Axioms

23 pages, 382 KiB

Open AccessArticle

Fractional Step Scheme to Approximate a Non-Linear Second-Order Reaction–Diffusion Problem with Inhomogeneous Dynamic Boundary Conditions

by Constantin Fetecău and Costică Moroşanu

Axioms 2023, 12(4), 406; https://doi.org/10.3390/axioms12040406 - 21 Apr 2023

Cited by 2 | Viewed by 1332

Abstract

Two main topics are addressed in the present paper, first, a rigorous qualitative study of a second-order reaction–diffusion problem with non-linear diffusion and cubic-type reactions, as well as inhomogeneous dynamic boundary conditions. Under certain assumptions about the input data: [...] Read more.

Two main topics are addressed in the present paper, first, a rigorous qualitative study of a second-order reaction–diffusion problem with non-linear diffusion and cubic-type reactions, as well as inhomogeneous dynamic boundary conditions. Under certain assumptions about the input data:

g_{_{d}} (t, x)

,

g_{_{f r}} (t, x)

,

U_{0} (x)

and

ζ_{0} (x)

, we prove the well-posedness (the existence, a priori estimates, regularity and uniqueness) of a solution in the space

W_{p}^{1, 2} (Q) \times W_{p}^{1, 2} (Σ)

. Here, we extend previous results, enabling new mathematical models to be more suitable to describe the complexity of a wide class of different physical phenomena of life sciences, including moving interface problems, material sciences, digital image processing, automatic vehicle detection and tracking, the spread of an epidemic infection, semantic image segmentation including U-Net neural networks, etc. The second goal is to develop an iterative splitting scheme, corresponding to the non-linear second-order reaction–diffusion problem. Results relating to the convergence of the approximation scheme and error estimation are also established. On the basis of the proposed numerical scheme, we formulate the algorithm alg-frac_sec-ord_dbc, which represents a delicate challenge for our future works. The benefit of such a method could simplify the process of numerical computation. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Calculus)

14 pages, 331 KiB

Open AccessArticle

The Cauchy–Optimal Stability Results for Cauchy–Jensen Additive Mappings in the Fuzzy Banach Space and the Unital Fuzzy Banach Space

by Zahra Eidinejad, Reza Saadati and Hari M. Srivastava

Axioms 2023, 12(4), 405; https://doi.org/10.3390/axioms12040405 - 21 Apr 2023

Viewed by 892

Abstract

In this article, we apply a new class of fuzzy control functions to approximate a Cauchy additive mapping in fuzzy Banach space (FBS). Further, considering the unital FBS (UFBS), we will investigate the isomorphisms defined in this space. By introducing several specific functions [...] Read more.

In this article, we apply a new class of fuzzy control functions to approximate a Cauchy additive mapping in fuzzy Banach space (FBS). Further, considering the unital FBS (UFBS), we will investigate the isomorphisms defined in this space. By introducing several specific functions and choosing the optimal control function from among these functions, we evaluate the Cauchy–Optimal stability (C–O-stability) for all defined mappings. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications IV)

19 pages, 1057 KiB

Open AccessArticle

Multi-Objective Optimization of the Robustness of Complex Networks Based on the Mixture of Weighted Surrogates

by Junfeng Nie, Zhuoran Yu and Junli Li

Axioms 2023, 12(4), 404; https://doi.org/10.3390/axioms12040404 - 21 Apr 2023

Viewed by 1371

Abstract

Network robustness is of paramount importance. Although great progress has been achieved in robustness optimization using single measures, such networks may still be vulnerable to many attack scenarios. Consequently, multi-objective network robustness optimization has recently garnered greater attention. A complex network structure plays [...] Read more.

Network robustness is of paramount importance. Although great progress has been achieved in robustness optimization using single measures, such networks may still be vulnerable to many attack scenarios. Consequently, multi-objective network robustness optimization has recently garnered greater attention. A complex network structure plays an important role in both node-based and link-based attacks. In this paper, since multi-objective robustness optimization comes with a high computational cost, a surrogate model is adopted instead of network controllability robustness in the optimization process, and the Dempster–Shafer theory is used for selecting and mixing the surrogate models. The method has been validated on four types of synthetic networks, and the results show that the two selected surrogate models can effectively assist the multi-objective evolutionary algorithm in finding network structures with improved controllability robustness. The adaptive updating of surrogate models during the optimization process leads to better results than the selection of two surrogate models, albeit at the cost of longer processing times. Furthermore, the method demonstrated in this paper achieved better performance than existing methods, resulting in a marked increase in computational efficiency. Full article

(This article belongs to the Special Issue Complex Networks, Evolutionary Computation and Machine Learning)

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17 pages, 900 KiB

Open AccessArticle

Aperiodic Sampled-Data Control for Anti-Synchronization of Chaotic Nonlinear Systems Subject to Input Saturation

by Meixuan Li and Yingjie Fan

Axioms 2023, 12(4), 403; https://doi.org/10.3390/axioms12040403 - 21 Apr 2023

Cited by 2 | Viewed by 1134

Abstract

This paper studies the aperiodic sampled-data (SD) control anti-synchronization issue of chaotic nonlinear systems under the effects of input saturation. At first, to describe the simultaneous existence of the aperiodic SD pattern and the input saturation, a nonlinear closed-loop system model is established. [...] Read more.

