Lie Theory and its Applcations in Physics III

One-parameter semigroups related to abstract quantum models of Calogero type - Rösler, Margit (Infinite dimensional harmonic analysis (Kyoto 1999))

The computation of lyapunov exponents via spatial integration with application to blowout bifurcations - Aston, Philip J.; Dellnitz, Michael (Computer Methods in Applied Mechanics and Engineering)

On the Approximation of Complicated Dynamical Behavior - Dellnitz, Michael; Junge, Oliver (SIAM Journal on Numerical Analysis)

Computation of Essential Molecular Dynamics by Subdivision Techniques - Deuflhard, Peter; Dellnitz, Michael; Junge, Oliver; Schütte, Christof (Computational Molecular Dynamics: Challenges, Methods, Ideas)

Visualizing Complicated Dynamics - Bürkle, David; Dellnitz, Michael; Junge, Oliver; Rumpf, Martin; Spielberg, Michael (Proceedings of Visualization 99)

An uncertainty principle for the Dunkl transform - Rösler, Margit (Bulletin of the Australian Mathematical Society)
This note presents an analogue of the classical Heisenberg-Weyl uncertainty principle for the Dunkl transform on ℝ N . Its proof is based on expansions with respect to generalised Hermite functions.
Keywords: General Mathematics

Positivity of Dunkl’s intertwining operator - Rösler, Margit (Duke Mathematical Journal)
Keywords: General Mathematics

Partial Characters and Signed Quotient Hypergroups - Rösler, Margit; Voit, Michael (Canadian Journal of Mathematics)
Abstract If G is a closed subgroup of a commutative hypergroup K , then the coset space K / G carries a quotient hypergroup structure. In this paper, we study related convolution structures on K / G coming fromdeformations of the quotient hypergroup structure by certain functions on K which we call partial characters with respect to G . They are usually not probability-preserving, but lead to so-called signed hypergroups on K / G . A first example is provided by the Laguerre convolution on [0, ∞[, which is interpreted as a signed quotient hypergroup convolution derived from the Heisenberg group. Moreover, signed hypergroups associated with the Gelfand pair ( U ( n , 1), U ( n )) are discussed.
Keywords: General Mathematics

On Polynomial Decompositions - Klüners, Jürgen (Journal of Symbolic Computation)
We present a new polynomial decomposition which generalizes the functional and homogeneous bivariate decomposition of irreducible monic polynomials in one variable over the rationals. With these decompositions it is possible to calculate the roots of an imprimitive polynomial by solving polynomial equations of lower degree.
Keywords: Computational Mathematics; Algebra and Number Theory

Computing Automorphisms of Abelian Number Fields - Klüners, Jürgen; Acciaro, Vincenzo (Mathematics of Computation)
Let L = ℚ(α) be an abelian number field of degree n. Most algorithms for computing the lattice of subfields of L require the computation of all the conjugates of α. This is usually achieved by factoring the minimal polynomial mα(x) of α over L. In practice, the existing algorithms for factoring polynomials over algebraic number fields can handle only problems of moderate size. In this paper we describe a fast probabilistic algorithm for computing the conjugates of α, which is based on p-adic techniques. Given mα(x) and a rational prime p which does not divide the discriminant disc(mα(x)) of mα(x), the algorithm computes the Frobenius automorphism of p in time polynomial in the size of p and in the size of mα(x). By repeatedly applying the algorithm to randomly chosen primes it is possible to compute all the conjugates of α.

Representations, Characters, and Spherical Functions Associated to Causal Symmetric Spaces - Hilgert, Joachim; Krötz, B. (J. Funct. Anal.)

Guillemin, V., E. Lerman, S. Sternberg. Symplectic fibration and Multiplicity Diagrams (Cambridge University Press, 1996) - Hilgert, Joachim (JBer. DMV)

A Conjugacy Theorem for Symmetric Spaces - Hilgert, Joachim; Neumann , A.; Ólafsson, G. (Math. Annalen)

Weighted Bergman Spaces Associated to Causal Symmetric Spaces - Hilgert, Joachim; Krötz, B. (Manus. Math.)

Positive Definite Spherical Functions on Olshanskii Domains - Hilgert, Joachim; Neeb, K.-H. (Trans. AMS.)

An uncertainty principle for Hankel transforms - Rösler, Margit (Proceedings of the American Mathematical Society)

An adaptive subdivision technique for the approximation of attractors and invariant measures - Dellnitz, Michael; Junge, Oliver (Computing and Visualization in Science)

Biorthogonal polynomials associated with reflection groups and a formula of Macdonald - Rösler, Margit; Voit, Michael (Journal of Computational and Applied Mathematics)
Keywords: Applied Mathematics; Computational Mathematics

Markov Processes Related with Dunkl Operators - Rösler, Margit; Voit, Michael (Advances in Applied Mathematics)
Keywords: Applied Mathematics

Projective transformations of a plane - Burban, Igor; Duma, W. (U sviti Mathematyky)

Projective transformations of a plane and three-dimensional space - Burban, Igor; Duma, W. (U sviti Mathematyky)

Rational parameterization of algebraic curves - Burban, Igor (U sviti Mathematyky)

Invariant Cones in Real Representations - Hilgert, Joachim; Neeb, K.-H. (Positivity in Lie Theory: Open Problems)

Reproducing Kernels on Vector Bundles - Hilgert, Joachim; Bertram, W. (Lie Theory and its Applications in Physics)

Hardy Spaces and Analytic Continuation of Bergman Spaces - Hilgert, Joachim; Bertram, W. (Bull. Math. Soc. Francaise)

Poisson Lie Groups and Non-Linear Convexity Theorems - Hilgert, Joachim; Neeb, K.-H. (Math. Nachr.)

