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| hammack book of proof: Book of Proof Richard H. Hammack, 2016-01-01 This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity. |
| hammack book of proof: How to Prove It Daniel J. Velleman, 2006-01-16 Many students have trouble the first time they take a mathematics course in which proofs play a significant role. This new edition of Velleman's successful text will prepare students to make the transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. The author shows how complex proofs are built up from these smaller steps, using detailed 'scratch work' sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets. To give students the opportunity to construct their own proofs, this new edition contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software. No background beyond standard high school mathematics is assumed. This book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and of course mathematicians. |
| hammack book of proof: Proofs from THE BOOK Martin Aigner, Günter M. Ziegler, 2013-04-17 The (mathematical) heroes of this book are perfect proofs: brilliant ideas, clever connections and wonderful observations that bring new insight and surprising perspectives on basic and challenging problems from Number Theory, Geometry, Analysis, Combinatorics, and Graph Theory. Thirty beautiful examples are presented here. They are candidates for The Book in which God records the perfect proofs - according to the late Paul Erdös, who himself suggested many of the topics in this collection. The result is a book which will be fun for everybody with an interest in mathematics, requiring only a very modest (undergraduate) mathematical background. For this revised and expanded second edition several chapters have been revised and expanded, and three new chapters have been added. |
| hammack book of proof: Mathematical Proofs Gary Chartrand, Albert D. Polimeni, Ping Zhang, 2013 This book prepares students for the more abstract mathematics courses that follow calculus. The author introduces students to proof techniques, analyzing proofs, and writing proofs of their own. It also provides a solid introduction to such topics as relations, functions, and cardinalities of sets, as well as the theoretical aspects of fields such as number theory, abstract algebra, and group theory. |
| hammack book of proof: Proofs Jay Cummings, 2021-01-19 This textbook is designed for students. Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. The proofs are not terse, and aim for understanding over economy. Furthermore, dozens of proofs are preceded by scratch work or a proof sketch to give students a big-picture view and an explanation of how they would come up with it on their own.This book covers intuitive proofs, direct proofs, sets, induction, logic, the contrapositive, contradiction, functions and relations. The text aims to make the ideas visible, and contains over 200 illustrations. The writing is relaxed and conversational, and includes periodic attempts at humor.This text is also an introduction to higher mathematics. This is done in-part through the chosen examples and theorems. Furthermore, following every chapter is an introduction to an area of math. These include Ramsey theory, number theory, topology, sequences, real analysis, big data, game theory, cardinality and group theory.After every chapter are pro-tips, which are short thoughts on things I wish I had known when I took my intro-to-proofs class. They include finer comments on the material, study tips, historical notes, comments on mathematical culture, and more. Also, after each chapter's exercises is an introduction to an unsolved problem in mathematics.In the first appendix we discuss some further proof methods, the second appendix is a collection of particularly beautiful proofs, and the third is some writing advice. |
| hammack book of proof: Real Analysis Jay Cummings, 2019-07-15 This textbook is designed for students. Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. The proofs are not terse, and aim for understanding over economy. Furthermore, dozens of proofs are preceded by scratch work or a proof sketch to give students a big-picture view and an explanation of how they would come up with it on their own. Examples often drive the narrative and challenge the intuition of the reader. The text also aims to make the ideas visible, and contains over 200 illustrations. The writing is relaxed and includes interesting historical notes, periodic attempts at humor, and occasional diversions into other interesting areas of mathematics. The text covers the real numbers, cardinality, sequences, series, the topology of the reals, continuity, differentiation, integration, and sequences and series of functions. Each chapter ends with exercises, and nearly all include some open questions. The first appendix contains a construction the reals, and the second is a collection of additional peculiar and pathological examples from analysis. The author believes most textbooks are extremely overpriced and endeavors to help change this.Hints and solutions to select exercises can be found at LongFormMath.com. |
