Vanishing of the Kontsevich Integrals of the Wheels

We prove that the Kontsevich integrals (in the sense of the formality theorem) of all even wheels are equal to zero. These integrals appear in the approach to the Duflo formula via the formality theorem. The result means that for any finite-dimensional Lie algebra g, and for invariant polynomials f, g [S ^·(g)]^g one has f · g = f * g, where * is the Kontsevich star product, corresponding to the Kirillov–Poisson structure on g^*. We deduce this theorem form the result contained in math.QA/0010321 on the deformation quantization with traces.     

Analysis of nonlinear dynamics on arbitrary geometries with the Virtual Cell

The Virtual Cell is a modeling tool that allows biologists and theorists alike to specify and simulate cell-biophysical models on arbitrarily complex geometries. The framework combines an intuitive, front-end graphical user interface that runs in a web browser, sophisticated server-side numerical algorithms, a database for storage of models and simulation results, and flexible visualization capabilities. In this paper, we present an overview of the capabilities of the Virtual Cell, and, for the first time, the detailed mathematical formulation used as the basis for spatial computations. We also present summaries of two rather typical modeling projects, in order to illustrate the principal capabilities of the Virtual Cell. (c) 2001 American Institute of Physics.     

Linear discrete operators on the disk algebra

Let be the disk algebra. In this paper we address the following question: Under what conditions on the points do there exist operators such that

and , , for every ? Here the convergence is understood in the sense of norm in . Our first result shows that if satisfy Carleson condition, then there exists a function such that , . This is a non-trivial generalization of results of Somorjai (1980) and Partington (1997). It also provides a partial converse to a result of Totik (1984). The second result of this paper shows that if are required to be projections, then for any choice of the operators do not converge to the identity operator. This theorem generalizes the famous theorem of Faber and implies that the disk algebra does not have an interpolating basis.

     

An angle in Minkowski space

A new angular measure in Minkowski space is introduced. It is defined for a cone of any dimension, is additive and invariant under invertible linear transformations. For the dimension 2, it has a clear interpretation as an amount of rotation in the Minkowski space from a direction to another one.     

On rational isotropic congruences of lines

The aim of this paper is to show a way to find an explicit parametrization of rational isotropic congruences of lines in Euclidean three-space It will be shown that also the focal surfaces of these congruences admit a rational parametrization. Furthermore, the close relation of isotropic congruences of lines to minimal surfaces will be used to find the related polynomial minimal surfaces.     

Cell Complexities in Hyperplane Arrangements

We derive improved bounds on the complexity, i.e., the total number of faces of all dimensions, of many cells in arrangements of hyperplanes in higher dimensions, and use these bounds to obtain a very simple proof of an earlier bound, due to Aronov, Matousek, and Sharir, on the sum of squares of cell complexities in such an arrangement.     

A blood vessel exposed to ultrasound: a mathematical simulation of the temperature field

In this article we present a mathematical simulation of the temperature field in and around a blood vessel when it is sonicated by a focused ultrasound beam. A simplified geometry is considered: a cylindrical blood vessel is embedded in tissue parallel to a flat skin surface. The ultrasound transducer is placed on the skin above the blood vessel, perpendicular to the skin surface. The 3D geometry of the problem is simplified by transformation, which maps the domain into a parallelepiped. A computational algorithm and computer program were developed. The simulation provides the conditions for successful occlusion of a blood vessel and demonstrates the significant role of the blood flow rate on the temperature difference between the vessel wall and the surrounding tissue. Comparing the predictions with published experimental data tested the validity of the method.     

Josephson plasma resonance and phonon anomalies in trilayer Bi2Sr2Ca2Cu3O10

The far-infrared (FIR) c axis conductivity of a Bi2223 crystal has been measured by ellipsometry. Below T(c) a strong absorption band develops near 500 cm(-1), corresponding to a transverse Josephson plasmon. The related increase in FIR spectral weight leads to a giant violation of the Ferrell-Glover-Tinkham sum rule. The gain in c axis kinetic energy accounts for a sizable part of the condensation energy. We also observe phonon anomalies which suggest that the Josephson currents lead to a drastic variation of the local electric field within the block of closely spaced CuO2 planes.     

Data storage in optical fibers and reconstruction by use of low-coherence spectral interferometry

We demonstrate optical data storage in optical fibers and reconstruction by use of low-coherence spectral interferometry. The information was stored by means of writing fiber Bragg gratings with different central wavelengths at different locations of the fiber. We need only a single short pulse is needed to read all the stored data. The maximum theoretical reconstruction rate that can be obtained with our technique is 10 Tbits/s. Our storage technique can be useful for identifying users in optical communication networks.