Dissertations / Theses on the topic 'Integrability'
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Clarke, Daniel. "Integrability in submanifold geometry." Thesis, University of Bath, 2012. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.558890.
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This thesis concerns the relationship of submanifold geometry, in both the smooth and discrete sense, to representation theory and the theory of integrable systems. We obtain Lie theoretic generalisations of the transformation theory of projectively and Lie applicable surfaces, and M�obius-flat submanifolds of the conformal n-sphere. In the former case, we propose a discretisation. We develop a projective approach to centro-ane hypersurfaces, analogous to the conformal approach to submanifolds in spaceforms. This yields a characterisation of centro-ane hypersurfaces amongst M�obius-flat projective hypersurfaces using polynomial conserved quantities. We also propose a discretisation of curved flats in symmetric spaces. After developing the transformation theory for this, we see how Darboux pairs of discrete isothermicnets arise as discrete curved flats in the symmetric space of opposite point pairs. We show how discrete curves in the 2-sphere fit into this framework.
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Young, Charles Alastair Stephen. "Integrability and symmetric spaces." Thesis, University of Cambridge, 2005. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.614914.
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Coletta, Meredith L. Hicks R. Andrew. "Integrability in optical design /." Philadelphia, Pa. : Drexel University, 2009. http://hdl.handle.net/1860/3079.
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Engbrant, Fredrik. "Supersymmetric Quantum Mechanics and Integrability." Thesis, Uppsala universitet, Teoretisk fysik, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-173301.
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This master’s thesis investigates the relationship between supersymmetry and integrability in quantum mechanics. This is done by finding a suitable way to systematically add more supersymmetry to the system. Adding more super- symmetry will give constraints on the potential which will lead to an integrable system. A possible way to explore the integrability of supersymmetric quantum mechanics was introduced in a paper by Crombrugghe and Rittenberg in 1983, their method has been used as well as another approach based on expanding a N = 1 system by introducing complex structures. N = 3 or more supersymmetry is shown to give an integrable system.
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Scott, Daniel R. D. "Separation of variables and integrability." Thesis, University of Cambridge, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.389963.
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Chen, Y.-C. "Anti-integrability in Lagrangian systems." Thesis, University of Cambridge, 2002. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.597512.
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Three examples of application of the anti-integrability concept in Lagrangian systems are proved, concerning the continuation of a class of trajectories from the anti-integrable limit. All three examples were proposed by Robert S. MacKay. The first example arises in adiabatically perturbed systems. With an assumption that the adiabatic Poincaré-Melnikov function has simple zeros, we constructed a variational functional whose critical points give rise to a sequence of homoclinic trajectories for the unperturbed Lagrangian in the adiabatic limit but a sequence of multi-bump trajectories under perturbations. We found there is a compact set, which is a Cantor set, such that the Poincaré map induced by the phase flow restricting to it is conjugate to the Bernoulli shift, in our case, with three symbols. Hence the approach of the anti-integrability to the adiabatically perturbed problems is equivalent to the one which combines the Poincaré-Melnikov method and the Birkhoff-Smale theory. The second example occurs in the Sinai billiard system. The anti-integrable limit is the limit when the radius of the scatterer-disc goes down to zero, and the system becomes "δ-billiards". The orbits of the δ-billiards are the anti-integrable orbits which are piecewise straight lines joining zero-radius discs to discs, and are easily obtained. Under some non-degeneracy conditions, we proved all anti-integrable orbits can be continued to the small radius case, and found that any periodic orbit has infinitely many homoclinic orbits as well as heteroclinic orbits to any others. These exists a compact set, which is also a Cantor set, such that the billiard map restricted to it is conjugate to a subshift of finite type with an arbitrarily given number of symbols. We studied in the third example when the scatterers are approximated by repulsive potentials such as the Coulomb potential ε/r, where ε and r are non-negative numbers and r is the distance from the potential centre. In the Coulomb potential case, the anti-integrable limit is the ε → 0, and the system becomes the δ-billiard system. Then we found that the results in the Sinai billiards also hold here when ε > 0 but small. More general type of repulsive potentials were also investigated and a sufficient condition under which anti-integrable trajectories persist was given.
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Zhao, Peng. "Integrability in supersymmetric gauge theories." Thesis, University of Cambridge, 2013. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.648125.
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Katsimpouri, Despoina. "Integrability in two-dimensional gravity." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät, 2015. http://dx.doi.org/10.18452/17316.
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In dieser Arbeit untersuchen wir Gravitations- und Supergravitationssysteme, die in zwei Dimensionen vollständig integrabel sind. Dies sind Theorien, zu denen auch die einsteinsche Gravitation zählt, die bei dimensionaler Reduktion auf drei Dimensionen, die Form eines nichtlinearen $\s$-Models für den Materieteil annehmen und als Zielmannigfaltigkeit den Cosetraum $\mathrm{G}/\mathrm{K}$ haben. Ausgehend von der einsteinschen Gravitation betrachten wir insbesondere die Klasse der stationären und axialsymmetrischen Lösungen. Dabei untersuchen wir das lineare System (Lax-Paar), das den nichtlinearen Feldgleichungen der Vakuumsgravitation entspricht, wie es von Belinski-Zakharov (BZ) und Breitenlohner-Maison (BM) formuliert wurde. Die Existenz des linearen Systems zeigt die Integrabilität des zweidimensionalen Systems und ist inversen Streumethoden zugänglich, wie in zwei unterschiedlichen Ansätzen von BZ und BM gezeigt. Aus der unendlich-dimensionalen Symmetrie, die mit den zweidimensionalen Gleichungen assoziiert ist, ergibt sich die sogenannte Gerochgruppe. Der BM-Ansatz ermöglicht eine direkte Implementierung der Gerochgruppe und der Erzeugung von physikalisch interessanten Lösungen im Solitonensektor auf manifest gruppentheoretischer Weise. Aus diesem Grund ist zu erwarten, dass es in einem breiteren Spektrum von Cosetmodellen angewendet werden kann. In dieser Arbeit konzentrieren wir uns auf diesen Ansatz und erweitern ihn um die STU-Supergravitation, wobei entsprechende technische Änderungen im BM-Lösungserzeugungsalgorithmus erforderlich werden. Basierend auf diesen Änderungen, diskutieren wir auch eine Verallgemeinerung auf andere Fälle. Wir testen die Anwendbarkeit der BM inversen Streumethode, indem wir explizit folgende Lösungen konstruieren: die Kerr-NUT Lösung der einsteinschen Gravitation, die Vier-Ladungs-Lösung eines schwarzen Lochs innerhalb der STU Supergravitation von Cvetic und Youm und die einfach rotierende JMaRT Lösung.In this thesis, we study gravity and supergravity systems that become completely integrable in two dimensions. Including Einstein gravity, these systems are theories that upon dimensional reduction to three dimensions assume the form of a non-linear $\s$-model for the matter part, with target manifold a coset space $\mathrm{G}/\mathrm{K}$. Starting from Einstein gravity and focusing on the class of stationary axisymmetric solutions, we study the linear system (Lax pair) associated with the non-linear field equations of vacuum gravity as formulated by Belinski - Zakharov (BZ) and Breitenlohner-Maison (BM). The existence of the linear system exhibits the integrability of the two-dimensional system and is amenable to inverse scattering methods as shown in two different approaches by BZ and BM. The infinite dimensional symmetry associated with the two-dimensional equations gives rise to the so-called Geroch group. The BM approach allows for a direct implementation of the Geroch group and the generation of physically interesting solutions in the soliton sector in a manifestly group theoretic way. For this reason, it is expected to apply to a broader set of coset models. Throughout this work, we concentrate on this approach and extend it to STU supergravity, where appropriate technical modifications were required in the BM solution generation algorithm. Based on these modifications, we also discuss a generalization to other set-ups. We test the applicability of the BM inverse scattering method by explicitly constructing the Kerr-NUT solution of Einstein gravity and within STU supergravity, the four-charge black hole solution of Cvetic and Youm as well as the singly rotating JMaRT solution.
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Gahramanov, Ilmar. "Superconformal indices, dualities and integrability." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät, 2016. http://dx.doi.org/10.18452/17568.
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In dieser Arbeit behandeln wir exakte, nicht-perturbative Ergebnisse, die mithilfe der superkonformen Index-Technik, in supersymmetrischen Eichtheorien mit vier Superladungen (d. h. N=1 Supersymmetrie in vier Dimensionen und N=2 in drei Dimensionen) gewonnen wurden. Wir benutzen die superkonforme Index-Technik um mehrere Dualitäts Vermutungen in supersymmetrischen Eichtheorien zu testen. Wir führen Tests der dreidimensionalen Spiegelsymmetrie und Seiberg ähnlicher Dualitäten durch. Das Ziel dieser Promotionsarbeit ist es moderne Fortschritte in nicht-perturbativen supersymmetrischen Eichtheorien und ihre Beziehung zu mathematischer Physik darzustellen. Im Speziellen diskutieren wir einige interessante Identitäten der Integrale, denen einfache und hypergeometrische Funktionen genügen und ihren Bezug zu supersymmetrischen Dualitäten in drei und vier Dimensionen. Methoden der exakten Berechnungen in supersymmertischen Eichtheorien sind auch auf integrierbare statistische Modelle anwendbar. Dies wird im letzten Kapitel der vorliegenden Arbeit behandelt.In this thesis we discuss exact, non-perturbative results achieved using superconformal index technique in supersymmetric gauge theories with four supercharges (which is N = 1 supersymmetry in four dimensions and N = 2 supersymmetry in three). We use the superconformal index technique to test several duality conjectures for supersymmetric gauge theories. We perform tests of three-dimensional mirror symmetry and Seiberg-like dualities. The purpose of this thesis is to present recent progress in non-perturbative supersymmetric gauge theories in relation to mathematical physics. In particular, we discuss some interesting integral identities satisfied by basic and elliptic hypergeometric functions and their relation to supersymmetric dualities in three and four dimensions. Methods of exact computations in supersymmetric theories are also applicable to integrable statistical models, which we discuss in the last chapter of the thesis.
