On vector variational-like inequalities and vector optimization problems with (G, α)-invexity

The aim of this paper is to study the relationship among Minty vector variational-like inequality problem, Stampacchia vector variational-like inequality problem and vector optimization problem involving (G, α)-invex functions. Furthermore, we establish equivalence among the solutions of weak formulations of Minty vector variational-like inequality problem, Stampacchia vector variational-like inequality problem and weak efficient solution of vector optimization problem under the assumption of (G, α)-invex functions. Examples are provided to elucidate our results.

Auxiliary principle and three-step iterative algorithms for generalized set-valued strongly nonlinear mixed variational-like inequalities

An auxiliary principle technique to study a class of generalized set-valued strongly nonlinear mixed variational-like inequalities is extended. The existence and uniqueness of the solution of the auxiliary problem for the generalized set-valued strongly nonlinear mixed variational-like inequalities are proved, a novel and innovative three-step iterative algorithm to compute approximate solution is constructed, and the existence of the solution of the generalized set-valued strongly nonlinear mixed variational-like inequality is shown using the auxiliary principle iterative sequences generated by the algorithm technique. The convergence of three-step is also proved.

ALGORITHM OF SOLUTIONS FOR MIXEDNONLINEAR VARIATIONAL-LIKE INEQUALITIES IN REFLEXIVE BANACH SPACE

In this paper, the author studies a class of mixed nonlinear variational-like inequalities in reflexive Banach space. By applying a minimax inequality obtained by the author, some existence uniqueness theorems of solutions for the mixed nonlinear variational-like inequalities are proved. Next, by applying the auxiliary problem technique, rite author suggests an innovative iterative algorithm to compute the approximate solutions of the mixed nonlinear variational-like inequalities. Finally, the convergence criteria is also discussed.

EXISTENCE OF SOLUTIONS FOR GENERALIZEDQUASI-VARIATIONAL-LIKE INEQUALITIES

In this paper, some existence theorems of solutions for a class of generalized quasi-variational-like inequalities with discontinuous mappings ape proved under paracompact setting in topological vector spaces. These theorems unify, improve and generalize many recent results.

Existence of Solutions to Generalized Vector Quasi-variational-like Inequalities with Set-valued Mappings

In this paper, we introduce and study a class of generalized vector quasivariational-like inequality problems, which includes generalized nonlinear vector variational inequality problems, generalized vector variational inequality problems and generalized vector variational-like inequality problems as special cases. We use the maximal element theorem with an escaping sequence to prove the existence results of a solution for generalized vector quasi-variational-like inequalities without any monotonicity conditions in the setting of locally convex topological vector space.

Existence and Algorithm of Solutions for Generalized Mixed Implicit Quasi-Variational-Like Inequalities

By applying the auxiliary variational principle technique, the existence of solutions for a new class of generalized mixed implicit quasi-variational-like inequalities and the convergence criteria of a new iterative algorithm to compute approximate solutions are proved in Hilbert spaces. The obtained result is a improvement over and generalization of the main theorem proposed by Ding.

An Existence Theorem of Solutions for the System of Generalized Vector Quasi-Variational-Like Inequalities

In this paper, we introduce and study the system of generalized vector quasi-variational-like inequalities in Hausdorff topological vector spaces, which include the system of vector quasi-variational-like inequalities, the system of vector variational-like inequalities, the system of vector quasi-variational inequalities, and several other systems as special cases. Moreover, a number of C-diagonal quasiconvexity properties are proposed for set-valued maps, which are natural generalizations of the g-diagonal quasiconvexity for real functions. Together with an application of continuous selection and fixed-point theorems, these conditions enable us to prove unified existence results of solutions for the system of generalized vector quasi-variational-like inequalities. The results of this paper can be seen as extensions and generalizations of several known results in the literature.

