Formulas

How exactly did political power operate in early medieval Europe? Taking Alsace as his focus, Hans Hummer offers an intriguing new case study on localised and centralised power and the relationship between the two from c. 600–1000. Providing a panoramic survey of the sources from the region, which include charters, notarial formulas, royal instruments, and Old High German literature, he untangles the networks of monasteries and kin groups which made up the political landscape of Alsace, and shows the significance of monastic control in shaping that landscape. He also investigates this local structure in light of comparative evidence from other regions. He tracks the emergence of the distinctive local order during the seventh century to its eventual decline in the late tenth century in the face of radical monastic reform. Highly original and well balanced, this 2006 work is of interest to all students of medieval political structures.

Presenting important trends in the field of stochastic analysis, this collection of thirteen articles provides an overview of recent developments and new results. Written by leading experts in the field, the articles cover a wide range of topics, ranging from an alternative set-up of rigorous probability to the sampling of conditioned diffusions. Applications in physics and biology are treated, with discussion of Feynman formulas, intermittency of Anderson models and genetic inference. A large number of the articles are topical surveys of probabilistic tools such as chaining techniques, and of research fields within stochastic analysis, including stochastic dynamics and multifractal analysis. Showcasing the diversity of research activities in the field, this book is essential reading for any student or researcher looking for a guide to modern trends in stochastic analysis and neighbouring fields.

The theory of Hardy spaces has close connections to many branches of mathematics including Fourier analysis, harmonic analysis, singular integrals, potential theory and operator theory, and has found essential applications in robust control engineering. For each application, the ability to represent elements of these classes by series or integral formulas is of utmost importance. This self-contained text provides an introduction to a wide range of representation theorems and provides a complete description of the representation theorems with direct proofs for both classes of Hardy spaces: Hardy spaces of the open unit disc and Hardy spaces of the upper half plane. With over 300 exercises, many with accompanying hints, this book is ideal for those studying Advanced Complex Analysis, Function Theory or Theory of Hardy Spaces. Advanced undergraduate and graduate students will find the book easy to follow, with a logical progression from basic theory to advanced research.

The theory of Hardy spaces has close connections to many branches of mathematics including Fourier analysis, harmonic analysis, singular integrals, potential theory and operator theory, and has found essential applications in robust control engineering. For each application, the ability to represent elements of these classes by series or integral formulas is of utmost importance. This self-contained text provides an introduction to a wide range of representation theorems and provides a complete description of the representation theorems with direct proofs for both classes of Hardy spaces: Hardy spaces of the open unit disc and Hardy spaces of the upper half plane. With over 300 exercises, many with accompanying hints, this book is ideal for those studying Advanced Complex Analysis, Function Theory or Theory of Hardy Spaces. Advanced undergraduate and graduate students will find the book easy to follow, with a logical progression from basic theory to advanced research.

By analysing original sources and evaluating conceptual frameworks, this book discusses the idea proclaimed in the Preamble to the Constitution that Australia is a federal commonwealth. Taking careful account of the influence which the American, Canadian and Swiss Constitutions had upon the framers of the Australian Constitution, the author shows how the framers wrestled with the problem of integrating federal ideas with inherited British traditions and their own experiences of parliamentary government. In so doing, the book explains how the Constitution came into being in the context of the groundswell of federal ideas then sweeping the English-speaking world. In advancing an original argument about the relationship between the formation of the Constitution, the representative institutions, configurations of power and amending formulas contained therein, light is shed on the terms and structure of the Constitution and a range of problems associated with its interpretation and practica

By analysing original sources and evaluating conceptual frameworks, this book discusses the idea proclaimed in the Preamble to the Constitution that Australia is a federal commonwealth. Taking careful account of the influence which the American, Canadian and Swiss Constitutions had upon the framers of the Australian Constitution, the author shows how the framers wrestled with the problem of integrating federal ideas with inherited British traditions and their own experiences of parliamentary government. In so doing, the book explains how the Constitution came into being in the context of the groundswell of federal ideas then sweeping the English-speaking world. In advancing an original argument about the relationship between the formation of the Constitution, the representative institutions, configurations of power and amending formulas contained therein, light is shed on the terms and structure of the Constitution and a range of problems associated with its interpretation and practica

Caring for newborns and infants in the first year of life can be anxiety-provoking and challenging, especially for the emergency physician. This book addresses the most common complaints that an emergency provider may have to evaluate and treat. It is divided into three sections for quick access to a variety of medical conditions. Section I covers the basics of neonate care in the emergency room and includes the general exam, airway management, and sedation. Section II focuses on specific common complaints, such as failure to thrive, fever, and seizures. Section III covers management of specific systems, including cardiology, dermatology, and toxicology. The Appendixes include handy pediatric formulas, immunization schedules, and easy-to-use algorithms and tables. A practical resource for a wide range of health providers, from medical students, residents, and Fellows to attending physicians.

As a partner to Volume 1: Dimensional Continuous Models, this monograph provides a self-contained introduction to algebro-geometric solutions of completely integrable, nonlinear, partial differential-difference equations, also known as soliton equations. The systems studied in this volume include the Toda lattice hierarchy, the Kac-van Moerbeke hierarchy, and the Ablowitz-Ladik hierarchy. An extensive treatment of the class of algebro-geometric solutions in the stationary as well as time-dependent contexts is provided. The theory presented includes trace formulas, algebro-geometric initial value problems, Baker-Akhiezer functions, and theta function representations of all relevant quantities involved. The book uses basic techniques from the theory of difference equations and spectral analysis, some elements of algebraic geometry and especially, the theory of compact Riemann surfaces. The presentation is constructive and rigorous, with ample background material provided in various appen

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

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

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

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.

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

The author of The Fortune-Telling Book unlocks the magic and power of dreams, using a dictionary format to help readers explore the meaning of these messages from the unconscious. 40,000 first printin

A new edition of a classic gathering of garden recipes offers advice on making delicious herbal teas, creating special soaps, designing a uniquely beautiful garden, and much more. Reprint.

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

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

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

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

How to formulate, compound, and manufacture industrial detergents. Contains 300 formulas to review and study, along with the author's detailed notes on each one.

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

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

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