{"id":"https://openalex.org/W7225144167","doi":"https://doi.org/10.5281/zenodo.23910675","title":"From Matrix Exponentials to Nonlinear Functions: A Common Structural View of Fourier Analysis, Hyperbolic Transformations, and Neural Learning (v1.1-en)","display_name":"From Matrix Exponentials to Nonlinear Functions: A Common Structural View of Fourier Analysis, Hyperbolic Transformations, and Neural Learning (v1.1-en)","publication_year":2026,"publication_date":"2026-11-03","ids":{"openalex":"https://openalex.org/W7225144167","doi":"https://doi.org/10.5281/zenodo.23910675"},"language":"en","primary_location":{"id":"doi:10.5281/zenodo.23910675","doi":"https://doi.org/10.5281/zenodo.23910675","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23910675","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article-journal"},"volume":null,"type":"article","indexed_in":["datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://doi.org/10.5281/zenodo.23910675","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5133993365","display_name":"Chih-Ming Wang","orcid":"https://orcid.org/0009-0001-3651-1422","observed_orcids":["https://orcid.org/0009-0001-3651-1422"]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Wang, Chih-Ming","raw_affiliation_strings":["Independent Researcher"],"raw_orcid":"https://orcid.org/0009-0001-3651-1422","affiliations":[{"raw_affiliation_string":"Independent Researcher","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5133993365"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":0,"citation_normalized_percentile":{"value":0.83028043,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.6035000085830688,"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"}},"topics":[{"id":"https://openalex.org/T10320","display_name":"Neural Networks and Applications","score":0.6035000085830688,"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/T11210","display_name":"Mathematical Analysis and Transform Methods","score":0.07259999960660934,"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/T13806","display_name":"Mathematical and Computational Methods","score":0.05689999833703041,"subfield":{"id":"https://openalex.org/subfields/2611","display_name":"Modeling and Simulation"},"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/matrix-exponential","display_name":"matrix exponential","score":0.9629999995231628},{"id":"https://openalex.org/keywords/matrix-generators","display_name":"matrix generators","score":0.5989999771118164},{"id":"https://openalex.org/keywords/fourier-synthesis","display_name":"Fourier synthesis","score":0.5070000290870667},{"id":"https://openalex.org/keywords/tanh-function","display_name":"tanh function","score":0.7459999918937683},{"id":"https://openalex.org/keywords/hyperbolic-transformations","display_name":"hyperbolic transformations","score":0.7379999756813049},{"id":"https://openalex.org/keywords/riccati-equation","display_name":"Riccati equation","score":0.8949999809265137},{"id":"https://openalex.org/keywords/projective-coordinates","display_name":"projective coordinates","score":0.7760000228881836},{"id":"https://openalex.org/keywords/logistic-function","display_name":"logistic function","score":0.6140000224113464},{"id":"https://openalex.org/keywords/lorentz-boost","display_name":"Lorentz boost","score":0.9990000128746033},{"id":"https://openalex.org/keywords/tensor-products","display_name":"tensor products","score":0.5419999957084656},{"id":"https://openalex.org/keywords/leibniz-rule","display_name":"Leibniz rule","score":0.9440000057220459},{"id":"https://openalex.org/keywords/nonlinear-functions","display_name":"nonlinear