Luck Theory
This book is an original--the first-ever treatment of the mathematics of Luck. Setting out from the principle that luck can be measured by the gap between reasonable expectation and eventual realization, the book develops step-by-step a mathematical theory that accommodates the entire range of our pre-systematic understanding of the way in which luck functions in human affairs. In so moving from explanatory exposition to mathematical treatment, the book provides a clear and accessible account of the way in which luck assessment enters into the calculations of rational decision theory.
Machine Learning with PyTorch and Scikit-Learn
This book of the bestselling and widely acclaimed Python Machine Learning series is a comprehensive guide to machine and deep learning using PyTorch's simple to code framework.Purchase of the print or Kindle book includes a free eBook in PDF format.Key FeaturesLearn applied machine learning with a solid foundation in theoryClear, intuitive explanations take you deep into the theory and practice of Python machine learningFully updated and expanded to cover PyTorch, transformers, XGBoost, graph neural networks, and best practicesBook DescriptionMachine Learning with PyTorch and Scikit-Learn is a comprehensive guide to machine learning and deep learning with PyTorch. It acts as both a step-by-step tutorial and a reference you'll keep coming back to as you build your machine learning systems.Packed with clear explanations, visualizations, and examples, the book covers all the essential machine learning techniques in depth. While some books teach you only to follow instructions, with this machine learning book, we teach the principles allowing you to build models and applications for yourself.Why PyTorch?PyTorch is the Pythonic way to learn machine learning, making it easier to learn and simpler to code with. This book explains the essential parts of PyTorch and how to create models using popular libraries, such as PyTorch Lightning and PyTorch Geometric.You will also learn about generative adversarial networks (GANs) for generating new data and training intelligent agents with reinforcement learning. Finally, this new edition is expanded to cover the latest trends in deep learning, including graph neural networks and large-scale transformers used for natural language processing (NLP).This PyTorch book is your companion to machine learning with Python, whether you're a Python developer new to machine learning or want to deepen your knowledge of the latest developments.What you will learnExplore frameworks, models, and techniques for machines to 'learn' from dataUse scikit-learn for machine learning and PyTorch for deep learningTrain machine learning classifiers on images, text, and moreBuild and train neural networks, transformers, and boosting algorithmsDiscover best practices for evaluating and tuning modelsPredict continuous target outcomes using regression analysisDig deeper into textual and social media data using sentiment analysisWho this book is forIf you have a good grasp of Python basics and want to start learning about machine learning and deep learning, then this is the book for you. This is an essential resource written for developers and data scientists who want to create practical machine learning and deep learning applications using scikit-learn and PyTorch.Before you get started with this book, you'll need a good understanding of calculus, as well as linear algebra.Table of ContentsGiving Computers the Ability to Learn from DataTraining Simple Machine Learning Algorithms for ClassificationA Tour of Machine Learning Classifiers Using Scikit-LearnBuilding Good Training Datasets - Data PreprocessingCompressing Data via Dimensionality ReductionLearning Best Practices for Model Evaluation and Hyperparameter TuningCombining Different Models for Ensemble LearningApplying Machine Learning to Sentiment AnalysisPredicting Continuous Target Variables with Regression AnalysisWorking with Unlabeled Data - Clustering AnalysisImplementing a Multilayer Artificial Neural Network from Scratch(N.B. Please use the Look Inside option to see further chapters)
Modeling Biomaterials
Farago, O., A Beginner's Short Guide to Membrane Biophysics.- Misailidis, G., Ferenc, J., Tsiairis, C., Self-Organization of Tissues through Biochemical and Mechanical Signals.- Righi, M., Balbi, V., Foundations of Viscoelasticity and Application to Soft Tissue Mechanics.- Klika, V., Modeling of Biomaterials as an Application of the Theory of Mixtures.- Miller, R., et al., Modeling Biomechanics in the Healthy and Diseased Heart.- Chabiniok, R., et al., Translational Cardiovascular Modeling: Tetralogy of Fallot and Modeling of Diseases.
Distribution of Statistical Observables for Anomalous and Nonergodic Diffusions
This book investigates statistical observables for anomalous and nonergodic dynamics, focusing on the dynamical behaviors of particles modelled by non-Brownian stochastic processes in the complex real-world environment.
Dimension Groups and Dynamical Systems
This book is the first self-contained exposition of the fascinating link between dynamical systems and dimension groups. The authors explore the rich interplay between topological properties of dynamical systems and the algebraic structures associated with them, with an emphasis on symbolic systems, particularly substitution systems. It is recommended for anybody with an interest in topological and symbolic dynamics, automata theory or combinatorics on words. Intended to serve as an introduction for graduate students and other newcomers to the field as well as a reference for established researchers, the book includes a thorough account of the background notions as well as detailed exposition - with full proofs - of the major results of the subject. A wealth of examples and exercises, with solutions, serve to build intuition, while the many open problems collected at the end provide jumping-off points for future research.
