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Summary: The present book contains 14 papers accepted and published in the Special Issue of the MDPI Mathematics journal, entitled "Fuzzy Logic and Soft Computing - Dedicated to the Centenary of the Birth of Lotfi A. Zadeh (1921-2017)", which covers a wide range of topics connected to the theory and applications of fuzzy logic and soft computing. More precisely, these topics include: mathematical programming problems with coefficients and/or decision variables expressed by fuzzy numbers; soft computing methods in a fuzzy environment; fuzzy inner product spaces; M-hazy vector spaces; interval ranges of fuzzy sets; arithmetic operations of fuzzy sets; gradual numbers; Interval-valued fuzzy soft sets; interval-valued fuzzy soft topology; preference relationships; L-fuzzy sub-effect algebras; Fuzzy differential subordinations; m-polar fuzzy set in semigroups; practical use of the marks; the fuzzy logic approach for decision-making systems in management control; multigranulation rough sets; the multigranulation roughness of an intuitionistic fuzzy set; decision-making algorithms, etc. We hope that this book will be useful for those who work in the domains of fuzzy logic and soft computing or for those who want to familiarize themselves with the most advanced knowledge in the field of fuzzy mathematics.

"This new and expanded edition is intended to help candidates prepare for entrance examinations in mathematics and scientific subjects, including STEP (Sixth Term Examination Paper). STEP is an examination used by Cambridge Colleges for conditional offers in mathematics. They are also used by some other UK universities and many mathematics departments recommend that their applicants practice on the past papers even if they do not take the examination. Advanced Problems in Mathematics bridges the gap between school and university mathematics, and prepares students for an undergraduate mathematics course. The questions analysed in this book are all based on past STEP questions and each question is followed by a comment and a full solution. The comments direct the reader’s attention to key points and put the question in its true mathematical context. The solutions point students to the methodology required to address advanced mathematical problems critically and independently. This book is a must read for any student wishing to apply to scientific subjects at university level and for anyone interested in advanced mathematics. "This new and expanded edition is intended to help candidates prepare for entrance examinations in mathematics and scientific subjects, including STEP (Sixth Term Examination Paper). STEP is an examination used by Cambridge Colleges for conditional offers in mathematics. They are also used by some other UK universities and many mathematics departments recommend that their applicants practice on the past papers even if they do not take the examination. Advanced Problems in Mathematics bridges the gap between school and university mathematics, and prepares students for an undergraduate mathematics course. The questions analysed in this book are all based on past STEP questions and each question is followed by a comment and a full solution. The comments direct the reader’s attention to key points and put the question in its true mathematical context. The solutions point students to the methodology required to address advanced mathematical problems critically and independently. This book is a must read for any student wishing to apply to scientific subjects at university level and for anyone interested in advanced mathematics.

Do math more quickly and with more confidence — with less reliance on paper, apps, and calculators. For people who automatically run to the nearest calculator, Idiot's Guides: Speed Math teaches tips, tricks, and straightforward methods to doing math at a fast — and accurate — rate. Practice examples easily illustrate how even the most math-shy person can better perform calculations.

The book deals with Gerbert of Aurillac (Pope Sylvester II) and his investigations in the field of theoretical (philosophy of numbers) and practical (counting) arithmetic. The book represents the comprehensive inquiry of both these aspects of arithmetic in his thought. The analysed sources are Gerbert's so-called scientific letters written to his friends, colleagues or pupils, and also including some of his other texts. On this basis, attention is paid to arithmetic as the mother of all sciences and as the path to wisdom (e.g., the so-called Saltus Gerberti, relation between arithmetic and other disciplines of the quadrivium, etc.) and also to practical arithmetic (e.g., the introduction of Hindu-Arabic numerals and the re-introduction of a new form of abacus to the Latin Christian West).

This volume introduces some recent developments in Arithmetic Geometry over local fields. Its seven chapters are centered around two common themes: the study of Drinfeld modules and non-Archimedean analytic geometry. The notes grew out of lectures held during the research program'Arithmetic and geometry of local and global fields'which took place at the Vietnam Institute of Advanced Study in Mathematics (VIASM) from June to August 2018. The authors, leading experts in the field, have put great effort into making the text as self-contained as possible, introducing the basic tools of the subject. The numerous concrete examples and suggested research problems will enable graduate students and young researchers to quickly reach the frontiers of this fascinating branch of mathematics.

