Digital library
Textbooks, references and open learning resources.
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Linear Algebra Done Right
Sheldon Axler · Springer · 2024
Now available in Open Access, this best-selling textbook for a second course in linear algebra is aimed at undergraduate math majors and graduate students. The fourth edition gives an expanded treatment of the singular value decomposition and its consequences. It includes a new chapter on multilinear algebra, treating bilinear forms, quadratic forms, tensor products, and an approach to determinants via alternating multilinear forms. This new edition also increases the use of the minimal polynomial to provide cleaner proofs of multiple results. Also, over 250 new exercises have been added. The novel approach taken here banishes determinants to the end of the book. The text focuses on the central goal of linear algebra: understanding the structure of linear operators on finite-dimensional vector spaces. The author has taken unusual care to motivate concepts and simplify proofs. A variety of interesting exercises in each chapter helps students understand and manipulate the objects of linear algebra. Beautiful formatting creates pages with an unusually student-friendly appearance in both print and electronic versions. No prerequisites are assumed other than the usual demand for suitable mathematical maturity. The text starts by discussing vector spaces, linear independence, span, basis, and dimension. The book then deals with linear maps, eigenvalues, and eigenvectors. Inner-product spaces are introduced, leading to the finite-dimensional spectral theorem and its consequences. Generalized eigenvectors are then used to provide insight into the structure of a linear operator.
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Statistics Minus The Math
Nathan Favero · Nathan Favero · 2024
This book was largely adapted from the public domain resource Online Statistics Education: A Multimedia Course of Study (https://onlinestatbook.com Project Leader: David M. Lane, Rice University). A huge thanks to David Lane and his colleagues at Rice University for their creation of this wonderful resource. I use footnotes throughout to indicate precisely where the various sections of each chapter came from. Chapters 11-13 (as well as Section 3.5, Section 4.2.2, most of Section 4.2.3, and Section 9.3.1) were written by me (Nathan). This book is meant to be a free resource and is licensed under CC BY 4.0. You’re welcome to share or adapt it, as long as you provide attribution to any work of mine that you use. This book was made using Quarto and is hosted using GitHub Pages. There are still formatting inconsistencies, and this is ever a work in progress. If you find errors, feel free to reach out (find updated contact info here: https://nathanfavero.com) so I can correct them for the next version I publish.
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Geometry with an Introduction to Cosmic Topology
Michael Hitchman · Michael P. Hitchman · 2020
Motivated by questions in cosmology, the open-content text Geometry with an Introduction to Cosmic Topology uses Mobius transformations to develop hyperbolic, elliptic, and Euclidean geometry - three possibilities for the global geometry of the universe. The text, written for students who have taken vector calculus, also explores the interplay between the shape of a space and the type of geometry it admits. Geometry is suitable for a semester course in non-Euclidean geometry or as a guide to independent study, with over 200 exercises and several essays on topics including the history of geometry, parallax and curvature, and research aimed at determining the shape of the universe.
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A Cool Brisk Walk Through Discrete Mathematics
Stephen Davies · University of Mary Washington · 2023
A Cool Brisk Walk Through Discrete Mathematics - and its companion site "allthemath" - are completely-and-forever-free-and-open-source educational materials dedicated to the mathematics that budding computer science practitioners actually need to know. They feature the fun and addictive teaching of award-winning lecturer Dr. Stephen Davies of the University of Mary Washington in Fredericksburg, Virginia!
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Basic Statistics Using R for Crime Analysis
Jaeyoung Choi · The Pennsylvania Alliance for Design of Open Textbooks (PA-ADOPT) · 2024
Limited access to subscription-based statistical software poses obstacles when students want to apply the skills they acquired in college. Although students may learn programs like SPSS or Stata while at the university, they often find themselves unable to continue using these programs after graduation, making their acquired skills obsolete. As an open-source software program, R offers a solution to this challenge. It is freely accessible to anyone, including students, after they graduate. Therefore, I decided to write a freely available book for those interested in becoming crime analysts, focusing on learning statistics without delving too deeply into mathematics. Moreover, this book emphasizes practical applications by utilizing R for data analysis, ensuring students can develop relevant skills beyond the university. I hope that students can easily follow the instructions in this book and replicate the same outcomes using the provided data. This practical experience will demonstrate the value of statistics and R, ideally inspiring students to further their learning in these areas.
