Formulas

This book provides a comprehensive account of a key (and perhaps the most important) theory upon which the Taylor–Wiles proof of Fermat's last theorem is based. The book begins with an overview of the theory of automorphic forms on linear algebraic groups and then covers the basic theory and results on elliptic modular forms, including a substantial simplification of the Taylor–Wiles proof by Fujiwara and Diamond. It contains a detailed exposition of the representation theory of profinite groups (including deformation theory), as well as the Euler characteristic formulas of Galois cohomology groups. The final chapter presents a proof of a non-abelian class number formula and includes several new results from the author. The book will be of interest to graduate students and researchers in number theory (including algebraic and analytic number theorists) and arithmetic algebraic geometry.

Providing a fundamental introduction to all aspects of modern plasma chemistry, this book describes mechanisms and kinetics of chemical processes in plasma, plasma statistics, thermodynamics, fluid mechanics and electrodynamics, as well as all major electric discharges applied in plasma chemistry. Fridman considers most of the major applications of plasma chemistry, from electronics to thermal coatings, from treatment of polymers to fuel conversion and hydrogen production and from plasma metallurgy to plasma medicine. It is helpful to engineers, scientists and students interested in plasma physics, plasma chemistry, plasma engineering and combustion, as well as chemical physics, lasers, energy systems and environmental control. The book contains an extensive database on plasma kinetics and thermodynamics and numerical formulas for practical calculations related to specific plasma-chemical processes and applications. Problems and concept questions are provided, helpful in courses relate

The Riemann zeta function is one of the most studied objects in mathematics, and is of fundamental importance. In this book, based on his own research, Professor Motohashi shows that the function is closely bound with automorphic forms and that many results from there can be woven with techniques and ideas from analytic number theory to yield new insights into, and views of, the zeta function itself. The story starts with an elementary but unabridged treatment of the spectral resolution of the non-Euclidean Laplacian and the trace formulas. This is achieved by the use of standard tools from analysis rather than any heavy machinery, forging a substantial aid for beginners in spectral theory as well. These ideas are then utilized to unveil an image of the zeta-function, first perceived by the author, revealing it to be the main gem of a necklace composed of all automorphic L-functions. In this book, readers will find a detailed account of one of the most fascinating stories in the develo

This book examines how national law is treated in WTO law, both in the WTO treaty and dispute settlement cases. The WTO treaty contains a set of far-reaching obligations establishing a systemic and constitutional framework of interaction between WTO law and national law. WTO dispute settlement operates as an international layer of judicial review of national laws and administrative, judicial or quasi-judicial measures. Consequently, much of the WTO dispute settlement decisions and rulings relate in different ways to Members' national laws. Yet, up until the publication of this book, there was no systematic analysis of this vastly important subject. This book provides a thorough map of an increasingly complex field. In doing so, it extends the enquiry beyond well-known formulas and combines practical analysis with principled discussion of how the treatment of national law in international law can and should ensure effectiveness of international rules and promote good governance within n

Peter Clemoes brings a lifetime's close study of Anglo-Saxon texts to this appreciation of Old English poetry, with an alternative interpretation which relates the poetry to both the entire Anglo-Saxon way of thinking and the structures of its society. Clemoes proposes a dynamic principle of Old English poetry, very different from the common notion of formulas slotted into poems for stylistic variation. In extended discussions of particular poems and images as well as of changes in language, he shows how the poetic medium became a vehicle for increasing transformation to Christian literacy and to that religion's conceptions of the natural world, morality, and individuality. Carefully thought out and elegantly written, this book is also accessible to students: its numerous quotations are accompanied by modern English translations.

