Mathematics for Data Science and Artificial Intelligence
This book provides a comprehensive foundation in the mathematical tools essential for modern data science and machine learning. It blends core subjects such as linear algebra, calculus, probability, statistics, optimization, and numerical methods with real-world applications. Readers explore matrix operations, eigenvalues, and dimensionality reduction techniques like PCA and t-SNE. Optimization is covered through gradient-based methods and regularization strategies. Probability theory, Bayes' theorem, and statistical inference form the basis for modeling uncertainty. Information theory concepts like entropy, cross-entropy, and KL divergence are applied to learning and feature selection. Efficient computational methods are introduced using Python/Numpy implementations. Advanced topics include graph theory for network analysis and stochastic models such as Markov chains and ARIMA for time series forecasting. This book bridges theory and practice, offering step-by-step problem-solving, coding exercises, and a deep understanding of the mathematical backbone driving AI and data science.
A Course in Game Theory
Game theory is a fascinating subject. We all know many entertaining games, such as chess, poker, tic-tac-toe, bridge, baseball, computer games - the list is quite varied and almost endless. In addition, there is a vast area of economic games, discussed in Myerson (1991) and Kreps (1990), and the related political games [Ordeshook (1986), Shubik (1982), and Taylor (1995)]. The competition between firms, the conflict between management and labor, the fight to get bills through congress, the power of the judiciary, war and peace negotiations between countries, and so on, all provide examples of games in action. There are also psychological games played on a personal level, where the weapons are words, and the payoffs are good or bad feelings [Berne (1964)]. There are biological games, the competition between species, where natural selection can be modeled as a game played between genes [Smith (1982)]. There is a connection between game theory and the mathematical areas of logic and computer science. One may view theoretical statistics as a two-person game in which nature takes the role of one of the players, as in Blackwell and Girshick (1954) and Ferguson (1968).Games are characterized by a number of players or decision makers who interact, possibly threaten each other and form coalitions, take actions under uncertain conditions, and finally receive some benefit or reward or possibly some punishment or monetary loss. In this text, we present various mathematical models of games and study the phenomena that arise. In some cases, we will be able to suggest what courses of action should be taken by the players. In others, we hope simply to be able to understand what is happening in order to make better predictions about the future.
Handbook of Algebra and Algebraic Topology
Covering all aspects of Elementary Algebra, it will surely serve as the complete elementary textbook for studying different concepts.The subject is presented in its simplest, most essential form, but with many explanatory details, figures and examples.The content of the book is sequentially presented in the book for easy and comprehensive learning.This book forms a strong basis for students and teachers engaged in the field of mathematics especially algebra and algebraic topology.
Introduction to Finite Elements in Engineering
"Introduction to Finite Elements in Engineering" provides a comprehensive introduction to the fundamentals of finite element analysis and its application in engineering problem-solving. Through collaboration with experts, this book covers topics such as mathematical foundations, element formulations, numerical techniques, software tools, and practical applications in structural, thermal, and fluid analysis. With theoretical explanations and practical examples, readers gain insights into understanding and applying FEA concepts in engineering practice. Bridging theory and application, this book empowers readers to leverage the power of computational simulation to innovate and advance engineering design and analysis.
Modern Trigonometry
"Modern Trigonometry" provides a comprehensive overview of trigonometric principles, techniques, and applications in contemporary mathematics and scientific inquiry. Through collaboration with experts, this book covers topics such as trigonometric functions, identities, equations, vectors, complex numbers, Fourier analysis, and applications in fields such as physics, engineering, computer graphics, and signal processing. With theoretical foundations and practical applications, readers gain a thorough understanding of modern trigonometry. Bridging theory and application, this book empowers readers to appreciate the beauty and utility of trigonometric concepts in the modern age.
Fundamentals of Coordinate Geometry
Coordinate geometry is the study of geometric figures by plotting them in the coordinate axes.The content of the book is sequentially presented in the book for easy and comprehensive learning.This book forms a strong basis for students, teachers, researchers, and person engaged in the field of mathematics.