![Statistics Through an Equity Lens [Revised Edition]](/api/files/covers/otl-1625.webp)
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Statistics Through an Equity Lens [Revised Edition]
Yvonne Anthony · ROTEL · 2023
This Open Educational Resource (OER) carries a significant responsibility by presenting statistics through an equity lens. The metaphor of a lens is used intentionally–as the glasses one wears can have a profound effect on what one sees. The book encourages further inspection of the ways in which data is collected, interpreted, and analyzed on a variety of social justice issues, such as health disparities, hunger and food insecurity, homelessness, behavioral health (mental health and substance use), and incarceration of males of color. It also attempts to reveal how the misuse of data can reinforce inequities, for example, by stigmatizing people and labeling neighborhoods as high poverty, violent, and having poor educational opportunities. Whether an intended or unintended consequence, irresponsible data use can contribute to racist impressions of people and communities.
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Statistics for Ecologists: A Frequentist and Bayesian Treatment of Modern Regression Models
John Fieberg · University of Minnesota Libraries Publishing · 2024
Ecological data pose many challenges to statistical inference. Most data come from observational studies rather than designed experiments; observational units are frequently sampled repeatedly over time, resulting in multiple, non-independent measurements; response data are often binary (e.g., presence-absence data) or non-negative integers (e.g., counts), and therefore, the data do not fit the standard assumptions of linear regression (Normality, independence, and constant variance). This book will familiarize readers with modern statistical methods that address these complexities using both frequentist and Bayesian frameworks.
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Counting Rocks! An Introduction to Combinatorics
Henry Adams; Kelly Emmrich; Maria Gillespie; Shannon Golden · Henry Adams, Rachel Pries, and Maria Gillespie · 2023
This textbook, Counting Rocks!, is the written component of an interactive introduction to combinatorics at the undergradaute level. Throughout the text, we link to videos where we describe the material and provide examples; see the Youtube playlist on the Colorado State University (CSU) Mathematics YouTube channel. The major topics in this text are counting problems (Chapters 1-4), proof techniques (Chapter 5), recurrence relations and generating functions (Chapters 6-7), and graph theory (Chapters 8-12). The material and the problems we include are standard for an undergraduate combinatorics course. In this text, one of our goals was to describe the mathematical structures underlying problems in combinatorics. For example, we separate the description of sequences, permutations, sets and multisets in Chapter 3. In addition to the videos, we would like to highlight some other features of this book. Most chapters contain an investigation section, where students are led through a series of deeper problems on a topic. In several sections, we show students how to use the free online computing software SAGE in order to solve problems; this is especially useful for the problems on recurrence relations. We have included many helpful figures throughout the text, and we end each chapter (and many of the sections) with a list of exercises of varying difficulty.
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Linear Transformations on Vector Spaces
Scott Kaschner; Amber Russell · PALNI · 2023
Linear Transformations on Vector Spaces serves primarily as a textbook for undergraduate Linear Algebra courses. While standard Linear Algebra books begin by focusing on solving systems of linear equations and associated procedural skills, our book begins by developing a conceptual framework for the topic using the central objects, vector spaces and linear transformations. It covers the same concepts, skills, and, applications as conventional texts in a one-semester course, but students walk away with a much richer and more useful mastery of the topics. The book is structured to facilitate the implementation of the flipped classroom. The text features a continuous narrative to illuminate the big picture of the material and is written to help students develop their textbook reading skills. Also, there are “Explorations” scattered throughout each section; these are quick first examples intended for students to complete while reading before class meetings. Additional materials include section overview homework assignments and worksheets that can be used for in-class practice.
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Introduction to Probability
John Baxter · John R. Baxter · 2026
This is an introduction to probability theory, designed for self-study. It covers the same topics as the one-semester introductory courses which I taught at the University of Minnesota, with some extra discussion for reading on your own. The reasons which underlie the rules of probability are emphasized. Probability theory is certainly useful. But how does it feel to study it? Well, like other areas of mathematics, probability theory contains elegant concepts, and it gives you a chance to exercise your ingenuity, which is often fun. But in addition, randomness and probability are part of our experience in the real world, present everywhere and yet still somewhat mysterious. This gives the subject of probability a special interest.
