We give a simple proof and strengthening of a uniqueness theorem for functions in the extended Selberg class.

Let A = (aij), B = (bjj), be two n-square matrices over the complex numbers. Then the n-square matrix H = (hjj) = ij(aijb) is called the Hadamard product of A and B, H = AoB, [l; p. 174]. Let the n2 - square matrix K = A⊗B denote the Kronecker product of A and B.

In this paper the following result is proved.

Let n > 0 and k ≥ 0 be integers with n — 2k ≥ 1. Given any n — k integers

there is a non-empty subset of indices I ⊂ {1, 2,…, n — k} such that the sum Σi∊I ≡ 0(mod n) if at most n — 2k of the integers (1) lie in the same residue class modulo n.

The result is best possible if n ≥ 3k — 2 in the sense that if "at most n — 2k" is replaced by "at most n — 2k + 1" the result becomes false. This can be seen by taking aj = 1 for 1 ≤ j ≤ n — 2k + 1 and aj = 2 for n —2k + 2 ≤ j ≤ n — k, noting that the number of 2's here is n — k — (n — 2k + 1)= k — 1 ≤ n — 2k + 1.

In this paper we give necessary and sufficient conditions under which the group of units of a group ring of a finite group G over a field K does not contain a free subgroup of rank 2.

Some extensions of this results to infinite nilpotent and FC groups are also considered.

Let R be an integral domain. An element u of the quotient field of R is said to be pseudo-integral over R if uIv ⊆ Iv for some nonzero finitely generated ideal I of R. The set of all pseudo-integral elements forms an integrally closed (but not necessarily pseudo-integrally closed) overling R ofR. It is shown that , where X is a family of indeterminates; pseudo-integrality is analyzed in rings of the form D + M; and an example is given to show that pseudo-integrality does not behave well with respect to localization.

Fan ([2, Theorem 2]) has proved the following theorem:

Let K be a nonempty compact convex set in a normed linear space X. For any continuous map f from K into X, there exists a point u∈K such that

In this note, we prove that the above theorem is true for a continuous condensing map defined on a closed ball in a Banach space. We also prove that it is true for a continuous condensing map defined on a closed convex bounded subset of a Hilbert space.

A well known theorem of van der Waerden [1] states that given any two positive integers k and t, there exists a positive integer m such that in every distribution of the numbers 1,2, …, m into k classes, at least one class contains an arithmetic progression of t + 1 terms. Other proofs and generalizations of this theorem have been given by Griinwald [2], Witt [3] and Lukomskaya [4]. The last mentioned proof appears in the booklet of Khinchin “Three pearls of number theory” in which van der Waerden's theorem plays the role of the first pearl.

Given a set of n points, with each pair of distinct points joined by a line that is oriented towards exactly one of the points, then the resulting configuration is called a (roun-drobin) tournament. A tournament is reducible if the points can be separated into two non-empty subsets, A and B, such that every line that joins a point in A to a point in B is oriented towards the point in B. If a tournament is not reducible it is called irreducible. The object of this note is to derive an approximation for P(n), the probability that a tournament on n point, chosen at random from the set of possible ones, will be irreducible. p(1)=1, by definition.

Criteria for local uniform rotundity and midpoint local uniform rotundity in Orlicz-Lorentz spaces with the Luxemburg norm are given. Strict K-monotonicity and Kadec-Klee property are also discussed.

This paper is about the topologies arising from statistical coincidence on locally finite point sets in locally compact Abelian groups $G$. The first part defines a uniform topology (autocorrelation topology) and proves that, in effect, the set of all locally finite subsets of $G$ is complete in this topology. Notions of statistical relative denseness, statistical uniform discreteness, and statistical Delone sets are introduced.

The second part looks at the consequences of mixing the original and autocorrelation topologies, which together produce a new Abelian group, the autocorrelation group. In particular the relation between its compactness (which leads then to a $G$-dynamical system) and pure point diffractivity is considered. Finally for generic regular model sets it is shown that the autocorrelation group can be identified with the associated compact group of the cut and project scheme that defines it. For such a set the autocorrelation group, as a $G$-dynamical system, is a factor of the dynamical local hull.

The purpose of this note is to provide integral inequalities which are related to Hardy's ([2] and [3, Theorem 330]). This latter result we state as

Theorem 1. Let p>1, r≠1, and ƒ(x) be nonnegative and Lebesgue integrable on [0, a] or [a, ∞] for every a>0, according as r> 1 or r< 1. If F(x) is defined by

1

then

2

unless f≡0. The constant is the best possible.

