Algebra

This book comprises select peer-reviewed articles submitted for the proceedings of the International Conference on Mathematics and Computing (ICMC 2022), held by the School of Advanced Sciences, Vellore Institute of Technology, Vellore, India, in association with Ramanujan Mathematical Society, India, Cryptology Research Society of India and Society for Electronic Transactions and Security, India, from 6-8 January 2022. With an aim to identify the existing challenges in the areas of mathematics and computing, the book emphasizes the importance of establishing new methods and algorithms to address these challenges. The book includes topics on diverse applications of cryptology, network security, cyber security, block chain, IoT, mobile network, data analytics, applied algebra, mathematical analysis, mathematical modelling, fluid dynamics, fractional calculus, multi-optimization, integral equations, dynamical systems, numerical analysis and scientific computing. Divided into five major parts-applied algebra and analysis, fractional calculus and integral equations, mathematical modelling and fluid dynamics, numerical analysis, and computer science and applications-the book is a useful resource for students, researchers and faculty as well as practitioners.

WE DELVE DEEP INTO THE SCIENCE AND INTER-RELATIONSHIPS OF FOUR FAMOUS MATHEMATICAL CONSTANTS: EULER'S NUMBER, e, THE LUDOLPHINE CONSTANT, ( pi ), PYTHAGORAS' CONSTANT AND THE RATIO OF PHIDIAS ( phi ). ALONG THE WAY WE ASSESS THE USEFULNESS OF DIVERSE METHODS OF COMPUTING THESE NUMBERS, TECHNIQUES DATING FROM THE BRONZE AGE TO THE TWENTY-FIRST CENTURY. FROM THE EARLIEST DAYS OF VISIBLE LIFE TO OUR OWN TIMES, TINY ANIMALS HAVE PLOWED AND BURROWED THE DEEP SEA FLOOR IN SYSTEMATIC, GEOMETRICAL PATHS. THEY OFTEN SEEMED TO HAVE CONFORMED TO THE MATHEMATICS OF OUR FAMOUS NUMBERS! CAN THIS REALLY BE TRUE? WE CONSIDER THE EVIDENCE. THIS SECOND EDITION, CORRECTED AND EXTENDED, IS PROFUSELY ILLUSTRATED AND HAS ORIGINAL RESEARCH, ALGEBRAIC DERIVATIONS, FULL ACADEMIC REFERENCES, AND A COMPREHENSIVE INDEX. "FOUR FAMOUS NUMBERS" WILL APPEAL TO ENTHUSIASTS, UNDERGRADUATES AND TEACHERS IN HIGHER EDUCATION.

This book discusses major theories and applications of fuzzy soft multisets and their generalization which help researchers get all the related information at one place. The primary objective of this book is to help bridge the gap to provide a textbook on the theories in fuzzy soft multisets and their applications in real life. It is targeted to researchers and students working in the field of fuzzy set theory, multiset theory, soft set theory and their applications. Uncertainty, vagueness and the representation of imperfect knowledge have been a problem in many fields of research, including artificial intelligence, network and communication, signal processing, machine learning, computer science, information technology, as well as medical science, economics, environments and engineering. There are many mathematical tools for dealing with uncertainties. They include fuzzy set theory, multiset theory, soft set theory and soft multiset theory.

ALGEBRA & TRIGONOMETRY SUPER REVIEWNeed help with algebra and trigonometry? Want a quick review or refresher for class? This is the book for you!CONCISE SUBJECT REVIEWCovers the material students typically learn in an introductory algebra and trigonometry course. Clear, easy-to-understand format makes learning easier.PACKED WITH PRACTICETopic-level questions with detailed explanations let you practice what you've learned and build your algebra and trigonometry skills.TEST WHAT YOU'VE LEARNEDEnd-of-chapter quizzes reinforce key concepts, so you'll be ready for any algebra and trigonometry problem you encounter on your next quiz or test.

""Statics: By Algebraic And Graphic Methods"" is a book written by Lewis Jerome Johnson and originally published in 1903. The book provides an in-depth study of the principles of statics, which is the branch of mechanics that deals with the study of forces and their effects on bodies at rest. The book is divided into two parts. The first part of the book focuses on algebraic methods of solving problems in statics. It covers topics such as the composition and resolution of forces, moments, equilibrium, and the analysis of trusses and frames. The second part of the book focuses on graphic methods of solving problems in statics. It covers topics such as force polygons, equilibrium polygons, and the graphical analysis of trusses and frames.The book is written in a clear and concise manner, making it easy for readers to understand the concepts and principles of statics. It is also filled with numerous examples and problems that help readers to apply the concepts they have learned. Overall, ""Statics: By Algebraic And Graphic Methods"" is a valuable resource for students and professionals in the fields of engineering, physics, and mathematics who are interested in gaining a deeper understanding of the principles of statics.Intended Primarily For Students Of Engineering And Architecture.This scarce antiquarian book is a facsimile reprint of the old original and may contain some imperfections such as library marks and notations. Because we believe this work is culturally important, we have made it available as part of our commitment for protecting, preserving, and promoting the world's literature in affordable, high quality, modern editions, that are true to their original work.

This scarce antiquarian book is a facsimile reprint of the original. Due to its age, it may contain imperfections such as marks, notations, marginalia and flawed pages. Because we believe this work is culturally important, we have made it available as part of our commitment for protecting, preserving, and promoting the world's literature in affordable, high quality, modern editions that are true to the original work.

This textbook provides an introduction to fundamental concepts of algebra at upper undergraduate to graduate level, covering the theory of rings, fields and modules, as well as the representation theory of finite groups.Throughout the book, the exposition relies on universal constructions, making systematic use of quotients and category theory ¿ whose language is introduced in the first chapter. The book is divided into four parts. Parts I and II cover foundations of rings and modules, field theory and generalities on finite group representations, insisting on rings of polynomials and their ideals. Part III culminates in the structure theory of finitely generated modules over Dedekind domains and its applications to abelian groups, linear maps, and foundations of algebraic number theory. Part IV is an extensive study of linear representations of finite groups over fields of characteristic zero, including graded representations and graded characters as well as a final chapter on the Drinfeld¿Lusztig double of a group algebra, appearing for the first time in a textbook at this level.Based on over twenty years of teaching various aspects of algebra, mainly at the École Normale Supérieure (Paris) and at Peking University, the book reflects the audiences of the author's courses. In particular, foundations of abstract algebra, like linear algebra and elementary group theory, are assumed of the reader. Each of the of four parts can be used for a course ¿ with a little ad hoc complement on the language of categories. Thanks to its rich choice of topics, the book can also serve students as a reference throughout their studies, from undergraduate to advanced graduate level.

The geometric construction of the regular 17, 257 and even the 65 537-gon are given in complete details, including programming codes. The theory of geometrical constructions and its connection to Galois theory is treated in detail. Later parts deal with totally positive real numbers as sums of squares, cyclotomic polynomials,Chebychev polynomials, Gaussian periods, Galois theory, and more related topics. The solvability of equations is treated as far as it is related to geometric constructions, and tailored to prove the steps needed is that context, as simple as possible. Much less known is the relation between the lunes of Hippocrates and transcendental numbers to which the final section is dedicated.

This scarce antiquarian book is a selection from Kessinger Publishing's Legacy Reprint Series. Due to its age, it may contain imperfections such as marks, notations, marginalia and flawed pages. Because we believe this work is culturally important, we have made it available as part of our commitment to protecting, preserving, and promoting the world's literature. Kessinger Publishing is the place to find hundreds of thousands of rare and hard-to-find books with something of interest for everyone!

This collection of four short courses looks at group representations, graph spectra, statistical optimality, and symbolic dynamics, highlighting their common roots in linear algebra. Aimed at researchers and beginning Ph.D. students, it includes copious exercises, notes, and references, leading the reader from the basics to high-level applications.

