Kolmogorov's Theorem Research Articles

Riesz and Kolmogorov inequality for harmonic quasiregular mappings

Let K⩾1 and p∈(1,2]. We obtain an asymptotically sharp constant c(K,p) as K→1 in the inequality‖ℑf‖p≤c(K,p)‖ℜf‖p, where f∈hp is a K-quasiregular harmonic mapping in the unit disk belonging to the Hardy space hp. This result holds under the conditions arg⁡(f(0))∈(−π2p,π2p) and f(D)∩(−∞,0)=∅. Our findings improve a recent result by Liu and Zhu [12]. Additionally, we extend this result to K-quasiregular harmonic mappings in the unit ball in Rn. Finally, we consider the Kolmogorov theorem for quasiregular harmonic mappings in the plane.

On application of Gaussian kernels and Laplace functions combined with Kolmogorov's theorem for approximation of functions of several variables

A special class of approximations of measurable functions of several variables on the unit coordinate cube is investigated. The class is constructed on the base of Kolmogorov's theorem (in version by Sprecher–Golubkov) stating that a continuous function $f$ of several variables can be represented as a finite superposition of continuous single-variable functions — so called outer functions (which depend of $f$) and inner one $\Psi$ (which is independent of $f$ and is monotone). In the case of continuous functions $f$ the class under study is obtained with outer functions approximated by linear combinations of quadratic exponentials (also known as Gaussian functions or Gaussian kernels) and with the inner function $\Psi$ approximated by Laplace functions. As is known, a measurable function $f$ can be approximated by a continuous one (up to a set of small measure) with the help of classical Luzin's theorem. The effectiveness of such approach is based on assertions that, firstly, the Mexican hat mother wavelet on any fixed bounded interval can be approximated as accurately as desired by a linear combination of two Gaussian functions, and, secondly, that a continuous monotone function on such an interval can be approximated as accurately as desired by a linear combination of translations and dilations of the Laplace integral (in other words, Laplace functions). It is proved that the class of approximations under study is dense everywhere in the class of continuous multivariable functions on the coordinate cube. For the case of continuous and piecewise continuous functions of two variables, numerical results are presented that confirm the effectiveness of approximations of the studied class.

On the Kolmogorov constant explicit form in the theory of discrete-time stochastic branching systems

AbstractWe consider a discrete-time population growth system called the Bienaymé–Galton–Watson stochastic branching system. We deal with a noncritical case, in which the per capita offspring mean $m\neq1$ . The famous Kolmogorov theorem asserts that the expectation of the population size in the subcritical case $m<1$ on positive trajectories of the system asymptotically stabilizes and approaches ${1}/\mathcal{K}$ , where $\mathcal{K}$ is called the Kolmogorov constant. The paper is devoted to the search for an explicit expression of this constant depending on the structural parameters of the system. Our argumentation is essentially based on the basic lemma describing the asymptotic expansion of the probability-generating function of the number of individuals. We state this lemma for the noncritical case. Subsequently, we find an extended analogue of the Kolmogorov constant in the noncritical case. An important role in our discussion is also played by the asymptotic properties of transition probabilities of the Q-process and their convergence to invariant measures. Obtaining the explicit form of the extended Kolmogorov constant, we refine several limit theorems of the theory of noncritical branching systems, showing explicit leading terms in the asymptotic expansions.

Fine asymptotic expansion of the ODE's flow

We study the dynamics of the flow X solution to the ODE: X′(t,x)=b(X(t,x)) with X(0,x)=x∈Rd, where b is a regular Zd-periodic vector field in Rd. We provide conditions on b to get the fine asymptotic expansion: |X(t,x)−x−tζ(x)|≤M<∞. To this end, we try to express X(t,x)−x−tζ(x) as Φ(X(t,x))−Φ(x), which yields the desired expansion when Φ is bounded. Then, assuming that the 2D Kolmogorov theorem and some extension for d>2 hold, we establish several regimes depending on the commensurability of the rotation vectors ζ(x) for which the expansion of X is valid. Moreover, we prove that for any 2D flow with a non vanishing smooth b inducing a unique incommensurable rotation vector ξ, the fine expansion holds in R2 if, and only if, ξ1/ξ2 is a Diophantine number. The case where ξ is commensurable is also investigated. Finally, several examples illustrate the different results, including the case of a vanishing b which blows up the asymptotic expansion in some direction. In particular, the case of some Euler flows is investigated.

