Bibliographies thématiques / Time-Scales calculus

Littérature scientifique sur le sujet « Time-Scales calculus »

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Publié le 25 mai 2024

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Articles de revues sur le sujet "Time-Scales calculus"

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Khan, A. R., F. Mehmood et M. A. Shaikh. « Обобщение неравенств Островского на временных шкалах ». Владикавказский математический журнал 25, no 3 (25 septembre 2023) : 98–110. http://dx.doi.org/10.46698/q4172-3323-1923-j.

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The idea of time scales calculus’ theory was initiated and introduced by Hilger (1988) in his PhD thesis order to unify discret and continuous analysis and to expend the discrete and continous theories to cases ``in between''. Since then, mathematical research in this field has exceeded more than 1000 publications and a lot of applications in the fields of science, i.e., operations research, economics, physics, engineering, statistics, finance and biology. Ostrowski proved an inequality to estimate the absolute deviation of a differentiable function from its integral mean. This result was obtained by applying the Montgomery identity. In the present paper we derive a generalization of the Montgomery identity to the various time scale versions such as discrete case, continuous case and the case of quantum calculus, by obtaining this generalization of Montgomery identity we would prove our results about the generalization of the Ostrowski inequalities (without weighted case) to the several time scales such as discrete case, continuous case and the case of quantum calculus and recapture the several published results of different authors of various papers and thus unify corresponding discrete version and continuous version. Similarly we would also derive our results about the generalization of the Ostrowski inequalities (weighted case) to the different time scales such as discrete case and continuous case and recapture the different published results of several authors of various papers and thus unify corresponding discrete version and continuous version. Moreover, we would use our obtained results (without weighted case) to the case of quantum calculus.
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Torrest, Delfim F. M. « The variational calculus on time scales ». International Journal for Simulation and Multidisciplinary Design Optimization 4, no 1 (janvier 2010) : 11–25. http://dx.doi.org/10.1051/ijsmdo/2010003.

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Yaslan, İsmail. « Beta-Fractional Calculus on Time Scales ». Journal of Fractional Calculus and Nonlinear Systems 4, no 2 (27 décembre 2023) : 48–60. http://dx.doi.org/10.48185/jfcns.v4i2.877.

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Sahir, Muhammad Jibril Shahab. « Uniformity of dynamic inequalities constituted on time Scales ». Engineering and Applied Science Letters 3, no 4 (24 octobre 2020) : 19–27. http://dx.doi.org/10.30538/psrp-easl2020.0048.

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In this article, we present extensions of some well-known inequalities such as Young's inequality and Qi's inequality on fractional calculus of time scales. To find generalizations of such types of dynamic inequalities, we apply the time scale Riemann-Liouville type fractional integrals. We investigate dynamic inequalities on delta calculus and their symmetric nabla results. The theory of time scales is utilized to combine versions in one comprehensive form. The calculus of time scales unifies and extends some continuous forms and their discrete and quantum inequalities. By applying the calculus of time scales, results can be generated in more general form. This hybrid theory is also extensively practiced on dynamic inequalities.
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Malinowska, Agnieszka B., et Natália Martins. « The Second Noether Theorem on Time Scales ». Abstract and Applied Analysis 2013 (2013) : 1–14. http://dx.doi.org/10.1155/2013/675127.

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We extend the second Noether theorem to variational problems on time scales. As corollaries we obtain the classical second Noether theorem, the second Noether theorem for theh-calculus and the second Noether theorem for theq-calculus.
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Sahir, Muhammad Jibril Shahab. « Coordination of Classical and Dynamic Inequalities Complying on Time Scales ». European Journal of Mathematical Analysis 3 (3 février 2023) : 12. http://dx.doi.org/10.28924/ada/ma.3.12.

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In this research article, we present extensions of some classical inequalities such as Schweitzer, Pólya–Szegö, Kantorovich and Greub–Rheinboldt inequalities of fractional calculus on time scales. To investigate generalizations of such types of classical inequalities, we use the time scales Riemann–Liouville type fractional integrals. We explore dynamic inequalities on delta calculus and their symmetric nabla versions. A time scale is an arbitrary nonempty closed subset of the real numbers. The theory of time scales is applied to combine results in one comprehensive form. The calculus of time scales unifies and extends continuous versions and their discrete and quantum analogues. By using the calculus of time scales, results are presented in more general form. This hybrid theory is also widely applied on dynamic inequalities.
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Ganie, Javid Ahmad, et Renu Jain. « THE SUMUDU TRANSFORM ON DISCRETE TIME SCALES ». Jnanabha 51, no 02 (2021) : 58–67. http://dx.doi.org/10.58250/jnanabha.2021.51208.