The COMPLETE GUIDE TO MIDDLE SCHOOL MATH Book Grades 6-8:PRE-ALGEBRA is created by American Math Academy to complete middle school mathematics, which includes:¿30 Topics with Detailed Summaries¿30 Challenging Tests¿30 Worksheets¿Total 800+ Practice QuestionsThis book brings together everything you need to know for the Middle school math-PRE-ALGEBRA. It will help you to cover all the math topics.CHAPTER I ARITHMETIC ¿The Number System¿Order of Operations ¿Prime & Composite Numbers ¿Divisibility Rules ¿Least Common Multiple & Greatest Common Factor¿Absolute Value¿Fractions & Operations with Fractions ¿Decimal Numbers ¿Rounding Numbers ¿Laws of Exponents ¿Laws of Radicals ¿Scientific Notation CHAPTER II ALGEBRA ¿ Algebraic Expressions ¿Equations with Two Variables ¿Solving Equations & Inequalities ¿Ratios, Proportional Relations & Variations¿Functions ¿Linear Equations & Slope ¿Unit Rate & Percentages CHAPTER III GEOMETRY ¿Angles ¿Distance & Midpoint ¿Triangles & Type of Triangles ¿Similarity Theorem ¿Pythagorean Theorem ¿Coordinate Plane ¿Area & Perimeter ¿Circles, Circumference, & Area VolumeCHAPTER IV PROBABILITY & STATISTICS ¿Mean, Median, Mode, & Range ¿Probability ¿Challenge Tests Answers Keys

ESSENTIAL PRACTICE WORKBOOK FOR MASTERING ALGEBRA 1 is created by American Math Academy to complete Algebra 1 mathematics, which includes:27 Topics with Detailed Summaries27 Challenging Tests4 Mixed Review Test1 Pretest1 final TestTotal 900+ Practice QuestionsThis book brings together everything you need to know for the ALGEBRA 1. It will help you to cover all the math topics.PretestOrder of OperationsFractions and Operations with FractionsIntegers and Operations with IntegersExponents and Law of ExponentsAbsolute Value and InequalitiesLaws of RadicalsCoordinate PlaneFactors & Multiples (GCF and LCM)Scientific NotationRatio, Proportions and VariationsUnit Rate and PercentMixed Review Test IMixed Review Test IISolving 2-Steps EquationsSolving Equations with Variable in Both SidesProperties of Algebraic Equations and Simplifying EquationsSolving Equations Involving Parallel and Perpendicular LinesSolving Systems of Equations by Substitution & EliminationFactoring Quadratic EquationsSolving Quadratic Equations by Formula and Complete SquareAdding and Subtracting PolynomialsMultiplying and Dividing PolynomialsSolving Equations with Algebraic FractionsSimplifying Rational ExpressionsFunction Notation and Inverse FunctionOperations with FunctionsMean, Median and ModeSlope and Slope Intercept FormDistance and MidpointPythagorean TheoremMixed Review Test IIIMixed Review Test IVFinal Test

Algebra II Essential Practice Workbook for the High School is created by American Math Academy to complete Algebra II Essential Practice Workbook, which includes:31 Topics with Detailed Summaries31 Exercises Test with solutions1 Sample Test1 Final TestOver 450+ Practice Questions with SolutionsThis book brings together everything you need to know for the Algebra II. It will help you to cover all the Algebra II topics

The finite generation theorem is a major achievement of modern algebraic geometry. Based on the minimal model theory, it states that the canonical ring of an algebraic variety defined over a field of characteristic zero is a finitely generated graded ring. This graduate-level text is the first to explain this proof. It covers the progress on the minimal model theory over the last 30 years, culminating in the landmark paper on finite generation by Birkar-Cascini-Hacon-McKernan. Building up to this proof, the author presents important results and techniques that are now part of the standard toolbox of birational geometry, including Mori's bend and break method, vanishing theorems, positivity theorems and Siu's analysis on multiplier ideal sheaves. Assuming only the basics in algebraic geometry, the text keeps prerequisites to a minimum with self-contained explanations of terminology and theorems.

