $$\begin{aligned} \langle j_+(t_1,x_1,i) j_+(t_2,x_2,j) \rangle_{\beta, \mu}^{\rm conn} \equiv \qquad \qquad \qquad \qquad \nonumber \\ \langle j_+(t_1,x_1,i) j_+(t_2,x_2,j) \rangle_{\beta, \mu} - \langle j_+(t_1,x_1,i)\rangle_{\beta, \mu}\langle j_+(t_2,x_2,j) \rangle_{\beta, \mu}\, . \label{N1}\end{aligned}$$ After some algebra one finds $$\begin{aligned} \langle j_+(t_1,x_1,i) j_+(t_2,x_2,j) \rangle_{\beta, \mu}^{\rm conn} \equiv \qquad \qquad \qquad \qquad \nonumber \\ \left (\frac{v}{2 v_f \zeta_+}\right )^2 \Biggl \{\frac{1}{\beta_i^2 \sinh^2\left [\frac{\pi}{\beta_i}(vt_{12}+ x_{12}) -\ri \varepsilon \right ]} \delta_{ij} + \sum_{l=1}^n \S_{il} \frac{1}{\beta_l^2 \sinh^2\left [\frac{\pi}{\beta_l}(vt_{12}- x_{12}) - \ri \varepsilon \right ]} \S_{lj}^t \quad \nonumber \\ -\S_{ij} \frac{1}{\beta_j^2 \sinh^2\left [\frac{\pi}{\beta_j}(vt_{12}-{\widetilde{x}}_{12}) -\ri \varepsilon \right ]} - \frac{1}{\beta_i^2 \sinh^2\left [\frac{\pi}{\beta_i} (vt_{12}+{\widetilde{x}}_{12}) -\ri \varepsilon \right ]} \S^t_{ij} \Biggr \} \, . \quad \label{N2}\end{aligned}$$ One easily verifies that the equilibrium limit ($\beta_i \to \beta$ for all $i$) of (\[N2\]) satisfies the KMS condition $$\langle j_+(t_1,x_1,i) [\varrho_{s+\ri \beta} j_+](t_2,x_2,j) \rangle_{\beta, \mu}^{\rm conn} = \langle [\varrho_{s} j_+](t_2,x_2,j) j_+(t_1,x_1,i) \rangle_{\beta, \mu}^{\rm conn}\, , \label{currKMS1}$$ where the KMS automorphism $\varrho$ acts on $j_+$ as follows, $$[\varrho_s j_+](t,x,i) = j_+(t+s/v,x,i) \, . \label{currKMS2}$$ The explicit expression (\[N2\]) contains fundamental physical information about the NESS. First of all, since[^6] $$\langle j_+(t_1,x_1,i_1) j_+(t_2,x_2,i_2)\rangle_{\beta,\mu}^{\rm conn} \not = \\ \overline {\langle j_+(-t_1,x_1,i_1) j_+(-t_2,x_2,i_2)\rangle}{}_{\beta,\mu}^{\rm conn} \label{cc2}$$ the NESS breaks down [*time reversal*]{} invariance, even if the junction interaction preserves it, i.e. if $\S=\S^t$ [@Bellazzini:2009nk]. Nevertheless, [*time translation*]{} invariance is preserved, which allows one to use the conventional definition [@bb-00] of [*noise power*]{} $$P_{ij}(\beta; x_1, x_2 ; \omega) \equiv \int_{-\infty}^\infty {{\rm d}}t\, \e^{i \omega t} \, \langle j_x(t,x_1,i) j_x(0,x_2,j) \rangle_{\beta, \mu}^{\rm conn}\, . \label{N3}$$ Eq. (\[N3\]) defines a complex matrix whose entries can be expressed [@PBM] in terms of the hypergeometric function ${}_2F_1$, namely $$\begin{aligned} P_{ij}(\beta; x_1, x_2 ; \omega) = \left (\frac{v}{2 v_F \zeta_+}\right )^2 \bigl \{ \bigl [F_-(\omega, \beta_i, x_{12}) - F_+(\omega, \beta_i,x_{12})\bigr ] \beta_i^{-1} \delta_{ij} + \qquad \qquad \nonumber \\ \sum_{l=1}^n\S_{il}\beta_l^{-1}\bigl [F_-(\omega, \beta_l, -x_{12}) - F_+(\omega, \beta_l, -x_{12})\bigr ]\S^t_{lj} - \qquad \qquad \qquad \qquad \nonumber \\ \S_{ij}\beta_j^{-1}\bigl [F_-(\omega, \beta_j,-{\widetilde{x}}_{12}) - F_+(\omega, \beta_j,-{\widetilde{x}}_{12})\bigr ] - \bigl [F_-(\omega, \beta_i, {\widetilde{x}}_{12}) - F_+(\omega, \beta_i,{\widetilde{x}}_{12})\bigr ] \beta_i^{-1} \S^t_{ij} \bigl \}\, , \nonumber \\ \label{A1}\end{aligned}$$ with $$F_\pm (\omega, \beta, x) = \frac{\e^{\pm 2\pi x/\beta }}{\ri \omega \beta \pm 2\pi v}\; {}_2F_1 \left (2, 1\pm \frac{\ri \omega \beta}{2 \pi v}, 2 \pm \frac{\ri \omega \beta}{2 \pi v}, \e^{\pm 2\pi x/\beta }\right )\, . \label{A2}$$ From (\[A1\], \[A2\]) one can deduce the zero-frequency limit ([*zero-frequency noise power*]{}) $$P_{ij}(\beta) \equiv \lim_{\omega \to 0^+} P_{ij}(\beta; x_1, x_2 ; \omega) \, . \label{N4}$$ Using $$\lim_{\omega \to 0^+} \left [F_- (\omega; \beta, x) - F_+ (\omega; \beta, x)\right ] = \frac{1}{2\pi v}\, , \label{N5}$$ one gets $$P_{ij}(\beta) = \frac{G(g_-,\k)}{v_F} \left (\beta_i^{-1}\delta_{ij} - \S_{ij}\beta_j^{-1} - \beta_i^{-1} \S^t_{ij} + \sum_{l=1}^n \S_{il}\beta_l^{-1} \S^t_{lj} \right )\, , \label{N6}$$ where $G$, defined by (\[ct3\]), captures the dependence (see Fig.\[fig7\]) of the noise on the statistical parameter $\k$. As expected, $P_{ij}(\beta)$ turns out to be a $x_{1,2}$-independent real symmetric matrix. If the electric charge is conserved ($\S \in O_{\bf v}$), the noise power (\[N6\]) satisfies in addition the Kirchhoff rule $$\sum_{i=1}^n P_{ij}(\beta ) = \sum_{j=1}^n P_{ij}(\beta ) = 0\, . \label{N7}$$ The expression (\[N6\]) admits the typical Johnson-Nyquist $\beta^{-1}$ behavior and shows the non-trivial interplay between the different temperatures and the scattering matrix. For example, in the two-terminal case with $\S=\S^+$ one finds $$P^+ = \frac{G}{v_Fk_B}\left(\begin{array}{cc}T_1(1-\cos \theta)^2 +T_2 \sin^2 \theta & (T_1-T_2)(1-\cos \theta)\sin \theta \\ (T_1-T_2)(1-\cos \theta)\sin \theta & T_2(1-\cos \theta)^2 + T_1\sin^2 \theta \\ \end{array} \right)\, , \label{N8}$$ where $T=(k_B\beta)^{-1}$ is the absolute temperature and $k_B$ is the Boltzmann constant. The eigenvalues of $P^+$ $$p^+_i= \frac{2G}{v_Fk_B}(1-\cos \theta)T_i\, \geq 0\, , \qquad i=1,2 \label{N9}$$ are nonnegative in agreement with the positivity of the two-point function (\[N1\]). Analogous result holds for $P^-$ corresponding to $\S^-$. Outlook and conclusions ======================= In this paper we constructed and investigated an exactly solvable model of a non-equilibrium Luttinger junction. The basic points of our approach are: (i) : a scale invariant point-like interaction, which is described by a scattering matrix $\S$ and drives the system away from equilibrium; (ii) : a representation generated by a NESS $\Omega_{\beta, \mu}$, which encodes the point-like interaction in the chiral fields ${\varphi_{i,Z}}$; (iii) : an exact operator solution of the TL model (in terms of ${\varphi_{i,Z}}$) on a star graph with the current splitting boundary condition in the vertex; (iv) : an extension of the conventional fermion Luttinger liquid to anyon statistics. Combining these ingredients, we derived the basic correlation functions in the state $\Omega_{\beta, \mu}$. The essential characteristic features of these functions are: (a) : the non-equilibrium two-point anyon correlations are products of $\S$-dependent powers of equilibrium correlations at the temperatures and chemical potentials of the heat baths, connected to the leads; (b) : accordingly, the corresponding momentum space distribution is the convolution of equilibrium anyon distributions at different temperatures and chemical potentials; (c) : the Fourier transform of the leading terms in the large distance expansion of the anyon correlations gives Cauchy-Lorentz distributions, which after convolution reproduce themselves with appropriate width and median; (d) : in the critical limit one has a $c=1$ conformal field theory with $\S$-dependent anomalous dimensions, which are explicitly derived; (e) : the expected breakdown of time reversal invariance is manifest in the current-current correlator. We investigated in detail the energy and charge transport in the junction for all values of the statistical parameter. The energy is conserved for $\S \in O(n)$, which covers both possibilities of a junction without and with electric charge dissipation. In the latter case we determined the exact expression for the charge flow leaving or entering the junction. The connected current-current correlation is a linear combination of hypergeometric functions. The associated zero-frequency noise power depends linearly on the temperatures. Our investigation above has been focused essentially on the critical properties of anyon Luttinger liquids away from equilibrium. It will be interesting to study the noncritical aspects as well. The generalization of the results of this paper beyond the Luttinger liquid paradigm, when the nonlinearity of the dispersion relation becomes essential, is also a challenging open problem. We thank B. Douçot and I. Safi for an inspiring discussion, which stimulated our interest in non-equilibrium Luttinger liquids. M.M. would like also to thank the Laboratoire de Physique Théorique d’Annecy-le-Vieux for the kind hospitality during the preparation of the manuscript. Large space separation asymptotics ================================== The behavior of the correlators (\[panc1\],\[panc2\]) at large large space separation $|x_{12}| \gg \beta_i$ is encoded in $$\begin{aligned} C_{11}(x,i;\beta,\mu) &=& \left (\frac{1}{2\ri}\right )^\k \left (\frac{2\pi}{\beta_i}\right )^\k L\left (x;\frac{\pi \k}{\beta_i},-\mu_i\right ) + \cdots \, , \label{ld1} \\ C_{22}(x,i;\beta,\mu) &=& \left (\frac{1}{2\ri}\right )^\k \prod_{k=1}^n \left (\frac{2\pi}{\beta_k}\right )^\k L\left (x;\frac{\pi \k\S^2_{ik}}{\beta_k},-\S_{ik}\mu_k\right ) +\cdots \, , \label{ld2}\end{aligned}$$ where $x\gg \beta_i >0$, the dots stand for sub-leading contributions and $$L(x;\gamma,\mu) \equiv \frac{1}{2\pi} \e^{-\ri \mu x - \gamma x}\, , \quad x>0\, . \label{ld3}$$ The Fourier transform $${\widehat{L}}(p;\gamma,\mu) \equiv \int_{-\infty}^\infty {{\rm d}}x\, \e^{\ri px} L(x;\gamma,\mu) = \frac{1}{\pi} \frac{\gamma}{\gamma^2 + (p-\mu)^2} \label{ld4}$$ is the familiar Cauchy-Lorentz distribution[^7], where $\gamma$ is the half width at half maximum and $\mu$ is the statistical median. 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[^4]: The $\ri \varepsilon$ prescription, adopted throughout the paper, indicates as usual the weak limit $\varepsilon \to 0^+$. [^5]: Without loss of generality we assume in what follows $\tau \geq 0$ and $\tau \not= \pm \sigma$. [^6]: The bar indicates complex conjugation. [^7]: Known also as non-relativistic Breit-Wigner distribution.

TIT/HEP–499\ [hep-th/0307206]{}\ July, 2003\ [\ ]{}\ [ **Minoru Eto $^{a}$**]{} [^1], [ **Nobuhito Maru $^{b}$**]{} [^2]  and   [ **Norisuke Sakai $^{a}$**]{} [^3] 1.5em [ *$^{a}$Department of Physics, Tokyo Institute of Technology\ Tokyo 152-8551, JAPAN\ and\ $^{b}$Theoretical Physics Laboratory\ RIKEN (The Institute of Physical and Chemical Research)\ 2-1 Hirosawa, Wako, Saitama 351-0198, JAPAN* ]{} [**Abstract**]{}\ Introduction ============ In the brane-world scenario [@LED; @RS1; @RS2], our four-dimensional world is to be realized on topological defects such as walls. To obtain realistic unified theories beyond the standard model, supersymmetry (SUSY) has been most useful [@DGSW]. Moreover, SUSY helps to construct topological defects like walls as BPS states [@WittenOlive] that preserve part of SUSY. For a realistic model, understanding SUSY breaking has been an important problem, which is addressed in the SUSY brane-world scenario extensively [@BULK]–[@MSSS]. Models have been constructed that realize one such idea : coexistence of BPS and anti-BPS walls produces SUSY breaking automatically [@MSSS]. In particular, the SUSY breaking effects are suppressed exponentially as a function of distance between walls. On the other hand, non-BPS multi-wall configurations are not protected by SUSY and need not be stable. Such non-BPS wall configurations was successfully stabilized by introducing topological quantum numbers, such as a winding number [@MSSS2; @SakaiSugisaka]. The physical reason behind the stability is simple : a BPS wall and an anti-BPS wall with winding numbers generally exert repulsion, which then pushes each other at anti-podal points of the compactified dimension. One of the most attractive models in the brane-world scenario is the model with the warped metric [@RS1; @RS2]. A possible solution of the gauge hierarchy problem was proposed in the two brane-model [@RS1], and a localization of graviton on a single brane was found even in a noncompact space [@RS2] at the cost of fine-tuning between bulk cosmological constant and boundary cosmological constant at orbifold fixed points. Supersymmetrization of the thin-wall model has also been constructed in five dimensions [@ABN]–[@FLP]. It is natural to ask if the infinitely thin branes in these models can be replaced by physical smooth wall configurations made out of scalar fields [@CGR]–[@SkTo]. We have succeeded in constructing BPS as well as non-BPS solutions in the ${\cal N}=1$ supergravity coupled with a chiral scalar multiplet in four dimensions [@EMSS]. A similar BPS solution has also been constructed in five-dimensional supergravity [@AFNS; @Eto:2003ut]. In the limit of vanishing gravitational coupling $\kappa \rightarrow 0$, our model reduces to the model having the exact solution of non-BPS multi-walls [@MSSS2]. Therefore the model is likely to be stable thanks to the winding number near the weak gravity limit. However, we need to address the issue of stability in the presence of gravity, since the radius of the extra dimension is now a dynamical variable which might introduce instability into the model. There have been a number of works to analyze the stability of the infinitely thin wall [@GiLa]–[@CsabaCsaki], especially in the presence of a stabilizing mechanism due to Goldberger and Wise [@GoWi]. The purpose of our paper is to study the stability of the model with winding number in the presence of gravity and to analyze the mass spectrum of fluctuations on the BPS and non-BPS solutions. We find that there are zero modes of transverse traceless fluctuations localized on the wall which play the role of the graviton in our world on the wall. The BPS solution has also gravitino zero mode which is localized on the wall and forms a supermultiplet with the graviton under the surviving supergravity transformation with the Killing spinor of the BPS solution. We obtain that the BPS solution has no other zero modes, and no tachyonic fluctuations. For instance, we find that possible additional massless tensor and scalar modes are either gauge degrees of freedom or unphysical (the mode function is not normalizable). As for the non-BPS solution, we find that another possible zero modes of the transverse traceless fluctuations of metric can be gauged away and that there exists no zero mode other than the graviton localized on the wall. To obtain a concrete estimate of the mass spectrum, we need to use approximations. We use small width approximation where the width $\Lambda^{-1}$ of the wall is small compared to the radius $R$ of compactified extra dimension. We find that the non-BPS solution has no tachyonic fluctuations in spite of the dynamical role played by the radius of the compactified dimension. Tensor as well as scalar fluctuations have massive modes, without any tachyons. This result shows that our non-BPS solution is stable without introducing an additional stabilizing mechanism such as the Goldberger-Wise mechanism [@GoWi]. The lightest massive scalar mode is usually called radion. We can evaluate the mass of the radion on our non-BPS background at least for $R \gg \Lambda^{-1}$, where $R$ is the radius of the compactified dimension and $\Lambda^{-1}$ is the width of the wall. We find that the mass squared of the radion is given by $$m^2_0 \propto \Lambda^2 e^{-\pi R \Lambda} \label{eq:radion-mass1}$$ It is interesting to note that the mass scale is given by the inverse wall width $\Lambda$, and that it becomes exponentially light as a function of the distance $\pi R$ between the two walls. This behavior is precisely the same as the previous model in the global SUSY case [@MSSS2]. Modes of fermions including gravitino are also analyzed. We find that the Nambu-Goldstone modes can be reproduced in the limit of vanishing gravitational coupling both for bosonic and fermionic modes. Our BPS solution has a smooth limit of thin walls where it reproduces the Randall-Sundrum model [@EMSS]. In the original Randall-Sundrum model, the fine-tuning was necessary between the boundary and the bulk cosmological constants. However, the necessary relation between bulk and boundary cosmological constants is now an automatic consequence of the equation of motion of scalar fields and Einstein equation in our model. We no longer need to impose a fine-tuning on input parameters of the model. Sec.2 summarizes our model and solutions briefly. Sec.3 separates various bosonic modes with respect to the surviving Lorentz symmetry (tensor and scalar modes) and addresses the question of stability of the BPS solution. Sec.4 discusses the stability of non-BPS solution and evaluates the mass of the radion. Sec.5 deals with the fermionic modes. The gauge fixing to the Newton gauge is justified in Appendix A, and some illustrative cases of potential in the conformal coordinate are worked out in Appendix B. Brief review of BPS domain wall in SUGRA ======================================== Lagrangian and BPS equations ----------------------------- We consider a chiral multiplet containing scalar $\phi$ and fermion $\chi$ with the minimal kinetic term and the superpotential $P$, and the gravity multiplet containing vielbein $e_m{^{\underline{a}}}$ and gravitino $\psi_m{^{\alpha}}$. The local Lorentz vector indices are denoted by letters with the underline as $\underline{a}$, and the vector indices transforming under general coordinate transformations are denoted by Latin letters as $m, n=0, \dots, 3$. The left(right)-handed spinor indices[^4] are denoted by undotted (dotted) indices as ${\alpha} ({\dot \alpha})$. Then the $\mathcal{N}=1$ supergravity Lagrangian is given in four-dimensional spacetime as [@WessBagger] $$\begin{aligned} e^{-1}\mathcal{L} &=& - \frac{1}{2\kappa^2}R + \varepsilon^{klmn}\bar\psi_k\bar\sigma_l \tilde{\mathcal{D}}_m\psi_n \nonumber\\ &&- g^{mn}\partial_m\phi^*\partial_n\phi - {\rm e}^{\kappa^2\phi^*\phi} \left(|D_\phi P|^2 - 3\kappa^2|P|^2\right) -i \bar\chi\bar\sigma^m\mathcal{D}_m\chi \nonumber\\ && - \frac{\sqrt{2}}{2}\kappa \left(\partial_n\phi^*\chi\sigma^m\bar\sigma^n\psi_m + \partial_n\phi\bar\chi\bar\sigma^m\sigma^n \bar\psi_m\right) \nonumber\\ &&+ \frac{\kappa^2}{4} \left(i\varepsilon^{klmn}\psi_k\sigma_l\bar\psi_m + \psi_m\sigma^n\bar\psi^m\right)\chi\sigma_n\bar\chi - \frac{\kappa^2}{8}\chi\chi\bar\chi\bar\chi \nonumber\\ &&- {\rm e}^{\frac{\kappa^2}{2}\phi^*\phi}\bigg[ \kappa^2\left(P^*\psi_m\sigma^{mn}\psi_n + P\bar\psi_m\bar\sigma^{mn}\bar\psi_n\right)\nonumber\\ &&+\frac{i\kappa}{\sqrt{2}} \left(D_\phi P\chi\sigma^m\bar\psi_m + D_{\phi^*}P^*\bar\chi\bar\sigma^m\psi_m\right) \nonumber\\ && + \frac{1}{2} \left(\mathcal{D}_\phi D_\phi P \chi\chi + \mathcal{D}_{\phi^*}D_{\phi^*}P^*\bar\chi\bar\chi\right) \bigg], \label{SUGRA_Lag}\end{aligned}$$ where the gravitational coupling $\kappa$ is the inverse of the four-dimensional Planck mass $M_{\rm Pl}$, $g_{mn}$ is the metric of the spacetime and $e$ is the determinant of the vierbein $e_m{^{\underline{a}}}$. The generalized supergravity covariant derivatives are defined as follows : $$\begin{aligned} \label{covder} \begin{array}{rll} \mathcal{D}_m\chi &= &\partial_m\chi + \chi\omega_m - \dfrac{i\kappa^2}{2}{\rm Im}\! \left[\phi^*\partial_m\phi\right]\chi,\\ \tilde{\mathcal{D}}_m\psi_n &= &\partial_m\psi_n + \psi_n\omega_m + \dfrac{i\kappa^2}{2}{\rm Im}\! \left[\phi^*\partial_m\phi\right]\psi_n,\\ D_\phi P &= &\partial_\phi P + \kappa^2\phi^*P,\\ \mathcal{D}_\phi D_\phi P &= &\partial_\phi^2P + 2\kappa^2\phi^*D_\phi P - \kappa^4\phi^{*2}P, \end{array}\end{aligned}$$ where $\omega_m$ is the spin connection and we use the notation ${\rm Im}\!\left[X\right] \equiv \dfrac{X-X^*}{2i}$ in what follows. The scalar potential in the supergravity Lagrangian (\[SUGRA\_Lag\]) is given by $$\begin{aligned} V(\phi,\phi^*) = {\rm e}^{\kappa^2\phi^*\phi} \left(|D_\phi P|^2 - 3\kappa^2|P|^2\right). \label{eq:scalar-potential}\end{aligned}$$ The above Lagrangian (\[SUGRA\_Lag\]) is invariant under the supergravity transformation : $$\begin{aligned} \begin{array}{rll} \delta_\zeta e_m{^{\underline{a}}} &= &i\kappa\left(\zeta\sigma^{\underline{a}}\bar\psi_m + \bar\zeta\bar\sigma^{\underline{a}}\psi_m\right),\\ \delta_\zeta \psi_m &= &2\kappa^{-1}\mathcal{D}_m\zeta + i\kappa{\rm e}^{\frac{\kappa^2}{2}\phi^*\phi} P\sigma_m\bar\zeta - \dfrac{i\kappa}{2}\sigma_{mn}\zeta\chi\sigma^n\bar\chi - \dfrac{i\kappa^2}{2}{\rm Im}\! \left[\phi^*\delta_\zeta\phi\right]\psi_m,\\ \delta_\zeta \phi &= &\sqrt{2}\ \zeta\chi,\\ \delta_\zeta \chi &= &i\sqrt{2}\ \sigma^m\bar\zeta \hat{\mathcal{D}}_m\phi - \sqrt{2}\ {\rm e}^{\frac{\kappa^2}{2}\phi^*\phi} D_{\phi^*}P^*\zeta + \dfrac{i\kappa^2}{2}{\rm Im}\! \left[\phi^*\delta_\zeta\phi\right]\chi, \end{array} \label{eq:SUGRAtransf}\end{aligned}$$ where $\zeta$ is a local SUSY transformation parameter and the covariant derivatives are given by $$\begin{aligned} \begin{array}{rll} \hat{\mathcal{D}}_m\phi &= &\partial_m\phi - \dfrac{\sqrt{2}}{2} \kappa\bar\psi_m\bar\chi,\\ \mathcal{D}_m\zeta &= &\partial_m\zeta + \zeta\omega_m + \dfrac{i\kappa^2}{2}{\rm Im}\! \left[\phi^*\partial_m\phi\right]\zeta. \end{array}\end{aligned}$$ Next we turn to derive the equations of motion for solutions which depend on only one “extra” coordinate $x^2=y$ under the warped metric Ansatz $$\begin{aligned} ds^2 = g_{mn}dx^mdx^n = {\rm e}^{2A(y)}\eta_{\mu\nu}dx^\mu dx^\nu + dy^2 \quad (\mu,\nu = 0,1,3),\label{warped_metric}\end{aligned}$$ where Greek indices $\mu=0, 1, 3$ denote three-dimensional vector transforming under general coordinate transformations, and $\eta_{\mu\nu} = {\rm diag}(-,+,+)$ denotes three dimensional flat spacetime metric. All the geometrical quantities can be written in terms of the function $A(y)$ in the warp factor and its derivatives with respect to the extra coordinate $y$. For later convenience, we write formulas in general $D$ space-time dimensions in the following : 1. vierbein $$\begin{aligned} e_m{^{\underline{a}}} = {\rm diag}\left({\rm e}^A,\ {\rm e}^A,\ 1,\ {\rm e}^A\right),\quad e_{\underline{a}}{^m} = {\rm diag}\left({\rm e}^{-A},\ {\rm e}^{-A},\ 1,\ {\rm e}^{-A}\right),\end{aligned}$$ 2. spin connection $$\begin{aligned} (\chi \omega_m)_\alpha = {1 \over 2} \omega_{m\underline{ab}} \left(\sigma^{\underline{ab}}\right)_\alpha{}^\beta \chi_\beta, \qquad \omega_{m\underline{ab}} = \dot{A}\left(\delta_{\underline{a}}{^2}e_{\underline{b}m} - \delta_{\underline{b}}{^2}e_{\underline{a}m}\right), \label{eq:spin-connec}\end{aligned}$$ 3. Ricci tensor $$\begin{aligned} R_{mn} = {\rm e}^{2A}\left( \ddot{A} + (D-1) \dot{A}^2\right) \eta_{\mu\nu}\delta_m{^\mu}\delta_n{^\nu} + (D-1) \left( \ddot{A} + \dot{A}^2\right)\delta_m{^2}\delta_n{^2},\end{aligned}$$ where a dot denotes a derivative with respect to $y$, $\dot A\equiv dA/dy$, and we turn off all the fermionic fields as a tree level solution. The energy momentum tensor is given in terms of the scalar potential $V(\phi, \phi^*)$ in (\[eq:scalar-potential\]) $$\begin{aligned} T_{mn} = \partial_m\phi^*\partial_n\phi + \partial_m\phi\partial_n\phi^* - g_{mn}\left(g^{kl}\partial_k\phi^*\partial_l\phi + V(\phi,\phi^*)\right).\end{aligned}$$ Plugging these into the Einstein equation[^5] $R_{mn} = - \kappa^2\tilde{T}_{mn}$, we obtain $$\begin{aligned} \ddot{A} = -{2\over D-2}\kappa^2 \dot\phi^*\dot\phi,\quad \dot{A}^2 = \frac{2\kappa^2}{(D-1)(D-2)}\left( \dot\phi^*\dot\phi-V(\phi,\phi^*)\right). \label{Einstein_eq}\end{aligned}$$ The field equation for the scalar $\phi$ in the chiral multiplet takes the form : $$\begin{aligned} \ddot\phi + (D-1) \dot{A}\dot\phi = \frac{\partial V}{\partial\phi^*}. \label{field_eq}\end{aligned}$$ Notice that only two out of the three equations in Eqs.(\[Einstein\_eq\]) and (\[field\_eq\]) are independent (assuming only one real component, say the real part of the scalar field $\phi$ is nontrivial in the solution). Any one of three equations are automatically satisfied if others are satisfied. It is well known that special type of solutions for these nonlinear second order differential equations are obtained as solutions of a set of the first order differential equations, the so-called BPS equations which guarantees the partial conservation of SUSY. Similarly to the global SUSY case, the BPS equations can be derived from the half SUSY condition where we parametrize the conserved SUSY parameter as $$\begin{aligned} \zeta(y) = {\rm e}^{i\theta(y)} \sigma^{\underline{2}}\bar\zeta(y). \label{eq:half-susy}\end{aligned}$$ That is, we demand that the bosonic configuration should satisfy $\delta_\zeta\chi = \delta_\zeta\psi_m=0$ for the parameter $\zeta(y)$ in Eq.(\[eq:half-susy\]). The BPS equations for the metric are derived from the condition for the gravitino. From $m=\mu=0, 1, 3$ components the first order equation for the warp factor $A$ is derived : $$\begin{aligned} 0 = \delta_\zeta\psi_\mu = \kappa^{-1}{\rm e}^A \left[{\rm e}^{i\theta}\dot{A} + i\kappa^2{\rm e}^{\frac{\kappa^2}{2}\phi^*\phi} P\right]\sigma_{\underline{\mu}}\bar\zeta ,\end{aligned}$$ $$\begin{aligned} \dot{A} = - i\kappa^2{\rm e}^{-i\theta} {\rm e}^{\frac{\kappa^2}{2}\phi^*\phi}P.\label{BPS_A}\end{aligned}$$ From $m=2$ component we find the first order equation for the Killing spinor $\zeta$ : $$\begin{aligned} 0 = \delta_\zeta\psi_2 = 2\kappa^{-1} \left[\dot\zeta + \frac{i\kappa^2}{2}{\rm Im}\! \left[\phi^*\dot\phi \right]\zeta + \frac{i\kappa^2}{2}{\rm e}^{-i\theta} {\rm e}^{\frac{\kappa^2}{2}\phi^*\phi}P\zeta\right].\end{aligned}$$ Rewriting the half SUSY condition as $\zeta_{\underline{\alpha}} ={\rm e}^{\frac{i}{2}\left(\theta + \frac{\pi}{2}\right)} |\zeta_{\underline{\alpha}}|$ and substituting it into the above equation, we find [@CGR] $$\begin{aligned} \dot{|\zeta_{\underline{\alpha}}|} = \frac{\dot{A}}{2} |\zeta_{\underline{\alpha}}|,\quad \dot\theta = - \kappa^2{\rm Im}\!\left[\phi^*\dot\phi \right].\label{BPS_kill}\end{aligned}$$ On the other hand, the first order equation for the matter field $\phi$ is derived from the half SUSY condition for the matter fermion $\chi$ : $$\begin{aligned} \dot\phi = -i{\rm e}^{i\theta}{\rm e}^{\frac{\kappa^2}{2} \phi^*\phi}D_{\phi^*}P^*.\label{BPS_phi}\end{aligned}$$ Eq.(\[BPS\_A\]), (\[BPS\_kill\]) and (\[BPS\_phi\]) are collectively called BPS equations. One can easily show that solutions of the BPS equations satisfy the equations of motion (\[Einstein\_eq\]) and (\[field\_eq\]). Notice that the Eq.(\[BPS\_phi\]) and the second equation of Eq.(\[BPS\_kill\]) do not contain the metric, so we can solve this as if the scalar field decouples from gravity. Once the configuration of the scalar field $\phi$ and the phase $\theta$ are determined, the warp factor $A$ is obtained from Eq.(\[BPS\_A\]). Finally, the Killing spinor $\zeta$ is also determined from the first equation of Eq.(\[BPS\_kill\]). Exact BPS solution ------------------ Recently, we found the exact BPS solutions for the periodic model in SUGRA [@EMSS], by allowing the gravitational correction for the superpotential as follows $$\begin{aligned} P(\phi) = {\rm e}^{-\frac{\kappa^2}{2}\phi^2} \times \frac{\Lambda^3}{g^2} \sin\frac{g}{\Lambda}\phi,\label{P_mod}\end{aligned}$$ where $\Lambda$ is a coupling with unit mass dimension and $g$ is a dimensionless coupling. We introduced this modification for the superpotential in SUGRA to maintain the periodicity of the model with the aid of the Kähler transformation. This modification for the superpotential gives SUSY vacua which do not depend on the gravitational coupling $\kappa$. This was crucial for us to obtain the exact BPS solutions in SUGRA. The superpotential (\[P\_mod\]) yields the following scalar potential : $$\begin{aligned} V = \frac{\Lambda^4}{g^2} {\rm e}^{2\kappa^2({\rm Im}\left[\phi\right])^2} \left[ \left|\cos\frac{g}{\Lambda}\phi - \frac{2i\kappa^2\Lambda}{g}{\rm Im}\!\left[\phi\right] \ \sin\frac{g}{\Lambda}\phi\right|^2 - \frac{3\kappa^2\Lambda^2}{g^2} \left|\sin\frac{g}{\Lambda}\phi\right|^2 \right].\end{aligned}$$ The SUSY vacua are determined from the condition $D_\phi P = 0$. For the above modified superpotential we find that the SUSY vacua are periodically distributed at $\phi = \dfrac{\Lambda}{g}\left(\dfrac{\pi}{2} + n\pi\right),\ (n\in\mathbb{Z})$ on the real axis in the complex $\phi$ plane. In order to determine the scalar field configuration, we need to solve the second equation of Eq.(\[BPS\_kill\]) for $\theta$ together with the equation for scalar field : $$\begin{aligned} \dot\phi = - i{\rm e}^{i\theta}{\rm e}^{i\kappa^2\phi^*{\rm Im}[\phi]} \frac{\Lambda^2}{g} \left[\cos\frac{g}{\Lambda}\phi^* + \frac{2i\Lambda\kappa^2}{g}{\rm Im}[\phi] \sin\frac{g}{\Lambda}\phi^*\right]. \label{eq:matterBPSeq}\end{aligned}$$ To solve Eqs.(\[BPS\_kill\]) and (\[eq:matterBPSeq\]), we choose $\phi_{\rm I} \equiv {\rm Im}[\phi] = 0$ and $\theta = \pm\dfrac{\pi}{2}$ at a point, say $y=y_i$ as an initial condition for the imaginary part $\phi_{\rm I}(y)$ of the scalar field and the phase $\theta(y)$. Then these equations tell that $\dot\phi_{\rm I} = \dot\theta = 0$ at $y=y_i$. Therefore we find $\phi_{\rm I} = 0$ and $\theta = \pm\dfrac{\pi}{2}$ at any $y$. At this stage, only the real part of $\phi$ has a nontrivial configuration in the extra dimension $y$. We shall call those scalar fields that have nontrivial configurations as a function of the coordinate of extra dimension, as “active” scalar fields. The scalar potential along $\phi_{\rm I}=0$ surface is given by the following potential $V_{\rm R}$ for the real part $\phi_{\rm R} \equiv\dfrac{\phi+\phi^*}{2}$ of the scalar field : $$\begin{aligned} V_{\rm R}(\phi_{\rm R}) = \frac{\Lambda^4}{g^2} \left[\cos^2\frac{g}{\Lambda}\phi_{\rm R} - \frac{3\kappa^2\Lambda^2}{g^2} \sin^2\frac{g}{\Lambda}\phi_{\rm R}\right]. \label{eq:real-scalar-pot}\end{aligned}$$ It has been shown that the following form of scalar potential with a real “superpotential” $\hat P(\phi_{\rm R})$ of a real scalar field $\phi_{\rm R}$ ensures the existence of a stable AdS vacuum in gravity theories in $D$ dimensions [@Boucher; @PKT] : $$\begin{aligned} V_{\rm R} = {D-2 \over 2}\left[{D-2 \over 2} \left(\frac{d\hat P}{d\phi_{\rm R}}\right)^2 -(D-1)\kappa^2\hat P^2\right],\label{potential_ads}\end{aligned}$$ if there is a critical point in $\hat P(\phi_R)$, even though supersymmetry is not required in this form. Let us note that our scalar potential is compatible with the above form of the scalar potential. In our case, the “superpotential” is given by $$\begin{aligned} \hat P(\phi_{\rm R}) = \frac{\Lambda^3}{g^2}\sin \frac{g}{\Lambda}\phi_{\rm R} .\end{aligned}$$ Since this $\hat P$ has critical points at $\phi_R=\dfrac{\Lambda}{g} \left(\dfrac{\pi}{2}+n\pi\right)$, our scalar potential $V_{\rm R}(\phi_{\rm R})$ in Eq.