Geodesic
Inversion of mapping and inverse problems
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 9 (2012), pp. 382-432.

Voir la notice de l'article provenant de la source Math-Net.Ru

We expose some selected results in finite-dimensional mapping theory and their connections with theory of inverse problems for evolution equations. We obtain new representations of solutions of multidimensional nonlinear evolution equations and those of their coefficients.
Keywords: inverse problems, ill-posed problems, representations of solutions, systems of differential equations, finite-dimensional mapping.
Mots-clés : evolution equations 
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Yu. E. Anikonov; N. B. Aiupova; V. G. Bardakov; V. P. Golubyatnikov; M. V. Neshchadim. Inversion of mapping and inverse problems. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 9 (2012), pp. 382-432. http://geodesic.mathdoc.fr/item/SEMR_2012_9_a28/

[1] Yu.E.Anikonov, “Obratnye zadachi dlya nelineinykh uravnenii i sinergetika”, DAN, 315 (1990), 355–388

[2] Yu.E.Anikonov, Multidimensional Inverse and Ill-Posed Problems for Differential Equations, VSP, Utrecht, 1995 | MR

[3] Yu.E.Anikonov, M.V.Neschadim, Analiticheskie predstavleniya reshenii ryada obratnykh zadach matematicheskoi fiziki, Preprint N 218, IM SO RAN, Novosibirsk, 2009

[4] Yu.E.Anikonov, Formulas in Inverse and Ill-Posed Problems, VSP, Utrecht, 1997 | MR

[5] Yu.E.Anikonov, N.B.Ayupova, Formuly dlya reshenii i koeffitsientov differentsialnykh uravnenii 2-go poryadka i obratnye zadachi, Preprint N 165, IM SO RAN, Novosibirsk, 2005, 58 pp. | MR

[6] V.P.Golubyatnikov, Uniqueness Questions in Reconstruction of Multidimensional Objects from Tomography-Type Projection Data, VSP, Utrecht, 2000

[7] van den Essen A., Seven lectures on polynomial automorphisms. Automorphisms of affine spaces, Kluwer academic publishers, 1995 | MR

[8] U.U.Umirbaev, I.P.Shestakov, “Podalgebry i avtomorfizmy kolets mnogochlenov”, DAN, 386 (2002), 745–748 | MR | Zbl

[9] A. I. Kostrikin, Vvedenie v algebru, Ucheb. dlya vuzov, v. I, Osnovy algebry, FIZMATLIT, M., 2004

[10] E. B. Vinberg, Lineinye predstavleniya grupp, Nauka, M., 1985

[11] E.A.Gorin, V.Ya. Lin, “Algebraicheskie uravneniya s nepreryvnymi koeffitsientami i nekotorye voprosy algebraicheskoi teorii kos”, Matem. sb., 78 (1969), 579–610 | MR | Zbl

[12] L.Khermander, Vvedenie v teoriyu funktsii neskolkikh kompleksnykh peremennykh, Mir, M., 1968

[13] L.A. Aizenberg, A.P.Yuzhakov, Integralnye predstavleniya i vychety v mnogomernom kompleksnom analize, Nauka, Novosibirsk, 1979

[14] Yu.E.Anikonov, Yu.V.Krivtsov, M.V.Neschadim, “Konstruktivnye metody v nelineinykh zadachakh teorii upravleniya”, Sib. zhurnal industrialnoi matematiki, 13 (2010), 30–45

[15] Yu.E. Anikonov, M.V.Neschadim, Vetvyaschiesya protsessy, otobrazheniya i obratnye zadachi, Preprint 247, IM SO RAN, 2010

[16] A.N.Tikhonov, V.Ya.Arsenin, Metody resheniya nekorrektnykh zadach, Nauka, M., 1974

[17] Yu.E. Anikonov, Nekotorye metody issledovaniya mnogomernykh obratnykh zadach dlya differentsialnykh uravnenii, Nauka, Novosibirsk, 1978

[18] Yu.E. Anikonov, “O geometricheskikh metodakh issledovaniya obratnykh zadach I”, Matematicheskie problemy geofiziki, vypusk II, VTs SO AN SSSR, Novosibirsk, 1971, 7–53

[19] Yu.E. Anikonov, “Teorema edinstvennosti dlya operatornogo uravneniya”, Matematicheskie problemy geofiziki, vypusk II, VTs SO AN SSSR, Novosibirsk, 1971, 54–56

[20] S.P.Kapitsa, “Matematicheskaya model rosta naseleniya mira”, Mat. modelirovanie, 4 (1992), 56–74

[21] S.P.Kapitsa, Skolko lyudei zhilo, zhivet i budet zhit na Zemle, Nauka, M., 1999

[22] A.S.Malkov, A.V.Korotaev, D.A.Khalturina, “Matematicheskaya model rosta naseleniya Zemli, ekonomiki, tekhnologii i obrazovaniya”, Novoe v sinergetike. Novaya realnost. Novye problemy. Novoe pokolenie, Nauka, M., 2007, 148–185

[23] Yu.E.Anikonov, N.B.Ayupova, M.V.Neschadim, Predstavleniya reshenii i koeffitsientov sistem evolyutsionnykh uravnenii, Preprint N 276, IM SO RAN, Novosibirsk, 2012

[24] Yu.E.Anikonov, M.V.Neschadim, “Ob analiticheskikh metodakh v teorii obratnykh zadach dlya parabolicheskikh uravnenii”, Vestnik NGU, 11 (2011)), 3–15

