Papers

We introduce an explicit construction of the non-diagonal and diagonal Wigner distributions for the homogeneous but anisotropic Taub and Kantowski-Sachs cosmological models within the framework of phase space deformation quantization. Conventional canonical quantization of these models via the Wheeler-DeWitt equation is inherently plagued by factor ordering ambiguities. To circumvent these issues, we employ the totally symmetric Weyl quantization map and the Moyal star product. By means of a canonical separation of the Hamiltonian constraint, we are able to resolve the formal convergence problems typically associated with the star product. Furthermore, to establish a rigorous connection with conventional quantum cosmology, we calculate the standard wave functions directly from the diagonal Wigner distributions, recovering the exact physical states in terms of modified Bessel functions in both cases.

In this paper, we investigate the effective resistance on the graph $G_N^{(r)}$, which is obtained by deleting all edges corresponding to circular distances $\{\pm1, \pm2, \dots, \pm r\}$ from the complete graph $K_N$. We utilize the cyclic symmetry of the graph to diagonalize the Laplacian matrix via the discrete Fourier basis and derive a finite trigonometric sum representation for the effective resistance between two vertices at distance $\ell$. Specifically, we treat the cases $r=1$ and $r=2$ in detail and provide explicit formulas. For the case of $r=1$, we use Fourier analysis to rederive the closed form in terms of Bejaia and Pisa numbers given by Chair. For the case of $r=2$, we show that the denominator reduces to a quadratic polynomial with complex roots and introduce a generalized Bejaia-Pisa-type complex sequence. Using this sequence, we provide some closed forms for the effective resistance and various related formulas.

We prove isoperimetric-type inequalities for complex-valued pluriharmonic functions in the unit polydisc $\mathbb{U}^n\subset\mathbb{C}^n$. Denote by $h^p(\mathbb{U}^n)$ and $b^p_{\mathbf q}(\mathbb{U}^n)$, respectively, the pluriharmonic Hardy space and the pluriharmonic weighted Bergman space in $\mathbb{U}^n$, where $\mathbf q=(q_1,\ldots,q_n)>-\mathbf1$. For $m\in\mathbb{N}$, $m\geq2$, write $\mathbf{m-2}=(m-2,\ldots,m-2)$ and let \[ d\mu_{\mathbf{m-2}}(z) =\frac{(m-1)^n}{\pi^n} \prod_{k=1}^n \left[(1-|z_k|^2)^{m-2}\,dx_kdy_k\right], \qquad z_k=x_k+iy_k. \] We prove that if $1<p_1,\ldots,p_m<\infty$ and $f_j\in h^{p_j}(\mathbb{U}^n)$, then \[ \int_{\mathbb{U}^n}\prod_{j=1}^m|f_j(z)|^{p_j}\, d\mu_{\mathbf{m-2}}(z) \leq \prod_{j=1}^m \left[ \frac{\sqrt2\cos\left(\frac{\pi}{2mp_j}\right)} {\sqrt{1-|\cos(\pi/p_j)|}} \right]^{p_j} \prod_{j=1}^m \|f_j\|_{h^{p_j}(\mathbb{U}^n)}^{p_j}. \] In particular, \[ \|f\|_{b^{mp}_{\mathbf{m-2}}(\mathbb{U}^n)} \leq \frac{\sqrt2\cos\left(\frac{\pi}{2mp}\right)} {\sqrt{1-|\cos(\pi/p)|}} \|f\|_{h^p(\mathbb{U}^n)}. \] For $p>2$, we refine this in the form \[ \|f\|_{b^{mp}_{\mathbf{m-2}}(\mathbb{U}^n)} \leq \left[ \sqrt2\cos\left(\frac{\pi}{4m}\right) \right]^{2n/p} \|f\|_{h^p(\mathbb{U}^n)}. \] Consequently, the obtained constant in the diagonal inclusion tends to $1$ as $p\to\infty$, for fixed $m$ and $n$. When $m=2$ and $n=1$, the latter estimate coincides with best-known planar estimate. Explicit lower bounds at $p=2$, together with the dimension-free upper estimate, show that the optimal diagonal constants converge to $\sqrt2$ as $m\to\infty$, uniformly in the dimension.

We establish the central limit theorem and the invariance principle for the inhomogeneous Diophantine approximations. The proof employs the cumulant method, which was developed by Björklund and Gorodnik to prove the central limit theorem in the homogeneous setting. Our approach also relies on the effective mixing of expanding translates for high-order correlations on the affine lattice space, extending the previous result by Kim.

