CR Runge Sets on Hypersurface Graphs

This work contains an improvement of earlier results of Boggess and Dwilewicz regarding global approximation of CR functions by entire functions in the case of hypersurface graphs. In this work, we show that if ω, an open subset of a real hypersurface in ℂ ⁿ , can be graphed over a convex subset in ℝ²ⁿ⁻¹, then ω is CR-Runge in the sense that continuous CR functions on ω can be approximated by entire functions on ℂ ⁿ in the compact open topology of ω. Examples are presented to show that this approximation result does not hold for graphed CR submanifolds in higher codimension. R. Dwilewicz is partially supported by the Polish Science Foundation (KBN) grant N201 019 32/805.     

Almost isometric embedding between metric spaces

We investigate the relations ofalmost isometric embedding and ofalmost isometry between metric spaces. These relations have several appealing features. For example, all isomorphism types of countable dense subsets of ∝ form exactly one almostisometry class, and similarly with countable dense subsets of Uryson's universal separable metric spaceU. We investigate geometric, set-theoretic and model-theoretic aspects of almost isometry and of almost isometric embedding. The main results show that almost isometric embeddability behaves in the category ofseparable metric spaces differently than in the category of general metric spaces. While in the category of general metric spaces the behavior of universality resembles that in the category of linear orderings —namely, no universal structure can exist on a regular λ > ℵ₁ below the continuum—in the category of separable metric spaces universality behaves more like that in the category of graphs, that is, a small number of metric separable metric spaces on an uncountable regular λ<2^ℵ ₀ may consistently almost isometrically embed all separable metric spaces on λ. Research of the first author was supported by an Israeli Science foundation grant no. 177/01. Research of the second author was supported by the United States-Israel Binational Science Foundation. Publication 827.     

A model in which there are Jech-Kunen trees but there are no Kurepa trees

By an ω₁-tree we mean a tree of power ω₁ and height ω₁. We call an ω₁-tree a Jech-Kunen tree if it hask-many branches for somek strictly between ω₁ and 2^ω1. In this paper we construct the models of CH plus 2^ω1 > ω₂, in which there are Jech-Kunen trees and there are no Kurepa trees. The research of the first author was partially supported by the Basic Research Fund, Israeli Academy of Science, Publ. No. 466.     

On the weight-spectrum of a compact space

The weight-spectrumSp(w, X) of a spaceX is the set of weights of all infinite closed subspaces ofX. We prove that ifκ>ω is regular andX is compactT ₂ withω(X)≥κ then some λ withκ≤λ≤2^<κ is inSp(ω, X). Under CH this implies that the weight spectrum of a compact space can not omitω ₁, and thus solves problem 22 of [M]. Also, it is consistent with 2^ω=c being anything it can be that every countable closed setT of cardinals less thanc withω ∈ T satisfiesSp(w, X)=T for some separable compact LOTSX. This shows the independence from ZFC of a conjecture made in [AT]. Research supported by OTKA grant no. 1908.     

Chains of baire class 1 functions and various notions of special trees

Following Laczkovich we consider the partially ordered setB ₁(ℝ) of Baire class 1 functions endowed with the pointwise order, and investigate the order types of the linearly ordered subsets. Answering a question of Komjáth and Kunen we show (inZFC) that special Aronszajn lines are embeddable intoB ₁(ℝ). We also show that under Martin's Axiom a linearly ordered set ℒ with |ℒ| < 2^ω is embeddable intoB ₁(ℝ) iff ℒ does not contain a copy of ω₁ or ω _* ¹ . We present aZFC example of a linear order of size 2^ω showing that this characterisation is not valid for orders of size continuum. These results are obtained using the notion of a compact-special tree; that is, a tree that is embeddable into the class of compact subsets of the reals partially ordered under reverse inclusion. We investigate how this notion is related to the well-known notion of an ℝ-special tree and also to some other notions of specialness. Partially supported by Hungarian Scientific Foundation grant no. 37758, 49786 and F 43620. The second author's research for this paper was partially supported by NSERC of Canada.     