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Debernardi, Pinos Alberto. "Convergence and integrability of fourier transforms." Doctoral thesis, Universitat Autònoma de Barcelona, 2018. http://hdl.handle.net/10803/463030.
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El propòsit d'aquesta tesi és el d'estudiar dos tipus de problema diferents per a certes transformades de Fourier.
Primer investiguem la convergència uniforme d'integrals sinusoidals en una i dos dimensions. Per a dur a terme aquesta investigació, utilitzem una condicio de monotonia general, recentment introduïda, tot desenvolupant aquesta teoria en concordança amb les nostres necessitats. Com a resultats principals, obtenim condicions necessàries i suficients que les funcions monòtones generals han de satisfer per tal de poder assegurar la convergència uniforme de les seves respectives transformades sinusoidals (en una i dues dimensions).
En segon lloc, estudiem la convergència puntual i uniforme de les transformades de Hankel amb pesos, a través de l'estudi de les condicions variacionals, d'integració i de magnitud de les funcions involucrades, amb especial èmfasi en les condicions variacionals. També utilitzem l'esmentada condició de monotonia general, que ens permet traduir condicions variacionals de les funcions en condicions d'integrabilitat o magnitud de les mateixes. Donem condicions suficients per a la convergència puntual, mentre que per a la convergència uniforme, també en donem de necessàries, quan és possible. En els casos en els quals només podem donar condicions suficients per a la convergència uniforme, també comentem l'optimalitat d'aquestes.
Finalment, considerem transformades de Fourier generalitzades, i estudiem condicions necessàries i suficients per tal de garantir desigualtats de normes amb pesos entre funcions i les seves transformades. Les desigualtats de normes amb pesos es poden considerar com a versions quantitatives del principi d'incertesa. Donem especial rellevància a les desigualtats amb pesos del tipus funció potencial i les transformades sinusoidals, cosinusoidals, de Hankel, i de Struve. També utilitzem la condició de monotonia general en aquest problema, que ens permet obtenir condicions necessàries i suficients menys restrictives per poder garantir desigualtats de normes amb pesos.The purpose of this dissertation is to study two different kind of problems for certain types of Fourier transforms. First, we investigate the uniform convergence of one and two-dimensional sine transforms. To this end, we make use of a general monotonicity condition that has been recently introduced, and develop the theory further according to our needs. We mainly obtain necessary and sufficient conditions on general monotone functions for the uniform convergence of their respective (single and double) sine integrals. Secondly, we study pointwise and uniform convergence of weighted Hankel transforms through an approach that consists on studying the variational, integrability, and magnitude conditions of the involved functions, with special emphasis on variational conditions. Here we also use the aforementioned general monotonicity, which allows us to translate from variational conditions to magnitude/integrability conditions of the functions. For the pointwise convergence only sufficient conditions are obtained, whilst for the uniform convergence, it is sometimes possible to obtain necessary and sufficient conditions. In the case when only sufficient conditions for the uniform convergence are given, the sharpness of those are discussed. Finally, we consider generalized Fourier transforms and study necessary and sufficient conditions for weighted norm inequalities between functions and their transforms to hold. Weighted norm inequalities can be considered as quantitative uncertainty principle relations. We particularly focus on inequalities with power weights and the sine, cosine, Hankel, and Struve transforms. We also make use of the general monotonicity condition in this problem, which allows us to obtain less restrictive necessary and sufficient for the weighted norm inequalities to hold.
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Evans, N. Wyn. "Separability and integrability in stellar dynamics." Thesis, University of Cambridge, 1988. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.315028.
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Chen, H. Y. "On integrability in gauge/string correspondence." Thesis, University of Cambridge, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.597543.
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In this Thesis, we present some direct quantitative tests for AdS/CFT correspondence using the newly discovered “integrability”. We shall begin by explaining the ideas of algebraic Bethe ansatz and scattering matrix in both gauge and string theories, and discussions how they enable us to extract the spectra of the scaling dimensions of the gauge invariant operators or the energies of the dual string states. In chapter 3, we explicitly apply the Bethe ansatz techniques in thermodynamically limit to the so-called β-deformation of ℵ = 4 SYM, and extract the one-loop anomalous dimensions for its long gauge invariant operators; we also construct the corresponding string solutions in the dual background, and show their energies precisely match with the gauge theory results. In the second part of Thesis, we consider a new asymptotic limit in both gauge and string theories, in such limit the elementary excitations are known as “magnons”; we shall also switch our focus to the building element of Bethe Ansatz, the scattering matrix between magnons. In chapter 4, we apply the magnon scattering matrix and explain how additional stable bound states can appear; both elementary magnons and their bound states have exact expressions for the dispersion relations. The classical string configuration dual to magnon bound state in gauge theory is constructed in chapter 5, where the connection between string sigma model on R x S3 and integrable complex sine-Gordon model is exploited. In chapter 6, the scattering matrix between the magnon bound states is considered via bootstrap method, and in the semiclassical limit, the result coincides with the scattering matrix between complex sine-Gordon solitons. This provides the direct verification for the proposed all-loop magnon scattering matrix and the lowest order “dressing factor”. In chapter 7, by applying the extended residual symmetry algebra psu(2|2)2 x ℝ3, we classify all possible magnon bound states in terms of the constituent fields in ℵ = 4 SYM and justify the exactness of their dispersion relation.
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Grossberg, Michael David. "Complete integrability and geometrically induced representations." Thesis, Massachusetts Institute of Technology, 1991. http://hdl.handle.net/1721.1/43161.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1991.GRSN 579612
Vita.
Includes bibliographical references (leaves 92-93).
by Michael David Grossberg.
Ph.D.
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Ferreira, Gonçalves Helena Daniela. "2-microlocal spaces with variable integrability." Universitätsverlag der Technischen Universität Chemnitz, 2017. https://monarch.qucosa.de/id/qucosa%3A20972.
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In this work we study several important properties of the 2-microlocal Besov and Triebel-Lizorkin spaces with variable integrability. Due to the richness of the weight sequence used to measure smoothness, this scale of function spaces incorporates a wide range of function spaces, of which we mention the spaces with variable smoothness.
Within the existing characterizations of these spaces, the characterization via smooth atoms is undoubtedly one of the most used when it comes to obtain new results in varied directions. In this work we make use of such characterization to prove several embedding results, such as Sobolev, Franke and Jawerth embeddings, and also to study traces on hyperplanes.
Despite the considerable benefits of resorting on the smooth atomic decomposition, there are still some limitations when one tries to use it in order to prove some specific results, such as pointwise multipliers and diffeomorphisms assertions. The non-smooth atomic characterization proved in this work overcome these problems, due to the weaker conditions of the (non-smooth) atoms. Moreover, it also allows us to give an intrinsic characterization of the 2-microlocal Besov and Triebel-Lizorkin spaces with variable integrability on the class of regular domains, in which connected bounded Lipschitz domains are included.In dieser Arbeit untersuchen wir einige wichtige Eigenschaften der 2-microlokalen Besov und Triebel-Lizorkin Räume mit variabler Integrabilität. Weil die Glattheit hier mit einer reicher Gewichtsfolge gemessen wird, beinhaltet diese Skala von Funktionsräumen eine große Anzahl von Funktionsräumen, von denen wir die Räume mit variabler Glattheit erwähnen. Innerhalb der vorhandenen Charakterisierungen dieser Räume ist die Charakterisierung mit glatten Atomen zweifellos eine der am häufigsten verwendeten, um neue Ergebnisse in verschiedenen Richtungen zu erhalten. In dieser Arbeit verwenden wir eine solche Charakterisierung, um mehrere Einbettungsergebnisse zu bewiesen, wie Sobolev-Einbettungen und Einbettungen vom Franke-Jawerth Typ, und auch Spurresultate zu untersuchen. Trotz der beträchtlichen Vorteile des Rückgriffs auf die glatte Atomaren-Zerlegung gibt es immer noch einige Einschränkungen, wenn man versucht, sie zu verwenden, um einige spezifische Ergebnisse zu beweisen, wie beispielsweise punktweise Multiplikatoren und Diffeomorphismen-Assertionen. Die nichtglatte atomare Charakterisierung, die wir in dieser Arbeit beweisen, überwindet diese Probleme aufgrund der schwächeren Bedingungen von (nichtglatten) Atomen. Außerdem erlaubt es uns, eine Intrinsische Charakterisierung der 2-mikrolokalen Besov- und Triebel-Lizorkin-Räume mit variabler Integrabilität auf regulärer Gebieten zu geben.
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Zenkov, Dmitry V. "Integrability and stability of nonholonomic systems /." The Ohio State University, 1998. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487951214939032.