functions","score":0.6679999828338623},{"id":"https://openalex.org/keywords/neural-learning","display_name":"neural learning","score":0.6679999828338623},{"id":"https://openalex.org/keywords/spectral-representation","display_name":"spectral representation","score":0.7559999823570251},{"id":"https://openalex.org/keywords/linear-differential-equations","display_name":"linear differential equations","score":0.5799999833106995},{"id":"https://openalex.org/keywords/two-dimensional-flow","display_name":"two-dimensional flow","score":0.5849999785423279},{"id":"https://openalex.org/keywords/higher-order-derivatives","display_name":"higher-order derivatives","score":0.5080000162124634},{"id":"https://openalex.org/keywords/dimensionality","display_name":"dimensionality","score":0.5460000038146973},{"id":"https://openalex.org/keywords/hyperbolic-trajectories","display_name":"hyperbolic trajectories","score":0.3199999928474426}],"concepts":[{"id":"https://openalex.org/C92047909","wikidata":"https://www.wikidata.org/wiki/Q204034","display_name":"Hyperbolic function","level":2,"score":0.6585000157356262},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6171000003814697},{"id":"https://openalex.org/C106487976","wikidata":"https://www.wikidata.org/wiki/Q685816","display_name":"Matrix (chemical analysis)","level":2,"score":0.5896999835968018},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5389999747276306},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.4726000130176544},{"id":"https://openalex.org/C102519508","wikidata":"https://www.wikidata.org/wiki/Q6520159","display_name":"Fourier transform","level":2,"score":0.4542999863624573},{"id":"https://openalex.org/C527412718","wikidata":"https://www.wikidata.org/wiki/Q855395","display_name":"Interpretation (philosophy)","level":2,"score":0.42410001158714294},{"id":"https://openalex.org/C136119220","wikidata":"https://www.wikidata.org/wiki/Q1000660","display_name":"Algebra over a field","level":2,"score":0.4198000133037567},{"id":"https://openalex.org/C5667645","wikidata":"https://www.wikidata.org/wiki/Q217255","display_name":"Lorentz transformation","level":2,"score":0.4163999855518341},{"id":"https://openalex.org/C14036430","wikidata":"https://www.wikidata.org/wiki/Q3736076","display_name":"Function (biology)","level":2,"score":0.41530001163482666},{"id":"https://openalex.org/C45473103","wikidata":"https://www.wikidata.org/wiki/Q851503","display_name":"Riccati equation","level":3,"score":0.39879998564720154},{"id":"https://openalex.org/C129844170","wikidata":"https://www.wikidata.org/wiki/Q41299","display_name":"Quadratic equation","level":2,"score":0.3970000147819519},{"id":"https://openalex.org/C195906000","wikidata":"https://www.wikidata.org/wiki/Q1191722","display_name":"Matrix exponential","level":3,"score":0.3905999958515167},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.3831000030040741},{"id":"https://openalex.org/C151376022","wikidata":"https://www.wikidata.org/wiki/Q168698","display_name":"Exponential function","level":2,"score":0.37959998846054077},{"id":"https://openalex.org/C108598597","wikidata":"https://www.wikidata.org/wiki/Q124255","display_name":"Conic section","level":2,"score":0.37380000948905945},{"id":"https://openalex.org/C2776359362","wikidata":"https://www.wikidata.org/wiki/Q2145286","display_name":"Representation (politics)","level":3,"score":0.3637999892234802},{"id":"https://openalex.org/C203024314","wikidata":"https://www.wikidata.org/wiki/Q1365258","display_name":"Fourier analysis","level":3,"score":0.32249999046325684},{"id":"https://openalex.org/C70984080","wikidata":"https://www.wikidata.org/wiki/Q2619102","display_name":"Hyperbolic