IslandScholar aggregator
http://islandscholar.ca
Aggregated results of: Maxim BurkeenLiftings 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
Entire 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
Punctually countable coverings by means of negligible sets
http://islandscholar.ca/islandora/object/ir:ir-batch6-1672
Models in which every nonmeager set is nonmeager in a nowhere dense Cantor set
http://islandscholar.ca/islandora/object/ir:ir-batch6-1654
We prove that it is relatively consistent with ZFC that in any perfect Polish space, for every nonmeager set A there exists a nowhere dense Cantor set C such that A boolean AND C is nonmeager in C. We also examine variants of this result and establish a measure theoretic analog.Various products of category densities and liftings
http://islandscholar.ca/islandora/object/ir:782
A note on measurability and almost continuity
http://islandscholar.ca/islandora/object/ir:ir-batch6-1671
Liftings for Haar measure on (0,1)K
http://islandscholar.ca/islandora/object/ir:ir-batch6-1667
We call E subset-of-or-is-equal-to {0,1}kappa projective if for some countable A subset-of-or-is-equal-to kappa there is an E(A) subset-of-or-is-equal-to {0,1}A such that E = E(A) x {0,1}kappa/A and E(A) is a projective subset of the Cantor set {0,1}A. We construct a model where Haar measure on {0,1}kappa has no projective lifting (and in particular no Baire lifting) for any kappa greater-than-or-equal-to omega.Liftings for noncomplete probability spaces
http://islandscholar.ca/islandora/object/ir:ir-batch6-1664
The current state of knowledge concerning liftings for noncomplete probability spaces is discussed. This is a somewhat expanded version of the author's talk given at the 1991 Summer Conference on General Topology and Applications in Honor of Mary Ellen Rudin and Her Work.Shelah's pcf theory and its applications
http://islandscholar.ca/islandora/object/ir:ir-batch6-1669
This is a survey paper giving a self-contained account of Shelah's theory of the pcf function pcf(a) = {cf(PI a/D, < D): D is an ultrafilter on a}, where a is a set of regular cardinals such that \a\ < min(a). We also give several applications of the theory to cardinal arithmetic, the existence of Jonsson algebras, and partition calculus.Linear liftings for noncomplete probability spaces
http://islandscholar.ca/islandora/object/ir:ir-batch6-1666
We show that it is consistent with ZFC that L(infinity)(Y, B, upsilon) has no linear lifting for many non-complete probability spaces (Y, B, upsilon), in particular for Y = [0, 1]A, B = Borel subsets of Y, upsilon = usual Radon measure on B.Sets of range uniqueness for classes of continuous functions
http://islandscholar.ca/islandora/object/ir:ir-batch6-1659
Diamond, Pomerance and Rubel (1981) proved that there are subsets M of the complex plane such that for any two entire functions f and g if f[M] = g[M], then f = g. Baraducci and Dikranjan showed in 1993 that the continuum hypothesis (CH) implies the existence of a similar set M subset of R for the class C-n( R) of continuous nowhere constant functions from R to R, while it follows from the results of Burke and Ciesielski (1997) and Ciesielski and Shelah that the existence of such a set is not provable in ZFC. In this paper we will show that for several well-behaved subclasses of C( R), including the class D-1 of differentiable functions and the class AC of absolutely continuous functions, a set M with the above property can be constructed in ZFC. We will also prove the existence of a set M subset of R with the dual property that for any f; g is an element of C-n( R) if f(-1) [M] = g(-1) [M], then f = g.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.