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Mathematical Methods for the Magnetohydrodynamics of Liquid Metals
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780198566656.jpg" alt="Mathematical Methods for the Magnetohydrodynamics of Liquid Metals"/><br/></td><td><dl><dt>Author:</dt><dd>Jean-Frédéric Gerbeau, Claude Le Bris, Tony Lelièvre</dd><dt>ISBN:</dt><dd>9780198566656</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Mathematical Physics</dd><dt>DOI:</dt><dd>10.1093/acprof:oso/9780198566656.001.0001</dd><dt>Published in print:</dt><dd>2006</dd><dt>Published Online:</dt><dd>2007-09-01</dd></dl></td></tr></table><p>This text focuses on mathematical and numerical techniques for the simulation of magnetohydrodynamic phenomena, with an emphasis on the magnetohydrodynamics of liquid metals, on two-fluid flows, and on a prototypical industrial application. The approach is a highly mathematical one, based on the rigorous analysis of the equations at hand, and a solid numerical analysis of the discretization methods. Up-to-date techniques, both on the theoretical side and the numerical side, are introduced to deal with the nonlinearities of the multifluid magnetohydrodynamics equations. At each stage of the exposition, examples of numerical simulations are provided, first on academic test cases to illustrate the approach, next on benchmarks well documented in the professional literature, and finally on real industrial cases. The simulation of aluminium electrolysis cells is used as a guideline throughout the book to motivate the study of a particular setting of the magnetohydrodynamics equations.</p>Jean-Frédéric Gerbeau, Claude Le Bris, and Tony Lelièvre2007-09-01Turbulence
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780198722588.jpg" alt="TurbulenceAn Introduction for Scientists and Engineers"/><br/></td><td><dl><dt>Author:</dt><dd>Peter Davidson</dd><dt>ISBN:</dt><dd>9780198722588</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Applied Mathematics, Mathematical Physics</dd><dt>DOI:</dt><dd>10.1093/acprof:oso/9780198722588.001.0001</dd><dt>Published in print:</dt><dd>2015</dd><dt>Published Online:</dt><dd>2015-08-20</dd></dl></td></tr></table><p>This book presents the subject of turbulence. The aim of the book is to bridge the gap between the elementary, heuristic accounts of turbulence and the more rigorous accounts given. Throughout, the book combines the maximum of physical insight with the minimum of mathematical detail. This second edition covers a decade of advancement in the field, streamlining the original content while updating the sections where the subject has moved on. The expanded content includes large-scale dynamics, stratified & rotating turbulence, the increased power of direct numerical simulation, two-dimensional turbulence, Magnetohydrodynamics, and turbulence in the core of the Earth.</p>Peter Davidson2015-08-20Differential Geometry
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780199605880.jpg" alt="Differential GeometryBundles, Connections, Metrics and Curvature"/><br/></td><td><dl><dt>Author:</dt><dd>Clifford Henry Taubes</dd><dt>ISBN:</dt><dd>9780199605880</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Geometry / Topology, Mathematical Physics</dd><dt>DOI:</dt><dd>10.1093/acprof:oso/9780199605880.001.0001</dd><dt>Published in print:</dt><dd>2011</dd><dt>Published Online:</dt><dd>2013-12-17</dd></dl></td></tr></table><p>Bundles, connections, metrics, and curvature are the ‘lingua franca’ of modern differential geometry and theoretical physics. Many of the tools used in differential topology are introduced and the basic results about differentiable manifolds, smooth maps, differential forms, vector fields, Lie groups, and Grassmanians are all presented here. Other material covered includes the basic theorems about geodesics and Jacobi fields, the classification theorem for flat connections, the definition of characteristic classes, and also an introduction to complex and Kähler geometry. The book uses many of the classical examples from, and applications of, the subjects it covers, in particular those where closed form expressions are available, to bring abstract ideas to life.</p>Clifford Henry Taubes2013-12-17The Universality of the Radon Transform
