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International audience; In a previous paper, we have described the construction of an automaton from a rational expression which has the property that the automaton built from an expression which is itself computed from a co-deterministic automaton by the state elimination method is co-deterministic. It turned out that the definition on which the construction is based was inappropriate, and thus the proof of the property was flawed. We give here the correct definition of the broken derived terms of an expression which allow to define the automaton and the detailed full proof of the property.

The spectral gap conjecture of M. van den Berg [2, formula (65)] asserts that λ2 − λ1 ≥ 3π for all convex euclidean domains of diameter 1, where λ1 and λ2 denote the first two eigenvalues of the Dirichlet Laplacian. Notice that equality holds for the 1-dimensional unit interval, which can be regarded also as a degenerate n-dimensional rectangular box. The gap estimate is conjectured to hold more generally for Schrodinger operators with convex potentials, under Dirichlet boundary conditions; see the work of S.-T. Yau and collaborators [9, 11]. This Schrodinger gap conjecture was proved some time ago in 1 dimension by R. Lavine [8], and more recently in all dimensions by B. Andrews and J. Clutterbuck [1]. The proof in this journal by H. Donnelly [3] of the original gap conjecture in 2 dimensions (for the Dirichlet Laplacian with zero potential) is not correct. The Editors of Mathematische Zeitschrift have asked us to describe the flaws in the proof, in order to clarify the state of the literature. Donnelly’s approach to the problem is a natural one: first perform a shape optimization to rule out a non-degenerate minimizing domain, and then analyze the spectral gap for a sequence of domains degenerating to an interval, with the help of results by D. Jerison [5]. (For some history on this approach, and on the gap conjecture more generally, see the report on the AIM meeting “Low Eigenvalues of Laplace and Schrodinger Operators” [10], especially page 12 of the open problems list.) The error lies in the proof of the shape optimization step, as we now explain. Donnelly wishes to prove that no minimizing domain can exist for

These notes are a companion to the article "Lorentz spacetimes of constant curvature" by Geoffrey Mess, which was first written in 1990 but never published. Mess' paper will appear together with these notes in a forthcoming issue of Geometriae Dedicata., Comment: 26 pages

In this note, we improve some results of Granville and Soundararajan on the distribution of values of the truncated random Euler product L ( 1 , X ; y ) : = ∏ p ⩽ y ( 1 − X ( p ) / p ) −1 , where the X ( p ) are independent random variables, taking the values ±1 with equal probability p / 2 ( p + 1 ) and 0 with probability 1 / ( p + 1 ) .

International audience

In this paper we determine all Padovan numbers that are palindromic concatenations of two distinct repdigits.

In this short paper, we construct a unipotent nearby cycle functor and show a p-adic analogue of Beilinson’s equivalence comparing two derived categories: the derived category of holonomic arithmetic $${\mathcal {D}}$$ -modules and the derived category of arithmetic $${\mathcal {D}}$$ -modules whose cohomologies are holonomic.

International audience; In this short paper we prove that the almost Schur theorem, introduced by De Lellis and Topping, is true on 4-dimensional Riemannian manifolds of nonnegative scalar curvature and discuss some related problems on other dimensional manifolds.

The control of the surface of water in a long canal by means of a wavemaker is investigated. The fluid motion is governed by the Korteweg-de Vries equation in Lagrangian coordinates. The null controllability of the elevation of the fluid surface is obtained thanks to a Carleman estimate and some weighted inequalities. The global uncontrollability is also established.

International audience; In this paper, we present an algebraic attack against the Flurry and Curry block ciphers [12,13]. Usually, algebraic attacks against block ciphers only require one message/ciphertext pair to be mounted. In this paper, we investigate a different approach. Roughly, the idea is to generate an algebraic system from the knowledge of several well chosen correlated message/ciphertext pairs. Flurry and Curry are two families of ciphers which fully parametrizable and having a sound design strategy against the most common statistical attacks; i.e. linear and differential attacks. These ciphers are then targets of choices for algebraic attacks. It turns out that our new approach permits to go one step further in the (algebraic) cryptanalysis of difficult instances of Flurry and Curry. To explain the behavior of our attack, we have established an interesting connection between algebraic attacks and high order differential cryptanalysis [32]. From extensive experiments, we estimate that our approach – that we will call ”algebraic-high order differential” cryptanalysis – is polynomial when the Sbox is a power function. As a proof of concept, we have been able to break Flurry/Curry – up to 8 rounds – in few hours. We have also investigated the more difficult (and interesting case) of the inverse function. For such function, we have not been able to bound precisely the theoretical complexity, but our experiments indicate that our approach permits to obtain a significant practical gain. We have attacked Flurry/Curry using the inverse Sbox up to 8 rounds.

