Logic Primer by 
Advanced Mathematics : An Invitation in Preparation for Graduate School by
This exploration of a selection of fundamental topics and general purpose tools provides a roadmap to undergraduate students who yearn for a deeper dive into many of the concepts and ideas they have been encountering in their classes, whether their motivation is pure curiosity or preparation for graduate studies. The topics intersect a wide range of areas encompassing both pure and applied mathematics. The emphasis and style of the book are motivated by the goal of developing self-reliance and independent mathematical thought. Mathematics requires both intuition and common sense as well as rigorous, formal argumentation. This book attempts to showcase both, simultaneously encouraging readers to develop their own insights and understanding and the adoption of proof writing skills. The most satisfying proofs/arguments are fully rigorous and completely intuitive at the same time.
Wu Wenjun : A Biography in Pictures by
This book depicts the fascinating life story of Wu Wenjun, a renowned mathematician who made significant contributions in the field of topology, ancient Chinese mathematics, and mathematics mechanization. He was a recipient of the Highest Science and Technology Award, the highest scientific award in China, as well as the Shaw Prize in Mathematics. Through vivid illustrations and eloquent writing, this book recounts rarely known anecdotes and significant events from Wu Wenjun's life from his childhood, education, and scientific career, offering insights into his life values.
Hex : A Playful Introduction by
This book offers a gentle introduction to Hex, the classic board game created by Piet Hein and popularized by John Nash and Martin Gardner. The first three chapters cover rules, basic strategy, and history. The remaining eight chapters cover a variety of topics: mathematical properties (there are no draws, the first player can win, the acute corner is a losing first move), the related game of Y, winning strategies for small boards, how computers play Hex, an analysis of Random-Move Hex (where one or both players move randomly) and Dark Hex (the imperfect information version of the game, where you can't see your opponent's moves). Did we mention puzzles? There are puzzles in every chapter, with solutions. This book is intended for anyone interested in playing board games or learning some recreational mathematics. It is written for a wide audience and will be enjoyed equally by general readers and professional mathematicians. The book could be used as a textbook or companion resource for a topics course on recreational mathematics or game theory or as a source for undergraduate research questions.
Fundamentals of Bayesian Epistemology by
Bayesian ideas have recently been applied across such diverse fields as philosophy, statistics, economics, psychology, artificial intelligence, and legal theory. Fundamentals of Bayesian Epistemology examines epistemologists' use of Bayesian probability mathematics to represent degrees ofbelief. Michael G. Titelbaum provides an accessible introduction to the key concepts and principles of the Bayesian formalism, enabling the reader both to follow epistemological debates and to see broader implicationsVolume 1 begins by motivating the use of degrees of belief in epistemology. It then introduces, explains, and applies the five core Bayesian normative rules: Kolmogorov's three probability axioms, the Ratio Formula for conditional degrees of belief, and Conditionalization for updating attitudes overtime. Finally, it discusses further normative rules (such as the Principal Principle, or indifference principles) that have been proposed to supplement or replace the core five.
Finite Difference Methods for Nonlinear Evolution Equations by
Nonlinear evolution equations are widely used to describe nonlinear phenomena in natural and social sciences. However, they are usually quite difficult to solve in most instances. This book introduces the finite difference methods for solving nonlinear evolution equations. The main numerical analysis tool is the energy method. This book covers the difference methods for the initial-boundary value problems of twelve nonlinear partial differential equations. They are Fisher equation, Burgers' equation, regularized long-wave equation, Korteweg-de Vries equation, Camassa-Holm equation, Schrödinger equation, Kuramoto-Tsuzuki equation, Zakharov equation, Ginzburg-Landau equation, Cahn-Hilliard equation, epitaxial growth model and phase field crystal model. This book is a monograph for the graduate students and science researchers majoring in computational mathematics and applied mathematics. It will be also useful to all researchers in related disciplines.