(\[eq:real-scalar-pot\]) has these critical points as stable AdS vacua. The remaining BPS equations for the active scalar field and the warp factor are of the form : $$\begin{aligned} \dot\phi_{\rm R} = \pm \frac{d\hat P}{d\phi_{\rm R}} = \pm \frac{\Lambda^2}{g} \cos\frac{g}{\Lambda}\phi_{\rm R},\quad \dot{A} = \mp \kappa^2\hat P = \mp \frac{\kappa^2\Lambda^3}{g^2} \sin\frac{g}{\Lambda}\phi_{\rm R}. \label{BPS_eq_R}\end{aligned}$$ Let us solve these BPS equations by choosing a SUSY vacuum $\phi_{\rm R} =\dfrac{\Lambda}{g}\left(\mp(-1)^n\dfrac{\pi}{2}+n\pi\right)$ as an initial condition at $y=-\infty$. We shall consider the solution for the BPS equations (\[BPS\_eq\_R\]) with the sign correlated to the sign of the initial condition at $y=-\infty$. The exact BPS solutions are found to be of the form : $$\begin{aligned} \phi_{\rm R} = \frac{\Lambda}{g}\left[(-1)^n \left\{2\tan^{-1}{\rm e}^{\pm\Lambda(y-y_0)} - \frac{\pi}{2}\right\} + n\pi\right],\quad {\rm e}^{A} = \left[\cosh\Lambda(y-y_0) \right]^{-\frac{k}{\Lambda}}, \label{BPS_sol}\end{aligned}$$ where $k\equiv \dfrac{\kappa^2\Lambda^3}{g^2}$ is the inverse of the curvature radius of the AdS spacetime at infinity. These solutions interpolate between the two SUSY vacua, from $\phi_{\rm R} = \dfrac{\Lambda}{g}\left(\mp(-1)^n \dfrac{\pi}{2} + n\pi\right)$ at $y = -\infty$ to $\phi_{\rm R} = \dfrac{\Lambda}{g}\left(\pm(-1)^n\dfrac{\pi}{2} + n\pi\right)$ at $y = +\infty$. We denote $y_0$ the modulus parameter of these solutions and we suppress an integration constant for $A$ which amounts to an irrelevant normalization constant of metric. Eq.(\[BPS\_kill\]) determines the Killing spinors which has two real Grassmann parameters $\epsilon_1, \epsilon_2$ corresponding to the two conserved SUSY directions on the BPS solution[^6] : $$\begin{aligned} \zeta = {\rm e}^{\frac{i}{2}\left(\theta + \frac{\pi}{2}\right)} {\rm e}^{\frac{A}{2}}\times \left(\begin{array}{c} \epsilon_1 \\ \epsilon_2 \end{array}\right), \label{eq:Killingspinor}\end{aligned}$$ $$\begin{aligned} {\rm e}^{\frac{i}{2}\left(\theta + \frac{\pi}{2}\right)} {\rm e}^{\frac{A}{2}} = \left\{ \begin{array}{ll} i\left[\cosh\Lambda(y-y_0)\right]^{ -\frac{k}{2\Lambda}}, \quad&{\rm for}\ \theta = \dfrac{\pi}{2},\\ \left[\cosh\Lambda(y-y_0)\right]^{ -\frac{k}{2\Lambda}}, \quad&{\rm for}\ \theta = -\dfrac{\pi}{2}. \end{array} \right.\end{aligned}$$ Our model has a smooth limit of thin walls where it reproduces the Randall-Sundrum model [@EMSS]. Notice that we do not need any fine-tuning of input parameters of the model, in contrast to the original Randall-Sundrum model. The necessary fine-tuning between bulk and boundary cosmological constants is now an automatic consequence of the equation of motion of scalar fields and Einstein equation in our model. non-BPS solution ---------------- Assuming that only single real scalar field $\phi_{\rm R}$ has nontrivial classical configuration, the equations (\[Einstein\_eq\]) and (\[field\_eq\]) reduce to $$\begin{aligned} \ddot{A} =-{2\kappa^2 \over D-2}\dot\phi_{\rm R}^2 ,\qquad \dot{A}^2 = \frac{2\kappa^2}{(D-1)(D-2)} \left(\dot\phi_{\rm R}^2 -V_{\rm R}\right), \qquad \ddot\phi_{\rm R} + (D-1) \dot{A}\dot\phi_{\rm R} = \frac{1}{2}\frac{dV_{\rm R}}{d\phi_{\rm R}}. \label{eom_real_phi}\end{aligned}$$ It has been shown that the above set of coupled second order differential equations is equivalent to the following set of nonlinear differential equations [@DFGK; @SkTo]. Given the scalar potential $V_{\rm R}(\phi_{\rm R})$, we should find a real function $W(\phi_{\rm R})$ by solving the following first order nonlinear differential equation $${d W(\phi_{\rm R}) \over d\phi_{\rm R}} = \pm {2 \over D-2} \sqrt{V_{\rm R}(\phi_{\rm R})+ {(D-1)(D-2) \over 2}\kappa^2W^2(\phi_{\rm R})} . \label{eq:nonlinear-eq}$$ Then $\phi_{\rm R}(y)$ and $A(y)$ are obtained by solving the following two first order differential equations $$\dot\phi_{\rm R}(y)= {D-2 \over 2}\dfrac{d W(\phi_{\rm R})}{d \phi_{\rm R}}, \qquad \dot A(y) = - \kappa^2W(\phi_{\rm R}) . \label{phi-A}$$ If we choose the “superpotential” $\hat P$ as a real function $W$, (\[eq:nonlinear-eq\]) and (\[phi-A\]) are satisfied by the scalar potential (\[potential\_ads\]) and the BPS equations (\[BPS\_eq\_R\]). Therefore these set of first order nonlinear differential equations includes all the BPS solutions as part of the solutions. However, it is important to realize that (\[eq:nonlinear-eq\]) and (\[phi-A\]) are equivalent to the set of Einstein equation and the scalar filed equation, and hence give all the non-BPS solutions as well. We have been able to construct non-BPS multi-wall solutions to the Einstein equation (\[Einstein\_eq\]) and the field equation (\[field\_eq\]) using the above method of nonlinear equations [@EMSS]. We have also found that BPS solutions are the only solution that do not encounter singularities at any finite $y$. To obtain any other regular solution, especially non-BPS solutions, negative cosmological constant has to be introduced at some boundary. Since we are interested in periodic array of walls where extra dimension can be identified as a torus $S^1$ with possible division by discrete groups (orbifolds), we introduced the cosmological constant and obtained a number of interesting non-BPS solutions [@EMSS]. The above nonlinear differential equation (\[eq:nonlinear-eq\]) gives a set of solution curves which fill once and only once the entire $(\phi_{\rm R}, W)$ plane except forbidden regions defined by $V_{\rm R}+(D-1)(D-2)\kappa^2W^2/2 \le 0$. Let us denote the solution curve starting from an initial condition $W_0$ at $\phi_{\rm R} = \phi_{{\rm R},0}$ as $W(\phi_{\rm R}; (\phi_{{\rm R},0}, W_0))$. A boundary cosmological constant $\lambda_i$ at $y_i$ gives a jump of derivative of the function $A(y)$ in the warp factor. Let us denote the value of the scalar field at the boundary $y_i$ as $\phi_{{\rm R},i}$. Eq.(\[phi-A\]) shows that this jump of $A(y)$ is satisfied by cutting the solution curve and jump to another solution curve at $\phi_{{\rm R},i}$ with the constraint $$\lambda_i =2 \left(W (\phi_{{\rm R},i}+\epsilon)-W (\phi_{{\rm R},i}-\epsilon)\right) . \label{eq:cosm_const_W}$$ Since we are interested in minimum amount of inputs at boundaries, we wish to implement only the boundary cosmological constant without any boundary potential for scalar fields $\phi_{\rm R}$, contrary to many other approaches characteristic of the Goldberger-Wise type of the stabilization mechanism [@GoWi], [@DFGK], [@SkTo]. Therefore we need to maintain the derivative $dW/d\phi_{\rm R}$ to be smoothly connected at the boundary. Since Eq.(\[eq:nonlinear-eq\]) gives the same value of derivative $dW/d\phi_{\rm R}$ for $\pm W$, we can connect the solution curve at any value of $\phi_{{\rm R},i}$ if we switch from a solution curve going through $W, \phi_{{\rm R},i}$ to another one going through $-W, \phi_{{\rm R},i}$. Eq.(\[eq:cosm\_const\_W\]) gives the necessary cosmological constant at this boundary as $\lambda=4W(\phi_{{\rm R},i})$. There may be other possibilities to connect the solution curves, but this is the simplest possibility that covers many interesting situations. To be definite, we shall consider walls that have simple symmetry property under the parity $Z_2$ : $\phi_{\rm R} \rightarrow -\phi_{\rm R}$. Let us start a solution curve going through $\phi_{\rm R}=0, W_0>0$. Then the solution curve goes above the forbidden region. To obtain a non-BPS solution which is odd under the $Z_2$ transformation, we place a boundary at $\phi_{\rm R}=0$ with a positive cosmological constant by an amount $\lambda_0=4W_0>0$. On the other hand, we can place a boundary at any $\phi_{\rm R}>0$ with a negative cosmological constant $\lambda_1=-4W(\phi_{{\rm R},1}, (\phi_{\rm R}=0, W_0))$. However, we can obtain a multi-wall solution that have simple transformation property under the $Z_2$ by placing another boundary at integer multiple of $\phi_{\rm R}=\Lambda \pi/(2g)$. If we place the first boundary at the vacuum point $\phi_{\rm R}=\Lambda \pi/(2g)$, we obtain a simplest model in the sense that the energy density at the second boundary at $\phi_{\rm R}=\Lambda \pi/(2g)$ is purely made of negative cosmological constant $$\lambda = -4W(\phi_{\rm R}=\Lambda \pi/(2g)). \label{eq:negaive-cosm-pi/2}$$ The magnitude of this negative cosmological constant becomes the same as the total energy of the wall centered at $\phi_{\rm R}=0$ in the limit of large separation of two boundaries. Since the solution admits $S^1/(Z_2\times Z_2)$ symmetry, we call the coordinate at the second boundary $y=\pi R/2$. The behavior of this non-BPS solutions in the $W, \phi$ plane is illustrated in Fig.\[fig:half\_wind\_sol\](a). The corresponding function $A(y)$ in the warp factor is illustrated in Fig.\[fig:half\_wind\_sol\](b), where one should note that $A(y)$ is linear near the second boundary at $y=\pi R/2$, showing that only the boundary cosmological constant exists apart from the bulk cosmological constant there. As another solution, we can place the second boundary at $\phi_{\rm R}=\Lambda \pi/g$, where the active scalar field $\phi_{\rm R}$ develops another wall configuration. In this case, the negative cosmological constant $-4W(\phi_{\rm R}=\Lambda \pi/g)$ placed at the second boundary has magnitude which becomes twice the total energy of the wall centered at $\phi_{\rm R}=0$ in the limit of large separation of two boundaries. The behavior of this non-BPS solutions in the $W, \phi$ plane is illustrated in Fig.\[fig:wind\_sol\](a). The corresponding function $A(y)$ in the warp factor is illustrated in Fig.\[fig:wind\_sol\](b), where one should note that the function $A(y)$ has additional kink behavior deviating from the linear exponent near the second boundary at $y=\pi R$, showing that there is an additional smooth positive energy density centered around the boundary besides the negative boundary cosmological constant in contrast to the previous $S^1/(Z_2\times Z_2)$ example in Fig.\[fig:half\_wind\_sol\]. Bosonic Fluctuation and the BPS Solution ======================================== A Bogomolo’nyi bound has been derived for the energy density of the BPS domain walls in $\mathcal{N}=1$ SUGRA in four-dimensional spacetime [@CGR]. They used the generalized Israel-Nester-Witten tensor, which was originally applied to a simple proof of the positive ADM mass conjecture in general relativity. However, the ADM mass may not be well-defined for domain walls, since they are extended to infinity. Therefore it is presumably still useful to check that there is really stability of the fluctuation on our wall configuration even in the case of BPS solutions. We shall present a general formalism to analyze the modes and their stability, and then apply it to the fluctuations around the BPS background configurations in this section. The equations and procedures obtained in this section can also be used to the non-BPS background solutions with appropriate additional inputs, which is dealt with in Sec.\[sc:stability=nonBPS\]. Mode equations for the bosonic sector ------------------------------------- We start with the metric perturbation in the Newton gauge [@TanakaMontes], [@CsabaCsaki] : $$\begin{aligned} ds^2 = {\rm e}^{2A}\left(\eta_{\mu\nu}+h^{\rm TT}_{\mu\nu} + 2B\eta_{\mu\nu}\right)dx^\mu dx^\nu + (1-2(D-3)B)dy^2, \label{eq:newton-gauge}\end{aligned}$$ where $h^{\rm TT}_{\mu\nu}$ is transverse traceless $\eta^{\mu\nu}h^{\rm TT}_{\mu\nu}=0, \; \partial^\mu h^{\rm TT}_{\mu\nu}=0$. Some details for the procedure of this gauge fixing are given in Appendix A. This gauge is useful since the linearized equations become very simple. The linearized Einstein equations in $D$ space-time dimensions ($D=4$ in our specific model) read : $$\begin{aligned} {2} &\left({\rm e}^{-2A}\square_{D-1} + \partial_y^2 + (D-1)\dot{A}\partial_y \right)h^{\rm TT}_{\mu\nu} = 0,\label{Newton_1}\\ &\left({\rm e}^{-2A}\square_{D-1} + \partial_y^2 + (3D-5)\dot{A}\partial_y + 2(D-3)\left((D-1)\dot{A}^2 + \ddot{A}\right) \right)B = -{2\kappa^2\over D-2}\frac{dV_{\rm R}}{d\phi_{\rm R}} \varphi_{\rm R}, \label{Newton_2}\\ &\left(\partial_y + (D-3)\dot{A} \right)B = - {2\kappa^2\over D-2}\kappa^2\dot\phi_{\rm R} \varphi_{\rm R},\label{Newton_3}\end{aligned}$$ where the first line comes from the traceless part of $(\mu,\nu)$ component of the linearized Einstein equations, the second line from the trace part of $(\mu,\nu)$ and the last from $(\mu,2)$ component. The $(2,2)$ component of the linearized equation is not shown, since it can be derived from Eqs.(\[Newton\_1\])-(\[Newton\_3\]). The linearized field equations give : $$\begin{aligned} {2} &\left({\rm e}^{-2A}\square_{D-1} + \partial_y^2 + (D-1)\dot{A}\partial_y - \frac{1}{2}\frac{d^2V_{\rm R}}{d\phi_{\rm R}^2} \right)\varphi_{\rm R} = -2(D-2) \dot\phi_{\rm R}\partial_yB - (D-3)\frac{dV_{\rm R}}{d\phi_{\rm R}}B,\label{active}\\ &\left({\rm e}^{-2A}\square_{D-1} + \partial_y^2 + (D-1)\dot{A}\partial_y - \frac{1}{2}\frac{\partial^2V}{\partial \phi_{\rm I}^2} \bigg|_{\rm background}\right)\varphi_{\rm I} = 0,\label{inert}\end{aligned}$$ where $\varphi_{\rm R(I)}$ denotes the real (imaginary) part of the fluctuation of the scalar field $\phi=\phi_{\rm R}+\varphi_{\rm R}+i\varphi_{\rm I}$ around the background field configuration $\phi_{\rm R}$. Notice that the solutions of the linearized Einstein equation automatically satisfy the linearized field equations for the active scalar field $\varphi_{\rm R}$. Therefore, the Eqs.(\[Newton\_1\])– (\[inert\]) constitute the full set of independent linearized equations for the fields $h^{\rm TT}_{\mu\nu},\ B,\ \varphi_{\rm R}$ and $\varphi_{\rm I}$. Tensor perturbation : localized massless graviton {#sc:tensot-perturb} ------------------------------------------------- First we show that the linearized equation for the transverse traceless mode (graviton) given in Eq.(\[Newton\_1\]) can be brought into a Schrödinger form. It can again be rewritten into a form of the supersymmetric quantum mechanics (SQM) which ensures the stability of the system. For that purpose we change the coordinate $y$ into the conformally flat coordinate $z$ defined as $$\begin{aligned} dz\equiv{\rm e}^{-A(y)}dy, \qquad ds^2={\rm e}^{2A(y)}\left( \eta_{\mu\nu}dx^\mu dx^\nu + dz^2 \right). \end{aligned}$$ We also redefine the field as $\tilde{h}^{\rm TT}_{\mu\nu} \equiv {\rm e}^{{D-2\over 2}A}h^{\rm TT}_{\mu\nu}$. In the following we use prime to denote a derivative in terms of $z$. Then the linearized equation (\[Newton\_1\]) becomes $$\begin{aligned} \square_{D-1}\tilde{h}^{\rm TT}_{\mu\nu}(x,z) = \left[-\partial_z^2 + \mathcal{V}_t(z)\right] \tilde{h}^{\rm TT}_{\mu\nu}(x,z), \qquad \mathcal{V}_t(z) = \left({D-2\over 2}\right)^2A'{^2} + {D-2\over 2}A'', \label{eq:schrod-tt}\end{aligned}$$ where $\mathcal{V}_t(z)$ is the potential in this “Schrödinger” type equation. For our BPS background solution (\[BPS\_sol\]) the Schrödinger potential takes the form : $$\begin{aligned} \mathcal{V}_t(y) = \big[\cosh\Lambda(y-y_0)\big]^{-\frac{2k}{\Lambda}} \left[-\frac{k\Lambda}{\cosh^2\Lambda(y-y_0)} + 2k^2\tanh^2\Lambda(y-y_0)\right], \label{sch_tensor}\end{aligned}$$ where $4T^3\equiv 4g^{-2}\Lambda^3$ is the tension of the wall and $k=\kappa^2T^3$. Although our model contains three parameters $\Lambda,\ g$ and $\kappa$, this potential depends on only two parameters $k$ and $\Lambda$. If we take the thin wall limit where $\Lambda\rightarrow\infty$ fixing $4T^3$, we obtain (putting $y_0=0$) $$\begin{aligned} \frac{\Lambda}{\cosh^2\Lambda y(z)} \rightarrow 2\delta(z),\quad \tanh^2\Lambda y(z) \rightarrow 1,\quad \left[\cosh\Lambda y(z) \right]^{-\frac{k}{\Lambda}} \rightarrow \frac{1}{\left(k|z|+1\right)^2},\end{aligned}$$ with $kz={\rm sgn}(y){\rm e}^{k|y|}-1$. Thus the Schrödinger potential (\[sch\_tensor\]) becomes precisely the potential of the Randall-Sundrum model : $$\begin{aligned} \mathcal{V}_t(z) \rightarrow \frac{2k^2}{\left(k|z| + 1\right)^2} - 2k\delta(z).\end{aligned}$$ We find that the part of action quadratic in $\tilde{h}^{\rm TT}_{\mu\nu}$ has no $z$ dependent weight $$\begin{aligned} S \sim \int dzd^{D-1}x\ \eta^{\mu\rho} \eta^{\nu\lambda}\tilde{h}^{\rm TT}_{\mu\nu} \left(\square_{D-1}+\partial_z^2 -\frac{1}{2}\mathcal{V}_t\right) \tilde{h}^{\rm TT}_{\rho\lambda}, \label{eq:quadratic-h-action}\end{aligned}$$ in conformity with the absence of the linear term [@CEHS] in $\partial_z$ in the Shrödinger type equation (\[eq:schrod-tt\]). We stress that this is written in terms of the conformal coordinate $z$ and the redefined field $\tilde{h}^{\rm TT}_{\mu\nu}$. Defining mode equations by eigenvalue equations $\mathcal{H}_t \psi_n(z)= \left[-\partial_z^2 + \mathcal{V}_t(z)\right] \psi_n(z)=m_n^2\psi_n(z)$ with mass squared eigenvalues $m_n^2$, and assuming mode functions $\psi_{n}(z)$ to form a complete set, the transverse traceless fields can be expanded into a set of effective fields $\hat{h}^{{\rm TT} (n)}_{\mu\nu}(x)$ $$\begin{aligned} \tilde{h}^{\rm TT}_{\mu\nu}(x,z) = \sum_n \hat{h}^{{\rm TT} (n)}_{\mu\nu}(x)\psi_{n}(z). \end{aligned}$$ Then the above quadratic action (\[eq:quadratic-h-action\]) becomes $$\begin{aligned} S \sim \sum_{n, k}\int dz\ \psi_n(z) \psi_k(z) \ \cdot \int d^{D-1}x\ \eta^{\mu\rho} \eta^{\nu\lambda} \hat{h}^{\rm TT(n)}_{\mu\nu} \left[\square_{D-1}-m_k^2\right] \hat{h}^{\rm TT(k)}_{\rho\lambda} . \end{aligned}$$ Therefore the inner product for the mode function $\psi(z)$ should be defined as as $$\begin{aligned} \langle\psi_1|\psi_2\rangle = \int dz\ \psi_1(z)\psi_2(z), \label{eq:inner-prod}\end{aligned}$$ for which the usual intuition of quantum mechanics works. The Hamiltonian $\mathcal{H}_t$ can now be expressed in a SQM form as follows $$\begin{aligned} \mathcal{H}_t = Q_t^\dagger Q_t, \qquad Q_t\equiv - \partial_z + {D-2 \over 2}A', \qquad Q_t^\dagger \equiv \partial_z + {D-2 \over 2}A', \label{eq:SQM-TT}\end{aligned}$$ where the “supercharge” $Q_t$ and $Q_t^\dagger$ are adjoint of each other at least for BPS background where no boundary condition has to be imposed. Therefore the Hamiltonian $\mathcal{H}_t$ is a nonnegative definite Hermitian operator[^7], and its eigenvalues are nonnegative definite. Therefore we can conclude that the tensor perturbation has no tachyonic modes which destabilize the background field configurations at least for BPS solutions. There are two possible zero modes in the tensor perturbation. One is the state which is annihilated by $Q_t|\tilde{h}^{\rm TT(+)}_{\mu\nu}\rangle=0$, and another is the state defined as $Q_t^\dagger\left(Q_t|\tilde{h}^{\rm TT(-)}_{\mu\nu} \rangle\right)=0$ where $Q_t|\tilde{h}^{\rm TT(-)}_{\mu\nu}\rangle\neq0$. Then zero modes are of the form : $$\begin{aligned} {2} &\tilde{h}^{\rm TT(0)}_{\mu\nu}(x,z) = \hat{h}^{\rm TT(+)}_{\mu\nu} (x)\ {\rm e}^{{D-2 \over 2}A(z)} + \hat{h}^{\rm TT(-)}_{\mu\nu} (x)\ {\rm e}^{{D-2 \over 2}A(z)}\int dz\ {\rm e}^{-(D-2)A(z)}, \label{eq:tensor-zero-mode}\end{aligned}$$ where $A(z) = A(y(z))$. Notice that we must verify the normalizability of the wave-function to obtain a physical massless effective field in the case of noncompact space such as our BPS background. In the case of non-BPS background, the boundary condition has to be verified, which we shall consider in Sec.\[sc:stability=nonBPS\]. The first mode $\hat{h}^{\rm TT(+)}_{\mu\nu}$ in Eq.(\[eq:tensor-zero-mode\]) is normalizable if $\int dz\ {\rm e}^{(D-2)A(z)} < \infty$, corresponds to the graviton which is localized at the wall with a positive energy density. namely, if ${\rm e}^{(D-2)A}$ falls off faster than $|z|^{-1}$ [@CEHS]. For our BPS solution (\[BPS\_sol\]) the asymptotic behavior of the warp factor ${\rm e}^A$ is of order $|z|^{-1}$. Therefore we obtain a normalizable massless transverse traceless mode $\hat{h}^{\rm TT(+)}_{\mu\nu}$ which gives the physical graviton localized on the wall. On the other hand, the second term $\hat{h}^{\rm TT(-)}_{\mu\nu}$ in Eq.(\[eq:tensor-zero-mode\]) is not normalizable and is unphysical since $\left({\rm e}^{D-2\over 2}A\int dz\ {\rm e}^{-(D-2)A}\right)^2\sim |z|^4$ at $|z| \rightarrow \infty$ for our BPS solution. If there exists a regulator brane with a negative tension at some $y$, this mode can become normalizable and localizes at the negative tension brane in contrast to the graviton. If there are no bulk scalar fields (contrary to our model) as in the original Randall-Sundrum model of single wall, this zero mode corresponds to the physical massless field which was called radion in Ref.[@Charmousis:1999rg]. Our specific four-dimensional model of non-BPS wall gives a three-dimensional effective theory on the wall. Transverse traceless mode of graviton in three dimensions has no dynamical degree of freedom except possible topological modes. However, our formalism and analysis can be applied at each step to general $D$-dimensional theories, once we obtain the relevant non-BPS solutions in such theories. In that respect, we believe that our findings should still be useful. ![[]{data-label="V_tensor"}](V_tensor.eps){width="6.5cm"} The Schödinger potential can always be expressed in terms of $y$, but is difficult in terms of $z$ explicitly[^8], since it is generally difficult to solve $dz = {\rm e}^{-A}dy$ explicitly. If we express the potential in terms of $y$, we obtain a volcano type potential as shown in Fig.\[V\_tensor\]. The width of the well is $\sim 2\Lambda^{-1}$ and the depth is $\sim k\Lambda$. Next we turn to analysis of the massive Kaluza-Klein (KK) mode. There are no modes with negative mass squared in the tensor perturbation, as we have already shown. Since the Schrödinger potential (\[sch\_tensor\]) vanishes asymptotically ($z=\pm\infty$), all the massive KK modes are continuum scattering states with eigenvalues $m^2>0$. In order to examine the mode functions of the massive KK modes, we look into the region far from the wall, namely $\Lambda|y|\gg1$. Since ${\rm e}^{A} \simeq {\rm e}^{-k|y|}$, $kz\simeq {\rm sgn}(y){\rm e}^{k|y|}-1$, we find that the Schrödinger potential becomes $$\begin{aligned} \mathcal{V}_t(z) \simeq \frac{2k^2}{\left(k|z| + 1\right)^2} \qquad \left(\Lambda|y|\gg1\right). \label{large_y_tensor}\end{aligned}$$ This happens to be the same potential as that in the Randall-Sundrum single wall model [@RS2], in spite of different spacetime dimensions. The wave functions of the continuum massive modes for this potential are known to be expressed as linear combinations of Bessel functions at the region far from the wall [@RS2]. The active scalar perturbation {#sc:active-scalar} ------------------------------- Next we study the perturbation of the active scalar field $\varphi_{\rm R}$. Notice that the fluctuation $\varphi_{\rm R}$ around the active scalar field background $\phi_{\rm R}$ can be reduced to the trace part $B$ of the metric perturbation through Eq.(\[Newton\_3\]). Therefore we mainly concentrate on the trace (scalar) part of the metric perturbation $B$ in what follows. The linearized equation which contains only $B$ can be derived by combining Eq.(\[Newton\_2\]) and (\[Newton\_3\]) and using the background field equation : $$\begin{aligned} \left[{\rm e}^{-2A}\square_{D-1} + \partial_y^2 + \left((D-3)\dot{A}-2\frac{\ddot\phi_{\rm R}} {\dot\phi_{\rm R}} \right)\partial_y + 2(D-3)\left(\ddot{A}-\dot{A} \frac{\ddot\phi_{\rm R}}{\dot\phi_{\rm R}}\right) \right]B=0.\label{B_eq}\end{aligned}$$ In order to transform this into the Schrödinger form, we change the coordinate from $y$ to $z$ and redefine the field as $\tilde{B}\equiv{\rm e}^{{D-2 \over 2}A} \phi_{\rm R}'{^{-1}}B$. Substituting this into Eq.(\[B\_eq\]), we find the Schödinger type equation for the scalar perturbation; $$\begin{aligned} \mathcal{H}_e \tilde{B} \equiv \left[-\partial_z^2 + \mathcal{V}_e(z)\right]\tilde{B} = \square_{D-1}\tilde{B}, \label{Ham_scalar}\end{aligned}$$ where the Schrödinger potential $\mathcal{V}_e(z)$ is defined by $$\begin{aligned} \mathcal{V}_e(z) \equiv - \frac{\phi_{\rm R}'''}{\phi_{\rm R}'} + 2 \left(\frac{\phi_{\rm R}''}{\phi_{\rm R}'}\right)^2 +(D-4){A''\phi_R'' \over \phi_R'} - {3D-10 \over 2}A'' + \left({D-2 \over 2}\right)^2A'{^2}. \label{eq:scalar-fluc-pot}\end{aligned}$$ Similarly to the tensor perturbation, the inner-product for the scalar perturbation $B$ should be defined in terms of the conformal coordinate $z$ and the redefined field $\tilde{B}$. Plugging our solution (\[BPS\_sol\]) into this, we find $$\begin{aligned} \mathcal{V}_e = \left[\cosh\Lambda (y-y_0)\right]^{-\frac{2k}{\Lambda}} \left[ \Lambda^2 + k\Lambda \left(1+\frac{1}{\cosh^2\Lambda (y-y_0)}\right) \right]. \label{Sch_scalar}\end{aligned}$$ ![[]{data-label="V_scalar"}](V_scalar.eps){width="6.5cm"} We stress that $\mathcal{V}_e$ can be expressed in terms of $y$, but not in terms of $z$, since it is generally difficult to solve $dz={\rm e}^{-A}dy$. This potential $\mathcal{V}_e$ has the following properties : i) it is positive definite, ii) it vanishes asymptotically at infinity, and iii) the height of $\mathcal{V}_e$ is of order $\Lambda^2$ as shown in Fig.\[V\_scalar\]. From i) , it follows that there are no tachyonic modes since the wave function of such modes will necessarily diverge either at $y=\infty$ or $y=-\infty$. Therefore we can conclude that the background configuration (\[BPS\_sol\]) is stable under the active scalar perturbation. From ii), it follows that the spectrum of the massive modes is continuous starting from zero. From iii), the potential diverges at any finite point $y$ in the thin wall limit $(\Lambda\rightarrow\infty)$. Though we can not find the exact solutions for the massive KK modes, zero modes can be found by rewriting the Hamiltonian (\[Ham\_scalar\]) into SQM form as follows : $$\begin{aligned} \mathcal{H}_e = Q_e^\dagger Q_e, \quad\quad Q_e \equiv -\partial_z + \left[\log\left({\rm e}^{-{D-2 \over 2}A} \dfrac{A'}{\phi_{\rm R}'}\right)\right]', \quad Q_e^\dagger \equiv \partial_z + \left[\log\left({\rm e}^{-{D-2 \over 2}A} \dfrac{A'}{\phi_{\rm R}'}\right)\right]'. \label{eq:SQM-scalar}\end{aligned}$$ To show this, we use the identity $\phi_{\rm R}'\left(A'''-2A'A''\right) = 2\phi_{\rm R}''\left(A''-A'{^2}\right)$. Similarly to the tensor perturbation there are two zero modes of $\mathcal{H}_e$ : $$\begin{aligned} {2} &\tilde{B}^{(0)}(x,z) = \hat{B}^{(+)}(x)\ \frac{A'}{\phi_{\rm R}'} {\rm e}^{-{D-2 \over 2}A} + \hat{B}^{(-)}(x)\ \frac{A'}{\phi_{\rm R}'} {\rm e}^{-{D-2 \over 2}A} \int dz\ \frac{\phi_{\rm R}'{^2}}{A'{^2}} {\rm e}^{(D-2)A} . \label{zero_scalar}\end{aligned}$$ Both zero modes are unphysical by the following reasons. The first term is unphysical, in the sense that this is eliminated by a gauge transformation preserving the Newton gauge (\[eq:newton-gauge\]) : $$\begin{aligned} \xi_2 = \hat{B}^{(+)}(x){{\rm e}^{-(D-3)A} \over D-3}, \quad \xi_\mu = - \hat{B}^{(+)}_{,\mu}(x) {{\rm e}^{2A} \over D-3}\int dy\ {\rm e}^{-(D-1)A}, \label{gauge_transf_newton}\end{aligned}$$ where $\xi_m$ is an infinitesimal coordinate transformation parameters. The transformation law is given in Appendix A. The second term is unphysical since it diverges at infinity and is not normalizable as illustrated in Fig.\[V\_scalar\_z\]. Next we turn to analysis for the massive KK modes. As we have mentioned above, massive modes are continuous from zero. Similarly to the tensor perturbation, properties of mode functions can be examined by analyzing the behavior of the potential in the region far from the wall. In the region where $|y|\Lambda\gg1$, the Schrödinger potential (\[Sch\_scalar\]) becomes : $$\begin{aligned} \mathcal{V}_e(z) \simeq \frac{\Lambda^2 + k\Lambda}{\left(k|z| +1\right)^2}. \label{large_y_scalar}\end{aligned}$$ This potential is very similar to the Schrödinger potential (\[large\_y\_tensor\]) for the tensor perturbation. Therefore all the massive modes are given by a linear combination of Bessel functions asymptotically at $|z|\rightarrow\infty$. Although these two Schrödinger potentials (\[large\_y\_tensor\]) and (\[large\_y\_scalar\]) have the same $z$ dependence asymptotically $|y|\Lambda\gg1$, their behaviors in the thin wall limit are very different. The potential (\[large\_y\_tensor\]) depends only on $k$ (fixed in the thin-wall limit), but not on $\Lambda$. On the other hand, the potential (\[large\_y\_scalar\]) is proportional to polynomials in $\Lambda$. Therefore, the latter diverges in thin wall limit whereas the former is finite. This can be understood as follows. The perturbation of the trace part of the metric $B$ is related to the active scalar field perturbation $\varphi_{\rm R}$ through Eq.(\[Newton\_3\]). Since all the massive KK modes associated with the active scalar field become infinitely heavy in the thin wall limit, the massive KK modes for the perturbation of the trace part of the metric freeze simultaneously. In this limit only the tensor perturbations remain which correspond to the known modes of the RS model[^9] . The zero modes $\hat{B}^{(+)}(x)\dfrac{A'}{\phi_{\rm R}'} {\rm e}^{-{D-2 \over 2}A}$ of the fluctuation of the trace part of the metric $B$ in Eq.(\[zero\_scalar\]) can be translated into the perturbation of the active scalar field $\varphi_{\rm R}$ by means of Eq.(\[Newton\_3\]) : $$\begin{aligned} \varphi_{\rm R}^{(0)}(x,y) = \hat\varphi_{\rm R}^{(+)}(x)\ \dot\phi_{\rm R} {\rm e}^{-(D-3)A} \rightarrow \hat\varphi_{\rm R}^{(+)}(x)\ \dot\phi_{\rm R},\quad \left(\kappa\rightarrow0\right).\end{aligned}$$ where $\hat\varphi_{\rm R}^{(+)}(x)\equiv (D-2)\hat{B}_{\rm R}^{(+)}(x)/2$. In weak gravity limit $(\kappa\rightarrow0)$, ${\rm e}^{A}$ reduces to a constant. Then we find that this zero mode is localized on the wall and that it corresponds to the Nambu-Goldstone boson corresponding to the spontaneously broken translational invariance. Analysis for the perturbation about $\phi_{\rm I}$ --------------------------------------------------- In our tree level solution the imaginary part of the scalar field $\phi$ vanishes identically and does not contribute to the energy momentum tensor. Therefore it does not affect the spacetime geometry. We shall call scalar fields with no nontrivial field configuration as inert field. In the linear order of perturbations we found that the fluctuation $\varphi_{\rm I}$ of this inert field decouples from any other fluctuations, as shown in Eq.(\[inert\]). In order to find the spectrum of $\varphi_{\rm I}$, we first bring Eq.(\[inert\]) into a Schrödinger form by changing the coordinate from $y$ to $z$ and redefining the field $\tilde\varphi_{\rm I}\equiv {\rm e}^{{D-2 \over 2}A}\varphi_{\rm I}$. Then we obtain $$\begin{aligned} \mathcal{H}_{\rm I}\tilde\varphi_I \equiv \left[-\partial_z^2 + \mathcal{V}_{\rm I}(z)\right]\tilde\varphi_{\rm I} = \square_{D-1}\tilde\varphi_{\rm I}, \quad \mathcal{V_{\rm I}}(z) \equiv \mathcal{V}_t(z) + {\rm e}^{2A}\dfrac{1}{2} \dfrac{\partial^2V}{\partial\phi_{\rm I}^2}\bigg|. \label{eq:inert-hamilton}\end{aligned}$$ where ${\cal V}_t(z)$ is the potential for transverse traceless part of the metric defined in Eq.