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

A lower bound on the fidelity between two states in terms of their Bures distance

Fidelity is a fundamental and ubiquitous concept in quantum information theory. Fuchs-van de Graaf's inequalities deal with bounding fidelity from above and below. In this paper, we give a lower bound on the quantum fidelity between two states in terms of their Bures distance.Comment: 5 pages, LaTeX, we have corrected some errors appearing in the original manuscript. We have partially fixed the gap in the proof of the previous versions. A new method towards the conjectured inequality in the present version is being expecte

Asymptotic Behavior of Solutions of a Free Boundary Problem Modelling the Growth of Tumors with Stokes Equations

We study a free boundary problem modelling the growth of non-necrotic tumors with fluid-like tissues. The fluid velocity satisfies Stokes equations with a source determined by the proliferation rate of tumor cells which depends on the concentration of nutrients, subject to a boundary condition with stress tensor effected by surface tension. It is easy to prove that this problem has a unique radially symmetric stationary solution. By using a functional approach, we prove that there exists a threshold value $\gamma_*>0$ for the surface tension coefficient $\gamma$, such that in the case $\gamma>\gamma_*$ this radially symmetric stationary solution is asymptotically stable under small non-radial perturbations, whereas in the opposite case it is unstable

Remarks on the Sequential Products

In this paper, we show that those sequential products which were proposed by Liu and Shen and Wu in [J. Phys. A: Math. Theor. {\bf 42}, 185206 (2009), J. Phys. A: Math. Theor. {\bf 42}, 345203 (2009)] are just unitary equivalent to the sequential product $A\circ B=A^{\frac{1}{2}}BA^{\frac{1}{2}}$

A generalized family of discrete PT-symmetric square wells

N-site-lattice Hamiltonians H are introduced and perceived as a set of systematic discrete approximants of a certain PT-symmetric square-well-potential model with the real spectrum and with a non-Hermiticity which is localized near the boundaries of the interval. Its strength is controlled by one, two or three parameters. The problem of the explicit construction of a nontrivial metric which makes the theory unitary is then addressed. It is proposed and demonstrated that due to the not too complicated tridiagonal-matrix form of our input Hamiltonians the computation of the metric is straightforward and that its matrix elements prove obtainable, non-numerically, in elementary polynomial forms.Comment: 21 pages, 2 figure

Asymptotic Stability of Stationary Solutions of a Free Boundary Problem Modeling the Growth of Tumors with Fluid Tissues

This paper aims at proving asymptotic stability of the radial stationary solution of a free boundary problem modeling the growth of nonnecrotic tumors with fluid-like tissues. In a previous paper we considered the case where the nutrient concentration $\sigma$ satisfies the stationary diffusion equation $\Delta\sigma=f(\sigma)$, and proved that there exists a threshold value $\gamma_*>0$ for the surface tension coefficient $\gamma$, such that the radial stationary solution is asymptotically stable in case $\gamma>\gamma_*$, while unstable in case $\gamma<\gamma_*$. In this paper we extend this result to the case where $\sigma$ satisfies the non-stationary diffusion equation \epsln\partial_t\sigma=\Delta\sigma-f(\sigma). We prove that for the same threshold value $\gamma_*$ as above, for every $\gamma>\gamma_*$ there is a corresponding constant \epsln_0(\gamma)>0 such that for any 0<\epsln<\epsln_0(\gamma) the radial stationary solution is asymptotically stable with respect to small enough non-radial perturbations, while for $0<\gamma<\gamma_*$ and \epsln sufficiently small it is unstable under non-radial perturbations

A Survey of Dynamical Matrices Theory

In this note, we survey some elementary theorems and proofs concerning dynamical matrices theory. Some mathematical concepts and results involved in quantum information theory are reviewed. A little new result on the matrix representation of quantum operation are obtained. And best separable approximation for quantum operations is presented.Comment: 22 pages, LaTe

Samuel multiplicities and Browder Spectrum of Operator Matrices

we show that the definitions of some classes of semi-Fredholm operators, which use the language of algebra and first introduced by X. Fang in [8], are equivalent to that of some well-known operator classes. For example, the concept of shift-like semi-Fredholm operator on Hilbert space coincide with that of upper semi-Browder operator. For applications of Samuel multiplicities we characterize the sets of $\bigcap_{C\in B(K,\,H)}\sigma_{ab}(M_{C}),\bigcap_{C\in B(K,\,H)}\sigma_{sb}(M_{C})$ and $\bigcap_{C\in B(K,\,H)}\sigma_{b}(M_{C}),$ respectively, where M_{C}=({array}{cc}A&C 0&B {array}) denotes a 2-by-2 upper triangular operator matrix acting on the Hilbert space $H\oplus K$.Comment: 1

Unified (r,s)-relative entropy

In this paper, we introduce and study unified $(r,s)$-relative entropy and quantum unified $(r,s)$-relative entropy, in particular, our main results of quantum unified $(r,s)$-relative entropy are established on the separable complex Hilbert spaces. Moreover, the entanglement-measure of states due to the quantum unified $(r,s)$-relative entropy is considered, too. Our results improved a uncorrect statement on the monotone property of entanglement-measure function

A lower bound of quantum conditional mutual information

In this paper, a lower bound of quantum conditional mutual information is obtained by employing the Peierls-Bogoliubov inequality and Golden Thompson inequality. Comparison with the bounds obtained by other researchers indicates that our result is independent of any measurements. It may give some new insights over squashed entanglement and perturbations of Markov chain states.Comment: 11 pages, LaTeX, published version. The missed second author is also adde