A Comparison Process for Mouse Pairs by
This book proves some important new theorems in the theory of canonical inner models for large cardinal hypotheses, a topic of central importance in modern set theory. In particular, the author completes the theory of Fine Structure and Iteration Trees (FSIT) by proving a comparison theorem for mouse pairs parallel to the FSIT comparison theorem for pure extender mice, and then using the underlying comparison process to develop a fine structure theory for strategy mice. Great effort has been taken to make the book accessible to non-experts so that it may also serve as an introduction to the higher reaches of inner model theory. It contains a good deal of background material, some of it unpublished folklore, and includes many references to the literature to guide further reading. An introductory essay serves to place the new results in their broader context. This is a landmark work in inner model theory that should be in every set theorist's library.
The Price of Cake : And 99 Other Classic Mathematical Riddles by
Ingeniously designed mathematical riddles to delight armchair-and seasoned-mathematicians, in high school and beyond. In The Price of Cake, brothers Clement and Guillaume Deslandes have assembled a uniquely rich and accessible collection of mathematical riddles, organized by difficulty and accompanied by unique hand-drawn illustrations. The challenges are exceptional in both their range and their precision. There are no "tricks" here, just good math to test your skills. While any student will be thrilled to encounter fresh presentations of classic conundrums such as Simpson's paradox, Hilbert's infinite hotel, Monty Hall's paradox, and Seven Bridges of Königsberg, seasoned solvers will also appreciate challenging original puzzles using the Axiom of Choice and Conway's Soldiers problem. The Deslandes' lucid and elegant solutions offer new insights, as they seamlessly connect the skills applied in each problem to broader theoretical revelations. Moreover, as Fields Medalist Cedric Villani writes in his preface, these riddles are much more than just exercises. They capture "a certain dose of decorum and mystery," and to decipher them will require both imagination and tenacity. To aid you on this journey, the authors provide hints to each puzzle, detailed proofs and solutions, and a series of Math Refreshers to revisit important concepts that are used to arrive at those answers. Entertaining and rigorous in equal measure, The Price of Cake is bound to captivate mathematics enthusiasts and students of all levels.
Harmonic Analysis on Hypergroups : Approximation and Stochastic Sequences by
This book gives a monographic presentation of the abstract harmonic analysis of hypergroups, while combining it with applied topics of spectral analysis, approximation by orthogonal expansions and stochastic sequences. Hypergroups are locally compact Hausdorff spaces equipped with a convolution, an involution and a unit element. Related algebraic structures had already been studied by Frobenius around 1900. Their axiomatic characterization in harmonic analysis was later developed in the 1970s. Hypergoups naturally emerge in seemingly different application areas as time series analysis, probability theory and theoretical physics. The book presents harmonic analysis on commutative and polynomial hypergroups as well as weakly stationary random fields and sequences thereon. For polynomial hypergroups, difference equations and stationary sequences are also considered. At greater extent than in the existing literature, the book compiles a comprehensive list of hypergroups, in particular of polynomial hypergroups. With an eye on readers at advanced undergraduate and graduate level, the proofs are worked out in careful detail.
Numerical Analysis on Time Scales by
Mathematical models cannot be solved using the traditional analytical methods for dynamic equations on time scales. These models must be dealt with using computational methods. This textbook introduces numerical methods for initial value problems for dynamic equations on time scales. Hands-on examples utilizing MATLAB and practical problems illustrate a wide variety of solution techniques.
