| 1. |
- Andersson, Anders, 1957-
(författare)
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Numerical Conformal mappings for regions Bounded by Smooth Curves
- 2006
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)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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| 2. |
- Andersson, Anders, 1957-
(författare)
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Numerical Conformal Mappings for Waveguides
- 2009
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)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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| 3. |
- Baladi, V., et al.
(författare)
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Almost sure rates of mixing for i.i.d. unimodal maps
- 2002
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Ingår i: Annales Scientifiques de l'Ecole Normale Supérieure. - 0012-9593 .- 1873-2151. ; 35:1, s. 77-126
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Tidskriftsartikel (refereegranskat)abstract
- It has been known since the pioneering work of Jakobson and subsequent work by Benedicks and Carleson and others that a positive measure set of quadratic maps admit an absolutely continuous invariant measure. Young and Keller-Nowicki proved exponential decay of its correlation functions. Benedicks and Young [8], and Baladi and Viana [4] studied stability of the density and exponential rate of decay of the Markov chain associated to i.i.d. small perturbations. The almost sure statistical proper-ties of the sample stationary measures of i.i.d. itineraries are more difficult to estimate than the averaged statistics. Adapting to random systems, on the one hand partitions associated to hyperbolic times due to Alves [I], and on the other a probabilistic coupling method introduced by Young [26] to study rates of mixing, we prove stretched exponential upper bounds for the almost sure rates of mixing.
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| 4. |
- Baladi V,, et al.
(författare)
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Almost sure rates of mixing for I.I.D. unimodal maps (vol 35, pg 117, 2002)
- 2003
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Ingår i: Annales Scientifiques de l'Ecole Normale Supérieure. - 0012-9593 .- 1873-2151. ; 36:2, s. 319-322
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Tidskriftsartikel (refereegranskat)abstract
- The definition of the return time (p. 117) and the beginning of the proof of Proposition 8.3 of our paper in Vol. 35 of Ann. Scient. Ec. Norm. Sup. (2002) are not correct. We give an amended version which shows that none of the statements are affected. We take this opportunity to correct some other mistakes (without consequences), e.g. in Sublemma 7.2(3) and Lemma 7.10. Published by Editions scientifiques et medicales Elsevier SAS
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| 5. |
- Baladi, V., et al.
(författare)
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Decay of random correlation functions for unimodal maps
- 2000
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Ingår i: Reports on mathematical physics. - 0034-4877 .- 1879-0674. ; 46:2-Jan, s. 15-26
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Tidskriftsartikel (refereegranskat)abstract
- Since the pioneering results of Jakobson and subsequent work by Benedicks-Carleson and others, it is known that quadratic maps f(a) (x) = a - x(2) admit a unique absolutely continuous invariant measure for a positive measure set of parameters a. For topologically mixing f(a), Young and Keller-Nowicki independently proved exponential decay of correlation functions for this a.c.i.m. and smooth observables. We consider random compositions of small perturbations f + omega (t), with f = f(a) or another unimodal map satisfying certain nonuniform hyperbolicity axioms, and omega (t) chosen independently and identically in [-epsilon, epsilon]. Baladi-Viana showed exponential mixing of the associated Markov chain, i.e., averaging over all random itineraries. We obtain stretched exponential bounds for the random correlation functions of Lipschitz observables for the sample measure mu (omega), of almost every itinerary.
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| 6. |
- Baladi, Viviane, et al.
(författare)
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Whitney-Holder continuity of the SRB measure for transversal families of smooth unimodal maps
- 2015
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Ingår i: Inventiones Mathematicae. - : Springer Science and Business Media LLC. - 0020-9910 .- 1432-1297. ; 201:3, s. 773-844
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Tidskriftsartikel (refereegranskat)abstract
- We consider families of nondegenerate unimodal maps. We study the absolutely continuous invariant probability (SRB) measure of , as a function of on the set of Collet-Eckmann (CE) parameters: Upper bounds: Assuming existence of a transversal CE parameter, we find a positive measure set of CE parameters , and, for each , a set of polynomially recurrent parameters containing as a Lebesgue density point, and constants , , so that, for every -Holder function , In addition, for all , the renormalisation period of satisfies , and there are uniform bounds on the rates of mixing of for all with . If , the set contains almost all CE parameters. Lower bounds: Assuming existence of a transversal mixing Misiurewicz-Thurston parameter , we find a set of CE parameters accumulating at , a constant , and a function , so that C vertical bar t - t(0)vertical bar(1/2) >= vertical bar integral A(0)d mu(t) - integral A(0)d mu(t0) vertical bar >= C-1 vertical bar t - t(0)vertical bar(1/2), for all t is an element of Delta(MT)'.
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| 7. |
- Benedicks, Michael, et al.
(författare)
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Beurling archive on the web
- 2018
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Ingår i: Notices of the American Mathematical Society. - : American Mathematical Society (AMS). - 0002-9920 .- 1088-9477. ; 65:5
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Tidskriftsartikel (refereegranskat)
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| 8. |
- Benedicks, Michael, Professor, et al.
(författare)
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Coexistence Phenomena in the Henon Family
- 2023
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Ingår i: Bulletin of the Brazilian Mathematical Society. - : Springer Nature. - 1678-7544 .- 1678-7714. ; 54:3
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Tidskriftsartikel (refereegranskat)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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| 9. |
- Benedicks, Michael, Professor, et al.
(författare)
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Expansion properties of double standard maps
- 2023
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Ingår i: Ergodic Theory and Dynamical Systems. - : Cambridge University Press (CUP). - 0143-3857 .- 1469-4417. ; 43:8, s. 2549-2588
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Tidskriftsartikel (refereegranskat)abstract
- For the family of double standard maps we investigate the structure of the space of parameters a when and when. In the first case the maps have a critical point, but for a set of parameters of positive Lebesgue measure there is an invariant absolutely continuous measure for. In the second case there is an open non-empty set of parameters for which the map is expanding. We show that as, the set accumulates on many points of in a regular way from the measure point of view.
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| 10. |
- Benedicks, Michael, et al.
(författare)
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Kneading sequences for double standard maps
- 2009
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Ingår i: Fundamenta Mathematicae. - : Institute of Mathematics, Polish Academy of Sciences. - 0016-2736 .- 1730-6329. ; 206, s. 61-75
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Tidskriftsartikel (refereegranskat)abstract
- We investigate the symbolic dynamics for the double standard maps of the circle onto itself, given by f(a,b) (x) = 2x + a + (b/pi) sin(2 pi x) (mod 1), where b = 1 and a is a real parameter, 0 <= a < 1
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