The section opens with a review well written by a giant in the field of numerical solution of partial differential equations, Roland Glowinski. In this featured review, he analyzes the book Think Before You Compute: A Prelude to Computational Fluid Dynamics, by E. J. Hinch, in much detail and contributes his own thoughts on appraisal or possibilities for improvement to every chapter of the book. The reviewer's verdict on the book is: “The book is a gem.” The following three reviewed books, written in an earlier and different world, take on a whole new relevance against the background of the current pandemic situation. First, we have the book by Irina Kareva and Georgy Karev, Modeling Evolution of Heterogeneous Populations: Theory and Applications. This book is diligently reviewed by Karen Page, who rates the book as “enlightening" even if it still contains small annoying errors. The following book, Delay-Adaptive Linear Control, by Yang Zhu and Miroslav Krstic, also seems to anticipate some aspects of the current modeling and control of the COVID-19 pandemic. This book is reviewed in detail by Bozenna Pasik-Duncan and is classified as an important contribution to control in delay systems. For the classification and mining of numerous medical data, such as those being generated right now, Mehrshad Sadria and Anita Layton suggest reading Bioinformatics, edited by Andreas D. Baxevanis, Gary D. Bader, and David S. Wishart, and recommend it as an “easy to read, greatly informative, and highly timely” book. Finally, I would like to mention the book by Zhendong Luo and Goong Chen on Proper Orthogonal Decomposition Methods for Partial Differential Equations, which is reviewed by the expert Stefan Volkwein, who praises the book as providing “a really good recipe for the application of the POD method to many nonstationary PDEs,” and also the book Basic Calculus of Planetary Orbits and Interplanetary Flight: The Missions of the Voyagers, Cassini, and Juno, by Alexander J. Hahn, that allows the reader to escape the current narrow earthly confinement in mathematical thoughts and delve into interplanetary explorations.

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistoryPublished online: 04 November 2021Publication DataISSN (print): 0036-1445ISSN (online): 1095-7200Publisher: Society for Industrial and Applied MathematicsCODEN: siread

The use of probabilistic ideas by applied mathematicians has seen a continued increase in recent decades. Probability now appears frequently at the modeling stage. There is widespread interest in investigating the effects of noise and uncertainty. Probabilistic algorithms are routinely applied with much success to the solution of deterministic problems (think of Monte Carlo quadrature or stochastic gradient descent). Markov chains give a simple, widely used tool to describe systems that evolve randomly. They are also a very popular topic in applied mathematics courses. At times \(0\), \(1\), \(2\), łdots, a Markov chain jumps from its current state \(x\) to a randomly chosen new state \(y\); the set of possible states is discrete. The requirement that the jumping times be uniformly spaced is a clear limitation in many situations, and this shortcoming is avoided by considering continuous-time Markov chains. In the continuous-time setting, if the chain has reached state \(x\) at time \(t_i\), it will jump to the randomly chosen \(y\) at time \(t_i+1=t_i+\tau_i\), where the waiting time \(\tau_i>0\) is itself random. Markov chains in continuous time are featured in many applications, including chemistry, ecology, and epidemiology. In a chemical system containing \(n\) species \(S_1\), łdots, \(S_n\), each state corresponds to vector \((c_1\), łdots, \( c_n)\) where \(c_j\) is the number of molecules of species \(S_j\). The species may take part in a number of chemical reactions, let us say \(S_1+2S_2 \rightarrow S_3\), \(2S_1+3S_3 \rightarrow S_4+2S_6\), and so on. At random times, one or another of the \(m\) possible reactions will take place and the state will change. Such a stochastic, molecular approach typically provides a more accurate description of the system than deterministic treatments where the concentrations of the different species are regarded as continuous variables that evolve according to a set of differential equations. This is particularly true in systems where, for some species, the number \(c_j\) is low, as is the case in many biological reactions. The following Survey and Review paper, “Stationary Distributions of Continuous-Time Markov Chains: A Review of Theory and Truncation-Based Approximations,” has been written by Juan Kuntz, Philipp Thomas, Guy-Bart Stan, and Mauricio Barahona. Section 2 provides a reader-friendly introduction to continuous-time Markov chains and their stationary distributions; these are important because they determine the long-time behavior of the chain. A salient future is that the authors present, in an accessible way, results that do not assume the chain to be irreducible (roughly speaking, irreducibility means that the chain is not the juxtaposition of two or more smaller chains that do not talk to each other; irreducibility simplifies the math, but is not a reasonable hypothesis in some applications). After that, the authors concentrate on how to compute invariant distributions. The paper contains numerical results, an extensive bibliography, and a detailed discussion of open problems.

