Let S(Rn) be the space of Schwartz class functions. The dual space of S′(Rn), S(Rn), is called the temperate distributions. In this article, we call them distributions. For 1 ≤ p ≤ ∞, let FLp(Rn) = {f:∈ Lp(Rn)}, then we know that FLp(Rn) ⊂ S′(Rn), for 1 ≤ p ≤ ∞. Let U be open and bounded in Rn−1 and let M = {(x, ψ(x));x ∈ U} be a smooth hypersurface of Rn with non-zero Gaussian curvature. It is easy to see that any bounded measure σ on Rn−1 supported in U yields a distribution T in Rn, supported in M, given by the formula

The paper deals with the free convection flow along a vertical plate moving arbitrarily in its own plane. The basic equations of the boundary-layer flow and heat transfer are linearized and the first two approximations are considered. The first approximation is the case of steady-state free convection flow while the second approximation is the response of the fluid velocity and temperature fields to the motion of the plate for which limiting solutions are obtained by the Laplace transform technique in two regions; namely, for large times and for small times. The particular case when the plate is given an impulsive start at t = 0 is investigated in detail. It is shown how the skin friction and the rate of heat transfer at the plate respond to the motion of the plate.

If f: X × Y → Z is a map, then the classical Hopf construction associates to f a map hf: X * Y → ΣZ, where X * Y is the join of X and Y and ΣZ the suspension of Z. Since X * Y has the homotopy type of Σ(X Λ Y), the suspension of the smash product of X and Y, the homotopy class of hf can be regarded as an element Hf ↦ [Σ(X Λ Y), ΣZ]. Now elements of [Σ(X Λ Y), ] are in one to one correspondence with homotopy classes in the group [σ(X Λ Y), ΣZ] which are trivial on the suspension of the wedge Σ(X ≷ Y).

1. Introduction. If G is a locally compact Hausdorff topological group, and μ is (left) Haar measure on G, then we denote by ℬ(G) the class of all Borel subsets of G having finite measure, and by VG the set {μ(E): E ∊ ℬ(G)} of real numbers. The product set function of G, ΦG: VG × VG → VG, is defined (see (4) and (5)) by

and, for each u, v ∈ VG, we call a pair (E, F) of Borel subsets of G a critical (u, v)-pair, if μ(E) = u, μ(F) = v, and μ*(EF) = ΦG(u, v). We denote the class of all critical (u, v)-pairs by and we write ℰG for .

It is familiar that if on an algebraic surface there is an exceptional curve, that is an irreducible rational curve of virtual grade − 1 when no points of it are assigned as base points, and if there is on the surface a canonical system containing some actual curves, so that pg ≥ 1, then the exceptional curve is a fixed constituent of every curve of the canonical system, generally a simple constituent, and in that case has no intersections with the residual constituent. More generally, if there is on the surface a reducible exceptional curve, i.e. a set of curves which can be transformed into the neighbourhoods of a family of simple points (some of which are in the neighbourhoods of others) on a surface birationally equivalent to the given one, then the canonical system has as a fixed constituent of all its curves at least that combination of the curves which corresponds to the sum of the total neighbourhoods of the points, and generally just this combination, in which case this fixed part has no intersection with the residual variable part.

Let ℘ be a Weierstrass elliptic function satisfying the differential equation

and let ζ(z) be the associated Weierstrass ζ-function satisfying (z) = −℘(z). Corresponding to a pair of fundamental periods ω1, ω2 of ℘(z), there is a pair of quasi-periods η1, η2 of ζ(z) defined by

and we have ηi = 2ζ(ωi/2) for i = 1, 2.

We show that a monotone Lagrangian L in ${\mathbb{C}}{\mathbb{P}}^n$ of minimal Maslov number n + 1 is homeomorphic to a double quotient of a sphere, and thus homotopy equivalent to ${\mathbb{R}}{\mathbb{P}}^n$. To prove this we use Zapolsky’s canonical pearl complex for L over ${\mathbb{Z}}$, and twisted versions thereof, where the twisting is determined by connected covers of L. The main tool is the action of the quantum cohomology of ${\mathbb{C}}{\mathbb{P}}^n$ on the resulting Floer homologies.

On a (pseudo-)Riemannian manifold (${\mathcal M}$, g), some fields of endomorphisms i.e. sections of End(T${\mathcal M}$) may be parallel for g. They form an associative algebra $\mathfrak e$, which is also the commutant of the holonomy group of g. As any associative algebra, $\mathfrak e$ is the sum of its radical and of a semi-simple algebra $\mathfrak s$. Here we study $\mathfrak s$: it may be of eight different types, including the generic type $\mathfrak s$ = ${\mathbb R}$ Id, and the Kähler and hyperkähler types $\mathfrak s$ ≃ ${\mathbb C}$ and $\mathfrak s$ ≃ ${\mathbb H}$. This is a result on real, semi-simple algebras with involution. For each type, the corresponding set of germs of metrics is non-empty; we parametrize it. We give the constraints imposed to the Ricci curvature by parallel endomorphism fields.

