Generalizations of Heilbronn’s triangle problem

For given integers d,j≥2 and any positive integers n, distributions of n points in the d-dimensional unit cube [0,1]^d are investigated, where the minimum volume of the convex hull determined by j of these n points is large. In particular, for fixed integers d,k≥2 the existence of a configuration of n points in [0,1]^d is shown, such that, simultaneously for j=2,…,k, the volume of the convex hull of any j points among these n points is Ω(1/n^{(j−1)/(1+|d−j+1|)}). Moreover, a deterministic algorithm is given achieving this lower bound, provided that d+1≤j≤k.     

Combined approach of isotope mass balance and hydrological water balance methods to constrain the sources of lake water as exemplified on the small dimictic lake Silbersee,northern Germany

Stable isotopes of hydrogen and oxygen are often used for water balance calculations of lakes. We present an approach combining the lake water balance with an isotope mass balance to constrain the sources and sinks of the water of a small dimictic lake subjected to eutrophication. Meteorological and hydraulic data in combination with measured isotope signatures of the different water compartments enabled to assess the degree of surface water/groundwater interaction and the amount of overland flow into the lake. Groundwater could be excluded as a lake water source, as its water level was always below the lake water level. In the absence of a channelled inflow, precipitation and overland flow were the remaining options, whereby the latter was only active during periods of exceptionally high rainfall. While the groundwater signatures adjacent to the lake showed an influence of lake water, the lake water balance itself indicated that the associated volumetric water loss to groundwater is rather negligible. In the present case, only a combined assessment of hydrological and isotopic data allowed for an accurate characterization of the studied lake and a quantification of its water sources and sinks, highlighting the importance of using more than one methodological approach for such a purpose.     

An Indefinite Metric Model for Interacting Quantum Fields on Globally Hyperbolic Space-Times

We consider a relativistic ansatz for the vacuum expectation values (VEVs) of a quantum field on a globally hyperbolic space-time which is motivated by certain Euclidean field theories. The Yang-Feldman asymptotic condition w.r.t. an in-field in a quasi-free representation of the canonic commutation relations (CCR) leads to a solution of this ansatz for the VEVs. A GNS-like construction on a non-degenerate inner product space then gives local, covariant quantum fields with indefinite metric on a globally hyperbolic space-time. The non-trivial scattering behavior of quantum fields is analyzed by construction of the out-fields and calculation of the scattering matrix. A new combined effect of non-trivial quantum scattering and non-stationary gravitational forces is described for this model, as quasi-free in- fields are scattered to out-fields which form a non-quasi-free representations of the CCR. The asymptotic condition, on which the construction is based, is verified for the concrete example of de Sitter space-time. Communicated by Klaus Fredenhagen submitted 03/05/02, accepted: 18/07/03     

Development and Modeling of a Continuous Dispersion Copolymerization Process in the Presence of a Stabilizing Agent Elaborated in situ

Summary: This work deals with the development of a process for the radical copolymerization of acrylonitrile and styrene in a dispersed medium. This process was carried out in a continuous stirred tank reactor, in the presence of a stabilizing agent produced in situ during the polymerization. The continuous phase is a polyol. Besides all elementary chemical mechanisms related to the copolymerization and to the synthesis and grafting of the stabilizing agent, this process involves several complex physical phenomena. A tendency model of the whole process was developed, using the corresponding mass balances and thermodynamics. Its unknown parameters were identified by use of an evolutionary algorithm and experimental data resulting from an adapted experimental strategy. This model was then validated and allowed to predict monomers and transfer agent conversions, amounts of solids and average molar masses, versus the operating conditions.     

A Characterisation of Locality in Momentum Space

It is proved that a Poincaré invariant Wightman function which fulfils the spectral property and can be defined at sharp times, is local if and only if the integration over both the energy variables of a commutator in momentum space is a polynomial in the momentum conjugated to the spacial difference variable of the commutator with distributional coefficients depending on the remaining energy and momentum variables. Using this characterisation of locality in momentum space, the locality of a sequence of Wightman functions with nontrivial scattering behavior (associated to some quantum field in indefinite metric) can be proved by explicit calculations. We compare the above characterisation of locality with the classical integral representation method of Jost, Lehmann and Dyson.