Nonlinear Analysis and Optimization with Applications
Nonlinear analysis has wide and significant applications in many areas of mathematics, including functional analysis, variational analysis, nonlinear optimization, convex analysis, nonlinear ordinary and partial differential equations, dynamical system theory, mathematical economics, game theory, signal processing, control theory, data mining, and so forth. Optimization problems have been intensively investigated, and various feasible methods in analyzing convergence of algorithms have been developed over the last half century. In this Special Issue, we will focus on the connection between nonlinear analysis and optimization as well as their applications to integrate basic science into the real world.
Operators Between Sequence Spaces and Applications
1. Introduction.- 2. Matrix Domains.- 3. Operators between Matrix Domains.- 4. Statistical Convergence.- 5. Inclusion Equations.- 6. Sequence Space Equations.- 7. Solvability.
Analysis of Longitudinal Data with Examples
Development in methodology on longitudinal data is fast. Currently, there are a lack of intermediate /advanced level textbooks which introduce students and practicing statisticians to the updated methods on correlated data inference. This book will present a discussion of the modern approaches to inference, including the links between the theories of estimators and various types of efficient statistical models including likelihood-based approaches. The theory will be supported with practical examples of R-codes and R-packages applied to interesting case-studies from a number of different areas. Key Features: -Includes the most up-to-date methods -Use simple examples to demonstrate complex methods -Uses real data from a number of areas -Examples utilize R code
Real Analysis and Infinity
Real Analysis and Infinity presents the essential topics for a first course in real analysis with an emphasis on the role of infinity in all of the fundamental concepts. After introducing sequences of numbers, it develops the set of real numbers in terms of Cauchy sequences of rational numbers, and uses this development to derive the important properties of real numbers like completeness. The book then develops the concepts of continuity, derivative, and integral, and presents the theory of infinite sequences and series of functions. Topics discussed are wide-ranging and include the convergence of sequences, definition of limits and continuity via converging sequences, and the development of derivative. The proofs of the vast majority of theorems are presented and pedagogical considerations are given priority to help cement the reader's knowledge. Preliminary discussion of each major topic is supplemented with examples and diagrams, and historical asides. Examples follow most major results to improve comprehension, and exercises at the end of each chapter help with the refinement of proof and calculation skills.
Statistical Methods for Mediation, Confounding and Moderation Analysis Using R and SAS
This book introduces general definitions of third-variable effects that are adaptable to all different types of response (categorical or continuous), exposure, or third-variables. Readers of all disciplines familiar with introductory statistics will find this a valuable resource for analysis.
Modelling and Machine Learning Methods for Bioinformatics and Data Science Applications
Mathematical modeling is routinely used in physical and engineering sciences to help understand complex systems and optimize industrial processes. Mathematical modeling differs from Artificial Intelligence because it does not exclusively use the collected data to describe an industrial phenomenon or process, but it is based on fundamental laws of physics or engineering that lead to systems of equations able to represent all the variables that characterize the process. Conversely, Machine Learning methods require a large amount of data to find solutions, remaining detached from the problem that generated them and trying to infer the behavior of the object, material or process to be examined from observed samples. Mathematics allows us to formulate complex models with effectiveness and creativity, describing nature and physics. Together with the potential of Artificial Intelligence and data collection techniques, a new way of dealing with practical problems is possible. The insertion of the equations deriving from the physical world in the data-driven models can in fact greatly enrich the information content of the sampled data, allowing to simulate very complex phenomena, with drastically reduced calculation times. Combined approaches will constitute a breakthrough in cutting-edge applications, providing precise and reliable tools for the prediction of phenomena in biological macro/microsystems, for biotechnological applications and for medical diagnostics, particularly in the field of precision medicine.
Calculus: The Elements
This book provides a full and clear account of the essentials of calculus, presented in an engaging style that is both readable and mathematically precise. Concepts and central ideas are emphasized throughout. Physical examples and interpretations play a leading role, and alternative approaches to fundamental ways of thinking help the student develop the intuitive understanding so important in science and engineering. Many questions and problems, with detailed solutions, encourage active reading and independent thought. Usable either as a basic classroom text or as a supplement that will give the reader a grasp of calculus as a whole, the book is also ideally suited for self-study.
Iterative Learning Control for Equations with Fractional Derivatives and Impulses
Infectious Diseases and Our Planet
This book features recent research in mathematical modeling of indirectly and directly transmitted infectious diseases in humans, animals, and plants. It compiles nine not previously published studies that illustrate the dynamic spread of infectious diseases, offering a broad range of models to enrich understanding. It demonstrates the capability of mathematical modeling to capture disease spread and interaction dynamics as well as the complicating factors of various evolutionary processes. In addition, it presents applications to real-world disease control by commenting on key parameters and dominant pathways related to transmission. While aimed at early-graduate level students, the book can also provide insights to established researchers in that it presents a survey of current topics and methodologies in a constantly evolving field.
