We are concerned with the existence proof of solution of the Falkner-Skan equation

1

subject to boundary conditions

2

The first existence and uniqueness proof based on a fixed point theorem was given by Weyl [4] in 1942, with the added assumption that f' > 0. In 1960, Coppel [1] proved the existence (and uniqueness with the assumption 0 < f' < 1) by considering trajectories in the three-dimensional phase space.

The following theorem is due to Walsh [2]. For another proof see [1].

Theorem A. All the zeros of the polynomial p(z) = a0 + a1z + … an-1zn-1+zn lie on the disk.

where M = Σj = 2n|an-j|1/j

This note is intended as a commentary on a part of the paper [1] by Gulliksen, Ribenboim, and Viswanathan. It takes its inspiration from a colloquium talk by P. Ribenboim.

Our aim is a partial generalization of the theorem of Cohen-Seidenberg (cf. [2], IX. 1, Propositions 9 and 10) to noncommutative algebras.

Definition. Let A be a ring with 1 in the center of another ring B (with the same 1). B is said to be integral over A, if each element b ∊ B satisfies an integral equation

It is shown that no pair of orthogonal Steiner triple systems of order 9 exists.

If p is a polynomial in one real variable and p(x) = p(-x) then p has only even powers of x and is thus a polynomial in x2. If p is a polynomial in n variables and p(x1, …, xn) = p(y1, …, yn) when x12 + … + xn2 = y12+ … + yn2 then p is a polynomial in q where q(x1, …, xn) = x12 + … + xn2.

The theorem stated below is due to R. Balbes. The present proof is direct; it uses only the following two well-known facts: (i) Let K be a category of algebras, and let free algebras exist in K; then an algebra is projective if and only if it is a retract of a free algebra, (ii) Let F be a free distributive lattice with basis {xi | i ∊ I}; then ∧(xi | i ∊ J0) ≤ ∨(xi | i ∊ J1) implies J0∩J1≠ϕ. Note that (ii) implies (iii): If for J0 ⊆ I, a, b ∊ F, ∧(xi | i ∊ J0)≤a ∨ b, then ∧ (xi | i ∊ J0)≤ a or b.

Consider the linear ordinary differential equation

1

where x ∊ En, the n-dimensional Euclidean space and A is an n × n constant matrix. Using a matrix result of Sylvester and a stability result of Perron, Lyapunov [4] established the following theorem which is basic in the stability theory of ordinary differential equations:

Theorem (Lyapunov). The following three statements are equivalent:

(I) The spectrum σ(A) of A lies in the negative half plane.

(II) Equation (1) is exponentially stable, i.e. there exist μ, K>0 such that every solution x(t) of (1) satisfies

2

where ∥ ∥ denotes the Euclidean norm.

(III) There exists a positive definite symmetric matrix Q, i.e. Q=Q* and there exist q1,q2>0 such that

3

satisfying

4

where I is the identity matrix.