Why Transfinite Set-Theory is Wrong and an Alternative Theory of Numeric Structures
Set Theory is a formalization of the existence and fundamental properties of mathematical objects as collections of elements and/or elements included in collections. Its formulation is so basic and comprehensive that it has been postulated as the foundation of all mathematics. Perhaps, the major achievement of Set Theory is that, after being criticized by many reputable mathematicians and philosophers since its appearance, it is now commonly accepted as the primary explanation of the most basic components of mathematics: numbers; and not only the numbers we have needed or we may ever need but all the numbers that could potentially exist. In Set Theory, an infinite sequence of numbers exists not as the mere projection of a construction algorithm but as a complete and self-identical mathematical object: a set. In Set Theory the words infinite and infinity do not refer to the property of growing endlessly (potential infinity) but to a definite magnitude; a number; the actual infinity. As a result of such a conception, Set Theory arrives at the conclusion that there exist infinitely many infinities, each one with a different value. The set of postulates, proofs and theorems used to justify the existence of such infinities is commonly known as Transfinite Set-Theory. The first part of this work shows how some of the properties and theorems applied to infinite sets, in Set Theory, necessarily lead to internal and fundamental contradictions under classical logic, even when the idea of actual infinity is accepted. Throughout the second part, motivated by the necessity of an alternative to Transfinite Set-Theory, due to the incapacity of such a theory to explain some of the findings shown in the first part (especially the proof of the existence of as many rational as irrational numbers), the author develops a theory to provide a better understanding of infinite sequences of numbers.
Methods for the Summation of Series
This book presents methods for the summation of infinite and finite series and the related identities and inversion relations. The summation includes the column sums and row sums of lower triangular matrices. The convergence of the summation of infinite series is considered.
Introductory Banach Space Operators
This book provides a concise introduction to the analysis of Banach space operators within the framework of Hilbert space theory. The guiding notion in this approach is that of operator properties. At the same time the notion of function of an operator is emphasized. The formal aspects of these concepts are explained in all chapters. Each chapter consists of varying sections and exercises at the end of each chapter.
Introduction to Iso- Euclidean Geometry
The main purpose in this book is to represent some recent researches of Santilli iso-mathematics in the area of the plane geometry. This book is devoted to the iso-plane geometry. It summarizes the most recent contributions in this area. The book is intended for senior undergraduate students and beginning graduate students of engineering and science courses. The book contains five chapters. The chapters in the book are pedagogically organized. Each chapter concludes with a section with practical problems. In Chapter 1 we introduce iso-real numbers with one and several iso-units. They are defined the basic operations with them and they are deducted some of their basic properties. In the chapter they are defined iso-matrices, iso-determinants and iso-trigonometric functions. Chapter 2 deals with straight iso-lines. It is defined iso-angle between two iso-vectors. They are introduced iso-lines and they are deducted the main equations of iso-lines. They are given criteria for iso-perpendicularity and iso-parallel of iso-lines. In Chapter 3 we introduce iso-motions: iso-reflections, iso-rotations, iso-translations and iso-glide iso-reflections. Chapter 4 is devoted on iso-circles. They are given the iso- parametric iso-representations of the iso-circles. In Chapter 5 they are introduced iso-parabolas, iso-ellipses and iso-hyperbolas. The aim of this book is to present a clear and well-organized treatment of the concept behind the development of iso-mathematics as well as solution techniques. The text material of this book is presented in a readable and mathematically solid format.
A Compendium of Musical Mathematics
The purpose of this book is to provide a concise introduction to the mathematical theory of music, opening each chapter to the most recent research. Despite the complexity of some sections, the book can be read by a large audience. Many examples illustrate the concepts introduced. The book is divided into 9 chapters.In the first chapter, we tackle the question of the classification of chords and scales. Chapter 2 is a mathematical presentation of David Lewin's Generalized Interval Systems. Chapter 3 offers a new theory of diatonicity in equal-tempered universes. Chapter 4 presents the Neo-Riemannian theories based on the work of David Lewin, Richard Cohn and Henry Klumpenhouwer. Chapter 5 is devoted to the application of word combinatorics to music. Chapter 6 studies the rhythmic canons and the tessellation of the line. Chapter 7 is devoted to serial knots. Chapter 8 presents combinatorial designs and their applications to music. The last chapter, chapter 9, is dedicated to the study of tuning systems.
Discrete-Valued Time Series
The analysis and modeling of time series has been an active research area for more than 100 years, with the main focus on time series having a continuous range consisting of real numbers or real vectors. It took until the 1980s for the first papers on discrete-valued time series to appear. In the 2000s, a rapid increase in research activity was noted, but only in the last few years was a certain maturity and consolidation of the area of discrete-valued time series observed. This reprint is a collection of articles on a wide range of topics on discrete-valued time series (especially count time series), covering stochastic models and methods for their analysis, univariate and multivariate time series, applications of time series methods to risk analysis, statistical process control, and many more. The proposed approaches and concepts are thoroughly discussed and illustrated with several real-world data examples.
Essays on Coding Theory
Critical coding techniques have developed over the past few decades for data storage, retrieval and transmission systems, significantly mitigating costs for governments and corporations that maintain server systems containing large amounts of data. This book surveys the basic ideas of these coding techniques, which tend not to be covered in the graduate curricula, including pointers to further reading. Written in an informal style, it avoids detailed coverage of proofs, making it an ideal refresher or brief introduction for students and researchers in academia and industry who may not have the time to commit to understanding them deeply. Topics covered include fountain codes designed for large file downloads; LDPC and polar codes for error correction; network, rank metric, and subspace codes for the transmission of data through networks; post-quantum computing; and quantum error correction. Readers are assumed to have taken basic courses on algebraic coding and information theory.
Volatility Estimation
These notes have been written with the precisely purpose of summarizing the more often encountered and implemented volatility estimation techniques, to describe the realized volatility surface and its term structure, for example in developing Option Pricing libraries. The common and accepted assumptions behind the random fashion, that each quoted and traded asset follows, there are stochastic differential equations (SDEs) characterized by two main terms: one is the drift and the other one is the diffusion term or volatility. If the drift term is set uniquely by the definition of the martingale measure, imposing the drift's value under such risk neutral measure, to be equal to the free interest rate; on the other side, the diffusion term or volatility is not estimated or defined uniquely. Indeed, the latter is estimated involving several different approaches, that over the time have been developed, trying to catch a better fit with the observed options' quotation.
Polytopes and Graphs
This book introduces convex polytopes and their graphs, alongside the results and methodologies required to study them. It guides the reader from the basics to current research, presenting many open problems to facilitate the transition. The book includes results not previously found in other books, such as: the edge connectivity and linkedness of graphs of polytopes; the characterisation of their cycle space; the Minkowski decomposition of polytopes from the perspective of geometric graphs; Lei Xue's recent lower bound theorem on the number of faces of polytopes with a small number of vertices; and Gil Kalai's rigidity proof of the lower bound theorem for simplicial polytopes. This accessible introduction covers prerequisites from linear algebra, graph theory, and polytope theory. Each chapter concludes with exercises of varying difficulty, designed to help the reader engage with new concepts. These features make the book ideal for students and researchers new to the field.
Understanding the Language of Mathematics
Alexander Firestone always wanted to be a teacher but felt that in order to know what was important to teach, he should be out in the real world to see what he was able to do with his present education. Upon graduation from the University, he secured a position as a Research Physicist working on new types of rocket propulsion for deep space exploration. In the first week, he realized that his present education ill-equipped him as a problem-solver working on new ideas. This was the beginning of What, How, and Why. After successfully working on the projects he was assigned, he realized he was ready for teaching. Over the last 50 years he has used his teaching and classroom experiences as a laboratory, developing What, How, and Why learning.I still get telephone calls this very evening (student from Westmount College in Christchurch) former students wanting to know how I'm doing and sharing their classroom experiences with me as a teacher. That was nearly 40 years ago. I'm a very passionate teacher who has taught for over 40 years and still teaches casually full-time. I am probably the oldest Mathematics teacher in Australia who is a passionate Mathematics teacher and is still able to teach full time. My teaching positions include classroom teacher, HOD mathematics, Principal, University lecturer in China. I have over 8 years of part-time experience doing post-graduate university study on What, How, and Why. Three in China and five at Griffith University in Queensland. I have presented papers and given Talks at International Education Conferences in Australia, and New Zealand.
The Daring Invention of Logarithm Tables
In the early 17th century, both Jost B羹rgi and John Napier dared to invent a logarithm table whose construction required tens of thousands of computing steps. These tables reduced computing effort for multiplication and division by an order of magnitude. Indeed, their invention launched a computing revolution that continues to this day. The book, which is the color edition of the original black and white edition published in 2020, tells the story of B羹rgi's and Napier's work, and how Henry Briggs built on Napier's idea, creating a table of logarithms that was easier to use. John Napier and Henry Briggs described their methods in detail; distribution of their results was widespread. In contrast, Jost B羹rgi did not leave detailed records of his work. Just a few copies of his table and terse handwritten instructions for its use have survived. To fill this gap, the book reconstructs B羹rgi's thinking leading up to his table. The reader looks over his shoulder, so to speak, and learns how B羹rgi came upon the idea, how he decided on the specific format of the table, and how his instructions should be interpreted. And so the reader experiences the magic of the invention of logarithms. The final chapters examine the question "Who invented logarithms?". For centuries, few people were aware of B羹rgi's work; John Napier was considered to be the sole inventor. This changed at the middle of the 19th century when Jost B羹rgi's work became more widely known. Since then there has been extensive debate whether B羹rgi should be considered an independent co-inventor. Careful parsing of the history of logarithm going back to Archimedes of antiquity then reveals that, without doubt, John Napier and Jost B羹rgi are independent co-inventors of logarithms.
Harmonic Numbers and Open problems in Series
Problems involving harmonic numbers are so strange in solution way and those approaches or results are attracting the attention of students interested in mathematics as well as mathematicians and engineers, as there are many areas of application. In scientific and technological calculations that occur in various fields of mathematics and engineering, computational problems associated with various mathematical constants are often encountered, some of which are important tools for solving scientific and technological problems. There have been many open problems that have not been addressed before, due to the high level of scientific theory and the high performance of computer-based computing tools. Nevertheless, the world of mathematics still has a lot of problems to be discovered, so new and diverse methods are needed to solve these problems. This book contains open problems and challenging problems presented in several mathematical reference books and add their new generalizations written by us.
Primes and Particles
Many philosophers, physicists, and mathematicians have wondered about the remarkable relationship between mathematics with its abstract, pure, independent structures on one side, and the wilderness of natural phenomena on the other. Famously, Wigner found the "effectiveness" of mathematics in defining and supporting physical theories to be unreasonable, for how incredibly well it worked. Why, in fact, should these mathematical structures be so well-fitting, and even heuristic in the scientific exploration and discovery of nature? This book argues that the effectiveness of mathematics in physics is reasonable. The author builds on useful analogies of prime numbers and elementary particles, elementary structure kinship and the structure of systems of particles, spectra and symmetries, and for example, mathematical limits and physical situations. The two-dimensional Ising model of a permanent magnet and the proofs of the stability of everyday matter exemplify such effectiveness, and the power of rigorous mathematical physics. Newton is our original model, with Galileo earlier suggesting that mathematics is the language of Nature.
Supportive Foundation for High School Mathematics
This book is written to help students acquire a solid foundation in high school mathematics and enable them to develop problem-solving skills. Considerable effort has been devoted to ensure that the text is understandable and covers areas where learners have common difficulty. Numerous carefully selected examples are provided with detailed solutions to illustrate the mathematical concepts and enhance students understanding of the subject matter. The book has five chapters: (1) Relations and graphs of equations; (2) Functions with emphasis on linear, quadratic, and polynomial functions; (3) Exponential and logarithmic functions; (4) Topics of geometry that are basic to high school; (5) Trigonometric functions, trigonometric equations, and applications involving triangles.Undoubtedly, students who work through this manuscript will find that the book presents a comprehensive foundation to successfully understand high school mathematics. The other book of the author, namely, "Supportive Foundation for Basic Algebra" provides a comprehensive approach to the fundamental concepts and techniques of algebra and offers a useful supplementary background for enhanced mathematical competence.
Data and Process Visualisation for Graphic Communication
This book guides the reader through the process of graphic communication with a particular focus on representing data and processes. It considers a variety of common graphic communication scenarios among those that arise most frequently in practical applications. The book is organized in two parts: representing data (Part I) and representing processes (Part II). The first part deals with the graphical representation of data. It starts with an introductory chapter on the types of variables, then guides the reader through the most common data visualization scenarios - i.e.: representing magnitudes, proportions, one variable as a function of the other, groups, relations, bivariate, trivariate and geospatial data. The second part covers various tools for the visual representation of processes; these include timelines, flow-charts, Gantt charts and PERT diagrams. In addition, the book also features four appendices which cover cross-chapter topics: mathematics and statistics review, Matplotlib primer, color representation and usage, and representation of geospatial data. Aimed at junior and senior undergraduate students in various technical, scientific, and economic fields, this book is also a valuable aid for researchers and practitioners in data science, marketing, entertainment, media and other fields.
Calculus for Beginners
Foundations of Calculus: A Beginner's Comprehensive GuideCalculus for Beginners stands out as a pioneering educational tool, designed to demystify the complexities of calculus for novices. This book is crafted to cater to the needs of beginners, providing a clear and thorough foundation in calculus concepts, principles, and applications. Its unique structure and comprehensive coverage make it an indispensable resource for anyone looking to grasp the fundamentals of calculus.Empowering Learners with a Multifaceted ApproachCentral to this guide is its multifaceted approach to teaching calculus, which addresses the subject's inherent complexity and variations through a meticulously designed curriculum. Each chapter unfolds logically, guiding learners from basic principles to more intricate concepts. This pedagogical strategy is augmented by a wealth of resources aimed at reinforcing understanding and facilitating practical application.Highlights of the Book: Interactive Learning Experience: For each topic covered, the book includes a QR code and a dedicated link to an accompanying online course. This feature provides learners with instant access to detailed lessons, examples, exercises, video lessons, and worksheets, creating a rich, interactive learning environment.Comprehensive Curriculum: The book covers all fundamental aspects of calculus, including limits, derivatives, integrals, and differential equations, ensuring a solid foundation in each area. The content is presented in a clear, understandable language, making complex concepts accessible to beginners.Practical Application and Examples: Real-world applications and examples are woven throughout the text, illustrating how calculus principles are applied in various fields. This approach not only enhances comprehension but also demonstrates the relevance of calculus in solving practical problems.Enhanced Learning Tools: Each section is supplemented with exercises and worksheets, allowing learners to practice and apply what they have learned. Video lessons offer visual explanations of complex topics, catering to different learning styles.Accessible Solutions: A complete set of solutions for all exercises and questions is provided, enabling learners to check their work and understand the rationale behind each answer. This feedback mechanism is crucial for self-assessment and improvement. Calculus for Beginners is more than just a textbook; it is a comprehensive learning system that integrates traditional teaching methods with innovative digital resources. By offering a blend of written content, interactive online components, and practical exercises, this book ensures that learners not only understand calculus concepts but also know how to apply them in real-world contexts. Whether you are a student, a professional seeking to refresh your knowledge, or a curious mind venturing into the world of calculus for the first time, this guide offers the tools and guidance necessary to master calculus with confidence and ease. Ideal for self-study and classroom usage!
