Axioms

2022

Jump to: 2021, 2020, 2019, 2018

16 pages, 312 KiB

Open AccessArticle

A New First Order Expansion Formula with a Reduced Remainder

by Joel Chaskalovic and Hessam Jamshidipour

Axioms 2022, 11(10), 562; https://doi.org/10.3390/axioms11100562 - 17 Oct 2022

Cited by 3 | Viewed by 1386

Abstract

This paper is devoted to a new first order Taylor-like formula, where the corresponding remainder is strongly reduced in comparison with the usual one, which appears in the classical Taylor’s formula. To derive this new formula, we introduce a linear combination of the [...] Read more.

This paper is devoted to a new first order Taylor-like formula, where the corresponding remainder is strongly reduced in comparison with the usual one, which appears in the classical Taylor’s formula. To derive this new formula, we introduce a linear combination of the first derivative of the concerned function, which is computed at

n + 1

equally spaced points between the two points, where the function has to be evaluated. We show that an optimal choice of the weights in the linear combination leads to minimizing the corresponding remainder. Then, we analyze the Lagrange

P_{1}

- interpolation error estimate and the trapezoidal quadrature error, in order to assess the gain of the accuracy we obtain using this new Taylor-like formula. Full article

11 pages, 1027 KiB

Open AccessArticle

Some Probabilistic Generalizations of the Cheney–Sharma and Bernstein Approximation Operators

by Seng Huat Ong, Choung Min Ng, Hong Keat Yap and Hari Mohan Srivastava

Axioms 2022, 11(10), 537; https://doi.org/10.3390/axioms11100537 - 8 Oct 2022

Cited by 5 | Viewed by 1661

Abstract

The objective of this paper is to give some probabilistic derivations of the Cheney, Sharma, and Bernstein approximation operators. Motivated by these probabilistic derivations, generalizations of the Cheney, Sharma, and Bernstein operators are defined. The convergence property of the Bernstein generalization is established. [...] Read more.

The objective of this paper is to give some probabilistic derivations of the Cheney, Sharma, and Bernstein approximation operators. Motivated by these probabilistic derivations, generalizations of the Cheney, Sharma, and Bernstein operators are defined. The convergence property of the Bernstein generalization is established. It is also shown that the Cheney–Sharma operator is the Szász–Mirakyan operator averaged by a certain probability distribution. Full article

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20 pages, 3960 KiB

Open AccessArticle

Mixture of Akash Distributions: Estimation, Simulation and Application

by Anum Shafiq, Tabassum Naz Sindhu, Showkat Ahmad Lone, Marwa K. H. Hassan and Kamsing Nonlaopon

Axioms 2022, 11(10), 516; https://doi.org/10.3390/axioms11100516 - 29 Sep 2022

Cited by 3 | Viewed by 1638

Abstract

In this paper, we propose a two-component mixture of Akash model (TC-MAM). The behavior of TC-MAM distribution has been presented graphically. Moment-based measures, including skewness, index of dispersion, kurtosis, and coefficient of variation, have been determined and hazard rate functions are presented graphically. [...] Read more.

In this paper, we propose a two-component mixture of Akash model (TC-MAM). The behavior of TC-MAM distribution has been presented graphically. Moment-based measures, including skewness, index of dispersion, kurtosis, and coefficient of variation, have been determined and hazard rate functions are presented graphically. The probability generating function, Mills ratio, characteristic function, cumulants, mean time to failure, and factorial moment generating function are all statistical aspects of the mixed model that we explore. Furthermore, we figure out the relevant parameters of the mixture model using the most suitable methods, such as least square, weighted least square, and maximum likelihood mechanisms. Findings of simulation experiments to examine behavior of these estimates are graphically presented. Finally, a set of data taken from the real world is examined in order to demonstrate the new model’s practical perspectives. All of the metrics evaluated favor the new model and the superiority of proposed distribution over mixture of Lindley, Shanker, and exponential distributions. Full article

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22 pages, 3889 KiB

Open AccessArticle

Arctan-Based Family of Distributions: Properties, Survival Regression, Bayesian Analysis and Applications

by Omid Kharazmi, Morad Alizadeh, Javier E. Contreras-Reyes and Hossein Haghbin

Axioms 2022, 11(8), 399; https://doi.org/10.3390/axioms11080399 - 12 Aug 2022

Cited by 5 | Viewed by 2171

Abstract

In this paper, a new class of the continuous distributions is established via compounding the arctangent function with a generalized log-logistic class of distributions. Some structural properties of the suggested model such as distribution function, hazard function, quantile function, asymptotics and a useful [...] Read more.

In this paper, a new class of the continuous distributions is established via compounding the arctangent function with a generalized log-logistic class of distributions. Some structural properties of the suggested model such as distribution function, hazard function, quantile function, asymptotics and a useful expansion for the new class are given in a general setting. Two special cases of this new class are considered by employing Weibull and normal distributions as the parent distribution. Further, we derive a survival regression model based on a sub-model with Weibull parent distribution and then estimate the parameters of the proposed regression model making use of Bayesian and frequentist approaches. We consider seven loss functions, namely the squared error, modified squared error, weighted squared error, K-loss, linear exponential, general entropy, and precautionary loss functions for Bayesian discussion. Bayesian numerical results include a Bayes estimator, associated posterior risk, credible and highest posterior density intervals are provided. In order to explore the consistency property of the maximum likelihood estimators, a simulation study is presented via Monte Carlo procedure. The parameters of two sub-models are estimated with maximum likelihood and the usefulness of these sub-models and a proposed survival regression model is examined by means of three real datasets. Full article

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16 pages, 322 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

On the Bias in Confirmatory Factor Analysis When Treating Discrete Variables as Ordinal Instead of Continuous

by Alexander Robitzsch

Axioms 2022, 11(4), 162; https://doi.org/10.3390/axioms11040162 - 1 Apr 2022

Cited by 8 | Viewed by 3149

Abstract

Confirmatory factor analysis is some of the most widely used statistical techniques in the social sciences. Frequently, variables (i.e., items) stemming from questionnaires are analyzed. Two competing approaches for estimating confirmatory factor analysis can be distinguished. First, ordinal variables could be treated as [...] Read more.

Confirmatory factor analysis is some of the most widely used statistical techniques in the social sciences. Frequently, variables (i.e., items) stemming from questionnaires are analyzed. Two competing approaches for estimating confirmatory factor analysis can be distinguished. First, ordinal variables could be treated as in the case of continuous variables using Pearson correlations, and maximum likelihood estimation method would be applied. Second, an ordinal factor analysis based on polychoric correlations can be fitted. In the majority of the psychometric literature, there is a preference for the ordinal factor analysis based on polychoric correlations because the continuous treatment of variables results in biased factor loadings and biased factor correlations. This article argues that it is not legitimate to speak about bias when comparing the two competing factor analytic approaches because it depends on how true model parameters are defined. This decision can be made individually by a researcher. It is shown in simulation studies and analytical derivations that treating variables ordinally using polychoric correlations instead of continuous using Pearson correlations can also lead to biased estimates of factor loadings and factor correlations. Consequently, it should only be stated that different model parameters are defined in a continuous and an ordinal treatment, and one approach should not generally be preferred over the other. Full article

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22 pages, 353 KiB

Open AccessArticle

Approximate Methods for Calculating Singular and Hypersingular Integrals with Rapidly Oscillating Kernels

by Ilya Boykov, Vladimir Roudnev and Alla Boykova

Axioms 2022, 11(4), 150; https://doi.org/10.3390/axioms11040150 - 24 Mar 2022

Viewed by 1969

Abstract

The article is devoted to the issue of construction of an optimal with respect to order passive algorithms for evaluating Cauchy and Hilbert singular and hypersingular integrals with oscillating kernels. We propose a method for estimating lower bound errors of quadrature formulas for [...] Read more.

The article is devoted to the issue of construction of an optimal with respect to order passive algorithms for evaluating Cauchy and Hilbert singular and hypersingular integrals with oscillating kernels. We propose a method for estimating lower bound errors of quadrature formulas for singular and hypersingular integral evaluation. Quadrature formulas were constructed for implementation of the obtained estimates. We constructed quadrature formulas and estimated the errors for hypersingular integrals with oscillating kernels. This method is based on using similar results obtained for singular integrals. Full article

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12 pages, 633 KiB

Open AccessArticle

Some Notes for Two Generalized Trigonometric Families of Distributions

by Maria T. Vasileva

Axioms 2022, 11(4), 149; https://doi.org/10.3390/axioms11040149 - 24 Mar 2022

Cited by 4 | Viewed by 1936

Abstract

The paper deals with two general families of cumulative distribution functions based on arctangent function. We provide analysis of the error of the best one-sided Hausdorff approximation for some special cases of these families. We obtain precious estimates for the value of the [...] Read more.

The paper deals with two general families of cumulative distribution functions based on arctangent function. We provide analysis of the error of the best one-sided Hausdorff approximation for some special cases of these families. We obtain precious estimates for the value of the Hausdorff distance that can be used as an additional criterion in practice. Further, the family of recurrence generated adaptive functions is constructed and investigated. All new results are illustrated with suitable numerical experiments. Simple dynamic software modules show applicability of Hausdorff approximation. Full article

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23 pages, 531 KiB

Open AccessArticle

The Effect of Financial Market Factors on House Prices: An Expected Utility Three-Asset Approach

by Yehui Wang, Jianxu Liu, Zhaolin Qiu and Songsak Sriboonchitta

Axioms 2022, 11(4), 145; https://doi.org/10.3390/axioms11040145 - 22 Mar 2022

Cited by 1 | Viewed by 2128

Abstract

This study aimed to theoretically identify the impact factors of the financial market on house prices. Developed upon the two-asset model and with the consideration of risky financial assets, our three-asset model reveals a new derivation of house prices. Compared with the two-asset [...] Read more.

This study aimed to theoretically identify the impact factors of the financial market on house prices. Developed upon the two-asset model and with the consideration of risky financial assets, our three-asset model reveals a new derivation of house prices. Compared with the two-asset model, the newly emerged term is similar to the Sharpe

β

; therefore, it is a risk premium term. Based on China’s 2001–2018 panel data, theoretical derivations are examined. However, the short-term effect of this risk term on house prices is practically small. Given the nonlinear pattern, the long-term effect of the risk term is checked by repeated stochastic simulation. The results imply the following: (i) real house prices are nonlinearly affected by three financial market factors, namely, the expected financial market return, financial market volatility, and the correlation between housing and financial markets; (ii) the correlation determines the signs and the significance of the effects of the other two factors; and (iii) the naturally changed correlation causes periodic house price fluctuations. Therefore, to stabilize real house prices, it is recommended that the government control the money flow between the two markets. Full article

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6 pages, 235 KiB

Open AccessArticle

On the Validity of Onsager Reciprocal Relations (ORR) for Heat Transfer in Anisotropic Solids

by Raj Kumar Arya and George D. Verros

Axioms 2022, 11(3), 104; https://doi.org/10.3390/axioms11030104 - 25 Feb 2022

Cited by 1 | Viewed by 2569

Abstract

In this work, we investigate the validity of axioms such as Onsager Reciprocal Relations (ORR) for heat transfer in irreversible thermodynamics close to equilibrium. We show that the ORR for this case could be directly derived by introducing the widely accepted concept of [...] Read more.