The unitary-triangular (QR) factorization of linear algebra may be used to robustly and efficiently solve a linear system. Toward a comparable numerical method to solve a polynomial system of higher degree, this paper proposes an any-degree unitary-triangular (Qr) factorization, which for a degree-one system reduces to the QR factorization. The work develops a tensor framework, i.e., codesigned algebra and software, where polynomial system coefficients are represented by a vector-shaped sparse tensor, a multidimensional array whose number of Ricci-notation indices, called the tensor degree, equals the highest monomial degree of the system. With the proposed Qr factorization, the coefficient tensor decomposes into a product of unitary and triangular factors that, in general, also have Ricci-notation indices and sparse entries. The unitary factor defines a unitary transform, a generalization of the related linear algebra concept to tensor algebra, that can triangularize a polynomial system while preserving its solution set, whether zero- or positive-dimensional. The work extends the author's Ricci-notation tensor framework, providing new algebra and new software to model, construct, and factorize polynomial systems in this manner. After applying the approach to numerically triangularize two zero-dimensional systems, chosen for educational value, results are compared to the Gröbner-basis (GB) method for triangularizing polynomial systems symbolically. One problem is of degree three, with three equations and unknowns, and the other of degree two, with four equations and unknowns. Although it resembles GB triangularization, the proposed Qr factorization has a completely different pedigree associated with numerical methods.

We establish Extreme Value Distributions for the closest encounter between trajectories generated by different maps defined in the same reference phase space. For a class of strongly mixing maps, we show that the limit distribution depends on the length of the different trajectories and the co-dimension of the associated invariant measures. It is also modulated by an Extremal Index, that informs on the tendency of nearby points to diverge along with the evolution of their respective dynamics, serving as an indicator of their compatibility. We give a formula for this quantity for a class of chaotic maps of the interval and for the co-dimension in the case when the respective measures admit densities with isolated zeros and singularities. We present diverse examples of systems satisfying these assumptions and compute the different parameters modulating the limit distribution.

We derive several applications of the path-minimality theorem for adjacency $p$-energy proved in the companion paper. First, we prove the sharp inequality $$ \mathcal E_p^+(G)\ge \mathcal E_p^+(P_n), $$ where $P_n$ is the path on $n$ vertices, in three settings: connected bipartite graphs for every real $p\ge2$, all connected graphs for every odd integer $p\ge3$, and all connected graphs for $p=4$. Second, using subdivision graphs, we prove path-minimality for Laplacian and signless Laplacian-type spectral sums, including power sums, Estrada-type quantities, resolvent energies, and thresholded tails. Third, we prove an edge-count second-order stop-loss comparison for the signless Laplacian above the threshold $2$. This yields the sharp line-graph inequality $$ \mathcal E_p^+(\mathcal L(G))\ge \mathcal E_p^+(P_m) $$ for every connected graph $G$ with $m$ edges and every real $p\ge2$.

This paper studies difference-of-convex (DC) optimization problems through smoothing descent techniques. In particular, we introduce the difference of high-order Moreau envelopes (HOME-DC) and establish its fundamental and differential properties. Approximating the underlying proximal points, we generate an inexact first-order oracle for HOME-DC and characterize its accuracy guarantees. Building upon this oracle, we propose a class of inexact descent methods for minimizing DC functions and provide a convergence analysis. The proposed framework extends the applicability of envelope-based optimization techniques to a broad class of structured nonconvex problems while accommodating inexact solutions to subproblems. Preliminary numerical experiments on a sparse clustering problem demonstrate the approach's practical potential and support the theoretical findings.

We investigate the algebraic nature of constants arising from Chebyshev equiripple (minimax) polynomial approximation of $L_p$ norms on $[0,1]$. For the Euclidean case $\sqrt{1+t^2}$, we compute equiripple solutions from degree~1 through~8 and determine exact minimal polynomials and Galois groups for degrees~1 and~2 in both absolute and relative error formulations. We find a sharp phase transition: the degree-1 constants are solvable by radicals (Galois groups $C_4$, $D_4$), while the degree-2 constants provably are not (Galois groups $S_{12}$, $S_{10}\times C_2$). We explain this transition by a structural dichotomy between decoupling and coupling of critical points, and extend the analysis to $L_3$ norms, where the minimal polynomial degree jumps to~246. A general impossibility result follows from Hilbert's irreducibility theorem. We also develop a theory of piecewise equiripple approximation with jointly optimized breakpoints, proving that each doubling of the number of subintervals gains $n+1$ bits of accuracy at no additional arithmetic cost. These results establish a previously unobserved connection between Chebyshev approximation theory and the non-solvability of algebraic equations.

