Using the canonical boundary reduction, suggested by Feng Kang, coupled with the finite element method, this paper gives the numerical solutions of the harmonic boundary-value problem over the domain with crack or concave angle. When the coupling is conforming, convergence and error estimates are obtained. This coupling removes the limitation of the canonical boundary reduction to some typical domains, and avoids the shortcoming of the classical finite element method, because of which the accuracy is damaged seriously and the approximate solution does not reflect the behaviour of the solution near the singularity. Numerical calculations have verified those conclusions.

In this paper the problem proposed by Kuhn on the presence of a monotonicity property related to the Kuhn's algorithm for finding roots of a polynomials is solved in the affirmative. Furthermore, an estimate of the threshold number D in the above-mentioned monotonicity problem expressed in terms of the complex coefficients of the polynomial is obtained.

In this paper we establish the Haar and Walsh systems on a triangle. These systems are complete in $L_2(\Delta)$. The uniform convergence of the Haar-Fourier series and the uniform convergence by group of the Walsh-Fourier series for any continuous function are proved.  

Let $A$ and $B$ be $m×n$ and $p×n$ complex matrices respectively. This paper, as a continuation of the author's papers [7], discusses perturbation bounds for the generalized singular values of the Matrix-pair{$A$,$B$} in the case of rank$\left( \begin{array}{c} A \\ B \\ \end{array} \right)$＜$n$.

Let $P,Q \subset L_1(X,\Sigma,\mu)$ and $q(x)>0$ a. e. in $X$ for all $q\in Q$. Define $R=\{p/q:p\in P,q\in Q\}$. In this paper we discuss an $L_1$ minimization problem of a nonnegative function $E(z,x)$, i.e. we wish to find a minimum of the functional $\phi(r)=\int _X qE(r,x)d\mu$ form $r=p/q\in R$. For such a problem we have established the complete characterizations of its minimum and of uniqueness of its minimum, when both $P,Q$ are arbitrary convex subsets.

The convergence rate of the boundary penalty finite element method is discussed for a model Poisson equation with inhomogeneous Dirichlet boundary conditions and a sufficiently smooth solution. It is proved that an optimal convergence rate can be achieved which agrees with the rate obtained recently in the numerical experiments by Utku and Carey.

A general theory for nonlinear implicit one-step schemes for solving initial value problems for ordinary differential equations is presented in this paper. The general expansion of "symmetric" implicit one-step schemes having second-order is derived and stability and convergence are studied. As examples, some geometric schemes are given.
Based on previous work of the first author on a generalization of means, a fourth-order nonlinear implicit one-step scheme is presented for solving equations with steep gradients. Also, a hybrid method based on the GMS and a fourth-order linear scheme is discussed. Some numerical results are given.  

A k-step, (k+2)th order two-stage implicit hybrid method which has all the advantages of Enright's method but not its principal disadvantages is proposed. A "simple" approach to estimate the local truncation error is developed. Preliminary numerical results indicate that the hybrid method compares favorably with Enright's method.