Preface This is a first draft of a free (as in speech, not as in beer) (although it is free as in beer as well) undergraduate number theory textbook. It was used for Math 319 at Colorado State University – Pueblo in the spring semester of 2014. Thanks are hereby offered to the students in that class — Megan Bissell, Tennille Candelaria, Ariana Carlyle, Michael Degraw, Daniel Fisher, Aaron Griffin, Lindsay Harder, Graham Harper, Helen Huang, Daniel Nichols, and Arika Waldrep — who offered many useful suggestions and found numerous typos. I am also grateful to the students in my Math 242 Introduction to Mathematical Programming class in that same spring semester of 2014 — Stephen Ciruli, Jamen Cox, Graham Harper, Joel Kienitz, Matthew Klamm, Christopher Martin, Corey Sullinger, James Todd, and Shelby Whalen — whose various programming projects produced code that I adapted to make some of the figures and examples in the text.
The author gratefully acknowledges the work An Introductory Course in Elementary Number Theory by Wissam Raji [see www.saylor.org/books/] from which this was initially adapted. Raji's text was released under the Creative Commons CC BY 3.0 license, see creativecommons.org/licenses/by/3.0. This work is instead released under a CC BY-SA 4.0 license, see creativecommons.org/licenses/by-sa/4.0. (The difference is that if you build future works off of this one, you must also release your derivative works with a license that allows further remixes over which you have no control.)
be sure to check out the materials that @poritzj has shared at his website, incl. all you wanted to know about cryptography but were afraid to ask:https://t.co/SdxbbNlNsT
Yet Another Introductory Number Theory Textbook (Cryptology Emphasis Version) — CC-licensed! #Domains19 https://t.co/HsWU5gxvmM
— Laura Gibbs (@OnlineCrsLady) June 10, 2019
Andrew Jordan reviews Peter Woit's Quantum Theory, Groups and Representations and finds much to admire.
I also don’t think I’ve ever come across the journal Inference before, but it looks quite nice in terms of content and editorial.
Affordable education. Transparent science. Accessible scholarship.
These ideals are slowly becoming a reality thanks to the open education, open science, and open access movements. Running separate—if parallel—courses, they all share a philosophy of equity, progress, and justice. This book shares the stories, motives, insights, and practical tips from global leaders in the open movement.
To be published by Cambridge University Press in April 2018.
Upon publication this book will be available for purchase through Cambridge University Press and other standard distribution channels. Please see the publisher's web page to pre-order the book or to obtain further details on its publication date.
A draft, pre-publication copy of the book can be found below. This draft copy is made available for personal use only and must not be sold or redistributed.
This largely self-contained book on the theory of quantum information focuses on precise mathematical formulations and proofs of fundamental facts that form the foundation of the subject. It is intended for graduate students and researchers in mathematics, computer science, and theoretical physics seeking to develop a thorough understanding of key results, proof techniques, and methodologies that are relevant to a wide range of research topics within the theory of quantum information and computation. The book is accessible to readers with an understanding of basic mathematics, including linear algebra, mathematical analysis, and probability theory. An introductory chapter summarizes these necessary mathematical prerequisites, and starting from this foundation, the book includes clear and complete proofs of all results it presents. Each subsequent chapter includes challenging exercises intended to help readers to develop their own skills for discovering proofs concerning the theory of quantum information.
Heterodox Academy has produced a new book based on John Stuart Mill’s famous essay On Liberty to make it accessible for the 21st century. Here’s what makes our edition special:
1) It’s just the second chapter (out of 5), because that chapter gives the best arguments ever made for the importance of free speech and viewpoint diversity;
2) We have reduced that chapter by 50% to remove repetitions and historical references that would be obscure today, producing a very readable 7000 word essay;
3) Editors Richard Reeves (a biographer of Mill) and Jon Haidt (a social psychologist) have written a brief introduction to link Mill and his time to the issues of our time, and
4) Artist Dave Cicirelli has created 16 gorgeous original illustrations that amplify the power of Mill’s metaphors and arguments.
