{"id":"https://openalex.org/W1984582452","doi":"https://doi.org/10.1090/s0025-5718-00-01282-5","title":"Period of the power generator and small values of Carmichael\u2019s function","display_name":"Period of the power generator and small values of Carmichael\u2019s function","publication_year":2000,"publication_date":"2000-07-13","ids":{"openalex":"https://openalex.org/W1984582452","doi":"https://doi.org/10.1090/s0025-5718-00-01282-5","mag":"1984582452"},"language":"lv","primary_location":{"id":"doi:10.1090/s0025-5718-00-01282-5","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-00-01282-5","pdf_url":"https://www.ams.org/mcom/2001-70-236/S0025-5718-00-01282-5/S0025-5718-00-01282-5.pdf","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":true,"oa_status":"bronze","oa_url":"https://www.ams.org/mcom/2001-70-236/S0025-5718-00-01282-5/S0025-5718-00-01282-5.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5066799294","display_name":"John Friedlander","orcid":"https://orcid.org/0000-0001-9011-0972"},"institutions":[{"id":"https://openalex.org/I185261750","display_name":"University of Toronto","ror":"https://ror.org/03dbr7087","country_code":"CA","type":"education","lineage":["https://openalex.org/I185261750"]}],"countries":["CA"],"is_corresponding":true,"raw_author_name":"John Friedlander","raw_affiliation_strings":["Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada","Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada ("],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada","institution_ids":["https://openalex.org/I185261750"]},{"raw_affiliation_string":"Department of Mathematics, University of Toronto, Toronto, Ontario M5S 3G3, Canada (","institution_ids":["https://openalex.org/I185261750"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5065324760","display_name":"Carl Pomerance","orcid":"https://orcid.org/0000-0002-0960-1692"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Carl Pomerance","raw_affiliation_strings":["Department of Fundamental Mathematics, Bell Labs, Murray Hill, New Jersey 07974-0636"],"affiliations":[{"raw_affiliation_string":"Department of Fundamental Mathematics, Bell Labs, Murray Hill, New Jersey 07974-0636","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5030033231","display_name":"Igor E. Shparlinski","orcid":"https://orcid.org/0000-0002-5246-9391"},"institutions":[{"id":"https://openalex.org/I99043593","display_name":"Macquarie University","ror":"https://ror.org/01sf06y89","country_code":"AU","type":"education","lineage":["https://openalex.org/I99043593"]}],"countries":["AU"],"is_corresponding":false,"raw_author_name":"Igor Shparlinski","raw_affiliation_strings":["Department of Computing, Macquarie University, Sydney, New South Wales 2109, Australia"],"affiliations":[{"raw_affiliation_string":"Department of Computing, Macquarie University, Sydney, New South Wales 2109, Australia","institution_ids":["https://openalex.org/I99043593"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":["https://openalex.org/A5066799294"],"corresponding_institution_ids":["https://openalex.org/I185261750"],"apc_list":null,"apc_paid":null,"fwci":6.4023,"has_fulltext":true,"cited_by_count":66,"citation_normalized_percentile":{"value":0.96721679,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":89,"max":98},"biblio":{"volume":"70","issue":"236","first_page":"1591","last_page":"1605"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11017","display_name":"Chaos-based Image/Signal