In analytic geometry, Bézout’s theorem stated the number of intersection points of two algebraic curves and Fulton introduced the intersection multiplicity of two curves at some point in local case. It is meaningful to give the exact expression of the intersection multiplicity of two curves at some point. In this paper, we mainly express the intersection multiplicity of two curves at some point in and under fold point, where . First, we give a sufficient and necessary condition for the coincidence of the intersection multiplicity of two curves at some point and the smallest degree of the terms of these two curves in . Furthermore, we show that two different definitions of intersection multiplicity of two curves at a point in
在解析几何中，bsamzout定理表述了两条代数曲线交点的个数，Fulton引入了两条曲线在局部情况下某点的交点多重性。给出两条曲线交点多重度的精确表达式是有意义的。在本文中，我们主要表示两条曲线在r2和A中某点的交点多重性折叠点下的k2，其中char K = 0。首先，给出了两条曲线相交多重性在某点重合的充要条件以及这两条曲线项在r2中的最小次。此外,证明了a K 2中两点曲线交点多重性的两种不同定义是等价的，并给出了确切的定义两条曲线在A K 2中某点处的交点多重性的表达式。
{"title":"The Intersection Multiplicity of Intersection Points over Algebraic Curves","authors":"Kailing Lai, Fanning Meng, Huanqi He","doi":"10.1155/2023/6346685","DOIUrl":"https://doi.org/10.1155/2023/6346685","url":null,"abstract":"<jats:p>In analytic geometry, Bézout’s theorem stated the number of intersection points of two algebraic curves and Fulton introduced the intersection multiplicity of two curves at some point in local case. It is meaningful to give the exact expression of the intersection multiplicity of two curves at some point. In this paper, we mainly express the intersection multiplicity of two curves at some point in <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\">\u0000 <msup>\u0000 <mrow>\u0000 <mi mathvariant=\"double-struck\">R</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </msup>\u0000 </math>\u0000 </jats:inline-formula> and <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\">\u0000 <msubsup>\u0000 <mrow>\u0000 <mi mathvariant=\"double-struck\">A</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>K</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </msubsup>\u0000 </math>\u0000 </jats:inline-formula> under fold point, where <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\">\u0000 <mtext>char</mtext>\u0000 <mfenced open=\"(\" close=\")\" separators=\"|\">\u0000 <mrow>\u0000 <mi>K</mi>\u0000 </mrow>\u0000 </mfenced>\u0000 <mo>=</mo>\u0000 <mn>0</mn>\u0000 </math>\u0000 </jats:inline-formula>. First, we give a sufficient and necessary condition for the coincidence of the intersection multiplicity of two curves at some point and the smallest degree of the terms of these two curves in <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M4\">\u0000 <msup>\u0000 <mrow>\u0000 <mi mathvariant=\"double-struck\">R</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </msup>\u0000 </math>\u0000 </jats:inline-formula>. Furthermore, we show that two different definitions of intersection multiplicity of two curves at a point in <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M5\">\u0000 <msubsup>\u0000 <mrow>\u0000 ","PeriodicalId":301406,"journal":{"name":"Int. J. Math. Math. Sci.","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129641899","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}
S. Okyere, J. Ackora-Prah, S. Abdullah, S. Adarkwa, Frank Kofi Owusu, Kwame Bonsu, Mary Osei Fokuo, M. Yeboah
Fractional-order derivative modeling continues to receive great interest among researchers across the globe. In this study, Tuberculosis-COVID-19 coinfection is studied using Atangana–Baleanu fractional-order derivatives defined in Caputo sense. We confirmed the existence and singularity of the solution and investigated the model’s equilibrium points. Additionally, we examined the model’s stability in terms of the Ulam–Hyers and generalized Ulam–Hyers stability criteria. The basic reproduction number R 0 was calculated using the next-generation matrix approach. We also looked into the model’s disease-free equilibrium point’s regional stability. Numerical scheme for simulating the fractional-order system with Mittag–Leffler Kernels are presented. Numerical simulations are given to validate the model. Results of the simulation showed a decline in the number of COVID-19 infections within the population when the fractional operator was reduced.