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780198509783.jpg" alt="The Universality of the Radon Transform"/><br/></td><td><dl><dt>Author:</dt><dd>Leon Ehrenpreis</dd><dt>ISBN:</dt><dd>9780198509783</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Mathematical Physics</dd><dt>DOI:</dt><dd>10.1093/acprof:oso/9780198509783.001.0001</dd><dt>Published in print:</dt><dd>2003</dd><dt>Published Online:</dt><dd>2007-09-01</dd></dl></td></tr></table><p>Radon showed how to write arbitrary functions in Rn in terms of the characteristic functions (delta functions) of hyperplanes. This idea leads to various generalizations. For example, R can be replaced by a more general group and “plane” can be replaced by other types of geometric objects. All this is for the “nonparametric” Radon transform. For the parametric Radon transform, this book parametrizes the points of the geometric objects, leading to differential equations in the parameters because the Radon transform is overdetermined. Such equations were first studied by F. John. This book elaborates on them and puts them in a general framework.</p>Leon Ehrenpreis2007-09-01The Porous Medium Equation
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780198569039.jpg" alt="The Porous Medium EquationMathematical Theory"/><br/></td><td><dl><dt>Author:</dt><dd>Juan Luis Vazquez</dd><dt>ISBN:</dt><dd>9780198569039</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Mathematical Physics</dd><dt>DOI:</dt><dd>10.1093/acprof:oso/9780198569039.001.0001</dd><dt>Published in print:</dt><dd>2006</dd><dt>Published Online:</dt><dd>2007-09-01</dd></dl></td></tr></table><p>The heat equation is one of the three classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it come to be fairly well understood. This book provides a presentation of the mathematical theory of the nonlinear heat equation usually called the Porous Medium Equation (PME). This equation appears in a number of physical applications, such as to describe processes involving fluid flow, heat transfer, or diffusion. Other applications have been proposed in mathematical biology, lubrication, boundary layer theory, and other fields. Each chapter contains a detailed introduction and is supplied with a section of notes, providing comments, historical notes or recommended reading, and exercises.</p>Juan Luis Vazquez2007-09-01Feynman's Operational Calculus and Beyond
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780198702498.jpg" alt="Feynman's Operational Calculus and BeyondNoncommutativity and Time-Ordering"/><br/></td><td><dl><dt>Author:</dt><dd>Gerald W Johnson, Michel L. Lapidus, Lance Nielsen</dd><dt>ISBN:</dt><dd>9780198702498</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Mathematical Physics, Numerical Analysis</dd><dt>DOI:</dt><dd>10.1093/acprof:oso/9780198702498.001.0001</dd><dt>Published in print:</dt><dd>2015</dd><dt>Published Online:</dt><dd>2015-09-17</dd></dl></td></tr></table><p>This book provides an abstract theory of Feynman’s operational calculus for functions of (typically) noncommuting operators. Although it is inspired by Feynman’s original heuristic suggestions and time-ordering (or disentangling) rules in his seminal 1951 paper, as is made clear in the introduction (Chapter 1) and elsewhere in the text, the theory developed in this book also goes well beyond them in a number of directions which were not anticipated in Feynman’s work. In particular, the work presented in this volume is oriented towards dealing with abstract and (typically) noncommuting linear operators acting on some Banach space, rather than operators arising from some variety of path integration. Some of the key structures developed in this volume enable us to obtain, in some sense, an appropriate abstract substitute for a generalized functional integral associated with the Feynman operational calculus attached to a given n-tuple of pairs {(Aj,μj)}j=1n of typically noncommuting bounded operators Aj and probability measures μj, for j = 1, …, n and n ≥ 2.</p>Gerald W Johnson, Michel L. Lapidus, and Lance Nielsen2015-09-17Applied Shape Optimization for Fluids
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780199546909.jpg" alt="Applied Shape Optimization for Fluids"/><br/></td><td><dl><dt>Author:</dt><dd>Bijan Mohammadi, Olivier Pironneau</dd><dt>ISBN:</dt><dd>9780199546909</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Mathematical Physics</dd><dt>DOI:</dt><dd>10.1093/acprof:oso/9780199546909.001.0001</dd><dt>Published in print:</dt><dd>2009</dd><dt>Published Online:</dt><dd>2010-02-01</dd></dl></td></tr></table><p>The fields of computational fluid dynamics (CFD) and optimal shape design (OSD) have received considerable attention in the recent past, and are of practical importance for many engineering applications. This book deals with shape optimization problems for fluids, with the equations needed for their understanding (Euler and Navier Strokes, but also those for microfluids) and with the numerical simulation