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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.

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

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).

Any germ of a complex analytic space is equipped with two natural metrics: the {\it outer metric} induced by the hermitian metric of the ambient space and the {\it inner metric}, which is the associated riemannian metric on the germ. We show that minimal surface singularities are Lipschitz normally embedded (LNE), i.e., the identity map is a bilipschitz homeomorphism between outer and inner metrics, and that they are the only rational surface singularities with this property., This paper is a major revision of the 2015 version. It now builds on the paper arXiv:1806.11240 by the same authors which gives a general characterization of Lipschitz normally embedded surface singularities

The main subject of this paper is equilibrium problems on an unbounded conductor $\Sigma$ of the complex plane in the presence of a weakly admissible external field. An admissible external field $Q$ on $\Sigma$ satisfies, along with other mild conditions, the following growth property at infinity: $$\lim_{|x| \rightarrow \infty}(Q(x) - \log |x|) = +\infty.$$ This condition guarantees the existence and uniqueness of the equilibrium measure in the presence of $Q$, and the compactness of its support. In the last 10-15 years, several papers have dealt with weakly admissible external fields, in the sense that $Q$ satisfies a weaker condition at infinity, namely, $$\exists M\in(-\infty,\infty],\quad\liminf_{|x| \rightarrow \infty}(Q(x) - \log |x|) = M.$$ Under this last assumption, there still exists a unique equilibrium measure in the external field $Q$, but the support need not be a compact subset of $\Sigma$ anymore. In most examples considered in the literature the support is indeed unbounded. Our main goal in this paper is to illustrate this topic by means of a simple class of external fields on the real axis created by a pair of attractive and repellent charges in the complex plane, and to study the dynamics of the associated equilibrium measures as the strength of the charges evolves. As one of our findings, we exhibit configurations where the support of the equilibrium measure in a weakly admissible external field is a compact subset of the real axis. To achieve our goal, we extend some results from potential theory, known for admissible external fields, to the weakly admissible case. These new results may be of independent interest. Finally, the so--called signed equilibrium measure is an important tool in our analysis. Its relationship with the (positive) equilibrium measure is also explored., Comment: To appear in Journal of Approximation Theory

This is a continuation of a previous paper by the same authors. In the former paper, it was proved that in order to obtain local uniformization for valuations centered on local domains, it is enough to prove it for rank one valuations. In this paper, we extend this result to the case of valuations centered on rings which are not necessarily integral domains and may even contain nilpotents.

In this paper we refine the main result of a previous paper of the author with Grimeland on derived invariants of surface algebras. We restrict to the case where the surface is a torus with one boundary component and give an easily computable derived invariant for such surface algebras. This result permits to give answers to open questions on gentle algebras: it provides examples of gentle algebras with the same AG-invariant (in the sense of Avella-Alaminos and Geiss) that are not derived equivalent and gives a partial positive answer to a conjecture due to Bobi\'nski and Malicki on gentle $2$-cycles algebras., Comment: 22 pages, a mistake concerning the computation of the mapping class group has been fixed, version 3: 25 pages, to appear in Algebras and Representation Theory

In a paper with Jean-Paul Dufour in 1999 \cite{DufourZung-Nambu1999}, we gave a classification of linear Nambu structures, and obtained linearization results for Nambu structures with a nondegenerate linear part. There was a case left open in \cite{DufourZung-Nambu1999}, namely the case of smooth linearization of Nambu structures with a Type 1 hyperbolic linear part which satisfies a natural signature condition. In this paper, we will show that such hyperbolic Nambu structures are also smoothly linearizable. We will also give a strong version of the analytic linearization theorem in the analytic case, improving a result obtained in \cite{DufourZung-Nambu1999}., 1st version, 12 pages

In a recent paper, the authors proved that, under natural assumptions on the first marginal, the Monge problem in $ {\mathbb{R}^d} $ for the cost given by a general norm admits a solution. Although the basic idea of the proof is simple, it involves some complex technical results. Here we will give a proof of the result in the simpler case of a uniformly convex norm, and we will also use very recent results by Ahmad, Kim, and McCann. This allows us to reduce the technical burdens while still giving the main ideas of the general proof. The proof of the density of the transport set in the particular case considered in this paper is original. Bibliography: 22 titles.