| hammack book of proof: Proofs and Fundamentals Ethan D. Bloch, 2013-12-01 In an effort to make advanced mathematics accessible to a wide variety of students, and to give even the most mathematically inclined students a solid basis upon which to build their continuing study of mathematics, there has been a tendency in recent years to introduce students to the for mulation and writing of rigorous mathematical proofs, and to teach topics such as sets, functions, relations and countability, in a transition course, rather than in traditional courses such as linear algebra. A transition course functions as a bridge between computational courses such as Calculus, and more theoretical courses such as linear algebra and abstract algebra. This text contains core topics that I believe any transition course should cover, as well as some optional material intended to give the instructor some flexibility in designing a course. The presentation is straightforward and focuses on the essentials, without being too elementary, too exces sively pedagogical, and too full to distractions. Some of features of this text are the following: (1) Symbolic logic and the use of logical notation are kept to a minimum. We discuss only what is absolutely necessary - as is the case in most advanced mathematics courses that are not focused on logic per se. |
| hammack book of proof: Understanding Analysis Stephen Abbott, 2012-12-06 Understanding Analysis outlines an elementary, one-semester course designed to expose students to the rich rewards inherent in taking a mathematically rigorous approach to the study of functions of a real variable. The aim of a course in real analysis should be to challenge and improve mathematical intuition rather than to verify it. The philosophy of this book is to focus attention on the questions that give analysis its inherent fascination. Does the Cantor set contain any irrational numbers? Can the set of points where a function is discontinuous be arbitrary? Are derivatives continuous? Are derivatives integrable? Is an infinitely differentiable function necessarily the limit of its Taylor series? In giving these topics center stage, the hard work of a rigorous study is justified by the fact that they are inaccessible without it. |
| hammack book of proof: Introductory Discrete Mathematics V. K. Balakrishnan, 1996-01-01 This concise, undergraduate-level text focuses on combinatorics, graph theory with applications to some standard network optimization problems, and algorithms. Geared toward mathematics and computer science majors, it emphasizes applications, offering more than 200 exercises to help students test their grasp of the material and providing answers to selected exercises. 1991 edition. |
| hammack book of proof: The Art of Proof Matthias Beck, Ross Geoghegan, 2010-08-17 The Art of Proof is designed for a one-semester or two-quarter course. A typical student will have studied calculus (perhaps also linear algebra) with reasonable success. With an artful mixture of chatty style and interesting examples, the student's previous intuitive knowledge is placed on solid intellectual ground. The topics covered include: integers, induction, algorithms, real numbers, rational numbers, modular arithmetic, limits, and uncountable sets. Methods, such as axiom, theorem and proof, are taught while discussing the mathematics rather than in abstract isolation. The book ends with short essays on further topics suitable for seminar-style presentation by small teams of students, either in class or in a mathematics club setting. These include: continuity, cryptography, groups, complex numbers, ordinal number, and generating functions. |
| hammack book of proof: Mathematical Reasoning Theodore A. Sundstrom, 2003 Focusing on the formal development of mathematics, this book demonstrates how to read and understand, write and construct mathematical proofs. It emphasizes active learning, and uses elementary number theory and congruence arithmetic throughout. Chapter content covers an introduction to writing in mathematics, logical reasoning, constructing proofs, set theory, mathematical induction, functions, equivalence relations, topics in number theory, and topics in set theory. For learners making the transition form calculus to more advanced mathematics. |
| hammack book of proof: Introduction to Mathematical Structures and Proofs Larry J. Gerstein, 1996-04-04 This acclaimed book aids the transition from lower-division calculus to upper-division courses in linear and abstract algebra, real and complex analysis, number theory, topology and more, with examples, images, exercises and a solution manual for instructors. |
