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In this paper, we study existence of solutions to the scalar additive Jump problem and the Riemann boundary value problems in the context of vectorial Clifford analysis on domains with fractal boundaries. A reduction procedure is applied with great effectiveness to find the solution of the problems. [ABSTRACT FROM AUTHOR]

Several new network information dimension definitions have been proposed in recent decades, expanding the scope of applicability of this seminal tool. This paper proposes a new definition based on Deng entropy and d-summability (a concept from geometric measure theory). We will prove to what extent the new formulation will be useful in the theoretical and applied points of view. [ABSTRACT FROM AUTHOR]

In this paper, we combine the fractional ψ−$$ \psi - $$hyperholomorphic function theory with the fractional calculus with respect to another function. As a main result, a fractional Borel–Pompeiu type formula related to a fractional ψ−$$ \psi - $$Fueter operator with respect to a vector‐valued function is proved. [ABSTRACT FROM AUTHOR]

This paper is a continuation of our work [J. O. González Cervantes and J. Bory Reyes, A quaternionic fractional Borel–Pompeiu type formula, Fractal30(1) (2022) 2250013], where we introduced a fractional operator calculus related to a fractional ψ -Fueter operator in the one-dimensional Riemann–Liouville derivative sense in each direction of the quaternionic structure, that depends on an additional vector of complex parameters with fractional real parts. This allowed us also to study a pair of lower order fractional operators and prove the associated analogues of both Stokes and Borel–Pompieu formulas for holomorphic functions in two complex variables. [ABSTRACT FROM AUTHOR]

Decision trees are decision support data mining tools that create, as the name suggests, a tree-like model. The classical C4.5 decision tree, based on the Shannon entropy, is a simple algorithm to calculate the gain ratio and then split the attributes based on this entropy measure. Tsallis and Renyi entropies (instead of Shannon) can be employed to generate a decision tree with better results. In practice, the entropic index parameter of these entropies is tuned to outperform the classical decision trees. However, this process is carried out by testing a range of values for a given database, which is time-consuming and unfeasible for massive data. This paper introduces a decision tree based on a two-parameter fractional Tsallis entropy. We propose a constructionist approach to the representation of databases as complex networks that enable us an efficient computation of the parameters of this entropy using the box-covering algorithm and renormalization of the complex network. The experimental results support the conclusion that the two-parameter fractional Tsallis entropy is a more sensitive measure than parametric Renyi, Tsallis, and Gini index precedents for a decision tree classifier. [ABSTRACT FROM AUTHOR]

In theoretical setting, associated with a fractional ψ -Fueter operator that depends on an additional vector of complex parameters with fractional real parts, this paper establishes a fractional analog of Borel–Pompeiu formula as a first step to develop a fractional ψ -hyperholomorphic function theory and the related operator calculus. [ABSTRACT FROM AUTHOR]

This article deals with the study of electromagnetic waves equations and the Lorentz condition, as emergent properties of Maxwell's system in the context of systems theory. To do this, the wave equations and the Helmholtz equation are first deduced. Using the displaced Dirac operator, which is closely related to the main vector calculation operators, it is possible to establish a direct connection between the solutions of the Maxwell time-harmonic system and two quaternion equations. Also, the application of the Lorentz condition to transform the time-harmonic Maxwell system into a simple quaternion equation based on the scalar and vector potentials is exposed. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to combine the (q,q′)$$ \left(q,{q}^{\prime}\right) $$‐calculus in the quaternionic context, which is proposed via two kinds of (q,q′)$$ \left(q,{q}^{\prime}\right) $$‐operators, with the theory of slice regular functions. Specifically, we shall work in suitable subclasses of slice regular functions in which the (q,q′)$$ \left(q,{q}^{\prime}\right) $$‐operators can be related with the slice derivative. The paper presents some results such as an integral formula and series expansion. [ABSTRACT FROM AUTHOR]

