**The Nuts And Bolts Of Proofs An Introduction To Mathematical Proofs**

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### The Nuts and Bolts of Proofs

- Author : Antonella Cupillari
- Publisher : Academic Press
- Release Date : 2012-01-05
- Total pages : 297
- ISBN : 9780123822178
- File Size : 28,6 Mb
- Total Download : 184
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The Nuts and Bolts of Proofs instructs students on the primary basic logic of mathematical proofs, showing how proofs of mathematical statements work. The text provides basic core techniques of how to read and write proofs through examples. The basic mechanics of proofs are provided for a methodical approach in gaining an understanding of the fundamentals to help students reach different results. A variety of fundamental proofs demonstrate the basic steps in the construction of a proof and numerous examples illustrate the method and detail necessary to prove various kinds of theorems. New chapter on proof by contradiction New updated proofs A full range of accessible proofs Symbols indicating level of difficulty help students understand whether a problem is based on calculus or linear algebra Basic terminology list with definitions at the beginning of the text

### The Nuts and Bolts of Proofs

- Author : Antonella Cupillari
- Publisher : Academic Press
- Release Date : 2005-09-08
- Total pages : 192
- ISBN : 9780080537900
- File Size : 21,6 Mb
- Total Download : 772
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Download The Nuts and Bolts of Proofs in PDF, Epub, and Kindle

The Nuts and Bolts of Proof instructs students on the basic logic of mathematical proofs, showing how and why proofs of mathematical statements work. It provides them with techniques they can use to gain an inside view of the subject, reach other results, remember results more easily, or rederive them if the results are forgotten.A flow chart graphically demonstrates the basic steps in the construction of any proof and numerous examples illustrate the method and detail necessary to prove various kinds of theorems. * The "List of Symbols" has been extended. * Set Theory section has been strengthened with more examples and exercises. * Addition of "A Collection of Proofs"

### A Transition to Proof

- Author : Neil R. Nicholson
- Publisher : CRC Press
- Release Date : 2019-03-21
- Total pages : 323
- ISBN : 9780429535475
- File Size : 37,9 Mb
- Total Download : 634
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Download A Transition to Proof in PDF, Epub, and Kindle

A Transition to Proof: An Introduction to Advanced Mathematics describes writing proofs as a creative process. There is a lot that goes into creating a mathematical proof before writing it. Ample discussion of how to figure out the "nuts and bolts'" of the proof takes place: thought processes, scratch work and ways to attack problems. Readers will learn not just how to write mathematics but also how to do mathematics. They will then learn to communicate mathematics effectively. The text emphasizes the creativity, intuition, and correct mathematical exposition as it prepares students for courses beyond the calculus sequence. The author urges readers to work to define their mathematical voices. This is done with style tips and strict "mathematical do’s and don’ts", which are presented in eye-catching "text-boxes" throughout the text. The end result enables readers to fully understand the fundamentals of proof. Features: The text is aimed at transition courses preparing students to take analysis Promotes creativity, intuition, and accuracy in exposition The language of proof is established in the first two chapters, which cover logic and set theory Includes chapters on cardinality and introductory topology

### The Nuts and Bolts of Proofs

- Author : Antonella Cupillari
- Publisher : Academic Press
- Release Date : 2001
- Total pages : 166
- ISBN : 0121994511
- File Size : 33,5 Mb
- Total Download : 415
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This book leads readers through a progressive explanation of what mathematical proofs are, why they are important, and how they work, along with a presentation of basic techniques used to construct proofs. The Second Edition presents more examples, more exercises, a more complete treatment of mathematical induction and set theory, and it incorporates suggestions from students and colleagues. Since the mathematical concepts used are relatively elementary, the book can be used as a supplement in any post-calculus course. This title has been successfully class-tested for years. There is an index for easier reference, a more extensive list of definitions and concepts, and an updated bibliography. An extensive collection of exercises with complete answers are provided, enabling students to practice on their own. Additionally, there is a set of problems without solutions to make it easier for instructors to prepare homework assignments. * Successfully class-tested over a number of years * Index for easy reference * Extensive list of definitions and concepts * Updated biblography

