A readable introduction to the subject of calculus on arbitrary surfaces or manifolds. Accessible to readers with knowledge of basic calculus and linear algebra. Sections include series of problems to reinforce concepts.

An analysis of large data sets from an empirical and geometric viewpoint

Data reduction is a rapidly emerging field with broad applications in essentially all fields where large data sets are collected and analyzed. Geometric Data Analysis is the first textbook to focus on the geometric approach to this problem of developing and distinguishing subspace and submanifold techniques for low-dimensional data representation. Understanding the geometrical nature of the data under investigation is presented as the key to identifying a proper reduction technique.

Focusing on the construction of dimensionality-reducing mappings to reveal important geometrical structure in the data, the sequence of chapters is carefully constructed to guide the reader from the beginnings of the subject to areas of current research activity. A detailed, and essentially self-contained, presentation of the mathematical prerequisites is included to aid readers from a broad variety of backgrounds. Other topics discussed in Geometric Data Analysis include:

• The Karhunen-Loeve procedure for scalar and vector fields with extensions to missing data, noisy data, and data with symmetry
• Nonlinear methods including radial basis functions (RBFs) and backpropa-gation neural networks
• Wavelets and Fourier analysis as analytical methods for data reduction
• Expansive discussion of recent research including the Whitney reduction network and adaptive bases codeveloped by the author
• And much more

The methods are developed within the context of many real-world applications involving massive data sets, including those generated by digital imaging systems and computer simulations of physical phenomena. Empirically based representations are shown to facilitate their investigation and yield insights that would otherwise elude conventional analytical tools.

Expositions of quantitative methods and algorithms for biological data tend to be scattered through the technical literature, often across different fields, and are thus awkward to assimilate. This book documents one example of this: the relationship between the cell biology idea of metabolic networks and the mathematical idea of polyhedral cones. Such cones can be used to describe the set of steady-state admissible fluxes through metabolic networks, and consequently have become important constructs in the field of microbiology. Via convex cone concepts, fundamental objects called elementary flux modes (EFMs) can be described mathematically. The fundamental algorithm of this relationship is the double description method, which has an extended history in the field of computational geometry. This monograph addresses its relatively recent use in the context of cellular metabolism. Metabolic Networks, Elementary Flux Modes, and Polyhedral Cones: Addresses important topics in the mathematical description of metabolic activity that have not previously appeared in unified form. Introduces a central topic of mathematical systems biology in a manner accessible to nonmathematicians with some mathematical and computational experience. Presents a careful study of the double description method, a fundamental algorithm of computational geometry, in the context of metabolic analysis. The core audience for this book includes mathematicians, engineers, and biologists interested in cell metabolism. Computational geometers will also find it of interest.

This two-volume set containts parts I and II. Each volume is a collection of articles written in memory of Boris Dubrovin (1950-2019). The authors express their admiration for his remarkable personality and for the contributions he made to mathematical physics. For many of the authors, Dubrovin was a friend, colleague, inspiring mentor, and teacher. The contributions are split into two parts: ``Integrable Systems'' and ``Quantum Theories and Algebraic Geometry'', reflecting the areas of main scientific interests of Dubrovin. Chronologically, these interests may be divided into several parts: integrable systems, integrable systems of hydrodynamic type, WDVV equations (Frobenius manifolds), isomonodromy equations (flat connections), and quantum cohomology. The articles included in the first part are more or less directly devoted to these areas (primarily with the first three listed above). The second part contains articles on quantum theories and algebraic geometry and is less directly connected with Dubrovin's early interests.

Rational thinking as exemplified in mathematical cognition is immensely important in the modern world. This book documents how a group of three eighth-grade girls developed specific group practices typical of such thinking in an online educational experience. A longitudinal case study tracks the team through eight hour-long sessions, following the students' meaning-making processes through their mutual chat responses preserved in computer logs coordinated with their geometric actions. The examination of data focuses on key areas of the team's development: its effective team collaboration, its productive mathematical discourse, its enacted use of dynamic-geometry tools, and its ability to identify and construct dynamic-geometry dependencies. This detailed study of group cognition serves as a paradigmatic example of computer-supported collaborative learning, incorporating a unique model of human-computer interaction analysis applied to the use of innovative educational technology. A valuable resource for researchers, instructors, and students alike, it offers concrete suggestions for improving educational practice.

A leading pioneer in the field offers practical applications of this innovative science. Peters describes complex concepts in an easy-to-follow manner for the non-mathematician. He uses fractals, rescaled range analysis and nonlinear dynamical models to explain behavior and understand price movements. These are specific tools employed by chaos scientists to map and measure physical and now, economic phenomena.