Linear liftings for non-complete probability spaces

We show that it is consistent with ZFC thatL ^∞(Y,B,ν) has no linear lifting for many non-complete probability spaces (Y,B,ν), in particular forY=[0,1]^A,B=Borel subsets ofY, ν=usual Radon measure onB. Research supported by UPEI Senate Grant no 602101, by the Research Institute for Mathematical Sciences at Bar-Ilan University, and by the Landau Center for Mathematical Research in Analysis (supported by the Minerva Foundation). The author would like to thank the organizers of the Winter Institute on the Set Theory of the Reals for their hospitality while part of this research was being carried out. Partially supported by the Foundation for Basic Research of the Israel Academy of Science. Publication number 437.     

On scattered Eberlein compact spaces

We prove that scattered Eberlein compacta of Cantor-Bendixson height at most ω + 1 are Uniform Eberlein compact spaces (ω + 1 is optimal for this result). For a set X and n ∈ ω, by σ _n (2^X) we denote the subspace of the product 2^X consisting of all characteristic functions of sets of cardinality ≥ n. We give an example of an Eberlein compactum K of weight ω _ω and of Cantor-Bendixson height 3 which cannot be embedded into any σ _n (2^X). Research of the first author supported by NSERC of Canada. Murray Bell died on December 9, 2001. Research of the second author supported by KBN grants 2 P03A 011 15 and 2 P03A 004 23. The main part of this research was done while the second author was visiting the University of Manitoba in 2000. He expresses his gratitude to the Department of Mathematics of U.M. for its hospitality.     

The distribution of the root degree of a random permutation

Given a permutation ω of {1, …, n}, let R(ω) be the root degree of ω, i.e. the smallest (prime) integer r such that there is a permutation σ with ω = σ ^r . We show that, for ω chosen uniformly at random, R(ω) = (lnlnn − 3lnlnln n + O ^p (1))⁻¹ lnn, and find the limiting distribution of the remainder term. Research supported in part by NSF grants CCR-0225610, DMS-0505550 and ARO grant W911NF-06-1-0076. Research supported by NSF grant DMS-0406024.     

Pseudoradial Spaces: Results and Problems

In this paper it is given a survey of principal results (old and new) concerning the class of pseudoradial spaces. In this class cardinal invariants and their inequalities are considered. The behaviour of pseudoradial spaces under the operations of taking topological products and subspaces are examined and a typical proof is given. A particular attention is dedicated to the so called “small cardinals” in connection with pseudoradiality. Pseudoradiality of 2^{ω 2} is also examined. It is proved that pseudoradiality can be ω¹ productive for spaces of weight at most ω². Finally, several open problems are presented. This work was supported by the National Group “Real Analysis, Measure Theory with Applications to Economy” of the Italian Ministery of Education, University and Research.     

Fundamental groups of branched covering spaces ofS 3

Summary Given a knotK⊂S ³, it is known a standard method for constructing a 4-coloured graph representing the closed orientable 3-manifoldM=M(K, d, ω) which is thed-fold covering space ofS ³ branched overK and associated to the transitived-representation ω of the knot group. In this paper we obtain a presentation of the fundamental group ofM, directly from the Wirtinger presentation of the knot group and from the transitived-representation ω.

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Riassunto Dato un nodoK⊂S ³, è noto un metodo standard per costruire un grafo 4-colorato rappresentante la 3-varietà chiusa ed orientabileM=M(K, d, ω) che è lo spazio di rivestimento diS ³ ramificato suK ed associato allad-rappresentazione transitiva ω del gruppo del nodo. In questo articolo si ottiene una presentazione del gruppo fondamentale diM, direttamente dalla presentazione di Wirtinger del gruppo del nodo e dallad-rappresentazione transitiva ω.

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Work performed under the auspicies of the G.N.S.A.G.A. of the C.N.R. (National Research Council of Italy) and financially supported by M.P.I. (project ?Geometria delle Varietà differenziabili?).     

Convex decompositions in the plane and continuous pair colorings of the irrationals

We address the structure of nonconvex closed subsets of the Euclidean plane. A closed subsetS⊆ℝ² which is not presentable as a countable union of convex sets satisfies the following dichotomy:

Vaught’s conjecture for some meager groups

AssumeG is a superstable group ofM-rank 1 and the division ring of pseudo-endomorphisms ofG is a prime field. We prove a relative Vaught’s conjecture for Th(G). When additionallyU(G) =ω, this yields Vaught’s conjecture for Th(G). Research supported by KBN grant 2 P03A 006 09.     