Geometric tomography deals with the retrieval of information about a geometric object from data concerning its projections (shadows) on planes or cross-sections by planes. It is a geometric relative of computerized tomography, which reconstructs an image from X-rays of a human patient. The subject overlaps with convex geometry and employs many tools from that area, including some formulas from integral geometry. It also has connections to discrete tomography, geometric probing in robotics and to stereology. This comprehensive study provides a rigorous treatment of the subject. Although primarily meant for researchers and graduate students in geometry and tomography, brief introductions, suitable for advanced undergraduates, are provided to the basic concepts. More than 70 illustrations are used to clarify the text. The book also presents 66 unsolved problems. Each chapter ends with extensive notes, historical remarks, and some biographies. This edition includes numerous updates and imp

Geometric tomography deals with the retrieval of information about a geometric object from data concerning its projections (shadows) on planes or cross-sections by planes. It is a geometric relative of computerized tomography, which reconstructs an image from X-rays of a human patient. The subject overlaps with convex geometry and employs many tools from that area, including some formulas from integral geometry. It also has connections to discrete tomography, geometric probing in robotics and to stereology. This comprehensive study provides a rigorous treatment of the subject. Although primarily meant for researchers and graduate students in geometry and tomography, brief introductions, suitable for advanced undergraduates, are provided to the basic concepts. More than 70 illustrations are used to clarify the text. The book also presents 66 unsolved problems. Each chapter ends with extensive notes, historical remarks, and some biographies. This edition includes numerous updates and imp

Neutron scattering is an extremely powerful tool in the study of elemental excitations in condensed matter. This book provides a practical guide to basic techniques using a triple-axis spectrometer. Introductory chapters summarize useful scattering formulas and describe the components of a spectrometer, followed by a comprehensive discussion of the resolution function and focusing effects. Later sections include simple examples of phonon and magnon measurements, and an analysis of spurious effects in both inelastic and elastic measurements, and how to avoid them. Finally, polarization analysis techniques and their applications are covered. This guide will allow graduate students and experienced researchers new to neutron scattering to make the most efficient use of their experimental time.

This book is a comprehensive study of the Radon transform, which operates on a function by integrating it over hyperplanes. The book begins with an elementary and graphical introduction to the Radon transform, tomography and CT scanners, followed by a rigorous development of the basic properties of the Radon transform. Next the author introduces Grassmann manifolds in the study of the k-plane transform (a version of the Radon transform) which integrates over k-dimensional planes rather than hyperplanes. The remaining chapters are concerned with more advanced topics, such as the attenuated Radon transform and generalized Radon transforms defined by duality of homogeneous spaces and double fibrations. Questions of invertibility and the range of the Radon transform are dealt with and inversion formulas are developed with particular attention to functions on L2 spaces and some discussion of the case of Lp spaces.

This book is an introductory textbook on the physical processes occurring in the Earth's radiation belts. The presentation is at the advanced undergraduate or first year graduate level, and it is appropriate for students who intend to work in some aspect of magnetospheric physics. The treatment is quantitative and provides the mathematical basis for original work in this subject. The equations describing the motion of energetic ions and electrons in the geomagnetic field are derived from basic principles, and concepts such as magnetic field representations, guiding centre motion, adiabatic invariance, and particle distribution functions are presented in a detailed and accessible manner. Relevant experimental techniques are reviewed and a summary is given of the intensity and energy spectra of the particle populations in the Earth's radiation belts. Problem sets are included as well as appendices of tables, graphs and frequently used formulas.

The interacting boson-fermion model has become in recent years the standard model for the description of atomic nuclei with an odd number of protons and/or neutrons. This book describes the mathematical framework on which the interacting boson-fermion model is built and presents applications to a variety of situations encountered in nuclei. The book addresses both the analytical and the numerical aspects of the problem. The analytical aspect requires the introduction of rather complex group theoretic methods, including the use of graded (or super) Lie algebras. The first (and so far only) example of supersymmetry occurring in nature is also discussed. The book is the first comprehensive treatment of the subject and will appeal to both theoretical and experimental physicists. The large number of explicit formulas for level energies, electromagnetic transition rates and intensities of transfer reactions presented in the book provide a simple but detailed way to analyse experimental data.

