Abstract In this paper, we mainly prove a congruence conjecture of M. Apagodu [3] and a supercongruence conjecture of Z.-W. Sun [25] .

We give some arithmetic-geometric interpretations of the moments M_2[a_1], M_1[a_2], and M_1[s_2] of the Sato-Tate group of an abelian variety A defined over a number field by relating them to the ranks of the endomorphism ring and Neron-Severi group of A.

Abstract Let U 2 n denote the quasi-split unitary group over 2n variables with respect to a quadratic extension E / F of p-adic fields. In this short note, we relate GL n ( F ) -distinction of ladder representations of GL n ( E ) with irreducibility of its Siegel parabolic induction in U 2 n .

For a certain full additive subcategory X of an additive category A, one defines the lower extension groups in relative homological algebra. We show that these groups are isomorphic to the suspended Hom groups in the Verdier quotient category of the bounded homotopy category of A by that of X. Alternatively, these groups are isomorphic to the negative cohomological groups of the Hom complexes in the

It is shown that, given a representation of a quiver over a finite field, one can check in polynomial time whether it is absolutely indecomposable.

Riemann's non-differentiable function is a celebrated example of a continuous but almost nowhere differentiable function. There is strong numeric evidence that one of its complex versions represents a geometric trajectory in experiments related to the binormal flow or the vortex filament equation. In this setting, we analyse certain geometric properties of its image in $\mathbb{C}$. The objective of

In this paper we give necessary and sufficient conditions on the compatibility of a $k$th order homogeneous linear elliptic differential operator $\mathbb{A}$ and differential constraint $\mathcal{C}$ for solutions of \begin{align*} \mathbb{A} u=f\quad\text{subject to}\quad \mathcal{C} f=0\quad\text{ in }\mathbb{R}^n \end{align*} to satisfy the estimates \begin{align*} \|D^{k-j}u\|_{L^{\frac{n}{n-

Abstract In this note, we introduce a class of maximal plurisubharmonic functions and use that class to prove some properties of maximal plurisubharmonics functions.

Abstract In this work, we introduce a two-dimensional domain predator-prey model with strong Allee effect and investigate the Turing instability and the phenomena of the emergence of patterns. The occurrence of the Turing instability is ensured by the conditions that are procured by using the stability analysis of local equilibrium points. The amplitude equations (for supercritical case cubic Stuart–Landau

Abstract We study the integral equation u ( x ) = ∫ R n u p ( y ) | x − y | n − α ∫ R n u q ( z ) | y − z | n − β d z d y , x ∈ R n , where 0 α , β n and p + q = n + α + 2 β n − α . We prove that all positive L 2 n n − α ( R n ) solutions to the equation are radially symmetric and monotone decreasing about some point, and we classify all such solutions when p + 1 = q = n + β n − α . As a consequence

Symmetric varieties are normal equivariant open embeddings of symmetric homogeneous spaces and they are interesting examples of spherical varieties. The principal goal of this article is to study the rigidity under Kahler deformations of smooth projective symmetric varieties with Picard number one.

Resume Dans une Note anterieure, le premier auteur a donne une formule locale explicite pour les integrales orbitales semi-simples associees au Casimir. Dans cette Note, nous etendons cette formule a tous les elements du centre de l'algebre enveloppante de l'algebre de Lie consideree.

Abstract We first establish, through a Berry–Esseen-type bound, the asymptotic normality of a local linear estimate of the regression function in a fixed design setting when the errors are stationary isotropic spatial random fields. On the other hand, we investigate the weak convergence of an empirical estimate of the variance of these errors in a general α-mixing setting.

Abstract We consider the problem of the nonparametric estimation in a functional regression model Y = r ( X ) + e , with Y a real random variable response and X representing a functional variable taking values in a semi-metric space. The aim of this note is to find conditions of admissibility of Stein-type estimators of such a model under a class of balanced loss functions. Our method is to compare

Abstract We establish the continuity of a surface as a function of its first two fundamental forms for several Frechet topologies, which include in particular those of the space W loc 1 , p for the first fundamental form and of the space L loc p for the second fundamental form, for any p > 2 .

