A remark on periodic solutions of singular Hamiltonian systems

In this note we study the existence of non-collision periodic solutions for singular Hamiltonian systems with weak force. In particular for potential where D is a compact C³-surface in we prove the existence of a non-collision periodic solution.     

A remark on almost periodic transition operators

Results of Rosenblatt on almost periodic transition operators are extended to the reducible case.     

A remark on cuspidal local systems

In this note we show that all reductive groups are clean in characteristic ?3. In characteristic 2 there are two cuspidal local systems (one for F₄ and one for E₈) which can not be handled by our method.     

Tchebyshev systems that cannot be transformed into Markov systems

The linear hull of a Tchebyshev system is called a Haar-space. A basis f₁,...,f_n of an n-dimensional Haar-space is called a Markov basis if f₁,...,f_i form a Tchebyshev system for each i=l,...,n. It is shown by suitable examples that for all n3 there exist Haar-spaces without a Markov basis.     

A remark on doubly critical elliptic systems

We obtain positive solutions for some doubly critical elliptic systems via variational methods. This result improves some existence results of Abdellaoui, Felli and Peral (Calc Var 34:97–137 2009). Our arguments are completely different from those in (Calc Var 34:97–137 2009).     

Tchebyshev Posets

We construct for each $n$ an Eulerian partially ordered set $T_n$ of rank $n+1$ whose $ce$-index provides a non-commutative generalization of the $n$th Tchebyshev polynomial. We show that the order complex of each $T_n$ is shellable, homeomorphic to a sphere, and that its face numbers minimize the expression $\max_{|x|\leq 1} |\sum_{j=0}^n (f_{j-1}/f_{n-1})\cdot 2^{-j}\cdot (x-1)^j|$ among the $f$-vectors of all $(n-1)$-dimensional simplicial complexes. The duals of the posets constructed have a recursive structure similar to face lattices of simplices or cubes, offering the study of a new special class of Eulerian partially ordered sets to test the validity of Stanleys conjecture on the non-negativity of the $cd$-index of all Gorenstein$^*$ posets.     

A remark on partly dissipative reaction diffusion systems on IRn

The long time behavior of the solutions of some partly dissipative reaction diffusion systems is studied. We prove the existence of a compact （L^2 × L^2 - H^1 × L^2） attractor for a partly dissipative reaction diffusion system in Rn. This improves a previous result obtained by A. Rodrigues-Bernal and B. Wang concerning the existence of a compact （L^2 × L^2 - L^2 × L^2） attractor for the same system.     

A remark on the period of the periodic solution in the Lotka-Volterra system

An important consideration in the nonlinear predator-prey problem of Lotka—Volterra type is the determination of the period. This paper gives a general expression for the period in terms of the given parameters in the Lotka-Volterra system. We also discuss the qualitative behavior of the period related to the energy level of the Lotka-Volterra system.     

A remark on global existence of solutions of shadow systems

Let ?? be an open, bounded domain in ${\mathbb{R}^n\;(n \in \mathbb{N})}$ with smooth boundary ???. Let p, q, r, d ₁, ?? be positive real numbers and s be a non-negative number which satisfies ${0 < \frac{p-1}{r} < \frac{q}{s+1}}$ . We consider the shadow system of the well-known Gierer?CMeinhardt system: $$ \left \{ \begin{array}{l@{\quad}l} \displaystyle{u_t = d_1\Delta u - u + \frac{u^p}{\xi^q}}, & \quad {\rm in}\;\Omega \times (0,T), \\ \displaystyle{\tau \xi_t = -\xi + \frac{1}{|\Omega|} \int\nolimits_\Omega\frac{u^r}{\xi^s} {\rm d}x}, & \quad {\rm in}\;(0,T), \\ \displaystyle{\frac{\partial u}{\partial \nu} =0}, & \quad {\rm on}\;\partial \Omega \times (0,T), \\ \displaystyle{\xi(0) = \xi_0 >0 , \quad u(\cdot,0) = u_0(\cdot)} \geq 0 & \quad {\rm in}\;\Omega. \end{array} \right. $$ We prove that solutions of this system exist globally in time under some conditions on the coefficients. Our results are based on a priori estimates of the solutions and improve the global existence results of Li and Ni in [4].     

A remark on sets of determining elements for reaction-diffusion systems

For a class of systems of parabolic equations, conditions represented by a finite set of linear functionals on the phase space that uniquely determine the long-time behavior of solutions are found. The cases in which it is sufficient to define these determining functionals only on a part of the components of the state vector are singled out. As examples, systems describing the Belousov-Zhabotinsky reaction and the two-dimensional Navier-Stokes equations are considered. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 774–784, May, 1998.     

A remark on global existence of solutions of shadow systems

Let Ω be an open, bounded domain in \mathbbRⁿ  (n ? \mathbbN){\mathbb{R}^n\;(n \in \mathbb{N})} with smooth boundary ∂Ω. Let p, q, r, d ₁, τ be positive real numbers and s be a non-negative number which satisfies 0 < \fracp-1r < \fracqs+1{0 < \frac{p-1}{r} < \frac{q}{s+1}}. We consider the shadow system of the well-known Gierer–Meinhardt system:

关于代换系统拓扑序列熵的注记

Goodman证明了对两符号等长代换系统, 如果代换规则中0和1对应的词只有一个位置不同,那么对应的代换系统为null的, 即此系统沿着任意正整数序列的序列熵均为0. 在本文中, 我们针对系统的结构特征, 通过考察因子系统, 给出了此经典结果的另外一种证明. 同时, 对此类代换系统沿着给定序列的复杂性, 我们得到了比Goodman更为精确的估计.     

A remark on Mackey-functors

In the following note we characterize the category of Mackey functors from a categoryC, satisfying a few assumptions, to a categoryD as the category of functors from Sp(C), the category of spans inC, toD which preserve finite products. This caracterization permits to apply all results on categories of functors preserving a given class of limits to the case of Mackey-functors.     

A remark on accessibility

This note obtains some characteristics of accessibility and Kato’s chaos. Applying these results, an accessible dynamical system whose product system is not accessible is constructed, giving a negative answer to a question in [Li R, Wang H, Zhao Y. Kato’s chaos in duopoly games. Chaos Solit Fract 2016;84:69–72]. Besides, it is proved that every transitive interval self-map is accessible.     

A remark on focal points of recessive solutions of discrete symplectic systems

We investigate nonoscillatory and controllable symplectic difference systems. We show that the recessive solution of such a system at +∞ has the same number of focal points (counting multiplicities) as the recessive solution at −∞.     

A remark on quasi-isometries

We show that if is an quasi-isometry, with , defined on the unit ball of , then there is an affine isometry with where is a universal constant. This result is sharp.