RSS feeds
РусскийEnglish
Issue 12, 2024
Volume year page
Issues Authors PACS Subscription For authors

Issues

 / 

2023

 / 

March

←  →

Reviews of topical problems


Nonlocal (fractional-differential) model of cosmic ray transport in the interstellar medium

†  a, ‡  b, §  c
a Ul'yanovsk State University, ul. L. Tolstogo 42, Ulyanovsk, 432700, Russian Federation
b Lebedev Physical Institute, Russian Academy of Sciences, Leninsky prosp. 53, Moscow, 119991, Russian Federation
c Moscow Institute of Physics and Technology (National Research University), Institutskii per. 9, Dolgoprudny, Moscow Region, 141701, Russian Federation

Mathematical modeling of the propagation of charged particles in the interstellar medium is a key aspect of the physics of cosmic rays. Improvements in measuring systems and enhancement of statistics via the results of extraterrestrial astronomy stimulate regular revisions of the computational models. Most of the currently used approaches are based on the local model of normal diffusion, which is attracting growing criticism. Reservations about this model arise because, first, it is in conflict with the relativistic principle of maximum speed and, second, it ignores specific spatial correlations in the properties of the turbulent interstellar medium. In this regard, in a previous review article by the first author (Physics-Uspekhi 56 1074 (2013)), some arguments were adduced in favor of a nonlocal model of cosmic ray transport based on fractional differential operators. In this review, we continue expounding on the physical and mathematical properties of the fractional differential theory, with more attention focused on the procedure for incorporating the features of the interstellar medium into nonlocal operators. New results obtained in the framework of this approach are presented, compared, and discussed.

Fulltext pdf (3.6 MB)
Fulltext is also available at DOI: 10.3367/UFNe.2021.12.039118
Keywords: cosmic rays, nonlocal model, fractional differential operator, turbulence, fractal medium, energy spectrum, anisotropy
PACS: 02.50.−r, 05.40.Fb, 98.70.Sa (all)
DOI: 10.3367/UFNe.2021.12.039118
URL: https://ufn.ru/en/articles/2023/3/a/
001096987300001
2-s2.0-85182874677
2023PhyU...66..221U
Citation: Uchaikin V V, Erlykin A D, Sibatov R T "Nonlocal (fractional-differential) model of cosmic ray transport in the interstellar medium" Phys. Usp. 66 221–262 (2023)
BibTexBibNote ® (generic)BibNote ® (RIS)MedlineRefWorks

Received: 16th, July 2021, revised: 6th, October 2021, 3rd, December 2021

Оригинал: Учайкин В В, Ерлыкин А Д, Сибатов Р Т «Нелокальная (дробно-дифференциальная) модель переноса космических лучей в межзвёздной среде» УФН 193 233–278 (2023); DOI: 10.3367/UFNr.2021.12.039118

References (183) ↓ Cited by (2) Similar articles (20)

