In this paper, we give necessary and sufficient conditions on the compatibility of a kth-order homogeneous linear elliptic differential operator A and differential constraint C for solutions toAu=fsubject toCf=0 in Rn to satisfy the estimates‖Dk−ju‖Lnn−j(Rn)⩽c‖f‖L1(Rn) for j∈{1,…,min⁡{k,n−1}} and‖Dk−nu‖L∞(Rn)⩽c‖f‖L1(Rn) when k≥n.

We establish the continuity of a surface as a function of its first two fundamental forms for several Fréchet topologies, which include in particular those of the space Wloc1,p for the first fundamental form and of the space Llocp for the second fundamental form, for any p>2.

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 M2[a1], M1[a2], and M1[s2] 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 Néron–Severi group of A.

Let U2n 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 GLn(F)-distinction of ladder representations of GLn(E) with irreducibility of its Siegel parabolic induction in U2n.

For certain full additive subcategories 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 cohomology groups of the Hom complexes in the dg quotient category A/X, where both A and X are viewed as dg categories concentrated in degree zero.

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 C. The objective of this note is to assert that the Hausdorff dimension of its image is no larger than 4/3 and that it has nowhere a tangent.

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

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 equation and for subcritical quintic Stuart–Landau equation) are derived appropriate for each case by using the method of multiple time scale and show that the system supports patterns like squares, stripes, mixed-mode patterns, spots and hexagonal patterns. We obtain the asymptotic solutions to the model close to the onset instability based on the amplitude equations. Finally, numerically simulations tell how cross-diffusion plays an important role in the emergence of patterns.

We study the integral equationu(x)=∫Rnup(y)|x−y|n−α∫Rnuq(z)|y−z|n−βdzdy,x∈Rn, where 0<α,β

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 Kähler deformations of smooth projective symmetric varieties with Picard number one.

Dans une Note antérieure, le premier auteur a donné une formule locale explicite pour les intégrales orbitales semi-simples associées au Casimir. Dans cette Note, nous étendons cette formule à tous les éléments du centre de l'algèbre enveloppante de l'algèbre de Lie considérée.

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.

We consider the problem of the nonparametric estimation in a functional regression model Y=r(X)+ε, 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 the risk with that obtained in the case of a quadratic loss.

The statistical problem of bridging is closely associated with the problem of heterogeneity in dose-finding studies. There are some distinctive features in the case of bridging which need to be considered if efficient estimation of the maximum tolerated dose (MTD) is to be accomplished. The case of two distinct populations is considered. Extensions to several populations are, at least in principle, straightforward although, in practice, likely to be awkward and infrequently encountered. The goal is to make efficient use of information gained in one study in the context of a second study. Since working models are typically misspecified it is not possible to just add a further parameter to deal with an added source of variability. Le problème statistique de l’association croisée est intimement lié au problème de l’hétérogénéité dans les études de dosages. Quelques aspects spécifiques dans le cas de l’association croisée sont à considérer si une estimation efficace de la dose maximum tolérée (DMT) doit être obtenue. Le cas de deux populations distinctes est envisagé. Les extensions au cas de plusieurs populations sont, au moins en principe, directes, mais en pratique vraisemblablement gauches et peu fréquentes. Le but est d’utiliser de façon efficace l’information tirée d’une premiére étude dans le contexte d’une seconde. Comme les modèles sont en général mal spécifiés, il n’est pas possible de se contenter d’ajouter un paramètre pour traiter une nouvelle source d’aléa.