DISCRETE FRIEDRICHS' AND KORN'S INEQUALITIES: PART I

The use of the finite finite element method leads to replace the initial domain by an approach- ing domain.In part I,the uniform validity of the friedrichs’and Korn’s inequalities in these ap- proaching domains will be proved under some appropriate assumptions.

Mapping and measuring urban-rural inequalities in accessibility to social infrastructures

Equal access to social infrastructures is a fundamental prerequisite for sustainable development,but has long been a great challenge worldwide.Previous studies have primarily focused on the accessibility to social infras-tructures in urban areas across various scales,with less attention to rural areas,where inequality can be more severe.Particularly,few have investigated the disparities of accessibility to social infrastructures between urban and rural areas.Here,using the Changsha-Zhuzhou-Xiangtan urban agglomeration,China,as an example,we investigated the inequality of accessibility in both urban and rural areas,and further compared the urban-rural difference.Accessibility was measured by travel time of residents to infrastructures.We selected four types of social infrastructures including supermarkets,bus stops,primary schools,and health care,which were funda-mentally important to both urban and rural residents.We found large disparities in accessibility between urban and rural areas,ranging from 20 min to 2 h.Rural residents had to spend one to two more hours to bus stops than urban residents,and 20 min more to the other three types of infrastructures.Furthermore,accessibility to multiple infrastructures showed greater urban-rural differences.Rural residents in more than half of the towns had no access to any infrastructure within 15 min,while more than 60%of the urban residents could access to all infrastructures within 15 min.Our results revealed quantitative accessibility gap between urban and rural areas and underscored the necessity of social infrastructures planning to address such disparities.

Does digitalization mitigate regional inequalities?Evidence from China

Regional inequality significantly influences sustainable development and human well-being.In China,there exists pronounced regional disparities in economic and digital advancements;however,scant research delves into the interplay between them.By analyzing the economic development and digitalization gaps at regional and city levels in China,extending the original Cobb-Douglas production function,this study aims to evaluate the impact of digitalization on China's regional inequality using seemingly unrelated regression.The results indicate a greater emphasis on digital inequality compared to economic disparity,with variable coefficients of 0.59 for GDP per capita and 0.92 for the digitalization index over the past four years.However,GDP per capita demonstrates higher spatial concentration than digitalization.Notably,both disparities have shown a gradual reduction in recent years.The southeastern region of the Hu Huanyong Line exhibits superior levels and rates of economic and digital advancement in contrast to the northwestern region.While digitalization propels economic growth,it yields a nuanced impact on achieving balanced regional development,encompassing both positive and negative facets.Our study highlights that the marginal utility of advancing digitalization is more pronounced in less developed regions,but only if the government invests in the digital infrastructure and education in these areas.This study's methodology can be utilized for subsequent research,and our findings hold the potential to the government's regional investment and policy-making.

Is it possible to run after positive affective experiences instead of health benefits without tackling social inequalities?

We recently read the opinion piece titled“Why people should run after positive affective experiences instead of health benefits”,which was carried out by Maltagliati et al.^(1)and published in the Journal of Sport and Health Science.The article started an important discussion about a physical activity(PA)promotion approach that encourages the use of affective experience to PA activity.Although we agree with the authors’point of view and believe that its message can generate an important insight for PA promotion,the authors presented a scenario where behavior change seems to be exclusively dependent on the individual.Recognizing the existence of other barriers to an active lifestyle(e.g.,social inequalities)could enhance the authors’suggested approach because many people live in conditions that reduce their opportunities to engage in affective experiences related to PA;furthermore,affective experiences can vary depending on individual context.

Understanding the educational inequalities in suicide attempts and their mediators:a Mendelian randomisation study

