In the stream-line motion of fluid in a curved pipe the primary motion along the line of the pipe is accompanied by a secondary motion in the plane of the cross-section. The secondary motion decreases the rate of flow produced by a given pressure gradient and causes an outward movement of the region where the primary motion is greatest. It is difficult to deduce these consequences from the exact equations of motion, but it is easy to do so if it is assumed that the actual secondary motion is replaced by a uniform stream; conditions in the central part of the section mainly determines the motion and here the secondary motion is approximately a uniform stream. The appropriate velocity of the stream can be determined from the relation that has been found experimentally between the rate of flow in a curved pipe and the pressure gradient.

One of the formulations of the prime number theorem is the statement that the number of primes in an interval (n, n + h], averaged over n ≤ N, tends to the limit λ, when N and h tend to infinity in such a way that h ∼ λ log N, with λ a positive constant.

In this paper we give a proof of the long-standing Upper-bound Conjecture for convex polytopes, which states that, for 1 ≤ j < d < v, the maximum possible number of j-faces of a d-polytope with v vertices is achieved by a cyclic polytope C(v, d).

It was remarked by Liouville in 1844 that there is an obvious limit to the accuracy with which algebraic numbers can be approximated by rational numbers; if α is an algebraic number of degree n (at least 2) then

for all rational numbers h/q, where A is a positive number depending only on α.

The purpose of this paper is to show how a sieve method which has had many applications to problems involving rational primes can be modified to derive new results on Gaussian primes (or, more generally, prime ideals in algebraic number fields). One consequence of our main theorem (Theorem 2 below) is the following result on rational primes.

This paper is motivated by Davenport’s problem and the subsequent work regarding badly approximable points in submanifolds of a Euclidean space. We study the problem in the area of twisted Diophantine approximation and present two different approaches. The first approach shows that, under a certain restriction, any countable intersection of the sets of weighted badly approximable points on any non-degenerate ${\mathcal{C}}^{1}$ submanifold of $\mathbb{R}^{n}$ has full dimension. In the second approach, we introduce the property of isotropically winning and show that the sets of weighted badly approximable points are isotropically winning under the same restriction as above.

We consider an elliptic self-adjoint first-order differential operator $L$ acting on pairs (2-columns) of complex-valued half-densities over a connected compact three-dimensional manifold without boundary. The principal symbol of the operator $L$ is assumed to be trace-free and the subprincipal symbol is assumed to be zero. Given a positive scalar weight function, we study the weighted eigenvalue problem for the operator $L$ . The corresponding counting function (number of eigenvalues between zero and a positive $\unicode[STIX]{x1D706}$ ) is known to admit, under appropriate assumptions on periodic trajectories, a two-term asymptotic expansion as $\unicode[STIX]{x1D706}\rightarrow +\infty$ and we have recently derived an explicit formula for the second asymptotic coefficient. The purpose of this paper is to establish the geometric meaning of the second asymptotic coefficient. To this end, we identify the geometric objects encoded within our eigenvalue problem—metric, non-vanishing spinor field and topological charge—and express our asymptotic coefficients in terms of these geometric objects. We prove that the second asymptotic coefficient of the counting function has the geometric meaning of the massless Dirac action.

We analyse the behaviour of the spectrum of the system of Maxwell equations of electromagnetism, with rapidly oscillating periodic coefficients, subject to periodic boundary conditions on a“macroscopic” domain $(0,T)^{3},T>0.$ We consider the case where the contrast between the values of the coefficients in different parts of their periodicity cell increases as the period of oscillations $\unicode[STIX]{x1D702}$ goes to zero. We show that the limit of the spectrum as $\unicode[STIX]{x1D702}\rightarrow 0$ contains the spectrum of a “homogenized” system of equations that is solved by the limits of sequences of eigenfunctions of the original problem. We investigate the behaviour of this system and demonstrate phenomena not present in the scalar theory for polarized waves.