{"id":"https://openalex.org/W1803424325","doi":"https://doi.org/10.1090/mcom/2968","title":"Rank deficiencies and bifurcation into affine subspaces for separable parameterized equations","display_name":"Rank deficiencies and bifurcation into affine subspaces for separable parameterized equations","publication_year":2015,"publication_date":"2015-06-02","ids":{"openalex":"https://openalex.org/W1803424325","doi":"https://doi.org/10.1090/mcom/2968","mag":"1803424325"},"language":"en","primary_location":{"id":"doi:10.1090/mcom/2968","is_oa":false,"landing_page_url":"https://doi.org/10.1090/mcom/2968","pdf_url":null,"source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","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/A5034630970","display_name":"Yun-Qiu Shen","orcid":"https://orcid.org/0000-0002-3397-3322"},"institutions":[{"id":"https://openalex.org/I52669646","display_name":"Western Washington University","ror":"https://ror.org/05wn7r715","country_code":"US","type":"education","lineage":["https://openalex.org/I52669646"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Yun-Qiu Shen","raw_affiliation_strings":["Department of Mathematics, Western Washington University, Bellingham, Washington 98225-9063"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Western Washington University, Bellingham, Washington 98225-9063","institution_ids":["https://openalex.org/I52669646"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5082401348","display_name":"Tjalling Ypma","orcid":"https://orcid.org/0000-0001-8154-500X"},"institutions":[{"id":"https://openalex.org/I52669646","display_name":"Western Washington University","ror":"https://ror.org/05wn7r715","country_code":"US","type":"education","lineage":["https://openalex.org/I52669646"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Tjalling Ypma","raw_affiliation_strings":["Department of Mathematics, Western Washington University, Bellingham, Washington 98225-9063"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Western Washington University, Bellingham, Washington 98225-9063","institution_ids":["https://openalex.org/I52669646"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I52669646"],"apc_list":null,"apc_paid":null,"fwci":0.7058,"has_fulltext":false,"cited_by_count":3,"citation_normalized_percentile":{"value":0.72544212,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":94},"biblio":{"volume":"85","issue":"297","first_page":"271","last_page":"293"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9994999766349792,"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"}},"topics":[{"id":"https://openalex.org/T10792","display_name":"Matrix Theory and Algorithms","score":0.9994999766349792,"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/T10963","display_name":"Advanced Optimization Algorithms Research","score":0.9977999925613403,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"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/T12661","display_name":"Iterative Methods for Nonlinear Equations","score":0.9954000115394592,"subfield":{"id":"https://openalex.org/subfields/2612","display_name":"Numerical Analysis"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7986337542533875},{"id":"https://openalex.org/keywords/linear-subspace","display_name":"Linear subspace","score":0.724494457244873},{"id":"https://openalex.org/keywords/parameterized-complexity","display_name":"Parameterized complexity","score":0.7120480537414551},{"id":"https://openalex.org/keywords/separable-space","display_name":"Separable space","score":0.69401615858078},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5711554288864136},{"id":"https://openalex.org/keywords/rank","display_name":"Rank (graph theory)","score":0.5598071217536926},{"id":"https://openalex.org/keywords/affine-transformation","display_name":"Affine