of these problems. It presents the state of the art in shape optimization for an extended range of applications involving fluid flows. Automatic differentiation, approximate gradients, unstructured mesh adaptation, multi-model configurations, and time-dependent problems are introduced, and their implementation into the industrial environments of aerospace and automobile equipment industry explained and illustrated. With the increases in the power of computers in industry since the first edition of this book, methods which were previously unfeasible have begun giving results, namely evolutionary algorithms, topological optimization methods, and level set algorithms. In this edition, these methods have been treated in separate chapters, but the book remains primarily one on differential shape optimization.</p>Bijan Mohammadi and Olivier Pironneau2010-02-01Hyperbolic Dynamics and Brownian Motion
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780199654109.jpg" alt="Hyperbolic Dynamics and Brownian MotionAn Introduction"/><br/></td><td><dl><dt>Author:</dt><dd>Jacques Franchi, Yves Le Jan</dd><dt>ISBN:</dt><dd>9780199654109</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Mathematical Physics</dd><dt>DOI:</dt><dd>10.1093/acprof:oso/9780199654109.001.0001</dd><dt>Published in print:</dt><dd>2012</dd><dt>Published Online:</dt><dd>2013-01-24</dd></dl></td></tr></table><p>The idea of this book is to illustrate an interplay between distinct domains of mathematics. Firstly, this book provides an introduction to hyperbolic geometry, based on the Lorentz group PSO(1, d) and its Iwasawa decomposition, commutation relations and Haar measure, and on the hyperbolic Laplacian. The Lorentz group plays a role in relativistic space–time analogous to rotations in Euclidean space. Hyperbolic geometry is the geometry of the unit pseudo-sphere. The boundary of hyperbolic space is defined as the set of light rays. Special attention is given to the geodesic and horocyclic flows. This book presents hyperbolic geometry via special relativity to benefit from physical intuition. Secondly, this book introduces some basic notions of stochastic analysis: the Wiener process, Itô's stochastic integral and Itô calculus. The book studies linear stochastic differential equations on groups of matrices, and diffusion processes on homogeneous spaces. Spherical and hyperbolic Brownian motions, diffusions on stable leaves, and relativistic diffusion are constructed. Thirdly, quotients of hyperbolic space under a discrete group of isometries are introduced, and form the framework in which some elements of hyperbolic dynamics are presented, especially the ergodicity of the geodesic and horocyclic flows. An analysis is given of the chaotic behaviour of the geodesic flow, using stochastic analysis methods. The main result is Sinai's central limit theorem. Some related results (including a construction of the Wiener measure) which complete the expositions of hyperbolic geometry and stochastic calculus are given in the appendices.</p>Jacques Franchi and Yves Le Jan2013-01-24The Statistical Mechanics of Interacting Walks, Polygons, Animals and Vesicles
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780199666577.jpg" alt="The Statistical Mechanics of Interacting Walks, Polygons, Animals and Vesicles"/><br/></td><td><dl><dt>Author:</dt><dd>E.J. Janse van Rensburg</dd><dt>ISBN:</dt><dd>9780199666577</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Mathematical Physics</dd><dt>DOI:</dt><dd>10.1093/acprof:oso/9780199666577.001.0001</dd><dt>Published in print:</dt><dd>2015</dd><dt>Published Online:</dt><dd>2015-08-20</dd></dl></td></tr></table><p>This book is an account of the theory and mathematical approaches in polymer entropy, with particular emphasis on mathematical approaches to directed and undirected lattice models. Results in the scaling and critical behaviour of models of directed and undirected models of self-avoiding walks, paths, polygons, animals and networks are presented. The general theory of tricritical scaling is reviewed in the context of models of lattice clusters, and the existence of a thermodynamic limit in these models is discussed in general and for particular models. Mathematical approaches based on subadditive and convex functions, generating function methods and percolation theory are used to analyse models of adsorbing, collapsing and pulled walks and polygons in the hypercubic and in the hexagonal lattice. These methods show the existence of thermodynamic limits, pattern theorems, phase diagrams and critical points and give results on topological properties such as knotting and writhing in models of lattice polygons. The use of generating function methods and scaling in directed models is comprehensively reviewed in relation to scaling and phase behaviour in models of directed paths and polygons, including Dyck paths and models of convex polygons. Monte Carlo methods for the self-avoiding walk are discussed, with particular emphasis on dynamic algorithms such as the pivot and BFACF algorithms, and on kinetic growth algorithms such as the Rosenbluth algorithms and its variants, including the PERM, GARM and GAS algorithms.</p>E.J. Janse van Rensburg2015-08-20On the Topology and Future Stability of the Universe
