{"id":3655,"date":"2014-11-20T11:32:52","date_gmt":"2014-11-20T07:32:52","guid":{"rendered":"http:\/\/www.imm.az\/exp\/?page_id=3655"},"modified":"2019-02-05T10:47:27","modified_gmt":"2019-02-05T06:47:27","slug":"ismayilov-vugar-elman","status":"publish","type":"page","link":"https:\/\/www.imm.az\/exp\/personnel\/ismayilov-vugar-elman\/","title":{"rendered":"Ismayilov Vugar Elman"},"content":{"rendered":"

Basic scientific achievements<\/strong><\/p>\n

1) Necessary and sufficient conditions for the representation of multivariate functions by linear combinations of ridge functions were obtained;<\/p>\n

2) A Chebyshev type theorem was proved for sums of ridge functions to be extremal to a given continuous function;<\/p>\n

3) Explicit formulas for an exact computation of the approximation error and construction of best approximating function were obtained in problems of approximation of multivariate functions by ridge functions and univariate functions in both continuous and square-integrable metrics;<\/p>\n

4) It was shown that if each continuous function defined on a compact Hausdorff space is represented by linear superpositions, then all functions on this space possess such representation;<\/p>\n

5) The problem of multivariate approximation theory associated with Golomb’s theorem was solved.<\/p>\n

The list of publications at the 5 years published<\/strong><\/p>\n

    \n
  1. (with N. Guliyev) On the approximation by single hidden layer feedforward neural networks with fixed weights,\u00a0Neural Networks<\/em>98<\/strong>(2018), 296-304,\u00a0https:\/\/doi.org\/10.1016\/j.neunet.2017.12.007<\/a><\/li>\n
  2. A note on the criterion for a best approximation by superpositions of functions,\u00a0Studia Mathematica<\/em>240\u00a0<\/strong>(2018), no. 2, 193-199,\u00a0https:\/\/doi.org\/10.4064\/sm170314-9-4<\/a><\/li>\n
  3. (with A. Asgarova) On the representation by sums of algebras of continuous functions,\u00a0Comptes Rendus Mathematique<\/em>355\u00a0<\/strong>(2017), no. 9, 949-955,\u00a0https:\/\/doi.org\/10.1016\/j.crma.2017.09.015<\/a><\/li>\n
  4. A note on the equioscillation theorem for best ridge function approximation,\u00a0Expositiones Mathematicae<\/em>35\u00a0<\/strong>(2017), no. 3, 343-349,\u00a0https:\/\/doi.org\/10.1016\/j.exmath.2017.05.003<\/a><\/li>\n
  5. (with A. Asgarova) Diliberto\u2013Straus algorithm for the uniform approximation by a sum of two algebras,\u00a0Proceedings – Mathematical Sciences<\/em>127\u00a0<\/strong>(2017), no. 2, 361-374,\u00a0http:\/\/dx.doi.org\/10.1007\/s12044-017-0337-4<\/a><\/li>\n
  6. (with E. Savas) Measure theoretic results for approximation by neural networks with limited weights,\u00a0Numerical Functional Analysis and Optimization<\/em>38<\/strong>(2017), no. 7, 819-830,\u00a0http:\/\/dx.doi.org\/10.1080\/01630563.2016.1254654<\/a><\/li>\n
  7. Approximation by sums of ridge functions with fixed directions, (Russian)\u00a0Algebra i Analiz<\/em>\u00a028<\/strong>\u00a0(2016), no. 6,\u00a020\u201369,\u00a0http:\/\/mi.mathnet.ru\/eng\/aa1513<\/a>\u00a0English transl.\u00a0St. Petersburg Mathematical Journal<\/em>\u00a028<\/strong>\u00a0(2017), 741-772,\u00a0https:\/\/doi.org\/10.1090\/spmj\/1471<\/a>\n
      \n
    1. On the uniqueness of representation by linear superpositions,\u00a0Ukrainskii Matematicheskii Zhurnal<\/em>68<\/strong>(2016), no. 12, 1620-1628. English transl.\u00a0Ukrainian Mathematical Journal<\/em>\u00a068\u00a0<\/strong>(2017), no. 12, 1874-1883,\u00a0https:\/\/doi.org\/10.1007\/s11253-017-1335-5<\/a><\/li>\n
    2. (with N. Guliyev)\u00a0A single hidden layer feedforward network with only one neuron in the hidden layer can approximate any univariate function,\u00a0Neural Computation\u00a0<\/em>28<\/strong>(2016), no. 7,\u00a01289\u20131304,\u00a0http:\/\/dx.doi.org\/10.1162\/NECO_a_00849<\/a><\/li>\n
    3. (with R. Aliev) On a smoothness problem in ridge function representation,\u00a0Advances in Applied Mathematics<\/em>73<\/strong>(2016), 154\u2013169,\u00a0http:\/\/dx.doi.org\/10.1016\/j.aam.2015.11.002<\/a><\/li>\n
    4. Approximation by ridge functions and neural networks with a bounded number of neurons,\u00a0Applicable Analysis<\/em>94<\/strong>(2015), no. 11, 2245-2260,\u00a0http:\/\/dx.doi.org\/10.1080\/00036811.2014.979809<\/a><\/li>\n
    5. On the approximation by neural networks with bounded number of neurons in hidden layers,\u00a0Journal of Mathematical Analysis and Applications<\/em>417\u00a0<\/strong>(2014), no. 2, 963\u2013969,\u00a0http:\/\/dx.doi.org\/10.1016\/j.jmaa.2014.03.092<\/a><\/li>\n
    6. (with A. Pinkus) Interpolation on lines by ridge functions,\u00a0Journal of Approximation Theory<\/em>175<\/strong>(2013), 91-113,\u00a0http:\/\/dx.doi.org\/10.1016\/j.jat.2013.07.010<\/a><\/li>\n
    7. Approximation by neural networks with weights varying on a finite set of directions,\u00a0Journal of Mathematical Analysis and Applications<\/em>389<\/strong>(2012), Issue 1, 72-83,\u00a0http:\/\/dx.doi.org\/10.1016\/j.jmaa.2011.11.037<\/a><\/li>\n
    8. A note on the representation of continuous functions by linear superpositions,\u00a0Expositiones Mathematicae<\/em>30<\/strong>(2012), Issue 1, 96-101,\u00a0http:\/\/dx.doi.org\/10.1016\/j.exmath.2011.07.005<\/a><\/li>\n
    9. On the theorem of M Golomb,\u00a0Proceedings – Mathematical Sciences<\/em>119<\/strong>(2009), no. 1, 45-52,\u00a0http:\/\/dx.doi.org\/10.1007\/s12044-009-0005-4<\/a><\/li>\n
    10. On the representation by linear superpositions,\u00a0Journal of Approximation Theory<\/em>151<\/strong>(2008), Issue 2 , 113-125,\u00a0http:\/\/dx.doi.org\/10.1016\/j.jat.2007.09.003<\/a><\/li>\n
    11. On the approximation by compositions of fixed multivariate functions with univariate functions,\u00a0Studia Mathematica<\/em>\u00a0183<\/strong>\u00a0(2007), 117-126,\u00a0http:\/\/dx.doi.org\/10.4064\/sm183-2-2<\/a>\n
        \n
      1. On the best L\u2082 approximation by ridge functions,\u00a0Applied Mathematics E-Notes<\/em>,\u00a07<\/strong>(2007), 71-76,\u00a0http:\/\/www.math.nthu.edu.tw\/~amen\/<\/a><\/li>\n
      2. Representation of multivariate functions by sums of ridge functions,\u00a0Journal of Mathematical Analysis and Applications<\/em>331<\/strong>(2007), Issue 1, 184-190,\u00a0http:\/\/dx.doi.org\/10.1016\/j.jmaa.2006.08.076<\/a><\/li>\n
      3. Characterization of an extremal sum of ridge functions,\u00a0Journal of Computational and Applied Mathematics<\/em>205<\/strong>(2007), Issue 1, 105-115,\u00a0http:\/\/dx.doi.org\/10.1016\/j.cam.2006.04.043<\/a><\/li>\n<\/ol>\n

