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Amdeberhan's conjectures on the enumeration, the average size, and the largest size of $(n,n+1)$-core partitions with distinct parts have motivated many research on this topic. Recently, Straub and Nath-Sellers obtained formulas for the numbers of $(n, dn-1)$ and $(n, dn+1)$-core partitions with distinct parts, respectively. Let $X_{s,t}$ be the size of a uniform random $(s,t)$-core partition with distinct parts when $s$ and $t$ are coprime to each other. Some explicit formulas for the $k$-th moments $\mathbb{E} [X_{n,n+1}^k]$ and $\mathbb{E} [X_{2n+1,2n+3}^k]$ were given by Zaleski and Zeilberger when $k$ is small. Zaleski also studied the expectation and higher moments of $X_{n,dn-1}$ and conjectured some polynomiality properties concerning them in arXiv:1702.05634. Motivated by the above works, we derive several polynomiality results and asymptotic formulas for the $k$-th moments of $X_{n,dn+1}$ and $X_{n,dn-1}$ in this paper, by studying the beta sets of core partitions. In particular, we show that these $k$-th moments are asymptotically some polynomials of n with degrees at most $2k$, when $d$ is given and $n$ tends to infinity. Moreover, when $d=1$, we derive that the $k$-th moment $\mathbb{E} [X_{n,n+1}^k]$ of $X_{n,n+1}$ is asymptotically equal to $\left(n^2/10\right)^k$ when $n$ tends to infinity. The explicit formulas for the expectations $\mathbb{E} [X_{n,dn+1}]$ and $\mathbb{E} [X_{n,dn-1}]$ are also given. The $(n,dn-1)$-core case in our results proves several conjectures of Zaleski on the polynomiality of the expectation and higher moments of $X_{n,dn-1}$., Comment: This paper has been accepted for publication in SCIENCE CHINA Mathematics

International audience; Let A * denote the free monoid generated by a finite nonempty set A. In this paper we introduce a new measure of complexity of languages L ⊆ A * defined in terms of the semigroup structure on A *. For each L ⊆ A * , we define its cost c(L) as the infimum of all real numbers α for which there exist a language S ⊆ A * with p S (n) = O(n α) and a positive integer k with L ⊆ S k. We also define the cost dimension d c (L) as the infimum of the set of all positive integers k such that L ⊆ S k for some language S with p S (n) = O(n c(L)). We are primarily interested in languages L given by the set of factors of an infinite word x = x 0 x 1 x 2 • • • ∈ A N of zero topological entropy, in which case c(L) < +∞. We establish the following characterisation of words of linear factor complexity: Let x ∈ A N and L = Fac(x) be the set of factors of x. Then p x (n) = Θ(n) if and only c(L) = 0 and d c (L) = 2. In other words, p x (n) = O(n) if and only if Fac(x) ⊆ S 2 for some language S ⊆ A + of bounded complexity (meaning lim sup p S (n) < +∞). In general the cost of a language L reflects deeply the underlying combinatorial structure induced by the semigroup structure on A *. For example, in contrast to the above characterisation of languages generated by words of sublinear complexity, there exist non factorial languages L of complexity p L (n) = O(log n) (and hence of cost equal to 0) and of cost dimension +∞. In this paper we investigate the cost and cost dimension of languages defined by infinite words of zero topological entropy. We establish the existence of words of cost zero and finite cost dimension having arbitrarily high polynomial complexity. In contrast we also show that for each α > 2 there exist infinite words x of positive cost and of complexity p x (n) = O(n α).

In this paper, we study the quotient and “blow-up” of graph-of-groups [Formula: see text] and of their automorphisms [Formula: see text]. We show that the existence of such a blow-up of any [Formula: see text], relative to a given family of “local” graph-of-groups isomorphisms [Formula: see text] depends crucially on the [Formula: see text]-conjugacy class of the correction term [Formula: see text] for any edge [Formula: see text] of [Formula: see text], where [Formula: see text]-conjugacy is a new but natural concept introduced here. As an application, we obtain a criterion as to whether a partial Dehn twist can be blown up relative to local Dehn twists, to give an actual Dehn twist. The results of this paper are also used crucially in the follow-up papers [Lustig and Ye, Normal form and parabolic dynamics for quadratically growing automorphisms of free groups, arXiv:1705.04110v2; Ye, Partial Dehn twists of free groups relative to local Dehn twists — A dichotomy, arXiv:1605.04479 ; When is a polynomially growing automorphism of [Formula: see text] geometric, arXiv:1605.07390 ].

