Expand all Collapse all | Results 1 - 7 of 7 |
1. CMB 2012 (vol 57 pp. 178)
Quasiconvexity and Density Topology We prove that if $f:\mathbb{R}^{N}\rightarrow \overline{\mathbb{R}}$ is
quasiconvex and $U\subset \mathbb{R}^{N}$ is open in the density topology, then
$\sup_{U}f=\operatorname{ess\,sup}_{U}f,$ while
$\inf_{U}f=\operatorname{ess\,inf}_{U}f$
if and only if the equality holds when $U=\mathbb{R}^{N}.$ The first (second)
property is typical of lsc (usc) functions and, even when $U$ is an ordinary
open subset, there seems to be no record that they both hold for all
quasiconvex functions.
This property ensures that the pointwise extrema of $f$ on any nonempty
density open subset can be arbitrarily closely approximated by values of $f$
achieved on ``large'' subsets, which may be of relevance in a variety of
issues. To support this claim, we use it to characterize the common points
of continuity, or approximate continuity, of two quasiconvex functions that
coincide away from a set of measure zero.
Keywords:density topology, quasiconvex function, approximate continuity, point of continuity Categories:52A41, 26B05 |
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 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 |
4. CMB 2009 (vol 52 pp. 544)
Intuitionistic Fuzzy $\gamma$-Continuity This paper introduces the concepts of
fuzzy $\gamma$-open sets and fuzzy $\gamma$-continuity
in intuitionistic fuzzy topological spaces. After defining the fundamental
concepts of intuitionistic fuzzy sets and intuitionistic fuzzy topological
spaces, we present intuitionistic fuzzy $\gamma$-open sets and
intuitionistic fuzzy $\gamma$-continuity and other results related
topological concepts.
Keywords:intuitionistic fuzzy set, intuitionistic fuzzy point, intuitionistic fuzzy topological space, intuitionistic fuzzy $\gamma$-open set, intuitionistic fuzzy $\gamma$-\continuity, intuitionistic fuzzy $\gamma$-closure ($\gamma$-interior) Categories:54A40, 54A20, 54F99 |
5. CMB 2007 (vol 50 pp. 434)
MKZ Type Operators Providing a Better Estimation on $[1/2,1)$ In the present paper, we introduce a modification of the Meyer-K\"{o}nig and
Zeller (MKZ) operators which preserve the test functions $f_{0}(x)=1$ and
$f_{2}(x)=x^{2}$, and we show that this modification provides a better estimation
than the classical MKZ operators on the interval $[\frac{1}{2},1)$ with
respect to the modulus of continuity and the Lipschitz class functionals.
Furthermore, we present the $r-$th order generalization of our operators and
study their approximation properties.
Keywords:Meyer-KÃ¶nig and Zeller operators, Korovkin type approximation theorem, modulus of continuity, Lipschitz class functionals Categories:41A25, 41A36 |
6. CMB 1999 (vol 42 pp. 118)
Points of Weak$^\ast$-Norm Continuity in the Unit Ball of the Space $\WC(K,X)^\ast$ For a compact Hausdorff space with a dense set of isolated points, we
give a complete description of points of weak$^\ast$-norm continuity
in the dual unit ball of the space of Banach space valued functions
that are continuous when the range has the weak topology. As an
application we give a complete description of points of weak-norm
continuity of the unit ball of the space of vector measures when
the underlying Banach space has the Radon-Nikodym property.
Keywords:Points of weak$^\ast$-norm continuity, space of vector valued weakly continuous functions, $M$-ideals Categories:46B20, 46E40 |
7. CMB 1998 (vol 41 pp. 49)
Stability of weighted darma filters We study the stability of linear filters associated with certain types of
linear difference equations with variable coefficients. We show that
stability is determined by the locations of the poles of a rational transfer
function relative to the spectrum of an associated weighted shift operator.
The known theory for filters associated with constant-coefficient difference
equations is a special case.
Keywords:Difference equations, adaptive $\DARMA$ filters, weighted shifts,, stability and boundedness, automatic continuity Categories:47A62, 47B37, 93D25, 42A85, 47N70 |