Weak Commutativity Research Articles

Non-Compact Inaudibility of Weak Symmetry and Commutativity via Generalized Heisenberg Groups

Two Riemannian manifolds are said to be isospectral if there exists a unitary operator which intertwines their Laplace-Beltrami operator. In this paper, we prove in the non-compact setting the inaudibility of the weak symmetry property and the commutative property using an isospectral pair of 23 dimensional generalized Heisenberg groups.

ஜ ஜ Near Semi Rings

Objectives: In the widened area of near semi rings, this article speaks about the interventions of a specific sub structure. Methods: Heading towards a more unique functionality, the chosen sub structure is intervened with regularity, homomorphism and also checked for weak commutative idempotency. Findings: The desired sub-structure is descried to have been preserved under homomorphism and is also found to be isomorphic to sub direct products of sub-directly irreducible sub structures. Novelty: Few descriptions and differences on weak commutative regularity and regular idempotency has been expounded. Keen and expanded utterance of Pierce Decomposition near semi ring functionalities have made the desired near semi ring to be closer to nil but not exactly a deeper nil near semi ring. Keywords: Regularity, Homomorphism, Weak commutativity, Idempotency, Sub­directly irreducible, Pierce decomposition

Finiteness conditions for the weak commutativity group and related constructions

The weak commutativity group χ ( G ) is generated by two isomorphic groups G and G φ subject to the relations [ g , g φ ] = 1 for all g ∈ G . We establish a finiteness criterion for the subgroups of χ ( G ) in terms of the set T χ ( G ) = { [ g 1 , g 2 φ ] | g i ∈ G } . We also obtain sufficient conditions under which some specific (local) classes of groups are invariant under the operator χ. Moreover, we prove that if G is a locally finite group with exp ( G ) = n , then χ ( G ) is locally finite and has finite n-bounded exponent. Communicated by Mark Lewis

Weak commutativity, virtually nilpotent groups, and Dehn functions

The group \mathfrak{X}(G) is obtained from G\ast G by forcing each element g in the first free factor to commute with the copy of g in the second free factor. We make significant additions to the list of properties that the functor \mathfrak{X} is known to preserve. We also investigate the geometry and complexity of the word problem for \mathfrak{X}(G) . Subtle features of \mathfrak{X}(G) are encoded in a normal abelian subgroup W<\mathfrak{X}(G) that is a module over \mathbb{Z} Q , where Q= H_1(G,\mathbb{Z}) . We establish a structural result for this module and illustrate its utility by proving that \mathfrak{X} preserves virtual nilpotence, the Engel condition, and growth type – polynomial, exponential, or intermediate. We also use it to establish isoperimetric inequalities for \mathfrak{X}(G) when G lies in a class that includes Thompson's group F and all non-fibred Kähler groups. The word problem is soluble in \mathfrak{X}(G) if and only if it is soluble in G . The Dehn function of \mathfrak{X}(G) is bounded below by a cubic polynomial if G maps onto a non-abelian free group.

Integral Type Contraction in Dislocated Metric Spaces

In this paper, by using the notion of compatibility, weak compatibility, occasionally weak compatibility and commutativity, we establish some common fixed point theorems for six mappings satisfying integral type contractive conditions in complete dislocated metric spaces. Our work improves and extends some earlier results in the literature.

On weak commutativity in 𝑝-groups

Abstract The weak commutativity group χ ⁢ ( G ) \chi(G) is generated by two isomorphic groups 𝐺 and G φ G^{\varphi} subject to the relations [ g , g φ ] = 1 [g,g^{\varphi}]=1 for all g ∈ G g\in G . We present new bounds for the exponent of χ ⁢ ( G ) \chi(G) and its sections, when 𝐺 is a finite 𝑝-group.

Coincidence and Common Fixed Point Theorems for Hybrid Mappings

Abstract In this paper, we prove a new coincidence and common fixed point theorem of hybrid mappings using the C-class function and T -weak commutativity. Finally, we give an example to illustrate our result.

The Weak Commutation Matrices of Matrix with Duplicate Entries in its Secondary Diagonal

The Weak Commutation Matrices of Matrix with Duplicate Entries in its Secondary Diagonal

On the Bieri–Neumann–Strebel–Renz invariants of the weak commutativity construction $\mathfrak{X}(G)$

For a finitely generated group G , we calculate the Bieri–Neumann–Strebel–Renz invariant \Sigma^1(\mathfrak{X}(G)) for the weak commutativity construction \mathfrak{X}(G) . Identifying S(\mathfrak{X}(G)) with S(\mathfrak{X}(G) /\allowbreak W(G)) , we show \Sigma^2(\mathfrak{X}(G),\mathbb{Z}) \subseteq \Sigma^2(\mathfrak{X}(G)/ W(G),\mathbb{Z}) and \Sigma^2(\mathfrak{X}(G)) \subseteq \Sigma^2(\mathfrak{X}(G)/ W(G)) , that are equalities when W(G) is finitely generated, and we explicitly calculate \Sigma^2(\mathfrak{X}(G)/ W(G),\mathbb{Z}) and \Sigma^2(\mathfrak{X}(G)/ W(G)) in terms of the \Sigma -invariants of G . We calculate completely the \Sigma -invariants in dimensions 1 and 2 of the group \nu(G) and show that if G is finitely generated group with finitely presented commutator subgroup then the non-abelian tensor square G \otimes G is finitely presented.