In this work, we investigate the validity of axioms such as Onsager Reciprocal Relations (ORR) for heat transfer in irreversible thermodynamics close to equilibrium. We show that the ORR for this case could be directly derived by introducing the widely accepted concept of heat transfer coefficients into the entropy production rate and by assuming that the thermal conductivity coefficients are uniquely defined. It is believed that this work can not only be used for pedagogical purposes but may also be generalized to other processes beyond heat transfer, thus leading to a generalized framework for transport phenomena and irreversible thermodynamics. Full article

14 pages, 1787 KiB

Open AccessArticle

Generalized Beta Prime Distribution Applied to Finite Element Error Approximation

by Joël Chaskalovic and Franck Assous

Axioms 2022, 11(3), 84; https://doi.org/10.3390/axioms11030084 - 22 Feb 2022

Cited by 1 | Viewed by 1716

Abstract

In this paper, we propose a new family of probability laws based on the Generalized Beta Prime distribution to evaluate the relative accuracy between two Lagrange finite elements

P_{k_{1}}

and

P_{k_{2}}, (k_{1} < k_{2})

[...] Read more.

In this paper, we propose a new family of probability laws based on the Generalized Beta Prime distribution to evaluate the relative accuracy between two Lagrange finite elements

P_{k_{1}}

and

P_{k_{2}}, (k_{1} < k_{2})

. Usually, the relative finite element accuracy is based on the comparison of the asymptotic speed of convergence, when the mesh size h goes to zero. The new probability laws we propose here highlight that there exists, depending on h, cases where the

P_{k_{1}}

finite element is more likely accurate than the

P_{k_{2}}

element. To confirm this assertion, we highlight, using numerical examples, the quality of the fit between the statistical frequencies and the corresponding probabilities, as determined by the probability law. This illustrates that, when h goes away from zero, a finite element

P_{k_{1}}

may produce more precise results than a finite element

P_{k_{2}}

, since the probability of the event “

P_{k_{1}}

is more accurate than

P_{k_{2}}

” becomes greater than 0.5. In these cases, finite element

P_{k_{2}}

is more likely overqualified. Full article

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9 pages, 920 KiB

Open AccessArticle

Some New Results for a Class of Multivalued Interpolative Kannan-Type Contractions

by Nabanita Konwar, Rekha Srivastava, Pradip Debnath and Hari Mohan Srivastava

Axioms 2022, 11(2), 76; https://doi.org/10.3390/axioms11020076 - 15 Feb 2022

Cited by 2 | Viewed by 2578

Abstract

In this paper, we introduce the notion of multivalued interpolative Kannan-type contractions. We also introduce a more general version of this notion by relaxing the degrees of freedom of the powers arising in the contractive condition. Gaba et al. (2021) recently pointed out [...] Read more.

In this paper, we introduce the notion of multivalued interpolative Kannan-type contractions. We also introduce a more general version of this notion by relaxing the degrees of freedom of the powers arising in the contractive condition. Gaba et al. (2021) recently pointed out a significant error in the paper of Gaba and Karapinar (2019), showing that a particular type of generalized interpolative Kannan-type contraction does not posses a fixed point in general in a complete metric space. Thus, the study of generalized Kannan-type mappings remains an interesting and mathematically challenging area of research. The main aim of this article is to address such existing results for multivalued mappings. We also investigate common fixed points for this type of contractions. Our results extend and unify some existing results in the literature. Full article

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10 pages, 608 KiB

Open AccessArticle

Logarithm of a Non-Singular Complex Matrix via the Dunford–Taylor Integral

by Diego Caratelli and Paolo Emilio Ricci

Axioms 2022, 11(2), 51; https://doi.org/10.3390/axioms11020051 - 27 Jan 2022

Cited by 1 | Viewed by 2439

Abstract

Using the Dunford–Taylor integral and a representation formula for the resolvent of a non-singular complex matrix, we find the logarithm of a non-singular complex matrix applying the Cauchy’s residue theorem if the matrix eigenvalues are known or a circuit integral extended to a [...] Read more.

Using the Dunford–Taylor integral and a representation formula for the resolvent of a non-singular complex matrix, we find the logarithm of a non-singular complex matrix applying the Cauchy’s residue theorem if the matrix eigenvalues are known or a circuit integral extended to a curve surrounding the spectrum. The logarithm function that can be found using this technique is essentially unique. To define a version of the logarithm with multiple values analogous to the one existing in the case of complex variables, we introduce a definition for the argument of a matrix, showing the possibility of finding equations similar to those of the scalar case. In the last section, numerical experiments performed by the first author, using the computer algebra program Mathematica©, confirm the effectiveness of this methodology. They include the logarithm of matrices of the fifth, sixth and seventh order. Full article

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55 pages, 544 KiB

Open AccessArticle

An Index for Graphs and Graph Groupoids

by Ilwoo Cho and Palle Jorgensen

Axioms 2022, 11(2), 47; https://doi.org/10.3390/axioms11020047 - 25 Jan 2022

Cited by 3 | Viewed by 2298

Abstract

In this paper, we consider certain quantities that arise in the images of the so-called graph-tree indexes of graph groupoids. In text, the graph groupoids are induced by connected finite-directed graphs with more than one vertex. If a graph groupoid

G

G contains [...] Read more.

In this paper, we consider certain quantities that arise in the images of the so-called graph-tree indexes of graph groupoids. In text, the graph groupoids are induced by connected finite-directed graphs with more than one vertex. If a graph groupoid

G

G contains at least one loop-reduced finite path, then the order of

G

is infinity; hence, the canonical groupoid index

[G : K]

of the inclusion

K \subseteq G

is either ∞ or 1 (under the definition and a natural axiomatization) for the graph groupoids

K

of all “parts” K of G. A loop-reduced finite path generates a semicircular element in graph groupoid algebra. Thus, the existence of semicircular systems acting on the free-probabilistic structure of a given graph G is guaranteed by the existence of loop-reduced finite paths in

G

. The non-semicircularity induced by graphs yields a new index-like notion called the graph-tree index

Γ

of

G

. We study the connections between our graph-tree index and non-semicircular cases. Hence, non-semicircularity also yields the classification of our graphs in terms of a certain type of trees. As an application, we construct towers of graph-groupoid-inclusions which preserve the graph-tree index. We further show that such classification applies to monoidal operads. Full article

8 pages, 683 KiB

Open AccessCommunication

Approximate Flow Friction Factor: Estimation of the Accuracy Using Sobol’s Quasi-Random Sampling

by Pavel Praks and Dejan Brkić

Axioms 2022, 11(2), 36; https://doi.org/10.3390/axioms11020036 - 19 Jan 2022

Cited by 5 | Viewed by 2771

Abstract

The unknown friction factor from the implicit Colebrook equation cannot be expressed explicitly in an analytical way, and therefore to simplify the calculation, many explicit approximations can be used instead. The accuracy of such approximations should be evaluated only throughout the domain of [...] Read more.

The unknown friction factor from the implicit Colebrook equation cannot be expressed explicitly in an analytical way, and therefore to simplify the calculation, many explicit approximations can be used instead. The accuracy of such approximations should be evaluated only throughout the domain of interest in engineering practice where the number of test points can be chosen in many different ways, using uniform, quasi-uniform, random, and quasi-random patterns. To avoid picking points with undetected errors, a sufficient minimal number of such points should be chosen, and they should be distributed using proper patterns. A properly chosen pattern can minimize the required number of testing points that are sufficient to detect maximums of the error. The ability of the Sobol quasi-random vs. random distribution of testing points to capture the maximal relative error using a sufficiently small number of samples is evaluated. Sobol testing points that are quasi-randomly distributed can cover the domain of interest more evenly, avoiding large gaps. Sobol sequences are quasi-random and are always the same, which allows the exact repetition of scientific results. Full article

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2021

Jump to: 2022, 2020, 2019, 2018

9 pages, 321 KiB

Open AccessArticle

Baire-Type Properties in Metrizable c₀(Ω, X)

by Salvador López-Alfonso, Manuel López-Pellicer and Santiago Moll-López

Axioms 2022, 11(1), 6; https://doi.org/10.3390/axioms11010006 - 23 Dec 2021

Cited by 2 | Viewed by 2403

Abstract

Ferrando and Lüdkovsky proved that for a non-empty set

Ω

and a normed space X, the normed space

c_{0} (Ω, X)

is barrelled, ultrabornological, or unordered Baire-like if and only if X is, respectively, barrelled, ultrabornological, or unordered [...] Read more.

Ferrando and Lüdkovsky proved that for a non-empty set

Ω

and a normed space X, the normed space

c_{0} (Ω, X)

is barrelled, ultrabornological, or unordered Baire-like if and only if X is, respectively, barrelled, ultrabornological, or unordered Baire-like. When X is a metrizable locally convex space, with an increasing sequence of semi-norms

\{{∥.∥}_{n} \in N\}

defining its topology, then

c_{0} (Ω, X)

is the metrizable locally convex space over the field

K

(of the real or complex numbers) of all functions

f : Ω \to X

such that for each

ε > 0

and

n \in N

the set

\{ω \in Ω : {∥f (ω)∥}_{n} > ε\}

is finite or empty, with the topology defined by the semi-norms

{∥f∥}_{n} = sup \{{∥f (ω)∥}_{n} : ω \in Ω\}

,

n \in N

. Kąkol, López-Pellicer and Moll-López also proved that the metrizable space

c_{0} (Ω, X)

is quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class p if and only if X is, respectively, quasi barrelled, barrelled, ultrabornological, bornological, unordered Baire-like, totally barrelled, and barrelled of class p. The main result of this paper is that the metrizable

c_{0} (Ω, X)

is baireled if and only if X is baireled, and its proof is divided in several lemmas, with the aim of making it easier to read. An application of this result to closed graph theorem, and two open problems are also presented. Full article

7 pages, 278 KiB

Open AccessArticle

Applying Set Theory

by Saharon Shelah

Axioms 2021, 10(4), 329; https://doi.org/10.3390/axioms10040329 - 30 Nov 2021

Viewed by 2410

Abstract

We prove some results in set theory as applied to general topology and model theory. In particular, we study

ℵ_{1}

-collectionwise Hausdorff, Chang Conjecture for logics with Malitz-Magidor quantifiers and monadic logic of the real line by odd/even Cantor sets. Full article

11 pages, 256 KiB

Open AccessArticle

A Relation-Theoretic Metrical Fixed Point Theorem for Rational Type Contraction Mapping with an Application

by Asik Hossain, Faizan Ahmad Khan and Qamrul Haq Khan

Axioms 2021, 10(4), 316; https://doi.org/10.3390/axioms10040316 - 23 Nov 2021

Cited by 8 | Viewed by 2339

Abstract

In this article, we discuss the relation theoretic aspect of rational type contractive mapping to obtain fixed point results in a complete metric space under arbitrary binary relation. Furthermore, we provide an application to find a solution to a non-linear integral equation. Full article

38 pages, 961 KiB

Open AccessArticle

Spatial Statistical Models: An Overview under the Bayesian Approach

by Francisco Louzada, Diego Carvalho do Nascimento and Osafu Augustine Egbon

Axioms 2021, 10(4), 307; https://doi.org/10.3390/axioms10040307 - 17 Nov 2021

Cited by 14 | Viewed by 9047

Abstract

Spatial documentation is exponentially increasing given the availability of Big Data in the Internet of Things, enabled by device miniaturization and data storage capacity. Bayesian spatial statistics is a useful statistical tool to determine the dependence structure and hidden patterns in space [...] Read more.

