{"id":"https://openalex.org/W2946874714","doi":"https://doi.org/10.1134/s0361768819020105","title":"Canonical Representation of Polynomial Expressions with Indices","display_name":"Canonical Representation of Polynomial Expressions with Indices","publication_year":2019,"publication_date":"2019-03-01","ids":{"openalex":"https://openalex.org/W2946874714","doi":"https://doi.org/10.1134/s0361768819020105","mag":"2946874714"},"language":"en","primary_location":{"id":"doi:10.1134/s0361768819020105","is_oa":false,"landing_page_url":"https://doi.org/10.1134/s0361768819020105","pdf_url":null,"source":{"id":"https://openalex.org/S150218641","display_name":"Programming and Computer Software","issn_l":"0361-7688","issn":["0361-7688","1608-3261"],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/P4310320267","host_organization_name":"Pleiades Publishing","host_organization_lineage":["https://openalex.org/P4310320267","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Pleiades Publishing","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Programming and Computer Software","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5109216605","display_name":"G. B. Shpiz","orcid":null},"institutions":[{"id":"https://openalex.org/I19880235","display_name":"Lomonosov Moscow State University","ror":"https://ror.org/010pmpe69","country_code":"RU","type":"education","lineage":["https://openalex.org/I19880235"]}],"countries":["RU"],"is_corresponding":true,"raw_author_name":"G. Shpiz","raw_affiliation_strings":["Moscow State University, 119991, Moscow, Russia"],"affiliations":[{"raw_affiliation_string":"Moscow State University, 119991, Moscow, Russia","institution_ids":["https://openalex.org/I19880235"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5000058748","display_name":"A. P. Kryukov","orcid":"https://orcid.org/0000-0002-1624-6131"},"institutions":[{"id":"https://openalex.org/I19880235","display_name":"Lomonosov Moscow State University","ror":"https://ror.org/010pmpe69","country_code":"RU","type":"education","lineage":["https://openalex.org/I19880235"]}],"countries":["RU"],"is_corresponding":false,"raw_author_name":"A. Kryukov","raw_affiliation_strings":["Moscow State University, 119991, Moscow, Russia"],"affiliations":[{"raw_affiliation_string":"Moscow State University, 119991, Moscow, Russia","institution_ids":["https://openalex.org/I19880235"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5109216605"],"corresponding_institution_ids":["https://openalex.org/I19880235"],"apc_list":null,"apc_paid":null,"fwci":0.1439,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.30701754,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":93},"biblio":{"volume":"45","issue":"2","first_page":"81","last_page":"87"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T12303","display_name":"Tensor decomposition and applications","score":0.9955999851226807,"subfield":{"id":"https://openalex.org/subfields/2605","display_name":"Computational Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T12303","display_name":"Tensor decomposition and applications","score":0.9955999851226807,"subfield":{"id":"https://openalex.org/subfields/2605","display_name":"Computational Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11435","display_name":"Polynomial and algebraic computation","score":0.9901999831199646,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9782000184059143,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6679068803787231},{"id":"https://openalex.org/keywords/algebra-over-a-field","display_name":"Algebra over a field","score":0.6303436160087585},{"id":"https://openalex.org/keywords/homogeneous-space","display_name":"Homogeneous space","score":0.6256788969039917},{"id":"https://openalex.org/keywords/polynomial","display_name":"Polynomial","score":0.5434823036193848},{"id":"https://openalex.org/keywords/multiplicative-function","display_name":"Multiplicative function","score":0.5327401757240295},{"id":"https://openalex.org/keywords/representation","display_name":"Representation (politics)","score":0.5205098986625671},{"id":"https://openalex.org/keywords/expression","display_name":"Expression (computer science)","score":0.5142292976379395},{"id":"https://openalex.org/keywords/canonical-form","display_name":"Canonical form","score":0.47898435592651367},{"id":"https://openalex.org/keywords/curvature","display_name":"Curvature","score":0.45457950234413147},{"id":"https://openalex.org/keywords/tensor","display_name":"Tensor (intrinsic definition)","score":0.4437597990036011},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.4271990656852722},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.1417587697505951},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.11936330795288086},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.09206759929656982}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6679068803787231},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.6303436160087585},{"id":"https://openalex.org/C96469262","wikidata":"https://www.wikidata.org/wiki/Q1324364","display_name":"Homogeneous space","level":2,"score":0.6256788969039917},{"id":"https://openalex.org/C90119067","wikidata":"https://www.wikidata.org/wiki/Q43260","display_name":"Polynomial","level":2,"score":0.5434823036193848},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.5327401757240295},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.5205098986625671},{"id":"https://openalex.org/C90559484","wikidata":"https://www.wikidata.org/wiki/Q778379","display_name":"Expression (computer science)","level":2,"score":0.5142292976379395},{"id":"https://openalex.org/C204707403","wikidata":"https://www.wikidata.org/wiki/Q1152398","display_name":"Canonical form","level":2,"score":0.47898435592651367},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.45457950234413147},{"id":"https://openalex.org/C155281189","wikidata":"https://www.wikidata.org/wiki/Q3518150","display_name":"Tensor (intrinsic definition)","level":2,"score":0.4437597990036011},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4271990656852722},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.1417587697505951},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.11936330795288086},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.09206759929656982},{"id":"https://openalex.org/C94625758","wikidata":"https://www.wikidata.org/wiki/Q7163","display_name":"Politics","level":2,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","level":0,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C199360897","wikidata":"https://www.wikidata.org/wiki/Q9143","display_name":"Programming language","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1134/s0361768819020105","is_oa":false,"landing_page_url":"https://doi.org/10.1134/s0361768819020105","pdf_url":null,"source":{"id":"https://openalex.org/S150218641","display_name":"Programming and Computer Software","issn_l":"0361-7688","issn":["0361-7688","1608-3261"],"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/P4310320267","host_organization_name":"Pleiades Publishing","host_organization_lineage":["https://openalex.org/P4310320267","https://openalex.org/P4310319965"],"host_organization_lineage_names":["Pleiades Publishing","Springer Nature"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Programming and Computer Software","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":18,"referenced_works":["https://openalex.org/W1482693440","https://openalex.org/W1529584516","https://openalex.org/W1566679152","https://openalex.org/W1979058129","https://openalex.org/W1994699824","https://openalex.org/W2008351685","https://openalex.org/W2014825433","https://openalex.org/W2044843196","https://openalex.org/W2060578797","https://openalex.org/W2072638007","https://openalex.org/W2096715245","https://openalex.org/W2131097667","https://openalex.org/W2251196829","https://openalex.org/W2583113001","https://openalex.org/W3098221122","https://openalex.org/W3099027509","https://openalex.org/W3103905489","https://openalex.org/W4394951875"],"related_works":["https://openalex.org/W4318195686","https://openalex.org/W2975999359","https://openalex.org/W4300711749","https://openalex.org/W4287173569","https://openalex.org/W2796561647","https://openalex.org/W4280497496","https://openalex.org/W4313513739","https://openalex.org/W3119693376","https://openalex.org/W3182422617","https://openalex.org/W2246967737"],"abstract_inverted_index":{"Computer":[0],"algebra":[1],"methods":[2,43],"are":[3,29],"widely":[4],"employed":[5],"in":[6,71],"various":[7,67],"branches":[8],"of":[9,16,23,34,62,69,92,99,114,125],"mathematics,":[10],"physics,":[11],"and":[12,80],"other":[13],"sciences.":[14],"Simplification":[15],"algebraic":[17],"expressions":[18,28,46,98,124],"with":[19,47,64,76,101],"indices":[20,48,70,103],"is":[21,54],"one":[22],"the":[24,30,60,108,112,129],"important":[25],"problems.":[26],"Tensor":[27],"most":[31],"typical":[32],"example":[33],"these":[35],"expressions.":[36],"This":[37,86],"paper":[38,87],"briefly":[39],"describes":[40],"some":[41,115],"basic":[42],"for":[44,95,123],"reducing":[45],"to":[49,66],"canonical":[50,93],"form.":[51],"The":[52],"focus":[53],"placed":[55],"on":[56],"taking":[57],"into":[58],"account":[59],"properties":[61],"symmetries":[63,74],"respect":[65],"permutations":[68],"elementary":[72],"symbols,":[73],"associated":[75],"renaming":[77],"summation":[78],"indices,":[79],"general":[81],"linear":[82],"relationships":[83],"among":[84],"them.":[85],"also":[88],"gives":[89],"a":[90],"definition":[91],"representation":[94],"polynomial":[96],"(multiplicative)":[97],"variables":[100],"abstract":[102],"that":[104],"results":[105],"from":[106],"averaging":[107],"initial":[109],"expression":[110],"over":[111],"action":[113],"finite":[116],"group":[117],"(signature":[118],"stabilizer).":[119],"In":[120],"practice,":[121],"e.g.,":[122],"Riemann":[126],"curvature":[127],"tensors,":[128],"proposed":[130],"algorithms":[131],"show":[132],"high":[133],"efficiency.":[134]},"counts_by_year":[{"year":2021,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}