Strongly meager sets can be quite big

AssumeCH. There exists a strongly meager setX⊆2^ω and a continuous functionF: 2^ω → 2^ω such thatF″ (X)=2^ω. The analogous statement for the strong measure zero, the notion dual to strongly meager, is false. The first author was partially supported by NSF grant DMS 9971282 and the Alexander von Humboldt Foundation. The second author was partially supported by grant BW 5100-5-0231-2.     

Some remarks on a question of D. H. Fremlin regarding ε-density

We show the relative consistency of ℵ₁ satisfying a combinatorial property considered by David Fremlin (in the question DU from his list) in certain choiceless inner models. This is demonstrated by first proving the property is true for Ramsey cardinals. In contrast, we show that in ZFC, no cardinal of uncountable cofinality can satisfy a similar, stronger property. The questions considered by D. H. Fremlin are if families of finite subsets of ω₁ satisfying a certain density condition necessarily contain all finite subsets of an infinite subset of ω₁, and specifically if this and a stronger property hold under MA + ?CH. Towards this we show that if MA + ?CH holds, then for every family ? of ℵ₁ many infinite subsets of ω₁, one can find a family ? of finite subsets of ω₁ which is dense in Fremlins sense, and does not contain all finite subsets of any set in ?. We then pose some open problems related to the question. Received: 2 June 1999 / Revised version: 2 February 2000 / Published online: 18 July 2001     

Splitting ρκλ into maximally many stationary sets

Letκ >ω be a regular cardinal and λ >κ a cardinal. Solovay’s classical result for κ[So] led Menas [Me] to conjecture that a stationary subset ofP _κλ would split into λ^<κ stationary set of size κ⁺ (see[BT]), the conjecture implies that the size is (κ⁺)^<κ as well. Part of this work was done during the author’s stay at Boston University as one of the Japanese Overseas Research Fellows. He gratefully acknowledge Professor Akihiro Kanamori’s hospitality. He also wishes to thank members of the set theory seminar at Waseda University for their interest at the early stage.     

The size of epimorphic extensions

A sharp bound is given for the size of epimorphic extensions in categories of models defined over elementary logic andL _κκ where κ is strongly compact. For fragments ofL _ω1ω an example is given of a category which has a countable model with epimorphic extensions whose cardinalities approach and include the first measurable cardinal. If no measurable cardinal exists then this category has a countable model with epimorphic extensions of unbounded cardinality. This work was supported in part by the National Research Council of Canada under grant numbers A8599, A5603 and A8190. Presented by J. D. Monk.     

Combinatorial dimension and random sets

Having defined the combinatorial dimension of an arbitrary subset of a finite dimensional lattice, for every αε(1,2) we produce a set in N² whose dimension equals α. Research partially supported by NSF Grant #MCS8002716. Research supported by a University of Connecticut Research Grant (Sept. 1980).     

Estimates of the weak distance between finite-dimensional Banach spaces

We prove a variant of a theorem of N. Alon and V. D. Milman. Using it we construct for everyn-dimensional Banach spacesX andY a measure space Ω and two operator-valued functionsT: Ω→L(X, Y),S: Ω→L(Y, X) so that ∫_Ω S(ω)oT(ω)dω is the identity operator inX and ∫_Ω||S(ω)||·||T(ω)||dω=O(n ^α ) for some absolute constantα<1. We prove also that any subset of the unitn-cube which is convex, symmetric with respect to the origin and has a sufficiently large volume possesses a section of big dimension isomorphic to ak-cube. Research supported in part by a grant of the Israel Academy of Sciences.     

An extension theorem for separable Banach spaces

We construct a totally disconnected ω^*, norming subsetF of the unit ballB ^* of an arbitrary separable Banach space,X, and an operator fromC(F) toC(B^*) that “amost” commutes with the natural embeddings ofX. This is used to give a new proof of Milutin's theorem and to prove some new results on complemented subspaces ofC[0, 1] with separable dual. In particular we show that a complemented subspace ofC(ω^ω), is either isomorphic toC(ω^ω) or toc _u.