Compressed Chebyshev Polynomials and Multiple-Angle Formulas II
The material of this publication can be regarded as the continuation of the author's previous publication (ISBN: 978-620-0-62498-7) with the same title. It can also be regarded as a supplement, containing additional tables, longer listings than the ones given before, and additional figures. In this second part the factorization of the compressed Chebyshev polynomials via cyclotomic pre-polynomials are mainly emphasized. It is shown that the factorization of the compressed Chebyshev polynomials of all kinds and the factorization of certain very simple polynomials are fully synchron with each other in the meaning that all factors of the formers are cyclotomic pre-polynomials, while the factors of the latters are the corresponding cyclotomic polynomials. Some of the tables summarize the most important general properties and identities (among them the factorizational formulas) with the four kinds of Chebyshev polynomials and their modified variants. These tables apply to arbitrary positive integer n in general. Another set of tables are listings for the values of n between 1 and 20. A long table contains the parallel listing of cyclotomic pre-polynomials and cyclotomic polynomials.
Operator Theory and Differential Equations
This volume features selected papers from The Fifteenth International Conference on Order Analysis and Related Problems of Mathematical Modeling, which was held in Vladikavkaz, Russia, on 15 - 20th July 2019. Intended for mathematicians specializing in operator theory, functional spaces, differential equations or mathematical modeling, the book provides a state-of-the-art account of various fascinating areas of operator theory, ranging from various classes of operators (positive operators, convolution operators, backward shift operators, singular and fractional integral operators, partial differential operators) to important applications in differential equations, inverse problems, approximation theory, metric theory of surfaces, the Hubbard model, social stratification models, and viscid incompressible fluids.
Statistics
Data are everywhere. This book introduces, in a conversational style, the foundational techniques and concepts necessary to permit the data to tell their story. You'll learn not only about the mechanics of basic data analyses but also about when to use the methods and about the statistical concepts relevant to the meaning of the results. Several key statistical concepts are introduced early and reinforced often: the importance of randomization in collecting data, statistical inference and three distinct types (statements about cause/effect, making predictions and forecasts, and generalizing from samples to populations), interpreting the results of an analysis, and common misconceptions. Designed to be used in a one-semester course in introductory statistics, the text comes complete with problem sets for each section. Contents: Chapter 1: An Introduction to Data Chapter 2: Summarizing Data Chapter 3: Producing Data Chapter 4: Probability Chapter 5: Probability Distributions for Random Sampling Chapter 6: Hypothesis Testing Chapter 7: Confidence Intervals Chapter 8: Inference About Two Groups
World and Logic
What is the relationship between the world and logic, between intuition and language, between objects and their quantitative determinations? Rationalists, on the one hand, hold that the world is structured in a rational way. Representationalists, on the other hand, assume that language, logic, and mathematics are only the means to order and describe the intuitively given world. In World and Logic, Jens Lemanski takes up three surprising arguments from Arthur Schopenhauer's hitherto undiscovered Berlin Lectures, which concern the philosophy of language, logic, and mathematics. Based on these arguments, Lemanski develops a new position entitled 'rational representationalism': the world is always structured by human beings according to linguistic, logical, and mathematical principles, but the basic vocabulary of these structural descriptions already contains metaphors taken from the world around us.
Soar to Excellence Primary 1 Mathematics 1A
The Soar to Excellence Primary 1 Mathematics 1A book follows Singapore's Ministry of Education syllabus to equip seven-year-old students with a solid foundation in Primary 1 Mathematics in a simple and effective way. This book is also suitable for students in other countries who wish to learn Mathematics based upon Singapore's renowned curriculum design and academic rigour. In each topic, students are taken through carefully structured concepts and examples to help them gain a clear understanding of the subject. Furthermore, practice questions help reinforce what they have learnt. Well-suited as an enrichment workbook or teaching material, here is a fuss-free resource for parents, tutors, and students, to help students Soar to Excellence.
Game Theory
This book is based on the concept of game theory with real-life examples and non-mathematical analysis. It examines individual as well as social decision-making scenarios via a variety of GAEORY techniques. Game theory is a vast and unique field, a very promising and interesting field of study. As a game theorist, my cognitive analysis decodes that, ""GAEORY"" will be a universal subject, theory of generation and disciple of next century, the main research analysis of this book to establish game theory as a subject. Research scholar S. J. Tripathi give it to name that is GAEORY. This book, the first time briefly discussed GAEORY (Game theory as a separate subject), propounded by the author.
