This book covers the basic topics in geometry (including trigonometry) that are accessible and valuable to senior high school and university students. It also includes material that are very useful for problem solving in mathematical competitions, from relatively easy to advanced levels, including the International Mathematical Olympiad.

This book covers the basic topics in geometry (including trigonometry) that are accessible and valuable to senior high school and university students. It also includes materials that are very useful for problem solving in mathematical competitions, from relatively easy to advanced levels, including the International Mathematical Olympiad.

Approaches to the recovery of three-dimensional information on a biological object, which are often formulated or implemented initially in an intuitive way, are concisely described here based on physical models of the object and the image-formation process. Both three-dimensional electron microscopy and X-ray tomography can be captured in the same mathematical framework, leading to closely-related computational approaches, but the methodologies differ in detail and hence pose different challenges. The editors of this volume, Gabor T. Herman and Joachim Frank, are experts in the respective methodologies and present research at the forefront of biological imaging and structural biology. Computational Methods for Three-Dimensional Microscopy Reconstruction will serve as a useful resource for scholars interested in the development of computational methods for structural biology and cell biology, particularly in the area of 3D imaging and modeling.

"Differential Geometry" offers a concise introduction to some basic notions of modern differential geometry and their applications to solid mechanics and physics.

Concepts such as manifolds, groups, fibre bundles and groupoids are first introduced within a purely topological framework. They are shown to be relevant to the description of space-time, configuration spaces of mechanical systems, symmetries in general, microstructure and local and distant symmetries of the constitutive response of continuous media.

Once these ideas have been grasped at the topological level, the differential structure needed for the description of physical fields is introduced in terms of differentiable manifolds and principal frame bundles. These mathematical concepts are then illustrated with examples from continuum kinematics, Lagrangian and Hamiltonian mechanics, Cauchy fluxes and dislocation theory.

This book will be useful for researchers and graduate students in science and engineering.

This book aims to propose implementations and applications of Fractional Order Systems (FOS). It is well known that FOS can be applied in control applications and systems modeling, and their effectiveness has been proven in many theoretical works and simulation routines. A further and mandatory step for FOS real world utilization is their hardware implementation and applications on real systems modeling. With this viewpoint, introductive chapters on FOS are included, on the definition of stability region of Fractional Order PID Controller and Chaotic FOS, followed by the practical implementation based on Microcontroller, Field Programmable Gate Array, Field Programmable Analog Array and Switched Capacitor. Another section is dedicated to FO modeling of Ionic Polymeric Metal Composite (IPMC). This new material may have applications in robotics, aerospace and biomedicine.

This book gives a modern differential geometric treatment of linearly nonholonomically constrained systems. It discusses in detail what is meant by symmetry of such a system and gives a general theory of how to reduce such a symmetry using the concept of a differential space and the almost Poisson bracket structure of its algebra of smooth functions. The above theory is applied to the concrete example of Carathodory's sleigh and the convex rolling rigid body. The qualitative behavior of the motion of the rolling disk is treated exhaustively and in detail. In particular, it classifies all motions of the disk, including those where the disk falls flat and those where it nearly falls flat.

The geometric techniques described in this book for symmetry reduction have not appeared in any book before. Nor has the detailed description of the motion of the rolling disk. In this respect, the authors are trail-blazers in their respective fields.

Convex geometry is at once simple and amazingly rich. While the classical results go back many decades, during that previous to this book's publication in 1999, the integral geometry of convex bodies had undergone a dramatic revitalization, brought about by the introduction of methods, results and, most importantly, new viewpoints, from probability theory, harmonic analysis and the geometry of finite-dimensional normed spaces. This book is a collection of research and expository articles on convex geometry and probability, suitable for researchers and graduate students in several branches of mathematics coming under the broad heading of 'Geometric Functional Analysis'. It continues the Israel GAFA Seminar series, which is widely recognized as the most useful research source in the area. The collection reflects the work done at the program in Convex Geometry and Geometric Analysis that took place at MSRI in 1996.

