Expand all Collapse all | Results 1 - 14 of 14 |
1. CMB Online first
Limited Sets and Bibasic Sequences Bibasic sequences are used to study relative weak compactness
and relative norm compactness of limited sets.
Keywords:limited sets, $L$-sets, bibasic sequences, the Dunford-Pettis property Categories:46B20, 46B28, 28B05 |
2. CMB Online first
On Convolutions of Convex Sets and Related Problems We prove some results concerning covolutions, the
additive energy and sumsets of convex sets and its generalizations. In
particular, we show that if a set $A=\{a_1,\dots,a_n\}_\lt \subseteq
\mathbb R$ has
the property that for every fixed
$1\leqslant d\lt n,$ all differences $a_i-a_{i-d}$, $d\lt i\lt n,$ are distinct, then
$|A+A|\gg |A|^{3/2+c}$ for a constant $c\gt 0.$
Keywords:convex sets, additive energy, sumsets Category:11B99 |
3. CMB Online first
On $3$-manifolds with Torus- or Klein-bottle Category Two A subset $W$ of a closed manifold $M$ is $K$-contractible, where $K$
is a torus or Kleinbottle, if the inclusion $W\rightarrow M$ factors
homotopically through a map to $K$. The image of $\pi_1 (W)$ (for any
base point) is a subgroup of $\pi_1 (M)$ that is isomorphic to a
subgroup of a quotient group of $\pi_1 (K)$. Subsets of $M$ with this
latter property are called $\mathcal{G}_K$-contractible. We obtain a
list of the closed $3$-manifolds that can be covered by two open
$\mathcal{G}_K$-contractible subsets. This is applied to obtain a list
of the possible closed prime $3$-manifolds that can be covered by two
open $K$-contractible subsets.
Keywords:Lusternik--Schnirelmann category, coverings of $3$-manifolds by open $K$-contractible sets Categories:57N10, 55M30, 57M27, 57N16 |
4. CMB 2012 (vol 57 pp. 240)
Addendum to ``Limit Sets of Typical Homeomorphisms'' Given an integer $n \geq 3$,
a metrizable compact topological $n$-manifold $X$ with boundary,
and a finite positive Borel measure $\mu$ on $X$,
we prove that for the typical homeomorphism $f : X \to X$,
it is true that for $\mu$-almost every point $x$ in $X$ the restriction of
$f$ (respectively of $f^{-1}$) to the omega limit set $\omega(f,x)$
(respectively to the alpha limit set $\alpha(f,x)$) is topologically
conjugate to the universal odometer.
Keywords:topological manifolds, homeomorphisms, measures, Baire category, limit sets Categories:37B20, 54H20, 28C15, 54C35, 54E52 |
5. CMB 2011 (vol 55 pp. 418)
Maximal Sets of Pairwise Orthogonal Vectors in Finite Fields Given a positive integer $n$, a finite field $\mathbb{F}_q$ of $q$ elements
($q$ odd), and a non-degenerate symmetric bilinear form $B$ on
$\mathbb{F}_q^n$, we determine the largest possible cardinality of pairwise
$B$-orthogonal subsets $\mathcal{E} \subseteq \mathbb{F}_q^n$, that is, for
any two vectors $\mathbf{x}, \mathbf{y} \in \mathcal{E}$, one has $B
(\mathbf{x}, \mathbf{y}) = 0$.
Keywords:orthogonal sets, zero-distance sets Category:05B25 |
6. CMB 2011 (vol 56 pp. 354)
The Sizes of Rearrangements of Cantor Sets A linear Cantor set $C$ with zero Lebesgue measure is associated with
the countable collection of the bounded complementary open intervals. A
rearrangment of $C$ has the same lengths of its complementary
intervals, but with different locations. We study the Hausdorff and packing
$h$-measures and dimensional properties of the set of all rearrangments of
some given $C$ for general dimension functions $h$. For each set of
complementary lengths, we construct a Cantor set rearrangement which has the
maximal Hausdorff and the minimal packing $h$-premeasure, up to a constant.
We also show that if the packing measure of this Cantor set is positive,
then there is a rearrangement which has infinite packing measure.
Keywords:Hausdorff dimension, packing dimension, dimension functions, Cantor sets, cut-out set Categories:28A78, 28A80 |
7. CMB 2011 (vol 55 pp. 523)
The Milnor-Stasheff Filtration on Spaces and Generalized Cyclic Maps The concept of $C_{k}$-spaces is introduced, situated at an
intermediate stage between $H$-spaces and $T$-spaces. The
$C_{k}$-space corresponds to the $k$-th Milnor-Stasheff filtration on
spaces. It is proved that a space $X$ is a $C_{k}$-space if and only
if the Gottlieb set $G(Z,X)=[Z,X]$ for any space $Z$ with ${\rm cat}\,
Z\le k$, which generalizes the fact that $X$ is a $T$-space if and
only if $G(\Sigma B,X)=[\Sigma B,X]$ for any space $B$. Some results
on the $C_{k}$-space are generalized to the $C_{k}^{f}$-space for a
map $f\colon A \to X$. Projective spaces, lens spaces and spaces with
a few cells are studied as examples of $C_{k}$-spaces, and
non-$C_{k}$-spaces.
