Products of symmetric spaces, and k-networks

As is well known, every product of symmetric spaces need not be symmetric. For symmetric spaces X and Y, in terms of their balls, we give characterizations for the product X×Y to be symmetric under X and Y having certain k-networks, or Y being semi-metric.     

Finite graphs have unique symmetric products

Let X be a metric continuum and let Fn(X) be the nth symmetric product of X (Fn(X) is the hyperspace of nonempty subsets of X with at most n points). In this paper we prove that if Fn(X) is homeomorphic to Fn(Y), where X is a finite graph and Y is a continuum, then X is homeomorphic to Y.     

A characteristic property of the exponential distribution

The following characterization of the exponential distribution is given: Under suitable conditions on the random variables X and Y, X is exponentially distributed if and only if E[min{X, Y}]=E(X)P(X<Y).     

Algebras of real-valued continuous functions in product spaces

Let {X_i:iϵI} be an arbitrary family of spaces, we say that the cartesian product X has the approximation property when C(X) coincides with the Algebra on X generated by the functions which depend on one variable. In this paper we study the problem of characterizing topologically when an arbitrary product space has the approximation property. We prove that if X is an uncountable pseudo-ℵ₁-compact P-space, then X×Y has the approximation property if, and only if, X×Y is pseudo-ℵ₁-compact. As a corollary we obtain the following characterization for P-spaces: Let X and Y be P-spaces, then X×Y has the approximation property if, and only if, X or Y is countable or X×Y is pseudo-ℵ₁-compact.     

On the product of two topological spaces

The thrust of the paper is toward answering “What conditions on X and Y are sufficient to insure that X × Y has property Q?” Important results by section are 3. Suppose X is subcompact (has a σ-discrete closed cover of compact subsets). If X is collectionwise normal and Y is paracompact (metacompact, screenable), then X × Y is paracompact (metacompact, screenable). If X is T₂ and Y is subparacompact, then X × Y is subparacompact. 4. Let X be metrizable and Y have the G_δ-property. If Y is screenable (metacompact, subparacompact), then X × Y is screenable (metacompact, subparacompact). 5. The product of a T₂, paracompact, sequential space with a countably compact, normal space is collectionwise normal. 6. X × Y is an F-product if X is T₂, paracompact, and sequential while Y is countably compact and normal, or if X is collectionwise normal and subcompact while Y is paracompact and normal.     

A note on a set-valued extension property

Principal result: Suppose Y is metrizable. Then: (a) if X is metrizable and A⊂X is closed, then every continuous g:A→Y extends to an l.s.c. ψ:X→K(Y); (b) Y satisfies (a) for all paracompact X if and only if Y is completely metrizable.     

Extremal continua: A class of non-separating subcontinua

The notion of a terminal continuum, as defined by D.E. Bennett and J.B. Fugate, is used to introduce extremal continua, a class of non-separating subcontinua of a continuum. An extremal continuum can be characterized as a proper subcontinuum Y of a metric continuum X with the property that Y contains a point of irreducibility of each irreducible subcontinuum of X that meets Y. If Y is an extremal subcontinuum of X, then Y does not separate any subcontinuum of X containing Y; moreover, if Y is a proper subcontinuum of X and Y does not cut any subcontinuum of X containing Y, then Y is extremal in X.     

The partially ordered set of one-point extensions

A space Y is called an extension of a space X if Y contains X as a dense subspace. Two extensions of X are said to be equivalent if there is a homeomorphism between them which fixes X point-wise. For two (equivalence classes of) extensions Y and Y^′ of X let Y?Y^′ if there is a continuous function of Y^′ into Y which fixes X point-wise. An extension Y of X is called a one-point extension of X if Y?X is a singleton. Let P be a topological property. An extension Y of X is called a P-extension of X if it has P.One-point P-extensions comprise the subject matter of this article. Here P is subject to some mild requirements. We define an anti-order-isomorphism between the set of one-point Tychonoff extensions of a (Tychonoff) space X (partially ordered by ?) and the set of compact non-empty subsets of its outgrowth βX?X (partially ordered by ⊆). This enables us to study the order-structure of various sets of one-point extensions of the space X by relating them to the topologies of certain subspaces of its outgrowth. We conclude the article with the following conjecture. For a Tychonoff spaces X denote by U(X) the set of all zero-sets of βX which miss X.

Conjecture. For locally compact spaces X and Y the partially ordered sets(U(X),⊆)and(U(Y),⊆)are order-isomorphic if and only if the spacesclβX(βX?υX)andclβY(βY?υY)are homeomorphic.     

Applications of balanced pairs

Let (X, Y) be a balanced pair in an abelian category. We first introduce the notion of cotorsion pairs relative to (X, Y), and then give some equivalent characterizations when a relative cotorsion pair is hereditary or perfect. We prove that if the X-resolution dimension of Y (resp. Y-coresolution dimension of X) is finite, then the bounded homotopy category of Y (resp. X) is contained in that of X (resp. Y). As a consequence, we get that the right X-singularity category coincides with the left Y-singularity category if the X-resolution dimension of Y and the Y-coresolution dimension of X are finite.     

Local compactness and Hewitt realcompactifications of products II

We generalize and refine results from the author's paper [18]. For a completely regular Hausdorff space X, υX denotes the Hewitt realcompactification of X. It is proved that if υ(X×Y)=υX×υY for any metacompact subparacompact (or m-paracompact) space Y, then X is locally compact. A P(n)-space is a space in which every intersection of less than n open sets is open. A characterization of those spaces X such that υ (X×Y = υX×υY for any (metacompact) P(n)-space Y is also obtained.     

