A modified k-deck of a graph is obtained by removing k edges in all possible ways and adding k (not necessarily new) edges in all possible ways. Krasikov and Roditty used these decks to give an independent proof of Müller’s result on the edge reconstructibility of graphs. They asked if a k-edge deck could be constructed from its modified k-deck. In this paper, we solve the problem when k = 1. W...

We introduce osp(m|n) spin Calogero-Sutherland models and find that the models have the symmetry of osp(m|n) half-loop algebra or Yangian of osp(m|n) if and only if the coupling constant of the model equals to 2 m−n−4 . Mathematics Subject Classifications (2000): 70H06, 81R12, 17B70.

A new one-dimensional model, the multicomponent Calogero model of B N-type confined in the harmonic potential, is introduced. The Lax pair of this model is determined, and then a set of functionally independent conserved operators are constructed. Moreover, the energy spectrums of the above model are obtained by three different methods.

An interesting observation was reported by Corrigan-Sasaki that all the frequencies of small oscillations around equilibrium are “quantised” for Calogero and Sutherland (C-S) systems, typical integrable multi-particle dynamics. We present an analytic proof by applying recent results of Loris-Sasaki. Explicit forms of ‘classical’ and quantum eigenfunctions are presented for C-S systems based on ...

We consider algebraic geometrical properties of the integrable billiard on a quadric Q with elastic impacts along another quadric confocal to Q. These properties are in sharp contrast with those of the ellipsoidal Birkhoff billiards in Rn. Namely, generic complex invariant manifolds are not Abelian varieties, and the billiard map is no more algebraic. A Poncelet-like theorem for such system is ...

In [LX2], we introduce a class of integrable spin Calogero-Moser systems associated with the classical dynamical r-matrices with spectral parameter. Here the main purpose is to give explicit solutions of several factorization problems on infinite dimensional Lie groupoids which will allow us to write down the solutions of these integrable models.

In 1947, Lehmer conjectured that the Ramanujan τ -function τ(m) never vanishes for all positive integers m, where the τ(m) are the Fourier coefficients of the cusp form ∆24 of weight 12. Lehmer verified the conjecture in 1947 for m < 214928639999. In 1973, Serre verified up to m < 10, and in 1999, Jordan and Kelly for m < 22689242781695999. The theory of spherical t-design, and in particular th...

We introduce osp(m|n) spin Calogero models and find that the models have the symmetry of osp(m|n) half-loop algebra if and only if the coupling constant of the model equals to 2 m−n−4 . Mathematics Subject Classifications (2000): 70H06, 81R12, 17B70.

We introduce a deformed product construction for simple polytopes in terms of lowertriangular block matrix representations. We further show how Gale duality can be employed for the construction and for the analysis of deformed products such that specified faces (e.g. all the k-faces) are “strictly preserved” under projection. Thus, starting from an arbitrary neighborly simplicial (d−2)-polytope...

Acknowledgements I would like to thank the following people. • Prof. Dr. Johannes Buchmann, for a fascinating thesis. • Dipl.-Math. Richard Lindner and Dipl.-Inf. Markus Rückert, for inspiring scientific guidance. Warranty I hereby warrant that the content of this thesis is the direct result of my own work and that any use made in it of published or unpublished material is fully and correctly r...