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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)
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)
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)
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)