if (pp<min) { dis-; } else { swap(dis,min); for (inc=min+1; inc<=dis;) { if (prec(min,dis)) dis-; else if (prec(dis,min)) swap(dis-,min); else swap(dis,inc++); } min++; } } Figure 1: The minima algorithm, in C heuristically there is no reason to keep track of the hulls of two diierent sets. Perhaps a simpler but still provably good algorithm can found, particularly for d 3. Appendix The trivial...

In the preprint [3], Goldston, Pintz, and Yıldırım established, among other things, (0) lim inf n→∞ pn+1 − pn log pn = 0, with pn the nth prime. In the present article, which is essentially self-contained, we shall develop a simplified account of the method used in [3]. We include a short expository last section. Key word: Prime number. 1. Basic lemma. In this section we shall prove an asymptot...

We review the broad band properties of X-ray detected hotspots in radio galaxies and quasars. We show that their collective spectral properties can be unified in a framework involving frequency dependent relativistic beaming and varying orientations to the observer’s line of sight. The simplest dynamic model consistent with this picture is a slowing-down relativistic flow downstream from the ho...

We present our initial results on the CO rotational spectral line energy distribution (SLED) of the J to J−1 transitions from J = 4 up to 13 from Herschel SPIRE spectroscopic observations of 65 luminous infrared galaxies (LIRGs) in the Great Observatories All-Sky LIRG Survey (GOALS). The observed SLEDs change on average from one peaking at J ≤ 4 to a broad distribution peaking around J ∼ 6−7 as...

Based on a recent paper by Takhtajan, we propose a formulation of 2D quantum gravity whose basic object is the Liouville action on the Riemann sphere Σ0,m+n with both parabolic and elliptic points. The identification of the classical limit of the conformal Ward identity with the Fuchsian projective connection on Σ0,m+n implies a relation between conformal weights and ramification indices. This ...

Abstract Study question What is the clinical utility and associated outcomes of mosaic whole chromosome or segmental aneuploidies detected using concurrent copy-number genotyping analysis in PGT-A cycles? Summary answer Although high-level whole-chromosome mosaicism linked to reduced sustained implantation, it has limited significance cycles when co-evaluated with other clinical/ embryological ...

This paper examines simultaneous diophantine approximations to coordinates of certain points on a product of elliptic curves. Specifically, let p(z) be a Weierstrass elliptic function with algebraic invariants and complex multiplication. Suppose that 0 is cubic over the "field of multiplications" of p(z) and that u 6 C such that f = (p{u), p(/3u), p(/32u)) is defined. We study approximations to...

The energy-energy correlation function of the two-dimensional Ising model with weakly fluctuating random bonds is evaluated in the large scale limit. Two correlation lengths exist in contrast to one correlation length in the pure 2D Ising model: one is finite whereas the other is divergent at the critical points. The corresponding exponent of the divergent correlation length is νe = 1/2 in cont...