A Tour Of Subriemannian Geometries Their Geodesics And Applications

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A Tour of Subriemannian Geometries, Their Geodesics and Applications

Author : Richard Montgomery
Publisher : American Mathematical Soc.
ISBN 13 : 0821841653
Total Pages : 282 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis A Tour of Subriemannian Geometries, Their Geodesics and Applications by : Richard Montgomery

Download or read book A Tour of Subriemannian Geometries, Their Geodesics and Applications written by Richard Montgomery and published by American Mathematical Soc.. This book was released on 2002 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: Subriemannian geometries can be viewed as limits of Riemannian geometries. They arise naturally in many areas of pure (algebra, geometry, analysis) and applied (mechanics, control theory, mathematical physics) mathematics, as well as in applications (e.g., robotics). This book is devoted to the study of subriemannian geometries, their geodesics, and their applications. It starts with the simplest nontrivial example of a subriemannian geometry: the two-dimensional isoperimetric problem reformulated as a problem of finding subriemannian geodesics. Among topics discussed in other chapters of the first part of the book are an elementary exposition of Gromov's idea to use subriemannian geometry for proving a theorem in discrete group theory and Cartan's method of equivalence applied to the problem of understanding invariants of distributions. The second part of the book is devoted to applications of subriemannian geometry. In particular, the author describes in detail Berry's phase in quantum mechanics, the problem of a falling cat righting herself, that of a microorganism swimming, and a phase problem arising in the $N$-body problem. He shows that all these problems can be studied using the same underlying type of subriemannian geometry. The reader is assumed to have an introductory knowledge of differential geometry. This book that also has a chapter devoted to open problems can serve as a good introduction to this new, exciting area of mathematics.

An Introduction To The Geometry Of Stochastic Flows

Author : Fabrice Baudoin
Publisher : World Scientific
ISBN 13 : 1783260580
Total Pages : 152 pages
Book Rating : 4.7/5 (832 download)

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Book Synopsis An Introduction To The Geometry Of Stochastic Flows by : Fabrice Baudoin

Download or read book An Introduction To The Geometry Of Stochastic Flows written by Fabrice Baudoin and published by World Scientific. This book was released on 2004-11-10 with total page 152 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book aims to provide a self-contained introduction to the local geometry of the stochastic flows. It studies the hypoelliptic operators, which are written in Hörmander's form, by using the connection between stochastic flows and partial differential equations.The book stresses the author's view that the local geometry of any stochastic flow is determined very precisely and explicitly by a universal formula referred to as the Chen-Strichartz formula. The natural geometry associated with the Chen-Strichartz formula is the sub-Riemannian geometry, and its main tools are introduced throughout the text./a

An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem

Author : Luca Capogna
Publisher : Springer Science & Business Media
ISBN 13 : 3764381337
Total Pages : 235 pages
Book Rating : 4.7/5 (643 download)

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Book Synopsis An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem by : Luca Capogna

Download or read book An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem written by Luca Capogna and published by Springer Science & Business Media. This book was released on 2007-08-08 with total page 235 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book gives an up-to-date account of progress on Pansu's celebrated problem on the sub-Riemannian isoperimetric profile of the Heisenberg group. It also serves as an introduction to the general field of sub-Riemannian geometric analysis. It develops the methods and tools of sub-Riemannian differential geometry, nonsmooth analysis, and geometric measure theory suitable for attacks on Pansu's problem.

Hodge Theory, Complex Geometry, and Representation Theory

Author : Robert S. Doran
Publisher : American Mathematical Soc.
ISBN 13 : 0821894153
Total Pages : 330 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Hodge Theory, Complex Geometry, and Representation Theory by : Robert S. Doran

Download or read book Hodge Theory, Complex Geometry, and Representation Theory written by Robert S. Doran and published by American Mathematical Soc.. This book was released on 2014 with total page 330 pages. Available in PDF, EPUB and Kindle. Book excerpt: Contains carefully written expository and research articles. Expository papers include discussions of Noether-Lefschetz theory, algebraicity of Hodge loci, and the representation theory of SL2(R). Research articles concern the Hodge conjecture, Harish-Chandra modules, mirror symmetry, Hodge representations of Q-algebraic groups, and compactifications, distributions, and quotients of period domains.

