Chen and Grätzer [3], [4] have had great success in describing the properties of Stone lattices by way of representing them as triples. Their triple representation has recently been generalized to distributive pseudo-complemented lattices by Katriňák [6]. By varying the approach slightly Katriňák [5] has been able to obtain a triple representation for distributive pseudo-complemented semilattices that enabled him to characterize semilattices from distributive pseudo-complemented semilattices through to Stone lattices and Brouwer lattices in an elegant unified manner.

It is shown that (n being an integer) any non-trivial finite linear space with n2- 1 points, all of degree at most n+1, is embeddable in a finite projective plane of order n. This generalizes a theorem of Bose and Shrikhande and settles the unsolved case n = 6.

Let L be a Lie algebra over a field κ of any characteristic, and consider the lattice ℒ(L) of all subalgebras of L. In this paper we prove that if L and M are lattice isomorphic Lie algebras, over a field of any characteristic, and L′ and M′ are nilpotent, then the difference between the orders of solvability of L and M differs by at most one.

The purpose of this paper is to improve a result in [2] on the automorphisms of the semigroup D = D(E) of all (Fréchet)-differentiable mappings of a real Banach space E into itself.

In this note we examine the relationships between a subnormal shift, the measure its moment sequence generates, and those of a large family of weighted shifts associated with the original shift. We examine the effects on subnormality of adding a new weight or changing a weight. We also obtain formulas for evaluating point mass at the origin for the measure associated with the shift. In addition, we examine the relationship between the measure associated with a subnormal shift and those of a family of shifts substantially different from the original shift.

A tensor norm β=(βn)∞n=1 is smooth if the natural correspondence where 𝕂=ℝ or ℂ, is always an isometric isomorphism. In this paper we study the representation of multi-ideals and of ideals of multilinear forms by smooth tensor norms.

Let R denote the set of real numbers, B the σ-field of all Borel subsets of R. A homogeneous Markov Chain with state space a Borel subset Ω of R is a sequence {an}, n≧ 0, of random variables, taking values in Ω, with one-step transition probabilities P(1) (ξ, A) defined by for each choice of ξ, ξ0, …, ξn−1 in ω and all Borel subsets A of ω The fact that the right-hand side of (1.1) does not depend on the ξi, 0 ≧ i > n, is of course the Markovian property, the non-dependence on n is the homogeneity of the chain.

We define in an axiomatic fashion a Coxeter datum for an arbitrary Coxeter group $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}W$. This Coxeter datum will specify a pair of reflection representations of $W$ in two vector spaces linked only by a bilinear pairing without any integrality or nondegeneracy requirements. These representations are not required to be embeddings of $W$ in the orthogonal group of any vector space, and they give rise to a pair of inter-related root systems generalizing the classical root systems of Coxeter groups. We obtain comparison results between these nonorthogonal root systems and the classical root systems. Further, we study the equivalent of the Tits cone in these nonorthogonal representations.

In a series of papers K. D. Magill, Jr. (see [1] and its references) has proved that, in various semigroups of mappings on topological spaces, every automorphism is inner, where an automorphism φ of a semigroup A is a bijection of A such that for all ƒ and g in A, and it is said to be inner if there exists a bijection h ∈ A such that h−1 (the inverse of h) belongs to A and for every ƒ ∈ A.

A thin Lie algebra is a Lie algebra graded over the positive integers satisfying a certain narrowness condition. We describe several cyclic grading of the modular Hamiltonian Lie algebras H(2: n; ω2) (of dimension one less than a power of p) from which we construct infinite-dimensional thin Lie algebras. In the process we provide an explicit identification of H(2: n; ω2) with a Block algebra. We also compute its second cohomology group and its derivation algebra (in arbitrary prime characteristic).

A technique for the estimation of domains of local stability for difference equations is discussed. A Liapunov function is used in the estimation. Sharper results are possible if there is only one type of nonlinearity. when open Liapunov surfaces are possible. An example of the technique is given.

Degeneracies occur with increasing frequency in some large scale linear programming problems, but with a simple change to the (revised) simplex method, resulting stationarity of the algorithm can be reduced. The method introduced here may also prevent cycling; neither the lexicographic refinement of Dantzig, Orden and Wolfe nor the perturbation technique of Charnes may be required to prevent cycling.

