In this paper, we establish an infinite series expansion of Leray–Trudinger inequality, which is closely related with Hardy inequality and Moser Trudinger inequality. Our result extends early results obtained by Mallick and Tintarev [A. Mallick and C. Tintarev. An improved Leray-Trudinger inequality. Commun. Contemp. Math. 20 (2018), 17501034. OP 21] to the case with many logs. It should be pointed out that our result is about series expansion of Hardy inequality under the case $p=n$, which case is not considered by Gkikas and Psaradakis in [K. T. Gkikas and G. Psaradakis. Optimal non-homogeneous improvements for the series expansion of Hardy's inequality. Commun. Contemp. Math. doi:10.1142/S0219199721500310]. However, we can't obtain the optimal form by our method.

In this paper, we develop an extremely simple method to establish the sharpened Adams-type inequalities on higher-order Sobolev spaces $W^{m,\frac {n}{m}}(\mathbb {R}^n)$ in the entire space $\mathbb {R}^n$, which can be stated as follows: Given $\Phi \left ( t\right ) =e^{t}-\underset {j=0}{\overset {n-2}{\sum }} \frac {t^{j}}{j!}$ and the Adams sharp constant $\beta _{n,m}$. Then,

$$ \begin{align*}\sup_{\|\nabla^mu\|_{\frac{n}{m}}^{\frac{n}{m}}+\|u\|_{\frac{n}{m}}^{\frac{n}{m}}\leq1}\int_{\mathbb{R}^n}\Phi\Big(\beta_{n,m} (1+\alpha \|u\|_{\frac{n}{m}}^{\frac{n}{m}} )^{\frac{m}{n-m}}|u|^{\frac{n}{n-m}}\Big)dx<\infty, \end{align*} $$

$0<\alpha <1$

$\alpha $

$\alpha \ge 1$

Our argument avoids applying the complicated blow-up analysis often used in the literature to deal with such sharpened inequalities.

Although research has shown that older nursing home residents can benefit from caring relationships with nurse aides, few studies have explored their dyadic, evolving relationship dynamics. Using a dyadic perspective, this study simultaneously explores caring relationships among older residents and nurse aides in Shanghai. In a government-sponsored nursing home in Shanghai, 20 matched resident–nurse aide dyads participated in semi-structured, in-depth interviews (N = 40). We performed thematic analysis to interpret and conceptualise the evolving caring relationships within dyads. Four types emerged during the evolution of caring relationships across the 20 dyads: (a) sharing strong rapport, (b) respecting each other, (c) hesitant responding, and (d) keeping emotional distance. Upon placement, all the residents kept emotional distance from nurse aides, and their assigned nurse aides provided care-giving by following nursing home regulations. As time passed, nurse aides began to create a family environment and tried to interact with residents on an emotional level; however, residents’ attitudes varied. The caring relationships in some dyads evolved as rapport and respect emerged, while others remained hesitant and distant. This suggests that residents and nurse aides prioritised caring relationships differently in terms of autonomy preservation and safety protection, respectively. This study sheds light on nursing home practice to facilitate building caring relationships between residents and nurse aides.

The early identification and prediction of hand-foot-and-mouth disease (HFMD) play an important role in the disease prevention and control. However, suitable models are different in regions due to the differences in geography, social economy factors. We collected data associated with daily reported HFMD cases and weather factors of Zibo city in 2010~2019 and used the generalised additive model (GAM) to evaluate the effects of weather factors on HFMD cases. Then, GAM, support vectors regression (SVR) and random forest regression (RFR) models are used to compare predictive results. The annual average incidence was 129.72/100 000 from 2010 to 2019. Its distribution showed a unimodal trend, with incidence increasing from March, peaking from May to September. Our study revealed the nonlinear relationship between temperature, rainfall and relative humidity and HFMD cases and based on the predictive result, the performances of three models constructed ranked in descending order are: SVR > GAM> RFR, and SVR has the smallest prediction errors. These findings provide quantitative evidence for the prediction of HFMD for special high-risk regions and can help public health agencies implement prevention and control measures in advance.