International audience; This paper concerns the simulation of a class of nonlinear discrete-time systems under a set of initial conditions described by an ellipsoid. We derive a procedure allowing the propagation of such ellipsoids through time, which makes it possible to set a guaranteed hard bound on the evolution of the state of the system for all the possible initial conditions. At the end of the paper, we show an application of the method through three academics examples, two of which are taken from the theory of fractals.

International audience; The purpose of this paper is to prove, under mild conditions, the asymptotic normality of the rank estimator of the slope parameter of a simple linear regression model with stationary associated errors. This result follows from a uniform linearity property for linear rank statistics that we establish under general conditions on the dependence of the errors. We prove also a tightness criterion for weighted empirical process constructed from associated triangular arrays. This criterion is needed for the proofs which are based on that of Koul [Behavior of robust estimators in the regression model with dependent errors. Ann Stat. 1977;5(4):681–699] and of Louhichi [Louhichi S. Weak convergence for empirical processes of associated sequences. Ann Inst Henri Poincaré Probabilités Statist. 2000;36(5):547–567].

In this paper we study the moderate deviations for the magnetization of critical Curie-Weiss model. Chen, Fang and Shao considered a similar problem for non-critical model by using Stein method. By direct and simple arguments based on Laplace method, we provide an explicit formula of the error and deduce a Cram\'er-type result., Comment: 12 pages, accepted for publication in Electronic Communication in Probability

In this paper, we start the investigation of a new natural approach to "unifying" noncommutative gauge field theories (NCGFT) based on approximately finite-dimensional ($AF$) $C^*$-algebras. The defining inductive sequence of an $AF$ $C^*$-algebra is lifted to enable the construction of a sequence of NCGFT of Yang-Mills-Higgs types. The present paper focus on derivation-based noncommutative field theories. A mathematical study of the ingredients involved in the construction of a NCGFT is given in the framework of $AF$ $C^*$-algebras: derivation-based differential calculus, modules, connections, metrics and Hodge $\star$-operators, Lagrangians... Some physical applications concerning mass spectra generated by Spontaneous Symmetry Breaking Mechanisms (SSBM) are proposed using numerical computations for specific situations., Comment: 42 pages, 20 figures

In this paper we consider kinetically constrained models (KCM) on $\mathbb Z^2$ with general update families $\mathcal U$. For $\mathcal U$ belonging to the so-called "critical class" our focus is on the divergence of the infection time of the origin for the equilibrium process as the density of the facilitating sites vanishes. In a recent paper Mar\^ech\'e and two of the present authors proved that if $\mathcal U$ has an infinite number of "stable directions", then on a doubly logarithmic scale the above divergence is twice the one in the corresponding $\mathcal U$-bootstrap percolation. Here we prove instead that, contrary to previous conjectures, in the complementary case the two divergences are the same. In particular, we establish the full universality partition for critical $\mathcal U$. The main novel contribution is the identification of the leading mechanism governing the motion of infected critical droplets. It consists of a peculiar hierarchical combination of mesoscopic East-like motions., Comment: 36 pages, 11 figures, expanded and revised presentation, added roadmaps for sections 4 and 5, added and improved figures

Erratum, 11 July 2022: This is an updated version of the original paper in which the notion of reparametrization category was incorrectly axiomatized. Details on the changes to the original paper are provided in the Appendix. A reparametrization category is a small topologically enriched semimonoidal category such that the semimonoidal structure induces a structure of a semigroup on objects, such that all spaces of maps are contractible and such that each map can be decomposed (not necessarily in a unique way) as a tensor product of two maps. A Moore flow is a small semicategory enriched over the biclosed semimonoidal category of enriched presheaves over a reparametrization category. We construct the q-model category of Moore flows. It is proved that it is Quillen equivalent to the q-model category of flows. This result is the first step to establish a zig-zag of Quillen equivalences between the q-model structure of multipointed $d$-spaces and the q-model structure of flows., Comment: 39 pages; Erratum to Compositionality 3, 3, 2021, in which the notion of reparametrization category was incorrectly axiomatized

We consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term. In some earlier work (Abdelhedi-Zaag JDE 2021), we constructed a blow-up solution for that equation, and showed that it blows up (at least) at the origin. We also derived the so called "intermediate blow-up profile". In this paper, we prove the single point blow-up property and determine the final blow-up profile.

International audience; In this paper, we prove a stability estimate of the conductivity parameters identification problem in cardiac electrophysiology. The propagation of the electrical wave in the heart is described by the monodomain model coupled to an elliptic equation describing the diffusion of the electrical wave in the whole body. Our result concerns both heart and torso conductivity parameters. The main difficulty that we solve in this paper is related to the transmission conditions between the heart and the torso. We first, establish Carleman estimates for the coupled heart-torso system. Then, using these estimates and the Bukhgeim and Klibanov approach, we prove a Lipschitz stability estimate of cardiac and torso conductivity parameters.

