On Powerful Normal Subgroups of a p-Group

 In this paper we investigate the action of a p-group G on its powerful normal subgroup N. We shall mainly be concerned to find conditions which guarantee that G acts uniserially on N. We shall also study what consequences uniserial action on N has on the structure of N. (Received 21 August 1998; in revised form 28 June 1999)     

Modulus of continuity of harmonic functions

LetG be a bounded plane domain, the diameters of whose boundary components have a fixed positive lower bound. Letu be harmonic inG and continuous in the closureG ofG. Suppose that the modulus of continuity ofu on the boundary ofG is majorized by a function of a suitable type. We shall then obtain upper bounds for the modulus of continuity ofu inG. Further, we shall show that in some situations these estimates cannot be essentially improved. We shall also consider the same problem for certain bounded domains in space. Research partially supported by the U.S. National Science Foundation. AMS (1980) Classification. Primary 31A05.     

Localic Krull dimension

We shall define localic Krull dimension for topological spaces. In particular, a space X has the localic Krull dimension n if n is the greatest number such that X can be mapped, via a continuous and open map, onto the n-chain seen as an Alexandroff space. We shall discuss the applications of this concept in obtaining topological completeness results in modal logic. We shall also show how the localic Krull dimension is related to the Krull dimension in ring theory. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Heisenberg uniqueness pairs and a theorem of Beurling and Malliavin

Let T denote the unit circle in the plane. For various simple sets Λ in the plane we shall study the question whether (T,Λ) is a Heisenberg uniqueness pair. For example, we shall consider the cases where Λ is a circle or a union of two straight lines. We shall also use a theorem of Beurling and Malliavin.     

Remarks on the Cauchy problem for first order hyperbolic systems with constant coefficient principal part

Abstract In this article we shall introduce the results obtained in [16], i.e., we shall give a necessary and sufficient condition that the Cauchy problem for first order hyperbolic systems with constant coefficient principal part is C^∞ well-posed under the maximal rank condition (see the condition (R) below). We shall also give a simple sufficient condition without any assumptions on the rank. Keywords: Hyperbolic system, Cauchy problem, Constant coefficient principal part     

H‐self‐adjoint matrices. Application to radiosity

In computer graphics, in the radiosity context, a linear system Φx=b must be solved and there exists a diagonal positive matrix H such that H Φ is symmetric. In this article, we extend this property to complex matrices: we are interested in matrices which lead to Hermitian matrices under premultiplication by a Hermitian positive‐definite matrix H. We shall prove that these matrices are self‐adjoint with respect to a particular innerproduct defined on ?ⁿ. As a result, like Hermitian matrices, they have real eigenvalues and they are diagonalizable. We shall also show how to extend the Courant–Fisher theorem to this class of matrices. Finally, we shall give a new preconditioning matrix which really improves the convergence speed of the conjugate gradient method used for solving the radiosity problem. Copyright © 2002 John Wiley & Sons, Ltd.     

Mutual subordination of multivariate stationary processes over any locally compact abelian group

Summary Our purpose is to extend Kolmogorov's theorem [5, Th. 10] on mutual subordination for univariate weakly stationary stochastic processes over the (discrete) group of integers to multivariate processes over any (Hausdorff) locally compact abelian (lca) group. This extension is given in Theorems (1.12) and (3.4) below. We shall lean heavily on the joint paper [10] on the decomposition of matricial measures, to which the present paper may be regarded as a sequel.In Section 1 of the paper we shall define and prove theorems on the concept of E-subordination, where E is a projection-valued measure. In Section 2 we shall examine the structure of stationary processes over an lca group. In Section 3 we shall consider the concept of subordination of stationary processes. Finally in Section 4, we shall apply our subordination theorems to deduce that matrixvalued functions in L ₂ on the unit circle having no negative frequencies have a constant rank a.e. (Lebesgue) (4.2), a theorem of F. and M. Riesz (4.3), and a theorem on wandering subspaces due to Robertson [9] (4.4).The author is grateful to Prof. P. R. Masani for his generous help in formulating this paper.     

Some calculations on the conditional densities of well-admissible measures on linear spaces

When a probability measurem on a topological vector spaceE is well-admissible in a directionk E, the conditional law in the directionk given the other directions is absolutely continuous with respect to the Lebesgue measure. We shall prove that its density function is differentiable (in the sense precised below) and we shall calculate their derivatives. We given then two applications of such calculations.     

Approximate Optimal Control and Stability of Nonlinear Finite- and Infinite-Dimensional Systems

We consider first nonlinear systems of the formx=A(x)x+B(x)u together with a standard quadratic cost functional and replace the system by a sequence of time-varying approximations for which the optimal control problem can be solved explicitly. We then show that the sequence converges. Although it may not converge to a global optimal control of the nonlinear system, we also consider a similar approximation sequence for the equation given by the necessary conditions of the maximum principle and we shall see that the first method gives solutions very close to the optimal solution in many cases. We shall also extend the results to parabolic PDEs which can be written in the above form on some Hilbert space.     