| hammack book of proof: Geometry Illuminated Matthew Harvey, 2015-09-25 Geometry Illuminated is an introduction to geometry in the plane, both Euclidean and hyperbolic. It is designed to be used in an undergraduate course on geometry, and as such, its target audience is undergraduate math majors. However, much of it should be readable by anyone who is comfortable with the language of mathematical proof. Throughout, the goal is to develop the material patiently. One of the more appealing aspects of geometry is that it is a very visual subject. This book hopes to takes full advantage of that, with an extensive use of illustrations as guides. Geometry Illuminated is divided into four principal parts. Part 1 develops neutral geometry in the style of Hilbert, including a discussion of the construction of measure in that system, ultimately building up to the Saccheri-Legendre Theorem. Part 2 provides a glimpse of classical Euclidean geometry, with an emphasis on concurrence results, such as the nine-point circle. Part 3 studies transformations of the Euclidean plane, beginning with isometries and ending with inversion, with applications and a discussion of area in between. Part 4 is dedicated to the development of the Poincaré disk model, and the study of geometry within that model. While this material is traditional, Geometry Illuminated does bring together topics that are generally not found in a book at this level. Most notably, it explicitly computes parametric equations for the pseudosphere and its geodesics. It focuses less on the nature of axiomatic systems for geometry, but emphasizes rather the logical development of geometry within such a system. It also includes sections dealing with trilinear and barycentric coordinates, theorems that can be proved using inversion, and Euclidean and hyperbolic tilings. |
| hammack book of proof: Discrete Structures, Logic, and Computability James L. Hein, 2001 Discrete Structure, Logic, and Computability introduces the beginning computer science student to some of the fundamental ideas and techniques used by computer scientists today, focusing on discrete structures, logic, and computability. The emphasis is on the computational aspects, so that the reader can see how the concepts are actually used. Because of logic's fundamental importance to computer science, the topic is examined extensively in three phases that cover informal logic, the technique of inductive proof; and formal logic and its applications to computer science. |
| hammack book of proof: How to Think About Analysis Lara Alcock, 2014-09-25 Analysis (sometimes called Real Analysis or Advanced Calculus) is a core subject in most undergraduate mathematics degrees. It is elegant, clever and rewarding to learn, but it is hard. Even the best students find it challenging, and those who are unprepared often find it incomprehensible at first. This book aims to ensure that no student need be unprepared. It is not like other Analysis books. It is not a textbook containing standard content. Rather, it is designed to be read before arriving at university and/or before starting an Analysis course, or as a companion text once a course is begun. It provides a friendly and readable introduction to the subject by building on the student's existing understanding of six key topics: sequences, series, continuity, differentiability, integrability and the real numbers. It explains how mathematicians develop and use sophisticated formal versions of these ideas, and provides a detailed introduction to the central definitions, theorems and proofs, pointing out typical areas of difficulty and confusion and explaining how to overcome these. The book also provides study advice focused on the skills that students need if they are to build on this introduction and learn successfully in their own Analysis courses: it explains how to understand definitions, theorems and proofs by relating them to examples and diagrams, how to think productively about proofs, and how theories are taught in lectures and books on advanced mathematics. It also offers practical guidance on strategies for effective study planning. The advice throughout is research based and is presented in an engaging style that will be accessible to students who are new to advanced abstract mathematics. |
| hammack book of proof: A Concise Introduction to Pure Mathematics Martin Liebeck, 2018-09-03 Accessible to all students with a sound background in high school mathematics, A Concise Introduction to Pure Mathematics, Fourth Edition presents some of the most fundamental and beautiful ideas in pure mathematics. It covers not only standard material but also many interesting topics not usually encountered at this level, such as the theory of solving cubic equations; Euler’s formula for the numbers of corners, edges, and faces of a solid object and the five Platonic solids; the use of prime numbers to encode and decode secret information; the theory of how to compare the sizes of two infinite sets; and the rigorous theory of limits and continuous functions. New to the Fourth Edition Two new chapters that serve as an introduction to abstract algebra via the theory of groups, covering abstract reasoning as well as many examples and applications New material on inequalities, counting methods, the inclusion-exclusion principle, and Euler’s phi function Numerous new exercises, with solutions to the odd-numbered ones Through careful explanations and examples, this popular textbook illustrates the power and beauty of basic mathematical concepts in number theory, discrete mathematics, analysis, and abstract algebra. Written in a rigorous yet accessible style, it continues to provide a robust bridge between high school and higher-level mathematics, enabling students to study more advanced courses in abstract algebra and analysis. |
