Voronoi diagrams for a set of geometric objects is a partition of the plane (or space in higher dimensions) into disjoint regions each dominated by some given object under a predetermined criterion. In this paper we are interested in various measures associated with criteria on goodness of an input line segment with respect to each point in the plane as the ``point of view.'' These measures basically show how well a segment or information displayed on the segment can be seen from the point. Mathematically, the measures are defined in terms of the shapes of the triangle determined by the point and the line segment. We study the combinatorial and algorithmic complexities of those Voronoi diagrams. We also study an associated optimization problem: find a point that maximizes the smallest measure value over the measures with respect to all the given line segments. We give sufficient conditions for an optimal point to lie on a Voronoi edge and present a heuristic optimization algorithm for those measures having this property.

A method for numerical solution of Volterra integral equations of the second kind with a weakly singular kernel based on the double exponential (DE) transformation is proposed. In this method we first express the approximate solution in the form of a Sinc expansion based on the double exponential transformation by Takahasi and Mori in 1974 followed by collocation at the Sinc points. We also apply the DE formula to the kernel integration. In every sample equation a numerical solution with very high accuracy is obtained and a nearly exponential convergence rate $\exp(-cM/{\log M})$, $c>0$ in the error is observed where $M$ is a parameter representing the number of terms in the Sinc expansion. We compared the result with the one based on the single exponential (SE) transformation by Riley in 1992 which made us confirm the high efficiency of the present method.

For least squares problems of minimizing $\|\boldsymbol{b}-A\boldsymbol{x}\|_2$ where $A$ is a large sparse $m\times n$ ($m\ge n$) matrix, the common method is to apply the conjugate gradient method to the normal equation $A^{\mathrm{T}} A\boldsymbol{x}=A^{\mathrm{T}} \boldsymbol{b}$. However, the condition number of $A^{\mathrm{T}} A$ is square of that of $A$, and convergence becomes problematic for severely ill-conditioned problems even with preconditioning. In this paper, we propose two methods for applying the GMRES method to the least squares problem by using an $n\times m$ matrix $B$. We give the necessary and sufficient condition that $B$ should satisfy in order that the proposed methods give a least squares solution. Then, for implementations for $B$, we propose an incomplete QR decomposition IMGS($l$). Numerical experiments showed that the simplest case $l=0$ gives the best results, and converges faster than previous methods for severely ill-conditioned problems. The preconditioner IMGS(0) is equivalent to the case $B=(\diag (A^{\mathrm{T}} A))^{-1} A^{\mathrm{T}}$, so $(\diag (A^{\mathrm{T}} A))^{-1} A^{\mathrm{T}}$ was the best preconditioner among IMGS($l$) and Jennings' IMGS($\tau$). On the other hand, CG-IMGS(0) was the fastest for well-conditioned problems.

This paper is concerned with theoretical error estimates for a sampling formula with the sinc-Gaussian kernel. Qian et al. have recently given an error estimate for the class of band-limited functions by Fourier-analytic approach. In contrast, we adopt in this paper a complex-analytic approach to derive an error estimate for a wider class of functions including unbounded functions on $\mathbf{R}$. Part of the result of Qian et al. can be derived from ours as an immediate corollary. Computational results show a fairly good agreement with our theoretical analysis.

The behavior of solution polygons to generalized crystalline curvature flow is discussed. The conditions to guarantee that the solution polygon keeps its admissibility as long as enclosed area of solution polygon is positive are clarified. We also show that the solution polygon becomes ``almost convex'' before the extinction time.