Pub Date : 2024-04-10DOI: 10.9734/arjom/2024/v20i4793
Daniel Sankei, Loyford Njagi, Josephine Mutembei
The Strong Goldbach's conjecture, a fundamental problem in Number Theory, asserts that every even integer greater than 2 can be expressed as the sum of two prime numbers. Despite significant efforts over centuries, this conjecture remains unproven, challenging the core of mathematics. The known algorithms for attempting to prove or verify the conjecture on a given interval [a,b] consist of finding two sets of primes Pi and Pj such that Pi+Pj cover all the even numbers in the interval [a,b]. However, the traditional definition of an even number as 2n for n ∈ ℕ (where ℕ is the set of natural numbers), has not provided mathematicians with a straightforward method to obtain all Goldbach partitions for any even number of this form. This paper introduces a novel approach to the problem, utilizing all odd partitions of an even number of a new formulation of the form Eij = ni + nj + (nj - ni)n or alln ∈ ℕ. By demonstrating that there exist at least a pair of prime numbers in these odd partitions, the fact that the sum of any two prime numbers is even and there exists infinitely many prime numbers, this paper provides a compelling proof of the conjecture. This breakthrough not only solves a long-standing mathematical mystery but also sheds light on the structure of prime numbers.
{"title":"A Detailed Proof of the Strong Goldbach Conjecture Based on Partitions of a New Formulation of a Set of Even Numbers","authors":"Daniel Sankei, Loyford Njagi, Josephine Mutembei","doi":"10.9734/arjom/2024/v20i4793","DOIUrl":"https://doi.org/10.9734/arjom/2024/v20i4793","url":null,"abstract":"The Strong Goldbach's conjecture, a fundamental problem in Number Theory, asserts that every even integer greater than 2 can be expressed as the sum of two prime numbers. Despite significant efforts over centuries, this conjecture remains unproven, challenging the core of mathematics. The known algorithms for attempting to prove or verify the conjecture on a given interval [a,b] consist of finding two sets of primes Pi and Pj such that Pi+Pj cover all the even numbers in the interval [a,b]. However, the traditional definition of an even number as 2n for n ∈ ℕ (where ℕ is the set of natural numbers), has not provided mathematicians with a straightforward method to obtain all Goldbach partitions for any even number of this form. This paper introduces a novel approach to the problem, utilizing all odd partitions of an even number of a new formulation of the form Eij = ni + nj + (nj - ni)n or alln ∈ ℕ. By demonstrating that there exist at least a pair of prime numbers in these odd partitions, the fact that the sum of any two prime numbers is even and there exists infinitely many prime numbers, this paper provides a compelling proof of the conjecture. This breakthrough not only solves a long-standing mathematical mystery but also sheds light on the structure of prime numbers.","PeriodicalId":281529,"journal":{"name":"Asian Research Journal of Mathematics","volume":"225 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140720044","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}
Pub Date : 2024-04-09DOI: 10.9734/arjom/2024/v20i4792
Mogoi N. Evans, A. Wanjara, Samuel B. Apima
This paper explores norm-attainability of orthogonal polynomials in Sobolev spaces, investigating properties like existence, uniqueness, and convergence. It establishes the convergence of these polynomials in Sobolev spaces, addressing orthogonality preservation and derivative behaviors. Spectral properties, including Sturm-Liouville eigenvalue problems, are analyzed, enhancing the understanding of these polynomials. The study incorporates fundamental concepts like reproducing kernels, Riesz representations, and Bessel’s inequality. Results contribute to the theoretical understanding of orthogonal polynomials, with potential applications in diverse mathematical and computational contexts.