Dyadic and circular convolution can both be computed in $O(N\log N)$ time using the Hadamard transform and the FFT-computed discrete Fourier transform (DFT), respectively. The Hadamard transform is preferable for its real-valued sign flips, yet its substitution for the DFT introduces algebraic error. We present three complementary results that characterize this error. First, we identify exact error cancellation: two input and two output positions are universally error-free, and no reordering of the output can eliminate this error. Second, the error operator is nearly full rank, while its null space has only logarithmic dimension. Third, the expected error is governed by a single alignment scalar, with a closed-form expression obtained by averaging over random filters. In general, the substitution error asymptotically doubles the output energy, except for filters in the universal zero-error subspace, which incur no error. Collectively, these results show that the substitution error is structured, predictable, and governed by alignment.

The time-domain Boundary Element Method (BEM) for linear elastodynamics with vanishing initial conditions is considered. Spatial discretization uses standard low-order boundary elements, while temporal discretization employs the generalized Convolution Quadrature (gCQ) method. The gCQ framework requires evaluating BEM matrices in the Laplace domain at several complex frequencies along a chosen contour, producing a three-dimensional tensor with one spatial matrix slice per frequency. To reduce storage and computational cost, a low rank approximation of the tensor is computed using 3D-Adaptive Cross Approximation (3D-ACA), extending the classical ACA to handle both the additional frequency dimension and the tensorial structure of elastodynamics. Within each frequency slice, the BEM matrices are further compressed using either the classical ACA algorithm using the $\mathcal{H}$-matrix approach or a Chebyshev interpolation based Fast Multipole Method (FMM). A comparative study of all proposed methods is carried out using two academic examples, and the structural vibration of an induction machine is analyzed.

In this paper, we introduce the notion of the generalized (b,c) EP inverse within the framework of a *-Banach algebra. This concept emerges as a logical extension of the weak group inverse and EP-like property, which is applicable to complex matrices and bounded linear operators in Hilbert spaces. We provide its characterizations in relation to its associated decomposition and the generalized Drazin inverse. A polar-like property for the generalized (b,c) EP inverse is presented. Furthermore, we reveal an intrinsic connection between the generalized (b,b) EP inverse and the converse law for the generalized group inverse.

Lunar positioning, navigation, and timing (PNT) is moving from concept to hardware, ESA's Moonlight/LCNS, NovaMoon reference stations, LunaNet, and Coordinated Lunar Time, all reducing to one estimation core: fix the orbits and clocks of the lunar infrastructure and tie them to an Earth/inertial frame. We ask which measurements make a surface station's absolute position observable, and prove the answer. In a snapshot batch fit, the internal observables (station-to-satellite and inter-satellite ranging plus clock-sync) constrain only relative geometry and leave a six-dimensional rigid-body datum defect: three translations and three rotations of the cluster. The clocks are fully observable, so the defect is purely positional, and closing it needs a tie to the Earth frame. Two such ties exist and are not interchangeable. An indirect tie (Earth-to-satellite ranging through the constellation) reaches the station only when the satellite geometry is rich; a direct tie (an Earth-baseline VLBI delay to the station beacon) fixes it regardless. This gives a conditional design law, not a single number: VLBI restores absolute observability when the constellation cannot supply it, and merely sharpens the bound when it can. For a sparse three-satellite constellation the station lies in the null space of the Fisher information until VLBI is added, reaching a Cramer-Rao bound of 20.1 m; for a rich six-satellite constellation VLBI tightens the bound from 23.2 m to 9.7 m. A single-epoch baseline informs at most two of three axes, so the datum closes at three non-collinear Earth stations. The Gauss-Newton estimator attains the bound (efficiency 1.02), with a 91x median station-error improvement in the sparse regime. The FIM/CRLB engine is validated against NumPy and published closed forms; the lunar application stays modelled, every figure deterministic and reproducible.

We classify a reciprocal degree-five quadrinomial family over the quadratic extension F_{q^2}, where q is an odd prime power. The family has four terms, coefficients in F_q, and a coefficient constraint that makes the induced rational function on the unit circle highly structured. The classification has two sharply different branches. When q is congruent to 1 modulo 4, infinite families occur and are governed by two quadratic-character conditions on the parameter b. When q is congruent to 3 modulo 4, a square-class obstruction converts the problem into a character-sum problem on a conic. A Weil-bound argument eliminates all large fields in this branch, and finite verification leaves only the sporadic fields q = 7, 19, 23. The result is a complete classification of the nondegenerate members of the family for all odd prime powers q.