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The study of dynamic equations on time scale is an area of mathematics that has recently received a lot of attention. This type of calculus has been created in order to unify the study of discrete and continuous analysis. Integral transforms play a crucial role in analysis in solving differential and difference equations. In this paper, we introduce the Sumudu transform on two different time scales by using the theory of time scale calculus. Finally, we employ these definitions to derive the results like, convolution, delay(shift) and inversion on these discrete time scales. Further, these results coincide with those in continuous case.
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Sahir, M. J. S. « Объединение классических и динамических неравенств, возникающих при анализе временных масштабов ». Вестник КРАУНЦ. Физико-математические науки, no 4 (29 décembre 2020) : 26–36. http://dx.doi.org/10.26117/2079-6641-2020-33-4-26-36.

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In this paper, we present an extension of dynamic Renyi’s inequality on time scales by using the time scale Riemann–Liouville type fractional integral. Furthermore, we find generalizations of the well–known Lyapunov’s inequality and Radon’s inequality on time scales by using the time scale Riemann–Liouville type fractional integrals. Our investigations unify and extend some continuous inequalities and their corresponding discrete analogues. В этой статье мы представляем расширение динамического неравенства Реньи на шкалы времени с помощью дробного интеграла типа Римана-Лиувилля. Кроме того, мы находим обобщения хорошо известного неравенства Ляпунова и неравенства Радона на шкалах времени с помощью дробных интегралов типа Римана-Лиувилля на шкале. Наши исследования объединяют и расширяют некоторые непрерывные неравенства и соответствующие им дискретные аналоги.
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Zhao, Dafang, et Tongxing Li. « On conformable delta fractional calculus on time scales ». Journal of Mathematics and Computer Science 16, no 03 (15 septembre 2016) : 324–35. http://dx.doi.org/10.22436/jmcs.016.03.03.

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Seiffertt, John. « Adaptive Resonance Theory in the time scales calculus ». Neural Networks 120 (décembre 2019) : 32–39. http://dx.doi.org/10.1016/j.neunet.2019.08.010.

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Thèses sur le sujet "Time-Scales calculus"

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Ferreira, Rui Alexandre Cardoso. « Calculus of variations on time scales and discrete fractional calculus ». Doctoral thesis, Universidade de Aveiro, 2010. http://hdl.handle.net/10773/2921.

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Doutoramento em Matemática
Estudamos problemas do cálculo das variações e controlo óptimo no contexto das escalas temporais. Especificamente, obtemos condições necessárias de optimalidade do tipo de Euler–Lagrange tanto para lagrangianos dependendo de derivadas delta de ordem superior como para problemas isoperimétricos. Desenvolvemos também alguns métodos directos que permitem resolver determinadas classes de problemas variacionais através de desigualdades em escalas temporais. No último capítulo apresentamos operadores de diferença fraccionários e propomos um novo cálculo das variações fraccionário em tempo discreto. Obtemos as correspondentes condições necessárias de Euler– Lagrange e Legendre, ilustrando depois a teoria com alguns exemplos.
We study problems of the calculus of variations and optimal control within the framework of time scales. Specifically, we obtain Euler–Lagrange type equations for both Lagrangians depending on higher order delta derivatives and isoperimetric problems. We also develop some direct methods to solve certain classes of variational problems via dynamic inequalities. In the last chapter we introduce fractional difference operators and propose a new discrete-time fractional calculus of variations. Corresponding Euler–Lagrange and Legendre necessary optimality conditions are derived and some illustrative examples provided.
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Dryl, Monika. « Calculus of variations on time scales and applications to economics ». Doctoral thesis, Universidade de Aveiro, 2014. http://hdl.handle.net/10773/12869.

