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Sibirsk. Mat. Zh., 2009, Volume 50, Number 5, Pages 967–986 (Mi smj2024)  

This article is cited in 11 scientific papers (total in 11 papers)

Ned sets on a hyperplane

V. V. Aseev

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: Under study are the sets in $\mathbb R^n$ (NED sets) each of which does not affect the conformal capacity of any condenser with connected plates disjoint from this set. These sets are removable singularities of quasiconformal mappings, which explains our interest in them. For compact sets on a hyperplane we obtain a geometric criterion of the NED property; we point out a simple sufficient condition for an NED set in terms of the connected attainability of its points from its complement in the hyperplane. For compact sets on a hypersphere we obtain a criterion for an NED set in terms of the reduced module at a pair of points in its complement. We establish that a compact set on a hypersphere $S$, removable for the capacity in at least one spherical ring concentric with $S$ and containing $S$, is an NED set.

Keywords: module of a family of curves, NED set, quasiconformal mapping, removable singularity, capacity of a condenser, reduced generalized module, capacity defect, attainable boundary point.

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English version:
Siberian Mathematical Journal, 2009, 50:5, 760–775

Bibliographic databases:

UDC: 517.54
Received: 15.02.2008

Citation: V. V. Aseev, “Ned sets on a hyperplane”, Sibirsk. Mat. Zh., 50:5 (2009), 967–986; Siberian Math. J., 50:5 (2009), 760–775

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Yu. V. Dymchenko, V. A. Shlyk, “Sufficiency of broken lines in the modulus method and removable sets”, Siberian Math. J., 51:6 (2010), 1028–1042  mathnet  crossref  mathscinet  isi
    2. F. I. Ivanov, V. A. Shlyk, “Null-sets for the extremal lengths”, J. Math. Sci. (N. Y.), 178:2 (2011), 163–169  mathnet  crossref
    3. P. A. Pugach, V. A. Shlyk, “Generalized capacities and polyhedral surfaces”, J. Math. Sci. (N. Y.), 178:2 (2011), 201–218  mathnet  crossref
    4. Yu. V. Dymchenko, V. A. Shlyk, “Sufficiency of Polyhedral Surfaces in the Modulus Method and Removable Sets”, Math. Notes, 90:2 (2011), 204–217  mathnet  crossref  crossref  mathscinet  isi
    5. Yu. V. Dymchenko, V. A. Shlyk, “Some properties of the capacity and module of a polycondenser and removable sets”, J. Math. Sci. (N. Y.), 184:6 (2012), 709–715  mathnet  crossref
    6. P. A. Pugach, V. A. Shlyk, “Removable sets for the generalized module of surface's family”, J. Math. Sci. (N. Y.), 184:6 (2012), 755–769  mathnet  crossref
    7. V. A. Shlyk, “The spherical symmetrization and NED-sets on a hyperplane”, J. Math. Sci. (N. Y.), 193:1 (2013), 145–150  mathnet  crossref  mathscinet
    8. P. A. Pugach, V. A. Shlyk, “Piecewise linear approximation and polyhedral surfaces”, J. Math. Sci. (N. Y.), 200:5 (2014), 617–623  mathnet  crossref
    9. V. N. Dubinin, “On the reduced modulus of the complex sphere”, Siberian Math. J., 55:5 (2014), 882–892  mathnet  crossref  mathscinet  isi
    10. V. A. Shlyk, A. A. Yakovlev, “Modules of space configuration and removable sets”, J. Math. Sci. (N. Y.), 225:6 (2017), 1022–1031  mathnet  crossref  mathscinet
    11. N. V. Abrosimov, A. D. Mednykh, I. A. Mednykh, A. V. Tetenov, “Vladislavu Vasilevichu Aseevu — 70 let”, Sib. elektron. matem. izv., 14 (2017), A43–A57  mathnet  mathscinet  isi
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