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This thesis by Célestin Kurujyibwami focuses on the group classification of linear Schrödinger equations using the algebraic method. It addresses the unsolved problem of classifying these equations with complex potentials, which are significant in fields like quantum mechanics, scattering theory, and optics. The work systematically solves the classification problem for one and two-dimensional spaces by identifying eight families of potentials and their maximal Lie invariance algebras. The thesis also reviews Lie symmetries and differential equations, applying the direct method to determine the equivalence groupoid and equivalence group of the class. Intended for researchers and students in mathematics and applied mathematics, the thesis contributes to the understanding of differential equations in complex systems.

Schrödinger equation. --- Lie groups. --- Schrödinger equation --- Lie groups

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The synthesis of symplectic geometry, the calculus of variations and control theory offered in this book provides a crucial foundation for the understanding of many problems in applied mathematics. Focusing on the theory of integrable systems, this book introduces a class of optimal control problems on Lie groups, whose Hamiltonians, obtained through the Maximum Principle of optimality, shed new light on the theory of integrable systems. These Hamiltonians provide an original and unified account of the existing theory of integrable systems. The book particularly explains much of the mystery surrounding the Kepler problem, the Jacobi problem and the Kovalevskaya Top. It also reveals the ubiquitous presence of elastic curves in integrable systems up to the soliton solutions of the non-linear Schroedinger's equation. Containing a useful blend of theory and applications, this is an indispensable guide for graduates and researchers in many fields, from mathematical physics to space control.

Control theory. --- Geometry, Differential. --- Hamiltonian systems. --- Lie groups. --- Manifolds (Mathematics)

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Ce travail en deux volumes donne la preuve de la stabilisation de la formule des trace tordue. Stabiliser la formule des traces tordue est la méthode la plus puissante connue actuellement pour comprendre l'action naturelle du groupe des points adéliques d'un groupe réductif, tordue par un automorphisme, sur les formes automorphes de carré intégrable de ce groupe. Cette compréhension se fait en réduisant le problème, suivant les idées de Langlands, à des groupes plus petits munis d'un certain nombre de données auxiliaires; c'est ce que l'on appelle les données endoscopiques. L'analogue non tordu a été résolu par J. Arthur et dans ce livre on suit la stratégie de celui-ci. Publier ce travail sous forme de livre permet de le rendre le plus complet possible. Les auteurs ont repris la théorie de l'endoscopie tordue développée par R. Kottwitz et D. Shelstad et par J.-P. Labesse. Ils donnent tous les arguments des démonstrations même si nombre d'entre eux se trouvent déjà dans les travaux d'Arthur concernant le cas de la formule des traces non tordue. Ce travail permet de rendre inconditionnelle la classification que J. Arthur a donnée des formes automorphes de carré intégrable pour les groupes classiques quasi-déployés, c’était pour les auteurs une des principales motivations pour l’écrire. Cette première partie comprend les chapitres préparatoires (I-V).

Number theory --- Ordered algebraic structures --- Topological groups. Lie groups --- Mathematics --- topologie (wiskunde) --- wiskunde --- getallenleer --- topologie

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Lie groups --- p-adic groups --- Lie, Groupes de. --- Groupes p-adiques.

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Exploring the interplay between deep theory and intricate computation, this volume is a compilation of research and survey papers in number theory, written by members of the Women In Numbers (WIN) network, principally by the collaborative research groups formed at Women In Numbers 3, a conference at the Banff International Research Station in Banff, Alberta, on April 21-25, 2014. The papers span a wide range of research areas: arithmetic geometry; analytic number theory; algebraic number theory; and applications to coding and cryptography. The WIN conference series began in 2008, with the aim of strengthening the research careers of female number theorists. The series introduced a novel research-mentorship model: women at all career stages, from graduate students to senior members of the community, joined forces to work in focused research groups on cutting-edge projects designed and led by experienced researchers. The goals for Women In Numbers 3 were to establish ambitious new collaborations between women in number theory, to train junior participants about topics of current importance, and to continue to build a vibrant community of women in number theory. Forty-two women attended the WIN3 workshop, including 15 senior and mid-level faculty, 15 junior faculty and postdocs, and 12 graduate students.

Mathematics. --- Topological groups. --- Lie groups. --- Geometry. --- Number theory. --- Number Theory. --- Topological Groups, Lie Groups. --- Number theory --- Topological Groups. --- Mathematics --- Euclid's Elements --- Groups, Topological --- Continuous groups --- Number study --- Numbers, Theory of --- Algebra --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups