Computing real definite representations of Helton–Vinnikov theorem

Abstract The Helton–Vinnikov theorem produces a real definite representation for a hyperbolic ternary form. The construction of real symmetric linear matrices relies on the Klein prime form and Riemann theta function associated to the Jacobian variety of the hyperbolic curve. Plaumann et al. present...
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Gespeichert in:

Autor*in:	Chien, Mao-Ting [verfasserIn] Nakazato, Hiroshi

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2022

Schlagwörter:	Real definite representation Riemann theta function Abel–Jacobi map Numerical range

Anmerkung:	© Tusi Mathematical Research Group (TMRG) 2022

Übergeordnetes Werk:	Enthalten in: Banach journal of mathematical analysis - Mashhad, Iran : BMRG, 2007, 16(2022), 2 vom: 17. Jan.
Übergeordnetes Werk:	volume:16 ; year:2022 ; number:2 ; day:17 ; month:01

Links:	Volltext

DOI / URN:	10.1007/s43037-021-00169-z

Katalog-ID:	SPR045991162

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520			\|a Abstract The Helton–Vinnikov theorem produces a real definite representation for a hyperbolic ternary form. The construction of real symmetric linear matrices relies on the Klein prime form and Riemann theta function associated to the Jacobian variety of the hyperbolic curve. Plaumann et al. presented Maple code to perform the Helton–Vinnikov formula numerically. They reported an unknown discrepancy appeared on the real symmetric linear matrices, and a choice problem of homology basis for computing the period matrix. In this paper, we fix the bug of their code which causes the discrepancy, and propose a method to improve the performance of their code based on the Vinnikov standard form of the canonical basis. Furthermore, numerical experiments are presented to illustrate the discrepancy and the performance of the proposed techniques.
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allfields	10.1007/s43037-021-00169-z doi (DE-627)SPR045991162 (SPR)s43037-021-00169-z-e DE-627 ger DE-627 rakwb eng Chien, Mao-Ting verfasserin (orcid)0000-0001-5608-9487 aut Computing real definite representations of Helton–Vinnikov theorem 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Tusi Mathematical Research Group (TMRG) 2022 Abstract The Helton–Vinnikov theorem produces a real definite representation for a hyperbolic ternary form. The construction of real symmetric linear matrices relies on the Klein prime form and Riemann theta function associated to the Jacobian variety of the hyperbolic curve. Plaumann et al. presented Maple code to perform the Helton–Vinnikov formula numerically. They reported an unknown discrepancy appeared on the real symmetric linear matrices, and a choice problem of homology basis for computing the period matrix. In this paper, we fix the bug of their code which causes the discrepancy, and propose a method to improve the performance of their code based on the Vinnikov standard form of the canonical basis. Furthermore, numerical experiments are presented to illustrate the discrepancy and the performance of the proposed techniques. Real definite representation (dpeaa)DE-He213 Riemann theta function (dpeaa)DE-He213 Abel–Jacobi map (dpeaa)DE-He213 Numerical range (dpeaa)DE-He213 Nakazato, Hiroshi aut Enthalten in Banach journal of mathematical analysis Mashhad, Iran : BMRG, 2007 16(2022), 2 vom: 17. Jan. (DE-627)549633782 (DE-600)2395499-1 1735-8787 nnns volume:16 year:2022 number:2 day:17 month:01 https://dx.doi.org/10.1007/s43037-021-00169-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 16 2022 2 17 01
spelling	10.1007/s43037-021-00169-z doi (DE-627)SPR045991162 (SPR)s43037-021-00169-z-e DE-627 ger DE-627 rakwb eng Chien, Mao-Ting verfasserin (orcid)0000-0001-5608-9487 aut Computing real definite representations of Helton–Vinnikov theorem 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Tusi Mathematical Research Group (TMRG) 2022 Abstract The Helton–Vinnikov theorem produces a real definite representation for a hyperbolic ternary form. The construction of real symmetric linear matrices relies on the Klein prime form and Riemann theta function associated to the Jacobian variety of the hyperbolic curve. Plaumann et al. presented Maple code to perform the Helton–Vinnikov formula numerically. They reported an unknown discrepancy appeared on the real symmetric linear matrices, and a choice problem of homology basis for computing the period matrix. In this paper, we fix the bug of their code which causes the discrepancy, and propose a method to improve the performance of their code based on the Vinnikov standard form of the canonical basis. Furthermore, numerical experiments are presented to illustrate the discrepancy and the performance of the proposed techniques. Real definite representation (dpeaa)DE-He213 Riemann theta function (dpeaa)DE-He213 Abel–Jacobi map (dpeaa)DE-He213 Numerical range (dpeaa)DE-He213 Nakazato, Hiroshi aut Enthalten in Banach journal of mathematical analysis Mashhad, Iran : BMRG, 2007 16(2022), 2 vom: 17. Jan. (DE-627)549633782 (DE-600)2395499-1 1735-8787 nnns volume:16 year:2022 number:2 day:17 month:01 https://dx.doi.org/10.1007/s43037-021-00169-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 16 2022 2 17 01
allfields_unstemmed	10.1007/s43037-021-00169-z doi (DE-627)SPR045991162 (SPR)s43037-021-00169-z-e DE-627 ger DE-627 rakwb eng Chien, Mao-Ting verfasserin (orcid)0000-0001-5608-9487 aut Computing real definite representations of Helton–Vinnikov theorem 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Tusi Mathematical Research Group (TMRG) 2022 Abstract The Helton–Vinnikov theorem produces a real definite representation for a hyperbolic ternary form. The construction of real symmetric linear matrices relies on the Klein prime form and Riemann theta function associated to the Jacobian variety of the hyperbolic curve. Plaumann et al. presented Maple code to perform the Helton–Vinnikov formula numerically. They reported an unknown discrepancy appeared on the real symmetric linear matrices, and a choice problem of homology basis for computing the period matrix. In this paper, we fix the bug of their code which causes the discrepancy, and propose a method to improve the performance of their code based on the Vinnikov standard form of the canonical basis. Furthermore, numerical experiments are presented to illustrate the discrepancy and the performance of the proposed techniques. Real definite representation (dpeaa)DE-He213 Riemann theta function (dpeaa)DE-He213 Abel–Jacobi map (dpeaa)DE-He213 Numerical range (dpeaa)DE-He213 Nakazato, Hiroshi aut Enthalten in Banach journal of mathematical analysis Mashhad, Iran : BMRG, 2007 16(2022), 2 vom: 17. Jan. (DE-627)549633782 (DE-600)2395499-1 1735-8787 nnns volume:16 year:2022 number:2 day:17 month:01 https://dx.doi.org/10.1007/s43037-021-00169-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 16 2022 2 17 01
allfieldsGer	10.1007/s43037-021-00169-z doi (DE-627)SPR045991162 (SPR)s43037-021-00169-z-e DE-627 ger DE-627 rakwb eng Chien, Mao-Ting verfasserin (orcid)0000-0001-5608-9487 aut Computing real definite representations of Helton–Vinnikov theorem 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Tusi Mathematical Research Group (TMRG) 2022 Abstract The Helton–Vinnikov theorem produces a real definite representation for a hyperbolic ternary form. The construction of real symmetric linear matrices relies on the Klein prime form and Riemann theta function associated to the Jacobian variety of the hyperbolic curve. Plaumann et al. presented Maple code to perform the Helton–Vinnikov formula numerically. They reported an unknown discrepancy appeared on the real symmetric linear matrices, and a choice problem of homology basis for computing the period matrix. In this paper, we fix the bug of their code which causes the discrepancy, and propose a method to improve the performance of their code based on the Vinnikov standard form of the canonical basis. Furthermore, numerical experiments are presented to illustrate the discrepancy and the performance of the proposed techniques. Real definite representation (dpeaa)DE-He213 Riemann theta function (dpeaa)DE-He213 Abel–Jacobi map (dpeaa)DE-He213 Numerical range (dpeaa)DE-He213 Nakazato, Hiroshi aut Enthalten in Banach journal of mathematical analysis Mashhad, Iran : BMRG, 2007 16(2022), 2 vom: 17. Jan. (DE-627)549633782 (DE-600)2395499-1 1735-8787 nnns volume:16 year:2022 number:2 day:17 month:01 https://dx.doi.org/10.1007/s43037-021-00169-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 16 2022 2 17 01
allfieldsSound	10.1007/s43037-021-00169-z doi (DE-627)SPR045991162 (SPR)s43037-021-00169-z-e DE-627 ger DE-627 rakwb eng Chien, Mao-Ting verfasserin (orcid)0000-0001-5608-9487 aut Computing real definite representations of Helton–Vinnikov theorem 2022 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © Tusi Mathematical Research Group (TMRG) 2022 Abstract The Helton–Vinnikov theorem produces a real definite representation for a hyperbolic ternary form. The construction of real symmetric linear matrices relies on the Klein prime form and Riemann theta function associated to the Jacobian variety of the hyperbolic curve. Plaumann et al. presented Maple code to perform the Helton–Vinnikov formula numerically. They reported an unknown discrepancy appeared on the real symmetric linear matrices, and a choice problem of homology basis for computing the period matrix. In this paper, we fix the bug of their code which causes the discrepancy, and propose a method to improve the performance of their code based on the Vinnikov standard form of the canonical basis. Furthermore, numerical experiments are presented to illustrate the discrepancy and the performance of the proposed techniques. Real definite representation (dpeaa)DE-He213 Riemann theta function (dpeaa)DE-He213 Abel–Jacobi map (dpeaa)DE-He213 Numerical range (dpeaa)DE-He213 Nakazato, Hiroshi aut Enthalten in Banach journal of mathematical analysis Mashhad, Iran : BMRG, 2007 16(2022), 2 vom: 17. Jan. (DE-627)549633782 (DE-600)2395499-1 1735-8787 nnns volume:16 year:2022 number:2 day:17 month:01 https://dx.doi.org/10.1007/s43037-021-00169-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 16 2022 2 17 01
language	English
source	Enthalten in Banach journal of mathematical analysis 16(2022), 2 vom: 17. Jan. volume:16 year:2022 number:2 day:17 month:01
sourceStr	Enthalten in Banach journal of mathematical analysis 16(2022), 2 vom: 17. Jan. volume:16 year:2022 number:2 day:17 month:01
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Real definite representation Riemann theta function Abel–Jacobi map Numerical range
isfreeaccess_bool	false
container_title	Banach journal of mathematical analysis
authorswithroles_txt_mv	Chien, Mao-Ting @@aut@@ Nakazato, Hiroshi @@aut@@
publishDateDaySort_date	2022-01-17T00:00:00Z
hierarchy_top_id	549633782
id	SPR045991162
language_de	englisch
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author	Chien, Mao-Ting
spellingShingle	Chien, Mao-Ting misc Real definite representation misc Riemann theta function misc Abel–Jacobi map misc Numerical range Computing real definite representations of Helton–Vinnikov theorem
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topic_title	Computing real definite representations of Helton–Vinnikov theorem Real definite representation (dpeaa)DE-He213 Riemann theta function (dpeaa)DE-He213 Abel–Jacobi map (dpeaa)DE-He213 Numerical range (dpeaa)DE-He213
topic	misc Real definite representation misc Riemann theta function misc Abel–Jacobi map misc Numerical range
topic_unstemmed	misc Real definite representation misc Riemann theta function misc Abel–Jacobi map misc Numerical range
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title	Computing real definite representations of Helton–Vinnikov theorem
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title_full	Computing real definite representations of Helton–Vinnikov theorem
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author_browse	Chien, Mao-Ting Nakazato, Hiroshi
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author-letter	Chien, Mao-Ting
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title_sort	computing real definite representations of helton–vinnikov theorem
title_auth	Computing real definite representations of Helton–Vinnikov theorem
abstract	Abstract The Helton–Vinnikov theorem produces a real definite representation for a hyperbolic ternary form. The construction of real symmetric linear matrices relies on the Klein prime form and Riemann theta function associated to the Jacobian variety of the hyperbolic curve. Plaumann et al. presented Maple code to perform the Helton–Vinnikov formula numerically. They reported an unknown discrepancy appeared on the real symmetric linear matrices, and a choice problem of homology basis for computing the period matrix. In this paper, we fix the bug of their code which causes the discrepancy, and propose a method to improve the performance of their code based on the Vinnikov standard form of the canonical basis. Furthermore, numerical experiments are presented to illustrate the discrepancy and the performance of the proposed techniques. © Tusi Mathematical Research Group (TMRG) 2022
