Some elements of a theory of multidimensional complex variables: part I. general theory

Martin, E. Dale

doi:10.1016/0016-0032(89)90024-0

Cited by 4 publications

(17 citation statements)

References 18 publications

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“…It is easy to see that this equation is the same as the condition for g to be A -differentiable with g' = / . Since / i s C 1 , we see that g is C 2 . Finally, if we have such a g,…”

Section: Second Derivatives Of A-differentiable Functionsmentioning

confidence: 90%

“…In view of Corollary 4.5, we see that in most cases the properties of the solutions of grad (w) = M T grad (v) found in [1] are consequences of stronger results about /1-differentiable functions. For an explicit example, let us look at the simplest nontrivial complex case, where the algebra is A = C[t]/(t) 2 . Let s = X + t, a generator of the algebra; its matrix in the basis 1, t is M =…”

Section: 5 Let a Be Frobenius And Suppose It Decomposes As The Promentioning

confidence: 99%

“…The smallest Frobenius algebra A not generated by one element is R[t, u]/(t 2 , u 2 ), and we can conclude by writing out the conclusion of Corollary 4.3 in this case. We have a basis 1 = e 0 , e x , e 2 , e 3 with e 2 x =e\ = Q and e x e 2 = e 3 . The function t/;(£ ;c,e,) = x 3 makes A Frobenius.…”

Section: Other Examplesmentioning

confidence: 99%

“…They discovered many special properties of these functions; in particular, the functions decompose into functions depending only on the variables in linear subspaces corresponding to the various primary components of M. Furthermore, even when we work with real rather than complex variables, the functions from a primary component corresponding to a conjugate pair of complex eigenvalues are real analytic. The first step in their treatment is to observe that w (when it exists) is determined by v up to constant, and thus the significant restriction is on v alone; explicitly, it is the set of differential equations MV 2 T . For example, if /0 -1 \ the dimension is 2 and M = I I, the condition on v is the Laplace equation.…”

Section: Introductionmentioning

confidence: 99%

“…Unfortunately, the theory of differentiable functions on such algebras is not very well known. (For instance, the recent works [2] and [4] are just rediscoveries of special cases.) The subject actually goes back to a paper published in 1893 by G. Scheffers (a student of Sophus Lie); a little more about its history can be found in [5] and [7].…”

Section: Introductionmentioning

confidence: 99%

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Differentiable functions on algebras and the equation grad (w) = M grad (v)

Waterhouse

1992

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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SynopsisLet U be a convex open set in a finite-dimensional commutative real algebra A. Consider A-differentiable functions f: U → A. When they are C2 as functions of their real variables, their A-derivatives are again A-differentiable, and they have second-order Taylor expansions. The real components of such functions then have second derivatives for which the A-multiplications are self-adjoint. When A is a Frobenius algebra, that condition (a system of second-order differential equations) actually forces a real function on U to be a component of some such f. If v is a function of n real variables, and M is a constant matrix, then the requirement that M∇(u) should equal ∇(w) for some w usually falls into this setting for a suitable A, and the quite special properties of such v, w can be deduced from known properties of A-differentiable functions.

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“…It is easy to see that this equation is the same as the condition for g to be A -differentiable with g' = / . Since / i s C 1 , we see that g is C 2 . Finally, if we have such a g,…”

Section: Second Derivatives Of A-differentiable Functionsmentioning

confidence: 90%

Section: 5 Let a Be Frobenius And Suppose It Decomposes As The Promentioning

confidence: 99%

Section: Other Examplesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations