A -Differentiability over Associative Algebras

Julio Cesar Avila (), Martín Eduardo Frías-Armenta and Elifalet López-González
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Julio Cesar Avila: Tecnologico de Monterrey, Escuela de Ingeniería y Ciencias, Ave. Eugenio Garza Sada 2 501 Sur, Col. Tecnológico, Monterrey, N.L., México, 64 700
Martín Eduardo Frías-Armenta: Departamento de Matemáticas, Universidad de Sonora, Blvrd. Rosales y Luis Encinas S/N, Col. Centro, Hermosillo 83 000, Sonora, Mexico
Elifalet López-González: Extensión Multidisciplinaria de la UACJ en Cuauhtémoc, Universidad Autónoma de Ciudad Juárez, Carretera Cuauhtémoc-Anáhuac, Col. Ejido Anáhuac Km. 3.5 S/N, Mpio. de Cuauhtémoc 31 600, Chihuahua, Mexico

Mathematics, 2025, vol. 13, issue 10, 1-21

Abstract: The unital associative algebra structure A on R n allows for defining elementary functions and functions defined by convergent power series. For these, the usual derivative has a simple form even for higher-order derivatives, which allows us to have the A -calculus. Thus, we introduce A -differentiability. Rules for A -differentiation are obtained: a product rule, left and right quotients, and a chain rule. Convergent power series are A -differentiable, and their A -derivatives are the power series defined by their A -derivatives. Therefore, we use associative algebra structures to calculate the usual derivatives. These calculations are carried out without using partial derivatives, but only by performing operations in the corresponding algebras. For f ( x ) = x 2 , we obtain d f x ( v ) = v x + x v , and for f ( x ) = x − 1 , d f x ( v ) = − x − 1 v x − 1 . Taylor approximations of order k and expansion by the Taylor series are performed. The pre-twisted differentiability for the case of non-commutative algebras is introduced and used to solve families of quadratic ordinary differential equations.

Keywords: associative algebras; Fréchet differentiability; Taylor approximations; Clifford algebras (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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