Recursion relations for the extended Krylov subspace method

The evaluation of matrix functions of the form f(A)v, where A is a large sparse or structured symmetric matrix, f is a nonlinear function, and v is a vector, is frequently subdivided into two steps: first an orthonormal basis of an extended Krylov subspace of fairly small dimension is determined, and then a projection onto this subspace is evaluated by a method designed for small problems. This paper derives short recursion relations for orthonormal bases of extended Krylov subspaces of the type K^m,mi+1(A)=span{A^-m+1v,…,A^-1v,v,Av,…,A^miv}, m=1,2,3,…, with i a positive integer, and describes applications to the evaluation of matrix functions and the computation of rational Gauss quadrature rules.     

On the local asymptotics of Faber polynomials

We study a local asymptotics of the (generalized) Faber polynomials at the boundary of the associated domain, under certain mild smoothness conditions on the weight function and geometric conditions on the boundary. The main result exhibits how this asymptotics depends on the corners at the boundary. Its proof is based on the continuity properties of the Visser-Ostrowski quotient at the corners.

     

Pascal k-elminated functional matrix and eulerian numbers

In this paper we shall first introduce the Pascal k-eliminated functional matrices P_n,k[xyz] and CP_n,k[xyz]. Then, using these matrices we obtain several important combinatorial identities. Finally, using the matrix inversion of P_n,k[xyz] and CP_n,k[xyz], we derive an interesting formula for Eulerian numbers [7]     

On the Right-Definite and Left-Definite Spectral Theory of the Legendre Polynomials

In this paper, we further develop the left-definite and right-definite spectral theory associated with the self-adjoint differential operator A in L²(-1,1), generated from the classical second-order Legendre differential equation, having the sequence of Legendre polynomials as eigenfunctions. Specifically, we determine the first three left-definite spaces associated with the pair (L²(-1,1),A). As a consequence of these results, we determine the explicit domain of both the associated left-definite operator A₁, first observed by Everitt, and the self-adjoint operator A^1/2. In addition, we give a new characterization of the domain D(A) of A and, as a corollary, we present a new proof of the Everitt-Mari result which gives optimal global smoothness of functions in D(A).     

Perfect Baer Subplane Partitions and Three-Dimensional Flag-Transitive Planes

The classification of perfectBaer subplane partitions of PG(2, q²) is equivalentto the classification of 3-dimensional flag-transitive planeswhose translation complements contain a linear cyclic group actingregularly on the line at infinity. Since all known flag-transitiveplanes admit a translation complement containing a linear cyclicsubgroup which either acts regularly on the points of the lineat infinity or has two orbits of equal size on these points,such a classification would be a significant step towards theclassification of all 3-dimensional flag-transitive planes. Usinglinearized polynomials, a parametric enumeration of all perfectBaer subplane partitions for odd q is described.Moreover, a cyclotomic conjecture is given, verified by computerfor odd prime powers q < 200, whose truth would implythat all perfect Baer subplane partitions arise from a constructionof Kantor and hence the corresponding flag-transitive planesare all known.     

On stability and stabilization of some discrete dynamical systems

The paper formulates effective and nonimprovable stability conditions for a linear difference system involving 2 integer delays. The used technique combines algorithm of the discrete D‐decomposition method with some procedures of the polynomial theory. Contrary to the related existing results, the derived conditions are fully explicit with respect to both delays, which enables their simple applicability in various scientific and engineering areas. As an illustration, we show their importance in delayed feedback controls of discrete dynamical systems, with a particular emphasis put on stabilization of unstable steady states of the discrete logistic map.     

Varying weights for orthogonal polynomials with monotonically varying recurrence coefficients

We consider orthogonal polynomials , where n is the degree of the polynomial and N is a discrete parameter. These polynomials are orthogonal with respect to a varying weight W_N which depends on the parameter N and they satisfy a recurrence relation with varying recurrence coefficients which we assume to be varying monotonically as N tends to infinity. We establish the existence of the limit and link this limit to an external field for an equilibrium problem in logarithmic potential theory.