Lie Theory and its Applcations in Physics II

Positivity in Lie Theory

Wachstumsverhalten von Lösungen der $\overline{\partial}$-Gleichung auf Pseudo-Siegel-Gebieten - Hesse, Kerstin

Almost Invariant Sets in Chua's Circuit - Dellnitz, Michael; Junge, Oliver (International Journal of Bifurcation and Chaos)
Recently multilevel subdivision techniques have been introduced in the numerical investigation of complicated dynamical behavior. We illustrate the applicability and efficiency of these methods by a detailed numerical study of Chua's circuit. In particular we will show that there exist two regions in phase space which are almost invariant in the sense that typical trajectories stay inside each of these sets on average for quite a long time.

Exploring invariant sets and invariant measures - Dellnitz, Michael; Hohmann, Andreas; Junge, Oliver; Rumpf, Martin (Chaos: An Interdisciplinary Journal of Nonlinear Science)

An adaptive method for the approximation of the generalized cell mapping - Guder, Rabbijah; Dellnitz, Michael; Kreuzer, Edwin (Chaos, Solitons & Fractals)

A subdivision algorithm for the computation of unstable manifolds and global attractors - Dellnitz, Michael; Hohmann, Andreas (Numerische Mathematik)

An Uncertainty Principle for Ultraspherical Expansions - Rösler, Margit; Voit, Michael (Journal of Mathematical Analysis and Applications)
Keywords: Applied Mathematics; Analysis

KANT V4 - DABERKOW, M.; FIEKER, C.; Klüners, Jürgen; POHST, M.; ROEGNER, K.; SCHÖRNIG, M.; WILDANGER, K. (Journal of Symbolic Computation)
The software packageKANT V4for computations in algebraic number fields is now available in version 4. In addition a new user interface has been released. We will outline the features of this new software package.
Keywords: Computational Mathematics; Algebra and Number Theory

On Computing Subfields - Klüners, Jürgen; Pohst, Michael (Journal of Symbolic Computation)
The purpose of this article is to determine all subfields ℚ(β) of fixed degree of a given algebraic number field ℚ(α). It is convenient to describe each subfield by a pair (h,g) of polynomials in ℚ[t] resp. Z[t] such thatgis the minimal polynomial of β = h(α). The computations are done in unramifiedp-adic extensions and use information concerning subgroups of the Galois group of the normal closure of ℚ(α) obtained from the van der Waerden criterion.
Keywords: Computational Mathematics; Algebra and Number Theory

Über die Berechnung von Automorphismen und Teilkörpern algebraischer Zahlkörper (Dissertation) - Klüners, Jürgen

Singular Unitary Highest Weight Representations via Geometric Quantization - Hilgert, Joachim (Reports on Math. Physics)

Causal Symmetric Spaces, Geometry and Harmonic Analysis

Vaisman, I. Lectures on the Geometry of Poisson Manifolds (Birkhäuser, Boston, 1994) - Hilgert, Joachim (JBer. DMV)

The Computation of Unstable Manifolds Using Subdivision and Continuation - Dellnitz, Michael; Hohmann, Andreas (Nonlinear Dynamical Systems and Chaos)

Euler’s theorem and map color problems - Burban, Igor (U sviti Mathematyky)

Solutions of equations in integers - Burban, Igor (U sviti Mathematyky)

Unitary Highest Weight Representations via the Orbit Method I: The Scalar Case - Hilgert, Joachim; Neeb, K.-H.; Orsted, B. (Acta Appl.. Math.)

Conal Heisenberg Algebras and Associated Hilbert Spaces - Hilgert, Joachim; Neeb, K.-H.; Orsted, B. (J. reine angew. Math.)

Orthogonal Lie Algrebras with Cone Potential - Hilgert, Joachim; Neeb, K.-H. (Commun. in Alg.)

Spherical Functions on Olshanskii Spaces - Hilgert, Joachim; Neeb, K.-H. (J. Funct. Anal.)

Knapp, A.W. Lie Groups Beyond an Introduction (Birkhäuser, Boston, 1996) - Hilgert, Joachim (Zentralblatt für Math.)

Symmetry Breaking Bifurcations of Chaotic Attractors - Aston, Philip J.; Dellnitz, Michael (International Journal of Bifurcation and Chaos)
In an array of coupled oscillators, synchronous chaos may occur in the sense that all the oscillators behave identically although the corresponding motion is chaotic. When a parameter is varied this fully symmetric dynamical state can lose its stability, and the main purpose of this paper is to investigate which type of dynamical behavior is expected to be observed once the loss of stability has occurred. The essential tool is a classification of Lyapunov exponents based on the symmetry of the underlying problem. This classification is crucial in the derivation of the analytical results but it also allows an efficient computation of the dominant Lyapunov exponent associated with each symmetry type. We show how these dominant exponents determine the stability of invariant sets possessing various instantaneous symmetries, and this leads to the idea of symmetry breaking bifurcations of chaotic attractors. Finally, the results and ideas are illustrated for several systems of coupled oscillators.