Expand all Collapse all | Results 1 - 13 of 13 |
1. CMB 2013 (vol 57 pp. 364)
How Lipschitz Functions Characterize the Underlying Metric Spaces Let $X, Y$ be metric spaces and $E, F$ be Banach spaces. Suppose that
both $X,Y$ are realcompact, or both $E,F$ are realcompact.
The zero set of a vector-valued function $f$ is denoted by $z(f)$.
A linear bijection $T$ between local or generalized Lipschitz vector-valued function spaces
is said to preserve zero-set containments or nonvanishing functions
if
\[z(f)\subseteq z(g)\quad\Longleftrightarrow\quad z(Tf)\subseteq z(Tg),\]
or
\[z(f) = \emptyset\quad \Longleftrightarrow\quad z(Tf)=\emptyset,\]
respectively.
Every zero-set containment preserver, and every nonvanishing function preserver when
$\dim E =\dim F\lt +\infty$, is a weighted composition operator
$(Tf)(y)=J_y(f(\tau(y)))$.
We show that the map $\tau\colon Y\to X$ is a locally (little) Lipschitz homeomorphism.
Keywords:(generalized, locally, little) Lipschitz functions, zero-set containment preservers, biseparating maps Categories:46E40, 54D60, 46E15 |
2. CMB 2011 (vol 56 pp. 55)
Cliquishness and Quasicontinuity of Two-Variable Maps We study the existence of continuity points for mappings
$f\colon X\times Y\to Z$ whose $x$-sections $Y\ni y\to f(x,y)\in Z$ are
fragmentable and $y$-sections $X\ni x\to f(x,y)\in Z$ are
quasicontinuous, where $X$ is a Baire space and $Z$
is a metric space. For the factor $Y$, we consider two
infinite ``point-picking'' games $G_1(y)$ and $G_2(y)$ defined respectively
for each $y\in Y$ as follows: in the $n$-th
inning, Player I gives a dense set $D_n\subset Y$, respectively, a dense open set $D_n\subset Y$. Then
Player II picks a point $y_n\in D_n$;
II wins if $y$ is in the closure of ${\{y_n:n\in\mathbb N\}}$, otherwise
I wins. It is shown that
(i) $f$ is
cliquish
if II has a winning strategy in $G_1(y)$ for every $y\in Y$, and (ii) $
f$ is quasicontinuous if
the $x$-sections of $f$ are continuous and the set of $y\in Y$
such that II has a winning strategy in $G_2(y)$ is dense in $Y$. Item (i) extends substantially
a result of Debs and item (ii) indicates that
the problem of Talagrand on separately continuous maps has a positive answer for a wide
class of ``small'' compact spaces.
Keywords:cliquishness, fragmentability, joint continuity, point-picking game, quasicontinuity, separate continuity, two variable maps Categories:54C05, 54C08, 54B10, 91A05 |
3. 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 |
4. CMB 2011 (vol 54 pp. 607)
Lightness of Induced Maps and Homeomorphisms An example is given of a map $f$ defined between arcwise connected continua such that $C(f)$ is light and
$2^{f}$ is not light, giving a negative answer to a question of Charatonik and Charatonik. Furthermore, given a positive
integer $n$, we study when the lightness of the induced map $2^{f}$ or $C_n(f)$ implies that $f$ is a
homeomorphism. Finally, we show a result in relation with the lightness of $C(C(f))$.
Keywords:light maps, induced maps, continua, hyperspaces Categories:54B20, 54E40 |
5. CMB 2010 (vol 53 pp. 466)
Separating Maps between Spaces of Vector-Valued Absolutely Continuous Functions In this paper we give a description of separating or disjointness preserving linear bijections on spaces of vector-valued absolutely continuous functions defined on compact subsets of the real line. We obtain that they are continuous and biseparating in the finite-dimensional case. The infinite-dimensional case is also studied.
Keywords:separating maps, disjointness preserving, vector-valued absolutely continuous functions, automatic continuity Categories:47B38, 46E15, 46E40, 46H40, 47B33 |
6. CMB 2009 (vol 52 pp. 18)
Harmonicity of Holomorphic Maps Between Almost Hermitian Manifolds In this paper we study holomorphic maps between almost Hermitian
manifolds. We obtain a new criterion for the harmonicity of such
holomorphic maps, and we deduce some applications to horizontally
conformal holomorphic submersions.
Keywords:almost Hermitian manifolds, harmonic maps, harmonic morphism Categories:53C15, 58E20 |
7. CMB 2008 (vol 51 pp. 448)
Stability of Biharmonic Legendrian Submanifolds in Sasakian Space Forms Biharmonic maps are defined as critical points of the bienergy.
Every harmonic map is a stable biharmonic map.
In this article, the stability of nonharmonic
biharmonic Legendrian submanifolds in Sasakian space forms is discussed.
