Preprint Series 2003

2003:26
D. Leviatan and
V. Temlyakov
We study nonlinear mterm approximation with regard to a redundant dictionary D in a Banach space. It is known that in the case of Hilbert space H the Pure Greedy Algorithm (or, more generally, the Weak Greedy Algorithm) provides for each $f\in{H}$ and any dictionary D an expansion ...
[Full Abstract] 
2003:25
M. CamposPinto,
A. Cohen, and
P. Petrushev
The smoothness of the solutions of 1D scalar conservation laws is investigated and it is shown that if the initial value has smoothness of order $\alpha$ in $L^q$ with $\alpha>1$ and $q=\frac{1}{\alpha}$, this smoothness is preserved at any time $t>0$ for the graph of ...
[Full Abstract] 
2003:24
L. Székely and
H. Wang
This paper characterizes binary trees with n leaves, which have the greatest number of subtrees. These binary trees coincide with those which we shown by Fischermann et al. [2] and Jelen and Triesch [3] to minimize the Wiener index.
[Full Abstract] 
2003:23
S. Brenner,
K. Wang, and
J. ZhaoPoincaréFriedrichs inequalities for piecewise H^{ 2} functions (file not available)
(Numerical Functional Analysis and Optimization 25 (2004), 463478)
PoincaréFriedrichs inequalities are derived for piecewise H^{2} functions on two dimensional domains. These inequalities can be applied to classical nonconforming finite element methods, mortar methods and discontinuous Galerkin methods.
[Full Abstract] 
2003:22
J. Griggs,
C. Killian, and
C. Savage
We show that symmetric Venn diagrams for n sets exist for every prime n, settling an open question. Until this time, $n=11$ was the largest prime for which the existence of such diagrams had been proven, a result of Peter Hamburger. We show that the problem can be reduced ...
[Full Abstract] 
2003:21
B. Popov and
O. Trifonov
A class of nonoscillatory numerical methods for solving nonlinear scalar conservation laws in one space dimension is considered. Nonoscillatory schemes are based on minmod limiters and the standard second order representatives are the staggered NessyahuTadmor scheme and the usual TVD2 scheme. It is well known that the $L _ p ...
[Full Abstract] 
2003:20
R. Kozrev
Greedy algorithms in ridge approximation (gridge algorithms) are considered. Functions from the Gaussian weighted Hilbert space L^{2} are approximated by linear combinations of ridge functions. The construction is iterative. On each step one more ridge function is added to the preceeding combination. This ridge function is selected greedily from ...
[Full Abstract] 
2003:19
J. Zhao
Multigrid Vcycle and Fcycle algorithms for the biharmonic problem using the HCT element are studied in the paper. We show that the contraction numbers can be uniformly improved by increasing the number of smoothing steps.
[Full Abstract] 
2003:18
P. Bechler,
R. DeVore,
A. Kamont,
G. Petrova, and
P. Wojtaszczyk
Let $BV=BV(IR^d)$ be the space of functions of bounded variation on $IR^d$ with $d\geq2$. Let $\psi _ \lambda,$$\lambda\in\Delta,$ be a wavelet basis of compactly supported functions normalized in $BV,$ i.e. $\psi _ \lambda _ {BV(IR^d)}=1,$$\lambda\in\Delta ...
[Full Abstract] 
2003:17
B. Kashin and
V. Temlyakov
We prove new estimates for the entropy numbers of classes of multivariate functions with bounded mixed derivative. It is known that the investigation of these classes requires development of new techniques comparing to the univariate classes. In this paper we continue to develop the technique based on estimates of volumes ...
[Full Abstract] 
2003:16
K. Oskolkov
The double trigonometric series with the hyperbolic phase
$$U(x):=\sum^\infty _ {m=1}\sum^\infty _ {n=1}\frac{e^{2\pi{mnx}}}{\pi{mn}}$$
is studied. Complete descriptions of the MHOconvergence (summability) sets of the sinseries $FU(x),$ and the cosseries $RU(x)$ are established. The MHOsum ...
[Full Abstract] 
2003:15
L. Székely
Pach, Spencer, and Toth showed that for a simple graph on n vertices and e edges, if $e>4n,$ and the girth of the graph exceeds 2r ($r > 0$ integer), then $cr(G)\geq{c _ r\frac{e^{r+2}}{n^{r+1}}}$. We give a simple new ...
[Full Abstract] 
2003:14
F. Shahrokhi,
O. Sykora,
L. Székely, and
I. Vrto
A convex drawing of an nvertex graph $G=(V(G),E(G))$ is a drawing in which the vertices are placed on the corners of a convex ngon in the plane and each edge is drawn using one straight line segment. We derive a general lower bound on ...
