| 1. |
- 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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| 2. |
- Elfverson, Daniel, et al.
(författare)
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Multiscale methods for problems with complex geometry
- 2017
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier BV. - 0045-7825 .- 1879-2138. ; 321, s. 103-123
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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.
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| 3. |
- Khesin, Boris, et al.
(författare)
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Newton's Equation on Diffeomorphisms and Densities
- 2017
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Ingår i: Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics). - Cham : Springer International Publishing. - 1611-3349 .- 0302-9743. - 9783319684451 ; 10589, s. 873-873
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Konferensbidrag (refereegranskat)abstract
- We develop a geometric framework for Newton-type equations on the infinite-dimensional configuration space of probability densities. It can be viewed as a second order analogue of the "Otto calculus" framework for gradient flow equations. Namely, for an n-dimensional manifold M we derive Newton's equations on the group of diffeomorphisms Diff(M) and the space of smooth probability densities Dens(M), as well as describe the Hamiltonian reduction relating them. For example, the compressible Euler equations are obtained by a Poisson reduction of Newton's equation on Diff(M) with the symmetry group of volume-preserving diffeomorphisms, while the Hamilton-Jacobi equation of fluid mechanics corresponds to potential solutions. We also prove that the Madelung transform between Schrodinger-type and Newton's equations is a symplectomorphism between the corresponding phase spaces T* Dens(M) and PL2 (M, C). This improves on the previous symplectic submersion result of von Renesse [1]. Furthermore, we prove that the Madelung transform is a Kahler map provided that the space of densities is equipped with the (prolonged) Fisher-Rao information metric and describe its dynamical applications. This geometric setting for the Madelung transform sheds light on the relation between the classical Fisher-Rao metric and its quantum counterpart, the Bures metric. In addition to compressible Euler, Hamilton-Jacobi, and linear and nonlinear Schrodinger equations, the framework for Newton equations encapsulates Burgers' inviscid equation, shallow water equations, two-component and mu-Hunter-Saxton equations, the Klein-Gordon equation, and infinite-dimensional Neumann problems.
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| 4. |
- Nyagahakwa, Venuste, et al.
(författare)
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Sets with the Baire property in topologies formed from a given topology and ideals of sets
- 2017
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Ingår i: Questions and answers in General Topology. - Osaka : Symposium of General Topology / Iso Shinpojumu. - 0918-4732. ; 35:1, s. 59-76
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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.
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| 5. |
- Pejlare, Johanna, 1976
(författare)
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On the relationships between the geometric and the algebraic ideas in Duhre’s textbooks of mathematics, as reflected via Book II of Euclid’s Elements
- 2017
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Ingår i: “DIG WHERE YOU STAND” 4. Proceedings of the Fourth International Conference on the History of Mathematics Education. 23-26 September, 2015. Torino, Italy. - Roma : Edizioni Nuova Cultura. - 9788868128647
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Konferensbidrag (refereegranskat)abstract
- The present article explores the relationships between the geometric and algebraic ideas presented in Anders Gabriel Duhre’s mathematics textbooks. Of particular interest is Book II of Euclid’s Elements as presented by Duhre in his textbook on geometry from 1721. We consider in detail Duhre’s two versions of Proposition II.5, dealing with straight lines cut into equal and unequal parts, as well as the two proofs of the propositions that he presents. Duhre’s formulations are slightly different from traditional geometric formulations, as he moved away from a purely geometrical context towards an algebraic one. Duhre established Proposition II.5 using algebra in Descartes’ notation as well as in the notation of Wallis and Oughtred. Duhre ́s reason for introducing algebra in Book II of Euclid’s Elements was to obtain convenience in calculations, as well as the possibility to generalize results to different kinds of quantities.
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| 6. |
- Modin, Klas, 1979
(författare)
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Geometry of Matrix Decompositions Seen Through Optimal Transport and Information Geometry
- 2017
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Ingår i: Journal of Geometric Mechanics. - : American Institute of Mathematical Sciences (AIMS). - 1941-4889 .- 1941-4889 .- 1941-4897. ; 9:3, s. 335-390
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Tidskriftsartikel (refereegranskat)abstract
- The space of probability densities is an infinite-dimensional Riemannian manifold, with Riemannian metrics in two flavors: Wasserstein and Fisher--Rao. The former is pivotal in optimal mass transport (OMT), whereas the latter occurs in information geometry---the differential geometric approach to statistics. The Riemannian structures restrict to the submanifold of multivariate Gaussian distributions, where they induce Riemannian metrics on the space of covariance matrices. Here we give a systematic description of classical matrix decompositions (or factorizations) in terms of Riemannian geometry and compatible principal bundle structures. Both Wasserstein and Fisher-Rao geometries are discussed. The link to matrices is obtained by considering OMT and information geometry in the category of linear transformations and multivariate Gaussian distributions. This way, OMT is directly related to the polar decomposition of matrices, whereas information geometry is directly related to the QR, Cholesky, spectral, and singular value decompositions. We also give a coherent description of gradient flow equations for the various decompositions; most flows are illustrated in numerical examples. The paper is a combination of previously known and original results. As a survey it covers the Riemannian geometry of OMT and polar decompositions (smooth and linear category), entropy gradient flows, and the Fisher--Rao metric and its geodesics on the statistical manifold of multivariate Gaussian distributions. The original contributions include new gradient flows associated with various matrix decompositions, new geometric interpretations of previously studied isospectral flows, and a new proof of the polar decomposition of matrices based an entropy gradient flow.
