Laplace Equations for Real Semisimple Associative Algebras of Dimension 2, 3 or 4.

Cook, James S.; Leslie, W. Spencer; Nguyen, Minh L.; Zhang, Bailu

doi:10.1007/978-1-4614-9332-7_8

Cited by 2 publications

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“…We also do not understand the complete connection between the A-Laplacian introduced by W.S. Leslie in [6] and the generalized A-Laplace equations of Wagner. It might be possible to prove something interesting in terms of the Wirtinger calculus.…”

Section: Integrationmentioning

confidence: 90%

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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: Integrationmentioning

confidence: 90%

“…4 this is not a hash-tag 5 Note, R × R we have 1 = (1, 1) and in R 2×2 we have 1 = 1 0 0 1 thus the usual standard bases for R 2 or R 2×2 do not include the identity of the algebra. 6 when β = {e 1 , . .…”

Section: Calculus On a Normed Linear Spacementioning

confidence: 99%

Introduction to $\mathcal{A}$-Calculus

Cook¹

2017

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“…Properties of logarithms over many algebras are studied in [2]. The reader may find the many explicit examples in [4] a helpful supplement to our current work.…”

Section: Further Backgroundmentioning

confidence: 99%

Introduction to the Theory of $\mathcal{A}$-ODEs

BeDell,

Cook

2017

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We study the theory of ordinary differential equations over a commutative finite dimensional real associative unital algebra A. We call such problems A-ODEs. If a function is real differentiable and its differential is in the regular representation of A then we say the function is A-differentiable. In this paper, we prove an existence and uniqueness theorem, derive Abel's formula for the Wronskian and establish the existence of a fundamental solution set for many A-ODEs. We show the Wronskian of a fundamental solution set cannot be a divisor of zero. Three methods to solve nondegenerate constant coefficient A-ODE are given. First, we show how zero-divisors complicate solution by factorization of operators. Second, isomorphisms to direct product are shown to produce interesting solutions. Third, our extension technique is shown to solve any nondegenerate A-ODE; we find a fundamental solution set by selecting the component functions of the exponential on the characteristic extension algebra. The extension technique produces all of the elementary functions seen in the usual analysis by a bit of abstract algebra applied to the appropriate exponential function. On the other hand, we show how zero-divisors destroy both existence and uniqueness in degenerate A-ODEs. We also study the Cauchy Euler problem for A-Calculus and indicate how we may solve first order A-ODEs. introduction and overviewWe use A to denote a real unital associative algebra of finite dimension. Elements of A are known as A-numbers. We study calculus where real numbers have been replaced by A-numbers. The resulting calculus we refer to as A-calculus. Our typical goal is to find theorems which apply to as large a class of real commutative associative algebras as possible.In this paper we study the elementary theory of ordinary differential equations over A. In particular, this means the differential equation, or system of differential equations, involve a set of dependent variables all of which depend on a single independent A-variable. We call such differential equations A-ODEs.

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Laplace Equations for Real Semisimple Associative Algebras of Dimension 2, 3 or 4.

Cited by 2 publications

References 8 publications

Introduction to $\mathcal{A}$-Calculus

Introduction to $\mathcal{A}$-Calculus

Introduction to the Theory of $\mathcal{A}$-ODEs

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