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Projective Geometry and Projective Metrics (Dover Books on Mathematics #Volume 3)
by Herbert Busemann Paul J. KellyThe basic results and methods of projective and non-Euclidean geometry are indispensable for the geometer, and this book--different in content, methods, and point of view from traditional texts--attempts to emphasize that fact. Results of special theorems are discussed in detail only when they are needed to develop a feeling for the subject or when they illustrate a general method. On the other hand, an unusual amount of space is devoted to the discussion of the fundamental concepts of distance, motion, area, and perpendicularity.Topics include the projective plane, polarities and conic sections, affine geometry, projective metrics, and non-Euclidean and spatial geometry. Numerous figures appear throughout the text, which concludes with a bibliography and index.
Projektive Geometrie der Ebene: Ein klassischer Zugang mit interaktiver Visualisierung (Masterclass)
by Stefan LiebscherDieses Buch bietet eine Einf#65533;hrung in die projektive Geometrie, wobei algebraische Details auf ein f#65533;r die Beweise n#65533;tiges Minimum beschr#65533;nkt werden. Um die Sachverhalte noch zeichnerisch darstellen zu k#65533;nnen, konzentrieren wir uns auf die reelle projektive Ebene. Zentrales Thema sind Kegelschnitte und ihre Beziehungen - hier zeigt sich die durch den projektiven Zugang erreichbare Klarheit besonders deutlich. Wir werden Geometrie betreiben, ohne zu messen. Auch wollen wir verstehen, inwiefern die euklidische Ebene - also unsere #65533;bliche geometrische Vorstellungswelt - ein singul#65533;rer Grenzfall ist und wie uns das helfen kann, geometrische Sachverhalte zu verstehen. Viele der besprochenen Sachverhalte k#65533;nnen mit einer interaktiven Applikation auf der Webseite des Autors visualisiert und nachvollzogen werden.
Prokhorov and Contemporary Probability Theory
by Albert N. Shiryaev Ernst L. Presman S. R. VaradhanThe role of Yuri Vasilyevich Prokhorov as a prominent mathematician and leading expert in the theory of probability is well known. Even early in his career he obtained substantial results on the validity of the strong law of large numbers and on the estimates (bounds) of the rates of convergence, some of which are the best possible. His findings on limit theorems in metric spaces and particularly functional limit theorems are of exceptional importance. Y.V. Prokhorov developed an original approach to the proof of functional limit theorems, based on the weak convergence of finite dimensional distributions and the condition of tightness of probability measures. The present volume commemorates the 80th birthday of Yuri Vasilyevich Prokhorov. It includes scientific contributions written by his colleagues, friends and pupils, who would like to express their deep respect and sincerest admiration for him and his scientific work.
Prolate Spheroidal Wave Functions of Order Zero
by Andrei Osipov Vladimir Rokhlin Hong XiaoProlate Spheroidal Wave Functions (PSWFs) are the eigenfunctions of the bandlimited operator in one dimension. As such, they play an important role in signal processing, Fourier analysis, and approximation theory. While historically the numerical evaluation of PSWFs presented serious difficulties, the developments of the last fifteen years or so made them as computationally tractable as any other class of special functions. As a result, PSWFs have been becoming a popular computational tool. The present book serves as a complete, self-contained resource for both theory and computation. It will be of interest to a wide range of scientists and engineers, from mathematicians interested in PSWFs as an analytical tool to electrical engineers designing filters and antennas.
