A special structure of the two level orthogonal array of order 12 is given. By using this structure and the method of constructing mixed-level orthogonal arrays presented by Zhang, Lu and Pang (1999), a lot of mixed-level orthogonal arrays of run size 108 are obtained. Especially, we obtain an orthogonal array of size 108 with 11 6-level columns.

We use an asymptotic expansion to study the behavior of American-style interest rate caplets and floorlets close to expiry, under the assumption that interest rates obey a mean-reverting random walk as given by the Vasicek model. Series solutions are obtained for the location of the free boundary and the price of the option for both the caplet and floorlet.

This paper deals with a slight improvement on the results of the 1-D semilinear Schrödinger equations with quadratic nonlinearities. We study the local well-posedness of the initial value problem in particular function spaces containing the Sobolev spaces $H^s$ with for the nonlinearity $u\overline{u}$, and with for $u^2$ or ${\overline{u}}^2$, in which the local well-posedness was proved by Kenig, Ponce and Vega. Our improvement lies in the estimate of the Fourier restriction norm with a homogeneous weight ${\left|\xi \right|}^s$. It makes the behavior of the initial data at $\xi =0$ in the phase space less restrictive.

We study the global existence and large time asymptotic behavior of solutions to the initial-boundary value problem for the nonlinear nonlocal Schrödinger equation on a segment $\left(0,a\right)$

(0.1) $$\{\begin{array}{c}{u}_t+i|u{|}^{2}u+\mathbb{K}u=0,t>0,x\in (0,a)\\ u(x,0)=u_0(x),x\in (0,a),\end{array}$$

where the pseudodifferential operator $\mathbb{K}$ has the dissipation propery and the symbol of order $\alpha \in (0,1)$. We prove that if the initial data $u_0\in {\mathbf{L}}^{\infty }$ are small, then there exists a unique solution $u\in \mathbf{C}([0,\infty );{\mathbf{L}}^{\infty })$ of the initial-boundary value problem (0.1) Moreover there exists a function $A\in {\mathbf{L}}^{\infty }$ such that the solution has the following large time asymptotics

$$u(x,t)=A(x)t^{-\frac{1}{\alpha }}\text{Λ}\left(\frac{x}{t^{\frac{1}{\alpha }}}\right)+O\left(t^{-\frac{1+\delta }{\alpha }}\right),$$

where $\text{Λ}(x)=\frac{1}{2\text{πi}}{\displaystyle {\int }_{-i\infty }^{i\infty }{e}^{-z^{\alpha }+z\text{x}}dz}$.