Kolmogorovs Theorem

Kolmogorov s Theorem Any real-valued continuous function f defined on an N-dimensional cube can be represented as 

While Kolmogorov s original theorem has subsequently been improved upon a number of times, the most interesting formulation for our purposes is due to Hecht-Nielson, who rephrased the theorem in the language of neural nets [hecht87]. 

Kolmogorov s Theorem (Reformulated by Hecht-Nielson) Any real-valued continuous function f defined on an N-dimensional cube can be implemented by a three layered neural network consisting of 2N -)-1 neurons in the hidden layer with transfer functions from the input to the hidden layer and (f from all of 

Kolmogorov s theorem thus effectively states that a three-layer net with N 2N -)-1) neurons using continuously increasing nonlinear transfer functions can compute any continuous function of N variables. Unfortunately, the theorem tells us nothing about how to select the required transfer functions or set the weights in our net. 

A related theorem is due to Cybenko [cyb88], who showed that, provided that there are enough neurons in the layer, one hidden layer is enough to compute any continuous function to an arbitrary degree of accuracy. 

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Giorgilli, A. and Locatelli, U. (1997a). Kolmogorov theorem and classical perturbation theory. ZAMP, 48 220-261. 

The Kolmogorov consistency theorem [gnto88] asserts that any set of self- and mutually- consistent probability functions Pj, j = 1,2,..., jV may be extended to a unique shift-invariant measure on F,... 

Below, we will define a canonical procedure for constructing probabilities of blocks of arbitrary lengths consistent with a given block probability function P. The Kolmogorov consistency theorem will then allow us to use this set of finite block probabilities to define a measure on the set of infinite configurations, F. 

The Local Structure Operator By the Kolmogorov consistency theorem, we can use the Bayesian extension of Pn to define a measure on F. This measure -called the finite-block measure, /i f, where N denotes the order of the block probability function from which it is derived by Bayesian extension - is defined by assigning t.o each cylinder c Bj) = 5 G F cti = 6i, 0 2 = 62, , ( j — bj a value equal to the probability of its associated block ... 

A detailed discussion of the implications of Kolmogorov s theorem to multilayered nets is given by Kurkova [kurk92]. 

Kolmogorov, A. N., 114,139,159 Konigs thorem applied to Bernoulli method, 81 Koopman, B., 307 Roster, G.F., 727,768 Kraft theorem, 201 Kronig-Penney problem, 726 antiferromagnetic, 747 Krylov-Bogoliubov method, 359 Krylov method, 73 Krylov, N., 322 Kuhn, W. H., 289,292,304 Kuratowski s theorem, 257... 

The phase space is partitioned into cells co of diameter 5. In the limit of an arbitrarily fine partition, the entropy per unit time tends to the Kolmogorov-Sinai entropy per unit time which is equal to the sum of positive Lyapunov exponents by Pesin theorem [16] ... 

According to dynamical systems theory, the escape rate is given by the difference (92) between the sum of positive Lyapunov exponents and the Kolmogorov-Sinai entropy. Since the dynamics is Hamiltonian and satisfies Liouville s theorem, the sum of positive Lyapunov exponents is equal to minus the sum of negative ones ... 

The above situation is the same as for the celebrated theorem of Kolmogorov-Arnold-Moser (KAM)—that is, the problem of small denominators. The convergence can be proved for sufficiently nonresonant combinations of the vibrational frequencies [31]. In other words, when tori of the vibrational motions on the NHIM Mq are sufficiently nonresonant, they survive under small perturbations. 

We have obtained several interesting results from the theorem If the period of the external transformation is much longer than the relaxation time, then thermodynamic entropy production is proportional to the ratio of the period and relaxation time. The relaxation time is proportional to the inverse of the Kolmogorov-Sinai entropy for small strongly chaotic systems. Thermodynamic entropy production is proportional to the inverse of the dynamical entropy [11]. On the other hand, thermodynamic entropy production is proportional to the dynamical entropy when the period of the external transformation is much shorter than the relaxation time. Furthermore, we found fractional scaling of the excess heat for long-period external transformations, when the system has longtime correlation such as 1 /fa noise. Since excess heat is measured as the area of a hysteresis loop [12], these properties can be confirmed in experiments. 

In addition to computer simulations, what drives the research in this direction is elaborated perturbation theories developed almost simultaneously. In particular, the Kolmogorov-Arnold-Moser (KAM) theorem, which has shown the existence of invariant tori under a small perturbation to completely inte-grable systems, and the Nekhoroshev theorem, which has proved exponentially long-time stability of trajectories close to completely integrable ones, are landmarks in this field. Although a lot of works have been done, there still remain unsolved important questions, and the Hamiltonian system is being studied as one of important branches in the theory of dynamical systems [3-5]. 

Abstract These lectures are devoted to the main results of classical perturbation theory. We start by recalling the methods of Hamiltonian dynamics, the problem of small divisors, the series of Lindstedt and the method of normal form. Then we discuss the theorem of Kolmogorov with an application to the Sun-Jupiter-Saturn problem in Celestial Mechanics. Finally we discuss the problem of long-time stability, by discussing the concept of exponential stability as introduced by Moser and Littlewood and fully exploited by Nekhoroshev. The phenomenon of superexponential stability is also recalled. 

This section is devoted to the statement of the theorem of Kolmogorov on the persistence of Invariant tori. We first state the theorem and recall in some detail the formal method invented by Kolmogorov. Then we show how the original scheme can be rearranged in the form of a constructive algorithm, suitable for an explicit calculation via algebraic manipulations. Finally we shall apply this method to the problem of three bodies. 

A first application of the normal form method is the proof of the theorem of Kolmogorov on the persistence of invariant tori. We outline here the basic scheme proposed in Kolmogorov s original paper. 

Applying the theorem of Kolmogorov to a real system is not an easy matter. Due to the very strong requests on the smallness of the parameter e (that we have not reported for simplicity in the statement of theorem 4), even using the best available analytical estimates it is typical to end up with ridiculous results. Thus, we resort to the computer-assisted methods of proof. To this end, we consider the so called secular dynamics for the motion of Jupiter and Saturn this will be our model. We proceed in four steps, with the aid of an algebraic manipulator. 

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