{"title":"Retracted: Probability Density Evolution Algorithm for Stochastic Dynamical Systems Based on Fractional Calculus","authors":"Journal of Mathematics","doi":"10.1155/2023/9783817","DOIUrl":"https://doi.org/10.1155/2023/9783817","url":null,"abstract":"<jats:p />","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"39 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85234506","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Retracted: Analysis and Risk Assessment of Corporate Financial Leverage Using Mobile Payment in the Era of Digital Technology in a Complex Environment","authors":"Journal of Mathematics","doi":"10.1155/2023/9852708","DOIUrl":"https://doi.org/10.1155/2023/9852708","url":null,"abstract":"<jats:p />","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"54 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80057930","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this study, for graph with connected components (also for connected nonbipartite and connected bipartite graphs) and a real number , we found generalized and improved bounds for the sum of -th powers of Laplacian and signless Laplacian eigenvalues of . Consequently, we also generalized and improved results on incidence energy and Laplacian energy-like invariant .
在这项研究中，对于具有r连通分量的图Γ(也适用于连通非二部图和连通二部图)，且实数ε≠0,1得到了Γ的拉普拉斯特征值和无符号拉普拉斯特征值ε -幂和的广义和改进界。因此,我们还推广和改进了有关入射能E和拉普拉斯类能不变量L E L的结果。
{"title":"(Generalized) Incidence and Laplacian-Like Energies","authors":"A. D. Maden, M. T. Rahim","doi":"10.1155/2023/6205632","DOIUrl":"https://doi.org/10.1155/2023/6205632","url":null,"abstract":"<jats:p>In this study, for graph <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\">\u0000 <mi mathvariant=\"normal\">Γ</mi>\u0000 </math>\u0000 </jats:inline-formula> with <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\">\u0000 <mi>r</mi>\u0000 </math>\u0000 </jats:inline-formula> connected components (also for connected nonbipartite and connected bipartite graphs) and a real number <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\">\u0000 <mi>ε</mi>\u0000 <mfenced open=\"(\" close=\")\" separators=\"|\">\u0000 <mrow>\u0000 <mo>≠</mo>\u0000 <mn>0,1</mn>\u0000 </mrow>\u0000 </mfenced>\u0000 </math>\u0000 </jats:inline-formula>, we found generalized and improved bounds for the sum of <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M4\">\u0000 <mi>ε</mi>\u0000 </math>\u0000 </jats:inline-formula>-th powers of Laplacian and signless Laplacian eigenvalues of <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M5\">\u0000 <mi mathvariant=\"normal\">Γ</mi>\u0000 </math>\u0000 </jats:inline-formula>. Consequently, we also generalized and improved results on incidence energy <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M6\">\u0000 <mfenced open=\"(\" close=\")\" separators=\"|\">\u0000 <mrow>\u0000 <mi mathvariant=\"normal\">I</mi>\u0000 <mi mathvariant=\"normal\">E</mi>\u0000 </mrow>\u0000 </mfenced>\u0000 </math>\u0000 </jats:inline-formula> and Laplacian energy-like invariant <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M7\">\u0000 <mfenced open=\"(\" close=\")\" separators=\"|\">\u0000 <mrow>\u0000 <mi mathvariant=\"normal\">L</mi>\u0000 <mi mathvariant=\"normal\">E</mi>\u0000 <mi mathvariant=\"normal\">L</mi>\u0000 </mrow>\u0000 </mfenced>\u0000 </math>\u0000 </jats:inline-formula>.</jats:p>","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84895309","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Tabraiz, Zeeshan Saleem Mufti, Muhammad Nauman Aslam, N. Saleem, H. Hosseinzadeh
Fuzzy graphs have many applications not only in mathematics but also in any field of science where the concept of fuzziness is involved. The notion of fuzziness is suitable in any environment, which favor to predicts the problem and solve this problem in a decent way. As compared to crisp theory, fuzzy graphs are a more beneficial and powerful tool to get better accuracy and precision due to their fuzziness property. A topological index is a numerical value which characterizes the properties of the graph. Topological indices were basically developed for chemical structures, but these are also used for general graphs as well. In chemical graph theory, topological indices are used to extract the chemical properties of the graphs. These indices are also well studied in fuzzy environment. Applications of fuzzy graphs are found in medicines, telecommunications, traffic light control, and many more. Our aim is to find these fuzzy topological indices for flower graphs to strengthen the concepts of fuzziness in general graphs. In this paper, some novel results for f m × r flower graphs are achieved.
{"title":"Fuzzy Computational Analysis of Flower Graph via Fuzzy Topological Indices","authors":"A. Tabraiz, Zeeshan Saleem Mufti, Muhammad Nauman Aslam, N. Saleem, H. Hosseinzadeh","doi":"10.1155/2023/8077729","DOIUrl":"https://doi.org/10.1155/2023/8077729","url":null,"abstract":"Fuzzy graphs have many applications not only in mathematics but also in any field of science where the concept of fuzziness is involved. The notion of fuzziness is suitable in any environment, which favor to predicts the problem and solve this problem in a decent way. As compared to crisp theory, fuzzy graphs are a more beneficial and powerful tool to get better accuracy and precision due to their fuzziness property. A topological index is a numerical value which characterizes the properties of the graph. Topological indices were basically developed for chemical structures, but these are also used for general graphs as well. In chemical graph theory, topological indices are used to extract the chemical properties of the graphs. These indices are also well studied in fuzzy environment. Applications of fuzzy graphs are found in medicines, telecommunications, traffic light control, and many more. Our aim is to find these fuzzy topological indices for flower graphs to strengthen the concepts of fuzziness in general graphs. In this paper, some novel results for \u0000 \u0000 \u0000 \u0000 f\u0000 \u0000 \u0000 m\u0000 ×\u0000 r\u0000 \u0000 \u0000 \u0000 flower graphs are achieved.","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"55 36 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88495771","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Complicated uncertainties arising in the multicriteria decision-making (MCDM) problems that show distinct possible satisfaction of the subjects to favorable and equally unfavorable parameters with varying preferences require reliable decision-making under comprehensive mathematical tools. For such complications, this work aims to develop a novel fuzzy parameterized possibility fuzzy bipolar soft set model as a fuzzy parameterized bipolar soft extension of possibility fuzzy sets. The proposed model efficiently depicts the possibility of fuzzy belongingness of alternatives under fuzzy parameterized bipolar parameters (or attributes). The respective operations and properties such as subset, complement, union, and intersection are presented along with their numerical illustrations. Two logical operations namely “AND” and “OR” operations followed by two corresponding MCDM algorithms have been developed and implemented. Furthermore, similarity measures between fuzzy parameterized possibility fuzzy bipolar soft sets are proposed and applied to an agricultural land selection scenario. Finally, a comparative analysis of current work with existing ones is discussed in detail to show the eminent quality of the proposed work over them.
{"title":"Novel MCDM Methods and Similarity Measures for Extended Fuzzy Parameterized Possibility Fuzzy Soft Information with Their Applications","authors":"Ghous Ali","doi":"10.1155/2023/5035347","DOIUrl":"https://doi.org/10.1155/2023/5035347","url":null,"abstract":"Complicated uncertainties arising in the multicriteria decision-making (MCDM) problems that show distinct possible satisfaction of the subjects to favorable and equally unfavorable parameters with varying preferences require reliable decision-making under comprehensive mathematical tools. For such complications, this work aims to develop a novel fuzzy parameterized possibility fuzzy bipolar soft set model as a fuzzy parameterized bipolar soft extension of possibility fuzzy sets. The proposed model efficiently depicts the possibility of fuzzy belongingness of alternatives under fuzzy parameterized bipolar parameters (or attributes). The respective operations and properties such as subset, complement, union, and intersection are presented along with their numerical illustrations. Two logical operations namely “AND” and “OR” operations followed by two corresponding MCDM algorithms have been developed and implemented. Furthermore, similarity measures between fuzzy parameterized possibility fuzzy bipolar soft sets are proposed and applied to an agricultural land selection scenario. Finally, a comparative analysis of current work with existing ones is discussed in detail to show the eminent quality of the proposed work over them.","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"22 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89369907","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Photovoltaic thermal systems (PV/T) are devices used to collect both electrical and thermal energies from solar energy. By passing a coolant through flow channels that are connected to the PV/T systems, the temperature of the cells is reduced to enhance their electrical efficiency. Therefore, this study aims at investigating a photovoltaic thermal system via the transport of hybrid nanofluids based on Cu-Al2O3/water. We assume that the flow channel can be considered in two dimensions and is composed of the silicon panel, absorber, and flow channel. The flow channel consists of a cavity along the absorber with a fixed length and a certain height. This will be a combined conduction and convection problem, with conduction occurring on the top two layers of the silicon panel and absorber. Modeling and simulation problems are performed using COMSOL Multiphysics 5.6. The aspect ratio from inlet height to cavity height is defined by Ar, and the volume fraction of Al2O3 is taken double that of Cu. The cell efficiency is analyzed by performing a parametric study by altering the Reynold number (100–1000), inlet temperature (50°C–450°C), the volume fraction of copper (0.01%–10%), and the aspect ratio (0.5, 0.7, 0.9, and 1). It is found that increasing the inlet temperature and aspect ratio decreases the cell efficiency while increasing the Reynolds number and volume fraction increases it. The maximum efficiency of the cell, about 6%, is achieved when the inlet temperature is 50°C, the volume fraction of copper is 10%, Re = 1000, and Ar = 0.5. It was also concluded that when the volume fraction of copper is 0.1, the increase in Reynolds number from 100 to 1000 is improving the cell efficiency by 0.5%. On the other hand, when increasing the volume fraction of copper from minimum to maximum at Re = 1000, the cell efficiency is increasing by 0.3%.
