456 research outputs found
The square negative correlation property for generalized Orlicz balls
Antilla, Ball and Perissinaki proved that the squares of coordinate functions
in are negatively correlated. This paper extends their results to
balls in generalized Orlicz norms on R^n. From this, the concentration of the
Euclidean norm and a form of the Central Limit Theorem for the generalized
Orlicz balls is deduced. Also, a counterexample for the square negative
correlation hypothesis for 1-symmetric bodies is given.
Currently the CLT is known in full generality for convex bodies (see the
paper "Power-law estimates for the central limit theorem for convex sets" by B.
Klartag), while for generalized Orlicz balls a much more general result is true
(see "The negative association property for the absolute values of random
variables equidistributed on a generalized Orlicz ball" by M. Pilipczuk and J.
O. Wojtaszczyk). While, however, both aforementioned papers are rather long,
complicated and technical, this paper gives a simple and elementary proof of,
eg., the Euclidean concentration for generalized Orlicz balls.Comment: 10 page
Multivariate integration in C^\infty([0,1]^d) is not strongly tractable
It has long been known that the multivariate integration problem for the unit
ball in is intractable for fixed finite . H. Wo\'zniakowski
has recently conjectured that this is true even if . This paper
establishes a partial result in this direction. We prove that the multivariate
integration problem, for infinitely differential functions all of whose
variables are bounded by one, is not strongly tractable.Comment: 6 page
A simpler proof of the negative association property for absolute values of measures tied to generalized Orlicz balls
Negative association for a family of random variables means that for
any coordinate--wise increasing functions we have \E
f(X_{i_1},...,X_{i_k}) g(X_{j_1},...,X_{j_l}) \leq \E f(X_{i_1},...,X_{i_k}) \E
g(X_{j_1},...,X_{j_l}) for any disjoint sets of indices , . It
is a way to indicate the negative correlation in a family of random variables.
It was first introduced in 1980s in statistics, and brought to convex geometry
in 2005 to prove the Central Limit Theorem for Orlicz balls.
The paper gives a relatively simple proof of negative association of absolute
values for a wide class of measures tied to generalized Orlicz balls, including
the uniform measures on generalized Orlicz balls.Comment: 16 page
Conditional quasi-greedy bases in Hilbert and Banach spaces
We show that, for quasi-greedy bases in Hilbert spaces, the associated
conditionality constants grow at most as , for some
, answering a question by Temlyakov. We show the optimality of this
bound with an explicit construction, based on a refinement of the method of
Olevskii. This construction leads to other examples of quasi-greedy bases with
large in Banach spaces, which are of independent interest
On nonatomic Banach lattices and Hardy spaces
We are interested in the question when a Banach space with an
unconditional basis is isomorphic (as a Banach space) to an order-continuous
nonatomic Banach lattice. We show that this is the case if and only if is
isomorphic as a Banach space with . This and results of J. Bourgain
are used to show that spaces are not isomorphic to nonatomic
Banach lattices. We also show that tent spaces introduced in \cite{4} are
isomorphic to Rad
The negative association property for the absolute values of random variables equidistributed on a generalized Orlicz ball
Random variables equidistributed on convex bodies have received quite a lot
of attention in the last few years. In this paper we prove the negative
association property (which generalizes the subindependence of coordinate
slabs) for generalized Orlicz balls. This allows us to give a strong
concentration property, along with a few moment comparison inequalities. Also,
the theory of negatively associated variables is being developed in its own
right, which allows us to hope more results will be available. Moreover, a
simpler proof of a more general result for balls is given.Comment: 44 pages (sorry
Data Assimilation and Sampling in Banach spaces
This paper studies the problem of approximating a function in a Banach
space from measurements , , where the are linear
functionals from . Most results study this problem for classical Banach
spaces such as the spaces, , and for the unit
ball of a smoothness space in . Our interest in this paper is in the model
classes , with and a finite dimensional
subspace of , which consists of all such that . These model classes, called {\it approximation sets}, arise
naturally in application domains such as parametric partial differential
equations, uncertainty quantification, and signal processing.
A general theory for the recovery of approximation sets in a Banach space is
given. This theory includes tight a priori bounds on optimal performance, and
algorithms for finding near optimal approximations. We show how the recovery
problem for approximation sets is connected with well-studied concepts in
Banach space theory such as liftings and the angle between spaces. Examples are
given that show how this theory can be used to recover several recent results
on sampling and data assimilation
On the structure of Lipschitz-free spaces
In this note we study the structure of Lipschitz-free Banach spaces. We show
that every Lipschitz-free Banach space over an infinite metric space contains a
complemented copy of . This result has many consequences for the
structure of Lipschitz-free Banach spaces. Moreover, we give an example of a
countable compact metric space such that is not isomorphic to a
subspace of and we show that whenever is a subset of , then
is weakly sequentially complete; in particular, does not embed
into .Comment: The only change in the latest version is the grant information of the
second named author. Previous version contained a false proof of Theorem 1.
This is corrected now. We have also added some remarks and changed Question
1, because we observed that the answer to the previous question is negative
due to a result of P. L. Kaufmann. The paper has been accepted in Proc. Amer.
Math. So
Beyond O*(2^n) in domination-type problems
In this paper we provide algorithms faster than O*(2^n) for several
NP-complete domination-type problems. More precisely, we provide: an algorithm
for CAPACITATED DOMINATING SET that solves it in O(1.89^n), a branch-and-reduce
algorithm solving LARGEST IRREDUNDANT SET in O(1.9657^n) time and a simple
iterative-DFS algorithm for SMALLEST INCLUSION-MAXIMAL IRREDUNDANT SET that
solves it in O(1.999956^n) time.
We also provide an exponential approximation scheme for CAPACITATED
DOMINATING SET. All algorithms require polynomial space. Despite the fact that
the discussed problems are quite similar to the DOMINATING SET problem, we are
not aware of any published algorithms solving these problems faster than the
obvious O*(2^n) solution prior to this paper.Comment: Submitted to STACS 201
On left democracy function
We continue the study undertaken in \cite{GHN} of left democracy function
h_l(N)=\inf_{#\Lambda=N}\left\|\sum_{n\in \Lambda_N} x_n\right\| of an
unconditional basis in a Banach space . We provide an example of a basis
with non-doubling. Then we show that for bases with non-doubling
the greedy projection is not optimal. Together with results from \cite{GHN} and
improved by C. Cabrelli, G. Garrig\'os, E. Hernandez and U. Molter we get that
the basis is greedy if and only if the greedy projection is optimal
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