This paper studies the aperiodic sampled-data (SD) control anti-synchronization issue of chaotic nonlinear systems under the effects of input saturation. At first, to describe the simultaneous existence of the aperiodic SD pattern and the input saturation, a nonlinear closed-loop system model is established. Then, to make the anti-synchronization analysis, a relaxed sampling-interval-dependent Lyapunov functional (RSIDLF) is constructed for the resulting closed-loop system. Thereinto, the positive definiteness requirement of the RSIDLF is abandoned. Due to the indefiniteness of RSIDLF, the discrete-time Lyapunov method (DTLM) then is used to guarantee the local stability of the trivial solutions of the modeled nonlinear system. Furthermore, two convex optimization schemes are proposed to expand the allowable initial area (AIA) and maximize the upper bound of the sampling period (UBSP). Finally, two examples of nonlinear systems are provided to illustrate the superiority of the RSIDLF method over the previous methods in expanding the AIA and enlarging the UBSP. Full article

(This article belongs to the Topic Advances in Nonlinear Dynamics: Methods and Applications)

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22 pages, 3141 KiB

Open AccessArticle

Default Prediction with Industry-Specific Default Heterogeneity Indicators Based on the Forward Intensity Model

by Zhengfang Ni, Minghui Jiang and Wentao Zhan

Axioms 2023, 12(4), 402; https://doi.org/10.3390/axioms12040402 - 21 Apr 2023

Viewed by 1135

Abstract

When predicting the defaults of a large number of samples in a region, this will be affected by industry default heterogeneity. To build a credit risk model that is more suitable for Chinese-listed firms, which have highly industry-specific default heterogeneity, we extend the [...] Read more.

When predicting the defaults of a large number of samples in a region, this will be affected by industry default heterogeneity. To build a credit risk model that is more suitable for Chinese-listed firms, which have highly industry-specific default heterogeneity, we extend the forward intensity model to predict the defaults of Chinese-listed firms with information about the default heterogeneity of industries. Compared with the original model, we combine the Bayes approach with the forward intensity model to generate time-varying industry-specific default heterogeneity indicators. Our model can capture co-movements of different industries that cannot be observed based on the original forward intensity model so that the model can flexibly adjust the firm’s PD according to the industry. In addition, we also consider the impact of default heterogeneity in other industries by studying the influence of the level and trends of other industries’ default heterogeneity on a firm’s credit risk. Finally, we compute PDs for 4476 firms from January 2001 to December 2019 for 36 prediction horizons. The extended model improves the prediction accuracy ratios both for the in-sample and out-of-sample firm’s PDs for all 36 horizons. Almost all the accuracy ratios of the prediction horizons’ PDs are increased by more than 6%. In addition, our model also reduces the gap between the aggregated PDs and the realized number of defaults. Our industry-specific default heterogeneity indicator is helpful to improve the model’s performance, especially for predicting defaults in a large portfolio, which is of significance for credit risk management in China and other regions. Full article

(This article belongs to the Special Issue Big Data Analytics and Mathematical Methods in Digital Economy)

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14 pages, 313 KiB

Open AccessArticle

On the Fixed Circle Problem on Metric Spaces and Related Results

by Nabil Mlaiki, Nihal Özgür, Nihal Taş and Dania Santina

Axioms 2023, 12(4), 401; https://doi.org/10.3390/axioms12040401 - 20 Apr 2023

Cited by 2 | Viewed by 1124

Abstract

The fixed-circle issue is a geometric technique that is connected to the study of geometric characteristics of certain points, and that are fixed by the self-mapping of either the metric space or of the generalized space. The fixed-disc problem is a natural resultant [...] Read more.

The fixed-circle issue is a geometric technique that is connected to the study of geometric characteristics of certain points, and that are fixed by the self-mapping of either the metric space or of the generalized space. The fixed-disc problem is a natural resultant that arises as a direct outcome of this problem. In this study, our goal is to examine new classes of self-mappings that meet a new particular sort of contraction in a metric space. The common geometrical characteristic of the set of fixed points of any element of these classes is that a circle or even a disc, that is either termed the fixed circle or even the fixed disc of the appropriate self-map, is included within that set. In order to accomplish this, we establish two new classifications of contraction mapping:

F_{c}

-contractive mapping and

F_{c}

-expanding mapping. In the investigation of neural networks, activation functions with either fixed circles (or even fixed discs) are observed frequently. This demonstrates how successful our results with the fixed-circle (respectively, the fixed-disc) model were. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics IV)

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13 pages, 1294 KiB

Open AccessArticle

Two Novel Computational Techniques for Solving Nonlinear Time-Fractional Lax’s Korteweg-de Vries Equation

by Nidhish Kumar Mishra, Mashael M. AlBaidani, Adnan Khan and Abdul Hamid Ganie

Axioms 2023, 12(4), 400; https://doi.org/10.3390/axioms12040400 - 20 Apr 2023

Cited by 9 | Viewed by 968

Abstract

This article investigates the seventh-order Lax’s Korteweg–de Vries equation using the Yang transform decomposition method (YTDM) and the homotopy perturbation transform method (HPTM). The physical phenomena that emerge in physics, engineering and chemistry are mathematically expressed by this equation. For instance, the KdV [...] Read more.

This article investigates the seventh-order Lax’s Korteweg–de Vries equation using the Yang transform decomposition method (YTDM) and the homotopy perturbation transform method (HPTM). The physical phenomena that emerge in physics, engineering and chemistry are mathematically expressed by this equation. For instance, the KdV equation was constructed to represent a wide range of physical processes involving the evolution and interaction of nonlinear waves. In the Caputo sense, the fractional derivative is considered. We employed the Yang transform, the Adomian decomposition method and the homotopy perturbation method to obtain the solution to the time-fractional Lax’s Korteweg–de Vries problem. We examined and compared a particular example with the actual result to verify the approaches. By utilizing these methods, we can construct recurrence relations that represent the solution to the problem that is being proposed, and we are then able to present graphical representations that enable us to visually examine all of the results in the proposed case for different fractional order values. Furthermore, the results of the current approach exhibit a good correlation with the precise solution to the problem being studied. Furthermore, the present study offers an example of error analysis. The numerical outcomes obtained by applying the provided approaches demonstrate that the techniques are easy to use and have superior computational performance. Full article

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

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19 pages, 928 KiB

Open AccessArticle

Properties of Convex Fuzzy-Number-Valued Functions on Harmonic Convex Set in the Second Sense and Related Inequalities via Up and Down Fuzzy Relation

by Muhammad Bilal Khan, Željko Stević, Abdulwadoud A. Maash, Muhammad Aslam Noor and Mohamed S. Soliman

Axioms 2023, 12(4), 399; https://doi.org/10.3390/axioms12040399 - 20 Apr 2023

Viewed by 880

Abstract

In this paper, we provide different variants of the Hermite–Hadamard (

H \cdot H

) inequality using the concept of a new class of convex mappings, which is referred to as up and down harmonically

s

-convex fuzzy-number-valued functions (

U D H

[...] Read more.

In this paper, we provide different variants of the Hermite–Hadamard (

H \cdot H

) inequality using the concept of a new class of convex mappings, which is referred to as up and down harmonically

s

-convex fuzzy-number-valued functions (

U D H

s

-convex

F N V M

) in the second sense based on the up and down fuzzy inclusion relation. The findings are confirmed with certain numerical calculations that take a few appropriate examples into account. The results deal with various integrals of the

\frac{2 ρ σ}{ρ + σ}

type and are innovative in the setting of up and down harmonically

s

-convex fuzzy-number-valued functions. Moreover, we acquire classical and new exceptional cases that can be seen as applications of our main outcomes. In our opinion, this will make a significant contribution to encouraging more research. Full article

(This article belongs to the Special Issue Editorial Board Members’ Collection Series: Fuzzy Modeling, Optimization and Computational Intelligence)

20 pages, 8061 KiB

Open AccessArticle

Reconstructing Loads in Nanoplates from Dynamic Data

by Alexandre Kawano and Antonino Morassi

Axioms 2023, 12(4), 398; https://doi.org/10.3390/axioms12040398 - 20 Apr 2023

Viewed by 898

Abstract

It was recently proved that the knowledge of the transverse displacement of a nanoplate in an open subset of its mid-plane, measured for any interval of time, allows for the unique determination of the spatial components [...] Read more.

It was recently proved that the knowledge of the transverse displacement of a nanoplate in an open subset of its mid-plane, measured for any interval of time, allows for the unique determination of the spatial components

{f_{m} (x, y)}_{m = 1}^{M}

of the transverse load

\sum_{m = 1}^{M} g_{m} (t) f_{m} (x, y)

, where

M \geq 1

and

{g_{m} (t)}_{m = 1}^{M}

is a known set of linearly independent functions of the time variable. The nanoplate mechanical model is built within the strain gradient linear elasticity theory, according to the Kirchhoff–Love kinematic assumptions. In this paper, we derive a reconstruction algorithm for the above inverse source problem, and we implement a numerical procedure based on a finite element spatial discretization to approximate the loads

{f_{m} (x, y)}_{m = 1}^{M}

. The computations are developed for a uniform rectangular nanoplate clamped at the boundary. The sensitivity of the results with respect to the main parameters that influence the identification is analyzed in detail. The adoption of a regularization scheme based on the singular value decomposition turns out to be decisive for the accuracy and stability of the reconstruction. Full article

(This article belongs to the Special Issue Mathematical Methods in Waves-Based Inverse Problems at Different Scales)

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16 pages, 365 KiB

Open AccessArticle

On the Existence of Solutions to a Boundary Value Problem via New Weakly Contractive Operator

by Rhoda Chiroma, Trad Alotaibi, Mohammed Shehu Shagari, Awad A. Bakery, OM Kalthum S. K. Mohammed and Arafa O. Mustafa

Axioms 2023, 12(4), 397; https://doi.org/10.3390/axioms12040397 - 20 Apr 2023

Cited by 1 | Viewed by 972

Abstract

In this paper, the notion of generalized quasi-weakly contractive operators in metric-like spaces is introduced, and new conditions for the existence of fixed points for such mappings are investigated. A non-trivial example which highlights the novelty of our principal idea is constructed. It [...] Read more.

In this paper, the notion of generalized quasi-weakly contractive operators in metric-like spaces is introduced, and new conditions for the existence of fixed points for such mappings are investigated. A non-trivial example which highlights the novelty of our principal idea is constructed. It is observed comparatively that the proposed concepts herein subsume some important results in the corresponding literature. As an application, one of our obtained findings is utilized to setup novel criteria for the existence of solutions to two-point boundary value problems of a second order differential equation. To attract new researchers in the directions examined in this article, a significant number of corollaries are pointed out and discussed. Full article

(This article belongs to the Special Issue Fixed Point Theory and Its Related Topics IV)

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12 pages, 289 KiB

Open AccessArticle

Fixed-Point Theorems for Nonlinear Contraction in Fuzzy-Controlled Bipolar Metric Spaces

by Gunaseelan Mani, Arul Joseph Gnanaprakasam, Santosh Kumar, Ozgur Ege and Manuel De la Sen

Axioms 2023, 12(4), 396; https://doi.org/10.3390/axioms12040396 - 19 Apr 2023

Cited by 1 | Viewed by 783

Abstract

In this paper, we introduce the concept of fuzzy-controlled bipolar metric space and prove some fixed-point theorems in this space. Our results generalize and expand some of the literature’s well-known results. We also provide some applications of our main results to integral equations. [...] Read more.

In this paper, we introduce the concept of fuzzy-controlled bipolar metric space and prove some fixed-point theorems in this space. Our results generalize and expand some of the literature’s well-known results. We also provide some applications of our main results to integral equations. Full article

(This article belongs to the Special Issue Nonlinear Functional Analysis in Natural Sciences)

19 pages, 1079 KiB

Open AccessArticle

Multi-Objective ABC-NM Algorithm for Multi-Dimensional Combinatorial Optimization Problem

by Muniyan Rajeswari, Rajakumar Ramalingam, Shakila Basheer, Keerthi Samhitha Babu, Mamoon Rashid and Ramar Saranya

Axioms 2023, 12(4), 395; https://doi.org/10.3390/axioms12040395 - 19 Apr 2023

Cited by 2 | Viewed by 935

Abstract

This article addresses the problem of converting a single-objective combinatorial problem into a multi-objective one using the Pareto front approach. Although existing algorithms can identify the optimal solution in a multi-objective space, they fail to satisfy constraints while achieving optimal performance. To address [...] Read more.

