We carry out an in-depth study of some domination and smoothing properties of linear operators and of their role within the theory of eventually positive operator semigroups. On the one hand, we prove that, on many important function spaces, they imply compactness properties. On the other hand, we show that these conditions can be omitted in a number of Perron–Frobenius type spectral theorems. We furthermore prove a Kreĭn–Rutman type theorem on the existence of positive eigenvectors and eigenfunctionals under certain eventual positivity conditions.

We investigate surjective solutions of the functional equation

$$\begin{eqnarray}\displaystyle \{\Vert f(x)+f(y)\Vert ,\Vert f(x)-f(y)\Vert \}=\{\Vert x+y\Vert ,\Vert x-y\Vert \}\quad (x,y\in X), & & \displaystyle \nonumber\end{eqnarray}$$

$f:X\rightarrow Y$

${\mathcal{L}}^{\infty }(\unicode[STIX]{x1D6E4})$

${\mathcal{L}}^{\infty }(\unicode[STIX]{x1D6E4})$

The maximal left ideal, the maximal right ideal and the maximal two-sided ideal are all characterized for quite general semigroups of endomorphisms. These results are then applied to such semigroups as the semigroup of all continuous selfmaps of a topological space, the semigroup of all closed selfmaps of a topological space, and the semigroup of all linear endomorphisms of a vector space.

We shall work with the Laguerre measure on and the associated Laplacian ℒα, by means of which the Laguerre semigroup is defined. Our main result is a multiplier theorem, saying that a functions of ℒα which is of Laplace transform type defines an operator of weak type (1, 1) for the Laguerre measure. Our starting point is the well-known relationship between the Laguerre and Ornstein-Uhlenbeck semigroups.

A new reverse of Bessel's inequality for orthornormal families in real or complex inner product space is obtained. Applications for some Grüss type results are also provided.

Let ℚp denote the p-adic numbers. We consider curves in defined by p-adic polynomials of one p-adic variable. We show that maximal averages along these curves are bounded, where 1 < q < ∞.

This paper, in conjunction with its algebraic sequel, aims to provide a foundation, long outstanding, to the theory of quasigroups obeying the law x(xy) = yx , otherwise known as Stein quasigroups.

This self-contained paper characterises those locally convex spaces whose (weakly) precompact (respectively, compact) subsets are metrisable. Applications and examples are provided. Our approach also applies to get Cascales-Orihuela's, Valdivia's and Robertson's metrisation theorems for (pre)compact sets.

The structure of inverse semigroup extensions of one inverse semigroup R by any other is analyzed in the case where R is a semilattice. Both a representation and method of construction are given. A brief preliminary examination is made of a certain class of congruences, on inverse semigroups, which are intimately related to such extensions.

We characterise all the perfect k-error correcting codes that can be defined on the graph associated with the Towers of Hanoi puzzle. In particular, a short proof for the existence of 1-error correcting code on such a graph is given.

Real ideals of the ring ℜL of real-valued continuous functions on a completely regular frame L are characterized in terms of cozero elements, in the manner of the classical case of the rings C(X). As an application, we show that L is realcompact if and only if every free maximal ideal of ℜL is hyper-real—which is the precise translation of how Hewitt defined realcompact spaces, albeit under a different appellation. We also obtain a frame version of Mrówka’s theorem that characterizes realcompact spaces.

Let a(z) and b(w) be two rational functions in z or W with algebraic coefficients, where a(0) = 0 and let

Assume that 0 < |z| < 1, that z is not a pole of for n ≥ 0, that w is neither a pole nor a zero of b(w, n) for n ≥ 1, and that the series

for fixed w is a transcendental function of z. Then, if z and W are algebraic numbers, f(z, w) is a transcendental number.

We prove that a Küchle fourfold $X$ of type d3 has a multiplicative Chow–Künneth decomposition. We present some consequences for the Chow ring of $X$.

We calculate the dimensions of using Erokhin's work on Niemeier lattices and geometric methods involving the hyperelliptic locus.

The Fibonacci group F(2, 7) has been known to be cyclic of order 29 for about five years. This was first established by computer coset enumerations which exhibit only the result, without supporting proofs. The working in a coset enumeration actually contains proofs of many relations that hold in the group. A hand proof that F(2, 7) is cyclic of order 29, based on the working in computer coset enumerations, is presented here.

Iséki's BCK-algebras form a quasivariety of groupoids and a finite BCK-algebra must satisfy the identity (En): xyn = xyn+1, for a suitable positive integer n. The class of BCK-algebras which satisfy (En) is a variety which has strongly equationally definable principal congruences, congruence-3-distributivity, and congruence-3-permutability. Thus, a finite BCK-algebra generates a 3-based variety of BCK-algebras. The variety of bounded commutative BCK-algebras which satisfy (En) is generated by n finite algebras, each of which is semiprimal.

We introduce a definition of generalised solutions of the Hessian equation Sm(D2u) = f in a convex set ω ⊂ ℝn, where Sm(D2u) denotes the m-th symmetric function of the eigenvalues of D2u, f ∈ Lp(ω), p ≥ 1, and m ∈ {1, …, n}. Such a definition is given in the class of semi-convex functions, and it extends the definition of convex generalised solutions for the Monge-Ampère equation. We prove that semiconvex weak solutions are solutions in the sense of the present paper.

We prove the existence of an order-preserving function on a class of preordered topological spaces that are inductive limits of preordered spaces. Some applications to economics are given.

In a previous paper (Acta Arith. 21 (1972), 89–97), I had proved that the sum of the absolute values of the coefficients of the mth transformation polynomial Fm (u, v) of the Weber modular function j(ω) of level 1 is not greater than 2(36n+57)2n when m = 2n is a power of 2. The aim of the present paper is to give an analogous bound for the case of general m. This upper bound is much less good and of the form where c > 0 is an absolute constant which can be determined effectively. It seems probable that also in the general case an upper bound of the form eO(m10gm) should hold, but I have not so far succeeded in proving such a result.

The purpose of this paper is to introduce a nonlinear scalarisation function for solving a class of implicit vector equilibrium problems. We prove a scalarisation lemma to show the relation between the implicit vector equilibrium problem and the nonlinear scalarisation function. Then we derive some new existence theorems for solutions of implicit vector equilibrium problems, using the scalarisation lemma and the FKKM theorem.