We study binary classification problems whose decision sets are given by definable sets in o-minimal expansions of the real field. Motivated by cell decomposition of definable sets, we introduce traceable sets as a classical proxy for definable decision regions and analyze their approximation by ReLU neural networks. Under uniform bounds on the number of connected components and suitable $C^m$ extensions for the boundary functions, we prove that characteristic functions of traceable subsets of $[-1/2,1/2]^n$ can be approximated in $L^p$ to accuracy $\varepsilon>0$ by ReLU neural networks of size $\mathcal{O}(\varepsilon^{-p(n-1)/m})$, with depth independent of $\varepsilon$ and polynomially bounded weights. This establishes quantitative approximation rates for certain definable collections in o-minimal structures using ReLU neural networks. The same approach also yields the stated approximation rates for a subclass of definable maps $[-1/2,1/2]^n \to \mathbb{R}$. We then combine the approximation capabilities with entropy estimates for ReLU neural network classes to obtain statistical learning rates for empirical risk minimization with hinge loss. For $N$ uniformly distributed samples, the resulting classifiers achieve expected misclassification error of order $N^{-m/(m+pn-p)}$ up to an arbitrarily small polynomial loss.

We introduce a new mathematical constant $\lambda_\infty = 0.674036183193696139936660007576508455780\ldots$ (OEIS A396695), defined as the unique solution in $(1/4,+\infty)$ of $h(x) := \sum_{p \text{ prime}} 1/(xp^2-p+1) = 1$. This equation arises as the $n\to\infty$ limit of the secular equation of rank-$n$ truncations of the infinite prime LCM matrix $\mathcal{L}[p_i,p_j]=1/\mathrm{lcm}(p_i,p_j)$, where $\mathcal{L}=D+vv^T$. Viewed as a compact self-adjoint operator on $\ell^2(\mathcal{P})$, $\mathcal{L}$ has spectral radius $\rho(\mathcal{L})=\lambda_\infty$ (Theorem 5.4). The corresponding integer LCM matrix satisfies $\lambda_{\max}(W_N^*)/N\to\lambda_\infty$ (Theorem 5.8), a prime-indexed counterpart to $\lambda_{\max}(M_N)/N\to\zeta(2)=\pi^2/6$ for the integer divisor matrix. We prove that $h$ is real-analytic, strictly decreasing and strictly convex on $(1/4,+\infty)$, ensuring existence and uniqueness of $\lambda_\infty$. We compute 500 decimal digits of $\lambda_\infty$, certified by rigorous error bounds and independently verified by six computational runs, including a fully independent recomputation (Run E, PARI-GP) and a machine-verifiable Arb interval certificate (Run F) covering 505 digits. Extensive PSLQ and LLL searches find no minimal polynomial of degree $\le 8$ satisfied by $\lambda_\infty$ (at 560 decimal digits), and no integer relation against catalogs of up to 31 classical constants. The arithmetic nature of $\lambda_\infty$ remains open.

We establish the generalized canonical bundle formula for generalized lc-trivial fibrations without the assumption on the nef part in the complex analytic setting. We also record the corresponding algebraic statement.

Let $\Omega$ be the superspace ring of regular differential forms on the affine space $\mathbb{C}^n$. If $G \subseteq GL_n(\mathbb{C})$ is a complex reflection group, the {\em $G$-superspace coinvariant ring} is the quotient $SR_G := \Omega_n/SI_G$ where $SI_G \subseteq \Omega$ is the ideal generated by $G$-invariants with vanishing constant term. We study this ring when $G = \mathbb{Z}_r \wr \mathfrak{S}_n$ is the group of $r$-colored permutation matrices. We prove a conjecture of Sagan and Swanson on a monomial basis for $SR_G$ and give an Operator Theorem description of its inverse system. We also give a combinatorial model for the ungraded and exterior-graded structure of $SR_G$ as a $G$-module.

