{"id":"https://openalex.org/W2950958653","doi":"https://doi.org/10.1137/16m1091988","title":"A Spacial Gradient Estimate for Solutions to the Heat Equation on Graphs","display_name":"A Spacial Gradient Estimate for Solutions to the Heat Equation on Graphs","publication_year":2019,"publication_date":"2019-01-01","ids":{"openalex":"https://openalex.org/W2950958653","doi":"https://doi.org/10.1137/16m1091988","mag":"2950958653"},"language":"en","primary_location":{"id":"doi:10.1137/16m1091988","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1091988","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","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/A5029310944","display_name":"Paul Horn","orcid":"https://orcid.org/0000-0003-3022-9036"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Paul Horn","raw_affiliation_strings":[],"affiliations":[]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5029310944"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.9881,"has_fulltext":false,"cited_by_count":9,"citation_normalized_percentile":{"value":0.85311292,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"33","issue":"2","first_page":"958","last_page":"975"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied 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/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied 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/T12351","display_name":"Geometry and complex manifolds","score":0.9861000180244446,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"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/T10194","display_name":"Nonlinear Partial Differential Equations","score":0.9829000234603882,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"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":0.7842556238174438},{"id":"https://openalex.org/keywords/heat-equation","display_name":"Heat equation","score":0.7405516505241394},{"id":"https://openalex.org/keywords/curvature","display_name":"Curvature","score":0.6293079853057861},{"id":"https://openalex.org/keywords/bounded-function","display_name":"Bounded function","score":0.5898990631103516},{"id":"https://openalex.org/keywords/corollary","display_name":"Corollary","score":0.5683683156967163},{"id":"https://openalex.org/keywords/manifold","display_name":"Manifold (fluid mechanics)","score":0.5185856819152832},{"id":"https://openalex.org/keywords/space","display_name":"Space (punctuation)","score":0.48984190821647644},{"id":"https://openalex.org/keywords/indifference-graph","display_name":"Indifference graph","score":0.4747063219547272},{"id":"https://openalex.org/keywords/degree","display_name":"Degree (music)","score":0.44668352603912354},{"id":"https://openalex.org/keywords/riemannian-manifold","display_name":"Riemannian manifold","score":0.4232654869556427},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.38502436876296997},{"id":"https://openalex.org/keywords/pure-mathematics","display_name":"Pure mathematics","score":0.34272897243499756},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.3280057907104492},{"id":"https://openalex.org/keywords/discrete-mathematics","display_name":"Discrete mathematics","score":0.32095766067504883},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.2240438461303711},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.13698026537895203}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7842556238174438},{"id":"https://openalex.org/C202787564","wikidata":"https://www.wikidata.org/wiki/Q6510488","display_name":"Heat equation","level":2,"score":0.7405516505241394},{"id":"https://openalex.org/C195065555","wikidata":"https://www.wikidata.org/wiki/Q214881","display_name":"Curvature","level":2,"score":0.6293079853057861},{"id":"https://openalex.org/C34388435","wikidata":"https://www.wikidata.org/wiki/Q2267362","display_name":"Bounded function","level":2,"score":0.5898990631103516},{"id":"https://openalex.org/C2780012671","wikidata":"https://www.wikidata.org/wiki/Q1343870","display_name":"Corollary","level":2,"score":0.5683683156967163},{"id":"https://openalex.org/C529865628","wikidata":"https://www.wikidata.org/wiki/Q1790740","display_name":"Manifold (fluid mechanics)","level":2,"score":0.5185856819152832},{"id":"https://openalex.org/C2778572836","wikidata":"https://www.wikidata.org/wiki/Q380933","display_name":"Space (punctuation)","level":2,"score":0.48984190821647644},{"id":"https://openalex.org/C74133993","wikidata":"https://www.wikidata.org/wiki/Q3115472","display_name":"Indifference graph","level":3,"score":0.4747063219547272},{"id":"https://openalex.org/C2775997480","wikidata":"https://www.wikidata.org/wiki/Q586277","display_name":"Degree (music)","level":2,"score":0.44668352603912354},{"id":"https://openalex.org/C2779593128","wikidata":"https://www.wikidata.org/wiki/Q632814","display_name":"Riemannian