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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 presents both long-standing and recent mathematical results from this field in a uniform way. It focuses on exact analytic formulas for reconstructing a function or a vector field from data of integrals over lines, rays, circles, arcs, parabolas, hyperbolas, planes, hyperplanes, spheres, and paraboloids. The book also addresses range characterizations and collects necessary definitions and elementary facts from geometry and analysis. Coverage is motivated by both applications and pure mathematics.

This book treats the extending structures problem in the context of groups, Lie/Leibniz algebras, associative algebras and Poisson/Jacobi algebra. This monograph offers the reader an incursion into the extending structures problem which provides a common ground for studying both the extension problem and the factorization problem.

This is the first book to tackle the topological fixed point theory for block operator matrices with nonlinear entries in Banach spaces and algebras. The authors present several extensions of Schauder¿s and Krasnosel¿skii¿s fixed point theorems to the class of weakly compact operators acting on Banach spaces and algebras. They also address under which conditions a 2×2 block operator matrix with single- and multi-valued nonlinear entries will have a fixed point. In addition, the book describes applications of fixed point theory to diverse equations.

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.

The second edition of this book has a new title that more accurately reflects the table of contents. Over the past few years, many new results have been proven in the field of partial differential equations. This edition takes those new results into account, in particular the study of nonautonomous operators with unbounded coefficients, which has received great attention. Additionally, this edition is the first to use a unified approach to contain the new results in a singular place.

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 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.