Bounds on cohomology and Castelnuovo-Mumford regularity

The Castelnuovo-Mumford regularity reg(X) of a projective scheme X was introduced by Mumford by generalizing ideas of Castelnuovo. The interest in this concept stems partly from the fact that X is m-regular if and only if for every p \geq 0 the minimal generators of the p-th syzygy module of the defining ideal I of X occur in degree \leq m + p. There are some bounds in the case that X is a locally Cohen-Macaulay scheme. The aim of this paper is to extend and improve these results for so-called (k,r)-Buchsbaum schemes. In order to prove our theorems, we need to apply a spectral sequence. We conclude by describing two sharp examples and open problems.Comment: LaTeX, 18 page

Towards a theory of arithmetic degrees

The aim of this paper is to start a systematic investigation of the arithmetic degree of projective schemes as introduced by D. Bayer and D. Mumford. One main theme concerns itself with the behaviour of this arithmetic degree under hypersurface sections. The notion of arithmetic degree involves the new concept of length-multiplicity of embedded primary ideals. Therefore it is much harder to control the arithmetic degree under a hypersurface section than in the case for the classical degree theory. Nevertheless it has important and interesting applications. We describe such applications to the Castelnuovo-Mumford regularity and to Bezout-type theorems.Comment: LaTeX, 14 page

Half-open Penning trap with efficient light collection for precision laser spectroscopy of highly charged ions

We have conceived, built and operated a 'half-open' cylindrical Penning trap for the confinement and laser spectroscopy of highly charged ions. This trap allows fluorescence detection employing a solid angle which is about one order of magnitude larger than in conventional cylindrical Penning traps. At the same time, the desired electrostatic and magnetostatic properties of a closed-endcap cylindrical Penning trap are preserved in this congfiuration. We give a detailed account on the design and confinement properties, a characterization of the trap and show first results of light collection with in-trap produced highly charged ions

Switchable Magnetic Bottles and Field Gradients for Particle Traps

Versatile methods for the manipulation of individual quantum systems, such as confined particles, have become central elements in current developments in precision spectroscopy, frequency standards, quantum information processing, quantum simulation, and alike. For atomic and some subatomic particles, both neutral and charged, a precise control of magnetic fields is essen- tial. In this paper, we discuss possibilities for the creation of specific magnetic field configurations which find appli- cation in these areas. In particular, we pursue the idea of a magnetic bottle which can be switched on and off by transition between the normal and the superconducting phase of a suitable material in cryogenic environments, for example in trap experiments in moderate magnetic fields. Methods for a fine-tuning of the magnetic field and its linear and quadratic components in a trap are presented together with possible applications

Ultrafine particles over Germany – an aerial survey

Ultrafine particles in the atmosphere may have important climate and health effects. As they are below visible size and not visible for remote sensing techniques, the majority of observations thus come from ground-based measurements. Some of those observations indicate elevated sources for ultrafine particles. Here we present for the first time airborne measurements of number concentration and size distributions of ultrafine particles along defined flight paths across Germany, allowing to derive background concentrations and to identify major single sources. A significant impact of fossil fuel–related emissions on background and maximum concentrations was found. Maxima reaching up to 90 000 particles cm−3 were encountered in plumes of single large sources extending over more than 200 km. Modelling shows that about 10–40 % of Germany were continuously affected by such plumes. Regional-scale transport and boundary layer dynamics were identified as major factors controlling spatial and temporal patterns of size and number distributions

Trapped Ion Oscillation Frequencies as Sensors for Spectroscopy

The oscillation frequencies of charged particles in a Penning trap can serve as sensors for spectroscopy when additional field components are introduced to the magnetic and electric fields used for confinement. The presence of so-called “magnetic bottles” and specific electric anharmonicities creates calculable energy-dependences of the oscillation frequencies in the radiofrequency domain which may be used to detect the absorption or emission of photons both in the microwave and optical frequency domains. The precise electronic measurement of these oscillation frequencies therefore represents an optical sensor for spectroscopy. We discuss possible applications for precision laser and microwave spectroscopy and their role in the determination of magnetic moments and excited state life-times. Also, the trap-assisted measurement of radiative nuclear de-excitations in the X-ray domain is discussed. This way, the different applications range over more than 12 orders of magnitude in the detectable photon energies, from below μeV in the microwave domain to beyond MeV in the X-ray domain

Off-diagonal long-range order for the free Bose gas via the Feynman--Kac formula

We consider the path-integral representation of the ideal Bose gas under various boundary conditions. We show that Bose--Einstein condensation occurs at the famous critical density threshold, by proving that its $1$-particle-reduced density matrix exhibits off-diagonal long-range order above that threshold, but not below. Our proofs are based on the well-known Feynman--Kac formula and a representation in terms of a crucial Poisson point process. Furthermore, in the condensation regime, we derive a law of large numbers with strong concentration for the number of particles in short loops. In contrast to the situation for free boundary conditions, where the entire condensate sits in just one loop, for all other boundary conditions we obtain the limiting Poisson--Dirichlet distribution for the collection of the lengths of all long loops. Our proofs are new and purely probabilistic (apart from a standard eigenvalue expansion), using elementary tools like Markov's inequality, Poisson point processes, combinatorial formulas for cardinalities of particular partition sets, and asymptotics for random walks with Pareto-distributed steps.Comment: 34 page