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  • Kurujyibwami, Celestin (författare)
  • Admissible transformations and the group classification of Schrödinger equations
  • 2017
  • Doktorsavhandling (övrigt vetenskapligt)abstract
    • 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.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.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.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.
  • Andersson, Fredrik K., et al. (författare)
  • The mathematics of internet search engines
  • 2008
  • Ingår i: Acta Applicandae Mathematicae - An International Survey Journal on Applying Mathematics and Mathematical Applications. - Springer. - 0167-8019. ; 104:2, s. 211-242
  • Tidskriftsartikel (refereegranskat)abstract
    • This article presents a survey of techniques for ranking results in search engines, with emphasis on link-based ranking methods and the PageRank algorithm. The problem of selecting, in relation to a user search query, the most relevant documents from an unstructured source such as the WWW is discussed in detail. The need for extending classical information retrieval techniques such as boolean searching and vector space models with link-based ranking methods is demonstrated. The PageRank algorithm is introduced, and its numerical and spectral properties are discussed. The article concludes with an alternative means of computing PageRank, along with some example applications of this new method.
  • Lundengård, Karl, 1987- (författare)
  • Generalized Vandermonde matrices and determinants in electromagnetic compatibility
  • 2017
  • Licentiatavhandling (övrigt vetenskapligt)abstract
    • 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.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.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.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.
  • Arnlind, Joakim, et al. (författare)
  • Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras
  • 2011
  • Ingår i: Journal of Mathematical Physics. - 0022-2488. ; 52:12, s. 123502
  • Tidskriftsartikel (refereegranskat)abstract
    • As n-ary operations, generalizing Lie and Poisson algebras, arise in many different physical contexts, it is interesting to study general ways of constructing explicit realizations of such multilinear structures. Generically, they describe the dynamics of a physical system, and there is a need of understanding their quantization. Hom-Nambu-Lie algebras provide a framework that might be an appropriate setting in which n-Lie algebras (n-ary Nambu-Lie algebras) can be deformed, and their quantization studied. We present a procedure to construct (n + 1)-ary Hom-Nambu-Lie algebras from n-ary Hom-Nambu-Lie algebras equipped with a generalized trace function. It turns out that the implications of the compatibility conditions, that are necessary for this construction, can be understood in terms of the kernel of the trace function and the range of the twisting maps. Furthermore, we investigate the possibility of defining (n + k)-Lie algebras from n-Lie algebras and a k-form satisfying certain conditions. (C) 2011 American Institute of Physics. [doi:10.1063/1.3653197]
  • Elfverson, Daniel, et al. (författare)
  • Multiscale methods for problems with complex geometry
  • 2017
  • Ingår i: Computer Methods in Applied Mechanics and Engineering. - 00457825. ; 321, s. 103-123
  • Tidskriftsartikel (refereegranskat)abstract
    • We propose a multiscale method for elliptic problems on complex domains, e.g. domains with cracks or complicated boundary. For local singularities this paper also offers a discrete alternative to enrichment techniques such as XFEM. We construct corrected coarse test and trail spaces which takes the fine scale features of the computational domain into account. The corrections only need to be computed in regions surrounding fine scale geometric features. We achieve linear convergence rate in the energy norm for the multiscale solution. Moreover, the conditioning of the resulting matrices is not affected by the way the domain boundary cuts the coarse elements in the background mesh. The analytical findings are verified in a series of numerical experiments.
  • Nyagahakwa, Venuste, et al. (författare)
  • Sets with the Baire property in topologies formed from a given topology and ideals of sets
  • 2017
  • Ingår i: Questions and answers in General Topology. - Osaka : Symposium of General Topology / Iso Shinpojumu. - 0918-4732. ; 35:1, s. 59-76
  • Tidskriftsartikel (refereegranskat)abstract
    • Let X be a set, τ1, τ2 topologies on X and Bp(X, τi) the family of all subsets of X possessing the Baire property in (X, τi), i = 1, 2. In this paper we study conditions on τ1 and τ2 that imply a relationship (for example, inclusion or equality) between the families Bp(X, τ1) and Bp(X, τ2). We are mostly interested in the case where the topology τ2 is formed with the help of a local function defined by the topology τ1 and an ideal of sets I on X. We also consider several applications of the local function defined by the usual topology on the reals and the ideal of all meager sets there, for proving some known facts.
  • Gerlee, Philip, 1980-, et al. (författare)
  • Scientific Models
  • 2016
  • Bok (övrigt vetenskapligt)
  • Vasilis, Jonatan, 1981- (författare)
  • Harmonic measures
  • 2010
  • Doktorsavhandling (övrigt vetenskapligt)
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