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 $s\ge 470.$

We prove an asymptotic formula for the mean-square average of L-functions associated with subgroups of characters of sufficiently large size. Our proof relies on the study of certain character sums ${\mathcal{A}}(p,d)$ recently introduced by E. Elma, where p ≥ 3 is prime and d ≥ 1 is any odd divisor of p − 1. We obtain an asymptotic formula for ${\mathcal{A}}(p,d),$ which holds true for any odd divisor d of p − 1, thus removing E. Elma’s restrictions on the size of d. This answers a question raised in Elma’s paper. Our proof relies on both estimates on the frequency of large character sums and techniques from the theory of uniform distribution. As an application, in the range $1\leq d\leq\frac{\log p}{3\log\log p}$, we obtain a significant improvement $h_{p,d}^- \leq 2(\frac{(1+o(1))p}{24})^{m/4}$ over the trivial bound $h_{p,d}^- \ll (\frac{dp}{24} )^{m/4}$ on the relative class numbers of the imaginary number fields of conductor $p\equiv 1\mod{2d}$ and degree $m=(p-1)/d$, where d ≥ 1 is odd.

In this paper we address the following shape optimization problem: find the planar domain of least area, among the sets with prescribed constant width and inradius. In the literature, the problem is ascribed to Bonnesen, who proposed it in \cite{BF}. In the present work, we give a complete answer to the problem, providing an explicit characterization of optimal sets for every choice of width and inradius. These optimal sets are particular Reuleaux polygons., Comment: to appear in Journal Ecole Polytechnique

A class $\mathcal{F}$ of graphs has the induced Erd\H{o}s-P\'osa property if there exists a function $f$ such that for every graph $G$ and every positive integer $k$, $G$ contains either $k$ pairwise vertex-disjoint induced subgraphs that belong to $\mathcal{F}$, or a vertex set of size at most $f(k)$ hitting all induced copies of graphs in $\mathcal{F}$. Kim and Kwon (SODA'18) showed that for a cycle $C_{\ell}$ of length $\ell$, the class of $C_{\ell}$-subdivisions has the induced Erd\H{o}s-P\'osa property if and only if $\ell\le 4$. In this paper, we investigate whether or not the class of $H$-subdivisions has the induced Erd\H{o}s-P\'osa property for other graphs $H$. We completely settle the case when $H$ is a forest or a complete bipartite graph. Regarding the general case, we identify necessary conditions on $H$ for the class of $H$-subdivisions to have the induced Erd\H{o}s-P\'osa property. For this, we provide three basic constructions that are useful to prove that the class of the subdivisions of a graph does not have the induced Erd\H{o}s-P\'osa property. Among remaining graphs, we prove that if $H$ is either the diamond, the $1$-pan, or the $2$-pan, then the class of $H$-subdivisions has the induced Erd\H{o}s-P\'osa property., Comment: 39 pages

We prove a quantitative version of the curve selection lemma. Denoting by $s,d,k$ a bound on the number, the degree and the number of variables of the polynomials describing a semi-algebraic set $S$ and a point $x$ in $\bar S$, we find a semi-algebraic path starting at $x$ and entering in $S$ with a description of degree $(O(d)^{3k+3},O(d)^{k})$ (using a precise definition of the description of a semi-algebraic path and its degree given in the paper). As a consequence, we prove that there exists a semi-algebraic path starting at $x$ and entering in $S$, such that the degree of the Zariski closure of the image of this path is bounded by $O(d)^{4k+3}$, improving a result of Jelonek and Kurdyka. We also give an algorithm for describing the real isolated points of $S$ whose complexity is bounded by $s^{2 k+1}d^{O(k)}$ improving a result of Le, Safey el Din, and de Wolff., Comment: Final version to appear in Mathematische Zeitschrift

Let f be a post-critically finite endomorphism (PCF map for short) on $${\mathbb {P}}^2$$ , let $$J_1$$ denote the Julia set and let $$J_2$$ denote the support of the measure of maximal entropy. In this paper we show that: 1. $$J_1\setminus J_2$$ is contained in the union of the (finitely many) basins of critical component cycles and stable manifolds of sporadic super-saddle cycles. 2. For every $$x\in J_2$$ which is not contained in the stable manifold of a sporadic super-saddle cycle, there is no Fatou disk containing x. Here sporadic means that the super-saddle cycle is not contained in a critical component cycle. Under the additional assumption that all branches of PC(f) are smooth and intersect transversally, we show that there is no sporadic super-saddle cycle. Thus in this case $$J_1\setminus J_2$$ is contained in the union of the basins of critical component cycles, and for every $$x\in J_2$$ there is no Fatou disk containing x. As consequences of our result: 1.We answer some questions of Fornaess-Sibony about the non-wandering set for PCF maps on $${\mathbb {P}}^2$$ with no sporadic super-saddle cycles. 2. We give a new proof of de Thelin’s laminarity of the Green currents in $$J_1\setminus J_2$$ for PCF maps on $${\mathbb {P}}^2$$ . 3. We show that for PCF maps on $${\mathbb {P}}^2$$ an invariant compact set is expanding if and only if it does not contain critical points, and we obtain characterizations of PCF maps on $${\mathbb {P}}^2$$ which are expanding on $$J_2$$ or satisfy Axiom A.