This article addresses the problem of converting a single-objective combinatorial problem into a multi-objective one using the Pareto front approach. Although existing algorithms can identify the optimal solution in a multi-objective space, they fail to satisfy constraints while achieving optimal performance. To address this issue, we propose a multi-objective artificial bee colony optimization algorithm with a classical multi-objective theme called fitness sharing. This approach helps the convergence of the Pareto solution set towards a single optimal solution that satisfies multiple objectives. This article introduces multi-objective optimization with an example of a non-dominated sequencing technique and fitness sharing approach. The experimentation is carried out in MATLAB 2018a. In addition, we applied the proposed algorithm to two different real-time datasets, namely the knapsack problem and the nurse scheduling problem (NSP). The outcome of the proposed MBABC-NM algorithm is evaluated using standard performance indicators such as average distance, number of reference solutions (NRS), overall count of attained solutions (TNS), and overall non-dominated generation volume (ONGV). The results show that it outperforms other algorithms. Full article

(This article belongs to the Special Issue Advances in Combinatorial Optimization and Discrete Mathematics with Applications)

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21 pages, 313 KiB

Open AccessArticle

Nonuniform Dichotomy with Growth Rates of Skew-Evolution Cocycles in Banach Spaces

by Ariana Găină, Mihail Megan and Rovana Boruga (Toma)

Axioms 2023, 12(4), 394; https://doi.org/10.3390/axioms12040394 - 18 Apr 2023

Cited by 1 | Viewed by 849

Abstract

This paper presents integral charaterizations for nonuniform dichotomy with growth rates and their correspondents for the particular cases of nonuniform exponential dichotomy and nonuniform polynomial dichotomy of skew-evolution cocycles in Banach spaces. The connections between these three concepts are presented. Full article

(This article belongs to the Special Issue Differential Equations and Dynamical Systems—Theory and Applications)

18 pages, 1382 KiB

Open AccessArticle

A Quasiconformal-Based Geometric Model for Craniofacial Analysis and Its Application

by Ming-Hei Wong, Meixi Li, King-Man Tam, Hoi-Man Yuen, Chun-Ting Au, Kate Ching-Ching Chan, Albert Martin Li and Lok-Ming Lui

Axioms 2023, 12(4), 393; https://doi.org/10.3390/axioms12040393 - 18 Apr 2023

Viewed by 1429

Abstract

We address the problem of craniofacial morphometric analysis using geometric models, which has important clinical applications for the diagnosis of syndromes associated with craniofacial dysmorphologies. In this work, a novel geometric model is proposed to analyze craniofacial structures based on local curvature information [...] Read more.

We address the problem of craniofacial morphometric analysis using geometric models, which has important clinical applications for the diagnosis of syndromes associated with craniofacial dysmorphologies. In this work, a novel geometric model is proposed to analyze craniofacial structures based on local curvature information and Teichmüller mappings. A key feature of the proposed model is that its pipeline starts with few two-dimensional images of the human face captured at different angles, from which the three-dimensional craniofacial structure can be reconstructed. The 3D surface reconstruction from 2D images is based on a modified 3D morphable model (3DMM) framework. Geometric quantities around important feature landmarks according to different clinical applications can then be computed on each three-dimensional craniofacial structure. Together with the Teichmüller mapping, the landmark-based Teichmüller curvature distances (LTCDs) for every classes can be computed, which are further used for three-class classification. A composite score model is used and the parameter optimization is carried out to further improve the classification accuracy. Our proposed model is applied to study the craniofacial structures of children with and without the obstructive sleep apnoea (OSA). Sixty subjects, with accessible multi-angle photography and polysomnography (PSG) data, are divided into three classes based on the severity of OSA. Using our proposed model, our proposed model achieves a high 90% accuracy, which outperforms other existing models. This demonstrates the effectiveness of our proposed geometric model for craniofacial analysis. Full article

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26 pages, 1010 KiB

Open AccessArticle

Theoretical Validation of New Two-Dimensional One-Variable-Power Copulas

by Christophe Chesneau

Axioms 2023, 12(4), 392; https://doi.org/10.3390/axioms12040392 - 18 Apr 2023

Cited by 1 | Viewed by 1013

Abstract

One of the most effective ways to illustrate the relationship between two quantitative variables is to describe the corresponding two-dimensional copula. This approach is acknowledged as practical, nonredundant, and computationally manageable in the context of data analysis. Modern data, however, contain a wide [...] Read more.

One of the most effective ways to illustrate the relationship between two quantitative variables is to describe the corresponding two-dimensional copula. This approach is acknowledged as practical, nonredundant, and computationally manageable in the context of data analysis. Modern data, however, contain a wide variety of dependent structures, and the copulas now in use may not provide the best model for all of them. As a result, researchers seek to innovate by building novel copulas with appealing properties that are also based on original methodologies. The foundations are theoretical; for a copula to be validated, it must meet specific requirements, which frequently dictate the constraints that must be placed on the relevant parameters. In this article, we make a contribution to the understudied field of one-variable-power copulas. We first identify the specific assumptions that, in theory, validate copulas of such nature. Some other general copulas and inequalities are discussed. Our general results are illustrated with numerous examples depending on two or three parameters. We also prove that strong connections exist between our assumptions and well-established distributions. To highlight the importance of our findings, we emphasize a particular two-parameter, one-variable-power copula that unifies the definition of some other copulas. We reveal its versatile shapes, related functions, various symmetry, Archimedean nature, geometric invariance, copula ordering, quadrant dependence, tail dependence, correlations, and distribution generation. Numerical tables and graphics are produced to support some of these properties. The estimation of the parameters based on data is discussed. As a complementary contribution, two new, intriguing one-variable-power copulas beyond the considered general form are finally presented and studied. Full article

(This article belongs to the Special Issue Statistical Modeling of Modern Multivariate Data)

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14 pages, 749 KiB

Open AccessArticle

Dynamics of a Double-Impulsive Control Model of Integrated Pest Management Using Perturbation Methods and Floquet Theory

by Fahad Al Basir, Jahangir Chowdhury and Delfim F. M. Torres

Axioms 2023, 12(4), 391; https://doi.org/10.3390/axioms12040391 - 18 Apr 2023

Cited by 1 | Viewed by 1143

Abstract

We formulate an integrated pest management model to control natural pests of the crop through the periodic application of biopesticide and chemical pesticides. In a theoretical analysis of the system pest eradication, a periodic solution is found and established. All the system variables [...] Read more.