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| 7. |
- Andersson, Mats, 1957, et al.
(författare)
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Segre numbers, a generalized King formula, and local intersections
- 2017
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Ingår i: Journal für die Reine und Angewandte Mathematik. - : Walter de Gruyter GmbH. - 1435-5345 .- 0075-4102. ; 728:728, s. 105-136
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Tidskriftsartikel (refereegranskat)abstract
- Let $\mathcal{J}$ be an ideal sheaf on a reduced analytic space $X$ with zero set $Z$.We show that the Lelong numbers of the restrictions to $Z$ of certain generalized Monge–Ampère products $(dd^c \log |f|^2)^k$, where $f$ is a tuple of generators of $\mathcal{J}$, coincide with theso-called Segre numbers of $\mathcal{J}$, introduced independently by Tworzewski, Achilles–Manaresi,and Gaffney–Gassler. More generally we show that these currents satisfy a generalization ofthe classical King formula that takes into account fixed and moving components of Vogelcycles associated with $\mathcal{J}$. A basic tool is a new calculus for products of positive currents ofBochner–Martinelli type. We also discuss connections to intersection theory.
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| 8. |
- Gimperlein, Heiko, et al.
(författare)
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Nonclassical Spectral Asymptotics and Dixmier Traces: from Circles to Contact Manifolds
- 2017
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Ingår i: Forum of Mathematics, Sigma. - : Cambridge University Press (CUP). - 2050-5094. ; 5, s. Article no e3 -
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Tidskriftsartikel (refereegranskat)abstract
- We consider the spectral behavior and noncommutative geometry of commutators [P, f], where P is an operator of order 0 with geometric origin and f a multiplication operator by a function. When f is Holder continuous, the spectral asymptotics is governed by singularities. We study precise spectral asymptotics through the computation of Dixmier traces; such computations have only been considered in less singular settings. Even though a Weyl law fails for these operators, and no pseudodifferential calculus is available, variations of Connes' residue trace theorem and related integral formulas continue to hold. On the circle, a large class of nonmeasurable Hankel operators is obtained from Holder continuous functions f, displaying a wide range of nonclassical spectral asymptotics beyond the Weyl law. The results extend from Riemannian manifolds to contact manifolds and noncommutative tori.
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| 9. |
- Johansson, Petter, et al.
(författare)
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A Ronkin type function for coamoebas
- 2017
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Ingår i: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 27:1, s. 643-670
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Tidskriftsartikel (refereegranskat)abstract
- The Ronkin function plays a fundamental role in the theory of amoebas. We introduce an analogue of the Ronkin function in the setting of coamoebas. It turns out to be closely related to a certain toric arrangement known as the shell of the coamoeba and we use our Ronkin type function to obtain some properties of it.
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| 10. |
- McCormick, Stephen
(författare)
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The Asymptotically Flat Scalar-Flat Yamabe Problem with Boundary
- 2017
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Ingår i: Journal of Geometric Analysis. - : SPRINGER. - 1050-6926 .- 1559-002X. ; 27:3, s. 2269-2277
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Tidskriftsartikel (refereegranskat)abstract
- We consider two cases of the asymptotically flat scalar-flat Yamabe problem on a non-compact manifold with inner boundary in dimension n >= 3. First, following arguments of Cantor and Brill in the compact case, we show that given an asymptotically flat metric g, there is a conformally equivalent asymptotically flat scalar-flat metric that agrees with g on the boundary. We then replace the metric boundary condition with a condition on the mean curvature: given a function f on the boundary that is not too large, we show that there is an asymptotically flat scalar-flat metric, conformally equivalent to g whose boundary mean curvature is given by f. The latter case involves solving an elliptic PDE with critical exponent using the method of sub- and supersolutions. Both results require the usual assumption that the Sobolev quotient is positive.
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