In a recent paper (see [2]), Orrin Frink introduced a method to provide Hausdorff compactifications for Tychonoff or completely regular T1 spaces X. His method utilized the notion of a normal base. A normal base ℒ for the closed sets of a space X is a base which is a disjunctive ring of sets, disjoint members of which may be separated by disjoint complements of members of ℒ.

Let A be an associative algebra over the field F. We denote by ℒ(A) the lattice of all subalgebras of A. By an ℒ-isomorphism (lattice isomorphism) of the algebra A onto an algebra B over the same field, we mean an isomorphism of ℒ(A) onto ℒ(B). We investigate the extent to which the algebra B is determined by the assumption that it is ℒ-isomorphic to a given algebra A. In this paper, we are mainly concerned with the case in which A is a finite- dimensional semi-simple algebra.

Let Q(x1, …, xn) be an indefinite quadratic form in n-variables with real coefficients, determinant D ≠ 0 and signature (r, s), r+s = n. Then it is known (e.g. see Blaney [2]) that there exist constants Γr, s depending only on r and s such for any real numbers c1, …, cn we can find integers x1, …, xn satisfying

The role played by the Möbius function of the lattice of all partitions of a set in the theory of k-statistics and their generalisations is pointer out and the main results conscerning these statistics are drived. The definitions and formulae for the expansion of products of generalished k-statistics are presented from this viewpoint and applied to arrays of random variables whos moments satisfy stitable symmentry constraints. Applications of the theory are given including the calculation of (joint) cumulants of k-statistics, the minimum variace estimation of (generalised) moments and the asymptotic behaviour of generalised k-statistics viewed as (reversed) martingales.

As a continuation of previous work of the first author with Ranjbar [‘A variational inequality in complete CAT(0) spaces’, J. Fixed Point Theory Appl. 17 (2015), 557–574] on a special form of variational inequalities in Hadamard spaces, in this paper we study equilibrium problems in Hadamard spaces, which extend variational inequalities and many other problems in nonlinear analysis. In this paper, first we study the existence of solutions of equilibrium problems associated with pseudo-monotone bifunctions with suitable conditions on the bifunctions in Hadamard spaces. Then, to approximate an equilibrium point, we consider the proximal point algorithm for pseudo-monotone bifunctions. We prove existence of the sequence generated by the algorithm in several cases in Hadamard spaces. Next, we introduce the resolvent of a bifunction in Hadamard spaces. We prove convergence of the resolvent to an equilibrium point. We also prove $\triangle$-convergence of the sequence generated by the proximal point algorithm to an equilibrium point of the pseudo-monotone bifunction and also the strong convergence under additional assumptions on the bifunction. Finally, we study a regularization of Halpern type and prove the strong convergence of the generated sequence to an equilibrium point without any additional assumption on the pseudo-monotone bifunction. Some examples in fixed point theory and convex minimization are also presented.

An algorithm for summing the series , where the coefficients an are assumed known, and the quantities pn, satisfy a linear three term recurrence relation, has been given by Clenshaw [1]. If we suppose that the pn satisfy the recurrence relation where αn and βn are, in general, functions of n, then PN may be found by constructing a sequence {bn} for n = N(−1)0, where the b satisfy the inhomogeneous recurrence relation with the conditions, The sum PN is then given by This result can be readily verified by multiplying each side of equation (1.2) by pn, summing from n = 0 to N, and making use of equations (1.1) and (1.3).

An operator is said to be nice if its conjugate maps extreme points of the dual unit ball to extreme points. The classical Banach-Stone Theorem says that an isometry from a space of continuous functions on a compact Hausdorff space onto another such space is a weighted composition operator. One common proof of this result uses the fact that an isometry is a nice operator. We use extreme point methods and the notion of centralizer to characterize nice operators as operator weighted compositions on subspaces of spaces of continuous functions with values in a Banach space. Previous characterizations of isometries from a subspace M of C0( Q, X) into C0(K, Y) require Y to be strictly convex, but we are able to obtain some results without that assumption. Important use is made of a vector-valued version of the Choquet Boundary. We also characterize nice operators from one function module to another.

We study the genus theory for function fields which is the analogue of the classical genus theory developed by Hasse and Fröhlich.

The notion of (p, k)-epi mappings is introduced. The properties of such mappings are studied and the results obtained are applied to some differential equations.

The set of all nilpotent elements in an IFP near-ring is characterized and necessary and sufficient conditions for the set of all nilpotent elements of an IFP near-ring to form an ideal are obtained.

In the present paper we are concerned with Schröder's functional equation , where ϕ(x) is the unknown function and is a number between 0 and 1: . We shall prove a theorem which generalizes some earlier results on convex solutions of the Schröder equation [4], [5].

