Liouville-type results for stationary maps of a class of functional related to pullback metrics

We study a generalized functional related to the pullback metrics (3). We derive the first variation formula which yield stationary maps. We introduce the stress–energy tensor which is naturally linked to conservation law and yield the monotonicity formula via the coarea formula and the comparison theorem in Riemannian geometry. A version of this monotonicity inequalities enables us to derive some Liouville type results. Also, we investigate the constant Dirichlet boundary value problems and the generalized Chern type results for tension field equation with respect to this functional.     

一类连续半鞅的极大不等式

本文主要研究了一类连续半鞅的极大不等式.利用伊藤公式和Lenglart控制定理,得到了它们的极大不等式,推广了文献[9]的主要结果.     

New convolutions and their applicability to integral equations of Wiener-Hopf plus Hankel type

We propose four new convolutions exhibiting convenient factorization properties associated with two finite interval integral transformations of Fourier-type together with their norm inequalities. Moreover, we study the solvability of a class of integral equations of Wiener-Hopf plus Hankel type (on finite intervals) with the help of the factorization identities of such convolutions. Fourier-type series are used to produce the solution formula of such equations, and a Shannon-type sampling formula is also obtained.     

The Median Formula and Related Inequalities in n—simplex

5l.TheMedianFormulaAseveryoneknows:in6ABC,letal=BC,a2=CA,a3=AB,;)11,)n2,n13aremedian1inesseparatelyonthethreeedgesBC,CA,AB,tI1entheequalityistrueasfollowsTheequality(l)iscalledmedianformulaofQABC,andwecanobtainfrom(l)Inthispaper,wesha1lextendtheequality(l)and(2)to,l-simplexin)l-EuclideanspaceH.Itisobtainedthemedianformulaof)I-simplex,andusingthisformula.wegetsomein-equalities.Themedianformulaofn-simp1exisobtainedasfollows:TheoremlFork=1,2,--',It 1andl相似文献   

A nonlinear stochastic heat equation: Hölder continuity and smoothness of the density of the solution

In this paper, we establish a version of the Feynman–Kac formula for multidimensional stochastic heat equation driven by a general semimartingale. This Feynman–Kac formula is then applied to study some nonlinear stochastic heat equations driven by nonhomogeneous Gaussian noise: first, an explicit expression for the Malliavin derivatives of the solutions is obtained. Based on the representation we obtain the smooth property of the density of the law of the solution. On the other hand, we also obtain the Hölder continuity of the solutions.     

Transportation inequalities for stochastic delay evolution equations driven by fractional Brownian motion

We discuss stochastic functional partial differential equations and neutral partial differential equations of retarded type driven by fractional Brownian motion with Hurst parameter H>1/2. Using the Girsanov transformation argument, we establish the quadratic transportation inequalities for the law of the mild solution of those equations driven by fractional Brownian motion under the L² metric and the uniform metric.     

Reflected BSDEs with nonpositive jumps,and controller-and-stopper games

We study a class of reflected backward stochastic differential equations with nonpositive jumps and upper barrier. Existence and uniqueness of a minimal solution are proved by a double penalization approach under regularity assumptions on the obstacle. In a suitable regime switching diffusion framework, we show the connection between our class of BSDEs and fully nonlinear variational inequalities. Our BSDE representation provides in particular a Feynman–Kac type formula for PDEs associated to general zero-sum stochastic differential controller-and-stopper games, where control affects both drift and diffusion term, and the diffusion coefficient can be degenerate. Moreover, we state a dual game formula of this BSDE minimal solution involving equivalent change of probability measures, and discount processes. This gives in particular a new representation for zero-sum stochastic differential controller-and-stopper games.     

On first order interpolation inequalities with weights on the Heisenberg group

In this paper, sufficient and necessary conditions for the first order interpolation inequalities with weights on the Heisenberg group are given. The necessity is discussed by polar coordinates changes of the Heisenberg group. Establishing a class of Hardy type inequalities via a new representation formula for functions and Hardy-Sobolev type inequalities by interpolation, we derive the sufficiency. Finally, sharp constants for Hardy type inequalities are determined.     