The Logic of Partitions
This book is an introduction to the logic of partitions on a set as well as the (quantum) logic of partitions (direct-sum decompositions or DSDs) on a vector space. Partitions of a set are categorically dual to subsets of a set. Thus the logic of partitions is, in that sense, the dual to the Boolean logic of subsets (usually presented as the special case of propositional logic).Since partitions can be seen as the inverse image partitions of random variables or numerical attributes, partition logic is the logic of random variables or numerical attributes (abstracted from the actual values). On the lattice of partitions of an arbitrary unstructured set, there is a rich algebraic structure of dual operations of implication and co-implication - resembling a non-distributive version of Heyting and co-Heyting algebras.
Four Famous Numbers
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.
Strategic Applications of Measurement Technologies and Instrumentation
Measurement techniques form the basis of scientific, engineering, and industrial innovations. The methods and instruments of measurement for different fields are constantly improving, and it's necessary to address not only their significance but also the challenges and issues associated with them. Strategic Applications of Measurement Technologies and Instrumentation is a collection of innovative research on the methods and applications of measurement techniques in medical and scientific discoveries, as well as modern industrial applications. The book is divided into two sections with the first focusing on the significance of measurement strategies in physics and biomedical applications and the second examining measurement strategies in industrial applications. Highlighting a range of topics including material assessment, measurement strategies, and nanoscale materials, this book is ideally designed for engineers, academicians, researchers, scientists, software developers, graduate students, and industry professionals.
Nonlinear Control Systems with Recent Advances and Applications
In the rapidly evolving landscape of nonlinear control systems, this reprint stands as a beacon of knowledge, showcasing the remarkable progress made over the last few decades. With a focus on design methodologies and their applications, this text employs various mathematical tools to address the myriad challenges inherent in nonlinearly controlled systems.This reprint extends its reach beyond traditional boundaries, presenting applications of nonlinear control across diverse fields such as energy, health care, robotics, biology, and big data research. As technology continues to advance, nonlinear control emerges as a critical player in shaping the future of theory and technology adoption across these domains.Despite the wealth of the existing literature, synthesizing control strategies for a broader class of nonlinear systems, especially those integrated with emerging technologies in communication and computation, remains a formidable task. This reprint addresses this gap, providing a cutting-edge collection of articles that push the boundaries of both theoretical background and practical applications. With its emphasis on novel developments and the broader class of applications, this reprint opens doors to new possibilities, making it a must-read for anyone seeking to navigate the intricate challenges of nonlinear control systems in the 21st century.
Comparative Perspectives on Inquiry-Based Science Education
The core practice of professional scientists is inquiry, often referred to as research. If educators are to prepare students for a role in the professional scientific and technological community, exposing them to inquiry-based learning is essential. Despite this, inquiry-based teaching and learning (IBTL) remains relatively rare, possibly due to barriers that teachers face in deploying it or to a lack of belief in the teaching community that inquiry-based learning is effective. Comparative Perspectives on Inquiry-Based Science Education examines stories and experiences from members of an international science education project that delivered learning resources based around guided inquiry for students to a wide range of schools in 12 different countries in order to identify key themes that can provide useful insights for student learning, teacher support, and policy formulation at the continental level. The book provides case studies across these 12 different settings that enable readers to compare and contrast both practice and policy issues with their own contexts while accessing a cutting-edge model of professional development. It is designed for educators, instructional designers, administrators, principals, researchers, policymakers, practitioners, and students seeking current and relevant research on international education and education strategies for science courses.
