http://www.rssboard.org/rss-specification 720 https://escholarship.org/uc/ucdavismath_faculty/rss Recent eScholarship items from Faculty Fri, 15 May 2026 06:54:10 +0000 https://escholarship.org/uc/item/7r57q2kp Simulations of undulatory swimming in viscoelastic fluids with large amplitude gaits show concentration of polymer elastic stress at the tips of the swimmers. We use a series of related theoretical investigations to probe the origin of these concentrated stresses. First the polymer stress is computed analytically at a given oscillating extensional stagnation point in a viscoelastic fluid. The analysis identifies a Deborah number (De) dependent Weissenberg number (Wi) transition below which the stress is linear in Wi, and above which the stress grows exponentially in Wi. Next, stress and velocity are found from numerical simulations in an oscillating 4-roll mill geometry. The stress from these simulations is compared with the theoretical calculation of stress in the decoupled (given flow) case, and similar stress behavior is observed. The flow around tips of a time-reversible flexing filament in a viscoelastic fluid is shown to exhibit an oscillating extension along particle trajectories,... https://escholarship.org/uc/item/7r57q2kp Thu, 13 Jun 2019 00:00:00 +0000 Thomases, Becca https://orcid.org/0000-0001-8502-8915 Guy, Robert D https://escholarship.org/uc/item/6674s3b3 Lectures on quasi-isometric rigidity https://escholarship.org/uc/item/6674s3b3 Thu, 13 Jun 2019 00:00:00 +0000 Kapovich, M https://orcid.org/0000-0001-5475-8755 https://escholarship.org/uc/item/5mr0v8z2 In his 1985 paper Sullivan sketched a proof of his structural stability theorem for differentiable group actions satisfying certain expansion-hyperbolicity axioms. In this paper we relax Sullivan's axioms and introduce a notion of "meandering hyperbolicity" for group actions on geodesic metric spaces. This generalization is substantial enough to encompass actions of certain non-hyperbolic groups, such as actions of "uniform lattices" in semisimple Lie groups on flag manifolds. At the same time, our notion is sufficiently robust and we prove that meandering-hyperbolic actions are still structurally stable. We also prove some basic results on meandering-hyperbolic actions and give other examples of such actions. https://escholarship.org/uc/item/5mr0v8z2 Thu, 13 Jun 2019 00:00:00 +0000 Kapovich, Michael Kim, Sungwoon Lee, Jaejeong https://escholarship.org/uc/item/5mc3m2mn In a GWAS study of 32,438 adults, the authors discovered five novel loci for intracranial volume and confirmed two known signals. Variants for intracranial volume were also related to childhood and adult cognitive function and to Parkinson's disease, and enriched near genes involved in growth pathways, including PI3K-AKT signaling. https://escholarship.org/uc/item/5mc3m2mn Thu, 13 Jun 2019 00:00:00 +0000 Adams, Hieab HH Hibar, Derrek P Chouraki, Vincent Stein, Jason L Nyquist, Paul A Rentería, Miguel E Trompet, Stella Arias-Vasquez, Alejandro Seshadri, Sudha Desrivières, Sylvane Beecham, Ashley H Jahanshad, Neda Wittfeld, Katharina Van der Lee, Sven J Abramovic, Lucija Alhusaini, Saud Amin, Najaf Andersson, Micael Arfanakis, Konstantinos Aribisala, Benjamin S Armstrong, Nicola J Athanasiu, Lavinia Axelsson, Tomas Beiser, Alexa Bernard, Manon Bis, Joshua C Blanken, Laura ME Blanton, Susan H Bohlken, Marc M Boks, Marco P Bralten, Janita Brickman, Adam M Carmichael, Owen Chakravarty, M Mallar Chauhan, Ganesh Chen, Qiang Ching, Christopher RK Cuellar-Partida, Gabriel Braber, Anouk Den Doan, Nhat Trung Ehrlich, Stefan Filippi, Irina Ge, Tian Giddaluru, Sudheer Goldman, Aaron L Gottesman, Rebecca F Greven, Corina U Grimm, Oliver Griswold, Michael E Guadalupe, Tulio Hass, Johanna https://orcid.org/0000-0002-9019-4464 Haukvik, Unn K Hilal, Saima Hofer, Edith Hoehn, David Holmes, Avram J Hoogman, Martine Janowitz, Deborah Jia, Tianye Kasperaviciute, Dalia Kim, Sungeun Klein, Marieke Kraemer, Bernd Lee, Phil H Liao, Jiemin Liewald, David CM Lopez, Lorna M Luciano, Michelle Macare, Christine Marquand, Andre Matarin, Mar Mather, Karen A Mattheisen, Manuel Mazoyer, Bernard McKay, David R McWhirter, Rebekah Milaneschi, Yuri Mirza-Schreiber, Nazanin Muetzel, Ryan L Maniega, Susana Muñoz Nho, Kwangsik Nugent, Allison C Loohuis, Loes M Olde Oosterlaan, Jaap Papmeyer, Martina Pappa, Irene Pirpamer, Lukas Pudas, Sara Pütz, Benno Rajan, Kumar B Ramasamy, Adaikalavan Richards, Jennifer S Risacher, Shannon L Roiz-Santiañez, Roberto Rommelse, Nanda Rose, Emma J Royle, Natalie A Rundek, Tatjana Sämann, Philipp G Satizabal, Claudia L https://escholarship.org/uc/item/5js648vn We prove a Morse lemma for regular quasigeodesics in nonpositively curved symmetric spaces and euclidean buildings. We apply it to give a new coarse geometric characterization of Anosov subgroups of the isometry groups of such spaces simply as undistorted subgroups which are uniformly regular. https://escholarship.org/uc/item/5js648vn Thu, 13 Jun 2019 00:00:00 +0000 Kapovich, Michael https://orcid.org/0000-0001-5475-8755 Leeb, Bernhard Porti, Joan https://escholarship.org/uc/item/5j79p8ng Hierarchical graph Laplacian eigen transforms https://escholarship.org/uc/item/5j79p8ng Thu, 13 Jun 2019 00:00:00 +0000 Irion, Jeff Saito, Naoki https://orcid.org/0000-0001-5234-4719 https://escholarship.org/uc/item/5cp2c1bd Genome-wide association studies are used to identify common genetic variants that affect the structure of selected subcortical regions of the human brain; their identification provides insight into the causes of variability in brain development and may help to determine mechanisms of neuropsychiatric dysfunction. https://escholarship.org/uc/item/5cp2c1bd Thu, 13 Jun 2019 00:00:00 +0000 Hibar, Derrek P Stein, Jason L Renteria, Miguel E Arias-Vasquez, Alejandro Desrivières, Sylvane Jahanshad, Neda Toro, Roberto Wittfeld, Katharina Abramovic, Lucija Andersson, Micael Aribisala, Benjamin S Armstrong, Nicola J Bernard, Manon Bohlken, Marc M Boks, Marco P Bralten, Janita Brown, Andrew A Mallar Chakravarty, M Chen, Qiang Ching, Christopher RK Cuellar-Partida, Gabriel den Braber, Anouk Giddaluru, Sudheer Goldman, Aaron L Grimm, Oliver Guadalupe, Tulio Hass, Johanna https://orcid.org/0000-0002-9019-4464 Woldehawariat, Girma Holmes, Avram J Hoogman, Martine Janowitz, Deborah Jia, Tianye Kim, Sungeun Klein, Marieke Kraemer, Bernd Lee, Phil H Olde Loohuis, Loes M Luciano, Michelle Macare, Christine Mather, Karen A Mattheisen, Manuel Milaneschi, Yuri Nho, Kwangsik Papmeyer, Martina Ramasamy, Adaikalavan Risacher, Shannon L Roiz-Santiañez, Roberto Rose, Emma J Salami, Alireza Sämann, Philipp G Schmaal, Lianne Schork, Andrew J Shin, Jean Strike, Lachlan T Teumer, Alexander van Donkelaar, Marjolein MJ van Eijk, Kristel R Walters, Raymond K Westlye, Lars T Whelan, Christopher D Winkler, Anderson M Zwiers, Marcel P Alhusaini, Saud Athanasiu, Lavinia Ehrlich, Stefan Hakobjan, Marina MH Hartberg, Cecilie B Haukvik, Unn K Heister, Angelien JGAM Hoehn, David Kasperaviciute, Dalia Liewald, David CM Lopez, Lorna M Makkinje, Remco RR Matarin, Mar Naber, Marlies AM Reese McKay, D Needham, Margaret Nugent, Allison C Pütz, Benno Royle, Natalie A Shen, Li Sprooten, Emma Trabzuni, Daniah van der Marel, Saskia SL van Hulzen, Kimm JE Walton, Esther Wolf, Christiane Almasy, Laura Ames, David Arepalli, Sampath Assareh, Amelia A Bastin, Mark E Brodaty, Henry Bulayeva, Kazima B Carless, Melanie A Cichon, Sven Corvin, Aiden Curran, Joanne E Czisch, Michael https://escholarship.org/uc/item/5b83d5dn We consider generalizations of Schuetzenberger's promotion operator on the set L of linear extensions of a finite poset of size n. This gives rise to a strongly connected graph on L. By assigning weights to the edges of the graph in two different ways, we study two Markov chains, both of which are irreducible. The stationary state of one gives rise to the uniform distribution, whereas the weights of the stationary state of the other has a nice product formula. This generalizes results by Hendricks on the Tsetlin library, which corresponds to the case when the poset is the anti-chain and hence L=S_n is the full symmetric group. We also provide explicit eigenvalues of the transition matrix in general when the poset is a rooted forest. This is shown by proving that the associated monoid is R-trivial and then using Steinberg's extension of Brown's theory for Markov chains on left regular bands to R-trivial monoids. https://escholarship.org/uc/item/5b83d5dn Thu, 13 Jun 2019 00:00:00 +0000 Ayyer, Arvind Klee, Steven Schilling, Anne https://orcid.org/0000-0002-2601-7340 https://escholarship.org/uc/item/57t0f8cm A multiple scale model of the nonlinearly coupled KdV equations is established to predict mechanism of interaction of equatorial Rossby waves and barotropic waves in certain case. Analytically, predicted precursor radiation is a centrosymmetric object and is shown in excellent quantitative agreement with numerical simulations; furthermore, the multiple scale model elucidates the salient mechanisms of the interaction of solitary waves and the mechanism for radiation. While the atmosphere-ocean science community is very interested in theoretical studies of tropical wave interactions and in developing reduced dynamical models that can explain some key features of equatorial phenomena, our analytic predictions quantitively explain formation of radiation during interaction in Biello's model beyond qualitative level. https://escholarship.org/uc/item/57t0f8cm Thu, 13 Jun 2019 00:00:00 +0000 Li, Yezheng Biello, Joseph A Li, Yerong https://escholarship.org/uc/item/4s0047d7 The true slime mold Physarum polycephalum exhibits a vast array of sophisticated manipulations of its intracellular cytoplasm. Growing microplasmodia of Physarum have been observed to adopt an elongated tadpole shape, then contract in a rhythmic, traveling wave pattern that resembles peristaltic pumping. This contraction drives a fast flow of non-gelated cytoplasm along the cell longitudinal axis. It has been hypothesized that this flow of cytoplasm is a driving factor in generating motility of the plasmodium. In this work, we use two different mathematical models to investigate how peristaltic pumping within Physarum may be used to drive cellular motility. We compare the relative phase of flow and deformation waves predicted by both models to similar phase data collected from in vivo experiments using Physarum plasmodia. The first is a PDE model based on a dimensional reduction of peristaltic pumping within a finite length chamber. The second is a more sophisticated computational... https://escholarship.org/uc/item/4s0047d7 Thu, 13 Jun 2019 00:00:00 +0000 Lewis, Owen L Guy, Robert D https://escholarship.org/uc/item/4f58m5rx The support