Smale's Conjecture on Mean Values of Polynomials and Electrostatics

2000 Mathematics Subject Classification: Primary 30C10, 30C15, 31B35.

Some Coefficient Estimates for Polynomials on the Unit Interval

2000 Mathematics Subject Classification: 26C05, 26C10, 30A12, 30D15, 42A05, 42C05.

Predegree Polynomials of Plane Configurations in Projective Space

2000 Mathematics Subject Classification: 14N10, 14C17.

Letter to the Editor. Remarks on Some Inequalities for Polynomials

MSC 2010: 30A10, 30C10, 30C80, 30D15, 41A17.

Even and Old Overdetermined Strata for Degree 6 Hyperbolic Polynomials

2000 Mathematics Subject Classification: 12D10.

Hyperbolic Fourth-R Quadratic Equation and Holomorphic Fourth-R Polynomials

MSC 2010: 30C10, 32A30, 30G35

Integral Representations of Functional Series with Members Containing Jacobi Polynomials

MSC 2010: Primary 33C45, 40A30; Secondary 26D07, 40C10

Density of Polynomials in the L^2 Space on the Real and the Imaginary Axes and in a Sobolev Space

2000 Mathematics Subject Classification: 41A10, 30E10, 41A65.

Special Classes of Orthogonal Polynomials and Corresponding Quadratures of Gaussian Type

MSC 2010: 33C47, 42C05, 41A55, 65D30, 65D32

Weighted Approximation by a Class of Bernstein-Type Operators

AMS classification: 41A36, 41A10, 41A25, 41Al7.

Monogenic polynomials of four variables with binomial expansion

In the recent past one of the main concern of research in the field of Hypercomplex Function Theory in Clifford Algebras was the development of a variety of new tools for a deeper understanding about its true elementary roots in the Function Theory of one Complex Variable. Therefore the study of the space of monogenic (Clifford holomorphic) functions by its stratification via homogeneous monogenic polynomials is a useful tool. In this paper we consider the structure of those polynomials of four real variables with binomial expansion. This allows a complete characterization of sequences of 4D generalized monogenic Appell polynomials by three different types of polynomials. A particularly important case is that of monogenic polynomials which are simply isomorphic to the integer powers of one complex variable and therefore also called pseudo-complex powers.

Paläobiogeographische Aspekte am Beispiel der Inklusen des Baltischen Bernsteins unter besonderer Berücksichtigung der Steinfliegen (Insecta: Plecoptera) und ein Unterrichtsentwurf zum Thema Baltischer Bernstein und seine Inklusen für den Biologieunterricht verschiedener Schulstufen

Der erste Teil der Dissertation befasst sich mit einem Forschungsprojekt zum Thema Paläobiogeographie der Steinfliegen des Baltischen Bernsteins. Nach einer Untersuchung von über 200 Steinfliegeninklusen des 40-50 Millionen Jahre alten Baltischen Bernsteins konnten vier Neubeschreibungen zu den bisher existierenden 14 hinzugefügt werden. Bei einem Vergleich der Verbreitungshistorie rezenter Gattungen wurden auf der Basis des Aktualitätsprinzips mögliche Verbreitungsrouten der Steinfliegengattungen des Baltischen Bernsteins über die kreidezeitliche Bering Landbrücke aufgestellt. Ausgewählte Aspekte des Forschungsprojekts stellten im zweiten, didaktischen Teil der Dissertation die Basis eines für die Mittelstufe entwickelten Projektes zum Thema Bernstein und seine Inklusen dar. Das Bernsteinprojekt wurde am zdi-Schülerlabor der Universität zu Köln mit mehrern Schulklassen durchgeführt und auf der Basis der im Vorfeld formulierten Forschungsfragen formativ evaluiert.

Clifford-Hermite polynomials in fractional Clifford analysis

In this paper we generalize radial and standard Clifford-Hermite polynomials to the new framework of fractional Clifford analysis with respect to the Riemann-Liouville derivative in a symbolic way. As main consequence of this approach, one does not require an a priori integration theory. Basic properties such as orthogonality relations, differential equations, and recursion formulas, are proven.

Multivariate orthogonal polynomials and integrable systems

Multivariate orthogonal polynomials in D real dimensions are considered from the perspective of the Cholesky factorization of a moment matrix. The approach allows for the construction of corresponding multivariate orthogonal polynomials, associated second kind functions, Jacobi type matrices and associated three term relations and also Christoffel-Darboux formulae. The multivariate orthogonal polynomials, their second kind functions and the corresponding Christoffel-Darboux kernels are shown to be quasi-determinants as well as Schur complements of bordered truncations of the moment matrix; quasi-tau functions are introduced. It is proven that the second kind functions are multivariate Cauchy transforms of the multivariate orthogonal polynomials. Discrete and continuous deformations of the measure lead to Toda type integrable hierarchy, being the corresponding flows described through Lax and Zakharov-Shabat equations; bilinear equations are found. Varying size matrix nonlinear partial difference and differential equations of the 2D Toda lattice type are shown to be solved by matrix coefficients of the multivariate orthogonal polynomials. The discrete flows, which are shown to be connected with a Gauss-Borel factorization of the Jacobi type matrices and its quasi-determinants, lead to expressions for the multivariate orthogonal polynomials and their second kind functions in terms of shifted quasi-tau matrices, which generalize to the multidimensional realm, those that relate the Baker and adjoint Baker functions to ratios of Miwa shifted tau-functions in the 1D scenario. In this context, the multivariate extension of the elementary Darboux transformation is given in terms of quasi-determinants of matrices built up by the evaluation, at a poised set of nodes lying in an appropriate hyperplane in R^D, of the multivariate orthogonal polynomials. The multivariate Christoffel formula for the iteration of m elementary Darboux transformations is given as a quasi-determinant. It is shown, using congruences in the space of semi-infinite matrices, that the discrete and continuous flows are intimately connected and determine nonlinear partial difference-differential equations that involve only one site in the integrable lattice behaving as a Kadomstev-Petviashvili type system. Finally, a brief discussion of measures with a particular linear isometry invariance and some of its consequences for the corresponding multivariate polynomials is given. In particular, it is shown that the Toda times that preserve the invariance condition lay in a secant variety of the Veronese variety of the fixed point set of the linear isometry.