{"title":"Analysis of Turberculosis-COVID-19 Coinfection Using Fractional Derivatives","authors":"S. Okyere, J. Ackora-Prah, S. Abdullah, S. Adarkwa, Frank Kofi Owusu, Kwame Bonsu, Mary Osei Fokuo, M. Yeboah","doi":"10.1155/2023/2831846","DOIUrl":"https://doi.org/10.1155/2023/2831846","url":null,"abstract":"Fractional-order derivative modeling continues to receive great interest among researchers across the globe. In this study, Tuberculosis-COVID-19 coinfection is studied using Atangana–Baleanu fractional-order derivatives defined in Caputo sense. We confirmed the existence and singularity of the solution and investigated the model’s equilibrium points. Additionally, we examined the model’s stability in terms of the Ulam–Hyers and generalized Ulam–Hyers stability criteria. The basic reproduction number \u0000 \u0000 \u0000 \u0000 R\u0000 \u0000 \u0000 0\u0000 \u0000 \u0000 \u0000 was calculated using the next-generation matrix approach. We also looked into the model’s disease-free equilibrium point’s regional stability. Numerical scheme for simulating the fractional-order system with Mittag–Leffler Kernels are presented. Numerical simulations are given to validate the model. Results of the simulation showed a decline in the number of COVID-19 infections within the population when the fractional operator was reduced.","PeriodicalId":301406,"journal":{"name":"Int. J. Math. Math. Sci.","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124869321","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 manuscript, we study the properties and equivalence results for different forms of quadratic functional equations. Using the Brzdȩk and Ciepliński fixed-point approach, we investigate the generalized Hyers–Ulam–Rassias stability for the 3-variables quadratic functional equation in the setting of 2-Banach space. Also, we obtain some hyperstability results for the 3-variables quadratic functional equation. The results obtained in this paper extend several known results of the literature to the setting of 2-Banach space.
{"title":"Equivalence for Various Forms of Quadratic Functional Equations and Their Generalized Hyers-Ulam-Rassias Stability","authors":"Ravi Sharma, S. Chandok","doi":"10.1155/2023/1721273","DOIUrl":"https://doi.org/10.1155/2023/1721273","url":null,"abstract":"In this manuscript, we study the properties and equivalence results for different forms of quadratic functional equations. Using the Brzdȩk and Ciepliński fixed-point approach, we investigate the generalized Hyers–Ulam–Rassias stability for the 3-variables quadratic functional equation in the setting of 2-Banach space. Also, we obtain some hyperstability results for the 3-variables quadratic functional equation. The results obtained in this paper extend several known results of the literature to the setting of 2-Banach space.","PeriodicalId":301406,"journal":{"name":"Int. J. Math. Math. Sci.","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124578098","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 prove that the characteristic of a minimal ideal and a minimal generalized ideal, which is meant to be one of minimal left ideal, minimal right ideal, bi-ideal, quasi-ideal, and m , n -ideal in a ring, is either zero or a prime number p . When the characteristic is zero, then the minimal ideal (minimal generalized ideal) as additive group is torsion-free, and when the characteristic is p , then every element of its additive group has order p . Furthermore, we give some properties for minimal ideals and for generalized ideals which depend on their characteristics.
本文证明了环上的极小理想和极小广义理想(即极小左理想、极小右理想、双理想、拟理想和m, n -理想中的一个)的特征是零或素数p。当特征为零时，作为加性群的最小理想(最小广义理想)是无扭的，当特征为p时，其加性群的每个元素都是p阶的。进一步给出了最小理想和广义理想的一些性质，这些性质取决于它们的特征。
{"title":"The Characteristic of the Minimal Ideals and the Minimal Generalized Ideals in Rings","authors":"Rigena Sema, P. Petro, K. Hila","doi":"10.1155/2023/6478767","DOIUrl":"https://doi.org/10.1155/2023/6478767","url":null,"abstract":"In this paper, we prove that the characteristic of a minimal ideal and a minimal generalized ideal, which is meant to be one of minimal left ideal, minimal right ideal, bi-ideal, quasi-ideal, and \u0000 \u0000 \u0000 \u0000 m\u0000 ,\u0000 n\u0000 \u0000 \u0000 \u0000 -ideal in a ring, is either zero or a prime number \u0000 \u0000 p\u0000 \u0000 . When the characteristic is zero, then the minimal ideal (minimal generalized ideal) as additive group is torsion-free, and when the characteristic is \u0000 \u0000 p\u0000 \u0000 , then every element of its additive group has order \u0000 \u0000 p\u0000 \u0000 . Furthermore, we give some properties for minimal ideals and for generalized ideals which depend on their characteristics.","PeriodicalId":301406,"journal":{"name":"Int. J. Math. Math. Sci.","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124294896","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}
Stated choice experiments are increasingly becoming popular due to their ability to optimize information gain with limited resources. Many designs have been developed for the selection of various attributes and their levels to form choice sets. One such design is blocked fractional factorial design (BFFD). Stated choice experiments for symmetric attributes of 4 choice sets of sizes 2 and 4 and 8 choice sets of sizes 2 and 4 were developed using BFFDs. Generators for the stated choice set of sizes 2 and 4 with resolution three, four, and five were developed. The alias structures and confounding effects for the designs were derived, as well as their clear effects if any for estimation. The A -efficiency was used to compute the efficiencies of the proposed designs since it has better statistical properties. The computed efficiencies for the proposed designs reveal that 4 choice sets of size 4 designs are more efficient. Finally, a practical application of the proposed method was carried out for four choice sets of size 4 using 2 V 5 − 1 − 2 design with attributes and levels of service quality in public transport.
