Counting Smooth Solutions to the Equation A+B=C

This paper studies integer solutions to the Diophantine equation A+B=C in which none of A, B, C have a large prime factor. We set H(A, B,C) = max(|A|, |B|, |C|), and consider primitive solutions (gcd}(A, B, C)=1) having no prime factor p larger than (log H(A, B,C))^K, for a given finite K. On the assumption that the Generalized Riemann hypothesis (GRH) holds, we show that for any K > 8 there are infinitely many such primitive solutions having no prime factor larger than (log H(A, B, C))^K. We obtain in this range an asymptotic formula for the number of such suitably weighted primitive solutions.Comment: 35 pages latex; v2 corrected misprint

Optimal multiqubit operations for Josephson charge qubits

We introduce a method for finding the required control parameters for a quantum computer that yields the desired quantum algorithm without invoking elementary gates. We concentrate on the Josephson charge-qubit model, but the scenario is readily extended to other physical realizations. Our strategy is to numerically find any desired double- or triple-qubit gate. The motivation is the need to significantly accelerate quantum algorithms in order to fight decoherence.Comment: 4 pages, 5 figure

Pure point diffraction implies zero entropy for Delone sets with uniform cluster frequencies

Delone sets of finite local complexity in Euclidean space are investigated. We show that such a set has patch counting and topological entropy 0 if it has uniform cluster frequencies and is pure point diffractive. We also note that the patch counting entropy is 0 whenever the repetitivity function satisfies a certain growth restriction.Comment: 16 pages; revised and slightly expanded versio

Photoactivatable genetically encoded calcium indicators for targeted neuronal imaging.

Circuit mapping requires knowledge of both structural and functional connectivity between cells. Although optical tools have been made to assess either the morphology and projections of neurons or their activity and functional connections, few probes integrate this information. We have generated a family of photoactivatable genetically encoded Ca(2+) indicators that combines attributes of high-contrast photolabeling with high-sensitivity Ca(2+) detection in a single-color protein sensor. We demonstrated in cultured neurons and in fruit fly and zebrafish larvae how single cells could be selected out of dense populations for visualization of morphology and high signal-to-noise measurements of activity, synaptic transmission and connectivity. Our design strategy is transferrable to other sensors based on circularly permutated GFP (cpGFP)