# Juan Bory-ReyesInstituto Politécnico Nacional | IPN · Department of Systems Engineering. SEPI-ESIME-Zac

Juan Bory-Reyes

Ph.D. (1988), Dr.Sc. (2008)

Engage in original research, teach postgraduate courses, mentor and supervisor of Master’s and Ph.D students.

## About

239

Publications

32,377

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1,466

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Introduction

His research interest includes: first order linear PDEs in complex, hypercomplex and Clifford analysis as well as mathematical modelling in medical physics, mechanical statistics, online learning and complex networks. He has papers co-authored by scientists from 14 countries.
Life-Member of the International Society for Analysis, its Applications and Computation, Member of the Mexican National Researchers System and Emeritus-Member of the Cuban Society of Mathematics and Computation.

Additional affiliations

December 1982 - June 2014

Education

September 2006 - December 2008

September 1984 - December 1988

September 1977 - June 1982

## Publications

Publications (239)

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 certai...

In this paper we present a hyperholomorphic (associated to the Helmholtz equation) approach to the Riemann-Hilbert boundary value problem (RHBVP for short) in domains of ${\mathbb R}^2$ with h-summable boundaries. We apply our results to Maxwell’s system
and study an electromagnetic RHBVP for the case time-harmonic. The study is based on a reformul...

The quantification of learning acquisition in a blended and online course is still slightly explored from the complex systems lens. The fractional online learning rate (fOLR) using fractional integrals is introduced. The notion of fOLR is based on the nonlinearity of the individual students learning pathway network, built from Learning Management S...

We give some characterizations of Lipschitz type spaces of slice regular functions in the unit ball of the skew field of quaternions with prescribed modulus of continuity.

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.

We first prove a Cauchy's integral theorem and Cauchy type formula for certain inhomogeneous Cimmino system from quaternionic analysis perspective. The second part of the paper directs the attention towards some applications of the mentioned results, dealing in particular with four kinds of weighted Bergman spaces, reproducing kernels, projection a...

The purpose of this paper is to establish a Borel-Pompeiu type formula induced from a fractional bicomplex $(\vartheta,\varphi)-$weighted Cauchy-Riemann operator, where the weights are two hyperbolic orthogonal bicomplex functions and the fractionality is understand in the Riemann-Liouville sense.

The purpose of this paper is to solve a kind of the Riemann–Hilbert boundary value problem for ( φ , ψ ) {(\varphi,\psi)} -harmonic functions, which are linked with the use of two orthogonal bases of the Euclidean space ℝ m {\mathbb{R}^{m}} . We approach this problem using the language of Clifford analysis for obtaining an explicit expression of th...

This paper is a continuation of our work [J. O. González Cervantes and J. Bory Reyes, A quaternionic fractional Borel–Pompeiu type formula, Fractal 30(1) (2022) 2250013], where we introduced a fractional operator calculus related to a fractional [Formula: see text]-Fueter operator in the one-dimensional Riemann–Liouville derivative sense in each di...

En esta plática se habla de la extensión de sistemas de funciones iteradas a espacios métricos hiperbólico valuados, así como su aplicación sobre una versión del juego del caos con probabilidades sobre el plano hiperbólico.

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...

In theoretical setting, associated with a fractional $\psi-$Fueter operator that depends on an additional vector of complex parameters with fractional real parts, this paper establishes a fractional analogue of Borel-Pompeiu formula as a first step to develop a fractional $\psi-$hyperholomorphic function theory and the related operator calculus.

In this work, we introduce a fractional generalization of the classical Moisil-Teodorescu operator that provides a concise notation for presenting a mathematical formulation of physical systems in fractional space from various branches of science and engineering. The method used in this article, called the Stillinger’s formalism, is combined in a n...

The history of Bergman spaces goes back to the book [4] in the early fifties by S. Bergman, where the first systematic treatment of the subject was given, and since then there have been a lot of papers devoted to this area. Some standard works here are [5], [8], [16], [29] and the references therein, which contain a broad summary and historical not...