We formulate an integrated pest management model to control natural pests of the crop through the periodic application of biopesticide and chemical pesticides. In a theoretical analysis of the system pest eradication, a periodic solution is found and established. All the system variables are proved to be bounded. Our main goal is then to ensure that pesticides are optimized, in terms of pesticide concentration and pesticide application frequency, and that the optimum combination of pesticides is found to provide the most benefit to the crop. By using Floquet theory and the small amplitude perturbation method, we prove that the pest eradication periodic solution is locally and globally stable. The acquired results establish a threshold time limit for the impulsive release of various controls as well as some valid theoretical conclusions for effective pest management. Furthermore, after a numerical comparison, we conclude that integrated pest management is more effective than single biological or chemical controls. Finally, we illustrate the analytical results through numerical simulations. Full article

(This article belongs to the Special Issue Mathematical Methods in the Applied Sciences)

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14 pages, 1643 KiB

Open AccessArticle

Stability Switching in Lotka-Volterra and Ricker-Type Predator-Prey Systems with Arbitrary Step Size

by Shamika Kekulthotuwage Don, Kevin Burrage, Kate J. Helmstedt and Pamela M. Burrage

Axioms 2023, 12(4), 390; https://doi.org/10.3390/axioms12040390 - 17 Apr 2023

Cited by 1 | Viewed by 1216

Abstract

Dynamical properties of numerically approximated discrete systems may become inconsistent with those of the corresponding continuous-time system. We present a qualitative analysis of the dynamical properties of two-species Lotka-Volterra and Ricker-type predator-prey systems under discrete and continuous settings. By creating an arbitrary time [...] Read more.

Dynamical properties of numerically approximated discrete systems may become inconsistent with those of the corresponding continuous-time system. We present a qualitative analysis of the dynamical properties of two-species Lotka-Volterra and Ricker-type predator-prey systems under discrete and continuous settings. By creating an arbitrary time discretisation, we obtain stability conditions that preserve the characteristics of continuous-time models and their numerically approximated systems. Here, we show that even small changes to some of the model parameters may alter the system dynamics unless an appropriate time discretisation is chosen to return similar dynamical behaviour to what is observed in the corresponding continuous-time system. We also found similar dynamical properties of the Ricker-type predator-prey systems under certain conditions. Our results demonstrate the need for preliminary analysis to identify which dynamical properties of approximated discretised systems agree or disagree with the corresponding continuous-time systems. Full article

(This article belongs to the Special Issue Mathematical Modeling with Differential Equations)

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25 pages, 539 KiB

Open AccessArticle

Efficiency of Orthogonal Matching Pursuit for Group Sparse Recovery

by Chunfang Shao, Xiujie Wei, Peixin Ye and Shuo Xing

Axioms 2023, 12(4), 389; https://doi.org/10.3390/axioms12040389 - 17 Apr 2023

Viewed by 944

Abstract

We propose the Group Orthogonal Matching Pursuit (GOMP) algorithm to recover group sparse signals from noisy measurements. Under the group restricted isometry property (GRIP), we prove the instance optimality of the GOMP algorithm for any decomposable approximation norm. Meanwhile, we show the robustness [...] Read more.

We propose the Group Orthogonal Matching Pursuit (GOMP) algorithm to recover group sparse signals from noisy measurements. Under the group restricted isometry property (GRIP), we prove the instance optimality of the GOMP algorithm for any decomposable approximation norm. Meanwhile, we show the robustness of the GOMP under the measurement error. Compared with the P-norm minimization approach, the GOMP is easier to implement, and the assumption of

γ

-decomposability is not required. The simulation results show that the GOMP is very efficient for group sparse signal recovery and significantly outperforms Basis Pursuit in both scalability and solution quality. Full article

(This article belongs to the Special Issue Applied Mathematical Modeling and Optimization)

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12 pages, 431 KiB

Open AccessArticle

A Numerical Approach of Handling Fractional Stochastic Differential Equations

by Iqbal M. Batiha, Ahmad A. Abubaker, Iqbal H. Jebril, Suha B. Al-Shaikh and Khaled Matarneh

Axioms 2023, 12(4), 388; https://doi.org/10.3390/axioms12040388 - 17 Apr 2023

Cited by 5 | Viewed by 1511

Abstract

This work proposes a new numerical approach for dealing with fractional stochastic differential equations. In particular, a novel three-point fractional formula for approximating the Riemann–Liouville integrator is established, and then it is applied to generate approximate solutions for fractional stochastic differential equations. Such [...] Read more.

This work proposes a new numerical approach for dealing with fractional stochastic differential equations. In particular, a novel three-point fractional formula for approximating the Riemann–Liouville integrator is established, and then it is applied to generate approximate solutions for fractional stochastic differential equations. Such a formula is derived with the use of the generalized Taylor theorem coupled with a recent definition of the definite fractional integral. Our approach is compared with the approximate solution generated by the Euler–Maruyama method and the exact solution for the purpose of verifying our findings. Full article

(This article belongs to the Special Issue Stochastic Processes: Theory and Application)

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18 pages, 2678 KiB

Open AccessArticle

Application of the Optimal Homotopy Asymptotic Approach for Solving Two-Point Fuzzy Ordinary Differential Equations of Fractional Order Arising in Physics

by Ali Fareed Jameel, Dulfikar Jawad Hashim, Nidal Anakira, Osama Ababneh, Ahmad Qazza, Abedel-Karrem Alomari and Khamis S. Al Kalbani

Axioms 2023, 12(4), 387; https://doi.org/10.3390/axioms12040387 - 17 Apr 2023

Cited by 1 | Viewed by 1045

Abstract

This work focuses on solving and analyzing two-point fuzzy boundary value problems in the form of fractional ordinary differential equations (FFOBVPs) using a new version of the approximation analytical approach. FFOBVPs are useful in describing complex scientific phenomena that include heritable characteristics and [...] Read more.

