The complete characterization of the minimum size supertail in a subspace partition
Let q be a prime power and let n be a positive integer. Let V=V(n,q) denote the vector space of dimension n over Fq. A subspace partitionP of V is a collection of subspaces of V with the property that each nonzero vector is in exactly one of the subspaces in P. Suppose that d1,…,dk are the different...
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| Veröffentlicht in: | Linear algebra and its applications 2018-12, Vol.559, p.172-180 |
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| creator | Năstase, Esmeralda L. Sissokho, Papa A. |
| description | Let q be a prime power and let n be a positive integer. Let V=V(n,q) denote the vector space of dimension n over Fq. A subspace partitionP of V is a collection of subspaces of V with the property that each nonzero vector is in exactly one of the subspaces in P. Suppose that d1,…,dk are the different dimensions, in increasing order, that occur in the subspace partition P. For any integer s, with 2≤s≤k, the ds-supertailS of P is the collection of all subspaces X∈P such that dimX |
| doi_str_mv | 10.1016/j.laa.2018.09.007 |
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Let V=V(n,q) denote the vector space of dimension n over Fq. A subspace partitionP of V is a collection of subspaces of V with the property that each nonzero vector is in exactly one of the subspaces in P. Suppose that d1,…,dk are the different dimensions, in increasing order, that occur in the subspace partition P. For any integer s, with 2≤s≤k, the ds-supertailS of P is the collection of all subspaces X∈P such that dimX<ds. It was shown that |S|≥σq(ds,ds−1), where σq(ds,ds−1) denotes the minimum number of subspaces over all subspace partitions of V(ds,q) in which the largest subspace has dimension ds−1. Moreover, it was shown that if ds≥2ds−1 and equality holds in the previous bound on |S|, then the union of the subspaces in S forms a subspace. This characterization was also conjectured to hold if ds<2ds−1. This conjecture was recently proved in certain cases. In this paper, we use a much simpler approach to completely settle this conjecture.</description><identifier>ISSN: 0024-3795</identifier><identifier>EISSN: 1873-1856</identifier><identifier>DOI: 10.1016/j.laa.2018.09.007</identifier><language>eng</language><publisher>Amsterdam: Elsevier Inc</publisher><subject>Collection ; Integral equations ; Linear algebra ; Partitions (mathematics) ; Subspace partition ; Subspaces ; Supertail of a subspace partition ; Vector space ; Vector space partition</subject><ispartof>Linear algebra and its applications, 2018-12, Vol.559, p.172-180</ispartof><rights>2018 Elsevier Inc.</rights><rights>Copyright American Elsevier Company, Inc. 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In this paper, we use a much simpler approach to completely settle this conjecture.</description><subject>Collection</subject><subject>Integral equations</subject><subject>Linear algebra</subject><subject>Partitions (mathematics)</subject><subject>Subspace partition</subject><subject>Subspaces</subject><subject>Supertail of a subspace partition</subject><subject>Vector space</subject><subject>Vector space partition</subject><issn>0024-3795</issn><issn>1873-1856</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2018</creationdate><recordtype>article</recordtype><recordid>eNp9kE1LxDAQhoMouK7-AG8Fz60z6bZJ8SSLX7DgwfUc0nTKpvTLJCu4v94s69nTzMD7zAwPY7cIGQKW913Wa51xQJlBlQGIM7ZAKfIUZVGeswUAX6W5qIpLduV9BwArAXzBPrY7Ssw0zD2F2Oy00yaQswcd7DQmU5uEGBjsaIf9kHh7oMTvZ3JB2z6xY6LjWPtZG0pm7YI9UtfsotW9p5u_umSfz0_b9Wu6eX95Wz9uUpNzCGmT56WkVteyNiRKXpRVzVEXlZBVo2FVmJoa4Jq3DVYoZVNUyHmB0ArZtijyJbs77Z3d9LUnH1Q37d0YTyqOuUCUIPKYwlPKuMl7R62anR20-1EI6uhOdSq6U0d3CioV3UXm4cRQfP_bklPeWBoNNdaRCaqZ7D_0LzEodyc</recordid><startdate>20181215</startdate><enddate>20181215</enddate><creator>Năstase, Esmeralda L.</creator><creator>Sissokho, Papa A.</creator><general>Elsevier Inc</general><general>American Elsevier Company, Inc</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20181215</creationdate><title>The complete characterization of the minimum size supertail in a subspace partition</title><author>Năstase, Esmeralda L. ; Sissokho, Papa A.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c320t-d3368efab8bce762569b21a59789da045cbed02a2fd19188d59122510f78ff173</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2018</creationdate><topic>Collection</topic><topic>Integral equations</topic><topic>Linear algebra</topic><topic>Partitions (mathematics)</topic><topic>Subspace partition</topic><topic>Subspaces</topic><topic>Supertail of a subspace partition</topic><topic>Vector space</topic><topic>Vector space partition</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Năstase, Esmeralda L.</creatorcontrib><creatorcontrib>Sissokho, Papa A.</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Linear algebra and its applications</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Năstase, Esmeralda L.</au><au>Sissokho, Papa A.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>The complete characterization of the minimum size supertail in a subspace partition</atitle><jtitle>Linear algebra and its applications</jtitle><date>2018-12-15</date><risdate>2018</risdate><volume>559</volume><spage>172</spage><epage>180</epage><pages>172-180</pages><issn>0024-3795</issn><eissn>1873-1856</eissn><abstract>Let q be a prime power and let n be a positive integer. Let V=V(n,q) denote the vector space of dimension n over Fq. A subspace partitionP of V is a collection of subspaces of V with the property that each nonzero vector is in exactly one of the subspaces in P. Suppose that d1,…,dk are the different dimensions, in increasing order, that occur in the subspace partition P. For any integer s, with 2≤s≤k, the ds-supertailS of P is the collection of all subspaces X∈P such that dimX<ds. It was shown that |S|≥σq(ds,ds−1), where σq(ds,ds−1) denotes the minimum number of subspaces over all subspace partitions of V(ds,q) in which the largest subspace has dimension ds−1. Moreover, it was shown that if ds≥2ds−1 and equality holds in the previous bound on |S|, then the union of the subspaces in S forms a subspace. This characterization was also conjectured to hold if ds<2ds−1. This conjecture was recently proved in certain cases. 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| source | Elsevier ScienceDirect Journals; Elektronische Zeitschriftenbibliothek - Frei zugängliche E-Journals |
| subjects | Collection Integral equations Linear algebra Partitions (mathematics) Subspace partition Subspaces Supertail of a subspace partition Vector space Vector space partition |
| title | The complete characterization of the minimum size supertail in a subspace partition |
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