(\[eq:schrod-tt\]). To obtain more concrete informations on the spectrum, we need to examine properties of each model. For our model we find $$\begin{aligned} \frac{1}{2}\frac{\partial^2V}{\partial\phi_{\rm I}^2}\bigg| &=& \Lambda^2 + \frac{\kappa^2\Lambda^4}{g^2} \left(1+2\cos^2\frac{g}{\Lambda}\phi_{\rm R}\right) - \frac{2\kappa^4\Lambda^6}{g^4} \sin^2\frac{g}{\Lambda}\phi_{\rm R} \label{eq:inert-potential}\end{aligned}$$ We shall discuss generic property of this inert scalar for the non-BPS background in Sec.\[sc:stability=nonBPS\]. If we choose the BPS solution as our background, we can rewrite the potential by using the BPS equations (\[BPS\_eq\_R\]) $$\begin{aligned} \frac{1}{2}\frac{\partial^2V}{\partial\phi_{\rm I}^2}\bigg| &=& \Lambda^2 + 2k\Lambda - 2\ddot{A} - 2\dot{A}^2,\end{aligned}$$ Then the Schrödinger potential $\mathcal{V}_{\rm I}$ takes the form : $$\begin{aligned} \mathcal{V}_{\rm I} = A'{^2} - A'' + {\rm e}^{2A}\left(\Lambda^2 + 2k\Lambda\right).\end{aligned}$$ We illustrate $\mathcal{V}_I$ in terms of $y$ in Fig.\[V\_inert\]. For vanishing gravitational coupling $\kappa \rightarrow 0$, $\mathcal{V}_{\rm I}$ reduces to a constant $\Lambda^2$, which agrees with the model of global SUSY in Ref.[@EMSS]. On the other hand, the potential $\mathcal{V}_{\rm I}$ acquires regions of negative values when $\kappa$ becomes large. ![[]{data-label="V_inert"}](V_inert.eps){width="6.5cm"} In the case of the BPS background, we can show that there are no tachyonic modes in this inert scalar sector with the aid of the SQM. Let us introduce a supercharge as $Q_{\rm I} = -\partial_z - A'$ and $Q_{\rm I}^\dagger=\partial_z - A'$. Then the Hamiltonian $\mathcal{H}_{\rm I}$ can be rewritten as $$\begin{aligned} \mathcal{H}_{\rm I} = Q_{\rm I}^\dagger Q_{\rm I} + {\rm e}^{2A}\left(\Lambda^2 + 2k\Lambda\right).\end{aligned}$$ The first term is a nonnegative definite Hermitian operator and the second term is never negative. Therefore, we can conclude that eigenvalues of $\mathcal{H}_{\rm I}$ are always nonnegative and there are no tachyonic mode. Stability of Non-BPS multi-Walls {#sc:stability=nonBPS} ================================= For non-BPS solutions, the positivity of the energy of the fluctuation and the associated stability is entirely nontrivial. In the limit of vanishing gravitational coupling, however, our supergravity model reduces to a global SUSY model that has been shown to be stable [@EMSS]. Since the mass gap in the global SUSY model should not disappear even if we switch on the gravitational coupling infinitesimally, the massive scalar fluctuations in the global SUSY model should remain massive at least for small gravitational coupling. On the other hand, we need to watch out a possible new tachyonic instability associated with the metric fluctuations. As for the transverse traceless part of the metric, we have already shown that there are two possible zero mode candidates $ \hat{h}^{\rm TT(+)}_{\mu\nu} (x)\ {\rm e}^{A(z)}$ and $\hat{h}^{\rm TT(-)}_{\mu\nu} (x)\ {\rm e}^{A(z)}\int dz\ {\rm e}^{-2A(z)}$ in Eq.(\[eq:tensor-zero-mode\]). In the non-BPS solution, we no longer need to worry about the normalizability of the wave function. Instead, we need to satisfy the boundary condition imposed by the presence of the boundary cosmological constants. To impose the boundary condition, we have to use coordinate system which is more more appropriate to specify the position of the boundary. This is achieved by going to the Gaussian normal coordinates [@TanakaMontes]. We have been using the Newton gauge to study the mass spectrum in the Shrödinger type equation. We can follow the argument in Ref.[@TanakaMontes] to obtain general coordinate transformations $\xi_m$ from the Newton gauge to the Gaussian normal gauge : $$\xi_2(x,y)={1 \over 2}\int_{0}^{y} dy' h_{22} + \bar \xi_2^{(\pm)}(x),$$ $$\xi_\mu(x,y)= -{1 \over 2}\int_{y^{(\pm)}}^ydy'e^{-2A} \int_{y^{(\pm)}}^{y'}dy'' h_{22, \mu} -\bar \xi_{2,\mu}^{(\pm)}\int_{y^{(\pm)}}^ydy'e^{-2A} + \bar \xi_\mu^{(\pm)}(x),$$ where $\bar \xi_2^{(\pm)}, \bar \xi_\mu^{(\pm)}$ depend on $x$ only. Boundary conditions in the Newton gauge are found to be [@TanakaMontes] $$\left[\partial_y h_{\mu\nu}^{TT} +2e^{-2A}\bar \xi_{2,\mu,\nu}^{(\pm)} \right]_{y^{(\pm)}-0}^{y^{(\pm)}+0} =0 , \label{const_tensor}$$ $$\left[\left(\partial_y+\dot A\right) h_{22} + 2\ddot{A} \bar \xi_2^{(\pm)} \right]_{y^{(\pm)}-0}^{y^{(\pm)}+0} =0. \label{const_scalar}$$ We find that the former mode $ \hat{h}^{\rm TT(+)}_{\mu\nu} (x)\ {\rm e}^{A(z)}$ receives no constraint from the boundary condition, and is still a physical massless mode localized on the wall which should be regarded as the graviton in the effective theory. The latter mode $\hat{h}^{\rm TT(-)}_{\mu\nu} (x)\ {\rm e}^{A(z)}\int dz\ {\rm e}^{-2A(z)}$ is constrained by the boundary condition (\[const\_tensor\]). Since the constraint (\[const\_tensor\]) and (\[const\_scalar\]) relate $\bar \xi_2^{(\pm)}$ to $\hat h^{(-)}_{\mu\nu}$ and $\hat h_{22}(x)$, this mode should be classified as a scalar type perturbation. On the other hand, $\hat h^{(-)}_{\mu\nu}$ can be gauged away through the gauge transformation (\[gauge\_transf\_newton\]), which preserves the Newton gauge[^10]. Therefore this mode becomes unphysical in the presence of the bulk scalar field like in our model. Since $\bar \xi_2$ corresponds to the physical distance between the two branes [@Charmousis:1999rg], our system automatically incorporates the stabilization mechanism without an additional bulk scalar fields. In the thin wall limit, the wall scalar field freezes out and the fluctuation of the scalar field ceases to be related to the scalar perturbation of the metric, so that $\hat h_{\mu\nu}^{(-)}$ can no longer be gauged away. As a result of restoration of $\hat h_{\mu\nu}^{(-)}$, any distance between two walls is admitted as classical solutions and the model becomes meta-stable. To evaluate the mass spectrum of massive modes, we use the small width approximation. Then the asymptotic behavior of the potential gives the wave functions expressed by means of the Bessel functions as in the Randall-Sundrum model. These eigenvalue spectrum is approximately equally spaced just like the plane wave solutions. These (almost) continuum modes should give the corrections to the effects of localized graviton, similarly to the Randall-Sundrum model. Therefore we obtain a massless graviton localized on the wall and a tower of massive KK modes for transverse traceless part of the metric. Since the active scalar field $\varphi_{\rm R}$ exhibits a mass gap without any tachyon, we expect that there should be no tachyonic instability at least for small enough gravitational coupling. We have found in Eq.(\[Newton\_3\]) that the active scalar field $\varphi_{\rm R}$ can be reduced to the trace part of the metric $B$. This implies that there should be no tachyon in the transverse traceless mode as well at least for small gravitational coupling, since both degrees of freedom represent one and the same dynamical degree of freedom. In fact, we have observed that the potential ${\cal V}_{e}$ defined in Eq.(\[eq:scalar-fluc-pot\]) is everywhere positive and has no tachyon. There are two possible zero mode candidates as given in Eq.(\[zero\_scalar\]). However, both of them are unphysical by the following reason. The first one can be eliminated by a gauge transformation. The second one has now no problem of normalization, since the extra dimension is now a finite interval, but it cannot satisfy the boundary conditions [@TanakaMontes]. To evaluate the mass of the lightest scalar particle, which is usually called radion, we use the thin-wall approximation where the wall width is assumed to be small compared to the radius of compactification $R$. To make this approximation, we separate the potential (\[eq:scalar-fluc-pot\]) for the trace part of the metric fluctuation into unperturbed and perturbation as $$\mathcal{V}_e(z) \equiv \mathcal{V}_e^{(0)}(z) + \mathcal{V}_e^{(1)}(z) ,$$ $$\mathcal{V}_e^{(0)}(z) = \phi_{\rm R}' e^{{D-4 \over 2}A}\left({ e^{-{D-4 \over 2}A} \over \phi_{\rm R}'}\right)'', \qquad \mathcal{V}_e^{(1)}(z) = {D-3 \over D-2} 2 \kappa^2 \left(\phi_{\rm R}'\right)^2,$$ where we considered in $D$-dimensions instead of $4$ dimensions. The zero-th order eigenfunction for the lowest eigenvalue is found to be $$\tilde B^{(0)}(z)= {{\cal N} \over {\rm e}^{{D-2 \over 2}A}\phi_{\rm R}'} ,$$ with the vanishing eigenvalue $(m_0^{(0)})^2=0$ and a normalization factor ${\cal N}$. The first order eigenfunction is given by $$\tilde B^{(1)}(z)= {{\cal N} \over {\rm e}^{{D-4 \over 2}A}\phi_{\rm R}^{'}} \int^z dz' {\rm e}^{(D-4)A} \phi_{\rm R}^{'2} \int^{z'} dz'' {\rm e}^{-(D-4)A} \phi_{\rm R}^{'-2} \left(\mathcal{V}_e^{(1)}(z) -(m_0^{(1)})^2\right) ,$$ where the first correction to the mass squared eigenvalue is denoted as $(m_0^{(1)})^2$. By using Eq.(\[Newton\_3\]), the trace part of the metric can be transformed into the active scalar fluctuation $\varphi_{\rm R}$ as $$\varphi_0^{(1)}(y)=-{D-2 \over \kappa^2} {\cal N} \dot\phi_{0} \int_{0}^{y_{(+)}} dy' \left({D-3 \over D-2}\kappa^2 {\rm e}^{-(D-3)A} -{(m_0^{(1)})^2{\rm e}^{-(D-1)A} \over 2 \dot\phi_{\rm R}^{2}} \right) ,$$ where the position of the wall is at $y=0$ and the boundary with the negative cosmological constant is $y_{(+)}$. To satisfy the correct boundary conditions, we have to require that this first order eigenfunction $\varphi_0^{(1)}$ should vanish at the boundary [@TanakaMontes]. This determines the first order mass squared as $$(m_0^{(1)})^2=2\kappa^2 \frac{D-3}{D-2} \frac{\displaystyle \int_0^{y_{(+)}} dy\ {\rm e}^{-A}}{ \displaystyle \int_{0}^{y_{(+)}} dy\ \dfrac{{\rm e}^{-3A}}{\dot \phi_{0}^2}},$$ Taking $D=4$ as in our model, and applying to the case of the first example of non-BPS background in Eq.(\[eq:negaive-cosm-pi/2\]) with the symmetry $Z_2\times Z_2$, we should identify $y_{(+)}=\pi R/2$ and obtain $$m^2_0 \approx 8\Lambda^2 e^{-(1+\alpha^2)\pi \Lambda R} \left(1+{3 \over 2}\alpha^2\right) 2^{\alpha^2}, \qquad \alpha^2 \equiv {\kappa^2 T^3 \over \Lambda }, \label{eq:radion-mass}$$ where $4T^3$ is the tension (energy density) of the wall. For the other background solution with the $Z_2\times Z_2$ symmetry, we should identify the boundary with the negative cosmological constant as $y_{(+)}=\pi R$, and obtain the same result. It is interesting to note that the mass scale is given by the inverse wall width $\Lambda$, and that it becomes exponentially light as a function of the distance between the walls, even though the radion mass receives a complicated gravitational corrections. It is appropriate to fix the wall tension and the gravitational coupling in taking the small width limit $\Lambda R \rightarrow \infty$. Then we obtain a simple mass formula in the limit [^11] $$m^2_0 \approx 8\Lambda^2 e^{-\pi \Lambda R} \rightarrow 0 . \label{eq:radion-mass2}$$ This characteristic feature of lightest massive scalar fluctuation is precisely the same as the global SUSY case [@MSSS2]. The lightest massive mode in that case results from the fact that two walls have no communication when they are far apart, and the translation zero modes of each wall becomes massless as the separation between walls goes to infinity. The mass spectrum of the inert scalar fluctuations $\varphi_{\rm I}$ is determined by the Schrördinger form of the eigenvalue problem (\[eq:inert-hamilton\]) with the potential $\mathcal{V_{\rm I}}(z)$. The potential has the same term as the transverse traceless mode $\mathcal{V_{\rm t}}(z)$ with an additional term $\frac{1}{2}\frac{\partial^2V}{\partial\phi_{\rm I}^2}\bigg|$ in Eq.(\[eq:inert-potential\]) which is nonnegative definite provided the gravitational coupling is not too strong $\kappa \le {g \over \Lambda}$ $$\begin{aligned} \frac{1}{2}\frac{\partial^2V}{\partial\phi_{\rm I}^2}\bigg| &=& \Lambda^2\left(1+3\frac{\kappa^2\Lambda^2}{g^2}\right) - 2\frac{\kappa^4\Lambda^4}{g^2} \left(1+ \frac{\kappa^2\Lambda^2}{g^2}\right) \sin^2\frac{g}{\Lambda}\phi_{\rm R} \nonumber \\ &\ge & \Lambda^2\left(1-\frac{\kappa^2\Lambda^2}{g^2}\right) \left(1+2\frac{\kappa^2\Lambda^2}{g^2}\right) \ge 0 . \label{eq:inert-positivity}\end{aligned}$$ Therefore inert scalar does not produce any additional tachyonic instability. Fermions ======== In the previous two sections, we focused on the stability of BPS and Non-BPS wall configurations and studied its fluctuations. In this section, we turn to the fermionic part of the model and study its fluctuation. We shall consider only the BPS solutions for simplicity, since it allows massless gravitino, whereas the non-BPS solutions do not. The part of the Lagrangian (\[SUGRA\_Lag\]) quadratic in fermion fields (with arbitrary powers of bosons) can be rewritten as $$\begin{aligned} \label{fermionlag} e^{-1}{\cal L}_{{\rm fermion}}^{{\rm quadratic}} &=& -i {\rm e}^{-A} {\bar{\chi}}{\bar{\sigma}}^{\underline{\mu}} {\cal D}_\mu \chi -i {\bar{\chi}}{\bar{\sigma}}^{\underline{2}} {\cal D}_2 \chi + \varepsilon^{\underline{\kappa 2 \mu \nu}} {\rm e}^{-3A} {\bar{\psi}}_{\kappa} {\bar{\sigma}}_{\underline{2}} \tilde{{\cal D}}_\mu \psi_{\nu} + \vep^{\underline{\kappa \lambda 2 \nu}} {\rm e}^{-2A} {\bar{\psi}}_{\kappa} {\bar{\sigma}}_{\underline{\lambda}} \tilde{{\cal D}}_2 \psi_{\nu} \nonumber \\ && + \vep^{\underline{2 \lambda \mu \nu}} {\rm e}^{-2A} {\bar{\psi}}_{2} {\bar{\sigma}}_{\underline{\lambda}} \tilde{{\cal D}}_{\mu} \psi_{\nu} + \vep^{\underline{\kappa \lambda \mu 2}} {\rm e}^{-2A} {\bar{\psi}}_{\kappa} {\bar{\sigma}}_{\underline{\lambda}} \tilde{{\cal D}}_{\mu} \psi_{2} \nonumber \\ && -\frac{\kappa}{\sqrt{2}} {\rm e}^{-A} \dot{\phi}^* \chi \sig^{\underline{\mu}} {\bar{\sigma}}^{\underline{2}} \psi_\mu - \frac{\kappa}{\sqrt{2}} {\rm e}^{-A} \dot{\phi} {\bar{\chi}}{\bar{\sigma}}^{\underline{\mu}} \sig^{\underline{2}} {\bar{\psi}}_\mu + \frac{\kappa}{\sqrt{2}} \dot{\phi}^* \chi \psi_2 + \frac{\kappa}{\sqrt{2}} \dot{\phi} {\bar{\chi}}{\bar{\psi}}_2 \nonumber \\ && - \kappa^2 {\rm e}^{{\kappa^2 \over 2}\phi^*\phi} \biggl[ P^* {\rm e}^{-2A} \psi_\mu \sig^{\underline{\mu \nu}} \psi_\nu + P {\rm e}^{-2A} {\bar{\psi}}_\mu {\bar{\sigma}}^{\underline{\mu \nu}} {\bar{\psi}}_\nu + 2 {\rm e}^{-A} (P^* \psi_\mu \sig^{\underline{\mu 2}} \psi_2 + P {\bar{\psi}}_\mu \sig^{\underline{\mu 2}} {\bar{\psi}}_2) \nonumber \\ && + {\rm e}^{-A} \frac{i\kappa}{\sqrt{2}} ( D_\phi P \chi \sig^{\underline{\mu}} \bar{\psi}_\mu + D_{\phi^*} P^* {\bar{\chi}}{\bar{\sigma}}^{\underline{\mu}} \psi_\mu) + \frac{i\kappa}{\sqrt{2}} (D_\phi P \chi \sig^{\underline{2}} {\bar{\psi}}_2 + D_{\phi^*} P^* {\bar{\chi}}{\bar{\sigma}}^{\underline{2}} \psi_2) \nonumber \\ && + {\frac{1}{2}}( {\cal D}_\phi D_\phi P \chi^2 + {\cal D}_{\phi^*} D_{\phi^*} P^* {\bar{\chi}}^2) \biggr]. \label{eq:fermion-lag}\end{aligned}$$ The terms in the fourth line is quadratic in gravitino without any derivatives, which can be regarded as mass terms for gravitino. We find that they are $Z_2$ odd under $\phi \rightarrow -\phi$. In this respect, our model provides an explicit realization of the condition to have a smooth limit of vanishing width of the wall [@BCY] and in agreement with one version of the five-dimensional supergravity on the orbifold [@FLP]. For our modified superpotential $P$ given in Eq.(\[P\_mod\]), $D_\phi P$ and ${\cal D}_\phi D_\phi P$ are of the form: $$\begin{aligned} D_\phi P &=& {\rm e}^{-\frac{\kappa^2}{2}\phi^2} \left[\kappa^2 (\phi^* - \phi) \frac{\Lambda^3}{g^2} {\rm sin} \frac{g}{\Lambda} \phi + \frac{\Lambda^2}{g} {\rm cos}\frac{g}{\Lambda}\phi \right],\\ {\cal D}_\phi D_\phi P &=& 2 \kappa^2 (\phi^*-\phi) \partial_\phi P - \left(\kappa^2 + \kappa^4 (\phi^2 - \phi^{*2}) + \frac{g^2}{\Lambda^2}\right)P, \\ &=&{\rm e}^{-\frac{\kappa^2}{2}\phi^2} \left[ -\left(\kappa^2 + \frac{g^2}{\Lambda^2} - \kappa^4 (\phi - \phi^*)^2\right) \frac{\Lambda^3}{g^2} {\rm sin}\frac{g}{\Lambda} \phi + 2 \kappa^2 (\phi^* - \phi) \frac{\Lambda^2}{g} {\rm cos}\frac{g}{\Lambda} \phi \right]. \label{eq:DDP}\end{aligned}$$ Gravitino --------- In this subsection, we will explore a massless gravitino which is a superpartner of the massless localized graviton under the SUGRA transformation with the conserved Killing spinor (\[eq:Killingspinor\]). Before studying equations of motion for gravitino, we will supertransform the wave function of the localized massless graviton to find conditions that the physical gravitino should satisfy. Let us focus on SUGRA transformation law for vierbein in Eq.(\[eq:SUGRAtransf\]), $$\delta_\zeta e_m{^{\underline{a}}} = i \kappa \left( \zeta \sigma^{\underline{a}} \bar{\psi}_m + \bar{\zeta} \bar{\sigma}^{\underline{a}} \psi_m \right). \label{vtrf}$$ The preserved SUSY along the Killing spinor $\zeta{(K)}$ in Eq.(\[eq:Killingspinor\]) with $\theta=\pi/2$ is given by $$\begin{aligned} \zeta^\alpha{(K)} &=& -i \bar{\zeta}_{\dot{\alpha}}(K) \bar{\sigma}^{2 \dot{\alpha} \alpha} = i {\rm e}^{A/2} [\epsilon_2, -\epsilon_1], \\ \bar{\zeta}_{\dot{\alpha}}(K) &=& -i \zeta^\alpha{(K)} \sigma^2_{\alpha \dot{\alpha}} = i{\rm e}^{A/2} [-\epsilon_1, -\epsilon_2]. \label{eq:Killing-spinor}\end{aligned}$$ Denoting the fluctuations $h_{mn}$ of the metric around the background spacetime metric $g_{mn}^{\rm background}\equiv diag({\rm e}^{2A}\eta_{\mu\nu}, 1)$ as $g_{mn}=g_{mn}^{\rm background} + h_{mn}$, the following linearized 3D SUGRA transformations with the Killing spinor $\zeta(K)$ are obtained for the metric fluctuations $\delta h_{mn} =\delta(e_m{}^{\underline{a}} e_{n \underline{a}}) =\delta e_m{}^{\underline{a}} e_{n \underline{a}} + e_{m\underline{a}} \delta e_{n}{}^{\underline{a}}$ $$\begin{aligned} \label{3dmntrf} \delta_{\zeta(K)} h_{\mu \nu} &=& i \kappa {\rm e}^A \zeta(K) (\sigma_{\underline{\mu}} \bar{\psi}_\nu -i \sigma^{\underline{2}} \bar{\sigma}_{\underline{\mu}} \psi_\nu + \sigma_{\underline{\nu}} \bar{\psi}_\mu -i \sigma^{\underline{2}} \bar{\sigma}_{\underline{\nu}} \psi_\mu), \\ \label{3d2mtrf} \delta_{\zeta(K)} h_{2 \mu} &=& i \kappa \zeta(K) \left\{ \sigma_{\underline{2}} \bar{\psi}_\mu + i \psi_\mu + (\sigma_{\underline{\mu}} \bar{\psi}_2 -i\sigma^{\underline{2}} \bar{\sigma}_{\underline{\mu}} \psi_2) {\rm e}^A \right\}, \\ \label{3d22trf} \delta_{\zeta(K)} h_{22} &=& 2i \kappa \zeta(K) (\sigma_2 \bar{\psi}_2 +i \psi_2). \end{aligned}$$ In sect.\[sc:tensot-perturb\], we have imposed the gauge fixing condition (Newton Gauge) for graviton $$h_{22}= - \frac{1}{3} {\rm e}^{-2A}\eta^{\mu\nu}h_{\mu\nu} \equiv - \frac{1}{3}h, \label{eq:gauge-fix-graviton1}$$ $$h_{2\mu}=0. \label{eq:gauge-fix-graviton2}$$ We can algebraically decompose $h_{\mu\nu}$ into traceless part ${\rm e}^{2A}h_{\mu\nu}^{TT}$ and trace part $h$. We have found that the localized graviton zero mode is contained in the traceless part $$\begin{aligned} \eta^{\mu \nu} h_{\mu \nu} = 0. \label{eq:traceless}\end{aligned}$$ Equation of motion shows that the localized graviton zero mode also satisfies the transverse condition: $$\begin{aligned} \eta^{\lambda \mu} \partial_\lambda h_{\mu \nu} = 0. \label{eq:transverse}\end{aligned}$$ The matter fermion of course do not have the graviton zero mode : $\varphi=0$. It is useful to decompose Weyl spinors in four dimensions into two 2-component Majorana spinors in three dimensions. For instance gravitinos $\psi_m$ are decomposed into two 2-component Majorana spinor-vectors $\psi_m^{(1)}$ and $\psi_m^{(2)}$ (real and imaginary part of the Weyl spinor-vector) $$\begin{aligned} \label{majo1} \psi_{m\alpha}^{(1)} \equiv \psi_{m\alpha} - i \sigma^2_{\alpha\dot\alpha} \bar{\psi}_m^{\dot\alpha} =-i\sigma^2_{\alpha\dot\alpha}\bar\psi_m^{(1)\dot\alpha},\end{aligned}$$ $$\begin{aligned} \label{majo2} \psi^{(2)}_{m\alpha} \equiv \psi_{m\alpha} + i \sigma^2_{\alpha\dot\alpha} \bar{\psi}_m^{\dot\alpha} =i\sigma^2_{\alpha\dot\alpha}\bar\psi_m^{(2)\dot\alpha}. \end{aligned}$$ Similarly to the traceless and trace part decomposition of graviton (symmetric tensor), gravitino (vector-spinor) can also be algebraically decomposed into its traceless part $\psi_\mu^T$ and trace part $\bar \psi$ as $$\psi_\mu=\psi_\mu^T - {1 \over 3}\sigma_{\underline{\mu}}\bar\psi, \qquad \bar \sigma^{\underline{\mu}}\psi_\mu^T=0, \qquad \bar \sigma^{\underline{\mu}}\psi_\mu=\bar\psi.$$ Let us make a SUGRA transformations of the physical state conditions (\[eq:gauge-fix-graviton1\])–(\[eq:transverse\]) for gravitons with the conserved Killing spinor $\zeta(K)$ in Eq.(\[eq:Killing-spinor\]). The SUGRA transformations with $\zeta(K)$ of Eqs.(\[eq:gauge-fix-graviton1\]), (\[eq:gauge-fix-graviton2\]) (\[eq:traceless\]) give $$\begin{aligned} 0 = \delta_{\zeta(K)} \left( h_{22} + \frac{1}{3} h\right) = 2i \kappa \zeta(K) \left( \sigma_{\underline{2}} \bar{\psi}_2^{(1)} + \frac{1}{3} {\rm e}^{-A}\sigma^{\underline{\mu}} \bar\psi_\mu^{(2)} \right), \label{eq:h22gauge-trans}\end{aligned}$$ $$\begin{aligned} 0 &=& \delta_{\zeta(K)} h_{2\mu} = i \kappa \zeta \left( \sigma_{\underline{2}} \bar{\psi}_\mu^{(1)} + {\rm e}^A \sigma_{\underline{\mu}} \bar{\psi}_2^{(2)} \right). \label{eq:gauge-fix-graviton2(2)}\end{aligned}$$ $$\begin{aligned} 0 &=& \delta_{\zeta(K)} \eta^{\mu \nu} h_{\mu \nu} = 2i\kappa {\rm e}^{-A} \zeta \sigma^{\underline{\mu}} \bar{\psi}_{\mu}^{(2)}. \label{eq:traceless-gauge-tr}\end{aligned}$$ These result suggest the most natural gauge fixing condition for local gauge SUGRA transformations $$\begin{aligned} \psi_2 =0, \label{eq:psi2-gauge}\end{aligned}$$ which can always be chosen. Then, the above gauge fixing conditions (\[eq:h22gauge-trans\])–(\[eq:traceless-gauge-tr\]) are translated as $\bar\psi_\mu^{(1)} = 0$ and the traceless condition for $\psi^{(2)}$ $$\begin{aligned} \sigma^{\underline{\mu}}\bar\psi_\mu = 0. \label{gravitino3}\end{aligned}$$ Therefore we expect[^12] that the localized massless gravitino should be contained in the traceless part of $\psi_\mu^{(2)}$. The SUGRA transformation of the remaining condition (\[eq:transverse\]) gives the transverse condition for the $\psi_\mu^{(2)}$. Similarly to the graviton case, the localized gravitino should not have matter component $$\chi=0. \label{eq:no-matter-fermion}$$ Let us now examine the equations of motion for gravitino $\psi_\mu$ coupled with the matter fermion $\chi$, which are obtained by varying the action (\[eq:fermion-lag\]). If we impose the conditions (\[eq:psi2-gauge\]), (\[gravitino3\]), and (\[eq:no-matter-fermion\]) on the gravitino equations of motion, we obtain $$\begin{aligned} 0 &=& {\rm e}^{-3A} \vep^{\underline{\mu 2 \rho \nu}} {\bar{\sigma}}_{\underline{2}} \partial_\rho \psi_\nu +{\rm e}^{-2A} \left( - {\frac{1}{2}}\dot{A} \vep^{\underline{\mu 2 \rho \nu}} {\bar{\sigma}}_{\underline{\rho}} \psi_\nu + \vep^{\underline{\mu \rho 2 \nu}} {\bar{\sigma}}_{\underline{\rho}} \partial_2 \psi_\nu - \kappa^2{\rm e}^{{\kappa^2 \over 2}\phi^*\phi} P \bar\sigma^{\mu\nu}{\bar{\psi}}_\nu \right) \nonumber \\ &=& {\rm e}^{-3A} \vep^{\underline{\mu 2 \rho \nu}} {\bar{\sigma}}_{\underline{2}} \partial_\rho \psi_\nu +{\rm e}^{-2A} \left[ - \vep^{\underline{\mu 2 \rho \nu}} {\bar{\sigma}}_{\underline{\rho}} \left(\partial_2 +{\frac{1}{2}}\dot{A}\right) \psi_\nu + \dot{A} \eta^{\mu\nu}{\bar{\psi}}_\nu \right], \end{aligned}$$ where we have used the BPS equation (\[BPS\_A\]) for background fields. Possible zero mode should give a vanishing eigenvalue for the operator in the parenthesis : $$\begin{aligned} 0 &=& - \vep^{\underline{\mu 2 \rho \nu}} {\bar{\sigma}}_{\underline{\rho}} \left(\partial_2 +{\frac{1}{2}}\dot{A}\right) \psi_\nu + \dot{A} \eta^{\mu\nu}{\bar{\psi}}_\nu =-i \eta^{\mu\nu} \bar{\sigma}^{\underline{2}} \left(\partial_2 + {\dot{A}\over 2}\right) \psi_\nu + \dot{A} \eta^{\mu\nu}{\bar{\psi}}_\nu \label{eom1}\end{aligned}$$ where $\vep^{\underline{\mu \rho 2 \nu}} {\bar{\sigma}}_{\underline{\rho}}= i ({\bar{\sigma}}^{\underline{\mu}} \sig^{\underline{2}} {\bar{\sigma}}^{\underline{\nu}} - \eta^{\underline{\mu \nu}} {\bar{\sigma}}^{\underline{2}})$ is used in the second equality. In terms of the 2-component Majorana spinors (\[majo1\]) and (\[majo2\]), we obtain $$\begin{aligned} \left(\partial_2 + \frac{3}{2}\dot A\right)\bar\psi^{(1)TT}_\mu + \left(-\partial_2 + \frac{1}{2}\dot A\right)\bar\psi^{(2)TT}_\mu = 0.\end{aligned}$$ Since $\psi^{(1)}_\mu = 0$, we obtain $$\begin{aligned} \left(-\partial_2 + \frac{1}{2}\dot A\right)\bar\psi^{(2)TT}_\mu = 0.\end{aligned}$$ Therefore, we find the gravitino zero mode in the transverse traceless part of the 2-component Majorana vector-spinor $\bar\psi^{(2)TT}_\mu$ with the wave function $$\begin{aligned} \bar\psi^{(2)TT}_\mu(y) = {\rm e}^{\frac{A(y)}{2}}.\end{aligned}$$ Now we see that the localized massless gravitino wave function is in precise agreement with that expected from the preserved SUGRA transformation with the Killing spinor $\zeta(K)$ : $$\begin{aligned} e_{\mu}{^{\underline{a}}} \sim \zeta(K) \sigma^{\underline{a}}\psi_{\mu}.\end{aligned}$$ Since the wave function of the graviton and the Killing spinor are $e_\mu{^{\underline{a}}} \sim {\rm e}^{A}$ and $\zeta(K) = {\rm e}^{\frac{A}{2}}$, we find $\psi_\mu \sim {\rm e}^{\frac{A}{2}}$. Matter Fermion -------------- In this subsection, we study the fluctuation of matter fermion. By varying the Lagrangian (\[eq:fermion-lag\]) with respect to $\chi$, we obtain the equation of motion for matter fermion $\chi$. Using the gauge choice $\psi_2 = {\bar{\psi}}_2 = 0$ to the equation of motion, we find $$\begin{aligned} 0 &=& -i{\rm e}^{A} \bar\sig^{\underline{\mu}} \partial_\mu {\chi} -\frac{3}{2}i \dot{A} {\bar{\sigma}}_2 {\chi} -i {\bar{\sigma}}^{\underline{2}} \partial_2 {\chi} -{\rm e}^{\frac{\kappa^2}{2}\phi^* \phi} {\cal D}_{\phi^*} D_{\phi^*} P^* \bar{{\chi}} \nonumber \\ &&- \frac{\kappa}{\sqrt{2}} \dot{\phi} {\bar{\sigma}}^{\underline{\mu}} \sig^{\underline{2}} {\bar{\psi}}_\mu -{\rm e}^{\frac{\kappa^2}{2}\phi^* \phi} \frac{i\kappa}{\sqrt{2}} D_{\phi^*} P^* {\bar{\sigma}}^{\underline{\mu}} \psi_\mu . \end{aligned}$$ The second line gives the mixing term between the trace part of gravitino $\bar\psi=\bar\sigma^\mu\psi_\mu$ and the matter fermion $\chi$. Using the BPS equation (\[BPS\_phi\]), the mixing term can be rewritten as $$\begin{aligned} - \frac{\kappa}{\sqrt{2}} \dot{\phi} {\bar{\sigma}}^{\underline{\mu}} \sig^{\underline{2}} {\bar{\psi}}_\mu -\frac{i\kappa}{\sqrt{2}} \dot{\phi} {\bar{\sigma}}^{\underline{\mu}} \psi_\mu = -\frac{i\kappa}{\sqrt{2}} \dot{\phi} {\bar{\sigma}}^{\underline{\mu}}\left(\psi_\mu - i\sig^{\underline{2}} \bar \psi_\mu\right) = -\frac{i\kappa}{\sqrt{2}} \dot{\phi} {\bar{\sigma}}^{\underline{\mu}}\psi_\mu^{(1)} ,\end{aligned}$$ where we used the 2-component Majorana spinor notation defined in Eqs.(\[majo1\]), (\[majo2\]). Since the mixing occurs only with $\psi_\mu^{(1)}$, it is also useful to decompose matter fermions into 2-component Majorana spinors, similarly to Eqs.(\[majo1\]), (\[majo2\]). Then the matter equation of motion is decomposed into two parts with opposite transformation property under the charge conjugation $\sigma^2$ $$-i{\rm e}^{A} \bar\sig^{\underline{\mu}} \partial_\mu {\chi}^{(2)} = i {\bar{\sigma}}^{\underline{2}} \left[ \partial_2 +\frac{3}{2} \dot{A} -{\rm e}^{\frac{\kappa^2}{2}\phi^* \phi} {\cal D}_{\phi^*} D_{\phi^*} P^* \right] {\chi}^{(1)} ,$$ $$-i{\rm e}^{A} \bar\sig^{\underline{\mu}} \partial_\mu {\chi}^{(1)} =i {\bar{\sigma}}^{\underline{2}} \left[ \partial_2 +\frac{3}{2} \dot{A} +{\rm e}^{\frac{\kappa^2}{2}\phi^* \phi} {\cal D}_{\phi^*} D_{\phi^*} P^* \right] {\chi}^{(2)} -i\sqrt2 \kappa \dot{\phi} \bar \sigma^{\underline{\mu}} \psi_\mu^{(1)} .$$ It is now clear that we have a zero mode consisting of purely $\chi^{(1)}$ : $$-i{\rm e}^{A} \bar\sig^{\underline{\mu}} \partial_\mu {\chi}^{(1)} =0 , \qquad \chi^{(2)}=0, \qquad \psi_\mu=0$$ The zero mode wave function for matter fermion is given by $$\left[ \partial_2 +\frac{3}{2} \dot{A} -{\rm e}^{\frac{\kappa^2}{2}\phi^* \phi} {\cal D}_{\phi^*} D_{\phi^*} P^* \right] {\chi}^{(1)} =0 ,$$ whose solution is given by $${\chi}^{(1)}_0 \sim {\rm e}^{-3A/2} {\rm exp} \left[ \int dy \ {\rm e}^{\frac{\kappa^2}{2}\phi^* \phi} {\cal D}_\phi D_\phi P \right], \label{matterzero}$$ Using (\[eq:DDP\]), the integral in (\[matterzero\]) in BPS case reads $$\begin{aligned} \label{intm32} \int dy \ {\rm e}^{\frac{\kappa^2}{2}\phi^* \phi} {\cal D}_\phi D_\phi P &=& -(\kappa^2 + \frac{g^2}{\Lambda^2}) \frac{\Lambda^3}{g^2} \int dy \ {\rm sin} \frac{g}{\Lambda} \phi =-(1+\kappa^2 \frac{\Lambda^2}{g^2} ) \Lambda \int dy \ {\rm tanh}(\Lambda y) \nonumber \\ &=& -(1+\kappa^2 \frac{\Lambda^2}{g^2} ) {\rm log}({\rm cosh}(\Lambda y)) .\end{aligned}$$ In the second equality, $P={\rm e}^{-\kappa^2 \phi^2/2} \frac{\Lambda^3}{g^2}{\rm sin}(g\phi/\Lambda)$ in Eq.(\[P\_mod\]) is substituted and $\phi = \phi^*$ is taken into account. In the last equality, the BPS solution $\phi = \frac{\Lambda}{g} \left( 2{\rm tan}^{-1}e^{\Lambda(y-y_0)} - \frac{\pi}{2} \right)$ in Eq.(\[BPS\_sol\]) with $n=0$ is considered. Then the zero mode wave function of matter fermion $\chi^{(1)}$ is given by $$\begin{aligned} \label{mf01} \chi^{(1)}_0 &\sim& {\rm e}^{-3A/2} \left[{\rm cosh}(\Lambda y) \right]^{-\left[1 + \kappa^2 \frac{\Lambda^2}{g^2} \right]}. \end{aligned}$$ In the weak gravity limit, the zero mode of matter fermion reduces to the Nambu-Goldstone fermion associated with the spontaneously broken SUSY [@MSSS2] $$\begin{aligned} \label{mf01-1} \chi^{(1)}_0 \to \frac{1}{{\rm cosh}(\Lambda y)}, \qquad \kappa \to 0 . \end{aligned}$$ As expected in the global SUSY limit, the wave function is localized at the wall where two out of four SUSY are broken. Let us note, however, that this zero mode of matter fermion should be unphysical except at $\kappa =0$ limit. For any finite values of $\kappa$, it should be possible to gauge away this zero mode, precisely analogously to the zero modes $\hat{B}^{(+)}$ in Eq.(\[zero\_scalar\]) in the matter scalar sector in sect.