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The concept of a value of a coalitional game, in the spirit ofR.J. Aumann and L.S. Shapley, is extended to the case of games withfuzzy coalitions, providing new and heuristically meaningful insightsinto the game theoretical context, which have some significant impacton balance and equilibria analysis in a cooperative environment.
Using the suggestive and philosophical power of the concept of fuzzysets introduced by L.A. Zadeh we develop the mathematical machinery oftriangular norm-based measures, i.e. valuations preserving binaryoperations induced by triangular norms on [0,1].
Our results not only prove the existence of Aumann--Shapley values forlarge classes of games with fuzzy coalitions satisfying certaindifferentiability conditions, but also allow the extension of thedomain of such values to games with crisp coalitions, and theapplication to real life situations such as rate problems for servicesin bulk.
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Generalising classical concepts of probability theory, theinvestigation of operator (semi)-stable laws as possible limitdistributions of operator-normalized sums of i.i.d. random variable onfinite-dimensional vector space started in 1969. Currently, thistheory is still in progress and promises interesting applications.Parallel to this, similar stability concepts for probabilities ongroups were developed during recent decades. It turns out that theexistence of suitable limit distributions has a strong impact on thestructure of both the normalizing automorphisms and the underlyinggroup. Indeed, investigations in limit laws led to contractablegroups and - at least within the class of connected groups- to homogeneous groups, in particular to groups that aretopologically isomorphic to a vector space. Moreover, it has beenshown that (semi)-stable measures on groups have a vector spacecounterpart and vice versa.
The purpose of this book is to describe the structure of limit lawsand the limit behaviour of normalized i.i.d. random variables ongroups and on finite-dimensional vector spaces from a common point ofview. This will also shed a new light on the classical situation.Chapter 1 provides an introduction to stability problems on vectorspaces. Chapter II is concerned with parallel investigations forhomogeneous groups and in Chapter III the situation beyond homogeneousLie groups is treated. Throughout, emphasis is laid on the descriptionof features common to the group- and vector space situation.
Chapter I can be understood by graduate students with some backgroundknowledge in infinite divisibility. Readers of Chapters II and III areassumed to be familiar with basic techniques from probability theoryon locally compact groups. ... Read more

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This unique book on the subject addresses fundamental problems and will be the standard reference for a long time to come. The authors have different scientific origins and combine these successfully, creating a text aimed at graduate students and researchers that can be used for courses and seminars.

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All you wnated to know but did not dare to ask
This book contains everything one needs about the transformations of the Wiener measure under the perturbation of identity-type mappings. It is particularly useful to graduate students in probability, analysis and theoretical physics. It is a pity that there is so few ad about it. ... Read more

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Symmetries and invariance principles play an important role in various branches of mathematics. This book deals with measures having weak symmetry properties. Even mild conditions ensure that all invariant Borel measures on a second countable locally compact space can be expressed as images of specific product measures under a fixed mapping. The results derived in this book are interesting for their own and, moreover, a number of carefully investigated examples underline and illustrate their usefulness and applicability for integration problems, stochastic simulations and statistical applications. ... Read more

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This book presents the theory of integration over surfaces inabstract topological vector space. Applications of the theory indifferent fields, such as infinite dimensional distributions anddifferential equations (including boundary value problems), stochasticprocesses, approximation of functions, and calculus of variation on aBanach space, are treated in detail.
Audience: This book will be of interest to specialists infunctional analysis, and those whose work involves measure andintegration, probability theory and stochastic processes, partialdifferential equations and mathematical physics. ... Read more

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W.K. ALLARD: On the first variation of area and generalized mean curvature.- F.J. ALMGREN Jr.: Geometric measure theory and elliptic variational problems.- E. GIUSTI: Minimal surfaces with obstacles.- J. GUCKENHEIMER: Singularities in soap-bubble-like and soap-film-like surfaces.- D. KINDERLEHRER:The analyticity of the coincidence set in variational inequalities.- M. MIRANDA: Boundaries of Caciopoli sets in the calculus of variations.- L. PICCININI: De Giorgi’s measure and thin obstacles. ... Read more

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Young measures are presented in a general setting which includes finite and for the first time infinite dimensional spaces: the fields of applications of Young measures (Control Theory, Calculus of Variations, Probability Theory...) are often concerned with problems in infinite dimensional settings.

The theory of Young measures is now well understood in a finite dimensional setting, but open problems remain in the infinite dimensional case. We provide several new results in the general frame, which are new even in the finite dimensional setting, such as characterizations of convergence in measure of Young measures (Chapter 3) and compactness criteria (Chapter 4). These results are established under a different form (and with fewer details and developments) in recent papers by the same authors. We also provide new applications to Visintin and Reshetnyak type theorems (Chapters 6 and 8), existence of solutions to differential inclusions (Chapter 7), dynamical programming (Chapter 8) and the Central Limit Theorem in locally convex spaces (Chapter 9).