This work focuses on solving and analyzing two-point fuzzy boundary value problems in the form of fractional ordinary differential equations (FFOBVPs) using a new version of the approximation analytical approach. FFOBVPs are useful in describing complex scientific phenomena that include heritable characteristics and uncertainty, and obtaining exact or close analytical solutions for these equations can be challenging, especially in the case of nonlinear problems. To address these difficulties, the optimal homotopy asymptotic method (OHAM) was studied and extended in a new form to solve FFOBVPs. The OHAM is known for its ability to solve both linear and nonlinear fractional models and provides a straightforward methodology that uses multiple convergence control parameters to optimally manage the convergence of approximate series solutions. The new form of the OHAM presented in this work incorporates the concepts of fuzzy sets theory and some fractional calculus principles to include fuzzy analysis in the method. The steps of fuzzification and defuzzification are used to transform the fuzzy problem into a crisp problem that can be solved using the OHAM. The method is demonstrated by solving and analyzing linear and nonlinear FFOBVPs at different values of fractional derivatives. The results obtained using the new form of the fuzzy OHAM are analyzed and compared to those found in the literature to demonstrate the method’s efficiency and high accuracy in the fuzzy domain. Overall, this work presents a feasible and efficient approach for solving FFOBVPs using a new form of the OHAM with fuzzy analysis. Full article

(This article belongs to the Special Issue Mathematical Methods in the Applied Sciences)

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24 pages, 15344 KiB

Open AccessArticle

Developing a Deep Learning-Based Defect Detection System for Ski Goggles Lenses

by Dinh-Thuan Dang and Jing-Wein Wang

Axioms 2023, 12(4), 386; https://doi.org/10.3390/axioms12040386 - 17 Apr 2023

Cited by 5 | Viewed by 1599

Abstract

Ski goggles help protect the eyes and enhance eyesight. The most important part of ski goggles is their lenses. The quality of the lenses has leaped with technological advances, but there are still defects on their surface during manufacturing. This study develops a [...] Read more.

Ski goggles help protect the eyes and enhance eyesight. The most important part of ski goggles is their lenses. The quality of the lenses has leaped with technological advances, but there are still defects on their surface during manufacturing. This study develops a deep learning-based defect detection system for ski goggles lenses. The first step is to design the image acquisition model that combines cameras and light sources. This step aims to capture clear and high-resolution images on the entire surface of the lenses. Next, defect categories are identified, including scratches, watermarks, spotlight, stains, dust-line, and dust-spot. They are labeled to create the ski goggles lenses defect dataset. Finally, the defects are automatically detected by fine-tuning the mobile-friendly object detection model. The mentioned defect detection model is the MobileNetV3 backbone used in a feature pyramid network (FPN) along with the Faster-RCNN detector. The fine-tuning includes: replacing the default ResNet50 backbone with a combination of MobileNetV3 and FPN; adjusting the hyper-parameter of the region proposal network (RPN) to suit the tiny defects; and reducing the number of the output channel in FPN to increase computational performance. Our experiments demonstrate the effectiveness of defect detection; additionally, the inference speed is fast. The defect detection accuracy achieves a mean average precision (mAP) of 55%. The work automatically integrates all steps, from capturing images to defect detection. Furthermore, the lens defect dataset is publicly available to the research community on GitHub. The repository address can be found in the Data Availability Statement section. Full article

(This article belongs to the Special Issue Various Deep Learning Algorithms in Computational Intelligence)

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17 pages, 3610 KiB

Open AccessArticle

A New Perspective for Multivariate Time Series Decision Making through a Nested Computational Approach Using Type-2 Fuzzy Integration

by Martha Ramirez and Patricia Melin

Axioms 2023, 12(4), 385; https://doi.org/10.3390/axioms12040385 - 17 Apr 2023

Viewed by 1057

Abstract

The integration of key indicators from the results of the analysis of time series represents a constant challenge within organizations; this could be mainly due to the need to establish the belonging of each indicator within a process, geographic region or category. This [...] Read more.

The integration of key indicators from the results of the analysis of time series represents a constant challenge within organizations; this could be mainly due to the need to establish the belonging of each indicator within a process, geographic region or category. This paper thus illustrates how both primary and secondary indicators are relevant for decision making, and why they need to be integrated by making new final fuzzy indicators. Thus, our proposal consists of a type-2 fuzzy integration of multivariate time series, such as OECD country risk classification, inflation, population and gross national income (GNI) by using multiple type-1 fuzzy inference systems to perform time series classification tasks. Our contribution consists of the proposal to integrate multiple nested type-1 fuzzy inference systems using a type-2 fuzzy integration. Simulation results show the advantages of using the proposed method for the fuzzy classification of multiple time series. This is done in order so the user can have tools that allow them to understand the environment and generate comparative analyses of multiple information sources, and finally use it during the process prior to decision making considering the main advantage of modeling the inherent uncertainty. Full article

(This article belongs to the Special Issue Advances in Fuzzy MCDM, Hybrid Methods, Fuzzy Number Ranking and Their Applications)

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22 pages, 1406 KiB

Open AccessArticle

Barrier Options and Greeks: Modeling with Neural Networks

by Nneka Umeorah, Phillip Mashele, Onyecherelam Agbaeze and Jules Clement Mba

Axioms 2023, 12(4), 384; https://doi.org/10.3390/axioms12040384 - 17 Apr 2023

Cited by 2 | Viewed by 2023

Abstract

This paper proposes a non-parametric technique of option valuation and hedging. Here, we replicate the extended Black–Scholes pricing model for the exotic barrier options and their corresponding Greeks using the fully connected feed-forward neural network. Our methodology involves some benchmarking experiments, which result [...] Read more.