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Boman, Frode. "Integrability of Boltzmann's discontinuous gravitational system." Thesis, KTH, Fysik, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-297603.
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A dynamical system originally invented by Boltzmann has had recent developments. The system consists of a particle in a gravitational potential with an added centrifugal force, which is subject to reflection against a wall that separates the system from the gravitational center. The recent developments are with regards to the integrability of the system in the special case of vanishing centrifugal term. The purpose of this essay is to explicate these developments.Ett dynamiskt system, ursprungligen uppfunnet av Boltzmann, har nyligen sett utvecklingar. Systemet består av en partikel i en gravitationspotential med en tillagd centrifugalkraft, som reflekterar vid kontakt med en vägg som skiljer partikeln och gravitationscentrumet. De nya utvecklingarna är inom systemets integrabilitet i det specialfall att centrifugalkraften är borttagen. Syftet med denna uppsats är att explicera dessa framtaganden.
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Contatto, Felipe. "Vortices, Painlevé integrability and projective geometry." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/275099.
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GaugThe first half of the thesis concerns Abelian vortices and Yang-Mills theory. It is proved that the 5 types of vortices recently proposed by Manton are actually symmetry reductions of (anti-)self-dual Yang-Mills equations with suitable gauge groups and symmetry groups acting as isometries in a 4-manifold. As a consequence, the twistor integrability results of such vortices can be derived. It is presented a natural definition of their kinetic energy and thus the metric of the moduli space was calculated by the Samols' localisation method. Then, a modified version of the Abelian–Higgs model is proposed in such a way that spontaneous symmetry breaking and the Bogomolny argument still hold. The Painlevé test, when applied to its soliton equations, reveals a complete list of its integrable cases. The corresponding solutions are given in terms of third Painlevé transcendents and can be interpreted as original vortices on surfaces with conical singularity. The last two chapters present the following results in projective differential geometry and Hamiltonians of hydrodynamic-type systems. It is shown that the projective structures defined by the Painlevé equations are not metrisable unless either the corresponding equations admit first integrals quadratic in first derivatives or they define projectively flat structures. The corresponding first integrals can be derived from Killing vectors associated to the metrics that solve the metrisability problem. Secondly, it is given a complete set of necessary and sufficient conditions for an arbitrary affine connection in 2D to admit, locally, 0, 1, 2 or 3 Killing forms. These conditions are tensorial and simpler than the ones in previous literature. By defining suitable affine connections, it is shown that the problem of existence of Killing forms is equivalent to the conditions of the existence of Hamiltonian structures for hydrodynamic-type systems of two components.
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Bonini, Alfredo <1986>. "Supersymmetric 4d gauge theories and Integrability." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2018. http://amsdottorato.unibo.it/8707/1/Bonini_Alfredo_Tesi.pdf.
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Taking advantage of the integrable structures emergent in the theory, non-local observables
such as null polygonal Wilson loops are studied in 4d planar N = 4 Super Yang-Mills. Their
duality with the 4d gluon scattering amplitudes makes the analysis even more interesting.
The so-called Pentagon approach, an application of the Operator Product Expansion (OPE)
method to the null polygonal Wilsol loops, makes possible a non-perturbative evaluation of
these objects. They are recast as an OPE series over the 2d GKP flux-tube excitations, a
description reminescent of the QCD flux-tube stretching between quarks. The integrability
of the flux-tube allows us to evaluate the series, in principle, for any value of the coupling
constant. From this analysis, several results have been obtained. In the strong coupling
regime we reproduced the TBA-like equations expected from the minimal area problem
in string theory, finding agreement with the AdS/CFT prediction. In this respect, of
fundamental importance is the emergence of effective bound states between elementary
fermionic excitations. Along the way, we encountered some intriguing analogies between
these null polygonal Wilson loops and the Nekrasov instanton partition function Z for
N = 2 theories. Furthermore, a new non-perturbative enhancement of the classical string
argument has been confirmed, stemming from the dynamics of the string in the five sphere
S5 and described by the non-linear σ-model O(6). Some properties of a fundamental
building block of the OPE series, the SU(4) structure of the form factors of a specific twist
operator P, have been analysed. This SU(4) matrix part is given a representation in terms
of rational functions, organized in a Young tableaux pattern.
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Tureli, Sina. "Integrability of Continuous Tangent Sub-bundles." Doctoral thesis, SISSA, 2015. http://hdl.handle.net/20.500.11767/4876.
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In this thesis, the main aim is to study the integrability properties of continuous tangent sub-bundles, especially those that arise in the study of dynamical systems. After the introduction and examples part we start by studying integrability of such sub-bundles under different regularity and dynamical assumptions. Then we formulate a continuous version of the classical Frobenius theorem and state some applications to such bundles, to ODE and PDE. Finally we close of by stating some ongoing work related to interactions between integrability, sub-Riemannian geometry and contact geometry.
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Ter-Braak, Floris. "Perturbed KdV equations and their integrability properties." Thesis, Durham University, 2018. http://etheses.dur.ac.uk/12550/.
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In this thesis we investigate the integrability properties of the regularized long-wave (RLW) equation and modified regularized long-wave (mRLW) equation as perturbations of the integrable Korteweg-de Vries (KdV) equation. We study various properties of numerical mRLW three-soliton scattering and compare these with the corresponding RLW soliton solutions. We find that the numerical mRLW solitons behave much like integrable solitons in the sense that the only result of the three-soliton interaction is the phase shift each soliton experiences, which is approximately equal to the sum of pairwise phase shifts. Furthermore, we investigate the so-called quasi-integrability properties of these RLW and mRLW simulations. Using both analytical and numerical methods, we argue that these models possess an infinite amount of asymptotically conserved charges, i.e., quasi-conserved charges, which are observed in multi-soliton interactions. Finally, we also simulate numerical RLW and mRLW solutions in the presence of additional perturbing terms. This allows us to study soliton-radiation interactions and we find that for certain perturbations, these interactions preserve the quasi-conservation laws to a certain extend.
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Fontanella, Andrea. "Black horizons and integrability in string theory." Thesis, University of Surrey, 2018. http://epubs.surrey.ac.uk/849271/.
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This thesis is devoted to the study of geometric aspects of black holes and integrable structures in string theory. In the first part, symmetries of the horizon and its bulk extension will be investigated. We investigate the horizon conjecture beyond the supergravity approximation, by considering alpha prime corrections of heterotic supergravity in perturbation theory, and show that standard global techniques can no longer be applied. A sufficient condition to establish the horizon conjecture will be identified. As a consequence of our analysis, we find a no-go theorem for AdS2 backgrounds in heterotic theory. The bulk extension of a prescribed near-horizon geometry will then be considered in various theories. The horizon fields will be expanded at first order in the radial coordinate. The moduli space of radial deformations will be proved to be finite dimensional, by showing that the moduli must satisfy elliptic PDEs. In the second part, geometric aspects and spectral properties of integrable anti-de Sitter backgrounds will be discussed. We formulate a Bethe ansatz in AdS2 x S2 x T6 type IIB superstring, overcoming the problem of the lack of pseudo-vacuum state affecting this background. In AdS3 x S3 x T4 type IIB superstring, we show that the S-matrix is annihilated by the boost generator of the q-deformed Poincarè superalgebra, and interpret this condition as a parallel equation for the S-matrix with respect to a connection on a fibre bundle. This hints that the algebraic problem associated with the scattering process can be geometrically rewritten. This allows us to propose a Universal S-matrix.
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Stepanchuk, Andrej. "Aspects of integrability in string sigma-models." Thesis, Imperial College London, 2015. http://hdl.handle.net/10044/1/28904.
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The recent success in applying integrability-based methods to study examples of gauge/gravity dualities in highly (super)symmetric settings motivates the question of whether such methods can be carried over to more physical and less symmetric cases. In this thesis we consider two such examples of string sigma-models, which interpolate between integrable or solvable limits. First we consider classical string motion on curved p-brane backgrounds for which the sigma-model interpolates between the integrable flat space and AdS(k)xS(k) coset or WZW sigma-models. We find that while the equations for particle (i.e. geodesic) motion are integrable in these backgrounds, the equations for extended string motion are not. The second example we consider is string theory on AdS3xS3xT4 with mixed Ramond-Ramond (R-R) and Neveu-Schwarz-Neveu-Schwarz (NS-NS) 3-form fluxes, which interpolates between the integrable pure R-R and the pure NS-NS theory that can be solved using CFT methods. The dispersion relation and S-matrix for world-sheet excitations, which are the essential ingredients in solving for the string spectrum, are only partially fixed by integrability and symmetry arguments. By constructing the mixed flux generalisation of the dyonic giant magnon soliton, which we show can be interpreted as a bound-state of excitations, we determine the dispersion relation for massive excitations. We also construct the mixed flux generalisation of the folded string on AdS3xS1 and show that, at leading order in large angular momentum on AdS3, its energy is given by the pure R-R expression with the string tension rescaled by the R-R flux coefficient. Further, we derive the bound-state S-matrix and its 1-loop correction by considering the scattering of dyonic giant magnons and plane waves. From this we deduce the semiclassical and 1-loop dressing phases in the massive sector S-matrix, which we find to agree with recent proposals.
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Lloyd, Thomas. "Advancing integrability for strings in AdS3/CFT2." Thesis, City University London, 2016. http://openaccess.city.ac.uk/14883/.