For a long time, all thought there was only one geometry — Euclidean geometry. Nevertheless, in the 19th century, many non-Euclidean geometries were discovered. It took almost two millennia to do this. This was the major mathematical discovery and advancement of the 19th century, which changed understanding of mathematics and the work of mathematicians providing innovative insights and tools for mathematical research and applications of mathematics.A similar event happened in arithmetic in the 20th century. Even longer than with geometry, all thought there was only one conventional arithmetic of natural numbers — the Diophantine arithmetic, in which 2+2=4 and 1+1=2. It is natural to call the conventional arithmetic by the name Diophantine arithmetic due to the important contributions to arithmetic by Diophantus. Nevertheless, in the 20th century, many non-Diophantine arithmetics were discovered, in some of which 2+2=5 or 1+1=3. It took more than two millennia to do this. This discovery has even more implications than the discovery of new geometries because all people use arithmetic.This book provides a detailed exposition of the theory of non-Diophantine arithmetics and its various applications. Reading this book, the reader will see that on the one hand, non-Diophantine arithmetics continue the ancient tradition of operating with numbers while on the other hand, they introduce extremely original and innovative ideas.

This book collects more than thirty contributions in memory of Wolfgang Schwarz, most of which were presented at the seventh International Conference on Elementary and Analytic Number Theory (ELAZ), held July 2014 in Hildesheim, Germany. Ranging from the theory of arithmetical functions to diophantine problems, to analytic aspects of zeta-functions, the various research and survey articles cover the broad interests of the well-known number theorist and cherished colleague Wolfgang Schwarz (1934-2013), who contributed over one hundred articles on number theory, its history and related fields. Readers interested in elementary or analytic number theory and related fields will certainly find many fascinating topical results among the contributions from both respected mathematicians and up-and-coming young researchers. In addition, some biographical articles highlight the life and mathematical works of Wolfgang Schwarz.

This book develops arithmetic without the induction principle, working in theories that are interpretable in Raphael Robinson's theory Q. Certain inductive formulas, the bounded ones, are interpretable in Q. A mathematically strong, but logically very weak, predicative arithmetic is constructed.Originally published in 1986.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

In the introduction to the first volume of The Arithmetic of Elliptic Curves (Springer-Verlag, 1986), I observed that'the theory of elliptic curves is rich, varied, and amazingly vast,'and as a consequence,'many important topics had to be omitted.'I included a brief introduction to ten additional topics as an appendix to the first volume, with the tacit understanding that eventually there might be a second volume containing the details. You are now holding that second volume. it turned out that even those ten topics would not fit Unfortunately, into a single book, so I was forced to make some choices. The following material is covered in this book: I. Elliptic and modular functions for the full modular group. II. Elliptic curves with complex multiplication. III. Elliptic surfaces and specialization theorems. IV. Neron models, Kodaira-Neron classification of special fibers, Tate's algorithm, and Ogg's conductor-discriminant formula. V. Tate's theory of q-curves over p-adic fields. VI. Neron's theory of canonical local height functions.

This Lecture Notes volume is the fruit of two research-level summer schools jointly organized by the GTEM node at Lille University and the team of Galatasaray University (Istanbul):'Geometry and Arithmetic of Moduli Spaces of Coverings (2008)'and'Geometry and Arithmetic around Galois Theory (2009)'. The volume focuses on geometric methods in Galois theory. The choice of the editors is to provide a complete and comprehensive account of modern points of view on Galois theory and related moduli problems, using stacks, gerbes and groupoids. It contains lecture notes on étale fundamental group and fundamental group scheme, and moduli stacks of curves and covers. Research articles complete the collection.​