### Introduction to Advanced Mathematics: A Guide to Understanding Proofs

- Author : Connie M. Campbell
- Publisher : Cengage Learning
- Release Date : 2011-01-01
- Total pages : 144
- ISBN : 0547165382
- File Size : 54,6 Mb
- Total Download : 914
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Download Introduction to Advanced Mathematics: A Guide to Understanding Proofs in PDF, Epub, and Kindle

This text offers a crucial primer on proofs and the language of mathematics. Brief and to the point, it lays out the fundamental ideas of abstract mathematics and proof techniques that students will need to master for other math courses. Campbell presents these concepts in plain English, with a focus on basic terminology and a conversational tone that draws natural parallels between the language of mathematics and the language students communicate in every day. The discussion highlights how symbols and expressions are the building blocks of statements and arguments, the meanings they convey, and why they are meaningful to mathematicians. In-class activities provide opportunities to practice mathematical reasoning in a live setting, and an ample number of homework exercises are included for self-study. This text is appropriate for a course in Foundations of Advanced Mathematics taken by students who've had a semester of calculus, and is designed to be accessible to students with a wide range of mathematical proficiency. It can also be used as a self-study reference, or as a supplement in other math courses where additional proofs practice is needed. Important Notice: Media content referenced within the product description or the product text may not be available in the ebook version.

### Understanding Mathematical Proof

- Author : John Taylor,Rowan Garnier
- Publisher : CRC Press
- Release Date : 2016-04-19
- Total pages : 414
- ISBN : 9781466514911
- File Size : 38,7 Mb
- Total Download : 837
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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

### Proofs and Refutations

- Author : Imre Lakatos,Lakatos Imre
- Publisher : Cambridge University Press
- Release Date : 1976
- Total pages : 190
- ISBN : 0521290384
- File Size : 24,8 Mb
- Total Download : 561
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Proofs and Refutations is for those interested in the methodology, philosophy and history of mathematics.

### Advanced Calculus

- Author : Thomas P. Dence,Joseph B. Dence
- Publisher : Academic Press
- Release Date : 2009-12-02
- Total pages : 392
- ISBN : 0080959326
- File Size : 15,5 Mb
- Total Download : 688
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Download Advanced Calculus in PDF, Epub, and Kindle

Advanced Calculus explores the theory of calculus and highlights the connections between calculus and real analysis – providing a mathematically sophisticated introduction to functional analytical concepts. The text is interesting to read and includes many illustrative worked-out examples and instructive exercises, and precise historical notes to aid in further exploration of calculus. It covers exponential function, and the development of trigonometric functions from the integral. The text is designed for a one-semester advanced calculus course for advanced undergraduates or graduate students. Appropriate rigor for a one-semester advanced calculus course Presents modern materials and nontraditional ways of stating and proving some results Includes precise historical notes throughout the book outstanding feature is the collection of exercises in each chapter Provides coverage of exponential function, and the development of trigonometric functions from the integral

### Engineering Writing by Design

- Author : Edward J. Rothwell,Michael J. Cloud
- Publisher : CRC Press
- Release Date : 2017-09-29
- Total pages : 201
- ISBN : 9781498700955
- File Size : 14,8 Mb
- Total Download : 470
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Download Engineering Writing by Design in PDF, Epub, and Kindle

Engineers are smart people. Their work is important, which is why engineering material should be written as deliberately and carefully as it will be read. Engineering Writing by Design: Creating Formal Documents of Lasting Value demonstrates how effective writing can be achieved through engineering-based thinking. Based on the authors’ combined experience as engineering educators, the book presents a novel approach to technical writing, positioning formal writing tasks as engineering design problems with requirements, constraints, protocols, standards, and customers (readers) to satisfy. Specially crafted for busy engineers and engineering students, this quick-reading, conversational text: Describes how to avoid logical fallacies and use physical reasoning to catch mistakes in claims Covers the essentials of technical grammar and style as well as the elements of mathematical exposition Emphasizes the centrality of the target audience, and thus the need for clear and concise prose Engineering Writing by Design: Creating Formal Documents of Lasting Value addresses the specific combination of thinking and writing skills needed to succeed in modern engineering. Its mantra is: to write like an engineer, you must think like an engineer. Featuring illustrative examples, chapter summaries and exercises, quick-reference tables, and recommendations for further reading, this book is packed with valuable tips and information practicing and aspiring engineers need to become effective writers.

