In the era of precision medicine, biomarker plays a vital role in drug clinical trials. It helps select the patients more likely to respond to the therapy and increases the possibility of success of the trial. Model selection is critical in the development of the algorithm. Traditional model selection metrics ignore two clinical utilities of the biomarker in drug clinical trials, one is its ability to distinguish positive and negative patients in terms of treatment effect and another is the total cost of the biomarker-based drug clinical trial. We proposed a new model selection metric that estimates the above two clinical utilities of biomarker detection algorithms without the need for a real drug clinical trial. In the simulation, we will compare the proposed metric with the widely used ROC-based metric in selecting the optimal cutoff value for the model and discuss which one to choose under various circumstances.
{"title":"A New Model Selection Metric for Biomarker Detection Algorithms and Tools","authors":"Bo Feng, Yubo Sun, B. Zee","doi":"10.1155/2023/8263804","DOIUrl":"https://doi.org/10.1155/2023/8263804","url":null,"abstract":"In the era of precision medicine, biomarker plays a vital role in drug clinical trials. It helps select the patients more likely to respond to the therapy and increases the possibility of success of the trial. Model selection is critical in the development of the algorithm. Traditional model selection metrics ignore two clinical utilities of the biomarker in drug clinical trials, one is its ability to distinguish positive and negative patients in terms of treatment effect and another is the total cost of the biomarker-based drug clinical trial. We proposed a new model selection metric that estimates the above two clinical utilities of biomarker detection algorithms without the need for a real drug clinical trial. In the simulation, we will compare the proposed metric with the widely used ROC-based metric in selecting the optimal cutoff value for the model and discuss which one to choose under various circumstances.","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"25 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81676186","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}
Muhammad Nadeem, I. Siddique, Md. Ashraful Alam, Waqas Ali
The most recent advancements in algebra and graph theory enable us to ask a straightforward question: what practical use does this graph connected with a mathematical system have in the real world? With the use of algebraic approaches, we may now tackle a wide range of graph theory-related problems. We compute loop-involutions and nonself-loop-involutions of flexible weak inverse property loops. Importantly, diameter, radius, and eccentricity of this loop structure’s inverse graph have all been calculated in a broad manner. Some topological indices, as an application of inverse graph, are also given at the end of the paper.
{"title":"A New Graphical Representation of the Old Algebraic Structure","authors":"Muhammad Nadeem, I. Siddique, Md. Ashraful Alam, Waqas Ali","doi":"10.1155/2023/4333301","DOIUrl":"https://doi.org/10.1155/2023/4333301","url":null,"abstract":"The most recent advancements in algebra and graph theory enable us to ask a straightforward question: what practical use does this graph connected with a mathematical system have in the real world? With the use of algebraic approaches, we may now tackle a wide range of graph theory-related problems. We compute loop-involutions and nonself-loop-involutions of flexible weak inverse property loops. Importantly, diameter, radius, and eccentricity of this loop structure’s inverse graph have all been calculated in a broad manner. Some topological indices, as an application of inverse graph, are also given at the end of the paper.","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"41 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74340347","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}
With the gradual progress of interest rate marketization, China’s interest rate fluctuates more and more frequently, and the range is also growing. As a result, more and more implicit options are embedded in commercial banks’ balance sheets, which brings new challenges to commercial banks’ interest rate risk management. On the basis of identifying implicit options and theoretically analyzing the mechanism, fuzzy MCS method is used to calculate C e f f and D e f f when implicit options exist, and compared with the traditional duration value and traditional convexity value when implicit options do not exist, further analyzing how implicit options affect the interest rate risk.
随着利率市场化的逐步推进，中国利率的波动越来越频繁，幅度也越来越大。因此，越来越多的隐性期权嵌入到商业银行的资产负债表中，这给商业银行的利率风险管理带来了新的挑战。在对隐式期权进行识别和理论分析机理的基础上，采用模糊MCS方法计算隐式期权存在时的C e f和D e f，并与不存在隐式期权时的传统久期值和传统凸度值进行比较，进一步分析隐式期权对利率风险的影响。
{"title":"An Empirical Study on the Influence of Embedded Option on Interest Rate Risk Based on Fuzzy Monte Carlo Simulation","authors":"Enlin Tang, Zebin Liu","doi":"10.1155/2023/3966972","DOIUrl":"https://doi.org/10.1155/2023/3966972","url":null,"abstract":"With the gradual progress of interest rate marketization, China’s interest rate fluctuates more and more frequently, and the range is also growing. As a result, more and more implicit options are embedded in commercial banks’ balance sheets, which brings new challenges to commercial banks’ interest rate risk management. On the basis of identifying implicit options and theoretically analyzing the mechanism, fuzzy MCS method is used to calculate \u0000 \u0000 \u0000 \u0000 C\u0000 \u0000 \u0000 e\u0000 f\u0000 f\u0000 \u0000 \u0000 \u0000 and \u0000 \u0000 \u0000 \u0000 D\u0000 \u0000 \u0000 e\u0000 f\u0000 f\u0000 \u0000 \u0000 \u0000 when implicit options exist, and compared with the traditional duration value and traditional convexity value when implicit options do not exist, further analyzing how implicit options affect the interest rate risk.","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"82 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88598263","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 paper, we present some of the foundational concepts of split-complex number theory such as split-complex divison, gcd, and congruencies. Also, we prove that Euler’s theorem is still true in the case of split-complex integers, and we use this theorem to present a split-complex version of the RSA algorithm which is harder to be broken than the classical version. On the other hand, we study some algebraic properties of split-complex matrices, where we present the formula of computing the exponent of a split-complex matrix e X with a novel algorithm to represent a split-complex matrix X by a split-complex diagonal matrix, which is known as the diagonalization problem. In addition, many examples were illustrated to clarify the validity of our work.
