The transitivity of induced maps

For a metric continuum X, we consider the hyperspaces X² and C(X) of the closed and nonempty subsets of X and of subcontinua of X, respectively, both with the Hausdorff metric. For a given map we investigate the transitivity of the induced maps and . Among other results, we show that if X is a dendrite or a continuum of type λ and is a map, then C(f) is not transitive. However, if X is the Hilbert cube, then there exists a transitive map such that f² and C(f) are transitive.     

Exponentiable monomorphisms in categories of domains

We give a characterization of exponentiable monomorphisms in the categories of ω-complete posets, of directed complete posets and of continuous directed complete posets as those monotone maps f that are convex and that lift an element (and then a queue) of any directed set (ω-chain in the case of ) whose supremum is in the image of f (Theorem 1.9). Using this characterization, we obtain that a monomorphism f:X→B in (, ) exponentiable in w.r.t. the Scott topology is exponentiable also in (, ). We prove that the converse is true in the category , but neither in , nor in .     

The Farahat-Higman ring of wreath products and Hilbert schemes

We study the structure constants of the class algebra of the wreath products Γ_n associated to an arbitrary finite group Γ with respect to the basis of conjugacy classes. We show that a suitable filtration on gives rise to the graded ring with non-negative integer structure constants independent of n (some of which are computed), which are then encoded in a Farahat-Higman ring . The real conjugacy classes of Γ come to play a distinguished role and are treated in detail in the case when Γ is a subgroup of . The above results provide new insight to the cohomology rings of Hilbert schemes of points on a quasi-projective surface X.     

Compactifications of the homeomorphism group of a graph

Let Γ be a countable locally finite graph and let H(Γ) and H₊(Γ) denote the homeomorphism group of Γ with the compact-open topology and its identity component. These groups can be embedded into the space of all closed sets of Γ×Γ with the Fell topology, which is compact. Taking closure, we have natural compactifications and . In this paper, we completely determine the topological type of the pair and give a necessary and sufficient condition for this pair to be a (Q,s)-manifold. The pair is also considered for simple examples, and in particular, we find that has homotopy type of RP³. In this investigation we point out a certain inaccuracy in Sakai-Uehara's preceding results on for finite graphs Γ.     

Continuous selections and σ-spaces

Assume that X⊆R?Q, and each clopen-valued lower semicontinuous multivalued map has a continuous selection . Our main result is that in this case, X is a σ-space. We also derive a partial converse implication, and present a reformulation of the Scheepers Conjecture in the language of continuous selections.     

Quasi sure analysis of local times of anticipating smooth semimartingales

Let be the anticipating smooth semimartingale and be its generalized local time. In this paper, we give some estimates about the quasi sure property of Xt and its quadratic variation process t〈X〉. We also study the fractional smoothness of and prove that the quadratic variation process of can be constructed as the quasi sure limit of the form , where is a sequence of subdivisions of [a,b], , i=0,1,…,n².     

On the topological Helly theorem

The main result of this paper is the following theorem, related to the missing link in the proof of the topological version of the classical result of Helly: Let be any family of simply connected compact subsets of R² such that for every i,j∈{0,1,2} the intersections Xi∩Xj are path connected and is nonempty. Then for every two points in the intersection there exists a cell-like compactum connecting these two points, in particular the intersection is a connected set.     

Almost isometries and orthogonality

Given any two almost orthonormal systems A and of unit vectors in a Hilbert space X of the same cardinality, there exists a norm preserving almost isometry F of X so that . It follows that to every almost orthonormal system there corresponds a system of almost coefficient functionals.     

Variations of selective separability

A space X is selectively separable if for every sequence of dense subspaces of X one can select finite Fn⊂Dn so that is dense in X. In this paper selective separability and variations of this property are considered in two special cases: Cp spaces and dense countable subspaces in κ².     

A new majorization between functions, polynomials, and operator inequalities

Let P₊ be the set of all non-negative operator monotone functions defined on [0,∞), and put . Then and . For a function and a strictly increasing function h we write if is operator monotone. If and and if and , then . We will apply this result to polynomials and operator inequalities. Let and be non-increasing sequences, and put for t≧a₁ and for t≧b₁. Then v₊?u₊ if m≦n and : in particular, for a sequence of orthonormal polynomials, (p_n-1)₊?(p_n)₊. Suppose 0<r,p and s=0 or 1≦s≦1+p/r. Then 0≦A≦B implies for 0<α≦r/(p+r).     

Holomorphic functions and quasiconformal mappings with smooth moduli

We discuss the implication , where f is a holomorphic function (resp., a quasiconformal mapping) on a domain (resp., ) and Λ_ω(G) is the Lipschitz space associated with a majorant ω.     

Geodesic currents and Teichmüller space

Consider a hyperbolic surface X of infinite area. The Liouville map assigns to any quasiconformal deformation of X a measure on the space of geodesics of the universal covering X? of X. We show that the Liouville map is a homeomorphism from the Teichmüller space onto its image, and that the image is closed and unbounded. The set of asymptotic rays to consists of all bounded measured laminations on X. Hence, the set of projective bounded measured laminations is a natural boundary for . The action of the quasiconformal mapping class group on continuously extends to this boundary for .     

Controlled Hahn-Mazurkiewicz Theorem and some new dimension functions of Peano continua

Given a metric Peano continuum X we introduce and study the Hölder Dimension there is a -Hölder onto map of X as well as its topological counterpart is an admissible metric for X}. We show that for each convex metric continuum X the dimension Hö-dim(X) equals the fractal dimension of X. The topological Hölder dimension Hö-dim(Mn) of the n-dimensional universal Menger cube Mn equals n. On the other hand, there are 1-dimensional rim-finite Peano continua X with arbitrary prescribed Hö-dim(X)?1.