Bøger i Chapman & Hall/CRC Monographs and Research Notes in Mathematics serien

In knot theory, diagrams of a given canonical genus can be described by means of a finite number of patterns ("generators"). This book presents a self-contained account of the canonical genus: the genus of knot diagrams. The author explores recent research on the combinatorial theory of knots and supplies proofs for a number of theorems. He gives a detailed structure theorem for canonical Seifert surfaces of a given genus and covers applications, such as the braid index of alternating knots and hyperbolic volume.

This book provides a thorough introduction to the theory of nonlinear PDEs with a variable exponent, particularly those of elliptic type. It presents the most important variational methods for elliptic PDEs described by nonhomogeneous differential operators and containing one or more power-type nonlinearities with a variable exponent. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear elliptic equations as well as their applications to various processes arising in the applied sciences.

This work focuses on the Cremona groups of ranks 2 and 3 and describes the beautiful appearances of the icosahedral group A5 in them. The authors present interesting results that highlight the beauty of icosahedral symmetries of the variety V5. The book surveys known facts about surfaces with an action of A5, explores A5-equivariant geometry of the quintic del Pezzo threefold V5, and gives a proof of its A5-birational rigidity.

This introductory book presents the main themes, techniques, and applications of geometric integrators for researchers in mathematics, physics, astronomy, and chemistry who are already familiar with numerical tools for solving differential equations. It also offers a bridge from traditional training in the numerical analysis of differential equations to understanding recent and advanced research literature on numerical geometric integration. Readers can reproduce the figures and results given in the text using the MATLAB® programs and model files available online.

Bringing together research that was otherwise scattered throughout the literature, this book collects the main results on the conditions for the existence of large algebraic substructures. Many examples illustrate lineability, dense-lineability, spaceability, algebrability, and strong algebrability in different areas of mathematics, including real and complex analysis. The book presents general techniques for discovering lineability in its diverse degrees, incorporates assertions with their corresponding proofs, and provides exercises in every chapter.

This book explains how mathematical tools can be used to solve problems in signal processing. Assuming an advanced undergraduate- or graduate-level understanding of mathematics, this second edition contains new chapters on convolution and the vector DFT, plane-wave propagation, and the BLUE and Kalman filters. It expands the material on Fourier analysis to three new chapters to provide additional background information, presents real-world examples of applications that demonstrate how mathematics is used in remote sensing, and includes robust appendices and problems for classroom use.

This book presents algebraic, combinatorial, and computational methods for studying monomial algebras and their ideals, including Stanley¿Reisner rings, monomial subrings, Ehrhart rings, and blowup algebras. Along with revising all existing chapters, this edition includes four new chapters that focus on the algebraic properties of blowup algebras in combinatorial optimization problems of clutters and hypergraphs. It also contains two new chapters that explore the algebraic and combinatorial properties of the edge ideal of clutters and hypergraphs.

This book provides a broad introduction to the mathematics of difference equations and their applications. Many worked examples illustrate how to calculate both exact and approximate solutions to special classes of difference equations. Along with more problems and an expanded bibliography, this edition includes two new chapters on special topics (such as discrete Cauchy¿Euler equations) and the application of difference equations to complex problems arising in the mathematical modeling of phenomena in engineering and the natural and social sciences.

This book shows how four types of higher-order nonlinear evolution PDEs have many commonalities through their special quasilinear degenerate representations. The authors present a unified approach to deal with these quasilinear PDEs, describe many properties of the equations, and examine traditional questions of existence/nonexistence, uniqueness/nonuniqueness, global asymptotics, regularizations, shock-wave theory, and various blow-up singularities. The book illustrates how complex PDEs are used in a variety of applications and describes new nonlinear phenomena for the equations.

This book brings together a number of important iterative algorithms for medical imaging, optimization, and statistical estimation. It incorporates recent work that has not appeared in other books and draws on the author¿s considerable research in the field, including his recently developed class of SUMMA algorithms. Related to sequential unconstrained minimization methods, the SUMMA class includes a wide range of iterative algorithms well known to researchers in various areas, such as statistics and image processing.

Modeling and Inverse Problems in the Presence of Uncertainty collects recent researchΓÇöincluding the authorsΓÇÖ own substantial projectsΓÇöon uncertainty propagation and quantification. It covers two sources of uncertainty: where uncertainty is present primarily due to measurement errors and where uncertainty is present due to the modeling formulation itself. After a useful review of relevant probability and statistical concepts, the book summarizes mathematical and statistical aspects of inverse problem methodology, including ordinary, weighted, and generalized least-squares formulations. It then discusses asymptotic theories, bootstrapping, and issues related to the evaluation of correctness of assumed form of statistical models. The authors go on to present methods for evaluating and comparing the validity of appropriateness of a collection of models for describing a given data set, including statistically based model selection and comparison techniques. They also explore recent results on the estimation of probability distributions when they are embedded in complex mathematical models and only aggregate (not individual) data are available. In addition, they briefly discuss the optimal design of experiments in support of inverse problems for given models. The book concludes with a focus on uncertainty in model formulation itself, covering the general relationship of differential equations driven by white noise and the ones driven by colored noise in terms of their resulting probability density functions. It also deals with questions related to the appropriateness of discrete versus continuum models in transitions from small to large numbers of individuals.With many examples throughout addressing problems in physics, biology, and other areas, this book is intended for applied mathematicians interested in deterministic and/or stochastic models and their interactions. It is also s

Although the theory behind solitary waves of strain shows that they hold promise in nondestructive testing and other applications, an enigma has long persisted - the absence of observable solitary waves in practice. This work explores how to construct a powerful deformation pulse in a waveguide without plastic flow or fracture.

Adding new results that have appeared in the last 15 years, this second edition provides an easy way for researchers to locate an inequality by name or subject. This edition offers an up-to-date, alphabetical listing of each inequality with a short statement of the result, some comments, references to related inequalities, and sources of information on proofs and other details. It includes more than 100 new inequalities, a new name index, and an updated bibliography that contains URLs for important references.