Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)2942610.22363/2658-4670-2021-29-4-306-336PersonalIn Memory of Vladimir GerdtEdneralVictor F.<p>Candidate of Physical and Mathematical Sciences, Senior Researcher of Skobeltsyn Institute of Nuclear Physics</p>edneral@theory.sinp.msu.ruhttps://orcid.org/0000-0002-5125-0603Lomonosov Moscow State UniversityPeoples’ Friendship University of Russia (RUDN University)1211202129430633612112021Copyright © 2021, Edneral V.F.2021<p style="text-align: justify;">Center for Computational Methods in Applied Mathematics of RUDN, Professor V.P. Gerdt, whose passing was a great loss to the scientific center and the computer algebra community. The article provides biographical information about V.P. Gerdt, talks about his contribution to the development of computer algebra in Russia and the world. At the end there are the authors personal memories of V.P. Gerdt.</p>computer algebraquantum computingmimetic methodspolynomial computer algebra methodsкомпьютерная алгебраквантовые вычислениямиметические методыметоды полиномиальной компьютерной алгебры<p>1. Introduction The name of Vladimir Gerdt is widely known among computer algebra community. Many years he was a professor at the Joint Institute for Nuclear Research (JINR), where he was the head of the Group of Algebraic and Quantum Computations (http://compalg.jinr.ru/CAGroup), and an organizer of many mathematical conferences. A few years ago, he was invited to head the Scientific Center for Computational Methods in Applied Mathematics founded in RUDN university. His passing was a great loss to the entire community. 2. Biography Vladimir Gerdt was born in Engels near Saratov. He earned his M.Sc. in Theoretical Physics from Saratov State University in 1971, his Ph.D. in Theoretical and Mathematical Physics from JINR in 1976, and his D.Sc. in Mathematics and Computer Science from JINR in 1992. In 1997 he got the scientific title Professor in Mathematics and Computer Science by Edneral V.F., 2021 This work is licensed under a Creative Commons Attribution 4.0 International License http://creativecommons.org/licenses/by/4.0/ specialty Application of Computer Techniques, Mathematical Modelling and Mathematical Methods to Scientific Research. After his M.Sc. Vladimir Gerdt worked in JINR until his death in January 5, 2021. He began as an engineer-programmer (1971-1975), then he worked as a junior researcher (1975-1977) at the JINR Department of Radiation Safety where a software for neutron spectroscopy was developed. In 1977 he moved to the JINR Laboratory of Computing Techniques and Automation renamed in 2000 as Laboratory of Information Technologies, where he worked as a researcher (1977-1980) and as a senior researcher (1980-1983), and since 1983 as the head of the research group on computer algebra. Vladimir Gerdt worked abroad for several years, in Lille and Aachen, using Russian, English, German and French in his work. Figure 1. Vladimir Gerdt in his office. Dubna, 1998 3. Professional activities V. Gerdt prepared 243 scientific articles, he edited 10 books. His latest researches are devoted to the construction of involutive monomial bases and to the discretizations of incompressible Navier-Stokes equations. His last huge article was published in ArXiv in September 2020. Vladimir was the referee at journals and organizations: - Journal of Symbolic Computation; - Programming and Computer Software; - Physics of Particles and Nuclei Letters; - Russian Foundation for Basic Research; - Russian Science Foundation. Vladimir was a member of: - Association for Computing Machinery (ACM); - ACM Special Interest Group on Symbolic and Algebraic Manipulation (SIGSAM); - Editorial Board of Journal of Symbolic Computation (Academic Press); - Advisory Board of Computer Science Journal of Moldova; - Special Computer Algebra Group of German Societies on Computer Science. Vladimir took part in the coordination of the international research projects: - he was adjoint coordinator of the INTAS-93-0030 project Computer Algebra, Symbolic and Combinatorial Tools in Differential Algebra and Differential Equations, with impact in Fundamental Physics and Control Theory with 10 research teams from EC countries and 7 research teams from NIS countries; - scientific coordinator of cluster A: Computer Assisted Mathematics of the