Chapter 1 - Elementary Arithmetic Chapter 2 - Subtraction and Addition Chapter 3 - Multiplication Chapter 4 - Division Chapter 5 - Binary Numeral System Chapter 6 - Cube (Algebra) and Decimal Chapter 7 - Equality and Finger Binary Chapter 8 - Fraction and Negative Number Chapter 9 - Least Common Multiple and Parity of Zero

Solve your math troubles with DeMYSTiFieD If you cannot tell the difference between your Roman and Arabic numerals, or if when someone asks'what is pi'you say'delicious,'you need Everyday Math DeMYSTiFieD, Second Edition, to unravel these fundamental concepts and theories at your own pace. This practical guide eases you into basic math, starting with counting and simple operations. As you progress, you will master essential concepts such as division, converting decimals into fractions, determining volume, and more. You will learn to measure capital gains and losses as well as apply percentages in the real world. Detailed examples make it easy to understand the material, and end-of- chapter quizzes and a final exam help reinforce key ideas. It's a no-brainer! You'll learn about: Decimals Proportions Prime numbers Surface area Powers of 10 Graphs English vs. metric units Simple enough for a beginner but challenging enough for an advanced student, Everyday Math DeMYSTiFieD, Second Edition, helps you master this essential subject.

Chapter 1 - Introduction to Arithmetic Chapter 2 - Rounding Chapter 3 - Addition and Subtraction Chapter 4 - Multiplication and Division Chapter 5 - Modular Arithmetic and Fundamental Theorem of Arithmetic Chapter 6 - Introduction to Factorization Chapter 7 - Integer Factorization Chapter 8 - Greatest Common Divisor and Singular Value Decomposition Chapter 9 - Eigendecomposition of a Matrix and Cholesky Decomposition Chapter 10 - Quadratic Sieve and Shor's Algorithm

Chapter 1 - Elementary Arithmetic Chapter 2 - Subtraction and Addition Chapter 3 - Multiplication Chapter 4 - Division Chapter 5 - Binary Numeral System Chapter 6 - Cube (Algebra) and Decimal Chapter 7 - Equality and Finger Binary Chapter 8 - Elementary Algebra Chapter 9 - Quadratic Equation Chapter 10 - Linear Equation Chapter 11 - System of Linear Equations Chapter 12 - Polynomial and Simultaneous Equations

Chapter 1 - Introduction to Numbers Chapter 2 - Types of Numbers Chapter 3 - Negative and Non-negative Numbers Chapter 4 - Infinity Chapter 5 - Mathematical Constant Chapter 6 - Transcendental Number and Geometry of Numbers Chapter 7 - Transcendence Theory and Analytic Number Theory

Chapter 1 - Introduction to Arithmetic Chapter 2 - Rounding Chapter 3 - Addition and Subtraction Chapter 4 - Multiplication and Division Chapter 5 - Modular Arithmetic and Fundamental Theorem of Arithmetic Chapter 6 - Arithmetic Function Chapter 7 - Interval Arithmetic

There's an unspoken weak spot in our society most adults can't do basic math. A recent U.S. Department of Education study found that American adults are largely unable to calculate tips, understand their mortgage payments, or even figure out their weekly salary when told their hourly wage. Though we live in a world where numbers, percentages and statistics are tossed at us daily - a world that expects us to make sense of these figures - 87% of American adults are deemed non-proficient at math, according to a recent Education Department study. But there is hope. As the scope of this problem has become more publicized, community colleges have moved in to help, offering courses to help adults gain math proficiency. And programs and books have been created to address the issue, too. Still, most adults need to start their math journey is back at the beginning, and few resources have been developed to address the earliest math skills. Singing Turtle Press, publisher of the award-winning Algebra Survival Guide, is coming to the rescue, though, with a new book that helps adults master foundational math concepts. Math Basics Made Easy, a new offering in eBook format, teaches adults the four fundamental math operations: adding, subtracting, multiplying and dividing. These four skill areas are the foundation stones of mathematics, says co-author Steve Slavin. Because most Americans have never mastered these skills, they haven't gotten very far in math. And that has closed the door to many high-paying careers. Slavin should know about these problems. He has devoted his life to teaching math, and to writing books to help people grasp math. Slavin believes that no matter how far behind a person may be in math, it's never too late to learn math, even if one must go back to the very first grade. Math Basics Made Easy presents early math concepts in a format that works for adults. The book offers brief, easily understood lessons with worked-out sample problems, followed by a collection of practice problems for the reader. Answers are provided to all practice problems, so readers get instant feedback on how well they are progressing. Within each of the four content areas - adding, subtracting, multiplying and dividing - concepts are taught sequentially, easiest concepts first, most challenging topics last. The book is heavy on practice because Slavin believes that we learn math by doing math. The benefit, though, is that anyone who works his way through this book will learn the basics of math and thereby put himself in a position to move on to more challenging topics. In today's challenging economic times, Math Basics Made Easy offers a gateway to better opportunities for today's job-seeker. Slavin's co-author, Ginny Crisonino, also teaches math daily, serving as Mathematics Chair at Union County College, Cranford, NJ. The book also gives anyone who never fully understood math's basics a chance to master them, for the book's message is that it's never too late to learn. A solid self-teaching guide, Math Basics Made Easy offers solid encouragement and competent instruction to anyone with the desire to master math's basic skills.