In this paper, we present diverse new metric properties that prox-regular sets shared with convex ones. At the heart of our work lie the Legendre-Fenchel transform and complements of balls. First, we show that a connected prox-regular set is completely determined by the Legendre-Fenchel transform of a suitable perturbation of its indicator function. Then, we prove that such a function is also the right tool to extend, to the context of prox-regular sets, the famous connection between the distance function and the support function of a convex set. On the other hand, given a prox-regular set, we examine the intersection of complements of open balls containing the set. We establish that the distance of a point to a prox-regular set is the maximum of the distances of the point from boundaries of all such complements separating the set and the point. This is in the line of the known result expressing the distance from a convex set in terms of separating hyperplanes. To the best of our knowledge, these results are new in the literature and show that the class of prox-regular sets have good properties known in convex analysis. Mathematics Subject Classification (2010) 49J52 · 49J53

Let $${ \rm I\!K}$$ be an ultrametric complete valued field and $${ \rm I\!E}$$ be an ultrametric space. We examine some Banach algebras $$S$$ of bounded continuous functions from $${ \rm I\!E}$$ to $${ \rm I\!K}$$ with the use of ultrafilters, particularly the relation of stickness. We recall and deepen results obtained in a previous paper by N. Mainetti and the third author concerning the whole algebra $$\cal A$$ of all bounded continuous functions from $${ \rm I\!E}$$ to $${ \rm I\!K}$$ . Every maximal ideal of finite codimension of $$\cal A$$ is of codimension $$1$$ and we show that this property also holds for every algebra $$S$$ , provided $${ \rm I\!K}$$ is perfect. If $$S$$ admits the uniform norm on $${ \rm I\!E}$$ as its spectral norm, then every maximal ideal is the kernel of only one multiplicative semi-norm, the Shilov boundary is equal to the whole multiplicative spectrum and the Banaschewski compactification of $${ \rm I\!E}$$ is homeomorphic to the multiplicative spectrum of $$S$$ .

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.

Consider the sequence $$ \{{F}_{n}\}_{n \ge 0} $${Fn}n≥0 of Fibonacci numbers defined by $${F}_0=0$$F0=0, $${F}_1 =1$$F1=1, and $${F}_{{n}+2}= {F}_{{n}+1}+ {F}_{n} $$Fn+2=Fn+1+Fn for all $$ n\ge 0 $$n≥0. In this paper, we find all integers c having at least two representations as a difference between a Fibonacci number and a power of 3.

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

This paper introduces a new method to solve the problem of the approximation of the diagonal for face-coherent families of polytopes. We recover the classical cases of the simplices and the cubes and we solve it for the associahedra, also known as Stasheff polytopes. We show that it satisfies an easy-to-state cellular formula. For the first time, we endow a family of realizations of the associahedra (the Loday realizations) with a topological and cellular operad structure; it is shown to be compatible with the diagonal maps., Comment: 19 pages, 7 figures. Comments are welcome

In the $r$-neighbour bootstrap process on a graph $G$, vertices are infected (in each time step) if they have at least $r$ already-infected neighbours. Motivated by its close connections to models from statistical physics, such as the Ising model of ferromagnetism, and kinetically constrained spin models of the liquid-glass transition, the most extensively-studied case is the two-neighbour bootstrap process on the two-dimensional grid $[n]^2$. Around 15 years ago, in a major breakthrough, Holroyd determined the sharp threshold for percolation in this model, and his bounds were subsequently sharpened further by Gravner and Holroyd, and by Gravner, Holroyd and Morris. In this paper we strengthen the lower bound of Gravner, Holroyd and Morris by proving that the critical probability $p_c\big( [n]^2,2 \big)$ for percolation in the two-neighbour model on $[n]^2$ satisfies \[p_c\big( [n]^2,2 \big) = \frac{\pi^2}{18\log n} - \frac{\Theta(1)}{(\log n)^{3/2}}\,.\] The proof of this result requires a very precise understanding of the typical growth of a critical droplet, and involves a number of technical innovations. We expect these to have other applications, for example, to the study of more general two-dimensional cellular automata, and to the $r$-neighbour process in higher dimensions., Comment: 53 pages, 6 figures, 1 appendix

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/

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 $$\mathbb{R}^{2}_{+}$$.

