It is shown that a quadratic system with a degenerate critical point has at most one limit cycle.

Let BO, BSO and BSpin be the classifying spaces for the infinite orthogonal, infinite special orthogonal and infinite spinor groups respectively. It is well known that their integral cohomology rings have torsion only of order 2. In this paper we present an elementary proof that for the 7-connective cover of BO, BO〈8〉, the integral cohomology ring H* (BO〈8〉; Z) too has torsion only of order 2. The method follows that of Borel and Hirzebruch and a result of Wu concerning the Steenrod reduced mod p operation for an odd prime p on the Pontrjagin classes.

In a previous paper, the author sought to classify those solutions of second order nonlinear ordinary differential equations which can be expressed as sums of solutions of related equations. In that paper one sub-class of solutions was overlooked. This paper is to remedy that defect.

We discuss purity and pure essentialness of abelian groups in a topos of sheaves on a locale and show that purity is not a local property. We prove that is divisible if and only if it is pure in every extension, and give an example of a category in winch absolutely pure does not imply divisible. We discuss uniform abelian groups and show that each AU uniform in Ab does not imply that A is uniform in

Banaschewski showed that the pure subgroups of are exactly of the type for the different . We show that is essential in if and only if U is dense in , Finally, we characterise as complete boolean algebras the locales for which the only pure and essential subgroup of is .

The main result of this paper is that there exist non-principal left ideals in a certain twisted group algebra A of the infinite dihedral group ≤ a, b | b−1ab = a−1, b2 = 1 ≥ over the field R of real numbers: namely in the A defined by b−l ab = a−1, b2 = −1, and λa = aλ, λb = bλ for all real λ.

The motivation comes from the study (in a series of papers by Berman and the author) of finitely generated torsion-free RG-modules for groups G which have an infinite cyclic subgroup of finite index. In a sense, this amounts to studying modules over (full matrix algebras over) a finite set of R-algebras [namely, for the groups in question, these algebras take on the role played by R, C and H (the real quaternions) in the theory of real representations of finite groups]. For all but two algebras in that finite set, satisfying results have been obtained by exploiting the fact that each of them is either a ring with zero divisors or a principal left ideal ring. The other two are known to have no zero divisors. One of them is the present A. The point of the main result is that new ideas will be needed for understanding A-modules.

A number of subsidiary results are concerned with convenient generating sets for left ideals in A.

In this paper I prove that if a semigroup S is stable then ∧L(S) and ∧R(S) (the Rhodes expansions), and ∧+(SA) (the iteration of those expansions) are also stable. I also prove that if S is stable and has a J-depth function then these expansions also have a J-depth functon. More generally, if X →→ S is a J*-surmorphism and if S is stable and has a J-depth function then X has a J-depth function. All these results are needed for the structure theory of semigroups which are stable and have a J-depth function.

The techniques used were originally developed by the author to prove that ∧+(SA) is finite if S is finite (later Rhodes found a much more direct proof of that result).

We characterise relative weak compactness in σBM(∑, X), the space of sigma-additive, X-valued measures of bounded variation, where X is a Banach space.

Let λ1, λ2, λ3 be non-zero reals, not all of the same sign and such that at least one ratio λi/λj is irrational. Then it is proved that for any given integer k ≥ 1 and real η, the inequaltiy

is solvable for every ε > 0. More general and sharper results are also proved.

We investigate the set of limit points of the continued fractions

where x1, x2, … is a given sequence of positive integers. We show that this set is closed, and that it may include any given countable subset of [0, 1] if the integers xk are chosen appropriately. Our main result, which has applications in transcendence theory, is that the sequence of continued fractions has no rational limit point when the sequence {xk} of partial quotients is bounded.

Let A be a semisimple Banach algebra with ‖ · ‖, which is a dense subalgebra of a semisimple Banach algebra B with | · | such that ‖ · ‖ majorises | · | on A. The purpose of this paper is to investigate the annihilator property between the algebras A and B.

Let (Xn)n≥1 be a sequence of random variables with zero means and uniformly bounded variences. Let τn be the stopping time defined on a given Brownian motion (Bt)t≥0, B0 = 0, by Dubins' method such that B(τn) has the same distribution as Xn. We prove that Xn converging to 0 in distribution implies that τn converges to 0 in probability. Examples are presented to illustrate the result is the best possible.

In this paper certain estimates of the rate of convergence of triangular matrix means of the Fourier Stieltjes series and its conjugate series are obtained.

A complete description is given of rings isomorphic to their unbounded left (right) ideals. The same problem for 2-sided ideals remains open.

A uniform space can be given a boundedly compact compatible psuedometric if and only if it is uniformly locally compact and second countable and has a countable base for its entourages.

A geometrically aberrant Banach space with uniformly normal structure, a modification of Brown's example, is given.

Using the Iwasawa structure theorem for connected locally compact Hausdorff groups we show that every locally compact Hausdorff group G is homeomorphic to Rn × K × D, where n is a non-negative integer, K is a compact group and D is a discrete group. This makes recent results on cardinal numbers associated with the topology of locally compact groups more transparent. For abelian G, we note that the dual group, Ĝ, is homeomorphic to This leads us to the relationship card G = ω0(Ĝ) + 2ω0(G), where ω (respectively, ω0) denotes the weight (respectively local weight) of the topological group. From this classical results such as card G = 2 card Ĝ for compact Hausdorff abelian groups, and ω(G) = ω(Ĝ) for general locally compact Hausdorff abelian groups are easily derived.

In the present paper the asymptotic and globally asymptotic stability of the zero solution of systems with impulses are investigated. For this purpose piecewise continuous auxiliary functions are used which are analogous to Lyapunov's functions. The theorem of Marachkov on the asymptotic stability of systems without impulses is generalised. The results obtained are formulated in four theorems.

Let I = [0, 1] and let f be an unimodal expanding map in C0(I, I). If f has an expanding constant for some integers m ≧ 0 and n ≧ 1, where λn is the unique positive zero of the polynomial x2n+1 − 2x2n−1 −1, then we show that f has a periodic point of period 2m(2n + 1). The converse of the above result is trivially false. The condition in the above result is the best possible in the sense that we cannot have the same conclusion if the number λn is replaced by any smaller positive number and the generalisation of the above result to arbitrary piecewise monotonic expanding maps in C0(I, I) is not possible.

The method of nondiscrete mathematical induction is used to find sharp error bounds for Newton's method. We assume only that the operator has Hölder continuous derivatives. In the case when the Fréchet-derivative of the operator satisfies a Lipschitz condition, our results reduce to the ones obtained by Ptak and Potra in 1972.

Bear's characterisation of central-by-finite groups as groups possessing a finite covering by abelian subgroups if the starting point for this invertigation. We characterise groups with a finite covering by 2-Engel subgroups as groups for which the subgroup of right 2-Engel elements has finite index; and the groups having a finite covering by normal 2-Engel subgroups are exactly the 3-Engel groups among those having a finite covering by 2-Engel subgroups. The second centre of a group having a finite covering by class two subgroups does not necessarily have finite index. However, a group has a finite covering by subgroups in a variety containing all cyclic groups if the margin of thes variety in the group has finite index.