Animals use stereo sampling of odor concentration to localize sources and follow odor trails. We analyze the dynamics of a bilateral model that depends on the simultaneous comparison between odor concentrations detected by left and right sensors. The general model consists of three differential equations for the positions in the plane and the heading. When the odor landscape is an infinite trail, then we reduce the dynamics to a planar system whose dynamics have just two fixed points. Using an integrable approximation (for short sensors) we estimate the basin of attraction. In the case of a radially symmetric landscape, we again can reduce the dynamics to a planar system, but the behavior is considerably richer with multi-stability, isolas, and limit cycles. As in the linear trail case, there is also an underlying integrable system when the sensors are short. In odor landscapes that consist of multiple spots and trail segments, we find periodic and chaotic dynamics and characterize the behavior on trails with gaps and that turn corners.

Polynomial approximations constructed using a least-squares approach form a ubiquitous technique in numerical computation. One of the simplest ways to generate data for least-squares problems is wi...

In mechanics, one often describes microscopic processes such as those leading to friction between relative interfaces using macroscopic variables (relative velocity, temperature, etc.) in order to avoid models of intangible complexity. As a consequence, such macroscopic models are frequently nonsmooth, a prominent example being the Coulomb law of friction. In many cases, these models are perfectly adequate for engineering purposes. Formally, however, since the Fundamental Theorem of Existence and Uniqueness does not apply to these situations, one generally expects that these models possess forward nonuniqueness of solutions. Consequently, numerical computations of such systems might possibly unknowingly discard certain solutions. In this paper, we try to shed further light on this issue by studying solutions of a simple friction oscillator subject to stiction friction. The stiction law is a simple nonsmooth model of friction that is a modification of Coulomb based on the fundamental observation that the dynamic friction force, when the mass is in motion, is smaller than the static friction force during stick. The resulting piecewise smooth vector field of this discontinuous model does not follow the classical Filippov convention, and the concept of a Filippov solution cannot be used. Furthermore, some Carathéodory solutions, i.e., absolutely continuous solutions satisfying the differential equation in a weaker sense, are nonphysical. Therefore, we introduce the concept of stiction solutions. These are the Carathéodory solutions that are physically relevant, i.e., the ones that follow the stiction law. However, we find that some of the stiction solutions are forward nonunique in subregions of the slip onset. We call these solutions singular, in contrast to the regular stiction solutions that are forward unique. In order to further understanding of the nonunique dynamics, we then introduce a general regularization of the model. This gives a singularly perturbed problem that captures the main features of the original discontinuous problem. Using geometric singular perturbation theory, we identify a repelling slow manifold that separates the forward slipping from the forward sticking solutions, leading to high sensitivity to the initial conditions. On this slow manifold we find canard trajectories that have the physical interpretation of delaying the slip onset. Most interestingly, we find that these new solutions do not correspond to stiction solutions in the piecewise-smooth limit, and are therefore seemingly nonphysical, yet they are robust and appear generically in the class of regularizations we consider. Finally, we show that the regularized problem has a family of periodic orbits interacting with the canards. We observe that this family has a saddle stability and that it connects, in the rigid body limit, the two regular, slip-stick branches of the discontinuous problem, which are otherwise disconnected.