| hammack book of proof: Handbook of Product Graphs Richard Hammack, Wilfried Imrich, Sandi Klavžar, 2011-06-06 This handbook examines the dichotomy between the structure of products and their subgraphs. It also features the design of efficient algorithms that recognize products and their subgraphs and explores the relationship between graph parameters of the product and factors. Extensively revised and expanded, this second edition presents full proofs of many important results as well as up-to-date research and conjectures. It illustrates applications of graph products in several areas and contains well over 300 exercises. Supplementary material is available on the book's website. |
| hammack book of proof: An Introduction to Abstract Mathematics Robert J. Bond, William J. Keane, 1999 The goal of this book is to show students how mathematicians think and to glimpse some of the fascinating things they think about. Bond and Keane develop students' ability to do abstract mathematics by teaching the form of mathematics in the context of real and elementary mathematics. Students learn the fundamentals of mathematical logic; how to read and understand definitions, theorems, and proofs; and how to assimilate abstract ideas and communicate them in written form. Students will learn to write mathematical proofs coherently and correctly. |
| hammack book of proof: Transition to Higher Mathematics Bob A. Dumas, John Edward McCarthy, 2007 This book is written for students who have taken calculus and want to learn what real mathematics is. |
| hammack book of proof: A First Course in Logic Mark Verus Lawson, 2018-12-07 A First Course in Logic is an introduction to first-order logic suitable for first and second year mathematicians and computer scientists. There are three components to this course: propositional logic; Boolean algebras; and predicate/first-order, logic. Logic is the basis of proofs in mathematics — how do we know what we say is true? — and also of computer science — how do I know this program will do what I think it will? Surprisingly little mathematics is needed to learn and understand logic (this course doesn't involve any calculus). The real mathematical prerequisite is an ability to manipulate symbols: in other words, basic algebra. Anyone who can write programs should have this ability. |
| hammack book of proof: An Introduction to Mathematical Reasoning Peter J. Eccles, 1997-12-11 ÍNDICE: Part I. Mathematical Statements and Proofs: 1. The language of mathematics; 2. Implications; 3. Proofs; 4. Proof by contradiction; 5. The induction principle; Part II. Sets and Functions: 6. The language of set theory; 7. Quantifiers; 8. Functions; 9. Injections, surjections and bijections; Part III. Numbers and Counting: 10. Counting; 11. Properties of finite sets; 12. Counting functions and subsets; 13. Number systems; 14. Counting infinite sets; Part IV. Arithmetic: 15. The division theorem; 16. The Euclidean algorithm; 17. Consequences of the Euclidean algorithm; 18. Linear diophantine equations; Part V. Modular Arithmetic: 19. Congruences of integers; 20. Linear congruences; 21. Congruence classes and the arithmetic of remainders; 22. Partitions and equivalence relations; Part VI. Prime Numbers: 23. The sequence of prime numbers; 24. Congruence modulo a prime; Solutions to exercises. |
| hammack book of proof: Introduction to Mathematical Thinking Keith J. Devlin, 2012 Mathematical thinking is not the same as 'doing math'--unless you are a professional mathematician. For most people, 'doing math' means the application of procedures and symbolic manipulations. Mathematical thinking, in contrast, is what the name reflects, a way of thinking about things in the world that humans have developed over three thousand years. It does not have to be about mathematics at all, which means that many people can benefit from learning this powerful way of thinking, not just mathematicians and scientists.--Back cover. |
| hammack book of proof: Understanding Mathematical Proof John Taylor, Rowan Garnier, 2016-04-19 The notion of proof is central to mathematics yet it is one of the most difficult aspects of the subject to teach and master. In particular, undergraduate mathematics students often experience difficulties in understanding and constructing proofs.Understanding Mathematical Proof describes the nature of mathematical proof, explores the various techn |
| hammack book of proof: A Radical Approach to Real Analysis David M. Bressoud, 2007-04-12 Second edition of this introduction to real analysis, rooted in the historical issues that shaped its development. |
| hammack book of proof: Lectures on the Curry-Howard Isomorphism Morten Heine Sørensen, Pawel Urzyczyn, 2006-07-04 The Curry-Howard isomorphism states an amazing correspondence between systems of formal logic as encountered in proof theory and computational calculi as found in type theory. For instance,minimal propositional logic corresponds to simply typed lambda-calculus, first-order logic corresponds to dependent types, second-order logic corresponds to polymorphic types, sequent calculus is related to explicit substitution, etc.The isomorphism has many aspects, even at the syntactic level:formulas correspond to types, proofs correspond to terms, provability corresponds to inhabitation, proof normalization corresponds to term reduction, etc.But there is more to the isomorphism than this. For instance, it is an old idea---due to Brouwer, Kolmogorov, and Heyting---that a constructive proof of an implication is a procedure that transformsproofs of the antecedent into proofs of the succedent; the Curry-Howard isomorphism gives syntactic representations of such procedures. The Curry-Howard isomorphism also provides theoretical foundations for many modern proof-assistant systems (e.g. Coq).This book give an introduction to parts of proof theory and related aspects of type theory relevant for the Curry-Howard isomorphism. It can serve as an introduction to any or both of typed lambda-calculus and intuitionistic logic.Key features- The Curry-Howard Isomorphism treated as common theme- Reader-friendly introduction to two complementary subjects: Lambda-calculus and constructive logics- Thorough study of the connection between calculi and logics- Elaborate study of classical logics and control operators- Account of dialogue games for classical and intuitionistic logic- Theoretical foundations of computer-assisted reasoning· The Curry-Howard Isomorphism treated as the common theme.· Reader-friendly introduction to two complementary subjects: lambda-calculus and constructive logics · Thorough study of the connection between calculi and logics.· Elaborate study of classical logics and control operators.· Account of dialogue games for classical and intuitionistic logic.· Theoretical foundations of computer-assisted reasoning |
| hammack book of proof: Halfway Human Carolyn Ives Gilman, 2010-02 The best science fiction novel I've read in a long time-St. Louis Post-Dispatch **** A Starflight Reader nominee for best book of the year **** Tedla is a 'bland, ' an asexual class of people that exist only to serve their fellow beings. **** Val is an expert on alien cultures but has never seen a bland before. They come together after Tedla is found light-years away from its home planet-alone, isolated and suicidal. Val's mission is to help Tedla recover. But the more she learns about the beautiful alien being, the more she discovers about the torment Tedla and its kind suffer on their planet. **** Little does the rest of the universe know of the hidden world of the blands, a world that hides shocking secrets and unspeakable crimes. **** Halfway Human is a mesmerizing look at an intricately created alien world which is strange and distant, yet hauntingly familiar. **** Beauty, pain, wit, and wisdom all suffuse this powerful novel, as it uses imagined futures to reveal our own world in starkest clarity...and yes, with a whisper of hope. -Locus |
| hammack book of proof: The Prodigal Prophet Timothy Keller, 2018-10-02 An angry prophet. A feared and loathsome enemy. A devastating storm. And the surprising message of a merciful God to his people. The story of Jonah is one of the most well-known parables in the Bible. It is also the most misunderstood. Many people, even those who are nonreligious, are familiar with Jonah: A rebellious prophet who defies God and is swallowed by a whale. But there's much more to Jonah's story than most of us realize. In The Prodigal Prophet, pastor and New York Times bestselling author Timothy Keller reveals the hidden depths within the book of Jonah. Keller makes the case that Jonah was one of the worst prophets in the entire Bible. And yet there are unmistakably clear connections between Jonah, the prodigal son, and Jesus. Jesus in fact saw himself in Jonah. How could one of the most defiant and disobedient prophets in the Bible be compared to Jesus? Jonah's journey also doesn't end when he is freed from the belly of the fish. There is an entire second half to his story--but it is left unresolved within the text of the Bible. Why does the book of Jonah end on what is essentially a cliffhanger? In these pages, Timothy Keller provides an answer to the extraordinary conclusion of this biblical parable--and shares the powerful Christian message at the heart of Jonah's story. |
| hammack book of proof: Classes of Directed Graphs Jørgen Bang-Jensen, Gregory Gutin, 2018-06-18 This edited volume offers a detailed account of the theory of directed graphs from the perspective of important classes of digraphs, with each chapter written by experts on the topic. Outlining fundamental discoveries and new results obtained over recent years, this book provides a comprehensive overview of the latest research in the field. It covers core new results on each of the classes discussed, including chapters on tournaments, planar digraphs, acyclic digraphs, Euler digraphs, graph products, directed width parameters, and algorithms. Detailed indices ease navigation while more than 120 open problems and conjectures ensure that readers are immersed in all aspects of the field. Classes of Directed Graphs provides a valuable reference for graduate students and researchers in computer science, mathematics and operations research. As digraphs are an important modelling tool in other areas of research, this book will also be a useful resource to researchers working in bioinformatics, chemoinformatics, sociology, physics, medicine, etc. |
