Download or read book A Course in Arithmetic written by J-P. Serre and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 126 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is divided into two parts. The first one is purely algebraic. Its objective is the classification of quadratic forms over the field of rational numbers (Hasse-Minkowski theorem). It is achieved in Chapter IV. The first three chapters contain some preliminaries: quadratic reciprocity law, p-adic fields, Hilbert symbols. Chapter V applies the preceding results to integral quadratic forms of discriminant ± I. These forms occur in various questions: modular functions, differential topology, finite groups. The second part (Chapters VI and VII) uses "analytic" methods (holomor phic functions). Chapter VI gives the proof of the "theorem on arithmetic progressions" due to Dirichlet; this theorem is used at a critical point in the first part (Chapter Ill, no. 2.2). Chapter VII deals with modular forms, and in particular, with theta functions. Some of the quadratic forms of Chapter V reappear here. The two parts correspond to lectures given in 1962 and 1964 to second year students at the Ecole Normale Superieure. A redaction of these lectures in the form of duplicated notes, was made by J.-J. Sansuc (Chapters I-IV) and J.-P. Ramis and G. Ruget (Chapters VI-VII). They were very useful to me; I extend here my gratitude to their authors.
Download or read book A First Course in Modular Forms written by Fred Diamond and published by Springer Science & Business Media. This book was released on 2006-03-30 with total page 462 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book introduces the theory of modular forms, from which all rational elliptic curves arise, with an eye toward the Modularity Theorem. Discussion covers elliptic curves as complex tori and as algebraic curves; modular curves as Riemann surfaces and as algebraic curves; Hecke operators and Atkin-Lehner theory; Hecke eigenforms and their arithmetic properties; the Jacobians of modular curves and the Abelian varieties associated to Hecke eigenforms. As it presents these ideas, the book states the Modularity Theorem in various forms, relating them to each other and touching on their applications to number theory. The authors assume no background in algebraic number theory and algebraic geometry. Exercises are included.
Download or read book A Course in Mathematical Logic for Mathematicians written by Yu. I. Manin and published by Springer Science & Business Media. This book was released on 2009-10-13 with total page 389 pages. Available in PDF, EPUB and Kindle. Book excerpt: 1. The ?rst edition of this book was published in 1977. The text has been well received and is still used, although it has been out of print for some time. In the intervening three decades, a lot of interesting things have happened to mathematical logic: (i) Model theory has shown that insights acquired in the study of formal languages could be used fruitfully in solving old problems of conventional mathematics. (ii) Mathematics has been and is moving with growing acceleration from the set-theoretic language of structures to the language and intuition of (higher) categories, leaving behind old concerns about in?nities: a new view of foundations is now emerging. (iii) Computer science, a no-nonsense child of the abstract computability theory, has been creatively dealing with old challenges and providing new ones, such as the P/NP problem. Planning additional chapters for this second edition, I have decided to focus onmodeltheory,the conspicuousabsenceofwhichinthe ?rsteditionwasnoted in several reviews, and the theory of computation, including its categorical and quantum aspects. The whole Part IV: Model Theory, is new. I am very grateful to Boris I. Zilber, who kindly agreed to write it. It may be read directly after Chapter II. The contents of the ?rst edition are basically reproduced here as Chapters I–VIII. Section IV.7, on the cardinality of the continuum, is completed by Section IV.7.3, discussing H. Woodin’s discovery.
Download or read book A Course in Number Theory and Cryptography written by Neal Koblitz and published by Springer Science & Business Media. This book was released on 2012-09-05 with total page 245 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a substantially revised and updated introduction to arithmetic topics, both ancient and modern, that have been at the centre of interest in applications of number theory, particularly in cryptography. As such, no background in algebra or number theory is assumed, and the book begins with a discussion of the basic number theory that is needed. The approach taken is algorithmic, emphasising estimates of the efficiency of the techniques that arise from the theory, and one special feature is the inclusion of recent applications of the theory of elliptic curves. Extensive exercises and careful answers are an integral part all of the chapters.
