A number of factors have come together in the last couple of decades to define the emerging interdisciplinary field of structural molecular biology. First, there has been the considerable growth in our ability to obtain atomic-resolution structural data for biological molecules in general, and proteins in particular. This is a result of advances in technique, both in x-ray crystallography, driven by the development of electronic detectors and of synchrotron radiation x-ray sources, and by the development ofNMR techniques which allow for inference of a three-dimensional structure of a protein in solution. Second, there has been the enormous development of techniques in DNA engineering which makes it possible to isolate and clone specific molecules of interest in sufficient quantities to enable structural measurements. In addition, the ability to mutate a given amino acid sequence at will has led to a new branch of biochemistry in which quantitative measurements can be made assessing the influence of a given amino acid on the function of a biological molecule. A third factor, resulting from the exponential increase in computing power available to researchers, has been the emergence of a growing body of people who can take the structural data and use it to build atomic-scale models of biomolecules in order to try and simulate their motions in an aqueous environment, thus helping to provide answers to one of the most basic questions of molecular biology: the relation of structure to function.

The main topics reflect the fields of mathematics in which Professor O.A. Ladyzhenskaya obtained her most influential results.

One of the main topics considered in the volume is the Navier-Stokes equations. This subject is investigated in many different directions. In particular, the existence and uniqueness results are obtained for the Navier-Stokes equations in spaces of low regularity. A sufficient condition for the regularity of solutions to the evolution Navier-Stokes equations in the three-dimensional case is derived and the stabilization of a solution to the Navier-Stokes equations to the steady-state solution and the realization of stabilization by a feedback boundary control are discussed in detail. Connections between the regularity problem for the Navier-Stokes equations and a backward uniqueness problem for the heat operator are also clarified.

Generalizations and modified Navier-Stokes equations modeling various physical phenomena such as the mixture of fluids and isotropic turbulence are also considered. Numerical results for the Navier-Stokes equations, as well as for the porous medium equation and the heat equation, obtained by the diffusion velocity method are illustrated by computer graphs.

Some other models describing various processes in continuum mechanics are studied from the mathematical point of view. In particular, a structure theorem for divergence-free vector fields in the plane for a problem arising in a micromagnetics model is proved. The absolute continuity of the spectrum of the elasticity operator appearing in a problem for an isotropic periodic elastic medium with constant shear modulus (the Hill body) is established. Time-discretizationproblems for generalized Newtonian fluids are discussed, the unique solvability of the initial-value problem for the inelastic homogeneous Boltzmann equation for hard spheres, with a diffusive term representing a random background acceleration is proved and some qualitative properties of the solution are studied. An approach to mathematical statements based on the Maxwell model and illustrated by the Lavrent'ev problem on the wave formation caused by explosion welding is presented. The global existence and uniqueness of a solution to the initial boundary-value problem for the equations arising in the modelling of the tension-driven Marangoni convection and the existence of a minimal global attractor are established. The existence results, regularity properties, and pointwise estimates for solutions to the Cauchy problem for linear and nonlinear Kolmogorov-type operators arising in diffusion theory, probability, and finance, are proved. The existence of minimizers for the energy functional in the Skyrme model for the low-energy interaction of pions which describes elementary particles as spatially localized solutions of nonlinear partial differential equations is also proved.

Several papers are devoted to the study of nonlinear elliptic and parabolic operators. Versions of the mean value theorems and Harnack inequalities are studied for the heat equation, and connections with the so-called growth theorems for more general second-order elliptic and parabolic equations in the divergence or nondivergence form are investigated. Additionally, qualitative properties of viscosity solutions of fully nonlinear partial differential inequalities of elliptic and degenerate elliptic type areclarified. Some uniqueness results for identification of quasilinear elliptic and parabolic equations are presented and the existence of smooth solutions of a class of Hessian equations on a compact Riemannian manifold without imposing any curvature restrictions on the manifold is established.

This volume presents the refereed proceedings of the Conference in Operator The ory in Honour of Moshe Livsic 80th Birthday, held June 29 to July 4, 1997, at the Ben-Gurion University of the Negev (Beer-Sheva, Israel) and at the Weizmann In stitute of Science (Rehovot, Israel). The volume contains papers in operator theory and its applications (understood in a very wide sense), many of them reflecting, 1 directly or indirectly, a profound impact of the work of Moshe Livsic. Moshe (Mikhail Samuilovich) Livsic was born on July 4, 1917, in the small town of Pokotilova near Uman, in the province of Kiev in the Ukraine; his family moved to Odessa when he was four years old. In 1933 he enrolled in the Department of Physics and Mathematics at the Odessa State University, where he became a student of M. G. Krein and an active participant in Krein's seminar - one of the centres where the ideas and methods of functional analysis and operator theory were being developed. Besides M. G. Krein, M. S. Livsic was strongly influenced B. Va. Levin, an outstanding specialist in the theory of analytic functions. A by deep understanding of operator theory as well as function theory and a penetrating search of connections between the two, were to become one of the landmarks of M. S. Livsic's work. M. S. Livsic defended his Ph. D.