In this paper, we study the behaviour of the shortest distance between orbits and show that under some rapidly mixing conditions, the decay of the shortest distance depends on the correlation dimension. For irrational rotations, we prove a different behaviour depending on the irrational exponent of the angle of the rotation. For random processes, this problem corresponds to the longest common substring problem. We extend the result of Arratia and Waterman on sequence matching to $\alpha$-mixing processes with exponential decay., Comment: Final version. Accepted for publication in Advances in Mathematics

In set theory without the axiom of Choice ZF, we prove that for every commutative field IK, the following statement D_{\IK}: "On every non null IK-vector space, there exists a non null linear form" implies the existence of a IK-linear extender on every vector subspace of a $\IK$-vector space. This solves a question raised in \cite{Mo09}. In the second part of the paper, we generalize our results in the case of spherically complete ultrametric valued fields, and show that Ingleton's statement is equivalent to the existence of "isometric linear extenders".

The infinite pigeonhole principle for 2-partitions asserts the existence, for every set $A$, of an infinite subset of $A$ or of its complement. In this paper, we develop a new notion of forcing enabling a fine analysis of the computability-theoretic features of the pigeonhole principle. We deduce various consequences, such as the existence, for every set $A$, of an infinite subset of it or its complement of non-high degree. We also prove that every $\Delta^0_3$ set has an infinite low${}_3$ solution and give a simpler proof of Liu's theorem that every set has an infinite subset in it or its complement of non-PA degree., Comment: 20 pages

The first two authors [Proc. Lond. Math. Soc. (3) {\bf 114}(1):1--34, 2017] classified the behaviour near zero for all positive solutions of the perturbed elliptic equation with a critical Hardy--Sobolev growth $$-\Delta u=|x|^{-s} u^{2^\star(s)-1} -\mu u^q \hbox{ in }B\setminus\{0\},$$ where $B$ denotes the open unit ball centred at $0$ in $\mathbb{R}^n$ for $n\geq 3$, $s\in (0,2)$, $2^\star(s):=2(n-s)/(n-2)$, $\mu>0$ and $q>1$. For $q\in (1,2^\star-1)$ with $2^\star=2n/(n-2)$, it was shown in the op. cit. that the positive solutions with a non-removable singularity at $0$ could exhibit up to three different singular profiles, although their existence was left open. In the present paper, we settle this question for all three singular profiles in the maximal possible range. As an important novelty for $\mu>0$, we prove that for every $q\in (2^\star(s) -1,2^\star-1)$ there exist infinitely many positive solutions satisfying $|x|^{s/(q-2^\star(s)+1)}u(x)\to \mu^{-1/(q-2^\star(s)+1)}$ as $|x|\to 0$, using a dynamical system approach. Moreover, we show that there exists a positive singular solution with $\liminf_{|x|\to 0} |x|^{(n-2)/2} u(x)=0$ and $\limsup_{|x|\to 0} |x|^{(n-2)/2} u(x)\in (0,\infty)$ if (and only if) $q\in (2^\star-2,2^\star-1)$., Comment: Mathematische Annalen, to appear

In 1994, Becker conjectured that if $F(z)$ is a $k$-regular power series, then there exists a $k$-regular rational function $R(z)$ such that $F(z)/R(z)$ satisfies a Mahler-type functional equation with polynomial coefficients where the initial coefficient satisfies $a_0(z)=1$. In this paper, we prove Becker's conjecture in the best-possible form; we show that the rational function $R(z)$ can be taken to be a polynomial $z^\gamma Q(z)$ for some explicit non-negative integer $\gamma$ and such that $1/Q(z)$ is $k$-regular., Comment: 19 pages

