Mathematical foundations Books

Book SynopsisTrade Review"Offers a clear and beautiful progression from addition to modern number theory."--Math-Blog "The authors did a remarkable job in making some aspects of modern number theory very accessible to readers with only a minimal knowledge of mathematics, say a student who had a first calculus course. However, also mathematicians who do not have number theory as their main focus will enjoy this book."--Adhemar Bultheel, European Mathematical Society "Ash and Gross do a masterful job of leading students from finite sums to modular forms and to the forefront of modern number theory... This is an excellent piece of mathematical writing."--Choice "[A]n accessible and fun introduction to modular forms... [Summing It Up] is engaging and conversational, without losing accuracy or essential rigor."--Dominic Lanphier, American Mathematical MonthlyTable of Contents*Frontmatter, pg. i*CONTENTS, pg. vii*PREFACE, pg. xi*ACKNOWLEDGMENTS, pg. xv*INTRODUCTION: WHAT THIS BOOK IS ABOUT, pg. 1*CHAPTER 1. PROEM, pg. 11*CHAPTER 2. SUMS OF TWO SQUARES, pg. 22*CHAPTER 3. SUMS OF THREE AND FOUR SQUARES, pg. 32*CHAPTER 4. SUMS OF HIGHER POWERS: WARING'S PROBLEM, pg. 37*CHAPTER 5. SIMPLE SUMS, pg. 42*CHAPTER 6. SUMS OF POWERS, USING LOTS OF ALGEBRA, pg. 50*CHAPTER 7. INFINITE SERIES, pg. 73*CHAPTER 8. CAST OF CHARACTERS, pg. 96*CHAPTER 9. ZETA AND BERNOULLI, pg. 103*CHAPTER 10. COUNT THE WAYS, pg. 110*CHAPTER 11. THE UPPER HALF-PLANE, pg. 127*CHAPTER 12. MODULAR FORMS, pg. 147*CHAPTER 13. HOW MANY MODULAR FORMS ARE THERE?, pg. 160*CHAPTER 14. CONGRUENCE GROUPS, pg. 179*CHAPTER 15. PARTITIONS AND SUMS OF SQUARES REVISITED, pg. 186*CHAPTER 16. MORE THEORY OF MODULAR FORMS, pg. 201*CHAPTER 17. MORE THINGS TO DO WITH MODULAR FORMS: APPLICATIONS, pg. 213*BIBLIOGRAPHY, pg. 225*INDEX, pg. 227

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Book SynopsisEmphasizes the historical development of number theory, describing methods, theorems, and proofs in the contexts in which they originated, and providing an accessible introduction to one of the subjects in mathematics. This title includes helpful hints for when students are unsure of how to get started on a given problem.Trade Review"An excellent contribution to the list of elementary number theory textbooks. Number theory, it is true, has as rich a history as any branch of mathematics, and Watkins has done terrific work in integrating the stories of the people behind this subject with the traditional topics of elementary number theory. There is more than enough material here for a one-semester course, and while this is standard for textbooks at this level, the added historical and biographical material--which cover mathematical developments and people well into the 20th century--are well worth the increased weight of the text."--Mark Bollman, MAA Reviews

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Book SynopsisDesign Techniques for Origami Tessellations is both a collection of origami tessellations and a manual to design them.This book begins by explaining general design methods, the history and definitions of origami tessellations, and the geometric features of flat origami, before moving on to introduce a brand-new design method: the twist-based design method. This method generates base parts that connect twist patterns (that can be folded with a twist) without using a lattice. Therefore, it can generate base parts such as regular pentagons, which cannot be generated with more conventional methods, and can generate new origami tessellations connected to them.Features: No proofs or formulas in the text and minimal jargon. Suitable for readers with a roughly middle school to high school level of mathematical background. Web application implementing the method described in this book is available, allowing the readers to design tTable of Contents0. Origami and Traditional Tessellation Patterns. 0.1. Background of Origami Tessellations. 0.2 Crease Patterns. 0.3 Basic Geometry of Flat-Foldable Crease Patterns. 0.4 Folded State of Crease Patterns. 0.5 Patterns for Twist-Folding. 0.6 Tessellations. 0.7. Tips for Making Beautiful Folds. 1.Folding on Square Grid. 1.1. Square Twist-Patterns. 1.2. Isosceles Right Triangle Twist Pattern. 1.3. Checker Base. 1.4. Changing folded shape. 1.5. Crease patterns as connectable tile. Appendix 1: Pixel Arts Composed of Origami Tessellation. 2. Folding on Equilateral Triangle Grid. 2.1. Equilateral Triangle Twist-Patterns. 2.2. Regular Hexagon Twist-Patterns. 2.3. Right Triangle Twist-Patterns. Column 1: Grid and Twist Pattern. 3. Connecting Triangle Twist Patterns. 3.1. Creating Triangle Twist Patterns. 3.2. Connecting Triangle Twist Patterns. 3.3. Design for Regular Polygon Patterns. Column 02: How to use Triangle Twist Pattern Maker. Appendix 2: Changing Length of Pleat Base. 4.Connecting of Different Base Parts. 4.1. Connectable Side of Boundary. 4.2. Regular Tessellations. 4.3. Tessellation with Equilateral Polygons. 4.4. Combining Crease Patterns Having Different Guide Sides. Appendix 3: Condition that Boundaries are Folded into Similar Shape. 5. Generating Aesthetic Origami Tessellations. 5.1. Origami Tessellations Regarding as Positive-Negative Pattern. 5.2. Parallel Moving Faces by Flat Folding. 5.3. Design for Origami Tessellations Regarding as Positive-Negative Pattern. Appendix 4: Deformation of crease pattern using pleat bases. 6. Folding Bellows. 6.1. Folding Parallel Lines. 6.2. Bent Bellows. 6.3. Periodic Bellows. 6.4. Bending Irregular Bellows. Column 03: Origami Tessellation Design Software “Tess”. 7. Application of Twist Pattern Design Method. 7.1. Reconstructing Guide from Given Origami Tessellation. 7.2. Fractal Origami Tessellations and Guides. 7.3. Guide with Gaps. Column 4: Connecting 3D Origami Arts and Origami Tessellations.

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Book SynopsisDescriptive set theory has been one of the main areas of research in set theory for almost a century. It includes a wide variety of examples, more than 400 exercises, and applications, in order to illustrate the general concepts and results of the theory.Table of ContentsI Polish Spaces.- 1. Topological and Metric Spaces.- 1.A Topological Spaces.- 1.B Metric Spaces.- 2. Trees.- 2.A Basic Concepts.- 2.B Trees and Closed Sets.- 2.C Trees on Produtcs.- 2.D Leftmost Branches.- 2.E Well-founded Trees and Rank.- 2.F The Well-founded Part of a Tree.- 2.G The Kleene-Brouwer Ordering.- 3. Polish Spaces.- 3.A Definitions and Examples.- 3.B Extensions of Continuous Functions and Homeomorphisms.- 3.C Polish Subspaces of Polish Spaces.- 4. Compact Metrizable Spaces.- 4.A Basic Facts.- 4.B Examples.- 4.C A Universality Property of the Hilbert Cube.- 4.D Continuous Images of the Cantor Space.- 4.E The Space of Continuous Functions on a Compact Space.- 4.F The Hyperspace of Compact Sets.- 5. Locally Compact Spaces.- 6. Perfect Polish Spaces.- 6.A Embedding the Cantor Space in Perfect Polish Spaces.- 6.B The Cantor-Bendixson Theorem.- 6.C Cantor-Bendixson Derivatives and Ranks.- 7.Zero-dimensional Spaces.- 7.A Basic Facts.- 7.B A Topological Characterization of the Cantor Space.- 7.C A Topological Characterization of the Baire Space.- 7.D Zero-dimensional Spaces aa Subspaces of the Baire Space.- 7.F Polish Spaces as Continuous Images of the Baire Space.- 7.F Closed Subsets Homcomorphic to the Baire Space.- 8. Baire Category.- 8.A Meager Sets.- 8.B Baire Spaces.- 8.C Choquet Games and Spaces.- 8.D Strong Choquet Games and Spaces.- 8.E A Characterization of Polish Spaces.- 8.F Sets with the Baire Property.- 8.G Localization.- 8.H The Banach-Mazur Game.- 8.I Baire Measurable Functions.- 8.J Category Quantifiers.- 8.K The Kuratowski-Ulam Theorem.- 8.L Some Applications.- 8.M Separate and Joint Continuity.- 9. Polish Groups.- 9.A Metrizable and Polish Groups.- 9.B Examples of Polish Groups.- 9.C Basic Facts about Baire Groups and Their Actions.- 9.D Universal Polish Groups.- II Borel Sets.- 10. Measurable Spaces and Functions.- 10.A Sigma-Algebras and Their Generators.- 10.B Measurable Spaces and Functions.- 11. Borel Sets and Functions.- 11.A Borel Sets in Topological Spaces.- 11.B The Borel Hierarchy.- 11.C Borel Functions.- 12. Standard Borel Spaces.- 12.A Borel Sets and Functions in Separable Metrizable Spaces.- 12.B Standard Borel Spaces.- 12.C The Effros Borel Space.- 12.D An Application to Selectors.- 12.E Further Examples.- 12.F Standard Borel Groups.- 13. Borel Sets as Clopen Sets.- 13.A Turning Borel into Clopen Sets.- 13.B Other Representations of Borel Sets.- 13.C Turning Borel into Continuous Functions.- 14. Analytic Sets and the Separation Theorem.- 14.A Basic Facts about Analytic Sets.- 14.B The Lusin Separation Theorem.- 14.C Sousliri’s Theorem.- 15. Borel Injections and Isomorphisms.- 15.A Borel Injective Images of Borel Sets.- 15.B The Isomorphism Theorem.- 15.C Homomorphisms of Sigma-Algebras Induced by Point Maps.- 15.D Some Applications to Group Actions.- 16. Borel Sets and Baire Category.- 16.A Borel Definability of Category Notions.- 16.B The Vaught Transforms.- 16.C Connections with Model Theory.- 16.D Connections with Cohen’s Forcing Method.- 17. Borel Sets and Measures.- 17.A General Facts on Measures.- 17.B Borel Measures.- 17.C Regularity and Tightness of Measures.- 17.D Lusin’s Theorem on Measurable Functions.- 17.E The Space of Probability Borel Measures.- 17.F The Isomorphism Theorem for Measures.- 18. Uniformization Theorems.- 18.A The Jankov, von Neumann Uniformization Theorem.- 18.B “Large Section” Uniformization Results.- 18.C “Small Section” Uniformization Results.- 18.D Selectors and Transversals.- 19. Partition Theorems.- 19.A Partitions with a Comeager or Non-meager Piece.- 19.B A Ramsey Theorem for Polish Spaces.- 19.C The Galvin-Prikry Theorem.- 19.D Ramsey Sets and the Ellentuck Topology.- 19.E An Application to Banach Space Theory.- 20. Borel Determinacy.- 20.A Infinite Games.- 20.B Determinacy of Closed Games.- 20.C Borel Determinacy.- 20.D Game Quantifiers.