陈述选择实验越来越受欢迎，因为它们能够在有限的资源下优化信息增益。已经开发了许多设计，用于选择各种属性及其级别，以形成选择集。其中一种设计是阻断分数因子设计(BFFD)。利用bffd对4个大小为2和4的选择集和8个大小为2和4的选择集进行了对称属性的陈述选择实验。为尺寸为2和4、分辨率为3,4,5的所述选择集开发了生成器。推导了设计的混叠结构和混淆效应，以及它们的清晰效应(如果有的话)。由于A -效率具有更好的统计特性，因此采用A -效率来计算所提出设计的效率。设计效率的计算结果表明，4个尺寸为4的设计选择集效率更高。最后，利用公共交通服务质量属性和水平的2 V 5−1−2设计，对4个大小为4的选择集进行了实际应用。
{"title":"Efficient Blocked Fractional Factorial Designs for Stated Choice Experiments of Size Two and Four","authors":"Emmanuel Amoah, J. Dioggban, A. Jackson","doi":"10.1155/2023/2515605","DOIUrl":"https://doi.org/10.1155/2023/2515605","url":null,"abstract":"Stated choice experiments are increasingly becoming popular due to their ability to optimize information gain with limited resources. Many designs have been developed for the selection of various attributes and their levels to form choice sets. One such design is blocked fractional factorial design (BFFD). Stated choice experiments for symmetric attributes of 4 choice sets of sizes 2 and 4 and 8 choice sets of sizes 2 and 4 were developed using BFFDs. Generators for the stated choice set of sizes 2 and 4 with resolution three, four, and five were developed. The alias structures and confounding effects for the designs were derived, as well as their clear effects if any for estimation. The \u0000 \u0000 A\u0000 \u0000 -efficiency was used to compute the efficiencies of the proposed designs since it has better statistical properties. The computed efficiencies for the proposed designs reveal that 4 choice sets of size 4 designs are more efficient. Finally, a practical application of the proposed method was carried out for four choice sets of size 4 using \u0000 \u0000 \u0000 \u0000 2\u0000 \u0000 \u0000 V\u0000 \u0000 \u0000 5\u0000 −\u0000 1\u0000 −\u0000 2\u0000 \u0000 \u0000 \u0000 design with attributes and levels of service quality in public transport.","PeriodicalId":301406,"journal":{"name":"Int. J. Math. Math. Sci.","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129355841","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 aim of this research is to study the effects of thermal radiation and chemical reaction on hydromagnetic fluid flow in a cylindrical collapsible tube with an obstacle. The fluid flow is governed by continuity, momentum, energy, and concentration equations. Similarity transformation has been used to convert the obtained PDEs into ODEs. The collocation method has been used to numerically solve the ODEs. The method has been implemented in MATLAB using the bvp4c inbuilt function. The effects of the nondimensional parameters on velocity, temperature, and concentration have been presented graphically. Additionally, the skin-friction coefficient, the Nusselt number, and the Sherwood number have been discussed and are presented in a tabular form. The findings demonstrated that increasing the Reynolds number causes a rise in the fluid temperature and velocity. The fluid velocity decreases as the Hartmann number and the weight of the obstacle increase but increases with increasing Grashof numbers. The temperature of the fluid increases as the radiation parameter, or Eckert number, increases, but decreases as the Prandtl number increases. As the Soret number rises, so do the fluid’s temperature and concentration distribution. With an increase in the unsteadiness parameter, the fluid velocity and the concentration distribution decrease, whereas the opposite is seen in temperature. As the Schmidt number, the concentration Grashof number, and the chemical reaction parameter increase, the fluid’s concentration decreases. There is an increase in skin-friction coefficient with increasing Prandtl number, Eckert number, Soret number, thermal Grashof number, concentration Grashof number, thermal radiation parameter, Hartmann number, and unsteadiness parameter, while a decrease is observed with increasing Reynolds number. The Nusselt number increases with an increase in the Prandtl number, Eckert number, thermal radiation parameter, Hartmann number, and unsteadiness parameter. A slight decrease in the Nusselt number has been observed with increasing thermal Grashof number. The Sherwood number decreases with increasing Prandtl number, chemical reaction parameter, and thermal radiation parameter but increases with increasing Schmidt number, Eckert number, and Soret number. The research has the potential for a wide range of applications including but not limited to the medical field and other physical sciences.
