Affine and toric hyperplane arrangements

We extend the Billera-Ehrenborg-Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For arrangements on the torus, we also generalize Zaslavsky's fundamental results on the number of regions.Comment: 32 pages, 4 figure

Dynamics of Entanglement in One-Dimensional Spin Systems

We study the dynamics of quantum correlations in a class of exactly solvable Ising-type models. We analyze in particular the time evolution of initial Bell states created in a fully polarized background and on the ground state. We find that the pairwise entanglement propagates with a velocity proportional to the reduced interaction for all the four Bell states. Singlet-like states are favored during the propagation, in the sense that triplet-like states change their character during the propagation under certain circumstances. Characteristic for the anisotropic models is the instantaneous creation of pairwise entanglement from a fully polarized state; furthermore, the propagation of pairwise entanglement is suppressed in favor of a creation of different types of entanglement. The ``entanglement wave'' evolving from a Bell state on the ground state turns out to be very localized in space-time. Further support to a recently formulated conjecture on entanglement sharing is given.Comment: 25 pages, 21 figures; revte

Comparative analysis of serine protease-related genes in the honey bee genome: possible involvement in embryonic development and innate immunity

We have identified 44 serine protease (SP) and 13 serine protease homolog (SPH) genes in the genome of Apis mellifera. Most of these genes encode putative secreted proteins, but four SPs and three SPHs may associate with the plasma membrane via a transmembrane region. Clip domains represent the most abundant non-catalytic structural units in these SP-like proteins −12 SPs and six SPHs contain at least one clip domain. Some of the family members contain other modules for protein–protein interactions, including disulphide-stabilized structures (LDL(r)A, SRCR, frizzled, kringle, Sushi, Wonton and Pan/apple), carbohydrate-recognition domains (C-type lectin and chitin-binding), and other modules (such as zinc finger, CUB, coiled coil and Sina). Comparison of the sequences with those from Drosophila led to a proposed SP pathway for establishing the dorsoventral axis of honey bee embryos. Multiple sequence alignments revealed evolutionary relationships of honey bee SPs and SPHs with those in Drosophila melanogaster, Anopheles gambiae, and Manduca sexta. We identified homologs of D. melanogaster persephone, M. sexta HP14, PAP-1 and SPH-1. A. mellifera genome includes at least five genes for potential SP inhibitors (serpin-1 through -5) and three genes of SP putative substrates (prophenoloxidase, spätzle-1 and spätzle-2). Quantitative RT-PCR analyses showed an elevation in the mRNA levels of SP2, SP3, SP9, SP10, SPH41, SPH42, SP49, serpin-2, serpin-4, serpin-5, and spätzle-2 in adults after a microbial challenge. The SP41 and SP6 transcripts significantly increased after an injection of Paenibacillus larva, but there was no such increase after injection of saline or Escherichia coli. mRNA levels of most SPs and serpins significantly increased by 48 h after the pathogen infection in 1st instar larvae. On the contrary, SP1, SP3, SP19 and serpin-5 transcript levels reduced. These results, taken together, provide a framework for designing experimental studies of the roles of SPs and related proteins in embryonic development and immune responses of A. mellifera

Hopf algebras and Markov chains: Two examples and a theory

The operation of squaring (coproduct followed by product) in a combinatorial Hopf algebra is shown to induce a Markov chain in natural bases. Chains constructed in this way include widely studied methods of card shuffling, a natural "rock-breaking" process, and Markov chains on simplicial complexes. Many of these chains can be explictly diagonalized using the primitive elements of the algebra and the combinatorics of the free Lie algebra. For card shuffling, this gives an explicit description of the eigenvectors. For rock-breaking, an explicit description of the quasi-stationary distribution and sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes will only appear on the version on Amy Pang's website, the arXiv version will not be updated.