QuoteThe Chicago PMI dropped -10.5 points in August, to 47.1, the lowest level since December 2025 and returning to contraction in business activity.
QuoteIntroduction
This text explains how to use mathematical models and methods to analyze problems that arise in computer science. Proofs play a central role in this work because the authors share a belief with most mathematicians that proofs are essential for genuine understanding. Proofs also play a growing role in computer science; they are used to certify that software and hardware will always behave correctly, something that no amount of testing can do.
Simply put, a proof is a method of establishing truth. Like beauty, "truth" sometimes depends on the eye of the beholder, and it should not be surprising that what constitutes a proof differs among fields. For example, in the judicial system, legal truth is decided by a jury based on the allowable evidence presented at trial. In the business world, authoritative truth is specified by a trusted person or organization, or maybe just your boss. In fields such as physics or biology, scientific truth is confirmed by experiment. In statistics, probable truth is established by statistical analysis of sample data.
Philosophical[./i] proof involves careful exposition and persuasion typically based on a series of small, plausible arguments. The best example begins with "Cogito ergo sum," a Latin sentence that translates as "I think, therefore I am." This phrase comes from the beginning of a 17th century essay by the mathematician/philosopher, René Descartes, and it is one of the most famous quotes in the world: do a web search for it, and you will be flooded with hits.
Deducing your existence from the fact that you're thinking about your existence is a pretty cool and persuasive-sounding idea. However, with just a few more lines of argument in this vein, Descartes goes on to conclude that there is an infinitely beneficent God. Whether or not you believe in an infinitely beneficent God, you'll probably agree that any very short "proof" of God's infinite beneficence is bound to be far-fetched. So even in masterful hands, this approach is not reliable. Mathematics has its own specific notion of "proof."
Definition. A mathematical proof of a proposition is a chain of logical deductions leading to the proposition from a base set of axioms.
The three key ideas in this definition are highlighted: proposition, logical deductioni, and axiom. Chapter 1 examines these three ideas along with some basic ways of organizing proofs. Chapter 2 introduces the Well Ordering Principle, a basic method of proof; later, Chapter 5 introduces the closely related proof method of induction.
If you're going to prove a proposition, you'd better have a precise understanding of what the proposition means. To avoid ambiguity and uncertain definitions in ordinary language, mathematicians use language very precisely, and they often express propositions using logical formulas; these are the subject of Chapter 3.
The first three Chapters assume the reader is familiar with a few mathematical concepts like sets and functions. Chapters 4 and 8 offer a more careful look at such mathematical data types, examining in particular properties and methods for proving things about infinite sets. Chapter 7 goes on to examine recursively defined data types.
QuotePreface
This book helps students to master the material of a standard US undergraduate first course in Linear Algebra.
The material is standard in that the subjects covered are Gaussian reduction, vector spaces, linear maps, determinants, and eigenvalues and eigenvectors. Another standard is book's audience: sophomores or juniors, usually with a background of at least one semester of calculus. The help that it gives to students comes from taking a developmental approach — this book's presentation emphasizes motivation and naturalness, using many examples as well as extensive and careful exercises.
The developmental approach is what most recommends this book so I will elaborate. Courses at the beginning of a mathematics program focus less on theory and more on calculating. Later courses ask for mathematical maturity: the ability to follow different types of arguments, a familiarity with the themes that underlie many mathematical investigations such as elementary set and function facts, and a capacity for some independent reading and thinking. Some programs have a separate course devoted to developing maturity and some do not. In either case, a Linear Algebra course is an ideal spot to work on this transition. It comes early in a program so that progress made here pays off later but also comes late enough that students are serious about mathematics. The material is accessible, coherent, and elegant. There are a variety of argument styles, including direct proofs, proofs by contradiction, and proofs by induction. And, examples are plentiful.
Helping readers start the transition to being serious students of mathematics requires taking the mathematics seriously so all of the results here are proved. On the other hand, we cannot assume that students have already arrived and so in contrast with more advanced texts this book is filled with examples, often quite detailed.
Some books that assume a not-yet-sophisticated reader begin with extensive computations of linear systems, matrix multiplications, and determinants. Then, when vector spaces and linear maps finally appear and definitions and proofs start, the abrupt change can bring students to an abrupt stop. While this book begins with linear reduction, from the start we do more than compute. The first chapter includes proofs showing that linear reduction gives a correct and complete solution set. Then, with the linear systems work as motivation so that the study of linear combinations is natural, the second chapter starts with the definition of a real vector space. In the schedule below this happens at the start of the third week.
Another example of this book's emphasis on motivation and naturalness is that the third chapter on linear maps does not begin with the definition of homomorphism. Instead it begins with the definition of isomorphism, which is natural: students themselves observe that some spaces are "the same" as others. After that, the next section takes the reasonable step of isolating the operationpreservation idea to define homomorphism. This approach loses mathematical slickness but it is a good trade because it gives to students a large gain in sensibility.
A student progresses most in mathematics while doing exercises. In this book problem sets start with simple checks and range up to reasonably involved proofs. Since instructors usually assign about a dozen exercises I have tried to put two dozen in each set, thereby giving a selection. There are even a few that are puzzles taken from various journals, competitions, or problems collections. These are marked with a '?' and as part of the fun I have retained the original wording as much as possible.
That is, as with the rest of the book the exercises are aimed to both build an ability at, and help students experience the pleasure of, doing mathematics. Students should see how the ideas arise and should be able to picture themselves doing the same type of work.