Starting from the foundations, the author presents an almost entirely
self-contained treatment of differentiable spaces of nonpositive
curvature, focusing on the symmetric spaces in which every geodesic lies
in a flat Euclidean space of dimension at least two. The book builds to
a discussion of the Mostow Rigidity Theorem and its generalizations, and
concludes by exploring the relationship in nonpositively curved spaces
between geometric and algebraic properties of the fundamental group.
This introduction to the geometry of symmetric spaces of non-compact
type will serve as an excellent guide for graduate students new to the
material, and will also be a useful reference text for mathematicians
already familiar with the subject.

The statistical theory of shape is a relatively new topic and is generating a great deal of interest and comment by statisticians, engineers and computer scientists. Mathematically, ‘shape’ is the geometrical information required to describe an object when location, scale and rotational effects are removed. The theory was pioneered by Professor David Kendall to solve practical problems concerning shape. This text presents an elegant account of the theory of shape that has evolved from Kendall’s work. Features include:

• A comprehensive account of Kendall’s shape spaces
• A variety of topological and geometric invariants of these spaces
• Emphasis on the mathematical aspects of shape analysis
• Coverage of the mathematical issues for a wide range of applications

The early chapters provide all the necessary background information, including the history and applications of shape theory. The authors then go on to analyse the topic, in brilliant detail, in a variety of different shape spaces. Kendall’s own procedures for visualising distributions of shapes and shape processes are covered at length. Implications from other branches of mathematics are explored, along with more advanced applications, incorporating statistics and stochastic analysis. Applied statisticians, applied mathematicians, engineers and computer scientists working and researching in the fields of archaeology, astronomy, biology, geography and physical chemistry will find this book of great benefit. The theories presented are used today in a wide range of subjects from archaeology through to physics, and will provide fascinating reading to anyone engaged in such research. Visit our web page! http://www.wiley.com/

A one-stop reference to fractional factorials and related orthogonal arrays.
Presenting one of the most dynamic areas of statistical research, this book offers a systematic, rigorous, and up-to-date treatment of fractional factorial designs and related combinatorial mathematics. Leading statisticians Aloke Dey and Rahul Mukerjee consolidate vast amounts of material from the professional literature--expertly weaving fractional replication, orthogonal arrays, and optimality aspects. They develop the basic theory of fractional factorials using the calculus of factorial arrangements, thereby providing a unified approach to the study of fractional factorial plans. An indispensable guide for statisticians in research and industry as well as for graduate students, Fractional Factorial Plans features:
* Construction procedures of symmetric and asymmetric orthogonal arrays.
* Many up-to-date research results on nonexistence.
* A chapter on optimal fractional factorials not based on orthogonal arrays.
* Trend-free plans, minimum aberration plans, and search and supersaturated designs.
* Numerous examples and extensive references.

The Shape of Space, Third Edition maintains the standard of excellence set by the previous editions. This lighthearted textbook covers the basic geometry and topology of two- and three-dimensional spaces-stretching students' minds as they learn to visualize new possibilities for the shape of our universe. Written by a master expositor, leading researcher in the field, and MacArthur Fellow, its informal exposition and engaging exercises appeal to an exceptionally broad audience, from liberal arts students to math undergraduate and graduate students looking for a clear intuitive understanding to supplement more formal texts, and even to laypeople seeking an entertaining self-study book to expand their understanding of space. Features of the Third Edition: Full-color figures throughout "Picture proofs" have replaced algebraic proofs Simpler handles-and-crosscaps approach to surfaces Updated discussion of cosmological applications Intuitive examples missing from many college and graduate school curricula About the Author: Jeffrey R. Weeks is a freelance geometer living in Canton, New York. With support from the U.S. National Science Foundation, the MacArthur Foundation and several science museums, his work spans pure mathematics, applications in cosmology and-closest to his heart-exposition for the general public.

This volume presents lecture notes based on the author's courses on Lie algebras and the solution of Hilbert's fifth problem. In chapter 1, "Lie Algebras," the structure theory of semi-simple Lie algebras in characteristic zero is presented, following the ideas of Killing and Cartan. Chapter 2, "The Structure of Locally Compact Groups," deals with the solution of Hilbert's fifth problem given by Gleason, Montgomery, and Zipplin in 1952.

