On the index of depth stability of symbolic powers of cover ideals of graphs

Let $G$ be a graph with $n$ vertices and let $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over a field $\mathbb{K}$. Assume that $I(G)$ and $J(G)$ denote the edge ideal and the cover ideal of $G$, respectively. We provide a combinatorial upper bound for the index of depth stability of symbolic powers of $J(G)$. As a consequence, we compute the depth of symbolic powers of cover ideals of fully clique-whiskered graphs. Meanwhile, we determine a class of graphs $G$ with the property that the Castelnuovo--Mumford regularity of $S/I(G)$ is equal to the induced matching number of $G$.Comment: arXiv admin note: substantial text overlap with arXiv:1709.0388

The Effect of Moment of Inertia on the Liquids in Centrifugal Microfluidics

The flow of liquids in centrifugal microfluidics is unidirectional and dominated by centrifugal and Coriolis forces (i.e., effective only at T-junctions). Developing mechanisms and discovering efficient techniques to propel liquids in any direction other than the direction of the centrifugal force has been the subject of a large number of studies. The capillary force attained by specific surface treatments, pneumatic energy, active and passive flow reciprocation and Euler force have been previously introduced in order to manipulate the liquid flow and push it against the centrifugal force. Here, as a new method, the moment of inertia of the liquid inside a chamber in a centrifugal microfluidic platform is employed to manipulate the flow and propel the liquid passively towards the disc center. Furthermore, the effect of the moment of inertia on the liquid in a rectangular chamber is evaluated, both in theory and experiments, and the optimum geometry is defined. As an application of the introduced method, the moment of inertia of the liquid is used in order to mix two different dyed deionized (DI) waters; the mixing efficiency is evaluated and compared to similar mixing techniques. The results show the potential of the presented method for pumping liquids radially inward with relatively high flow rates (up to 23 mm3/s) and also efficient mixing in centrifugal microfluidic platforms