In this note we discuss a type of matrix ring that has nice properties concerning the injectivity and quasi-injectivity of one-sided ideals.

New reverses of the Schwarz inequality in inner product spaces that incorporate the classical Klamkin-McLenaghan result for the case of positive n-tuples are given. Applications for Lebesgue integrals are also provided.

We shall discuss boundedness and compactness of the products of composition and differentiation between Hardy spaces.

We prove the existence of universal functions for mappings Tn: C([0,1]) → Lp([0,1]), 0 < p < 1, with Tn(f) → f′ (n → ∞) on certain subsets of C1([0,1]). As an application we conclude that there are continuous functions f ∈ C([0,1]), such that the derivatives of the Bernstein polynomials form a dense subset of Lp([0,1]) for each 0 < p < 1.

We provide a wide class of Möbius inversion formulas in terms of the generalised Möbius functions and their application to the setting of the Selberg multiplicative functions.

We consider a Helgason-type Fourier transform on SL2(ℝ) and prove various results on L1-harmonic analysis on the full group analogous to those on symmetric spaces.

The purpose of this article is to construct families of elliptic curves E over finite fields F so that the groups of F-rational points of E are cyclic, by using a representation of the modular invariant function by a generator of a modular function field associated with the modular group Γ0(N), where N = 5, 7 or 13.

Consider a bounded linear operator T between Banach spaces ℬ, ℬ′ which can be decomposed into direct sums ℬ = ℬ1 ⌖ ℬ2, ℬ′ = ℬ1′ ⌖ ℬ2′. Such linear operator can be represented by a 2 × 2 operator matrix of the form where Tij ∈ ℒ(ℬj, ℒi′) i, j = 1, 2. (By ℒ(ℬj, ℒi′) we denote the space of bounded linear operators acting from ℬj to ℬi′ (i, j = 1, 2).) The map GT from L (B1, B2) into the set of closed affine subspaces of ℒ(ℬ1′ ℬ2′), defined by is called a linear fractional relation associated with T.

Such relations can be considered as a generalisation of linear fractional transformations which were studied by many authors and found many applications. Many traditional and recently discovered areas of application of linear fractional transformations would benefit from a better understanding of the behaviour of linear fractional relations. The present paper is devoted to analogues of the Liouville theorem “a bounded entire function is constant” for linear fractional relations.

We study multiplicative character sums taken on the values of a non-homogeneous Beatty sequence where α,β ∈ ℝ, and α is irrational. In particular, our bounds imply that for every fixed ε > 0, if p is sufficiently large and p½+ε ≤ N ≤ p, then among the first N elements of ℬα,β, there are N/2+o(N) quadratic non-residues modulo p. When more information is available about the Diophantine properties of α, then the error term o(N) admits a sharper estimate.

Some elementary inequalities providing upper bounds for the difference of the norm and the numerical radius of a bounded linear operator on Hilbert spaces under appropriate conditions are given.

The convolution sums and are evaluated for all n ∈ ℕ.

We present some of the group theoretic properties of reversing symmetry groups, and classify their structure in simple cases that occur frequently in several well-known groups of dynamical systems.

We introduce a class of β − v-unfavourable spaces, which contains some known classes of β-unfavourable spaces for topological games of Choquet type. It is proved that every β − v-unfavourable space X is a Namioka space, that is for any compact space Y and any separately continuous function f : x × Y → ℝ there exists a dense in XGδ-set A ⊆ X such that f is jointly continuous at each point of A × Y.

In this paper, nonlinear minimax problems are discussed. Using Sequential Quadratic Programming and the generalised monotone line search technique, we propose a new algorithm for solving degenerate minimax problems. At each iteration of the proposed algorithm, a search direction is obtained by solving a new Quadratic Programming problem which always has a solution. Global convergence can be obtained without the regularity condition of linear independence. Finally, some numerical experiments are reported.

We construct a family of representations of an arbitrary variant Sa of a semigroup S, induced by a given representation of S, and investigate properties of such representations and their kernels.

The length of every pair {A, B} of n×n complex matrices is at most 2n − 2, if n ≤ 5. That is, for n ≤ 5, the (possibly empty) words in A, B of length at most 2n − 2 span the unital algebra  generated by A, B. For every positive integer m there exist m×m complex matrices C, D such that the length of the pair {C, D} is 2m – 2.

We use exponential sums to obtain new lower bounds on the number of distinct distances defined by all pairs of points (a, b) ∈ A × B for two given sets where is a finite field of q elements and n ≥ 1 is an integer.

We consider the logistic equation −Δu = a (x) u − b (x) uq on all of RN with a (x)/|x|γ and b (x)/|x|τ bounded away from 0 and infinity for all large |x|, where γ > −2, τ ∈ (−∞, ∞). We show that this problem has a unique positive solution. This considerably improves some earlier results. The main new technique here is a Safonov type iteration argument. The result can also be proved by a technique introduced by Marcus and Veron, and the two different techniques are compared.

In this short paper we show that if (X, ∥ · ∥) is a Banach space, Γ a norming set for X and C is a nonempty, bounded and Γ sequentially compact subset of X, then in C the Γ-Opial condition for nets is equivalent to the Γ-Opial condition.