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  • Xia, Jiacheng, 1991- (författare)
  • Vector-valued Eisenstein series of congruence types and their products
  • 2019
  • Licentiatavhandling (övrigt vetenskapligt)abstract
    • Historically, Kohnen and Zagier connected modular forms with period polynomials, and as a consequence of this association concluded that the products of at most two Eisenstein series span all spaces of classical modular forms of level 1. Later Borisov and Gunnells among other authors extended the result to higher levels. We consider this problem for vector-valued modular forms, establish the framework of congruence types and obtain the structure of the space of vector-valued Eisenstein series using tools from representation theory. Based on this development and historic results, we show that the space of vector-valued modular forms of certain weights and any congruence type can be spanned by the invariant vectors of that type tensor at most two Eisenstein series.
  • Kurujyibwami, Celestin (författare)
  • Admissible transformations and the group classification of Schrödinger equations
  • 2017
  • Doktorsavhandling (övrigt vetenskapligt)abstract
    • <p>We study admissible transformations and solve group classification problems for various classes of linear and nonlinear Schrödinger equations with an arbitrary number n of space variables.</p><p>The aim of the thesis is twofold. The first is the construction of the new theory of uniform seminormalized classes of differential equations and its application to solving group classification problems for these classes. Point transformations connecting two equations (source and target) from the class under study may have special properties of semi-normalization. This makes the group classification of that class using the algebraic method more involved. To extend this method we introduce the new notion of uniformly semi-normalized classes. Various types of uniform semi-normalization are studied: with respect to the corresponding equivalence group, with respect to a proper subgroup of the equivalence group as well as the corresponding types of weak uniform semi-normalization. An important kind of uniform semi-normalization is given by classes of homogeneous linear differential equations, which we call uniform semi-normalization with respect to linear superposition of solutions.</p><p>The class of linear Schrödinger equations with complex potentials is of this type and its group classification can be effectively carried out within the framework of the uniform semi-normalization. Computing the equivalence groupoid and the equivalence group of this class, we show that it is uniformly seminormalized with respect to linear superposition of solutions. This allow us to apply the version of the algebraic method for uniformly semi-normalized classes and to reduce the group classification of this class to the classification of appropriate subalgebras of its equivalence algebra. To single out the classification cases, integers that are invariant under equivalence transformations are introduced. The complete group classification of linear Schrödinger equations is carried out for the cases n = 1 and n = 2.</p><p>The second aim is to study group classification problem for classes of generalized nonlinear Schrödinger equations which are not uniformly semi-normalized. We find their equivalence groupoids and their equivalence groups and then conclude whether these classes are normalized or not. The most appealing classes are the class of nonlinear Schrödinger equations with potentials and modular nonlinearities and the class of generalized Schrödinger equations with complex-valued and, in general, coefficients of Laplacian term. Both these classes are not normalized. The first is partitioned into an infinite number of disjoint normalized subclasses of three kinds: logarithmic nonlinearity, power nonlinearity and general modular nonlinearity. The properties of the Lie invariance algebras of equations from each subclass are studied for arbitrary space dimension n, and the complete group classification is carried out for each subclass in dimension (1+2). The second class is successively reduced into subclasses until we reach the subclass of (1+1)-dimensional linear Schrödinger equations with variable mass, which also turns out to be non-normalized. We prove that this class is mapped by a family of point transformations to the class of (1+1)-dimensional linear Schrödinger equations with unique constant mass.</p>
  • Strömberg, Ann-Brith, 1961-, et al. (författare)
  • Mixed-Integer Linear Optimization: Primal–Dual Relations and Dual Subgradient and Cutting-Plane Methods
  • 2020
  • Ingår i: Numerical Nonsmooth Optimization. - 978-3-030-34910-3 ; s. 499-547
  • Bokkapitel (övrigt vetenskapligt)abstract
    • This chapter presents several solution methodologies for mixed-integer linear optimization, stated as mixed-binary optimization problems, by means of Lagrangian duals, subgradient optimization, cutting-planes, and recovery of primal solutions. It covers Lagrangian duality theory for mixed-binary linear optimization, a problem framework for which ultimate success—in most cases—is hard to accomplish, since strong duality cannot be inferred. First, a simple conditional subgradient optimization method for solving the dual problem is presented. Then, we show how ergodic sequences of Lagrangian subproblem solutions can be computed and used to recover mixed-binary primal solutions. We establish that the ergodic sequences accumulate at solutions to a convexified version of the original mixed-binary optimization problem. We also present a cutting-plane approach to the Lagrangian dual, which amounts to solving the convexified problem by Dantzig–Wolfe decomposition, as well as a two-phase method that benefits from the advantages of both subgradient optimization and Dantzig–Wolfe decomposition. Finally, we describe how the Lagrangian dual approach can be used to find near optimal solutions to mixed-binary optimization problems by utilizing the ergodic sequences in a Lagrangian heuristic, to construct a core problem, as well as to guide the branching in a branch-and-bound method. The chapter is concluded with a section comprising notes, references, historical downturns, and reading tips.
  • Balkanova, Olga, et al. (författare)
  • Convolution formula for the sums of generalized Dirichlet L-functions
  • 2019
  • Ingår i: Revista Matematica Iberoamericana. - 0213-2230. ; 35:7, s. 1973-1995
  • Tidskriftsartikel (refereegranskat)abstract
    • Using the Kuznetsov trace formula, we prove a spectral decomposition for the sums of generalized Dirichlet L-functions. Among applications are an explicit formula relating norms of prime geodesics to moments of symmetric square L-functions and an asymptotic expansion for the average of central values of generalized Dirichlet L-functions.
