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Axioms, Volume 13, Issue 2 (February 2024) – 62 articles

Cover Story (view full-size image): In this, we propose a method able to model fractional behaviors with convolution models involving non-singular kernels rather than the usual fractional calculus-based models. If the fractional behavior is a pure power law function, a particular rational kernel made of alternate poles and zeros permits accurate fitting on a time range. This fitting solution, similar to the one commonly employed in the frequency domain to approximate fractional integrator behavior, is performed here in the time domain; its leads to fewer gain and phase oscillations for a given number of poles and zeros. This approach also permits physical interpretation to be performed in terms of the delay distributions close to the probabilistic interpretation of the fractional behaviors involved in phenomena such as diffusion, adsorption or aggregation. View this paper
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21 pages, 381 KiB  
Article
The Fourier–Legendre Series of Bessel Functions of the First Kind and the Summed Series Involving 1F2 Hypergeometric Functions That Arise from Them
by Jack C. Straton
Axioms 2024, 13(2), 134; https://doi.org/10.3390/axioms13020134 - 19 Feb 2024
Cited by 1 | Viewed by 853
Abstract
The Bessel function of the first kind JNkx is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind INkx. The purpose of these expansions in Legendre polynomials was not an [...] Read more.
The Bessel function of the first kind JNkx is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind INkx. The purpose of these expansions in Legendre polynomials was not an attempt to rival established numerical methods for calculating Bessel functions but to provide a form for JNkx useful for analytical work in the area of strong laser fields, where analytical integration over scattering angles is essential. Despite their primary purpose, one can easily truncate the series at 21 terms to provide 33-digit accuracy that matches the IEEE extended precision in some compilers. The analytical theme is furthered by showing that infinite series of like-powered contributors (involving  1F2 hypergeometric functions) extracted from the Fourier–Legendre series may be summed, having values that are inverse powers of the eight primes 1/2i3j5k7l11m13n17o19p multiplying powers of the coefficient k. Full article
19 pages, 410 KiB  
Article
Analytical Solution of Generalized Bratu-Type Fractional Differential Equations Using the Homotopy Perturbation Transform Method
by Ghaliah Alhamzi, Aafrin Gouri, Badr Saad T. Alkahtani and Ravi Shanker Dubey
Axioms 2024, 13(2), 133; https://doi.org/10.3390/axioms13020133 - 19 Feb 2024
Viewed by 773
Abstract
In this study, we present the generalized form of the higher-order nonlinear fractional Bratu-type equation. In this generalization, we deal with a generalized fractional derivative, which is quite useful from an application point of view. Furthermore, some special cases of the generalized fractional [...] Read more.
In this study, we present the generalized form of the higher-order nonlinear fractional Bratu-type equation. In this generalization, we deal with a generalized fractional derivative, which is quite useful from an application point of view. Furthermore, some special cases of the generalized fractional Bratu equation are recognized and examined. To solve these nonlinear differential equations of fractional order, we employ the homotopy perturbation transform method. This work presents a useful computational method for solving these equations and advances our understanding of them. We also plot some numerical outcomes to show the efficiency of the obtained results. Full article
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16 pages, 318 KiB  
Article
An Algorithm Based on Compute Unified Device Architecture for Estimating Covering Functionals of Convex Bodies
by Xiangyang Han, Senlin Wu and Longzhen Zhang
Axioms 2024, 13(2), 132; https://doi.org/10.3390/axioms13020132 - 19 Feb 2024
Viewed by 745
Abstract
In Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a longstanding open problem from Convex and Discrete Geometry, it is essential to estimate covering functionals of convex bodies effectively. Recently, He et al. and Yu et al. provided two deterministic global [...] Read more.
In Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a longstanding open problem from Convex and Discrete Geometry, it is essential to estimate covering functionals of convex bodies effectively. Recently, He et al. and Yu et al. provided two deterministic global optimization algorithms having high computational complexity for this purpose. Since satisfactory estimations of covering functionals will be sufficient in Zong’s program, we propose a stochastic global optimization algorithm based on CUDA and provide an error estimation for the algorithm. The accuracy of our algorithm is tested by comparing numerical and exact values of covering functionals of convex bodies including the Euclidean unit disc, the three-dimensional Euclidean unit ball, the regular tetrahedron, and the regular octahedron. We also present estimations of covering functionals for the regular dodecahedron and the regular icosahedron. Full article
(This article belongs to the Special Issue Advances in Convex Geometry and Analysis)
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16 pages, 305 KiB  
Article
Study of Uniqueness and Ulam-Type Stability of Abstract Hadamard Fractional Differential Equations of Sobolev Type via Resolvent Operators
by Khellaf Ould Melha, Abdelhamid Mohammed Djaouti, Muhammad Amer Latif and Vaijanath L. Chinchane
Axioms 2024, 13(2), 131; https://doi.org/10.3390/axioms13020131 - 19 Feb 2024
Viewed by 1766
Abstract
This paper focuses on studying the uniqueness of the mild solution for an abstract fractional differential equation. We use Banach’s fixed point theorem to prove this uniqueness. Additionally, we examine the stability properties of the equation using Ulam’s stability. To analyze these properties, [...] Read more.
