关于指数、Neuman-Sándor和二次平均的一个精确双向不等式

本文证明了双向不等式αI(a; b)+(1-α )Q(a; b) < M(a; b) < βI(a; b)+(1-β)Q(a; b) 对所有不相等的正实数a和b成立当且仅当α≥1/2 和β≤[e(√2log(1+√2)-1)]/[(√2e-2) log(1+√2)]=0:4121…,其中I(a; b), M(a; b)和Q(a; b)分别表示a和b的指数平均、Neuman-Sándor平均和二次平均.     

Amenability,Poincaré级数算子与Teichmüller距离

设Y→X为双曲型Riemann曲面X的覆盖映射,它诱导了X与Y的Teichmller空间之间的一个映射T(X)→T(Y),得到了i)覆盖Y→X为amenable的充分必要条件是诱导映射T(X)→T(Y)对于Teichmller度量为整体等距;ii)诱导映射T(X)→T(Y)为收缩的充分必要条件是对于任意的[μ]∈T(X),都有||_[μ]||<1,其中_[μ]为Poincar级数算子;iii)诱导映射T(X)→T(Y)在T(X)上不是一致收缩的.     

胰岛素及其类似物的圆二色性

本文研究了胰岛素与其类似物DPI,DHPI,DOI及工字肽的紫外圆二色性。胰岛素的CD谱呈双负峰,但[θ]₂₀₇负值偏大。类似物的谱与此相似。[θ]₂₂₂的负值从DPI到DOI逐个下降;[θ]₂₀₇与胰岛素相同,极值稍蓝移。从工字肽的谱设想,其中有一结构使胰岛素的[θ]₂₀₇负值偏大。 从CD谱看,类似物的构象相似,但比胰岛素松散。这就支持了胰岛素中受体结合部位的设想。 从表面活性剂对这些分子的CD谱的影响看,它对n-π^*和π-π^*跃迁的影响不同。     

关于Hayman方向

本文获得如下定理:设f(z)为开平面上λ(0<λ<+∞)级亚纯函数,则至少存在一条由原点引出的半直线(B):arg z=θ_?(0≤θ₀<2π),使得对于任意正数ε,任意正整数k及任意两个开平面上级小于λ的亚纯函数a(z)及b(z),只要b(z)—a^(k)(z)?0就有 lim log{n(r,θ₀,ε,f=a(z))+n(r,θ₀,ε,f^(k)=b(z))}/logr=λ。     

满足Denjoy猜测极值情况的整函数(Ⅱ)

本文主要证明了如下结果: 设ρ级整函数f(x)具有k(1≤k<+∞)个判别有穷渐近值。如果k=2ρ,则有 1) f(z)不能有有穷亏值, 2) 对任意值θ,0≤θ<2π,或者半直线argz=θ是f(x)的一条Julia方向,或者有...     

Drasin问题和Mues猜测

设f(z)为开平面上的超越亚纯函数,则 (i)∑δ(a,f)+∑δ(b,f^k)≤3,(k=1,2,…). a∈C b∈C (ii)∑δ(b,f^k)≤1,k≥K, b∈C这里K是仅依赖于f(z)的正数.本文并得到了使(i)中等号成立的充分必要条件.     

关于ff＇的辐角分布

设f为一整函数,下级为μ,级为λ,若ff′-1的零点均聚集于q(<∞)条射线上,则λ必为有穷,并且
(i) 当q=1时,λ≤1/2;
(ii)当q=2与μ≥1时,这两条射线之夹角必等于π/λ;
(iii)当q≥2时,至少存在两条射线,其夹角不超过π/λ。     

亚纯函数的幅角分布与增长性

设 f(z)为下级μ<+∞的平面内的亚纯函数,argz=θ_k(k=1,2,…,m;1≤m <+∞;0≤θ₁<θ₂<…<θ_m<2π,θ_m+1=θ_1+2π为平面内m条射线,使得对任意的ε>0及X=0,∞有 这里ρ为一任意给定的非负实数.如果f⁽¹⁾(z)(l≥0)具有一个有穷非零亏值 a,则f(z)的级λ≥max(π/ωρ)其中ω=min (θ_k+1-θ_k).     