Introduction to Geometric Control

Author : Yuri Sachkov
Publisher : Springer Nature
ISBN 13 : 3031020707
Total Pages : 176 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis Introduction to Geometric Control by : Yuri Sachkov

Download or read book Introduction to Geometric Control written by Yuri Sachkov and published by Springer Nature. This book was released on 2022-07-02 with total page 176 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text is an enhanced, English version of the Russian edition, published in early 2021 and is appropriate for an introductory course in geometric control theory. The concise presentation provides an accessible treatment of the subject for advanced undergraduate and graduate students in theoretical and applied mathematics, as well as to experts in classic control theory for whom geometric methods may be introduced. Theory is accompanied by characteristic examples such as stopping a train, motion of mobile robot, Euler elasticae, Dido's problem, and rolling of the sphere on the plane. Quick foundations to some recent topics of interest like control on Lie groups and sub-Riemannian geometry are included. Prerequisites include only a basic knowledge of calculus, linear algebra, and ODEs; preliminary knowledge of control theory is not assumed. The applications problems-oriented approach discusses core subjects and encourages the reader to solve related challenges independently. Highly-motivated readers can acquire working knowledge of geometric control techniques and progress to studying control problems and more comprehensive books on their own. Selected sections provide exercises to assist in deeper understanding of the material. Controllability and optimal control problems are considered for nonlinear nonholonomic systems on smooth manifolds, in particular, on Lie groups. For the controllability problem, the following questions are considered: controllability of linear systems, local controllability of nonlinear systems, Nagano–Sussmann Orbit theorem, Rashevskii–Chow theorem, Krener's theorem. For the optimal control problem, Filippov's theorem is stated, invariant formulation of Pontryagin maximum principle on manifolds is given, second-order optimality conditions are discussed, and the sub-Riemannian problem is studied in detail. Pontryagin maximum principle is proved for sub-Riemannian problems, solution to the sub-Riemannian problems on the Heisenberg group, the group of motions of the plane, and the Engel group is described.

Foliations in Cauchy-Riemann Geometry

Author : Elisabetta Barletta
Publisher : American Mathematical Soc.
ISBN 13 : 0821843044
Total Pages : 270 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Foliations in Cauchy-Riemann Geometry by : Elisabetta Barletta

Download or read book Foliations in Cauchy-Riemann Geometry written by Elisabetta Barletta and published by American Mathematical Soc.. This book was released on 2007 with total page 270 pages. Available in PDF, EPUB and Kindle. Book excerpt: The authors study the relationship between foliation theory and differential geometry and analysis on Cauchy-Riemann (CR) manifolds. The main objects of study are transversally and tangentially CR foliations, Levi foliations of CR manifolds, solutions of the Yang-Mills equations, tangentially Monge-Ampere foliations, the transverse Beltrami equations, and CR orbifolds. The novelty of the authors' approach consists in the overall use of the methods of foliation theory and choice of specific applications. Examples of such applications are Rea's holomorphic extension of Levi foliations, Stanton's holomorphic degeneracy, Boas and Straube's approximately commuting vector fields method for the study of global regularity of Neumann operators and Bergman projections in multi-dimensional complex analysis in several complex variables, as well as various applications to differential geometry. Many open problems proposed in the monograph may attract the mathematical community and lead to further applications of

The Geometry of Filtering

Author : K. David Elworthy
Publisher : Springer Science & Business Media
ISBN 13 : 303460176X
Total Pages : 179 pages
Book Rating : 4.0/5 (346 download)

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Book Synopsis The Geometry of Filtering by : K. David Elworthy

