Marshall Hall has proved that every real number is representable as the sum of two continued fractions with partial quotients at most 4. This implies that for any real β1, β2 there exists a real α such that

for all integers x > 0 and y, where C is a positive constant. In this note I prove a generalization to r numbers β2, …, βr. The case r = 2 implies a result similar to Marshall Hall's but with a larger number (71) in place of 4.

Let f = f(x, y, z) be a positive definite form of the type

where x, y, z are integral valued variables, and the coefficients a, …, t are integers whose highest common factor is 1. As the determinant of such a form may be fractional, I define

and

thus — C is the discriminant of the binary form f(x, y, 0), and the necessary and sufficient condition for f to be positive definite is that a > 0, C > 0, and d > 0.

Abstract. Bose-Chaudhuri-Hocquenghem and Justesen codes are used to pack equa spheres in n–dimensional Euclidean space with density Δ satisfying

for all sufficiently large n of the form m 2m, where m is a power of 4. These appear to be the densest packings yet constructed in high dimensional space.

The main goal of this paper is to prove that the classical theorem of local inversion for functions extends in finite dimension to everywhere differentiable functions. As usual, a theorem of implicit functions can be deduced from this “Local Inversion Theorem”. The deepest part of the local inversion theorem consists of showing that a differentiable function with non-vanishing Jacobian determinant is locally one-to-one. In turn, this fact allows one to extend the Darboux property of derivative functions on ℝ (the range of the derivative is an interval) to the Jacobian function Df of a differentiable function, under the condition that this Jacobian function does not vanish. It is also proved that these results are no longer true in infinite dimension. These results should be known in whole or part, but references to a complete proof could not be found.

A graph G (finite, undirected, and without loops or multiple lines) is n-connected if the removal of fewer than n points from G neither disconnects it nor reduces it to the trivial graph consisting of a single point. We present in this note a sufficient set of conditions on the degrees (valences) of the points of a graph G so that G is n-connected.

§1. Preliminaries. Let X be a stable subordinator in R, B a subset of R and A a time set. In this paper we shall consider the Hausdorff dimension properties of the random sets

A method of assessing the accuracy of an approximation to the steady two-dimensional flow of viscous incompressible liquid past a flat plate is developed. The approximate solution, with stream-function ψ1 is closely related to the boundary-layer solution, and is expressed in terms of the function used in that theory. The conditions at the surface of the plate and at infinity are exactly satisfied by ψ1 but the equation of (finite) motion is not exactly satisfied. However, ψ1 is an exact solution of a problem in fluid motion, if a body force, of appropriate magnitude depending on ψ1 is assumed to act on the fluid. The magnitude of this force provides a criterion of the accuracy of ψ1, and some use has been made [5] of this means of estimating accuracy. This criterion is, in effect, only qualitative, since it is not possible to make a numerical estimate of the effect on the motion of a body force which is non-conservative and acts over the whole field of flow.

We prove a stability version of the Prékopa–Leindler inequality.

In [CKR], Chan, Kim and Raghavan determine all universal positive ternary integral quadratic forms over real quadratic number fields. In this context, universal means that the form represents all totally positive elements of the ring of integers of the underlying field. This generalizes the usage of the term introduced by Dickson for the case of the ring of rational integers [D]. In the present paper, we will continue the investigation of quadratic forms with this property, considering positive quaternary forms over totally real number fields. The main goal of the paper is to prove that if E is a totally real number field of odd degree over the field of rational numbers, then there are at most finitely many inequivalent universal positive quaternary quadratic forms over the ring of integers of E. In fact, the stronger result will be proved that this finiteness holds for those forms which represent all totally positive multiples of any fixed totally positive integer. The necessity of the assumption of oddness of the degree of the extension for a general result of this type can be seen from the existence of universal ternary forms over certain real quadratic fields (for example, the sum of three squares over the field ℚ(√5), as first shown by Maass [M]).

It is known that if Λ is a self-dual lattice in ℝn, then

.

If equality holds the lattice is called extremal. In this paper we find all the extremal lattices: there are unique lattices in dimensions 1, 2, 3, 4, 5, 6, 7, 8, 12, 14, 15, 23, 24 and no others.

