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3 pages, 193 KiB

Open AccessEditorial

Approximation Theory and Related Applications

by Yurii Kharkevych

Axioms 2022, 11(12), 736; https://doi.org/10.3390/axioms11120736 - 16 Dec 2022

Cited by 24 | Viewed by 1519

Abstract

The theory of approximation of functions is one of the central branches of mathematical analysis [...] Full article

(This article belongs to the Special Issue Approximation Theory and Related Applications)

Research

Jump to: Editorial

13 pages, 297 KiB

Open AccessArticle

Supersymmetric Polynomials and a Ring of Multisets of a Banach Algebra

by Iryna Chernega and Andriy Zagorodnyuk

Axioms 2022, 11(10), 511; https://doi.org/10.3390/axioms11100511 - 27 Sep 2022

Cited by 4 | Viewed by 1191

Abstract

In this paper, we consider rings of multisets consisting of elements of a Banach algebra. We investigate the algebraic and topological structures of such rings and the properties of their homomorphisms. The rings of multisets arise as natural domains of supersymmetric functions. We [...] Read more.

In this paper, we consider rings of multisets consisting of elements of a Banach algebra. We investigate the algebraic and topological structures of such rings and the properties of their homomorphisms. The rings of multisets arise as natural domains of supersymmetric functions. We introduce a complete metrizable topology on a given ring of multisets and extend some known results about structures of the rings to the general case. In addition, we consider supersymmetric polynomials and other supersymmetric functions related to these rings. This paper contains a number of examples and some discussions. Full article

(This article belongs to the Special Issue Approximation Theory and Related Applications)

13 pages, 295 KiB

Open AccessArticle

Pointwise Wavelet Estimations for a Regression Model in Local Hölder Space

by Junke Kou, Qinmei Huang and Huijun Guo

Axioms 2022, 11(9), 466; https://doi.org/10.3390/axioms11090466 - 10 Sep 2022

Cited by 2 | Viewed by 1338

Abstract

This paper considers an unknown functional estimation problem in a regression model with multiplicative and additive noise. A linear wavelet estimator is first constructed by a wavelet projection operator. The convergence rate under the pointwise error of linear wavelet estimators is studied in [...] Read more.

This paper considers an unknown functional estimation problem in a regression model with multiplicative and additive noise. A linear wavelet estimator is first constructed by a wavelet projection operator. The convergence rate under the pointwise error of linear wavelet estimators is studied in local Hölder space. A nonlinear wavelet estimator is provided by the hard thresholding method in order to obtain an adaptive estimator. The convergence rate of the nonlinear estimator is the same as the linear estimator up to a logarithmic term. Finally, it should be pointed out that the convergence rates of two wavelet estimators are consistent with the optimal convergence rate on pointwise nonparametric estimation. Full article

(This article belongs to the Special Issue Approximation Theory and Related Applications)

10 pages, 2758 KiB

Open AccessCommunication

Selection of Appropriate Symbolic Regression Models Using Statistical and Dynamic System Criteria: Example of Waste Gasification

by Pavel Praks, Marek Lampart, Renáta Praksová, Dejan Brkić, Tomáš Kozubek and Jan Najser

Axioms 2022, 11(9), 463; https://doi.org/10.3390/axioms11090463 - 8 Sep 2022

Cited by 5 | Viewed by 2256

Abstract

In this paper, we analyze the interpretable models from real gasification datasets of the project “Centre for Energy and Environmental Technologies” (CEET) discovered by symbolic regression. To evaluate CEET models based on input data, two different statistical metrics to quantify their accuracy are [...] Read more.

In this paper, we analyze the interpretable models from real gasification datasets of the project “Centre for Energy and Environmental Technologies” (CEET) discovered by symbolic regression. To evaluate CEET models based on input data, two different statistical metrics to quantify their accuracy are usually used: Mean Square Error (MSE) and the Pearson Correlation Coefficient (PCC). However, if the testing points and the points used to construct the models are not chosen randomly from the continuum of the input variable, but instead from the limited number of discrete input points, the behavior of the model between such points very possibly will not fit well the physical essence of the modelled phenomenon. For example, the developed model can have unexpected oscillatory tendencies between the used points, while the usually used statistical metrics cannot detect these anomalies. However, using dynamic system criteria in addition to statistical metrics, such suspicious models that do fit well-expected behavior can be automatically detected and abandoned. This communication will show the universal method based on dynamic system criteria which can detect suitable models among all those which have good properties following statistical metrics. The dynamic system criteria measure the complexity of the candidate models using approximate and sample entropy. The examples are given for waste gasification where the output data (percentage of each particular gas in the produced mixture) is given only for six values of the input data (temperature in the chamber in which the process takes place). In such cases instead, to produce expected simple spline-like curves, artificial intelligence tools can produce inappropriate oscillatory curves with sharp picks due to the known tendency of symbolic regression to produce overfitted and relatively more complex models if the nature of the physical model is simple. Full article