The theory of integer partitions is a subject of enduring interest. A major research area in its own right, it has found numerous applications, and celebrated results such as the Rogers-Ramanujan identities make it a topic filled with the true romance of mathematics. The aim in this introductory textbook is to provide an accessible and wide ranging introduction to partitions, without requiring anything more of the reader than some familiarity with polynomials and infinite series. Many exercises are included, together with some solutions and helpful hints. The book has a short introduction followed by an initial chapter introducing Euler's famous theorem on partitions with odd parts and partitions with distinct parts. This is followed by chapters titled: Ferrers Graphs, The Rogers-Ramanujan Identities, Generating Functions, Formulas for Partition Functions, Gaussian Polynomials, Durfee Squares, Euler Refined, Plane Partitions, Growing Ferrers Boards, and Musings.

The theory of integer partitions is a subject of enduring interest. A major research area in its own right, it has found numerous applications, and celebrated results such as the Rogers-Ramanujan identities make it a topic filled with the true romance of mathematics. The aim in this introductory textbook is to provide an accessible and wide ranging introduction to partitions, without requiring anything more of the reader than some familiarity with polynomials and infinite series. Many exercises are included, together with some solutions and helpful hints. The book has a short introduction followed by an initial chapter introducing Euler's famous theorem on partitions with odd parts and partitions with distinct parts. This is followed by chapters titled: Ferrers Graphs, The Rogers-Ramanujan Identities, Generating Functions, Formulas for Partition Functions, Gaussian Polynomials, Durfee Squares, Euler Refined, Plane Partitions, Growing Ferrers Boards, and Musings.

The focus of this book is on algebro-geometric solutions of completely integrable nonlinear partial differential equations in (1+1)-dimensions, also known as soliton equations. Explicitly treated integrable models include the KdV, AKNS, sine-Gordon, and Camassa-Holm hierarchies as well as the classical massive Thirring system. An extensive treatment of the class of algebro-geometric solutions in the stationary as well as time-dependent contexts is provided. The formalism presented includes trace formulas, Dubrovin-type initial value problems, Baker-Akhiezer functions, and theta function representations of all relevant quantities involved. The book uses techniques from the theory of differential equations, spectral analysis, and elements of algebraic geometry (most notably, the theory of compact Riemann surfaces). The presentation is rigorous, detailed, and self-contained, with ample background material provided in various appendices. Detailed notes for each chapter together with an exh

Myths inspired Greek and Roman artists to rise to the challenge of conveying flowing narratives in static form. This book describes the different ways painters, sculptors and other artists explored and exploited the dense forest of myth. It explains how formulas were devised for certain stories; how these could be adapted, developed and even transferred to other contexts; how one myth could be distinguished from another - or confused with it; how myths related to daily life or political propaganda; and the influence of evolving tastes. Written in a lively and accessible style, fully illustrated with examples drawn from a wide range of media, Images of Myths in Classical Antiquity provides fresh and stimulating insights into the representation of myths in Greek and Roman art.

Neutron scattering is an extremely powerful tool in the study of elemental excitations in condensed matter. This book provides a practical guide to basic techniques using a triple-axis spectrometer. Introductory chapters summarize useful scattering formulas and describe the components of a spectrometer, followed by a comprehensive discussion of the resolution function and focusing effects. Later sections include simple examples of phonon and magnon measurements, and an analysis of spurious effects in both inelastic and elastic measurements, and how to avoid them. Finally, polarization analysis techniques and their applications are covered. This guide will allow graduate students and experienced researchers new to neutron scattering to make the most efficient use of their experimental time.

Numerical analysis is an increasingly important link between pure mathematics and its application in science and technology. This textbook provides an introduction to the justification and development of constructive methods that provide sufficiently accurate approximations to the solution of numerical problems, and the analysis of the influence that errors in data, finite-precision calculations, and approximation formulas have on results, problem formulation and the choice of method. It also serves as an introduction to scientific programming in MATLAB, including many simple and difficult, theoretical and computational exercises. A unique feature of this book is the consequent development of interval analysis as a tool for rigorous computation and computer assisted proofs, along with the traditional material.

