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Keywords = (left-, right-) pre-commutative

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21 pages, 296 KB  
Article
A-Differentiability over Associative Algebras
by Julio Cesar Avila, Martín Eduardo Frías-Armenta and Elifalet López-González
Mathematics 2025, 13(10), 1619; https://doi.org/10.3390/math13101619 - 15 May 2025
Viewed by 444
Abstract
The unital associative algebra structure A on Rn 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, [...] Read more.
The unital associative algebra structure A on Rn 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)=x2, we obtain dfx(v)=vx+xv, and for f(x)=x1, dfx(v)=x1vx1. 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. Full article
(This article belongs to the Special Issue Applications of Differential Equations in Sciences)
7 pages, 223 KB  
Article
On Pre-Commutative Algebras
by Hee Sik Kim, Joseph Neggers and Sun Shin Ahn
Mathematics 2019, 7(4), 336; https://doi.org/10.3390/math7040336 - 8 Apr 2019
Cited by 9 | Viewed by 2404
Abstract
In this paper, we introduce the notions of generalized commutative laws in algebras, and investigate their relations by using Smarandache disjointness. Moreover, we show that every pre-commutative B C K -algebra is bounded. Full article
(This article belongs to the Special Issue General Algebraic Structures)
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