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  Mathematical Proofs: A Transition to Advanced Mathematics

  Gary Chartrand, Albert D. Polimeni, and Ping Zhang

  Mathematical Proofs: A Transition to Advanced Mathematics, 4th Edition introduces students to proof techniques, analyzing proofs, and writing proofs of their own that are not only mathematically correct but clearly written. Written in a student-friendly manner, it provides a solid introduction to such topics as relations, functions, and cardinalities of sets, as well as optional excursions into fields such as number theory, combinatorics, and calculus. The exercises receive consistent praise from users for their thoughtfulness and creativity. They help students progress from understanding and analyzing proofs and techniques to producing well-constructed proofs independently. This book is also an excellent reference for students to use in future courses when writing or reading proofs.

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  The Fascinating World of Graph Theory

  Arthur Benjamin, Gary Chartrand, and Ping Zhang

  The fascinating world of graph theory goes back several centuries and revolves around the study of graphs--mathematical structures showing relations between objects. With applications in biology, computer science, transportation science, and other areas, graph theory encompasses some of the most beautiful formulas in mathematics--and some of its most famous problems. For example, what is the shortest route for a traveling salesman seeking to visit a number of cities in one trip? What is the least number of colors needed to fill in any map so that neighboring regions are always colored differently? Requiring readers to have a math background only up to high school algebra, this book explores the questions and puzzles that have been studied, and often solved, through graph theory. In doing so, the book looks at graph theory's development and the vibrant individuals responsible for the field's growth.

  Introducing graph theory's fundamental concepts, the authors explore a diverse plethora of classic problems such as the Lights Out Puzzle, the Minimum Spanning Tree Problem, the Königsberg Bridge Problem, the Chinese Postman Problem, a Knight's Tour, and the Road Coloring Problem. They present every type of graph imaginable, such as bipartite graphs, Eulerian graphs, the Petersen graph, and trees. Each chapter contains math exercises and problems for readers to savor.

  An eye-opening journey into the world of graphs, this book offers exciting problem-solving possibilities for mathematics and beyond.

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  Graphs & Digraphs

  Ping Zhang

  Graphs & Digraphs masterfully employs student-friendly exposition, clear proofs, abundant examples, and numerous exercises to provide an essential understanding of the concepts, theorems, history, and applications of graph theory. Fully updated and thoughtfully reorganized to make reading and locating material easier for instructors and students, the Sixth Edition of this bestselling, classroom-tested text: Adds more than 160 new exercises Presents many new concepts, theorems, and examples Includes recent major contributions to long-standing conjectures such as the Hamiltonian Factorization Conjecture, 1-Factorization Conjecture, and Alspachs Conjecture on graph decompositions Supplies a proof of the perfect graph theorem Features a revised chapter on the probabilistic method in graph theory with many results integrated throughout the text At the end of the book are indices and lists of mathematicians¿ names, terms, symbols, and useful references. There is also a section giving hints and solutions to all odd-numbered exercises. A complete solutions manual is available with qualifying course adoption. Graphs & Digraphs, Sixth Edition remains the consummate text for an advanced undergraduate level or introductory graduate level course or two-semester sequence on graph theory, exploring the subjects fascinating history while covering a host of interesting problems and diverse applications.

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  Advanced Calculus: Theory and Practice

  John Srdjan Petrovic

  Suitable for a one- or two-semester course, Advanced Calculus: Theory and Practice expands on the material covered in elementary calculus and presents this material in a rigorous manner. The text improves studentse(tm) problem-solving and proof-writing skills, familiarizes them with the historical development of calculus concepts, and helps them understand the connections among different topics.

  The book takes a motivating approach that makes ideas less abstract to students. It explains how various topics in calculus may seem unrelated but in reality have common roots. Emphasizing historical perspectives, the text gives students a glimpse into the development of calculus and its ideas from the age of Newton and Leibniz to the twentieth century. Nearly 300 examples lead to important theorems as well as help students develop the necessary skills to closely examine the theorems. Proofs are also presented in an accessible way to students.

  By strengthening skills gained through elementary calculus, this textbook leads students toward mastering calculus techniques. It will help them succeed in their future mathematical or engineering studies.