### Engineering Speaking by Design

- Author : Edward J. Rothwell,Michael J. Cloud
- Publisher : CRC Press
- Release Date : 2017-12-19
- Total pages : 164
- ISBN : 9781498705783
- File Size : 53,6 Mb
- Total Download : 188
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From the Authors of Engineering Writing by Design: Creating Formal Documents of Lasting Value Engineering presentations are often a topic of frustration. Engineers complain that they don't enjoy public speaking, and that they don't know how to address audiences with varying levels of technical knowledge. Their colleagues complain about the state of information transfer in the profession. Non-engineers complain that engineers are boring and talk over everybody’s heads. Although many public speaking books exist, most concentrate on surface issues, failing to distinguish the formal oral technical presentation from general public speaking. Engineering Speaking by Design: Delivering Technical Presentations with Real Impact targets the formal oral technical presentation skills needed to succeed in modern engineering. Providing clear and concise instruction supported by illustrative examples, the book explains how to avoid logical fallacies (both formal and informal), use physical reasoning to catch mistakes in claims, master the essentials of presentation style, conquer the elements of mathematical exposition, and forge a connection with the audience. Each chapter ends with a convenient checklist, bulleted summary, and set of exercises. A solutions manual is available with qualifying course adoption. Yet the book’s most unique feature is its conceptual organization around the engineering design process. This is the process taught in most engineering survey courses: understand the problem, collect relevant information, generate alternative solutions, choose a preferred solution, refine the chosen solution, and so on. Since virtually all engineers learn and practice this process, it is so familiar that it can be applied seamlessly to formal oral technical presentations. Thus, Engineering Speaking by Design: Delivering Technical Presentations with Real Impact is inherently valuable in that it shows engineers how to leverage what they already know. The book’s mantra is: if you can think like an engineer, you can speak like an engineer.

### Linear Optimization and Duality

- Author : Craig A. Tovey
- Publisher : CRC Press
- Release Date : 2020-12-15
- Total pages : 555
- ISBN : 9781439887479
- File Size : 49,8 Mb
- Total Download : 527
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Download Linear Optimization and Duality in PDF, Epub, and Kindle

Linear Optimization and Dualiyy: A Modern Exposition departs from convention in significant ways. Standard linear programming textbooks present the material in the order in which it was discovered. Duality is treated as a difficult add-on after coverage of formulation, the simplex method, and polyhedral theory. Students end up without knowing duality in their bones. This text brings in duality in Chapter 1 and carries duality all the way through the exposition. Chapter 1 gives a general definition of duality that shows the dual aspects of a matrix as a column of rows and a row of columns. The proof of weak duality in Chapter 2 is shown via the Lagrangian, which relies on matrix duality. The first three LP formulation examples in Chapter 3 are classic primal-dual pairs including the diet problem and 2-person zero sum games. For many engineering students, optimization is their first immersion in rigorous mathematics. Conventional texts assume a level of mathematical sophistication they don’t have. This text embeds dozens of reading tips and hundreds of answered questions to guide such students. Features Emphasis on duality throughout Practical tips for modeling and computation Coverage of computational complexity and data structures Exercises and problems based on the learning theory concept of the zone of proximal development Guidance for the mathematically unsophisticated reader About the Author Craig A. Tovey is a professor in the H. Milton Stewart School of Industrial and Systems Engineering at Georgia Institute of Technology. Dr. Tovey received an AB from Harvard College, an MS in computer science and a PhD in operations research from Stanford University. His principal activities are in operations research and its interdisciplinary applications. He received a Presidential Young Investigator Award and the Jacob Wolfowitz Prize for research in heuristics. He was named an Institute Fellow at Georgia Tech, and was recognized by the ACM Special Interest Group on Electronic Commerce with the Test of Time Award. Dr. Tovey received the 2016 Golden Goose Award for his research on bee foraging behavior leading to the development of the Honey Bee Algorithm.