manifold","level":2,"score":0.4232654869556427},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.38502436876296997},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.34272897243499756},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3280057907104492},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.32095766067504883},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.2240438461303711},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.13698026537895203},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","level":0,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C127413603","wikidata":"https://www.wikidata.org/wiki/Q11023","display_name":"Engineering","level":0,"score":0.0},{"id":"https://openalex.org/C78519656","wikidata":"https://www.wikidata.org/wiki/Q101333","display_name":"Mechanical engineering","level":1,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/16m1091988","is_oa":false,"landing_page_url":"https://doi.org/10.1137/16m1091988","pdf_url":null,"source":{"id":"https://openalex.org/S117679221","display_name":"SIAM Journal on Discrete Mathematics","issn_l":"0895-4801","issn":["0895-4801","1095-7146"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Discrete Mathematics","raw_type":"journal-article"},{"id":"pmh:oai:digitalcommons.du.edu:math_faculty-1029","is_oa":false,"landing_page_url":"https://digitalcommons.du.edu/math_faculty/31","pdf_url":null,"source":{"id":"https://openalex.org/S4377196520","display_name":"Digital Commons - DU (University of Denver)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I131651094","host_organization_name":"University of Denver","host_organization_lineage":["https://openalex.org/I131651094"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Mathematics: Faculty Scholarship","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G872736033","display_name":null,"funder_award_id":"H98230-15-1-0258","funder_id":"https://openalex.org/F4320311089","funder_display_name":"National Security Agency"}],"funders":[{"id":"https://openalex.org/F4320311089","display_name":"National Security Agency","ror":"https://ror.org/0047bvr32"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":20,"referenced_works":["https://openalex.org/W1582534285","https://openalex.org/W1968781546","https://openalex.org/W1970224775","https://openalex.org/W1971118512","https://openalex.org/W1989957502","https://openalex.org/W2000124739","https://openalex.org/W2046098168","https://openalex.org/W2063015130","https://openalex.org/W2064759795","https://openalex.org/W2069789834","https://openalex.org/W2079509727","https://openalex.org/W2087667141","https://openalex.org/W2334471843","https://openalex.org/W2619527024","https://openalex.org/W2690240663","https://openalex.org/W2963046031","https://openalex.org/W2963584630","https://openalex.org/W2963972560","https://openalex.org/W2964259716","https://openalex.org/W4247653541"],"related_works":["https://openalex.org/W2376029318","https://openalex.org/W2388301251","https://openalex.org/W2013770701","https://openalex.org/W3013941779","https://openalex.org/W4289492338","https://openalex.org/W3035440551","https://openalex.org/W2921922410","https://openalex.org/W1920430386","https://openalex.org/W2546668543","https://openalex.org/W2952208990"],"abstract_inverted_index":{"The":[0],"study":[1],"of":[2,5,24,36,41,57,66,121,142],"positive":[3,89],"solutions":[4,44,91],"the":[6,29,32,37,46,101,108,140,143],"heat":[7,47,58,90,96,128],"equation":[8,48],"$\\frac{\\partial}{\\partial":[9],"\\alpha}":[10],"u":[11],"=":[12],"\\Delta":[13],"u$,":[14],"on":[15,148],"both":[16,115],"manifolds":[17],"and":[18,100,118],"graphs,":[19],"gives":[20],"an":[21],"analytic":[22],"way":[23],"extracting":[25],"geometric":[26],"information":[27],"about":[28,139],"object.":[30],"In":[31,80],"manifold":[33,109],"case,":[34],"one":[35],"most":[38],"effective":[39],"ways":[40],"studying":[42],"how":[43,95],"to":[45,51,111,114],"evolve":[49],"is":[50,131],"derive":[52,84],"a":[53,85,125,127],"local":[54],"\u201cgradient":[55],"estimate\u201d":[56],"change,":[59],"using":[60],"curvature":[61,67],"lower":[62],"bounds.":[63],"Recently,":[64],"notions":[65],"for":[68,78,88],"graphs":[69,117,120],"have":[70],"been":[71],"developed":[72],"which":[73],"enable":[74],"proving":[75],"similar":[76],"estimates":[77],"graphs.":[79,149],"this":[81],"article,":[82],"we":[83],"gradient":[86],"estimate":[87],"that":[92],"considers":[93],"only":[94],"varies":[97],"in":[98,107,135],"space":[99],"time":[102,145],"derivative.":[103],"This":[104,134],"result,":[105],"due":[106],"case":[110],"Hamilton,":[112],"applies":[113],"finite":[116],"infinite":[119],"bounded":[122],"degree.":[123],"As":[124],"corollary,":[126],"comparison":[129],"theorem":[130],"also":[132],"developed.":[133],"turn":[136],"yields":[137],"results":[138],"mixing":[141],"continuous":[144],"random":[146],"walk":[147]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":1},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":2}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}