| 1. |
- Andersson, Anders, 1957-
(author)
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Numerical Conformal Mappings for Waveguides
- 2009
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Doctoral thesis (other academic/artistic)abstract
- Acoustic or electro-magnetic scattering in a waveguide with varying direction and cross-section can be re-formulated as a two-dimensional scattering problem, provided that the variations take place in only one dimension at a time. By using the so-called Building Block Method, it is possible to construct the scattering properties of a combination of scatterers when the properties of each scatterer are known. Hence, variations in the waveguide geometry or in the boundary conditions can be treated one at a time. Using the Building Block Method, the problem takes the form of the Helmholtz equation for stationary waves in a waveguide of infinite length and with smoothly varying geometry and boundary conditions. A conformal mapping is used to transform the problem into a corresponding problem in a straight horizontal waveguide, and by expanding the field in Fourier trigonometric series, the problem can be reformulated as an infinite-dimensional ordinary differential equation. From this, numerically solvable differential equations for the reflection and transmission operators are derived. To be applicable in the Building Block Method, the numerical conformal mapping must be constructed such that the direction of the boundary curve can be controlled. At the channel ends ,it is an indispensable requirement, that the two boundary curves are (at least) asymptotically parallel and straight. Furthermore, to achieve bounded operators in the differential equations, the boundary curves must satisfy different regularity conditions, depending on the boundary conditions. In this work, several methods to accomplish such conformal mappings are presented. The Schwarz–Christoffel mapping, which is a natural starting point and for which also efficient numerical software exists, can be modified in different ways in order to achieve polygons with rounded corners. We present algorithms by which the parameters in the mappings can be determined after such modifications. We show also how the unmodified Schwarz–Christoffel mapping can be used for regions with a smooth boundary. This is done by constructing an appropriate outer polygon to the considered region.Finally, we introduce one method that is not Schwarz–Christoffel-related, by showing how one of the so-called zipper algorithms can be used for waveguides. Keywords: waveguides, building block method, numerical conformalmappings, Schwarz–Christoffel mapping, rounded corners method, approximate curve factors, outer polygon method, boundary curvature, zipper method, geodesic algorithm, acoustic wave scattering, electro-magnetic wave scattering
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| 2. |
- Winckler, Björn
(author)
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Renormalization of Lorenz Maps
- 2011
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Doctoral thesis (other academic/artistic)abstract
- This thesis is a study of the renormalization operator on Lorenz αmaps with a critical point. Lorenz maps arise naturally as first-return maps for three-dimensional geometric Lorenz flows. Renormalization is a tool for analyzing the microscopic geometry of dynamical systems undergoing a phase transition. In the first part we develop new tools to study the limit set of renormalization for Lorenz maps whose combinatorics satisfy a long return condition. This combinatorial condition leads to the construction of a relatively compact subset of Lorenz maps which is essentially invariant under renormalization. From here we can deduce topological properties of the limit set (e.g. existence of periodic points of renormalization) as well as measure theoretic properties of infinitely renormalizable maps (e.g. existence of uniquely ergodic Cantor attractors). After this, we show how Martens’ decompositions can be used to study the differentiable structure of the limit set of renormalization. We prove that each point in the limit set has a global two-dimensional unstable manifold which is a graph and that the intersection of an unstable manifold with the domain of renormalization is a Cantor set. All results in this part are stated for arbitrary real critical exponents α> 1. In the second part we give a computer assisted proof of the existence of a hyperbolic fixed point for the renormalization operator on Lorenz maps of the simplest possible nonunimodal combinatorial type. We then show how this can be used to deduce both universality and rigidity for maps with the same combinatorial type as the fixed point. The results in this part are only stated for critical exponenta α= 2.
