scholarly journals Projective and Non-Projective Varieties of Topological Decomposition of Groups with Embeddings

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 450 ◽  
Author(s):  
Susmit Bagchi
Keyword(s):  
Fixed Point ◽  
Direct Product ◽  
Semidirect Product ◽  
Group Action ◽  
Topological Spaces ◽  
Local Homeomorphism ◽  
Projective Varieties ◽  
Semidirect Products ◽  
The Reformation ◽  
The Stability

In general, the group decompositions are formulated by employing automorphisms and semidirect products to determine continuity and compactification properties. This paper proposes a set of constructions of novel topological decompositions of groups and analyzes the behaviour of group actions under the topological decompositions. The proposed topological decompositions arise in two varieties, such as decomposition based on topological fibers without projections and decomposition in the presence of translated projections in topological spaces. The first variety of decomposition introduces the concepts of topological fibers, locality of group operation and the partitioned local homeomorphism resulting in formulation of transitions and symmetric surjection within the topologically decomposed groups. The reformation of kernel under decomposed homeomorphism and the stability of group action with the existence of a fixed point are analyzed. The first variety of decomposition does not require commutativity maintaining generality. The second variety of projective topological decomposition is formulated considering commutative as well as noncommutative projections in spaces. The effects of finite translations of topologically decomposed groups under projections are analyzed. Moreover, the embedding of a decomposed group in normal topological spaces is formulated in this paper. It is shown that Schoenflies homeomorphic embeddings preserve group homeomorphism in the decomposed embeddings within normal topological spaces. This paper illustrates that decomposed group embedding in normal topological spaces is separable. The applications aspects as well as parametric comparison of group decompositions based on topology, direct product and semidirect product are included in the paper.

Semigroup Compactifications of Direct and Semidirect Products

1983 ◽  
Vol 26 (2) ◽  
pp. 233-240 ◽  
Author(s):  
Paul Milnes
Keyword(s):  
Direct Product ◽  
Semidirect Product ◽  
Classical Result ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Direct Products ◽  
Almost Periodic ◽  
Semidirect Products ◽  
Semigroup Compactifications ◽  
Necessary And Sufficient

AbstractA classical result of I. Glicksberg and K. de Leeuw asserts that the almost periodic compactification of a direct product S × T of abelian semigroups with identity is (canonically isomorphic to) the direct product of the almost periodic compactiflcations of S and T. Some efforts have been made to generalize this result and recently H. D. Junghenn and B. T. Lerner have proved a theorem giving necessary and sufficient conditions for an F-compactification of a semidirect product S⊗σT to be a semidirect product of compactiflcations of S and T. A different such theorem is presented here along with a number of corollaries and examples which illustrate its scope and limitations. Some behaviour that can occur for semidirect products, but not for direct products, is exposed

A fixed point approach to the stability of sextic Lie *-derivations

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4933-4944
Author(s):  
Dongseung Kang ◽  
Heejeong Koh
Keyword(s):  
Functional Equation ◽  
Fixed Point ◽  
General Solution ◽  
Single Case ◽  
Fixed Point Method ◽  
Point Method ◽  
Fixed Point Approach ◽  
The Stability ◽  
The Given ◽  
Lie Derivations

We obtain a general solution of the sextic functional equation f (ax+by)+ f (ax-by)+ f (bx+ay)+ f (bx-ay) = (ab)2(a2 + b2)[f(x+y)+f(x-y)] + 2(a2-b2)(a4-b4)[f(x)+f(y)] and investigate the stability of sextic Lie *-derivations associated with the given functional equation via fixed point method. Also, we present a counterexample for a single case.