manifold","level":3,"score":0.3199999928474426},{"id":"https://openalex.org/C155281189","wikidata":"https://www.wikidata.org/wiki/Q3518150","display_name":"Tensor (intrinsic definition)","level":2,"score":0.31690001487731934},{"id":"https://openalex.org/C2780992000","wikidata":"https://www.wikidata.org/wiki/Q17016113","display_name":"Generator (circuit theory)","level":3,"score":0.31060001254081726},{"id":"https://openalex.org/C166437778","wikidata":"https://www.wikidata.org/wiki/Q50695","display_name":"Quadratic function","level":3,"score":0.3052999973297119},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.296099990606308},{"id":"https://openalex.org/C11210021","wikidata":"https://www.wikidata.org/wiki/Q1520713","display_name":"Linearization","level":3,"score":0.29440000653266907},{"id":"https://openalex.org/C55862258","wikidata":"https://www.wikidata.org/wiki/Q1017238","display_name":"Hyperbolic coordinates","level":5,"score":0.2799000144004822},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.2766999900341034},{"id":"https://openalex.org/C68827188","wikidata":"https://www.wikidata.org/wiki/Q674533","display_name":"Inverse trigonometric functions","level":2,"score":0.2531000077724457}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.5281/zenodo.23910675","doi":"https://doi.org/10.5281/zenodo.23910675","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23910675","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article-journal"},{"id":"doi:10.5281/zenodo.23910676","doi":"https://doi.org/10.5281/zenodo.23910676","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23910676","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article-journal"}],"best_oa_location":{"id":"doi:10.5281/zenodo.23910675","doi":"https://doi.org/10.5281/zenodo.23910675","is_oa":true,"landing_page_url":"https://doi.org/10.5281/zenodo.23910675","pdf_url":null,"source":{"id":"https://openalex.org/S4306400562","display_name":"Zenodo (CERN European Organization for Nuclear Research)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I67311998","host_organization_name":"European Organization for Nuclear Research","host_organization_lineage":["https://openalex.org/I67311998"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"article-journal"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":0,"referenced_works":[],"related_works":["https://openalex.org/W7225204635","https://openalex.org/W7225164919","https://openalex.org/W7225201596","https://openalex.org/W109686687","https://openalex.org/W6963698001","https://openalex.org/W1599038387","https://openalex.org/W6983397590","https://openalex.org/W1985940179","https://openalex.org/W4249186845","https://openalex.org/W2925089915"],"abstract_inverted_index":{"This":[0,238],"research":[1],"note":[2,72,239],"explores":[3],"how":[4,97,135],"elementary":[5],"linear":[6,84],"generators,":[7],"matrix":[8,29],"exponentials,":[9],"and":[10,22,53,65,69,118,121,128,156,170,200,225],"projective":[11,58],"readouts":[12],"lead":[13],"to":[14,76,162],"familiar":[15],"nonlinear":[16],"functions.":[17],"Beginning":[18],"with":[19,159],"Fourier":[20,120],"synthesis":[21,152],"a":[23,33,62,82,91,98,201],"tanh":[24,122,173,194],"function":[25,79],"unit,":[26],"it":[27],"derives":[28],"flows":[30,49],"generated":[31],"by":[32],"two-dimensional":[34],"family":[35],"satisfying":[36],"A2":[37,175],"=":[38,42,176,178],"\u03b5I":[39],"for":[40],"\u03b5":[41],"\u22121,":[43],"0,":[44],"or":[45],"+1.":[46],"The":[47,71,146],"planar":[48],"give":[50],"circular,":[51],"linear,":[52],"hyperbolic":[54,89],"trajectories,":[55],"while":[56],"the":[