of a vector is the number of nonzero components. We show that given anintegral m×n matrix A, the integer linear optimization problem maxfcT x: Ax = b; x = 0; x 2 Znghas an optimal solution whose support is bounded by 2m log(2pmkAk1), where kAk1 is the largestabsolute value of an entry of A. Compared to previous bounds, the one presented here is independentof the objective function. We furthermore provide a nearly matching asymptotic lower bound on thesupport of optimal solutions. https://escholarship.org/uc/item/4f58m5rx Thu, 13 Jun 2019 00:00:00 +0000 Aliev, I De Loera, JA https://orcid.org/0000-0002-9556-1112 Eisenbrand, F Oertel, T Weismantel, R https://escholarship.org/uc/item/3tn924mz We extend several notions and results from the classical Patterson-Sullivan theory to the setting of Anosov subgroups of higher rank semisimple Lie groups, working primarily with invariant Finsler metrics on associated symmetric spaces. In particular, we prove the equality between the Hausdorff dimensions of flag limit sets, computed with respect to a suitable Gromov (pre-)metric on the flag manifold, and the Finsler critical exponents of Anosov subgroups. https://escholarship.org/uc/item/3tn924mz Thu, 13 Jun 2019 00:00:00 +0000 Dey, Subhadip Kapovich, Michael https://escholarship.org/uc/item/3kc4r7t6 We study the geometry and dynamics of discrete infinite covolume subgroups of higher rank semisimple Lie groups. We introduce and prove the equivalence of several conditions, capturing "rank one behavior'' of discrete subgroups of higher rank Lie groups. They are direct generalizations of rank one equivalents to convex cocompactness. We also prove that our notions are equivalent to the notion of Anosov subgroup, for which we provide a closely related, but simplified and more accessible reformulation, avoiding the geodesic flow of the group. We show moreover that the Anosov condition can be relaxed further by requiring only non-uniform unbounded expansion along the (quasi)geodesics in the group. A substantial part of the paper is devoted to the coarse geometry of these discrete subgroups. A key concept which emerges from our analysis is that of Morse quasigeodesics in higher rank symmetric spaces, generalizing the Morse property for quasigeodesics in Gromov hyperbolic spaces.... https://escholarship.org/uc/item/3kc4r7t6 Thu, 13 Jun 2019 00:00:00 +0000 Kapovich, Michael Leeb, Bernhard Porti, Joan https://escholarship.org/uc/item/31k9s95q Cells employing amoeboid motility exhibit repetitive cycles of rapid expansion and contraction and apply coordinated traction forces to their environment. Although aspects of this process are well studied, it is unclear how the cell controls the coordination of cell length changes with adhesion to the surface. Here, we develop a simple model to mechanistically explain the emergence of periodic changes in length and spatiotemporal dynamics of traction forces measured in chemotaxing unicellular amoeba, Dictyostelium discoideum. In contrast to the biochemical mechanisms that have been implicated in the coordination of some cellular processes, we show that many features of amoeboid locomotion emerge from a simple mechanochemical model. The mechanism for interaction with the environment in Dictyostelium is unknown and thus, we explore different cell-environment interaction models to reveal that mechanosensitive adhesions are necessary to reproduce the spatiotemporal adhesion patterns.... https://escholarship.org/uc/item/31k9s95q Thu, 13 Jun 2019 00:00:00 +0000 Copos, Calina A Walcott, Sam del Álamo, Juan C Bastounis, Effie Mogilner, Alex Guy, Robert D https://escholarship.org/uc/item/30d0p416 We provide a unified framework to compute the stationary distribution of any finite irreducible Markov chain or equivalently of any irreducible random walk on a finite semigroup S. Our methods use geometric finite semigroup theory via the Karnofsky–Rhodes and the McCammond expansions of finite semigroups with specified generators; this does not involve any linear algebra. The original Tsetlin library is obtained by applying the expansions to P(n), the set of all subsets of an n element set. Our set-up generalizes previous groundbreaking work involving left-regular bands (or R-trivial bands) by Brown and Diaconis, extensions to R-trivial semigroups by Ayyer, Steinberg, Thiéry and the second author, and important recent work by Chung and Graham. The Karnofsky–Rhodes expansion of the right Cayley graph of S in terms of generators yields again a right Cayley graph. The McCammond expansion provides normal forms for elements in the expanded S. Using our previous results with Silva based... https://escholarship.org/uc/item/30d0p416 Thu, 13 Jun 2019 00:00:00 +0000 Rhodes, John Schilling, Anne https://orcid.org/0000-0002-2601-7340 https://escholarship.org/uc/item/2wv4q3j1 We prove noncoherence of certain families of lattices in the isometry group of the hyperbolic n-space for n greater than 3. For instance, every nonuniform arithmetic lattice in SO(n,1) is noncoherent, provided that n is at least 6. https://escholarship.org/uc/item/2wv4q3j1 Thu, 13 Jun 2019 00:00:00 +0000 Kapovich, Michael Potyagailo, Leonid Vinberg, Ernest https://escholarship.org/uc/item/2qz417kk A major challenge in matrix-metalloproteinase (MMP) target validation and MMP-inhibitor-drug development for anti-cancer clinical trials is to better understand their complex roles (often competing with each other) in tumor progression. While there is extensive research on the growth-promoting effects of MMPs, the growth-inhibiting effects of MMPs has not been investigated thoroughly. So we develop a continuum model of tumor growth and invasion including chemotaxis and haptotaxis in order to examine the complex interaction between the tumor and its host microenvironment and to explore the inhibiting influence of the gradients of soluble fragments of extracellular matrix (ECM) density on tumor growth and morphology. Previously, it was shown both computationally (in one spatial dimension) and experimentally that the chemotactic pull due to soluble ECM gradients is anti-invasive, contrary to the traditional view of the role of chemotaxis in malignant invasion [1]. With two-dimensional... https://escholarship.org/uc/item/2qz417kk Thu, 13 Jun 2019 00:00:00 +0000 Nargis, Nurun N Aldredge, Ralph C https://orcid.org/0000-0002-2418-2035 Guy, Robert D https://orcid.org/0000-0002-0602-7015 https://escholarship.org/uc/item/1x84j5s5 In this paper, we prove a quantitative version of the Tits alternative for negatively pinched manifolds $X$. Precisely, we prove that a nonelementary discrete isometry subgroup of $\mathrm{Isom}(X)$ generated by two non-elliptic isometries $g$, $f$ contains a free subgroup of rank $2$ generated by isometries $f^N , h$ of uniformly bounded word length. Furthermore, we show that this free subgroup is convex-cocompact when $f$ is hyperbolic. https://escholarship.org/uc/item/1x84j5s5 Thu, 13 Jun 2019 00:00:00 +0000 Dey, Subhadip Kapovich, Michael Liu, Beibei https://escholarship.org/uc/item/1tw883m1 We describe structure of quasihomomorphisms from arbitrary groups to discrete groups. We show that all quasihomomorphisms are “constructible”, i.e., are obtained via certain natural operations from homomorphisms to some groups and quasihomomorphisms to abelian groups. We illustrate this theorem by describing quasihomomorphisms to certain classes of groups. For instance, every unbounded quasihomomorphism to a torsion-free hyperbolic group H is either a homomorphism to a subgroup of H or is a quasihomomorphism to an infinite cyclic subgroup of H. https://escholarship.org/uc/item/1tw883m1 Thu, 13 Jun 2019 00:00:00 +0000 Fujiwara, Koji Kapovich, Michael https://orcid.org/0000-0001-5475-8755 https://escholarship.org/uc/item/01c0p4jn The reviewers contacted by the editors to evaluate this work have been unable to confirm that the main results are correct. Flaws that were identified by the reviewers in earlier versions of the paper have been addressed by the author. Although it is possible that future research will uncover a significant mistake in this paper or show that the conclusions are in error, I believe that publishing it may benefit the readership of the Journal and stimulate further work in mathematical physics on an important topic. https://escholarship.org/uc/item/01c0p4jn Thu, 13 Jun 2019 00:00:00 +0000 Nachtergaele, Bruno https://orcid.org/0000-0002-7835-3776 https://escholarship.org/uc/item/95j4s9r6 We propose several common extensions of the classes of Anosov subgroups and geometrically finite Kleinian groups among discrete subgroups of semisimple Lie groups. We relativize various dynamical and coarse geometric characterizations of Anosov subgroups given in our earlier work, extending the class from intrinsically hyperbolic to relatively hyperbolic subgroups. We prove implications and equivalences between the various relativizations. https://escholarship.org/uc/item/95j4s9r6 Fri, 14 Sep 2018 00:00:00 +0000 Kapovich, Michael Leeb, Bernhard https://escholarship.org/uc/item/858382hb © 2018 World Scientific Publishing Co. Pte. Ltd. This chapter aims at giving an introduction to the notion of quantum curves. The main purpose is to describe the new discovery of the relation between the following two disparate subjects: one is the topological recursion, that has its origin in random matrix theory and has been effectively applied to many enumerative geometry problems; and the other is the quantization of Hitchin spectral curves associated with Higgs bundles. Our emphasis is on explaining the motivation and examples. Concrete examples of the direct relation between Hitchin spectral curves and enumeration problems are given. A general geometric framework of quantum curves is also discussed. https://escholarship.org/uc/item/858382hb Fri, 14 Sep 2018 00:00:00 +0000 Dumitrescu, O Mulase, M https://escholarship.org/uc/item/80w4w0h3 Answering a question by Margulis we prove that the conclusion of Selberg's Lemma fails for discrete isometry groups of negatively curved Hadamard manifolds. https://escholarship.org/uc/item/80w4w0h3 Fri, 14 Sep 2018 00:00:00 +0000 Kapovich, Michael https://escholarship.org/uc/item/67f779w5 A challenge for chromosome segregation in all domains of life is the formation of catenated progeny chromosomes, which arise during replication as a consequence of the interwound strands of the DNA double helix. Topoisomerases play a key role in DNA unlinking both during and at the completion of replication. Here we report that chromosome unlinking can instead be accomplished by multiple rounds of site‐specific recombination. We show that step‐wise, site‐specific recombination by XerCD‐dif or Cre‐loxP can unlink bacterial chromosomes in vivo, in reactions that require KOPS‐guided