| hammack book of proof: Book of Numbers Joshua Cohen, 2015-06-09 NATIONAL BESTSELLER • “A wheeling meditation on the wired life, on privacy, on what being human in the age of binary code might mean” (The New York Times), from the Pulitzer Prize–winning author of The Netanyahus NAMED ONE OF THE TEN BEST BOOKS OF THE YEAR BY VULTURE AND ONE OF THE BEST BOOKS OF THE YEAR BY NPR AND THE WALL STREET JOURNAL “Shatteringly powerful . . . I cannot think of anything by anyone in [Cohen’s] generation that is so frighteningly relevant and composed with such continuous eloquence. There are moments in it that seem to transcend our impasse.”—Harold Bloom The enigmatic billionaire founder of Tetration, the world’s most powerful tech company, hires a failed novelist, Josh Cohen, to ghostwrite his memoirs. The mogul, known as Principal, brings Josh behind the digital veil, tracing the rise of Tetration, which started in the earliest days of the Internet by revolutionizing the search engine before venturing into smartphones, computers, and the surveillance of American citizens. Principal takes Josh on a mind-bending world tour from Palo Alto to Dubai and beyond, initiating him into the secret pretext of the autobiography project and the life-or-death stakes that surround its publication. Insider tech exposé, leaked memoir-in-progress, international thriller, family drama, sex comedy, and biblical allegory, Book of Numbers renders the full range of modern experience both online and off. Embodying the Internet in its language, it finds the humanity underlying the virtual. Featuring one of the most unforgettable characters in contemporary fiction, Book of Numbers is an epic of the digital age, a triumph of a new generation of writers, and one of those rare books that renew the idea of what a novel can do. Praise for Book of Numbers “The Great American Internet Novel is here. . . . Book of Numbers is a fascinating look at the dark heart of the Web. . . . A page-turner about life under the veil of digital surveillance . . . one of the best novels ever written about the Internet.”—Rolling Stone “A startlingly talented novelist.”—The Wall Street Journal “Remarkable . . . dazzling . . . Cohen’s literary gifts . . . suggest that something is possible, that something still might be done to safeguard whatever it is that makes us human.”—Francine Prose, The New York Review of Books |
| hammack book of proof: A Basic Course in Real Analysis Ajit Kumar, S. Kumaresan, 2014-01-10 Based on the authors’ combined 35 years of experience in teaching, A Basic Course in Real Analysis introduces students to the aspects of real analysis in a friendly way. The authors offer insights into the way a typical mathematician works observing patterns, conducting experiments by means of looking at or creating examples, trying to understand the underlying principles, and coming up with guesses or conjectures and then proving them rigorously based on his or her explorations. With more than 100 pictures, the book creates interest in real analysis by encouraging students to think geometrically. Each difficult proof is prefaced by a strategy and explanation of how the strategy is translated into rigorous and precise proofs. The authors then explain the mystery and role of inequalities in analysis to train students to arrive at estimates that will be useful for proofs. They highlight the role of the least upper bound property of real numbers, which underlies all crucial results in real analysis. In addition, the book demonstrates analysis as a qualitative as well as quantitative study of functions, exposing students to arguments that fall under hard analysis. Although there are many books available on this subject, students often find it difficult to learn the essence of analysis on their own or after going through a course on real analysis. Written in a conversational tone, this book explains the hows and whys of real analysis and provides guidance that makes readers think at every stage. |
| hammack book of proof: Basic Mathematics Serge Lang, 1988-01 |
| hammack book of proof: Common Ground? Anthony M. Orum, Zachary Neal, 2009-09-10 Public spaces have long been the focus of urban social activity, but investigations of how public space works often adopt only one of several possible perspectives, which restricts the questions that can be asked and the answers that can be considered. In this volume, Anthony Orum and Zachary Neal explore how public space can be a facilitator of civil order, a site for power and resistance, and a stage for art, theatre, and performance. They bring together these frequently unconnected models for understanding public space, collecting classic and contemporary readings that illustrate each, and synthesizing them in a series of original essays. Throughout, they offer questions to provoke discussion, and conclude with thoughts on how these models can be combined by future scholars of public space to yield more comprehensive understanding of how public space works. |
