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19 pages, 603 KiB

Open AccessArticle

Chebyshev Polynomials in the Physics of the One-Dimensional Finite-Size Ising Model: An Alternative View and Some New Results

by Nicholay S. Tonchev and Daniel Dantchev

Condens. Matter 2024, 9(4), 53; https://doi.org/10.3390/condmat9040053 - 2 Dec 2024

Viewed by 1413

Abstract

For studying the finite-size behavior of the Ising model under different boundary conditions, we propose an alternative to the standard transfer matrix technique approach based on Abelès theorem and Chebyshev polynomials. Using it, one can easily reproduce the known results for periodic boundary [...] Read more.

For studying the finite-size behavior of the Ising model under different boundary conditions, we propose an alternative to the standard transfer matrix technique approach based on Abelès theorem and Chebyshev polynomials. Using it, one can easily reproduce the known results for periodic boundary conditions concerning the Lee–Yang zeros, the exact position-space renormalization-group transformation, etc., and can extend them by deriving new results for antiperiodic boundary conditions. Note that in the latter case, one has a nontrivial order parameter profile, which we also calculate, where the average value of a given spin depends on the distance from the seam with the opposite bond in the system. It is interesting to note that under both boundary conditions, the one-dimensional case exhibits Schottky anomaly. Full article

(This article belongs to the Section Condensed Matter Theory)

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16 pages, 373 KiB

Open AccessArticle

Analytical Solutions to the Unsteady Poiseuille Flow of a Second Grade Fluid with Slip Boundary Conditions

by Evgenii S. Baranovskii

Polymers 2024, 16(2), 179; https://doi.org/10.3390/polym16020179 - 7 Jan 2024

Cited by 10 | Viewed by 2189

Abstract

This paper deals with an initial-boundary value problem modeling the unidirectional pressure-driven flow of a second grade fluid in a plane channel with impermeable solid walls. On the channel walls, Navier-type slip boundary conditions are stated. Our aim is to investigate the well-posedness [...] Read more.

This paper deals with an initial-boundary value problem modeling the unidirectional pressure-driven flow of a second grade fluid in a plane channel with impermeable solid walls. On the channel walls, Navier-type slip boundary conditions are stated. Our aim is to investigate the well-posedness of this problem and obtain its analytical solution under weak regularity requirements on a function describing the velocity distribution at initial time. In order to overcome difficulties related to finding classical solutions, we propose the concept of a generalized solution that is defined as the limit of a uniformly convergent sequence of classical solutions with vanishing perturbations in the initial data. We prove the unique solvability of the problem under consideration in the class of generalized solutions. The main ingredients of our proof are a generalized Abel criterion for uniform convergence of function series and the use of an orthonormal basis consisting of eigenfunctions of the related Sturm–Liouville problem. As a result, explicit expressions for the flow velocity and the pressure in the channel are established. The constructed analytical solutions favor a better understanding of the qualitative features of time-dependent flows of polymer fluids and can be applied to the verification of relevant numerical, asymptotic, and approximate analytical methods. Full article

(This article belongs to the Special Issue Polymer Physics: From Theory to Experimental Applications)

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13 pages, 286 KiB

Open AccessArticle

A Weighted Generalization of Hardy–Hilbert-Type Inequality Involving Two Partial Sums

by Bicheng Yang and Shanhe Wu

Mathematics 2023, 11(14), 3212; https://doi.org/10.3390/math11143212 - 21 Jul 2023

Cited by 2 | Viewed by 1025

Abstract

In this paper, we address Hardy–Hilbert-type inequality by virtue of constructing weight coefficients and introducing parameters. By using the Euler–Maclaurin summation formula, Abel’s partial summation formula, and differential mean value theorem, a new weighted Hardy–Hilbert-type inequality containing two partial sums can be proven, [...] Read more.