Promising Practices for Strengthening the Regional STEM Workforce Development Ecosystem
by Engineering Medicine National Academies of SciencesU.S. strength in science, technology, engineering, and mathematics (STEM) disciplines has formed the basis of innovations, technologies, and industries that have spurred the nation’s economic growth throughout the last 150 years. Universities are essential to the creation and transfer of new knowledge that drives innovation. This knowledge moves out of the university and into broader society in several ways — through highly skilled graduates (i.e. human capital); academic publications; and the creation of new products, industries, and companies via the commercialization of scientific breakthroughs. Despite this, our understanding of how universities receive, interpret, and respond to industry signaling demands for STEM-trained workers is far from complete. Promising Practices for Strengthening the Regional STEM Workforce Development Ecosystem reviews the extent to which universities and employers in five metropolitan communities (Phoenix, Arizona; Cleveland, Ohio; Montgomery, Alabama; Los Angeles, California; and Fargo, North Dakota) collaborate successfully to align curricula, labs, and other undergraduate educational experiences with current and prospective regional STEM workforce needs. This report focuses on how to create the kind of university-industry collaboration that promotes higher quality college and university course offerings, lab activities, applied learning experiences, work-based learning programs, and other activities that enable students to acquire knowledge, skills, and attributes they need to be successful in the STEM workforce. The recommendations and findings presented will be most relevant to educators, policy makers, and industry leaders.
Promoting Statistical Practice and Collaboration in Developing Countries
by O. Olawale Awe"Rarely, but just often enough to rebuild hope, something happens to confound my pessimism about the recent unprecedented happenings in the world. This book is the most recent instance, and I think that all its readers will join me in rejoicing at the good it seeks to do. It is an example of the kind of international comity and collaboration that we could and should undertake to solve various societal problems. This book is a beautiful example of the power of the possible. [It] provides a blueprint for how the LISA 2020 model can be replicated in other fields. Civil engineers, or accountants, or nurses, or any other profession could follow this outline to share expertise and build capacity and promote progress in other countries. It also contains some tutorials for statistical literacy across several fields. The details would change, of course, but ideas are durable, and the generalizations seem pretty straightforward. This book shows every other profession where and how to stand in order to move the world. I urge every researcher to get a copy!" —David Banks from the Foreword Promoting Statistical Practice and Collaboration in Developing Countries provides new insights into the current issues and opportunities in international statistics education, statistical consulting, and collaboration, particularly in developing countries around the world. The book addresses the topics discussed in individual chapters from the perspectives of the historical context, the present state, and future directions of statistical training and practice, so that readers may fully understand the challenges and opportunities in the field of statistics and data science, especially in developing countries. Features • Reference point on statistical practice in developing countries for researchers, scholars, students, and practitioners • Comprehensive source of state-of-the-art knowledge on creating statistical collaboration laboratories within the field of data science and statistics • Collection of innovative statistical teaching and learning techniques in developing countries Each chapter consists of independent case study contributions on a particular theme that are developed with a common structure and format. The common goal across the chapters is to enhance the exchange of diverse educational and action-oriented information among our intended audiences, which include practitioners, researchers, students, and statistics educators in developing countries.
Proof Complexity (Encyclopedia of Mathematics and its Applications #170)
by Jan KrajíčekProof complexity is a rich subject drawing on methods from logic, combinatorics, algebra and computer science. This self-contained book presents the basic concepts, classical results, current state of the art and possible future directions in the field. It stresses a view of proof complexity as a whole entity rather than a collection of various topics held together loosely by a few notions, and it favors more generalizable statements. Lower bounds for lengths of proofs, often regarded as the key issue in proof complexity, are of course covered in detail. However, upper bounds are not neglected: this book also explores the relations between bounded arithmetic theories and proof systems and how they can be used to prove upper bounds on lengths of proofs and simulations among proof systems. It goes on to discuss topics that transcend specific proof systems, allowing for deeper understanding of the fundamental problems of the subject.
Proof Patterns
by Mark JoshiThis innovative textbook introduces a new pattern-based approach to learning proof methods in the mathematical sciences. Readers will discover techniques that will enable them to learn new proofs across different areas of pure mathematics with ease. The patterns in proofs from diverse fields such as algebra, analysis, topology and number theory are explored. Specific topics examined include game theory, combinatorics and Euclidean geometry, enabling a broad familiarity. The author, an experienced lecturer and researcher renowned for his innovative view and intuitive style, illuminates a wide range of techniques and examples from duplicating the cube to triangulating polygons to the infinitude of primes to the fundamental theorem of algebra. Intended as a companion for undergraduate students, this text is an essential addition to every aspiring mathematician's toolkit.
