{"id":"https://openalex.org/W3142746000","doi":"https://doi.org/10.1109/tit.1956.1056822","title":"The axis-crossing intervals of random functions","display_name":"The axis-crossing intervals of random functions","publication_year":1956,"publication_date":"1956-12-01","ids":{"openalex":"https://openalex.org/W3142746000","doi":"https://doi.org/10.1109/tit.1956.1056822","mag":"3142746000"},"language":"en","primary_location":{"id":"doi:10.1109/tit.1956.1056822","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.1956.1056822","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","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/A5069892918","display_name":"J. A. McFadden","orcid":"https://orcid.org/0000-0001-7221-0400"},"institutions":[{"id":"https://openalex.org/I4399598358","display_name":"United States Navy","ror":"https://ror.org/03cs53d16","country_code":null,"type":"funder","lineage":["https://openalex.org/I1330347796","https://openalex.org/I3130687028","https://openalex.org/I4399598358"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"J. McFadden","raw_affiliation_strings":["U.S. Naval Ordnance Laboratory, Silver Spring, MD, USA"],"affiliations":[{"raw_affiliation_string":"U.S. Naval Ordnance Laboratory, Silver Spring, MD, USA","institution_ids":["https://openalex.org/I4399598358"]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5069892918"],"corresponding_institution_ids":["https://openalex.org/I4399598358"],"apc_list":null,"apc_paid":null,"fwci":2.944,"has_fulltext":false,"cited_by_count":59,"citation_normalized_percentile":{"value":0.91601563,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"2","issue":"4","first_page":"146","last_page":"150"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.995199978351593,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T11152","display_name":"Stochastic processes and statistical mechanics","score":0.995199978351593,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/T10282","display_name":"Financial Risk and Volatility Modeling","score":0.9675999879837036,"subfield":{"id":"https://openalex.org/subfields/2003","display_name":"Finance"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9503999948501587,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/function","display_name":"Function (biology)","score":0.515217125415802},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.34714430570602417},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3329387605190277},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.32534492015838623},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.2504799962043762},{"id":"https://openalex.org/keywords/biology","display_name":"Biology","score":0.061151713132858276}],"concepts":[{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.515217125415802},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.34714430570602417},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3329387605190277},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.32534492015838623},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.2504799962043762},{"id":"https://openalex.org/C86803240","wikidata":"https://www.wikidata.org/wiki/Q420","display_name":"Biology","level":0,"score":0.061151713132858276},{"id":"https://openalex.org/C78458016","wikidata":"https://www.wikidata.org/wiki/Q840400","display_name":"Evolutionary biology","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tit.1956.1056822","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.1956.1056822","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Information Theory","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":4,"referenced_works":["https://openalex.org/W1965827398","https://openalex.org/W1995842970","https://openalex.org/W2005047575","https://openalex.org/W2114509411"],"related_works":["https://openalex.org/W2899084033","https://openalex.org/W2748952813","https://openalex.org/W2051487156","https://openalex.org/W2073681303","https://openalex.org/W2390279801","https://openalex.org/W2358668433","https://openalex.org/W2386767533","https://openalex.org/W3098582471","https://openalex.org/W2390720117","https://openalex.org/W2385614099"],"abstract_inverted_index":{"For":[0],"an":[1],"arbitrary":[2],"random":[3,192],"process":[4],"<tex":[5,12,26,34,44,51,59,83,94,99,112,121,132,138,148,155,164,173,178],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[6,13,27,35,45,52,60,84,95,100,113,122,133,139,149,156,165,174,179],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">\\xi(t)</tex>":[7,36],"there":[8],"exists":[9],"a":[10,106,118,197],"function":[11,57],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">x(t)</tex>":[14,28,61],"which":[15],"may":[16],"be":[17],"obtained":[18],"by":[19,189,196],"infinite":[20],"clipping.":[21,67],"The":[22],"axis":[23],"crossings":[24],"of":[25,33,47,58,75,93,147,172],"are":[29,187],"identical":[30],"with":[31,199],"those":[32],".":[37,104,153,184],"This":[38],"paper":[39],"relates":[40],"the":[41,55,64,72,90,144,169],"probability":[42],"density":[43],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">P(\\tau)</tex>":[46,175],"axis-crossing":[48],"intervals":[49],"to":[50,82],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">\\gamma(\\tau)</tex>":[53,134],",":[54,62,88,167],"autocorrelation":[56,65],"i.e.,":[63,89],"after":[66],"It":[68],"is":[69,80,108,135,159,176],"shown":[70],"that":[71,111],"expected":[73],"number":[74],"zeros":[76],"per":[77],"unit":[78],"time":[79],"proportional":[81],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">\\gamma":[85,96,157,180],"\\prime":[86,181,182],"(0+)</tex>":[87],"right-hand":[91],"derivative":[92],"(\\tau)</tex>":[97,158,183],"at":[98],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">\\tau":[101],"=":[102,115],"0</tex>":[103,116],"Next":[105],"theorem":[107],"proved,":[109],"stating":[110],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">P(\\tau)":[114],"over":[117,143],"finite":[119],"range":[120,146],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">0":[123],"\\leq":[124],"\\tau":[125,141,151],"&lt;":[126],"T</tex>":[127],"if":[128,131],"and":[129,195],"only":[130],"linear":[136,161],"in":[137],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">\\mid":[140,150],"\\mid</tex>":[142,152],"corresponding":[145],"If":[154],"nearly":[160],"for":[162,202],"small":[163],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">\\tau</tex>":[166],"then":[168],"initial":[170],"behavior":[171],"like":[177],"These":[185],"results":[186,201],"illustrated":[188],"some":[190],"simple,":[191],"square-wave":[193],"models":[194],"comparison":[198],"Rice's":[200],"Gaussian":[203],"noise.":[204]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2022,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}