This paper focuses on studying the uniqueness of the mild solution for an abstract fractional differential equation. We use Banach’s fixed point theorem to prove this uniqueness. Additionally, we examine the stability properties of the equation using Ulam’s stability. To analyze these properties, we consider the involvement of Hadamard fractional derivatives. Throughout this study, we put significant emphasis on the role and properties of resolvent operators. Furthermore, we investigate Ulam-type stability by providing examples of partial fractional differential equations that incorporate Hadamard derivatives. Full article
11 pages, 1326 KiB  
Article
Entropy of Difference: A New Tool for Measuring Complexity
by Pasquale Nardone and Giorgio Sonnino
Axioms 2024, 13(2), 130; https://doi.org/10.3390/axioms13020130 - 19 Feb 2024
Viewed by 725
Abstract
We propose a new tool for estimating the complexity of a time series: the entropy of difference (ED). The method is based solely on the sign of the difference between neighboring values in a time series. This makes it possible to describe the [...] Read more.
We propose a new tool for estimating the complexity of a time series: the entropy of difference (ED). The method is based solely on the sign of the difference between neighboring values in a time series. This makes it possible to describe the signal as efficiently as prior proposed parameters, such as permutation entropy (PE) or modified permutation entropy (mPE). Firstly, this method reduces the size of the sample that is necessary to estimate the parameter value, and secondly it enables the use of the Kullback–Leibler divergence to estimate the “distance” between the time series data and random signals. Full article
(This article belongs to the Section Mathematical Physics)
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17 pages, 1459 KiB  
Article
On the Exact Solution of a Scalar Differential Equation via a Simple Analytical Approach
by Nada A. M. Alshomrani, Abdelhalim Ebaid, Faten Aldosari and Mona D. Aljoufi
Axioms 2024, 13(2), 129; https://doi.org/10.3390/axioms13020129 - 19 Feb 2024
Viewed by 806
Abstract
The existence of the advance parameter in a scalar differential equation prevents the application of the well-known standard methods used for solving classical ordinary differential equations. A simple procedure is introduced in this paper to remove the advance parameter from a special kind [...] Read more.
The existence of the advance parameter in a scalar differential equation prevents the application of the well-known standard methods used for solving classical ordinary differential equations. A simple procedure is introduced in this paper to remove the advance parameter from a special kind of first-order scalar differential equation. The suggested approach transforms the given first-order scalar differential equation to an equivalent second-order ordinary differential equation (ODE) without the advance parameter. Using this method, we are able to construct the exact solution of both the transformed model and the given original model. The exact solution is obtained in a wave form with specified amplitude and phase. Furthermore, several special cases are investigated at certain values/relationships of the involved parameters. It is shown that the exact solution in the absence of the advance parameter reduces to the corresponding solution in the literature. In addition, it is declared that the current model enjoys various kinds of solutions, such as constant solutions, polynomial solutions, and periodic solutions under certain constraints of the included parameters. Full article
(This article belongs to the Special Issue Difference, Functional, and Related Equations)
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9 pages, 952 KiB  
Article
The Average Sentinel of the Heat Equation with an Unknown Reaction
by Houria Selatnia, Abdelhamid Ayadi and Imad Rezzoug
Axioms 2024, 13(2), 128; https://doi.org/10.3390/axioms13020128 - 19 Feb 2024
Viewed by 731
Abstract
In this paper, we analyze the identification of the amount of pollutant discharged problem by each source in a heat system when the dynamics of the state are governed by a parameterized unknown operator. In this way, we introduce the notion of average [...] Read more.
In this paper, we analyze the identification of the amount of pollutant discharged problem by each source in a heat system when the dynamics of the state are governed by a parameterized unknown operator. In this way, we introduce the notion of average sentinel. The decomposition method is used to solve the equation of this problem, the gradient method is used to calculate the averaged control, and the combination of the two methods is used to estimate the pollution terms. Numerical example is given to confirm this result. Full article
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18 pages, 4766 KiB  
Article
Widest Path in Networks with Gains/Losses
by Javad Tayyebi, Mihai-Lucian Rîtan and Adrian Marius Deaconu
Axioms 2024, 13(2), 127; https://doi.org/10.3390/axioms13020127 - 18 Feb 2024
Viewed by 805
Abstract
In this paper, the generalized widest path problem (or generalized maximum capacity problem) is studied. This problem is denoted by the GWPP. The classical widest path problem is to find a path from a source (s) to a sink (t) with the highest [...] Read more.
