In the few days prior to eye-opening in mice, the excitatory drive underlying waves switches from cholinergic to glutamatergic. Here, we describe the unique synaptic and spatiotemporal properties of waves generated by the retina's glutamatergic circuits. First, knockout mice lacking vesicular glutamate transporter type 1 do not have glutamatergic waves, but… (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)
Before vision, a transient network of recurrently connected cholinergic interneurons, called starburst amacrine cells (SACs), generates spontaneous retinal waves. Despite an absence of robust inhibition, cholinergic retinal waves initiate infrequently and propagate within finite boundaries. Here, we combine a variety of electrophysiological and imaging… (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)
In the few weeks prior to the onset of vision, the retina undergoes a dramatic transformation. Neurons migrate into position and target appropriate synaptic partners to assemble the circuits that mediate vision. During this period of development, the retina is not silent but rather assembles and disassembles a series of transient circuits that use distinct… (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) .
The strength and dynamics of synaptic transmission are determined, in part, by the presynaptic action potential (AP) waveform at the nerve terminal. The ion channels that shape the synaptic AP waveform remain essentially unknown for all but a few large synapses amenable to electrophysiological interrogation. The Drosophila neuromuscular junction (NMJ) is a… (More)