{"title":"On Some Novel Results about Split-Complex Numbers, the Diagonalization Problem, and Applications to Public Key Asymmetric Cryptography","authors":"Mehmet Merkepci, Mohammad Abobala","doi":"10.1155/2023/4481016","DOIUrl":"https://doi.org/10.1155/2023/4481016","url":null,"abstract":"In this paper, we present some of the foundational concepts of split-complex number theory such as split-complex divison, gcd, and congruencies. Also, we prove that Euler’s theorem is still true in the case of split-complex integers, and we use this theorem to present a split-complex version of the RSA algorithm which is harder to be broken than the classical version. On the other hand, we study some algebraic properties of split-complex matrices, where we present the formula of computing the exponent of a split-complex matrix \u0000 \u0000 \u0000 \u0000 e\u0000 \u0000 \u0000 X\u0000 \u0000 \u0000 \u0000 with a novel algorithm to represent a split-complex matrix \u0000 \u0000 X\u0000 \u0000 by a split-complex diagonal matrix, which is known as the diagonalization problem. In addition, many examples were illustrated to clarify the validity of our work.","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"29 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79084425","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 graph theory has wide important applications in various other types of sciences. In chemical graph theory, we have many topological polynomials for a graph G through which we can compute many topological indices. Topological indices are numerical values and descriptors which are used to quantify the physiochemical properties and bioactivities of the chemical graph. In this paper, we compute Hosoya polynomial, hyper-Wiener index, Tratch–Stankevitch–Zefirov index, Harary index, Schultz polynomial, Gutman polynomial, Schultz index, and Gutman index of generalized Petersen graphs P n , 1 and P n , 2 .
{"title":"Hosoya, Schultz, and Gutman Polynomials of Generalized Petersen Graphs <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\">\u0000 <mi>P</mi>\u0000 <mfenced open=\"(\" close=\")\" separators=\"|\">\u0000 <mrow>\u0000 <mi>n</mi>\u0000 ","authors":"Ramy S. Shaheen, Suhail Mahfud, Qays Alhawat","doi":"10.1155/2023/7341285","DOIUrl":"https://doi.org/10.1155/2023/7341285","url":null,"abstract":"The graph theory has wide important applications in various other types of sciences. In chemical graph theory, we have many topological polynomials for a graph \u0000 \u0000 G\u0000 \u0000 through which we can compute many topological indices. Topological indices are numerical values and descriptors which are used to quantify the physiochemical properties and bioactivities of the chemical graph. In this paper, we compute Hosoya polynomial, hyper-Wiener index, Tratch–Stankevitch–Zefirov index, Harary index, Schultz polynomial, Gutman polynomial, Schultz index, and Gutman index of generalized Petersen graphs \u0000 \u0000 P\u0000 \u0000 \u0000 n\u0000 ,\u0000 1\u0000 \u0000 \u0000 \u0000 and \u0000 \u0000 P\u0000 \u0000 \u0000 n\u0000 ,\u0000 2\u0000 \u0000 \u0000 \u0000 .","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83107520","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 current progressive fusion methods for digital images have poor denoising performance, which leads to a decrease in image quality after progressive fusion. Therefore, a new method for digital image progressive fusion was proposed based on discrete cosine transform, and its effectiveness was verified through experiments. The experimental results show that the proposed method has a PSNR value higher than 42.13 db in image fusion, both of which are higher than the comparison method, and the fusion effect comparison also has higher image quality. In terms of fusion time, the time of the research method is lower than that of the comparison method when the data volume is between 10 and 100, while in the comparison of structural similarity, the structural similarity of the image fused by the research method is always higher than 0.83. Overall, the fusion method proposed in the study results in higher image quality and is effective in progressive digital image fusion, which is of great significance for practical digital image fusion.
