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A Course in Differential Geometry
This textbook for second-year graduate students is intended as an introduction to differential geometry with principal emphasis on Riemannian geometry. Chapter I explains basic definitions and gives the proofs of the important theorems of Whitney and Sard. Chapter II deals with vector fields and differential forms. Chapter III addresses integration of vector fields and p-plane fields. Chapter IV develops the notion of connection on a Riemannian manifold considered as a means to define parallel transport on the manifold. The author also discusses related notions of torsion and curvature, and gives a working knowledge of the covariant derivative. Chapter V specializes on Riemannian manifolds b...
Some Nonlinear Problems in Riemannian Geometry
This book deals with such important subjects as variational methods, the continuity method, parabolic equations on fiber bundles, ideas concerning points of concentration, blowing-up technique, geometric and topological methods. It explores important geometric problems that are of interest to many mathematicians and scientists but have only recently been partially solved.
Nonlinear Analysis on Manifolds. Monge-Ampère Equations
This volume is intended to allow mathematicians and physicists, especially analysts, to learn about nonlinear problems which arise in Riemannian Geometry. Analysis on Riemannian manifolds is a field currently undergoing great development. More and more, analysis proves to be a very powerful means for solving geometrical problems. Conversely, geometry may help us to solve certain problems in analysis. There are several reasons why the topic is difficult and interesting. It is very large and almost unexplored. On the other hand, geometric problems often lead to limiting cases of known problems in analysis, sometimes there is even more than one approach, and the already existing theoretical stu...
The Voices of Nature
Songs, barks, roars, hoots, squeals, and growls: exploring the mysteries of how animals communicate by sound What is the meaning of a bird’s song, a baboon’s bark, an owl’s hoot, or a dolphin’s clicks? In The Voices of Nature, Nicolas Mathevon explores the mysteries of animal sound. Putting readers in the middle of animal soundscapes that range from the steamy heat of the Amazon jungle to the icy terrain of the Arctic, Mathevon reveals the amazing variety of animal vocalizations. He describes how animals use sound to express emotion, to choose a mate, to trick others, to mark their territory, to call for help, and much more. What may seem like random chirps, squawks, and cries are ac...
Classical Groups and Geometric Algebra
“Classical groups”, named so by Hermann Weyl, are groups of matrices or quotients of matrix groups by small normal subgroups. Thus the story begins, as Weyl suggested, with “Her All-embracing Majesty”, the general linear group $GL_n(V)$ of all invertible linear transformations of a vector space $V$ over a field $F$. All further groups discussed are either subgroups of $GL_n(V)$ or closely related quotient groups. Most of the classical groups consist of invertible linear transformations that respect a bilinear form having some geometric significance, e.g., a quadratic form, a symplectic form, etc. Accordingly, the author develops the required geometric notions, albeit from an algebrai...
Elements of Dimensionality Reduction and Manifold Learning
Dimensionality reduction, also known as manifold learning, is an area of machine learning used for extracting informative features from data for better representation of data or separation between classes. This book presents a cohesive review of linear and nonlinear dimensionality reduction and manifold learning. Three main aspects of dimensionality reduction are covered: spectral dimensionality reduction, probabilistic dimensionality reduction, and neural network-based dimensionality reduction, which have geometric, probabilistic, and information-theoretic points of view to dimensionality reduction, respectively. The necessary background and preliminaries on linear algebra, optimization, an...
Resolution of Singularities
The notion of singularity is basic to mathematics. In algebraic geometry, the resolution of singularities by simple algebraic mappings is truly a fundamental problem. It has a complete solution in characteristic zero and partial solutions in arbitrary characteristic. The resolution of singularities in characteristic zero is a key result used in many subjects besides algebraic geometry, such as differential equations, dynamical systems, number theory, the theory of $\mathcal{D}$-modules, topology, and mathematical physics. This book is a rigorous, but instructional, look at resolutions. A simplified proof, based on canonical resolutions, is given for characteristic zero. There are several pro...
Advances in the Study of Behavior
Advances in the Study of Behavior was initiated over 40 years ago to serve the increasing number of scientists engaged in the study of animal behavior. That number is still expanding. This volume makes another important "contribution to the development of the field" by presenting theoretical ideas and research to those studying animal behavior and to their colleagues in neighboring fields. Advances in the Study of Behavior is now available online at ScienceDirect full-text online from volume 30 onward.
New Research on Acoustics
Acoustics is the science concerned with the production, control, transmission, reception, and effects of sound. Its origins began with the study of mechanical vibrations and the radiation of these vibrations through mechanical waves, and still continue today. Research was done to look into the many aspects of the fundamental physical processes involved in waves and sound and into possible applications of these processes in modern life. The study of sound waves also leads to physical principles that can be applied to the study of all waves. The broad scope of acoustics as an area of interest and endeavour can be ascribed to a variety of reasons. First, there is the ubiquitous nature of mechanical radiation, generated by natural causes and by human activity. Then, there is the existence of the sensation of hearing, of the human vocal ability, of communication via sound, along with the variety of psychological influences sound has on those who hear it. Such areas as speech, music, sound recording and reproduction.
