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warrant that the content of this thesis is the direct result of my own work and that any use made in it of published or unpublished material is fully and correctly r...

We give an affirmative answer to a question of Gorelic [5], by showing it is consistent, relative to the existence of large cardinals, that there is a proper class of cardinals α with cf(α) = ω1 and αω > α.

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 R. 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 k...

2. The Carlitz-Witt suite 5 2.1. The classical ghost-Witt equivalence theorem . . . . . . . . . . . . . 5 2.2. Classical Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.3. The Carlitz ghost-Witt equivalence theorem . . . . . . . . . . . . . . 9 2.4. Carlitz-Witt vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.5. F -modules . . . . . . . . . . . . . . ...

We solve Lehmer’s problem for a class of polynomials arising from Hermitian matrices over the Eisenstein and Gaussian integers, that is, we show that all such polynomials have Mahler measure at least Lehmer’s number τ0 = 1.17628 . . . .

It is shown that from some solutions of generalized Knizhnik-Zamolodchikov equations one can construct eigenfunctions of the Calogero-Sutherland-Moser Hamiltonians with exchange terms, which are characterized by any given permutational symmetry under particle exchange. This generalizes some results previously derived by Matsuo and Cherednik for the ordinary Calogero-Sutherland-Moser Hamiltonians.

The canonical transformation and its unitary counterpart which relate the rational Calogero–Moser system to the free one are constructed. Lloyd’s Tercentenary Fellow. On leave from: Department of Theoretical Physics, University of Lódź, ul. Pomorska 149/153, 90-236 Lódź, Poland. E-mail: [email protected]

More than 30 years ago, Hedetniemi made a conjecture which says that the categorical product of two n-chromatic graphs is still n-chromatic. The conjecture is still open, despite many different approaches from different point of views. This article surveys methods and partial results; and discuss problems related to or motivated by this conjecture.

We prove that every tree of order n 5 with three leaves is (n + 1)-unavoidable. More precisely, we prove that every tree A with three leaves of order n is contained in every tournament T of order n + 1 except if (T ; A) is (R 5 ; S + 3) or its dual, where R 5 is the regular tournament on ve vertices and S + 3 is the outstar of degree three, that is, the tree consisting of a root dominating thre...