On the metric properties of multimodal interval maps and <Emphasis Type="Italic">C</Emphasis><Superscript>2</Superscript> density of Axiom A

In this paper, we shall prove that Axiom A maps are dense in the space of C² interval maps (endowed with the C² topology). As a step of the proof, we shall prove real and complex a priori bounds for (first return maps to certain small neighborhoods of the critical points of) real analytic multimodal interval maps with non-degenerate critical points. We shall also discuss rigidity for interval maps without large bounds. Mathematics Subject Classification (2000) Primary 37E05; Secondary 37F25     

Remarks on annihilators preserving congruence relations

In this note we shall give some results on annihilators preserving congruence relations, or AP-congruences, in bounded distributive lattices. We shall give some new characterizations, and a topological interpretation of the notion of annihilator preserving congruences introduced in [JANOWITZ, M. F.: Annihilator preserving congruence relations of lattices, Algebra Universalis 5 (1975), 391–394]. As an application of these results, we shall prove that the quotient of a quasicomplemented lattice by means of a AP-congruence is a quasicomplemented lattice. Similarly, we will prove that the quotient of a normal latttice by means of a AP-congruence is also a normal lattice.     

Weak‐quasi‐Stone algebras

In this paper we shall introduce the variety WQS of weak‐quasi‐Stone algebras as a generalization of the variety QS of quasi‐Stone algebras introduced in [9]. We shall apply the Priestley duality developed in [4] for the variety N of ¬‐lattices to give a duality for WQS. We prove that a weak‐quasi‐Stone algebra is characterized by a property of the set of its regular elements, as well by mean of some principal lattice congruences. We will also determine the simple and subdirectly irreducible algebras (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Functions of translation type and functorial properties of Segal algebras I

In [9]H. Reiter introduced functions of translation type on a locally compact group to show some functorial properties of the spaceS (G) of Schwartz-Bruhat functions. We shall investigate functorial properties of the Segal algebraP ¹ (G) which is defined by means of a more general version of functions of translation type without using a norm. Furthermore we shall introduce a new Segal algebraE ¹ (G) that is closely related to these functorial properties.     

Efficacy of cones on topological vector spaces and application to common fixed points of multifunctions

Let (E, τ) be a topological vector space and P a cone in E. We shall define a topology τ _P on E so that (E, τ _P ) is a normable topological vector space and P is a normal cone with normal constant M = 1. Then by using the norm, we shall give some results about common fixed points of two multifunctions on cone metric spaces.     

On finite loops with nilpotent inner mapping groups

We shall show how the nilpotency class of a finite loop Q is determined by the properties of a nilpotent inner mapping group. We also show that a classical result by Baer on the structure of abelian finite capable groups holds for Moufang loops of odd order.     

Abel transform on PSL(2, ℝ) and some of its applications

We shall investigate the use of Abel transform on PSL₂(ℝ) as a tool beyond K-biinvariant setup, discuss its properties and show some applications.     

Lower bounds for the maximal Lyapunov exponent

Upper bounds for the maximal Lyapunov exponent,E, of a sequence of matrix-valued random variables are easy to come by asE is the infimum of a real-valued sequence. We shall show that under irreducibility conditions similar to those needed to prove the Perron-Frobenius theorem, one can find sequences which increase toE. As a byproduct of the proof we shall see that we may replace the matrix norm with the spectral radius when computingE in such cases. Finally, a sufficient condition for transience of random walk in a random environment is given.     

The narrow class groups of the ℤ<Subscript>17</Subscript>- and ℤ<Subscript>19</Subscript>-extensions over the rational field

Let p be either 17 or 19, let ℤ _p denote the ring of p-adic integers, and let l be a prime number which is a primitive root modulo p ². We shall prove, with the help of a computer, that the l-class group of the ℤ _p -extension over the rational field is trivial. We shall also prove the triviality of the narrow 2-class group of the same ℤ _p -extension.     

On finitely generated multiplication modules

We shall prove that if M is a finitely generated multiplication module and Ann(M) is a finitely generated ideal of R, then there exists a distributive lattice M such that Spec(M) with Zariski topology is homeomorphic to Spec(M) to Stone topology. Finally we shall give a characterization of finitely generated multiplication R-modules M such that Ann(M) is a finitely generated ideal of R.     

Counting finite index subgroups and the P. Hall enumeration principle

We apply the P. Hall enumeration principle to count the number of subgroups of a given index in the free pro-p group and the free abelian group. We shall present an infinite family of non-isomorphic pro-p groups with the same zeta function.