\[sc:active-scalar\]. In fact we can see that the $A$ dependence (warp factor) of the zero modes of active scalar $\hat{B}^{(+)}(x)$ and the matter fermion $\chi^{(1)}_0(x)$ agrees with the surviving SUSY transformation generated by the Killing spinor $\zeta(K)$, and will form a supermultiplet under the surviving SUGRA, since $\phi_{{\rm R}}^{(0)}(y) \sim e^{-A}, \chi^{(1)}_0(y) \sim {\rm e}^{-3A/2}$ and $\zeta(y) \sim {\rm e}^{A/2}$ $$\begin{aligned} \delta_\zeta \phi(x,y)_{{\rm R}} = \sqrt{2} \zeta(x,y) \chi^{(1)}(x,y). \end{aligned}$$ On the other hand, ${\chi}^{(2)}$ should contain another Nambu-Goldstone fermion corresponding to the SUSY charges broken by the negative tension brane, if we consider non-BPS multi-wall configurations. Noting that the mixing term is suppressed by the Planck scale $M_P$, the zero mode equation of motion for ${\chi}^{(2)}$ in weak gravity limit $\kappa \to 0$ is given by $$\begin{aligned} 0 &=& \partial_2 {\chi}^{(2)} + {\rm e}^{\frac{\kappa^2}{2}\phi^* \phi} {\cal D}_\phi D_\phi P {\chi}^{(2)}, \\ &\to& \partial_2 {\chi}^{(2)} - \Lambda {\rm sin}\frac{g}{\Lambda} \phi~{\chi}^{(2)} = \partial_2 {\chi}^{(2)} - \Lambda {\rm tanh}(\Lambda y) {\chi}^{(2)}, \quad (\kappa \to 0) \end{aligned}$$ where BPS solution $\phi = \frac{\Lambda}{g} \left( 2{\rm tan}^{-1}e^{\Lambda(y-y_0)} - \frac{\pi}{2} \right)$ is substituted in the last equality, thus the zero mode wave function becomes $${\chi}^{(2)}_0 \to {\rm cosh}(\Lambda y)~(\kappa \to 0),$$ which is not normalizable and hence unphysical even in the limit of $\kappa\rightarrow 0$. We know from the exact solution of the non-BPS two-wall solution[@MSSS2], that this wave function results when taking the limit of large radius to obtain the BPS solution. In that limit, SUSY broken on the second wall at $y=\pi R$ is restored and the corresponding Nambu-Goldstone fermion, which was localized on the second brane, becomes non-normalizable and unphysical. This is precisely our zero mode wave function $\chi^{(2)}_0$. Acknowledgements {#acknowledgements .unnumbered} ================ We thank Daisuke Ida, Kazuya Koyama, Tetsuya Shiromizu, and Takahiro Tanaka for useful discussions in several occasions. One of the authors (M.E.) gratefully acknowledges support from the Iwanami Fujukai Foundation. This work is supported in part by Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology, Japan No.13640269 (NS) and by Special Postdoctoral Researchers Program at RIKEN (NM). Appendix A ========== In this appendix we show the gauge fixing to the Newton gauge. The most general fluctuation around the background metric (\[warped\_metric\]) takes the form : $$\begin{aligned} ds^2 &=& {\rm e}^{2A}\left(\eta_{\mu\nu} + h^{\rm T}_{\mu\nu} + 2\eta_{\mu\nu}B\right)dx^\mu dx^\nu + 2 f_{\mu}\ dx^\mu dy + \left(1 - 2C\right)dy^2,\end{aligned}$$ where the trace part of the $(\mu,\nu)$ component of fluctuations is denoted as $2\eta_{\mu\nu}B$ and the traceless part is denoted as $h^{\rm T}_{\mu\nu}$. $f_\mu$ denotes the fluctuation of $(\mu,2)$ component and $-2C$ denotes the fluctuation of $(2,2)$ component. After a tedious calculation we find the linearized Ricci tensors : $$\begin{aligned} R^{(1)}_{\mu\nu} &\!\!\!=&\!\!\! \frac{1}{2}{\rm e}^{2A}\left({\rm e}^{-2A}\square_{D-1} + \partial_y^2 + (D-1)\dot{A}\partial_y + 2(D-1)\dot{A}^2 + 2\ddot{A} \right) h^{\rm T}_{\mu\nu} - h^{\rm T}_{(\mu\rho,\nu)}{^{,\rho}}\nonumber\\ &\!\!\!&\!\!\! + \eta_{\mu\nu}{\rm e}^{2A}\left({\rm e}^{-2A}\square_{D-1} + \partial_y^2 + 2(D-1)\left(\dot{A}\partial_y + \dot{A}^2 \right) + 2\ddot{A} \right)B + (D-3)B_{,\mu,\nu}\nonumber\\ &\!\!\!&\!\!\! - \eta_{\mu\nu}\dot{A} f_\rho{^{,\rho}} - \left(\partial_y + (D-3)\dot{A} \right)f_{(\mu,\nu)} + \eta_{\mu\nu}{\rm e}^{2A}\left(\dot{A}\partial_y + 2\ddot{A} + 2(D-1)\dot{A}^2 \right)C - C_{,\mu,\nu}, \\\ \nonumber\\ R^{(1)}_{\mu2} &\!\!\!=&\!\!\! - \frac{1}{2}\left(-{\rm e}^{-2A}\square_{D-1} - 2(D-1)\dot{A}^2 - 2\ddot{A} \right)f_{\mu} - \frac{1}{2}{\rm e}^{-2A}f_{\rho,\mu}{^{,\rho}}\nonumber\\ &\!\!\!&\!\!\! + (D-2)\dot{A} C_{,\mu} + (D-2)\partial_y B_{,\mu} - \frac{1}{2}\partial_y h^{\rm T}_{\mu\rho}{^{,\rho}}, \\\ \nonumber\\ R^{(1)}_{22} &\!\!\!=&\!\!\! (D-1)\left(\partial_y^2 + 2\dot{A}\partial_y \right)B - {\rm e}^{-2A}\partial_y f_{\rho}{^{,\rho}} - \left({\rm e}^{-2A}\square_{D-1} - (D-1)\dot{A}\partial_y \right)C,\end{aligned}$$ where we define $B_{,\mu} = \partial_\mu B$, $h^{\rm T}_{(\mu\rho,\nu)} = \dfrac{1}{2}\left(h^{\rm T}_{\mu\rho,\nu} + h^{\rm T}_{\nu\rho,\mu}\right)$, $f_\rho{^{,\rho}} = \eta^{\rho\lambda}\partial_\lambda f_\rho$ and $\square_{D-1} = \eta^{\rho\lambda}\partial_\rho\partial_\lambda$. We also find the linearized energy momentum tensor as follows : $$\begin{aligned} \tilde{T}^{(1)}_{\mu\nu} &\!\!\!=&\!\!\! {2 \over D-2}{\rm e}^{2A}\left[ V_{\rm R}h^{\rm T}_{\mu\nu} + \eta_{\mu\nu}\left(2V_{\rm R}B + \frac{dV_{\rm R}}{d\phi_{\rm R}}\varphi_{\rm R}\right)\right],\\ \tilde{T}^{(1)}_{\mu2} &\!\!\!=&\!\!\! 2\dot\phi_{\rm R} \varphi_{{\rm R},\mu} + {2 \over D-2}V_{\rm R}f_\mu,\\ \tilde{T}^{(1)}_{22} &\!\!\!=&\!\!\! 4 \dot\phi_{\rm R}\partial_y \varphi_{\rm R} + {2 \over D-2}\frac{dV_{\rm R}}{d\phi_{\rm R}}\varphi_{\rm R} - {4 \over D-2}V_{\rm R}C,\end{aligned}$$ where $\varphi_{\rm R}$ is the fluctuation around the background active scalar field $\phi_{\rm R}$. Notice that the fluctuation $\varphi_{\rm I}$ about the background configuration for the imaginary part $\phi_{\rm I}$ decouples from any other fields in linear order of the fluctuations. We can obtain the linearized Einstein equations by plugging these into $R^{(1)}_{mn} = -\kappa^2\tilde{T}^{(1)}_{mn}$. The above results are the most general in the sense that we do not fix any gauge for the fluctuations. As a next step, we wish to fix the gauge that simplifies the linearized equations. The “Newton” gauge is known as a candidate of such a gauge [@TanakaMontes; @CsabaCsaki]. The gauge transformation laws for the fluctuations are of the form : $$\begin{aligned} \delta h^{\rm T}_{\mu\nu} = - \hat\xi_{(\mu,\nu)} + \dfrac{2}{D-1}\eta_{\mu\nu}\hat\xi_{\rho}{^{,\rho}},\quad \delta B = - \dot{A} \xi_2 - \dfrac{1}{D-1}\hat\xi_{\rho}{^{,\rho}}, \nonumber\\ \delta f_{\mu} = - {\rm e}^{2A}\partial_y \hat{\xi}_{\mu} - \xi_{2,\mu},\quad \delta C = \partial_y \xi_{2},\quad \delta\varphi_{\rm R} = - \dot\phi_{\rm R} \xi_2,\end{aligned}$$ where $\xi_m$ is an infinitesimal coordinate transformation parameter $\delta x_m \equiv \xi_m$ and $\hat\xi_\mu \equiv {\rm e}^{-2A}\xi_\mu$. Using these four gauge freedom, we fix $f_\mu = 0$ and $(D-3)B = C$. The residual gauge transformation should satisfy $$\begin{aligned} \partial_y \hat\xi_\mu + {\rm e}^{-2A}\xi_{2,\mu}=0,\quad \left(\partial_y +(D-3)\dot{A} \right)\xi_2 = -\dfrac{D-3}{D-1}\hat\xi_\rho{^{,\rho}}. \label{residual}\end{aligned}$$ In this gauge the linearized Einstein equations take the form : $$\begin{aligned} {2} \!\!\!&\frac{1}{2}{\rm e}^{2A} \left({\rm e}^{-2A}\square_{D-1} + \partial_y^2 + (D-1)\dot{A}\partial_y + 2(D-1)\dot{A}^2 + 2\ddot{A} \right)h^{\rm T}_{\mu\nu} - h^{\rm T}_{(\mu\rho,\nu)}{^{,\rho}} + \frac{1}{D-1}\eta_{\mu\nu}h^{\rm T}_{\rho\lambda} {^{,\rho,\lambda}} \nonumber \\ \!\!\!& = - {2 \over D-2}\kappa^2 {\rm e}^{2A} V_{\rm R}h^{\rm T}_{\mu\nu}, \label{1st_gauge_1} \\ \!\!\!&{\rm e}^{2A}\left({\rm e}^{-2A}\square_{D-1} + \partial_y^2 + (3D-5)\dot{A}\partial_y + 2(D-2)\left((D-1)\dot{A}^2 + \ddot{A}\right) \right)B - \frac{1}{D-1}h^{\rm T}_{\rho\lambda}{^{,\rho,\lambda}} \nonumber \\ \!\!\!&= -{2 \over D-2}\kappa^2{\rm e}^{2A} \left(2V_{\rm R}B + \frac{dV_{\rm R}} {d\phi_{\rm R}}\varphi_{\rm R}\right),\label{1st_gauge_2}\\ \!\!\!&(D-2)\left(\partial_y + (D-3)\dot{A} \right)B_{,\mu} - \frac{1}{2}\partial_y h^{\rm T}_{\mu\rho}{^{,\rho}} = -2\kappa^2\dot\phi_{\rm R} \varphi_{{\rm R},\mu},\label{1st_gauge_3'} \\ \!\!\!&\left(-(D-3){\rm e}^{-2A}\square_{D-1} + (D-1)\partial_y^2 + (D-1)^2\dot{A}\partial_y \right)B \nonumber \\ \!\!\!& = -2\kappa^2\left( 2\dot\phi_{\rm R}\partial_y \varphi_{\rm R} + \frac{1}{D-2}\frac{dV_{\rm R}}{d\phi_{\rm R}} \varphi_{\rm R} - {2(D-3) \over D-2}V_{\rm R}B \right),\label{1st_gauge_4'}\end{aligned}$$ where the Eq.(\[1st\_gauge\_1\]) is the traceless part of the $(\mu,\nu)$ component whereas the Eq.(\[1st\_gauge\_2\]) is the trace part of it. Denoting $D(x,y) = (D-2)\left(\partial_y+(D-3)\dot{A}\right)B + 2\kappa^2\dot\phi_{\rm R} \varphi_{\rm R}$, the Eq.(\[1st\_gauge\_3’\]) is rewritten as $$\begin{aligned} D_{,\mu}(x,y) = \dfrac{1}{2}\partial_y h^{\rm T}_{\mu\rho}{^{,\rho}}.\label{1st_gauge_3}\end{aligned}$$ Summing the background Einstein equation Eq.(\[1st\_gauge\_2\]) multiplied by $(D-3){\rm e}^{-2A}$ and Eq.(\[1st\_gauge\_4’\]) gives $\left(\partial_y + (D-1)\dot{A}\right)D(x,y) = \dfrac{D-3}{2(D-1)}{\rm e}^{-2A} h^{\rm T}_{\rho\lambda}{^{,\rho,\lambda}}$. Then we find $$\begin{aligned} \displaystyle D(x,y) = E(x)\ {\rm e}^{-(D-1)A(y)} + \dfrac{D-3}{2(D-1)}{\rm e}^{-(D-1)A}\int dy\ {\rm e}^{(D-3)A} h^{\rm T}_{\rho\lambda}{^{,\rho,\lambda}} \label{1st_gauge_4}\end{aligned}$$ where $E(x)$ is an arbitrary function of $x$. At this stage, the linearized Einstein equations are Eq.(\[1st\_gauge\_1\]), (\[1st\_gauge\_2\]), (\[1st\_gauge\_3\]) and (\[1st\_gauge\_4\]). Next we attempt to eliminate the longitudinal mode of $h^{\rm T}_{\mu\nu}$ by using the residual gauge freedom. That is, we wish to set $v_\mu \equiv h^{\rm T}_{\mu\rho}{^{,\rho}} = 0$. For that purpose we first derive the equations of motion for $v_\mu$. Taking a divergence of Eq.(\[1st\_gauge\_1\]), we find $$\begin{aligned} \left(\partial_y + (D-1)\dot{A} \right)\partial_yv_\mu = \frac{D-3}{D-1} {\rm e}^{-2A}v_\rho{^{,\rho}}{_{,\mu}}.\label{eq_v}\end{aligned}$$ This equation can be solved as follows : i) taking divergence, we can determine $v_\rho{^{,\rho}}$, ii) regarding the solution $v_\rho{^{,\rho}}$ as a source, we can determine $v_\mu$. The gauge transformation law of $v_\mu$ takes the form $$\begin{aligned} \delta v_\mu = - \square_{D-1} \hat\xi_\mu - \frac{D-3}{D-1}\hat\xi_{\rho,\mu}{^{,\rho}}.\label{delta_v}\end{aligned}$$ We want to set $0 = v_\mu + \delta v_\mu$ by using the gauge transformation (\[delta\_v\]) whose $\hat\xi_\mu$ satisfies the condition (\[residual\]) for the residual gauge transformation. Notice that the equations for $v_\mu$ and $\delta v_\mu$ are identical second order differential equations since the gauge transformation law consistent with the gauge condition (\[residual\]) does not change the form of the equation (\[eq\_v\]). We can also verify this from the condition (\[residual\]) straightforwardly as follows. From Eq.(\[residual\]) we find the identity ${\rm e}^{-2A}\square_{D-1}\xi_2 = {D-1 \over D-3}\partial_y\left(\partial_y+ (D-3)\dot{A}\right)\xi_2$. Combining this and Eq.(\[delta\_v\]), we find $\delta v_\rho{^{,\rho}} = {2(D-2)(D-1) \over (D-3)^2} {\rm e}^{2A}\left(\partial_y+(D-1)\dot{A}\right) \partial_y \left(\partial_y+(D-3)\dot{A}\right) \xi_2$ and $\partial_y\delta v_\mu = {2(D-2) \over D-3}\partial_y \left(\partial_y + (D-3)\dot{A}\right)\xi_{2,\mu}$. Hence, $\delta v_\mu$ satisfies just the same equation as Eq.(\[eq\_v\]) : $$\begin{aligned} \left(\partial_y + (D-1)\dot{A}\right) \partial_y\delta v_\mu = \frac{D-3}{D-1}{\rm e}^{-2A} \delta v_\rho{^{,\rho}}{_{,\mu}}.\end{aligned}$$ Therefore, $v_\mu$ can be eliminated in the gauge, if we can set at a given $y=y_0$ surface $$\begin{aligned} v_\mu = - \delta v_\mu,\quad \partial_y v_\mu = - \partial_y \delta v_\mu.\label{tt_condition}\end{aligned}$$ To clear matters, we introduce new functions $\Lambda_\mu(x) \equiv \hat\xi_\mu(x,y_0)$, $\Xi_\mu(x) \equiv \left(\partial_y \hat\xi_\mu\right)(x,y_0)$, $\Gamma(x) \equiv \xi_2(x,y_0)$, $\Delta(x) \equiv \left(\partial_y \xi_2\right)(x,y_0)$ which are defined at $y=y_0$ surface. In terms of these functions Eq.(\[tt\_condition\]) can be rewritten as $$\begin{aligned} \square_{D-1}\Lambda_\mu + \frac{D-3}{D-2}\Lambda_{\rho,\mu}{^{,\rho}} = \mathcal{A}_\mu,\quad \square_{D-1}\Xi_\mu + \frac{D-3}{D-2}\Xi_{\rho,\mu}{^{,\rho}} = \mathcal{B}_\mu,\label{const_surface}\end{aligned}$$ where $\mathcal{A}_\mu(x) \equiv v_\mu(x,y_0)$ and $\mathcal{B}_\mu(x) \equiv \left(\partial_y v_\mu\right)(x,y_0)$. Similarly, the gauge condition (\[residual\]) at $y=y_0$ surface can be rewritten as $$\begin{aligned} \Xi_{\mu} + {\rm e}^{-2A}\Gamma_{,\mu} = 0,\quad \Delta + (D-3)\dot{A}\Gamma = -\frac{D-3}{D-1}\Lambda_\rho{^{,\rho}}.\label{residual2}\end{aligned}$$ $\Lambda_\mu$ and $\Xi_\mu$ can be determined similarly to the Eq.(\[eq\_v\]). Next, we determine $\Gamma$ from the first equation of Eq.(\[residual2\]). However, this equation does not necessarily have a solution for a general function $\Xi_\mu$. To see this in detail, plug this into the second equation of Eq.(\[const\_surface\]) and we obtain $\square_{D-1}\Gamma_{,\mu} = - \dfrac{D-2}{2D-5}{\rm e}^{2A}\mathcal{B}_\mu$. This equation can be solved if and only if $\mathcal{B}_\mu$ is expressed as a gradient of some function. In our case we obtain $\mathcal{B}_\mu = 2\partial_\mu D$ from Eq.(\[1st\_gauge\_3\]). Hence, a solution $\Gamma$ of the first equation of Eq.(\[residual2\]) exists. At the end, $\Delta$ is determined from the second equation of Eq.(\[residual2\]). In this gauge, we obtain $D(x,y) = E{\rm e}^{-(D-1)A}$ where $E$ is a constant from Eq.(\[1st\_gauge\_3\]) and (\[1st\_gauge\_4\]). We set $E=0$ since we require that the fluctuations $B$ and $\varphi_{\rm R}$ should vanish at infinity $|x|\rightarrow\infty$ on the wall. Thus we established our gauge choice (Newton gauge) and the constraints for the residual gauge transformations are (\[residual\]) and $$\begin{aligned} \square_{D-1}\hat\xi_\mu + \frac{D-3}{D-1} \hat\xi_\rho{^{,\rho}}{_{,\mu}}=0.\end{aligned}$$ Appendix B ========== For a special case where $k\Lambda^{-1}$ is an integer[^13], we can express the Schrödinger potential in terms of $z$ explicitly. As an illustrative example, let us take $k=\Lambda$, where we find (putting $y_0=0$) ${\rm e}^{A}=(\cosh ky)^{-1}$, $z=k^{-1}\sinh ky$, $A(z)=-\dfrac{1}{2}\log\left(1+k^2z^2\right)$. The Schrödinger potential for the tensor perturbation takes the form : $$\begin{aligned} \mathcal{V}_t(z) = - \frac{k^2(1-2k^2z^2)}{(1+k^2z^2)^2}.\end{aligned}$$ There remains only one parameter controlling both the width of the wall and the magnitude of the gravitational coupling, similarly to Ref.[@Gremm:1999pj]. 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We show this in Appendix B [^9]: Generically speaking, the fluctuations of the inert scalar field $\varphi_{\rm I}$ can be an exception depending on the potential, although the inert scalar $\varphi_{\rm I}$ in our model is also frozen in the thin wall limit. [^10]: For simplicity, we have assumed $Z_2$ parity of the metric perturbation to be even. [^11]: This mass squared is factor two larger compared to the value of lightest massive scalar in the global SUSY model[@MSSS2]. We have not understood this discrepancy. [^12]: One should have in mind that it is desirable to choose a gauge fixing condition for SUGRA transformations to become a supertransformation under $\zeta(K)$ of the gauge fixing condition for general coordinate transformations. However, it may not be logically mandatory. [^13]: Then we can no longer take the thin wall limit of $\Lambda\rightarrow \infty$ with $k$ fixed.

--- abstract: | Given a class $\clC$ of models, a binary relation $\clR$ between models, and a model-theoretic language $L$, we consider the modal logic and the modal algebra of the theory of $\clC$ in $L$ where the modal operator is interpreted via $\clR$. We discuss how modal theories of $\clC$ and $\clR$ depend on the model-theoretic language, their Kripke completeness, and expressibility of the modality inside $L$. We calculate such theories for the submodel and the quotient relations. We prove a downward Löwenheim–Skolem theorem for first-order language expanded with the modal operator for the extension relation between models. Keywords: *modal logic, modal algebra, robust modal theory, logic of submodels, logic of quotients, logic of forcing, provability logic, model-theoretic logic* author: - 'Denis I. Saveliev' - 'Ilya B. Shapirovsky' bibliography: - 'modmod.bib' title: 'On modal logics of model-theoretic relations[^1]' --- Introduction {#introduction .unnumbered} ============ We consider modal systems in which the modal operator is interpreted via a binary relation on a class of models. Many instances of such systems can be found in the literature. During the last years, modal logics of various relations between models of set theory have been studied, see, e.g., [@Hamkins03; @HamkLowe; @BlockLowe2015; @ForcStruct2015; @innerMods2016]. A well established area in provability logic deals with modal axiomatizations of relations between models of arithmetic (and between arithmetic theories), see, e.g., [@Shavrukov1988; @Berarducci1990; @Ignatiev93; @VisserBigFirst; @Visser14; @Henk2015; @HamkinsArithmeticPotentialism2018]. In another extensively studied area modalities are interpreted by relations between Kripke and temporal models, see, e.g., [@Veltman96; @agostino_hollenberg_2000; @TemporalSubst2013] or the monograph [@vanBenthem2014]. In [@BarwiseVBenthem1999], the consequence along an abstract relation between models is studied, which is closely related to our consideration. Let $f$ be a unary operation on sentences of a model-theoretic language $L$, and $T$ a set of sentences of $L$ (e.g., the set of theorems in a given calculus, or the set of sentences valid in a given class of models). Using the propositional modal language, one can consider the following “fragment” of $T$: variables are evaluated by sentences of $L$, and $f$ interprets the modal operator; the [*modal theory of $f$ on $T$*]{}, or just the [*$f$-fragment of $T$*]{}, is defined as the set of those modal formulas which are in $T$ under every valuation. A well-known example of this approach is a complete modal axiomatization of formal provability in Peano arithmetic given by Solovay [@Solovay1976]. Another important example is the theorem by Hamkins and Löwe axiomatizing the modal logic of forcing (introduced earlier by Hamkins in [@Hamkins03]) where the modal operator expresses satisfiability in forcing extensions [@HamkLowe]. Both these modal systems have good semantic and algorithmic properties; in particular, they have the finite model property, are finitely axiomatizable, and hence decidable. These examples inspire the following observation. Let $\clC$ be an arbitrary class of models of the same signature, $T=\Th^L(\clC)$ the theory of $\clC$ in a model-theoretic language $L$, and $\clR$ a binary relation on $\clC$. Assuming that the satisfiability in $\clR$-images of models in $\clC$ can be expressed by an operation $f$ on sentences of $L$, i.e., for every sentence $\vf$ of $L$, and every $\stA\in\clC$, > $\stA\mo f(\varphi)$ (“$\varphi$ is possible at $\stA$”) iff $\stB\mo\vf$ for some $\mathfrak B$ with $\stA\,\clR\,\stB$, we can define the [*modal theory of $\clR$ in $L$*]{} as the $f$-fragment of $T$. In the general frame semantics, this modal theory is characterized by an enormous structure $ (\mathcal C,\mathcal R,\mathcal C_\vf: \vf\text{ is a~sentence of~}L) $ where $\clC_\vf$ is the class of models in $\clC$ validating $\vf$. We can also define the [*modal Lindenbaum algebra of $\Th^L(\clC)$ and $\clR$*]{}, i.e., the Boolean algebra of sentences of $L$ modulo the equivalence on $\clC$, endowed with the modal operator induced by $f$.[^2] In Section \[sec:defs\], we provide formal definitions and basic semantic tools for such modal theories. In particular, the algebra of $\Th^L(\clC)$ and $\clR$ can be represented as the modal algebra of a general frame consisting of complete theories $\{\Th^L(\stA):\stA\in\clC\}$. We use this in Section \[sec:sub\], where we calculate modal logics of the submodel relation $\supmod$; to express the satisfiability in submodels, we use second-order language. In Section \[sec:upanddown\],[^3] we discuss the situation when the $\clR$-satisfiability is not expressible in a language $K$ (for example, this situation is typical when $K$ is first-order). In this case the modal algebra of $\Th^K(\clC)$ can be defined as the subalgebra of the modal algebra of $\Th^L(\clC)$ generated by the sentences of $K$, for any $L$ stronger than $K$ where the $\clR$-satisfiability on $\clC$ is expressible: the resulting modal algebra (and hence, its modal logic) does not depend on the way how we extend the language $K$. Under a natural assumption on $\clC$ and $\clR$, such an $L$ can always by constructed, and hence, [*the*]{} modal algebra of $\Th^K(\clC)$ and $\clR$ is well-defined for arbitrary $K$ (however, the resulting modal logic is not necessarily a “fragment” of the theory $\Th^K(\clC)$ anymore). Then we consider the finitary first-order language expanded with the modal operator for the extension relation $\submod$ and prove a version of the downward Löwenheim–Skolem theorem for this language. In general, modal theories of $\clR$ depend on the model-theoretic language we consider. We say that a modal theory of $\clR$ is [*robust*]{} iff making the language stronger does not alter this theory (intuitively, the robust theory can be considered as the “true” modal logic of the model-theoretic relation $\mathcal R$). We discuss this notion in Section \[sec:rob\]. In Theorem \[thm: robust logicGOOD\], we show that under a certain assumption on $\clC$ and $\clR$, the robust logic is Kripke complete. Then we use this theorem to describe robust theories of the quotient and the submodel relations on certain natural classes. A preliminary report on some results in Sections \[sec:defs\] and \[sec:sub\] can be found in [@SavelievShapirovsky2016]. Preliminaries ============= To simplify reading, as a rule we denote the syntax of our object languages differently: we use inclined letters if they are related to model theory ($x,y,\ldots$ for individual variables, $\varphi,\psi,\ldots$ for formulas), and upright letters if they are related to modal logic ($\mathsf p,\mathsf q,\ldots$ for propositional variables, $\upvarphi,\uppsi,\ldots$ for formulas). #### Model-theoretic languages The languages we use for model theory are model-theoretic languages in sense of [@BarwiseFeferman] (where they are called “model-theoretic logics”). We are pointing out here only two of their features, which will be essential for the further investigations: if $L$ is a model-theoretic language under our consideration, then: - Satisfiability in $L$ is preserved under isomorphisms; - $L$ includes $L_{\omega,\omega}$, the standard first-order language with usual finitary connectives and quantifiers. Also, we assume that $L$ is a set unless otherwise specified. For a model $\stA$, $\Th^L(\stA)$, or just $\Th(\stA)$, denotes its theory in $L$, i.e., the set of all sentences of $L$ holding in $\stA$; likewise for a class of models. #### Modal logic {#subs:basicML} By a [*modal logic*]{} we shall mean a normal propositional unimodal logic (see, e.g., [@BRV-ML] or [@CZ]). Modal formulas are built from a countable set $\PV$ of propositional variables $\mathsf{p},\mathsf{q},\ldots$, the Boolean connectives, and the modal connectives $\Di$, $\Box$. We assume that $\bot$, $\imp$, $\Di$ are basic, and others are abbreviations; in particular, $\Box$ abbreviates $\neg\,\Di\,\neg$. Let $\ML$ denote the set of modal formulas. A set of modal formulas is called a [*normal modal logic*]{} iff it contains all classical tautologies, two following modal axioms: $$\neg\,\Di\bot \;\;\text{and}\;\; \Di(\mathsf p\vee\mathsf q)\to\Di\mathsf p\vee\Di\mathsf q,$$ and is closed under three following rules of inference: $$\text{Modus Ponens}\; \frac{\mvf,\;\mvf\to\mpsi}{\mpsi}, \;\; \text{Substitution}\; \frac{\mvf(\mathsf p)}{\mvf(\mpsi)}, \;\; \text{Monotonicity}\; \frac{\mvf\to\mpsi}{\Di\mvf\to\Di\mpsi}.$$ A [*general frame*]{} $\frF$ is a triple $(W,R,\clV)$ where $W$ is a nonempty set, $R\subseteq W\times W$, and $\clV$ is a subalgebra of the powerset algebra $\clP(W)$ closed under $R^{-1}$ (i.e., such that $R^{-1}(A)=\{x\in W:\exists y\in A\,\,x R y\}$ is in $\clV$ whenever $A$ is in $\clV$). By a [*Kripke frame*]{} $(W,R)$ we mean $(W,R,\clP(W))$. We shall often use the term [*frame*]{} for a general frame (and never for a Kripke frame). A [*modal algebra*]{} is a Boolean algebra endowed with a unary operation that distributes w.r.t. finite disjunctions. The [*modal algebra of a frame*]{} $(W,R,\clV)$ is $(\clV,R^{-1})$. A [*Kripke model*]{} $\mathfrak M$ on a frame $\mathfrak F=(W,R,\clV)$ is a pair $(\mathfrak F,\theta)$ where $\theta:\PV\to\clV$ is a [*valuation*]{}. The [*truth relation*]{} $\mM,x\mo \mvf$ (“$\mvf$ is true at $x$ in $\mM$”) is defined in the standard way (see, e.g., [@CZ]); in particular, $\mM,x\mo\Di\mvf$ iff $\mM,y\mo\mvf$ for some $y$ with $xRy$. A formula $\mvf\in\MF$ is [*true in a model $\mM$*]{} iff it is true at every $x$ in $\mM$, and $\mvf$ is [*valid in a frame $\frF$*]{} iff it is true in every model on $\mathfrak F$. A modal formula $\mvf$ is [*valid in a modal algebra*]{} $\stA$ iff $\mvf=\top$ holds in $\stA$. In particular, it follows that if $\stA$ is the modal algebra of a frame $\frF$, then $\mvf$ is valid in $\stA$ iff $\mvf$ is valid in $\frF$. The set $\MLog(\frF)$ of all valid in $\frF$ formulas is called the ([*modal*]{}) [*logic of the frame $\frF$*]{}; likewise for algebras. It is well-known (see, e.g., [@BRV-ML] or [@CZ]) that a set of formulas is a consistent normal logic iff it is the logic of a general frame iff it is the logic of a non-trivial modal algebra. In our paper we also consider modal logics of general frames $(\mathcal C,\mathcal R,\mathcal V)$ with a proper class $\mathcal C$; see Remarks \[rem:claclaclass1\] and \[rem:claclaclassGeneral\] in the next section. ${{{\mathrm{S4}}}}$ is the smallest logic containing the formulas $\Di\Di\mathsf p\imp\Di\mathsf p$ and $\mathsf p\imp\Di\mathsf p$, ${{{\mathrm{S4.2}}}}$ is ${{{\mathrm{S4}}}}+\Di\Box\mathsf p\;{\rightarrow}\;\Box\Di\mathsf p$, and ${{{\mathrm{S4.2.1}}}}$ is $ {{{\mathrm{S4}}}}+ \Box\Di\mathsf p\;{\leftrightarrow}\;\Di\Box\mathsf p $, where $\Lambda+\mvf$ is the smallest logic that includes $\Lambda\cup\{\mvf\}$. Modal theories of model-theoretic relations {#sec:defs} =========================================== #### Definitions Fix a signature $\Omega$ and a language $L=L(\Omega)$ based on $\Omega$. Let $L_s$ be the set of all sentences of $L$. Consider a unary operation $f$ on sentences of $L$. A ([*propositional*]{}) [*valuation in $L$*]{} is a map from $\PV$ to $L_s$. A valuation $\vl$ extends to the set $\MF$ of modal formulas as follows: $$\begin{aligned} { \| \bot \|} &= \bot, \\ { \| \mvf\to\mpsi \|} &= { \| \mvf \|}\to{ \| \mpsi \|}, \\ { \| \Di\mvf \|} &= f({ \| \mvf \|}).\end{aligned}$$ Given a set of sentences $T\subseteq L_s$, we define the [*modal theory of $f$ on $T$*]{} $\mathrm{MTh}(T,f)$ as the set of modal formulas $\mvf$ such that for every valuation $\vl$ in $L$, ${ \| \mvf \|}$ is in $T$. Thus, modal formulas can be viewed as axiom schemas (for sentences) and we can think of the modal theory of $f$ on $T$ as a fragment of $T$. Let $L$ be the usual first-order finitary language $L_{\omega,\omega}$. 1\. Let $\Omega$ be the signature of arithmetic and $f(\varphi)$ express the consistency of a sentence $\varphi$ in Peano arithmetic ${\mathrm{PA}}$. By the well-known Solovay’s results [@Solovay1976], the modal theory of $f$ on $\PA$ is the Gödel–Löb logic ${\mathrm{GL}}$, and the modal theory of $f$ on the true arithmetic ${\mathrm{TA}}$, the set of all sentences that are true in the standard model of arithmetic, is the (quasi-normal but not normal) Solovay logic ${\mathrm{S}}$. A survey on provability logics can be found in [@ArtemovBekl04Prov]. 2\. Let $\Omega$ be the signature of set theory (i.e., $\Omega=\{\in\}$) and $f(\varphi)$ express that $\varphi$ holds in a generic extension. By Hamkins and Löwe [@HamkLowe], the modal theory of $f$ on ${\mathrm{ZFC}}$ is ${\mathrm{S4.2}}$. In our work, we are interested in the case when $T$ is the theory of some class $\clC$ of models of $\Omega$. An easy observation shows that in this case the modal theory is closed under the rules of Modus Ponens and Substitution. Notice that it is not necessarily closed under Monotonicity, as the instance of the Solovay logic ${\mathrm{S}}$ shows.[^4] However, as we shall see, the modal theory is a normal logic whenever $f$ expresses the satisfiability in images of a binary relation between models. Let $\clR$ be a binary relation on $\clC$ (or, perhaps, on a larger class $\clC'\supseteq\clC$). The [*$\clR$-satisfiability on $\clC$ is expressible in*]{} $L$ iff there exists $f:L_s\to L_s$ such that for every sentence $\vf\in L_s$ and every model $\stA\in\clC$, $$\label{eq:r-sat} \stA\vDash f(\vf) \;\Iff\; \exists\,\stB\in\clC\; (\stA\,\clR\,\stB\;\&\;\stB\vDash\vf).$$ Some examples of such expressibility and non-expressibility will be given below. The following is straightforward: If $f$ and $f'$ both express the $\mathcal R$-satisfiability on $\clC$, and $T=\Th(\clC)$, then $\mathrm{MTh}(T,f)=\mathrm{MTh}(T,f').$ Thus, in this case, $\mathrm{MTh}(T,f)$ does not depend on the choice of $f$. \[def:exprMTH\] Let the $\mathcal R$-satisfiability be expressible in $L$ on a class $\mathcal C$ of models. The [*modal theory of $(\mathcal C,\mathcal R)$ in $L$*]{}, denoted by $\MTh^L(\mathcal C,\mathcal R)$, is the set of modal formulas $\mvf$ such that $\clC\mo{ \| \mvf \|}$ for every propositional valuation in $L$. In this section, we just write $\mathrm{MTh}(\clC,\clR)$ assuming $L$ is fixed. Given a propositional valuation $\vl$ in $L$, consider $\mM=(\clC,\clR,\theta)$, the enormous “Kripke model” with $\theta(\mathsf p)=\{\stA\in\clC:\stA\mo{ \| \mathsf p \|}\}$. By a straightforward induction on a modal formula $\mvf$, for every $\stA$ in $\clC$ $$\stA\mo{ \| \mvf \|} \;\Iff\; \mM,\stA\mo\mvf,$$ where $\mo$ on the left-hand and on the right-hand side of the equivalence denotes the satisfiability relation in model theory and the truth relation of a modal formula in a Kripke model, respectively. It follows that $$\clC\mo{ \| \mvf \|} \;\Iff\; \mM,\stA\mo\mvf\text{ for all }\stA\in\clC.$$ Let $\Mod(\psi)$ be the class of models of $\psi\in\L_s$, and $\clC_\psi=\Mod(\psi)\cap \clC$. Then validity of ${ \| \mvf \|}$ in $\clC$ for all propositional valuations in $L$ can be considered as validity of the modal formula $\mvf$ in an enormous “general frame of models” $(\clC,\clR,\clV)$ where $\clV$ “consists” of $\clC_\psi$ with $\psi\in L_s$. \[rem:claclaclass1\] The collection $\mathcal V$ looks like a “class of classes”. In fact, however, $\mathcal V$ is a usual class defined by a formula of set theory with two parameters: if $\mathcal C$, $L_s$ are defined by formulas $\varPhi$, $\varPhi'$, respectively, then what we understand by $\mathcal V$ is the class of pairs $(\mathfrak A,\psi)$ defined by a formula $\varUpsilon$ that is constructed from $\varPhi$, $\varPhi'$ and expresses the satisfaction relation between models in $\mathcal C$ and sentences in $L_s$. Thus subclasses $\mathcal C_\psi$ of $\clC$ playing the role of “elements” of $\mathcal V$ are in fact defined by $\varUpsilon$ with a fixed second argument $\psi$. This allows us to formally extend the definition of validity in a (set) frame to the case of the “frame of models” $(\clC,\clR,\clV)$. It follows that the modal logic $\MLog(\mathcal C,\mathcal R,\mathcal V)$, i.e., the set of all modal formulas that are valid in $(\mathcal C,\mathcal R,\mathcal V)$, coincides with the modal theory of $(\mathcal C,\mathcal R)$ in $L$. Namely, we have \[thm:big-as-general\] If the $\mathcal R$-satisfiability on $\mathcal C$ is expressible in $L$, then $$\label{eq:big-as-general} \mathrm{MTh}(\mathcal C,\mathcal R)= \MLog(\mathcal C,\mathcal R,\mathcal V).