It is a well-known result of W. Magnus [3] that there is a faithful matrix representation for metabelian groups i.e the groups satisfying the law ‘x, y; u, v’. The work in this paper arose in an attempt by the author to find a faithful matrix representation for centre-by-metabelian groups i.e. the groups satisfying the law ‘ x, y; u, v].

We establish a one-to-one “group-like” correspondence between congruences on a free monoid X* and so-called positively self-conjugate inverse submonoids of the polycyclic monoid P(X). This enables us to translate many concepts in semigroup theory into the language of inverse semigroups.

A module M over a ring R is κ-projective, κ a cardinal, if M is projective relative to all exact sequence of R-modules 0 → A → B → C → 0 such that C has a generating set of cardinality less than κ. A structure theorem for κ-projective modules over Dedekind domains is proven, and the κ-projectivity of M is related to properties of ExtR (M, ⊕ R). Using results of S. Chase, S. Shelah and P. Eklof, the existence of non-projective и1-projective modules is shown to undecidable, while both the Continuum Hypothesis and its denial (Plus Martin's Axiom) imply the existence of a reduced И0-projective Z-module which is not free.

In order to classify solvable groups Philip Hall introduced in 1939 the concept of isoclinism. Subsequently he defined a more general notion called isologism. This is so to speak isoclinism with respect to a certain variety of groups. The equivalence relation isologism partitions the class of all groups into families. The present paper is concerned with the internal structure of these families.

In this paper we shall extend the classical theory of Morita equivalence to semigroups with local units. We shall use the concept of a Morita context to rediscover the Rees theorem and to characterise completely 0-simple and regular bisimple semigroups.

We consider a meromorphic function of finite lower order that has ∞ as its deficient value or as its Borel exceptional value. We prove that the set of limiting directions of its Julia set must have a definite range of measure.

The relation between the fractional integral operator and the fractional maximal operator is investigated in the framework of Morrey spaces. Applications to the Fefferman–Phong and the Olsen inequalities are also included.

The greatest lower bound (in fact, ½) is found of constants k such that for an infinity of rationals p/q.

We present an operator algebraic approach to Wigner's unitary-antiunitary theorem using some classical results from ring theory. To show how effective this approach is, we prove a generalization of this celebrated theorem for Hilbert modules over matrix algebras. We also present a Wigner-type result for maps on prime C*-algebras.

Let Sn denote the sphere of all points in Euclidean space Rn + 1 at a distance of 1 from the origin and Dn + 1 the ball of all points in Rn + 1 at a distance not exceeding 1 from the origin The space X is said to be aspherical if for every n ≧ 2 and every continuous mapping: f: Sn → X, there exists a continuous mapping g: Dn + 1 → X with restriction to the subspace Sn equal to f. Thus, the only homotopy group of X which might be non-zero is the fundamental group τ1(X, *) ≅ G. If X is also a cell-complex, it is called a K(G, 1). If X and Y are K(G, l)'s, then they have the same homotopy type, and consequently

A Latin square is considered to be a set of n2 cells with three block systems. An automorphisni is a permutation of the cells which preserves each block system. The automorphism group of a Latin Square necessarily has at least 4 orbits on unordered pairs of cells if n < 2. It is shown that there are exactly 4 orbits if and only if the square is the composition table of an elementary abelian 2-group or the cclic group of order 3.

Let ${\mathcal{S}}$ be the family of analytic and univalent functions $f$ in the unit disk $\mathbb{D}$ with the normalization $f(0)=f^{\prime }(0)-1=0$, and let $\unicode[STIX]{x1D6FE}_{n}(f)=\unicode[STIX]{x1D6FE}_{n}$ denote the logarithmic coefficients of $f\in {\mathcal{S}}$. In this paper we study bounds for the logarithmic coefficients for certain subfamilies of univalent functions. Also, we consider the families ${\mathcal{F}}(c)$ and ${\mathcal{G}}(c)$ of functions $f\in {\mathcal{S}}$ defined by

$$\begin{eqnarray}\text{Re}\biggl(1+{\displaystyle \frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}}\biggr)>1-{\displaystyle \frac{c}{2}}\quad \text{and}\quad \text{Re}\biggl(1+{\displaystyle \frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}}\biggr)<1+{\displaystyle \frac{c}{2}},\quad z\in \mathbb{D},\end{eqnarray}$$

