Consider Hardy’s inequalities with general weight ϕ for functions nonzero on the boundary. By an integral identity in C¹($\overline \Omega $$), define Hilbert spaces H_k ¹ (Ω, ϕ) called Sobolev-Hardy spaces with weight ϕ. As a corollary of this identity, Hardy’s inequalities with weight ϕ in C¹ ($\overline \Omega $$) 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 H¹ ₁ (Ω).

We prove the uniqueness of weak solutions of the time-dependent 3-D Ginzburg-Landau model for superconductivity with (Ψ₀, A₀) ∈ L²(Ω) initial data under the hypothesis that (Ψ, A) ∈ C([0, T]; L³(Ω)) using the Lorentz gauge.

In this paper, we estimate the supremum of Perelman’s λ-functional λ_M(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 λ_M(g₁) = λ_M(g₀), then $Vol_{g_1 } $$ (M) ⩾ $Vol_{g_0 } $$ (M). Moreover, the equality holds if and only if g₁ is also a Kähler-Einstein metric with negative scalar curvature. (ii) If {g_t}, t ∈ [−1, 1], is a family of Einstein metrics on M with initial metric g₀, then g_t is a Kähler-Einstein metric with negative scalar curvature.

In this paper, we construct the fractional generalized Lévy random fields (FGLRF) as tempered white noise functionals. We find that this white noise approach is very effective in investigating the properties of these fields. Under some conditions, the fractional Lévy fields in the usual sense are obtained. In addition, we also present a method to construct the anisotropic fractional generalized Lévy random fields (AFGLRF).

A new algorithm for solving a shifted skew-Hermitian linear system is presented and the corresponding convergence theorem is given. Some numerical examples are provided to demonstrate the algorithm.

In this paper, we consider the binomial tree method for pricing perpetual American and perpetual Bermudan options. The closed form solutions of these discrete models are solved. Explicit formulas for the optimal exercise boundary of the perpetual American option is obtained. A nonlinear equation that is satisfied by the optimal exercise boundaries of the perpetual Bermudan option is found.

Let F(x) be a distribution function supported on [0, ∞) with an equilibrium distribution function F_e(x). In this paper we pay special attention to the hazard rate function r_e(x) of F_e(x), which is also called the equilibrium hazard rate (E.H.R.) of F(x). By the asymptotic behavior of r_e(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 − θ, where Z and θ 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 r_e(x).

We characterize the symbol functions so that the associated commutators with symbol functions and the Hilbert transform are bounded on Lipschitz space Λ_α ^p, where 1 < p < ∞ and 0 < α < 1/p. Properties of such symbols are also discussed.

The group of Hopf algebra automorphisms for a finite-dimensional semisimple cosemisimple Hopf algebra over a field k was considered by Radford and Waterhouse. In this paper, the groups of Hopf algebra automorphisms for two classes of pointed Hopf algebras are determined. Note that the Hopf algebras we consider are not semisimple Hopf algebras.

In this paper, we assume that the surplus of an insurer follows a Lévy risk process and the insurer would invest its surplus in a risky asset, whose prices are modeled by a geometric Brownian motion. It is shown that the ruin probabilities (by a jump or by oscillation) of the resulting surplus process satisfy certain integro-differential equations.