| 11. |
- 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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| 12. |
- Brandes, Julia, et al.
(författare)
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On generating functions in additive number theory, II: lower-order terms and applications to PDEs
- 2021
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Ingår i: Mathematische Annalen. - : Springer Science and Business Media LLC. - 0025-5831 .- 1432-1807. ; 379, s. 347-76
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Tidskriftsartikel (refereegranskat)abstract
- We obtain asymptotics for sums of the form Sigma(p)(n=1) e(alpha(k) n(k) + alpha(1)n), involving lower order main terms. As an application, we show that for almost all alpha(2) is an element of [0, 1) one has sup(alpha 1 is an element of[0,1)) | Sigma(1 <= n <= P) e(alpha(1)(n(3) + n) + alpha(2)n(3))| << P3/4+epsilon, and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schrodinger and Airy equations.
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| 13. |
- Ericsson, Niklas, 1968
(författare)
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A Framework for Approximation of the Stokes Equations in an Axisymmetric Domain
- 2021
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Ingår i: Computational Methods in Applied Mathematics. - : Walter de Gruyter GmbH. - 1609-4840 .- 1609-9389. ; 21:4
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Tidskriftsartikel (refereegranskat)abstract
- We develop a framework for solving the stationary, incompressible Stokes equations in an axisymmetric domain. By means of Fourier expansion with respect to the angular variable, the three-dimensional Stokes problem is reduced to an equivalent, countable family of decoupled two-dimensional problems. By using decomposition of three-dimensional Sobolev norms, we derive natural variational spaces for the two-dimensional problems, and show that the variational formulations are well-posed. We analyze the error due to Fourier truncation and conclude that, for data that are sufficiently regular, it suffices to solve a small number of two-dimensional problems. © 2021 Ericsson, published by De Gruyter 2021.
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| 14. |
- Gourevitch, D., et al.
(författare)
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EULERIANITY OF FOURIER COEFFICIENTS OF AUTOMORPHIC FORMS
- 2021
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Ingår i: Representation Theory. - : American Mathematical Society (AMS). - 1088-4165. ; 25, s. 481-507
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Tidskriftsartikel (refereegranskat)abstract
- We study the question of Eulerianity (factorizability) for Fourier coefficients of automorphic forms, and we prove a general transfer theorem that allows one to deduce the Eulerianity of certain coefficients from that of another coefficient. We also establish a `hidden' invariance property of Fourier coefficients. We apply these results to minimal and next-to-minimal automorphic representations, and deduce Eulerianity for a large class of Fourier and Fourier-Jacobi coefficients. In particular, we prove Eulerianity for parabolic Fourier coefficients with characters of maximal rank for a class of Eisenstein series in minimal and next-to-minimal representations of groups of ADE-type that are of interest in string theory.
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| 15. |
- Kim, Inkang, et al.
(författare)
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Toledo invariant of lattices in SU(2,1) via symmetric square
- 2020
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Ingår i: Journal of Mathematical Physics. - : AIP Publishing. - 1089-7658 .- 0022-2488. ; 61:11
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we address the issue of the quaternionic Toledo invariant to study the character variety of two-dimensional complex hyperbolic uniform lattices into SU(n, 2), n ≥ 4. We construct four distinct representations to prove that the character variety contains at least seven distinct components. We also show the existence of holomorphic horizontal lift to various period domains of SU(n, 2).
