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Multiscroll attractors by switching systems

ERIC CAMPOS CANTON JUAN GONZALO BARAJAS RAMIREZ GUALBERTO CELESTINO SOLIS PERALES ALEJANDRO RICARDO FEMAT FLORES (2010)

"In this paper, we present a class of three-dimensional dynamical systems having multiscrolls which we call unstable dissipative systems (UDSs). The UDSs are dissipative in one of its components but unstable in the other two. This class of systems is constructed with a switching law to display various multiscroll strange attractors. The multiscroll strange attractors result from the combination of several unstable "one-spiral" trajectories by means of switching. Each of these trajectories lies around a saddle hyperbolic stationary point. Thus, we describe how a piecewise-linear switching system yields multiscroll attractors, symmetric or asymmetric, with chaotic behavior."

Article

Chaos Differential equations; Nonlinear dynamical systems; Synchronisation CIENCIAS FÍSICO MATEMÁTICAS Y CIENCIAS DE LA TIERRA

Experimental multi-scroll attractor driven by switched systems

ISAAC CAMPOS CANTON ERIC CAMPOS CANTON RAEL EDUARDO BALDERAS NAVARRO (2017)

"This article deals with an electronic implementation of a 3-D dynamical system that comprises multiple scrolls and is regarded as unstable dissipative system. Such a system is dissipative in one of its components but unstable in the other two. The proposed electronic circuit is implemented with resistors, capacitors and comparators and has the capability to generate two or three scrolls"

Article

Phase portrait Analog electronic Differential equations Operational Amplifier CIENCIAS FÍSICO MATEMÁTICAS Y CIENCIAS DE LA TIERRA FÍSICA

Análisis histórico-epistemológico en la educación matemática.

FABIAN WILFRIDO ROMERO FONSECA FLOR MONSERRAT RODRIGUEZ VASQUEZ SARA MARCELA HENAO SALDARRIAGA (2017)

Researchers in the field of Mathematics Education have pointed out the importance of performing historicalepistemological studies of mathematical concepts; there is even a wide discussion about the contributions of the history of mathematics to the teaching-learning processes of concepts. Partly, it responds to the problematic of the consideration of the concepts without a historical contextualization. In this article, we show two investigations whose objective was to make a historical-epistemological study on the construction of mathematical concepts, namely the ordinary differential equations and Fourier trigonometric series. Besides, the methodology of historical research will be discussed.

Article

epistemology differential equations fourier series HUMANIDADES Y CIENCIAS DE LA CONDUCTA PEDAGOGÍA OTRAS ESPECIALIDADES PEDAGÓGICAS

Node adaptive domain decomposition method by radial basis functions

PABLO GONZALEZ CASANOVA JOSE ANTONIO MUÑOZ GOMEZ GUSTAVO RODRIGUEZ GOMEZ (2008)

During the last years, there has been increased interest in developing efficient radial basis function (RBF) algorithms to solve partial differential problems of great scale. In this article, we are interested in solving large PDEs problems, whose solution presents rapid variations. Our main objective is to introduce a RBF dynamical domain decomposition algorithm which simultaneously performs a node adaptive strategy. This algorithm is based on the RBFs unsymmetric collocation setting. Numerical experiments performed with the multiquadric kernel function, for two stationary problems in two dimensions are presented.

Article

Domain decomposition Meshfree methods Node adaptive Partial differential equations Radial basis function CIENCIAS FÍSICO MATEMÁTICAS Y CIENCIAS DE LA TIERRA MATEMÁTICAS CIENCIA DE LOS ORDENADORES

The theorem existence and uniqueness of the solution of a fractional differential equation

El teorema de existencia y unicidad de la solución de una ecuación diferencial fraccionaria

JUAN MARTINEZ ORTIZ LETICIA RAMIREZ HERNANDEZ (2013)

This article aims to demonstrate, using the Picard-Banach theorem, the proof of the theorem-based existence and uniqueness of the solution of an fractional orden differential equation with a fractional Caputo-type derivative

El objetivo de este artículo es demostrar, usando el teorema de Picard-Banach, el teorema de existencia y unicidad de la solución de una ecuación diferencial de orden fraccionario con derivada tipo Caputo

Article

INGENIERÍA Y TECNOLOGÍA Fractional differential equations Theorem existence and uniqueness Theorem Picard-Banach Ecuaciones diferenciales fraccionarias Teorema de existencia y unicidad Teorema de Picard-Banach

Solution of finite element problems using hybrid parallelization with MPI and OpenMP

Solución de problemas de elemento finito utilizando paralelización híbrida con MPI y OpenMP

JOSE MIGUEL VARGAS FELIX SALVADOR BOTELLO RIONDA (2012)