Introduction to Modern Algebra
It covers a range of topics related to the study of algebra, from the basics of set theory to advanced topics such as Galois Theory and representation theory. The book begins with an overview of modern algebra and its importance in various fields of mathematics and also covers the history and development of ring theory, including its applications in algebraic geometry and number theory.The book covers the basics of set theory, including the axioms of set theory and cardinality and introduces module theory, which is a generalization of vector spaces, and its applications in various fields of mathematics.The book discusses the applications of commutative algebra in algebraic geometry and algebraic number theory and also includes a discussion of current research topics in group theory.The book covers the history and development of Galois Theory and discusses the importance of representation theory in various fields of mathematics, including physics and geometry.It is suitable for undergraduate and graduate students as well as researchers in mathematics and related fields.
Essentials of Nonlinearity Bifurcation and Chaos Theory
"Essentials of Nonlinearity, Bifurcation, and Chaos Theory" provides a concise yet comprehensive overview of these fundamental concepts in dynamical systems theory. This book introduces readers to the principles of nonlinear dynamics, bifurcation theory, and chaos theory, exploring their applications across various scientific disciplines. Through clear explanations and illustrative examples, readers gain insight into the behavior of nonlinear systems, including their stability, bifurcation points, and chaotic dynamics. Whether you're a student, researcher, or practitioner in fields such as physics, engineering, biology, or economics, this book serves as an essential guide to understanding the intriguing phenomena of nonlinearity, bifurcation, and chaos and their implications for modeling and analysis in complex systems.
Fractional Order PID and ADR Controls
This book explores the design and analysis of fractional-order and active disturbance rejection control, examining both the theoretical foundations and their practical applications.It covers fractional-order proportional-integral-derivative (PID) control, fractional-order active disturbance rejection (ADR) control, and the combined fractional-order PID-ADR control. The book begins with an analysis of the three-parameter fractional-order PID controller, demonstrating its application to the permanent magnet synchronous motor (PMSM) speed servo system, due to its comprehensive inclusion of proportional, integral, and differential elements. It then delves into active disturbance rejection control and periodic disturbance compensation, comparing the performance of each controller based on various parameters. This comparison enables readers to critically evaluate the advantages and limitations of each approach before implementation. Offering a thorough guide to fractional-order and active disturbance rejection control, the book also includes numerical methods for assessing and developing these systems.The book will be of particular interest to professionals working with numerical methods, fractional-order systems and control, PID controller, active disturbance rejection, control design, and production and is especially relevant to those in mechanical, industrial, and electrical engineering.
Beyond Signals - Exploring Revolutionary Fourier Transform Applications
Fourier Transform is a fundamental mathematical framework that has revolutionized numerous scientific and technological domains. Beyond Signals - Exploring Revolutionary Fourier Transform Applications presents an in-depth analysis of its profound influence on modern research and industry. This volume explores advanced applications in signal processing, spectroscopy, quantum mechanics, biomedical imaging, nanomaterials, and renewable energy, illustrating how Fourier techniques enable precise data interpretation and system optimization. The book integrates theoretical foundations with practical implementations, offering insights into its role in material characterization, sensor technology, and computational modeling. Authored by distinguished experts, including Dr. Muhammad Bilal Tahir-recognized for his contributions to nanomaterials, optoelectronics, and applied physics-this work is a comprehensive resource for researchers, engineers, and scholars. By bridging classical theories with emerging advancements, Beyond Signals - Exploring Revolutionary Fourier Transform Applications highlights the transformative potential of Fourier Transform methodologies in solving complex scientific and engineering challenges.
The Geometry of Equilibrium
James Clerk Maxwell is one of the giants of scientific thought, and whilst his groundbreaking contributions to electromagnetism and statistical physics are well known, his profound insights into the theory of structures are appreciated less widely. Maxwell's approach was deeply geometrical, and this richly illustrated book reveals his astute perception of the remarkable dualities that exist between the form of a structure and the forces it can carry, with understandings that will surprise contemporary readers. Early chapters introduce the background in which Maxwell was working, followed by contributions by leading researchers describing the latest applications of these ideas. Subsequent chapters introduce the many subtopics that this work embraces. The book ends with Maxwell's original papers on structural mechanics, each annotated to highlight and explain the ideas therein. This is a wonderful resource for mathematicians, scientists, engineers, and designers to enter this rich and underexplored aspect of the genius of Maxwell.