Forthcoming (2017) in Journal of Theoretical Probability; Our interest in this paper is to explore limit theorems for various geometric function-als of excursion sets of isotropic Gaussian random fields. In the past, limit theorems have been proven for various geometric functionals of excursion sets/sojourn times (see [4, 13, 14, 18, 22, 25] for a sample of works in such settings). The most recent addition being [6] where a central limit theorem (CLT) for Euler-Poincaré characteristic of the excursions set of a Gaussian random field is proven under appropriate conditions. In this paper, we obtain a CLT for some global geometric functionals, called the Lipschitz-Killing curvatures of excursion sets of Gaussian random fields in an appropriate setting.

In this paper we study the geodesic flow for a particular class of Riemannian non-compact manifolds with variable pinched negative sectional curvature. For a sequence of invariant measures we are able to prove results relating the loss of mass and bounds on the measure entropies. We compute the entropy contribution of the cusps. We develop and study the corresponding thermodynamic formalism. We obtain certain regularity results for the pressure of a class of potentials. We prove that the pressure is real analytic until it undergoes a phase transition, after which it becomes constant. Our techniques are based on the one side on symbolic methods and Markov partitions and on the other on geometric techniques and approximation properties at level of groups., 34 pages, 4 figures. In this new version we have improved the organization of the paper and the clarity of some statements

International audience; We study change-points tests based on U-statistics for absolutely regular observations. Our method avoids some technical assumptions on the data and the kernel. The asymptotic properties of the U-statistics are studied under the null hypothesis, under fixed alternatives and under a sequence of local alternatives. The asymptotic distributions of the test statistics under the null hypothesis and under the local alternatives are given explicitly and the tests are shown to be consistent. A small set of simulations is done for evaluating the performance of the tests in detecting changes in the mean, variance and autocorrelation of some simple time series.

International audience; Recently, Iwahori-Hecke algebras were associated to Kac-Moody groups over non-Archimedean local fields. In a previous paper, we introduced principal series representations for these algebras and partially generalized Kato's irreducibility criterion. In this paper, we study how some of these representations decompose when they are reducible and deduce information on the irreducible representations of these algebras.

The paper is devoted to generic translation flows corresponding to Abelian differentials on flat surfaces of arbitrary genus $g\ge 2$. These flows are weakly mixing by the Avila-Forni theorem. In genus 2, the Hölder property for the spectral measures of these flows was established in our papers [10,12]. Recently Forni [17], motivated by [10], obtained Hölder estimates for spectral measures in the case of surfaces of arbitrary genus. Here we combine Forni's idea with the symbolic approach of [10] and prove Hölder regularity for spectral measures of flows on random Markov compacta, in particular, for translation flows in all genera., Corrects the formulation of Proposition 5.4 in the published version (reversed order of quantifiers); the results and proofs are unchanged, with a minor rewording

Building on the weak CSR approach developed in a previous paper by Merlet, Nowak and Sergeev, we establish new bounds for the periodicity threshold of the powers of a tropical matrix. According to that approach, bounds on the ultimate periodicity threshold take the form of T=max(T_1,T_2), where T_1 is a bound on the time after which the weak CSR expansion starts to hold and T_2 is a bound on the time after which the first CSR term starts to dominate. The new bounds on T_1 and T_2 established in this paper make use of the cyclicity of the associated graph and the (tropical) factor rank of the matrix, which leads to much improved bounds in favorable cases. For T_1, in particular, we obtain new extensions of bounds of Schwarz, Kim and Gregory-Kirkland-Pullman, previously known as bounds on exponents of digraphs. For similar bounds on T_2, we introduce the novel concept of walk reduction threshold and establish bounds on it that use both cyclicity and factor rank., Comment: 37 pages

For the Helmholtz equation posed in the exterior of a Dirichlet obstacle, we prove that if there exists a family of quasimodes (as is the case when the exterior of the obstacle has stable trapped rays), then there exist near-zero eigenvalues of the standard variational formulation of the exterior Dirichlet problem (recall that this formulation involves truncating the exterior domain and applying the exterior Dirichlet-to-Neumann map on the truncation boundary). Our motivation for proving this result is that a) the finite-element method for computing approximations to solutions of the Helmholtz equation is based on the standard variational formulation, and b) the location of eigenvalues, and especially near-zero ones, plays a key role in understanding how iterative solvers such as the generalised minimum residual method (GMRES) behave when used to solve linear systems, in particular those arising from the finite-element method. The result proved in this paper is thus the first step towards rigorously understanding how GMRES behaves when applied to discretisations of high-frequency Helmholtz problems under strong trapping (the subject of the companion paper [Marchand, Galkowski, Spence, Spence, 2021]).