{"title":"Numerical Investigation of Electrical Efficiency with the Application of Hybrid Nanofluids for Photovoltaic Thermal Systems Contained in a Cavity Channel","authors":"A. A. Memon, M. A. Memon, M. Haque","doi":"10.1155/2023/5465847","DOIUrl":"https://doi.org/10.1155/2023/5465847","url":null,"abstract":"Photovoltaic thermal systems (PV/T) are devices used to collect both electrical and thermal energies from solar energy. By passing a coolant through flow channels that are connected to the PV/T systems, the temperature of the cells is reduced to enhance their electrical efficiency. Therefore, this study aims at investigating a photovoltaic thermal system via the transport of hybrid nanofluids based on Cu-Al2O3/water. We assume that the flow channel can be considered in two dimensions and is composed of the silicon panel, absorber, and flow channel. The flow channel consists of a cavity along the absorber with a fixed length and a certain height. This will be a combined conduction and convection problem, with conduction occurring on the top two layers of the silicon panel and absorber. Modeling and simulation problems are performed using COMSOL Multiphysics 5.6. The aspect ratio from inlet height to cavity height is defined by Ar, and the volume fraction of Al2O3 is taken double that of Cu. The cell efficiency is analyzed by performing a parametric study by altering the Reynold number (100–1000), inlet temperature (50°C–450°C), the volume fraction of copper (0.01%–10%), and the aspect ratio (0.5, 0.7, 0.9, and 1). It is found that increasing the inlet temperature and aspect ratio decreases the cell efficiency while increasing the Reynolds number and volume fraction increases it. The maximum efficiency of the cell, about 6%, is achieved when the inlet temperature is 50°C, the volume fraction of copper is 10%, Re = 1000, and Ar = 0.5. It was also concluded that when the volume fraction of copper is 0.1, the increase in Reynolds number from 100 to 1000 is improving the cell efficiency by 0.5%. On the other hand, when increasing the volume fraction of copper from minimum to maximum at Re = 1000, the cell efficiency is increasing by 0.3%.","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"3 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87951762","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Ginzburg–Landau (GL) equation and the Ginzburg–Landau couple system are important models in the study of superconductivity and superfluidity. This study describes the q-homotopy analysis transform method (q-HATM) as a powerful technique for solving nonlinear problems, which has been successfully used with a set of mathematical models in physics, engineering, and biology. We apply the q-HATM to solve the Ginzburg–Landau equation and the Ginzburg–Landau coupled system and derive analytical solutions in terms of the q-series. Also, we investigate the convergence and accuracy of the obtained solutions. Our results show that q-HATM is a reliable and promising approach for solving nonlinear differential equations and provides a valuable tool for researchers in the field of superconductivity. Several graphs have been presented for the solutions obtained utilizing different levels of the fractional-order derivative and at various points in time.
{"title":"Solving Nonlinear Fractional Models in Superconductivity Using the q-Homotopy Analysis Transform Method","authors":"K. Ali, M. Maneea, M. Mohamed","doi":"10.1155/2023/6647375","DOIUrl":"https://doi.org/10.1155/2023/6647375","url":null,"abstract":"The Ginzburg–Landau (GL) equation and the Ginzburg–Landau couple system are important models in the study of superconductivity and superfluidity. This study describes the q-homotopy analysis transform method (q-HATM) as a powerful technique for solving nonlinear problems, which has been successfully used with a set of mathematical models in physics, engineering, and biology. We apply the q-HATM to solve the Ginzburg–Landau equation and the Ginzburg–Landau coupled system and derive analytical solutions in terms of the q-series. Also, we investigate the convergence and accuracy of the obtained solutions. Our results show that q-HATM is a reliable and promising approach for solving nonlinear differential equations and provides a valuable tool for researchers in the field of superconductivity. Several graphs have been presented for the solutions obtained utilizing different levels of the fractional-order derivative and at various points in time.","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"25 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90768720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Selecting the most suitable activation function is a critical factor in the effectiveness of deep learning models, as it influences their learning capacity, stability, and computational efficiency. In recent years, the Gaussian error linear unit (GELU) activation function has emerged as a dominant method, surpassing traditional functions such as the rectified linear unit (ReLU) in various applications. This study presents a rigorous mathematical investigation of the GELU activation function, exploring its differentiability, boundedness, stationarity, and smoothness properties in detail. In addition, we conduct an extensive experimental comparison of the GELU function against a broad range of alternative activation functions, utilizing a residual convolutional network trained on the CIFAR-10, CIFAR-100, and STL-10 datasets as the empirical testbed. Our results demonstrate the superior performance of GELU compared to other activation functions, establishing its suitability for a wide range of deep learning applications. This comprehensive study contributes to a more profound understanding of the underlying mathematical properties of GELU and provides valuable insights for practitioners aiming to select activation functions that optimally align with their specific objectives and constraints in deep learning.