$c\in (0,3]$

$c\in (0,1]$

$|\unicode[STIX]{x1D6FE}_{n}|$

$n=1,2,3$

$f$

${\mathcal{F}}(c)$

${\mathcal{G}}(c)$

• ∙ If $f\in {\mathcal{F}}(-1/2)$, then $|\unicode[STIX]{x1D6FE}_{n}|\leq 1/n(1-(1/2^{n+1}))$ for $n\geq 1$, and

  $$\begin{eqnarray}\mathop{\sum }_{n=1}^{\infty }|\unicode[STIX]{x1D6FE}_{n}|^{2}\leq {\displaystyle \frac{\unicode[STIX]{x1D70B}^{2}}{6}}+{\displaystyle \frac{1}{4}}~\text{Li}_{2}\biggl({\displaystyle \frac{1}{4}}\biggr)-\text{Li}_{2}\biggl({\displaystyle \frac{1}{2}}\biggr),\end{eqnarray}$$
  where $\text{Li}_{2}(x)$ denotes the dilogarithm function.

• ∙ If $f\in {\mathcal{G}}(c)$, then $|\unicode[STIX]{x1D6FE}_{n}|\leq c/2n(n+1)$ for $n\geq 1$.

We determine all conjugacy classes of maximal p-local subgroups of the Monster for p ≠ 2.

The problem of the survival of a single mutant in a haploid genetic population when there exists selection is considered for a type of population model in which the generations ar overlapping. The results are compared with the previous work of Fisherand others for other models. The need is stressed for a solution of the same problem in a diploid population with general phenotypic selection coefficients.

We construct the freest idempotent-pure expansion of an inverse semigroup, generalizing an expansion of Margolis and Meakin for the group case. We also generalize the Birget-Rhodes prefix expansion to inverse semigroups with an application to partial actions of inverse semigroups. In the process of generalizing the latter expansion, we are led to a new class of idempotent-pure homomorphisms which we term F-morphisms. These play the same role in the theory of idempotent-pure homomorphisms that F-inverse monoids play in the theory of E-unitary inverse semigroups.

We study the boundedness for multilinear fractional integrals on spaces as Morrey spaces and Lipschitz spaces.

The problem of storage in an infinite dam with a continuous release has been studied by a number of authors ([5], [3], [2]), who have formulated it in probabilistic terms by supposing the input to be a continuous time stochastic process. These authors have encountered difficulties which they have overcome by regarding the continuous time problem as a limit of discrete time analogues. analogues. The purpose of this paper is to suggest that these difficulties are the result of an unfortunate specification of the problem, and to show that the adoption of a slightly different (and more realistic) formulation avoids the difficulties and allows a treatment which does not have recourse to discrete time analogues.

The present paper deals with the Clifford groups in the case p = 2. For the most part, it runs parallel to the previous paper I ([1]) on the case p > 2, and a number of proofs are therefore either given in outline or omitted. A general introduction to both papers is given in I, § 1.

Suppose S is a regular semigroup and E is its set of idempotents. If E is subsemigroup of S, then S has been called orthodox and studied recently by Hall [3], Meakin [6], and Yamada [8]. In this paper we assume that E is not (necessarily) a subsemigroup of S and consider the subsemigroup generated by E, denoted <E>. If E denotes the set of all elements of S which can be written E, denoted <E>. If E denotes the set of all elements of S which can be written as the product of n (not necessarily distinct) idempotents of S, then . We show that <E> is always a regular subsemigroup of S and investigate relationships between it and S. The case where <E> = S is of particular interest to us; such semigroups will be referred to as idempotent-generated regular semi- groups.

We will consider a reflexive Banach space 𝔅, with real or complex scalars, and a bounded operator in 𝔅 with a real spectrum.