Phase Transition of Degeneracy in Minor-Closed Families

Given an infinite family ${\mathcal G}$ of graphs and a monotone property ${\mathcal P}$, an (upper) threshold for ${\mathcal G}$ and ${\mathcal P}$ is a "fastest growing" function $p: \mathbb{N} \to [0,1]$ such that $\lim_{n \to \infty} \Pr(G_n(p(n)) \in {\mathcal P})= 1$ for any sequence $(G_n)_{n \in \mathbb{N}}$ over ${\mathcal G}$ with $\lim_{n \to \infty}\lvert V(G_n) \rvert = \infty$, where $G_n(p(n))$ is the random subgraph of $G_n$ such that each edge remains independently with probability $p(n)$. In this paper we study the upper threshold for the family of $H$-minor free graphs and for the graph property of being $(r-1)$-degenerate, which is one fundamental graph property with many applications. Even a constant factor approximation for the upper threshold for all pairs $(r,H)$ is expected to be very difficult by its close connection to a major open question in extremal graph theory. We determine asymptotically the thresholds (up to a constant factor) for being $(r-1)$-degenerate for a large class of pairs $(r,H)$, including all graphs $H$ of minimum degree at least $r$ and all graphs $H$ with no vertex-cover of size at most $r$, and provide lower bounds for the rest of the pairs of $(r,H)$. The results generalize to arbitrary proper minor-closed families and the properties of being $r$-colorable, being $r$-choosable, or containing an $r$-regular subgraph, respectively

On invariants of constant pp-mean curvature surfaces in the Heisenberg group H1H_{1}

A fundamental goal of geometry of submanifolds is to find fascinating and significant classical examples. In this paper, depending on the theory we established in [5], along with the approach we provided for constructing constant $p$-mean curvature surfaces, we found some interesting examples of constant $p$-mean curvature surfaces. In particular, we give a complete description for rotationally invariant surfaces of constant $p$-mean curvature, including the geometric meaning of the energy $E$ (1.6}.Comment: 31 pages, 1 figure and 2 tables. Submitted. arXiv admin note: text overlap with arXiv:2101.1178

Influence of China's 2009 healthcare reform on the utilisation of continuum of care for maternal health services: evidence from two cross-sectional household surveys in Shaanxi Province.

BACKGROUND: Continuum of care for maternal health services (CMHS) is a proven approach to improve health and safety for mothers and newborns. This study aims to explore the influence of China's 2009 healthcare reform on improving the CMHS utilisation. METHODS: This population-based cross-sectional quantitative study included 2332 women drawn from the fourth and fifth National Health Service Surveys of Shaanxi Province, conducted in 2008 and 2013 respectively, before and after China's 2009 healthcare reform. A generalised linear mixed model (GLMM) was applied to analyse the influence of this healthcare reform on utilisation of CMHS. Concentration curves, concentration indexes and its decomposition method were used to analyse the equity of changes in utilisation. RESULTS: This study showed post-reform CMHS utilisation was higher in both rural and urban women than the CMHS utilisation pre-reform (according to China's policy defining CMHS). The rate of CMHS utilisation increased from 24.66 to 41.55% for urban women and from 18.31 to 50.49% for rural women (urban: χ2 = 20.64, P < 0.001; rural: χ2 = 131.38, P < 0.001). This finding is consistent when the WHO's definition of CMHS is applied for rural women after reform (12.13% vs 19.26%; χ2 = 10.99, P = 0.001); for urban women, CMHS utilisation increased from 15.70 to 20.56% (χ2 = 2.57, P = 0.109). The GLMM showed that the rate of CMHS utilisation for urban women post-reform was five times higher than pre-reform rates (OR = 5.02, 95%CL: 1.90, 13.31); it was close to 15 times higher for rural women (OR = 14.70, 95%CL: 5.43, 39.76). The concentration index for urban women decreased from 0.130 pre-reform (95%CI: - 0.026, 0.411) to - 0.041 post-reform (95%CI: - 0.096, 0.007); it decreased from 0.104 (95%CI: - 0.012, 0.222) to 0.019 (95%CI: - 0.014, 0.060) for rural women. The horizontal inequity index for both groups of women also decreased (0.136 to - 0.047 urban and 0.111 to 0.019 for rural). CONCLUSIONS: China's 2009 healthcare reform has positively influenced utilisation rates and equity of CMHS's utilisation among both urban and rural women in Shaanxi Province. Addressing economic and educational attainment gaps between the rich and the poor may be effective ways to improve the persistent health inequities for rural women