{"title":"Effects of Thermal Radiation and Chemical Reaction on Hydromagnetic Fluid Flow in a Cylindrical Collapsible Tube with an Obstacle","authors":"Nictor Mwamba, J. Abonyo, K. Awuor","doi":"10.1155/2023/9991376","DOIUrl":"https://doi.org/10.1155/2023/9991376","url":null,"abstract":"The aim of this research is to study the effects of thermal radiation and chemical reaction on hydromagnetic fluid flow in a cylindrical collapsible tube with an obstacle. The fluid flow is governed by continuity, momentum, energy, and concentration equations. Similarity transformation has been used to convert the obtained PDEs into ODEs. The collocation method has been used to numerically solve the ODEs. The method has been implemented in MATLAB using the bvp4c inbuilt function. The effects of the nondimensional parameters on velocity, temperature, and concentration have been presented graphically. Additionally, the skin-friction coefficient, the Nusselt number, and the Sherwood number have been discussed and are presented in a tabular form. The findings demonstrated that increasing the Reynolds number causes a rise in the fluid temperature and velocity. The fluid velocity decreases as the Hartmann number and the weight of the obstacle increase but increases with increasing Grashof numbers. The temperature of the fluid increases as the radiation parameter, or Eckert number, increases, but decreases as the Prandtl number increases. As the Soret number rises, so do the fluid’s temperature and concentration distribution. With an increase in the unsteadiness parameter, the fluid velocity and the concentration distribution decrease, whereas the opposite is seen in temperature. As the Schmidt number, the concentration Grashof number, and the chemical reaction parameter increase, the fluid’s concentration decreases. There is an increase in skin-friction coefficient with increasing Prandtl number, Eckert number, Soret number, thermal Grashof number, concentration Grashof number, thermal radiation parameter, Hartmann number, and unsteadiness parameter, while a decrease is observed with increasing Reynolds number. The Nusselt number increases with an increase in the Prandtl number, Eckert number, thermal radiation parameter, Hartmann number, and unsteadiness parameter. A slight decrease in the Nusselt number has been observed with increasing thermal Grashof number. The Sherwood number decreases with increasing Prandtl number, chemical reaction parameter, and thermal radiation parameter but increases with increasing Schmidt number, Eckert number, and Soret number. The research has the potential for a wide range of applications including but not limited to the medical field and other physical sciences.","PeriodicalId":301406,"journal":{"name":"Int. J. Math. Math. Sci.","volume":"28 5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131150992","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}
Ababi Hailu Ejere, G. Duressa, M. Woldaregay, T. G. Dinka
The objective of this research work is to develop and analyse a numerical scheme for solving singularly perturbed parabolic reaction-diffusion problems with large spatial delay. The presence of the small positive parameter on the term with the highest order of derivative exhibits two strong boundary layers in the solution of the problem, and the large delay term gives rise to a strong interior layer. The layers’ behavior makes it difficult to solve the problem analytically. To treat such a problem, we developed a numerical scheme using the Crank–Nicolson method in the time direction and the central difference method in the spatial direction via nonstandard finite difference methods on uniform meshes. Stability and convergence analyses for the obtained scheme have been established, which show that the developed numerical scheme is uniformly convergent. To confirm the theoretical analysis, model numerical examples are considered and demonstrated.