DNA translocation by FtsK. Furthermore, we show that overexpression of a cytoplasmic FtsK derivative is sufficient to allow chromosome unlinking by XerCD‐dif recombination when either subunit of TopoIV is inactivated. We conclude that FtsK acts in vivo to simplify chromosomal topology as Xer recombination interconverts monomeric and dimeric chromosomes. https://escholarship.org/uc/item/67f779w5 Fri, 14 Sep 2018 00:00:00 +0000 Grainge, Ian Bregu, Migena Vazquez, Mariel https://orcid.org/0000-0001-8328-6806 Sivanathan, Viknesh Ip, Stephen CY Sherratt, David J https://escholarship.org/uc/item/52m5t4gs The immersed boundary method is a widely used mixed Eulerian/Lagrangian framework for simulating the motion of elastic structures immersed in viscous fluids. In this work, we consider a poroelastic immersed boundary method in which a fluid permeates a porous, elastic structure of negligible volume fraction, and extend this method to include stress relaxation of the material. The porous viscoelastic method presented here is validated for a prescribed oscillatory shear and for an expansion driven by the motion at the boundary of a circular material by comparing numerical solutions to an analytical solution of the Maxwell model for viscoelasticity. Finally, an application of the modelling framework to cell biology is provided: passage of a cell through a microfluidic channel. We demonstrate that the rheology of the cell cytoplasm is important for capturing the transit time through a narrow channel in the presence of a pressure drop in the extracellular fluid. https://escholarship.org/uc/item/52m5t4gs Fri, 14 Sep 2018 00:00:00 +0000 COPOS, CALINA A GUY, ROBERT D https://escholarship.org/uc/item/51f0d82j Difference topology is an experimental technique that can be used to unveil the topological structure adopted by two or more DNA segments in a stable protein-DNA complex. Difference topology has also been used to detect intermediates in a reaction pathway and to investigate the role of DNA supercoiling. In the present article, we review difference topology as applied to the Mu transpososome. The tools discussed can be applied to any stable nucleoprotein complex. https://escholarship.org/uc/item/51f0d82j Fri, 14 Sep 2018 00:00:00 +0000 Darcy, Isabel K Vazquez, Mariel https://orcid.org/0000-0001-8328-6806 https://escholarship.org/uc/item/29b5v2hp We prove Lieb-Robinson bounds for a general class of lattice fermion systems. By making use of a suitable conditional expectation onto subalgebras of the CAR algebra, we can apply the Lieb-Robinson bounds much in the same way as for quantum spin systems. We preview how to obtain the spectral flow automorphisms and to prove stability of the spectral gap for frustration-free gapped systems satisfying a Local Topological Quantum Order condition. https://escholarship.org/uc/item/29b5v2hp Fri, 14 Sep 2018 00:00:00 +0000 Nachtergaele, Bruno https://orcid.org/0000-0002-7835-3776 Sims, Robert Young, Amanda https://escholarship.org/uc/item/1px69959 We prove an analogue of Klein combination theorem for Anosov subgroups by using a local-to-global principle for Morse quasigeodesics. https://escholarship.org/uc/item/1px69959 Fri, 14 Sep 2018 00:00:00 +0000 Dey, Subhadip Kapovich, Michael https://orcid.org/0000-0001-5475-8755 Leeb, Bernhard https://escholarship.org/uc/item/0mt6k7m9 We discuss the role of compact symmetry groups, G, in the classification of gapped ground state phases of quantum spin systems. We consider two representations of G on infinite subsystems. First, in arbitrary dimensions, we show that the ground state spaces of models within the same G-symmetric phase carry equivalent representations of the group for each finite or infinite sublattice on which they can be defined and on which they remain gapped. This includes infinite systems with boundaries or with non-trivial topologies. Second, for two classes of one-dimensional models, by two different methods, for G=SU(2) in one, and G\subset SU(d), in the other we construct explicitly an `excess spin' operator that implements rotations of half of the infinite chain on the GNS Hilbert space of the ground state of the full chain. Since this operator is constructed as the limit of a sequence of observables, the representation itself is, in principle, experimentally observable. We claim that... https://escholarship.org/uc/item/0mt6k7m9 Fri, 14 Sep 2018 00:00:00 +0000 Bachmann, S Nachtergaele, B https://orcid.org/0000-0002-7835-3776 https://escholarship.org/uc/item/0j98n6t5 © 2018, Mathematical Sciences Publishers. All Rights reserved. We give a geometric interpretation of the maximal Satake compactification of symmetric spaces X=G/K of noncompact type, showing that it arises by attaching the horofunction boundary for a suitable G-invariant Finsler metric on X. As an application, we establish the existence of natural bordifications, as orbifolds-with-corners, of locally symmetric spaces X/Γ for arbitrary discrete subgroups Γ https://escholarship.org/uc/item/0j98n6t5 Fri, 14 Sep 2018 00:00:00 +0000 Kapovich, M Leeb, B https://escholarship.org/uc/item/0bv9t4c8 We propose methods to efficiently approximate and denoise signals sampled on the nodes of graphs using our overcomplete multiscale transforms/basis dictionaries for such graph signals: the hierarchical graph Laplacian eigen transform (HGLET) and the generalized HaarWalsh transform (GHWT). These can be viewed as generalizations of the hierarchical discrete cosine transform and the Haar-Walsh wavelet packet transform, respectively, from regularly sampled signals to graph signals. Both of these transforms generate dictionaries containing an immense number of choosable bases, and in order to select a particular basis most suitable for a given task, we have generalized the best basis algorithm from classical signal processing. After briefly reviewing these transforms and the best basis algorithm, we precisely prove their efficiency in approximating graph signals belonging to discrete analogs of the space of Hlder continuous functions and the Besov spaces. Then, we validate their effectiveness... https://escholarship.org/uc/item/0bv9t4c8 Fri, 14 Sep 2018 00:00:00 +0000 Irion, Jeff Saito, Naoki https://orcid.org/0000-0001-5234-4719 https://escholarship.org/uc/item/062077zh When attempting to develop wavelet transforms for graphs and networks, some researchers have used graph Laplacian eigenvalues and eigenvectors in place of the frequencies and complex exponentials in the Fourier theory for regular lattices in the Euclidean domains. This viewpoint, however, has a fundamental flaw: on a general graph, the Laplacian eigenvalues cannot be interpreted as the frequencies of the corresponding eigenvectors. In this paper, we discuss this important problem further and propose a new method to organize those eigenvectors by defining and measuring 'natural' distances between eigenvectors using the Ramified Optimal Transport Theory followedby embedding them into a low-dimensional Euclidean domain. We demonstrate its effectiveness using a synthetic graph as well as a dendritic tree of a retinal ganglioncell of a mouse. https://escholarship.org/uc/item/062077zh Fri, 14 Sep 2018 00:00:00 +0000 Saito, Naoki https://orcid.org/0000-0001-5234-4719 https://escholarship.org/uc/item/9w04t1t0 We generalize the topological recursion of Eynard-Orantin (JHEP 0612:053, 2006; Commun Number Theory Phys 1:347-452, 2007) to the family of spectral curves of Hitchin fibrations. A spectral curve in the topological recursion, which is defined to be a complex plane curve, is replaced with a generic curve in the cotangent bundle T*C of an arbitrary smooth base curve C. We then prove that these spectral curves are quantizable, using the new formalism. More precisely, we construct the canonical generators of the formal h{stroke}-deformation family of D modules over an arbitrary projective algebraic curve C of genus greater than 1, from the geometry of a prescribed family of smooth Hitchin spectral curves associated with the SL(2, ℂ)-character variety of the fundamental group π1(C). We show that the semi-classical limit through the WKB approximation of these h{stroke}-deformed D modules recovers the initial family of Hitchin spectral curves. © 2014 Springer Science+Business Media Dordrecht. https://escholarship.org/uc/item/9w04t1t0 Mon, 14 May 2018 00:00:00 +0000 Dumitrescu, O Mulase, M https://escholarship.org/uc/item/9k771949 Product Vacua and Boundary State Models in -Dimensions https://escholarship.org/uc/item/9k771949 Mon, 14 May 2018 00:00:00 +0000 Bachmann, S Hamza, E Nachtergaele, B https://orcid.org/0000-0002-7835-3776 Young, A https://escholarship.org/uc/item/9d327011 We prove that for any affine variety S defined over Q there exist Shephard and Artin groups G such that a Zariski open subset U of S is biregular isomorphic to a Zariski open subset of the character variety X(G, PO(3)) = Hom(G, PO(3))//PO(3). The subset U contains all real points of S. As an application we construct new examples of finitely-presented groups which are not fundamental groups of smooth complex algebraic varieties. © 1998 Publications mathématiques de l'I.H.É.S. https://escholarship.org/uc/item/9d327011 Mon, 14 May 2018 00:00:00 +0000 Kapovich, Michael https://orcid.org/0000-0001-5475-8755 Millson, John J https://escholarship.org/uc/item/9cb006k5 We examine the phase response properties of half-center oscillators (HCOs) that are modeled by a pair of Morris-Lecar-type neurons connected by strong fast inhibitory synapses. We find that the two basic mechanisms for half-center oscillations, "release" and "escape", give rise to strikingly different phase response curves (PRCs). Release-type HCOs are most sensitive to perturbations delivered to cells at times when they are about to transition from the active to the suppressed state, and PRCs are dominated by a large negative peak (phase delays) at corresponding phases. On the other hand, escape-type HCOs are most sensitive to perturbations delivered to cells at times when they are about to transition from the suppressed to the active state, and PRCs are dominated by a large positive peak (phase advances) at corresponding phases. By analyzing the phase space structure of Morris-Lecar-type HCO models with fast synaptic dynamics, we identify the dynamical mechanisms underlying... https://escholarship.org/uc/item/9cb006k5 Mon, 14 May 2018 00:00:00 +0000 Zhang, Calvin Lewis, Timothy J https://escholarship.org/uc/item/98k4c5zk Since the first Hodgkin and Huxley ion channel model was described in the 1950s, there has been an explosion in mathematical models to describe ion channel function. As experimental data has become richer, models have concomitantly been improved to better represent ion channel kinetic processes, although