We consider the equation

εy″ + p(x)y′ + q(x)y = 0, where ε is a small positive parameter and p vanishes in the interval. Two asymptotic forms of solution are obtained and a rigorous estimate is made of the difference between the exact solutions and the asymptotic forms.

“Fuzzy theories” and “distributive laws” are used to define “fuzzy systems” in an arbitrary category. The resulting minimal realization theory provides new insights even in classical cases (so that, for non-deterministic sequential machines, the minimal realization problem is reformulated in terms of the structure of join-irreducibles in finite lattices). The definition of “fuzzy theory” is of independent interest and meshes well with philosophical aspects of fuzzy set theory.

A semigroup is said to be eventually regular if each of its elements has some power that is regular. Regular and group-bound semigroups are each eventually regular. Idempotent-surjective semigroups are semigroups such that all idempotent congruence classes contain idempotents; eventually regular semigroups are idempotent-surjective. Many results for regular semigroups also hold for eventually regular semigroups or even for idempotent-surjective semigroups and so in particular are also valid for group-bound semigroups. Lallement's lemma is generalized to eventually regular semigroups and the maximum idempotent-separating congruence on such a semigroup is found. Other congruences are considered and the results obtained are applied to yield results on biordered sets.

In this paper we show that there do not exist any real hypersurfaces in a complex two dimensional Grassmannians G2(ℂn+2) with parallel shape operator.

The following conditions for a group G are investigated:

(i) maximal class n subgroups are normal,

(ii) normal closures of elements have nilpotency class n at most,

(iii) normal closures are n–Engel groups,

(iv) G is an (n+1 )-Engel group.

Each of these conditions is a consequence of the preceding one. The second author has shown previously that all conditions are equivalent for n = 1. Here the question is settled for n = 2 as follows: conditions (ii), (iii) and (iv) are equivalent. The class of groups defined by (i) is not closed under homomorphisms, and hence (i) does not follow from the other conditions.

This paper establishes new criteria for meromorphic starlike univalent functions of the form

Further property preserving integrals are considered.

We introduce the concept of a variety of topological groups and of a free topological group F(X, ) of on a topological space X as generalizations of the analogous concepts in the theory of varieties of groups. Necessary and sufficient conditions for F(X, ) to exist are given and uniqueness is proved. We say the topological group FM,(X) is moderately free on X if its topology is maximal and it is algebraically free with X as a free basis. We show that FM(X) is a free topological group of the variety it generates and that if FM(X) is in then it is topologically isomorphic to a quotient group of F(X, ). It is also shown how well known results on free (free abelian) topological groups can be deduced. In the algebraic theory there are various equivalents of a free group of a variety. We examine the relationships between the topological analogues of these. In the appendix a result similar to the Stone-Čech compactification is proved.

Each metacyclic p–group has a natural canonical presentation which is easily derived from the usual presentation. Parameters in the canonical presentation measure how far the group is from splitting, and from being either commutative or dihedral. The structure of the groups is discussed.

Consider the continuous mappings f from a compact real interval to itself. We show that when f has a positive topological entropy (or equivalently, when f has a cycle of order ≠ 2n, n = 0, 1, 2, …) then f has a more complex behaviour than chaoticity in the sense of Li and Yorke: something like strong or uniform chaoticity, distinguishable on a certain level ɛ > 0. Recent results of the second author then imply that any continuous map has exactly one of the following properties: It is either strongly chaotic or every trajectory is approximable by cycles. Also some other conditions characterizing chaos are given.

Characterizations of efficiency and proper efficiency are given for classes of multiple objective fractional optimization problems. These results are then applied to the case of multiple objective fractional linear problems. A dual problem is given for the multiple objective fractional problem and it is shown that for a properly efficient primal solution the dual solution is also properly efficient.

We obtain new sufficient conditions for the oscillation of all solutions of the neutral differential equation with variable coefficients

where P, Q, R ∈ C([t0, ∞), R+), r ∈ (0, ∞) and τ, σ ∈ [0, ∞). Our results improve several known results in papers by: Chuanxi and Ladas; Lalli and Zhang; Wei; Ruan.

The main goal of the paper is the construction of a triangular mapping F of the square with zero topological entropy, possessing a minimal set M such that F|M is a strongly chaotic homeomorphism, as well as other properties that are impossible for continuous maps on an interval.

To do this we define a parametric class of triangular maps on Q × I, where Q is an infinite minimal set on the interval, which are extendable to continuous triangular maps F: I2 → I2. This class can be used to create other examples.

A graph G is called a $(P_{\geq n},k)$ -factor-critical covered graph if for any $Q\subseteq V(G)$ with $|Q|=k$ and any $e\in E(G-Q)$ , $G-Q$ has a $P_{\geq n}$ -factor covering e. We demonstrate that (i) a $(k+1)$ -connected graph G with at least $k+3$ vertices is a $(P_{\geq 3},k)$ -factor-critical covered graph if its toughness $t(G)>{(2+k)}/{3}$ ; (ii) a $(k+2)$ -connected graph G is a $(P_{\geq 3},k)$ -factor-critical covered graph if its isolated toughness $I(G)>{(5+k)}/{3}$ . Furthermore, we show that the conditions on $t(G)$ and $I(G)$ are sharp.

Let R be a ring with prime radical P. The main theorems of this paper are (1) The following conditions are equivalent.: 1) R is a commutative ring in which every element is the sum of two idempotents; 2) R is a ring in which every element is the sum of two commuting idempotents; 3) R satisfies the identity x3 = x. (2) If R is a PI-ring in which every element is the sum of two idempotents, then R/P satisfies the identity x3 = x. (3) Let R be a semi-perfect ring in which every element is the sum of two idempotents. If RRR is quasi-projective, then R is a finite direct sum of copies of GF(2) and/or GF(3).

One of the main problems in Seiberg-Witten theory is to find (SW)-basic classes and their invariants for a given smooth 4-manifold. The rational blow-down procedure introduced by Fintushel and Stern is one way to compute these invariants for some smooth 4-manifolds. In this paper, we extend their results to the general case. That is, we find (SW)-basic classes and Seiberg-Witten invariants for generalised rational blow-down 4-manifolds by using index computations.

We begin the study of Hankel matrices whose entries are logarithmic coefficients of univalent functions and give sharp bounds for the second Hankel determinant of logarithmic coefficients of convex and starlike functions.

We prove here that the (saturated) formation generated by a finite soluble group has only finitely many (saturated) subformations. This answers a question asked by Professor W. Gaschütz. Some partial results are also given in the case of a formation generated by an arbitrary finite group.

Let CL(X) denote the nonempty closed subsets of a metric space X. We answer the following question: in which spaces X is the Kuratowski convergence of a sequence {Cn} in CL(X) to a nonempty closed set C equivalent to the pointwise convergence of the distance functions for the sets in the sequence to the distance function for C ? We also obtain some related results from two general convergence theorems for equicontinuous families of real valued functions regarding the convergence of graphs and epigraphs of functions in the family.

Normed linear spaces possessing the euclidean space property that every bounded closed convex set is an intersection of closed balls, are characterised as those with dual ball having weak* denting points norm dense in the unit sphere. A characterisation of Banach spaces whose duals have a corresponding intersection property is established. The question of the density of the strongly exposed points of the ball is examined for spaces with such properties.