{"title":"Digital Image Progressive Fusion Method Based on Discrete Cosine Transform","authors":"Jie Chen","doi":"10.1155/2023/9905604","DOIUrl":"https://doi.org/10.1155/2023/9905604","url":null,"abstract":"The current progressive fusion methods for digital images have poor denoising performance, which leads to a decrease in image quality after progressive fusion. Therefore, a new method for digital image progressive fusion was proposed based on discrete cosine transform, and its effectiveness was verified through experiments. The experimental results show that the proposed method has a PSNR value higher than 42.13 db in image fusion, both of which are higher than the comparison method, and the fusion effect comparison also has higher image quality. In terms of fusion time, the time of the research method is lower than that of the comparison method when the data volume is between 10 and 100, while in the comparison of structural similarity, the structural similarity of the image fused by the research method is always higher than 0.83. Overall, the fusion method proposed in the study results in higher image quality and is effective in progressive digital image fusion, which is of great significance for practical digital image fusion.","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"33 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89800777","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}
Y. Tashkandy, W. Emam, G. Cordeiro, M. M. Ali, K. Aidi, H. Yousof, M. Ibrahim
This paper introduces and studies a unique probability distribution. The maximum likelihood estimation, the ordinary least squares, the weighted least squares, and the Anderson–Darling estimation methods all take into account a number of financial risk indicators, including the value-at-risk, tail-value-at-risk, tail variance, tail mean-variance, and mean excess loss function. These four approaches were used in a simulation study and an application to insurance claims data for the actuarial evaluation. The well-known Nikulin–Rao–Robson statistic is taken into consideration for distributional validation under the whole set of data. Three complete actual datasets and a simulation study are used to evaluate the Nikulin–Rao–Robson test statistic. An updated version of the Nikulin–Rao–Robson statistic is taken into consideration for censored distributional validation. Three censored actual datasets and a thorough simulation analysis are used to evaluate the novel Nikulin–Rao–Robson test statistic.
{"title":"Distributional Censored and Uncensored Validation Testing under a Modified Test Statistic with Risk Analysis and Assessment","authors":"Y. Tashkandy, W. Emam, G. Cordeiro, M. M. Ali, K. Aidi, H. Yousof, M. Ibrahim","doi":"10.1155/2023/8852528","DOIUrl":"https://doi.org/10.1155/2023/8852528","url":null,"abstract":"This paper introduces and studies a unique probability distribution. The maximum likelihood estimation, the ordinary least squares, the weighted least squares, and the Anderson–Darling estimation methods all take into account a number of financial risk indicators, including the value-at-risk, tail-value-at-risk, tail variance, tail mean-variance, and mean excess loss function. These four approaches were used in a simulation study and an application to insurance claims data for the actuarial evaluation. The well-known Nikulin–Rao–Robson statistic is taken into consideration for distributional validation under the whole set of data. Three complete actual datasets and a simulation study are used to evaluate the Nikulin–Rao–Robson test statistic. An updated version of the Nikulin–Rao–Robson statistic is taken into consideration for censored distributional validation. Three censored actual datasets and a thorough simulation analysis are used to evaluate the novel Nikulin–Rao–Robson test statistic.","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"34 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91352648","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 article, our goal is to study the system of extended nonlinear mixed variational-like inequalities (in short, SENMVLI) with a nonconvex functional in the setting of real Hilbert spaces and discuss the existence of solution of our considered problem. We propose a three-step iterative algorithm to calculate the approximate solutions of SENMVLI and investigate the convergence analysis as well as stability analysis of the proposed algorithm. Furthermore, we also study the proximal dynamical system for SENMVLI and prove that the trajectory of the solution of the extended proximal dynamical system converges globally exponentially to a unique solution of SENMVLI. Our suggested iterative algorithm and results have become the significant improvement, enhancement, and generalization of many previously known results in the literature.
{"title":"Solvability of the System of Extended Nonlinear Mixed Variational-Like Inequalities and Proximal Dynamical System","authors":"Iqbal Ahmad, Mijanur Rahaman","doi":"10.1155/2023/7414514","DOIUrl":"https://doi.org/10.1155/2023/7414514","url":null,"abstract":"In this article, our goal is to study the system of extended nonlinear mixed variational-like inequalities (in short, SENMVLI) with a nonconvex functional in the setting of real Hilbert spaces and discuss the existence of solution of our considered problem. We propose a three-step iterative algorithm to calculate the approximate solutions of SENMVLI and investigate the convergence analysis as well as stability analysis of the proposed algorithm. Furthermore, we also study the proximal dynamical system for SENMVLI and prove that the trajectory of the solution of the extended proximal dynamical system converges globally exponentially to a unique solution of SENMVLI. Our suggested iterative algorithm and results have become the significant improvement, enhancement, and generalization of many previously known results in the literature.","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"91 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85847394","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 paper, we apply kernel methods to deep convolutional neural network (DCNN) to improve its nonlinear ability. DCNNs have achieved significant improvement in many computer vision tasks. For an image classification task, the accuracy comes to saturation when the depth and width of network are enough and appropriate. The saturation accuracy will not rise even by increasing the depth and width. We find that improving nonlinear ability of DCNNs can break through the saturation accuracy. In a DCNN, the former layer is more inclined to extract features and the latter layer is more inclined to classify features. Therefore, we apply kernel methods at the last fully connected layer to implicitly map features to a higher-dimensional space to improve nonlinear ability so that the network achieves better linear separability. Also, we name the network as fully kernected neural networks (fully connected neural networks with kernel methods). Our experiment result shows that fully kernected neural networks achieve higher classification accuracy and faster convergence rate than baseline networks.