We prove that in the Wasserstein space built over R the subset of measures that does not charge the non-differentiability set of convex functions is not displacement convex. This completes the study of Gigli on the geometric structure of measures meeting the sharp hypothesis of the refined version of Brenier’s Theorem. Optimal transport is nowadays a central tool in many fields of analysis, differential geometry and probability theory (see e.g. the books by Villani [11, 12]). It takes its origin in the problem of Monge, asking for the shortest way to displace an amount of soil from one place of the Euclidean space to a heap of soil at another place. This problem induces a very natural distance between probability measures – the Wasserstein distance – where the measures represent the heaps of soil. Wasserstein distance is somewhat complementary with the Lebesgue L norms because it corresponds to an “horizontal” displacement: in first approximation, for two localized probability measures on R, the Wasserstein distance is the distance between their barycenters. Brenier’s Theorem [4] on monotone rearrangement of maps of R has become the very core of the theory of optimal transport. It gives a representation of the optimal transport map in term of gradient of convex functions. A very enlightening heuristic on (P2(R ),W2) is proposed in [7] where it appears with an infinite differential structure and the Wasserstein distance is seen as a Riemannian-like distance. This point of view has raised a lot of developments, among which the gradient flow theory presented in [2]. In [6], Gigli explores the Riemannian-like structure and proposes to think of the measures meeting the hypothesis of a refined version of Brenier’s Theorem as the regular points of the Wasserstein space. In this paper we show that the set of those transport-regular measures is not geodesically convex (Theorems 2.2 and 2.5). It is quite surprising because it is well-known that the subset made of absolutely continuous measures (the most notorious transport-regular measures) is geodesically convex. It answers a question suggested by Gigli [6, Remark 2.12]. The paper is built as follows: we first recall some main results on the quadratic Monge(-Kantorovich) problem. The key idea of our result can be found in Lemma 1.6 while Proposition 1.8 gives a way, together with Proposition 1.7, to characterize transport-regular measures. In the second (and last) part, we prove the main theorem (Theorem 2.2) and its generalization (Theorem 2.5) thanks to explicit constructions. 2010 Mathematics Subject Classification. Primary 28A75.

This paper explains the recent developments on the symplectic theory of Hamiltonian completely integrable systems on symplectic 4-manifolds, compact or not. One fundamental ingredient of these developments has been the understanding of singular affine structures. These developments make use of results obtained by many authors in the second half of the twentieth century, notably Arnold, Duistermaat and Eliasson, of which we also give a concise survey. As a motivation, we present a collection of remarkable results proven in the early and mid 1980s in the theory of Hamiltonian Lie group actions by Atiyah, Guillemin-Sternberg and Delzant among others, and which inspired many people, including the authors, to work on more general Hamiltonian systems. The paper concludes discussing a spectral conjecture for quantum integrable systems., 40 pages

In this paper we consider the following Timoshenko system: in (0, L) × ℝ + {ρ 1 ϕ tt -k 1 (ϕ x +ψ) x =0 ρ 2 ψ tt - k 2 ψ xx + ∫ t 0 g(t - τ)(a(x)ψ x (τ)) x dτ+k 1 (ϕ x + ψ) + b(x)h(ψ t ) =0 in (0, L) × ℝ + with Dirichlet boundary conditions and initial data where a, b, g and h are specific functions and ρ 1 , ρ 2 , κ 1 , κ 2 and L are given positive constants. We establish a general stability estimate using the multiplier method and some properties of convex functions. Without imposing any growth condition on h at the origin, we show that the energy of the system is bounded above by a quantity, depending on g and h, which tends to zero as time goes to infinity. Our estimate allows us to consider a large class of functions h with general growth at the origin. We use some examples (known in the case of wave equations and Maxwell system) to show how to derive from our general estimate the polynomial, exponential or logarithmic decay. The results of this paper improve and generalize some existing results in the literature and generate some interesting open problems.

Let R be a real closed field. The Pierce–Birkhoff conjecture says that any piecewise polynomial function f on Rn can be obtained from the polynomial ring R[x1,..., xn] by iterating the operations of maximum and minimum. The purpose of this paper is threefold. First, we state a new conjecture, called the Connectedness conjecture, which asserts, for every pair of points \({{\alpha,\beta\in\,{\rm {Sper}}\ R[x_1,\ldots,x_n]}}\) , the existence of connected sets in the real spectrum of R[x1,..., xn], satisfying certain conditions. We prove that the Connectedness conjecture implies the Pierce–Birkhoff conjecture. Secondly, we construct a class of connected sets in the real spectrum which, though not in itself enough for the proof of the Pierce–Birkhoff conjecture, is the first and simplest example of the sort of connected sets we really need, and which constitutes the first step in our program for a proof of the Pierce–Birkhoff conjecture in dimension greater than 2. Thirdly, we apply these ideas to give two proofs that the Connectedness conjecture (and hence also the Pierce–Birkhoff conjecture in the abstract formulation) holds locally at any pair of points \({{\alpha,\beta\in\,{\rm {Sper}}\ R[x_1,\ldots,x_n]}}\) , one of which is monomial. One of the proofs is elementary while the other consists in deducing this result as an immediate corollary of the main connectedness theorem of this paper.