El Método de Elemento Finito (FEM, por sus siglas en inglés) es utilizado para resolver problemas como la deformación de sólidos o la difusión de calor en dominios con geometrías complejas. Este tipo de geometrías requiere de discretizaciones con millones de elementos, lo que equivale a resolver sistemas de ecuaciones con matrices dispersas de decenas o cien-tos de millones de variables. La meta es utilizar clústeres de computadoras para resolver estos sistemas. El método de solución utilizado es la subestructuración de Schur. Utilizando ésta es posible dividir un sistema grande de ecuaciones en muchos pequeños para resolv-erse más eficientemente. Este método permite la paralelización. La MPI (Message Passing Interface, Interfaz para Paso de Mensajes) es utilizada para distribuir los sistemas de ecu-aciones a resolver en cada computadora del cluster. Cada sistema de ecuaciones es resuelta utilizando un solver implementado con OpenMP como método de paralelización local

The Finite Element Method (FEM) is used to solve problems like solid deformation and heat diffusion in domains with complex geometries. This kind of geometries requires discretization with millions of elements; this is equivalent to solve systems of equations with

sparse matrices and tens or hundreds of millions of variables. The aim is to use computer clusters to solve these systems. The solution method used is Schur substructuration.

Using it is possible to divide a large system of equations into many small ones to solve them more efficiently. This method allows parallelization. MPI (Message Passing Interface) is used to distribute the systems of equations to solve each one in a computer of a cluster.

Each system of equations is solved using a solver implemented to use OpenMP as a local parallelization method.

Article

CIENCIAS FÍSICO MATEMÁTICAS Y CIENCIAS DE LA TIERRA Cómputo en paralelo Matrices dispersas Solvers lineales Ecuaciones diferenciales parciales Método de Elemento Finito MPI (Interfaz de Paso de Mensajes) Parallel computing Sparse matrices Linear solvers Partial differential equations Finite Element Method MPI (Message Passing Interface)

Solución numérica de ecuaciones en derivadas parciales mediante funciones radiales

JOSE ANTONIO MUÑOZ GOMEZ (2007)

La presente tesis trata sobre la solución numérica de ecuaciones en derivadas

parciales mediante funciones de base radial. En la primera parte investigamos

el orden de convergencia en h-c para una ecuación dependiente del tiempo de

tipo convección-difusión en una dimensión. Con base en la función radial multicuádrica, y con un esquema implícito y explicito en la discretización temporal,

observamos una tasa de convergencia exponencial en h-c, en donde el coeficiente

de la exponencial es reducido conforme incrementamos el número de Péclet.

Adicionalmente, mostramos numéricamente que el valor óptimo del parámetro

c decrece monotónicamente conforme el coeficiente de difusión es reducido.

En general, cuando utilizamos funciones de base radial para resolver ecuaciones

en derivadas parciales, la matriz resultante es por lo general densa y mal

condicionada. Por lo cual, el uso de métodos directos es aplicable solo a problemas

de moderado tamaño. En la segunda parte de este trabajo abordamos

dicho problema empleando descomposición de dominio con nodos distribuidos

uniformemente. La estrategia propuesta es aplicada a un problema dependiente

del tiempo en 2-dimensiones. Empleando un solo procesador, observamos una

disminución lineal en el tiempo de procesamiento conforme incrementamos el

número de particiones, por lo tanto, el esquema propuesto puede ser aplicado

a problemas de gran escala con cúmulos de computadoras.

El incremento uniforme en la densidad de los nodos induce una disminución

en el error de aproximación; sin embargo, aún con cúmulos de computadoras la

estrategia de refinamiento global de nodos es un método computacionalmente

ineficiente. En problemas en donde existen capas límite, zonas de alto gradiente

o una gran variación espacial en la solución, es conveniente aproximar dichos

problemas con un esquema de refinamiento local de nodos. La idea de refinamiento

local, consiste en densificar el número de nodos en las regiones en donde

se requiere de mayor exactitud. Con base en el error de interpolación local y

el esquema de celda×celda, se obtiene un método eficiente para el esquema de

refinamiento local con funciones de base radial. Este esquema es probado en

distintas ecuaciones diferenciales parciales en una y dos dimensiones, mostrando

la efectividad del método propuesto. El esquema desarrollado no requiere

de una malla para el proceso de refinamiento y puede extenderse a tres o más

dimensiones con fronteras complejas.

Finalmente, basándose en el es

Doctoral thesis

Numerical analysis Partial differential equations Interpolation CIENCIAS FÍSICO MATEMÁTICAS Y CIENCIAS DE LA TIERRA MATEMÁTICAS ANÁLISIS NUMÉRICO ECUACIONES DIFERENCIALES EN DERIVADAS PARCIALES

Helmholtz Theorems, Gauge Transformations, General Covariance and the Empirical Meaning of Gauge Conditions

Andrew Chubykalo (2016)

It is well known that the use of Helmholtz decomposition theorem for static vector fields C : R3 → R3 ,

when applied to the time dependent vector fields E : R4 → R3 , B : R4 → R3 which represent the

electromagnetic field, allows us to obtain instantaneous-like solutions all along R3 . For this reason,

some people thought (see e.g. [1] and references therein) that the Helmholtz theorem cannot

be applied to time dependent vector fields and some modification is wanted in order to get the retarded

solutions. However, the use of the Helmholtz theorem for static vector fields is correct even

for time dependent vector fields (see, e.g. [2]), so a relation between the solutions was required, in

such a way that a retarded solution can be transformed in an instantaneous one, and conversely.

