Geodesic
Normalized form and resonances of matrix-valued functions of two variables
Problemy fiziki, matematiki i tehniki, no. 4 (2017), pp. 33-39 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Matrix-functions that arise when solving systems of differential equations with Delta-shaped coefficients are investigated. The process of reducing the matrix-function $F(\mu,\varepsilon)$ is considered depending on two variables to the normal form by means of the matrix functions G and T such that their elements belong to a ring wide then the ring containing elements of $F(\mu,\varepsilon)$. The explicit form of the main term of expansion $[F(\mu,\varepsilon)]^{-1}$ in the case of matrices of dimension $2$ is found explicitly. The cases of resonance for systems with delta-coefficients are revealed.
Keywords: matrix-function, normalized form, resonance, ring, main term of expansion. 
@article{PFMT_2017_4_a5,
     author = {A. B. Antonevich and M. G. Kot},
     title = {Normalized form and resonances of matrix-valued functions of two variables},
     journal = {Problemy fiziki, matematiki i tehniki},
     pages = {33--39},
     year = {2017},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PFMT_2017_4_a5/}
}
TY  - JOUR
AU  - A. B. Antonevich
AU  - M. G. Kot
TI  - Normalized form and resonances of matrix-valued functions of two variables
JO  - Problemy fiziki, matematiki i tehniki
PY  - 2017
SP  - 33
EP  - 39
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/PFMT_2017_4_a5/
LA  - ru
ID  - PFMT_2017_4_a5
ER  - 
%0 Journal Article
%A A. B. Antonevich
%A M. G. Kot
%T Normalized form and resonances of matrix-valued functions of two variables
%J Problemy fiziki, matematiki i tehniki
%D 2017
%P 33-39
%N 4
%U http://geodesic.mathdoc.fr/item/PFMT_2017_4_a5/
%G ru
%F PFMT_2017_4_a5
A. B. Antonevich; M. G. Kot. Normalized form and resonances of matrix-valued functions of two variables. Problemy fiziki, matematiki i tehniki, no. 4 (2017), pp. 33-39. http://geodesic.mathdoc.fr/item/PFMT_2017_4_a5/

[1] M.I. Vishik, L.A. Lyusternik, “Resheniya nekotorykh zadach o vozmuschenii v sluchae matrits i samosopryazhennykh i nesamosopryazhennykh differentsialnykh uravnenii”, Uspekhi matem. nauk, 15:3(93) (1960), 3–80

[2] T.A. Romanchuk, “Yavlenie rezonansa dlya matrichno-znachnykh funktsii”, Vestsi NAN Belarusi. Ser. fiz.-mat. navuk, 2008, no. 2, 8–16

[3] A.B. Antonevich, T.A. Romanchuk, Uravneniya s delta-obraznymi koeffitsientami: metod konechnomernykh approksimatsii, LAP LAMBERT, 2012

[4] T. Kato, Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972, 740 pp.

[5] A.G. Kurosh, Kurs vysshei algebry, Nauka, M., 1965, 431 pp.

[6] E.V. Vinberg, Kurs algebry, 2-e izd., ispr. i dop., Izd-vo Faktorial Press, M., 2001, 544 pp.

[7] V.T. Borukhov, “Kriterii obratimosti lineinykh statsionarnykh mnogomernykh sistem”, Avtomatika i telemekhanika. AN SSSR, 1978, no. 11, 5–11

[8] F.R. Gantmakher, Teoriya matrits, 5-e izd., Fizmatlit, M., 2004, 560 pp.

[9] M.G. Kot, “O rezolventnoi skhodimosti operatorov, approksimiruyuschikh sistemu uravnenii s $\delta$-obraznymi koeffitsientami”, Vestnik BGU. Seriya 1. Fizika. Matematika. Informatika, 2015, 111–117