The aim of this paper is to present and study nonlinear bivariate C1 quadratic spline quasi-interpolants on uniform criss-cross triangulations for the approximation of piecewise smooth functions. Indeed, by using classical spline quasi-interpolants, the Gibbs phenomenon appears when approximating near discontinuities. Here, we use weighted essentially non-oscillatory techniques to modify classical quasi-interpolants in order to avoid oscillations near discontinuities and maintain high-order accuracy in smooth regions. We study the convergence properties of the proposed quasi-interpolants and we provide some numerical and graphical tests confirming the theoretical results.
{"title":"Nonlinear 2D C1 Quadratic Spline Quasi-Interpolants on Triangulations for the Approximation of Piecewise Smooth Functions","authors":"Francesc Aràndiga, Sara Remogna","doi":"10.3390/axioms12101002","DOIUrl":"https://doi.org/10.3390/axioms12101002","url":null,"abstract":"The aim of this paper is to present and study nonlinear bivariate C1 quadratic spline quasi-interpolants on uniform criss-cross triangulations for the approximation of piecewise smooth functions. Indeed, by using classical spline quasi-interpolants, the Gibbs phenomenon appears when approximating near discontinuities. Here, we use weighted essentially non-oscillatory techniques to modify classical quasi-interpolants in order to avoid oscillations near discontinuities and maintain high-order accuracy in smooth regions. We study the convergence properties of the proposed quasi-interpolants and we provide some numerical and graphical tests confirming the theoretical results.","PeriodicalId":53148,"journal":{"name":"Axioms","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135366994","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the explainable artificial intelligence (XAI) field, an algorithm or a tool can help people understand how a model makes a decision. And this can help to select important features to reduce computational costs to realize high-performance computing. But existing methods are usually used to visualize important features or highlight active neurons, and few of them show the importance of relationships between features. In recent years, some methods based on a white-box approach have taken relationships between features into account, but most of them can only work on some specific models. Although methods based on a black-box approach can solve the above problems, most of them can only be applied to tabular data or text data instead of image data. To solve these problems, we propose a local interpretable model-agnostic explanation approach based on feature relationships. This approach combines the relationships between features into the interpretation process and then visualizes the interpretation results. Finally, this paper conducts a lot of experiments to evaluate the correctness of relationships between features and evaluates this XAI method in terms of accuracy, fidelity, and consistency.
{"title":"Interpretable Model-Agnostic Explanations Based on Feature Relationships for High-Performance Computing","authors":"Zhouyuan Chen, Zhichao Lian, Zhe Xu","doi":"10.3390/axioms12100997","DOIUrl":"https://doi.org/10.3390/axioms12100997","url":null,"abstract":"In the explainable artificial intelligence (XAI) field, an algorithm or a tool can help people understand how a model makes a decision. And this can help to select important features to reduce computational costs to realize high-performance computing. But existing methods are usually used to visualize important features or highlight active neurons, and few of them show the importance of relationships between features. In recent years, some methods based on a white-box approach have taken relationships between features into account, but most of them can only work on some specific models. Although methods based on a black-box approach can solve the above problems, most of them can only be applied to tabular data or text data instead of image data. To solve these problems, we propose a local interpretable model-agnostic explanation approach based on feature relationships. This approach combines the relationships between features into the interpretation process and then visualizes the interpretation results. Finally, this paper conducts a lot of experiments to evaluate the correctness of relationships between features and evaluates this XAI method in terms of accuracy, fidelity, and consistency.","PeriodicalId":53148,"journal":{"name":"Axioms","volume":"6 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135405302","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the optimality conditions for set optimization problems with set criterion. Firstly, we establish a few important properties of the Minkowski difference for sets. Then, we introduce the generalized second-order lower radial epiderivative for a set-valued maps by Minkowski difference, and discuss some of its properties. Finally, by virtue of the generalized second-order lower radial epiderivatives and the generalized second-order radial epiderivatives, we establish the necessary optimality conditions and sufficient optimality conditions of approximate Benson proper efficient solutions and approximate weakly minimal solutions of unconstrained set optimization problems without convexity conditions, respectively. Some examples are provided to illustrate the main results obtained.
