104 research outputs found

    Integral expressions for derivations of multiarrangements

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    The construction of an explicit basis for a free multiarrangement is not easy in general. Inspired by the integral expressions for quasi-invariants of quantum Calogero-Moser systems, we present integral expressions for specific bases of certain multiarrangements. Our construction covers the cases of three lines in dimension 22 (previously examined by Wakamiko) and free multiarrangements associated with complex reflection groups (Hoge, Mano, R\"ohrle, Stump). Furthermore, we propose a conjectural basis for the module of logarithmic vector fields of the extended Catalan arrangement of type B2B_2.Comment: 21 page

    Ehrhart quasi-polynomials of almost integral polytopes

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    A lattice polytope translated by a rational vector is called an almost integral polytope. In this paper we investigate Ehrhart quasi-polynomials of almost integral polytopes. We study the relationship between the shape of the polytopes and algebraic properties of the Ehrhart quasi-polynomials. In particular, we prove that lattice zonotopes and centrally symmetric lattice polytopes are characterized by Ehrhart quasi-polynomials of their rational translations.Comment: ver 3: revisions on presentation

    Primary Structure and Conformation of a Tetrodotoxin-Binding Protein in the Hemolymph of Non-Toxic Shore Crab <i>Hemigrapsus sanguineus</i>

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    Tetrodotoxin (TTX)-binding proteins are present in toxic TTX-bearing animals, such as pufferfish and gastropods. These may prevent autotoxicity. However, TTX-binding proteins are also found in the nontoxic marine shore crab, Hemigrapsus sanguineus. Here, we isolated the TTX-binding protein, HSTBP (Hemigrapsus sanguineus TTX-binding protein), from the hemolymph of H. sanguineus and elucidated its primary structure using cDNA cloning. HSTBP, a 400 kDa acidic glycoprotein by gel filtration high-performance liquid chromatography, comprises 3 subunits, 88 kDa (subunit-1), 65 kDa (subunit-2), and 26 kDa (subunit-3) via sodium dodecyl sulfate-polyacrylamide gel electrophoresis under reduced conditions. The open reading frame of the cDNA comprises 5049 base pairs encoding 1683 amino acid residues, and the mature protein contains 1650 amino acid residues from Arg34 to Ser1683. The three subunits are arranged in tandem in the following order: subunit-3 (Arg34-Gln261), subunit-1 (Asp262-Phe1138), and subunit-2 (Val1139-Ser1683). A BLAST homology search showed weak similarity of HSTBP to clotting proteins of crustaceans (29–40%). SMART analysis revealed a von Willebrand factor (vWF)-type (⇒delete hyphen) D domain at Phe1387-Gly1544. We confirmed that the recombinant protein of HSTBP subunit-2 containing the vWF-type (⇒delete hyphen) D domain bound to TTX at a molecular ratio of 1:1

    Free reflection multiarrangements and quasi-invariants

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    To a complex reflection arrangement with an invariant multiplicity function one can relate the space of logarithmic vector fields and the space of quasi-invariants, which are both modules over invariant polynomials. We establish a close relation between these modules. Berest-Chalykh freeness results for the module of quasi-invariants lead to new free complex reflection multiarrangements. K. Saito's primitive derivative gives a linear map between certain spaces of quasi-invariants. We also establish a close relation between non-homogeneous quasi-invariants for root systems and logarithmic vector fields for the extended Catalan arrangements. As an application, we prove the freeness of Catalan arrangements corresponding to the non-reduced root system BCNBC_N.Comment: 26 pages; small change

    The Primitive Derivation and Discrete Integrals

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    The modules of logarithmic derivations for the (extended) Catalan and Shi arrangements associated with root systems are known to be free. However, except for a few cases, explicit bases for such modules are not known. In this paper, we construct explicit bases for type AA root systems. Our construction is based on Bandlow-Musiker's integral formula for a basis of the space of quasiinvariants. The integral formula can be considered as an expression for the inverse of the primitive derivation introduced by K. Saito. We prove that the discrete analogues of the integral formulas provide bases for Catalan and Shi arrangements

    G-Tutte Polynomials and Abelian Lie Group Arrangements

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    For a list A of elements in a finitely generated abelian group Gamma and an abelian group G, we introduce and study an associated G-Tutte polynomial, defined by counting the number of homomorphisms from associated finite abelian groups to G. The G-Tutte polynomial is a common generalization of the (arithmetic) Tutte polynomial for realizable (arithmetic) matroids, the characteristic quasi-polynomial for integral arrangements, Branden-Moci's arithmetic version of the partition function of an abelian group-valued Potts model, and the modified Tutte-Krushkal-Renhardy polynomial for a finite CW complex. As in the classical case, G-Tutte polynomials carry topological and enumerative information (e.g., the Euler characteristic, point counting, and the Poincare polynomial) of abelian Lie group arrangements. We also discuss differences between the arithmetic Tutte and the G-Tutte polynomials related to the axioms for arithmetic matroids and the (non-)positivity of coefficients

    Feynman graphs and hyperplane arrangements defined over F-1

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    Motivated by some computations of Feynman integrals and certain conjectures on mixed Tate motives, Bejleri and Marcolli posed questions about the F-1-structure (in the sense of torification) on the complement of a hyperplane arrangement, especially for an arrangement defined in the space of cycles of a graph. In this paper, we prove that an arrangement has an F-1-structure if and only if it is Boolean. We also prove that the arrangement in the cycle space of a graph is Boolean if and only if the cycle space has a basis consisting of cycles such that any two of them do not share edges. (C) 2021 Elsevier B.V. All rights reserved

    Enhancing rare variant interpretation in inherited arrhythmias through quantitative analysis of consortium disease cohorts and population controls.

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    PURPOSE: Stringent variant interpretation guidelines can lead to high rates of variants of uncertain significance (VUS) for genetically heterogeneous disease like long QT syndrome (LQTS) and Brugada syndrome (BrS). Quantitative and disease-specific customization of American College of Medical Genetics and Genomics/Association for Molecular Pathology (ACMG/AMP) guidelines can address this false negative rate. METHODS: We compared rare variant frequencies from 1847 LQTS (KCNQ1/KCNH2/SCN5A) and 3335 BrS (SCN5A) cases from the International LQTS/BrS Genetics Consortia to population-specific gnomAD data and developed disease-specific criteria for ACMG/AMP evidence classes-rarity (PM2/BS1 rules) and case enrichment of individual (PS4) and domain-specific (PM1) variants. RESULTS: Rare SCN5A variant prevalence differed between European (20.8%) and Japanese (8.9%) BrS patients (p = 5.7 × 10-18) and diagnosis with spontaneous (28.7%) versus induced (15.8%) Brugada type 1 electrocardiogram (ECG) (p = 1.3 × 10-13). Ion channel transmembrane regions and specific N-terminus (KCNH2) and C-terminus (KCNQ1/KCNH2) domains were characterized by high enrichment of case variants and >95% probability of pathogenicity. Applying the customized rules, 17.4% of European BrS and 74.8% of European LQTS cases had (likely) pathogenic variants, compared with estimated diagnostic yields (case excess over gnomAD) of 19.2%/82.1%, reducing VUS prevalence to close to background rare variant frequency. CONCLUSION: Large case-control data sets enable quantitative implementation of ACMG/AMP guidelines and increased sensitivity for inherited arrhythmia genetic testing