Spatial documentation is exponentially increasing given the availability of Big Data in the Internet of Things, enabled by device miniaturization and data storage capacity. Bayesian spatial statistics is a useful statistical tool to determine the dependence structure and hidden patterns in space through prior knowledge and data likelihood. However, this class of modeling is not yet well explored when compared to adopting classification and regression in machine-learning models, in which the assumption of the spatiotemporal independence of the data is often made, that is an inexistent or very weak dependence. Thus, this systematic review aims to address the main models presented in the literature over the past 20 years, identifying the gaps and research opportunities. Elements such as random fields, spatial domains, prior specification, the covariance function, and numerical approximations are discussed. This work explores the two subclasses of spatial smoothing: global and local. Full article

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16 pages, 344 KiB

Open AccessArticle

On the Discrete Weibull Marshall–Olkin Family of Distributions: Properties, Characterizations, and Applications

by Jiju Gillariose, Oluwafemi Samson Balogun, Ehab M. Almetwally, Rehan Ahmad Khan Sherwani, Farrukh Jamal and Joshin Joseph

Axioms 2021, 10(4), 287; https://doi.org/10.3390/axioms10040287 - 30 Oct 2021

Cited by 12 | Viewed by 2244

Abstract

In this article, we introduce a new flexible discrete family of distributions, which accommodates wide collection of monotone failure rates. A sub-model of geometric distribution or a discrete generalization of the exponential model is proposed as a special case of the derived family. [...] Read more.

In this article, we introduce a new flexible discrete family of distributions, which accommodates wide collection of monotone failure rates. A sub-model of geometric distribution or a discrete generalization of the exponential model is proposed as a special case of the derived family. Besides, we point out a comprehensive record of some of its mathematical properties. Two distinct estimation methods for parameters estimation and two different methods for constructing confidence intervals are explored for the proposed distribution. In addition, three extensive Monte Carlo simulations studies are conducted to assess the advantages between estimation methods. Finally, the utility of the new model is embellished by dint of two real datasets. Full article

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19 pages, 319 KiB

Open AccessArticle

Some New Post-Quantum Integral Inequalities Involving Twice (p, q)-Differentiable ψ-Preinvex Functions and Applications

by Miguel Vivas-Cortez, Muhammad Uzair Awan, Sadia Talib, Artion Kashuri and Muhammad Aslam Noor

Axioms 2021, 10(4), 283; https://doi.org/10.3390/axioms10040283 - 29 Oct 2021

Cited by 4 | Viewed by 1947

Abstract

The main motivation of this article is derive a new post-quantum integral identity using twice (p, q)-differentiable functions. Using the identity as an auxiliary result, we will obtain some new variants of Hermite–Hadamard’s inequality essentially via the class of ψ-preinvex functions. To [...] Read more.

The main motivation of this article is derive a new post-quantum integral identity using twice (p, q)-differentiable functions. Using the identity as an auxiliary result, we will obtain some new variants of Hermite–Hadamard’s inequality essentially via the class of ψ-preinvex functions. To support our results, we offer some applications to a special means of positive real numbers and twice (p, q)-differentiable functions that are in absolute value bounded as well. Full article

17 pages, 775 KiB

Open AccessArticle

Is Football/Soccer Purely Stochastic, Made Out of Luck, or Maybe Predictable? How Does Bayesian Reasoning Assess Sports?

by Leonardo Barrios Blanco, Paulo Henrique Ferreira, Francisco Louzada and Diego Carvalho do Nascimento

Axioms 2021, 10(4), 276; https://doi.org/10.3390/axioms10040276 - 26 Oct 2021

Cited by 2 | Viewed by 3550

Abstract

Predicting the game score is a well-explored duty, using mathematical/statistical models. Nonetheless, by adopting a Bayesian methodology, this study aimed to estimate probabilistically the Chilean Premier League teams’ position, considering them a hierarchical structure. This approach enabled the evaluation of the main Chilean [...] Read more.

Predicting the game score is a well-explored duty, using mathematical/statistical models. Nonetheless, by adopting a Bayesian methodology, this study aimed to estimate probabilistically the Chilean Premier League teams’ position, considering them a hierarchical structure. This approach enabled the evaluation of the main Chilean championship that provides the major soccer players for the national team. Thus, a countable (Poisson) regression structure was considered to explain each match as a combination of home advantage, added to the power of attack and defense of each team and considering their performance in the championship as an independent game. We were able to quantify the relationship across the defense and attack of each team and, in addition, were able to group/verify the performance of the entirety of the 2020 Chilean Premier League. For the model validation, we saved the last five games for the model prediction and we found that, in this league, the teams presented a statistical significance in the attack factors, which influences the scores (goals); however, all the teams showed low defense power and we have also found that playing at home or away did not present a game advantage. Our model was able to predict the Chilean league position table, with precision on the top five positions, and from the 6–11 positions there was a small shift (close performance in the championship) caused by the similarity of the expected number of goals, which implied the same position on the rank. This type of model has been shown to be very competitive for the soccer championship prediction. Full article

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17 pages, 792 KiB

Open AccessArticle

A Meta-Analysis for Simultaneously Estimating Individual Means with Shrinkage, Isotonic Regression and Pretests

by Nanami Taketomi, Yoshihiko Konno, Yuan-Tsung Chang and Takeshi Emura

Axioms 2021, 10(4), 267; https://doi.org/10.3390/axioms10040267 - 20 Oct 2021

Cited by 11 | Viewed by 3030

Abstract

Meta-analyses combine the estimators of individual means to estimate the common mean of a population. However, the common mean could be undefined or uninformative in some scenarios where individual means are “ordered” or “sparse”. Hence, assessments of individual means become relevant, rather than [...] Read more.

Meta-analyses combine the estimators of individual means to estimate the common mean of a population. However, the common mean could be undefined or uninformative in some scenarios where individual means are “ordered” or “sparse”. Hence, assessments of individual means become relevant, rather than the common mean. In this article, we propose simultaneous estimation of individual means using the James–Stein shrinkage estimators, which improve upon individual studies’ estimators. We also propose isotonic regression estimators for ordered means, and pretest estimators for sparse means. We provide theoretical explanations and simulation results demonstrating the superiority of the proposed estimators over the individual studies’ estimators. The proposed methods are illustrated by two datasets: one comes from gastric cancer patients and the other from COVID-19 patients. Full article

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31 pages, 421 KiB

Open AccessArticle

Can You Hear the Shape of a Market? Geometric Arbitrage and Spectral Theory

by Simone Farinelli and Hideyuki Takada

Axioms 2021, 10(4), 242; https://doi.org/10.3390/axioms10040242 - 28 Sep 2021

Cited by 2 | Viewed by 2489

Abstract

Utilizing gauge symmetries, the Geometric Arbitrage Theory reformulates any asset model, allowing for arbitrage by means of a stochastic principal fibre bundle with a connection whose curvature measures the “instantaneous arbitrage capability”. The cash flow bundle is the associated vector bundle. The zero [...] Read more.

Utilizing gauge symmetries, the Geometric Arbitrage Theory reformulates any asset model, allowing for arbitrage by means of a stochastic principal fibre bundle with a connection whose curvature measures the “instantaneous arbitrage capability”. The cash flow bundle is the associated vector bundle. The zero eigenspace of its connection Laplacian parameterizes all risk-neutral measures equivalent to the statistical one. A market satisfies the No-Free-Lunch-with-Vanishing-Risk (NFLVR) condition if and only if 0 is in the discrete spectrum of the Laplacian. The Jarrow–Protter–Shimbo theory of asset bubbles and their classification and decomposition extend to markets not satisfying the NFLVR. Euler’s characteristic of the asset nominal space and non-vanishing of the homology group of the cash flow bundle are both topological obstructions to NFLVR. Full article

7 pages, 229 KiB

Open AccessArticle

Revisiting a Classic Identity That Implies the Rogers–Ramanujan Identities II

by Hei-Chi Chan

Axioms 2021, 10(4), 239; https://doi.org/10.3390/axioms10040239 - 27 Sep 2021

Cited by 1 | Viewed by 1689

Abstract

We give a new proof of an identity due to Ramanujan. From this identity, he deduced the famous Rogers–Ramanujan identities. We prove this identity by establishing a simple recursion

J_{k} = q^{k} J_{k - 1}

, where [...] Read more.

We give a new proof of an identity due to Ramanujan. From this identity, he deduced the famous Rogers–Ramanujan identities. We prove this identity by establishing a simple recursion

J_{k} = q^{k} J_{k - 1}

, where

| q | < 1

. This is a sequel to our recent work. Full article

9 pages, 247 KiB

Open AccessArticle

Mellin Transform of Logarithm and Quotient Function with Reducible Quartic Polynomial in Terms of the Lerch Function

by Robert Reynolds and Allan Stauffer

Axioms 2021, 10(3), 236; https://doi.org/10.3390/axioms10030236 - 21 Sep 2021

Viewed by 1703

Abstract

A class of definite integrals involving a quotient function with a reducible polynomial, logarithm and nested logarithm functions are derived with a possible connection to contact problems for a wedge. The derivations are expressed in terms of the Lerch function. Special cases are [...] Read more.

A class of definite integrals involving a quotient function with a reducible polynomial, logarithm and nested logarithm functions are derived with a possible connection to contact problems for a wedge. The derivations are expressed in terms of the Lerch function. Special cases are also derived in terms fundamental constants. The majority of the results in this work are new. Full article

16 pages, 314 KiB

Open AccessArticle

Statistical Riemann and Lebesgue Integrable Sequence of Functions with Korovkin-Type Approximation Theorems

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

Axioms 2021, 10(3), 229; https://doi.org/10.3390/axioms10030229 - 16 Sep 2021

Cited by 9 | Viewed by 2038

Abstract

In this work we introduce and investigate the ideas of statistical Riemann integrability, statistical Riemann summability, statistical Lebesgue integrability and statistical Lebesgue summability via deferred weighted mean. We first establish some fundamental limit theorems connecting these beautiful and potentially useful notions. Furthermore, based [...] Read more.

In this work we introduce and investigate the ideas of statistical Riemann integrability, statistical Riemann summability, statistical Lebesgue integrability and statistical Lebesgue summability via deferred weighted mean. We first establish some fundamental limit theorems connecting these beautiful and potentially useful notions. Furthermore, based upon our proposed techniques, we establish the Korovkin-type approximation theorems with algebraic test functions. Finally, we present two illustrative examples under the consideration of positive linear operators in association with the Bernstein polynomials to exhibit the effectiveness of our findings. Full article

16 pages, 1881 KiB

Open AccessArticle

A Family of the r-Associated Stirling Numbers of the Second Kind and Generalized Bernoulli Polynomials

by Paolo Emilio Ricci, Rekha Srivastava and Pierpaolo Natalini

Axioms 2021, 10(3), 219; https://doi.org/10.3390/axioms10030219 - 9 Sep 2021

Cited by 5 | Viewed by 3056

Abstract

In this article, we derive representation formulas for a class of r-associated Stirling numbers of the second kind and examine their connections with a class of generalized Bernoulli polynomials. Herein, we use the Blissard umbral approach and the familiar Bell polynomials. Links [...] Read more.

In this article, we derive representation formulas for a class of r-associated Stirling numbers of the second kind and examine their connections with a class of generalized Bernoulli polynomials. Herein, we use the Blissard umbral approach and the familiar Bell polynomials. Links with available literature on this subject are also pointed out. The extension to the bivariate case is discussed. Full article

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17 pages, 2266 KiB

Open AccessArticle

Exponentially Convergent Galerkin Method for Numerical Modeling of Lasing in Microcavities with Piercing Holes

by Alexander O. Spiridonov, Anna I. Repina, Ilya V. Ketov, Sergey I. Solov’ev and Evgenii M. Karchevskii

Axioms 2021, 10(3), 184; https://doi.org/10.3390/axioms10030184 - 11 Aug 2021

Cited by 3 | Viewed by 1933

Abstract

The paper investigates an algorithm for the numerical solution of a parametric eigenvalue problem for the Helmholtz equation on the plane specially tailored for the accurate mathematical modeling of lasing modes of microring lasers. The original problem is reduced to a nonlinear eigenvalue [...] Read more.