In this paper, a new algorithm to select the relevant nodes — those that maintain the cohesion of the network — of the complex network is presented. The experiments on most of the real complex networks show that the proposed approach outperforms centrality measures as node degree, PageRank algorithm and betweenness centrality. The rationale of the algorithm for extracting relevant nodes is to discover the self-similarity of the network. As seen in the algorithm, throughout the extraction sequence of relevant nodes, differences are advised with node degree, PageRank algorithm and betweenness centrality. Finally, empirical evidence is considered to show that complex network robustness is a nonlinear function of the small-worldness measure. [ABSTRACT FROM AUTHOR]

The Moisil‐Teodorescu operator is considered to be a good analogue of the usual Cauchy–Riemann operator of complex analysis in the framework of quaternionic analysis and it is a square root of the scalar Laplace operator in ℝ3. In the present work, a general quaternionic structure is developed for the local fractional Moisil–Teodorescu operator in Cantor‐type cylindrical and spherical coordinate systems. Furthermore, in order to reveal the capacity and adaptability of the methods, we show two examples for the Helmholtz equation with local fractional derivatives on the Cantor sets by making use of the local fractional Moisil–Teodorescu operator. [ABSTRACT FROM AUTHOR]

The paper provides integral representations for solutions to a certain first order partial differential equation natural arising in the factorization of the Lamé—Navier system with the help of Clifford analysis techniques. These representations look like in spirit to the Borel—Pompeiu and Cauchy integral formulas both in three and higher dimensional setting. [ABSTRACT FROM AUTHOR]

We construct new examples of parametric iterated function systems converging to some fractal shapes. The main goal is the study of the continuous growth and the rate of change of the attractor of the corresponding parametrization. [ABSTRACT FROM AUTHOR]

In this paper, we present examples of one-parametric iterated function systems converging to the standard middle-third Cantor set. The main goal is the study of the continuous growth of the attractor of the corresponding parametrization. [ABSTRACT FROM AUTHOR]

Iterated function systems provide the most fundamental framework to create many fascinating fractal sets. They have been extensively studied when the functions are affine transformations of Euclidean spaces. This paper investigates the iterated function systems consisting of affine transformations of the hyperbolic number plane. We show that the basics results of the classical Hutchinson–Barnsley theory can be carried over to construct fractal sets on hyperbolic number plane as its unique fixed point. We also discuss about the notion of hyperbolic derivative of an hyperbolic-valued function and then we use this notion to get some generalization of cookie-cutter Cantor sets in the real line to the hyperbolic number plane. [ABSTRACT FROM AUTHOR]

We obtain some analogues of the Hilbert formulas on the unit circle for a-hyperholomorphic function theory when a is a complex number. Such formulas relate a pair of components of the boundary value of an a -hyperholomorphic function in the unit circle with the other one. Furthermore, the corresponding Poincaré-Bertrand formula for the a - hyperholomorphic singular integrals in the unit circle is presented. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the construction of new Cantor like sets in the hyperbolic plane. Also, we study the arithmetic sum of two of these Cantor like sets, as well as of those previously introduced in the literature. An hyperbolization, in the sense of Gromov, of the commutative ring of hyperbolic numbers is also given. Finally, we present the construction of a Cantor-type set as hyperbolic boundary. [ABSTRACT FROM AUTHOR]

Simple Summary: The fractional ( q , q ′ )-information dimension for complex networks is introduce, and a dual version of the ( q , q ′ )-entropy, called ( q , q ′ )-extropy, is proposed. Experiments reveal that the fractional ( q , q ′ )-information dimension is less than the classical one (based on Shannon entropy) for both real-world and synthetic networks. This article introduces a new fractional approach to the concept of information dimensions in complex networks based on the ( q , q ′ )-entropy proposed in the literature. The q parameter measures how far the number of sub-systems (for a given size ε) is from the mean number of overall sizes, whereas q ′ (the interaction index) measures when the interactions between sub-systems are greater ( q ′ > 1 ), lesser ( q ′ < 1 ), or equal to the interactions into these sub-systems. Computation of the proposed information dimension is carried out on several real-world and synthetic complex networks. The results for the proposed information dimension are compared with those from the classic information dimension based on Shannon entropy. The obtained results support the conjecture that the fractional ( q , q ′ )-information dimension captures the complexity of the topology of the network better than the information dimension. [ABSTRACT FROM AUTHOR]