- 21. Games People Play.- 21.A The *-Games.- 21.B Unfolding.- 21.C The Banach-Mazur or **-Games.- 21.D The General Unfolded Banach-Mazur Games.- 21.E Wadge Games.- 21.F Separation Games and Hurewicz’s Theorem.- 21.G Turing Degrees.- 22. The Borel Hierarchy.- 22. A Universal Sets.- 22.B The Borel versus the Wadge Hierarchy.- 22.C Structural Properties.- 22.D Additional Results.- 22.E The Difference Hierarchy.- 23. Some Examples.- 23.A Combinatorial Examples.- 23.B Classes of Compact Sets.- 23.C Sequence Spaces.- 23.D Classes of Continuous Functions.- 23.E Uniformly Convergent Sequences.- 23.F Some Universal Sets.- 23.G Further Examples.- 24. The Baire Hierarchy.- 24.A The Baire Classes of Functions.- 24.B Functions of Baire Class 1.- III Analytic Sets.- 25. Representations of Analytic Sets.- 25.A Review.- 25.B Analytic Sets in the Baire Space.- 25.C The Souslin Operation.- 25.D Wellordered Unions and Intersections of Borel Sets.- 25. E Analytic Sets as Open Sets in Strong Choquet Spaces.- 26. Universal and Complete Sets.- 26.A Universal Analytic Sets.- 26.B Analytic Determinacy.- 26.C Complete Analytic Sets.- 26.D Classification up to Borel Isomorphism.- 27. Examples.- 27.A The Class of Ill-founded Trees.- 27.B Classes of Closed Sets.- 27.C Classes of Structures in Model Theory.- 27.D Isomorphism.- 27.E Some Universal Sets.- 27.F Miscellanea.- 28. Separation Theorems.- 28.A The Lusin Separation Theorem Revisited.- 28.B The Novilcov Separation Theorem.- 28.C Borel Sets with Open or Closed Sections.- 28.D Some Special Separation Theorems.- 28.E “Hurewicz-Type” Separation Theorems.- 29. Regularity Properties.- 29.A The Perfect Set Property.- 29.B Measure. Category, and Ramsey.- 29.C A Closure Property for the Souslin Operation.- 29.D The Class of C-Sets.- 29.E Analyticity of “Largeness” Conditions on Analytic Sets.- 30. Capacities.- 30.A The Basic Concept.- 30.B Examples.- 30.C The Choquet Capacitability Theorem.- 31. Analytic Well-founded Relations.- 31.A Bounds on Ranks of Analytic Well-founded Relations.- 31.B The Kunen-Martin Theorem.- IV Co-Analytic Sets.- 32. Review.- 32.A Basic Facts.- 32.B Representations of Co-Analytic Sets.- 32.C Regularity Properties.- 33. Examples.- 33.A Well-founded Trees and Wellorderings.- 33.B Classes of Closed Sets.- 33.C Sigma-ldoals of Compact Sets.- 33.D Differentiable Functions.- 33.E Everywhere Convergence.- 33.F Parametrizing Baire Class 1 Functions.- 33.G A Method for Proving Completeness.- 33.H Singular Functions.- 33.I Topological Examples.- 33.J Homeomorphisms of Compact Spaces.- 33.K Classes of Separable Banach Spaces.- 33.L Other Examples.- 34. Co-Analytic Ranks.- 34.A Ranks and Prewellorderings.- 34.B Ranked Classes.- 34.C Co-Analytic Ranks.- 34.D Derivatives.- 34.E Co-Analytic Ranks Associated with Borel Derivatives.- 34.F Examples.- 35. Rank Theory.- 35.A Basic Properties of Ranked Classes.- 35.B Parametrizing Bi-Analytic and Borel Sets.- 35.C Reflection Theorems.- 35.D Boundedness Properties of Ranks.- 35.E The Rank Method.- 35.F The Strategic Uniformization Theorem.- 35.G Co-Analytic Families of Closed Sets and Their Sigma-Ideals.- 35.H Borel Sots with F? and K? Sections.- 36. Scales and Uniformiiatiou.- 36.A Kappa-Souslin Sets.- 36.B Scales.- 36.C Sealed Classes and Urniformization.- 36.D The Novikov-Kondô Uniformization Theorem.- 36.E Regularity Properties of Uniformizing Functions.- 36.F Uniforniizing Co-Analytic Sets with Large Sections.- 36.G Examples of Co-Analytic Scales.- V Projective Sets.- 37. The Projective Hierarchy.- 37.A Basic Facts.- 37.B Examples.- 38. Projective Determinacy.- 38.A The Second Level of the Projective Hierarchy.- 38.B Projective Determinacy.- 38.C Regularity Properties.- 39. The Periodicity Theorems.- 39.A Periodicity in the Projective Hierarchy.- 39.B The First Periodicity Theorem.- 39.C The Second Periodicity Theorem.- 39.D The Third Periodicity Theorem.- 40. Epilogue.- 40.A Extensions of the Projective Hierarchy.- 40.B Effective Descriptive Set Theory.- 40.C Large Cardinals.- 40.D Connections to Other Areas of Mathematics.- Appendix A. Ordinals and Cardinals.- Appendix B. Well-founded Relations.- Appendix C. On Logical Notation.- Notes and Hints.- References.- Symbols and Abbreviations.

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Book SynopsisA Stitch in Line: Mathematics and One-Stitch Sashiko provides readers with instructions for creating hitomezashi items with minimum outlay. The reader is guided through the practical steps involved in creating each design, and then the mathematics which underpins it is explained in a friendly, accessible way. This is a fantastic book for anyone who is interested in recreational mathematics and/or fibre arts and can be a useful resource for teaching and learning mathematical concepts in a fun and engaging format. Features Numerous full-colour photographs of hitomezashi stitch patterns which have been mathematically designed. Suitable for readers of all mathematical levels and backgrounds no prior knowledge is automatically assumed. A compressed encoding for recording and designing hitomezashi patterns to be stitched or drawn. Accessible explanations and explorations of mathematical concepts inherent in, or illustr

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Book SynopsisThe magnum opus of one of the world’s leading origami artists, the second edition of Origami Design Secrets reveals the underlying concepts of origami and how to create original origami designs. Containing step-by-step instructions for 26 models, this book is not just an origami cookbook or list of instructions—it introduces the fundamental building blocks of origami, building up to advanced methods such as the combination of uniaxial bases, the circle/river method, and tree theory. With corrections and improved illustrations, this new expanded edition also covers uniaxial box pleating, introduces the new design technique of hex pleating, and describes methods of generalizing polygon packing to arbitrary angles.With coverage spanning the foundations of origami construction and advanced methods using both paper and pencil and custom-built free software, Origami Design Secrets helps readers cultivate the intuition and skills necessary to develop their own designs. It takes them beyond merely following a recipe to crafting a work of art. Trade ReviewI am in complete agreement with the questions and thoughts expressed in his book. … This book is perfect for math geeks who aim to apply their math in a very creative way.—OrigamiBlog.com, January 2012Robert Lang made some substantial changes — much to the joy of many origami enthusiasts. The first edition impressed with 594 pages full of high-quality content going into techniques, mathematics, and folding instructions. The second edition tops that by an extra 176 pages, an extra 30% of content! … Altogether, this wealth of new content definitely makes it worth buying the second edition — even if you already have the first edition. As to if you don’t have either edition: I’d recommend going for the second one.—Sara Adams, HappyFolding.com, December 2011The good news is that a second edition of Robert Lang’s Origami Design Secrets is available. The bad news is you’re going to want to buy it, even if you have the first edition. … It’s a tribute to both the author and the publisher (CRC Press) that such major updates have been allowed. This book is without doubt the best guide to creating origami that has ever been published and I suspect it will retain the crown forever. … Quite simply, all creative folders should own this book if they have any interest in developing their skills. Even if they have no such interest, they should buy it — it’s that good.—Nick Robinson, origami artist and author, on his blog, September 2011Praise for the First Edition:The first part of his plan was to write the book he’d been contemplating while still at JDS Uniphase — Origami Design Secrets, which was published in 2003 and lays out the underlying principles of origami and design techniques.—Susan Orlean, The New Yorker, February 2007 When Robert J. Lang, a laser physicist, talks with passion about origami — the Japanese art of paper folding — the line between play and discovery also completely dissolves. Mr. Lang is what might be called an origamist (see www.langorigami.com). He wrote the book Origami Design Secrets: Mathematical Methods for an Ancient Art and has created tarantulas, delicate herons, 12-spined shells and big-horned elk out of single, uncut, folded sheets of paper. —Edward Rothstein, The New York Times, April 2006This book is a synthesis of origami technique, history, and instructions, with very thoughtful and clear explanations for almost every aspect of the art of folding and designing origami.—Sarah Gourlie, Math Horizons, January 2006Lang chose to strike a balance between a book that describes origami design algorithmically and one that appeals to the origami community … . For mathematicians and origamists alike, Lang’s expository approach introduces the reader to technical aspects of folding and the mathematical models with clarity and good humor … Origami Design Secrets … is highly recommended for mathematicians and students alike who want to view, explore, wrestle with open problems in, or even try their own hand at the complexity of origami model design.—Thomas C. Hull, The Mathematical Intelligencer, March 2005This magisterial work, splendidly produced, covers all aspects of the art and science.—SIAM Book Review, November 2004For most origami enthusiasts, the ancient Japanese art of folding and making creases is a mere hobby, and a grand pursuit for those giddily obsessed with the mathematics (or aesthetics) of how mere paper can be contorted into objects of beauty and interest. For Robert J. Lang, however, origami has become a life’s pursuit …—Chad Berndtson, The Patriot Ledger, November 2004… finally, the secrets of an origami master are revealed! It feels like Lang has taken you on as an apprentice as he teaches you his techniques, stepping you through examples of real origami designs and their development.—Erik Demaine, October 2003There is something for everyone here, whether beginner or expert, left-brained or right, specialist or dilettante.—Gail Anderson, Engineering & Science, April 2003I am in complete agreement with the questions and thoughts expressed in his book. … This book is perfect for math geeks who aim to apply their math in a very creative way.—OrigamiBlog.com, January 2012Robert Lang made some substantial changes — much to the joy of many origami enthusiasts. The first edition impressed with 594 pages full of high-quality content going into techniques, mathematics, and folding instructions. The second edition tops that by an extra 176 pages, an extra 30% of content! … Altogether, this wealth of new content definitely makes it worth buying the second edition — even if you already have the first edition. As to if you don’t have either edition: I’d recommend going for the second one.—Sara Adams, HappyFolding.com, December 2011The good news is that a second edition of Robert Lang’s Origami Design Secrets is available. The bad news is you’re going to want to buy it, even if you have the first edition. … It’s a tribute to both the author and the publisher (CRC Press) that such major updates have been allowed. This book is without doubt the best guide to creating origami that has ever been published and I suspect it will retain the crown forever. … Quite simply, all creative folders should own this book if they have any interest in developing their skills. Even if they have no such interest, they should buy it — it’s that good.—Nick Robinson, origami artist and author, on his blog, September 2011Praise for the First Edition:The first part of his plan was to write the book he’d been contemplating while still at JDS Uniphase — Origami Design Secrets, which was published in 2003 and lays out the underlying principles of origami and design techniques.—Susan Orlean, The New Yorker, February 2007When Robert J. Lang, a laser physicist, talks with passion about origami — the Japanese art of paper folding — the line between play and discovery also completely dissolves. Mr. Lang is what might be called an origamist (see www.langorigami.com). He wrote the book Origami Design Secrets: Mathematical Methods for an Ancient Art and has created tarantulas, delicate herons, 12-spined shells and big-horned elk out of single, uncut, folded sheets of paper. —Edward Rothstein, The New York Times, April 2006This book is a synthesis of origami technique, history, and instructions, with very thoughtful and clear explanations for almost every aspect of the art of folding and designing origami.—Sarah Gourlie, Math Horizons, January 2006Lang chose to strike a balance between a book that describes origami design algorithmically and one that appeals to the origami community … . For mathematicians and origamists alike, Lang’s expository approach introduces the reader to technical aspects of folding and the mathematical models with clarity and good humor … Origami Design Secrets … is highly recommended for mathematicians and students alike who want to view, explore, wrestle with open problems in, or even try their own hand at the complexity of origami model design.—Thomas C. Hull, The Mathematical Intelligencer, March 2005This magisterial work, splendidly produced, covers all aspects of the art and science.—SIAM Book Review, November 2004For most origami enthusiasts, the ancient Japanese art of folding and making creases is a mere hobby, and a grand pursuit for those giddily obsessed with the mathematics (or aesthetics) of how mere paper can be contorted into objects of beauty and interest. For Robert J. Lang, however, origami has become a life’s pursuit …—Chad Berndtson, The Patriot Ledger, November 2004… finally, the secrets of an origami master are revealed! It feels like Lang has taken you on as an apprentice as he teaches you his techniques, stepping you through examples of real origami designs and their development.—Erik Demaine, October 2003There is something for everyone here, whether beginner or expert, left-brained or right, specialist or dilettante.—Gail Anderson, Engineering & Science, April 2003Table of ContentsIntroduction. The Building Blocks of Origami. Elephant Design. Traditional Bases. Splitting Points. Grafting. Pattern Grafting. Tiling. Circle Packing. Molecules. Tree Theory. Box Pleating. Uniaxial Box Pleating. Polygon Packing. Hybrid Bases. References. Glossary. Index.