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In this thesis we develop techniques of integrability in the study of dualities between two-dimensional conformal field theories and theories of closed strings on three-dimensional Anti-de Sitter background geometries. For several years after integrability was first applied to the 3d/2d dualities, it was an unanswered question how to incorporate the so-called \massless modes" of these theories into the integrability machinery. Here we tackle this problem in several contexts. We show that in the classical integrable description of closed strings the implementation of the string Virasoro constraints needs to be modified for geometries with multiple factors where massless modes are present. We show further that with the correct implementation of the Virasoro constraints, massless modes can be included in integrability techniques for obtaining quantum corrections to physical quantities such as the energies of string solutions. Lastly, we consider the scattering of fundamental string excitations and derive all-loop expressions for the scattering matrix that includes both massless and massive excitations.
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Groha, Stefan. "Weak integrability breaking and full counting statistics." Thesis, University of Oxford, 2018. http://ora.ox.ac.uk/objects/uuid:9ea5d98c-0aa6-4ea3-a6b7-2e413c24811d.
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In this thesis two questions of equilibrium and non-equilibrium properties in many-body quantum mechanical systems are investigated. The first part is focused on probability distributions of quantum observables in many-body quantum systems. After an introduction to probability distributions, full counting statistics and the transverse field Ising model we give a brief overview over related experiments and then derive an expression for the probability distribution of the transverse field magnetization of a finite subsystem in any Gaussian state. We study the probability distribution in ground and thermal states as well as in a non-equilibrium setting after a quantum quench. We find an analytic expression for the time evolution after the quench and compare to numerics. The second part of the thesis is concerned with the stability of exact quasi-particle excitations of an integrable model after weak integrability breaking perturbations are introduced. For this we first discuss the stability of excitations in integrable systems and then give an introduction to the Heisenberg XXX-model in a magnetic field. After constructing exact excitations we calculate the decay rate in leading order perturbation theory using methods of integrability.
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Spalding, Kathryn. "Growth and integrability in multi-valued dynamics." Thesis, Loughborough University, 2018. https://dspace.lboro.ac.uk/2134/33483.
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This thesis is focused on the problem of growth and integrability in multi-valued dynamics generated by $SL_2 (\mathbb{Z})$ actions. An important example is given by Markov dynamics on the cubic surface $$x^2+ y^2 +z^2 = 3xyz,$$ generating all the integer solutions of this celebrated Diophantine equation, known as Markov triples. To study the growth problem of Markov numbers we use the binary tree representation. This allows us to define the Lyapunov exponents $\Lambda (x)$ as the function of the paths on this tree, labelled by $x \in \mathbb{R}P^1$. We prove that $\Lambda (x)$ is a $PGL_2 (\mathbb{Z})$-invariant function, which is zero almost everywhere but takes all values in $\left[ 0, \ln \varphi \right]$ (where $\varphi$ denotes the golden ratio). We also show that this function is monotonic, and that its restriction to the Markov-Hurwitz set of most irrational numbers is convex in the Farey parametrisation. We also study the growth problem for integer binary quadratic forms using Conway's topograph representation. It is proven that the corresponding Lyapunov exponent $\Lambda_Q(x) = 2 \Lambda(x)$ except for the paths along the Conway river. Finally, we study the tropical version of the Markov dynamics on the tropical version of the Cayley cubic proposed by Adler and Veselov, and show that it is semi-conjugated to the standard action of $SL_2(\mathbb{Z})$ on a torus. This implies the dynamics is ergodic, with the Lyapunov exponent and entropy given by the logarithm of the spectral radius of the corresponding matrix.
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Bombardelli, Diego <1980>. "Aspects of Integrability in Gauge/String Correspondence." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2009. http://amsdottorato.unibo.it/2244/1/Bombardelli_Diego_tesi.pdf.
Full textAbstract:
In this thesis, we present our work about some generalisations of ideas, techniques and physical interpretations typical for integrable models to one of the
most outstanding advances in theoretical physics of nowadays: the AdS/CFT correspondences.
We have undertaken the problem of testing this conjectured duality under various points of view, but with a clear starting point - the integrability - and with a clear
ambitious task in mind: to study the finite-size effects in the energy spectrum of certain string solutions on a side and in the anomalous dimensions of the gauge
theory on the other. Of course, the final desire woul be the exact comparison between these two faces of the gauge/string duality.
In few words, the original part of this work consists in application of well known integrability technologies, in large parte borrowed by the study of relativistic
(1+1)-dimensional integrable quantum field theories, to the highly non-relativisic and much complicated case of the thoeries involved in the recent conjectures of AdS5/CFT4 and AdS4/CFT3 corrspondences.
In details, exploiting the spin chain nature of the dilatation operator of N = 4 Super-Yang-Mills theory, we concentrated our attention on one of the most
important sector, namely the SL(2) sector - which is also very intersting for the QCD understanding - by formulating a new type of nonlinear integral equation (NLIE) based on a previously guessed asymptotic Bethe Ansatz. The solutions
of this Bethe Ansatz are characterised by the length L of the correspondent spin chain and by the number s of its excitations. A NLIE allows one, at least in principle, to make analytical and numerical calculations for arbitrary values of these parameters. The results have been rather exciting. In the important regime of high Lorentz spin, the NLIE clarifies how it reduces to a linear integral
equations which governs the subleading order in s, o(s0). This also holds in the regime with L ! 1, L/ ln s finite (long operators case). This region of parameters has been particularly investigated in literature especially because of an intriguing limit into the O(6) sigma model defined on the string side.
One of the most powerful methods to keep under control the finite-size spectrum of an integrable relativistic theory is the so called thermodynamic Bethe Ansatz (TBA). We proposed a highly non-trivial generalisation of this technique to the non-relativistic case of AdS5/CFT4 and made the first steps in order to determine
its full spectrum - of energies for the AdS side, of anomalous dimensions for the CFT one - at any values of the coupling constant and of the size.
At the leading order in the size parameter, the calculation of the finite-size corrections is much simpler and does not necessitate the TBA. It consists in deriving
for a nonrelativistc case a method, invented for the first time by L¨uscher to compute the finite-size effects on the mass spectrum of relativisic theories. So, we
have formulated a new version of this approach to adapt it to the case of recently found classical string solutions on AdS4 × CP3, inside the new conjecture of an
AdS4/CFT3 correspondence. Our results in part confirm the string and algebraic curve calculations, in part are completely new and then could be better understood
by the rapidly evolving developments of this extremely exciting research field.
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Bombardelli, Diego <1980>. "Aspects of Integrability in Gauge/String Correspondence." Doctoral thesis, Alma Mater Studiorum - Università di Bologna, 2009. http://amsdottorato.unibo.it/2244/.
Full textAbstract:
In this thesis, we present our work about some generalisations of ideas, techniques and physical interpretations typical for integrable models to one of the
most outstanding advances in theoretical physics of nowadays: the AdS/CFT correspondences.
We have undertaken the problem of testing this conjectured duality under various points of view, but with a clear starting point - the integrability - and with a clear
ambitious task in mind: to study the finite-size effects in the energy spectrum of certain string solutions on a side and in the anomalous dimensions of the gauge
theory on the other. Of course, the final desire woul be the exact comparison between these two faces of the gauge/string duality.
In few words, the original part of this work consists in application of well known integrability technologies, in large parte borrowed by the study of relativistic
(1+1)-dimensional integrable quantum field theories, to the highly non-relativisic and much complicated case of the thoeries involved in the recent conjectures of AdS5/CFT4 and AdS4/CFT3 corrspondences.
In details, exploiting the spin chain nature of the dilatation operator of N = 4 Super-Yang-Mills theory, we concentrated our attention on one of the most
important sector, namely the SL(2) sector - which is also very intersting for the QCD understanding - by formulating a new type of nonlinear integral equation (NLIE) based on a previously guessed asymptotic Bethe Ansatz. The solutions
of this Bethe Ansatz are characterised by the length L of the correspondent spin chain and by the number s of its excitations. A NLIE allows one, at least in principle, to make analytical and numerical calculations for arbitrary values of these parameters. The results have been rather exciting. In the important regime of high Lorentz spin, the NLIE clarifies how it reduces to a linear integral
equations which governs the subleading order in s, o(s0). This also holds in the regime with L ! 1, L/ ln s finite (long operators case). This region of parameters has been particularly investigated in literature especially because of an intriguing limit into the O(6) sigma model defined on the string side.
One of the most powerful methods to keep under control the finite-size spectrum of an integrable relativistic theory is the so called thermodynamic Bethe Ansatz (TBA). We proposed a highly non-trivial generalisation of this technique to the non-relativistic case of AdS5/CFT4 and made the first steps in order to determine
its full spectrum - of energies for the AdS side, of anomalous dimensions for the CFT one - at any values of the coupling constant and of the size.
At the leading order in the size parameter, the calculation of the finite-size corrections is much simpler and does not necessitate the TBA. It consists in deriving
for a nonrelativistc case a method, invented for the first time by L¨uscher to compute the finite-size effects on the mass spectrum of relativisic theories. So, we
have formulated a new version of this approach to adapt it to the case of recently found classical string solutions on AdS4 × CP3, inside the new conjecture of an
AdS4/CFT3 correspondence. Our results in part confirm the string and algebraic curve calculations, in part are completely new and then could be better understood
by the rapidly evolving developments of this extremely exciting research field.