Applications and computers. The point of view taken here, that students should think of Linear Algebra as about vector spaces and linear maps, is not taken to the complete exclusion of others. Applications and computing are interesting and vital aspects of the subject. Consequently each of this book's chapters closes with a few topics in those areas. They are brief enough that an instructor can do one in a day's class or can assign them as independent or small-group projects. Most simply give a reader a taste of the subject, discuss how Linear Algebra comes in, point to some further reading, and give a few exercises. Whether they figure formally in a course or not these help readers see for themselves that Linear Algebra is a tool that a professional must master.
QuoteFOREWORD
When we started Burgundy, we had two main goals: to focus on earning long-term absolute returns for our clients (without taking excessive risks), and to look after our clients extremely well. These two goals may sound simple, but they are not simple to achieve. In fact, many investment firms fall well short of these goals because they focus on short-term relative results and too often take clients for granted.
Our investment approach embodies a long-term view. We invest in good companies when we can buy them for less than they are worth. Then we wait for the market to recognize the value of these companies. We try to invest in companies that have the ability to compound their value over time so that the investments grow accordingly. It takes time and patience; more importantly, it takes original, intensive research to identify companies with strong investment potential. We have always made our investment decisions this way and we will continue to follow this investment path in the future.
Over the past 25 years, we've shared our thinking on important business and investment topics with clients through our newsletter, The View from Burgundy. We've covered a range of topics, from specific companies to overall economic policy and external factors that affect the growth and progress of companies. We've examined issues affecting investors and advocated on behalf of investors. We've also debunked conventional thinking and explored contrarian positions that are irksome to parties stuck on mainstream thinking. The View has often been provocative in nature.
The View has been a critical part of our client discourse and relations. I cannot count the number of times that clients have referred to or called me about a specific issue or topic. I know that issues of The View are often cited in corporate boardrooms. It's gratifying to know that the information is useful, provoking discussion and conversation with our clients and industry leaders around the world.
I am proud to introduce the second edition compilation of The View from Burgundy, going back to 1993. The second edition is an update of our original compilation, published in 2007, and includes 14 additional issues of The View up to the end of 2016. Our style of writing has evolved over the years, but our investment philosophy has remained constant over a quarter-century. By consistently adhering to this philosophy, we continue to hold those same goals set 25 years ago, and remain focused first and foremost on you, our clients.
Yours truly,
Tony Arrell,
Chairman and Chief Executive Officer
QuotePreface
This text aims to introduce select topics in discrete mathematics at a level appropriate for first- or second-year undergraduate math and computer science majors, especially those who intend to teach middle and high school mathematics. The book began as a set of notes for the Discrete Mathematics course at the University of Northern Colorado. This course serves both as a survey of the topics in discrete math and as the "bridge" course for math majors, as UNC does not offer a separate "introduction to proofs" course. As this course has evolved to support our computer science major, so has the text. The current version of the book is intended to support inquiry-based teaching for understanding that is so crucial for future teachers, while also providing the necessary mathematical foundation and application-based motivation for computer science students. While teaching the course in Spring 2024 using an early version of this edition, I was pleasantly surprised by how many students reported that they, for the first time, saw how useful math could be in the "real world." I hope that this experience can be replicated in other classes using this text.
This book is intended to be used in a class taught using problem-oriented or inquiry-based methods. Each section begins with a preview of the content that includes an open-ended Investigate! motivating question, as well as a structured preview activity. The preview activities are carefully scaffolded to provide an entry-point to the section's topic and to prime students to engage deeply in the material. Depending on the pace of the class, I have found success assigning only the section preview before class, using the preview activity as in-class group work, or assigning the entire section to be read before class (each section concludes with a small set of reading questions that can be assigned to encourage students to actually read). For those readers using this book for self-study, the organization of the sections will hopefully mimic the style of a rich inquiry-based classroom.
The topics covered in this text were chosen to match the needs of the students I teach at UNC. The main areas of study are logic and proof, graph theory, combinatorics, and sequences. Induction is covered at the end of the chapter on sequences. Discrete structures are introduced "as needed", but a more thorough treatment of sets and functions is included as a separate chapter, which can be studied independent of the other content. The final chapter covers two additional topics: generating functions and number theory.
While I believe this selection and order of topics is optimal, you should feel free to skip around to what interests you. There are occasionally examples and exercises that rely on earlier material, but I have tried to keep these to a minimum, and they usually can either be skipped or understood without too much additional study. If you are an instructor, you can also create a custom version by editing the PreTeXt source to fit your needs.
QuoteAbstract: This paper provides a review of the Fractal Market Hypothesis (FMH) focusing on financial times series analysis. In order to put the FMH into a broader perspective, the Random Walk and Efficient Market Hypotheses are considered together with the basic principles of fractal geometry. After exploring the historical developments associated with different financial hypotheses, an overview of the basic mathematical modelling is provided. The principal goal of this paper is to consider the intrinsic scaling properties that are characteristic for each hypothesis. In regard to the FMH, it is explained why a financial time series can be taken to be characterised by a 1/t1−1/γ scaling law, where γ > 0 is the Lévy index, which is able to quantify the likelihood of extreme changes in price differences occurring (or otherwise). In this context, the paper explores how the Lévy index, coupled with other metrics, such as the Lyapunov Exponent and the Volatility, can be combined to provide long-term forecasts. Using these forecasts as a quantification for risk assessment, short-term price predictions are considered using a machine learning approach to evolve a nonlinear formula that simulates price values. A short case study is presented which reports on the use of this approach to forecast Bitcoin exchange rate values.
Quote1 Introduction
Fractals are intricate structures that exhibit self-similarity across different scales. They are observed in both physics and finance, describing patterns ranging from natural formations like coastlines and clouds to stock market price fluctuations.