  1. Uchaikin V V Phys. Usp. 56 1074 (2013); Uchaikin V V Usp. Fiz. Nauk 183 1175 (2013)
  2. Ginzburg V L Usp. Fiz. Nauk 51 343 (1953)
  3. Ginzburg V L, Syrovatskii S I The Origin Of Cosmic Rays (Oxford: Pergamon Press, 1964); Translated from Russian, Ginzburg V L, Syrovatskii S I Proiskhozhdenie Kosmicheskikh Luchei (Moscow-Leningrad: Izd. AN SSSR, 1963)
  4. Parker E N Planet. Space Sci. 13 9 (1965)
  5. Tjus J B, Merten L Phys. Rep. 872 1 (2020)
  6. Chuvilgin L G, Ptuskin V S Astron. Astrophys. 279 278 (1993)
  7. Zybin K P, Istomin Ya N Sov. Phys. JETP 62 479 (1985); Zybin K P, Istomin Ya N Zh. Eksp. Teor. Fiz. 89 836 (1985)
  8. Shalchi A, Kourakis I Phys. Plasmas 14 092903 (2007)
  9. Shalchi A, Kourakis I Astron. Astrophys. 470 405 (2007)
  10. Osborne J L, Ptuskin V S Sov. Astron. Lett. 13 413 (1987); Osborne J L, Ptuskin V S Pis’ma Astron. Zh. 13 980 (1987)
  11. Ptuskin V S, Soutoul A Astron. Astrophys. 237 445 (1990)
  12. Peebles P J E The Large-Scale Structure Of The Universe (Princeton, NJ: Princeton Univ. Press, 1980)
  13. Uchaikin V, Sibatov R Fractional Kinetics In Space: Anomalous Transport Models (Singapore: World Scientific, 2018)
  14. Dorman L I Cosmic Ray Interactions, Propagation, And Acceleration In Space Plasmas (Astrophysics and Space Science Library) Vol. 339 (Dordrecht: Springer, 2006)
  15. Uchaikin V V J. Exp. Theor. Phys. 97 810 (2003); Uchaikin V V Zh. Eksp. Teor. Fiz. 124 903 (2003)
  16. Uchaikin V V J. Appl. Math. Phys. 3 (2) 187 (2015)
  17. Borgani S Phys. Rep. 251 1 (1995)
  18. Baryshev Yu, Teerikorpi P Discovery Of Cosmic Fractals (River Edge, NJ: World Scientific, 2002)
  19. Combes F Celestial Mech. Dyn. Astron. 72 91 (1998)
  20. Combes F "Astrophysical fractals: interstellar medium and galaxies" The Chaotic Universe. Proc. of the Second ICRA Network Workshop, Rome, Pescara, Italy, 1-5 February 1999 (Advanced Series in Astrophysics and Cosmology) Vol. 10 (Eds V G Gurzadyan, R Ruffini) (Singapore: World Scientific, 2000) p. 143
  21. Uchaikin V V, Korobko D A, Gismyatov I F Russ. Phys. J. 40 711 (1997); Uchaikin V V, Korobko D A, Gismyatov I F Izv. Vyssh. Uchebn. Zaved. Fiz. (8) 7 (1997)
  22. Parisi G, Frisch U A Turbulence And Predictability In Geophysical Fluid Dynamics And Climate Dynamics (Proc. of the Intern. School of Physics "Enrico Fermi", Course 88, Eds M Ghil, R Benzi, G Parisi) (Amsterdam: North-Holland, 1985)
  23. Zelenyi L M, Milovanov A V Geomagn. Aeron. 33 (4) 18 (1993)
  24. Beck C, Schlögl F Thermodynamics Of Chaotic Systems: An Introduction (Cambridge: Cambridge Univ. Press, 1993)
  25. Milovanov A V et al Cosmic Res. 34 415 (1996); Milovanov A V et al Kosmich. Issled. 34 451 (1996)
  26. Badii R, Politi A Complexity: Hierarchical Structures And Scaling In Physics (Cambridge: Cambridge Univ. Press, 1997)
  27. Meakin P Fractals, Scaling, And Growth Far From Equilibrium (Cambridge: Cambridge Univ. Press, 1998)
  28. Harte D Multifractals: Theory And Applications (Boca Raton, FL: Chapman and Hall. CRC, 2001)
  29. Zelenyi L M, Milovanov A V Phys. Usp. 47 749 (2004); Zelenyi L M, Milovanov A V Usp. Fiz. Nauk 174 809 (2004)
