Commutator Map Research Articles

Correspondence of Fixed-Point Theorem in \(T_2, T_3\)-SPACE

Fixed-point theory (FPT) has lot of applications not only in the field of mathematics but also in various other disciplines. Fixed Point Theorem presents that if T:X \(\to\) X is a contraction mapping on a complete metric space (X, d) then there exists a unique fixed point in X. FPT is also essential in game theory, in this case Brower Fixed Point has an application in game theory specifically in non-cooperative games and existence of Equilibrium. In particular, a game is a set of actions done by the participants defined by a set of rules. This is commonly described using mathematical concepts, which offers a concrete model to describe a variety of situations. On the other hand, the separation axioms Ti, i = 0,1,2,3,4 are vital properties that describes the topological spaces T0,T1,T2,T3 and T4. It is noted that a T3 - space is a generalized version of T2- space and since various results on application of fixed point theory in game theory on an arbitrary locally convex T2 - space has been established, in this study we sort to extend this concept to the general T3 - space. The utilization of a symmetric property of Hausdorff space established that if two continuous commutative mappings are defined on a T3 - space, then the two maps achieves unique fixed points.

Analyzing Stability and Data Dependence Notions by a Novel Jungck-Type Iteration Method

Finding the ideal circumstances for a mapping to have a fixed point is the fundamental goal of fixed point theory. These criteria can also be used for the structure under investigation. One of this theory’s most well-known theorems, Banach’s fixed point theorem, has been expanded adopting various methods, making it possible to conduct numerous research studies. Thanks to the Jungck-Contraction Theorem, which has been proven through commutative mappings, many fixed point theorems have been obtained using classical fixed point iteration methods and newly defined methods. This study aims to investigate the convergence, stability, convergence rate, and data dependency of the new four-step fixed-point iteration method. Nontrivial examples are also included to support some of the results herein.

Weakly Commutative Mappings and Common Fixed-Point Theorem in Digital Metric Space

Weakly Commutative Mappings and Common Fixed-Point Theorem in Digital Metric Space

A generalization of the common fixed point theorem for normally 2-generalized hybrid mappings in Hilbert spaces

In this paper, we prove a common fixed point theorem for commutative nonlinear mappings that jointly satisfy a certain condition. A required condition is given as a convex combination of those of a well-known class of nonlinear mappings. From the main theorem of this paper, the common fixed point theorem for nonexpansive mappings, generalized hybrid mappings, and normally 2-generalized hybrid mappings are uniformly derived as corollaries. Our approach expands the applicable range of mappings for existing fixed point theorems to be effective. Using specific mappings, we illustrate the effectiveness.

Some Results on Best Approximations without star-shaped

In this paper, we give two theorems in Banach space. The first one is about the existence of a fixed point for generalized affine mapping, and a class of relatively nonexpansive commutative mappings and then we use it in the proof of theorem about existence of best approximation.

Generalized common fixed point theorem for generalized hybrid mappings in Hilbert spaces

Abstract In this article, we prove a common fixed point theorem for commutative nonlinear mappings that jointly satisfy a certain condition. From the main theorem, a common fixed point theorem for commutative generalized hybrid mappings is derived as a special case. Our novel approach significantly expands the applicable range of mappings for well-known fixed point theorems to be effective. Examples are presented to explicitly illustrate this contribution.

Functional equations with multiple recursive terms

In this paper, we study a functional equation for generating functions of the form f(z) = g(z) sum _{i=1}^M p_i f(alpha _i(z)) + K(z), viz. a recursion with multiple recursive terms. We derive and analyze the solution of this equation for the case that the alpha _i(z) are commutative contraction mappings. The results are applied to a wide range of queueing, autoregressive and branching processes.

A Mean Convergence Theorem without Convexity for Finite Commutative Nonlinear Mappings in Reflexive Banach Spaces

This paper investigates the Bregman version of the Takahashi-type generic 2-generalized nonspreading mapping which includes the generic 2-generalized Bregman nonspreading mapping as a special case. Relative to the attractive points of nonlinear mapping, the Baillon-type nonlinear mean convergence theorem for finite commutative generic 2-generalized Bregman nonspreading mappings without the convexity assumption is proved in the setting of reflexive Banach spaces. Using this result, some new and well-known nonlinear mean convergence theorems for the finite generic generalized Bregman nonspreading mapping, the 2-generalized Bregman nonspreading mapping and the normally 2-generalized hybrid mapping, among others, are established. Our results extend and generalize many corresponding ones announced in the literature.

COMMON FIXED POINT THEOREMS FOR COMMUTATIVE, WEAKLY COMMUTATIVE AND COMPATIBLE MAPPINGS IN MULTIPLICATIVE CONE METRIC SPACE

n this paper, we introduce the notions of commutative, weakly commutative, and compatible mappings in multiplicative cone metric space and prove common fixed point theorems for these mappings with multiplicative normal cone setting. Also, we give an example to show the validity of our results.

On commutator fibers over regular semisimple and central elements of SLn

We study fibers of the commutator map on general linear maps using representation theory. Especially, we give detailed information on dimension and geometric connectedness of those fibers over regular semisimple and central elements.

Flat Connections and the Commutator Map for SU(2)

Abstract We study the topology of the SU(2)-representation variety of the compact oriented surface of genus 2 with one boundary component about which the holonomy is a generator of the center of SU(2).

