According to the still unproved conjecture of Borsuk [1] a bounded subset A of the Euclidean n-space En is a union of n + 1 sets of diameters less than the diameter D of A. Since A can be imbedded in a set of constant width D, [2], it may be assumed that A is already of constant width. If in addition A is smooth, i. e., if through every point of its boundary ∂A there passes one and only one support plane of A, then the truth of Borsuk′s conjecture can be proved very easily [3]. The question arises whether Borsuk′s conjecture holds also for arbitrary smooth convex bodies, not merely for those of constant width. Since it is not known whether a smooth convex body K can be imbedded in a smooth set of constant width D, the answer is not immediate. In this note we show that the answer is affirmative.

If one considers the theories of Hurewicz, - Serre - or other fibrations in the categories of topological or pointed topological spaces, one can see that many of the fundamental theorems can be formulated and proven in the general case of categories for which certain functors and natural transformations are given. And, since fibrations may be defined either by a cylinder or a path space construction, we shall give in an analogous way two different definitions of fibrations in the general case.

In [3], the author proved that a ring R with identity is right noetherian and right injective if and only if R is a direct sum of a finite number of uniform right ideals, which are completely primary in the sense of that paper. In this paper, we shall determine the structure of such rings in the case where the sum of the isomorphic uniform components are twosided ideals. The ring is found to be a direct sum of total matrix rings over local rings.

If f(x) is a polynomial with integral coefficients then the integer r is said to be a residue of f(x) modulo an integer m if the congruence

is soluble for x; otherwise r is termed a non-residue. When m is a prime p, Mord ell [4] has shown that the least nonnegative residue ℓ of f(x) (mod p) satisfies

The space co consists of the sequences tending to zero with addition and scalar multiplication defined coordinate-wise and with the sup norm. The space ℓ1 consists of the sequences b = (bi) under coordinate»wise arithmetic for which

The different results proved in this paper do not have very much in common. Since they all deal with the location of the zeros of a polynomial, we have decided to put them in one place. Improving upon a classical result of Cauchy we obtain in § 2 a circle containing all the zeros of a polynomial. In § 3 we obtain an extension of the well known theorem of Enestrőm and Kakeya concerning the zeros of a polynomial whose coefficients are non-negative and monotonie.

Throughout this paper the arithmetic functions L(n) and w(n) denote respectively the number and product of the distinct prime divisors of the integer n > 1, with L(1) = 0 and w(1) = 1. Also let

We recall that an arithmetic function f(n) is said to be multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever (m, n) = 1, where (m, n) denotes as usual the greatest common divisor of m and n.

In [1] J. O. C. Ezeilo asks whether the equation

1

has periodic solutions for a ≠ 0. Since (1) has a two-dimensional space of solutions of period 2π if sin x is approximated by x, it is plausible to conclude, by analogy with x″ + sin x = 0, that (1) does have periodic solutions. However, when one applies the standard theory of perturbation of periodic solutions (treating a as small, see [2]), one finds that the only real periodic solutions obtainable in this manner are the trivial ones x(t, a) = nπ for some integer n.

Let f(t) be integrable L(-π, π) and periodic with period 2 π and let

1.1

be its Fourier series. The series

1.2

obtained by term by term differentiation of the series (1.1) is called the derived Fourier series of f.

The pk+2 - integral of James [2] is strong enough to integrate a trigonometric series of the form

1.1

which is summable (C, k) in [0,2π], provided an extra condition holds involving the conjugate series

1.2

Considering series with coefficients o(n), Taylor [5] constructed an integral (the AP-integral) which successfully integrates series of the form (1. 1) which are Abel summable provided an extra condition holds involving the Abel means of the conjugate series (1.2).

As is well known, every real function is the pointwise (uniform) limit of a sequence of functions with a finite (countable) range of values. Monna [5] and Kvačko [4] suggested some extensions of this theorem to functions with values in a separable metric space. In the present note we give some further generalizations, with an emphasis on uniform approximations which have many applications in the generalized theory of measure and integration. In particular, we consider measurable abstract functions (mappings).

In [6] J. Rainwater obtained the following theorem.

Theorem. Let N be a normed linear space, {xn} a bounded sequence of elements in N and X ∊ N. for each extreme point f of the unit ball of N✶, then {xn} converges weakly to x.

Now let X be a compact Hausdorff space and H a linear subspace of C(X) (all real-valued continuous functions on X ) which separates the points of X and contains the constant functions. If x∊X, then MX(H) denotes the set of positive linear functionals μ on C(X) such that μ(h) = h(x) for all h in H.

In a number of papers [1] - [7], successively simpler proofs were given for the following inequality of Opial [1], in case p=1.

Theorem 1. If x(t) is absolutely continuous with x(0)=0, then for any p ≧ 0,

(1)

Equality holds only if x(t) = Kt for some constant K.

In [l] Melzak has posed the following problem: “A plane finite set Xn consists of n ≥ 3 points and contains together with any two points a third one, equidistant from them. Does Xn exist for every n ? Must it consist of points lying on some two concentric circles (one of which may reduce to a point)? How many distinct (that is, not similar) Xn are there for a given n ? …” We shall here provide a construction for uncountably many Xn for every n > 4, and a counterexample to the second question above.