01.01.02 (Differential equations, dynamical systems, and optimal control)
Birth date:
17.09.1964
E-mail:
Keywords:
group analysis, kinetic equations, momentum system, wave equation in nonhomogeneous media,
eiconal,
differential equation on network,
oscillation spectral theorems,
Green function of boundary value problem,
sign-regularity of discontinuous boundary value problem,
summarising of Fourie series in the order of coefficients decreasing,
constructive non-standard analysis.
Subject:
Theory of oscillation spectral properties. The analogue of Kellogg oscillation theorem is obtained for integral operators with discontinuous kernels and with integrating by measure is obtained. The effective conditions of sign-regularity (of Kalafaty type) of discontinuous boundary value problems are discovered. Theory of differential equations on networks. For forth order equations on the network the maximum principle is discovered. The system of determinating conditions for Green"s function of the forth order equation is found, the positivity if Green function proved. The method of the reduction of the problem on the network to a problem on an interval developed. Summarising of Fourie series. The formula of the maximal coeffitient in Fourie series and the formula of its number is obtained. Wave propagation in nonhomogeneous media. The formula of propagating waves for one-dimentional media is obtained. The group classification of eiconal equation for non-homogeneous and anisotropic medium fulfilled. Constructive non-standard analysis. The set of hyperreal values allowing the analysis of singularly perturbed equations is made constractively.Group analysis of kinetic equations and the problem of the closure of the momentum system.
Biography
Graduated from Voronezh State University (Mathematical Department) in 1986. Candidate (Ph.D.) thesis "Boundary value problems with singularities" — 1990. About 20 referred publications.
Since 1996 — sci. secr. of Voronezh Spring Mathematical Schools and of Voronezh Winter Schools on the Function Theory.
Main publications:
Borovskikh A. V., Pokornyi Yu. V., “Sistemy Chebysheva–Khaara v teorii razryvnykh yader Kelloga”, Uspekhi matem. nauk, 49:3 (1994), 3–42
Borovskikh A.V., “Uravneniya eikonala dlya neodnorodnoi
i anizotropnoi sredy”, Uravneniya matematicheskoi fiziki, Sovremennaya matematika i ee prilozheniya, 64, 18–38
A. V. Borovskikh, “Geometry of the Lie group: Riemann and Ricci tensors and normal forms of Lie algebras”, TMF (to appear)
2.
A. V. Borovskikh, “Lie group geometry. Invariant metrics and dynamical systems, dual algebra, and their applications in the group analysis of a one-dimensional kinetic equation”, Theoret. and Math. Phys., 217:1 (2023), 1528–1540
3.
K. S. Platonova, A. V. Borovskikh, “Group analysis of the one-dimensional Boltzmann
equation. Invariants and the problem of moment system closure”, Theoret. and Math. Phys., 208:3 (2021), 1165–1181
4.
K. S. Platonova, A. V. Borovskikh, “Group analysis of the Boltzmann and Vlasov equations”, Theoret. and Math. Phys., 203:3 (2020), 794–823
5.
A. V. Borovskikh, K. S. Platonova, “Group analysis of the one-dimensional Boltzmann equation: IV. Complete group classification in the general case”, Theoret. and Math. Phys., 201:2 (2019), 1612–1642
6.
K. S. Platonova, A. V. Borovskikh, “Group analysis of the one-dimensional Boltzmann equation: III. Condition for the moment quantities to be physically meaningful”, Theoret. and Math. Phys., 195:3 (2018), 886–915
7.
I. V. Astashova, A. V. Borovskikh, V. V. Bykov, A. N. Vetokhin, A. Yu. Goritskii, N. A. Izobov, Yu. S. Ilyashenko, T. O. Kapustina, V. V. Kozlov, A. A. Kon'kov, I. V. Matrosov, V. V. Palin, N. Kh. Rozov, M. S. Romanov, I. N. Sergeev, E. V. Radkevich, O. S. Rozanova, I. V. Filimonova, A. V. Filinovskii, G. A. Chechkin, A. S. Shamaev, T. A. Shaposhnikova, “Scientific heritage of Vladimir Mikhailovich Millionshchikov”, J. Math. Sci. (N. Y.), 210:2 (2015), 115–134
8.
A. V. Borovskikh, “Eikonal equation for anisotropic media”, J. Math. Sci. (N. Y.), 197:2 (2014), 248–289
Borovskikh A.V., “Formulas of boundary control of an inhomogeneous string: I”, Differential Equations, 43:1 (2007), 69-95
25.
Borovskikh A.V., “Formulas of boundary control of an inhomogeneous string: II”, Differential Equations, 43:5 (2007), 656-666
26.
