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Game theory is the science of interaction. This textbook, derived from courses taught by the author and developed over several years, is a comprehensive, straightforward introduction to the mathematics of non-cooperative games. It teaches what every game theorist should know: the important ideas and results on strategies, game trees, utility theory, imperfect information, and Nash equilibrium. This detailed and lively text requires minimal mathematical background and includes many examples, exercises, and pictures. It is suitable for self-study or introductory courses in mathematics, computer science, or economics departments.
Reflections of Alan Turing : A Relative Story by Dermot TuringEveryone knows the story of the codebreaker and computer science pioneer Alan Turing. Except ... When Dermot Turing is asked about his famous uncle, people want to know more than the bullet points of his life. They want to know everything- was Alan Turing actually a codebreaker? What did he make of artificial intelligence? What is the significance of Alan Turing's trial, his suicide, the Royal Pardon, the £50 note and the film The Imitation Game? In Reflections of Alan Turing, Dermot strips off the layers to uncover the real story. It's time to discover a fresh legacy of Alan Turing for the twenty-first century.
Call Number: QA29.T8 T783 2021
An Introduction to Partial Differential Equations (with Maple) : A Concise Course by Zhilin LiThis book is designed for undergraduate or beginning level graduate students, and students from interdisciplinary areas including engineers, and others who need to use partial differential equations, Fourier series, Fourier and Laplace transforms. The prerequisite is a basic knowledge of calculus, linear algebra, and ordinary differential equations. The textbook aims to be practical, elementary, and reasonably rigorous; the book is concise in that it describes fundamental solution techniques for first order, second order, linear partial differential equations for general solutions, fundamental solutions, solution to Cauchy (initial value) problems, and boundary value problems for different PDEs in one and two dimensions, and different coordinates systems. Analytic solutions to boundary value problems are based on Sturm-Liouville eigenvalue problems and series solutions. The book is accompanied with enough well tested Maple files and some Matlab codes that are available online. The use of Maple makes the complicated series solution simple, interactive, and visible. These features distinguish the book from other textbooks available in the related area.
Call Number: QA374 .L428 2022
Philosophy of Mathematics : Classic and Contemporary Studies by Ahmet CevikPhilosophy of Mathematics: Classic and Contemporary Studies explores the foundations of mathematical thought. The aim of this book is to encourage young mathematicians to think about the philosophical issues behind fundamental concepts and about different views on mathematical objects and mathematical knowledge. With this new approach, the author rekindles an interest in philosophical subjects surrounding the foundations of mathematics. He offers the mathematical motivations behind the topics under debate. He introduces various philosophical positions ranging from the classic views to more contemporary ones, including subjects which are more engaged with mathematical logic.
Call Number: QA8.4 .C48 2022
Crossed Modules by Friedrich WagemannThis book presents material in two parts. Part one provides an introduction to crossed modules of groups, Lie algebras and associative algebras with fully written out proofs and is suitable for graduate students interested in homological algebra. In part two, more advanced and less standard topics such as crossed modules of Hopf algebra, Lie groups, and racks are discussed as well as recent developments and research on crossed modules.
Call Number: QA169 .W34 2021
Infinity Operads and Monoidal Categories with Group Equivariance by Donald Y. YauThis monograph provides a coherent development of operads, infinity operads, and monoidal categories, equipped with equivariant structures encoded by an action operad. A group operad is a planar operad with an action operad equivariant structure. In the first three parts of this monograph, we establish a foundation for group operads and for their higher coherent analogues called infinity group operads. Examples include planar, symmetric, braided, ribbon, and cactus operads, and their infinity analogues. For example, with the tools developed here, we observe that the coherent ribbon nerve of the universal cover of the framed little 2-disc operad is an infinity ribbon operad. In Part 4 we define general monoidal categories equipped with an action operad equivariant structure and provide a unifying treatment of coherence and strictification for them. Examples of such monoidal categories include symmetric, braided, ribbon, and coboundary monoidal categories, which naturally arise in the representation theory of quantum groups and of coboundary Hopf algebras and in the theory of crystals of finite dimensional complex reductive Lie algebras.