The paper investigates an algorithm for the numerical solution of a parametric eigenvalue problem for the Helmholtz equation on the plane specially tailored for the accurate mathematical modeling of lasing modes of microring lasers. The original problem is reduced to a nonlinear eigenvalue problem for a system of Muller boundary integral equations. For the numerical solution of the obtained problem, we use a trigonometric Galerkin method, prove its convergence, and derive error estimates in the eigenvalue and eigenfunction approximation. Previous numerical experiments have shown that the method converges exponentially. In the current paper, we prove that if the generalized eigenfunctions are analytic, then the approximate eigenvalues and eigenfunctions exponentially converge to the exact ones as the number of basis functions increases. To demonstrate the practical effectiveness of the algorithm, we find geometrical characteristics of microring lasers that provide a significant increase in the directivity of lasing emission, while maintaining low lasing thresholds. Full article

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22 pages, 2250 KiB

Open AccessArticle

A Method for Visualizing Posterior Probit Model Uncertainty in the Early Prediction of Fraud for Sustainability Development

by Shih-Hsien Tseng and Tien Son Nguyen

Axioms 2021, 10(3), 178; https://doi.org/10.3390/axioms10030178 - 4 Aug 2021

Cited by 1 | Viewed by 2361

Abstract

Corporate fraud is not only curtailed investors’ rights and privileges but also disrupts the overall market economy. For this reason, the formulation of a model that could help detect any unusual market fluctuations would be essential for investors. Thus, we propose an early [...] Read more.

Corporate fraud is not only curtailed investors’ rights and privileges but also disrupts the overall market economy. For this reason, the formulation of a model that could help detect any unusual market fluctuations would be essential for investors. Thus, we propose an early warning system for predicting fraud associated with financial statements based on the Bayesian probit model while examining historical data from 1999 to 2017 with 327 businesses in Taiwan to create a visual method to aid in decision making. In this study, we utilize a parametric estimation via the Markov Chain Monte Carlo (MCMC). The result show that it can reduce over or under-confidence within the decision-making process when standard logistic regression is utilized. In addition, the Bayesian probit model in this study is found to offer more accurate calculations and not only represent the prediction value of the responses but also possible ranges of these responses via a simple plot. Full article

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16 pages, 2546 KiB

Open AccessArticle

Water Particles Monitoring in the Atacama Desert: SPC Approach Based on Proportional Data

by Anderson Fonseca, Paulo Henrique Ferreira, Diego Carvalho do Nascimento, Rosemeire Fiaccone, Christopher Ulloa-Correa, Ayón García-Piña and Francisco Louzada

Axioms 2021, 10(3), 154; https://doi.org/10.3390/axioms10030154 - 13 Jul 2021

Cited by 14 | Viewed by 3748

Abstract

Statistical monitoring tools are well established in the literature, creating organizational cultures such as Six Sigma or Total Quality Management. Nevertheless, most of this literature is based on the normality assumption, e.g., based on the law of large numbers, and brings limitations towards [...] Read more.

Statistical monitoring tools are well established in the literature, creating organizational cultures such as Six Sigma or Total Quality Management. Nevertheless, most of this literature is based on the normality assumption, e.g., based on the law of large numbers, and brings limitations towards truncated processes as open questions in this field. This work was motivated by the register of elements related to the water particles monitoring (relative humidity), an important source of moisture for the Copiapó watershed, and the Atacama region of Chile (the Atacama Desert), and presenting high asymmetry for rates and proportions data. This paper proposes a new control chart for interval data about rates and proportions (symbolic interval data) when they are not results of a Bernoulli process. The unit-Lindley distribution has many interesting properties, such as having only one parameter, from which we develop the unit-Lindley chart for both classical and symbolic data. The performance of the proposed control chart is analyzed using the average run length (ARL), median run length (MRL), and standard deviation of the run length (SDRL) metrics calculated through an extensive Monte Carlo simulation study. Results from the real data applications reveal the tool’s potential to be adopted to estimate the control limits in a Statistical Process Control (SPC) framework. Full article

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19 pages, 707 KiB

Open AccessArticle

Inertial Accelerated Algorithm for Fixed Point of Asymptotically Nonexpansive Mapping in Real Uniformly Convex Banach Spaces

by Murtala Haruna Harbau, Godwin Chidi Ugwunnadi, Lateef Olakunle Jolaoso and Ahmad Abdulwahab

Axioms 2021, 10(3), 147; https://doi.org/10.3390/axioms10030147 - 3 Jul 2021

Cited by 6 | Viewed by 2346

Abstract

In this work, we introduce a new inertial accelerated Mann algorithm for finding a point in the set of fixed points of asymptotically nonexpansive mapping in a real uniformly convex Banach space. We also establish weak and strong convergence theorems of the scheme. [...] Read more.

In this work, we introduce a new inertial accelerated Mann algorithm for finding a point in the set of fixed points of asymptotically nonexpansive mapping in a real uniformly convex Banach space. We also establish weak and strong convergence theorems of the scheme. Finally, we give a numerical experiment to validate the performance of our algorithm and compare with some existing methods. Our results generalize and improve some recent results in the literature. Full article

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11 pages, 286 KiB

Open AccessArticle

Explicit Formulas for Some Infinite ₃F₂(1)-Series

by Kwang-Wu Chen

Axioms 2021, 10(2), 125; https://doi.org/10.3390/axioms10020125 - 19 Jun 2021

Cited by 7 | Viewed by 2293

Abstract

We establish two recurrence relations for some Clausen’s hypergeometric functions with unit argument. We solve them to give the explicit formulas. Additionally, we use the moments of Ramanujan’s generalized elliptic integrals to obtain these recurrence relations. Full article

10 pages, 273 KiB

Open AccessArticle

B-Fredholm Spectra of Drazin Invertible Operators and Applications

by Elvis Aponte, Jhixon Macías, José Sanabria and José Soto

Axioms 2021, 10(2), 111; https://doi.org/10.3390/axioms10020111 - 2 Jun 2021

Cited by 8 | Viewed by 3069

Abstract

In this article, we consider Drazin invertible operators for study of the relationship between their B-Fredholm spectra and the transfer between some of the spectral properties defined through B-Fredholm spectra of this class of operators. Among other results, we investigate the [...] Read more.

In this article, we consider Drazin invertible operators for study of the relationship between their B-Fredholm spectra and the transfer between some of the spectral properties defined through B-Fredholm spectra of this class of operators. Among other results, we investigate the transfer of generalized a-Weyl’s theorem from T to their Drazin inverse S, if it exists. Full article

18 pages, 368 KiB

Open AccessArticle

A Self-Adaptive Algorithm for the Common Solution of the Split Minimization Problem and the Fixed Point Problem

by Nattakarn Kaewyong and Kanokwan Sitthithakerngkiet

Axioms 2021, 10(2), 109; https://doi.org/10.3390/axioms10020109 - 30 May 2021

Cited by 2 | Viewed by 2344

Abstract

In this paper, a new self-adaptive step size algorithm to approximate the solution of the split minimization problem and the fixed point problem of nonexpansive mappings was constructed, which combined the proximal algorithm and a modified Mann’s iterative method with the inertial extrapolation. [...] Read more.

In this paper, a new self-adaptive step size algorithm to approximate the solution of the split minimization problem and the fixed point problem of nonexpansive mappings was constructed, which combined the proximal algorithm and a modified Mann’s iterative method with the inertial extrapolation. The strong convergence theorem was provided in the framework of Hilbert spaces and then proven under some suitable conditions. Our result improved related results in the literature. Moreover, some numerical experiments were also provided to show our algorithm’s consistency, accuracy, and performance compared to the existing algorithms in the literature. Full article

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14 pages, 311 KiB

Open AccessArticle

Note on the Equivalence of Special Norms on the Lebesgue Space

by Maksim V. Kukushkin

Axioms 2021, 10(2), 64; https://doi.org/10.3390/axioms10020064 - 16 Apr 2021

Cited by 1 | Viewed by 1868

Abstract

In this paper, we consider a norm based on the infinitesimal generator of the shift semigroup in a direction. The relevance of such a focus is guaranteed by an abstract representation of a uniformly elliptic operator by means of a composition of the [...] Read more.

In this paper, we consider a norm based on the infinitesimal generator of the shift semigroup in a direction. The relevance of such a focus is guaranteed by an abstract representation of a uniformly elliptic operator by means of a composition of the corresponding infinitesimal generator. The main result of the paper is a theorem establishing equivalence of norms in functional spaces. Even without mentioning the relevance of this result for the constructed theory, we claim it deserves to be considered itself. Full article

13 pages, 297 KiB

Open AccessFeature PaperArticle

Coefficients of a Comprehensive Subclass of Meromorphic Bi-Univalent Functions Associated with the Faber Polynomial Expansion

by Hari Mohan Srivastava, Ahmad Motamednezhad and Safa Salehian

Axioms 2021, 10(1), 27; https://doi.org/10.3390/axioms10010027 - 27 Feb 2021

Cited by 10 | Viewed by 2512

Abstract

In this paper, we introduce a new comprehensive subclass

Σ_{B} (λ, μ, β)

of meromorphic bi-univalent functions in the open unit disk

U

. We also find the upper bounds for the initial Taylor-Maclaurin coefficients [...] Read more.

In this paper, we introduce a new comprehensive subclass

Σ_{B} (λ, μ, β)

of meromorphic bi-univalent functions in the open unit disk

U

. We also find the upper bounds for the initial Taylor-Maclaurin coefficients

| b_{0} |

,

| b_{1} |

and

| b_{2} |

for functions in this comprehensive subclass. Moreover, we obtain estimates for the general coefficients

| b_{n} | (n ≧ 1)

for functions in the subclass

Σ_{B} (λ, μ, β)

by making use of the Faber polynomial expansion method. The results presented in this paper would generalize and improve several recent works on the subject. Full article

7 pages, 249 KiB

Open AccessArticle

Fixed Points of Some Asymptotically Regular Multivalued Mappings Satisfying a Kannan-Type Condition

by Pradip Debnath, Zoran D. Mitrović and Hari Mohan Srivastava

Axioms 2021, 10(1), 24; https://doi.org/10.3390/axioms10010024 - 25 Feb 2021

Cited by 11 | Viewed by 2637

Abstract

In this paper, we establish some existence of fixed-point results for some asymptotically regular multivalued mappings satisfying Kannan-type contractive condition without assuming compactness of the underlying metric space or continuity of the mapping. Full article

30 pages, 685 KiB

Open AccessArticle

Decomposing the Krohn-Rhodes Form of Electroencephalography (EEG) Signals Using Jordan-Chevalley Decomposition Technique

by Amirul Aizad Ahmad Fuad and Tahir Ahmad

Axioms 2021, 10(1), 10; https://doi.org/10.3390/axioms10010010 - 18 Jan 2021

Cited by 2 | Viewed by 2528

Abstract

This paper explores how electroencephalography (EEG) signals in the Krohn-Rhodes form can be decomposed further using the Jordan-Chevalley decomposition technique. First, the recorded EEG signals of a seizure were transformed into a set of matrices. Each of these matrices was decomposed into its [...] Read more.

This paper explores how electroencephalography (EEG) signals in the Krohn-Rhodes form can be decomposed further using the Jordan-Chevalley decomposition technique. First, the recorded EEG signals of a seizure were transformed into a set of matrices. Each of these matrices was decomposed into its elementary components using the Krohn-Rhodes decomposition method. The components were then further decomposed into semisimple and nilpotent matrices using the Jordan-Chevalley decomposition. These matrices—which are the extended building blocks of elementary EEG signals—provide evidence that the EEG signals recorded during a seizure contain patterns similar to that of prime numbers. Full article

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2020

Jump to: 2022, 2021, 2019, 2018

15 pages, 325 KiB

Open AccessArticle

Slice Holomorphic Functions in the Unit Ball Having a Bounded L-Index in Direction

by Andriy Bandura, Maria Martsinkiv and Oleh Skaskiv

Axioms 2021, 10(1), 4; https://doi.org/10.3390/axioms10010004 - 30 Dec 2020

Cited by 7 | Viewed by 2208

Abstract

Let

b \in C^{n} \ {0}

be a fixed direction. We consider slice holomorphic functions of several complex variables in the unit ball, i.e., we study functions that are analytic in the intersection of every slice [...] Read more.