Borovskikh A.V., Rozov N.Kh., “O materialnoi i simvolnoi kulture v shkolnom obrazovanii”, Teoriya i praktika innovatsionnoi deyatelnosti v obrazovanii, MAKS Press, Moskva, 2007, 14-17
27.
Borovskikh A.V., Reikhani E., Rozov N.Kh., “Razvitie geometricheskogo myshleniya shkolnikov”, Teoriya i praktika innovatsionnoi deyatelnosti v obrazovanii, MAKS Press, Moskva, 2007, 28-46
28.
Borovskikh A.V., Reikhani E., Rozov N.Kh., “Uzly v shkole”, Teoriya i praktika innovatsionnoi deyatelnosti v obrazovanii, MAKS Press, Moskva, 2007, 47-62
A. V. Borovskikh, “The two-dimensional eikonal equation”, Siberian Math. J., 47:5 (2006), 813–834
32.
Borovskikh A.V., “Equivalence groups of eikonal equation and classes of equivalent equations”, Bulletin of NSU, 2006, no. 4, 3-42
33.
A. V. Borovskikh, “Equivalence groupes of eikonal equation and classes of equivalent equations”, Vestnik, Quart. J. of Novosibirsk State Univ., Series: Math., mech. and informatics, 6:4 (2006), 3–42
34.
Borovskikh A.V., “An expression of the Riemann function of the wave equation in an Inhomogeneous medium via the transport coefficients”, Differential equations, 41:6 (2005), 895-896
35.
Borovskikh A.V., “Group analysis of the two-dimentional eikonal equation”, Differential equations, 41:6 (2005), 899
A. V. Borovskikh, “Group classification of the eikonal equation for a 3-dimensional inhomogeneous medium”, Sb. Math., 195:4 (2004), 479–520
38.
Borovskikh A.V., “Investigation of Relaxation Oscillations with the Use of Constructive Nonstandard Analysis: I.”, Differential Equations, 40:3 (2004), 309-317
39.
Borovskikh A.V., “Investigation of Relaxation Oscillations with the Use of Constructive Nonstandard Analysis: II”, Differential Equations, 40:4 (2004), 491-501
40.
Borovskikh A.V., “Method of Propagating Waves for a One-Dimensional Inhomogeneous Medium”, Journal of Mathematical Sciences, 127:5 (2005), 2135-2158
41.
Borovskikh A.V., “The Moving Source Phantom in Geometric Optics of Inhomogeneous Media”, Differential Equations, 40:7 (2004), 927-933
42.
Borovskikh A.V., “On a certain Luzin problem”, Analysis. Mathematica, 30:4 (2004), 259-287
43.
A. V. Borovskikh, “Sign Regularity Conditions for Discontinuous Boundary-Value Problems”, Math. Notes, 74:5 (2003), 607–618
44.
Pokornyi Y.V., Pryadiev V.L., Borovskikh A.V., “The wave equation on a spatial network”, Doklady Mathematics, 67:1 (2003), 10-12
45.
Borovskikh A.V., “The eiconal equation in an inhomogeneous medium”, Doklady Mathematics, 68:1 (2003), 103-106
46.
Borovskikh A.V., “Group Classification of Eikonal Equations for the Wave Equation in Inhomogeneous Media”, Journal of Mathematical Sciences, 129:1 (2005), 3537-3549
47.
Borovskikh A.V., “A Traveling Wave Formula in a One-Dimensional Inhomogeneous Medium”, Differential Equations, 38:6 (2002), 796-807
48.
Borovskikh A.V., “Group classification of eikonal equation for an inhomogeneous medium”, Differential Equations, 38:11 (2002), 1669-1670
49.
Borovskikh A.V., “Geomerty of fronts and rays in a medium with linear and quadratic velocity function”, Differential Equations, 38:11 (2002), 1673
50.
Borovskikh A.V., “Application of constructive non-standard analysis to investigation of a system of connected Van-der-Pol equations”, Differential Equations, 38:11 (2002), 1678-1679
51.
Pokornyi Yu.V., Borovskikh A.V., “On defects of axiomatics of Green's function”, Doklady Mathematics, 65:3 (2002), 399-403
52.
Pokornyi Yu.V., Borovskikh A.V., “Differential equations on networks (geometrical graphs)”, Journal of Mathematical Sciences, 119:6 (2004), 691-718
53.
Borovskikh A.V., Lazarev K.P., “Differential equations of the forth order on geometrical graphs”, Journal of Mathematical Sciences, 119:6 (2004), 719-738
54.
Pokornyi Yu.V., Borovskikh A.V., “Connection between the Green's function and the influence function for the non-classical problems”, Journal of Mathematical Sciences, 119:6 (2004), 739-768
55.
