Schroedinger Operators on Regular Metric Trees with Long Range Potentials: Weak Coupling Behavior

Consider a regular $d$-dimensional metric tree $\Gamma$ with root $o$. Define the Schroedinger operator $-\Delta - V$, where $V$ is a non-negative, symmetric potential, on $\Gamma$, with Neumann boundary conditions at $o$. Provided that $V$ decays like $x^{-\gamma}$ at infinity, where $1 < \gamma \leq d \leq 2, \gamma \neq 2$, we will determine the weak coupling behavior of the bottom of the spectrum of $-\Delta - V$. In other words, we will describe the asymptotical behavior of $\inf \sigma(-\Delta - \alpha V)$ as $\alpha \to 0+

Hardy inequalities for p-Laplacians with Robin boundary conditions

In this paper we study the best constant in a Hardy inequality for the p-Laplace operator on convex domains with Robin boundary conditions. We show, in particular, that the best constant equals $((p-1)/p)^p$ whenever Dirichlet boundary conditions are imposed on a subset of the boundary of non-zero measure. We also discuss some generalizations to non-convex domains

Weak perturbations of the p-Laplacian

We consider the p-Laplacian in R^d perturbed by a weakly coupled potential. We calculate the asymptotic expansions of the lowest eigenvalue of such an operator in the weak coupling limit separately for p>d and p=d and discuss the connection with Sobolev interpolation inequalities.Comment: 20 page

EFFECT OF STERILIZATION ON MECHANICAL PROPERTIES OF COLLAGEN-BASED COMPOSITE TUBES

In this study, composite tubes were manufactured from biological collagenous matrix and reinforcing polyester mesh. The effect of sterilization on mechanical properties of this structure was evaluated using inflation-extension tests. Samples were exposed to two types of sterilization (ethylene oxide and gamma irradiation). The control (non-sterilized) samples were also tested. The closed thick walled tube model was used in order to compute stresses within sterilized and control specimens. It was found that the process of sterilization (especially irradiation) dramatically affects the final mechanical properties of the material. These findings should be taken into account when such collagenous material is assumed to be used in tissue engineering

Stability of the magnetic Schr\"odinger operator in a waveguide

The spectrum of the Schr\"odinger operator in a quantum waveguide is known to be unstable in two and three dimensions. Any enlargement of the waveguide produces eigenvalues beneath the continuous spectrum. Also if the waveguide is bent eigenvalues will arise below the continuous spectrum. In this paper a magnetic field is added into the system. The spectrum of the magnetic Schr\"odinger operator is proved to be stable under small local deformations and also under small bending of the waveguide. The proof includes a magnetic Hardy-type inequality in the waveguide, which is interesting in its own

Eigenvalue estimates for Schrödinger operators on metric trees

We consider Schrödinger operators on radial metric trees and prove Lieb–Thirring and Cwikel–Lieb–Rozenblum inequalities for their negative eigenvalues. The validity of these inequalities depends on the volume growth of the tree. We show that the bounds are valid in the endpoint case and reflect the correct order in the weak or strong coupling limit

Opening angle of human saphenous vein

In this study, the residual strain was evaluated for human saphenous vein. Rings of the vein from four donors (two male and two female; age 62±5 years) were radially cut to obtain the opening angle (α) of the tissue. It was found that the average opening angle (α) 45°±18° (mean±SD). Then, the intraluminal distribution of circumferential stress was computed for one donor in order to verify the uniform stress hypothesis (opening angle is homogenizing the stress distribution across the wall thickness). The results suggests that α obtained from experiments is close to the value of opening angle which homogenizes the stress distribution across the wall thickness determined from simulations

Eigenvalue estimates for Schroedinger operators on metric trees

We consider Schroedinger operators on regular metric trees and prove Lieb-Thirring and Cwikel-Lieb-Rozenblum inequalities for their negative eigenvalues. The validity of these inequalities depends on the volume growth of the tree. We show that the bounds are valid in the endpoint case and reflect the correct order in the weak or strong coupling limit