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| 3. |
- Andersson, Anders, 1957-
(author)
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Numerical Conformal mappings for regions Bounded by Smooth Curves
- 2006
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Licentiate thesis (other academic/artistic)abstract
- Inom många tillämpningar används konforma avbildningar för att transformera tvådimensionella områden till områden med enklare utseende. Ett exempel på ett sådant område är en kanal av varierande tjocklek begränsad av en kontinuerligt deriverbar kurva. I de tillämpningar som har motiverat detta arbete, är det viktigt att dessa egenskaper bevaras i det område en approximativ konform avbildning producerar, men det är också viktigt att begränsningskurvans riktning kan kontrolleras, särkilt i kanalens båda ändar. Denna avhandling behandlar tre olika metoder för att numeriskt konstruera konforma avbildningar mellan ett enkelt standardområde, företrädesvis det övre halvplanet eller enhetscirkeln, och ett område begränsat av en kontinuerligt deriverbar kurva, där begränsningskurvans riktning kan kontrolleras, exakt eller approximativt. Den första metoden är en utveckling av en idé, först beskriven av Peter Henrici, där en modifierad Schwarz-Christoffel-avbildning avbildar det övre halvplanet konformt på en polygon med rundade hörn. Med utgångspunkt i denna idé skapas en algoritm för att konstruera avbildningar på godtyckliga områden med släta randkurvor. Den andra metoden bygger också den på Schwarz-Christoffel-avbildningen, och utnyttjar det faktum att om enhetscirkeln eller halvplanet avbildas på en polygon kommer ett område Q i det inre av dessa, som till exempel en cirkel med centrum i origo och radie mindre än 1, eller ett område i övre halvplanet begränsat av två strålar, att avbildas på ett område R i det inre av polygonen begränsat av en slät kurva. Vi utvecklar en metod för att hitta ett polygonalt område P, utanför det Omega som man önskar att skapa en avbildning för, sådant att den Schwarz-Christoffel-avbildning som avbildar enhetscirkeln eller halvplanet på P, avbildar Q på Omega. I båda dessa fall används tangentpolygoner för att numeriskt bestämma den önskade avbildningen. Slutligen beskrivs en metod där en av Don Marshalls så kallade zipper-algoritmer används för att skapa en avbildning mellan det övre halvplanet och en godtycklig kanal, begränsad av släta kurvor, som i båda ändar går mot oändligheten som räta parallella linjer.
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| 4. |
- Johansson, Carl Fredrik, 1979-
(author)
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Random Loewner Chains
- 2010
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Doctoral thesis (other academic/artistic)abstract
- This thesis contains four papers and two introductory chapters. It is mainly devoted to problems concerning random growth models related to the Loewner differential equation. In Paper I we derive a rate of convergence of the Loewner driving function for loop-erased random walk to Brownian motion with speed 2 on the unit circle, the Loewner driving function for radial SLE(2). Thereby we provide the first instance of a formal derivation of a rate of convergence for any of the discrete models known to converge to SLE. In Paper II we use the known convergence of (radial) loop-erased random walk to radial SLE(2) to prove that the scaling limit of loop-erased random walk excursion in the upper half plane is chordal SLE(2). Our proof relies on a version of Wilson’s algorithm for weighted graphs together with a Beurling-type hitting estimate for random walk excursion. We also establish and use the convergence of the radial SLE path to the chordal SLE path as the bulk point tends to a boundary point. In the final section we sketch how to extend our results to more general domains. In Paper III we prove an upper bound on the optimal Hölder exponent for the chordal SLE path parameterized by capacity and thereby establish the optimal exponent as conjectured by J. Lind. We also give a new proof of the lower bound. Our proofs are based on sharp estimates of moments of the derivative of the inverse SLE map. In particular, we improve an estimate of G. F. Lawler. In Paper IV we consider radial Loewner evolutions driven by unimodular Lévy processes. We rescale the hulls of the evolution by capacity, and prove that the weak limit of the rescaled hulls exists. We then study a random growth model obtained by driving the Loewner equation with a compound Poisson process with two real parameters: the intensity of the underlying Poisson process and a localization parameter of the Poisson kernel which determines the jumps. A particular choice of parameters yields a growth process similar to the Hastings-Levitov HL(0) model. We describe the asymptotic behavior of the hulls with respect to the parameters, showing that growth tends to become localized as the jump parameter increases. We obtain deterministic evolutions in one limiting case, and Loewner evolution driven by a unimodular Cauchy process in another. We also show that the Hausdorff dimension of the limiting rescaled hulls is equal to 1.
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| 5. |
- Karagulyan, Davit, 1989-
(author)
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Certain results on the Möbius disjointness conjecture
- 2017
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Doctoral thesis (other academic/artistic)abstract
- We study certain aspects of the Möbius randomness principle and more specifically the Möbius disjointness conjecture of P. Sarnak. In paper A we establish this conjecture for all orientation preserving circle homeomorphisms and continuous interval maps of zero entropy. In paper B we show, that for all subshifts of finite type with positive topological entropy the Möbius disjointness does not hold. In paper C we study a class of three-interval exchange maps arising from a paper of Bourgain and estimate its Hausdorff dimension. In paper D we consider the Chowla and Sarnak conjectures and the Riemann hypothesis for abstract sequences and study their relationship.