Let Ω⊂ℝm$$ \Omega \subset {\mathrm{\mathbb{R}}}^m $$ be a bounded regular domain, let ∂x_$$ {\partial}_{\underset{\_}{x}} $$ be the standard Dirac operator in ℝm$$ {\mathrm{\mathbb{R}}}^m $$, and let ℝ0,m$$ {\mathrm{\mathbb{R}}}_{0,m} $$ be the Clifford algebra constructed over the quadratic space ℝ0,m$$ {\mathrm{\mathbb{R}}}^{0,m} $$. For k∈{1,...,m}$$ k\in \left\{1,\dots, m\right\} $$ fixed, ℝ0,m(k)$$ {\mathrm{\mathbb{R}}}_{0,m}^{(k)} $$ denotes the space of k$$ k $$‐vectors in ℝ0,m$$ {\mathrm{\mathbb{R}}}_{0,m} $$. In the framework of Clifford analysis, we consider two boundary value problems for a second‐order elliptic system of partial differential equations of the form ∂x_Fk∂x_=fk$$ {\partial}_{\underset{\_}{x}}{F}_k{\partial}_{\underset{\_}{x}}={f}_k $$ in Ω$$ \Omega $$, where fk$$ {f}_k $$ is a smooth k$$ k $$‐vector valued function. The boundary conditions of the problems contain the inner and outer products of the k$$ k $$‐vector solution Fk$$ {F}_k $$ with both the Dirac operator and the normal vector to ∂Ω$$ \mathrm{\partial \Omega } $$, ensuring the well‐posedness for the problems. Investigation of the spectral properties of the sandwich operator ∂x_(.)∂x_$$ {\partial}_{\underset{\_}{x}}(.){\partial}_{\underset{\_}{x}} $$ is considered by using the Fredholm theory. Finally, it is shown that satisfactory problem‐solving properties, in general, fail when we replace the standard Dirac operator by those, obtained via unusual orthogonal bases of ℝm$$ {\mathrm{\mathbb{R}}}^m $$. [ABSTRACT FROM AUTHOR]

The main goal of this paper is to reveal the effect of crude oil price bubbling on the price volatility dynamics. For this purpose, the time series of volatility at different horizons are mapped into a model of kinetic roughening of interface growing in a stochastic environment. In this way, we found that the volatility dynamics obeys the Family-Viscek dynamic scaling ansatz. Although during the period from January 2, 1986 to July 25, 2014 the volatility remains a slightly anti-persistent, the dynamic exponent is found to be quite different during different regimes of price evolution. Accordingly, we define the intrinsic time of price volatility and metric of volatility horizons. This allows us to construct the Langevin-type equation governing the volatility dynamics during bubble and non-bubble periods. The data analysis indicates that the bubbling does not affect the intrinsic time of volatility, but strongly affect the metric of volatility horizons. In this regard, numerical data suggest the existence of two universal metrics characterizing the volatility dynamics during the bubble and non-bubble regimes of crude oil price evolution, respectively. The results of this work help us to get a further insight into the dynamics of crude oil price volatility. [ABSTRACT FROM AUTHOR]

One way to deal with physical problems on nowhere differentiable fractals is the mapping of these problems into the corresponding problems for continuum with a proper fractal metric. On this way different definitions of the fractal metric were suggested to account for the essential fractal features. In this work we develop the metric differential vector calculus in a three-dimensional continuum with a non-Euclidean metric. The metric differential forms and Laplacian are introduced, fundamental identities for metric differential operators are established and integral theorems are proved by employing the metric version of the quaternionic analysis for the Moisil–Teodoresco operator, which has been introduced and partially developed in this paper. The relations between the metric and conventional operators are revealed. It should be emphasized that the metric vector calculus developed in this work provides a comprehensive mathematical formalism for the continuum with any suitable definition of fractal metric. This offers a novel tool to study physics on fractals. [ABSTRACT FROM AUTHOR]