Abstract In this paper, we present two characterizations of the sequences of Kummer hypergeometric polynomials B a , b , n ( x ) and Kummer hypergeometric polynomials of the second kind K a , b , n ( x ) , which are respectively defined by the exponential generating functions: e x t M ( a , a + b ; t ) = ∑ n = 0 ∞ B a , b , n ( x ) t n n ! and e x t U ( a , a + b ; t ) = ∑ n = 0 ∞ K a , b , n ( x )

Abstract We prove that any q-automatic completely multiplicative function f : N → C essentially coincides with a Dirichlet character. This answers a question of J.-P. Allouche and L. Goldmakher and confirms a conjecture of J. Bell, N. Bruin and M. Coons for completely multiplicative functions. Further, assuming GRH, the methods allow us to replace completely multiplicative functions with multiplicative

Abstract In this paper, we establish some combinatorial identities involving harmonic numbers via the package Sigma , by which we confirm some conjectural congruences of Z.-W. Sun. For example, for any prime p > 3 , we have ∑ k = 0 ( p − 3 ) / 2 ( 2 k k ) 2 ( 2 k + 1 ) 16 k H k ( 2 ) ≡ − 7 B p − 3 ( mod p ) , ∑ k = 1 p − 1 ( 2 k k ) 2 k 16 k H 2 k ( 2 ) ≡ B p − 3 ( mod p ) , ∑ k = 1 ( p − 1 ) / 2 (

Abstract In this note, we study a sum–product estimate over matrix rings M n ( F q ) . More precisely, for A ⊂ M n ( F q ) , we have • if | A ∩ G L n ( F q ) | ≤ | A | / 2 , then max ⁡ { | A + A | , | A A | } ≫ min ⁡ { | A | q , | A | 3 q 2 n 2 − 2 n } ; • if | A ∩ G L n ( F q ) | ≥ | A | / 2 , then max ⁡ { | A + A | , | A A | } ≫ min ⁡ { | A | 2 3 q n 2 3 , | A | 3 / 2 q n 2 2 − 1 4 } . We also will

Abstract Proportions of zeros in character tables of finite groups are dense in [ 0 , 1 ] .

Abstract Let T be a positive plurisubharmonic (psh for short) current of bidegree ( k , k ) on a neighborhood Ω of 0 in C N = C n × C m ( n = N − m ⩾ k ), B be a Borel subset of L : = { 0 } × C m such that B ⋐ Ω . Taking ( z , t ) ∈ C n × C m , we define a C 2 positive semi-exhaustive psh function on Ω, ( z , t ) ↦ φ ( z ) , such that log ⁡ φ is also psh on the open set { φ > 0 } and consider ( z

In this article, we establish the existence of a family of hypersurfaces $(\Gamma (t))_{0< t \leq T}$ which evolve by the vanishing mean curvature flow in Minkowski space and which as $t$ tends to~$0$ blow up towards a hypersurface which behaves like the Simons cone at infinity. This issue amounts to investigate the singularity formation for a second order quasilinear wave equation. Our constructive

In this paper we consider the inverse scattering problem for the Schr{o}dinger operator with short-range electric potential. We prove in dimension n $\geq$ 2 that the knowledge of the scattering operator determines the electric potential and we establish H{o}lder-type stability in determining the short range electric potential.

Abstract Let T ≡ ( T 1 , ⋯ , T n ) be a commuting n-tuple of operators on a Hilbert space H , and let T i ≡ V i P ( 1 ≤ i ≤ n ) be its canonical joint polar decomposition (i.e. P : = T 1 ⁎ T 1 + ⋯ + T n ⁎ T n , ( V 1 , ⋯ , V n ) a joint partial isometry, and ⋂ i = 1 n ker ⁡ T i = ⋂ i = 1 n ker ⁡ V i = ker ⁡ P ). The spherical Aluthge transform of T is the (necessarily commuting) n-tuple T ˆ : = ( P

Abstract We establish a version of a localization formula for equivariant η-invariants by combining an extension of Goette's result on the comparison of two types of equivariant η-invariants and a localization formula in differential K-theory for S 1 -actions. An important step is to construct a pre-λ-ring structure in differential K-theory.

Abstract Let F be the category of functors that send a finite-dimensional vector space over F 2 to a vector space over F 2 . In this note, we describe the first extension groups between some exponential functors such as Ext F 1 ( S ⁎ , Λ ⁎ ) , Ext F 1 ( S 4 ⁎ , Λ ⁎ ) , and Ext F 1 ( S 4 ⁎ , S 4 ⁎ ) , where S ⁎ , Λ ⁎ , S 4 ⁎ are the symmetric power, the exterior power, and the truncated symmetric power

Abstract For a semisimple Lie group G satisfying the equal rank condition, the most basic family of unitary irreducible representations is the discrete series found by Harish-Chandra. In this paper, we study some of the branching laws for these when restricted to a subgroup H of the same type by combining the classical results with the recent work of T. Kobayashi. We analyze aspects of having differential

In this note, we propose a new relative entropy combination of the methods developed by P.--E. Jabin and Z.~Wang [Inventiones (2018)] and by S. Serfaty [Proc. Int. Cong. of Math, (2018) and references therein] to treat more general kernels in mean field limit theory. This new relative entropy may be understood as introducing appropriate weights in the relative entropy developed by P.-E. Jabin and Z

Abstract Bostan and Namah (Remarks on bounded solutions of steady Hamilton–Jacobi equations, C. R. Acad. Sci. Paris, Ser. I 347(15–16) (2009) 873–878) proved that constant functions are the only bounded solutions to H ( D u ) = H ( 0 ) when H is superlinear and strictly convex. In this short note, we present a proof other than that of Bostan and Namah for equations that can be easily applied to some

Abstract Let V 5 be the del Pezzo 3-fold defined by the 6-dimensional linear section of the Grassmannian variety Gr ( 2 , 5 ) under the Plucker embedding. In this paper, we present an explicit birational relation of compactifications of degree-two rational curves (i.e. conics) in V 5 . By a product, we obtain the virtual Poincare polynomial of compactified moduli spaces.