Encryption","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/T11017","display_name":"Chaos-based Image/Signal Encryption","score":0.9995999932289124,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"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/T11130","display_name":"Coding theory and cryptography","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T12162","display_name":"Cellular Automata and Applications","score":0.992900013923645,"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/algorithm","display_name":"Algorithm","score":0.43233722448349},{"id":"https://openalex.org/keywords/materials-science","display_name":"Materials science","score":0.3297906517982483},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.15372276306152344}],"concepts":[{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.43233722448349},{"id":"https://openalex.org/C192562407","wikidata":"https://www.wikidata.org/wiki/Q228736","display_name":"Materials science","level":0,"score":0.3297906517982483},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.15372276306152344}],"mesh":[],"locations_count":3,"locations":[{"id":"doi:10.1090/s0025-5718-00-01282-5","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-00-01282-5","pdf_url":"https://www.ams.org/mcom/2001-70-236/S0025-5718-00-01282-5/S0025-5718-00-01282-5.pdf","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"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.153.9184","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.153.9184","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.ams.org/mcom/2001-70-236/S0025-5718-00-01282-5/S0025-5718-00-01282-5.pdf","raw_type":"text"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.501.311","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.501.311","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://www.math.dartmouth.edu/~carlp/PDF/paper124.pdf","raw_type":"text"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-00-01282-5","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-00-01282-5","pdf_url":"https://www.ams.org/mcom/2001-70-236/S0025-5718-00-01282-5/S0025-5718-00-01282-5.pdf","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"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G884384748","display_name":null,"funder_award_id":"A5123","funder_id":"https://openalex.org/F4320334593","funder_display_name":"Natural Sciences and Engineering Research Council of Canada"}],"funders":[{"id":"https://openalex.org/F4320334593","display_name":"Natural Sciences and Engineering Research Council of Canada","ror":"https://ror.org/01h531d29"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W1984582452.pdf","grobid_xml":"https://content.openalex.org/works/W1984582452.grobid-xml"},"referenced_works_count":29,"referenced_works":["https://openalex.org/W1493612669","https://openalex.org/W1503851905","https://openalex.org/W1546597647","https://openalex.org/W1556140814","https://openalex.org/W1574486381","https://openalex.org/W1576538821","https://openalex.org/W1577566258","https://openalex.org/W1608356921","https://openalex.org/W1632088469","https://openalex.org/W1660562555","https://openalex.org/W1999537317","https://openalex.org/W2014937115","https://openalex.org/W2018738738","https://openalex.org/W2025977722","https://openalex.org/W2085759780","https://openalex.org/W2088772065","https://openalex.org/W2112846031","https://openalex.org/W2117420234","https://openalex.org/W2123087644","https://openalex.org/W2123355614","https://openalex.org/W2154021956","https://openalex.org/W2168453574","https://openalex.org/W2317183192","https://openalex.org/W2330778380","https://openalex.org/W3028729063","https://openalex.org/W4229801863","https://openalex.org/W4241202974","https://openalex.org/W4249797500","https://openalex.org/W6636398258"],"related_works":["https://openalex.org/W4246450666","https://openalex.org/W4388998267","https://openalex.org/W2898370298","https://openalex.org/W4390401159","https://openalex.org/W2744391499","https://openalex.org/W3120461830","https://openalex.org/W4230250635","https://openalex.org/W3041790586","https://openalex.org/W2018879842","https://openalex.org/W3144504424"],"abstract_inverted_index":{"Consider":[0],"the":[1,64,70,80,93,100,117,126,142,145,172,193,208,211,235,251,255,271,284,287,292],"pseudorandom":[2],"number":[3],"generator<disp-formula":[4],"content-type=\"math/mathml\"><mml:math":[5,66,73,82,95,102,108,134,147,158,186,201,215,274,298,309,332,360,364,379],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[6,67,74,83,96,103,109,135,148,159,187,202,216,275,299,310,333,361,365,380],"alttext=\"u":[7,75],"Subscript":[8,13,28,151],"n":[9,14,29,36,152],"Baseline":[10,19,30,153,400],"identical-to":[11],"u":[12,27,150],"minus":[15,33,223,385],"1":[16,34,38,222,397],"Superscript":[17,161,221,396],"e":[18],"left-parenthesis":[20,277,315,335,342,384,387],"mod":[21],"m":[22,32,278,336,368],"right-parenthesis":[23,220,320,337,395],"comma":[24,35,39,41,111,189,204],"0":[25,76],"less-than-or-equal-to":[26,31,367,369],"equals":[37,77,137],"2":[40],"ellipsis":[42],"comma\"><mml:semantics><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>\u2261</mml:mo><mml:msubsup><mml:mi>u</mml:mi><mml:mrow":[43],"class=\"MJX-TeXAtom-ORD\"><mml:mi>n</mml:mi><mml:mo>\u2212</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>e</mml:mi></mml:msubsup><mml:mspace":[44],"width=\"0.667em\"/><mml:mo":[45],"stretchy=\"false\">(</mml:mo><mml:mi>mod</mml:mi><mml:mspace":[46],"width=\"0.333em\"/><mml:mi>m</mml:mi><mml:mo":[47],"stretchy=\"false\">)</mml:mo><mml:mo>,</mml:mo><mml:mspace":[48],"width=\"1em\"/><mml:mn>0</mml:mn><mml:mo>\u2264</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>\u2264</mml:mo><mml:mi>m</mml:mi><mml:mo>\u2212</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mspace":[49],"width=\"1em\"/><mml:mi>n</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>\u2026</mml:mo><mml:mo>,</mml:mo></mml:mrow><mml:annotation":[50],"encoding=\"application/x-tex\">\\begin{equation*}":[51],"u_n\\equiv":[52],"u_{n-1}^e\\pmod":[53],"{m},\\quad":[54],"0\\le":[55],"u_n\\le":[56],"m-1,\\quad":[57],"n=1,2,\\ldots":[58],",":[59],"\\end{equation*}</mml:annotation></mml:semantics></mml:math></disp-formula>where":[60],"we":[61,304,330],"are":[62],"given":[63],"modulus<inline-formula":[65,94],"alttext=\"m\"><mml:semantics><mml:mi>m</mml:mi><mml:annotation":[68,97,300,362],"encoding=\"application/x-tex\">m</mml:annotation></mml:semantics></mml:math></inline-formula>,":[69],"initial":[71],"value<inline-formula":[72],"theta\"><mml:semantics><mml:mrow><mml:msub><mml:mi>u</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mi>\u03d1</mml:mi></mml:mrow><mml:annotation":[78],"encoding=\"application/x-tex\">u_0=\\vartheta</mml:annotation></mml:semantics></mml:math></inline-formula>and":[79],"exponent<inline-formula":[81],"alttext=\"e\"><mml:semantics><mml:mi>e</mml:mi><mml:annotation":[84],"encoding=\"application/x-tex\">e</mml:annotation></mml:semantics></mml:math></inline-formula>.":[85],"One":[86],"case":[87,194],"of