(This article belongs to the Special Issue Approximation Theory and Related Applications)

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20 pages, 349 KiB

Open AccessArticle

Entire Symmetric Functions on the Space of Essentially Bounded Integrable Functions on the Union of Lebesgue-Rohlin Spaces

by Taras Vasylyshyn and Kostiantyn Zhyhallo

Axioms 2022, 11(9), 460; https://doi.org/10.3390/axioms11090460 - 7 Sep 2022

Cited by 7 | Viewed by 1453

Abstract

The class of measure spaces which can be represented as unions of Lebesgue-Rohlin spaces with continuous measures contains a lot of important examples, such as

R^{n}

for any

n \in N

with the Lebesgue measure. In this work we consider symmetric functions [...] Read more.

The class of measure spaces which can be represented as unions of Lebesgue-Rohlin spaces with continuous measures contains a lot of important examples, such as

R^{n}

for any

n \in N

with the Lebesgue measure. In this work we consider symmetric functions on Banach spaces of all complex-valued integrable essentially bounded functions on such unions. We construct countable algebraic bases of algebras of continuous symmetric polynomials on these Banach spaces. The completions of such algebras of polynomials are Fréchet algebras of all complex-valued entire symmetric functions of bounded type on the abovementioned Banach spaces. We show that each such Fréchet algebra is isomorphic to the Fréchet algebra of all complex-valued entire symmetric functions of bounded type on the complex Banach space of all complex-valued essentially bounded functions on

[0, 1]

. Full article

(This article belongs to the Special Issue Approximation Theory and Related Applications)

16 pages, 335 KiB

Open AccessArticle

Approximation for the Ratios of the Confluent Hypergeometric Function

Φ_{D}^{(N)}

by the Branched Continued Fractions

by Tamara Antonova, Roman Dmytryshyn and Roman Kurka

Axioms 2022, 11(9), 426; https://doi.org/10.3390/axioms11090426 - 24 Aug 2022

Cited by 6 | Viewed by 1586

Abstract

The paper deals with the problem of expansion of the ratios of the confluent hypergeometric function of N variables

Φ_{D}^{(N)} (a, \bar{b}; c; \bar{z})

into the branched continued fractions (BCF) of the [...] Read more.

The paper deals with the problem of expansion of the ratios of the confluent hypergeometric function of N variables

Φ_{D}^{(N)} (a, \bar{b}; c; \bar{z})

into the branched continued fractions (BCF) of the general form with N branches of branching and investigates the convergence of these BCF. The algorithms of construction for BCF expansions of confluent hypergeometric function

Φ_{D}^{(N)}

ratios are based on some given recurrence relations for this function. The case of nonnegative parameters

a, b_{1}, \dots, b_{N - 1}

and positive c is considered. Some convergence criteria for obtained BCF with elements in

R^{N}

and

C^{N}

are established. It is proven that these BCF converge to the functions which are an analytic continuation of the above-mentioned ratios of function

Φ_{D}^{(N)} (a, \bar{b}; c; \bar{z})

in some domain of

C^{N}

. Full article

(This article belongs to the Special Issue Approximation Theory and Related Applications)

13 pages, 856 KiB

Open AccessArticle

The Relationship between Fuzzy Reasoning Methods Based on Intuitionistic Fuzzy Sets and Interval-Valued Fuzzy Sets

by Minxia Luo, Wenling Li and Hongyan Shi

Axioms 2022, 11(8), 419; https://doi.org/10.3390/axioms11080419 - 20 Aug 2022

Cited by 4 | Viewed by 1691

Abstract

Two important basic inference models of fuzzy reasoning are Fuzzy Modus Ponens (FMP) and Fuzzy Modus Tollens (FMT). In order to solve FMP and FMT problems, the full implication triple I algorithm, the reverse triple I algorithm and the Subsethood Inference Subsethood (SIS [...] Read more.

Two important basic inference models of fuzzy reasoning are Fuzzy Modus Ponens (FMP) and Fuzzy Modus Tollens (FMT). In order to solve FMP and FMT problems, the full implication triple I algorithm, the reverse triple I algorithm and the Subsethood Inference Subsethood (SIS for short) algorithm are proposed, respectively. Furthermore, the existing reasoning algorithms are extended to intuitionistic fuzzy sets and interval-valued fuzzy sets according to different needs. The purpose of this paper is to study the relationship between intuitionistic fuzzy reasoning algorithms and interval-valued fuzzy reasoning algorithms. It is proven that there is a bijection between the solutions of intuitionistic fuzzy triple I algorithm and the interval-valued fuzzy triple I algorithm. Then, there is a bijection between the solutions of intuitionistic fuzzy reverse triple I algorithm and the interval-valued fuzzy reverse triple I algorithm. At the same time, it is shown that there is also a bijection between the solutions of intuitionistic fuzzy SIS algorithm and interval-valued fuzzy SIS algorithm. Full article