In this paper, the generalized widest path problem (or generalized maximum capacity problem) is studied. This problem is denoted by the GWPP. The classical widest path problem is to find a path from a source (s) to a sink (t) with the highest capacity among all possible s-t paths. The GWPP takes into account the presence of loss/gain factors on arcs as well. The GWPP aims to find an s-t path considering the loss/gain factors while satisfying the capacity constraints. For solving the GWPP, three strongly polynomial time algorithms are presented. Two algorithms only work in the case of losses. The first one is less efficient than the second one on a CPU, but it proves to be more efficient on large networks if it parallelized on GPUs. The third algorithm is able to deal with the more general case of losses/gains on arcs. An example is considered to illustrate how each algorithm works. Experiments on large networks are conducted to compare the efficiency of the algorithms proposed. Full article
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23 pages, 357 KiB  
Article
Existence and Properties of the Solution of Nonlinear Differential Equations with Impulses at Variable Times
by Huifu Xia, Yunfei Peng and Peng Zhang
Axioms 2024, 13(2), 126; https://doi.org/10.3390/axioms13020126 - 18 Feb 2024
Viewed by 720
Abstract
In this paper, a class of nonlinear ordinary differential equations with impulses at variable times is considered. The existence and uniqueness of the solution are given. At the same time, modifying the classical definitions of continuous dependence and Gâteaux differentiability, some results on [...] Read more.
In this paper, a class of nonlinear ordinary differential equations with impulses at variable times is considered. The existence and uniqueness of the solution are given. At the same time, modifying the classical definitions of continuous dependence and Gâteaux differentiability, some results on the continuous dependence and Gâteaux differentiable of the solution relative to the initial value are also presented in a new topology sense. For the autonomous impulsive system, the periodicity of the solution is given. As an application, the properties of the solution for a type of controlled nonlinear ordinary differential equation with impulses at variable times is obtained. These results are a foundation to study optimal control problems of systems governed by differential equations with impulses at variable times. Full article
(This article belongs to the Special Issue Advances in Difference Equations)
22 pages, 6832 KiB  
Article
A Statistical Model for Count Data Analysis and Population Size Estimation: Introducing a Mixed Poisson–Lindley Distribution and Its Zero Truncation
by Gadir Alomair, Razik Ridzuan Mohd Tajuddin, Hassan S. Bakouch and Amal Almohisen
Axioms 2024, 13(2), 125; https://doi.org/10.3390/axioms13020125 - 17 Feb 2024
Viewed by 1223
Abstract
Count data consists of both observed and unobserved events. The analysis of count data often encounters overdispersion, where traditional Poisson models may not be adequate. In this paper, we introduce a tractable one-parameter mixed Poisson distribution, which combines the Poisson distribution with the [...] Read more.
Count data consists of both observed and unobserved events. The analysis of count data often encounters overdispersion, where traditional Poisson models may not be adequate. In this paper, we introduce a tractable one-parameter mixed Poisson distribution, which combines the Poisson distribution with the improved second-degree Lindley distribution. This distribution, called the Poisson-improved second-degree Lindley distribution, is capable of effectively modeling standard count data with overdispersion. However, if the frequency of the unobserved events is unknown, the proposed distribution cannot be directly used to describe the events. To address this limitation, we propose a modification by truncating the distribution to zero. This results in a tractable zero-truncated distribution that encompasses all types of dispersions. Due to the unknown frequency of unobserved events, the population size as a whole becomes unknown and requires estimation. To estimate the population size, we develop a Horvitz–Thompson-like estimator utilizing truncated distribution. Both the untruncated and truncated distributions exhibit desirable statistical properties. The estimators for both distributions, as well as the population size, are asymptotically unbiased and consistent. The current study demonstrates that both the truncated and untruncated distributions adequately explain the considered medical datasets, which are the number of dicentric chromosomes after being exposed to different doses of radiation and the number of positive Salmonella. Moreover, the proposed population size estimator yields reliable estimates. Full article
(This article belongs to the Special Issue Methods and Applications of Advanced Statistical Analysis)
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10 pages, 285 KiB  
Article
Application of the Concept of Statistical Causality in Integrable Increasing Processes and Measures
by Dragana Valjarević, Vladica Stojanović and Aleksandar Valjarević
Axioms 2024, 13(2), 124; https://doi.org/10.3390/axioms13020124 - 17 Feb 2024
Viewed by 680
Abstract
In this paper, we investigate an application of the statistical concept of causality, based on Granger’s definition of causality, on raw increasing processes as well as on optional and predictable measures. A raw increasing process is optional (predictable) if the bounded (left-continuous) process [...] Read more.
In this paper, we investigate an application of the statistical concept of causality, based on Granger’s definition of causality, on raw increasing processes as well as on optional and predictable measures. A raw increasing process is optional (predictable) if the bounded (left-continuous) process X, associated with the measure μA(X), is self-caused. Also, the measure μA(X) is optional (predictable) if an associated process X is self-caused with some additional assumptions. Some of the obtained results, in terms of self-causality, can be directly applied to defining conditions for an optional stopping time to become predictable. Full article
(This article belongs to the Special Issue Stochastic and Statistical Analysis in Natural Sciences)
14 pages, 592 KiB  
Article
On Equivalence Operators Derived from Overlap and Grouping Functions
by Lei Du, Yingying Xu, Haifeng Song and Songsong Dai
Axioms 2024, 13(2), 123; https://doi.org/10.3390/axioms13020123 - 17 Feb 2024
Viewed by 675
Abstract
This paper introduces the concept of equivalence operators based on overlap and grouping functions where the associativity property is not strongly required. Overlap functions and grouping functions are weaker than positive and continuous t-norms and t-conorms, respectively. Therefore, these equivalence operators do not [...] Read more.