小区间上的素变数三角和估计Ⅰ

设α是实数,x≥A≥2,e(θ)=e^2πiθ,以及A(n)是Mangoldt 函数。本文主要证明以下结论(见定理1):设δ是任给的正数,x^91/96+ε≤A≤x。那么,对任给的正数c,一定存在正数c₁,使当A^-1log^cx≤|α|≤(log x)^-c1)时,A(n)e(nα)<<A(log x)^-c     

同时鉴定斜长石成分和结构状态的X射线粉末法

将2θ(204)-2θ(400)角度差(称为E函数),与△θ₁=2θ(131)-2θ(131),△θ₂=2θ(241)-2θ(241),Γ=2θ(131)+2θ(220)-4θ(131)或△θ₁-△θ₂等角度差相配合,可以对An₀—An₇0范围内的斜长石An含量和结构状态作出合理的鉴定.将E=2θ(204)-2θ(400)CuK_α分别投于△θ₁—An(mol％),△θ₂—An(mol％),Γ—An(mol％)和△θ₁—△θ₂—An(mol％)图中,即得到四个鉴定图(图1—4).运用本文的方法,在一未知斜长石的x射线粉末衍射图中测定出所需反射的2θ值,再将有关的2θ差值投于相应的鉴定图中,便可容易地同时鉴定出该斜长石的An含量和结构状态.     

Remarks on differential Harnack inequalities

In this paper, we give a generalization of (global and local) differential Harnack inequalities for heat equations obtained by Li and Xu [J.F. Li, X.J. Xu, Differential Harnack inequalities on Riemannian manifolds I: linear heat equation, Adv. Math. 226 (5) (2011) 4456–4491] and Baudoin and Garofalo [F. Baudoin, N. Garofalo, Perelman’s entropy and doubling property on Riemannian manifolds, J. Geom. Anal. 21 (2011) 1119–1131]. From this we can derive new Harnack inequalities and new lower bounds for the associated heat kernel. Also we provide some new entropy formulas with monotonicity.     

Lyapunov conditions for Super Poincaré inequalities

We show how to use Lyapunov functions to obtain functional inequalities which are stronger than Poincaré inequality (for instance logarithmic Sobolev or F-Sobolev). The case of Poincaré and weak Poincaré inequalities was studied in [D. Bakry, P. Cattiaux, A. Guillin, Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré, J. Funct. Anal. 254 (3) (2008) 727-759. Available on Mathematics arXiv:math.PR/0703355, 2007]. This approach allows us to recover and extend in a unified way some known criteria in the euclidean case (Bakry and Emery, Wang, Kusuoka and Stroock, …).     

Best constants in the Hardy-Rellich inequalities and related improvements

We consider Hardy-Rellich inequalities and discuss their possible improvement. The procedure is based on decomposition into spherical harmonics, where in addition various new inequalities are obtained (e.g. Rellich-Sobolev inequalities). We discuss also the optimality of these inequalities in the sense that we establish (in most cases) that the constants appearing there are the best ones. Next, we investigate the polyharmonic operator (Rellich and higher order Rellich inequalities); the difficulties arising in this case come from the fact that (generally) minimizing sequences are no longer expected to consist of radial functions. Finally, the successively use of the Rellich inequalities lead to various new higher order Rellich inequalities.     

Weak logarithmic Sobolev inequalities and entropic convergence

In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincaré inequalities, general Beckner inequalities, etc.). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semi-group. In particular, we exhibit an example where Poincaré inequality can not be used for deriving entropic convergence whence weak logarithmic Sobolev inequality ensures the result.      