RESUMEN En el presente artículo se estudian las ecuaciones de ondas electromagnéticas y la condición de Lorentz, como propiedades emergentes del sistema de Maxwell en el contexto de la Teoría de Sistemas. Para este fin, se deducen las ecuaciones de ondas y la ecuación de Helmholtz. Haciendo uso del operador de Dirac desplazado y su estrecha relació...

In this note we establish a necessary and sufficient condition for solvability of the homogeneous Riemann boundary problem with infinity index on a rectifiable open curve. The index of the problem we deal with considers the influence of the requirement of the solutions of the problem, the degree of non-smoothness of the curve at the endpoints as we...

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 share many clinical symptoms with severe pneumonia, including fever, fatigue, dry cough, and respiratory distress...

We present a brief review of results on the homogeneous Riemann boundary value problem for analytic functions with very general boundary data.

SEMINARIO ABIERTO ESIME ZACATENCO

In this paper, we aim to discuss the Noether property of the Riemann boundary value
problems in a Banach algebra of continuous functions over simple closed curves and
its direct approximate solution through approximation of the principal coefficient,
establishing a bound for the error of approximate solution of the problem to the exact
solution.

In this paper, we define two types of partitions of an hyperbolic interval: weak and strong. Strong partitions enables us to define, in a natural way, a notion of hyperbolic valued functions of bounded variation and hyperbolic analogue of Riemann-Stieltjes integral. We prove a deep relation between both concepts like it occurs in the context of rea...

Quaternionic analysis offers a function theory focused on the concept of $\psi-$hyperholomorphic functions defined as null solutions of the $\psi-$Fueter operator, where $\psi$ is an arbitrary orthogonal base (called structural set) of $\mathbb H^4$. The main goal of the present paper is to extend the results given in \cite{BG2}, where a fractional...

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 $\mathbb H$. In this work we deals with a well-known $(\theta, u)-$hyperholomorphic $\mathbb H-$valued functions class related to elements of the kernel of...

This paper aims at proving the boundedness property of multidimensional singular integral operators associated with \((\varphi ,\psi )\)-harmonic functions, which are connected by the use of two orthogonal basis of the Euclidean space \({{\mathbb {R}}}^m\). Besides, necessary and sufficient conditions for the solvability of the \({\overline{\partia...

We consider the behavior of generalized Laplacian vector fields on a Jordan domain of R3 with fractal boundary. Our approach is based on properties of the Teodorescu transform and suitable extension of the vector fields. Specifically, the present article addresses the decomposition problem of a Hölder continuous vector field on the boundary (also c...

In this paper, based on a proposed notion of generalized conjugate
harmonic pairs in the framework of complex Clifford analysis,
necessary and sufficient conditions for the solvability of inhomogeneous
perturbed generalized Moisil–Teodorescu systems in higher dimensional
Euclidean spaces are proved. As an application, we derive corresponding
solvab...

In the present work we obtain some analogues of the Hilbert formulas on the unit circle
for iterated Cauchy-Riemann operator in one-dimensional complex analysis involving
higher order Lipschitz classes. Furthermore, a Poincaré-Bertrand formula related to
the corresponding singular iterated Cauchy integral over the boundary of a smoothly
bounded dom...

A two-parameter (namely, \vec{\alpha} and \vec{\beta}) fractional Tsallis information dimensions of complex networks based on q−logarithm is introduced. The meanings assigned to such parameters are the quantification of the interaction among the elements (nodes) that are part of the same sub-system (sub-network) and the interaction among the sub-sy...

In this paper, we mainly consider the Riemann boundary value problems for lower dimensional non-commutative Clifford algebras valued monogenic functions. The solutions are given in an explicit way and concrete examples are presented to illustrate the results.

