Threshold singularities of the spectral shift function for geometric perturbations of magnetic Hamiltonians

We consider the 3D Schr\"odinger operator $H_0$ with constant magnetic field $B$ of scalar intensity $b>0$, and its perturbations $H_+$ (resp., $H_-$) obtained by imposing Dirichlet (resp., Neumann) conditions on the boundary of the bounded domain $\Omega_{\rm in} \subset {\mathbb R}^3$. We introduce the Krein spectral shift functions $\xi(E;H_\pm,H_0)$, $E \geq 0$, for the operator pairs $(H_\pm,H_0)$, and study their singularities at the Landau levels $\Lambda_q : = b(2q+1)$, $q \in {\mathbb Z}_+$, which play the role of thresholds in the spectrum of $H_0$. We show that $\xi(E;H_+,H_0)$ remains bounded as $E \uparrow \Lambda_q$, $q \in {\mathbb Z}_+$ being fixed, and obtain three asymptotic terms of $\xi(E;H_-,H_0)$ as $E \uparrow \Lambda_q$, and of $\xi(E;H_\pm,H_0)$ as $E \downarrow \Lambda_q$. The first two terms are independent of the perturbation while the third one involves the {\em logarithmic capacity} of the projection of $\Omega_{\rm in}$ onto the plane perpendicular to $B$.Comment: 35 page

Advances on Testing C-Planarity of Embedded Flat Clustered Graphs

We show a polynomial-time algorithm for testing c-planarity of embedded flat clustered graphs with at most two vertices per cluster on each face.Comment: Accepted at GD '1

Modeling of the behavior of AC undervoltage relays during voltage dips

This paper proposes a dynamic model for the behavior of an AC undervoltage protection relay during voltage dips. The behavior is expressed by means of standardized voltage tolerance curves and highly depends on the point on-wave (phase angle) of the dip initiation. Using the model, it can be explained that AC undervoltage relays are much more sensitive to voltage dips of e.g. 50% Urated than to short interruptions (0% Urated), caused by the magnetic behavior. The model takes into account the electrical, magnetic and mechanical system equations. An experimental set-up with a programmable power source as dipgenerator is build to validate the model. Finally, a complete weaving machine is represented to analyze the influence towards voltage dip immunity by adding an undervoltage protection relay