Topic
Absolute continuity
About: Absolute continuity is a research topic. Over the lifetime, 2622 publications have been published within this topic receiving 64577 citations.
Papers published on a yearly basis
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Papers
Book•
1 Jan 1964
TL;DR: The real and complex number system as discussed by the authors is a real number system where the real number is defined by a real function and the complex number is represented by a complex field of functions.
Abstract: Chapter 1: The Real and Complex Number Systems Introduction Ordered Sets Fields The Real Field The Extended Real Number System The Complex Field Euclidean Spaces Appendix Exercises Chapter 2: Basic Topology Finite, Countable, and Uncountable Sets Metric Spaces Compact Sets Perfect Sets Connected Sets Exercises Chapter 3: Numerical Sequences and Series Convergent Sequences Subsequences Cauchy Sequences Upper and Lower Limits Some Special Sequences Series Series of Nonnegative Terms The Number e The Root and Ratio Tests Power Series Summation by Parts Absolute Convergence Addition and Multiplication of Series Rearrangements Exercises Chapter 4: Continuity Limits of Functions Continuous Functions Continuity and Compactness Continuity and Connectedness Discontinuities Monotonic Functions Infinite Limits and Limits at Infinity Exercises Chapter 5: Differentiation The Derivative of a Real Function Mean Value Theorems The Continuity of Derivatives L'Hospital's Rule Derivatives of Higher-Order Taylor's Theorem Differentiation of Vector-valued Functions Exercises Chapter 6: The Riemann-Stieltjes Integral Definition and Existence of the Integral Properties of the Integral Integration and Differentiation Integration of Vector-valued Functions Rectifiable Curves Exercises Chapter 7: Sequences and Series of Functions Discussion of Main Problem Uniform Convergence Uniform Convergence and Continuity Uniform Convergence and Integration Uniform Convergence and Differentiation Equicontinuous Families of Functions The Stone-Weierstrass Theorem Exercises Chapter 8: Some Special Functions Power Series The Exponential and Logarithmic Functions The Trigonometric Functions The Algebraic Completeness of the Complex Field Fourier Series The Gamma Function Exercises Chapter 9: Functions of Several Variables Linear Transformations Differentiation The Contraction Principle The Inverse Function Theorem The Implicit Function Theorem The Rank Theorem Determinants Derivatives of Higher Order Differentiation of Integrals Exercises Chapter 10: Integration of Differential Forms Integration Primitive Mappings Partitions of Unity Change of Variables Differential Forms Simplexes and Chains Stokes' Theorem Closed Forms and Exact Forms Vector Analysis Exercises Chapter 11: The Lebesgue Theory Set Functions Construction of the Lebesgue Measure Measure Spaces Measurable Functions Simple Functions Integration Comparison with the Riemann Integral Integration of Complex Functions Functions of Class L2 Exercises Bibliography List of Special Symbols Index
7,509 citations
Book•
1 Jan 1987
TL;DR: In this article, the General Theory of Stochastic Processes, Semimartingales, and Stochastically Integrals is discussed and the convergence of Processes with Independent Increments is discussed.
Abstract: I. The General Theory of Stochastic Processes, Semimartingales and Stochastic Integrals.- II. Characteristics of Semimartingales and Processes with Independent Increments.- III. Martingale Problems and Changes of Measures.- IV. Hellinger Processes, Absolute Continuity and Singularity of Measures.- V. Contiguity, Entire Separation, Convergence in Variation.- VI. Skorokhod Topology and Convergence of Processes.- VII. Convergence of Processes with Independent Increments.- VIII. Convergence to a Process with Independent Increments.- IX. Convergence to a Semimartingale.- X. Limit Theorems, Density Processes and Contiguity.- Bibliographical Comments.- References.- Index of Symbols.- Index of Terminology.- Index of Topics.- Index of Conditions for Limit Theorems.
6,483 citations
TL;DR: A novel class of information-theoretic divergence measures based on the Shannon entropy is introduced, which do not require the condition of absolute continuity to be satisfied by the probability distributions involved and are established in terms of bounds.
