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

Abstract

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.

𝐴-differentiability and 𝐴-analyticity

“…and we can conclude as in the proof of Theorem 2.3 of [4]; i.e., since F is Adifferentiable, we have…”

“…We say that f is A-differentiable of class C r and set f ∈ C r A (U ) if for every a ∈ U, f (a), f (a), ..., f (r) (a) exist and are continuous. For further properties on Adifferentiability we refer to [2,3,4].…”

A Weierstrass representation theorem for Lorentz surfaces

“…The particular case of the algebra L was studied in [2,11,17,19] in the 1950s. Recently there has been a renewed interest in the Lorentz numbers analysis [1,5,15,16,20,23,24].…”

“…However, if f is L-differentiable in then f is differentiable in the ordinary sense in . There is no regularity property for L-differentiable maps and there exist L-differentiable maps of any class [5,8,23,24]. The following is a well-known fact from the theory of paracomplex maps.…”

Differentiability and Analyticity

“…We say that f is A-differentiable of class C r and set f ∈ C r A (U ) if for every a ∈ U, f (a), f (a), ..., f (r) (a) exist and they are continuous. For further properties on A-differentiability we send to [2,3,4].…”