VUGAR E. ISMAILOV
Institute of Mathematics and Mechanics
9, B. Vahabzadeh str., Az-1141, Baku, Azerbaijan
e-mail: vugaris@mail.ru, vugaris@gmail.com
Curriculum Vitae
Date and place of birth: September 26, 1971; Salyan, Azerbaijan
Marital Status: Married
Citizenship: Azerbaijani
Education
1988 - 1993 Baku State University, Mathematics and Mechanics Department; student
1995 - 1998 Institute of Mathematics and Mechanics, NAS of Azerbaijan; post graduate student
Academic degrees and ranks
2000 PhD in Mathematics
Thesis: Approximation of functions of several variables by sums of functions of fewer variables
2014 Doctor of Mathematics (Habilitation)
Thesis: Approximation by ridge functions with fixed directions
2017 Professor of Mathematics
Professional experience
2013 - present Institute of Mathematics and Mechanics, NAS of Azerbaijan Chair of the Department of Function Theory
2006 - 2013 Institute of Mathematics and Mechanics, NAS of Azerbaijan Senior Researcher
2008 (April-July) - Technion - Israel Institute of Technology, Haifa, Israel Visiting Professor
2007 (April-July) - Technion - Israel Institute of Technology, Haifa, Israel Visiting Professor
2005 - 2006 Baku State University Associate Professor
2001 - 2006 Institute of Mathematics and Mechanics, NAS of Azerbaijan Researcher
1999 - 2001 Institute of Mathematics and Mechanics, NAS of Azerbaijan Junior Researcher
Research Interests
Approximation Theory, Functional Analysis
Particular Interests: ridge functions, radial functions, neural networks, linear superpositions, algebras of continuous functions, approximation algorithms, functional equations
Membership in Scientific Associations
American Mathematical Society
Papers in refereed journals
(with A. Ismayilova) On the Kolmogorov neural networks, Neural Networks 176 (2024), Paper No. 106333, https://doi.org/10.1016/j.neunet.2024.106333
Approximation error of single hidden layer neural networks with fixed weights, Information Processing Letters 185 (2024), Paper No. 106467, doi.org/10.1016/j.ipl.2023.106467
(with A.Kh. Asgarova and A.A. Huseynli) A Chebyshev-type alternation theorem for best approximation by a sum of two algebras, Proceedings of the Edinburgh Mathematical Society (2) 66 (2023), no. 4, 971-978, doi.org/10.1017/S0013091523000494
(with R.A. Aliev and A.A. Asgarova) The double difference property for the class of locally Hölder continuous functions, Moscow Mathematical Journal 22 (2022), no. 3, 393-400, http://www.mathjournals.org/mmj/2022-022-003/2022-022-003-002.html
(with R.A. Aliev and A.A. Asgarova) On the representation by bivariate ridge functions, Ukrainian Mathematical Journal 73 (2021), no. 5, 675-685, https://doi.org/10.1007/s11253-021-01952-9
(with A.Kh. Asgarova) A Chebyshev-type theorem characterizing best approximation of a continuous function by elements of the sum of two algebras, (Russian) Mat. Zametki 109 (2021), no. 1, 19-26; English transl. in Mathematical Notes 109 (2021), 15-20, https://doi.org/10.1134/S0001434621010028
(with R.A. Aliev) A representation problem for smooth sums of ridge functions, Journal of Approximation Theory 257 (2020), 105448, 13 pp, https://doi.org/10.1016/j.jat.2020.105448
(with R.A. Aliev and A.A. Asgarova) On the Hölder continuity in ridge function representation, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 45 (2019), no. 1, 31-40, http://proc.imm.az/volumes/45-1/45-01-03.pdf
Computing the approximation error for neural networks with weights varying on fixed directions, Numerical Functional Analysis and Optimization 40 (2019), no. 12, 1395-1409, https://doi.org/10.1080/01630563.2019.1605523
(with R.A. Aliev and A.A. Asgarova) A note on continuous sums of ridge functions, Journal of Approximation Theory 237 (2019), 210-221, https://doi.org/10.1016/j.jat.2018.09.006
(with N. Guliyev) Approximation capability of two hidden layer feedforward neural networks with fixed weights, Neurocomputing 316 (2018), 262-269, https://doi.org/10.1016/j.neucom.2018.07.075
(with N. Guliyev) On the approximation by single hidden layer feedforward neural networks with fixed weights, Neural Networks 98 (2018), 296-304, https://doi.org/10.1016/j.neunet.2017.12.007