Mathematical Logic
Presents an introduction to of formal mathematical logic and set theoryPresents simple yet nontrivial results in modern model theoryProvides introductory remarks to all results, including a historical background
Differential Equations
The book concerns with solving about 650 ordinary and partial differential equations. Each equation has at least one solution and each solution has at least one coloured graph. The coloured graphs reveal different features of the solutions. Some graphs are dynamical as for Clairaut differential equations. Thus, one can study the general and the singular solutions. All the equations are solved by Mathematica. The first chapter contains mathematical notions and results that are used later through the book. Thus, the book is self-contained that is an advantage for the reader. The ordinary differential equations are treated in Chapters 2 to 4, while the partial differential equations are discussed in Chapters 5 to 10. The book is useful for undergraduate and graduate students, for researchers in engineering, physics, chemistry, and others. Chapter 9 treats parabolic partial differential equations while Chapter 10 treats third and higher order nonlinear partial differential equations, both with modern methods. Chapter 10 discusses the Korteweg-de Vries, Dodd-Bullough-Mikhailov, Tzitzeica-Dodd-Bullough, Benjamin, Kadomtsev-Petviashvili, Sawada-Kotera, and Kaup-Kupershmidt equations.
C∞-Algebraic Geometry with Corners
Schemes in algebraic geometry can have singular points, whereas differential geometers typically focus on manifolds which are nonsingular. However, there is a class of schemes, 'C∞-schemes', which allow differential geometers to study a huge range of singular spaces, including 'infinitesimals' and infinite-dimensional spaces. These are applied in synthetic differential geometry, and derived differential geometry, the study of 'derived manifolds'. Differential geometers also study manifolds with corners. The cube is a 3-dimensional manifold with corners, with boundary the six square faces. This book introduces 'C∞-schemes with corners', singular spaces in differential geometry with good notions of boundary and corners. They can be used to define 'derived manifolds with corners' and 'derived orbifolds with corners'. These have applications to major areas of symplectic geometry involving moduli spaces of J-holomorphic curves. This work will be a welcome source of information and inspiration for graduate students and researchers working in differential or algebraic geometry.
Analytic Combinatorics in Several Variables
Discrete structures model a vast array of objects ranging from DNA sequences to internet networks. The theory of generating functions provides an algebraic framework for discrete structures to be enumerated using mathematical tools. This book is the result of 25 years of work developing analytic machinery to recover asymptotics of multivariate sequences from their generating functions, using multivariate methods that rely on a combination of analytic, algebraic, and topological tools. The resulting theory of analytic combinatorics in several variables is put to use in diverse applications from mathematics, combinatorics, computer science, and the natural sciences. This new edition is even more accessible to graduate students, with many more exercises, computational examples with Sage worksheets to illustrate the main results, updated background material, additional illustrations, and a new chapter providing a conceptual overview.
A Course in Combinatorics and Graphs
This compact textbook consists of lecture notes given as a fourth-year undergraduate course of the mathematics degree at the Universitat Polit癡cnica de Catalunya, including topics in enumerative combinatorics, finite geometry, and graph theory. This text covers a single-semester course and is aimed at advanced undergraduates and masters-level students. Each chapter is intended to be covered in 6-8 hours of classes, which includes time to solve the exercises. The text is also ideally suited for independent study. Some hints are given to help solve the exercises and if the exercise has a numerical solution, then this is given. The material covered allows the reader with a rudimentary knowledge of discrete mathematics to acquire an advanced level on all aspects of combinatorics, from enumeration, through finite geometries to graph theory. The intended audience of this book assumes a mathematical background of third-year students in mathematics, allowing for a swifter useof mathematical tools in analysis, algebra, and other topics, as these tools are routinely incorporated in contemporary combinatorics. Some chapters take on more modern approaches such as Chapters 1, 2, and 9. The authors have also taken particular care in looking for clear concise proofs of well-known results matching the mathematical maturity of the intended audience.
From Computational Logic to Computational Biology
Alfredo Ferro's impact on information technology has traversed diverse domains, encompassing Computational Logic, Data Mining, Bioinformatics, and Complex Systems. After first studying Mathematics at the University of Catania, he received a Ph.D. in Computer Science from NYU in 1981, working under the supervision of Jacob Theodor (Jack) Schwartz. He returned to the University of Catania where he established the Computer Science undergraduate program, served as the coordinator of the Ph.D. program in Computer Science, cofounded the Ph.D. program in Biology, Human Genetics, and Bioinformatics, and retired as a full professor in 2021.Alfredo's academic career as a computer scientist is characterized by two distinct research phases: Computational Logic until approximately 1995, followed by a notable focus on Data Mining and Bioinformatics. The contributions in this volume reflect the quality and the scope of his personal and collaborative successes.He also taught andinspired many excellent scientists. A pioneering initiative was to establish summer schools for Ph.D. students in 1989, leading to the so-called Lipari School, now the J.T. Schwartz International School for Scientific Research, where Alfredo continues to serve as director. This prestigious series includes schools focused on Computer Science, Complex Systems, and Computational Biology, featuring world-class scientists as lecturers and mentors.
