Expand all Collapse all | Results 1 - 19 of 19 |
1. CJM Online first
Addendum to ``Nearly Countable Dense Homogeneous Spaces' This paper provides an addendum to M. HruÅ¡Ã¡k
and J. van Mill ``Nearly countable dense homogeneous spaces.''
Canad. J. Math., published online 2013-03-08
http://dx.doi.org/10.4153/CJM-2013-006-8.
Keywords:countable dense homogeneous, nearly countable dense homogeneous, Effros Theorem, Vaught's conjecture Categories:54H05, 03E15, 54E50 |
2. CJM Online first
Non-tame Mice from Tame Failures of the Unique Branch Hypothesis In this paper, we show that the failure of the unique branch
hypothesis (UBH) for tame trees
implies that in some homogenous generic extension of $V$ there is a
transitive model $M$ containing $Ord \cup \mathbb{R}$ such that
$M\vDash \mathsf{AD}^+ + \Theta \gt \theta_0$. In particular, this
implies the existence (in $V$) of a non-tame mouse. The results of
this paper significantly extend J. R. Steel's earlier results
for tame trees.
Keywords:mouse, inner model theory, descriptive set theory, hod mouse, core model induction, UBH Categories:03E15, 03E45, 03E60 |
3. CJM Online first
Nearly Countable Dense Homogeneous Spaces We study separable metric spaces with few types of countable dense
sets. We present a structure theorem for locally compact spaces
having precisely $n$ types of countable dense sets: such a space
contains a subset $S$ of size at most $n{-}1$ such that $S$ is
invariant under
all homeomorphisms of $X$ and $X\setminus S$ is countable dense
homogeneous. We prove that every Borel space having fewer than $\mathfrak{c}$
types of
countable dense sets is Polish. The natural question of whether every
Polish space has either countably many or $\mathfrak{c}$ many types of
countable
dense sets, is shown to be closely related to Topological Vaught's
Conjecture.
Keywords:countable dense homogeneous, nearly countable dense homogeneous, Effros Theorem, Vaught's conjecture Categories:54H05, 03E15, 54E50 |
4. CJM 2013 (vol 66 pp. 303)
Haar Null Sets and the Consistent Reflection of Non-meagreness A subset $X$ of a Polish group $G$ is called Haar null if there exists
a Borel set $B \supset X$ and Borel probability measure $\mu$ on $G$ such that
$\mu(gBh)=0$ for every $g,h \in G$.
We prove that there exist a set $X \subset \mathbb R$ that is not Lebesgue null and a
Borel probability measure $\mu$ such that $\mu(X + t) = 0$ for every $t \in
\mathbb R$.
This answers a question from David Fremlin's problem list by showing
that one cannot simplify the definition of a Haar null set by leaving out the
Borel set $B$. (The answer was already known assuming the Continuum
Hypothesis.)
This result motivates the following Baire category analogue. It is consistent
with $ZFC$ that there exist an abelian Polish group $G$ and a Cantor
set $C \subset G$ such that for every non-meagre set $X \subset G$ there exists a $t
\in G$ such that $C \cap (X + t)$ is relatively non-meagre in $C$. This
essentially generalises results of BartoszyÅski and Burke-Miller.
Keywords:Haar null, Christensen, non-locally compact Polish group, packing dimension, Problem FC on Fremlin's list, forcing, generic real Categories:28C10, 03E35, 03E17, , , , , 22C05, 28A78 |
5. CJM 2012 (vol 64 pp. 1378)
On Weakly Tight Families Using ideas from Shelah's recent proof that a completely
separable maximal almost disjoint family exists when
$\mathfrak{c} \lt {\aleph}_{\omega}$, we construct a weakly tight family
under the hypothesis $\mathfrak{s} \leq \mathfrak{b} \lt
{\aleph}_{\omega}$.
The case when $\mathfrak{s} \lt \mathfrak{b}$
is handled in $\mathrm{ZFC}$ and does not require $\mathfrak{b} \lt {\aleph}_{\omega}$,
while an additional PCF type hypothesis, which holds when $\mathfrak{b} \lt
{\aleph}_{\omega}$ is used to treat the case $\mathfrak{s} = \mathfrak{b}$. The notion of
a weakly tight family is a natural weakening of the well studied
notion of a Cohen indestructible maximal almost disjoint family. It
was introduced by HruÅ¡Ã¡k and GarcÃa
Ferreira, who applied it to the KatÃ©tov order on almost
disjoint families.
