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26. CMB 2001 (vol 44 pp. 385)

Ballantine, Cristina M.
 A Hypergraph with Commuting Partial Laplacians Let $F$ be a totally real number field and let $\GL_{n}$ be the general linear group of rank $n$ over $F$. Let $\mathfrak{p}$ be a prime ideal of $F$ and $F_{\mathfrak{p}}$ the completion of $F$ with respect to the valuation induced by $\mathfrak{p}$. We will consider a finite quotient of the affine building of the group $\GL_{n}$ over the field $F_{\mathfrak{p}}$. We will view this object as a hypergraph and find a set of commuting operators whose sum will be the usual adjacency operator of the graph underlying the hypergraph. Keywords:Hecke operators, buildingsCategories:11F25, 20F32

27. CMB 2000 (vol 43 pp. 406)

Borwein, David
 Weighted Mean Operators on $l_p$ The weighted mean matrix $M_a$ is the triangular matrix $\{a_k/A_n\}$, where $a_n > 0$ and $A_n := a_1 + a_2 + \cdots + a_n$. It is proved that, subject to $n^c a_n$ being eventually monotonic for each constant $c$ and to the existence of $\alpha := \lim \frac{A_n}{na_n}$, $M_a \in B(l_p)$ for $1 < p < \infty$ if and only if $\alpha < p$. Keywords:weighted means, operators on $l_p$, norm estimatesCategories:47B37, 47A30, 40G05

28. CMB 2000 (vol 43 pp. 496)

Xu, Yuan
 Harmonic Polynomials Associated With Reflection Groups We extend Maxwell's representation of harmonic polynomials to $h$-harmonics associated to a reflection invariant weight function $h_k$. Let $\CD_i$, $1\le i \le d$, be Dunkl's operators associated with a reflection group. For any homogeneous polynomial $P$ of degree $n$, we prove the polynomial $|\xb|^{2 \gamma +d-2+2n}P(\CD)\{1/|\xb|^{2 \gamma +d-2}\}$ is a $h$-harmonic polynomial of degree $n$, where $\gamma = \sum k_i$ and $\CD=(\CD_1,\ldots,\CD_d)$. The construction yields a basis for $h$-harmonics. We also discuss self-adjoint operators acting on the space of $h$-harmonics. Keywords:$h$-harmonics, reflection group, Dunkl's operatorsCategories:33C50, 33C45

29. CMB 1999 (vol 42 pp. 129)

Baker, Andrew
 Hecke Operations and the Adams $E_2$-Term Based on Elliptic Cohomology Hecke operators are used to investigate part of the $\E_2$-term of the Adams spectral sequence based on elliptic homology. The main result is a derivation of $\Ext^1$ which combines use of classical Hecke operators and $p$-adic Hecke operators due to Serre. Keywords:Adams spectral sequence, elliptic cohomology, Hecke operatorsCategories:55N20, 55N22, 55T15, 11F11, 11F25

30. CMB 1998 (vol 41 pp. 129)

Lee, Young Joo
 Pluriharmonic symbols of commuting Toeplitz type operators on the weighted Bergman spaces A class of Toeplitz type operators acting on the weighted Bergman spaces of the unit ball in the $n$-dimensional complex space is considered and two pluriharmonic symbols of commuting Toeplitz type operators are completely characterized. Keywords:Pluriharmonic functions, Weighted Bergman spaces, Toeplitz type operators.Categories:47B38, 32A37

31. CMB 1998 (vol 41 pp. 10)

Borwein, David
 Simple conditions for matrices to be bounded operators on $l_p$ The two theorems proved yield simple yet reasonably general conditions for triangular matrices to be bounded operators on $l_p$. The theorems are applied to N\"orlund and weighted mean matrices. Keywords:Triangular matrices, NÃ¶rlund matrices, weighted means, operators, on $l_p$.Categories:47B37, 47A30, 40G05
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