(This article belongs to the Special Issue Approximation Theory and Related Applications)

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18 pages, 307 KiB

Open AccessArticle

On Some Generalizations of Reverse Dynamic Hardy Type Inequalities on Time Scales

by Ahmed A. El-Deeb and Clemente Cesarano

Axioms 2022, 11(7), 336; https://doi.org/10.3390/axioms11070336 - 11 Jul 2022

Cited by 1 | Viewed by 1330

Abstract

In the present paper, we prove some new reverse type dynamic inequalities on

T

. Our main inequalities are proved by using the chain rule and Fubini’s theorem on time scales

T

. Our results extend some existing results in the literature. As [...] Read more.

In the present paper, we prove some new reverse type dynamic inequalities on

T

. Our main inequalities are proved by using the chain rule and Fubini’s theorem on time scales

T

. Our results extend some existing results in the literature. As special cases, we obtain some new discrete inequalities, quantum inequalities and integral inequalities. Full article

(This article belongs to the Special Issue Approximation Theory and Related Applications)

10 pages, 281 KiB

Open AccessArticle

Approximate Optimal Control for a Parabolic System with Perturbations in the Coefficients on the Half-Axis

by Olena A. Kapustian, Oleksiy V. Kapustyan, Anton Ryzhov and Valentyn Sobchuk

Axioms 2022, 11(4), 175; https://doi.org/10.3390/axioms11040175 - 14 Apr 2022

Cited by 10 | Viewed by 1722

Abstract

In this paper, we use the averaging method to find an approximate solution in the optimal control problem of a parabolic system with non-linearity of the form

f (t / ε, y)

on an infinite time interval. Full article

(This article belongs to the Special Issue Approximation Theory and Related Applications)

28 pages, 423 KiB

Open AccessArticle

BMO and Asymptotic Homogeneity

by Vladimir Gutlyanskii, Vladimir Ryazanov, Evgeny Sevost’yanov and Eduard Yakubov

Axioms 2022, 11(4), 171; https://doi.org/10.3390/axioms11040171 - 12 Apr 2022

Cited by 3 | Viewed by 1661

Abstract

First, we prove that the BMO condition by John–Nirenberg leads in the natural way to the asymptotic homogeneity at the origin of regular homeomorphic solutions of the degenerate Beltrami equations. Then, on this basis we establish a series of criteria for the existence [...] Read more.

First, we prove that the BMO condition by John–Nirenberg leads in the natural way to the asymptotic homogeneity at the origin of regular homeomorphic solutions of the degenerate Beltrami equations. Then, on this basis we establish a series of criteria for the existence of regular homeomorphic solutions of the degenerate Beltrami equations in the whole complex plane with asymptotic homogeneity at infinity. These results can be applied to the fluid mechanics in strongly anisotropic and inhomogeneous media because the Beltrami equation is a complex form of the main equation of hydromechanics. Full article

(This article belongs to the Special Issue Approximation Theory and Related Applications)

12 pages, 280 KiB

Open AccessArticle

Approximation Properties of the Generalized Abel-Poisson Integrals on the Weyl-Nagy Classes

by Inna Kal’chuk and Yurii Kharkevych

Axioms 2022, 11(4), 161; https://doi.org/10.3390/axioms11040161 - 1 Apr 2022

Cited by 31 | Viewed by 2045

Abstract

Asymptotic equalities are obtained for the least upper bounds of approximations of functions from the classes

W_{β, \infty}^{r}

by the generalized Abel-Poisson integrals

P_{γ} (δ), 0 < γ \leq 2,

for the case [...] Read more.

Asymptotic equalities are obtained for the least upper bounds of approximations of functions from the classes

W_{β, \infty}^{r}

by the generalized Abel-Poisson integrals

P_{γ} (δ), 0 < γ \leq 2,

for the case

r > γ

in the uniform metric, which provide the solution to the Kolmogorov–Nikol’skii problem for the given method of approximation on the Weyl-Nagy classes. Full article

(This article belongs to the Special Issue Approximation Theory and Related Applications)

11 pages, 319 KiB

Open AccessArticle

Some Korovkin-Type Approximation Theorems Associated with a Certain Deferred Weighted Statistical Riemann-Integrable Sequence of Functions

by Hari Mohan Srivastava, Bidu Bhusan Jena and Susanta Kumar Paikray

Axioms 2022, 11(3), 128; https://doi.org/10.3390/axioms11030128 - 12 Mar 2022

Cited by 4 | Viewed by 2296

Abstract

Here, in this article, we introduce and systematically investigate the ideas of deferred weighted statistical Riemann integrability and statistical deferred weighted Riemann summability for sequences of functions. We begin by proving an inclusion theorem that establishes a relation between these two potentially useful [...] Read more.