these improvements have generally resulted in more model complexity and an increase in the number of parameters necessary to populate the models. Models have also been developed to explicitly model drug interactions with ion channels. Recent models of drug-channel interactions account for the discrete kinetics of drug interaction with distinct ion channel state conformations, as it has become clear that such interactions underlie complex emergent kinetics such as use-dependent block. Here, we describe an approach for developing a model for ion channel drug interactions. The method describes the process of extracting rate constants from experimental electrophysiological function... https://escholarship.org/uc/item/98k4c5zk Mon, 14 May 2018 00:00:00 +0000 Moreno, Jonathan D Lewis, Timothy J Clancy, Colleen E https://escholarship.org/uc/item/95m842mb Motivated by Bland's linear programming (LP) generalization of the renowned Edmonds-Karp efficient refinement of the Ford-Fulkerson maximum flow algorithm, we analyze three closely related natural augmentation rules for LP and integer-linear programming (ILP) augmentation algorithms. For all three rules and in both contexts, LP and ILP, we bound the number of augmentations. Extending Bland's "discrete steepest-descent" augmentation rule (i.e., choosing directions with the best ratio of cost improvement per unit 1-norm length, and then making maximal augmentations in such directions) from LP to ILP, we (i) show that the number of discrete steepest-descent augmentations is bounded by the number of elements in the Graver basis of the problem matrix and (ii) give the first strongly polynomial-time algorithm for N-fold ILP. For LP, two of the rules can suffer from a "zig-zagging" phenomenon, and so in those cases we apply the rules more subtly to achieve good bounds. Our results improve... https://escholarship.org/uc/item/95m842mb Mon, 14 May 2018 00:00:00 +0000 De Loera, Jesús A https://orcid.org/0000-0002-9556-1112 Hemmecke, Raymond Lee, Jon https://escholarship.org/uc/item/95d5t7zz Consider the following partial “sorting algorithm” on permutations: take the first entry of the permutation in one-line notation and insert it into the position of its own value. Continue until the first entry is 1. This process imposes a forest structure on the set of all permutations of size n, where the roots are the permutations starting with 1 and the leaves are derangements. Viewing the process in the opposite direction towards the leaves, one picks a fixed point and moves it to the beginning. Despite its simplicity, this “fixed point forest” exhibits a rich structure. In this paper, we consider the fixed point forest in the limit n → ∞ and show using Stein’s method that at a random permutation the local structure weakly converges to a tree defined in terms of independent Poisson point processes. We also show that the distribution of the length of the longest path from a random permutation to a leaf converges to the geometric distribution with mean e − 1, and the length... https://escholarship.org/uc/item/95d5t7zz Mon, 14 May 2018 00:00:00 +0000 Johnson, Tobias Schilling, Anne https://orcid.org/0000-0002-2601-7340 Slivken, Erik https://escholarship.org/uc/item/9535d55z © 2017, Springer Science+Business Media New York. We establish a bijection between the set of rigged configurations and the set of tensor products of Kirillov–Reshetikhin crystals of type Dn(1) in full generality. We prove the invariance of rigged configurations under the action of the combinatorial R-matrix on tensor products and show that the bijection preserves certain statistics (cocharge and energy). As a result, we establish the fermionic formula for type Dn(1). In addition, we establish that the bijection is a classical crystal isomorphism. https://escholarship.org/uc/item/9535d55z Mon, 14 May 2018 00:00:00 +0000 Okado, M Sakamoto, R Schilling, A https://orcid.org/0000-0002-2601-7340 Scrimshaw, T https://escholarship.org/uc/item/92p61887 Frogs produce two distinct yet highly coordinated ventilatory behaviors, buccal and lung. Lung ventilation occurs in short episodes, interspersed with periods of buccal ventilation. Recent data suggests that two brainstem oscillators are involved in generating these behaviors, one primarily responsible for buccal ventilation, the other for lung. Here we use a modeling approach to demonstrate that the episodic pattern of lung ventilation might be an emergent property of the coupling between the oscillators, and may not require a perturbing input from another, as yet unidentified but previously postulated, neuronal oscillator. © 2004 Elsevier B.V. All rights reserved. https://escholarship.org/uc/item/92p61887 Mon, 14 May 2018 00:00:00 +0000 Bose, Amitabha Lewis, Timothy J Wilson, Richard JA https://escholarship.org/uc/item/8t4175pw We propose an efficient nonlinear approximation scheme using the Polyharmonic Local Sine Transform (PHLST) of Saito and Remy combined with an algorithm to tile a given image automatically and adaptively according to its local smoothness and singularities. To measure such local smoothness, we introduce the so-called local Besov indices of an image, which is based on the pointwise modulus of smoothness of the image. Such an adaptive tiling of an image is important for image approximation using PHLST because PHLST stores the corner and boundary information of each tile and consequently it is wasteful to divide a smooth region of a given image into a set of smaller tiles. We demonstrate the superiority of the proposed algorithm using Antarctic remote sensing images over the PHLST using the uniform tiling. Analysis of such images including their efficient approximation and compression has gained its importance due to the global climate change. © 2014 American Institute of Mathematical... https://escholarship.org/uc/item/8t4175pw Mon, 14 May 2018 00:00:00 +0000 Zhang, Zhihua Saito, Naoki https://orcid.org/0000-0001-5234-4719 https://escholarship.org/uc/item/8rq33480 We construct the quantum curve for the Gromov-Witten theory of the complex projective line. https://escholarship.org/uc/item/8rq33480 Mon, 14 May 2018 00:00:00 +0000 Dunin-Barkowski, Petr Mulase, Motohico Norbury, Paul Popolitov, Alexander Shadrin, Sergey https://escholarship.org/uc/item/8b53k9s2 © 2015 World Scientific Publishing Company. We develop a general theory of Markov chains realizable as random walks on R-trivial monoids. It provides explicit and simple formulas for the eigenvalues of the transition matrix, for multiplicities of the eigenvalues via Möbius inversion along a lattice, a condition for diagonalizability of the transition matrix and some techniques for bounding the mixing time. In addition, we discuss several examples, such as Toom-Tsetlin models, an exchange walk for finite Coxeter groups, as well as examples previously studied by the authors, such as nonabelian sandpile models and the promotion Markov chain on posets. Many of these examples can be viewed as random walks on quotients of free tree monoids, a new class of monoids whose combinatorics we develop. https://escholarship.org/uc/item/8b53k9s2 Mon, 14 May 2018 00:00:00 +0000 Ayyer, A Schilling, A https://orcid.org/0000-0002-2601-7340 Steinberg, B Thiéry, NM https://escholarship.org/uc/item/8784b7nr We prove that the discreteness problem for two-generated nonelementary subgroups of SL(2, C) is undecidable in the Blum-Shub-Smale (BSS) computability model. https://escholarship.org/uc/item/8784b7nr Mon, 14 May 2018 00:00:00 +0000 Kapovich, Michael https://orcid.org/0000-0001-5475-8755 https://escholarship.org/uc/item/86v883p8 © 2015, Australian National University. All rights reserved. In this paper, we extend work of the first author on a crystal structure on rigged con_gurations of simply-laced type to all non-exceptional affine types using the technology of virtual rigged con_gurations and crystals. Under the bijection between rigged configurations and tensor products of Kirillov-Reshetikhin crystals specialized to a single tensor factor, we obtain a new tableaux model for Kirillov-Reshetikhin crystals. This is related to the model in terms of Kashiwara-Nakashima tableaux via a filling map, generalizing the recently discovered filling map in typeD(1)n https://escholarship.org/uc/item/86v883p8 Mon, 14 May 2018 00:00:00 +0000 Schilling, A https://orcid.org/0000-0002-2601-7340 Scrimshawy, T https://escholarship.org/uc/item/81g179cj Understanding the folding of the human genome is a key challenge of modern structural biology. The emergence of chromatin conformation capture assays (e.g., Hi-C) has revolutionized chromosome biology and provided new insights into the three dimensional structure of the genome. The experimental data are highly complex and need to be analyzed with quantitative tools. It has been argued that the data obtained from Hi-C assays are consistent with a fractal organization of the genome. A key characteristic of the fractal globule is the lack of topological complexity (knotting or inter-linking). However, the absence of topological complexity contradicts results from polymer physics showing that the entanglement of long linear polymers in a confined volume increases rapidly with the length and with decreasing volume. In vivo and in vitro assays support this claim in some biological systems. We simulate knotted lattice polygons confined inside a sphere and demonstrate that their contact... https://escholarship.org/uc/item/81g179cj Mon, 14 May 2018 00:00:00 +0000 Arsuaga, Javier Jayasinghe, Reyka G Scharein, Robert G Segal, Mark R Stolz, Robert H Vazquez, Mariel https://orcid.org/0000-0001-8328-6806 https://escholarship.org/uc/item/7z04t51g An analytic, asymptotic approximation of the nonlinear steady-state equations for viscoelastic creeping flow, modeled by the Oldroyd-B equations with polymer stress diffusion, is derived. Near the extensional stagnation point the flow stretches and aligns polymers along the outgoing streamlines of the stagnation point resulting in a stress-island, or birefringent strand. The polymer stress diffusion coefficient is used, both as an asymptotic parameter and a regularization parameter. The structure of the singular part of the polymer stress tensor is a Gaussian aligned with the incoming streamline of the stagnation point a smoothed δ-distribution whose width is proportional to the square-root of the diffusion coefficient. The amplitude of the stress island scales with the Wiessenberg number, and although singular in the limit of vanishing diffusion, it is integrable in the cross stream direction due to its vanishing width; this yields a convergent secondary flow. The leading order... https://escholarship.org/uc/item/7z04t51g Mon, 14 May 2018 00:00:00 +0000 Biello, Joseph A Thomases, Becca