A Long-term Prediction of Covid-19 by Building a Transition Probability Matrix

The outbreak of the covid-19 pandemic caused a devastating scenario in the human lives for a long period of time. Since the outbreak of the pandemic, many researchers have published papers on the prediction of covid-19 using machine learning techniques, artificial intelligence, statistical techniques such as regression analysis, Markov chain models etc. In this paper a transition probability matrix is introduced by building a first order Markov chain. This first order Markov chain is built by generating sequences from the data. Later with the help of Chapman Kolmogorov theorem and ergodic theorem, higher order transition probabilities and limiting probabilities are obtained.

Approximation by linear combinations of translates in invariant Banach spaces of tempered distributions via Tauberian conditions

This paper describes an approximation theoretic approach to the problem of completeness of a set of translates of a “Tauberian generator”, which is an integrable function whose Fourier transform does not vanish. This is achieved by the construction of finite rank operators, whose range is contained in the linear span of the translates of such a generator, and which allow uniform approximation of the identity operator over compact sets of certain Banach spaces (B,‖⋅‖B). The key assumption is availability of a double module structure on (B,‖⋅‖B), meaning the availability of sufficiently many smoothing operators (via convolution) and also pointwise multipliers, allowing localization of its elements. This structure is shared by a wide variety of function spaces and allows us to make explicit use of the Riesz–Kolmogorov Theorem characterizing compact subsets in such Banach spaces.The construction of these operators is universal with respect to large families of such Banach spaces, i.e. they do not depend on any further information concerning the particular Banach space. As a corollary we conclude that the linear span of the set of the translates of such a Tauberian generator is dense in any such space (B,‖⋅‖B). Our work has been inspired by a completeness result of V. Katsnelson which was formulated in the context of specific Hilbert spaces within this family and Gaussian generators.

Riesz–Kolmogorov Type Compactness Criteria in Function Spaces with Applications

We present forms of the classical Riesz–Kolmogorov theorem for compactness that are applicable in a wide variety of settings. In particular, our theorems apply to classify the precompact subsets of the Lebesgue space $$L^2$$ , Paley–Wiener spaces, weighted Bargmann–Fock spaces, and a scale of weighted Besov–Sobolev spaces of holomorphic functions that includes weighted Bergman spaces of general domains as well as the Hardy space and the Dirichlet space. We apply the compactness criteria to characterize the compact Toeplitz operators on the Bergman space, deduce the compactness of Hankel operators on the Hardy space, and obtain general umbrella theorems.

The Price of My House. Kolmogorov's Theorem

The Price of My House. Kolmogorov's Theorem

An Analogy of the Carleson–Hunt Theorem with Respect to Vilenkin Systems

In this paper we discuss and prove an analogy of the Carleson–Hunt theorem with respect to Vilenkin systems. In particular, we use the theory of martingales and give a new and shorter proof of the almost everywhere convergence of Vilenkin–Fourier series of fin L_p(G_m) for p>1 in case the Vilenkin system is bounded. Moreover, we also prove sharpness by stating an analogy of the Kolmogorov theorem for p=1 and construct a function fin L_1(G_m) such that the partial sums with respect to Vilenkin systems diverge everywhere.

Modeling of Chaotic Processes by Means of Antisymmetric Neural ODEs

The main goal of this work is to construct an algorithm for modeling chaotic processes using special neural ODEs with antisymmetric matrices (antisymmetric neural ODEs) and power activation functions (PAFs). The central part of this algorithm is to design a neural ODEs architecture that would guarantee the generation of a stable limit cycle for a known time series. Then, one neuron is added to each equation of the created system until the approximating properties of this system satisfy the well-known Kolmogorov theorem on the approximation of a continuous function of many variables. In addition, as a result of such an addition of neurons, the cascade of bifurcations that allows generating a chaotic attractor from stable limit cycles is launched. We also consider the possibility of generating a homoclinic orbit whose bifurcations lead to the appearance of a chaotic attractor of another type. In conclusion, the conditions under which the found attractor adequately simulates the chaotic process are discussed. Examples are given.

