A Weak Trudinger-Moser Inequality with a Singular Weight on a Compact Riemannian Surface

Let(∑,g)be a compact Riemannian surface,pj∈∑,βj>-1,forj=1,..i,m.Denoteβ=min{0,β1……Bm}.Let H∈C^(0)(∑)be a positivefunction and h(x)=H(x)(dg(x,pj))^(2βj),where dg(x,pj)denotes the geodesic distance between x and p;for each j=1,...,m.In this paper,using a method of blow-up analysis,we prove that the functional J(u)=1/2∫∑|ΔgU|^(2)dV_(g)+8π(1+β)1/volg(∑)∫∑udvg-8π(1+β)log∫_(∑)he^(U)dv_(g)is bounded from below on the Sobolev space w^(1,2)(g).

A Weighted Singular Trudinger-Moser Inequality

In this paper,we obtained the extremal function for a weighted singular Trudinger-Moser inequality by blow-up analysis in the Euclidean space R^(2).This extends recent results of Hou(J.Inequal.Appl.,2018)and similar result was proved by Zhu(Sci.China Math.,2021).

Extremal Functions for an Improved Trudinger-Moser Inequality Involving L^(P)-Norm in R^(n)

Let W^(1,n)(R^(n))be the standard Sobolev space.For any T>0 and p>n>2,we denote■Define a norm in W^(1,n)(R^(n))by■where 0≤α<λ_(n,p).Using a rearrangement argument and blow-up analysis,we will prove■can be attained by some function u_(0)∈W^(1,n)(R^(n))∩C^(1)(R^(n))with ||u_(0)||_(n,p)=1,here a_(n)=n■_(n-1)^(1/n-1) and ■_(n-1) is the measure of the unit sphere in R^(n).

Energy Estimate Related to a Hardy-Trudinger-Moser Inequality

Let Bi be a unit disc of R^2,and H be a completion of C0^∞(B1)under the norm||u||^2H=∫B1(|■u|^2-u^2/(1-|x|^2)^2)dx.Using blow-up analysis,Wang-Ye[1]proved existence of extremals for a Hardy-Trudinger-Moser inequality.In particular,the supremum u∈,H,^sup||u|,H≤|1∫B1^e4πu2dx can be attained by some function u0∈H with||u0|H=1|This was improved by the author and Zhu[2]to a version involving the first eigenvalue of the Hardy-Laplacian operator-△-1/(1-|x|^2)^2.In this note,the results of[1,2]will be reproved by the method of energy estimate,which was recently developed by Malchiodi-Martinazzi[3]and Mancini-Martinazzi[4].

A generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularities

In this paper, using the method of blow-up analysis, we establish a generalized Trudinger-Moser inequality on a compact Riemannian surface with conical singularities. Precisely, let(Σ, D) be such a surface∑with divisor D =Σ_(i=1)~mβ_(ipi), where β_i >-1 and p_i ∈Σ for i = 1,..., m, and g be a metric representing D.Denote b_0 = 4π(1 + min_(1≤i≤mβ_i). Suppose ψ : Σ→ R is a continuous function with ∫_Σψdv_g ≠0 and define■Then for any α∈ [0, λ_1^(**)(Σ, g)), we have■When b > b0, the integrals■are still finite, but the supremum is infinity. Moreover, we prove that the extremal function for the above inequality exists.

A critical Trudinger-Moser inequality involving a degenerate potential and nonlinear Schrodinger equations

The classical critical Trudinger-Moser inequality in R^(2)under the constraint∫_(R_(2))(|▽u|^(2)+|u|^(2))dx≤1 was established through the technique of blow-up analysis or the rearrangement-free argument:for anyτ>0,it holds that sup u∈H^(1)(r^(2)fR^(2)(τ|U|^(2)+|▽u|^(2)dx≤1∫_(R^(2)(E^4π|u|^(2)-1)dx≤C(τ)<+∞)))and 4πis sharp.However,if we consider the less restrictive constraint∫_(R_(2))(|▽u|^(2)+|u|^(2))dx≤1,where V(x)is nonnegative and vanishes on an open set in R^(2),it is unknown whether the sharp constant of the Trudinger-Moser inequality is still 4π.The loss of a positive lower bound of the potential V(x)makes this problem become fairly nontrivial.The main purpose of this paper is twofold.We will first establish the Trudinger-Moser inequality sup u∈H^(1)(r^(2)fR^(2)(τ|U|^(2)+|▽u|^(2)dx≤1∫_(R^(2)(E^4π|u|^(2)-1)dx≤C(τ)<+∞)))when V is nonnegative and vanishes on an open set in R^(2).As an application,we also prove the existence of ground state solutions to the following Sciridinger equations with critical exponeitial growth:-Δu+V(x)u=f u)in R^(2),(0.1)where V(x)≥0 and vanishes on an open set of R^(2)and f has critical exponential growth.Having a positive constant lower bound for the potential V(x)(e.g.,the Rabinowitz type potential)has been the standard assumption when one deals with the existence of solutions to the above Schrodinger equations when the nonlinear term has the exponential growth.Our existence result seems to be the first one without this standard assumption.

