In this paper, we discuss how to use the critical point theory to study the existence of gap solitons for periodic discrete nonlinear Schrödinger equations. An open problem proposed by Professor Alexander Pankov is solved.

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Summary Background Tinea capitis is still common in developing countries, such as China. Its pathogen spectrum varies across regions and changes over time. Objectives This study aimed to clarify the current epidemiological characteristics and pathogen spectrum of tinea capitis in China. Methods A multicentre, prospective descriptive study involving 29 tertiary hospitals in China was conducted. From August 2019 to July 2020, 611 patients with tinea capitis were enrolled. Data concerning demography, risk factors and fungal tests were collected. When necessary, the pathogens were further identified by morphology or molecular sequencing in the central laboratory. Results Among all enrolled patients, 74·1% of the cases were in patients aged 2–8 years. The children with tinea capitis were mainly boys (56·2%) and were more likely than adults to have a history of animal contact (57·4% vs. 35·3%, P = 0·012) and zoophilic dermatophyte infection (73·5% vs. 47%). The adults were mainly female (83%) and were more likely than children to have anthropophilic agent infection (53% vs. 23·9%). The most common pathogen was zoophilic Microsporum canis (354, 65·2%), followed by anthropophilic Trichophyton violaceum (74, 13·6%). In contrast to the eastern, western and northeastern regions, where zoophilic M. canis predominated, anthropophilic T. violaceum predominated in central China (69%, P < 0·001), where the patients had the most tinea at other sites (20%) and dermatophytosis contact (26%) but the least animal contact (39%). Microsporum ferrugineum was the most common anthropophilic agent in the western area, especially in Xinjiang province. Conclusions Boys aged approximately 5 years were the most commonly affected group. Dermatologists are advised to pay more attention to the different transmission routes and pathogen spectra in different age groups from different regions.