In this paper, we mainly prove the following conjectures of Z.-W. Sun (J. Number Theory 133 (2013), 2914–2928): let $p>2$ be a prime. If $p=x^2+3y^2$ with $x,y\in \mathbb {Z}$ and $x\equiv 1\ ({\rm {mod}}\ 3)$, then

\[ x\equiv\frac14\sum_{k=0}^{p-1}(3k+4)\frac{f_k}{2^k}\equiv\frac12\sum_{k=0}^{p-1}(3k+2)\frac{f_k}{({-}4)^k}\ ({\rm{mod}}\ p^2), \]

$p\equiv 1\pmod 3$

\[ \sum_{k=0}^{p-1}\frac{f_k}{2^k}\equiv\sum_{k=0}^{p-1}\frac{f_k}{({-}4)^k}\ ({\rm{mod}}\ p^3), \]

$f_n=\sum _{k=0}^n\binom {n}k^3$

$n$

The integrality of the numbers $A_{n,m}={(2n)!(2m)!}/{n!m!(n+m)!}$ was observed by Catalan as early as 1874 and Gessel named $A_{n,m}$ the super Catalan numbers. The positivity of the q-super Catalan numbers (q-analogue of the super Catalan numbers) was investigated by Warnaar and Zudilin [‘A q-rious positivity’, Aequationes Math. 81 (2011), 177–183]. We prove the divisibility of sums of q-super Catalan numbers, which establishes a q-analogue of Apagodu’s congruence involving super Catalan numbers.

This study aims to understand the epidemiological characteristics of SARS-CoV-2 infection in the paediatric population during the outbreak of the Omicron variant in Shanghai. We retrospectively analysed the population-based epidemiological characteristics and clinical outcome of SARS-CoV-2 Omicron variant infection in children in Minhang District, Shanghai, based on the citywide surveillance system during the outbreak period in 2022 (March to May). During this time, a total of 63,969 cases of SARS-CoV-2 infection were notified in Minhang District, out of which 4,652 (7.3%) were children and adolescents <18 years. The incidence rate of SARS-CoV-2 infections in children was 153 per 10,000. Of all paediatric cases, 50% reported to be clinically symptomatic within 1–3 days after PCR confirmation by parents or themselves, with 36.3% and 18.9% of paediatric cases reporting fever and cough. Also, 58.4% of paediatric cases had received at least one dose of the COVID-19 vaccine and 52.1% had received two doses of the COVID-19 vaccination. Our findings are informative for the implementation of appropriate measures to protect children from the threat of SARS-CoV-2 infection.

Menaquinone-7 (MK-7), a multipotent vitamin K2, possesses a wide range of biological activities, a precise curative effect and excellent safety. A simple and rapid LC-APCI-MS/MS method for the determination of MK-7 in human plasma with single liquid–liquid extraction (LLE) extraction and 4·5-min analysis time has been developed and validated. Four per cent bovine serum albumin (BSA) was used as surrogate matrix for standard curves and endogenous baseline subtraction. This method was reproducible and reliable and was used to analyse of MK-7 in human plasma. The endogenous circadian rhythm and bioavailability of MK-7 were investigated in two randomised single-dose, open, one-way clinical trials (Study I and Study II). A total of five healthy male subjects were enrolled in Study I and 12 healthy male subjects in Study II. Single-dose (1 mg) of MK-7 was given to each subject under fasting condition, and all eligible subjects were given a restricting VK2 diet for 4 d prior to drug administration and during the trial. The experiment results of Study I demonstrated that endogenous MK-7 has no circadian rhythm in individuals. Both studies showed MK-7 are absorbed with peak plasma concentrations at about 6 h after intake and has a very long half-life time.

