It is known that Kloosterman sums with prime denominator p take real values, so one can define a sequence of signs of such sums. Several pseudorandom properties of this sequence have recently been studied by Fouvry, Michel, Rivat and Sárközy. Here we use one of their results to estimate a certain important characteristic of this sequence which is also of cryptographic interest.

We show that any residue class λ modulo p can be represented in the form n1! +…+ nℓ! ≡ λ (mod p) with ℓ = O((log p)3 log log p).

We give a sufficient condition for an operator to be chaotic and we use this condition to show that, in the Banach space C0[0, ∞) the operator (τλ, cf)(t) = λf(t + c) (with λ > 1 and c > 0) is chaotic, with every n ∈ ℕ being a period for this operator. We also describe a technique to construct, explicitly, hypercyclic functions for this operator.

Let G be a graph, and f: G → G be a continuous map with periodic points. In this paper we show that the following five statements are equivalent.

(1) f is equicontinuous.

(2) There exists some positive integer N such that fN is uniformly convergent.

(3) f is S-equicontinuous for some positive integer sequence S = {n1 < n2 < …}.

(4) Ω(x, f) = ω(x, f) for every x ∈ G.

(5) is a periodic map.

The index and period of an element a of a finite semigroup are the smallest values of m ≥ 1 and r ≥ 1 such that am+r = am. An element with index m and period 1 is called an m-potent element. For an element α of a finite full transformation semigroup with index m and period r, a unique factorisation α = σβ such that Shift(σ) ∩ Shift(β) = ∅ is obtained, where σ is a permutation of order r and β is an m-potent. Some applications of this factorisation are given.

The purpose of the paper is to show that weak order convergence of a net of Dedekind complete Riesz space-valued σ-measures is uniform over uniformly bounded, uniformly equicontinuous classes of functions. The paper ends with generalizing Ulam's theorem for tightness of positive, finite Borel measures to Riesz space-valued σ-measures.

We look for elliptic curves featuring rational points whose coordinates form two arithmetic progressions, one for each coordinate. A constructive method for creating such curves is shown, for lengths up to 5.

In this paper, we show that Jacobi's two-square and two-triangular number theorems are immediate consequence of Ramanujan's continued fraction identities.

In this paper, we consider farthest points and the farthest distance map of a closed bounded set in a Banach space. We show, inter alia, that a strictly convex Banach space has the Mazur intersection property for weakly compact sets if and only if every such set is the closed convex hull of its farthest points, and recapture a classical result of Lau in a broader set-up. We obtain an expression for the subdifferential of the farthest distance map in the spirit of Preiss' Theorem which in turn extends a result of Westphal and Schwartz, showing that the subdifferential of the farthest distance map is the unique maximal monotone extension of a densely defined monotone operator involving the duality map and the farthest point map.

We investigate Carlson type inequalities for finite sums, that is, inequalities of the form to hold for some constant C independent of the finite, non-zero set a1,…,am of non-negative numbers. We find constants C which are strictly smaller than the sharp constants in the corresponding infinite series case. Moreover, corresponding results for integrals over bounded intervals are given and a case with any finite number of factors on the right-hand side is proved.

This paper studies the problem of numerical integration over the unit sphere S2 ⊆ ℝ3 for functions in the Sobolev space H3/2(S2). We consider sequences Qm(n), n ∈ ℕ, of cubature (or numerical integration) rules, where Qm(n) is assumed to integrate exactly all (spherical) polynomials of degree ≤ n, and to use m = m(n) values of f. The cubature weights of all rules Qm(n) are assumed to be positive, or alternatively the sequence Qm(n), n ∈ ℕ, is assumed to have a certain local regularity property which involves the weights and the points of the rules Qm(n), n ∈ ℕ. Under these conditions it is shown that the worst-case (cubature) error, denoted by E3/2 (Qm(n)), for all functions in the unit ball of the Hilbert space H3/2 (S2) satisfies the estimate E3/2 (Qm(n)) ≤ c n−3/2, where the constant c is a universal constant for all sequences of positive weight cubature rules. For a sequence Qm(n), n ∈ ℕ, of cubature rules that satisfies the alternative local regularity property the constant c may depend on the sequence Qm(n), n ∈ ℕ. Examples of cubature rules that satisfy the assumptions are discussed.

We consider a generalised Cahn-Hilliard system with elasticity based on constitutives laws proposed by Gurtin, with a logarithmic free energy. We obtain some results on the existence and uniqueness of solutions.

In this paper we study a general variational inequality model with set-valued quasimonotone operators, a model which includes several variational inequalities and equilibrium problems. We establish unifying conditions for existence of solutions in a topological vector space setting. Applications to parametric equilibrium models and to a contact problem are given.

The main objective of this paper is a study of some new generalisations of Hilbert and Hardy-Hilbert type inequalities involving non-conjugate parameters. We prove general forms of multiple Hilbert-type inequalities, and we also introduce multiple inequalities of Hardy-Hilbert type with non-conjugate parameters.

In this note we show that k-convex functions on ℝn are twice differentiable almost everywhere for every positive integer k > n/2. This generalises Alexsandrov's classical theorem for convex functions.

In this paper we adopt the notion of orthogonality in Banach spaces introduced by the author in [6]. There, the author showed that in any two-dimensional subspace F of E, every nonzero element admits at most one orthogonal direction. The problem of existence of such orthogonal direction was not addressed before. Our main purpose in this paper is the investigation of this problem in the case where E is a real Banach space. As a result we obtain a characterisation of Hilbert spaces stating that, if in every two-dimensional subspace F of E every nonzero element admits an orthogonal direction, then E is isometric to a Hilbert space. We conclude by presenting some open problems.

We study the behaviour of solutions in a neighbourhood of the origin for a certain type of non-autonomous system of partial differential equations whose linear approximation is non autonomous.

A ring is called Bezout if every finitely generated ideal is principal. We show that every ideal of a commutative Bezout ring R is flat if and only if every submodule of a free R-module is flat. Using this theorem we obtain Neville's theorem.

This article describes the cohomology spaces of any Lie algebra containing a Lie algebra of Heisenberg type (whose cohomology was studied by Santharoubane) as an ideal of codimension 1. For instance, the twisted standard filiform Lie algebras are of this kind. We give an explicit formula for the Betti numbers of this Lie algebra, and use this to describe new families of algebras whose Betti numbers do not behave unimodally.

In this paper we examine some properties of the Carter subgroups in the group of units of certain associative algebras. A description of the Carter subgroups in the case of a solvable associative algebra is obtained. Moreover, given an associative algebra A, we study relationships between the Cartan subalgebras of the Lie algebra associated with A and the Carter subgroups of the group of units of A.