Some Properties of the Calogero-Sutherland Model with Reflections

We prove that the Calogero-Sutherland Model with reflections (the BC_N model) possesses a property of duality relating the eigenfunctions of two Hamiltonians with different coupling constants. We obtain a generating function for their polynomial eigenfunctions, the generalized Jacobi polynomials. The symmetry of the wave-functions for certain particular cases (associated to the root systems of the classical Lie groups B_N, C_N and D_N) is also discussed.Comment: 16 pages, harvmac.te

On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures

This note, mostly expository, is devoted to Poincar{\'e} and log-Sobolev inequalities for a class of Boltzmann-Gibbs measures with singular interaction. Such measures allow to model one-dimensional particles with confinement and singular pair interaction. The functional inequalities come from convexity. We prove and characterize optimality in the case of quadratic confinement via a factorization of the measure. This optimality phenomenon holds for all beta Hermite ensembles including the Gaussian unitary ensemble, a famous exactly solvable model of random matrix theory. We further explore exact solvability by reviewing the relation to Dyson-Ornstein-Uhlenbeck diffusion dynamics admitting the Hermite-Lassalle orthogonal polynomials as a complete set of eigenfunctions. We also discuss the consequence of the log-Sobolev inequality in terms of concentration of measure for Lipschitz functions such as maxima and linear statistics.Comment: Minor improvements. To appear in Geometric Aspects of Functional Analysis -- Israel Seminar (GAFA) 2017-2019", Lecture Notes in Mathematics 225

An (inverse) Pieri formula for Macdonald polynomials of type C

We give an explicit Pieri formula for Macdonald polynomials attached to the root system C_n (with equal multiplicities). By inversion we obtain an explicit expansion for two-row Macdonald polynomials of type C.Comment: 31 pages, LaTeX, to appear in Transformation Group

Explicit formulas for the generalized Hermite polynomials in superspace

We provide explicit formulas for the orthogonal eigenfunctions of the supersymmetric extension of the rational Calogero-Moser-Sutherland model with harmonic confinement, i.e., the generalized Hermite (or Hi-Jack) polynomials in superspace. The construction relies on the triangular action of the Hamiltonian on the supermonomial basis. This translates into determinantal expressions for the Hamiltonian's eigenfunctions.Comment: 19 pages. This is a recasting of the second part of the first version of hep-th/0305038 which has been splitted in two articles. In this revised version, the introduction has been rewritten and a new appendix has been added. To appear in JP

TPMS: a set of utilities for querying collections of gene trees

Background: The information in large collections of phylogenetic trees is useful for many comparative genomic studies. Therefore, there is a need for flexible tools that allow exploration of such collections in order to retrieve relevant data as quickly as possible. Results: In this paper, we present TPMS (Tree Pattern-Matching Suite), a set of programs for handling and retrieving gene trees according to different criteria. The programs from the suite include utilities for tree collection building, specific tree-pattern search strategies and tree rooting. Use of TPMS is illustrated through three examples: systematic search for incongruencies in a large tree collection, a short study on the Coelomata/Ecdysozoa controversy and an evaluation of the level of support for a recently published Mammal phylogeny. Conclusion: TPMS is a powerful suite allowing to quickly retrieve sets of trees matching complex patterns in large collection or to root trees using more rigorous approaches than the classical midpoint method. As it is made of a set of command-line programs, it can be easily integrated in any sequence analysis pipeline for an automated use

Asymptotics of Selberg-like integrals: The unitary case and Newton's interpolation formula

We investigate the asymptotic behavior of the Selberg-like integral $$\frac1{N!}\int_{[0,1]^N}x_1^p\prod_{i<j}(x_i-x_j)^2\prod_ix_i^{a-1}(1-x_i)^{b-1}dx_i$$, as $N\to\infty$ for different scalings of the parameters $a$ and $b$ with $N$. Integrals of this type arise in the random matrix theory of electronic scattering in chaotic cavities supporting $N$ channels in the two attached leads. Making use of Newton's interpolation formula, we show that an asymptotic limit exists and we compute it explicitly

A new multivariable 6-psi-6 summation formula

By multidimensional matrix inversion, combined with an A_r extension of Jackson's 8-phi-7 summation formula by Milne, a new multivariable 8-phi-7 summation is derived. By a polynomial argument this 8-phi-7 summation is transformed to another multivariable 8-phi-7 summation which, by taking a suitable limit, is reduced to a new multivariable extension of the nonterminating 6-phi-5 summation. The latter is then extended, by analytic continuation, to a new multivariable extension of Bailey's very-well-poised 6-psi-6 summation formula.Comment: 16 page

Quantum Calogero-Moser Models: Integrability for all Root Systems

The issues related to the integrability of quantum Calogero-Moser models based on any root systems are addressed. For the models with degenerate potentials, i.e. the rational with/without the harmonic confining force, the hyperbolic and the trigonometric, we demonstrate the following for all the root systems: (i) Construction of a complete set of quantum conserved quantities in terms of a total sum of the Lax matrix (L), i.e. (\sum_{\mu,\nu\in{\cal R}}(L^n)_{\mu\nu}), in which ({\cal R}) is a representation space of the Coxeter group. (ii) Proof of Liouville integrability. (iii) Triangularity of the quantum Hamiltonian and the entire discrete spectrum. Generalised Jack polynomials are defined for all root systems as unique eigenfunctions of the Hamiltonian. (iv) Equivalence of the Lax operator and the Dunkl operator. (v) Algebraic construction of all excited states in terms of creation operators. These are mainly generalisations of the results known for the models based on the (A) series, i.e. (su(N)) type, root systems.Comment: 45 pages, LaTeX2e, no figure

Telomere lengths in human oocytes, cleavage stage embryos and blastocysts

Telomeres are repeated sequences that protect the ends of chromosomes and harbour DNA-repair proteins. Telomeres shorten during each cell division in the absence of telomerase. When telomere length becomes critically short, cell senescence occurs. Telomere length therefore reflects both cellular ageing and capacity for division. We have measured telomere length in human germinal vesicle (GV) oocytes and pre-implantation embryos, by quantitative fluorescence in-situ hybridisation (Q-FISH), providing baseline data towards our hypothesis that telomere length is a marker of embryo quality. The numbers of fluorescent foci suggest that extensive clustering of telomeres occurs in mature GV stage oocytes, and in pre-implantation embryos. When calculating average telomere length by assuming that each signal presents one telomere, the calculated telomere length decreased from the oocyte to the cleavage stages, and increased between the cleavage stages and the blastocyst (11.12 vs 8.43 vs 12.22kb respectively, p<0.001). Other methods of calculation, based upon expected maximum and minimum numbers of telomeres, confirm that telomere length in blastocysts is significantly longer than cleavage stages. Individual blastomeres within an embryo showed substantial variation in calculated average telomere length. This study implies that telomere length changes according to the stage of pre-implantation embryo development