Das Buch zeigt angehenden und bereits praktizierenden Lehrkräften Wege zu einem guten, substanziellen und kompetenzorientierten Geometrieunterricht in der Grundschule auf. Dazu bietet es sowohl theoretisches Hintergrundwissen zur fachlichen Orientierung als auch vielfältige, didaktisch detailliert aufbereitete Anregungen zur praktischen Umsetzung an. Die vorliegende Neuauflage ist eine vollständige Neubearbeitung und Aktualisierung des erfolgreichen Standardwerks von Marianne Franke † zur Didaktik der Geometrie in der Grundschule. Die vorliegende Überarbeitung durch Simone Reinhold bezieht verstärkt jüngere Ergebnisse aus der geometriedidaktischen Forschung ein. Zu allen grundschulrelevanten Inhaltsbereichen der ebenen und räumlichen Geometrie wurden zudem inhaltliche Ergänzungen und Hinweise aus den eigenen langjährigen Unterrichtserfahrungen in der Grundschule aufgenommen. Durch zahlreiche Aktivitäten möchte dieses Buch dazu anregen, Kindern Freude an geometrischen Erkundungen zu vermitteln. Reichhaltige geometrische Lernumgebungen sollen dazu beitragen, räumliche Fähigkeiten von Kindern im Grundschulalter zu entwickeln und den Erwerb geometrischen Wissens zu fördern.

A new threshold for the existence of weak solutions to the incompressible Euler equations To gain insight into the nature of turbulent fluids, mathematicians start from experimental facts, translate them into mathematical properties for solutions of the fundamental fluids PDEs, and construct solutions to these PDEs that exhibit turbulent properties. This book belongs to such a program, one that has brought convex integration techniques into hydrodynamics. Convex integration techniques have been used to produce solutions with precise regularity, which are necessary for the resolution of the Onsager conjecture for the 3D Euler equations, or solutions with intermittency, which are necessary for the construction of dissipative weak solutions for the Navier-Stokes equations. In this book, weak solutions to the 3D Euler equations are constructed for the first time with both non-negligible regularity and intermittency. These solutions enjoy a spatial regularity index in L^2 that can be taken as close as desired to 1/2, thus lying at the threshold of all known convex integration methods. This property matches the measured intermittent nature of turbulent flows. The construction of such solutions requires technology specifically adapted to the inhomogeneities inherent in intermittent solutions. The main technical contribution of this book is to develop convex integration techniques at the local rather than global level. This localization procedure functions as an ad hoc wavelet decomposition of the solution, carrying information about position, amplitude, and frequency in both Lagrangian and Eulerian coordinates.

Meyer's Geometry and Its Applications, Second Edition, combines traditional geometry with current ideas to present a modern approach that is grounded in real-world applications. It balances the deductive approach with discovery learning, and introduces axiomatic, Euclidean geometry, non-Euclidean geometry, and transformational geometry. The text integrates applications and examples throughout and includes historical notes in many chapters.
The Second Edition of Geometry and Its Applications is a significant text for any college or university that focuses on geometry's usefulness in other disciplines. It is especially appropriate for engineering and science majors, as well as future mathematics teachers.
* Realistic applications integrated throughout the text, including (but not limited to):
- Symmetries of artistic patterns
- Physics
- Robotics
- Computer vision
- Computer graphics
- Stability of architectural structures
- Molecular biology
- Medicine
- Pattern recognition
* Historical notes included in many chapters
* Instructor's Manual with solutions available for all adopters of the text

The origins of the word problem are in group theory, decidability and complexity. But through the vision of M. Gromov and the language of filling functions, the topic now impacts the world of large-scale geometry. This book contains accounts of many recent developments in Geometric Group Theory and shows the interaction between the word problem and geometry continues to be a central theme. It contains many figures, numerous exercises and open questions.

Borel’s Conjecture entered the mathematics arena in 1919 as an innocuous remark about sets of real numbers in the context of a new covering property introduced by Émile Borel. In the 100 years since, this conjecture has led to a remarkably rich adventure of discovery in mathematics, producing independent results and the discovery of countable support iterated forcing, developments in infinitary game theory, deep connections with infinitary Ramsey Theory, and significant impact on the study of topological groups and topological covering properties. The papers in this volume present a broad introduction to the frontiers of research that has been spurred on by Borel’s 1919 conjecture and identify fundamental unanswered research problems in the field. Philosophers of science and historians of mathematics can glean from this collection some of the typical trends in the discovery, innovation, and development of mathematical theories.

Winner, Euler Book Prize, awarded by the Mathematical Association of America. With over 200 full color photographs, this non-traditional, tactile introduction to non-Euclidean geometries also covers early development of geometry and connections between geometry, art, nature, and sciences. For the crafter or would-be crafter, there are detailed instructions for how to crochet various geometric models and how to use them in explorations. New to the 2nd Edition; Daina Taimina discusses her own adventures with the hyperbolic planes as well as the experiences of some of her readers. Includes recent applications of hyperbolic geometry such as medicine, architecture, fashion & quantum computing.

