Cauchy-Difference Conservative Vector Fields for Dimension Two and Three

Document Type

Article

Publication Date

1994

Department

Department of Mathematical Sciences

Abstract

Let G be an abelian group and X a vector space over the rationals. If Φ: G → X its 1st Cauchy difference is the function K2Φ:G2 → X defined by K2Φ(x1, x2) = Φ(x1 + x2) − Φ(x1) − Φ(x2) and in general for n = 2, 3, … the (n − 1)th Cauchy difference of Φ is the function, KnΦ: Gn → X, defined by where ϕ ≠ ⊆In = {1, 2, …, n} and xJ = Σj∈JxJ. If Ψ: Gn → X(n = 2, 3, …) then its i-th partial Cauchy difference of order r(r = 2, 3, …), Kr(i)Ψ: Gn+r+1 → X, is its Cauchy difference of order r − 1 with respect to its i-th variable with all the other variables held fixed. For n = 2 we have K21Ψ(x1, x2, x3) = Ψ(x1+ x2, x3) − Ψ(x1, x3) − Ψ(x2, x3) and K22Ψ(x1; x2, x3) = Ψ(x1, x2+ x3) − Ψ(x1, x2) − Ψ(x1, x3). In this paper if ƒ = < ƒ1, … ƒn > : Gn → xn the solution of K2(i)ƒj = K2(j)ƒi (i ≠ j) is given for n = 2 and 3.

Publication Title

Results in Mathematics

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