| hammack book of proof: Adolescent Rationality and Development David Moshman, 2011-03-01 Frequently cited in scholarly books and journals and praised by students, this book focuses on developmental changes and processes in adolescence rather than on the details and problems of daily life. Major developmental changes associated with adolescence are identified. Noted for its exceptionally strong coverage of cognitive, moral, and social development, this brief, inexpensive book can be used independently or as a supplement to other texts on adolescence. Highlights of the new edition include: expanded coverage of thinking and reasoning. a new chapter on metacognition and epistemic cognition. expanded coverage of controversies concerning the foundations of morality. a new chapter on moral principles and perspective taking. a new chapter on the relation of personal and social identity. a new chapter addressing current controversies concerning the rationality, maturity, and brains of adolescents. more detail on key studies and methodologies and boldfaced key terms and a glossary to highlight and clarify key concepts. Rather than try to cover everything about adolescence at an elementary level, this book presents and builds on the core issues in the scholarly literature, thus encouraging deeper levels of understanding. The book opens with an introduction to the concepts of adolescence, rationality, and development and then explores the three foundational literatures of adolescent development - cognitive development, moral development, and identity formation. The book concludes with a more general account of rationality and development in adolescence and beyond. Appropriate for advanced undergraduate and graduate courses on adolescence or adolescent development offered by departments of psychology, educational psychology, or human development, this brief text is also an ideal supplement for courses on social and/or moral development, cognitive development, or lifespan development. The book is also appreciated by scholars interested in connections across standard topics and research programs. Prior knowledge of psychology is not assumed. |
| hammack book of proof: Contests in Higher Mathematics Gabor J Szekely, 2012-09-30 One of the most effective ways to stimulate students to enjoy intellectual efforts is the scientific competition. In 1894 the Hungarian Mathematical and Physical Society introduced a mathematical competition for high school students. The success of high school competitions led the Mathematical Society to found a college level contest, named after Miklós Schweitzer. The problems of the Schweitzer Contests are proposed and selected by the most prominent Hungarian mathematicians. This book collects the problems posed in the contests between 1962 and 1991 which range from algebra, combinatorics, theory of functions, geometry, measure theory, number theory, operator theory, probability theory, topology, to set theory. The second part contains the solutions. The Schweitzer competition is one of the most unique in the world. The experience shows that this competition helps to identify research talents. This collection of problems and solutions in several fields in mathematics can serve as a guide for many undergraduates and young mathematicians. The large variety of research level problems might be of interest for more mature mathematicians and historians of mathematics as well. |
| hammack book of proof: Applied Linear Algebra Peter J. Olver, Chehrzad Shakiban, 2018-05-30 This textbook develops the essential tools of linear algebra, with the goal of imparting technique alongside contextual understanding. Applications go hand-in-hand with theory, each reinforcing and explaining the other. This approach encourages students to develop not only the technical proficiency needed to go on to further study, but an appreciation for when, why, and how the tools of linear algebra can be used across modern applied mathematics. Providing an extensive treatment of essential topics such as Gaussian elimination, inner products and norms, and eigenvalues and singular values, this text can be used for an in-depth first course, or an application-driven second course in linear algebra. In this second edition, applications have been updated and expanded to include numerical methods, dynamical systems, data analysis, and signal processing, while the pedagogical flow of the core material has been improved. Throughout, the text emphasizes the conceptual connections between each application and the underlying linear algebraic techniques, thereby enabling students not only to learn how to apply the mathematical tools in routine contexts, but also to understand what is required to adapt to unusual or emerging problems. No previous knowledge of linear algebra is needed to approach this text, with single-variable calculus as the only formal prerequisite. However, the reader will need to draw upon some mathematical maturity to engage in the increasing abstraction inherent to the subject. Once equipped with the main tools and concepts from this book, students will be prepared for further study in differential equations, numerical analysis, data science and statistics, and a broad range of applications. The first author’s text, Introduction to Partial Differential Equations, is an ideal companion volume, forming a natural extension of the linear mathematical methods developed here. |