Keywords:maximal almost disjoint family, cardinal invariants Categories:03E17, 03E15, 03E35, 03E40, 03E05, 03E50, 03E65 |
6. CJM 2012 (vol 65 pp. 222)
Distance Sets of Urysohn Metric Spaces A metric space $\mathrm{M}=(M;\operatorname{d})$ is {\em homogeneous} if for every
isometry $f$ of a finite subspace of $\mathrm{M}$ to a subspace of
$\mathrm{M}$ there exists an isometry of $\mathrm{M}$ onto
$\mathrm{M}$ extending $f$. The space $\mathrm{M}$ is {\em universal}
if it isometrically embeds every finite metric space $\mathrm{F}$ with
$\operatorname{dist}(\mathrm{F})\subseteq \operatorname{dist}(\mathrm{M})$. (With
$\operatorname{dist}(\mathrm{M})$ being the set of distances between points in
$\mathrm{M}$.)
A metric space $\boldsymbol{U}$ is an {\em Urysohn} metric space if
it is homogeneous, universal, separable and complete. (It is not
difficult to deduce
that an Urysohn metric space $\boldsymbol{U}$ isometrically embeds
every separable metric space $\mathrm{M}$ with
$\operatorname{dist}(\mathrm{M})\subseteq \operatorname{dist}(\boldsymbol{U})$.)
The main results are: (1) A characterization of the sets
$\operatorname{dist}(\boldsymbol{U})$ for Urysohn metric spaces $\boldsymbol{U}$.
(2) If $R$ is the distance set of an Urysohn metric space and
$\mathrm{M}$ and $\mathrm{N}$ are two metric spaces, of any
cardinality with distances in $R$, then they amalgamate disjointly to
a metric space with distances in $R$. (3) The completion of every
homogeneous, universal, separable metric space $\mathrm{M}$ is
homogeneous.
Keywords:partitions of metric spaces, Ramsey theory, metric geometry, Urysohn metric space, oscillation stability Categories:03E02, 22F05, 05C55, 05D10, 22A05, 51F99 |
7. CJM 2012 (vol 64 pp. 1182)
PFA$(S)[S]$: More Mutually Consistent Topological Consequences of $PFA$ and $V=L$ Extending the work of Larson and Todorcevic,
we show there
is a model of set theory in which normal spaces are collectionwise
Hausdorff if they are either first countable or locally compact, and
yet there are no first countable $L$-spaces or compact
$S$-spaces. The model is one of the form PFA$(S)[S]$, where $S$
is a coherent Souslin tree.
Keywords:PFA$(S)[S]$, proper forcing, coherent Souslin tree, locally compact, normal, collectionwise Hausdorff, supercompact cardinal Categories:54A35, 54D15, 54D20, 54D45, 03E35, 03E57, 03E65 |
8. CJM 2012 (vol 65 pp. 485)
Filters in C*-Algebras In this paper we analyze states on C*-algebras and
their relationship to filter-like structures of projections and
positive elements in the unit ball. After developing the basic theory
we use this to investigate the Kadison-Singer conjecture, proving its
equivalence to an apparently quite weak paving conjecture and the
existence of unique maximal centred extensions of projections coming
from ultrafilters on the natural numbers. We then prove that Reid's
positive answer to this for q-points in fact also holds for rapid
p-points, and that maximal centred filters are obtained in this case.
We then show that consistently such maximal centred filters do not
exist at all meaning that, for every pure state on the Calkin algebra,
there exists a pair of projections on which the state is 1, even
though the state is bounded strictly below 1 for projections below
this pair. Lastly we investigate towers, using cardinal invariant
equalities to construct towers on the natural numbers that do and do
not remain towers when canonically embedded into the Calkin algebra.
Finally we show that consistently all towers on the natural numbers
remain towers under this embedding.
Keywords:C*-algebras, states, Kadinson-Singer conjecture, ultrafilters, towers Categories:46L03, 03E35 |
9. CJM 2011 (vol 63 pp. 1416)
MAD Saturated Families and SANE Player We throw some light on the question: is there a MAD family
(a maximal family of infinite subsets of $\mathbb{N}$, the intersection of any
two is finite) that is saturated (completely separable \emph{i.e.,} any
$X \subseteq \mathbb{N}$ is
included in a finite union of members of the family \emph{or} includes a
member (and even continuum many members) of the family).
