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卷 77, 编号 3 (2022)
- 年: 2022
- 文章: 9
- URL: https://journals.rcsi.science/0042-1316/issue/view/7528
Elements of hyperbolic theory on an infinite-dimensional torus
摘要
On the infinite-dimensional torus $\mathbb{T}^{\infty}=E/2\pi\mathbb{Z}^{\infty}$, where $E$ is an infinite-dimensional Banach torus and $\mathbb{Z}^{\infty}$ is an abstract integer lattice, a special class of diffeomorphisms $\operatorname{Diff}(\mathbb{T}^{\infty})$ is considered. It consists of the maps $G\colon\mathbb{T}^{\infty}\to\mathbb{T}^{\infty}$ whose differentials $DG$ and $D(G^{-1})$ are uniformly bounded and uniformly continuous on $\mathbb{T}^{\infty}$. For diffeomorphisms in $\operatorname{Diff}(\mathbb{T}^{\infty})$ elements of hyperbolic theory are presented systematically, starting with definitions and some auxiliary facts and ending by more advanced results. The latter include a criterion for hyperbolicity, a theorem on the $C^1$-roughness of hyperbolicity for diffeomorphisms in $\operatorname{Diff}(\mathbb{T}^{\infty})$, the Hadamard–Perron theorem, as well as a fundamental result of hyperbolic theory, the fact that each Anosov diffeomorphism $G\in\operatorname{Diff}(\mathbb{T}^{\infty})$ has a stable and an unstable invariant foliation.Bibliography: 34 titles.
Uspekhi Matematicheskikh Nauk. 2022;77(3):3-72
3-72
Feynman checkers: towards algorithmic quantum theory
摘要
We survey and develop the most elementary model of electron motion introduced by Feynman. In this game, a checker moves on a checkerboard by simple rules, and we count the turns. Feynman checkers are also known as a one-dimensional quantum walk or an Ising model at imaginary temperature. We solve mathematically a Feynman problem from 1965, which was to prove that the discrete model (for large time, small average velocity, and small lattice step) is consistent with the continuum one. We study asymptotic properties of the model (for small lattice step and large time) improving the results due to Narlikar from 1972 and to Sunada and Tate from 2012.For the first time we observe and prove concentration of measure in the small-lattice-step limit. We perform the second quantization of the model.We also present a survey of known results on Feynman checkers.Bibliography: 53 titles.
Uspekhi Matematicheskikh Nauk. 2022;77(3):73-160
73-160
Nikolai Nikolaevich Konstantinov (obituary)
Uspekhi Matematicheskikh Nauk. 2022;77(3):161-170
161-170
On estimating the local error of a numerical solution of a parametrized Cauchy problem
Uspekhi Matematicheskikh Nauk. 2022;77(3):171-172
171-172
Spectrum of a convolution operator with potential
Uspekhi Matematicheskikh Nauk. 2022;77(3):173-174
173-174
Infinite sets can be Ramsey in the Chebyshev metric
Uspekhi Matematicheskikh Nauk. 2022;77(3):175-176
175-176
Roots of the characteristic equation for the symplectic groupoid
Uspekhi Matematicheskikh Nauk. 2022;77(3):177-178
177-178
Viktor Stepanovich Kulikov (on his 70th birthday)
Uspekhi Matematicheskikh Nauk. 2022;77(3):179-181
179-181
Evgenii Vital'evich Shchepin (on his 70th birthday)
Uspekhi Matematicheskikh Nauk. 2022;77(3):182-192
182-192