We give a Riemann–Hilbert approach to the theory of matrix orthogonal polynomials. We will focus on the algebraic aspects of the problem, obtaining difference and differential relations satisfied by the corresponding orthogonal polynomials. We will show that in the matrix case there is some extra freedom that allows us to obtain a family of ladder operators, some of them of 0-th order, somethin...

It has been shown by Lemke that if a matrix is copositive plus on IR n , then feasibility of the corresponding Linear Complementarity Problem implies solvability. In this article we show, under suitable conditions, that feasibility of a Generalized Linear Complementarity Problem (i.e., deened over a more general closed convex cone in a real Hilbert Space) implies solvability whenever the operat...

The matrix representation of the set ∆(d), d = (d1, d2, d3), of the integers which are unrepresentable by d1, d2, d3 is found. The diagrammatic procedure of calculation of the generating function Φ ( d; z ) for the set ∆(d) is developed. The Frobenius number F ( d ) , genus G ( d ) and Hilbert series H(d; z) of a graded subring for non–symmetric and symmetric semigroups S ( d ) are found. The u...

We analyse an initial-boundary value problem for the mKdV equation on a finite interval by expressing the solution in terms of the solution of an associated matrix Riemann-Hilbert problem in the complex k-plane. This Riemann-Hilbert problem has explicit (x, t)-dependence and it involves certain functions of k referred to as “spectral functions”. Some of these functions are defined in terms of t...

The following results (and one or two others like them) are proved in [3]; the continuous-parameter analogues will appear in [4] and some applications will be discussed in [5]. The proofs are based on Sz.-Nagy's group of theorems [9] which show how a discreteor continuous-parameter semigroup of contraction operators on Hilbert space can be "dilated" to a similar semigroup (actually a group) of ...

A basic overview of quantum theory fundamentals (pure states, density matrix, observables) is presented, then density matrices in finite dimensional Hilbert spaces are examined in more detail. Then the problem of optimal quantum state determination is dealt with. This leads to definition of mutually unbiased bases. An explicit form of such bases for prime dimensions is given. An alternative pro...

In this contribution we solve the following problem. Let Hnm be a Hilbert space of dimension nm, and let A be a positive semidefinite self-adjoint linear operator on Hnm. Under which conditions on the spectrum has A a positive partial transpose (is PPT) with respect to any partition Hn⊗Hm of the space Hnm as a tensor product of an n-dimensional and an m-dimensional Hilbert space? We show that t...

The classical phase of the matrix model of 11–dimensional M–theory is complex, infinite–dimensional Hilbert space. As a complex manifold, the latter admits a continuum of nonequivalent, complex–differentiable structures that can be placed in 1–to–1 correspondence with families of coherent states in the Hilbert space of quantum states. The moduli space of nonbiholomorphic complex structures on c...

the concept of soft sets, introduced by molodtsov [20] is a mathematicaltool for dealing with uncertainties, that is free from the difficultiesthat have troubled the traditional theoretical approaches. in this paper, weapply the notion of the soft sets of molodtsov to the theory of hilbert algebras.the notion of soft hilbert (abysmal and deductive) algebras, soft subalgebras,soft abysms and sof...

in this paper, we investigate duality of modular g-riesz bases and g-riesz basesin hilbert c*-modules. first we give some characterization of g-riesz bases in hilbert c*-modules, by using properties of operator theory. next, we characterize the duals of a giveng-riesz basis in hilbert c*-module. in addition, we obtain sucient and necessary conditionfor a dual of a g-riesz basis to be again a g...