This volume contains papers presented at the meeting Deformation Theory, Symplectic Geometry and Applications, held in Ascona, June 17-21, 1996. The contents touch upon many frontier domains of modern mathematics, mathematical physics and theoretical physics and include authoritative, state-of-the-art contributions by leading scientists. New and important developments in the fields of symplectic geometry, deformation quantization, noncommutative geometry (NCG) and Lie theory are presented. Among the subjects treated are: quantization of general Poisson manifolds; new deformations needed for the quantization of Nambu mechanics; quantization of intersection cardinalities; quantum shuffles; new types of quantum groups and applications; quantum cohomology; strong homotopy Lie algebras; finite- and infinite-dimensional Lie groups; and 2D field theories and applications of NCG to gravity coupled with the standard model. Audience: This book will be of interest to researchers and post-graduate students of mathematical physics, global analysis, analysis on manifolds, topological groups, nonassociative rings and algebras, and Lie algebras.

In the past few decades, Magnetic Resonance Imaging (MRI) has become an indispensable tool in modern medicine, with MRI systems now available at every major hospital in the developed world. But for all its utility and prevalence, it is much less commonly understood and less readily explained than other common medical imaging techniques. Unlike optical, ultrasonic, X-ray (including CT), and nuclear medicine-based imaging, MRI does not rely primarily on simple transmission and/or reflection of energy, and the highest achievable resolution in MRI is orders of magnitude smaller that the smallest wavelength involved. In this book, MRI will be explained with emphasis on the magnetic fields required, their generation, their concomitant electric fields, the various interactions of all these fields with the subject being imaged, and the implications of these interactions to image quality and patient safety. Classical electromagnetics will be used to describe aspects from the fundamental phenomenon of nuclear precession through signal detection and MRI safety. Simple explanations and Illustrations combined with pertinent equations are designed to help the reader rapidly gain a fundamental understanding and an appreciation of this technology as it is used today, as well as ongoing advances that will increase its value in the future. Numerous references are included to facilitate further study with an emphasis on areas most directly related to electromagnetics.

This, the fourth volume of the handbook "Integrals and Series", contains tables of the direct Laplace transforms and includes results set forth in books of a similar kind and in periodical literature. All the tables are arranged in two columns - originals f(x) and corresponding images F(p). The Laplace transformation is extensively used in various problems of pure and applied mathematics. Particularly widespread and effective is its application to problems arising in the theory of operational calculus and its applications, embracing the most diverse branches of science and technology. An important advantage of methods using the Laplace transformation lies in the possibility of compiling tables of various elementary and special functions commonly encountered in applications. A number of Laplace transforms are expressed in terms of Meijer G-function. When combined with the table of special cases of the G-function, these formulae make it possible to obtain Laplace transforms of various elementary and special functions of mathematical physics.

The seminal 1970 Moscow thesis of Grigoriy A. Margulis, published for the first time. Entitled "On Some Aspects of the Theory of Anosov Systems", it uses ergodic theoretic techniques to study the distribution of periodic orbits of Anosov flows. The thesis introduces the "Margulis measure" and uses it to obtain a precise asymptotic formula for counting periodic orbits. This has an immediate application to counting closed geodesics on negatively curved manifolds. The thesis also contains asymptotic formulas for the number of lattice points on universal coverings  of compact manifolds of negative curvature. The thesis is complemented by a survey by Richard Sharp, discussing more recent developments in the theory of periodic orbits for hyperbolic flows, including the results obtained in the light of Dolgopyat's breakthroughs on bounding transfer operators and rates of mixing.

The book is concerned with mathematical modelling of supersonic and hyper sonic flows about bodies. Permanent interest in this topic is stimulated, first of all, by aviation and aerospace engineering. The designing of aircraft and space vehicles requires a more precise prediction of the aerodynamic and heat transfer characteristics. Together with broadening of the flight condition range, this makes it necessary to take into account a number of gas dynamic and physical effects caused by rarefaction, viscous-inviscid interaction, separation, various physical and chemical processes induced by gas heating in the intensive bow shock wave. The flow field around a body moving at supersonic speed can be divided into three parts, namely, shock layer, near wake including base flow, and far wake. The shock layer flow is bounded by the bow shock wave and the front and lat eral parts of the body surface. A conventional approach to calculation of shock layer flows consists in a successive solution of the inviscid gas and boundary layer equations. When the afore-mentioned effects become important, implementation of these models meets difficulties or even becomes impossible. In this case, one has to use a more general approach based on the viscous shock layer concept."

