Algebrization of Nonautonomous Differential Equations

Alcorta-García, María Aracelia; Frías-Armenta, Martín Eduardo; Grimaldo-Reyna, María Esther; López‐González, Elifalet

doi:10.1155/2015/632150

Cited by 3 publications

(3 citation statements)

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“…Lorch's work [30] is also interesting. Lorch's concept of differentiation over an algebra was foundational for the recent papers [16] and [17] where it is shown how to solve certain ordinary differential equations via an algebra substitution. In particular, the concept of multiple conjugates as shown in [16] is very helpful and it was the inspiration for the generalized Wirtinger calculus we present in this paper.…”

Section: Historymentioning

confidence: 99%

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Introduction to $\mathcal{A}$-Calculus

Cook¹

2017

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Let A denote an n-dimensional associative algebra over R. This paper gives an introductory exposition of calculus over A. An A-differentiable function f : A → A is one for which the differential is right-A-linear. The basis-dependent correspondence between right-A-linear maps and the regular representation of real matrices is discussed in detail. The requirement that the Jacobian matrix of a function fall in the regular representation of A gives n 2 −n generalized A-CR equations. In contrast, some authors use a deleted-difference quotient to describe differentiability over an algebra. We compare these concepts of differentiability over an algebra and prove they are equivalent in the semisimple commutative case. We also show how difference quotients are ill-equipt to study calculus over a nilpotent algebra.Cauchy Riemann equations are elegantly captured by the Wirtinger calculus as ∂f ∂ z = 0. We generalize this to any commutative unital associative algebra. Instead of one conjugate, we need n − 1 conjugate variables. Our construction modifies that given by Alvarez-Parrilla, Frías-Armenta, López-González and Yee-Romero in [16].We derive Taylor's Theorem over an algebra. Following Wagner, we show how Generalized Laplace equations are naturally seen from the multiplication table of an algebra. We show how d'Alembert's solution to the wave-equation can be derived from the function theory of an appropriate algebra. The inverse problem of A-calculus is introduced and we show how the Tableau for an A-differentiable function has a rather special form. The integral over an algebra is also studied. We find the usual elementary topological results about closed and exact forms generalize nicely to our integral. Generalizations of the First and Second Fundamental Theorems of Calculus are given. This paper should be accessible to undergraduates with a firm foundation in linear algebra and calculus. Some background in complex analysis would also be helpful.

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Section: Historymentioning

confidence: 99%

“…How do we choose A? In [16] and [17] the authors study how to modify certain ODEs in terms of A-calculus. In contrast, our study on the inverse problem has centered around systems of PDEs.…”

Section: An Approach To the Inverse Problemmentioning

confidence: 99%

Introduction to $\mathcal{A}$-Calculus

Cook¹

2017

Preprint

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“…En este trabajo se analizaron los artículos de A. Alvarez-Parrilla [4] y de Alcorta-García [5] referentes a la A-algebrización, en los cuales propusieron el método de algebrización de ecuaciones diferenciales, el cual se utiliza para encontrar la solución de algunos sistemas de ecuaciones diferenciales.…”

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El método de algebrización de sistemas de ecuaciones diferenciales

Flores

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On geodesibility of algebrizable planar vector fields

Frías-Armenta

López‐González

2017

Bol. Soc. Mat. Mex.

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Recently, the geodesibility of planar vector fields, which are algebrizable (differentiable in the sense of Lorch for some associative and commutative unital algebra), has been established. In this paper, we consider algebrizable three-dimensional vector fields, for which we give rectifications and Riemannian metrics under which they are geodesible. Furthermore, for each of these vector fields F we give two first integrals h 1 and h 2 such that the integral curves of F are locally defined by the intersections of the level surfaces of h 1 and h 2 .

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Algebrization of Nonautonomous Differential Equations

Cited by 3 publications

References 9 publications

Introduction to $\mathcal{A}$-Calculus

Introduction to $\mathcal{A}$-Calculus

El método de algebrización de sistemas de ecuaciones diferenciales

On geodesibility of algebrizable planar vector fields

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