{"title":"Fully Kernected Neural Networks","authors":"W. Zhang, Zhi Han, Xi’ai Chen, Baicheng Liu, Huidi Jia, Yandong Tang","doi":"10.1155/2023/1539436","DOIUrl":"https://doi.org/10.1155/2023/1539436","url":null,"abstract":"In this paper, we apply kernel methods to deep convolutional neural network (DCNN) to improve its nonlinear ability. DCNNs have achieved significant improvement in many computer vision tasks. For an image classification task, the accuracy comes to saturation when the depth and width of network are enough and appropriate. The saturation accuracy will not rise even by increasing the depth and width. We find that improving nonlinear ability of DCNNs can break through the saturation accuracy. In a DCNN, the former layer is more inclined to extract features and the latter layer is more inclined to classify features. Therefore, we apply kernel methods at the last fully connected layer to implicitly map features to a higher-dimensional space to improve nonlinear ability so that the network achieves better linear separability. Also, we name the network as fully kernected neural networks (fully connected neural networks with kernel methods). Our experiment result shows that fully kernected neural networks achieve higher classification accuracy and faster convergence rate than baseline networks.","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"57 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83582566","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}
Maha E. Qura, Mohammed Alqawba, Mashail M. Al Sobhi, A. Afify
One of the important features of generalized distribution is its ability and flexibility to model real-life data in several applied fields such as medicine, engineering, and survival analysis, among others. In this paper, a flexible four-parameter Lomax extension called the alpha-power power-Lomax (APPLx) distribution is introduced. The APPLx distribution is analytically tractable, and it can be used quite effectively for real-life data analysis. Key mathematical properties of the APPLx distribution including mode, moments, stress-strength reliability, quantile and generating functions, and order statistics are presented. The APPEx parameters are estimated by using eight classical estimation methods. Extensive simulation studies are provided to explore the performance of the proposed estimation methods and to provide a guideline for practitioners and engineers to choose the best estimation method. Three real-life datasets from applied fields are fitted to assess empirically the flexibility of the APPLx distribution. The APPLx distribution shows greater flexibility as compared to the McDonal–Lomax, Fréchet Topp–Leone Lomax, transmuted Weibull–Lomax, Kumaraswamy–Lomax, beta exponentiated-Lomax, Weibull–Lomax, Burr-X Lomax, Lomax–Weibull, odd exponentiated half-logistic Lomax, and alpha-power Lomax distributions.
{"title":"A Novel Extended Power-Lomax Distribution for Modeling Real-Life Data: Properties and Inference","authors":"Maha E. Qura, Mohammed Alqawba, Mashail M. Al Sobhi, A. Afify","doi":"10.1155/2023/6661792","DOIUrl":"https://doi.org/10.1155/2023/6661792","url":null,"abstract":"One of the important features of generalized distribution is its ability and flexibility to model real-life data in several applied fields such as medicine, engineering, and survival analysis, among others. In this paper, a flexible four-parameter Lomax extension called the alpha-power power-Lomax (APPLx) distribution is introduced. The APPLx distribution is analytically tractable, and it can be used quite effectively for real-life data analysis. Key mathematical properties of the APPLx distribution including mode, moments, stress-strength reliability, quantile and generating functions, and order statistics are presented. The APPEx parameters are estimated by using eight classical estimation methods. Extensive simulation studies are provided to explore the performance of the proposed estimation methods and to provide a guideline for practitioners and engineers to choose the best estimation method. Three real-life datasets from applied fields are fitted to assess empirically the flexibility of the APPLx distribution. The APPLx distribution shows greater flexibility as compared to the McDonal–Lomax, Fréchet Topp–Leone Lomax, transmuted Weibull–Lomax, Kumaraswamy–Lomax, beta exponentiated-Lomax, Weibull–Lomax, Burr-X Lomax, Lomax–Weibull, odd exponentiated half-logistic Lomax, and alpha-power Lomax distributions.","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":"37 1","pages":""},"PeriodicalIF":0.5,"publicationDate":"2023-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77039948","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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