Asymptotic and boundary integral formulae for the computational reconstruction of small conductivity imperfections

In this paper we investigate the electrostatic problem of determining conductivity profiles from the knowledge of boundary currents and voltages. We obtain an improved estimate for the voltage potential of a two-dimensional conductor having finitely many circular inclusions and piecewise constant conductivity profile. We derive an asymptotic expansion for the voltage potential in terms of the reference voltage potential and the location, size, and conductivity of the inhomogeneities. This representation is used to formulate the nonlinear least squares problem for estimating the location and size of the inhomogeneities. Required boundary data for the voltage potential are generated numerically by solving a system of integral equations. Computational experiments are presented to demonstrate the effectiveness of our identification procedure.     

Theory of Valuations on Manifolds: A Survey

This is a non-technical survey of a recent theory of valuations on manifolds constructed in [A10], [A11], [AF] and [A12], and actually a guide to this series of articles. We also review some recent related results obtained by a number of people. Received: February 2006, Revision: June 2006, Accepted: June 2006     

Partitioning Balls into Topologically Equivalent Pieces

---

Partitioning Balls into Topologically Equivalent Pieces

---

     

On Nodal Sets for Dirac and Laplace Operators

We prove that the nodal set (zero set) of a solution of a generalized Dirac equation on a Riemannian manifold has codimension 2 at least. If the underlying manifold is a surface, then the nodal set is discrete. We obtain a quick proof of the fact that the nodal set of an eigenfunction for the Laplace-Beltrami operator on a Riemannian manifold consists of a smooth hypersurface and a singular set of lower dimension. We also see that the nodal set of a Δ-harmonic differential form on a closed manifold has codimension 2 at least; a fact which is not true if the manifold is not closed. Examples show that all bounds are optimal. Received: 28 October 1996 / Accepted: 3 March 1997     

Proton relaxation times in human red bone marrow by volume selective magnetic resonance spectroscopy

In hematological diseases the composition of red bone marrow shows alterations. The relaxation timesT ₁ andT ₂ of water and lipids in human hemopoietic bone marrow of 14 normal volunteers and 10 patients with acute leukemia and bone marrow carcinosis are determined using a double spin echo spectroscopy sequencein vivo. The volumes of interest (VOI) of (13 mm)³ in the center of vertebral bodies are examined using different measurement parameters. ForT ₁ measurements an inversion-recovery method is used.T ₂ is evaluated from spectra with differentTE. T ₁ (water) is found in a range between 1000 and 1700 ms,T ₁ (lipids) in a range between 260 and 320 ms in healthy volunteers.T ₂ (water) is determined between 32 and 65 ms. In some cases phase distortions of the water signals occur in the spectra. Water flow within the VOI may be a possible reason.T ₂ (lipids) is evaluated between 73 and 91 ms. The patients with acute leukemia exhibit clearly reduced lipid signals in their spectra. Lipid relaxation times could not be determined in these cases.T ₂ (water) is prolonged in acute leukemia to 51–98 ms.T ₁ (water) was not significantly different from values of healthy volunteers in our measurements. Results are discussed in comparison to relaxometric data from imaging and STEAM spectroscopic methods of other authors.     

A Note on an Input Control Model of a Predictive Regression Interval

A well-known problem of prediction in linear regression models is to find a confidence interval for the random value of the dependent variable when the values of the independent variables are given. Such a situation may arise in economic quality control models when the independent variables are costly inputs and the dependent variable is some measure of quality or production. In such a circumstance, an important control objective may be to find values for the inputs that will maximize the lower limit of the prediction confidence interval for a fixed budget, or alternatively, to minimize the cost of the inputs for a fixed lower limit of the confidence interval. In this paper, we shall show that global optima can be found using known algorithms. The special case of simple linear regression is discussed and an illustrative example is provided.     

Stability of translating solutions to mean curvature flow

We prove stability of rotationally symmetric translating solutions to mean curvature flow. For initial data that converge spatially at infinity to such a soliton, we obtain convergence for large times to that soliton without imposing any decay rates. The authors are members of SFB 647/B3 “Raum – Zeit – Materie: Singularity Structure, Long-time Behaviour and Dynamics of Solutions of Non-linear Evolution Equations”.     

A characterization of certain irreducible symplectic 4-folds

 We give a characterization of irreducible symplectic fourfolds which are given as Hilbert scheme of points on a K3 surface. Received: 26 June 2002 / Revised version: 9 September 2002 Published online: 14 February 2003 Mathematics Subject Classification (2000): Primary 14J32, Secondary 32Q20     

A proof of Culter’s theorem on the existence of periodic orbits in polygonal outer billiards

We prove a recent theorem by C. Culter every polygonal outer billiard in the affine plane has a periodic trajectory.