Advance Numerical Techniques to Solve Linear and Nonlinear Differential Equations
Real-world issues can be translated into the language and concepts of mathematics with the use of mathematical models. This book provides these real-world examples, explores research challenges in numerical treatment, and demonstrates how to create new numerical methods for resolving problems.
Engaging Young Students in Mathematics through Competitions - World Perspectives and Practices
Engaging Young Students in Mathematics through Competitions presents a wide range of topics relating to mathematics competitions and their meaning in the world of mathematical research, teaching and entertainment. Following the earlier two volumes, contributors explore a wide variety of fascinating problems of the type often presented at mathematics competitions. In this new third volume, many chapters are directly related to the challenges involved in organizing competitions under Covid-19, including many positive aspects resulting from the transition to online formats. There are also sections devoted to background information on connections between the mathematics of competitions and their organization, as well as the competitions' interplay with research, teaching and more.The various chapters are written by an international group of authors involved in problem development, many of whom were participants of the 9th Congress of the World Federation of National Mathematics Competitions in Bulgaria in 2022. Together, they provide a deep sense of the issues involved in creating such problems for competition mathematics and recreational mathematics.
Local Mathematics for Local Physics
The language of the universe is mathematics, but how exactly do you know that all parts of the universe 'speak' the same language? Benioff builds on the idea that the entity that gives substance to both mathematics and physics is the fundamental field, called the 'value field'. While exploring this idea, he notices the similarities that the value field shares with several mysterious phenomena in modern physics: the Higgs field, and dark energy.The author first introduces the concept of the value field and uses it to reformulate the basic framework of number theory, calculus, and vector spaces and bundles. The book moves on to find applications to classical field theory, quantum mechanics and gauge theory. The last two chapters address the relationship between theory and experiment, and the possible physical consequences of both the existence and non-existence of the value field. The book is open-ended, and the list of open questions is certainly longer than the set of proposed answers.Paul Benioff, a pioneer in the field of quantum computing and the author of the first quantum-mechanical description of the Turing machine, devoted the last few years of his life to developing a universal description in which mathematics and physics would be on equal footing. He died on March 29, 2022, his work nearly finished. The final editing was undertaken by Marek Czachor who, in the editorial afterword, attempts to place the author's work in the context of a shift in the scientific paradigm looming on the horizon.
Sharpening Everyday Mental/Thinking Skills Through Mathematics Problem Solving and Beyond
Mathematics is a subject taught from kindergarten through to high school, and yet it is the one subject that most adults are almost proud to admit to not having been very good at, and, therefore, tend to avoid it where they can. However, one of the key factors in mathematics is its ability to enable us to solve everyday problems. When we consider 'the worst-case scenario' of the situation, it is analogous to solving a mathematical problem by considering extremes. Or, we might consider the best path to take from point A to point B, where geometric relationships can be helpful. This book is intended to demonstrate a variety of neglected aspects of mathematics, in order to demonstrate the power and beauty of the field of mathematics beyond where most people, students, and teachers believe is possible.The chapters of the book explore a multitude of topics: unusual arithmetic calculations and shortcuts, entertaining and instructional problem-solving strategies, unusual applications of algebra, and how geometry allows us to better appreciate physical relationships. Only a basic mathematical knowledge is needed to understand these topics and problems; however, the book also demonstrates that, armed with even this level of understanding, our mathematical skills far exceed what we learned at school! The final chapter is the most challenging, and explores a curious problem-solving technique.
Sharpening Everyday Mental/Thinking Skills Through Mathematics Problem Solving and Beyond
Mathematics is a subject taught from kindergarten through to high school, and yet it is the one subject that most adults are almost proud to admit to not having been very good at, and, therefore, tend to avoid it where they can. However, one of the key factors in mathematics is its ability to enable us to solve everyday problems. When we consider 'the worst-case scenario' of the situation, it is analogous to solving a mathematical problem by considering extremes. Or, we might consider the best path to take from point A to point B, where geometric relationships can be helpful. This book is intended to demonstrate a variety of neglected aspects of mathematics, in order to demonstrate the power and beauty of the field of mathematics beyond where most people, students, and teachers believe is possible.The chapters of the book explore a multitude of topics: unusual arithmetic calculations and shortcuts, entertaining and instructional problem-solving strategies, unusual applications of algebra, and how geometry allows us to better appreciate physical relationships. Only a basic mathematical knowledge is needed to understand these topics and problems; however, the book also demonstrates that, armed with even this level of understanding, our mathematical skills far exceed what we learned at school! The final chapter is the most challenging, and explores a curious problem-solving technique.