We prove that it is hard to prove the consistency of the negation:
(i) if $2^{\aleph_0} \lt \aleph_\omega$, then there is such a family;
(ii) if there is no such family, then some situation
related to pcf holds whose consistency is large (and if ${\mathfrak a}_* \gt
\aleph_1$ even unknown);
(iii) if, \emph{e.g.,} there is no inner model with measurables,
\emph{then} there is such a family.
Keywords:set theory, MAD families, pcf, the continuum Categories:03E05, 03E04, 03E17 |
10. CJM 2010 (vol 62 pp. 481)
Elements of Algebraic Geometry and the Positive Theory of Partially Commutative Groups The first main result of the paper is a criterion for a partially commutative group $\mathbb G$ to be a domain. It allows us to reduce the study of algebraic sets over $\mathbb G$ to the study of irreducible algebraic sets, and reduce the elementary theory of $\mathbb G$ (of a coordinate group over $\mathbb G$) to the elementary theories of the direct factors of $\mathbb G$ (to the elementary theory of coordinate groups of irreducible algebraic sets). Then we establish normal forms for quantifier-free formulas over a non-abelian directly indecomposable partially commutative group $\mathbb H$. Analogously to the case of free groups, we introduce the notion of a generalised equation and prove that the positive theory of $\mathbb H$ has quantifier elimination and that arbitrary first-order formulas lift from $\mathbb H$ to $\mathbb H\ast F$, where $F$ is a free group of finite rank. As a consequence, the positive theory of an arbitrary partially commutative group is decidable.
Categories:20F10, 03C10, 20F06 |
11. CJM 2009 (vol 61 pp. 604)
First Countable Continua and Proper Forcing Assuming the Continuum Hypothesis,
there is a compact, first countable, connected space of weight $\aleph_1$
with no totally disconnected perfect subsets.
Each such space, however, may be destroyed by
some proper forcing order which does not add reals.
Keywords:connected space, Continuum Hypothesis, proper forcing, irreducible map Categories:54D05, 03E35 |
12. CJM 2008 (vol 60 pp. 88)
Nilpotent Conjugacy Classes in $p$-adic Lie Algebras: The Odd Orthogonal Case We will study the following question: Are nilpotent conjugacy
classes of reductive Lie algebras over $p$-adic fields
definable? By definable, we mean definable by a formula in Pas's
language. In this language, there are no field extensions and no
uniformisers. Using Waldspurger's parametrization, we answer in the
affirmative in the case of special orthogonal Lie algebras
$\mathfrak{so}(n)$ for $n$ odd, over $p$-adic fields.
Categories:17B10, 03C60 |
13. CJM 2007 (vol 59 pp. 575)
Cardinal Invariants of Analytic $P$-Ideals We study the cardinal invariants of analytic $P$-ideals, concentrating on the
ideal $\mathcal{Z}$ of asymptotic density zero. Among other results we prove
$ \min\{ \mathfrak{b},\cov\ (\mathcal{N})
\} \leq\cov^{\ast}(\mathcal{Z}) \leq\max\{
\mathfrak{b},\non(\mathcal{N}) \right\}.
$
Categories:03E17, 03E40 |
14. CJM 2006 (vol 58 pp. 768)
Decomposability of von Neumann Algebras and the Mazur Property of Higher Level The decomposability
number of a von Neumann algebra $\m$ (denoted by $\dec(\m)$) is the
greatest cardinality of a family of pairwise orthogonal non-zero
projections in $\m$. In this paper, we explore the close
connection between $\dec(\m)$ and the cardinal level of the Mazur
property for the predual $\m_*$ of $\m$, the study of which was
initiated by the second author. Here, our main focus is on
those von Neumann algebras whose preduals constitute such
important Banach algebras on a locally compact group $G$ as the
group algebra $\lone$, the Fourier algebra $A(G)$, the measure
algebra $M(G)$, the algebra $\luc^*$, etc. We show that for
any of these von Neumann algebras, say $\m$, the cardinal number
$\dec(\m)$ and a certain cardinal level of the Mazur property of $\m_*$
are completely encoded in the underlying group structure. In fact,
they can be expressed precisely by two dual cardinal
invariants of $G$: the compact covering number $\kg$ of $G$ and
the least cardinality $\bg$ of an open basis at the identity of
$G$. We also present an application of the Mazur property of higher
level to the topological centre problem for the Banach algebra
$\ag^{**}$.