Abstract Algebra
Abstract Algebra: An Interactive Approach is a new concept in learning modern algebra. Each chapter in the textbook has a corresponding interactive Mathematica notebook and an interactive SageMath workbook which can be used either in the classroom, or outsideof the classroom.
AP Calculus Study Guide AP 微积分 AB/BC 指导用书
本书是一部系统全面的微积分教程，涵盖了从基础概念到高级应用的完整内容。书中包括了极限与连续性、微分及其基本性质、复合函数的微分、积分及其累积变化、微分方程和积分应用等主题。内容结构清晰，循序渐进，辅以大量的例题和解析，帮助学生理解并掌握微积分的核心概念及解题技巧，特别适用于准备AP微积分的学习者。这部书为学生提供了深入的理论知识和实用的数学工具，使其能更好地理解和应用微积分于实际问题中。This book is a systematic and comprehensive calculus tutorial that covers a complete range of content from basic concepts to advanced applications. The book includes topics such as limits and continuity, differentiation and its basic properties, differentiation of composite functions, integration and its cumulative change, differential equations, and applications of integration. With a clear content structure and a step-by-step approach, it is supplemented by a large number of examples and analyses to help students understand and master the core concepts of calculus and problem-solving techniques. It is especially suitable for learners preparing for AP Calculus. This book provides students with in-depth theoretical knowledge and practical mathematical tools, enabling them to better understand and apply calculus to practical problems.
HESI Math Workbook
Over 200 HESI A2(R) practice questions, prepared by a dedicated team of exam experts, with detailed answer key, Math shortcuts, tips and tricks, tutorials and multiple choice strategies! You have arrived!Here is everything you need to pass the HESI(R) Math! We have helped thousands of students and we can help you!Complete HESI Math workbook, prepared by exam experts, with over 200 questions. Basic MathSolving Word ProblemsSolve Algebra EquationsMetric ConversionData InterpretationWhy Us?The Complete Test Preparation Team has been publishing high quality study materials since 2005, with a catalogue of over 145 titles, in English, French and Chinese, as well as ESL curriculum for all levels.You have come to the right place.If you want to spend your valuable study time where it will help you the most - we've got you covered today and tomorrow.You can pass the HESI with a great score! Here is how! You already know that Practice Makes Perfect! The more questions you see, and the more you practice - the better your HESI score! HESI A2(R) Math practice questions, easy-to-read tutorials explaining everything in plain language, exam tips and tricks, math shortcuts, and multiple choice strategies! Everything you need, complied by a dedicated team of experts with everything you need all in one place! Please note that HESI(R) is a registered trademark of the Health Education Systems Inc., which was not involved in the production of, and does not endorse, this product. Practice Really Does Make Perfect! The more questions you see, the more likely you are to pass the test. And between our study guide and practice tests, you'll have over 200 practice questions that cover every category. Our practice test questions have been developed by our dedicated team of experts. All the material in the study guide, including every practice question, are designed to engage your critical thinking skills needed to pass the test! Heard it all before? Maybe you have heard this kind of thing before, and don't feel you need it. Maybe you are not sure if you are going to buy this book. Remember though, it only a few percentage points divide the PASS from the FAIL students! Even if our test tips increase your score by a few percentage points, isn't that worth it?