Call Number: QA169 .Y39 2022
An Introduction to Proof Theory : Normalization, Cut-elimination, and Consistency Proofs by Paolo Mancosu; Sergio Galvan; Richard ZachAn Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details of proofs worked out and examples and exercises to aid the reader's understanding. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The firsthalf covers topics in structural proof theory, including the Godel-Gentzen translation of classical into intuitionistic logic (and arithmetic), natural deduction and the normalization theorems (for both NJ and NK), the sequent calculus, including cut-elimination and mid-sequent theorems, and variousapplications of these results. The second half examines ordinal proof theory, specifically Gentzen's consistency proof for first-order Peano Arithmetic. The theory of ordinal notations and other elements of ordinal theory are developed from scratch, and no knowledge of set theory is presumed. Theproof methods needed to establish proof-theoretic results, especially proof by induction, are introduced in stages throughout the text. Mancosu, Galvan, and Zach's introduction will provide a solid foundation for those looking to understand this central area of mathematical logic and the philosophyof mathematics.
Call Number: QA9.54 .M36 2021
Life Is Simple : How Occam's Razor Set Science Free and Shapes the Universe by Johnjoe McFaddenA biologist argues that simplicity is the guiding principle of the universe Centuries ago, the principle of Ockham's razor changed our world by showing simpler answers to be preferable and more often true. In Life Is Simple, scientist Johnjoe McFadden traces centuries of discoveries, taking us from a geocentric cosmos to quantum mechanics and DNA, arguing that simplicity has revealed profound answers to the greatest mysteries. This is no coincidence. From the laws that keep a ball in motion to those that govern evolution, simplicity, he claims, has shaped the universe itself. And in McFadden's view, life could only have emerged by embracing maximal simplicity, making the fundamental law of the universe a cosmic form of natural selection that favors survival of the simplest. Recasting both the history of science and our universe's origins, McFadden transforms our understanding of ourselves and our world.
Call Number: Q175 .M4168 2021
Transformational Change Efforts : Student Engagement in Mathematics Through an Institutional Network for Active Learning by Wendy M. Smith; Matthew Voigt; April Strom; David C. Webb; W. Gary Martin (Editors)The purpose of this handbook is to help launch institutional transformations in mathematics departments to improve student success. We report findings from the Student Engagement in Mathematics through an Institutional Network for Active Learning (SEMINAL) study. SEMINAL's purpose is to help change agents, those looking to (or currently attempting to) enact change within mathematics departments and beyond--trying to reform the instruction of their lower division mathematics courses in order to promote high achievement for all students. SEMINAL specifically studies the change mechanisms that allow postsecondary institutions to incorporate and sustain active learning in Precalculus to Calculus 2 learning environments. Out of the approximately 2.5 million students enrolled in collegiate mathematics courses each year, over 90% are enrolled in Precalculus to Calculus 2 courses. Forty-four percent of mathematics departments think active learning mathematics strategies are important for Precalculus to Calculus 2 courses, but only 15percnt; state that they are very successful at implementing them. Therefore, insights into the following research question will help with institutional transformations: What conditions, strategies, interventions and actions at the departmental and classroom levels contribute to the initiation, implementation, and institutional sustainability of active learning in the undergraduate calculus sequence (Precalculus to Calculus 2) across varied institutions? This book is published in cooperation with CBMS.
Call Number: QA20.W43 T73 2021
The Semantic Conception of Logic by Gil Sagi; Jack Woods (Editors)This collection of new essays presents cutting-edge research on the semantic conception of logic, the invariance criteria of logicality, grammaticality, and logical truth. Contributors explore the history of the semantic tradition, starting with Tarski, and its historical applications, while central criticisms of the tradition, and especially the use of invariance criteria to explain logicality, are revisited by the original participants in that debate. Other essays discuss more recent criticism of the approach, and researchers from mathematics and linguistics weigh in on the role of the semantic tradition in their disciplines. This book will be invaluable to philosophers and logicians alike.
