Einstein's Tutor by 
Quipu : Decorated Permutation Representations of Finite Groups by
This book studies dihedral groups, dicyclic groups, other finite subgroups of the 3-dimensional sphere, and the 2-fold extensions of the symmetric group on 4 letters from the point of view of decorated string diagrams of permutations. These are our metaphorical quipu. As you might expect, the book is replete with illustrations. In (almost) all cases, explicit diagrams for the elements of the group are given. The exception is the binary icosahedral group in which only the generators and relations are exhibited.
The Waltz of Reason : The Entanglement of Mathematics and Philosophy by
"Let no one ignorant of geometry enter here," Plato warned would-be philosophers. Mathematician Karl Sigmund agrees. In The Waltz of Reason, he shows how mathematics and philosophy together have shaped our understanding of space, chance, logic, cooperation, voting, and the social contract. Sigmund shows how game theory is integral to moral philosophy, how statistics shaped the meaning of reason, and how the search for a logical basis for math leads to deep questions about the nature of truth itself. But this is no dry tome: Sigmund's wit and humor shine as brightly as his erudition. The Waltz of Reason is an engrossing history of ideas as vibrant as a ballroom full of dancers, one that empowers as it entertains, following the complex and occasionally dizzying steps of the thinkers who have molded our thought and founded our world.
Pragmatic Mathematics for Scientists and Engineers by
This is a textbook on basic to intermediate mathematics for undergraduate students majoring in the physical sciences and engineering. Many chapters, covering topics like Green's functions, calculus of variations, and functions of a complex variable, are well-suited for graduate classes. Additionally, researchers can benefit from the book as a mathematical refresher for their professional work.The book provides readers with a fundamental understanding of underlying principles, using derivations based more on mathematical intuition rather than exposing them to multiple theorems, proofs, and lemmas. Each chapter includes highly relevant examples with detailed solutions and explanations, promoting a practical application of knowledge to real problems in the physical sciences. For the convenience of both students and instructors, there are end-of-chapter exercises with answers that can be easily utilized for assignments.The book is not a replacement for calculus textbooks, but rather a guide to the mathematics most relevant to the physical sciences and engineering.In conclusion, this book can be readily adapted for upper-level undergraduate and graduate classes, particularly those focusing on mathematical methods for students in physical sciences, applied mathematics, and engineering majors.
Advanced Algebra by
This book evolved from our lectures in the advanced algebra courses at Nanjing University. It is intended for use by instructors and undergraduate students in a one-year advanced algebra course.The topics covered in this book consist of integers and polynomials, determinants and matrices, linear systems, linear spaces, linear maps, λ-matrices, quadratic forms, inner product spaces, and bilinear forms.There are sufficient well-selected exercises of a wide range to provide ample practice, expand coverage of topics treated in the text, and challenge the strongest students.Our objective is to prepare students with a solid foundation for pursuing advanced mathematics. We hope that the material presented here will interest the students and help them ask their own questions, look for their own examples, and discover their own proofs.
the Relay Race to Infinity : Developments in Mathematics from Euclid to Fermat by
Why were most historically important mathematicians wealthy? Why were they often lawyers and many had pastors for fathers? Why were original results sometimes discovered by two mathematicians independently within a short time of each other? Why did the Italian Fibonacci, speak Arabic? It all began a couple of years ago, when one of the authors started to write short biographies of important historical mathematicians for the teaching journal Australian Primary Mathematics Classroom. It was felt that teachers generally knew very little about the way the subject developed or the people who developed it. And it was felt that historical knowledge would help them see how the subject progressed and enable them to fit in with the historical episodes that would be of interest to students. Clearly, the book that developed contains mathematics up to the 17th century, but we are keen to set the subject in those times, to try to give short biographies of the people involved, as well as provide a perspective of the events that led up to the times and led up to the mathematics. Importantly, it is shown that the maths enterprise was not undertaken by a small few, but worked like a relay race. One or a few might take up an idea and develop it, but it often gets only so far. Later, others would take up the idea, the baton, and the relay race to find results continues.
Introduction to the Calculus of Variations by
The calculus of variations is one of the oldest subjects in mathematics, and it is very much alive and still evolving. Besides its mathematical importance and its links to other branches of mathematics, such as geometry or differential equations, it is widely used in physics, engineering, economics and biology.This book serves both as a guide to the expansive existing literature and as an aid to the non-specialist -- mathematicians, physicists, engineers, students or researchers -- in discovering the subject's most important problems, results and techniques. Despite the aim of addressing non-specialists, mathematical rigor has not been sacrificed; most of the theorems are either fully proved or proved under more stringent conditions.This new edition offers an entirely new chapter, as well as the addition of several new exercises. The book, containing a total of 147 exercises with detailed solutions, is well designed for a course at both undergraduate and graduate levels.
Aspects of Harmonic Analysis on Locally Compact Abelian Groups by
The Fourier transform is a "tool" used in engineering and computer vision to model periodic phenomena. Starting with the basics of measure theory and integration, this book delves into the harmonic analysis of locally compact abelian groups. It provides an in-depth tour of the beautiful theory of the Fourier transform based on the results of Gelfand, Pontrjagin, and Andre Weil in a manner accessible to an undergraduate student who has taken linear algebra and introductory real analysis. Highlights of this book include the Bochner integral, the Haar measure, Radon functionals, the theory of Fourier analysis on the circle, and the theory of the discrete Fourier transform. After studying this book, the reader will have the preparation necessary for understanding the Peter-Weyl theorems for complete, separable Hilbert algebras, a key theoretical concept used in the construction of Gelfand pairs and equivariant convolutional neural networks.
