A method of finding critical points in a half space is developed. It is then applied to the study of semilinear boundary value problems, and used to determine conditions which lead to multiple non-trivial solutions.

We study the variance of the fluctuations in the number of lattice points in a ball and in a thin spherical shell of large radius centred at a Diophantine point.

An (r,r+1)-factor of a graph G is a spanning subgraph H such that dH(v)∈{r,r+1} for all vertices v∈𝒱(G). If G is expressed as the union of edge-disjoint (r,r+1)-factors, then this expression is an (r,r+1)-factorization of G. Let μ(r) be the smallest integer with the property that if G is a regular loopless multigraph of degree d with d≥μ(r), then G has an (r,r+1)-factorization. It is shown that if r is even. The proof employs a novel list-coloring approach. Together with known results, this shows that μ(r)=r2+1 if r is odd and if r is even.

The temporal evolution of nonlinear, incompressible ensembles is examined first for the one-dimensional Burgers' equation and then for the incompressible, unsteady Navier-Stokes equations. It is shown that local closure of the averaged problem can be obtained for finite ensembles of Burgers' equation in the limit as the number of moments tends to infinity. This limit behaviour is verified via direct numerical computations for the onedimensional inviscid and viscous Burgers' equation. Closure is found to occur at reasonably low order. It is shown that this technique can be extended to obtain a local closure of the convective terms of the Navier-Stokes Reynoldsaveraged equations.

Hyperplane mean values of non-negative subharmonic functions have been studied in many papers, of which [2] and [3] are examples. Recently, Armitage [1] began a study of hyperplane means of non-negative superharmonic functions. One of his results [1, Theorem 2] shows that, if w is a positive superharmonic function on

and n/(n + 1) p < 1, then

as t → ∞, while another [1, Theorem 1] shows that, if 0 < p ≤ n/(n + l), then the integral in (1) is always infinite. However, he did not present a complete analogue of the result of Flett and Brawn [2, 3], which states that, if Ф: [0, ∞ [ → [0, ∞ [ is a non-decreasing convex function such that Ф(u)/u → 0 as u → 0 +, and w is a nonnegative subharmonic function on Rn × ]0, ∞ [, then under certain conditions on the size of w, the integral mean

tends to zero as t → ∞. In this note we present an analogue for superharmonic functions of the above result, in which the mean M(Ф(w); t) is shown to tend to infinity with t, provided that Ф(u)/u → ∞ as u → 0 +, and which therefore generalizes (1). It might be expected that, in dealing with the superharmonic case, the function Ф would have to be concave, so that Ф(w) would also be superharmonic. It turns out that this condition is unnecessary.

In a recent book, L. H. Loomis has obtained the “Bohr compactification” of a topological group, in terms of almost periodic functions, by applying the representation theory of commutative B-algebras. It is simpler, and perhaps more natural, to consider this matter from the point of view of comparative topology; we can then obtain a more general result, in that the discussion is no longer restricted to the case of numerically valued (or even vector-valued) functions.

The problem dealt with in this paper was suggested by Dr. E. C. Dade and communicated to me, at Stockholm in August 1962, by Dr. Taussky Todd. It may be stated as follows. Consider the equation

where f is a form (i.e., a homogeneous polynomial) with coefficients in some ring R of algebraic integers. An obviously necessary condition for the solubility of (1) with the xi in R is that the coefficients of f should have no common factor, except for units of R. Now let S be a ring which is an algebraic extension of R, and let us try to solve (1) with the xi in S. The condition just mentioned remains necessary; and Dade has proved in [1] that for some S (depending on R and f) it is sufficient. His theorem is valid for forms of any degree but is merely an existence theorem as far as S is concerned. The problem is to do better for the special case in which f is of degree 2 and R the ring of rational integers; and more precisely, to show if possible that S can be taken to be a quadratic extension R(√q) of R, with an integer q which could be estimated in terms of the coefficients of f.