We establish local Gevrey regularity for the weak solution to parametric divergence-form diffusion elliptic PDEs, assuming the diffusion coefficient itself possesses local Gevrey parametric regularity over a non-compact domain. Here "local Gevrey regularity" means that the regularity is determined in a neighborhood of each parametric point and depends on that point. Explicit bounds on parametric derivatives of the weak solution are proved in the $H^1$-norm. Building on this local Gevrey framework, we develop a novel theoretical treatment of the dimension truncation error for infinite-dimensional integration. We prove convergence rates of quasi-Monte Carlo quadrature for the finite-dimensional Gaussian-weighted integration of integrands having local Gevrey regularity. Together with the finite element discretization error, the errors arising from dimension truncation and quasi-Monte Carlo quadrature yield a complete error analysis and convergence rates for the fully discrete approximation of a bounded linear functional of the weak solution. Numerical experiments confirming the theoretical convergence rates are presented.

We review the notion of homotopy of quantum master actions in geometric Batalin-Vilkovisky formalism. Then we construct new examples of such homotopies, coming from renormalization group flow and non-infinitesimal changes of gauge fixing. Finally, we use the field redefinitions given by these homotopies to construct spans of quantum master actions with isomorphic effective actions.

The Blaschke-Lebesgue theorem states that the Reuleaux triangle has the smallest area among planar convex bodies of a fixed constant width. We study how small bodies of constant width can be on the unit sphere $\mathbb S^n$ when $n$ is large. For a spherical convex body $K\subset \mathbb S^n$ of constant width $w\in(0,\pi)$, its relative effective radius is \[ \left(\frac{\mu_n(K)}{\mu_n(\mathbb B^n(w/2))}\right)^{1/n}, \] where $\mu_n$ is the spherical $n$-measure and $\mathbb B^n(w/2)$ is a geodesic ball of radius $w/2$. Let $\sigma_n(w)$ be the infimum of the relative effective radius over all spherical bodies of constant width $w$. Define $\underline{\sigma}(w)=\liminf_{n\to\infty}\sigma_n(w)$ and $\overline{\sigma}(w)=\limsup_{n\to\infty}\sigma_n(w)$. For each fixed $w\in(0,\pi)$, we prove non-trivial bounds \[ 0<\sigma_{\ell}(w)\le \underline{\sigma}(w)\le \overline{\sigma}(w)\le \sigma_u(w)<1, \] where $\sigma_\ell(w)$ and $\sigma_u(w)$ are defined in terms of $w$ either explicitly or through a root of a quartic equation. The upper bounds are obtained by constructing small spherical bodies of constant width: for $w\le\pi/2$ by a spherical version of the recent Arman-Bondarenko-Nazarov-Prymak-Radchenko Euclidean example, and for $w>\pi/2$ by spherical duality. The lower bounds combine a spherical adaptation of Schramm's illumination argument with a Gaussian autocorrelation method for the associated convex cones.

With the emergence of various tensor data, tensor completion from partial measurements has attracted widespread attention in data science and signal processing. Total Variation (TV) has been widely used as an effective regularization technique for tensor completion; however, theoretical studies on TV regularization in this context remain limited. In this work, we present a rigorous theoretical analysis of TV regularization for tensor completion. Specifically, we consider tensor completion under exponential-family noise, which generalizes the standard settings such as Gaussian and Poisson tensor completion. To handle exponential-family tensor completion, we propose a family of dual-TV (DTV) regularizers based on the transformed L1 function, which simultaneously capture sparsity and low-rank structures in the gradient tensor. Moreover, we establish the theoretical upper bounds on the recovery error of the proposed estimator. In certain cases, these upper bounds can attain the convergence order of $\mathcal{O}\big( n_3 r_t\big(\max_{k} s_k^2\big) \log\big((n_1+n_2)n_3\big) /n \big)$, and the minimax lower bound analysis is further presented to show that the upper-bounds can approach the lower bound with the gap of order $\mathcal{O}(\max_k s_k^2/max(n_1, n_2))$ up to a logarithmic factor. Finally, multiple groups of experiments on synthetic, image and video tensor data sets are conducted to support our theoretical results and demonstrate the effectiveness of our method.

Lyapunov exponents are fundamental invariants in smooth ergodic theory describing the asymptotic infinitesimal behavior along typical orbits. This text aims to explain how and why to control Lyapunov exponents using entropy for smooth surface diffeomorphisms. It fits into the framework of our recent joint works with Sylvain CROVISIER and Omri SARIG. We will focus especially on the continuity property of exponents for measures near the maximal entropy measure, by presenting a simplified version of the original argument. Our exposition is geared towards advanced students and researchers in dynamics that are not necessarily familiar with smooth ergodic theory.

Generalized Fofana spaces were recently introduced as generalizations of Fofana spaces and Nakai's generalized Morrey spaces. In this paper, we establish the boundedness properties of the following operators in these spaces: fractional integral operators, fractional maximal operators and generalized fractional integral operators. As a consequence, we obtain generalized Olsen-type inequalities involving the Riesz potential and generalized fractional integral operators.

