Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

In recent years, technological progress created a great need for complex mathematical models. Many practical problems can be formulated using optimization theory and they hope to obtain an optimal solution. In most cases, such optimal solution can not be found. So, non-convex optimization problems (arising, e.g., in variational calculus, optimal control, nonlinear evolutions equations) may not possess a classical minimizer because the minimizing sequences have typically rapid oscillations. This behavior requires a relaxation of notion of solution for such problems; often we can obtain such a relaxation by means of Young measures. This monograph is a self-contained book which gathers all theoretical aspects related to the defining of Young measures (measurability, disintegration, stable convergence, compactness), a book which is also a useful tool for those interested in theoretical foundations of the measure theory. It provides a complete set of classical and recent compactness results in measure and function spaces. The book is organized in three chapters: The first chapter covers background material on measure theory in abstract frame. In the second chapter the measure theory on topological spaces is presented. Compactness results from the first two chapters are used to study Young measures in the third chapter. All results are accompanied by full demonstrations and for many of these results different proofs are given. All statements are fully justified and proved.

Spaces of measures. --- Measure theory. --- Mathematical optimization. --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Lebesgue measure --- Measurable sets --- Measure of a set --- Algebraic topology --- Integrals, Generalized --- Measure algebras --- Rings (Algebra) --- Measures, Spaces of --- Function spaces --- Measure theory --- Topological spaces --- Bounded Measure. --- Functional Analysis. --- Measure Space. --- Topological Space. --- Weak Compactness. --- Young Measure.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Higher category theory is generally regarded as technical and forbidding, but part of it is considerably more tractable: the theory of infinity-categories, higher categories in which all higher morphisms are assumed to be invertible. In Higher Topos Theory, Jacob Lurie presents the foundations of this theory, using the language of weak Kan complexes introduced by Boardman and Vogt, and shows how existing theorems in algebraic topology can be reformulated and generalized in the theory's new language. The result is a powerful theory with applications in many areas of mathematics. The book's first five chapters give an exposition of the theory of infinity-categories that emphasizes their role as a generalization of ordinary categories. Many of the fundamental ideas from classical category theory are generalized to the infinity-categorical setting, such as limits and colimits, adjoint functors, ind-objects and pro-objects, locally accessible and presentable categories, Grothendieck fibrations, presheaves, and Yoneda's lemma. A sixth chapter presents an infinity-categorical version of the theory of Grothendieck topoi, introducing the notion of an infinity-topos, an infinity-category that resembles the infinity-category of topological spaces in the sense that it satisfies certain axioms that codify some of the basic principles of algebraic topology. A seventh and final chapter presents applications that illustrate connections between the theory of higher topoi and ideas from classical topology.

Algebraic geometry --- Topology --- Toposes --- Categories (Mathematics) --- Categories (Mathematics). --- Toposes. --- Algebra --- Mathematics --- Physical Sciences & Mathematics --- Category theory (Mathematics) --- Topoi (Mathematics) --- Algebra, Homological --- Algebra, Universal --- Group theory --- Logic, Symbolic and mathematical --- Functor theory --- Adjoint functors. --- Associative property. --- Base change map. --- Base change. --- CW complex. --- Canonical map. --- Cartesian product. --- Category of sets. --- Category theory. --- Coequalizer. --- Cofinality. --- Coherence theorem. --- Cohomology. --- Cokernel. --- Commutative property. --- Continuous function (set theory). --- Contractible space. --- Coproduct. --- Corollary. --- Derived category. --- Diagonal functor. --- Diagram (category theory). --- Dimension theory (algebra). --- Dimension theory. --- Dimension. --- Enriched category. --- Epimorphism. --- Equivalence class. --- Equivalence relation. --- Existence theorem. --- Existential quantification. --- Factorization system. --- Functor category. --- Functor. --- Fundamental group. --- Grothendieck topology. --- Grothendieck universe. --- Group homomorphism. --- Groupoid. --- Heyting algebra. --- Higher Topos Theory. --- Higher category theory. --- Homotopy category. --- Homotopy colimit. --- Homotopy group. --- Homotopy. --- I0. --- Inclusion map. --- Inductive dimension. --- Initial and terminal objects. --- Inverse limit. --- Isomorphism class. --- Kan extension. --- Limit (category theory). --- Localization of a category. --- Maximal element. --- Metric space. --- Model category. --- Monoidal category. --- Monoidal functor. --- Monomorphism. --- Monotonic function. --- Morphism. --- Natural transformation. --- Nisnevich topology. --- Noetherian topological space. --- Noetherian. --- O-minimal theory. --- Open set. --- Power series. --- Presheaf (category theory). --- Prime number. --- Pullback (category theory). --- Pushout (category theory). --- Quillen adjunction. --- Quotient by an equivalence relation. --- Regular cardinal. --- Retract. --- Right inverse. --- Sheaf (mathematics). --- Sheaf cohomology. --- Simplicial category. --- Simplicial set. --- Special case. --- Subcategory. --- Subset. --- Surjective function. --- Tensor product. --- Theorem. --- Topological space. --- Topology. --- Topos. --- Total order. --- Transitive relation. --- Universal property. --- Upper and lower bounds. --- Weak equivalence (homotopy theory). --- Yoneda lemma. --- Zariski topology. --- Zorn's lemma.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