Background Educational inequalities in suicide have become increasingly prominent over the past decade.Elucidating modifiable risk factors that serve as intermediaries in the impact of low educational attainment on suicide has the potential to reduce health disparities.Aims To examine the risk factors that mediate the relationship between educational attainment and suicide attempts and quantify their contributions to the mediation effect.Methods We conducted a two-sample Mendelian randomisation(MR)analysis to estimate the causal effect of educational attainment on suicide attempts,utilising genome-wide association study summary statistics from the Integrative Psychiatric Research(iPSYCH;6024 cases and 44240 controls)and FinnGen(8978 cases and 368299 controls).We systematically evaluated 42 putative mediators within the causal pathway connecting reduced educational attainment to suicide attempts and employed two-step and multivariable MR to quantify the proportion of the mediated effect.Results In the combined analysis of iPSYCH and FinnGen,each standard deviation(SD)decrease in genetically predicted educational attainment(equating to 3.4 years of education)was associated with a 105%higher risk of suicide attempts(odds ratio(OR):2.05;95%confidence interval(Cl):1.81 to 2.31).0f the 42 risk factors analysed,the two-step MR identified five factors that mediated the association between educational attainment and suicide attempts.The respective proportions of mediation were 47%(95%Cl:29%to 66%)for smoking behaviour,36%(95%Cl:0%to 84%)for chronic pain,49%(95%Cl:36%to 61%)for depression,35%(95%Cl:12%to 59%)for anxiety and 26%(95%Cl:18%to 34%)for insomnia.Multivariable MR implicated these five mediators collectively,accounting for 68%(95%Cl:40%to 96%)of the total effect.Conclusions This study identified smoking,chronic pain and mental disorders as primary intervention targets for attenuating suicide risk attributable to lower educational levels in the European population.

Entanglement polygon inequalities for a class of mixed states

The study on the entanglement polygon inequality of multipartite systems has attracted much attention.However,most of the results are on pure states.Here we consider the property for a class of mixed states,which are the reduced density matrices of generalizedW-class states in multipartite higher dimensional systems.First we show the class of mixed states satisfies the entanglement polygon inequalities in terms of Tsallis-q entanglement,then we propose a class of tighter inequalities for mixed states in terms of Tsallis-q entanglement.At last,we get an inequality for the mixed states,which can be regarded as a relation for bipartite entanglement.

Ostrowski’s Type Inequalities for Generalized(h,m)−Preinvex Functions with Its Applications

A new concept generalized(h,m)−preinvex function on Yang’s fractal sets is proposed.Some Ostrowski’s type inequalities with two parameters for generalized(h,m)−preinvex function are established,where three local fractional inequalities involving generalized midpoint type,trapezoid type and Simpson type are derived as consequences.Furthermore,as some applications,special means inequalities and numerical quadratures for local fractional integrals are discussed.

STRONGLY CONVERGENT INERTIAL FORWARD-BACKWARD-FORWARD ALGORITHM WITHOUT ON-LINE RULE FOR VARIATIONAL INEQUALITIES

This paper studies a strongly convergent inertial forward-backward-forward algorithm for the variational inequality problem in Hilbert spaces.In our convergence analysis,we do not assume the on-line rule of the inertial parameters and the iterates,which have been assumed by several authors whenever a strongly convergent algorithm with an inertial extrapolation step is proposed for a variational inequality problem.Consequently,our proof arguments are different from what is obtainable in the relevant literature.Finally,we give numerical tests to confirm the theoretical analysis and show that our proposed algorithm is superior to related ones in the literature.

Global prevalence and gender inequalities in at least 60 min of self-reported moderate-to-vigorous physical activity 1 or more days per week:An analysis with 707,616 adolescents

Background:Guidelines recommend that adolescents should accumulate an average of 60 min per day of moderate-to-vigorous physical activity(MVPA).However,using only this cut-off could hide important information.For instance,from a population-level point of view,increasing physical activity for those with no or low physical activity could provide more health benefits than increasing physical activity for those with intermediate levels.Also,including a more sensitive cut-point of≥1 days per week could be an additional strategy for identifying those with low access/opportunities for physical activity practice.Thus,the current study aims to estimate the prevalence of≥60 min of MVPA≥1 days per week among adolescents globally,and to describe any relevant gender inequalities.Methods:We used representative datasets from 146 countries/territories collected between 2003 and 2019.MVPA was self-reported.Participants were grouped into younger(≤14 years old)and older(>14 years old)adolescents.Crude Poisson regression models were used to identify the relative differences in≥60 min of MVPA≥1 days per week between boys and girls,and random-effects meta-analysis models were used to identify the pooled estimates.Analyses were stratified by country and region.Results:Approximately 80%of both younger and older adolescents reported≥60 min of MVPA≥1 days per week.This prevalence was≥94%in Europe and Central Asia and North America,while the estimates for the other regions were<77%.The prevalence of≥60 min of MVPA≥1 days per week was higher among boys than girls,with the largest differences occurring among the oldest adolescents(Prevalence ratio_(≤14y)=1.04(95%confidence interval(95%CI)):1.03-1.04)vs.Prevalence ratio_(>14y)=1.09(95%CI:1.08-1.10)).Conclusion:Approximately 8 out of 10 adolescents reported accumulating≥60 min of MVPA≥1 days per week,with notable differences between regions.Gender differences were observed in several countries,especially among the oldest adolescents.Priorities for physical activity promotion among adolescents should include increasing access/opportunities for physical activity among those who do not achieve≥60 min of MVPA≥1 days per week and reducing gender inequalities.

