Expected density of complex zeros of random hyperbolic polynomials
There are many known asymptotic estimates for the expected number of real zeros of polynomial H n ( z) = η 1 cosh ζ z + η 2 cosh 2ζ z + ⋯ + η n cosh nζ z, where η j , j = 1, 2, 3, …, n is a sequence of independent random variables. This paper provides the asymptotic formula for the expected density...
Saved in:
| Published in: | Applied mathematics letters Vol. 15; no. 4; pp. 389 - 393 |
|---|---|
| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Oxford
Elsevier Ltd
01.05.2002
Elsevier |
| Subjects: | |
| ISSN: | 0893-9659, 1873-5452 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Abstract | There are many known asymptotic estimates for the expected number of real zeros of polynomial
H
n
(
z) = η
1 cosh ζ
z + η
2 cosh 2ζ
z + ⋯ + η
n
cosh
nζ
z, where η
j
,
j = 1, 2, 3, …,
n is a sequence of independent random variables. This paper provides the asymptotic formula for the expected density of complex zeros of
H
n
(
z), where η
j
=
a
j
+
ib
j
and
a
j
and
b
j
,
j = 1, 2, 3, …,
n are sequences of independent normally distributed random variables. It is shown that this asymptotic formula for the density of complex zeros remains invariant for other types of polynomials, for instance random trigonometric polynomials, previously studied. |
|---|---|
| AbstractList | There are many known asymptotic estimates for the expected number of real zeros of polynomial
H
n
(
z) = η
1 cosh ζ
z + η
2 cosh 2ζ
z + ⋯ + η
n
cosh
nζ
z, where η
j
,
j = 1, 2, 3, …,
n is a sequence of independent random variables. This paper provides the asymptotic formula for the expected density of complex zeros of
H
n
(
z), where η
j
=
a
j
+
ib
j
and
a
j
and
b
j
,
j = 1, 2, 3, …,
n are sequences of independent normally distributed random variables. It is shown that this asymptotic formula for the density of complex zeros remains invariant for other types of polynomials, for instance random trigonometric polynomials, previously studied. |
| Author | Farahmand, K. Grigorash, A. |
| Author_xml | – sequence: 1 givenname: K. surname: Farahmand fullname: Farahmand, K. email: K.Farahmand@ulst.ac.uk – sequence: 2 givenname: A. surname: Grigorash fullname: Grigorash, A. email: AV.Grigorash@ulst.ac.uk |
| BackLink | http://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=13546620$$DView record in Pascal Francis |
| BookMark | eNqFkEtLAzEUhYNUsK3-BGE2gi5G85g8ZiVa6gMKLtR1SPPAyMxkSIp0-uvNtKJLV_dwOfdw7jcDky50FoBzBK8RROzmFYqalDWj9SVEVxCiSpTkCEyR4KSkFcUTMP21nIBZSp8QQlITMQX3y21v9caawtgu-c1QBFfo0PaN3RY7G0MaF1F1JrTFx9DbuA6N10UfmqELrVdNOgXHLg979jPn4P1h-bZ4Klcvj8-Lu1WpseCbUhjETC2QMJyvLRGVw4pSZpzCNTS6Mo5nyRwmjgitGcrScIe5xkwJQ8gcXBxye5W0alwupX2SffStioNEhFaMYZh99ODTuX2K1v1ZoByByT0wOdKQEMk9MDnm3x7ubH7iy9sok_a209b4mBFJE_w_Cd8rYHUQ |
| Cites_doi | 10.1090/S0002-9947-97-01766-2 10.1090/S0273-0979-1995-00571-9 10.1090/S0002-9947-1995-1308023-8 10.1090/S0002-9939-1995-1264810-1 10.1090/S0002-9904-1943-07912-8 10.1090/S0002-9939-1988-0955018-1 10.1023/A:1021697221011 10.1080/07362999908809619 |
| ContentType | Journal Article |
| Copyright | 2002 2002 INIST-CNRS |
| Copyright_xml | – notice: 2002 – notice: 2002 INIST-CNRS |
| DBID | 6I. AAFTH AAYXX CITATION IQODW |
| DOI | 10.1016/S0893-9659(01)00148-3 |
