Let $\mathbb{N}$ be the set of all non-negative integers. For any integer r and m, let $r+m\mathbb{N}=\{r+mk: k\in\mathbb{N}\}$. For $S\subseteq \mathbb{N}$ and $n\in \mathbb{N}$, let $R_{S}(n)$ denote the number of solutions of the equation $n=s+s'$ with $s, s'\in S$ and $s \lt s'$. Let $r_{1}, r_{2}, m$ be integers with $0 \lt r_{1} \lt r_{2} \lt m$ and $2\mid r_{1}$. In this paper, we prove that there exist two sets C and D with $C\cup D=\mathbb{N}$ and $C\cap D=(r_{1}+m\mathbb{N})\cup (r_{2}+m\mathbb{N})$ such that $R_{C}(n)=R_{D}(n)$ for all $n\in\mathbb{N}$ if and only if there exists a positive integer l such that $r_{1}=2^{2l+1}-2, r_{2}=2^{2l+1}-1, m=2^{2l+2}-2$.

High prevalence of long COVID symptoms has emerged as a significant public health concern. This study investigated the associations between three doses of COVID-19 vaccines and the presence of any and ≥3 types of long COVID symptoms among people with a history of SARS-CoV-2 infection in Hong Kong, China. This is a secondary analysis of a cross-sectional online survey among Hong Kong adult residents conducted between June and August 2022. This analysis was based on a sub-sample of 1,542 participants with confirmed SARS-CoV-2 infection during the fifth wave of COVID-19 outbreak in Hong Kong (December 2021 to April 2022). Among the participants, 40.9% and 16.1% self-reported having any and ≥3 types of long COVID symptoms, respectively. After adjusting for significant variables related to sociodemographic characteristics, health conditions and lifestyles, and SARS-CoV-2 infection, receiving at least three doses of COVID-19 vaccines was associated with lower odds of reporting any long COVID symptoms comparing to receiving two doses (adjusted odds ratio [AOR]: 0.69, 95% CI: 0.54, 0.87, P = .002). Three doses of inactivated and mRNA vaccines had similar protective effects against long COVID symptoms. It is important to strengthen the coverage of COVID-19 vaccination booster doses, even in the post-pandemic era.

Timing of food intake is an emerging aspect of nutrition; however, there is a lack of research accurately assessing food timing in the context of the circadian system. The study aimed to investigate the relation between food timing relative to clock time and endogenous circadian timing with adiposity and further explore sex differences in these associations among 151 young adults aged 18–25 years. Participants wore wrist actigraphy and documented sleep and food schedules in real time for 7 consecutive days. Circadian timing was determined by dim-light melatonin onset (DLMO). The duration between last eating occasion and DLMO (last EO-DLMO) was used to calculate the circadian timing of food intake. Adiposity was assessed using bioelectrical impedance analysis. Of the 151 participants, 133 were included in the statistical analysis finally. The results demonstrated that associations of adiposity with food timing relative to circadian timing rather than clock time among young adults living in real-world settings. Sex-stratified analyses revealed that associations between last EO-DLMO and adiposity were significant in females but not males. For females, each hour increase in last EO-DLMO was associated with higher BMI by 0·51 kg/m2 (P = 0·01), higher percent body fat by 1·05 % (P = 0·007), higher fat mass by 0·99 kg (P = 0·01) and higher visceral fat area by 4·75 cm2 (P = 0·02), whereas non-significant associations were present among males. The findings highlight the importance of considering the timing of food intake relative to endogenous circadian timing instead of only as clock time.

The Chengba site is the only city site dated from the late Warring States Period in eastern Sichuan Province, China. New discoveries of artefacts and structures at the site enable exploration of the regional role and management of counties that were established at this time by the central government.

