On the concept of optimality interval
The approximants to regular continued fractions constitute best approximations to the numbers they converge to in two ways known as the first and the second kind. This property of continued fractions provides a solution to Gosper's problem of the batting average: if the batting average of a bas...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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Wiley
2002-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Online Access: | http://dx.doi.org/10.1155/S0161171202011420 |
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| _version_ | 1849306676394459136 |
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| author | Lluís Bibiloni Pelegrí Viader Jaume Paradís |
| author_facet | Lluís Bibiloni Pelegrí Viader Jaume Paradís |
| author_sort | Lluís Bibiloni |
| collection | DOAJ |
| description | The approximants to regular continued fractions constitute
best approximations to the numbers they converge to in two ways known as the first and the second kind. This property of continued fractions provides a solution to Gosper's problem of the batting average: if the batting average of a baseball player is 0.334, what is the minimum number of times he has been at bat? In this paper, we tackle somehow the inverse question: given a rational number P/Q, what is the set of all numbers for which P/Q is a best approximation of one or the other kind? We prove that in both cases these optimality sets are intervals and we give a precise description of their endpoints. |
| format | Article |
| id | doaj-art-25eeaabba39d46b5accb9174dbcf6cbf |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 2002-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-25eeaabba39d46b5accb9174dbcf6cbf2025-08-20T03:55:00ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04252002-01-0130955956710.1155/S0161171202011420On the concept of optimality intervalLluís Bibiloni0Pelegrí Viader1Jaume Paradís2Facultat de Ciències de l'Educació, Universidad Autònoma de Barcelona, 08193 Bellaterra, Barcelona, SpainDepartament d'Economia i Empresa, Universidad Pompeu Fabra, Ramon Trias Fargas 25-27, Barcelona 08005, SpainDepartament d'Economia i Empresa, Universidad Pompeu Fabra, Ramon Trias Fargas 25-27, Barcelona 08005, SpainThe approximants to regular continued fractions constitute best approximations to the numbers they converge to in two ways known as the first and the second kind. This property of continued fractions provides a solution to Gosper's problem of the batting average: if the batting average of a baseball player is 0.334, what is the minimum number of times he has been at bat? In this paper, we tackle somehow the inverse question: given a rational number P/Q, what is the set of all numbers for which P/Q is a best approximation of one or the other kind? We prove that in both cases these optimality sets are intervals and we give a precise description of their endpoints.http://dx.doi.org/10.1155/S0161171202011420 |
| spellingShingle | Lluís Bibiloni Pelegrí Viader Jaume Paradís On the concept of optimality interval International Journal of Mathematics and Mathematical Sciences |
| title | On the concept of optimality interval |
| title_full | On the concept of optimality interval |
| title_fullStr | On the concept of optimality interval |
| title_full_unstemmed | On the concept of optimality interval |
| title_short | On the concept of optimality interval |
| title_sort | on the concept of optimality interval |
| url | http://dx.doi.org/10.1155/S0161171202011420 |
| work_keys_str_mv | AT lluisbibiloni ontheconceptofoptimalityinterval AT pelegriviader ontheconceptofoptimalityinterval AT jaumeparadis ontheconceptofoptimalityinterval |