{"title":"A Parameter-Uniform Numerical Scheme for Solving Singularly Perturbed Parabolic Reaction-Diffusion Problems with Delay in the Spatial Variable","authors":"Ababi Hailu Ejere, G. Duressa, M. Woldaregay, T. G. Dinka","doi":"10.1155/2023/7215106","DOIUrl":"https://doi.org/10.1155/2023/7215106","url":null,"abstract":"The objective of this research work is to develop and analyse a numerical scheme for solving singularly perturbed parabolic reaction-diffusion problems with large spatial delay. The presence of the small positive parameter on the term with the highest order of derivative exhibits two strong boundary layers in the solution of the problem, and the large delay term gives rise to a strong interior layer. The layers’ behavior makes it difficult to solve the problem analytically. To treat such a problem, we developed a numerical scheme using the Crank–Nicolson method in the time direction and the central difference method in the spatial direction via nonstandard finite difference methods on uniform meshes. Stability and convergence analyses for the obtained scheme have been established, which show that the developed numerical scheme is uniformly convergent. To confirm the theoretical analysis, model numerical examples are considered and demonstrated.","PeriodicalId":301406,"journal":{"name":"Int. J. Math. Math. Sci.","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131288002","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, two new versions of the Schur algorithm for computing the matrix exponential of an n × n complex matrix A are presented. Instead of the Schur form, these algorithms use the Jordan–Schur form of A . The Jordan–Schur form is found by less computation and it is determined more reliable than the reduction to Jordan form since it is obtained using only unitary similarity transformations. In contrast to the known methods, the diagonal blocks of the matrix exponential are obtained by using finite Taylor series. This improves the accuracy and avoids the decisions made about the termination of the series expansion. The off-diagonal blocks of the exponential are determined by modifications of the Schur–Parlett or Schur–Fréchet method, which takes advantage of the Jordan–Schur form of the matrix. The numerical features of the new algorithms are discussed, revealing their advantages and disadvantages in comparison with the other methods for computing the matrix exponential. Computational experiments show that using the new algorithms, the matrix exponential is determined in certain cases with higher accuracy than some widely used methods, however, at the price of an increase in the computational cost which is of order n 4 . It is shown that the Jordan–Schur algorithms for computing the matrix exponential are appropriate for matrices with multiple eigenvalues and are especially efficient in cases of large Weyr characteristics.
{"title":"Jordan-Schur Algorithms for Computing the Matrix Exponential","authors":"P. Petkov","doi":"10.1155/2023/4908185","DOIUrl":"https://doi.org/10.1155/2023/4908185","url":null,"abstract":"In this paper, two new versions of the Schur algorithm for computing the matrix exponential of an \u0000 \u0000 n\u0000 ×\u0000 n\u0000 \u0000 complex matrix \u0000 \u0000 A\u0000 \u0000 are presented. Instead of the Schur form, these algorithms use the Jordan–Schur form of \u0000 \u0000 A\u0000 \u0000 . The Jordan–Schur form is found by less computation and it is determined more reliable than the reduction to Jordan form since it is obtained using only unitary similarity transformations. In contrast to the known methods, the diagonal blocks of the matrix exponential are obtained by using finite Taylor series. This improves the accuracy and avoids the decisions made about the termination of the series expansion. The off-diagonal blocks of the exponential are determined by modifications of the Schur–Parlett or Schur–Fréchet method, which takes advantage of the Jordan–Schur form of the matrix. The numerical features of the new algorithms are discussed, revealing their advantages and disadvantages in comparison with the other methods for computing the matrix exponential. Computational experiments show that using the new algorithms, the matrix exponential is determined in certain cases with higher accuracy than some widely used methods, however, at the price of an increase in the computational cost which is of order \u0000 \u0000 \u0000 \u0000 n\u0000 \u0000 \u0000 4\u0000 \u0000 \u0000 \u0000 . It is shown that the Jordan–Schur algorithms for computing the matrix exponential are appropriate for matrices with multiple eigenvalues and are especially efficient in cases of large Weyr characteristics.","PeriodicalId":301406,"journal":{"name":"Int. J. Math. Math. Sci.","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123092023","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 model is proposed for two-way tables of measurement data containing outliers. The two independent variables are categorical and error-free. Neither missing values nor replication is present. The model consists of the sum of a customary additive part that can be fit using least squares and a part that is composed of outliers. Recommendations are made for methods for identifying cells containing outliers and fitting the model. A graph of the observations is used to determine the outliers’ locations. For all cells containing an outlier, replacement values are determined simultaneously using a classical missing-data tool. The result is called the adjusted table. The inserted values are such that, when a mean-based fitting of the adjusted table is performed, the residuals in those cells are zero. The outlying portion of the observation in each of those cells is the difference of the observation and the replacement value. In this way, outliers are removed from further analyses of the adjusted table. This is particularly helpful because outliers can greatly contaminate and alter computations and conclusions. Subsequently, the causes of the outliers might be determined, and statistical estimation and testing can be implemented on the adjusted table.