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Doutoramento em Matemática
We consider some problems of the calculus of variations on time scales. On the beginning our attention is paid on two inverse extremal problems on arbitrary time scales. Firstly, using the Euler-Lagrange equation and the strengthened Legendre condition, we derive a general form for a variation functional that attains a local minimum at a given point of the vector space. Furthermore, we prove a necessary condition for a dynamic integro-differential equation to be an Euler-Lagrange equation. New and interesting results for the discrete and quantum calculus are obtained as particular cases. Afterwards, we prove Euler-Lagrange type equations and transversality conditions for generalized infinite horizon problems. Next we investigate the composition of a certain scalar function with delta and nabla integrals of a vector valued field. Euler-Lagrange equations in integral form, transversality conditions, and necessary optimality conditions for isoperimetric problems, on an arbitrary time scale, are proved. In the end, two main issues of application of time scales in economic, with interesting results, are presented. In the former case we consider a firm that wants to program its production and investment policies to reach a given production rate and to maximize its future market competitiveness. The model which describes firm activities is studied in two different ways: using classical discretizations; and applying discrete versions of our result on time scales. In the end we compare the cost functional values obtained from those two approaches. The latter problem is more complex and relates to rate of inflation, p, and rate of unemployment, u, which inflict a social loss. Using known relations between p, u, and the expected rate of inflation π, we rewrite the social loss function as a function of π. We present this model in the time scale framework and find an optimal path π that minimizes the total social loss over a given time interval.
Consideramos alguns problemas do cálculo das variações em escalas temporais. Primeiramente, demonstramos equações do tipo de Euler-Lagrange e condições de transversalidade para problemas de horizonte infinito generalizados. De seguida, consideramos a composição de uma certa função escalar com os integrais delta e nabla de um campo vetorial. Presta-se atenção a problemas extremais inversos para funcionais variacionais em escalas de tempo arbitrárias. Começamos por demonstrar uma condição necessária para uma equação dinâmica integro-diferencial ser uma equação de Euler-Lagrange. Resultados novos e interessantes para o cálculo discreto e quantum são obtidos como casos particulares. Além disso, usando a equação de Euler-Lagrange e a condição de Legendre fortalecida, obtemos uma forma geral para uma funcional variacional atingir um mínimo local, num determinado ponto do espaço vetorial. No final, duas aplicações interessantes em termos económicos são apresentadas. No primeiro caso, consideramos uma empresa que quer programar as suas políticas de produção e de investimento para alcançar uma determinada taxa de produção e maximizar a sua competitividade no mercado futuro. O outro problema é mais complexo e relaciona a inflação e o desemprego, que inflige uma perda social. A perda social é escrita como uma função da taxa de inflação p e a taxa de desemprego u, com diferentes pesos. Em seguida, usando as relações conhecidas entre p, u, e a taxa de inflação esperada π, reescrevemos a função de perda social como uma função de π. A resposta é obtida através da aplicação do cálculo das variações, a fim de encontrar a curva ótima π que minimiza a perda social total ao longo de um determinado intervalo de tempo.
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Bastos, Nuno Rafael de Oliveira. « Fractional calculus on time scales - Cálculo fraccional em escalas temporais ». Doctoral thesis, Universidade de Aveiro, 2012. http://hdl.handle.net/10773/8566.

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Doutoramento em Matemática
Introduzimos um cálculo das variações fraccional nas escalas temporais ℤ e (hℤ)!. Estabelecemos a primeira e a segunda condição necessária de optimalidade. São dados alguns exemplos numéricos que ilustram o uso quer da nova condição de Euler–Lagrange quer da nova condição do tipo de Legendre. Introduzimos também novas definições de derivada fraccional e de integral fraccional numa escala temporal com recurso à transformada inversa generalizada de Laplace.
We introduce a discrete-time fractional calculus of variations on the time scales ℤ and (ℎℤ)!. First and second order necessary optimality conditions are established. Some numerical examples illustrating the use of the new Euler— Lagrange and Legendre type conditions are given. We also give new definitions of fractional derivatives and integrals on time scales via the inverse generalized Laplace transform.
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McMahon, Chris. « Calculus of Variations on Time Scales and Its Applications to Economics ». TopSCHOLAR®, 2008. http://digitalcommons.wku.edu/theses/370.