Pommerenke initiated the study of linearly invariant families of locally schlicht holomorphic functions defined on the unit disk The concept of linear invariance has proved fruitful in geometric function theory. One aspect of Pommerenke's work is the extension of certain results from classical univalent function theory to linearly invariant functions. We propose a definition of a related concept that we call hyperbolic linear invariance for locally schlicht holomorphic functions that map the unit disk into itself. We obtain results for hyperbolic linearly invariant functions which generalize parts of the theory of bounded univalent functions. There are many similarities between linearly invariant functions and hyperbolic linearly invariant functions, but some new phenomena also arise in the study of hyperbolic linearly invariant functions.

In a recent paper [7] the author considered, among other things, the integral transform where is the fundamental solution of the heat equation There we gave a physical interpretation of the transform (1.1). Here we shall choose a slightly different interpretation, more convenient for our present purposes. If then u(O, t) = f(t). That is, the function f(t) defined by equation (1.1) is the temperature at the origin (x = 0) of an infinite bar along the x-axis t seconds after it was at a temperature defined by the equation .

In [2], John Cossey and Sheila Oates Macdonald give a basis for the set of laws of PSL(2, 5) — the simple group of order 60 — and with one extreme exception the laws of their basis involve at most two variables. They raise the problem of finding a basis in which all of the laws involve only a small number of variables, and remark that they have shown that five variables will suffice. Here we give a basis consisting entirely of two variable laws.

As a natural generalisation of $q$-Schur algebras associated with the Hecke algebra ${\mathcal{H}}_{r,R}$ (of the symmetric group), we introduce the queer $q$-Schur superalgebra associated with the Hecke–Clifford superalgebra ${\mathcal{H}}_{r,R}^{\mathsf{c}}$, which, by definition, is the endomorphism algebra of the induced ${\mathcal{H}}_{r,R}^{\mathsf{c}}$-module from certain $q$-permutation modules over ${\mathcal{H}}_{r,R}$. We will describe certain integral bases for these superalgebras in terms of matrices and will establish the base-change property for them. We will also identify the queer $q$-Schur superalgebras with the quantum queer Schur superalgebras investigated in the context of quantum queer supergroups and provide a constructible classification of their simple polynomial representations over a certain extension of the field $\mathbb{C}(\mathbf{v})$ of complex rational functions.

We give continuous separation theorems for convex sets in a real linear space equipped with a norm that can assume the value infinity. In such a space, it may be impossible to continuously strongly separate a point $p$ from a closed convex set not containing $p$, that is, closed convex sets need not be weakly closed. As a special case, separation in finite-dimensional extended normed spaces is considered at the outset.

Let $(X,d,\unicode[STIX]{x1D707})$ be a metric measure space endowed with a distance $d$ and a nonnegative, Borel, doubling measure $\unicode[STIX]{x1D707}$. Let $L$ be a nonnegative self-adjoint operator on $L^{2}(X)$. Assume that the (heat) kernel associated to the semigroup $e^{-tL}$ satisfies a Gaussian upper bound. In this paper, we prove that for any $p\in (0,\infty )$ and $w\in A_{\infty }$, the weighted Hardy space $H_{L,S,w}^{p}(X)$ associated with $L$ in terms of the Lusin (area) function and the weighted Hardy space $H_{L,G,w}^{p}(X)$ associated with $L$ in terms of the Littlewood–Paley function coincide and their norms are equivalent. This improves a recent result of Duong et al. [‘A Littlewood–Paley type decomposition and weighted Hardy spaces associated with operators’, J. Geom. Anal.26 (2016), 1617–1646], who proved that $H_{L,S,w}^{p}(X)=H_{L,G,w}^{p}(X)$ for $p\in (0,1]$ and $w\in A_{\infty }$ by imposing an extra assumption of a Moser-type boundedness condition on $L$. Our result is new even in the unweighted setting, that is, when $w\equiv 1$.

We examine positive semigroups acting on Banach lattices and operator algebras. In the lattice framework we characterize strict positivity and strict ordering of holomorphic semigroups by irreducibility criteria. In the algebraic setting we derive ergodic criteria for irreducibility and discuss various aspects of strict positivity. Finally we examine invariant states of a C*-dynamical system in which the automorphism group is replaced by a strongly positive semigroup. We demonstrate that ergodic states are characterized by a cluster property despite the absence of a covariant implementation law for the semigroup.

Let K be a field and G a finite group with subgroup H. We say that (G H) is a K-free pair if whenever M is a finitely generated KG-module whose restriction MH is a free KH-module, then M is a free KG-module. In this paper pairs of groups with this property will be investigated.