In this paper, K denotes a complete, non-trivially valued, non-archimedean field. The class (lα, lα) of infininite matrices transforming sequences over K in lα to sequences in lα is characterized. Further a Mercerian theorem is proved in the context of the Banach algebra (lα, lα), α ≥ 1 and finally a Steinhaus type result is proved for the space lα. In the case of ℝ or ℀, on the other hand, the best known result so far seems to be a characterization of positive matrix transformations of the class (lα, lβ), ∞ > α ≥ β > 1.

For a modulus function φ, we define the Hardy-Orlicz space H (φ). Two main questions are discussed in this paper. First, when is a linear map mg: H (φ) → H (φ), mg (f) = g.f an isometry? Second, when is H (φ) = H1?

This paper considers an aspect of the general problem of how the structure of a group influences the structure of its automorphisms group. A recent result of Beisiegel shows that if P is a p-group then the central automorphisms group of P has no normal subgroups of order prime to p. So, roughly speaking, most of the central automorphisms are of p-power order. This generalizes an old result of Hopkins that if Aut P is abelian (so every automorphisms is central), then Aut P is a p-group.

This paper uses a different approach to consider the case when P is a π-group. It is shown that the central automorphism group of P has a normal. π′-subgroup only if P has an abelian direct factor whose automorphism group has such a subgroup.

A method of analysing the general linear regression model is described, for the case where the observations are correlated. For many applications the correlations are structured, with neighbouring observations being more strongly correlated than those some distance apart in time or space. Such correlation structures may often be assumed to belong to some class of models indexed by a small number of parameters. Estimation and inference procedures which are able to cope with a wide range of correlation models, are described and the methods are applied to problems which occur in biometry.

Let K (m,M) denote the class of functions regular and satisfying |1 + zf″(z)/f′(z)− m| < M in |z| < 1, where |m−1| < M ≤ m. Recently, R.K. Pandey and G. P. Bhargava have shown that if f ε K (m,M), then the function du also belongs to K (m,M) provided α is a complex number satisfying the inequality |α| ≤ (1−b)/2, where b = (m-1)/M. In this paper we show by a counterexample that their inequality is in general wrong, and prove a corrected version of their result. We show that F ε K (m,M) provided that α is a real number satisfying −φ ≤ α ≤1, φ = (M−|m−1|)/(M + |m−1|), or a complex number satisfying |α| ≤ φ. In both cases the bounds for α are sharp.

It is shown that the free orthomodular word problem is solvable. Since the free orthomodular lattice L0 on countable many generators has, as a partial subalgebra, every finite partial orthomodular lattice P, which is contained in some orthomodular lattice as a partial subalgebra, it is sufficient to prove Evans embedding property for these P only. The construction of the finite orthomodular lattice containing P as a partial subalgebra has and can be done outside of L0. It uses the coatom extension for ortholattices.

We consider Hardy's classical inequality for Cesaro averages and show that a reversed version exists if we restrict ourselves to monotone sequences. Some consequences of this result as well as an integral version are also obtained.

In terms of an arbitrary limit process T, defined abstractly for real functions, we define in a novel way a T-continuous integral of Perron type, admitting mean value theorems, integration by parts and the analogue of the Marcinkiewicz theorem for the ordinary Perron integral. The integral is shown to include, as particular cases, the various known continuous, approximately continuous, cesàro-continuous, mean-continuous and proximally Cesàro-continuous integrals of Perron and Denjoy types. An interesting generalization of the classical Lebesgue decomposition theorem is also obtained.

We prove the existence of nontrivial periodic solutions of some nonlinear ordinary differential equations with time-dependent coefficients using variational methods.

It is shown that if the ring A is left Artinian and L1 and L2 are left ideals of AL1 is nilpotent modulo L2 if L1 is nil modulo L2.

In this note we prove the uniqueness of a projection onto a given subspace with strictly contractive complement. We also show that, if one completely contractive projection is invariant under another, then the two commute.

In this paper we study the additive group structure of subdirectly irreducible rings and their hearts. We give and example of a torsion-free, non-reduced abelian group which is not the underlying additive group of any associative subdirectly irreducible ring. It is a counterexample to a theorem in Feigelstock's book “Additive Groups of Rings.”

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.

In this brief note, we prove the following: Let R be a semi-primitive ring. Suppose that for each pair x, y ε R there exist positive integers m = m (x,y) and n = n (x,y) such that either [xm,(xy) n − (yx) n] = 0 or [xm,(xy) n + (yx) n] = 0. Then R is commutative.

In this paper we employ quasi-continuous multifunctions and introduce almost Baire class 1 multifunctions in order to generalize a theorem of Kuratowski and also to answer a question posed by him concerning Baire class 1 multifunctions. We also show that certain multifunctions can be decomposed into mutually singular multifunctions.

The concept of dependence of subgraphs of a plane graph is defined, as a measure of how much they overlap. It is shown that if M is a 3-connected plane graph, then the number of copies of M in a plane graph which are dependent on a given copy is bounded above by a constant c (M). The number of copies of M in any n-vertex plane graph is at most nc (m).