Borovskikh A.V., “Konstruktivnyi nestandartnyi analiz”, Analiticheskie i chislennye metody v matematike i mekhanike. Tr. XXII konf. molodykh uchenykh mekh.-mat. f-ta MGU. (Moskva, 17-22 aprelya 2000 g.), izd-voMGU, Moskva, 2001, 28-31
A. V. Borovskikh, K. P. Lazarev, Yu. V. Pokornyi, “On the Kellogg kernels in discontinuous problems”, Proc. Steklov Inst. Math., 211 (1995), 92–108
73.
Borovskikh A.V., “Ob ekstremumakh razryvnykh sistem Chebysheva i Markova”, Tez. dokl. shkoly “Sovremennye metody teorii funktsii i smezhnye problemy prikladnoi matematiki i mekhaniki” (Voronezh, 25 yanvarya – 1 fevralya 1995 g.), VGU, Voronezh, 1995, 48
74.
Borovskikh A.V., Kolmykov V.A., “Integrirovanie po diskretnym meram na grafakh”, Tez. dokl. shkoly “Sovremennye metody teorii funktsii i smezhnye problemy prikladnoi matematiki i mekhaniki” (Voronezh, 25 yanvarya – 1 fevralya 1995 g.), VGU, Voronezh, 1995, 49
Borovskikh A.V., Burlutskaya M.Sh., “O differentsirovanii i integrirovanii po mere na abstraktnykh lineinykh kontinuumakh i o kvazichislakh”, Tez.dokl. II mezhdun. konferentsii “Algebraicheskie, veroyatnostnye, geometricheskie, kombinatornye i funktsionalnye metody v teorii chisel” (Voronezh, 25-30 sentyabrya 1995 g.), VGU, Voronezh, 1995, 25
81.
Borovskikh A.V., Mustafokulov R., Lazarev K.P., Pokornyi Yu.V., “A class of fourth-order differential equations on a spatial net”, Doklady Mathematics, 52:3 (1995), 433-435
82.
A. V. Borovskikh, Yu. V. Pokornyi, “Chebyshev–Haar systems in the theory of discontinuous Kellogg kernels”, Russian Math. Surveys, 49:3 (1994), 1–42
83.
Borovskikh A.V., Lazarev K.P., “Matematicheskaya model sterzhnevoi sistemy”, Tez.dokl. shkoly “Sovremennye problemy mekhaniki i matematicheskoi fiziki” (Voronezh, 21-28 yanvarya 1994 g..), VGU, Voronezh, 1994, 17
Borovskikh A.V., “Group analysis of eikonal equation for non-homogeneous and anisotropic medium”, Abstr. of Intern. Conf. “Symmetry in non-linear physics” (Kiev, 21-27 iyunya 2009 g.), IM NAN Ukrainy, Kiev, 2009, po ssylke http://imath.kiev.ua/~appmath/
Borovskikh A.V., “Illusion of the moving source in geometric optics of non-homogeneous medium”, International Seminar “Days on Diffraction”. Abstracts. (S.-Peterburg, 24-27 iyunya 2003 g.), S.-Pb otdelenie MIRAN, S.-Peterburg, 2003, 17
122.
Borovskikh A.V., “Summirovanie primera Feiera v poryadke ubyvaniya koeffitsientov”, Tez. dokl. shkoly “Sovremennye problemy teorii funktsii i ikh prilozheniya” (Saratov, 27 yanvarya – 3 fevralya 2002 g.), Saratovskii gosuniversitet, Saratov, 2002, 36
123.
Borovskikh A.V., Lazarev K.P., “Printsip maksimuma dlya differentsialnykh uravnenii chetvertogo poryadka na grafakh”, Tez. dokl. shkoly “Sovremennye problemy teorii funktsii i ikh prilozheniya” (Saratov, 27 yanvarya – 3 fevralya 2002 g.), Saratovskii gosuniversitet, Saratov, 2002, 37
Borovskikh A.V., Rozov N.Kh., “O bednom protsente zamolvite slovo...”, Matematika v shkole, 2010, no. 3, 3-15
144.
Borovskikh A.V., Rozov N.Kh., “Nekotorye mysli o pedagogicheskikh realiyakh”, Vestnik Uralskogo instituta ekonomiki, upravleniya i prava., 2010, no. 4, 92-97
145.
Borovskikh A.V., Perov A.I., “O nekotorykh metodicheskikh printsipakh uchebnika po obyknovennym differentsialnym uravneniyam”, Matematika v vysshem obrazovanii, 2008, no. 6, 45-49
146.
Rozov N.Kh., Reikhani E., Borovskikh A.V., V'zlite v uchilische. Urotsi za razvivane na prostranstvenogo mislene, Fondatsiya “Mlada B'lgariya”, Sofiya, 2008 , 162 pp., (na bolgarskom yazyke)
147.
Rozov N.Kh, Reihani E., Borovskikh A.V., Knots at school. Lessons of the development of the spatial thinking, KDU, Moscow, 2007
148.
