In this paper we are concerned with a class of p(x)-Kirchhoff equation where the non-linearity has non-standard growth and contains a bi-non-local term. We prove, by using variational methods (Mountain Pass Theorem and Ekeland Variational Principle), several results on the existence of positive solutions.

Let G be a finitely generated (f.g.) torsion-free nilpotent group. Then the group algebra k[G] of G over a field k is a Noetherian domain and hence has a classical division ring of fractions, denoted by k(G). Recently, the division algebras k(G) and, somewhat more generally, division algebras generated by f.g. nilpotent groups have been studied in [3] and [5]. These papers are concerned with the question to what extent the division algebra determines the group under consideration. Here we continue the study of the division algebras k(G) and investigate their Gelfand–Kirillov (GK–) transcendence degree.

We give an asymptotic bound for the size of the n-generated relatively free semigroup in the variety generated by all combinatorial strictly 0-simple semigroups.

It is well-known (see e.g. [1, p. 5]) that a class ℳ of (not necessarily associative) rings is the semisimple class for some radical class, relative to some universal class if and only if it has the following properties:

(a)if ℳ, then every non-zero ideal I of Rhas a non-zero homomorphic image I/J∈ℳ.

(b) If R∈ but R∉ℳ, then R has a non-zero ideal I∈, where ℳ = {K ∈ | every non-zero K/H∉ℳ}. In fact ℳ is the radical class whose semisimple class is ℳ. On the other hand, if ℘ is a radical class, then ℐ℘ = {K∈/ if I is a non-zero ideal of K, then I∉℘} is its semisimple class. If a class ℳ is hereditary (that is, when R∈ℳ, then all its ideals are in ℳ), it clearly satisfies (a), but there do exist non-hereditary semisimple classes (see [2]). The condition (satisfied in all associative or alternative classes) is that ℘ is hereditary for a radical class ℘ if and only if ℘(I) ⊆ ℘(R) for all ideals I of all rings R∈ [3, Lemma 2, p. 595].

It is shown that the creation operator is the only (up to a multiplicative constant) injective weighted shift all of whose translations (or at least one) are still injective weighted shifts regardless of what the weight sequences and the bases are. A similar result is true for the annihilation operator as well as for the Heisenberg and Schrödinger couples.

A compact bordered Klein surface of genus g ≥ 2 has maximal symmetry [4] if its automorphism group is of order 12(g − 1), the largest possible. An M*-group [8] acts on a bordered surface with maximal symmetry. The first important result about these groups was that they must have a certain partial presentation [8, p. 5]. However, research has tended to focus more on the surfaces with maximal symmetry than on the M*-groups, and results about these groups typically deal with existence.

Many special cases of the equation x2+C= yn where x and y are positive integers and n≥3 have been considered over the years, but most results for general n are of fairly recent origin. The earliest reference seems to be an assertion by Fermat that he had shown that when C=2, n=3, the only solutions are given by x = 5, y = 3; a proof was published by Euler [1]. The first result for general n is due to Lebesgue [2] who proved that when C = 1 there are no solutions. Nagell [4] generalised Fermat's result and proved that for C = 2 the equation has no solution other than x = 5, y = 3, n = 3. He also showed [5] that for C = 4 the equation has no solution except x = 2, y = 2, n = 3 and x = 11, y = 5, n = 3, and claims in [6] to have dealt with the case C = 5. The case C = -1 was solved by Chao Ko, and an account appears in [3], pp. 302–304.

The concept of a metaplectic form was introduced about 40 years ago by T. Kubota. He showed how Jacobi-Legendre symbols of arbitrary order give rise to characters of arithmetic groups. Metaplectic forms are the automorphic forms with these characters. Kubota also showed how higher analogues of the classical theta functions could be constructed using Selberg's theory of Eisenstein series. Unfortunately many aspects of these generalized theta series are still unknown, for example, their Fourier coefficients. The analogues in the case of function fields over finite fields can in principle be calculated explicitly and this was done first by J. Hoffstein in the case of a rational function field. Here we shall return to his calculations and clarify a number of aspects of them, some of which are important for recent developments.

All sets considered will be finite, and |x| will denote the cardinal number of the set X.

Let = (Ai:i∈I) be a family of subsets of a set E. A subset E′ ⊆ E is called a transversal of if there exists a bijection σ:E′→ I such that e ∈ Aσ(e) (e ∈ E′). According to a well-known theorem of P. Hall [2], the familyhas a transversal if and only iffor every subset I′ of I. Ford and Fulkerson [1] obtained (as a special case of a more general theorem) an analogous criterion for the existence of a common transversal (CT) of two families. We may state their result in the following terms.

Using the Hilbert Uniqueness Method, we study the problem of exact controllability in Neumann boundary conditions for problems of transmission of the wave equation. We prove that this system is exactly controllable for all initial states in L2(Ω)×(H1(Ω))′.

In this paper, we give a characterization of Clifford tori and in a unit sphere S n+1 (1). Our results extend the results due to Cheng and Yau [4], and Wang and Xia [11].

The problem of Josephus and the forty Jewsis well known [1, 3]. In its most general form, this problem is equivalent to the problem of m-enumeration of a set, as described below.

Define the ordered set

We choose and remove cyclically, from left to right, each with element of Zn until the set is exhausted. The chosen elements are ordered into a new ordered set

which is therefore a permutation of Zn, obtained by what we call m-enumeration of theset Zn.

Although the classes of regular simple semigroups and simple semigroups without idempotents are evidently at opposite ends of the spectrum of simple semigroups, their theories involve some interesting connections. Jones [5] has obtained analogues of the bicyclic semigroup for simple semigroups without idempotents. Megyesi and Pollák [7] have classified all combinatorial simple principal ideal semigroups on two generators, showing that all are homomorphic images of one such semigroup Po which has no idempotents.

We offer a Lebesgue-type decomposition of a representable functional on a *-algebra into absolutely continuous and singular parts with respect to another. Such a result was proved by Zs. Szűcs due to a general Lebesgue decomposition theorem of S. Hassi, H.S.V. de Snoo, and Z. Sebestyén concerning non-negative Hermitian forms. In this paper, we provide a self-contained proof of Szűcs' result and in addition we prove that the corresponding absolutely continuous parts are absolutely continuous with respect to each other.

Suppose that $c$ is a linear operator acting on an $n$-dimensional complex Hilbert Space $H$, and let $\tau$ denote the normalized trace on $B(H)$. Set $b_1 = (c+c^*)/2$ and $b_2 = (c-c^*)/2i$, and write $B$ for the spectral scale of $\{b_1, b_2\}$ with respect to $\tau$. We show that $B$ contains full information about $W_k(c)$, the $k$-numerical range of $c$ for each $k = 1,\dots,n$. This is in addition to the matrix pencil information that has been described in previous papers. Thus both types of information are contained in the geometry of a single 3-dimensional compact, convex set. We then use spectral scales to prove a new fact about $W_k(c)$. We show in Theorem 3.4 that the point $\lambda$ is a singular point on the boundary of $W_k(c)$ if and only if $\lambda$ is an isolated extreme point of $W_k(c)$: i.e. it is the end point of two line segments on the boundary of $W_k(c)$. In this case $\lambda = (n/k)\tau(cz)$, where $z$ is a central projection in the algebra generated by $c$ and the identity. In addition we show how the general theory of the spectral scale may be used to derive some other known properties of the $k$-numerical range.

Let х denote a primitive character modulo k. Using two different representations for Dirichlet L-functions, Apostol [1] recently derived a representation for

involving the sums

where m is a positive integer. Furthermore, if х(r) = (r|p)the residue class character modulo the odd prime p, he derived a representation for Mm(х) involving the sums

In this paper we give a characterization of the central elements of the algebra A** for a class of weakly sequentially complete Banach algebras A and present a detailed study of several questions related to Arens regularity of the algebra A.

1991 Mathematics Subject Classification Primary: 43A10, 46B20, 46J99.

Let X be a finite set and let (X) be the full transformation semigroup on X, i.e. the set of all mappings from X into X, the semigroup operation being composition of mappings. This paper aims to characterize those elements of (X) which have square roots. An easily verifiable necessary condition, that of being quasi-square, is found in Theorem 2, and in Theorems 4 and 5 we find necessary and sufficient conditions for certain special elements of (X). The property of being compatibly amenable is shown in Theorem 7 to be equivalent for all elements of (X) to the possession of a square root.

A construction of differential constraints compatible with the Gibbons-Tsarev equation is considered. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the classical determining equations that are used in searching for Lie operators. We introduce the notion of an invariant solution under an involutive distribution and give sufficient conditions for existence of such solutions.

Given a nonincreasing function f : ℤ≥ 0 \{0} → ℤ≥ 0 such that (i) f(k) − f(k + 1) ≤ 1 for all k ≥ 1 and (ii) if a = f(1) and b = limk → ∞f(k), then |f−1(a)| ≤ |f−1(a − 1)| ≤ ··· ≤ |f−1(b + 1)|, a system of generators of a monomial ideal I ⊂ K[x1, . . ., xn] for which depth S/Ik = f(k) for all k ≥ 1 is explicitly described. Furthermore, we give a characterization of triplets of integers (n, d, r) with n > 0, d ≥ 0 and r > 0 with the properties that there exists a monomial ideal I ⊂ S = K[x1, . . ., xn] for which limk→∞ depth S/Ik = d and dstab(I) = r, where dstab(I) is the smallest integer k0 ≥ 1 with depth S/Ik0 = depth S/Ik0+1 = depth S/Ik0+2 = ···.