The hyperspace of the regions below continuous maps with the fell topology

For a Tychonoff space X,we use ↓USC F(X) and ↓C F(X) to denote the families of the hypographs of all semi-continuous maps and of all continuous maps from X to I = [0,1] with the subspace topologies of the hyperspace Cld F(X × I) consisting of all non-empty closed sets in X × I endowed with the Fell topology.In this paper,we shall show that there exists a homeomorphism h:↓USC F(X) → Q = [1,1] ω such that h(↓CF(X))=c0 = {(xn)∈Q| lim n→∞ x n = 0} if and only if X is a locally compact separable metrizable space and the set of isolated points is not dense in X.     

METRIC ENTROPY OF HOMEOMORPHISM ON NON-COMPACT METRIC SPACE

Let T : X → X be a uniformly continuous homeomorphism on a non-compact metric space (X, d). Denote by X* = X ∪ {x*} the one point compactification of X and T * : X* → X* the homeomorphism on X* satisfying T *|X = T and T *x* = x*. We show that their topological entropies satisfy hd(T, X) ≥ h(T *, X*) if X is locally compact. We also give a note on Katok’s measure theoretic entropy on a compact metric space.     

On the Equi-nuclearity of Roe Algebras of Metric Spaces

The authors define the equi-nuclearity of uniform Roe algebras of a family of metric spaces. For a discrete metric space X with bounded geometry which is covered by a family of subspaces {Xi}∞i=1, if {Cu*(Xi)}∞i=1 are equi-nuclear and under some proper gluing conditions, it is proved that Cu*(X) is nuclear. Furthermore, it is claimed that in general, the coarse Roe algebra C*(X) is not nuclear.     

Permanence of Metric Sparsification Property under Finite Decomposition Complexity

The notions of metric sparsification property and finite decomposition com- plexity are recently introduced in metric geometry to study the coarse Novikov conjecture and the stable Borel conjecture. In this paper, it is proved that a metric space X has finite decomposition complexity with respect to metric sparsification property if and only if X itself has metric sparsification property. As a consequence, the authors obtain an alterna- tive proof of a very recent result by Guentner, Tessera and Yu that all countable linear groups have the metric sparsification property and hence the operator norm localization property.     

A topological position of the set of continuous maps in the set of upper semicontinuous maps

Let (X, ρ) be a metric space and ↓USCC(X) and ↓CC(X) be the families of the regions below all upper semi-continuous compact-supported maps and below all continuous compact-supported maps from X to I = [0, 1], respectively. With the Hausdorff-metric, they are topological spaces. In this paper, we prove that, if X is an infinite compact metric space with a dense set of isolated points, then (↓USCC(X), ↓CC(X)) ≈ (Q, c0 ∪ (Q \ Σ)), i.e., there is a homeomorphism h :↓USCC(X) → Q such that h(↓CC(X)) = c0 ∪ (Q \ Σ...     

COMPLETE ACCUMULATION POINTS AND COMPACTNESS OF FUZZY SETS

Let A be a fuzzy set in a fuzzy topological space (X,τ) and β∈(0, 1]. We denote by ζ_x~a the fuzzy point defined by ζ_x~a(x)=a and ζ_x~a(y)=0 for eaoh y≠x, where a is called the height of ζ_x~a. A subfamily u of τ is called an open β-cover of A if each fuzzy point in A with height β is quasi-coincident with ([1]) some member of u. By a β-subcover of the open β-cover u of A is meant any subfamily of u that is also an open β-cover of A. A is called β-compact in (X, τ) if every open β-cover     

The hyperspace of the regions below of all lattice-value continuous maps and its Hilbert cube compactification

Let L be a continuous semilattice. We use USC(X, L) to denote the family of all lower closed sets including X × {0} in the product space X × AL and ↓1 C(X,L) the one of the regions below of all continuous maps from X to AL. USC(X, L) with the Vietoris topology is a topological space and ↓C(X, L) is its subspace. It will be proved that, if X is an infinite locally connected compactum and AL is an AR, then USC(X, L) is homeomorphic to [-1,1]ω. Furthermore, if L is the product of countably many intervals, then ↓ C(X, L) is homotopy dense in USC(X,L), that is, there exists a homotopy h : USC(X,L) × [0,1] →USC(X,L) such that h0 = idUSC(X,L) and ht(USC(X,L)) C↓C(X,L) for any t > 0. But ↓C(X, L) is not completely metrizable.     