Keywords:biharmonic maps, Sasakian manifolds, Legendrian submanifolds Categories:53C42, 53C40 |
8. CMB 2005 (vol 48 pp. 614)
On Finite-to-One Maps Let $f\colon X\to Y$ be a $\sigma$-perfect $k$-dimensional surjective
map of metrizable spaces such that $\dim Y\leq m$. It is shown that
for every positive integer $p$ with $ p\leq m+k+1$ there exists a
dense $G_{\delta}$-subset ${\mathcal H}(k,m,p)$ of $C(X,\uin^{k+p})$
with the source limitation topology such that each fiber of
$f\triangle g$, $g\in{\mathcal H}(k,m,p)$, contains at most
$\max\{k+m-p+2,1\}$ points. This result
provides a proof the following conjectures of
S. Bogatyi, V. Fedorchuk and J. van Mill.
Let $f\colon X\to Y$ be a $k$-dimensional map between compact
metric spaces with $\dim Y\leq m$. Then:
\begin{inparaenum}[\rm(1)]
\item there exists a map
$h\colon X\to\uin^{m+2k}$ such that $f\triangle h\colon X\to
Y\times\uin^{m+2k}$ is 2-to-one provided $k\geq 1$;
\item there exists a
map $h\colon X\to\uin^{m+k+1}$ such that $f\triangle h\colon X\to
Y\times\uin^{m+k+1}$ is $(k+1)$-to-one.
\end{inparaenum}
Keywords:finite-to-one maps, dimension, set-valued maps Categories:54F45, 55M10, 54C65 |
9. CMB 2002 (vol 45 pp. 97)
Invariant Measures and Natural Extensions We study ergodic properties of a family of interval maps that are
given as the fractional parts of certain real M\"obius
transformations. Included are the maps that are exactly
$n$-to-$1$, the classical Gauss map and the Renyi or backward
continued fraction map. A new approach is presented for deriving
explicit realizations of natural automorphic extensions and their
invariant measures.
Keywords:Continued fractions, interval maps, invariant measures Categories:11J70, 58F11, 58F03 |
10. CMB 2001 (vol 44 pp. 266)
Extension of Maps to Nilpotent Spaces We show that every compactum has cohomological dimension $1$ with respect
to a finitely generated nilpotent group $G$ whenever it has cohomological
dimension $1$ with respect to the abelianization of $G$. This is applied
to the extension theory to obtain a cohomological dimension theory condition
for a finite-dimensional compactum $X$ for extendability of every map from
a closed subset of $X$ into a nilpotent $\CW$-complex $M$ with finitely
generated homotopy groups over all of $X$.
Keywords:cohomological dimension, extension of maps, nilpotent group, nilpotent space Categories:55M10, 55S36, 54C20, 54F45 |
11. CMB 2000 (vol 43 pp. 294)
Fixed Points of Commuting Holomorphic Maps Without Boundary Regularity We identify a class of domains of $\C^n$ in which any two commuting
holomorphic self-maps have a common fixed point.
Keywords:Holomorphic self-maps, commuting functions, fixed points, Wolff point, Julia's Lemma Categories:32A10, 32A40, 32H15, 32A30 |
12. CMB 1999 (vol 42 pp. 274)
The Bockstein Map is Necessary We construct two non-isomorphic nuclear, stably finite,
real rank zero $C^\ast$-algebras $E$ and $E'$ for which
there is an isomorphism of ordered groups
$\Theta\colon \bigoplus_{n \ge 0} K_\bullet(E;\ZZ/n) \to
\bigoplus_{n \ge 0} K_\bullet(E';\ZZ/n)$ which is compatible
with all the coefficient transformations. The $C^\ast$-algebras
$E$ and $E'$ are not isomorphic since there is no $\Theta$
as above which is also compatible with the Bockstein operations.
By tensoring with Cuntz's algebra $\OO_\infty$ one obtains a pair
of non-isomorphic, real rank zero, purely infinite $C^\ast$-algebras
with similar properties.
Keywords:$K$-theory, torsion coefficients, natural transformations, Bockstein maps, $C^\ast$-algebras, real rank zero, purely infinite, classification Categories:46L35, 46L80, 19K14 |
13. CMB 1998 (vol 41 pp. 442)
A Mountain Pass to the Jacobian Conjecture. This paper presents an approach to injectivity theorems via the
Mountain Pass Lemma and raises an open question. The main result
of this paper (Theorem~1.1) is proved by means of the Mountain Pass
Lemma and states that if the eigenvalues of $F' (\x)F' (\x)^{T}$
are uniformly bounded away from zero for $\x \in \hbox{\Bbbvii
R}^{n}$, where $F \colon \hbox{\Bbbvii R}^n \rightarrow
\hbox{\Bbbvii R}^n$ is a class $\cC^{1}$ map, then $F$ is
injective. This was discovered in a joint attempt by the authors
to prove a stronger result conjectured by the first author: Namely,
that a sufficient condition for injectivity of class $\cC^{1}$ maps
$F$ of $\hbox{\Bbbvii R}^n$ into itself is that all the eigenvalues
of $F'(\x)$ are bounded away from zero on $\hbox{\Bbbvii
R}^n$. This is stated as Conjecture~2.1. If true, it would imply
(via {\it Reduction-of-Degree}) {\it injectivity of polynomial
maps} $F \colon \hbox{\Bbbvii R}^n \rightarrow \hbox{\Bbbvii R}^n$
{\it satisfying the hypothesis}, $\det F'(\x) \equiv 1$, of the
celebrated Jacobian Conjecture (JC) of Ott-Heinrich Keller. The
paper ends with several examples to illustrate a variety of cases
and known counterexamples to some natural questions.
Keywords:Injectivity, ${\cal C}^1$-maps, polynomial maps, Jacobian Conjecture, Mountain Pass Categories:14A25, 14E09 |