In this paper, we study the algebraic relations satisfied by the solutions of q-difference equations and their transforms with respect to an auxiliary operator. Our main tools are the parametrized Galois theories developed in Hardouin and Singer (Math Ann 342(2):333–377, 2008) and Ovchinnikov and Wibmer (Int Math Res Not 12:3962–4018, 2015). The first part of this paper is concerned with the case where the auxiliary operator is a derivation, whereas the second part deals with a $$\mathbf {q}$$ -difference operator. In both cases, we give criteria to guarantee the algebraic independence of a series, solution of a q-difference equation, with either its successive derivatives or its $$\mathbf {q}$$ -transforms. We apply our results to q-hypergeometric series.

In this paper we present a theoretical background for a coupled analytical–numerical approach to model a crack propagation process in two-dimensional bounded domains. The goal of the coupled analytical–numerical approach is to obtain the correct solution behaviour near the crack tip by help of the analytical solution constructed by using tools of complex function theory and couple it continuously with the finite element solution in the region far from the singularity. In this way, crack propagation could be modelled without using remeshing. Possible directions of crack growth can be calculated through the minimization of the total energy composed of the potential energy and the dissipated energy based on the energy release rate. Within this setting, an analytical solution of a mixed boundary value problem based on complex analysis and conformal mapping techniques is presented in a circular region containing an arbitrary crack path. More precisely, the linear elastic problem is transformed into a Riemann–Hilbert problem in the unit disk for holomorphic functions. Utilising advantages of the analytical solution in the region near the crack tip, the total energy could be evaluated within short computation times for various crack kink angles and lengths leading to a potentially efficient way of computing the minimization procedure. To this end, the paper presents a general strategy of the new coupled approach for crack propagation modelling. Additionally, we also discuss obstacles in the way of practical realisation of this strategy.

International audience; We analyze a system of partial differential equations that model a potential mean field game of controls, briefly MFGC. Such a game describes the interaction of infinitely many negligible players competing to optimize a personal value function that depends in aggregate on the state and, most notably, control choice of all other players. A solution of the system corresponds to a Nash Equilibrium, a group optimal strategy for which no one player can improve by altering only their own action. We investigate the second order, possibly degenerate, case with non-strictly elliptic diffusion operator and local coupling function. The main result exploits potentiality to employ variational techniques to provide a unique weak solution to the system, with additional space and time regularity results under additional assumptions. New analytical subtleties occur in obtaining a priori estimates with the introduction of an additional coupling that depends on the state distribution as well as feedback.

International audience; This paper exposes full analytical solutions of a plane, quasi-static but large transformation of a Timoshenko beam. The problem is first re-formulated in the form of a Cauchy initial value problem where load (force and moment) is prescribed at one-end and kinematics (translation, rotation) at the other one. With such formalism solutions are explicit for any load and existence, unicity and regularity of the solution of the problem are proven. Therefore analytical post-buckling solutions were found with different regimes driven explicitly by two invariants of the problem. The paper presents how these solutions of a Cauchy initial value problem may help tackle (1) boundary problem, where physical quantities (of load, position or section orientation) are prescribed at both ends (2) problem of quasi-static instabilities. In particular several problems of bifurcation are explicitly formulated in case of buckling or catastrophe.

The paper addresses the problem of defining families of ordered sequences $\{x_i\}_{i\in N}$ of elements of a compact subset $X$ of $R^d$ whose prefixes $X_n=\{x_i\}_{i=1}^{n}$, for all orders $n$, have good space-filling properties as measured by the dispersion (covering radius) criterion. Our ultimate aim is the definition of incremental algorithms that generate sequences $X_n$ with small optimality gap, i.e., with a small increase in the maximum distance between points of $X$ and the elements of $X_n$ with respect to the optimal solution $X_n^\star$. The paper is a first step in this direction, presenting incremental design algorithms with proven optimality bound for one-parameter families of criteria based on coverings and spacings that both converge to dispersion for large values of their parameter. The examples presented show that the covering-based method outperforms state-of-the-art competitors, including coffee-house, suggesting that it inherits from its guaranteed 50\% optimality gap., Comment: 28 pages, 13 figures

This paper proposes two novel families of non-stationary subdivision schemes with a shape parameter of hierarchically and efficiently generating mixed hyperbolic/trigonometric curves of order 3 and 4. An analysis of convergence and smoothness of the proposed schemes is established by using the asymptotic equivalence method. This paper also discusses the bivariate tensor-product subdivision scheme for the surface modeling on the regular meshes. The numerical results they produce are very encouraging. In this context, the performance of our algorithms has been exposed by considering examples, which illustrate how the shape parameter and the control points are assigned to reproduce such analytic curves and surfaces using tensor-product notion.