  30. Kolmogorov A N J. Fluid Mech. 13 (1) 82 (1962)
  31. Novikov E A, Styuart R V Izv. Akad. Nauk SSSR. Geofiz. (3) 408 (1964)
  32. Zybin K P, Sirota V A Phys. Rev. E 88 043017 (2013)
  33. Frisch U Turbulence: The Legacy Of A.N. Kolmogorov (Cambridge: Cambridge Univ. Press, 1995); Translated into Russian, Frisch U Turbulentnost’. Nasledie A.N. Kolmogorova (Moscow: Fazis, 1998) p. 155
  34. Boffetta G, Mazzino A, Vulpiani A J. Phys. A 41 363001 (2008)
  35. Stell G Lect. Appl. Mathem. 27 109 (1991)
  36. Uchaikin V, Gusarov G J. Math. Phys. 38 2453 (1997)
  37. Uchaikin V, Gusarov G J. Math. Sci. 83 439 (1997)
  38. Coleman P H, Pietronero L Phys. Rep. 213 311 (1992)
  39. Uchaikin V V Gen. Relat. Grav. 36 1689 (2004)
  40. Uchaikin V V Grav. Cosmol. 10 5 (2004)
  41. Uchaikin V V, Zolotarev V M Chance And Stability: Stable Distributions And Their Applications (New York: Walter de Gruyter, 1999)
  42. Uchaikin V V Method Of Fractional Derivatives (Ulyanovsk: Artishok, 2008)
  43. Davis A B, Xu F J. Comput. Theor. Transport 43 474 (2014)
  44. Gorbatskii V G Gazodinamicheskie Neustoichivosti V Astrofizicheskikh Sistemakh (Gas-Dynamic Instabilities In Astrophysical Systems) (St. Petersburg: Izd. Sankt-Peterburgskogo Univ., 1999)
  45. Gorbatskii V G, Tarakanov P A Astrophysics 41 53 (1998); Gorbatskii V G, Tarakanov P A Astrofiz. 41 81 (1998)
  46. Dudorov A E Sov. Astron. 35 342 (1991); Dudorov A E Astron. Zh. 68 695 (1991)
  47. Berezinskii V S et al Astrophysics Of Cosmic Rays (Ed. V L Ginzburg) (Amsterdam: North-Holland, 1990); Translated from Russian, Berezinskii V S et al Astrofizika Kosmicheskikh Luchei (Ed. V L Ginzburg) (Moscow: Nauka, 1990)
  48. Inoue T, Yamazaki R, Inutsuka S Astrophys. J. 695 825 (2009)
  49. Stone J M, Norman M L Astrophys J. Suppl. 80 753 (1992)
  50. Clarke D A Astrophys. J. 457 291 (1996)
  51. Gaite J Europhys. Lett. 71 332 (2005)
  52. Burkhart B et al Astrophys. J. 771 (2) 123 (2013)
  53. Hayakawa S Cosmic Ray Physics: Nuclear And Astrophysical Aspects (Interscience Monographs and Texts in Physics and Astronomy) Vol. 22 (New York: Wiley-Interscience, 1969)
  54. Kol’chuzhkin A M, Uchaikin V V Vvedenie V Teoriyu Prokhozhdeniya Chastits Cherez Veshchestvo (Introduction To The Theory Of The Transit Of Particles Through Matter) (Moscow: Atomizdat, 1978)
  55. Fermi E Phys. Rev. 75 1169 (1949)
  56. Strömgren B Astrophys. J. 108 242 (1948)
  57. Uchaikin V V, Korobko D A Tech. Phys. 49 532 (2004); Uchaikin V V, Korobko D A Zh. Tekh. Fiz. 74 (5) 12 (2004)
  58. Uchaikin V V Phys. Usp. 46 821 (2003); Uchaikin V V Usp. Fiz. Nauk 46 847 (2003)
  59. Uchaikin V V, Korobko D A Tech. Phys. Lett. 25 435 (1999); Uchaikin V V, Korobko D A Pis’ma Zh. Tekh. Fiz. 25 (11) 34 (1999)
  60. Isliker H, Vlahos L Phys. Rev. E 67 026413 (2003)
  61. Dorman L I Eksperimental’nye I Teoreticheskie Osnovy Astrofiziki Kosmicheskikh Luchei (Experimental And Theoretical Foundations Of Astrophysics Of Cosmic Rays) (Moscow: Nauka, 1975)
  62. Uchaikin V V, UchaikD V Obozrenie Priklad. Promyshlen. Matem. 9 477 (2002)
  63. Bohm D The Many Body Problem, Les Houches, Session 1958 (London: Wiley, 1959) p. 401; Translated into Russian, Bohm D Obshchaya Teoriya Kollektivnykh Peremennykh ((Moscow: Mir, 1964)
