Several arguments have been proposed some years ago, attempting to prove the impossibility of defining Lorentz-invariant elements of reality. I find that a sufficient condition for the existence of elements of reality, introduced in these proofs, is in fact also used as a necessary condition. I argue that Lorentz-invariant elements of reality can be defined but, as Vaidman pointed out, they won...

We define analogs of the Jucys-Murphy elements for the affine Temperley-Lieb algebra and give their explicit expansion in terms of the basis of planar Brauer diagrams. These Jucys-Murphy elements are a family of commuting elements in the affine Temperley-Lieb algebra, and we compute their eigenvalues on the generic irreducible representations. We show that they come from Jucys-Murphy elements i...

‎Let $G$ be a finite non-abelian group with center $Z(G)$‎. ‎The non-commuting graph of $G$ is a simple undirected graph whose vertex set is $Gsetminus Z(G)$ and two vertices $x$ and $y$ are adjacent if and only if $xy ne yx$‎. ‎In this paper‎, we compute Laplacian energy of the non-commuting graphs of some classes of finite non-abelian groups‎..

let $g$ be a non-abelian group. the non-commuting graph $gamma_g$ of $g$ is defined as the graph whose vertex set is the non-central elements of $g$ and two vertices are joined if and only if they do not commute.in this paper we study some properties of $gamma_g$ and introduce $n$-regular $ac$-groups. also we then obtain a formula for szeged index of $gamma_g$ in terms of $n$, $|z(g)|$ and $|g|...

The commuting graph C(G,X), where G is a group and X is a subset of G, is the graph with vertex set X and distinct vertices being joined by an edge whenever they commute. Here the diameter of C(G,X) is studied when G is a symmetric group and X a conjugacy class of elements of order 3.

The commuting graph C(G,X) , where G is a group and X a subset of G, has X as its vertex set with two distinct elements of X joined by an edge when they commute in G. Here the diameter and disc structure of C(G,X) is investigated when G is the symmetric group and X a conjugacy class of G.

A semigroup is said to be power centralized if for any pair of elements there exists a power of the first one commuting with the second one. We describe the structure of power centralized groups and semigroups. In particular, we present several characterizations of periodic semigroups with central idempotents.

A Coxeter group element w is fully commutative if any reduced expression for w can be obtained from any other via the interchange of commuting generators. For example, in the symmetric group of degree n, the number of fully commutative elements is the nth Catalan number. The Coxeter groups with finitely many fully commutative elements can be arranged into seven infinite families An , Bn , Dn , ...

The exact scattering solutions of the Klein-Gordon equation in cylindrically symmetric field are constructed as eigenfunctions of a complete set of commuting operators. The matrix elements and the corresponding differential scattering cross-section are calculated. Properties of the pair production at various limits are discussed.

The Murphy operators in the Hecke algebra Hn of type A are explicit commuting elements, whose symmetric functions are central in Hn. In [8] I defined geometrically a homomorphism from the Homfly skein C of the annulus to the centre of each algebra Hn, and found an element Pm in C, independent of n, whose image, up to an explicit linear combination with the identity of Hn, is the mth power sum o...