The description for this book, Surface Area. (AM-35), Volume 35, will be forthcoming.

Surfaces. --- Absolute continuity. --- Addition. --- Admissible set. --- Arc length. --- Axiom. --- Axiomatic system. --- Bearing (navigation). --- Bounded variation. --- Calculus of variations. --- Circumference. --- Compact space. --- Complex analysis. --- Concentric. --- Connected space. --- Continuous function (set theory). --- Continuous function. --- Corollary. --- Countable set. --- Covering set. --- Curve. --- Derivative. --- Diameter. --- Differentiable function. --- Differential geometry. --- Direct proof. --- Dirichlet integral. --- Disjoint sets. --- Empty set. --- Equation. --- Equicontinuity. --- Existence theorem. --- Existential quantification. --- Function (mathematics). --- Functional analysis. --- Geometry. --- Hausdorff measure. --- Homeomorphism. --- Homotopy. --- Infimum and supremum. --- Integral geometry. --- Intersection number (graph theory). --- Interval (mathematics). --- Iterative method. --- Jacobian. --- Lebesgue integration. --- Lebesgue measure. --- Limit (mathematics). --- Limit point. --- Limit superior and limit inferior. --- Linearity. --- Line–line intersection. --- Locally compact space. --- Mathematician. --- Mathematics. --- Measure (mathematics). --- Metric space. --- Morphism. --- Natural number. --- Nonparametric statistics. --- Orientability. --- Parameter. --- Parametric equation. --- Parametric surface. --- Partial derivative. --- Potential theory. --- Radon–Nikodym theorem. --- Representation theorem. --- Representation theory. --- Right angle. --- Semi-continuity. --- Set function. --- Set theory. --- Sign (mathematics). --- Smoothness. --- Space-filling curve. --- Subset. --- Summation. --- Surface area. --- Tangent space. --- Theorem. --- Topological space. --- Topology. --- Total order. --- Total variation. --- Uniform convergence. --- Unit square.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This book gives a comprehensive and self-contained introduction to the theory of symmetric Markov processes and symmetric quasi-regular Dirichlet forms. In a detailed and accessible manner, Zhen-Qing Chen and Masatoshi Fukushima cover the essential elements and applications of the theory of symmetric Markov processes, including recurrence/transience criteria, probabilistic potential theory, additive functional theory, and time change theory. The authors develop the theory in a general framework of symmetric quasi-regular Dirichlet forms in a unified manner with that of regular Dirichlet forms, emphasizing the role of extended Dirichlet spaces and the rich interplay between the probabilistic and analytic aspects of the theory. Chen and Fukushima then address the latest advances in the theory, presented here for the first time in any book. Topics include the characterization of time-changed Markov processes in terms of Douglas integrals and a systematic account of reflected Dirichlet spaces, and the important roles such advances play in the boundary theory of symmetric Markov processes. This volume is an ideal resource for researchers and practitioners, and can also serve as a textbook for advanced graduate students. It includes examples, appendixes, and exercises with solutions.