This book offers an introduction to the theory of groupoids and their representations encompassing the standard theory of groups. Using a categorical language, developed from simple examples, the theory of finite groupoids is shown to knit neatly with that of groups and their structure as well as that of their representations is described. The book comprises numerous examples and applications, including well-known games and puzzles, databases and physics applications. Key concepts have been presented using only basic notions so that it can be used both by students and researchers interested in the subject. Category theory is the natural language that is being used to develop the theory of groupoids. However, categorical presentations of mathematical subjects tend to become highly abstract very fast and out of reach of many potential users. To avoid this, foundations of the theory, starting with simple examples, have been developed and used to study the structure of finite groups and groupoids. The appropriate language and notions from category theory have been developed for students of mathematics and theoretical physics. The book presents the theory on the same level as the ordinary and elementary theories of finite groups and their representations, and provides a unified picture of the same. The structure of the algebra of finite groupoids is analysed, along with the classical theory of characters of their representations. Unnecessary complications in the formal presentation of the subject are avoided. The book offers an introduction to the language of category theory in the concrete setting of finite sets. It also shows how this perspective provides a common ground for various problems and applications, ranging from combinatorics, the topology of graphs, structure of databases and quantum physics.

Curious Curves is self-contained and unified in presentation. This book is suitable for a topics course, capstone course, or senior seminar; it is also intended for independent study by students and others interested in mathematics.Curves can often provide a better representation of natural phenomena than do the figures of classical geometry. Thus the content - presented with an emphasis on the geometric intuition characteristic of the study of curves - is highly relevant not only for people working in mathematics, but also those in other sciences. The explanations are detailed and illustrative to capture the interest of the reader, as well as complete to provide the necessary background information needed to go further into the subject.

Spencer Bloch's 1979 Duke lectures, a milestone in modern mathematics, have been out of print almost since their first publication in 1980, yet they have remained influential and are still the best place to learn the guiding philosophy of algebraic cycles and motives. This edition, now professionally typeset, has a new preface by the author giving his perspective on developments in the field over the past 30 years. The theory of algebraic cycles encompasses such central problems in mathematics as the Hodge conjecture and the Bloch-Kato conjecture on special values of zeta functions. The book begins with Mumford's example showing that the Chow group of zero-cycles on an algebraic variety can be infinite-dimensional, and explains how Hodge theory and algebraic K-theory give new insights into this and other phenomena.

Many of the modern variational problems in topology arise in different but overlapping fields of scientific study: mechanics, physics and mathematics. In this work, Professor Fomenko offers a concise and clean explanation of some of these problems (both solved and unsolved), using current methods and analytical topology. The author's skillful exposition gives an unusual motivation to the theory expounded, and his work is recommended reading for specialists and nonspecialists alike, involved in the fields of physics and mathematics at both undergraduate and graduate levels.

This book unravels the mystery of Geometry in Origami with a unique approach: 64 Polyhedra designs, each made from a single square sheet of paper, no cuts, no glue; each polyhedron the largest possible from the starting size of square and each having an ingenious locking mechanism to hold its shape.

The author covers the five Platonic solids (cube, tetrahedron, octahedron, icosahedron and dodecahedron). There are ample variations with different color patterns and sunken sides. Dipyramids and Dimpled Dipyramids, unexplored before this in Origami, are also covered. There are a total of 64 models in the book. All the designs have an interesting look and a pleasing folding sequence and are based on unique mathematical equations.