[Full Abstract] 
2003:13
R. Kellogg and
M. Stynes
A singularly perturbed convectiondiffusion problem posed on the unit square is considered. Its solution may have exponential and parabolic boundary layers, and corner singularities may also be present. Pointwise bounds on the solution and its derivatives are derived. The dependence of these bounds on the small diffusion coefficient, on the ...
[Full Abstract] 
2003:12
S. BrennerDiscrete Sobolev and Poincaré inequalities for piecewise polynomial functions (file not available)
(Electronic Transactions on Numerical Analysis 18 (2004), 4248)
Discrete Sobolev and Poincaré inequalities are derived for piecewise polynomial functions on two dimensional domains. These inequalities can be applied to classical nonconforming finite element methods and discontinuous Galerkin methods.
[Full Abstract] 
2003:11
K. Oskolkov
The probability density function is studied for the onedimensional quantum particle whose motion is defined by the Schrodinger equation
$$\frac{\delta\psi}{\delta{t}}=\frac{1}{2\pi{i}}\frac{\delta^2\psi}{\delta{x^2}},\,\,\,\,\psi(f;t,x)\Big  _ {t=0}=f(x),$$
with the periodic initial ...
[Full Abstract] 
2003:10
S. Brenner and
L. SungC^{0} interior penalty methods for fourth order elliptic boundary value problems on polygonal domains (file not available)
(Journal of Scientific Computing 22/23 (2005), 83118)
$C^0$ interior penalty methods for fourth order elliptic boundary value problems on polygonal domains are analyzed in this paper. A postprocessing procedure that can generate $C^1$ approximate solutions from the $C^0$ approximate solutions is presented. New $C^0$ interior ...
[Full Abstract] 
2003:09
K. Fadimba and
R. Sharpley
We study the numerical approximation of the Saturation Equation which arises in the formulation of two phase fluid flow through porous media, idealized as either a convex bounded polyhedral domain or a domain with smooth boundary. This equation is degenerate and the solutions are not guaranteed to be sufficiently smooth ...
[Full Abstract] 
2003:08
P. Binev,
W. Dahmen, and
R. DeVore
Adaptive Finite Element Methods for numerically solving elliptic equations are used often in practice. Only recently [12], [17] have these methods been shown to converge. However, this convergence analysis says nothing about the rates of convergence of these methods and therefore does, in principle, not guarantee yet any numerical advantages ...
[Full Abstract] 
2003:07
K. Oskolkov
The divergence set of the trigonometric series
$$S(x):=\sum^\infty _ {n=1}\frac{\{nx\}}{n};\ \ T(x):=\sum^\infty _ {n=1}d(n)\frac{\sin2\pi{nx}}{\pi{n}};\ \ U(x):=\sum^\infty _ {m=1}\sum^\infty _ {n=1}\frac{\sin2\pi{mnx}}{\pi{mn ...
[Full Abstract] 
2003:06
V. Temlyakov
We study convergence and rate of convergence of expansions of elements in a Banach space X into series with regard to a given dictionary D. For convenience we assume that D is symmetric: $g\in{D}$ implies $g\in{D}$. The primary goal of this paper is to study representations ...
[Full Abstract] 
2003:05
K. Oskolkov
The goal of this paper is to study the convergence of the double trigonometric series
$$S(x) := 3D\sum _ {m=3D _ 1}^{\infty} \sum _ {n=3D _ 1}^{\infty} \frac{\sin mnx}{m^2 + n^2}$$
We prove that this series converges for all real x, and ...
[Full Abstract] 
2003:04
J. Zhao
Multigrid Vcycle and Fcycle algorithms for the biharmonic problem using the Morley element are studied in the paper. We show that the contraction numbers can be uniformly improved by increasing the number of smoothing steps.
[Full Abstract] 
2003:03
S. BrennerAnalysis of twodimensional FETIDP preconditioners by the standard additive Schwarz framework (file not available)
(Electronic Transactions on Numerical Analysis 16 (2003), 165185)
FETIDP preconditioners for twodimensional elliptic boundary value problems with heterogeneous coefficients are analyzed by the standard additive Schwarz framework. It is shown that the condition number of the preconditioned system for both secondorder and fourthorder problems is bounded by $C(1+\ln ...
[Full Abstract] 
2003:02
D. Leviatan and
V. Temlyakov
We study nonlinear mterm approximation with regard to a redundant dictionary D in a Hilbert space H. It is known that the Pure Greedy Algorithm (or, more generally, the Weak Greedy Algorithm) provides for each $f\in{H}$ and any dictionary D an expansion into a series
$$f=\sum _ ...
[Full Abstract]  2003:01 K. Oskolkov