Hyperinsulinaemia and insulin resistance have been proposed to be associated with mortality risk, and diet can modulate insulin response. However, whether dietary patterns with high insulinaemic potential are associated with mortality remains unknown. We prospectively examined the associations between hyperinsulinaemic diets and the risk of total and cause-specific mortality in a large nationally representative population. Dietary factors were assessed by 24-h recalls. Two empirical dietary indices for hyperinsulinaemia (EDIH) and insulin resistance (EDIR) were developed to identify food groups most predictive of biomarkers for hyperinsulinaemia (C-peptide and insulin) and insulin resistance (homoeostatic model assessment for insulin resistance), respectively. Deaths from date of the first dietary interview until 31 December 2015 were identified by the National Death Index. Multivariable hazard ratios (HR) and 95 % CI were calculated using Cox regression models. During a median follow-up of 7·8 years, 4904 deaths were documented among 40 074 participants. For EDIH, the multivariable-adjusted HR (comparing extreme quintiles) were 1·20 (95 % CI 1·09, 1·32, P-trend<0·001) for overall mortality and 1·41 (95 % CI 1·15, 1·74, P-trend = 0·002) for CVD mortality. Similar associations were observed for EDIR with HR of 1·18 (95 % CI 1·07, 1·29, P-trend < 0·001) for total and 1·35 (95 % CI 1·09, 1·67, P-trend = 0·005) for CVD mortality. After further adjustments for BMI and diabetes, these positive associations were somewhat attenuated. Our findings suggested that diets with higher insulinaemic potential are associated with increased risk of overall and CVD-specific mortality.

Technology that develops rapidly has profoundly affected the business field and reshaped some behaviours of corporations, and the discussion on startup risk-taking behaviour in the new era is still insufficient. Based on social network theory and social capital theory, this article studies how social networks and entrepreneurial ecosystems support startup risk-taking behaviour. This article cuts into this issue through the perspective of coopetition. Based on 737 responses, this article employs regression and fuzzy-set qualitative comparative analysis to explore the relationships between networks, ecosystem coopetition, and risk-taking behaviour. Results indicate that networks and coopetition may stimulate startup risk-taking behaviour, and coopetition may weaken the impacts of networks. There are replacement effects between different characteristics of networks, and there are several configurations, which may lead to high-level risk-taking. This article may help us understand startup risk-taking behaviour in the digital era and the positive impacts of ecosystems.

Direct numerical simulation of spanwise-rotation-driven flow transitions in viscoelastic plane Couette flow from a drag-reduced inertial to a drag-enhanced elasto-inertial turbulent flow state followed by full relaminarization is reported for the first time. Specifically, this novel flow transition begins with a drag-reduced inertial turbulent flow state at a low rotation number $0\leqslant Ro \leqslant 0.1$, and then transitions to a rotation/polymer-additive-driven drag-enhanced inertial turbulent regime, $0.1\leqslant Ro \leqslant 0.3$. In turn, the flow transitions to a drag-enhanced elasto-inertial turbulent state, $0.3\leqslant Ro \leqslant 0.9$, and eventually relaminarizes at $Ro=1$. In addition, two novel rotation-dependent drag enhancement mechanisms are proposed and substantiated. (1) The formation of large-scale roll cells results in enhanced convective momentum transport along with significant polymer elongation and stress generated in the extensionally dominated flow between adjacent roll cells at $Ro\leqslant 0.2$. (2) Coriolis-force-generated turbulent vortices cause strong incoherent transport and homogenization of significant polymer stress in the bulk via their vortical circulations at $Ro=0.5 - 0.9$.

This paper explores the short-run impact of work resumption, extensively launched on February 10, 2020 in China, on air quality after the subsiding of COVID-19. Utilizing the data of 1012 air-quality monitoring sites in 233 cities derived from the Real-time Release Air Quality Platform and the difference-in-differences method, we find that alternative measures of air quality index in non-Hubei provinces increase significantly, compared with those in Hubei province which was temporarily not allowed work resumption due to the severity of epidemic. Specifically, our results reveal a rise in AQI of 11.28 per cent, in PM2.5 of 12.47 per cent, in PM10 of 10.49 per cent, and in NO2 of 23.64 per cent, relative to the baseline mean. Moreover, the deterioration of air quality is found to be caused by intracity rather than intercity migration.

Convolutional sequence to sequence (CNN seq2seq) models have met success in abstractive summarization. However, their outputs often contain repetitive word sequences and logical inconsistencies, limiting the practicality of their application. In this paper, we find the reasons behind the repetition problem in CNN-based abstractive summarization through observing the attention map between the summaries with repetition and their corresponding source documents and mitigate the repetition problem. We propose to reduce the repetition in summaries by attention filter mechanism (ATTF) and sentence-level backtracking decoder (SBD), which dynamically redistributes attention over the input sequence as the output sentences are generated. The ATTF can record previously attended locations in the source document directly and prevent the decoder from attending to these locations. The SBD prevents the decoder from generating similar sentences more than once via backtracking at test. The proposed model outperforms the baselines in terms of ROUGE score, repeatedness, and readability. The results show that this approach generates high-quality summaries with minimal repetition and makes the reading experience better.

