Normal families and fixed points of iterates

Let F be a family of holomorphic functions and let K be a constant less than 4. Suppose that for all f in F the second iterate of f does not have fixed points for which the modulus of the multiplier is greater than K. We show that then F is normal. This is deduced from a result about the multipliers of iterated polynomials.Comment: 5 page

A mathematical model of critical jet heights causing droplets splashing in BOF steelmaking

It is important to understand the physical interaction between top-blown oxygen jet and liquid bath in basic oxygen steelmaking furnaces (BOF). In this study, cold model experiments were carried out to investigate the cavity depth, diameter and the instability at the gas-liquid interface. Images of the cavities were captured by a high-speed video camera to study cavity performances. A modified judging equation of the gas-liquid interface instability was developed by the critical jet flow and the critical jet height at a determined jet diameter. The critical parameters were in good agreement with experimental measurements

A mathematical model of critical jet heights causing droplets splashing in BOF steelmaking

It is important to understand the physical interaction between top-blown oxygen jet and liquid bath in basic oxygen steelmaking furnaces (BOF). In this study, cold model experiments were carried out to investigate the cavity depth, diameter and the instability at the gas-liquid interface. Images of the cavities were captured by a high-speed video camera to study cavity performances. A modified judging equation of the gas-liquid interface instability was developed by the critical jet flow and the critical jet height at a determined jet diameter. The critical parameters were in good agreement with experimental measurements

Gender difference in the association between education and schizophrenia in Chinese adults

Background: Improving education level was evidenced to decrease the risk of schizophrenia, but whether this strength of education role depends on gender is not. This study aimed to investigate whether there was gender difference in the association between education and schizophrenia in Chinese adults. Methods: Data were obtained from the Second China National Sample Survey on Disability in 2006, including 1,909,205 participants aged 18 years or older. Schizophrenia was ascertained according to the International Statistical Classification of Diseases, Tenth Revision. Logistics regression models were fitted to examine the combined effect of gender and education on schizophrenia. Results: The lifetime prevalence of schizophrenia in female groups was higher than in male groups, with 0.44% (95%CI: 0.42-0.45%) and 0.36% (95%CI: 0.35-0.37%), respectively. Compared with schizophrenia male patients, more females with schizophrenia experienced severe or extreme difficulty in understanding and communicating. However, more males with schizophrenia suffered from severe or extreme difficulty in the function of daily activities. The combined effect of education and schizophrenia was statistically significant, indicating that, as the level of education increased, schizophrenia risk of females decreased faster than the risk of males. Conclusions: This study showed that additional years of education associated with lower risk of schizophrenia, and this association was stronger in females than in males. As education elevated, the risk of schizophrenia decreased more for women than for men. The findings indicate that improving education level may have an effect on reducing the gender disparities in mental health of China. Actions to prevent schizophrenia and address its gender disparities will require attention to the improving educational opportunities