Daniela Calvetti - Böcker

The book of nature, according to Galilei, is written in the language of mat- matics. The nature of mathematics is being exact, and its exactness is und- lined by the formalism used by mathematicians to write it. This formalism, characterized by theorems and proofs, and syncopated with occasional l- mas, remarks and corollaries, is so deeply ingrained that mathematicians feel uncomfortable when the pattern is broken, to the point of giving the - pression that the attitude of mathematicians towards the way mathematics should be written is almost moralistic. There is a de?nition often quoted, “A mathematician is a person who proves theorems”, and a similar, more alchemistic one, credited to Paul Erd? os, but more likely going back to Alfr´ ed R´ enyi,statingthat“Amathematicianisamachinethattransformsco?eeinto 1 theorems ”. Therefore it seems to be the form, not the content, that char- terizes mathematics, similarly to what happens in any formal moralistic code wherein form takes precedence over content. This book is deliberately written in a very di?erent manner, without a single theorem or proof. Since morality has its subjective component, to pa- phrase Manuel Vasquez Montalban, we could call it Ten Immoral Mathemat- 2 ical Recipes . Does the lack of theorems and proofs mean that the book is more inaccurate than traditional books of mathematics? Or is it possibly just a sign of lack of co?ee? This is our ?rst open question. Exactness is an interesting concept.

Interesting, real-world mathematical modeling problems are complex and can usually be studied at different scales. The scale at which the investigation is carried out is one of the factors that determines the type of mathematics most appropriate to describe the problem. The book concentrates on two modeling paradigms: the macroscopic, in which the authors describe phenomena in terms of time evolution via ordinary differential equations, and the microscopic, which requires knowledge of random events and probability. The text emphasizes the development of computational skills to construct predictive models and analyze the results. To elucidate the concepts, a wealth of examples and portions of MATLAB? code used by the authors are included.Computational Mathematical Modeling: An Integrated Approach Across Scales:Is designed for classroom use, has been extensively tested by the authors, and has homework problems carefully designed to develop students’ computational skills.Is based on an unorthodox combination of deterministic and probablistic methodologies that are naturally bridged through examples.Painlessly introduces students to advanced themes in a natural progression.Includes suggestions for further reading.

Designed for a proof-based course on linear algebra, this rigorous and concise textbook intentionally introduces vector spaces, inner products, and vector and matrix norms before Gaussian elimination and eigenvalues so students can quickly discover the singular value decomposition (SVD)—arguably the most enlightening and useful of all matrix factorizations. Gaussian elimination is then introduced after the SVD and the four fundamental subspaces and is presented in the context of vector spaces rather than as a computational recipe. This allows the authors to use linear independence, spanning sets and bases, and the four fundamental subspaces to explain and exploit Gaussian elimination and the LU factorization, as well as the solution of overdetermined linear systems in the least squares sense and eigenvalues and eigenvectors. This unique textbook also includes examples and problems focused on concepts rather than the mechanics of linear algebra. The problems at the end of each chapter and in an associated website encourage readers to explore how to use the notions introduced in the chapter in a variety of ways. Additional problems, quizzes, and exams will be posted on an accompanying website and updated regularly. The Less Is More Linear Algebra of Vector Spaces and Matrices is for students and researchers interested in learning linear algebra who have the mathematical maturity to appreciate abstract concepts that generalize intuitive ideas. The early introduction of the SVD makes the book particularly useful for those interested in using linear algebra in applications such as scientific computing and data science. It is appropriate for a first proof-based course in linear algebra.

The once esoteric idea of embedding scientific computing into a probabilistic framework, mostly along the lines of the Bayesian paradigm, has recently enjoyed wide popularity and found its way into numerous applications.