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Polimetrics: A Stata Companion to Introduction to Political Science Research Methods
Josh Franco · Open Political Science (OPoliSci) · 2021
Polimetrics: A Stata Companion, authored by Dr. Josh Franco, is an Open Education Resource workbook licensed CC BY-NC and designed as a Stata companion to Introduction to Political Science Research Methods. This workbook provides a tour of the Stata software, an introduction to cross-sectional, time series, and panel data, and an introduction to a variety of models. I review models where the outcome is linear, binary, ordinal, categorical, and count. Additionally, I have an interpretation chapter on survival models. Each “Models” chapter has a similar organizational structure: about, estimated time, what is the model, how are models run in Stata, how do we interpret the model results, and a real-world example of model results in a Creative Commons licensed, peer-reviewed journal article. Additionally, mini-assignment instructions and a rubric are included so students can practice their interpretation skills.
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Classical Numerical Methods in Scientific Computing
Jos van Kan; Guus Segal; Fred Vermolen; Hans Kraaijevanger · TU Delft Open · 2023
Partial differential equations are paramount in mathematical modelling with applications in engineering and science. The book starts with a crash course on partial differential equations in order to familiarize the reader with fundamental properties such as existence, uniqueness and possibly existing maximum principles. The main topic of the book entails the description of classical numerical methods that are used to approximate the solution of partial differential equations. The focus is on discretization methods such as the finite difference, finite volume and finite element method. The manuscript also makes a short excursion to the solution of large sets of (non)linear algebraic equations that result after application of discretization method to partial differential equations. The book treats the construction of such discretization methods, as well as some error analysis, where it is noted that the error analysis for the finite element method is merely descriptive, rather than rigorous from a mathematical point of view. The last chapters focus on time integration issues for classical time-dependent partial differential equations. After reading the book, the reader should be able to derive finite element methods, to implement the methods and to judge whether the obtained approximations are consistent with the solution to the partial differential equations. The reader will also obtain these skills for the other classical discretization methods. Acquiring such fundamental knowledge will allow the reader to continue studying more advanced methods like meshfree methods, discontinuous Galerkin methods and spectral methods for the approximation of solutions to partial differential equations.
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Numerical Methods in Scientific Computing
Jos van Kan; Guus Segal; Fred Vermolen · TU Delft Open · 2023
This is a book about numerically solving partial differential equations occurring in technical and physical contexts and the authors have set themselves a more ambitious target than to just talk about the numerics. Their aim is to show the place of numerical solutions in the general modeling process and this must inevitably lead to considerations about modeling itself. Partial differential equations usually are a consequence of applying first principles to a technical or physical problem at hand. That means, that most of the time the physics also have to be taken into account especially for validation of the numerical solution obtained. This book aims especially at engineers and scientists who have ’real world’ problems. It will concern itself less with pesky mathematical detail. For the interested reader though, we have included sections on mathematical theory to provide the necessary mathematical background. Since this treatment had to be on the superficial side we have provided further reference to the literature where necessary.
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Introducing Mathematical Biology
Alex Best · The University of Sheffield · 2023
Mathematical modelling plays an increasingly important role in almost any area of life sciences, and this interactive textbook focuses on the areas of population ecology, infectious diseases, immunology and cell dynamics, gene networks and pharmacokinetics. It is aimed at anyone who is interested in learning about how to model biological systems, including undergraduate and postgraduate mathematics students who have not studied mathematical biology before, life-sciences students with an interest in modelling, and post-16 mathematics students interested in university-level material. Some mathematical knowledge is assumed, and the mathematical models used are all in the form of ordinary differential equations.
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Foundations of Biomedical Science: Quantitative Literacy: Theory and Problems
Julian Pakay · La Trobe University · 2023
Foundations of Biomedical Science: Quantitative Literacy Theory and Problems is designed to help students develop the fundamental mathematical and quantitative literacy required to navigate and interpret evidence-based Biomedical data. This will provide students with the skills and confidence to habitually question any quantitative data they come across and to use these skills to make informed judgements regarding their veracity.
