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A Semigroup Approach to Harmonic Maps   总被引:1,自引:0,他引:1  
We present a semigroup approach to harmonic maps between metric spaces. Our basic assumption on the target space (N,d) is that it admits a barycenter contraction, i.e. a contracting map which assigns to each probability measure q on N a point b(q) in N. This includes all metric spaces with globally nonpositive curvature in the sense of Alexandrov as well as all metric spaces with globally nonpositive curvature in the sense of Busemann. It also includes all Banach spaces.The analytic input comes from the domain space (M,) where we assume that we are given a Markov semigroup (pt)t>0. Typical examples come from elliptic or parabolic second-order operators on Rn, from Lévy type operators, from Laplacians on manifolds or on metric measure spaces and from convolution operators on groups. In contrast to the work of Korevaar and Schoen (1993, 1997), Jost (1994, 1997), Eells and Fuglede (2001) our semigroups are not required to be symmetric.The linear semigroup acting, e.g., on the space of bounded measurable functions u:MR gives rise to a nonlinear semigroup (Pt*)t acting on certain classes of measurable maps f:MN. We will show that contraction and smoothing properties of the linear semigroup (pt)t can be extended to the nonlinear semigroup (Pt*)t, for instance, LpLq smoothing, hypercontractivity, and exponentially fast convergence to equilibrium. Among others, we state existence and uniqueness of the solution to the Dirichlet problem for harmonic maps between metric spaces. Moreover, for this solution we prove Lipschitz continuity in the interior and Hölder continuity at the boundary.Our approach also yields a new interpretation of curvature assumptions which are usually required to deduce regularity results for the harmonic map flow: lower Ricci curvature bounds on the domain space are equivalent to estimates of the L1-Wasserstein distance between the distribution of two Brownian motions in terms of the distance of their starting points; nonpositive sectional curvature on the target space is equivalent to the fact that the L1-Wasserstein distance of two distributions always dominates the distance of their barycenters.Dedicated to the memory of Professor Dr. Heinz Bauer  相似文献
2.
Closed kernel systems of the coalition matrix turn out to correspond to cones of games on which the core correspondence is additive and on which the related barycentric solution is additive, stable and continuous. Different perfect cones corresponding to closed kernel systems are described. Received: December 2001/Revised: July 2002 RID="*" ID="*"  This note contains the new results, which were presented by the first author in an invited lecture at the XIV Italian Meeting on Game Theory and Applications in Ischia, July 2001. The lecture was dedicated to Irinel Dragan on the occasion of his seventieth birthday.  相似文献
3.
Let F be a mapping from a metric space into the family of all m-dimensional affine subsets of a Banach space X. We present a Helly-type criterion for the existence of a Lipschitz selection f of the set-valued mapping F, i.e., a Lipschitz continuous mapping satisfying . The proof of the main result is based on an inductive geometrical construction which reduces the problem to the existence of a Lipschitz (with respect to the Hausdorff distance) selector SX(m) defined on the family of all convex compacts in X of dimension at most m. If X is a Hilbert space, then the classical Steiner point of a convex body provides such a selector, but in the non-Hilbert case there is no known way of constructing such a point. We prove the existence of a Lipschitz continuous selector for an arbitrary Banach space X. The proof is based on a new result about Lipschitz properties of the center of mass of a convex set.  相似文献
4.
介绍了二次曲面的切线的几个性质 ,把二次曲线的类似性质推广到了二次曲面 .  相似文献
5.
The Besson–Courtois–Gallot theorem is proven for noncompact finite volume Riemannian manifolds. In particular, no bounded geometry assumptions are made. This proves the minimal entropy conjecture for nonuniform rank one lattices. This research was partially supported by an NSF Postdoctoral Fellowship. Received: June 2004; Revision: January 2006; Accepted: March 2006  相似文献
6.
We develop a theory of harmonic maps f:MN between singular spaces M and N. The target will be a complete metric space (N,d) of nonpositive curvature in the sense of A. D. Alexandrov. The domain will be a measurable space (M,) with a given Markov kernel p(x,dy) on it. Given a measurable map f:MN, we define a new map Pf:MN in the following way: for each xM, the point Pf(x)N is the barycenter of the probability measure p(x,f –1(dy)) on N. The map f is called harmonic on DM if Pf=f on D. Our theory is a nonlinear generalization of the theory of Markov kernels and Markov chains on M. It allows to construct harmonic maps by an explicit nonlinear Markov chain algorithm (which under suitable conditions converges exponentially fast). Many smoothing and contraction properties of the linear Markov operator P M,R carry over to the nonlinear Markov operator P=P M,N . For instance, if the underlying Markov kernel has the strong Lipschitz Feller property then all harmonic maps will be Lipschitz continuous.  相似文献
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We present a new method for finding positive solutions of nonlinear elliptic equations, which are non-homogeneous and asymptotically linear at infinity, by using projections on a Pohozaev manifold rather than the Nehari manifold associated with the problem.  相似文献
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Let S be a denumerable state space and let P be a transition probability matrix on S. If a denumerable set M of nonnegative matrices is such that the sum of the matrices is equal to P, then we call M a partition of P.  相似文献
9.
本文在研究多因素数据重心法的基础上,进一步提出滑动数据重心预测方法,该方法是对原始样本数据提出了一种新的数据处理方法,大大降低了由于历史数据组中的异常点对预测结果产生的破坏性。通过建立我国钢材消费量与国内生产总值(GDP)的计量动态模型对该方法与多因素数据重心预测法进行对比研究。同时利用时间序列自回归AR(p)对计量动态模型的初级预测结果进行差值校正,并将该方法应用于我国2015年、2020年的钢材消费量预测。对比研究表明该方法使得预测结果更加精确、稳健。  相似文献
10.
The article deals with generalizations of the inequalities for convex functions on the triangle. The Jensen and the Hermite-Hadamard inequality are included in the study. Considering a convex function on the triangle, we obtain a generalization of the Jensen-Mercer inequality, and a refinement of the Hermite-Hadamard inequality.  相似文献
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