On the disposition of the prime numbers
Academic Paper from the year 2021 in the subject Mathematics - Analysis, grade: 2.00, language: English, abstract: In this paper, a mathematical structure is formed which consists of a matrix of all of the odd numbers. It then follows that the matrix contains all of the prime numbers with the exception of 2. In conjunction with a prime number counting function denoted by pi(x) we can determine the number of primes in any row of the matrix and their location using the prime number search method developed, which uses the prime number counting function data compiled from various sources by A V Kulsha. The columns of the matrix are of infinite extent and in some there are either one or no prime numbers. In a column where there is a single prime number, this without exception appears in the first row. The matrix shows that there are prime numbers which can never be twinned and that it is not possible, with the exception of 3,5,7 that three consecutive odd numbers can be prime. Further it is demonstrated, in some of the rows, even although the distribution of the primes is random, that it is possible to predict the next prime, and in some instances the prime following it, thus predicting a set of twins. Moreover in many cases the gap between consecutive primes can be predicted. The work, in particular the Discussion contains numerous examples . Any odd number can, in principle be located precisely within the matrix. In the Appendix it is shown how, using only the prime number counting function data, to determine the primality (or otherwise) of any odd number without examining forensically the number itself. It is noted that, in contradistinction to formulae which predict particular prime numbers, such as those of Mersenne and Sophie Germain, the analysis presented here makes no distinction between any of the 'types' of primes.
An Introduction to Infinite Products
This text provides a detailed presentation of the main results for infinite products, as well as several applications. The target readership is a student familiar with the basics of real analysis of a single variable and a first course in complex analysis up to and including the calculus of residues. The book provides a detailed treatment of the main theoretical results and applications with a goal of providing the reader with a short introduction and motivation for present and future study. While the coverage does not include an exhaustive compilation of results, the reader will be armed with an understanding of infinite products within the course of more advanced studies, and, inspired by the sheer beauty of the mathematics. The book will serve as a reference for students of mathematics, physics and engineering, at the level of senior undergraduate or beginning graduate level, who want to know more about infinite products. It will also be of interest to instructors who teach courses that involve infinite products as well as mathematicians who wish to dive deeper into the subject. One could certainly design a special-topics class based on this book for undergraduates. The exercises give the reader a good opportunity to test their understanding of each section.
Research in Mathematics and Public Policy
This volume features a variety of research projects at the intersection of mathematics and public policy. The topics included here fall in the areas of cybersecurity and climate change, two broad and impactful issues that benefit greatly from mathematical techniques. Each chapter in the book is a mathematical look into a specific research question related to one of these issues, an approach that offers the reader insight into the application of mathematics to important public policy questions. The articles in this volume are papers inspired by a Workshop for Women in Mathematics and Public Policy, held January 22-25, 2019 at the Institute for Pure and Applied Mathematics and the Luskin Center at the University of California, Los Angeles. The workshop was created to promote and develop women at all levels of their careers as researchers in mathematics and public policy. The idea was modeled after other successful Research Collaboration Conferences for Women, where junior and senior women come together at week-long conferences held at mathematics institutes to work on pre-defined research projects. The workshop focused on how mathematics can be used in public policy research and was designed to foster collaborative networks for women to help address the gender gap in mathematics and science.
Mathematics (Education) in the Information Age
This book brings together ideas from experts in cognitive science, mathematics, and mathematics education to discuss these issues and to present research on how mathematics and its learning and teaching are evolving in the Information Age. Given the ever-broadening trends in Artificial Intelligence and the processing of information generally, the aim is to assess their implications for how math is evolving and how math should now be taught to a generation that has been reared in the Information Age. It will also look at the ever-spreading assumption that human intelligence may not be unique--an idea that dovetails with current philosophies of mind such as posthumanism and transhumanism. The role of technology in human evolution has become critical in the contemporary world. Therefore, a subgoal of this book is to illuminate how humans now use their sophisticated technologies to chart cognitive and social progress. Given the interdisciplinary nature of the chapters, this will be of interest to all kinds of readers, from mathematicians themselves working increasingly with computer scientists, to cognitive scientists who carry out research on mathematics cognition and teachers of mathematics in a classroom.
Sequents and Trees
This textbook offers a detailed introduction to the methodology and applications of sequent calculi in propositional logic. Unlike other texts concerned with proof theory, emphasis is placed on illustrating how to use sequent calculi to prove a wide range of metatheoretical results. The presentation is elementary and self-contained, with all technical details both formally stated and also informally explained. Numerous proofs are worked through to demonstrate methods of proving important results, such as the cut-elimination theorem, completeness, decidability, and interpolation. Other proofs are presented with portions left as exercises for readers, allowing them to practice techniques of sequent calculus.After a brief introduction to classical propositional logic, the text explores three variants of sequent calculus and their features and applications. The remaining chapters then show how sequent calculi can be extended, modified, andapplied to non-classical logics, including modal, intuitionistic, substructural, and many-valued logics.Sequents and Trees is suitable for graduate and advanced undergraduate students in logic taking courses on proof theory and its application to non-classical logics. It will also be of interest to researchers in computer science and philosophers.