| DatabaseName | ScienceDirect Open Access Titles Elsevier:ScienceDirect:Open Access CrossRef Pascal-Francis |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EISSN | 1873-5452 |
| EndPage | 393 |
| ExternalDocumentID | 13546620 10_1016_S0893_9659_01_00148_3 S0893965901001483 |
| GroupedDBID | --K --M .~1 0R~ 1B1 1RT 1~. 1~5 23M 4.4 457 4G. 5GY 5VS 6I. 7-5 71M 8P~ 9JN AACTN AAEDT AAEDW AAFTH AAIAV AAIKJ AAKOC AALRI AAOAW AAQFI AAQXK AAXUO ABAOU ABEFU ABFNM ABJNI ABMAC ABVKL ABXDB ABYKQ ACAZW ACDAQ ACGFS ACNNM ACRLP ADBBV ADEZE ADMUD ADTZH AEBSH AECPX AEKER AENEX AEXQZ AFFNX AFKWA AFTJW AGHFR AGUBO AGYEJ AHHHB AHJVU AIEXJ AIGVJ AIKHN AITUG AJBFU AJOXV ALMA_UNASSIGNED_HOLDINGS AMFUW AMRAJ ARUGR ASPBG AVWKF AXJTR AZFZN BJAXD BKOJK BLXMC CS3 EBS EFJIC EFLBG EJD EO8 EO9 EP2 EP3 F5P FDB FEDTE FGOYB FIRID FNPLU FYGXN G-2 G-Q GBLVA HVGLF HZ~ IHE IXB J1W JJJVA KOM M26 M41 MHUIS MO0 N9A NCXOZ O-L O9- OAUVE OK1 OZT P-8 P-9 P2P PC. Q38 R2- RIG RNS ROL RPZ SDF SDG SDP SES SEW SPC SPCBC SST SSW SSZ T5K UHS WUQ XPP ZMT ~G- 9DU AATTM AAXKI AAYWO AAYXX ABWVN ACLOT ACRPL ACVFH ADCNI ADNMO ADVLN AEIPS AEUPX AFJKZ AFPUW AGQPQ AIGII AIIUN AKBMS AKRWK AKYEP ANKPU APXCP CITATION EFKBS ~HD AFXIZ AGCQF AGRNS BNPGV IQODW SSH |
| ID | FETCH-LOGICAL-c287t-8d16d9818d77be384f2a556dfa290dc4df7fa26f23f38cc616f2d7f27c26a8d33 |
| ISICitedReferencesCount | 0 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000175347700001&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0893-9659 |
| IngestDate | Mon Jul 21 09:13:41 EDT 2025 Sat Nov 29 07:15:31 EST 2025 Fri Feb 23 02:17:34 EST 2024 |
| IsDoiOpenAccess | true |
| IsOpenAccess | true |
| IsPeerReviewed | true |
| IsScholarly | true |
| Issue | 4 |
| Keywords | Density of zeros Real roots Adler's theorem Random hyperbolic polynomials Random algebraic polynomials Coordinate transform Complex roots Random trigonometric polynomials Jacobian of transformation Number of complex zeros |
| Language | English |
| License | http://www.elsevier.com/open-access/userlicense/1.0 CC BY 4.0 |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-c287t-8d16d9818d77be384f2a556dfa290dc4df7fa26f23f38cc616f2d7f27c26a8d33 |
| OpenAccessLink | https://dx.doi.org/10.1016/S0893-9659(01)00148-3 |
| PageCount | 5 |
| ParticipantIDs | pascalfrancis_primary_13546620 crossref_primary_10_1016_S0893_9659_01_00148_3 elsevier_sciencedirect_doi_10_1016_S0893_9659_01_00148_3 |
| PublicationCentury | 2000 |
| PublicationDate | 2002-05-01 |
| PublicationDateYYYYMMDD | 2002-05-01 |
| PublicationDate_xml | – month: 05 year: 2002 text: 2002-05-01 day: 01 |
| PublicationDecade | 2000 |
| PublicationPlace | Oxford |
| PublicationPlace_xml | – name: Oxford |
| PublicationTitle | Applied mathematics letters |
| PublicationYear | 2002 |
| Publisher | Elsevier Ltd Elsevier |
| Publisher_xml | – name: Elsevier Ltd – name: Elsevier |
| References | Bharucha-Reid, Sambandham (BIB8) 1986 Farahmand (BIB9) 1998 Ibragimov, Zeitouni (BIB5) 1997; 349 Farahmand (BIB11) 1999; 17 Shepp, Vanderbei (BIB4) 1995 Kac (BIB1) 1943; 49 Edelman, Kostlan (BIB3) 1995; 32 Wilkins (BIB2) 1988; 103 Farahmand (BIB10) 1995; 123 Farahmand, Grigorash (BIB6) 1999; 12 Adler (BIB7) 1981 Wilkins (10.1016/S0893-9659(01)00148-3_BIB2) 1988; 103 Bharucha-Reid (10.1016/S0893-9659(01)00148-3_BIB8) 1986 Farahmand (10.1016/S0893-9659(01)00148-3_BIB9) 1998 Shepp (10.1016/S0893-9659(01)00148-3_BIB4) 1995 Farahmand (10.1016/S0893-9659(01)00148-3_BIB10) 1995; 123 Kac (10.1016/S0893-9659(01)00148-3_BIB1) 1943; 49 Ibragimov (10.1016/S0893-9659(01)00148-3_BIB5) 1997; 349 Farahmand (10.1016/S0893-9659(01)00148-3_BIB11) 1999; 17 Edelman (10.1016/S0893-9659(01)00148-3_BIB3) 1995; 32 Farahmand (10.1016/S0893-9659(01)00148-3_BIB6) 1999; 12 Adler (10.1016/S0893-9659(01)00148-3_BIB7) 1981 |
