Let T be a bounded linear operator defined on a Hilbert space H. An element z∈H is called a least squares solution of the equation

if . It is easily shown that z is a least squares solution of (1) if and only if z satisfies the normal equation

If the samples of sizes n1, n2,…, nq+1 are available from (q+1) normal populations with different mean vectors μα, α=l, 2,…, q+1 and the same covariance matrix ∑ and if x’=(x1, x2, …, xp) denotes the vector of p variates on which the measurements are made, then we obtain the following multivariate analysis of variance table.

Recently, in the Research Problems of Canadian Mathematical Bulletin, Vol. 14, No. 4, 1971, there appeared a problem which asks “Is a prime Von Neumann regular ring pimitive?” While we are not able to settle this question one way or the other, we prove that in a Von Neumann regular ring, there is a maximal annihilator right ideal if and only if there is a minimal right ideal.

In [2: p. 415], P. Gabriel proves that if R is a ring with 1 and S is a non-empty multiplicative set such that 0∉S, then S-1R exists if and only if for every pair (a, s)∈R×S, there is a pair (b, t)∈R×S such that at=sb and if s1a=0 for some s1 in S then as2=0 for some s2 in S. The purpose of this note is to give a self contained elementary proof of Gabriel’s result.

Let p be a rational prime and n a positive integer ≥2. We denote by an(p) the least positive integral value of a for which the polynomial xn+x+a is irreducible (mod p), and set

(1)

The numerical range W(T) of a bounded linear operator T on a Hilbert space H is defined by

W(T) is always a convex subset of the plane [1] and clearly W(T) is bounded since it is contained in the ball of radius ‖T‖ about the origin. Which non-empty convex bounded subsets of the plane are the numerical range of an operator? The theorem we prove below shows that every non-empty convex bounded subset of the plane is W(T) for some T.

A standard example of a matrix for which the computation of eigenvectors and eigenvalues is very awkward is the matrix A with ars=1/(r+s—1), 1≤r≤n, 1 ≤s≤n. It is therefore of interest that A commutes with a tridiagonal matrix.

The Hermite-Fejér interpolation polynomial Hn[f] of degree ≤2n—1 is defined by

(1)

Where

(2)

are the zeroes of Chebyshev polynomial of first kind Tn(x)=cos n(arc cos x).