$$ Consequently, $\MTh(\mathcal C,\mathcal R)$ is a normal logic. \[prop:extendingexpr\] Assume that $f$ expresses the $\clR$-satisfiability on $\clC$ in $L$. 1. If $\clD\subseteq \clC$ is $\clR$-upward closed (i.e., $\stA\in \clD\;\&\;\stA\,\clR\,\stB\;\&\;\stB\in\clC$ implies $\stB\in\clD$), then the $\clR$-satisfiability on $\clD$ is expressible in $L$ by $f$. 2. If $\psi$ is a sentence of $L$, then the $\clR$-satisfiability on $\clC\cap\Mod(\psi)$ is expressible in $L$ by $g:\vf\mapsto f(\vf\wedge\psi)$. Let $\clR^*(\stA)$ denote $\bigcup_{n<\omega} \clR^n(\stA)$, the least $\clR$-closed $\clD\subseteq\clC$ containing $\stA$. From (\[eq:big-as-general\]) and the generated subframe construction (see, e.g., [@CZ Section 8.5]), we obtain \[cor:gener\] If the $\clR$-satisfiability on $\clC$ is expressible in $L$, then $ \MTh(\clC,\clR)\,=\, \bigcap\limits_{\mathfrak A\in\clC} \MTh(\clR^*(\mathfrak A),\clR). $ #### Frames of theories Assume that the $\clR$-satisfiability is expressible on $\clC$ in $L$ by some $f:L_s\to L_s$. We have observed in Theorem \[thm:big-as-general\] that $\MTh(\mathcal C,\mathcal R)$ can be viewed as the modal logic of a general frame of models. Now we provide other semantic characterizations of $\MTh(\mathcal C,\mathcal R)$. Put $\Ths=\{\Th(\stA): \stA\in\clC\}$. Note that $\Ths$ is a set since $L$ is assumed to be a set. For theories $T_1,T_2 \in \Ths$, let $$\begin{aligned} T_1\,\clR_{\Ths}\,T_2 &\tiff& \EE\,\stA_1,\stA_2\in\clC\,\; (\stA_1\mo T_1\;\&\;\stA_2\mo T_2\;\&\;\stA_1\clR\,\stA_2), \\ T_1\,\Rmax\,T_2 &\tiff& \AA\vf\in T_2\;\,f(\vf)\in T_1. $$ Observe that $\Rmax$ does not depend on the choice of $f$, and $\clR_{\Ths}\subseteq\Rmax$. For $\vf\in L_s$, put $\Ths_\vf=\{T\in\Ths:\vf\in T\}$. Clearly, $\Ths_{\vf\con\psi}=\Ths_{\vf}\cap\Ths_{\psi}$ and $\Ths_{\neg\,\vf}=\Ths\setminus\Ths_{\vf}$. Consider an arbitrary binary relation $\filtR$ such that $$\clR_{\Ths}\subseteq\filtR\subseteq\Rmax.$$ It follows from the definitions that $\filtR^{-1}(\Ths_\vf)=\Ths_{f(\vf)}$. Therefore, $$\clA=\{\Ths_{\vf}:\vf\in L_s\}$$ is a subalgebra of the powerset algebra $\clP(\Ths)$ closed under $\filtR^{-1}$, and $(\Ths,\filtR,\clA)$ is a general frame. We call $(\Ths,\clR_{\Ths},\clA)$ and $(\Ths,\Rmax,\clA)$ the [*minimal*]{} and the [*maximal frames of theories for $\clC,\clR$, and $L$*]{}. The relation $\filtR$ can be viewed as a [*filtration of $\clR$*]{} (see, e.g., [@Goldblatt92 Section 4]). The frame $(\Ths,\Rmax,\clA)$ is known as the [*refinement*]{} of $(\clC,\clR,\mathcal V)$ (cf. [@CZ Chapter 8]). For sentences $\vf,\psi$, put $\vf\approx\psi$ iff $\clC\mo\vf\lra \psi$. Let $L/{\approx}$ be the Lindenbaum algebra of the theory $\Th(\clC)$, i.e., the set $L_s$ of sentences of $L$ modulo $\approx$ with operations induced by Boolean connectives. We have $f(\vf)\approx f(\psi)$ whenever $\vf\approx\psi$, hence, $f$ induces the operation $f_\approx$ on $L/{\approx}$. It is easy to see that $(L/{\approx},f_\approx)$ is a modal algebra. It is called the [*modal (Lindenbaum) algebra of the language $L$ on $(\clC,\clR)$*]{}. The above arguments yield \[thm:minmax-ths\] If $\clR_{\Ths}\subseteq \filtR\subseteq\Rmax$, then the modal algebras $(L/{\approx},f_\approx)$ and $(\clA,\filtR^{-1})$ are isomorphic. The isomorphism takes the $\approx$-class of a sentence $\vf$ to $\Ths_{\vf}$. \[thm:maintoolnew\] Assume that the $\clR$-satisfiability on $\clC$ is expressible in $L$. Let $\clR_{\Ths}\subseteq \filtR \subseteq \Rmax$. Then $$\MTh(\clC,\clR)= \MLog(\Ths,\filtR,\clA)= \MLog(L/{\approx},f_\approx).$$ Every valuation $\vl$ in $L$ can be viewed as a valuation $\theta$ in the frame $\frF=(\Ths,\filtR,\clA)$, and vice versa. By induction on $\mvf$, for every $\stA$ in $\clC$ we have $ \stA\mo{ \| \mvf \|}\tiff (\frF,\theta){,}\Th(\stA)\mo\mvf. $ Thus $\MTh(\clC,\clR)=\MLog(\Ths,\filtR,\clA)$. The second equality immediately follows from Theorem \[thm:minmax-ths\]. \[prop: finite frame\] If $\Ths$ is finite, then $\mathrm{MTh}(\mathcal C,\mathcal R)$ is the logic of the Kripke frame $(\Ths,\clR_{\Ths})$. In this case $\clA=\clP(\Ths)$. The family $\Ths$ of complete theories can be viewed as the quotient of $\clC$ by the $L$-equivalence $\equiv$ where $\stA\equiv \stB$ iff $\Th(\stA)=\Th(\stB)$. Theorem \[thm:maintoolnew\] can be generalized for the case of any equivalence $\sim$ on $\clC$ finer than $\equiv$; in particular, it holds for the isomorphism equivalence $\isom$ on models, or for the equivalence in a stronger model-theoretic language. Namely, we let: $$\begin{aligned} \,[\stA]_\sim\clR_\sim[\stB]_\sim &\,\Iff\,& \exists\,\mathfrak A'\sim\mathfrak A\; \exists\,\mathfrak B'\sim\mathfrak B\;\, \mathfrak A'\,\mathcal R\,\mathfrak B', \\ \,[\stA]_\sim\Rsimmax[\stB]_\sim &\Iff& \forall\,\varphi\in L_s\;\, (\mathfrak B\vDash\varphi\,\Rightarrow\, \mathfrak A\vDash f(\varphi));\end{aligned}$$ the algebra of valuations is defined as the collection $\clV_\sim$ “consisting” of classes $\clC_\psi/{\sim}$ for $\psi\in L_s$. (Again, since $[\mathfrak A]_\sim$ are in general proper classes, $\clV_\sim$ looks like a “class of classes of classes” but actually is nothing but a three-parameter formula; cf. Remarks \[rem:claclaclass1\] and \[rem:claclaclassGeneral\].) As in the proof of Theorem \[thm:maintoolnew\], for $\clR_\sim\subseteq\filtR\subseteq \Rsimmax$ one can obtain that $$\label{eq:quot} \MTh(\clC,\clR)=\MLog(\clC/{\sim},\filtR,\clV_\sim).$$ To the best of our knowledge, Theorem \[thm:maintoolnew\] (or its analogue (\[eq:quot\])) has never been formulated explicitly before, although similar constructions related to frames of arithmetic theories were considered earlier (V.Yu. Shavrukov, an unpublished note, 2013; [@Henk2015 Remark 3]). We conclude this section with a continuation of Remark \[rem:claclaclass1\]. \[rem:claclaclassGeneral\] It is possible to give a general definition of frames that are classes and their logics (in our metatheory which is assumed to be $\ZFC$ where “classes” are shorthands for formulas). We outline the idea and postpone details for a further paper. Below capital Greek letters $\varPhi,\varPsi,\ldots$ denote formulas of the metatheory. Assume that $\mathcal C$, $\mathcal L$, $\mathcal R$ are (arbitrary) classes defined by formulas $\varPhi$, $\varPhi'$, $\varPsi$, respectively: $\mathcal C=\{x:\varPhi(x)\}$, $\mathcal L=\{y:\varPhi'(y)\}$, and $\mathcal R=\{(x,v):\varPsi(x,v)\}$. Let us say that a class $\mathcal V$ forms a *class modal algebra* (admissible for $\mathcal C$, $\mathcal L$, $\mathcal R$) iff it is defined by a formula $\varUpsilon$ that fulfills the conditions expressing that classes $\mathcal C_y=\{x:\varUpsilon(x,y)\}$ indexed by $y$ in $\mathcal L$ play the role of “elements” of $\mathcal V$. Namely, $\varUpsilon$ implies that all $\clC_y$ are subclasses of $\mathcal C$ and their collection is closed under Boolean operations and the modal operator given by $\mathcal R$; e.g., the latter is expressed as follows: $$\forall y\,\exists z\,\forall x\; (\varUpsilon(x,z)\:\&\:\varPhi'(y) \;\Leftrightarrow\; \varPhi(x)\:\&\:\exists v\: (\varPsi(x,v)\:\&\:\varUpsilon(v,y))).$$ Further, let us say that $\theta$ is a *valuation* of propositional variables in $\mathcal V$ iff $\theta$ is a (set) function with ${\mathrm{dom}}(\theta)=\PV$ and ${\mathrm{ran}}(\theta)\subseteq\mathcal L$, and that $\mathfrak F=(\mathcal C,\mathcal R,\mathcal V)$ is a *class general frame* and $\mathfrak M=(\mathcal C,\mathcal R,\mathcal V,\theta)$ a *class Kripke model* on $\mathfrak F$. To define the *truth* of modal formulas $\upvarphi$ at a point $x\in\mathcal C$ in the model $\mathfrak M$, denoted by $\mathfrak M,x\vDash\upvarphi,$ we first extend $\theta$ to a suitable $\bar\theta$ with ${\mathrm{dom}}(\bar\theta)=\MF$ and ${\mathrm{ran}}(\theta)\subseteq\mathcal L$. It can be shown that such an extension exists and is unique up to the equivalence $\sim$ on $\mathcal L$ defined by letting $y\sim z$ iff $ \forall x\, (\varUpsilon(x,y)\Leftrightarrow\varUpsilon(x,z)). $ Then we let $$\begin{aligned} \mathfrak M,x\vDash\upvarphi \;\Leftrightarrow\; \varUpsilon(x,\bar\theta(\upvarphi)).\end{aligned}$$ This notion of truth has the expected properties; e.g., we have $$\begin{aligned} \mathfrak M,x\vDash\Diamond\upvarphi &\;\Leftrightarrow\; \exists v\:(\varPsi(x,v)\:\&\: \mathfrak M,v\vDash\upvarphi).\end{aligned}$$ A formula $\upvarphi\in\MF$ is *true* in $\mathfrak M$ iff $ \forall x\,(\varPhi(x)\Rightarrow \mM,x \vDash\upvarphi), $ and *valid* in $\frF$ iff it is true in all models $\mathfrak M$ on $\frF$. The *modal logic $\MLog(\frF)$ of the class frame $\frF$* consists of those $\upvarphi\in\MF$ that are valid in $\frF$. It can be verified that this logic is a set defined by a $\ZFC$-formula constructed from the formulas $\varPhi,\varPhi',\varPsi,\varUpsilon$, and it is normal. By using formulas with additional parameters, one can imitate higher order class algebras and their modal logics. Logics of submodels {#sec:sub} =================== In this part, we apply Theorem \[thm:maintoolnew\] to calculate the modal theory of the [*submodel relation*]{} on the class of all models of a given signature. #### Expressing the satisfiability {#subsec:relativ} Given models $\mathfrak A$ and $\mathfrak B$ of a signature $\Omega$, let $\mathfrak A\supmod\mathfrak B$ mean “$\mathfrak A$ contains $\mathfrak B$ as a submodel”. As the initial step, we find a model-theoretic language for expressing the $\sqsupseteq$-satisfiability. Observe first that first-order languages are generally too weak for this. \[prop:submod-non-exprr\] If $\Omega$ contains a predicate symbol of arity $\ge2$, then the $\sqsupseteq$-satisfiability is not expressible in $L_{\omega,\omega}$ and moreover, in the infinitary language $L_{\infty,\omega}$. We can suppose w.l.g. that $\Omega$ contains a binary predicate symbol $<$ (otherwise mimic it by a predicate symbol of a bigger arity by fixing other arguments). Toward a contradiction, assume that some $f$ mapping the class of $L_{\infty,\omega}$-sentences into itself expresses the $\sqsupseteq$-satisfiability. Let $\varphi$ be an obvious $L_{\omega,\omega}$-sentence saying that there exists a $<$-minimal element, and let $\psi$ be the sentence $\neg\,f(\neg\,\varphi)$. Then $\psi$ says that each submodel has a $<$-minimal element (thus whenever $\Omega$ has no functional symbols then $\psi$ says that $<$ is well-founded). Let $\kappa$ be such that $\psi\in L_{\kappa,\omega}$. It follows from Karp’s theorem (see, e.g., [@rosenstein1982linear Theorem 14.29]) that there are models $\mathfrak A_0$ and $\mathfrak B_0$ of $\Omega_0=\{<\}$ such that $\mathfrak A_0$ is isomorphic to an ordinal while $\mathfrak B_0$ is not, and $\mathfrak A_0\equiv_{L_{\kappa,\omega}}\!\mathfrak B_0$. Add a $<$-last element to each of the models $\mathfrak A_0$ and $\mathfrak B_0$ and check that the resulting models $\mathfrak A_1$ and $\mathfrak B_1$ remain $L_{\kappa,\omega}$-equivalent (e.g., by using [@rosenstein1982linear Lemma 14.24]). Expand $\mathfrak A_1$ and $\mathfrak B_1$ to models $\mathfrak A$ and $\mathfrak B$ of $\Omega$, respectively, by interpreting each predicate symbol other than $<$ by the empty set, each functional symbol of positive arity by the projection onto the first argument, and each constant symbol by the $<$-last element of the model. It is easy to see that in both $\mathfrak A$ and $\mathfrak B$ any formula of $\Omega$ is equivalent to a formula of $\Omega_0$; so we still have $\mathfrak A\equiv_{L_{\kappa,\omega}}\!\mathfrak B$. On the other hand, in both models every subset forms a submodel whenever it contains the $<$-last element of the whole model, whence it easily follows that $\mathfrak A\vDash\psi$ and $\mathfrak B\vDash\neg\,\psi$. A contradiction. However, second-order language suffices to express the $\sqsupseteq$-satisfiability by the relativization argument (see, e.g., [@QuantHand14-2007 p. 242]). Given a second-order formula $\varphi$ and a unary predicate variable $U$ that does not occur in $\varphi$, let $\varphi^U$ be the relativization of $\varphi$ to $U$ defined in the standard way; in particular, if $P$ and $F$ are second-order predicate functional variables of arity $n$, then - $(\exists P\,\varphi)^U$ is $ \exists P\, \bigl(\forall x_0\ldots\forall x_{n-1}\, \bigl(P(x_0,\ldots,x_{n-1})\to \bigwedge_{i<n}U(x_i)\bigr)\wedge\varphi^U\bigr), $ - $(\exists F\,\varphi)^U$ is $ \exists F\, \bigl(\forall x_0\ldots\forall x_{n-1}\, \bigl(\bigwedge_{i<n}U(x_i)\to U(F(x_0,\ldots,x_{n-1}))\bigr) \wedge\varphi^U\bigr). $ Let $\psi(U)$ be the formula expressing that $U$ is a submodel, i.e., saying that the interpretation of $U$ is non-empty and is closed under interpretations of functional symbols in $\Omega$. Then the map $\varphi\mapsto\exists U(\psi(U)\wedge\varphi^U)$ expresses the $\sqsupseteq$-satisfiability on the class of all models of $\Omega$. In view of Proposition \[prop:extendingexpr\], we obtain: \[prop:submod-quot-exprr\] Let $\kappa=|\{F\in\Omega:F$ is a functional symbol$\}|.$ - Let $\mathcal C$ be a class of models of $\Omega$ closed under submodels. Then the $\sqsupseteq$-satisfiability is expressible on $\mathcal C$ in $L^{2}_{\lambda,\omega}$ whenever $\lambda>\kappa$ and $\lambda\ge\omega$. - Let $T$ be a set of sentences of $L^{2}_{\mu,\omega}$ of $\Omega$. Then the $\sqsupseteq$-satisfiability is expressible on the class $Mod(T)$ in $L^{2}_{\lambda,\omega}$ whenever $\lambda>\max(\kappa,|T|)$ and $\lambda\geq\mu$. These results on expressibility can be refined in several directions. In particular, the first statement of Proposition \[prop:submod-quot-exprr\] remains true for monadic language $L^{2}_{\lambda,\omega}$; the assumption $\lambda>\kappa$ is necessary; for details and further results, see [@Saveliev2019]. #### Axiomatization Henceforth in this section we assume that $L$ expresses the $\supmod$-satisfiability on the class of models under consideration. The next easy result is soundness for modal theories of the submodel relation. \[thm:sqsup-soundness\] Let $\clC$ be a class of $\Omega$-models closed under submodels. Then $\MTh^L(\clC,\supmod)$ is a normal modal logic including $\mathrm{S4}$. If moreover, $\Omega$ contains a constant symbol, then $\MTh^L(\clC,\supmod)$ includes $\mathrm{S4.2.1}$. Let $\stA\in\clC$, $\vl$ a valuation in $L$. Trivially, $\stA\mo{ \| \Di\Di \mathsf{p}\imp \Di \mathsf{p} \|}$ and $\stA\mo{ \| \mathsf{p}\imp \Di \mathsf{p} \|}$. If $\Omega$ contains a constant symbol, consider the submodel $\stB$ of $\stA$ generated by constants. It is straightforward that in this case ${\stA\mo { \| \Box\Di \mathsf{p} \|}} \tiff \stB\mo { \| \Di \mathsf{p} \|} \tiff \stB\mo { \| \Box \mathsf{p} \|} \tiff \stA \mo { \| \Di \Box \mathsf{p} \|}.$ We are going to prove completeness. Let $Q_n$ be the lexicographic product of $(n^{<n},\subseteq)$ (an $n$-ramified tree of height $n$) and $(n,n\!\times\!n)$ (a cluster of size $n$). Thus for $s,t\in n^{<n}$ and $i,j\in n$, in $Q_n$ we have $$(s,i)\le(t,j) \;\Iff\; s\subseteq t,$$ so $Q_n$ is a pre-tree which is $n$-ramified, has height $n$ and clusters of size $n$ at each point. Let also $Q_n'$ be the ordered sum of $Q_n$ and a reflexive singleton, thus $Q'_n$ adds to $Q_n$ an extra top element. The following fact is standard (see, e.g., [@CZ p. 563]). \[prop:preetreeS4\] Let $\mvf$ be a modal formula. If $\mvf\notin{{{\mathrm{S4}}}}$, then $\mvf$ is not valid in $Q_n$ for some $n>0$. If $\mvf\notin{{{\mathrm{S4.2.1}}}}$, then $\mvf$ is not valid in $Q'_n$ for some $n>0$. Let $\equiv$ be the $L$-equivalence. For a model $\mathfrak A$, let $\Sub(\stA)$ be the set of all its submodels, $\Sub(\stA)_\equiv$ abbreviate $\Sub(\stA)/{\equiv}$. \[thm:models-An\] Let $\Omega$ have a functional symbol of arity $\geq 2$. For every positive $n<\omega$, there exists a model $\mathfrak A_n$ of $\Omega$ such that $$\bigl(\Sub(\mathfrak A_n)_\equiv,\sqsupseteq_\equiv\bigr) \text{ is isomorphic to } \left\{ \begin{array}{ll} Q_n & \text{ if $\Omega$ has no constant symbols}, \\ Q'_n& \text{ otherwise}. \end{array} \right.$$ First, suppose that $\Omega$ has no constant symbols. Without loss of generality we may assume that $\Omega$ has a binary functional symbol; we write $\,\cdot\;$ for it. Fix $n\geq 1$. Let $X_n=n^{<n}\times \omega$. We define the model $\mathfrak{A}_n=(X_n,\cdot\,,\dots)$ of $\Omega$ as follows. Let $E$ be any injective map from $n^{<n}$ into $\omega$. For $s,t\in n^{<n}$ and $i,j\in\omega$, we put $$\begin{aligned} (s,i)\cdot(t,j)= \left\{ \begin{array}{ll} (s,i+1) &\text{if }s=t,\,i=j, \\ (s^{\conc}({i\bmod n}),j) &\text{if }s=t,\,j<i,\,|s|<n-1, \\ (s,j+E(s)) &\text{if }s=t,\,j=i+1,\,i\equiv_n\!0, \\ (\inf\{s,t\},\inf\{i,j\}) &\textrm{otherwise}, \end{array} \right.\end{aligned}$$ where ${}^{\conc}$ denotes the concatenation, $|s|$ the length of $s$, $\bmod$ the remainder, and $\equiv_n$ the congruence modulo $n$. For other operations $F$ in $\mathfrak{A}_n$ we put $F((s,i),\ldots)=(s,i)$ and take the relations in $\mathfrak{A}_n$ to be empty. For $(s,i)\in X_n$ let $ X_n(s,i)= \{(t,j)\in X_n:(s,i)\preceq(t,j)\} $, where we let $(s,i)\preceq(t,j)$ iff $s\subseteq t$ and $i\le j$. \[lem:Aui\] An $X\subseteq X_n$ is the universe of a submodel of $\mathfrak{A}_n$ iff $X=X_n(s,i)$ for some $(s,i)\in X_n$. Straightforward from the definition of $\mathfrak{A}_n$. Let $\mathfrak{A}_n(s,i)$ be the submodel of $\mathfrak{A}_n$ with the universe $X_n(s,i)$. \[lem:isom\] Let $(s,i),(s,j)\in X_n$. If $i\equiv_n\!j$, then the models $\mathfrak{A}_n(s,i)$ and $\mathfrak{A}_n(s,j)$ are isomorphic. The map $(t,l)\mapsto(t,l+n)$ is an isomorphism between $\mathfrak{A}_n(s,i)$ and $\mathfrak{A}_n(s,i+n)$. We define $p^0(x)$ as $x$, and $p^{k+1}(x)$ as $(p^{k}(x))\cdot (p^{k}(x))$. Then for $k<\omega$ $$\mathfrak{A}_n\vDash(t,j)=p^k(s,i) \quad \text{iff}\quad s=t \text{ and } j=i+k.$$ For $s\in n^{<n}$, let $\varphi_s(x)$ be the following one-parameter formula: $$x\cdot p(x)=p^{E(s)+1}(x).$$ \[lem:x1\] Let $\mathfrak{A}$ be a submodel of $\mathfrak{A}_n$ and $(t,i)$ an element of $\mathfrak{A}$. Then $\mathfrak{A}\vDash\varphi_s(t,i)$ iff $s=t$ and $i\equiv_n\!0$. We have $p(t,i)=(t,i+1)$. By the definition, $(t,i)\cdot(t,i+1)=(t,i+E(t)+1)$ if $i\equiv_n\!0$, and $(t,i)\cdot(t,i+1)=(t,i)$ otherwise. \[lem:x2\] Let $\mathfrak{A}$ be a submodel of $\mathfrak{A}_n$. For every $s\in n^{<n}$, we have: [$\mathfrak{A}\vDash\exists x\,\varphi_s(x)$]{} iff $(s,i)$ is in $\mathfrak{A}$ for some $i\in\omega$. The ‘only if’ part is immediate from Lemma \[lem:x1\]. For the ‘if’ part we use Lemmas \[lem:x1\] and \[lem:Aui\]. For $S\subseteq n^{<n}$, let $\chi_S$ be the sentence $ \bigwedge_{s\in S}\exists x\,\varphi_s(x) \;\wedge\, \bigwedge_{s\notin S}\neg\,\exists x\,\varphi_s(x), $ and let $\chi_{\ge s}$ be $\chi_S$ for $S=\{t\in n^{<n}:s\subseteq t\}$. \[lem:x3\] Let $\mathfrak{A}$ be a submodel of $\mathfrak{A}_n$. Then $\mathfrak{A}\vDash\chi_{\ge s}$ iff $\mathfrak{A}=\mathfrak{A}_n(s,i)$ for some $i\in\omega$. Follows from Lemmas \[lem:x2\] and \[lem:Aui\]. Let $\psi(x)$ be the formula $\neg\,\exists y\,(x=p(y)).$ Then $$\mathfrak A_n(s,i)\vDash\psi(t,j) \text{ iff } j=i.$$ For $s\in n^{<n}$ and $k<n$, let $\chi_{s,k}$ be the following sentence: $$\exists x\, \bigl(\varphi_s(p^{n-k}(x))\wedge\psi(x)\bigr) \wedge \chi_{\ge s}.$$ \[lem:character\] For every submodel $\mathfrak{A}$ of $\mathfrak{A}_n$ and every $k<n$, we have $\mathfrak{A}\vDash\chi_{s,k}$ iff $\mathfrak{A}=\mathfrak{A}_n(s,i)$ for some $i$ such that $i\equiv_n k$. Follows from Lemmas \[lem:x3\] and \[lem:x1\]. \[lem:comp-important\] Let $\mathfrak{A},\mathfrak{B}$ be submodels of $\mathfrak{A}_n$. The following are equivalent: 1. $\mathfrak{A}$ and $\mathfrak{B}$ are isomorphic, 2. $\mathfrak{A}$ and $\mathfrak{B}$ are $L$-equivalent, 3. $\mathfrak{A}$ and $\mathfrak{B}$ are elementarily equivalent, i.e., $L_{\omega,\omega}$-equivalent, 4. $\mathfrak{A}=\mathfrak{A}_n(s,i)$ and $\mathfrak{B}=\mathfrak{A}_n(s,j)$ for some $s\in n^{<n}$ and $i,j<\omega$ such that $i\equiv_n j$. The implications [(i)$\,\Imp\,$(ii)]{} and [(ii)$\,\Imp\,$(iii)]{} are immediate from our basic assumptions on model-theoretic languages. The crucial step [(iii)$\,\Imp\,$(iv)]{} follows from Lemmas \[lem:Aui\] and \[lem:character\]. Finally, [(iv)$\,\Imp\,$(i)]{} holds by Lemma \[lem:isom\]. From Lemma \[lem:Aui\] we conclude that $(\Sub(\mathfrak{A}_n),\sqsupseteq)$ is isomorphic to $(X_n,\preceq)$. Now it follows that $(\Sub(\mathfrak{A}_n)_\equiv,\sqsupseteq_\equiv)$ is isomorphic to $Q_n$, as required. For the case when $\Omega$ has constant symbols, we add a new element $c$ to $X_n$ and define the model $\mathfrak{A}'_n$ on the set $X_n\cup\{c\}$. We extend the above defined operation $\cdot$ by letting $c\cdot x=x\cdot c=c$ for all $x$; all constant symbols in $\Omega$ are interpreted by $c$. The same arguments as above prove that $(\Sub(\mathfrak{A}'_n)_\equiv,\sqsupseteq_\equiv)$ is isomorphic to $Q'_n$. This completes the proof of Theorem \[thm:models-An\]. Now the completeness result follows: \[thm:sqsup-completeness\] Let $\Omega$ contain a functional symbol of arity $\ge 2$, and let $\mathcal C$ be the class of all models of $\Omega$. Then $$\mathrm{MTh}^L(\mathcal C,\sqsupseteq)= \left\{ \begin{array}{ll} \mathrm{S4} &\text{ if $\Omega$~has no constant symbols}, \\ \mathrm{S4.2.1} &\text{ otherwise}. \end{array} \right.$$ We have soundness by Theorem \[thm:sqsup-soundness\]. On the other hand, if $\stA\in \clC$, then $\mathrm{MTh}^L(\mathcal C,\sqsupseteq)$ is contained in the logic of the Kripke frame $(\Sub(\stA)_\equiv,\sqsupseteq_\equiv)$ by Corollaries \[cor:gener\] and \[prop: finite frame\]. Now completeness follows from Theorem \[thm:models-An\]. Notice that the binary operation used in the proof of Theorem \[thm:models-An\] is not associative. Modal axiomatizations of $\sqsupseteq$ in second-order language on semigroups, monoids, and groups are open questions. Recall that Theorem \[thm:sqsup-completeness\] was formulated under the assumption that $L$ expresses the $\sqsupseteq$-satisfiability. In the next section we discuss how to define modal theories without such requirements. The logics calculated in Theorem \[thm:sqsup-completeness\] do not depend on $L$, while in general, modal theories depend on a chosen model-theoretic language; we shall discuss this in Section \[sec:rob\]. Inexpressible modalities {#sec:upanddown} ======================== In this section, we discuss the situation when the $\clR$-satisfiability is not expressible in a model-theoretic language. #### Definition The following question was raised by one of the reviewers on an earlier version of the paper: what is the modal theory of $(\clC,\clR)$ if the $\clR$-satisfiability on $\clC$ is not expressible in a given language? In particular, what is the modal theory of the relation $\supmod$ in the first-order case? Another natural question concerns the definition of modal theories in the case of $\submod$. Assume that $L$ is a language stronger than $K$, and the $\clR$-satisfiability on $\clC$ is expressible in $L$. In this case, $\MTh^{K}(\clC,\clR)$ can be naturally defined as the logic of the subalgebra of the modal algebra of $L$ on $(\clC,\clR)$ generated by the sentences of $K$. Let us provide details. First, we define an interim notion $\MTh^{\KL}(\clC,\clR)$. By Theorem \[thm:maintoolnew\], $\MTh^{L}(\clC,\clR)$ is the logic of the Lindenbaum modal algebra $\clA(L)=(L/{\approx},f^L_\approx)$. Let $\clA(\KL)$ be the subalgebra of $\clA(L)$ generated by the sentences of $K$ (more formally, $\clA(\KL)$ is generated by the set $\{[\vf]^L_\approx: \vf\in K_s\}$, where $[\vf]^L_\approx=\{\psi\in L_s: \clC\mo \vf\lra\psi\}$). Define $\MTh^{\KL}(\clC,\clR)$ as the modal logic of $\clA(\KL)$. (Formally, we have defined $\MTh^{\KL}(\clC,\clR)$ for the case $K\subseteq L$, but the same construction works for the case when $K$ is weaker than $L$ in $\clC$, i.e., if every $K$-definable subclass of $\clC$ is definable in $L$.) It is immediate that $\MTh^{\KL}(\clC,\clR)$ is a normal logic, which includes $\MTh^L(\clC,\clR)$. Another simple (but important) observation is that $\MTh^{\KL}(\clC,\clR)$ does not depend on the choice of $L$. Indeed, if $M$ is another language stronger than $K$, which expresses the $\clR$-satisfiability on $\clC$, then the algebras $\clA(\KL)$ and $\clA({M[K]})$ are isomorphic: their elements can be thought as classes of models obtained from definable in $K$ subclasses of $\clC$ via Boolean operations and $\clR^{-1}$. Thus we have \[prop:unspeak\] If languages $L$ and $M$ express the $\clR$-satisfiability on $\clC$ and are stronger than $K$, then: 1. the algebras $\clA(\KL)$ and $\clA({M[K]})$ are isomorphic, and so 2. $\MTh^{\KL}(\clC,\clR)=\MTh^{{M[K]}}(\clC,\clR)$. \[def:secondMain\] If the $\clR$-satisfiability on $\clC$ is expressible in some language $L$ stronger than $K$, put $\MTh^K(\clC,\clR)=\MTh^{\KL}(\clC,\clR)$. Trivially, $\MTh^{{L[L]}}(\clC,\clR)=\MTh^L(\clC,\clR)$ whenever $L$ expresses the $\clR$-satisfiability on $\clC$. Thus this definition generalizes Definition \[def:exprMTH\]. Note that, however, the modal logic $\MTh^K(\clC,\clR)$ is not necessarily a “fragment” of the theory $\Th^K(\clC)$ anymore. Once we do not require the $\clR$-satisfiability to be expressible in the language, we can give an “external” definition of modal theory and modal algebra for $\clC$, $\clR$, and $K$. Let $K_\Di$ be the set of modal [*$K$-sentences*]{}, which are built from sentences of $K$ using Boolean connectives and $\Di$; namely, they are expressions of form $\mvf(\psi_1,\ldots,\psi_n)$ where $\mvf(\mathsf p_1,\ldots,\mathsf p_n)$ is a modal formula and $\psi_i$ are sentences of $K$. Such languages are regularly used in the context of modal logics of relations between models of arithmetic or set theory.[^5] Given $\clC$ and $\clR$, the $K_\Di$-satisfaction relation is defined in the straightforward way; in particular, $\stA\mo\Di\mvf(\psi_1,\ldots,\psi_k)$ iff $\stB\mo\mvf(\psi_1,\ldots,\psi_k)$ for some $\stB\in\clC$ with $\stA\,\clR\,\stB$. Assume that $\clR$ and $\clC$ satisfy the following natural condition: for all $\stA,\stA',\stB$ in $\clC$, $$\label{eq:isom-bisim} \stA\isom\stA'\;\&\;\stA\:\clR\:\stB \;\Rightarrow\; \EE\stB'\in\clC\; (\stB'\isom\stB\;\&\;\stA'\,\clR\,\stB')$$ (this condition says that the isomorphism equivalence $\isom$ is a [*bisimulation*]{} w.r.t. $\clR$ on $\clC$). In this case the $K_\Di$-satisfaction relation is preserved under isomorphisms, and so $K_\Di$ satisfies our basic assumptions on model-theoretic languages. The operation on sentences of $K_{\Di}$ that takes $\vf$ to $\Di\vf$ expresses the $\clR$-satisfiability on $\clC$ in $K_{\Di}$. Hence, we can apply the constructions described in Section \[sec:defs\] to $K_\Di$. In particular, $\vf\approx \psi$ implies $\Di\vf\approx\Di\psi$ (recall that $\vf\approx\psi$ means $\clC\mo\vf\lra\psi$), so the connective $\Di$ induces the operation $\Di_\approx$ on $K_\Di/{\approx}$, and $\clA(K_\Di)=(K_\Di/{\approx},\Di_\approx)$ is a modal algebra. It is immediate that $\clA(K_\Di)=\clA(K_\Di[K])$, and hence $$\MTh^{K}(\clC,\clR)=\MTh^{K_\Di}(\clC,\clR)$$ by Definition \[def:secondMain\]. Therefore, assuming that $\clC$ and $\clR$ satisfy (\[eq:isom-bisim\]), we obtain the following generalization of our previous definitions for arbitrary $K$: \[def:external-sat\] Let $\clC$ and $\clR$ satisfy (\[eq:isom-bisim\]). The [*modal (Lindenbaum) algebra of a language $K$ on $(\clC,\clR)$*]{} is the modal algebra $({K_\Di/{\approx}},\Di_\approx)$. The [*modal theory $\MTh^K(\clC,\clR)$ of $(\clC,\clR)$ in $K$*]{} is the modal logic of this algebra. Let us emphasize that natural classes of models with relations between them always satisfy (\[eq:isom-bisim\]); in particular, so are the instances discussed in our paper. One might say that $\clR$ is a [*model theoretic relation on*]{} $\clC$ iff property (\[eq:isom-bisim\]) holds. E.g., the submodel relation on the class of models of a given signature is model-theoretic; hence, Theorem \[thm:sqsup-completeness\] remains true for arbitrary model-theoretic language. #### Downward Löwenheim–Skolem theorem Our next result provides a version of the downward Löwenheim–Skolem theorem for first-order language enriched with the modal operator for the extension relation between models. For a cardinal $\kappa$, put $\clC_{\leq \kappa}=\{\stA\in\clC:|\stA|\leq\kappa\}$. \[thm:submotapprox\] Let $K$ be $L_{\omega,\omega}$ based on a signature $\Omega$, let $K_\Di$ be $K$ enriched with the modal operator for the extension relation on models $\sqsubseteq$, and let $\clC$ be an elementary (in $K$) class of models. For every $\kappa\geq\omega+|\Omega|$, the following statements hold: 1. Let $X$ be a set of elements of $\stA\in\clC$, and let $\lambda$ be a cardinal such that $|X|+\kappa\leq\lambda\leq|\stA|$. Then $\stA$ has a submodel $\stB$ of cardinality $\lambda$ such that $\Th^{K_\Di}(\stA)=\Th^{K_\Di}(\stB)$ and $\stB$ contains $X$. 2. $ \{\Th^{K_\Di}(\stA):\stA\in\clC\}= \{\Th^{K_\Di}(\stA):\stA\in\clC_{\leq\kappa}\}. $ 3. The modal algebras of $K$ on $(\clC,\submod)$ and on $(\clC_{\leq\kappa},\submod)$ coincide. 4. $ \MTh^K(\clC,\submod)= \MTh^K(\clC_{\leq \kappa},\submod). $ First, we provide a general observation about any $\clC$ and $\clR$ satisfying the condition (\[eq:isom-bisim\]), and arbitrary $K$. Let $\equiv$ be the $K$-equivalence on $\clC$, and let $\equiv_\Di$ be the $K_\Di$-equivalence on $\clC$. Hence, $\stA\equiv\stB$ means that $\Th^K(\stA)=\Th^K(\stB)$, and $\stA\equiv_\Di\stB$ that $\Th^{K_\Di}(\stA)=\Th^{K_\Di}(\stB)$. We recall that $\Th^{K_\Di}(\stA)$ depends not only on $\stA$ but also on $\clC$ and $\clR$. \[prop:equive-str\] Assume that $\clC$ and $\clR$ satisfy  (\[eq:isom-bisim\]). Let $\equiv$ be a bisimulation w.r.t. $\clR$ on $\clC$, i.e., for all $\stA,\stA',\stB\in\clC$ we have $$\label{eq:equive-bisim} \stA\equiv\stA'\;\&\;\stA\:\clR\:\stB \;\Rightarrow\; \EE\stB'\in\clC\; (\stB'\equiv\stB\;\&\;\stA'\,\clR\,\stB').