This paper introduces the concept of equivalence operators based on overlap and grouping functions where the associativity property is not strongly required. Overlap functions and grouping functions are weaker than positive and continuous t-norms and t-conorms, respectively. Therefore, these equivalence operators do not necessarily satisfy certain properties, such as associativity and the neutrality principle. In this paper, two models of fuzzy equivalence operators are obtained by the composition of overlap functions, grouping functions and fuzzy negations. Their main properties are also studied. Full article
(This article belongs to the Special Issue New Perspectives in Fuzzy Sets and Its Applications)
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15 pages, 261 KiB  
Article
On the Noteworthy Properties of Tangentials in Cubic Structures
by Vladimir Volenec and Ružica Kolar-Šuper
Axioms 2024, 13(2), 122; https://doi.org/10.3390/axioms13020122 - 16 Feb 2024
Viewed by 778
Abstract
The cubic structure, a captivating geometric structure, finds applications across various areas of geometry through different models. In this paper, we explore the significant characteristics of tangentials in cubic structures of ranks 0, 1, and 2. Specifically, in the cubic structure of rank [...] Read more.
The cubic structure, a captivating geometric structure, finds applications across various areas of geometry through different models. In this paper, we explore the significant characteristics of tangentials in cubic structures of ranks 0, 1, and 2. Specifically, in the cubic structure of rank 2, we derive the Hessian configuration (123,164) of points and lines. Finally, we introduce and investigate the de Vries configuration of points and lines in a cubic structure. Full article
(This article belongs to the Special Issue Discrete Curvatures and Laplacians)
10 pages, 250 KiB  
Article
Lax Pairs for the Modified KdV Equation
by Georgy I. Burde
Axioms 2024, 13(2), 121; https://doi.org/10.3390/axioms13020121 - 14 Feb 2024
Viewed by 839
Abstract
Multi-parameter families of Lax pairs for the modified Korteweg-de Vries (mKdV) equation are defined by applying a direct method developed in the present study. The gauge transformations, converting the defined Lax pairs to some simpler forms, are found. The direct method and its [...] Read more.
Multi-parameter families of Lax pairs for the modified Korteweg-de Vries (mKdV) equation are defined by applying a direct method developed in the present study. The gauge transformations, converting the defined Lax pairs to some simpler forms, are found. The direct method and its possible applications to other types of evolution equations are discussed. Full article
(This article belongs to the Special Issue Advances in Differential Geometry and Mathematical Physics)
23 pages, 354 KiB  
Article
Integral Representations over Finite Limits for Quantum Amplitudes
by Jack C. Straton
Axioms 2024, 13(2), 120; https://doi.org/10.3390/axioms13020120 - 14 Feb 2024
Viewed by 761
Abstract
We extend previous research to derive three additional M-1-dimensional integral representations over the interval [0,1]. The prior version covered the interval [0,]. This extension applies to products of M Slater orbitals, since they [...] Read more.
We extend previous research to derive three additional M-1-dimensional integral representations over the interval [0,1]. The prior version covered the interval [0,]. This extension applies to products of M Slater orbitals, since they (and wave functions derived from them) appear in quantum transition amplitudes. It enables the magnitudes of coordinate vector differences (square roots of polynomials) |x1x2|=x122x1x2cosθ+x22 to be shifted from disjoint products of functions into a single quadratic form, allowing for the completion of its square. The M-1-dimensional integral representations of M Slater orbitals that both this extension and the prior version introduce provide alternatives to Fourier transforms and are much more compact. The latter introduce a 3M-dimensional momentum integral for M products of Slater orbitals (in M separate denominators), followed in many cases by another set of M-1-dimensional integral representations to combine those denominators into one denominator having a single (momentum) quadratic form. The current and prior methods are also slightly more compact than Gaussian transforms that introduce an M-dimensional integral for products of M Slater orbitals while simultaneously moving them into a single (spatial) quadratic form in a common exponential. One may also use addition theorems for extracting the angular variables or even direct integration at times. Each method has its strengths and weaknesses. We found that these M-1-dimensional integral representations over the interval [0,1] are numerically stable, as was the prior version, having integrals running over the interval [0,], and one does not need to test for a sufficiently large upper integration limit as one does for the latter approach. For analytical reductions of integrals arising from any of the three, however, there is the possible drawback for large M of there being fewer tabled integrals over [0,1] than over [0,]. In particular, the results of both prior and current representations have integration variables residing within square roots asarguments of Macdonald functions. In a number of cases, these can be converted to Meijer G-functions whose arguments have the form (ax2+bx+c)/x, for which a single tabled integral exists for the integrals from running over the interval [0,] of the prior paper, and from which other forms can be found using the techniques given therein. This is not so for integral representations over the interval [0,1]. Finally, we introduce a fourth integral representation that is not easily generalizable to large M but may well provide a bridge for finding the requisite integrals for such Meijer G-functions over [0,1]. Full article
16 pages, 301 KiB  
Article
A Global Minimizer for Mass-Constrained Problem Revisited
by Chun-Fei Long and Gui-Dong Li
Axioms 2024, 13(2), 118; https://doi.org/10.3390/axioms13020118 - 13 Feb 2024
Viewed by 786
Abstract
We investigate the existence of solutions to the scalar field equation Δu=g(u)λuinRN, with mass constraint [...] Read more.