This arXived paper has two independant parts, that are improved and corrected versions of different parts of a single paper once named "On equations in relatively hyperbolic groups". The first part is entitled "Existential questions in (relatively) hyperbolic groups". We study there the existential theory of torsion free hyperbolic and relatively hyperbolic groups, in particular those with virtually abelian parabolic subgroups. We show that the satisfiability of systems of equations and inequations is decidable in these groups. In the second part, called "Finding relative hyperbolic structures", we provide a general algorithm that recognizes the class of groups that are hyperbolic relative to abelian subgroups.

In two fundamental classical papers, Masur and Veech have independently proved that the Teichmueller geodesic flow acts ergodically on each connected component of each stratum of the moduli space of quadratic differentials. It is therefore interesting to have a classification of the ergodic components. Veech has proved that these strata are not necessarily connected. In a recent work, Kontsevich and Zorich have completely classified the components in the particular case where the quadratic differentials are given by the global square of Abelian differentials. Here we are interested in the complementary case. In a previous paper, we have described some particular component, namely the hyperelliptic connected components, and showed that some strata are non-connected. In this paper, we give the general classification theorem: up to four exceptional cases in low genera, the strata of meromorphic quadratic differentials are either connected, or have exactly two connected components. In this last case, one component is hyperelliptic, the other not. Our proof is based on a new approach of the so-called Jenkins-Strebel differential. We will present and use the notion of generalized permutations., 49 pages, 12 figures, submitted. major revision, typos corrected

This paper is devoted to the study of multi-dimensional travelling wave solution for a thermo-diffusive model, describing the propagation of curved flames in an infinite cylinder. The linear dependence of the components of the reaction rate together with the existence of an ignition temperature ensure that the corresponding linearized operator does not satisfy the Fredholm property. A direct consequence is that solvability conditions for the linearized operator are not known and classical methods of nonlinear analysis cannot be directly applied. We prove in this paper existence results of such travelling waves, by first introducing a suitable re-formulation of the equations and then by choosing suitable weighted spaces that allows us to move the essential spectrum away from zero. Copyright © 2006 John Wiley & Sons, Ltd.

In earlier papers the authors studied finite pseudorandom binary sequences, and they constructed sequences with strong pseudorandom properties. In these earlier constructions multiplicative characters were used. In this paper a new construction is presented which utilizes properties of additive characters. These new sequences can be computed fast, they are well-distributed relative to arithmetic progressions and their correlations of “small” order are “small”, but the price paid for the fast computation is that the correlations of “large” order can be “large”.

A first aim of this paper is to answer, in the case of real ruled surfaces $X_l$ of base $\mathbb{C}P^1$, with $l \geq 2$, a question of V. A. Rokhlin: is it true that the equivariant isotopy class does not suffice to distinguish the connected components of the space of smooth real algebraic curves of $X_l$? A second aim is to prove that there exist in these surfaces some real schemes realized by real flexible curves but not by smooth real algebraic curves. These two results of real algebraic geometry are deduced from the following comparison theorem: when $m = l + 2k$, with $k > 0$, the discriminants of the surface $X_m$ are deduced from those of the surface $X_l$ via weighted homotheties. All these results are obtained from a study of a deformation of ruled surfaces.The paper is written in French.2000 Mathematical Subject Classification: 14H10, 14J26, 14P25.

Résumé: Cet article présente une étude combinatoire du monoïde Chinois, un monoïde ternaire proche du monoïde plaxique, fondé sur le schéma cba≡bca≡cab. Un algorithme proche de l'algorithme de Schensted nous permet de caractériser les classes d'équivalence et d'exhiber une section du monoïde. Nous énonçons également une correspondance de Robinson–Schensted pour le monoïde Chinois avant de nous intéresser au calcul du cardinal de certaines classes. Ce travail a permis de développer de nouveaux outils combinatoires. Entre autres, nous avons trouvé un plongement de chacune des classes d'équivalence dans la plus grande classe. Quant à la dernière partie de cet article, elle présente l'étude des relations de conjugaison. This paper presents a combinatorial study of the Chinese monoid, a ternary monoid related to the plactic monoid and based on the relation scheme cba≡bca≡cab. An algorithm similar to Schensted's algorithm yields a characterization of the equivalence classes and a cross-section theorem. We also establish a Robinson–Schensted correspondence for the Chinese monoid before computing the order of specific Chinese classes. For this work, we had to develop some new combinatorial tools. Among other things we discovered an embedding of every equivalence class in the largest one. Finally, the end of this paper is devoted to the study of conjugacy classes.