Download or read book A Course in Universal Algebra written by S. Burris and published by Springer. This book was released on 2011-10-21 with total page 276 pages. Available in PDF, EPUB and Kindle. Book excerpt: Universal algebra has enjoyed a particularly explosive growth in the last twenty years, and a student entering the subject now will find a bewildering amount of material to digest. This text is not intended to be encyclopedic; rather, a few themes central to universal algebra have been developed sufficiently to bring the reader to the brink of current research. The choice of topics most certainly reflects the authors' interests. Chapter I contains a brief but substantial introduction to lattices, and to the close connection between complete lattices and closure operators. In particular, everything necessary for the subsequent study of congruence lattices is included. Chapter II develops the most general and fundamental notions of uni versal algebra-these include the results that apply to all types of algebras, such as the homomorphism and isomorphism theorems. Free algebras are discussed in great detail-we use them to derive the existence of simple algebras, the rules of equational logic, and the important Mal'cev conditions. We introduce the notion of classifying a variety by properties of (the lattices of) congruences on members of the variety. Also, the center of an algebra is defined and used to characterize modules (up to polynomial equivalence). In Chapter III we show how neatly two famous results-the refutation of Euler's conjecture on orthogonal Latin squares and Kleene's character ization of languages accepted by finite automata-can be presented using universal algebra. We predict that such "applied universal algebra" will become much more prominent.
Download or read book The New Math written by Christopher James Phillips and published by University of Chicago Press. This book was released on 2015 with total page 242 pages. Available in PDF, EPUB and Kindle. Book excerpt: An era of sweeping cultural change in America, the postwar years saw the rise of beatniks and hippies, the birth of feminism, and the release of the first video game. This book examines the rise and fall of the new math as a marker of the period's political and social ferment.
Download or read book A Course in Computational Algebraic Number Theory written by Henri Cohen and published by Springer Science & Business Media. This book was released on 2013-04-17 with total page 556 pages. Available in PDF, EPUB and Kindle. Book excerpt: A description of 148 algorithms fundamental to number-theoretic computations, in particular for computations related to algebraic number theory, elliptic curves, primality testing and factoring. The first seven chapters guide readers to the heart of current research in computational algebraic number theory, including recent algorithms for computing class groups and units, as well as elliptic curve computations, while the last three chapters survey factoring and primality testing methods, including a detailed description of the number field sieve algorithm. The whole is rounded off with a description of available computer packages and some useful tables, backed by numerous exercises. Written by an authority in the field, and one with great practical and teaching experience, this is certain to become the standard and indispensable reference on the subject.
Download or read book Arithmetic Algebraic Geometry written by Brian David Conrad and published by American Mathematical Soc.. This book was released on with total page 588 pages. Available in PDF, EPUB and Kindle. Book excerpt: The articles in this volume are expanded versions of lectures delivered at the Graduate Summer School and at the Mentoring Program for Women in Mathematics held at the Institute for Advanced Study/Park City Mathematics Institute. The theme of the program was arithmetic algebraic geometry. The choice of lecture topics was heavily influenced by the recent spectacular work of Wiles on modular elliptic curves and Fermat's Last Theorem. The main emphasis of the articles in the volume is on elliptic curves, Galois representations, and modular forms. One lecture series offers an introduction to these objects. The others discuss selected recent results, current research, and open problems and conjectures. The book would be a suitable text for an advanced graduate topics course in arithmetic algebraic geometry.
