{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:43:44Z","timestamp":1760233424431,"version":"build-2065373602"},"reference-count":43,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2021,1,18]],"date-time":"2021-01-18T00:00:00Z","timestamp":1610928000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>This paper explores how electroencephalography (EEG) signals in the Krohn-Rhodes form can be decomposed further using the Jordan-Chevalley decomposition technique. First, the recorded EEG signals of a seizure were transformed into a set of matrices. Each of these matrices was decomposed into its elementary components using the Krohn-Rhodes decomposition method. The components were then further decomposed into semisimple and nilpotent matrices using the Jordan-Chevalley decomposition. These matrices\u2014which are the extended building blocks of elementary EEG signals\u2014provide evidence that the EEG signals recorded during a seizure contain patterns similar to that of prime numbers.<\/jats:p>","DOI":"10.3390\/axioms10010010","type":"journal-article","created":{"date-parts":[[2021,1,19]],"date-time":"2021-01-19T04:55:31Z","timestamp":1611032131000},"page":"10","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Decomposing the Krohn-Rhodes Form of Electroencephalography (EEG) Signals Using Jordan-Chevalley Decomposition Technique"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0819-4922","authenticated-orcid":false,"given":"Amirul Aizad","family":"Ahmad Fuad","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, Skudai 81310, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2675-7952","authenticated-orcid":false,"given":"Tahir","family":"Ahmad","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, Skudai 81310, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,18]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"296","DOI":"10.1212\/WNL.0000000000003509","article-title":"Prevalence and incidence of epilepsy","volume":"88","author":"Fiest","year":"2017","journal-title":"Neurology"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"796","DOI":"10.1046\/j.1528-1157.2001.10401.x","article-title":"A proposed diagnostic scheme for people with epileptic seizures and with epilepsy: Report of the ILAE task force on classification and terminology","volume":"42","author":"Jerome","year":"2001","journal-title":"Epilepsia"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.jneumeth.2015.03.018","article-title":"Tensor decomposition of EEG signals: A brief review","volume":"248","author":"Cong","year":"2015","journal-title":"J. Neurosc. Methods"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Yakovleva, T.V., Kutepov, I.E., Krysko, A.V., Erofeev, N.P., Yaroshenko, T.Y., Saltykova, O.A., Kirichenko, A.V., Zhigalov, M.V., Papkova, I.V., and Krysko, V.A. (2019). Wavelet Analysis of EEG Signals in Epilepsy Patients, Atlantis Press.","DOI":"10.2991\/ahcs.k.191206.015"},{"key":"ref_5","unstructured":"Ahmad, T., Ahmad, R.S., Zakaria, F., and Yun, L.L. (2000). Development of detection model for neuromagnetic fields. Proceedings of the BIOMED 2000, University of Malaya."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1080\/09720502.2008.10700565","article-title":"Fuzzy topographic topological mapping for localisation simulated multiple current sources of MEG","volume":"11","author":"Ahmad","year":"2008","journal-title":"J. Interdiscip. Math."},{"key":"ref_7","first-page":"35","article-title":"Homeomorphisms of fuzzy topographic topological mapping (FTTM)","volume":"21","author":"Ahmad","year":"2005","journal-title":"MATEMATIKA Malays. J. Ind. Appl. Math."},{"key":"ref_8","unstructured":"Barja, A.O. (2015). Hierarchical Complexity of Krohn-Rhodes Decomposition of EEG Signals during Epileptic Seizure. [Ph.D. Thesis, Universiti Teknologi Malaysia]."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"540","DOI":"10.3844\/ajassp.2010.540.544","article-title":"EEG signals during epileptic seizure as a semigroup of upper triangular matrices","volume":"7","author":"Binjadhnan","year":"2010","journal-title":"Am. J. Appl. Sci."},{"key":"ref_10","first-page":"423","article-title":"Sequence of fuzzy topographic topological mapping","volume":"4","author":"Jamaian","year":"2008","journal-title":"Malays. J. Fundam. Appl. Sci."},{"key":"ref_11","first-page":"139","article-title":"On properties of the graph of fuzzy topographic topological mapping","volume":"9","author":"Sayed","year":"2013","journal-title":"Malays. J. Fundam. Appl. Sci."},{"key":"ref_12","unstructured":"Yun, L.L. (2006). Group-Like Algebraic Structures of Fuzzy Topographic Topological Mapping for Solving Neuromagnetic Inverse Problem. [Ph.D. Thesis, Universiti Teknologi Malaysia]."},{"key":"ref_13","first-page":"651","article-title":"Embodiment and memories\u2014Algebras of time and history for autobiographic agents","volume":"98","author":"Nehaniv","year":"1998","journal-title":"Cybern. Syst."},{"key":"ref_14","unstructured":"Binjadhnan, F.A.M. (2011). Krohn-Rhodes Decomposition for Electroencephalography Signals during Epileptic Seizure. [Ph.D. Thesis, Universiti Teknologi Malaysia]."},{"key":"ref_15","unstructured":"Rosen, K.H. (2014). Elementary Number Theory, Pearson Education."},{"key":"ref_16","first-page":"E4446","article-title":"Unexpected biases in the distribution of consecutive primes","volume":"113","author":"Robert","year":"2016","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"189","DOI":"10.4169\/math.mag.86.3.189","article-title":"Feedback, control, and the distribution of prime numbers","volume":"86","author":"Marshall","year":"2013","journal-title":"Math. Mag."