https://orcid.org/0000-0001-8502-8915 https://escholarship.org/uc/item/79b5x84m © 2018, Mathematical Sciences Publishers. All rights reserved. For noncompact semisimple Lie groups G with finite center, we study the dynamics of the actions of their discrete subgroups Γ < G on the associated partial flag manifolds G/P. Our study is based on the observation, already made in the deep work of Benoist, that they exhibit also in higher rank a certain form of convergence-type dynamics. We identify geometrically domains of proper discontinuity in all partial flag manifolds. Under certain dynamical assumptions equivalent to the Anosov subgroup condition, we establish the cocompactness of the T-action on various domains of proper discontinuity, in particular on domains in the full flag manifold G/B. In the regular case (eg of B-Anosov subgroups), we prove the nonemptiness of such domains if G has (locally) at least one noncompact simple factor not of the type A1, B2or G2by showing the nonexistence of certain ball packings of the visual boundary. https://escholarship.org/uc/item/79b5x84m Mon, 14 May 2018 00:00:00 +0000 Kapovich, M Leeb, B Porti, J https://escholarship.org/uc/item/715632rq We study mirror symmetry for orbifold Hurwitz numbers. We show that the Laplace transform of orbifold Hurwitz numbers satisfy a differential recursion, which is then proved to be equivalent to the integral recursion of Eynard and Orantin with spectral curve given by the r-Lambert curve. We argue that the r-Lambert curve also arises in the infinite framing limit of orbifold Gromov-Witten theory of [C3/(Z/rZ)]. Finally, we prove that the mirror model to orbifold Hurwitz numbers admits a quantum curve. © 2014 Lehigh University. https://escholarship.org/uc/item/715632rq Mon, 14 May 2018 00:00:00 +0000 Bouchard, Vincent Serrano, Daniel Hernández Liu, Xiaojun Mulase, Motohico https://escholarship.org/uc/item/6zg865h3 We generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, $\mathbb{C}$) to geometrically infinite discrete isometry subgroups in the case of rank 1 symmetric spaces, and, under the assumption of bounded torsion, to the case of negatively pinched Hadamard manifolds. Every such geometrically infinite isometry subgroup $\Gamma$ has a set of nonconical limit points with cardinality of continuum. https://escholarship.org/uc/item/6zg865h3 Mon, 14 May 2018 00:00:00 +0000 Kapovich, Michael Liu, Beibei https://escholarship.org/uc/item/6vm3n48c Braid moves in commutation classes of the symmetric group https://escholarship.org/uc/item/6vm3n48c Mon, 14 May 2018 00:00:00 +0000 Schilling, A https://orcid.org/0000-0002-2601-7340 Thiery, NM White, G Williams, N https://escholarship.org/uc/item/6t79m3nh We prove realizability theorems for vector-valued polynomial mappings, real-algebraic sets and compact smooth manifolds by moduli spaces of planar linkages. We also establish a relation between universality theorems for moduli spaces of mechanical linkages and projective arrangements. © 2002 Elsevier Science Ltd. All rights reserved. https://escholarship.org/uc/item/6t79m3nh Mon, 14 May 2018 00:00:00 +0000 Kapovich, Michael https://orcid.org/0000-0001-5475-8755 Millson, John J https://escholarship.org/uc/item/6r46c4sg We prove that all arithmetic lattices in O (n, 1), n≥ 4, n ≠ 7, are noncoherent. We lso establish noncoherence of uniform arithmetic lattices of the simplest type in SU (2, 1), n ≥ 2, and of uniform lattices in SU(2, 1) which have infinite abelianization. https://escholarship.org/uc/item/6r46c4sg Mon, 14 May 2018 00:00:00 +0000 Kapovich, M https://escholarship.org/uc/item/6qc6j30v A fundamental challenge in neuroscience is to understand how biologically salient motor behaviors emerge from properties of the underlying neural circuits. Crayfish, krill, prawns, lobsters, and other long-tailed crustaceans swim by rhythmically moving limbs called swimmerets. Over the entire biological range of animal size and paddling frequency, movements of adjacent swimmerets maintain an approximate quarter-period phase difference with the more posterior limbs leading the cycle. We use a computational fluid dynamics model to show that this frequency-invariant stroke pattern is the most effective and mechanically efficient paddling rhythm across the full range of biologically relevant Reynolds numbers in crustacean swimming. We then show that the organization of the neural circuit underlying swimmeret coordination provides a robust mechanism for generating this stroke pattern. Specifically, the wave-like limb coordination emerges robustly from a combination of the half-center... https://escholarship.org/uc/item/6qc6j30v Mon, 14 May 2018 00:00:00 +0000 Zhang, Calvin Guy, Robert D https://orcid.org/0000-0002-0602-7015 Mulloney, Brian Zhang, Qinghai Lewis, Timothy J https://escholarship.org/uc/item/6bs706xg We present a multivariate generating function for all n×n nonnegative integral matrices with all row and column sums equal to a positive integer t, the so called semi-magic squares. As a consequence we obtain formulas for all coefficients of the Ehrhart polynomial of the polytope B n of n×n doubly-stochastic matrices, also known as the Birkhoff polytope. In particular we derive formulas for the volumes of B n and any of its faces. © 2008 Springer Science+Business Media, LLC. https://escholarship.org/uc/item/6bs706xg Mon, 14 May 2018 00:00:00 +0000 De Loera, JA Liu, F Yoshida, R https://escholarship.org/uc/item/69p8q00w © 2016, Springer Science+Business Media New York. In this note we give an overview of some of our recent work on Anosov representations of discrete groups into higher rank semisimple Lie groups. https://escholarship.org/uc/item/69p8q00w Mon, 14 May 2018 00:00:00 +0000 Kapovich, M Leeb, B Porti, J https://escholarship.org/uc/item/666982kh We study algebraic algorithms for expressing the number of non-negative integer solutions to a unimodular system of linear equations as a function of the right hand side. Our methods include Todd classes of toric varieties via Gröbner bases, and rational generating functions as in Barvinok's algorithm. We report polyhedral and computational results for two special cases: counting contingency tables and Kostant's partition function. https://escholarship.org/uc/item/666982kh Mon, 14 May 2018 00:00:00 +0000 De Loera, JA Sturmfels, B https://escholarship.org/uc/item/641060t5 Let θ : π1(R) → PSL(2, ℂ) be a homomorphism of the fundamental group of an oriented, closed surface R of genus exceeding one. We will establish the following theorem. THEOREM. Necessary and sufficient for θ to be the monodromy representation associated with a complex projective stucture on R, either unbranched or with a single branch point of order 2, is that θ(π1(R)) be nonelementary. A branch point is required if and only if the representation θ does not lift to SL(2, ℂ). https://escholarship.org/uc/item/641060t5 Mon, 14 May 2018 00:00:00 +0000 Gallo, Daniel Kapovich, Michael https://orcid.org/0000-0001-5475-8755 Marden, Albert https://escholarship.org/uc/item/5xk368c6 © 2014 © The Author(s) 2014. Published by Oxford University Press. All rights reserved. We lift the parabolic quantum Bruhat graph (QBG) into the Bruhat order on the affine Weyl group and into Littelmann's poset on level-zero weights. We establish a quantum analog of Deodhar's Bruhat-minimum lift from a parabolic quotient of the Weyl group. This result asserts a remarkable compatibility of the QBG on the Weyl group, with the cosets for every parabolic subgroup. Also, we generalize Postnikov's lemma from the QBG to the parabolic one; this lemma compares paths between two vertices in the former graph. The results in this paper will be applied in a second paper to establish a uniform construction of tensor products of one-column Kirillov-Reshetikhin (KR) crystals, and the equality, for untwisted affine root systems, between the Macdonald polynomial with t set to zero and the graded character of tensor products of one-column KR modules. https://escholarship.org/uc/item/5xk368c6 Mon, 14 May 2018 00:00:00 +0000 Lenart, C Naito, S Sagaki, D Schilling, A https://orcid.org/0000-0002-2601-7340 Shimozono, M https://escholarship.org/uc/item/5v8503xj This book is an exposition of the current state of research of affine Schubert calculus and $k$-Schur functions. This text is based on a series of lectures given at a workshop titled "Affine Schubert Calculus" that took place in July 2010 at the Fields Institute in Toronto, Ontario. The story of this research is told in three parts: 1. Primer on $k$-Schur Functions 2. Stanley symmetric functions and Peterson algebras 3. Affine Schubert calculus https://escholarship.org/uc/item/5v8503xj Mon, 14 May 2018 00:00:00 +0000 Lam, Thomas Lapointe, Luc Morse, Jennifer Schilling, Anne https://orcid.org/0000-0002-2601-7340 Shimozono, Mark Zabrocki, Mike https://escholarship.org/uc/item/5qs3v0rs The symmetric eigenvalue decomposition and the singular value decomposition (SVD) are fundamental matrix decompositions with many applications. Conventional algorithms for computing these decompositions are suboptimal in view of recent trends in computer architectures which require minimizing communication together with arithmetic costs. Spectral divide-and-conquer algorithms, which recursively decouple the problem into two smaller subproblems, can achieve both requirements. Such algorithms can be constructed with the polar decomposition playing two key roles: it forms a bridge between the symmetric eigendecomposition and the SVD, and its connection to the matrix sign function naturally leads to spectral-decoupling. For computing the polar decomposition, the scaled Newton and QDWH iterations are two of the most popular algorithms, as they are backward stable and converge in at most nine and six iterations, respectively. Following this framework, we develop a higher-order variant... https://escholarship.org/uc/item/5qs3v0rs Mon, 14 May 2018 00:00:00 +0000 Nakatsukasa, Yuji Freund, Roland W https://escholarship.org/uc/item/5d18n90g We encode the binomials belonging to the toric ideal IAassociated with an integral d×n matrix A using a short sum of rational functions as introduced by Barvinok (Math. Operations Research 19 (1994) 769) and Barvinok and Woods (J. Amer. Math. Soc. 16 (2003) 957). Under the assumption that d and n are fixed, this representation allows us to compute a universal Gröbner basis and the reduced Gröbner basis of the ideal IA, with respect to any term order, in time polynomial in the size of the input. We also derive a polynomial time algorithm for normal form computations which replaces in this new encoding the usual reductions typical of the division algorithm. We describe other applications, such as the computation of Hilbert series of normal semigroup rings, and we indicate