{"title":"Modeling and Fitting Two-Way Tables Containing Outliers","authors":"D. Farnsworth","doi":"10.1155/2023/6352058","DOIUrl":"https://doi.org/10.1155/2023/6352058","url":null,"abstract":"A model is proposed for two-way tables of measurement data containing outliers. The two independent variables are categorical and error-free. Neither missing values nor replication is present. The model consists of the sum of a customary additive part that can be fit using least squares and a part that is composed of outliers. Recommendations are made for methods for identifying cells containing outliers and fitting the model. A graph of the observations is used to determine the outliers’ locations. For all cells containing an outlier, replacement values are determined simultaneously using a classical missing-data tool. The result is called the adjusted table. The inserted values are such that, when a mean-based fitting of the adjusted table is performed, the residuals in those cells are zero. The outlying portion of the observation in each of those cells is the difference of the observation and the replacement value. In this way, outliers are removed from further analyses of the adjusted table. This is particularly helpful because outliers can greatly contaminate and alter computations and conclusions. Subsequently, the causes of the outliers might be determined, and statistical estimation and testing can be implemented on the adjusted table.","PeriodicalId":301406,"journal":{"name":"Int. J. Math. Math. Sci.","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115283882","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, the effects of chemical reaction and Joule heating on MHD generalized Couette flow between two vertical porous plates with induced magnetic field and Newtonian heating/cooling have been investigated. The mathematical model used for the MHD generalized Couette flow takes into account the effect of viscous dissipation. The system of nonlinear partial differential equations governing the flow was solved numerically using the finite difference method. The resulting numerical schemes are simulated in MATLAB to obtain the profiles of the flow variables such as velocity, induced magnetic field, temperature, and concentration profiles graphically. Also, the effects of the flow parameters on the skin-friction coefficient, Nusselt number, and Sherwood number are obtained and discussed numerically through tabular forms. The findings show that an increase in the chemical reaction parameter leads to a decrease in the concentration profiles. Also, increase in the Joule heating parameter and heat generation parameter leads to an increase in the temperature profiles. Induced magnetic field profiles increase with an increase in Reynold’s number. The findings of this study are important due to its application in developing a variety of chemical technologies, including polymer manufacturing, MHD pumps, food processing, chemical catalytic reactors, astronomy, MHD flow meters, and lubrication.
{"title":"Effects of Chemical Reaction and Joule Heating on MHD Generalized Couette Flow between Two Parallel Vertical Porous Plates with Induced Magnetic Field and Newtonian Heating/Cooling","authors":"Jumanne Mng’ang’a","doi":"10.1155/2023/9134811","DOIUrl":"https://doi.org/10.1155/2023/9134811","url":null,"abstract":"In this article, the effects of chemical reaction and Joule heating on MHD generalized Couette flow between two vertical porous plates with induced magnetic field and Newtonian heating/cooling have been investigated. The mathematical model used for the MHD generalized Couette flow takes into account the effect of viscous dissipation. The system of nonlinear partial differential equations governing the flow was solved numerically using the finite difference method. The resulting numerical schemes are simulated in MATLAB to obtain the profiles of the flow variables such as velocity, induced magnetic field, temperature, and concentration profiles graphically. Also, the effects of the flow parameters on the skin-friction coefficient, Nusselt number, and Sherwood number are obtained and discussed numerically through tabular forms. The findings show that an increase in the chemical reaction parameter leads to a decrease in the concentration profiles. Also, increase in the Joule heating parameter and heat generation parameter leads to an increase in the temperature profiles. Induced magnetic field profiles increase with an increase in Reynold’s number. The findings of this study are important due to its application in developing a variety of chemical technologies, including polymer manufacturing, MHD pumps, food processing, chemical catalytic reactors, astronomy, MHD flow meters, and lubrication.","PeriodicalId":301406,"journal":{"name":"Int. J. Math. Math. Sci.","volume":"64 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125091344","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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