The Maker-Breaker domination game is played on a graph $G$ by Dominator and Staller. The players alternatively select a vertex of $G$ that was not yet chosen in the course of the game. Dominator wins if at some point the vertices he has chosen form a dominating set. Staller wins if Dominator cannot form a dominating set. In this paper we introduce the Maker-Breaker domination number $\gamma_{{\rm MB}}(G)$ of $G$ as the minimum number of moves of Dominator to win the game provided that he has a winning strategy and is the first to play. If Staller plays first, then the corresponding invariant is denoted $\gamma_{{\rm MB}}'(G)$. Comparing the two invariants it turns out that they behave much differently than the related game domination numbers. The invariant $\gamma_{{\rm MB}}(G)$ is also compared with the domination number. Using the Erd\H{o}s-Selfridge Criterion a large class of graphs $G$ is found for which $\gamma_{{\rm MB}}(G) > \gamma(G)$ holds. Residual graphs are introduced and used to bound/determine $\gamma_{{\rm MB}}(G)$ and $\gamma_{{\rm MB}}'(G)$. Using residual graphs, $\gamma_{{\rm MB}}(T)$ and $\gamma_{{\rm MB}}'(T)$ are determined for an arbitrary tree. The invariants are also obtained for cycles and bounded for union of graphs. A list of open problems and directions for further investigations is given., Comment: 20 pages, 5 figures

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 $K$.

Pre-print of a publication in "Annales math\'ematiques du Qu{\'e}bec". Let $k$ be a totally real number field and let $k_\infty$ be its cyclotomic $\mathbb{Z}_p$-extension for $p$ totally split in $k$. This text completes our article entitled: "Approche $p$-adique de la conjecture de Greenberg pour les corps totalement r\'eels" (Annales Math\'ematiques Blaise Pascal 2017), by means of heuristics on the $p$-adic behavior of the norms, in $k_n/k$, of the ideals in $k_\infty$ ; indeed, this conjecture (on the nullity of the invariants $\lambda$ et $\mu$ of Iwasawa) depends of images in the torsion group ${\mathcal T}_k$ of the Galois group of the maximal abelian $p$-ramified pro-$p$-extension of $k$, thus of Artin symbols in a finite extension $F/k$ obtained by Galois descent of ${\mathcal T}_k$. An assumption of distribution of these norms implies $\lambda=\mu=0$. Several statistics and numerical examples in the quadratic case confirm the probable exactness of such properties which constitute the fundamental obstruction for a proof of Greenberg's conjecture in the sole context of Iwasawa's theory., Comment: in French, Completes with numerical computations and heuristics our previous paper arXiv:1611.09592 on Greenberg's conjecture. Annales math{\'e}matiques du Quebec, A para{\^i}tre

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$\unicode[STIX]{x1D702}\in (\frac{32}{17},2.1426\cdots \,)$.

In this paper, we provide refined sufficient conditions for the quadratic Chabauty method to produce a finite set of points, with the conditions on the rank of the Jacobian replaced by conditions on the rank of a quotient of the Jacobian plus an associated space of Chow-Heegner points. We then apply this condition to prove the finiteness of this set for any modular curves $X_{\mathrm{ns} }^+ (N)$ and $X_0 ^+ (N)$ of genus at least 2 with N prime. The proof relies on the existence of a quotient of their Jacobians whose Mordell-Weil rank is equal to its dimension (and at least 2), which is proven via analytic estimates for orders of vanishing of L-functions of modular forms, thanks to a Kolyvagin-Logachev type result., Comment: 51 pages, comments welcome

This paper deals with graph classes characterization and recognition. A popular way to characterize a graph class is to list a minimal set of forbidden induced subgraphs. Unfortunately this strategy usually does not lead to an efficient recognition algorithm. On the other hand, many graph classes can be efficiently recognized by techniques based on some interesting orderings of the nodes, such as the ones given by traversals. We study specifically graph classes that have an ordering avoiding some ordered structures. More precisely, we consider what we call patterns on three nodes, and the recognition complexity of the associated classes. In this domain, there are two key previous works. Damashke started the study of the classes defined by forbidden patterns, a set that contains interval, chordal and bipartite graphs among others. On the algorithmic side, Hell, Mohar and Rafiey proved that any class defined by a set of forbidden patterns can be recognized in polynomial time. We improve on these two works, by characterizing systematically all the classes defined sets of forbidden patterns (on three nodes), and proving that among the 23 different classes (up to complementation) that we find, 21 can actually be recognized in linear time. Beyond this result, we consider that this type of characterization is very useful, leads to a rich structure of classes, and generates a lot of open questions worth investigating., Third version version. 38 pages

International audience; In this paper we define and study a filtration (G_s) on the algebra of values at algebraic points of analytic continuations of G-functions: G_s is the set of values at algebraic points in the disk of convergence of all G-functions sum_n a_n z^n for which there exist some positive integers b and c such that d_{s b}^n c^{n+1} a_n is an algebraic integer for any n, where d_n = lcm(1, 2,. .. , n). We study the situation at the boundary of the disk of convergence, and using transfer results from analysis of singularities we deduce that constants in G_s appear in the asymptotic expansion of such a sequence (a_n).