Methods of Geometry in the Theory of Partial Differential Equations
Mathematical models are used to describe the essence of the real world, and their analysis induces new predictions filled with unexpected phenomena.In spite of a huge number of insights derived from a variety of scientific fields in these five hundred years of the theory of differential equations, and its extensive developments in these one hundred years, several principles that ensure these successes are discovered very recently.This monograph focuses on one of them: cancellation of singularities derived from interactions of multiple species, which is described by the language of geometry, in particular, that of global analysis.Five objects of inquiry, scattered across different disciplines, are selected in this monograph: evolution of geometric quantities, models of multi-species in biology, interface vanishing of d - δ systems, the fundamental equation of electro-magnetic theory, and free boundaries arising in engineering.The relaxation of internal tensions in these systems, however, is described commonly by differential forms, and the reader will be convinced of further applications of this principle to other areas.
Mathematics for Fisheries Economics
The institute (ICAR-CIFE) has continuously adopted itself to meet the current needs, methodological challenges, and quality enhancement in fisheries education with the advances in information technology. The increasing application of mathematics to various branches/courses of economics in the past decade has made it necessary for economists to have an elementary knowledge of mathematics. To this end, an attempt has been made to bring out a practical manual on mathematics for catering to needs of fisheries students, most of whom having a non-mathematical background during their tertiary education. This book has been structured to solve problems on a wide range of topics such as matrix, determinants, limit & continuity, differentiation, integration, partial differentiation, etc. It is hoped that the book brought out will serve as a useful source of knowledge to the students and also will help them in solving various types of mathematical problems.
Ancilla to the Pre-Socratic Philosophers
Dive into the intellectual origins of Western philosophy with "Ancilla to the Pre-Socratic Philosophers." This comprehensive translation brings to life the fragments of wisdom left by the ancient Greek thinkers, spanning from the enigmatic Orpheus to the brilliant minds of Thales, Pythagoras, Heracleitus, Zeno, and Democritus. Delve into the essence of their philosophies through meticulously translated quotations, discovering the lost books they authored and gaining insight into the perspectives of contemporaneous authors on their beliefs.This invaluable reference provides a window into the foundations of philosophical thought, offering not only the known but also the obscure and debated fragments. From brief biographies to spurious quotes, this book paints a vivid picture of the Pre-Socratic era, making it an essential companion for scholars and a captivating read for anyone intrigued by the roots of Western philosophical inquiry.
The World Through the Lens of Mathematics
This amazing book aims to shatter the barrier between students and mathematics. By encouraging students to look at mathematics from a different perspective, build a bridge between their surroundings and mathematics and, at the same time, enrich them with the culture, history, customs, and geography of different parts of the world.
Fractional Integrals and Derivatives
This Special Issue is devoted to some serious problems that the Fractional Calculus (FC) is currently confronted with and aims at providing some answers to the questions like "What are the fractional integrals and derivatives?", "What are their decisive mathematical properties?", "What fractional operators make sense in applications and why?'', etc. In particular, the "new fractional derivatives and integrals" and the models with these fractional order operators are critically addressed. The Special Issue contains both the surveys and the research contributions. A part of the articles deals with foundations of FC that are considered from the viewpoints of the pure and applied mathematics, and the system theory. Another part of the Special issue addresses the applications of the FC operators and the fractional differential equations. Several articles devoted to the numerical treatment of the FC operators and the fractional differential equations complete the Special Issue.
Logos and Alogon
This book is a philosophical study of mathematics, pursued by considering and relating two aspects of mathematical thinking and practice, especially in modern mathematics, which, having emerged around 1800, consolidated around 1900 and extends to our own time, while also tracing both aspects to earlier periods, beginning with the ancient Greek mathematics. The first aspect is conceptual, which characterizes mathematics as the invention of and working with concepts, rather than only by its logical nature. The second, Pythagorean, aspect is grounded, first, in the interplay of geometry and algebra in modern mathematics, and secondly, in the epistemologically most radical form of modern mathematics, designated in this study as radical Pythagorean mathematics. This form of mathematics is defined by the role of that which beyond the limits of thought in mathematical thinking, or in ancient Greek terms, used in the book's title, an alogon in the logos of mathematics. The outcomeof this investigation is a new philosophical and historical understanding of the nature of modern mathematics and mathematics in general. The book is addressed to mathematicians, mathematical physicists, and philosophers and historians of mathematics, and graduate students in these fields.
Elementary School Test Materials 2022-2023
This book contains one year of mathleague.org elementary school tests and answer keys: 7 qualifying-level test sets, one district-level test set, one state-level test set, one national-level test set, and one international-level test set. Each test set contains a Sprint, Target, Team, and Number Sense test.
2022-2023 High School Contest Materials
This book contains one year of mathleague.org high school tests and answer keys: 7 qualifying-level test sets, one state-level test set, one national-level test set, and one international-level test set. Each test set contains a Sprint, Target, Team, and Relay test; three of the sets contain a Power Question.
The Logica Yearbook 2022
This volume of the Logica Yearbook series brings together articles presented at the annual international symposium Logica 2022, Tepl獺, the Czech Republic. The articles range over mathematical and philosophical logic, history and philosophy of logic, and the analysis of natural language.