A. P. Morse's Set Theory and Analysis by
This volume explores A.P. Morse's (1911-1984) development of a formal language for writing mathematics, his application of that language in set theory and mathematical analysis, and his unique perspective on mathematics. The editor brings together a variety of Morse's works in this compilation, including Morse's book A Theory of Sets, Second Edition (1986), in addition to material from another of Morse's publications, Web Derivatives, and notes for a course on analysis from the early 1950s. Because Morse provided very little in the way of explanation in his written works, the editor's commentary serves to outline Morse's goals, give informal explanations of Morse's formal language, and compare Morse's often unique approaches to more traditional approaches. Minor corrections to Morse's previously published works have also been incorporated into the text, including some updated axioms, theorems, and definitions. The editor's introduction thoroughly details the corrections and changes made and provides readers with valuable insight on Morse's methods. A.P. Morse's Set Theory and Analysis will appeal to graduate students and researchers interested in set theory and analysis who also have an interest in logic. Readers with a particular interest in Morse's unique perspective and in the history of mathematics will also find this book to be of interest.
Glimpses of Soliton Theory : The Algebra and Geometry of Nonlinear PDEs by
Solitons are nonlinear waves which behave like interacting particles. When first proposed in the 19th century, leading mathematical physicists denied that such a thing could exist. Now they are regularly observed in nature, shedding light on phenomena like rogue waves and DNA transcription. Solitons of light are even used by engineers for data transmission and optical switches. Furthermore, unlike most nonlinear partial differential equations, soliton equations have the remarkable property of being exactly solvable. Explicit solutions to those equations provide a rare window into what is possible in the realm of nonlinearity. Glimpses of Soliton Theory reveals the hidden connections discovered over the last half-century that explain the existence of these mysterious mathematical objects. It aims to convince the reader that, like the mirrors and hidden pockets used by magicians, the underlying algebro-geometric structure of soliton equations provides an elegant explanation of something seemingly miraculous. Assuming only multivariable calculus and linear algebra, the book introduces the reader to the KdV Equation and its multisoliton solutions, elliptic curves and Weierstrass ℘-functions, the algebra of differential operators, Lax Pairs and their use in discovering other soliton equations, wedge products and decomposability, the KP Hierarchy, and Sato's theory relating the Bilinear KP Equation to the geometry of Grassmannians. Notable features of the book include: careful selection of topics and detailed explanations to make the subject accessible to undergraduates, numerous worked examples and thought-provoking exercises, footnotes and lists of suggested readings to guide the interested reader to more information, and use of MathematicaR to facilitate computation and animate solutions. The second edition refines the exposition in every chapter, adds more homework exercises and projects, updates references, and includes new examples involving non-commutative integrable systems. Moreover, the chapter on KdV multisolitons has been greatly expanded with new theorems providing a thorough analysis of their behavior and decomposition.
Inference and Learning from Data by
This extraordinary three-volume work, written in an engaging and rigorous style by a world authority in the field, provides an accessible, comprehensive introduction to the full spectrum of mathematical and statistical techniques underpinning contemporary methods in data-driven learning and inference. The first volume, Foundations, establishes core topics in inference and learning, and prepares readers for studying their practical application. The second volume, Inference, introduces readers to cutting-edge techniques for inferring unknown variables and quantities. The final volume, Learning, provides a rigorous introduction to state-of-the-art learning methods. A consistent structure and pedagogy is employed throughout all three volumes to reinforce student understanding, with over 1280 end-of-chapter problems (including solutions for instructors), over 600 figures, over 470 solved examples, datasets and downloadable Matlab code. Unique in its scale and depth, this textbook sequence is ideal for early-career researchers and graduate students across many courses in signal processing, machine learning, statistical analysis, data science and inference.
Lost in the Math Museum : A Survival Story by
"But when I turned the handle on the door, suddenly the buzzing went crazy. I slapped my hands over my ears, when I should have jerked the door shut. It flew open, and I was face-to-face with the Weierstrass function. It was the ugliest function I could imagine, with kinks, and kinks on kinks and kinks on those. And it was shrieking in its buzz-like way, vibrating all over like a plucked string. I stood there, frozen for just a second, and then I was sprinting after the others, with the wild frantic buzzing right behind me.'' From the twisted imagination of best-selling author Colin Adams (Zombies & Calculus, The Knot Book) comes this tale of sixteen-year-old Kallie trying to escape death at the hands of the exhibits in a mathematics museum. Kallie crosses paths with Carl Gauss, Bertrand Russell, Sophie Germain, G. H. Hardy, and John von Neumann, as she tries to save herself, her dad, and his colleague Maria from the deadly Hairy Ball theorem, the harrowing Hilbert Hotel, the bisecting Ham Sandwich machine, and a variety of other mathematical menaces. It's a wild romp through a mathematical bestiary featuring the bizarre, the exotic, and the counterintuitive. You'll never think of math the same way again.
