We obtain an estimate of free entropy of generators in a type II1-factor M which has a subfactor N of finite index with a subalgebra P = P1∨P2 ⊂ N where P1 = R1∩P , P2 = R2∩P are diffuse, R1,R2 ⊂ P are mutually commuting hyperfinite subfactors, and an abelian subalgebra A ⊂ N such that the correspondence PL 2(N , τ)A is M-weakly contained in a subcorrespondence PHA of PL 2(M, τ)A, generated by ...

‎suppose $n$ is a fixed positive integer‎. ‎we introduce the relative n-th non-commuting graph $gamma^{n} _{h,g}$‎, ‎associated to the non-abelian subgroup $h$ of group $g$‎. ‎the vertex set is $gsetminus c^n_{h,g}$ in which $c^n_{h,g} = {xin g‎ : ‎[x,y^{n}]=1 mbox{~and~} [x^{n},y]=1mbox{~for~all~} yin h}$‎. ‎moreover‎, ‎${x,y}$ is an edge if $x$ or $y$ belong to $h$ and $xy^{n}eq y^{n}x$ or $x...

Three well-known solutions of the Gaudin equation are obtained under a set of standard assumptions. By relaxing one of these assumptions we introduce a class of mutually commuting Hamiltonians based on a different solution of the Gaudin equation. Application of the algebraic Bethe ansatz technique to diagonalize these Hamiltonians reveals a new infinite dimensional complex Lie algebra.

We analyze a class of dynamics of open quantum systems which is governed by the dynamical map mutually commuting at different times. Such evolution may be effectively described via spectral analysis of the corresponding time dependent generators. We consider both Markovian and non-Markovian cases.

We show that if P is a lattice polytope in the nonnegative orthant of R and χ is a coloring of the lattice points in the orthant such that the color χ(a+b) depends only on the colors χ(a) and χ(b), then the number of colors of the lattice points in the dilation nP of P is for large n given by a polynomial (or, for rational P , by a quasipolynomial). This unifies a classical result of Ehrhart an...

the non commuting graph of a non-abelian finite group $g$ is defined as follows: its vertex set is $g-z(g)$ and two distinct vertices $x$ and $y$ are joined by an edge if and only if the commutator of $x$ and $y$ is not the identity. in this paper we prove some new results about this graph. in particular we will give a new proof of theorem 3.24 of [2]. we also prove that if $g_1$, $g_2$, ..., $...

We say that two graphs G1 and G2 with the same vertex set commute if their adjacency matrices commute. In this paper, we find all integers n such that the complete bipartite graph Kn, n is decomposable into commuting perfect matchings or commuting Hamilton cycles. We show that there are at most n−1 linearly independent commuting adjacency matrices of size n; and if this bound occurs, then there...

The $q$-Onsager algebra $O_q$ is presented by two generators $W_0$, $W_1$ and relations, called the $q$-Dolan/Grady relations. Recently Baseilhac Koizumi introduced a current $\mathcal A_q$ for $O_q$. Soon afterwards, Shigechi gave presentation of We show that these give PBW basis A_q$. Using this basis, we isomorphic to $O_q \otimes \mathbb F \lbrack z_1, z_2, \ldots \rbrack$, where $\mathbb F...

We study a Leech homology of a locally bounded free partially commutative monoid M(E, I). Given a contravariant natural system of abelian groups F on M(E, I) we build a precubical set T (E, I) with a homological system of abelian groups F and prove that the Leech homology groups Hn(M(E, I), F ) are isomorphic to the cubical homology groups Hn(T (E, I), F ), n > 0. As a consequence we have confi...

We define basics of (4, 4) 2D harmonic superspace with two independent sets of SU (2) harmonic variables and apply it to construct new superfield actions of (4, 4) supersymmetric two-dimensional sigma models with torsion and mutually commuting left and right complex structures, as well as of their massive deformations. We show that the generic off-shell sigma model action is the general action ...