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Uspekhi Mat. Nauk, 2004, Volume 59, Issue 1(355), Pages 11–24 (Mi umn698)  

This article is cited in 15 scientific papers (total in 15 papers)

On Hilbert's thirteenth problem and related questions

A. G. Vitushkin

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Hilbert's thirteenth problem involves the study of solutions of algebraic equations. The object is to obtain a complexity estimate for an algebraic function. As of now, the problem remains open. There are only a few partial algebraic results in this connection, but at the same time the problem has stimulated a series of studies in the theory of functions with their subsequent applications. The most brilliant result in this cycle is Kolmogorov's theorem on superpositions of continuous functions.

DOI: https://doi.org/10.4213/rm698

Full text: PDF file (260 kB)
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English version:
Russian Mathematical Surveys, 2004, 59:1, 11–25

Bibliographic databases:

UDC: 517.51+512
MSC: Primary 28B40; Secondary 68Q30, 26B45, 41A46, 68P30
Received: 17.06.2003

Citation: A. G. Vitushkin, “On Hilbert's thirteenth problem and related questions”, Uspekhi Mat. Nauk, 59:1(355) (2004), 11–24; Russian Math. Surveys, 59:1 (2004), 11–25

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Beiu V., Zawadzki A., “On Kolmogorov's superpositions: Novel gates and circuits for nanoelectronics?”, Proceedings of the International Joint Conference on Neural Networks, 2005, 651–656  crossref  isi
    2. V. Ya. Lin, “Algebraic functions, configuration spaces, Teichmüller spaces, and new holomorphically combinatorial invariants”, Funct. Anal. Appl., 45:3 (2011), 204–224  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. V. A. Krasikov, T. M. Sadykov, “On the analytic complexity of discriminants”, Proc. Steklov Inst. Math., 279 (2012), 78–92  mathnet  crossref  mathscinet  isi  elib  elib
    4. I. A. Antipova, E. N. Mikhalkin, “Analytic continuations of a general algebraic function by means of Puiseux series”, Proc. Steklov Inst. Math., 279 (2012), 3–13  mathnet  crossref  mathscinet  isi  elib  elib
    5. Beloshapka V.K., “Analytical Complexity: Development of the Topic”, Russ. J. Math. Phys., 19:4 (2012), 428–439  crossref  mathscinet  zmath  isi  elib
    6. Goldenbaum M. Boche H. Stanczak S., “Harnessing Interference for Analog Function Computation in Wireless Sensor Networks”, IEEE Trans. Signal Process., 61:20 (2013), 4893–4906  crossref  mathscinet  isi  elib
    7. V. K. Beloshapka, “A Seven-Dimensional Family of Simple Harmonic Functions”, Math. Notes, 98:6 (2015), 867–871  mathnet  crossref  crossref  mathscinet  isi  elib
    8. Valery K. Beloshapka, “Three families of functions of complexity one”, Zhurn. SFU. Ser. Matem. i fiz., 9:4 (2016), 416–426  mathnet  crossref
    9. Stepanova M., “On rational functions of first-class complexity”, Russ. J. Math. Phys., 23:2 (2016), 251–256  crossref  mathscinet  zmath  isi  elib  scopus
    10. Zusmanovich P., “On the last question of Stefan Banach”, Expo. Math., 34:4 (2016), 454–466  crossref  mathscinet  zmath  isi  scopus
    11. Gromov M., “Geometric, algebraic, and analytic descendants of Nash isometric embedding theorems”, Bull. Amer. Math. Soc., 54:2 (2017), 173–245  crossref  mathscinet  zmath  isi  scopus
    12. V. K. Beloshapka, “Analytic Complexity: Gauge Pseudogroup, Its Orbits, and Differential Invariants”, Proc. Steklov Inst. Math., 298 (2017), 51–59  mathnet  crossref  crossref  isi  elib
    13. V. K. Beloshapka, “Simple solutions of three equations of mathematical physics”, Trans. Moscow Math. Soc., 2018, 187–200  mathnet  crossref  elib
    14. V. K. Beloshapka, “O slozhnosti differentsialno-algebraicheskogo opisaniya klassov analiticheskoi slozhnosti”, Matem. zametki, 105:3 (2019), 323–331  mathnet  crossref  elib
    15. Wei Zh., Kim S., Choi B., Kim D., “Multivariate Skew Normal Copula For Asymmetric Dependence: Estimation and Application”, Int. J. Inf. Technol. Decis. Mak., 18:1 (2019), 365–387  crossref  isi  scopus
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