In this paper, we prove a conjecture of Aharoni and Howard on the existence of rainbow (transversal) matchings in sufficiently large families $\mathcal F_1,\ldots, \mathcal F_s$ of tuples in $\{1,\ldots, n\}^k$, provided $s\ge 470.$

In this paper, we provide asymptotic upper bounds on the complexity in two (closely related) situations. We confirm for the total doubling coverings and not only for the chains the expected bounds of the form [Formula: see text] This is done in a rather general setting, i.e. for the [Formula: see text]-complement of a polynomial zero-level hypersurface [Formula: see text] and for the regular level hypersurfaces [Formula: see text] themselves with no assumptions on the singularities of [Formula: see text]. The coefficient [Formula: see text] is the ambient dimension [Formula: see text] in the first case and [Formula: see text] in the second case. However, the question of a uniform behavior of the coefficient [Formula: see text] remains open. As a second theme, we confirm in arbitrary dimension the upper bound for the number of a-charts covering a real semi-algebraic set [Formula: see text] of dimension [Formula: see text] away from the [Formula: see text]-neighborhood of a lower dimensional set [Formula: see text], with bound of the form [Formula: see text] holding uniformly in the complexity of [Formula: see text]. We also show an analogue for level sets with parameter away from the [Formula: see text]-neighborhood of a low dimensional set. More generally, the bounds are obtained also for real subanalytic and real power-subanalytic sets.

This paper constructs a combinatorial model for all postcritically finite rational maps arising as the Newton's method of a complex polynomial. This model is used in [LMS] to give a combinatorial classification of postcritically finite Newton maps of any degree.

In this paper, we give a very general criterion for elimination of imaginaries using an abstract independent relation. We also study germs of definable functions at certain well-behaved invariant types. Finally we apply these tools to the elimination of imaginaries in bounded pseudo-p-adically closed fields., Comment: Minor corrections. Acknowledgement of support by ValCoMo (ANR-13-BS01-0006) added

This paper builds on two covering Hopf algebras of the Hopf algebra QSym of quasi-symmetric functions, with linear bases parameterized by compositions. One is the Malvenuto-Reutenauer Hopf algebra SSym of permutations, mapped onto QSym by taking descents of permutations. The other one is the recently introduced Hopf algebra RQSym of weak quasi-symmetric functions, mapped onto QSym by extracting compositions from weak compositions. We extend these two surjective Hopf algebra homomorphisms into a commutative diagram by introducing a Hopf algebra HSym, linearly spanned by signed permutations from the hyperoctahedral groups, equipped with the shifted quasi-shuffle product and deconcatenation coproduct. Extracting a permutation from a signed permutation defines a Hopf algebra surjection form HSym to SSym and taking a suitable descent from a signed permutation defines a linear surjection from HSym to RQSym. The notion of signed $P$-partitions from signed permutations is introduced which, by taking generating functions, gives fundamental weak quasi-symmetric functions and sends the shifted quasi-shuffle product to the product of the corresponding generating functions. Together with the existing Hopf algebra surjections from SSym and RQSym to QSym, we obtain a commutative diagram of Hopf algebras revealing the close relationship among compositions, weak compositions, permutations and signed permutations., Comment: 25 pages, 2 figures

Grothendieck’s theorem asserts that every continuous linear operator from $$\ell _1$$ to $$\ell _2$$ is absolutely (1, 1)-summing. This kind of result is commonly called coincidence result. In this paper we investigate coincidence results in the multilinear setting, showing how the cotype of the spaces involved affect such results. The special role played by $$\ell _1$$ spaces is also investigated with relation to interpolation of tensor products. In particular, an open problem on the interpolation of m injective tensor products is solved.

International audience; In this paper we study the existence and regularity of solutions to the following Dirichlet problem        −div(a(x)|∇u| p−2 ∇u) + u|u| r−1 = f (x) u θ in Ω, u > 0 in Ω, u = 0 on ∂Ω proving that the lower order term u|u| r−1 has some regularizing effects on the solutions. Mathematics Subject Classification (2010). 35J66, 35J75

For a generalized continuous-state branching process with non-vanishing diffusion part, finite expectation and a directed (“left-to-right”) interaction, we construct the height process of its forest of genealogical trees. The connection between this height process and the population size process is given by an extension of the second Ray–Knight theorem. This paper generalizes earlier work of the two last authors which was restricted to the case of continuous branching mechanisms. Our approach is different from that of Berestycki et al. (Probab Theory Relat Fields 172:725–788, 2018). There the diffusion part of the population process was allowed to vanish, but the class of interactions was more restricted.