Call Number: QA9.7 .S46 2021
George Spencer Brown's "Design with the NOR" : With Related Essays by Steffen Roth; Markus Heidingsfelder; Lars Clausen; Klaus Brønd Laursen (Editors)George Spencer Brown, a polymath and author ofLaws of Form, brought together mathematics, electronics, engineering and philosophy to form an unlikely bond. This book investigates Design with NOR, the title of the yet unpublished 1961 typescript by Spencer Brown. The typescript formed through the author's experiences as technical engineer and developer of a new form of switching algebra for Mullard Equipment Ltd., a British manufacturer of electronic components, and is published here for the first time. Related essays contextualise the typescript drawing on a variety backgrounds from mathematics and engineering to philosophy and sociology, and thus invite readers to a reverse-engineering of both the form and its laws.
Call Number: QA9 .G46 2021
A Course in Hodge Theory : With Emphasis on Multiple Integrals by Hossein MovasatiHodge theory-one of the pillars of modern algebraic geometry-is a deep theory with many applications and open problems, the most enigmatic of which is the Hodge conjecture, one of the Clay Institute's seven Millennium Prize Problems. Hodge theory is also famously difficult to learn, requiring training in many different branches of mathematics. The present volume begins with an examination of the precursors of Hodge theory: first, the studies of elliptic and abelian integrals by Cauchy, Abel, Jacobi, and Riemann, among many others; and then the studies of two-dimensional multiple integrals by Poincaré and Picard. Thenceforth, the focus turns to the Hodge theory of affine hypersurfaces given by tame polynomials, for which many tools from singularity theory, such as Brieskorn modules and Milnor fibrations, are used. Another aspect of this volume is its computational presentation of many well-known theoretical concepts, such as the Gauss-Manin connection, homology of varieties in terms of vanishing cycles, Hodge cycles, Noether-Lefschetz, and Hodge loci. All are explained for the generalized Fermat variety, which for Hodge theory boils down to a heavy linear algebra. Most of the algorithms introduced here are implemented in Singular, a computer algebra system for polynomial computations. Finally, the author introduces a few problems, mainly for talented undergraduate students, which can be understood with a basic knowledge of linear algebra. The origins of these problems may be seen in the discussions of advanced topics presented throughout this volume.
The Ten Equations That Rule the World : And How You can Use Them Too by David SumpterIs there a secret formula for getting rich? For going viral? For deciding how long to stick with your current job, Netflix series, or even relationship? This book is all about the equations that make our world go round. Ten of them, in fact. They are integral to everything from investment banking to betting companies and social media giants. And they can help you to increase your chance of success, guard against financial loss, live more healthfully, and see through scaremongering. They are known by only the privileged few - until now. With wit and clarity, mathematician David Sumpter shows that it isn't the technical details that make these formulas so successful. It is the way they allow mathematicians to view problems from a different angle - a way of seeing the world that anyone can learn. Empowering and illuminating,The Ten Equations shows how math really can change your life.
Call Number: QA93 .S86 2021
All the Math You Missed : But Need to Know for Graduate School by Thomas A. GarrityBeginning graduate students in mathematical sciences and related areas in physical and computer sciences and engineering are expected to be familiar with a daunting breadth of mathematics, but few have such a background. This bestselling book helps students fill in the gaps in their knowledge. Thomas A. Garrity explains the basic points and a few key results of all the most important undergraduate topics in mathematics, emphasizing the intuitions behind the subject. The explanations are accompanied by numerous examples, exercises and suggestions for further reading that allow the reader to test and develop their understanding of these core topics. Featuring four new chapters and many other improvements, this second edition of All the Math You Missed is an essential resource for advanced undergraduates and beginning graduate students who need to learn some serious mathematics quickly.
