| 1. |
- Berndtsson, Bo, 1950
(författare)
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Probability measures associated to geodesics in the space of kähler metrics
- 2018
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Ingår i: Springer Proceedings in Mathematics and Statistics. - Cham : Springer International Publishing. - 2194-1017 .- 2194-1009. ; 269, s. 395-419
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Konferensbidrag (refereegranskat)abstract
- We associate certain probability measures on R to geodesics in the space HL of positively curved metrics on a line bundle L, and to geodesics in the finite dimensional symmetric space of hermitian norms on H0(X, kL). We prove that the measures associated to the finite dimensional spaces converge weakly to the measures related to geodesics in HL as k goes to infinity. The convergence of second order moments implies a recent result of Chen and Sun on geodesic distances in the respective spaces, while the convergence of first order moments gives convergence of Donaldson’s Z-functional to the Aubin–Yau energy. We also include a result on approximation of infinite dimensional geodesics by Bergman kernels which generalizes work of Phong and Sturm.
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| 2. |
- Xia, Jiacheng, 1991
(författare)
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Vector-valued Eisenstein series of congruence types and their products
- 2019
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)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.
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| 3. |
- Kurujyibwami, Celestin
(författare)
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Admissible transformations and the group classification of Schrödinger equations
- 2017
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)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.
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| 4. |
- Berman, Robert, 1976
(författare)
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On the strict convexity of the K-energy
- 2019
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Ingår i: Pure and Applied Mathematics Quarterly. - 1558-8599 .- 1558-8602. ; 15:4, s. 983-999
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Tidskriftsartikel (refereegranskat)abstract
- Let (X, L) be a polarized projective complex manifold. We show, by a simple toric one-dimensional example, that Mabuchi's K-energy functional on the geodesically complete space of bounded positive (1, 1)-forms in c(1)(L), endowed with the Mabuchi-Donaldson-Semmes metric, is not strictly convex modulo automorphisms. However, under some further assumptions the strict convexity in question does hold in the toric case. This leads to a uniqueness result saying that a finite energy minimizer of the K-energy (which exists on any toric polarized manifold (X, L) which is uniformly K-stable) is uniquely determined modulo automorphisms under the assumption that there exists some minimizer with strictly positive curvature current.
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| 5. |
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| 6. |
- Wang, Siyang, et al.
(författare)
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CONVERGENCE OF FINITE DIFFERENCE METHODS FOR THE WAVE EQUATION IN TWO SPACE DIMENSIONS
- 2018
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Ingår i: Mathematics of Computation. - : American Mathematical Society (AMS). - 0025-5718 .- 1088-6842. ; 87:314, s. 2737-2763
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Tidskriftsartikel (refereegranskat)abstract
- When using a finite difference method to solve an initial-boundary-value 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.
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| 7. |
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Algebra, Geometry, and Mathematical Physics 2010
- 2012
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Ingår i: Journal of Physics: Conference Series. - : IOP Publishing. - 1742-6596 .- 1742-6588.
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Samlingsverk (redaktörskap) (övrigt vetenskapligt/konstnärligt)abstract
- This proceedings volume presents results obtained by the participants of the 6th Baltic–Nordic workshop 'Algebra, Geometry, and Mathematical Physics (AGMP-6)' held at the Sven Lovén Centre for Marine Sciences in Tjärnö, Sweden on October 25–30, 2010. The Baltic–Nordic Network AGMP 'Algebra, Geometry, and Mathematical Physics' http://www.agmp.eu was created in 2005 on the initiative of two Estonian universities and two Swedish universities: Tallinn University of Technology represented by Eugen Paal (coordinator of the network), Tartu University represented by Viktor Abramov, Lund University represented by Sergei Silvestrov, and Chalmers University of Technology and the University of Gothenburg represented by Alexander Stolin. The goal was to promote international and interdisciplinary cooperation between scientists and research groups in the countries of the Baltic–Nordic region in mathematics and mathematical physics, with special emphasis on the important role played by algebra and geometry in modern physics, engineering and technologies. The main activities of the AGMP network consist of a series of regular annual international workshops, conferences and research schools. The AGMP network also constitutes an important educational forum for scientific exchange and dissimilation of research results for PhD students and Postdocs. The network has expanded since its creation, and nowadays its activities extend beyond countries in the Baltic–Nordic region to universities in other European countries and participants from elsewhere in the world. As one of the important research-dissimilation outcomes of its activities, the network has a tradition of producing high-quality research proceedings volumes after network events, publishing them with various international publishers.
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| 8. |
- Lundengård, Karl, 1987-
(författare)
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Generalized Vandermonde matrices and determinants in electromagnetic compatibility
- 2017
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)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.
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| 9. |
- Arnlind, Joakim, et al.
(författare)
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Construction of n-Lie algebras and n-ary Hom-Nambu-Lie algebras
- 2011
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Ingår i: Journal of Mathematical Physics. - : AIP Publishing. - 0022-2488 .- 1089-7658. ; 52:12, s. 123502-
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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]
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| 10. |
- Balkanova, Olga, 1988, et al.
(författare)
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Prime geodesic theorem in the 3-dimensional hyperbolic space
- 2019
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Ingår i: Transactions of the American Mathematical Society. - : American Mathematical Society (AMS). - 0002-9947 .- 1088-6850. ; 372:8, s. 5355-5374
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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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