Soft drink consumption has become a highly controversial public health issue. Given the pattern of consumption in China, sugar-sweetened beverage is the main type of soft drink consumed. Due to containing high levels of fructose, a soft drink may have a deleterious effect on handgrip strength (HGS) due to oxidative stress, inflammation and insulin resistance. However, few studies show an association between soft drink consumption and HGS in adults. We aimed to investigate the association between soft drink consumption and longitudinal changes in HGS among a Chinese adult population. A longitudinal population-based cohort study (5-year follow-up, median: 3·66 years) was conducted in Tianjin, China. A total of 11 125 participants (56·7 % men) were enrolled. HGS was measured using a handheld digital dynamometer. Soft drink consumption (mainly sugar-containing carbonated beverages) was measured at baseline using a validated FFQ. ANCOVA was used to evaluate the association between soft drink consumption and annual change in HGS or weight-adjusted HGS. After adjusting for multiple confounding factors, the least square means (95 % CI) of annual change in HGS across soft drink consumption frequencies were −0·70 (–2·49, 1·09) for rarely drinks, −0·82 (–2·62, 0·97) for < 1 cup/week and −0·86 (–2·66, 0·93) for ≥ 1 cup/week (Pfor trend < 0·05). Likewise, a similar association was observed between soft drink consumption and annual change in weight-adjusted HGS. The results indicate that higher soft drink consumption was associated with faster HGS decline in Chinese adults.

In this paper, we study the singular boundary value problem

$$ \begin{align*} \begin{cases} \Delta_\infty^h u=\lambda f(x,u,Du) \quad &\mathrm{in}\; \Omega, \\ u>0\quad &\mathrm{in}\; \Omega,\\ u=0 \quad &\mathrm{on} \;\partial\Omega, \end{cases} \end{align*} $$

where $\lambda>0$ is a parameter, $h>1$ and $\Delta _\infty ^h u=|Du|^{h-3} \langle D^2uDu,Du \rangle $ is the highly degenerate and h-homogeneous operator related to the infinity Laplacian. The nonlinear term $f(x,t,p):\Omega \times (0,\infty )\times \mathbb {R}^{n}\rightarrow \mathbb {R}$ is a continuous function and may exhibit singularity at $t\rightarrow 0^{+}$. We establish the comparison principle by the double variables method for the general equation $\Delta _\infty ^h u=F(x,u,Du)$ under some conditions on the term $F(x,t,p)$. Then, we establish the existence of viscosity solutions to the singular boundary value problem in a bounded domain based on Perron’s method and the comparison principle. Finally, we obtain the existence result in the entire Euclidean space by the approximation procedure. In this procedure, we also establish the local Lipschitz continuity of the viscosity solution.

Based on the path encoding pulse compression teleology, a novel method for obtaining high-power microwave (HPM) pulse with ultrahigh repetition frequency is proposed in this paper. The mechanism of the path encoding pulse compression teleology is first introduced. And then, the obtained HPM pulse is analyzed. Theoretical analysis shows that the peak power of MW level and the repetition frequency of MHz level for the generated HPM pulse can be easily reached. To demonstrate the effectiveness of this method for obtaining HPM pulse with ultrahigh repetition frequency characteristic, a HPM-obtaining experiment was carried out based on an S-band microwave source. The HPM pulses with the width of 1 ns, 2 ns, and 3 ns are studied, respectively. The measured results show that the HPM pulse with the power higher than 100 kW and the repetition frequency of 250 kHz at the frequency of 2.856 GHz is easily obtained. The repetition frequency of the generated HPM pulse can be easily changed. Because the pulse with the power higher than 100 kW and the repetition frequency of several hundreds of kHz is obtained for the first time, this type of pulse will have a broad prospect of application in the communication, radar, and electronic countermeasure fields. In addition, the effect experiment of interfering communication and control links was carried out by utilizing the ultrahigh repetition frequency characteristic of the generated HPM pulse. Also, the experiment results show the feasibility of this pulse for interfering the communication and control links.

Abnormalities in the zona pellucida (ZP) adversely affect oocyte maturation, embryo development and pregnancy outcomes. However, the assessment of severity is challenging. To evaluate the effects of different degrees of ZP abnormalities on embryo development and clinical outcomes, in total, 590 retrieval cycles were scored and divided into four categories (control, mild, moderate and severe) based on three parameters: perivitelline space, percentage of immature oocytes and percentage of oocytes with abnormal morphology. As the severity of abnormal ZP increased, both the number of retrieved oocytes and mature oocytes decreased. The fertilization rate did not differ significantly among groups. The rates of embryo cleavage and day-3 high-quality embryos in the mild group and the moderate group did not vary significantly between the two groups but were significantly higher than those in the severe group. The blastulation rates of the abnormal ZP groups were similar; however, they were lower than those of the control group. Moreover, the cycle cancellation rate of the severe abnormal ZP group was as high as 66.20%, which was significantly higher than that of the other three groups. Although the rates of cumulative clinical pregnancy and live births were lower than those in the control group, they were comparable among the abnormal ZP groups. There were no differences in the neonatal outcomes of the different groups. Together, ZP abnormalities show various degrees of severity, and in all patients regardless of the degree of ZP abnormalities who achieve available embryos, there will be an opportunity to eventually give birth.

