Abstract. This book provides an in-depth treatment, including full proofs, of the development of fast matrix multiplication algorithms following the discovery of Strassen’s algorithm or, more precisely, Winograd’s algorithm a year earlier, although the latter cannot be applied recursively. The development is described up to Schönhage’s tau theorem. The author places special emphasis on exact algorithms for multiplying matrices of small formats, which are particularly well suited for designing practically relevant algorithms through recursive application. In addition, the book concludes with a range of applications, such as computing other linear algebra problems in matrix multiplication time.
Reading a book about the history of mathematics is never a waste of time, especially if you are a working mathematician, whether applied or pure. Obviously, there are many excellent monographs devoted to specific periods such as nineteenth-century mathematics, or to specific subjects such as mathematical analysis, and these serve specialists very well. The book under review is more ambitious and attempts to cover the whole history of mathematics under the additional assumption that the intended reader is a layperson, not necessarily a professional mathematician. Remarkably, this ambitious task is carried out in just over 200 pages organized into ten chapters, complemented by a bibliography, a name index, and 32 short appendices containing mathematical gems.
Statistical learning has become an essential component across a wide range of research fields. Among the many textbooks that lay the foundations of theory and practice, Matthias Schonlau’s Applied Statistical Learning: With Case Studies in Stata is a valuable addition—not only for undergraduate and graduate students and course instructors, but also for applied researchers, particularly those who use Stata for data analysis.
Probability Theory: The Logic of Science by E. T. Jaynes is a highly ambitious, unusual, and personal book. Its goal is no less than to lay the foundation for how to reason under uncertainty, not only in science, but generally. Inspired by Polya’s Mathematics and Plausible Reasoning, throughout his career Jaynes pursued a program of understanding the rules of plausible reasoning that would be broadly applicable across domains. Starting not from axioms, but from the “desiderata,” or desirable properties, that one would like to have in a quantitative system of reasoning, Jaynes arrives at the traditional rules of probability theory, namely, the sum and product rules. In so doing, he argues that probability theory is not constrained to any discipline, but constitutes a widely applicable extension of traditional logical systems, wherein statements have truth values that have a quantifiable uncertainty. Jaynes viewed probability theory as a kind of “theory of everything” for reasoning. The book contains an extensive and impressive collection of examples and discussions aimed at underlining this idea and showing the ways in which it aligns or not with traditional views.
Deciding whether a political districting plan was distorted by a hidden agenda, or whether it dilutes the voting power of some group, requires a neutral baseline for comparison. Remarkably, all nine U.S. Supreme Court justices have now signed on to decisions finding that computational methods can provide key evidence. Today, the leading approaches to the benchmarking of districting plans are based on the use of spanning trees for sampling graph partitions. We present a new reversible recombination algorithm and rigorously prove its fundamental properties. Furthermore, we argue for a canonical sampling distribution called the spanning tree distribution that is well adapted to redistricting and provides a principled foundation for comparing and validating methods. Together with a highly efficient (and open-source) implementation that can generate and handle large datasets, this work provides the most powerful null model to date for the gerrymandering problem, meeting an urgent democratic challenge with sound scientific methodology.