In this paper, we address Hardy–Hilbert-type inequality by virtue of constructing weight coefficients and introducing parameters. By using the Euler–Maclaurin summation formula, Abel’s partial summation formula, and differential mean value theorem, a new weighted Hardy–Hilbert-type inequality containing two partial sums can be proven, which is a further generalization of an existing result. Based on the obtained results, we provide the equivalent statements of the best possible constant factor related to several parameters. Also, we illustrate how the inequalities obtained in the main results can generate some new Hardy–Hilbert-type inequalities. Full article

(This article belongs to the Special Issue Recent Trends in Convex Analysis and Mathematical Inequalities)

19 pages, 345 KiB

Open AccessArticle

Abstract Evolution Equations with an Operator Function in the Second Term

by Maksim V. Kukushkin

Axioms 2022, 11(9), 434; https://doi.org/10.3390/axioms11090434 - 26 Aug 2022

Cited by 4 | Viewed by 1555

Abstract

In this paper, having introduced a convergence of a series on the root vectors in the Abel-Lidskii sense, we present a valuable application to the evolution equations. The main issue of the paper is an approach allowing us to principally broaden conditions imposed [...] Read more.

In this paper, having introduced a convergence of a series on the root vectors in the Abel-Lidskii sense, we present a valuable application to the evolution equations. The main issue of the paper is an approach allowing us to principally broaden conditions imposed upon the second term of the evolution equation in the abstract Hilbert space. In this way, we come to the definition of the function of an unbounded non-selfadjoint operator. Meanwhile, considering the main issue we involve an additional concept that is a generalization of the spectral theorem for a non-selfadjoint operator. Full article

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

26 pages, 427 KiB

Open AccessArticle

Boundedness of Some Paraproducts on Spaces of Homogeneous Type

by Xing Fu

Mathematics 2021, 9(20), 2591; https://doi.org/10.3390/math9202591 - 15 Oct 2021

Viewed by 1596

Abstract

Let

(X, d, μ)

be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the author develops a partial theory of paraproducts

{Π_{j}}_{j = 1}^{3}

defined via approximations [...] Read more.

Let

(X, d, μ)

be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the author develops a partial theory of paraproducts

{Π_{j}}_{j = 1}^{3}

defined via approximations of the identity with exponential decay (and integration 1), which are extensions of paraproducts defined via regular wavelets. Precisely, the author first obtains the boundedness of

Π_{3}

on Hardy spaces and then, via the methods of interpolation and the well-known

T (1)

theorem, establishes the endpoint estimates for

{Π_{j}}_{j = 1}^{3}

. The main novelty of this paper is the application of the Abel summation formula to the establishment of some relations among the boundedness of

{Π_{j}}_{j = 1}^{3}

, which has independent interests. It is also remarked that, throughout this article,

μ

is not assumed to satisfy the reverse doubling condition. Full article

(This article belongs to the Special Issue Recent Developments of Function Spaces and Their Applications I)

13 pages, 289 KiB

Open AccessArticle

Uniqueness of Solutions of the Generalized Abel Integral Equations in Banach Spaces

by Chenkuan Li and Hari M. Srivastava

Fractal Fract. 2021, 5(3), 105; https://doi.org/10.3390/fractalfract5030105 - 31 Aug 2021

Cited by 10 | Viewed by 2233

Abstract

This paper studies the uniqueness of solutions for several generalized Abel’s integral equations and a related coupled system in Banach spaces. The results derived are new and based on Babenko’s approach, Banach’s contraction principle and the multivariate Mittag–Leffler function. We also present some [...] Read more.

This paper studies the uniqueness of solutions for several generalized Abel’s integral equations and a related coupled system in Banach spaces. The results derived are new and based on Babenko’s approach, Banach’s contraction principle and the multivariate Mittag–Leffler function. We also present some examples for the illustration of our main theorems. Full article

(This article belongs to the Special Issue Operators of Fractional Calculus and Their Multidisciplinary Applications)

13 pages, 327 KiB

Open AccessArticle

Advantages of the Discrete Stochastic Arithmetic to Validate the Results of the Taylor Expansion Method to Solve the Generalized Abel’s Integral Equation

by Eisa Zarei and Samad Noeiaghdam

Symmetry 2021, 13(8), 1370; https://doi.org/10.3390/sym13081370 - 28 Jul 2021

Cited by 5 | Viewed by 2025

Abstract

The aim of this paper is to apply the Taylor expansion method to solve the first and second kinds Volterra integral equations with Abel kernel. This study focuses on two main arithmetics: the FPA and the DSA. In order to apply the DSA, [...] Read more.

