Failure of the Weierstrass Preparation Theorem in quasi-analytic Denjoy-Carleman rings

It is shown that Denjoy-Carleman quasi-analytic rings of germs of functions in two or more variables fail to satisfy the Weierstrass Preparation Theorem. The result is proven via a non-extension theorem.Comment: 17 pages; accepted at the Advances in Mathematic

On the injectivity of the circular Radon transform arising in thermoacoustic tomography

The circular Radon transform integrates a function over the set of all spheres with a given set of centers. The problem of injectivity of this transform (as well as inversion formulas, range descriptions, etc.) arises in many fields from approximation theory to integral geometry, to inverse problems for PDEs, and recently to newly developing types of tomography. The article discusses known and provides new results that one can obtain by methods that essentially involve only the finite speed of propagation and domain dependence for the wave equation.Comment: To appear in Inverse Problem

On planar Sobolev L-p(m)-extension domains

For each m \u3e = 1 and p \u3e 2 we characterize bounded simply connected Sobolev L-p(m)-extension domains Omega subset of R-2. Our criterion is expressed in terms of certain intrinsic subhyperbolic metrics in Omega. Its proof is based on a series of results related to the existence of special chains of squares joining given points x and y in Omega. An important geometrical ingredient for obtaining these results is a new Square Separation Theorem . It states that under certain natural assumptions on the relative positions of a point x and a square S subset of Omega there exists a similar square Q subset of Omega which touches S and has the property that x and S belong to distinct connected components of Omega \ Q. (C) 2015 Elsevier Inc. All rights reserved

Adaptive cluster expansion for the inverse Ising problem: convergence, algorithm and tests

We present a procedure to solve the inverse Ising problem, that is to find the interactions between a set of binary variables from the measure of their equilibrium correlations. The method consists in constructing and selecting specific clusters of variables, based on their contributions to the cross-entropy of the Ising model. Small contributions are discarded to avoid overfitting and to make the computation tractable. The properties of the cluster expansion and its performances on synthetic data are studied. To make the implementation easier we give the pseudo-code of the algorithm.Comment: Paper submitted to Journal of Statistical Physic

Earthquake source parameters and scaling relationships in Hungary (central Pannonian basin)

Abstract Fifty earthquakes that occurred in Hungary (central part of the Pannonian basin) with local magnitude ML ranging from 0.8 to 4.5 have been analyzed. The digital seismograms used in this study were recorded by six permanent broad-band stations and twenty short-period ones at hypocentral distances between 10 and 327 km. The displacement spectra for P- and SH-waves were analyzed according to Brune’s source model. Observed spectra were corrected for path-dependent attenuation effects using an independent regional estimate of the quality factor QS. To correct spectra for near-surface attenuation, the k parameterwas calculated, obtaining it fromwaveforms recorded at short epicentral distances. The values of the k parameter vary between 0.01 to 0.06 s with a mean of 0.03 s for P-waves and between 0.01 to 0.09 s with a mean of 0.04 s for SH-waves. After correction for attenuation effects, spectral parameters (corner frequency and low-frequency spectral level) were estimated by a grid search algorithm. The obtained seismic moments range from4.21×1011 to 3.41×1015 Nm (1.7≤Mw ≤4.3). The source radii are between 125 and 1343 m. Stress drop values vary between 0.14 and 32.4 bars with a logarithmic mean of 2.59 bars (1 bar = 105 Pa). From the results, a linear relationship between local andmomentmagnitudes has been established. The obtained scaling relations show slight evidence of self-similarity violation. However, due to the high scatter of our data, the existence of self-similarity cannot be excluded