applications to enumerative combinatorics, integer programming, and statistics. © 2004 Elsevier Ltd. All rights reserved. https://escholarship.org/uc/item/5d18n90g Mon, 14 May 2018 00:00:00 +0000 De Loera, JA Haws, D Hemmecke, R Huggins, P Sturmfels, B Yoshida, R https://escholarship.org/uc/item/5c82z72n We define two general classes of nonabelian sandpile models on directed trees (or arborescences) as models of nonequilibrium statistical phenomena. These models have the property that sand grains can enter only through specified reservoirs, unlike the well-known abelian sandpile model. In the Trickle-down sandpile model, sand grains are allowed to move one at a time. For this model, we show that the stationary distribution is of product form. In the Landslide sandpile model, all the grains at a vertex topple at once, and here we prove formulas for all eigenvalues, their multiplicities, and the rate of convergence to stationarity. The proofs use wreath products and the representation theory of monoids. https://escholarship.org/uc/item/5c82z72n Mon, 14 May 2018 00:00:00 +0000 Ayyer, Arvind Schilling, Anne https://orcid.org/0000-0002-2601-7340 Steinberg, Benjamin Thiéry, Nicolas M https://escholarship.org/uc/item/5bv869t1 We prove variations of Carathéodory’s, Helly’s and Tverberg’s theorems where the sets involved are measured according to continuous functions such as the volume or diameter. Among our results, we present continuous quantitative versions of Lovász’s colorful Helly’s theorem, Bárány’s colorful Carathéodory’s theorem, and the colorful Tverberg’s theorem. https://escholarship.org/uc/item/5bv869t1 Mon, 14 May 2018 00:00:00 +0000 De Loera, Jesús A https://orcid.org/0000-0002-9556-1112 La Haye, Reuben N Rolnick, David Soberón, Pablo https://escholarship.org/uc/item/50x6b862 We analyze Weierstrass cycles and tautological rings in moduli spaces of smooth algebraic curves and in moduli spaces of integral algebraic curves with embedded disks with special attention to moduli spaces of curves having genus less than or equal to 6. In particular, we show that our general formula gives a good estimate for the dimension of Weierstrass cycles for low genera. https://escholarship.org/uc/item/50x6b862 Mon, 14 May 2018 00:00:00 +0000 Liou, Jia-Ming Frank Schwarz, Albert Xu, Renjun https://escholarship.org/uc/item/4xp7v6zj The central curve of a linear program is an algebraic curve specified by linear and quadratic constraints arising from complementary slackness. It is the union of the various central paths for minimizing or maximizing the cost function over any region in the associated hyperplane arrangement. We determine the degree, arithmetic genus and defining prime ideal of the central curve, thereby answering a question of Bayer and Lagarias. These invariants, along with the degree of the Gauss image of the curve, are expressed in terms of the matroid of the input matrix. Extending work of Dedieu, Malajovich and Shub, this yields an instance-specific bound on the total curvature of the central path, a quantity relevant for interior-point methods. The global geometry of central curves is studied in detail. © 2012 SFoCM. https://escholarship.org/uc/item/4xp7v6zj Mon, 14 May 2018 00:00:00 +0000 De Loera, Jesús A https://orcid.org/0000-0002-9556-1112 Sturmfels, Bernd Vinzant, Cynthia https://escholarship.org/uc/item/4vv4w2pc It is widely recognized that the three-dimensional (3D) architecture of eukaryotic chromatin plays an important role in processes such as gene regulation and cancer-driving gene fusions. Observing or inferring this 3D structure at even modest resolutions had been problematic, since genomes are highly condensed and traditional assays are coarse. However, recently devised high-throughput molecular techniques have changed this situation. Notably, the development of a suite of chromatin conformation capture (CCC) assays has enabled elicitation of contacts-spatially close chromosomal loci-which have provided insights into chromatin architecture. Most analysis of CCC data has focused on the contact level, with less effort directed toward obtaining 3D reconstructions and evaluating the accuracy and reproducibility thereof. While questions of accuracy must be addressed experimentally, questions of reproducibility can be addressed statistically-the purpose of this paper. We use a constrained... https://escholarship.org/uc/item/4vv4w2pc Mon, 14 May 2018 00:00:00 +0000 Segal, Mark R Xiong, Hao Capurso, Daniel Vazquez, Mariel https://orcid.org/0000-0001-8328-6806 Arsuaga, Javier https://orcid.org/0000-0002-4979-8215 https://escholarship.org/uc/item/4vb7215r © 2014 by De Gruyter 2014. We provide details for Gromov's proof of Stallings' theorem on groups with infinitely many ends using harmonic functions. The main technical result of the paper is a compactness theorem for a certain family of harmonic functions. https://escholarship.org/uc/item/4vb7215r Mon, 14 May 2018 00:00:00 +0000 Kapovich, M https://escholarship.org/uc/item/4d28920b We present a new set of axioms for 2D TQFT formulated on the category of cell graphs with edge-contraction operations as morphisms. We construct a functor from this category to the endofunctor category consisting of Frobenius algebras. Edge-contraction operations correspond to natural transformations of endofunctors, which are compatible with the Frobenius algebra structure. Given a Frobenius algebra A, every cell graph determines an element of the symmetric tensor algebra defined over the dual space A⁎. We show that the edge-contraction axioms make this assignment depending only on the topological type of the cell graph, but not on the graph itself. Thus the functor generates the TQFT corresponding to A. https://escholarship.org/uc/item/4d28920b Mon, 14 May 2018 00:00:00 +0000 Dumitrescu, Olivia Mulase, Motohico https://escholarship.org/uc/item/48r5h3gd We present a Rainbow Ramsey version of the well-known Ramsey-type theorem of Richard Rado. We use new techniques from the Geometry of Numbers. We also disprove two conjectures proposed in the literature. https://escholarship.org/uc/item/48r5h3gd Mon, 14 May 2018 00:00:00 +0000 De Loera, JA https://orcid.org/0000-0002-9556-1112 La Haye, RN Montejano, A Oliveros, D Roldán-Pensado, E https://escholarship.org/uc/item/4827g1tw © 2015 Elsevier Ltd. This paper studies systems of polynomial equations that provide information about orientability of matroids.First, we study systems of linear equations over F2, originally alluded to by Bland and Jensen in their seminal paper on weak orientability. The Bland-Jensen linear equations for a matroid M have a solution if and only if M is weakly orientable. We use the Bland-Jensen system to determine weak orientability for all matroids on at most nine elements and all matroids between ten and twelve elements having rank three. Our experiments indicate that for small rank, about half the time, when a simple matroid is not orientable, it is already non-weakly orientable, and further this may happen more often as the rank increases. Thus, about half of the small simple non-orientable matroids of rank three are not representable over fields having order congruent to three modulo four. For binary matroids, the Bland-Jensen linear systems provide a practical way to check... https://escholarship.org/uc/item/4827g1tw Mon, 14 May 2018 00:00:00 +0000 De Loera, JA Lee, J Margulies, S Miller, J https://escholarship.org/uc/item/47t065k4 The study of the graph diameter of polytopes is a classical open problem in polyhedral geometry and the theory of linear optimization. In this paper we continue the investigation initiated in [5] by introducing a vast hierarchy of generalizations to the notion of graph diameter. This hierarchy provides some interesting lower bounds for the usual graph diameter. After explaining the structure of the hierarchy and discussing these bounds, we focus on clearly explaining the differences and similarities among the many diameter notions of our hierarchy. Finally, we fully characterize the hierarchy in dimension two. It collapses into fewer categories, for which we exhibit the ranges of values that can be realized as diameters. https://escholarship.org/uc/item/47t065k4 Mon, 14 May 2018 00:00:00 +0000 Borgwardt, S De Loera, JA https://orcid.org/0000-0002-9556-1112 Finhold, E https://escholarship.org/uc/item/44c3g3xm In earlier work, the authors obtained formulas for the probability in the asymmetric simple exclusion process that the mth particle from the left is at site x at time t. They were expressed in general as sums of multiple integrals and, for the case of step initial condition, as an integral involving a Fredholm determinant. In the present work, these results are generalized to the case where the mth particle is the left-most one in a contiguous block of L particles. The earlier work depended in a crucial way on two combinatorial identities, and the present work begins with a generalization of these identities to general L. https://escholarship.org/uc/item/44c3g3xm Mon, 14 May 2018 00:00:00 +0000 Tracy, Craig A https://orcid.org/0000-0002-3915-2568 Widom, Harold https://escholarship.org/uc/item/3xs9z3m3 In this paper, we generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2,C) to geometrically infinite discrete subgroups Γ of isometries of negatively pinched Hadamard manifolds X. We then generalize a theorem of Bishop to prove that every discrete geometrically infinite isometry subgroup Γ has a set of nonconical limit points with the cardinality of the continuum. https://escholarship.org/uc/item/3xs9z3m3 Mon, 14 May 2018 00:00:00 +0000 Kapovich, Michael https://orcid.org/0000-0001-5475-8755 Liu, Beibei https://escholarship.org/uc/item/3t23p33z The thalamus provides fundamental input to the neocortex. This input activates inhibitory interneurons more strongly than excitatory neurons, triggering powerful feedforward inhibition. We studied the mechanisms of this selective neuronal activation using a mouse somatosensory thalamocortical preparation. Notably, the greater responsiveness of inhibitory interneurons was not caused by their distinctive intrinsic properties but was instead produced by synaptic mechanisms. Axons from the thalamus made stronger and more frequent excitatory connections onto inhibitory interneurons than onto excitatory cells. Furthermore, circuit dynamics allowed feedforward inhibition to suppress responses in excitatory cells more effectively than in interneurons. Thalamocortical excitatory currents rose quickly in interneurons, allowing them to fire action potentials before significant feedforward inhibition emerged. In contrast, thalamocortical excitatory currents rose slowly in excitatory cells,... https://escholarship.org/uc/item/3t23p33z Mon, 14 May 2018 00:00:00 +0000 