关于单形一个对偶式的商榷与两个新的对偶式

讨论了n维欧氏空间E^n中n维单形不等式的对偶式．利用距离几何理论与解析方法，建立了n维单形两个不等式的对偶式，指出了最近所建立的单形Finsler-Hadwiger不等式的A维对偶式是错误的．     

Itô-Wiener Chaos Expansion with Exact Residual and Correlation, Variance Inequalities

We give a formula of expanding the solution of a stochastic differential equation (abbreviated as SDE) into a finite Itô-Wiener chaos with explicit residual. And then we apply this formula to obtain several inequalities for diffusions such as FKG type inequality, variance inequality and a correlation inequality for Gaussian measure. A simple proof for Houdré-Kagan's variance inequality for Gaussian measure is also given.     

从积分几何的观点看几何不等式——纪念李国平院士吴新谋教授诞辰100周年

该文先介绍一些中国数学家在几何不等式方面的工作.作者用积分几何中著名的Poincarè公式及Blaschke公式估计一随机凸域包含另一域的包含测度, 得到了经典的等周不等式和Bonnesen -型不等式.还得到了一些诸如对称混合等周不等式、Minkowski -型和Bonnesen -型对称混合等似不等式在内的一些新的几何不等式.最后还研究了Gage -型等周不等式以及Ros -型等周不等式.     

半直线上L~p-Poincaré不等式最优常数的估计

运用分析方法给出半直线上测度有限时在Dirichlet边界条件下L~p-Poincaré不等式最优常数的变分公式,并运用迭代方法结合变分公式得到了最优常数的显式估计.     

Hilbert重级数定理的一个新推广

通过引入一个适当的对数函数建立了一种新的Hilbert型不等式.利用Euler-Maclaurin求和公式对权函数进行了估计.证明了常数因子π2r＋1Er是最佳的,其中Er是Euler数.作为应用,给出了一些互相等价的不等式.     

The adapted solution and comparison theorem for backward stochastic differential equations with Poisson jumps and applications

This paper deals with a class of backward stochastic differential equations with Poisson jumps and with random terminal times. We prove the existence and uniqueness result of adapted solution for such a BSDE under the assumption of non-Lipschitzian coefficient. We also derive two comparison theorems by applying a general Girsanov theorem and the linearized technique on the coefficient. By these we first show the existence and uniqueness of minimal solution for one-dimensional BSDE with jumps when its coefficient is continuous and has a linear growth. Then we give a general Feynman-Kac formula for a class of parabolic types of second-order partial differential and integral equations (PDIEs) by using the solution of corresponding BSDE with jumps. Finally, we exploit above Feynman-Kac formula and related comparison theorem to provide a probabilistic formula for the viscosity solution of a quasi-linear PDIE of parabolic type.     

一个Hilbert型不等式及其等价形式的加强推广

本文研究了Hilbert型级数不等式.利用改进的Euler-Maclaurin求和公式及权系数的方法,并引进两对共轭指数,获得一个具有最佳常数系数的Hilbert型不等式,该不等式是联系若干Hilbert型不等式的加强推广式.作为应用,给出该不等式的等价形式及若干有意义的特殊结果.     

Eikonal型方程粘性解的表达式

对于圆锥型和棱锥型Hamiltonian的Eikonal型方程,本文给出了一种几何方法,得出其初值问题解的表达式并且说明由此式给出的解为原初值问题的粘性解．首先用一个凸函数序列逼近Eikonal型方程中的Hamiltonian,再由Hopf-Lax公式给出方程序列的粘性解,最后证明了该粘性解序列会收敛到Eikonal方程的粘性解．     

离散型Carlson's不等式的一些推广及改进

In this paper, we consider Carlson type inequalities and discuss their possible improvement. First, we obtain two different types of generalizations of discrete Carlson's inequality by using the Hlder inequality and the method of real analysis, then we combine the obtained results with a summation formula of infinite series and some Mathieu type inequalities to establish some improvements of discrete Carlson's inequality and some Carlson type inequalities which are equivalent to the Mathieu type inequalities. Finally, we prove an integral inequality that enables us to deduce an improvement of the Nagy-Hardy-Carlson inequality.     

泰勒公式及泰勒级数之妙用

泰勒公式及泰勒级数是非常重要的数学工具,除了读者熟知的应用方面外,在其他问题的解决中也有妙用.举例介绍了应用泰勒公式及泰勒级数解决判断级数的敛散性、证明与积分有关的不等式等问题.     

关于一类差分不等式的估计和计算

主要讨论一类差分不等式,研究了它们的估计和计算,给出估计公式和算法,推广了文[1]的结果.