$$ Then $\equiv$ and $\equiv_\Di$ coincide on $\clC$. By the standard bisimulation argument: an easy induction on the construction of $\mpsi(\mpv_1,\ldots,\mpv_n)$ shows that, whenever $\stA\equiv\stA'$ and $\vf_1,\ldots,\vf_n\in K_s$ then we have $ \stA\mo\mpsi(\vf_1,\ldots,\vf_n) \,\Iff\, \stA'\mo\mpsi(\vf_1,\ldots,\vf_n). $ \[prop:equive-bisim\] Let $K=L_{\omega,\omega}$. If $\clC$ is an elementary class, then $\equiv$ is a bisimulation w.r.t. $\sqsubseteq$ on $\clC$. Pick $\stA,\stA',\stB$ in $\clC$. If $\stA\equiv\stA'$, then there exists an ultrafilter $D$ such that the ultrapowers $\stA_D$ and $\stA'_D$ are isomorphic, due to the Keisler–Shelah isomorphism theorem (see, e.g., [@chang1990model Theorem 6.1.15]). If $\stA\submod\stB$, then $\stA_D$ is embeddable in $\stB_D$, the ultrapower of $\stB$. Since $\stA'$ is embeddable in $\stA'_D$, we obtain that $\stA'$ is embeddable in $\stB_D$, and thus $\stA'\submod\stB'$ for some $\stB'\isom\stB_D$. But then we have $\stB'\in\clC$ and $\stB'\equiv\stB$, which completes the proof. By Propositions \[prop:equive-str\] and \[prop:equive-bisim\], $\equiv$ and $\equiv_\Di$ coincide on $\clC$, i.e., for all $\stA',\stA$ in $\clC$ we have: $$\label{eq:cool1} \text{$\stA'$ and $\stA$ are $K$-equivalent} \;\Iff\; \textrm{$\stA'$ and $\stA$ are $K_\Di$-equivalent}.$$ Now the first statement of the theorem follows from the downward Löwenheim–Skolem theorem for the first-order case: $\stA$ has an elementary submodel $\stB$ of cardinality $\lambda$ such that $\stB$ contains $X$ (see, e.g., [@chang1990model Theorem 3.1.6]); hence $\stA\equiv_\Di\stB$ by (\[eq:cool1\]). Let $(\Ths,\submod_\Ths,\clA)$ and $(\Ths^\kpp,\submod^\kpp_\Ths,\clA^\kpp)$ be the minimal frames of theories for $\clC$, $\submod$, $K_\Di$ and for $\clC_{\leq \kappa}$, $\submod$, $K_\Di$, respectively. Let us show that these two structures are equal. We have $\Ths=\Ths^\kpp$, the second statement of the theorem, as an immediate corollary of the first one. Suppose that $T_1\submod_\Ths T_2$, that is, $T_1=\Th^{K_\Di}(\stA)$ and $T_2=\Th^{K_\Di}(\stB)$ for some $\stA,\stB\in\clC$ with $\stA\submod\stB$. Then $\stA$ has a submodel $\stA'$ of cardinality $\leq\kappa$ with $\stA'\equiv_\Di\stA$ (indeed, if the cardinality of $\stA$ is less than $\kappa$ then we put $\stA'=\stA$, otherwise we use the first statement of the theorem). Likewise, $\stB$ has a submodel $\stB'$ of cardinality $\leq\kappa$ such that $\stB'\equiv_\Di\stB$ and $\stB'$ contains the universe of $\stA'$. Clearly, $\stA'$ is submodel of $\stB'$. It follows that $T_1\submod^\kpp_\Ths T_2$. Hence $\submod^\kpp$ includes $\submod$. The converse inclusion is obvious, so $\submod^\kpp$ equals $\submod$. Let $\vf\in K_\Di$. Since $\Ths=\Ths^\kpp$, it is immediate that $\{T\in\Ths:\vf\in T\}=\{T\in\Ths^\kpp:\vf\in T\}$. Hence, $\clA^\kpp=\clA$. Thus $ (\Ths,\submod_\Ths,\clA)= (\Ths^\kpp,\submod^\kpp_\Ths,\clA^\kpp) $. Now the third and consequently, the forth, statements are immediate from Definition \[def:external-sat\] and Theorem \[thm:minmax-ths\]. This result is based on the interplay between the downward Löwenheim–Skolem property of first-order language and the fact that $\equiv$ is a bisimulation w.r.t. $\sqsubseteq$. The property of $\equiv$ being a bisimulation w.r.t. a given $\mathcal R$ seems interesting enough per se. Contrary to the case of the relation $\submod$ (Proposition \[prop:equive-bisim\]), we have: Let $\Omega$ contain a predicate symbol of arity $\ge2$. Then the (usual) elementary equivalence is not a bisimulation w.r.t. $\sqsupseteq$ on the class of models of $\Omega$. Assume w.l.g. $\Omega$ contains a single binary predicate symbol ${\le}$, and let $\equiv$ denote $\equiv_{L_{\omega,\omega}}$. Let $\mathfrak A,\mathfrak A',\mathfrak B$ be the linearly ordered sets $\mathbb Q\cdot\mathbb Z$, $\mathbb Z$, $\mathbb Q$, respectively (where $\,\cdot\,$ denotes the lexicographical multiplication). We have $\mathfrak A\equiv\mathfrak A'$ (this fact can be established by using an Eurenfeucht–Fraïssé game, see, e.g., [@marker2002modelIntro Proposition 2.4.10]) and $\stB$ is embeddable in $\stA$. However, every $\mathfrak B'$ satisfying $\mathfrak B'\equiv\mathfrak B$ is a dense linearly ordered set without end-points. Clearly, no such $\mathfrak B'$ can at the same time be embeddable in $\mathfrak A'$. Let us say that $\clR$ is [*image-closed under ultraproducts*]{} iff for every $\mathfrak A$, $(\mathfrak B_i)_{i\in I}$, and ultrafilter $D$ over $I$, if $\mathfrak A\,\clR\,\mathfrak B_i$ for all $i\in I$ then $\mathfrak A\,\clR\prod_D\mathfrak B_i$ (e.g., $\mathfrak A\,\mathcal R\,\mathfrak B$ may mean that, up to isomorphism, $\mathfrak B$ is an extension of $\mathfrak A$). Let $\clC$ and $\clR$ satisfy $(\ref{eq:isom-bisim})$, let $\clR$ be image-closed under ultraproducts on $\clC$, and let $K=L_{\omega,\omega}$. If $\equiv$ and $\equiv_\Di$ coincide on $\clC$, then $\equiv$ is a bisimulation w.r.t. $\clR$ on $\clC$. Let $\mathfrak A,\mathfrak A',\mathfrak B$ in $\clC$ be such that $\mathfrak A\equiv\mathfrak A'$ and $\mathfrak A\,\clR\,\mathfrak B$. Observe that for all $\varphi\in K_s$ we have: $$\begin{aligned} \mathfrak B\vDash\varphi &\;\Rightarrow\; \mathfrak A\vDash\Di\varphi \\ &\;\Rightarrow\; \mathfrak A'\vDash\Di\varphi \;\;\;(\text{since }\mathfrak A\equiv_\Di\mathfrak A'\,) \\ &\;\Rightarrow\; \mathfrak A'\,\clR\,\mathfrak B_\varphi \;\&\; \mathfrak B_\varphi\vDash\varphi \text{ for some }\mathfrak B_\vf\in\clC.\end{aligned}$$ Let $T=Th^K(\mathfrak B)$. For any $\varGamma\in\mathcal P_{\omega}(T)$ choose a model $\mathfrak B_{\varGamma}$ in $\clC$ as in the observation above, i.e., such that $\mathfrak A'\,\clR\,\mathfrak B_{\varGamma}$ and $\mathfrak B_{\varGamma}\vDash\bigwedge\varGamma$. Pick any ultrafilter $D$ extending the centered family of sets $ S_\varphi= \{\varGamma\in\mathcal P_{\omega}(T): \varphi\in\varGamma\} $ for all $\varphi\in T$ (i.e., a [*fine*]{} ultrafilter over $\mathcal P_{\omega}(T)$). Then the ultraproduct $\mathfrak B'=\prod_D\mathfrak B_\varGamma$ satisfies all $\varphi\in T$, thus $\mathfrak B'\vDash T$. Moreover, since $\clR$ is image-closed under ultraproducts, we have $\mathfrak A'\,\clR\,\mathfrak B'$. This completes the proof. It suffices to assume that $\clR$ is image-closed under ultraproducts by ultrafilters over $|K|$ only. The proposition remains true for $K=L_{\kappa,\lambda}$ with any strongly compact $\kappa$. Robustness and Kripke completeness {#sec:rob} ================================== The logics of submodels calculated in Section \[sec:sub\] do not depend on choosing particular language (Theorem \[thm:sqsup-completeness\]). However, in general, modal theories depend on $L$. \[ex:non-rob1\] Let $L$ be the first-order language $L_{\omega,\omega}$, $T$ the theory of dense linear orders without end-points. Trivially, the $\supmod$-satisfiability is expressible on $\clC=\Mod(T)$ in $L$: put $f(\vf)=\vf$. Then $\MTh^L(\clC,\supmod)$ contains the “trivial” formula $\mathsf{p} \leftrightarrow \Di \mathsf{p}$. Obviously, this formula is falsified if $L$ is second-order. Henceforth we assume that $\clC$ and $\clR$ satisfy (\[eq:isom-bisim\]). \[prop:language-monot\] Let $L$ and $K$ be two languages. Then $L\subseteq K$ implies $ \mathrm{MTh}^{L}(\mathcal C,\mathcal R) \supseteq \mathrm{MTh}^{K}(\mathcal C,\mathcal R). $ Follows from Definition \[def:external-sat\], since on $(\clC,\clR)$, the modal algebra of $L$ is a subalgebra of the algebra of $K$. One can think that $L$ describes the properties of $(\mathcal C,\mathcal R)$ in a robust way whenever the modal theory does not change under strengthening the language. Thus, we shall say that $\MTh^L(\mathcal C,\mathcal R)$ is [*robust*]{} iff for every language $K\supseteq L$ we have $$\mathrm{MTh}^{K}(\mathcal C,\mathcal R)= \mathrm{MTh}^{L}(\mathcal C,\mathcal R).$$ Intuitively, the robust theory can be considered as a “true” modal logic of the model-theoretic relation $\mathcal R$ on $\mathcal C$. By Definition \[def:external-sat\] and Theorem \[thm:maintoolnew\], modal theories are logics of general frames. The following construction shows that, under certain assumptions, robust theories are logics of Kripke frames (i.e., they are [*Kripke complete*]{}). A relation $\clS$ is said to be [*image-set*]{} iff for every $\stA$ the image $\clS(\stA)=\{\stB:\stA\,\clS\,\stB\}$ is a set; $\clS$ is [*image-set on*]{} $\clC$ iff its restriction to $\clC$ is image-set. Recall that $\mathfrak A\isom\mathfrak B$ means that $\mathfrak A$ and $\mathfrak B$ are isomorphic, and that $\,[\stA]_\isom\clR_\isom[\stB]_\isom$ iff $ \exists\,\mathfrak A'\isom\mathfrak A\, \exists\,\mathfrak B'\isom\mathfrak B\; (\mathfrak A'\,\mathcal R\,\mathfrak B'). $ Let $\mathcal C_\isom$ abbreviate $\mathcal C/{\isom}$. Also, recall that $\clR^*(\stA)$ denotes $\bigcup_{n<\omega}\clR^n(\stA)$, the least $\clR$-closed $\clD\subseteq\clC$ containing $\stA$. \[prop:sup-isom\] If $\clR_{\isom}$ is image-set on $\clC_\isom$, then $\MTh^L(\clC,\clR) \supseteq \,\bigcap\limits_{\mathfrak A\in\clC} \MLog(\clR^*(\mathfrak A)_{\isom},\clR_\isom).$ Let $\stA\in\clC$. The inclusion $ \MTh^{L}(\clR^*(\stA),\clR)\supseteq \MLog(\clR^*(\mathfrak A)_{\isom},\clR_\isom) $ follows from (\[eq:quot\]) at the end of Section \[sec:defs\]. Now we apply Corollary \[cor:gener\]. \[thm: robust logicGOOD\] If $\clR_{\isom}$ is image-set on $\clC_\isom$, then $\MTh^L(\clC,\clR)$ is robust iff $ \MTh^L(\clC,\clR)= \bigcap\limits_{\mathfrak A\in\clC} \MLog(\clR^*(\mathfrak A)_{\isom},\clR_\isom). $ $(\Rightarrow)$ Put $\Lambda=\MTh^L(\clC,\clR)$. In view of Proposition \[prop:sup-isom\], we only have to show that $\Lambda$ is valid in the Kripke frame $(\clR^*(\mathfrak A)_{\isom},\clR_\isom)$ for every $\stA\in\clC$. Fix $\stA\in\clC$ and put $Y=\clR^*(\stA)$. For a language $M$, let $({\Ths^M}\!,\clR_{\Ths^M},{\clA^M})$ be the minimal frame of its theories in the class $Y$. Since $\clR_{\isom}$ is image-set on $\clC_\isom$, the quotient $Y_{\isom}$ is a set. Hence we can choose a language $K$ stronger than $L$ and such that the $K$-equivalence and the isomorphism relation coincide on $Y$. Observe that for every $M$ stronger than $K$, the Kripke frames $({\Ths^M}\!,\clR_{\Ths^M})$, $({\Ths^K}\!,\clR_{\Ths^K})$, and $(Y_{\isom},\clR_\isom)$ are isomorphic. We choose such $M$ that $$\begin{aligned} &\AA\, T\in{\Ths^K}\: \EE\,\vf_T\in M_s\: \AA\,\stA\in Y\; \bigl(\stA\mo\vf_T\,\Iff\,\Th^K(\stA)=T\bigr), &\!\text{and moreover} \\ &\AA\,\clS\subseteq{\Ths^K}\: \EE\,\vf_\clS\in M_s\: \AA\,\stA\in Y\; \bigl(\stA\mo\vf_\clS\,\Iff\,\Th^K(\stA)\in\clS\bigr).&\end{aligned}$$ It follows that ${\clA^M}$ is $\mathcal P(\Ths^M)$, the powerset of $\Ths^M$, and so $(\Ths^M\!,\clR_{\Ths^M},{\clA^M})$ is a Kripke frame. By Theorem \[thm:maintoolnew\], $\MTh^M(Y,\clR)$ is the logic of this frame. Therefore, $\MTh^M(Y,\clR)$ is the logic of the Kripke frame $(Y_{\isom},\clR_\isom)$. Since $\Lambda$ is robust, we have $\Lambda = \MTh^M(\clC,\clR)$. By Corollary \[cor:gener\], $\Lambda \subseteq \MTh^M(Y,\clR)$. $(\Leftarrow)$ Immediate from Propositions \[prop:language-monot\] and \[prop:sup-isom\]. Hence, when $\clR_{\isom}$ is image-set on $\clC_\isom$, this theorem describes a unique modal logic, the [*robust theory of $(\clC,\clR)$*]{}. More generally, if $\mathcal K$ is a class of model-theoretic languages, a modal theory $\MTh^L(\mathcal C,\mathcal R)$ is *robust in* $\mathcal K$ iff it coincides with $\MTh^K(\mathcal C,\mathcal R)$ for every language $K\supseteq L$ in $\mathcal K$. It is easy to see that there exists at most one robust theory of $(\mathcal C,\mathcal R)$ whenever $\mathcal K$ is directed, and exactly one such theory whenever $\mathcal K$ is countably directed, in the sense that for every $\{K_n\in\mathcal K:n\in\omega\}$ there is $K\in\mathcal K$ which is stronger than any $K_n$. Notice that even if $\clC$ is a set, Theorem \[thm: robust logicGOOD\] does not mean that the robust theory of $(\clC,\clR)$ is the logic of the Kripke frame $(\clC,\clR)$. Let $\stA$ be the algebra $(2^{<\omega},\inf,{\mathrm l},{\mathrm r})$ with the binary operation $\inf(x,y)=x\cap y$ and the unary operations ${\mathrm l}(x)=x^\conc 0$, ${\mathrm r}(x)=x^\conc 1$, where ${}^{\conc}$ denotes the concatenation. It is easy to see that $(\Sub(\stA),\supmod)$, the structure of submodels of $\stA$, is isomorphic to the binary tree $\frT_2=(2^{<\omega},\subseteq)$. The modal logic of the Kripke frame $\frT_2$ is known to be ${{{\mathrm{S4}}}}$ (see [@GoldMink1980]). However, for every $L$, the modal theory $\MTh^L(\Sub(\stA),\supmod)$ is the trivial logic given by the axiom $\mathsf{p}\leftrightarrow\Di\mathsf{p}$ because every submodel of $\stA$ is isomorphic to $\stA$. #### More completeness results {#par:morecompl} We apply Theorem \[thm: robust logicGOOD\] to describe robust theories of the quotient and the submodel relations on certain natural classes. The following fact most probably was known since 1970s. \[prop:MedvedevTop\] The logic of the Kripke frame $(\mathcal P(\omega),\subseteq)$ is $\mathrm{S4.2.1}$. It is not difficult to construct a family $\clV\subset\mathcal P(\omega)$ such that $\clV$ is downward-closed (i.e., $U_1\subseteq U_2\in\clV$ implies $U_1\in \clV$) and there exists a p-morphism of $(\clV,\subseteq)$ onto the binary tree $\frT_2=(2^{<\omega},\subseteq)$. (E.g., let $f$ be a bijection $\omega\to 2^{<\omega}$, and let $\clV$ consist of $U\subset \omega$ such that $f(U)$ is a finite chain in $\frT_2$; the required p-morphism takes $U$ to the greatest element of this chain.) This p-morphism obviously extends to the p-morphism of $(\clP(\omega),\subseteq)$ onto $\frT_2'$, the ordered sum of $\frT_2$ and a reflexive singleton. Since $\MLog(\frT_2')=\mathrm{S4.2.1}$ (see [@ShehtMink83]), it follows that ${\MLog(\mathcal P(\omega),\subseteq)}$ is included in $\mathrm{S4.2.1}$. The converse inclusion is clear. \[thm:submRob\] Let $\Omega$ contain a functional symbol and a constant symbol, and let $\mathcal C$ be the class of all models of $\Omega$. Then the robust theory of $(\mathcal C,\sqsupseteq)$ is $\mathrm{S4.2.1}$. Soundness is straightforward. Let us check the converse inclusion. Let a signature $\Omega_0$ consist of a single unary functional symbol $F$; models of this signature are called [*unars*]{}. For any $n\in\omega{\setminus}\{0\}$, let $\mathfrak A_n$ be a cycle of length $n$, i.e., a one-generated unar of cardinality $n$ that satisfies $F^n(x)=x$. As easy to see, each $\mathfrak A_n$ has no proper submodels. Let also $\mathfrak A$ be the disjoint sum of all these $\mathfrak A_n$. Clearly, $\mathfrak A$ is a countably generated unar, and any its submodel is given by a nonempty set $S\subseteq\omega$, and the map taking $S$ to the disjoint sum of $\mathfrak A_n$ for all $n\in S$ is an isomorphism between $(\mathcal P(\omega){\setminus}\{\emptyset\},\supseteq)$ and the frame $(Sub(\mathfrak A),\sqsupseteq)$ of submodels of $\stA$. Moreover, all submodels of $\mathfrak A$ are pairwise non-isomorphic, and hence, the Kripke frame $(Sub(\mathfrak A)_{\isom},\sqsupseteq_{\isom})$ is also isomorphic to $(\mathcal P(\omega){\setminus}\{\emptyset\},\supseteq)$. Without loss of generality we may assume that $F$ is the only functional symbol in $\Omega$ (we can always mimic a unary functional symbol by any functional symbol of arity $\geq 1$). Expanding $\Omega_0$ to $\Omega$, let $\mathfrak A'$ be the model of $\Omega$ obtained from $\mathfrak A$ by adding a single extra point $a$ which interprets all constant symbols and assuming $F(a)=a$. Then the Kripke frame $(Sub(\mathfrak A')_{\isom},\sqsupseteq_{\isom})$ is isomorphic to $(\mathcal P(\omega),\supseteq)$. By Theorem \[thm: robust logicGOOD\], the robust theory of $(\mathcal C,\sqsupseteq)$ is included in the logic $\MLog(\mathcal P(\omega),\supseteq)$, which is $\mathrm{S4.2.1}$ by Proposition \[prop:MedvedevTop\]. It follows from the above proof that for the class $\mathcal C$ of unars without constant symbols, the robust theory of $(\mathcal C,\sqsupseteq)$ is included in the intersection of the logics of [*Skvortsov frames*]{} $(\clP(\kappa){\setminus}\{\emptyset\},\supseteq)$ for all $\kappa$. However, it is unclear how the “soundness” part of the proof can be obtained. This question is connected to a long-standing open problem about properties of the modal Medvedev logic, see, e.g., [@ShehtMedv1990]. \[rem:compareTheorems\] The proof of Theorem \[thm:submRob\] is much simpler than the proof of Theorem \[thm:sqsup-completeness\], and covers more signatures in the case with constants. However, it is unclear whether the completeness result holds for second- or first-order language. In other words, we do not know whether the resulting theories are robust in the case of second- or first-order language. Let $\mathfrak B\le\mathfrak A$ mean that $\mathfrak B$ is a quotient of $\mathfrak A$ (for the definition of quotients of arbitrary models, see [@Malcev73 Section 2.4]), and $ Quot(\mathfrak A)=\{\mathfrak B: \mathfrak B\le\mathfrak A\}. $ Quotients are, up to isomorphism, images under strong homomorphisms, hence the logic of quotients is the same that the logic of strong homomorphic images. \[thm:quot\] Let $\Omega$ have a functional symbol of arity $\ge1$, and $\mathcal C$ the class of all models of $\Omega$. Then the robust theory of $(\mathcal C,\geq)$ is $\mathrm{S4.2.1}$. Soundness is straightforward. In particular, every model $\stA$ has a unique single-point quotient $\stB$, and for any valuation $\vl$ in $L$ we have: ${\stA\mo{ \| \Box\Di \mathsf{p} \|}}$ iff $\stB\mo{ \| \Di \mathsf{p} \|}$ iff $\stB\mo{ \| \Box \mathsf{p} \|}$ iff $\stA \mo { \| \Di \Box \mathsf{p} \|}.$ By Theorem \[thm: robust logicGOOD\], the robust theory of $(\mathcal C,\ge)$ is included in the logic $(Quot(\mathfrak A)_{\isom},\ge_{\isom})$ for any model $\stA$ of $\Omega$. We shall construct $\mathfrak A$ such that $\MLog(Quot(\mathfrak A)_\isom,\ge_\isom)\subseteq \mathrm{S4.2.1}$. It suffices to handle the case when $\Omega$ consists of a single unary functional symbol $F$. As in Theorem \[thm:submRob\], let $\mathfrak A_n$ be a one-generated unar forming the cycle of length $n$. It is easy to see that all quotients of $\mathfrak A_n$ are (up to isomorphism) exactly $\mathfrak A_m$ for $n$ divisible by $m$; in particular, if $p$ is prime, $Quot(\mathfrak A_p)$ is $\{\mathfrak A_1,\mathfrak A_p\}$. Let $\mathfrak A$ be the disjoint sum of the unars $\mathfrak A_p$ for $p=1$ or $p$ prime. Let $P$ denote the set of primes. Up to isomorphism, every quotient $\stB$ of $\stA$ is the disjoint sum of unars $\{\stA_p:p\in S\}$ and also $n$ copies of $\stA_1$ for some $S\subseteq P$ and $n\leq 1+|P{\setminus}S|$; we put $h(\stB)=h([\stB]_\isom])=S$ and $g(\stB)=n$. We claim that $h$ is a p-morphism of $(Quot(\mathfrak A)_\isom,\ge_\isom)$ onto $(\clP(P),\supseteq)$. Surjectivity and monotonicity are straightforward. Assume that $h([\stB]_\isom)=S$ and pick any $S'\subseteq S$. Let $\stB'$ be the disjoint sum of unars $\{\stA_p:p\in S'\}$ and $g(\stB)$ copies of $\stA_1$. Then $\stB'$ is isomorphic to a quotient of $\stB$, and $h([\stB']_\isom)=S'$, as required. It follows that $\MLog(Quot(\mathfrak A)_\isom,\ge_\isom)$ is included in $\MLog(\clP(P),\supseteq)$. By Proposition \[prop:MedvedevTop\], the latter logic is ${{{\mathrm{S4.2.1}}}}$, which completes the proof. This theorem remains true for (not necessarily strong) homomorphic images as well. That ${{{\mathrm{S4.2.1}}}}$ includes the modal theory of homomorphisms follows from the proof above (the signature of $\stA$ does not have predicate symbols). And ${{{\mathrm{S4.2.1}}}}$ is sound since $\mathcal C$ has a single-point model that gives a top element in $\mathcal C/{\isom}$. \[ren:quot\] Similarly to the case of the submodel relation, the quotient relation is expressible in an appropriate second-order language. Given $\Omega$ and $\lambda>|\{F\in\Omega:F$ is a functional symbol$\}|$, for every $\varphi$ of $L^{2}_{\lambda,\omega}$ in $\Omega$ we pick a fresh binary predicate variable $U$ and define $\varphi_U$ by induction on $\varphi$: if $t_0,\ldots,t_{n-1}$ are terms in $\Omega$ and $P(t_0,\ldots,t_{n-1})$ is an atomic formula, then - $P(t_0,\ldots,t_{n-1})_U$ is $\exists x_0\ldots\exists x_{n-1} \bigl(\bigwedge_{i<n}U(x_i,t_i) \wedge P(x_0,\ldots,x_{n-1})\bigr) $ where $x_i$ are fresh first-order variables; if $F$ is a functional variable, then - $(\exists F\,\varphi)_U$ is $ \exists F\,(\text{$U$~is a~congruence for~$F$} \wedge\varphi_U); $ we let the operation distribute w.r.t. Boolean connectives and quantifiers over first-order and predicate variables (e.g., $(\exists P\,\varphi\bigr)_U$ is $\exists P\,\varphi_U$ for every predicate variable $P$). If $\chi(U)$ says that the interpretation of $U$ is a congruence for interpretations of all functional symbols in $\Omega$, then the map $\varphi\mapsto\exists U(\chi(U)\wedge\varphi_U)$ expresses the $\geq$-satisfiability on $\Omega$-models. However, in second-order language, we do not know whether the modal theory of quotients is ${{{\mathrm{S4.2.1}}}}$. So far we have calculated modal theories only for classes consisting of all models of a given signature. In the following examples, we describe robust modal theories of classes consisting of models of a non-trivial theory; we only outline ideas and postpone complete proofs for a further paper. Let $\mathcal D$ be the class of dense linearly ordered sets without end-points. It was observed in Example \[ex:non-rob1\] that the modal theory $\MTh^{L_{\omega,\omega}}(\mathcal D,\sqsupseteq)$ is trivial: this is the logic of a reflexive singleton axiomatized by the formula $\mathsf{p}\leftrightarrow\Diamond\mathsf{p}$. On the other hand, the robust theory of $(\mathcal D,\sqsupseteq)$ is $\mathrm{S4.2.1}$. By the classical Cantor theorem, every two countable orders in $\mathcal D$ are isomorphic (see, e.g., [@rosenstein1982linear Theorem 2.8]). Therefore, $(\mathcal D_{\isom},\sqsupseteq_{\isom})$ has a top element (the order type of rationals), so its logic includes $\mathrm{S4.2.1}$ (in spite of the fact that we do not have constant symbols in the signature). To prove the converse inclusion, in view of Corollary \[cor:gener\], it suffices to show that S4.2.1 includes the logic of a generated subframe of the frame $(\mathcal D_{\isom},\sqsupseteq_{\isom})$. Given $S\subseteq\mathbb R$, let $D_S=\{[X]_{\isom}:X\subseteq S$ is dense in $\mathbb R\}$. (Note that $[X]_{\isom}$ is the order type of $X$.) First we observe that, whenever $S\subseteq\mathbb R$ is dense, then $(D_S,\sqsupseteq_{\isom})$ forms an upper cone of $(\mathcal D_{\isom},\sqsupseteq_{\isom})$. Hence it suffices to find $S$ such that the logic of $(D_S,\sqsupseteq_{\isom})$ coincides with S4.2.1. For this, we use the following result by Sierpiński: there are two disjoint sets $E,F$ of reals both dense in $\mathbb R$, having cardinality $|E|=|F|=2^{\aleph_0}$, and such that $f(E)\not\subseteq E\cup F$ for any non-identity order embedding $f:\mathbb R\to\mathbb R$ (see, e.g., [@rosenstein1982linear Chapter 9, §2]). Let $S=E\cup F$, and for any dense subset $X$ of $S$, let $\pi([X]_\isom)=A$ if $X=E\cup A$ for some $A\subseteq F$, and $\pi([X]_\isom)=\emptyset$ otherwise. It can be verified that $\pi$ is a well-defined map and moreover, a p-morphism of $(D_S,\sqsupseteq_{\isom})$ onto $(\mathcal P(F),\supseteq)$. It induces a p-morphism of $(D_S,\sqsupseteq_{\isom})$ onto $(\mathcal P(\omega),\supseteq)$. Applying Proposition \[prop:MedvedevTop\], we conclude that $\MLog(D_S,\sqsupseteq_{\isom})$ is $\mathrm{S4.2.1}$, as required. Let $\mathcal L$ and $\mathcal O$ be the classes of linearly ordered sets and partially ordered sets, respectively. The robust theories of $(\mathcal L,\sqsupseteq)$ and $(\mathcal O,\sqsupseteq)$ also coincide with $\mathrm{S4.2.1}$. To see this, we note that there exists a p-morphism of $(\mathcal L,\sqsupseteq)$ onto $(\mathcal D,\sqsupseteq)$ (condensing scattered segments, see, e.g., [@rosenstein1982linear Chapter 4]) and that $(\mathcal L,\sqsupseteq)$ is an upper cone of $(\mathcal O,\sqsupseteq)$. This induces the same relationships between $(\mathcal D_\isom\sqsupseteq_\isom)$, $(\mathcal L_\isom,\sqsupseteq_\isom)$, and $(\mathcal O_\isom,\sqsupseteq_\isom)$. Let $\clC$ be the class of modal algebras. Similarly to the proof of Theorem \[thm:submRob\], one can show that the robust theory of $(\clC,\quot)$ is ${{{\mathrm{S4.2.1}}}}$. Namely, let $\frF$ be the disjoint sum of Kripke frames $\frF_n=(n,n\times n)$, and $\stA$ be the modal algebra of $\frF$. One can construct a p-morphism from $(Quot(\stA)_\isom,\quot_\isom)$ onto $(\clP(\omega),\supseteq)$. Thus, ${\mathrm{S4.2.1}}$ is the robust theory of the quotient relation on the class $\clC$ of modal algebras. We have the same axiomatization in the case when $\clC$ is the class of ${\mathrm{S4}}$-algebras or ${\mathrm{S5}}$-algebras, because $\stA$ is an ${{{\mathrm{S5}}}}$-algebra. (More generally, this holds if $\clC$ contains $\stA$ and is closed under quotients.) The axiomatization in the case when $\clC$ is the class of Boolean algebras is an open question. \[ex:freegr\] Let $\mathcal C$ be the class of free groups. Let $\mathbb F_\kappa$ be a $\kappa$-generated free group. All $\mathbb F_\kappa$ with $2\le\kappa\le\aleph_0$ are pairwise embeddable and non-isomorphic, while on all other $\mathbb F_\kappa$ their ordering by embedding coincides with their ordering by rank (the least cardinality of generators). Therefore, the Kripke frame $(\mathcal C_\isom,\sqsupseteq_\isom)$ has the top (corresponding to $\mathbb F_1$), a countable cluster immediate below the top (corresponding to $\mathbb F_\kappa$’s with $2\le\kappa\le\aleph_0$), and a structure isomorphic to $(Ord,\ge)$ (corresponding to $\mathbb F_\kappa$’s with $\kappa\ge\aleph_1$). By Theorem \[thm: robust logicGOOD\], it follows that the robust theory of $(\mathcal C,\sqsupseteq)$ is the modal logic of ordered sums $(\alpha,\geq)+S+S_0$ for $\alpha\in Ord$, a countable cluster $S$, and a singleton $S_0$. By the standard filtration technique, this logic has the finite model property (more precisely, we may assume that $\alpha$ and $S$ are finite) and decidable. The class $\clC$ is closed under $\sqsupseteq$ as any subgroup of a free group is free by the classical Nielsen–Schreier theorem. Hence, $\sqsupseteq$-satisfiability on $\mathcal C$ is expressible in $L^{2}_{\omega,\omega}$ by Proposition \[prop:submod-quot-exprr\]. We do not know whether the modal theory of $\sqsupseteq$ on $\mathcal C$ in $L^{2}_{\omega,\omega}$ is robust. We remark that in the first-order case the theory is not robust. Let $\equiv$ be the usual elementary equivalence. Observe that $\mathbb F_1\isom\mathbb Z\not\equiv\mathbb F_2$ (obvious), and $\mathbb F_\kappa\equiv\mathbb F_2$ for all $\kappa\ge2$ (by recently proved famous Tarski’s conjecture; see, e.g., [@ElementaryGroups Chapter 9]). It follows that $(\mathcal C_\equiv,\sqsupseteq_\equiv)$ is isomorphic to the ordinal $2$ (with the top corresponding to $\mathbb F_1$ and the bottom to other free groups). Moreover, the logic $\MTh^{L_{\omega,\omega}}(\clC,\sqsupseteq)$ is the logic of the ordinal 2. This follows from Corollary \[prop: finite frame\] in view of the following observation made by one the reviewers on an earlier version of the paper: the function $f:L_s\to L_s$ defined by letting $$f(\vf)=\left\{ \begin{array}{ll} \top & \text{ if } \mathbb F_1\mo \vf,\\ \vf & \text{ if } \mathbb F_1\not\mo\vf\text{ but }\mathbb F_2\mo\vf,\\ \bot & \text{ otherwise} \end{array} \right.$$ expresses the $\sqsupseteq$-satisfiability on $\clC$. This observation can be generalized as follows: if the elementary equivalence $\equiv$ is a bisimulation w.r.t. $\clR$ on $\clC$ and $\clC_\equiv$ is finite, then the $\clR$-satisfiability on $\clC$ is first-order expressible. Even in the case of all models of a given signature, the axiomatization of the robust theory can be a very difficult problem. Consider the theories of the relation $\supmod$ on the class $\clC$ of all models of a signature $\Omega$ consisting of unary predicates and perhaps some constants. The easiest is the degenerated case when $\Omega$ has no symbols other than constants. The theory of submodels on this class is ${\mathrm{Grz.3}}$, the Grzegorczyk logic with the linearity axiom. As well-known, this is the logic of conversely well-ordered sets, or of the set $\omega$ with the converse order, or else of all finite linearly ordered sets. Furthermore, these theories include the logic ${\mathrm{Grz}}$ iff $\Omega$ contains finitely many predicates. We announce that in the case of $n<\omega$ predicates, the robust modal theory coincides with the modal logic of the direct product of $2^n$ copies of $\omega$ with the converse order if the signature has constants, and with the logic of this structure without the top element otherwise. These logics have the finite model property, decrease as $n$ grows, and their intersection coincides with $\MLog(\clP_{\omega}(\omega){\setminus}\{\emptyset\},\supseteq)$, the modal Medvedev logic. Despite this clear semantical description, no complete axiomatizations for these logics are known. #### Acknowledgements. We are grateful to the reviewers for their multiple suggestions and questions on earlier versions of the paper. We are also grateful to Lev Beklemishev, Philip Kremer, Fedor Pakhomov, Vladimir Shavrukov, Valentin Shehtman, Albert Visser for their attention to our work, useful comments, and discussions. The work on this paper was supported by the Russian Science Foundation under grant 16-11-10252 and carried out at Steklov Mathematical Institute of Russian Academy of Sciences. [^1]: The work on this paper was supported by the Russian Science Foundation under grant 16-11-10252 and carried out at Steklov Mathematical Institute of Russian Academy of Sciences. [^2]: Modal algebras of theories are broadly used in provability logic, where they are called [*Magari*]{} (or [*diagonizable*]{}) [*algebras*]{}; they are known to keep a lot of information about theories containing arithmetic (see, e.g., [@ShavrukovPhD93]). [^3]: A significant part of Sections \[sec:upanddown\] and \[sec:rob\] was motivated by reviewers’ questions on an earlier version of this paper. [^4]: Let us also mention the logic of [*pure provability*]{} introduced by Buss in [@buss1990], which is not closed under the rule of substitution. [^5]: In [@HamkinsArithmeticPotentialism2018], the language $K_\Di$ is called [*partial potentialist*]{}.