The Six Pillars of Calculus : Business Edition by
The Six Pillars of Calculus is a conceptual and practical introduction to differential and integral calculus for use in a one- or two-semester course. By boiling calculus down to six common-sense ideas, the text invites students to make calculus an integral part of how they view the world. Each pillar is introduced by tackling and solving a challenging, realistic problem. This engaging process of discovery encourages students to wrestle with the material and understand the reasoning behind the techniques they are learning--to focus on when and why to use the tools of calculus, not just on how to apply formulas. Modeling and differential equations are front and center. Solutions begin with numerical approximations; derivatives and integrals emerge naturally as refinements of those approximations. Students use and modify computer programs to reinforce their understanding of each algorithm. The Business Edition of the Six Pillars series has been extensively field-tested at the University of Texas. It features hundreds of examples and problems designed specifically for business students. The core ideas are introduced by modeling market penetration of a new product, tracking changes in the national debt, and maximizing the profit of a business. Along the way, students learn about present value, consumer and producer surplus, amortization, and probability.
From Model Reduction to Efficient Predictive Control with Guarantees by
High-dimensional dynamic models frequently occur in many practical problems. Using these models is often computationally intractable; for instance, analysis via repeated simulations or control design. In this book, several algorithms relying on model reduction are proposed to deal with the high-dimensionality of these models. The main contribution is a novel model predictive control scheme, which uses reduced models for linear time-invariant systems. An intermediate result is a generalized bound for the error between the high-dimensional and the reduced model, while simulating the reduced model. This error bounding system is included in the model predictive control scheme in order to guarantee 1) asymptotic stability, 2) satisfaction of hard input and state constraints, 3) a bound for the cost functional value, and 4) minimization of the infinite horizon cost functional for the high-dimensional model. For discrete-time models, it is shown that the optimization problem of the model predictive control scheme can be reformulated as a second-order cone program. The applicability of the proposed methods is demonstrated by means of a nonisothermal tubular chemical reactor. A further contribution is a model reduction procedure, which approximates the input-output map of continuous-time nonlinear ordinary differential equations. This method allows to preserve the location and local exponential stability of multiple steady states.
Moduli Stacks of Étale ([phi], [gamma])-Modules and the Existence of Crystalline Lifts by
A foundational account of a new construction in the p-adic Langlands correspondence Motivated by the p-adic Langlands program, this book constructs stacks that algebraize Mazur's formal deformation rings of local Galois representations. More precisely, it constructs Noetherian formal algebraic stacks over Spf Zp that parameterize étale (ϕ, Γ)-modules; the formal completions of these stacks at points in their special fibres recover the universal deformation rings of local Galois representations. These stacks are then used to show that all mod p representations of the absolute Galois group of a p-adic local field lift to characteristic zero, and indeed admit crystalline lifts. The book explicitly describes the irreducible components of the underlying reduced substacks and discusses the relationship between the geometry of these stacks and the Breuil-Mézard conjecture. Along the way, it proves a number of foundational results in p-adic Hodge theory that may be of independent interest.
Geometry and Discrete Mathematics : A Selection of Highlights by
In the two-volume set "A Selection of Highlights" we present basics of mathematics in an exciting and pedagogically sound way. This volume examines many fundamental results in geometry and discrete mathematics along with their proofs and their history. The second edition includes a new chapter on topological data analysis and an enhanced chapter on graph theory for solving further classical problems such as the Traveling Salesman Problem.