{"title":"Optimality Conditions for Approximate Solutions of Set Optimization Problems with the Minkowski Difference","authors":"Yuhe Zhang, Qilin Wang","doi":"10.3390/axioms12101001","DOIUrl":"https://doi.org/10.3390/axioms12101001","url":null,"abstract":"In this paper, we study the optimality conditions for set optimization problems with set criterion. Firstly, we establish a few important properties of the Minkowski difference for sets. Then, we introduce the generalized second-order lower radial epiderivative for a set-valued maps by Minkowski difference, and discuss some of its properties. Finally, by virtue of the generalized second-order lower radial epiderivatives and the generalized second-order radial epiderivatives, we establish the necessary optimality conditions and sufficient optimality conditions of approximate Benson proper efficient solutions and approximate weakly minimal solutions of unconstrained set optimization problems without convexity conditions, respectively. Some examples are provided to illustrate the main results obtained.","PeriodicalId":53148,"journal":{"name":"Axioms","volume":"2005 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135366701","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work we show that if iterates of a nonexpansive self-mapping of a complete metric with a graph converge uniformly on a subset of the space, then this convergence is stable under the presence of small computational errors.
{"title":"Two Convergence Results for Inexact Orbits of Nonexpansive Operators in Metric Spaces with Graphs","authors":"Alexander J. Zaslavski","doi":"10.3390/axioms12100999","DOIUrl":"https://doi.org/10.3390/axioms12100999","url":null,"abstract":"In this work we show that if iterates of a nonexpansive self-mapping of a complete metric with a graph converge uniformly on a subset of the space, then this convergence is stable under the presence of small computational errors.","PeriodicalId":53148,"journal":{"name":"Axioms","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135413233","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Miodrag Mateljević, Nikola Mutavdžić, Adel Khalfallah
In this paper we investigate the solutions of the so-called α¯-Poisson equation in the complex plane. In particular, we will give sufficient conditions for Lipschitz continuity of such solutions. We also review some recently obtained results. As a corollary, we can restate results for harmonic and (p,q)-harmonic functions.
{"title":"Lipschitz Continuity for Harmonic Functions and Solutions of the α¯-Poisson Equation","authors":"Miodrag Mateljević, Nikola Mutavdžić, Adel Khalfallah","doi":"10.3390/axioms12100998","DOIUrl":"https://doi.org/10.3390/axioms12100998","url":null,"abstract":"In this paper we investigate the solutions of the so-called α¯-Poisson equation in the complex plane. In particular, we will give sufficient conditions for Lipschitz continuity of such solutions. We also review some recently obtained results. As a corollary, we can restate results for harmonic and (p,q)-harmonic functions.","PeriodicalId":53148,"journal":{"name":"Axioms","volume":"40 8","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135413537","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the literature, it is usually assumed that Leibniz described proof for the Fundamental Theorem of Calculus (FTC) in 1693. However, did he really prove it? If the answer is no from today’s perspective, are there works in which Leibniz introduced arguments that can be understood as formulations and justifications of the FTC? In order to answer this question, we used a historiographic methodology with expert triangulation. From the study of Leibniz’s manuscripts describing the inverse problem of tangents and its relationship with the quadrature problem, we found evidence of a geometrical argument from which the FTC can be inferred. We present this argument using technological resources and modern notation. This result can be used to teach the FTC due to the existence of dynamic and geometrical software, which makes it suitable for the classroom. Moreover, it provides another interpretation of the FTC complementary to the interpretation using Riemann sums.
