29 research outputs found

    Cyclic disjointness of Hamiltonian tours

    Get PDF

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

    Get PDF
    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Effects of Repulsive Coupling in Ensembles of Excitable Elements

    Get PDF
    Die vorliegende Arbeit behandelt die kollektive Dynamik identischer Klasse-I-anregbarer Elemente. Diese können im Rahmen der nichtlinearen Dynamik als Systeme nahe einer Sattel-Knoten-Bifurkation auf einem invarianten Kreis beschrieben werden. Der Fokus der Arbeit liegt auf dem Studium aktiver Rotatoren als Prototypen solcher Elemente. In Teil eins der Arbeit besprechen wir das klassische Modell abstoßend gekoppelter aktiver Rotatoren von Shinomoto und Kuramoto und generalisieren es indem wir höhere Fourier-Moden in der internen Dynamik der Rotatoren berĂŒcksichtigen. Wir besprechen außerdem die mathematischen Methoden die wir zur Untersuchung des Aktive-Rotatoren-Modells verwenden. In Teil zwei untersuchen wir Existenz und StabilitĂ€t periodischer Zwei-Cluster-Lösungen fĂŒr generalisierte aktive Rotatoren und beweisen anschließend die Existenz eines Kontinuums periodischer Lösungen fĂŒr eine Klasse Watanabe-Strogatz-integrabler Systeme zu denen insbesondere das klassische Aktive-Rotatoren-Modell gehört und zeigen dass (i) das Kontinuum eine normal-anziehende invariante Mannigfaltigkeit bildet und (ii) eine der auftretenden periodischen Lösungen Splay-State-Dynamik besitzt. Danach entwickeln wir mit Hilfe der Averaging-Methode eine Störungstheorie fĂŒr solche Systeme. Mit dieser können wir RĂŒckschlĂŒsse auf die asymptotische Dynamik des generalisierten Aktive-Rotatoren-Modells ziehen. Als Hauptergebnis stellen wir fest dass sowohl periodische Zwei-Cluster-Lösungen als auch Splay States robuste Lösungen fĂŒr das Aktive-Rotatoren-Modell darstellen. Wir untersuchen außerdem einen "StabilitĂ€tstransfer" zwischen diesen Lösungen durch sogenannte Broken-Symmetry States. In Teil drei untersuchen wir Ensembles gekoppelter Morris-Lecar-Neuronen und stellen fest, dass deren asymptotische Dynamik der der aktiven Rotatoren vergleichbar ist was nahelegt dass die Ergebnisse aus Teil zwei ein qualitatives Bild fĂŒr solch kompliziertere und realistischere Neuronenmodelle liefern.We study the collective dynamics of class I excitable elements, which can be described within the theory of nonlinear dynamics as systems close to a saddle-node bifurcation on an invariant circle. The focus of the thesis lies on the study of active rotators as a prototype for such elements. In part one of the thesis, we motivate the classic model of repulsively coupled active rotators by Shinomoto and Kuramoto and generalize it by considering higher-order Fourier modes in the on-site dynamics of the rotators. We also discuss the mathematical methods which our work relies on, in particular the concept of Watanabe-Strogatz (WS) integrability which allows to describe systems of identical angular variables in terms of Möbius transformations. In part two, we investigate the existence and stability of periodic two-cluster states for generalized active rotators and prove the existence of a continuum of periodic orbits for a class of WS-integrable systems which includes, in particular, the classic active rotator model. We show that (i) this continuum constitutes a normally attracting invariant manifold and that (ii) one of the solutions yields splay state dynamics. We then develop a perturbation theory for such systems, based on the averaging method. By this approach, we can deduce the asymptotic dynamics of the generalized active rotator model. As a main result, we find that periodic two-cluster states and splay states are robust periodic solutions for systems of identical active rotators. We also investigate a 'transfer of stability' between these solutions by means of so-called broken-symmetry states. In part three, we study ensembles of higher-dimensional class I excitable elements in the form of Morris-Lecar neurons and find the asymptotic dynamics of such systems to be similar to those of active rotators, which suggests that our results from part two yield a suitable qualitative description for more complicated and realistic neural models

    Fragments d'Optimisation Différentiable - Théories et Algorithmes

    Get PDF
    MasterLecture Notes (in French) of optimization courses given at ENSTA (Paris, next Saclay), ENSAE (Paris) and at the universities Paris I, Paris VI and Paris Saclay (979 pages).Syllabus d’enseignements délivrés à l’ENSTA (Paris, puis Saclay), à l’ENSAE (Paris) et aux universités Paris I, Paris VI et Paris Saclay (979 pages)

    Notes in Pure Mathematics & Mathematical Structures in Physics

    Full text link
    These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

    Electronic Journal of Qualitative Theory of Differential Equations 2021

    Get PDF

    Abstract Algebra: Theory and Applications

    Get PDF
    Tom Judson\u27s Abstract Algebra: Theory and Applications is an open source textbook designed to teach the principles and theory of abstract algebra to college juniors and seniors in a rigorous manner. Its strengths include a wide range of exercises, both computational and theoretical, plus many nontrivial applications. Rob Beezer has contributed complementary material using the open source system, Sage.An HTML version on the PreText platform is available here. The first half of the book presents group theory, through the Sylow theorems, with enough material for a semester-long course. The second-half is suitable for a second semester and presents rings, integral domains, Boolean algebras, vector spaces, and fields, concluding with Galois Theory.https://scholarworks.sfasu.edu/ebooks/1022/thumbnail.jp

    Combinatorial and Additive Number Theory Problem Sessions: '09--'19

    Full text link
    These notes are a summary of the problem session discussions at various CANT (Combinatorial and Additive Number Theory Conferences). Currently they include all years from 2009 through 2019 (inclusive); the goal is to supplement this file each year. These additions will include the problem session notes from that year, and occasionally discussions on progress on previous problems. If you are interested in pursuing any of these problems and want additional information as to progress, please email the author. See http://www.theoryofnumbers.com/ for the conference homepage.Comment: Version 3.4, 58 pages, 2 figures added 2019 problems on 5/31/2019, fixed a few issues from some presenters 6/29/201

    Adaptations of neutrality tests

    Get PDF
    Most of the genetic variation observed within a biological species is generally thought to be evolutionary “neutral” in the sense that it is irrelevant for an individuum whether its genome contains one particular variant or another. Evolutionary biologists, and in the case of the human species anthropologists and medical scientists as well, are by contrast interested in variants which do influence on an individual’s survival and/or its ability to reproduce. Population geneticists try to find such variants by purely statistical methods in the form of tests on neutrality or shortly neutrality tests. In this thesis four publications are reprinted and discussed which are concerned with modifications of existing neutrality tests. Three of them deal with a class of tests relying on the so-called site frequency spectrum. It was shown previously that some of these tests, originally designed on models of constant population size, can be adapted to allow for changes in population size. This is generalized in the first publication to all tests of similar structure. Another aspect of these tests is that they are ignorant with respect to which variant in a sample might evolve non-neutrally. If instead a particular variant is suspected a priori, the tests have to allow for this information by conditioning on the existence of a variant with the observed frequency. The second and third article introduce the concept of a conditional frequency spectrum and derive its first resp. second moments which are necessary for an appropriate extension of the above-mentioned class of tests. The fourth article presents an algorithmic improvement of a neutrality test of a different kind. Here, primarily computational speed was of concern, in order to bear comparison with competing software. Solely applications on human data are presented, which is available in unrivalled abundance, owing to several large-scale genotyping and sequencing projects. The applicability of neutrality tests, however, is not confined to any particular species
    corecore