The aim of this paper is to apply the Taylor expansion method to solve the first and second kinds Volterra integral equations with Abel kernel. This study focuses on two main arithmetics: the FPA and the DSA. In order to apply the DSA, we use the CESTAC method and the CADNA library. Using this method, we can find the optimal step of the method, the optimal approximation, the optimal error, and some of numerical instabilities. They are the main novelties of the DSA in comparison with the FPA. The error analysis of the method is proved. Furthermore, the main theorem of the CESTAC method is presented. Using this theorem we can apply a new termination criterion instead of the traditional absolute error. Several examples are approximated based on the FPA and the DSA. The numerical results show the applications and advantages of the DSA than the FPA. Full article

11 pages, 256 KiB

Open AccessArticle

Uniqueness of Abel’s Integral Equations of the Second Kind with Variable Coefficients

by Chenkuan Li and Joshua Beaudin

Symmetry 2021, 13(6), 1064; https://doi.org/10.3390/sym13061064 - 13 Jun 2021

Cited by 5 | Viewed by 2515

Abstract

This paper studies the uniqueness of the solutions of several of Abel’s integral equations of the second kind with variable coefficients as well as an in-symmetry system in Banach spaces

L (Ω)

and [...] Read more.

This paper studies the uniqueness of the solutions of several of Abel’s integral equations of the second kind with variable coefficients as well as an in-symmetry system in Banach spaces

L (Ω)

and

L (Ω) \times L (Ω)

, respectively. The results derived are new and original, and can be applied to solve the generalized Abel’s integral equations and obtain convergent series as solutions. We also provide a few examples to demonstrate the use of our main theorems based on convolutions, the gamma function and the Mittag–Leffler function. Full article

(This article belongs to the Special Issue Applied Mathematics and Fractional Calculus)

15 pages, 338 KiB

Open AccessFeature PaperReview

Series in Le Roy Type Functions: A Set of Results in the Complex Plane—A Survey

by Jordanka Paneva-Konovska

Mathematics 2021, 9(12), 1361; https://doi.org/10.3390/math9121361 - 12 Jun 2021

Cited by 9 | Viewed by 2042

Abstract

This study is based on a part of the results obtained in the author’s publications. An enumerable family of the Le Roy type functions is considered herein. The asymptotic formula for these special functions in the cases of ‘large’ values of indices, that [...] Read more.

This study is based on a part of the results obtained in the author’s publications. An enumerable family of the Le Roy type functions is considered herein. The asymptotic formula for these special functions in the cases of ‘large’ values of indices, that has been previously obtained, is provided. Further, series defined by means of the Le Roy type functions are considered. These series are studied in the complex plane. Their domains of convergence are given and their behaviour is investigated ‘near’ the boundaries of the domains of convergence. The discussed asymptotic formula is used in the proofs of the convergence theorems for the considered series. A theorem of the Cauchy–Hadamard type is provided. Results of Abel, Tauber and Littlewood type, which are analogues to the corresponding theorems for the classical power series, are also proved. At last, various interesting particular cases of the discussed special functions are considered. Full article

(This article belongs to the Special Issue Special Functions with Applications to Mathematical Physics)

16 pages, 507 KiB

Open AccessFeature PaperArticle

Robust H_∞ Loop Shaping Controller Synthesis for SISO Systems by Complex Modular Functions

by Ahmad Taher Azar, Fernando E. Serrano and Nashwa Ahmad Kamal

Math. Comput. Appl. 2021, 26(1), 21; https://doi.org/10.3390/mca26010021 - 3 Mar 2021

Cited by 2 | Viewed by 2982

Abstract

In this paper, a loop shaping controller design methodology for single input and a single output (SISO) system is proposed. The theoretical background for this approach is based on complex elliptic functions which allow a flexible design of a SISO controller considering that [...] Read more.