University Physics with Modern Physics, Technology Update, Thirteenth Edition continues to set the benchmark for clarity and rigor combined with effective teaching and research-based innovation. The Thirteenth Edition Technology Update contains QR codes throughout the textbook, enabling students to use their smartphone or tablet to instantly watch interactive videos about relevant demonstrations or problem-solving strategies. University Physics is known for its uniquely broad, deep, and thoughtful set of worked examples-key tools for developing both physical understanding and problem-solving skills. The Thirteenth Edition revises all the Examples and Problem-solving Strategies to be more concise and direct while maintaining the Twelfth Edition's consistent, structured approach and strong focus on modeling as well as math. To help students tackle challenging as well as routine problems, the Thirteenth Edition adds Bridging Problems to each chapter, which pose a difficult, multiconcept problem and provide a skeleton solution guide in the form of questions and hints. The text's rich problem sets-developed and refined over six decades-are upgraded to include larger numbers of problems that are biomedically oriented or require calculus. The problem-set revision is driven by detailed student-performance data gathered nationally through MasteringPhysics (R), making it possible to fine-tune the reliability, effectiveness, and difficulty of individual problems. Complementing the clear and accessible text, the figures use a simple graphic style that focuses on the physics. They also incorporate explanatory annotations-a technique demonstrated to enhance learning. This package consists of: University Physics with Modern Physics Technology Update, Volume 1 (Chapters 1-20), Thirteenth Edition

The topics of this set of student-oriented books are presented in a discursive style that is readable and easy to follow. Numerous clearly stated, completely worked out examples together with carefully selected problem sets with answers are used to enhance students' understanding and manipulative skill. The goal is to help students feel comfortable and confident in using advanced mathematical tools in junior, senior, and beginning graduate courses.

The Edexcel A level Lab Books support students in completing the A level Core Practical requirements. This lab book includes: all the instructions students need to perform the Core Practicals, consistent with our A level online teaching resources writing frames for students to record their results and reflect on their work CPAC Skills Checklists, so that students can track the practical skills they have learned, in preparation for their exams practical skills practice questions a full set of answers. This lab book is designed to help students to: structure their A level lab work to ensure that they cover the Core Practical assessment criteria track their progress in the development of A level practical skills create a record of all of the Core Practical work they will have completed, in preparation for revision.

The theory of complex functions is a strikingly beautiful and powerful area of mathematics. Some particularly fascinating examples are seemingly complicated integrals which are effortlessly computed after reshaping them into integrals along contours, as well as apparently difficult differential and integral equations, which can be elegantly solved using similar methods. To use them is sometimes routine but in many cases it borders on an art. The goal of the book is to introduce the reader to this beautiful area of mathematics and to teach him or her how to use these methods to solve a variety of problems ranging from computation of integrals to solving difficult integral equations. This is done with a help of numerous examples and problems with detailed solutions.

Learn how to think like a physicist from a Nobel laureate and "one of the greatest minds of the twentieth century" (New York Review of Books) with these six classic and beloved lessons It was Richard Feynman's outrageous and scintillating method of teaching that earned him legendary status among students and professors of physics. From 1961 to 1963, Feynman delivered a series of lectures at the California Institute of Technology that revolutionized the teaching of physics around the world. Six Easy Pieces, taken from these famous Lectures on Physics, represent the most accessible material from the series. In these classic lessons, Feynman introduces the general reader to the following topics: atoms, basic physics, energy, gravitation, quantum mechanics, and the relationship of physics to other topics. With his dazzling and inimitable wit, Feynman presents each discussion with a minimum of jargon. Filled with wonderful examples and clever illustrations, Six Easy Pieces is the ideal introduction to the fundamentals of physics by one of the most admired and accessible physicists of modern times. "If one book was all that could be passed on to the next generation of scientists it would undoubtedly have to be Six Easy Pieces."- John Gribbin, New Scientist

An account of Newton's life and work.

Modern Physics, Second Edition provides a clear, precise, and contemporary introduction to the theory, experiment, and applications of modern physics. Ideal for both physics majors and engineers, this eagerly awaited second edition puts the modern back into modern physics courses. Pedagogical features throughout the text focus the reader on the core concepts and theories while offering optional, more advanced sections, examples, and cutting-edge applications to suit a variety of students and courses. Critically acclaimed for his lucid style, in the second edition, Randy Harris applies the same insights into recent developments in physics, engineering, and technology.

This is the best seller in this market. It provides a comprehensive introduction to complex variable theory and its applications to current engineering problems. It is designed to make the fundamentals of the subject more easily accessible to students who have little inclination to wade through the rigors of the axiomatic approach. Modeled after standard calculus books-both in level of exposition and layout-it incorporates physical applications throughout the presentation, so that the mathematical methodology appears less sterile to engineering students.

This brilliant CGP book covers all the maths skills needed in AS and A-Level Physics (the use of maths is required for up to 40% of the marks in the final exams and assessments). It explains Calculations, Geometry, Trigonometry, Graph Skills and Handling Date, with clear study notes and step-by-step examples in the context of Physics. And to make sure you've really got to grips with it all, there are practice questions for each topic - with answers included at the back of the book.