This volume contains 21 articles written by leading experts in the fields of algebraic and arithmetic geometry. The treated topics range over a variety of themes, including moduli spaces of curves and abelian varieties, algebraic cycles, vector bundles and coherent sheaves, curves over finite fields, and algebraic surfaces, among others. The volume originates from the conference “Geometry and Arithmetic”, which was held on the island of Schiermonnikoog in The Netherlands in September 2010.

Number theory is a branch of mathematics which draws its vitality from a rich historical background. It is also traditionally nourished through interactions with other areas of research, such as algebra, algebraic geometry, topology, complex analysis and harmonic analysis. More recently, it has made a spectacular appearance in the field of theoretical computer science and in questions of communication, cryptography and error-correcting codes. Providing an elementary introduction to the central topics in number theory, this book spans multiple areas of research. The first part corresponds to an advanced undergraduate course. All of the statements given in this part are of course accompanied by their proofs, with perhaps the exception of some results appearing at the end of the chapters. A copious list of exercises, of varying difficulty, are also included here. The second part is of a higher level and is relevant for the first year of graduate school. It contains an introduction to elliptic curves and a chapter entitled “Developments and Open Problems”, which introduces and brings together various themes oriented toward ongoing mathematical research. Given the multifaceted nature of number theory, the primary aims of this book are to: - provide an overview of the various forms of mathematics useful for studying numbers - demonstrate the necessity of deep and classical themes such as Gauss sums - highlight the role that arithmetic plays in modern applied mathematics - include recent proofs such as the polynomial primality algorithm - approach subjects of contemporary research such as elliptic curves - illustrate the beauty of arithmetic The prerequisites for this text are undergraduate level algebra and a little topology of Rn. It will be of use to undergraduates, graduates and phd students, and may also appeal to professional mathematicians as a reference text.

Is this the right book for me?Train your brain to be quicker, sharper and more acute by challenging yourself with these puzzles and games. This book does much more than give you the skills to tackle maths with confidence - instead it shows you how, by learning to solve practical problems and perfecting your mental arithmetic, you can strengthen all your key thinking skills and astonish your friends and family. This is the ultimate mental workout - and the only one to show you how these fun and diverting number games will actually make you smarter, quicker and more acute than any of your peers.Number Training Your Brain includes:Part 1 - Thinking about numbersChapter 1: Mathematical minds: the story so farChapter 2: Adding skill, subtracting stressChapter 3: Multiply your mind power: divide and conquerChapter 4: Bits and piecesPart two - Thinking about everythingChapter 5: Shape up your thinkingChapter 6: Measurable improvementsChapter 7: Chances to shineChapter 8: The formula for successChapter 9: Think like a mathematicianChapter 10: Big maths, big challengesLearn effortlessly with a new easy-to-read page design and interactive features: Not got much time?One, five and ten-minute introductions to key principles to get you started.Author insightsLots of instant help with common problems and quick tips for success, based on the author's many years of experience.Test yourselfTests in the book and online to keep track of your progress.Extend your knowledgeExtra online articles to give you a richer understanding of the subject.Five things to rememberQuick refreshers to help you remember the key facts.Try thisInnovative exercises illustrate what you've learnt and how to use it.