Download or read book A New Course in Geometry written by Andrew Walker and published by Orient Blackswan. This book was released on 1969 with total page 508 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Connecting Arithmetic to Algebra written by Susan Jo Russell and published by Heinemann Educational Books. This book was released on 2011 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: "To truly engage in mathematics is to become curious and intrigued about regularities and patterns, then describe and explain them. A focus on the behavior of the operations allows students starting in the familiar territory of number and computation to progress to true engagement in the discipline of mathematics." -Susan Jo Russell, Deborah Schifter, and Virginia Bastable Algebra readiness: it's a topic of concern that seems to pervade every school district. How can we better prepare elementary students for algebra? More importantly, how can we help all children, not just those who excel in math, become ready for later instruction? The answer lies not in additional content, but in developing a way of thinking about the mathematics that underlies both arithmetic and algebra. Connecting Arithmetic to Algebra invites readers to learn about a crucial component of algebraic thinking: investigating the behavior of the operations. Nationally-known math educators Susan Jo Russell, Deborah Schifter, and Virginia Bastable and a group of collaborating teachers describe how elementary teachers can shape their instruction so that students learn to: *notice and describe consistencies across problems *articulate generalizations about the behavior of the operations *develop mathematical arguments based on representations to explain why such generalizations are or are not true. Through such work, students become familiar with properties and general rules that underlie computational strategies-including those that form the basis of strategies used in algebra-strengthening their understanding of grade-level content and at the same time preparing them for future studies. Each chapter is illustrated by lively episodes drawn from the classrooms of collaborating teachers in a wide range of settings. These provide examples of posing problems, engaging students in productive discussion, using representations to develop mathematical arguments, and supporting both students with a wide range of learning profiles. Staff Developers: Available online, the Course Facilitator's Guide provides math leaders with tools and resources for implementing a Connecting Arithmetic to Algebra workshop or preservice course. For information on the PD course offered through Mount Holyoke College, download the flyer.
Download or read book Principles of Mathematics Book 1 Set written by Katherine Loop and published by Master Books. This book was released on 2016-09-02 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Katherine Loop has done the remarkable! She has written a solid math course with a truly Biblical worldview. This course goes way beyond the same old Christian math course that teaches math with a few Scriptures sprinkled in and maybe some church-based word problems. This course truly transforms the way we see math. Katherine makes the argument that math is not a neutral subject as most have come to believe. She carefully lays the foundation of how math points to our Creator, the God of the Bible. The nature of God, His Creation, and even the Gospel itself is seen through the study of math. Katherine does a marvelous job of revealing His Glory in this one-of-a-kind math course. Katherine Loop's Principles of Mathematics Biblical Worldview Curriculum is a first of its kind. It takes math to a whole new level students and parents are going to love. It is a guaranteed faith grower!
Download or read book Number Theory and Geometry An Introduction to Arithmetic Geometry written by Álvaro Lozano-Robledo and published by American Mathematical Soc.. This book was released on 2019-03-21 with total page 506 pages. Available in PDF, EPUB and Kindle. Book excerpt: Geometry and the theory of numbers are as old as some of the oldest historical records of humanity. Ever since antiquity, mathematicians have discovered many beautiful interactions between the two subjects and recorded them in such classical texts as Euclid's Elements and Diophantus's Arithmetica. Nowadays, the field of mathematics that studies the interactions between number theory and algebraic geometry is known as arithmetic geometry. This book is an introduction to number theory and arithmetic geometry, and the goal of the text is to use geometry as the motivation to prove the main theorems in the book. For example, the fundamental theorem of arithmetic is a consequence of the tools we develop in order to find all the integral points on a line in the plane. Similarly, Gauss's law of quadratic reciprocity and the theory of continued fractions naturally arise when we attempt to determine the integral points on a curve in the plane given by a quadratic polynomial equation. After an introduction to the theory of diophantine equations, the rest of the book is structured in three acts that correspond to the study of the integral and rational solutions of linear, quadratic, and cubic curves, respectively. This book describes many applications including modern applications in cryptography; it also presents some recent results in arithmetic geometry. With many exercises, this book can be used as a text for a first course in number theory or for a subsequent course on arithmetic (or diophantine) geometry at the junior-senior level.
Download or read book Discrete Mathematics written by Oscar Levin and published by Createspace Independent Publishing Platform. This book was released on 2016-08-16 with total page 342 pages. Available in PDF, EPUB and Kindle. Book excerpt: This gentle introduction to discrete mathematics is written for first and second year math majors, especially those who intend to teach. The text began as a set of lecture notes for the discrete mathematics course at the University of Northern Colorado. This course serves both as an introduction to topics in discrete math and as the "introduction to proof" course for math majors. The course is usually taught with a large amount of student inquiry, and this text is written to help facilitate this. Four main topics are covered: counting, sequences, logic, and graph theory. Along the way proofs are introduced, including proofs by contradiction, proofs by induction, and combinatorial proofs. The book contains over 360 exercises, including 230 with solutions and 130 more involved problems suitable for homework. There are also Investigate! activities throughout the text to support active, inquiry based learning. While there are many fine discrete math textbooks available, this text has the following advantages: It is written to be used in an inquiry rich course. It is written to be used in a course for future math teachers. It is open source, with low cost print editions and free electronic editions.