Attractive point and nonlinear ergodic theorems without convexity in reflexive Banach spaces

In this paper, we prove attractive point theorem for two commutative generic 2-generalized Bregman nonspreading mappings in reflexive Banach spaces. Also, a nonlinear ergodic theorem of Baillon type without convexity assumption on the aforementioned mappings is proved in the space. Our results improve and generalize many results announced recently in the literature.

(m,q)-isometric and (m,∞)-isometric tuples of commutative mappings on a metric space

In this paper, we introduce new concepts of (m,q)-isometries and (m,?)-isometries tuples of commutative mappings on metrics spaces. We discuss the most interesting results concerning this class of mappings obtained form the idea of generalizing the (m,q)-isometries and (m,?)-isometries for single mappings. In particular, we prove that if T = (T1,..., Tn) is an (m,q)-isometric commutative and power bounded tuple, then T is a (1,q)-isometric tuple. Moreover, we show that if T = (T1,...,Td) is an (m,?)- isometric commutative tuple of mappings on a metric space (E,d), then there exists a metric d? on E such that T is a (1,?)-isometric tuple on (E,d?).

Flatness of the Commutator Map Over SLn

Abstract Let $K$ be any field, and let $n$ be a positive integer. If we denote by $\xi _{\textrm{SL}_n}\colon \textrm{SL}_n\times \textrm{SL}_n\to \textrm{SL}_n$ the commutator morphism over $K$, then $\xi _{\textrm{SL}_n}$ is flat over the complement of the center of $\textrm{SL}_n$.

The first common fixed point theorem for commutative set-valued mappings

In this paper, we establish a common fixed point theorem for commutative set-valued convex mappings. We also prove the existence of a fixed point in a continuously expanding set under a non-convex upper semicontinuous set-valued mapping. Applying this result, we develop a common fixed point theorem for a family of non-convex set-valued upper semicontinuous mappings which provides an answer to a question of Lau and Yao.

Fixed Points in A Complex Valued Metric Space

Objectives/Aim: To find fixed points for integral types of contraction mappings in a complex valued metric spaces. Methods: We used mathmatical analysis to prove the results as a generalization of weakly contraction self mappings defined in a complex metric space. Findings: We are finding by weakly integral types of contraction self mappings that there are common fixed points. In this paper we used commutative self mappings for our conclusions . Application/ Improvements: In future it can also be used for the exestense and uniquence of the differential equations solutions. Keywords: Complex Valued Metric Space, Contraction Mapping, Fixed Point, Integral Type, Weakly Commuting

The homotopy types of Sp(3)-gauge groups

Let Gk denote the gauge group of the principal Sp(3)-bundle over S4 with first symplectic Pontryagin class k∈H4S4=Z. We show that when localised at an odd prime there is a homotopy equivalence Gk≃Gl if and only if (21,k)=(21,l). We also give bounds on the number of 2-local homotopy types amongst the Gk. Our results follow from a thorough examination of the commutator map c:Sp(1)∧Sp(3)→Sp(3) and we determine its order to be either 4⋅3⋅7=84, 8⋅3⋅7=168 or 16⋅3⋅7=336. In particular its order at odd primes is completely determined.

Drinfeld orbifold algebras for symmetric groups

Drinfeld orbifold algebras are a type of deformation of skew group algebras generalizing graded Hecke algebras of interest in representation theory, algebraic combinatorics, and noncommutative geometry. In this article, we classify all Drinfeld orbifold algebras for symmetric groups acting by the natural permutation representation. This provides, for nonabelian groups, infinite families of examples of Drinfeld orbifold algebras that are not graded Hecke algebras. We include explicit descriptions of the maps recording commutator relations and show there is a one-parameter family of such maps supported only on the identity and a three-parameter family of maps supported only on 3-cycles and 5-cycles. Each commutator map must satisfy properties arising from a Poincaré–Birkhoff–Witt condition on the algebra, and our analysis of the properties illustrates reduction techniques using orbits of group element factorizations and intersections of fixed point spaces.

Existence and structure of the common fixed points based on TVS

In this paper, we investigate the common fixed point property for commutative nonexpansive mappings on $\tau$-compact convex sets in normed and Banach spaces, where $\tau$ is a Hausdorff topological vector space topology that is weaker than the norm topology. As a consequence of our main results, we obtain that the set of common fixed points of any commutative family of nonexpansive self-mappings of a nonempty $clm$-compact (resp. weak* compact) convex subset $C$ of $L_1(\mu)$ with a $\sigma$-finite $\mu$ (resp. the James space $J_0$) is a nonempty nonexpansive retract of $C$.

Code loops in dimension at most 8

Code loops are certain Moufang 2-loops constructed from doubly even binary codes that play an important role in the construction of local subgroups of sporadic groups. More precisely, code loops are central extensions of the group of order 2 by an elementary abelian 2-group V in the variety of loops such that their squaring map, commutator map and associator map are related by combinatorial polarization and the associator map is a trilinear alternating form.Using existing classifications of trilinear alternating forms over the field of 2 elements, we enumerate code loops of dimension d=dim⁡(V)≤8 (equivalently, of order 2d+1≤512) up to isomorphism. There are 767 code loops of order 128, and 80826 of order 256, and 937791557 of order 512.