INTAS-93-0893 project ERSIM-FSU Cooperative Network in Informatics and Applied Mathematics with 10 research teams in EC countries and 10 research teams from NIS countries. Vladimir Gerdt paid great attention to teaching. He gave 24 lecture courses for students and young scientists. Under his supervision 10 master theses were prepared, 9 Ph.D. theses were defended. He was the scientific consultant of Yuri Blinkovs thesis for Doctorship of Sciences. Figure 2. Vladimir with students. Dubna, 2002 4. Vladimir Gerdt and computer algebra 4.1. At the beginning of computer algebra Vladimir was one of the first who started computer algebra usage in the USSR in the 70th. This activity was supported by Academic Dmitry Shirkov and Professor Nikolay Govorun. In the early 80th the Joined Institute for Nuclear Research (JINR, Dubna) bought the computer CDC-6500. It was powerful enough for the implementation of the universal computer algebra systems. Professor Tony Hearn kindly passed the REDUCE system to the JINR during his visit to Dubna. Professor Gerdt with colleagues took a large part in its implementation in the institute and assisted in spreading the REDUCE in the scientific centers of the USSR. Vladimir got the First JINR Prize (1986) for the Research on Installation, Development and Application of Program Systems for Symbolic Computation on Mainframe Computers. Vladimir was on Committees of many conferences. The main of them are: - International Symposium on Symbolic and Algebraic Computation (ISSAC); - Conference on Applications of Computer Algebra (ACA); - Polynomial Computer Algebra (PCA); - Computer Algebra in Scientific Computing (CASC), Vladimir was one of its founders. Now CASC-2021 is the 23rd conference in this series. It takes place in Sochi (Russia). Figure 3. Foundators of the CASC Profs. Vladimir Gerdt and Ernst Mayr, Armenia, 2010 4.2. Partial differential equations A large cycle of works by Vladimir Gerdt was devoted to the study of the compatibility of systems of partial differential equations (PDEs) by means of computer algebra. The key to solving the problem was the CauchyKovalevskaya theorem, which reduces the study of the solvability of some classes of systems of partial differential equations to the study of the compatibility of a system of algebraic equations for the coefficients of the corresponding power series. Theoretical research on the compatibility of systems of nonlinear differential equations in general form was started at the beginning of the 20th century by Riquier [1], Janet [2], and Thomas [3]. V.P. Gerdt told us about the long months he spent in the 1980s searching and studying these far from well-known works written in various European languages. Riquier proposed a complete ordering for partial derivatives, using which he distinguished some of the derivatives, called principal ones, with respect to which the system of PDEs can be resolved. The remaining derivatives, called parametric, leave arbitrariness in the solution and affect the setting of the initial conditions. As a result, a theory was constructed containing the Cauchy- Kovalevskaya theorem as a special case. Along the way of algorithmization of these results, Janet introduced the partition of independent variables into multiplicative and non-multiplicative for the principal derivatives. Thomas generalized the Riquier-Janet approach over the case of nonlinear algebraic equations with respect to the principal derivative. He showed how to check the consistency of a system or to split it into subsystems in a finite number of steps (Thomas decomposition). These works, at first, gave rise to a modern theory, which makes it possible to investigate the compatibility of systems of partial differential equations and carry out their decomposition into subsystems (Thomas decomposition), created by V.P. Gerdt together with D. Roberts and very elegantly inscribed in the theory of differential rings. Later, the theory was used to create the DifferentialThomas package, recently implemented in Maple (https: //www.maplesoft.com). A monograph by D. Roberts is devoted to this issue [4]. 