A two-dimensional fluid layer of height d is confined laterally by rigid sidewalls distance 2Ld apart, where L, the semi-aspect ratio of the layer, is large. Constant temperatures are maintained at the upper and lower boundaries while at the sidewalls it is assumed that the horizontal heat flux has magnitude λ. If λ = 0 (perfect insulation) a finite amplitude motion sets in when the Rayleigh number R reaches a critical value Rc, but in part I (Daniels 1977) it was shown that if λ = O(L−1) this bifurcation (in a state diagram of amplitude versus Rayleigh number) is displaced into a single stable solution in the region |R – Rc| = O(L−2), representing a smooth increase in amplitude of the cellular motion with Rayleigh number. All other solutions (or “secondary modes”) in this region were shown to be unstable. In the present paper an examination of the two intermediate regimes λ − O(L−5/2) and λ = O(L−2) is carried out, to trace the location of an additional stable solution in the form of a secondary mode, which stems from Rc when λ = 0, and which in the limit as λL2 → ∞ is shown to be removed from the region ߋR − Rc| = 0(L−2), consistent with the results of I.

Let C be a compact convex set in En, not necessarily containing interior points. The Steiner point of K has been defined (see Shephard [6]) as

The idea of obtaining bounds on the order of approximation of algebraic functions by rational functions apparently goes back to Maillet [8]. It is very easy, using an argument analogous to that used by Liouville for algebraic numbers, to see that if w(z) is any algebraic function of degree n ≥ 2 over Q[z] then for all pairs of polynomials r(z) and s(z) with s(z) ≢ 0 we have

The work to be described here, with the fruit of previous investigations in [1] and [2], yields the following general theorem.

Assume that n points P1,…,Pn are distributed independently and uniformly in the triangle with vertices (0, 1), (0, 0), and (1, 0). Consider the convex hull of (0, 1), P1,…,Pn, and (1, 0). The vertices of the convex hull form a convex chain. Let be the probability that the convex chain consists – apart from the points (0, 1) and (1, 0) – of exactly k of the points P1,…,Pn. Bárány, Rote, Steiger, and Zhang [3] proved that . The values of are determined for k = 1,…,n − 1, and thus the distribution of the number of vertices of a random convex chain is obtained. Knowing this distribution provides the key to the answer of some long-standing questions in geometrical probability.

We provide new quantitative versions of Helly’s theorem. For example, we show that for every family $\{P_{i}:i\in I\}$ of closed half-spaces in $\mathbb{R}^{n}$ such that $P=\bigcap _{i\in I}P_{i}$ has positive volume, there exist $s\leqslant \unicode[STIX]{x1D6FC}n$ and $i_{1},\ldots ,i_{s}\in I$ such that

$$\begin{eqnarray}\operatorname{vol}_{n}(P_{i_{1}}\cap \cdots \cap P_{i_{s}})\leqslant (Cn)^{n}\operatorname{vol}_{n}(P),\end{eqnarray}$$

$\unicode[STIX]{x1D6FC},C>0$

Define $r_{4}(N)$ to be the largest cardinality of a set $A\subset \{1,\ldots ,N\}$ that does not contain four elements in arithmetic progression. In 1998, Gowers proved that

$$\begin{eqnarray}r_{4}(N)\ll N(\log \log N)^{-c}\end{eqnarray}$$

$c>0$

$$\begin{eqnarray}r_{4}(N)\ll N\text{e}^{-c\sqrt{\log \log N}}.\end{eqnarray}$$

$$\begin{eqnarray}r_{4}(N)\ll N(\log N)^{-c},\end{eqnarray}$$

In 1979 Montgomery and Vaughan [3] showed that for each k > 0

the sum being over the non-principal characters χ modulo q. As a corollary they deduced that, if 0 < θ < 1 then there is a constant c(θ) such that at least θφ(q) of the non-principal characters χ modulo q satisfy

Some examples of the bending of a plane elastic plate by transverse forces applied at isolated points are first considered. The plate is infinite and is bounded internally by a circular edge along which it is clamped; simple expressions are found for the displacement.

The analogous hydrodynamical problem is that of the steady flow of viscous incompressible liquid past a fixed circular cylinder; the equation for the stream function is in the same form as the equation for the displacement of a plate due to a distributed force of amount Z per unit area. The inertia terms in the hydrodynamical problem correspond to Z. In slow motion there is no stream function with the correct form at infinity, because in this case the inertia terms are ignored so that in the plate problem Z = 0. The effect of a transverse force system can be most simply illustrated by supposing that the forces are concentrated at isolated points; in the corresponding stream functions a simple form of allowance for inertia is therefore made, and they display some of the features of steady flow past a cylinder.

In this paper we prove

Theorem 1. If N is a natural number and I is an interval of N consecutive integers, then there is a 1–1 correspondencef: {1, 2,…, N} → I such that (i, f(i)) = 1 for 1 ≤ i ≤ N.