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  Mathematical Proofs: A Transition to Advanced Mathematics

  Gary Chartrand, Albert Polimeni, and Ping Zhang

  Mathematical Proofs: A Transition to Advanced Mathematics, Third Edition, prepares students for the more abstract mathematics courses that follow calculus. Appropriate for self-study or for use in the classroom, this text introduces students to proof techniques, analyzing proofs, and writing proofs of their own. Written in a clear, conversational style, this book 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. It is also a great reference text that students can look back to when writing or reading proofs in their more advanced courses.

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  Introduction to Mathematical Statistics

  Robert V. Hogg, Joseph W. McKean, and Allen T. Craig

  Introduction to Mathematical Statistics, Seventh Edition, offers a proven approach designed to provide you with an excellent foundation in mathematical statistics. Ample examples and exercises throughout the text illustrate concepts to help you gain a solid understanding of the material.

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  Graphs & Diagraphs

  Gary Chartrand, Ping Zhang, and Linda Lesniak

  Continuing to provide a carefully written, thorough introduction, Graphs & Digraphs, Fifth Edition expertly describes the concepts, theorems, history, and applications of graph theory. Nearly 50 percent longer than its bestselling predecessor, this edition reorganizes the material and presents many new topics.

  New to the Fifth Edition

  • New or expanded coverage of graph minors, perfect graphs, chromatic polynomials, nowhere-zero flows, flows in networks, degree sequences, toughness, list colorings, and list edge colorings
  • New examples, figures, and applications to illustrate concepts and theorems
  • Expanded historical discussions of well-known mathematicians and problems
  • More than 300 new exercises, along with hints and solutions to odd-numbered exercises at the back of the book
  • Reorganization of sections into subsections to make the material easier to read
  • Bolded definitions of terms, making them easier to locate

  Despite a field that has evolved over the years, this student-friendly, classroom-tested text remains the consummate introduction to graph theory. It explores the subject’s fascinating history and presents a host of interesting problems and diverse applications.

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  Modern Classical Homotopy Theory

  Jeffrey Strom

  The core of classical homotopy theory is a body of ideas and theorems that emerged in the 1950s and was later largely codified in the notion of a model category. This core includes the notions of fibration and cofibration; CW complexes; long fiber and cofiber sequences; loop spaces and suspensions; and so on. Brown's representability theorems show that homology and cohomology are also contained in classical homotopy theory. This text develops classical homotopy theory from a modern point of view, meaning that the exposition is informed by the theory of model categories and that homotopy limits and colimits play central roles. The exposition is guided by the principle that it is generally preferable to prove topological results using topology (rather than algebra). The language and basic theory of homotopy limits and colimits make it possible to penetrate deep into the subject with just the rudiments of algebra. The text does reach advanced territory, including the Steenrod algebra, Bott periodicity, localization, the Exponent Theorem of Cohen, Moore, and Neisendorfer, and Miller's Theorem on the Sullivan Conjecture. Thus the reader is given the tools needed to understand and participate in research at (part of) the current frontier of homotopy theory. Proofs are not provided outright. Rather, they are presented in the form of directed problem sets. To the expert, these read as terse proofs; to novices they are challenges that draw them in and help them to thoroughly understand the arguments.

  From Amazon.com

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  Discrete Mathematics

  Ping Zhang and Gary Chartrand

  Chartrand and Zhang's Discrete Mathematics presents a clearly written, student-friendly introduction to discrete mathematics. The authors draw from their background as researchers and educators to offer lucid discussions and descriptions fundamental to the subject of discrete mathematics. Unique among discrete mathematics textbooks for its treatment of proof techniques and graph theory, topics discussed also include logic, relations and functions (especially equivalence relations and objective functions), algorithms and analysis of algorithms, introduction to number theory, combinatorics (counting, the Pascal triangle, and the binomial theorem), discrete probability, partially ordered sets, lattices and Boolean algebras, cryptography, and finite-state machines. This highly versatile text provides mathematical background used in a wide variety of disciplines, including mathematics and mathematics education, computer science, biology, chemistry, engineering, communications, and business. Some of the major features and strengths of this textbook: Numerous carefully explained examples and applications facilitate learning, More than 1,600 exercises, ranging from elementary to challenging, are included with hints/answers to all odd-numbered exercises, Descriptions of proof techniques are accessible and lively, Students benefit from the historical discussions throughout the textbook, An Instructor's Solutions Manual contains complete solutions to all exercises.