57,77,87,105,136,163,197,204,207,219,230],"coordinate":[59],"Y/X":[60,180],"obeys":[61],"Riccati":[63,182],"equation":[64],"yields":[66],"tan(t),":[67],"t,":[68],"tanh(t).":[70],"relates":[73],"these":[74],"representations":[75,133],"logistic":[78],"and,":[80],"as":[81],"separate":[83],"interpretation":[85],"of":[86,114,153,233],"same":[88,137],"generator,":[90],"Lorentz":[92,188],"boost.":[93],"It":[94],"then":[95],"explains":[96],"generator":[99],"acts":[100],"on":[101,131],"tensor-product":[102],"slots":[103],"through":[104],"Leibniz":[106,192],"rule,":[107],"distinguishes":[108],"that":[109],"induced":[110],"representation":[111],"from":[112],"tensors":[113],"higher":[115],"input":[116],"derivatives,":[117],"compares":[119],"differentiation.":[123],"A":[124],"short":[125],"neural-learning":[126],"example":[127],"an":[129,149],"appendix":[130],"spectral":[132],"illustrate":[134],"structures":[138],"can":[139],"be":[140],"viewed":[141],"in":[142,214,218],"different":[143],"mathematical":[144],"settings.":[145,227],"purpose":[147],"is":[148,229],"accessible,":[150],"checkable":[151],"established":[154],"results":[155],"their":[157],"relationships,":[158],"particular":[160],"attention":[161],"distinctions":[164],"among":[165],"dimension,":[166],"tensor":[167],"order,":[168],"representation,":[169],"dynamics.":[171],"\u672c\u7814\u7a76\u672d\u8a18\u5f9e\u5085\u7acb\u8449\u6a21\u614b\u8207":[172],"\u51fd\u6578\u55ae\u5143\u51fa\u767c,\u56de\u5230\u81ea\u7136\u6307\u6578\u3001\u7dda\u6027\u5fae\u5206\u65b9\u7a0b\u8207\u77e9\u9663\u751f\u6210\u5143,\u8aaa\u660e\u4e0d\u540c\u7684\u51fd\u6578\u8868\u793a\u5982\u4f55\u7531\u7c21\u55ae\u7684\u4f5c\u7528\u95dc\u4fc2\u5f62\u6210\u3002\u85c9\u7531\u6eff\u8db3":[174],"\u03b5I(\u03b5":[177],"\u22121\u30010\u3001+1)\u7684\u4e8c\u7dad\u751f\u6210\u5143,\u9010\u6b65\u63a8\u5c0e\u5713\u5468\u3001\u76f4\u7dda\u8207\u96d9\u66f2\u7dda\u6027\u6d41;\u518d\u4ee5\u5206\u91cf\u6bd4\u503c":[179],"\u4f5c\u70ba\u5c04\u5f71\u5ea7\u6a19,\u5f97\u5230":[181],"\u65b9\u7a0b\u53ca":[183],"tan(t)\u3001t\u3001tanh(t)":[184],"\u7b49\u975e\u7dda\u6027\u51fd\u6578\u3002\u540c\u6642\u6bd4\u8f03":[185],"Logistic":[186],"\u8868\u793a\u8207":[187],"boost":[189],"\u6240\u5c55\u73fe\u7684\u4e0d\u540c\u6578\u5b78\u8b80\u6cd5\u3002":[190],"\u5f8c\u7e8c\u4ee5":[191],"\u4e58\u7a4d\u5fae\u5206\u6cd5\u5247\u63a8\u5c0e\u5f35\u91cf\u7a4d\u4e0a\u7684\u8a98\u5c0e\u751f\u6210\u5143,\u8fa8\u660e\u5176\u8207\u8f38\u5165\u9ad8\u968e\u5c0e\u6578\u5f35\u91cf\u7684\u5340\u5225,\u4e26\u6bd4\u8f03\u5085\u7acb\u8449\u8207":[193],"\u7684\u5c0e\u6578\u7d50\u69cb\u3002\u795e\u7d93\u5b78\u7fd2\u8207\u983b\u8b5c\u8868\u793a\u50c5\u4f5c\u70ba\u5ef6\u4f38\u89c0\u5bdf\u7a97\u53e3\u3002\u5168\u6587\u5f37\u8abf\u57fa\u790e\u6578\u5b78\u7684\u9010\u6b65\u63a8\u5c0e\u8207\u8de8\u9818\u57df\u95dc\u4fc2\u7684\u6e05\u6670\u95e1\u91cb,\u4ee5\u5354\u52a9\u8b80\u8005\u7406\u89e3\u7dad\u5ea6\u3001\u5f35\u91cf\u968e\u6578\u3001\u8868\u793a\u65b9\u5f0f\u8207\u6f14\u5316\u4f5c\u7528\u4e4b\u9593\u7684\u5dee\u7570\u3002":[195],"Files:":[196],"English":[198],"PDF":[199],"ZIP":[202],"containing":[203],"LaTeX":[205],"source,":[206],"Markdown":[208],"manuscript":[209,234],"(with":[210],"one":[211],"notation":[212],"fix":[213],"Section":[215],"4.1,":[216],"noted":[217],"README),":[220],"PNG/SVG":[221],"figures,":[222],"revision":[223],"notes,":[224],"build":[226],"v1.1-en":[228],"Zenodo":[231],"release":[232],"version":[235],"v0.2-r2":[236],"(2026-11-03).":[237],"has":[240],"not":[241],"been":[242],"peer":[243],"reviewed.":[244]},"counts_by_year":[],"updated_date":"2026-11-05T12:25:35.769452","created_date":"2026-11-04T00:00:00"}