All Minus One is ideal for use in college courses, advanced high school classes, or in any organization in which people would benefit from productive disagreement. We offer free and paid versions of the book below.
h/t Claire Lehmann
A wonderful resource for every highschooler & uni student: a condensed version of JS Mill’s arguments for free speech, with accompanying illustrations.
If you have kids, give it to them. (Also it’s free)https://t.co/GWmwffTcmH
— Claire Lehmann (@clairlemon) April 13, 2018
This book is an invitation to discover advanced topics in category theory through concrete, real-world examples. It aims to give a tour: a gentle, quick introduction to guide later exploration. The tour takes place over seven sketches, each pairing an evocative application, such as databases, electric circuits, or dynamical systems, with the exploration of a categorical structure, such as adjoint functors, enriched categories, or toposes. No prior knowledge of category theory is assumed. [.pdf]
The main purpose of this blog is to share updates about the open-access, open-source textbook Understanding Linear Algebra. Though work is continuing on this project, the HTML version of the text is now freely available, the forthcoming PDF version will also be free, and low-cost print options will be provided. The PreTeXt source code will be posted on GitHub as well.
Homotopy type theory is a new branch of mathematics that combines aspects of several different fields in a surprising way. It is based on a recently discovered connection between homotopy theory and type theory. It touches on topics as seemingly distant as the homotopy groups of spheres, the algorithms for type checking, and the definition of weak ∞-groupoids. Homotopy type theory offers a new “univalent” foundation of mathematics, in which a central role is played by Voevodsky’s univalence axiom and higher inductive types. The present book is intended as a first systematic exposition of the basics of univalent foundations, and a collection of examples of this new style of reasoning — but without requiring the reader to know or learn any formal logic, or to use any computer proof assistant. We believe that univalent foundations will eventually become a viable alternative to set theory as the “implicit foundation” for the unformalized mathematics done by most mathematicians.
A Course in Game Theory presents the main ideas of game theory at a level suitable for graduate students and advanced undergraduates, emphasizing the theory's foundations and interpretations of its basic concepts. The authors provide precise definitions and full proofs of results, sacrificing generalities and limiting the scope of the material in order to do so. The text is organized in four parts: strategic games, extensive games with perfect information, extensive games with imperfect information, and coalitional games. It includes over 100 exercises.
Seymour Papert’s Mindstorms was published by Basic Books in 1980, and outlines his vision of children using computers as instruments for learning. A second edition, with new Forewords by John Sculley and Carol Sperry, was published in 1993. The book remains as relevant now as when first published almost forty years ago.
The Media Lab is grateful to Seymour Papert’s family for allowing us to post the text here. We invite you to add your comments and reflections.
If you are interested in purchasing the print edition of Mindstorms, please visit Basic Books.
This short introduction to category theory is for readers with relatively little mathematical background. At its heart is the concept of a universal property, important throughout mathematics. After a chapter introducing the basic definitions, separate chapters present three ways of expressing universal properties: via adjoint functors, representable functors, and limits. A final chapter ties the three together. For each new categorical concept, a generous supply of examples is provided, taken from different parts of mathematics. At points where the leap in abstraction is particularly great (such as the Yoneda lemma), the reader will find careful and extensive explanations.
My friend Tom Leinster has written a great introduction to that wonderful branch of math called category theory! It’s free:
It starts with the basics and it leads up to a trio of related concepts, which are all ways of talking about universal properties.
Huh? What’s a ‘universal property’?
In category theory, we try to describe things by saying what they do, not what they’re made of. The reason is that you can often make things out of different ingredients that still do the same thing! And then, even though they will not be strictly the same, they will be isomorphic: the same in what they do.
A universal property amounts to a precise description of what an object does.
Universal properties show up in three closely connected ways in category theory, and Tom’s book explains these in detail:
through representable functors (which are how you actually hand someone a universal property),
through limits (which are ways of building a new object out of a bunch of old ones),
through adjoint functors (which give ways to ‘freely’ build an object in one category starting from an object in another).
If you want to see this vague wordy mush here transformed into precise, crystalline beauty, read Tom’s book! It’s not easy to learn this stuff – but it’s good for your brain. It literally rewires your neurons.