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| 6. |
- Laestadius, Andre, 1984-
(author)
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Foundation of Density Functionals in the Presence of Magnetic Field
- 2014
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Doctoral thesis (other academic/artistic)abstract
- This thesis contains four articles related to mathematical aspects of Density Functional Theory.In Paper A, the theoretical justification of density methods formulated with current densities is addressed. It is shown that the set of ground-states is determined by the ensemble-representable particle and paramagnetic current density. Furthermore, it is demonstrated that the Schrödinger equation with a magnetic field is not uniquely determined by its ground-state solution. Thus, a wavefunction may be the ground-state of two different Hamiltonians, where the Hamiltonians differ by more than a gauge transformation. This implies that the particle and paramagnetic current density do not determine the potentials of the system and, consequently, no Hohenberg-Kohn theorem exists for Current Density Functional Theory formulated with the paramagnetic current density. On the other hand, by instead using the particle density as data, we show that the scalar potential in the system's Hamiltonian is determined for a fixed magnetic field. This means that the Hohenberg-Kohn theorem continues to hold in the presence of a magnetic field, if the magnetic field has been fixed.Paper B deals with N-representable density functionals that also depend on the paramagnetic current density. Here the Levy-Lieb density functional is generalized to include the paramagnetic current density. It is shown that a wavefunction exists that minimizes the "free" Hamiltonian subject to the constraints that the particle and paramagnetic current density are held fixed. Furthermore, a convex and universal current density functional is introduced and shown to equal the convex envelope of the generalized Levy-Lieb density functional. Since this functional is convex, the problem of finding the particle and paramagnetic current density that minimize the energy is related to a set of Euler-Lagrange equations.In Paper C, an N-representable Kohn-Sham approach is developed that also include the paramagnetic current density. It is demonstrated that a wavefunction exists that minimizes the kinetic energy subject to the constraint that only determinant wavefunctions are considered, as well as that the particle and paramagnetic current density are held fixed. Using this result, it is then shown that the ground-state energy can be obtained by minimizing an energy functional over all determinant wavefunctions that have finite kinetic energy. Moreover, the minimum is achieved and this determinant wavefunction gives the ground-state particle and paramagnetic current density.Lastly, Paper D addresses the issue of a Hohenberg-Kohn variational principle for Current Density Functional Theory formulated with the total current density. Under the assumption that a Hohenberg-Kohn theorem exists formulated with the total current density, it is shown that the map from particle and total current density to the vector potential enters explicitly in the energy functional to be minimized. Thus, no variational principle as that of Hohenberg and Kohn exists for density methods formulated with the total current density.
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| 7. |
- Parra, Rodrigo, 1979-
(author)
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Equidistribution towards the Green current in complex dynamics
- 2011
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Doctoral thesis (other academic/artistic)abstract
- Given a holomorphic self-map of complex projective space of de-gree larger than one, we prove that there exists a finite collection oftotally invariant algebraic sets with the following property: given anypositive closed (1,1)-current of mass 1 with no mass on any element of this family, the sequence of normalized pull-backs of the current converges to the Green current. Under suitable geometric conditions on the collection of totally invariant algebraic sets, we prove a sharper equidistribution result.
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| 8. |
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| 9. |
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| 10. |
- Benedicks, Michael, Professor, et al.
(author)
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Coexistence Phenomena in the Henon Family
- 2023
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In: Bulletin of the Brazilian Mathematical Society. - : Springer Nature. - 1678-7544 .- 1678-7714. ; 54:3
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Journal article (peer-reviewed)abstract
- We study the classical H & eacute;non family fa,b : (x, y) i? (1 - ax(2) + y, bx), 0 < a < 2, 0 < b < 1, and prove that given an integer k = 1, there is a set of parameters Ek of positive two-dimensional Lebesgue measure so that fa,b, for (a, b) ? E-k, has at least k attractive periodic orbits and one strange attractor of the type studied in Benedicks and Carleson (Ann Math (2) 133(1):73-169, 1991). A corresponding statement also holds for the H & eacute;non-like families of Mora and Viana (Acta Math 171:1-71, 1993), and we use the techniques of Mora and Viana (1993) to study homoclinic unfoldings also in the case of the original H & eacute;non maps. The final main result of the paper is the existence, within the classical H & eacute;non family, of a positive Lebesgue measure set of parameters whose corresponding maps have two coexisting strange attractors.
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