Submanifold Theory : Beyond an Introduction by Marcos Dajczer; Ruy TojeiroThis book provides a comprehensive introduction to Submanifold theory, focusing on general properties of isometric and conformal immersions of Riemannian manifolds into space forms. One main theme is the isometric and conformal deformation problem for submanifolds of arbitrary dimension and codimension. Several relevant classes of submanifolds are also discussed, including constant curvature submanifolds, submanifolds of nonpositive extrinsic curvature, conformally flat submanifolds and real Kaehler submanifolds. This is the first textbook to treat a substantial proportion of the material presented here. The first chapters are suitable for an introductory course on Submanifold theory for students with a basic background on Riemannian geometry. The remaining chapters could be used in a more advanced course by students aiming at initiating research on the subject, and are also intended to serve as a reference for specialists in the field.
Call Number: QA649 .D35 2019
Differential Equations : Techniques, Theory, and Applications by Barbara D. MacCluer; Paul Bourdon; Thomas L. KrieteDifferential Equations: Techniques, Theory, and Applications is designed for a modern first course in differential equations either one or two semesters in length. The organization of the book interweaves the three components in the subtitle, with each building on and supporting the others. Techniques include not just computational methods for producing solutions to differential equations, but also qualitative methods for extracting conceptual information about differential equations and the systems modeled by them. Theory is developed as a means of organizing, understanding, and codifying general principles. Applications show the usefulness of the subject as a whole and heighten interest in both solution techniques and theory. Formal proofs are included in cases where they enhance core understanding; otherwise, they are replaced by informal justifications containing key ideas of a proof in a more conversational format. Applications are drawn from a wide variety of fields: those in physical science and engineering are prominent, of course, but models from biology, medicine, ecology, economics, and sports are also featured. The 1,400+ exercises are especially compelling. They range from routine calculations to large-scale projects. The more difficult problems, both theoretical and applied, are typically presented in manageable steps. The hundreds of meticulously detailed modeling problems were deliberately designed along pedagogical principles found especially effective in the MAA study Characteristics of Successful Calculus Programs, namely, that asking students to work problems that require them to grapple with concepts (or even proofs) and do modeling activities is key to successful student experiences and retention in STEM programs. The exposition itself is exceptionally readable, rigorous yet conversational. Students will find it inviting and approachable. The text supports many different styles of pedagogy from traditional lecture to a flipped classroom model. The availability of a computer algebra system is not assumed, but there are many opportunities to incorporate the use of one.
Call Number: QA371 .M1625 2019
Open Problems in Arithmetic Algebraic Geometry by Frans Oort (Editor)This book originated in the idea that open problems act as crystallization points in mathematical research. Mathematical books usually deal with fully developed theories. But here we present work at an earlier stage—when challenging questions can give new directions to mathematical research.
In mathematics, significant progress is often made by looking at the underlying structures of open problems and discovering new directions that are developed to find solutions. In that process, the search for finding the "true" nature of the problem at hand is the impetus for our thoughts. It is only much later, in retrospect, that we see the "flow of mathematics"—from problem to theory and new insight. This is the gist of the present volume.
Call Number: QA242.5 .O64 2019
Introduction to Complex Analysis by Michael E. TaylorIn this text, the reader will learn that all the basic functions that arise in calculus--such as powers and fractional powers, exponentials and logs, trigonometric functions and their inverses, as well as many new functions that the reader will meet--are naturally defined for complex arguments. Furthermore, this expanded setting leads to a much richer understanding of such functions than one could glean by merely considering them in the real domain. For example, understanding the exponential function in the complex domain via its differential equation provides a clean path to Euler's formula and hence to a self-contained treatment of the trigonometric functions. Complex analysis, developed in partnership with Fourier analysis, differential equations, and geometrical techniques, leads to the development of a cornucopia of functions of use in number theory, wave motion, conformal mapping, and other mathematical phenomena, which the reader can learn about from material presented here. This book could serve for either a one-semester course or a two-semester course in complex analysis for beginning graduate students or for well-prepared undergraduates whose background includes multivariable calculus, linear algebra, and advanced calculus.