This book provides theoretical concepts and applications of fractals and multifractals to a broad range of audiences from various scientific communities, such as petroleum, chemical, civil and environmental engineering, atmospheric research, and hydrology. In the first chapter, we introduce fractals and multifractals from physics and math viewpoints. We then discuss theory and practical applications in detail. In what follows, in chapter 2, fragmentation process is modeled using fractals. Fragmentation is the breaking of aggregates into smaller pieces or fragments, a typical phenomenon in nature. In chapter 3, the advantages and disadvantages of two- and three-phase fractal models are discussed in detail. These two kinds of approach have been widely applied in the literature to model different characteristics of natural phenomena. In chapter 4, two- and three-phase fractal techniques are used to develop capillary pressure curve models, which characterize pore-size distribution of porous media. Percolation theory provides a theoretical framework to model flow and transport in disordered networks and systems. Therefore, following chapter 4, in chapter 5 the fractal basis of percolation theory and its applications in surface and subsurface hydrology are discussed. In chapter 6, fracture networks are shown to be modeled using fractal approaches. Chapter 7 provides different applications of fractals and multifractals to petrophysics and relevant area in petroleum engineering. In chapter 8, we introduce the practical advantages of fractals and multifractals in geostatistics at large scales, which have broad applications in stochastic hydrology and hydrogeology. Multifractals have been also widely applied to model atmospheric characteristics, such as precipitation, temperature, and cloud shape. In chapter 9, these kinds of properties are addressed using multifractals. At watershed scales, river networks have been shown to follow fractal behavior. Therefore, the applications of fractals are addressed in chapter 10. Time series analysis has been under investigations for several decades in physics, hydrology, atmospheric research, civil engineering, and water resources. In chapter 11, we therefore, provide fractal, multifractal, multifractal detrended fluctuation analyses, which can be used to study temporal characterization of a phenomenon, such as flow discharge at a specific location of a river. Chapter 12 addresses signals and again time series using a novel fractal Fourier analysis. In chapter 13, we discuss constructal theory, which has a perspective opposite to fractal theories, and is based on optimizationof diffusive exchange. In the case of river drainages, for example, the constructal approach begins at the divide and generates headwater streams first, rather than starting from the fundamental drainage pattern.

For closed manifolds, there is a highly elaborated theory of number-valued invariants, attached to the underlying manifold, structures and differential operators. On open manifolds, nearly all of this fails, with the exception of some special classes. The goal of this monograph is to establish for open manifolds, structures and differential operators an applicable theory of number-valued relative invariants. This is of great use in the theory of moduli spaces for nonlinear partial differential equations and mathematical physics. The book is self-contained: in particular, it contains an outline of the necessary tools from nonlinear Sobolev analysis.

A Sampler of Useful Computational Tools for Applied Geometry, Computer Graphics, and Image Processing shows how to use a collection of mathematical techniques to solve important problems in applied mathematics and computer science areas. The book discusses fundamental tools in analytical geometry and linear algebra. It covers a wide range of topics, from matrix decomposition to curvature analysis and principal component analysis to dimensionality reduction. Written by a team of highly respected professors, the book can be used in a one-semester, intermediate-level course in computer science. It takes a practical problem-solving approach, avoiding detailed proofs and analysis. Suitable for readers without a deep academic background in mathematics, the text explains how to solve non-trivial geometric problems. It quickly gets readers up to speed on a variety of tools employed in visual computing and applied geometry.

This classic text serves as a tool for self-study; it is also used as a basic text for undergraduate courses in differential geometry. The author's ability to extract the essential elements of the theory in a lucid and concise fashion allows the student easy access to the material and enables the instructor to add emphasis and cover special topics. The extraordinary wealth of examples within the exercises and the new material, ranging from isoperimetric problems to comments on Einstein's original paper on relativity theory, enhance this new edition.

This volume contains research and expository papers on recent advances in foliations and Riemannian geometry. Some of the topics covered in this volume include: topology, geometry, dynamics and analysis of foliations, curvature, submanifold theory, Lie groups and harmonic maps.Among the contributions, readers may find an extensive survey on characteristic classes of Riemannian foliations offering also new results, an article showing the uniform simplicity of certain diffeomorphism groups, an exposition of convergences of contact structures to foliations from the point of view of Thurston's and Thurston-Bennequin's inequalities, a discussion about Fatou-Julia decompositions for foliations and a description of singular Riemannian foliations on spaces without conjugate points.Papers on submanifold theory focus on the existence of graphs with prescribed mean curvature and mean curvature flow for spacelike graphs, isometric and conformal deformations and detailed surveys on totally geodesic submanifolds in symmetric spaces, cohomogeneity one actions on hyperbolic spaces and rigidity of geodesic spheres in space forms. Geometric realizability of curvature tensors and curvature operators are also treated in this volume with special attention to the affine and the pseudo-Riemannian settings. Also, some contributions on biharmonic maps and submanifolds enrich the scope of this volume in providing an overview of different topics of current interest in differential geometry.