Call Number: QA37.3 .G37 2021
Essentials of Tropical Combinatorics by Michael JoswigThe goal of this book is to explain, at the graduate student level, connections between tropical geometry and optimization. Building bridges between these two subject areas is fruitful in two ways. Through tropical geometry optimization algorithms become applicable to questions in algebraic geometry. Conversely, looking at topics in optimization through the tropical geometry lens adds an additional layer of structure. The author covers contemporary research topics that are relevant for applications such as phylogenetics, neural networks, combinatorial auctions, game theory, and computational complexity. This self-contained book grew out of several courses given at Technische Universitat Berlin and elsewhere, and the main prerequisite for the reader is a basic knowledge in polytope theory. It contains a good number of exercises, many examples, beautiful figures, as well as explicit tools for computations using $\texttt{{polymake}}$.
Call Number: QA582 .J67 2021
Discovering Abstract Algebra by John K. OsoinachDiscovering Abstract Algebra takes an inquiry-based learning approach to the subject, leading students to discover for themselves its main themes and techniques. Concepts are introduced conversationally through extensive examples and student investigation before being formally defined. Students will develop skills in carefully making statements and writing proofs, while they simultaneously build a sense of ownership over the ideas and results. The book has been extensively tested and reinforced at points of common student misunderstanding or confusion, and includes a wealth of exercises at a variety of levels. The contents were deliberately organized to follow the recommendations of the MAA's 2015 Curriculum Guide. The book is ideal for a one- or two-semester course in abstract algebra, and will prepare students well for graduate-level study in algebra.
Call Number: QA162 .O86 2021
An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows by Alessio Figalli; Federico GlaudoThis book provides a self-contained introduction to optimal transport, and it is intended as a starting point for any researcher who wants to enter into this beautiful subject. The presentation focuses on the essential topics of the theory: Kantorovich duality, existence and uniqueness of optimal transport maps, Wasserstein distances, the JKO scheme, Otto's calculus, and Wasserstein gradient flows. At the end, a presentation of some selected applications of optimal transport is given. The book is suitable for a course at the graduate level and also includes an appendix with a series of exercises along with their solutions.
Call Number: QA402.6 .F51 2021
The Mathematical Principles of Natural Philosophy by Issac Newton; C. R. Leedham-Green (Edited and Translated by)Newton's Principia is perhaps the second most famous work of mathematics, after Euclid's Elements. Originally published in 1687, it gave the first systematic account of the fundamental concepts of dynamics, as well as three beautiful derivations of Newton's law of gravitation from Kepler's laws of planetary motion. As a book of great insight and ingenuity, it has raised our understanding of the power of mathematics more than any other work. This heavily annotated translation of the third and final edition (1726) of the Principia will enable any reader with a good understanding of elementary mathematics to easily grasp the meaning of the text, either from the translation itself or from the notes, and to appreciate some of its significance. All forward references are given to illuminate the structure and unity of the whole, and to clarify the parts. The mathematical prerequisites for understanding Newton's arguments are given in a brief appendix.
Game Theory Basics by 
Reflections of Alan Turing : A Relative Story by Everyone knows the story of the codebreaker and computer science pioneer Alan Turing. Except ... When Dermot Turing is asked about his famous uncle, people want to know more than the bullet points of his life. They want to know everything- was Alan Turing actually a codebreaker? What did he make of artificial intelligence? What is the significance of Alan Turing's trial, his suicide, the Royal Pardon, the £50 note and the film The Imitation Game? In Reflections of Alan Turing, Dermot strips off the layers to uncover the real story. It's time to discover a fresh legacy of Alan Turing for the twenty-first century.
An Introduction to Partial Differential Equations (with Maple) : A Concise Course by This book is designed for undergraduate or beginning level graduate students, and students from interdisciplinary areas including engineers, and others who need to use partial differential equations, Fourier series, Fourier and Laplace transforms. The prerequisite is a basic knowledge of calculus, linear algebra, and ordinary differential equations. The textbook aims to be practical, elementary, and reasonably rigorous; the book is concise in that it describes fundamental solution techniques for first order, second order, linear partial differential equations for general solutions, fundamental solutions, solution to Cauchy (initial value) problems, and boundary value problems for different PDEs in one and two dimensions, and different coordinates systems. Analytic solutions to boundary value problems are based on Sturm-Liouville eigenvalue problems and series solutions. The book is accompanied with enough well tested Maple files and some Matlab codes that are available online. The use of Maple makes the complicated series solution simple, interactive, and visible. These features distinguish the book from other textbooks available in the related area.