Keywords:Gottlieb sets for maps, L-S category, T-spaces Categories:55P45, 55P35 |
8. CMB 2011 (vol 55 pp. 487)
Weighted Model Sets and their Higher Point-Correlations Examples of distinct weighted model sets with equal $2,3,4, 5$-point
correlations are given.
Keywords:model sets, correlations, diffraction Categories:52C23, 51P05, 74E15, 60G55 |
9. CMB 2011 (vol 55 pp. 225)
Limit Sets of Typical Homeomorphisms Given an integer $n \geq 3$, a metrizable compact
topological $n$-manifold $X$ with boundary, and a finite positive Borel
measure $\mu$ on $X$, we prove that for the typical homeomorphism
$f \colon X \to X$, it is true that for $\mu$-almost every point $x$ in $X$
the limit set $\omega(f,x)$ is a Cantor set of Hausdorff dimension zero,
each point of $\omega(f,x)$ has a dense orbit in $\omega(f,x)$, $f$ is
non-sensitive at each point of $\omega(f,x)$, and the function
$a \to \omega(f,a)$ is continuous at $x$.
Keywords:topological manifolds, homeomorphisms, measures, Baire category, limit sets Categories:37B20, 54H20, 28C15, 54C35, 54E52 |
10. CMB 2007 (vol 50 pp. 579)
$p$-Radial Exceptional Sets and Conformal Mappings For $p>0$ and for a given set $E$ of type $G_{\delta}$ in the boundary
of the unit disc $\partial\mathbb D$ we construct a holomorphic function
$f\in\mathbb O(\mathbb D)$ such that
\[
\int_{\mathbb D\setminus[0,1]E}|ft|^{p}\,d\mathfrak{L}^{2}<\infty\]
and\[
E=E^{p}(f)=\Bigl\{ z\in\partial\mathbb D:\int_{0}^{1}|f(tz)|^{p}\,dt=\infty\Bigr\} .\]
In particular if a set $E$ has a measure equal to zero, then a function
$f$ is constructed as integrable with power $p$ on the unit disc $\mathbb D$.
Keywords:boundary behaviour of holomorphic functions, exceptional sets Categories:30B30, 30E25 |
11. CMB 2007 (vol 50 pp. 123)
Simultaneous Approximation and Interpolation on Arakelian Sets We extend results of P.~M. Gauthier, W. Hengartner and
A.~A. Nersesyan
on simultaneous approximation and interpolation
on Arakelian sets.
Keywords:Arakelian's theorem,, Arakelian sets Category:30E10 |
12. CMB 2006 (vol 49 pp. 256)
A Bernstein--Walsh Type Inequality and Applications A Bernstein--Walsh type inequality for $C^{\infty }$ functions of several
variables is derived, which then is applied to obtain analogs and
generalizations of the following classical theorems: (1) Bochnak--Siciak
theorem: a $C^{\infty }$\ function on $\mathbb{R}^{n}$ that is real
analytic on every line is real analytic; (2) Zorn--Lelong theorem: if a
double power series $F(x,y)$\ converges on a set of lines of positive
capacity then $F(x,y)$\ is convergent; (3) Abhyankar--Moh--Sathaye theorem:
the transfinite diameter of the convergence set of a divergent series is
zero.
Keywords:Bernstein-Walsh inequality, convergence sets, analytic functions, ultradifferentiable functions, formal power series Categories:32A05, 26E05 |
13. CMB 2005 (vol 48 pp. 580)
Exceptional Sets in Hartogs Domains Assume that $\Omega$ is a Hartogs domain in $\mathbb{C}^{1+n}$,
defined as $\Omega=\{(z,w)\in\mathbb{C}^{1+n}:|z|<\mu(w),w\in H\}$, where $H$ is an open set in
$\mathbb{C}^{n}$ and $\mu$ is a continuous function with positive values in $H$ such that $-\ln\mu$
is a strongly plurisubharmonic function in $H$. Let $\Omega_{w}=\Omega\cap(\mathbb{C}\times\{w\})$.
For a given set $E$ contained in $H$ of the type $G_{\delta}$ we construct a holomorphic function
$f\in\mathbb{O}(\Omega)$ such that
\[
E=\Bigl\{ w\in\mathbb{C}^{n}:\int_{\Omega_{w}}|f(\cdot\,,w)|^{2}\,d\mathfrak{L}^{2}=\infty\Bigr\}.
\]
Keywords:boundary behaviour of holomorphic functions,, exceptional sets Category:30B30 |
14. CMB 2002 (vol 45 pp. 483)
Diffraction of Weighted Lattice Subsets A Dirac comb of point measures in Euclidean space with bounded
complex weights that is supported on a lattice $\varGamma$ inherits
certain general properties from the lattice structure. In
particular, its autocorrelation admits a factorization into a
continuous function and the uniform lattice Dirac comb, and its
diffraction measure is periodic, with the dual lattice
$\varGamma^*$ as lattice of periods. This statement remains true
in the setting of a locally compact Abelian group whose topology
has a countable base.
Keywords:diffraction, Dirac combs, lattice subsets, homometric sets Categories:52C07, 43A25, 52C23, 43A05 |