In this paper we are concerned with upper bounds for the sums

where

and Δ2(n) is written simply as Δ(n). These functions were introduced by Hooley [4] and applied in a novel way to problems related to Waring's, and in Diophantine approximation. Thus Hooley deduced from his result about S2(x) that for any irrational θ, real γ, and fixed ε > 0, the inequality

holds for infinitely many n. His result for S3(x) led to a proof that

where r8(n) denotes the number of representations of n as the sum of eight positive cubes.

The effects on a boundary layer of thickness O(LR−1/2) (where L is a typical streamwise lengthscale, and R is the Reynolds number) of a small unsteady hump at the wall is considered. The hump is of height O(LR−5/8) and length O(LR−3/8), and outside the boundary layer is potential flow. Three different regimes of unsteadiness parameter are considered, leading to a description of the flow over the complete spectrum for this size of excrescence.

For α≥0 and β≥0 we denote by K (α, β) the Kaplan classes of functions f analytic and non-zero in the open unit disk U = {z: |z| < 1} such that f ∈ K(α, β), if, and only if, for θ1 < θ2 < θ1 + 2π and 0 < r < 1,

Where

The notion of stability at infinity for an infinite finitely presented group with one end was introduced in [14], and for groups stable at infinity, the end invariant e was defined and studied for some (non-trivial) direct products. In this paper we study the corresponding problem for extensions.

It is well known that the classical Gauss sum, normalized by the square-root number of terms, takes only finitely many values. If one restricts the range of summation to a subinterval, a much richer structure emerges. We prove a limit law for the value distribution of such incomplete Gauss sums. The limit distribution is given by the distribution of a certain family of periodic functions. Our results complement Oskolkov’s pointwise bounds for incomplete Gauss sums as well as the limit theorems for quadratic Weyl sums (theta sums) due to Jurkat and van Horne and the second author.

Based on a construction method introduced by Bourgain and Delbaen, we give a general definition of a Bourgain–Delbaen space and prove that every infinite-dimensional separable ${\mathcal{L}}_{\infty }$ -space is isomorphic to such a space. Furthermore, we provide an example of a ${\mathcal{L}}_{\infty }$ and asymptotic $c_{0}$ space not containing $c_{0}$ .

Let ℋ be the system of all continuous increasing functions h(t), denned for t ≥ 0, with h(0) = 0 and h(t)>0 for t > 0. Let Ω be a separable metric space. Then, for each h of ℋ, we may introduce a Hausdorff measure into Ω, by taking

where d(Fi) denotes the diameter of Fi, and where the infimum is taken over all sequences {Fi} of closed sets, covering E and having diameters less than δ. We introduce a natural partial order in the system of these Hausdorff measures by writing j < h, if j, h are functions of ℋ and

Let fn be a positive-definite n-ary quadratic form, with real coefficients. By the minimum of fn, denoted by min fn, is meant as usual the least value of fn(x1, …, xn) for integers xi not all 0. A minimum point of fn is a point x = (x1, …, xn), with integer coordinates, at which fn takes its minimum value. Let Δ (> 0) be the determinant of a set of n minimum points of fn; then in [1] it was proved that

where γn is the Hermite constant. Enough is known about γn to deduce from (1.1), as in [1], that

In 1891 Victor Eberhard proved the following theorem concerning the number pk(P) of k-gonal facets of simple polytopes P, [4].

Eberhard's Theorem. For each k ≥ 3, k ≠ 6, let pk be a non-negative integer. Then there exists a simple 3-polytope P such that pk(P) = pk (k ≠ 6), if, and only if,

Let pn denote the n-th prime number, and let

It follows from the prime number theorem that E ≤ 1. Erdős used a sieve argument to show that E < 1, and Rankin and Ricci gave the explicit estimates E ≤ 57/59 and E ≤ 15/16 respectively. A more powerful approach used by Hardy and Little-wood and by Rankin depended on hypotheses about the zeros of L-functions until Bombieri's Theorem was found. With its aid Bombieri and Davenport [1] proved that