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Brini, Andrea. "Duality and integrability in topological string theory." Doctoral thesis, SISSA, 2009. http://hdl.handle.net/20.500.11767/4929.
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Little, Steven. "INTEGRABILITY OF A SINGULARLY PERTURBED MODEL DESCRIBING GRAVITY WATER WAVES ON A SURFACE OF FINITE DEPTH." Master's thesis, University of Central Florida, 2008. http://digital.library.ucf.edu/cdm/ref/collection/ETD/id/3285.
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Our work is closely connected with the problem of splitting of separatrices (breaking of homoclinic orbits) in a singularly perturbed model describing gravity water waves on a surface of finite depth. The singularly perturbed model is a family of singularly perturbed fourth-order nonlinear ordinary differential equations, parametrized by an external parameter (in addition to the small parameter of the perturbations). It is known that in general separatrices will not survive a singular perturbation. However, it was proven by Tovbis and Pelinovsky that there is a discrete set of exceptional values of the external parameter for which separatrices do survive the perturbation. Since our family of equations can be written in the Hamiltonian form, the question is whether or not survival of separatrices implies integrability of the corresponding equation. The complete integrability of the system is examined from two viewpoints: 1) the existence of a second first integral in involution (Liouville integrability), and 2) the existence of single-valued, meromorphic solutions (complex analytic integrability). In the latter case, a singular point analysis is done using the technique given by Ablowitz, Ramani, and Segur (the ARS algorithm) to determine whether the system is of Painlevé-type (P-type), lacking movable critical points. The system is shown by the algorithm to fail to be of P-type, a strong indication of nonintegrability.M.S.
Department of Mathematics
Sciences
Mathematical Science MS
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Jogia, Danesh Michael Mathematics & Statistics Faculty of Science UNSW. "Algebraic aspects of integrability and reversibility in maps." Publisher:University of New South Wales. Mathematics & Statistics, 2008. http://handle.unsw.edu.au/1959.4/40947.
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We study the cause of the signature over finite fields of integrability in two dimensional discrete dynamical systems by using theory from algebraic geometry. In particular the theory of elliptic curves is used to prove the major result of the thesis: that all birational maps that preserve an elliptic curve necessarily act on that elliptic curve as addition under the associated group law. Our result generalises special cases previously given in the literature. We apply this theorem to the specific cases when the ground fields are finite fields of prime order and the function field $mathbb{C}(t)$. In the former case, this yields an explanation of the aforementioned signature over finite fields of integrability. In the latter case we arrive at an analogue of the Arnol'd-Liouville theorem. Other results that are related to this approach to integrability are also proven and their consequences considered in examples. Of particular importance are two separate items: (i) we define a generalization of integrability called mixing and examine its relation to integrability; and (ii) we use the concept of rotation number to study differences and similarities between birational integrable maps that preserve the same foliation. The final chapter is given over to considering the existence of the signature of reversibility in higher (three and four) dimensional maps. A conjecture regarding the distribution of periodic orbits generated by such maps when considered over finite fields is given along with numerical evidence to support the conjecture.
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Blom, Jonas. "Topics in dynamical systems : integrability and power control /." Stockholm : Tekniska högsk, 1999. http://www.lib.kth.se/abs99/blom0924.pdf.
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MacIntyre, Alistair. "On the integrability of the sine-Gordon system." Thesis, Durham University, 1997. http://etheses.dur.ac.uk/5011/.
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This thesis investigates the integrability of the sine-Gordon system of nonlinear partial differential equations when the dependent variables are subject to some very particular boundary conditions. In chapter 1 the sine-Gordon system is introduced and, with N ϵ Z, P, Q ϵ R, the sets of initial-boundary value problems A(_N) and B(_P,Q) are defined. In the set A(_N) at the spatial variable x is unbounded and the boundary conditions are fixed by initially choosing the topological charge N. This set of problems is the one usually associated with the sine-Gordon system. In the set B(_P,Q) the spatial coordinate is constrained to the semi-line (-oo,0) and there exists two boundary parameters P,Q ϵ R to be chosen a priori. It is the study of this second set of initial-boundary value problems for arbitrary P, Q which forms all the original work of this dissertation. The study presented here is primarily concerned with the development of three separate inverse scattering methods for solving these sets of initial-boundary value problems. The first of these is developed in chapter 3 and is applicable to a subset of the problems in A(_N). The method is the one usually associated with the sine-Gordon system and studies the asymptotics of the initial data as x → ±oo. It is included in this thesis for completeness and as background for the original material which follows. Next, in chapters 4 and 5, the inverse scattering methods appropriate to initial-boundary value problems in subsets of B(_P,O) and B(_P,Q#O) are constructed. In these cases it is important to realise that it is only possible to study the asymptotics of the initial data as x → -oo. Once these three methods have been formulated they are used to find soliton solutions and infinite sets of integrals of motion for these boundary value problems. When a boundary is present at x = 0 the interaction of the solitons with this boundary is studied. These topics are addressed in chapter 6. Finally in chapter 7 the question of the integrability of both sets of problems is addressed. By interpreting the various inverse scattering methods in terms of canonical coordinate transformations of phase space it is seen that the existence of such methods can be viewed as a constructive proof of the integrability of these boundary value problems.
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Alhily, Shatha Sami Sejad. "Higher integrability of the gradient of conformal maps." Thesis, University of Sussex, 2013. http://sro.sussex.ac.uk/id/eprint/45885/.
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Alves, Victor César Costa [UNESP]. "Painlevé Integrability and mixed P_III-P_V system solutions." Universidade Estadual Paulista (UNESP), 2017. http://hdl.handle.net/11449/149963.
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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)
O presente trabalho trata de um abordagem de aplicações em física dos métodos matemáticos de integrabilidade de Painlevé, por outro lado também aborda o formalismo de hierarquias integráveis e o modelo de 2M-bosons onde são usados métodos de equações diferenciais bem como um método para soluções usando aproximantes de Padé.
The current work aims at applications of mathematical methods of Painlevé integrability in physics, on the other side it also approaches the integrable hierarchies formalism and the 2M-bose model where differential equations methods are used as well as a method for solutions using Padé approximants.
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Alves, Victor César Costa. "Painlevé Integrability and mixed P_III-P_V system solutions /." São Paulo, 2017. http://hdl.handle.net/11449/149963.
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Orientador: Abraham Hirsz ZimermanAbstract: The current work aims at applications of mathematical methods of Painlevé integrability in physics, on the other side it also approaches the integrable hierarchies formalism and the 2M-bose model where differential equations methods are used as well as a method for solutions using Padé approximants.
Resumo: O presente trabalho trata de um abordagem de aplicações em física dos métodos matemáticos de integrabilidade de Painlevé, por outro lado também aborda o formalismo de hierarquias integráveis e o modelo de 2M-bosons onde são usados métodos de equações diferenciais bem como um método para soluções usando aproximantes de Padé.
Mestre
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Pittelli, Antonio. "Dualities and integrability in low dimensional AdS/CFT." Thesis, University of Surrey, 2016. http://epubs.surrey.ac.uk/812577/.
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Scientific abstract In this dissertation we perform a series of study concerning dualities and integrability properties underlying the AdS3/CFT2 and AdS2/CFT1 correspondences. These are particularly interesting because symmetry do not constrain the dynamics of AdS3 and AdS3 superstrings in the same way as in the higher dimensional instances of AdS/CFT, allowing for novel phenomena such as the presence of massless worldsheet modes or non-coset fermions. We will investigate the self-duality of Green–Schwarz supercoset sigma models on AdSd⇥Sd⇥Sd (d = 2, 3), whose isometry supergroups are (d − 1) copies of the exceptional Lie supergroup D(2, 1; ↵). Our main finding is that additional complex T-dualities along one of the spheres Sd are needed to map the superstring action to itself. Importantly, this proves dual superconformal symmetry of their CFT duals via AdS/CFT. Dual superconformal symmetry is strictly related to integrability, which we study in depth for both AdS3 and AdS2 superstrings. Indeed, we will derive the exact S-matrix conjectured to be related to the massive modes of type IIB AdS2 ⇥ S2 ⇥ T6 superstrings. This S-matrix is psuc(1|1) invariant and it was found to be in perfect agreement with the tree-level result following from string perturbation theory. We also unveil the Yangian algebra ensuring the integrability of the AdS2⇥S2⇥T6 superstring in the planar limit, Y[psu(1|1)c], as well as its secret symmetries. By using the RTT realisation, we provide two di↵erent representations of the Hopf algebra: one is reminiscent of the Yangian underlying AdS5/CFT4, but it is not of evaluation type. The other representation, obtained from co-commutativity, is instead of evaluation type. We explore two limits of the S-matrix for AdS2/CFT1: one is the classical r-matrix, which is the first non-trivial order in the 1/g expansion, with g being the e↵ective tension in the AdS2 ⇥ S2 ⇥ T6 superstring action. In this limit, corresponding to classical strings, we found that secret symmetries not only are present, but also essential to formulate the classical rmatrix in a universal, representation independent form. On the other hand, the limit g ! 0, corresponding to the weakly coupled CFT1, shows that the dual integrable model is described by an e↵ective theory of free fermions on a periodic spin-chain if g = 0, while one obtains a non-trivial spin chain of XYZ type if g 6= 0. Finally, we investigate Yangian and secret symmetries for AdS3 type IIB superstring backgrounds, verifying the persistence of such structures in AdS3/CFT2. Especially, we find that the antipode map, related to crossing symmetry, exchanges in a non-trivial way left and right generators of Y[psu(1|1)2c], the Yangian underlying the integrability of AdS3 superstrings.