Download or read book The Geometry of Filtering written by K. David Elworthy and published by Springer Science & Business Media. This book was released on 2010-11-27 with total page 179 pages. Available in PDF, EPUB and Kindle. Book excerpt: Filtering is the science of nding the law of a process given a partial observation of it. The main objects we study here are di usion processes. These are naturally associated with second-order linear di erential operators which are semi-elliptic and so introduce a possibly degenerate Riemannian structure on the state space. In fact, much of what we discuss is simply about two such operators intertwined by a smooth map, the \projection from the state space to the observations space", and does not involve any stochastic analysis. From the point of view of stochastic processes, our purpose is to present and to study the underlying geometric structure which allows us to perform the ltering in a Markovian framework with the resulting conditional law being that of a Markov process which is time inhomogeneous in general. This geometry is determined by the symbol of the operator on the state space which projects to a symbol on the observation space. The projectible symbol induces a (possibly non-linear and partially de ned) connection which lifts the observation process to the state space and gives a decomposition of the operator on the state space and of the noise. As is standard we can recover the classical ltering theory in which the observations are not usually Markovian by application of the Girsanov- Maruyama-Cameron-Martin Theorem. This structure we have is examined in relation to a number of geometrical topics.

Parabolic Geometries I

Author : Andreas Čap
Publisher : American Mathematical Society
ISBN 13 : 1470478226
Total Pages : 642 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Parabolic Geometries I by : Andreas Čap

Download or read book Parabolic Geometries I written by Andreas Čap and published by American Mathematical Society. This book was released on 2024-07-29 with total page 642 pages. Available in PDF, EPUB and Kindle. Book excerpt: Parabolic geometries encompass a very diverse class of geometric structures, including such important examples as conformal, projective, and almost quaternionic structures, hypersurface type CR-structures and various types of generic distributions. The characteristic feature of parabolic geometries is an equivalent description by a Cartan geometry modeled on a generalized flag manifold (the quotient of a semisimple Lie group by a parabolic subgroup). Background on differential geometry, with a view towards Cartan connections, and on semisimple Lie algebras and their representations, which play a crucial role in the theory, is collected in two introductory chapters. The main part discusses the equivalence between Cartan connections and underlying structures, including a complete proof of Kostant's version of the Bott–Borel–Weil theorem, which is used as an important tool. For many examples, the complete description of the geometry and its basic invariants is worked out in detail. The constructions of correspondence spaces and twistor spaces and analogs of the Fefferman construction are presented both in general and in several examples. The last chapter studies Weyl structures, which provide classes of distinguished connections as well as an equivalent description of the Cartan connection in terms of data associated to the underlying geometry. Several applications are discussed throughout the text.

Partial Differential Equations and Spectral Theory

Author : Michael Demuth
Publisher : Springer Science & Business Media
ISBN 13 : 303480024X
Total Pages : 351 pages
Book Rating : 4.0/5 (348 download)

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Book Synopsis Partial Differential Equations and Spectral Theory by : Michael Demuth

Download or read book Partial Differential Equations and Spectral Theory written by Michael Demuth and published by Springer Science & Business Media. This book was released on 2011-02-01 with total page 351 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume collects six articles on selected topics at the frontier between partial differential equations and spectral theory, written by leading specialists in their respective field. The articles focus on topics that are in the center of attention of current research, with original contributions from the authors. They are written in a clear expository style that makes them accessible to a broader audience. The articles contain a detailed introduction and discuss recent progress, provide additional motivation, and develop the necessary tools. Moreover, the authors share their views on future developments, hypotheses, and unsolved problems.

Around the Research of Vladimir Maz'ya I

Author : Ari Laptev
Publisher : Springer Science & Business Media
ISBN 13 : 1441913416
Total Pages : 414 pages
Book Rating : 4.4/5 (419 download)

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Book Synopsis Around the Research of Vladimir Maz'ya I by : Ari Laptev

Download or read book Around the Research of Vladimir Maz'ya I written by Ari Laptev and published by Springer Science & Business Media. This book was released on 2009-12-02 with total page 414 pages. Available in PDF, EPUB and Kindle. Book excerpt: The fundamental contributions of Professor Maz'ya to the theory of function spaces and especially Sobolev spaces are well known and often play a key role in the study of different aspects of the theory, which is demonstrated, in particular, by presented new results and reviews from world-recognized specialists. Sobolev type spaces, extensions, capacities, Sobolev inequalities, pseudo-Poincare inequalities, optimal Hardy-Sobolev-Maz'ya inequalities, Maz'ya's isocapacitary inequalities in a measure-metric space setting and many other actual topics are discussed.

