An integral representation for the Bessel form

AbstractWe deal with integral representation problems of the Bessel form. Suitable formulations are obtained, but they are not proved for all values of the parameter. Generalizations to modified classical forms are possible

On the substitution rule for Lebesgue-Stieltjes integrals

We show how two change-of-variables formulae for Lebesgue-Stieltjes integrals generalize when all continuity hypotheses on the integrators are dropped. We find that a sort of "mass splitting phenomenon" arises.Comment: 6 page

The impact of Stieltjes' work on continued fractions and orthogonal polynomials

Stieltjes' work on continued fractions and the orthogonal polynomials related to continued fraction expansions is summarized and an attempt is made to describe the influence of Stieltjes' ideas and work in research done after his death, with an emphasis on the theory of orthogonal polynomials

Multipeakons and a theorem of Stieltjes

A closed form of the multi-peakon solutions of the Camassa-Holm equation is found using a theorem of Stieltjes on continued fractions. An explicit formula is obtained for the scattering shifts.Comment: 6 page

The indeterminate moment problem for the qq-Meixner polynomials

For a class of orthogonal polynomials related to the $q$-Meixner polynomials corresponding to an indeterminate moment problem we give a one-parameter family of orthogonality measures. For these measures we complement the orthogonal polynomials to an orthogonal basis for the corresponding weighted $L^2$-space explicitly. The result is proved in two ways; by a spectral decomposition of a suitable operator and by direct series manipulation. We discuss extensions to explicit non-positive measures and the relation to other indeterminate moment problems for the continuous $q^{-1}$-Hahn and $q$-Laguerre polynomials.Comment: 26 page

Relations for zeros of special polynomials associated to the Painleve equations

A method for finding relations for the roots of polynomials is presented. Our approach allows us to get a number of relations for the zeros of the classical polynomials and for the roots of special polynomials associated with rational solutions of the Painleve equations. We apply the method to obtain the relations for the zeros of several polynomials. They are: the Laguerre polynomials, the Yablonskii - Vorob'ev polynomials, the Umemura polynomials, the Ohyama polynomials, the generalized Okamoto polynomials, and the generalized Hermite polynomials. All the relations found can be considered as analogues of generalized Stieltjes relations.Comment: 17 pages, 5 figure

Polynomials Associated with Equilibria of Affine Toda-Sutherland Systems

An affine Toda-Sutherland system is a quasi-exactly solvable multi-particle dynamics based on an affine simple root system. It is a `cross' between two well-known integrable multi-particle dynamics, an affine Toda molecule and a Sutherland system. Polynomials describing the equilibrium positions of affine Toda-Sutherland systems are determined for all affine simple root systems.Comment: 9 page

On connectivity in the central nervous systeem : a magnetic resonance imaging study

Brain function has long been the realm of philosophy, psychology and psychiatry and since the mid 1800s, of histopathology. Through the advent of magnetic imaging in the end of the last century, an in vivo visualization of the human brain became available. This thesis describes the development of two unique techniques, imaging of diffusion of water protons and manganese enhanced imaging, that both allow for the depiction of white matter tracts. The reported studies show, that these techniques can be used for a three-dimensional depiction of fiber bundles and that quantitative measures reflecting fiber integrity and neuronal function can be extracted from such data. In clinical applications, the potential use of the developed methods is illustrated in human gliomas, as measure for fiber infiltration, and in spinal cord injury, to monitor potential neuroprotective and __regenerative medication.UBL - phd migration 201