  64. Jullien M-C, Paret J, Tabeling P Phys. Rev. Lett. 82 2872 (1999)
  65. Tchen C-M Phys. Rev. 93 4 (1954)
  66. Heisenberg W Z. Phys. 124 628 (1948)
  67. Monin A S, Yaglom A M Statistical Fluid Mechanics: Mechanics Of Turbulence Vol. 2 (Cambridge, MA: MIT Press, 1975) p. 874
  68. Elmegreen B G Astrophys. J. 853 88 (2018)
  69. Zybin K P, Sirota V A Phys. Usp. 58 556 (2015); Zybin K P, Sirota V A Usp. Fiz. Nauk 185 593 (2015)
  70. Xu S, Lazarian A Astrophys. J. 894 63 (2020)
  71. Maiti S et al Astrophys. J. 926 94 (2022); Maiti S et al arXiv:2108.01936
  72. Lagutin A A, Nikulin Yu A, Uchaikin V V Preprint AGU-2000/4 (Barnaul: Altaiskii Gos. Univ., 2000)
  73. Laguti A A, Nikulin Yu A, Uchaikin V V Nucl. Phys. B Proc. Suppl. 97 (1) 267 (2001)
  74. Bird G A Molecular Gas Dynamics (Oxford: Clarendon Press, 1976); Translated into Russian, Bird G A Molekulyarnaya Gazovaya Dinamika (Moscow: Mir, 1981)
  75. Filippov B V, Il’Inskaya G B Sov. Phys. Dokl. 10 915 (1966); Filippov B V, Il’Inskaya G B Dokl. Akad. Nauk SSSR 164 768 (1965)
  76. Lagutin A A, Raikin R I, Tyumentsev A G Izv. Altaiskogo Gos. Univ. (22) 6 (2004)
  77. Robitaille J-F et al Astron. Astrophys. 641 A138 (2020)
  78. Kraus A L, Hillenbrand L A Astrophys. J. 686 L111 (2008)
  79. Elia D et al Mon. Not. R. Astron. Soc. 481 509 (2018)
  80. Khalil A et al Astrophys. J. Suppl. 165 512 (2006)
  81. Vazquez-Semadeni E Astrophys. J. 423 681 (1994)
  82. Lis D C et al Astrophys. J. 463 623 (1996)
  83. Hily-Blant P, Falgarone E, Pety J Astron. Astrophys. 481 367 (2008)
  84. Federrath C et al Astron. Astrophys. 512 A81 (2010)
  85. Bertram E et al Mon. Not. R. Astron. Soc. 451 3679 (2015)
  86. Chappell D, Scalo J Astrophys. J. 551 712 (2001)
  87. Muzy J F, Bacry E, Arneodo A Phys. Rev. E 47 875 (1993)
  88. Milovanov A V, Rasmussen J J Phys. Lett. A 378 1492 (2014)
  89. Uchaikin V V Fract. Calc. Appl. Anal. 19 1306 (2016)
  90. Klimas A J, Sandri G Astrophys. J. 169 41 (1971)
  91. Klimas A J, Sandri G Astrophys. J. 180 937 (1973)
  92. Moraal H Space Sci. Rev. 176 299 (2013)
  93. Mörters P, Peres Yu Brownian Motion (Cambridge: Cambridge Univ. Press, 2010)
  94. Bardou F et al Phys. Rev. Lett. 72 203 (1994)
  95. Balescu R Plasma Phys. Control. Fusion 42 (12B) B1-B13 (2000)
  96. Uchaikin V V J. Exp. Theor. Phys. 88 1155 (1999); Uchaikin V V Zh. Eksp. Teor. Fiz. 115 2113 (1999)
  97. Ragot B R, Kirk J G Astron. Astrophys. 327 432 (1997)
  98. Uchaikin V V JETP Lett. 91 105 (2010); Uchaikin V V Pis’ma Zh. Eksp. Teor. Fiz. 91 115 (2010)
  99. Lagutin A A, Uchaikin V V Nucl. Instrum. Meth. Phys. Res. B 201 212 (2003)
  100. Samko S G, Kilbas A A, Marichev O I Fractional Integrals And Derivatives: Theory And Applications (New York: Gordon and Breach, 1993)
  101. Uchaikin V V Fractional Derivatives For Physicists And Engineers (Berlin: Springer, 2013)
  102. Uchaikin V V, Sibatov R T J. Phys. A 44 145501 (2011)
  103. Saichev A I, Zaslavsky G M Chaos 7 753 (1997)
  104. Kolokoltsov V, Korolev V, Uchaikin V J. Math. Sci. 105 2569 (2001)
  105. Uchaikin V V Chem. Phys. 284 507 (2002)