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Book SynopsisWhen John Napier published his invention of logarithms in 1614 he was announcing one of the greatest advances in the history of mathematics, and log tables were used universally until the mid 1970s. With his Rabdologia, an ingenious calculating tool composed of numbered rods which came to be known as 'Napier's Bones', he enabled people in the marketplace to do multiplication sums without knowing any multiplication tables. Perhaps the most extraordinary thing about this most extrordinary man was that his great inventions were made without the stimulus of talking to other mathematicians in mainstream Europe. Working away in comparative isolation in a tower house in Scotland, Napier produced methods of calculation that literally changed lives all over the world. He is the father of the slide-rule and the grandfather of today's calculators. Despite his achievements, he remains curiously uncelebrated, and this absorbing story of his life aims to give John Napier his true status. This new edition has been redesigned in a new format and has a new cover.Trade ReviewReview of first edition: 'What a wonderful little book; it is beautifully written and has wonderful photographs and illustrations ... Moreover it accomplishes its purpose, to give us a glimpse into the nature and times of John Napier.' History of Mathematics NewsletterTable of ContentsAcknowledgements 1 An Astonishing World 2 A Privileged Beginning 3 A Very Young Student 4 Travel was not for the Faint hearted 5 The Student comes Home 6 A Country Laird or a Sorcerer? 7 Weapons against the Spaniards 8 Logarithms - The Quantum Leap 9 The World's First Pocket Calculator 10 Up amongst the Greats Selected Bibliography

