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1.
Hypersurfaces with constant scalar curvature   总被引:38,自引:0,他引:38  
2.
The authors consider the length, , of the longest increasing subsequence of a random permutation of numbers. The main result in this paper is a proof that the distribution function for , suitably centered and scaled, converges to the Tracy-Widom distribution of the largest eigenvalue of a random GUE matrix. The authors also prove convergence of moments. The proof is based on the steepest descent method for Riemann-Hilbert problems, introduced by Deift and Zhou in 1993 in the context of integrable systems. The applicability of the Riemann-Hilbert technique depends, in turn, on the determinantal formula of Gessel for the Poissonization of the distribution function of .

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3.
拟二次系统的广义焦点量与极限环分枝   总被引:15,自引:0,他引:15       下载免费PDF全文
刘一戎 《数学学报》2002,45(4):671-682
本文给出了拟二次系统的前18个奇点量和可积性条件,由此统一解决了几类实平面微分自治系统的初奇点、高次奇点以及无穷远点的中心焦点判定与极限环分枝问题.  相似文献
4.
5.
Blow-up of solutions of nonlinear wave equations in three space dimensions   总被引:11,自引:0,他引:11  
Let u(x,t) be a solution, uA|u|p for xIR3, t0 where is the d'Alembertian, and A, p are constants with A>0, 10–|x–x0|, if the initial data u(x,0), ut(x,0) have their support in the ball |x–x0|t0. In particular global solutions of u=A|u|p with initial data of compact support vanish identically. On the other hand for A>0, p>1+2 global solutions of u=A|u|p exist, if the initial data are of compact support and u is sufficiently small in a suitable norm. For p=2 the time at which u becomes infinite is of order u–2.Dedicated to Hans Lewy and Charles B. Morrey, Jr.The research for this paper was performed at the Courant Institute and supported by the Office of Naval Research under Contract No. N00014-76-C-0301. Reproduction in whole or part is permitted for any purpose of the United States Government.  相似文献
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7.
PIECEWISE SEMIALGEBRAIC SETS   总被引:7,自引:0,他引:7  
Semialgebraic sets are objects which are truly a special feature of real algebraic geometry. This paper presents the piecewise semialgebraic set, which is the subset of Rn satisfying a boolean combination of multivariate spline equations and inequalities with real coefficients. Moreover, the stability under projection and the dimension of C^μ piecewise semialgebraic sets are also discussed.  相似文献
8.
Stirling数的概率表示和应用   总被引:7,自引:0,他引:7       下载免费PDF全文
孙平  王天明 《数学学报》1998,41(2):281-290
本文证明了第二类Stirling数S(n,k)和第一类Stirling数s(n,k)都是常见的随机变量和的矩,从而使概率论的方法和技巧,在组合和式计算、恒等式的发现与证明以及渐近估计等方面得到许多新的结果.  相似文献
9.
Discrete convex analysis   总被引:6,自引:0,他引:6  
A theory of “discrete convex analysis” is developed for integer-valued functions defined on integer lattice points. The theory parallels the ordinary convex analysis, covering discrete analogues of the fundamental concepts such as conjugacy, subgradients, the Fenchel min-max duality, separation theorems and the Lagrange duality framework for convex/nonconvex optimization. The technical development is based on matroid-theoretic concepts, in particular, submodular functions and exchange axioms. Sections 1–4 extend the conjugacy relationship between submodularity and exchange ability, deepening our understanding of the relationship between convexity and submodularity investigated in the eighties by A. Frank, S. Fujishige, L. Lovász and others. Sections 5 and 6 establish duality theorems for M- and L-convex functions, namely, the Fenchel min-max duality and separation theorems. These are the generalizations of the discrete separation theorem for submodular functions due to A. Frank and the optimality criteria for the submodular flow problem due to M. Iri-N. Tomizawa, S. Fujishige, and A. Frank. A novel Lagrange duality framework is also developed in integer programming. We follow Rockafellar’s conjugate duality approach to convex/nonconvex programs in nonlinear optimization, while technically relying on the fundamental theorems of matroid-theoretic nature.  相似文献
10.
On condition numbers and the distance to the nearest ill-posed problem   总被引:5,自引:0,他引:5  
Summary The condition number of a problem measures the sensitivity of the answer to small changes in the input. We call the problem ill-posed if its condition number is infinite. It turns out that for many problems of numerical analysis, there is a simple relationship between the condition number of a problem and the shortest distance from that problem to an ill-posed one: the shortest distance is proportional to the reciprocal of the condition number (or bounded by the reciprocal of the condition number). This is true for matrix inversion, computing eigenvalues and eigenvectors, finding zeros of polynomials, and pole assignment in linear control systems. In this paper we explain this phenomenon by showing that in all these cases, the condition number satisfies one or both of the diffrential inequalitiesm·2DM·2, where D is the norm of the gradient of . The lower bound on D leads to an upper bound 1/m(x) on the distance. fromx to the nearest ill-posed problem, and the upper bound on D leads to a lower bound 1/(M(X)) on the distance. The attraction of this approach is that it uses local information (the gradient of a condition number) to answer a global question: how far away is the nearest ill-posed problem? The above differential inequalities also have a simple interpretation: they imply that computing the condition number of a problem is approximately as hard as computing the solution of the problem itself. In addition to deriving many of the best known bounds for matrix inversion, eigendecompositions and polynomial zero finding, we derive new bounds on the distance to the nearest polynomial with multiple zeros and a new perturbation result on pole assignment.  相似文献
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