transformation","score":0.559234082698822},{"id":"https://openalex.org/keywords/bifurcation","display_name":"Bifurcation","score":0.5216979384422302},{"id":"https://openalex.org/keywords/bifurcation-theory","display_name":"Bifurcation theory","score":0.43545204401016235},{"id":"https://openalex.org/keywords/reachability","display_name":"Reachability","score":0.42697668075561523},{"id":"https://openalex.org/keywords/subspace-topology","display_name":"Subspace topology","score":0.42601048946380615},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.34556710720062256},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.26545628905296326},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.18740162253379822},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.12668609619140625},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.12254318594932556}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7986337542533875},{"id":"https://openalex.org/C12362212","wikidata":"https://www.wikidata.org/wiki/Q728435","display_name":"Linear subspace","level":2,"score":0.724494457244873},{"id":"https://openalex.org/C165464430","wikidata":"https://www.wikidata.org/wiki/Q1570441","display_name":"Parameterized complexity","level":2,"score":0.7120480537414551},{"id":"https://openalex.org/C70710897","wikidata":"https://www.wikidata.org/wiki/Q680081","display_name":"Separable space","level":2,"score":0.69401615858078},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5711554288864136},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.5598071217536926},{"id":"https://openalex.org/C92757383","wikidata":"https://www.wikidata.org/wiki/Q382497","display_name":"Affine transformation","level":2,"score":0.559234082698822},{"id":"https://openalex.org/C2781349735","wikidata":"https://www.wikidata.org/wiki/Q4904824","display_name":"Bifurcation","level":3,"score":0.5216979384422302},{"id":"https://openalex.org/C85075877","wikidata":"https://www.wikidata.org/wiki/Q1154357","display_name":"Bifurcation theory","level":4,"score":0.43545204401016235},{"id":"https://openalex.org/C136643341","wikidata":"https://www.wikidata.org/wiki/Q1361526","display_name":"Reachability","level":2,"score":0.42697668075561523},{"id":"https://openalex.org/C32834561","wikidata":"https://www.wikidata.org/wiki/Q660730","display_name":"Subspace topology","level":2,"score":0.42601048946380615},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.34556710720062256},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.26545628905296326},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.18740162253379822},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.12668609619140625},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.12254318594932556},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1090/mcom/2968","is_oa":false,"landing_page_url":"https://doi.org/10.1090/mcom/2968","pdf_url":null,"source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":32,"referenced_works":["https://openalex.org/W218234981","https://openalex.org/W357530330","https://openalex.org/W1498887287","https://openalex.org/W1499481339","https://openalex.org/W1529663478","https://openalex.org/W1530314303","https://openalex.org/W1565524949","https://openalex.org/W1596321532","https://openalex.org/W1603340857","https://openalex.org/W1972632202","https://openalex.org/W1981745143","https://openalex.org/W1981943156","https://openalex.org/W2003434190","https://openalex.org/W2018089423","https://openalex.org/W2063576047","https://openalex.org/W2063886834","https://openalex.org/W2067439591","https://openalex.org/W2071860648","https://openalex.org/W2076950429","https://openalex.org/W2079177133","https://openalex.org/W2079371292","https://openalex.org/W2081696580","https://openalex.org/W2090636411","https://openalex.org/W2090963365","https://openalex.org/W2165914592","https://openalex.org/W2232452261","https://openalex.org/W2798909945","https://openalex.org/W3031689935","https://openalex.org/W4214513274","https://openalex.org/W4233654647","https://openalex.org/W4250716583","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W1985684185","https://openalex.org/W1524827711","https://openalex.org/W3184847830","https://openalex.org/W1803424325","https://openalex.org/W2065530191","https://openalex.org/W4286911979","https://openalex.org/W4229082471","https://openalex.org/W218234981","https://openalex.org/W1636753615","https://openalex.org/W2165914592"],"abstract_inverted_index":{"Many":[0],"applications":[1],"lead":[2],"to":[3],"separable":[4],"parameterized":[5],"equations":[6],"of":[7],"the":[8],"form":[9],"<inline-formula":[10,97,132,168,243,302,337],"content-type=\"math/mathml\">":[11,98,133,169,244,303,338],"<mml:math":[12,99,134,170,245,304,339],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[13,100,135,171,246,305,340],"alttext=\"upper":[14,172,341],"F":[15],"left-parenthesis":[16,26,34,174,184,248],"y":[17,27,35,175,249],"comma":[18,20,28,36,176,250],"mu":[19,29,37,177,251],"z":[21,31],"right-parenthesis":[22,30,38,178,189,252],"identical-to":[23],"upper":[24,104,139,142,181,185,191,255,258,309],"A":[25,173],"plus":[32,187,260],"b":[33],"equals":[39],"0\">":[40],"<mml:semantics>":[41,108,144,193,262,311,346],"<mml:mrow>":[42,109,145,194,263,312,347],"<mml:mi>F</mml:mi>":[43],"<mml:mo":[44,53,59,66,71,78,196,203,216,221,265,272],"stretchy=\"false\">(</mml:mo>":[45,60,72,197,217,266],"<mml:mi>y</mml:mi>":[46,61,73,110,198,267],"<mml:mo>,</mml:mo>":[47,51,62,74,199,268],"<mml:mi>":[48,63,75,200,269,313],"\u03bc":[49,64,76,201,270,314],"</mml:mi>":[50,65,77,202,271,315],"<mml:mi>z</mml:mi>":[52,68,146],"stretchy=\"false\">)</mml:mo>":[54,67,79,204,222,273],"<mml:mo>":[55,111,147,205,223,274,316],"\u2261":[56],"</mml:mo>":[57,113,149,207,225,276,318],"<mml:mi>A</mml:mi>":[58,195],"<mml:mo>+</mml:mo>":[69,219,286],"<mml:mi>b</mml:mi>":[70,264],"<mml:mo>=</mml:mo>":[80],"<mml:mn>0</mml:mn>":[81],"</mml:mrow>":[82,119,122,155,158,213,227,229,282,288,290,323,324,351],"<mml:annotation":[83,123,159,230,291,325,352],"encoding=\"application/x-tex\">F(y,\\mu":[84],",z)":[85],"\\equiv":[86],"A(y,":[87],"\\mu":[88,90,232,293],")z+b(y,":[89],")=0</mml:annotation>":[91],"</mml:semantics>":[92,128,164,239,298,330],"</mml:math>":[93,129,165,240,299,331],"</inline-formula>":[94,130,166,241,300,332],",":[95,131,167,242],"where":[96],"alttext=\"y":[101],"element-of":[102,137,179,253,307],"double-struck":[103,138,180,254,308],"R":[105,140,182,256],"Superscript":[106,141,183,257],"n\">":[107,261,345],"\u2208":[112,148,206,275,317],"<mml:msup>":[114,150,208,277],"<mml:mrow":[115,151,209,214,278,283,319],"class=\"MJX-TeXAtom-ORD\">":[116,152,210,215,279,284,320],"<mml:mi":[117,153,211,280,321],"mathvariant=\"double-struck\">R</mml:mi>":[118,154,212,281,322],"<mml:mi>n</mml:mi>":[120,220,287,350],"</mml:msup>":[121,157,228,289],"encoding=\"application/x-tex\">y":[124],"\\in":[125,161,234,295,327],"\\mathbb":[126,162,235,296,328],"R^n</mml:annotation>":[127],"alttext=\"z":[136],"N\">":[143,192],"<mml:mi>N</mml:mi>":[156,218,226,285,348],"encoding=\"application/x-tex\">z":[160],"R^N</mml:annotation>":[163],"N":[186,259,342],"n":[188],"times":[190],"\u00d7":[224],"encoding=\"application/x-tex\">A(y,":[231],")":[233,294],"{R}^{(N+n)":[236],"\\times":[237],"N}</mml:annotation>":[238],"alttext=\"b":[247],"encoding=\"application/x-tex\">b(y,":[292],"{R}^{N+n}</mml:annotation>":[297],"and":[301],"alttext=\"mu":[306],"R\">":[310],"encoding=\"application/x-tex\">\\mu":[326],"R</mml:annotation>":[329],"is":[333],"a":[334],"parameter.":[335],"Typically":[336],"greater-than":[343,344],"<mml:mo>&gt;&gt;</mml:mo>":[349],"encoding=\"application/x-tex\">N":[353],"&gt;&gt;n</mml:annotation>":[354]},"counts_by_year":[{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2017,"cited_by_count":1}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}
