"On the engineers' new toolbox" or Analog Circuit Design, using Symbolic Analysis, Computer Algebra, and Elementary Network Transformations

In this paper, by way of three examples - a fourth order low pass active RC filter, a rudimentary BJT amplifier, and an LC ladder - we show, how the algebraic capabilities of modern computer algebra systems can, or in the last example, might be brought to use in the task of designing analog circuits.Comment: V1: documentclass IEEEtran, 7 pages, 6 figures. Re-release of the printed version, with some minor typographical errors correcte

Time-dependent shortest paths in bounded treewidth graphs

We present a proof that the number of breakpoints in the arrival function between two terminals in graphs of treewidth $w$ is $n^{O(\log^2 w)}$ when the edge arrival functions are piecewise linear. This is an improvement on the bound of $n^{\Theta(\log n)}$ by Foschini, Hershberger, and Suri for graphs without any bound on treewidth. We provide an algorithm for calculating this arrival function using star-mesh transformations, a generalization of the wye-delta-wye transformations

Electrical Reduction, Homotopy Moves, and Defect

We prove the first nontrivial worst-case lower bounds for two closely related problems. First, $\Omega(n^{3/2})$ degree-1 reductions, series-parallel reductions, and $\Delta$Y transformations are required in the worst case to reduce an $n$-vertex plane graph to a single vertex or edge. The lower bound is achieved by any planar graph with treewidth $\Theta(\sqrt{n})$. Second, $\Omega(n^{3/2})$ homotopy moves are required in the worst case to reduce a closed curve in the plane with $n$ self-intersection points to a simple closed curve. For both problems, the best upper bound known is $O(n^2)$, and the only lower bound previously known was the trivial $\Omega(n)$. The first lower bound follows from the second using medial graph techniques ultimately due to Steinitz, together with more recent arguments of Noble and Welsh [J. Graph Theory 2000]. The lower bound on homotopy moves follows from an observation by Haiyashi et al. [J. Knot Theory Ramif. 2012] that the standard projections of certain torus knots have large defect, a topological invariant of generic closed curves introduced by Aicardi and Arnold. Finally, we prove that every closed curve in the plane with $n$ crossings has defect $O(n^{3/2})$, which implies that better lower bounds for our algorithmic problems will require different techniques.Comment: 27 pages, 15 figure