In this paper, the structural and operator-theoretic properties of contracting perimeters of triangles mapping (CPTM) within the generalized topological framework of complete $b$-metric spaces with coefficient $s \geq 1$, is systematically investigated. Extending recent foundational advancements from classical metric spaces, we explore the architectural interplay between multi-point perimetric constraints and path-wise orbital stability under two distinct structural framework. First, assuming the minimal exclusion of periodic orbits of prime period two, we prove that the higher-order iterates $f^{n}$ of an CPTM behave as graphic contractions for all indices satisfying the condition $sq^{n} < 1$. This classifies the operator as a weakly Picard operator and yields a unified existence and cardinality theorem establishing that the fixed-point set satisfies $1 \leq |\mathrm{Fix}(f)| \leq 2$.\\ Second, in the alternative configuration where the operator does possess a periodic orbit of prime period two, we resolve a significant structural gap under the parameter condition $sq^{2} < 1$. We demonstrate that the higher even iterates $f^{2n}$ collapse into continuous graphic contractions, proving that the mapping possesses exactly two periodic points which form a single, isolated 2-cycle. Throughout our proofs, we rigorously navigate the analytical challenges arising from the potential simultaneous non-continuity of the $b$-metric function by relying strictly on sequential tracking inequalities. Finally, we present concrete analytical examples, including a shift map on a discrete metric space, to show that the class of CPTM is strictly larger than the class of graphic contractions, thereby demonstrating the sharpness and optimality of the obtained parameter conditions.

We consider the space $C^1[0, 1]$ of continuously differentiable functions on the closed unit interval $[0, 1]$ and the space $\operatorname{Lip}[0, 1]$ of Lipschitz continuous functions on $[0, 1]$, equipped with the norms \begin{align*} \|f\|_{\sigma, p} = \begin{cases} \sqrt[p]{|f(0)|^p + \|f'\|_\infty^p} & (1 \le p < \infty), \\ \max\{\, |f(0)|, \|f'\|_\infty \,\} & (p = \infty). \end{cases} \end{align*} We show that every surjective isometry on the positive part of the unit sphere extends to a surjective complex-linear isometry on the entire space. As a corollary, every such isometry also extends to an isometric order isomorphism on the real subspaces $C^1_{\mathbb{R}}[0, 1]$ and $\operatorname{Lip}_{\mathbb{R}}[0, 1]$.

Throttling is a graph optimization problem, where the throttling number of a graph is the minimum sum or minimum product of the number of vertices in an initial set and the time required to complete a certain graph operation. A Nordhaus-Gaddum bound refers to an upper or lower bound of the sum or product of a graph parameter together with that of its complement. In this paper, we study the Nordhaus-Gaddum sum and product bounds of the various throttling numbers (sum throttling and product throttling with or without initial cost). Graph operations considered are standard zero forcing, positive semidefinite forcing, power domination, and Cops and Robbers.

We revisit the calculation of the von Neumann (or "entanglement") entropy of a subsystem of a pure quantum state, under the assumption that the latter is drawn at random from a uniform distribution on the full Hilbert space. We derive simple and closed expressions for all power moments, from which the moments of the entropy can be computed by simple differentiation. Our approach (different from the usual one based on random matrix theory and Laguerre polynomials) makes use of Schur-Weyl duality and the character theory of the symmetric group $S_N$ . The paper is self-contained, providing all the necessary mathematical background.

We present some developments in the study of chaotic dynamics following the solution of a conjecture of Newhouse on the measures maximizing the entropy of smooth surface diffeomorphisms. We focus on strong positive recurrence, a generalization of the classical Anosov-Smale theory of uniform hyperbolicity introduced in a joint work with Sylvain Crovisier and Omri Sarig. This new property is general enough to be satisfied by all smooth surface diffeomorphisms with positive entropy, yet it still ensures many quantitative properties such as exponential mixing or limit theorems for regular functions. We also present some open problems, including its abundance (or not) in higher dimensions.

In this paper we study the potential measures and the Laplace transforms of the occupation times of a refracted-reflected spectrally negative Lévy process when the process is observed at the arrival epochs of two independent Poisson processes. In this case, the rates of observing the underlying process differ in time which deviates from the classical theory of Poissonian observations. Explicit expressions for the so-called Poissonian potential measures and the Poissonian occupation times are derived in terms of (known) scale functions. Other fluctuation identities are also derived.