Examples and Problems in Advanced Calculus
This book includes over 500 most challenging exercises and problems in calculus. Topical problems and exercises are discussed on set theory, numbers, functions, limits and continuity, derivative, integral calculus, Rolle's theorem, mean value theorem, optimization problems, sequences and series. All the seven chapters recall important definitions, theorems and concepts, making this book immensely valuable to undergraduate students of engineering, mathematics, statistics, computer science and basic sciences.
Nonlinear Dynamics, Chaos, and Complexity
Chapter 1. Professor Valentin Afraimovich by Alexandra Afraimovich Chapter 2.The need for more integration between machine learning and neuroscience by Adrian Hernandez, Jose M. Amigo Chapter 3.Quasiperiodic Route to Transient Chaos in Vibroimpact System by Victor Bazhenov, Olga Pogorelova, Tatiana Postnikova Chapter 4. Modeling Ensembles of Nonlinear Dynamic Systems in Ultrawideb and Active Wireless Direct Chaotic Networks by A.S. Dmitriev, R.Yu. Yemelyanov, M.Yu. Gerasimov, Yu.V. Andreyev Chapter 5.Verification of Biomedical Processes with Anomalous Diffusion, Transport and Interaction of Species by Messoud Efendiev, Vitali Vougalter Chapter6.Chaos-based communication using isochronal synchronization: considerations about the synchronization manifold byJ.M.V. Grzybowski, E.E.N. Macau, T. Yoneyama Chapter 7.A sequential order of periodic motions in a 1-dimensional, time-delay, dynamical system by Siyuan Xing and Albert C. J. Luo Chapter 8.On the Geometric Approach to Transformations of the Coordinates of Inertial Frames of Reference by A.A. Talyshev Chapter 9.Corpuscular models of information transfer in a random environment by V.V. Uchaikin Chapter 10. Kinetic equation for systems with resonant captures and scatterings by A.V. Artemyev, A.I. Neishtadt, A.A. Vasiliev Chapter 11. Solvability In The Sense Of Sequences For Some Non-Fredholm Operators In Higher Dimensions by Vitali Vougalter, Vitaly Volpert
Mathematical Modelling in Real Life Problems
This book is intended to be a useful contribution for the modern teaching of applied mathematics, educating Industrial Mathematicians that will meet the growing demand for such experts. It covers many applications where mathematics play a fundamental role, from biology, telecommunications, medicine, physics, finance and industry. It is presented in such a way that can be useful in Modelation, Simulation and Optimization courses, targeting master and PhD students. Its content is based on many editions from the successful series of Modelling Weeks organized by the European Consortium of Mathematics in Industry (ECMI). Each chapter addresses a particular problem, and is written in a didactic way, providing the description of the problem, the particular way of approaching it and the proposed solution, along with the results obtained.
Modal View of Atmospheric Variability
This book reviews the theory and applications of the normal-mode functions in numerical weather prediction and weather and climate dynamics. The normal-mode functions, the eigensolutions of the linearized primitive equations describing the evolution of atmospheric winds and mass variables, have been used for a long time. They have played an important role in the development of data assimilation schemes and the initialization of numerical weather prediction models. Chapters also present how the normal modes can be applied to many theoretical and numerical problems in the atmospheric sciences, such as equatorial wave dynamics, baroclinic instability, energy transfers, and predictability across scales.
Lie Models in Topology
Since the birth of rational homotopy theory, the possibility of extending the Quillen approach - in terms of Lie algebras - to a more general category of spaces, including the non-simply connected case, has been a challenge for the algebraic topologist community. Despite the clear Eckmann-Hilton duality between Quillen and Sullivan treatments, the simplicity in the realization of algebraic structures in the latter contrasts with the complexity required by the Lie algebra version. In this book, the authors develop new tools to address these problems. Working with complete Lie algebras, they construct, in a combinatorial way, a cosimplicial Lie model for the standard simplices. This is a key object, which allows the definition of a new model and realization functors that turn out to be homotopically equivalent to the classical Quillen functors in the simply connected case. With this, the authors open new avenues for solving old problems and posing new questions. This monograph is the winner of the 2020 Ferran Sunyer i Balaguer Prize, a prestigious award for books of expository nature presenting the latest developments in an active area of research in mathematics.