A note on the criterion for a best approximation by superpositions of functions, Studia Mathematica 240 (2018), no. 2, 193-199, https://doi.org/10.4064/sm170314-9-4
(with A.Kh. Asgarova) On the representation by sums of algebras of continuous functions, Comptes Rendus Mathematique 355 (2017), no. 9, 949-955, https://doi.org/10.1016/j.crma.2017.09.015
A note on the equioscillation theorem for best ridge function approximation, Expositiones Mathematicae 35 (2017), no. 3, 343-349, https://doi.org/10.1016/j.exmath.2017.05.003
(with A.Kh. Asgarova) Diliberto–Straus algorithm for the uniform approximation by a sum of two algebras, Proceedings - Mathematical Sciences 127 (2017), no. 2, 361-374, http://dx.doi.org/10.1007/s12044-017-0337-4
(with E. Savas) Measure theoretic results for approximation by neural networks with limited weights, Numerical Functional Analysis and Optimization 38 (2017), no. 7, 819-830, http://dx.doi.org/10.1080/01630563.2016.1254654
Approximation by sums of ridge functions with fixed directions, (Russian) Algebra i Analiz 28 (2016), no. 6, 20–69, http://mi.mathnet.ru/eng/aa1513 English transl. St. Petersburg Mathematical Journal 28 (2017), 741-772, https://doi.org/10.1090/spmj/1471
On the uniqueness of representation by linear superpositions, Ukrainskii Matematicheskii Zhurnal 68 (2016), no. 12, 1620-1628. English transl. Ukrainian Mathematical Journal 68 (2017), no. 12, 1874-1883, https://doi.org/10.1007/s11253-017-1335-5
(with N. Guliyev) A single hidden layer feedforward network with only one neuron in the hidden layer can approximate any univariate function, Neural Computation 28 (2016), no. 7, 1289–1304, http://dx.doi.org/10.1162/NECO_a_00849
(with R. Aliev) On a smoothness problem in ridge function representation, Advances in Applied Mathematics 73 (2016), 154--169, http://dx.doi.org/10.1016/j.aam.2015.11.002
(with R. Aliev and T. Shahbalayeva) On the representation by sums of ridge functions, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 41 (2015), no. 2, 106--118, http://proc.imm.az/volumes/41-2/41-02-10.pdf
Alternating algorithm for the approximation by sums of two compositions and ridge functions, Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 41 (2015), no. 1, 146--152, http://www.proc.imm.az/volumes/41-1/41-01-15.pdf
Approximation by ridge functions and neural networks with a bounded number of neurons, Applicable Analysis 94 (2015), no. 11, 2245-2260,
On the approximation by neural networks with bounded number of neurons in hidden layers, Journal of Mathematical Analysis and Applications 417 (2014), no. 2, 963--969, http://dx.doi.org/10.1016/j.jmaa.2014.03.092
(with A. Pinkus) Interpolation on lines by ridge functions, Journal of Approximation Theory 175 (2013), 91-113, http://dx.doi.org/10.1016/j.jat.2013.07.010
A review of some results on ridge function approximation, Azerbaijan Journal of Mathematics 3 (2013), no.1, 3-51, http://azjm.org/index.php/azjm/article/view/143
Approximation by neural networks with weights varying on a finite set of directions, Journal of Mathematical Analysis and Applications 389 (2012), Issue 1, 72-83, http://dx.doi.org/10.1016/j.jmaa.2011.11.037
A note on the representation of continuous functions by linear superpositions, Expositiones Mathematicae 30 (2012), Issue 1, 96-101, http://dx.doi.org/10.1016/j.exmath.2011.07.005
Approximation by Neural Networks with a Restricted Set of Weights, book chapter, in: Advances in Mathematics Research (published by NOVA Science Publishers, USA), 2011, Chapter 6, pp. 193-206.
Approximation capabilities of neural networks with weights from two directions, Azerbaijan Journal of Mathematics 1 (2011), 122-129.
On the proximinality of ridge functions, Sarajevo Journal of Mathematics 5(17) (2009), no. 1, 109-118, http://www.anubih.ba/Journals/vol.5,no-1,y09/12revIsmailov[2].pdf
On the theorem of M Golomb, Proceedings - Mathematical Sciences 119 (2009), no. 1, 45-52, http://dx.doi.org/10.1007/s12044-009-0005-4
On the representation by linear superpositions, Journal of Approximation Theory 151 (2008), Issue 2 , 113-125, http://dx.doi.org/10.1016/j.jat.2007.09.003
On the approximation by weighted ridge functions, Analele Universitatii de Vest din Timisoara, Ser. Math.-Inform. 46 (2008), 75-83.
