Recent years have seen renewed interest in the six and eight-vertex model at rational coupling values, that is when the crossing parameter is evaluated at roots of unity. Deguchi, Fabricius and McCoy pointed out that the extra degeneracies in the spectrum of the six-vertex transfer matrix can be understood in terms of an affine symmetry algebra in certain commensurate spin-sectors [1]. This rai...

This paper presents a parallel algorithm to solve the eigenvalue problem for non-symmetric matrices. The idea of homotopy is used to generate initial approximations, then the Aberth method and our modiied Aberth method are used to nd simultaneously all the eigenvalues. The advantage of this approach is that multiple eigenvalues can be handled. In addition, the multiplicities of the eigenvalues ...

Nešetřil and Ossona de Mendez introduced the notion of first order convergence, which unifies the notions of convergence for sparse and dense graphs. They asked whether if (Gi)i∈N is a sequence of graphs with M being their first order limit and v is a vertex of M , then there exists a sequence (vi)i∈N of vertices such that the graphs Gi rooted at vi converge to M rooted at v. We show that this ...

Let A be an n × n real matrix. An eigenvector of A is a vector such that Ax is parallel to x; in other words, Ax = λx for some real or complex number λ. This number λ is called the eigenvalue of A belonging to eigenvector v. Clearly λ is an eigenvalue iff the matrix A − λI is singular, equivalently, iff det(A − λI) = 0. This is an algebraic equation of degree n for λ, and hence has n roots (wit...

Recently it has been shown that elements of the unitary matrix determined by a quantum circuit can be computed by counting the number of common roots in the ˇnite ˇeld Z2 for a certain set of multivariate polynomials over Z2. In this paper we present a C# package that allows a user to assemble a quantum circuit and to generate the multivariate polynomial set associated with the circuit. The gen...