In this paper, we discuss a necessary and sufficient condition for common fixed points of two nonexpansive mappings. We then prove a convergence theorem to a common fixed point. Finally, we discuss the existence of a nonexpansive retraction onto the set of common fixed points of nonexpansive mappings. In these theorems, we do not assume the strict (uniform) convexity of the norm of the Banach space.

We show that the concepts of shadowing and (c, h−) inverse shadowing of circle homeomorphisms are equivalent.

In this paper we introduce and consider a class of multi-valued and single-valued operators of generalised monotone type. We proved a new general lemma on the convergence of real sequences and some new convergence theorems for the Ishikawa and Mann iteration processes with errors to the unique fixed point of such operators, which are not necessarily Lipschitz operators. Our results generalise, improve, and extend several recent results.

The universal localisation RΓ(s) at a semiprime ideal S of a left Noetherian ring R was defined and studied by P. M. Cohn. In this note we investigate the interaction between the universal localisation RΓ(s), the Ore localisation at S, and the torsion-theoretic localisation at the injective envelope E(R/S) of the module R(R/S).

We introduce the concept of Cartan decomposition relative to a Cartan subalgebra H in the sense of Y. Billig and A. Pianzola for involutive complex Lie algebras L of arbitrary dimension. If L has such a decomposition and is infinite dimensional and simple, we show it is *-isomorphic to a direct limit of classical finite dimensional simple involutive Lie algebras of the same type A, B, C, or D.

We prove a general theorem on dissections of quotients of theta-functions. As corollaries, we establish six q-series identities that were conjectured by M.D. Hirschhorn:

In this paper, we give several kinds of characterisations of isomorphic authentication codes by examining a correspondence between optimal authentication codes and some combinatorial designs. The isomorphism classes of some kinds of authentication codes are also enumerated.

The half-angle formulæ, familiar from trigonometry, can be used to compute the polar decomposition of the operator on l2(ℤ) of convolution by δ0 + δ1. This calculation is extended here to a non-commutative setting by computing the polar decomposition of certain convolution operators on the spaces of square integrable functions of free groups.

Sufficient conditions are provided in order that some classes of finite soluble groups, defined by properties of the Sylow normalisers, are saturated formations.

Let Hi(i = 1, 2, …, n), be closed operators in a Banach space X. The generalised initialvalue problem

of the abstract Cauchy problem is studied. We show that the uniqueness of solution u: [0, T1] × [0, T2] × … × [0, Tn] → X of this n-abstract Cauchy problem is closely related to C0-n-parameter semigroups of bounded linear operators on X. Also as another application of C0-n-parameter semigroups, we prove that many n-parameter initial value problems cannot have a unique solution for some initial values.

New results related to the Boas–Bellman generalisation of Bessel' inequality in inner product spaces are given.

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.

We study a quantitative approach to weak noncompactness of operators under the Cobos-Peetre polygon interpolation method for Banach N-tuples. In the case of operators acting between two J-spaces or two K-spaces obtained by this method we prove logarithmically convex-type inequalities for certain operator seminorm vanishing on the subspace of weakly compact operators. Geometrically speaking, in these estimates only some triangles inscribed in the polygon are involved. For operators acting from a J-space to a K-space we prove logarithmically convex-type estimates where all polygon vertices are included. In particular, the estimates obtained here give the new proofs of the results showing the relation between distribution of weakly compact operators among polygon vertices and weak compactness of operators under interpolation.

Let p > 1 real. We Doob-Meyer-decompose Lp(ℙ)-valued positive submartingales such that the martingale and predictable parts are also in Lp(ℙ). We give two variants of such a decomposition. The first one handles also not necessarily right continuous submartingales, since its proof is as discrete in its nature as Doob's archaically decomposition. The second decomposition acts in Lp (ℝ × Ω ℬ ⊗ ℱ, μ ⊗ ℙ) for some finite measure μ on ℝ.

We characterise those inverse semigroups whose proper(non-isomorphic) homomorphic images are all groups. We also show that the bicyclic semigroup is the only such semigroup in certain cases.

We demonstrate fast algorithm to resolve local singularities of algebraic curves. The algorithm is based on the monomial transform and is independent of any other coordinate change. Two new invariants are introduced to gauge the singularities and sharply control the number of algorithmic steps. Our algorithm is applicable to both real and complex domains.

Let T be a tree, f: T → T be a continuous map. We show that if f is pointwise chain recurrent (that is, every point of T is chain recurrent under f), then either fan is identity or fan is turbulent if Fix(f) ∩ End(T) = ∅ or else fan−1 is identity or fan−1 is turbulent if Fix(f) ∩ End(T) ≠  . Here n denotes the number of endpoints of T and, an denotes the minimal common multiple of 2,3,…,n.

We show how to modify any Bratteli diagram E for an AF-algebra A to obtain a Bratteli diagram KE for A whose graph algebra C*(KE) contains both A and C*(E) as full corners.

It is well known that in general the Jacobi–Perron algorithm (a multi-dimensional analogue of the continued fraction algorithm) may or might not acknowledge the dependence over ℚ of its arguments 1, α1,…, αn by truncating itself down to fewer arguments from some step onwards (if so, the algorithm is said to display an ‘interruption’). We show here that if n = 2 then 1, α1, α2 are linearly dependent over ℚ of and only if the Jacobi–Perron Algorithm displays an interruption. We give examples showing this is not so for any n ≥ 3.

In this article, the existence of travelling wave solutions for premixed laminar flames in a model of slow, “constant density” combustion is studied. The model is governed by a simple system of an exothermic chemical reaction in a gas, via the reaction rate function, which is very natural, as we do not impose the assumption of its continuity. The existence of travelling waves is demonstrated and they are shown to be specific heteroclinic orbits of a three dimensional system of ordinary differential equations, connecting the unburned state points to a burned state point. The existence of these solutions is based on some general topological arguments in ordinary differential equations.