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This primer on averaging theorems provides a practical toolbox for applied mathematicians, physicists, and engineers seeking to apply the well-known mathematical theory to real-world problems. With a focus on practical applications, the book introduces new approaches to dissipative and Hamiltonian resonances and approximations on timescales longer than 1/¿.Accessible and clearly written, the book includes numerous examples ranging from elementary to complex, making it an excellent basic reference for anyone interested in the subject. The prerequisites have been kept to a minimum, requiring only a working knowledge of calculus and ordinary and partial differential equations (ODEs and PDEs).In addition to serving as a valuable reference for practitioners, the book could also be used as a reading guide for a mathematics seminar on averaging methods. Whether you're an engineer, scientist, or mathematician, this book offers a wealth of practicaltools and theoretical insights to help you tackle a range of mathematical problems.

This book develops a class of graded finite element methods to solve singular elliptic boundary value problems in two- and three-dimensional domains. It provides an approachable and self-contained presentation of the topic, including both the mathematical theory and numerical tools necessary to address the major challenges imposed by the singular solution. Moreover, by focusing upon second-order equations with constant coefficients, it manages to derive explicit results that are accessible to the broader computation community. Although written with mathematics graduate students and researchers in mind, this book is also relevant to applied and computational mathematicians, scientists, and engineers in numerical methods who may encounter singular problems.

This primer on mathematics formalisation provides a rapid, hands-on introduction to proof verification in Lean.After a quick introduction to Lean, the basic techniques of human-readable formalisation are introduced, illustrated by simple examples on maps, induction and real numbers. Subsequently, typical design options are discussed and brought to life through worked examples in the setting of simplicial complexes (a higher-dimensional generalisation of graph theory). Finally, the book demonstrates how current research in algebraic and geometric topology can be formalised by means of suitable abstraction layers.Informed by the author's recent teaching and research experience, this book allows students and researchers to quickly get started with formalising and checking their proofs. The core material of the book is accessible to mathematics students with basic programming skills. For the final chapter, familiarity with elementary category theory and algebraic topology is recommended.

This monograph provides a concise overview of nonlinear internal wave theory. It serves as a self-contained reference for both students of mathematics as well as scientific professionals by presenting the material in two parts, isolating the narrative analysis from the mathematical detail. This unique format allows the text to remain accessible to oceanographers and researchers outside of mathematics by presenting a range of relevant theories on their own terms. Conversely, it enables applied mathematicians to understand how the conversation between mathematics and sciences proceeds in a field that has developed through a combination of the two. In addition, the text is supplemented by hands-on Matlab software, as the book incorporates a collection of working codes that enable readers to reproduce all theoretical figures in the text, with modification potential to fit a range of applications including a number of mini-projects outlined throughout the text.

This monograph demonstrates a new approach to the classical mode decomposition problem through nonlinear regression models, which achieve near-machine precision in the recovery of the modes.

The book lies at the interface of mathematics, social media analysis, and data science. The new approach advocates a paradigm shift for modeling information diffusion in online social networks and lays the theoretical groundwork for many spatio-temporal modeling problems in the big-data era.

Graph theory, homology theory, and the theory of covering maps are employed to introduce the notion of the topological crystal which retains, in the abstract, all the information on the connectivity of atoms in the crystal.