Wasserstein-Divergence transportation inequalities and polynomial concentration inequalities

In this paper, we study Wasserstein-Divergence transportation inequalities which are the generalization of classical transportation inequalities. We present sufficient and necessary conditions for them separately, which coincide in the limit case. Using this kind of inequalities, we establish polynomial concentration inequalities for probability measures with no exponential moments.     

Kirwan-Novikov inequalities on a manifold with boundary

We extend the Novikov Morse-type inequalities for closed 1-forms in 2 directions. First, we consider manifolds with boundary. Second, we allow a very degenerate structure of the critical set of the form, assuming only that the form is non-degenerated in the sense of Kirwan. In particular, we obtain a generalization of a result of Floer about the usual Morse inequalities on a manifold with boundary. We also obtain an equivariant version of our inequalities.

Our proof is based on an application of the Witten deformation technique. The main novelty here is that we consider the neighborhood of the critical set as a manifold with a cylindrical end. This leads to a considerable simplification of the local analysis. In particular, we obtain a new analytic proof of the Morse-Bott inequalities on a closed manifold.

     

On the extreme inequalities of infinite group problems

Infinite group relaxations of integer programs (IP) were introduced by Gomory and Johnson (Math Program 3:23–85, 1972) to generate cutting planes for general IPs. These valid inequalities correspond to real-valued functions defined over an appropriate infinite group. Among all the valid inequalities of the infinite group relaxation, extreme inequalities are most important since they are the strongest cutting planes that can be obtained within the group-theoretic framework. However, very few properties of extreme inequalities of infinite group relaxations are known. In particular, it is not known if all extreme inequalities are continuous and what their relations are to extreme inequalities of finite group problems. In this paper, we describe new properties of extreme functions of infinite group problems. In particular, we study the behavior of the pointwise limit of a converging sequence of extreme functions as well as the relations between extreme functions of finite and infinite group problems. Using these results, we prove for the first time that a large class of discontinuous functions is extreme for infinite group problems. This class of extreme functions is the generalization of the functions given by Letchford and Lodi (Oper Res Lett 30(2):74–82, 2002), Dash and Günlük (Proceedings 10th conference on integer programming and combinatorial optimization. Springer, Heidelberg, pp 33–45 (2004), Math Program 106:29–53, 2006) and Richard et al. (Math Program 2008, to appear). We also present several other new classes of discontinuous extreme functions. Surprisingly, we prove that the functions defining extreme inequalities for infinite group relaxations of mixed integer programs are continuous. S.S. Dey and J.-P.P. Richard was supported by NSF Grant DMI-03-48611.     

Entropy and set cardinality inequalities for partition‐determined functions

A new notion of partition‐determined functions is introduced, and several basic inequalities are developed for the entropies of such functions of independent random variables, as well as for cardinalities of compound sets obtained using these functions. Here a compound set means a set obtained by varying each argument of a function of several variables over a set associated with that argument, where all the sets are subsets of an appropriate algebraic structure so that the function is well defined. On the one hand, the entropy inequalities developed for partition‐determined functions imply entropic analogues of general inequalities of Plünnecke‐Ruzsa type. On the other hand, the cardinality inequalities developed for compound sets imply several inequalities for sumsets, including for instance a generalization of inequalities proved by Gyarmati, Matolcsi and Ruzsa (2010). We also provide partial progress towards a conjecture of Ruzsa (2007) for sumsets in nonabelian groups. All proofs are elementary and rely on properly developing certain information‐theoretic inequalities. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 40, 399–424, 2012     

Hardy inequalities with non-standard remainder terms

Improved Hardy inequalities, involving remainder terms, are established both in the classical and in the limiting case. The relevant remainders depend on a suitable distance from the families of the “virtual” extremals.     

Variational inequalities, coincidence theory, and minimax inequalities

New fixed-point theorems in Fréchet spaces are used to establish new variational inequalities, coincidence results, analytic alternatives, and minimax inequalities.