This book presents, in a clear and structured way, the set function \mathcal{T} and how it evolved since its inception by Professor F. Burton Jones in the 1940s. It starts with a very solid introductory chapter, with all the prerequisite material for navigating through the rest of the book. It then gradually advances towards the main properties, Decomposition theorems, \mathcal{T}-closed sets, continuity and images, to modern applications. The set function \mathcal{T} has been used by many mathematicians as a tool to prove results about the semigroup structure of the continua, and about the existence of a metric continuum that cannot be mapped onto its cone or to characterize spheres. Nowadays, it has been used by topologists worldwide to investigate open problems in continuum theory. This book can be of interest to both advanced undergraduate and graduate students, and to experienced researchers as well. Its well-defined structure make this book suitable not only for self-study but also as support material to seminars on the subject. Its many open problems can potentially encourage mathematicians to contribute with further advancements in the field.

There has been an increasing interest in the statistical analysis of geometric objects and structures in many branches of science and engineering in recent years. The aim of this book is to present these statistical methods for practical use by non-mathematicians by outlining the mathematical ideas rather than concentrating on detailed proofs. The clarity of exposition ensures that the book will be a valuable resource for researchers and practitioners in many scientific disciplines who wish to use these methods in their work. In particular, the book is suited to materials scientists, geologists, environmental scientists, and biologists.

This textbook introduces the special theory of relativity at a level which is accessible to undergraduate students and even high school students with a strong foundation in algebra. The presentation emphasizes clean algebraic and geometrical methods, visualized with plenty of illustrations, resulting in a textbook that is modern and serious yet accessible. Replete with many solved exercises and copious spacetime diagrams, this book will help students develop relativistic intuition when encountering the subject for the first time. The emphasis on geometric methods, combined with the pedagogically appealing k-calculus approach, makes this book ideal for a self-contained course on special relativity or as supplementary reading for modern physics courses. It will also appeal to high schoolers with a strong math background who want to get ahead.

This is the second of three volumes that, together, give an exposition of the mathematics of grades 9-12 that is simultaneously mathematically correct and grade-level appropriate. The volumes are consistent with CCSSM (Common Core State Standards for Mathematics) and aim at presenting the mathematics of K-12 as a totally transparent subject. The first part of this volume is devoted to the study of standard algebra topics: quadratic functions, graphs of equations of degree 2 in two variables, polynomials, exponentials and logarithms, complex numbers and the fundamental theorem of algebra, and the binomial theorem. Having translations and the concept of similarity at our disposal enables us to clarify the study of quadratic functions by concentrating on their graphs, the same way the study of linear functions is greatly clarified by knowing that their graphs are lines. We also introduce the concept of formal algebra in the study of polynomials with complex coefficients. The last three chapters in this volume complete the systematic exposition of high school geometry that is consistent with CCSSM. These chapters treat the geometry of the triangle and the circle, ruler and compass constructions, and a general discussion of axiomatic systems, including non-Euclidean geometry and the celebrated work of Hilbert on the foundations. This book should be useful for current and future teachers of K-12 mathematics, as well as for some high school students and for education professionals.

The second, revised edition of this book covers all aspects of non-uniform rational B-splines necessary to design geometry in a computer-aided environment. Basic B-spline features, curve and surface algorithms, and state-of-the-art geometry tools are all discussed. Detailed code for design algorithms and computational tricks are covered, too, in a lucid, easy-to-understand style, with a minimum of mathematics and using numerous worked examples. The book is a must for students, researchers, and implementors whose work involves the use of splines.

The classic Heath translation, in a completely new layout with plenty of space and generous margins. An affordable but sturdy sewn hardcover student and teacher edition in one volume, with minimal notes and a new index/glossary.

This book is an introduction to the theory of calculus in the style of inquiry-based learning. The text guides students through the process of making mathematical ideas rigorous, from investigations and problems to definitions and proofs. The format allows for various levels of rigour as negotiated between instructor and students, and the text can be of use in a theoretically oriented calculus course or an analysis course that develops rigor gradually. Material on topology (e.g. of higher dimensional Euclidean spaces) and discrete dynamical systems can be used as excursions within a study of analysis or as a more central component of a course. The themes of bisection, iteration, and nested intervals form a common thread throughout the text. The book is intended for students who have studied some calculus and want to gain a deeper understanding of the subject through an inquiry-based approach.

This volume contains the proceedings of the ICM 2018 satellite school and workshop $K$-theory conference in Argentina. The school was held from July 16-20, 2018, in La Plata, Argentina, and the workshop was held from July 23-27, 2018, in Buenos Aires, Argentina. The volume showcases current developments in $K$-theory and related areas, including motives, homological algebra, index theory, operator algebras, and their applications and connections. Papers cover topics such as $K$-theory of group rings, Witt groups of real algebraic varieties, coarse homology theories, topological cyclic homology, negative $K$-groups of monoid algebras, Milnor $K$-theory and regulators, noncommutative motives, the classification of $C^*$-algebras via Kasparov's $K$-theory, the comparison between full and reduced $C^*$-crossed products, and a proof of Bott periodicity using almost commuting matrices.