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.

We show the existence, in c0 ⊗γc0, of a shrinking, subsymmetric basic sequence (zi) having no c0 subsequences. In particular, the coefficient functionals for (zi) are weakly null and the subspace fails to have the Dunford-Pettis property.

Let G be a locally compact group and H be a compact subgroup of G. Using a general criterion established by Neufang [‘A unified approach to the topological centre problem for certain Banach algebras arising in abstract harmonic analysis’, Arch. Math.82(2) (2004), 164–171], we show that the Banach algebra L1(G/H) is strongly Arens irregular for a large class of locally compact groups.

We prove that every sufficiently large even integer can be represented as the sum of two squares of primes, four cubes of primes and 28 powers of two. This improves the result obtained by Liu and Lü [‘Two results on powers of 2 in Waring–Goldbach problem’, J. Number Theory 131(4) (2011), 716–736].

We present some inequalities for the mappings defined by Dragomir [‘Two mappings in connection to Hadamard’s inequalities’, J. Math. Anal. Appl.167 (1992), 49–56]. We analyse known inequalities connected with these mappings using a recently developed method connected with stochastic orderings and Stieltjes integrals. We show that some of these results are optimal and others may be substantially improved.

We consider some generalisations of the Cahn—Hilliard equation based on constitutive equations derived by M. Gurtin in (1996) with a logarithmic free energy. Compared to the classical Cahn—Hilliard equation (see [4, 5]), these models take into account the work of internal microforces and the anisotropy of the material. We obtain the existence and uniqueness of solutions results and then prove the existence of finite dimensional attractors.

Upper and lower bounds are obtained for the left tail of the normed limit W0 of a supercritical branching process with varying environment, that is, for P(W0 < x) for small x. Two types of process are dealt with—öttcher type and Schröder type—which between them cover “most” processes with zero extinction probability.

Kang and Liu [‘On supersolvability of factorized finite groups’, Bull. Math. Sci.3 (2013), 205–210] investigate the structure of finite groups that are products of two supersoluble groups. The goal of this note is to give a correct proof of their main theorem.

We complete the determination of the groups of positive deficiency which occur as lattices in connected Lie groups. The torsion free groups among them are 3-mainfold groups. We show that any other torsion free 3-manifold group which is such a lattice is the group of an aspherical closed geometric 3-manifold.

It has been shown that the existence of a (left) approximate identity in the tensor product A ⊗ B of Banach algebras A and B, where α is an admissible algebra norm on A ⊗ B, implies the existence of approximate identities in A and B. The question has been raised as to whether the boundedness of the approximate identity in A ⊗αB implies the boundedness of the approximate identities in A and B. This paper answers the question affirmatively with a being the greatest cross-norm.