Download or read book Introduction to Modular Forms written by Serge Lang and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 267 pages. Available in PDF, EPUB and Kindle. Book excerpt: From the reviews: "This book gives a thorough introduction to several theories that are fundamental to research on modular forms. Most of the material, despite its importance, had previously been unavailable in textbook form. Complete and readable proofs are given... In conclusion, this book is a welcome addition to the literature for the growing number of students and mathematicians in other fields who want to understand the recent developments in the theory of modular forms." #Mathematical Reviews# "This book will certainly be indispensable to all those wishing to get an up-to-date initiation to the theory of modular forms." #Publicationes Mathematicae#
Download or read book The Math Myth written by Andrew Hacker and published by New Press, The. This book was released on 2010-05-25 with total page 257 pages. Available in PDF, EPUB and Kindle. Book excerpt: A New York Times–bestselling author looks at mathematics education in America—when it’s worthwhile, and when it’s not. Why do we inflict a full menu of mathematics—algebra, geometry, trigonometry, even calculus—on all young Americans, regardless of their interests or aptitudes? While Andrew Hacker has been a professor of mathematics himself, and extols the glories of the subject, he also questions some widely held assumptions in this thought-provoking and practical-minded book. Does advanced math really broaden our minds? Is mastery of azimuths and asymptotes needed for success in most jobs? Should the entire Common Core syllabus be required of every student? Hacker worries that our nation’s current frenzied emphasis on STEM is diverting attention from other pursuits and even subverting the spirit of the country. Here, he shows how mandating math for everyone prevents other talents from being developed and acts as an irrational barrier to graduation and careers. He proposes alternatives, including teaching facility with figures, quantitative reasoning, and understanding statistics. Expanding upon the author’s viral New York Times op-ed, The Math Myth is sure to spark a heated and needed national conversation—not just about mathematics but about the kind of people and society we want to be. “Hacker’s accessible arguments offer plenty to think about and should serve as a clarion call to students, parents, and educators who decry the one-size-fits-all approach to schooling.” —Publishers Weekly, starred review
Download or read book p adic Numbers written by Fernando Q. Gouvea and published by Springer Science & Business Media. This book was released on 2013-06-29 with total page 285 pages. Available in PDF, EPUB and Kindle. Book excerpt: p-adic numbers are of great theoretical importance in number theory, since they allow the use of the language of analysis to study problems relating toprime numbers and diophantine equations. Further, they offer a realm where one can do things that are very similar to classical analysis, but with results that are quite unusual. The book should be of use to students interested in number theory, but at the same time offers an interesting example of the many connections between different parts of mathematics. The book strives to be understandable to an undergraduate audience. Very little background has been assumed, and the presentation is leisurely. There are many problems, which should help readers who are working on their own (a large appendix with hints on the problem is included). Most of all, the book should offer undergraduates exposure to some interesting mathematics which is off the beaten track. Those who will later specialize in number theory, algebraic geometry, and related subjects will benefit more directly, but all mathematics students can enjoy the book.
Download or read book Advanced Topics in the Arithmetic of Elliptic Curves written by Joseph H. Silverman and published by Springer Science & Business Media. This book was released on 2013-12-01 with total page 482 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the introduction to the first volume of The Arithmetic of Elliptic Curves (Springer-Verlag, 1986), I observed that "the theory of elliptic curves is rich, varied, and amazingly vast," and as a consequence, "many important topics had to be omitted." I included a brief introduction to ten additional topics as an appendix to the first volume, with the tacit understanding that eventually there might be a second volume containing the details. You are now holding that second volume. it turned out that even those ten topics would not fit Unfortunately, into a single book, so I was forced to make some choices. The following material is covered in this book: I. Elliptic and modular functions for the full modular group. II. Elliptic curves with complex multiplication. III. Elliptic surfaces and specialization theorems. IV. Neron models, Kodaira-Neron classification of special fibers, Tate's algorithm, and Ogg's conductor-discriminant formula. V. Tate's theory of q-curves over p-adic fields. VI. Neron's theory of canonical local height functions.