4.3. Polynomial computer algebra One of the most important achievements of algebra in the XX century was the creation of the theory of Grbner bases, which made it possible to study problems from the theory of polynomial rings and algebraic geometry using a computer [5]. The main obstacle to the application of this technique is the cost of calculating these bases according to the Buchberger algorithm, therefore, the development of more efficient methods for finding Grbner bases has been and remains an urgent problem of computer algebra. The key idea of the theory of Grbner bases is the division of a polynomial into polynomials generating a certain ideal</p>[C. Riquier, Les Systèmes d’Equations aux Dérivées Partielles. Paris: Gauthier-Villars, 1910.][M. Janet, “Systèmes d’équations aux dérivées partielles,” Journals de mathématiques, 8e série, vol. 3, pp. 65-151, 1920.][J. Thomas, Differential systems. New York: American Mathematical Society, 1937.][D. Robertz, Formal algorithmic elimination for PDEs. Springer, 2014.][D. Cox, J. Little, and D. O’Shea, Ideals, varieties, and algorithms, 3rd ed. Springer, 2007.][F. S. Macaulay, “Some properties of enumeration in the theory of modular systems,” Proceedings of the London Mathematical Society, vol. s2-26, no. 1, pp. 531-555, 1927. DOI: 10.1112/plms/s2-26.1.531.][A. Y. Zharkov and Y. A. Blinkov, “Involution approach to solving systems of algebraic equations,” in Proceedings of the 1993 International IMACS Symposium on Symbolic Computation. Laboratoire d’Informatique Fondamentale de Lille, France, 1993, pp. 11-16.][A. Y. Zharkov, “Involutive polynomial bases: General case,” in Preprint JINR E5-94-224. Dubna, 1994.][A. Y. Zharkov and Y. A. Blinkov, “Algorithm for constructing involutive bases of polynomial ideal,” in International Conference on Interval and Computer-Algebraic Methods in Science and Engineering. St-Petersburg, 1994, pp. 258-260.][A. Y. Zharkov and Y. A. Blinkov, “Involutive bases of zero-dimensional ideals,” in Preprint JINR E5-94-318. Dubna, 1994.][V. P. Gerdt, “Gröbner bases and involutive methods for algebraic and differential equations,” in Computer Algebra in Science and Engineering. Singapore: World Scientific, 1995, pp. 117-137.][J. Apel, “A Gröbner approach to involutive bases,” Journal of Symbolic Computation, vol. 19, no. 5, pp. 441-458, 1995.][V. P. Gerdt and Y. A. Blinkov, “Involutive polynomial bases,” in Publication IT-95-271. Laboratoire d’Informatique Fondamentale de Lille, 1995.][V. P. Gerdt and Y. A. Blinkov, “Involutive Bases of Polynomial Ideals,” in Preprint-Nr.1/1996. Naturwissenschaftlich-Theoretisches Zentrum, University of Leipzig, 1996.][V. P. Gerdt, “Gröbner bases and involutive methods for algebraic and differential equations,” Mathematical and computer modelling, vol. 25, no. 8-9, pp. 75-90, 1997. DOI: 10.1016/S0895-7177(97)00060-5.][V. P. Gerdt and Y. A. Blinkov, “Involutive bases of polynomial ideals,” in Preprint JINR E5-97-3. Dubna, 1997.][V. P. Gerdt, “Involutive divisions in mathematica: Implementation and some applications,” in Proceedings of the Rhein Workshop on Computer Algebra. Institute for Algorithms and Scientific Computing, GMD-SCAI, 1998, pp. 74-91.][V. P. Gerdt and Y. A. Blinkov, “Involutive divisions of monomials,” Programming and Computer Software, vol. 24, no. 6, pp. 283-285, 1998.][Y. A. Blinkov, “Division and algorithms in the ideal membership problem [Deleniye i algoritmy v zadache o prinadlezhnosti k idealu],” Izvestija Saratovskogo universiteta, vol. 1, no. 2, pp. 156-167, 2001, In Russian.][K. Lipnikov, G. Manzini, and M. Shashkov, “Mimetic finite difference method,” Journal of Computational Physics, vol. 257, pp. 1163-1227, 2014. DOI: 10.1016/j.jcp.2013.07.031.][B. Koren, R. Abgrall, P. Bochev, J. Frank, and B. Perot, “Physicscompatible numerical methods,” Journal of Computational Physics, vol. 257, no. Part B, pp. 1039-1524, 2014. DOI: 10.1016/j.jcp.2013.10.015.][M. Shashkov, Conservative finite difference methods. Boca Raton: CRC Press, 1996.][D. Arnold, P. Bonchev, R. Lehoucq, R. Nikolaides, and M. Shashkov, Compatible spatial discretizations. Springer-Verlag, Berlin, 2006.][L. B. da Veiga, K. Lipnikov, and G. Manzini, The mimetic finite difference method for elliptic problems. Springer, 2014, vol. 11.][J. E. Castillo and G. F. Miranda, Mimetic discretization methods. Chapman and Hall/CRC, 2013.][V. P. Gerdt, D. Robertz, and Y. A. Blinkov, “Strong Consistency and Thomas Decomposition of Finite Difference Approximations to Systems of Partial Differential Equations,” 2020. arXiv: 2009.01731[cs.SC].]