[25] Yu.E.Anikonov, M.V.Neschadim, “Ob analiticheskikh metodakh v teorii obratnykh zadach dlya giperbolicheskikh uravnenii. I”, Sibirskii zhurnal industrialnoi matematiki, 14 (2011), 27–39 | MR | Zbl

[26] Yu.E.Anikonov, M.V.Neschadim, “Ob analiticheskikh metodakh v teorii obratnykh zadach dlya giperbolicheskikh uravnenii. II”, Sibirskii zhurnal industrialnoi matematiki, 14 (2011), 28–33 | Zbl

[27] M.A.Lavrentev, B.V.Shabat, Problemy gidrodinamiki i ikh matematicheskie modeli, Nauka, M., 1977

[28] S.A.Makhov, “Matematicheskoe modelirovanie mirovoi dinamiki i ustoichivogo razvitiya na primere modeli Forrestera”, Novoe v sinergetike. Novaya realnost. Novye problemy. Novoe pokolenie, Nauka, M., 2007

[29] A.S.Malkov, G.G.Malinetskii, D.S.Chernavskii, “Modelirovanie razvitiya agrarnykh obschestv s pozitsii nelineinoi dinamiki”, Novoe v sinergetike. Novaya realnost. Novye problemy. Novoe pokolenie, Nauka, M., 2007

[30] V.I. Arnold, A.I.Varchenko, S.M.Gusein-Zade, Osobennosti differentsiruemykh otobrazhenii, Nauka, M., 1982

[31] B.A. Sevastyanov, Vetvyaschiesya protsessy, Nauka, M., 1971

[32] M. Kuczma, Functional equations in a single variable, Warszawa, 1968 | MR

[33] L.Lyung, Identifikatsiya sistem. Teoriya dlya polzovatelya, Nauka, M., 1991

[34] Congwen Liu, Lizhong Peng, “Generalized Helgason-Fourier transforms associated to variants of the Laplace-Beltrami operators on the unit ball in $\mathbb R^n$”, Indiana University Math. Journal, 58 (2009), 1457–1491 | DOI | MR

[35] M.I.Lighthill, G.B.Witham, “On kinematic waves”, Proc. of Royal Society, A229 (1955), 281–293 | DOI | MR

[36] V.I.Arnold, Matematicheskie metody klassicheskoi mekhaniki, Nauka, M., 1979

[37] S. Leibovich, A.R. Seebass (eds.), Nonlinear waves, Cornell university press, Ithaca–London, 1974 | MR

[38] R.Bott, “Nondegenerate critical manifolds”, Annals of math., 60 (1954), 248–261 | DOI | MR | Zbl

[39] D.Montgomery, H.Samelson, L.Zippin, “Singular points of a compact transformation group”, Annals of math., 63 (1956), 1–9 | DOI | MR | Zbl

[40] V.Guillemin, S.Sternberg, “Convexity properties of the moment mappings”, Inventions Mathematicae, 67 (1982), 491–513 | DOI | MR | Zbl

[41] V.P.Golubyatnikov, “Nekotorye kogomotopicheskie svoistva prostranstv Toma”, Sib. mat. zhurnal, 27 (1986), 201–205

[42] S.P.Novikov, “Gamiltonov formalizm i mnogoznachnyi analog teorii Morsa”, UMN, 37 (1982), 3–49 | MR

[43] A.S.Mischenko, A.T.Fomenko, “Integrirovanie gamiltonovykh sistem s nekommutativnymi simmetriyami”, Trudy seminara po vektornomu i tenzornomu analizu, 20, MGU, 1981, 5–54

[44] Yu.E.Anikonov, “Formuly dlya reshenii nekotorykh obratnykh zadach dlya evolyutsionnykh uravnenii”, DAN SSSR, 319 (1991), 1117–1119 | Zbl

[45] Yu.E.Anikonov, “Formuly v mnogomernoi obratnoi zadache dlya evolyutsionnogo uravneniya”, DAN RAN, 334 (1994), 263–264

[46] Yu.E.Anikonov, M.P.Vishnevskii, “Reduktsiya mnogomernykh obratnykh zadach k nachalno-kraevym zadacham v prostranstvakh Gilberta”, Sib. mat. zhurnal, 35 (1994), 495–514 | MR | Zbl

[47] Yu.E.Anikonov, M.P.Vishnevskii, “Formuly v obratnykh zadachakh dlya evolyutsionnogo uravneniya”, Sib. mat. zhurnal, 37 (1996), 963–976 | MR | Zbl

[48] N.B. Ayupova and V.P.Golubyatnikov, “Inverse problems for evolution equations and matrix Fourier transform”, Journal of Inverse and Ill-posed Problems, 5 (1997), 401–409 | DOI | MR | Zbl

[49] N.B.Ayupova and V.P.Golubyatnikov, “On formal solutions to multidimensional evolution equations”, Sib. Advances in Mathematics, 8 (1998), 21–40 | MR | Zbl

[50] Yu.E.Anikonov, “Formuly v odnoi zadache poiska simvola lineinogo operatora”, Sibirskie elektronnye matematicheskie izvestiya, 8 (2011), 163–167 | MR

[51] L.I.Ronkin, Vvedenie v teoriyu tselykh funktsii mnogikh peremennykh, Nauka, M., 1971