We investigate the existence of solutions to the scalar field equation Δu=g(u)λuinRN, with mass constraint RN|u|2dx=a>0,uH1(RN). Here, N3; g is a continuous function satisfying the conditions of the Berestycki–Lions type; λ is a Lagrange multiplier. Our results supplement and generalize some of the results in L. Jeanjean, S.-S. Lu, Calc. Var. Partial Differential Equations. 61 (2022), Paper No. 214, 18, and J. Hirata, K. Tanaka, Adv. Nonlinear Stud. 19 (2019), 263–290. Full article
(This article belongs to the Section Mathematical Analysis)
20 pages, 565 KiB  
Article
A Hybrid Rule-Based Rough Set Approach to Explore Corporate Governance: From Ranking to Improvement Planning
by Kao-Yi Shen
Axioms 2024, 13(2), 119; https://doi.org/10.3390/axioms13020119 - 11 Feb 2024
Viewed by 824
Abstract
This research introduces a rule-based decision-making model to investigate corporate governance, which has garnered increasing attention within financial markets. However, the existing corporate governance model developed by the Security and Future Institute of Taiwan employs numerous indicators to assess listed stocks. The ultimate [...] Read more.
This research introduces a rule-based decision-making model to investigate corporate governance, which has garnered increasing attention within financial markets. However, the existing corporate governance model developed by the Security and Future Institute of Taiwan employs numerous indicators to assess listed stocks. The ultimate ranking hinges on the number of indicators a company meets, assuming independent relationships between these indicators, thereby failing to reveal contextual connections among them. This study proposes a hybrid rough set approach based on multiple rules induced from a decision table, aiming to overcome these constraints. Additionally, four sample companies from Taiwan undergo evaluation using this rule-based model, demonstrating consistent rankings with the official outcome. Moreover, the proposed approach offers a practical application for guiding improvement planning, providing a basis for determining improvement priorities. This research introduces a rule-based decision model comprising ten rules, revealing contextual relationships between indicators through if–then decision rules. This study, exemplified through a specific case, also provides insights into utilizing this model to strengthen corporate governance by identifying strategic improvement priorities. Full article
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37 pages, 3336 KiB  
Article
Estimating the Rate of Mutation to a Mutator Phenotype
by Isaac Vázquez-Mendoza, Erika E. Rodríguez-Torres, Mojgan Ezadian, Lindi M. Wahl and Philip J. Gerrish
Axioms 2024, 13(2), 117; https://doi.org/10.3390/axioms13020117 - 11 Feb 2024
Viewed by 913
Abstract
A mutator is a variant in a population of organisms whose mutation rate is higher than the average mutation rate in the population. For genetic and population dynamics reasons, mutators are produced and survive with much greater frequency than anti-mutators (variants with a [...] Read more.
A mutator is a variant in a population of organisms whose mutation rate is higher than the average mutation rate in the population. For genetic and population dynamics reasons, mutators are produced and survive with much greater frequency than anti-mutators (variants with a lower-than-average mutation rate). This strong asymmetry is a consequence of both fundamental genetics and natural selection; it can lead to a ratchet-like increase in the mutation rate. The rate at which mutators appear is, therefore, a parameter that should be of great interest to evolutionary biologists generally; for example, it can influence: (1) the survival duration of a species, especially asexual species (which are known to be short-lived), (2) the evolution of recombination, a process that can ameliorate the deleterious effects of mutator abundance, (3) the rate at which cancer appears, (4) the ability of pathogens to escape immune surveillance in their hosts, (5) the long-term fate of mitochondria, etc. In spite of its great relevance to basic and applied science, the rate of mutation to a mutator phenotype continues to be essentially unknown. The reasons for this gap in our knowledge are largely methodological; in general, a mutator phenotype cannot be observed directly, but must instead be inferred from the numbers of some neutral “marker” mutation that can be observed directly: different mutation-rate variants will produce this marker mutation at different rates. Here, we derive the expected distribution of the numbers of the marker mutants observed, accounting for the fact that some of the mutants will have been produced by a mutator phenotype that itself arose by mutation during the growth of the culture. These developments, together with previous enhancements of the Luria–Delbrück assay (by one of us, dubbed the “Jones protocol”), make possible a novel experimental protocol for estimating the rate of mutation to a mutator phenotype. Simulated experiments using biologically reasonable parameters that employ this protocol show that such experiments in the lab can give us fairly accurate estimates of the rate of mutation to a mutator phenotype. Although our ability to estimate mutation-to-mutator rates from simulated experiments is promising, we view this study as a proof-of-concept study and an important first step towards practical empirical estimation. Full article
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16 pages, 830 KiB  
Article
Robustness Analysis for Sundry Disturbed Open Loop Dynamics Using Robust Right Coprime Factorization
by Yuanhong Xu and Mingcong Deng
Axioms 2024, 13(2), 116; https://doi.org/10.3390/axioms13020116 - 09 Feb 2024
Viewed by 772
Abstract
In this paper, the robustness of a system with sundry disturbed open loop dynamics is investigated by employing robust right coprime factorization (RRCF). These sundry disturbed open loop dynamics are present not only in the feed forward path, but also within the feedback [...] Read more.