Topological Left Amenability of Semidirect Products

1981 ◽  
Vol 24 (1) ◽  
pp. 79-85 ◽  
Author(s):  
H. D. Junghenn
Keyword(s):  
Large Class ◽  
Semidirect Product ◽  
Locally Compact ◽  
Semidirect Products ◽  
Topological Semigroups

AbstractLet S and T be locally compact topological semigroups and a semidirect product. Conditions are determined under which topological left amenability of S and T implies that of , and conversely. The results are used to show that for a large class of semigroups which are neither compact nor groups, various notions of topological left amenability coincide.

scholarly journals On the stability of set-valued functional equations with the fixed point alternative

2012 ◽  
Vol 2012 (1) ◽  
pp. 81 ◽  
Author(s):  
Hassan Kenary ◽  
Hamid Rezaei ◽  
Yousof Gheisari ◽  
Choonkil Park
Keyword(s):  
Fixed Point ◽  
Functional Equations ◽  
The Stability

Existence of Periodic Solution for Equation of Motion of Simple Beams With Harmonically Variable Length

1995 ◽  
Author(s):  
Ebrahim Esmailzadeh ◽  
Gholamreza Nakhaie-Jazar ◽  
Bahman Mehri
Keyword(s):  
Periodic Solution ◽  
Fixed Point ◽  
Nonlinear Function ◽  
Axial Direction ◽  
Equation Of Motion ◽  
The Other ◽  
Sufficient Condition ◽  
Vibrating String ◽  
The Stability ◽  
Special Case

Abstract The transverse vibrating motion of a simple beam with one end fixed while driven harmonically along its axial direction from the other end is investigated. For a special case of zero value for the rigidity of the beam, the system reduces to that of a vibrating string with the corresponding equation of its motion. The sufficient condition for the periodic solution of the beam is then derived by means of the Green’s function and Schauder’s fixed point theorem. The criteria for the stability of the system is well defined and the condition for which the performance of the beam behaves as a nonlinear function is stated.

scholarly journals Further Investigations on the Dynamics and Multistability Coexisted in a Memory-Based Cobweb Model

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
S. S. Askar
Keyword(s):  
Fixed Point ◽  
Equilibrium Point ◽  
Profit Maximization ◽  
Period Doubling ◽  
Speed Of Adjustment ◽  
Discrete Dynamic ◽  
Cobweb Model ◽  
Numerical Studies ◽  
Market Clearing Price ◽  
The Stability

Based on a nonlinear demand function and a market-clearing price, a cobweb model is introduced in this paper. A gradient mechanism that depends on the marginal profit is adopted to form the 1D discrete dynamic cobweb map. Analytical studies show that the map possesses four fixed points and only one attains the profit maximization. The stability/instability conditions for this fixed point are calculated and numerically studied. The numerical studies provide some insights about the cobweb map and confirm that this fixed point can be destabilized due to period-doubling bifurcation. The second part of the paper discusses the memory factor on the stabilization of the map’s equilibrium point. A gradient mechanism that depends on the marginal profit in the past two time steps is adopted to incorporate memory in the model. Hence, a 2D discrete dynamic map is constructed. Through theoretical and numerical investigations, we show that the equilibrium point of the 2D map becomes unstable due to two types of bifurcations that are Neimark–Sacker and flip bifurcations. Furthermore, the influence of the speed of adjustment parameter on the map’s equilibrium is analyzed via numerical experiments.

scholarly journals Topological characterizations of amenability and congeniality of bases

2020 ◽  
Vol 21 (1) ◽  
pp. 1
Author(s):  
Sergio R. López-Permouth ◽  
Benjamin Stanley
Keyword(s):  
Direct Product ◽  
Topological Spaces ◽  
Countable Dimension ◽  
One To One ◽  
Natural Way

<div>We provide topological interpretations of the recently introduced notions of amenability and congeniality of bases of innite dimensional algebras. In order not to restrict our attention only to the countable dimension case, the uniformity of the topologies involved is analyzed and therefore the pertinent ideas about uniform topological spaces are surveyed.</div><div><p>A basis B over an innite dimensional F-algebra A is called amenable if F<sup>B</sup>, the direct product indexed by B of copies of the eld F, can be made into an A-module in a natural way. (Mutual) congeniality is a relation that serves to identify cases when different amenable bases yield isomorphic A-modules.</p><p>(Not necessarily mutual) congeniality between amenable bases yields an epimorphism of the modules they induce. We prove that this epimorphism is one-to-one only if the congeniality is mutual, thus establishing a precise distinction between the two notions.</p></div>

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