Books: 'Complete equational theories'

1

Schapira, Pierre. Index theorem for elliptic pairs. Paris: Société mathématique de France, 1994.

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2

Pipkin, A. C. A course on integral equations. New York: Springer-Verlag, 1991.

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3

Pipkin, A. C. A course on integral equations. New York: Springer-Verlag, 1991.

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4

Biase, Fausto. Fatou Type Theorems: Maximal Functions and Approach Regions. Boston, MA: Birkhäuser Boston, 1997.

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5

Alekseev, V. B. Abel's theorem in problems and solutions based on the lectures of professor V.I. Arnold. Dordrecht: Kluwer Academic, 2003.

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6

Alekseev, V. B. Abel's theorem in problems and solutions based on the lectures of professor V.I. Arnold. Boston: Kluwer Academic Publishers, 2004.

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7

Sottile, Frank. Real solutions to equations from geometry. Providence, R.I: American Mathematical Society, 2011.

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8

Furstenberg, Harry. Ergodic theory and fractal geometry. Providence, Rhode Island: Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society, 2014.

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9

Southeast Geometry Seminar (15th 2009 University of Alabama at Birmingham). Geometric analysis, mathematical relativity, and nonlinear partial differential equations: Southeast Geometry Seminars Emory University, Georgia Institute of Technology, University of Alabama, Birmingham, and the University of Tennessee, 2009-2011. Edited by Ghomi Mohammad 1969-. Providence, Rhode Island: American Mathematical Society, 2013.

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10

Wood, John C. Harmonic maps and differential geometry: A harmonic map fest in honour of John C. Wood's 60th birthday, September 7-10, 2009, Cagliari, Italy. Providence, R.I: American Mathematical Society, 2011.

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11

Mann, Peter. Vector Calculus. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0034.

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This chapter gives a non-technical overview of differential equations from across mathematical physics, with particular attention to those used in the book. It is a common trend in physics and nature, or perhaps just the way numbers and calculus come together, that, to describe the evolution of things, most theories use a differential equation of low order. This chapter is useful for those with no prior knowledge of the differential equations and explains the concepts required for a basic exposition of classical mechanics. It discusses separable differential equations, boundary conditions and initial value problems, as well as particular solutions, complete solutions, series solutions and general solutions. It also discusses the Cauchy–Lipschitz theorem, flow and the Fourier method, as well as first integrals, complete integrals and integral curves.

12

Mann, Peter. Canonical & Gauge Transformations. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0018.

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In this chapter, the Hamilton–Jacobi formulation is discussed in two parts: from a generating function perspective and as a variational principle. The Poincaré–Cartan 1-form is derived and solutions to the Hamilton–Jacobi equations are discussed. The canonical action is examined in a fashion similar to that used for analysis in previous chapters. The Hamilton–Jacobi equation is then shown to parallel the eikonal equation of wave mechanics. The chapter discusses Hamilton’s principal function, the time-independent Hamilton–Jacobi equation, Hamilton’s characteristic function, the rectification theorem, the Maupertius action principle and the Hamilton–Jacobi variational problem. The chapter also discusses integral surfaces, complete integral hypersurfaces, completely separable solutions, the Arnold–Liouville integrability theorem, general integrals, the Cauchy problem and de Broglie–Bohm mechanics. In addition, an interdisciplinary example of medical imaging is detailed.

13

Mann, Peter. Classical Electromagnetism. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198822370.003.0027.

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In this chapter, Noether’s theorem as a classical field theory is presented and the properties of variations are again discussed for fields (i.e. field variations, space variations, time variations, spacetime variations), resulting in the Noether condition. Quasisymmetries and spontaneous symmetry breaking are discussed, as well as local symmetry and global symmetry. Following these definitions, Noether’s first theorem and Noether’s second theorem are developed. The classical Schrödinger field is investigated and the key equations of classical mechanics are summarised into a single Lagrangian. Symmetry properties of the field action and equations of motion are then compared. The chapter discusses the energy–momentum tensor, the Klein–Utiyama theorem, the Liouville equation and the Hamilton–Jacobi equation. It also discusses material science, special orthogonal groups and complex scalar fields.

14

Morawetz, Klaus. Nonequilibrium Quantum Hydrodynamics. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198797241.003.0015.