Resume En considerant des homotopies preservant la stratification, on obtient une notion naturelle d'homotopie pour les espaces stratifies. Dans cette note, on presente des invariants d'homotopie stratifiee, les groupes d'homotopie stratifies. On montre que ces groupes d'homotopie stratifies verifient un analogue stratifie au theoreme de Whitehead. Comme illustration, on presente un invariant de nœud

Abstract The coset diagram for each orbit under the action of the modular group on Q ( n ) ⁎ = Q ( n ) ∪ { ∞ } contains a circuit C i . For any α ∈ Q ( n ) , the path leading to the circuit C i and the circuit itself are obtained through continued fractions in this paper. We show that the structure of the continued fractions of a reduced quadratic irrational element is weaved with the structure or

Abstract The aim of this paper is to give some sufficient conditions, called generalized good-λ conditions, to obtain the weighted Lorentz comparisons between two measurable functions. Moreover, we also present several applications of these results for the gradient estimates of solutions to quasilinear elliptic problems in weighted Lorentz spaces.

Abstract We study the terminate distribution of a martingale whose square function is bounded. We obtain sharp estimates for the exponential and p-moments, as well as for the distribution function itself. The proofs are based on the elaboration of the Burkholder method and on the investigation of certain locally concave functions.

We present a short proof of the Alexandrov-Fenchel inequalities for mixed volumes of convex bodies.

Abstract We give a new integral formula for the centro-projective area of a convex body, which was first defined by Berck–Bernig–Vernicos. We then use the formula to prove that it is bounded from above by the centro-projective area of an ellipsoid and that equality occurs if and only if the convex set is an ellipsoid.

Abstract We explicitly compute the homotopy groups of the topological spaces B Sp ( Z ) + , B O ∞ , ∞ ( Z ) + , and B O ∞ ( Z ) + .

Abstract Let G be a second countable locally compact group with type-I left regular representation, G ˆ its dual and K = ( K π ) π ∈ G ˆ a specific measurable field of operators. In this paper, we investigate an inversion formula for L p ( G ) . Let 1 p , r ≤ 2 , 1 p + 1 q = 1 s + 1 r = 1 , and F p : L p ∩ L 1 ( G ) ⟶ L q ( G ˆ ) be defined by F p ( f ) π = π ( f ) K π 1 q . The map F p extends uniquely

Abstract For a pair of conjugate trigonometrical polynomials C ( t ) = ∑ j = 1 N a j cos ⁡ j t , S ( t ) = ∑ j = 1 N a j sin ⁡ j t with real coefficients and normalization a 1 = 1 the following extremal value is found: sup a 2 , … , a N ⁡ min t ⁡ { C ( t ) : S ( t ) = 0 } = − 1 4 sec 2 ⁡ π N + 2 . An application of this result in geometric complex analysis is shown. Several conjectures for a number

Abstract Both optimal transport and minimal surfaces have received much attention in recent years. We show that the methodology introduced by Kantorovich on the Monge problem can, surprisingly, be adapted to questions involving least area, e.g., Plateau-type problems as investigated by Federer.

We show that on every ${\sf RCD}$ spaces it is possible to introduce, by a distributional-like approach, a Riemann curvature tensor. Since after the works of Petrunin and Zhang-Zhu we know that finite dimensional Alexandrov spaces are ${\sf RCD}$ spaces, our construction applies in particular to the Alexandrov setting. We conjecture that an ${\sf RCD}$ space is Alexandrov if and only if the sectional

We prove that for a given flat surface with conical singularities, any pair of geometric triangulations can be connected by a chain of flips.

Abstract In this article, we prove that compact simple Lie groups SO ( n ) ( n > 12 ) admit at least two left-invariant Einstein metrics that are not geodesic orbit, which gives a positive answer to a problem recently posed by Nikonorov.

In this note, we investigate the continuity in law with respect to the Hurst index of the exponential functional of the fractional Brownian motion. Based on the techniques of Malliavin's calculus, we provide an explicit bound on the Kolmogorov distance between two functionals with different Hurst indexes.