In this paper we prove that a generalization of complex Π-operator in Clifford analysis, obtained by the use of two orthogonal bases of a Euclidean space, possesses several mapping and invertibility properties, as studied before for quaternion-valued functions as well as in the standard Clifford analysis setting. We improve and generalize most of those previous results in this direction and additionally other consequent results are presented. In particular, the expression of the jump of the generalized Π-operator across the boundary of the domain is obtained as well as an estimate for the norm of the Π-operator is given. At the end an application of the generalized Π-operator to the solution of Beltrami equations is studied where we give conditions for a solution to realize a local and global homeomorphism. [ABSTRACT FROM AUTHOR]

Quaternionic analysis is regarded as a broadly accepted branch of classical analysis referring to many different types of extensions of the Cauchy-Riemann equations to the quaternion skew field [ABSTRACT FROM AUTHOR]

A theory of quaternion-valued functions, called hyperholomorphic, of two real variables has long been established. This theory is in the same relation to the two dimensional Helmholtz equation as the usual one-dimensional complex analysis is to the Laplace equation in R 2 . In this work we define a new Cauchy integral for domains with fractal boundary illustrating its applications and usage to study the jump and Dirichlet type boundary value problems in a fractal domain. [ABSTRACT FROM AUTHOR]

Abstract: Solvability conditions for some Dirichlet type boundary value problems in the framework of Hermitian Clifford analysis are established. [Copyright &y& Elsevier]

We establish an upper bound for the norm of a fractal Hilbert transform in the space of Hölder analytic functions in the sense of Douglis. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this work is to further extend the notion of d-summability due to Harrison and Norton in the beginning of the 1990s. Explicit examples are given to illustrate how our notion can be applied to describe the geometry of a simply connected bounded open subset of in a more delicate manner than the latter one. Applications on the solvability conditions for the Riemann boundary value problems for analytic functions over closed curves merely required to be summable in the generalized sense are described. [Copyright &y& Elsevier]

Recently the basic setting has been established for the development of quaternionic Hermitian Clifford analysis, a theory centred around the simultaneous null solutions, calledq-Hermitian monogenic functions, of four Hermitian Dirac operators in a quaternionic Clifford algebra setting. Borel–Pompeiu and Cauchy integral formulae have been established in this framework by means of a (4 × 4) circulant matrix approach. By means of the matricial quaternionic Hermitian Cauchy kernel involved in these formulae, a quaternionic Hermitian Cauchy integral may be defined. The subsequent study of the boundary limits of this Cauchy integral then leads to the definition of a quaternionic Hermitian Hilbert transform. These integral transforms are studied in this article. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study a class of boundary value problems for a first order linear partial differential equation (all of whose solutions are harmonic functions), which is called the Cimmino system. With the help of the one-to-one correspondence between the theory of quaternion valued hyperholomorphic functions and that of Cimmino system’s solutions, necessary and sufficient conditions for the solvability of the non-homogeneous Cimmino system coupled by the boundary conditions are derived and its general solution is explicitly described. [Copyright &y& Elsevier]

Suppose that Ω is a bounded domain of ℝ with a fractal boundary Γ and let ℝ be the real Clifford algebra constructed over the quadratic space ℝ. Replacing the fractal dimensions of Γ with conditions of approximating character we will characterize the monogenicity of a ℝ-valued function F in the interior and exterior of Ω, in terms of its boundary value f = F|. Moreover, our geometric perspective allows for generalizations of certain two-sided monogenic extension results to a wide class of domains. [ABSTRACT FROM AUTHOR]

