We improve the intuition and some ideas of G. D. Faulkner and M. C. Vipera presented in the article “Remainders of compactifications and their relation to a quotient lattice of the topology” [Proc. Amer. Math. Soc. 122 (1994)] in connection with the question about internal conditions for locally compact spaces X and Y under which βX/X ≅ βY/Y or, more generally, under which the remainders of compactifications of X belong to the collection of the remainders of compactifications of Y. We point out the reason why the quotient lattice of the topology considered by Faulkner and Vipera cannot lead to a satisfactory answer to the above question. We replace their lattice by the qoutient lattice of a new equivalence relation on a Wallman base in order to describe a method of constructing a Wallman-type compactification which allows us to deduce more complete solutions to the problems investigated by Faulkner and Vipera.

Assume that $(G,+)$ is a commutative semigroup, $\unicode[STIX]{x1D70F}$ is an endomorphism of $G$ and an involution, $D$ is a nonempty subset of $G$ and $(H,+)$ is an abelian group uniquely divisible by two. We prove that if $D$ is ‘sufficiently large’, then each function $g:D\rightarrow H$ satisfying $g(x+y)+g(x+\unicode[STIX]{x1D70F}(y))=2g(x)$ for $x,y\in D$ with $x+y,x+\unicode[STIX]{x1D70F}(y)\in D$ can be extended to a unique solution $f:G\rightarrow H$ of the generalised Jensen functional equation $f(x+y)+f(x+\unicode[STIX]{x1D70F}(y))=2f(x)$.

A Littlewood–Paley operator associated with the reflection part of the Dunkl operator is introduced and proved to be of type $(p,p)$ for $1<p<\infty$, based on boundedness of a generalised vector-valued singular integral. This fills a gap for $2<p<\infty$ concerning the boundedness of a $g$-function in the Dunkl setting. The paper also supplies new proofs for $1<p<\infty$ on the $(p,p)$ boundedness of various $g$-functions associated with the Dunkl operator.

A review of the development of estimates for Lebesgue constants associated with Lagrange interpolation on the one hand, and estimates for the rate of convergence of Hermite-Fejér interpolation on the other hand, provides a historical perspective for the following surprising, close link between these apparently diverse concepts. Denoting by Λn (T) the Lebesgue constant of order n and by Δn (T) the maximum interpolation error for functions of class Lip 1 by Hexmite-Fejér interpolation polynomials of degree not exceeding 2n − 1, based on the zeros of the Chebyshev polynomial of first kind, we discover that, for even values of n, Λn(T) = n Δn(T).

This note is to indicate a nonsemiprime ring R such that the smallest supernilpotent (respectively special) radical containing the ring R is an atom of the lattice of all supernilpotent (respectively special) radicals. This gives a positive answer to Puczylowski's and Roszkowska's question.

A new sharp general Ostrowski type inequality in L∞ norm is established. Some special cases are discussed.

The problem of finding geodesics that avoid certain obstacles in negatively curved manifolds has been studied in different situations. In this note we give a generalization of the unclouding theorem of J. Parkkonen and F. Paulin: there is a constant s0=1.534 such that for any Hadamard manifold M with curvature ≤−1 and for any family of disjoint balls or horoballs {Ca}a∈A and for any point p∈M−⋃ a∈ACa if we shrink these balls uniformly by s0 one can always find a geodesic ray emanating from p that avoids the shrunk balls. It will be shown that in the theorem above one can replace the balls by arbitrary convex sets.

Forster [‘Finite-to-one maps’, J. Symbolic Logic68 (2003), 1251–1253] showed, in Zermelo–Fraenkel set theory, that if there is a finite-to-one map from ${\mathcal{P}}(A)$, the set of all subsets of a set $A$, onto $A$, then $A$ must be finite. If we assume the axiom of choice (AC), the cardinalities of ${\mathcal{P}}(A)$ and the set $S(A)$ of permutations on $A$ are equal for any infinite set $A$. In the absence of AC, we cannot make any conclusion about the relationship between the two cardinalities for an arbitrary infinite set. In this paper, we give a condition that makes Forster’s theorem, with ${\mathcal{P}}(A)$ replaced by $S(A)$, provable without AC.

For each λ ≥ 1 a class of Banach spaces φλ is defined. Isometric results are obtained on the equivalence between a measure of non-compactness and the essential norm of a linear operator defined on a φλ space. Best values of λ for the classical Banach spaces and for spaces with unconditional basis are investigated. For the space c of convergent sequences the non-existence of a λ-unconditional basis with λ < 2 is deduced.

In this note, we prove that the Gauss–Picard modular group PU(2,1;Θ1) has Property (FA). Our result gives a positive answer to a question by Stover [‘Property (FA) and lattices in SU(2,1)’, Internat. J. Algebra Comput.17 (2007), 1335–1347] for the group PU(2,1;Θ1).

We use a unified approach to study the boundedness of fractional integral operators on $\unicode[STIX]{x1D6FC}$-modulation spaces and find sharp conditions for boundedness in the full range.

We obtain some explicit formulae for series of the type

where f is an entire function of exponential type τ, bounded on the real exis (and satisfying in the first case). These series are expressed in terms of the derivatives of f and Bernoulli numbers. We examine the case where f is a trigonometric polynomial which lead us, in particular, to a new representation of the associated Fejér mean.

In this paper we provide some bounds for the quantity $\Vert f(y)-f(x)\Vert$, where $f:D\rightarrow \mathbb{C}$ is an analytic function on the domain $D\subset \mathbb{C}$ and $x$, $y\in {\mathcal{B}}$, a Banach algebra, with the spectra $\unicode[STIX]{x1D70E}(x)$, $\unicode[STIX]{x1D70E}(y)\subset D$. Applications for the exponential and logarithmic functions on the Banach algebra ${\mathcal{B}}$ are also given.

Let A be a hermitian transformation on an n-dimensional unitary space En, with proper values a1 ≥ … ≥ an. Let M be a proper subspace of En. Suppose b1 ≥ … ≥ bh are the proper values of A/M and c1 ≥ … ≥ ck are the proper values of . Let i1. < … < ir and j1 < … < jr be sequences of positive integers, with ir ≤ k and jr ≤ h. then

.

This is a special case of one of the Thompson-Freede theorems which is proved by use of certain invariants.

Let $A$ be a subset of $\mathbb{N}$ , the set of all nonnegative integers. For an integer $h\geq 2$ , let $hA$ be the set of all sums of $h$ elements of $A$ . The set $A$ is called an asymptotic basis of order $h$ if $hA$ contains all sufficiently large integers. Otherwise, $A$ is called an asymptotic nonbasis of order $h$ . An asymptotic nonbasis $A$ of order $h$ is called a maximal asymptotic nonbasis of order $h$ if $A\cup \{a\}$ is an asymptotic basis of order $h$ for every $a\notin A$ . In this paper, we construct a sequence of asymptotic nonbases of order $h$ for each $h\geq 2$ , each of which is not a subset of a maximal asymptotic nonbasis of order $h$ .

In this paper, existence criteria for multiple solutions of periodic boundary value problems for the first-order difference equation are established by using the Leggett–Williams multiple fixed point theorem and fixed point theorem of cone expansion and compression. Two examples are also given to illustrate the main results.

For $n\in \mathbb{Z}$ and $A\subseteq \mathbb{Z}$, define $r_{A}(n)$ and ${\it\delta}_{A}(n)$ by $r_{A}(n)=\#\{(a_{1},a_{2})\in A^{2}:n=a_{1}+a_{2},a_{1}\leq a_{2}\}$ and ${\it\delta}_{A}(n)=\#\{(a_{1},a_{2})\in A^{2}:n=a_{1}-a_{2}\}$. We call $A$ a unique representation bi-basis if $r_{A}(n)=1$ for all $n\in \mathbb{Z}$ and ${\it\delta}_{A}(n)=1$ for all $n\in \mathbb{Z}\setminus \{0\}$. In this paper, we prove that there exists a unique representation bi-basis $A$ such that $\limsup _{x\rightarrow \infty }A(-x,x)/\sqrt{x}\geq 1/\sqrt{2}$.

A graph is called arc-regular if its full automorphism group acts regularly on its arc set. In this paper, we completely determine all the arc-regular Frobenius metacirculants of prime valency.

Let K1 and K2 be extension fields over a field K with char K = p > 0. Assume L = K1(x1) = K2(x2) ⊃ K where xi is transcendental over Ki, for i = 1, 2. In this paper we prove that if K1 is a perfect field, then K1 = K2.