--- abstract: 'The fabrication of artificial pinning structures allows a new generation of experiments which can probe the properties of vortex arrays by forcing them to flow in confined geometries. We discuss the theoretical analysis of such experiments in both flux liquids and flux solids, focusing on the Corbino disk geometry. In the liquid, these experiments can probe the critical behavior near a continuous liquid-glass transition. In the solid, they probe directly the onset of plasticity.' address: | Physics Department, Syracuse University,\ Syracuse, NY 13244-1130, U.S.A. author: - 'M. Cristina Marchetti [^1]' title: 'Driven vortices in confined geometry: the Corbino disk' --- INTRODUCTION ============ In the mixed state of type-II superconductors the magnetic field is concentrated in an array of flexible flux bundles that, much like ordinary matter, can form crystalline, liquid and glassy phases.[@CN97; @blatter94] In clean systems the vortex solid melts into a flux liquid via a first order phase transition.[@CN97] If the barriers to vortex line crossing are high, a rapidly cooled vortex liquid can bypass the crystal phase and get trapped in a metastable polymer-like glass phase.[@nelson_review] The diversity of vortex structures is further increased by pinning from material disorder, which leads to a variety of novel glasses. Disorder-driven glass transitions are continuous, with diverging correlation lengths and universal critical behavior.[@ffh; @drnvv] Of particular interest is the dynamics of the vortex array in the various phases and in the proximity of a phase transition. In the liquid phase the vortex array flows yielding a linear resistivity. In the presence of large scale spatial inhomogeneities, the liquid flow can be highly nonlocal due to interactions and entanglement.[@MCMDRN90; @huse] The correlation length controlling the nonlocality of the flow grows with the liquid shear viscosity, which becomes large as the liquid freezes. At a continuous liquid-glass transition this correlation length diverges with a universal critical exponent. In the solid phase the vortex array moves as a single elastic object under uniform drive, provided the shear stresses are not too large. In the presence of strong spatial inhomogeneities, plastic flow occurs for large drives (or even for vanishingly small drives in a glassy solid) and the response is always nonlinear.[@argonne] The dynamical correlation length can be identified with the separation between free dislocations and diverges at a continuous melting transition. Probing spatial velocity correlations can therefore give information on vortex dynamics within a given phase, as well as on the nature of the phase transitions connecting the various phases. As for ordinary matter, the shear rigidity of the vortex array can be probed by forcing the vortices to flow in confined geometries. [@MCMDRN90; @MCMDRN99] This type of experiments was pioneered by Kes and collaborators to study the shear rigidity of the two-dimensional vortex liquid near freezing in thin films.[@kes] More recently, patterned irradiation of cuprate superconductors with heavy ions has made it possible to create samples with controlled distributions of damage tracks.[@pastoriza] We recently showed that an analysis of such experiments that combines an inhomogeneous scaling theory with the hydrodynamics of viscous flux liquids can be used to infer the critical behavior near a continuous glass transition, as well as to distinguish between continuous transitions, such as that to a Bose glass, and nonequilibrium transition to a polymer-like glass driven by interaction and entanglement.[@MCMDRN99]. ![The field profile, $E(r)(2\pi t/\rho_fI)$, in the liquid annulus of an irradiated Corbino disk. The inner and outer radii are $R_1=2\mu m$ and $R_2=12\mu m$, and $\xi=1\mu m$. The dashed line is the $\sim 1/r$ field profile in an uncorrelated liquid, with $\xi=0$. Inset: a sketch of the disk [–]{} the Bose glass contacts are not shown.](fig1.eps "fig:"){width="2.3in" height="2.in"} \[corbino1\] Large scale spatial inhomogeneities can also be introduced in the flow, even in the absence of pinning, by applying a driving force with controlled spatial gradients, as done recently by the Argonne group using the Corbino disk geometry.[@argonne] In this paper we illustrate the analysis of spatially inhomogeneous vortex motion in both the liquid and the solid using the Corbino disk as a prototype of a novel class of experiments exploiting the effect of geometry to study the dynamics of vortex matter. LIQUID FLOW IN CHANNELS ======================= In the Corbino disk, with magnetic field along the disk axis ($z$ direction), a uniform radial current density of magnitude $J(r)=I/(2\pi t r)$ is introduced in the sample by injecting current at the center and removing it at the outer circumference of the disk (inset of Fig. 1). The current drives the vortices to move in circles about the axis. In the flux liquid, the dynamics on scales larger than the intervortex spacing, $a_0$, is described by hydrodynamic equations for the flow velocity ${\bf v}({\bf r})$, which determines the local field from flux motion, ${\bf E}=n_0\phi_0{\bf\hat{z}}\times{\bf v}({\bf r})/c$, with $n_0=1/a_0^2$. For simple geometries like the Corbino disk, where the current is spatially homogeneous in the $z$ direction, hydrodynamics reduces to a single equation, [@MCMDRN90; @MCMDRN99] $$\label{hydro} -\gamma{\bf v}+\eta\nabla^2_\perp{\bf v}={1\over c}n_0\phi_0{\bf{\hat{z}}}\times{\bf J}({\bf r}),$$ where $\gamma(T,H)$ is the friction, $\eta(T,H)$ is the viscosity controlling the viscous drag from interactions and entanglement, and the term on the right hand side is the Lorentz force density driving flux motion. It is instructive to rewrite Eq. (\[hydro\]) as an equation for the local field,[@MCMDRN90; @MCMDRN99] $$\label{viscousE} -\xi^2\nabla^2_\perp{\bf E}+{\bf E}=\rho_f{\bf J},$$ with $\xi=\sqrt{\eta/\gamma}$ the viscous correlation length and $\rho_f=(n_0\phi_0/c)^2/\gamma$ the flux flow resistivity. If the viscous force is negligible, Eq. (\[viscousE\]) is simply Ohm’s law and the radial field is $E_0(r)=(\rho_f I/2\pi t)(1/r)$. To probe the viscous drag, it is necessary to force large scale spatial inhomogeneities in the flow. This may be achieved by suitable pinning boundaries. As an example, we imagine selectively irradiating a cylindrical central region and an outer annular region of the disk to obtain the structure sketched in the inset of Fig. 2. Here the vortices in the heavily irradiated central and outer regions (shaded) are in the Bose glass phase, while vortices in the unirradiated (white) annular region are in the flux liquid phase. A radial current drives tangential flow in the resistive flux liquid annulus, which is impeded by the “Bose-glass contacts” at the boundaries. The field profile obtained by solving Eq. (\[viscousE\]) with no-slip boundary conditions [@MCMDRN99] is spatially inhomogeneous on length $\xi$, as shown in Fig. 1. One can probe this profile and extract $\xi$ by placing a string of radial contacts at $r_n$, for $n=1,2,3,...$, and measuring the voltage $V_{n+1,n}$ across each successive pair (inset of Fig. 2). If the viscosity is small ($\xi<<d$), the voltage decreases logarithmically as one moves from the inner to the outer contacts, as in a freely flowing uncorrelated liquid, where $V^0_{n,+1,n}=(\rho_fI/2\pi t)\ln(r_{n+1}/r_n)$. When $\xi$ grows, the onset of rigidity in the liquid becomes apparent (Fig. 2). An elastic vortex solid would rotate as a rigid object under the radial drive, with $v(r)\sim r$ and $V_{n+1,n}^s=(\rho_f I/2\pi tR_2^2)(r_{n+1}^2-r_n^2)$, for $R_2>>R_1$. Indeed for $\xi\geq d$, $V_{n+1,n}$ is no longer monotonic with $n$ and it exhibits a solid-like growth with $n$ within a boundary layer of width $\xi$. {width="2.5in" height="2.2in"} \[corbino\] The resistance per unit thickness of the disk, $\rho_R(T,R_1,R_2)=\Delta V(R_2,R_1)/(I/2\pi t)$, is $$\label{rhoR} \rho_R(T,R_1,R_2)=\rho_f(T){\cal F}\Big({R_1\over\xi},{R_2\over\xi}\Big),$$ with ${\cal F}(x_1,x_2)$ a function determined by the geometry that is obtained from the solution of the hydrodynamic equation. If $\xi<<R_1,R_2$, viscous effect are not important and $\rho_R\approx\rho_f\ln(R_2/R_1)\sim 1/\gamma$. Conversely, when the viscous length exceeds the sample size ($\xi>>R_1,(R_2-R_1)$), then $\rho_R\approx \rho_f(R_2^2 8\pi\xi^2)\sim1/\eta$, for $R_2>>R_1$. When the effects of geometry are negligible, resistivity measurements directly probe the friction, $\gamma$, while measurements in channels narrow compared to $\xi$ probe the flux liquid viscosity, $\eta$. Experiments in artificial pinning structures can therefore be used to infer both $\gamma$ and $\eta$. The behavior of these two parameters near a phase transition can in fact be used to identify the transition, as summarized in table 1. Transition $\gamma(T)$ $\eta(T)$ ---------------------------------- ---------------------- ----------------------- Bose Glass [@drnvv] $t^{(1-z)\nu_\perp}$ $t^{-(z+1)\nu_\perp}$ Vortex Glass [@ffh] $t^{(2-z)\nu_\perp}$ $t^{-z\nu_\perp}$ “Polymer” Glass [@nelson_review] finite $e^{a/t}$ First order freezing [@CN97] finite jumps to $\infty$ Continuous freezing [@kes] finite $ e^{c/t^{0.369...}}$ : Behavior of $\gamma$ and $\eta$ near various transitions of vortex matter. Here $t=(T-T_0)/T_0$, with $T_0$ the relevant transition temperature in each case.[]{data-label="table:1"} \ Of particular interest is the case of continuous glass transitions, which are characterized by universal critical behavior.[@ffh; @drnvv] Experiments of the type just described are especially powerful in this case as they can be used to map out the critical behavior.[@MCMDRN99] One can probe the Bose glass transition by lightly irradiating the liquid annular region, and then lowering the temperature at constant field from $T_{BG}^{\rm annulus}<T<T_{BG}^{\rm contacts}$ to $T\rightarrow T_{BG}^{\rm annulus}$. Most physical properties near the transition can be described via a scaling theory in terms of diverging correlation lengths perpendicular and parallel to the field direction, $\xi_\perp(T)\sim|T-T_{BG}|^{-\nu_\perp}$ and $\xi_\parallel(T)\sim|T-T_{BG}|^{-\nu_\parallel}$, with $\nu_\parallel=2\nu_\perp$, and a diverging correlation time, $\tau\sim l_\perp^z\sim|T-T_{BG}|^{-z\nu_\perp}$.[@drnvv] Scaling can then be used to relate physical quantities to these diverging length and time scales. In particular, the friction coefficient $\gamma$ that determines the bulk flux flow resistivity $\rho_f(T)$ is predicted to diverge as $T\rightarrow T_{BG}^+$ as $\rho_f\sim |T-T_{BG}|^{-\nu_\perp(z-2)}$. [@drnvv] As shown recently by Marchetti and Nelson, when flux flow in confined geometries is analyzed by combining hydrodynamics with the the Bose glass scaling theory [–]{} generalized to the spatially inhomogeneous case [–]{} the Bose glass correlation length $\xi_\perp$ is naturally identified with the viscous length $\xi$.[@MCMDRN99] It then follows that the liquid shear viscosity diverges at $T_{BG}$ as $\eta\sim |T-T_{BG}|^{-z\nu_\perp}$. Furthermore, the scaling of the finite-geometry resistivity displayed in Eq. \[rhoR\] is a general property of the vortex liquid near a continuous glass transition. Hydrodynamics yields the [*precise*]{} form of the scaling function, which depends on the experimental geometry and can be found in [@MCMDRN99] for the Corbino disk. PLASTIC FLOW IN DRIVEN SOLIDS ============================= As shown in recent experiments by the Argonne group, the Corbino disk geometry can also be used to study the onset of plastic flow in a driven solid.[@argonne] In this case we consider an unirradiated disk, where the vortex array has a clear melting transition. Below melting the vortex solid moves as a rigid body, with $v(r)\sim r$, and the voltage grows as $r^2$. The $\sim 1/r$ dependence of the driving force yields, however, large elastic deformations of the medium, described by the solution of $$c_{66}\nabla^2{\bf u}={1\over c}n_0\phi_0{\bf{\hat{z}}}\times{\bf J}({\bf r}),$$ with free boundary conditions ($c_{66}$ is the shear modulus of the lattice, assumed incompressible). Elastic deformations yield a finite shear stress, $\sigma_{r\phi}(r)=(n_0\phi_0 I/4\pi c t)\big[1+2\ln(R_2/R_1)R_1^2/r^2\big]$, that can unbind dislocations from bound pairs. Assuming for simplicity that vortices are straight along the field direction, the energy of a pair of dislocations of opposite Burgers vectors, separated by a distance $x$, is $U_0(x)=(c_{66}ta_0^2/\pi)\big[\ln(x/a_0)-\cos^2\theta\big]-2E_ct$, with $\theta$ the angle between ${\bf x}$ and ${\bf b}$ and $E_c\approx c_{66}a_0^2$ the core energy per unit length of an edge dislocation. An applied stress pulls the two dislocations in opposite directions.[@bruinsma] Ignoring climb, and assuming the spatial variations of the stress field are on scales large compared to $x\sim a_0$, the interaction energy is now $U(x)=U_0(x)-a_0x\sigma_{r\phi}(r)$, with $r$ the radial location of the center of mass of the pair. The applied stress lowers the barrier that confines the bound pair. In the simplest model, unbinding occurs where the location $x_B$ of the barrier, defined by $\big[\partial U/\partial x\big]_{x=x_B}=0$, becomes comparable to $a_0$, or $x_B=c_{66}a_0/(2\pi\sigma_{r\phi}(r))\approx a_0$. By solving for $r$, one obtains the critical radius $R_M(I)$ where shear-induced dislocation unbinding occurs, yielding slippage of neighboring planes of the vortex lattice, $R_M\approx R_1\sqrt{2\ln(R_2/R_1)I/(I_0-I)}$, for $R_2>>R_1$, with $I_0=2cc_{66}t/(n_0\phi_0)$ a maximum current above which the entire vortex solid shear melts. The melting radius $R_M$ increases with current, indicating that, since the stresses are largest near the axis of the disk, “shear-induced melting” occurs first in circular layers close to the axis. This behavior is qualitatively consistent with the observations by the Argonne group.[@argonne] The simple model described here suggests that at high fields the current scale $I_0$ is independent of field. A more detailed calculation incorporating field and temperature dependence will be described elsewhere.[@pb] [9]{} G.W. Crabtree, and D.R. Nelson, Physics Today [**50**]{} (1997) 38, and references therein. G. Blatter, [*et al.*]{}, Rev. Mod. Phys. [**66**]{} (1994) 1125. D.R. Nelson, in [*Phenomenology and Applications of High Temperature Superconductors*]{}, edited by K. Bedek [*et al.*]{} (Addison-Wesley, New York, 1992), Sec. 4.2. D.S. Fisher, M.P.A. Fisher, and D.A. Huse, Phys. Rev. B [**43**]{}, 130 (1991). D.R. Nelson, and V.M. Vinokur, Phys. Rev. Lett. [**68**]{}, 2398 (1992); Phys. Rev. B [**48**]{}, 13 060 (1993). M.C. Marchetti, and D.R. Nelson, Phys. Rev. B [**42**]{}, 9938 (1990); Physica C [**174**]{}, 40 (1991). D.A. Huse and S.N. Majumdar, Phys. Rev. Lett. [**71**]{}, 2473 (1993). D. López [*et al.*]{}, Phys. Rev. Lett. [**82**]{}, 1277 (1999); G.W. Crabtree, [*et al.*]{}, in [*Proceedings of MOS 99*]{}, J. Low Temp. Phys. [**117**]{}, 1999. M.C. Marchetti and D.R. Nelson, Phys. Rev. B [**59**]{}, 13624 (1999); Physica C, to appear. M.H. Theunissen, E. Van der Drift, and P.H. Kes, Phys. Rev. Lett. [**77**]{}, 159 (1996). H. Pastoriza, and P.H. Kes, Phys. Rev. Lett. [**75**]{}, 3525 (1995). R. Bruinsma, B.I. Halperin, and A. Zippelius, Phys. Rev. [**25**]{} (1982) 579. P. Benetatos, and M.C. Marchetti, in preparation. [^1]: This work was done in collaboration with D.R. Nelson and P. Benetatos and was supported by the NSF through grants DMR-9730678 and DMR-9805818.

--- abstract: 'We discuss the soundness of the scaling functional (SF) approach proposed by Aubouy Guiselin and Raphaël (Macromolecules **29**, 7261 (1996)) to describe polymeric interfaces. In particular, we demonstrate that this approach is a variational theory. We emphasis the role of SF theory as an important link between ground-state theories suitable to describe adsorbed layers, and “classical” theories for polymer brushes.' author: - Manoel Manghi - Miguel Aubouy title: | Validity of the scaling functional approach for polymer interfaces\ as a variational theory --- Introduction ============ Polymer interfaces are layers made of polymeric chains in direct contact with a boundary which may be a solid/liquid, liquid/liquid interface or a more complex surface such as a membrane. Because they have applications in such diverse fields as colloid stabilization, coating, tribology, galenic, they have been the subject of active research since the 80’s both from a fundamental and applied point of view. At present, there are two well established self-consistent-field (SCF) theories to describe polymer layers. They both start from the partition function of an ensemble of chains in contact with the interface treated in mean-field, but they soon proceed in a marked different way. Eventually, they become very different type of theories, depending on whether the chains are reversibly adsorbed, and there is an adsorbed state which dominates the solution of the Schrödinger equation associated (ground state dominance (GSD) theories [@holbook; @Semenov-Joanny]), or they are end-tethered to a repulsive surface (so-called ”brushes”), and the path integral is dominated by the classical solution (classical theories [@Semenov; @MWC; @Zhulina]). Because the two types of theories are very different in spirit, there is a conceptual gap for intermediate cases. In other words, there is no mean-field theory available to describe both adsorption and grafting of polymers within the same formalism. Such case arises, e.g., when chains are grafted onto an attractive surface. In principle, at least, one should be able to go in a continuous way from adsorbed-like to brush-like layers by tuning the amount of chains per unit surface. A tentative to bridge such gap was proposed in a series of paper where the so-called Scaling Functional (SF) approach is developed [@AGR; @se]. This is an approach where the layer of monodisperse adsorbed chains ($N$ monomers of size $a$) is considered as a thermodynamic ensemble of interacting loops and tails. These loops are polydisperse in size, and the main tool is the “loop size profile”, $S$, such that $$S(n)=S_{0}\int_{n}^{N}P(u)du, \label{defS}$$ where $P$ is the statistical distribution of loop sizes in monomer units, and $S_{0}$ is the total number (per $cm^{2}$) of loops. The free-energy (per $cm^2$) of the layer of chains is written as: $$\begin{aligned} \mathcal{F}\{S\} &\cong& \frac{k_BT}{a^2} \int_0^N \left\{k[a^2S(n)]^{\beta}\right. \nonumber \\ &+& \left.[-a^2S'(n)] \ln \left[-\frac{S'(n)}{S_0}\right]\right\} dn, \label{fenergie}\end{aligned}$$ where $k\cong 1$ is a constant, $k_BT$ is the thermal energy and $S'(n)=dS/dn$. The first term in rhs of Eq. (\[fenergie\]) accounts for loop interactions (which depend on solvent conditions through the value of the exponent $\beta$, see Table \[table\]). The second term in the rhs of Eq. (\[fenergie\]) is the usual entropy associated with a set of polydisperse objects. Similarly, the extension of the layer is computed following $$L\{S\} \cong a\int_0^N[a^2S(n)]^{\alpha}dn ,$$ where the exponent $\alpha$ is given in Table \[table\]. In the SF approach, the layer of chains is actually described as a polydisperse polymer brush (the role of the chains being played here by the “pseudo-loops”, i.e. half loops) *plus* an entropic term which stems for the fact that the size distribution is not fixed by any external operator, but the system of loops is in thermodynamic equilibrium. type of solvent good $\Theta$ melt “mean-field” ----------------- ------ ---------- ------ -------------- $\alpha$ 1/3 1/2 1 1/3 $\beta$ 11/6 2 3 5/3 : \[table\] Values of the scaling exponents for the layer thickness and the free energy. If we impose monodisperse pseudo-loops ($P(u)=\delta (u-N)$) and $S_0=\sigma$, the grafting density, we immediately recover the standard results for polymer brushes. In good solvent conditions, these are: the extension $L\cong aN(a^2\sigma)^{1/3}$, the free energy $\mathcal{F}\cong k_BTN(a^2\sigma)^{11/6}$ and the volume fraction of monomers $\Phi \cong (a^2\sigma)^{2/3}$. On the other hand, if we let the polydispersity free to minimize the thermodynamical potential (with $S_0=a^{-2}$ to account for attraction), we recover the results found for reversibly adsorbed chains. In good solvent conditions, we find that the volume fraction of monomer scales as $\Phi(z)\cong (a/z)^{4/3}$, and the extension as $L\cong aN^{3/5}$. Such idea proved to be successful in describing many different kinds of polymer layers (grafted, reversibly adsorbed [@AGR], irreversibly adsorbed [@Guiselin]), whatever the solvent quality (good solvent, $\Theta$-solvent and melt, i.e. no solvent). The approach was further expanded to the cases of convex interfaces [@AGRspheres]. The success of this phenomenological approach lead us to address the status of Eq. (\[fenergie\]). The SF approach is so far an elegant model but not a theory because Eq. (\[fenergie\]) is not deduced from first principles, and the set of approximations involved is not explicited. Recently, the SF approach was applied to the issue of surface tension of polymeric liquids [@ManoPRL; @ManoMacromol; @ManoColloid]. Here again, the SF approach proved to be successful in reproducing the experimental features in great detail. However, because the results presented in Ref. [@ManoPRL] are different from the results of the self-consistent field theory on the same issue, it seems important to clarify the soundness of the SF approach. This question is addressed here in some detail. The SF approach raises two questions essentially: *a)* is it sound ? *b)* is it valid ? The first question addresses the status of the SF approach, the second has to do with the validity of the results that we will find by using it. Obviously, these two issues are linked. Because “sound” is sometimes used for “crude” or “inaccurate”, it is useful to carefully explain what we mean by “sound” and “valid” before we start arguing. As it stands, the SF approach is a phenomenological description. This is useful on issues where we do not have any theory available. On the other hand, suppose we are in a position to compare a phenomenological approach to a theory on the same issue. The theory will always prevail. If the two results are in agreement, this is fine, but then the phenomenology is a trick to understand qualitatively the issue, and essentially does not bring new features. If, on the contrary, the two results are different, there is always the suspicion that the phenomenological approach is a good idea extrapolated to an issue where this idea is too simple, and therefore, the result is wrong. We simply say “the approach is not sound”. Accuracy then is less relevant. The debate is quite different when we have to compare two theories on the same issue. If somehow we were able to deduce the SF approach from first principles, and therefore prove that this is a theory, then the question of soundness is resolved. Of course this would be done within approximations, and the theory may be crude or inaccurate to treat the issue, but it is sound. Then the debate over accuracy is essential to evaluate the results. We see that the status of the SF approach is the first question to be addressed, and depending on the answer, the debate over validity will be different. In Section \[status\], we deduce the effective free energy, Eq. (\[fenergie\]), from first principles. In doing so, we demonstrate that the SF approach is indeed a variational theory for polymer layers. Then we are lead to ask the second question: is it valid ? Such task involves comparing the results found with SF theory to SCF theories both at a formal level, and at the level of the results. In Section \[validity\], we address this question. Status \[status\] ================= Variational free energy ----------------------- We consider a set of $N_C$ monodisperse, linear, neutral chains in contact with a solid plane (area $\Sigma $). We assume that the layer is uniform in the directions parallel to the surface. Our starting point is the partition function, $\mathcal{Z}$, of the chains, each characterized by the path $z_i(n)$, where $z_i$ is the position normal to the surface and $n$ is the curvilinear index ($1\leq i\leq N_C$): $$\mathcal {Z} =\prod_i^{N_C} \int_0^{\infty} dz_i(0) \int_0^{\infty} dz_i(N)\int \mathcal{D}\{z_i\}\exp \left[ - \frac{{\mathcal H}}{k_BT}\right]$$ where the effective Hamiltonian, $\mathcal{H}$, is the sum of an elastic (entropic) contribution, $$\mathcal{H}_{\mathrm{el}}=\frac32\frac{k_BT}{a^2}\sum_i^{N_C}\int_0^N \left(\frac{dz_i}{dn}\right)^2 dn$$ and an excluded-volume (two-body) interaction with parameter $v$, $$\mathcal{H}_{\mathrm{ex}}=\frac{vk_{B}T}{2\Sigma}\sum_{i,j}^{N_C} \int_0^N\int_0^N\delta \left(z_i(n)-z_j(n')\right) dndn'$$ where $\delta$ is the Dirac distribution. We limit ourselves to two-body interactions and thus neglect interactions of further order. This is not valid for a $\Theta$-solvent (where $v=0$), but as we shall see in Section \[solvent\], the correct free energy for this type of solvent is easily introduced afterwards. The volume fraction at distance $z$ writes $$\phi(z)=\frac{1}{\Sigma}\sum_{i=1}^{N_C}\int_0^N\delta(z-z_i(n))dn. \label{philoc}$$ ![\[figure\] As far as the Hamiltonian is concerned, each chain in contact with the boundary (a) with associated path $z(n)$ ($1<n<N$), is formally equivalent to the set of pseudo-loops (b) obtained by cutting the loops into two equal pieces with associated paths $\{z_{\alpha}(n),1<n<m_{\alpha}\}$.](figure1.eps) Regardless of the particular microscopic situation that is realized, we can always decompose the chain into loops and tails, and rewrite $\mathcal{H}$ accordingly. This amounts to cut the integrals into smaller pieces, by identifying the monomers either in contact with the surface or at the top of the loops and that we note $n_{i,\alpha}$. Each piece corresponds to the complete path of a loop or a tail. The “cutting” scheme is described in Fig. \[figure\]. We implicitly assume that the loops are symmetric, which comes from the translation invariance parallel to the solid surface. Hence mathematically, these identified monomers have a “null velocity”: $dz_i/dn|_{n_{i,\alpha}}=0$ for each $i$ and $\alpha$. The chain $i$ is then cut in $N_i$ pieces of size $m_{i,\alpha}=n_{i,\alpha}-n_{i,\alpha-1}$ where $1\leq\alpha\leq N_i$ with $\sum_{\alpha }m_{i,\alpha}=N$. For tails, we consider the full path from the extreme monomer to the first monomer in direct contact with the surface, as expected. The loop are cut into two pieces of equal length, which we shall call pseudo-loops. Clearly, as far as mathematics is concerned, tails and pseudo-loops are similar objects : these are chain segments starting at the surface and ending somewhere in the solution with no velocity at these extreme monomers (cf. Fig. \[figure\]). For that reason, we shall not distinguish between tails and pseudo-loops in the rest of the letter, and refer to both of them as “pseudo-loops”. As is obvious, such decomposition *a)* is always possible, *b)* is unambiguous, *c)* let the partition function $\mathcal{Z}$ identical without any approximation, provided that we supplement the cutting procedure by the constraint (later referred to as $\mathcal{C}$), that the free extremities of the chain *segments* originating from the same loop should be at the same height $z$. Then $\left(m_{i,\alpha}, \left\{z_{i,\alpha}\right\}\right)_{\alpha=1,N_i}$ designates the set of sizes and paths of pseudo-loops for chain $i$, and we rewrite the Hamiltonian: $$\mathcal{H} = \frac{3}{2}\frac{k_{B}T}{a^{2}}\sum_{i=1}^{N_{C}}\sum_{\alpha =1}^{N_{i}}\int_{0}^{m_{i,\alpha }}\left( \frac{dz_{i,\alpha }}{dn}\right) ^{2}dn + \frac{vk_{B}T}{2\Sigma }\sum_{i,j=1}^{N_{C}}\sum_{\alpha ,\beta =1}^{N_{i}}\int_{0}^{m_{i,\alpha }}\int_{0}^{m_{j,\beta }}\delta \left( z_{i,\alpha }(n)-z_{j,\beta }(n^{\prime })\right) dndn^{\prime } . \label{Hdécomposé2}$$ Computing exactly the partition function of the system with the Hamiltonian Eq. (\[Hdécomposé2\]) is clearly out of reach. Rather, we implement the variational principle which necessitates two steps [@variationnel]. First, we need to choose a trial probability such that $\mathcal{P}_{T}$ is a good approximation of the actual probability, $\mathcal{P}= \mathcal{Z}^{-1} \exp [-\mathcal{H}/k_{B}T]$, but nevertheless allows for analytical calculations. Second, we approximate the exact free-energy, $\mathcal{F}$, of the system by the extremum of the functional $\mathcal{F}_{\mathrm{var}}\{\mathcal{P}_{T}\} = \left\langle \mathcal{H}\right\rangle_{\mathcal{P}_{T}}+ k_BT\left\langle \ln\mathcal{P}_{T}\right\rangle_{\mathcal{ P}_{T}}$. Of these two steps, the second one is the simplest because it is purely a matter of calculation. Only the first one is significant as regard to the physics, since the success of the variational theory lies in finding an appropriate trial function. The guess $\mathcal{P}_{T}$ is a functional form with free unspecified parameters. By minimizing $\mathcal{F}_{\mathrm{var}}\{\mathcal{P}_{T}\}$ with respect to these parameters, we will obtain $\mathcal{P}_{T}$ with the chosen functional form that best approximates $\mathcal{P}$. This is what ultimately controls the difference between $\mathcal{F}$ and the approximation $\mathcal{F}_{\mathrm{var}}$. Note that the choice of the ensemble of functions over which we shall perform the minimization is arbitrary. It is a guess, not an approximation which could be somehow quantified *a priori*. Our guess for $\mathcal{P}_{T}$ is: $$\mathcal{P}_{T}\left(\left\{(m_{i,\alpha}, \{z_{i,\alpha}\})_{\alpha=1,N_i}\right\}_{i=1,N_C}\right)= \prod_{i=1}^{N_C}\prod_{\alpha=1}^{N_i}P(m_{i,\alpha},\{z_{i,\alpha}\}), \label{defP}$$ where $\int_0^N P(m_{i,\alpha},\{z_{i,\alpha}\})dm=1$. Equation (\[defP\]) is a mean-field type of approximation for the pseudo-loops since their probability distributions are decorellated (hypothesis *A*). Furthermore, we assume that *the path $\{z_{i,\alpha}\}$ is the same for all the pseudo-loops and is noted $\{z\}$* (hypothesis *B*). Because $P(m_{i,\alpha},\{z_{i,\alpha}\})$ does not depend on the particular pseudo-loop that is considered, we can drop the indexes and write $P(m,\{z\})$. Hence the probability distribution reads $\mathcal{P}_{T}=P(m,\{z\})^{B}$ where $B=\sum_{i=1}^{N_C}N_i$ is the number of pseudo-loops at the interface. The crucial point is that $\mathcal{P}_{T}$ does no more depend on the complete set of sizes and path, $\left\{ m_{i,\alpha },\left\{ z_{i,\alpha }\right\} \right\} $, but only on *a)* the size of the pseudo-loop, $m$, and on *b)* the path $z$, chosen to be the same for all pseudo-loops. Importantly, the constraint $\mathcal{C}$ is automatically fulfilled with our approximation since two pseudo-loops originating from the same loop have the same size, $m$, and thus terminate at the same height $z(m)$. Then, the system is described by two functions: $P(m)$, the probability that we have a pseudo-loop of size $m$, and $z(n)$, the path of the chain segments. Hence, the trial free energy is obtained by minimizing $\mathcal{F}_{\mathrm{var}}$ with respect to changes in $z$ and $P$ (later, we will find it more convenient to work with $S$, rather than $P$). With (\[defP\]), we find $$\left\langle\mathcal{H}_{\mathrm{ex}}\right\rangle_{\mathcal{P}_{T}} = \frac12 \frac{v\Sigma}{a^6} k_BT \int dz\Phi^2(z) \label{Hex}$$ where $$\begin{aligned} \Phi(z) &\equiv& \left\langle\phi(z)\right\rangle_{\mathcal{P}_{T}} \nonumber \\ &=& a^3S_0\int_0^N dmP(m)\int_0^m\delta(z-z(n))dn \label{phim}\end{aligned}$$ and $$\left\langle \mathcal{H}_{\mathrm{el}}\right\rangle _{\mathcal{P}_{T}}=\frac32 \frac{k_BT}{a^2}\Sigma S_0 \int_0^Ndm P(m)\int_0^m\dot{z}^2(n)dn \label{Hel}$$ where $B=\Sigma S_0$ (hence $S_0$ is the “grafting density” of pseudo-loops), and $\dot{z}=dz/dn$. Similarly, the entropic part of $\mathcal{F}_{\mathrm{var}}$ is found to be $$k_BT\left\langle \ln \mathcal{P}_{T}\right\rangle_{\mathcal{P}_{T}} = k_BT\Sigma S_0\int_0^NdmP(m)\ln P(m).$$ Combining all these results and integrating by parts and using Eq. (\[defS\]), we find $$\begin{aligned} \frac{{\mathcal F}_{\mathrm{var}}(\{S\},\{\dot{z}\})}{k_BT\Sigma} &=& \int_0^N \left\{\frac{3}{2a^2}\dot{z}^2(n)S(n) + \frac{v}{2}\frac{S^2(n)}{\dot{z}(n)} \right. \nonumber\\ &-& \left. S'(n)\ln\left(-\frac{S'(n)}{S_0}\right) \right\}dn . \label{centralresult}\end{aligned}$$ Note that $\Phi(z)=S(n(z))/\dot{z}$. Equation (\[centralresult\]) is the central result of this letter which we now discuss. To get the best approximation, we minimize Eq. (\[centralresult\]) with respect to $\dot{z}$ which yields $\dot{z}=\left(va^{2}/6\right)^{1/3} S^{1/3}$, and when this result is introduced back into Eq. (\[centralresult\]), we find Eq. (\[fenergie\]) with $\beta =5/3$ and $k=\frac{3.6^{1/3}}{4}\left(v/a^3\right)^{2/3}$. We thus find the mean-field version of our effective free energy Eq. (\[fenergie\]) with a numerical coefficient, $k$, of order 1. The formal derivation presented here brings an interesting remark. In the early developpements of the SF theory, the entropic part in Eq. (\[fenergie\]) was introduced (and interpreted) as a contribution arising from combinatorial arrangements of pseudo-loops at the surface: the presence of the interface breaks down the symmetry of the solution and these monomers in contact with the surface become *distinguishable*. We see that the entropic term in Eq. (\[centralresult\]) is formally that contribution arising from the entropy of the trial probability. Generalization to other solvent conditions \[solvent\] ------------------------------------------------------ The generalization to other solvent conditions, i.e. good solvent, $\Theta$-solvent and melt, has been done in other references [@AGR; @se] and deserves some comments. In the case of a melt, the excluded volume interactions are screened at all scales, and our mean-field approximation for pseudo-loops is automatically verified. The probability distribution is then related to the Green function of a chain by $P(m)\propto G(0,z(m);m)$ where $z(m)$ is self-consistently determined *via* the constraint $\dot{z}(n)=S_0\int_n^NP(m)dm$ ($\phi(z)=1$ everywhere in an incompressible melt). For a good solvent, the osmotic Eq. (\[Hel\]) and elastic term Eq. (\[Hex\]) are easily renormalized, following the des Cloiseaux law [@desCloiseauxlaw], and using semi-dilute blobs [@PGGbook]. However, the approximation which consists in neglecting correlations between pseudo-loops is *a priori* not verified. Thus, the transformation of $k_BT\langle\ln{\mathcal P}_T\rangle_{{\mathcal P}_T}$ in $k_BT\Sigma S_0\int_0^Ndm P(m)\ln P(m)$ is not justified. However, correlations between monomers inside the same pseudo-loop are taken into account through the blob renormalization. Hence we have demonstrated that the SF approach is a variational theory, and Eq. (\[fenergie\]) is sound. Validity \[validity\] ===================== Of course, that the SF theory is sound (in the sense that it is deduced from first principles) does not guarantee at all that it is accurate, or even simply valid to describe polymeric layers. This is because we have made approximations whose range of validity remains to be examined. *A priori*, we could distinguish three different point of view to discuss the issue of accuracy: *a)* internal, *b)* external and *c)* experimental. Internal estimate of accuracy ----------------------------- Internal means that we are able to estimate the error that we have made in approximating the initial Hamiltonian, and thus propose an internal criterium of validity, very much like the Lifchitz criterium of validity for mean-field theories. This requires that we define a relevant parameter which would quantify the difference between the initial and the approximated Hamiltonian, i.e. the two assumption that we made. Concerning hypothesis *A*, we know that the mean-field approximation for the loops is not valid in good solvent conditions. This implies that the last term of Eq. (\[centralresult\]) is wrong. However, the renormalization with semi-dilute blobs of the first two terms takes into account the swelling of the pseudo-loops (hence correlations between monomers) on scales smaller than the pseudo-loop sizes. Thus for loops at least larger than one blob size, the excluded volume interactions are screened and this loops are decorellated. Hence, the entropic term of Eq. (\[centralresult\]) is justified for a large number of the pseudo-loops and even if it is not fully satisfying, this is the best way we can take into account these correlations unless we are lead to use renormalization group theory, which has been done for one chain but not for many chains [@Eisenriegler]. The hypothesis *B* is the crudest assumption in our theory. We assume that all pseudo-loops have the same *mean* path $z(n)$. It is easy to show that for a melt, we find by minimization $z_{\mathrm{eq}}(n) \simeq n^{1/2}$, which is the best variational approximation with our probability distribution Eq. (\[defP\]). This result is quite similar to the Flory theorem $R \simeq aN^{1/2}$ for the extension of a polymer chain in a melt. Of course this result is valid for large $n$, since for a random walk, fluctuations around this value is proportional to $n^{-1/2}$. This result may not be valid for small loops. However, with variational theories, the estimate of this error is impossible. External estimate of accuracy ----------------------------- External means that we compare the SF theory with another theory. For polymeric layers, the obvious candidate is SCF theories. *A priori*, there are two ways to do that: *a)* a formal comparison, *b)* a comparison of the results that we obtain on a given issue. A formal comparison is simple when the two theories have a common language. Unfortunately, this is not the case for SCF theories and the SF theory. The former is deduced from the initial Hamiltonian through a mean-field type of approximation for monomer-monomer correlations which is then applied to the problem of polymer at interfaces, whereas the latter proceeds in *first* rewriting the Hamiltonian for chains at interfaces and *then* using a mean-field approximation for pseudo-loops. Because of this different order for these two steps, we do not know the way to formally compare SCF and SF theories. Then we are left with comparing the results. There are two issues where such comparison is possible: *a)* brushes in the infinite stretching limit, “mean-field” solvent conditions, and *b)* reversibly adsorbed layers, “mean-field” solvent conditions. This issues are conceptually important because we know exactly the solution of the SCF theory in the asymptotic limit $N\rightarrow\infty$. ### Brushes As shown by Netz and Schick [@Netz] and Li and Witten [@Li], the theory of polymer brushes proposed simultaneously by Milner, Witten, and Cates (MWC) and Zhulina *et al.* in Refs. [@MWC; @Zhulina], which consists in keeping the classic path in the partition function, can also be considered as a variational approach. However, the trial probability is different, and the layer is described by two functions: $g$, such that $g(z_0)dz_0$ is the probability that the chain free extremity belongs to the interval $[z_0,z_0+dz_0]$, and $e$, such that $e(z,z_0)=|dz/dn|$ is the extension at position $z$ for a chain whose free-extremity is situated at $z_0$. Paths (described by $e$) are chosen such as polymers are grafted at one end (with grafting density $\sigma $), i.e. $\int_0^{\infty}\frac{dz_0}{e(z,z_0)}=N$ (which leads to the so-called equal time argument). The variational free energy (per $cm^2$) is [@Netz]: $$\begin{aligned} \frac{\mathcal{F}_{\mathrm{MWC}}}{k_BT} &=& \frac{v}{2}\int_0^{\infty}\Phi^2(z)dz \nonumber \\ &+& \sigma \int_0^{\infty}dz_0 g(z_0)\int_0^{z_0}\frac{3}{2a^2}e(z,z_0)dz \nonumber \\ &+& \sigma \int_0^{\infty}g(z_0)\ln [g(z_0)]dz_0, \label{NRJSchick}\end{aligned}$$ with $\Phi(z)=\sigma \int_z^{\infty}dz_0 \frac{g(z_0)}{e(z,z_0)}$. Note that in the context of brushes, the entropic contribution in Eq. (\[NRJSchick\]), which is similar to that in Eq. (\[centralresult\]), is the entropy of the chain end distribution [@Netz; @g1; @g2]. Simple arguments show that the first two terms in the rhs of Eq. (\[NRJSchick\]) scale as $N(a^2\sigma)^{5/3}$, whereas $\int g(z_0)\ln [g(z_0)]dz_0\sim 1$. Hence, in the strong stretching limit, $N(a^2\sigma)^{2/3}\gg 1$, the entropic contribution to $\mathcal{F}_{\mathrm{MWC}}$ is negligible [@Netz]. However, this term is conceptually important and has a physical signification since $e(z_0,z_0)$ is the tension sustained by the free chain-ends. Hence, we see that Eq. (\[centralresult\]) and Eq. (\[NRJSchick\]) are formally very close, but the choices for, respectively, $z(n)$ and $e(z,z_0)$ are different. To compare the SF theory with the MWC theory, we concentrate on monodisperse brushes (hence the entropic contribution in Eq. (\[centralresult\]) disappears) in the strong stretching limit (hence, we neglect the entropic contribution in Eq. (\[NRJSchick\])). We find in equilibrium $\mathcal{F}_{\mathrm{MWC}}^*=0.892\,\mathcal{F}^*$. We see that the extremum of $\mathcal{F}_{\mathrm{MWC}}^*$ is lower and according to the variational criterium, the MWC theory is a better approximation of the exact free energy. See [@Li; @milner] for a thorough discussion of this difference. It is related to the different choices for the paths where the MWC choice (i.e. the equal time argument) is less restrictive. The reason is that in the SF theory for brushes, we impose an additional constraint: all chain free extremities are situated in the outer edge of the layer, in a fashion similar to the Flory approach (or the Alexander-de Gennes, which is similar in spirit but introduces the correct scaling exponents). Formally, this amounts to impose a delta type of function for $g$, a restriction motivated by our desire to keep the SF theory tractable in a wider range of situations. Eventually, we find the same results for $L$ and $\mathcal{F}^*$ at the scaling level, although the description of the volume fraction profile is more accurate in the MWC theory. ### Adsorbed layers Presumably, the case of reversibly adsorbed polymers is more significant for our purpose since our variational approach is based on a “loop description”, which is justified for the homogeneous adsorption. If we go to reversible adsorption, we have to turn our attention to GSD theory. Although desirable, it is not so simple to compare the SF theory with GSD theories. There are two reasons for this: *a)* the GSD theory uses the analogy between the partition function $\mathcal{Z}$ and the Green propagator in quantum mechanics, which does not allow a description in “polymer trajectories”; *b)* in this theory, the free energy is expressed in terms of the mean monomer concentration $\Phi(z)$, a quantity not simply related to our probability density $P(m).