Stability Analysis of Neural Networks
1. Introduction.- 2. LMI-Based Stability Criteria for BAM Neural Networks.- 3. Exponential Stability Criteria for Uncertain Inertial BAM Neural Networks.- 4. Exponential Stability of Impulsive Cohen-Grossberg BAM Neural Networks.- 5. Exponential Stability of Recurrent Neural Networks with Impulsive and Stochastic Effects.- 6. Stability of Markovian Jumping Stochastic Impulsive Uncertain BAM Neural Networks.- 7. Global Robust Exponential Stability of Stochastic Neutral-Type Neural Networks.- 8. Exponential Stability of Discrete-Time Cellular Uncertain BAM Neural Networks.- 9. Exponential Stability of Discrete-Time Stochastic Impulsive BAM Neural Networks.- 10. Stability of Discrete-Time Stochastic Quaternion-Valued Neural Networks.- 11. Robust Finite-Time Passivity of Markovian Jump Discrete-Time BAM Neural Networks.- 12 Robust Stability of Discrete-Time Stochastic Genetic Regulatory Networks.
In the Tradition of Thurston
This book consists of 16 surveys on Thurston's work and its later development. The authors are mathematicians who were strongly influenced by Thurston's publications and ideas. The subjects discussed include, among others, knot theory, the topology of 3-manifolds, circle packings, complex projective structures, hyperbolic geometry, Kleinian groups, foliations, mapping class groups, Teichm羹ller theory, anti-de Sitter geometry, and co-Minkowski geometry. The book is addressed to researchers and students who want to learn about Thurston's wide-ranging mathematical ideas and their impact. At the same time, it is a tribute to Thurston, one of the greatest geometers of all time, whose work extended over many fields in mathematics and who had a unique way of perceiving forms and patterns, and of communicating and writing mathematics.
Flatland
First published in 1884, Edwin A. Abbott's classic novella is many things: a biting social satire on Victorian society, an exhortation to literally think outside the box (or, if you're a privileged Spacelander, outside the cube), and above all a testament to the universal quest for knowledge and the inherent beauty and purity of mathematical truth. Flatland tells the story of "A SQUARE," a mathematician living in the Second dimension, known as Flatland, whose perceptions of his Universe are turned upside down when he is visited by a Sphere from the Third dimension, known as Spaceland. "A SQUARE" also muses on life in Lineland and Pointland. Flatland is one of the very few novels about math and philosophy that can appeal to almost any layperson. Published in 1880, this short fantasy takes us to a completely flat world of two physical dimensions where all the inhabitants are geometric shapes, and who think the planar world of length and width that they know is all there is. But one inhabitant discovers the existence of a third physical dimension, enabling him to finally grasp the concept of a fourth dimension. Watching our Flatland narrator, we begin to get an idea of the limitations of our own assumptions about reality, and we start to learn how to think about the confusing problem of higher dimensions. The book is also quite a funny satire on society and class distinctions of Victorian England.
The Integration of Functions of a Single Variable
G. H. Hardy (7 Feb 1877 - 1 Dec 1947) was an eccentric British mathematician who worked extensively in mathematical analysis and analytical number theory alongside J.E Littlewood.He is perhaps even better known for his adoption and mentoring of the self-taught Indian mathematical genius, Srinivasa Ramanujan.Hardy wanted his work to be referred to as pure mathematics rather than applied mathematics. In his view, mathematics was not something to be used in social destruction and to fulfill political purposes.
Mathematics in Middle and High School
We have developed this book - Problems in Middle and High School Mathematics: Solving Techniques Edition II - to help middle and high school students acquire a deeper understanding of mathematics and prepare themselves for Olympiads and contests in this field, as well as having high levels of success in future related university courses. The wish is to instill in our students the art of reasoning and the power of evaluating and solving mathematical problems at a competitive level. However, this work can be counted on with assisting many more students (even those less inclined towards mathematics) with understanding proper methods of solving mathematical problems. We believe this work to be filling a gap in the area of high school mathematics in North-America, in terms of the material it covers, as well as the depth and rigor of the problems. While common elements with other works of this type do exist, we have tried to enrich its content through the addition of special features, important problems, and diversified solutions. Intending to often present to students multiple approaches in solving a problem, each chapter presents necessary core knowledge, problems grouped on thematic and solving ideas - as a result of a lengthy pedagogical career - with examples solved in great detail (sometimes with multiple solutions), followed by similar problems with answers in the back. Each chapter contains exercises that increase in difficulty, from solving simple algebraic expressions to problems up to par with major mathematics competitions and Olympiads. At the end of each chapter, we have included problems we have published in the "Mathematical Olympiads Correspondence Program" (http: //www.math.ca/Competitions/MOCP/) in Ontario, Canada. Occasionally, some problems at the end of each chapter may appear very hard to solve. However, my thirty-year experience in teaching mathematics and my interactions with students enabled me to demonstrate the solutions in a clear, easy to understand manner. This second edition contains new problems, solutions and chapters that are sure to cover any Canadian high school curriculum and help students excel in their math-related courses. Additionally, the expertise gained from solving difficult problems through completing problem sets in this edition can be applied to many other courses and even daily life. We are confident that any students who pick up this book and solve one set of exercises can expand their knowledge, fuel their interest, and excel in their mathematics courses. We also can guarantee that solving problem sets within this book will be some of the best post-secondary preparation for students looking to pursue a degree in the field of mathematics. We also hope this work will become instrumental to mathematics teachers working both in the private and public education systems. Mihai Rosu rosumihai@yahoo.ca Vlad Rosu vladrosu20@gmail.com
Quantum Key Distribution Using FPGA
This book presents a study and analysis to investigate of a secret-key sharing in quantum technique and try to increase the key length and security level in the sifting key stage. Enhancements in sifting key stage and error correction stage are done by repeating the QKD protocol and encoding the key bits before sending them by adding a C_QUBITS technique. After that, 60% base probability is used to increase the protocol gain. The prototype is implemented with the default FPGA clock period which is 20 ns; the performance analysis shows a good enhancement in the speed of generation which is less than 20 ns and the number of used hardware pins is just 3 pins.