In this paper, we shall be interested in solving Dirichlet-type problems for solutions of certain classes of Beltrami equations, to be called β-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-...

It is generally well understood the legitimate action of the Moisil-Theodoresco operator, over a quaternionic valued function defined on $\mathbb{R}^3$ (sum of a scalar and a vector field) in Cartesian coordinates, but it does not so in any orthogonal curvilinear coordinate system. This paper sheds some new light on the technical aspect of the subj...

In this article, a new algorithm to select the relevant nodes —those that maintain the cohesion of the network— of the complex network is presented. The result shows that the proposed approach outperforms degree, PageRank and betweenness in most of the several real complex networks. The rationale of the algorithm is to discover the self-similarity...

In this paper, we provide extensions to hyperbolic numbers plane of the classical Chaos game algorithm and the Shannon entropy. Both notions connected with that of probability with values in hyperbolic number, introduced by Alpay et al. [Kolmogorov’s axioms for probabilities with values in hyperbolic numbers, Adv. Appl. Clifford Algebras 27(2) (201...

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 $\mathbb{R}^3$. In the present work, a general quaternionic structure is developed for the local fractional Moisil–Teodorescu...

The paper provides integral representations for solutions to a certain first order partial
differential equation natural arising in the factorization of the Lame-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 dimensiona...

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 one...

The classical and modified equations of Kolmogorov-Johnson-Mehl-Avrami are compared with the equations of conventional Gompertz and Montijano-Bergues-Bory-Gompertz, in the frame of growth kinetics of tumors. For this, different analytical and numerical criteria are used to demonstrate the similarity between them, in particular the distance of Hausd...

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 deriv...

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.

In this paper we consider functions satisfying the sandwich equation $\pux f\pux=0$, where $\pux$ stands for the Dirac operator in $\R^m$. Such functions are referred as {\emph{inframonogenic}} and represent an extension of the monogenic functions, i.e., null solutions of $\pux$.
In particular for odd $m$, we prove that a $C^2$-function is both in...

In this paper we derive a Cauchy integral representation formula for the solutions of the iterated sandwich equation \(\partial _{{\underline{x}}}^{2k-1}f\partial _{{\underline{x}}}=0\), where k is a positive integer and \(\partial _{{\underline{x}}}\) stands for the Dirac operator in the Euclidean space \({{\mathbb {R}}}^m\). We call these solutio...

We investigate an electromagnetic Dirichlet type problem for the 2D quaternionic time-harmonic Maxwell system over a great generality of fractal closed type curves, which bound Jordan domains in R2. The study deals with a novel approach of h-summability condition for the curves, which would be extremely irregular and deserve to be considered fracta...

In this paper we consider the problem of reconstructing solutions to a generalized
Moisil–Teodorescu system in Jordan domains of R3 with rectifiable boundary. In
order to determine conditions for existence of solutions to the problem we embed the
system in an appropriate generalized quaternionic setting.

In this work, we propose a biquaternionic reformulation of a fractional monochromatic Maxwell system. Additionally, some examples are given to illustrate how the quaternionic fractional approach emerges in linear hydrodynamics and elasticity.

In this article, a box-covering Tsallis information dimension is introduced, and the physical interpretation of this new dimension has been assigned. Moreover, based on the introduced parameter q , a characterization of non-extensive networks is stated, allowing the classification according to super-extensive (q ≺ 1), sub-extensive (q 1) or extensi...

Se muestra como se puede generar un concepto análogo de entropía para el caso de probabilidades hiperbólco valuadas. Así como una introducción a una variante del juego del caos con números hiperbólcos

In this paper sufficient conditions for continuous and compact embeddings of generalized higher order Lipschitz classes on a compact subset of n-dimensional real Euclidean spaces are obtained.

Background
Different equations have been used to describe and understand the growth kinetics of undisturbed malignant solid tumors. The aim of this paper is to propose a new formulation of the Gompertz equation in terms of different parameters of a malignant tumor: the intrinsic growth rate, the deceleration factor, the apoptosis rate, the number o...

