Rational points of the group of components¶of a Néron model

The structure of component groups of Néron models has been investigated on several occasions. Here we admit non-separably closed residue fields and are interested in the subgroup of rational points or, in other terms, in the subgroup of geometrically connected components of a Néron model. We consider Néron models of abelian varieties and of algebraic tori and give detailed computations in the case of Jacobians of curves. Received: 4 May 1998 / Revised version: 12 October 1998     

Exact Estimates of N widths and One-sided N-widths For a Set of Convolution Classes of Periodic Functions

Let E~ ( ,H_n)_q be the one-sided approximation by n-dimensional subspacein metric L_q to the set of functions and d_m~ ( H_n)_q be the one-sidedn-widths of in L_q. Let     

The existence therems for a class of sublinear elliptic equations in ℝ N

An existence theorem for the solution to the equation

Turán Number of the Family Consisting of a Blow-up of a Cycle and a Blow-up of a Star

Let F={H₁,...,H_k}(k> 1) be a family of graphs.The Turán number of the family F is the maximum number of edges in an n-vertex {H₁,...,H_k)-free graph,denoted by ex(n,F) or ex(n,{H₁,H₂,...,H_k}).The blow-up of a graph H is the graph obtained from H by replacing each edge in H by a clique of the same size where the new vertices of the cliques are all different.In this paper we determine the Turán number of the family cons...     

Positivity and Negativity of Solutions to a Schrödinger Equation in ℝ N

Weak L ² -solutions u of the Schrödinger equation, –u + q(x) u – u = f(x) in L ² , are represented by a Fourier series using spherical harmonics in order to prove the following strong maximum and anti-maximum principles in (N 2): Let ₁ denote the positive eigenfunction associated with the principal eigenvalue ₁ of the Schrödinger operator . Assume that the potential q(x) is radially symmetric and grows fast enough near infinity, and f is a `sufficiently smooth' perturbation of a radially symmetric function, f 0 and 0 f / C const a.e. in . Then u is ₁-positive for - < < ₁ (i.e., u c ₁ with c const > 0) and ₁-negative for ₁ < < ₁ + (i.e., u –c₁ with c const > 0), where > 0 is a number depending on f. The constant c > 0 depends on both and f.     

Extension of a Theorem of Carleson-Duren

1. Introduction A theorem of Carleson,as generalized by Duren characterizes those positive measure μ on the unit disc U={z∈C:|z|<1} for which the H~p norm domiates the L~q(μ) norm of elements of H~p. Later on, Hasting proved an analogous results with H~p replaced by A~p, the Bergman space of fuctions f     

Regularity of the inverse of a homeomorphism of a Sobolev–Orlicz space

Given a homeomorphism ? ∈ W _M ¹ , we determine the conditions that guarantee the belonging of the inverse of ? in some Sobolev–Orlicz space W _F ¹ . We also obtain necessary and sufficient conditions under which a homeomorphism of domains in a Euclidean space induces the bounded composition operator of Sobolev–Orlicz spaces defined by a special class of N-functions. Using these results, we establish requirements on a mapping under which the inverse homeomorphism also induces the bounded composition operator of another pair of Sobolev–Orlicz spaces which is defined by the first pair.     

On Sums of Two Squares of Primes and a Cube of a Prime

In this paper we are concerned with the exceptional set of the sum of two squares of primesand a cube of a prime P;+p;+p;.Noting that竹三1 or 3(mod 6)is a necessary conditionfor the solvability of the equation n=P}+P;+P;(see【1]),we define E(N)=Card{n:n≤N,礼∈三and n≠P;+Pi+P;for any Pi,1≤i s 3), (1)where三={n:n三1 or 3(mod 6)). This and similar problems have been studied by a number of authors.In 1937,Davenportand Heilbronn[2J proved that if后2 3 is an odd integer then almost all posi…     

New integrable systems as a limit of the elliptic SL(N, ?) top

We consider the scaling limit of an elliptic top. This limit is a combination of a scaling of the elliptic top variables, an infinite shift of the spectral parameter, and the trigonometric limit. We give general necessary constraints on the scaling of the variables and examples of such a degeneracy. A certain subclass of limit systems is integrable in the Liouville sense, which can also be shown directly.     

On Sums of a Prime, and a Square of Prime, and a κ-power of Prime

In 1923, Hardy and Littlewood[1] conjectured that each integer n can be written asp+m12+ m22 = n,and Linnik[2,3] proved that this conjecture is true. But if these mi with i = 1,2 are restricted to primes Pi, the corresponding result is out of reach at present. We consider the following Diophantine equation     

Inequality for the Moment of a Function of a Random Variable

ＩｎｅｑｕａｌｉｔｙｆｏｒｔｈｅＭｏｍｅｎｔｏｆａＦｕｎｃｔｉｏｎｏｆａＲａｎｄｏｍＶａｒｉａｂｌｅ￥ＬｉＢａｉｎｉａｎＨｕＳｈｕｈｅ（李柏年，胡舒合）（ＡｎｈｕｉＩｎｓｔｉｔｕｔｅｏｆＦｉｎａｎｃｅａｎｄＴｒａｄｅ）（ＡｎｈｕｉＵｎｉｖｅｒｓｉｔｙ...     

A Generalization of a Theorem of Liao

Let X be a metrizable space and let φ：R× X → X be a continuous flow on X. For any given {φt}-invariant Borel probability measure, this paper presents a {φt}-invariant Borel subset of X satisfying the requirements of the classical ergodic theorem for the contiImous flow （X, {φt}）. The set is more restrictive than the ones in the literature, but it might be more useful and convenient, particularly for non-uniformly hyperbolic systems and skew-product flows.     

On a Zero-Sum Generalization of a Variation of Schur’s Equation

Let be a positive integer, and let denote the cyclic group of residues modulo m. Furthermore, let denote the minimum integer N such that for every function there exist m integers satisfying and (and ). It is shown that for every odd prime m. Daniel Schaal: Partially supported by a South Dakota Governor’s 2010 Individual Research Seed Grant.     

Multi-dimensional versions of a formula of Popoviciu

In this paper, an explicit formulation for multivariate truncated power functions of degree one is given firstly. Based on multivariate truncated power functions of degree one, a formulation is presented which counts the number of non-negative integer solutions of s×(s 1) linear Diophantine equations and it can be considered as a multi-dimensional versions of the formula counting the number of non-negative integer solutions of ax by = n which is given by Popoviciu in 1953.