In the Late Neoarchaean, the lithosphere of the North China Craton (NCC) experienced a strong extensional event, which is of great significance for understanding the evolution of the continental crust in the Precambrian. In this study, a suite of mafic dykes from Shandong province in the northeastern NCC were investigated to determine the nature, timing and source of rift-related magma activities using zircon U–Pb data, whole-rock geochemistry and Nd–Hf isotopes. Zircon U–Pb dating of four dolerites by laser ablation – inductively coupled plasma – mass spectrometry (LA-ICP-MS) yielded weighted mean 207Pb/206Pb ages in the range 2509 ± 6.1 to 2537 ± 6.2 Ma (2σ, 95 % confidence interval). The mafic dykes are classified as alkaline rocks based on their K2O + Na2O contents (6.78–7.21 wt %) and belong to the shoshonitic series according to their K2O contents (3.23–3.36 wt %). The dolerites show low concentrations of light rare earth elements ((La/Yb)N between 7.17 and 8.55), positive Eu anomalies (Eu/Eu* between 1.12 and 1.27), positive Ba, K, Pb, Sr, Eu, Dy and Lu anomalies, and depleted U, Nb, Pr, Ta, P, Nd and Ti anomalies. The dykes are characterized by low initial (87Sr/86Sr)i (∼0.6969), positive εNd(t) values (0.2–0.8) and εHf(t) values (0.5–8.6) and relatively old mean Nd and Hf model age (2.73 Ga). Collectively, the data suggest that the mafic dykes were derived from the partial melting (10–20 %) of an isotope-depleted garnet–lherzolite mantle source that was hybridized through interaction with subducted lower crustal material. The parental magmas of these dykes underwent a certain number of crustal contaminations during magma ascent. The mafic magmatism represented in the form of the dyke swarms is considered to be a response to widespread lithospheric extension which affected the NCC at c. 2.5 Ga during the Neoarchaean.

Collaborative planning for multiple hypersonic vehicles can effectively improve operational effectiveness. Time coordination is one of the main forms of cooperation among multi-hypersonic glide vehicles, and time cooperation trajectory optimisation is a key technology that can significantly increase the success rate of flight missions. However, it is difficult to obtain satisfactory time as a constraint condition during trajectory optimisation. To solve this problem, a multilayer Perceptrona is trained and adopted in a time-decision module, whose input is a four-dimensional vector selected according to the trajectory characteristics. Additionally, the MLP will be capable of determining the optimal initial heading angle of each aircraft to reduce unnecessary manoeuvering performance consumption in the flight mission. Subsequently, to improve the cooperative flight performance of hypersonic glide vehicles, the speed-dependent angle-of-attack and bank command were designed and optimised using the Artificial Bee Colony algorithm. The final simulation results show that the novel strategy proposed in this study can satisfy terminal space constraints and collaborative time constraints simultaneously. Meanwhile, each aircraft saves an average of 13.08% flight range, and the terminal speed is increased by 315.6m/s compared to the optimisation results of general purpose optimal control software (GPOPS) tools.

There is still controversy about optimal dietary iodine intake as the Universal Salt Iodization policy enforcement in China. A modified iodine balance study was thus conducted to explore the suitable iodine intake in Chinese adult males using the iodine overflow hypothesis. In this study, thirty-eight apparently healthy males (19·1 (sd 0·6) years) were recruited and provided with designed diets. After the 14-d iodine depletion, daily iodine intake gradually increased in the 30-d iodine supplementation, consisting of six stages and each of 5 d. All foods and excreta (urine, faeces) were collected to examine daily iodine intake, iodine excretion and the changes of iodine increment in relation to those values at stage 1. The dose–response associations of iodine intake increment with excretion increment were fitted by the mixed effects models, as well as with retention increment. Daily iodine intake and excretion were 16·3 and 54·3 μg/d at stage 1, and iodine intake increment increased from 11·2 μg/d at stage 2 to 118·0 μg/d at stage 6, while excretion increment elevated from 21·5 to 95·0 μg/d. A zero iodine balance was dynamically achieved as 48·0 μg/d of iodine intake. The estimated average requirement and recommended nutrient intake were severally 48·0 and 67·2 μg/d, which could be corresponded to a daily iodine intake of 0·74 and 1·04 μg/kg per d. The results of our study indicate that roughly half of current iodine intakes recommendation could be enough in Chinese adult males, which would be beneficial for the revision of dietary reference intakes.