Let

b \in C^{n} \ {0}

be a fixed direction. We consider slice holomorphic functions of several complex variables in the unit ball, i.e., we study functions that are analytic in the intersection of every slice

{z^{0} + t b : t \in C}

with the unit ball

B^{n} = {z \in C^{:} | z | : = \sqrt{{| z |}_{1}^{2} + \dots + {| z_{n} |}^{2}} < 1}

for any

z^{0} \in B^{n}

. For this class of functions, there is introduced a concept of boundedness of L-index in the direction

b

, where

L : B^{n} \to R_{+}

is a positive continuous function such that

L (z) > \frac{β | b |}{1 - | z |},

where

β > 1

is some constant. For functions from this class, we describe a local behavior of modulus of directional derivatives on every ’circle’

{z + t b : | t | = r / L (z)}

with

r \in (0; β],

t \in C,

z \in C^{n}

. It is estimated by the value of the function at the center of the circle. Other propositions concern a connection between the boundedness of L-index in the direction

b

of the slice holomorphic function F and the boundedness of

l_{z}

-index of the slice function

g_{z} (t) = F (z + t b)

with

l_{z} (t) = L (z + t b) .

In addition, we show that every slice holomorphic and joint continuous function in the unit ball has a bounded L-index in direction in any domain compactly embedded in the unit ball and for any continuous function

L : B^{n} \to R_{+} .

Full article

18 pages, 1059 KiB

Open AccessArticle

Approximate Solutions of the Model Describing Fluid Flow Using Generalized ρ-Laplace Transform Method and Heat Balance Integral Method

by Mehmet Yavuz and Ndolane Sene

Axioms 2020, 9(4), 123; https://doi.org/10.3390/axioms9040123 - 24 Oct 2020

Cited by 61 | Viewed by 4199

Abstract

This paper addresses the solution of the incompressible second-grade fluid models. Fundamental qualitative properties of the solution are primarily studied for proving the adequacy of the physical interpretations of the proposed model. We use the Liouville-Caputo fractional derivative with its generalized version that [...] Read more.

This paper addresses the solution of the incompressible second-grade fluid models. Fundamental qualitative properties of the solution are primarily studied for proving the adequacy of the physical interpretations of the proposed model. We use the Liouville-Caputo fractional derivative with its generalized version that gives more comprehensive physical results in the analysis and investigations. In this work, both the

ρ

-Laplace homotopy transform method (

ρ

-LHTM) and the heat balance integral method (HBIM) are successfully combined to solve the fractional incompressible second-grade fluid differential equations. Numerical simulations and their physical interpretations of the mentioned incompressible second-grade fluid model are ensured to illustrate the main findings. It is also proposed that one can recognize the differences in physical analysis of diffusions such as ballistic diffusion, super diffusion, and subdiffusion cases by considering the impact of the orders

ρ

and

φ

. Full article

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8 pages, 235 KiB

Open AccessArticle

Convergence of Weak*-Scalarly Integrable Functions

by Noureddine Sabiri and Mohamed Guessous

Axioms 2020, 9(3), 112; https://doi.org/10.3390/axioms9030112 - 22 Sep 2020

Cited by 2 | Viewed by 2095

Abstract

Let

(Ω, F, μ)

be a complete probability space, E a separable Banach space and

E^{'}

the topological dual vector space of E. We present some compactness results in

L_{E^{'}}^{1} [E]

, the Banach [...] Read more.

Let

(Ω, F, μ)

be a complete probability space, E a separable Banach space and

E^{'}

the topological dual vector space of E. We present some compactness results in

L_{E^{'}}^{1} [E]

, the Banach space of weak*-scalarly integrable

E^{'}

-valued functions. As well we extend the classical theorem of Komlós to the bounded sequences in

L_{E^{'}}^{1} [E]

. Full article

28 pages, 351 KiB

Open AccessArticle

The Existence and Uniqueness of an Entropy Solution to Unilateral Orlicz Anisotropic Equations in an Unbounded Domain

by Omar Benslimane, Ahmed Aberqi and Jaouad Bennouna

Axioms 2020, 9(3), 109; https://doi.org/10.3390/axioms9030109 - 17 Sep 2020

Cited by 13 | Viewed by 2623

Abstract

The purpose of this work is to prove the existence and uniqueness of a class of nonlinear unilateral elliptic problem

(P)

in an arbitrary domain, managed by a low-order term and non-polynomial growth described by an N-uplet of N-function [...] Read more.

The purpose of this work is to prove the existence and uniqueness of a class of nonlinear unilateral elliptic problem

(P)

in an arbitrary domain, managed by a low-order term and non-polynomial growth described by an N-uplet of N-function satisfying the

Δ_{2}

-condition. The source term is merely integrable. Full article

8 pages, 253 KiB

Open AccessArticle

Star-Shapedness of \({\mathcal N}\)-Structures in Euclidean Spaces

by Kyoung Ja Lee, Seok-Zun Song and Young Bae Jun

Axioms 2020, 9(3), 107; https://doi.org/10.3390/axioms9030107 - 12 Sep 2020

Viewed by 2442

Abstract

The notions of (quasi, pseudo) star-shaped sets are introduced, and several related properties are investigated. Characterizations of (quasi) star-shaped sets are considered. The translation of (quasi, pseudo) star-shaped sets are discussed. Unions and intersections of quasi star-shaped sets are conceived. Conditions for a [...] Read more.

The notions of (quasi, pseudo) star-shaped sets are introduced, and several related properties are investigated. Characterizations of (quasi) star-shaped sets are considered. The translation of (quasi, pseudo) star-shaped sets are discussed. Unions and intersections of quasi star-shaped sets are conceived. Conditions for a quasi (or, pseudo) star-shaped set to be a star-shaped set are provided. Full article

9 pages, 251 KiB

Open AccessArticle

A Proof of Komlós Theorem for Super-Reflexive Valued Random Variables

by Abdessamad Dehaj and Mohamed Guessous

Axioms 2020, 9(3), 106; https://doi.org/10.3390/axioms9030106 - 11 Sep 2020

Cited by 3 | Viewed by 2412

Abstract

We give a geometrical proof of Komlós’ theorem for sequences of random variables with values in super-reflexive Banach space. Our approach is inspired by the elementary proof given by Guessous in 1996 for the Hilbert case and uses some geometric properties of smooth [...] Read more.

We give a geometrical proof of Komlós’ theorem for sequences of random variables with values in super-reflexive Banach space. Our approach is inspired by the elementary proof given by Guessous in 1996 for the Hilbert case and uses some geometric properties of smooth spaces. Full article

9 pages, 236 KiB

Open AccessArticle

A New Common Fixed Point Theorem for Three Commuting Mappings

by Meryeme El Harrak and Ahmed Hajji

Axioms 2020, 9(3), 105; https://doi.org/10.3390/axioms9030105 - 11 Sep 2020

Cited by 1 | Viewed by 2206

Abstract

In the present paper, we propose a common fixed point theorem for three commuting mappings via a new contractive condition which generalizes fixed point theorems of Darbo, Hajji and Aghajani et al. An application is also given to illustrate our main result. Moreover, [...] Read more.

In the present paper, we propose a common fixed point theorem for three commuting mappings via a new contractive condition which generalizes fixed point theorems of Darbo, Hajji and Aghajani et al. An application is also given to illustrate our main result. Moreover, several consequences are derived, which are generalizations of Darbo’s fixed point theorem and a Hajji’s result. Full article

8 pages, 228 KiB

Open AccessEditor’s ChoiceArticle

Global Optimization and Common Best Proximity Points for Some Multivalued Contractive Pairs of Mappings

by Pradip Debnath and Hari Mohan Srivastava

Axioms 2020, 9(3), 102; https://doi.org/10.3390/axioms9030102 - 7 Sep 2020

Cited by 21 | Viewed by 2754

Abstract

In this paper, we study a problem of global optimization using common best proximity point of a pair of multivalued mappings. First, we introduce a multivalued Banach-type contractive pair of mappings and establish criteria for the existence of their common best proximity point. [...] Read more.

In this paper, we study a problem of global optimization using common best proximity point of a pair of multivalued mappings. First, we introduce a multivalued Banach-type contractive pair of mappings and establish criteria for the existence of their common best proximity point. Next, we put forward the concept of multivalued Kannan-type contractive pair and also the concept of weak

Δ

-property to determine the existence of common best proximity point for such a pair of maps. Full article

23 pages, 1040 KiB

Open AccessArticle

Feedback Diagram Application for the Generation and Solution of Linear Differential Equations Solvable by Quadrature

by Romeo Pascone and Cathryn Callahan

Axioms 2020, 9(3), 91; https://doi.org/10.3390/axioms9030091 - 29 Jul 2020

Viewed by 2232

Abstract

A novel method for generating and providing quadrature solutions to families of linear, second-order, ordinary differential equations is presented in this paper. It is based upon a comparison of control system feedback diagrams—one representing the system and equation under study and a second [...] Read more.

A novel method for generating and providing quadrature solutions to families of linear, second-order, ordinary differential equations is presented in this paper. It is based upon a comparison of control system feedback diagrams—one representing the system and equation under study and a second equalized to it and providing solutions. The resulting Riccati equation connection between them is utilized to generate and solve groups of equations parameterized by arbitrary functions and constants. This method also leads to a formal solution mechanism for all second-order linear differential equations involving an infinite series of integrals of each equation’s Schwarzian derivative. The practicality of this mechanism is strongly dependent on the series rates of and allowed regions for convergence. The feedback diagram method developed is shown to be equivalent to a comparable method based on the differential equation’s normal form and another relying upon the grouping of terms for a reduction of the equation order, but augmenting their results. Applications are also made to the Helmholtz equation. Full article

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10 pages, 4581 KiB

Open AccessArticle

Mindfulness Model Using Polariton Oscillation in Plasmonic Circuit for Human Performance Management

by Senee Suwandee, Arumona Edward Arumona, Kanad Ray, Phichai Youplao and Preecha Yupapin

Axioms 2020, 9(3), 76; https://doi.org/10.3390/axioms9030076 - 8 Jul 2020

Cited by 2 | Viewed by 3546

Abstract

We have proposed that human life is formed on a space and time function relationship basis, which is distorted after fertilization in the ovum, from which growth is generated by a space–time distortion against the universe’s gravity. A space–time distortion’s reduction can be [...] Read more.

We have proposed that human life is formed on a space and time function relationship basis, which is distorted after fertilization in the ovum, from which growth is generated by a space–time distortion against the universe’s gravity. A space–time distortion’s reduction can be managed by space and time separation, which is known as mindfulness. A space–time distortion in human cells is configured by a polariton traveling in a gold grating film, which can be employed to investigate mindfulness characteristics. Mindfulness is the steady state of the time function of energy after the separation. Energy levels of mindfulness based on polariton aspects are categorized by a quantum number (n), which can be reduced to be a two-level system called Rabi oscillation by a successive filtering method. We have assumed a cell space–time distortion can reduce to reach the original state, which is the stopping state. Mindfulness with a certain frequency energy level of n = 2 was achieved. Several techniques in the practice of mindfulness based on successive filtering called meditation are given and explained, where the required levels of the mindfulness state can be achieved. The criteria of the proposed method are a low energy level (n) and high frequency (f) outputs, which can apply to having a working performance improvement. Full article

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9 pages, 275 KiB

Open AccessArticle

Fixed Point Results under Generalized c-Distance in Cone b-Metric Spaces Over Banach Algebras

by Ataollah Arabnia Firozjah, Hamidreza Rahimi, Manuel De la Sen and Ghasem Soleimani Rad

Axioms 2020, 9(1), 31; https://doi.org/10.3390/axioms9010031 - 21 Mar 2020

Cited by 3 | Viewed by 2546

Abstract

In this work, we define the concept of a generalized c-distance in cone b-metric spaces over a Banach algebra and introduce some its properties. Then, we prove the existence and uniqueness of fixed points for mappings satisfying weak contractive conditions such [...] Read more.

