Expand all Collapse all | Results 1 - 7 of 7 |
1. CJM 2011 (vol 63 pp. 689)
Higher Rank Wavelets A theory of higher rank multiresolution analysis is given in the
setting of abelian multiscalings. This theory enables the
construction, from a higher rank MRA, of finite wavelet sets
whose multidilations have translates forming an orthonormal basis
in $L^2(\mathbb R^d)$. While tensor products of uniscaled MRAs provide
simple examples we construct many nonseparable higher rank
wavelets. In particular we construct \emph{Latin square
wavelets} as rank~$2$ variants of Haar wavelets. Also we construct
nonseparable scaling functions for rank $2$ variants of Meyer
wavelet scaling functions, and we construct the associated
nonseparable wavelets with compactly supported Fourier transforms.
On the other hand we show that compactly supported scaling
functions for biscaled MRAs are necessarily separable.
Keywords: wavelet, multi-scaling, higher rank, multiresolution, Latin squares Categories:42C40, 42A65, 42A16, 43A65 |
2. CJM 2008 (vol 60 pp. 334)
Low-Pass Filters and Scaling Functions for Multivariable Wavelets We show that a characterization of scaling functions for
multiresolution analyses given by Hern\'{a}ndez and Weiss and that a
characterization of low-pass filters given by Gundy both hold for
multivariable multiresolution analyses.
Keywords:multivariable multiresolution analysis, low-pass filter, scaling function Categories:42C40, 60G35 |
3. CJM 2007 (vol 59 pp. 332)
Endomorphism Rings of Finite Global Dimension For a commutative local ring $R$, consider (noncommutative)
$R$-algebras $\Lambda$ of the form $\Lambda = \operatorname{End}_R(M)$
where $M$ is a reflexive $R$-module with nonzero free direct summand.
Such algebras $\Lambda$ of finite global dimension can be viewed as
potential substitutes for, or analogues of, a resolution of
singularities of $\operatorname{Spec} R$. For example, Van den Bergh
has shown that a three-dimensional Gorenstein normal
$\mathbb{C}$-algebra with isolated terminal singularities has a
crepant resolution of singularities if and only if it has such an
algebra $\Lambda$ with finite global dimension and which is maximal
Cohen--Macaulay over $R$ (a ``noncommutative crepant resolution of
singularities''). We produce algebras
$\Lambda=\operatorname{End}_R(M)$ having finite global dimension in
two contexts: when $R$ is a reduced one-dimensional complete local
ring, or when $R$ is a Cohen--Macaulay local ring of finite
Cohen--Macaulay type. If in the latter case $R$ is Gorenstein, then
the construction gives a noncommutative crepant resolution of
singularities in the sense of Van den Bergh.
Keywords:representation dimension, noncommutative crepant resolution, maximal Cohen--Macaulay modules Categories:16G50, 16G60, 16E99 |
4. CJM 2005 (vol 57 pp. 61)
On Operators with Spectral Square but without Resolvent Points Decompositions of spectral type are
obtained for closed Hilbert space operators with empty resolvent
set, but whose square has closure which is spectral. Krein space
situations are also discussed.
Keywords:unbounded operators, closed operators,, spectral resolution, indefinite metric Categories:47A05, 47A15, 47B40, 47B50, 46C20 |
5. CJM 2002 (vol 54 pp. 634)
Frames and Single Wavelets for Unitary Groups We consider a unitary representation of a discrete countable abelian
group on a separable Hilbert space which is associated to a cyclic
generalized frame multiresolution analysis. We extend Robertson's
theorem to apply to frames generated by the action of the group.
Within this setup we use Stone's theorem and the theory of projection
valued measures to analyze wandering frame collections. This yields a
functional analytic method of constructing a wavelet from a
generalized frame multi\-resolution analysis in terms of the frame
scaling vectors. We then explicitly apply our results to the action
of the integers given by translations on $L^2({\mathbb R})$.
Keywords:wavelet, multiresolution analysis, unitary group representation, frame Categories:42C40, 43A25, 42C15, 46N99 |
6. CJM 2000 (vol 52 pp. 1149)
Canonical Resolution of a Quasi-ordinary Surface Singularity We describe the embedded resolution of an irreducible quasi-ordinary
surface singularity $(V,p)$ which results from applying the canonical
resolution of Bierstone-Milman to $(V,p)$. We show that this process
depends solely on the characteristic pairs of $(V,p)$, as predicted
by Lipman. We describe the process explicitly enough that a resolution
graph for $f$ could in principle be obtained by computer using only
the characteristic pairs.
Keywords:canonical resolution, quasi-ordinary singularity Categories:14B05, 14J17, 32S05, 32S25 |
7. CJM 2000 (vol 52 pp. 123)
An Algorithm for Fat Points on $\mathbf{P}^2 Let $F$ be a divisor on the blow-up $X$ of $\pr^2$ at $r$ general
points $p_1, \dots, p_r$ and let $L$ be the total transform of a
line on $\pr^2$. An approach is presented for reducing the
computation of the dimension of the cokernel of the natural map
$\mu_F \colon \Gamma \bigl( \CO_X(F) \bigr) \otimes \Gamma \bigl(
\CO_X(L) \bigr) \to \Gamma \bigl( \CO_X(F) \otimes \CO_X(L) \bigr)$
to the case that $F$ is ample. As an application, a formula for
the dimension of the cokernel of $\mu_F$ is obtained when $r = 7$,
completely solving the problem of determining the modules in
minimal free resolutions of fat point subschemes\break
$m_1 p_1 + \cdots + m_7 p_7 \subset \pr^2$. All results hold for
an arbitrary algebraically closed ground field~$k$.
Keywords:Generators, syzygies, resolution, fat points, maximal rank, plane, Weyl group Categories:13P10, 14C99, 13D02, 13H15 |