| hammack book of proof: Proof Positive Neil Nedley, 1999 A must for all wanting to use natural means for preventing or treating high blood pressure, blocked arteries, cancer, chronic fatigue, diabetes, osteoporosis, and many other afflictions. Provides information that minimizes the use of prescription drugs, diet fads, and their accompanying side effects. Highly illustrated in full color, this tome of information is designed to be readable and easy-to-understand. Singular case studies, which can be misleading, are not used. Instead, the results of a host of scientific studies conducted around the world are cited, many of which involve large groups of individuals with widely varying lifestyles. Many topics are covered such as how to strengthen the immune system, overcome addictions, increase reasoning ability, cope with stress, and enhance children's mental and physical potential. |
| hammack book of proof: Methods of Applied Mathematics Francis Begnaud Hildebrand, 1992-01-01 This book offers engineers and physicists working knowledge of a number of mathematical facts and techniques not commonly treated in courses in advanced calculus, but nevertheless extremely useful when applied to typical problems. Explores linear algebraic equations, quadratic and Hermitian forms, operations with vectors and matrices, the calculus of variations, more. Includes annotated problems and exercises. |
| hammack book of proof: Proof in Mathematics James Franklin, Albert Daoud, 2010 |
| hammack book of proof: Doing Mathematics Steven Galovich, 2007 Prepare for success in mathematics with DOING MATHEMATICS: AN INTRODUCTION TO PROOFS AND PROBLEM SOLVING! By discussing proof techniques, problem solving methods, and the understanding of mathematical ideas, this mathematics text gives you a solid foundation from which to build while providing you with the tools you need to succeed. Numerous examples, problem solving methods, and explanations make exam preparation easy. |
| hammack book of proof: A Course in Linear Algebra David B. Damiano, John B. Little, 1988-08-01 |
Beth M. Hammack - Wikipedia
Elizabeth Morgan Hammack (born 1971/1972) is the 12th president and CEO of the Federal Reserve Bank of Cleveland. [1][2] Prior to that, Hammack worked at Goldman Sachs for three …
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Cleveland Fed taps Goldman Sachs veteran Beth Hammack as …
May 29, 2024 · The Federal Reserve Bank of Cleveland announced Wednesday that Beth M. Hammack will be its next president and chief executive officer after current president Loretta …
Beth M. Hammack - Federal Reserve History
An accomplished thought leader, Hammack has worked closely with US policymakers as chair of the Treasury Borrowing Advisory Committee, offering presentations on macroeconomic and …
Office of the President - Federal Reserve Bank of Cleveland
Beth M. Hammack is the president and chief executive officer of the Federal Reserve Bank of Cleveland.
Goldman Sachs partner Beth Hammack to succeed Mester as …
May 29, 2024 · Hammack will take office officially on Aug. 21. The Cleveland Fed president plays an important role this year as a voter on the rate-setting Federal Open Market Committee. A …
Beth M. Hammack - Goldman Sachs
Earlier in her career, Ms. Hammack was a market maker on the IRP desk, where she traded a variety of instruments, focused primarily on options and later agencies. She joined Goldman …
FX Blue - Calendar - Fed's Hammack speech - United States
May 9, 2025 · Beth M. Hammack has been president and chief executive officer of the Federal Reserve Bank of Cleveland since August 2024. She participates in the formulation of US …
Beth M. Hammack - Wikipedia
Elizabeth Morgan Hammack (born 1971/1972) is the 12th president and CEO of the Federal Reserve Bank of Cleveland. [1][2] Prior to that, Hammack worked at Goldman Sachs for three …
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Beth M. Hammack - Federal Reserve Bank of Cleveland
Beth M. Hammack is the president and chief executive officer of the Federal Reserve Bank of Cleveland, one of 12 regional Reserve Banks in the Federal Reserve System.
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Shop ENO Hammocks - explore our collection of premium hammocks, perfect for camping, relaxing, or backyard lounging. Trusted by adventurers worldwide. Free shipping on $49+.
Cleveland Fed taps Goldman Sachs veteran Beth Hammack as …
May 29, 2024 · The Federal Reserve Bank of Cleveland announced Wednesday that Beth M. Hammack will be its next president and chief executive officer after current president Loretta …
Beth M. Hammack - Federal Reserve History
An accomplished thought leader, Hammack has worked closely with US policymakers as chair of the Treasury Borrowing Advisory Committee, offering presentations on macroeconomic and …
Office of the President - Federal Reserve Bank of Cleveland
Beth M. Hammack is the president and chief executive officer of the Federal Reserve Bank of Cleveland.
Goldman Sachs partner Beth Hammack to succeed Mester as …
May 29, 2024 · Hammack will take office officially on Aug. 21. The Cleveland Fed president plays an important role this year as a voter on the rate-setting Federal Open Market Committee. A …
Beth M. Hammack - Goldman Sachs
Earlier in her career, Ms. Hammack was a market maker on the IRP desk, where she traded a variety of instruments, focused primarily on options and later agencies. She joined Goldman …
FX Blue - Calendar - Fed's Hammack speech - United States
May 9, 2025 · Beth M. Hammack has been president and chief executive officer of the Federal Reserve Bank of Cleveland since August 2024. She participates in the formulation of US …
