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.
Finding exact Ramsey numbers is a problem typically restricted to relatively small graphs. The flag algebra method introduced by Razborov in 2007 was developed to find asymptotic results for very large graphs, so it seems that the method should not be suitable for finding small Ramsey numbers. But this intuition is wrong, and we will develop a technique to do just that in this paper. We find new upper bounds for many small graph and hypergraph Ramsey numbers. As a result, we prove several exact values. The main power of the method relies on utilizing semidefinite programming to find certificates that are sums of squares. We hope that this technique will be adapted to address other questions for smaller graphs with the flag algebra method.
Daniel B. Forger’s Biological Rhythms is an engaging, compact, and intellectually generous exploration of the clocks that govern life. Written for the MIT Press “Essential Knowledge” series, the book aims to offer a rigorous but accessible introduction to the science of circadian and other biological rhythms. Forger succeeds admirably, distilling a wide swath of chronobiology—spanning physiology, neuroscience, endocrinology, and applied mathematics—into a volume that is approachable, fascinating, and often surprisingly personal.
If you’ve ever wondered how math and biology can play complex, yet harmonious jazz, this book is your backstage pass. Non-Local Cell Adhesion Models: Symmetries and Bifurcations in 1- D takes you on an intellectually thrilling journey starting from a rather (apparently) humble but powerfully insightful heuristic model introduced by Armstrong, Painter, and Sherratt. Despite its seemingly simple beginnings, this model, a nonlocal nonlinear PDE, is a heavyweight champion in applied mathematics capable of mimicking certain cell sorting experiments in the fascinating and fundamental realm of cell adhesion.
This valuable and unique book delivers a comprehensive lecture on a wide range of control theory issues in relation to matrix computing. Individual problems are illustrated with examples of sufficient dimensionality to ensure they can be manually recalculated, while still illustrating all the intricacies of the relevant calculations and algorithms. The book also contains numerous drawings and diagrams that clarify the various issues.
Linear algebra is often viewed as one of the most foundational courses in a mathematics or computer science curriculum, yet it is also one that can intimidate students with its abstract formalism and steep learning curve. In Linear Algebra: A Problem-Centered Approach, Róbert Freud reimagines the subject by presenting it not as a procession of theorems and proofs, but as an unfolding narrative of problems, motivations, and applications. Published as part of the AMS’s Pure and Applied Undergraduate Texts series, this book brings together the rigor of traditional mathematics with the accessibility and playfulness of the Hungarian problem-solving tradition.
Stellarators are devices used in plasma physics to confine very hot plasmas (i.e., ionized gases) with magnetic fields to sustain nuclear fusion reactions. Fusion is the Sun’s energy source, and the achievement of sustained fusion on Earth has been studied for several decades as a promising source of clean and safe energy. Unlike tokamaks, which use a combination of simple magnetic fields and plasma current to cage the plasma, stellarators rely solely on external magnetic fields. This has potential advantages for sustained fusion energy production, but requires the design of complicated magnetic fields and expensive-to-build, complex electromagnetic coils.
Control in Finite and Infinite Dimension is an excellent textbook based on many years of in-depth teaching experience, as well as on the author’s expertise in the field. It provides a concise and (mostly) self-contained introduction to mathematical control theory, for both finite- and infinite-dimensional systems. It is written at the level of a Master’s/Ph.D. program, but for more experienced researchers it can also serve as a good overview of the basic results and the main tools used in the field, as well as material for lectures in specialised courses on the topic. It covers the most important parts of the classical control theory: controllability, observability and their duality, optimal controls, and stabilization (the latter two only in the finite-dimensional case).
The two volumes of Feller’s Introduction to Probability are classic and comprehensive. They span a huge range of topics, starting from measure theory and basic probability distributions and moving on to topics such as Markov chains and the central limit theorem. The books are very well written and accessible. It seems that most of the theory that is being employed in 2025 by applied modelers and analysts is touched on in these books in some way. Of course, the emphases are different. Much of the foundational material on probability theory has not changed a lot since Feller’s two volumes, including basic measure theory, the law of large numbers and the central limit theorem, conditional probabilities, and theorems such as the Borel Cantelli Lemma. In contrast, it seems that Feller spends a lot of time surveying a range of special probability models and distributions whereas more modern treatments would perhaps spend more time in surveying general analytic techniques. This is probably partly due to the fact that numerical simulation techniques were not nearly as powerful when the book was written, so scholars tended to focus more on specific tractable models.
"One of the ways to help make computer science respectable is to show that it is deeply rooted in history ...""(Donald E. Knuth, Comm. ACM, 15 (1972), p. 671). A great many of the "respectable"" modern numerical methods proceed iteratively, and we give an overview of them in the final section 11. Teaching and learning science from a historical perspective also leads to a ``respectable"" deeper understanding. The first problems requiring iterative processes were square-root calculations in Babylon, Greece, and India. More complicated problems such as sine tables in the Arabic, Indian, and medieval calculations, including Kepler's Problem, were performed with fixed point iterations. With Newton, Raphson, and Simpson we enter the ``respectable"" realm of methods based on derivatives. Mourraille and Cayley contribute geometric insights in both R and C, while Fourier, Cauchy, and Kantorovich provide rigorous error estimations. Surprisingly, even linear problems became interesting for very large dimensions, beginning with the work of Gauss, Seidel, Young, Richardson, and Krylov to domain decomposition and multigrid methods. We explain all of these methods and illustrate them using the "Montreal test problem."
