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...

Top-k queries allow end-users to focus on the most important (top-k) answers amongst those which satisfy the query. In traditional databases, a user defined score function assigns a score value to each tuple and a top-k query returns k tuples with the highest score. In uncertain database, top-k answer depends not only on the scores but also on the membership probabilities of tuples. Several top...

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.

Iosevich and Senger (2008) showed that if a subset of the d-dimensional vector space over a finite field is large enough, then it contains many k-tuples of mutually orthogonal vectors. In this note, we provide a graph theoretic proof of this result.

In this paper we recover an [m,C]-isometric operators and (m,C)-isometric commuting tuples of on a Hilbert space studied respectively in [11] [16], introduce the class [m,C]-isometries for tuple operators. This is generalization spaces. A S = (S1,..., Sp) ? B(H)p said to be p-tuple if ?m (S,C):= j=0 (?1)m?j (m j) (? |?|=j j!/?! CS?CS?)=0 some positive integer m conjugation C. We consider multi-...

We prove that if a subset of the d-dimensional vector space over the ring of integers modulo pr is large enough, then the number of k-tuples of mutually orthogonal vectors in this set is close to its expected value.

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...