IslandScholar aggregator
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Aggregated results of: Maxim BurkeenEntire functions mapping uncountable dense sets of reals onto each other monotonically
http://islandscholar.ca/islandora/object/ir:13949
No Description[Reviews of the following books Forcings with ideals and simple forcing notions by Moti Gitik & Saharon Shelah; More on simple forcing Notions and forcing with ideals by M. Gitik, & S. Shelah; Real-valued-measurable cardinals by D. H. Fremin]
http://islandscholar.ca/islandora/object/ir:13978
Liftings for noncomplete probability spaces
http://islandscholar.ca/islandora/object/ir:13944
Approximation by entire functions in the construction of order-isomorphisms and large cross-sections
http://islandscholar.ca/islandora/object/ir:21376
No DescriptionParadoxical decompositions of planar sets of positive outer measure Paradoxical decompositions of planar sets
http://islandscholar.ca/islandora/object/ir:13940
Approximation and interpolation by large entire cross-sections of second category sets in Rn+1
http://islandscholar.ca/islandora/object/ir:9780
In [M.R. Burke, Large entire cross-sections of second category sets in Rn+1Rn+1, Topology Appl. 154 (2007) 215–240], a model was constructed in which for any everywhere second category set A⊆Rn+1A⊆Rn+1 there is an entire function f:Rn→Rf:Rn→R which cuts a large section through A in the sense that {x∈Rn:(x,f(x))∈A}{x∈Rn:(x,f(x))∈A} is everywhere second category in RnRn. Moreover, the function f can be taken so that its derivatives uniformly approximate those of a given CNCN function g in the sense of a theorem of Hoischen. In the theory of the approximation of CNCN functions by entire functions, it is often possible to insist that the entire function interpolates the restriction of the CNCN function to a closed discrete set. In the present paper, we show how to incorporate a closed discrete interpolation set into the above mentioned theorem. When the set being sectioned is sufficiently definable, an absoluteness argument yields a strengthening of the Hoischen theorem in ZFC. We get in particular the following: Suppose g:Rn→Rg:Rn→R is a CNCN function, ε:Rn→Rε:Rn→R is a positive continuous function, T⊆RnT⊆Rn is a closed discrete set, and G⊆Rn+1G⊆Rn+1 is a dense GδGδ set. Let A⊆RnA⊆Rn be a countable dense set disjoint from T and for each x∈Ax∈A, let Bx⊆RBx⊆R be a countable dense set. Then there is a function f:Rn→Rf:Rn→R which is the restriction of an entire function Cn→CCn→C such that the following properties hold. (a) For all multi-indices α of order at most N and all x∈Rnx∈Rn, |(Dαf)(x)−(Dαg)(x)|<ε(x)|(Dαf)(x)−(Dαg)(x)|<ε(x), and moreover (Dαf)(x)=(Dαg)(x)(Dαf)(x)=(Dαg)(x) when x∈Tx∈T. (b) For each x∈Ax∈A, f(x)∈Bxf(x)∈Bx. (c) {x∈Rn:(x,f(x))∈G}{x∈Rn:(x,f(x))∈G} is a dense GδGδ set in RnRn.Approximation and interpolation by entire functions with restriction of the values of the derivatives
http://islandscholar.ca/islandora/object/ir:20427
No DescriptionHechler’s theorem for the null ideal
http://islandscholar.ca/islandora/object/ir:13936
[Review of the book Introduction to Cardinal Arithmetic by M. Holz, K. Steffens, and E. Weitz]
http://islandscholar.ca/islandora/object/ir:13954
Large entire cross-sections of second category sets in Rn+1
http://islandscholar.ca/islandora/object/ir:ir-batch6-1652
By the Kuratowski-Ulam theorem, if A subset of Rn+1 = R-n x R is a Borel set which has second category intersection with every ball (i.e., is "everywhere second category"), then there is a y is an element of R such that the section A boolean AND (R-n x {y}) is everywhere second category in R-n x {y}. If A is not Borel, then there may not exist a large cross-section through A, even if the section does not have to be flat. For example, a variation on a result of T. Bartoszynski and L. Halbeisen shows that there is an everywhere second category set A subset of Rn+1 such that for any polynomial p in n variables, A boolean AND graph(p) is finite. It is a classical result that under the Continuum Hypothesis, there is an everywhere second category set L in Rn+1 which has only countably many points in any first category set. In particular, L boolean AND graph(f) is countable for any continuous function f : R-n -> R. We prove that it is relatively consistent with ZFC that for any everywhere second category set A in Rn+1, there is a function f : R-n -> R which is the restriction to R-n of an entire function on C-n and is such that, relative to graph(f), the set A n graph(f) is everywhere second category. For any collection of less than 2(N)0 sets A, the function f can be chosen to work for all sets A in the collection simultaneously. Moreover, given a nonnegative integer k, a function g: R-n -> R of class C-k and a positive continuous function epsilon:R-n -> R, we may choose f so that for all multiindices alpha of order at most k and for all X is an element of R-n, vertical bar D-alpha g(x) - D(alpha)g(x)vertical bar <epsilon(x). The method builds on fundamental work of K. Ciesielski and S. Shelah which provides, for everywhere second category sets in 2(omega) x 2(omega), large sections which are the graphs of homeomorphisms of 2(omega). K. Ciesielski and T. Natkaniec adapted the Ciesielski-Shelah result for subsets of R x R and proved the existence in this setting of large sections which are increasing homeormorphisms of R. The technique used in this paper extends to functions of several variables an approach developed for functions of a single variable in previous related work of the author. (c) 2006 Elsevier B.V. All rights reserved.A proof of Hechler's theorem on embedding N-1-directed sets cofinally into ((ωω, <∗)