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"093401","DOI":"10.1088\/1742-5468\/aad6be","article-title":"Uncovering multiscale order in the prime numbers via scattering","volume":"2018","author":"Torquato","year":"2018","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"135002","DOI":"10.1088\/1751-8121\/ab0588","article-title":"Hidden multiscale order in the primes","volume":"52","author":"Torquato","year":"2019","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"115001","DOI":"10.1088\/1751-8121\/aaa52a","article-title":"The structure factor of primes","volume":"51","author":"Zhang","year":"2018","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1016\/S0960-0779(03)00433-8","article-title":"Toward a dynamical model for prime numbers","volume":"20","author":"Bonanno","year":"2004","journal-title":"Chaos Solitons Fractals"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"23","DOI":"10.1016\/j.chaos.2007.10.017","article-title":"The distribution of prime numbers: The solution comes from dynamical processes and genetic algorithms","volume":"37","author":"Iovane","year":"2008","journal-title":"Chaos Solitons Fractals"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"950","DOI":"10.1016\/j.chaos.2007.12.004","article-title":"The set of prime numbers: Symmetries and supersymmetries of selection rules and asymptotic behaviours","volume":"37","author":"Iovane","year":"2008","journal-title":"Chaos Solitons Fractals"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1953","DOI":"10.1016\/j.chaos.2008.07.047","article-title":"The set of prime numbers: Multiscale analysis and numeric accelerators","volume":"41","author":"Iovane","year":"2009","journal-title":"Chaos Solitons Fractals"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1344","DOI":"10.1016\/j.chaos.2008.04.060","article-title":"The set of primes: Towards an optimized algorithm, prime generation and validation, and asymptotic consequences","volume":"41","author":"Iovane","year":"2009","journal-title":"Chaos Solitons Fractals"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"110029","DOI":"10.1016\/j.chaos.2020.110029","article-title":"Fractals and discrete dynamics associated to prime numbers","volume":"139","year":"2020","journal-title":"Chaos Solitons Fractals"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1002\/cplx.1040","article-title":"Prime number selection of cycles in a predator-prey model","volume":"6","author":"Goles","year":"2001","journal-title":"Complexity"},{"key":"ref_28","first-page":"7","article-title":"On Jordan-Chevalley decomposition","volume":"1","author":"Bajorska","year":"2011","journal-title":"Zeszyty Nauk. Mat. Stosow.\/Politech. \u015al\u0105ska"},{"key":"ref_29","unstructured":"Zakaria, F. (2008). Dynamic Profiling of EEG Data during Seizure Using Fuzzy Information Space. [Ph.D. Thesis, Universiti Teknologi Malaysia]."},{"key":"ref_30","first-page":"370","article-title":"The ten-twenty electrode system of the international federation","volume":"10","author":"Jasper","year":"1958","journal-title":"Electroencephalogr. Clin. Neurophysiol."},{"key":"ref_31","first-page":"3","article-title":"The ten-twenty electrode system of the international federation. the internanional federation of clinical nenrophysiology","volume":"52","author":"Klem","year":"1999","journal-title":"Electroencephalogr. Clin. Neurophysiol. Suppl."},{"key":"ref_32","first-page":"1","article-title":"EEG signal classification for brain computer interface applications","volume":"7","author":"Ochoa","year":"2002","journal-title":"Ecole Polytech. Fed. Lausanne"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Malmivuo, J., and Plonsey, R. (1995). Bioelectromagnetism: Principles and Applications of Bioelectric and Biomagnetic Fields, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780195058239.001.0001"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Robu, A.D., Salge, C., Nehaniv, C.L., and Polani, D. (2017). Time as it could be measured in artificial living systems. Artificial Life Conference Proceedings, MIT Press.","DOI":"10.7551\/ecal_a_061"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"450","DOI":"10.1090\/S0002-9947-1965-0188316-1","article-title":"Algebraic theory of machines. I. Prime decomposition theorem for finite semigroups and machines","volume":"116","author":"Krohn","year":"1965","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"128","DOI":"10.2307\/1970558","article-title":"Complexity of finite semigroups","volume":"88","author":"Krohn","year":"1968","journal-title":"Ann. Math."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1016\/0022-4049(84)90092-6","article-title":"Almost finite expansions of arbitrary semigroups","volume":"32","author":"Birget","year":"1984","journal-title":"J. Pure Appl. Algebra"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"791","DOI":"10.1142\/S0218196702001206","article-title":"Holonomy embedding of arbitrary stable semigroups","volume":"12","author":"Elston","year":"2002","journal-title":"Int. J. Algebra Comput."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"127","DOI":"10.1016\/0022-4049(88)90043-6","article-title":"Prime decomposition theorem for arbitrary semigroups: General holonomy decomposition and synthesis theorem","volume":"55","author":"Henckell","year":"1988","journal-title":"J. Pure Appl. Algebra"},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Kambites, M., and Steinberg, B. (2007). Wreath Product Decompositions for Triangular Matrix Semigroups, World Scientific.","DOI":"10.1142\/9789812708700_0010"},{"key":"ref_41","doi-asserted-by":"crossref","unstructured":"Howie, J.M. (1995). Fundamentals of Semigroup Theory, Oxford University Press. Number 12.","DOI":"10.1093\/oso\/9780198511946.001.0001"},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Carrell, J.B. (2017). The Structure Theory of Linear Mappings, Springer.","DOI":"10.1007\/978-0-387-79428-0_10"},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Axler, S.J. 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