Some Equivalent Forms of Bernoulli’s Inequality: A Survey

The main purpose of this paper is to link some known inequalities which are equivalent to Bernoulli’s inequality.

Global inequality in built-up land per capita and its change trajectories between 1985 and 2020

Built-up land has increased rapidly in recent decades,thus aggravating the competition for land between multiple uses.The increase in urban land can be decomposed into changes in population and changes in built-up land per person.Yet,it is unclear how this decomposition differs by country and how this decomposition changes over time.Moreover,we don’t know whether these changes affect the inequality in built-up land per capita.Here,we analyze the global distribution of built-up land per capita in the year 2020,as well as the changes therein between 1985 and 2020.We find that built-up land per capita in 2020 differs by an order of magnitude between countries,ranging from 15 m^(2) per person in Ethiopia to 734 m^(2) per person in Australia.Moreover,we find a wide range of different change trajectories,including both increases and decreases in built-up land per capita and in total population.As the total area of urban land increased in all countries,decreases in urban land consumption reflect a situation where the population increases faster than the total amount of urban land.We also find a large inequality in urban land consumption across countries,as indicated by a Gini index of 0.47 in 1985,decreasing only slightly to 0.45 in 2020.These findings suggest the need for a regionally differentiated approach to reduce urban land take,focusing first on mitigating further increases in those countries that already have a high urban land consumption.

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.

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.

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.

An analysis of inequality in physical health status of women in India:2015-2021

Objective:We examine how inequality in women's health has changed in India and its major states between 2015 and 2021.We also attempt to identify the sources of any change in this inequality using a decomposition analysis.In order to suggest suitable policies for reducing this inequality,we attempt to identify the factors that are strongly associated with it.Methods:We construct an index of women's health from five indicators using the principal component analysis.Based on this index we measure inequality in women's health using the coefficient of variation and the Theil index.To identify the sources of change in this inequality,we apply Theil decomposition technique.Results:We find that inequality in women's health has increased significantly in India between 2015 and 2021.During this period,this inequality has increased significantly in six states,while it has decreased significantly in five states.The decomposition analysis suggests that the increase in this inequality between 2015 and 2021 has been driven by an increase in interstate inequality.We also find a strong positive association between inequality in women's health and inequality in public health amenities and social development.Conclusion:We suggest that for reducing inequality in women's health in developing countries like India,policies should focus on eliminating disparities in income,female education and provision of public health amenities.

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.

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.

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.

Revisiting Income Inequality in China from an Evolving Productivity Perspective

Income inequality in China has evolved substantially amid rising productivity since reform and opening up in the late 1970s.Based on the group decomposition equation for Gini coefficient,this paper estimates China’s inequality possibility frontier(IPF)and the inequality extraction rate(IER).Results indicate that:(i)From 1978 to 2017,China’s IPF continuously expanded amid improving productivity,and the maximum feasible Gini coefficient rose from 0.2281 to 0.8446.(ii)Meanwhile,China’s overall IER decreased from 123%to 55%.More specifically,China’s IER fell sharply over the period 1978-1980,stabilized in the period from the mid-1980s to 2012,and further declined after the 18th CPC National Congress in 2012.Currently,55%of China’s maximum feasible inequality has been converted into actual inequality.The correlation between inequality and productivity is recognized in academia but seldom explored in the literature.To fill this void,this paper empirically measures inequality in light of productivity development.Our research conclusions explain why yawning income gaps in China have been tolerated since reform and opening up,and offer empirical evidence for setting income distribution policies according to economic development in the new era.

BURKHOLDER-GUNDY-DAVIS INEQUALITY IN MARTINGALE HARDY SPACES WITH VARIABLE EXPONENT

In this article, by extending classical Dellacherie＇s theorem on stochastic se- quences to variable exponent spaces, we prove that the famous Burkholder-Gundy-Davis in- equality holds for martingales in variable exponent Hardy spaces. We also obtain the variable exponent analogues of several martingale inequalities in classical theory, including convexity lemma, Chevalier＇s inequality and the equivalence of two kinds of martingale spaces with predictable control. Moreover, under the regular condition on σ-algebra sequence we prove the equivalence between five kinds of variable exponent martingale Hardy spaces.

Necessary and Sufficient Conditions of Doubly Weighted Hardy-Littlewood-Sobolev Inequality

Using product and convolution theorems on Lorentz spaces, we characterize the sufficient and necessary conditions which ensure the validity of the doubly weighted Hardy-Littlewood-Sobolev inequality. It should be pointed out that we con- sider whole ranges of p and q, i.e., 0 〈 p ≤∞ and 0 〈 q ≤∞.