The Education section of SIAM Review presents three papers in this issue. In the first paper Robert M. Corless and Leili Rafiee Sevyeri discuss “The Runge Example for Interpolation and Wilkinson's Examples for Rootfinding.” The authors use several classical examples in numerical analysis to discuss propagation of error and the role of sensitivity and the conditioning of the functions subject to numerical procedures. The first example was brought to the fore by Carl Runge in 1901 when exploring the approximation error occurring in polynomial interpolation. The effect illustrated by this example is also referred to as Runge's phenomenon. It consists of oscillation at the edges of the domain of the function, which is observed when the function is approximated by interpolation using a high-degree polynomial over a uniform grid. This example plays an important role and is used in many numerical analysis textbooks because it illustrates pitfalls and limitations of high-degree polynomial interpolation. It reveals that using higher degree polynomials does not always improve the accuracy of the approximation. The second example is a classical polynomial, introduced by J. H. Wilkinson in his paper “The Evaluation of the Zeros of Ill-Conditioned Polynomials. Part I" [Numer. Math., 1 (1959), pp. 150--166]. It illustrates a difficulty when finding the root of a polynomial, which stems from the sensitivity of the polynomial at the location of the roots, with respect to perturbations in its coefficients. Wilkinson's polynomial is also frequently used in the context of eigenvalue calculation. In that case, computing eigenvalues of a matrix by first calculating the coefficients of the matrix's characteristic polynomial and then calculating its roots may result in a very ill-conditioned problem even if the original problem is well conditioned. The characteristic polynomial can be very sensitive to perturbations in its coefficients near its roots. The authors bring the theory of backward error analysis and conditioning to the reader's attention. This approach explains the phenomena of the presented examples and is in harmony with the numerical experience thus far. The second paper is titled “Mathematics + Cancer: An Undergraduate `Bridge' Course in Applied Mathematics.” It is presented by Tracy L. Stepien, Eric J. Kostelich, and Yang Kuang. The authors share their insightful experience with a one-semester modeling course based on mathematical models for cancer occurrence and growth, as well as models describing the effect of certain cancer treatments. The targeted audience is undergraduate students who have completed standard courses on calculus, ordinary differential equations, linear algebra, and introductory probability and statistics. The offer of such a course supports various goals. It helps develop modeling skills for students in applied mathematics while at the same time exposing students to many mathematical ideas and enabling them to make informed choices about future specialization in the various branches of applied mathematics and statistics. Additionally, the students come in contact with compelling scientific as well as social problems and see the impact of applied mathematics on real life. Finally, a research component in the course provides invaluable experience to the enrolled students. The paper discusses the structure and organization of the course. The main portion of it provides an overview of various modules whose suggested length is one or two weeks. The topics encompass statistical models regarding the occurrence of cancer depending on age, the growth of the population of cancer cells, the evolution of resistance to treatment, tumor dynamics at the cellular and tissue levels, the measurement and assessment of treatment efficacy, short-term forecasts of tumor progression, the use of statistics in designing clinical trials, and many others. Finally, the public health perspective is brought into the picture: how much cancer might be preventable by a healthy life style, how much is due to environmental exposure, and how much might be due to hereditary reasons or random mutations. The reader will find a sample syllabus, homework problem sets, and computer lab descriptions in the supplementary materials. The authors themselves aim to give students experience in reading research papers and do not use textbooks. However, they mention suitable textbooks for those teachers who would prefer to follow one. Extensive references to relevant literature are included. Finally, the authors point out that their course might be used as a template for other courses of similar type based on topics drawn from other areas of the mathematical sciences. The third contribution is the paper by Antônio Neto on “Matrix Analysis and Omega Calculus.” The first ideas giving rise to Omega calculus were presented by MacMahon in his book Combinatory Analysis published by Cambridge University Press in 1915--1916. The main subject there is the solution of linear Diophantine systems composed of equalities and inequalities; in this context, the author involves results on partitions of natural numbers. In that framework, the so-called Omega operator is applied systematically to each equation and inequality. The final outcome of the procedure is a generating function describing all the solutions of the Diophantine system. The methodology is referred to as MacMahon's partition analysis (MPA) or Omega calculus. Frequently, the latter is identified with the Omega package, which contains numerical methods based on MPA. In the current paper, Antônio Neto presents an extension of Omega operator calculus to matrix-valued functions. The focus is placed on the matrix exponential functions, which have many applications. The extended Omega operator is defined based on the matrix exponential series and the Neumann series with the use of the Frobenius norm for square matrices. Application of the Omega operator to the theory of differential equations, to graph theory, and to other areas is discussed. The paper contains a compact and easy-to-apply method to compute multiple integrals involving matrix exponentials based on MPA. The author stipulates that the presented definition and method generalize and unify previous work. The material is accessible to advanced undergraduate students in applied mathematics who have knowledge in calculus including Taylor series of scalar functions, various norms, matrix operations and inversion, and matrix exponentials.

This paper is a gentle but rigorous introduction to quantum computing intended for discrete mathematicians. Starting from a small set of assumptions on the behavior of quantum computing devices, we analyze their main characteristics, stressing the differences with classical computers, and finally describe two well-known algorithms (Simon's algorithm and Grover's algorithm) using the formalism developed in previous sections. This paper does not touch on the physics of the devices, and therefore does not require any notion of quantum mechanics. Numerical examples based on an implementation of Grover's algorithm using open-source software are provided.

This is an elementary introduction to the Hodge Laplacian on a graph, a higher-order generalization of the graph Laplacian. We will discuss basic properties including cohomology and Hodge theory. The main feature of our approach is simplicity, requiring only knowledge of linear algebra and graph theory. We have also isolated the algebra from the topology to show that a large part of cohomology and Hodge theory is nothing more than the linear algebra of matrices satisfying $AB = 0$. For the remaining topological aspect, we cast our discussion entirely in terms of graphs as opposed to less familiar topological objects like simplicial complexes.

We present an elementary proof of a generalization of Kirchhoff's matrix tree theorem to directed, weighted graphs. The proof is based on a specific factorization of the Laplacian matrices associat...