On Theories : Logical Empiricism and the Methodology of Modern Physics by
A renowned philosopher's final work, illuminating how the logical empiricist tradition has failed to appreciate the role of actual experiments in forming its philosophy of science. The logical empiricist treatment of physics dominated twentieth-century philosophy of science. But the logical empiricist tradition, for all it accomplished, does not do justice to the way in which empirical evidence functions in modern physics. In his final work, the late philosopher of science William Demopoulos contends that philosophers have failed to provide an adequate epistemology of science because they have failed to appreciate the tightly woven character of theory and evidence. As a consequence, theory comes apart from evidence. This trouble is nowhere more evident than in theorizing about particle and quantum physics. Arguing that we must consider actual experiments as they have unfolded across history, Demopoulos provides a new epistemology of theories and evidence, albeit one that stands on the shoulders of giants. On Theories finds clarity in Isaac Newton's suspicion of mere "hypotheses." Newton's methodology lies in the background of Jean Perrin's experimental investigations of molecular reality and of the subatomic investigations of J. J. Thomson and Robert Millikan. Demopoulos extends this account to offer novel insights into the distinctive nature of quantum reality, where a logico-mathematical reconstruction of Bohrian complementarity meets John Stewart Bell's empirical analysis of Einstein's "local realism." On Theories ultimately provides a new interpretation of quantum probabilities as themselves objectively representing empirical reality.
Philosophy of Science : Perspectives from Scientists by
This book describes the framework of a new theory of science. Over the last hundred years, philosophy of science developed its theory based on philosophers' perception of what science is and what scientists do. It did not address the basic question of what scientists care about. This book examines the conventional theories of philosophy of science from a completely different point of view and describes the most difficult problems that scientists face and the way scientific inquiry is conducted.
Introduction to Matrix Theory : With Applications in Economics and Engineering by
Linear algebra and matrix theory are among the most important and most frequently applied branches of mathematics. They are especially important in solving engineering and economic models, where either the model is assumed linear, or the nonlinear model is approximated by a linear model, and the resulting linear model is examined. This book covers a one-semester upper division course or a two-semester lower division course on the subject. The second edition will be an extended and modernized version of the first edition, including new theoretical topics and applications from fields other than economics.
Spectral Flow : A Functional Analytic and Index-Theoretic Approach by
This is the first treatment entirely dedicated to an analytic study of spectral flow for paths of selfadjoint Fredholm operators, possibly unbounded or understood in a semifinite sense. The importance of spectral flow for homotopy and index theory is discussed in detail. Applications concern eta-invariants, the Bott-Maslov and Conley-Zehnder indices, Sturm-Liouville oscillation theory, the spectral localizer and bifurcation theory.
BCC-Algebras by
First book on BCC algebras and first to gather the research in one place. Appeals to researchers interested in algebra, the largest topic group of mathematics researchers. No competition on this topic.
Iwasawa Theory and Its Perspective by
Iwasawa theory began in the late 1950s with a series of papers by Kenkichi Iwasawa on ideal class groups in the cyclotomic tower of number fields and their relation to p-adic L-functions. The theory was later generalized by putting it in the context of elliptic curves and modular forms. The main motivation for writing this book was the need for a total perspective of Iwasawa theory that includes the new trends of generalized Iwasawa theory. Another motivation of this book is an update of the classical theory for class groups taking into account the changed point of view on Iwasawa theory. The goal of this first part of the two-part publication is to explain the theory of ideal class groups, including its algebraic aspect (the Iwasawa class number formula), its analytic aspect (Leopoldt-Kubota L-functions), and the Iwasawa main conjecture, which is a bridge between the algebraic and the analytic aspects. The second part of the book will be published as a separate volume in the same series, Mathematical Surveys and Monographs of the American Mathematical Society.
