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27 pages, 957 KiB

Open AccessArticle

Complex-Valued Multivariate Neural Network (MNN) Approximation by Parameterized Half-Hyperbolic Tangent Function

by Seda Karateke

Mathematics 2025, 13(3), 453; https://doi.org/10.3390/math13030453 - 29 Jan 2025

Viewed by 773

This paper deals with a family of normalized multivariate neural network (MNN) operators of complex-valued continuous functions for a multivariate context on a box of

R^{\bar{N}}

\bar{N} \in N

. Moreover, we consider the case of approximation employing iterated [...] Read more.

This paper deals with a family of normalized multivariate neural network (MNN) operators of complex-valued continuous functions for a multivariate context on a box of

R^{\bar{N}}

\bar{N} \in N

. Moreover, we consider the case of approximation employing iterated MNN operators. In addition, pointwise and uniform convergence results are obtained in Banach spaces thanks to the multivariate versions of trigonometric and hyperbolic-type Taylor formulae on the corresponding feed-forward neural networks (FNNs) based on one or more hidden layers. Full article

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

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27 pages, 326 KiB

Open AccessArticle

Parametrized Half-Hyperbolic Tangent Function-Activated Complex-Valued Neural Network Approximation

by George A. Anastassiou and Seda Karateke

Symmetry 2024, 16(12), 1568; https://doi.org/10.3390/sym16121568 - 23 Nov 2024

Cited by 1 | Viewed by 816

Abstract

In this paper, we create a family of neural network (NN) operators employing a parametrized and deformed half-hyperbolic tangent function as an activation function and a density function produced by the same activation function. Moreover, we consider the univariate quantitative approximations by complex-valued neural network (NN) operators of complex-valued functions on a compact domain. Pointwise and uniform convergence results on Banach spaces are acquired through trigonometric, hyperbolic, and hybrid-type hyperbolic–trigonometric approaches. Full article

(This article belongs to the Special Issue Functional Analysis, Fractional Operators and Symmetry/Asymmetry: Second Edition)

13 pages, 294 KiB

Open AccessArticle

A Variety of Weighted Opial-Type Inequalities with Applications for Dynamic Equations on Time Scales

by Barakah Almarri, Samer D. Makarish and Ahmed A. El-Deeb

Symmetry 2023, 15(5), 1039; https://doi.org/10.3390/sym15051039 - 8 May 2023

Viewed by 1799

Abstract

Using higher order delta derivatives on time scales, we demonstrated a few dynamic inequalities of the Opial type in this paper. Our findings expanded upon and generalised earlier findings in the literature. Furthermore, we give the discrete and continuous inequalities as special cases. At the end of this paper, we apply our results to study the behaviour of the solution of an initial value problem. In selecting the best ways to solve dynamic inequalities, symmetry is crucial. Full article

(This article belongs to the Section Mathematics)

16 pages, 302 KiB

Open AccessArticle

General Opial Type Inequality and New Green Functions

by Ana Gudelj, Kristina Krulić Himmelreich and Josip Pečarić

Axioms 2022, 11(6), 252; https://doi.org/10.3390/axioms11060252 - 26 May 2022

Viewed by 1937

Abstract

In this paper we provide many new results involving Opial type inequalities. We consider two functions—one is convex and the other is concave—and prove a new general inequality on a measure space

(Ω, Σ, μ)

. We give an [...] Read more.

In this paper we provide many new results involving Opial type inequalities. We consider two functions—one is convex and the other is concave—and prove a new general inequality on a measure space

(Ω, Σ, μ)

. We give an new result involving four new Green functions. Our results include Grüss and Ostrowski type inequalities related to the generalized Opial type inequality. The obtained inequalities are of Opial type because the integrals contain the function and its integral representation. They are not a direct generalization of the Opial inequality. Full article

(This article belongs to the Special Issue Special Functions and Polynomials: Theory, Practice, Applications, and Modeling)

18 pages, 331 KiB

Open AccessArticle

New Weighted Opial-Type Inequalities on Time Scales for Convex Functions

by Ahmed A. El-Deeb and Dumitru Baleanu

Symmetry 2020, 12(5), 842; https://doi.org/10.3390/sym12050842 - 21 May 2020

Cited by 8 | Viewed by 2612

Abstract

Our work is based on the multiple inequalities illustrated in 1967 by E. K. Godunova and V. I. Levin, in 1990 by Hwang and Yang and in 1993 by B. G. Pachpatte. With the help of the dynamic Jensen and Hölder inequality, we generalize a number of those inequalities to a general time scale. In addition to these generalizations, some integral and discrete inequalities will be obtained as special cases of our results. Full article

(This article belongs to the Special Issue Composite Structures with Symmetry)

9 pages, 225 KiB

Open AccessArticle

On Opial’s Type Integral Inequalities

by Chang-Jian Zhao

Mathematics 2019, 7(4), 375; https://doi.org/10.3390/math7040375 - 25 Apr 2019

Cited by 2 | Viewed by 2446

Abstract

In the article we establish some new Opial’s type inequalities involving higher order partial derivatives. These new inequalities, in special cases, yield Agarwal-Pang’s, Pachpatte’s and Das’s type inequalities. Full article

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