Comments on ‘On Hadamard powers of polynomials’
Let f ( x ) = a n x n + a n - 1 x n - 1 + ⋯ + a 1 x + a 0 be a polynomial with real positive coefficients and p ∈ R . The p th Hadamard power of f is the polynomial f [ p ] ( x ) : = a n p x n + a n - 1 p x n - 1 + ⋯ + a 1 p x + a 0 p . We give sufficient conditions for f [ p ] to be a Hurwitz polyn...
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| Veröffentlicht in: | Mathematics of control, signals, and systems signals, and systems, 2017-09, Vol.29 (3), p.1, Article 16 |
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| container_title | Mathematics of control, signals, and systems |
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| creator | Białas, Stanisław Białas-Cież, Leokadia |
| description | Let
f
(
x
)
=
a
n
x
n
+
a
n
-
1
x
n
-
1
+
⋯
+
a
1
x
+
a
0
be a polynomial with real positive coefficients and
p
∈
R
. The
p
th Hadamard power of
f
is the polynomial
f
[
p
]
(
x
)
:
=
a
n
p
x
n
+
a
n
-
1
p
x
n
-
1
+
⋯
+
a
1
p
x
+
a
0
p
. We give sufficient conditions for
f
[
p
]
to be a Hurwitz polynomial (i.e., to be a stable polynomial) for all
p
>
p
0
or
p
<
p
1
with some positive
p
0
and negative
p
1
(without any assumption about stability of
f
). Theorem 5 by Gregor and Tišer (Math Control Signals Syst 11:372–378,
1998
) asserts that if
f
is a stable polynomial with positive coefficients then
f
[
p
]
is stable for every
p
≥
1
. We construct a counterexample to this statement. |
| doi_str_mv | 10.1007/s00498-017-0202-0 |
| format | Article |
| fullrecord | <record><control><sourceid>proquest_cross</sourceid><recordid>TN_cdi_proquest_journals_1943842717</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>1943842717</sourcerecordid><originalsourceid>FETCH-LOGICAL-c316t-bae189764653964a7e50fde0c8b9c1dd208a4960b5c1b2a1762fdf58c386fd923</originalsourceid><addsrcrecordid>eNp1kM1KAzEUhYMoWKsP4G7AdfTeJJOfpRS1QqEbBXchM5NIS2emJi3SXR9DX69PYsq4cOPqHrjn3J-PkGuEWwRQdwlAGE0BFQUGjMIJGaHgJS2lfjslIzCcUYFGnJOLlJYAgFLhiPBJ37a-26Si74rD_mveFVPXuNbFplj3nz7mRshqtev6duFW6bD_viRnISt_9VvH5PXx4WUypbP50_PkfkZrjnJDK-dRGyWFLLmRwilfQmg81LoyNTYNA-2EkVCVNVbMoZIsNKHUNdcyNIbxMbkZ5q5j_7H1aWOX_TZ2eaXNj3AtmEKVXTi46tinFH2w67jI9-8sgj2ysQMbm9nYIxsLOcOGTMre7t3HP5P_Df0AAChnIg</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype><pqid>1943842717</pqid></control><display><type>article</type><title>Comments on ‘On Hadamard powers of polynomials’</title><source>SpringerLink Journals - AutoHoldings</source><creator>Białas, Stanisław ; Białas-Cież, Leokadia</creator><creatorcontrib>Białas, Stanisław ; Białas-Cież, Leokadia</creatorcontrib><description>Let
f
(
x
)
=
a
n
x
n
+
a
n
-
1
x
n
-
1
+
⋯
+
a
1
x
+
a
0
be a polynomial with real positive coefficients and
p
∈
R
. The
p
th Hadamard power of
f
is the polynomial
f
[
p
]
(
x
)
:
=
a
n
p
x
n
+
a
n
-
1
p
x
n
-
1
+
⋯
+
a
1
p
x
+
a
0
p
. We give sufficient conditions for
f
[
p
]
to be a Hurwitz polynomial (i.e., to be a stable polynomial) for all
p
>
p
0
or
p
<
p
1
with some positive
p
0
and negative
p
1
(without any assumption about stability of
f
). Theorem 5 by Gregor and Tišer (Math Control Signals Syst 11:372–378,
1998
) asserts that if
f
is a stable polynomial with positive coefficients then
f
[
p
]
is stable for every
p
≥
1
