A closer look at semistability for singular principal bundles

We substantially refine the theory of singular principal bundles introduced in a former paper. In particular, we show that we need only honest singular principal bundles in our compactification. These are objects which carry the structure of a rational principal bundle in the sense of Ramanathan. Moreover, we arrive at a much simpler semistability condition. In the case of a semisimple group, this is just the Gieseker-version of Ramanathan's semistability condition for the corresponding rational principal $G$-bundle.Comment: To appear in the International Mathematics Research Notices. V2: Minor correction

Global Boundedness for Decorated Sheaves

An important classification problem in Algebraic Geometry deals with pairs (\E,\phi), consisting of a torsion free sheaf \E and a non-trivial homomorphism \phi\colon (\E^{\otimes a})^{\oplus b}\lra\det(\E)^{\otimes c}\otimes \L on a polarized complex projective manifold (X,\O_X(1)), the input data $a$, $b$, $c$, \L as well as the Hilbert polynomial of \E being fixed. The solution to the classification problem consists of a family of moduli spaces ${\cal M}^\delta:={\cal M}^{\delta-\rm ss}_{a/b/c/L/P}$ for the $\delta$-semistable objects, where \delta\in\Q[x] can be any positive polynomial of degree at most $\dim X-1$. In this note we show that there are only finitely many distinct moduli spaces among the ${\cal M}^\delta$ and that they sit in a chain of "GIT-flips". This property has been known and proved by ad hoc arguments in several special cases. In our paper, we apply refined information on the instability flag to solve this problem. This strategy is inspired by the fundamental paper of Ramanan and Ramanathan on the instability flag.Comment: To appear in the International Mathematics Research Notices. V2: A few typos corrected (notably in the definition of semistability in the introduction); Expanded Introductio

The initial temporal evolution of a feedback dynamo for Mercury

Various possibilities are currently under discussion to explain the observed weakness of the intrinsic magnetic field of planet Mercury. One of the possible dynamo scenarios is a dynamo with feedback from the magnetosphere. Due to its weak magnetic field Mercury exhibits a small magnetosphere whose subsolar magnetopause distance is only about 1.7 Hermean radii. We consider the magnetic field due to magnetopause currents in the dynamo region. Since the external field of magnetospheric origin is antiparallel to the dipole component of the dynamo field, a negative feedback results. For an alpha-omega-dynamo two stationary solutions of such a feedback dynamo emerge, one with a weak and the other with a strong magnetic field. The question, however, is how these solutions can be realized. To address this problem, we discuss various scenarios for a simple dynamo model and the conditions under which a steady weak magnetic field can be reached. We find that the feedback mechanism quenches the overall field to a low value of about 100 to 150 nT if the dynamo is not driven too strongly

A general notion of coherent systems

We look at coherent systems for decorated vector bundles and propose a notion of semistability. In the special case of tensor powers, we will examine this notion more closely. In particular, we will construct moduli spaces with the help of geometric invariant theory. It is an interesting aspect that ampleness of the linearization in the geometric invariant theory construction yields a bound on the stability parameter for coherent systems

Entropy and Long range correlations in literary English

Recently long range correlations were detected in nucleotide sequences and in human writings by several authors. We undertake here a systematic investigation of two books, Moby Dick by H. Melville and Grimm's tales, with respect to the existence of long range correlations. The analysis is based on the calculation of entropy like quantities as the mutual information for pairs of letters and the entropy, the mean uncertainty, per letter. We further estimate the number of different subwords of a given length $n$. Filtering out the contributions due to the effects of the finite length of the texts, we find correlations ranging to a few hundred letters. Scaling laws for the mutual information (decay with a power law), for the entropy per letter (decay with the inverse square root of $n$) and for the word numbers (stretched exponential growth with $n$ and with a power law of the text length) were found.Comment: 8 page

An algorithm for correcting CoRoT raw light curves

We introduce the CoRoT detrend algorithm (CDA) for detrending CoRoT stellar light curves. The algorithm CDA has the capability to remove random jumps and systematic trends encountered in typical CoRoT data in a fully automatic fashion. Since enormous jumps in flux can destroy the information content of a light curve, such an algorithm is essential. From a study of 1030 light curves in the CoRoT IRa01 field, we developed three simple assumptions which upon CDA is based. We describe the algorithm analytically and provide some examples of how it works. We demonstrate the functionality of the algorithm in the cases of CoRoT0102702789, CoRoT0102874481, CoRoT0102741994, and CoRoT0102729260. Using CDA in the specific case of CoRoT0102729260, we detect a candidate exoplanet around the host star of spectral type G5, which remains undetected in the raw light curve, and estimate the planetary parameters to be Rp=6.27Re and P=1.6986 days.Comment: 8 pages, 13 figure

Microscopic magnetic modeling for the SS=1/2 alternating chain compounds Na3_3Cu2_2SbO6_6 and Na2_2Cu2_2TeO6_6

The spin-1/2 alternating Heisenberg chain system Na$_3$Cu$_2$SbO$_6$ features two relevant exchange couplings: $J_{1a}$ within the structural Cu$_2$O$_6$ dimers and $J_{1b}$ between the dimers. Motivated by the controversially discussed nature of $J_{1a}$, we perform extensive density-functional-theory (DFT) calculations, including DFT+$U$ and hybrid functionals. Fits to the experimental magnetic susceptibility using high-temperature series expansions and quantum Monte Carlo simulations yield the optimal parameters $J_{1a}$ = $-$217 K and $J_{1b}$ = 174 K with the alternation ratio $\alpha = J_{1a}/J_{1b} \simeq$ $-$1.25. For the closely related system Na$_2$Cu$_2$TeO$_6$, DFT yields substantially enhanced $J_{1b}$, but weaker $J_{1a}$. The comparative analysis renders the buckling of the chains as the key parameter altering the magnetic coupling regime. Numerical simulation of the dispersion relations of the alternating chain model clarify why both antiferromagnetic and ferrromagnetic $J_{1a}$ can reproduce the experimental magnetic susceptibility data.Comment: published version: 11 pages, 8 figures, 5 tables + Supplemental materia