On the index of finite determinacy of vector fields with resonances

In this paper we study normal forms for a class of germs of 1-resonant vector fields on R^n with mutually different eigenvalues which may admit extraneous resonance relations. We give an estimation on the index of finite determinacy from above as well as the essentially simplified polynomial normal forms for such vector fields. In the case that a vector field has a zero eigenvalue, the result leads to an interesting corollary, a linear dependence of the derivatives of the hyperbolic variables on the central variable.     

Non-linear determinacy of minimum wave speed for a Lotka-Volterra competition model

For a reaction-diffusion system that serves as a 2-species Lotka-Volterra diffusive competition model, suppose that the corresponding reaction system has one stable boundary equilibrium and one unstable boundary equilibrium. Then it is well known that there exists a positive number c^?, called the minimum wave speed, such that, for each c larger than or equal to c^?, the reaction-diffusion system has a positive traveling wave solution of wave speed c connecting these two equilibria if and only if c?c^?. It has been shown that the minimum wave speed for this system is identical to another important quantity - the asymptotical speed of population spread towards the stable equilibrium. Hence to find the minimum wave speed c^? not only is of the interest in mathematics but is of the importance in application. It has been conjectured that the minimum wave speed can be determined by studying the eigenvalues of the unstable equilibrium, called the linear determinacy. In this paper we will show that the conjecture on the linear determinacy is not true in general.     

某类实解析与复解析分歧问题的R-有限决定及K-有限决定问题的注记

该文给出了某类实解析与复解析分歧问题是不能R-有限决定的．并利用上述结果证明了一类分岐问题是不能K-有限决定的．同时给出了一个K-有限决定但不能R-有限决定的个例子,它说明在分岐问题中R-有限决定与K-有限决定不是互为充要的条件．     

Notes on Smooth Map Germs

ＮｏｔｅｓｏｎＳｍｏｏｔｈＭａｐＧｅｒｍｓＹｕＬｉ（余立）；ＺｈａｎｇＧｕｏｂｉｎ（张国滨）（ＺｈａｎｊｉａｎｇＴｅａｃｈｅｒｓＣｏｌｌｅｇｅ，Ｃｈｉｎａ）Ａｂｓｔｒａｃｔ：Ｉｎｔｈｉｓｐａｐｅｒ，ｗｅｓｔｕｄｙｔｈｅｄｅｔｅｒｍｉｎａｃｙｏｆｒｅａ...     

Qualitatively invertible matrices

Our purpose is to describe the properties of the class of n×n matrices such that some or all of the signs of the nonzero elements of the inverse matrix can be determined based only upon the knowledge of the signs of the matrix being inverted. Applications of such matrices can be found in the study of the testibility of economic theory, the stability of economic systems and the general ‘quality’ of solutions to large computer models.     

Elementary inductive dichotomy: Separation of open and clopen determinacies with infinite alternatives

We introduce a new axiom called inductive dichotomy, a weak variant of the axiom of inductive definition, and analyze the relationships with other variants of inductive definition and with related axioms, in the general second order framework, including second order arithmetic, second order set theory and higher order arithmetic. By applying these results to the investigations on the determinacy axioms, we show the following. (i) Clopen determinacy is consistency-wise strictly weaker than open determinacy in these frameworks, except second order arithmetic; this is an enhancement of Schweber–Hachtman separation of open and clopen determinacy into the consistency-wise separation. (ii) Hausdorff–Kuratowski hierarchy of differences of opens is faithfully reflected by the hierarchy of consistency strengths of corresponding parameter-free determinacies in the aforementioned frameworks; this result is valid also in second order arithmetic only except clopen determinacy.     

关于映射芽在A和K的一些子群下的有限决定性

本文利用乘积积分理论给出了映射芽在Ａ和Ｋ的一些子群下有限决定的充分必要条件     

Spreading and generalized propagating speeds of discrete KPP models in time varying environments

The current paper deals with spatial spreading and front propagating dynamics for spatially discrete KPP (Kolmogorov, Petrovsky and Paskunov) models in time recurrent environments, which include time periodic and almost periodic environments as special cases. The notions of spreading speed interval, generalized propagating speed interval, and traveling wave solutions are first introduced, which are proper modifications of those introduced for spatially continuous KPP models in time almost periodic environments. Among others, it is then shown that the spreading speed interval in a given direction is the minimal generalized propagating speed interval in that direction. Some important upper and lower bounds for the spreading and generalized propagating speed intervals are provided. When the environment is unique ergodic and the so called linear determinacy condition is satisfied, it is shown that the spreading speed interval in any direction is a singleton (called the spreading speed), which equals the classical spreading speed if the environment is actually periodic. Moreover, in such a case, a variational principle for the spreading speed is established and it is shown that there is a front of speed c in a given direction if and only if c is greater than or equal to the spreading speed in that direction.      

Traveling waves for a reaction–diffusion–advection predator–prey model

In this paper we study a reaction–diffusion–advection predator–prey model in a river. The existence of predator-invasion traveling wave solutions and prey-spread traveling wave solutions in the upstream and downstream directions is established and the corresponding minimal wave speeds are obtained. While some crucial improvements in theoretical methods have been established, the proofs of the existence and nonexistence of such traveling waves are based on Schauder’s fixed-point theorem, LaSalle’s invariance principle and Laplace transform. Based on theoretical results, we investigate the effect of the hydrological and biological factors on minimal wave speeds and hence on the spread of the prey and the invasion of the predator in the river. The linear determinacy of the predator–prey Lotka–Volterra system is compared with nonlinear determinacy of the competitive Lotka–Volterra system to investigate the mechanics of linear and nonlinear determinacy.     

A GAME-THEORETIC PROOF OF ANALYTIC RAMSEY THEOREM

We give a simple game-theoretic proof of Silver's theorem that every analytic set is Ramsey. A set P of subsets of ω is called Ramsey if there exists an infinite set H such that either all infinite subsets of H are in P or all out of P. Our proof clarifies a strong connection between the Ramsey property of partitions and the determinacy of infinite games.