In this paper, the robustness of a system with sundry disturbed open loop dynamics is investigated by employing robust right coprime factorization (RRCF). These sundry disturbed open loop dynamics are present not only in the feed forward path, but also within the feedback loop. In such a control framework, the nominal plant is firstly right coprime factorized and a feed forward and a feedback controllers are designed based on Bezout identity to ensure the overall stability. Subsequently, considering the sundry disturbed open loop dynamics, a new condition formulated as a disturbed Bezout identity is put forward to achieve the closed loop stability of the system, even in the presence of disturbances existing in sundry open loops, where in the feedback loop a disturbed identity operator is defined. This approach guarantees the system robustness if a specific inequality condition is satisfied. And, it should be noted that the proposed approach is applicable to both linear and nonlinear systems with sundry disturbed open loop dynamics. Simulations demonstrate the effectiveness of our methodology. Full article
(This article belongs to the Special Issue Advances in Mathematical Methods in Optimal Control and Applications)
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15 pages, 269 KiB  
Article
A SIR Epidemic Model Allowing Recovery
by Anthony G. Pakes
Axioms 2024, 13(2), 115; https://doi.org/10.3390/axioms13020115 - 08 Feb 2024
Viewed by 752
Abstract
The deterministic SIR model for disease spread in a closed population is extended to allow infected individuals to recover to the susceptible state. This extension preserves the second constant of motion, i.e., a functional relationship of susceptible and removed numbers, [...] Read more.
The deterministic SIR model for disease spread in a closed population is extended to allow infected individuals to recover to the susceptible state. This extension preserves the second constant of motion, i.e., a functional relationship of susceptible and removed numbers, S(t) and R(t), respectively. This feature allows a substantially complete elucidation of qualitative properties. The model exhibits three modes of behaviour classified in terms of the sign of S(0), the initial value of the epidemic curve. Model behaviour is similar to that of the SIS model if S(0)>0 and to the SIR model if S(0)<0. The separating case is completely soluble and S(t) is constant-valued. Long-term outcomes are determined for all cases, together with determination of the rate of convergence. Determining the shape of the epidemic curve motivates an investigation of curvature properties of all three state functions and quite complete results are obtained that are new, even for the SIR model. Finally, the second threshold theorem for the SIR model is extended in refined and generalised forms. Full article
(This article belongs to the Topic Mathematical Modeling)
8 pages, 221 KiB  
Article
Hermite–Hadamard–Mercer Inequalities Associated with Twice-Differentiable Functions with Applications
by Muhammad Aamir Ali, Thanin Sitthiwirattham, Elisabeth Köbis and Asma Hanif
Axioms 2024, 13(2), 114; https://doi.org/10.3390/axioms13020114 - 08 Feb 2024
Viewed by 731
Abstract
In this work, we initially derive an integral identity that incorporates a twice-differentiable function. After establishing the recently created identity, we proceed to demonstrate some new Hermite–Hadamard–Mercer-type inequalities for twice-differentiable convex functions. Additionally, it demonstrates that the recently introduced inequalities have extended certain [...] Read more.
In this work, we initially derive an integral identity that incorporates a twice-differentiable function. After establishing the recently created identity, we proceed to demonstrate some new Hermite–Hadamard–Mercer-type inequalities for twice-differentiable convex functions. Additionally, it demonstrates that the recently introduced inequalities have extended certain pre-existing inequalities found in the literature. Finally, we provide applications to the newly established inequalities to verify their usefulness. Full article
(This article belongs to the Special Issue Recent Advances in Fractional Calculus)
15 pages, 5987 KiB  
Article
Optimizing Controls to Track Moving Targets in an Intelligent Electro-Optical Detection System
by Cheng Shen, Zhijie Wen, Wenliang Zhu, Dapeng Fan and Mingyuan Ling
Axioms 2024, 13(2), 113; https://doi.org/10.3390/axioms13020113 - 08 Feb 2024
Viewed by 837
Abstract
Electro-optical detection systems face numerous challenges due to the complexity and difficulty of targeting controls for “low, slow and tiny” moving targets. In this paper, we present an optimal model of an advanced n-step adaptive Kalman filter and gyroscope short-term integration weighting fusion [...] Read more.