The target backsheath field acceleration mechanism is one of the main mechanisms of laser-driven proton acceleration (LDPA) and strongly depends on the comprehensive performance of the ultrashort ultra-intense lasers used as the driving sources. The successful use of the SG-II Peta-watt (SG-II PW) laser facility for LDPA and its applications in radiographic diagnoses have been manifested by the good performance of the SG-II PW facility. Recently, the SG-II PW laser facility has undergone extensive maintenance and a comprehensive technical upgrade in terms of the seed source, laser contrast and terminal focus. LDPA experiments were performed using the maintained SG-II PW laser beam, and the highest cutoff energy of the proton beam was obviously increased. Accordingly, a double-film target structure was used, and the maximum cutoff energy of the proton beam was up to 70 MeV. These results demonstrate that the comprehensive performance of the SG-II PW laser facility was improved significantly.

In this paper, we study the asymptotic profiles of positive solutions for diffusive logistic equations. The aim is to study the sharp effect of linear growth and nonlinear function. Both the classical reaction-diffusion equation and nonlocal dispersal equation are investigated. Our main results reveal that the linear and nonlinear parts of reaction term play quite different roles in the study of positive solutions.

Adolescence is a significant period for the formation of relationship networks and the development of internalizing problems. With a sample of Chinese adolescents (N = 3,834, 52.01% girls, Mage = 16.68 at Wave 1), the present study aimed to identify the configuration of adolescents’ relationship qualities from four important domains (i.e., relationship quality with mother, father, peers, and teachers) and how distinct profiles were associated with the development of internalizing problems (indicated by depressive and anxiety symptoms) across high school years. Latent profile analysis identified a five-profile configuration with four convergent profiles (i.e., relationship qualities with others were generally good or bad) and one “Father estrangement” profile (i.e., the relationship quality with others were relatively good but that with father was particularly poor). Further conditional latent growth curve analysis indicated the “Father estrangement” profile was especially vulnerable to an increase in the internalizing problems as compared with other relationship profiles. This study contributes to understanding the characteristics of interpersonal relationship qualities and their influences on adolescent internalizing problems in a non-Western context. Results were further discussed from a culturally specific perspective.

The selection of high-quality sperms is critical to intracytoplasmic sperm injection, which accounts for 70–80% of in vitro fertilization (IVF) treatments. So far, sperm screening is usually performed manually by clinicians. However, the performance of manual screening is limited in its objectivity, consistency, and efficiency. To overcome these limitations, we have developed a fast and noninvasive three-stage method to characterize morphology of freely swimming human sperms in bright-field microscopy images using deep learning models. Specifically, we use an object detection model to identify sperm heads, a classification model to select in-focus images, and a segmentation model to extract geometry of sperm heads and vacuoles. The models achieve an F1-score of 0.951 in sperm head detection, a z-position estimation error within ±1.5 μm in in-focus image selection, and a Dice score of 0.948 in sperm head segmentation, respectively. Customized lightweight architectures are used for the models to achieve real-time analysis of 200 frames per second. Comprehensive morphological parameters are calculated from sperm head geometry extracted by image segmentation. Overall, our method provides a reliable and efficient tool to assist clinicians in selecting high-quality sperms for successful IVF. It also demonstrates the effectiveness of deep learning in real-time analysis of live bright-field microscopy images.