A Remark on Hardy-Trudinger-Moser Inequality

Let IB be the unit disc in R^2,H be the completion of C∞0(B)under the norm||u||H=(∫B|▽u|^2dx-∫Bu^2/(1-|x^2|^2dx)^1/2,■u∈C∞0(B).Using blow-up analysis,we prove that for anyγ≤4π,the supremum sup u∈H,||u||1,h≤1∫Beγu^2dx can be attained by some function u0∈H with||u0||1,h=1,where is a decreasingly nonnegative,radially symmetric function,and satisfies a coercive cond让ion.Namely there exists a constantδ>0 satisfying||u||21,h=||u||2H-∫Bhu^2dx≥δ||u||H^2,■u∈H.This extends earlier results of Wang-Ye[1]and Yang-Zhu[2].

A Trudinger-Moser Inequality Involving Lp-norm on a Closed Riemann Surface

In this paper,using the method of blow-up analysis,we obtained a Trudinger–Moser inequality involving L^(p)-norm on a closed Riemann surface and proved the existence of an extremal function for the corresponding Trudinger–Moser functional.

A Singular Trudinger-Moser Inequality in Hyperbolic Space

In this paper, we establish a singular Trudinger-Moser inequality for the whole hyperbolic space H^n：u∈W^1,n（H^n）,^sup,fH^n｜ H^nu｜^ndμ≤1∫H^n ea｜u｜n/n-1-∑^n-2a^k｜u｜nk/n-1 k=0 k!/ρβ dμ〈∞ a/an＋β/n≤1,where α 〉 0,β E [0,n）, ρ and dμ are the distance function and volume element of H^n respectively.

负幂次对数加权的Trudinger-Moser不等式

本文研究具有负幂次对数加权的Trudinger-Moser不等式.通过建立了一个径向引理,利用著名的Leckband泛函不等式证明了Calanchi和Ruf(2015)得到的对数加权Trudinger-Moser不等式在负幂次情形依然成立.

Cartan-Hadamard流形上关于Lorentz范数的Trudinger-Moser不等式

本文研究了Cartan-Hadamard流形上带Lorentz范数的Trudinger-Moser不等式.利用了相关格林函数的逐点估计以及O’Neil不等式,我们得到了该不等式的最佳常数,推广了相应欧氏空间上的结果.

Heisenberg群中带奇异权的最优临界Hardy-Trudinger-Moser不等式

本文建立了Heisenberg群中有界域和一般无界域上带奇异权的最优临界Hardy-Trudinger-Moser不等式.克服临界Hardy不等式和奇异权函数带来的困难,利用带奇异权的Trudinger-Moser不等式和一些基本估计建立了有界域上一般的带奇异权的临界Hardy-Trudinger-Moser不等式,并通过选取适当的Moser函数得到了最佳常数.最后,利用分割积分区域的方法得到了一般无界域上带奇异权的最优临界Hardy-Trudinger-Moser不等式.

Fermions: Spin, Hidden Variables, Violation of Bell’s Inequality and Quantum Entanglement

Using real fields instead of complex ones, it was recently claimed, that all fermions are made of pairs of coupled fields (strings) with an internal tension related to mutual attraction forces, related to Planck’s constant. Quantum mechanics is described with real fields and real operators. Schrodinger and Dirac equations then are solved. The solution to Dirac equation gives four, real, 2-vectors solutions ψ1=(U1D1)ψ2=(U2D2)ψ3=(U3D3)ψ4=(U4D4)where (ψ1,ψ4) are coupled via linear combinations to yield spin-up and spin-down fermions. Likewise, (ψ2,ψ3) are coupled via linear combinations to represent spin-up and spin-down anti-fermions. For an incoming entangled pair of fermions, the combined solution is Ψin=c1ψ1+c4ψ4where c1and c4are some hidden variables. By applying a magnetic field in +Z and +x the theoretical results of a triple Stern-Gerlach experiment are predicted correctly. Then, by repeating Bell’s and Mermin Gedanken experiment with three magnetic filters σθ, at three different inclination angles θ, the violation of Bell’s inequality is proven. It is shown that all fermions are in a mixed state of spins and the ratio between spin-up to spin-down depends on the hidden variables.