Borovskikh A.V., Perov A.I., Lectures on ordinary differential equations, 1, RC “Regular and chaotic dynamics” Institute of Computer Research, Moscow-Izhevsk, 2004 , 540 pp.
149.
Astakhova I.F., Borovskikh A.V., Podbolotov B.N., Lektsii pro FoxPro, Izd-vo TOO MP “Elist”, Voronezh, 1996 , 199 pp.
Information matherials
150.
Borovskikh A.V., Rozov N.Kh., Shamaev A.S., “Pervyi opyt organizatsii pedagogicheskoi sektsii na konferentsii, posvyaschennoi pamyati I.G.Petrovskogo”, Matematika v vysshem obrazovanii, 2008, no. 8, 9-10
151.
Borovskikh A.V., Il'in V.A., Pokornyi Yu.V., ““Pontryagin Readings – XV” at Voronezh Mathematical School “Modern methods in Qualitative Theory of Boundary Value Problems””, Differential Equations, 40:11 (2004), 1665-1666
152.
P. L. Ul'yanov, Yu. V. Pokornyi, A. V. Borovskikh, “Contemporary methods in theory of functions and adjacent problems”, Uspekhi Mat. Nauk, 58:4(352) (2003), 177–178
153.
Il'in V.A., Pokornyi Yu.V., Borovskikh A.V., ““Pontryagin Readings – XIV” at Voronezh Mathematical School “Modern methods in Qualitative Theory of Boundary Value Problems””, Differential Equations, 39:11 (2003), 1661-1662
154.
V. A. Il'in, Yu. V. Pokorny, A. V. Borovskikh, Differ. Uravn., 39:11 (2003), 1574
155.
Yu. V. Pokornyi, P. L. Ul'yanov, A. V. Borovskikh, “Voronezh Winter Mathematical School “Modern Methods in Theory of Functions and Adjacent Problems””, Uspekhi Mat. Nauk, 56:4(340) (2001), 185–186
156.
P. L. Ul'yanov, Yu. V. Pokornyi, A. V. Borovskikh, ““Modern Methods in Theory of Functions and Adjacent Problems” on Voronezh Winter Mathematical School”, Uspekhi Mat. Nauk, 55:2(332) (2000), 189–190
Personalia
157.
I. V. Astashova, A. V. Borovskikh, V. V. Bykov, A. N. Vetokhin, A. Goritsky, N. V. Denisova, Yu. S. Ilyashenko, T. O. Kapustina, A. A. Kon'kov, V. V. Palin, E. V. Radkevich, V. V. Rogachev, O. S. Rozanova, M. S. Romanov, I. N. Sergeev, I. V. Filimonova, A. V. Filinovskii, G. A. Chechkin, A. S. Shamaev, M. V. Shamolin, T. A. Shaposhnikova, Tr. Semim. im. I. G. Petrovskogo, 33, 2023, 3–7
158.
I. V. Astashova, A. V. Borovskikh, G. Yu. Riznichenko, “In memory of Rozov Nikolai Khristovich”, Computer Research and Modeling, 12:6 (2020), 1265–1267
I. V. Astashova, A. V. Borovskikh, V. V. Bykov, A. N. Vetokhin, V. V. Kozlov, V. A. Kondratiev, A. A. Kon'kov, N. Kh. Rozov, I. N. Sergeev, V. N. Chubarikov, A. T. Fomenko, A. S. Shamaev, “To memory of Vladimir Mikhailovich Millionshchikov”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 2010, no. 2, 71–72
161.
I. V. Astashova, L. A. Bagirov, A. V. Borovskikh, V. V. Bykov, A. N. Vetokhin, A. Yu. Goritskii, G. V. Grishina, N. V. Denisova, Yu. S. Ilyashenko, T. O. Kapustina, V. V. Kozlov, A. A. Kon'kov, I. V. Matrosov, V. M. Millionshchikov, V. A. Nikishkin, E. V. Radkevich, O. S. Rozanova, È. R. Rozendorn, N. Kh. Rozov, V. A. Sadovnichii, V. S. Samovol, I. N. Sergeev, I. V. Filimonova, A. V. Filinovskii, A. F. Filippov, T. S. Khachlaev, G. A. Chechkin, A. S. Shamaev, T. A. Shaposhnikova, “Vladimir Alexandrovich Kondratiev on the 70th anniversary of his birth”, J. Math. Sci. (N. Y.), 143:4 (2007), 3183–3197
Содержание математического образования: математика для не-математиков A. V. Borovskikh VI International Conference "Function Spaces. Differential Operators. Problems of Mathematical Education", dedicated to the centennial anniversary of the corresponding member of Russian Academy of Sciences, academician of European Academy of Sciences L.D. Kudryavtsev November 16, 2023 16:30