A state i is said to be transient if fii < 1; and it is said to be persistent if fii = 1. Denote by q = (q 1, q t 2, . . . , q t n) the state probability vector (the distribution of the chain at time t), to be a row vector whose i-th component is the probability that the Markov chain is in state i at time t. A distribution π is a stationary distribution for a Markov chain with transition matrix...

When the parameter of deformation q is a root of unity, the centre of Uq(sl(N)) contains, besides the usual q-deformed Casimirs, a set of new generators, which are basically the m-th powers of all the Cartan generators of Uq(sl(N)). All these central elements are however not independent. In this letter, generalising the well-known case of Uq(sl(2)), we explicitly write polynomial relations sati...

We discuss a family of operators which commute or anti-commute with the twisted transfer matrix of the six-vertex model at q being roots of unity: q = 1. The operators commute with the Hamiltonian of the XXZ spin chain under the twisted boundary conditions, and they are valid also for the inhomogeneous case. For the case of the anti-periodic boundary conditions, we show explicitly that the oper...

We present a detailed study of the representations of the algebra of functions on the quantum group GLq(n). A q-analouge of the root system is constructed for this algebra which is then used to determine explicit matrix representations of the generators of this algebra. At the end a q-boson realization of the generators of GLq(n) is given.

We study the transfer matrix of the 8 vertex model with an odd number of lattice sites N. For systems at the root of unity points η = mK/L with m odd the transfer matrix is known to satisfy the famous “TQ” equation where Q(v) is a specifically known matrix. We demonstrate that the location of the zeroes of this Q(v) matrix is qualitatively different from the case of even N and in particular the...

We discuss some aspects of the representation theory of the deformed Virasoro algebra Virp,q. In particular, we give a proof of the formula for the Kac determinant and then determine the center of Virp,q for q a primitive N -th root of unity. We derive explicit expressions for the generators of the center in the limit t = qp → ∞ and elucidate the connection to the Hall-Littlewood symmetric func...

The m-th root of the diagonal of the upper triangular matrix Rm in the QR decomposition of AXB = QmRm converges and the limit is given by the moduli of the eigenvalues of X with some ordering, where A,B,X ∈ Cn×n are nonsingular. The asymptotic behavior of the strictly upper triangular part of Rm is discussed. Some computational experiments are discussed.

In this paper, we focus on deriving low-diameter networks, beginning with D = 2, the next best value to that of the complete network, and proceeding to somewhat larger (constant) values leading to more economical networks. We show that perfect difference networks (PDNs), which are based on the mathematical notion of perfect difference sets, offer a diameter of 2 in an asymptotically optimal man...

In this paper, we present a numerical method to solve the palindromic quadratic eigenvalue problem (PQEP) (λA+λQ+A)z = 0 arising from the vibration analysis of high speed trains, where A, Q ∈ Cn×n have special structures: both Q and A are m × m block matrices with each block being k × k, and moreover they are complex symmetric, block tridiagonal and block Toeplitz, and also A has only one nonze...