| References_xml | – volume: 32 start-page: 1 year: 1995 end-page: 37 ident: BIB3 article-title: How many zeros of a random polynomial area real? publication-title: Bull. Amer. Math. Soc. – volume: 123 start-page: 1887 year: 1995 end-page: 1892 ident: BIB10 article-title: Level crossings of a random polynomial with hyperbolic elements publication-title: Proc. Amer. Math. Soc. – year: 1986 ident: BIB8 publication-title: Random Polynomials – volume: 103 start-page: 1249 year: 1988 end-page: 1258 ident: BIB2 article-title: An asymptotic expansion for the expected number of real zeros of a random polynomial publication-title: Proc. Amer. Math. Soc. – year: 1995 ident: BIB4 article-title: The complex zeros of random polynomials publication-title: Trans. Amer. Math. Soc. – volume: 17 start-page: 553 year: 1999 end-page: 562 ident: BIB11 article-title: Real zeros of a random hyperbolic polynomial publication-title: Stoch. Anal. and Appl. – year: 1981 ident: BIB7 publication-title: The Geometry of Random Fields – year: 1998 ident: BIB9 publication-title: Topics in Random Polynomials – volume: 349 start-page: 2427 year: 1997 end-page: 2441 ident: BIB5 article-title: On roots of random polynomials publication-title: Trans. Amer. Math. Soc. – volume: 49 start-page: 314 year: 1943 end-page: 320 ident: BIB1 article-title: On the average number of real roots of a random algebraic equation publication-title: Bull. Amer. Math. Soc. – volume: 12 start-page: 1037 year: 1999 end-page: 1044 ident: BIB6 article-title: Complex zeros of algebraic polynomial with non-zero mean random coefficients publication-title: Journal of Theoretical Probability – volume: 349 start-page: 2427 year: 1997 ident: 10.1016/S0893-9659(01)00148-3_BIB5 article-title: On roots of random polynomials publication-title: Trans. Amer. Math. Soc. doi: 10.1090/S0002-9947-97-01766-2 – volume: 32 start-page: 1 year: 1995 ident: 10.1016/S0893-9659(01)00148-3_BIB3 article-title: How many zeros of a random polynomial area real? publication-title: Bull. Amer. Math. Soc. doi: 10.1090/S0273-0979-1995-00571-9 – year: 1981 ident: 10.1016/S0893-9659(01)00148-3_BIB7 – year: 1998 ident: 10.1016/S0893-9659(01)00148-3_BIB9 – year: 1995 ident: 10.1016/S0893-9659(01)00148-3_BIB4 article-title: The complex zeros of random polynomials publication-title: Trans. Amer. Math. Soc. doi: 10.1090/S0002-9947-1995-1308023-8 – volume: 123 start-page: 1887 year: 1995 ident: 10.1016/S0893-9659(01)00148-3_BIB10 article-title: Level crossings of a random polynomial with hyperbolic elements publication-title: Proc. Amer. Math. Soc. doi: 10.1090/S0002-9939-1995-1264810-1 – year: 1986 ident: 10.1016/S0893-9659(01)00148-3_BIB8 – volume: 49 start-page: 314 year: 1943 ident: 10.1016/S0893-9659(01)00148-3_BIB1 article-title: On the average number of real roots of a random algebraic equation publication-title: Bull. Amer. Math. Soc. doi: 10.1090/S0002-9904-1943-07912-8 – volume: 103 start-page: 1249 year: 1988 ident: 10.1016/S0893-9659(01)00148-3_BIB2 