In this paper, a loop shaping controller design methodology for single input and a single output (SISO) system is proposed. The theoretical background for this approach is based on complex elliptic functions which allow a flexible design of a SISO controller considering that elliptic functions have a double periodicity. The gain and phase margins of the closed-loop system can be selected appropriately with this new loop shaping design procedure. The loop shaping design methodology consists of implementing suitable filters to obtain a desired frequency response of the closed-loop system by selecting appropriate poles and zeros by the Abel theorem that are fundamental in the theory of the elliptic functions. The elliptic function properties are implemented to facilitate the loop shaping controller design along with their fundamental background and contributions from the complex analysis that are very useful in the automatic control field. Finally, apart from the filter design, a PID controller loop shaping synthesis is proposed implementing a similar design procedure as the first part of this study. Full article

(This article belongs to the Special Issue Advances of Modern Control Systems and Robotic Applications)

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13 pages, 280 KiB

Open AccessArticle

On Smoothness of the Solution to the Abel Equation in Terms of the Jacobi Series Coefficients

by Maksim V. Kukushkin

Axioms 2020, 9(3), 81; https://doi.org/10.3390/axioms9030081 - 17 Jul 2020

Cited by 3 | Viewed by 2076

Abstract

In this paper, we continue our study of the Abel equation with the right-hand side belonging to the Lebesgue weighted space. We have improved the previously known result— the existence and uniqueness theorem formulated in terms of the Jacoby series coefficients that gives [...] Read more.

In this paper, we continue our study of the Abel equation with the right-hand side belonging to the Lebesgue weighted space. We have improved the previously known result— the existence and uniqueness theorem formulated in terms of the Jacoby series coefficients that gives us an opportunity to find and classify a solution by virtue of an asymptotic of some relation containing the Jacobi series coefficients of the right-hand side. The main results are the following—the conditions imposed on the parameters, under which the Abel equation has a unique solution represented by the series, are formulated; the relationship between the values of the parameters and the solution smoothness is established. The independence between one of the parameters and the smoothness of the solution is proved. Full article

(This article belongs to the Special Issue Fractional Calculus, Wavelets and Fractals)

18 pages, 831 KiB

Open AccessArticle

The New New-Nacci Method for Calculating the Roots of a Univariate Polynomial and Solution of Quintic Equation in Radicals

by Ilija Tanackov, Ivan Pavkov and Željko Stević

Mathematics 2020, 8(5), 746; https://doi.org/10.3390/math8050746 - 8 May 2020

Cited by 3 | Viewed by 3256

Abstract

An arbitrary univariate polynomial of nth degree has n sequences. The sequences are systematized into classes. All the values of the first class sequence are obtained by Newton’s polynomial of nth degree. Furthermore, the values of all sequences for each class [...] Read more.

An arbitrary univariate polynomial of nth degree has n sequences. The sequences are systematized into classes. All the values of the first class sequence are obtained by Newton’s polynomial of nth degree. Furthermore, the values of all sequences for each class are calculated by Newton’s identities. In other words, the sequences are formed without calculation of polynomial roots. The New-nacci method is used for the calculation of the roots of an nth-degree univariate polynomial using radicals and limits of successive members of sequences. In such an approach as is presented in this paper, limit play a catalytic–theoretical role. Moreover, only four basic algebraic operations are sufficient to calculate real roots. Radicals are necessary for calculating conjugated complex roots. The partial limitations of the New-nacci method may appear from the decadal polynomial. In the case that an arbitrary univariate polynomial of nth degree (n ≥ 10) has five or more conjugated complex roots, the roots of the polynomial cannot be calculated due to Abel’s impossibility theorem. The second phase of the New-nacci method solves this problem as well. This paper is focused on solving the roots of the quintic equation. The method is verified by applying it to the quintic polynomial with all real roots and the Degen–Abel polynomial, dating from 1821. Full article

(This article belongs to the Section E: Applied Mathematics)

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20 pages, 286 KiB

Open AccessArticle

Means as Improper Integrals

by John E. Gray and Andrew Vogt

Mathematics 2019, 7(3), 284; https://doi.org/10.3390/math7030284 - 20 Mar 2019

Viewed by 3002

Abstract

The aim of this work is to study generalizations of the notion of the mean. Kolmogorov proposed a generalization based on an improper integral with a decay rate for the tail probabilities. This weak or Kolmogorov mean relates to the weak law of [...] Read more.