The Rikitake two-disc system [10] has been proposed as a model of the Earth's magnetic dipole. Numerical integration of the system by Rikitake[10] and Allan [1] showed the phenomenon of polarity reversal which is seen in terrestrial paleomagnetic data [6]. Reversals in the simulations occurred at irregular and apparently unpredictable time intervals, in accord with the capricious character of the real data, though the agreement was qualitative rather than quantitative.

Let p be a prime and let f(x) be a quartic polynomial with integral coefficients. I consider the problem of estimating the least non-negative non-residue k of f(x) (modp) (I omit the mod p hereafter), for large primes p, so f(x) ≡ r has a solution for

but not for r = k. The same problem for cubics has been considered by Mordell ((1)), who showed that

as p → ∞, where the constant implied in the O-symbol is independent of the coefficients of the cubic. In fact a more detailed examination of Mordell's proof gives the better estimate

It is the purpose of this paper to show that this same estimate also holds for quartic polynomials.

Let f(z) = u(z) + iv(z) be a function regular for |z| < 1 with real and imaginary parts u(z) and u(z) respectively. The following theorems have been proved by Miss M. L. Cartwright(1) and (2).

Theorem A. if v(0) = 0, 0 < α < ∞ and

for 0 ≤ r < 1, 0 ≤ θ < 2π then

for 0 ≤ r < 1, 0 ≤ θ < 2π

In (1) A. H. Boers has studied the properties of n-PA rings, i.e. rings where any n-product (any product of n factors) is independent of its bracketing; a private communication is quoted in that paper to the effect that the n-PA rings were defined by Y. Ilamed starting from problems in mathematical physics.

A claim of H. H. Teh on Archimedean full orders ((2), ‘only if’ of Theorem 2, p. 480) fails. His statement (p. 478, line 5) that the direct product of n copies of the additive rational numbers admits no full order of Archimedean rank more than n is proved without Hahn's embedding theorem. A few other remarks are made.

A classical result of Pólya states that 2z is the slowest growing transcendental entire function taking integer values on the non-negative integers. Langley generalised this result to show that 2z is the slowest growing transcendental function in the closed right half-plane Ω = {z ∈ : Re(z) ≥ 0} taking integer values on the non-negative integers. Let E be a subset of the Gaussian integers in the open right half-plane with positive lower density and let f be an analytic function in Ω taking values in the Gaussian integers on E. Then in this paper we prove that if f does not grow too rapidly, then f must be a polynomial. More precisely, there exists L > 0 such that if either the order of growth of f is less than 2 or the order of growth is 2 and the type is less than L, then f is a polynomial.

We study moduli spaces of d-dimensional manifolds with embedded particles and discs, which we refer to as decorations. These spaces admit a model in which points are unparametrised d-dimensional manifolds in $\mathbb{R}^\infty$ with particles and discs constrained to it. We compare this to the space of d-dimensional manifolds in $\mathbb{R}^\infty$ with particles and discs that are no longer constrained, i.e. the decorations are decoupled. We show that under certain conditions these spaces cannot be distinguished by homology groups within a range. This generalises work by Bödigheimer–Tillmann for oriented surfaces to different tangential structures and also to higher dimensional manifolds. We also extend this result to moduli spaces with more general submanifolds as decorations and specialise in the case of decorations being embedded circles.

In this study of complete, or integrally closed, ideals in a two-dimensional regular local ring (R, m), Zariski established a one-to-one correspondence between prime divisors of R, i.e. rank 1 discrete valuations v birationally dominating R with residue field of transcendence degree 1 over R/m, and m-primary simple complete ideals Iv in R; cf. [17] and [18]. In this correspondence, the blow-up of such an ideal has unique exceptional prime and the localization at this prime is the valuation ring of a prime divisor of R. In this paper, we will study such ideals in a more general setting, so we begin by recalling some definitions and background results.

In Ewald's theory of crystal optics, based on classical principles, an optical wave through a crystal lattice of polarizable atoms (idealized as isotropic oscillators) is a self-sustaining system of vibrations so constituted that, on the one hand, the electric moment of each atom is caused to oscillate by the electromagnetic field and, on the other hand, the electromagnetic field is itself the resultant field due to the superposition of the dipole waves produced by the lattice atoms. Born extended the theory to the case of movable lattice ions and showed that in an optical wave the electromagnetic field is so coupled to the lattice vibrations that each lattice ion vibrates in phase with the local field which, as in Ewald's case, is itself produced by the vibrating ions. In Born's original theory, the motion of the lattice particles had to be treated by classical mechanics. It is shown that the results of the quantum-mechanical treatment of the lattice motion agrees exactly with the classical theory. Not only is the induced current at each lattice point in phase with the local field, but the magnitude of the current is also identical with the classical value, completely independent of whichever vibrational state the crystal might be in.

We study the question of whether the Morava K-theory of the classifying space of an elementary abelian group V is a permutation module (in either of two distinct senses, defined below) for the automorphism group of V. We use Brauer characters and computer calculations. Our algorithm for finding permutation submodules of modules for p-groups may be of independent interest.

In an earlier paper I considered the problem of determining those subgroups of SL (2, ℤ) consisting of all matrices

satisfying a linear congruence