Dirichlet Series and Holomorphic Functions in High Dimensions by Andreas Defant; Domingo García; Manuel Maestre; Pablo Sevilla-PerisOver 100 years ago Harald Bohr identified a deep problem about the convergence of Dirichlet series, and introduced an ingenious idea relating Dirichlet series and holomorphic functions in high dimensions. Elaborating on this work, almost twnety years later Bohnenblust and Hille solved the problem posed by Bohr. In recent years there has been a substantial revival of interest in the research area opened up by these early contributions. This involves the intertwining of the classical work with modern functional analysis, harmonic analysis, infinite dimensional holomorphy and probability theory as well as analytic number theory. New challenging research problems have crystallized and been solved in recent decades. The goal of this book is to describe in detail some of the key elements of this new research area to a wide audience. The approach is based on three pillars: Dirichlet series, infinite dimensional holomorphy and harmonic analysis.
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An Invitation to Computational Homotopy by Graham EllisAn Invitation to Computational Homotopy is an introduction to elementary algebraic topology for those with an interest in computers and computer programming. It expertly illustrates how the basics of the subject can be implemented on a computer through its focus on fully-worked examplesdesigned to develop problem solving techniques.The transition from basic theory to practical computation raises a range of non-trivial algorithmic issues which will appeal to readers already familiar with basic theory and who are interested in developing computational aspects. The book covers a subset of standard introductory material onfundamental groups, covering spaces, homology, cohomology and classifying spaces as well as some less standard material on crossed modules.These topics are covered in a way that hints at potential applications of topology in areas of computer science and engineering outside the usual territory of pure mathematics, and also in a way that demonstrates how computers can be used to perform explicit calculations within the domain of purealgebraic topology itself. The initial chapters include in-depth examples from data mining, biology and digital image analysis, while the later chapters cover a range of computational examples on the cohomology of classifying spaces that are likely beyond the reach of a purely paper-and-pen approachto the subject.An Invitation to Computational Homotopy serves as a self-contained and informal introduction to these topics and their implementation in the sphere of computer science. Written in a dynamic and engaging style, it skilfully showcases a range of useful machine computations, and will serve as aninvaluable aid to graduate students working with algebraic topology.
Model-Based Clustering and Classification for Data Science : with Applications in R by Charles Bouveyron; Gilles Celeux; T. Brendan Murphy; Adrian E. RafteryCluster analysis finds groups in data automatically. Most methods have been heuristic and leave open such central questions as: how many clusters are there? Which method should I use? How should I handle outliers? Classification assigns new observations to groups given previously classified observations, and also has open questions about parameter tuning, robustness and uncertainty assessment. This book frames cluster analysis and classification in terms of statistical models, thus yielding principled estimation, testing and prediction methods, and sound answers to the central questions. It builds the basic ideas in an accessible but rigorous way, with extensive data examples and R code; describes modern approaches to high-dimensional data and networks; and explains such recent advances as Bayesian regularization, non-Gaussian model-based clustering, cluster merging, variable selection, semi-supervised and robust classification, clustering of functional data, text and images, and co-clustering. Written for advanced undergraduates in data science, as well as researchers and practitioners, it assumes basic knowledge of multivariate calculus, linear algebra, probability and statistics.