It is generally accepted that high-oleic crops have at least 70% oleate. As compared to their normal-oleic counterparts, oil and food products made from high-oleic peanut have better keeping quality and are much healthier. Therefore, high-oleic peanut is well recognized by processors and consumers. However, owing to the limited availability of high-oleic donors, most present-day high-oleic peanut varietal releases merely have F435 type FAD2 mutations. Through screening of a mutagenized peanut population of 15L46, a high-yielding peanut line with desirable elliptical oblong large seeds, using near infrared model for predicting oleate content in individual single seeds, high-oleic peanut mutants were identified. Sequencing FAD2A and FAD2B of the mutants along with the wild type revealed that these mutants possessed G448A FAD2A (F435 type FAD2A mutation) and G558A FAD2B (non-F435 type FAD2B mutation). Expression of the wild and mutated type FAD2B in yeast verified that the functional mutation contributed to the high-oleic phenotype in these mutants. The mutants provided additional high-oleic donors to peanut quality improvement.

Viruses completely rely on the energy and metabolic systems of host cells for life activities. Viral infections usually lead to cytopathic effects and host diseases. To date, there are still no specific clinical vaccines or drugs against most viral infections. Therefore, understanding the molecular and cellular mechanisms of viral infections is of great significance to prevent and treat viral diseases. A variety of viral infections are related to the p38 MAPK signalling pathway, and p38 is an important host factor in virus-infected cells. Here, we introduce the different signalling pathways of p38 activation and then summarise how different viruses induce p38 phosphorylation. Finally, we provide a general summary of the effect of p38 activation on virus replication. Our review provides integrated data on p38 activation and viral infections and describes the potential application of targeting p38 as an antiviral strategy.

This manuscript details the strategy employed for categorising food items based on their processing levels into the four NOVA groups. Semi-quantitative food frequency questionnaires (FFQs) from the Nurses’ Health Studies (NHS) I and II, the Health Professionals Follow-up Study (HPFS) and the Growing Up Today Studies (GUTS) I and II cohorts were used. The four-stage approach included: (i) the creation of a complete food list from the FFQs; (ii) assignment of food items to a NOVA group by three researchers; (iii) checking for consensus in categorisation and shortlisting discordant food items; (iv) discussions with experts and use of additional resources (research dieticians, cohort-specific documents, online grocery store scans) to guide the final categorisation of the short-listed items. At stage 1, 205 and 315 food items were compiled from the NHS and HPFS, and the GUTS FFQs, respectively. Over 70 % of food items from all cohorts were assigned to a NOVA group after stage 2. The remainder were shortlisted for further discussion (stage 3). After two rounds of reviews at stage 4, 95⋅6 % of food items (NHS + HPFS) and 90⋅7 % items (GUTS) were categorised. The remaining products were assigned to a non-ultra-processed food group (primary categorisation) and flagged for sensitivity analyses at which point they would be categorised as ultra-processed. Of all items in the food lists, 36⋅1 % in the NHS and HPFS cohorts and 43⋅5 % in the GUTS cohorts were identified as ultra-processed. Future work is needed to validate this approach. Documentation and discussions of alternative approaches for categorisation are encouraged.

Chronic inflammation exerts pleiotropic effects in the aetiology and progression of chronic obstructive pulmonary disease (COPD). Glucosamine is widely used in many countries and may have anti-inflammatory properties. We aimed to prospectively evaluate the association of regular glucosamine use with incident COPD risk and explore whether such association could be modified by smoking in the UK Biobank cohort, which recruited more than half a million participants aged 40–69 years from across the UK between 2006 and 2010. Cox proportional hazards models with adjustment for potential confounding factors were used to calculate hazard ratios (HR) as well as 95 % CI for the risk of incident COPD. During a median follow-up of 8·96 years (interquartile range 8·29–9·53 years), 9016 new-onset events of COPD were documented. We found that the regular use of glucosamine was associated with a significantly lower risk of incident COPD with multivariable adjusted HR of 0·80 (95 % CI, 0·75, 0·85; P < 0·001). When subgroup analyses were performed by smoking status, the adjusted HR for the association of regular glucosamine use with incident COPD were 0·84 (0·73, 0·96), 0·84 (0·77, 0·92) and 0·71 (0·62, 0·80) among never smokers, former smokers and current smokers, respectively. No significant interaction was observed between glucosamine use and smoking status (P for interaction = 0·078). Incident COPD could be reduced by 14 % to 84 % through a combination of regular glucosamine use and smoking cessation.