In this work, we define the concept of a generalized c-distance in cone b-metric spaces over a Banach algebra and introduce some its properties. Then, we prove the existence and uniqueness of fixed points for mappings satisfying weak contractive conditions such as Han–Xu-type contraction and Cho-type contraction with respect to this distance. Our assertions are useful, since we remove the continuity condition of the mapping and the normality condition for the cone. Several examples are given to support the main results. Full article

3 pages, 190 KiB

Open AccessEditorial

Mathematical Analysis and Applications II

by Hari M. Srivastava

Axioms 2020, 9(1), 16; https://doi.org/10.3390/axioms9010016 - 6 Feb 2020

Cited by 1 | Viewed by 2205

Abstract

Web Site: http://www [...] Full article

2019

Jump to: 2022, 2021, 2020, 2018

9 pages, 266 KiB

Open AccessArticle

Repeated Derivatives of Hyperbolic Trigonometric Functions and Associated Polynomials

by Giuseppe Dattoli, Silvia Licciardi, Rosa Maria Pidatella and Elio Sabia

Axioms 2019, 8(4), 138; https://doi.org/10.3390/axioms8040138 - 6 Dec 2019

Cited by 1 | Viewed by 3114

Abstract

Elementary problems as the evaluation of repeated derivatives of ordinary transcendent functions can usefully be treated with the use of special polynomials and of a formalism borrowed from combinatorial analysis. Motivated by previous researches in this field, we review the results obtained by [...] Read more.

Elementary problems as the evaluation of repeated derivatives of ordinary transcendent functions can usefully be treated with the use of special polynomials and of a formalism borrowed from combinatorial analysis. Motivated by previous researches in this field, we review the results obtained by other authors and develop a complementary point of view for the repeated derivatives of

sec (.)

,

tan (.)

and for their hyperbolic counterparts. Full article

13 pages, 266 KiB

Open AccessEditor’s ChoiceArticle

Approximation Properties of an Extended Family of the Szász–Mirakjan Beta-Type Operators

by Hari Mohan Srivastava, Gürhan İçöz and Bayram Çekim

Axioms 2019, 8(4), 111; https://doi.org/10.3390/axioms8040111 - 10 Oct 2019

Cited by 27 | Viewed by 3496

Abstract

Approximation and some other basic properties of various linear and nonlinear operators are potentially useful in many different areas of researches in the mathematical, physical, and engineering sciences. Motivated essentially by this aspect of approximation theory, our present study systematically investigates the approximation [...] Read more.

Approximation and some other basic properties of various linear and nonlinear operators are potentially useful in many different areas of researches in the mathematical, physical, and engineering sciences. Motivated essentially by this aspect of approximation theory, our present study systematically investigates the approximation and other associated properties of a class of the Szász-Mirakjan-type operators, which are introduced here by using an extension of the familiar Beta function. We propose to establish moments of these extended Szász-Mirakjan Beta-type operators and estimate various convergence results with the help of the second modulus of smoothness and the classical modulus of continuity. We also investigate convergence via functions which belong to the Lipschitz class. Finally, we prove a Voronovskaja-type approximation theorem for the extended Szász-Mirakjan Beta-type operators. Full article

12 pages, 277 KiB

Open AccessArticle

Slice Holomorphic Functions in Several Variables with Bounded L-Index in Direction

by Andriy Bandura and Oleh Skaskiv

Axioms 2019, 8(3), 88; https://doi.org/10.3390/axioms8030088 - 26 Jul 2019

Cited by 15 | Viewed by 3033

Abstract

In this paper, for a given direction

b \in C^{n} \ {0}

we investigate slice entire functions of several complex variables, i.e., we consider functions which are entire on a complex line [...] Read more.

In this paper, for a given direction

b \in C^{n} \ {0}

we investigate slice entire functions of several complex variables, i.e., we consider functions which are entire on a complex line

{z^{0} + t b : t \in C}

for any

z^{0} \in C^{n}

. Unlike to quaternionic analysis, we fix the direction

b

. The usage of the term slice entire function is wider than in quaternionic analysis. It does not imply joint holomorphy. For example, it allows consideration of functions which are holomorphic in variable

z_{1}

and continuous in variable

z_{2} .

For this class of functions there is introduced a concept of boundedness of L-index in the direction

b

where

L : C^{n} \to R_{+}

is a positive continuous function. We present necessary and sufficient conditions of boundedness of L-index in the direction. In this paper, there are considered local behavior of directional derivatives and maximum modulus on a circle for functions from this class. Also, we show that every slice holomorphic and joint continuous function has bounded L-index in direction in any bounded domain and for any continuous function

L : C^{n} \to R_{+} .

Full article

21 pages, 312 KiB

Open AccessArticle

On a New Class of Laplace-Type Integrals Involving Generalized Hypergeometric Functions

by Wolfram Koepf, Insuk Kim and Arjun K. Rathie

Axioms 2019, 8(3), 87; https://doi.org/10.3390/axioms8030087 - 26 Jul 2019

Cited by 13 | Viewed by 3793

Abstract

In the theory of generalized hypergeometric functions, classical summation theorems for the series

_{2} F_{1}

,

_{3} F_{2}

,

_{4} F_{3}

,

_{5} F_{4}

and

_{7} F_{6}

play a key role. Very recently, Masjed-Jamei and Koepf established generalizations [...] Read more.

In the theory of generalized hypergeometric functions, classical summation theorems for the series

_{2} F_{1}

,

_{3} F_{2}

,

_{4} F_{3}

,

_{5} F_{4}

and

_{7} F_{6}

play a key role. Very recently, Masjed-Jamei and Koepf established generalizations of the above-mentioned summation theorems. Inspired by their work, the main objective of the paper is to provide a new class of Laplace-type integrals involving generalized hypergeometric functions

_{p} F_{p}

for

p = 2, 3, 4, 5

and 7 in the most general forms. Several new and known cases have also been obtained as special cases of our main findings. Full article

11 pages, 258 KiB

Open AccessArticle

Generalized Hyers–Ulam Stability of the Additive Functional Equation

by Yang-Hi Lee and Gwang Hui Kim

Axioms 2019, 8(2), 76; https://doi.org/10.3390/axioms8020076 - 25 Jun 2019

Cited by 4 | Viewed by 3291

Abstract

We will prove the generalized Hyers–Ulam stability and the hyperstability of the additive functional equation f(x₁ + y₁, x₂ + y₂, …, x_n + y_n) = f(x₁, x [...] Read more.

We will prove the generalized Hyers–Ulam stability and the hyperstability of the additive functional equation f(x₁ + y₁, x₂ + y₂, …, x_n + y_n) = f(x₁, x₂, … x_n) + f(y₁, y₂, …, y_n). By restricting the domain of a mapping f that satisfies the inequality condition used in the assumption part of the stability theorem, we partially generalize the results of the stability theorems of the additive function equations. Full article

12 pages, 276 KiB

Open AccessArticle

On Almost b-Metric Spaces and Related Fixed Point Results

by Nabil Mlaiki, Katarina Kukić, Milanka Gardašević-Filipović and Hassen Aydi

Axioms 2019, 8(2), 70; https://doi.org/10.3390/axioms8020070 - 1 Jun 2019

Cited by 9 | Viewed by 3632

Abstract

In this manuscript, we introduce almost b-metric spaces and prove modifications of fixed point theorems for Reich and Hardy–Rogers type contractions. We present an approach generalizing some fixed point theorems to the case of almost b-metric spaces by reducing almost b [...] Read more.

In this manuscript, we introduce almost b-metric spaces and prove modifications of fixed point theorems for Reich and Hardy–Rogers type contractions. We present an approach generalizing some fixed point theorems to the case of almost b-metric spaces by reducing almost b-metrics to the corresponding b-metrics. Later, we show that this approach can not work for all kinds of contractions. To confirm this, we present a proof in which the contraction condition is such that it cannot be reduced to corresponding b-metrics. Full article

10 pages, 774 KiB

Open AccessArticle

(p, q)-Hermite–Hadamard Inequalities for Double Integral and (p, q)-Differentiable Convex Functions

by Julalak Prabseang, Kamsing Nonlaopon and Jessada Tariboon

Axioms 2019, 8(2), 68; https://doi.org/10.3390/axioms8020068 - 28 May 2019

Cited by 24 | Viewed by 3466

Abstract

The aim of this paper is to establish some new

(p, q)

-calculus of Hermite–Hadamard inequalities for the double integral and refinements of the Hermite–Hadamard inequality for

(p, q)

-differentiable convex functions. [...] Read more.

The aim of this paper is to establish some new

(p, q)

-calculus of Hermite–Hadamard inequalities for the double integral and refinements of the Hermite–Hadamard inequality for

(p, q)

-differentiable convex functions. Full article

12 pages, 276 KiB

Open AccessArticle

Some New Results Involving the Generalized Bose–Einstein and Fermi–Dirac Functions

by Rekha Srivastava, Humera Naaz, Sabeena Kazi and Asifa Tassaddiq

Axioms 2019, 8(2), 63; https://doi.org/10.3390/axioms8020063 - 21 May 2019

Cited by 2 | Viewed by 3254

Abstract

In this paper, we obtain a new series representation for the generalized Bose–Einstein and Fermi–Dirac functions by using fractional Weyl transform. To achieve this purpose, we obtain an analytic continuation for these functions by generalizing the domain of Riemann zeta functions from [...] Read more.

In this paper, we obtain a new series representation for the generalized Bose–Einstein and Fermi–Dirac functions by using fractional Weyl transform. To achieve this purpose, we obtain an analytic continuation for these functions by generalizing the domain of Riemann zeta functions from

(0 < ℜ (s) < 1)

to

(0 < ℜ (s) < μ) .

This leads to fresh insights for a new generalization of the Riemann zeta function. The results are validated by obtaining the classical series representation of the polylogarithm and Hurwitz–Lerch zeta functions as special cases. Fractional derivatives and the relationship of the generalized Bose–Einstein and Fermi–Dirac functions with Apostol–Euler–Nörlund polynomials are established to prove new identities. Full article

10 pages, 262 KiB

Open AccessArticle

A Short Note on Integral Transformations and Conversion Formulas for Sequence Generating Functions

by Maxie D. Schmidt

Axioms 2019, 8(2), 62; https://doi.org/10.3390/axioms8020062 - 19 May 2019

Cited by 1 | Viewed by 4362

Abstract

The purpose of this note is to provide an expository introduction to some more curious integral formulas and transformations involving generating functions. We seek to generalize these results and integral representations which effectively provide a mechanism for converting between a sequence’s ordinary and [...] Read more.