. We construct a counterexample to this statement.</description><identifier>ISSN: 0932-4194</identifier><identifier>EISSN: 1435-568X</identifier><identifier>DOI: 10.1007/s00498-017-0202-0</identifier><language>eng</language><publisher>London: Springer London</publisher><subject>Communications Engineering ; Control ; Control stability ; Mathematics ; Mathematics and Statistics ; Mechatronics ; Networks ; Original Article ; Polynomials ; Robotics ; Systems Theory</subject><ispartof>Mathematics of control, signals, and systems, 2017-09, Vol.29 (3), p.1, Article 16</ispartof><rights>The Author(s) 2017</rights><rights>Mathematics of Control, Signals, and Systems is a copyright of Springer, 2017.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c316t-bae189764653964a7e50fde0c8b9c1dd208a4960b5c1b2a1762fdf58c386fd923</citedby><cites>FETCH-LOGICAL-c316t-bae189764653964a7e50fde0c8b9c1dd208a4960b5c1b2a1762fdf58c386fd923</cites><orcidid>0000-0002-1207-2541</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00498-017-0202-0$$EPDF$$P50$$Gspringer$$Hfree_for_read</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00498-017-0202-0$$EHTML$$P50$$Gspringer$$Hfree_for_read</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Białas, Stanisław</creatorcontrib><creatorcontrib>Białas-Cież, Leokadia</creatorcontrib><title>Comments on ‘On Hadamard powers of polynomials’</title><title>Mathematics of control, signals, and systems</title><addtitle>Math. Control Signals Syst</addtitle><description>Let
f
(
x
)
=
a
n
x
n
+
a
n
-
1
x
n
-
1
+
⋯
+
a
1
x
+
a
0
be a polynomial with real positive coefficients and
p
∈
R
. The
p
th Hadamard power of
f
is the polynomial
f
[
p
]
(
x
)
:
=
a
n
p
x
n
+
a
n
-
1
p
x
n
-
1
+
⋯
+
a
1
p
x
+
a
0
p
. We give sufficient conditions for
f
[
p
]
to be a Hurwitz polynomial (i.e., to be a stable polynomial) for all
p
>
p
0
or
p
<
p
1
with some positive
p
0
and negative
p
1
(without any assumption about stability of
f
). Theorem 5 by Gregor and Tišer (Math Control Signals Syst 11:372–378,
1998
) asserts that if
f
is a stable polynomial with positive coefficients then
f
[
p
]
is stable for every
p
≥
1
. We construct a counterexample to this statement.</description><subject>Communications Engineering</subject><subject>Control</subject><subject>Control stability</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Mechatronics</subject><subject>Networks</subject><subject>Original Article</subject><subject>Polynomials</subject><subject>Robotics</subject><subject>Systems Theory</subject><issn>0932-4194</issn><issn>1435-568X</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2017</creationdate><recordtype>article</recordtype><sourceid>C6C</sourceid><sourceid>ABUWG</sourceid><sourceid>AFKRA</sourceid><sourceid>AZQEC</sourceid><sourceid>BENPR</sourceid><sourceid>CCPQU</sourceid><sourceid>DWQXO</sourceid><sourceid>GNUQQ</sourceid><recordid>eNp1kM1KAzEUhYMoWKsP4G7AdfTeJJOfpRS1QqEbBXchM5NIS2emJi3SXR9DX69PYsq4cOPqHrjn3J-PkGuEWwRQdwlAGE0BFQUGjMIJGaHgJS2lfjslIzCcUYFGnJOLlJYAgFLhiPBJ37a-26Si74rD_mveFVPXuNbFplj3nz7mRshqtev6duFW6bD_viRnISt_9VvH5PXx4WUypbP50_PkfkZrjnJDK-dRGyWFLLmRwilfQmg81LoyNTYNA-2EkVCVNVbMoZIsNKHUNdcyNIbxMbkZ5q5j_7H1aWOX_TZ2eaXNj3AtmEKVXTi46tinFH2w67jI9-8sgj2ysQMbm9nYIxsLOcOGTMre7t3HP5P_Df0AAChnIg</recordid><startdate>20170901</startdate><enddate>20170901</enddate><creator>Białas, Stanisław</creator><creator>Białas-Cież, Leokadia</creator><general>Springer London</general><general>Springer Nature B.V</general><scope>C6C</scope><scope>AAYXX</scope><scope>CITATION</scope><scope>3V.