Asymptotic behavior of a free boundary problem for the growth of multi-layer tumors in necrotic phase

In this paper we study a free boundary problem for the growth of multi-layer tumors in necrotic phase. The tumor region is strip-like and divided into necrotic region and proliferating region with two free boundaries. The upper free boundary is tumor surface and governed by a Stefan condition. The lower free boundary is the interface separating necrotic region from proliferating region, its evolution is implicit and intrinsically governed by an obstacle problem. We prove that the problem has a unique flat stationary solution, and there exists a positive constant $\gamma_*$, such that the flat stationary solution is asymptotically stable for cell-to-cell adhesiveness $\gamma>\gamma_*$, and unstable for $0<\gamma<\gamma_*$.Comment: 22 page

Non-disturbance criteria of quantum measurements

Using the general sequential product proposed by Shen and Wu in [J. Phys. A: Math. Theor. 42, 345203, 2009], we derive three criteria for describing non-disturbance between quantum measurements that may be unsharp with such new sequential products, which generalizes Gudder's results

Duistermaat-Heckman measure and the mixture of quantum states

In this paper, we present a general framework to solve a fundamental problem in Random Matrix Theory (RMT), i.e., the problem of describing the joint distribution of eigenvalues of the sum \bsA+\bsB of two independent random Hermitian matrices \bsA and \bsB. Some considerations about the mixture of quantum states are basically subsumed into the above mathematical problem. Instead, we focus on deriving the spectral density of the mixture of adjoint orbits of quantum states in terms of Duistermaat-Heckman measure, originated from the theory of symplectic geometry. Based on this method, we can obtain the spectral density of the mixture of independent random states. In particular, we obtain explicit formulas for the mixture of random qubits. We also find that, in the two-level quantum system, the average entropy of the equiprobable mixture of $n$ random density matrices chosen from a random state ensemble (specified in the text) increases with the number $n$. Hence, as a physical application, our results quantitatively explain that the quantum coherence of the mixture monotonously decreases statistically as the number of components $n$ in the mixture. Besides, our method may be used to investigate some statistical properties of a special subclass of unital qubit channels.Comment: 40 pages, 10 figures, LaTeX, the final version accepted for publication in J. Phys.