We continue our study of the problem of mixing for a class of PDEs with very degenerate noise. As we established earlier, the uniqueness of stationary measure and its exponential stability in the dual-Lipschitz metric holds under the hypothesis that the unperturbed equation has exactly one globally stable equilibrium point. In this paper, we relax that condition, assuming only global controllability to a given point. It is proved that the uniqueness of a stationary measure and convergence to it are still valid, whereas the rate of convergence is not necessarily exponential. The result is applicable to randomly forced parabolic-type PDEs, provided that the deterministic part of the external force is in general position, ensuring a regular structure for the attractor of the unperturbed problem. The proof uses a new idea that reduces the verification of a stability property to the investigation of a conditional random walk.

A hole in a graph is an induced cycle of length at least $4$, and an antihole is the complement of an induced cycle of length at least $4$. A hole or antihole is long if its length is at least $5$. For an integer $k$, the $k$-prism is the graph consisting of two cliques of size $k$ joined by a matching. The complexity of Maximum (Weight) Independent Set (MWIS) in long-hole-free graphs remains an important open problem. In this paper we give a polynomial time algorithm to solve MWIS in long-hole-free graphs with no $k$-prism (for any fixed integer $k$), and a subexponential algorithm for MWIS in long-hole-free graphs in general. As a special case this gives a polynomial time algorithm to find a maximum weight clique in perfect graphs with no long antihole, and no hole of length $6$. The algorithms use the framework of minimal chordal completions and potential maximal cliques.

In the present paper, we give a definition of the quantum group $$\mathbf {U}_\upsilon (\mathfrak {sl}(S^1))$$ of the circle $$S^1:={\mathbb {R}}/{\mathbb {Z}}$$, and its fundamental representation. Such a definition is motivated by a realization of a quantum group $$\mathbf {U}_\upsilon (\mathfrak {sl}(S^1_{\mathbb {Q}}))$$ associated to the rational circle $$S^1_{\mathbb {Q}}:={\mathbb {Q}}/{\mathbb {Z}}$$ as a direct limit of $$\mathbf {U}_\upsilon (\widehat{\mathfrak {sl}}(n))$$’s, where the order is given by divisibility of positive integers. The quantum group $$\mathbf {U}_\upsilon (\mathfrak {sl}(S^1_{\mathbb {Q}}))$$ arises as a subalgebra of the Hall algebra of coherent sheaves on the infinite root stack $$X_\infty $$ over a fixed smooth projective curve X defined over a finite field. Via this Hall algebra approach, we are able to realize geometrically the fundamental and the tensor representations, and a family of symmetric tensor representations, depending on the genus $$g_X$$, of $$\mathbf {U}_\upsilon (\mathfrak {sl}(S^1_{\mathbb {Q}}))$$. Moreover, we show that $$\mathbf {U}_\upsilon (\widehat{\mathfrak {sl}}(+\infty ))$$ and $$\mathbf {U}_\upsilon (\widehat{\mathfrak {sl}}(\infty ))$$ are subalgebras of $$\mathbf {U}_\upsilon (\mathfrak {sl}(S^1_{\mathbb {Q}}))$$. As proved by T. Kuwagaki in the appendix, the quantum group $$\mathbf {U}_\upsilon (\mathfrak {sl}(S^1))$$ naturally arises as well in the mirror dual picture, as a Hall algebra of constructible sheaves on the circle $$S^1$$.

Let N be a closed oriented k-dimensional submanifold of the (k+2)-dimensional sphere; denote its complement by C(N). Denote by x the 1-dimensional cohomology class in C(N), dual to N. The Morse-Novikov number of C(N) is by definition the minimal possible number of critical points of a regular Morse map f from C(N) to a circle, such that f belongs to x. In the first part of this paper we study the case when N is the twist frame spun knot associated to an m-knot K. We obtain a formula which relates the Morse-Novikov numbers of N and K and generalizes the classical results of D. Roseman and E.C. Zeeman about fibrations of spun knots. In the second part we apply the obtained results to the computation of Morse-Novikov numbers of surface-links in 4-sphere., 13 pages

We explain why Theorem B in the original article does not follow from the main result of this paper (Theorem A). While we conjecture that Theorem B should nevertheless be true, in this erratum we prove a slightly weaker version of it.