Introductory Differential Equations
**2025 Textbook and Academic Authors Association (TAA) McGuffey Longevity Award Winner** Introductory Differential Equations, Sixth Edition provides the foundations to assist students in learning not only how to read and understand differential equations, but also how to read technical material in more advanced texts as they progress through their studies. The book's accessible explanations and many robust sample problems are appropriate for a first semester course in introductory ordinary differential equations (including Laplace transforms), for a second course in Fourier series and boundary value problems, and for students with no background on the subject.
Partial Clones of Terms
Words are strings of letters from a fixed alphabet. Sets of words are said to be formal languages. Natural languages, but also programming languages, are examples of such formal languages. On the set of all words the concatenation is a binary associative operation which produces a new word from any two given words. Therefore, there is a semigroup defined on the set of all words on an alphabet. This semigroup is an algebra of type (2), i.e., it has one binary operation satisfying the associative identity. Many properties of words and formal languages can be described by the algebraic properties of the word semigroup. To get languages of more expressive power, words can be generalized to terms using one more alphabet consisting of operation symbols. The combination of n+1 terms to a new term can be described by an (n+1)-ary superposition operation. This superposition operation satisfies the superassociative identity, a generalization of the associative identity. A clone is a multi-based algebraic structure with (n+1)-ary superposition operations as fundamental operations and satisfying the superassociative itentity. Clones of terms take over the role of word semigroups and describe the properties of terms and sets of terms which are also called tree languages. In this book we generalize the superposition operations to partial many-sorted operations. If the superassociative law is satisfied as a weak identity we obtain partial clones. The properties of several important kinds of terms such as linear terms and linear tree languages can be described by partial clones.
A Promenade in Mathematical Proofs with Comprehensive Review of Proof Techniques
A Promenade in Mathematical Proofs with Comprehensive Review of Proof Techniques is designed to support students as they advance their mathematical knowledge, bridging the gap from basic arithmetic and calculus to a deeper understanding of mathematical concepts through proof writing. It emphasizes the importance of being able to articulate mathematical ideas and solutions effectively, asserting that proof writing is as crucial as the subject matter itself. The book is structured to build upon the reader's knowledge systematically, starting with mathematical writing and progressing through logic, set theory, proof techniques, and more advanced topics such as sequences, relations, and functions. Each chapter concludes with practice problems and solutions to reinforce learning. A Promenade in Mathematical Proofs with Comprehensive Review of Proof Techniques is suitable for upper-division undergraduate courses in mathematics, particularly those focusing on advanced mathematical concepts and proof-based learning. Courses in fields such as computer science that require a strong foundation in mathematical reasoning and proof techniques would also benefit from this material.
Documentary review of the Pythagorean Theorem throughout history
Geometry has two great treasures: one is the theorem of Pythagoras; the other is the division of a line into an extreme and an average proportion. We may compare the first to a measure of gold; the second we may call a precious jewel. Jonannes Kepler (1596)The Pythagorean theorem, as Kepler (1596) says, is one of the greatest treasures of Geometry, it states that "in every right triangle the measure of the hypotenuse squared is equal to the sum of the measures of the squares of the legs". Anyone who has attended high school has been in contact with this theorem, whether they remember it or not. The general objective of this work is to analyze the proofs and demonstrations of the Pythagorean Theorem throughout the History of Mathematics, with the purpose of creating didactic sequences for its teaching at the Middle Level.
Human and Artificial Intelligence
Although tremendous advances have been made in recent years, many real-world problems still cannot be solved by machines alone. Hence, the integration of Human Intelligence and Artificial Intelligence is needed. However, several challenges make this integration complex. The aim of this Special Issue was to provide a large and varied collection of high-level contributions presenting novel approaches and solutions to address the above issues. This Special Issue contains 14 papers (13 research papers and 1 review paper) that deal with various topics related to human-machine interactions and cooperation. Most of these works concern different aspects of recommender systems, which are among the most widespread decision support systems. The domains covered range from healthcare to movies and from biometrics to cultural heritage. However, there are also contributions on vocal assistants and smart interactive technologies. In summary, each paper included in this Special Issue represents a step towards a future with human-machine interactions and cooperation. We hope the readers enjoy reading these articles and may find inspiration for their research activities.
Proven Impossible
In mathematics, it simply is not true that 'you can't prove a negative'. Many revolutionary impossibility theorems reveal profound properties of logic, computation, fairness and the universe, and form the mathematical background of new technologies and Nobel prizes. But to fully appreciate these theorems and their impact on mathematics and beyond, you must understand their proofs. This book is the first to present these proofs for a broad, lay audience. It fully develops the simplest rigorous proofs found in the literature, reworked to contain less jargon and notation, and more background, intuition, examples, explanations, and exercises. Amazingly, all of the proofs in this book involve only arithmetic and basic logic - and are elementary, starting only from first principles and definitions. Very little background knowledge is required, and no specialized mathematical training - all you need is the discipline to follow logical arguments and a pen in your hand.
