We present an outline of the problem of irrotational compressible flow past an airfoil at speeds that lie somewhere between those of the supersonic flight of the Concorde and the subsonic flight of commercial airlines. The problem is simplified and the important role of modifying the equations with physics terms is examined.

We give a criterion in order that an affine variety defined over any field has a complete intersection (ci.) embedding into some affine space. Moreover we give an example of a smooth real curve C all of whose embeddings into affine spaces are c.i.; nevertheless it has an embedding into ℝ3 which cannot be realized as a c.i. by polynomials.

We show that any W*-continuous semi-group in L∞ is L1-norm continuous. As an application we prove the n-dimensional local ergodic theorem in L∞. We also note that any bounded additive process in L∞ is absolutely continuous.

For n = 1 this local theorem improves those of R. Sato [14] and D. Feyel [6] and for n ≥ 1 it generalizes M. Lin's ones which hold for positive operators [12].

A topological space X is said to be somewhat normal provided that for each open cover is a normal cover of X. We show that a completely regular somewhat normal space need not be normal, thereby answering a question of W. M. Fleischman. We note that a collectionwise normal somewhat normal space need not be almost 2-fully normal, as had previously been asserted, and that neither the perfect image nor the perfect preimage of a somewhat normal space has to be somewhat normal.

A commutative semigroup G of continuous, selfmappings on (X, d) is called proximately nonexpansive on X if for every x in X and every (β > 0, there is a member g in G such that d(fg(x),fg(y)) ≤ (1 + β) d (x, y) for every f in G and y in X. For a uniformly convex Banach space it is shown that if G is a commutative semigroup of continuous selfmappings on X which is proximately nonexpansive, then a common fixed point exists if there is an x0 in X such that its orbit G(x0) is bounded. Furthermore, the asymptotic center of G(x0) is such a common fixed point.

An asymptotic approximation is obtained, as k → ∞, for the integral

where Φ is the cumulative distribution function for a standard normal random variable, and L is a positive constant. The problem is motivated by a question in statistics, and an outline of'the application is given. Similar methods may be used to approximate other integrals involving the normal distribution.

Let G be the group of integers of a p-series field. A class {E(θ)} of perfect null subsets of G is introduced and classified into M-sets and U-sets according to the arithmetical nature of the field element θ. This is analogous to the well-known classification, involving Pisot numbers, of certain Cantor sets on the circle.

Let Tk = T1#...#T1 T1 = Sl x Sl, Uk = ℝP2#... #ℝP2, and G is a finite group. We prove (1) Every free action of G on Ul + 2 lifts to a free action of G on the orientable two fold cover Tl+1 → Ul+1 and (2) The minimum k such that can act freely on Tk is ml((l - 2)/2) + 1 if m = 2 or l is even and ml((l - 1)/2) + 1 otherwise.

Let L be a finitely generated field extension of a field K. The order of inseparability of L/K is the minimum of {n|[L:S] = pn where S is a separable extension of K}. If V is a subfield of L/K, then its order of inseparability is less than or equal to that of L/K. This paper examines the question of when there are unique minimal subfields of order of inseparability n — j, 0 ≤ j ≤ n.

The abelian groups which are the additive groups of only finitely many non-isomorphic (associative) nilpotent rings are studied. Progress is made toward a complete classification of these groups. In the torsion free case, the actual number of non-isomorphic nilpotent rings these groups support is obtained.

The first result gives a (modest) improvement of the best general bound known to date for the rank of the intersection U ∩ V of two finite-rank subgroups of a free group F in terms of the ranks of U and V. In the second result it is deduced from that bound that if A is a finite-rank subgroup of F and B < F is non-cyclic, then the index of A ∩ B in B, if finite, is less than 2(rank(A) - 1), whence in particular if rank (A) = 2, then B ≤ A. (This strengthens a lemma of Gersten.) Finally a short proof is given of Stallings' result that if U, V (as above) are such that U ∩ V has finite index in both U and V, then it has finite index in their join 〈U, V〉.

Let ω(n) denote the number of distinct prime divisors of a positive integer n. Then we define and where are primes and r ≥ 2. Similarly denote by the number of divisors of n and let be defined by where are the divisors of n. We prove that there exists constants A and B such that and

Formulae for the nilpotency class of the p-sylow subgroups of GL(n, q) where (p, q) = 1 are derived. These formulae are used in author's following paper: “On the other pα qβ theorem of Burnside”.

Let A be a finite dimensional algebra (not necessarily associative) over a field, whose automorphism group acts transitively. It is shown that K = GF(2) and A is a Kostrikin algebra. The automorphism group is determined to be a semi-direct product of two cyclic groups. The number of such algebras is also calculated.

Identities containing the Gaussian binomial coefficient are obtained. From these results, we deduce some identities of the combinatorial analysis which contain the binomial coefficient

Within the framework of a-differentiability, introduced by H. R. Fischer in locally convex spaces, sufficient conditions for an inverse mapping theorem between Fréchet spaces are established.

The question of which degree 0 stable cohomology operations for connective K-theory localized at a prime p arise from operations for non-connective K-theory is investigated. A necessary and sufficient condition is established, and an example of a connective operation not arising in this way is constructed.