{"title":"Analysis of Norm-Attainability and Convergence Properties of Orthogonal Polynomials in Weighted Sobolev Spaces","authors":"Mogoi N. Evans, A. Wanjara, Samuel B. Apima","doi":"10.9734/arjom/2024/v20i4792","DOIUrl":"https://doi.org/10.9734/arjom/2024/v20i4792","url":null,"abstract":"This paper explores norm-attainability of orthogonal polynomials in Sobolev spaces, investigating properties like existence, uniqueness, and convergence. It establishes the convergence of these polynomials in Sobolev spaces, addressing orthogonality preservation and derivative behaviors. Spectral properties, including Sturm-Liouville eigenvalue problems, are analyzed, enhancing the understanding of these polynomials. The study incorporates fundamental concepts like reproducing kernels, Riesz representations, and Bessel’s inequality. Results contribute to the theoretical understanding of orthogonal polynomials, with potential applications in diverse mathematical and computational contexts.","PeriodicalId":281529,"journal":{"name":"Asian Research Journal of Mathematics","volume":"15 6","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140724303","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}
Pub Date : 2024-03-29DOI: 10.9734/arjom/2024/v20i3790
R. S. M. Kularathna, N. Kajan, T. Jeyamugan, S. Thilaganathan
In this paper, we applied the modified two-dimensional differential transform method to solve Laplace equation. Laplace equation is one of Elliptic partial differential equations. These kinds of differential equations have specific applications models of physics and engineering. We consider four models with two Dirichlet and two Neumann boundary conditions. The simplicity of this method compared to other iteration methods is shown here. It is worth mentioning that here only a few number of iterations are required to reach the closed form solutions as series expansions of some known functions.
{"title":"Solution of Laplace Equation by Modified Differential Transform Method","authors":"R. S. M. Kularathna, N. Kajan, T. Jeyamugan, S. Thilaganathan","doi":"10.9734/arjom/2024/v20i3790","DOIUrl":"https://doi.org/10.9734/arjom/2024/v20i3790","url":null,"abstract":"In this paper, we applied the modified two-dimensional differential transform method to solve Laplace equation. Laplace equation is one of Elliptic partial differential equations. These kinds of differential equations have specific applications models of physics and engineering. We consider four models with two Dirichlet and two Neumann boundary conditions. The simplicity of this method compared to other iteration methods is shown here. It is worth mentioning that here only a few number of iterations are required to reach the closed form solutions as series expansions of some known functions.","PeriodicalId":281529,"journal":{"name":"Asian Research Journal of Mathematics","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140367399","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}
Pub Date : 2024-02-28DOI: 10.9734/arjom/2024/v20i2782
Sheetal Yadav, Manoj Ughade, D. Singh, Manoj Kumar Shukla
((lambda), (alpha))- interpolative Kannan contraction, ((lambda), (alpha), (beta))- interpolative Kannan contraction, ((lambda), (alpha), (beta), (gamma))- interpolative Riech contraction and ((lambda), (alpha), (beta))- interpolative Dass-Gupta rational contraction are presented in this study. Furthermore, we prove a few fixed-point theorems for interpolative contractions in complete A-metric spaces. These theorems also extend and apply to an A-metric setting several interesting results from metric fixed-point theory.
((lambda),((α))) - 插值卡南收缩， ((lambda),(α),((β)) - 插值卡南收缩， ((lambda),(α),(β)、(（gamma）)-插值里奇收缩和（（lambda）, （（alpha）, （（beta））-插值达斯-古普塔有理收缩在本研究中被提出。此外，我们还证明了完全 A 度量空间中内插收缩的几个定点定理。这些定理还将公设定点理论中几个有趣的结果扩展并应用到了 A 度量环境中。
{"title":"Fixed Point Theorems for Kannan Interpolative, Riech Interpolative and Dass-Gupta Interpolative Rational type Contractions in A-Metric Spaces","authors":"Sheetal Yadav, Manoj Ughade, D. Singh, Manoj Kumar Shukla","doi":"10.9734/arjom/2024/v20i2782","DOIUrl":"https://doi.org/10.9734/arjom/2024/v20i2782","url":null,"abstract":"((lambda), (alpha))- interpolative Kannan contraction, ((lambda), (alpha), (beta))- interpolative Kannan contraction, ((lambda), (alpha), (beta), (gamma))- interpolative Riech contraction and ((lambda), (alpha), (beta))- interpolative Dass-Gupta rational contraction are presented in this study. Furthermore, we prove a few fixed-point theorems for interpolative contractions in complete A-metric spaces. These theorems also extend and apply to an A-metric setting several interesting results from metric fixed-point theory.","PeriodicalId":281529,"journal":{"name":"Asian Research Journal of Mathematics","volume":"19 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140420077","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}
Pub Date : 2024-02-21DOI: 10.9734/arjom/2024/v20i1781
Tejaskumar R., A. Ismayil
Kathiresan and Marimuthu were the pioneers of superior distance in graphs. The same authors put forth the concept of superior domination in 2008. Superior distance is the shortest walk between any two vertices including their respective neighbours. The minimum superior dominating energy is defined by the sum of the eigenvalues and it is obtained from the minimum superior dominating matrix . The minimum superior dominating energy for star and crown graphs are computed. Properties of eigenvalues of minimum superior dominating matrix for star, cocktail party, complete and crown graphs are discussed. Results related to upper and lower bounds of minimum superior dominating energy for star, cocktail party, complete and crown graphs are stated and proved.
