On R∞ and Q∞-manifolds

We give a characterization of manifolds modeled on $\text{R}{}^{\mathrm{\infty }}\text{= dir lim}$ or $\text{R}{}^{\mathrm{n}}\text{Q}{}^{\mathrm{\infty }}\text{=dir lim Q}{}^{\mathrm{n}}$, where Q is the Hilbert cube, and elementary short proofs of the Open Embedding Theorem for these manifolds and the following theorem generalizing the Stability Theorem: Each fine homotopy equivalence between these manifolds is a near homeomorphism. Moreover we establish the Open Embedding Approximation Theorem.     

Cube-approximating bounded wavelet sets in

We prove that for any real expansive matrix , there exists a bounded -dilation wavelet set in the frequency domain (the inverse Fourier transform of whose characteristic function is a band-limited single wavelet in the time domain ). Moreover these wavelet sets can approximate a cube in arbitrarily. This result improves Dai, Larson and Speegle's result about the existence of (basically unbounded) wavelet sets for real expansive matrices.

     

On Homeomorphisms of Effective Topological Spaces

We study effective presentations and homeomorphisms of effective topological spaces. By constructing a functor from the category of computable models into the category of effective topological spaces, we show in particular that there exist homeomorphic effective topological spaces admitting no hyperarithmetical homeomorphism between them and there exist effective topological spaces whose autohomeomorphism group has the cardinality of the continuum but whose only hyperarithmetical autohomeomorphism is trivial. It is also shown that if the group of autohomeomorphisms of a hyperarithmetical topological space has cardinality less than 2 then this group is hyperarithmetical. We introduce the notion of strong computable homeomorphism and solve the problem of the number of effective presentations of T ₀-spaces with effective bases of clopen sets with respect to strong homeomorphisms.     

Limiting Laws for Entrance Times of Critical Mappings of a Circle

A renormalization group transformation R ₁ has a single stable point in the space of the analytic circle homeomorphisms with a single cubic critical point and with the rotation number (the golden mean). Let a homeomorphism T be the C ¹-conjugate of . We let denote the sequence of distribution functions of the time of the kth entrance to the nth renormalization interval for the homeomorphism T. We prove that for any , the sequence has a finite limiting distribution function , which is continuous in , and singular on the interval [0,1]. We also study the sequence for k>1.     

Convergence of Double Fourier Series after a Change of Variable

In this paper, we prove that for any compact set there exists a homeomorphism of the closed interval such that for an arbitrary function f the Fourier series of the function F(x,y) = f((x),(y)) converges uniformly on simultaneously over rectangles, over spheres, and over triangles.     

关于非线性边值问题几个存在性定理的新结果

本文运用同胚理论,研究微分方程边值问题解的存在性与唯一性,得到两个基本定理,推广了Ｂｒｏｗｎ（ｉｎＡｎｎａｌｉ．Ｍａｔ．Ｐｕｒａ．Ａｐｐｌｉ．１９７５,１０６：２０５～２１４）的结论,取消了有界性假设,并把我们的结果用于有限维的情况,考虑了非线性守恒系统在扰动情况下（Ｎｅｗｔｏｎ类运动方程）周期解的存在性与唯一性问题·把同胚用于这类问题的研究目前还是新的     

Locally injective maps in O-minimal structures without poles are surjective

If is continuous and locally injective, then is in fact surjective and a homeomorphism, provided is definable in an o-minimal expansion without poles of the ordered additive group of real numbers; `without poles' means that every one-variable definable function is locally bounded. Some general properties of definable maps in o-minimal expansions of ordered abelian groups without poles are also established.

     

On extension of isometries in (F)-spaces

An (F)-spaceE is said to be locally midpoint constricted (in short, Imp-constricted) if there exists some>0 such thatD(A/2) <D(A) for every subsetA ofE with 0<D(A), whereD(A) denotes the diameter ofA. Our main result goes as follow: LetE be an Imp-constricted (F)-space andU an open connected subset ofE. Assume thatT:U F is an isometry (i.e., a distance-preserving map) which mapsU onto an open subset of the (F)-spaceF. ThenT can be extended to an affine homeomorphism fromE toF. Also, some other results about the question whether each isometry between two (F)-spaces is affine are obtained.     

Manifolds of support sets of chebyshev polynomials

We consider a parameterization of the set of polynomialsT _n(E, x) whose deviation from zero is the least on a systemE consisting of several intervals on the real axis. We point out a new way for obtaining the equations which describe the boundary of the maximum set of the least deviationE ⁺ ⊃E. We describe the geometry of the variety of all possible setsE ⁺; this manifold is embedded in the moduli space of hyperelliptic curves with real branch points. Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 828–836, June, 2000.     

关于超空间上Whitney映射的一个问题

本文研究了超空间上的Ｗｎｉｔｎｅｙ映射,给出了它的一些性质和一个反例,从而解决了文献［１］中的一个问题．