This paper proposes a non-parametric technique of option valuation and hedging. Here, we replicate the extended Black–Scholes pricing model for the exotic barrier options and their corresponding Greeks using the fully connected feed-forward neural network. Our methodology involves some benchmarking experiments, which result in an optimal neural network hyperparameter that effectively prices the barrier options and facilitates their option Greeks extraction. We compare the results from the optimal NN model to those produced by other machine learning models, such as the random forest and the polynomial regression; the output highlights the accuracy and the efficiency of our proposed methodology in this option pricing problem. The results equally show that the artificial neural network can effectively and accurately learn the extended Black–Scholes model from a given simulated dataset, and this concept can similarly be applied in the valuation of complex financial derivatives without analytical solutions. Full article

(This article belongs to the Special Issue Various Deep Learning Algorithms in Computational Intelligence)

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27 pages, 613 KiB

Open AccessArticle

A Compound Class of Inverse-Power Muth and Power Series Distributions

by Leonardo Barrios-Blanco, Diego I. Gallardo, Héctor J. Gómez and Marcelo Bourguignon

Axioms 2023, 12(4), 383; https://doi.org/10.3390/axioms12040383 - 16 Apr 2023

Viewed by 1201

Abstract

This paper introduces the inverse-power Muth power series model, which is a composition of the inverse-power Muth and the class of power series distributions. The use of the Bell distribution in this context is emphasized for the first time in the literature. Probability [...] Read more.

This paper introduces the inverse-power Muth power series model, which is a composition of the inverse-power Muth and the class of power series distributions. The use of the Bell distribution in this context is emphasized for the first time in the literature. Probability density, survival and hazard functions are studied, as well as their moments. Using the stochastic representation of the model, the maximum-likelihood estimators are implemented by the use of the expectation-maximization algorithm, while standard errors are calculated using Oakes’ method. Monte Carlo simulation studies are conducted to show the performance of the maximum-likelihood estimators in finite samples. Two applications to real datasets are shown, where our proposal is compared with some models based on power series compositions. Full article

(This article belongs to the Special Issue Probability, Statistics and Estimation)

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30 pages, 457 KiB

Open AccessArticle

Infinite Series and Logarithmic Integrals Associated to Differentiation with Respect to Parameters of the Whittaker W_κ,μ(x) Function II

by Alexander Apelblat and Juan Luis González-Santander

Axioms 2023, 12(4), 382; https://doi.org/10.3390/axioms12040382 - 16 Apr 2023

Viewed by 995

Abstract

In the first part of this investigation, we considered the parameter differentiation of the Whittaker function

M_{κ, μ} (x)

. In this second part, first derivatives with respect to the parameters of the Whittaker function

W_{κ, μ} (x)

are [...] Read more.

In the first part of this investigation, we considered the parameter differentiation of the Whittaker function

M_{κ, μ} (x)

. In this second part, first derivatives with respect to the parameters of the Whittaker function

W_{κ, μ} (x)

are calculated. Using the confluent hypergeometric function, these derivatives can be expressed as infinite sums of quotients of the digamma and gamma functions. Furthermore, it is possible to obtain these parameter derivatives in terms of infinite integrals, with integrands containing elementary functions (products of algebraic, exponential, and logarithmic functions), from the integral representation of

W_{κ, μ} (x)

. These infinite sums and integrals can be expressed in closed form for particular values of the parameters. Finally, an integral representation of the integral Whittaker function

{wi}_{κ, μ} (x)

and its derivative with respect to

κ

, as well as some reduction formulas for the integral Whittaker functions

{Wi}_{κ, μ} (x)

and

{wi}_{κ, μ} (x)

, are calculated. Full article

(This article belongs to the Special Issue Dedicated to Professor Hari Mohan Srivastava on the Occasion of His 82nd Birthday)

29 pages, 433 KiB

Open AccessArticle

Infinite Series and Logarithmic Integrals Associated to Differentiation with Respect to Parameters of the Whittaker M_κ_,_μ(x) Function I

by Alexander Apelblat and Juan Luis González-Santander

Axioms 2023, 12(4), 381; https://doi.org/10.3390/axioms12040381 - 16 Apr 2023

Viewed by 1045

Abstract

In this paper, first derivatives of the Whittaker function

M_{κ, μ} (x)

are calculated with respect to the parameters. Using the confluent hypergeometric function, these derivarives can be expressed as infinite sums of quotients of the digamma and gamma functions. Moreover, [...] Read more.

In this paper, first derivatives of the Whittaker function

M_{κ, μ} (x)

are calculated with respect to the parameters. Using the confluent hypergeometric function, these derivarives can be expressed as infinite sums of quotients of the digamma and gamma functions. Moreover, from the integral representation of

M_{κ, μ} (x)

it is possible to obtain these parameter derivatives in terms of finite and infinite integrals with integrands containing elementary functions (products of algebraic, exponential, and logarithmic functions). These infinite sums and integrals can be expressed in closed form for particular values of the parameters. For this purpose, we have obtained the parameter derivative of the incomplete gamma function in closed form. As an application, reduction formulas for parameter derivatives of the confluent hypergeometric function are derived, along with finite and infinite integrals containing products of algebraic, exponential, logarithmic, and Bessel functions. Finally, reduction formulas for the Whittaker functions

M_{κ, μ} (x)

and integral Whittaker functions

{Mi}_{κ, μ} (x)

and

{mi}_{κ, μ} (x)

are calculated. Full article

(This article belongs to the Special Issue Dedicated to Professor Hari Mohan Srivastava on the Occasion of His 82nd Birthday)

16 pages, 326 KiB

Open AccessArticle

Pascu-Rønning Type Meromorphic Functions Based on Sălăgean-Erdély–Kober Operator

by Sheza M. El-Deeb, Gangadharan Murugusundaramoorthy, Kaliappan Vijaya and Alhanouf Alburaikan

Axioms 2023, 12(4), 380; https://doi.org/10.3390/axioms12040380 - 16 Apr 2023

Cited by 1 | Viewed by 884

Abstract

In the present investigation, we introduce a new class of meromorphic functions defined in the punctured unit disk

Δ^{*} : = {ϑ \in C : 0 < | ϑ | < 1}

by making use of the Erdély–Kober operator [...] Read more.