An odd-quadratic Leibniz superalgebra is a (left or right) Leibniz superalgebra with an odd, supersymmetric, non-degenerate and invariant bilinear form. In this paper, we prove that a left (resp. right) Leibniz superalgebra that carries this structure is symmetric (meaning that it is simultaneously a left and a right Leibniz superalgebra). Moreover, we show that any non-abelian (left or right) Leibniz superalgebra does not possess simultaneously a quadratic and an odd-quadratic structure. Further, we obtain an inductive description of odd-quadratic Leibniz superalgebras using the procedure of generalized odd double extension and we reduce the study of this class of Leibniz superalgebras to that of odd-quadratic Lie superalgebras. Finally, several non-trivial examples of odd-quadratic Leibniz superalgebras are included.

This paper is concerned with the propagation dynamics of a predator–prey system with nonlocal dispersal. We obtain a threshold phenomenon for the invasion of the predator into the habitat of the aborigine prey. It turns out that this threshold is the so-called spreading speed of the predator as well as the minimal wave speed of traveling wave solutions connecting the predator-free state to a nontrivial state.

We consider the class of $\lambda$-concave bodies in $\mathbb R^{n+1}$; that is, convex bodies with the property that each of their boundary points supports a tangent ball of radius $1/\lambda$ that lies locally (around the boundary point) inside the body. In this class we solve a reverse isoperimetric problem: we show that the convex hull of two balls of radius $1/\lambda$ (a sausage body) is a unique volume minimizer among all $\lambda$-concave bodies of given surface area. This is in a surprising contrast to the standard isoperimetric problem for which, as it is well-known, the unique maximizer is a ball. We solve the reverse isoperimetric problem by proving a reverse quermassintegral inequality, the second main result of this paper., Comment: 1 figure

For the random interval partition of $[0,1]$ generated by the uniform stick-breaking scheme known as GEM$(1)$, let $u_k$ be the probability that the first $k$ intervals created by the stick-breaking scheme are also the first $k$ intervals to be discovered in a process of uniform random sampling of points from $[0,1]$. Then $u_k$ is a renewal sequence. We prove that $u_k$ is a rational linear combination of the real numbers $1, \zeta(2), \ldots, \zeta(k)$ where $\zeta$ is the Riemann zeta function, and show that $u_k$ has limit $1/3$ as $k \to \infty$. Related results provide probabilistic interpretations of some multiple zeta values in terms of a Markov chain derived from the interval partition. This Markov chain has the structure of a weak record chain. Similar results are given for the GEM$(\theta)$ model, with beta$(1,\theta)$ instead of uniform stick-breaking factors, and for another more algebraic derivation of renewal sequences from the Riemann zeta function., Comment: 25 pages. This paper is published by https://www.ams.org/journals/tran/2019-371-08/S0002-9947-2018-07516-X/

Given a connected real Lie group and a contractible homogeneous proper $G$--space $X$ furnished with a $G$--invariant volume form, a real valued volume can be assigned to any representation $\rho\colon \pi_1(M)\to G$ for any oriented closed smooth manifold $M$ of the same dimension as $X$. Suppose that $G$ contains a closed and cocompact semisimple subgroup, it is shown in this paper that the set of volumes is finite for any given $M$. From a perspective of model geometries, examples are investigated and applications with mapping degrees are discussed., Comment: 34 pages

In this paper, we show the existence and multiplicity of positive solutions of the fractional Kirchhoff system $$\begin{aligned} {\left\{ \begin{array}{ll}\mathcal {L}_{M}(u) = {\lambda }f(x)|u|^{q-2}{u} + \frac{2{\alpha }}{{\alpha }+{\beta }} |u|^{\alpha -2} \,u|v|^{\beta} &{}\quad \mathrm{in}\, \Omega ,\\ \mathcal {L}_M(v) = {\mu }g(x)|v|^{q-2}v + \frac{2{\beta }}{{\alpha }+{\beta }}|u|^{\alpha} \,|v|^{\beta -2}v &{}\quad \mathrm{in}\, \Omega ,\\ \quad \;\;\; u = v = 0 &{}\quad \mathrm{on}\, {\partial }{\Omega }, \end{array}\right. } \end{aligned}$$ where $$\mathcal{L}_{M}(u) = M \big(\int_{\Omega} |(-{\Delta})^\frac{s}{2}u|^{2}dx\big) (-\Delta)^{s}u$$ is a double non-local operator due to a Kirchhoff term $$M(t) = a + bt$$ with a, b > 0 and the fractional Laplacian $$(-\Delta)^{s}, s \in (0,1)$$. We consider that $$\Omega$$ is an open and bounded domain in $$\mathbb{R}^{N}, 2s 0$$ are real parameters, $$1 < q < 2, \alpha,\beta \geq 2$$ and $$\alpha + \beta = 2_{s}^{*} = 2N/(N-2s)$$ is a fractional critical exponent. Using the idea of Nehari manifold technique and a compactness result based on the classical idea of the Brezis–Lieb lemma, we prove the existence of at least two positive solutions for $$(\lambda,\mu)$$ lying in a suitable subset of $$\mathbb{R}^{2}_{+}$$.