abstractGer	Abstract The Helton–Vinnikov theorem produces a real definite representation for a hyperbolic ternary form. The construction of real symmetric linear matrices relies on the Klein prime form and Riemann theta function associated to the Jacobian variety of the hyperbolic curve. Plaumann et al. presented Maple code to perform the Helton–Vinnikov formula numerically. They reported an unknown discrepancy appeared on the real symmetric linear matrices, and a choice problem of homology basis for computing the period matrix. In this paper, we fix the bug of their code which causes the discrepancy, and propose a method to improve the performance of their code based on the Vinnikov standard form of the canonical basis. Furthermore, numerical experiments are presented to illustrate the discrepancy and the performance of the proposed techniques. © Tusi Mathematical Research Group (TMRG) 2022
abstract_unstemmed	Abstract The Helton–Vinnikov theorem produces a real definite representation for a hyperbolic ternary form. The construction of real symmetric linear matrices relies on the Klein prime form and Riemann theta function associated to the Jacobian variety of the hyperbolic curve. Plaumann et al. presented Maple code to perform the Helton–Vinnikov formula numerically. They reported an unknown discrepancy appeared on the real symmetric linear matrices, and a choice problem of homology basis for computing the period matrix. In this paper, we fix the bug of their code which causes the discrepancy, and propose a method to improve the performance of their code based on the Vinnikov standard form of the canonical basis. Furthermore, numerical experiments are presented to illustrate the discrepancy and the performance of the proposed techniques. © Tusi Mathematical Research Group (TMRG) 2022
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title_short	Computing real definite representations of Helton–Vinnikov theorem
url	https://dx.doi.org/10.1007/s43037-021-00169-z
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author2	Nakazato, Hiroshi
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up_date	2024-07-03T19:37:27.661Z
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fullrecord_marcxml	<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">SPR045991162</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20230507171742.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">220119s2022 xx \|\|\|\|\|o 00\| \|\|eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s43037-021-00169-z</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)SPR045991162</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(SPR)s43037-021-00169-z-e</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Chien, Mao-Ting</subfield><subfield code="e">verfasserin</subfield><subfield code="0">(orcid)0000-0001-5608-9487</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Computing real definite representations of Helton–Vinnikov theorem</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2022</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">Computermedien</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Online-Ressource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© Tusi Mathematical Research Group (TMRG) 2022</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract The Helton–Vinnikov theorem produces a real definite representation for a hyperbolic ternary form. The construction of real symmetric linear matrices relies on the Klein prime form and Riemann theta function associated to the Jacobian variety of the hyperbolic curve. Plaumann et al. presented Maple code to perform the Helton–Vinnikov formula numerically. They reported an unknown discrepancy appeared on the real symmetric linear matrices, and a choice problem of homology basis for computing the period matrix. In this paper, we fix the bug of their code which causes the discrepancy, and propose a method to improve the performance of their code based on the Vinnikov standard form of the canonical basis. Furthermore, numerical experiments are presented to illustrate the discrepancy and the performance of the proposed techniques.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Real definite representation</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Riemann theta function</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Abel–Jacobi map</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Numerical range</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Nakazato, Hiroshi</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Banach journal of mathematical analysis</subfield><subfield code="d">Mashhad, Iran : BMRG, 2007</subfield><subfield code="g">16(2022), 2 vom: 17. 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Computing real definite representations of Helton–Vinnikov theorem

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