An approach is presented that allows quantifying the three dimensional magnetization pattern of a magnetic nanoobject from measured two dimensional Magnetic Force Microscopy (MFM) data. This is based on a MFM deconvolution approach, which quantitatively determines the effective surface charges, on a micromagnetic calculation of the total magnetic charges at and below the sample surface, and on a projection of the lower lying charges onto the sample surface for a comparison of the such obtained effective surface charges with the experimentally determined ones. Thus, by making use of the depth sensitivity of MFM and by applying a quantitative contrast analysis, we are able to reconstruct the inhomogeneous magnetization state at the end of individual cylindrical Fe52Co48 nanowires arranged in a triangular array. As a result, we prove the existence of a magnetic vortex state at their ends.
Variograms and covariance functions are key tools in geostatistics. However, various properties, characterizations, and decomposition theorems have been established for covariance functions only. We present analogous results for variograms and explore the connections with covariance functions. Our findings include criteria for covariance functions on intervals, and we apply them to exponential models, fractional Brownian motion, and locally polynomial covariances. In particular, we characterize isotropic locally polynomial covariance functions of degree 3.
Most of the literature on spatio-temporal covariance models proposes structures that are positive in the whole domain. However, problems of physical, biological or medical nature need covariance models allowing for negative values or oscillations from positive to negative values. In this paper we propose an easy-to-implement and interpretable class of models that admits this type of covariances. We show particular analytical examples that may be of interest in the biometrical context.
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