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Possible isolation number of a matrix over nonnegative integers
Let ℤ + be the semiring of all nonnegative integers and A an m × n matrix over ℤ + . The rank of A is the smallest k such that A can be factored as an m × k matrix times a k × n matrix. The isolation number of A is the maximum number of nonzero entries in A such that no two are in any row or any col...
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| Published in: | Czechoslovak mathematical journal 2018-12, Vol.68 (4), p.1055-1066 |
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| Main Authors: | , , |
| Format: | Article |
| Language: | English |
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| container_end_page | 1066 |
| container_issue | 4 |
| container_start_page | 1055 |
| container_title | Czechoslovak mathematical journal |
| container_volume | 68 |
| creator | Beasley, LeRoy B. Jun, Young Bae Song, Seok-Zun |
| description | Let ℤ
+
be the semiring of all nonnegative integers and
A
an
m
×
n
matrix over ℤ
+
. The rank of
A
is the smallest
k
such that
A
can be factored as an
m
×
k
matrix times a
k
×
n
matrix. The isolation number of
A
is the maximum number of nonzero entries in
A
such that no two are in any row or any column, and no two are in a 2 × 2 submatrix of all nonzero entries. We have that the isolation number of
A
is a lower bound of the rank of
A
. For
A
with isolation number
k
, we investigate the possible values of the rank of
A
and the Boolean rank of the support of
A
. So we obtain that the isolation number and the Boolean rank of the support of a given matrix are the same if and only if the isolation number is 1 or 2 only. We also determine a special type of
m
×
n
matrices whose isolation number is
m
. That is, those matrices are permutationally equivalent to a matrix
A
whose support contains a submatrix of a sum of the identity matrix and a tournament matrix. |
| doi_str_mv | 10.21136/CMJ.2018.0068-17 |
| format | article |
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+
be the semiring of all nonnegative integers and
A
an
m
×
n
matrix over ℤ
+
. The rank of
A
is the smallest
k
such that
A
can be factored as an
m
×
k
matrix times a
k
×
n
matrix. The isolation number of
A
is the maximum number of nonzero entries in
A
such that no two are in any row or any column, and no two are in a 2 × 2 submatrix of all nonzero entries. We have that the isolation number of
A
is a lower bound of the rank of
A
. For
A
with isolation number
k
, we investigate the possible values of the rank of
A
and the Boolean rank of the support of
A
. So we obtain that the isolation number and the Boolean rank of the support of a given matrix are the same if and only if the isolation number is 1 or 2 only. We also determine a special type of
m
×
n
matrices whose isolation number is
m
. That is, those matrices are permutationally equivalent to a matrix
A
whose support contains a submatrix of a sum of the identity matrix and a tournament matrix.</description><identifier>ISSN: 0011-4642</identifier><identifier>EISSN: 1572-9141</identifier><identifier>DOI: 10.21136/CMJ.2018.0068-17</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Analysis ; Boolean algebra ; Convex and Discrete Geometry ; Integers ; Lower bounds ; Mathematical Modeling and Industrial Mathematics ; Mathematics ; Mathematics and Statistics ; Ordinary Differential Equations</subject><ispartof>Czechoslovak mathematical journal, 2018-12, Vol.68 (4), p.1055-1066</ispartof><rights>Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2018</rights><rights>Copyright Springer Science & Business Media 2018</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c311t-a05a16eec66b73e840363f0fd31c9276c5200aea3e4ec89c1e747b9c6c5fa3de3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27923,27924</link.rule.ids></links><search><creatorcontrib>Beasley, LeRoy B.</creatorcontrib><creatorcontrib>Jun, Young Bae</creatorcontrib><creatorcontrib>Song, Seok-Zun</creatorcontrib><title>Possible isolation number of a matrix over nonnegative integers</title><title>Czechoslovak mathematical journal</title><addtitle>Czech Math J</addtitle><description>Let ℤ
+
be the semiring of all nonnegative integers and
A
an
m
×
n
matrix over ℤ
+
. The rank of
A
is the smallest
k
such that
A
can be factored as an
m
×
k
matrix times a
k
×
n
matrix. The isolation number of
A
is the maximum number of nonzero entries in
A
such that no two are in any row or any column, and no two are in a 2 × 2 submatrix of all nonzero entries. We have that the isolation number of
A
is a lower bound of the rank of
A
. For
A
