In the last two decades, the homotopy continuation method has been developed into a reliable and ecient numerical algorithm for solving all isolated zeros of polynomial systems. During the last few years, a major computational breakthrough has emerged in the area. Based on the Bernshtein theory on root count, the polyhedral homotopy is established to considerably reduce the number of homotopy paths that need to be traced to nd all the isolated roots, making the method much more powerful. This article reports the most recent development of this new method along with future considerations.

Many interconnection networks can be constructed with line digraph iterations. In this paper, we will establish a general theorem on diameter vulnerability based on the line digraph iteration which improves and generalizes several existing results in the literature.

For each product of random matrices, there associated an invariant measure on the projective space. The convergence to the invariant measure is exponentially fast. In this paper we give uniform estimates on the exponential convergence when the distribution of the random matrices depends on a compact set of parameters.

A Steinhaus matrix is a symmetric 0-1 matrix $[a_{i,j}]_{n\times n}$ such that $a_{i,i}=0$ for $0 \le i \le n-1$ and $a_{i,j}\equiv (a_{i-1,j-1}+a_{i-1,j})(\mbox{mod} 2)$ for $1 \le i \lt j \le n-1$. A Steinhaus graph is a graph whose adjacency matrix is a Steinhaus matrix. In this paper, we present a new characterization of bipartite Steinhaus graphs.

We use the parametric approach and exact penalty function to establish the Karush-Kuhn-Tucker type necessary and sucient conditions for an ɛ-optimum of nondierentiable fractional objective function subject to nondierentiable convex inequality constraint, linear equality constraints and abstract constraints. Subsequently, these optimality criteria are utilized as a basis for constructing one dual problem, and duality theorems are presented.

We provide convergence theorems for a fast iterative method to solve nonlinear operator equations in a Banach space. The same method under stronger conditions was found to be of order four, under standard Newton-Kantorovich type assumptions. The monotone convergence of this method in a partially ordered topological space is also examined here.

A nonconforming weak residual error estimator is presented and analyzed for nite element solutions of linear elliptic partial dierential equations. The treatment of the ux jumps across element edges is of special interest. The estimator is obtained by solving local residual problems which do not explicitly involve the jumps and do not require boundary conditions. The estimator handles both interior and edge residuals on each element by a suitable construction of the basis functions for the local problems. Together with the previous conforming estimator, the weak residual error estimation, without the ux jumps, can thus be applied to both odd- and even-order nite element approximations.

Let $\cal F$ be a left Gabriel topology on a ring $R$ and $\cal X$ be a special class of left $R$-modules (for example, the class of all quasi-continuous left $R$-modules in $\sigma[M]$, etc.). Suppose that all left $R$-modules in $\cal X$ are $\cal F$-injective. Then, it is proved in this paper that a left $R$-module $M$ is $\cal F$-co-semisimple (that is, every $\cal F$-cocritical left $R$-module $C$ in $\sigma[M]$ is dense in its $M$-injective hull) if and only if every $\cal F$-torsionfree $\cal F$-finitely cogenerated left $R$-module $N$ in $\sigma[M]$ is dense in its some essential extensions which are in $\cal X$. As a corollary we show that a left $R$-module $M$ is co-semisimple if and only if every finitely cogenerated left $R$-module in $\sigma [M]$ is continuous (or quasi-continuous, or direct-injective, etc.)

In this paper we provide an algorithm for computing Betti numbers and ranks of syzygy of manageable modules with respect to an R-regular sequence of a Noetherian local ring R when certain conditions are met. This method can be applied to many interesting cases and used to verify Horrocks\' conjecture. The problem of calculating Betti numbers is reduced to calculating ranks of certain matrices with a special form over a eld. Our approach is characteristic free.