{"id":"https://openalex.org/W2005832942","doi":"https://doi.org/10.1137/040618916","title":"Lattice points on circles and discrete velocity models for the Boltzmann equation","display_name":"Lattice points on circles and discrete velocity models for the Boltzmann equation","publication_year":2006,"publication_date":"2006-01-01","ids":{"openalex":"https://openalex.org/W2005832942","doi":"https://doi.org/10.1137/040618916","mag":"2005832942"},"language":"en","primary_location":{"id":"doi:10.1137/040618916","is_oa":false,"landing_page_url":"https://doi.org/10.1137/040618916","pdf_url":null,"source":{"id":"https://openalex.org/S74141806","display_name":"SIAM Journal on Mathematical Analysis","issn_l":"0036-1410","issn":["0036-1410","1095-7154"],"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 Mathematical Analysis","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/A5000398362","display_name":"Laura Fainsilber","orcid":null},"institutions":[{"id":"https://openalex.org/I66862912","display_name":"Chalmers University of Technology","ror":"https://ror.org/040wg7k59","country_code":"SE","type":"education","lineage":["https://openalex.org/I66862912"]}],"countries":["SE"],"is_corresponding":true,"raw_author_name":"Laura Fainsilber","raw_affiliation_strings":["DEPARTMENT OF MATHEMATICS, CHALMERS UNIVERSITY OF TECHNOLOGY, SE-412 96 GOTHENBURG, SWEDEN"],"affiliations":[{"raw_affiliation_string":"DEPARTMENT OF MATHEMATICS, CHALMERS UNIVERSITY OF TECHNOLOGY, SE-412 96 GOTHENBURG, SWEDEN","institution_ids":["https://openalex.org/I66862912"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5053501812","display_name":"P\u00e4r Kurlberg","orcid":"https://orcid.org/0000-0003-4734-5092"},"institutions":[{"id":"https://openalex.org/I66862912","display_name":"Chalmers University of Technology","ror":"https://ror.org/040wg7k59","country_code":"SE","type":"education","lineage":["https://openalex.org/I66862912"]}],"countries":["SE"],"is_corresponding":false,"raw_author_name":"P\u00e4r Kurlberg","raw_affiliation_strings":["DEPARTMENT OF MATHEMATICS, CHALMERS UNIVERSITY OF TECHNOLOGY, SE-412 96 GOTHENBURG, SWEDEN"],"affiliations":[{"raw_affiliation_string":"DEPARTMENT OF MATHEMATICS, CHALMERS UNIVERSITY OF TECHNOLOGY, SE-412 96 GOTHENBURG, SWEDEN","institution_ids":["https://openalex.org/I66862912"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5085499157","display_name":"Bernt Wennberg","orcid":null},"institutions":[{"id":"https://openalex.org/I66862912","display_name":"Chalmers University of Technology","ror":"https://ror.org/040wg7k59","country_code":"SE","type":"education","lineage":["https://openalex.org/I66862912"]}],"countries":["SE"],"is_corresponding":false,"raw_author_name":"Bernt Wennberg","raw_affiliation_strings":["DEPARTMENT OF MATHEMATICS, CHALMERS UNIVERSITY OF TECHNOLOGY, SE-412 96 GOTHENBURG, SWEDEN"],"affiliations":[{"raw_affiliation_string":"DEPARTMENT OF MATHEMATICS, CHALMERS UNIVERSITY OF TECHNOLOGY, SE-412 96 GOTHENBURG, SWEDEN","institution_ids":["https://openalex.org/I66862912"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5000398362"],"corresponding_institution_ids":["https://openalex.org/I66862912"],"apc_list":null,"apc_paid":null,"fwci":1.8034,"has_fulltext":false,"cited_by_count":45,"citation_normalized_percentile":{"value":0.81604177,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":99},"biblio":{"volume":"37","issue":"6","first_page":"1903","last_page":"1922"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11012","display_name":"Gas Dynamics and Kinetic Theory","score":0.9991000294685364,"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/T11012","display_name":"Gas Dynamics and Kinetic Theory","score":0.9991000294685364,"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/T12404","display_name":"Mathematical Approximation and Integration","score":0.9945999979972839,"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/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9936000108718872,"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":0.8338289260864258},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.5783223509788513},{"id":"https://openalex.org/keywords/lattice","display_name":"Lattice (music)","score":0.5540362000465393},{"id":"https://openalex.org/keywords/collision","display_name":"Collision","score":0.5180287957191467},{"id":"https://openalex.org/keywords/multiplicative-function","display_name":"Multiplicative function","score":0.4750502109527588},{"id":"https://openalex.org/keywords/gaussian","display_name":"Gaussian","score":0.4661010503768921},{"id":"https://openalex.org/keywords/boltzmann-equation","display_name":"Boltzmann equation","score":0.45245277881622314},{"id":"https://openalex.org/keywords/bhatnagar\u2013gross\u2013krook-operator","display_name":"Bhatnagar\u2013Gross\u2013Krook operator","score":0.4519854784011841},{"id":"https://openalex.org/keywords/monte-carlo-method","display_name":"Monte Carlo method","score":0.26100611686706543},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.14070859551429749},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.10816067457199097},{"id":"https://openalex.org/keywords/direct-simulation-monte-carlo","display_name":"Direct simulation Monte Carlo","score":0.0926831066608429}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8338289260864258},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.5783223509788513},{"id":"https://openalex.org/C2781204021","wikidata":"https://www.wikidata.org/wiki/Q6497091","display_name":"Lattice (music)","level":2,"score":0.5540362000465393},{"id":"https://openalex.org/C121704057","wikidata":"https://www.wikidata.org/wiki/Q352070","display_name":"Collision","level":2,"score":0.5180287957191467},{"id":"https://openalex.org/C42747912","wikidata":"https://www.wikidata.org/wiki/Q1048447","display_name":"Multiplicative function","level":2,"score":0.4750502109527588},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.4661010503768921},{"id":"https://openalex.org/C165995430","wikidata":"https://www.wikidata.org/wiki/Q891653","display_name":"Boltzmann equation","level":2,"score":0.45245277881622314},{"id":"https://openalex.org/C102727969","wikidata":"https://www.wikidata.org/wiki/Q4901423","display_name":"Bhatnagar\u2013Gross\u2013Krook operator","level":5,"score":0.4519854784011841},{"id":"https://openalex.org/C19499675","wikidata":"https://www.wikidata.org/wiki/Q232207","display_name":"Monte Carlo method","level":2,"score":0.26100611686706543},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.14070859551429749},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.10816067457199097},{"id":"https://openalex.org/C164916747","wikidata":"https://www.wikidata.org/wiki/Q4382753","display_name":"Direct simulation Monte Carlo","level":4,"score":0.0926831066608429},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.0},{"id":"https://openalex.org/C24890656","wikidata":"https://www.wikidata.org/wiki/Q82811","display_name":"Acoustics","level":1,"score":0.0},{"id":"https://openalex.org/C122592724","wikidata":"https://www.wikidata.org/wiki/Q3855635","display_name":"Dynamic Monte Carlo method","level":3,"score":0.0},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0},{"id":"https://openalex.org/C38652104","wikidata":"https://www.wikidata.org/wiki/Q3510521","display_name":"Computer security","level":1,"score":0.0}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1137/040618916","is_oa":false,"landing_page_url":"https://doi.org/10.1137/040618916","pdf_url":null,"source":{"id":"https://openalex.org/S74141806","display_name":"SIAM Journal on Mathematical Analysis","issn_l":"0036-1410","issn":["0036-1410","1095-7154"],"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 Mathematical Analysis","raw_type":"journal-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.63.6962","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.63.6962","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.chalmers.se/~kurlberg/eprints/circle.pdf","raw_type":"text"},{"id":"pmh:oai:publications.lib.chalmers.se:21395","is_oa":false,"landing_page_url":"http://publications.lib.chalmers.se/publication/21395-lattice-points-on-circles-and-discrete-velocity-models-for-the-boltzmann-equation","pdf_url":null,"source":{"id":"https://openalex.org/S4377196470","display_name":"Chalmers