with isolation number
k
, we investigate the possible values of the rank of
A
and the Boolean rank of the support of
A
. So we obtain that the isolation number and the Boolean rank of the support of a given matrix are the same if and only if the isolation number is 1 or 2 only. We also determine a special type of
m
×
n
matrices whose isolation number is
m
. That is, those matrices are permutationally equivalent to a matrix
A
whose support contains a submatrix of a sum of the identity matrix and a tournament matrix.</description><subject>Analysis</subject><subject>Boolean algebra</subject><subject>Convex and Discrete Geometry</subject><subject>Integers</subject><subject>Lower bounds</subject><subject>Mathematical Modeling and Industrial Mathematics</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Ordinary Differential Equations</subject><issn>0011-4642</issn><issn>1572-9141</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2018</creationdate><recordtype>article</recordtype><sourceid/><recordid>eNpFkMFKAzEQhoMoWKsP4G3B864zSTbZPYkUrUpFD3oO2XS2bNkmdbMtPr6pFTwNzHzM__Mxdo1QcEShbmevLwUHrAoAVeWoT9gES83zGiWesgkAYi6V5OfsIsY1AAiU1YTdvYcYu6anrIuht2MXfOZ3m4aGLLSZzTZ2HLrvLOzTwgfvaZWYfaL9SCsa4iU7a20f6epvTtnn48PH7ClfvM2fZ_eL3AnEMbdQWlRETqlGC6okCCVaaJcCXc21ciUHsGQFSXJV7ZC01E3t0qG1Ykliym6Of7dD-NpRHM067AafIg1HCWVdKlUnih-puB06n_r9UwjmV5RJosxBlDmIMqjFD5zYW6E</recordid><startdate>20181201</startdate><enddate>20181201</enddate><creator>Beasley, LeRoy B.</creator><creator>Jun, Young Bae</creator><creator>Song, Seok-Zun</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope/></search><sort><creationdate>20181201</creationdate><title>Possible isolation number of a matrix over nonnegative integers</title><author>Beasley, LeRoy B. ; Jun, Young Bae ; Song, Seok-Zun</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c311t-a05a16eec66b73e840363f0fd31c9276c5200aea3e4ec89c1e747b9c6c5fa3de3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2018</creationdate><topic>Analysis</topic><topic>Boolean algebra</topic><topic>Convex and Discrete Geometry</topic><topic>Integers</topic><topic>Lower bounds</topic><topic>Mathematical Modeling and Industrial Mathematics</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Ordinary Differential Equations</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Beasley, LeRoy B.</creatorcontrib><creatorcontrib>Jun, Young Bae</creatorcontrib><creatorcontrib>Song, Seok-Zun</creatorcontrib><jtitle>Czechoslovak mathematical journal</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Beasley, LeRoy B.</au><au>Jun, Young Bae</au><au>Song, Seok-Zun</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Possible isolation number of a matrix over nonnegative integers</atitle><jtitle>Czechoslovak mathematical journal</jtitle><stitle>Czech Math J</stitle><date>2018-12-01</date><risdate>2018</risdate><volume>68</volume><issue>4</issue><spage>1055</spage><epage>1066</epage><pages>1055-1066</pages><issn>0011-4642</issn><eissn>1572-9141</eissn><abstract>Let ℤ
+
be the semiring of all nonnegative integers and
A
an
m
×
n
matrix over ℤ
+
. The rank of
A
is the smallest
k
such that
A
can be factored as an
m
×
k
matrix times a
k
×
n
matrix. The isolation number of
A
is the maximum number of nonzero entries in
A
such that no two are in any row or any column, and no two are in a 2 × 2 submatrix of all nonzero entries. We have that the isolation number of
A
is a lower bound of the rank of
A
. For
A
with isolation number
k
, we investigate the possible values of the rank of
A
and the Boolean rank of the support of
A
. So we obtain that the isolation number and the Boolean rank of the support of a given matrix are the same if and only if the isolation number is 1 or 2 only. We also determine a special type of
m
×
n
matrices whose isolation number is
m
. That is, those matrices are permutationally equivalent to a matrix
A
whose support contains a submatrix of a sum of the identity matrix and a tournament matrix.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.21136/CMJ.2018.0068-17</doi><tpages>12</tpages><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0011-4642 |
| ispartof | Czechoslovak mathematical journal, 2018-12, Vol.68 (4), p.1055-1066 |
| issn | 0011-4642 1572-9141 |
| language | eng |
| recordid | cdi_proquest_journals_2140595669 |
| source | Springer Nature |
| subjects | Analysis Boolean algebra Convex and Discrete Geometry Integers Lower bounds Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Ordinary Differential Equations |
| title | Possible isolation number of a matrix over nonnegative integers |
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