It is well known (see e.g. [1]) that the Cauchy functional equation

characterizes the function f:x→exx.

It was mentioned in [2D that the functions f:x→A exp(αx2m), g:x→exx/A can be characterized by the equation

(1)

but the proof was done only for m=l which was considerably simple.

A divisor d of a positive integer n is a unitary divisor if d and n/d are relatively prime. An integer is said to be unitary perfect if it equals the sum of its proper unitary divisors. Subbarao and Warren [2] gave the first four unitary perfect numbers: 6, 60, 90 and 87360. In 1969,1 reported [3] that

is also unitary perfect. The purpose of this paper is to show that this last number, which for brevity we denote by W, is indeed the next unitary perfect number after 87360.

By determining the spectrum of a particular set of equations of type 〈2, 2, 0, 0,0〉 it is shown that a positive integer n is the cardinality of the endomorphism monoid of a universal algebra of the form if and only if n is square.

The Hardy space Hp=Hp(U), where 1≤p≤∞ and U is the unit disc |z|<1, is shown to be an algebra under the product

Moreover for each p, 1≤p≤∞, there exists a constant Cp such that ‖f*g‖p,≤ Cp‖f‖p ‖g‖p.

For a continuously differentiable function g defined on an interval [α,β], define ||g|| to be the uniform norm of g, i.e. ||g||=sup ∊[α,β]|g(x)|. Define ||g||1, by ||g||1=max||g||, ||g'||}. We call the norm ||.||x the function-derivative norm.

Let us consider the well-known Hermite-Fejér interpolating process on the interval [—1,1] i.e. let

(1.1)

(sometimes we omit the second indices),

(1.2)

In certain systems analysis ([1], [2], [3]), it is essential to invert the n-dimensional Laplace transform and specify the inverse image at a single variable t.

The first author [2] calls a Hausdorff locally convex (abbreviated to l.c.) space (E, u) countably barrelled if each σ(E′, E)-bounded subset of E′ which is the countable union of equicontinuous subsets of E′, is itself equicontinuous.

Let B be the set of positive integers prime to d which are representable by some primitive, positive, integral binary quadratic form of discriminant d. It is the purpose of this note to show that the following asymptotic estimate for the number of integers in B less than or equal to x can be proved using only elementary arguments:

(1)

where c1 is the positive constant given in (17) below. (Using the deeper methods of complex analysis James [2] has proved this result with the error term ((log x)-1/2) replacing ((log log x)-1). Heupel [1] using a transcendental method as in James [2] improved this to ((log x)-1).)

We recall that a Dirac delta function δ(x) in the real number system is the idealization of a function that vanishes outside a "short" interval and satisfies It is conceived as a function δ for which δ(0)=+ ∞, δ(t)=0 if t≠0, and This function should possess the "sifting property" for any continuous function f. Even though certain sequences of functions are used, via a limit operation, to approximate a Dirac delta function (for details, see [3] and [4]), no function in has these properties.

In his paper [2] O. N. Golovin introduced the notion of a neutral polyverbal operation on groups, of which Moran's verbal operations [4] and Gruenberg's and Šmelkin's operations [3,5] are special cases. The free and direct multiplication we shall call here trivial operations. It is known [5,4] that Mal'cev's postulate does not hold for nontrivial verbal operations but the postulate of associativity does.

In [5] R. C. Lyndon gave the first proof of the Vaught conjecture: that if a, b9 and c are elements of a free group F such that a2b2=c2, then ab=ba. Lyndon's proof has been followed by many alternative proofs and generalizations [1, 2, 3, 4, 6, 8, 9, 10, 11, 13, 14] all of which involve rather long combinatorial arguments or group theoretical arguments of a noncombinatorial nature.This note provides a short, purely combinatorial proof of the Vaught conjecture.

In this note we show that every (not necessarily commutative) integral domain R has a quotient ring which, although need not be a field, has the property that all of its right invariant elements are units. As an application this shows that every PRI (principal right ideal) domain can be embedded in a simple PRI domain which is, in general, not a field.

Let S be a locally compact Borel subsemigroup of a locally compact semigroup G. It is shown that the algebra of all "absolutely continuous' measures on S is isometrically order isomorphic to the algebra of all measures in M(G) which are "concentrated" and "absolutely continuous" on S.

In this note we give a simple formula for the nth prime number. Let pn denote the nth prime number (p1=2, p2 = 3, etc.). We shall show that pn is given by the following formula.

In [3], Montgomery proved that if h is a pointwise periodic homeomorphism of a connected manifold without boundary onto itself, then h is periodic. Kaul generalized this result in [2] by showing that if X is a connected metrizable manifold without boundary and if (X, T)is a transformation group with T countable such that T is pointwise periodic, then T is periodic.

In a paper several years ago, Faith [2] proved an extension of a well-known theorem of Kaplansky [4]. His proof, even for the division ring case, was somewhat complicated. Using an old trick of Brauer [1] we show how Faith's theorem follows from Kaplansky's immediately.

Dans cette note, nous employons la terminologie et les définitions de [1] et [3]. A un sous-espace vectoriel W d'un espace vectoriel V pré-ordonné par un cône convexe pointé V+ avec élément unité e, nous associons la semi-norme suivante: pour x ϵ V, pw(x)=inf{ε>0|∃ωε ϵ W tel que -ωε-εe≤x≤ωε+εe}. Posons J(W)={x|pw(x)=0}={x ϵ V|∀ε > 0, ∃ωε ϵ W, -ωε-εe≤x≤ωε+εe}. J(W) est un idéal.

A family of sets is said to have property if there exists a set B such that B ∩ F ≠ Ø and B ⊅ F for every F ∊ . Such a B will be called suitable with respect to F. It is known (see [3]) that for each positive integer n there exists a family of sets satisfying the following conditions:

1. (a) |F| = n for each F ∊

2. (b) |F ∩ G| ≤ for F, G ∊ F ≠ G

3. (c) does not have property

Let X1, X2, X3 … be i.i.d., Sn their nth partial sum with Sn = 0; Suppose that