Vol 74, No 5 (2019)

Year: 2019
Articles: 10
URL: https://journals.rcsi.science/0042-1316/issue/view/7512

Chebyshev centres, Jung constants, and their applications

Alimov A.R., Tsar'kov I.G.

Abstract

The approximation of concrete function classes is the most common subject in the theory of approximations of functions. An important particular case of this is the problem of the Chebyshev centre and radius. As it turns out, this problem is not only a special case of the Kolmogorov width problem, but it is also related in a mysterious way to other important characteristics and results in the theory of functions and other more general branches of analysis and geometry. The aim of the present study is to give a survey of the current state of this problem and to discuss its possible applications.Bibliography: 169 titles.

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Uspekhi Matematicheskikh Nauk. 2019;74(5):3-82

3-82

(Rus)

(JATS XML)

Krotov method for optimal control of closed quantum systems

Morzhin O.V., Pechen A.N.

Abstract

The mathematics of optimal control of quantum systems is of great interest in connection with fundamental problems of physics as well as with existing and prospective applications to quantum technologies. One important problem is the development of methods for constructing controls for quantum systems. One of the commonly used methods is the Krotov method, which was initially proposed outside of quantum control theory in articles by Krotov and Feldman (1978, 1983). This method was used to develop a novel approach to finding optimal controls for quantum systems in [64] (Tannor, Kazakov, and Orlov, 1992), [65] (Somloi, Kazakov, and Tannor, 1993), and in many other works by various scientists. Our survey discusses mathematical aspects of this method for optimal control of closed quantum systems. It outlines various modifications with different forms of the improvement function (for example, linear or linear-quadratic), different constraints on the control spectrum and on the admissible states of the quantum system, different regularisers, and so on. The survey describes applications of the Krotov method to controlling molecular dynamics and Bose–Einstein condensates, and to quantum gate generation. This method is compared with the GRAPE (GRadient Ascent Pulse Engineering) method, the CRAB (Chopped Random-Basis) method, and the Zhu–Rabitz and Maday–Turinici methods.Bibliography: 158 titles.

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Uspekhi Matematicheskikh Nauk. 2019;74(5):83-144

83-144

(Rus)

(JATS XML)

Circle problem and the spectrum of the Laplace operator on closed 2-manifolds

Popov D.A.

Abstract

In this survey the circle problem is treated in the broad sense, as the problem of the asymptotic properties of the quantity $P(x)$, the remainder term in the circle problem. A survey of recent results in this direction is presented. The main focus is on the behaviour of $P(x)$ on short intervals. Several conjectures on the local behaviour of $P(x)$ which lead to a solution of the circle problem are presented. A strong universality conjecture is stated which links the behaviour of $P(x)$ with the behaviour of the second term in Weyl's formula for the Laplace operator on a closed Riemannian 2-manifold with integrable geodesic flow.Bibliography: 43 titles.

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Uspekhi Matematicheskikh Nauk. 2019;74(5):145-162

145-162

(Rus)

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Evgenii Solomonovich Golod (obituary)

Artamonov V.A., Buchstaber V.M., Vinberg E.B., Kuz'min L.V., Kulikov V.S., Latyshev V.N., Mikhalev A.V., Ol'shanskii A.Y., Orlov D.O., Parshin A.N., Piontkovskii D.I.

Uspekhi Matematicheskikh Nauk. 2019;74(5):163-169

163-169