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The aim of this monograph is to give a thorough and self-contained account of functions of (generalized) bounded variation, the methods connected with their study, their relations to other important function classes, and their applications to various problems arising in Fourier analysis and nonlinear analysis. In the first part the basic facts about spaces of functions of bounded variation and related spaces are collected, the main ideas which are useful in studying their properties are presented, and a comparison of their importance and suitability for applications is provided, with a particular emphasis on illustrative examples and counterexamples. The second part is concerned with (sometimes quite surprising) properties of nonlinear composition and superposition operators in such spaces. Moreover, relations with Riemann-Stieltjes integrals, convergence tests for Fourier series, and applications to nonlinear integral equations are discussed. The only prerequisite for understanding this book is a modest background in real analysis, functional analysis, and operator theory. It is addressed to non-specialists who want to get an idea of the development of the theory and its applications in the last decades, as well as a glimpse of the diversity of the directions in which current research is moving. Since the authors try to take into account recent results and state several open problems, this book might also be a fruitful source of inspiration for further research.

Functions of bounded variation. --- Bounded variables, Functions of --- Bounded variation, Functions of --- BV functions --- Functions of bounded variables --- Functions of real variables --- Boundary Value Problem. --- Bounded Variation. --- Continuity Properties. --- Fourier Analysis. --- Monotonicity Properties. --- Nonlinear Composition Operators. --- Nonlinear Integral Equation.

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Economists have long sought to learn the effect of a "treatment" on some outcome of interest, just as doctors do with their patients. A central practical objective of research on treatment response is to provide decision makers with information useful in choosing treatments. Often the decision maker is a social planner who must choose treatments for a heterogeneous population--for example, a physician choosing medical treatments for diverse patients or a judge choosing sentences for convicted offenders. But research on treatment response rarely provides all the information that planners would like to have. How then should planners use the available evidence to choose treatments? This book addresses key aspects of this broad question, exploring and partially resolving pervasive problems of identification and statistical inference that arise when studying treatment response and making treatment choices. Charles Manski addresses the treatment-choice problem directly using Abraham Wald's statistical decision theory, taking into account the ambiguity that arises from identification problems under weak but justifiable assumptions. The book unifies and further develops the influential line of research the author began in the late 1990s. It will be a valuable resource to researchers and upper-level graduate students in economics as well as other social sciences, statistics, epidemiology and related areas of public health, and operations research.

Estimation theory. --- Social choice. --- Social sciences --- Statistical methods. --- "Objective Function. --- Bayes Rules. --- Binary Covariates. --- Bounding Expected Welfare. --- Concave Monotonicity. --- Covariate Information. --- Empirical Success Rules. --- Hoeffding theorem. --- Hypothesis Testing. --- Maximin Criterion. --- Nonutilitarian Objective Functions. --- Optimal Treatment Rule. --- Semi-Monotone Response. --- Sentencing. --- Social Interactions. --- Untenable Assumptions. --- conditional empirical success. --- external validity. --- instrumental variable. --- quasi-optimal". --- shrinkage estimators. --- variable selection procedures.

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Simple games are mathematical structures inspired by voting systems in which a single alternative, such as a bill, is pitted against the status quo. The first in-depth mathematical study of the subject as a coherent subfield of finite combinatorics--one with its own organized body of techniques and results--this book blends new theorems with some of the striking results from threshold logic, making all of it accessible to game theorists. Introductory material receives a fresh treatment, with an emphasis on Boolean subgames and the Rudin-Keisler order as unifying concepts. Advanced material focuses on the surprisingly wide variety of properties related to the weightedness of a game. A desirability relation orders the individuals or coalitions of a game according to their influence in the corresponding voting system. As Taylor and Zwicker show, acyclicity of such a relation approximates weightedness--the more sensitive the relation, the closer the approximation. A trade is an exchange of players among coalitions, and robustness under such trades is equivalent to weightedness of the game. Robustness under trades that fit some restrictive exchange pattern typically characterizes a wider class of simple games--for example, games for which some particular desirability order is acyclic. Finally, one can often describe these wider classes of simple games by weakening the total additivity of a weighting to obtain what is called a pseudoweighting. In providing such uniform explanations for many of the structural properties of simple games, this book showcases numerous new techniques and results.

