Strong Feller properties for distorted Brownian motion with reflecting boundary condition and an application to continuous N-particle systems with singular interactions

Assuming that Ω⊂Rn, n?2, is an open, relatively compact set with boundary ∂Ω of Lebesgue measure zero we prove strong Feller properties for a class of distorted Brownian motions in with reflecting boundary condition. Dirichlet form techniques give the existence of a weak solution to the corresponding stochastic differential equation for quasi all starting points in the sense of the associated martingale problem. Combining this result with the strong Feller properties we can construct a weak solution for specified starting points. If Ω has C²-boundary the construction works for all starting points, where the drift term is not singular, even on the boundary. But also for a certain class of sets with less smooth boundary our approach works for all points in Ω, where the drift term is not singular, and at least some points from ∂Ω. Our techniques allow very singular drift terms. This enables us to construct continuous N-particle gradient stochastic dynamics in cuboids Λ⊂Rd, d∈N, with reflecting boundary condition and singular interactions for dN?2. We can start the stochastic dynamics in all initial configurations having at most one particle in ∂Λ, provided ∂Λ is locally smooth there.     

Immunoassay as an analytical tool in agricultural biotechnology

Immunoassays for biotechnology engineered proteins are used by AgBiotech companies at numerous points in product development and by feed and food suppliers for compliance and contractual purposes. Although AgBiotech companies use the technology during product development and seed production, other stakeholders from the food and feed supply chains, such as commodity, food, and feed companies, as well as third-party diagnostic testing companies, also rely on immunoassays for a number of purposes. The primary use of immunoassays is to verify the presence or absence of genetically modified (GM) material in a product or to quantify the amount of GM material present in a product. This article describes the fundamental elements of GM analysis using immunoassays and especially its application to the testing of grains. The 2 most commonly used formats are lateral flow devices (LFD) and plate-based enzyme-linked immunosorbent assays (ELISA). The main applications of both formats are discussed in general, and the benefits and drawbacks are discussed in detail. The document highlights the many areas to which attention must be paid in order to produce reliable test results. These include sample preparation, method validation, choice of appropriate reference materials, and biological and instrumental sources of error. The article also discusses issues related to the analysis of different matrixes and the effects they may have on the accuracy of the immunoassays.     

Fusenes and benzenoids with perfect matchings

In this paper, a method to constructively enumerate fusenes and benzenoids with perfect matchings is given. It is based on an algorithm for generating all fusenes and benzenoids and introduces new restrictions into the generation process that avoid the generation of structures without perfect matchings.      

Scaling limit of fluctuations for the equilibrium Glauber dynamics in continuum

The Glauber dynamics investigated in this paper are spatial birth and death processes in a continuous system having a grand canonical Gibbs measure of Ruelle type as an invariant measure. We prove that such processes, when appropriately scaled, have as scaling limit a generalized Ornstein-Uhlenbeck process. First we prove convergence of the corresponding Dirichlet forms. This convergence requires only very weak assumptions. The interaction potential ? only has to be stable (S), integrable (I), and we have to assume the low activity high temperature regime. Under a slightly stronger integrability condition (I_∞) and a conjecture on the Percus-Yevick equation we even can prove strong convergence of the corresponding generators. Finally, we prove that the scaled processes converge in law. Here the hardest part is to show tightness of the scaled processes (note that the processes only have càdlàg sample path). For the proof we have to assume that the interaction potential is positive (P). The limiting process then is identified via the associated martingale problem. For this the above mentioned strong convergence of generators is essential.     

Hypocoercivity for degenerate Kolmogorov equations and applications to the Langevin dynamics

We present an extension of the powerful Hilbert space hypocoercivity method that was developed originally by Dolbeault, Mouhot and Schmeiser. We focus attention on including important domain issues that have not been considered before. The setting can be used to provide a complete elaboration of the hypocoercivity theorem for the degenerate Langevin dynamics. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Analysis of Stochastic Fiber Lay-Down Models: Geometry and Convergence to Equilibrium of the Basic Model

The so-called fiber lay-down models arise in the production process of nonwovens. We introduce the generalized version of the basic fiber lay-down model which can precisely be formulated in abstract form as some manifold-valued stochastic differential equation. An important criterion for the quality of the nonwoven material is how the solution to the associated Fokker-Planck equation converges towards its stationary state. Especially, one is interested in determining the speed of convergence. Here we present some results concerning the long-time behavior by using classical stochastic methods as well as modern analytic methods from the theory of hypocoercivity. Demanding mathematical difficulties arising since the equation is degenerate. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Universal structural phase transition in network glasses

Raman and Mössbauer spectroscopy provide evidence for a transition from a molecular cluster network at x=0 to a continuous network at x=0.35 in Ge_1?xSn_xSe₂ alloy glasses. The nature of this morphological transition involves a reformation of molecular cluster surfaces in the heterogeneous phase to yield a homogeneous phase. The transition is believed to be a universal property of the easy glass formers and can be effected in one of several ways.     

Construction of N-Particle Langevin Dynamics for <Emphasis Type="Italic">H</Emphasis><Superscript>1,∞</Superscript>-Potentials via Generalized Dirichlet Forms

We construct an N-particle Langevin dynamics on a cuboid region in with periodic boundary condition, i.e., a diffusion process solving the Langevin equation with periodic boundary condition in the sense of the corresponding martingale problem. Our approach works for general H ^1,∞ potentials allowing N-particle interactions and external forces. Of course, the corresponding forces are not necessarily continuous. Since the generator of the dynamics is non-sectorial, for the construction we use the theory of generalized Dirichlet forms.      

The Outer Oblique Boundary Problem of Potential Theory

In this article, we prove existence and uniqueness results for solutions to the outer oblique boundary problem for the Poisson equation under very weak assumptions on boundary, coefficients, and inhomogeneities. The main tools are the Kelvin transformation and the solution operator for the regular inner problem, provided in Grothaus and Raskop (Stochastics 2006; 78(4):233–257). Moreover we prove regularization results for the weak solutions of both the inner and the outer problem. We investigate the nonadmissible direction for the oblique vector field, state results with stochastic inhomogeneities, and provide a Ritz–Galerkin approximation. The results are applicable to problems from Geomathematics (see, e.g., Freeden and Maier (Electronic Trans. Numer. Anal. 2002; 14:40–62 and Bauer, An Alternative Approach to the Oblique Derivative Problem in Potential Theory, Shaker Verlag, Aachen, 2004)).