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Petrovsky lacuna 1Ivan Petrovsky 2

ReferencesArticle Sources and Contributors 5Image Sources, Licenses and Contributors 6

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Petrovsky lacuna 1

Petrovsky lacuna

Petrovsky lacunas are similar to the spacesbetween shock waves of a supersonic object.

In mathematics, a Petrovsky lacuna, named for the Russianmathematician I. G. Petrovsky, is a region where the fundamentalsolution of a linear hyperbolic partial differential equation vanishes.They were studied by Petrovsky (1945) who found topologicalconditions for their existence.

Petrovsky's work was generalized and updated by Atiyah, Bott, andGårding (1970, 1973).

References

• Atiyah, Michael Francis (1966-1968), "Hyperbolic differentialequations and algebraic geometry (after Petrowsky)" [1], SéminaireBourbaki, Vol. 10, Paris: Société Mathématique de France,pp. 87–99, MR1610456, Zbl 0201.12501.

• Atiyah, Michael Francis; Bott, Raoul; Gårding, Lars (1970), "Lacunas for hyperbolic differential operators withconstant coefficients. I" [2], Acta Mathematica 124: 109–189, doi:10.1007/BF02394570, MR0470499,Zbl 0191.11203.

• Atiyah, Michael Francis; Bott, Raoul; Gårding, Lars (1973), "Lacunas for hyperbolic differential operators withconstant coefficients. II" [3], Acta Mathematica 131: 145–206, doi:10.1007/BF02392039, MR0470500,Zbl 0266.35045.

• Petrovsky, I.G. (1945), "On the diffusion of waves and the lacunas for hyperbolic equations" [4] (in English),Recueil Mathématique (Matematicheskii Sbornik) [5] 17 (59) (3): 289–368, MR16861, Zbl 0061.21309.

References[1] http:/ / www. numdam. org/ item?id=SB_1966-1968__10__87_0[2] http:/ / www. springerlink. com/ content/ k53520330t634p71/[3] http:/ / www. springerlink. com/ content/ 073332016w411637/[4] http:/ / mi. mathnet. ru/ eng/ msb/ v59/ i3/ p289[5] http:/ / www. mathnet. ru/ msb

Ivan Petrovsky 2

Ivan Petrovsky

Ivan G. PetrovskyBorn 18 January 1901

Sevsk, Russian Empire

Died 15 January 1973 (aged 71)Moscow, USSR

Institutions Moscow State UniversitySteklov Institute of Mathematics

Alma mater Moscow State University

Doctoral advisor Dmitri Egorov

Doctoral students Olga LadyzhenskayaYevgeniy LandisOlga OleinikSergei GodunovAleksei Filippov

Ivan Georgievich Petrovsky, also Petrovskii (Russian: Ива́н Гео́ргиевич Петро́вский) (18 January 1901 – 15January 1973) was a Soviet mathematician in the field of partial differential equations, and studied Petrovskylacunas. He greatly contributed to solution of Hilbert's 19th and 16th problems. He also worked on the boundaryvalue problem, probability, and topology of algebraic curves and surfaces.Petrovsky was a student of Dmitri Egorov. Among his students were Olga Ladyzhenskaya, Yevgeniy Landis, OlgaOleinik and Sergei Godunov.Petrovsky taught at Steklov Institute of Mathematics. He was a member of the Soviet Academy of Sciences since1946 and was awarded Hero of Socialist Labor in 1969. He was the president of Moscow State University (1951 —1973) and the head of the International Congress of Mathematicians (Moscow, 1966). He is buried in the cemeteryof the Novodevichy Convent in Moscow.

Ivan Petrovsky 3

Grave of Ivan Petrovsky in the cemetery of the Novodevichy Convent in Moscow

Selected Publications

• Petrovsky, I. G. (1937), "Über dasCauchysche Problem für Systeme vonpartiellen Differentialgleichungen" [1] (inGerman), Recueil Mathématique(Matematicheskii Sbornik) [5] 2 (44) (5):815–870, JFM 63.0466.03,Zbl 0018.40503.

• Petrovsky, I. G. (1939), "Sur l'analyticitédes solutions des systèmes d'équationsdifférentielles" [2] (in French), RecueilMathématique (Matematicheskii Sbornik)[5] 5 (47) (1): 3–70, JFM 65.0405.02,MR1425, Zbl 0022.22601.