H. Weyl in 1921 demonstrated that for a connected manifold of dimension greater than $1$, if two Riemannian metrics are conformal and have the same geodesics up to a reparametrization, then one metric is a constant scaling of the other one. In the present paper, we investigate the analogous property for sub-Riemannian metrics. In particular, we prove that the analogous statement, called the Weyl projective rigidity, holds either in real analytic category for all sub-Riemannian metrics on distributions with a specific property of their complex abnormal extremals, called minimal order, or in smooth category for all distributions such that all complex abnormal extremals of their nilpotent approximations are of minimal order. This also shows, in real analytic category, the genericity of distributions for which all sub-Riemannian metrics are Weyl projectively rigid and genericity of Weyl projectively rigid sub-Riemannian metrics on a given bracket generating distributions. Finally, this allows us to get analogous genericity results for projective rigidity of sub-Riemannian metrics, i.e.when the only sub-Riemannian metric having the same sub-Riemannian geodesics, up to a reparametrization, with a given one, is a constant scaling of this given one. This is the improvement of our results on the genericity of weaker rigidity properties proved in recent paper arXiv:1801.04257[math.DG]., Comment: 18 pages

We present a solution of the algebraic version of Birkhoff Conjecture on integrable billiards. Namely we show that every polynomially integrable real bounded convex planar billiard with smooth boundary is an ellipse. We extend this result to billiards with piecewise-smooth and not necessarily convex boundary on arbitrary two-dimensional surface of constant curvature: plane, sphere, Lobachevsky (hyperbolic) plane; each of them being modeled as a plane or a (pseudo-) sphere in $\mathbb R^3$ equipped with appropriate quadratic form. Namely, we show that a billiard is polynomially integrable, if and only if its boundary is a union of confocal conical arcs and appropriate geodesic segments. We also present a complexification of these results. These are joint results of Mikhail Bialy, Andrey Mironov and the author. The proof is split into two parts. The first part is given by Bialy and Mironov in their two joint papers. They considered the tautological projection of the boundary to $\mathbb{RP}^2$ and studied its orthogonal-polar dual curve, which is piecewise algebraic, by S.V.Bolotin's theorem. By their arguments and another Bolotin's theorem, it suffices to show that each non-linear complex irreducible component of the dual curve is a conic. They have proved that all its singularities and inflection points (if any) lie in the projectivized zero locus of the corresponding quadratic form on $\mathbb C^3$. The present paper provides the second part of the proof: we show that each above irreducible component is a conic and finish the solution of the Algebraic Birkhoff Conjecture in constant curvature., To appear in the Journal of the European Mathematical Society (JEMS), 69 pages, 2 figures. A shorter proof of Theorem 4.24. Minor precisions and misprint corrections

In the first part of this paper, we express the generalized Bessel function associated with dihedral systems and a constant multiplicity function as a infinite series of confluent Horn functions. The key ingredient leading to this expression is an extension of an identity involving Gegenbauer polynomials proved in a previous paper by the authors, together with the use of the Poisson kernel for these polynomials. In particular, we derive an integral representation of this generalized Bessel function over the standard simplex. The second part of this paper is concerned with even dihedral systems and boundary values of one of the variables. Still assuming that the multiplicity function is constant, we obtain a Laplace-type integral representation of the corresponding generalized Bessel function, which extends to all even dihedral systems a special instance of the Laplace-type integral representation proved in \cite{Amr-Dem}., Comment: To appear in Ramanujan Journal of Maths

There are two major routes to address the ubiquitous family of inverse problems appearing in signal and image processing, such as denoising or deblurring. A first route relies on Bayesian modeling, where prior probabilities are used to embody models of both the distribution of the unknown variables and their statistical dependence with the observed data. The estimation process typically relies on the minimization of an expected loss (e.g. minimum mean squared error, or MMSE). The second route has received much attention in the context of sparse regularization and compressive sensing: it consists in designing (often convex) optimization problems involving the sum of a data fidelity term and a penalty term promoting certain types of unknowns (e.g., sparsity, promoted through an 1 norm). Well known relations between these two approaches have lead to some widely spread misconceptions. In particular, while the so-called Maximum A Posterori (MAP) estimate with a Gaussian noise model does lead to an optimization problem with a quadratic data-fidelity term, we disprove through explicit examples the common belief that the converse would be true. It has already been shown [7, 9] that for denoising in the presence of additive Gaussian noise, for any prior probability on the unknowns, MMSE estimation can be expressed as a penalized least squares problem, with the apparent characteristics of a MAP estimation problem with Gaussian noise and a (generally) different prior on the unknowns. In other words, the variational approach is rich enough to build all possible MMSE estimators associated to additive Gaussian noise via a well chosen penalty. We generalize these results beyond Gaussian denoising and characterize noise models for which the same phenomenon occurs. In particular, we prove that with (a variant of) Poisson noise and any prior probability on the unknowns, MMSE estimation can again be expressed as the solution of a penalized least squares optimization problem. For additive scalar denois-ing the phenomenon holds if and only if the noise distribution is log-concave. In particular, Laplacian denoising can (perhaps surprisingly) be expressed as the solution of a penalized least squares problem. In the multivariate case, the same phenomenon occurs when the noise model belongs to a particular subset of the exponential family. For multivariate additive denoising, the phenomenon holds if and only if the noise is white and Gaussian., Comment: This work and the companion paper [10] are dedicated to the memory of Mila Nikolova, who passed away prematurely in June 2018. Mila dedicated much of her energy to bring the technical content to completion during the spring of 2018. The first author did his best to finalize the papers as Mila would have wished. He should be held responsible for any possible imperfection in the final manuscript

