A generalisation of the fractional Brownian field based on non-Euclidean norms

We explore a generalisation of the L´evy fractional Brownian field on the Euclidean space based on replacing the Euclidean norm with another norm. A characterisation result for admissible norms yields a complete description of all self-similar Gaussian random fields with stationary increments. Several integral representations of the introduced random fields are derived. In a similar vein, several non-Euclidean variants of the fractional Poisson field are introduced and it is shown that they share the covariance structure with the fractional Brownian field and converge to it. The shape parameters of the Poisson and Brownian variants are related by convex geometry transforms, namely the radial pth mean body and the polar projection transforms.

Newton-Cartan supergravity with torsion and Schrödinger supergravity

We derive a torsionfull version of three-dimensional N=2 Newton-Cartan supergravity using a non-relativistic notion of the superconformal tensor calculus. The “superconformal” theory that we start with is Schrödinger supergravity which we obtain by gauging the Schrödinger superalgebra. We present two non-relativistic N=2 matter multiplets that can be used as compensators in the superconformal calculus. They lead to two different off-shell formulations which, in analogy with the relativistic case, we call “old minimal” and “new minimal” Newton-Cartan supergravity. We find similarities but also point out some differences with respect to the relativistic case.

The Gronwall form of the Schrodinger Equation for Helium

The 1937 paper of Gronwall which concerns an alternative form for the Schrodinger Equation of the 2-electron Helium problem is re-derived in a (hopefully) transparent (possibly pedestrian) manner.

Frobenius Solution for Legendre's Equation, Rodrigue's Formula and Normalization

The Frobenius solution to Legendre/s equation is developed in detail as is Rodrigue's formula, which is employed to normalize Legendre polynomials.

Continued Fraction Solutions to Hermite's, Legendre's and Laguerre's Differential Equation

The continued fraction method for solving differential equations is illustrated using three famous differential equations used in quantum chemistry.

The Wave Equation for Stringed Instruments

An elementary derivation of the wave equation as applied to violin strings is given.

Guessing Solutions to the H-atom Schrodinger Equation

In teaching elementary quantum chemistry, the concept of eigenfunctionality is explored using the H-atom's Hamiltonian and various guessed functions. This is done in Cartesian coordinates, in Spherical Polar coordinates, and in Confocal Elliptical coordinates.

Interim analysis of clinical trials: Simulation studies of fractional Brownian motion

Interim clinical trial monitoring procedures were motivated by ethical and economic considerations. Classical Brownian motion (Bm) techniques for statistical monitoring of clinical trials were widely used. Conditional power argument and α-spending function based boundary crossing probabilities are popular statistical hypothesis testing procedures under the assumption of Brownian motion. However, it is not rare that the assumptions of Brownian motion are only partially met for trial data. Therefore, I used a more generalized form of stochastic process, called fractional Brownian motion (fBm), to model the test statistics. Fractional Brownian motion does not hold Markov property and future observations depend not only on the present observations but also on the past ones. In this dissertation, we simulated a wide range of fBm data, e.g., H = 0.5 (that is, classical Bm) vs. 0.5< H <1, with treatment effects vs. without treatment effects. Then the performance of conditional power and boundary-crossing based interim analyses were compared by assuming that the data follow Bm or fBm. Our simulation study suggested that the conditional power or boundaries under fBm assumptions are generally higher than those under Bm assumptions when H > 0.5 and also matches better with the empirical results. ^

Element concentration and GDGT fractional abundances of sediment core GeoB3910-2

In dieser Arbeit wurde das Zeitintervall zwischen 20 und 10 ka vor heute einschließlich des Heinrichevent 1 und der Younger Dryas am Kern GeoB 3910-2 neu untersucht. An organischen Parametern, basierend auf der Verteilung von bakteriellen GDGTs, und Elementkonzentrationen wurde eine Rekonstruktion der klimatischen Bedingungen und Veränderungen im Hinterland von NO Brasilien durchgeführt. Es zeigt sich, dass sich die durchschnittliche Landtemperatur gleich der Oberflächenwassertemperatur verhält und im Gegensatz zu den Phasen von H6 bis H2 dem antarktischen Erwärmungstrend ab 17 ka vor heute folgt. Weiterhin konnte gezeigt werden, dass durch die südwärts Verlagerung der ITCZ während H1 und der YD die Niederschläge in NO Brasilen intensiviert wurden, was eine Ausbreitung der Flüsse und Änderung der Erosionsgebiete zur Folge hatte.

Fractional abundance and indices for brGDGTs and crenarchaeol in dust samples, surface water samples and surface sediment samples

Branched glycerol dialkyl glycerol tetraethers (brGDGTs) are membrane lipids produced by soil bacteria and occur in near coastal marine sediments as a result of soil organic matter input. Their abundance relative to marine-derived crenarchaeol, quantified in the BIT index, generally decreases offshore. However, in distal marine sediments, low relative amounts of brGDGTs can often still be observed. Sedimentary in situ production as well as dust input have been suggested as potential, though as yet not well constrained, sources. In this study brGDGT distributions in dust were examined and compared with those in distal marine sediments. Dust was sampled along the equatorial West African coast and brGDGTs were detected in most of the samples, albeit in low abundance. Their degree of methylation and cyclisation, expressed in the MBT' (methylation index of branched tetraethers) and DC (degree of cyclisation) indices, respectively, were comparable with those for African soils, their presumed source. Comparison of DC index values for brGDGTS in global soils, Congo deep-sea river fan sediments and dust with those of distal marine sediments clearly showed, however, that distal marine sediments had significantly higher values. This distinctive distribution is suggestive of sedimentary in situ production as a source of brGDGTs in marine sediments, rather than dust input. The presence of in situ produced brGDGTs in marine sediments means that caution should be exercised when applying the MBT'-CBT palaeothermometer to sediments with low BIT index values, i.e. < 0.1, based on our dataset.

A fractional vegetation cover remote sensing product on pan-arctic scale with link to geotiff image

The paper presents first results of a pan-boreal scale land cover harmonization and classification. A methodology is presented that combines global and regional vegetation datasets to extract percentage cover information for different vegetation physiognomy and barren for the pan-arctic region within the ESA Data User Element Permafrost. Based on the legend description of each land cover product the datasets are harmonized into four LCCS (Land Cover Classification System) classifiers which are linked to the MODIS Vegetation Continuous Field (VCF) product. Harmonized land cover and Vegetation Continuous Fields products are combined to derive a best estimate of percentage cover information for trees, shrubs, herbaceous and barren areas for Russia. Future work will concentrate on the expansion of the developed methodology to the pan-arctic scale. Since the vegetation builds an isolation layer, which protects the permafrost from heat and cold temperatures, a degradation of this layer due to fire strongly influences the frozen conditions in the soil. Fire is an important disturbance factor which affects vast processes and dynamics in ecosystems (e.g. biomass, biodiversity, hydrology, etc.). Especially in North Eurasia the fire occupancy has dramatically increased in the last 50 years and has doubled in the 1990s with respect to the last five decades. A comparison of global and regional fire products has shown discrepancies between the amounts of burn scars detected by different algorithms and satellite data.