Jakow Trachtenberg believed that everyone is born with phenomenal abilities to calculate, he devised a set of rules that allows every child to make multiplication, division, addition, subtraction and square-root calculations with unerring accuracy and at remarkable speed.

Algebra, Arithmetic, and Geometry: In Honor of Yu. I. Manin consists of invited expository and research articles on new developments arising from Manin's outstanding contributions to mathematics.

The theory of elliptic curves is distinguished by its long history and by the diversity of the methods that have been used in its study. This book treats the arithmetic theory of elliptic curves in its modern formulation, through the use of basic algebraic number theory and algebraic geometry. The book begins with a brief discussion of the necessary algebro-geometric results, and proceeds with an exposition of the geometry of elliptic curves, the formal group of an elliptic curve, and elliptic curves over finite fields, the complex numbers, local fields, and global fields. Included are proofs of the Mordell–Weil theorem giving finite generation of the group of rational points and Siegel's theorem on finiteness of integral points. For this second edition of The Arithmetic of Elliptic Curves, there is a new chapter entitled Algorithmic Aspects of Elliptic Curves, with an emphasis on algorithms over finite fields which have cryptographic applications. These include Lenstra's factorization algorithm, Schoof's point counting algorithm, Miller's algorithm to compute the Tate and Weil pairings, and a description of aspects of elliptic curve cryptography. There is also a new section on Szpiro's conjecture and ABC, as well as expanded and updated accounts of recent developments and numerous new exercises. The book contains three appendices: Elliptic Curves in Characteristics 2 and 3, Group Cohomology, and a third appendix giving an overview of more advanced topics.

Visual thinking - visual imagination or perception of diagrams and symbol arrays, and mental operations on them - is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? By examining the many kinds of visual representation in mathematics and the diverse ways in which they are used, Marcus Giaquinto argues that visual thinking in mathematics is rarely just a superfluous aid; it usually has epistemological value, often as a means of discovery. Drawing from philosophical work on the nature of concepts and from empirical studies of visual perception, mental imagery, and numerical cognition, Giaquinto explores a major source of our grasp of mathematics, using examples from basic geometry, arithmetic, algebra, and real analysis. He shows how we can discern abstract general truths by means of specific images, how synthetic a priori knowledge is possible, and how visual means can help us grasp abstract structures. Visual Thinking in Mathematics reopens the investigation of earlier thinkers from Plato to Kant into the nature and epistemology of an individual's basic mathematical beliefs and abilities, in the new light shed by the maturing cognitive sciences. Clear and concise throughout, it will appeal to scholars and students of philosophy, mathematics, and psychology, as well as anyone with an interest in mathematical thinking.

Master math at your own pace!Does working with numbers often frustrate you? Do you need to brush up on your basic math skills? Do you feel math stands between you and your career goals, or a better grade at school?Quick Arithmetic, Third Edition is the quickest and easiest way to teach yourself the basic math skills you need to advance on the job or in school. Using cartoons and a clear writing style, this practical guide provides a fresh start for learning or reviewing how to work with whole numbers, fractions, decimals, and percentages. The book's proven self-teaching approach allows you to work at your own pace and learn only the material you need. Previews and objectives at the beginning of each section help you determine your particular needs, while self-tests, practice problems, and a final exam let you measure your progress and reinforce what you've learned.For anyone who has ever felt intimidated by a page of numbers, Quick Arithmetic, Third Edition has the answers!

Fundamental Concepts of Mathematics, 2nd Edition provides an account of some basic concepts in modern mathematics. The book is primarily intended for mathematics teachers and lay people who wants to improve their skills in mathematics. Among the concepts and problems presented in the book include the determination of which integral polynomials have integral solutions; sentence logic and informal set theory; and why four colors is enough to color a map. Unlike in the first edition, the second edition provides detailed solutions to exercises contained in the text. Mathematics teachers and people who want to gain a thorough understanding of the fundamental concepts of mathematics will find this book a good reference.