{"title":"The Minimum Superior Dominating Energy of Graphs","authors":"Tejaskumar R., A. Ismayil","doi":"10.9734/arjom/2024/v20i1781","DOIUrl":"https://doi.org/10.9734/arjom/2024/v20i1781","url":null,"abstract":"Kathiresan and Marimuthu were the pioneers of superior distance in graphs. The same authors put forth the concept of superior domination in 2008. Superior distance is the shortest walk between any two vertices including their respective neighbours. The minimum superior dominating energy is defined by the sum of the eigenvalues and it is obtained from the minimum superior dominating matrix . The minimum superior dominating energy for star and crown graphs are computed. Properties of eigenvalues of minimum superior dominating matrix for star, cocktail party, complete and crown graphs are discussed. Results related to upper and lower bounds of minimum superior dominating energy for star, cocktail party, complete and crown graphs are stated and proved.","PeriodicalId":281529,"journal":{"name":"Asian Research Journal of Mathematics","volume":"15 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140444868","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}
Pub Date : 2024-01-23DOI: 10.9734/arjom/2024/v20i1777
Jackson K. Njenga, I. C. Kipchirchir
This research work seeks to analysis the mortality trend experienced in Kenya over the sample period 1950 to 2021 using a multidimensional modeling framework. Life table functions, namely; life expectancy, survival function and age at death distribution are applied to depict mortality characteristics. Life expectancy and survival rate have significantly improved. There has been a shift in mortality status from a high mortality population, to an intermediate stage and mortality risk factors have increased across age. Mortality concentration curve and index within the Lorenz curve and Gini coefficient framework are used to analyze the lifespan inequality. Lifespan inequality is high with negligible improvements over time. Gompertz force of mortality is then estimated, which is statistically significant at 5% level. Deaths at exact age 25 is about 35 per ten thousand, with the rate death rate increasing by 6.09% per year starting from age 25. Under the assumptions of stable population, where the mortality and fertility functions are independent of time, Malthusian parameter is estimated which is less than zero for selected years. Kenya is a shrinking population and death rate decrease with increase in Malthusian parameter. Finally, to model long-term mortality rate forecast, Lee-Carter model is estimated. The model is statistically significant at 5% level explaining 78.4% of the variations. Expected life expectancy at a given age is projected to increase, with life expectancy at birth in 2030 and 2071 being 65.6 and 70.5 years respectively.