The asymptotic study of partial differential equations in cylinders becoming unbounded in one or several directions has known important developments in the last years, especially thanks to the works of Michel Chipot and his collaborators, see for example Chipot ( $$\ell $$ goes to plus infinity, Birkauser Verlag, Basel, 2002, Asymptotic issues for some partial differential equations, Imperial College Press, London, 2016), Chipot and Mardare (J Math Pures Appl 104:921–941, 2015), Chipot and Rougirel (Discrete Contin Dyn Syst Ser B 1(3):319–338, 2001), Chipot et al. (Asympt Anal 85:199–227, 2013), Chipot and Yeressian (C R Acad Sci Paris Ser I 346:21–26, 2008), Guesmia (Nonlinear Anal 70(9):3320–3331, 2009) and Xie (in: Recent advances on elliptic and parabolic issues, Proceedings of the 2004 Swiss-Japanese Seminar, World Scientific, 2006). In this paper, we prove the convergence to the solution of a linear elliptic problem on an infinite cylinder of the solutions of the same problem taken on larger and larger truncations of the cylinder. Following the methods introduced in Chipot and Yeressian (2008) and Chipot and Mardare (2015), we generalize the result of convergence found in Chipot and Yeressian (2008) for the case where the data is not necessarily independent of the coordinate along the axis of the cylinder. We also consider the non-homogenous Dirichlet problem instead of the homogenous one.

International audience; In 2008, Kedlaya and Umans designed the first multivariate multi-point evaluation algorithm over finite fields with an asymptotic complexity that can be made arbitrarily close to linear. However, it remains a major challenge to make their algorithm efficient for practical input sizes. In this paper, we revisit and improve their algorithm, while keeping this ultimate goal in mind. In addition we sharpen the known complexity bounds for modular composition of univariate polynomials over finite fields.

International audience; This paper is concerned with qualitative properties of solutions to nonlocal reaction-diffusion equations of the form$$ \int_{\mathbb{R}^N\setminus K} J(x-y)\,\big( u(y)-u(x) \big)\,\D y+f(u(x))=0, \quad x\in\R^N\setminus K,$$set in a perforated open set $\mathbb{R}^N\setminus K$, where $K\subset\mathbb{R}^N$ is a bounded compact ``obstacle" and $f$ is a bistable nonlinearity. When $K$ is convex, we prove some Liouville-type results for solutions satisfying some asymptotic limiting conditions at infinity. We also establish a robustness result, assuming slightly relaxed conditions on $K$.

Denote by$\mathbb{P}$the set of all prime numbers and by$P(n)$the largest prime factor of positive integer$n\geq 1$with the convention$P(1)=1$. In this paper, we prove that, for each$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$, there is a constant$c(\unicode[STIX]{x1D702})>1$such that, for every fixed nonzero integer$a\in \mathbb{Z}^{\ast }$, the set$$\begin{eqnarray}\{p\in \mathbb{P}:p=P(q-a)\text{ for some prime }q\text{ with }p^{\unicode[STIX]{x1D702}}has relative asymptotic density one in$\mathbb{P}$. This improves a similar result due to Banks and Shparlinski [‘On values taken by the largest prime factor of shifted primes’,J. Aust. Math. Soc.82(2015), 133–147], Theorem 1.1, which requires$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.0606\cdots \,)$in place of$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$.

This paper explores the structure groups $G_{(X,r)}$ of finite non-degenerate set-theoretic solutions $(X,r)$ to the Yang-Baxter equation. Namely, we construct a finite quotient $\overline{G}_{(X,r)}$ of $G_{(X,r)}$, generalizing the Coxeter-like groups introduced by Dehornoy for involutive solutions. This yields a finitary setting for testing injectivity: if $X$ injects into $G_{(X,r)}$, then it also injects into $\overline{G}_{(X,r)}$. We shrink every solution to an injective one with the same structure group, and compute the rank of the abelianization of $G_{(X,r)}$. We show that multipermutation solutions are the only involutive solutions with diffuse structure group; that only free abelian structure groups are biorderable; and that for the structure group of a self-distributive solution, the following conditions are equivalent: biorderable, left-orderable, abelian, free abelian, torsion free., 32 pages. Final version. Accepted for publication in Proc. Edinburgh Math. Soc

In the present paper, we generalize the well-known Hensel’s lifting lemma to any continuous function f : Z p → Z p . This answers a question posed by Axelsson and Khrennikov (2016) who showed the validity of Hensel’s lemma for 1- and for p α -Lipschitz functions. For the statement and the proof, we introduce a suitable generalization of the original van der Put series. We use the concept of approximability of continuous functions to give numerical examples.