Electro-optical detection systems face numerous challenges due to the complexity and difficulty of targeting controls for “low, slow and tiny” moving targets. In this paper, we present an optimal model of an advanced n-step adaptive Kalman filter and gyroscope short-term integration weighting fusion (nKF-Gyro) method with targeting control. A method is put forward to improve the model by adding a spherical coordinate system to design an adaptive Kalman filter to estimate target movements. The targeting error formation is analyzed in detail to reveal the relationship between tracking controller feedback and line-of-sight position correction. Based on the establishment of a targeting control coordinate system for tracking moving targets, a dual closed-loop composite optimization control model is proposed. The outer loop is used for estimating the motion parameters and predicting the future encounter point, while the inner loop is used for compensating the targeting error of various elements in the firing trajectory. Finally, the modeling method is substituted into the disturbance simulation verification, which can monitor and compensate for the targeting error of moving targets in real time. The results show that in the optimal model incorporating the nKF-Gyro method with targeting control, the error suppression was increased by up to 36.8% compared to that of traditional KF method and was 25% better than that of the traditional nKF method. Full article
(This article belongs to the Special Issue Control Theory and Control Systems: Algorithms and Methods)
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13 pages, 211 KiB  
Article
Convergence Results for Contractive Type Set-Valued Mappings
by Alexander J. Zaslavski
Axioms 2024, 13(2), 112; https://doi.org/10.3390/axioms13020112 - 07 Feb 2024
Viewed by 868
Abstract
In this work, we study an iterative process induced by a contractive type set-valued mapping in a complete metric space and show its convergence, taking into account computational errors. Full article
(This article belongs to the Special Issue Research on Fixed Point Theory and Application)
13 pages, 3864 KiB  
Article
Fog, Friction, and Failure in Organized Conflict: A Formal Study
by Rodrick Wallace
Axioms 2024, 13(2), 111; https://doi.org/10.3390/axioms13020111 - 06 Feb 2024
Viewed by 837
Abstract
Organized conflict, while confined by the laws of physics—and, under profound strategic incompetence, by the Lanchester equations—is not a physical process but rather an extended exchange between cognitive entities that have been shaped by path-dependent historical trajectories and cultural traditions. Cognition itself is [...] Read more.
Organized conflict, while confined by the laws of physics—and, under profound strategic incompetence, by the Lanchester equations—is not a physical process but rather an extended exchange between cognitive entities that have been shaped by path-dependent historical trajectories and cultural traditions. Cognition itself is confined by the necessity of duality, with an underlying information source constrained by the asymptotic limit theorems of information and control theories. We introduce the concept of a ‘basic underlying probability distribution’ characteristic of the particular cognitive process studied. The dynamic behavior of such systems is profoundly different for ‘thin-tailed’ and ‘fat-tailed’ distributions. The perspective permits the construction of new probability models that may provide useful statistical tools for the analysis of observational and experimental data associated with organized conflict, and, in some measure, for its management. Full article
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16 pages, 271 KiB  
Article
C-S and Strongly C-S Orthogonal Matrices
by Xiaoji Liu, Ying Liu and Hongwei Jin
Axioms 2024, 13(2), 110; https://doi.org/10.3390/axioms13020110 - 05 Feb 2024
Viewed by 770
Abstract
In this paper, we present a new concept of the generalized core orthogonality (called the C-S orthogonality) for two generalized core invertible matrices A and B. A is said to be C-S orthogonal to B if ASB=0 and [...] Read more.
In this paper, we present a new concept of the generalized core orthogonality (called the C-S orthogonality) for two generalized core invertible matrices A and B. A is said to be C-S orthogonal to B if ASB=0 and BAS=0, where AS is the generalized core inverse of A. The characterizations of C-S orthogonal matrices and the C-S additivity are also provided. And, the connection between the C-S orthogonality and C-S partial order has been given using their canonical form. Moreover, the concept of the strongly C-S orthogonality is defined and characterized. Full article
(This article belongs to the Special Issue Advances in Linear Algebra with Applications)
20 pages, 346 KiB  
Article
Linear Programming-Based Fuzzy Alternative Ranking Order Method Accounting for Two-Step Normalization for Comprehensive Evaluation of Digital Economy Development in Provincial Regions
by Huiling Xiang, Hafiz Muhammad Athar Farid and Muhammad Riaz
Axioms 2024, 13(2), 109; https://doi.org/10.3390/axioms13020109 - 05 Feb 2024
Viewed by 865
Abstract
As digital technologies continue to reshape economic landscapes, the comprehensive evaluation of digital economy (DE) development in provincial regions becomes a critical endeavor. This article proposes a novel approach, integrating the linear programming method, fuzzy logic, and the alternative ranking order method accounting [...] Read more.
As digital technologies continue to reshape economic landscapes, the comprehensive evaluation of digital economy (DE) development in provincial regions becomes a critical endeavor. This article proposes a novel approach, integrating the linear programming method, fuzzy logic, and the alternative ranking order method accounting for two-step normalization (AROMAN), to assess the multifaceted facets of DE growth. The primary contribution of the AROMAN is the coupling of vector and linear normalization techniques in order to produce accurate data structures that are subsequently utilized in calculations. The proposed methodology accommodates the inherent uncertainties and complexities associated with the evaluation process, offering a robust framework for decision-makers. The linear programming aspect optimizes the weightings assigned to different evaluation criteria, ensuring a dynamic and context-specific assessment. By incorporating fuzzy logic, the model captures the vagueness and imprecision inherent in qualitative assessments, providing a more realistic representation of the DE’s multifaceted nature. The AROMAN further refines the ranking process, considering the interdependencies among the criteria and enhancing the accuracy of the evaluation. In order to ascertain the efficacy of the suggested methodology, a case study is undertaken pertaining to provincial areas, showcasing its implementation in the evaluation and a comparison of DE progress in various geographical settings. The outcomes illustrate the capacity of the model to produce perceptive and implementable insights for policymakers, thereby enabling them to make well-informed decisions and implement focused interventions that promote the expansion of the DE. Moreover, managerial implications, theoretical limitations, and a comparative analysis are also given of the proposed method. Full article
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17 pages, 292 KiB  
Article
Tractability of Approximation of Functions Defined over Weighted Hilbert Spaces
by Huichao Yan and Jia Chen
Axioms 2024, 13(2), 108; https://doi.org/10.3390/axioms13020108 - 05 Feb 2024
Viewed by 846
Abstract
We investigate L2-approximation problems in the worst case setting in the weighted Hilbert spaces H(KRd,α,γ) with weights Rd,α,γ under parameters [...] Read more.