The Proof of Sobolev Embedding Inequality

We consider the problem about the space embedded by the space and the embedding inequality. With the HSlder inequality and interpolation inequality, we give the proof of the space embedding theorem and the space holder embedding theorem.

An Operator Inequality for Three Operators

As a generalization of grand Furuta inequality,recently Furuta obtain:If A≥ B≥0 with A>0,then for t∈[0,1]and p1,p2,p3,p4≥1, A t 2[A- t 2{A t 2(A/ t 2 Bp 1A /t2 )p 2A t 2}p 3A /t2 ]p 4A t 2 1 [{(p1/t)p2+t}p3-t]p4+t]≤A. In this paper,we generalize this result for three operators as follow:If A≥B≥C≥0 with B>0,t∈[0,1]and p1,p2,···,p2n/1,p2n≥1 for a natural number n.Then the following inequalities hold for r≥t, A1/t+r≥ [A r 2[B /t 2{B t 2······[B /t 2{B t 2(B /t 2 ←B /t 2 n times Bt 2 n/1 times by turns Cp 1B /t 2)p 2B t 2}p 3B /t 2]p 4···B t 2}p 2n/1B /t 2 B /t 2 n times Bt 2 n/1 times by turns→ ]p 2nA r 2] 1/t+r q[2n]+r/t, where q[2n]≡{···[{[(p1/t)p2+t]p3/t}p4+t]p5/···/t}p2n+t /t and t alternately n times appear .

Application Of Jessen＇s Type Inequality For Positive C0-Semigroup Of Operators

Recently in [4], the Jessen＇s type inequality for normalized positive C0-semigroups is obtained. In this note, we present few results of this inequality, yielding Holder＇s Type and Minkowski＇s type inequalities for corresponding semigroup. Moreover, a Dresher＇s type inequality for two-parameter family of means, is also proved.

Exponential Generalization of Newman's Inequality and Klamkin's Inequality

Exponential generalizations of Newman＇s inequality and Klamkin＇s inequality are established by the Wang Wan-lan＇s inequality, and they are extended to the cases involving general elementary symmetric functions. As an application, some new inequalities for a simplex are established. In addition, an open problem is posed.

Some Refinement of Holder’s and Its Reverse Inequality

Holder’s inequality, its refinement, and reverse have received considerable attention in the theory of mathematical analysis and differential equations. In this paper, we give some refinements of Holder’s inequality and its reverse using a simple analytical technique of algebra and calculus. Our results show many results related to holder’s inequality as special cases of the inequalities presented.

A New Approach to Measurement of Regional Inequality in Particular Directions

Regional inequality is a core issue in geography,and it can be measured by several approaches and indexes.However,the global inequality measures can not reflect regional characteristics in terms of spatiality and non-mobility,as well as correctly explore regional inequality in particular directions.Although conventional between-group inequality indexes can measure the inequality in particular directions,they can not reflect the reversals of regional patterns and changes of within-group patterns.Therefore,we set forth a new approach to measure regional inequality in particular directions,which is applicable to geographic field.Based on grouping,we established a new index to measure regional inequality in particular directions named Particular Direction Inequality index(PDI index),which is comprised of between-group inequality of all data and between-group average gap.It can reflect regional spatiality and non-mobility,judge the main direction of regional inequality,and capture the changes and reversals of regional patterns.We used the PDI index to measure the changes of regional inequality from 1952 to 2009 in China.The results show that:1) the main direction of China's regional inequality was between coastal areas and inland areas;the increasing extent of inequality between coastal areas and inland areas was higher than the global inequality;2) the PDI index can measure the between-region average gap,and is more sensitive to evolution of within-region patterns;3) the inequality between the northern China and the southern China has been decreasing from 1952 to 2009 and was reversed in 1994 and 1995.

GENERALIZED ROSENTHAL'S INEQUALITY FOR BANACH-SPACE-VALUED MARTINGALES

A generalized Rosenthal＇s inequality for Banach-space-valued martingales is proved, which extends the corresponding results in the previous literatures and characterizes the p-uniform smoothness and q-uniform convexity of the underlying Banach space. As an application of this inequality, the strong law of large numbers for Banach-space-valued martingales is also given.

THE HADAMARD INEQUALITY FOR CONVEX FUNCTION VIA FRACTIONAL INTEGRALS

In this paper, we establish several inequalities for some differantiable mappings that are connected with the Riemann-Liouville fractional integrals. The analysis used in the proofs is fairly elementary.