In the present investigation, we introduce a new class of meromorphic functions defined in the punctured unit disk

Δ^{*} : = {ϑ \in C : 0 < | ϑ | < 1}

by making use of the Erdély–Kober operator

I_{ς, ϱ}^{τ, κ}

which unifies well-known classes of the meromorphic uniformly convex function with positive coefficients. Coefficient inequalities, growth and distortion inequalities, in addition to closure properties are acquired. We also set up a few outcomes concerning convolution and the partial sums of meromorphic functions in this new class. We additionally state some new subclasses and its characteristic houses through specializing the parameters that are new and no longer studied in association with the Erdély–Kober operator thus far. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications IV)

29 pages, 1913 KiB

Open AccessArticle

A New Extension of the Kumaraswamy Exponential Model with Modeling of Food Chain Data

by Eman A. Eldessouky, Osama H. Mahmoud Hassan, Mohammed Elgarhy, Eid A. A. Hassan, Ibrahim Elbatal and Ehab M. Almetwally

Axioms 2023, 12(4), 379; https://doi.org/10.3390/axioms12040379 - 16 Apr 2023

Cited by 6 | Viewed by 1589

Abstract

Statistical models are useful in explaining and forecasting real-world occurrences. Various extended distributions have been widely employed for modeling data in a variety of fields throughout the last few decades. In this article we introduce a new extension of the Kumaraswamy exponential (KE) [...] Read more.

Statistical models are useful in explaining and forecasting real-world occurrences. Various extended distributions have been widely employed for modeling data in a variety of fields throughout the last few decades. In this article we introduce a new extension of the Kumaraswamy exponential (KE) model called the Kavya–Manoharan KE

(K M K E)

distribution. Some statistical and computational features of the

K M K E

distribution including the quantile (

Q U A

) function, moments (

M O_{m}

s), incomplete

M O_{m}

s (

I N M O_{m}

s), conditional

M O_{m}

s (

C O M O_{m}

s) and

M O_{m}

generating functions are computed. Classical maximum likelihood and Bayesian estimation approaches are employed to estimate the parameters of the KMKE model. The simulation experiment examines the accuracy of the model parameters by employing Bayesian and maximum likelihood estimation methods. We utilize two real datasets related to food chain data in this work to demonstrate the importance and flexibility of the proposed model. The new KMKE proposed distribution is very flexible, more so than numerous well-known distributions. Full article

(This article belongs to the Special Issue Mathematical Modelling in Sustainable Global Supply Chain Management)

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13 pages, 424 KiB

Open AccessArticle

Reinsurance Policy under Interest Force and Bankruptcy Prohibition

by Yangmin Zhong and Huaping Huang

Axioms 2023, 12(4), 378; https://doi.org/10.3390/axioms12040378 - 16 Apr 2023

Viewed by 1237

Abstract

In this paper, we solve an optimal reinsurance problem in the mathematical finance area. We assume that the surplus process of the insurance company follows a controlled diffusion process and the constant interest rate is involved in the financial model. During the whole [...] Read more.

In this paper, we solve an optimal reinsurance problem in the mathematical finance area. We assume that the surplus process of the insurance company follows a controlled diffusion process and the constant interest rate is involved in the financial model. During the whole optimization period, the company has a choice to buy reinsurance contract and decide the reinsurance retention level. Meanwhile, the bankruptcy at the terminal time is not allowed. The aim of the optimization problem is to minimize the distance between the terminal wealth and a given goal by controlling the reinsurance proportion. Using the stochastic control theory, we derive the Hamilton-Jacobi-Bellman equation for the optimization problem. Via adopting the technique of changing variable as well as the dual transformation, an explicit solution of the value function and the optimal policy are shown. Finally, several numerical examples are shown, from which we find several main factors that affect the optimal reinsurance policy. Full article

(This article belongs to the Special Issue A Themed Issue on Mathematical Inequalities, Analytic Combinatorics and Related Topics in Honor of Professor Feng Qi)

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12 pages, 323 KiB

Open AccessArticle

A Note on Finite Coarse Shape Groups

by Ivan Jelić and Nikola Koceić-Bilan

Axioms 2023, 12(4), 377; https://doi.org/10.3390/axioms12040377 - 14 Apr 2023

Cited by 1 | Viewed by 771

Abstract

In this paper, we investigate properties concerning some recently introduced finite coarse shape invariants—the k-th finite coarse shape group of a pointed topological space and the k-th relative finite coarse shape group of a pointed topological pair. We define the notion [...] Read more.

In this paper, we investigate properties concerning some recently introduced finite coarse shape invariants—the k-th finite coarse shape group of a pointed topological space and the k-th relative finite coarse shape group of a pointed topological pair. We define the notion of finite coarse shape group sequence of a pointed topological pair

(X, X_{0}, x_{0})

as an analogue of homotopy and (coarse) shape group sequences and show that for any pointed topological pair, the corresponding finite coarse shape group sequence is a chain. On the other hand, we construct an example of a pointed pair of metric continua whose finite coarse shape group sequence fails to be exact. Finally, using the aforementioned pair of metric continua together with a pointed dyadic solenoid, we show that finite coarse-shape groups, in general, differ from both shape and coarse-shape groups. Full article

(This article belongs to the Special Issue Advances in General Topology and Its Application)

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Axioms, Volume 12, Issue 4 (April 2023) – 90 articles

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