On quantum field theory in gravitational background

We discuss quantum fields on Riemannian space-time. A principle of local definiteness is introduced which is needed beyond equations of motion and commutation relations to fix the theory uniquely. It also allows us to formulate local stability. In application to a region with a time-like Killing vector field and horizons it yields the value of the Hawking temperature. The concept of vacuum and particles in a non-stationary metric is treated in the example of the Robertson-Walker metric and some remarks on detectors in non-inertial motion are added.     

Comments on Mielnik's generalized (non linear) quantum mechanics

We discuss a model of non linear quantum mechanics in which the wave equation satisfies the homogeneity condition (2.1). It is argued that in this model the set of (mixed) states is a simplex.Dedicated to Professor Günther Ludwig on the occasion of his sixtieth birthday     

Exact solutions of discrete master equations in terms of continued fractions

We present the continued fraction solution for the stationary probability of discrete master equations of one-variable processes. After we elucidate the method for simple birth and death processes we focus the study on processes which introduce at least two-particle jumps. Consequently, these processes do in general not obey a detailed balance condition. The outlined method applies as well to solutions of eigenmodes of the stochastic operator. Further we derive explicit continued fraction solutions for the Laplace transform of conditional probabilities. All the various continued fraction coefficients are given directly in terms of the transition rates and they obey recursion relations. The method is illustrated for the stationary solution of a simple nonlinear chemical reaction scheme originated by Nicolis.     

Qualitative properties of modified equations

Suppose that a consistent one-step numerical method of orderr is applied to a smooth system of ordinary differential equations.Given any integer m 1, the method may be shown to be of orderr + m as an approximation to a certain modified equation. Ifthe method and the system have a particular qualitative propertythen it is important to determine whether the modified equationsinherit this property. In this article, a technique is introducedfor proving that the modified equations inherit qualitativeproperties from the method and the underlying system. The techniqueuses a straightforward contradiction argument applicable toarbitrary one-step methods and does not rely on the detailedstructure of associated power series expansions. Hence the conclusionsapply, but are not restricted, to the case of Runge-Kutte methods.The new approach unifies and extends results of this type thathave been derived by other means: results are presented forintegral preservation, reversibility, inheritance of fixed points.Hamiltonian problems and volume preservation. The techniquealso applies when the system has an integral that the methodpreserves not exactly, but to order greater than r. Finally,a negative result is obtained by considering a gradient systemand gradient numerical method possessing a global property thatis not shared by the associated modified equations.     

Mimicking of Chondrocyte Microenvironment Using In Situ Forming Dendritic Polyglycerol Sulfate‐Based Synthetic Polyanionic Hydrogels

A stable polymeric network that mimics the highly polyanionic extracellular cartilage matrix still remains a great challenge. The main aim of this study is to present the synthesis of dendritic polyglycerol sulfate (dPGS)‐based in situ forming hydrogels using strain promoted azide‐alkyne cycloaddition reactions. A real time rheological study has been used to characterize the hydrogel properties. The viability of encapsulated human chondrocytes in the different hydrogels are monitored using live‐dead staining. Furthermore, type I and II collagen gene have been analyzed. Hydrogels with elastic moduli ranging from 1 to 5 kPa have been prepared by varying the dPGS amount. The chondrocyte viability in dPGS hydrogels is found to be higher than in pure PEG and alginate‐based hydrogels after 21 d. The higher cell viability in the dPGS engineered hydrogels can be explained by the fact that dPGS can interact with different proteins responsible for cell growth and proliferation.