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## Laplace transform of the survival probability under Sparre Andersen model

### Applied Mathematics-A Journal of Chinese Universities (2007-03-01) 22: 109-118 , March 01, 2007

In this paper a class of risk processes in which claims occur as a renewal process is studied. A clear expression for Laplace transform of the survival probability is well given when the claim amount distribution is Erlang distribution or mixed Erlang distribution. The expressions for moments of the time to ruin with the model above are given.

## Existence of three-solutions for second-order differential equations with nonlinear boundary value conditions

### Applied Mathematics-A Journal of Chinese Universities (2002-06-01) 17: 135-144 , June 01, 2002

The paper deals with the existence of three-solutions for the second-order differential equations with nonlinear boundary value conditions
$$\begin{gathered} x'' = f(t,x,x'), t \in \left[ {a,b} \right], \hfill \\ g_1 (x(a),x'(a)) = 0, g_2 (x(b), x'(b)) = 0, \hfill \\ \end{gathered} $$
where *f*:[*a,b*] × *R*^{1} × *R*^{1} → *R*^{1}, *g*_{i}:*R*^{1} × *R*^{1} → *R*^{1} (*i*=1,2) are continuous functions. The methods employed are the coincidence degree theory. As an application, the sufficient conditions under which there are arbitrary odd solutions for the BVP are obtained.

## Some remarks on Wente’s inequality and the Lorentz-Sobolev space

### Applied Mathematics-A Journal of Chinese Universities (2016-09-01) 31: 355-361 , September 01, 2016

In this note we consider Wente’s type inequality on the Lorentz-Sobolev space. If
$$\nabla f \in {L^{{p_1},{q_1}}}{\left( R \right)^n},\;G \in {L^{{p_2},{q_2}}}{\left( R \right)^n}$$
and div *G* ≡ 0 in the sense of distribution where
$$\frac{1}
{{p_1 }} + \frac{1}
{{p_2 }} = \frac{1}
{{q_1 }} + \frac{1}
{{q_2 }} = 1,1 < p_1 ,p_2 < \infty ,$$
it is known that *G* · ∇*f* belongs to the Hardy space H^{1} and furthermore
$${\left\| {G \cdot \nabla f} \right\|_{{H^1}}} \leqslant C{\left\| {\nabla f} \right\|_{{L^{{p_1},{q_1}}}\left( {{R^2}} \right)}}{\left\| G \right\|_{{L^{{p_2},{q_2}}}\left( {{R^2}} \right)}}$$
. Reader can see [9] Section 4.

Here we give a new proof of this result. Our proof depends on an estimate of a maximal operator on the Lorentz space which is of some independent interest. Finally, we use this inequality to get a generalisation of Bethuel’s inequality.

## Diagrams of Hopf algebras with the Chevalley property

### Applied Mathematics-A Journal of Chinese Universities (2016-09-01) 31: 367-378 , September 01, 2016

In this paper, we study non-cosemisimple Hopf algebras through their underlying coalgebra structure. We introduce the concept of the maximal pointed subcoalgebra/Hopf subalgebra. For a non-cosemisimple Hopf algebra *A* with the Chevalley property, if its diagram is a Nichols algebra, then the diagram of its maximal pointed Hopf subalgebra is also a Nichols algebra. When *A* is of finite dimension, we provide a necessary and sufficient condition for *A*’s diagram equaling the diagram of its maximal pointed Hopf subalgebra.

## Convergence rates in the strong laws for a class of dependent random fields

### Applied Mathematics-A Journal of Chinese Universities (2003-06-01) 18: 209-213 , June 01, 2003

By using a Rosenthal type inequality established in this paper, the complete convergence rates in the strong laws for a class of dependent random fields are discussed. And the result obtained extends those for *ρ*^{−}-mixing random fields, *ρ**-mixing random fields and negatively associated fields.

## L p estimates for the Schrödinger type operators

### Applied Mathematics-A Journal of Chinese Universities (2011-12-01) 26: 412-424 , December 01, 2011

Let *L*_{k} = (−Δ)^{k} + *V*^{k} be a Schrödinger type operator, where *k* ≥ 1 is a positive integer and *V* is a nonnegative polynomial. We obtain the *L*^{p} estimates for the operators ∇^{2k}*L*_{k}^{−1}
and ∇^{k}*L*_{k}^{−1/2}
.

## An inexact lagrange-newton method for stochastic quadratic programs with recourse

### Applied Mathematics-A Journal of Chinese Universities (2004-06-01) 19: 229-238 , June 01, 2004

In this paper, two-stage stochastic quadratic programming problems with equality constraints are considered. By Monte Carlo simulation-based approximations of the objective function and its first (second) derivative, an inexact Lagrange-Newton type method is proposed. It is showed that this method is globally convergent with probability one. In particular, the convergence is local superlinear under an integral approximation error bound condition. Moreover, this method can be easily extended to solve stochastic quadratic programming problems with inequality constraints.

## A generalized projection-successive linear equations algorithm for nonlinearly equality and inequality constrained optimization and its rate of convergence

### Applied Mathematics-A Journal of Chinese Universities (1997-09-01) 12: 343-354 , September 01, 1997

In this paper, a new superlinearly convergent algorithm is presented for optimization problems with general nonlinear equality and inequality constraints. Comparing with other methods for these problems, the algorithm has two main advantages. First, it doesn ’ t solve any quadratic programming (QP), and its search directions are determined by the generalized projection technique and the solutions of two systems of linear equations. Second, the sequential points generated by the algorithm satisfy all inequality constraints and its step-length is computed by the straight line search. The algorithm is proved to possess global and superlinear convergence.

## Tests of covariance matrix by using projection pursuit and bootstrap method

### Applied Mathematics-A Journal of Chinese Universities (1998-09-01) 13: 309-322 , September 01, 1998

Testing equality of covariance matrix has long been an interesting issue in statistics inference. To overcome the sparseness of data points in high-dimensional space and deal with the general cases, the author suggests several projection pursuit type statistics. Some results on the limiting distributions of the statistics are obtained. Some properties of bootstrap approximation are investigated. Furthermore, for computational reasons an approximation for the statistics based on number-theoretic method is applied. Several simulation experiments are performed.

## Estimating of order-restricted location parameters of two-exponential distributions under multiple type-II censoring

### Applied Mathematics-A Journal of Chinese Universities (1999-03-01) 14: 51-56 , March 01, 1999

In this article, Bayes estimation of location parameters under restriction is brought forth. Since Bayes estimator is closely connected with the first value of order statistics that can be observed, it is possible to consider “complete data” method, through which the pseudo-value of first order statistics and pseudo-right censored samples can be obtained. Thus the results under Type- II right censoring can be used directly to get more accurate estimators by Bayes method.