Let G be a finite group and let H≤G. We refer to |H||CG(H)| as the Chermak–Delgado measure ofH with respect to G. Originally described by Chermak and Delgado, the collection of all subgroups of G with maximal Chermak–Delgado measure, denoted 𝒞𝒟(G), is a sublattice of the lattice of all subgroups of G. In this paper we note that if H∈𝒞𝒟(G) then H is subnormal in G and prove that if K is a second finite group then 𝒞𝒟(G×K)=𝒞𝒟(G)×𝒞𝒟(K) . We additionally describe the 𝒞𝒟(G≀Cp) where G has a nontrivial centre and p is an odd prime and determine conditions for a wreath product to be a member of its own Chermak–Delgado lattice. We also examine the behaviour of centrally large subgroups, a subset of the Chermak–Delgado lattice.

This paper studies the problem of numerical integration over the unit sphere S2 ⊆ ℝ3 for functions in the Sobolev space H3/2(S2). We consider sequences Qm(n), n ∈ ℕ, of cubature (or numerical integration) rules, where Qm(n) is assumed to integrate exactly all (spherical) polynomials of degree ≤ n, and to use m = m(n) values of f. The cubature weights of all rules Qm(n) are assumed to be positive, or alternatively the sequence Qm(n), n ∈ ℕ, is assumed to have a certain local regularity property which involves the weights and the points of the rules Qm(n), n ∈ ℕ. Under these conditions it is shown that the worst-case (cubature) error, denoted by E3/2 (Qm(n)), for all functions in the unit ball of the Hilbert space H3/2 (S2) satisfies the estimate E3/2 (Qm(n)) ≤ c n−3/2, where the constant c is a universal constant for all sequences of positive weight cubature rules. For a sequence Qm(n), n ∈ ℕ, of cubature rules that satisfies the alternative local regularity property the constant c may depend on the sequence Qm(n), n ∈ ℕ. Examples of cubature rules that satisfy the assumptions are discussed.

This paper centres around the variety 0 of distributive Ockham algebras, and those subvarieties of 0 which are generated by a single finite subdirectly irreducible algebra A. We use H.A. Priestley's duality for bounded distributive lattices throughout. First, intrinsic descriptions of the duals of certain finite subdirectly irreducibles are given; these are later used to determine projectives in the dual categories. Next, left adjoints to the forgetful functors from 0 and Var(A) into bounded distributive lattices are obtained, thereby allowing us to describe all free algebras and coproducts of arbitrary algebras. Finally, by applying the duality, we characterize injectivity in Var(A) for each finite subdirectly irreducible algebra A.

We study the class of Tychonoff topological spaces such that the free Abelian topological group A(X) is reflexive (satisfies the Pontryagin-van Kampen duality). Every such X must be totally path-disconnected and (if it is pseudocompact) must have a trivial first cohomotopy group π1(X). If X is a strongly zero-dimensional space which is either metrisable or compact, then A(X) is reflexive.

In this paper, we investigate the value distribution of difference polynomials and prove some difference analogues of results of Hayman and the Brück conjecture.

A closed form expression is given for the eigenvalues and eigenvectors of a symmetric tridiagonal matrix of odd order whose diagonal elements are all equal and whoes superdiagonal elements alternate between the values c and d. An implicit formula is given for the even order case.

We shall give an elementary proof of a characterisation for the Bloch space due to Holland and Walsh, and obtain analogous characterisations for the little Bloch space and Besov spaces of analytic functions on the unit disk in the complex plane.

In this paper we improve and generalise a result of J. Clunie by proving that if f(z) is a transcendental integral function with only zeros of order at least k + 1, then f(k)(z) assumes every finite non-zero complex value infinitely often. Also, the related criterion for normality of a family of holomorphic functions is given, and the value distribution of f2 + afk is discussed.

In this paper we propose an approach to classifying a subclass of filiform Leibniz algebras. This subclass arises from the naturally graded filiform Lie algebras. We reconcile and simplify the structure constants of such a class. In the arbitrary fixed dimension case an effective algorithm to control the behavior of the structure constants under adapted transformations of basis is presented. In one particular case, the precise formulas for less than 10 dimensions are given. We provide a computer program in Maple that can be used in computations as well.

In this paper, we consider the growth of meromorphic solutions of nonlinear differential equations of the form L (f) + P (z, f) = h (z), where L (f) denotes a linear differential polynomial in f, P (z, f) is a polynomial in f, both with small meromorphic coefficients, and h (z) is a meromorphic function. Specialising to L (f) − p (z) fn = h (z), where p (z) is a small meromorphic function, we consider the uniqueness of meromorphic solutions with few poles only. Our results complement earlier ones due to C.-C. Yang.

Recently, Visintin gave conditions under which weak convergence in L1 (T;RN) implies strong convergence. Here we analyze such results in terms of associated young measures and present an extension to L1 (T;Ξ), where Ξ is a separable reflexive Banach space.

In this note a general solution of the problem of the characterisation of quadratic functionals posed by Vukman is given.

In this paper we introduce two classes, namely Γp(m, M) and Σp(m, M), of functions

regular and p-valent in D − {0} where D = {z : |z| < 1}. We show that, for suitable choices of real constants α, β and γ, the integral operators of the form

map into , where is the class of p-valent meromorphically starlike functions of order ρ, 0 ≤ ρ < 1. For the classes Γp(m, M) and Σp(m, M), we obtain class preserving integral operators of the form

with suitable restrictions on real constants α and γ.

Our results generalize almost all known results obtained so far in this direction.

This paper uses Clarke’s generalised directional derivative to describe several types of invexity, pseudoinvexity and quasiinvexity at a point of a nonlinear function. Direct implications of the relations existing between the various types of invexity and generalised invexity are presented, as well as a block diagram of these implications. In particular, similar results in the class of quasiconvex functions are obtained.

Let E be a barrelled space with dual F ≠ E*. It is shown that F has uncountable codimension in E*. If M is a vector subspace of E* of countable dimension with M ∩ F = {o}, the topology τ(E, F+M) is called a countable enlargement of τ(E, F). The results of the two previous papers are extended: it is proved that a non-barrelled countable enlargement always exists, and sufficient conditions for the existence of a barrelled countable enlargement are established, to include cases where the bounded sets may all be finite dimensional. An example of this case is given, derived from Amemiya and Kōmura; some specific and general classes of spaces containing a dense barrelled vector subspace of codimension greater than or equal to c are discussed.

Using a theorem of Jacques Lewin, the zero divisor question is solved for group rings of supersolvable groups.

The main purpose of this work is to give a general and elementary treatment of the fixed point and the coincidence problems for compositions of set-valued maps with not necessarily locally convex domains and to display, once more, the central rôle played by the selection property.

We investigate the structure of the C*-algebras 𝒪ρ constructed by Doplicher and Roberts from the intertwining operators between the tensor powers of a representation ρ of a compact group. We show that each Doplicher-Roberts algebra is isomorphic to a corner in the Cuntz-Krieger algebra 𝒪A of a {0,1}-matrix A = Aρ associated to ρ. When the group is finite, we can then use Cuntz's calculation of the K-theory of 𝒪A to compute K*(𝒪ρ).

It is shown that if a nonconstant entire function f(z) and its derivative f(k)(z), share two finite values (counting multiplicities), then f(z) = f(k)(z), where k is a positive integer.

This paper is concerned with finding periodic solutions of differential delay systems

and

where ri (i = 1, 2,…, n − 1) are positive constants. By using the theory of Hamiltonian systems, we obtain some sufficient conditions under which these systems have many periodic solutions with known periods.