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Wolf, Martin. "On supertwistor geometry and integrability in super gauge theory." [S.l.] : [s.n.], 2006. http://deposit.ddb.de/cgi-bin/dokserv?idn=981911811.
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Mukanov, Askhat. "Integrability of Fourier transforms, general monotonicity, and related problems." Doctoral thesis, Universitat Autònoma de Barcelona, 2018. http://hdl.handle.net/10803/463043.
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El propòsit d'aquesta tesi és el d'estudiar les propietats d'integrabilitat i convergència de sèries i transformades de Fourier. Els resultats principals són els següents: 1. Incestiguem les propietats d'integrabilitat de sèries trigonomètriques amb coeficients que satisfan una condició de monotonia general i demostrem resultats del tipus Hardy-Littlewood, és a dir, equivalències entre normes de les sumes de sèries trigonomètriques i normes amb pesos dels seus coeficients de Fourier. Demostrem aquestes equivalències en espais de Lorentz i espais de Lebesgue amb pesos. 2. Estudiem propietats de suavitat de funcions que poden ser representades per mitjà de sèries trigonomètriques amb coefficients que satisfan una condició de monotonia general. Es demostra una equivalència del Lp-mòdul de suavitat d'aquestes funcions i les sumes amb pesos dels seus coeficients de Fourier. 3. Obtenim versions multi-dimensionals de teoremes de tipus Boas en relació a les propietats d'integrabilitat de transformades de Fourier de funcions que són monòtones en totes les variable. 4. Finalment, estudiem criteris per a la convergència uniforme de sèries trigonomètriques amb coefficients que satisfan una condició de monotonia general. En particular, generalitzem el conegut criteri de Chaundy-Jolliffe per a la convergència uniforme de sèries sinusoidals i obtenim el resultat corresponent per sèries cosinusoidals. A més, provem condicions necessàries i suficients per tal que les sumes partials de Fourier d'aquestes sèries tinguin un cert ordre de convergència.This thesis is devoted to the study of integrability and convergence properties of Fourier series and transforms. The main results are the following. 1. We investigate the integrability properties of trigonometric series with general monotone coefficients and prove the Hardy-Littlewood-type results, i.e., equivalences of the norms of sums of trigonometric series and weighted norms of their Fourier coefficients. We prove such equivalences for the Lorentz and weighted Lebesgue spaces. Here we deal with the trigonometric series with general monotone coefficients. 2. We study the smoothness properties of functions that can be represented by trigonometric series with general monotone coefficients. The equivalence of the Lp-modulus of smoothness of such functions and weighted sums of their Fourier coefficients is proved. 3. We obtain the multidimensional versions of Boas-type theorem on integrability properties of the Fourier transforms of monotone in each variable functions. 4. Finally, we study criteria for the uniform convergence of trigonometric series with general monotone coefficients. In particular, we generalize the well-known Chaundy-Jolliffee criterion for the uniform convergence of sine series and obtain the corresponding result for cosine series. Moreover, we prove necessary and sufficient conditions for partial Fourier sums of such series to have certain convergence rate.
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Ward, Chloe. "Discrete integrability and nonlinear recurrences with the laurent property." Thesis, University of Kent, 2013. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.655224.
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In this thesis we consider four families of nonlinear recurrences which can be shown to either fit into Fomin and Zelevinsky's framework of cluster algebras, or the more general setting of Laurent phenomenon algebras given recently by Lam and Pylyavskyy. It then follows that each family of recurrences we study possesses the Laurent property. Our main interest lies in the linearisability and Liouville integrability of the maps defined by these families. We prove that three of the families are linearisable. Firstly, we study examples arising in the context of cluster algebras and provide a detailed survey of recent results of Fordy and Hone, with the aim to develop the understanding of Liouville integrability for odd order examples of this type. Following this, we extend the results of Heideman and Hogan, to show that their family of nonlinear recurrences is linearisable for general initial data. The third order example from this family of recurrences admits a different generalisation of a new family of nonlinear recurrences for which we also show the general case to be linearisable. We also present a connection with the dressing chain which provides a generating function for the first integrals for recurrences of this type. Lastly we study a family of Somos-type recurrences which is not linearisable. However we present the method of finding the Lax representation from which we can generate first integrals and show that the examples of recurrences studied here, arising in the context of cluster algebras, are Liouville integrable.
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Kagan, David. "Integrability and string theory : σ-models and spin chains." Thesis, University of Cambridge, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.613236.
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Lesame, William Mphepeng. "A study of integrability conditions for irrotational dust spacetimes." Doctoral thesis, University of Cape Town, 1998. http://hdl.handle.net/11427/21328.
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Bibliography: pages 139-145.This thesis examines consistency conditions for fluid solutions of the field equations of general relativity. The exact non-linear dynamic equations for a generic irrotational dust spacetime are consistent. To analyse conditions characterizing pure gravity waves, linearization instability in general relativity and consistency of the so-called "silent universes", further exact conditions are imposed locally on irrotational dust. These are classified into Class II conditions, which change evolution equations into constraint equations, and Class I and III conditions, which do not doso-rather they add a new constraint, leaving the propagation equations unchanged in form. Class I conditions are imposed on terms in the constraint equations, while Class II and III conditions are imposed on terms in the evolution equations. In the Class I case it is shown that for irrotational dust space times the divergence-free magnetic Weyl tensor and the divergence-free electric Weyl tensor (necessary conditions for gravity waves interacting with matter), both imply integrability conditions in the exact non-linear case. The integrability conditions for the divergence-free magnetic Weyl tensor are identically satisfied in the linearized perturbation case, but are non-trivial in the exact non-linear case. This leads to a linearization instability in these models. The integrability conditions for the divergence-free electric Weyltensor are non-trivial in both the linear and non-linear cases. The Class II case focuses on irrotational silent cosmological dust models characterized by vanishing magnetic Weyl tensor and vanishing electric Weyl tensor. In both these models there exist a series of integrability conditions that need to be satisfied. Integrability conditions for the zero magnetic Weyl tensor condition hold identically for linearized case, but are non-trivial in the exact non-linear case. Thus there is also a linearization instability. The zero electric Weyl tensor condition leads to a chain of non-trivial integrability conditions in both the linear and non-linear cases. Because of the complexity of the integrability conditions, it is highly unlikely that there is a large class of models in both the silent zero magnetic Weyl tensor case and the silent zero electric Weyl tensor case.
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Granet, Étienne. "Advanced integrability techniques and analysis for quantum spin chains." Thesis, Université Paris-Saclay (ComUE), 2019. http://www.theses.fr/2019SACLS239.
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Dans cette thèse sont principalement étudiés des systèmes quantiques intégrables critiques avec l’ansatz de Bethe qui ont la propriété particulière d’être non-unitaires ou non-compacts. Ceci concerne des modèles de physique statistique non-locaux tels que la percolation, mais aussi par exemple les systèmes désordonnés. Ce manuscrit présente à la fois des études détaillées de la limite continue de modèles intégrables sur réseau, et développe de nouvelles techniques pour étudier cette correspondance. Dans une première partie nous étudions en détail la limite continue de chaînes de superspins non-unitaires (et parfois non-compactes) qui ont une symétrie orthosymplectique. Nous montrons qu’il s’agit de modèles sigma sur supersphère en calculant leur spectre avec la théorie des champs, avec l’ansatz de Bethe, et numériquement. Leur non-unitarité autorise une brisure spontanée de symétrie habituellement interdite par le théorème de Mermin-Wagner. Leur caractère de perturbation marginale d’une théorie conforme des champs logarithmique est particulièrement étudié. Nous établissons également une correspondance précise entre le spectre et des configurations de boucles avec intersections, et obtenons de nouveaux exposants critiques pour les chemins non-recouvrants compacts ainsi que leurs corrections logarithmiques multiplicatives. Cette étude fut par ailleurs l’occasion de développer une nouvelle méthode pour calculer le spectre d’excitation d’une chaîne de spin quantique critique à partir de l’ansatz de Bethe, incluant les corrections logarithmiques, également en présence de racines de Bethe dites ’en chaînes’, et qui évite les méthodes de Wiener-Hopf et les équations intégrales non-linéaires. Dans une deuxième partie nous abordons l’influence d’un champ magnétique sur une chaîne de spin quantique et montrons que des séries convergentes peuvent être obtenues pour plusieurs quantités physiques telles que l’aimantation acquise ou les exposants critiques, dont les coefficients peuvent être calculés efficacement par récurrence. La structure de ces relations de récurrence permet d’étudier génériquement le spectre d’excitation, et elles sont applicables y compris dans certains cas où les racines de Bethe sont sur une courbe dans le plan complexe. Nous espérons que l’étude de la continuation analytique de ces séries puisse être utile pour les chaînes non-compactes. Par ailleurs, nous montrons que les fluctuations à l’intérieur de la courbe arctique du modèle à six vertex avec conditions aux bords de type mur sont décrites par un champ Gaussien libre avec une constante de couplage dépendant de la position, qui peut être calculée à partir de l’énergie libre de la chaîne XXZ avec une torsion imaginaire dans un champ magnétiqueThis thesis mainly deals with integrable quantum critical systems that exhibit peculiar features such as non-unitarity or non-compactness, through the technology of Bethe ansatz. These features arise in non-local statistical physics models such as percolation, but also in disordered systems for example. The manuscript both presents detailed studies of the continuum limit of finite-size lattice integrable models, and develops new techniques to study this correspondence. In a first part we study in great detail the continuum limit of non-unitary (and sometimes non-compact) super spin chains with orthosymplectic symmetry which is shown to be supersphere sigma models, by computing their spectrum from field theory, from the Bethe ansatz, and numerically. The non-unitarity allows for a spontaneous symmetry breaking usually forbidden by the Mermin-Wagner theorem. The fact that they are marginal perturbations of a Logarithmic Conformal Field Theory is particularly investigated. We also establish a precise correspondence between the spectrum and intersecting loops configurations, and derive new critical exponents for fully-packed trails, as well as their multiplicative logarithmic corrections. During this study we developed a new method to compute the excitation spectrum of a critical quantum spin chain from the Bethe ansatz, together with their logarithmic corrections, that is also applicable in presence of so-called ’strings’, and that avoids Wiener-Hopf and Non-Linear Integral Equations. In a second part we address the problem of the behavior of a spin chain in a magnetic field, and show that one can derive convergent series for several physical quantities such as the acquired magnetization or the critical exponents, whose coefficients can be efficiently and explicitely computed recursively using only algebraic manipulations. The structure of the recurrence relations permits to study generically the excitation spectrum content – moreover they are applicable even to some cases where the Bethe roots lie on a curve in the complex plane. It is our hope that the analytic continuation of such series might be helpful the study non-compact spin chains, for which we give some flavour. Besides, we show that the fluctuations within the arctic curve of the six-vertex model with domain-wall boundary conditions are captured by a Gaussian free field with space-dependent coupling constant that can be computed from the free energy of the periodic XXZ spin chain with an imaginary twist and in a magnetic field
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Dartois, Stephane. "Random Tensor models : Combinatorics, Geometry, Quantum Gravity and Integrability." Thesis, Sorbonne Paris Cité, 2015. http://www.theses.fr/2015USPCD104/document.