The main purpose of this work is to prove that the higher order Lipschitz classes behave invariant under the action of a singular integral operator which arise naturally in polymonogenic Clifford algebra valued function theory.

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.

In this paper a bicomplex analogue of a Cauchy type integral in case of a hyperbolic curve of integration is studied. We derive the corresponding Plemelj–Sokhotski formulas for their limit values for all densities satisfying a BC\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepa...

This work addresses the design and experimental evaluation of a passive suspension system on the handlebar of a two-wheel tractor to reduce vibrations affecting the hand-arm-shoulder of the farmer. The existing handlebar element is analyzed and evaluated via numerical and experimental modal analysis to determine a proper passive suspension system b...

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 iterated function systems consisting of affine transformations of the hyperbolic number plane. We show that basics res...

Suppose $\Omega$ is a bounded domain in $\R^n$ with boundary $\Gamma$ and let $\cW$ be a non-homogeneous differential form harmonic in $\Omega$ and H\"older-continuous in $\Omega\cup\Gamma$. In this paper we study and obtain some necessary and sufficient conditions for the self-conjugateness of $\cW$ in terms of its boundary value $\cW|_\Gamma=\ome...

The Pennes bioheat transfer equation is the most used model to calculate the temperature induced in a tumor when physical therapies like electrochemical treatment, electrochemotherapy and/or radiofrequency are applied. In this work, a modification of the Pennes bioheat equation to study the temperature distribution induced by any electrode array in...

Several studies have focused on identifying the significant behavioral predictors of learning performances in web-based courses by examining the log data variables of learning management systems, including time spent on lectures, the number of assignments submitted, and so forth. However, such studies fail to quantify the impact of emotional, motiv...

The Riemann boundary value problem (RBVP to shorten notation) in the complex plane, for different classes of functions and curves, is still widely used in mathematical physics and engineering. For instance, in elasticity theory, hydro and aerodynamics, shell theory, quantum mechanics, theory of orthogonal polynomials, and so on. In this paper, we p...

Unfortunately, the name of the communicating editor has been published incorrectly. The correct name is Rafał Abłamowicz

Let $\mathbb R_{0, m+1}^{(s)}$ be the space of $s$-vectors ($0\le s\le m+1$) in the Clifford algebra $\mathbb R_{0, m+1}$ constructed over the quadratic vector space $\mathbb R^{0, m+1}$, let $r, p, q\in\mathbb N$ with $0\le r\le m+1$, $0\le p\le q$ and $r+2q\le m+1$ and let $\mathbb R_{0, m+1}^{(r,p,q)}=\sum_{j=p}^q\bigoplus\mathbb R_{0, m+1}^{(r+...

In past two decades a wide range of complex systems, spanning many different disciplines, have been structured in the form of networks. Network dimension is a crucial concept to understand not only net- work topology, but also dynamical processes on networks. From the perspective of the box covering, volume dimension, information dimension, and cor...

We prove that a Cauchy transform naturally arising for bimonogenic Clifford algebra valued functions theory behaves invariant on the higher order Lipschitz classes and obtain a Plemelj–Privalov theorem for the related singular integral transform. Moreover, we obtain an upper bound for the norm of such an operator in the pathological case of fractal...

In a series of recent papers, a harmonic and hypercomplex function theory in superspace has been established and amply developed. In this paper, we address the problem of establishing Cauchy integral formulae in the framework of Hermitian Clifford analysis in superspace. This allows us to obtain a successful extension of the classical Bochner-Marti...

The power law has been assumed as a universal model that describes the probability distribution of nodes degree of several real complex networks. Recently a debate has arisen over the validity of this assumption. This debate as encouraged us to carry out a statistical analysis — from the box covering approach — of the fit of five models, included t...

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 concernin...