Nonholonomic Mechanics and Control

Author : A.M. Bloch
Publisher : Springer
ISBN 13 : 1493930176
Total Pages : 582 pages
Book Rating : 4.4/5 (939 download)

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Book Synopsis Nonholonomic Mechanics and Control by : A.M. Bloch

Download or read book Nonholonomic Mechanics and Control written by A.M. Bloch and published by Springer. This book was released on 2015-11-05 with total page 582 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book explores connections between control theory and geometric mechanics. The author links control theory with a geometric view of classical mechanics in both its Lagrangian and Hamiltonian formulations, and in particular with the theory of mechanical systems subject to motion constraints. The synthesis is appropriate as there is a rich connection between mechanics and nonlinear control theory. The book provides a unified treatment of nonlinear control theory and constrained mechanical systems that incorporates material not available in other recent texts. The book benefits graduate students and researchers in the area who want to enhance their understanding and enhance their techniques.

Geometric Control And Nonsmooth Analysis: In Honor Of The 73rd Birthday Of H Hermes And Of The 71st Birthday Of R T Rockafellar

Author : Fabio Ancona
Publisher : World Scientific
ISBN 13 : 9814472522
Total Pages : 377 pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis Geometric Control And Nonsmooth Analysis: In Honor Of The 73rd Birthday Of H Hermes And Of The 71st Birthday Of R T Rockafellar by : Fabio Ancona

Download or read book Geometric Control And Nonsmooth Analysis: In Honor Of The 73rd Birthday Of H Hermes And Of The 71st Birthday Of R T Rockafellar written by Fabio Ancona and published by World Scientific. This book was released on 2008-07-08 with total page 377 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this volume is to provide a synthetic account of past research, to give an up-to-date guide to current intertwined developments of control theory and nonsmooth analysis, and also to point to future research directions.

Trends in Control Theory and Partial Differential Equations

Author : Fatiha Alabau-Boussouira
Publisher : Springer
ISBN 13 : 3030179494
Total Pages : 285 pages
Book Rating : 4.0/5 (31 download)

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Book Synopsis Trends in Control Theory and Partial Differential Equations by : Fatiha Alabau-Boussouira

Download or read book Trends in Control Theory and Partial Differential Equations written by Fatiha Alabau-Boussouira and published by Springer. This book was released on 2019-07-04 with total page 285 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book presents cutting-edge contributions in the areas of control theory and partial differential equations. Over the decades, control theory has had deep and fruitful interactions with the theory of partial differential equations (PDEs). Well-known examples are the study of the generalized solutions of Hamilton-Jacobi-Bellman equations arising in deterministic and stochastic optimal control and the development of modern analytical tools to study the controllability of infinite dimensional systems governed by PDEs. In the present volume, leading experts provide an up-to-date overview of the connections between these two vast fields of mathematics. Topics addressed include regularity of the value function associated to finite dimensional control systems, controllability and observability for PDEs, and asymptotic analysis of multiagent systems. The book will be of interest for both researchers and graduate students working in these areas.

Tunneling Estimates and Approximate Controllability for Hypoelliptic Equations

Author : Camille Laurent
Publisher : American Mathematical Society
ISBN 13 : 1470451387
Total Pages : 108 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Tunneling Estimates and Approximate Controllability for Hypoelliptic Equations by : Camille Laurent

Download or read book Tunneling Estimates and Approximate Controllability for Hypoelliptic Equations written by Camille Laurent and published by American Mathematical Society. This book was released on 2022-04-08 with total page 108 pages. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.