Publication Library (Chalmers University of Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I66862912","host_organization_name":"Chalmers University of Technology","host_organization_lineage":["https://openalex.org/I66862912"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text.Article.Journal.PeerReviewed"}],"best_oa_location":null,"sustainable_development_goals":[{"display_name":"Sustainable cities and communities","id":"https://metadata.un.org/sdg/11","score":0.4000000059604645}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":24,"referenced_works":["https://openalex.org/W174006204","https://openalex.org/W749116816","https://openalex.org/W1568671078","https://openalex.org/W1588244485","https://openalex.org/W1977795007","https://openalex.org/W1978535359","https://openalex.org/W1989597834","https://openalex.org/W1990352033","https://openalex.org/W1991174092","https://openalex.org/W1992331506","https://openalex.org/W2000415500","https://openalex.org/W2016713393","https://openalex.org/W2070656973","https://openalex.org/W2075557408","https://openalex.org/W2076597612","https://openalex.org/W2086112279","https://openalex.org/W2090161681","https://openalex.org/W2107967981","https://openalex.org/W2125505119","https://openalex.org/W2165319431","https://openalex.org/W2577105026","https://openalex.org/W4214662400","https://openalex.org/W4285719527","https://openalex.org/W4388288416"],"related_works":["https://openalex.org/W2007632118","https://openalex.org/W2062900651","https://openalex.org/W44857498","https://openalex.org/W2055632123","https://openalex.org/W1969064103","https://openalex.org/W1982767103","https://openalex.org/W2661863244","https://openalex.org/W2381588372","https://openalex.org/W613555516","https://openalex.org/W2147968878"],"abstract_inverted_index":{"The":[0,29,110],"construction":[1],"of":[2,18,43,58,116,136],"discrete":[3],"velocity":[4],"models":[5],"or":[6],"numerical":[7],"methods":[8],"for":[9,120,132],"the":[10,16,20,41,56,93,107,117,128,133],"Boltzmann":[11],"equation,":[12],"may":[13],"lead":[14],"to":[15,40],"necessity":[17],"computing":[19],"collision":[21,30,94],"operator":[22,31,95],"as":[23,100],"a":[24,36,101,113],"sum":[25,102],"over":[26,35,103],"lattice":[27,68,77,83,104],"points.":[28,78],"involves":[32],"an":[33,60],"integral":[34],"sphere,":[37],"which":[38],"corresponds":[39],"conservation":[42],"energy":[44],"and":[45,70,130,151],"momentum.":[46],"In":[47],"dimension":[48],"two":[49],"there":[50],"are":[51,88],"difficulties":[52],"even":[53],"in":[54,106,127],"proving":[55],"convergence":[57],"such":[59],"approximation":[61],"since":[62],"many":[63,74],"circles":[64,72,87],"contain":[65,73],"very":[66],"few":[67],"points,":[69],"some":[71],"badly":[75],"distributed":[76],"However,":[79],"by":[80,148],"showing":[81],"that":[82,92],"points":[84,105],"on":[85],"most":[86],"equidistributed":[89],"we":[90],"find":[91],"can":[96],"indeed":[97],"be":[98],"approximated":[99],"two-dimensional":[108],"case.":[109],"proof":[111,124],"uses":[112],"weak":[114],"form":[115],"Halberstam--Richert":[118],"inequality":[119],"multiplicative":[121],"functions":[122],"(a":[123],"is":[125],"given":[126],"paper),":[129],"estimates":[131],"angular":[134],"distribution":[135],"Gaussian":[137],"primes.":[138],"For":[139],"higher":[140],"dimensions,":[141],"this":[142],"result":[143],"has":[144],"already":[145],"been":[146],"obtained":[147],"Palczewski,":[149],"Schneider,":[150],"Bobylev":[152],"[SIAM":[153],"J.":[154],"Numer.":[155],"Anal.,":[156],"34":[157],"(1997),":[158],"pp.":[159],"1865-1883].":[160]},"counts_by_year":[{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":2},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":1},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":4},{"year":2016,"cited_by_count":8},{"year":2015,"cited_by_count":2},{"year":2014,"cited_by_count":3},{"year":2013,"cited_by_count":4},{"year":2012,"cited_by_count":1}],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}