International audience; In this paper we consider the transmission eigenvalue problem for Maxwell's equations corresponding to nonmagnetic inhomogeneities with contrast in electric permittivity that has fixed sign (only) in a neighborhood of the boundary. Following the analysis made by Robbiano in the scalar case we study this problem in the framework of semiclassical analysis and relate the transmission eigenvalues to the spectrum of a Hilbert-Schmidt operator. Under the additional assumption that the contrast is constant in a neighborhood of the boundary, we prove that the set of transmission eigenvalues is discrete, infinite and without finite accumulation points. A notion of generalized eigenfunctions is introduced and a denseness result is obtained in an appropriate solution space.; Nous considérons le problème de valeurs propres de transmission pour les équations de Maxwell où le contraste sur la permittivité électrique admet un signe fixe seulement sur un voisinage du bord du domaine. En suivant l’approche développée par Robbiano dans le cas scalaire, nous formulons le problème aux valeurs propres comme un problème spectral pour un opérateur d’Hilbert-Schmidt. Sous l’hypothèse supplémentaire que le contraste est constant au voisinage de la frontière nous montrons que l’ensemble des valeurs propres de transmission est discret sans points d’accumulation finis. Nous définissons une famille de fonctions propres généralisées pour laquelle un résultat de densité est obtenu dans un espace de solutions bien choisi.

Moser's theorem (1965) states that the diffeomorphism group of a compact manifold acts transitively on the space of all smooth positive densities with fixed volume. Here we describe the extension of this result to manifolds with corners. In particular we obtain Moser's theorem on simplices. The proof is based on Banyaga's paper (1974), where Moser's theorem is proven for manifolds with boundary. A cohomological interpretation of Banyaga's operator is given, which allows a proof of Lefschetz duality using differential forms., 9 pages; mistakes corrected, final accepted version

Mahler functions are power series f(x) with complex coefficients for which there exist a natural number n and an integer l ≥ 2 such that f(x), f(xl),..., $$f({x^{{\ell ^{n - 1}}}}),f({x^{{\ell ^n}}})$$ are linearly dependent over ℂ(x). The study of the transcendence of their values at algebraic points was initiated by Mahler around the’ 30s and then developed by many authors. This paper is concerned with some arithmetic aspects of these functions. In particular, if f(x) satisfies f(x) = p(x)f(xl) with p(x) a polynomial with integer coefficients, we show how the behaviour of f(x) mirrors on the polynomial p(x). We also prove some general results on Mahler functions in analogy with G-functions and E-functions.

We discuss how one uses the thermodynamic formalism to produce metrics on higher Teichmüller spaces. Our higher Teichmüller spaces will be spaces of Anosov representations of a word-hyperbolic group into a semi-simple Lie group. We begin by discussing our construction in the classical setting of the Teichmüller space of a closed orientable surface of genus at least 2, then we explain the construction for Hitchin components and finally we treat the general case. This paper surveys results of Bridgeman, Canary, Labourie and Sambarino, The pressure metric for Anosov representations, and discusses questions and open problems which arise.

Je ne peux pas deposer l'article en anglais au cause du probleme avec le copyright. Je depose donc, l'original russe; International audience; The algebraic version of the Birkhoff conjecture is solved completely for billiards with a piecewiseC2-smooth boundary on surfaces of constant curvature: Euclidean plane, sphere, and Lobachevsky plane.Namely, we obtain a complete classification of billiards for which the billiard geodesic flow has a nontrivialfirst integral depending polynomially on the velocity. According to this classification, every polynomiallyintegrable convex bounded planar billiard with C2-smooth boundary is an ellipse. This is a joint result ofM. Bialy, A.E. Mironov, and the author. The proof consists of two parts. The first part was given by Bialy andMironov in their two joint papers, where the result was reduced to an algebraic-geometric problem, whichwas partially studied there. The second part is the complete solution of the algebraic-geometric problem presentedbelow.

The Arnold conjecture states that a Hamiltonian diffeomorphism of a closed and connected symplectic manifold $$(M, \omega )$$ must have at least as many fixed points as the minimal number of critical points of a smooth function on M. It is well known that the Arnold conjecture holds for Hamiltonian homeomorphisms of closed symplectic surfaces. The goal of this paper is to provide a counterexample to the Arnold conjecture for Hamiltonian homeomorphisms in dimensions four and higher. More precisely, we prove that every closed and connected symplectic manifold of dimension at least four admits a Hamiltonian homeomorphism with a single fixed point.