On annealed elliptic Green function estimates

We consider a random, uniformly elliptic coefficient field $a$ on the lattice $\mathbb{Z}^d$. The distribution $\langle \cdot \rangle$ of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green function $G(t,x,y)$ satisfy optimal annealed estimates which are $L^2$ resp. $L^1$ in probability, i.e. they obtained bounds on $\langle |\nabla_x G(t,x,y)|^2 \rangle^{\frac{1}{2}}$ and $\langle |\nabla_x \nabla_y G(t,x,y)| \rangle$, see T. Delmotte and J.-D. Deuschel: On estimating the derivatives of symmetric diffusions in stationary random environments, with applications to the $\nabla\phi$ interface model, Probab. Theory Relat. Fields 133 (2005), 358--390. In particular, the elliptic Green function $G(x,y)$ satisfies optimal annealed bounds. In a recent work, the authors extended these elliptic bounds to higher moments, i.e. $L^p$ in probability for all $p<\infty$, see D. Marahrens and F. Otto: {Annealed estimates on the Green function}, arXiv:1304.4408 (2013). In this note, we present a new argument that relies purely on elliptic theory to derive the elliptic estimates (see Proposition 1.2 below) for $\langle |\nabla_x G(x,y)|^2 \rangle^{\frac{1}{2}}$ and $\langle |\nabla_x \nabla_y G(x,y)| \rangle$.Comment: 15 page

Global attractor for a Ginzburg-Landau type model of rotating Bose-Einstein condensates

We study the long time behavior of solutions to a nonlinear partial differential equation arising in the description of trapped rotating Bose-Einstein condensates. The equation can be seen as a hybrid between the well-known nonlinear Schr\"odinger/Gross-Pitaevskii equation and the Ginzburg-Landau equation. We prove existence and uniqueness of global in-time solutions in the physical energy space and establish the existence of a global attractor within the associated dynamics. We also obtain basic structural properties of the attractor and an estimate on its Hausdorff and fractal dimensions.Comment: 25 pages; some more typos fixed; additional references adde

Moment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations

We consider the corrector equation from the stochastic homogenization of uniformly elliptic finite-difference equations with random, possibly non-symmetric coefficients. Under the assumption that the coefficients are stationary and ergodic in the quantitative form of a Logarithmic Sobolev inequality (LSI), we obtain optimal bounds on the corrector and its gradient in dimensions $d \geq 2$. Similar estimates have recently been obtained in the special case of diagonal coefficients making extensive use of the maximum principle and scalar techniques. Our new method only invokes arguments that are also available for elliptic systems and does not use the maximum principle. In particular, our proof relies on the LSI to quantify ergodicity and on regularity estimates on the derivative of the discrete Green's function in weighted spaces.Comment: added applications, e.g. two-scale expansion, variance estimate of RV

Tin isotope analysis of tin ore deposits in Europe and Central Asia in view of the tin provenance in archaeological metal objects

The aim of this work is to establish an overview of the tin isotope ratios of cassiterite and stannite from various mineralized regions in Europe, the Mediterranean and Central Asia in order to assess the possibility to geochemically discriminate tin ore deposits, which could have been exploited in ancient times. The motivation for this study was to eventually relate the tin found in ancient bronze objects to specific tin ore deposits and thus to clarify the origin and distribution of the tin bronze technology in the Bronze Age of the so-called Old World. For this purpose, we determined 413 primary and secondary cassiterite and stannite samples from the major tin provinces in SW England and Ireland, the Saxonian-Bohemian province, the Iberian Peninsula, France, Italy, Serbia, Egypt and Central Asia. The tin isotope compositions were analysed in solution with a multi-collector inductive-coupled plasma mass spectrometer (MC-ICP-MS) in the Curt-Engelhorn-Zentrum Archäometrie in Mannheim. The samples mainly derive from granitic pegmatites and hydrothermal vein mineralizations of tin ore deposits associated with granite complexes in the Variscan and Asian fold belts. Overall, the isotope ratios in primary and secondary cassiterites are highly variable and range from δ124Sn/120Sn -0.82 to 0.85 ‰. This variation is observed in the tin ore samples from SW England which have an average δ124Sn/120Sn of 0.10 ± 0.59 ‰ (2SD). Among the tin provinces of the Variscan fold belt in Europe those of SW England and the Saxonian-Bohemian province (δ124Sn/120Sn = 0.12 ‰ ± 0.37) show the largest variations but the ranges of isotope ratios in both regions overlap to a large extent. Despite the large overlap, cassiterite from Spain (δ124Sn/120Sn = -0.07 ‰ ± 0.35) and France (δ124Sn/120Sn = -0.005 ‰ ± 0.31) tend to have on average lighter isotopic compositions than SW England, the Saxonian-Bohemian province or Portugal (δ124Sn/120Sn = 0.07 ‰ ± 0.40). However, the stannite samples from SW England and the Saxonian-Bohemian province have significantly lighter isotope ratios than the associated cassiterites. The tin ores from Central Asia exhibit the largest total variation of 1.94 ‰ ranging from -1.27 to 0.67 ‰ for δ124Sn/120Sn. This extent of fractionation is observed in cassiterites from Afghanistan and Uzbekistan. Afghanistan with its pegmatitic cassiterite has the lightest isotopic composition of all investigated areas with -0.38 ± 0.84 ‰ for δ124Sn/120Sn and, therefore, stands out as an identifiable source. Similar to the European stannites, the Asian stannites also have significantly lighter isotope compositions. Because of the large overlap and the highly variable isotope composition of cassiterites from all tin provinces a clearcut discrimination based on tin isotope ratios is difficult. But on a more detailed scale within each tin province it is possible to distinguish several mining or granite areas by their Sn isotope composition. However, it is also difficult to distinguish between different mineralization types

Optimal bilinear control of Gross-Pitaevskii equations

A mathematical framework for optimal bilinear control of nonlinear Schr\"odinger equations of Gross-Pitaevskii type arising in the description of Bose-Einstein condensates is presented. The obtained results generalize earlier efforts found in the literature in several aspects. In particular, the cost induced by the physical work load over the control process is taken into account rather then often used $L^2$- or $H^1$-norms for the cost of the control action. Well-posedness of the problem and existence of an optimal control is proven. In addition, the first order optimality system is rigorously derived. Also a numerical solution method is proposed, which is based on a Newton type iteration, and used to solve several coherent quantum control problems.Comment: 30 pages, 14 figure

Difficulties with the genetic improvement of feed efficiency in broiler lines using group cage information

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A hydrodynamic limit for chemotaxis in a given heterogeneous environment

In this paper, the first equation within a class of well-known chemotaxis systems is derived as a hydrodynamic limit from a stochastic interacting many particle system on the lattice. The cells are assumed to interact with attractive chemical molecules on a finite number of lattice sites, but they only directly interact among themselves on the same lattice site. The chemical environment is assumed to be stationary with a slowly varying mean, which results in a non-trivial macroscopic chemotaxis equation for the cells. Methodologically, the limiting procedure and its proofs are based on results by Koukkous (Stoch. Process. Appl. 84, 297–312, cite.​Kou99) and Kipnis and Landim (Scaling limits of interacting particle systems, cite.​KL99). Numerical simulations extend and illustrate the theoretical findings