Here, in this article, we introduce and systematically investigate the ideas of deferred weighted statistical Riemann integrability and statistical deferred weighted Riemann summability for sequences of functions. We begin by proving an inclusion theorem that establishes a relation between these two potentially useful concepts. We also state and prove two Korovkin-type approximation theorems involving algebraic test functions by using our proposed concepts and methodologies. Furthermore, in order to demonstrate the usefulness of our findings, we consider an illustrative example involving a sequence of positive linear operators in conjunction with the familiar Bernstein polynomials. Finally, in the concluding section, we propose some directions for future research on this topic, which are based upon the core concept of statistical Lebesgue-measurable sequences of functions. Full article

(This article belongs to the Special Issue Approximation Theory and Related Applications)

11 pages, 280 KiB

Open AccessArticle

Quasi-Density of Sets, Quasi-Statistical Convergence and the Matrix Summability Method

by Renata Masarova, Tomas Visnyai and Robert Vrabel

Axioms 2022, 11(3), 88; https://doi.org/10.3390/axioms11030088 - 23 Feb 2022

Cited by 1 | Viewed by 1946

Abstract

In this paper, we define the quasi-density of subsets of the set of natural numbers and show several of the properties of this density. The quasi-density

d_{p} (A)

of the set

A \subseteq N

is dependent on the sequence [...] Read more.

In this paper, we define the quasi-density of subsets of the set of natural numbers and show several of the properties of this density. The quasi-density

d_{p} (A)

of the set

A \subseteq N

is dependent on the sequence

p = (p_{n})

. Different sequences

(p_{n})

, for the same set

A

, will yield new and distinct densities. If the sequence

(p_{n})

does not differ from the sequence

(n)

in its order of magnitude, i.e.,

\lim_{n \to \infty} \frac{p_{n}}{n} = 1

, then the resulting quasi-density is very close to the asymptotic density. The results for sequences that do not satisfy this condition are more interesting. In the next part, we deal with the necessary and sufficient conditions so that the quasi-statistical convergence will be equivalent to the matrix summability method for a special class of triangular matrices with real coefficients. Full article

(This article belongs to the Special Issue Approximation Theory and Related Applications)

18 pages, 307 KiB

Open AccessArticle

Fixed Point Results on Partial Modular Metric Space

by Dipankar Das, Santanu Narzary, Yumnam Mahendra Singh, Mohammad Saeed Khan and Salvatore Sessa

Axioms 2022, 11(2), 62; https://doi.org/10.3390/axioms11020062 - 1 Feb 2022

Cited by 3 | Viewed by 2620

Abstract

In the present paper, we refine the notion of the partial modular metric defined by Hosseinzadeh and Parvaneh to eliminate the occurrence of discrepancies in the non-zero self-distance and triangular inequality. In support of this, we discuss non-trivial examples. Finally, we prove a [...] Read more.

In the present paper, we refine the notion of the partial modular metric defined by Hosseinzadeh and Parvaneh to eliminate the occurrence of discrepancies in the non-zero self-distance and triangular inequality. In support of this, we discuss non-trivial examples. Finally, we prove a common fixed-point theorem for four self-mappings in partial modular metric space and an application to our result; the existence of a solution for a system of Volterra integral equations is discussed. Full article

(This article belongs to the Special Issue Approximation Theory and Related Applications)

14 pages, 455 KiB

Open AccessArticle

Quadratic Lyapunov Functions for Stability of the Generalized Proportional Fractional Differential Equations with Applications to Neural Networks

by Ricardo Almeida, Ravi P. Agarwal, Snezhana Hristova and Donal O’Regan

Axioms 2021, 10(4), 322; https://doi.org/10.3390/axioms10040322 - 27 Nov 2021

Cited by 17 | Viewed by 2665

Abstract

A fractional model of the Hopfield neural network is considered in the case of the application of the generalized proportional Caputo fractional derivative. The stability analysis of this model is used to show the reliability of the processed information. An equilibrium is defined, [...] Read more.

A fractional model of the Hopfield neural network is considered in the case of the application of the generalized proportional Caputo fractional derivative. The stability analysis of this model is used to show the reliability of the processed information. An equilibrium is defined, which is generally not a constant (different than the case of ordinary derivatives and Caputo-type fractional derivatives). We define the exponential stability and the Mittag–Leffler stability of the equilibrium. For this, we extend the second method of Lyapunov in the fractional-order case and establish a useful inequality for the generalized proportional Caputo fractional derivative of the quadratic Lyapunov function. Several sufficient conditions are presented to guarantee these types of stability. Finally, two numerical examples are presented to illustrate the effectiveness of our theoretical results. Full article

(This article belongs to the Special Issue Approximation Theory and Related Applications)

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