2014, I, 254 p.
Hardcover
ISBN 978-3-319-14050-6
Due: March 14, 2015
Presents state-of-the-art advancements in the field of modular function theory
Provides a self-contained overview of the topic
Includes open problems, extensive bibliographic references, and suggestions for further development
This monograph provides a concise introduction to the main results and methods of the fixed point theory in modular function spaces. Modular function spaces are natural generalizations of both function and sequence variants of many important spaces like Lebesgue, Orlicz, Musielak-Orlicz, Lorentz, Orlicz-Lorentz, Calderon-Lozanovskii spaces, and others. In most cases, particularly in applications to integral operators, approximation and fixed point results, modular type conditions are much more natural and can be more easily verified than their metric or norm counterparts. There are also important results that can be proved only using the apparatus of modular function spaces. The material is presented in a systematic and rigorous manner that allows readers to grasp the key ideas and to gain a working knowledge of the theory. Despite the fact that the work is largely self-contained, extensive bibliographic references are included, and open problems and further development directions are suggested when applicable.
The monograph is targeted mainly at the mathematical research community but it is also accessible to graduate students interested in functional analysis and its applications. It could also serve as a text for an advanced course in fixed point theory of mappings acting in modular function spaces.?
Introduction.- Fixed Point Theory in Metric Spaces: An Introduction.- Modular Function Spaces.- Geometry of Modular Function Spaces.- Fixed Point Existence Theorems in Modular Function Spaces.- Fixed Point Construction Processes.- Semigroups of Nonlinear Mappings in Modular Function Spaces.- Modular Metric Spaces.
English translation of the revised 3rd edition of "5000 Jahre Geometrie"
2015, XII, 614 p. 315 illus., 165 illus. in color.
ISBN 978-3-0348-0897-2
Due: May 24, 2015
The presentation of geometry as developments in cultural history with the historical background
Tables with the historical and cultural developments and with the geometrical contents in each chapter
Exercises with a historical background
Illustrations and original texts from the eras of cultural history
The book provides a fascinating overview of geometrical ideas and perceptions all the way from primitive societies up to the mathematical as well as artistic concepts of the 20th century. Geometry is presented as developments in cultural history and its interaction with architecture, the visual arts, philosophy, science, and engineering:
Its origins in the ancient cultures along the Indus and Nile rivers and in Mesopotamia
The golden age of ancient Greece
In Islamic countries.
In India, China, Japan, and in the ancient American cultures
The developments in Europe during the Middle Ages and above all during the Renaissance
New approaches in the 17th and 18th centuries: descriptive and projective geometry, coordinate methods, and analytical geometry
The revolutionary findings of the 19th and 20th centuries: axiom systems, geometry as a theory with multiple structures.
Content Level ā Research
Keywords ā Archimedes - Durer - Escher - Euclid - Geometry
Related subjects ā Birkhauser History of Science - Birkhauser Mathematics
Series: Publications of the Scuola Normale Superiore, Vol. 17
Subseries: CRM Series
2015, Approx. 250 p.
Softcover
ISBN 978-88-7642-522-6
Covers very recent developments, partially unpublished at the time of the school
Covers the most exciting developments in this research area
In 2013, a school on Geometric Measure Theory and Real Analysis, organized by G. Alberti, C. De Lellis and myself, took place at the Centro De Giorgi in Pisa, with lectures by V. Bogachev, R. Monti, E. Spadaro and D. Vittone.
The book collects the notes of the courses. The courses provide a deep and up to date insight on challenging mathematical problems and their recent developments: infinite-dimensional analysis, minimal surfaces and isoperimetric problems in the Heisenberg group, regularity of sub-Riemannian geodesics and the regularity theory of minimal currents in any dimension and codimension.
Vladimir I. Bogachev: Sobolev classes on infinite-dimensional spaces.- Roberto Monti: Isoperimetric problem and minimal surfaces in the Heisenberg group.- Emanuele Spadaro: Regularity of higher codimension area minimizing integral currents.- Davide Vittone: The regularity problem for sub-Riemannian geodesics.
Series: Fields Institute Communications, Vol. 73
2015, XII, 368 p. 70 illus., 26 illus. in color.
Hardcover
ISBN 978-1-4939-2440-0
Due: April 14, 2015
Discusses various aspects of fluid mechanics in a geometric framework
Represents the state-of-the-art of research in fields of geometric mechanics
Contains discussions and applications of variational integrators for precise numerical integration of various dynamics
This book illustrates the broad range of Jerry Marsdenfs mathematical legacy in areas of geometry, mechanics, and dynamics, from very pure mathematics to very applied, but always with a geometric perspective. Each contribution develops its material from the viewpoint of geometric mechanics beginning at the very foundations, introducing readers to modern issues via illustrations in a wide range of topics. The twenty refereed papers contained in this volume are based on lectures and research performed during the month of July 2012 at the Fields Institute for Research in Mathematical Sciences, in a program in honor of Marsden's legacy.
The unified treatment of the wide breadth of topics treated in this book will be of interest to both experts and novices in geometric mechanics. Experts will recognize applications of their own familiar concepts and methods in a wide variety of fields, some of which they may never have approached from a geometric viewpoint. Novices may choose topics that interest them among the various fields and learn about geometric approaches and perspectives toward those topics that will be new for them as well.
A Global Version of the Koon-Marsden Jacobiator Formula (P. Balsiero).- Geometry of Image Registration (M. Bruveris, D.D. Holm).- Multisymplectic Geometry and Lie Groupoids (H. Bursztyn, A. Cabrera, D. Iglesias).- The Topology of Change (G. Chichilnisky).- Chaos in the Kepler Problem with Quadrupole Perturbations (G. Depetri, A. Saa).- Groups of Diffeomorphisms and Fluid Motion (D.G. Ebin).- Dual Pairs For Non-Abelian Fluids (F. Gay-Balmaz).- The Role of SE(d)-Reduction For Swimming in Stokes and Navier-Stokes Fluids (H.O. Jacobs).- Lagrangian Mechanics on Centered Semi-Direct Products (L. Colombo, H.O. Jacobs).- Vortices on Closed Surfaces (S. Boatto, J. Koiller).- The Geometry of Radiative Transfer (C. Lessig, A.L. Castro).- A Soothing Invisible Hand: Moderation Potentials in Optimal Control (D. Lewis).- The Local Description of Discrete Mechanics (J.C. Marrero, D. Martin de Diego, E. Martinez).- Keplerian Dynamics on the Heisenberg Group and Elsewhere (R. Montgomery, C. Shanbrom).- On the Completeness of Trajectories for Some Mechanical Systems (M. Sanchez).- Diffeomorphic Image Matching with Left-Invariant Metrics (T. Schmah, L. Risser, F.-X. Vialard).- Normal Forms for Lie Symmetric Cotangent Bundle Systems with Free and Proper Actions (T. Schmah, C. Stoica).- Polite Actions of Non-Compact Lie Groups (L. Bates, J. Sniaticki).- Geometric Computational Electrodynamics with Variational Integrators and Discrete Differential Forms (A. Stern, Y. Tong, M. Desbrun, J.E. Marsden).- Hamel's Formalism and Variational Integrators (K.R. Ball, D.V. Zenkov).