We improve W. Schmidt's lower bound for the slice (intersection of two halfspheres) discrepancy of point distributions on spheres and show that this estimate is up to a logarithmic factor best possible. It is shown that the slice and spherical cap discrepancies are equivalent for the definition of uniformly distributed sequences on spheres.

Let Ω(n) denote the number of prime factors of n, counted according to multiplicity. We shall consider the following question. Are there infinitely many natural numbers n for which Ω(n) = Ω(n + 1)? Erdős and Mirsky [4] have asked a closely related question concerning the divisor function d(n)—are there infinitely many n for which d(n) = d(n + 1)? The fact that Ω(n)is completely additive makes our problem slightly easier.

In many physical problems it is necessary to express a solution of Laplace's equation relative to one set of coordinates in terms of harmonics relative to another set. We term such a relationship an addition or shift formula. Well-known examples are the formulae for spherical harmonics [Hobson 1, p. 139] and cylindrical harmonics [Watson 2, p. 360]. In this paper we shall consider the problem for oblate spheroidal coordinate systems and obtain some addition theorems for the corresponding harmonics. These addition theorems are used to write down “two-centred” expansions of the Coulomb Green's function, and by a limiting process a new form of the “one-centred” expansion is obtained in a non-orthogonal coordinate system, closely related to oblate spheroidal coordinates. This expansion is applied to the evaluation of the Coulomb (or gravitational) energy of spheroidal distributions of charge (or mass) in which the surfaces of constant density are concentric similar spheroids, a situation which occurs in both nuclear physics and cosmology [Carlson 3]

In this paper we study the number of rational points on curves in an ensemble of abelian covers of the projective line: let $\ell$ be a prime, $q$ a prime power and consider the ensemble ${\mathcal{H}}_{g,\ell }$ of $\ell$-cyclic covers of $\mathbb{P}_{\mathbb{F}_{q}}^{1}$ of genus $g$. We assume that $q\not \equiv 0,1~\text{mod}~\ell$. If $2g+2\ell -2\not \equiv 0~\text{mod}~(\ell -1)\operatorname{ord}_{\ell }(q)$, then ${\mathcal{H}}_{g,\ell }$ is empty. Otherwise, the number of rational points on a random curve in ${\mathcal{H}}_{g,\ell }$ distributes as $\sum _{i=1}^{q+1}X_{i}$ as $g\rightarrow \infty$, where $X_{1},\ldots ,X_{q+1}$ are independent and identically distributed random variables taking the values $0$ and $\ell$ with probabilities $(\ell -1)/\ell$ and $1/\ell$, respectively. The novelty of our result is that it works in the absence of a primitive $\ell$th root of unity, the presence of which was crucial in previous studies.

We prove that the relation type of all high powers of an ideal in a Noetherian ring is either one or two. It is one exactly when some power of the ideal is locally principal.

Norms with moduli of smoothness of power type are constructed on spaces with the Radon-Nikodym property that admit pointwise Lipschitz bump functions with pointwise moduli of smoothness of power type. It is shown that no norms with pointwise moduli of rotundity of power type can exist on nonsuperreflexive spaces. A new smoothness characterization of spaces isomorphic to Hilbert spaces is given.

We prove that if 3|d, then the d-dimensional balls are m-divisible for every m large enough. In particular, the three-dimensional balls are m-divisible for every m≥22.

The Picone identity which occurs in the oscillation theory of second-order ordinary differential equations is

A study is made of the solutions of the differential equation f′″ + ff″ = 1 subject to the initial conditions f(0) = f′(0) = 0.

The purpose of this paper is to demonstrate that a number of properties of independence spaces are of finite character, thus making it possible to easily generalise known theorems for finite spaces, or matroids, to independence spaces on infinite sets.

Since Minkowski [29] gave his famous lattice point theorem for centrally symmetric convex bodies, a theorem that turned out to be of fundamental importance in number theory, much effort has been made to obtain tight estimates for the number of lattice points of a given lattice in convex bodies in terms of the basic quermass-integrals Wo,…, Wd, whose eminent role shows in Hadwiger's functional theorem [14, 15, 16, see also 17, p. 221–225]. (For the discrete analogues of Wo,…, Wd see [2].) The present paper is concerned with an upper estimate of this kind.