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Book SynopsisFamous mathematical constants include the ratio of circular circumference to diameter, p = 3.14 , and the natural logarithm base, e = 2.718 . Students and professionals can often name a few others, but there are many more buried in the literature and awaiting discovery. How do such constants arise, and why are they important? Here the author renews the search he began in his book Mathematical Constants, adding another 133 essays that broaden the landscape. Topics include the minimality of soap film surfaces, prime numbers, elliptic curves and modular forms, PoissonVoronoi tessellations, random triangles, Brownian motion, uncertainty inequalities, PrandtlBlasius flow (from fluid dynamics), Lyapunov exponents, knots and tangles, continued fractions, GaltonWatson trees, electrical capacitance (from potential theory), Zermelo''s navigation problem, and the optimal control of a pendulum. Unsolved problems appear virtually everywhere as well. This volume continues an outstanding scholarly atTrade Review'Like the best sequels, this one covers similar ground to the original but finds ways to stay fresh and interesting … any mathematician or math student who picks it up and spends a few minutes with it is likely to find something that is new and of interest to them. … Finch has once again written a collection of essays about a wide range of topics that I expect I will enjoy flipping through for another decade and a half until I look forward to having Volume III land on my desk.' Darren Glass, MAA reviews'This is a remarkable book … [which] can be thought of as a collection of essays that recount stories that are both successful and tangible.' Paul F. Bracken, MathSciNet'Some of the most intriguing formulas of mathematics (like those of Ramanujan) adorn this treasure trove of mathematical gems … Steven R. Finch's incredible labor of love, an encyclopedia of mathematical constants … contain a total of 269 meticulously documented essays from all fields of mathematics.' Osmo Peokonen, The Mathematical Intelligencer'Taken together, Mathematical Constants and Mathematical Constants II form a comprehensive and unique work that is a welcome addition to the mathematician's reference library.' Steven R. Finch, Notices of the AMS'Great care is taken about numerical results and the precise determination of constants. The choice of the material complements the first volume; overall, the topics seem also to be more advanced, but every now and then there is a little pearl which is indeed accessible at high school level. The text is certainly not intended for linear reading - although this might well be possible - but for eclectic readers who want to enjoy themselves and broaden their horizons, or for researchers who need information on a particular constant and further stepping stones.' Rene L. Schilling, The Mathematical Gazette'Great care is taken about numerical results and the precise determination of constants. The choice of the material complements the first volume; overall, the topics seem also to be more advanced, but every now and then there is a little pearl which is indeed accessible at high school level. The text is certainly not intended for linear reading - although this might well be possible - but for eclectic readers who want to enjoy themselves and broaden their horizons, or for researchers who need information on a particular constant and further stepping stones.' Rene L. Schilling, The Mathematical GazetteTable of Contents1. Number theory and combinatorics; 2. Inequalities and approximation; 3. Real and complex analysis; 4. Probability and stochastic processes; 5. Geometry and topology; Index.

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Book SynopsisMathematical Explorations follows on from the author''s previous book, Creative Mathematics, in the same series, and gives the reader experience in working on problems requiring a little more mathematical maturity. The author''s main aim is to show that problems are often solved by using mathematics that is not obviously connected to the problem, and readers are encouraged to consider as wide a variety of mathematical ideas as possible. In each case, the emphasis is placed on the important underlying ideas rather than on the solutions for their own sake. To enhance understanding of how mathematical research is conducted, each problem has been chosen not for its mathematical importance, but because it provides a good illustration of how arguments can be developed. While the reader does not require a deep mathematical background to tackle these problems, they will find their mathematical understanding is enriched by attempting to solve them.Trade Review'The broad array of mathematics that is covered in this text might provide a thought-provoking summary of topics and give students an opportunity to expand their horizons.' Mark Hunacek, MAA ReviewsTable of ContentsPreface; How to use this book; 1. Paying for parking; 2. Lengths and angles; 3. Magic squares; 4. Intersecting chords; 5. Crossing squares; 6. Repeated vector products; 7. A rolling disc; 8. Sums of powers of digits; 9. The metric dimension; 10. Primes and irreducible elements; 11. The symmetries of a quadrilateral; 12. Removing a vertex; 13. Squares within squares; 14. Catalan numbers; References; Index.

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Book SynopsisTake the mystery out of basic math with the latest edition of BarCharts' best-selling Math Review QuickStudy guide. With updated content and an additional panel of information, Math Review includes hard-to-remember formulas and properties, along with numerous examples and illustrations to improve understanding. This comprehensive math guide will assist you way beyond your high school and college years.