### Discrete Mathematics with Proof

- Author : Eric Gossett
- Publisher : John Wiley & Sons
- Release Date : 2009-06-22
- Total pages : 928
- ISBN : 9780470457931
- File Size : 36,9 Mb
- Total Download : 230
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A Trusted Guide to Discrete Mathematics with Proof?Now in a Newly Revised Edition Discrete mathematics has become increasingly popular in recent years due to its growing applications in the field of computer science. Discrete Mathematics with Proof, Second Edition continues to facilitate an up-to-date understanding of this important topic, exposing readers to a wide range of modern and technological applications. The book begins with an introductory chapter that provides an accessible explanation of discrete mathematics. Subsequent chapters explore additional related topics including counting, finite probability theory, recursion, formal models in computer science, graph theory, trees, the concepts of functions, and relations. Additional features of the Second Edition include: An intense focus on the formal settings of proofs and their techniques, such as constructive proofs, proof by contradiction, and combinatorial proofs New sections on applications of elementary number theory, multidimensional induction, counting tulips, and the binomial distribution Important examples from the field of computer science presented as applications including the Halting problem, Shannon's mathematical model of information, regular expressions, XML, and Normal Forms in relational databases Numerous examples that are not often found in books on discrete mathematics including the deferred acceptance algorithm, the Boyer-Moore algorithm for pattern matching, Sierpinski curves, adaptive quadrature, the Josephus problem, and the five-color theorem Extensive appendices that outline supplemental material on analyzing claims and writing mathematics, along with solutions to selected chapter exercises Combinatorics receives a full chapter treatment that extends beyond the combinations and permutations material by delving into non-standard topics such as Latin squares, finite projective planes, balanced incomplete block designs, coding theory, partitions, occupancy problems, Stirling numbers, Ramsey numbers, and systems of distinct representatives. A related Web site features animations and visualizations of combinatorial proofs that assist readers with comprehension. In addition, approximately 500 examples and over 2,800 exercises are presented throughout the book to motivate ideas and illustrate the proofs and conclusions of theorems. Assuming only a basic background in calculus, Discrete Mathematics with Proof, Second Edition is an excellent book for mathematics and computer science courses at the undergraduate level. It is also a valuable resource for professionals in various technical fields who would like an introduction to discrete mathematics.

### Introduction · to Mathematical Structures and · Proofs

- Author : Larry Gerstein
- Publisher : Springer Science & Business Media
- Release Date : 2013-11-21
- Total pages : 350
- ISBN : 9781468467086
- File Size : 24,6 Mb
- Total Download : 902
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This is a textbook for a one-term course whose goal is to ease the transition from lower-division calculus courses to upper-division courses in linear and abstract algebra, real and complex analysis, number theory, topology, combinatorics, and so on. Without such a "bridge" course, most upper division instructors feel the need to start their courses with the rudiments of logic, set theory, equivalence relations, and other basic mathematical raw materials before getting on with the subject at hand. Students who are new to higher mathematics are often startled to discover that mathematics is a subject of ideas, and not just formulaic rituals, and that they are now expected to understand and create mathematical proofs. Mastery of an assortment of technical tricks may have carried the students through calculus, but it is no longer a guarantee of academic success. Students need experience in working with abstract ideas at a nontrivial level if they are to achieve the sophisticated blend of knowledge, disci pline, and creativity that we call "mathematical maturity. " I don't believe that "theorem-proving" can be taught any more than "question-answering" can be taught. Nevertheless, I have found that it is possible to guide stu dents gently into the process of mathematical proof in such a way that they become comfortable with the experience and begin asking them selves questions that will lead them in the right direction.

### Independent Component Analysis

- Author : James V. Stone
- Publisher : MIT Press
- Release Date : 2004
- Total pages : 224
- ISBN : 0262693151
- File Size : 14,6 Mb
- Total Download : 987
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A fundamental problem in neural network research, as well as in many other disciplines, is finding a suitable representation of multivariate data, i.e. random vectors. For reasons of computational and conceptual simplicity, the representation is often sought as a linear transformation of the original data. In other words, each component of the representation is a linear combination of the original variables. Well-known linear transformation methods include principal component analysis, factor analysis, and projection pursuit. Independent component analysis (ICA) is a recently developed method in which the goal is to find a linear representation of nongaussian data so that the components are statistically independent, or as independent as possible. Such a representation seems to capture the essential structure of the data in many applications, including feature extraction and signal separation.

### The Way of Analysis

- Author : Robert S. Strichartz
- Publisher : Jones & Bartlett Learning
- Release Date : 2000
- Total pages : 764
- ISBN : 0763714976
- File Size : 16,7 Mb
- Total Download : 981
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Download The Way of Analysis in PDF, Epub, and Kindle

The Way of Analysis gives a thorough account of real analysis in one or several variables, from the construction of the real number system to an introduction of the Lebesgue integral. The text provides proofs of all main results, as well as motivations, examples, applications, exercises, and formal chapter summaries. Additionally, there are three chapters on application of analysis, ordinary differential equations, Fourier series, and curves and surfaces to show how the techniques of analysis are used in concrete settings.