Previous studies have found negative ageing narratives in the media during the COVID-19 pandemic. However, few have focused on compassionate ageism and how the news responded to the progression of the COVID-19 pandemic. We investigated (a) media themes of negative and compassionate ageism and (b) their relationships with COVID-19 parameters and the public health response. The sample included 1,197 articles relevant to COVID-19 and older people in Hong Kong published between January and December 2020. We used thematic analysis to identify themes from the news articles and structural equation modelling to explore these themes' relationship with the number of older people infected, effective reproduction number, number of COVID-19 deaths and public health response parallel in time. Pandemic-related variables were lagged for a day – the time needed to be reflected in the news. Two negative ageism themes portrayed older people as vulnerable to COVID-19 but counterproductive in combating the pandemic. Two compassionate ageism themes depicted older people as a homogenous group of passive assistance recipients. The theme blaming older people was associated with the number of confirmed infections (β = 0.418, p = 0.002) but vulnerability of older people was not associated with pandemic-related variables. The theme helping older people was negatively associated with the percentage of older people in confirmed infections (β = −0.155, p = 0.019). The theme resources available was negatively associated with confirmed infections (β = −0.342, p < 0.001) but positively associated with the Containment and Health Index (β = 0.217, p = 0.005). Findings suggested that negative and compassionate ageism were translated into narratives about older people in the media as the pandemic evolved but did not address the actual risk they faced. Media professionals should be aware of the potential negative and compassionate ageism prompted by the news agenda and promote adequate health behaviours and responses.

Coastal eutrophication and hypoxia remain a persistent environmental crisis despite the great efforts to reduce nutrient loading and mitigate associated environmental damages. Symptoms of this crisis have appeared to spread rapidly, reaching developing countries in Asia with emergences in Southern America and Africa. The pace of changes and the underlying drivers remain not so clear. To address the gap, we review the up-to-date status and mechanisms of eutrophication and hypoxia in global coastal oceans, upon which we examine the trajectories of changes over the 40 years or longer in six model coastal systems with varying socio-economic development statuses and different levels and histories of eutrophication. Although these coastal systems share common features of eutrophication, site-specific characteristics are also substantial, depending on the regional environmental setting and level of social-economic development along with policy implementation and management. Nevertheless, ecosystem recovery generally needs greater reduction in pressures compared to that initiated degradation and becomes less feasible to achieve past norms with a longer time anthropogenic pressures on the ecosystems. While the qualitative causality between drivers and consequences is well established, quantitative attribution of these drivers to eutrophication and hypoxia remains difficult especially when we consider the social economic drivers because the changes in coastal ecosystems are subject to multiple influences and the cause–effect relationship is often non-linear. Such relationships are further complicated by climate changes that have been accelerating over the past few decades. The knowledge gaps that limit our quantitative and mechanistic understanding of the human-coastal ocean nexus are identified, which is essential for science-based policy making. Recognizing lessons from past management practices, we advocate for a better, more efficient indexing system of coastal eutrophication and an advanced regional earth system modeling framework with optimal modules of human dimensions to facilitate the development and evaluation of effective policy and restoration actions.

We analyze a model in which an anomaly is unknown to arbitrageurs until its discovery, and test the model implications on both asset prices and arbitrageurs’ trading activities. Using data on 99 anomalies documented in the existing literature, we find that the discovery of an anomaly reduces the correlation between the returns of its decile-1 and decile-10 portfolios. This discovery effect is stronger if the aggregate wealth of hedge funds is more volatile. Finally, hedge funds increase (reverse) their positions in exploiting anomalies when their aggregate wealth increases (decreases), further suggesting that these discovery effects operate through arbitrage trading.