This book provides a comprehensive account of a key (and perhaps the most important) theory upon which the Taylor–Wiles proof of Fermat's last theorem is based. The book begins with an overview of the theory of automorphic forms on linear algebraic groups and then covers the basic theory and results on elliptic modular forms, including a substantial simplification of the Taylor–Wiles proof by Fujiwara and Diamond. It contains a detailed exposition of the representation theory of profinite groups (including deformation theory), as well as the Euler characteristic formulas of Galois cohomology groups. The final chapter presents a proof of a non-abelian class number formula and includes several new results from the author. The book will be of interest to graduate students and researchers in number theory (including algebraic and analytic number theorists) and arithmetic algebraic geometry.

The Riemann zeta function is one of the most studied objects in mathematics, and is of fundamental importance. In this book, based on his own research, Professor Motohashi shows that the function is closely bound with automorphic forms and that many results from there can be woven with techniques and ideas from analytic number theory to yield new insights into, and views of, the zeta function itself. The story starts with an elementary but unabridged treatment of the spectral resolution of the non-Euclidean Laplacian and the trace formulas. This is achieved by the use of standard tools from analysis rather than any heavy machinery, forging a substantial aid for beginners in spectral theory as well. These ideas are then utilized to unveil an image of the zeta-function, first perceived by the author, revealing it to be the main gem of a necklace composed of all automorphic L-functions. In this book, readers will find a detailed account of one of the most fascinating stories in the develo

This book presents an up-to-date, unified treatment of research in bounded arithmetic and complexity of propositional logic, with emphasis on independence proofs and lower bound proofs. The author discusses the deep connections between logic and complexity theory and lists a number of intriguing open problems. An introduction to the basics of logic and complexity theory is followed by discussion of important results in propositional proof systems and systems of bounded arithmetic. More advanced topics are then treated, including polynomial simulations and conservativity results, various witnessing theorems, the translation of bounded formulas (and their proofs) into propositional ones, the method of random partial restrictions and its applications, direct independence proofs, complete systems of partial relations, lower bounds to the size of constant-depth propositional proofs, the method of Boolean valuations, the issue of hard tautologies and optimal proof systems, combinatorics and

Pseudodifferential operators arise naturally in the solution of boundary problems for partial differential equations. The formalism of these operators serves to make the Fourier-Laplace method applicable for nonconstant coefficient equations. This book presents the technique of pseudodifferential operators and its applications, especially to the Dirac theory of quantum mechanics. The treatment uses 'Leibniz formulas' with integral remainders or as asymptotic series. A pseudodifferential operator may also be described by invariance under action of a Lie-group. The author discusses connections to the theory of C*-algebras, invariant algebras of pseudodifferential operators under hyperbolic evolution and the relation of the hyperbolic theory to the propagation of maximal ideals. This book will be of particular interest to researchers in partial differential equations and mathematical physics.

It is usually straightforward to calculate the result of a practical experiment in the laboratory. Estimating the accuracy of that result is often regarded by students as an obscure and tedious routine, involving much arithmetic. An estimate of the error is, however, an integral part of the presentation of the results of experiments. This textbook is intended for undergraduates who are carrying out laboratory experiments in the physical sciences for the first time. It is a practical guide on how to analyse data and estimate errors. The necessary formulas for performing calculations are given, and the ideas behind them are explained, although this is not a formal text on statistics. Specific examples are worked through step by step in the text. Emphasis is placed on the need to think about whether a calculated error is sensible. At first students should take this book with them to the laboratory, and the format is intended to make this convenient. The book will provide the necessary und

It is usually straightforward to calculate the result of a practical experiment in the laboratory. Estimating the accuracy of that result is often regarded by students as an obscure and tedious routine, involving much arithmetic. An estimate of the error is, however, an integral part of the presentation of the results of experiments. This textbook is intended for undergraduates who are carrying out laboratory experiments in the physical sciences for the first time. It is a practical guide on how to analyse data and estimate errors. The necessary formulas for performing calculations are given, and the ideas behind them are explained, although this is not a formal text on statistics. Specific examples are worked through step by step in the text. Emphasis is placed on the need to think about whether a calculated error is sensible. At first students should take this book with them to the laboratory, and the format is intended to make this convenient. The book will provide the necessary und

