Regularity of Bunimovich’s stadia

Stadia are popular models of chaotic billiards introduced by Bunimovich in 1974. They are analogous to dispersing billiards due to Sinai, but their fundamental technical characteristics are quite different. Recently many new results were obtained for various chaotic billiards, including sharp bounds on correlations and probabilistic limit theorems, and these results require new, more powerful technical apparatus. We present that apparatus here, in the context of stadia, and prove “regularity” properties.      

Domain separation by means of sign changing eigenfunctions of p-laplacians

We are interested in algorithms for constructing surfaces Γ of possibly small measure that separate a given domain ω into two regions of equal measure. Using the integral formula for the total gradient variation, we show that such separators can be constructed approximatively by means of sign changing eigenfunctions of the p-Laplacians,p→1 under humogeneous Neumann houndary conditions. These eigenfunctions are proven to be limtes of a steepest descent method applied to suitable norm quotients. Finally we use these ideas for the construction of separators on simplex grids.     

The asymptotic eigenfunctions of the operator ?D(x, y)? corresponding to Liouville metrics and waves on water captured by bottom irregularities

We construct a set of examples of bottom reliefs for which there exist captured waves corresponding to quasimodes of the wave operator ∇D(x, y)∇. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 414–422, September, 1998. The author thanks S. Yu. Dobrokhotov for stating the problem and numerous helpful discussions.     

在连续的时间系统中不存在Shilnikov型混沌

在n维的、时间连续的光滑系统中,得到了不存在同宿轨道和异宿轨道的条件.基于此结论并用一个基本实例,推断出如下结论:在多项式常微分方程系统中,有着以不存在同宿轨道和异宿轨道为特征的第4类混沌.     

Analytic description of statistics of spectra of quantum graphs

We discuss how to obtain exact and approximate distributions for various statistical characteristics of the spectra of quantum graphs using previously found exact solutions of the spectral problem. We indicate the relation between the appearing spectral decompositions and the theory of weakly dependent random variables and indicate the relation between the known limit theorems for trigonometric sums and the universal statistical properties of the spectra of quantum chaotic systems. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 1, pp. 38–66, July, 2008.     

An alternative proof of lower bounds for the first eigenvalue on manifolds

Recently, Andrews and Clutterbuck [1] gave a new proof of the optimal lower eigenvalue bound on manifolds via modulus of continuity for solutions of the heat equation. In this short note, we give an alternative proof of Theorem 2 in [1]. More precisely, following Ni's method (Section 6 of [5]), we give an elliptic proof of this theorem.     

Li-Yorke chaos in the system with impacts

The analogue of Li-Yorke chaos [T.Y. Li, J. Yorke, Period three implies chaos, Amer. Math. Monthly 87 (1975) 985-992] for a special initial value problem of a non-autonomous impulsive differential equation is developed.     

Bifurcation,chaos, and their control in a wheelset model

In this paper, we present an improved wheelset motion model with two degrees of freedom and study the dynamic behaviors of the system including the symmetry, the existence and uniqueness of the solution, continuous dependence on initial conditions, and Hopf bifurcation. The dynamic characteristics of the wheelset motion system under a nonholonomic constraint are investigated. These results generalize and improve some known results about the wheelset motion system. Meanwhile, based on multiple equilibrium analysis, calculation of Lyapunov exponents and Poincaré section, the chaotic behaviors of the wheelset system are discussed, which indicates that there are more complex dynamic behaviors in the railway wheelset system with higher order terms of Taylor series of trigonometric functions. This paper has also realized the chaos control and bifurcation control for the wheelset motion system by adaptive feedback control method and linear feedback control. The results show that the chaotic wheelset system and bifurcation wheelset system are all well controlled, whether by controlling the yaw angle and the lateral displacement or only by controlling the yaw angle. Numerical simulations are carried out to further verify theoretical analyses.     

Spreading of the free boundary of an ideal fluid in a vacuum

The diameter of a region occupied by an ideal fluid surrounded by vacuum will be shown to grow linearly in time provided the pressure is positive and there are no singularities. A family of explicit spherically symmetric, self-similar global solutions is constructed which illustrate the result in the compressible case.     

The Effect of Clone Template Parameters on the Spreading Speeds in CNNs

The aim of this paper is to investigate the effect of clone template parameters on the spreading speeds in cellular neural networks(CNNs). According to the property analysis of spreading speeds of monotone semiflows developed by Yu and Zhang [{\it European Journal of Applied Mathematics}, {\bf 31} (2020), 369-384], we investigate the sign of spreading speeds, continuity and limit cases with no propagation phenomena for CNNs with general output functions where each cell interacts with its 2-neighborhood cell.     

Primitive Ideals in the Algebra of Regular Functions on Quantum m × n Matrices

Mathematical Notes -