Keywords:Mazur property, predual of a von Neumann algebra, locally compact group and its cardinal invariants, group algebra, Fourier algebra, topological centre Categories:22D05, 43A20, 43A30, 03E55, 46L10 |
15. CJM 2005 (vol 57 pp. 1139)
Models in Which Every Nonmeager Set is Nonmeager in a Nowhere Dense Cantor Set 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 $A\cap C$ is nonmeager in
$C$. We also examine variants of this result and establish a
measure theoretic analog.
Keywords:Property of Baire, Lebesgue measure,, Cantor set, oracle forcing Categories:03E35, 03E17, 03E50 |
16. CJM 2005 (vol 57 pp. 471)
Small Coverings with Smooth Functions under the Covering Property Axiom In the paper we formulate a Covering Property Axiom, \psmP,
which holds in the iterated perfect set model,
and show that it implies the following facts,
of which (a) and (b) are the generalizations
of results of J. Stepr\={a}ns.
\begin{compactenum}[\rm(a)~~]
\item There exists a family $\F$ of less than continuum many $\C^1$
functions from $\real$ to $\real$ such that $\real^2$ is covered
by functions from $\F$, in the sense that for every $\la
x,y\ra\in\real^2$ there exists an $f\in\F$ such that either
$f(x)=y$ or $f(y)=x$.
\item For every Borel function $f\colon\real\to\real$ there exists a
family $\F$ of less than continuum many ``$\C^1$'' functions ({\em
i.e.,} differentiable functions with continuous derivatives, where
derivative can be infinite) whose graphs cover the graph of $f$.
\item For every $n>0$ and
a $D^n$ function $f\colon\real\to\real$ there exists
a family $\F$ of less than continuum many $\C^n$ functions
whose graphs cover the graph of $f$.
\end{compactenum}
We also provide the examples showing that in the above properties
the smoothness conditions are the best possible. Parts (b), (c),
and the examples are closely related to work of
A. Olevski\v{\i}.
Keywords:continuous, smooth, covering Categories:26A24, 03E35 |
17. CJM 1999 (vol 51 pp. 309)
Symmetric sequence subspaces of $C(\alpha)$, II If $\alpha$ is an ordinal, then the space of all ordinals less than or
equal to $\alpha$ is a compact Hausdorff space when endowed with the
order topology. Let $C(\alpha)$ be the space of all continuous
real-valued functions defined on the ordinal interval $[0,
\alpha]$. We characterize the symmetric sequence spaces which embed
into $C(\alpha)$ for some countable ordinal $\alpha$. A hierarchy
$(E_\alpha)$ of symmetric sequence spaces is constructed so that, for
each countable ordinal $\alpha$, $E_\alpha$ embeds into
$C(\omega^{\omega^\alpha})$, but does not embed into
$C(\omega^{\omega^\beta})$ for any $\beta < \alpha$.
Categories:03E13, 03E15, 46B03, 46B45, 46E15, 54G12 |
18. CJM 1997 (vol 49 pp. 1089)
Sets on which measurable functions are determined by their range We study sets on which measurable real-valued functions on a
measurable space with negligibles are determined by their range.
Keywords:measurable function, measurable space with negligibles, continuous image, set of range uniqueness (SRU) Categories:28A20, 28A05, 54C05, 26A30, 03E35, 03E50 |
19. CJM 1997 (vol 49 pp. 641)
Fine spectra and limit laws II First-order 0--1 laws. Using Feferman-Vaught techniques a condition on the fine
spectrum of an admissible class of structures is found
which leads to a first-order 0--1 law.
The condition presented is best possible in the
sense that if it is violated then one can find an admissible
class with the same fine spectrum which does not have
a first-order 0--1 law.
If the condition is satisfied (and hence we have a first-order %% 0--1 law)
Categories:03N45, 11N45, 11N80, 05A15, 05A16, 11M41, 11P81 |