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Operators in the Cowen-Douglas Class and Related Topics
Published in Kehe Zhu, Handbook of Analytic Operator Theory, 2019
is the reproducing kernel of H2(D2), restricting it to the hyper-surface z1 − z2 = 0 and using new coordinates u1=z1−z22,u2=z1+z22, we see that KQ(u,v)=1(1−uv¯)2 for u, v in {z1 − z2 = 0}. This is a multiple of the kernel function for the Bergman space Lhol2(D). Hence the quotient module is isometrically isomorphism to the Bergman module since multiplication by a constant doesn’t change the isomorphism class of a Hilbert module.
The history of Tutte–Whitney polynomials
Published in Joanna A. Ellis-Monaghan, Iain Moffatt, Handbook of the Tutte Polynomial and Related Topics, 2022
Consider the deletion–contraction relation (13.13), recast as a relation among graphic forms, in which any graph may be replaced by a linear combination of two graphs in any of three ways for each non-loop edge e: G→G\e+G/e, or G\e→G−G/e, or G/e→G−G\e. Starting with any graph, deletion–contraction can be applied repeatedly to give a sequence of graphic forms, leading eventually to a graphic form whose summands are all disjoint unions of bouquets (where a bouquet has a single vertex and some nonnegative integer number of loops). Tutte defines an elementary graph to be a bouquet with r≥0 loops. He denotes it by yr and its isomorphism class by yr. So the deletion–contraction process can transform the isomorphism class of any graph to a multivariate polynomial expression in the yr. Indeed, it can do the same for any graphic form at all. He constructs the ring R, obtained from the ring B of graphic forms by factoring out by the deletion–contraction relation.10 In other words, two graphic forms are equivalent if one can be transformed to the other by repeated application of the deletion–contraction relation.
Enumeration and Multicriteria Selection of Orthogonal Minimally Aliased Response Surface Designs
Published in Technometrics, 2020
Enumeration techniques have been around for designs other than RSDs. Bailey and Chigbu (1997) completely enumerated all nonisomorphic semi-Latin squares of size 4 × 4 containing 2, 3, and 4 plots. They generated a subset of designs and filter them to obtain one representative of each isomorphism class. Our approach in Section 2 avoids the generation of isomorphic designs. Harris, Hoffman, and Yarrow (1995) used an integer programming (IP) approach to obtain minimum-correlation Latin hypercubes. While we also use IP techniques, our goal is the enumeration of designs, rather than finding a single one. The most common use of enumeration techniques in experimental design is related to orthogonal arrays (OAs). Different enumeration procedures have been used. Bulutoglu and Margot (2008) used IP and symmetry reduction, while Schoen, Eendebak, and Nguyen (2010) and Eendebak and Schoen (2017) used a sequential extension algorithm. A number of papers exist about RSD generation, but their scope is different from ours. Draper and Lin (1990) generated small designs by selecting subsets of runs from standard designs. Xiao, Lin, and Bai (2012) found a recurrence formula to construct DSDs for an even number of factors.
Integer programming approaches to find row–column arrangements of two-level orthogonal experimental designs
Published in IISE Transactions, 2020
Nha Vo-Thanh, Peter Goos, Eric D. Schoen
The two-level treatment designs considered in this article are derived from orthogonal arrays. There may be many treatment designs with N runs, n treatment factors and strength t. Designs that can be obtained from each other by row permutations, column permutations and level permutations in the columns are statistically equivalent and belong to the same isomorphism class. We denote a catalog of two-level orthogonal arrays in which a single representative of every isomorphism class is included by OA, and we call this set the set of non-isomorphic designs with parameters N, n, and t. Also, we denote the two levels of each treatment factor by –1 and 1.
One-sided topological conjugacy of topological Markov shifts, continuous full groups and Cuntz–Krieger algebras
Published in Dynamical Systems, 2023
By [16], we know that the condition (ii) above is equivalent to the condition that and Hence the isomorphism class of the discrete group is completely characterized in terms of the isomorphism class of the Cuntz–Krieger algebra with , so that we have an infinite family of mutually non isomorphic finitely presented non-amenable infinite discrete groups.