{"title":"Modelling Mortality in Kenya","authors":"Jackson K. Njenga, I. C. Kipchirchir","doi":"10.9734/arjom/2024/v20i1777","DOIUrl":"https://doi.org/10.9734/arjom/2024/v20i1777","url":null,"abstract":"This research work seeks to analysis the mortality trend experienced in Kenya over the sample period 1950 to 2021 using a multidimensional modeling framework. Life table functions, namely; life expectancy, survival function and age at death distribution are applied to depict mortality characteristics. Life expectancy and survival rate have significantly improved. There has been a shift in mortality status from a high mortality population, to an intermediate stage and mortality risk factors have increased across age. Mortality concentration curve and index within the Lorenz curve and Gini coefficient framework are used to analyze the lifespan inequality. Lifespan inequality is high with negligible improvements over time. Gompertz force of mortality is then estimated, which is statistically significant at 5% level. Deaths at exact age 25 is about 35 per ten thousand, with the rate death rate increasing by 6.09% per year starting from age 25. Under the assumptions of stable population, where the mortality and fertility functions are independent of time, Malthusian parameter is estimated which is less than zero for selected years. Kenya is a shrinking population and death rate decrease with increase in Malthusian parameter. Finally, to model long-term mortality rate forecast, Lee-Carter model is estimated. The model is statistically significant at 5% level explaining 78.4% of the variations. Expected life expectancy at a given age is projected to increase, with life expectancy at birth in 2030 and 2071 being 65.6 and 70.5 years respectively.","PeriodicalId":281529,"journal":{"name":"Asian Research Journal of Mathematics","volume":"147 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140498083","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}
Pub Date : 2023-12-28DOI: 10.9734/arjom/2023/v19i12772
Henry Otoo, W. Obeng-Denteh, Lewis Brew
Oscillatory solutions play a pivotal role in understanding functional differential and integral equations, offering insights into the behaviour of these equations' solutions, and assisting in understanding their growth, stability, and convergence properties. This study establishes the oscillatory solution of a convolutional Volterra integral equation using mathematical proofs. Theorems for oscillatory solutions are proposed and proven based on well-defined assumptions, along with an illustrated example. The proofs presented herein reveal that the convolutional Volterra integral equation can exhibit oscillatory or non-oscillatory behavior, contingent upon the characteristics of the function within the integral.
{"title":"Oscillatory Solution of a Convolutional Volterra Integral Equation","authors":"Henry Otoo, W. Obeng-Denteh, Lewis Brew","doi":"10.9734/arjom/2023/v19i12772","DOIUrl":"https://doi.org/10.9734/arjom/2023/v19i12772","url":null,"abstract":"Oscillatory solutions play a pivotal role in understanding functional differential and integral equations, offering insights into the behaviour of these equations' solutions, and assisting in understanding their growth, stability, and convergence properties. This study establishes the oscillatory solution of a convolutional Volterra integral equation using mathematical proofs. Theorems for oscillatory solutions are proposed and proven based on well-defined assumptions, along with an illustrated example. The proofs presented herein reveal that the convolutional Volterra integral equation can exhibit oscillatory or non-oscillatory behavior, contingent upon the characteristics of the function within the integral.","PeriodicalId":281529,"journal":{"name":"Asian Research Journal of Mathematics","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139149324","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}
Pub Date : 2023-12-28DOI: 10.9734/arjom/2023/v19i12771
Atsu, J. U., Abong, A. A.
Using Gumbel's extreme value theory method, this study looked into the probability of the largest earthquakes for different return periods in Zimbabwe. The Advanced National Seismic System (ANSS), the Northern California Earthquake Data Centre website, and UC Berkeley in the United States provided the data used for this study. The natural earthquakes with Mb ≥ 4.0 that occurred in the study area between January 1, 1901 and December 31, 2001 (a period of 100 years) with a focal depth ranging from 0 to 700 km made up the selected data. The study area is between 150S - 220S and 250E -340E in coordinates. A total of 81 events were used in the investigation. According to the results, there is a 100% chance that earthquakes with a magnitude between 4.5 and 5.5 will occur during the time interval of approximately 1 to 14 years and a 9.5% chance that within the next 100 years, there will be an earthquake with a magnitude of Mb = 6.0 or higher. For magnitudes Mb = 6.0 and above, the return period is relatively long - roughly 50 – 701 years. This suggests that there is little chance that Zimbabwe will experience earthquakes larger than magnitude 6.0. Despite being low, it is impossible to forecast with certainty because earthquake forecasting and prediction is still a complex topic.