{"title":"Mathematical Analysis and Performance Evaluation of the GELU Activation Function in Deep Learning","authors":"Minhyeok Lee","doi":"10.1155/2023/4229924","DOIUrl":"https://doi.org/10.1155/2023/4229924","url":null,"abstract":"Selecting the most suitable activation function is a critical factor in the effectiveness of deep learning models, as it influences their learning capacity, stability, and computational efficiency. In recent years, the Gaussian error linear unit (GELU) activation function has emerged as a dominant method, surpassing traditional functions such as the rectified linear unit (ReLU) in various applications. This study presents a rigorous mathematical investigation of the GELU activation function, exploring its differentiability, boundedness, stationarity, and smoothness properties in detail. In addition, we conduct an extensive experimental comparison of the GELU function against a broad range of alternative activation functions, utilizing a residual convolutional network trained on the CIFAR-10, CIFAR-100, and STL-10 datasets as the empirical testbed. Our results demonstrate the superior performance of GELU compared to other activation functions, establishing its suitability for a wide range of deep learning applications. This comprehensive study contributes to a more profound understanding of the underlying mathematical properties of GELU and provides valuable insights for practitioners aiming to select activation functions that optimally align with their specific objectives and constraints in deep learning.","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"34 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79994029","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The S I R D V (Susceptible, Infected, Recovered, Death, Vaccinated) compartmental model along with time-varying parameters is used to model the spread of COVID-19 in the United States. Time-varying parameters account for changes in transmission rates, people’s behaviors, safety precautions, government regulations, the rate of vaccinations, and also the probabilities of recovery and death. By using a parameter estimation based on the simplex algorithm, the system of differential equations is able to match real COVID-19 data for infections, deaths, and vaccinations in the United States of America with relatively high precision. Autoregression is used to forecast parameters in order to forecast solutions. Van den Driessche’s next-generation approach for basic reproduction number agrees well across the entire time period. Analyses on sensitivity and elasticity are performed on the reproduction number with respect to transmission, exit, and natural death rates in order to observe the changes from a small change in parameter values. Model validation through the Akaike Information Criterion ensures that the model is suitable and optimal for modeling the spread of COVID-19.
S I R D V(易感、感染、康复、死亡、接种)区隔模型以及时变参数用于模拟COVID-19在美国的传播。时变参数考虑了传播率、人们的行为、安全预防措施、政府法规、疫苗接种率以及康复和死亡概率的变化。通过基于单纯形算法的参数估计，微分方程系统能够以相对较高的精度匹配美国COVID-19感染、死亡和疫苗接种的真实数据。为了预测解，采用自回归方法对参数进行预测。Van den Driessche的下一代基本繁殖数方法在整个时期都很一致。为了从参数值的微小变化中观察变化，对繁殖数与传播率、退出率和自然死亡率进行了敏感性和弹性分析。通过赤池信息准则对模型进行验证，确保模型适合和最优地建模COVID-19的传播。
{"title":"Modeling the Spread of COVID-19 Using Nonautonomous Dynamical System with Simplex Algorithm-Based Optimization for Time-Varying Parameters","authors":"Kevin Yotongyos, Somchai Sriyab","doi":"10.1155/2023/6156749","DOIUrl":"https://doi.org/10.1155/2023/6156749","url":null,"abstract":"The \u0000 \u0000 S\u0000 I\u0000 R\u0000 D\u0000 V\u0000 \u0000 (Susceptible, Infected, Recovered, Death, Vaccinated) compartmental model along with time-varying parameters is used to model the spread of COVID-19 in the United States. Time-varying parameters account for changes in transmission rates, people’s behaviors, safety precautions, government regulations, the rate of vaccinations, and also the probabilities of recovery and death. By using a parameter estimation based on the simplex algorithm, the system of differential equations is able to match real COVID-19 data for infections, deaths, and vaccinations in the United States of America with relatively high precision. Autoregression is used to forecast parameters in order to forecast solutions. Van den Driessche’s next-generation approach for basic reproduction number agrees well across the entire time period. Analyses on sensitivity and elasticity are performed on the reproduction number with respect to transmission, exit, and natural death rates in order to observe the changes from a small change in parameter values. Model validation through the Akaike Information Criterion ensures that the model is suitable and optimal for modeling the spread of COVID-19.","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"32 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83949922","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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