This Special Issue brings together the latest advancements across various facets of viscous and viscoelastic fluid flows. Encompassing a spectrum of contributions, the topics span from innovative numerical methods and sophisticated mathematical modeling to cutting-edge experimental research. In addition to providing insights into the current state of research in these domains, the issue aims to foster a comprehensive understanding of the intricate dynamics and behaviors exhibited by viscous and viscoelastic fluids.

In the realm of complex decision making, characterized by inherent incompleteness and uncertainty, the foundational work of Lotfi A. Zadeh on fuzzy set theory has been instrumental. The efficacy of classical fuzzy sets in addressing vagueness has prompted an exploration of various extensions, each catering to the intricacies of real-world decision-making problems. This reprint delves into an array of advanced fuzzy theories, including type-2 fuzzy sets, hesitant fuzzy sets, multivalued fuzzy sets, cubic sets, intuitionistic fuzzy sets, Pythagorean fuzzy sets, spherical fuzzy sets, neutrosophic sets, and more. The richness of these extensions reflects the dynamism of fuzzy theories in diverse decision-making applications.

Autonomous vehicles are increasingly prevalent, navigating both structured urban roads and challenging offroad scenes. At the core of these vehicles lie the planning and control modules, which are crucial for demonstrating the intelligence inherent in an autonomous driving system. The planning module is responsible for devising an open-loop trajectory, taking into account a variety of environmental restrictions, task-related demands, and vehicle-kinematics-related constraints, while the control module ensures adherence to this trajectory in a closed-loop manner. This adherence is vital in a range of conditions, including diverse weather scenarios, different driving situations, and in response to potential disturbances such as mechanical failures or cyber threats. In certain contexts, these modules are collectively referred to as 'control', with the planning component considered an open-loop controller. This Special Issue focuses on the latest research trends in planning and control methods for autonomous driving. It comprises 11 papers that cover a broad spectrum of applications, including occlusion-aware motion planning in warehouses, control strategies for articulated vehicles, cooperative trajectory planning for autonomous forklifts, and tracking control for underwater vehicles in the face of disturbances and uncertainties. These contributions collectively underscore the diverse and evolving nature of autonomous vehicle technology.

This innovative undergraduate textbook approaches number theory through the lens of abstract algebra. Written in an engaging and whimsical style, this text will introduce students to rings, groups, fields, and other algebraic structures as they discover the key concepts of elementary number theory. Inquiry-based learning (IBL) appears throughout the chapters, allowing students to develop insights for upcoming sections while simultaneously strengthening their understanding of previously covered topics. The text is organized around three core themes: the notion of what a ¿number¿ is, and the premise that it takes familiarity with a large variety of number systems to fully explore number theory; the use of Diophantine equations as catalysts for introducing and developing structural ideas; and the role of abstract algebra in number theory, in particular the extent to which it provides the Fundamental Theorem of Arithmetic for various new number systems. Other aspects of modern number theory ¿ including the study of elliptic curves, the analogs between integer and polynomial arithmetic, p-adic arithmetic, and relationships between the spectra of primes in various rings ¿ are included in smaller but persistent threads woven through chapters and exercise sets.Each chapter concludes with exercises organized in four categories: Calculations and Informal Proofs, Formal Proofs, Computation and Experimentation, and General Number Theory Awareness. IBL ¿Exploration¿ worksheets appear in many sections, some of which involve numerical investigations. To assist students who may not have experience with programming languages, Python worksheets are available on the book¿s website. The final chapter provides five additional IBL explorations that reinforce and expand what students have learned, and can be used as starting points for independent projects. The topics covered in these explorations are public key cryptography, Lagrange¿s four-square theorem, units and Pell¿s Equation, various cases of the solution to Fermat¿s Last Theorem, and a peek into other deeper mysteries of algebraic number theory.Students should have a basic familiarity with complex numbers, matrix algebra, vector spaces, and proof techniques, as well as a spirit of adventure to explore the ¿numberverse.¿