scholarly journals A proof of Kolmogorov's theorem

2003 ◽  
Vol 10 (1-2) ◽  
pp. 367-385 ◽  
Author(s):  
John Hubbard ◽  
Yulij Ilyashenko
Keyword(s):  
Kolmogorov's Theorem

NORMAL FORMS FOR FUZZY LOGIC — AN APPLICATION OF KOLMOGOROV’S THEOREM

1996 ◽  
Vol 04 (04) ◽  
pp. 331-349 ◽  
Author(s):  
VLADIK KREINOVICH ◽  
HUNG T. NGUYEN ◽  
DAVID A. SPRECHER
Keyword(s):  
Neural Networks ◽  
Fuzzy Logic ◽  
Normal Form ◽  
Normal Forms ◽  
Approximation Property ◽  
Universal Approximation ◽  
Approximation Result ◽  
Logical Operations ◽  
Kolmogorov's Theorem

This paper addresses mathematical aspects of fuzzy logic. The main results obtained in this paper are: 1. the introduction of a concept of normal form in fuzzy logic using hedges; 2. using Kolmogorov’s theorem, we prove that all logical operations in fuzzy logic have normal forms; 3. for min-max operators, we obtain an approximation result similar to the universal approximation property of neural networks.

Kolmogorov's Theorem Is Relevant

1991 ◽  
Vol 3 (4) ◽  
pp. 617-622 ◽  
Author(s):  
Věra Kůrková
Keyword(s):  
Neural Networks ◽  
Continuous Functions ◽  
Sigmoidal Function ◽  
Linear Combinations ◽  
Affine Functions ◽  
The One ◽  
Upper Estimate ◽  
Kolmogorov's Theorem

We show that Kolmogorov's theorem on representations of continuous functions of n-variables by sums and superpositions of continuous functions of one variable is relevant in the context of neural networks. We give a version of this theorem with all of the one-variable functions approximated arbitrarily well by linear combinations of compositions of affine functions with some given sigmoidal function. We derive an upper estimate of the number of hidden units.

A proof of Kolmogorov’s theorem on invariant tori using canonical transformations defined by the Lie method

1984 ◽  
Vol 79 (2) ◽  
pp. 201-223 ◽  
Author(s):  
G. Benettin ◽  
L. Galgani ◽  
A. Giorgilli ◽  
J. -M. Strelcyn
Keyword(s):  
Invariant Tori ◽  
Canonical Transformations ◽  
Kolmogorov's Theorem

scholarly journals Analytical properties of sample paths of some stochastic processes

2020 ◽  
pp. 11-15
Author(s):  
I. V. Rozora
Keyword(s):  
Covariance Function ◽  
Sufficient Conditions ◽  
Random Processes ◽  
Continuity Module ◽  
Classic Result ◽  
Sample Paths ◽  
Analytical Properties ◽  
Relevant Topic ◽  
Local Properties ◽  
Kolmogorov's Theorem

The study of the analytical properties of random processes and their functionals, without a doubt, was and remains the relevant topic of the theory of random processes. The first result from which the study of the local properties of random processes began is Kolmogorov’s theorem on sample continuity with probability one. The classic result for Gaussian random processes is Dudley’s theorem. This paper is devoted to the study of local properties of sample paths of random processes that can be represented as a sum of squares of Gaussian random processes. Such processes are called square-Gaussian. We investigate the sufficient conditions of sample continuity with probability 1 for square-Gaussian processes based on the convergence of entropy Dudley type integrals. The estimation of the distribution of the continuity module is studied for square-Gaussian random processes. It is considered in detail an example with an estimator (correlogram) of the covariance function of a Gaussian stationary random process. The conditions on continuity of correlogram’s trajectories with probability one are found and the distribution of the continuity module is also estimated.

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