• Petrovsky, I. G. (1945), "On the diffusionof waves and the lacunas for hyperbolicequations" [4] (in English), RecueilMathématique (Matematicheskii Sbornik) [5] 17 (59) (3): 289–368, MR16861, Zbl 0061.21309.

• Petrowsky, I. G. (1996), Oleinik, O. A., ed., Selected works. Part I: Systems of partial differential equations andalgebraic geometry [3], Classics of Soviet Mathematics, 5 (part 1), Amsterdam: Gordon and Breach Publishers,ISBN 2-88124-978-7, MR1677652, Zbl 0948.01042.

• Petrowsky, I. G. (1996), Oleinik, O. A., ed., Selected works. Part II: Differential equations and probability theory[4], Classics of Soviet Mathematics, 5 (part 2), Amsterdam: Gordon and Breach Publishers, ISBN 2-88124-979-5,MR1677648, Zbl 0948.01043.

References• Aleksandrov, P. S.; Arnol'd, V. I.; Gel'fand, I. M.; Kolmogorov, A. N.; Novikov, S. P.; Oleinik, O. A. (2001),

"Ivan Georgievich Petrovskii" [5], in Osipov, Yu. S.; Sadovnichii, V. A. (in English), Differential Equations andRelated Topics - dedicated to the 100th Anniversary of I.G. Petrovskii [6], Moscow: Lomonosov State Universityand Steklov Mathematical Institute, pp. 1–18, retrieved 1 September 2009. A very ample paper describingPetrovsky's scientific research, authored by friends, collaborators and pupils.

• Aleksandrov, P. S.; Oleinik, O. A. (1981), "On the eightieth anniversary of the birth of Ivan GeorgievichPetrovskii" [7] (in Russian), Uspekhi Matematicheskikh Nauk 36 (1(217)): 3–10, MR0608939, Zbl 0454.01022:the original paper translated in (Alexandrov & Oleinik 1996), and also in the Russian Mathematical Surveys,1981, 36:1, 1–8 [8].

• Alexandrov, P. S.; Oleinik, O. A. (1996), "Ivan Georgievich Petrowsky" [9], in Oleinik, O. A., Selected works.Part II: Differential equations and probability theory [4], Classics of Soviet Mathematics, 5 (part 2), Amsterdam:Gordon and Breach Publishers, pp. 1–9, ISBN 2-88124-979-9 , MR1677648, Zbl 0948.01043: an Englishtranslation of the paper (Aleksandrov & Oleinik 1981).

• Kolmogorov, A. N. (1996), "Ivan Georgievich Petrowsky" [10], in Oleinik, O. A., Selected works. Part I: Systemsof partial differential equations and algebraic geometry, Classics of Soviet Mathematics, 5 (part 1), Amsterdam:Gordon and Breach Publishers, pp. 1–3, ISBN 2-88124-978-7, MR1677652, Zbl 0948.01042

• Lui, S. H. (1997), "An Interview with Vladimir Arnol´d" [11], Notices of the American Mathematical Society 44 (4): 432–438, Zbl 0913.01024. An interview with Vladimir Igorevich Arnol'd containing several important

Ivan Petrovsky 4

historical details about his teachers and other great mathematicians he knew when he was first studying and thenworking at the MSU Faculty of Mechanics and Mathematics, including Ivan Petrowsky.

• Oleinik, O. A. (1996), "I. G. Petrowsky and Modern Mathematics" [12], in Oleinik, O. A., Selected works. Part I:Systems of partial differential equations and algebraic geometry [3], Classics of Soviet Mathematics, 5 (part 1),Amsterdam: Gordon and Breach Publishers, pp. 4–30, ISBN 2-88124-978-7, MR1677652, Zbl 0948.01042

• Gårding, L. (1996), "Ivan Georgievich Petrowsky and Partial Differential Equations" [13], in Oleinik, O. A.,Selected works. Part I: Systems of partial differential equations and algebraic geometry [3], Classics of SovietMathematics, 5 (part 1), Amsterdam: Gordon and Breach Publishers, pp. 31–39, ISBN 2-88124-978-7,MR1677652, Zbl 0948.01042.