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Hardy Inequalities on Homogeneous Groups : 100 Years of Hardy Inequalities by Michael Ruzhansky; Durvudkhan SuraganThis open access book provides an extensive treatment of Hardy inequalities and closely related topics from the point of view of Folland and Stein's homogeneous (Lie) groups. The place where Hardy inequalities and homogeneous groups meet is a beautiful area of mathematics with links to many other subjects. While describing the general theory of Hardy, Rellich, Caffarelli-Kohn-Nirenberg, Sobolev, and other inequalities in the setting of general homogeneous groups, the authors pay particular attention to the special class of stratified groups. In this environment, the theory of Hardy inequalities becomes intricately intertwined with the properties of sub-Laplacians and subelliptic partial differential equations. These topics constitute the core of this book and they are complemented by additional, closely related topics such as uncertainty principles, function spaces on homogeneous groups, the potential theory for stratified groups, and the potential theory for general Hörmander's sums of squares and their fundamental solutions. This monograph is the winner of the 2018 Ferran Sunyer i Balaguer Prize, a prestigious award for books of expository nature presenting the latest developments in an active area of research in mathematics. As can be attested as the winner of such an award, it is a vital contribution to literature of analysis not only because it presents a detailed account of the recent developments in the field, but also because the book is accessible to anyone with a basic level of understanding of analysis. Undergraduate and graduate students as well as researchers from any field of mathematical and physical sciences related to analysis involving functional inequalities or analysis of homogeneous groups will find the text beneficial to deepen their understanding.
Call Number: QA387 .R89 2019
Weirder Maths : at the Edge of the Possible by David Darling; Agnijo BanerjeeEven the most enthusiastic of maths students probably at one time wondered when exactly it would all prove useful in 'real life'. Well, maths reaches so far and wide through our world that, love it or hate it, we're all doing maths almost every minute of every day. David Darling and Agnijo Banerjee go in search of the perfect labyrinth, journey back to the second century in pursuit of 'bubble maths', reveal the weirdest mathematicians in history and transform the bewildering into the beautiful, delighting us once again.
Call Number: QA93 .D37 2019
Elementary Mathematical Models : an Accessible Development Without Calculus by Dan Kalman; Sacha Forgoston; Albert GoetzElementary Mathematical Models offers instructors an alternative to standard college algebra, quantitative literacy, and liberal arts mathematics courses. Presuming only a background of exposure to high school algebra, the text introduces students to the methodology of mathematical modeling, which plays a role in nearly all real applications of mathematics. A course based on this text would have as its primary goal preparing students to be competent consumers of mathematical modeling in their future studies.
Call Number: QA401 .K24 2019
Living Proof : Stories of Resilience Along the Mathematical Journey by Allison K. Henrich; Emille D. Lawrence; Matthew A. Pons; David George TaylorWow! This is a powerful book that addresses a long-standing elephant in the mathematics room. Many people learning math ask ``Why is math so hard for me while everyone else understands it?'' and ``Am I good enough to succeed in math?'' In answering these questions the book shares personal stories from many now-accomplished mathematicians affirming that ``You are not alone; math is hard for everyone'' and ``Yes; you are good enough.'' Along the way the book addresses other issues such as biases and prejudices that mathematicians encounter, and it provides inspiration and emotional support for mathematicians ranging from the experienced professor to the struggling mathematics student. --Michael Dorff, MAA President This book is a remarkable collection of personal reflections on what it means to be, and to become, a mathematician. Each story reveals a unique and refreshing understanding of the barriers erected by our cultural focus on ``math is hard.'' Indeed, mathematics is hard, and so are many other things--as Stephen Kennedy points out in his cogent introduction. This collection of essays offers inspiration to students of mathematics and to mathematicians at every career stage. --Jill Pipher, AMS President This book is published in cooperation with the Mathematical Association of America.
Call Number: QA11 .L7285 2019
A Singular Mathematical Promenade by Étienne GhysA stroll in the mathematical world. This is neither an elementary introduction to the theory of singularities, nor a specialized treatise containing many new theorems. The purpose of this little book is to invite the reader on a mathematical promenade. We pay a visit to Hipparchus, Newton and Gauss, but also to many contemporary mathematicians. We play with a bit of algebra, topology, geometry, complex analysis, combinatorics, and computer science. Hopefully, motivated undergraduates and more advanced mathematicians will enjoy some of these panoramas.