Languages and Automata : GAGTA Book 3 by
This reference discusses how automata and language theory can be used to understand solutions to solving equations in groups and word problems in groups. Examples presented include, how Fine scale complexity theory has entered group theory via these connections and how cellular automata, has been generalized into a group theoretic setting. Chapters written by experts in group theory and computer science explain these connections.
Ninety Years of zbMATH by
zbMATH Open, the world’s most comprehensive and longest-running abstracting and reviewing service in pure and applied mathematics, was founded by Otto Neugebauer in 1931. It celebrated its 90th anniversary by becoming an open access database. In December 2019, the Joint Science Conference (Gemeinsame Wissenschaftskonferenz) agreed that the Federal and State Governments of Germany would support FIZ Karlsruhe in transforming zbMATH into an open platform. In the future, zbMATH Open will link mathematical services and platforms so as to provide considerably more content for further research and collaborative work in mathematics and related fields.
This book explains how zbMATH Open has reacted to a rapidly changing digital era. Topics covered include: the linkage of zbMATH Open with different community platforms and digital maths libraries, the use of zbMATH Open as a bibliographical tool, API solutions, current advancements in author profiles, the indexing of mathematical software packages (swMATH), and issues concerning mathematical formula search in zbMATH Open. The authors also reflect on the gender publication gap in mathematics and focus on one of the central pillars of zbMATH Open: the community of reviewers.
Nonlinear Trending Time Series : Theory and Practice by
Trends often play a dominant role in many empirical time series data, but do we truly understand these trends? What challenges arise when dealing with complex, nonstationary trending time series? How do we model such trends in fields like economics, finance, and climate change?This book provides an overview of recently developed models for trending time series and introduces new nonlinear, nonparametric, and semiparametric methods. Specifically, it offers practical approaches to address the problem of endogeneity in linear trending regression models where the trend component is either weak or strong. Additionally, it proposes a testing procedure for identifying common trends across multiple time series and includes illustrative examples from economics, finance, and climate change.As a practical toolbox, this book equips analysts with essential methods for investigating trends in their research and applications.
Cluster Analysis : A Primer Using R by
Cluster analysis is a fundamental data analysis task that aims to group similar data points together, revealing the inherent structure and patterns within complex datasets. This book serves as a comprehensive and accessible guide, taking readers on a captivating journey through the foundational principles of cluster analysis.At its core, the book delves deeply into various clustering algorithms, covering partitioning methods, hierarchical methods, and advanced techniques such as mixture density-based clustering, graph clustering, and grid-based clustering. Each method is presented with clear, concise explanations, supported by illustrative examples and hands-on implementations in the R programming language -- a popular and powerful tool for data analysis and visualization.Recognizing the importance of cluster validation and evaluation, the book devotes a dedicated chapter to exploring a wide range of internal and external quality criteria, equipping readers with the necessary tools to assess the performance of clustering algorithms. For those eager to stay at the forefront of the field, the book also presents deep learning-based clustering methods, showcasing the remarkable capabilities of neural networks in uncovering hidden structures within complex, high-dimensional data. Whether you are a student seeking to expand your knowledge, a data analyst looking to enhance your toolbox, or a researcher exploring the frontiers of data analysis, this book will provide you with a solid foundation in cluster analysis and empower you to tackle a wide range of data-driven problems.
Algebraic Topology : A Toolkit by
This book is ideal as an introduction to algebraic topology and applied algebraic topology. It features a streamlined approach that covers basic categorical notions, simplicial, cellular, and singular homology, persistent homology, cohomology groups, cup products, Poincare Duality, homotopy theory, and spectral sequences. The focus is on examples and computations, and there are many end of chapter exercises and extensive student projects.
A Primer on Smooth Manifolds by
Differential Geometry is one of the major branches of current Mathematics, and it is an unavoidable language in modern Physics. The main characters in Differential Geometry are smooth manifolds: a class of geometric objects that locally behave like the standard Euclidean space.The book provides a first introduction to smooth manifolds, aimed at undergraduate students in Mathematics and Physics. The only prerequisites are the Linear Algebra and Calculus typically covered in the first two years. The presentation is as simple as possible, but it does not sacrifice the rigor.The lecture notes are divided into 10 chapters, with gradually increasing difficulty. The first chapters cover basic material, while the last ones present more sophisticated topics. The definitions, propositions, and proofs are complemented by examples and exercises. The exercises, which include part of the proofs, are designed to help the reader learn the language of Differential Geometry and develop their problem-solving skills in the area. The exercises are also aimed at promoting an active learning process. Finally, the book contains pictures which are useful aids for the visualization of abstract geometric situations. The lecture notes can be used by instructors as teaching material in a one-semester course on smooth manifolds.
P(X)-Bi-Laplacian : Application on Time-PDEs in Viscoelasticity by
The main subject of our book is to use the (p, p(x) and p(x))-bi-Laplacian operator in some partial differential systems, where we developed and obtained many results in quantitative and qualitative point of view.
Algebraic Topology : A Structural Introduction by
Algebraic Topology is a system and strategy of partial translations, aiming to reduce difficult topological problems to algebraic facts that can be more easily solved. The main subject of this book is singular homology, the simplest of these translations. Studying this theory and its applications, we also investigate its underlying structural layout - the topics of Homological Algebra, Homotopy Theory and Category Theory which occur in its foundation.This book is an introduction to a complex domain, with references to its advanced parts and ramifications. It is written with a moderate amount of prerequisites -- basic general topology and little else -- and a moderate progression starting from a very elementary beginning. A consistent part of the exposition is organised in the form of exercises, with suitable hints and solutions.It can be used as a textbook for a semester course or self-study, and a guidebook for further study.