International audience; A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three internally vertex-disjoint paths of length at least 2 between the same pair of distinct vertices. A wheel is a graph formed by a hole and a node that has at least 3 neighbors in the hole. In this series of papers we study the class of graphs that do not contain as an induced subgraph a theta nor a wheel. In Part II of the series we prove a decomposition theorem for this class, that uses clique cutsets and 2-joins. In this paper we use this decomposition theorem to solve several problems related to finding induced paths and cycles in our class.

International audience; In this paper, we consider the design of wavelet filters based on the Thiran common-factor approach proposed in Selesnick [2001]. This approach aims at building finite impulse response filters of a Hilbert-pair of wavelets serving as real and imaginary part of a complex wavelet. Unfortunately it is not possible to construct wavelets which are both finitely supported and analytic. The wavelet filters constructed using the common-factor approach are then approximately analytic. Thus, it is of interest to control their analyticity. The purpose of this paper is to first provide precise and explicit expressions as well as easily exploitable bounds for quantifying the analytic approximation of this complex wavelet. Then, we prove the existence of such filters enjoying the classical perfect reconstruction conditions, with arbitrarily many vanishing moments.

Fully revised. To appear in Nonlinearity; International audience; This paper extends the Madelung-Bohm formulation of quantum mechanics to describe the time-reversible interaction of classical and quantum systems. The symplectic geometry of the Madelung transform leads to identifying hybrid classical-quantum Lagrangian paths extending the Bohmian trajectories from standard quantum theory. As the classical symplectic form is no longer preserved, the nontrivial evolution of the Poincar\'e integral is presented explicitly. Nevertheless, the classical phase-space components of the hybrid Bohmian trajectory identify a Hamiltonian flow parameterized by the quantum coordinate and this flow is associated to the motion of the classical subsystem. In addition, the continuity equation of the joint classical-quantum density is presented explicitly. While the von Neumann density operator of the quantum subsystem is always positive-definite by construction, the hybrid density is generally allowed to be unsigned. However, the paper concludes by presenting an infinite family of hybrid Hamiltonians whose corresponding evolution preserves the sign of the probability density for the classical subsystem.

In a previous paper Chevallier et al. (2018), it has been shown that the mean-field limit of spatially extended Hawkes processes is characterized as the unique solution u ( t , x ) of a neural field equation (NFE). The value u ( t , x ) represents the membrane potential at time t of a typical neuron located in position x , embedded in an infinite network of neurons. In the present paper, we complement this result by studying the fluctuations of such a stochastic system around its mean field limit u ( t , x ) . Our first main result is a central limit theorem stating that the spatial distribution associated to these fluctuations converges to the unique solution of some stochastic differential equation driven by a Gaussian noise. In our second main result we show that the solutions of this stochastic differential equation can be well approximated by a stochastic version of the neural field equation satisfied by u ( t , x ) . To the best of our knowledge, this result appears to be new in the literature.

International audience; The paper deals with output feedback regulation of exponentially stable systems by an integral controller. We have recently proposed an appropriate Lyapunov functional to prove exponential stability of the closed-loop system. The approach is dedicated in this paper to hyperbolic systems and especially to the de Saint-Venant equations giving explicitly the gains to ensure an exponentially stabilized integral controller: the parameters expression is deduced directly of the Lyapunov functional based on the Forwarding approach. Numerical simulations illustrate this approach.

We prove a long exact sequence in KK-theory for both full and reduced amalgamated free products in the presence of conditional expectations. In the course of the proof, we established the KK-equivalence between the full amalgamated free product of two unital C*-algebras and a newly defined reduced amalgamated free product that is valid even for non GNS-faithful conditional expectations. Our results unify, simplify and generalize all the previous results obtained before by Cuntz, Germain and Thomsen., V.3, the paper has been splitted into two papers, this is the first part on amalgamated free products

In view of studying incompressible inviscid fluids, Brenier introduced in the late 80's a relaxation of a geodesic problem addressed by Arnold in 1966. Instead of inviscid fluids, the present paper is devoted to incompressible viscid fluids. A natural analogue of Brenier's problem is introduced, where generalized flows are no more supported by absolutely continuous paths, but by Brownian sample paths. It turns out that this new variational problem is an entropy minimization problem with marginal constraints entering the class of convex minimization problems. This paper explores the connection between this variational problem and Brenier's original problem. Its dual problem is derived and the general shape of its solution is described. Under the restrictive assumption that the pressure is a nice function, the kinematics of its solution is made explicit and its connection with the Navier-Stokes equation is established.

