Depth of powers of edge ideals of cycles and trees

Let $I$ be the edge ideal of a cycle of length $n \ge 5$ over a polynomial ring $S = \mathrm{k}[x_1,\ldots,x_n]$. We prove that for $2 \le t < \lceil (n+1)/2 \rceil$, $$\operatorname{depth} (S/I^t) = \lceil \frac{n -t + 1}{3} \rceil.$$ When $G = T_{\mathbf{a}}$ is a starlike tree which is the join of $k$ paths of length $a_1, \ldots, a_k$ at a common root $1$, we give a formula for the depth of powers of $I(T_{\mathbf{a}})$

Thermoresistance of p-Type 4H–SiC Integrated MEMS Devices for High-Temperature Sensing

There is an increasing demand for the development and integration of multifunctional sensing modules into power electronic devices that can operate in high temperature environments. Here, the authors demonstrate the tunable thermoresistance of p‐type 4H–SiC for a wide temperature range from the room temperature to above 800 K with integrated flow sensing functionality into a single power electronic chip. The electrical resistance of p‐type 4H–SiC is found to exponentially decrease with increasing temperature to a threshold temperature of 536 K. The temperature coefficient of resistance (TCR) shows a large and negative value from −2100 to −7600 ppm K−1, corresponding to a thermal index of 625 K. From the threshold temperature of 536–846 K, the electrical resistance shows excellent linearity with a positive TCR value of 900 ppm K−1. The authors successfully demonstrate the integration of p–4H–SiC flow sensing functionality with a high sensitivity of 1.035 μA(m s−1)−0.5 mW−1. These insights in the electrical transport of p–4H–SiC aid to improve the performance of p–4H–SiC integrated temperature and flow sensing systems, as well as the design consideration and integration of thermal sensors into 4H–SiC power electronic systems operating at high temperatures of up to 846 K

Edge of Infinity: The Clash between Edge Effect and Infinity Assumption for the Distribution of Charge on a Conducting Plate

We re-examine a familiar problem given in introductory physics courses, about determining the induced charge distribution on an uncharged ``infinitely-large'' conducting plate when placing parallel to it a uniform charged dielectric plate of the same size. We show that, no matter how large the plates are, the edge effect will always be strong enough to influence the charge distribution deep in the central region, which totally destroyed the infinity assumption (that the surface charge densities on the two sides are uniform and of opposite magnitudes). For a more detailed analysis, we solve Poisson's equation for a similar setting in two-dimensional space and obtain the exact charge distribution, helping us to understand what happens how charge distributes at the central, the asymptotic, and the edge regions

The Effect of Charge Discretization on the Electrical Field inside a Conductor

We show how the electrical field inside the conductor changes as a function of the number of charged-particles. We show that the non-vanishing electrical field is concentrated near the surface of the conductor, at a shallow depth on the same order of magnitude as the separation between charges. Our study has illustrated the effect of charge discretization on a fundamental emergent law of electrostatics

Micromixers-a review

Abstract This review reports the progress on the recent development of micromixers. The review first presents the different micromixer types and designs. Micromixers in this review are categorized as passive micromixers and active micromixers. Due to the simple fabrication technology and the easy implementation in a complex microfluidic system, passive micromixers will be the focus of this review. Next, the review discusses the operation points of the micromixers based on characteristic dimensionless numbers such as Reynolds number Re, Peclet number Pe, and in dynamic cases the Strouhal number St. The fabrication technologies for different mixer types are also analysed. Quantification techniques for evaluation of the performance of micromixers are discussed. Finally, the review addresses typical applications of micromixers