Multiplicative submodularity of a matrix's principal minor as a function of the set of its rows and some combinatorial applications

Let N be the set of rows of a positive semidefinite symmetric matrix, and denote by d(X), X?N, the determinant of the submatrix with X as the set of its rows and columns. It is proved that d(X)·d(Y)?d(X∪Yd(X∩Y. Some combinatorial applications of the inequality are also given.     

Weak similarities of metric and semimetric spaces

Let (X,d _X ) and (Y,d _Y ) be semimetric spaces with distance sets D(X) and D(Y), respectively. A mapping F:?X→Y is a weak similarity if it is surjective and there exists a strictly increasing f:?D(Y)→D(X) such that d _X =f°d _Y °(F?F). It is shown that the weak similarities between geodesic spaces are usual similarities and every weak similarity F:?X→Y is an isometry if X and Y are ultrametric and compact with D(X)=D(Y). Some conditions under which the weak similarities are homeomorphisms or uniform equivalences are also found.     

On shape of product spaces

It is known that if X is a compactum and Y is metrizable Sh₅(X × Y) is not generally determined by Sh₅(X) and Sh₅(Y), where Sh₅(Z) is the strong shape of Z in the sense of Borsuk. In this paper it is proved that Sh(X × Y) is uniquely determined by Sh(X) and Sh(Y), where Sh(Z) is the shape of Z in the sense of Fox. If X is an FANR and Y is an MANR, then X × Y is an MANR.     

The weak metric approximation property and ideals of operators

We study the weak metric approximation property introduced by Lima and Oja. We show that a Banach space X has the weak metric approximation property if and only if F(Y,X), the space of finite rank operators, is an ideal in W(Y,X∗∗), the space of weakly compact operators for all Banach spaces Y.     

A method for computing lowest eigenvalues of symmetric polynomial differential operators by semidefinite programming

A method for computing global minima of real multivariate polynomials based on semidefinite programming was developed by N.Z. Shor, J.B. Lasserre and P.A. Parrilo. The aim of this article is to extend a variant of their method to noncommutative symmetric polynomials in variables X and Y satisfying YX−XY=1 and X^*=X, Y^*=−Y. Global minima of such polynomials are defined and showed to be equal to minima of the spectra of the corresponding differential operators. We also discuss how to exploit sparsity and symmetry. Several numerical experiments are included. The last section explains how our theory fits into the framework of noncommutative real algebraic geometry.     

Shy couplings

A pair (X, Y) of Markov processes on a metric space is called a Markov coupling if X and Y have the same transition probabilities and (X, Y) is a Markov process. We say that a coupling is “shy” if inf _{t ≥ 0} dist(X _t , Y _t ) >  0 with positive probability. We investigate whether shy couplings exist for several classes of Markov processes.     

The regular part of a semigroup of transformations with restricted range

Let T(X) be the full transformation semigroup on the set X and let T(X,Y) be the semigroup consisting of all total transformations from X into a fixed subset Y of X. It is known that $$F(X, Y)=\{\alpha\in T(X, Y): X\alpha\subseteq Y\alpha\},$$ is the largest regular subsemigroup of T(X,Y) and determines Green??s relations on T(X,Y). In this paper, we show that F(X,Y)?T(Z) if and only if X=Y and |Y|=|Z|; or |Y|=1=|Z|, and prove that every regular semigroup S can be embedded in F(S ¹,S). Then we describe Green??s relations and ideals of F(X,Y) and apply these results to get all of its maximal regular subsemigroups when Y is a nonempty finite subset of X.     

Sufficient enlargements of minimal volume for finite-dimensional normed linear spaces

Let BY denote the unit ball of a normed linear space Y. A symmetric, bounded, closed, convex set A in a finite-dimensional normed linear space X is called a sufficient enlargement for X if, for an arbitrary isometric embedding of X into a Banach space Y, there exists a linear projection such that P(BY)⊂A. The main results of the paper: (1) Each minimal-volume sufficient enlargement is linearly equivalent to a zonotope spanned by multiples of columns of a totally unimodular matrix. (2) If a finite-dimensional normed linear space has a minimal-volume sufficient enlargement which is not a parallelepiped, then it contains a two-dimensional subspace whose unit ball is linearly equivalent to a regular hexagon.     

Products with a compact factor

In this paper we consider spaces X X Y, where Y is a compact Hausdorff space. Most of this paper is devoted to giving new, simplified proofs to some recent results concerning normality and map extension properties in some products. The theorems are of two types. First, we assume that the product X X Y is normal and deduce separation and covering properties for X, for example, that X must be v(Y)-collectionwise normal. Second, we assume that X has some special separation properties (namely, w(Y)-collectionwise normality) and deduce some map extension properties for X X Y. For example, if A and B are closed subsets of X and Y, respectively, then maps from A X B into the real line R can be extended to all of X X Y regardless of whether X X Y is normal or not. The proofs of all the theorems take advantage of the natural one-to-one correspondence between maps ⨍: X X Y → R and maps ⨍&#x0303;:X → C(Y).     

Hereditarily indecomposable subcontinua of the product of two Souslin arcs are metric

We prove that if each of X and Y is a Souslin arc (a Hausdorff arc that is the compactification of a connected Souslin line), then every hereditarily indecomposable subcontinuum of X×Y is metric.