Proof Technology in Mathematics Research and Teaching (Mathematics Education in the Digital Era #14)
by Gila Hanna Michael De Villiers David A. ReidThis book presents chapters exploring the most recent developments in the role of technology in proving. The full range of topics related to this theme are explored, including computer proving, digital collaboration among mathematicians, mathematics teaching in schools and universities, and the use of the internet as a site of proof learning. Proving is sometimes thought to be the aspect of mathematical activity most resistant to the influence of technological change. While computational methods are well known to have a huge importance in applied mathematics, there is a perception that mathematicians seeking to derive new mathematical results are unaffected by the digital era. The reality is quite different. Digital technologies have transformed how mathematicians work together, how proof is taught in schools and universities, and even the nature of proof itself. Checking billions of cases in extremely large but finite sets, impossible a few decades ago, has now become a standard method of proof. Distributed proving, by teams of mathematicians working independently on sections of a problem, has become very much easier as digital communication facilitates the sharing and comparison of results. Proof assistants and dynamic proof environments have influenced the verification or refutation of conjectures, and ultimately how and why proof is taught in schools. And techniques from computer science for checking the validity of programs are being used to verify mathematical proofs. Chapters in this book include not only research reports and case studies, but also theoretical essays, reviews of the state of the art in selected areas, and historical studies. The authors are experts in the field.
Proof Theory and Algebra in Logic (Short Textbooks in Logic)
by Hiroakira OnoThis book offers a concise introduction to both proof-theory and algebraic methods, the core of the syntactic and semantic study of logic respectively. The importance of combining these two has been increasingly recognized in recent years. It highlights the contrasts between the deep, concrete results using the former and the general, abstract ones using the latter. Covering modal logics, many-valued logics, superintuitionistic and substructural logics, together with their algebraic semantics, the book also provides an introduction to nonclassical logic for undergraduate or graduate level courses.The book is divided into two parts: Proof Theory in Part I and Algebra in Logic in Part II. Part I presents sequent systems and discusses cut elimination and its applications in detail. It also provides simplified proof of cut elimination, making the topic more accessible. The last chapter of Part I is devoted to clarification of the classes of logics that are discussed in the second part. Part II focuses on algebraic semantics for these logics. At the same time, it is a gentle introduction to the basics of algebraic logic and universal algebra with many examples of their applications in logic. Part II can be read independently of Part I, with only minimum knowledge required, and as such is suitable as a textbook for short introductory courses on algebra in logic.
Proof Theory: Second Edition (Dover Books on Mathematics #Volume 81)
by Gaisi TakeutiFocusing on Gentzen-type proof theory, this volume presents a detailed overview of creative works by author Gaisi Takeuti and other twentieth-century logicians. The text explores applications of proof theory to logic as well as other areas of mathematics. Suitable for advanced undergraduates and graduate students of mathematics, this long-out-of-print monograph forms a cornerstone for any library in mathematical logic and related topics.The three-part treatment begins with an exploration of first order systems, including a treatment of predicate calculus involving Gentzen's cut-elimination theorem and the theory of natural numbers in terms of Gödel's incompleteness theorem and Gentzen's consistency proof. The second part, which considers second order and finite order systems, covers simple type theory and infinitary logic. The final chapters address consistency problems with an examination of consistency proofs and their applications.