Abstract: A novel class of information-theoretic divergence measures based on the Shannon entropy is introduced. Unlike the well-known Kullback divergences, the new measures do not require the condition of absolute continuity to be satisfied by the probability distributions involved. More importantly, their close relationship with the variational distance and the probability of misclassification error are established in terms of bounds. These bounds are crucial in many applications of divergence measures. The measures are also well characterized by the properties of nonnegativity, finiteness, semiboundedness, and boundedness. >
5,055 citations
TL;DR: In this paper, the authors introduce sample path properties such as boundedness, continuity, and oscillations, as well as integrability, and absolute continuity of the path in the real line.
Abstract: Stable random variables on the real line Multivariate stable distributions Stable stochastic integrals Dependence structures of multivariate stable distributions Non-linear regression Complex stable stochastic integrals and harmonizable processes Self-similar processes Chentsov random fields Introduction to sample path properties Boundedness, continuity and oscillations Measurability, integrability and absolute continuity Boundedness and continuity via metric entropy Integral representation Historical notes and extensions.
4,604 citations
Book•
1 Jul 1984
TL;DR: In this article, a set of trajectories of Convex-Valued Differential Inclusions with Maximal Monotone Maps are described. But the complexity of the set of Trajectories of a differential inclusion is not discussed.
Abstract: 0 Background Notes- 1 Continuous Partitions of Unity- 2 Absolutely Continuous Functions- 3 Some Compactness Theorems- 4 Weak Convergence and Asymptotic Center of Bounded Sequences- 5 Closed Convex Hulls and the Mean-Value Theorem- 6 Lower Semicontinuous Convex Functions and Projections of Best Approximation- 7 A Concise Introduction to Convex Analysis- 1 Set-Valued Maps- 1 Set-Valued Maps and Continuity Concepts- 2 Examples of Set-Valued Maps- 3 Continuity Properties of Maps with Closed Convex Graph- 4 Upper Hemicontinuous Maps and the Convergence Theorem- 5 Hausdorff Topology- 6 The Selection Problem- 7 The Minimal Selection- 8 Chebishev Selection- 9 The Barycentric Selection- 10 Selection Theorems for Locally Selectionable Maps- 11 Michael's Selection Theorem- 12 The Approximate Selection Theorem and Kakutani's Fixed Point Theorem- 13 (7-Selectionable Maps- 14 Measurable Selections- 2 Existence of Solutions to Differential Inclusions- 1 Convex Valued Differential Inclusions- 2 Qualitative Properties of the Set of Trajectories of Convex-Valued Differential Inclusions- 3 Nonconvex-Valued Differential Inclusions- 4 Differential Inclusions with Lipschitzean Maps and the Relaxation Theorem- 5 The Fixed-Point Approach- 6 The Lower Semicontinuous Case- 3 Differential Inclusions with Maximal Monotone Maps- 1 Maximal Monotone Maps- 2 Existence and Uniqueness of Solutions to Differential Inclusions with Maximal Monotone Maps- 3 Asymptotic Behavior of Trajectories and the Ergodic Theorem- 4 Gradient Inclusions- 5 Application: Gradient Methods for Constrained Minimization Problems- 4 Viability Theory: The Nonconvex Case- 1 Bouligand's Contingent Cone- 2 Viable and Monotone Trajectories- 3 Contingent Derivative of a Set-Valued Map- 4 The Time Dependent Case- 5 A Continuous Version of Newton's Method- 6 A Viability Theorem for Continuous Maps with Nonconvex Images- 7 Differential Inclusions with Memory- 5 Viability Theory and Regulation of Controled Systems: The Convex Case- 1 Tangent Cones and Normal Cones to Convex Sets- 2 Viability Implies the Existence of an Equilibrium- 3 Viability Implies the Existence of Periodic Trajectories- 4 Regulation of Controled Systems Through Viability- 5 Walras Equilibria and Dynamical Price Decentralization- 6 Differential Variational Inequalities- 7 Rate Equations and Inclusions- 6 Liapunov Functions- 1 Upper Contingent Derivative of a Real-Valued Function- 2 Liapunov Functions and Existence of Equilibria- 3 Monotone Trajectories of a Differential Inclusion- 4 Construction of Liapunov Functions- 5 Stability and Asymptotic Behavior of Trajectories- Comments
1,388 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2025 | 22 |
| 2024 | 26 |
| 2023 | 48 |
| 2022 | 204 |
| 2021 | 114 |
| 2020 | 93 |