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We give a new explicit construction of n × N matrices satisfying the Restricted Isometry Property (RIP). Namely, for some ε > 0, large N and any n satisfying N 1−ε ≤ n ≤ N , we construct RIP matrices of order k ≥ n 1/2+ε and constant δ −ε. This overcomes the natural barrier k = O(n 1/2) for proofs based on small coherence, which are used in all previous… (More)

- KEVIN FORD
- 2006

We determine the order of magnitude of H(x, y, z), the number of integers n ≤ x having a divisor in (y, z], for all x, y and z. We also study H r (x, y, z), the number of integers n ≤ x having exactly r divisors in (y, z]. When r = 1 we establish the order of magnitude of H 1 (x, y, z) for all x, y, z satisfying z ≤ x 1/2−ε. For every r ≥ 2, C > 1 and ε >… (More)

- Kevin A Ford, John E Casida, +6 authors Mary C Wildermuth
- Proceedings of the National Academy of Sciences…
- 2010

Neonicotinoid insecticides control crop pests based on their action as agonists at the insect nicotinic acetylcholine receptor, which accepts chloropyridinyl- and chlorothiazolyl-analogs almost equally well. In some cases, these compounds have also been reported to enhance plant vigor and (a)biotic stress tolerance, independent of their insecticidal… (More)

We give a new explicit construction of n × N matrices satisfying the Restricted Isometry Property (RIP). Namely, for some ε > 0, large k and k 2−ε ≤ N ≤ k 2+ε , we construct RIP matrices of order k with n = O(k 2−ε). This overcomes the natural barrier n k 2 for proofs based on small coherence, which are used in all previous explicit constructions of RIP… (More)

We investigate the distribution of n − M (n) where M (n) = max { |a − b| : 1 ≤ a, b ≤ n − 1 and ab ≡ 1 (mod n)}. Exponential sums provide a natural tool for obtaining upper bounds on this quantity. Here we use results about the distribution of integers with a divisor in a given interval to obtain lower bounds on n − M (n). We also present some heuristic… (More)

- KEVIN FORD
- 2007

We give sharp, uniform estimates for the probability that the empirical distribution function for n uniform-[0, 1] random variables stays to one side of a given line.

We show that the equation φ(a) = σ(b) has infinitely many solutions, where φ is Euler's totient function and σ is the sum-of-divisors function. This proves a 50-year old conjecture of Erd˝ os. Moreover, we show that there are infinitely many integers n such that φ(a) = n and σ(b) = n each have more than n c solutions, for some c > 0. The proofs rely on the… (More)

We show that for a prime p the smallest a with a p−1 ≡ 1 (mod p 2) does not exceed (log p) 463/252+o(1) which improves the previous bound O((log p) 2) obtained by H. W. Lenstra in 1979. We also show that for almost all primes p the bound can be improved as (log p) 5/3+o(1) .

A new form of the equations of motion for a spacecraft with single gimbal control-moment gyros (CMG) is developed using a momentum approach. This set of four vector equations describing the rotational motion of the system is of order 2N + 7, where N is the number of CMGs. The control input is an N £ 1 column vector of torques applied to the gimbal axes. A… (More)

We enhance the efficient congruencing method for estimating Vinogradov's integral for moments of order 2s, with 1 s k 2 − 1. In this way, we prove the main conjecture for such even moments when 1 s 1 4 (k + 1) 2 , showing that the moments exhibit strongly diagonal behaviour in this range. There are improvements also for larger values of s, these finding… (More)