        22. Methods for computing the least deviation from the sums of functions of one variable, (Russian)\u00a0Sibirskii Matematicheskii Zhurnal<\/em>\u00a047<\/strong>\u00a0(2006), no. 5, 1076 -1082; translation in\u00a0Siberian Mathematical Journal<\/em>\u00a047<\/strong>\u00a0(2006), no. 5, 883\u2013888,\u00a0http:\/\/dx.doi.org\/10.1007\/s11202-006-0097-3<\/a><\/li>\n<\/ol>\n<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"

        Basic scientific achievements 1) Necessary and sufficient conditions for the representation of multivariate functions by linear combinations of ridge functions were obtained; 2) A Chebyshev type theorem was proved for sums of ridge functions to be extremal to a given continuous function; 3) Explicit formulas for an exact computation of the approximation error and construction […]<\/p>\n","protected":false},"author":6,"featured_media":0,"parent":2350,"menu_order":0,"comment_status":"open","ping_status":"open","template":"","meta":{"footnotes":""},"_links":{"self":[{"href":"https:\/\/www.imm.az\/exp\/wp-json\/wp\/v2\/pages\/3655"}],"collection":[{"href":"https:\/\/www.imm.az\/exp\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.imm.az\/exp\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.imm.az\/exp\/wp-json\/wp\/v2\/users\/6"}],"replies":[{"embeddable":true,"href":"https:\/\/www.imm.az\/exp\/wp-json\/wp\/v2\/comments?post=3655"}],"version-history":[{"count":2,"href":"https:\/\/www.imm.az\/exp\/wp-json\/wp\/v2\/pages\/3655\/revisions"}],"predecessor-version":[{"id":23346,"href":"https:\/\/www.imm.az\/exp\/wp-json\/wp\/v2\/pages\/3655\/revisions\/23346"}],"up":[{"embeddable":true,"href":"https:\/\/www.imm.az\/exp\/wp-json\/wp\/v2\/pages\/2350"}],"wp:attachment":[{"href":"https:\/\/www.imm.az\/exp\/wp-json\/wp\/v2\/media?parent=3655"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}