### How to Prove It

- Author : Daniel J. Velleman
- Publisher : Cambridge University Press
- Release Date : 2006-01-16
- Total pages : 401
- ISBN : 9780521861243
- File Size : 11,7 Mb
- Total Download : 908
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Download How to Prove It in PDF, Epub, and Kindle

This new edition of Daniel J. Velleman's successful textbook contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software.

### Ricci Flow and the Poincare Conjecture

- Author : John W. Morgan,Gang Tian
- Publisher : American Mathematical Soc.
- Release Date : 2007
- Total pages : 586
- ISBN : 0821843281
- File Size : 53,8 Mb
- Total Download : 338
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Download Ricci Flow and the Poincare Conjecture in PDF, Epub, and Kindle

For over 100 years the Poincare Conjecture, which proposes a topological characterization of the 3-sphere, has been the central question in topology. Since its formulation, it has been repeatedly attacked, without success, using various topological methods. Its importance and difficulty were highlighted when it was chosen as one of the Clay Mathematics Institute's seven Millennium Prize Problems. in 2002 and 2003 Grigory Perelman posted three preprints showing how to use geometric arguments, in particular the Ricci flow as introduced and studied by Hamilton, to establish the Poincare Conjecture in the affirmative. This book provides full details of a complete proof of the Poincare Conjecture following Perelman's three preprints. After a lengthy introduction that outlines the entire argument, the book is divided into four parts. The first part reviews necessary results from Riemannian geometry and Ricci flow, including much of Hamilton's work. The second part starts with Perelman's length function, which is used to establish crucial non-collapsing theorems. Then it discusses the classification of non-collapsed, ancient solutions to the Ricci flow equation. The third part concerns the existence of Ricci flow with surgery for all positive time and an analysis of the topological and geometric changes introduced by surgery. The last part follows Perelman's third preprint to prove that when the initial Riemannian 3-manifold has finite fundamental group, Ricci flow with surgery becomes extinct after finite time. The proofs of the Poincare Conjecture and the closely related 3-dimensional spherical space-form conjectu The existence of Ricci flow with surgery has application to 3-manifolds far beyond the Poincare Conjecture. It forms the heart of the proof via Ricci flow of Thurston's Geometrization Conjecture. Thurston's Geometrization Conjecture, which classifies all compact 3-manifolds, will be the subject of a follow-up article. The organization of the material in this book differs from that given by Perelman. From the beginning the authors present all analytic and geometric arguments in the context of Ricci flow with surgery. in addition, the fourth part is a much-expanded version of Perelman's third preprint; it gives the first complete and detailed proof of the finite-time extinction theorem. With the large amount of background material that is presented and the detailed versions of the central arguments, this book is suitable for all mathematicians from advanced graduate students to specialists in geometry and topology. Clay Mathematics Institute Monograph Series The Clay Mathematics Institute Monograph Series publishes selected expositions of recent developments, both in emerging areas and in older subjects transformed by new insights or unifying ideas. Information for our distributors: Titles in this series are co-published with the Clay Mathematics Institute (Cambridge, MA).

### A Transition to Advanced Mathematics

- Author : Douglas Smith,Maurice Eggen,Richard St.Andre
- Publisher : Unknown
- Release Date : 2010-06-01
- Total pages : 416
- ISBN : 0495826707
- File Size : 46,6 Mb
- Total Download : 682
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A TRANSITION TO ADVANCED MATHEMATICS, 7e, International Edition helps students make the transition from calculus to more proofs-oriented mathematical study. The most successful text of its kind, the 7th edition continues to provide a firm foundation in major concepts needed for continued study and guides students to think and express themselves mathematically—to analyze a situation, extract pertinent facts, and draw appropriate conclusions. The authors place continuous emphasis throughout on improving students' ability to read and write proofs, and on developing their critical awareness for spotting common errors in proofs. Concepts are clearly explained and supported with detailed examples, while abundant and diverse exercises provide thorough practice on both routine and more challenging problems. Students will come away with a solid intuition for the types of mathematical reasoning they'll need to apply in later courses and a better understanding of how mathematicians of all kinds approach and solve problems.

