Homogenization of a parabolic Dirichlet problem by a method of Dahlberg

Consider the linear parabolic operator in divergence form $$\mathcal{H} u =\partial_t u(X,t)-\text{div}(A(X)\nabla u(X,t)).$$ We employ a method of Dahlberg to show that the Dirichlet problem for $\mathcal{H}$ in the upper half plane is well-posed for boundary data in $L^p$, for any elliptic matrix of coefficients $A$ which is periodic and satisfies a Dini-type condition. This result allows us to treat a homogenization problem for the equation $\partial_t u_\varepsilon(X,t)-\text{div}(A(X/\varepsilon)\nabla u_\varepsilon(X,t))$ in Lipschitz domains with $L^p$-boundary data.Comment: 21 page

Noncommutative associative superproduct for general supersymplectic forms

We define a noncommutative and nonanticommutative associative product for general supersymplectic forms, allowing the explicit treatment of non(anti)commutative field theories from general nonconstant string backgrounds like a graviphoton field. We propose a generalization of deformation quantization a la Fedosov to superspace, which considers noncommutativity in the tangent bundle instead of base space, by defining the Weyl super product of elements of Weyl super algebra bundles. Super Poincare symmetry is not broken and chirality seems not to be compromised in our formulation. We show that, for a particular case, the projection of the Weyl super product to the base space gives rise the Moyal product for non(anti)commutative theories.Comment: 22 pages, revtex4. References added. Comments added. Includes additional theorem proof

Effect of graphene substrate on the SERS Spectra of Aromatic bifunctional molecules on metal nanoparticles

The design of molecular sensors plays a very important role within nanotechnology and especially in the development of different devices for biomedical applications. Biosensors can be classified according to various criteria such as the type of interaction established between the recognition element and the analyte or the type of signal detection from the analyte (transduction). When Raman spectroscopy is used as an optical transduction technique the variations in the Raman signal due to the physical or chemical interaction between the analyte and the recognition element has to be detected. Therefore any significant improvement in the amplification of the optical sensor signal represents a breakthrough in the design of molecular sensors. In this sense, Surface-Enhanced Raman Spectroscopy (SERS) involves an enormous enhancement of the Raman signal from a molecule in the vicinity of a metal surface. The main objective of this work is to evaluate the effect of a monolayer of graphene oxide (GO) on the distribution of metal nanoparticles (NPs) and on the global SERS enhancement of paminothiophenol (pATP) and 4-mercaptobenzoic acid (4MBA) adsorbed on this substrate. These aromatic bifunctional molecules are able to interact to metal NPs and also they offer the possibility to link with biomolecules. Additionally by decorating Au or Ag NPs on graphene sheets, a coupled EM effect caused by the aggregation of the NPs and strong electronic interactions between Au or Ag NPs and the graphene sheets are considered to be responsible for the significantly enhanced Raman signal of the analytes [1-2]. Since there are increasing needs for methods to conduct reproducible and sensitive Raman measurements, Grapheneenhanced Raman Scattering (GERS) is emerging as an important method [3].Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tech

Symplectic connections, Noncommutative Yang Mills theory and Supermembranes

In built noncommutativity of supermembranes with central charges in eleven dimensions is disclosed. This result is used to construct an action for a noncommutative supermembrane where interesting topological terms appear. In order to do so, we first set up a global formulation for noncommutative Yang Mills theory over general symplectic manifolds. We make the above constructions following a pure geometrical procedure using the concept of connections over Weyl algebra bundles on symplectic manifolds. The relation between noncommutative and ordinary supermembrane actions is discussed.Comment: 18 page

A Fractal Interaction Model for Winding Paths through Complex Distributions: Application to Soil Drainage Networks

Water interacts with soil through pore channels putting mineral constituents and pollutants into solution. The irregularity of pore boundaries and the heterogeneity of distribution of soil minerals and contaminants are, among others, two factors influencing that interaction and, consequently, the leaching of chemicals and the dispersion of solute throughout the soil. This paper deals with the interaction of irregular winding dragging paths through soil complex distributions. A mathematical modelling of the interplay between multifractal distributions of mineral/pollutants in soil and fractal pore networks is presented. A Ho¨lder path is used as a model of soil pore network and a multifractal measure as a model of soil complex distribution, obtaining a mathematical result which shows that the Ho¨lder exponent of the path and the entropy dimension of the distribution may be used to quantify such interplay. Practical interpretation and potential applications of the above result in the context of soil are discussed. Since estimates of the value of both parameters can be obtained from field and laboratory data, hopefully this mathematical modelling might prove useful in the study of solute dispersion processes in soil

Homogenization of a parabolic Dirichlet problem by a method of Dahlberg

Consider the linear parabolic operator in divergence form Hu := ∂tu(X, t) − div(A(X)∇u(X, t)). We employ a method of Dahlberg to show that the Dirichlet problem for H in the upper half plane is well-posed for boundary data in Lp, for any elliptic matrix of coefficients A which is periodic and satisfies a Dini-type condition. This result allows us to treat a homogenization problem for the equation ∂tuε(X, t) − div(A(X/ε)∇uε(X, t)) in Lipschitz domains with Lp-boundary data.Consider the linear parabolic operator in divergence form Hu := ∂tu(X, t) − div(A(X)∇u(X, t)). We employ a method of Dahlberg to show that the Dirichlet problem for H in the upper half plane is well-posed for boundary data in Lp, for any elliptic matrix of coefficients A which is periodic and satisfies a Dini-type condition. This result allows us to treat a homogenization problem for the equation ∂tuε(X, t) − div(A(X/ε)∇uε(X, t)) in Lipschitz domains with Lp-boundary data