A fuller history, with bibliography, is to be found in [1]

Introduction. Suppose that M i s a set in a cartesian product space X × Y. If X and Y are both R1 and M is a borel set, K. Kunugui [1] and W. J. Arsenin [2] have shown that those points which lie on sections {x} × Y which meet M in a nonempty Kσ set project onto a set in X which is the complement of an analytic set. Somewhat later, W. J. Arsenin and A. A. Lyapunov [3] reproduced the proof from [2], still apparently without being aware of Kunugui's results. Although, as we shall see later, Kunugui's proof seems to be incorrect, it does contain an ingenious lemma (Lemma 5 below). Actually [2] also contains this lemma in a disguised form that seems, to me at least, to be less satisfactory. One of the purposes of this article is to give a promised proof, see D. G. Larman [4] of Kunugui's theorem in slightly more general circumstances by combining ideas of Kunugui [1] with those of P. Novikoflf [5].

Le point de départ de ce travail est le résultat suivant.

THÉORÈME (I. Namioka). Soient X et Y deux espaces compacts et

Si f est continue quand on munit(Y) de la topologie de convergence simple alors X contient un Gδ dense en tout point duquel f reste continue quand on munit(Y) de la topologie de la convergence uniform.

A new criterion on Catalan's equation is proved by elementary means

This shows, without appealing either to the theory of linear forms in logarithms, or to any computation, that (C) has no solution (x, y, p, q) with min {p, q}≤41, except (3,2, 2, 3).

We prove a result related to the Erdős-Ginzburg-Ziv theorem: Let p and q be primes, α a positive integer, and m∈{pα, pαq}. Then for any sequence of integers c= {c1, c2,…, cn} there are at least

subsequences of length m, whose terms add up to 0 modulo m (Theorem 8). We also show why it is unlikely that the result is true for any m not of the form pα or pαq (Theorem 9).

The main object of this paper is to show that an indefinite quadratic form in four or more variables with integral coefficients represents all integers except those excluded by congruence considerations. The subject of the representation of integers by quadratic forms is one on which little is known, and I believe this result to be new, although I can deduce it in an elementary way from the classical theory.

In this paper, we consider an initial-value problem for the Korteweg–de Vries equation. The normalized Korteweg–de Vries equation considered is given by

$$\begin{equation*} u_{\tau }+u u_{x}+u_{xxx}=0, \quad -\infty <x<\infty ,\ \tau >0, \end{equation*}$$

$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}x$

$\tau $

$u(x,0) =u_{0}>0$

$x \le 0$

$u(x,0)=0$

$x>0$

$\tau $

We prove that if $k$ and $\ell$ are sufficiently large, then all the zeros of the weight $k+\ell$ cusp form $E_{k}(z)E_{\ell }(z)-E_{k+\ell }(z)$ in the standard fundamental domain lie on the boundary. We, moreover, find formulas for the number of zeros on the bottom arc with $|z|=1$, and those on the sides with $Re(z)=\pm 1/2$. One important ingredient of the proof is an approximation of the Eisenstein series in terms of the Jacobi theta function.

In the last three decades there appeared a number of elementary proofs of the prime number theorem (PNT) in the literature (see [3] for a survey). Most of these proofs are based, at least in part, on ideas from the original proof by Erdős [5] and Selberg [12]. In particular, one of the main ingredients of the Erdős-Selberg proof, Selberg's formula

(where p and q run through primes) appears, in some form, in almost all these proofs.

We give an asymptotic formula for the number of primes $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}p \le x$ of the form $p = [n_1^{c_1}] = \cdots = [n_d^{c_d}]$, where $c_1, \ldots, c_d$ are greater than 1 but “sufficiently close” to 1. This improves work of E. R. Sirota $(d=2)$ and W. Zhai $(d \ge 3)$.

Given a topological space X, we denote by Cp(X) the space of real-valued continuous functions on X, equipped with the topology of pointwise convergence.