  106. Uchaikin V V Physica A 255 65 (1998)
  107. Uchaikin V V, Sibatov R T Int. J. Mod. Phys. B 26 1230016 (2012)
  108. Magdziarz M, Zorawik T Phys. Rev. E 94 022130 (2016)
  109. Perri S, Zimbardo G Astrophys. J. Lett. 671 L177 (2007)
  110. Perri S, Zimbardo G J. Geophys. Res. Space Phys. 113 (A3) A03107 (2008)
  111. Perri S, Zimbardo G Astrophys. J. 693 L118 (2009)
  112. Uchaikin V, Sibatov R, Byzykchi A Commun. Appl. Industrial Math. 6 (1) e-479 (2014)
  113. Litvinenko Yu E, Effenberger F Astrophys. J. 796 125 (2014)
  114. Uchaikin V V, Sibatov R T, Byzykchi A N Bull. Russ. Acad. Sci. Phys. 79 592 (2015); Uchaikin V V, Sibatov R T, Byzykchi A N Izv. Ross. Akad. Nauk Fiz. 79 646 (2015)
  115. Zumofen G, J Klafter J Phys. Rev. E 47 851 (1993)
  116. Uchaikin V V, Sibatov R T Grav. Cosmol. 18 122 (2012)
  117. Perri S, Zimbardo G Adv. Space Res. 44 465 (2009)
  118. Zank G P, Matthaeus W H, Smith C W J. Geophys. Res. Space Phys. 101 (A8) 17093 (1996)
  119. Uchaikin V V, Sibatov R T Tech. Phys. Lett. 30 316 (2004); Uchaikin V V, Sibatov R T Pis’ma Zh. Tekh. Fiz. 30 (8) 27 (2004)
  120. Zimbardo G, Pommois P, Veltri P Astrophys. J. 639 L91 (2006)
  121. Perri S et al Atmosphere 12 508 (2021)
  122. Pucci F et al Mon. Not. R. Astron. Soc. 459 3395 (2016)
  123. Zaslavsky G M Phys. Rep. 371 461 (2002)
  124. Uchaikin V V Theor. Math. Phys. 115 496 (1998); Uchaikin V V Teor. Matem. Fiz. 115 154 (1998)
  125. Cartea Á, del-Castillo-Negrete D Phys. Rev. E 76 041105 (2007)
  126. Uchaikin V V JETP Lett. 92 200 (2010); Uchaikin V V Pis’ma Zh. Eksp. Teor. Fiz. 92 226 (2010)
  127. Uchaikin V V, Sibatov R T Chaos Solitons Fractals 102 197 (2017)
  128. Jokipii J R Astrophys. J. 143 961 (1966)
  129. Jokipii J R, Parker E N Astrophys. J. 155 777 (1969)
  130. Webb G M et al Astrophys. J. 651 211 (2006)
  131. Shalchi A, Kourakis I Phys. Plasmas 14 112901 (2007)
  132. Shalchi A et al Phys. Rev. E 80 066408 (2009)
  133. Lévy P Random Functions: General Theory With Special Reference To Laplacian Random Functions (Univ. Of California Publ. In Statistics, Vol. 1, No. 12) (Berkeley, CA: Univ. of California Press, 1953)
  134. Wang K G, Lung C W Phys. Lett. A 151 119 (1990)
  135. Lagutin A A, Uchaikin V V Proc. of the 27th Intern. Cosmic Ray Conf., 7-15 August 2001, Hamburg, Germany Vol. 5 Vol. 5 (Geneve: International Union of Pure and Applied Physics, 2001) p. 1900
  136. Blümer J, Engel R, Hörandel J R Prog. Part. Nucl. Phys. 63 293 (2009)
  137. Ptuskin V S Phys. Usp. 50 534 (2007); Ptuskin V S Usp. Fiz. Nauk 177 558 (2007)
  138. Lagutin A A, Strelnikov D V, Tyumentsev A G Proc. of 27th Intern. Cosmic Ray Conf., Hamburg, Germany, 2001 Vol. 5 (Hamburg: Copernicus Gesellschaft, 2001)
  139. Erlykin A D, Lagutin A A, Wolfendale A W Astropart. Phys. 19 351 (2003)
  140. Lagutin A A, Tyumentsev A G Izv. Altaiskogo Gos. Univ. (4(35)) 4 (2004)
  141. Ketabi N, Fatemi J Scientia Iranica 16 (3) 269 (2009)
  142. Kermani H A, Fatemi J South African J. Sci. 107 (1) 1 (2011)
  143. Doostmohammadi S, Fatemi S J Int. Scholarly Res. Not. 2012 673250 (2012)
  144. Genolini Y et al Astron. Astrophys. 600 A68 (2017)
