Pretentiously detecting power cancellation

Granville and Soundararajan have recently introduced the notion of pretentiousness in the study of multiplicative functions of modulus bounded by 1, essentially the idea that two functions which are similar in a precise sense should exhibit similar behavior. It turns out, somewhat surprisingly, that this does not directly extend to detecting power cancellation - there are multiplicative functions which exhibit as much cancellation as possible in their partial sums that, modified slightly, give rise to functions which exhibit almost as little as possible. We develop two new notions of pretentiousness under which power cancellation can be detected, one of which applies to a much broader class of multiplicative functions

The distribution of the Tamagawa ratio in the family of elliptic curves with a two-torsion point

In recent work, Bhargava and Shankar have shown that the average size of the $2$-Selmer group of an elliptic curve over $\mathbb{Q}$ is exactly $3$, and Bhargava and Ho have shown that the average size of the $2$-Selmer group in the family of elliptic curves with a marked point is exactly $6$. In contrast to these results, we show that the average size of the $2$-Selmer group in the family of elliptic curves with a two-torsion point is unbounded. In particular, the existence of a two-torsion point implies the existence of rational isogeny. A fundamental quantity attached to a pair of isogenous curves is the Tamagawa ratio, which measures the relative sizes of the Selmer groups associated to the isogeny and its dual. Building on previous work in which we considered the Tamagawa ratio in quadratic twist families, we show that, in the family of all elliptic curves with a two-torsion point, the Tamagawa ratio is essentially governed by a normal distribution with mean zero and growing variance

Very cold and massive cores near ISOSS J18364-0221: Implications for the initial conditions of high-mass star-formation

We report the discovery of two very cold and massive molecular cloud cores in the region ISOSS J18364-0221. The object has been identified by a systematic search for very early evolutionary stages of high-mass stars using the 170 micron ISOPHOT Serendipity Survey (ISOSS). Submm continuum and molecular line measurements reveal two compact cores within this region. The first core has a temperature of 16.5 K, shows signs of ongoing infall and outflows, has no NIR or MIR counterpart and is massive enough (M ~ 75 M_sun) to form at least one O star with an associated cluster. It is therefore considered a candidate for a genuine high-mass protostar and a high-mass analog to the Class 0 objects. The second core has an average gas and dust temperature of only ~ 12 K and a mass of M ~ 280 M_sun. Its temperature and level of turbulence are below the values found for massive cores so far and are suggested to represent the initial conditions from which high-mass star formation occurs.Comment: 9 pages, 6 figures, accepted for publication in the Astrophysical Journa

Theoretical Analysis of Biomolecular Systems: Computational Simulations, Core-set Markov State Models, Clustering, Molecular Docking

The analysis of the structural and the dynamical behavior of biomolecules is very important to under- stand their biological function, stability or physico-chemical properties. In this thesis, it is highlighted how different theoretical methods to characterize the aforementioned structural and dynamical properties can be used and combined, to obtain kinetic information or to detect biomolecule-ligand interactions. The basis for most of the analyses, performed in the course of this work, are molecular dynamics sim- ulations sampling the conformational space of the biomolecule of interest. Using molecular dynamics simulations, the remarkable stable water-soluble-binding-protein is examined first. On a theoretical ba- sis, structural modifications that can influence the stability of the protein are discussed. Additionally, by combining the simulations with a QM/MM optimization scheme and quantum chemical calculations, spectroscopical properties can be investigated. Markov State Models are applied frequently to capture the slow dynamics within simulation trajectories. They are based on a discretization of the conformational space. This discretization, however, introduces an error in the outcome of the analysis. The application of a core-set discretization can reduce this error. In this thesis, it is discussed how density-based cluster algorithms can be used to determine these core sets, and the application on linear and cyclic peptides is highlighted. The performance of a promising cluster algorithm is investigated and error sources in the construction of the Markov models are discussed. Finally, it is shown how molecular docking combined with molecular dynamics simulations can be used to determine the binding behavior of ligands towards biomolecules. In this context, the important in- teractions within the active site of an enzyme, and different binding modes of DNA intercalators are identified