International audience; This paper is devoted to the numerical analysis of a numerical scheme dedicated to the simulation of front advection (see https://hal.archives-ouvertes.fr/hal-02940407v1 for a preprint version presenting this scheme and some numerical results). The latter has been recently proposed and it is based on the ideas used for the Glimm's scheme. It relies on a two-step approach: a convection step is followed by a projection step which is based on a random choice. The main advantage of this scheme is that it applies to multi-dimensional problems. In the present paper a convergence result for this scheme is provided for a particular class of multi-dimensional problems. This work has been accepted for publication in IMA Journal of Numerical Analysis: https://doi.org/10.1093/imanum/drab053.

In this paper, we study the problem of constructing flat inputs for multi-output dynamical systems. The notion of flat inputs has been introduced by Waldherr and Zeitz in [30, 31] and can be seen as dual to that of flat outputs. In the single-output case, a flat input can be constructed if and only if the original dynamical system together with its output is observable. In the multi-output case, the observability is not necessary for the existence of flat inputs. The observable case has been treated in [31], where a system of linear algebraic equations has been proposed in order to determine the control vector fields associated to the flat inputs. The goal of this paper is to treat the unobservable case for multi-output dynamical systems. We start by discussing the case when the dynamical system together with the given output is observable and we present a generalization of the results of [31] by relating them with the notion of minimal differential weight. Then we give our main results. We consider the unobservable case for which locally, on an open and dense subset of $\mathbb R^n$ , we construct control vector fields $g_1 , \ldots, g_m$ such that the associated control system is flat. Finally, we explain how our results can be applied to private communication.

For two graphs $B$ and $H$ the strong Ramsey game $\mathcal{R}(B,H)$ on the board $B$ and with target $H$ is played as follows. Two players alternately claim edges of $B$. The first player to build a copy of $H$ wins. If none of the players win, the game is declared a draw. A notorious open question of Beck asks whether the first player has a winning strategy in $\mathcal{R}(K_n,K_k)$ in bounded time as $n\rightarrow\infty$. Surprisingly, in a recent paper Hefetz et al. constructed a $5$-uniform hypergraph $\mathcal{H}$ for which they proved that the first player does not have a winning strategy in $\mathcal{R}(K_n^{(5)},\mathcal{H})$ in bounded time. They naturally ask whether the same result holds for graphs. In this paper we make further progress in decreasing the rank. In our first result, we construct a graph $G$ (in fact $G=K_6\setminus K_4$) and prove that the first player does not have a winning strategy in $\mathcal{R}(K_n \sqcup K_n,G)$ in bounded time. As an application of this result we deduce our second result in which we construct a $4$-uniform hypergraph $G'$ and prove that the first player does not have a winning strategy in $\mathcal{R}(K_n^{(4)},G')$ in bounded time. This improves the result in the paper above. An equivalent formulation of our first result is that the game $\mathcal{R}(K_\omega\sqcup K_\omega,G)$ is a draw. Another reason for interest on the board $K_\omega\sqcup K_\omega$ is a folklore result that the disjoint union of two finite positional games both of which are first player wins is also a first player win. An amusing corollary of our first result is that at least one of the following two natural statements is false: (1) for every graph $H$, $\mathcal{R}(K_\omega,H)$ is a first player win; (2) for every graph $H$ if $\mathcal{R}(K_\omega,H)$ is a first player win, then $\mathcal{R}(K_\omega\sqcup K_\omega,H)$ is also a first player win., Comment: 17 pages, 48 figures; improved presentation, particularly in section 3

This paper is devoted to the abstract study of operational second-order differential equations of elliptic type with nonregular coefficient-operator boundary conditions in a non-commutative framework. The study is performed when the second member f belongs to $$L^{p}(0,1;X)$$, with general $$p\in ]1,+\infty [$$, X being a UMD Banach space. We give some new results by using semigroups and interpolation theory. Existence, uniqueness and optimal regularity of the classical and semi-classical solution are proved. This paper improves naturally the ones studied in the commutative case by Hammou et al. (Mediterr J Math, 1669–1683, 2015).