The model of FCFS infinite bipartite matching was introduced in caldentey-kaplan-weiss 2009. In this model there is a sequence of items that are chosen i.i.d. from $\mathcal{C}=\{c_1,\ldots,c_I\}$ and an independent sequence of items that are chosen i.i.d. from $\mathcal{S}=\{s_1,\ldots,s_J\}$, and a bipartite compatibility graph $G$ between $\mathcal{C}$ and $\mathcal{S}$. Items of the two sequences are matched according to the compatibility graph, and the matching is FCFS, each item in the one sequence is matched to the earliest compatible unmatched item in the other sequence. In adan-weiss 2011 a Markov chain associated with the matching was analyzed, a condition for stability was verified, a product form stationary distribution was derived and the rates $r_{c_i,s_j}$ of matches between compatible types $c_i$ and $s_j$ were calculated. In the current paper, we present several new results that unveil the fundamental structure of the model. First, we provide a pathwise Loynes' type construction which enables to prove the existence of a unique matching for the model defined over all the integers. Second, we prove that the model is dynamically reversible: we define an exchange transformation in which we interchange the positions of each matched pair, and show that the items in the resulting permuted sequences are again independent and i.i.d., and the matching between them is FCFS in reversed time. Third, we obtain product form stationary distributions of several new Markov chains associated with the model. As a by product, we compute useful performance measures, for instance the link lengths between matched items., Comment: 33 pages, 12 figures

The simple $GL(n,\mathbb {C})$ -modules are described by using semistandard Young tableaux. Any semistandard skew tableau can be transformed into a well defined semistandard tableau by a combinatorial operation, the Schutzenberger jeu de taquin. Associated to the classical Lie groups $SP(2n,\mathbb {C})$ , $SO(2n+1,\mathbb {C})$ , there are other notions of semistandard Young tableaux and jeux de taquin. In this paper, we study these various jeux de taquin, proving that each of them has a simple and explicit formulation as a step-by-step sliding. Any of these jeux de taquin is the restriction of the orthogonal one, associated to $SO(2n+1,\mathbb {C})$ .

Let X be a projective curve over $${\mathbb Q}$$ and $${t\in {\mathbb Q}(X)}$$ a non-constant rational function of degree $${n\ge 2}$$ . For every $${\tau \in {\mathbb Z}}$$ pick $${P_\tau \in X(\bar{\mathbb Q})}$$ such that $${t(P_\tau )=\tau }$$ . Dvornicich and Zannier proved that, for large N, the field $${\mathbb Q}(P_1, \ldots , P_N)$$ is of degree at least $$e^{cN/\log N}$$ over $${\mathbb Q}$$ , where $${c>0}$$ depends only on X and t. In this paper we extend this result, replacing $${\mathbb Q}$$ by an arbitrary number field.

Let Ω be a bounded and connected open subset of ℝ N with a Lipschitz-continuous boundary, the set Ω being locally on the same side of ∂Ω. A vector version of a fundamental lemma of J. L. Lions, due to C. Amrouche, the first author, L. Gratie and S. Kesavan, asserts that any vector field v = (ui) ∈ (D′(Ω)) N , such that all the components $$\frac{1}{2}({\partial _j}{v_i} + {\partial _i}{v_j})$$ , 1 ≤ i, j ≤ N, of its symmetrized gradient matrix field are in the space H−1(Ω), is in effect in the space (L2(Ω)) N . The objective of this paper is to show that this vector version of J. L. Lions lemma is equivalent to a certain number of other properties of interest by themselves. These include in particular a vector version of a well-known inequality due to J. Necas, weak versions of the classical Donati and Saint-Venant compatibility conditions for a matrix field to be the symmetrized gradient matrix field of a vector field, or a natural vector version of a fundamental surjectivity property of the divergence operator.

Let G be a graph of order n with minimum degree $$\delta (G) \geqslant \tfrac{n} {2} + 1$$ . Faudree and Li (2012) conjectured that for any pair of vertices x and y in G and any integer $$2 \leqslant k \leqslant \tfrac{n} {2}$$ , there exists a Hamiltonian cycle C such that the distance between x and y on C is k. In this paper, we prove that this conjecture is true for graphs of sufficiently large order. The main tools of our proof are the Regularity lemma of Szemeredi and the Blow-up lemma of Komlos et al. (1997).

International audience; Let S ⊆ N be a numerical semigroup with multiplicity m = min(S \ {0}), conductor c = max(N \ S) + 1 and minimally generated by e elements. Let L be the set of elements of S which are smaller than c. Wilf conjectured in 1978 that |L| is bounded below by c/e. We show here that if c ≤ 3m, then S satisfies Wilf's conjecture. Combined with a recent result of Zhai, this implies that the conjecture is asymptotically true as the genus g(S) = |N \ S| goes to infinity. One main tool in this paper is a classical theorem of Macaulay on the growth of Hilbert functions of standard graded algebras.