It is generalized the concept of iteration of a fractal, by means of the parametrization of Iterated Function System (S.F.I) that defines to the fractal and defining a S.F.I. of two dimensions, such that the first entrance is the original S.F.I. and the second represents the iteration. Two examples, the Koch curve and Cantor set, are shown

We obtain some analogues of the Hilbert formulas on the unit circle for α-hyperholomorphic function theory in R2 for α being a complex quaternionic number. The obtained formulas relate a pair of components of the boundary value of an α-hyperholomorphic function in the unit disc with the other pair of components and, hence, being analogous to the ca...

Currently in Mexico there is a great disparity between the production units of the agricultural sector since 80% of agricultural producers own less than five hectares with a production for sub-sistence. Therefore, the improvement and design of appropriate technologies for small farmers is an aspect that must be addressed. In this context, for desig...

A polyanalytic Cauchy transform over fractal curves is introduced and discussed, which allows us to obtain, in particular, integral representation formulas for smooth and polyanalytic functions in the presence of fractal boundaries.

Solutions of the sandwich equation , where stands for the first‐order differential operator (called Dirac operator) in the Euclidean space , are known as inframonogenic functions. These functions generalize in a natural way the theory of kernels associated with , the nowadays well‐known monogenic functions, and can be viewed also as a refinement of...

In this paper we are interested in finding solvability conditions for Riemann
and Dirichlet type boundary value problems with generalized Hölder-continuous
boundary data in the case of hyperanalytic functions defined in a domain of the complex
plane bounded by an h-summable closed curve.

The classical Beltrami system of elliptic equations generalizes the Cauchy Riemann equation in the complex plane
and offers the possibility to consider homogeneous system with no terms of zero order. The theory of Douglis-valued functions, called hyperanalytic functions, is special case of the above situation. In this note, we prove an analogue of...

We obtain some analogues of the Hilbert formulas on the unit circle for $\alpha$-hyperholomorphic function theory when $\alpha$ is a complex number. Such formulas relate a pair of components of the boundary value of an $\alpha$-hyperholomorphic function in the unit circle with the other one. Furthermore, the corresponding Poincare-Bertrand formula...

One of the most challenging problems of electrochemical therapy is the design and selection of suitable electrode array for cancer. The aim is to determine how two-dimensional spatial patterns of tissue damage, temperature, and pH induced in pieces of potato (Solanum tuberosum L., var. Mondial) depend on electrode array with circular, elliptical, p...

In this paper we prove that the higher order Lipschitz classes behave invariant under the action of a singular integral operator naturally arising in polyanalytic function theory. This result provides a generalization of the well-known theorem of Plemelj-Privalov.

The main goal of this article is to present a statistical study of decision tree learning algorithms based on the measures of different parametric entropies. Partial empirical evidence is presented to support the conjecture that the parameter adjusting of different entropy measures might bias the classification. Here, the receiver operating charact...

The Cimmino system offers a natural and elegant generalization to four-dimensional case of the Cauchy–Riemann system of first order complex partial differential equations. Recently, it has been proved that many facts from the holomorphic function theory have their extensions onto the Cimmino system theory. In the present work a Poincare–Bertrand fo...

The Electrochemical treatment can be used for local control of solid tumors in both preclinical and clinical studies. In this paper, an integrated analysis of the spatial distributions of the electric potential, electric field, temperature and pH together with the acidic and basic areas are computed, via Finite Element Methods, to improve the geome...

The main goal of this article is to develop a Management System for Merging Learning Objects (msMLO), which offers an approach that retrieves learning objects (LOs) based on students’ learning styles and term-based queries, which produces a new outcome with a better score. The msMLO faces the task of retrieving LOs via two steps: The first step ran...

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...

In Mexico, the technological development in the agricultural sector is an aspect that must be addressed, especially the design of agricultural machinery suitable for small farmers, who have limited resources to acquire large and sophisticated machinery developed in industrialized countries. In this context, this paper presents a literature review o...