We investigate L2-approximation problems in the worst case setting in the weighted Hilbert spaces H(KRd,α,γ) with weights Rd,α,γ under parameters 1γ1γ20 and 1<α1α2. Several interesting weighted Hilbert spaces H(KRd,α,γ) appear in this paper. We consider the worst case error of algorithms that use finitely many arbitrary continuous linear functionals. We discuss tractability of L2-approximation problems for the involved Hilbert spaces, which describes how the information complexity depends on d and ε1. As a consequence we study the strongly polynomial tractability (SPT), polynomial tractability (PT), weak tractability (WT), and (t1,t2)-weak tractability ((t1,t2)-WT) for all t1>1 and t2>0 in terms of the introduced weights under the absolute error criterion or the normalized error criterion. Full article
(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications)
10 pages, 267 KiB  
Article
A Class of Bounded Iterative Sequences of Integers
by Artūras Dubickas
Axioms 2024, 13(2), 107; https://doi.org/10.3390/axioms13020107 - 04 Feb 2024
Viewed by 1002
Abstract
In this note, we show that, for any real number τ[12,1), any finite set of positive integers K and any integer s12, the sequence of integers [...] Read more.
In this note, we show that, for any real number τ[12,1), any finite set of positive integers K and any integer s12, the sequence of integers s1,s2,s3, satisfying si+1siK if si is a prime number, and 2si+1τsi if si is a composite number, is bounded from above. The bound is given in terms of an explicit constant depending on τ,s1 and the maximal element of K only. In particular, if K is a singleton set and for each composite si the integer si+1 in the interval [2,τsi] is chosen by some prescribed rule, e.g., si+1 is the largest prime divisor of si, then the sequence s1,s2,s3, is periodic. In general, we show that the sequences satisfying the above conditions are all periodic if and only if either K={1} and τ[12,34) or K={2} and τ[12,59). Full article
63 pages, 1850 KiB  
Article
Parametric Expansions of an Algebraic Variety Near Its Singularities II
by Alexander D. Bruno and Alijon A. Azimov
Axioms 2024, 13(2), 106; https://doi.org/10.3390/axioms13020106 - 04 Feb 2024
Viewed by 751
Abstract
The paper is a continuation and completion of the paper Bruno, A.D.; Azimov, A.A. Parametric Expansions of an Algebraic Variety Near Its Singularities. Axioms 2023, 5, 469, where we calculated parametric expansions of the three-dimensional algebraic manifold Ω, which [...] Read more.
The paper is a continuation and completion of the paper Bruno, A.D.; Azimov, A.A. Parametric Expansions of an Algebraic Variety Near Its Singularities. Axioms 2023, 5, 469, where we calculated parametric expansions of the three-dimensional algebraic manifold Ω, which appeared in theoretical physics, near its 3 singular points and near its one line of singular points. For that we used algorithms of Nonlinear Analysis: extraction of truncated polynomials, using the Newton polyhedron, their power transformations and Formal Generalized Implicit Function Theorem. Here we calculate parametric expansions of the manifold Ω near its one more singular point, near two curves of singular points and near infinity. Here we use 3 new things: (1) computation in algebraic extension of the field of rational numbers, (2) expansions near a curve of singular points and (3) calculation of branches near infinity. Full article
(This article belongs to the Section Algebra and Number Theory)
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12 pages, 270 KiB  
Article
New Conditions for Testing the Oscillation of Solutions of Second-Order Nonlinear Differential Equations with Damped Term
by Asma Al-Jaser, Belgees Qaraad, Higinio Ramos and Stefano Serra-Capizzano
Axioms 2024, 13(2), 105; https://doi.org/10.3390/axioms13020105 - 04 Feb 2024
Cited by 1 | Viewed by 801
Abstract
This paper deals with the oscillatory behavior of solutions of a new class of second-order nonlinear differential equations. In contrast to most of the previous results in the literature, we establish some new criteria that guarantee the oscillation of all solutions of the [...] Read more.
This paper deals with the oscillatory behavior of solutions of a new class of second-order nonlinear differential equations. In contrast to most of the previous results in the literature, we establish some new criteria that guarantee the oscillation of all solutions of the studied equation without additional restrictions. Our approach improves the standard integral averaging technique to obtain simpler oscillation theorems for new classes of nonlinear differential equations. Two examples are presented to illustrate the importance of our findings. Full article
(This article belongs to the Special Issue Advances in Difference Equations)
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