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Contemporary Mathematics
Donna Kirk · OpenStax · 2023
Contemporary Mathematics is designed to meet the scope and sequence requirements for a liberal arts mathematics course. This resource provides stand-alone sections with a focus on showing relevance in the features as well as the examples, exercises, and exposition. Contemporary Mathematics integrates technology applications, projects, and highlights a diverse group of contributors to mathematics, statistics, and related fields. Print copies of this book will be available for the fall 2023 semester. You can sign up to be notified when print copies are available.
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An Intuitive, Interactive, Introduction to Biostatistics
Caitlin Ward; Collin Nolte · University of Iowa · 2023
"An Intuitive, Interactive Introduction to Biostatistics" is an introductory statistics textbook oriented towards towards undergraduate students in the health sciences. While covering the breadth of material typically presented in a first semester statistics course, including introductions to probability and distributions, study design, CLT, hypothesis testing, and inference, IIIB distinguishes itself with its focus on cultivating student intuition through the use of guided questions and interactive simulation-based applets. Written in R, this open-source text has been created with customizability in mind, offering instructors maximal flexibility in arranging and modifying the content.
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Introduction to Arithmetic Groups
Dave Morris · Deductive Press · 2015
Introduction to Arithmetic Groups
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Numerical Methods for Ordinary Differential Equations
Kees Vuik; Fred Vermolen; Martin van Gijzen · TU Delft Open · 2023
In this book we discuss several numerical methods for solving ordinary differential equations. We emphasize the aspects that play an important role in practical problems. We confine ourselves to ordinary differential equations with the exception of the last chapter in which we discuss the heat equation, a parabolic partial differential equation. The techniques discussed in the intro-ductory chapters, for instance interpolation, numerical quadrature and the solution to nonlinear equations, may also be used outside the context of differential equations. They have been in-cluded to make the book self-contained as far as the numerical aspects are concerned. Chapters, sections and exercises marked with a * are not part of the Delft Institutional Package. The numerical examples in this book were implemented in Matlab, but also Python or any other programming language could be used. A list of references to background knowledge and related literature can be found at the end of this book. Extra information about this course can be found at http://NMODE.ewi.tudelft.nl, among which old exams, answers to the exercises, and a link to an online education platform. We thank Matthias Moller for his thorough reading of the draft of this book and his helpful suggestions.
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COMPUTATIONAL PHYSICS: A Practical Introduction to Computational Physics and Scientific Computing (using C++)
Konstantinosn Anagnostopoulos · National Technical University of Athens · 2016
This book is an introduction to the computational methods used in physics, but also in other scientific fields. It is addressed to an audience that has already been exposed to the introductory level of college physics, usually taught during the first two years of an undergraduate program in science and engineering. It assumes no prior knowledge of numerical analysis, programming or computers and teaches whatever is necessary for the solution of the problems addressed in the text. It can be used as a textbook in introductory computational physics or scientific computing classes. The book starts with very simple problems in particle motion and ends with an in-depth discussion of advanced techniques used in Monte Carlo simulations in statistical mechanics. The level of instruction rises slowly, while discussing problems like the diffusion equation, electrostatics on the plane, quantum mechanics and random walks. All the material can be taught in two semesters, but a selection of topics can form the material of a one semester course. The book aims to provide the students with the background and the experience needed in order to advance to high performance computing projects in science and engineering. It puts emphasis on hands--on programming of numerical code but also on the production, analysis and interpretation of data. But it also tries to keep the students motivated by considering interesting applications in physics, like chaos, quantum mechanics, special relativity and the physics of phase transitions. There is a C++ and a Fortran edition for the core programming. Data analysis is performed using the powerful tools of the GNU/Linux environment. All the necessary software is open source and freely available. The book and the accompanying software are given under a Creative Commons License/GNU public License as a service to the community. It can be used freely as a whole, or any part of it, in any form, by anyone. There is no official distribution of hard copies, b