Geometry of Deep Learning
The focus of this book is on providing students with insights into geometry that can help them understand deep learning from a unified perspective. Rather than describing deep learning as an implementation technique, as is usually the case in many existing deep learning books, here, deep learning is explained as an ultimate form of signal processing techniques that can be imagined. To support this claim, an overview of classical kernel machine learning approaches is presented, and their advantages and limitations are explained. Following a detailed explanation of the basic building blocks of deep neural networks from a biological and algorithmic point of view, the latest tools such as attention, normalization, Transformer, BERT, GPT-3, and others are described. Here, too, the focus is on the fact that in these heuristic approaches, there is an important, beautiful geometric structure behind the intuition that enables a systematic understanding. A unified geometric analysis to understand the working mechanism of deep learning from high-dimensional geometry is offered. Then, different forms of generative models like GAN, VAE, normalizing flows, optimal transport, and so on are described from a unified geometric perspective, showing that they actually come from statistical distance-minimization problems.Because this book contains up-to-date information from both a practical and theoretical point of view, it can be used as an advanced deep learning textbook in universities or as a reference source for researchers interested in acquiring the latest deep learning algorithms and their underlying principles. In addition, the book has been prepared for a codeshare course for both engineering and mathematics students, thus much of the content is interdisciplinary and will appeal to students from both disciplines.
Anisotropy Across Fields and Scales
This open access book focuses on processing, modeling, and visualization of anisotropy information, which are often addressed by employing sophisticated mathematical constructs such as tensors and other higher-order descriptors. It also discusses adaptations of such constructs to problems encountered in seemingly dissimilar areas of medical imaging, physical sciences, and engineering. Featuring original research contributions as well as insightful reviews for scientists interested in handling anisotropy information, it covers topics such as pertinent geometric and algebraic properties of tensors and tensor fields, challenges faced in processing and visualizing different types of data, statistical techniques for data processing, and specific applications like mapping white-matter fiber tracts in the brain.The book helps readers grasp the current challenges in the field and provides information on the techniques devised to address them. Further, it facilitates the transfer of knowledge between different disciplines in order to advance the research frontiers in these areas.This multidisciplinary book presents, in part, the outcomes of the seventh in a series of Dagstuhl seminars devoted to visualization and processing of tensor fields and higher-order descriptors, which was held in Dagstuhl, Germany, on October 28-November 2, 2018.
Functions of Mathematics for the Liberal Arts
"Functions of Mathematics for the Liberal Arts" is designed to showcase the utility and beauty of mathematics for the college student majoring in the humanities or liberal arts. It uses a slow pace and informal style to explore not only its many uses but also its rich history as part of the human experience. The development of mathematics is showcased within a larger cultural context and explained as the evolution both as a true art from as well as a universal language for solving problems to improve society.
Hardy Operators on Euclidean Spaces and Related Topics
In many branches of mathematical analysis and mathematical physics, the Hardy operator and Hardy inequality are fundamentally important and have been intensively studied ever since the pioneer researches. This volume presents new properties of higher-dimensional Hardy operators obtained by the authors and their collaborators over the last decade. Its prime focus is on higher-dimensional Hardy operators that are based on the spherical average form.The key motivation for this monograph is based on the fact that the Hardy operator is generally smaller than the Hardy-Littlewood maximal operator, which leads to, on the one hand, the operator norm of the Hardy operator itself being smaller than the latter. On the other hand, the former characterizing the weight function class or function spaces is greater than the latter.