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Book SynopsisAdvances in Mathematics Research presents original studies on the leading edge of mathematics. Each article has been carefully selected in an attempt to present substantial research results across a broad spectrum. Chapter One summarizes the phase/current generalized measures of the entropy/information content in complex quantum states of molecular systems. Chapter Two reviews the current knowledge regarding Mavridis'' area (MA), with emphasis on the role of applied mathematics in its discovery, and aims to explore its mathematical expression. In Chapter Three a model of fractional difference has been defined by the author as a fractional Newton binomial with respect to the finite difference operator as parameter, therefore they obtained an alternative to fractional derivative, and further, as a by-product, they came across the so-called modified Riemann-Liouville derivative which ascribes a special role to the initial value of the considered function. Chapter Four presents some popular uses of exponential distribution in the context of ordered random variables. Chapter Five gives a comprehensive introduction to the Ricci flow on manifolds of dimension two which can be done in a reasonable fashion when the Euler characteristic is negative or zero. Chapter Six investigates some geometric properties by using the concepts of the geometric function theory and studies the convexity and star-like for the new operator.

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Book SynopsisIn the opening chapter by Victor Martinez-Lukacs, two kinds of matrices related to chemical problems are examined and an outline of their main properties about their eigenvalues is exhibited in order to demonstrate that all the ODE solutions are either stable or asymptotically stable. In chapter two by Ivan Kyrchei, the Cramer rules for the weighted Moore-Penrose solutions of left and right systems of quaternion linear equations are obtained. Next, in chapter three, Tadeusz Antczak showcases numerous sets of saddle point criteria for a new class of nonconvex non-smooth discrete minimax fractional programming problems. Marcia de F. B. Binelo, Airam T. Z. R. Sausen, Paulo S. Sausen, and Manuel O. Binelo provide a summary of electric mathematical models used for the prediction of batteries charge and discharge behaviour in chapter four. In chapter five, general methodology for the precise modelling and performance assessment of launch vehicles dedicated to microsatellites is proposed by M. Pontani, M. Palloney, and P. Teofilattoz. In chapter six, Nodari Vakhania exemplifies ties and relationships among some optimisation problems such as scheduling and transportation issues. In chapter seven, a geometry without using points in established by N. L. Bushwick, bringing the book to a close.

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Book Synopsis

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Book Synopsis

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Book Synopsis

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Book SynopsisProvides authoritative exposition, comprehensive survey, and fundamental research exploring the underlying unifying themes in the various areas of application of logic in artificial intelligence and computer science. The book assumes as background some mathematical sophistication.Trade ReviewReview of the first three volumes: `.. an essential acquisition for any library covering theoretical computer science and highly desirable for any researcher in the field.' Times Higher Education SupplementReview of the first three volumes: `... the first three volumes... represent a detailed and comprehensive exposition of the theoretical and computational features of a wide variety of classical and non-classical logics.' `... can be unreservedly recommended to AI practitioners with proficiency in logic and commitment to its role in the development of AI systems.' The Computer JournalTable of ContentsList of contributors ; 1.1 The role(s) of logic in artificial intelligence ; 1.2 First order logic ; 1.3 Methods and calculi for deduction ; 1.4 Deduction systems based on resolution ; 1.5 Equational reasoning and term rewriting systems ; 1.6 Basic modal logic ; 1.7 Logical features of Horn clauses ; Author index ; Subject index

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1 The Axiom of Extension.- 2 The Axiom of Specification.- 3 Unordered Pairs.- 4 Unions and Intersections.- 5 Complements and Powers.- 6 Ordered Pairs.- 7 Relations.- 8 Functions.- 9 Families.- 10 Inverses and Composites.- 11 Numbers.- 12 The Peano Axioms.- 13 Arithmetic.- 14 Order.- 15 The Axiom of Choice.- 16 Zorn's Lemma.- 17 Well Ordering.- 18 Transfinite Recursion.- 19 Ordinal Numbers.- 20 Sets of Ordinal Numbers.- 21 Ordinal Arithmetic.- 22 The Schröder-Bernstein Theorem.- 23 Countable Sets.- 24 Cardinal Arithmetic.- 25 Cardinal Numbers.

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Book Synopsis1 Naive Set Theory.- 1.1 What is a Set?.- 1.2 Operations on Sets.- 1.3 Notation for Sets.- 1.4 Sets of Sets.- 1.5 Relations.- 1.6 Functions.- 1.7 Well-Or der ings and Ordinals.- 1.8 Problems.- 2 The ZermeloFraenkel Axioms.- 2.1 The Language of Set Theory.- 2.2 The Cumulative Hierarchy of Sets.- 2.3 The ZermeloFraenkel Axioms.- 2.4 Classes.- 2.5 Set Theory as an Axiomatic Theory.- 2.6 The Recursion Principle.- 2.7 The Axiom of Choice.- 2.8 Problems.- 3 Ordinal and Cardinal Numbers.- 3.1 Ordinal Numbers.- 3.2 Addition of Ordinals.- 3.3 Multiplication of Ordinals.- 3.4 Sequences of Ordinals.- 3.5 Ordinal Exponentiation.- 3.6 Cardinality, Cardinal Numbers.- 3.7 Arithmetic of Cardinal Numbers.- 3.8 Regular and Singular Cardinals.- 3.9 Cardinal Exponentiation.- 3.10 Inaccessible Cardinals.- 3.11 Problems.- 4 Topics in Pure Set Theory.- 4.1 The Borel Hierarchy.- 4.2 Closed Unbounded Sets.- 4.3 Stationary Sets and Regressive Functions.- 4.4 Trees.- 4.5 Extensions of Lebesgue Measure.- 4.6 A ReTable of ContentsPreface; 1. Naïve Set Theory; 2. The Zermelo-Fraenkel Axioms; 3. Ordinal and Cardinal Numbers; 4. Topics in Pure Set Theory; 5. The Axiom of Constructibility; 6. Independence Proofs in Set Theory; 7. Non-Well-Founded Set Theory; Bibliography; Glossary of Symbols; Index