article-title: An asymptotic expansion for the expected number of real zeros of a random polynomial publication-title: Proc. Amer. Math. Soc. doi: 10.1090/S0002-9939-1988-0955018-1 – volume: 12 start-page: 1037 issue: 4 year: 1999 ident: 10.1016/S0893-9659(01)00148-3_BIB6 article-title: Complex zeros of algebraic polynomial with non-zero mean random coefficients publication-title: Journal of Theoretical Probability doi: 10.1023/A:1021697221011 – volume: 17 start-page: 553 issue: 4 year: 1999 ident: 10.1016/S0893-9659(01)00148-3_BIB11 article-title: Real zeros of a random hyperbolic polynomial publication-title: Stoch. Anal. and Appl. doi: 10.1080/07362999908809619 |
| SSID | ssj0003938 |
| Score | 1.6006509 |
| Snippet | There are many known asymptotic estimates for the expected number of real zeros of polynomial
H
n
(
z) = η
1 cosh ζ
z + η
2 cosh 2ζ
z + ⋯ + η
n
cosh
nζ
z,... |
| SourceID | pascalfrancis crossref elsevier |
| SourceType | Index Database Publisher |
| StartPage | 389 |
| SubjectTerms | Adler's theorem Complex roots Coordinate transform Density of zeros Exact sciences and technology Jacobian of transformation Mathematics Number of complex zeros Random algebraic polynomials Random hyperbolic polynomials Random trigonometric polynomials Real roots Sciences and techniques of general use |
| Title | Expected density of complex zeros of random hyperbolic polynomials |
| URI | https://dx.doi.org/10.1016/S0893-9659(01)00148-3 |
| Volume | 15 |
| WOSCitedRecordID | wos000175347700001&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| journalDatabaseRights | – providerCode: PRVESC databaseName: Elsevier SD Freedom Collection Journals 2021 customDbUrl: eissn: 1873-5452 dateEnd: 20180131 omitProxy: false ssIdentifier: ssj0003938 issn: 0893-9659 databaseCode: AIEXJ dateStart: 19950101 isFulltext: true titleUrlDefault: https://www.sciencedirect.com providerName: Elsevier |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1Jj9MwFLZKhwMjhFjFsIxy4ACqUpLY8XIsaBDrCGkG1Fvk2jGM1KZV0xkVfj3PsROnIJY5cIksS32x_L76fX55C0JPNDWUGm1injMcE0mSWMx4GguqKS0Zw7KpxPT5PTs-5tOp-DgYnLS5MBdzVlV8uxWr_6pqmANl29TZS6i7EwoTMAalwxPUDs9_UrwtXqwsj9Q2Nt0FXDSB4-V29L0Eo2gnwELp5WL0FW6h65ktDWy7NXyzKcpyXvcJa8tSF11513o0b1KAQuKIdSovfIDku3GI6Dn7AviqG7_NZLzjXshCMF97Cgkc26qDO0dm3oMG6Z1_2PUD8qYUu-aHv5zSzmFw0okGLm37BYikcW_iYJraz_E_WawujjCEqIGowooqkrRoxBT4CtrLWC74EO1N3hxN33YGGpbVGOj29SGx63lY09MkfebX8zvKcn0la_gjGdcBpUdLTm-iG_4-EU0cDm6hQVndRvsfgrbuoBctIiKPiGhpIo-IqEGEnXCIiAIioh4i7qJPr45OX76OfeuMWMEVeBNznVItgIxpxmYl5sRkMs-pNhJ2WSuiDYMhNRk2mCtFUxhqZjKmMiq5xvgeGlbLqryPIkUSQzWweikIyUUupTREC03yDCtVigM0bnenWLkKKcUf9XKAeLuHhad5jr4VgI-__fRwZ8_DC3FOKM2SB5ddzEN0LaD-ERpu1uflY3RVXWzO6vWhR84PApx6Og |
| linkProvider | Elsevier |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Expected+density+of+complex+zeros+of+random+hyperbolic+polynomials&rft.jtitle=Applied+mathematics+letters&rft.au=Farahmand%2C+K.&rft.au=Grigorash%2C+A.&rft.date=2002-05-01&rft.issn=0893-9659&rft.volume=15&rft.issue=4&rft.spage=389&rft.epage=393&rft_id=info:doi/10.1016%2FS0893-9659%2801%2900148-3&rft.externalDBID=n%2Fa&rft.externalDocID=10_1016_S0893_9659_01_00148_3 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0893-9659&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0893-9659&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0893-9659&client=summon |