Non-commutative Pieri operators on posets

We consider graded representations of the algebra NC of noncommutative symmetric functions on the Z-linear span of a graded poset P. The matrix coefficients of such a representation give a Hopf morphism from a Hopf algebra HP generated by the intervals of P to the Hopf algebra of quasi-symmetric functions. This provides a unified construction of quasi-symmetric generating functions from different branches of algebraic combinatorics, and this construction is useful for transferring techniques and ideas between these branches. In particular we show that the (Hopf) algebra of Billera and Liu related to Eulerian posets is dual to the peak (Hopf) algebra of Stembridge related to enriched P-partitions, and connect this to the combinatorics of the Schubert calculus for isotropic flag manifolds.Comment: LaTeX 2e, 22 pages Minor corrections, updated references. Complete and final version, to appear in issue of J. Combin. Th. Ser. A dedicated to G.-C. Rot

Efficient Quantum Tensor Product Expanders and k-designs

Quantum expanders are a quantum analogue of expanders, and k-tensor product expanders are a generalisation to graphs that randomise k correlated walkers. Here we give an efficient construction of constant-degree, constant-gap quantum k-tensor product expanders. The key ingredients are an efficient classical tensor product expander and the quantum Fourier transform. Our construction works whenever k=O(n/log n), where n is the number of qubits. An immediate corollary of this result is an efficient construction of an approximate unitary k-design, which is a quantum analogue of an approximate k-wise independent function, on n qubits for any k=O(n/log n). Previously, no efficient constructions were known for k>2, while state designs, of which unitary designs are a generalisation, were constructed efficiently in [Ambainis, Emerson 2007].Comment: 16 pages, typo in references fixe

A schlieren method for ultra-low angle light scattering measurements

We describe a self calibrating optical technique that allows to perform absolute measurements of scattering cross sections for the light scattered at extremely small angles. Very good performances are obtained by using a very simple optical layout similar to that used for the schlieren method, a technique traditionally used for mapping local refraction index changes. The scattered intensity distribution is recovered by a statistical analysis of the random interference of the light scattered in a half-plane of the scattering wave vectors and the main transmitted beam. High quality data can be obtained by proper statistical accumulation of scattered intensity frames, and the static stray light contributions can be eliminated rigorously. The potentialities of the method are tested in a scattering experiment from non equilibrium fluctuations during a free diffusion experiment. Contributions of light scattered from length scales as long as Lambda=1 mm can be accurately determined.Comment: 7 pages, 3 figure

Distributive Lattices, Affine Semigroups, and Branching Rules of the Classical Groups

We study algebras encoding stable range branching rules for the pairs of complex classical groups of the same type in the context of toric degenerations of spherical varieties. By lifting affine semigroup algebras constructed from combinatorial data of branching multiplicities, we obtain algebras having highest weight vectors in multiplicity spaces as their standard monomial type bases. In particular, we identify a family of distributive lattices and their associated Hibi algebras which can uniformly describe the stable range branching algebras for all the pairs we consider.Comment: 30 pages, extensively revise

Prostaglandins A1 and E1 influence gene expression in an established insect cell line (BCIRL-HzAM1 cells)

Prostaglandins (PGs) and other eicosanoids exert important physiological actions in insects and other invertebrates, including influencing ion transport and mediating cellular immune defense functions. Although these actions are very well documented, we have no information on the mechanisms of PGs actions in insect cells. Here we report on the outcomes of experiments designed to test our hypothesis that PGs modulate gene expression in an insect cell line established from pupal ovarian tissue of the moth Helicoverpa zea (BCIRL-HzAM1 cells). We treated cells with either PGA1 or PGE1 for 12 or 24 h then analyzed cell lysates by 2-D electrophoresis. Analysis of the gels by densitometry revealed substantial changes in protein expression in some of the protein spots we analyzed. These spots were processed for mass spectrometric analysis by MALDI TOF/TOF, which yielded in silico protein identities for all 34 spots. The apparent changes in three of the proteins were confirmed by semi-quantative PCR, showing that the changes in mRNA expression were reflected in changes in protein expression. The 34 proteins were sorted into six categories, protein actions, lipid metabolism, signal transduction, protection, cell functions and metabolism. The findings support the hypothesis that one mechanism of PG action in insect cells is the modulation of gene expression

LR characterization of chirotopes of finite planar families of pairwise disjoint convex bodies

We extend the classical LR characterization of chirotopes of finite planar families of points to chirotopes of finite planar families of pairwise disjoint convex bodies: a map \c{hi} on the set of 3-subsets of a finite set I is a chirotope of finite planar families of pairwise disjoint convex bodies if and only if for every 3-, 4-, and 5-subset J of I the restriction of \c{hi} to the set of 3-subsets of J is a chirotope of finite planar families of pairwise disjoint convex bodies. Our main tool is the polarity map, i.e., the map that assigns to a convex body the set of lines missing its interior, from which we derive the key notion of arrangements of double pseudolines, introduced for the first time in this paper.Comment: 100 pages, 73 figures; accepted manuscript versio

The nullcone in the multi-vector representation of the symplectic group and related combinatorics

We study the nullcone in the multi-vector representation of the symplectic group with respect to a joint action of the general linear group and the symplectic group. By extracting an algebra over a distributive lattice structure from the coordinate ring of the nullcone, we describe a toric degeneration and standard monomial theory of the nullcone in terms of double tableaux and integral points in a convex polyhedral cone.Comment: 21 pages, v2: title changed, typos and errors correcte

Products of Foldable Triangulations

Regular triangulations of products of lattice polytopes are constructed with the additional property that the dual graphs of the triangulations are bipartite. The (weighted) size difference of this bipartition is a lower bound for the number of real roots of certain sparse polynomial systems by recent results of Soprunova and Sottile [Adv. Math. 204(1):116-151, 2006]. Special attention is paid to the cube case.Comment: new title; several paragraphs reformulated; 23 page