{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,30]],"date-time":"2025-12-30T15:32:41Z","timestamp":1767108761719,"version":"build-2065373602"},"reference-count":61,"publisher":"MDPI AG","issue":"21","license":[{"start":{"date-parts":[[2023,11,4]],"date-time":"2023-11-04T00:00:00Z","timestamp":1699056000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001871","name":"Portuguese National Funds","doi-asserted-by":"publisher","award":["UIDB\/00066\/2020","UIDB\/50022\/2020"],"award-info":[{"award-number":["UIDB\/00066\/2020","UIDB\/50022\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Mathematics"],"abstract":"<jats:p>General variable-order fractional scale derivatives are introduced and studied. Both the stretching and the shrinking cases are considered for definitions of the derivatives of the GL type and of the Hadamard type. Their properties are deduced and discussed. Fractional variable-order systems of autoregressive\u2013moving-average type are introduced and exemplified. The corresponding transfer functions are obtained and used to find the corresponding impulse responses.<\/jats:p>","DOI":"10.3390\/math11214549","type":"journal-article","created":{"date-parts":[[2023,11,4]],"date-time":"2023-11-04T14:51:01Z","timestamp":1699109461000},"page":"4549","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Variable-Order Fractional Scale Calculus"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9388-4308","authenticated-orcid":false,"given":"Duarte","family":"Val\u00e9rio","sequence":"first","affiliation":[{"name":"IDMEC, Instituto Superior T\u00e9cnico, Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4270-3284","authenticated-orcid":false,"given":"Manuel D.","family":"Ortigueira","sequence":"additional","affiliation":[{"name":"NOVA School of Science and Technology, UNINOVA-CTS and LASI, NOVA University of Lisbon, Quinta da Torre, 2829-516 Caparica, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2023,11,4]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"4899","DOI":"10.1142\/S0217751X92002222","article-title":"The theory of scale relativity","volume":"7","author":"Nottale","year":"1992","journal-title":"Int. J. Mod. Phys."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"3275","DOI":"10.1109\/78.258073","article-title":"The scale representation","volume":"41","author":"Cohen","year":"1993","journal-title":"IEEE Trans. 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