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Plasmon dispersion in Al atk≦k F using electron energy loss spectroscopy

We find a quasi quadratic dispersion of the plasmon up to _p = 28 eV atk=1.77 Å^–1 with no levelling off contrary to other measurements. We usedk-scans at constant energy loss andE-scans at constant wave vector, which yield different dispersions of the plasmon peak maximum because of the finite halfwidth. The levelling off in plasmon dispersion observed previously is due to multiple scattering events, which obscure the plasmon dispersion inE-scans.We confirmed the anisotropy of the dispersion coefficient by investigating polycrystalline samples of different orientation.     

Surface enhanced Raman scattering (SERS): Annealing the silver substrate

Silver films, deposited at 120 K in UHV, were slowly warmed to room temperature, Characteristic intensity variations of the inelastic background (I), of disorder induced Raman scattering from phonons (II) and of the SERS signal from adsorbed pyridine have been observed. In the case of (I) and (II) these are explained by the bulk concentration of point defects, in the SERS case by annealing of atomic scale surface roughness.     

A higher-order large-scale regularity theory for random elliptic operators

We develop a large-scale regularity theory of higher order for divergence-form elliptic equations with heterogeneous coefficient fields a in the context of stochastic homogenization. The large-scale regularity of a-harmonic functions is encoded by Liouville principles: The space of a-harmonic functions that grow at most like a polynomial of degree k has the same dimension as in the constant-coefficient case. This result can be seen as the qualitative side of a large-scale C^k,α-regularity theory, which in the present work is developed in the form of a corresponding C^k,α-“excess decay” estimate: For a given a-harmonic function u on a ball B_R, its energy distance on some ball B_r to the above space of a-harmonic functions that grow at most like a polynomial of degree k has the natural decay in the radius r above some minimal radius r₀.

Though motivated by stochastic homogenization, the contribution of this paper is of purely deterministic nature: We work under the assumption that for the given realization a of the coefficient field, the couple (φ, σ) of scalar and vector potentials of the harmonic coordinates, where φ is the usual corrector, grows sublinearly in a mildly quantified way. We then construct “kth-order correctors” and thereby the space of a-harmonic functions that grow at most like a polynomial of degree k, establish the above excess decay, and then the corresponding Liouville principle.     

On the existence of $$\kappa $$-existentially closed groups

Using explicit constructions of the Weierstrass mock modular form and Eisenstein series coefficients, we obtain closed formulas for the generating functions of values of shifted convolution L-functions associated to certain elliptic curves. These identities provide a surprising relation between weight 2 newforms and shifted convolution L-values when the underlying elliptic curve has modular degree 1 with conductor N such that \(\text {genus}(X_0(N)) = 1\).     

On robust cross-validation for nonparametric smoothing

An essential problem in nonparametric smoothing of noisy data is a proper choice of the bandwidth or window width, which depends on a smoothing parameter $k$ . One way to choose $k$ based on the data is leave-one-out-cross-validation. The selection of the cross-validation criterion is similarly important as the choice of the smoother. Especially, when outliers are present, robust cross-validation criteria are needed. So far little is known about the behaviour of robust cross-validated smoothers in the presence of discontinuities in the regression function. We combine different smoothing procedures based on local constant fits with each of several cross-validation criteria. These combinations are compared in a simulation study under a broad variety of data situations with outliers and abrupt jumps. There is not a single overall best cross-validation criterion, but we find Boente-cross-validation to perform well in case of large percentages of outliers and the Tukey-criterion in case of data situations with jumps, even if the data are contaminated with outliers.