935 research outputs found
Probabilistic and average linear widths of weighted Sobolev spaces on the ball equipped with a Gaussian measure
Let , , denotes the weighted space of
functions on the unit ball with respect to weight
, and let be the weighted
Sobolev space on with a Gaussian measure . We investigate the
probabilistic linear -widths
and the -average linear
-widths
, and obtain their asymptotic
orders for all and
Prescribed matchings extend to Hamiltonian cycles in hypercubes with faulty edges
Ruskey and Savage asked the following question: Does every matching of
for extend to a Hamiltonian cycle of ? J. Fink showed
that the question is true for every perfect matching, and solved the Kreweras'
conjecture. In this paper we consider the question in hypercubes with faulty
edges. We show that every matching of at most edges can be extended
to a Hamiltonian cycle of for . Moreover, we can prove that
when and is nonempty this result still holds even if has
at most faulty edges with one exception.Comment: 16 pages, 10 figure
Bilinear Riesz means on the Heisenberg group
In this article, we investigate the bilinear Riesz means
associated to the sublaplacian on the Heisenberg group. We prove that the
operator is bounded from into for and when is large than a suitable smoothness index .
There are some essential differences between the Euclidean space and the
Heisenberg group for studying the bilinear Riesz means problem. We make use of
some special techniques to obtain a lower index
Wiener measure for Heisenberg group
In this paper, we build Wiener measure for the path space on the Heisenberg
group by using of the heat kernel corresponding to the sub-Laplacian and give
the definition of the Wiener integral. Then we give the Feynman-Kac formula.Comment: 14 page
Positive Cubature formulas and Marcinkiewicz-Zygmund inequalities on spherical caps
Let denote the space of all spherical polynomials of degree at most
on the unit sphere \sph of , and let denote
the usual geodesic distance between x, y\in \sph. Given a
spherical cap B(e,\al)=\{x\in\sph: d(x, e) \leq \al\}, (e\in\sph,
\text{$\al\in (0,\pi)$ is bounded away from $\pi$}), we define the metric
\rho(x,y):=\frac 1{\al} \sqrt{(d(x, y))^2+\al(\sqrt{\al-d(x,
e)}-\sqrt{\al-d(y,e)})^2},
where x, y\in B(e,\al). It is shown that given any \be\ge 1, and any finite subset \Ld of B(e,\al) satisfying the condition
\dmin_{\sub{\xi,\eta \in \Ld \xi\neq \eta}} \rho (\xi,\eta) \ge \f \da n with
\da\in (0,1], there exists a positive constant , independent of \al,
, \Ld and \da, such that, for any , \begin{equation*}
\sum_{\og\in \Ld} (\max_{x,y\in B_\rho (\og, \be\da/n)}|f(x)-f(y)|^p)
|B_\rho(\og, \da/n)| \le (C \dz)^p \int_{B(e,\al)} |f(x)|^p
d\sa(x),\end{equation*} where d\sa(x) denotes the usual Lebesgue measure on
\sph, B_\rho(x, r)=\Bl\{y\in B(e,\al): \rho(y,x)\leq r\Br\}, (r>0), and
\Bl|B_\rho(x, \f\da n)\Br|=\int_{B_{\rho}(x, \da/n)} d\sa(y) \sim \al
^{d}\Bl[ (\f{\da}n)^{d+1}+ (\f\da n)^{d} \sqrt{1-\f{d(x, e)}\al}\Br]. As a
consequence, we establish positive cubature formulas and Marcinkiewicz-Zygmund
inequalities on the spherical cap B(e,\al)
The intrinsic square function characterizations of weighted Hardy spaces
In this paper, we will study the boundedness of intrinsic square functions on
the weighted Hardy spaces for , where is a Muckenhoupt's
weight function. We will also give some intrinsic square function
characterizations of weighted Hardy spaces for .Comment: 17 page
Tractability of non-homogeneous tensor product problems in the worst case setting
We study multivariate linear tensor product problems with some special
properties in the worst case setting. We consider algorithms that use finitely
many continuous linear functionals. We use a unified method to investigate
tractability of the above multivariate problems, and obtain necessary and
sufficient conditions for strong polynomial tractability, polynomial
tractability, quasi-polynomial tractability, uniformly weak tractability,
-weak tractability, and weak tractability. Our results can apply to
multivariate approximation problems with kernels corresponding to Euler
kernels, Wiener kernels, Korobov kernels, Gaussian kernels, and analytic
Korobov kernels.Comment: 23 page
Boundedness of the bilinear Bochner-Riesz Means in the non-Banach triangle case
In this article, we investigate the boundedness of the bilinear Bochner-Riesz
means in the non-Banach triangle case. We improve the
corresponding results in [Bern] in two aspects: Our partition of the non-Banach
triangle is simpler and we obtain lower smoothness indices for various cases apart from
On filtered polynomial approximation on the sphere
This paper considers filtered polynomial approximations on the unit sphere
, obtained by truncating smoothly the
Fourier series of an integrable function with the help of a "filter" ,
which is a real-valued continuous function on such that
for and for . The resulting "filtered polynomial
approximation" (a spherical polynomial of degree ) is then made fully
discrete by approximating the inner product integrals by an -point cubature
rule of suitably high polynomial degree of precision, giving an approximation
called "filtered hyperinterpolation". In this paper we require that the filter
and all its derivatives up to are absolutely
continuous, while its right and left derivatives of order exist everywhere and are of bounded variation. Under
this assumption we show that for a function in the Sobolev space
, both approximations are of the
optimal order , in the first case for and in the second fully
discrete case for
Cayley Configuration Spaces of 1-dof Tree-decomposable Linkages, Part II: Combinatorial Characterization of Complexity
We continue to study Cayley configuration spaces of 1-dof linkages in 2D
begun in Part I of this paper, i.e. the set of attainable lengths for a
non-edge. In Part II, we focus on the algebraic complexity of describing
endpoints of the intervals in the set, i.e., the Cayley complexity.
Specifically, We focus on Cayley configuration spaces of a natural class of
1-dof linkages, called 1-dof tree-decomposable linkages. The underlying graphs
G satisfy the following: for some base non-edge f, G \cup f is
quadratic-radically solvable (QRS), meaning that G \cup f is minimally rigid,
and given lengths \bar{l} of all edges, the corresponding linkage (G \cup f,
\bar{l}) can be simply realized by ruler and compass starting from f. It is
clear that the Cayley complexity only depends on the graph G and possibly the
non-edge f. Here we ask whether the Cayley complexity depends on the choice of
a base non-edge f. We answer this question in the negative, thereby showing
that low Cayley complexity is a property of the graph G (independent of the
non-edge f).
Then, we give a simple characterization of graphs with low Cayley complexity,
leading to an efficient algorithmic characterization, i.e. an efficient
algorithm for recognizing such graphs.
Next, we show a surprising result that (graph) planarity is equivalent to low
Cayley complexity for a natural subclass of 1-dof triangle-decomposable graphs.
While this is a finite forbidden minor graph characterization of low Cayley
complexity, we provide counterexamples showing impossibility of such finite
forbidden minor characterizations when the above subclass is enlarged
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