The purpose of this note is to provide an expository introduction to some more curious integral formulas and transformations involving generating functions. We seek to generalize these results and integral representations which effectively provide a mechanism for converting between a sequence’s ordinary and exponential generating function (OGF and EGF, respectively) and vice versa. The Laplace transform provides an integral formula for the EGF-to-OGF transformation, where the reverse OGF-to-EGF operation requires more careful integration techniques. We prove two variants of the OGF-to-EGF transformation integrals from the Hankel loop contour for the reciprocal gamma function and from Fourier series expansions of integral representations for the Hadamard product of two generating functions, respectively. We also suggest several generalizations of these integral formulas and provide new examples along the way. Full article

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12 pages, 260 KiB

Open AccessArticle

Fixed Point Theorems through Modified ω-Distance and Application to Nontrivial Equations

by Tariq Qawasmeh, Abdalla Tallafha and Wasfi Shatanawi

Axioms 2019, 8(2), 57; https://doi.org/10.3390/axioms8020057 - 8 May 2019

Cited by 20 | Viewed by 3127

Abstract

In this manuscript, we utilize the concept of modified

ω

-distance mapping, which was introduced by Alegre and Marin [Alegre, C.; Marin, J. Modified

ω

-distance on quasi metric spaces and fixed point theorems on complete quasi metric spaces. Topol. Appl. 2016, [...] Read more.

In this manuscript, we utilize the concept of modified

ω

-distance mapping, which was introduced by Alegre and Marin [Alegre, C.; Marin, J. Modified

ω

-distance on quasi metric spaces and fixed point theorems on complete quasi metric spaces. Topol. Appl. 2016, 203, 120–129] in 2016 to introduce the notions of

(ω, φ)

-Suzuki contraction and generalized

(ω, φ)

-Suzuki contraction. We employ these notions to prove some fixed point results. Moreover, we introduce an example to show the novelty of our results. Furthermore, we introduce some applications for our results. Full article

15 pages, 238 KiB

Open AccessEditor’s ChoiceReview

Differential Equations for Classical and Non-Classical Polynomial Sets: A Survey

by Paolo Emilio Ricci

Axioms 2019, 8(2), 50; https://doi.org/10.3390/axioms8020050 - 25 Apr 2019

Cited by 2 | Viewed by 2956

Abstract

By using the monomiality principle and general results on Sheffer polynomial sets, the differential equation satisfied by several old and new polynomial sets is shown. Full article

6 pages, 257 KiB

Open AccessArticle

A Note on the Displacement Problem of Elastostatics with Singular Boundary Values

by Alfonsina Tartaglione

Axioms 2019, 8(2), 46; https://doi.org/10.3390/axioms8020046 - 19 Apr 2019

Cited by 4 | Viewed by 2624

Abstract

The displacement problem of linear elastostatics in bounded and exterior domains with a non-regular boundary datum

a

is considered. Precisely, if the elastic body is represented by a domain of class

C^{k} (k \geq 2)

of

R^{3}

and [...] Read more.

The displacement problem of linear elastostatics in bounded and exterior domains with a non-regular boundary datum

a

is considered. Precisely, if the elastic body is represented by a domain of class

C^{k} (k \geq 2)

of

R^{3}

and

a \in W^{2 - k - 1 / q, q} (\partial Ω)

,

q \in (1, + \infty)

, then it is proved that there exists a solution which is of class

C^{\infty}

in the interior and takes the boundary value in a well-defined sense. Moreover, it is unique in a natural function class. Full article

15 pages, 258 KiB

Open AccessArticle

Fixed Point Results in Partial Symmetric Spaces with an Application

by Mohammad Asim, A. Rauf Khan and Mohammad Imdad

Axioms 2019, 8(1), 13; https://doi.org/10.3390/axioms8010013 - 22 Jan 2019

Cited by 15 | Viewed by 3286

Abstract

In this paper, we first introduce the class of partial symmetric spaces and then prove some fixed point theorems in such spaces. We use one of the our main results to examine the existence and uniqueness of a solution for a system of [...] Read more.

In this paper, we first introduce the class of partial symmetric spaces and then prove some fixed point theorems in such spaces. We use one of the our main results to examine the existence and uniqueness of a solution for a system of Fredholm integral equations. Furthermore, we introduce an analogue of the Hausdorff metric in the context of partial symmetric spaces and utilize the same to prove an analogue of the Nadler contraction principle in such spaces. Our results extend and improve many results in the existing literature. We also give some examples exhibiting the utility of our newly established results. Full article

9 pages, 217 KiB

Open AccessArticle

A New Identity for Generalized Hypergeometric Functions and Applications

by Mohammad Masjed-Jamei and Wolfram Koepf

Axioms 2019, 8(1), 12; https://doi.org/10.3390/axioms8010012 - 18 Jan 2019

Cited by 5 | Viewed by 3317

Abstract

We establish a new identity for generalized hypergeometric functions and apply it for first- and second-kind Gauss summation formulas to obtain some new summation formulas. The presented identity indeed extends some results of the recent published paper (Some summation theorems for generalized [...] Read more.

We establish a new identity for generalized hypergeometric functions and apply it for first- and second-kind Gauss summation formulas to obtain some new summation formulas. The presented identity indeed extends some results of the recent published paper (Some summation theorems for generalized hypergeometric functions, Axioms, 7 (2018), Article 38). Full article

2018

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9 pages, 221 KiB

Open AccessArticle

Extended Partial S_b-Metric Spaces

by Aiman Mukheimer

Axioms 2018, 7(4), 87; https://doi.org/10.3390/axioms7040087 - 21 Nov 2018

Cited by 5 | Viewed by 3416

Abstract

In this paper, we introduce the concept of extended partial

S_{b}

-metric spaces, which is a generalization of the extended

S_{b}

-metric spaces. Basically, in the triangle inequality, we add a control function with some very interesting properties. These new metric [...] Read more.

In this paper, we introduce the concept of extended partial

S_{b}

-metric spaces, which is a generalization of the extended

S_{b}

-metric spaces. Basically, in the triangle inequality, we add a control function with some very interesting properties. These new metric spaces generalize many results in the literature. Moreover, we prove some fixed point theorems under some different contractions, and some examples are given to illustrate our results. Full article

2 pages, 158 KiB

Open AccessEditorial

Mathematical Analysis and Applications

by Hari M. Srivastava

Axioms 2018, 7(4), 82; https://doi.org/10.3390/axioms7040082 - 12 Nov 2018

Viewed by 2882

Abstract

Website: http://www.math.uvic.ca/faculty/harimsri/ [...] Full article

10 pages, 240 KiB

Open AccessArticle

On the Fixed-Circle Problem and Khan Type Contractions

by Nabil Mlaiki, Nihal Taş and Nihal Yılmaz Özgür

Axioms 2018, 7(4), 80; https://doi.org/10.3390/axioms7040080 - 8 Nov 2018

Cited by 34 | Viewed by 3770

Abstract

In this paper, we consider the fixed-circle problem on metric spaces and give new results on this problem. To do this, we present three types of

F_{C}

-Khan type contractions. Furthermore, we obtain some solutions to an open problem related to the [...] Read more.

In this paper, we consider the fixed-circle problem on metric spaces and give new results on this problem. To do this, we present three types of

F_{C}

-Khan type contractions. Furthermore, we obtain some solutions to an open problem related to the common fixed-circle problem. Full article

17 pages, 789 KiB

Open AccessArticle

Common Fixed Point Theorems for Generalized Geraghty (α,ψ,ϕ)-Quasi Contraction Type Mapping in Partially Ordered Metric-Like Spaces

by Haitham Qawaqneh, Mohd Noorani, Wasfi Shatanawi and Habes Alsamir

Axioms 2018, 7(4), 74; https://doi.org/10.3390/axioms7040074 - 25 Oct 2018

Cited by 17 | Viewed by 3777

Abstract

The aim of this paper is to establish the existence of some common fixed point results for generalized Geraghty

(α, ψ, ϕ)

-quasi contraction self-mapping in partially ordered metric-like spaces. We display an example and an application to show [...] Read more.

The aim of this paper is to establish the existence of some common fixed point results for generalized Geraghty

(α, ψ, ϕ)

-quasi contraction self-mapping in partially ordered metric-like spaces. We display an example and an application to show the superiority of our results. The obtained results progress some well-known fixed (common fixed) point results in the literature. Our main results cannot be specifically attained from the corresponding metric space versions. This paper is scientifically novel because we take Geraghty contraction self-mapping in partially ordered metric-like spaces via

α -

admissible mapping. This opens the door to other possible fixed (common fixed) point results for non-self-mapping and in other generalizing metric spaces. Full article

10 pages, 233 KiB

Open AccessArticle

New Bell–Sheffer Polynomial Sets

by Pierpaolo Natalini and Paolo Emilio Ricci

Axioms 2018, 7(4), 71; https://doi.org/10.3390/axioms7040071 - 8 Oct 2018

Cited by 6 | Viewed by 3365

Abstract

In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers, and several integer sequences related to them, have been studied. The method used in previous articles, and even in the present one, traces back to preceding results [...] Read more.

In recent papers, new sets of Sheffer and Brenke polynomials based on higher order Bell numbers, and several integer sequences related to them, have been studied. The method used in previous articles, and even in the present one, traces back to preceding results by Dattoli and Ben Cheikh on the monomiality principle, showing the possibility to derive explicitly the main properties of Sheffer polynomial families starting from the basic elements of their generating functions. The introduction of iterated exponential and logarithmic functions allows to construct new sets of Bell–Sheffer polynomials which exhibit an iterative character of the obtained shift operators and differential equations. In this context, it is possible, for every integer r, to define polynomials of higher type, which are linked to the higher order Bell-exponential and logarithmic numbers introduced in preceding papers. Connections with integer sequences appearing in Combinatorial analysis are also mentioned. Naturally, the considered technique can also be used in similar frameworks, where the iteration of exponential and logarithmic functions appear. Full article

13 pages, 809 KiB

Open AccessArticle

Periodically Forced Nonlinear Oscillatory Acoustic Vacuum

by Makrina Agaoglou, Michal Fečkan, Michal Pospíšil, Vassilis M. Rothos and Alexander F. Vakakis

Axioms 2018, 7(4), 69; https://doi.org/10.3390/axioms7040069 - 22 Sep 2018

Cited by 1 | Viewed by 3399

Abstract

In this work, we study the in-plane oscillations of a finite lattice of particles coupled by linear springs under distributed harmonic excitation. Melnikov-type analysis is applied for the persistence of periodic oscillations of a reduced system. Full article

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17 pages, 281 KiB

Open AccessArticle

Solutions to Abel’s Integral Equations in Distributions

by Chenkuan Li, Thomas Humphries and Hunter Plowman

Axioms 2018, 7(3), 66; https://doi.org/10.3390/axioms7030066 - 2 Sep 2018

Cited by 6 | Viewed by 4174

Abstract

The goal of this paper is to study fractional calculus of distributions, the generalized Abel’s integral equations, as well as fractional differential equations in the distributional space

D^{'} (R^{+})

based on inverse convolutional operators and Babenko’s approach. Furthermore, we [...] Read more.

The goal of this paper is to study fractional calculus of distributions, the generalized Abel’s integral equations, as well as fractional differential equations in the distributional space

D^{'} (R^{+})

based on inverse convolutional operators and Babenko’s approach. Furthermore, we provide interesting applications of Abel’s integral equations in viscoelastic systems, as well as solving other integral equations, such as

\int_{θ}^{π / 2} \frac{y (φ)}{\cos^{β} φ {(\cos θ - \cos φ)}^{α}} d φ = f (θ)

, and

\int_{0}^{\infty} x^{1 / 2} g (x) y (x + t) d x = f (t)

. Full article

16 pages, 856 KiB

Open AccessArticle

Interior Regularity Estimates for a Degenerate Elliptic Equation with Mixed Boundary Conditions

by Jean-Daniel Djida and Arran Fernandez

Axioms 2018, 7(3), 65; https://doi.org/10.3390/axioms7030065 - 1 Sep 2018

Cited by 3 | Viewed by 3901

Abstract

The Marchaud fractional derivative can be obtained as a Dirichlet-to–Neumann map via an extension problem to the upper half space. In this paper we prove interior Schauder regularity estimates for a degenerate elliptic equation with mixed Dirichlet–Neumann boundary conditions. The degenerate elliptic equation [...] Read more.

