Let K be the finite field of pn elements, and Zp its prime subfield. It was proved by Davenport [3] that if p > p0(n) and θ is any given generating element of K, then there exists an integer m such that θ + m is a primitive root of K.

In this paper X(t) denotes Brownian motion on the line 0 ≤ t < ∞, E is a compact subset of (0, ∞) and F a compact subset of ( -∞, ∞). Then δ(E, F) is the supremum of the numbers c such that

In [4, 5] some lower and upper bounds for δ were found in terms of dim E and dim F, and it seemed possible that δ(E, F) could be determined entirely by these constants; this much is false, as the examples in our last paragraph will demonstrate. Here we shall show that δ(E, F) depends on a certain metric character η(E × F). However, η is not calculated relative to the Euclidean metric: the set F must be compressed to compensate for the oscillations of most paths X(t). Fortunately, η(E × F) can be calculated for a large enough class of sets E and F, by means of sequences of integers, to test any conjectures (and disprove most of them). In passing from η to δ we present a slight variation of Frostman's theory [3II].

A uniqueness theorem is obtained for a theory of linear thermoelasticity which allows for“second sound”. Propagation of acceleration waves in an isotropic material is also discussed.

In 1944 and 1945 H. Hadwiger [1, 2] proved the following theorems.

Theorem A. Let En be covered by n + 1 closed sets. Then there is one of the sets, within which all distances are realized.

Theorem B. Let En be covered by 4n−3 closed sets that are all mutually congruent. Then all distances are realized within each set.

Here a distance d is realized within a set S, if there are points x, y in S at distance d apart.

Let a−N,…, aN be any complex numbers and let

If y1,y2,…,yM are any real numbers such that

one form of the large sieve inequality (see [1], Chapter 2 and the Appendix) asserts that

where yM+1 ≡ y1 (mod 1).

Let be a topological property. We say that the locally finite sum theorem holds for the property if the following is true:

“If {Fα : α ∈ Λ} be a locally finite closed covering of X such that each Fα possesses the property , then X possesses .” The above property is known as the locally finite sum theorem (referred to as LFST in the present note). The LFST has been of interest to many people for it holds for many interesting properties such as metrizability, paracompactness, normality, collectionwise normality, local compactness, stratifiability, the property of being a normal M-space etc. etc. In [10], a large number of properties for which the LFST holds have been noted. In the same paper, a general method for proving this has been obtained. It has been shown that if a property is such that it is preserved under finite-to-one, closed continuous maps and also preserved under disjoint topological sums, then the LFST holds for and the same has been used to establish the LFST for a large number of properties. In [9,10], several interesting consequences of the LFST have been obtained. In the present note, some more interesting consequences of it have been obtained in regular, normal, collectionwise normal and countably paracompact spaces. Also, the LFST has been established for some other topological properties

Heilbronn [5] proved that for any ε > 0 there exists C(ε) such that for any real θ and N ≥ 1 there is an integer x satisfying

where ‖α‖ denotes the difference between α and the nearest integer, taken positively. The result is uniform in θ and so analogous to Dirichlet's inequality for the fractional parts of nθ. The result has been generalized to simultaneous approximations by Danicic [1] and Ming-chit Liu [6]. Here we shall extend the result to any finite number of simultaneous approximations when x2 is replaced by a k-th power.

A tournament is a relational structure on the non-empty set T such that for x, y ∈ T exactly one of the three relations

holds. Here x → y expresses the fact that {x, y} ∈ → and we sometimes write this in the alternative form y ← x. Extending the notation to subsets of T we write A → B or B ← A if a → b holds for all pairs a, b with a ∈ A and b ∈ B. is a subtournament of , and is an extension of , if T′ ⊂ T and →′ is the restriction of → to T′; we will usually write 〈′, → 〉 instead of 〈 ′, → ′〉. In particular, if |T − T′| = k, we call a k-poinf extension of .

We consider positive-definite ternary quadratic forms with integer coefficients. Such a form, f, can be written in matrix notation as

Here x′ is the transpose of the column vector x = {x1, x2, x3) and a,ij = aji is the coefficient of xixj in f. Clearly det A is positive and even and so

is a negative integer.

