The distribution of spacings between quadratic residues

We study the distribution of spacings between squares modulo q, where q is square-free and highly composite, in the limit as the number of prime factors of q goes to infinity. We show that all correlation functions are Poissonian, which among other things, implies that the spacings between nearest neighbors, normalized to have unit mean, have an exponential distribution.Comment: 38 pages; introduction and section 6.2 revised, references updated. To appear in Duke Math. Journa

A generalization of Schur's theorem and its application to consecutive power residues

This article provides a proof of a generalization of Schur's theorem on the partition regularity of the equation x+y=z, which involves a divisibility condition. This generalization will be utilized to prove the existence of 'small' consecutive power residues modulo p, where p is a sufficiently large prime.Comment: 5 page

Permutations over cyclic groups

Generalizing a result in the theory of finite fields we prove that, apart from a couple of exceptions that can be classified, for any elements $a_1,...,a_m$ of the cyclic group of order $m$, there is a permutation $\pi$ such that $1a_{\pi(1)}+...+ma_{\pi(m)}=0$

Structure and function of the Rad9-binding region of the DNA-damage checkpoint adaptor TopBP1

TopBP1 is a scaffold protein that coordinates activation of the DNA-damage-checkpoint response by coupling binding of the 9-1-1 checkpoint clamp at sites of ssDNA, to activation of the ATR-ATRIP checkpoint kinase complex. We have now determined the crystal structure of the N-terminal region of human TopBP1, revealing an unexpected triple-BRCT domain structure. The arrangement of the BRCT domains differs significantly from previously described tandem BRCT domain structures, and presents two distinct sites for binding phosphopeptides in the second and third BRCT domains. We show that the site in the second but not third BRCT domain in the N-terminus of TopBP1, provides specific interaction with a phosphorylated motif at pSer387 in Rad9, which can be generated by CK2

Crucial stages of protein folding through a solvable model: predicting target sites for enzyme-inhibiting drugs

An exactly solvable model based on the topology of a protein native state is applied to identify bottlenecks and key-sites for the folding of HIV-1 Protease. The predicted sites are found to correlate well with clinical data on resistance to FDA-approved drugs. It has been observed that the effects of drug therapy are to induce multiple mutations on the protease. The sites where such mutations occur correlate well with those involved in folding bottlenecks identified through the deterministic procedure proposed in this study. The high statistical significance of the observed correlations suggests that the approach may be promisingly used in conjunction with traditional techniques to identify candidate locations for drug attacks.Comment: 12 pages, 5 figure