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Author : Stefano Bianchini,Sara Daneri
Publisher : Unknown
Page : 128 pages
File Size : 48,5 Mb
Release : 2018
Category : Decomposition (Mathematics)
ISBN : 1470442787
Author : Stefano Bianchini,Sara Daneri
Publisher : American Mathematical Soc.
Page : 112 pages
File Size : 45,9 Mb
Release : 2018-02-23
Category : Decomposition (Mathematics)
ISBN : 9781470427665
The authors consider the original strategy proposed by Sudakov for solving the Monge transportation problem with norm cost with , probability measures in and absolutely continuous w.r.t. . The key idea in this approach is to decompose (via disintegration of measures) the Kantorovich optimal transportation problem into a family of transportation problems in , where are disjoint regions such that the construction of an optimal map is simpler than in the original problem, and then to obtain by piecing together the maps . When the norm is strictly convex, the sets are a family of -dimensional segments determined by the Kantorovich potential called optimal rays, while the existence of the map is straightforward provided one can show that the disintegration of (and thus of ) on such segments is absolutely continuous w.r.t. the -dimensional Hausdorff measure. When the norm is not strictly convex, the main problems in this kind of approach are two: first, to identify a suitable family of regions on which the transport problem decomposes into simpler ones, and then to prove the existence of optimal maps. In this paper the authors show how these difficulties can be overcome, and that the original idea of Sudakov can be successfully implemented. The results yield a complete characterization of the Kantorovich optimal transportation problem, whose straightforward corollary is the solution of the Monge problem in each set and then in . The strategy is sufficiently powerful to be applied to other optimal transportation problems.
Author : Roman Vershynin
Publisher : Cambridge University Press
Page : 299 pages
File Size : 40,9 Mb
Release : 2018-09-27
Category : Business & Economics
ISBN : 9781108415194
An integrated package of powerful probabilistic tools and key applications in modern mathematical data science.
Author : Alessio Figalli
Publisher : Edizioni della Normale
Page : 0 pages
File Size : 43,7 Mb
Release : 2008-07-17
Category : Mathematics
ISBN : 8876423303
In this book we describe recent developments in the theory of optimal transportation, and some of its applications to fluid dynamics. Moreover we explore new variants of the original problem, and we try to figure out some common (and sometimes unexpected) features in this emerging variety of problems . In Chapter 1 we study the optimal transportation problem on manifolds with geometric costs coming from Tonelli Lagrangians, while in Chapter 2 we consider a generalization of the classical transportation problem called the optimal irrigation problem. Then, Chapter 3 is about the Brenier variational theory of incompressible flows, which concerns a weak formulation of the Euler equations viewed as a geodesic equation in the space of measure-preserving diffeomorphism. Chapter 4 is devoted to the study of regularity and uniqueness of solutions of Hamilton-Jacobi equations applying the Aubry-Mather theory. Finally, the last chapter deals with a DiPerna-Lions theory for martingale solutions of stochastic differential equations.
Author : Lawrence C. Evans,Wilfrid Gangbo
Publisher : American Mathematical Soc.
Page : 66 pages
File Size : 43,7 Mb
Release : 1999
Category : Mathematics
ISBN : 9780821809389
In this volume, the authors demonstrate under some assumptions on $f^+$, $f^-$ that a solution to the classical Monge-Kantorovich problem of optimally rearranging the measure $\mu{^+}=f^+dx$ onto $\mu^-=f^-dy$ can be constructed by studying the $p$-Laplacian equation $- \mathrm{div}(\vert DU_p\vert^{p-2}Du_p)=f^+-f^-$ in the limit as $p\rightarrow\infty$. The idea is to show $u_p\rightarrow u$, where $u$ satisfies $\vert Du\vert\leq 1,-\mathrm{div}(aDu)=f^+-f^-$ for some density $a\geq0$, and then to build a flow by solving a nonautonomous ODE involving $a, Du, f^+$ and $f^-$.
Author : Zhou Gang,Dan Knopf,Israel Michael Siga
Publisher : American Mathematical Soc.
Page : 78 pages
File Size : 51,7 Mb
Release : 2018-05-29
Category : Asymptotic expansions
ISBN : 9781470428402
Author : Terence Tao
Publisher : American Mathematical Soc.
Page : 306 pages
File Size : 52,7 Mb
Release : 2009
Category : Mathematics
ISBN : 9780821848838
Focuses on ergodic theory, combinatorics, and number theory. This book discusses a variety of topics, ranging from developments in additive prime number theory to expository articles on individual mathematical topics such as the law of large numbers and the Lucas-Lehmer test for Mersenne primes.
Author : Roelof Bruggeman,Youngju Choie,Nikolaos Diamantis
Publisher : American Mathematical Soc.
Page : 167 pages
File Size : 48,9 Mb
Release : 2018-05-29
Category : Algebraic topology
ISBN : 9781470428556
Author : Sebastiano Carpi,Yasuyuki Kawahigashi,Roberto Longo,Mihály Weiner
Publisher : American Mathematical Soc.