Vertical chutes and pipes are a common component of many industrial apparatus used in the transport and processing of powders and grains. Here, a typical arrangement is considered first in which a hopper at the top feeds the chute and a converging outlet at the bottom controls the mass flux. Discrete element method (DEM) simulations reveal that steady uniform flow is only observed for intermediate flow rates, with jamming and unsteady waves dominating slow flows and non-uniform wall detachment in fast flow. Focusing on the steady uniform regimes, a progressive idealisation is carried out by matching with equivalent DEM simulations in periodic cells. These investigations justify a one-dimensional continuum modelling of the problem and provide key test data. Novel exact solutions are derived here for vertical flow using a linear version of the ‘$\mu(I),\varPhi(I)$-rheology’, for which the bulk friction $\mu$ and steady solid volume fraction $\varPhi$ depend on the inertial number I. Despite not capturing the full nonlinear complexities, the solutions match important aspects of the DEM flow fields and reveal simple scaling laws linking many quantities of interest. In particular, this study clearly demonstrates a linear relation between the chute width and the size of the shear zones at the walls. This finding contrasts with previous works on purely quasi-static flow, which instead predict a roughly constant shear zone width, a difference which implies that finite-size effects are minimal for the inertial flows studied here.

We report on a three-dimensional direct numerical simulation study of flow structure and heat transport in the annular centrifugal Rayleigh–Bénard convection (ACRBC) system, with cold inner and hot outer cylinders corotating axially, for the Rayleigh number range $Ra \in [{10^6},{10^8}]$ and radius ratio range $\eta = {R_i}/{R_o} \in [0.3,0.9]$ ($R_i$ and $R_o$ are the radius of the inner and outer cylinders, respectively). This study focuses on the dependence of flow dynamics, heat transport and asymmetric mean temperature fields on the radius ratio $\eta$. For the inverse Rossby number $Ro^{-1} = 1$, as the Coriolis force balances inertial force, the flow is in the inertial regime. The mechanisms of zonal flow revolving in the prograde direction in this regime are attributed to the asymmetric movements of plumes and the different curvatures of the cylinders. The number of roll pairs is smaller than the circular roll hypothesis as the convection rolls are probably elongated by zonal flow. The physical mechanism of zonal flow is verified by the dependence of the drift frequency of the large-scale circulation (LSC) rolls and the space- and time-averaged azimuthal velocity on $\eta$. The larger $\eta$ is, the weaker the zonal flow becomes. We show that the heat transport efficiency increases with $\eta$. It is also found that the bulk temperature deviates from the arithmetic mean temperature and the deviation increases as $\eta$ decreases. This effect can be explained by a simple model that accounts for the curvature effects and the radially dependent centrifugal force in ACRBC.

Whereas the existing literature on the relationship between parental behavior and family business succession mainly focuses on parental behavior in the business domain, we highlight the importance of parental behavior in the family domain. Integrating attachment theory, the family business succession literature, and person-job fit literature, our study proposes a theoretical framework hypothesizing that general self-efficacy and perceived person-job fit mediate the association between perceived parental care (an underrepresented family-domain-specific parental behavior) and next-generation family members’ succession intentions. This framework is tested by data from two surveys and further verified by qualitative interviews of next-generation family members. Multivariate analysis results suggest that next-generation family members’ general self-efficacy and perceived person-job fit played a sequential-mediating role in the relationship between perceived parental care and next-generation family members’ succession intentions. Our interviews not only confirm these results but also reveal new insights, particularly into the specific Chinese context in the study of family business succession.

The article aims to estimate and forecast the transmissibility of shigellosis and explore the association of meteorological factors with shigellosis. The mathematical model named Susceptible–Exposed–Symptomatic/Asymptomatic–Recovered–Water/Food (SEIARW) was used to explore the feature of shigellosis transmission based on the data of Wuhan City, China, from 2005 to 2017. The study applied effective reproduction number (Reff) to estimate the transmissibility. Daily meteorological data from 2008 to 2017 were used to determine Spearman's correlation with reported new cases and Reff. The SEIARW model fit the data well (χ2 = 0.00046, p > 0.999). The simulation results showed that the reservoir-to-person transmission of the shigellosis route has been interrupted. The Reff would be reduced to a transmission threshold of 1.00 (95% confidence interval (CI) 0.82–1.19) in 2035. Reducing the infectious period to 11.25 days would also decrease the value of Reff to 0.99. There was a significant correlation between new cases of shigellosis and atmospheric pressure, temperature, wind speed and sun hours per day. The correlation coefficients, although statistically significant, were very low (<0.3). In Wuhan, China, the main transmission pattern of shigellosis is person-to-person. Meteorological factors, especially daily atmospheric pressure and temperature, may influence the epidemic of shigellosis.