Cruikshank, Scott J Lewis, Timothy J Connors, Barry W https://escholarship.org/uc/item/3hw5g673 We examine the influence of dendritic load on the firing dynamics of a spatially extended leaky integrate-and-fire (LIF) neuron that explicitly includes spiking dynamics. We obtain an exact analytical solution for this model and use it to derive a return map that completely captures the dynamics of the system. Using the map, we find that dendritic properties can significantly change the firing dynamics of the system. Under certain conditions, the addition of the dendrite can change the LIF model from type 1 excitability to type 2 excitability and induce bistability between periodic firing and the quiescent state. We identify the mechanism that causes the periodic behavior in the bistable regime as somatodendritic ping-pong. Furthermore, we use the return map to fully explore the model parameter space in order to find regions where this bistable behavior occurs. We then give physical interpretations of the dependence of the bistable behavior on model parameters. Finally, we demonstrate... https://escholarship.org/uc/item/3hw5g673 Mon, 14 May 2018 00:00:00 +0000 Schwemmer, Michael A Lewis, Timothy J https://escholarship.org/uc/item/3447s8vt © 2016 Elsevier Inc. In this article we prove the existence, uniqueness, and simplicity of a negative eigenvalue for a class of integral operators whose kernel is of the form |x−y|ρ, 0<ρ≤1, x,y∈[−a,a]. We also provide two different ways of producing recursive formulas for the Rayleigh functions (i.e., recursion formulas for power sums) of the eigenvalues of this integral operator when ρ=1, providing means of approximating this negative eigenvalue. These methods offer recursive procedures for dealing with the eigenvalues of a one-dimensional Laplacian with non-local boundary conditions which commutes with an integral operator having a harmonic kernel. The problem emerged in recent work by one of the authors [48]. We also discuss extensions in higher dimensions and links with distance matrices. https://escholarship.org/uc/item/3447s8vt Mon, 14 May 2018 00:00:00 +0000 Hermi, L Saito, N https://orcid.org/0000-0001-5234-4719 https://escholarship.org/uc/item/33h6h6cc We derive the spectral curves for q-part double Hurwitz numbers, r-spin simple Hurwitz numbers, and arbitrary combinations of these cases, from the analysis of the unstable (0, 1)-geometry. We quantize this family of spectral curves and obtain the Schrödinger equations for the partition function of the corresponding Hurwitz problems. We thus confirm the conjecture for the existence of quantum curves in these generalized Hurwitz number cases. https://escholarship.org/uc/item/33h6h6cc Mon, 14 May 2018 00:00:00 +0000 Mulase, M Shadrin, S Spitz, L https://escholarship.org/uc/item/30j0s5fh We explicitly calculate the triangle inequalities for the group PSO(8). Therefore we explicitly solve the eigenvalues of sum problem for this group (equivalently describing the side-lengths of geodesic triangles in the corresponding symmetric space for the Weyl chamber-valued metric). We then apply some computer programs to verify two basic questions/conjectures. First, we verify that the above system of inequalities is irredundant. Then, we verify the ``saturation conjecture'' for the decomposition of tensor products of finite-dimensional irreducible representations of Spin(8). Namely, we show that for any triple of dominant weights a, b, c such that a+b+c is in the root lattice, and any positive integer N, the tensor product of the irreducible representations V(a) and V(b) contains V(c) if and only if the tensor product of V(Na) and V(Nb) contains V(Nc). https://escholarship.org/uc/item/30j0s5fh Mon, 14 May 2018 00:00:00 +0000 Kapovich, M Kumar, S Millson, JJ https://escholarship.org/uc/item/3079k8f5 In Escherichia coli, complete unlinking of newly replicated sister chromosomes is required to ensure their proper segregation at cell division. Whereas replication links are removed primarily by topoisomerase IV, XerC/XerD-dif site-specific recombination can mediate sister chromosome unlinking in Topoisomerase IV-deficient cells. This reaction is activated at the division septum by the DNA translocase FtsK, which coordinates the last stages of chromosome segregation with cell division. It has been proposed that, after being activated by FtsK, XerC/XerD-dif recombination removes DNA links in a stepwise manner. Here, we provide a mathematically rigorous characterization of this topological mechanism of DNA unlinking. We show that stepwise unlinking is the only possible pathway that strictly reduces the complexity of the substrates at each step. Finally, we propose a topological mechanism for this unlinking reaction. https://escholarship.org/uc/item/3079k8f5 Mon, 14 May 2018 00:00:00 +0000 Shimokawa, Koya Ishihara, Kai Grainge, Ian Sherratt, David J Vazquez, Mariel https://orcid.org/0000-0001-8328-6806 https://escholarship.org/uc/item/2vc4921t Copyright © 2016 University College London. We continue our study of intermediate sums over polyhedra, interpolating between integrals and discrete sums, which were introduced by Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex. Math. Comp. 75 (2006), 1449-1466]. By well-known decompositions, it is sufficient to consider the case of affine cones s+c, where is an arbitrary real vertex and is a rational polyhedral cone. For a given rational subspace L, we define the intermediate generating functions SL(s+c)(ξ) by integrating an exponential function over all lattice slices of the affine cone s+c parallel to the subspace L and summing up the integrals. We expose the bidegree structure in parameters and ξ, which was implicitly used in the algorithms in our papers [Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra. Found. Comput. Math. 12 (2012), 435-469] and [Intermediate sums on polyhedra: computation and real... https://escholarship.org/uc/item/2vc4921t Mon, 14 May 2018 00:00:00 +0000 Baldoni, V Berline, N De Loera, JA Köppe, M Vergne, M https://escholarship.org/uc/item/2mb049gn The study of the diameter of the graph of polyhedra is a classical problem in the theory of linear programming. While transportation polytopes are at the core of operations research and statistics it is still unknown whether the Hirsch conjecture is true for general m×n-transportation polytopes. In earlier work the first three authors introduced a hierarchy of variations to the notion of graph diameter in polyhedra. This hierarchy provides some interesting lower bounds for the usual graph diameter. This paper has three contributions: First, we compare the hierarchy of diameters for the m×n-transportation polytopes. We show that the Hirsch conjecture bound of m+n−1 is actually valid in most of these diameter notions. Second, we prove that for 3×n-transportation polytopes the Hirsch conjecture holds in the classical graph diameter. Third, we show for 2×n-transportation polytopes that the stronger monotone Hirsch conjecture holds and improve earlier bounds on the graph diameter. https://escholarship.org/uc/item/2mb049gn Mon, 14 May 2018 00:00:00 +0000 Borgwardt, S De Loera, JA https://orcid.org/0000-0002-9556-1112 Finhold, E Miller, J https://escholarship.org/uc/item/2f1562bj © 2017, Mathematical Sciences Publishers. All rights reserved. We prove universality theorems (“Murphy’s laws”) for representation varieties of fundamental groups of closed 3-dimensional manifolds. We show that germs of SL(2)- representation schemes of such groups are essentially the same as germs of schemes over ℚ of finite type. https://escholarship.org/uc/item/2f1562bj Mon, 14 May 2018 00:00:00 +0000 Kapovich, M Millson, JJ https://escholarship.org/uc/item/2bj3g2sz © 2015 Elsevier Inc. All rights reserved. We developed a procedure of reducing the number of vertices and edges of a given tree, which we call the "tree simplification procedure," without changing its topological information. Our motivation for developing this procedure was to reduce computational costs of graph Laplacian eigenvalues of such trees. When we applied this procedure to a set of trees representing dendritic structures of retinal ganglion cells of a mouse and computed their graph Laplacian eigenvalues, we observed two "plateaux" (i.e., two sets of multiple eigenvalues) in the eigenvalue distribution of each such simplified tree. In this article, after describing our tree simplification procedure, we analyze why such eigenvalue plateaux occur in a simplified tree; explain such plateaux can occur in a more general graph if it satisfies a certain condition; and identify these two eigenvalues specifically as well as the lower bound to their multiplicity. https://escholarship.org/uc/item/2bj3g2sz Mon, 14 May 2018 00:00:00 +0000 Saito, N https://orcid.org/0000-0001-5234-4719 Woei, E https://escholarship.org/uc/item/29h0171c For any finite Coxeter group W, we introduce two new objects: its cutting poset and its biHecke monoid. The cutting poset, constructed using a generalization of the notion of blocks in permutation matrices, almost forms a lattice on W. The construction of the biHecke monoid relies on the usual combinatorial model for the 0-Hecke algebra H0(W), that is, for the symmetric group, the algebra (or monoid) generated by the elementary bubble sort operators. The authors previously introduced the Hecke group algebra, constructed as the algebra generated simultaneously by the bubble sort and antisort operators, and described its representation theory. In this paper, we consider instead the monoid generated by these operators. We prove that it admits |W| simple and projective modules. In order to construct the simple modules, we introduce for each w ∈ W a combinatorial module Tw whose support is the interval [1,w]R in right weak order. This module yields an algebra, whose representation... https://escholarship.org/uc/item/29h0171c Mon, 14 May 2018 00:00:00 +0000 Hivert, Florent Schilling, Anne https://orcid.org/0000-0002-2601-7340 Thiéry, Nicolas https://escholarship.org/uc/item/28m357wt It is well known that excitable media can sustain "fast" stable traveling pulses of excitation. In models for which analysis is tractable, the fast pulse solution is known to appear through a saddlenode bifurcation accompanied by a "slow" unstable traveling pulse. Furthermore, the uniform rest state is also a stable solution. It is generally assumed that the boundary between the basins of attractions of the rest state and fast pulse (i.e., threshold) consists of the stable manifold of the slow pulse. We use numerical experiments to explore this issue. Our results indicate that, near the saddle-node bifurcation, the stable manifold of the slow pulse does indeed act as the threshold between the rest state and fast pulse. However, further