Page : 85 pages
File Size : 53,6 Mb
Release : 2018-08-09
Category : Conformal invariants
ISBN : 9781470428587
The authors consider unitary simple vertex operator algebras whose vertex operators satisfy certain energy bounds and a strong form of locality and call them strongly local. They present a general procedure which associates to every strongly local vertex operator algebra V a conformal net AV acting on the Hilbert space completion of V and prove that the isomorphism class of AV does not depend on the choice of the scalar product on V. They show that the class of strongly local vertex operator algebras is closed under taking tensor products and unitary subalgebras and that, for every strongly local vertex operator algebra V, the map W↦AW gives a one-to-one correspondence between the unitary subalgebras W of V and the covariant subnets of AV.
Author : Matthew Harrison-Trainor
Publisher : American Mathematical Soc.
Page : 107 pages
File Size : 49,5 Mb
Release : 2018-05-29
Category : Angles (Geometry)
ISBN : 9781470428396
Author : Bo’az Klartag
Publisher : American Mathematical Soc.
Page : 77 pages
File Size : 55,8 Mb
Release : 2017-09-25
Category : Curvature
ISBN : 9781470425425
The localization technique from convex geometry is generalized to the setting of Riemannian manifolds whose Ricci curvature is bounded from below. In a nutshell, the author's method is based on the following observation: When the Ricci curvature is non-negative, log-concave measures are obtained when conditioning the Riemannian volume measure with respect to a geodesic foliation that is orthogonal to the level sets of a Lipschitz function. The Monge mass transfer problem plays an important role in the author's analysis.
Author : David Chataur,Martintxo Saralegi-Aranguren,Daniel Tanré
Publisher : American Mathematical Soc.
Page : 108 pages
File Size : 50,6 Mb
Release : 2018-08-09
Category : Electronic
ISBN : 9781470428877
Let X be a pseudomanifold. In this text, the authors use a simplicial blow-up to define a cochain complex whose cohomology with coefficients in a field, is isomorphic to the intersection cohomology of X, introduced by M. Goresky and R. MacPherson. The authors do it simplicially in the setting of a filtered version of face sets, also called simplicial sets without degeneracies, in the sense of C. P. Rourke and B. J. Sanderson. They define perverse local systems over filtered face sets and intersection cohomology with coefficients in a perverse local system. In particular, as announced above when X is a pseudomanifold, the authors get a perverse local system of cochains quasi-isomorphic to the intersection cochains of Goresky and MacPherson, over a field. We show also that these two complexes of cochains are quasi-isomorphic to a filtered version of Sullivan's differential forms over the field Q. In a second step, they use these forms to extend Sullivan's presentation of rational homotopy type to intersection cohomology.
Author : Filippo Santambrogio
Publisher : Birkhäuser
Page : 353 pages
File Size : 41,5 Mb
Release : 2015-10-17
Category : Mathematics
ISBN : 9783319208282
This monograph presents a rigorous mathematical introduction to optimal transport as a variational problem, its use in modeling various phenomena, and its connections with partial differential equations. Its main goal is to provide the reader with the techniques necessary to understand the current research in optimal transport and the tools which are most useful for its applications. Full proofs are used to illustrate mathematical concepts and each chapter includes a section that discusses applications of optimal transport to various areas, such as economics, finance, potential games, image processing and fluid dynamics. Several topics are covered that have never been previously in books on this subject, such as the Knothe transport, the properties of functionals on measures, the Dacorogna-Moser flow, the formulation through minimal flows with prescribed divergence formulation, the case of the supremal cost, and the most classical numerical methods. Graduate students and researchers in both pure and applied mathematics interested in the problems and applications of optimal transport will find this to be an invaluable resource.
Author : Sourav Chatterjee
Publisher : Springer
Page : 170 pages
File Size : 44,9 Mb
Release : 2017-08-31
Category : Mathematics
ISBN : 9783319658162
This book addresses the emerging body of literature on the study of rare events in random graphs and networks. For example, what does a random graph look like if by chance it has far more triangles than expected? Until recently, probability theory offered no tools to help answer such questions. Important advances have been made in the last few years, employing tools from the newly developed theory of graph limits. This work represents the first book-length treatment of this area, while also exploring the related area of exponential random graphs. All required results from analysis, combinatorics, graph theory and classical large deviations theory are developed from scratch, making the text self-contained and doing away with the need to look up external references. Further, the book is written in a format and style that are accessible for beginning graduate students in mathematics and statistics.
Author : Martin Grötschel,Gyula O.H. Katona
Publisher : Springer Science & Business Media
Page : 536 pages
File Size : 40,8 Mb
Release : 2010-05-28
Category : Mathematics
ISBN : 9783540852216
Discrete mathematics and theoretical computer science are closely linked research areas with strong impacts on applications and various other scientific disciplines. Both fields deeply cross fertilize each other. One of the persons who particularly contributed to building bridges between these and many other areas is László Lovász, a scholar whose outstanding scientific work has defined and shaped many research directions in the last 40 years. A number of friends and colleagues, all top authorities in their fields of expertise and all invited plenary speakers at one of two conferences in August 2008 in Hungary, both celebrating Lovász’s 60th birthday, have contributed their latest research papers to this volume. This collection of articles offers an excellent view on the state of combinatorics and related topics and will be of interest for experienced specialists as well as young researchers.