The minimal (faithful) degree μ(G) of a finite group G is the least positive integer n such that G ≲ Sn. Clearly if H ≤ G then μ(H) ≤ μ(G). However if N ◃ G then it is possible for μ(G/N) to be greater than μ(G); such groups G are here called exceptional. Properties of exceptional groups are investigated and several families of exceptional groups are given. For example it is shown that the smallest exceptional groups have order 32.

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.

The problem of exponential stability of the problem of transmission of the wave equation with lower-order terms is considered. Making use of the classical energy method and multiplier technique, we prove that this problem of transmission is exponentially stable.

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.

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.

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.

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.

We apply the Schwarz lemma to find general formulas for the third coefficient of Carathéodory functions dependent on a parameter in the closed unit polydisk. Next we find sharp estimates of the Hankel determinant $H_{2,2}$ and Zalcman functional $J_{2,3}$ over the class ${\mathcal{C}}{\mathcal{V}}$ of analytic functions $f$ normalised such that $\text{Re}\{(1-z^{2})f^{\prime }(z)\}>0$ for $z\in \mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$, that is, the subclass of the class of functions convex in the direction of the imaginary axis.

The paper introduces a graph theory variation of the general position problem: given a graph $G$, determine a largest set $S$ of vertices of $G$ such that no three vertices of $S$ lie on a common geodesic. Such a set is a max-gp-set of $G$ and its size is the gp-number $\text{gp}(G)$ of $G$. Upper bounds on $\text{gp}(G)$ in terms of different isometric covers are given and used to determine the gp-number of several classes of graphs. Connections between general position sets and packings are investigated and used to give lower bounds on the gp-number. It is also proved that the general position problem is NP-complete.

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).

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.

We show that a ruled surface of finite type in a Euclidean space is a cylinder on a curve of finite type or a helicoid in Euclidean 3-space.

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.

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.

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.

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.