  145. Wang S-H et al Phys. Rev. D 103 063035 (2021)
  146. Blasi P, Amato E J. Cosmol. Astropart. Phys. 2012 (01) 010 (2012)
  147. Blasi P Astron. Astrophys. Rev. 21 70 (2013)
  148. Ptuskin V Astropart. Phys. 39 44 (2012)
  149. Blasi P, Amato E J. Cosmol. Astropart. Phys. 2012 (01) 011 (2012)
  150. Faucher-Giguère C-A, Kaspi V M Astrophys. J. 643 332 (2006)
  151. Case G, Bhattacharya D Astron. Astrophys. Suppl. 120 437 (1996)
  152. Lagutin A A, Tyumentsev A G, Yushkov A V Nucl. Phys. B Proc. Suppl. 175 555 (2008)
  153. Lagutin A A, Strelnikov D V, Tyumentsev A G astro-ph/0107231
  154. Lagutin A A, Makarov V V, Tyumentsev A G astro-ph/0107253
  155. Sveshnikova L G Astron. Astrophys. 409 799 (2003)
  156. Lagutin A A et al Bull. Russ. Acad. Sci. Phys. 79 322 (2015); Lagutin A A et al Izv. Ross. Akad. Nauk Fiz. 79 354 (2015)
  157. Hörandel J, Kalmykov N N, Timokhin V V Astropart. Phys. 27 119 (2007)
  158. Sibatov R T, Uchaikin V V PoS 538 ICRC2015 (2015)
  159. Uchaikin V V, Sibatov R T J. Phys. Conf. Ser. 798 012029 (2017)
  160. Hörandel J R Astropart. Phys. 21 241 (2004)
  161. Berezinskii V S e al. Asrofizika Kosmicheskikh Luchei (Astrophysics Of Cosmic Rays, Ed. V L Ginzburg) (Moscow: Nauka, 1984)
  162. Ahlers M, Mertsch P Astrophys. J. Lett. 815 L2 (2015)
  163. Lagutin A A, Nikulin Yu A J. Exp. Theor. Phys. 81 825 (1995); Lagutin A A, Nikulin Yu A Zh. Eksp. Teor. Fiz. 108 1505 (1995)
  164. Erlykin A D, Machavariani S K, Wolfendale A W J. Phys. Conf. Ser. 1181 012037 (2019)
  165. Erlykin A D, Wolfendale A W J. Phys. G 23 979 (1997)
  166. Pavlyuchenko V P et al J. Phys. G 45 015202 (2018)
  167. Erlykin A D et al Proc. of the 34th Intern. Cosmic Ray Conf., ICRC’15, 30 July - 6 August, 2015, The Netherlands
  168. Case K M, Zweifel P F Linear Transport Theory (Reading, MA: Addison-Wesley Publ. Co., 1967)
  169. Cox D P, Reynolds R J Annu. Rev. Astron. Astrophys. 25 303 (1987)
  170. Maiz-Apellániz J Astrophys. J. Lett. 560 L83 (2001)
  171. Frisch P C Int. Astron. Union Colloq. 166 269 (1997)
  172. Opher M et al Nature 462 1036 (2009)
  173. Obukhov A M Dokl. Akad. Nauk SSSR 32 22 (1941)
  174. del-Castillo-Negrete D, Carreras B A, Lynch V E Phys. Rev. Lett. 94 065003 (2005)
  175. Klafter J, Blumen A, Shlesinger M F Phys. Rev. A 35 3081 (1987)
  176. Uchaikin V V JETP Lett. 91 105 (2010); Uchaikin V V Pis’ma Zh. Eksp. Teor. Fiz. 91 115 (2010)
  177. Duo S, Wang H, Zhang Y Discrete Continuous Dyn. Syst. B 24 231 (2019); Duo S, Wang H, Zhang Y arXiv:1711.06916
  178. Chen Z Q, Song R J. Funct. Analysis 226 (1) 90 (2005)
  179. Guan Q-Y, Ma Z-M Probab. Theory Related Fields 134 649 (2006)
  180. Silling S A J. Mech. Phys. Solids 48 (1) 175 (2000)
  181. Guan Q, Gunzburger M J. Comput. Appl. Math. 313 102 (2017)
  182. Uchaikin V V Mekhanika. Osnovy Mekhaniki Sploshnykh Sred (Mechanics. Fundamentals Of Continuum Mechanics) (Moscow: Lan’, 2017)
  183. Wendt H, Jaffard S, Abry P IEEE Statistical Signal Processing Workshop, 2012, August 5-8 (Piscataway, NJ: IEEE, 2012) p. 69-72
© 1918–2024 Uspekhi Fizicheskikh Nauk
Email: ufn@ufn.ru Editorial office contacts About the journal Terms and conditions