A Hilbert transform for Hölder continuous circulant (2 × 2) matrix functions, on the d-summable (or fractal) boundary Γ of a Jordan domain Ω in ℝ, has recently been introduced within the framework of Hermitean Clifford analysis. The main goal of the present paper is to estimate the Hölder norm of this Hermitean Hilbert transform. The expression for the upper bound of this norm is given in terms of the Hölder exponents, the diameter of Γ and a specific d-sum ( d > d) of the Whitney decomposition of Ω. The result is shown to include the case of a more standard Hilbert transform for domains with left Ahlfors-David regular boundary. [ABSTRACT FROM AUTHOR]

Given a Jordan domain of ℂ2, we consider the -problem and establish a necessary and sufficient condition for its solvability in terms of the existence of hyper-conjugate harmonic functions, a notion coming from quaternionic analysis. Besides, whenever the -problem is solvable we give its general solution in a quite explicit form. [ABSTRACT FROM PUBLISHER]

Abstract: As is the case for the Laplace operator, in Euclidean Clifford analysis also the Helmholtz operator can be factorized, more precisely by using perturbed Dirac operators. In this paper we consider the Helmholtz equation in a circulant matrix form in the context of Hermitian Clifford analysis. The aim is to introduce and study the corresponding inhomogeneous Hermitian Dirac operators, which will constitute a splitting of the traditional perturbed Dirac operators of the Euclidean Clifford analysis context. This will not only lead to special solutions of the Hermitian Helmholtz equation as such, but also to the study of boundary value problems of Riemann type for those solutions, which are, in fact, solutions of the Hermitian perturbed Dirac operators involved. [Copyright &y& Elsevier]

Abstract: Let Ω be a simply connected bounded domain in with boundary an Ahlfors David regular surface Γ and f be a continuous function on Γ. Ahlfors David regular surfaces include a broad range of those from smooth to piece-wise Liapunov and Lipschitz surfaces. Using intimate relation between holomorphic function theory of two complex variables and some version of quaternionic analysis we prove that the Bochner–Martinelli integralhas continuous limit values on Γ if the truncated integrals. converge uniformly with respect to z on Γ as ϵ →0. This allows us to discuss, in the last part of the note, a formula for the square of the singular Bochner–Martinelli integral on Ahlfors David regular surfaces. Our formula is in agreement with that of obtained for the context of piece-wise Liapunov surface of integration. [Copyright &y& Elsevier]

Abstract: In this paper we introduce an alternative way of defining the curvilinear Cauchy integral over non-rectifiable curves in the case of complex functions of one complex variable. Especially the jump behavior on the boundary is considered. As an application, solvability conditions of the Riemann boundary value problem are derived on very weak conditions to the boundary. Besides the complex case the consideration can be extended to the theory of Douglis algebra valued functions. [Copyright &y& Elsevier]

Let Ω ⊂ ℝ be a Jordan domain with d-summable boundary Γ. The main gol of this paper is to estimate the Hölder norm of a fractal version of the Hilbert transform in the Clifford analysis context acting from Hölder spaces of Clifford algebra valued functions defined on Γ. The explicit expression for the upper bound of the norm provided here is given in terms of the Hölder exponents, the diameter of Γ and certain d-sum ( d > d) of the Whitney decomposition of Ω. The result obtained is applied to standard Hilbert transform for domains with left Ahlfors-David regular surface. [ABSTRACT FROM AUTHOR]

Abstract: We are interested in finding solvability conditions for the Riemann boundary value problems for hyperanalytic functions in a simply connected bounded open subset of the complex plane whose boundary is merely required to be a d-summable closed curve. [Copyright &y& Elsevier]

We consider Hölder continuous circulant (2 x 2) matrix functions Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. defined on the fractal boundary Γ of a domain Ω in ℝ2n. The main goal is to study under which conditions such a function Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. can be decomposed as Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed., where the components Due to image rights restrictions, multiple line equation(s) cannot be graphically displayed. are extendable to H-monogenic functions in the interior and the exterior of Ω, respectively. H-monogenicity are a concept from the framework of Hermitean Clifford analysis, a higher-dimensional function theory centered around the simultaneous null solutions of two first-order vector-valued differential operators, called Hermitean Dirac operators. H-monogenic functions then are the null solutions of a (2 x 2) matrix Dirac operator, having these Hermitean Dirac operators as its entries; such matrix functions play an important role in the function theoretic development of Hermitean Clifford analysis. In the present paper a matricial Hermitean Téodorescu transform is the key to solve the problem under consideration. The obtained results are then shown to include the ones where domains with an Ahlfors-David regular boundary were considered. [ABSTRACT FROM AUTHOR]