A new infinite product ${{t}_{n}}$ was introduced by S. Ramanujan on the last page of his third notebook. In this paper, we prove Ramanujan’s assertions about ${{t}_{n}}$ by establishing new connections between the modular $j$-invariant and Ramanujan’s cubic theory of elliptic functions to alternative bases. We also show that for certain integers $n$, ${{t}_{n}}$ generates the Hilbert class field of $\mathbb{Q}\left( \sqrt{-n} \right)$. This shows that ${{t}_{n}}$ is a new class invariant according to H. Weber’s definition of class invariants.

A Dirac comb of point measures in Euclidean space with bounded complex weights that is supported on a lattice $\Gamma$ inherits certain general properties from the lattice structure. In particular, its autocorrelation admits a factorization into a continuous function and the uniformlattice Dirac comb, and its diffraction measure is periodic, with the dual lattice ${{\Gamma }^{*}}$ as lattice of periods. This statement remains true in the setting of a locally compact Abelian group whose topology has a countable base.

Following E. H. Feller [l], a ring R is called a duo ring if every one-sided ideal of R is a two-sided ideal.

In the first part of this paper, we give some properties of duo rings and we show that the set of the nilpotent elements of a duo ring R is an ideal, the intersection of the completely prime ideals of R.

It is easy to see that every duo ring is a subdirect sum of subdirectly irreducible duo rings. We give in the second part of this paper a characterization of the subdirectly irreducible duo rings. This characterization is quite similar to the characterization of the subdirectly irreducible commutative rings, due to N. H. McCoy [2], whose methods we use.

Let H = (Hi, j) (1 ≦ i, j ≦ n) be an nk × nk matrix with complex coefficients, where each Hi, j is itself a k × k matrix (n, k ≧ 2). Let |H| denote the determinant of H and let ∥H∥ = |(|H i, j|)| (1 ≦ i, j ≦ n ). The purpose of this note is to prove the following theorem.

Theorem. If H is positive definite Hermitian then |H| ≦∥H∥. Moreover, |H| = ∥H∥ if and only if Hi, j = 0 whenever i ≠ j.

The case n = 2 of this theorem is contained in [1].

Gromov introduced the concept of uniform embedding into Hilbert space and asked if every separable metric space admits a uniform embedding into Hilbert space. In this paper, we study uniform embedding into Hilbert space and answer Gromov’s question negatively.

We obtain sufficient conditions for the oscillation of all solutions of the linear delay differential equation with positive and negative coefficients

where

Extensions to neutral differential equations and some applications to the global asymptotic stability of the trivial solution are also given.

By an (incomplete) r × s latin rectangle is meant an r × s array such that (in some subset of the rs cells of the array) each of the cells is occupied by an integer from the set 1, 2, …, s and such that no integer from the set 1,2, …, s occurs in any row or column more than once. This definition requires that r≦s. If r=s we will replace the word rectangle by square. It is easy to see that for any n≧2 there is an incomplete n × 2n latin rectangle with 2n cells occupied which cannot be completed to a n × 2n latin rectangle. In this paper we prove the following theorem.

Theorem 1. An incomplete n × 2n latin rectangle with 2n—1 cells occupied can be completed to a n × 2n latin rectangle.

Nous donnons une expression finie pour la valeur L(1, X) dès lors que X est un caractère de Dirichlet modulo f ≥ 2, impair et non principal. Cette expression, valable même lorsque ce caractère n'est pas primitif, nous permet de généraliser au théorème 2 le résultat de H. Walum sur un comportement en moyenne de ces fonctions L (sa démonstration qui fait usage de sommes de Gauss ne semble pas pouvoir être adaptée au cas de caractères non primitifs.) Nous appliquons ces résultats à l'obtention de bornes pour le nombre de classes relatif des corps cyclotomiques: nous retrouvons celles de T. Metsänkylä et de K. Feng par une méthode nous permettant de les ensuite amender.

It is well known that the compactly supported wavelets cannot belong to the class ${{C}^{\infty }}\,(\text{R})\,\cap \,{{L}^{2}}\,(R)$. This is also true for wavelets with exponential decay. We show that one can construct wavelets in the class ${{C}^{\infty }}\,(\text{R})\,\cap \,{{L}^{2}}\,(R)$ that are “almost” of exponential decay and, moreover, they are band-limited. We do this by showing that we can adapt the construction of the Lemarié-Meyer wavelets $[\text{LM }\!\!]\!\!\text{ }$ that is found in $[\text{BSW}]$ so that we obtain band-limited, ${{C}^{\infty }}$-wavelets on $R$ that have subexponential decay, that is, for every $0<\varepsilon <1$, there exits ${{C}_{\in }}\,>\,0$ such that $|\psi (x)|\le {{C}_{\varepsilon }}{{e}^{-|x{{|}^{1-\varepsilon }}}}$, $x\in \text{R}$. Moreover, all of its derivatives have also subexponential decay. The proof is constructive and uses the Gevrey classes of functions.