Untwining multiple parameters at the exclusive zero-coincidence points with quantum control

In this paper we address a special case of ‘sloppy’ quantum estimation procedures which happens in the presence of intertwined parameters. A collection of parameters are said to be intertwined when their imprinting on the quantum probe that mediates the estimation procedure, is performed by a set of linearly dependent generators. Under this circumstance the individual values of the parameters can not be recovered unless one tampers with the encoding process itself. An example is presented by studying the estimation of the relative time-delays that accumulate along two parallel optical transmission lines. In this case we show that the parameters can be effectively untwined by inserting a sequence of balanced beam splitters (and eventually adding an extra phase shift on one of the lines) that couples the two lines at regular intervals in a setup that remind us a generalized Hong-Ou-Mandel interferometer. For the case of two time delays we prove that, when the employed probe is the frequency-correlated biphoton state, the untwining occurs in correspondence of exclusive zero-coincidence (EZC) point. Furthermore we show the statistical independence of two time delays and the optimality of the quantum Fisher information at the EZC point. Finally we prove the compatibility of this scheme by checking the weak commutativity condition associated with the symmetric logarithmic derivative operators.

Pólya–Carlson dichotomy for coincidence Reidemeister zeta functions via profinite completions

We consider coincidence Reidemeister zeta functions for tame endomorphism pairs of nilpotent groups of finite rank, shedding new light on the subject by means of profinite completion techniques.In particular, we provide a closed formula for coincidence Reidemeister numbers for iterations of endomorphism pairs of torsion-free nilpotent groups of finite rank, based on a weak commutativity condition, which derives from simultaneous triangularisability on abelian sections. Furthermore, we present results in support of a Pólya–Carlson dichotomy between rationality and a natural boundary for the analytic behaviour of the zeta functions in question.

Causality, crossing and analyticity in conformal field theories

Analyticity and crossing properties of four-point function are investigated in conformal field theories in the frameworks of Wightman axioms. A Hermitian scalar conformal field, satisfying the Wightman axioms, is considered. The crucial role of microcausality in deriving analyticity domains is discussed and domains of analyticity are presented. A pair of permuted Wightman functions are envisaged. The crossing property is derived by appealing to the technique of analytic completion for the pair of permuted Wightman functions. The operator product expansion of a pair of scalar fields is studied and analyticity property of the matrix elements of composite fields, appearing in the operator product expansion, is investigated. An integral representation is presented for the commutator of composite fields where microcausality is a key ingredient. Three fundamental theorems of axiomatic local field theories; namely, PCT theorem, the theorem proving equivalence between PCT theorem and weak local commutativity and the edge-of-the-wedge theorem are invoked to derive a conformal bootstrap equation rigorously.

On the Exponent of the Weak Commutativity Group $$\chi (G)$$

The weak commutativity group $\chi(G)$ is generated by two isomorphic groups $G$ and $G^{\varphi }$ subject to the relations $[g,g^{\varphi}]=1$ for all $g \in G$. The group $\chi(G)$ is an extension of $D(G) = [G,G^{\varphi}]$ by $G \times G$. We prove that if $G$ is a finite solvable group of derived length $d$, then $\exp(D(G))$ divides $\exp(G)^{d}$ if $|G|$ is odd and $\exp(D(G))$ divides $2^{d-1}\cdot \exp(G)^{d}$ if $|G|$ is even. Further, if $p$ is a prime and $G$ is a $p$-group of class $p-1$, then $\exp(D(G))$ divides $\exp(G)$. Moreover, if $G$ is a finite $p$-group of class $c\geq 2$, then $\exp(D(G))$ divides $\exp(G)^{\lceil \log_{p-1}(c+1)\rceil}$ ($p\geq 3$) and $\exp(D(G))$ divides $2^{\lfloor \log_2(c)\rfloor} \cdot \exp(G)^{\lfloor \log_2(c)\rfloor+1}$ ($p=2$).

On the Theory of Left/Right Almost Groups and Hypergroups with their Relevant Enumerations

This paper presents the study of algebraic structures equipped with the inverted associativity axiom. Initially, the definition of the left and the right almost-groups is introduced and afterwards, the study is focused on the more general structures, which are the left and the right almost-hypergroups and on their enumeration in the cases of order 2 and 3. The outcomes of these enumerations compared with the corresponding in the hypergroups reveal interesting results. Next, fundamental properties of the left and right almost-hypergroups are proved. Subsequently, the almost hypergroups are enriched with more axioms, like the transposition axiom and the weak commutativity. This creates new hypercompositional structures, such as the transposition left/right almost-hypergroups, the left/right almost commutative hypergroups, the join left/right almost hypergroups, etc. The algebraic properties of these new structures are analyzed and studied as well. Especially, the existence of neutral elements leads to the separation of their elements into attractive and non-attractive ones. If the existence of the neutral element is accompanied with the existence of symmetric elements as well, then the fortified transposition left/right almost-hypergroups and the transposition polysymmetrical left/right almost-hypergroups come into being.