Let X be a jump-diffusion process and X* its running supremum. In this paper we first show that for any t > 0, the law of the pair (X*t, Xt) has a density with respect to the Lebesgue measure. This allows us to show that for any t > 0, the law of the pair formed by the random variable Xt and the running supremum X*t of X at time t can be characterized as a weak solution of a partial differential equation concerning the distribution of the pair (X*t, Xt). Then we obtain an expression of the marginal density of X*t for all t > 0.

In a previous work, we stated a conjecture, called the Galois Brumer-Stark conjecture, that generalizes the (abelian) Brumer-Stark conjecture to Galois extensions. Other generalizations of the Brumer-Stark conjecture to non-abelian Galois extensions are due to Nickel. Nomura proved that the Brumer-Stark conjecture implies the weak non-abelian Brumer-Stark conjecture of Nickel when the group is monomial. In this paper, we use the methods of Nomura to prove that the Brumer-Stark conjecture implies the Galois Brumer-Stark conjecture for monomial groups in the semi-simple case.

International audience; We consider small polynomial deformations of integrable systems of the form $dF=0, F\in\mathbb{C}[x,y]$ and the first nonzero term $M_\mu$ of the displacement function $\Delta(t,\epsilon)=\sum_{i=\mu}M_i(t)\epsilon^i$ along a cycle $\gamma(t)\in F^{-1}(t)$. It is known that $M_\mu$ is an iterated integral of length at most $\mu$. The bound $\mu$ epends on the deformation of $dF$.In this paper we give a universal bound for the length of the iterated integral expressing the first nonzero term $\mu$ depending only on the geometry of the unperturbed system $dF=0$. The result generalizes the result of Gavrilov and Iliev providing a sufficient condition for $M_\mu$ to be given by an abelian integral, i.e., by an iterated integral of length 1. We conjecture that our bound is optimal.

In the present paper, we introduce a new approach, relying on the Galois theory of difference equations, to study the nature of the generating series of walks in the quarter plane. Using this approach, we are not only able to recover many of the recent results about these series, but also to go beyond them. For instance, we give for the first time hypertranscendency results, {\it i.e.}, we prove that certain of these generating series do not satisfy any nontrivial nonlinear algebraic differential equation with rational coefficients., To appear in Inventiones Mathematicae

As usual, denote by P(n) the largest prime factor of the integer n ⩾ 1 with the convention P(1) = 1. For 0 < θ < 1, define $${T_\theta }\left( x \right): = \left| {\left\{ {p \leqslant x:P\left( {p - 1} \right) \geqslant {p^\theta }} \right\}} \right|.$$ In this paper, we obtain a new lower bound for T θ (x) as x → ∞, which improves some recent results of Luca et al. (2015) and of Chen and Chen (2017). As a corollary, we partially prove a conjecture of Chen and Chen (2017) about the size of T θ (x).

In this paper, we explore the fixed point theory of $n$-valued maps using configuration spaces and braid groups, focussing on two fundamental problems, the Wecken property, and the computation of the Nielsen number. We show that the projective plane (resp.\ the $2$-sphere ${\mathbb S}^{2}$) has the Wecken property for $n$-valued maps for all $n\in {\mathbb N}$ (resp.\ all $n\geq 3$). In the case $n=2$ and ${\mathbb S}^{2}$, we prove a partial result about the Wecken property. We then describe the Nielsen number of a non-split $n$-valued map $\phi\colon\thinspace X \multimap X$ of an orientable, compact manifold without boundary in terms of the Nielsen coincidence numbers of a certain finite covering $q\colon\thinspace \widehat{X} \to X$ with a subset of the coordinate maps of a lift of the $n$-valued split map $\phi\circ q\colon\thinspace \widehat{X} \multimap X$., Comment: To appear in Science China Math

Let $P^+(n)$ denote the largest prime factor of the integer $n$ and $P_y^+(n)$ denote the largest prime factor $p$ of $n$ which satisfies $p\leqslant y$. In this paper, firstly we show that the triple consecutive integers with the two patterns $P^+(n-1)>P^+(n)P^+(n+1)$ have a positive proportion respectively. More generally, with the same methods we can prove that for any $J\in \mathbb{Z}, J\geqslant3$, the $J-$tuple consecutive integers with the two patterns $P^+(n+j_0)= \min\limits_{0\leqslant j\leqslant J-1}P^+(n+j)$ and $P^+(n+j_0)= \max\limits_{0\leqslant j\leqslant J-1}P^+(n+j)$ also have a positive proportion respectively. Secondly for $y=x^{\theta}$ with $0, Comment: in French