Solution of Initial-Boundary Value Problems
Methods for solving problems of mathematical physics can be divided into the following four classes. Analytical methods (the method of separation of variables, the method of characteristics, the method of Green's functions, etc.) methods have a relatively low degree of universality, i.e. focused on solving rather narrow classes of problems. Approximate analytical methods (projection, variational methods, small parameter method, operational methods, various iterative methods) are more versatile than analytical ones. Numerical methods (finite difference method, direct method, control volume method, finite element method, etc.) are very universal methods. Probabilistic methods (Monte Carlo methods) are highly versatile. Can be used to calculate discontinuous solutions. However, they require large amounts of calculations and, as a rule, they lose with the computational complexity of the above methods when solving such problems to which these methods are applicable. Comparing methods for solving problems of mathematical physics, it is impossible to give unconditional primacy to any of them. Any of them may be the best for solving problems of a certain class. The proposed method of moving nodes for boundary value problems of differential equations combines a combination of numerical and analytical methods. In this case, we can obtain, on the one hand, an approximately analytical solution of the problem, which is not related to the methods listed above. On the other hand, this method allows one to obtain compact discrete approximations of the original problem. Note that obtaining an approximately analytical solution of differential equations is based on numerical methods. The nature of numerical methods also allows obtaining an approximate analytical expression for solving differential equations
Carleman Estimates in Mean Field Games
This book provides a comprehensive exploration of Mean Field Games (MFG) theory, a mathematical framework for modeling the collective behavior of rational agents in complex systems. MFG theory can govern a range of societal phenomena, including finance, sociology, machine learning, and economics. The focus is on the system of two coupled nonlinear parabolic partial differential equations (PDEs) that define the Mean Field Games System. The book covers key theoretical topics such as solution stability and uniqueness, with a particular emphasis on Carleman estimates, which are used to estimate solution errors based on noise in the input data. It also introduces the theory of Ill-Posed and Inverse Problems within MFG theory. Both theoretical and numerical aspects of forward and inverse problems are explored through Carleman estimates, offering a rigorous foundation for researchers and practitioners in applied mathematics and related fields. This book offers a rigorous approach to Carleman estimates, a key element of Mean Field Games theory, making it an essential resource for researchers, graduate students, and professionals looking to apply this powerful framework to complex, real-world systems.
Multivariable Calculus
Foundations of Multivariable Functions: Provide a comprehensive introduction to functions of multiple variables, including definitions, graphical interpretations, and basic properties. Cover topics such as partial derivatives, gradient vectors, and the concept of differentiability for functions of several variables.Multiple Integration Techniques: Explore techniques for evaluating multiple integrals, including double and triple integrals. Discuss applications of multiple integrals in calculating areas, volumes, and other physical quantities. Include methods for changing variables, such as polar, cylindrical, and spherical coordinates.Vector Calculus and Field Theory: Examine key concepts in vector calculus, including vector fields, line integrals, surface integrals, and flux. Discuss fundamental theorems such as Green's Theorem, Stokes' Theorem, and the Divergence Theorem, and their applications in physics and engineering.Optimization and Constrained Optimization: Discuss methods for optimization in multivariable settings, including finding local and global extrema of functions. Cover techniques such as Lagrange multipliers for constrained optimization problems and the use of Hessian matrices for analyzing critical points.Applications and Advanced Topics: Address various applications of multivariable calculus in fields such as physics, engineering, economics, and computer science. Include advanced topics such as differential forms, manifold theory, and the applications of multivariable calculus in differential equations and dynamical systems.
Introductory Discrete Mathematics
Fundamental Concepts: Provide an overview of foundational concepts in discrete mathematics, including sets, relations, functions, and basic logic. Explain how these concepts form the basis for more advanced topics in discrete mathematics.Combinatorics and Counting: Explore combinatorial methods and counting principles, including permutations, combinations, the pigeonhole principle, and inclusion-exclusion principles. Discuss their applications in solving counting problems and understanding discrete structures.Graph Theory: Introduce the basics of graph theory, including definitions and properties of graphs, types of graphs (e.g., directed, undirected, weighted), and key concepts such as paths, cycles, connectivity, and graph coloring.Applications of Discrete Mathematics: Explore the practical applications of discrete mathematics in computer science, including its role in cryptography, network design, coding theory, and algorithm design. Discuss how discrete mathematics provides tools for solving real-world problems in these areas.