PCT theorem, Wightman axioms and conformal bootstrap

The axiomatic Wightman formulation for nonderivative conformal field theory is adopted to derive conformal bootstrap equation for the four-point function. The equivalence between PCT theorem and weak local commutativity, due to Jost plays a very crucial role in axiomatic field theory. The theorem is suitably adopted for conformal field theory to derive the desired equations in CFT. We demonstrate that the two Wightman functions are analytic continuation of each other.

Weak commutativity for pro-p groups

We define and study a pro-p version of Sidki’s weak commutativity construction. This is the pro-p group $$\mathfrak {X}_p(G)$$ generated by two copies G and $$G^{\psi }$$ of a pro-p group, subject to the defining relators $$[g,g^{\psi }]$$ for all $$g \in G$$ . We show for instance that if G is finitely presented or analytic pro-p, then $$\mathfrak {X}_p(G)$$ has the same property. Furthermore we study properties of the non-abelian tensor product and the pro-p version of Rocco’s construction $$\nu (H)$$ . We also study finiteness properties of subdirect products of pro-p groups. In particular we prove a pro-p version of the $$(n-1)-n-(n+1)$$ Theorem.

Weak commutativity and nilpotency

We continue the analysis of the weak commutativity construction for Lie algebras. This is the Lie algebra χ(g) generated by two isomorphic copies g and gψ of a fixed Lie algebra, subject to the relations [x,xψ]=0 for all x∈g. In this article we study the ideal L=L(g) generated by x−xψ for all x∈g. We obtain an (infinite) presentation for L as a Lie algebra, and we show that in general it cannot be reduced to a finite one. With this in hand, we study the question of nilpotency. We show that if g is nilpotent of class c, then χ(g) is nilpotent of class at most c+2, and this bound can improved to c+1 if g is 2-generated or if c is odd. We also obtain concrete descriptions of L(g) (and thus of χ(g)) if g is free nilpotent of class 2 or 3. Finally, using methods of Gröbner-Shirshov bases we show that the abelian ideal R(g)=[g,[L,gψ]] is infinite-dimensional if g is free of rank at least 3.

A Commingle between the Contractive Conditions of Type Jungck and a General Weak Commutativity Concept

In this work, we prove a common fixed point theorem of type Jungck under a generalized definition of weak commutativity and a well-known contractive inequality by commingling both conditions in the proof. Some open questions are also indicated and intimately connected (or not?) to the metric to be used.

Non-Abelian tensor square and related constructions of p-groups

Let $G$ be a group. We denote by $\nu(G)$ a certain extension of the non-abelian tensor square $[G,G^{\varphi}]$ by $G \times G$. We prove that if $G$ is a finite potent $p$-group, then $[G,G^{\varphi}]$ and the $k$-th term of the lower central series $\gamma_k(\nu(G))$ are potently embedded in $\nu(G)$ (Theorem A). Moreover, we show that if $G$ is a potent $p$-group, then the exponent $\exp(\nu(G))$ divides $p \cdot \exp(G)$ (Theorem B). We also study the weak commutativity construction of powerful $p$-groups (Theorem C).

A Construction of Lower-Bounded Generalized Twisted Modules for a Grading-Restricted Vertex (Super)Algebra

We give a general, direct and explicit construction of lower-bounded generalized twisted modules satisfying a universal property for a grading-restricted vertex (super)algebra V associated to an automorphism g of V. In particular, when g is the identity, we obtain lower-bounded generalized V-modules satisfying a universal property. Let W be a lower-bounded graded vector space equipped with a set of “generating twisted fields” and a set of “generator twist fields” satisfying a weak commutativity for generating twisted fields, a generalized weak commutativity for one generating twisted field and one generator twist field and some other properties that are relatively easy to verify. We first prove the convergence and commutativity of products of an arbitrary number of generating twisted fields, one twist generator field and an arbitrary number of generating fields for V. Then using the convergence and commutativity, we define a twisted vertex operator map for W and prove that W equipped with this twisted vertex operator map is a lower-bounded generalized g-twisted V-module. Using this result, we give an explicit construction of lower-bounded generalized g-twisted V-modules satisfying a universal property starting from vector spaces graded by weights, $${\mathbb {Z}}_{2}$$-fermion numbers and g-weights (eigenvalues of g) and real numbers corresponding to the lower bounds of the weights of the modules to be constructed. In particular, every lower-bounded generalized g-twisted V-module (every lower-bounded generalized V-module when g is the identity) is a quotient of such a universal lower-bounded generalized g-twisted V-module (a universal lower-bounded generalized V-module).