Philosophy of Mathematics : Classic and Contemporary Studies by Philosophy of Mathematics: Classic and Contemporary Studies explores the foundations of mathematical thought. The aim of this book is to encourage young mathematicians to think about the philosophical issues behind fundamental concepts and about different views on mathematical objects and mathematical knowledge. With this new approach, the author rekindles an interest in philosophical subjects surrounding the foundations of mathematics. He offers the mathematical motivations behind the topics under debate. He introduces various philosophical positions ranging from the classic views to more contemporary ones, including subjects which are more engaged with mathematical logic.
Crossed Modules by This book presents material in two parts. Part one provides an introduction to crossed modules of groups, Lie algebras and associative algebras with fully written out proofs and is suitable for graduate students interested in homological algebra. In part two, more advanced and less standard topics such as crossed modules of Hopf algebra, Lie groups, and racks are discussed as well as recent developments and research on crossed modules.
Infinity Operads and Monoidal Categories with Group Equivariance by This monograph provides a coherent development of operads, infinity operads, and monoidal categories, equipped with equivariant structures encoded by an action operad. A group operad is a planar operad with an action operad equivariant structure. In the first three parts of this monograph, we establish a foundation for group operads and for their higher coherent analogues called infinity group operads. Examples include planar, symmetric, braided, ribbon, and cactus operads, and their infinity analogues. For example, with the tools developed here, we observe that the coherent ribbon nerve of the universal cover of the framed little 2-disc operad is an infinity ribbon operad. In Part 4 we define general monoidal categories equipped with an action operad equivariant structure and provide a unifying treatment of coherence and strictification for them. Examples of such monoidal categories include symmetric, braided, ribbon, and coboundary monoidal categories, which naturally arise in the representation theory of quantum groups and of coboundary Hopf algebras and in the theory of crystals of finite dimensional complex reductive Lie algebras.
An Introduction to Proof Theory : Normalization, Cut-elimination, and Consistency Proofs by An Introduction to Proof Theory provides an accessible introduction to the theory of proofs, with details of proofs worked out and examples and exercises to aid the reader's understanding. It also serves as a companion to reading the original pathbreaking articles by Gerhard Gentzen. The firsthalf covers topics in structural proof theory, including the Godel-Gentzen translation of classical into intuitionistic logic (and arithmetic), natural deduction and the normalization theorems (for both NJ and NK), the sequent calculus, including cut-elimination and mid-sequent theorems, and variousapplications of these results. The second half examines ordinal proof theory, specifically Gentzen's consistency proof for first-order Peano Arithmetic. The theory of ordinal notations and other elements of ordinal theory are developed from scratch, and no knowledge of set theory is presumed. Theproof methods needed to establish proof-theoretic results, especially proof by induction, are introduced in stages throughout the text. Mancosu, Galvan, and Zach's introduction will provide a solid foundation for those looking to understand this central area of mathematical logic and the philosophyof mathematics.
Life Is Simple : How Occam's Razor Set Science Free and Shapes the Universe by A biologist argues that simplicity is the guiding principle of the universe Centuries ago, the principle of Ockham's razor changed our world by showing simpler answers to be preferable and more often true. In Life Is Simple, scientist Johnjoe McFadden traces centuries of discoveries, taking us from a geocentric cosmos to quantum mechanics and DNA, arguing that simplicity has revealed profound answers to the greatest mysteries. This is no coincidence. From the laws that keep a ball in motion to those that govern evolution, simplicity, he claims, has shaped the universe itself. And in McFadden's view, life could only have emerged by embracing maximal simplicity, making the fundamental law of the universe a cosmic form of natural selection that favors survival of the simplest. Recasting both the history of science and our universe's origins, McFadden transforms our understanding of ourselves and our world.