Mathematical Modeling and Control of COVID-19 and Its Variants
In this book, we developed a new hybrid deterministic model, we do this by considering several essential properties of the pandemic COVID-19, such as the existence of individuals tested positive for COVID-19 with severe, mild, or asymptomatic symptoms, and dividing infectious compartment into two essential compartments of hospitalized individuals and those in intensive care units. The content of this book is divided into 3 chapters: In the first chapter, we study the stability properties of the solutions of a proposed nonlinear mathematical model with nine compartments, namely, susceptible-exposed infectious with severe symptoms-infectious with mild symptoms-asymptomatic infectious-hospitalized intensive care unit-dead infectious corpses-recovered to investigate the current outbreak of coronavirus disease (COVID-19) in Morocco and beyond. In the second chapter, we employed the control theory procedure to better understand the ways to maintain the progression of the COVID-19 outbreak and its variant Omicron in a case study of Morocco by designing optimal disease intervention strategies.The third chapter is devoted to expand the proposed nonlinear model into a fractional-order COVID.
Exponential Sums, Hypergeometric Sheaves, and Monodromy Groups
An examination of some of the remarkable connections between group theory and arithmetic algebraic geometry over finite fields Exponential sums have been of great interest ever since Gauss, and their importance in analytic number theory goes back a century to Kloosterman. Grothendieck's creation of the machinery of l-adic cohomology led to the understanding that families of exponential sums give rise to local systems, while Deligne, who gave his general equidistribution theorem after proving the Riemann hypothesis part of the Weil conjectures, established the importance of the monodromy groups of these local systems. Deligne's theorem shows that the monodromy group of the local system incarnating a given family of exponential sums determines key statistical properties of the family of exponential sums in question. Despite the apparent simplicity of this relation of monodromy groups to statistical properties, the actual determination of the monodromy group in any particular situation is highly nontrivial and leads to many interesting questions. This book is devoted to the determination of the monodromy groups attached to various explicit families of exponential sums, especially those attached to hypergeometric sheaves, arguably the simplest local systems on G_m, and to some simple (in the sense of simple to write down) one-parameter families of one-variable sums. These last families turn out to have surprising connections to hypergeometric sheaves. One of the main technical advances of this book is to bring to bear a group-theoretic condition (S+), which, when it applies, implies very strong structural constraints on the monodromy group, and to show that (S+) does indeed apply to the monodromy groups of most hypergeometric sheaves.
Vector Calculus
It strives to present the material in a clear and accessible manner, with a focus on providing practical examples and applications to various fields such as physics, engineering, computer graphics, and more.Covers the essential concepts of vectors in three-dimensional Euclidean space and explains vector operations, including addition, subtraction, scalar multiplication, and dot product.Discusses partial derivatives, gradient vectors, directional derivatives, and tangent planes and also provides a comprehensive introduction to double and triple integrals.Demonstrates techniques for evaluating integrals using different coordinate systems and covers line integrals and their relationship to work and circulation.This book serves as a valuable resource for students studying mathematics, physics, engineering, and related disciplines. It also serves as an excellent reference for professionals seeking a concise and practical guide to vector calculus.
Advances in Fuzzy MCDM, Hybrid Methods, Fuzzy Number Ranking and Their Applications
This Special Issue of Axioms entitled "Advances in Fuzzy MCDM, Hybrid Methods, Fuzzy Number Ranking, and Their Applications" consists of a collection of ten papers written by eminent mathematicians and experts in their fields, covering numerous different areas of fuzzy MCDM, hybrid methods, ranking methods, and/or their applications. The objective of this Special Issue is to provide a platform for researchers to publish their recent work, delve deeper into various problems, and solve them mathematically.
Variance-Constrained Filtering for Stochastic Complex Systems
Connected Sets in Global Bifurcation Theory
This book explores the topological properties of connected and path-connected solution sets for nonlinear equations in Banach spaces, focusing on the distinction between these concepts. Building on Rabinowitz's dichotomy and classical results on Peano continua, the authors introduce "congestion points"--where connected sets fail to be weakly locally connected--and examine the extent to which their presence is compatible with path-connectedness. Through rigorous analysis and examples, the book provides new insights into global bifurcations. Structured into seven chapters, the book begins with an introduction to global bifurcation theory and foundational concepts in set theory and metric spaces. Subsequent chapters delve into connectedness, local connectedness, and congestion points, culminating in the construction of intricate examples that highlight the complexities of solution sets. The authors' careful selection of material and fluent writing style make this work a valuable resource for PhD students and experts in functional analysis and bifurcation theory.