On the approximation by compositions of fixed multivariate functions with univariate functions, Studia Mathematica 183 (2007), 117-126, http://dx.doi.org/10.4064/sm183-2-2
On the best L₂ approximation by ridge functions, Applied Mathematics E-Notes, 7 (2007), 71-76, http://www.math.nthu.edu.tw/~amen/
Representation of multivariate functions by sums of ridge functions, Journal of Mathematical Analysis and Applications 331 (2007), Issue 1, 184-190, http://dx.doi.org/10.1016/j.jmaa.2006.08.076
Characterization of an extremal sum of ridge functions, Journal of Computational and Applied Mathematics 205 (2007), Issue 1, 105-115, http://dx.doi.org/10.1016/j.cam.2006.04.043
On error formulas for approximation by sums of univariate functions, International Journal of Mathematics and Mathematical Sciences, volume 2006 (2006), Article ID 65620, 11 pages, http://dx.doi.org/10.1155/IJMMS/2006/65620
Methods for computing the least deviation from the sums of functions of one variable, (Russian) Sibirskii Matematicheskii Zhurnal 47 (2006), no. 5, 1076 -1082; translation in Siberian Mathematical Journal 47 (2006), no. 5, 883--888, http://dx.doi.org/10.1007/s11202-006-0097-3
On a theorem of approximation by sums g₁(x₁)+g₂(x₂)+...+g_{n}(x_{n}), Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 25 (2005), no. 4, 49-54.
On two-sided exact estimates for the best approximation by sums ϕ(x)+ψ(y) , Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 25 (2005), no. 1, 89-94.
On maximization principle of lightning bolts, (Russian) Vestnik of Baku State University 3 (2005), 57-63.
On some classes of bivariate functions characterized by formulas for the best approximation. Radovi Matematicki 13 (2004), 53-62, www.anubih.ba/Journals/vol-13,no-1,y04/07ismailov.pdf
On discontinuity of the best approximation of a continuous function. Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 23 (2003), no. 4, 57-60.
On behaviour of the best approximation as a function of an approximation set. Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 19 (2003), 113-116.
Theorem on lightning bolts for elementary domains. Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 17 (2002), 78-85.
On some geometrical conditions for the existence of the best approximating function of the form ϕ(x)+ψ(y). Trans. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 19(1999), no.1-2, 91-95.
On a characteristic property of a family of classes of best approximation. (Russian) Proc. Inst. Math. Mech. Acad. Sci. Azerb. 6 (1997), 74-82.
(with M-B. A. Babaev) Two-sided estimates for the best approximation in domains different from the parallelepiped. Functiones Et Approximatio Commentarii Mathematici 25 (1997), 121-128.
Two-sided estimates for the best approximation in domains consisting of a union of rectangles. (Russian) Izv. Akad. Nauk Azerb. Ser. Fiz-Tekh. Mat. Nauk 17(1996), no. 1-3, 109-114.
Books
Ridge functions and applications in neural networks. Mathematical Surveys and Monographs, 263. American Mathematical Society, Providence, RI, [2021], ©2021. ix+186 pp. ISBN: 978-1-4704-6765-4 https://bookstore.ams.org/surv-263
Editorship of Journals
Managing Editor, Proceedings of the Institute of Mathematics and Mechanics, NAS of Azerbaijan, 2014--
Editorial Board Member, Azerbaijan Journal of Mathematics, 2013--
Editorial Board Member, Transactions of the Institute of Mathematics and Mechanics, NAS of Azerbaijan, 2015--
Fellowships and Grants
2007 INTAS Fellowship (Ref. Nr. 06-1000015-6283)
2010 Azerbaijan Science Development Foundation Grant (EIF-2010-1(1)-40/07-1)
2013 SOCAR Science Foundation Grant (SOCAR EF-2013)
2013 Azerbaijan Science Development Foundation Grant (EIF-2013-9(15)-46/11/1)
Selected Talks
Oxford University, Functional Analysis Seminar -- "Represenation of C(X) as a sum of its subalgebras and some applications"
Technion - Israel Institute of Technology, Functional Analysis Seminar -- "Representation of multivariate functions by sums of ridge functions"
Tel-Aviv University, Approximation Theory Seminar -- "Approximation by linear combinations of ridge functions"