Preprint Series 2005

2005:18
F. Chung and
L. Lu
We examine a number of generalized and extended versions of concentration inequalities and martingale inequalities. These inequalities are effective for analyzing processes with quite general conditions as illustrated in an example for an infinite Polya process and webgraphs.
[Full Abstract] 
2005:17
V. Temlyakov
This paper addresses a problem of constructing and analysing estimators for the regression problem in supervised learning. Recently, there was a big interest in studying universal estimators. Universal means that the estimator does not depend on an a priori assumption that the regression function $f _ \rho$ belongs to some ...
[Full Abstract] 
2005:16
M. Gunzburger,
L. Ju, and
W. Zhao
A new triangular mesh adaptivity algorithm for elliptic PDEs that combines a posteriori error estimation with centroidal Voronoi/Delaunay tessellations of domains in two dimensions is proposed and tested. The ability of the first ingredient to detect local regions of large error and the ability of the second ingredient to ...
[Full Abstract] 
2005:15
G. Kerkyacharian,
D. Picard, and
V. Temlyakov
In this paper we study properties of bases that are important in nonlinear mterm approximation with regard to these bases. It is known that the univariate Haar basis is a greedy basis for $L _ p([0,1))$ , $1<p<\infty$. This means that a greedy type algorithm realizes nearly ...
[Full Abstract] 
2005:14
J. Griggs and
T. Jin
The problem of radio channel assignments with multiple levels of interference depending on distance can be modeled using graph theory. The authors previously introduced a model of labeling by real numbers. Given a graph G, possibly infinite, and real numbers $k _ 1,k _ 2,\ldots,$$k _ p ...
[Full Abstract] 
2005:13
J. Griggs and
T. Jin
Numerical channels must be assigned to each transmitter in a large regular array such that multiple levels of interference, which depend on the distance between transmitters, are avoided by sufficiently separating the channels. The goal is to find assignments that minimize the span of the labels used. Our previous paper ...
[Full Abstract] 
2005:12
F. Narcowich,
P. Petrushev, and
L. Ward
A discrete system of almost exponentially localized elements (needlets) on the ndimensional unit sphere S^{n} is constructed. It shown that the needlet system can be used for decomposition of Besov and TriebelLizorkin spaces on the sphere. As an application of Besov spaces on S^{n}, a Jackson estimate ...
[Full Abstract] 
2005:11
B. Popov and
O. Trifonov
Nonoscillatory schemes are widely used in numerical approximations of nonlinear conservation laws. The NessyahuTadmor (NT) scheme is an example of a second order scheme that is both robust and simple. In this paper, we prove a new stability property of the NT scheme based on the standard minmod reconstruction in ...
[Full Abstract] 
2005:10
P. Petrushev and
Y. Xu
As is well known the kernel of the orthogonal projector onto the polynomials of degree n in $L^2(w _ {\alpha,\beta},[1,1])$ with $w _ {\alpha,\beta}(t)=(1t)^\alpha(1+t)^\beta$ can be written in terms of Jacobi polynomials. It is shown that if the ...
[Full Abstract] 
2005:09
M. Steel and
L. Székely
A widelystudied model for generating sequences is to 'evolve' them on a tree according to a symmetric Markov process. We prove that model trees tend to be maximally "far apart" in terms of variational distance. Yet, almost paradoxically, tree reconstruction is successful for sequences that are short enough that the ...
[Full Abstract] 
2005:08
M. Steel and
L. Székely
A widelystudied model for generating binary sequences is to 'evolve' them on a tree according to a symmetric Markov process. We show that under this model distinguishing the true (model) tree from a false one is substantially "easier" (in terms of the sequence length needed) than determining the true tree ...
[Full Abstract] 
2005:07
F. Chung and
L. Lu
We consider the random graph model $G(w)$ for a given expected degree sequence $w=(w _ 1,w _ 2,\ldots,$$w _ n)$. If the expected average degree is strictly greater than 1, then almost surely the giant component in G of $G(w)$ has volume (i.e ...
[Full Abstract] 
2005:06
Q. Du and
L. Ju
In this paper, we study the finite volume method for numerical solution of a set of model partial differential equations defined on a smooth surface. The discretization is defined via a surface mesh consisting of piecewise planar triangles and piecewise polygons. We prove the optimal error estimates of the approximate ...
[Full Abstract] 
2005:05
V. Temlyakov
This paper addresses some problems of supervised learning in the setting formulated by Cucker and Smale. Supervised learning, or learningfromexamples, refers to a process that builds on the base of available data of inputs $x _ i$ and outputs $y _ i,i=1,\ldots,m$, a function that best ...
[Full Abstract] 
2005:04
F. Narcowich,
P. Petrushev, and
J. Ward
In this paper we wish to present a new class of tight frames on the sphere. These frames have excellent pointwise localization and approximation properties. These properties are based on pointwise localization of kernels arising in the spectral calculus for certain pseudodifferential operators, and on a positiveweight quadrature formula for ...
[Full Abstract] 
2005:03
K. Oskolkov
We study the fractal properties of the timedependent probability density function for the free quantum particle in a box, i. e. the squared magnitude of the solution of the Cauchy initial value problem for the Schrödinger equation with zero potential, and the periodic initial data. √(δ)families of the initial ...
[Full Abstract] 
2005:02
P. Petrushev
This article is a survey of some new (and old) results in nonlinear nterm spline approximation and related topics. A substantial part of the paper is devoted to the anisotropic spaces generated by sequences of nested triangulations of compact polygonal domains in R^{2} and their characterization via corresponding Franklin ...
[Full Abstract] 
2005:01
H. Wang and
G. Yu
The Wiener index is one of the main descriptors that correlate a chemical compound’s molecular graph with experimentally gathered data regarding the compound’s characteristics. A long standing conjecture on the Wiener index ([4], [5]) states that for any positive integer n (except numbers from a given 49 element ...
[Full Abstract]