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The goal of time scale research is to progress the development of a harmonized theory that is all encompassing of the more commonly known specialized forms. The main results of this paper is the presentation of the Ramsey model which can be written using both the A and V operators, and solved using the two separate theories of the calculus of variations on time scales. The next presentation will be of the solution of an adjustment model, for a specific form of a time scale, whose functional can only be optimized, using the existing theory, when written with the A operator. We will also develop certain elements of stochastic time scale calculus, in order to lay the groundwork necessary to develop the theory of stochastic calculus of variations on time scales.
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Hariz, Belgacem Khader. « Higher-order Embedding Formalism, Noether’s Theorem on Time Scales and Eringen’s Nonlocal Elastica ». Electronic Thesis or Diss., Pau, 2022. https://theses.hal.science/tel-03981833.

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En mathématiques, le calcul des variations est un ensemble de méthodes permettant la détermination de solutions à des problèmes d'optimisation des quantités traduites en termes de fonctionnelle. De nombreuses applications existent, notamment dans la recherche de courbes ou de surfaces minimales. Les systèmes dynamiques considérés sont de natures diverses (équations différentielles, intégrales ou stochastiques) et modélisent des problèmes d'origines multiples : aérospatiale, automobile, biologie, économie, médecine, etc. Le théorème de Noether présente un fort intérêt puisqu'il propose une loi de conservation explicite (traduisant souvent une quantité physique comme l'énergie totale ou le moment angulaire en mécanique classique) qui permet de réduire ou d'intégrer l'équation différentielle associée par quadrature. L'objectif de ma thèse contient de nombreux thèmes, dans le premier but nous allons : *) donner le théorème de Noether discret dans le cadre ”time scale” (Le formalisme lagrangien et hamiltonien). Le passage de la nature discrète à la nature continue de la structure la morphologie est d'un intérêt primordial en physique pour comprendre comment la microstructure peut influencer les propriétés macroscopiques du matériau à plus grande échelle. Ce passage peut être modélisé par un système discret appelé 'Hencky's chain' et l'équation du mouvement est donnée par des équations aux différences non linéaires et cette équation ne possède pas de Lagrangien. Le deuxième but nous allons : *) donner les structures lagrangienne, hamiltonienne via le facteur intégrant et trouver la solution analytique de l'équation non locale au sens d'Eringen (nonlocalité différentielle d'Eringen, 1983). Le troisième but nous allons : *) étudier l'existence des formulations variationnelles via le principe de Brezis Ekeland-Nayroles (Gery de Saxce) - application sur la formulation 4D développée par E. Rouhaud pour l'étude des déformations des matériaux *) développer des schémas numériques qui respectent certaines particularités. En particulier, un schéma permettent de mettre en œuvre la théorie 4D développée par E. Rouhaud. *) applications numériques et théoriques sur le problème des déformations des matériaux
The aim of this thesis is to deal with the connection between continuous and discrete versions of a given object. This connection can be studied in two different directions: one going from a continuous setting to a discrete analogue, and in a symmetric way, from a discrete setting to a continuous one. The first procedure is typically used in numerical analysis in order to construct numerical integrators and the second one is typical of continuous modeling for the study of micro-structured materials.In this manuscript, we focus our attention on three distinct problems. In the first part, we propose a general framework precising different ways to derive a discrete version of a differential equation called discrete embedding formalism.More precisely, we exhibit three main discrete associate: the differential, integral or variational structure in both classical and high-order approximations.The second part focuses on the preservation of symmetries for discrete versions of Lagrangian and Hamiltonian systems, i.e., the discrete analogue of Noether's theorem.Finally, the third part applies these results in mechanics, i.e., the problem studied by N. Challamel, Kocsis and Wang called Eringen's nonlocal elastica equation which can beobtained by the continualization method. Precisely, we construct a discrete version of Eringen's nonlocal elastica then we study the difference with Challamel's proposal
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Arslan, Aykut. « Discrete Fractional Hermite-Hadamard Inequality ». TopSCHOLAR®, 2017. http://digitalcommons.wku.edu/theses/1940.