The aim of this work is to study generalizations of the notion of the mean. Kolmogorov proposed a generalization based on an improper integral with a decay rate for the tail probabilities. This weak or Kolmogorov mean relates to the weak law of large numbers in the same way that the ordinary mean relates to the strong law. We propose a further generalization, also based on an improper integral, called the doubly-weak mean, applicable to heavy-tailed distributions such as the Cauchy distribution and the other symmetric stable distributions. We also consider generalizations arising from Abel–Feynman-type mollifiers that damp the behavior at infinity and alternative formulations of the mean in terms of the cumulative distribution and the characteristic function. Full article

13 pages, 1204 KiB

Open AccessArticle

The Decomposition Theorems of AG-Neutrosophic Extended Triplet Loops and Strong AG-(l, l)-Loops

by Xiaoying Wu and Xiaohong Zhang

Mathematics 2019, 7(3), 268; https://doi.org/10.3390/math7030268 - 15 Mar 2019

Cited by 22 | Viewed by 2618

Abstract

In this paper, some new properties of Abel Grassmann‘s Neutrosophic Extended Triplet Loop (AG-NET-Loop) were further studied. The following important results were proved: (1) an AG-NET-Loop is weakly commutative if, and only if, it is a commutative neutrosophic extended triplet (NETG); (2) every [...] Read more.

In this paper, some new properties of Abel Grassmann‘s Neutrosophic Extended Triplet Loop (AG-NET-Loop) were further studied. The following important results were proved: (1) an AG-NET-Loop is weakly commutative if, and only if, it is a commutative neutrosophic extended triplet (NETG); (2) every AG-NET-Loop is the disjoint union of its maximal subgroups. At the same time, the new notion of Abel Grassmann’s (l, l)-Loop (AG-(l, l)-Loop), which is the Abel-Grassmann’s groupoid with the local left identity and local left inverse, were introduced. The strong AG-(l, l)-Loops were systematically analyzed, and the following decomposition theorem was proved: every strong AG-(l, l)-Loop is the disjoint union of its maximal sub-AG-groups. Full article

(This article belongs to the Special Issue General Algebraic Structures)

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13 pages, 503 KiB

Open AccessArticle

The Structure Theorems of Pseudo-BCI Algebras in Which Every Element is Quasi-Maximal

by Xiaoying Wu and Xiaohong Zhang

Symmetry 2018, 10(10), 465; https://doi.org/10.3390/sym10100465 - 8 Oct 2018

Cited by 2 | Viewed by 3054

Abstract

For mathematical fuzzy logic systems, the study of corresponding algebraic structures plays an important role. Pseudo-BCI algebra is a class of non-classical logic algebras, which is closely related to various non-commutative fuzzy logic systems. The aim of this paper is focus on the [...] Read more.

For mathematical fuzzy logic systems, the study of corresponding algebraic structures plays an important role. Pseudo-BCI algebra is a class of non-classical logic algebras, which is closely related to various non-commutative fuzzy logic systems. The aim of this paper is focus on the structure of a special class of pseudo-BCI algebras in which every element is quasi-maximal (call it QM-pseudo-BCI algebras in this paper). First, the new notions of quasi-maximal element and quasi-left unit element in pseudo-BCK algebras and pseudo-BCI algebras are proposed and some properties are discussed. Second, the following structure theorem of QM-pseudo-BCI algebra is proved: every QM-pseudo-BCI algebra is a KG-union of a quasi-alternating BCK-algebra and an anti-group pseudo-BCI algebra. Third, the new notion of weak associative pseudo-BCI algebra (WA-pseudo-BCI algebra) is introduced and the following result is proved: every WA-pseudo-BCI algebra is a KG-union of a quasi-alternating BCK-algebra and an Abel group. Full article

(This article belongs to the Special Issue Discrete Mathematics and Symmetry)

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