加权的Friedrichs不等式

在加权Sobolov空间 Wm ,p(Ω ;w ,v{α} ) 中 ,讨论了加权的Friedrichs不等式 ,给出了加权Friedrichs不等式成立的充分和必要条件。

高阶微分算子在直和空间上的Friedrichs扩张

应用一阶对称微分系统及相应的高阶微分方程的基本理论,讨论了正则型高阶微分算子的最小算子在直和空间上的Friedrichs扩张,给出Friedrichs扩张的边条件形式.

两个奇型微分算子积的Friedrichs扩张

将偶数阶对称的微分方程转化成相关的Hamiltonian系统,再利用主解的性质,讨论了区间[a,+∞)和(-∞,+∞)上两个奇型微分算子积的Friedrichs扩张.

On Several New Inequalities

In this paper, we find some new inequalities by Redheffer inequality, Jordan inequality and Jensen inequality. As application, we generalize one result which was given in [1].

奇型Sturm-Liouville算子的Friedrichs扩张的辛几何刻划

通过复辛空间中完全的Lagrange子流形与自伴扩张的等价描述关系,对奇型的SturmLiouville算子的Friedrichs扩张域给出了辛几何形式的新刻划,并得到Friedrichs扩张的充分必要条件.

高阶微分算子在直和空间上的Friedrichs扩张的辛几何刻画

通过最大与最小算子域构造了一个辛空间,用辛空间中的完全Lagrangian子流形与对称微分算子自共轭扩张的一一对等关系,研究对称微分算子自共轭域的辛结构,从辛几何的角度给出直和空间上正则型高阶微分算子的Friedrichs扩张域的代数结构.

奇型微分算子幂的Friedrichs扩张

将一个偶数阶对称微分方程转化为Hamiltonian系统,在区间[a,+∞)上,证明了2n阶奇型拟微分算子幂的最小算子的Friedrichs扩张存在的边条件形式,即由2n阶对称系统的2n×2n阶基解矩阵的2n×n阶主解子矩阵给出的边条件形式.

NEW SIMPLE SMOOTH MERIT FUNCTION FOR BOX CONSTRAINED VARIATIONAL INEQUALITIES AND DAMPED NEWTON TYPE METHOD

By introducing a smooth merit function for the median function, a new smooth merit function for box constrained variational inequalities (BVIs) was constructed. The function is simple and has some good differential properties. A damped Newton type method was presented based on it.Global and local superlinear/quadratic convergence results were obtained under mild conditions, and the finite termination property was also shown for the linear BVIs. Numerical results suggest that the method is efficient and promising.

A Class of Inequalities Associated with Gram Determinant and Their Applications

In this paper, a class of inequalities associated with Gram determinant are established, as their application, a class of inequalities on the volume of the pedal simplex are got and Hadamard inequality is derived.

基于局部Lax-Friedrichs通量分裂格式的透视SFS

提出了一种基于局部Lax-Friedrichs通量分裂格式的快速扫描算法,用以解决透视投影下的SFS问题。先对透视投影SFS进行建模,将其转换为静态Hamilton-Jacobi方程,再结合局部Lax-Friedrichs通量分裂格式和快速扫描算法对静态Hamilton-Jacobi方程进行求解,从而得到物体表面。本文算法可以用于非凸Hamiltonian函数的情况,提高了局部分析能力,不需要对初始高度进行估计,且算法简单,易于实现。对合成图像和实际图像的实验表明本文算法可以得到较好的透视投影SFS的恢复结果。

Comparison of two kinds of approximate proximal point algorithms for monotone variational inequalities

This paper proposes two kinds of approximate proximal point algorithms （APPA） for monotone variational inequalities, both of which can be viewed as two extended versions of Solodov and Svaiter＇s APPA in the paper ＂Error bounds for proximal point subproblems and associated inexact proximal point algorithms＂ published in 2000. They are both prediction- correction methods which use the same inexactness restriction; the only difference is that they use different search directions in the correction steps. This paper also chooses an optimal step size in the two versions of the APPA to improve the profit at each iteration. Analysis also shows that the two APPAs are globally convergent under appropriate assumptions, and we can expect algorithm 2 to get more progress in every iteration than algorithm 1. Numerical experiments indicate that algorithm 2 is more efficient than algorithm 1 with the same correction step size,