|
General Hardy’s inequalities for functions nonzero on the boundary
CHEN Zhihui, CHEN Zhihui, SHEN Yaotian, SHEN Yaotian
Front. Math. China. 2007, 2 (2): 169-181.
https://doi.org/10.1007/s11464-007-0012-7
Consider Hardy s inequalities with general weight φ for functions nonzero on the boundary. By an integral identity in C1( ),define Hilbert spaces H1k(Ω, φ) called Sobolev-Hardy spaces with weight φ. As a corollary of this identity, Hardy s inequalities with weight φ in C1( ) follow. At last, by Hardy s inequalities with weight φ = 1, discuss the eigenvalue problem of the Laplace-Hardy operator with critical parameter ( N - 2) 2/4 in H 11(Ω).
Related Articles |
Metrics
|
|
Perelman’s λ-functional and Seiberg-Witten equations
FANG Fuquan, ZHANG Yuguang
Front. Math. China. 2007, 2 (2): 191-210.
https://doi.org/10.1007/s11464-007-0014-5
In this paper, we estimate the supremum of Perelman s λ-functional λM(g) on Riemannian 4-manifold (M,g) by using the Seiberg-Witten equations. Among other things, we prove that, for a compact Kähler-Einstein complex surface (M,J, g0) with negative scalar curvature, (i) if g1 is a Riemannian metric on M with λM(g1) = λM(g0), then Volg1(M) "e Volg0 (M). Moreover, the equality holds if and only if g1 is also a Kãhler-Einstein metric with negative scalar curvature. (ii) If {gt}, t " [-1, 1], is a family of Einstein metrics on M with initial metric g0, then gt is a Kãhler-Einstein metric with negative scalar curvature.
Related Articles |
Metrics
|
|
A wide class of heavy-tailed distributions and its applications
SU Chun, HU Zhishui, CHEN Yu, LIANG Hanying
Front. Math. China. 2007, 2 (2): 257-286.
https://doi.org/10.1007/s11464-007-0018-1
Let F(x) be a distribution function supported on [0,∞) with an equilibrium distribution function Fe(x). In this paper we pay special attention to the hazard rate function re(x) of Fe(x), which is also called the equilibrium hazard rate (E.H.R.) of F(x). By the asymptotic behavior of re(x) we give a criterion to identify F(x) to be heavy-tailed or light-tailed. Moreover, we introduce two subclasses of heavy-tailed distributions, i.e., M and M*, where M contains almost all the most important heavy-tailed distributions in the literature. Some further discussions on the closure properties of M and M* under convolution are given, showing that both of them are ideal heavy-tailed subclasses. In the paper we also study the model of independent difference ξ = Z - ?, where Z and ? are two independent and non-negative random variables. We give intimate relationships of the tail distributions of ξ and Z, as well as relationships of tails of their corresponding equilibrium distributions. As applications, we apply the properties of class M to risk theory. In the final, some miscellaneous problems and examples are laid, showing the complexity of characterizations on heavy-tailed distributions by means of re(x).
Related Articles |
Metrics
|
10 articles
|