Perturbed Proximal-Projection Methods for Nonlinear Mixed Variational-Like Inequalities

In this paper we introduce a new perturbed proximal-projection algorithm for finding the common element of the set of fixed points of non-expansive mappings and the set of solutions of nonlinear mixed variational-like inequalities. The convergence criteria of the iterative sequences generated by the new iterative algorithm is also given. Our approach and results generalize many known results in this field.

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.

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.

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.

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.

Generalized Nonlinear Implicit Variational-Like Inequalities

A new class of generalized nonlinear implicit variational-like inequality problems （for short, GNIVLIP） in the setting of locally convex topological vector spaces is introduced and studied in this paper. Under suitable conditions, some existence theorems of solutions for （GNIVLIP） are presented by using some fixed point theorems.

A RELAXED INERTIAL FACTOR OF THE MODIFIED SUBGRADIENT EXTRAGRADIENT METHOD FOR SOLVING PSEUDO MONOTONE VARIATIONAL INEQUALITIES IN HILBERT SPACES

In this paper,we investigate pseudomonotone and Lipschitz continuous variational inequalities in real Hilbert spaces.For solving this problem,we propose a new method that combines the advantages of the subgradient extragradient method and the projection contraction method.Some very recent papers have considered different inertial algorithms which allowed the inertial factor is chosen in[0;1].The purpose of this work is to continue working in this direction,we propose another inertial subgradient extragradient method that the inertial factor can be chosen in a special case to be 1.Under suitable mild conditions,we establish the weak convergence of the proposed algorithm.Moreover,linear convergence is obtained under strong pseudomonotonicity and Lipschitz continuity assumptions.Finally,some numerical illustrations are given to confirm the theoretical analysis.

NEW DOOB'S MAXIMAL INEQUALITIES FOR MARTINGALES

Let 1≤q≤∞,b be a slowly varying function and letΦ:[0,∞)■[0,∞)be an increasing convex function withΦ(0)=0 and■Φ(r)=∞.In this paper,we present a new class of Doob’s maximal inequality on Orlicz-Lorentz-Karamata spaces LΦ,q,b.The results are new,even for the Lorentz-Karamata spaces withΦ(t)=tp,the Orlicz-Lorentz spaces with b≡1,and weak Orlicz-Karamata spaces with q=∞in the framework of LΦ,q,b-Moreover,we obtain some even stronger qualitative results that can remove the△2-condition of Liu,Hou and Wang(Sci China Math,2010,53(4):905-916).

Generalization of Inequalities in Metric Spaces with Applications

In this paper, which serves as a continuation of earlier work, we generalize the idea of inequalities in metric spaces and use them to demonstrate that the incomplete metric space can be used to obtain a Banach space.

What in Fact Proves the Violation of the Bell-Type Inequalities?

A. Peres constructed an example of particles entangled in the state of spin singlet. He claimed to have obtained the CHSH inequality and concluded that the violation of this inequality shows that in a measurement in which some variables are tested, other variables, not tested, have no defined value. In the present paper is proved that the correct conclusion of the violation of the CHSH inequality is different. It is proved that the classical calculus of probabilities of test results, obeying the Kolmogorov axioms, is unfit for the quantum formalism, dominated by probability amplitudes.

HARNACK TYPE INEQUALITIES FOR SDES DRIVEN BY FRACTIONAL BROWNIAN MOTION WITH MARKOVIAN SWITCHING

In this paper, by constructing a coupling equation, we establish the Harnack type inequalities for stochastic differential equations driven by fractional Brownian motion with Markovian switching. The Hurst parameter H is supposed to be in(1/2, 1). As a direct application, the strong Feller property is presented.

A Note on Sharp Affine Poincaré-Sobolev Inequalities and Exact in Minimization of Zhang’s Energy on Bounded Variation and Exactness

As for the affine energy, Edir Junior and Ferreira Leite establish the existence of minimizers for particular restricted subcritical and critical variational issues on BV(Ω). Similar functionals exhibit deeper weak* topological traits including lower semicontinuity and affine compactness, and their geometry is non-coercive. Our work also proves the result that extremal functions exist for certain affine Poincaré-Sobolev inequalities.