  From Amazon.com

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  A 5-Year Study of the First Edition of the Core-Plus Mathematics Curriculum

  Harold Schoen, Steven Ziebarth, and Christian R. Hirsch

  A volume in Research in Mathematics Education Series Editor Barbara J. Dougherty, Iowa State University The study reported in this volume adds to the growing body of evaluation studies that focus on the use of NSF-funded Standards-based high school mathematics curricula. Most previous evaluations have studied the impact of field-test versions of a curriculum. Since these innovative curricula were so new at the time of many of these studies, students and teachers were relative novices in their use. These earlier studies were mainly one year or less in duration. Students in the comparison groups were typically from schools in which some classes used a Standards-based curriculum and other classes used a conventional curriculum, rather than using the Standards-based curriculum with all students as curriculum developers intended. This volume reports one of the first studies of the efficacy of Standards-based mathematics curricula with all of the following characteristics: · The study focused on fairly stable implementations of a first-edition Standards-based high school mathematics curriculum that was used by all students in each of three schools. · It involved students who experienced up to seven years of Standards-based mathematics curricula and instruction in middle school and high school. · It monitored students’ mathematical achievement, beliefs, and attitudes for four years of high school and one year after graduation. Prior to the study, many of the teachers had one or more years of experience teaching the Standards-based curriculum and/or professional development focusing on how to implement the curriculum well. · In the study, variations in levels of implementation of the curriculum are described and related to student outcomes and teacher behavior variables. Item data and all unpublished testing instruments from this study are available at www.wmich.edu/ cpmp/evaluation.html for use as a baseline of instruments and data for future curriculum evaluators or Core-Plus Mathematics users who may wish to compare results of new groups of students to those in the present study on common tests or surveys. Taken together, this volume, the supplement at the CPMP Web site, and the first edition Core-Plus Mathematics curriculum materials (samples of which are also available at the Web site) serve as a fairly complete description of the nature and impact of an exemplar of first edition NSF-funded Standards-based high school mathematics curricula as it existed and was implemented with all students in three schools around the turn of the 21st century. (description from amazon.com)

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  Chromatic Graph Theory

  Gary Chartrand and Ping Zhang

  Beginning with the origin of the four color problem in 1852, the field of graph colorings has developed into one of the most popular areas of graph theory. Introducing graph theory with a coloring theme, Chromatic Graph Theory explores connections between major topics in graph theory and graph colorings as well as emerging topics.

  This self-contained book first presents various fundamentals of graph theory that lie outside of graph colorings, including basic terminology and results, trees and connectivity, Eulerian and Hamiltonian graphs, matchings and factorizations, and graph embeddings. The remainder of the text deals exclusively with graph colorings. It covers vertex colorings and bounds for the chromatic number, vertex colorings of graphs embedded on surfaces, and a variety of restricted vertex colorings. The authors also describe edge colorings, monochromatic and rainbow edge colorings, complete vertex colorings, several distinguishing vertex and edge colorings, and many distance-related vertex colorings.

  With historical, applied, and algorithmic discussions, this text offers a solid introduction to one of the most popular areas of graph theory.

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  Mathematical Proofs: A Transition to Advanced Mathematics (2nd Edition)

  Gary Chartrand, Albert D. Polimeni, and Ping Zhang

  Mathematical Proofs: A Transition to Advanced Mathematics, 2/e,prepares students for the more abstract mathematics courses that follow calculus. This text introduces students to proof techniques and writing proofs of their own. As such, it is an introduction to the mathematics enterprise, providing solid introductions to relations, functions, and cardinalities of sets.KEY TOPICS: Communicating Mathematics, Sets, Logic, Direct Proof and Proof by Contrapositive, More on Direct Proof and Proof by Contrapositive, Existence and Proof by Contradiction, Mathematical Induction, Prove or Disprove, Equivalence Relations, Functions, Cardinalities of Sets, Proofs in Number Theory, Proofs in Calculus, Proofs in Group Theory.MARKET: For all readers interested in advanced mathematics and logic.

