We prove bispectral duality for the generalized Calogero–Moser–Sutherland systems related to configurations An,2(m), Cn(l,m). The trigonometric axiomatics of Baker–Akhiezer function is modified, the dual difference operators of rational Macdonald type and the Baker–Akhiezer functions related to both series are explicitly constructed.

We consider the physical properties of elementary excitations of the CalogeroSutherland (CS) model with SU(K) internal symmetry. From the results on the thermodynamics of this model, we obtain the charge, spin, and statistics of elementary excitations. Combining this knowledge and the known results on the dynamics in the spinless CS model, we propose the expression for the dynamical correlation...

The Andrews-Curtis conjecture claims that every balanced presentation of the trivial group can be reduced to the standard one by a sequence of \elementary transformations" which are Nielsen transformations augmented by arbitrary conjugations. It is a prevalent opinion that this conjecture is false; however, not many potential counterexamples are known. In this paper, we show that some of the pr...

We propose that multi-linear functions of relatively low degree over GF(2) may be goodcandidates for obtaining exponential lower bounds on the size of constant-depth Boolean cir-cuits (computing explicit functions). Specifically, we propose to move gradually from linearfunctions to multilinear ones, and conjecture that, for any t ≥ 2, some explicit t-linear functionsF : ({0, 1}n...

I n 1970, Gian-Carlo Rota posed a conjecture predicting a beautiful combinatorial characterization of linear dependence in vector spaces over any given finite field. We have recently completed a fifteen-year research program that culminated in a solution of Rota’s Conjecture. In this article we discuss the conjecture and give an overview of the proof. Matroids are a combinatorial abstraction of...

The Erdős-Faber-Lovász conjecture is the statement that every graph that is the union of n cliques of size n intersecting pairwise in at most one vertex has chromatic number n. Kahn and Seymour proved a fractional version of this conjecture, where the chromatic number is replaced by the fractional chromatic number. In this note we investigate similar fractional relaxations of the Erdős-Faber-Lo...

Let D be a bounded, convex domain in Cn, and suppose that φ :D→D is holomorphic. Assume that ψ : D → C is analytic, bounded away from zero toward the boundary of D, and not identically zero on the fixed point set of D. Suppose also that the weighted composition operator Wψ,φ given by Wψ,φ( f )= ψ( f ◦φ) is compact on a holomorphic, functional Hilbert space (containing the polynomial functions d...