Customer Reviews (1)

Good for Review
Like the other books of the Schaum's Outline series, this book goes through the theories and shows you how to work the problems step by step.Although this is very helpful, I like to be able to test my knowledge.I say this is good for review because you will remember as you read this.You will also have a good idea of whether you are doing the supplementary problems correct or not.Here, most the supplementary problems do not have answers.Those that do, just have the final answer rather than an illustration of how the writers got the answer.This is difficult if you are learning about real variables for the first time.

As well as giving you the basic concepts, this book also goes into Lebesgue integrals, Riemann integrals, and Fourier series.The explanations are fairly well written, and the examples are easy to follow. ... Read more

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This digital document is a journal article from Electoral Studies, published by Elsevier in 2007. The article is delivered in HTML format and is available in your Amazon.com Media Library immediately after purchase. You can view it with any web browser.

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A variety of methodologies have been employed in order to measure the positions of political parties on the issue of European Integration. This paper compares four types of measures of party position: reputational, textual, behavioral, and self reported indicators. These indicators are assessed in terms of their content validity, convergent validity, and, construct validity. All of these approaches yield plausible measures which correlate reasonably well with each other, although care must be taken with the creation of indices from the Comparative Manifesto Project dataset. However, these indicators are not interchangeable, and the results of substantive research on EU politics are dependent, to an alarming extent, on the choice of indicators. Careful attention should be paid to the choice of measures, and the theoretical justification of any measure used. Future researchers would be well advised to test whether their results are robust across alternative measures of party positions. ... Read more

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This book aims at restructuring some fundamentals in measure and integration theory. It centers around the ubiquitous task to produce appropriate contents and measures from more primitive data like elementary contents and elementary integrals. It develops the new approach started around 1970 by Topsoe and others into a systematic theory. The theory is much more powerful than the traditional means and has striking implications all over measure theory and beyond.

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This text takes advantage of recent developments in the theory of path integration and attempts to make a major paradigm shift in how the art of functional integration is practiced. The techniques developed in the work will prove valuable to graduate students and researchers in physics, chemistry, mathematical physics, and applied mathematics who find it necessary to deal with solutions to wave equations, both quantum and beyond.

A Modern Approach to Functional Integration offers insight into a number of contemporary research topics, which may lead to improved methods and results that cannot be found elsewhere in the textbook literature. Exercises are included in most chapters, making the book suitable for a one-semester graduate course on functional integration.

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Intégration is the sixth and last of the Books that form the core of the Bourbaki series; it draws abundantly on the preceding five Books, especially General Topology and  Topological Vector Spaces, making it a culmination of the core six. The power of the tool thus fashioned is strikingly displayed in Chapter II of the author's Théories Spectrales [MR  35 #4725], an exposition, in a mere 38 pages, of abstract harmonic analysis and the structure of locally compact abelian groups.

Customer Reviews (1)

Excellent Treatise on the topic.
This book is well written and nicely layed out. With a great exposition and any worked examples, it is all you need for self study.And with plenty of problems, and solutions or hints to each, it is a great companionfor any first year graduate student in mathematics.The scope of the bookis limited to its title (measure and integration), but you will find it agreat reference for many years! ... Read more

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High Quality Content by WIKIPEDIA articles! Noncommutative measure and integration refers to the theory of weights, states, and traces on von Neumann algebras (Takesaki 1979 v. 2 p. 141). If a von Neumann algebra acts on a Hilbert space containing a norm 1 vector v, then the functional a → (av,v) is a normal state. This construction can be reversed to give an action on a Hilbert space from a normal state: this is the GNS construction for normal states. Any factor has a trace such that the trace of a non-zero projection is non-zero and the trace of a projection is infinite if and only if the projection is infinite. Such a trace is unique up to rescaling. For factors that are separable or finite, two projections are equivalent if and only if they have the same trace. ... Read more

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In probability theory and statistics, the expected valuemathematical expectation, or mean,of a random variable is the integral of the random variable with respect to its probability measure.For discrete random variables this is equivalent to the probability-weighted sum of the possible values.For continuous random variables with a density function it is the probability density-weighted integral of the possible values.The term "expected value" can be misleading. It must not be confused with the "most probable value." The expected value is in general not a typical value that the random variable can take on. It is often helpful to interpret the expected value of a random variable as the long-run average value of the variable over many independent repetitions of an experiment. ... Read more