In this paper we are interested in the large time behavior of linear kinetic equations with heavy-tailed local equilibria. Our main contribution concerns the kinetic Lévy-Fokker-Planck equation, for which we adapt hypocoercivity techniques in order to show that solutions converge exponentially fast to the global equilibrium. Compared to the classical kinetic Fokker-Planck equation, the issues here concern the lack of symmetry of the non-local Lévy-Fokker-Planck operator and the understanding of its regularization properties. As a complementary related result, we also treat the case of the heavy-tailed BGK equation.; In this paper we are interested in the large time behavior of linear kinetic equations with heavy-tailed local equilib-ria. Our main contribution concerns the kinetic Lévy-Fokker-Planck equation, for which we adapt hypocoercivity techniques in order to show that solutions converge exponentially fast to the global equilibrium. Compared to the classical kinetic Fokker-Planck equation, the issues here concern the lack of symmetry of the non-local Lévy-Fokker-Planck operator and the understanding of its regularization properties. As a complementary related result, we also treat the case of the heavy-tailed BGK equation. Résumé Une note sur l'hypocoercivité pour leséquations cinétiques avecéquilibresà queue lourde. Dans cet article, on s'intéresse au comportement en temps long d'équations cinétiques linéaires dont leséquilibres locaux sontà queue lourde. Notre contribution principale concerne l'équation de Lévy-Fokker-Planck cinétique, pour laquelle nous adaptons des techniques d'hypocoercivité afin de démontrer la convergence exponentielle des solutions vers unéquilibre global. En comparant au cas de l'équation de Fokker-Planck cinétique classique, les enjeux ici sont liés au manque de symétrie de l'opérateur non-local de Lévy-Fokker-Planck età la compréhension de ses propriétés de régularisation. En complément de notre analyse, nous traitonségalement le cas de l'équation de BGKà queue lourde.

the density of the winding process in the complex projective line is worked out; International audience; In the first part of this paper, we derive explicit expressions of the semi-group densities of generalized stochastic areas arising from the Anti-de Sitter and the Hopf fibrations. Motivated by the number-theoretical connection between the Heisenberg group and Dirichlet series, we express the Mellin transform of the generalized stochastic area corresponding to the one-dimensional Anti de Sitter fibration as a series of Riemann Zeta function evaluated at integers. In the second part of the paper, we focus on winding processes around the origin in the Poincar\'e disc and in the complex projective line. More pricesely, we derive the fixed-time marginal density of the former process while we give a ultraspherical series expansion of the characteristic function of the latter.

International audience; In general, the group decompositions are formulated by employing automorphisms and semidirect products to determine continuity and compactification properties. This paper proposes a set of constructions of novel topological decompositions of groups and analyzes the behaviour of group actions under the topological decompositions. The proposed topological decompositions arise in two varieties, such as decomposition based on topological fibers without projections and decomposition in the presence of translated projections in topological spaces. The first variety of decomposition introduces the concepts of topological fibers, locality of group operation and the partitioned local homeomorphism resulting in formulation of transitions and symmetric surjection within the topologically decomposed groups. The reformation of kernel under decomposed homeomorphism and the stability of group action with the existence of a fixed point are analyzed. The first variety of decomposition does not require commutativity maintaining generality. The second variety of projective topological decomposition is formulated considering commutative as well as noncommutative projections in spaces. The effects of finite translations of topologically decomposed groups under projections are analyzed. Moreover, the embedding of a decomposed group in normal topological spaces is formulated in this paper. It is shown that Schoenflies homeomorphic embeddings preserve group homeomorphism in the decomposed embeddings within normal topological spaces. This paper illustrates that decomposed group embedding in normal topological spaces is separable. The applications aspects as well as parametric comparison of group decompositions based on topology, direct product and semidirect product are included in the paper.

The paper deals with a three-parameter family of special double confluent Heun equations that was introduced and studied by V.M.Buchstaber and S.I.Tertychnyi as an equivalent presentation of a model of overdamped Josephson junction in superconductivity. The parameters are $l,\lambda,\mu\in\mathbb R$. Buchstaber and Tertychnyi described those parameter values, for which the corresponding equation has a polynomial solution. They have shown that for $\mu\neq0$ this happens exactly when $l\in\mathbb N$ and the parameters $(\lambda,\mu)$ lie on an algebraic curve $\Gamma_l\subset\mathbb C^2_{(\lambda,\mu)}$ called the $l$-th spectral curve and defined as zero locus of determinant of a remarkable three-diagonal $l\times l$-matrix. They studied the real spectral curves and obtained important results with applications to phase-lock areas in model of Josephson junction, which is a family of dynamical systems on 2-torus. In the present paper we prove irreducibility of complex spectral curves. We also calculate their genera for $l\leqslant20$ and present a conjecture on general genus formula. We apply the irreducibility result to the phase-lock areas, which are those level sets of the rotation number function $\rho$ on the parameter space of the above-mentioned family of dynamical systems that have non-empty interiors. The family of their boundaries is a countable union of analytic surfaces. We show that, unexpectedly, its complexification is a complex analytic subset consisting of just four irreducible components, and we describe them. We present a Monotonicity Conjecture on the evolution of the phase-lock area portraits and a partial positive result towards its confirmation., Comment: To appear in the Journal of Dynamical and Control Systems. 50 pages, 17 figures. Minor editorial changes, new references added