On the Influence of the Lowest Terms on Dirichlet—Type Problems for Elliptic Systems Degenerate at a Point

We consider Dirichlet–type problems for weakly connected systems of elliptic equations of second order with a strong degeneracy at an inner point of the domain, when, in a neighborhood of this point, the asymptotics of a solution is additionally given. The form of the given asymptotics essentially depends on the properties of the coefficients at the first–order derivatives. We prove the existence and uniqueness of solutions of the problems considered in Hölder function classes.     

On the Dirichlet Problem with an Asymptotic Condition for an Elliptic System Strongly Degenerate at a Point

We consider a Dirichlettype problem for a system of elliptic equations of second order with a strong degeneracy at an inner point of the domain, when, in a neighborhood of this point, the principal term of the asymptotics of a solution is additionally given. We prove the existence and uniqueness of a solution of the problem considered in a weighted class of Hölder vector functions.     

A non-conforming finite element method with anisotropic mesh grading for the Stokes problem in domains with edges

The solution of the Stokes problem in three-dimensional domainswith edges has anisotropic singular behaviour which is treatednumerically by using anisotropic finite element meshes. Thevelocity is approximated by Crouzeix–Raviart (nonconformingP₁ ) elements and the pressure by piecewise constants. Thismethod is stable for general meshes (without minimal or maximalangle condition). Denoting by N_e the number of elements in themesh, the interpolation and consistency errors are of the optimalorder h N_e^–1/3 which is proved for tensor product meshes.As a by-product, we analyse also nonconforming prismatic elementswith P₁ [oplus ] span {x₃²} as the local space for the velocitywhere x₃ is the direction of the edge.     

Finitely Valued f-Modules, An Addendum

In an -group M with an appropriate operator set it is shown that the -value set (M) can be embedded in the value set (M). This embedding is an isomorphism if and only if each convex -subgroup is an -subgroup. If (M) has a.c.c. and M is either representable or finitely valued, then the two value sets are identical. More generally, these results hold for two related operator sets ₁ and ₂ and the corresponding -value sets and . If R is a unital -ring, then each unital -module over R is an f-module and has exactly when R is an f-ring in which 1 is a strong order unit.     

Statistical Inference in a Linear Model for Spatially Located Sensors and Random Input

In the paper we consider a random linear model for observations provided by spatially located sensors measuring signals coming from one source. For this model a set of sufficient and complete statistics are found, and it is shown that the maximum likelihood estimators of unknown parameters (characteristics of the source) are functions of those statistics. The problem of nonnegative estimators of variance components of the model is shortly discussed. Comparisons of the mean squared errors of several estimators are given. Numerical example concerning hunting for defects in solar cells is considered in details.     

BLOWUP OF SOLUTIONS TO THE CAUCHY PROBLEM FOR NONLINEARWAVE EQUATIONS

gi. IntroductionThis paper deals with solutionS of certain nonlinear wave equationS Of the formcorresponding to Antial conditionSwuersis the wave OPerstor.we are interested in showing the ~ up" Of solutions to (1.1)--(1.2). For that, wereIf ac ~ 1)(n ~ 1) > 2, global solutions of ~ equation subject to very general perturbationsof order p exist Provided the initial data are swhciently small (see I6] and references therein).We are also interested in esthaattw the take when "blow up" occurs. …     

On classification of polarized varieties with non-integral nef values

Let be an -dimensional normal projective variety with only Gorenstein, terminal, -factorial singularities. Let be an ample line bundle on . Let denote the nef value of . The classification of via the nef value morphism is given for the situations when satisfies or .

     

Densité des Points à Coordonnées Multiplicativement Indépendantes

Nous montrons que pour toute sous-variété algébrique d'un tore multiplicatif (non contenue dans un sous-groupe algébrique propre), on peut choisir un ensemble Zariski dense de points algébriques de hauteur contrôlée, dont toutes les coordonnées sont multiplicativement indépendantes. Cet énoncé précise et généralise un théorème de S. Zhang qui lie la hauteur projective d'une varété au minimum essentiel de la hauteur des points algébriques de celle-ci. En tenant compte d'un résultat précédent des auteurs sur le problème de Lehmer généralisé à un tore, nous en déduisons une minoration pour la hauteur normalisée d'une sous-variété d'un tore. Cette dernière est optimale à un «-prés» en le degré géométrique de la variété étudiée (confer une conjecture du second auteur avec P. Philippon).In this article, we prove that on any subvariety of a multiplicative torus which is not contained in a proper algebraic subgroup, one can find a Zariski dense set of algebraic points of small height whose coordinates are multiplicatively independent. This statement generalizes an earlier result of S. Zhang which links the projective height of a variety with the essential minimum of its algebraic points. Taking into account an earlier result of the authors on the Lehmer problem generalized to a multiplicative torus, one deduces a lower bound for the normalized height of subvarieties of multiplicative groups. This lower bound is optimal up to an in the geometric degree of the variety studied (confer a conjecture by the second author and P. Philippon).     

On the Representations of a Number as the Sum of Three Cubes and a Fourth or Fifth Power

Let R _k (n) denote the number of representations of a natural number n as the sum of three cubes and a kth power. In this paper, we show that R ₃ (n) n ^5/9+, and that R ₄ (n) n ^47/90+, where > 0 is arbitrary. This extends work of Hooley concerning sums of four cubes, to the case of sums of mixed powers. To achieve these bounds, we use a variant of the Selberg sieve method introduced by Hooley to study sums of two kth powers, and we also use various exponential sum estimates.     

A Note on a Boundary Value Problem

In this note, we prove that, for Robins boundary value problem, a unique solution exists if f_x(t, x, x), f_x(t, x, x), (t), and (t) are continuous, and f_x -(t), f_x -(t), 4(t) ² + ²(t) ++ 2(t), and 4(t) ² + ²(t) + 2(t).AMS Subject Classification (2000) 34B15