Let ${\rm\Lambda}(n)$ be the von Mangoldt function, $x$ be real and $2\leqslant y\leqslant x$. This paper improves the estimate for the exponential sum over primes in short intervals

$$\begin{eqnarray}S_{k}(x,y;{\it\alpha})=\mathop{\sum }_{x<n\leqslant x+y}{\rm\Lambda}(n)e(n^{k}{\it\alpha})\end{eqnarray}$$

$k\geqslant 3$

${\it\alpha}$

$n$

$s$

$k$

The problem concerning the distribution of the fractional parts of the sequence ank (k an integer exceeding one) was first considered by Hardy and Littlewood [6] and Weyl [20] earlier this century. This work was developed, with the focus on small fractional parts of the sequence, by Vinogradov [17], Heilbronn [13] and Danicic [2] (see [1]). Recently Heath-Brown [12] has improved the unlocalized versions of these results for k ≥ 6 (a slightly stronger result than Heath-Brown's for K = 8 is given on page 24 of [8]. The method mentioned there can, after some numerical calculation, improve Heath-Brown's result for 8 ≤ k ≤ 20, but still stronger results have recently been obtained by Dr. T. D. Wooley). The cognate problem regarding the sequence apk, where p denotes a prime, has also received some attention. In this situation even the case k = 1 proves to be difficult (see [9] and [14]). The first results in this field were given by Vinogradov (see Chapter 11 of [19] for the case k = 1, [18] for k ≥ 2). For k = 2 the best result to date has been supplied by Ghosh [5], and for ≥, by Harman (Theorem 3 in [9], building on the work in [7] and [8]). In this paper we shall improve the known results for 2 ≤ k ≤ 12. For larger k, Theorem 3 in [8] is more efficient. The theorem we prove is as follows.

We apply multigrade efficient congruencing to estimate Vinogradov’s integral of degree $k$ for moments of order $2s$, establishing strongly diagonal behaviour for $1\leqslant s\leqslant \frac{1}{2}k(k+1)-\frac{1}{3}k+o(k)$. In particular, as $k\rightarrow \infty$, we confirm the main conjecture in Vinogradov’s mean value theorem for a proportion asymptotically approaching $100\%$ of the critical interval $1\leqslant s\leqslant \frac{1}{2}k(k+1)$.

The quasigeostrophic equations describe large scale motion in the atmosphere and oceans at middle latitudes. Being considerably simpler than the primitive equations, they have been widely used for modelling atmospheric and oceanic circulation, and for studies of stability, frontogenesis and turbulence. A number of assertions have been made about these equations; first, that finite element approximate solutions converge in an open flow region [7]; secondly, that solutions depend discontinuously or nondeterministically on initial data [13]; and thirdly, that the energy wave number spectra of the solutions asymptote to the statistical equilibrium spectra of the spectrally truncated equations [12].

Abstract. In this paper is proved a formula for the number of neighbourly d-polytopes with d + 3 vertices, when d is odd.

Let En+1, for some integer n ≥ 0, be the (n + 1)-dimensional Euclidean space, and denote by Sn the standard n–sphere in En+1, . It is convenient to introduce the (–1)-dimensional sphere , where denotes the empty set. By an i-dimensional subsphere T of Sn, i = 0 n, we understand the intersection of Sn with some (i+1)-dimensional subspace of En+1. The affine hull of T always contains, with this definition, the origin of En+1. is the unique (–1)-dimensional subsphere of Sn. By the spherical hull, sph X, of a set , we understand the intersection of all subspheres of Sn containing X. Further we set dim X: = dim sph X. The interior, the boundary and the complement of an arbitrary set , with respect to Sn, shall be denoted by int X, bd X and cpl X. Finally we define the relative interior rel int X to be the interior of with respect to the usual topology sphZ . For each (n–1)-dimensional subsphere of Sn defines two closed hemispheres of Sn, whose common boundary it is. The two hemispheres of the sphere Sº are denned to be the two one-pointed subsets of Sº. A subset is called a closed (spherical) polytope, if it is the intersection of finitely many closed hemispheres, and, if, in addition, it does not contain a subsphere of Sn. is called an i-dimensional, relatively open polytope, , or shortly an i-open polytope, if there exists a closed polytope such that dim P = i and Q = rel int P. is called a closed polyhedron, if it is a finite union of closed polytopes P1 …, Pr. The empty set is the only (–1)-dimensional closed polyhedron of Sn. We denote by the set of all closed polyhedra of Sn. is called an i-open polyhedron, for some , if there are finitely many i-open polytopes Q1 …, Qr in Sn such that , and dim . By we denote the set of all i-open polyhedra. Clearly for all , and each i-dimensional subsphere of Sn, , belongs to and to , For each i-dimensional subsphere T of Sn, set . A map is defined by , for all , and, for all .

