Rational solutions of pairs of diagonal equations, one cubic and one quadratic

We obtain an essentially optimal estimate for the moment of order 32/3 of the exponential sum having argument $\alpha x^3+\beta x^2$. Subject to modest local solubility hypotheses, we thereby establish that pairs of diagonal Diophantine equations, one cubic and one quadratic, possess non-trivial integral solutions whenever the number of variables exceeds 10

Mean value estimates for odd cubic Weyl sums

We establish an essentially optimal estimate for the ninth moment of the exponential sum having argument $\alpha x^3+\beta x$. The first substantial advance in this topic for over 60 years, this leads to improvements in Heath-Brown's variant of Weyl's inequality, and other applications of Diophantine type

Solvable points on smooth projective varieties

We establish that smooth, geometrically integral projective varieties of small degree are not pointless in suitable solvable extensions of their field of definition, provided that this field is algebraic over $\Bbb Q$.Comment: 11 page

Multigrade efficient congruencing and Vinogradov's mean value theorem

We develop a multigrade enhancement of the efficient congruencing method to estimate Vinogradov's integral of degree $k$ for moments of order $2s$, thereby obtaining near-optimal estimates for $\tfrac{5}{8}k^2<s\le k^2-k+1$. There are numerous applications. In particular, when $k$ is large, the anticipated asymptotic formula in Waring's problem is established for sums of $s$ $k$th powers of natural numbers whenever $s>1.543k^2$. The asymptotic formula is also established for sums of $28$ fifth powers.Comment: 48pp; modest revisions in light of referee comment

Sums of three cubes, II

Estimates are provided for $s$th moments of cubic smooth Weyl sums, when $4\le s\le 8$, by enhancing the author's iterative method that delivers estimates beyond classical convexity. As a consequence, an improved lower bound is presented for the number of integers not exceeding $X$ that are represented as the sum of three cubes of natural numbers.Comment: 25 page

Approximating the main conjecture in Vinogradov's mean value theorem

We apply multigrade efficient congruencing to estimate Vinogradov's integral of degree $k$ for moments of order $2s$, establishing strongly diagonal behaviour for $1\le s\le \frac{1}{2}k(k+1)-\frac{1}{3}k+o(k)$. In particular, as $k\rightarrow \infty$, we confirm the main conjecture in Vinogradov's mean value theorem for 100% of the critical interval $1\le s\le \frac{1}{2}k(k+1)$.Comment: arXiv admin note: text overlap with arXiv:1310.844

Flight service evaluation of PRD-49/epoxy composite panels in wide-bodied commercial transport aircraft

Fairing panels were fabricated to evaluate the fabrication characteristics and flight service performance of PRD-49 (Kevlar-49) a composite reinforcing material and to compare it with the fiberglass which is currently in use. Panel configurations were selected to evaluate the PRD-49 with two resin matrix materials in sandwich and solid laminate construction. Left and right hand versions of these configurations were installed on L-1011's which will accumulate approximately 3000 flight hours per year per aircraft. The direct substitution of PRD-49 for fiberglass produced a twenty-six percent weight reduction on the panel configurations. Examination of these panels revealed that there was no visible difference between the PRD-49 and adjacent fiberglass panels

Relations between exceptional sets for additive problems

We describe a method for bounding the set of exceptional integers not represented by a given additive form in terms of the exceptional set corresponding to a subform. Illustrating our ideas with examples stemming from Waring's problem for cubes, we show, in particular, that the number of positive integers not exceeding N, that fail to have a representation as the sum of six cubes of natural numbers, is O(N^{3/7})

On Sums of Powers of Almost Equal Primes

We investigate the Waring-Goldbach problem of representing a positive integer $n$ as the sum of $s$ $k$th powers of almost equal prime numbers. Define $s_k=2k(k-1)$ when $k\ge 3$, and put $s_2=6$. In addition, put $\theta_2=\frac{19}{24}$, $\theta_3=\frac{4}{5}$ and $\theta_k=\frac{5}{6}$ $(k\ge 4)$. Suppose that $n$ satisfies the necessary congruence conditions, and put $X=(n/s)^{1/k}$. We show that whenever $s>s_k$ and $\varepsilon>0$, and $n$ is sufficiently large, then $n$ is represented as the sum of $s$ $k$th powers of prime numbers $p$ with $|p-X|\le X^{\theta_k+\varepsilon}$. This conclusion is based on a new estimate of Weyl-type specific to exponential sums having variables constrained to short intervals.Comment: 38 pages; in version 2 we have corrected a significant oversight in section 4 of the original version, leading to a slight adjustment of the admissible exponents for larger