Monogenic extension theorem of complex Clifford algebras-valued functions over a bounded domain with fractal boundary is obtained. The paper is dealing with the class of Hölder continuous functions. Applications to holomorphic functions theory of several complex variables as well as to that of the so-called biregular functions will be deduced directly from the isotonic approach. [ABSTRACT FROM AUTHOR]

This article is devoted to the study of a hypercomplex singular integral equation, which contains as special case the characteristic equation and other ones associated with it. The equation is solved in closed form by reduction to two Riemann boundary value problems for Douglis algebras-valued functions in generalized Holder spaces over a closed Jordan Carleson curve. [ABSTRACT FROM AUTHOR]

Abstract: This Note gives geometric conditions on a surface of so that the Hilbert transform on that surface in the framework of Clifford analysis defines a bounded operator in the Hölder continuous functions classes. This result provides a generalization of the well-known theorem of Plemelj and Privalov for curves in . To cite this article: R. Abreu Blaya et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009). [Copyright &y& Elsevier]

In this paper, we extend to the theory of functions of several complex variables, a theorem due to Davydov from classical complex analysis. We prove the following: if Ω⊂n is a bounded domain with boundary ∂Ω of finite (2n-1)-dimensional Hausdorff measure H2n-1 and f is a continuous complex-valued function on ∂Ω such that [image omitted] converges uniformly on ∂Ω as r→0, then the Bochner-Martinelli transform on Ω of f admits a continuous extension to ∂Ω and the Sokhotski-Plemelj formulae hold. For n=2, we briefly sketch how quaternionic analysis techniques may be used to give an alternative proof of the above result. [ABSTRACT FROM AUTHOR]

Euclidean Clifford analysis has become a well‐established theory of monogenic functions in higher‐dimensional Euclidean space with a variety of applications both inside and outside of mathematics. Noncommutativity of the geometric product in Clifford algebras leads to what are now known as inframonogenic functions, which are characterized by certain elliptic system associated to the orthogonal Dirac operator in ℝm. The main question we shall be concerned with is whether or not a higher‐order Lipschitz function on the boundary Γ of a Jordan domain Ω⊂ℝm can be decomposed into a sum of the two boundary values of a sectionally inframonogenic function with jump across Γ. To this end, a kind of Cauchy‐type integral and singular integral operator, very specific to the inframonogenic setting, are widely used. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to solve a kind of the Riemann–Hilbert boundary value problem for (φ , ψ) -harmonic functions, which are linked with the use of two orthogonal bases of the Euclidean space ℝ m . We approach this problem using the language of Clifford analysis for obtaining an explicit expression of the solution of the problem in a Jordan domain Ω ⊂ ℝ m with fractal boundary. Since our study is concerned with a second order differential operator, the boundary data are restricted to involve the higher order Lipschitz class Lip ⁡ (1 + α , Γ) . [ABSTRACT FROM AUTHOR]

In the present work we obtain some analogues of the Hilbert formulas on the unit circle and on the upper half-plane for the theory of solutions of a special case of the Beltrami equation in C to be referred as β -analytic functions. Furthermore, a Poincaré–Bertrand formula related to the β -Cauchy singular integral over a closed Jordan curve is derived and it is used to derive the corresponding Schwarz and Poisson formulas. [ABSTRACT FROM AUTHOR]

