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...

We generalize the notion of graded posets to what we call sign-graded (labeled) posets. We prove that the W -polynomial of a sign-graded poset is symmetric and unimodal. This extends a recent result of Reiner and Welker who proved it for graded posets by associating a simplicial polytopal sphere to each graded poset. By proving that the W -polynomials of sign-graded posets has the right sign at...

An exposition of the different definitions and approaches to quantum statistics is given, with emphasis in one-dimensional situations. Permutation statistics, scattering statistics and exclusion statistics are analyzed. The Calogero model, matrix model and spin chain models constitute specific realizations. E-mail: [email protected]