Game theory. --- B-medium coalition. --- Boolean function. --- Boolean subgame. --- Canadian system. --- Chow games. --- Chow parameters. --- EL-sequence. --- Gabelman game. --- Hahn-Banach theorem. --- Harvard Band. --- Luxembourg. --- abstention. --- adiabatic computing. --- adjacency of vertices. --- alternating cycle. --- antisymmetry. --- augmented trading matrix. --- automorphic coalitions. --- balanced game. --- basic class. --- bicameral join. --- bicameral meet. --- characteristic function form. --- classification of linear games. --- classified coalition. --- column union. --- complement symmetric relation. --- compound simple game. --- constant-sum games. --- count and account method. --- direct market. --- double appearance. --- downward closure. --- dual game. --- dual property. --- edge, of a graph. --- fair division. --- geometry of simple games. --- grant of certiorari. --- high failure. --- homogeneous games. --- hypergraph. --- identification of variables. --- interval game. --- k-asuramability. --- large coalition. --- linear games. --- linear graph. --- losing coalition. --- major player. --- master player. --- monotonicity. --- multipartition. --- multiweighting.

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Asymptotic differential algebra seeks to understand the solutions of differential equations and their asymptotics from an algebraic point of view. The differential field of transseries plays a central role in the subject. Besides powers of the variable, these series may contain exponential and logarithmic terms. Over the last thirty years, transseries emerged variously as super-exact asymptotic expansions of return maps of analytic vector fields, in connection with Tarski's problem on the field of reals with exponentiation, and in mathematical physics. Their formal nature also makes them suitable for machine computations in computer algebra systems.This self-contained book validates the intuition that the differential field of transseries is a universal domain for asymptotic differential algebra. It does so by establishing in the realm of transseries a complete elimination theory for systems of algebraic differential equations with asymptotic side conditions. Beginning with background chapters on valuations and differential algebra, the book goes on to develop the basic theory of valued differential fields, including a notion of differential-henselianity. Next, H-fields are singled out among ordered valued differential fields to provide an algebraic setting for the common properties of Hardy fields and the differential field of transseries. The study of their extensions culminates in an analogue of the algebraic closure of a field: the Newton-Liouville closure of an H-field. This paves the way to a quantifier elimination with interesting consequences.

Series, Arithmetic. --- Divergent series. --- Asymptotic expansions. --- Differential algebra. --- Algebra, Differential --- Differential fields --- Algebraic fields --- Differential equations --- Asymptotic developments --- Asymptotes --- Convergence --- Difference equations --- Divergent series --- Functions --- Numerical analysis --- Series, Divergent --- Series --- Arithmetic series --- Progressions, Arithmetic --- Equalizer Theorem. --- H-asymptotic couple. --- H-asymptotic field. --- H-field. --- Hahn Embedding Theorem. --- Hahn space. --- Johnson's Theorem. --- Krull's Principal Ideal Theorem. --- Kähler differentials. --- Liouville closed H-field. --- Liouville closure. --- Newton degree. --- Newton diagram. --- Newton multiplicity. --- Newton tree. --- Newton weight. --- Newton-Liouville closure. --- Riccati transform. --- Scanlon's extension. --- Zariski topology. --- algebraic differential equation. --- algebraic extension. --- angular component map. --- asymptotic couple. --- asymptotic differential algebra. --- asymptotic field. --- asymptotic relation. --- asymptotics. --- closed H-asymptotic couple. --- closure properties. --- coarsening. --- commutative algebra. --- commutative ring. --- compositional conjugation. --- constant. --- continuity. --- d-henselian. --- d-henselianity. --- decomposition. --- derivation. --- differential field extension. --- differential field. --- differential module. --- differential polynomial. --- differential-hensel. --- differential-henselian field. --- differential-henselianity. --- differential-valued extension. --- differentially closed field. --- dominant part. --- equivalence. --- eventual quantities. --- exponential integral. --- extension. --- filtered module. --- gaussian extension. --- grid-based transseries. --- henselian valued field. --- homogeneous differential polynomial. --- immediate extension. --- integral. --- integrally closed domain. --- linear differential equation. --- linear differential operator. --- linear differential polynomial. --- mathematics. --- maximal immediate extension. --- model companion. --- monotonicity. --- noetherian ring. --- ordered abelian group. --- ordered differential field. --- ordered set. --- pre-differential-valued field. --- pseudocauchy sequence. --- pseudoconvergence. --- quantifier elimination. --- rational asymptotic integration. --- regular local ring. --- residue field. --- simple differential ring. --- small derivation. --- special cut. --- specialization. --- substructure. --- transseries. --- triangular automorphism. --- triangular derivation. --- valuation topology. --- valuation. --- value group. --- valued abelian group. --- valued differential field. --- valued field. --- valued vector space.