Recent Progress on Topics of Ramanujan Sums and Cotangent Sums Associated with the Riemann Hypothesis
In this monograph, we study recent results on some categories of trigonometric/exponential sums along with various of their applications in Mathematical Analysis and Analytic Number Theory. Through the two chapters of this monograph, we wish to highlight the applicability and breadth of techniques of trigonometric/exponential sums in various problems focusing on the interplay of Mathematical Analysis and Analytic Number Theory. We wish to stress the point that the goal is not only to prove the desired results, but also to present a plethora of intermediate Propositions and Corollaries investigating the behaviour of such sums, which can also be applied in completely different problems and settings than the ones treated within this monograph.In the present work we mainly focus on the applications of trigonometric/exponential sums in the study of Ramanujan sums -- which constitute a very classical domain of research in Number Theory -- as well as the study of certain cotangent sums with a wide range of applications, especially in the study of Dedekind sums and a facet of the research conducted on the Riemann Hypothesis. For example, in our study of the cotangent sums treated within the second chapter, the methods and techniques employed reveal unexpected connections with independent and very interesting problems investigated in the past by R de la Bret癡che and G Tenenbaum on trigonometric series, as well as by S Marmi, P Moussa and J-C Yoccoz on Dynamical Systems.Overall, a reader who has mastered fundamentals of Mathematical Analysis, as well as having a working knowledge of Classical and Analytic Number Theory, will be able to gradually follow all the parts of the monograph. Therefore, the present monograph will be of interest to advanced undergraduate and graduate students as well as researchers who wish to be informed on the latest developments on the topics treated.
Hypergroup Theory
The book presents an updated study of hypergroups, being structured on 12 chapters in starting with the presentation of the basic notions in the domain: semihypergroups, hypergroups, classes of subhypergroups, types of homomorphisms, but also key notions: canonical hypergroups, join spaces and complete hypergroups. A detailed study is dedicated to the connections between hypergroups and binary relations, starting from connections established by Rosenberg and Corsini. Various types of binary relations are highlighted, in particular equivalence relations and the corresponding quotient structures, which enjoy certain properties: commutativity, cyclicity, solvability.A special attention is paid to the fundamental beta relationship, which leads to a group quotient structure. In the finite case, the number of non-isomorphic Rosenberg hypergroups of small orders is mentioned. Also, the study of hypergroups associated with relations is extended to the case of hypergroups associated to n-ary relations. Then follows an applied excursion of hypergroups in important chapters in mathematics: lattices, Pawlak approximation, hypergraphs, topology, with various properties, characterizations, varied and interesting examples. The bibliography presented is an updated one in the field, followed by an index of the notions presented in the book, useful in its study.
Memory Dump Analysis Anthology, Volume 13
This reference volume consists of revised, edited, cross-referenced, and thematically organized selected articles from Software Diagnostics Institute (DumpAnalysis.org + TraceAnalysis.org) and Software Diagnostics Library (former Crash Dump Analysis blog, DumpAnalysis.org/blog) about software diagnostics, root cause analysis, debugging, crash and hang dump analysis, software trace and log analysis written in December 2019 - July 2020 for software engineers developing and maintaining products on Windows and Linux platforms, quality assurance engineers testing software, technical support, escalation and site reliability engineers dealing with complex software issues, security and vulnerability researchers, reverse engineers, malware and memory forensics analysts. This volume is fully cross-referenced with volumes 1 - 12 and features: - 9 new crash dump analysis patterns with selected downloadable example memory dumps; - 15 new software trace and log analysis patterns; - Introduction to diagnostic analysis gestures; - Introduction to the category-theoretic view of debugging; - Lists of recommended category theory, number theory, and cybersecurity books.
Research Schools on Number Theory in India
This book is an attempt to describe the gradual development of the major schools of research on number theory in South India, Punjab, Mumbai, Bengal, and Bihar-including the establishment of Tata Institute of Fundamental Research (TIFR), Mumbai, a landmark event in the history of research of number theory in India. Research on number theory in India during modern times started with the advent of the iconic genius Srinivasa Ramanujan, inspiring mathematicians around the world. This book discusses the national and international impact of the research made by Indian number theorists. It also includes a carefully compiled, comprehensive bibliography of major 20th century Indian number theorists making this book important from the standpoint of historic documentation and a valuable resource for researchers of the field for their literature survey. This book also briefly discusses the importance of number theory in the modern world of mathematics, including applications of the results developed by indigenous number theorists in practical fields. Since the book is written from the viewpoint of the history of science, technical jargon and mathematical expressions have been avoided as much as possible.
Domain-Specific Languages of Mathematics
The main idea behind this book is to encourage readers to approach mathematical domains from a functional programming perspective: to identify the main functions and types involved and, when necessary, to introduce new abstractions; to give calculational proofs; to pay attention to the syntax of the mathematical expressions; and, finally, to organize the resulting functions and types in domain-specific languages. The book is recommended for developers who are learning mathematics and would like to use Haskell to make sense of definitions and theorems. It is also a book for the mathematically interested who wants to explore functional programming and domain-specific languages. The book helps put into perspective the domains of Mathematics and Functional Programming and shows how Computer Science and Mathematics are usefully taught together.