In this note we report on some numerical results regarding reverse flow solutions of the Falkner-Skan equation

on the half line 0 < t < ∞, which satisfy the boundary conditions

and

Let 2 = p1 < p2 < … be the sequence of consecutive prime numbers. Put dn = pn+l − pn. Turán and I proved [1] that the inequalities dn+1 > dn and dn+1 < dn both have infinitely many solutions. It is not known if dn = dn+l has infinitely many solutions. The answer is undoubtedly affirmative but the proof will probably be very difficult [2]. It was a great surprise and disappointment to us that we could not prove that dn+2 > dn+1 > dn has infinitely many solutions. We could not even prove that (− 1)n (dn+l − dn) changes sign infinitely often. It seems certain that the answer to both of these questions is affirmative and perhaps a simple proof can be found.

In this note I settle a question which arose out of my first paper under the above title (cf. [1]), where I considered the classgroup C(Z(Γ)) of the integral groupring Z(Γ) of a finite Abelian group Γ. This classgroup maps onto the classgroup C() of the maximal order of the rational groupring Q(Γ), and C() is the product of the ideal classgroups of the algebraic number fields which occur as components of Q(Γ) and is thus in a sense known. One is then interested in the kernel D(Z(Γ)) of C(Z(Γ)) → C() and in its order k(Γ). In [1] I proved that, for Γ a p-group, k(Γ) is a power of p. I also computed k(Γ) for small exponents. My computation used crucially the fact that, for the groups Γ considered, the groups of units of algebraic integers which occurred were finite, i.e. that the only number fields which turned up were Q and Q(n) with n4 = 1 or n6 = 1. The numerical results obtained led me to the question whether in fact k(Γ) tends to infinity with the order of Γ.

Let R be a Dedekind domain whose quotient field K is an algebraic number field, and let Λ be an R-order in a semisimple K-algebra A with 1. A Λ-lattice is a finitely generated R-torsionfree left Λ-module. We shall call a Λ-lattice M locally free of rank n if for each maximal ideal p of R, Mp is Λp,-free on n generators. (The subscript p denotes localization.) The (locally free) class group of Λ is the additive group C(Λ) generated by symbols

where

and where xM = 0 if and only if M is stably free (that is, M + Λ(k) ≅ Λ + Λ(k) for some k).

The study of similarity solutions of Prandtl's equations for the steady two dimensional flow of an incompressible fluid past a rigid wall leads to the equation

where the primes denote differentiation with respect to the independent variable t, and λ is a parameter. It was first obtained in 1930 by Falkner and Skan [3]. For its derivation we refer to Schlichting [6] here we merely note that the function f′(t) represents, after suitable normalization, the velocity parallel to the wall.

Let P be a simplicial d-polytope, and, for – 1 ≤ j < d, let fj(P) denote the number of j-faces of P (with f_1 (P) = 1). For k = 0, ..., [½d] – 1, we define

and conjecture that

gk(d + 1)(P) ≥ 0,

with equality in the k-th relation if and only if P can be subdivided into a simplicial complex, all of whose simplices of dimension at most d – k – 1 are faces of P. This conjecture is compared with the usual lower-bound conjecture, evidence in support of the conjecture is given, and it is proved that any linear inequality satisfied by the numbers fj(P) is a consequence of the linear inequalities given above.

An example will be given of a compact linear set K which is of zero or non-σ-finite measure for every translation-invariant Borel measure. This answers a question asked me by C. Dellacherie.

Let D denote the unit disc in the complex plane. If p and q are two complex numbers, let x(p, q) denote the chordal distance between p and q on the Riemann sphere. In particular, we have the formula

and

The following problem was posed by Paul Gauthier: if f(z) and g(z) are meromorphic functions in D such that Clunie [4] has answered this problem in the negative by constructing different meromorphic functions f(z) and g(z) with the desired property. However, the functions constructed by Clunie both have an infinity of poles in D. It is the purpose of this note to give an example of two analytic functions―which thus have no poles―which also give a negative answer to the problem of Gauthier.

In [5] Knowles proved the following.

Every compact Hausdorff space with no isolated points admits a non-atomic measure.

This note is concerned with the converse problem in a more general set up. Here we deal with certain properties of the family of completely regular spaces admitting no continuous measures. In §3 it is shown that this family contains spaces with no isolated points, thus theorem (1.1) does not generalize to completely regular spaces. In §4 a canonical decomposition of the compact members of the above family into discrete subspaces is obtained, and it is shown that these spaces are metrizable whenever they satisfy the first axiom of countability.