Systolic Geometry and Topology

Author : Mikhail Gersh Katz
Publisher : American Mathematical Soc.
ISBN 13 : 0821841777
Total Pages : 238 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Systolic Geometry and Topology by : Mikhail Gersh Katz

Download or read book Systolic Geometry and Topology written by Mikhail Gersh Katz and published by American Mathematical Soc.. This book was released on 2007 with total page 238 pages. Available in PDF, EPUB and Kindle. Book excerpt: The systole of a compact metric space $X$ is a metric invariant of $X$, defined as the least length of a noncontractible loop in $X$. When $X$ is a graph, the invariant is usually referred to as the girth, ever since the 1947 article by W. Tutte. The first nontrivial results for systoles of surfaces are the two classical inequalities of C. Loewner and P. Pu, relying on integral-geometric identities, in the case of the two-dimensional torus and real projective plane, respectively. Currently, systolic geometry is a rapidly developing field, which studies systolic invariants in their relation to other geometric invariants of a manifold. This book presents the systolic geometry of manifolds and polyhedra, starting with the two classical inequalities, and then proceeding to recent results, including a proof of M. Gromov's filling area conjecture in a hyperelliptic setting. It then presents Gromov's inequalities and their generalisations, as well as asymptotic phenomena for systoles of surfaces of large genus, revealing a link both to ergodic theory and to properties of congruence subgroups of arithmetic groups. The author includes results on the systolic manifestations of Massey products, as well as of the classical Lusternik-Schnirelmann category.

Analysis and Geometry in Several Complex Variables

Author : Shiferaw Berhanu
Publisher : American Mathematical Soc.
ISBN 13 : 1470422557
Total Pages : 194 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Analysis and Geometry in Several Complex Variables by : Shiferaw Berhanu

Download or read book Analysis and Geometry in Several Complex Variables written by Shiferaw Berhanu and published by American Mathematical Soc.. This book was released on 2017-01-17 with total page 194 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the proceedings of the workshop on Analysis and Geometry in Several Complex Variables, held from January 4–8, 2015, at Texas A&M University at Qatar, Doha, Qatar. This volume covers many topics of current interest in several complex variables, CR geometry, and the related area of overdetermined systems of complex vector fields, as well as emerging trends in these areas. Papers feature original research on diverse topics such as the rigidity of CR mappings, normal forms in CR geometry, the d-bar Neumann operator, asymptotic expansion of the Bergman kernel, and hypoellipticity of complex vector fields. Also included are two survey articles on complex Brunn-Minkowski theory and the regularity of systems of complex vector fields and their associated Laplacians.

Geometric Control Theory and Sub-Riemannian Geometry

Author : Gianna Stefani
Publisher : Springer
ISBN 13 : 331902132X
Total Pages : 385 pages
Book Rating : 4.3/5 (19 download)

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Book Synopsis Geometric Control Theory and Sub-Riemannian Geometry by : Gianna Stefani

Download or read book Geometric Control Theory and Sub-Riemannian Geometry written by Gianna Stefani and published by Springer. This book was released on 2014-06-05 with total page 385 pages. Available in PDF, EPUB and Kindle. Book excerpt: Honoring Andrei Agrachev's 60th birthday, this volume presents recent advances in the interaction between Geometric Control Theory and sub-Riemannian geometry. On the one hand, Geometric Control Theory used the differential geometric and Lie algebraic language for studying controllability, motion planning, stabilizability and optimality for control systems. The geometric approach turned out to be fruitful in applications to robotics, vision modeling, mathematical physics etc. On the other hand, Riemannian geometry and its generalizations, such as sub-Riemannian, Finslerian geometry etc., have been actively adopting methods developed in the scope of geometric control. Application of these methods has led to important results regarding geometry of sub-Riemannian spaces, regularity of sub-Riemannian distances, properties of the group of diffeomorphisms of sub-Riemannian manifolds, local geometry and equivalence of distributions and sub-Riemannian structures, regularity of the Hausdorff volume, etc.