By using a simple method based on the fractional integration by parts, we prove the existence and the Besov regularity of the density for solutions to stochastic differential equations driven by an additive Gaussian Volterra process. We assume weak regularity conditions on the drift. Several examples of Gaussian Volterra noises are discussed.

Abstract We present a filter-based approach to computing artificial viscosities for spectral element methods. A number of applications for this approach are presented.

Abstract Assuming the abc conjecture, Silverman proved that, for any given positive integer a ⩾ 2 , there are ≫ log ⁡ x primes p ⩽ x such that a p − 1 ≢ 1 ( mod p 2 ) . In this paper, we show that, for any given integers a ⩾ 2 and k ⩾ 2 , there still are ≫ log ⁡ x primes p ⩽ x satisfying a p − 1 ≢ 1 ( mod p 2 ) and p ≡ 1 ( mod k ) , under the assumption of the abc conjecture. This improves a recent

Abstract We prove well-posedness and regularity results for elliptic boundary value problems on certain singular domains that are conformally equivalent to manifolds with boundary and bounded geometry. Our assumptions are satisfied by the domains with a smooth set of singular cuspidal points, and hence our results apply to the class of domains with isolated oscillating conical singularities. In particular

Abstract The purpose of this note is to investigate the stabilization of a system of two wave equations coupled by velocities with only one localized damping. The main novelty in this note is that the waves are not necessarily propagating at same speed and the coupling coefficient is not assumed to be positive and small. Assume that the coupling region and the damping region intersect. We prove that

Abstract This paper deals with the chemotaxis-growth system: u t = Δ u − ∇ ⋅ ( u ∇ v ) + μ u ( 1 − u ) , v t = Δ v − v + w , τ w t + δ w = u in a smooth bounded domain Ω ⊂ R 3 with zero-flux boundary conditions, where μ, δ, and τ are given positive parameters. It is shown that the solution ( u , v , w ) exponentially stabilizes to the constant stationary solution ( 1 , 1 δ , 1 δ ) in the norm of L

In this paper, we extend and complete the classification of the generic singularities of the 3D-contact sub-Riemmanian conjugate locus in a neighbourhood of the origin.

Abstract We characterize homotopy classes of rational proper holomorphic Shilov maps from the unit disc to bounded symmetric domains of type I through rational proper holomorphic Shilov discs. This characterization generalizes results of D'Angelo–Huo–Xiao and D'Angelo–Lebl, where the codomains are the unit balls.

Abstract We correct a mistake that occurred in the proof of the main theorem of “Singular Hochschild cohomology via singularity categories” and some inaccuracies in the proof of the reconstruction theorem.

Abstract In this Note, we provide exponential inequalities for suprema of empirical processes with heavy tails on the left. Our approach is based on a martingale decomposition, associated with comparison inequalities over a cone of convex functions originally introduced by Pinelis. Furthermore, the constants in our inequalities are explicit.

This paper presents two new approaches for finding the homogenized coefficients of multiscale elliptic PDEs. Standard approaches for computing the homogenized coefficients suffer from the so-called resonance error, originating from a mismatch between the true and the computational boundary conditions. Our new methods, based on solutions of parabolic and elliptic cell-problems, result in an exponential

Abstract In this brief note, we introduce a non-symmetric mixed finite element formulation for Brinkman equations written in terms of velocity, vorticity, and pressure with non-constant viscosity. The analysis is performed by the classical Babuska–Brezzi theory, and we state that any inf–sup stable finite element pair for Stokes approximating velocity and pressure can be coupled with a generic discrete

Abstract In this Note, we formulate a sparse Krylov-based algorithm for solving large-scale linear systems of algebraic equations arising from the discretization of randomly parametrized (or stochastic) elliptic partial differential equations (SPDEs). We analyze the proposed sparse conjugate gradient (CG) algorithm within the framework of inexact Krylov subspace methods, prove its convergence and study

Resume Nous developpons une methode de type bissection pour calculer la distance a l'instabilite de polynomes matriciels quadratiques. Le calcul prend en compte les erreurs d'arrondi.

Abstract We consider the solution to a transmission problem at a thin layer interface of thickness e > 0 in a mechanical structure. We build a multi-scale expansion for that solution as e → 0 , which enables to replace the thin layer with an improved boundary condition and leads to optimal estimates for the remainders. This short note presents new results when a Dirichlet condition is imposed on the

Abstract A group G is said to be ( 2 , 3 ) -generated if it can be generated by an involution x and an element y of order three. For G a sporadic simple group, it was proved by the third author Woldar (1989) [26] that G is ( 2 , 3 ) -generated if and only if G ∉ { M 11 , M 22 , M 23 , M c L } . In this paper, we investigate all possible ( 2 , 3 ) -generations of Fischer's largest sporadic simple group