$ Indeed, the partition function of a chain having one end at $z$ and the other free, $\mathcal{Z}(N,z)$, in the SCF theory, is the solution of the Schrödinger equation : $\partial \mathcal{Z}/\partial N=\frac{a^2}{6}\partial^2 \mathcal{Z}/\partial z^2-U\mathcal{Z}$, where the external potential, $U$, is the sum of the attractive potential due to the surface, $U_{\mathrm{surf}}$, and the self-consistent potential, $U_{\mathrm{SCF}}$. For adsorbed chains, there is a ground state of negative energy $-\varepsilon Nk_BT$ which dominates the solution, and (in the limiting case where $\varepsilon N\gg 1$) the free energy approximates to: $$\mathcal{F}_{\mathrm{GSD}}=k_BT\int_0^{\infty}dz\left[\kappa(\Phi)\left( \frac{d\Phi}{dz}\right)^2 + U(z)\Phi(z)\right] , \label{GSD}$$ where $\kappa(\Phi)=a^2/(24\Phi)$. As shown by Lifshitz and des Cloiseaux [@Lifshitz; @des; @Cloiseaux], the square gradient term in Eq. (\[GSD\]) has essentially an entropic origin [@note], whereas the polymeric nature of the liquid can be neglected in the molecular field $U_{\mathrm{SCF}}(z)$ (which is estimated for a monomeric liquid). Then we are lead to think that the elastic and entropic part in Eq. (\[centralresult\]) are related to the square gradient term, but we are not able to rewrite the former as the latter at the moment. In the absence of any clue to formally compare Eqs. (\[centralresult\]) and (\[GSD\]), we shall compare their results for infinite chains and mean-field potential, a limit where the GSD theory happens to be exact. If we minimize the free energy Eq. (\[centralresult\]) with the boundary conditions $S(0)=a^{-2}$, $S(N\rightarrow\infty)=0$, we find $a^2 S_{\mathrm{eq}}(n)=k'^{3/2}/(n+k')^{3/2}$ where $k'=(3/(2k))^{4/9}$ which yields $\Phi(z)\sim z^{-2}$, essentially the solution found by minimizing Eq. (\[GSD\]). Similarly, we find that $\mathcal{F}^*\cong k_BT/a^2$ as with GSD theory. Hence, we find a very good agreement for infinite chains. That the agreement should be better (in the sense that both the scaling and the concentration profile are identical) for adsorption than for brushes reflects the validity of our initial assumption that all pseudo-loops have the same path. As explained in Ref. [@MWCpoly], for very polydisperse layers, we expect a stratification of the locations of the free chain ends above the surface. This is because the free ends of a long chain locates further away from the surface than that of a short chain to take advantage of a lower osmotic pressure (the concentration decreases away from the interface). In the continuum limit, this argument suggests that every pseudo-loops are similarly extended, and therefore validates our guess. Experimental estimate of accuracy --------------------------------- To evaluate the accuracy of a variational theory, the ultimate and major argument is to compare the value of the free energy at its minimum to experiments. The good candidate is thus the surface tension of polymeric liquids, $\gamma$. We have shown in Refs. [@ManoPRL; @ManoMacromol; @ManoColloid], that the SF approach allows the calculation of the variations of $\gamma(N)$ in very good agreement with experimental data found in the literature. This is a good test for the theory which has been done both for melts and semi-dilute solutions (in good solvent). It is important to note that the SCF theory in the GSD approximation leads to a different result for the melt surface tension. The finite chain correction in that case is proportional to $N^{-1}$. We found a larger correction in $\ln N/N^{1/2}$. An explanation of this discrepancy is that the SCF description relates the surface tension to the gradients in volume fraction which are localized in a very thin layer of thickness $a$ (indeed this approach is not valid for large gradients). We argue that this dependence comes from the chains reorganization on a larger layer of thickness the radius of gyration of a chain. In this layer, the volume fraction is constant. Thus it cannot be described by the SCF approach whereas the SF approach uses different tools, namely $z(n)$ and $S(n)$, which allows such a description. Hence, for adsorbed layers from a melt and a semi-dilute solution, we see that these two approaches are quite different. Concluding remarks ================== This article aims at clarifying the debate concerning the “soundness” of the Scaling Functional approach. In view of this, the demonstration that the SF approach is a variational theory is certainly the essential and most significant result of this article (Section \[status\]). But we think we have made clear a certain number of points (Section \[validity\]). These are: 1. The SF approach is a variational theory and therefore has the same epistemological status as SCF theories for brushes (“classical” solution) and adsorbed chains (GSD). Of course, the approximations made are different and each of these theories has a different range of validity. 2. Because the SF theory is a variational theory, we are not able to properly quantify the approximations that are involved, and therefore we are unable to define the range of validity of this theory. 3. There is no way that we know to formally compare the two theories because the first step in approximating the initial Hamiltonian are different. 4. Each time a direct comparison with SCF theory is possible, we find *a)* always the same scaling results, and *b)* sometimes the same analytical result. Thus we conclude that the SCF theory does not provide any argument against the SF theory. That the GSD approximation to the SCF theory is formally justified and quantifiable in “mean-field” solvent conditions does not guarantee that the result that we find is accurate for real solvent conditions and notably for the melt case. A description only in terms of volume fraction (see Eq. (\[GSD\])) comes also from a variational argument [@des; @Cloiseaux] and has not been quantitatively justified in real systems. In other words, the GSD in the limit $N\rightarrow\infty$, is the exact solution of the SCF theory, but still an approximate solution of the initial Hamiltonian. The crucial point regarding our approximations (cutting loops into two tails and describing all the pseudo-loops with the same path) is whether the distinction between loops and tails is important enough to modify the conclusions of a simple theory in which it is neglected. When we are in a position to directly evaluate the consequences of these approximations, we find that this distinction does not affect the scaling results. It is interesting to note that the distinction between loops and tails has been done self-consistently for the SCF theory but is put *ad hoc* for other type of solvents [@Semenov-Joanny]. Therefore we conclude that there is no valid argument to support that these approximations are not sound, provided that we remain at a scaling level of description. Finally, we assume in this approach that a large number of loops are formed at the interface. This impose both a sharp interface and the presence of many adsorbed chains. Therefore, this theory does not apply to single chain adsorption and to systems such as interfaces between incompatible polymers or diblock copolymers, for which other approaches based on the SCF theory have been developed [@Helfand; @Leibler]. As a conclusion, the SF theory proposes a compromise between a precise description of the polymeric layer, and a wide ranging scaling type of theory valid for arbitrary polymer layers, various solvent conditions and various geometries. 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--- author: - 'Hao Tong, Jialin Liu, XinYao' bibliography: - 'reference.bib' title: 'Algorithm Portfolio for Individual-based Surrogate-Assisted Evolutionary Algorithms' --- Introduction ============ Computationally expensive problems (CEPs) are very common in many real-world systems, requiring enormous computational resources to accomplish one fitness evaluation [@jin2018data]. For instance, one evaluation based on computational fluid dynamic simulations will cost several hours [@jin2009systems]. Obviously, canonical evolutionary algorithms are challenging to handle this kind of problems directly. To overcome this challenge, surrogate-assisted evolutionary algorithms (SAEAs) are developed by applying a much cheaper model to replace the actual expensive fitness evaluation process to reduce the computational cost [@jin2011surrogate]. Over the past decades, many efficient SAEAs has been proposed and applied into complex real-world applications, such as trauma system [@wang2016data]. Individual-based model control [@jin2005comprehensive] method is the most effective strategy that a few individuals will be re-evaluated by the actual function in each generation according to different criteria. For example, some criteria like expected improvements (EI), consider the fidelity of surrogate models and the quality of evaluated solutions simultaneously in global optimisation [@jones1998efficient]. On the other hand, recent works in individual-based SAEAs proposed new strategies to trade-off exploration and exploitation of optimisation, like active learning based model management [@wang2017committee] and Voronoi-based SAEA framework for very expensive problems [@hao2018voronoi]. They obtain a better solution in test problems and real-world applications compared with classical efficient global optimisation (EGO) [@jones1998efficient]. Even though many model management strategies in individual-based SAEAs are successful in the literature, no free lunch theorems indicate that there is no one best approach appropriate for every problem [@wolpert1997no]. For example, EGO is much more powerful than state-of-the-art algorithms in low-dimension cases and algorithm in [@yu2019generation] is an expert in multi-modal expensive problems while Voronoi based SAEA framework is good at uni-modal problems [@hao2018voronoi]. However, it is hard to determine the optimal algorithm for an unknown problem in practice. In order to address this challenge, algorithm portfolio is employed to reduce the risk of failing to optimise problems in multiple scenarios[@huberman1997economics]. We proposed two algorithm portfolio frameworks in this paper for individual-based SAEAs in very expensive problems [@hao2018voronoi]. The first framework is motivated from the population-based algorithm portfolio [@peng2010population], which runs all algorithm candidates simultaneously. In another framework, we employ the technique from reinforcement learning to select relatively “best” algorithm for every generation. Unlike the portfolio for Bayesian optimisation, we directly choose a method to search a solution for re-evaluation instead of generating several solutions simultaneously by different approaches and then evaluating one of them by actual fitness function [@shahriari2016taking]. The remainder of this paper is structured as follows. Section \[related-work\] will present some related work about algorithm portfolio. And then the detail of two algorithm portfolio frameworks will be introduced in Section \[algorithm portfolio\]. In Section \[experiments\], we will apply some state-of-the-art individual-based SAEAs to the proposed frameworks and test them in a series of benchmark problems. Finally, the paper will end with a brief conclusion and a discussion of future work in Section \[conclusion\]. Related work ============ Portfolio of evolutionary algorithm ----------------------------------- In the areas of evolutionary algorithms, algorithm portfolio is applied to increase the probability of finding a better solution by allocating computational resources to several complementary algorithms. The algorithm portfolio frameworks in the literature can be classified into two categories as the parallel-based framework and the sequential-based framework. For the parallel-based framework, all candidates will run simultaneously in multiple sub-processes. Population-based algorithm portfolio (PAP) is a typical example [@peng2010population], which allocates computational resources before the optimization according to the prior knowledge. Each algorithm has its own population and evolve independently, but the information is shared among different algorithms by migration strategy. Besides, other parallel-based portfolio frameworks like AMALGAM-SO [@jasper2009self] and the UMOEAs [@saber2014testing] collect the performance of algorithms during the optimisation process and allocate more resources to the better algorithm. On the other hand, the sequential based framework only runs one algorithm at most of the time during the process of optimisation. Different from the parallel-based algorithm portfolio, this kind of framework try to select the best algorithm in different optimisation stage. The multiple evolutionary algorithm (MultiEA) is one of the state-of-the-art sequential algorithm portfolio frameworks [@yuen2016algorithm]. It utilises the history convergence curve of each algorithm to predict its performance in the near future, and then the best algorithm will be selected to optimise the problem. Another typical sequential portfolio strategy worthy of mention is an online racing algorithm, max-race portfolio (MRP) [@tian2014online]. The best algorithm is selected by a statistical test on algorithms’ online performance and when enough statistical evidence indicates that one algorithm is significantly inferior to other algorithms, the worst one will be removed by framework permanently. Multi-armed bandit problem -------------------------- For a $K$-armed bandit problem, it is basically defined by random variables $\{X_{i, t}|i=1,2,...,K, t\in \mathbb{N}$} where each $X_{i, t}$ represents an independent and identical distribution with an unknown expectation $\mu_i$ for each arm of bandit machine in $t_{th}$ successful pull [@peter2002finite]. For any environment state, the action being taken at next time step is determined by a bandit policy $\pi$, which is learned according to the actions’ history rewards. The quality of a policy is measured by cumulative regret, which could be defined by Eq. : $$R_n = \mu^* n - \sum_{j=1}^{n} E [T_j(n)\mu_j] \label{regret}$$ where $\mu_j$ is the expected reward of arm $j$, $\mu^*$ is the expectation reward of optimal arm, i.e. $\mu^* \overset{\underset{\mathrm{def}}{}}{=} \max\limits_{1\leq j \leq K}\mu_j$ and $T_j(n)$ represents the number of times arm $j$ has been pulled over $n$ trails. The upper confidence bound (UCB) algorithm is a prevalent and effective method for multi-armed bandit problems to tackle the dilemma between exploitation and exploration [@peter2002finite]. In this paper, the UCB-Tuned (UCB-t) algorithm is applied for algorithm portfolio because there is no additional parameter requiring to be adjusted in the algorithm which is presented in Eq. : $$\pi_{j, n} = \overline {{\mu}} _j + \sqrt {\frac{{\ln {\rm{n}}}}{{{T_j(n)}}} \cdot \min \{ \frac{1}{4},{v_j}({T_j(n)})\} } \label{ucb-t}$$ and $${v_j}(s) = \frac{1}{s}\mathop \sum \limits_{\tau = 1}^s \mu _{j,\tau }^2 - \overline \mu _{j,s}^2 + \sqrt {\frac{{2\ln n}}{s}} \label{ucb-t1}$$ where $\bar{\mu}_j$ is the average reward of arm $j$ after $n$ trails. The policy will select the arm with maximal UCB value according to Eq. and for the next generation. In the literature, there has been some works about bandit framework for algorithm selection. Baudi[š]{} and Po[š]{}[í]{}k [@baudivs2014online] applied basic UCB in black box optimisation in which they defined the reward by introducing a log-rescaling method to process the raw fitness value. And the value rank [@fialho2010toward] as a method of reward definition is also compared in the experiments. The results show that the UCB algorithm is efficient in algorithm selection problems. Also in [@david2014differential], authors regarded the algorithm selection problem as a non-stationary bandit problem and applied UCB algorithm to be the decision policy. From this view, it is reasonable to consider the algorithm portfolio problem in the area of reinforcement learning and employ appropriate methods to construct the framework for individual-based SAEAs. Algorithm portfolio strategies {#algorithm portfolio} ============================== Individual-based SAEAs re-evaluate a few individuals at each generation and individuals being re-evaluated in the next generation is only determined by the current database. As a sequence, we will introduce two portfolio frameworks as parallel individual-based SAEAs and UCB for individual-based SAEAs which are motivated from two different aspects as reviewed previously. Parallel individual-based SAEAs ------------------------------- ![The diagram of the framework: Parallel individual-based SAEAs.[]{data-label="par-ibsaea"}](img/par.pdf) Similar with the algorithm portfolio for canonical evolutionary algorithms, it is intuitive to consider each individual-based SAEA as a simple evolutionary algorithm and embed them into the existing framework, like PAP or MultiEA. From this aspect, the parallel individual-based SAEAs (Par-IBSAEA) framework is proposed that all algorithm candidates run simultaneously at each generation. Nevertheless, it is more convenient than portfolio for canonical evolutionary algorithms because almost all individual-based SAEAs have the same algorithm structure and it does not require a particular design for each algorithm. A brief diagram for Par-IBSAEA with only three algorithm instances is presented in Figure \[par-ibsaea\] where the double solid line represents the interaction between individual-based SAEAs and the database. Considering the set of individual-based SAEAs $\mathcal{A} = \{A_1, A_2, ..., A_n\}$ with $n$ algorithms, each algorithm is independent with each other. The framework starts by the initialization of a database and all SAEAs use the same surrogate model constructed by samples in the database to search for the next re-evaluated solution. Due to different mechanisms for various algorithms, they will obtain several different solutions for re-evaluation as $\mathcal{D} = \{\mathbf{x}_1, \mathbf{x}_2,..., \mathbf{x}_k\}$, where the size of $k$ is not completely equal to $n$ because some individual-based SAEAs might get more than one solution. For example, VESAEA will re-evaluate two solutions in one generation when the optimisation in global search stage [@hao2018voronoi]. The database will be updated by adding evaluated solutions at the end of one iteration, and the whole algorithm will be stopped when the fitness evaluations are exhausted. UCB for individual-based SAEAs ------------------------------ ![The diagram of the framework: UCB for individual-based SAEAs.[]{data-label="ucb-ibsaea"}](img/ucb.pdf) On the other hand, the individual-based SAEA portfolio is regarded as a multi-armed bandit problem that an individual-based SAEA is considered as an arm and the quality of solutions could be used to measure the reward of actions. Then, we could apply the UCB algorithm to determine which algorithm is used in the next generation for CEPs. The key point in UCB for individual-based SAEAs (UCB-IBSAEA) is the definition of reward. According to the condition of UCB, the reward used in UCB policy must be in the range of $[0, 1]$. However, we can not obtain the actual bounds due to the problem solved is a black box. And the only information we could use is the evaluated individuals in the database. Therefore, the bounds could be estimated by the samples’ fitness in the database. Assuming all solutions’ fitness in database in $t_{th}$ generation is $\mathcal{Y}_t = \{y_1, y_2, ..., y_n\}$, the empirical upper bound $eUB_{t}$ and empirical lower bound $eLB_{t}$ could be estimated by Eq. : $$\left\{\begin{matrix} eUB_t = \max(\mathcal{Y}_t)\\ eLB_t = \min(\mathcal{Y}_t) \end{matrix}\right. \label{ebound}$$ and the empirical upper bound and lower bound will be updated after updating the database. As a sequence, the reward of one algorithm could be formulated as Eq. : $$\mu_{j, t} = \frac{eUB_{t}-f_{i, t}}{eUB_t-eLB_t} \label{reward}$$ where $f_{i,t}$ is the actual fitness of solution found by the algorithm in $t_{th}$ iteration. However, the performance of empirical bound estimated by Eq. will be bad due to the characteristic of the optimisation process. It is obvious that the convergence speed varies with the optimisation stage that the speed is much faster at the beginning and slows down when the optimization is near convergence. In order to improve the stability of the UCB policy, we introduce an online strategy to update the estimated bound by sliding window. The main idea is shown in Algorithm \[update-bound\]. Considering all evaluated solutions in the database after $t_{th}$ generation, a sliding window with the size of $sw$ selects best $sw$ solutions to form a subset. The $sw$ is set as $2d$ in this work, where $d$ is the dimension of a problem. If the minimal fitness in the subset is lower than the current $eLB$ which indicates the problem’s lower bound could be a much lower value, the $eLB$ will be updated by the newest minimal value. And if the maximal fitness in the subset is lower than current $eUB$ which means the optimisation has entered into another stage compared to the last stage, $eUB$ will be updated to refine the empirical bounds. Size of sliding window: $sw$\ Sort evaluated samples in database from best to worst;\ The top $sw$ best individuals $\mathcal{S} = \{(\mathbf{x}_1, y_1), ..., (\mathbf{x}_{sw}, y_{sw})\}$;\ Collect fitness of individuals in $\mathcal{S}$: $\mathcal{Y} = [y_1, y_2, ..., y_{sw}]$;\ After defining the algorithm’s reward, the UCB-IBSAEA framework is easy to be implemented which is presented in the Figure \[ucb-ibsaea\]. The UCB-t with no additional parameter in Eq. is used in the framework. The framework starts by initialising a database and then the UCB-IBSAEA will select one best algorithm $A_k$ from the algorithm pool $\mathcal{A} = \{A_1, A_2, ..., A_n\}$ according to the UCB-t policy for the following generation. The new solution will be added into the database after being evaluated by the actual fitness function. After that, the empirical bounds and reward information for each algorithm will be updated. Finally, the optimisation process will stop after running out of all fitness evaluations. Numerical experiments and analysis {#experiments} ================================== The performance of two proposed portfolio frameworks is studied in this section. We choose three state-of-the-art individual-based SAEAs to generate two portfolio instances and evaluate them on a set of benchmark functions. Experimental setting -------------------- In this work, very computationally expensive problems are considered so that we employ three efficient individual-based SAEAs to asses the efficacy of portfolio frameworks. Three algorithms are expert in different kinds of problems, and a brief introduction for them is presented below: - EGO-LCB: Efficient global optimisation with lower confidence bound [@jones1998efficient] is a very efficient algorithm for low-dimension expensive problems. - VESAEA: Voronoi-based efficient SAEA employed Voronoi diagram in SAEA framework to assist the local search process [@hao2018voronoi]. It is good at uni-modal problems even though there are some noises over the landscape. - GORS-SSLPSO: Frameworks are tested on five widely used problems with dimensions $d = 10, 20, 30$ as shown in Table \[test-problems\]. The maximal fitness evaluation is setting as $5d$ where $2d$ is used for initialisation, and all comparisons are based on 25 independent runs. In the following, the comparison between portfolio frameworks and three single SAEAs will be performed firstly and then we will analyse the performance of portfolio frameworks by comparing with another two frameworks: random selection framework and epsilon-greedy selection framework [@peter2014coco]. Comparative results with single algorithms ------------------------------------------ ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Problem D UCB-IBSAEA Par-IBSAEA GORS-SSLPSO VESAEA EGO-LCB --------- ---- ---------------------- ----------------------------------------------------------------------------------------------------------------------------------------------- ------------- -------- --------- sphere 10 1421.21 $\pm$ 939.89 **[1039.95 $\pm$ 712.93]{} & 5483.17 $\pm$ 2789.23 & 1878.56 $\pm$ 691.56 & 5728.52 $\pm$ 2791.72\ sphere & 20 & 2445.62 $\pm$ 1109.81 & **[2417.95 $\pm$ 938.53]{} & 9604.73 $\pm$ 2911.24 & 10906.42 $\pm$ 1487.33 & 44083.71 $\pm$ 7045.51\ sphere & 30 & 3606.77 $\pm$ 1377.47 & **[3091.30 $\pm$ 1171.53]{} & 15133.57 $\pm$ 4323.70 & 25736.52 $\pm$ 2554.90 & 81185.30 $\pm$ 8810.83\ rosenbrock & 10 & 209.05 $\pm$ 82.43 & 234.90 $\pm$ 112.40 & **[89.03 $\pm$ 46.93]{} & 238.17 $\pm$ 101.78 & 1127.73 $\pm$ 399.72\ rosenbrock & 20 & 349.14 $\pm$ 105.95 & 512.16 $\pm$ 144.50 & **[281.95 $\pm$ 314.65]{} & 936.38 $\pm$ 240.64 & 5281.28 $\pm$ 1460.20\ rosenbrock & 30 & 418.05 $\pm$ 86.88 & 596.34 $\pm$ 150.43 & **[311.92 $\pm$ 116.62]{} & 2419.77 $\pm$ 435.33 & 14763.02 $\pm$ 2247.72\ ackley & 10 & 19.46 $\pm$ 0.69 & 19.11 $\pm$ 1.39 & 19.52 $\pm$ 0.85 & 16.61 $\pm$ 1.89 & **[15.71 $\pm$ 5.35]{}\ ackley & 20 & 19.18 $\pm$ 0.58 & 19.27 $\pm$ 0.70 & 19.49 $\pm$ 0.51 & **[18.43 $\pm$ 0.74]{} & 20.48 $\pm$ 0.23\ ackley & 30 & **[19.06 $\pm$ 0.59]{} & 19.22 $\pm$ 0.50 & 19.52 $\pm$ 0.59 & 19.13 $\pm$ 0.37 & 20.75 $\pm$ 0.18\ griewank & 10 & 15.18 $\pm$ 12.21 & **[12.82 $\pm$ 9.30]{} & 78.07 $\pm$ 37.73 & 17.83 $\pm$ 5.11 & 46.91 $\pm$ 20.39\ griewank & 20 & 27.34 $\pm$ 12.03 & **[18.84 $\pm$ 5.62]{} & 143.75 $\pm$ 41.18 & 101.19 $\pm$ 11.65 & 386.30 $\pm$ 58.78\ griewank & 30 & 39.46 $\pm$ 13.21 & **[31.33 $\pm$ 12.98]{} & 209.52 $\pm$ 48.56 & 226.61 $\pm$ 15.51 & 714.12 $\pm$ 92.21\ rastrigin & 10 & 73.63 $\pm$ 16.69 & 79.21 $\pm$ 18.00 & **[55.79 $\pm$ 24.19]{} & 81.52 $\pm$ 14.91 & 101.07 $\pm$ 20.36\ rastrigin & 20 & 117.38 $\pm$ 30.23 & 152.84 $\pm$ 34.60 & **[86.66 $\pm$ 31.20]{} & 165.87 $\pm$ 34.84 & 271.99 $\pm$ 19.77\ rastrigin & 30 & 143.47 $\pm$ 29.35 & 176.62 $\pm$ 26.82 & **[114.12 $\pm$ 25.37]{} & 235.87 $\pm$ 53.24 & 467.43 $\pm$ 28.99\ & 2.07 & 2.20 & 2.73 & 3.33 & 4.67\ & Control method & 4-9-2 & 7-2-6 & 10-3-2 & 14-1-0\ & 2-9-4 & Control method & 6-3-6 & 9-4-2 & 14-1-0\ ****************************** ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- The results of two proposed frameworks and three algorithm candidates on benchmark problems over 25 independent runs are presented in Table \[result1\] where figures in each cell denote averaged best fitness and standard deviation, in which bold ones are the best results among five algorithms in one problem. The averaged ranking for algorithm portfolio and three SAEAs are listed on the third row from the bottom, from which we could find the proposed frameworks are much better than all single algorithms. Moreover, the last two rows provide the result of Wilcoxon test with a 0.05 significance level, in which the UCB-IBSAEA and Par-IBSAEA are control methods, respectively and the ‘win-draw-lose’ represents the control method is superior, not significantly different and inferior to the compared algorithm. Convergence profiles of all algorithms on 15 test problems are plotted in Figs. \[sphere-griewank\]-\[ackley\], where the x-axis ranges from $2D$ to $5D$ because the first $2D$ fitness evaluations are used for initialization, which is same for all algorithms in each run for fair comparison. The overall performance of an algorithm portfolio is significantly better than each single algorithm in Table \[result1\]. However, the performance of an algorithm portfolio in different problems has a significant difference. For example, in sphere and griewank problems, the portfolio dramatically improves the quality of the final solution. By contrast, they are not always the best in rosenbrock and rastrigin problems, but it still can get an acceptable result compared with the best single algorithm. The sphere function is a uni-modal problem and griewank function can also be regarded as a uni-modal problem with many small noises in the whole landscape. It is obvious in Table \[result1\] that the portfolio performs much better than every single algorithm on these kind of problems. In 10-dimension case, the portfolio is similar to the performance of VESAEA which is the best algorithm for this kind of problem among three candidates. ![Convergence curves of comparison algorithms on the Sphere and Griewank function with $D = 10,20,30$.[]{data-label="sphere-griewank"}](trend/sphere_10.pdf "fig:"){width=".45\linewidth"} ![Convergence curves of comparison algorithms on the Sphere and Griewank function with $D = 10,20,30$.[]{data-label="sphere-griewank"}](trend/griewank_10.pdf "fig:"){width=".45\linewidth"} ![Convergence curves of comparison algorithms on the Sphere and Griewank function with $D = 10,20,30$.[]{data-label="sphere-griewank"}](trend/sphere_20.pdf "fig:"){width=".45\linewidth"} ![Convergence curves of comparison algorithms on the Sphere and Griewank function with $D = 10,20,30$.[]{data-label="sphere-griewank"}](trend/griewank_20.pdf "fig:"){width=".45\linewidth"} ![Convergence curves of comparison algorithms on the Sphere and Griewank function with $D = 10,20,30$.[]{data-label="sphere-griewank"}](trend/sphere_30.pdf "fig:"){width=".45\linewidth"} ![Convergence curves of comparison algorithms on the Sphere and Griewank function with $D = 10,20,30$.[]{data-label="sphere-griewank"}](trend/griewank_30.pdf "fig:"){width=".45\linewidth"} Rosenbrock and rastrigin problems are both multi-modal problems which the performance of optimisation algorithms is likely to be hindered by attractive local optimums. GORS-SSLPSO is the most appropriate optimization algorithm for multi-modal problems among three algorithm candidates. And two algorithm portfolio frameworks are a little worse than GORS-SSLPSO, but performs better than other two algorithms as shown in convergence profiles in Figure \[rastrigin-rosenbrock\]. ![Convergence curves of comparison algorithms on the Rosenbrock and Rastrigin function with $D = 10,20,30$.[]{data-label="rastrigin-rosenbrock"}](trend/rosenbrock_10.pdf "fig:"){width=".45\linewidth"} ![Convergence curves of comparison algorithms on the Rosenbrock and Rastrigin function with $D = 10,20,30$.[]{data-label="rastrigin-rosenbrock"}](trend/rastrigin_10.pdf "fig:"){width=".45\linewidth"} ![Convergence curves of comparison algorithms on the Rosenbrock and Rastrigin function with $D = 10,20,30$.[]{data-label="rastrigin-rosenbrock"}](trend/rosenbrock_20.pdf "fig:"){width=".45\linewidth"} ![Convergence curves of comparison algorithms on the Rosenbrock and Rastrigin function with $D = 10,20,30$.[]{data-label="rastrigin-rosenbrock"}](trend/rastrigin_20.pdf "fig:"){width=".45\linewidth"} ![Convergence curves of comparison algorithms on the Rosenbrock and Rastrigin function with $D = 10,20,30$.[]{data-label="rastrigin-rosenbrock"}](trend/rosenbrock_30.pdf "fig:"){width=".45\linewidth"} ![Convergence curves of comparison algorithms on the Rosenbrock and Rastrigin function with $D = 10,20,30$.[]{data-label="rastrigin-rosenbrock"}](trend/rastrigin_30.pdf "fig:"){width=".45\linewidth"} In Ackley problems, although the superiority of algorithm portfolio is not obvious, portfolio still obtains relatively better result compared with the worst single algorithm. It is probably due to the little difference between three algorithms for this problem, and the performance of portfolio is restricted by the single algorithm’s ability. ![Convergence curves of comparison algorithms on the Ackley function with $D = 10,20,30$.[]{data-label="ackley"}](trend/ackley_10.pdf "fig:"){width="0.45\linewidth"} ![Convergence curves of comparison algorithms on the Ackley function with $D = 10,20,30$.[]{data-label="ackley"}](trend/ackley_20.pdf "fig:"){width="0.45\linewidth"} ![Convergence curves of comparison algorithms on the Ackley function with $D = 10,20,30$.[]{data-label="ackley"}](trend/ackley_30.pdf "fig:"){width="0.45\linewidth"} As introduced above, we could obtain the conclusion that two proposed frameworks could improve the performance of single algorithms in uni-modal problems and obtain the similar performance with the most appropriate algorithm candidate in complex multi-modal problems. For uni-modal problems, there is only one attractive global optimal solution over the whole landscape and the algorithm portfolio performs better than every single algorithm probably because various algorithms provide more diversity during the optimization process. VESAEA and EGO-LCB are good at exploiting the search space and GORS-SSLPSO could contribute more diversity so that they promote the performance of each other in uni-modal problems. But in the multi-modal problem which has many attractive local optimums, although the GORS-SSLPSO can provide much diversity, other two algorithms are easy to be trapped in the local optimal. As a result, the whole performance of the portfolio is hindered by VESAEA and EGO-LCB so that the result of the portfolio is slightly worse than the GORS-SSLPSO algorithm. Comparative results with other frameworks ----------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------- Problem D UCB-IBSAEA Par-IBSAEA RS EG50 --------- ---- ---------------------- ------------------------------------------------------------------------------------------------------------------- ---- ------ sphere 10 1421.21 $\pm$ 939.89 **[1039.95 $\pm$ 712.93]{} & 1178.42 $\pm$ 567.00 & 1280.76 $\pm$ 494.39\ sphere & 20 & 2445.62 $\pm$ 1109.81 & **[2417.95 $\pm$ 938.53]{} & 2598.02 $\pm$ 1144.57 & 4455.35 $\pm$ 1584.86\ sphere & 30 & 3606.77 $\pm$ 1377.47 & **[3091.30 $\pm$ 1171.53]{} & 3134.88 $\pm$ 995.80 & 4980.92 $\pm$ 1781.29\ rosenbrock & 10 & **[209.05 $\pm$ 82.43]{} & 234.90 $\pm$ 112.40 & 268.05 $\pm$ 143.15 & 264.17 $\pm$ 108.04\ rosenbrock & 20 & **[349.14 $\pm$ 105.95]{} & 512.16 $\pm$ 144.50 & 467.78 $\pm$ 174.07 & 561.89 $\pm$ 183.74\ rosenbrock & 30 & **[418.05 $\pm$ 86.88]{} & 596.34 $\pm$ 150.43 & 604.03 $\pm$ 134.11 & 721.90 $\pm$ 194.39\ ackley & 10 & 19.46 $\pm$ 0.69 & 19.11 $\pm$ 1.39 & **[18.66 $\pm$ 1.10]{} & 18.82 $\pm$ 1.35\ ackley & 20 & **[19.18 $\pm$ 0.58]{} & 19.27 $\pm$ 0.70 & 19.36 $\pm$ 0.57 & 19.36 $\pm$ 0.47\ ackley & 30 & **[19.06 $\pm$ 0.59]{} & 19.22 $\pm$ 0.50 & 19.27 $\pm$ 0.45 & 19.17 $\pm$ 0.42\ griewank & 10 & 15.18 $\pm$ 12.21 & **[12.82 $\pm$ 9.30]{} & 15.07 $\pm$ 9.69 & 20.56 $\pm$ 12.21\ griewank & 20 & 27.34 $\pm$ 12.03 & **[18.84 $\pm$ 5.62]{} & 20.12 $\pm$ 7.59 & 37.70 $\pm$ 19.87\ griewank & 30 & 39.46 $\pm$ 13.21 & **[31.33 $\pm$ 12.98]{} & 35.29 $\pm$ 9.84 & 50.00 $\pm$ 15.22\ rastrigin & 10 & **[73.63 $\pm$ 16.69]{} & 79.21 $\pm$ 18.00 & 84.20 $\pm$ 17.47 & 79.99 $\pm$ 18.15\ rastrigin & 20 & **[117.38 $\pm$ 30.23]{} & 152.84 $\pm$ 34.60 & 138.08 $\pm$ 23.11 & 155.38 $\pm$ 31.97\ rastrigin & 30 & **[143.47 $\pm$ 29.35]{} & 176.62 $\pm$ 26.82 & 175.89 $\pm$ 26.73 & 191.12 $\pm$ 40.51\ & 2.00 & 1.93 & 2.60 & 3.47\ & Control method & 4-9-2 & 5-8-2 & 9-6-0\ & 2-9-4 & Control method & 0-15-0 & 7-8-0\ ****************************** --------------------------------------------------------------------------------------------------------------------------------------------------------------------- To further analyse the efficacy of two proposed portfolio frameworks, we compare their performance with other two frameworks: random selection (RS) policy and epsilon-greedy policy with $\epsilon=0.5$ (EG50). RS policy randomly selects an algorithm at each generation while EG50 policy selects the “optimal” algorithm with probability 0.5 according to the cumulative reward and selects a random algorithm with probability 0.5 [@peter2014coco]. The comparative results on test problems are presented in Table \[result2\]. As shown in the result table, either UCB-IBSAEA or Par-IBSAEA obtain the best result among four frameworks except in 10-dimension ackley problem that RS gets a relatively better solution. Furtherly, the Par-IBSAEA performs better in sphere and griewank problem which are regarded as uni-modal problems. And UCB-IBSAEA is always the best framework among four frameworks in rosenbrock and rastrigin problems which are complex multi-modal problems. And the four frameworks also have the similar performance in Ackley function. Compared with UCB-IBSAEA, RS, EG50, Par-IBSAEA is the only parallel framework that running algorithm candidate in parallel. It is better than other three sequential frameworks in sphere and griewank problems. As discussed in the above subsection, algorithm portfolio increases the diversity during the optimization process by combining different kinds of algorithms. Par-IBSAEA uses all algorithm simultaneously at each generation, using the same database that algorithm candidates search the next re-evaluated solution from various aspects. This idea is similar to the negative correlation search (NCS) [@ke2016negative], which uses different agents to cover different regions of search space. However, if the number of algorithm candidates increases, the performance of Par-IBSAEA might deteriorate because the limited computational cost cannot afford the huge cost for parallelly running many algorithms simultaneously at each iteration. Although Par-IBSAEA is outstanding in uni-modal problems, UCB-IBSAEA is superior to other frameworks in multi-modal problems as shown in Table \[result2\]. It selects the appropriate algorithm at each generation according to the UCB policy in Eq. . The GORS-SSLPSO is the best algorithm among three algorithm candidates and UCB-IBSAEA is likely to detect the fact and allocate more computational budget to this algorithm. Take the 20-dimension rastrigin problem as an example, the behaviour of selection in each iteration of UCB-IBSAEA is plotted in Figure \[choice\]. We could find UCB-IBSAEA explores three algorithms at the early stage and then detect that GORS-SSLPSO is better than another two algorithms so that it mostly selects GORS-SSLPSO at the later stage. ![An illustration of selection behaviour for UCB-IBSAEA in 20-dimension rastrigin problem.[]{data-label="choice"}](img/choice.pdf) On the other hand, we find that there is no big difference between random selection policy and two proposed frameworks in Table \[result2\]. This is mainly because there are only three algorithms in our experiments that the best algorithm will be selected at a high probability. Furtherly, the randomness could provide diversity for portfolio framework and avoid framework being trapped in one action so that RS policy might obtain the best result than other frameworks, like 10-dimension ackley problem. Meanwhile, the difference among algorithm candidates is not very large so that random selection will not perform very badly. Risk analysis ------------- The risk of proposed framework for individual-based SAEAs will be analyzed in this subsection. The metric to measure algorithms’ risk is employed from the work in [@peng2010population], in which the comparative risk is estimated by comparing the quality of solutions obtained by two algorithms in a set of test problems with several runs. Considering benchmark problems $\mathcal{F} = \{f_k|k=1, 2, ..., n\}$ and algorithm constituents $\mathcal{A} = \{A_j|j=1,2, ..., m\}$, the probability of $A_i$ outperforming $A_j$ as $A_i \succ A_j$ can be calculated by the following equation: $$P(A_i \succ A_j) = \frac{1}{n}\sum_{k=1}^{n}P(q_{i, k} < q_{j, k} | f_k)$$ where the $q_{i, k}$ denotes the quality of solution obtained by $A_i$ in $f_k$. And $P(q_{i, k} < q_{j, k} | f_k)$ can be estimated by the following equation: $$P(q_{i, k} < q_{j, k} | f_k) = \frac{\sum_{s=1}^{s_i} \sum_{t=1}^{s_j} \mathcal{I} (y_{i, k, s} < y_{j, k, t}) }{s_i \times s_j} $$ where $y_{i, k, s}$ represents the fitness of solution obtained by $A_i$ in $s_{th}$ trial for $f_k$ problem, $s_i$, $s_j$ represent the number of trails of each algorithm for one problem and $\mathcal{I}(\cdot)$ denotes the indicator function. GORS-SSLPSO VESAEA EGO-LCB ------------ ------------- ------------- ------------- UCB-IBSAEA 0.60 - 0.40 0.75 - 0.25 0.94 - 0.06 Par-IBSAEA 0.55 - 0.45 0.72 - 0.28 0.94 - 0.06 RS 0.56 - 0.44 0.71 - 0.29 0.94 - 0.06 EG50 0.55 - 0.45 0.68 - 0.32 0.94 - 0.06 : The comparison risk of algorithm portfolio and single algorithm. The two figures in each cell stand for the probabilities that the portfolio and the individual-based SAEAs outperformed each other.[]{data-label="risk"} The comparative risk between portfolio frameworks and three SAEAs are shown in the Table \[risk\]. The four portfolio frameworks are all better than single SAEAs, in which the UCB-IBSAEA is slightly better than other three frameworks. Firstly, we can conclude that the portfolio framework could obviously reduce the risk of failing in optimizing problems. The performance of SAEAs could be promoted by combining various efficient algorithms so that the CEPs could be solved much better within limited computational resources. Secondly, we can find the proposed UCB-IBSAEA is more effective than other three frameworks in terms of optimisation risk so that the UCB-IBSAEA is more appropriate to be used for an unknown CEP. Conclusion ========== This work mainly focuses on individual-based SAEAs only for solving CEPs. In the future, the portfolio framework will be improved to suit for more general optimisation algorithms. On the other hand, it is much important to select appropriate algorithm candidates for algorithm portfolio which directly influence the framework’s performance, so it is valuable to do more research on how to choose constituent algorithms for algorithm portfolio in SAEAs. Acknowledgments {#acknowledgments .unnumbered} =============== This work was supported by the National Key R$\&$D Program of China (Grant No. 2017YFC0804003), the Program for Guangdong Introducing Innovative and Enterpreneurial Teams (Grant No. 2017ZT07X386), Shenzhen Peacock Plan (Grant No. KQTD2016112514355531), the Science and Technology Innovation Committee Foundation of Shenzhen (Grant No. ZDSYS201703031748284, Grant No. JCYJ20180504165652917 and Grant No. JCYJ20170817112421757) and the Program for University Key Laboratory of Guangdong Province (Grant No. 2017KSYS008).