We shall consider incomplete exponential sums of the shape

where q, a and h are integers satisfying 1 ≤ a < a + h ≤ q, f(x) is a function denned at least for the integers in the range of summation, and eq(t) is an abbreviation for e2πit/q.

1. Throughout this note p is a prime and θ = θ(x1, …, xn) a polynomial of degree 3, with integral coefficients and an integral constant term. The object is to study, by elementary methods, the cubic congruence θ(x1, … xn)≡0 (mod p). (1)

This paper presents an analysis of two canonical problems for the steady supersonic inviscid flow past a wing-body combination with the body nonlifting and the wings at a very small angle of attack. Pressure distributions in the vicinity of the lines separating the interaction regions from the remainder of the wing and body are determined from asymptotic expansions by modifying a method due to Friedlander, who considered the diffraction of sound pulses by a cylinder. In a manner similar to Keller's geometrical theory of diffraction the results of the canonical problem are directly applicable to the case where the cross section of the body is any closed convex curve which meets the wings at right angles. In the first canonical problem, the leading edge of the wings has no sweepback, that is, it is normal to the body surface. Here results are compared with those of an approach using boundary layer methods by Stewartson and of a Green's function method by Jones. In the second problem which is even more important in practice and has been neglected in the literature, the leading edge of the wings has positive sweepback; our results are valid provided that the leading edge of the wings is supersonic.

Let λ1, …, λ8 be any non-zero real numbers not all of the same sign and not all in rational ratios. According to a theorem of Davenport and Roth [3], given a real number κ, the inequality

has infinitely many solutions in positive integers for any ε > 0. Recently Liu, Ng and Tsang [5] gave a refinement of this result: for any δ > 0, the inequality

has infinitely many solutions in positive integers. In the present note we obtain a better exponent.

We show that if d≥4 is even, then one can find two essentially different convex bodies such that the volumes of their maximal sections, central sections, and projections coincide for all directions.

Let ${\it\mu}_{1},\ldots ,{\it\mu}_{s}$ be real numbers, with ${\it\mu}_{1}$ irrational. We investigate sums of shifted $k$th powers $\mathfrak{F}(x_{1},\ldots ,x_{s})=(x_{1}-{\it\mu}_{1})^{k}+\cdots +(x_{s}-{\it\mu}_{s})^{k}$. For $k\geqslant 4$, we bound the number of variables needed to ensure that if ${\it\eta}$ is real and ${\it\tau}>0$ is sufficiently large then there exist integers $x_{1}>{\it\mu}_{1},\ldots ,x_{s}>{\it\mu}_{s}$ such that $|\mathfrak{F}(\mathbf{x})-{\it\tau}|<{\it\eta}$. This is a real analogue to Waring’s problem. When $s\geqslant 2k^{2}-2k+3$, we provide an asymptotic formula. We prove similar results for sums of general univariate degree-$k$ polynomials.

We prove a family of sharp bilinear space–time estimates for the half-wave propagator $\text{e}^{\text{i}t\sqrt{-\unicode[STIX]{x1D6E5}}}$ . As a consequence, for radially symmetric initial data, we establish sharp estimates of this kind for a range of exponents beyond the classical range.

We obtain density estimates for subsets of the $n$-dimensional integer lattice lacking four-term matrix progressions. As a consequence, we show that a subset of the grid $\{1,2,\dots ,N\}^{2}$ lacking four corners in a square has size at most $\mathit{CN}^{2}(\log \log N)^{-c}$. Our proofs involve the density increment method of Roth [J. London Math. Soc.28 (1953), 104–109] and Gowers [Geom. Funct. Anal.11(3) (2001), 465–588], together with the $U^{3}$-inverse theorem of Green and Tao [Proc. Edinb. Math. Soc. (2) 51(1) (2008), 73–153].

In this paper we prove sharp estimates for generalized eigenvectors of Hermitian Jacobi matrices associated with the spectral parameter lying in a gap of their essential spectra. The estimates do not depend on the main diagonals of these matrices. The types of estimates obtained for bounded and unbounded gaps are different. These estimates extend the previous ones found in [J. Janas, S. Naboko and G. Stolz, Decay bounds on eigenfunctions and the singular spectrum of unbounded Jacobi matrices. Int. Math. Res. Not.4 (2009), 736–764], for the spectral parameter being outside the whole spectrum of Jacobi matrices. Examples illustrating optimality of our results are also given.