Criterion on remote clocks synchronization within a Heisenberg scaling accuracy

We propose a quantum method to judge whether two spatially separated clocks have been synchronized within a specific accuracy $\sigma$. If the measurement result of the experiment is obviously a nonzero value, the time difference between two clocks is smaller than $\sigma$; otherwise the difference is beyond $\sigma$. On sharing the 2$N$-qubit bipartite maximally entangled state in this scheme, the accuracy of judgement can be enhanced to $\sigma\sim{\pi}/{(\omega(N+1))}$. This criterion is consistent with Heisenberg scaling that can be considered as beating standard quantum limit, moreover, the unbiased estimation condition is not necessary.Comment: 5 pages, 1 figur

A Comprehensive Analysis on Repeating Fast Radio Bursts

About 900 fast radio burst (FRB) sources have been detected till now, among which 29 FRBs are found to burst out repeatedly. Although a firm connection between at least some FRBs and magnetars has been established, the trigger mechanism and radiation process in these enigma phenomena are still highly controversial. In this study, we build a sample of 16 repeating FRBs from which at least five bursts have been detected, including the most active four repeaters of FRBs 20121102A, 20180916B, 20190520B and 20201124A. Various key parameters of their bursts are collected from the literature, which include the trigger time, pulse width, dispersion measure (DM), Faraday rotation measure (RM), bandwidth, waiting time, peak flux and fluence. The distribution and time evolution of these parameters are investigated. Potential correlations between various parameter pairs are also extensively explored. The behaviors of different repeaters are then compared. It is found that the DM of FRB 20121102A seems to increase continuously on a long timescale. While the $DM$ of most repeaters varies in a narrow range of $\pm 3$ cm$^{-3}$ pc, FRB 20190520B is found to have a large variation range of $\pm 12$ cm$^{-3}$ pc. A linear correlation between $DM$ and the star formation rate is established. $RM$ evolves with time in a much more chaos behavior in different repeaters. Generally, the waiting time shows a bimodal distribution in each source. The implications of these features to the underlying physics are discussed.Comment: 47 pages, 51 figures, 8 table

Improved Time Bounds for All Pairs Non-decreasing Paths in General Digraphs

We present improved algorithms for solving the All Pairs Non-decreasing Paths (APNP) problem on weighted digraphs. Currently, the best upper bound on APNP is O~(n^{(9+omega)/4})=O(n^{2.844}), obtained by Vassilevska Williams [TALG 2010 and SODA\u2708], where omega<2.373 is the usual exponent of matrix multiplication. Our first algorithm improves the time bound to O~(n^{2+omega/3})=O(n^{2.791}). The algorithm determines, for every pair of vertices s, t, the minimum last edge weight on a non-decreasing path from s to t, where a non-decreasing path is a path on which the edge weights form a non-decreasing sequence. The algorithm proposed uses the combinatorial properties of non-decreasing paths. Also a slightly improved algorithm with running time O(n^{2.78}) is presented

Role of Extractable and Residual Organic Matter Fractions on Sorption of Phenanthrene in Sediments

Two sediments were demineralized and sequentially fractionated into extracted fractions [free lipid (FL), bound lipid (BL) and lignin (LG)] and residual fractions [free lipid free (FLF), bound lipid free (BLF) and lignin free (LGF)]. The sorption isotherms of phenanthrene (Phen) were examined to evaluate the importance of various fractions on sorption. A lignin extraction procedure was for the first time applied to separate the lignin or degraded lignin fraction from sediment organic matter (SOM). The extracted LG was similar to model lignin in terms of elemental ratios and sorption behavior. FL and LG fractions were quite important, as their contents were much higher than reported values. Phen sorption for the extracted fractions was almost linear, whereas that for the residual fractions was nonlinear, especially for LGF with n 0.56-0.63. As the different organic fractions were removed sequentially, sorption energy distribution on the residual sediment organic matter (SOM) became more heterogeneous. In addition, increasing sorption capacity for the residual fractions, except for BLF with its high polarity, suggested that more sorption sites on the SOM matrix became accessible to Phen. The sorption capacity for LGF was comparable to that of condensed SOM. The residual fraction LGF generally controlled the overall sorption at low Phen concentration, but the extractable fraction FL surpassed the former fraction at high Phen concentration, demonstrating the importance of condensed SUM in the sorption of hydrophobic organic compounds (HOCs) in sediments. (c) 2012 Elsevier Ltd. All rights reserved

Roundtrip Spanners with (2k-1) Stretch

A roundtrip spanner of a directed graph $G$ is a subgraph of $G$ preserving roundtrip distances approximately for all pairs of vertices. Despite extensive research, there is still a small stretch gap between roundtrip spanners in directed graphs and undirected graphs. For a directed graph with real edge weights in $[1,W]$, we first propose a new deterministic algorithm that constructs a roundtrip spanner with $(2k-1)$ stretch and $O(k n^{1+1/k}\log (nW))$ edges for every integer $k> 1$, then remove the dependence of size on $W$ to give a roundtrip spanner with $(2k-1)$ stretch and $O(k n^{1+1/k}\log n)$ edges. While keeping the edge size small, our result improves the previous $2k+\epsilon$ stretch roundtrip spanners in directed graphs [Roditty, Thorup, Zwick'02; Zhu, Lam'18], and almost matches the undirected $(2k-1)$-spanner with $O(n^{1+1/k})$ edges [Alth\"ofer et al. '93] when $k$ is a constant, which is optimal under Erd\"os conjecture.Comment: 12 page