The laplacian operator applied to the coordinates of a manifold provides the mean curvature vector. Manipulating the metric of the manifold or interpreting its coordinates in various ways provide useful tools for shape and image processing and representation. We will review some of these tools focusing on scale invariant geometry, curvature flow with respect to an embedding of the image manifold in a high dimensional space, and object segmentation by active contours defined via the shape laplacian...

The cutwidth is an important graph-invariant in circuit layout designs. The cutwidth of a graph G is the minimum value of the maximum number of overlap edges when G is embedded into a line. A caterpillar is a tree which yields a path when all its leaves are removed. An iterated caterpillar is a tree which yields a caterpillar when all its leaves are removed. In this paper we present an exact formula for the cutwidth of the iterated caterpillars.

We prove that the cutwidth of the r-dimensional mesh of d-ary trees is of order $\Theta \left(d^{(r-1)n+1}\right)$, which improves and generalizes previous results.

We study two topological properties of the 5-ary $n$-cube $Q_n^5$. Given two arbitrary distinct nodes $x$ and $y$ in $Q_n^5$, we prove that there exists an $x$-$y$ path of every length ranging from $2n$ to $5^n-1$, where $n\ge 2$. Based on this result, we prove that $Q_n^5$ is 5-edge-pancyclic by showing that every edge in $Q_n^5$ lies on a cycle of every length ranging from $5$ to $5^n$.

We study two topological properties of the 5-ary n-cube $Q_n^5$. Given two arbitrary distinct nodes x and y in $Q_n^5$, we prove that there exists an x-y path of every length ranging from 2n to 5n - 1, where n ≥ 2. Based on this result, we prove that $Q_n^5$ is 5-edge-pancyclic by showing that every edge in $Q_n^5$ lies on a cycle of every length ranging from 5 to 5n.

The Special Issue of Kybernetika is devoted to the publication of selected peer-reviewed articles submitted by the participants of the Czech-Japanese Seminar in Applied Mathematics 2008 which took place on September 1-7, 2008 in Takachi-ho and Miyazaki, Japan. The Czech-Japanese Seminar in Applied Mathematics 2008 was organized by the Department of Applied Physics, Faculty of Engineering, University of Miyazaki. It was the fourth meeting in the series of the Czech-Japanese Seminars in Applied Mathematics....