Dot product rearrangements
Let a=(an), x=(xn) denote nonnegative sequences; x=(xπ(n)) denotes the rearranged sequence determined by the permutation π, a⋅x denotes the dot product ∑anxn; and S(a,x) denotes {a⋅xπ:π is a permuation of the positive integers}. We examine S(a,x) as a subset of the nonnegative real line in certain s...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Wiley
1983-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
| Subjects: | |
| Online Access: | http://dx.doi.org/10.1155/S0161171283000368 |
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| _version_ | 1849306331897397248 |
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| author | Paul Erdos Gary Weiss |
| author_facet | Paul Erdos Gary Weiss |
| author_sort | Paul Erdos |
| collection | DOAJ |
| description | Let a=(an), x=(xn) denote nonnegative sequences; x=(xπ(n)) denotes the rearranged sequence determined by the permutation π, a⋅x denotes the dot product ∑anxn; and S(a,x) denotes {a⋅xπ:π is a permuation of the positive integers}. We examine S(a,x) as a subset of the nonnegative real line in certain special circumstances. The main result is that if an↑∞, then S(a,x)=[a⋅x,∞] for every xn↓≠0 if and only if an+1/an is uniformly bounded. |
| format | Article |
| id | doaj-art-2e80f1b6e009410db2a6b6d805855ca3 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1983-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-2e80f1b6e009410db2a6b6d805855ca32025-08-20T03:55:07ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251983-01-016340941810.1155/S0161171283000368Dot product rearrangementsPaul Erdos0Gary Weiss1Mathematics Institute, University of Cincinnati, Budapest, HungaryMathematics Institute, University of Cincinnati, Budapest, HungaryLet a=(an), x=(xn) denote nonnegative sequences; x=(xπ(n)) denotes the rearranged sequence determined by the permutation π, a⋅x denotes the dot product ∑anxn; and S(a,x) denotes {a⋅xπ:π is a permuation of the positive integers}. We examine S(a,x) as a subset of the nonnegative real line in certain special circumstances. The main result is that if an↑∞, then S(a,x)=[a⋅x,∞] for every xn↓≠0 if and only if an+1/an is uniformly bounded.http://dx.doi.org/10.1155/S0161171283000368dot productseries rearrangementsconditional convergence. |
| spellingShingle | Paul Erdos Gary Weiss Dot product rearrangements International Journal of Mathematics and Mathematical Sciences dot product series rearrangements conditional convergence. |
| title | Dot product rearrangements |
| title_full | Dot product rearrangements |
| title_fullStr | Dot product rearrangements |
| title_full_unstemmed | Dot product rearrangements |
| title_short | Dot product rearrangements |
| title_sort | dot product rearrangements |
| topic | dot product series rearrangements conditional convergence. |
| url | http://dx.doi.org/10.1155/S0161171283000368 |
| work_keys_str_mv | AT paulerdos dotproductrearrangements AT garyweiss dotproductrearrangements |