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 Sibirsk. Mat. Zh., 2011, Volume 52, Number 5, Pages 962–976 (Mi smj2250)

The integral geometry boundary determination problem for a pencil of straight lines

D. S. Anikonov, D. S. Konovalova

Sobolev Institute of Mathematics, Novosibirsk, Russia

Abstract: Under consideration is the problem of integrating finitely many functions over straight lines. Each function as well as the corresponding line is assumed unknown. The available information is the sum of integrals over all straight lines of a family of pencils in each of which the intersection of lines is a point of a given bounded open set in a finite-dimensional Euclidean space. Each integrand depends on a greater number of variables than the sum of the integrals. Hence, the conventional statement of the problem of determining the integrands becomes underspecified. In this situation we pose and study the problem of determining the discontinuity surfaces of the integrands. The uniqueness theorem is proven under the condition that these surfaces exist. The present article is a refinement of the previous studies of the authors and differs from them in [1–6] by not only some technical improvements but also the principally new fact that the integration is performed over an unknown set.

Keywords: singular integral, integral geometry, unknown boundary.

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English version:
Siberian Mathematical Journal, 2011, 52:5, 763–775

Bibliographic databases:

UDC: 517.958

Citation: D. S. Anikonov, D. S. Konovalova, “The integral geometry boundary determination problem for a pencil of straight lines”, Sibirsk. Mat. Zh., 52:5 (2011), 962–976; Siberian Math. J., 52:5 (2011), 763–775

Citation in format AMSBIB
\Bibitem{AniKon11} \by D.~S.~Anikonov, D.~S.~Konovalova \paper The integral geometry boundary determination problem for a~pencil of straight lines \jour Sibirsk. Mat. Zh. \yr 2011 \vol 52 \issue 5 \pages 962--976 \mathnet{http://mi.mathnet.ru/smj2250} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2908119} \transl \jour Siberian Math. J. \yr 2011 \vol 52 \issue 5 \pages 763--775 \crossref{https://doi.org/10.1134/S0037446611050016} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000298650500001} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-80155151878} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. G. Romanov, “Recovering jumps in X-ray tomography”, J. Appl. Industr. Math., 8:4 (2014), 582–593
2. Romanov V.G., “Reconstruction of Discontinuities in a Problem of Integral Geometry”, Dokl. Math., 90:3 (2014), 758–761
3. D. S. Anikonov, D. S. Konovalova, “An integral geometry underdetermined problem for a family of curves”, Siberian Math. J., 56:2 (2015), 217–230
4. Anikonov D.S. Konovalova D.S., “a Problem of Integral Geometry For a Family of Curves With Incomplete Data”, Dokl. Math., 92:2 (2015), 521–524
5. Nedergaard J.L., “Role Differences in Healthcare: Overcoming Borders Through Semiotic Skin Is the Basis For Communication”, Integr. Psychol. Behav. Sci., 53:2 (2019), 283–297
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