":[88,99,116,125,144,210,286,291,295],"particular":[89],"interest":[90],"is":[91,121,174,192,238,266],"when":[92],"encoding=\"application/x-tex\">m</mml:annotation></mml:semantics></mml:math></inline-formula>is":[98],"form<inline-formula":[101],"alttext=\"p":[104,110,188],"l\"><mml:semantics><mml:mrow><mml:mi>p</mml:mi><mml:mi>l</mml:mi></mml:mrow><mml:annotation":[105],"encoding=\"application/x-tex\">pl</mml:annotation></mml:semantics></mml:math></inline-formula>,":[106],"where<inline-formula":[107],"l\"><mml:semantics><mml:mrow><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>l</mml:mi></mml:mrow><mml:annotation":[112,190],"encoding=\"application/x-tex\">p,l</mml:annotation></mml:semantics></mml:math></inline-formula>are":[113],"different":[114],"primes":[115],"same":[118],"magnitude.":[119],"It":[120],"known":[122],"from":[123,376],"work":[124],"first":[127],"and":[128,258],"third":[129],"authors":[130],"that":[131,180,195,306],"for":[132,181,196,307],"moduli<inline-formula":[133],"alttext=\"m":[136,160],"p":[138,218],"l\"><mml:semantics><mml:mrow><mml:mi>m</mml:mi><mml:mo>=</mml:mo><mml:mi>p</mml:mi><mml:mi>l</mml:mi></mml:mrow><mml:annotation":[139],"encoding=\"application/x-tex\">m=pl</mml:annotation></mml:semantics></mml:math></inline-formula>,":[140],"if":[141],"period":[143,209],"sequence<inline-formula":[146],"alttext=\"left-parenthesis":[149,217],"right-parenthesis\"><mml:semantics><mml:mrow><mml:mo":[154],"stretchy=\"false\">(</mml:mo><mml:msub><mml:mi>u</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo":[155],"stretchy=\"false\">)</mml:mo></mml:mrow><mml:annotation":[156,281,352],"encoding=\"application/x-tex\">(u_n)</mml:annotation></mml:semantics></mml:math></inline-formula>exceeds<inline-formula":[157],"3":[162,399],"slash":[163,398],"4":[164],"plus":[165],"epsilon\"><mml:semantics><mml:msup><mml:mi>m</mml:mi><mml:mrow":[166],"class=\"MJX-TeXAtom-ORD\"><mml:mn>3</mml:mn><mml:mrow":[167],"class=\"MJX-TeXAtom-ORD\"><mml:mo>/</mml:mo></mml:mrow><mml:mn>4</mml:mn><mml:mo>+</mml:mo><mml:mi>\u03b5</mml:mi></mml:mrow></mml:msup><mml:annotation":[168],"encoding=\"application/x-tex\">m^{3/4+\\varepsilon":[169],"}</mml:annotation></mml:semantics></mml:math></inline-formula>,":[170],"then":[171],"sequence":[173],"uniformly":[175,239],"distributed.":[176,240],"We":[177,241],"show":[178,305],"rigorously":[179],"almost":[182,197],"all":[183,198,358],"choices":[184,199],"of<inline-formula":[185,200],"encoding=\"application/x-tex\">p,l</mml:annotation></mml:semantics></mml:math></inline-formula>it":[191],"alttext=\"theta":[203],"e\"><mml:semantics><mml:mrow><mml:mi>\u03d1</mml:mi><mml:mo>,</mml:mo><mml:mi>e</mml:mi></mml:mrow><mml:annotation":[205],"encoding=\"application/x-tex\">\\vartheta":[206],",e</mml:annotation></mml:semantics></mml:math></inline-formula>,":[207],"power":[212,236],"generator":[213,237],"exceeds<inline-formula":[214],"l":[219],"epsilon\"><mml:semantics><mml:mrow><mml:mo":[224],"stretchy=\"false\">(</mml:mo><mml:mi>p</mml:mi><mml:mi>l</mml:mi><mml:msup><mml:mo":[225],"stretchy=\"false\">)</mml:mo><mml:mrow":[226],"class=\"MJX-TeXAtom-ORD\"><mml:mn>1</mml:mn><mml:mo>\u2212</mml:mo><mml:mi>\u03b5</mml:mi></mml:mrow></mml:msup></mml:mrow><mml:annotation":[227],"encoding=\"application/x-tex\">(pl)^{1-\\varepsilon":[228],"}</mml:annotation></mml:semantics></mml:math></inline-formula>.":[229],"And":[230],"so,":[231],"in":[232],"this":[233],"case,":[234],"also":[242],"give":[243],"some":[244],"other":[245],"cryptographic":[246],"