On the Solutions of Three-Variable Frobenius-Related Problems Using Order Reduction Approach
This paper presents a new approach to determine the number of solutions of three-variable Frobenius-related problems and to find their solutions by using order reducing methods. Here, the order of a Frobenius-related problem means the number of variables appearing in the problem. We present two type...
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| Format: | Article |
| Language: | English |
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Wiley
2021-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/2021/6396792 |
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| author | Tian-Xiao He Peter J.-S. Shiue Rama Venkat |
| author_facet | Tian-Xiao He Peter J.-S. Shiue Rama Venkat |
| author_sort | Tian-Xiao He |
| collection | DOAJ |
| description | This paper presents a new approach to determine the number of solutions of three-variable Frobenius-related problems and to find their solutions by using order reducing methods. Here, the order of a Frobenius-related problem means the number of variables appearing in the problem. We present two types of order reduction methods that can be applied to the problem of finding all nonnegative solutions of three-variable Frobenius-related problems. The first method is used to reduce the equation of order three from a three-variable Frobenius-related problem to be a system of equations with two fixed variables. The second method reduces the equation of order three into three equations of order two, for which an algorithm is designed with an interesting open problem on solutions left as a conjecture. |
| format | Article |
| id | doaj-art-c92981e6884a482b822c44c237dfc77f |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2021-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-c92981e6884a482b822c44c237dfc77f2025-02-03T07:23:30ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252021-01-01202110.1155/2021/63967926396792On the Solutions of Three-Variable Frobenius-Related Problems Using Order Reduction ApproachTian-Xiao He0Peter J.-S. Shiue1Rama Venkat2Department of Mathematics, Illinois Wesleyan University, Bloomington, IL 61702-2900, USADepartment of Mathematical Sciences, University of Nevada, Las Vegas, NV 89154-4020, USAHoward R. Hughes College of Engineering, University of Nevada, Las Vegas, NV 89154-4020, USAThis paper presents a new approach to determine the number of solutions of three-variable Frobenius-related problems and to find their solutions by using order reducing methods. Here, the order of a Frobenius-related problem means the number of variables appearing in the problem. We present two types of order reduction methods that can be applied to the problem of finding all nonnegative solutions of three-variable Frobenius-related problems. The first method is used to reduce the equation of order three from a three-variable Frobenius-related problem to be a system of equations with two fixed variables. The second method reduces the equation of order three into three equations of order two, for which an algorithm is designed with an interesting open problem on solutions left as a conjecture.http://dx.doi.org/10.1155/2021/6396792 |
| spellingShingle | Tian-Xiao He Peter J.-S. Shiue Rama Venkat On the Solutions of Three-Variable Frobenius-Related Problems Using Order Reduction Approach International Journal of Mathematics and Mathematical Sciences |
| title | On the Solutions of Three-Variable Frobenius-Related Problems Using Order Reduction Approach |
| title_full | On the Solutions of Three-Variable Frobenius-Related Problems Using Order Reduction Approach |
| title_fullStr | On the Solutions of Three-Variable Frobenius-Related Problems Using Order Reduction Approach |
| title_full_unstemmed | On the Solutions of Three-Variable Frobenius-Related Problems Using Order Reduction Approach |
| title_short | On the Solutions of Three-Variable Frobenius-Related Problems Using Order Reduction Approach |
| title_sort | on the solutions of three variable frobenius related problems using order reduction approach |
| url | http://dx.doi.org/10.1155/2021/6396792 |
| work_keys_str_mv | AT tianxiaohe onthesolutionsofthreevariablefrobeniusrelatedproblemsusingorderreductionapproach AT peterjsshiue onthesolutionsofthreevariablefrobeniusrelatedproblemsusingorderreductionapproach AT ramavenkat onthesolutionsofthreevariablefrobeniusrelatedproblemsusingorderreductionapproach |