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91.
92.
Marco Grandis 《Applied Categorical Structures》1997,5(1):75-97
We study here a notion of simplicial satellites, as a first step towards a characterisation of simplicial derived functors, a problem unsolved since the latter were introduced.The problem comes from the fact that, in contrast with the abelian case, simplicial derived functors do not produce by themselves an exact sequence. Our solution consists in extending them to commutative k-cubes, for all k, forming thus an exact system of functors universal within the connected ones; or, in other words, a system of simplicial satellites. The tool we develop here for this extension is the homotopy kernel of a commutative k-dimensional cubic diagram, generalising the homotopy kernel of a map; its 2-dimensional version has already been proved essential in other homotopical topics. 相似文献
93.
Saburou Saitoh 《Proceedings of the American Mathematical Society》1997,125(12):3633-3639
Let be an arbitrary map from an abstract set into an abstract set . We shall use the theory of reproducing kernels to provide a general method for representing the inverse map in terms of . We give several concrete examples of this method.
94.
Learning function relations or understanding structures of data lying in manifolds embedded in huge dimensional Euclidean
spaces is an important topic in learning theory. In this paper we study the approximation and learning by Gaussians of functions
defined on a d-dimensional connected compact C
∞ Riemannian submanifold of which is isometrically embedded. We show that the convolution with the Gaussian kernel with variance σ provides the uniform approximation order of O(σ
s
) when the approximated function is Lipschitz s ∈(0, 1]. The uniform normal neighborhoods of a compact Riemannian manifold play a central role in deriving the approximation
order. This approximation result is used to investigate the regression learning algorithm generated by the multi-kernel least
square regularization scheme associated with Gaussian kernels with flexible variances. When the regression function is Lipschitz
s, our learning rate is (log2
m)/m)
s/(8 s + 4 d) where m is the sample size. When the manifold dimension d is smaller than the dimension n of the underlying Euclidean space, this rate is much faster compared with those in the literature. By comparing approximation
orders, we also show the essential difference between approximation schemes with flexible variances and those with a single
variance.
Supported partially by the Research Grants Council of Hong Kong [Project No. CityU 103405], City University of Hong Kong [Project
No. 7001983], National Science Fund for Distinguished Young Scholars of China [Project No. 10529101], and National Basic Research
Program of China [Project No. 973-2006CB303102]. 相似文献
95.
Xiaomei Wu 《分析论及其应用》2008,24(2):139-148
Let→b=(b1,b2,…,bm),bi∈∧βi(Rn),1≤I≤m,βi>0,m∑I=1βi=β,0<β<1,μΩ→b(f)(x)=(∫∞0|F→b,t(f)(x)|2dt/t3)1/2,F→b,t(f)(x)=∫|x-y|≤t Ω(x,x-y)/|x-y|n-1 mΠi=1[bi(x)-bi(y)dy.We consider the boundedness of μΩ,→b on Hardy type space Hp→b(Rn). 相似文献
96.
For any , a truncated symmetric α-stable process is a symmetric Lévy process in with a Lévy density given by for some constant c. In this paper we study the potential theory of truncated symmetric stable processes in detail. We prove a Harnack inequality
for nonnegative harmonic functions of these processes. We also establish a boundary Harnack principle for nonnegative functions
which are harmonic with respect to these processes in bounded convex domains. We give an example of a non-convex domain for
which the boundary Harnack principle fails.
The research of Panki Kim is supported by Research Settlement Fund for the new faculty of Seoul National University. The research
of Renming Song is supported in part by a joint US-Croatia grant INT 0302167. 相似文献
97.
Nicolas Th. Varopoulos 《Milan Journal of Mathematics》2007,75(1):1-60
In this paper I consider a class of non-standard singular integrals motivated by potential theoretic and probabilistic considerations.
The probabilistic applications, which are by far the most interesting part of this circle of ideas, are only outlined in Section
1.5: They give the best approximation of the solution of the classical Dirichlet problem in a Lipschitz domain by the corresponding solution by finite differences.
The potential theoretic estimate needed for this gives rise to a natural duality between the L
p
functions on the boundary ∂Ω and a class of functions A on Ω that was first considered by Dahlberg. The actual duality is given by ∫Ω
S f(x)A(x)dx = (f, A) where S f(x) = ∫∂Ω |x − y|1−n
f(y)dy is the Newtonian potential.
We can identify the upper half Lipschitz space with in the obvious way and express for an appropriate kernel K. It is the boundedness properties of the above (for , ) that is the essential part of this work. This relates with more classical (but still “rough”) singular integrals that have
been considered by Christ and Journé.
Lecture held in the Seminario Matematico e Fisico on March 14, 2005
Received: April 2007 相似文献
98.
Zhixiang Chen 《分析论及其应用》2007,23(4):325-333
The spherical approximation between two nested reproducing kernels Hilbert spaces generated from different smooth kernels is investigated. It is shown that the functions of a space can be approximated by that of the subspace with better smoothness. Furthermore, the upper bound of approximation error is given. 相似文献
99.
Hongwei Sun 《Frontiers of Mathematics in China》2007,2(3):455-465
Let H be a Hilbert space, A ∈ L(H), y ∈
, and y ∉ R(A). We study the behavior of the distance square between y and A(B
τ), defined as a functional F(τ), as the radius τ of the ball B
τ of H tends to ∞. This problem is important in estimating the approximation error in learning theory. Our main result is to estimate
the asymptotic behavior of F(τ) without the compactness assumption on the operator A. We also consider the Peetre K-functional and its convergence rates.
相似文献
100.
考虑了再生核希尔伯特空间连续线性泛函范数的表示,得到了用其范数平方等于该线性泛函连续两次作于再生核的简明表示.对于常见的Sobolev-Hibert空间而言,其再生核则可用截幂函数来表示,从而得到Sobolev-Hibert空间上连续线性泛函范数的简洁表示,以新视角解释和简化了文献中的现有结果. 相似文献