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1. CJM 2011 (vol 63 pp. 689)

Olphert, Sean; Power, Stephen C.
 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 squaresCategories:42C40, 42A65, 42A16, 43A65

2. CJM 2008 (vol 60 pp. 334)

Curry, Eva
 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 functionCategories:42C40, 60G35

3. CJM 2007 (vol 59 pp. 332)

Leuschke, Graham J.
 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 modulesCategories: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 metricCategories:47A05, 47A15, 47B40, 47B50, 46C20

5. CJM 2002 (vol 54 pp. 634)

Weber, Eric
 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, frameCategories:42C40, 43A25, 42C15, 46N99

6. CJM 2000 (vol 52 pp. 1149)

Ban, Chunsheng; McEwan, Lee J.
 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 singularityCategories:14B05, 14J17, 32S05, 32S25

7. CJM 2000 (vol 52 pp. 123)

Harbourne, Brian
 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 groupCategories:13P10, 14C99, 13D02, 13H15