{"title":"A Visualization in GeoGebra of Leibniz’s Argument on the Fundamental Theorem of Calculus","authors":"Weimar Muñoz, Olga Lucía León, Vicenç Font","doi":"10.3390/axioms12101000","DOIUrl":"https://doi.org/10.3390/axioms12101000","url":null,"abstract":"In the literature, it is usually assumed that Leibniz described proof for the Fundamental Theorem of Calculus (FTC) in 1693. However, did he really prove it? If the answer is no from today’s perspective, are there works in which Leibniz introduced arguments that can be understood as formulations and justifications of the FTC? In order to answer this question, we used a historiographic methodology with expert triangulation. From the study of Leibniz’s manuscripts describing the inverse problem of tangents and its relationship with the quadrature problem, we found evidence of a geometrical argument from which the FTC can be inferred. We present this argument using technological resources and modern notation. This result can be used to teach the FTC due to the existence of dynamic and geometrical software, which makes it suitable for the classroom. Moreover, it provides another interpretation of the FTC complementary to the interpretation using Riemann sums.","PeriodicalId":53148,"journal":{"name":"Axioms","volume":"53 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135366880","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The space, Fcc(R), of all fuzzy intervals in R cannot form a vector space. However, the space Fcc(R) maintains a vector structure by treating the addition of fuzzy intervals as a vector addition and treating the scalar multiplication of fuzzy intervals as a scalar multiplication of vectors. The only difficulty in taking care of Fcc(R) is missing the additive inverse element. This means that each fuzzy interval that is subtracted from itself cannot be a zero element in Fcc(R). Although Fcc(R) cannot form a vector space, we still can endow a norm on the space Fcc(R) by following its vector structure. Under this setting, many different types of open sets can be proposed by using the different types of open balls. The purpose of this paper is to study the topologies generated by these different types of open sets.
{"title":"Normed Space of Fuzzy Intervals and Its Topological Structure","authors":"Hsien-Chung Wu","doi":"10.3390/axioms12100996","DOIUrl":"https://doi.org/10.3390/axioms12100996","url":null,"abstract":"The space, Fcc(R), of all fuzzy intervals in R cannot form a vector space. However, the space Fcc(R) maintains a vector structure by treating the addition of fuzzy intervals as a vector addition and treating the scalar multiplication of fuzzy intervals as a scalar multiplication of vectors. The only difficulty in taking care of Fcc(R) is missing the additive inverse element. This means that each fuzzy interval that is subtracted from itself cannot be a zero element in Fcc(R). Although Fcc(R) cannot form a vector space, we still can endow a norm on the space Fcc(R) by following its vector structure. Under this setting, many different types of open sets can be proposed by using the different types of open balls. The purpose of this paper is to study the topologies generated by these different types of open sets.","PeriodicalId":53148,"journal":{"name":"Axioms","volume":"44 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135461723","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
To quantitatively identify internal wire breakage damage in mining wire ropes, a wire rope internal wire breakage signal identification method is proposed. First, the whale optimization algorithm is used to find the optimal value of the variational mode decomposition parameter [K, α] to obtain the optimal combination of the parameters, which reduces the signal noise with a signal-to-noise ratio of 29.29 dB. Second, the minimum envelope entropy of the noise reduction signal is extracted and combined with the time-domain features (maximum and minimum) and frequency-domain features (frequency–amplitude average, average frequency, average power) to form a fusion feature set. Finally, we use a particle swarm optimization–least squares support vector machine model to identify the internal wire breakage of wire ropes. The experimental results show that the method can effectively identify the internal wire rope breakage damage, and the average recognition rate is as high as 99.32%, so the algorithm can greatly reduce the system noise and effectively identify the internal damage signal of the wire rope, which is superior to a certain extent.