Is FS Tau B Driving an Asymmetric Jet?

FS Tau B is one of the few T Tauri stars that possess a jet and a counterjet as well as an optically-visible cavity wall. We obtained images and spectra of its jet-cavity system in the near-infrared H and K bands using Subaru/IRCS and detected the jet and the counterjet in the [Fe II] 1.644 \mu m line for the first time. Within the inner 2" the blueshifted jet is brighter, whereas beyond ~ 5" the redshifted counterjet dominates the [Fe II] emission. The innermost blueshifted knot is spectrally resolved to have a large line width of ~ 110 km/s, while the innermost redshifted knot appears spectrally unresolved. The velocity ratio of the jet to the counterjet is ~ 1.34, which suggests that FS Tau B is driving an asymmetric jet, similar to those found in several T Tauri Stars. Combining with optical observations in the literature, we showed that the blueshifted jet has lower density and higher excitation than the redshifted counterjet. We suggest that the asymmetry in brightness and velocity is the manifestation of a bipolar outflow driving at different mass-loss rates, while maintaining balance of linear momentum. A full explanation to the asymmetry in the FS Tau B system awaits detail modeling and further investigation of the kinematic structure of the wind-associated cavity walls.Comment: 14 pages, 2 figures, 1 table; accepted for publication in ApJ. Aspect ratio changes for Fig.1

Patient-controlled epidural Levobupicvacaine with or without Fentanyl for post-cesarean section pain relief

Purpose. The purpose of this study was to compare the analgesic properties of levobupivacaine with or without fentanyl for patient-controlled epidural analgesia after Cesarean section in a randomized, double-blinded study. Methods. We enrolled American Society of Anesthesiologists class I/II, full-term pregnant women at National Taiwan University Hospital who received patient-controlled epidural analgesia after Cesarean section between 2009 and 2010. Eighty women were randomly assigned into two groups. In group A, the 40 subjects received drug solutions made of 0.6 mg/ml levobupivacaine plus 2 mcg/ml fentanyl, and in group B the 40 subjects received 1 mg/ml levobupivacaine. Maintenance was self-administered boluses and a continuous background infusion. Results. There were no significant differences in the resting and dynamic pain scales and total volume of drug used between the two groups. Patient satisfaction was good in both groups. Conclusion. Our study showed that pure epidural levobupivacaine can provide comparative analgesic properties to the levobupivacaine-fentanyl combination after Cesarean section. Pure levobupivacaine may serve as an alternative pain control regimen to avoid opioid-related adverse events in parturients

Distributed nonnegative matrix factorization for web-scale dyadic data analysis on mapreduce

The Web abounds with dyadic data that keeps increasing by every single second. Previous work has repeatedly shown the usefulness of extracting the interaction structure inside dyadic data [21, 9, 8]. A commonly used tool in extracting the underlying structure is the matrix factorization, whose fame was further boosted in the Netflix challenge [26]. When we were trying to replicate the same success on real-world Web dyadic data, we were seriously challenged by the scal-ability of available tools. We therefore in this paper report our efforts on scaling up the nonnegative matrix factoriza-tion (NMF) technique. We show that by carefully partition-ing the data and arranging the computations to maximize data locality and parallelism, factorizing a tens of millions by hundreds of millions matrix with billions of nonzero cells can be accomplished within tens of hours. This result ef-fectively assures practitioners of the scalability of NMF on Web-scale dyadic data