On Alternative Approximating Distributions in the Multivariate Version of Kolmogorov's Second Uniform Limit Theorem

The aim of the present work is to show that recent results of the authors on the approximation of distributions of sums of independent summands by the infinitely divisible laws on convex polyhedra can be shown via an alternative class of approximating infinitely divisible distributions. We will also generalize the results to the infinite-dimensional case.

On the Szegő—Kolmogorov prediction theorem

The classical Szegő—Kolmogorov theorem characterizes the weights ω such that the family of exponentials with positive integer frequencies spans the whole weighted space \({L^2}(\mathbb{T},w)\) on the circle. Kolmogorov’s probabilistic interpretation of this result connects it with the possibility to ‘predict precisely the future from the past’ for the stationary stochastic processes with discrete time. We discuss the problem whether the prediction remains possible if some part of the ‘past’ is not known.

Compactness Characterizations of Commutators on Ball Banach Function Spaces

Let X be a ball Banach function space on \({\mathbb R}^{n}\). Let Ω be a Lipschitz function on the unit sphere of \({\mathbb R}^{n}\), which is homogeneous of degree zero and has mean value zero, and let TΩ be the convolutional singular integral operator with kernel Ω(⋅)/|⋅|n. In this article, under the assumption that the Hardy–Littlewood maximal operator \({\mathscr{M}}\) is bounded on both X and its associated space, the authors prove that the commutator [b, TΩ] is compact on X if and only if \(b\in \text {CMO }({\mathbb R}^{n})\). To achieve this, the authors mainly employ three key tools: some elaborate estimates, given in this article, on the norm of X about the commutators and the characteristic functions of some measurable subsets, which are implied by the assumed boundedness of \({\mathcal M}\) on X and its associated space as well as the geometry of \(\mathbb R^{n}\); the complete John–Nirenberg inequality in X obtained by Y. Sawano et al.; the generalized Fréchet–Kolmogorov theorem on X also established in this article. All these results have a wide range of applications. Particularly, even when \(X:=L^{p(\cdot )}({\mathbb R}^{n})\) (the variable Lebesgue space), \(X:=L^{\vec {p}}({\mathbb R}^{n})\) (the mixed-norm Lebesgue space), \(X:=L^{\Phi }({\mathbb R}^{n})\) (the Orlicz space), and \(X:=(E_{\Phi }^{q})_{t}({\mathbb R}^{n})\) (the Orlicz-slice space or the generalized amalgam space), all these results are new.

Optimization of lapping process parameters of CP-Ti based on PSO with mutation and BPNN

This work aims to improve the surface quality of commercially pure titanium (CP-Ti) with free alumina lapping fluid and establish the relationship between the main process parameters of lapping and roughness. On this basis, the optimal process parameters were searched by performing particle swarm optimization with mutation. First, free alumina lapping fluid was used to perform an L9(33) orthogonal experiment on CP-Ti to acquire data samples to train the neural network. At the same time, a BP neural network was created to fit the nonlinear functional relation among the lapping pressure P, spindle speed n, slurry flow Q and roughness Ra. Then, the range of the neuron numbers in the hidden layer of the neural network was determined by empirical formulas and the Kolmogorov theorem. On this basis, particle swarm optimization with mutation was used to search for the optimal process parameter configurations for lapping CP-Ti. The optimal process parameter configurations were used in the neural network to calculate the prediction value. Finally, the accuracy of the prediction was verified experimentally. The optimum process parameter configurations found by particle swarm optimization were as follows: the lapping pressure was 5 kPa, spindle speed was 60 r·min−1 and slurry flow was 50 ml·min−1. Then, the configurations were applied to a neural network to simulate prediction: the roughness was 0.1127 μm. The roughness obtained by experiments was 0.1134 μm. The error was 0.62%, which indicates that the well-trained neural network can achieve a good prediction when experimental data are missing. Applying the particle swarm optimization (PSO) algorithm with mutation to a neural network will obtain the optimal process parameter configurations, which can effectively improve the surface quality of CP-Ti lapped with free abrasive.

