Advanced Search

The integration to steady state of many initial value ODEs and PDEs using the forward Euler method can alternatively be considered as gradient descent for an associated minimization problem. Greedy algorithms such as steepest descent for determining the step size are as slow to reach steady state as is forward Euler integration with the best uniform step size. But other, much faster methods using bolder step size selection exist. Various alternatives are investigated from both theoretical and practical...

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 -cube $Q_n^5$. Given two arbitrary distinct nodes and in $Q_n^5$, we prove that there exists an - path of every length ranging from to 5 - 1, where ≥ 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 to 5.