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| 16. |
- Malik, Adam, 1991
(författare)
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Mathematical Modelling of Cell Migration and Polarization
- 2020
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Cell migration plays a fundamental role in both development and disease. It is a complex process during which cells interact with one another and with their local environment. Mathematical modelling offers tools to investigate such processes and can give insights into the underlying biological details, and can also guide new experiments. The first two papers of this thesis are concerned with modelling durotaxis, which is the phenomena where cells migrate preferentially up a stiffness gradient. Two distinct mechanisms which potentially drive durotaxis are investigated. One is based on the hypothesis that adhesion sites of migrating cells become reinforced and have a longer lifespan on stiffer substrates. The second mechanism is based on cells being able to generate traction forces, the magnitude of which depend on the stiffness of the substrate. We find that both mechanisms can indeed give rise to biased migration up a stiffness gradient. Our results encourages new experiments which could determine the importance of the two mechanisms in durotaxis. The third paper is devoted to a population-level model of cancer cells in the brain of mice. The model incorporates diffusion tensor imaging data, which is used to guide the migration of the cells. Model simulations are compared to experimental data, and highlights the model’s difficulty in producing irregular growth patterns observed in the experiments. As a consequence, the findings encourage further model development. The fourth paper is concerned with modelling cell polarization, in the absence of environmental cues, referred to as spontaneous symmetry breaking. Polarization is an important part of cell migration, but also plays a role during division and differentiation. The model takes the form of a reaction diffusion system in 3D and describes the spatio-temporal evolution of three forms of Cdc42 in the cell. The model is able to produce biologically relevant patterns, and numerical simulations show how model parameters influence key features such as pattern formation and time to polarization.
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| 17. |
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| 18. |
- 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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| 19. |
- Zhao, D., et al.
(författare)
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Low Phase-Rank Approximation
- 2022
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Ingår i: Linear Algebra and Its Applications. - : Elsevier BV. - 0024-3795. ; 639, s. 177-204
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Tidskriftsartikel (refereegranskat)abstract
- In this paper, we propose and solve low phase-rank approximation problems, which serve as a counterpart to the well-known low-rank approximation problem and the Schmidt-Mirsky theorem. It is well known that a nonzero complex number can be specified by its gain and phase, and while it is generally accepted that the gains of a matrix may be defined by its singular values, there is no widely accepted definition for its phases. In this work, we consider sectorial matrices, whose numerical ranges do not contain the origin, and adopt the canonical angles of such matrices as their phases. Similarly to the rank of a matrix being defined as the number of its nonzero singular values, we define the phase-rank of a sectorial matrix as the number of its nonzero phases. While a low-rank approximation problem is associated with the matrix arithmetic mean, it turns out that a natural parallel for the low phase-rank approximation problem is to use the matrix geometric mean to measure the approximation error. Importantly, we derive a majorization inequality between the phases of the geometric mean and the arithmetic mean of the phases, similarly to the Ky-Fan inequality for eigenvalues of Hermitian matrices. A characterization of the solutions to the proposed problem, with the same flavor as the Schmidt-Mirsky theorem, is then obtained in the case where both the objective matrix and the approximant are restricted to be positive-imaginary. In addition, we provide an alternative formulation of the low phase-rank approximation problem using geodesic distances between sectorial matrices. The two formulations give rise to the exact same set of solutions when the involved matrices are additionally assumed to be unitary. © 2022 Elsevier Inc.
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| 20. |
- Karlsson, Carl-Joar, 1995
(författare)
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Game theory and applications
- 2021
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- Individual-level interactions and decisions spread though populations and change the collective-level dynamics in an intricate way. Nevertheless, game theory is well suited for the unification of these viewpoints. This thesis introduces the works The strength of diversity and Decisions and disease: a mechanism for the evolution of cooperation, which show that games have broad applications in population dynamics modeling. In The strength of diversity we calculate the equilibrium strategies of the so-called game of teams. The game of teams is an individual-level competition between teams, and a team's strategy in this context is a distribution of strength over the team members. It turns out that the equilibrium strategies are flat distributions or 'alternating' flat distributions whenever there exists equilibrium strategies. In Decisions and disease: a mechanism for the evolution of cooperation we combine the classic SIR and SIS models from epidemiology with the prisoner's dilemma game. The transmission rate is computed as the average over defecting and cooperating individuals, and the individuals are subjected to a replicator equation that takes into account the portion of infectious members of the population. We compute the steady state solutions and interpret the results.
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