</scope><scope>7SC</scope><scope>7SP</scope><scope>7WY</scope><scope>7WZ</scope><scope>7XB</scope><scope>87Z</scope><scope>88I</scope><scope>8AL</scope><scope>8AO</scope><scope>8FD</scope><scope>8FE</scope><scope>8FG</scope><scope>8FK</scope><scope>8FL</scope><scope>ABJCF</scope><scope>ABUWG</scope><scope>AFKRA</scope><scope>ARAPS</scope><scope>AZQEC</scope><scope>BENPR</scope><scope>BEZIV</scope><scope>BGLVJ</scope><scope>CCPQU</scope><scope>DWQXO</scope><scope>FRNLG</scope><scope>F~G</scope><scope>GNUQQ</scope><scope>HCIFZ</scope><scope>JQ2</scope><scope>K60</scope><scope>K6~</scope><scope>K7-</scope><scope>L.-</scope><scope>L6V</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope><scope>M0C</scope><scope>M0N</scope><scope>M2P</scope><scope>M7S</scope><scope>P5Z</scope><scope>P62</scope><scope>PQBIZ</scope><scope>PQBZA</scope><scope>PQEST</scope><scope>PQQKQ</scope><scope>PQUKI</scope><scope>PTHSS</scope><scope>PYYUZ</scope><scope>Q9U</scope><scope>S0W</scope><orcidid>https://orcid.org/0000-0002-1207-2541</orcidid></search><sort><creationdate>20170901</creationdate><title>Comments on ‘On Hadamard powers of polynomials’</title><author>Białas, Stanisław ; Białas-Cież, Leokadia</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c316t-bae189764653964a7e50fde0c8b9c1dd208a4960b5c1b2a1762fdf58c386fd923</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2017</creationdate><topic>Communications Engineering</topic><topic>Control</topic><topic>Control stability</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Mechatronics</topic><topic>Networks</topic><topic>Original Article</topic><topic>Polynomials</topic><topic>Robotics</topic><topic>Systems Theory</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Białas, Stanisław</creatorcontrib><creatorcontrib>Białas-Cież, Leokadia</creatorcontrib><collection>Springer Nature OA Free Journals</collection><collection>CrossRef</collection><collection>ProQuest Central (Corporate)</collection><collection>Computer and Information Systems Abstracts</collection><collection>Electronics & Communications Abstracts</collection><collection>ABI/INFORM Collection</collection><collection>ABI/INFORM Global (PDF only)</collection><collection>ProQuest Central (purchase pre-March 2016)</collection><collection>ABI/INFORM Global (Alumni Edition)</collection><collection>Science Database (Alumni Edition)</collection><collection>Computing Database (Alumni Edition)</collection><collection>ProQuest Pharma Collection</collection><collection>Technology Research Database</collection><collection>ProQuest SciTech Collection</collection><collection>ProQuest Technology Collection</collection><collection>ProQuest Central (Alumni) (purchase pre-March 2016)</collection><collection>ABI/INFORM Collection (Alumni Edition)</collection><collection>Materials Science & Engineering Collection</collection><collection>ProQuest Central (Alumni Edition)</collection><collection>ProQuest Central UK/Ireland</collection><collection>Advanced Technologies & Aerospace Collection</collection><collection>ProQuest Central Essentials</collection><collection>ProQuest Central</collection><collection>Business Premium Collection</collection><collection>Technology Collection</collection><collection>ProQuest One Community College</collection><collection>ProQuest Central Korea</collection><collection>Business