On holomorphic immersions into Khler manifolds of constant holomorphic sectional curvature

We study holomorphic immersions f:X→M from a complex manifold X into a Kahler manifold of constant holomorphic sectional curvature M, i.e. a complex hyperbolic space form, a complex Euclidean space form, or the complex projective space equipped with the Fubini-Study metric. For X compact we show that the tangent sequence splits holomorphically if and only if f is a totally geodesic immersion. For X not necessarily compact we relate an intrinsic cohomological invariant p(X) on X, viz. the invariant defined by Gunning measuring the obstruction to the existence of holomorphic projective connections, to an extrinsic cohomological invariant v(f) measuring the obstruction to the holomorphic splitting of the tangent sequence. The two invariants p(X) and v(f) are related by a linear map on cohomology groups induced by the second fundamental form. In some cases, especially when X is a complex surface and M is of complex dimension 4, under the assumption that X admits a holomorphic projective connection we obtain     

A Class of Homogeneous Einstein Manifolds

A Riemannian manifold (M, g) is called Einstein manifold if its Ricci tensor satisfies r = c·g for some constant c. General existence results are hard to obtain, e.g., it is as yet unknown whether every compact manifold admits an Einstein metric. A natural approach is to impose additional homogeneous assumptions. M. Y. Wang and W. Ziller have got some results on compact homogeneous space G/H. They investigate standard homogeneous metrics, the metric induced by Killing form on G/H, and get some classification results. In this paper some more general homogeneous metrics on some homogeneous space G/H are studies, and a necessary and sufficient condition for this metric to be Einstein is given. The authors also give some examples of Einstein manifolds with non-standard homogeneous metrics.     

RETRACTED ARTICLE: Some applications of BP-theorem in approximation theory

In this paper we apply Bishop-Phelps property to show that if X is a Banach space and G X is the maximal subspace so that G⊥ = {x* ∈ X*|x*(y) = 0; y∈ G} is an L-summand in X*, then L1(Ω,G) is contained in a maximal proximinal subspace of L1(Ω,X).     

Saturation theorems for generalized Bernstein polynomials

In the present paper, we give the explicit formula of the principal part of n ∑ k=0 ([k]q -[n]qx)sxk n-k-1 ∏ m=0 (1-qmx) with respect to [n]q for any integer s and q ∈ (0,1]. And, using the expressions, we obtain saturation theorems for Bn(f,qn;x) approximating to f(x) ∈ C[0,1], 0 < qn ≤ 1, qn → 1.     

Compact and Loeb Hausdorff spaces in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {ZF}$\end{document} and the axiom of choice for families of finite sets

Given a set X, $\mathsf {AC}^{\mathrm{fin}(X)}$ denotes the statement: “$[X]^{<\omega }\backslash \lbrace \varnothing \rbrace$ has a choice set” and $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )$ denotes the family of all closed subsets of the topological space $\mathbf {2}^{X}$ whose definition depends on a finite subset of X. We study the interrelations between the statements $\mathsf {AC}^{\mathrm{fin}(X)},$ $\mathsf {AC}^{\mathrm{fin}([X]^{<\omega })},$ $\mathsf {AC}^{\mathrm{fin} (F_{n}(X,2))},$ $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$ and “$\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set”. We show:

• (i) $\mathsf {AC}^{\mathrm{fin}(X)}$ iff $\mathsf {AC}^{\mathrm{fin}([X]^{<\omega } )}$ iff $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set iff $\mathsf {AC}^{\mathrm{fin}(F_{n}(X,2))}$.
• (ii) $\mathsf {AC}_{\mathrm{fin}}$ ($\mathsf {AC}$ restricted to families of finite sets) iff for every set X, $\mathcal {C}_\mathrm{R}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set.
• (iii) $\mathsf {AC}_{\mathrm{fin}}$ does not imply “$\mathcal {K}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$ has a choice set($\mathcal {K}(\mathbf {X})$ is the family of all closed subsets of the space $\mathbf {X}$)
• (iv) $\mathcal {K}(\mathbf {2}^{X})\backslash \lbrace \varnothing \rbrace$ implies $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$ but $\mathsf {AC}^{\mathrm{fin}(X)}$ does not imply $\mathsf {AC}^{\mathrm{fin}(\mathcal {\wp }(X))}$.

We also show that “For every setX, “$\mathcal {K}\big (\mathbf {2}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set” iff “for every setX, $\mathcal {K}\big (\mathbf {[0,1]}^{X}\big )\backslash \lbrace \varnothing \rbrace$has a choice set” iff “for every product$\mathbf {X}$of finite discrete spaces,$\mathcal {K}(\mathbf {X})\backslash \lbrace \varnothing \rbrace$ has a choice set”.     

Cn中单位球上$mu$-Bloch函数的等价刻画

设$\mu$是$[0,1)$上的正规函数, 给出了${\bf C}^{\it n}$中单位球$B$上$\mu$-Bloch空间$\beta_{\mu}$中函数的几种刻画. 证明了下列条件是等价的: (1) $f\in \beta_{\mu}$; \ (2) $f\in H(B)$且函数$\mu(|z|)(1-|z|^{2})^{\gamma-1}R^{\alpha,\gamma}f(z)$ 在$B$上有界; (3) $f\in H(B)$ 且函数${\mu(|z|)(1-|z|^{2})^{M_{1}-1}\frac{\partial^{M_{1}} f}{\partial z^{m}}(z)}$ 在$B$上有界, 其中$|m|=M_{1}$; (4) $f\in H(B)$ 且函数${\mu(|z|)(1-|z|^{2})^{M_{2}-1}R^{(M_{2})}f(z)}$ 在$B$上有界.     

