ON THE RELATIVE POSITION OF LIMIT CYCLES FOR THE EQUATION OF TYPE $\[{(II)_{l = 0}}\]$

In this paper, we consider the relative position of limit cycles for the system $$\[\begin{array}{*{20}{c}} {\frac{{dx}}{{dt}} = \delta x - y + mxy - {y^2}}\{\frac{{dy}}{{dt}} = x + a{x^2}} \end{array}\]$$ under the condition $$\[a < 0,0 < \delta \le m,m \le \frac{1}{a} - a\]$$ The main result is as follows: (i)Under Condition (2), if $\[\delta = \frac{m}{2} + \frac{{{m^2}}}{{4a}} \equiv {\delta _0}\]$, then system $\[{(1)_{{\delta _0}}}\] $ has no limit cycles and on singular closed trajectory through a saddle point in the whole plane, (ii)Under condition (2), the foci 0 and R'' cannot be surrounded by the limit cycles of system (1) simultaneously.     

Archimedan Prüfer Structures Alias Filter Archimedean d-Semigroups

Inspired by the monograph of {Larsen/McCarthy} the author\break started a series of articles concerning abstract multiplicative ideal theory along the lines of [23]. In the present paper we turn to archimedean Pr\"ufer structures, that is to algebraic -lattices satisfying the two implications and . Since these properties imply commutativity we start from commutative structures without real loss of generality and give characterizations of arbitrary and also of special Archimedean Pr\"ufer structures.     

Limit sets for complete minimal immersions

In this paper we study the behaviour of the limit set of complete proper compact minimal immersions in a domain with the boundary We prove that the second fundamental form of the surface ∂G is nonnegatively defined at every point of the limit set of such immersions. A. Alarcón’s research is partially supported by MEC-FEDER Grant no. MTM2004-00160.     

Isometries and direct decompositions of pseudo MV-algebras

In the paper isometries in pseudo MV-algebras are investigated. It is shown that for every isometry f in a pseudo MV-algebra = (A, ⊕, ⁻, ^∼, 0, 1) there exists an internal direct decomposition of with commutative such that and for each x ∈ A. On the other hand, if is an internal direct decomposition of a pseudo MV-algebra = (A, ⊕, ⁻, ^∼, 0, 1) with commutative, then the mapping g: A → A defined by is an isometry in and .      

Nonlinear Parabolic Equations with Singularities in Colombeau Vector Spaces

We consider nonlinear parabolic equations with nonlinear non-Lipschitz＇s term and singular initial data like Dirac measure, its derivatives and powers. We prove existence-uniqueness theorems in Colombeau vector space yC^1,W^2,2（[0,T）,R^n）,n ≤ 3. Due to high singularity in a case of parabolic equation with nonlinear conservative term we employ the regularized derivative for the conservative term, in order to obtain the global existence-uniqueness result in Colombeau vector space yC^1,L^2（[0,T）,R^n）,n≤ 3.     

Fourier transforms related to a root system of rank 1

We introduce an algebra consisting of difference-reflection operators and multiplication operators that can be considered as a q = 1 analogue of Sahi's double affine Hecke algebra related to the affine root system of type . We study eigenfunctions of a Dunkl-Cherednik-type operator in the algebra , and the corresponding Fourier transforms. These eigenfunctions are nonsymmetric versions of the Wilson polynomials and the Wilson functions.     

图的平面Turán数和平面anti-Ramsey数

在所有顶点数为$n$且不包含图$G$作为子图的平面图中,具有最多边数的图的边数称为图$G$的平面Turán数,记为$ex_{_\mathcal{P}}（n,G）$。给定正整数$n$以及平面图$H$,用$\mathcal{T}_n （H）$来表示所有顶点数为$n$且不包含$H$作为子图的平面三角剖分图所组成的图集合。设图集合$\mathcal{T}_n （H）$中的任意平面三角剖分图的任意$k$边染色都不包含彩虹子图$H$,则称满足上述条件的$k$的最大值为图$H$的平面anti-Ramsey数,记作$ar_{_\mathcal{P}}（n,H）$。两类问题的研究均始于2015年左右,至今已经引起了广泛关注。全面地综述两类问题的主要研究成果,以及一些公开问题。     

A Characteristic Property of Injective Holomorphic Germs

Let X ₀ be the germ at 0 of a complex variety and let be a holomorphic germ. We say that f is pseudoimmersive if for any such that , we have . We prove that f is pseudoimmersive if and only if it is injective. Some results about the real case are also considered.     

Boundedness of solutions for polynomial potentials withC 2 time dependent coefficients

In this paper we prove the existence of invariant curves and thus stability for all time for a class of Hamiltonian systems with time dependent potentials: $$\frac{{d^2 x}}{{dt^2 }} + Vx(x,t) = 0,x \in R^1 $$ where $\begin{gathered} V(x,t) = \tfrac{1}{{2n + 2}}x^{2n + 2} + \Sigma _{j = 0}^{2n} \tfrac{{Pj(t)}}{{j + 1}}x^{j + 1} ,p_j (t + 1) = p_j (t),p_j \in C^2 ,2n \geqslant j \geqslant n + 1;p_j \in \\ C^1 ,n \geqslant j \geqslant 0,n \geqslant 1. \\ \end{gathered} $      

Realization of the mapping class group by homeomorphisms

In this paper, we show that the mapping class group of a closed surface can not be geometrically realized as a group of homeomorphisms of that surface. More precisely, let denote the standard projection of the group of homeomorphisms to the mapping class group of a closed surface M of genus g>5. We show that there is no homomorphism , such that is the identity. This answers a question by Thurston (see [11]). Mathematics Subject Classification (2000)  Primary 20H10, 37F30     

Twelve limit cycles around a singular point in a planar cubic-degree polynomial system

In this paper, we prove the existence of 12 small-amplitude limit cycles around a singular point in a planar cubic-degree polynomial system. Based on two previously developed cubic systems in the literature, which have been proved to exhibit 11 small-amplitude limit cycles, we applied a different method to show 11 limit cycles. Moreover, we show that one of the systems can actually have 12 small-amplitude limit cycles around a singular point. This is the best result so far obtained in cubic planar vector fields around a singular point.     

