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This two-volume textbook provides comprehensive coverage of partial differential equations, spanning elliptic, parabolic, and hyperbolic types in two and several variables. In this first volume, special emphasis is placed on geometric and complex variable methods involving integral representations. The following topics are treated: • integration and differentiation on manifolds • foundations of functional analysis • Brouwer's mapping degree • generalized analytic functions • potential theory and spherical harmonics • linear partial differential equations This new second edition of this volume has been thoroughly revised and a new section on the boundary behavior of Cauchy’s integral has been added. The second volume will present functional analytic methods and applications to problems in differential geometry. This textbook will be of particular use to graduate and postgraduate students interested in this field and will be of interest to advanced undergraduate students. It may also be used for independent study.

Differential equations, Partial. --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Integral representations. --- Representations, Integral --- Partial differential equations --- Mathematics. --- Partial differential equations. --- Physics. --- Partial Differential Equations. --- Mathematical Methods in Physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Math --- Science --- Algebraic number theory --- Crystallography, Mathematical --- Representations of groups --- Differential equations, partial. --- Mathematical physics. --- Physical mathematics --- Physics

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This comprehensive two-volume textbook presents the whole area of Partial Differential Equations - of the elliptic, parabolic, and hyperbolic type - in two and several variables. Special emphasis is put on the connection of PDEs and complex variable methods. In this first volume the following topics are treated: Integration and differentiation on manifolds, Functional analytic foundations, Brouwer's degree of mapping, Generalized analytic functions, Potential theory and spherical harmonics, Linear partial differential equations. While we solve the partial differential equations via integral representations in this volume, we shall present functional analytic solution methods in the second volume. This textbook can be chosen for a course over several semesters on a medium level. Advanced readers may study each chapter independently from the others.

Differential equations, Partial --- Integral representations --- Functional analysis --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Representations, Integral --- Algebraic number theory --- Crystallography, Mathematical --- Representations of groups --- Partial differential equations --- Differential equations, partial. --- Mathematical physics. --- Partial Differential Equations. --- Mathematical Methods in Physics. --- Physical mathematics --- Physics --- Mathematics --- Differential equations, Partial. --- Partial differential equations. --- Physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics

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This monograph presents the state of the art of convexity, with an emphasis to integral representation. The exposition is focused on Choquet's theory of function spaces with a link to compact convex sets. An important feature of the book is an interplay between various mathematical subjects, such as functional analysis, measure theory, descriptive set theory, Banach spaces theory and potential theory. A substantial part of the material is of fairly recent origin and many results appear in the book form for the first time. The text is self-contained and covers a wide range of applications. From the contents: Geometry of convex sets Choquet theory of function spaces Affine functions on compact convex sets Perfect classes of functions and representation of affine functions Simplicial function spaces Choquet's theory of function cones Topologies on boundaries Several results on function spaces and compact convex sets Continuous and measurable selectors Construction of function spaces Function spaces in potential theory and Dirichlet problem Applications

Functional analysis. --- Convex domains. --- Banach spaces. --- Potential theory (Mathematics) --- Integral representations. --- Representations, Integral --- Algebraic number theory --- Crystallography, Mathematical --- Representations of groups --- Green's operators --- Green's theorem --- Potential functions (Mathematics) --- Potential, Theory of --- Mathematical analysis --- Mechanics --- Functions of complex variables --- Generalized spaces --- Topology --- Convex regions --- Convexity --- Calculus of variations --- Convex geometry --- Point set theory --- Functional calculus --- Functional equations --- Integral equations --- Convex Analysis. --- Dirichlet Problem. --- Functional Analysis. --- Partial Differential Equation. --- Potential Theory. --- Functional analysis --- Convex domains --- Banach spaces --- Integral representations

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Emotions have emerged as a topic of interest across the disciplines, yet studies and findings on emotions tend to fall into two camps: body versus brain, nature versus nurture. Emotions as Bio-cultural Processes offers a unique collaboration across the biological/social divide—from psychology and neuroscience to cultural anthropology and sociology—as 15 noted researchers develop a common language, theoretical basis, and methodology for examining this most sociocognitive aspect of our lives. Starting with our evolutionary past and continuing into our modern world of social classes and norms, these multidisciplinary perspectives reveal the complex interplay of biological, social, cultural, and personal factors at work in emotions, with particular emphasis on the nuances involved in pride and shame. A sampling of the topics: The roles of the brain in emotional processing. Emotional development milestones in childhood. Social feeling rules and the experience of loss. Emotions as commodities? The management of feelings and the self-help industry. Honor and dishonor: societal and gender manifestations of pride and shame. Emotion regulation and youth culture. Pride and shame in the classroom. A volume of such wide and integrative scope as Emotions as Bio-cultural Processes should attract a large cohort of readers on both sides of the debate, among them emotion researchers, social and developmental psychologists, sociologists, social anthropologists, and others who analyze the links between humans that on the one hand differentiate us as individuals but on the other hand tie us to our socio-cultural worlds.

Emotions. --- Emotions --- Sociology --- Behavior and Behavior Mechanisms --- Anthropology, Cultural --- Social Sciences --- Psychiatry and Psychology --- Anthropology --- Anthropology, Education, Sociology and Social Phenomena --- Culture --- Psychology --- Feelings --- Human emotions --- Passions --- Associative algebras --- Integral representations --- Linear algebraic groups --- Algebraic groups, Linear --- Representations, Integral --- Algebras, Associative --- Associative algebras. --- Psychology. --- Clinical psychology. --- Developmental psychology. --- Personality. --- Social psychology. --- Developmental Psychology. --- Personality and Social Psychology. --- Clinical Psychology. --- Mathematical analysis --- Affect (Psychology) --- Affective neuroscience --- Apathy --- Pathognomy --- Finite groups. --- Integral representations. --- Linear algebraic groups. --- Finite groups --- Geometry, Algebraic --- Group theory --- Algebraic varieties --- Algebraic number theory --- Crystallography, Mathematical --- Representations of groups --- Groups, Finite --- Modules (Algebra) --- Algebra --- Consciousness. --- Psychology, clinical. --- Apperception --- Mind and body --- Perception --- Philosophy --- Spirit --- Self --- Development (Psychology) --- Developmental psychobiology --- Life cycle, Human --- Personal identity --- Personality psychology --- Personality theory --- Personality traits --- Personology --- Traits, Personality --- Individuality --- Persons --- Temperament --- Psychiatry --- Psychology, Applied --- Psychological tests --- Mass psychology --- Psychology, Social --- Human ecology --- Social groups --- Groupes algébriques linéaires