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  Introduction to Graph Theory

  Gary Chartrand

  Graph theory

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  Teachers Engaged in Research: Inquiry in Mathematics Classrooms, Grades 9-12

  Laura R. Van Zoest

  This book provides examples of the ways in which 9-12 grade mathematics teachers from across North America are engaging in research. It offers a glimpse of the questions that capture the attention of teachers, the methodologies that they use to gather data, and the ways in which they make sense of what they find. The focus of these teachers' investigations into mathematics classrooms ranges from students' understanding of content to pedagogical changes to social issues. Underlying the chapters is the common goal of enabling students to develop a deep understanding of the mathematics they learn in their classrooms. By opening their analysis of their classroom practice to our inspection, these courageous teachers have invited us to think along with them and to learn more about our own teaching as a result. By sharing their work, they have given the mathematics education community an important opportunity. Everyone who reads this book-teachers, researchers, teacher-researchers, policy makers, administrators, and others interested in mathematics education-can learn from the findings and the light that they shed on issues important to mathematics education. opportunity to step back and reflect on what can be learned about research from teachers who have engaged in the process. Areas of insight include: (a) the importance of collaboration and participation in communities that value research, (b) the potential of teacher research as a way to warrant teacher practice, (c) the power of video and other artifacts of teaching to support classroom inquiry, (d) connections between teaching and research, and (e) the publication process as professional development.

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  Techniques of Variational Analysis

  Jonathan M. Borwein and Qiji Zhu

  Borwein is an authority in the area of mathematical optimization, and his book makes an important contribution to variational analysis

  Provides a good introduction to the topic

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  Contemporary Mathematics in Context: A Unified Approach, Course 1, Part A, Student Edition

  Arthur F. Coxford, James T. Fey, Christian R. Hirsch, Harold L. Schoen, Gail Burrill, Eric W. Hart, Ann E. Watkins, Beth Ritsema, and Mary Jo Messenger

  Contemporary Mathematics in Context engages students in investigation-based, multi-day lessons organized around big ideas. Important mathematical concepts are developed in relevant contexts by students in ways that make sense to them. Courses 1, along with Courses 2 and 3, comprise a core curriculum that upgrades the mathematics experience for all your students. Course 4 is designed for all college-bound students. Developed with funding from the National Science Foundation, each course is the product of a four-year research, development, and evaluation process involving thousands of students in schools across the country.

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  Graphs of Groups on Surfaces: Interactions and Models

  Arthur T. White

  The book, suitable as both an introductory reference and as a text book in the rapidly growing field of topological graph theory, models both maps (as in map-coloring problems) and groups by means of graph imbeddings on surfaces. Automorphism groups of both graphs and maps are studied. In addition connections are made to other areas of mathematics, such as hypergraphs, block designs, finite geometries, and finite fields. There are chapters on the emerging subfields of enumerative topological graph theory and random topological graph theory, as well as a chapter on the composition of English church-bell music. The latter is facilitated by imbedding the right graph of the right group on an appropriate surface, with suitable symmetries. Throughout the emphasis is on Cayley maps: imbeddings of Cayley graphs for finite groups as (possibly branched) covering projections of surface imbeddings of loop graphs with one vertex. This is not as restrictive as it might sound; many developments in topological graph theory involve such imbeddings. The approach aims to make all this interconnected material readily accessible to a beginning graduate (or an advanced undergraduate) student, while at the same time providing the research mathematician with a useful reference book in topological graph theory. The focus will be on beautiful connections, both elementary and deep, within mathematics that can best be described by the intuitively pleasing device of imbedding graphs of groups on surfaces.

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  Basic Theory of Ordinary Differential Equations

  Po-Fang Hsieh and Yasutaka Sibuya

  Providing readers with the very basic knowledge necessary to begin research on differential equations with professional ability, the selection of topics here covers the methods and results that are applicable in a variety of different fields. The book is divided into four parts. The first covers fundamental existence, uniqueness, smoothness with respect to data, and nonuniqueness. The second part describes the basic results concerning linear differential equations, while the third deals with nonlinear equations. In the last part the authors write about the basic results concerning power series solutions. Each chapter begins with a brief discussion of its contents and history, and hints and comments for many problems are given throughout. With 114 illustrations and 206 exercises, the book is suitable for a one-year graduate course, as well as a reference book for research mathematicians.