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The balance equations resulting from the nonlocal kinetic equation are derived. They show besides the Landau-like quasiparticle contributions explicit two-particle correlated parts which can be interpreted as molecular contributions. It looks like as if two particles form a short-living molecule. All observables like density, momentum and energy are found as a conserving system of balance equations where the correlated parts are in agreement with the forms obtained when calculating the reduced density matrix with the extended quasiparticle functional. Therefore the nonlocal kinetic equation for the quasiparticle distribution forms a consistent theory. The entropy is shown to consist also of a quasiparticle part and a correlated part. The explicit entropy gain is proved to complete the H-theorem even for nonlocal collision events. The limit of Landau theory is explored when neglecting the delay time. The rearrangement energy is found to mediate between the spectral quasiparticle energy and the Landau variational quasiparticle energy.

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Batterman, Robert W. A Middle Way. Oxford University Press, 2021. http://dx.doi.org/10.1093/oso/9780197568613.001.0001.

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This book focuses on a method for exploring, explaining, and understanding the behavior of large many-body systems. It describes an approach to non-equilibrium behavior that focuses on structures (represented by correlation functions) that characterize mesoscale properties of the systems. In other words, rather than a fully bottom-up approach, starting with the components at the atomic or molecular scale, the “hydrodynamic approach” aims to describe and account for continuum behaviors by largely ignoring details at the “fundamental” level. This methodological approach has its origins in Einstein’s work on Brownian motion. He gave what may be the first instance of “upscaling” to determine an effective (continuum) value for a material parameter—the viscosity. His method is of a kind with much work in the science of materials. This connection and the wide-ranging interdisciplinary nature of these methods are stressed. Einstein also provided the first expression of a fundamental theorem of statistical mechanics called the Fluctuation-Dissipation theorem. This theorem provides the primary justification for the hydrodynamic, mesoscale methodology. Philosophical consequences include an argument to the effect that mesoscale parameters can be the natural variables for characterizing many-body systems. Further, the book offers a new argument for why continuum theories (fluid mechanics and equations for the bending of beams) are still justified despite completely ignoring the fact that fluids and materials have lower scale structure. The book argues for a middle way between continuum theories and atomic theories. A proper understanding of those connections can be had when mesoscales are taken seriously.

16

McMaster, Brian, and Aisling McCluskey. Integration with Complex Numbers. Oxford University Press, 2022. http://dx.doi.org/10.1093/oso/9780192846075.001.0001.

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This introductory text on complex analysis focuses on how to evaluate challenging improper (real) integrals, or their Cauchy principal values if need be, by associating them with (complex) contour integrals. On the way to this goal it explains in detail the basic arithmetic, algebra and analysis of complex numbers and functions: particularly the Cauchy–Riemann equations, Cauchy’s theorem, Cauchy’s integral formula, Taylor’s theorem, Laurent’s theorem and Cauchy’s residue theorem. Recognising that many non-specialist cohorts need to acquire skill and confidence in these techniques, great care is taken to allow time for consolidation of fundamental ideas before proceeding to more sophisticated ones, and stress is laid on worked examples to explain ideas and applications, informal diagrams to build insight, roughwork initial explorations to help seek out solution strategies and—above all—suites of exercises in which the learner can develop and reinforce competence: learning through doing being the hallmark of the working textbook. Substantial revision sections on real analysis and calculus are built into the text for learners who may require additional preparation. An appended final chapter addresses some more advanced topics, such as uniform convergence, that are relevant to why certain key theorems work. Specimen solutions for many exercises will be made available to instructors upon application to the publishers.

17

Hoveijn, I., S. A. van Gils, F. Takens, and H. W. Broer. Nonlinear Dynamical Systems and Chaos. Springer Basel AG, 2013.

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18

Horing, Norman J. Morgenstern. Non-Equilibrium Green’s Functions: Variational Relations and Approximations for Particle Interactions. Oxford University Press, 2018. http://dx.doi.org/10.1093/oso/9780198791942.003.0009.

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Chapter 09 Nonequilibrium Green’s functions (NEGF), including coupled-correlated (C) single- and multi-particle Green’s functions, are defined as averages weighted with the time-development operator U(t0+τ,t0). Linear conductivity is exhibited as a two-particle equilibrium Green’s function (Kubo-type formulation). Admitting particle sources (S:η,η+) and non-conservation of number, the non-equilibrium multi-particle Green’s functions are constructed with numbers of creation and annihilation operators that may differ, and they may be derived as variational derivatives with respect to sources η,η+ of a generating functional eW=TrU(t0+τ,t0)CS/TrU(t0+τ,t0)C. (In the non-interacting case this yields the n-particle Green’s function as a permanent/determinant of single-particle Green’s functions.) These variational relations yield a symmetric set of multi-particle Green’s function equations. Cumulants and the Linked Cluster Theorem are discussed and the Random Phase Approximation (RPA) is derived variationally. Schwinger’s variational differential formulation of perturbation theories for the Green’s function, self-energy, vertex operator, and also shielded potential perturbation theory, are reviewed. The Langreth Algebra arises from analytic continuation of integration of products of Green’s functions in imaginary time to the real-time axis with time-ordering along the integration contour in the complex time plane. An account of the Generalized Kadanoff-Baym Ansatz is presented.