The interacting boson-fermion model has become in recent years the standard model for the description of atomic nuclei with an odd number of protons and/or neutrons. This book describes the mathematical framework on which the interacting boson-fermion model is built and presents applications to a variety of situations encountered in nuclei. The book addresses both the analytical and the numerical aspects of the problem. The analytical aspect requires the introduction of rather complex group theoretic methods, including the use of graded (or super) Lie algebras. The first (and so far only) example of supersymmetry occurring in nature is also discussed. The book is the first comprehensive treatment of the subject and will appeal to both theoretical and experimental physicists. The large number of explicit formulas for level energies, electromagnetic transition rates and intensities of transfer reactions presented in the book provide a simple but detailed way to analyse experimental data.

Most salespeople and sales leaders who marvel at the consistent year in and year out performance of sales superstars are blind to the real reason for their success. Unwilling to accept that the foundational root of all success in sales is a fanatical focus on prospecting, they waste time tilting at windmills on their quixotic pursuit of fads, silver bullets, and secret formulas they believe will deliver them into arms of success with little effort. Fanatical ProspectingPlaybook will be an essential tool for leaders that aids and coaches sales people to become more effective prospectors. For salespeople, this book will explain the how and why behind the most important activity in sales and teach the skills and techniques required to become both effective and efficient at prospecting.

Written by Master Longevity Herbalist Scott Rauvers, author of 5 books on Longevity, and 2 Herbal Brand Name Longevity Supplements, this unique book lists at least 3 Stem Cell herbal longevity formula

This is a bundle of previously published eBook lessons.The Mastering Excel: Dashboard Bundle contains the following lessons:- Mastering Excel Formulas: SUM, SUMIF- Mastering Excel Formulas: VLOOKUP- M

There are formulas for things that exist. But what about formulas for the things that don’t exist? Increasingly the traditional ways of making our path in this world are irrelevant and there comes a p

Shoki Coe was among the first to speak of "contextualization" in theology. Coe argued that theology is not a reiteration of past formulas or doctrines but a response to the self-disclosing initiative

This is the only book dedicated to the Geometry of Polycentric Ovals. It includes problem solving constructions and mathematical formulas. For anyone interested in drawing or recognizing an oval, this

CCH's Corporate Controller's Handbook of Financial Management is a comprehensive source of practical solutions, strategies, techniques, procedures, and formulas covering all key aspects of accounting

This textbook aims to fill the gap between those that offer a theoretical treatment without many applications and those that present and apply formulas without appropriately deriving them. T

CCH's Corporate Controller's Handbook of Financial Management is a comprehensive source of practical solutions, strategies, techniques, procedures, and formulas covering all key aspects of accounting

A Concise Handbook of Mathematics, Physics, and Engineering Sciences takes a practical approach to the basic notions, formulas, equations, problems, theorems, methods, and laws that most frequently oc

In this volume algorithms are compared for rhumb line sailing (RLS) and great elliptic sailing (GES) calculations commonly used for route planning and route monitoring. Compact vector formulas are giv

Avicenna structured his Pharmacopia beginning with the formulas for compound medications arranged according to the various types of preparations that he found in the previous medical books translated

This text for advanced undergraduates emphasizes the logical connections of the subject. The derivations of formulas from the axioms do not make use of models of the hyperbolic plane until the axioms

Offers information on urban beekeeping, as well as twenty-one recipes for enjoying honey in baking and cooking, and five formulas for salves, beauty products, and beeswax candles.

Need help writing a high-scoring essay for the SAT or ACT exam? This expert guide will give you winning strategies, formulas, and content guidance to do just that. Go into the essay sections of these

The well respected and ever popular Fieser and Fieser series on reagents for organic synthesis provides concise descriptions, good structural formulas and selected examples of applications—providing r