Premium Collection (Alumni)</collection><collection>ABI/INFORM Global (Corporate)</collection><collection>ProQuest Central Student</collection><collection>SciTech Premium Collection</collection><collection>ProQuest Computer Science Collection</collection><collection>ProQuest Business Collection (Alumni Edition)</collection><collection>ProQuest Business Collection</collection><collection>Computer Science Database</collection><collection>ABI/INFORM Professional Advanced</collection><collection>ProQuest Engineering 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><collection>ABI/INFORM Global</collection><collection>Computing Database</collection><collection>Science Database</collection><collection>Engineering Database</collection><collection>Advanced Technologies & Aerospace Database</collection><collection>ProQuest Advanced Technologies & Aerospace Collection</collection><collection>ProQuest One Business</collection><collection>ProQuest One Business (Alumni)</collection><collection>ProQuest One Academic Eastern Edition (DO NOT USE)</collection><collection>ProQuest One Academic</collection><collection>ProQuest One Academic UKI Edition</collection><collection>Engineering Collection</collection><collection>ABI/INFORM Collection China</collection><collection>ProQuest Central Basic</collection><collection>DELNET Engineering & Technology Collection</collection><jtitle>Mathematics of control, signals, and systems</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Białas, Stanisław</au><au>Białas-Cież, Leokadia</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Comments on ‘On Hadamard powers of polynomials’</atitle><jtitle>Mathematics of control, signals, and systems</jtitle><stitle>Math. Control Signals Syst</stitle><date>2017-09-01</date><risdate>2017</risdate><volume>29</volume><issue>3</issue><spage>1</spage><pages>1-</pages><artnum>16</artnum><issn>0932-4194</issn><eissn>1435-568X</eissn><abstract>Let
f
(
x
)
=
a
n
x
n
+
a
n
-
1
x
n
-
1
+
⋯
+
a
1
x
+
a
0
be a polynomial with real positive coefficients and
p
∈
R
. The
p
th Hadamard power of
f
is the polynomial
f
[
p
]
(
x
)
:
=
a
n
p
x
n
+
a
n
-
1
p
x
n
-
1
+
⋯
+
a
1
p
x
+
a
0
p
. We give sufficient conditions for
f
[
p
]
to be a Hurwitz polynomial (i.e., to be a stable polynomial) for all
p
>
p
0
or
p
<
p
1
with some positive
p
0
and negative
p
1
(without any assumption about stability of
f
). Theorem 5 by Gregor and Tišer (Math Control Signals Syst 11:372–378,
1998
) asserts that if
f
is a stable polynomial with positive coefficients then
f
[
p
]
is stable for every
p
≥
1
. We construct a counterexample to this statement.</abstract><cop>London</cop><pub>Springer London</pub><doi>10.1007/s00498-017-0202-0</doi><orcidid>https://orcid.org/0000-0002-1207-2541</orcidid><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0932-4194 |
| ispartof | Mathematics of control, signals, and systems, 2017-09, Vol.29 (3), p.1, Article 16 |
| issn | 0932-4194 1435-568X |
| language | eng |
| recordid | cdi_proquest_journals_1943842717 |
| source | SpringerLink Journals - AutoHoldings |
| subjects | Communications Engineering Control Control stability Mathematics Mathematics and Statistics Mechatronics Networks Original Article Polynomials Robotics Systems Theory |
| title | Comments on ‘On Hadamard powers of polynomials’ |
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