THE LOGARITHMIC SOBOLEV INEQUALITY FOR A SUBMANIFOLD IN MANIFOLDS WITH ASYMPTOTICALLY NONNEGATIVE SECTIONAL CURVATURE

In this note,we prove a logarithmic Sobolev inequality which holds for compact submanifolds without a boundary in manifolds with asymptotically nonnegative sectional curvature.Like the Michale-Simon Sobolev inequality,this inequality contains a term involving the mean curvature.

Inequality of rural residents'income in China since the targeted poverty alleviation strategy:New trends,causes,and policy implications

Poverty and income inequality are problems faced by many countries including China.Since the implementation of Targeted Poverty Alleviation(TPA),absolute poverty has been eradicated,meanwhile,Chinese rural residents'income and inequality have experienced unprecedented new dynamics,but still,very few studies have focused on them.Hence,we attempt to examine the new trends in Chinese farmers'income inequality and to understand its causes during this period.We observe that China's urban-rural income ratio has been shrinking and the urban-rural inequality has been decreasing from 2014 to 2021.The reduction in the Theil index indicates a decrease in the regional inequality of farmers'income as well,and the decline in inter-regional inequality among eight economic zones contributes about 80%to increased equality.These new trends are caused by the fact that,with the TPA,the incomes of rural residents especially that in destitute areas,grew faster than that of urban residents and farmers in prosperous areas.The higher share of non-agricultural industry and agricultural mechanization level have significant positive impact on regional equality of farmers'income,while urban-rural dual structure has significant negative inhibitory effects.The lowest farmers'income and the highest inequality in the Northwest,and the slowest growth in the Northeast deserve more attention in rural Revitalization.From these findings,we propose four policy implications that would be applied to improve Chinese farmers'income equality,govern relative poverty,and achieve common prosperity in the post-poverty era.

Weak and Strong Convergence of Self Adaptive Inertial Subgradient Extragradient Algorithms for Solving Variational Inequality Problems

Many solutions of variational inequalities have been proposed,among which the subgradient extragradient method has obvious advantages.Two different algorithms are given for solving variational inequality problem in this paper.The problem we study is defined in a real Hilbert space and has L-Lipschitz and pseudomonotone condition.Two new algorithms adopt inertial technology and non-monotonic step size rule,and their convergence can still be proved when the value of L is not given in advance.Finally,some numerical results are designed to demonstrate the computational efficiency of our two new algorithms.

A Study of Triangle Inequality Violations in Social Network Clustering

Clustering a social network is a process of grouping social actors into clusters where intra-cluster similarities among actors are higher than inter-cluster similarities. Clustering approaches, i.e. , k-medoids or hierarchical, use the distance function to measure the dissimilarities among actors. These distance functions need to fulfill various properties, including the triangle inequality (TI). However, in some cases, the triangle inequality might be violated, impacting the quality of the resulting clusters. With experiments, this paper explains how TI violates while performing traditional clustering techniques: k-medoids, hierarchical, DENGRAPH, and spectral clustering on social networks and how the violation of TI affects the quality of the resulting clusters.

AGGREGATE SPECIAL FUNCTIONS TO APPROXIMATE PERMUTING TRI-HOMOMORPHISMS AND PERMUTING TRI-DERIVATIONS ASSOCIATED WITH A TRI-ADDITIVEψ-FUNCTIONAL INEQUALITY IN BANACH ALGEBRAS

In this paper,we define a new class of control functions through aggregate special functions.These class of control functions help us to stabilize and approximate a tri-additiveψ-functional inequality to get a better estimation for permuting tri-homomorphisms and permuting tri-derivations in unital C*-algebras and Banach algebras by the vector-valued alternative fixed point theorem.

The Compactness of Extremals for a Singular Hardy-Trudinger-Moser Inequality

Motivated by a recent work of Wang-Yang[19],we study the compactness of extremals[ug]for singular Hardy-Trudinger-Moser inequalities due to Hou[24].In particular,by the method of blow-up analysis,we conclude that,up to a subsequence,ug converges to an extremal in some sense asβtends to zero.

Further Improvement of Klamkin Inequality

In this paper, we use a geometric identity in the n-dimensional Euclidean space En and give the further improveme nt of Klamkin inequality in the space En.