The classical theorem of Müntz and Szász says that the span of

is dense in C[0,1] in the uniform norm if and only if . We prove that, if {λi} is lacunary, we can replace the underlying interval [0,1] by any set of positive measure. The key to the proof is the establishment of a bounded Remez-type inequality for lacunary Müntz systems. Namely if A ⊆ [0,1] and its Lebesgue measure µ(A) is at least ε > 0 then

where c depends only on ε and Λ (not on n and A) and where Λ:=infiλi+1/λi>1.

The base transformations which transform the chord system of one twisted cubic into the chord system of another.

Recently J. M. Osborn has investigated the structure of a simple commutative non-associative algebra with unity element satisfying a polynomial identity, (4), (5) and (6). From his work it seems likely that if such an algebra is of degree three or more it is necessarily power-associative. In (4) he establishes a hierarchy of identities with the property that each identity is satisfied by an algebra satisfying no preceding identity. Following (5), (6),the next identity to consider is

where a, c and h are elements of the ground field.

A star operation is defined and studied for the Steenrod algebra. Numerous product formulas of Steenrod operations are presented.

For what sequences {an} of points of the open unit disc D does there exist a constant k such that

for all bounded harmonic functions f on D?

1. The term Second Moment, which is already in frequent use, as applied to lines, areas and volumes, as well as masses, is preferable to the older term Moment of Inertia which properly applies only to masses.

If

represent the points A, B and C, D respectively and if

then A, B, C, D are four harmonic points on a circle. If 0 is the mid point of AB then OA2 = OC. OD and ∠COB = ∠DOB.

Let Tm, Vmn be Hermitean linear operators on complex Hilbert spaces Hm, m=1…k. A nonzero column vector satisfying

will be called an eigenvalue. This type of problem has been studied extensively by Atkinson [2] from the viewpoint of determinantal operators on the tensor product We shall connect his work with more recent investigations [5,7] of eigenvalue indices based on minimax principles for , which can be viewed as an operator on .

For a large class of C*-algebras including all von Neumann algebras, the central Haagerup tensor product of the multiplier algebra with itself has an isometric representation as completely bounded operators.

Suppose that un ≥ un+1 > 0, for n = 1, 2, … and that e1, e2, … are factors which make convergent.

Let where all the a's are positive, diverges, and Define En by the equation

For the case where an = 1 it has been shewn by Fuchs1 and Karamata1 that, under various conditions, En = 0(1) The object of this note is to extend some of their results.

If the elements of a symmetric matrix lie in the real field it is well known that the roots of its characteristic equation are real. This implies that the discriminant of that equation (i.e. the product of the squared differences of the roots) is a polynomial in the elements which is non-negative and the same must be true for the leading coefficients of all the other Sturm functions associated with the characteristic equation. One would expect that it should be possible to express them as a sum of squares. Conversely, such an expression would establish the reality of the roots.

Jacobsthal (4)has proved that the n×n matrix

is invertible with the inverse,

Here μ(x) denotes the Möbius function for positive integral x and is assumed to be 0 for other values; [x] has its usual meaning as the number of positive integers ≦x.

Provided that the weight function ω satisfies certain submultiplicative and decay conditions, the discrete convolution algebra ℓ1(ω) becomes a commutative radical Banach algebra with identity adjoined. There are obvious closed ideals in ℓ1(ω) and these are denoted standard ideals. Earlier results of Thomas, strengthened by Yakubovich and Domar, showed that if the weight ω is star-shaped then all closed ideals are standard. Consequently, the closed ideal generated by any element f in ℓ1(ω) must be standard.

The requirement that ω be star-shaped (essentially that ω(n)1/n must decrease to zero) is somewhat restrictive in that no local maxima of ω(n)1/n are allowed. We generalize this previous result to apply to the larger class of ε-star shaped weights (0 < ε ≤ 1) which allow such local maxima. If f is a non-zero element on ℓ1(ω) we let the integer α(f) = k0 denote the index of its first non-zero term. We introduce the concept of an ε-peak point for k0. If ε = 1 then ω is star-shaped in the usual sense and there are an infinite number of 1-peak points for any k0. Although this latter fact may fail if 0 < ε < 1, if ω(n)1/n tends to zero sufficiently quickly (dependent on k0 and ε) there will always be an infinite number of ε-peak points for k0.

Our main result is that if ω is an ε-star shaped weight, if f is an non-zero element of ℓ1(ω), if α(f) = k0, and if the number of ε-peak points for k0 is infinite, then the closed ideal generated by f is standard.

An infinite family of triangles, having a common pole (determined by three fixed lines through it), and polar (determined by three fixed points on it), and an allied family of conics with imaginary double contact.

Construction for pole of a line with reference to a triangle.