Let q be a power of 2. We show by representation theory that there exists a q × q unitary matrix of multiplicative order q + 1 whose powers generate q + 1 complex pairwise mutually unbiased bases in C. When q is a power of an odd prime, there is a q × q unitary matrix of multiplicative order q+1 whose first (q+1)/2 powers generate (q+1)/2 complex pairwise mutually unbiased bases. We also show h...

This paper studies quantum systems with a finite number of degrees of freedom in the context of non-extensive thermodynamics. A trial density matrix, obtained by heuristic methods, is proved to be the equilibrium density matrix. If the entropic parameter q is larger than 1 then existence of the trial equilibrium density matrix requires that q is less than some critical value qc which depends on...

The matrix equation X + ATX−1A = Q has been studied extensively when A and Q are real square matrices and Q is symmetric positive definite. The equation has positive definite solutions under suitable conditions, and in that case the solution of interest is the maximal positive definite solution. The same matrix equation plays an important role in Green’s function calculations in nano research, ...

!he incidence matrix If of points and d-flats in PG(t, q) is a (O,l)-matrix which may be considered a _trAx with entries from the field GF (q) , where q is a prime power, q'" p. Let r d (t, 9.) denote thr rank of If over (JF (q). In this paper, & fonml.a for r t 1 (t, q) and an upper bound for r (t,q), 1 S d S t-l, are given. In the case q =p, the 'bound on rd(t,p) is attained. Similar results ...

This work is concerned with the Macdonald q;t-analogue of the Kostka matrix. This matrix relates the two parameter Macdonald basis fP x; q;tg to the modiied Schur basis fS X1 , tg. The entries in this matrix, which h a v e come to be denoted by K ; q;t, h a v e been conjectured by Macdonald to be polynomials in q;t with positive i n tegral coeecients. Our main main result here is an algorithm f...

The star-matrix models are difficult to solve due to the multiple powers of the Vandermonde determinants in the partition function. We apply to these models a modified Q-matrix approach and we get results consistent with those obtained by other methods.As examples we study the inhomogenous gaussian model on Bethe tree and matrix q-Potts-like model. For the last model in the special cases q = 2 ...

This paper surveys mathematical properties of (layered-) mixed matrices with emphasis on irreducibility and block-triangular decomposition. A matrix A is a mixed matrix if A = Q + T , where Q is a “constant” matrix and T is a “generic” matrix (or formal incidence matrix) in the sense that the nonzero entries of T are algebraically independent parameters. A layered mixed (or LM-) matrix is a mix...

This note deals with the following problem: Let A be an n X n Hermitian matrix, and Q and 0 be two n X rn (n > m > 1) matrices both with orthonormal column vectors. How do the eigenvalues of the m X m Hermitian matrix Q”AQ differ from those of the m X m Hermitian matrix QHAQ? We give a positive answer to one of the unsolved problems raised recently by Sun. In what follows, we will consider the ...

The recent surge of interests in cognitive assessment has led to developments of novel statistical models for diagnostic classification. Central to many such models is the well-known Q-matrix, which specifies the item-attribute relationships. This article proposes a data-driven approach to identification of the Q-matrix and estimation of related model parameters. A key ingredient is a flexible ...