Number Theory and Geometry Through History
Developed from a course on the history of mathematics, the book is aimed at school teachers of mathematics who need to learn more about mathematics than its history, and in a way they can communicate to middle and high school students. The author hopes to overcome, through these teachers using this book, math phobia among these students.
Game Theory for Applied Econometricians
Over the last 30 years the practice and use of game theory has changed dramatically, yet textbooks continue to present game theory with algebraic formalism and toy models. This book, on the other hand, illustrates game theory concepts using real-world data and analyses problems with real policy implications.
Number Theory and Geometry Through History
Developed from a course on the history of mathematics, the book is aimed at school teachers of mathematics who need to learn more about mathematics than its history, and in a way they can communicate to middle and high school students. The author hopes to overcome, through these teachers using this book, math phobia among these students.
Unique Continuation Properties for Partial Differential Equations
This book provides a comprehensive and self-contained introduction to the study of the Cauchy problem and unique continuation properties for partial differential equations. Aimed at graduate and advanced undergraduate students, it bridges foundational concepts such as Lebesgue measure theory, functional analysis, and partial differential equations with advanced topics like stability estimates in inverse problems and quantitative unique continuation. By presenting detailed proofs and illustrative examples, the text equips readers with a deeper understanding of these fundamental topics and their applications in mathematical analysis. Designed to serve as both a learning resource and a reference, this book is particularly suited for those pursuing research in mathematical physics, inverse problems, or applied analysis.
Deep Learning in Personalized Music Emotion Recognition
Music has a unique power to evoke strong emotions in us--bringing us to tears, lifting us into ecstasy or triggering vivid memories. Often described as a universal language, it conveys feelings that transcend words. But are machines, too, able to understand this language and capture emotions conveyed in music? This book delves into the field of Musical Emotion Recognition (MER), aiming to develop a mathematical model to predict the emotional content of music. It explores the fundamentals of this interdisciplinary research area, including the relationship between music and emotions, mathematical representations of music and deep learning algorithms. Two MER models are developed and evaluated: one employing handcrafted audio features with a long short-term memory architecture and the other using embeddings from the pre-trained music understanding model MERT. Results show that MERT embeddings can enhance predictions compared to traditional handcrafted features. Additionally, driven by the subjectivity of musical emotions and the low inter-rater agreement of annotations, this book investigates personalized emotion recognition. The findings suggest that personalized models surpass the limitations of general MER systems and can even outperform a theoretically perfect general MER system.
Complex Analysis and Dynamics in One Variable with Applications
Geometric Deformations of Discriminants and Apparent Contours
Generalization of Some Inequalities for Rational Functions with Prescribed Poles
Elements of Algebraic Topology
With coverage of homology and cohomology theory, universal coefficient theorems, Kunneth theorem, duality in manifolds, and applications to classical theorems of point-set topology, this book is perfect for communicating complex topics and the fun nature of algebraic topology for beginners.
Introduction to Linear Algebra with Earth Science Applications
This is a Linear Algebra textbook specifically designed for students who are majoring in any Earth Science-related subjects like Geophysics and Atmospheric Sciences, and are also interested in Mathematics. With these target readers in mind, we set out to provide an adequate treatment of Linear Algebra concepts that enables them to tackle relevant Earth Science problems. It aims at striking a balance between mathematical theories and physical (mainly Earth Science) applications. For the first part, it contains a comprehensive review ofMatrix and Vector OperationsVector (Inner Product) Spaces and Linear TransformationsEigenvalues/Eigenvectors and Diagonalization, and moreMeanwhile, the topics of the second part include: Principal Component Analysis (PCA)Least-square Linear RegressionDiscrete Fourier Transform (DFT)Markov ChainsMatrix Factorization Methods such as Cholesky/L(D)U/SVDDynamical SystemsContinuum MechanicsData AssimilationIn each chapter, we focus on a selected Linear Algebra topic, motivated by Earth Science examples (if available) and supplemented with Python programming tutorials. At the end of each chapter, a number of exercises (answered, a total of 136) are given for elucidating the concepts and working on Earth Science projects. This book also provides several colored figures for illustration.