Proof Theory: Sequent Calculi and Related Formalisms (Discrete Mathematics and Its Applications)
by Katalin BimboAlthough sequent calculi constitute an important category of proof systems, they are not as well known as axiomatic and natural deduction systems. Addressing this deficiency, Proof Theory: Sequent Calculi and Related Formalisms presents a comprehensive treatment of sequent calculi, including a wide range of variations. It focuses on sequent calculi
Proof and Proving in Mathematics Education
by Gila Hanna Michael De VilliersOne of the most significant tasks facing mathematics educators is to understand the role of mathematical reasoning and proving in mathematics teaching, so that its presence in instruction can be enhanced. This challenge has been given even greater importance by the assignment to proof of a more prominent place in the mathematics curriculum at all levels. Along with this renewed emphasis, there has been an upsurge in research on the teaching and learning of proof at all grade levels, leading to a re-examination of the role of proof in the curriculum and of its relation to other forms of explanation, illustration and justification. This book, resulting from the 19th ICMI Study, brings together a variety of viewpoints on issues such as: The potential role of reasoning and proof in deepening mathematical understanding in the classroom as it does in mathematical practice. The developmental nature of mathematical reasoning and proof in teaching and learning from the earliest grades. The development of suitable curriculum materials and teacher education programs to support the teaching of proof and proving.The book considers proof and proving as complex but foundational in mathematics. Through the systematic examination of recent research this volume offers new ideas aimed at enhancing the place of proof and proving in our classrooms.
Proof and Proving in Mathematics Education: The 19th ICMI Study (New ICMI Study Series #15)
by Gila Hanna Michael De Villiers*THIS BOOK WILL SOON BECOME AVAILABLE AS OPEN ACCESS BOOK*One of the most significant tasks facing mathematics educators is to understand the role of mathematical reasoning and proving in mathematics teaching, so that its presence in instruction can be enhanced. This challenge has been given even greater importance by the assignment to proof of a more prominent place in the mathematics curriculum at all levels.Along with this renewed emphasis, there has been an upsurge in research on the teaching and learning of proof at all grade levels, leading to a re-examination of the role of proof in the curriculum and of its relation to other forms of explanation, illustration and justification. This book, resulting from the 19th ICMI Study, brings together a variety of viewpoints on issues such as: The potential role of reasoning and proof in deepening mathematical understanding in the classroom as it does in mathematical practice. The developmental nature of mathematical reasoning and proof in teaching and learning from the earliest grades. The development of suitable curriculum materials and teacher education programs to support the teaching of proof and proving.The book considers proof and proving as complex but foundational in mathematics. Through the systematic examination of recent research this volume offers new ideas aimed at enhancing the place of proof and proving in our classrooms.
Proof and the Art of Mathematics: Examples And Extensions
by Joel David HamkinsAn introduction to writing proofs, presented through compelling mathematical statements with interesting elementary proofs.This book offers an introduction to the art and craft of proof-writing. The author, a leading research mathematician, presents a series of engaging and compelling mathematical statements with interesting elementary proofs. These proofs capture a wide range of topics, including number theory, combinatorics, graph theory, the theory of games, geometry, infinity, order theory, and real analysis. The goal is to show students and aspiring mathematicians how to write proofs with elegance and precision.
Proof and the Art of Mathematics: Examples and Extensions
by Joel David HamkinsHow to write mathematical proofs, shown in fully-worked out examples.This is a companion volume Joel Hamkins's Proof and the Art of Mathematics, providing fully worked-out solutions to all of the odd-numbered exercises as well as a few of the even-numbered exercises. In many cases, the solutions go beyond the exercise question itself to the natural extensions of the ideas, helping readers learn how to approach a mathematical investigation. As Hamkins asks, "Once you have solved a problem, why not push the ideas harder to see what further you can prove with them?" These solutions offer readers examples of how to write a mathematical proofs. The mathematical development of this text follows the main book, with the same chapter topics in the same order, and all theorem and exercise numbers in this text refer to the corresponding statements of the main text.