Markov processes. --- Boundary value problems. --- Dirichlet problem. --- Beurling-Deny decomposition. --- Beurling-Deny formula. --- Brownian motions. --- Dirichlet forms. --- Dirichlet spaces. --- Douglas integrals. --- Feller measures. --- Hausdorff topological space. --- Markovian symmetric operators. --- Silverstein extension. --- additive functional theory. --- additive functionals. --- analytic concepts. --- analytic potential theory. --- boundary theory. --- countable boundary. --- decompositions. --- energy functional. --- extended Dirichlet spaces. --- fine properties. --- harmonic functions. --- harmonicity. --- hitting distributions. --- irreducibility. --- lateral condition. --- local properties. --- m-tight special Borel. --- many-point extensions. --- one-point extensions. --- part processes. --- path behavior. --- perturbed Dirichlet forms. --- positive continuous additive functionals. --- probabilistic derivation. --- probabilistic potential theory. --- quasi properties. --- quasi-homeomorphism. --- quasi-regular Dirichlet forms. --- recurrence. --- reflected Dirichlet spaces. --- reflecting Brownian motions. --- reflecting extensions. --- regular Dirichlet forms. --- regular recurrent Dirichlet forms. --- smooth measures. --- symmetric Hunt processes. --- symmetric Markov processes. --- symmetric Markovian semigroups. --- terminal random variables. --- time change theory. --- time changes. --- time-changed process. --- transience. --- transient regular Dirichlet forms.

Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

Intended for researchers in Riemann surfaces, this volume summarizes a significant portion of the work done in the field during the years 1966 to 1971.

Riemann surfaces --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Surfaces, Riemann --- Functions --- Congresses --- Differential geometry. Global analysis --- RIEMANN SURFACES --- congresses --- Congresses. --- MATHEMATICS / Calculus. --- Affine space. --- Algebraic function field. --- Algebraic structure. --- Analytic continuation. --- Analytic function. --- Analytic set. --- Automorphic form. --- Automorphic function. --- Automorphism. --- Beltrami equation. --- Bernhard Riemann. --- Boundary (topology). --- Canonical basis. --- Cartesian product. --- Clifford's theorem. --- Cohomology. --- Commutative diagram. --- Commutative property. --- Complex multiplication. --- Conformal geometry. --- Conformal map. --- Coset. --- Degeneracy (mathematics). --- Diagram (category theory). --- Differential geometry of surfaces. --- Dimension (vector space). --- Dirichlet boundary condition. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Eisenstein series. --- Euclidean space. --- Existential quantification. --- Explicit formulae (L-function). --- Exterior (topology). --- Finsler manifold. --- Fourier series. --- Fuchsian group. --- Function (mathematics). --- Generating set of a group. --- Group (mathematics). --- Hilbert space. --- Holomorphic function. --- Homeomorphism. --- Homology (mathematics). --- Homotopy. --- Hyperbolic geometry. --- Hyperbolic group. --- Identity matrix. --- Infimum and supremum. --- Inner automorphism. --- Intersection (set theory). --- Intersection number (graph theory). --- Isometry. --- Isomorphism class. --- Isomorphism theorem. --- Kleinian group. --- Limit point. --- Limit set. --- Linear map. --- Lorentz group. --- Mapping class group. --- Mathematical induction. --- Mathematics. --- Matrix (mathematics). --- Matrix multiplication. --- Measure (mathematics). --- Meromorphic function. --- Metric space. --- Modular group. --- Möbius transformation. --- Number theory. --- Osgood curve. --- Parity (mathematics). --- Partial isometry. --- Poisson summation formula. --- Pole (complex analysis). --- Projective space. --- Quadratic differential. --- Quadratic form. --- Quasiconformal mapping. --- Quotient space (linear algebra). --- Quotient space (topology). --- Riemann mapping theorem. --- Riemann sphere. --- Riemann surface. --- Riemann zeta function. --- Scalar multiplication. --- Scientific notation. --- Selberg trace formula. --- Series expansion. --- Sign (mathematics). --- Square-integrable function. --- Subgroup. --- Teichmüller space. --- Theorem. --- Topological manifold. --- Topological space. --- Uniformization. --- Unit disk. --- Variable (mathematics). --- Riemann, Surfaces de --- RIEMANN SURFACES - congresses --- Fonctions d'une variable complexe --- Surfaces de riemann