• Vol'pert, A. I. (1996), "Propagation of Waves Described by Nonlinear Parabolic Equations (a commentary onarticle 6)" [9], in Oleinik, O. A., Selected works. Part II: Differential equations and probability theory [4], Classicsof Soviet Mathematics, 5 (part 2), Amsterdam: Gordon and Breach Publishers, pp. 364–399,ISBN 2-88124-979-9 , MR1677648, Zbl 0948.01043

External links• Ivan Petrovsky [14] at the Mathematics Genealogy Project• O'Connor, John J.; Robertson, Edmund F., "Ivan Petrovsky" [15], MacTutor History of Mathematics archive,

University of St Andrews.• Short Biography of Petrowsky [16] – from the Moscow Mathematical Journal

References[1] http:/ / mi. mathnet. ru/ eng/ msb/ v44/ i5/ p815[2] http:/ / mi. mathnet. ru/ eng/ msb/ v47/ i1/ p3[3] http:/ / books. google. com/ ?id=hmGQE27hIJgC& printsec=frontcover#v=onepage& q=[4] http:/ / books. google. com/ ?id=FdCuk2YumowC& printsec=frontcover#v=onepage& q=[5] http:/ / www. math. msu. su/ department/ tffa/ petr100/ PETROVSK. PS[6] http:/ / www. math. msu. su/ department/ tffa/ petr100/[7] http:/ / mi. mathnet. ru/ eng/ umn/ v36/ i1/ p3[8] http:/ / dx. doi. org/ 10. 1070/ RM1981v036n01ABEH002539[9] http:/ / books. google. it/ books?id=GsJALnwAvm8C& pg=PA1& dq=Ivan+ Georgievich+ Petrowsky+ + Alexandrov+ Oleinik#v=onepage&

q=Ivan%20Georgievich%20Petrowsky%20%20Alexandrov%20Oleinik& f=false[10] http:/ / books. google. it/ books?id=hmGQE27hIJgC& printsec=frontcover#v=onepage& q=& f=false[11] http:/ / www. ams. org/ notices/ 199704/ arnold. pdf[12] http:/ / books. google. it/ books?id=hmGQE27hIJgC& pg=PA4& dq=I. G. + Petrowsky+ and+ modern+ mathematics#v=onepage& q=I. G.

%20Petrowsky%20and%20modern%20mathematics& f=true[13] http:/ / books. google. it/ books?id=i-uAucjb3h0C& pg=PA31& dq=Ivan+ Georgievich+ Petrowsky+ and+ Partial+ Differential+

Equations#v=onepage& q=Ivan%20Georgievich%20Petrowsky%20and%20Partial%20Differential%20Equations& f=false[14] http:/ / genealogy. math. ndsu. nodak. edu/ id. php?id=55146[15] http:/ / www-history. mcs. st-andrews. ac. uk/ Biographies/ Petrovsky. html[16] http:/ / www. ams. org/ distribution/ mmj/ vol1-4-2001/ dedication. html

Article Sources and Contributors 5

Article Sources and ContributorsPetrovsky lacuna  Source: http://en.wikipedia.org/w/index.php?oldid=533646862  Contributors: Adamdetx, Chillum, Daniele.tampieri, Giftlite, Michael Hardy, R.e.b., 2 anonymous edits

Ivan Petrovsky  Source: http://en.wikipedia.org/w/index.php?oldid=533647033  Contributors: Alex Bakharev, Blablabla, Carel.jonkhout, CaritasUbi, Daniele.tampieri, El Roih, Geschichte,Giftlite, Glane23, Harris7, JeanneMish, Jsqqq777, LarRan, Lockley, Masterpiece2000, Mhym, Nick Number, Omnipaedista, Pedron, R.e.b., Rapidtear, Salih, Sk741, Sodin, Vald, Zloyvolsheb, 7anonymous edits

Image Sources, Licenses and Contributors 6

Image Sources, Licenses and ContributorsImage:Schlierenfoto Mach 1-2 Pfeilflügel - NASA.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:Schlierenfoto_Mach_1-2_Pfeilflügel_-_NASA.jpg  License: Public Domain Contributors: NASAFile:I g petrovsky grave.png  Source: http://en.wikipedia.org/w/index.php?title=File:I_g_petrovsky_grave.png  License: Creative Commons Zero  Contributors: CaritasUbi

License 7

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