Proof in Geometry: With "Mistakes in Geometric Proofs" (Dover Books on Mathematics)
by A. I. Fetisov Ya. S. DubnovThis single-volume compilation of two books explores the construction of geometric proofs. In addition to offering useful criteria for determining correctness, it presents examples of faulty proofs that illustrate common errors. High-school geometry is the sole prerequisite.Proof in Geometry, the first in this two-part compilation, discusses the construction of geometric proofs and presents criteria useful for determining whether a proof is logically correct and whether it actually constitutes proof. It features sample invalid proofs, in which the errors are explained and corrected.Mistakes in Geometric Proofs, the second book in this compilation, consists chiefly of examples of faulty proofs. Some illustrate mistakes in reasoning students might be likely to make, and others are classic sophisms. Chapters 1 and 3 present the faulty proofs, and chapters 2 and 4 offer comprehensive analyses of the errors.
Proof!: How the World Became Geometrical
by Amir AlexanderA “lucid and convincingly argued” narrative of how ancient geometric principles continue to shape the contemporary world (Publishers Weekly).On a cloudy day in 1413, a balding young man stood at the entrance to the Cathedral of Florence, facing the ancient Baptistery across the piazza. As puzzled passers-by looked on, he raised a small painting to his face, then held a mirror in front of the painting. Few at the time understood what he was up to; even he barely had an inkling of what was at stake. But on that day, the master craftsman and engineer Filippo Brunelleschi would prove that the world and everything within it was governed by the ancient science of geometry.In Proof!, the award-winning historian Amir Alexander traces the path of the geometrical vision of the world as it coursed its way from the Renaissance to the present, shaping our societies, our politics, and our ideals. Geometry came to stand for a fixed and unchallengeable universal order, and kings, empire-builders, and even republican revolutionaries would rush to cast their rule as the apex of the geometrical universe. For who could doubt the right of a ruler or the legitimacy of a government that drew its power from the immutable principles of Euclidean geometry?From the elegant terraces of Versailles to the broad avenues of Washington, DC, and on to the boulevards of New Delhi and Manila, the geometrical vision was carved into the landscape of modernity. Euclid, Alexander shows, made the world as we know it possible.
Proof, Computation and Agency
by Johan Van Benthem Amitabha Gupta Rohit ParikhProof, Computation and Agency: Logic at the Crossroads provides an overview of modern logic and its relationship with other disciplines. As a highlight, several articles pursue an inspiring paradigm called 'social software', which studies patterns of social interaction using techniques from logic and computer science. The book also demonstrates how logic can join forces with game theory and social choice theory. A second main line is the logic-language-cognition connection, where the articles collected here bring several fresh perspectives. Finally, the book takes up Indian logic and its connections with epistemology and the philosophy of science, showing how these topics run naturally into each other.
Proof: The Art and Science of Certainty
by Adam KucharskiAn award-winning mathematician—he "made me smile and made me feel clever" (Peter Frankopan)—shows how we prove what&’s true, and what to do when we can&’t How do we establish what we believe? And how can we be certain that what we believe is true? And how do we convince other people that it is true? For thousands of years, from the ancient Greeks to the Arabic golden age to the modern world, science has used different methods—logical, empirical, intuitive, and more—to separate fact from fiction. But it all had the same goal: find perfect evidence and be rewarded with universal truth. As mathematician Adam Kucharski shows, however, there is far more to proof than axioms, theories, and laws: when demonstrating that a new medical treatment works, persuading a jury of someone&’s guilt, or deciding whether you trust a self-driving car, the weighing up of evidence is far from simple. To discover proof, we must reach into a thicket of errors and biases and embrace uncertainty—and never more so than when existing methods fail. Spanning mathematics, science, politics, philosophy, and economics, this book offers the ultimate exploration of how we can find our way to proof—and, just as importantly, of how to go forward when supposed facts falter.
