The discrete parts of approximately decidable sets in Euclidean spaces

It is shown that the classes of discrete parts, A ∩ ?^k, of approximately resp. weakly decidable subsets of Euclidean spaces, A ? ?^k, coincide and are equal to the class of ω‐r. e. sets which is well‐known as the first transfinite level in Ershov's hierarchy exhausting Δ⁰₂.     

The Hausdorff-Ershov Hierarchy in Euclidean Spaces

The topological arithmetical hierarchy is the effective version of the Borel hierarchy. Its class Δ^ta₂ is just large enough to include several types of pointsets in Euclidean spaces ℝ^k which are fundamental in computable analysis. As a crossbreed of Hausdorff's difference hierarchy in the Borel class Δ^B₂ and Ershov's hierarchy in the class Δ⁰₂ of the arithmetical hierarchy, the Hausdorff-Ershov hierarchy introduced in this paper gives a powerful classification within Δ^ta₂. This is based on suitable characterizations of the sets from Δ^ta₂ which are obtained in a close analogy to those of the Δ^B₂ sets as well as those of the Δ⁰₂ sets. A helpful tool in dealing with resolvable sets is contributed by the technique of depth analysis. On this basis, the hierarchy properties, in particular the strict inclusions between classes of different levels, can be shown by direct constructions of witness sets. The Hausdorff-Ershov hierarchy runs properly over all constructive ordinals, in contrast to Ershov's hierarchy whose denotation-independent version collapses at level ω². Also, some new characterizations of concepts of decidability for pointsets in Euclidean spaces are presented.     

Martingale convergence,series expansion of Gaussian elements,and strong law of large numbers in Fréchet spaces

The main result of this paper is the derivation of a convergence theorem for certain martingales with values in a separable Fréchet space F. It is shown that this result includes a well known theorem due to Chatterji. Moreover, the series expansion of zero-mean Gaussian elements with values in F and the strong law of large numbers for i.i.d. F-valued random elements also follow as applications of the main theorem.     

Dynamical behavior for a class of predator–prey system with general functional response and discontinuous harvesting policy

In this paper, we introduce a class of predator–prey system with general functional response, whose harvesting policy is modeled by a discontinuous function. Based on the differential inclusions theory, topological degree theory in set‐valued analysis and generalized Lyapunov approach, we analyze the existence, uniqueness and global asymptotic stability of positive periodic solution. In particular, a series of useful criteria on existence, uniqueness and global asymptotic stability of the positive equilibrium point are established for the autonomous system corresponding to the non‐autonomous biological and mathematical model with a discontinuous right‐hand side. Moreover, some new sufficient conditions are provided to guarantee the global convergence in measure of harvesting solution and convergence in finite time of any positive solution for the autonomous discontinuous biological system. The obtained results, which improve and generalize previous works on dynamical behavior in the literature, are of interest for understanding and designing biological system with not only continuous or even Lipschitz continuous but also discontinuous harvesting function. Finally, we give three examples with numerical simulations to show the applicability and effectiveness of our main results. Copyright © 2015 John Wiley & Sons, Ltd.     

Borel resolvability of compact spaces and their subspaces

The presence of disjoint dense (Borel) subsets in Tychonoff cubes, Borel subspaces of Tychonoff cubes, and dyadic compacta is examined. Several problems are stated. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 701–712, November, 1998. The author wishes to express his gratitude to V. Shchigolev for fruitful discussion of some questions touched upon in this paper and to the referee for valuable comments and suggestions concerning the style and the content. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-01619.     

AHP中互反判断矩阵一致性调整的新简易方法及交互式实现

提出Satty′s互反判断矩阵一致性调整的一种新的简易方法,并借助EXCEL电子表格的动态引用功能,实现了判断者在建立判断矩阵时可即时地进行一致性的互动式调整;不但改善了现有不合理的强制性调整方法,而且提供了一种有效辅助判断矩阵建立的途径.其中,借鉴欧式距离原理构造了"行和列偏离程度系数"来衡量一致性优化调整过程有效性,同时实现了权重向量相对于互反判断矩阵调整的灵敏度分析.最后通过一个具体实例演示方法的操作过程,并检验了其有效性.