applications,":[247],"namely,":[248],"to":[249,259,270],"ruling-out":[250],"cycling":[252],"attack":[253],"on":[254],"RSA":[256],"cryptosystem":[257],"so-called":[260],"time-release":[261],"crypto.":[262],"The":[263],"principal":[264],"tool":[265],"an":[267],"estimate":[268],"related":[269],"Carmichael":[272],"function<inline-formula":[273],"alttext=\"lamda":[276,334],"right-parenthesis\"><mml:semantics><mml:mrow><mml:mi>\u03bb</mml:mi><mml:mo":[279,347],"stretchy=\"false\">(</mml:mo><mml:mi>m</mml:mi><mml:mo":[280,348],"encoding=\"application/x-tex\">\\lambda":[282,353],"(m)</mml:annotation></mml:semantics></mml:math></inline-formula>,":[283],"size":[285],"largest":[288],"cyclic":[289],"subgroup":[290],"multiplicative":[293],"group":[294],"residues":[296],"modulo<inline-formula":[297],"encoding=\"application/x-tex\">m</mml:annotation></mml:semantics></mml:math></inline-formula>.":[301],"In":[302],"particular,":[303],"any<inline-formula":[308],"alttext=\"normal":[311],"upper":[312,318,339,345,370,389,393],"Delta":[313,346,390,394],"greater-than-or-equal-to":[314,338],"log":[316,317,391],"N":[319,340,382],"cubed\"><mml:semantics><mml:mrow><mml:mi":[321],"mathvariant=\"normal\">\u0394</mml:mi><mml:mo>\u2265</mml:mo><mml:mo":[322],"stretchy=\"false\">(</mml:mo><mml:mi>log</mml:mi><mml:mo>\u2061</mml:mo><mml:mi>log</mml:mi><mml:mo>\u2061</mml:mo><mml:mi>N</mml:mi><mml:msup><mml:mo":[323],"stretchy=\"false\">)</mml:mo><mml:mn>3</mml:mn></mml:msup></mml:mrow><mml:annotation":[324],"encoding=\"application/x-tex\">\\Delta":[325],"\\ge":[326],"(\\log":[327],"\\log":[328],"N)^3</mml:annotation></mml:semantics></mml:math></inline-formula>,":[329],"have<inline-formula":[331],"exp":[341,383],"negative":[343],"normal":[344,388,392],"stretchy=\"false\">)</mml:mo><mml:mo>\u2265</mml:mo><mml:mi>N</mml:mi><mml:mi>exp</mml:mi><mml:mo>\u2061</mml:mo><mml:mo":[349],"stretchy=\"false\">(</mml:mo><mml:mo>\u2212</mml:mo><mml:mi":[350],"mathvariant=\"normal\">\u0394</mml:mi><mml:mo":[351],"(m)\\ge":[354],"N\\exp":[355],"(-\\Delta":[356],")</mml:annotation></mml:semantics></mml:math></inline-formula>for":[357],"integers<inline-formula":[359],"encoding=\"application/x-tex\">m</mml:annotation></mml:semantics></mml:math></inline-formula>with<inline-formula":[363],"alttext=\"1":[366],"N\"><mml:semantics><mml:mrow><mml:mn>1</mml:mn><mml:mo>\u2264</mml:mo><mml:mi>m</mml:mi><mml:mo>\u2264</mml:mo><mml:mi>N</mml:mi></mml:mrow><mml:annotation":[371],"encoding=\"application/x-tex\">1\\le":[372],"m\\le":[373],"N</mml:annotation></mml:semantics></mml:math></inline-formula>,":[374],"apart":[375],"at":[377],"most<inline-formula":[378],"alttext=\"upper":[381],"0.69":[386],"right-parenthesis\"><mml:semantics><mml:mrow><mml:mi>N</mml:mi><mml:mi>exp</mml:mi><mml:mo>\u2061</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:mo>\u2212</mml:mo><mml:mn>0.69</mml:mn><mml:msup><mml:mrow><mml:mo>(</mml:mo><mml:mi":[401],"mathvariant=\"normal\">\u0394</mml:mi><mml:mi>log</mml:mi><mml:mo>\u2061</mml:mo><mml:mi":[402],"mathvariant=\"normal\">\u0394</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mrow":[403],"class=\"MJX-TeXAtom-ORD\"><mml:mn>1</mml:mn><mml:":[404]},"counts_by_year":[{"year":2024,"cited_by_count":3},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":1},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":1},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":5},{"year":2012,"cited_by_count":2}],"updated_date":"2026-04-22T08:38:42.863108","created_date":"2025-10-10T00:00:00"}