Proofiness
by Charles SeifeThe bestselling author of Zero shows how mathematical misinformation pervades-and shapes-our daily lives. According to MSNBC, having a child makes you stupid. You actually lose IQ points. Good Morning America has announced that natural blondes will be extinct within two hundred years. Pundits estimated that there were more than a million demonstrators at a tea party rally in Washington, D. C. , even though roughly sixty thousand were there. Numbers have peculiar powers-they can disarm skeptics, befuddle journalists, and hoodwink the public into believing almost anything. "Proofiness," as Charles Seife explains in this eye-opening book, is the art of using pure mathematics for impure ends, and he reminds readers that bad mathematics has a dark side. It is used to bring down beloved government officials and to appoint undeserving ones (both Democratic and Republican), to convict the innocent and acquit the guilty, to ruin our economy, and to fix the outcomes of future elections. This penetrating look at the intersection of math and society will appeal to readers of Freakonomics and the books of Malcolm Gladwell. .
Proofiness
by Charles SeifeThe bestselling author of Zero shows how mathematical misinformation pervades-and shapes-our daily lives. According to MSNBC, having a child makes you stupid. You actually lose IQ points. Good Morning America has announced that natural blondes will be extinct within two hundred years. Pundits estimated that there were more than a million demonstrators at a tea party rally in Washington, D.C., even though roughly sixty thousand were there. Numbers have peculiar powers-they can disarm skeptics, befuddle journalists, and hoodwink the public into believing almost anything. "Proofiness," as Charles Seife explains in this eye-opening book, is the art of using pure mathematics for impure ends, and he reminds readers that bad mathematics has a dark side. It is used to bring down beloved government officials and to appoint undeserving ones (both Democratic and Republican), to convict the innocent and acquit the guilty, to ruin our economy, and to fix the outcomes of future elections. This penetrating look at the intersection of math and society will appeal to readers of Freakonomics and the books of Malcolm Gladwell.
Proofs 101: An Introduction to Formal Mathematics
by Joseph KirtlandProofs 101: An Introduction to Formal Mathematics serves as an introduction to proofs for mathematics majors who have completed the calculus sequence (at least Calculus I and II) and a first course in linear algebra. The book prepares students for the proofs they will need to analyze and write the axiomatic nature of mathematics and the rigors of upper-level mathematics courses. Basic number theory, relations, functions, cardinality, and set theory will provide the material for the proofs and lay the foundation for a deeper understanding of mathematics, which students will need to carry with them throughout their future studies. Features Designed to be teachable across a single semester Suitable as an undergraduate textbook for Introduction to Proofs or Transition to Advanced Mathematics courses Offers a balanced variety of easy, moderate, and difficult exercises
Proofs and Algorithms
by Gilles DowekLogic is a branch of philosophy, mathematics and computer science. It studies the required methods to determine whether a statement is true, such as reasoning and computation. Proofs and Algorithms: Introduction to Logic and Computability is an introduction to the fundamental concepts of contemporary logic - those of a proof, a computable function, a model and a set. It presents a series of results, both positive and negative, - Church's undecidability theorem, Gödel's incompleteness theorem, the theorem asserting the semi-decidability of provability - that have profoundly changed our vision of reasoning, computation, and finally truth itself. Designed for undergraduate students, this book presents all that philosophers, mathematicians and computer scientists should know about logic.
Proofs and Fundamentals: A First Course in Abstract Mathematics
by Ethan D. BlochProofs and Fundamentals: A First Course in Abstract Mathematics” 2nd edition is designed as a "transition" course to introduce undergraduates to the writing of rigorous mathematical proofs, and to such fundamental mathematical ideas as sets, functions, relations, and cardinality. The text serves as a bridge between computational courses such as calculus, and more theoretical, proofs-oriented courses such as linear algebra, abstract algebra and real analysis. This 3-part work carefully balances Proofs, Fundamentals, and Extras. Part 1 presents logic and basic proof techniques; Part 2 thoroughly covers fundamental material such as sets, functions and relations; and Part 3 introduces a variety of extra topics such as groups, combinatorics and sequences. A gentle, friendly style is used, in which motivation and informal discussion play a key role, and yet high standards in rigor and in writing are never compromised.