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

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).

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.

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.

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.

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.

For non-negative integers n we determine the roots of the trinmial , with a ≠ 0, over a finite field of characteristic p.

Let κ be a field of characteristic 2. The author's previous results (Arch. Math. (1994)) are used to prove the excellence of quadratic extensions of κ. This in turn is used to determine the Witt kernel of a quadratic extension up to Witt equivalence. An example is given to show that Witt equivalence cannot be strengthened to isometry.

A group G is called a BCF-group if there is a positive integer κ such that |X : XG| ≤ κ for each subgroup X of G. The structure of BCF-groups has been studied by Buckley, Lennox, Neumann, Smith and Wiegold; they proved in particular that locally finite groups with the property BCF are Abelian-by-finite. As a group lattice version of this concept, we say that a group G is a BMF-group if there is a positive integer κ such that every subgroup X of G contains a modular subgroup Y of G for which the index |X : Y| is finite and the number of its prime divisors with multiplicity is bounded by κ (it is known that that such number can be characterised by purely lattice-theoretic considerations, and so it is invariant under lattice isomorphisms of groups). It is proved here that any locally finite BMF-group contains a subgroup of finite index with modular subgroup lattice.

A close connection between the strict convexity of the Day norm to the concept of the Gruenhage compacta is shown. As a byproduct we give an elementary characterisation of Gul'ko compacta in the sigma-product of lines and a more elementary proof of Mercourakis' renorming result for Vašák spaces.

We establish a correlation between the structures of a group of power automorphisms of some group and their mutual commutator subgroup and consider the consequences for the norm of a group, and for its capability.

Some new reverses of the Cauchy–Bunyakovsky–Schwarz inequality for n-tuples of real and complex numbers related to Cassels and Shisha–Mond results are given.

Zagier showed that the number of integer solutions to the Markoff equation with components bounded by T grows asymptotically like C(log T)2, where C is explicity computable. Rosenberger showed that there are only a finite number of equations ax2 + by2 + cz2 = dxyz with a, b, and c dividing d, and for which the equation admits an infinite number of integer solutions. In this paper, we generalise Zagier's techniques so that they may be applied to the Rosenberger equations. We also apply these techniques to the equations ax2 + by2 + cz2 = dxyz + 1.

We study the tensor norm defined by a sequence space λ and its minimal and maximal operator ideals associated in the sense of Defant and Floret. Our results extend the classical theory related to the tensor norms of Saphar [16]. They show the key role played by the finite dimensional structure of the ultrapowers of λ in this kind of problems.

In this paper we shall prove that for any 0 < d ≤ 2, holds for n ≥ 1.

As an application, we shall then show that the following recursively defined sequence

satisfies

The difference equation above originates from a heat conduction problem studied by Myshkis (J. Difference Equ. Appl. 3(1997), 89–91).