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)
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 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)
An old conjecture of Sierpi´nski asserts that for every integer k 2, there is a number m for which the equation φ(x) = m has exactly k solutions. Here φ is Euler's totient function. In 1961, Schinzel deduced this conjecture from his Hypothesis H. The purpose of this paper is to present an unconditional proof of Sierpi´nski's conjecture. The proof uses many… (More)
We investigate the distribution of the fractional parts of αγ, where α is a fixed non-zero real number and γ runs over the imaginary parts of the non-trivial zeros of the Riemann zeta function.
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) .
Let P (k) be the largest prime factor of the positive integer k. In this paper, we prove that the series n≥1 (log n) α P (2 n − 1) is convergent for each constant α < 1/2, which gives a more precise form of a result of C. L. Stewart of 1977.
We study the distribution of prime chains, which are sequences p 1 ,. .. , p k of primes for which p j+1 ≡ 1 (mod p j) for each j. We first give conditional upper bounds on the length of Cunningham chains, chains with p j+1 = 2p j +1 for each j. We give estimates for P (x), the number of chains with p k x (k variable), and P (x; p), the number of chains… (More)
The present study quantitatively compared the basilar dendritic/spine systems of lamina V pyramidal neurons across four hierarchically arranged regions of neonatal human neocortex. Tissue blocks were removed from four Brodmann's areas (BAs) in the left hemisphere of four neurologically normal neonates (mean age=41+/- 40 days): primary (BA4 and BA3-1-2),… (More)