{"title":"Estimation of the Probability of Earthquakes Return Period in Zimbabwe Using Gumbel’s Extreme Value Theory Method","authors":"Atsu, J. U., Abong, A. A.","doi":"10.9734/arjom/2023/v19i12771","DOIUrl":"https://doi.org/10.9734/arjom/2023/v19i12771","url":null,"abstract":"Using Gumbel's extreme value theory method, this study looked into the probability of the largest earthquakes for different return periods in Zimbabwe. The Advanced National Seismic System (ANSS), the Northern California Earthquake Data Centre website, and UC Berkeley in the United States provided the data used for this study. The natural earthquakes with Mb ≥ 4.0 that occurred in the study area between January 1, 1901 and December 31, 2001 (a period of 100 years) with a focal depth ranging from 0 to 700 km made up the selected data. The study area is between 150S - 220S and 250E -340E in coordinates. A total of 81 events were used in the investigation. According to the results, there is a 100% chance that earthquakes with a magnitude between 4.5 and 5.5 will occur during the time interval of approximately 1 to 14 years and a 9.5% chance that within the next 100 years, there will be an earthquake with a magnitude of Mb = 6.0 or higher. For magnitudes Mb = 6.0 and above, the return period is relatively long - roughly 50 – 701 years. This suggests that there is little chance that Zimbabwe will experience earthquakes larger than magnitude 6.0. Despite being low, it is impossible to forecast with certainty because earthquake forecasting and prediction is still a complex topic.","PeriodicalId":281529,"journal":{"name":"Asian Research Journal of Mathematics","volume":"52 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139151134","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}
Pub Date : 2023-12-23DOI: 10.9734/arjom/2023/v19i12769
Alshaimaa Awad, Saleh Omran
In this article, we introduced and focused our attention to some fixed point theorems using (psi)-contraction mapping on (mathit{C}^*)-algebra valued metric space. In particular, we established some Banach fixed point theorem as well as several extensions and generalizations of this theorem in (mathit{C}^*)-algebras valued metric spaces. Moreover, in order to illustrate the current results, some basic examples are presented and we gave an application on system linear operator equation by investigating the existence and uniqueness to the solution of this equation.
{"title":"Some Fixed Point Techniques using (psi)-contraction Mapping on the (mathit{C}^*)-algebra Valued Metric Space","authors":"Alshaimaa Awad, Saleh Omran","doi":"10.9734/arjom/2023/v19i12769","DOIUrl":"https://doi.org/10.9734/arjom/2023/v19i12769","url":null,"abstract":"In this article, we introduced and focused our attention to some fixed point theorems using (psi)-contraction mapping on (mathit{C}^*)-algebra valued metric space. In particular, we established some Banach fixed point theorem as well as several extensions and generalizations of this theorem in (mathit{C}^*)-algebras valued metric spaces. Moreover, in order to illustrate the current results, some basic examples are presented and we gave an application on system linear operator equation by investigating the existence and uniqueness to the solution of this equation.","PeriodicalId":281529,"journal":{"name":"Asian Research Journal of Mathematics","volume":"40 19","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139161940","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}
Pub Date : 2023-11-28DOI: 10.9734/arjom/2023/v19i11763
Stephen Kadedesya, L. Nyaga, J. Rimberia
In this paper, the suborbits and graphs associated with the action of direct product of two Alternating groups on the Cartesian product of two sets are studied. It is shown that the suborbits are self-paired and the associated graphs are undirected and regular with girth 3.
{"title":"On Suborbits and Graphs Associated with Action of Alternating Groups on Cartesian Product of Two Sets","authors":"Stephen Kadedesya, L. Nyaga, J. Rimberia","doi":"10.9734/arjom/2023/v19i11763","DOIUrl":"https://doi.org/10.9734/arjom/2023/v19i11763","url":null,"abstract":"In this paper, the suborbits and graphs associated with the action of direct product of two Alternating groups on the Cartesian product of two sets are studied. It is shown that the suborbits are self-paired and the associated graphs are undirected and regular with girth 3.","PeriodicalId":281529,"journal":{"name":"Asian Research Journal of Mathematics","volume":"24 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139222352","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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