Sufficient statistics for the Pareto distribution parameter

The task of estimating the parameters of the Pareto distribution, first of all, of an indicator of this distribution for a given sample, is relevant. This article establishes that for this estimate, it is sufficient to know the product of the sample elements. It is proved that this product is a sufficient statistic for the Pareto distribution parameter. On the basis of the maximum likelihood method the distribution degree indicator is estimated. It is proved that this estimate is biased, and a formula eliminating the bias is justified. For the product of the sample elements considered as a random variable the distribution function and probability density are found; mathematical expectation, higher moments, and differential entropy are calculated. The corresponding graphs are built. In addition, it is noted that any function of this product is a sufficient statistic, in particular, the geometric mean. For the geometric mean also considered as a random variable, the distribution function, probability density, and the mathematical expectation are found; the higher moments, and the differential entropy are also calculated, and the corresponding graphs are plotted. In addition, it is proved that the geometric mean of the sample is a more convenient sufficient statistic from a practical point of view than the product of the sample elements. Also, on the basis of the Rao–Blackwell–Kolmogorov theorem, effective estimates of the Pareto distribution parameter are constructed. In conclusion, as an example, the technique developed here is applied to the exponential distribution. In this case, both the sum and the arithmetic mean of the sample can be used as sufficient statistics.

A New Approach to Optimize the Relative Clearance for Cylindrical Joints Manufactured by FDM 3D Printing Using a Hybrid Genetic Algorithm Artificial Neural Network and Rational Function

Nowadays, FDM technology permits obtaining functional prototypes or even end parts. The process parameters, such as layer thickness, building orientation, fill density, type of support, etc., have great influence on the quality, functionality and behavior of the obtained parts during their lifetime. In this paper, we present a study concerning the possibilities of obtaining certain values for clearance in revolute joints of non-assembly mechanisms manufactured by FDM 3D Printing. To ensure the functioning of the assembly, one must know the relationship between the imposed and measured clearances by taking into account the significant input data. One way is to use the automat learning method with an artificial neuronal network (ANN). The data necessary for the training, testing, and validation of ANN were experimentally obtained, using a complete L 27 Taguchi experimental plan. A total of 27 samples were printed with different values of the following parameters: the infill density, the imposed clearance between the shaft and the hole, and the layer thickness. ANN architecture corresponds to the Hecht–Kolmogorov theorem. Genetic algorithms (GA) were used for the optimization of the output. The Neural Network Toolbox from MATLAB was used for training the network and a hybrid tool genetic algorithm artificial neural network (GA-ANN) was used to minimize the value of the absolute relative clearance (arc). The minimum value of the absolute relative clearance established by GA-ANN was 0.0385788. This value was validated experimentally, with a relative difference of 4%. We also introduced a rational function to approximate the correlation between the input and output parameters. This function fulfills some frontier conditions resulted from practice. In addition, the function may be used to establish the designed clearance in order to obtain an imposed one.

Weighted Fréchet–Kolmogorov Theorem and Compactness of Vector-Valued Multilinear Operators

In this paper, we establish a weighted version of the well-known Fréchet–Kolmogorov theorem, which holds for weights beyond \(A_\infty \). This weighted theory extends the previous known unsatisfactory results in the terms of relaxing the index to the natural range. As applications, we obtain the weighted compactness theory for the commutators of multilinear vector-valued Calderón–Zygmund type operators, including the commutators of multilinear Littlewood–Paley type operators. It is worthy to pointing out that the commutators we considered contain almost all the commutators formerly studied in this literature.

Finitely additive mixtures of probability measures

Let D be a linear space of real bounded functions and P:D→R a coherent functional. Also, let Q be a collection of coherent functionals on D. Under mild conditions, there is a finitely additive probability Π on the power set of Q such that P(f)=∫QQ(f)Π(dQ) for each f∈D. This fact has various consequences and such consequences are investigated in this paper. Three types of results are provided: (i) Existence of common extensions satisfying certain properties, (ii) Finitely additive mixtures of extreme points, (iii) Countably additive mixtures. Among other things, we obtain new versions of Kolmogorov's consistency theorem and de Finetti's representation theorem.

Products and projective limits of continuous valuations on T0 spaces

AbstractWe show analogues of the Daniell–Kolmogorov and Prohorov theorems on the existence of projective limits of measures, in the setting of continuous valuations on T0 topological spaces.

$$AC(\sigma)$$ spaces for polygonally inscribed curves

For certain families of compact subsets of the plane, the isomorphism class of the algebra of absolutely continuous functions on a set is completely determined by the homeomorphism class of the set. This is analogous to the Gelfand--Kolmogorov theorem for $C(K)$ spaces. In this paper we define a family of compact sets comprising finite unions of convex curves and show that this family has the `Gelfand--Kolmogorov' property.