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The present book focuses on that part of calculus of variations, optimization, nonlinear analysis and related applications which combines tools and methods from partial differential equations with geometrical techniques. More precisely, this work is devoted to nonlinear problems coming from different areas, with particular reference to those introducing new techniques capable of solving a wide range of problems. The book is a valuable guide for researchers, engineers and students in the field of mathematics, operations research, optimal control science, artificial intelligence, management science and economics.

Research & information: general --- Mathematics & science --- well posedness --- constrained variational control problem --- monotonicity --- pseudomonotonicity --- hemicontinuity --- multiple integral functional --- lower semicontinuity --- fractional differential equations --- fractional derivative of Riemann–Liouville type --- integral boundary value problems --- Green’s functions --- Guo–Krasnosel’skii fixed point theorem in cones --- sublinearity and superlinearity --- Arzelà-Ascoli Theorem --- multi-objective programming --- fractional transportation problem --- intuitionistic fuzzy set --- parametric programming --- convex function --- h-convex function --- Hermite–Hadamard inequality --- Caputo–Fabrizio fractional integral --- Jensen inequality --- Jensen–Mercer inequality --- multiobjective programs with vanishing constraints --- semidefinite programming --- convexificators --- nonsmooth analysis --- constraint qualifications --- interval-valued function --- Riemann integral --- LR-convex interval-valued function --- interval Hermite–Hadamard inequality --- interval Hermite–Hadamard–Fejér inequality --- Lieb concavity theorem --- deformed exponential --- Pick function --- convexity of matrix --- low carbon inventory --- discount --- payment in advance --- price-sensitive demand --- emission reduction --- advances of SDO --- applications of SDO --- metaheuristic optimization --- nature-inspired algorithms --- optimization problems --- spiral dynamics optimization --- spiral-inspired optimization algorithms --- spiral paths --- (p,s)-convex fuzzy-interval-valued function --- fuzzy Riemann integral --- Jensen type inequality --- Schur type inequality --- Hermite–Hadamard type inequality --- Hermite–Hadamard–Fejér type inequality --- inverse geometric problem --- Laplace equation --- method of fundamental solution --- least-square problem --- micro resonator --- fractal --- multistability --- safe jump --- hidden attractor --- chaos --- basin of attraction --- LR-Harmonically convexity --- fractional integral operator --- Hermite–Hadamard type inequalities --- multimodal multi-objective optimization --- manta ray foraging optimizer --- non-dominated solution --- crowing distance --- engineering design problem --- optimal power flow --- renewable energy sources --- improved chaos game optimization --- TD-TI controller --- load frequency control --- electrical vehicles --- n/a --- fractional derivative of Riemann-Liouville type --- Green's functions --- Guo-Krasnosel'skii fixed point theorem in cones --- Arzelà-Ascoli Theorem --- Hermite-Hadamard inequality --- Caputo-Fabrizio fractional integral --- Jensen-Mercer inequality --- interval Hermite-Hadamard inequality --- interval Hermite-Hadamard-Fejér inequality --- Hermite-Hadamard type inequality --- Hermite-Hadamard-Fejér type inequality --- Hermite-Hadamard type inequalities