It is shown that Bernstein Polynomials do not diminish the total “twist” of space curves in contrast to their length and curvature diminishing properties. This phenomenon is shown to be related to the fact that Bernstein Polynomials of a plane curve may have more inflections than the curve possesses.

Let A be a real n × n matrix and define the Hadamard ratio h(A) to be the absolute value of det A divided by the product of the Euclidean norms of the columns of A. It is shown that if A is a random variable whose distribution satisfies some simple symmetry properties then the random variable log h(A) has mean and variance . In particular, for each ε > 0, the probability that h(A) lies in the range tends to 1 as n tends to ∞.

In this paper we shall use matrix methods to obtain several generalizations of a well known result of Eneström and Kakeya about the location of the zeros of polynomials. We shall also obtain zero-free regions of polynomials having complex coefficients. Finally we prove some results concerning the zeros of a class of polynomials.

It is proved that a function is nondecreasing if it is Baire one and Darboux and fulfills Lusin’s condition (N), and if its derivative is non-negative for almost every point at which the function is derivable. Using this result, a process to formulate various results on the existence and the valuation of indeterminate forms via various monotonicity theorems is illustrated. In particular, the ordinary Bernoulli-L’Hospital rule and some of its variations obtained recenty by A. M. Ostrowski are generalized.

Let L be an a-implicative semilattice. We obtain a characterization of those elements which cover a. This gives a characterization of atoms in pseudocomplemented semilattices, and leads to various results on primes and irreducibles in semilattices. As an application, we prove that in a complete, atomistic lattice L, the following are equivalent (i) L is implicative (ii) L is (2, ∞) meet distributive (iii) each element of L is a meet of primes.

It is shown that if the isometric image of a linear subspace of Hilbert space operators is irreducible in a strong sense, then the isometry is either a multiplicative or anti-multiplicative map, possibly followed by multiplication by a unitary.

In this paper it is proved that a finite translation plane of order n ≡ 3 (mod 4) which contains an oval with the four point Pascalian property, or a finite dual translation plane of order n ≡ 3 (mod 4) which contains an oval with the four point Pascalian property, can be coordinatized by a commutative semifield.

Etant donné un anneau commutatif unitaire A et un ensemble d’indices infini I; on peut obtenir, par passage à la limite projective, essentiellement trois anneaux A1, A2, A3 de séries formelles en des indéterminées indexées par I et à coefficients dans A. On étudie alors les implications A factoriel ⇒ A1, A2, A3 factoriels et A de Krull ⇒ A1, A2, A3 de Krull. On donne à cet effet des conditions pour qu’une limite projective d’anneaux factoriels soit factorielle.

In this paper we give necessary and sufficient conditions under which the group of units of a group ring of a finite group G over a field K does not contain a free subgroup of rank 2.

Some extensions of this results to infinite nilpotent and FC groups are also considered.

Let K be a field of characteristic 0 and t an indeterminate. It is shown that the absolute Galois group of K(t) is the semi-direct product of a free profinite group with the absolute Galois group of K.

It is shown that if L is a Dedekind complete Riesz space equipped with a locally solid topology T defined by strongly (A, 0) Riesz pseudonorms, then order intervals of L are T-complete. This is an extension of a well known theorem of Nakano. The second part of the paper gives a necessary and sufficient condition for topological completeness of order intervals in a Dedekind σ-complete Riesz space which has a weak order unit and which is equipped with a locally solid σ-Fatou topology.

Given a Prüfer domain R and a prime ideal P in R, we study some conditions which force P to be a divisorial ideal of R. This paper extends some recent work of Huckaba and Papick.

In his Ph.D. thesis Allaway found all polynomials that can be represented as

and a0bn≠0. We solve the essentially equivalent problem of finding all symmetric polynomials with when are orthogonal with respect to dα(x). The polynomials are the continuous q-ultraspherical polynomials and some of their limiting cases.

During the last decade or so there has been a revival of interest in the analysis of error-bounds f(s)-S(s) for different classes of functions and their interpolatory splines of odd degree on a finite interval with variations on end conditions. Our object is to present a unified treatment of the asymptotic error expansion both for even and for odd degree interpolatory splines.

First we construct spreads consisting of analytic curves (circular arcs and segments), without points of finite multiplicity. Then we see that, in the sense of Baire categories, most such spreads have no points of finite multiplicity.

The following two theorems are proved. If R is an Archimedean conducive integral domain, then R is quasilocal and dim(R) ≤1. If each overring of an integral domain R has ascending chain condition on divisorial ideals, then the integral closure of R is a Dedekind domain. Both theorems sharpen results already known in the Noetherian case. The second theorem leads to a strengthened converse of the Krull-Akizuki Theorem. We also investigate the effect of restricting the hypothesis in the second theorem to the proper overrings of R.

The Franklin spline system in [0,1] has been generalized by Strömberg to a system in ℝn which is an unconditional basis in Hp(ℝn) for p > n/(n + m +1). Here the natural number m is the order of the system. For some of these values of p, it was known that the Hp quasi-norm is equivalent to a certain expression containing the coefficients of the function with respect to this basis. We prove this equivalence for all p > n/(n + m +1).

Let KG be the group ring of the group G over the field K and U(KG) its unit group. When G is finite we derive conditions which imply that every noncentral subnormal subgroup of U(KG) contains a free group of rank two. We also show that residual nilpotence of U(KG) coincides with nilpotence, this being no longer true if G is infinite.

We can answer partially the following question: when is G sub-normal in U(KG)?

Inverting the Castelnuovo bound in two ways, we show that for given integers p ≥ 0, d > 1, n > 1, we can find a smooth irreducible curve of genus p which contains a linear system of degree d and of maximal dimension relative to the given data p and d, and a smooth irreducible curve of genus p which contains a linear system of dimension n and of minimal degree relative to the data p and n.

The number of sets of integers which are realizable as block sizes of a pairwise balanced design of order n is between and ; in contrast, when the multiplicity of each block size is also specified, the number of multisets which can be realized is between and . Although this gives a reasonable bound on the number of multisets which can be realized, a good characterization is not likely to exist; deciding whether a multiset can be so realized is NP-complete.