A Nonhomogeneous Boundary Value Problem for the Boussinesq Equation on a Bounded Domain

In this paper, we study the well-posedness of an initial-boundary-value problem (IBVP) for the Boussinesq equation on a bounded domain,\begin{cases} &u_{tt}-u_{xx}+(u^2)_{xx}+u_{xxxx}=0,\quad x\in (0,1), \;\;t>0,\\ &u(x,0)=\varphi(x),\;\;\; u_t(x,0)=ψ(x),\\ &u(0,t)=h_1(t),\;\;\;u(1,t)=h_2(t),\;\;\;u_{xx}(0,t)=h_3(t),\;\;\;u_{xx}(1,t)=h_4(t).\\ \end{cases} It is shown that the IBVP is locally well-posed in the space $H^s (0,1)$ for any $s\geq 0$ with the initial data $\varphi,$ $\psi$ lie in $H^s(0,1)$ and $ H^{s-2}(0,1)$, respectively, and the naturally compatible boundary data $h_1,$ $h_2$ in the space $H_{loc}^{(s+1)/2}(\mathbb{R}^+)$, and $h_3 $, $h_4$ in the the space of $H_{loc}^{(s-1)/2}(\mathbb{R}^+)$ with optimal regularity.     

一类NMIFS吸引子

本文给出了NMIFS(Nonlinear Markov Iterated Function System)理论与构造NMIFS吸引子的方法,讨论了一类NMIFS吸引子的平衡向量测度和“矩”的递归计算,分析了NMIFS吸引子的结构特征．研究结果表明:对于MIFS,可以通过递归方法来计算矩M(i)(i=1,2...);而对于NMIFS,因M(i)的计算依赖于M(j)(j≥i),故不能直接计算M(i),而只能计算其近似值．     

Precise inequalities for norms of functions,third partial,second mixed,or directional derivatives

For functions f which are bounded throughout the plane R₂ together with the partial derivatives f(^3,0) f(_0,3), inequalities $$\left\| {f^{(1,1)} } \right\| \leqslant \sqrt[3]{3}\left\| f \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left\| {f^{(3,0)} } \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left\| {f^{(0,3)} } \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} ,\left\| {f_e^{(2)} } \right\| \leqslant \sqrt[3]{3}\left\| f \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left( {\left\| {f^{(3,0)} } \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left| {e_1 } \right| + \left\| {f^{(0,3)} } \right\|^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$3$}}} \left| {e_2 } \right|} \right)^2 ,$$ are established, where ∥?∥denotes the upper bound on R² of the absolute values of the corresponding function, andf _fe ⁽²⁾ is the second derivative in the direction of the unit vector e=(e₁, e₂). Functions are exhibited for which these inequalities become equalities.     

Banach 空间中极大单调算子扰动的值域

设X为实Banach空间, T:D(T)(?)X→2X*为极大单调算子, C: D(T)(?)X→X*为有界算子(未必连续),而C(T+J)-1为紧算子．本文在上述假设条件下,通过附加一定的边界条件应用Leray-Schauder度理论研究了下述包含关系:0∈(T+C)(D(T)∩ BQ(0)),0∈(T+C)(D(T)∩ BQ(0));以及S(?)R(T+C), intS(?)intR(T+C)(其中S(?) X*);B+D(?)R(T+C),int(B+D)(?)intR(T+C)(其中 B(?)X*,D(?)X*)的可解性,得出了一些新的结论．     

The Hodge–Poincaré polynomial of the moduli spaces of stable vector bundles over an algebraic curve

Let X be a nonsingular complex projective variety that is acted on by a reductive group G and such that ${X^{ss} \neq X_{(0)}^{s}\neq \emptyset}$ . We give formulae for the Hodge–Poincaré series of the quotient ${X_{(0)}^{s}/G}$ . We use these computations to obtain the corresponding formulae for the Hodge–Poincaré polynomial of the moduli space of properly stable vector bundles when the rank and the degree are not coprime. We compute explicitly the case in which the rank equals 2 and the degree is even.     

涉及分担函数的正规定则

设k为正整数,M为正数;F为区域D内的亚纯函数族,且其零点重级至少为k;h为D内的亚纯函数(h(z)≠0,∞),且h(z)的极点重级至多为k.若对任意给定的函数f∈F,f与f~((k))分担0,且f~((k))(z)-h(z)=0?|f(z)|≥M,则F在D内正规.