Recently there has been much interest in theories describing the behaviour of a mixture of two or more continua, which are allowed to diffuse through each other and which interact mechanically and thermally. In the present note we consider such a mixture consisting of two isotropic incompressible non-linear elastic solids; physically this may describe a composite of two rubber-like materials. Such a mixture possesses internal friction between the constituents when deformed (called diffusive resistance or diffusive force), which acts as a damping mechanism. This damping mechanism has a static part and a dynamic part, and we here show that the exact static solutions which exist for an isotropic incompressible non-linear elastic solid are such that the damping mechanism is expressible as a gradient, and hence are also controllable for such a mixture. We also discuss the corresponding dynamic solutions.

From a practical viewpoint, some experiments carried out on vulcanized synthetic rubber-like materials may involve a mixture rather than a pure solid.

The even dimensional C∞ manifold M2n is said to be symplectic, if there exists a 2-form Ω, defined everywhere on M such that

(i)Ω is closed, that is dΩ, = 0, and

(ii)Ωn ≠ 0.

Let K be a successor cardinal. Call an regularizing family (for short, a regularizing family) if Card ℱ = k and for each with Card . In [7], Prikry proved, assuming V = L, that if is an ultrafilter on K which is uniform (Card X = K, each ) then is regular (some forms a regularizing family).†

An old theorem of Pólya and Carlson [2] states that, if the power series has rational integer coefficients, positive radius of convergence, and can be continued analytically to a region that contains points outside the closed unit disc, then the function that the power series represents is rational. This result has been extended in a number of ways (cf. e.g. Petersson [4]). The present note gives a new extension based on a recent theorem of Güting [3]. My thanks to Professors Henry Helson and Raphael Robinson for introducing me to this subject.

A PV-number is defined to be an algebraic integer θ, of modulus greater than one, all of whose conjugates (excluding θ itself) lie inside the unit circle. Salem [1] has shown that the set S of PV-numbers forms a closed subset of the real line.

For a metric space <Ω, ρ> and a ‘measure function’ h, the Hausdorff measure mh on Ω is denned by applying Method II to the premeasure defined by τ(E) = h(d(E)), E ⊆ Ω, where

with d(Φ) = 0, is the diameter of E. The set function mh is then a metric outer measure. There are many variations on this definition producing measures also associated with the name Hausdorff. Here we are concerned with those measures which arise when there is a restriction on the sets E for which τ is defined. Such measures arise, for example, as net measures, Rogers [1]. Also we might find it useful to have τ defined only on disks, or only on squares, or only on rectangles with a given relation between vertical and horizontal sides.

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).

For a typical convex body in Ed a typical shadow boundary under parallel illumination has infinite (d - 2)-dimensional Hausdorff measurewhile having Hausdorff dimension d 2.

Recently the notion of elementary symmetrization attracted new attention in the field of convex sets (see [L; W]), and it was proved that Minkowski's “Quermassintegrale” are decreased by elementary symmetrization. On the other hand, the concept of Schwarz symmetrization for Borel functions gained new interest from its possible applications in the field of elliptic partial differential equations (see [S1, S2, HI, T1, T2, HY1, HY2, GL1, GL2]).

Let $p,q$ be primes such that $q|p-1$ and set $\unicode[STIX]{x1D6F7}=C_{p}\rtimes C_{q}$, $G=\unicode[STIX]{x1D6F7}\times C_{\infty }^{n}$ and $\unicode[STIX]{x1D6EC}=\mathbf{Z}[G]$, the integral group ring of $G$. By means of a fibre square decomposition, we show that stably free modules over $\unicode[STIX]{x1D6EC}$ are necessarily free.

We obtain new bounds of multivariate exponential sums with monomials, when the variables run over rather short intervals. Furthermore, we use the same method to derive estimates on similar sums with multiplicative characters to which previously known methods do not apply. In particular, in the case of multiplicative characters modulo a prime $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}p$ we break the barrier of $p^{1/4}$ for ranges of individual variables.