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780199680290.jpg" alt="On the Topology and Future Stability of the Universe"/><br/></td><td><dl><dt>Author:</dt><dd>Hans Ringström</dd><dt>ISBN:</dt><dd>9780199680290</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Mathematical Physics, Geometry / Topology</dd><dt>DOI:</dt><dd>10.1093/acprof:oso/9780199680290.001.0001</dd><dt>Published in print:</dt><dd>2013</dd><dt>Published Online:</dt><dd>2013-09-26</dd></dl></td></tr></table><p>The subject of the book is the topology and future stability of models of the universe. In standard cosmology, the universe is assumed to be spatially homogeneous and isotropic. However, it is of interest to know whether perturbations of the corresponding initial data lead to similar solutions or not. This is the question of stability. It is also of interest to know what the limitations on the global topology imposed by observational constraints are. These are the topics addressed in the book. The theory underlying the discussion is the general theory of relativity. Moreover, in the book, matter is modelled using kinetic theory. As background material, the general theory of the Cauchy problem for the Einstein–Vlasov equations is therefore developed.</p>Hans Ringström2013-09-26Thermoelasticity with Finite Wave Speeds
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<table><tr><td width="200px"><img width="150px" src="/view/covers/9780199541645.jpg" alt="Thermoelasticity with Finite Wave Speeds"/><br/></td><td><dl><dt>Author:</dt><dd>Józef Ignaczak, Martin Ostoja-Starzewski</dd><dt>ISBN:</dt><dd>9780199541645</dd><dt>Publisher:</dt><dd>Oxford University Press</dd><dt>Subjects:</dt><dd>Mathematics, Applied Mathematics, Mathematical Physics</dd><dt>DOI:</dt><dd>10.1093/acprof:oso/9780199541645.001.0001</dd><dt>Published in print:</dt><dd>2009</dd><dt>Published Online:</dt><dd>2010-02-01</dd></dl></td></tr></table><p>Generalized dynamic thermoelasticity is a vital area of research in continuum mechanics, free of the classical paradox of infinite propagation speeds of thermal signals in Fourier‐type heat conduction. Besides that paradox, the classical dynamic thermoelasticity theory offers either unsatisfactory or poor descriptions of a solid's response to a fast transient loading (say, due to short laser pulses) or at low temperatures. Several models were developed and intensively studied over the past four decades, and this book is the first monograph on the subject since the 1970s, aiming to provide a point of reference in the field. It focuses on dynamic thermoelasticity governed by hyperbolic equations, and, in particular, on the two leading theories: that of Lord‐Shulman (with one relaxation time), and that of Green‐Lindsay (with two relaxation times). While the resulting field equations are linear partial differential ones, the complexity of theories is due to the coupling of mechanical with thermal fields. The book is concerned with the mathematical aspects of both theories — existence and uniqueness theorems, domain of influence theorems, convolutional variational principles — as well as with the methods for various initial/boundary value problems. In the latter respect, following the establishment of the central equation of thermoelasticity with finite wave speeds, there are extensive presentations of: the exact, aperiodic‐in‐time solutions of Green‐Lindsay theory; Kirchhoff‐type formulas and integral equations in Green‐Lindsay theory; thermoelastic polynomials; moving discontinuity surfaces; and time‐periodic solutions. This is followed by a chapter on physical aspects of generalized thermoelasticity, with a review of several applications. The book closes with a chapter on a nonlinear hyperbolic theory of a rigid heat conductor for which a number of asymptotic solutions are obtained using a method of weakly nonlinear geometric optics. The book is augmented by an extensive bibliography.</p>Józef Ignaczak and Martin Ostoja-Starzewski2010-02-01