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Dans cette thèse nous explorons différentes facettes des modèles de tenseurs aléatoires. Les modèles de tenseurs aléatoires ont été introduits en physique dans le cadre de l'étude de la gravité quantique. En effet les modèles de matrices aléatoires, qui sont un cas particuliers de modèles de tenseurs, en sont une des origines. Ces modèles de matrices sont connus pour leur riche combinatoire et l'incroyable diversité de leurs propriétés qui les font toucher tous les domaines de l'analyse, la géométrie et des probabilités. De plus leur étude par les physiciens ont prouvé leur efficacité en ce qui concerne l'étude de la gravité quantique à deux dimensions. Les modèles de tenseurs aléatoires incarnent une généralisation possible des modèles de matrices. Comme leurs cousins, les modèles de matrices, ils posent questions dans les domaines de la combinatoire (comment traiter les cartes combinatoires d dimensionnelles ?), de la géométrie (comment contrôler la géométrie des triangulations générées ?) et de la physique (quel type d'espace-temps produisent-ils ? Quels sont leurs différentes phases ?). Cette thèse espère établir des pistes ainsi que des techniques d'études de ces modèles. Dans une première partie nous donnons une vue d'ensemble des modèles de matrices. Puis, nous discutons la combinatoire des triangulations en dimensions supérieures ou égales à trois en nous concentrant sur le cas tridimensionnelle (lequel est plus simple à visualiser). Nous définissons ces modèles et étudions certaines de leurs propriétés à l'aide de techniques combinatoires permettant de traiter les cartes d dimensionnelles. Enfin nous nous concentrons sur la généralisation de techniques issues des modèles de matrices dans le cas d'une famille particulières de modèles de tenseurs aléatoires. Ceci culmine avec le dernier chapitre de la thèse donnant des résultats partiels concernant la généralisation de la récurrence topologique de Eynard et Orantin à cette famille de modèles de tenseursIn this thesis manuscript we explore different facets of random tensor models. These models have been introduced to mimic the incredible successes of random matrix models in physics, mathematics and combinatorics. After giving a very short introduction to few aspects of random matrix models and recalling a physical motivation called Group Field Theory, we start exploring the world of random tensor models and its relation to geometry, quantum gravity and combinatorics. We first define these models in a natural way and discuss their geometry and combinatorics. After these first explorations we start generalizing random matrix methods to random tensors in order to describes the mathematical and physical properties of random tensor models, at least in some specific cases
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War, Khadim Mbacke. "Integrability of continuous bundles and applications to dynamical systems." Doctoral thesis, SISSA, 2016. http://hdl.handle.net/20.500.11767/4883.
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Auríchio, Vinícius Henrique. "Deformações quase-integráveis do modelo de Bullough-Dodd." Universidade de São Paulo, 2014. http://www.teses.usp.br/teses/disponiveis/76/76131/tde-08092014-153357/.
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Esta dissertação investiga uma particular deformação do modelo de Bullough-Dodd. Não se sabe se tais deformações são ou não integráveis, ainda que nossas simulações numéricas apresentem soluções solitônicas. Exploramos o conceito de quase-integrabilidade em mais esse contexto e mostramos um argumento analítico para a quase-conservação das cargas (isto é, as cargas variam com o tempo, mas seus valores inicial e final são os mesmos). Mesmo quando o argumento analítico não pode fazer previsões, nossas simulações mostram que as cargas apresentam o mesmo comportamento. Isso sugere que as deformações consideradas são integráveis e ainda há espaço para explorá-las.This dissertation investigates a particular deformation of the Bullough-Dodd model. It\'s unknown if such deformations are integrable or not, yet they present solitonic solutions which were obtained through numerical simulations. We further explore the concept of quasi-integrability in this context, showing analiticaly that at least for some sets of parameters, the charges are quasi-conserved (i.e. the charges vary over time, but it\'s initial and final values are the same). Even when the analitical argument can\'t predict what happens, our simulations show the same charge behaviour. This suggests that those deformations are integrable and can be further explored.
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Grosse, Harald, Karl-Georg Schlesinger, and grosse@doppler thp univie ac at. "A Suggestion for an Integrability Notion for Two Dimensional Spin." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1015.ps.
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SCOLERI, STEFANO. "PERTURBATIVE APPROACH TO INTEGRABILITY IN THREE-DIMENSIONAL CHERN-SIMONS THEORIES." Doctoral thesis, Università degli Studi di Milano, 2013. http://hdl.handle.net/2434/217569.
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One of the most fascinating discoveries of contemporary Theoretical Physics is
the AdS/CFT correspondence relating gauge theories to gravity theories. Soon after its formulation, tremendous developments allowed to obtain a deep comprehension of the four-dimensional N=4 SYM theory and, in particular, led to the discovery of integrable structures both in the gauge theory itself and in its string counterpart. In
the last few years, much attention was devoted to the study of supersymmetric Chern-Simons-matter theories in three dimensions. In this class of theories a distinguished role is played by the N=6 ABJM model which is a U(N) x U(N) superconformal
gauge theory. Indeed, in the large N limit, the ABJM theory has been conjectured to be the AdS/CFT dual description of M-theory on an
AdS4 x S7/Zk background and, for k << N << k^5, of type IIA string theory on
AdS4 × CP3. For this reason, soon after its discovery the ABJM model has quickly
become the ideal three-dimensional playground to study AdS/CFT as much as N=4 SYM has been in the four-dimensional case. Quite surprisingly, the ABJM model
seems to share a number of notable properties with N=4 SYM theory even though the two theories are a priori different in nature. One of the common features is provided by the fact that also in planar ABJM theory integrable structures naturally show up. In particular, a Bethe Ansatz approach to the computation of anomalous
dimensions is possible and a set of all-loop Bethe equations was formulated. These equations are very similar to those of SYM theory, however, starting at eight loops,
important new features are believed to appear, in particular in connection with the appearance of the dressing phase. With the aim to check such a picture, we analyze for the first time the form of the eight-loop dilatation operator in the SU(2) x SU(2) sector
and, adopting a perturbative approach based on superspace techniques, we directly extract the value of the unknown leading order coefficient of the dressing phase from supergraphs involving maximal interactions.
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Högner, Moritz. "Anti-self-dual fields and manifolds." Thesis, University of Cambridge, 2013. https://www.repository.cam.ac.uk/handle/1810/244668.