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This thesis is comprised of three main parts: The Hermite-Hadamard inequality on discrete time scales, the fractional Hermite-Hadamard inequality, and Karush-Kuhn- Tucker conditions on higher dimensional discrete domains. In the first part of the thesis, Chapters 2 & 3, we define a convex function on a special time scale T where all the time points are not uniformly distributed on a time line. With the use of the substitution rules of integration we prove the Hermite-Hadamard inequality for convex functions defined on T. In the fourth chapter, we introduce fractional order Hermite-Hadamard inequality and characterize convexity in terms of this inequality. In the fifth chapter, we discuss convexity on n{dimensional discrete time scales T = T1 × T2 × ... × Tn where Ti ⊂ R , i = 1; 2,…,n are discrete time scales which are not necessarily periodic. We introduce the discrete analogues of the fundamental concepts of real convex optimization such as convexity of a function, subgradients, and the Karush-Kuhn-Tucker conditions. We close this thesis by two remarks for the future direction of the research in this area.
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Kisela, Tomáš. « Basics of Qualitative Theory of Linear Fractional Difference Equations ». Doctoral thesis, Vysoké učení technické v Brně. Fakulta strojního inženýrství, 2012. http://www.nusl.cz/ntk/nusl-234025.

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Tato doktorská práce se zabývá zlomkovým kalkulem na diskrétních množinách, přesněji v rámci takzvaného (q,h)-kalkulu a jeho speciálního případu h-kalkulu. Nejprve jsou položeny základy teorie lineárních zlomkových diferenčních rovnic v (q,h)-kalkulu. Jsou diskutovány některé jejich základní vlastnosti, jako např. existence, jednoznačnost a struktura řešení, a je zavedena diskrétní analogie Mittag-Lefflerovy funkce jako vlastní funkce operátoru zlomkové diference. Dále je v rámci h-kalkulu provedena kvalitativní analýza skalární a vektorové testovací zlomkové diferenční rovnice. Výsledky analýzy stability a asymptotických vlastností umožňují vymezit souvislosti s jinými matematickými disciplínami, např. spojitým zlomkovým kalkulem, Volterrovými diferenčními rovnicemi a numerickou analýzou. Nakonec je nastíněno možné rozšíření zlomkového kalkulu na obecnější časové škály.
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Livres sur le sujet "Time-Scales calculus"

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Bohner, Martin, et Svetlin G. Georgiev. Multivariable Dynamic Calculus on Time Scales. Cham : Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-47620-9.

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Georgiev, Svetlin G. Fractional Dynamic Calculus and Fractional Dynamic Equations on Time Scales. Cham : Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-73954-0.

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Variational Calculus on Time Scales. Nova Science Publishers, Incorporated, 2018.

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Georgiev, Svetlin G. Variational Calculus on Time Scales. Nova Science Publishers, Incorporated, 2018.

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Bohner, Martin, et Svetlin G. Georgiev. Multivariable Dynamic Calculus on Time Scales. Springer, 2017.

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Bohner, Martin, et Svetlin G. Georgiev. Multivariable Dynamic Calculus on Time Scales. Springer, 2017.

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Bohner, Martin, et Svetlin G. Georgiev. Multivariable Dynamic Calculus on Time Scales. Springer, 2018.

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Hardy Type Inequalities on Time Scales. Springer, 2016.

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O'Regan, Donal, Ravi P. Agarwal et Samir H. Saker. Hardy Type Inequalities on Time Scales. Springer, 2016.

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O'Regan, Donal, Ravi P. Agarwal et Samir H. Saker. Hardy Type Inequalities on Time Scales. Springer, 2018.

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Chapitres de livres sur le sujet "Time-Scales calculus"

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Bohner, Martin, et Svetlin G. Georgiev. « Time Scales ». Dans Multivariable Dynamic Calculus on Time Scales, 1–22. Cham : Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-47620-9_1.

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Seiffertt, John, et Donald C. Wunsch. « The Time Scales Calculus ». Dans Evolutionary Learning and Optimization, 49–60. Berlin, Heidelberg : Springer Berlin Heidelberg, 2010. http://dx.doi.org/10.1007/978-3-642-03180-9_4.

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Georgiev, Svetlin G. « Calculus on Time Scales ». Dans Functional Dynamic Equations on Time Scales, 1–36. Cham : Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-15420-2_1.

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Bohner, Martin, et Allan Peterson. « The Time Scales Calculus ». Dans Dynamic Equations on Time Scales, 1–50. Boston, MA : Birkhäuser Boston, 2001. http://dx.doi.org/10.1007/978-1-4612-0201-1_1.

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Goodrich, Christopher, et Allan C. Peterson. « Calculus on Mixed Time Scales ». Dans Discrete Fractional Calculus, 353–414. Cham : Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-25562-0_5.

