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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]

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]

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]

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]