A conjecture regarding the logarithmic coefficients of univalent functions

One considers the class S of functions, regular and univalent in ¦Z¦<1 and normalized by the expansion f(z)=Z + C₂Z² +.... By the logarithmic coefficients of the function f (z) S one means the coefficients of the expansion Earlier, the author had formulated the following conjecture: for any function f(z) S, for each z (0,1) one has the inequality In this paper this conjecture is proved for spiral-shaped functions and for functions from S with real coefficients and under some additional assumptions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 125, pp. 135–143, 1983.     

Characterization of the Hilbert ball by its automorphism group

Let be a bounded, convex domain in a separable Hilbert space. The authors prove a version of the theorem of Bun Wong, which asserts that if such a domain admits an automorphism orbit accumulating at a strongly pseudoconvex boundary point, then it is biholomorphic to the ball. Key ingredients in the proof are a new localization argument using holomorphic peaking functions and the use of new ``normal families' arguments in the construction of the limit biholomorphism.

     

Near-perfect matrices

A 0, 1 matrixA isnear-perfect if the integer hull of the polyhedron {x0: Ax } can be obtained by adding one extra (rank) constraint. We show that in general, such matrices arise as the cliquenode incidence matrices of graphs. We give a colouring-like characterization of the corresponding class of near-perfect graphs which shows that one need only check integrality of a certain linear program for each 0, 1, 2-valued objective function. This in contrast with perfect matrices where it is sufficient to check 0, 1-valued objective functions. We also make the following conjecture: a graph is near-perfect if and only if sequentially lifting any rank inequality associated with a minimally imperfect graph results in the rank inequality for the whole graph. We show that the conjecture is implied by the Strong Perfect Graph Conjecture. (It is also shown to hold for graphs with no stable set of size eleven.) Our results are used to strengthen (and give a new proof of) a theorem of Padberg. This results in a new characterization of minimally imperfect graphs: a graph is minimally imperfect if and only if both the graph and its complement are near-perfect.The research has partially been done when the author visited Mathematic Centrum, CWI, Amsterdam, The Netherlands.     

Stability of a reverse isoperimetric inequality

In this note we will present a stability property of the reverse isoperimetric inequality newly obtained in [S.L. Pan, H. Zhang, A reverse isoperimetric inequality for convex plane curves, Beiträge Algebra Geom. 48 (2007) 303-308], which states that if K is a convex domain in the plane with perimeter p(K) and area a(K), then one gets , where denotes the oriented area of the domain enclosed by the locus of curvature centers of the boundary curve ∂K, and the equality holds if and only if K is a circular disc.     

Points of Positive Density for the Solution to a Hyperbolic SPDE

Using a slightly modified version of Aida–Kusuoka–Stroock's characterization of the points of strictly positive density for an arbitrary Wiener functional, we extend the theorem of Ben Arous–Léandre to solutions of hyperbolic SPDE's. Thus we show that the density f of the law of X_z is positive at y if and only if y can be achieved as S_z(h), where S(h) is the controlled equation corresponding to an element h of the Cameron–Martin space, and S(.)_z is a submersion at h. The proof depends on a convergence result for a sequence X^n, of perturbed processes (defined in terms of a non homogeneous linear interpolation of the Brownian sheet) to the solution X of the corresponding perturbed SPDE.     

A Friedrichs–Maz'ya inequality for functions of bounded variation

The aim of this short note is to give an alternative proof, which applies to functions of bounded variation in arbitrary domains, of an inequality by Maz'ya that improves Friedrichs inequality. A remarkable feature of such a proof is that it is rather elementary, if the basic background in the theory of functions of bounded variation is assumed. Nevertheless, it allows to extend all the previously known versions of this fundamental inequality to a completely general version. In fact the inequality presented here is optimal in several respects. As already observed in previous proofs, the crucial step is to provide conditions under which a function of bounded variation on a bounded open set, extended to zero outside, has bounded variation on the whole space. We push such conditions to their limits. In fact, we give a sufficient and necessary condition if the open set has a boundary with σ‐finite surface measure and a sufficient condition if the open set is fully arbitrary. Via a counterexample we show that such a general sufficient condition is sharp.     

A Faber-Krahn inequality for Robin problems in any space dimension

We prove a Faber-Krahn inequality for the first eigenvalue of the Laplacian with Robin boundary conditions, asserting that amongst all Lipschitz domains of fixed volume, the ball has the smallest first eigenvalue. We prove the result in all space dimensions using ideas from [M.-H. Bossel, C. R. Acad. Sci. Paris Sér. I Math. 302 (1986), 47–50], where a proof for smooth domains in the plane was given. The method does not involve the use of symmetrisation arguments. The results also imply variants of the Cheeger inequality for the first eigenvalue.     

带有阻尼项的偏泛函微分方程解的振动性

本文研究带有阻尼项的双曲型时滞偏微分方程 2 t2 u(x,t) +m(t) u t=a(t)△ u(x,t) +b(t)△ u(x,ρ(t) ) -q(t) f (u(x,σ(t) ) ,(x,t)∈ G≡Ω× R+ (1 )其中 ,R+=[0 ,+∞ ) ,Ω是一个具有逐段光滑边界的有界区域 .利用平均法和微分不等式方法得到方程 (1 )的若干新的振动准则 .     

