| 1. |
- Avelin, Helen, 1975-
(författare)
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Computations of Automorphic Functions on Fuchsian Groups
- 2007
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis consists of four papers which all deal with computations of automorphic functions on cofinite Fuchsian groups. In the first paper we develop an algorithm for numerical computation of the Eisenstein series. We focus in particular on the computation of the poles of the Eisenstein series. Using our numerical methods we study the spectrum of the Laplace-Beltrami operator as the surface is being deformed. Numerical evidence of the destruction of Γ0(5)-cusp forms is presented. In the second paper we use the algorithm described in the first paper. We present numerical investigations of the value distribution and distribution of Fourier coefficients of the Eisenstein series E(z;s) on arithmetic and non-arithmetic Fuchsian groups. Our numerics indicate a Gaussian limit value distribution for a real-valued rotation of E(z;s) as Re s=1/2, Im s → ∞ and also, on non-arithmetic groups, a complex Gaussian limit distribution for E(z;s) when Re s > 1/2 near 1/2 and Im s → ∞, at least if we allow Re s → 1/2 at some rate. Furthermore, on non-arithmetic groups and for fixed s with Re s ≥ 1/2 near 1/2, our numerics indicate a Gaussian limit distribution for the appropriately normalized Fourier coefficients.In the third paper we develop algorithms for computations of Green's function and its Fourier coefficients, Fn(z;s), on Fuchsian groups with one cusp. Also an analog of a Rankin-Selberg bound for Fn(z;s) is presented.In the fourth paper we use the algorithms described in the third paper. We present some examples of numerical investigations of the value distribution of the Green's function and of its Fourier coefficients on PSL(2,Z). We also discuss the appearance of pseudo cusp forms in a numerical experiment by Hejhal.
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| 2. |
- Barrio, Roberto, et al.
(författare)
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A database of rigorous and high-precision periodic orbits of the Lorenz model
- 2015
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Ingår i: Computer Physics Communications. - : Elsevier BV. - 0010-4655 .- 1879-2944. ; 194, s. 76-83
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Tidskriftsartikel (refereegranskat)abstract
- A benchmark database of very high-precision numerical and validated initial conditions of periodic orbits for the Lorenz model is presented. This database is a "computational challenge" and it provides the initial conditions of all periodic orbits of the Lorenz model up to multiplicity 10 and guarantees their existence via computer-assisted proofs methods, The orbits are computed using high-precision arithmetic and mixing several techniques resulting in 1000 digits of precision on the initial conditions of the periodic orbits, and intervals of size 10100 that prove the existence of each orbit. Program summary Program title: Lorenz-Database Catalogue identifier: AEWM_v1_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/AEWM_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 8515 No. of bytes in distributed program, including test data, etc.: 6964501 Distribution format: tar.gz Programming language: Data. Computer: Any computer. Operating system: Any. RAM: Database, no requirements Classification: 4.3, 4.12. Nature of problem: Database of all periodic orbits of the Lorenz model up to multiplicity 10 with 1000 precision digits. Solution method: Advanced search methods for locating unstable periodic orbits combined with the Taylor series method for multiple precision integration of ODEs and interval methods for providing Computer-Assisted proofs of the periodic orbits. Unusual features: The database gives 100 digits rigorously proved using Computer-Assisted techniques and 1000 digits using an optimal adaptive Taylor series method. Running time: Not Applicable.
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| 3. |
- Bartha, Ferenc A., et al.
(författare)
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Fixed points of a destabilized Kuramoto-Sivashinsky equation
- 2015
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Ingår i: Applied Mathematics and Computation. - : Elsevier BV. - 0096-3003 .- 1873-5649. ; 266, s. 339-349
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Tidskriftsartikel (refereegranskat)abstract
- We consider the family of destabilized Kuramoto-Sivashinsky equations in one spatial dimension u(t) + nu u(xxx) + beta u(xx) + gamma uu(x) - alpha u for alpha, nu >= 0 and beta, gamma is an element of R. For certain parameter values, shock like stationary solutions have been numerically observed. In this work we verify the existence of several such solutions using the framework of self consistent bounds and validated numerics. Published by Elsevier Inc.