This paper offers a self-contained exposition of the fundamental mathematical and computational tools for interpolation on the Grassmann manifold, including detailed derivations of geodesics and explicit formulations of the exponential and logarithmic maps. The presentation emphasizes intuition and draws continuous parallels with the Euclidean setting. This pedagogical approach facilitates the understanding of linear, piecewise linear, and high-order interpolation algorithms, as well as their extension to more general manifolds. Two numerical examples are finally used to illustrate the potential of these algorithms: one in the context of parametric model order reduction, and another drawn from stationary iterative methods for linear systems.
We present a variational framework for studying functions learned by deep neural networks with rectified linear unit nonlinearities. We introduce a function space built from compositions of functions of second-order Radon-domain bounded variation. The compositional form of these functions captures the structure of deep neural networks. We prove a representer theorem that shows that deep neural networks with finite width solve regularized data-fitting problems over this space. The critical width is controlled by the square of the number of training data. This perspective explains the effect of weight-decay regularization in neural network training, the importance of skip connections, and the role of sparsity in neural networks. By considering the function-space perspective, we provide sharp links between deep learning and variational methods.
This paper surveys recent progress in understanding the dynamics and loss landscape of the gradient flow equations associated with deep linear neural networks, i.e., the gradient descent training dynamics (in the limit when the step size goes to 0) of deep neural networks missing the activation functions and subject to quadratic loss functions. When formulated in terms of the adjacency matrix of the neural network, as is done in this paper, these gradient flow equations form a class of converging matrix ODEs which is nilpotent, polynomial, isospectral, and with conservation laws. A detailed description of the loss landscape shows that it is described in detail and is characterized by infinitely-many global minima and saddle points, both strict and nonstrict, but that it lacks local minima and maxima. The loss function itself is a positive semidefinite Lyapunov function for the gradient flow, and its level sets are unbounded invariant sets of critical points with critical values that correspond to the amount of singular values of the input-output data learnt by the gradient along a certain trajectory. The adjacency matrix representation we use in the paper allows us to highlight the existence of a quotient space structure in which each critical value of the loss function is represented only once, while all other critical points with the same critical value belong to the fiber associated to the quotient space. It also allows us to easily determine stable and unstable submanifolds at the saddle points, even when the Hessian fails to obtain them.
This issue's SIGEST paper, from the SIAM Journal on Computing (SICOMP), takes SIAM readers into the world of quantum computing, a world with its roots in physics that still probably is better known to many physicists and theoretical computer scientists than to a good portion of SIAM readers. Quantum computation is a form of computing based upon quantum mechanics, rather than the classical physics that conventional computers utilize. The distinction between conventional and quantum computers starts to become apparent at the most basic level of bits: whereas standard computers utilize binary bits that may have either the state 0 or 1, quantum computers are based upon “qubits” (quantum binary digits) that may have the state 0, 1, or a superposition of these two states with a complex number that specifies the probability for being in each state. Mathematically, the state of a quantum computer can change through a sequence of unitary transformations to the initial state. One reason for the great interest in quantum computation is that it has been shown that quantum computers can solve some important problems, such as the factorization of very large integers (which has important implications for cryptography), far more efficiently than we currently are able to solve these problems on conventional computers. The selected paper, “Adiabatic Quantum Computation Is Equivalent to Standard Quantum Computation” by Dorit Aharonov, Wim van Dam, Julia Kempe, Zeph Landau, Seth Lloyd, and Oded Regev, which was originally published in SICOMP in 2007, establishes an important theoretical result in the field of quantum computation. As the title indicates, it involves adiabatic quantum computation, a form of quantum computing that has attracted interest in recent years in part because it may offer promise in the effort to build effective quantum computers. Adiabatic quantum computation is distinctly different from standard quantum computation. In the standard model, computations are represented similarly to classical circuits, except that the circuits carry qubits instead of bits. In contrast, the adiabatic model is inspired by the adiabatic theorem in quantum mechanics which states that a system in its lowest energy or ground state will remain in that state if it is subjected to conditions that change sufficiently slowly. It already was known that standard quantum computers can efficiently simulate adiabatic quantum computers. The key contribution of this paper is to show the reverse: that adiabatic quantum computation can efficiently simulate standard quantum computation. In the words of the SICOMP editorial board in nominating this paper, “This is a surprising result that continues to be very influential.” It established that the two forms of quantum computation are theoretically equivalent, one implication of which is to bolster the potential practical importance of adiabatic quantum computation. The paper by Aharonov et al. is very nicely suited to SIGEST—it is important in its field, it is nicely and accessibly written, it offers a glimpse into an area of applied mathematics and computation that is of growing importance, and it touches on many areas of applied mathematics, including linear algebra, Markov chains, and optimization. We hope it will provide SIAM readers a glimpse of current theoretical research that may, some day, help lead to a brave new world of practical computation.
Nordic skiing provides fascinating opportunities for mathematical modeling studies that exploit methods and insights from physics, applied mathematics, data analysis, scientific computing, and sports science. A typical ski course winds over varied terrain with frequent changes in elevation and direction, and so its geometry is naturally described by a threedimensional space curve. The skier travels along a course under the influence of various forces, and their dynamics can be described using a nonlinear system of ordinary differential equations (ODEs) that are derived from Newton's laws of motion. We develop an algorithm for solving the governing equations that combines Hermite spline interpolation, numerical quadrature, and a high-order ODE solver. Numerical simulations are compared with measurements of skiers on actual courses to demonstrate the effectiveness of the model. Throughout, we aim to illustrate how elementary concepts from undergraduate courses in calculus and scientific computing can be applied to the study of real problems in sport, which we hope will provide stimulating examples for both instructors and students. At the same time, we demonstrate how these concepts are capable of providing novel insights into skiing that should also be of interest to sport scientists.