一反应扩散模型解的有限元方法和误差分析

１引言在地下水运移过程中,污染物（溶质）随地下水在含水层中运移,并常常发生各种化学反应．文献［１－３］等提出并论述了三种化学物质（如Ｍ１,Ｍ２和Ｍ３）之间发生的一类化学反应．文献［４,５］等建立和描述了这类反应的数学模型（Ｐ）．文献［６,７」的作者首次对模型（Ｐ）进行了理论上的定性分析,主要是利用上,下解方法,算子半群理论和Ｓｏｂｏｌｅｖ空间的般人定理等论证了模型（Ｐ）的整体古典解的存在唯一性和渐近性质,文［６,７］也讨论了整体解的极限性态和收敛性估计．此外,文［８,９］等也就模型（Ｐ）的一类特…     

Some properties of a class of symmetric functions

The Schur-convexity and Schur-geometric-convexity of a class of symmetric functions are investigated. As consequences some new proofs of the well-known Ky Fan's inequality and Shapiro's inequality are presented, respectively. We also give another proof of a problem posted by S. Gabler in [S. Gabler, Aufgabe 830, Elem. Math. 3 (1980) 124-125]. Some interesting matrix and geometric inequalities are established to show the applications of our results.     

Strong normalization of barrecursive terms without using infinite terms

In this paper a new proof of the strong normalization theorem (SN) for barrecursive terms is presented.The proof is based on a syntactical version of Howard's compactness of functionals of finite type (see [T, 2.8.6]). The proofs of Tait [Ta], Luckhardt [L], and Vogel [V] are all based on continuity. These proofs use infinite terms: ifT ₀,T ₁, ... is an infinite sequence of terms of type , then T ₀,T ₁, ... is an infinite term of type (0). The proof below does not make use of infinite terms.     

On Korn’s Inequality

For β ∈ R, the authors consider the evolution system in the unknown variables u and α { ttu＋ xxxxu＋ xxtα＋（β＋｜｜ xu｜｜L2^2） xxu=f, ttα- xxα- xxtα- xxtu=0} describing the dynamics of type III thermoelastic extensible beams, where the dissipation is entirely contributed by the second equation ruling the evolution of the thermal displacement α. Under natural boundary conditions, the existence of the global attractor of optimal regularity for the related dynamical system acting on the phase space of weak energy solutions is established.     

Sur les sous-variétés des tores

In Invent. Math. 126 (2000), pp. 513–545, we gave a proof of Lang's conjecture on Abelian varieties leading to an effective bound for the number of translates involved. We show here that the method can be extended to give a similar statement for the Mordell–Lang plus Bogomolov theorem proven by B. Poonen and independently by S. Zhang. We deal in detail with tori for which effective results have been obtained by J.-H. Evertse and H. P. Schlickewei; we improve on these mainly by providing polynomial bounds in the degree instead of doubly exponential ones. We also state a theorem for Abelian varieties. In both cases the strategy of proof is based on the approach of Mumford and Vojta–Faltings–Bombieri together with an effective Bogomolov property and therefore does not rely on either equidistribution nor subspace theorem arguments.     

与条形区域有关的双向单叶解析函数子类的系数估计

本文引入了条形区域内由Salagean算子刻画的解析函数新子类, 讨论了该函数子类的系数估计、Fekete-Szego不等式以及与该函数子类有关的双向单叶函数的系数估计, 所得结果推广了前人的一些工作.     

Remarks on a Hardy–Sobolev inequality

We compute the optimal constant for a generalized Hardy–Sobolev inequality, and using the product of two symmetrizations we present an elementary proof of the symmetries of some optimal functions. This inequality was motivated by a nonlinear elliptic equation arising in astrophysics. To cite this article: S. Secchi et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On the maximum of a Wiener process and its location

Summary Consider a Wiener process {W(t),t0}, letM(t)=max |W(s)| andv(t) be the location of the maximum of the absolute value of in [0,t] i.e.|W(v(t))|=M(t). We study the limit points of ( _t M(t), _t v(t)) ast where _t and _t are positive, decreasing normalizing constants. Moreover, a lim inf result is proved for the length of the longest flat interval ofM(t).Research supported by Hungarian National Foundation for Scientific Research Grant n. 1808     

On a Problem of Dobrowolski and Williams and the Pólya-Vinogradov Inequality

In [2] E. Dobrowolski and K.S. Williams considered a problem of obtaining estimates for the sum _n=a+1 ^a+N f(n),for a certain class of functions f. One specific application of their result is a new method for estimating character sums. In particular, they obtain a form of the Pólya-Vinogradov inequality with the constant 1/(2 log 2). In this note we improve their estimates and obtain, in particular, a form of the Pólya-Vinogradov inequality with the constant 1/(3 log 3). A nice feature of our estimate is that it is obtained by a very simple argument.     

m重似星树的谱半径

仅有一个顶点的度大于2的树称为似星树.
在一棵似星树的每个一度点粘接一棵似星树构成的图称为$m$重似星树.
Gutman 和L. Shi给出了似星树谱半径的一个界.
在本文中我们给出了另外一个更简洁的证明方法并做了深入的讨论,
同时给出了$m$重似星树谱半径的一个最好界.     

On the stability of Thomson’s vortex pentagon inside a circular domain

We investigate the stability problem for stationary rotation of five identical point vortices located at the vertices of a regular pentagon inside a circular domain. The main result is the proof of theorems which have been announced by the author in Doklady Physics (2004, vol. 49, no. 11, pp. 658–661).     

Jensen's inequality for a convex vector-valued function on an infinite-dimensional space

Jensen's inequality f(EX) ≤ Ef(X) for the expectation of a convex function of a random variable is extended to a generalized class of convex functions f whose domain and range are subsets of (possibly) infinite-dimensional linear topological spaces. Convexity of f is defined with respect to closed cone partial orderings, or more general binary relations, on the range of f. Two different methods of proof are given, one based on geometric properties of convex sets and the other based on the Strong Law of Large Numbers. Various conditions under which Jensen's inequality becomes strict are studied. The relation between Jensen's inequality and Fatou's Lemma is examined.