### Introduction to Mathematical Thinking

- Author : Keith J. Devlin
- Publisher : Unknown
- Release Date : 2012
- Total pages : 92
- ISBN : 0615653634
- File Size : 32,5 Mb
- Total Download : 908
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Download Introduction to Mathematical Thinking in PDF, Epub, and Kindle

In the twenty-first century, everyone can benefit from being able to think mathematically. This is not the same as "doing math." The latter usually involves the application of formulas, procedures, and symbolic manipulations; mathematical thinking is a powerful way of thinking about things in the world -- logically, analytically, quantitatively, and with precision. It is not a natural way of thinking, but it can be learned.Mathematicians, scientists, and engineers need to "do math," and it takes many years of college-level education to learn all that is required. Mathematical thinking is valuable to everyone, and can be mastered in about six weeks by anyone who has completed high school mathematics. Mathematical thinking does not have to be about mathematics at all, but parts of mathematics provide the ideal target domain to learn how to think that way, and that is the approach taken by this short but valuable book.The book is written primarily for first and second year students of science, technology, engineering, and mathematics (STEM) at colleges and universities, and for high school students intending to study a STEM subject at university. Many students encounter difficulty going from high school math to college-level mathematics. Even if they did well at math in school, most are knocked off course for a while by the shift in emphasis, from the K-12 focus on mastering procedures to the "mathematical thinking" characteristic of much university mathematics. Though the majority survive the transition, many do not. To help them make the shift, colleges and universities often have a "transition course." This book could serve as a textbook or a supplementary source for such a course.Because of the widespread applicability of mathematical thinking, however, the book has been kept short and written in an engaging style, to make it accessible to anyone who seeks to extend and improve their analytic thinking skills. Going beyond a basic grasp of analytic thinking that everyone can benefit from, the STEM student who truly masters mathematical thinking will find that college-level mathematics goes from being confusing, frustrating, and at times seemingly impossible, to making sense and being hard but doable.Dr. Keith Devlin is a professional mathematician at Stanford University and the author of 31 previous books and over 80 research papers. His books have earned him many awards, including the Pythagoras Prize, the Carl Sagan Award, and the Joint Policy Board for Mathematics Communications Award. He is known to millions of NPR listeners as "the Math Guy" on Weekend Edition with Scott Simon. He writes a popular monthly blog "Devlin's Angle" for the Mathematical Association of America, another blog under the name "profkeithdevlin", and also blogs on various topics for the Huffington Post.

### Handbook of Bolts and Bolted Joints

- Author : John Bickford
- Publisher : CRC Press
- Release Date : 1998-04-28
- Total pages : 911
- ISBN : 9781482273786
- File Size : 28,6 Mb
- Total Download : 610
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Presenting time-tested standard as well as reliable emerging knowledge on threaded fasteners and joints, this book covers how to select parts and materials, predict behavior, control assembly processes, and solve on-the-job problems. It examines key issues affecting bolting in the automotive, pressure vessel, petrochemical, aerospace, and structura

### Mathematics of Optimization: How to do Things Faster

- Author : Steven J. Miller
- Publisher : American Mathematical Soc.
- Release Date : 2017-12-20
- Total pages : 327
- ISBN : 9781470441142
- File Size : 26,5 Mb
- Total Download : 306
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Download Mathematics of Optimization: How to do Things Faster in PDF, Epub, and Kindle

Optimization Theory is an active area of research with numerous applications; many of the books are designed for engineering classes, and thus have an emphasis on problems from such fields. Covering much of the same material, there is less emphasis on coding and detailed applications as the intended audience is more mathematical. There are still several important problems discussed (especially scheduling problems), but there is more emphasis on theory and less on the nuts and bolts of coding. A constant theme of the text is the “why” and the “how” in the subject. Why are we able to do a calculation efficiently? How should we look at a problem? Extensive effort is made to motivate the mathematics and isolate how one can apply ideas/perspectives to a variety of problems. As many of the key algorithms in the subject require too much time or detail to analyze in a first course (such as the run-time of the Simplex Algorithm), there are numerous comparisons to simpler algorithms which students have either seen or can quickly learn (such as the Euclidean algorithm) to motivate the type of results on run-time savings.