Görtler vortices are thought to be the cause of transition in many fluid flows of practical importance. In this paper a review of the different stages of vortex growth is given. In the linear regime nonparallel effects completely govern this growth and parallel flow theories do not capture the essential features of the development of the vortices. A detailed comparison between the parallel and nonparallel theories is given and it is shown that at small vortex wavelengths the parallel flow theories have some validity; otherwise nonparallel effects are dominant. New results for the receptivity problem for Gortler vortices are given; in particular vortices induced by free-stream perturbations impinging on the leading edge of the wall are considered. It is found that the most dangerous mode of this type can be isolated and its neutral curve is determined. This curve agrees very closely with the available experimental data. A discussion of the different regimes of growth of nonlinear vortices is also given. Again it is shown that, unless the vortex wavelength is small, nonparallel effects are dominant. Some new results for nonlinear vortices of O(l) wavelengths are given and compared with experimental observations. The agreement between theory and experiment is shown to be excellent up to the point where unsteady effects become important. For small wavelength vortices the nonlinear regime is of particular interest since a strongly nonlinear theory can be developed there. Here the vortices can be large enough to drive the mean state which then adjusts itself to make all modes neutral. The breakdown of this nonlinear state into a three-dimensional time dependent flow is also discussed.

If a scattered compact space K is such that its ω1-th derived set K(ω1) is empty then the Banach space ℒ(K) admits an equivalent locally uniformly convex norm.

I propose a definition of “κ-Souslin operation”, for uncountable cardinals κ, which for certain applications in measure theory seems an appropriate generalization of the usual Souslin operation.

In 1941 Duffin and Schaeffer [2] considered the case k = 1 of the assertion that when the statement;

(Ak) “For almost all there are infinitely many natural numbers q for which there exist integers a1,…, ak such that (a1 … ak, q) = 1 and

where

holds, if, and only if,

diverges.

Of prime concern in this paper is the flow induced in a channel when a thermal wave moves along a boundary with topographical features. The principal result obtained is that the time-averaged flow in the channel is predominantly cellular in nature, which is qualitatively quite different from its unidirectional form when such structures are absent.

Nous estimons le module des sommes trigonométriques sur la variété de dimension n – s definie par s formes en n variables, avec une forme linéaire en exposant. Cela s'applique a l'étude de la distribution des points rationnels d'une telle variété definie sur un corps fini ou sur le corps des nombres rationnels.

The aim of this paper is to prove the following

THEOREM. Let M be a C∞ compact and strictly convex surface embedded in the euclidean space E3 or in the hyperbolic space H3. We suppose that all shadow-lines ofM are congruent. Then M is a euclidean 2-sphere or a hyperbolic 2-sphere respectively.

Let where a0≠0, m≥2, n = e1 + … + em and the ξi(1≤i≤m) are the distinct zeros of f in some algebraic closure of the p-adic field . Then K = (ξ1, ξ2, …, ξm) is a finite separable extension of and we denote by “ord” the unique extension of the (additive) p-adic valuation on to K, normalized with ord p = 1.

Atkinson, Young and Brezovich [1: 1983] gave a formula for the potential distribution due to a circular disc condenser with arbitrary spacing parameter к (the ratio of separation of the discs to their radius). This was simpler to calculate than the formulation which I gave in [8: 1949]; but unfortunately it fails to satisfy two requirements, as the present paper shows. Together with [8], this paper shows that the potential formulated in [8] satisfies all requirements.

Sets of the general form

where U is a subset of ℛk and is a family of subsets of U indexed by a set J, are common in the theory of Diophantine approximation [4, 7, 18, 19]. They are also closely connected with exceptional sets arising in analysis and with sets of “small divisors” in dynamical systems [1, 8, 15”. When J is the set of positive integers ℕ, the set Λ(ℱ) is of course the lim-sup of the sequence of sets Fj, j = 1, 2,… [11, p. 1]. We will also call sets of the form (1), with the more general index set J, lim-sup sets. When such lim-sup sets have Lebesgue measure zero, it is of interest to determine their Hausdorff dimension. It is usually difficult to obtain a good lower bound for the Hausdorff dimension (and it can be much harder to determine than an upper bound). In this paper we will obtain a lower bound for the dimension of lim-sup sets of the form (1) for a fairly general class of families ℕ which includes a range of results in the theory of Diophantine approximation. This lower bound depends explicitly on the geometric structure and distribution in U of the sets Fα in ℕ.

We determine what is the maximum possible (by volume) portion of the three-dimensional Euclidean space that can be occupied by a family of non-overlapping congruent circular cylinders of infinite length in both directions. We show that the ratio of that portion to the whole of the space cannot exceed π/√12 and it attains π/√12 when all cylinders are parallel to each other and each of them touches six others. In the terminology of the theory of packings and coverings, we prove that the space packing density of the cylinder equals π/√12, the same as the plane packing density of the circular disk.

A convex compact subset of ℝd is called a convex body. The (Euclidean) surface area and volume of a convex body K are denoted s(K) and v(K) respectively. The support function of a convex body K is denned by h(K, x) = maxy∈K xty and the polar dual of K is given by K0 = {x: |xty|1, y∈K}. Double vertical bars shall denote the Euclidean length of a vector , and S shall denote the unit sphere (the Euclidean unit ball): S = {x: ║x║≤1}. We use for the mixed volume

Let M be a convex body, i.e., a compact, convex set with non-empty interior, in n-dimensional Euclidean space En. A chord [a, b] of M is said to be an affine diameter of M, if, and only if, there exists a pair of (different) parallel supporting hyperplanes of that body, each containing one of the points a, b. The following result of Eggleston (cf. [1] and [2]) is well-known. A convex figure M Ì E2 is a triangle, if, and only if, each of its interior points belongs to exactly three affine diameters. In [3] this result is sharpened. A convex figure M Ì E2 is a triangle, if, and only if, each of its interior points belongs to at least two, but a finite number of affine diameters. A natural problem for the n-dimensional case, based on Eggleston's result, is the following (cf. also [4]). Is it true that the n-dimensional simplex is the only convex body in En such that through each interior point pass precisely 2n − l affine diameters? For the case of convex polytopes, i.e., convex bodies with a finite number of extreme points, we shall give a positive answer to this question.

Let

It is proved that for suitable a and b, n≥7, one can have Vn(An) = Vn(Bn) and for every (n–1)-dimensional subspace H of ℝn, where Bn is the unit ball of ℝn. This strengthens previous negative results on a problem of H. Busemann and C. M. Petty.

Let ci, and di, (1≤i≤s) be rational integers, and k and n be natural numbers. We shall consider the solubility over the p-adic integers ℤp of the pair of additive equations

Let K1, K2,…, Kd+1 be d+1 convex bodies in Ed, the interiors of which have no point in common. If mX denotes the measure of set X, we prove

and this inequality is best possible.

In the terminology of Birget and Rhodes [3], an expansion is a functor F from the category of semigroups into some special category of semigroups such that there is a natural transformation η from F to the identity functor for which ηs is surjective for every semigroup S. The three expansions introduced in [3] have proved to be of particular interest when applied to groups. In fact, as shown in [4], Ĝ(2) are isomorphic for any group G, is an E-unitary inverse monoid and the kernel of the homomorphism ηG is the minimum group congruence on . Furthermore, if G is the free group on A, then the “cut-down to generators” which is a subsemigroup of is the free inverse semigroup on A. Essentially the same result was given by Margolis and Pin [12].

Let Vo be a discrete real valuation of a field K and x an indeterminate. In 1936, MacLane [3] gave a method of constructing all real valuations of K(x) which are extensions of Vo. In this paper, we determine explicitly all rank 2 valuations of K(x) which extend Vo. One can thereby describe all rank 2 valuations of K(x, y) which are trivial on an arbitrary K; x, y being algebraically independent over the field K. The latter valuations have been considered by Zariski [5] in the case when K is an algebraically closed field of characteristic zero.

Let Q(x) = Q(x1, x2,…, xn) be a quadratic form with integer coefficients. Schinzel, Schickewei and Schmidt [9, Theorem 1] have shown that for any modulus m there exists a nonzero such that

and ║x║≤m(1/2)+(1/2(n-1)), where ║x║ = max |xi|. When m is a prime Heath-Brown [8] has obtained a nonzero solution of (1) with ║x║≤m1/2 log m. Yuan [10] has extended Heath-Brown's work to all finite fields. We have proved related results in [5] and [6]. In this paper we extend Heath-Brown's work to moduli which are a product of two primes. Throughout the paper we shall assume that n is even and n>2. For any odd prime p let

where det Q is the determinant of the integer matrix representing Q and is the Legendre symbol.

For each odd prime p there is a finite regular abstract 4-dimensional polytope of type {3, 3, p}. Its cells are simplices, and its vertex figures belong to an infinite family of regular polyhedra. We also give a geometric realization for these polytopes.