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| 4. |
- Belova, Anna, 1987-
(författare)
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Computational dynamics – real and complex
- 2017
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The PhD thesis considers four topics in dynamical systems and is based on one paper and three manuscripts.In Paper I we apply methods of interval analysis in order to compute the rigorous enclosure of rotation number. The described algorithm is supplemented with a method of proving the existence of periodic points which is used to check rationality of the rotation number.In Manuscript II we provide a numerical algorithm for computing critical points of the multiplier map for the quadratic family (i.e., points where the derivative of the multiplier with respect to the complex parameter vanishes).Manuscript III concerns continued fractions of quadratic irrationals. We show that the generating function corresponding to the sequence of denominators of the best rational approximants of a quadratic irrational is a rational function with integer coefficients. As a corollary we can compute the Lévy constant of any quadratic irrational explicitly in terms of its partial quotients.Finally, in Manuscript IV we develop a method for finding rigorous enclosures of all odd periodic solutions of the stationary Kuramoto-Sivashinsky equation. The problem is reduced to a bounded, finite-dimensional constraint satisfaction problem whose solution gives the desired information about the original problem. Developed approach allows us to exclude the regions in L2, where no solution can exist.
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| 6. |
- Dahne, Joel, 1994-
(författare)
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Computer Assisted Studies in Fluid Mechanics and Spectral Geometry
- 2024
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis contains four papers in the area of partial differential equations. The first two papers are related to spectral geometry and the last two papers to fluid mechanics. A common theme of the papers is that they make use of computer assisted methods in the proofs.Paper I concerns the computation of very precise enclosures of eigenvalues of the Laplace-Beltrami operator on spherical triangles. The interest in these eigenvalues comes from a connection with the combinatorial problem of studying discrete random walks.Paper II gives a concrete counterexample to Payne's nodal line conjecture. The conjecture is concerned with the existence of bounded planar domains for which the second eigenfunction of the Dirichlet Laplacian has a nodal line that doesn't touch the boundary. The paper gives an explicit domain for which it is proved that the nodal line doesn't touch the boundary.Paper III and IV both prove the existence of a certain type of solution known as a highest cusped traveling wave. Paper III deals with the Burgers-Hilbert equation and Paper IV with a family of fractional Korteweg-de Vries equations. The existence is asserted by constructing an explicit approximation of the solution and proving the existence of an exact solution nearby with the use of a fixed point formulation. The proof not only establishes the existence, but also determines the precise asymptotic behavior of the waves near the cusp.
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| 7. |
- Dahne, Joel, et al.
(författare)
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Enclosing all zeros of a system of analytic functions
- 2019
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Ingår i: Applied Mathematics and Computation. - : Elsevier. - 0096-3003 .- 1873-5649. ; 348, s. 513-522
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Tidskriftsartikel (refereegranskat)abstract
- We present a rigorous numerical method for location of simple zeros of a system of two analytic functions in a rectangular cuboid domain based on the logarithmic integral. We compare this to a simpler, also rigorous, method based on bisection. The latter is determined to be more efficient in the examples considered. This is mainly due to inefficient methods for computing the logarithmic integral occurring in the former method. (C) 2018 Elsevier Inc. All rights reserved.
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| 8. |
- Figueras, Jordi-Lluis, et al.
(författare)
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Computer-assisted techniques for the verification of the Chebyshev property of Abelian integrals
- 2013
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Ingår i: Journal of Differential Equations. - : Elsevier BV. - 0022-0396 .- 1090-2732. ; 254:8, s. 3647-3663
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Tidskriftsartikel (refereegranskat)abstract
- We develop techniques for the verification of the Chebyshev property of Abelian integrals. These techniques are a combination of theoretical results, analysis of asymptotic behavior of Wronskians, and rigorous computations based on interval arithmetic. We apply this approach to tackle a conjecture formulated by Dumortier and Roussarie in [F. Dumortier, R Roussarie, Birth of canard cycles, Discrete Contin. Dyn. Syst. 2 (2009) 723-781], which we are able to prove for q <= 2.
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