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Georgiev, Svetlin G. « Elements of the Time Scale Calculus ». Dans Integral Equations on Time Scales, 1–75. Paris : Atlantis Press, 2016. http://dx.doi.org/10.2991/978-94-6239-228-1_1.

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Bohner, Martin, et Svetlin G. Georgiev. « Partial Differentiation on Time Scales ». Dans Multivariable Dynamic Calculus on Time Scales, 303–447. Cham : Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-47620-9_6.

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Bohner, Martin, et Svetlin G. Georgiev. « Multiple Integration on Time Scales ». Dans Multivariable Dynamic Calculus on Time Scales, 449–515. Cham : Springer International Publishing, 2016. http://dx.doi.org/10.1007/978-3-319-47620-9_7.

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Bohner, Martin, Gusein Guseinov et Allan Peterson. « Introduction to the Time Scales Calculus ». Dans Advances in Dynamic Equations on Time Scales, 1–15. Boston, MA : Birkhäuser Boston, 2003. http://dx.doi.org/10.1007/978-0-8176-8230-9_1.

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Georgiev, Svetlin G. « Convolution on Time Scales ». Dans Fractional Dynamic Calculus and Fractional Dynamic Equations on Time Scales, 157–215. Cham : Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-73954-0_3.

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Actes de conférences sur le sujet "Time-Scales calculus"

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Kapcak, Sinan, et Ünal Ufuktepe. « Multivariable Calculus on Time Scales ». Dans 2011 International Conference on Computational Science and Its Applications (ICCSA). IEEE, 2011. http://dx.doi.org/10.1109/iccsa.2011.28.

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Girejko, Ewa, Agnieszka B. Malinowska et Delfim F. M. Torres. « A unified approach to the calculus of variations on time scales ». Dans 2010 Chinese Control and Decision Conference (CCDC). IEEE, 2010. http://dx.doi.org/10.1109/ccdc.2010.5498972.

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Niu, Haoyu, YangQuan Chen, Lihong Guo et Bruce J. West. « A New Triangle : Fractional Calculus, Renormalization Group, and Machine Learning ». Dans ASME 2021 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2021. http://dx.doi.org/10.1115/detc2021-70505.

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Abstract The emergence of the systematic study of complexity as a science has resulted from the growing recognition that the fundamental assumptions upon which Newtonian physics is based are not satisfied throughout most of science, e.g., time is not necessarily uniformly flowing in one direction, nor is space homogeneous. Herein we discuss how the fractional calculus (FC), renormalization group (RG) theory and machine learning (ML) have each developed independently in the study of distinct phenomena in which one or more of the underlying assumptions of Newtonian formalism is violated. FC has been shown to help us better understand complex systems, improve the processing of complex signals, enhance the control of complex networks, increase optimization performance, and even extend the enabling of the potential for creativity. RG allows one to investigate the changes of a dynamical system at different scales. For example, in quantum field theory, divergent parts of a calculation can lead to nonsensical infinite results. However, by applying RG, the divergent parts can be adsorbed into fewer measurable quantities, yielding finite results. To date, ML is a fashionable research topic and will probably remain so into the foreseeable future. How a model can learn efficiently (optimally) is always essential. The key to learnability is designing efficient optimization methods. Although extensive research has been carried out on the three topics separately, few studies have investigated the association triangle between the FC, RG, and ML. To initiate the study of their interdependence, herein the authors discuss the critical connections between them (Fig. 1). In the FC and RG, scaling laws reveal the complexity of the phenomena discussed. The authors emphasize that the FC’s and RG’s critical connection is the form of inverse power laws (IPL), and the IPL index provides a measure of the level of complexity. For FC and ML, the critical connections in big data, wherein variability, optimization, and non-local models are described. The authors introduce the derivative-free and gradient-based optimization methods and explain how the FC could contribute to these study areas. In the end, the association between the RG and ML is also explained. The mutual information, feature extraction, and locality are also discussed. Many of the cross-sectional studies suggest a connection between the RG and ML. The RG has a superficial similarity to deep neural networks (DNNs) structure in which one marginalizes over hidden degrees of freedom. The authors remark in the conclusions that the association triangle between FC, RG, and ML, form a stool on which the foundation to complexity science might comfortably sit for a wide range of future research topics.
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