It is well known that many of the field equations from theoretical physics (e.g. Einstein field equations, Maxwell's equations, Klein-Gordon equation) can be obtained from a variational principle with a suitably chosen Lagrange density. In the case of the Einstein equations the corresponding Lagrangian is degenerate (i.e., the associated Euler-Lagrange equations are of second order whereas in general these would be of fourth order), while in the cases of the Maxwell and Klein-Gordon equations the Lagrangian usually used is not degenerate.

We employ the Dyson's Lemma of Esnault and Viehweg to obtain a new and sharp formulation of Roth's Theorem on the approximation of algebraic numbers by algebraic numbers and apply our arguments to yield a refinement of the Davenport-Roth result on the number of exceptions to Roth's inequality and a sharpening of the Cugiani-Mahler theorem. We improve on the order of magnitude of the results rather than just on the constants involved.

We give two charaterizations of the Möbius invariant QK spaces, one in terms of a double integral and the other in terms of the mean oscillation in the Bergman metric. Both charaterizations avoid the use of derivatives. Our results are new even in the case of Qp.

The class of finite groups having a subgroup of order 4 which is its own centralizer has been studied by Suzuki [9], Gorenstein and Walter [6], and the present author [11]. The main purpose of this paper is to strengthen Theorem 5 of [11] by using an early result of Zassenhaus [12]. In particular, we find all groups of the class which are core-free, i.e. which have no nontrivial normal subgroup of odd order. As an application, we make a determination of a certain class of primitive permutation groups.

Special numerical formulae for the dilogarithm of powers of a base quantity have been shown to exist by Watson, Coxeter and others. Abel's equation for the dilogarithm was put in this “exponent” form, as a result of which a four-variable, symmetrical equation, also in exponent form, was deduced. From these equations a large number of special numerical results were produced, from which certain properties of a general structural nature emerged, and enabled two new results to be predicted.

The algebraic bases for these results can be grouped in trigonometric or non-trigonometric form, and for the former it seems to be necessary to examine the properties of the dilogarithm in the complex plane. Even so, there are some identities that seem to be outside the scope of the present methods.

It is speculated that certain factorization relations, which can be identified from the equations, may play a substantial role in the results; but so far no analytic derivation of this property has appeared possible.

Since it is no longer fashionable to publish per se results which depend only on algebraic manipulation, many useful and complicated results of quite general interest languish unseen hidden as lemmata in specialist papers. In particular, the theory of transcendental numbers is rich in ingenious techniques for evaluating determinants. These techniques are apparently not well known even to workers in the field of transcendental numbers, let alone to researchers on other areas where the results might find application. This note accordingly discusses a variety of interesting results on determinants and is to be viewed as an appendix to the encyclopaedic volumes of Muir (1911, 1933) which the researcher might approach in order to obtain information in this area. For reasons of motivation a brief mention is made of the context in which the determinants arise.

We consider solutions to the algebraic differential equation $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}f^nf'+Q_d(z,f)=u(z)e^{v(z)}$, where $Q_d(z,f)$ is a differential polynomial in $f$ of degree $d$ with rational function coefficients, $u$ is a nonzero rational function and $v$ is a nonconstant polynomial. In this paper, we prove that if $n\ge d+1$ and if it admits a meromorphic solution $f$ with finitely many poles, then

$$\begin{equation*} f(z)=s(z)e^{v(z)/(n+1)} \quad \mbox {and}\quad Q_d(z,f)\equiv 0. \end{equation*}$$

$f$

$f'p_k(f)+q_m(f)$

$\alpha $

$p_k(f), q_m(f)$

$f$

$k$

$m$

$k\ge m+1$

Sets of independence are studied for compact abelian hypergroups and they are used, along with Riesz products, to investigate lacunarity questions on the dual object. It is shown that bounded Stechkin sets are always Sidon and that every bounded infinite subset of the dual contains an infinite Sidon set which is also a Λ set. Independent sets are shown to always be Sidon and a necessary condition for Sidonicity is provided. A result of Pisier is used to show that for compact non-abelian groups Sidon and central Λ are equivalent. Several applications are provided, primarily to questions regarding lacunarity on compact groups.

The purpose of this paper is to complete results concerning the class ℋ of expansive homeomorphisms having the pseudo orbits tracing property on a compact metric space. We show that hyperbolic homeomorphisms introduced by Mañé in [8] are exactly those in the class ℋ then by the result of [12, 20] they form a class equal to the Smale space introduced by Ruelle in [18]. Next, assuming that the phase space is a smooth manifold, we show that a diffeomorphism is Anosov if and only if it is in the class ℋ and is a lower semi-continuity point of the map which assigns to any diffeomorphism the supremum of its expansive constants (possibly zero). Then we discuss the behavior of the dynamical systems generated by homeomorphisms from ℋ near their basic sets.