Let V = V2n be the Segre product variety of two n–dimensional complex projective spaces. Then any Cremona transformation of Sn into (regarded as an irreducible algebraic system of ∞n ordered pairs of points) is represented on V by an irreducible n-dimensional subvariety H which satisfies (on V) the algebraic equivalence

where Si, j is a subvariety of V and m1, …, mn–1 are positive integers. We call m1, …, mn–1 the characters of T noting that, numerically,

It was conjectured by Artin [1] that each non-zero integer a unequal to +1, −1 or a perfect square is a primitive root for infinitely many primes p. More precisely, denoting by Na(x) the number of primes p ≤ x for which a is a primitive root, he conjectured that

where c(a) is a positive constant. This conjecture has recently been proved by C. Hooley [2] under the assumption that the Riemann hypothesis holds for fields of the type .

In this note we remark on certain “universal fixed messages” for the Rivest-Shamir-Adleman cryptosystem [1], and we describe how in certain cases these universal fixed messages can play a role in an attempt to break the cryptosystem. This use of the universal fixed messages in an attempt to break the RSA cryptosystem† involves a certain prime factorization technique using the fixed messages. We characterize the situation in which such an attempt would be successful in practice, and we show that a user of the RSA cryptosystem can easily arrange to avoid this situation. Hence the use of the present technique does not pose a threat to the security of the RSA cryptosystem.

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].

We prove that Alexandrov spaces of non-negative curvature have Markov type 2 in the sense of Ball. As a corollary, any Lipschitz continuous map from a subset of an Alexandrov space of non-negative curvature into a 2-uniformly convex Banach space can be extended to a Lipschitz continuous map on the entire space.

We show that the conjecture of Kannan, Lovász, and Simonovits on isoperimetric properties of convex bodies and log-concave measures is true for log-concave measures of the form ρ(∣x∣B) dx on ℝn and ρ(t,∣x∣B) dx on ℝ1+n, where ∣x∣B is the norm associated to any convex body B already satisfying the conjecture. In particular, the conjecture holds for convex bodies of revolution.

§1. Introduction. In this paper we give a new proof of a theorem of Bombieri and Davenport [2, Theorem 1]. Let t(–k) = t(k) be real,

where e(u) = e2πiu. Let p and p' denote primes, k an integer, and define

Given a probability density, we estimate the rate of decay of the measure of the level sets of its evolutes by the Ornstein–Uhlenbeck semigroup. The rate is faster than what follows from the preservation of mass and Markov’s inequality.

We investigate the intersections of the curve $ \mathbb{R} \ni t\mapsto \zeta (\frac{1}{2} + \mathrm{i} t)$ with the real axis. We show unconditionally that the zeta function takes arbitrarily large positive and negative values on the critical line.

The number of self-complementary graphs and the number of self-complementary digraphs were expressed by Read [4] in terms of cycle indexes of the appropriate pair groups. These formulas for and , together with a modification of the method employed by Oberschelp [3] for graphs, can be used to obtain estimates for and and a bound on the error. For graph theoretic definitions not given here, we refer to the book [2].

Following A. S. Besicovitch [1] and L. J. Mordell [3], we say that a polygon is rational if the lengths of all its sides and diagonals are rational. Besicovitch proved that the set of all rational right-angled triangles is dense in the set of all right-angled triangles and that the set of all rational parallelograms is dense in the set of all parallelograms. Then Mordell showed that the set of all rational quadrilaterajs is dense in the set of all quadrilaterals.

Since my article McMullen [1980] has appeared, Professors S. S. Ryškov and B. B. Venkov have drawn my attention to two previously published papers. B. A. Venkov [1954] proves my main Theorem 1 (and its corollary Theorem 2) by methods apparently very similar to mine (1 have not checked all the details), while A. D. Aleksandrov [1954] generalizes Venkov's result to tilings of spaces of constant curvature by polytopes (not necessarily convex) congruent to ones in some finite collection. I am happy to acknowledge their priority.

The integral equation

occurs in the theory of the motion of an electrically conducting fluid, occupying the space between two electrodes at different potentials, in the presence of a strong magnetic fluid [Hunt and Malcolm 1]. In this theory a solution is required in which

and one of the principal properties of the solution to be determined is

It has been shown by Mirsky and Perfect [1] that the theorem of Rado [2], linking matroid theory and transversal theory, has important applications in combinatorial theory. In this note I use it to obtain necessary and sufficient conditions for two families of sets to have a common transversal containing a given set, and then I show how it may be used to obtain a variant of a well-known theorem that was obtained by Hoffman and Kuhn [3] using linear programming methods.