A new approach to target development for laboratory astrophysics experiments at high-power laser facilities is presented. With the dawn of high-power lasers, laboratory astrophysics has emerged as a field, bringing insight into physical processes in astrophysical objects, such as the formation of stars. An important factor for success in these experiments is targetry. To date, targets have mainly relied on expensive and challenging microfabrication methods. The design presented incorporates replaceable machined parts that assemble into a structure that defines the experimental geometry. This can make targets cheaper and faster to manufacture, while maintaining robustness and reproducibility. The platform is intended for experiments on plasma flows, but it is flexible and may be adapted to the constraints of other experimental setups. Examples of targets used in experimental campaigns are shown, including a design for insertion in a high magnetic field coil. Experimental results are included, demonstrating the performance of the targets.

Time preference reversal refers to systematic inconsistencies between preferences and bids for intertemporal options. From the two eye-tracking studies (N1 = 60, N2 = 110), we examined the underlying mechanisms of time preference reversal. We replicated the reversal effect in which individuals facing a pair of intertemporal options choose the smaller-sooner option but assign a higher value to the larger-later one. Results revealed that the mean fixation duration and the proportion of gaze time on the outcome attribute varied across the choice and bid tasks. In addition, time preference reversals correlated with individual differences in maximizing tendencies. Findings support the contingent weighting hypothesis and strategy compatibility hypothesis and allow for improved theoretical understanding of the potential mechanisms and processes involved in time preference reversals.

An increasing number of studies have evaluated the association between ultra-processed foods (UPF) consumption and metabolic disorders. However, the association between UPF intake and non-alcoholic fatty liver disease (NAFLD) remains unclear. In this study, we analysed data from 6545 participants who were recruited in National Health and Nutrition Examination Surveys 2011–2018. UPF were defined in light of the NOVA food classification system and divided into quartiles based on its proportion of total weight intake. Complex logistic regression models were used to assess the association between UPF and NAFLD. Mediation analyses were conducted to reveal underlying mediators. We found that NAFLD patients consumed more UPF than controls (925·92 ± 18·08 v. 812·70 ± 14·32 g/d, P < 0·001). Dietary intake of UPF (% weight) was negatively related to the Healthy Eating Index-2015 score (Spearman r = −0·32, P < 0·001). In the multivariable model, the highest quartile compared with the lowest, the OR (95 % CI) were 1·83 (1·33, 2·53) for NAFLD (OR per 10 % increment: 1·15; 95 % CI: 1·09, 1·22; P for trend < 0·001) and 1·52 (1·12, 2·07) for insulin resistance (OR per 10 % increment: 1·11; 95 % CI: 1·05, 1·18; P for trend = 0·002). Mediation analyses revealed that poor diet quality, high saturated fat and refined grain intake partly mediated the association between UPF and NAFLD. In conclusion, high UPF intake was associated with an increased risk of NAFLD in US adults. Further prospective studies are needed to verify these findings.

Meristems in land plants share conserved functions but develop highly variable structures. Meristems in seed-free plants, including ferns, usually contain one or a few pyramid-/wedge-shaped apical cells (ACs) as initials, which are lacking in seed plants. It remained unclear how ACs promote cell proliferation in fern gametophytes and whether any persistent AC exists to sustain fern gametophyte development continuously. Here, we uncovered previously undefined ACs maintained even at late developmental stages in fern gametophytes. Through quantitative live-imaging, we determined division patterns and growth dynamics that maintain the persistent AC in Sphenomeris chinensis, a representative fern. The AC and its immediate progenies form a conserved cell packet, driving cell proliferation and prothallus expansion. At the apical centre of gametophytes, the AC and its adjacent progenies display small dimensions resulting from active cell division instead of reduced cell expansion. These findings provide insight into diversified meristem development in land plants.