Poincar{\''e} Bifurcation from an Elliptic Hamiltonian of Degree Four with Two-saddle Cycle

In this paper, we consider Poincar{\''e} bifurcation from an elliptic Hamiltonian of degree four with two-saddle cycle. Based on the Chebyshev criterion, not only one case in the Li{\''e}nard equations of type $(3, 2)$ is discussed again in a different way from the previous ones, but also its two extended cases are investigated, where the perturbations are given respectively by adding $\varepsilon y(d_0 + d_2 v^{2n})\frac{\partial }{{\partial y}}$ with $ n\in \mathbb{N^+}$ and $\varepsilon y(d_0 + d_4 {v^4}+ d_2 v^{2n+4})\frac{\partial }{{\partial y}}$ with $n=-1$ or $ n\in \mathbb{N^+}$, for small $\varepsilon > 0$. For the above cases, we obtain all the sharp upper bound of the number of zeros for Abelian integrals, from which the existence of limit cycles at most via the first-order Melnikov functions is determined. Finally, one example of double limit cycles for the latter case is given.     

Existence and uniqueness of limit cycles on a cubic kolmogorov differential system in the Predator-Prey relation

In this paper, we analyse qualitatively a cubic Kolmogorov system: which is one of the mathematical models in ecology describing the interaction between Predator-Prey of two populations; and give the conditions of nonexistence, existence and uniqueness of limit cycles for three different cases.Fulfilled during engagement in advanced studies at the Institute of Mathematics, Academia Sinica.     

The Laplacian on $$ C(\overline \Omega) $$ with generalized Wentzell boundary conditions

In this note we prove that the Laplacian with generalized Wentzell boundary conditions on an open bounded regular domain in defined by generates an analytic semigroup of angle on for every > 0 and (for the definition of cf. (1.3)).Received: 13 July 2002     

Some existence and nonexistence theorems for compact graphs of constant mean curvature with boundary in parallel planes

Given H≥0 and bounded convex curves α₁, ...,⇌_n, α in the plane z=0 bounding domains D₁, …, D_n, D, respectively, with if i ∈ j and with D_i ⊂ D, we obtain several results proving the existence of a constanth depending only on H and on the geometry of the curves α_i, α such that the Dirichlet problem for the constant mean curvature H equation: where may accept or not a solution.     

Vanishing of the third simplicial cohomology group of

We show that . We first use the Connes-Tzygan exact sequence to prove that this is equivalent to the vanishing of the third cyclic cohomology group , where is the non-unital Banach algebra , and then prove that .

     

一类三次系统的极限环II

讨论一类三次系统$$\begin{array}{ll}&\dot{x}=-y(1-ax)(1-bx)+\delta x-lx^3,\\[1mm]&\dot{y}=x(1-c_1x)(1-c_2x)\end{array}$$的极限环问题.这一系统包括了在$a=c_1,~b=c_2$且$a=-b$或$a=c_1,~b=c_2$或$a=c_1$的限制下的系统.去掉了全部这些限制,得到的极限环存在唯一性定理比以前已得到的相关的定理更具广泛性.     

Theory of nonstationary flows of Kelvin-Voigt fluids

One proves the global unique solvability in class \(W_\infty ^1 (0,T;C^{2,d} (\bar \Omega ) \cap H(\Omega ))\) of the initial-boundary-value problem for the quasilinear system $$\frac{{\partial \vec \upsilon }}{{\partial t}} + \upsilon _k \frac{{\partial \vec \upsilon }}{{\partial x_k }} - \mu _1 \frac{{\partial \Delta \vec \upsilon }}{{\partial t}} - \int\limits_0^t {K(t - \tau )\Delta \vec \upsilon (\tau )d\tau + grad p = \vec f,di\upsilon \bar \upsilon = 0,\upsilon , > 0.}$$ This system described the nonstationary flows of the elastic-viscous Kelvin-Voigt fluids with defining relation $$\left( {1 + \sum\limits_{\ell = 1}^L {\lambda _\ell } \frac{{\partial ^\ell }}{{\partial t^\ell }}} \right)\sigma = 2\left( {v + \sum\limits_{m = 1}^{L + 1} {\user2{\ae }_m } \frac{{\partial ^m }}{{\partial t^m }}} \right)D,L = 0,1,2,...;\lambda _L ,\user2{\ae }_{L + 1} > 0.$$      

Periodic solutions of prescribed energy for a class of symmetric singular dynamical systems

We consider the following Hamiltonian system:

Perturbation Bounds for the Polar Factors

Let $A$, $\tilde{A}\in C^{m\times n}$, rank (A)=rank ($\tilde{A}$)=$n$. Suppose that $A=QH$ and $\tilde{A}=\tilde{Q}\tilde{H}$ are the polar decompositions of $A$ and $\tilde{A}$, respectively. It is proved that $$\|\tilde{Q}-Q\|_F\leq 2\|A^+\|_2\|\tilde{A}-A\|_F$$ and $$\|\tilde{H}-H\|_F\leq \sqrt{2}\|\tilde{A}-A\|_F$$ hold, where $A^+$ is the Moore-Penrose inverse of $A$, and $\| \|_2$ and $\| \|_F$ denote the spectral norm and the Frobenius norm, respectively.