Search results for "Category"

General Set-Up

2017

Fuzzy $$\varphi $$ -pseudometrics and Fuzzy $$\varphi $$ -pseudometric Spaces

2017

By replacing the axiom \(m(x,x,t) = 1\) for all \(x\in X, t>0\) in the definition of a fuzzy pseudometric in the sense of George-Veeramani with a weaker axiom \(m(x,x,t) = \varphi (t)\) for all \(x\in X, t>0\) where \(\varphi : {\mathbb R}^+ \rightarrow (0,1]\) is a non-decreasing function, we come to the concept of a fuzzy \(\varphi \)-pseudometric space. Basic properties of fuzzy \(\varphi \)-pseudometric spaces and their mappings are studied. We show also an application of fuzzy \(\varphi \)-pseudometrics in the words combinatorics.

Fittingmengen und lockettabschnitte

1990

Abstract The theory of Lockett sections is transferred from Fitting classes to Fitting sets. This in general works only partially; in some special groups (which I call “mobility” groups), however, among these the stable linear groups, a literal translation of the Fitting class theory is possible. As the groups relevant in outer Fitting pairs actually are mobility groups, a new way of deriving information on the Lockett section of a Fitting class arises. This is used to present a simplified, if nonsoluble, counter-example to Lockett's conjecture and to decide a related question. Also, an approach to generating Fitting classes is given.

Two Questions of L. A. Shemetkov on Critical Groups

1996

Throughout the paper we consider only finite groups. Let X be a class of groups. A group G is called s-critical for X , or simply X-critical, if G is not in X but all proper subgroups of G are in X. w Ž .x Ž . Following Doerk and Hawkes 3, VII, 6.1 , we denote Crit X the class s of all X-critical groups. Knowledge of the structure of the groups in Ž . Crit X for a class of groups X can often help one to obtain detailed s information for the structure of the groups belonging to X. Ž w Ž .x. O. J. Schmidt see 5, III, 5.2 studied the N-critical groups, where N is the formation of the nilpotent groups. These groups are also called w x Schmidt groups. In 2 , answering to a question posed by Shem…

Lp-Spaces

1998

For (X, ℜ, μ) a positive measure space, it has already been noted that μ - a.e. equality is an equivalence relation, and the relation ≤ μ-a.e. a preorder, on.This section studies the structure of the equivalence classes into which μ-a,e. equality partitions.Since the set X/X( ℜ) is always u-null (2.7.7 a)), only the function values on the set X(ℜ) have any significance when equivalence classes are formed: whether we form equivalence classes by partitioning or by partitioningX(ℜ) the resulting structures will be isomorphic. Nevertheless, it is natural to allow functions on an arbitrary X ⊃ X(ℜ). Our choice is to form μ-equivalence classes by partitioning the set X(ℜ). For arbitrary X ⊃ X(ℜ),…

Analytic extension of non quasi-analytic Whitney jets of Roumieu type

1997

Let (Mr)r∈ℕ0 be a logarithmically convex sequence of positive numbers which verifies M0 = 1 as well as Mr≥ 1 for every r ∈ ℕ and defines a non quasi-analytic class. Let moreover F be a closed proper subset of ℝn. Then for every function ƒ on ℝn belonging to the non quasi-analytic (Mr)-class of Roumieu type, there is an element g of the same class which is analytic on ℝnF and such that Dα ƒ(x) = Dαg(x) for every σ ∈ ƒ0n SBAP and x ∈ F.

Generators of Random Processes in Ultrametric Spaces and Their Spectra

2009

The L 2(\( \mathbb{S} \)) space of square integrable functions on an ultrametric space \( \mathbb{S} \) has rather specific structure. As a consequence in a natural way there appear in L 2(\( \mathbb{S} \)) the operators of which unitary counterparts in L 2(ℝn) would be difficult to construct. Such class of self-adjoint operators emerge from theory of random processes on ultrametric spaces. In this paper we collect known material on spectral properties of the generators of random processes on \( \mathbb{S}_B \) an ultrametric space of sequences. (The set of p-adic numbers is a subset of \( \mathbb{S}_B \).) Then we discuss structure of the eigenspaces of the generators.

On finite products of groups and supersolubility

2010

Two subgroups X and Y of a group G are said to be conditionally permutable in G if X permutes with Y(g) for some element g E G. i.e., XY(g) is a subgroup of G. Using this permutability property new criteria for the product of finite supersoluble groups to be supersoluble are obtained and previous results are recovered. Also the behaviour of the supersoluble residual in products of finite groups is studied.

Lifting paths on quotient spaces

2009

Abstract Let X be a compactum and G an upper semi-continuous decomposition of X such that each element of G is the continuous image of an ordered compactum. If the quotient space X / G is the continuous image of an ordered compactum, under what conditions is X also the continuous image of an ordered compactum? Examples around the (non-metric) Hahn–Mazurkiewicz Theorem show that one must place severe conditions on G if one wishes to obtain positive results. We prove that the compactum X is the image of an ordered compactum when each g ∈ G has 0-dimensional boundary. We also consider the case when G has only countably many non-degenerate elements. These results extend earlier work of the firs…

On a Linear Diophantine Problem of Frobenius: Extending the Basis

1998

LetXk={a1, a2, …, ak},k>1, be a subset of N such that gcd(Xk)=1. We shall say that a natural numbernisdependent(onXk) if there are nonnegative integersxisuch thatnhas a representationn=∑ki=1 xiai, elseindependent. The Frobenius numberg(Xk) ofXkis the greatest integer withnosuch representation. Selmer has raised the problem of extendingXkwithout changing the value ofg. He showed that under certain conditions it is possible to add an elementc=a+kdto the arithmetic sequencea,a+d,a+2d, …, a+(k−1) d, gcd(a, d)=1, without alteringg. In this paper, we give the setCof all independent numberscsatisfyingg(A, c)=g(A), whereAcontains the elements of the arithmetic sequence. Moreover, ifa>kthen we give …