Subnormal and quasinormal tensor product operators and generalized derivations on the Hilbert-Schmidt class will be characterized.

We introduce a new class of generalized arcwise connected functions and discuss their basic properties. Our generalization is illustrated by an example and an application is given for a mathematical programming problem involving this new class of functions.

Let f(x) be a continuous function from a compact real interval into itself with a periodic orbit of minimal period m, where m is not an integral power of 2. Then, by Sharkovskii's theorem, for every positive integer n with m → n in the Sharkovskii ordering defined below, a lower bound on the number of periodic orbits of f(x) with minimal period n is 1. Could we improve this lower bound from 1 to some larger number? In this paper, we give a complete answer to this question.

Let P(z) be a polynomial of degree n and P′(z) be its derivative. Given a zero of P′(z), we shall determine regions which contains at least one zero of P(z). In particular, it will be shown that if all the zeros of P(z) lie in |z| < 1 and W1, W2, …, Wn−1 are the zeros of P′(z), then each of the disks |(z/2)–wj| < ½ and |z–Wj| < 1, j = 1, 2, …, n−1 contains at least one zero of P(z). We shall also determine regions which contain at least one zero of the polynomials mP(z) + zP′(z) and P′(z) under some appropriate assumptions. Finally some other results of similar nature will be obtained.

Let L denote the subvariety lattice of the variety of distributive double p-algebras, that is, the lattice whose universe consists of all varieties of distributive double p-algebras and whose ordering is the inclusion relation. We prove in this paper that each proper filter in L is uncountable. Moreover, we prove that except for the trivial variety (the zero in L) and the variety of Boolean algebras (the unique atom in L) every other element of L, generated by a finite algebra, has infinitely many covers in L, among which at least one is not generated by any finite algebra. The former result strengthens a result of Urquhart who showed that the lattice L is uncountable. On the other hand, both of our results indicate a high complexity of the lattice L at least in comparison with the subvariety lattice of the variety of distributive p-algebras, since a result of Lee shows that the latter lattice forms a chain of type ω + 1 and every cover in it of the variety generated by a finite algebra is itself generated by a finite algebra.

A family of n k-subsets of the integers modulo ν are said to be supplementary difference sets if developing them by addition modulo ν leads to a balanced incomplete block design, and to be minimal if no proper subfamily leads to a balanced incomplete block design when developed modulo ν. In other words, the family of supplementary difference sets is minimal precisely when it leads to a balanced incomplete block design which cannot be partitioned into a union of proper subdesigns, each consisting of complete cyclic sets of ν blocks. We discuss the conditions under which such a balanced incomplete block design can be partitioned in some non-cyclic fashion into a union of proper subdesigns.

In 1978 Anderson and Gardner investigated semisimple classes and recently Buys and Gerber developed the theory of special radicals in Andrunakievich varieties. In this note we continue the study of radical theory in Andrunakievich varieties. Sharpening some of the results of Anderson and Gardner we prove versions of Sands' Theorem characterizing semisimple classes by regularity, coinductivity and being closed under extensions. In the proof we follow a new method which avoids calculations with defining identities of the variety. We generalize van Leeuwen's Theorem characterizing semisimple classes of hereditary radicals as classes being regular and closed under essential extensions and subdirect sums.

The purpose of this paper is to study the trajectory multifunction Φ(·) determined by the indefinite set valued integral of a measurable Banach space valued multifunction F(·), that is for all t ∈ [0, T], , where the set valued integral is interpreted in the sense of Aumann. We study the topological and algebraic properties of SΦ equaling the set of selectors of Φ(·) whose primitive is an integrable selector of F(·). We also determine several useful properties that Φ(·) possesses and finally we present some convergence and stability results using the Kuratowski-Mosco convergence of sets.

A new notion of α-connectedness (α-path connectedness) in general topological spaces is introduced and it is proved that for a real-valued function defined on a space with this property, the cardinality of the antipodal coincidence set is at least as large as the cardinal number α. In particular, in linear topological spaces, the antipodal coincidence set of a real-valued function has cardinality at least that of the continuum. This could be regarded as a treatment of some Borsuk-Ulam type results in the setting of general topology.

Let X be a non-empty set and let H(X) denote a Hibert space of complex-valued functions on X. Let T be a mapping from X to X and θ a mapping from X to C such that for all f in H(X), f ° T is in H(x) and the mappings CT taking f to f ° T and M taking f to θ.f are bounded linear operators on H(X). Then the operator CTMθ is called a weighted composition operator on H(X). This note is a report on the characterization of weighted composition operators on functional Hilbert spaces and the computation of the adjoint of such operators on L2 of an atomic measure space. Also the Fredholm criteria are discussed for such classes of operators.

New oscillation criteria for nonlinear differential equations with deviating arguments of the form

n even, are established.

Namioka proved that the following condition (*) given below holds, if X is Čech-complete and Y is a locally compact, σ-compact space.

(*) Let X and Y be spaces, Z be a metric space and let f: X × Y → Z be separately continuous. Then there is a dense, Gδ set A in X such that A × Y ⊂ C(f).

Following Christensen a space X is called Namioka if (*) is true for any compact space Y. In this paper we introduce and study a new class of spaces which is closely related to Namioka spaces. Namely, we say that a space Y is co-Namioka if (*) holds for any Namioka space X.

It is proved here that for two distribution functions with equal moments up to order n, the number of crossings is at least n, and if exactly n, the remaining odd or even moments (for n even or odd respectively) do not cross again. This both generalises and extends a number of previous results.

In this paper some results concerning the Cauchy functional equation, that is the functional equation f(x+y) = f(x) + f(y) in complex hermitian Banach *-algebras with an identity element are presented. As an application a generalization of Kurepa's extension of the Jordan-Neumann characterization of pre-Hilbert space is obtained.

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.

The paper justifies a method of bilateral approximations for finding the periodic solution of a non-linear system of differential equations with impulsive perturbations at fixed moments of time.

This note completes the determination of the basis numbers of the complete bipartite graphs.

Let m and n be coprime positive integers. The variety (consisting of all groups G such that for some normal subgroup H of G, H is abelian of exponent dividing m and G/H is abelian of exponent dividing n) and the variety both satisfy the following three laws:

all elements have order dividing mn;

the commutator of two mth powers has order dividing m;

the commutator of two nth powers has order dividing n.

It is proved that any law which holds in both these varieties (notably that commutators commute) is a consequence of the above three laws.