Far Eastern Mathematical Journal

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The generalized reduced modulus in spatial problems of the capacitorial tomography


V. V. Aseev

2007, issue 1-2, Ñ. 17–29


Abstract
The external problem of the spatial capacitorial tomography is considered. The notion of capacitorial defect of an object (a compact set) along Mo?bius directions in the space has been introduced. The criteria for the capacitorial invisibility of an object along the Mo?bius direction determined by a pair of points in the accessible region of the space has been obtained. The problem of upper estimates for the capacitorial defect along M\"{o}bius directions in the space, as well as it's connection with the notion of generalized reduced modulus by V.N. Dubinin, is there also considered in this paper.

Keywords:
condenser, conformal capacity, conformal modulus, modulus of a set of curves, capacitorial defect, capacitorial tomography, capacitorial invisibility, NED-set, eliminability along direction, Apollonian condenser, generalized reduced modulus

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References

[1] J. V\"{a}is\"{a}l\"{a}, “Lectures on n-dimensional quasiconformal mappings”, Lect. Notes Math., 228, Springer-Verlag, Berlin – Heidelberg – New York, 1971, 144 pp.
[2] B. Fuglede, “Extremal length and functional completion”, Acta Math., 98:3–4 (1957), 171–219.
[3] J. Hesse, “A p-extremal length and p-capacity equality”, Ark. mat., 13:1 (1975), 131–144.
[4] P. Caraman, “p-Capacity and p-modulus”, Symp. Math. Inst. naz. alta mat., 18 (1976), 455–484.
[5] V. A. Shlyk, “O ravenstve $p$-emkosti i $p$-modulya”, Sib. matem. zhurnal, 34:6 (1993), 216–221.
[6] F. W. Gehring, “Extremal length definitions for the conformal capacity of rings in space”, Michigan Math. J., 9 (1962), 137–150.
[7] V. A. Shlyk, “Normal'nye oblasti po Gretshu i topologicheskaya ustranimost' mnozhestva dlya prostranstvennyx gomeomorfizmov”, Dokl. AN SSSR, 112:3 (1988), 553–555.
[8] V. A. Shlyk, “Stroenie kompaktov, porozhdayushhix normal'nye oblasti i ustranimye osobennosti dlya prostranstva L1p(D)”, Matem. sb., 181:11 (1990), 1558–1572.
[9] V. A. Shlyk, “Normal'nye oblasti i ustranimye osobennosti”, Izv. AN, Ser. mat., 57:4 (1993), 92–117.
[10] J. V\"{a}is\"{a}l\"{a}, “On the null-sets for extremal lengths”, Ann. Acad. Sci. Fenn. Ser. A I, 322 (1962), 1–22.
[11] V. N. Dubinin, Emkosti kondensatorov v geometricheskoj teorii funkcij, Vladivostok, 2003, 116 s.
[12] F. W. Gehring, “Symmetrization of rings in space”, Trans. Amer. Math. Soc., 101:3 (1961), 499–519.
[13] G. D. Mostov, “Kvazikonformnye otobrazheniya v n-mernom prostranstve i zhestkost' giperbolicheskix prostranstvennyx form”, Matematika. Sb. perevodov, 16:5 (1972), 105–157.
[14] P. Caraman, n-Dimensional quasiconformal (QCf) mappings, Editura Acad. Roma?nia Abacus Press, Tundridge Wells, Kent, 1974, 554 pp.
[15] V. N. Dubinin, N. V. E'jrix, “Obobshhennyj privedennyj modul'”, Dal'nevost. matem. zhurnal, 3:2 (2002), 150–164.
[16] V. N. Dubinin, “Simmetrizaciya v geometricheskoj teorii funkcij kompleksnogo peremennogo”, Uspexi matem. nauk, 49:1 (1994), 3–76.
[17] G. M. Goluzin, Geometricheskaya teoriya funkcij kompleksnogo peremennogo, Nauka, M., 1966, 628 s.
[18] Dzh. Dzhenkins, Odnolistnye funkcii i konformnye otobrazheniya, Izd-vo IL, M., 1962, 268 s.
[19] V. V. Aseev, “Opisanie NED-mnozhestv, lezhashhix na gipersfere”, Sovremennye metody teorii kraevyx zadach, Mater. Voronezhskoj vesen. matem. shk. “Pontryaginskie chteniya – 7”, Centr.-Chernozemn. kn. izd-vo, Voronezh, 2006, 9.
[20] L. Ahlfors, A. Beurling, “Conformal invariants and function-theoretic null-sets”, Acta math., 83 (1950), 101–129.
[21] V. A. Shlyk, “O pronicaemosti ustranimyx mnozhestv”, Fundam. probl. mat. i mex., Mat. 41, MGU, M., 1994, 20–21.
[22] V. A. Shlyk, “Uslovie ?-obxvata dlya N-kompaktov”, Zap. nauch. semin. POMI, 196, 1991, 154–161.
[23] L. V. Kantorovich, G. P. Akilov, Funkcional'nyj analiz, M., 1984, 752 s.
[24] V. A. Shlyk, “Metod sopryazhennyx semejstv v teorii modulej”, Doklady AN SSSR, 306:2 (1989), 297–300.
[25] V. V. Aseev, “Moduli semejstv lokal'no kvazisimmetricheskix poverxnostej”, Sib. matem. Zhurnal, 30:3 (1989), 9–15.

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