In recent years, the old idea that gauge theories and string theories are equivalent has been implemented and developed in various ways, and there are by now various models where the string theory / gauge theory correspondence is at work. One of the most important examples of this correspondence relates Chern-Simons theory, a topological gauge theory in three dimensions which describes knot and three-manifold invariants, to topological string theory, which is deeply related to Gromov-Witten invariants. This has led to some surprising relations between three-manifold geometry and enumerative geometry. This book gives the first coherent presentation of this and other related topics. After an introduction to matrix models and Chern-Simons theory, the book describes in detail the topological string theories that correspond to these gauge theories and develops the mathematical implications of this duality for the enumerative geometry of Calabi-Yau manifolds and knot theory. It is written in a pedagogical style and will be useful reading for graduate students and researchers in both mathematics and physics willing to learn about these developments.

Since the 1950s control theory has established itself as a major mathematical discipline, particularly suitable for application in a number of research fields, including advanced engineering design, economics and the medical sciences. However, since its emergence, there has been a need to rethink and extend fields such as calculus of variations, differential geometry and nonsmooth analysis, which are closely tied to research on applications. Today control theory is a rich source of basic abstract problems arising from applications, and provides an important frame of reference for investigating purely mathematical issues. In many fields of mathematics, the huge and growing scope of activity has been accompanied by fragmentation into a multitude of narrow specialties. However, outstanding advances are often the result of the quest for unifying themes and a synthesis of different approaches. Control theory and its applications are no exception. Here, the interaction between analysis and geometry has played a crucial role in the evolution of the field. This book collects some recent results, highlighting geometrical and analytical aspects and the possible connections between them. Applications provide the background, in the classical spirit of mutual interplay between abstract theory and problem-solving practice.

It is impossible to trisect angles with straightedge and compass alone, but many people try and think they have succeeded. This book is about angle trisections and the people who attempt them. Its purposes are to collect many trisections in one place, inform about trisectors, to amuse the reader, and, perhaps most importantly, to reduce the number of trisectors. This book includes detailed information about the personalities of trisectors and their constructions. It can be read by anyone who has taken a high school geometry course.

This book pioneers a nonlinear Fredholm theory in a general class of spaces called polyfolds. The theory generalizes certain aspects of nonlinear analysis and differential geometry, and combines them with a pinch of category theory to incorporate local symmetries. On the differential geometrical side, the book introduces a large class of `smooth' spaces and bundles which can have locally varying dimensions (finite or infinite-dimensional). These bundles come with an important class of sections, which display properties reminiscent of classical nonlinear Fredholm theory and allow for implicit function theorems. Within this nonlinear analysis framework, a versatile transversality and perturbation theory is developed to also cover equivariant settings. The theory presented in this book was initiated by the authors between 2007-2010, motivated by nonlinear moduli problems in symplectic geometry. Such problems are usually described locally as nonlinear elliptic systems, and they have to be studied up to a notion of isomorphism. This introduces symmetries, since such a system can be isomorphic to itself in different ways. Bubbling-off phenomena are common and have to be completely understood to produce algebraic invariants. This requires a transversality theory for bubbling-off phenomena in the presence of symmetries. Very often, even in concrete applications, geometric perturbations are not general enough to achieve transversality, and abstract perturbations have to be considered. The theory is already being successfully applied to its intended applications in symplectic geometry, and should find applications to many other areas where partial differential equations, geometry and functional analysis meet. Written by its originators, Polyfold and Fredholm Theory is an authoritative and comprehensive treatise of polyfold theory. It will prove invaluable for researchers studying nonlinear elliptic problems arising in geometric contexts.