The current COVID-19 pandemic mainly affects the upper respiratory tract. People with COVID-19 report a wide range of symptoms, some of which are similar to those of common flu, such as sore throat and rhinorrhea. Additionally, COVID-19 shares many clinical symptoms with severe pneumonia, including fever, fatigue, dry cough, and respiratory distress. Several diagnostic strategies, such as the real-time polymerase chain reaction technique and computed tomography imaging, which are more costly than chest radiography, are employed as diagnostic tools. The purpose of this paper is to describe the role of the d-summable information dimension of X-ray images in differentiating several lesions and lung illnesses better than both fractal and information dimensions. The statistical analysis shows that the d-summable information dimension model better describes the information obtained from the X-ray images. Therefore, it is a more precise measure of complexity than the information and box-counting dimension. The results also show that the X-ray images of COVID-19 pneumonia reveal greater damage than those of tuberculosis, pneumonia, and various lung lesions, where the damage is minor or much focused. Because the d-summable information dimension increases as the image complexity decreases, it could pave the way to formulate a new measure to quantify the lung damage and assist the clinical diagnosis based on the area under the d-summable information model. In addition, the physical meaning of the ν parameter in the d-summable information dimension is given. [ABSTRACT FROM AUTHOR]

In this article, new information dimensions of complex networks are introduced underpinned by fractional order entropies proposed in the literature. This fractional approach of the concept of information dimension is applied to several real and synthetic complex networks, and the achieved results are analyzed and compared with the corresponding ones obtained using classic information dimension based on the Shannon entropy. In addition, we have investigated an extensive classification of the treated complex networks in correspondence with the fractional information dimensions. [ABSTRACT FROM AUTHOR]

In Mexico, approximately 504,000 students pursue a bachelor's degree by means of distance or blended programmes. However, only 42% of these students conclude their degree on time. In the context of blended learning, the focus of this research is to present a causal model, based on a theoretical framework, which describes the relationships concerning motivations, emotions, cognitive strategies, metacognitive strategies, and learning strategies, and their impact on learning performance. The results suggest that negative emotions play a meaningful role between expectancy (a component of motivation) and learning strategies. Also, the expectancy component of motivation positively influences metacognitive strategies. Concerning the relationship between cognition and metacognition, metacognitive strategies take preference concerning the relationship between metacognitive and learning strategies, supporting the theoretical hypothesis that metacognitive processes are on a higher plane than cognition, and affect cognitive process directly. Moreover, the learning outcomes are directly influenced by cognitive and learning strategies, but not by metacognitive ones. Similarly, motivation has direct effects on metacognitive and learning strategies but not on cognitive ones. Lay Description: What is currently known about the subject matter: Several studies describe separately the causal relationships among motivation, emotions, cognition, metacognition, and learning performance in various learning contexts (mainly in distance and face‐to‐face learning).Although the fuzzy boundaries of metacognition have been recognized and a well‐sounded definition has been established, the theoretical relations with motivation, emotions, and cognition have barely been confirmed. What this paper adds to this: From the systems theory point of view, motivation, emotions, cognition, and metacognition are emergent subsystems of the human mind. Thus, studying each of them separately cuts off the relationships that conform the whole system and gives us a limited view of it.This integrating idea is depicted in a causal model, based on a theoretical framework, which describes the relationships among motivations, emotions, cognitive strategies, metacognitive strategies, and learning strategies, and their impact on learning performance.This research offers practical evidence that supports the theoretical relation between motivation and metacognition. The implications of study findings for practitioners. The motivation (expectancy) increases the use of metacognitive and learning strategies; this finding has practical implications, because metacognition may be positively stimulated by knowing that an individual's efforts to learn (control of learning beliefs) and judgments about the individual's abilities to accomplish a given task (self‐efficacy)Moreover, the results suggest that metacognitive, cognitive, and learning strategies are tightly related, in a hierarchical structure where metacognition plays an important role.The causal relations from positive emotions to metacognitive, cognitive, and learning strategies were not significant. Thus, the impact of negative emotions on the reviewed learning content (which captures the computer‐assisted nature of blended learning) and overall grade was explored. Students that face several obstacles in reviewing the learning content have not enough control to overcome this situation, and frustration is instigated. Also, anxious students think in advance that they will may fail the course, and this thought negatively impacts on their overall grade. [ABSTRACT FROM AUTHOR]