{"title":"Detection of Internal Wire Broken in Mining Wire Ropes Based on WOA–VMD and PSO–LSSVM Algorithms","authors":"Pengbo Li, Jie Tian, Zeyang Zhou, Wei Wang","doi":"10.3390/axioms12100995","DOIUrl":"https://doi.org/10.3390/axioms12100995","url":null,"abstract":"To quantitatively identify internal wire breakage damage in mining wire ropes, a wire rope internal wire breakage signal identification method is proposed. First, the whale optimization algorithm is used to find the optimal value of the variational mode decomposition parameter [K, α] to obtain the optimal combination of the parameters, which reduces the signal noise with a signal-to-noise ratio of 29.29 dB. Second, the minimum envelope entropy of the noise reduction signal is extracted and combined with the time-domain features (maximum and minimum) and frequency-domain features (frequency–amplitude average, average frequency, average power) to form a fusion feature set. Finally, we use a particle swarm optimization–least squares support vector machine model to identify the internal wire breakage of wire ropes. The experimental results show that the method can effectively identify the internal wire rope breakage damage, and the average recognition rate is as high as 99.32%, so the algorithm can greatly reduce the system noise and effectively identify the internal damage signal of the wire rope, which is superior to a certain extent.","PeriodicalId":53148,"journal":{"name":"Axioms","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135511411","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we investigate a delayed matrix exponential and utilize it to derive a representation of solutions to a linear nonsingular delay problem with permutable matrices. To begin with, we present a novel definition of α-exponential stability for these systems. Subsequently, we put forward several adequate conditions to ensure the α-exponential stability of solutions for such delay systems. Moreover, by constructing a Grammian matrix that accounts for delays, we provide a criterion to determine the relative controllability of a linear problem. Additionally, we extend our analysis to nonlinear problems. Lastly, we offer several examples to verify the effectiveness of our theoretical findings.
{"title":"Exponential Stability and Relative Controllability of Nonsingular Conformable Delay Systems","authors":"Airen Zhou","doi":"10.3390/axioms12100994","DOIUrl":"https://doi.org/10.3390/axioms12100994","url":null,"abstract":"In this paper, we investigate a delayed matrix exponential and utilize it to derive a representation of solutions to a linear nonsingular delay problem with permutable matrices. To begin with, we present a novel definition of α-exponential stability for these systems. Subsequently, we put forward several adequate conditions to ensure the α-exponential stability of solutions for such delay systems. Moreover, by constructing a Grammian matrix that accounts for delays, we provide a criterion to determine the relative controllability of a linear problem. Additionally, we extend our analysis to nonlinear problems. Lastly, we offer several examples to verify the effectiveness of our theoretical findings.","PeriodicalId":53148,"journal":{"name":"Axioms","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135569244","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Amal Alshabanat, Eman Almoalim, Mohamed Jleli, Bessem Samet
Several phenomena from natural sciences can be described by partial differential equations of Sobolev-type. On the other hand, it was shown that in many cases, the use of fractional derivatives provides a more realistic model than the use of standard derivatives. The goal of this paper is to study the nonexistence of weak solutions to a time-fractional differential inequality of Sobolev-type. Namely, we give sufficient conditions for the nonexistence or equivalently necessary conditions for the existence. Our method makes use of the nonlinear capacity method, which consists in making an appropriate choice of test functions in the weak formulation of the problem. This technique has been employed in previous papers for some classes of time-fractional differential inequalities of Sobolev-type posed on the whole space RN. The originality of this work is that the considered problem is posed on an annulus domain, which leads to some difficulties concerning the choice of adequate test functions.
{"title":"A Time-Fractional Differential Inequality of Sobolev Type on an Annulus","authors":"Amal Alshabanat, Eman Almoalim, Mohamed Jleli, Bessem Samet","doi":"10.3390/axioms12100993","DOIUrl":"https://doi.org/10.3390/axioms12100993","url":null,"abstract":"Several phenomena from natural sciences can be described by partial differential equations of Sobolev-type. On the other hand, it was shown that in many cases, the use of fractional derivatives provides a more realistic model than the use of standard derivatives. The goal of this paper is to study the nonexistence of weak solutions to a time-fractional differential inequality of Sobolev-type. Namely, we give sufficient conditions for the nonexistence or equivalently necessary conditions for the existence. Our method makes use of the nonlinear capacity method, which consists in making an appropriate choice of test functions in the weak formulation of the problem. This technique has been employed in previous papers for some classes of time-fractional differential inequalities of Sobolev-type posed on the whole space RN. The originality of this work is that the considered problem is posed on an annulus domain, which leads to some difficulties concerning the choice of adequate test functions.","PeriodicalId":53148,"journal":{"name":"Axioms","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135569389","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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