away from the saddle-node bifurcation, a global bifurcation involving the heteroclinic connections between slow and fast pulses occurs. This bifurcation gives rise to an unstable periodic solution that has been referred to as a one-dimensional spiral... https://escholarship.org/uc/item/28m357wt Mon, 14 May 2018 00:00:00 +0000 Cytrynbaum, Eric N Lewis, Timothy J https://escholarship.org/uc/item/2849t7dv In this paper, we propose a new Markov chain which generalizes random-to-random shuffling on permutations to random-to-random shuffling on linear extensions of a finite poset of size $n$. We conjecture that the second largest eigenvalue of the transition matrix is bounded above by $(1+1/n)(1-2/n)$ with equality when the poset is disconnected. This Markov chain provides a way to sample the linear extensions of the poset with a relaxation time bounded above by $n^2/(n+2)$ and a mixing time of $O(n^2 \log n)$. We conjecture that the mixing time is in fact $O(n \log n)$ as for the usual random-to-random shuffling. https://escholarship.org/uc/item/2849t7dv Mon, 14 May 2018 00:00:00 +0000 Ayyer, Arvind Schilling, Anne https://orcid.org/0000-0002-2601-7340 Thiéry, Nicolas M https://escholarship.org/uc/item/27w4s14f We prove that for every finitely-presented group G there exists a 2-dimensional irreducible complex-projective variety W with the fundamental group G, so that all singularities of W are normal crossings and Whitney umbrellas. © 2013 Springer-Verlag Berlin Heidelberg. https://escholarship.org/uc/item/27w4s14f Mon, 14 May 2018 00:00:00 +0000 Kapovich, M https://escholarship.org/uc/item/27n8r19q © 2015 Elsevier Inc. The Immersed Boundary method is a simple, efficient, and robust numerical scheme for solving PDE in general domains, yet it only achieves first-order spatial accuracy near embedded boundaries. In this paper, we introduce a new high-order numerical method which we call the Immersed Boundary Smooth Extension (IBSE) method. The IBSE method achieves high-order accuracy by smoothly extending the unknown solution of the PDE from a given smooth domain to a larger computational domain, enabling the use of simple Cartesian-grid discretizations (e.g. Fourier spectral methods). The method preserves much of the flexibility and robustness of the original IB method. In particular, it requires minimal geometric information to describe the boundary and relies only on convolution with regularized delta-functions to communicate information between the computational grid and the boundary. We present a fast algorithm for solving elliptic equations, which forms the basis for simple,... https://escholarship.org/uc/item/27n8r19q Mon, 14 May 2018 00:00:00 +0000 Stein, DB Guy, RD Thomases, B https://escholarship.org/uc/item/22f42910 We study the effects of passive dendritic properties on the dynamics of neuronal oscillators. We find that the addition of a passive dendrite can sometimes have counterintuitive effects on firing frequency. Specifically, the addition of a hyperpolarized passive dendritic load can either increase, decrease, or have negligible effects on firing frequency. We use the theory of weak coupling to derive phase equations for "ball-and-stick" model neurons and two-compartment model neurons. We then develop a framework for understanding how the addition of passive dendrites modulates the frequency of neuronal oscillators. We show that the average value of the neuronal oscillator's phase response curves measures the sensitivity of the neuron's firing rate to the dendritic load, including whether the addition of the dendrite causes an increase or decrease in firing frequency. We interpret this finding in terms of to the slope of the neuronal oscillator's frequency-applied current curve. We... https://escholarship.org/uc/item/22f42910 Mon, 14 May 2018 00:00:00 +0000 Schwemmer, Michael A Lewis, Timothy J https://escholarship.org/uc/item/2037z49w This paper presents a new variation of Tverberg’s theorem. Given a discrete set S of Rd, we study the number of points of S needed to guarantee the existence of an m-partition of the points such that the intersection of the m convex hulls of the parts contains at least k points of S. The proofs of the main results require new quantitative versions of Helly’s and Carathéodory’s theorems. https://escholarship.org/uc/item/2037z49w Mon, 14 May 2018 00:00:00 +0000 De Loera, Jesus A https://orcid.org/0000-0002-9556-1112 La Haye, Reuben N Rolnick, David Soberón, Pablo https://escholarship.org/uc/item/1zw9r8c7 We prove that every RAAG (Right Angled Artin Group) embeds in the group of Hamiltonian symplectomorphisms of every symplectic manifold. © 2012 Springer Basel AG. https://escholarship.org/uc/item/1zw9r8c7 Mon, 14 May 2018 00:00:00 +0000 Kapovich, M https://escholarship.org/uc/item/1x30s3zh Homological index of a holomorphic 1-form on a complex-analytic variety with an isolated singular point is an analogue of the usual index of a 1-form on a non-singular manifold. One can say that it corresponds to the top Chern number of a manifold. We offer a definition of homological indices for collections of 1-forms on a (purely dimensional) complex-analytic variety with an isolated singular point corresponding to other Chern numbers. We also define new invariants of germs of complex-analytic varieties with isolated singular points related to ‘vanishing Chern numbers’ at them. https://escholarship.org/uc/item/1x30s3zh Mon, 14 May 2018 00:00:00 +0000 Gorsky, E https://orcid.org/0000-0002-9572-4938 Gusein‐Zade, SM https://escholarship.org/uc/item/1wc1h3wq If a torsion-free hyperbolic group G has 1-dimensional boundary ∂∞G, then ∂∞G is a Menger curve or a Sierpinski carpet provided G does not split over a cyclic group. When ∂∞G is a Sierpinski carpet we show that G is a quasi-convex subgroup of a 3-dimensional hyperbolic Poincaré duality group. We also construct a "topologically rigid" hyperbolic group G: any homeomorphism of ∂∞G is induced by an element of G. © 2000 Éditions scientifiques et médicales Elsevier SAS. https://escholarship.org/uc/item/1wc1h3wq Mon, 14 May 2018 00:00:00 +0000 Kapovich, Michael https://orcid.org/0000-0001-5475-8755 Kleiner, Bruce https://escholarship.org/uc/item/1v05d417 © 2017, Springer Science+Business Media New York. We establish the equality of the specialization Ewλ(x ; q; 0) of the nonsymmetric Macdonald polynomial Ewλ(x ; q; t) at t = 0 with the graded character gch Uw+(λ) of a certain Demazure-type submodule Uw+(λ) of a tensor product of “single-column” Kirillov–Reshetikhin modules for an untwisted affine Lie algebra, where λ is a dominant integral weight and w is a (finite) Weyl group element; this generalizes our previous result, that is, the equality between the specialization Pλ(x ; q; 0) of the symmetric Macdonald polynomial Pλ(x ; q; t) at t = 0 and the graded character of a tensor product of single-column Kirillov–Reshetikhin modules. We also give two combinatorial formulas for the mentioned specialization of nonsymmetric Macdonald polynomials: one in terms of quantum Lakshmibai–Seshadri paths and the other in terms of the quantum alcove model. https://escholarship.org/uc/item/1v05d417 Mon, 14 May 2018 00:00:00 +0000 Lenart, C Naito, S Sagaki, D Schilling, A https://orcid.org/0000-0002-2601-7340 Shimozono, M https://escholarship.org/uc/item/1s69r185 © 2016 World Scientific Publishing Company. We introduce semaphore codes associated to a Turing machine via resets. Semaphore codes provide an approximation theory for resets. In this paper, we generalize the set-up of our previous paper "Random walks on semaphore codes and delay de Bruijn semigroups" to the infinite case by taking the profinite limit of k-resets to obtain (-ω)-resets. We mention how this opens new avenues to attack the vs. NP problem. https://escholarship.org/uc/item/1s69r185 Mon, 14 May 2018 00:00:00 +0000 Rhodes, J Schilling, A https://orcid.org/0000-0002-2601-7340 Silva, PV https://escholarship.org/uc/item/1s132759 This paper transfers a randomized algorithm, originally used in geometric optimization, to computational problems in commutative algebra. We show that Clarkson's sampling algorithm can be applied to two problems in computational algebra: solving large-scale polynomial systems and finding small generating sets of graded ideals. The cornerstone of our work is showing that the theory of violator spaces of Gärtner et al. applies to polynomial ideal problems. To show this, one utilizes a Helly-type result for algebraic varieties. The resulting algorithms have expected runtime linear in the number of input polynomials, making the ideas interesting for handling systems with very large numbers of polynomials, but whose rank in the vector space of polynomials is small (e.g., when the number of variables and degree is constant). https://escholarship.org/uc/item/1s132759 Mon, 14 May 2018 00:00:00 +0000 De Loera, Jesús A https://orcid.org/0000-0002-9556-1112 Petrović, Sonja Stasi, Despina https://escholarship.org/uc/item/1rk725c0 The famous Doignon-Bell-Scarf theorem is a Helly-type result about the existence of integer solutions to systems of linear inequalities. The purpose of this paper is to present the following quantitative generalization: Given an integer k, we prove that there exists a constant c(n,k), depending only on the dimension n and k, such that if a polyhedron {x∈Rn: Ax≤b} contains exactly k integer points, then there exists a subset of the rows, of cardinality no more than c(n,k), defining a polyhedron that contains exactly the same k integer points. In this case c(n,0)=2n as in the original case of Doignon-Bell-Scarf for infeasible systems of inequalities. We work on both upper and lower bounds for the constant c(n,k) and discuss some consequences, including a Clarkson-style algorithm to find the l-th best solution of an integer program with respect to the ordering induced by the objective function. https://escholarship.org/uc/item/1rk725c0 Mon, 14 May 2018 00:00:00 +0000 Aliev, Iskander Bassett, Robert De Loera, Jesús A https://orcid.org/0000-0002-9556-1112 Louveaux, Quentin https://escholarship.org/uc/item/1947n9cf Fast-spiking (FS) cells in layer IV of the somatosensory cortex receive direct thalamocortical (TC) input and provide feed-forward inhibition onto layer IV excitatory cells. The level of synchronous firing of FS cells will affect the shape of this feed-forward output. Two factors that contribute to the synchrony are correlated TC input and electrical coupling between FS cells. Using a cell-pair model, we show that these two factors act synergistically to increase synchrony, and we examine the underlying mechanism. © 2006 Elsevier B.V. All rights reserved. https://escholarship.org/uc/item/1947n9cf Mon, 14 May 2018 00:00:00 +0000 Schneider, Abraham R Lewis, Timothy J Rinzel, John