19

Anderson, James A. Brain Theory. Oxford University Press, 2018. http://dx.doi.org/10.1093/acprof:oso/9780199357789.003.0012.

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What form would a brain theory take? Would it be short and punchy, like Maxwell’s Equations? Or with a clear goal but achieved by a community of mechanisms—local theories—to attain that goal, like the US Tax Code. The best developed recent brain-like model is the “neural network.” In the late 1950s Rosenblatt’s Perceptron and many variants proposed a brain-inspired associative network. Problems with the first generation of neural networks—limited capacity, opaque learning, and inaccuracy—have been largely overcome. In 2016, a program from Google, AlphaGo, based on a neural net using deep learning, defeated the world’s best Go player. The climax of this chapter is a fictional example starring Sherlock Holmes demonstrating that complex associative computation in practice has less in common with accurate pattern recognition and more with abstract high-level conceptual inference.

20

Oertel, Gerhard. Stress and Deformation. Oxford University Press, 1996. http://dx.doi.org/10.1093/oso/9780195095036.001.0001.

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Students of geology who may have only a modest background in mathematics need to become familiar with the theories of stress, strain, and other tensor quantities, so that they can follow, and apply to their own research, developments in modern, quantitative geology. This book, based on a course taught by the author at UCLA, can provide the proper introduction. Included throughout the eight chapters are 136 complex problems, advancing from vector algebra in standard and subscript notations, to the mathematical description of finite strain and its compounding and decomposition. Fully worked solutions to the problems make up the largest part of the book. With their help, students can monitor their progress, and geologists will be able to utilize subscript and matrix notations and formulate and solve tensor problems on their own. The book can be successfully used by anyone with some training in calculus and the rudiments of differential equations.

21

Pool, Robert. Beyond Engineering. Oxford University Press, 1997. http://dx.doi.org/10.1093/oso/9780195107722.001.0001.

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We have long recognized technology as a driving force behind much historical and cultural change. The invention of the printing press initiated the Reformation. The development of the compass ushered in the Age of Exploration and the discovery of the New World. The cotton gin created the conditions that led to the Civil War. Now, in Beyond Engineering, science writer Robert Pool turns the question around to examine how society shapes technology. Drawing on such disparate fields as history, economics, risk analysis, management science, sociology, and psychology, Pool illuminates the complex, often fascinating interplay between machines and society, in a book that will revolutionize how we think about technology. We tend to think that reason guides technological development, that engineering expertise alone determines the final form an invention takes. But if you look closely enough at the history of any invention, says Pool, you will find that factors unrelated to engineering seem to have an almost equal impact. In his wide-ranging volume, he traces developments in nuclear energy, automobiles, light bulbs, commercial electricity, and personal computers, to reveal that the ultimate shape of a technology often has as much to do with outside and unforeseen forces. For instance, Pool explores the reasons why steam-powered cars lost out to internal combustion engines. He shows that the Stanley Steamer was in many ways superior to the Model T--it set a land speed record in 1906 of more than 127 miles per hour, it had no transmission (and no transmission headaches), and it was simpler (one Stanley engine had only twenty-two moving parts) and quieter than a gas engine--but the steamers were killed off by factors that had little or nothing to do with their engineering merits, including the Stanley twins' lack of business acumen and an outbreak of hoof-and-mouth disease. Pool illuminates other aspects of technology as well. He traces how seemingly minor decisions made early along the path of development can have profound consequences further down the road, and perhaps most important, he argues that with the increasing complexity of our technological advances--from nuclear reactors to genetic engineering--the number of things that can go wrong multiplies, making it increasingly difficult to engineer risk out of the equation. Citing such catastrophes as Bhopal, Three Mile Island, the Exxon Valdez, the Challenger, and Chernobyl, he argues that is it time to rethink our approach to technology. The days are gone when machines were solely a product of larger-than-life inventors and hard-working engineers. Increasingly, technology will be a joint effort, with its design shaped not only by engineers and executives but also psychologists, political scientists, management theorists, risk specialists, regulators and courts, and the general public. Whether discussing bovine growth hormone, molten-salt reactors, or baboon-to-human transplants, Beyond Engineering is an engaging look at modern technology and an illuminating account of how technology and the modern world shape each other.

Books on the topic 'Complete equational theories'

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