For any N ⩾ 1 and sufficiently small ε > 0, we find a positive solution of a nonlinear elliptic equation

when lim|x| → ∞V(x) = m > 0 and some optimal conditions on f are satisfied. Furthermore, we investigate the asymptotic behaviour of the solution as ε → 0.

We give a new bound ω < 2.37369 for the exponent of complexity of matrix multiplication, giving a small improvement on the previous bound obtained by Coppersmith and Winograd. The proof involves an extension of the method used by these authors. We have attempted to make the exposition self-contained.

The Schrödinger operator on the half-line with periodic background potential perturbed by a certain potential of Wigner—von Neumann type is considered. The asymptotics of generalized eigenvectors for λ ϵ ℂ+ and on the absolutely continuous spectrum is established. The Weyl—Titchmarsh-type formula for this operator is proven.

We establish a symmetrization procedure in the context of general orthogonal expansions associated with a second-order differential operator L, a Laplacian. Combining with a unified conjugacy scheme from an earlier paper by Nowak and Stempak permits, using a suitable embedding, a differential-difference Laplacian to be associated with the initially given orthogonal system of eigenfunctions of L, so that the resulting extended conjugacy scheme has the natural classical shape. This means, in particular, that the related partial derivatives decomposing are skew-symmetric in an appropriate L2 space and they commute with Riesz transforms and conjugate Poisson integrals. The results also shed new light on the issue of defining higher-order Riesz transforms for general orthogonal expansions.

We consider a non-homogeneous generalized Burgers equation

Here, ν is small and positive, f is strongly convex and satisfies a growth assumption, while ηω is a space-smooth random ‘kicked’ forcing term. For any solution u of this equation, we consider the quasi-stationary regime, corresponding to t ⩾ 2. After taking the ensemble average, we obtain upper estimates and time-averaged lower estimates for a class of Sobolev norms of u. These estimates are of the form Cν-β with the same values of β for bounds from above and from below. They depend on η and f, but do not depend on the time t or the initial condition.

We find the best asymptotic lower bounds for the coefficient of the leading term of the ${L}_{1} $ norm of the two-dimensional axis-parallel discrepancy that can be obtained by Roth’s orthogonal function method among a large class of test functions. We use methods of combinatorics, probability, and complex and harmonic analysis.

A new class of integrals involving the confluent hypergeometric function 1F1(a;c;z) and the Riemann Ξ-function is considered. It generalizes a class containing some integrals of Ramanujan, Hardy and Ferrar and gives, as by-products, transformation formulae of the form F(z, α) = F(iz, β), where αβ = 1. As particular examples, we derive an extended version of the general theta transformation formula and generalizations of certain formulae of Ferrar and Hardy. A one-variable generalization of a well-known identity of Ramanujan is also given. We conclude with a generalization of a conjecture due to Ramanujan, Hardy and Littlewood involving infinite series of the Möbius function.

This paper is devoted to studying the asymptotic behaviour of a sequence of elliptic systems posed in a sequence of rough domains Ωn. The solutions un are assumed to satisfy un(x) ϵ Vn(x), where Vn(x) is a vectorial space depending on . This enables one to consider several types of boundary conditions posed in variable sets of the boundary. For some choices of the vectorial spaces Vn(x), our study provides, in particular, some classical results for the homogenization of Dirichlet elliptic problems in varying domains.

We define the algebraic Dirac induction map ${\mathrm{Ind} }_{D} $ for graded affine Hecke algebras. The map ${\mathrm{Ind} }_{D} $ is a Hecke algebra analog of the explicit realization of the Baum–Connes assembly map in the $K$ -theory of the reduced ${C}^{\ast } $ -algebra of a real reductive group using Dirac operators. The definition of ${\mathrm{Ind} }_{D} $ is uniform over the parameter space of the graded affine Hecke algebra. We show that the map ${\mathrm{Ind} }_{D} $ defines an isometric isomorphism from the space of elliptic characters of the Weyl group (relative to its reflection representation) to the space of elliptic characters of the graded affine Hecke algebra. We also study a related analytically defined global elliptic Dirac operator between unitary representations of the graded affine Hecke algebra which are realized in the spaces of sections of vector bundles associated to certain representations of the pin cover of the Weyl group. In this way we realize all irreducible discrete series modules of the Hecke algebra in the kernels (and indices) of such analytic Dirac operators. This can be viewed as a graded affine Hecke algebra analog of the construction of the discrete series representations of semisimple Lie groups due to Parthasarathy and to Atiyah and Schmid.

We reduce the openness conjecture of Demailly and Kollár on the singularities of plurisubharmonic functions to a purely algebraic statement.

Pursuit-evasion games, such as the game of Revolutionaries and Spies, are a simplified model for network security. In the game we consider in this paper, a team of r revolutionaries tries to hold an unguarded meeting consisting of m revolutionaries. A team of s spies wants to prevent this forever. For given r and m, the minimum number of spies required to win on a graph G is the spy number σ(G,r,m). We present asymptotic results for the game played on random graphs G(n,p) for a large range of p = p(n), r=r(n), and m=m(n). The behaviour of the spy number is analysed completely for dense graphs (that is, graphs with average degree at least n1/2+ε for some ε > 0). For sparser graphs, some bounds are provided.

Nous démontrons, sous la forme forte conjecturée par Peyre, la conjecture de Manin pour les surfaces de Châtelet dont les équations sont du type ${y}^{2} + {z}^{2} = P(x, 1)$, où $P$ est une forme binaire quartique à coefficients entiers irréductible sur $ \mathbb{Q} [i] $ ou produit de deux formes quadratiques à coefficients entiers irréductibles sur $ \mathbb{Q} [i] $. De plus, nous fournissons une estimation explicite du terme d’erreur de la formule asymptotique sous-jacente. Cela finalise essentiellement la validation de la conjecture de Manin pour l’ensemble des surfaces de Châtelet. La preuve s’appuie sur deux méthodes nouvelles, concernant, du part, les estimations en moyenne d’oscillations locales de caractères sur les diviseurs, et, d’autre part, les majorations de certaines fonctions arithmétiques de formes binaires.