The Low-Temperature Expansion of the Casimir-Polder Free Energy of an Atom with Graphene

We consider the low-temperature expansion of the Casimir-Polder free energy for an atom and graphene by using the Poisson representation of the free energy. We extend our previous analysis on the different relations between chemical potential μ and mass gap parameter m. The key role plays the dependence of graphene conductivities on the μ and m. For simplicity, we made the manifest calculations for zero values of the Fermi velocity. For μ>m, the thermal correction ∼T2, and for μ<m, we confirm the recent result of Klimchitskaya and Mostepanenko, that the thermal correction ∼T5. In the case of exact equality μ=m, the correction ∼T. This point is unstable, and the system falls to the regime with μ>m or μ<m. The analytical calculations are illustrated by numerical evaluations for the Hydrogen atom/graphene system.

The Hopf Lemma for the Schrödinger Operator

We prove the Hopf boundary point lemma for solutions of the Dirichlet problem involving the Schrödinger operator {-\Delta+V} with a nonnegative potential V which merely belongs to {L_{\mathrm{loc}}^{1}(\Omega)}. More precisely, if {u\in W_{0}^{1,2}(\Omega)\cap L^{2}(\Omega;V\mathop{}\!\mathrm{d}{x})} satisfies {-\Delta u+Vu=f} on Ω for some nonnegative datum {f\in L^{\infty}(\Omega)}, {f\not\equiv 0}, then we show that at every point {a\in\partial\Omega} where the classical normal derivative {\frac{\partial u(a)}{\partial n}} exists and satisfies the Poisson representation formula, one has {\frac{\partial u(a)}{\partial n}>0} if and only if the boundary value problem\begin{dcases}\begin{aligned} \displaystyle-\Delta v+Vv&\displaystyle=0&&% \displaystyle\phantom{}\text{in ${\Omega}$,}\\ \displaystyle v&\displaystyle=\nu&&\displaystyle\phantom{}\text{on ${\partial% \Omega}$,}\end{aligned}\end{dcases}involving the Dirac measure {\nu=\delta_{a}} has a solution. More generally, we characterize the nonnegative finite Borel measures ν on {\partial\Omega} for which the boundary value problem above has a solution in terms of the set where the Hopf lemma fails.

Self-regulating genes. Exact steady state solution by using Poisson representation

Systems biology studies the structure and behavior of complex gene regulatory networks. One of its aims is to develop a quantitative understanding of the modular components that constitute such networks. The self-regulating gene is a type of auto regulatory genetic modules which appears in over 40% of known transcription factors in E. coli. In this work, using the technique of Poisson Representation, we are able to provide exact steady state solutions for this feedback model. By using the methods of synthetic biology (P.E.M. Purnick and Weiss, R., Nature Reviews, Molecular Cell Biology, 2009, 10: 410–422) one can build the system itself from modules like this.

Poisson Representation of a Ewens Fragmentation Process

A simple explicit construction is provided of a partition-valued fragmentation process whose distribution on partitions of [n] = 1,. . .,nat time θ ≥ 0 is governed by the Ewens sampling formula with parameter θ. These partition-valued processes are exchangeable and consistent, asnvaries. They can be derived by uniform sampling from a corresponding mass fragmentation process defined by cutting a unit interval at the points of a Poisson process with intensity θx−1dx on/mathbbR+, arranged to beintensifying as θ increases.