Expand all Collapse all | Results 1 - 7 of 7 |
1. CJM 2011 (vol 64 pp. 1359)
Note on Cubature Formulae and Designs Obtained from Group Orbits In 1960,
Sobolev proved that for a finite reflection group $G$,
a $G$-invariant cubature formula is of degree $t$ if and only if
it is exact for all $G$-invariant polynomials of degree at most $t$.
In this paper,
we find some observations on invariant cubature formulas and Euclidean designs
in connection with the Sobolev theorem.
First, we give an alternative proof of
theorems by Xu (1998) on necessary and sufficient conditions
for the existence of cubature formulas with some strong symmetry.
The new proof is shorter and simpler compared to the original one by Xu, and
moreover gives a general interpretation of
the analytically-written conditions of Xu's theorems.
Second,
we extend a theorem by Neumaier and Seidel (1988) on
Euclidean designs to invariant Euclidean designs, and thereby
classify tight Euclidean designs obtained from
unions of the orbits of the corner vectors.
This result generalizes a theorem of Bajnok (2007) which classifies
tight Euclidean designs invariant under the Weyl group of type $B$
to other finite reflection groups.
Keywords:cubature formula, Euclidean design, radially symmetric integral, reflection group, Sobolev theorem Categories:65D32, 05E99, 51M99 |
2. CJM 2011 (vol 64 pp. 24)
Lower Order Terms of the Discrete Minimal Riesz Energy on Smooth Closed Curves We consider the problem of minimizing the energy of $N$ points
repelling each other on curves in $\mathbb{R}^d$ with the potential
$|x-y|^{-s}$, $s\geq 1$, where $|\, \cdot\, |$ is
the Euclidean norm. For a sufficiently smooth, simple, closed,
regular curve, we find the next order term in the asymptotics of the
minimal $s$-energy. On our way, we also prove that at
least for $s\geq 2$, the minimal pairwise distance in optimal configurations
asymptotically equals $L/N$, $N\to\infty$, where $L$ is the length
of the curve.
Keywords:minimal discrete Riesz energy, lower order term, power law potential, separation radius Categories:31C20, 65D17 |
3. CJM 2004 (vol 56 pp. 825)
Differentiability Properties of Optimal Value Functions Differentiability properties of optimal value functions associated with
perturbed optimization problems require strong assumptions. We consider such
a set of assumptions which does not use compactness hypothesis but which
involves a kind of coherence property. Moreover, a strict differentiability
property is obtained by using techniques of Ekeland and Lebourg and a result
of Preiss. Such a strengthening is required in order to obtain genericity
results.
Keywords:differentiability, generic, marginal, performance function, subdifferential Categories:26B05, 65K10, 54C60, 90C26, 90C48 |
4. CJM 2003 (vol 55 pp. 401)
Gaussian Estimates in Lipschitz Domains We give upper and lower Gaussian estimates for the diffusion kernel of a
divergence and nondivergence form elliptic operator in a Lipschitz domain.
On donne des estimations Gaussiennes pour le noyau d'une diffusion,
r\'eversible ou pas, dans un domaine Lipschitzien.
Categories:39A70, 35-02, 65M06 |
5. CJM 2001 (vol 53 pp. 1057)
Potential Theory in Lipschitz Domains We prove comparison theorems for the probability of life in a
Lipschitz domain between Brownian motion and random walks.
On donne des th\'eor\`emes de comparaison pour la probabilit\'e de
vie dans un domain Lipschitzien entre le Brownien et de marches
al\'eatoires.
Categories:39A70, 35-02, 65M06 |
6. CJM 1999 (vol 51 pp. 326)
Association Schemes for Ordered Orthogonal Arrays and $(T,M,S)$-Nets In an earlier paper~\cite{stinmar}, we studied a generalized Rao bound
for ordered orthogonal arrays and $(T,M,S)$-nets. In this paper,
we extend this to a coding-theoretic approach to ordered orthogonal
arrays. Using a certain association
scheme, we prove a MacWilliams-type theorem for linear ordered orthogonal
arrays and linear ordered codes as well as a linear programming bound
for the general case. We include some tables which compare this
bound against two previously known bounds for ordered orthogonal arrays.
Finally we show that, for even strength, the LP bound is always at
least as strong as the generalized Rao bound.
Categories:05B15, 05E30, 65C99 |
7. CJM 1997 (vol 49 pp. 944)
Approximation by multiple refinable functions We consider the shift-invariant space,
$\bbbs(\Phi)$, generated by a set $\Phi=\{\phi_1,\ldots,\phi_r\}$
of compactly supported distributions on $\RR$ when the vector of
distributions $\phi:=(\phi_1,\ldots,\phi_r)^T$ satisfies a system
of refinement equations expressed in matrix form as
$$
\phi=\sum_{\alpha\in\ZZ}a(\alpha)\phi(2\,\cdot - \,\alpha)
$$
where $a$ is a finitely supported sequence of $r\times r$ matrices
of complex numbers. Such {\it multiple refinable functions} occur
naturally in the study of multiple wavelets.
The purpose of the present paper is to characterize the {\it accuracy}
of $\Phi$, the order of the polynomial space contained in
$\bbbs(\Phi)$, strictly in terms of the refinement mask $a$. The
accuracy determines the $L_p$-approximation order of $\bbbs(\Phi)$ when
the functions in $\Phi$ belong to $L_p(\RR)$ (see Jia~[10]).
The characterization is achieved in terms of the eigenvalues and
eigenvectors of the subdivision operator associated with the mask $a$.
In particular, they extend and improve the results of Heil, Strang
and Strela~[7], and of Plonka~[16]. In addition, a
counterexample is given to the statement of Strang and Strela~[20]
that the eigenvalues of the subdivision operator determine the
accuracy. The results do not require the linear independence of
the shifts of $\phi$.
Keywords:Refinement equations, refinable functions, approximation, order, accuracy, shift-invariant spaces, subdivision Categories:39B12, 41A25, 65F15 |