  • Boucksom, S., et al. (författare)
  • Uniform K-stability and asymptotics of energy functionals in Kähler geometry
  • 2019
  • Ingår i: Journal of the European Mathematical Society. - 1435-9855. ; 21:9, s. 2905-2944
  • Tidskriftsartikel (refereegranskat)abstract
    • Consider a polarized complex manifold (X, L) and a ray of positive metrics on L defined by a positive metric on a test configuration for (X, L). For many common functionals in Kähler geometry, we prove that the slope at infinity along the ray is given by evaluating the non-Archimedean version of the functional (as defined in our earlier paper [BHJ17]) at the non-Archimedean metric on L defined by the test configuration. Using this asymptotic result, we show that coercivity of the Mabuchi functional implies uniform K-stability, as defined in [Der15, BHJ17]. As a partial converse, we show that uniform K-stability implies coercivity of the Mabuchi functional when restricted to Bergman metrics.
  • Wang, Siyang, et al. (författare)
  • 2018
  • Ingår i: Mathematics of Computation. - 0025-5718. ; 87:314, s. 2737-2763
  • Tidskriftsartikel (refereegranskat)abstract
    • When using a finite difference method to solve an initial-boundaryvalue problem, the truncation error is often of lower order at a few grid points near boundaries than in the interior. Normal mode analysis is a powerful tool to analyze the effect of the large truncation error near boundaries on the overall convergence rate, and has been used in many research works for different equations. However, existing work only concerns problems in one space dimension. In this paper, we extend the analysis to problems in two space dimensions. The two dimensional analysis is based on a diagonalization procedure that decomposes a two dimensional problem to many one dimensional problems of the same type. We present a general framework of analyzing convergence for such one dimensional problems, and explain how to obtain the result for the corresponding two dimensional problem. In particular, we consider two kinds of truncation errors in two space dimensions: the truncation error along an entire boundary, and the truncation error localized at a few grid points close to a corner of the computational domain. The accuracy analysis is in a general framework, here applied to the second order wave equation. Numerical experiments corroborate our accuracy analysis.
  • Björnberg, Jakob E., 1983-, et al. (författare)
  • Characterising random partitions by random colouring
  • 2020
  • Ingår i: Electronic Communications in Probability. - 1083-589X. ; 25
  • Tidskriftsartikel (refereegranskat)abstract
    • Let (X-1, X-2, ...) be a random partition of the unit interval [0, 1], i.e. X-i &gt;= 0 and Sigma(i &gt;= 1) X-i = 1, and let (epsilon(1), epsilon(2), ...) be i.i.d. Bernoulli random variables of parameter p is an element of (0, 1). The Bernoulli convolution of the partition is the random variable Z = Sigma(i &gt;= 1) epsilon X-i(i). The question addressed in this article is: Knowing the distribution of Z for some fixed p is an element of (0, 1), what can we infer about the random partition (X-1, X-2, ...)? We consider random partitions formed by residual allocation and prove that their distributions are fully characterised by their Bernoulli convolution if and only if the parameter p is not equal to 1/2.
  • Lundengård, Karl, 1987- (författare)
  • Generalized Vandermonde matrices and determinants in electromagnetic compatibility
  • 2017
  • Licentiatavhandling (övrigt vetenskapligt)abstract
    • <p>Matrices whose rows (or columns) consists of monomials of sequential powers are called Vandermonde matrices and can be used to describe several useful concepts and have properties that can be helpful for solving many kinds of problems. In this thesis we will discuss this matrix and some of its properties as well as a generalization of it and how it can be applied to curve fitting discharge current for the purpose of ensuring electromagnetic compatibility.</p><p>In the first chapter the basic theory for later chapters is introduced. This includes the Vandermonde matrix and some of its properties, history, applications and generalizations, interpolation and regression problems, optimal experiment design and modelling of electrostatic discharge currents with the purpose to ensure electromagnetic compatibility.</p><p>The second chapter focuses on finding the extreme points for the determinant for the Vandermonde matrix on various surfaces including spheres, ellipsoids, cylinders and tori. The extreme points are analysed in three dimensions or more.</p><p>The third chapter discusses fitting a particular model called the p-peaked Analytically Extended Function (AEF) to data taken either from a standard for electromagnetic compatibility or experimental measurements. More specifically the AEF will be fitted to discharge currents from the IEC 62305-1 and IEC 61000-4-2 standards for lightning protection and electrostatic discharge immunity as well as some experimentally measured data of similar phenomena.</p>
  • Balkanova, Olga, et al. (författare)
  • Prime geodesic theorem in the 3-dimensional hyperbolic space
  • 2019
  • Ingår i: Transactions of the American Mathematical Society. - 0002-9947 .- 1088-6850. ; 372:8, s. 5355-5374
  • Tidskriftsartikel (refereegranskat)abstract
    • For Γ a cofinite Kleinian group acting on H3, we study the prime geodesic theorem on M = Γ\H3, which asks about the asymptotic behavior of lengths of primitive closed geodesics (prime geodesics) on M. Let EΓ(X) be the error in the counting of prime geodesics with length at most log X. For the Picard manifold, Γ = PSL(2, Z[i]), we improve the classical bound of Sarnak, EΓ(X) = O(X5/3+e), to EΓ(X) = O(X13/8+e). In the process we obtain a mean subconvexity estimate for the Rankin-Selberg L-function attached to Maass-Hecke cusp forms. We also investigate the second moment of EΓ(X) for a general cofinite group Γ, and we show that it is bounded by O(X16/5+e).
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