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Book SynopsisA.- I Introduction.- II Syntax of First-Order Languages.- III Semantics of First-Order Languages.- IV A Sequent Calculus.- V The Completeness Theorem.- VI The Löwenheim-Skolem and the Compactness Theorem.- VII The Scope of First-Order Logic.- VIII Syntactic Interpretations and Normal Forms.- B.- IX Extensions of First-Order Logic.- X Limitations of the Formal Method.- XI Free Models and Logic Programming.- XII An Algebraic Characterization of Elementary Equivalence.- XIII Lindström's Theorems.- References.- Symbol Index.Trade Review“…the book remains my text of choice for this type of material, and I highly recommend it to anyone teaching a first logic course at this level.” – Journal of Symbolic LogicTable of ContentsPreface; Part A: 1. Introduction; 2. Syntax of First-Order Languages; 3. Semantics of first-Order Languages; 4. A Sequent Calculus; 5. The Completeness Theorem; 6. The Lowenheim-Skolem and the Compactness Theorem; 7. The Scope of First-Order Logic; 8. Syntactic Interpretations and Normal Forms; Part B: 9. Extensions of First-Order Logic; 10. Limitations of the Formal Method; 11. Free Models and Logic Programming; 12. An Algebraic Characterization of Elementary Equivalence; 13. Lindstroem's Theorems; References; Symbol Index; Subject Index

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Book SynopsisThe later parts of the book develop a formal theory of computation which integrates major themes of the classical theory and which is more directly applicable to problems in mathematics, numerical analysis, and scientific computing.Table of Contents1 Introduction.- 2 Definitions and First Properties of Computation.- 3 Computation over a Ring.- 4 Decision Problems and Complexity over a Ring.- 5 The Class NP and NP-Complete Problems.- 6 Integer Machines.- 7 Algebraic Settings for the Problem “P ? NP?”.- 8 Newton’s Method.- 9 Fundamental Theorem of Algebra: Complexity Aspects.- 10 Bézout’s Theorem.- 11 Condition Numbers and the Loss of Precision of Linear Equations.- 12 The Condition Number for Nonlinear Problems.- 13 The Condition Number in ?(H(d).- 14 Complexity and the Condition Number.- 15 Linear Programming.- 16 Deterministic Lower Bounds.- 17 Probabilistic Machines.- 18 Parallel Computations.- 19 Some Separations of Complexity Classes.- 20 Weak Machines.- 21 Additive Machines.- 22 Nonuniform Complexity Classes.- 23 Descriptive Complexity.- References.

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Book SynopsisAssumes only a familiarity with algebra at the beginning graduate level; Stresses applications to algebra; Illustrates several of the ways Model Theory can be a useful tool in analyzing classical mathematical structuresTrade ReviewFrom the reviews: MATHEMATICAL REVIEWS "This is an extremely fine graduate level textbook on model theory. There is a careful selection of topics…There is a strong focus on the meaning of model-theoretic concepts in mathematically interesting examples. The exercises touch on a wealth of beautiful topics." "This is an extremely fine graduate level textbook on model theory. There is a careful selection of topics, with a route leading to a substantial treatment of Hrushovski’s proof of the Mordell-Lang conjecture for function fields. … The exercises touch on a wealth of beautiful topics. … There is additional basic background in two appendices (on set theory and on real algebra)." (Dugald Macpherson, Mathematical Reviews, 2003 e) "Model theory is the branch of mathematical logic that examines what it means for a first-order sentence … to be true in a particular structure … . This is a text for graduate students, mainly aimed at those specializing in logic, but also of interest for mathematicians outside logic who want to know what model theory can offer them in their own disciplines. … it is one which makes a good case for model theory as much more than a tool for specialist logicians." (Gerry Leversha, The Mathematical Gazette, Vol. 88 (513), 2004) "The author’s intended audience for this high level introduction to model theory is graduate students contemplating research in model theory, graduate students in logic, and mathematicians who are not logicians but who are in areas where model theory has interesting applications. … The text is noteworthy for its wealth of examples and its desire to bring the student to the point where the frontiers of research are visible. … this book should be on the shelf of anybody with an interest in model theory." (J. M. Plotkin, Zentralblatt Math, Vol. 1003 (03), 2003)Table of ContentsIntroduction * Structures and Theories * Basic Techniques * Algebraic Examples * Realizing and Omitting Types * Indiscernibles * w-stable theoryes * w-stable groups * Geometry of strongly minmal sets * Appendix A: Set Theory * Appendix B: Real Algebra * References * Index

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