scholarly journals Duality of locally quasi-convex convergence groups

2021 ◽  
Vol 22 (1) ◽  
pp. 193
Author(s):  
Pranav Sharma
Keyword(s):  
Necessary Condition ◽  
Topological Groups ◽  
Pontryagin Duality ◽  
Character Group ◽  
Convergence Spaces ◽  
Convergence Group ◽  
Quasi Convexity

<p>In the realm of the convergence spaces, the generalisation of topological groups is the convergence groups, and the corresponding extension of the Pontryagin duality is the continuous duality. We prove that local quasi-convexity is a necessary condition for a convergence group to be c-reflexive. Further, we prove that every character group of a convergence group is locally quasi-convex.</p>

scholarly journals A Schreier theorem for free topological groups

1975 ◽  
Vol 13 (1) ◽  
pp. 121-127 ◽  
Author(s):  
Peter Nickolas
Keyword(s):  
Topological Group ◽  
Closed Subgroup ◽  
Open Subgroup ◽  
Necessary Condition ◽  
Topological Groups ◽  
Free Topological Group

M.I.Graev has shown that subgroups of free topological groups need not be free. Brown and Hardy, however, have proved that any open subgroup of the free topological group on a kw-space is again a free topological group: indeed, this is true for any closed subgroup for which a Schreier transversal can be chosen continuously. This note provides a proof of this result more direct than that of Brown and Hardy. An example is also given to show that the condition stated in the theorem is not a necessary condition for freeness of a subgroup. Finally, a sharpened version of a particular case of the theorem is obtained, and is applied to the preceding example.

Studies in harmonic analysis

1964 ◽  
Vol 60 (3) ◽  
pp. 465-516 ◽  
Author(s):  
N. Th. Varopoulos
Keyword(s):  
Harmonic Analysis ◽  
Complete Characterization ◽  
Locally Compact Groups ◽  
Topological Spaces ◽  
Compact Groups ◽  
Measure Algebra ◽  
Topological Groups ◽  
Inductive Limits ◽  
Locally Compact ◽  
Character Group

In this paper we shall be mainly concerned with the following three apparently widely differing questions.(a) What are the possible group topologies on an Abelian group that have a given, fixed continuous character group?In developing our theory, we are very strongly motivated by the duality theory of linear topological spaces and in particular by Mackey's theorem of that theory. This important result gives a complete characterization of all locally convex topologies on a linear space that have a given, fixed, separating dual space. The analogue of Mackey's theorem for groups, together with related results, is examined in sections 1 and 2 of part 2 of the paper.(b) What are the properties of topological groups that are denumerable inductive limits of locally compact groups? (See section 1 of part 1 of the paper for definitions.)Our aim here is to extend results known for locally compact groups to this larger class of groups. The topological study of these groups is carried out in section 3 of part 1 of the paper and the really deep results about their characters are proved in section 5 of part 3 of the paper, as applications of the theory developed in that part of the paper, which is a type of harmonic analysis for these groups.(c) What are the properties of certain algebras of measures of a locally compact group G, that strictly contain L1(G), and share most of the pleasing properties of L1(G), that is, they do not have any of the pathological features of the full measure algebra M(G) such as the Wiener–Pitt phenomenon or asymmetry?

scholarly journals On the Aspects of Enriched Lattice-valued Topological Groups and Closure of Lattice-valued Subgroups

2021 ◽  
Vol 14 (3) ◽  
pp. 949-968
Author(s):  
TMG Ahsanullah ◽  
Fawzi Al-Thukair
Keyword(s):  
Topological Group ◽  
Initial Structure ◽  
Topological Category ◽  
Topological Groups ◽  
Convergence Group ◽  
Closure Spaces

Starting with L as an enriched cl-premonoid, in this paper, we explore some categorical connections between L-valued topological groups and Kent convergence groups, where it is shown that every L-valued topological group determines a well-known Kent convergence group, and conversely, every Kent convergence group induces an L-valued topological group. Considering an L-valued subgroup of a group, we show that the category of L-valued groups, L-GRP has initial structure. Furthermore, we consider a category L-CLS of L-valued closure spaces, obtaining its relation with L-valued Moore closure, and provide examples in relation to L-valued subgroups that produce Moore collection. Here we look at a category of L-valued closure groups, L-CLGRP proving that it is a topological category. Finally, we obtain a relationship between L-GRP and L-TransTOLGRP, the category of L-transitive tolerance groups besides adding some properties of L-valued closures of L-valued subgroups on L-valued topological groups.

scholarly journals Embedding topological semigroups in topological groups

1970 ◽  
Vol 17 (2) ◽  
pp. 127-138 ◽  
Author(s):  
Sheila A. McKilligan
Keyword(s):  
Commutative Semigroup ◽  
Algebraic Structure ◽  
Necessary Conditions ◽  
Cancellative Semigroup ◽  
Necessary Condition ◽  
Topological Groups ◽  
Topological Semigroups

If we consider a semigroup, its algebraic structure may be such that it is isomorphic to a subsemigroup of a group, or is algebraically embeddable in a group. This problem was investigated in 1931 by Ore who obtained in (4) a set of necessary conditions for this embedding. A necessary condition is that the semigroup should be cancellative: for any a, x, y in the semigroup either xa = ya or ax = ay implies that x = y. Malcev in (3) showed that this was not sufficient. It is enough to note that his example was a non-commutative semigroup: a commutative cancellative semigroup is embeddable algebraically in a group.