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In this thesis we study anti–self–duality equations in four and eight dimensions on manifolds of special Riemannian holonomy, among these hyper–Kähler, Quaternion–Kähler and Spin(7)–manifolds. We first consider the octonionic anti–self–duality equations on manifolds with holonomy Spin(7). We construct explicit solutions to their symmetry reductions, the non–abelian Seiberg–Witten equations, with gauge group SU(2). These solutions are singular for flat and Eguchi–Hanson backgrounds, however we find a solution on a co–homogeneity one hyper–Kähler metric with a domain wall, and the solution is regular away from the wall. We then turn to Quaternion–Kähler four–manifolds, which are locally determined by one scalar function subject to Przanowski’s equation. Using twistorial methods we construct a Lax Pair for Przanowski’s equation, confirming its integrability. The Lee form of a compatible local complex structure gives rise to a conformally invariant differential operator, special cases of the associated generalised Laplace operator are the conformal Laplacian and the linearised Przanowski operator. Using recursion relations we construct a contour integral formula for perturbations of Przanowski’s function. Finally, we construct an algorithm to retrieve Przanowski’s function from twistor data. At last, we investigate the relationship between anti–self–dual Einstein metrics with non–null symmetry in neutral signature and pseudo–, para– and null–Kähler metrics. We classify real–analytic anti–self–dual null–Kähler metrics with a Killing vector that are conformally Einstein. This allows us to formulate a neutral signature version of Tod’s result, showing that around non-singular points all real–analytic anti–self–dual Einstein metrics with symmetry are conformally pseudo– or para–Kähler.
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Simon, i. Estrada Sergi. "On the Meromorphic Non-Integrability of Some Problems in Celestial Mechanics." Doctoral thesis, Universitat de Barcelona, 2007. http://hdl.handle.net/10803/2110.
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In this thesis, we present a proof of the meromorphic non-integrability for some problems arising from Celestial Mechanics, as well as a new necessary condition for partial integrability in a wider Hamiltonian setting. First of all, a simpler proof is added to those already existing for the Three-Body Problem with arbitrary masses. The N-Body Problem with equal masses is also proven non-integrable. Second of all, a further strengthening of a prior existing result allows us to detect obstructions to the existence of a single additional first integral for classical Hamiltonians with a homogeneous potential. Third of all, using the aforementioned new result, we have proven the non-existence of an additional integral both for the general Three-Body Problem (hence generalizing, in a certain sense, Bruns'Theorem) and for the equal-mass Problem for N=4,5,6. Fourth of all, finally, we have proven the non-integrability of Hill's problem using the most general instance of the Morales-Ramis Theorem.A varying degree of theoretical complexity was involved in these results. Indeed, the proofs involving the given instances of the N-Body Problem required nothing but the exploration of the eigenvalues of a given matrix, with the advantage of knowing four of them explicitly. Thus, not all variational equations were needed but those not corresponding to these four eigenvalues -- this is exactly what transpires from the system reduction and subsequent introduction of normal variational equations, as done by S. Ziglin, J. J. Morales-Ruiz, J.-P. Ramis, and others. Hill's Problem, however, required the whole variational system since only thanks to the special functions introduced in the process of variation of constants was it possible to assure the presence of obstructions to integrability.
These results appear to qualify differential Galois theory, and especially a new incipient theory stemming from it, as an amenable setting for the detection of obstructions to total and partial Hamiltonian integrability.
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Santallusia, Esvert Xavier. "Contribution to the center and integrability problems in planar vector fields." Doctoral thesis, Universitat de Lleida, 2017. http://hdl.handle.net/10803/402941.
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Aquesta tesi consta d'un primer capítol introductori, set capítols amb diferents resultats i una bibliografia.
El primer capítol conté la definició i els resultats previs necessaris per abordar la resta de la memòria. El capítol 2 i 3 estan molt relacionats. En el primer es descriu un mètode alternatiu per al còmput de les constants de Poincaré--Liapunov. A diferència de mètodes anteriors, el mètode presentat no requereix el càlcul d'integrals i d'ona de forma explícita les constants de Poincaré--Liapunov. En el tercer capítol es descriu com s'ha implementat aquest nou mètode i els resultats que d'ona per a sistemes quadràtics i sistemes amb termes no lineals cúbics homogenis.
El quart capítol es centra en equacions d'Abel i la seva integrabilitat. Es descriu la forma d'una integral primera que sigui algebraica en funcíó de les variables dependents i es donen múltiples exemples d'equacions d'Abel integrables en aquest sentit. En el cinquè
capítol també s'aborda el problema de la integrabilitat però per a equacions diferencials en el pla definides per funcions analítiques. Es fa un reescalat de les variables dependents i de la variable independent amb un paràmetre "epsilon" que està elevat a potències senceres (blow-up paramètric) de forma que el sistema resultant sigui analític en "epsilon". Es d'ona un mètode que aprofita que una integral primera, si existeix, ha de ser analítica en el paràmetre a fi de trobar condicions per a l'existència d'aquesta integral primera. D'aquesta manera es defineix el que s'anomenen variables essencials del sistema.
Els darrers tres capítols versen sobre les equacions d'Abel i el problema del centre. En general es consideren equacions d'Abel trigonomètriques. En el sisè capítol es donen algunes condicions necessàries i suficients per a que una equació d'Abel trigonomètrica definida per polinomis trigonomètrics de grau fins a 3 tingui un centre. Tots els exemples donats en aquest capítol tenen un centre universal. En el capítol setè es d'ona un exemple d'una equació d'Abel trigonomètrica definida per polinomis trigonomètrics de grau 3 que té un centre que no és universal. D'aquesta manera es resol un problema obert: determinar el grau més petit pel qual un equació d'Abel trigonomètrica amb centre no és de composició. El darrer capítol tracta equacions d'Abel trigonomètriques i polinomials i d'ona un compendi dels darrers resultats coneguts i conjectures sobre el problema del centre en aquestes equacions. També es donen exemples nous d'equacions d'Abel amb centre.Esta tesis consta de un primer capítulo introductorio, siete capítulos con diferentes resultados y una bibliografía. El primer capítulo contiene la definición y los resultados previos necesarios para abordar el resto de la memoria. Los capítulos 2 y 3 están muy relacionados. En el primero se describe un método alternativo para el cómputo de las constantes de Poincaré--Liapunov. A diferencia de métodos anteriores, el método presentado no requiere el cálculo de integrales y da de forma explícita las constantes de Poincaré--Liapunov. En el tercer capítulo se describe cómo se ha implementado este nuevo método y los resultados que da para sistemas cuadráticos y sistemas con términos no lineales cíbicos homogéneos. El cuarto capítulo se centra en ecuaciones de Abel y su integrabilidad. Se describe la forma de una integral primera que sea algebraica en función de las variables dependientes y se dan múltiples ejemplos de ecuaciones de Abel integrables en este sentido. En el quinto capítulo también se aborda el problema de la integrabilidad pero para ecuaciones diferenciales en el plano definidas por funciones analíticas. Se hace un reescalado de las variables dependientes y de la variable independiente con un parámetro "epsilon" que está elevado a poténcias enteras (blow-up paramétrico) de forma que el sistema resultante sea analítico en "epsilon". Se da un método que aprovecha que una integral primera, si existe, debe ser analítica en el parámetro con el fin de encontrar condiciones para la existéncia de esta integral primera. De esta manera se define lo que se llaman variables esenciales del sistema. Los últimos tres capítulos versan sobre las ecuaciones de Abel y el problema del centro. En general se consideran ecuaciones de Abel trigonométricas. En el sexto capítulo se dan algunas condiciones necesarias y suficientes para que una ecuación de Abel definida por polinomios trigonométricos de grado hasta 3 tenga un centro. Todos los ejemplos dados en este capítulo tienen un centro universal. En la capítulo séptimo se da un ejemplo de una ecuación de Abel definida por polinomios trigonométricos de grado 3 que tiene un centro que no es universal. De esta manera se resuelve un problema abierto: determinar el grado mas pequeño por el que una ecuación de Abel trigonométrica con centro no es de composición. El último capítulo trata ecuaciones de Abel trigonométricas y polinomiales y da un compendio de los últimos resultados conocidos y conjeturas sobre el problema del centro en estas ecuaciones. También se dan ejemplos nuevos de ecuaciones de Abel con centro.
This thesis consists of a first introductory chapter, seven chapters with different results and a bibliography. The first chapter contains the definition and the previous results necessary to address the rest of the memory. Chapters 2 and 3 are closely related. In the first one, an alternative method is described for the computation of the Poincaré--Liapunov constants. Unlike previous methods, the presented method does not require the computation of primitives and gives an explicit expression of the Poincaré--Liapunov constants. The third chapter describes how this new method has been implemented and the results that it gives for quadratic systems and systems with homogeneous, cubic, non-linear terms. The fourth chapter focuses on Abel equations and their integrability. We describe the form of a first integral that is algebraic in function of the dependent variables and give more examples of equations of Abel integrable from this point of view. The fifth chapter also discusses the integrability problem but for differential equations in the plane defined by analytical functions. A rescaling of the dependent and the independent variables with a parameter "epsilon" which is elevated to integer powers (parametrical blow up) so that the resulting system is analytical in "epsilon". A method is given that takes advantage that a first integral, if it exists, it must be analytical in the parameter in order to find conditions for the existence of this first integral. In this way we define what are called essential variables of the system. The last three chapters deal with Abel equations and the center problem. In general, we consider Abel trigonometric equations. In the sixth chapter some necessary and sufficient conditions for an Abel equation defined by trigonometric polynomials of degree up to 3 have a center are given. All the examples given in this chapter have a universal center. In the seventh chapter it is given an example of an Abel equation defined by trigonometric polynomials of degree 3 with a center which is not universal. In this way an open problem is solved: to determine the lowest degree such that a trigonometric Abel equation has a center which is not a composition center. The last chapter deals with trigonometric and polynomial Abel equations and gives a survey of the last known results and conjectures about the center problem for these equations. Besides some new examples of Abel differential equations with a center are given.