The Marchaud fractional derivative can be obtained as a Dirichlet-to–Neumann map via an extension problem to the upper half space. In this paper we prove interior Schauder regularity estimates for a degenerate elliptic equation with mixed Dirichlet–Neumann boundary conditions. The degenerate elliptic equation arises from the Bernardis–Reyes–Stinga–Torrea extension of the Dirichlet problem for the Marchaud fractional derivative. Full article

9 pages, 243 KiB

Open AccessArticle

Umbral Methods and Harmonic Numbers

by Giuseppe Dattoli, Bruna Germano, Silvia Licciardi and Maria Renata Martinelli

Axioms 2018, 7(3), 62; https://doi.org/10.3390/axioms7030062 - 1 Sep 2018

Cited by 7 | Viewed by 3433

Abstract

The theory of harmonic-based functions is discussed here within the framework of umbral operational methods. We derive a number of results based on elementary notions relying on the properties of Gaussian integrals. Full article

14 pages, 487 KiB

Open AccessArticle

Sub-Optimal Control in the Zika Virus Epidemic Model Using Differential Evolution

by Nonthamon Chaikham and Wannika Sawangtong

Axioms 2018, 7(3), 61; https://doi.org/10.3390/axioms7030061 - 23 Aug 2018

Cited by 2 | Viewed by 3616

Abstract

A dynamical model of Zika virus (ZIKV) epidemic with direct transmission, sexual transmission, and vertical transmission is developed. A sub-optimal control problem to counter against the disease is proposed including three controls: vector elimination, vector-to-human contact reduction, and sexual contact reduction. Each control [...] Read more.

A dynamical model of Zika virus (ZIKV) epidemic with direct transmission, sexual transmission, and vertical transmission is developed. A sub-optimal control problem to counter against the disease is proposed including three controls: vector elimination, vector-to-human contact reduction, and sexual contact reduction. Each control variable is discretized into piece-wise constant intervals. The problem is solved by Differential Evolution (DE), which is one of the evolutionary algorithm developed for optimization. Two scenarios, namely four time horizons and eight time horizons, are compared and discussed. The simulations show that models with controls lead to decreasing the number of patients as well as epidemic period length. From the optimal solution, vector elimination is the prioritized strategy for disease control. Full article

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19 pages, 894 KiB

Open AccessArticle

Some Identities for Euler and Bernoulli Polynomials and Their Zeros

by Taekyun Kim and Cheon Seoung Ryoo

Axioms 2018, 7(3), 56; https://doi.org/10.3390/axioms7030056 - 14 Aug 2018

Cited by 45 | Viewed by 4805 | Correction

Abstract

In this paper, we study some special polynomials which are related to Euler and Bernoulli polynomials. In addition, we give some identities for these polynomials. Finally, we investigate the zeros of these polynomials by using the computer. Full article

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11 pages, 273 KiB

Open AccessArticle

A New Type of Generalization on W—Asymptotically J _λ—Statistical Equivalence with the Number of α

by Hafize Gümüş and Nihal Demir

Axioms 2018, 7(3), 54; https://doi.org/10.3390/axioms7030054 - 2 Aug 2018

Cited by 1 | Viewed by 3309

Abstract

In our paper, by using the concept of

W -

asymptotically

J -

statistical equivalence of order

α

which has been previously defined, we present the definitions of

W -

asymptotically

J_{λ} -

statistical equivalence of order

α

,

W -

strongly [...] Read more.

In our paper, by using the concept of

W -

asymptotically

J -

statistical equivalence of order

α

which has been previously defined, we present the definitions of

W -

asymptotically

J_{λ} -

statistical equivalence of order

α

,

W -

strongly asymptotically

J_{λ} -

statistical equivalence of order

α

, and

W -

strongly Cesáro asymptotically

J -

statistical equivalence of order

α

where

0 < α \leq 1

. We also extend these notions with a sequence of positive real numbers,

p = (p_{k})

, and we investigate how our results change if

p

is constant. Full article

18 pages, 2902 KiB

Open AccessArticle

Some Exact Solutions to Non-Fourier Heat Equations with Substantial Derivative

by Konstantin Zhukovsky, Dmitrii Oskolkov and Nadezhda Gubina

Axioms 2018, 7(3), 48; https://doi.org/10.3390/axioms7030048 - 18 Jul 2018

Cited by 11 | Viewed by 4153

Abstract

One-dimensional equations of telegrapher’s-type (TE) and Guyer–Krumhansl-type (GK-type) with substantial derivative considered and operational solutions to them are given. The role of the exponential differential operators is discussed. The examples of their action on some initial functions are explored. Proper solutions are constructed [...] Read more.

One-dimensional equations of telegrapher’s-type (TE) and Guyer–Krumhansl-type (GK-type) with substantial derivative considered and operational solutions to them are given. The role of the exponential differential operators is discussed. The examples of their action on some initial functions are explored. Proper solutions are constructed in the integral form and some examples are studied with solutions in elementary functions. A system of hyperbolic-type inhomogeneous differential equations (DE), describing non-Fourier heat transfer with substantial derivative thin films, is considered. Exact harmonic solutions to these equations are obtained for the Cauchy and the Dirichlet conditions. The application to the ballistic heat transport in thin films is studied; the ballistic properties are accounted for by the Knudsen number. Two-speed heat propagation process is demonstrated—fast evolution of the ballistic quasi-temperature component in low-dimensional systems is elucidated and compared with slow diffusive heat-exchange process. The comparative analysis of the obtained solutions is performed. Full article

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20 pages, 253 KiB

Open AccessArticle

Some Summation Theorems for Generalized Hypergeometric Functions

by Mohammad Masjed-Jamei and Wolfram Koepf

Axioms 2018, 7(2), 38; https://doi.org/10.3390/axioms7020038 - 8 Jun 2018

Cited by 12 | Viewed by 4857

Abstract

Essentially, whenever a generalized hypergeometric series can be summed in terms of gamma functions, the result will be important as only a few such summation theorems are available in the literature. In this paper, we apply two identities of generalized hypergeometric series in [...] Read more.

Essentially, whenever a generalized hypergeometric series can be summed in terms of gamma functions, the result will be important as only a few such summation theorems are available in the literature. In this paper, we apply two identities of generalized hypergeometric series in order to extend some classical summation theorems of hypergeometric functions such as Gauss, Kummer, Dixon, Watson, Whipple, Pfaff–Saalschütz and Dougall formulas and also obtain some new summation theorems in the sequel. Full article

10 pages, 249 KiB

Open AccessArticle

Pre-Metric Spaces Along with Different Types of Triangle Inequalities

by Hsien-Chung Wu

Axioms 2018, 7(2), 34; https://doi.org/10.3390/axioms7020034 - 24 May 2018

Cited by 3 | Viewed by 4285

Abstract

The

T_{1}

-spaces induced by the pre-metric spaces along with many forms of triangle inequalities are investigated in this paper. The limits in pre-metric spaces are also studied to demonstrate the consistency of limit concept in the induced topologies. [...] Read more.

The

T_{1}

-spaces induced by the pre-metric spaces along with many forms of triangle inequalities are investigated in this paper. The limits in pre-metric spaces are also studied to demonstrate the consistency of limit concept in the induced topologies. Full article

13 pages, 281 KiB

Open AccessArticle

Yukawa Potential, Panharmonic Measure and Brownian Motion

by Antti Rasila and Tommi Sottinen

Axioms 2018, 7(2), 28; https://doi.org/10.3390/axioms7020028 - 1 May 2018

Cited by 3 | Viewed by 4625

Abstract

This paper continues our earlier investigation, where a walk-on-spheres (WOS) algorithm for Monte Carlo simulation of the solutions of the Yukawa and the Helmholtz partial differential equations (PDEs) was developed by using the Duffin correspondence. In this paper, we investigate the foundations behind [...] Read more.

This paper continues our earlier investigation, where a walk-on-spheres (WOS) algorithm for Monte Carlo simulation of the solutions of the Yukawa and the Helmholtz partial differential equations (PDEs) was developed by using the Duffin correspondence. In this paper, we investigate the foundations behind the algorithm for the case of the Yukawa PDE. We study the panharmonic measure, which is a generalization of the harmonic measure for the Yukawa PDE. We show that there are natural stochastic definitions for the panharmonic measure in terms of the Brownian motion and that the harmonic and the panharmonic measures are all mutually equivalent. Furthermore, we calculate their Radon–Nikodym derivatives explicitly for some balls, which is a key result behind the WOS algorithm. Full article

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13 pages, 316 KiB

Open AccessArticle

Subordination Properties for Multivalent Functions Associated with a Generalized Fractional Differintegral Operator

by Hanaa M. Zayed, Mohamed Kamal Aouf and Adela O. Mostafa

Axioms 2018, 7(2), 27; https://doi.org/10.3390/axioms7020027 - 24 Apr 2018

Cited by 6 | Viewed by 3884

Abstract

Using of the principle of subordination, we investigate some subordination and convolution properties for classes of multivalent functions under certain assumptions on the parameters involved, which are defined by a generalized fractional differintegral operator under certain assumptions on the parameters involved. Full article

12 pages, 302 KiB

Open AccessArticle

New Definitions about A I -Statistical Convergence with Respect to a Sequence of Modulus Functions and Lacunary Sequences

by Ömer Kişi, Hafize Gümüş and Ekrem Savas

Axioms 2018, 7(2), 24; https://doi.org/10.3390/axioms7020024 - 13 Apr 2018

Cited by 2 | Viewed by 4337

Abstract

In this paper, using an infinite matrix of complex numbers, a modulus function and a lacunary sequence, we generalize the concept of

I

-statistical convergence, which is a recently introduced summability method. The names of our new methods are

A^{I}

-lacunary statistical [...] Read more.

In this paper, using an infinite matrix of complex numbers, a modulus function and a lacunary sequence, we generalize the concept of

I

-statistical convergence, which is a recently introduced summability method. The names of our new methods are

A^{I}

-lacunary statistical convergence and strongly

A^{I}

-lacunary convergence with respect to a sequence of modulus functions. These spaces are denoted by

S_{θ}^{A} (I, F)

and

N_{θ}^{A} (I, F),

respectively. We give some inclusion relations between

S^{A} (I, F),

S_{θ}^{A} (I, F)

and

N_{θ}^{A} (I, F)

. We also investigate Cesáro summability for

A^{I}

and we obtain some basic results between

A^{I}

-Cesáro summability, strongly

A^{I}

-Cesáro summability and the spaces mentioned above. Full article

12 pages, 800 KiB

Open AccessArticle

Special Numbers and Polynomials Including Their Generating Functions in Umbral Analysis Methods

by Yilmaz Simsek

Axioms 2018, 7(2), 22; https://doi.org/10.3390/axioms7020022 - 1 Apr 2018

Cited by 14 | Viewed by 4215

Abstract

In this paper, by applying umbral calculus methods to generating functions for the combinatorial numbers and the Apostol type polynomials and numbers of order k, we derive some identities and relations including the combinatorial numbers, the Apostol-Bernoulli polynomials and numbers of order [...] Read more.

In this paper, by applying umbral calculus methods to generating functions for the combinatorial numbers and the Apostol type polynomials and numbers of order k, we derive some identities and relations including the combinatorial numbers, the Apostol-Bernoulli polynomials and numbers of order k and the Apostol-Euler polynomials and numbers of order k. Moreover, by using p-adic integral technique, we also derive some combinatorial sums including the Bernoulli numbers, the Euler numbers, the Apostol-Euler numbers and the numbers

y_{1} (n, k; λ)

. Finally, we make some remarks and observations regarding these identities and relations. Full article

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