On this paper we want to suggest, following most of the time the mathematical formalism of Woodside

in [3], that: 1) there are many Helmholtz decompositions, all equally consistent, 2) each one is

naturally related to a space-time structure, 3) when we use the Helmholtz decomposition for the

electromagnetic potentials it is equivalent to a gauge transformation, 4) there is a natural methodological

criterion for choosing the gauge according to the structure postulated for a global spacetime,

5) the Helmholtz decomposition is the manifestation at the level of the fields that a gauge is

involved. So, when we relate the retarded solution to the instantaneous one what we do is to change

the gauge and the space-time. And, if the Helmholtz decompositions are related to a space-time

structure, and are equivalent to gauge transformations, each gauge transformation is natural for a

specific space-time. In this way, a Helmholtz decomposition for Euclidean space is equivalent to

the Coulomb gauge and a Helmholtz decomposition for the Minkowski space is equivalent to the

Lorenz gauge. This leads us to consider that the theories defined by different gauges may be mathematically

equivalent, because they can be related by means of a gauge transformation, but they

are not empirically equivalent, because they have quite different observational consequences due

to the different space-time structure involved.

Producción Científica de la Universidad Autónoma de Zacatecas UAZ

Article

CIENCIAS FÍSICO MATEMÁTICAS Y CIENCIAS DE LA TIERRA Helmholtz Theorem Gauge Transformations Space-Time Transformations Symmetries of Differential Equations Underdetermination of Systems of Differential Equations Natural Covariance

Helmholtz Theorems, Gauge Transformations, General Covariance and the Empirical Meaning of Gauge Conditions

Andrew Chubykalo Augusto Espinoza (2016)

It is well known that the use of Helmholtz decomposition theorem for static vector fields C : R3 → R3 ,

when applied to the time dependent vector fields E : R4 → R3 , B : R4 → R3 which represent the

electromagnetic field, allows us to obtain instantaneous-like solutions all along R3 . For this reason,

some people thought (see e.g. [1] and references therein) that the Helmholtz theorem cannot

be applied to time dependent vector fields and some modification is wanted in order to get the retarded

solutions. However, the use of the Helmholtz theorem for static vector fields is correct even

for time dependent vector fields (see, e.g. [2]), so a relation between the solutions was required, in

such a way that a retarded solution can be transformed in an instantaneous one, and conversely.

On this paper we want to suggest, following most of the time the mathematical formalism of Woodside

in [3], that: 1) there are many Helmholtz decompositions, all equally consistent, 2) each one is

naturally related to a space-time structure, 3) when we use the Helmholtz decomposition for the

electromagnetic potentials it is equivalent to a gauge transformation, 4) there is a natural methodological

criterion for choosing the gauge according to the structure postulated for a global spacetime,

5) the Helmholtz decomposition is the manifestation at the level of the fields that a gauge is

involved. So, when we relate the retarded solution to the instantaneous one what we do is to change

the gauge and the space-time. And, if the Helmholtz decompositions are related to a space-time

structure, and are equivalent to gauge transformations, each gauge transformation is natural for a

specific space-time. In this way, a Helmholtz decomposition for Euclidean space is equivalent to

the Coulomb gauge and a Helmholtz decomposition for the Minkowski space is equivalent to the

Lorenz gauge. This leads us to consider that the theories defined by different gauges may be mathematically

equivalent, because they can be related by means of a gauge transformation, but they

are not empirically equivalent, because they have quite different observational consequences due

to the different space-time structure involved.

Producción Científica de la Universidad Autónoma de Zacatecas UAZ

Article

CIENCIAS FÍSICO MATEMÁTICAS Y CIENCIAS DE LA TIERRA Helmholtz Theorem Gauge Transformations Space-Time Transformations Symmetries of Differential Equations Underdetermination of Systems of Differential Equations Natural Covariance

Exceptional Smooth bol loops.

LARISSA SBITNEVA (2001)

One new class of smooth Bol loops, exceptional Bol loops, is introduced and studied. The

approach to the Campbell-Hausdorff formula is outlined. Bol-Bruck loops and Moufang

loops are exceptional which justifies our consideration.

Article

Differential equation Basic differential equations. Conditions of integrability Matrix form of differential equations. CIENCIAS FÍSICO MATEMÁTICAS Y CIENCIAS DE LA TIERRA MATEMÁTICAS