The art of origami, or paper folding, is carried out using a square piece of paper to obtain attractive figures of animals, flowers or other familiar figures. It is easy to see that origami has links with geometry. Creases and edges represent lines, intersecting creases and edges make angles, while the intersections themselves represent points. Because of its manipulative and experiential nature, origami could become an effective context for the learning and teaching of geometry. In this unique and original book, origami is an object of mathematical exploration. The activities in this book differ from ordinary origami in that no figures of objects result. Rather, they lead the reader to study the effects of the folding and seek patterns. The experimental approach that characterizes much of science activity can be recognized throughout the book, as the manipulative nature of origami allows much experimenting, comparing, visualizing, discovering and conjecturing. The reader is encouraged to fill in all the proofs, for his/her own satisfaction and for the sake of mathematical completeness. Thus, this book provides a useful, alternative approach for reinforcing and applying the theorems of high school mathematics.

This encyclopedia contains trigonometric identity proofs for some three hundred identities. The book is presented in the form of mathematical games for the reader's enjoyment and includes a concordance of trigonometric identities, enabling easy reference.Trig or Treat is a must-have for:* every student of trigonometry, to find the proofs for trigonometric identities that may be set in class or in exams;* every teacher of trigonometry, to have easy access to trigonometric identities of different levels of difficulty, to set assignments for different levels of students;* every library, to provide a ready resource for students* every aficionado of Sudoku, crossword puzzles, jigsaw puzzles, Tetris, and other games, who love an intellectual challenge.

This is a book about prime numbers, congruences, secret messages, and elliptic curves that you can read cover to cover. It grew out of undergr- uate courses that the author taught at Harvard, UC San Diego, and the University of Washington. The systematic study of number theory was initiated around 300B. C. when Euclid proved that there are in?nitely many prime numbers, and also cleverly deduced the fundamental theorem of arithmetic, which asserts that every positive integer factors uniquely as a product of primes. Over a thousand years later (around 972A. D. ) Arab mathematicians formulated the congruent number problem that asks for a way to decide whether or not a given positive integer n is the area of a right triangle, all three of whose sides are rational numbers. Then another thousand years later (in 1976), Di?e and Hellman introduced the ?rst ever public-key cryptosystem, which enabled two people to communicate secretely over a public communications channel with no predetermined secret; this invention and the ones that followed it revolutionized the world of digital communication. In the 1980s and 1990s, elliptic curves revolutionized number theory, providing striking new insights into the congruent number problem, primality testing, publ- key cryptography, attacks on public-key systems, and playing a central role in Andrew Wiles' resolution of Fermat's Last Theorem.

The focus of this book and its geometric notions is on real vector spaces X that are finite or infinite inner product spaces of arbitrary dimension greater than or equal to 2. It characterizes both euclidean and hyperbolic geometry with respect to natural properties of (general) translations and general distances of X. Also for these spaces X, it studies the sphere geometries of Moebius and Lie as well as geometries where Lorentz transformations play the key role. Proofs of newer theorems characterizing isometries and Lorentz transformations under mild hypotheses are included, such as for instance infinite dimensional versions of famous theorems of A.D. Alexandrov on Lorentz transformations. A real benefit is the dimension-free approach to important geometrical theories. New to this third edition is a chapter dealing with a simple and great idea of Leibniz that allows us to characterize, for these same spaces X, hyperplanes of euclidean, hyperbolic geometry, or spherical geometry, the geometries of Lorentz-Minkowski and de Sitter, and this through finite or infinite dimensions greater than 1. Another new and fundamental result in this edition concerns the representation of hyperbolic motions, their form and their transformations. Further we show that the geometry (P,G) of segments based on X is isomorphic to the hyperbolic geometry over X. Here P collects all x in X of norm less than one, G is defined to be the group of bijections of P transforming segments of P onto segments. The only prerequisites for reading this book are basic linear algebra and basic 2- and 3-dimensional real geometry. This implies that mathematicians who have not so far been especially interested in geometry could study and understand some of the great ideas of classical geometries in modern and general contexts.