Transformational Change Efforts : Student Engagement in Mathematics Through an Institutional Network for Active Learning by The purpose of this handbook is to help launch institutional transformations in mathematics departments to improve student success. We report findings from the Student Engagement in Mathematics through an Institutional Network for Active Learning (SEMINAL) study. SEMINAL's purpose is to help change agents, those looking to (or currently attempting to) enact change within mathematics departments and beyond--trying to reform the instruction of their lower division mathematics courses in order to promote high achievement for all students. SEMINAL specifically studies the change mechanisms that allow postsecondary institutions to incorporate and sustain active learning in Precalculus to Calculus 2 learning environments. Out of the approximately 2.5 million students enrolled in collegiate mathematics courses each year, over 90% are enrolled in Precalculus to Calculus 2 courses. Forty-four percent of mathematics departments think active learning mathematics strategies are important for Precalculus to Calculus 2 courses, but only 15percnt; state that they are very successful at implementing them. Therefore, insights into the following research question will help with institutional transformations: What conditions, strategies, interventions and actions at the departmental and classroom levels contribute to the initiation, implementation, and institutional sustainability of active learning in the undergraduate calculus sequence (Precalculus to Calculus 2) across varied institutions? This book is published in cooperation with CBMS.
The Semantic Conception of Logic by This collection of new essays presents cutting-edge research on the semantic conception of logic, the invariance criteria of logicality, grammaticality, and logical truth. Contributors explore the history of the semantic tradition, starting with Tarski, and its historical applications, while central criticisms of the tradition, and especially the use of invariance criteria to explain logicality, are revisited by the original participants in that debate. Other essays discuss more recent criticism of the approach, and researchers from mathematics and linguistics weigh in on the role of the semantic tradition in their disciplines. This book will be invaluable to philosophers and logicians alike.
George Spencer Brown's "Design with the NOR" : With Related Essays by George Spencer Brown, a polymath and author ofLaws of Form, brought together mathematics, electronics, engineering and philosophy to form an unlikely bond. This book investigates Design with NOR, the title of the yet unpublished 1961 typescript by Spencer Brown. The typescript formed through the author's experiences as technical engineer and developer of a new form of switching algebra for Mullard Equipment Ltd., a British manufacturer of electronic components, and is published here for the first time. Related essays contextualise the typescript drawing on a variety backgrounds from mathematics and engineering to philosophy and sociology, and thus invite readers to a reverse-engineering of both the form and its laws.
A Course in Hodge Theory : With Emphasis on Multiple Integrals by Hodge theory-one of the pillars of modern algebraic geometry-is a deep theory with many applications and open problems, the most enigmatic of which is the Hodge conjecture, one of the Clay Institute's seven Millennium Prize Problems. Hodge theory is also famously difficult to learn, requiring training in many different branches of mathematics. The present volume begins with an examination of the precursors of Hodge theory: first, the studies of elliptic and abelian integrals by Cauchy, Abel, Jacobi, and Riemann, among many others; and then the studies of two-dimensional multiple integrals by Poincaré and Picard. Thenceforth, the focus turns to the Hodge theory of affine hypersurfaces given by tame polynomials, for which many tools from singularity theory, such as Brieskorn modules and Milnor fibrations, are used. Another aspect of this volume is its computational presentation of many well-known theoretical concepts, such as the Gauss-Manin connection, homology of varieties in terms of vanishing cycles, Hodge cycles, Noether-Lefschetz, and Hodge loci. All are explained for the generalized Fermat variety, which for Hodge theory boils down to a heavy linear algebra. Most of the algorithms introduced here are implemented in Singular, a computer algebra system for polynomial computations. Finally, the author introduces a few problems, mainly for talented undergraduate students, which can be understood with a basic knowledge of linear algebra. The origins of these problems may be seen in the discussions of advanced topics presented throughout this volume.
