82 research outputs found

    Topics in Graph Theory: Extremal Intersecting Systems, Perfect Graphs, and Bireflexive Graphs

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    In this thesis we investigate three different aspects of graph theory. Firstly, we consider interesecting systems of independent sets in graphs, and the extension of the classical theorem of Erdos, Ko and Rado to graphs. Our main results are a proof of an Erdos-Ko-Rado type theorem for a class of trees, and a class of trees which form counterexamples to a conjecture of Hurlberg and Kamat, in such a way that extends the previous counterexamples given by Baber. Secondly, we investigate perfect graphs - specifically, edge modification aspects of perfect graphs and their subclasses. We give some alternative characterisations of perfect graphs in terms of edge modification, as well as considering the possible connection of the critically perfect graphs - previously studied by Wagler - to the Strong Perfect Graph Theorem. We prove that the situation where critically perfect graphs arise has no analogue in seven different subclasses of perfect graphs (e.g. chordal, comparability graphs), and consider the connectivity of a bipartite reconfiguration-type graph associated to each of these subclasses. Thirdly, we consider a graph theoretic structure called a bireflexive graph where every vertex is both adjacent and nonadjacent to itself, and use this to characterise modular decompositions as the surjective homomorphisms of these structures. We examine some analogues of some graph theoretic notions and define a “dual” version of the reconstruction conjecture

    An effective Chebotarev density theorem for families of number fields, with an application to â„“\ell-torsion in class groups

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    We prove a new effective Chebotarev density theorem for Galois extensions L/QL/\mathbb{Q} that allows one to count small primes (even as small as an arbitrarily small power of the discriminant of LL); this theorem holds for the Galois closures of "almost all" number fields that lie in an appropriate family of field extensions. Previously, applying Chebotarev in such small ranges required assuming the Generalized Riemann Hypothesis. The error term in this new Chebotarev density theorem also avoids the effect of an exceptional zero of the Dedekind zeta function of LL, without assuming GRH. We give many different "appropriate families," including families of arbitrarily large degree. To do this, we first prove a new effective Chebotarev density theorem that requires a zero-free region of the Dedekind zeta function. Then we prove that almost all number fields in our families yield such a zero-free region. The innovation that allows us to achieve this is a delicate new method for controlling zeroes of certain families of non-cuspidal LL-functions. This builds on, and greatly generalizes the applicability of, work of Kowalski and Michel on the average density of zeroes of a family of cuspidal LL-functions. A surprising feature of this new method, which we expect will have independent interest, is that we control the number of zeroes in the family of LL-functions by bounding the number of certain associated fields with fixed discriminant. As an application of the new Chebotarev density theorem, we prove the first nontrivial upper bounds for ℓ\ell-torsion in class groups, for all integers ℓ≥1\ell \geq 1, applicable to infinite families of fields of arbitrarily large degree.Comment: 52 pages. This shorter version aligns with the published paper. Note that portions of Section 8 of the longer v1 have been developed as a separate paper with identifier arXiv:1902.0200

    Dubrovin's approach to the FPU Problem

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    In the study of perturbed Hamiltonian systems, there is a theorem (due to Poincarè) that says this: non degenerate integrable Hamiltonian systems, under generic perturbation, loose all the first integrals in the analytic class. Along such line, methods to extend solutions and first integrals of the unperturbed system have increased. In particular, in recent years, Boris Dubrovin developed new techniques, for perturbed Hamiltonian PDEs of hyperbolic type, to extend solutions and first integrals from the unperturbed system to the perturbed one. One of the relevant cases, treated in the work [1], is that of a continuum version of a particle chain with a pair interaction potential phi(r) (known as the generic FPU problem). Dubrovin showed that, under suitable dispersive perturbations, all the first integral of this unperturbed system admit a deformation at the second order iff and, for special choose of constants, we obtain the Toda potential. This means that the integrable Toda chain plays a kind of unique role among the FPU systems. In the thesis, we extend this results also for generic perturbation, adding a potential psi(r) at the second order of the perturbation, and apply this techniques to the actual FPU chains, regarded as a perturbation of the Toda chain, to see if it's possible to extend integral at the second order or further. A condition on this last target is given. 1. B. Dubrovin, ``On universality of critical behaviour in Hamiltonian PDEs'', Amer. Math. Soc. Transl. 224 (2008) 59-109.ope

    Mathematical and computational studies of equilibrium capillary free surfaces

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    The results of several independent studies are presented. The general question is considered of whether a wetting liquid always rises higher in a small capillary tube than in a larger one, when both are dipped vertically into an infinite reservoir. An analytical investigation is initiated to determine the qualitative behavior of the family of solutions of the equilibrium capillary free-surface equation that correspond to rotationally symmetric pendent liquid drops and the relationship of these solutions to the singular solution, which corresponds to an infinite spike of liquid extending downward to infinity. The block successive overrelaxation-Newton method and the generalized conjugate gradient method are investigated for solving the capillary equation on a uniform square mesh in a square domain, including the case for which the solution is unbounded at the corners. Capillary surfaces are calculated on the ellipse, on a circle with reentrant notches, and on other irregularly shaped domains using JASON, a general purpose program for solving nonlinear elliptic equations on a nonuniform quadrilaterial mesh. Analytical estimates for the nonexistence of solutions of the equilibrium capillary free-surface equation on the ellipse in zero gravity are evaluated

    Arithmetic properties of curves and Jacobians

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    2020 Fall.Includes bibliographical references.This thesis is about algebraic curves and their Jacobians. The first chapter concerns Abhyankar's Inertia Conjecture which is about the existence of unramified covers of the affine line in positive characteristic with prescribed ramification behavior. The second chapter demonstrates the existence of a curve C for which a particular algebraic cycle, called the Ceresa cycle, is torsion in the Jacobian variety of C. The final chapter is a study of supersingular Hurwitz curves in positive characteristic
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