Coincidence of the upper Kuratowski topology with the co-compact topology on compact sets, and the Prohorov property

For a Hausdorff space X, let F be the hyperspace of all closed subsets of X and H a sublattice of F. Following Nogura and Shakhmatov, X is said to be H-trivial if the upper Kuratowski topology and the co-compact topology coincide on H. F-trivial spaces are the consonant spaces first introduced and studied by Dolecki, Greco and Lechicki. In this paper, we deal with K-trivial spaces and Fin-trivial space, where K and Fin are respectively the lattices of compact and of finite subsets of X. It is proved that if C_k(X) is a Baire space or more generally if X has ‘the moving off property’ of Gruenhage and Ma, then X is K-trivial. If X is countable, then C_p(X) is Baire if and only if X is Fin-trivial and all compact subsets of X are finite. As for consonant spaces, it turns out that every regular K-trivial space is a Prohorov space. This result remains true for any regular Fin-trivial space in which all compact subsets are scattered. It follows that every regular first countable space without isolated points, all compact subsets of which are countable, is Fin-nontrivial. Examples of K-trivial non-consonant spaces, of Fin-trivial K-nontrivial spaces and of countably compact Prohorov Fin-nontrivial spaces, are given. In particular, we show that all (generalized) Fréchet–Urysohn fans are K-trivial, answering a question by Nogura and Shakhmatov. Finally, we describe an example of a continuous open compact-covering mapping f :X→Y, where X is Prohorov and Y is not Prohorov, answering a long-standing question by Topsøe.     

Function-space compactifications of function spaces

If X and Y are Hausdorff spaces with X locally compact, then the compact-open topology on the set C(X,Y) of continuous maps from X to Y is known to produce the right function-space topology. But it is also known to fail badly to be locally compact, even when Y is locally compact. We show that for any Tychonoff space Y, there is a densely injective space Z containing Y as a densely embedded subspace such that, for every locally compact space X, the set C(X,Z) has a compact Hausdorff topology whose relative topology on C(X,Y) is the compact-open topology. The following are derived as corollaries: (1) If X and Y are compact Hausdorff spaces then C(X,Y) under the compact-open topology is embedded into the Vietoris hyperspace V(X×Y). (2) The space of real-valued continuous functions on a locally compact Hausdorff space under the compact-open topology is embedded into a compact Hausdorff space whose points are pairs of extended real-valued functions, one lower and the other upper semicontinuous. The first application is generalized in two ways.     

Factorization theorems for cohomological dimension

The well-known factorization theorems for covering dimension dim and compact Hausdorff spaces are here established for the cohomological dimension dim using a new characterization of dim In particular, it is proved that every mapping f: X → Y from a compact Hausdorff space X with to a compact metric space Y admits a factorization f = hg, where g: X → Z, h: Z → Y and Z is a metric compactum with . These results are applied to the well-known open problem whether . It is shown that the problem has a positive answer for compact Hausdorff spaces X if and only if it has a positive answer for metric compacta X.     

Isomorphisms in pro-categories

A morphism of a category which is simultaneously an epimorphism and a monomorphism is called a bimorphism. In (Dydak and Ruiz del Portal (Monomorphisms and epimorphisms in pro-categories, preprint)) we gave characterizations of monomorphisms (resp. epimorphisms) in arbitrary pro-categories, pro-C, where C has direct sums (resp. weak push-outs). In this paper, we introduce the notions of strong monomorphism and strong epimorphism. Part of their significance is that they are preserved by functors. These notions and their characterizations lead us to important classical properties and problems in shape and pro-homotopy. For instance, strong epimorphisms allow us to give a categorical point of view of uniform movability and to introduce a new kind of movability, the sequential movability. Strong monomorphisms are connected to a problem of K. Borsuk regarding a descending chain of retracts of ANRs. If f : X→Y is a bimorphism in the pointed shape category of topological spaces, we prove that f is a weak isomorphism and f is an isomorphism provided Y is sequentially movable and X or Y is the suspension of a topological space. If f : X→Y is a bimorphism in the pro-category pro-H₀ (consisting of inverse systems in H₀, the homotopy category of pointed connected CW complexes) we show that f is an isomorphism provided Y is sequentially movable.     

The homotopy Lie algebra of classifying spaces

Let X be a 1-connected CW-complex of finite type and L_x its rational homotopy Lie algebra. In this work, we show that there is a spectral sequence whose E² term is the Lie algebra Ext_ULx(Q, L_x), and which converges to the homotopy Lie algebra of the classifying space B autX. Moreover, some terms of this spectral sequence are related to derivations of L_x and to the Gottlieb group of X.     

On analytic and coanalytic function spaces Cp(X)

Under the assumption (V = L) we construct countable completely regular spaces X and Y such that the spaces C_p(X) and C_p(Y) of real-valued continuous functions on X and Y, equipped with the pointwise convergence topology, are analytic noncoanalytic and they are not homeomorphic. We also give analogous examples of coanalytic nonanalytic function spaces.     

Extension dimension for paracompact spaces

We prove existence of extension dimension for paracompact spaces. Here is the main result of the paper:

Theorem. Suppose X is a paracompact space. There is a CW complex K such that

(a) K is an absolute extensor of X up to homotopy,

(b) If a CW complex L is an absolute extensor of X up to homotopy, then L is an absolute extensor of Y up to homotopy of any paracompact space Y such that K is an absolute extensor of Y up to homotopy.

The proof is based on the following simple result (see Theorem 1.2).

Theorem. Let X be a paracompact space. Suppose a space Y is the union of a family {Y_s}_sS of its subspaces with the following properties:

(a) Each Y_s is an absolute extensor of X,

(b) For any two elements s and t of S there is uS such that Y_sY_tY_u.

If f :A→Y is a map from a closed subset A to Y such that A=_sSInt_A(f⁻¹(Y_s)), then f extends over X.

That result implies a few well-known theorems of classical theory of retracts which makes it of interest in its own.     

Efficient robust estimation of parameter in the random censorship model

For an open set Θ of ^k, let \s{P_θ: θ Θ\s} be a parametric family of probabilities modeling the distribution of i.i.d. random variables X₁,…, X_n. Suppose X_i's are subject to right censoring and one is only able to observe the pairs (min(X_i, Y_i), [X_i Y_i]), i = 1,…, n, where [A] denotes the indicator function of the event A, Y₁,…, Y_n are independent of X₁,…, X_n and i.i.d. with unknown distribution Q₀. This paper investigates estimation of the value θ that gives a fitted member of the parametric family when the distributions of X₁ and Y₁ are subject to contamination. The constructed estimators are adaptive under the semi-parametric model and robust against small contaminations: they achieve a lower bound for the local asymptotic minimax risk over Hellinger neighborhoods, in the Hájel—Le Cam sense. The work relies on Beran (1981). The construction employs some results on product-limit estimators.     

Some complex Grassmannian manifolds that do not fibre nontrivially

A finite CW complex X is said to be prime if, given a Hurewicz fibration F→E→B with E homotopy equivalent to X, and B and F homotopy equivalent to finite CW complexes, either B or F is contractible. We show that certain 3- and 4-plane complex Grassmanian manifolds are prime.     

Estimating the spectral measure of an extreme value distribution

Let (X₁, Y₁), (X₂, Y₂),…, (X_n, Y_n) be a random sample from a bivariate distribution function F which is in the domain of attraction of a bivariate extreme value distribution function G. This G is characterized by the extreme value indices and its spectral measure or angular measure. The extreme value indices determine both the marginals and the spectral measure determines the dependence structure. In this paper, we construct an empirical measure, based on the sample, which is a consistent estimator of the spectral measure. We also show for positive extreme value indices the asymptotic normality of the estimator under a suitable 2nd order strengthening of the bivariate domain of attraction condition.     

Some algebraic constructions in rational homotopy theory

In this note we describe constructions in the category of differential graded commutative algebras over the rational numbers Q which are analogs of the space F(X, Y) of continuous maps of X to Y, the component F(X, Y,ƒ) containing ƒ ε F(X, Y), fibrations, induced fibrations, the space Γ(π) of sections of a fibration π: E → X, and the component Γ(π,σ) containing σ ε Γ (π). As a focus, we address the problem of expressing π^*(F(X, Y, ƒ)) = Hom(π_*(F(X,Y, ƒ)),Q) in terms of differential graded algebra models for X and Y.     

Arc-smooth generalized universal covering spaces

Let Y be a path-connected subset of a CAT(0) space Z, allowing for a map to a 1-dimensional separable metric space X, such that the nontrivial point preimages of f form a null sequence of convex subsets of Z. Such Y need not be homotopy equivalent to a 1-dimensional space.

We prove that Y admits a generalized universal covering space, which we equip with an arc-smooth structure by consistently and continuously selecting one tight representative from each path homotopy class of Y. It follows that all homotopy groups of Y vanish in dimensions greater than 1.     

On Grauert–Riemenschneider Type Criteria

Let(X, ω) be a compact Hermitian manifold of complex dimension n. In this article,we first survey recent progress towards Grauert–Riemenschneider type criteria. Secondly, we give a simplified proof of Boucksom's conjecture given by the author under the assumption that the Hermitian metric ω satisfies ?■ω~l= for all l, i.e., if T is a closed positive current on X such that ∫_XT_(ac)~n 0, then the class {T } is big and X is Kahler. Finally, as an easy observation, we point out that Nguyen's result can be generalized as follows: if ?■ω = 0, and T is a closed positive current with analytic singularities,such that ∫_XT_(ac)~n 0, then the class {T} is big and X is Kahler.     

Limiting values of the variance and the moments of the dimension of a sum or intersection of random vector subspaces

Let W be an n-dimensional vector space over a field F; for each positive integer m, let the m-tuples (U₁, …, U_m) of vector subspaces of W be uniformly distributed; and consider the statistics X_m,1 dim_F(∑_i=1^m U_i) and X_m,2 dim_F (∩_i=1^m U_i). If F is finite of cardinality q, we determine lim E(X_m,1^k), and lim E(X_m,2^k), and hence, lim var(X_m,1) and lim var(X_m,2), for any k > 0, where the limits are taken as q → ∞ (for fixed n). Further, we determine whether these, and other related, limits are attained monotonically. Analogous issues are also addressed for the case of infinite F.     

First passage time for some stationary processes

For a 1-dependent stationary sequence {X_n} we first show that if u satisfies p₁=p₁(u)=P(X₁>u)0.025 and n>3 is such that 88np₁³1, then

P{max(X₁,…,X_n)u}=ν·μⁿ+O{p₁³(88n(1+124np₁³)+561)}, n>3,

where

ν=1−p₂+2p₃−3p₄+p₁²+6p₂²−6p₁p₂,μ=(1+p₁−p₂+p₃−p₄+2p₁²+3p₂²−5p₁p₂)⁻¹

with

p_k=p_k(u)=P{min(X₁,…,X_k)>u}, k1

and

|O(x)||x|.

From this result we deduce, for a stationary T-dependent process with a.s. continuous path {Y_s}, a similar, in terms of P{max_0skTY_s<u}, k=1,2 formula for P{max_0stY_su}, t>3T and apply this formula to the process Y_s=W(s+1)−W(s), s0, where {W(s)} is the Wiener process. We then obtain numerical estimations of the above probabilities.     

The equivariant Serre spectral sequence as an application of a spectral sequence of Spanier

E.H. Spanier (1992) has constructed, for a cohomology theory defined on a triangulated space and locally constant on each open simplex, a spectral sequence whose E₂-term consists of certain simplicial cohomology groups, converging to the cohomology of the space. In this paper we study a closed G-fibration ƒ: Y → X, where G is a finite group. We show that if the base-G-spaceX is equivariantly triangulated and Y is paracompact, then Spanier's spectral sequence yields an equivariant Serre spectral sequence for ƒ. The main point here is to identify the equivariant singular cohomology groups of X with appropriate simplicial cohomology groups of the orbit space X/G.     

Stable 2-pairs and (X,Y)-intersection graphs

Given two fixed graphs X and Y, the (X,Y)-intersection graph of a graph G is a graph where

1. each vertex corresponds to a distinct induced subgraph in G isomorphic to Y, and

2. two vertices are adjacent iff the intersection of their corresponding subgraphs contains an induced subgraph isomorphic to X.

This notion generalizes the classical concept of line graphs since the (K₁,K₂)-intersection graph of a graph G is precisely the line graph of G.

Let ( , respectively) denote the family of line graphs of bipartite graphs (bipartite multigraphs, respectively), and refer to a pair (X,Y) as a 2-pair if Y contains exactly two induced subgraphs isomorphic to X. Then and , respectively, are the smallest families amongst the families of (X,Y)-intersection graphs defined by so called hereditary 2-pairs and hereditary non-compact 2-pairs. Furthermore, they can be characterized through forbidden induced subgraphs. With this motivation, we investigate the properties of a 2-pair (X,Y) for which the family of (X,Y)-intersection graphs coincides with (or ). For this purpose, we introduce a notion of stability of a 2-pair and obtain the desired characterization for such stable 2-pairs. An interesting aspect of the characterization is that it is based on a graph determined by the structure of (X,Y).     

Coupling and harmonic functions in the case of continuous time Markov processes

Consider two transient Markov processes (X^v_t)_tεR·, (X^μ_t)_tεR· with the same transition semigroup and initial distributions v and μ. The probability spaces supporting the processes each are also assumed to support an exponentially distributed random variable independent of the process.

We show that there exist (randomized) stopping times S for (X^v_t), T for (X^μ_t) with common final distribution, L(X^v_S|S < ∞) = L(X^μ_T|T < ∞), and the property that for t < S, resp. t < T, the processes move in disjoint portions of the state space. For such a coupling (S, T) it is shown

where denotes the bounded harmonic functions of the Markov transition semigroup. Extensions, consequences and applications of this result are discussed.     

Instabilities of robot motion

Instabilities of robot motion are caused by topological reasons. In this paper we find a relation between the topological properties of a configuration space (the structure of its cohomology algebra) and the character of instabilities, which are unavoidable in any motion planning algorithm. More specifically, let X denote the space of all admissible configurations of a mechanical system. A motion planner is given by a splitting X×X=F₁F₂F_k (where F₁,…,F_k are pairwise disjoint ENRs, see below) and by continuous maps s_j :F_j→PX, such that Es_j=1_Fj. Here PX denotes the space of all continuous paths in X (admissible motions of the system) and E :PX→X×X denotes the map which assigns to a path the pair of its initial–end points. Any motion planner determines an algorithm of motion planning for the system. In this paper we apply methods of algebraic topology to study the minimal number of sets F_j in any motion planner in X. We also introduce a new notion of order of instability of a motion planner; it describes the number of essentially distinct motions which may occur as a result of small perturbations of the input data. We find the minimal order of instability, which may have motion planners on a given configuration space X. We study a number of specific problems: motion of a rigid body in R³, a robot arm, motion in R³ in the presence of obstacles, and others.     

The extreme set condition of a graph

Let G be a simple graph. The size of any largest matching in G is called the matching number of G and is denoted by ν(G). Define the deficiency of G, def(G), by the equation def(G)=|V(G)|−2ν(G). A set of points X in G is called an extreme set if def(G−X)=def(G)+|X|. Let c₀(G) denote the number of the odd components of G. A set of points X in G is called a barrier if c₀(G−X)=def(G)+|X|. In this paper, we obtain the following:

(1) Let G be a simple graph containing an independent set of size i, where i2. If X is extreme in G for every independent set X of size i in G, then there exists a perfect matching in G.

(2) Let G be a connected simple graph containing an independent set of size i, where i2. Then X is extreme in G for every independent set X of size i in G if and only if G=(U,W) is a bipartite graph with |U|=|W|i, and |Γ(Y)||U|−i+m+1 for any Y U, |Y|=m (1mi−1).

(3) Let G be a connected simple graph containing an independent set of size i, where i2. Then X is a barrier in G for every independent set X of size i in G if and only if G=(U,W) is a bipartite graph with |U|=|W|=i, and |Γ(Y)|m+1 for any Y U, |Y|=m (1mi−1).