The Ten Equations That Rule the World : And How You can Use Them Too by Is there a secret formula for getting rich? For going viral? For deciding how long to stick with your current job, Netflix series, or even relationship? This book is all about the equations that make our world go round. Ten of them, in fact. They are integral to everything from investment banking to betting companies and social media giants. And they can help you to increase your chance of success, guard against financial loss, live more healthfully, and see through scaremongering. They are known by only the privileged few - until now. With wit and clarity, mathematician David Sumpter shows that it isn't the technical details that make these formulas so successful. It is the way they allow mathematicians to view problems from a different angle - a way of seeing the world that anyone can learn. Empowering and illuminating,The Ten Equations shows how math really can change your life.
All the Math You Missed : But Need to Know for Graduate School by Beginning graduate students in mathematical sciences and related areas in physical and computer sciences and engineering are expected to be familiar with a daunting breadth of mathematics, but few have such a background. This bestselling book helps students fill in the gaps in their knowledge. Thomas A. Garrity explains the basic points and a few key results of all the most important undergraduate topics in mathematics, emphasizing the intuitions behind the subject. The explanations are accompanied by numerous examples, exercises and suggestions for further reading that allow the reader to test and develop their understanding of these core topics. Featuring four new chapters and many other improvements, this second edition of All the Math You Missed is an essential resource for advanced undergraduates and beginning graduate students who need to learn some serious mathematics quickly.
Essentials of Tropical Combinatorics by The goal of this book is to explain, at the graduate student level, connections between tropical geometry and optimization. Building bridges between these two subject areas is fruitful in two ways. Through tropical geometry optimization algorithms become applicable to questions in algebraic geometry. Conversely, looking at topics in optimization through the tropical geometry lens adds an additional layer of structure. The author covers contemporary research topics that are relevant for applications such as phylogenetics, neural networks, combinatorial auctions, game theory, and computational complexity. This self-contained book grew out of several courses given at Technische Universitat Berlin and elsewhere, and the main prerequisite for the reader is a basic knowledge in polytope theory. It contains a good number of exercises, many examples, beautiful figures, as well as explicit tools for computations using $\texttt{{polymake}}$.
Discovering Abstract Algebra by Discovering Abstract Algebra takes an inquiry-based learning approach to the subject, leading students to discover for themselves its main themes and techniques. Concepts are introduced conversationally through extensive examples and student investigation before being formally defined. Students will develop skills in carefully making statements and writing proofs, while they simultaneously build a sense of ownership over the ideas and results. The book has been extensively tested and reinforced at points of common student misunderstanding or confusion, and includes a wealth of exercises at a variety of levels. The contents were deliberately organized to follow the recommendations of the MAA's 2015 Curriculum Guide. The book is ideal for a one- or two-semester course in abstract algebra, and will prepare students well for graduate-level study in algebra.
An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows by This book provides a self-contained introduction to optimal transport, and it is intended as a starting point for any researcher who wants to enter into this beautiful subject. The presentation focuses on the essential topics of the theory: Kantorovich duality, existence and uniqueness of optimal transport maps, Wasserstein distances, the JKO scheme, Otto's calculus, and Wasserstein gradient flows. At the end, a presentation of some selected applications of optimal transport is given. The book is suitable for a course at the graduate level and also includes an appendix with a series of exercises along with their solutions.
The Mathematical Principles of Natural Philosophy by Newton's Principia is perhaps the second most famous work of mathematics, after Euclid's Elements. Originally published in 1687, it gave the first systematic account of the fundamental concepts of dynamics, as well as three beautiful derivations of Newton's law of gravitation from Kepler's laws of planetary motion. As a book of great insight and ingenuity, it has raised our understanding of the power of mathematics more than any other work. This heavily annotated translation of the third and final edition (1726) of the Principia will enable any reader with a good understanding of elementary mathematics to easily grasp the meaning of the text, either from the translation itself or from the notes, and to appreciate some of its significance. All forward references are given to illuminate the structure and unity of the whole, and to clarify the parts. The mathematical prerequisites for understanding Newton's arguments are given in a brief appendix.