Weak phase stiffness and mass divergence of superfluid in underdoped cuprates

Despite more than two decades of intensive investigations, the true nature of high temperature (high-$T_c$) superconductivity observed in the cuprates remains elusive to the researchers. In particular, in the so-called `underdoped' region, the overall behavior of superconductivity deviates $qualitatively$ from the standard theoretical description pioneered by Bardeen, Cooper and Schrieffer (BCS). Recently, the importance of phase fluctuation of the superconducting order parameter has gained significant support from various experiments. However, the microscopic mechanism responsible for the surprisingly soft phase remains one of the most important unsolved puzzles. Here, opposite to the standard BCS starting point, we propose a simple, solvable low-energy model in the strong coupling limit, which maps the superconductivity literally into a well-understood physics of superfluid in a special dilute bosonic system of local pairs of doped holes. In the prototypical material (La$_{1-\delta}$Sr$_\delta$)$_2$CuO$_4$, without use of any free parameter, a $d$-wave superconductivity is obtained for doping above $\sim 5.2\%$, below which unexpected incoherent $p$-wave pairs dominate. Throughout the whole underdoped region, very soft phases are found to originate from enormous mass enhancement of the pairs. Furthermore, a striking mass divergence is predicted that dictates the occurrence of the observed quantum critical point. Our model produces properties of the superfluid in good agreement with the experiments, and provides new insights into several current puzzles. Owing to its simplicity, this model offers a paradigm of great value in answering the long-standing challenges in underdoped cuprates

Comparing different accelerometer cut-points for sedentary time in children

Actigraph accelerometers are hypothesized to be valid measurements for assessing children\u27s sedentary time. However, there is considerable variation in accelerometer cut-points used. Therefore, we compared the most common accelerometer sedentary cut-points of children performing sedentary behaviors. Actigraph Actitrainer uniaxial accelerometers were used to measure children\u27s activity intensity (29 children, 5-11 years old) during different activities, namely playing computer games, nonelectronic sedentary games, watching television and playing outdoors. A structured protocol was the criterion for assessing the validity of four common cut-points (100, 300, 800, 1100 counts/minute). The median counts during all sedentary behaviors were below the lowest comparison cut-point of 100 cpm. The 75th percentile values for the sedentary behaviors were always below the cut-point of 300 cpm. Our results suggest that the cut-point of <100 cpm is the most appropriate

Small gaps between products of two primes

Let $q_n$ denote the $n^{th}$ number that is a product of exactly two distinct primes. We prove that $$\liminf_{n\to \infty} (q_{n+1}-q_n) \le 6.$$ This sharpens an earlier result of the authors (arXivMath NT/0506067), which had 26 in place of 6. More generally, we prove that if $\nu$ is any positive integer, then $$\liminf_{n\to \infty} (q_{n+\nu}-q_n) \le C(\nu) = \nu e^{\nu-\gamma} (1+o(1)).$$ We also prove several other results on the representation of numbers with exactly two prime factors by linear forms.Comment: 11N25 (primary) 11N36 (secondary