scispace - formally typeset
Search or ask a question
Topic

Bounded function

About: Bounded function is a research topic. Over the lifetime, 77295 publications have been published within this topic receiving 1321552 citations.


Papers
More filters
Journal ArticleDOI
TL;DR: In this paper, it was shown that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states, and systems with bounded solutions are shown to possess bounded numerical solutions.
Abstract: Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into consider­ably different states. Systems with bounded solutions are shown to possess bounded numerical solutions.

16,554 citations

Journal Article
TL;DR: In this article, the authors consider the problem of finding the components of the velocity at every point of a point with rectangular cartesian coordinates x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 8.
Abstract: §1. We shall denote by uα ( P ) = uα ( x 1, x 2, x 3, t ), α = 1, 2, 3, the components of velocity at the moment t at the point with rectangular cartesian coordinates x 1, x 2, x 3. In considering the turbulence it is natural to assume the components of the velocity uα ( P ) at every point P = ( x 1, x 2, x 3, t ) of the considered domain G of the four-dimensional space ( x 1, x 2, x 3, t ) are random variables in the sense of the theory of probabilities (cf. for this approach to the problem Millionshtchikov (1939) Denoting by Ᾱ the mathematical expectation of the random variable A we suppose that ῡ 2 α and (d uα /d xβ )2― are finite and bounded in every bounded subdomain of the domain G .

6,063 citations

Journal ArticleDOI
TL;DR: The condition of self-adjointness as discussed by the authors ensures that the eigenvalues of a Hamiltonian are real and bounded below, replacing this condition by the weaker condition of $\mathrm{PT}$ symmetry, one obtains new infinite classes of complex Hamiltonians whose spectra are also real and positive.
Abstract: The condition of self-adjointness ensures that the eigenvalues of a Hamiltonian are real and bounded below. Replacing this condition by the weaker condition of $\mathrm{PT}$ symmetry, one obtains new infinite classes of complex Hamiltonians whose spectra are also real and positive. These $\mathrm{PT}$ symmetric theories may be viewed as analytic continuations of conventional theories from real to complex phase space. This paper describes the unusual classical and quantum properties of these theories.

5,626 citations

Journal ArticleDOI
TL;DR: It is shown that standard multilayer feedforward networks with as few as a single hidden layer and arbitrary bounded and nonconstant activation function are universal approximators with respect to L p (μ) performance criteria, for arbitrary finite input environment measures μ.

5,593 citations

Book
14 Mar 1996
TL;DR: In this article, the authors define the Ball Sigma-Field and Measurability of Suprema and show that it is possible to achieve convergence almost surely and in probability.
Abstract: 1.1. Introduction.- 1.2. Outer Integrals and Measurable Majorants.- 1.3. Weak Convergence.- 1.4. Product Spaces.- 1.5. Spaces of Bounded Functions.- 1.6. Spaces of Locally Bounded Functions.- 1.7. The Ball Sigma-Field and Measurability of Suprema.- 1.8. Hilbert Spaces.- 1.9. Convergence: Almost Surely and in Probability.- 1.10. Convergence: Weak, Almost Uniform, and in Probability.- 1.11. Refinements.- 1.12. Uniformity and Metrization.- 2.1. Introduction.- 2.2. Maximal Inequalities and Covering Numbers.- 2.3. Symmetrization and Measurability.- 2.4. Glivenko-Cantelli Theorems.- 2.5. Donsker Theorems.- 2.6. Uniform Entropy Numbers.- 2.7. Bracketing Numbers.- 2.8. Uniformity in the Underlying Distribution.- 2.9. Multiplier Central Limit Theorems.- 2.10. Permanence of the Donsker Property.- 2.11. The Central Limit Theorem for Processes.- 2.12. Partial-Sum Processes.- 2.13. Other Donsker Classes.- 2.14. Tail Bounds.- 3.1. Introduction.- 3.2. M-Estimators.- 3.3. Z-Estimators.- 3.4. Rates of Convergence.- 3.5. Random Sample Size, Poissonization and Kac Processes.- 3.6. The Bootstrap.- 3.7. The Two-Sample Problem.- 3.8. Independence Empirical Processes.- 3.9. The Delta-Method.- 3.10. Contiguity.- 3.11. Convolution and Minimax Theorems.- A. Appendix.- A.1. Inequalities.- A.2. Gaussian Processes.- A.2.1. Inequalities and Gaussian Comparison.- A.2.2. Exponential Bounds.- A.2.3. Majorizing Measures.- A.2.4. Further Results.- A.3. Rademacher Processes.- A.4. Isoperimetric Inequalities for Product Measures.- A.5. Some Limit Theorems.- A.6. More Inequalities.- A.6.1. Binomial Random Variables.- A.6.2. Multinomial Random Vectors.- A.6.3. Rademacher Sums.- Notes.- References.- Author Index.- List of Symbols.

5,231 citations


Network Information
Related Topics (5)
Dynamical systems theory
37.1K papers, 1M citations
92% related
Eigenvalues and eigenvectors
51.7K papers, 1.1M citations
91% related
Upper and lower bounds
56.9K papers, 1.1M citations
91% related
Differential equation
88K papers, 2M citations
91% related
Partial differential equation
70.8K papers, 1.6M citations
90% related
Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20243
20234,319
20229,421
20214,210
20204,423
20194,004