http://islandscholar.ca/islandora/object/ir:ir-batch6-1661
We give a proof of Hechler's theorem that any aleph(1)-directed partial order can be embedded via a ccc forcing notion cofinally into omega(omega) ordered by eventual dominance, The proof relies on the standard forcing relation rather than the variant introduced by Hechler.Liftings and the property of Baire in locally compact groups
http://islandscholar.ca/islandora/object/ir:ir-batch6-1665
For each locally compact group G with Haar measure mu , we obtain the following results. The first is a version for group quotients of a classical result of Kuratowski and Ulam on first category subsets of the plane. The second is a strengthening of a theorem of Kupka and Prikry; we obtain it by a much simpler technique, building on work of Talagrand and Losert.Borel measurability of separately continuous functions, II
http://islandscholar.ca/islandora/object/ir:ir-batch6-1656
This paper continues the investigation begun in [M.R. Burke, Topology Appl. 129 (2003) 29-65] into the measurability properties of separately continuous functions. We sharpen several results from that paper. (1) If X is any product of countably compact Dedekind complete linearly ordered spaces, then there is a network for the norm topology on C(X) which is sigma-isolated in the topology of pointwise convergence. (2) If X is a nonseparable ccc space, then the evaluation map X x C-p(X) --> R is not a Baire function. (3) If X-i, i R is F sigma-measurable if and only if kappa less than or equal to c. (C) 2003 Elsevier B.V. All rights reserved.Characterizing uniform continuity with closure operations
http://islandscholar.ca/islandora/object/ir:ir-batch6-1663
The purpose of this paper is to determine which metric spaces X and Y are such that the uniformly continuous maps f : X --> Y are precisely the continuous maps between (X, tau1) and (Y, tau2) for some new topologies tau1 and tau2 on X and Y respectively.Simultaneous approximation and interpolation of increasing functions by increasing entire functions
http://islandscholar.ca/islandora/object/ir:783
Sets on which measurable functions are determined by their range
http://islandscholar.ca/islandora/object/ir:ir-batch6-1660
We study sets on which measurable real-valued functions on a measurable space with negligibles are determined by their range.Weakly dense subsets of the measure algebra
http://islandscholar.ca/islandora/object/ir:ir-batch6-1670
Continuous functions which take a somewhere dense set of values on every open set (vol 103, pg 95, 2000)--correction
http://islandscholar.ca/islandora/object/ir:ir-batch6-1657
No DescriptionCategory product densities and liftings
http://islandscholar.ca/islandora/object/ir:ir-batch6-1653
In this paper we investigate two main problems. One of them is the question on the existence of category liftings in the product of two topological spaces. We prove, that if X x Y is a Baire space, then, given (strong) category liftings rho and sigma on X and Y, respectively, there exists a (strong) category lifting pi on the product space such that pi is a product of rho and sigma and satisfies the following section property: [pi (E)](x)=sigma([pi(E)](x)) for all E subset of X x Y with Baire property and all x is an element of X. We give also an example, where some of the sections [pi(E)](y) must be without Baire property. Then, we investigate the existence of densities respecting coordinates on products of topological spaces, provided these products are Baire spaces. The densities are defined on sigma-algebras of sets with Baire property and select elements modulo the sigma-ideal of all meager sets. In all the problems the situation in the "category case" turns out to be much better, than in case of products of measure spaces. In particular, in every product there exists a canonical strong density being a product of the canonical densities in the factors and there exist (strong) densities respecting coordinates with given a priori marginal (strong) densities. (C) 2005 Elsevier B.V. All rights reserved.Continuous functions which take a somewhere dense set of values on every open set
http://islandscholar.ca/islandora/object/ir:ir-batch6-1658
We study the class of Tychonoff spaces that can be mapped continuously into R in such a way that the preimage of every nowhere dense set is nowhere dense. We show that every metric space without isolated points is in this class. We also give examples of spaces which have nowhere constant continuous maps into R and are not in this class. (C) 2000 Elsevier Science B.V. All rights reserved.