Concepts of differentiability and analyticity on certain classes of topological groups

1965 ◽  
Vol 61 (2) ◽  
pp. 347-379 ◽  
Author(s):  
G. A. Reid
Keyword(s):  
Fourier Transforms ◽  
Locally Compact Abelian Group ◽  
Discrete Groups ◽  
Topological Groups ◽  
Locally Compact ◽  
Character Group ◽  
Compact Abelian Groups ◽  
Irreducible Unitary Representations ◽  
Locally Convex ◽  
Spaces Of Analytic Functions

AbstractWe introduce the concepts of a local seminorm on a topological group and of a locally convex group, showing that discrete groups, locally compact Abelian groups and compact groups are locally convex, and that a topological vector space is locally convex as a linear space if and only if it is locally convex as a group. We show that notions of differentiability, analyticity and derivability can be defined for locally convex groups and that these notions are suitably related and well behaved. We prove that for a locally compact Abelian group G the Fourier transforms of measures of compact support on the character group Ĝ are analytic, and for G compact the coefficients of continuous irreducible unitary representations are. Using these spaces of analytic functions we define the basic concepts of a differential geometry.

CHANGE OF BASIS FOR LATTICE-VALUED CONVERGENCE GROUPS

2011 ◽  
Vol 07 (03) ◽  
pp. 453-469 ◽  
Author(s):  
T. M. G. AHSANULLAH ◽  
FAWZI AL-THUKAIR
Keyword(s):  
Topological Groups ◽  
Limit Groups ◽  
Convergence Group ◽  
The Impact

Using the idea of changing the basis-lattice, we investigate in this article the impact of change-of-basis to the notion of stratified lattice-valued generalized convergence group by the so-called functorial mechanism. We discuss here some of the subcategories of the category of stratified lattice-valued generalized convergence groups, such as, stratified lattice-valued Kent convergence groups, stratified lattice-valued limit groups, and look into the possible link between these objects with stratified lattice-valued neighborhood groups and stratified lattice-valued neighborhood topological groups when bases are changed. Moreover, with the help of the notion of stratified lattice-valued filter attributed to U. Höhle and A. Šostak, we introduce a category >HŠ-SL-FilGrp, of stratified lattice-valued filter groups, and study its relationship with other categories so far achieved in this paper.

scholarly journals On strongly reflexive topological groups

2001 ◽  
Vol 2 (2) ◽  
pp. 219 ◽  
Author(s):  
M. J. Chasco ◽  
E. Martin-Peinador
Keyword(s):  
Topological Group ◽  
Closed Subgroup ◽  
Dual Group ◽  
Point Of View ◽  
Topological Groups ◽  
Character Group ◽  
Abelian Topological Group ◽  
Analogous Statement ◽  
Perfect Duality ◽  
Metrizable Group

<p>An Abelian topological group G is strongly reflexive if every closed subgroup and every Hausdorff quotient of G and of its dual group G<sup>⋀</sup>, is reflexive.</p> <p>In this paper we prove the following: the annihilator of a closed subgroup of an almost metrizable group is topologically isomorphic to the dual of the corresponding Hausdorff quotient, and an analogous statement holds for the character group of the starting group. As a consequence of this perfect duality, an almost metrizable group is strongly reflexive just if its Hausdorff quotients, as well as the Hausdorff quotients of its dual, are reflexive. The simplification obtained may be significant from an operative point of view.</p>

The Infancy of Solar-Type Stars and its Relevance for the Origin of Life

1997 ◽  
Vol 161 ◽  
pp. 267-282 ◽  
Author(s):  
Thierry Montmerle
Keyword(s):  
Main Sequence ◽  
Solar Nebula ◽  
Circumstellar Disks ◽  
T Tauri Stars ◽  
Necessary Condition ◽  
Sequence Evolution ◽  
T Tauri ◽  
Solar Type ◽  
Optical Domain ◽  
The Origin Of Life

AbstractFor life to develop, planets are a necessary condition. Likewise, for planets to form, stars must be surrounded by circumstellar disks, at least some time during their pre-main sequence evolution. Much progress has been made recently in the study of young solar-like stars. In the optical domain, these stars are known as «T Tauri stars». A significant number show IR excess, and other phenomena indirectly suggesting the presence of circumstellar disks. The current wisdom is that there is an evolutionary sequence from protostars to T Tauri stars. This sequence is characterized by the initial presence of disks, with lifetimes ~ 1-10 Myr after the intial collapse of a dense envelope having given birth to a star. While they are present, about 30% of the disks have masses larger than the minimum solar nebula. Their disappearance may correspond to the growth of dust grains, followed by planetesimal and planet formation, but this is not yet demonstrated.

Universal ESEM

1993 ◽  
Vol 51 ◽  
pp. 786-787
Author(s):  
G.D. Danilatos
Keyword(s):  
Electron Microscope ◽  
Scanning Electron Microscope ◽  
High Vacuum ◽  
Natural Extension ◽  
Necessary Condition ◽  
Environmental Scanning ◽  
Detection Systems ◽  
Small Gain ◽  
Detection Medium ◽  
Scanning Electron

The environmental scanning electron microscope (ESEM) has evolved as the natural extension of the scanning electron microscope (SEM), both historically and technologically. ESEM allows the introduction of a gaseous environment in the specimen chamber, whereas SEM operates in vacuum. One of the detection systems in ESEM, namely, the gaseous detection device (GDD) is based on the presence of gas as a detection medium. This might be interpreted as a necessary condition for the ESEM to remain operational and, hence, one might have to change instruments for operation at low or high vacuum. Initially, we may maintain the presence of a conventional secondary electron (E-T) detector in a "stand-by" position to switch on when the vacuum becomes satisfactory for its operation. However, the "rough" or "low vacuum" range of pressure may still be considered as inaccessible by both the GDD and the E-T detector, because the former has presumably very small gain and the latter still breaks down.

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