A linear directed forest is a directed graph in which every component is a directed path. The linear arboricity la(D) of a digraph D is the minimum number of linear directed forests in D whose union covers all arcs of D. For every d-regular digraph D, Nakayama and Peroche conjecture that la(D) = d + 1. In this paper, we consider the linear arboricity for complete symmetric digraphs, regular digraphs with high directed girth and random regular digraphs and we improve some well-known results. Moreover, we propose a more precise conjecture about the linear arboricity for regular digraphs.

The present paper presents a counterexample to the sequentially weak density of smooth maps between two manifolds $M$ and $N$ in the Sobolev space $W^{1, p} (M, N)$, in the case $p$ is an integer. It has been shown that, if $p, Comment: 68 pages

The elastic energy of a planar convex body is defined by $E(\Om)=\frac 12\,\int\_{\partial\Om} k^2(s)\,ds$where $k(s)$ is the curvature of the boundary. In this paper we are interested in the minimization problemof $E(\Om)$ with a constraint on the inradius of $\Om$. By contrast with all the other minimization problemsinvolving this elastic energy (with a perimeter, area, diameter or circumradius constraints) for which thesolution is always the disk, we prove here that the solution of this minimization problem is not the disk and we completely characterizeit in terms of elementary functions., Equipe EDP et Applications

For a subset $A$ of the real line $\mathbb R$, Hattori space $H(A)$ is a topological space whose underlying point set is the reals $\mathbb R$ and whose topology is defined as follows: points from $A$ are given the usual Euclidean neighborhoods while remaining points are given the neighborhoods of the Sorgenfrey line. In this paper, among other things, we give conditions on $A$ which are sufficient and necessary for $H(A)$ to be respectively almost Cech-complete, Cech-complete, quasicomplete, Cech-analytic and weakly separated (in Tkacenko sense). Some of these results solve questions raised by V.A. Chatyrko and Y. Hattori.

In this paper, the spectrum of the following fourth order problem $$\begin{aligned} {\left\{ \begin{array}{ll} \varDelta ^2 u+\nu u=-\lambda \varDelta u &{}\quad \text {in } D_1,\\ u=\partial _r u= 0 &{}\quad \text {on } \partial D_1, \end{array}\right. } \end{aligned}$$ where $$D_1$$ is the unit ball in $${\mathbb {R}}^N$$ , is determined for $$\nu < 0$$ as well as the nodal properties of the corresponding eigenfunctions. In particular, we show that the first eigenvalue is simple and that the corresponding eigenfunction is radial and (up to a multiplicative factor) positive and decreasing with respect to the radius. This completes earlier results obtained for $$\nu \geqslant 0$$ (see Coster et al. in Positivity 19:843–875, 2015) and for $$\nu

Let $$\mathcal S$$ be an abelian group of automorphisms of a probability space $$(X, {\mathcal A}, \mu )$$ with a finite system of generators $$(A_1, \ldots , A_d).$$ Let $$A^{{\underline{\ell }}}$$ denote $$A_1^{\ell _1} \ldots A_d^{\ell _d}$$ , for $${{\underline{\ell }}}= (\ell _1, \ldots , \ell _d).$$ If $$(Z_k)$$ is a random walk on $${\mathbb {Z}}^d$$ , one can study the asymptotic distribution of the sums $$\sum _{k=0}^{n-1} \, f \circ A^{\,{Z_k(\omega )}}$$ and $$\sum _{{\underline{\ell }}\in {\mathbb {Z}}^d} {\mathbb {P}}(Z_n= {\underline{\ell }}) \, A^{\underline{\ell }}f$$ , for a function f on X. In particular, given a random walk on commuting matrices in $$SL(\rho , {\mathbb {Z}})$$ or in $${\mathcal M}^*(\rho , {\mathbb {Z}})$$ acting on the torus $${\mathbb {T}}^\rho $$ , $$\rho \ge 1$$ , what is the asymptotic distribution of the associated ergodic sums along the random walk for a smooth function on $${\mathbb {T}}^\rho $$ after normalization? In this paper, we prove a central limit theorem when X is a compact abelian connected group G endowed with its Haar measure (e.g., a torus or a connected extension of a torus), $$\mathcal S$$ a totally ergodic d-dimensional group of commuting algebraic automorphisms of G and f a regular function on G. The proof is based on the cumulant method and on preliminary results on random walks.