von Neumann’s trace inequality for Hilbert–Schmidt operators

von Neumann’s inequality in matrix theory refers to the fact that the Frobenius scalar product of two matrices is less than or equal to the scalar product of the respective singular values. Moreover, equality can only happen if the two matrices share a joint set of singular vectors, and this latter part is hard to find in the literature. We extend these facts to the separable Hilbert space setting, and provide a self-contained proof of the “latter part”.     

介观电路中量子纠缠的经典对应

从量子力学诞生日起,它的经典对应(或类比)一直是物理学家关心的话题.本文以介观电路量子化的框架中,带有互感的两个介观电容-电感(LC)电路为例,首次讨论了量子纠缠的经典类比(或对应)问题.先用有序算符内的积分理论证明其互感是产生量子纠缠的源头;再推导出求解特征频率的公式,就发现它与一个经典系统的小振动频率的表达式有相似之处,该经典系统组成如下:两个墙壁各连一个相同的弹簧,两个弹簧之间接着一个滑动小车可以在光滑的桌面上运动,小车挂有一根单摆.用分析力学求此系统的小振动频率,发现与上述介观电路的特征频率形式类似,单摆的摆动会造成小车来回振动,摆、小车和弹簧的互相牵制效应反映了小车和摆的"纠缠".     

Quantifier-free epistemic term-modal logic with assignment operator

In standard epistemic logic, the names and the existence of agents are usually assumed to be common knowledge implicitly. This is unreasonable for various applications in computer science and philosophy. Inspired by term-modal logic and assignment operators in dynamic logic, we introduce a lightweight modal predicate logic where names can be non-rigid, and the existence of agents can be uncertain. The language can handle various de dicto/de re distinctions in a natural way. We characterize the expressive power of our language, obtain complete axiomatisations of the logics over several classes of varying-domain/constant-domain epistemic models, and show their (un)decidability.     

Renormalization group second‐order approximation for singularly perturbed nonlinear ordinary differential equations

We consider a 2 time scale nonlinear system of ordinary differential equations. The small parameter of the system is the ratio ϵ of the time scales. We search for an approximation involving only the slow time unknowns and valid uniformly for all times at order O(ϵ²). A classical approach to study these problems is Tikhonov's singular perturbation theorem. We develop an approach leading to a higher order approximation using the renormalization group (RG) method. We apply it in 2 steps. In the first step, we show that the RG method allows for approximation of the fast time variables by their RG expansion taken at the slow time unknowns. Next, we study the slow time equations, where the fast time unknowns are replaced by their RG expansion. This allows to rigorously show the second order uniform error estimate. Our result is a higher order extension of Hoppensteadt's work on the Tikhonov singular perturbation theorem for infinite times. The proposed procedure is suitable for problems from applications, and it is computationally less demanding than the classical Vasil'eva‐O'Malley expansion. We apply the developed method to a mathematical model of stem cell dynamics.