Generalized differential equations satisfying Carathéodory type conditions

Group rings satisfying generalized Engel conditions

Let R be a commutative ring with unity of characteristic r≥0 and G be a locally finite group. For each x and y in the group ring RG define [x,y]=xy-yx and inductively via [x ,_( n+1)  y]=[[x ,_( n)  y]  , y]. In this paper we show that necessary and sufficient conditions for RG to satisfies [x^m(x,y)   ,_( n(x,y))  y]=0 is: 1) if r is a power of a prime p, then G is a locally nilpotent group an...

Spaces of Functions Satisfying Simple Differential Equations

In [6]–[9] the first author published an algorithm for the conversion of analytic functions for which derivative rules are given into their representing power series ∞ ∑ k=0 akz k at the origin and vice versa, implementations of which exist in Mathematica [19], (s. [9]), Maple [12] (s. [4]) and Reduce [5] (s. [13]). One main part of this procedure is an algorithm to derive a homogeneous linear ...

Permutable Entire Functions Satisfying Algebraic Differential Equations

We show that if f and g are transcendental entire functions such that f(g) = g(f), then f satisfies an algebraic differential equation if and only if g does.

The graphs satisfying conditions of Ore's type

Generalized Bilinear Differential Equations

We introduce a kind of bilinear differential equations by generalizing Hirota bilinear operators, and explore when the linear superposition principle can apply to the resulting generalized bilinear differential equations. Together with an algorithm using weights, two examples of generalized bilinear differential equations are computed to shed light on the presented general scheme for constructi...