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The indicator of contact boundaries for an integral geometry problem

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Abstract

We pose and study a rather particular integral geometry problem. In the two-dimensional space we consider all possible straight lines that cross some domain. The known data consist of the integrals over every line of this kind of an unknown piecewise smooth function that depends on both points of the domain and the variables characterizing the lines. The object we seek is the discontinuity curve of the integrand. This problem arose in the author’s previous research in X-ray tomography. In essence, it is a generalization of one mathematical aspect of flaw detection theory, but seems of interest in its own right. The main result of this article is the construction of a special function that can be unbounded only near the required curve. Precisely for this reason we call the function the indicator of contact boundaries. A uniqueness theorem for the solution follows rather easily from the property of indicators.

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Correspondence to D. S. Anikonov.

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Original Russian Text Copyright © 2008 Anikonov D. S.

The author was supported by the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-1440.2008.1), and the Interdisciplinary Integration Grants (No. 3 and No. 48; 2006) of the Siberian Division of the Russian Academy of Sciences.

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 49, No. 4, pp. 739–755, July–August, 2008.

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Anikonov, D.S. The indicator of contact boundaries for an integral geometry problem. Sib Math J 49, 587–600 (2008). https://doi.org/10.1007/s11202-008-0056-2

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