Prof. G. D. Birkhoff

On a Result of G. D. Birkhoff on Linear Differential Systems

We give a simple example to show that a result on the equivalent singular points of systems of ordinary linear differential equations due to G. D. Birkhoff [3; 5, pp. 252-257] needs amendment. In matrix notation, in which Y, P, etc. denote nXn matrix-valued functions of a complex variable z, this result is as follows. A. Result. Every linear differential system (1) Y'(z) = P(z)Y(z) with a singu...

Prof. Dr. Prof. Prof.

On the Vertices of the d-Dimensional Birkhoff Polytope

Let us denote by Ωn the Birkhoff polytope of n× n doubly stochastic matrices. As the Birkhoff–von Neumann theorem famously states, the vertex set of Ωn coincides with the set of all n× n permutation matrices. Here we consider a higherdimensional analog of this basic fact. Let Ω n be the polytope which consists of all tristochastic arrays of order n. These are n×n×n arrays with nonnegative entri...

Sub-Birkhoff

A fundamental result in rewriting is : E |= s ≈ t ⇐⇒1 E ` s = t ⇐⇒2 s↔E t 1. is Birkhoff’s theorem expressing that semantic consequence (|=) coincides with syntactic consequence (`). 2. expresses that equality in equational logic (=) coincides with convertibility in rewriting (↔∗), a property known as logicality. We show that the result can be extended to hold for several sub-equational logics ...

Topological Birkhoff

One of the most fundamental mathematical contributions of Garrett Birkhoff is the HSP theorem, which implies that a finite algebra B satisfies all equations that hold in a finite algebra A of the same signature if and only if B is a homomorphic image of a subalgebra of a finite power of A. On the other hand, if A is infinite, then in general one needs to take an infinite power in order to obtai...