Here’s what he wrote, over on the category theory mailing list:
My introductory textbook “Basic Category Theory” was published by Cambridge University Press in 2014. By arrangement with them, it’s now also free online:
It’s also freely editable, under a Creative Commons licence. For instance, if you want to teach a class from it but some of the examples aren’t suitable, you can delete them or add your own. Or if you don’t like the notation (and when have two category theorists ever agreed on that?), you can easily change the Latex macros. Just go the arXiv, download, and edit to your heart’s content.
There are lots of good introductions to category theory out there. The particular features of this one are:
• It’s short.
• It doesn’t assume much.
• It sticks to the basics.
Instagram filter used: Clarendon
Photo taken at: UCLA Bookstore
I just saw Emily Riehl‘s new book Category Theory in Context on the shelves for the first time. It’s a lovely little volume beautifully made and wonderfully typeset. While she does host a free downloadable copy on her website, the book and the typesetting is just so pretty, I don’t know how one wouldn’t purchase the physical version.
I’ll also point out that this is one of the very first in Dover’s new series Aurora: Dover Modern Math Originals. Dover has one of the greatest reprint collections of math texts out there, I wish them the best in publishing new works with the same quality and great prices as they always have! We need more publishers like this.
“contains significant amounts of material not well-explained elsewhere.”He expects to finish up the diagrams and publish it next year some time, potentially through Springer.
Advanced Data Analysis from an Elementary Point of View
by Cosma Rohilla Shalizi
This is a draft textbook on data analysis methods, intended for a one-semester course for advance undergraduate students who have already taken classes in probability, mathematical statistics, and linear regression. It began as the lecture notes for 36-402 at Carnegie Mellon University.
By making this draft generally available, I am not promising to provide any assistance or even clarification whatsoever. Comments are, however, welcome.
The book is under contract to Cambridge University Press; it should be turned over to the press before the end of 2015. A copy of the next-to-final version will remain freely accessible here permanently.
Table of contents:
I. Regression and Its Generalizations
- Regression Basics
- The Truth about Linear Regression
- Model Evaluation
- Smoothing in Regression
- The Bootstrap
- Weighting and Variance
- Additive Models
- Testing Regression Specifications
- Logistic Regression
- Generalized Linear Models and Generalized Additive Models
- Classification and Regression Trees
II. Distributions and Latent Structure
- Density Estimation
- Relative Distributions and Smooth Tests of Goodness-of-Fit
- Principal Components Analysis
- Factor Models
- Nonlinear Dimensionality Reduction
- Mixture Models
- Graphical Models
III. Dependent Data
- Time Series
- Spatial and Network Data
- Simulation-Based Inference
IV. Causal Inference
- Graphical Causal Models
- Identifying Causal Effects
- Causal Inference from Experiments
- Estimating Causal Effects
- Discovering Causal StructureAppendices
- Data-Analysis Problem Sets
- Reminders from Linear Algebra
- Big O and Little o Notation
- Taylor Expansions
- Multivariate Distributions
- Algebra with Expectations and Variances
- Propagation of Error, and Standard Errors for Derived Quantities
- chi-squared and the Likelihood Ratio Test
- Proof of the Gauss-Markov Theorem
- Rudimentary Graph Theory
- Information Theory
- Hypothesis Testing
- Writing R Functions
- Random Variable Generation
- Unified treatment of information-theoretic topics (relative entropy / Kullback-Leibler divergence, entropy, mutual information and independence, hypothesis-testing interpretations) in an appendix, with references from chapters on density estimation, on EM, and on independence testing
- More detailed treatment of calibration and calibration-checking (part II)
- Missing data and imputation (part II)
- Move d-separation material from “causal models” chapter to graphical models chapter as no specifically causal content (parts II and IV)?
- Expand treatment of partial identification for causal inference, including partial identification of effects by looking at all data-compatible DAGs (part IV)
- Figure out how to cut at least 50 pages
- Make sure notation is consistent throughout: insist that vectors are always matrices, or use more geometric notation?
- Move simulation to an appendix
- Move variance/weights chapter to right before logistic regression
- Move some appendices online (i.e., after references)?
(Text last updated 30 March 2016; this page last updated 6 November 2015)