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A curvature inequality is established for contractive commuting tuples of operators in the Cowen–Douglas class of rank defined on some bounded domain in . Properties of the extremal operators (that is, the operators which achieve equality) are investigated. Specifically, a substantial part of a well-known question due to R. G. Douglas involving these extremal operators, in the case of the unit disc, is answered.
We start from a discrete random variable, , defined on and taking on values with equal probability—any member of a certain family whose simplest member is the Rademacher random variable (with domain ), whose constant value on is . We create (via left-shifts) independent copies, , of and let . We let be the quantile of . If is Rademacher, the sequence is the equiprobable random walk on with domain . In the general case, follows a multinomial distribution and as varies over the family, the resulting family of multinomial distributions is sufficiently rich to capture the full generality of situations where the Central Limit Theorem applies.
The provide a representation of that is strong in that their sum is equal to pointwise. They represent only in distribution. Are there strong representations of ? We establish the affirmative answer, and our proof gives a canonical bijection between, on the one hand, the set of all strong representations with the additional property of being trim and, on the other hand, the set of permutations, , of , with the property that we call admissibility. Passing to sequences, , of admissible permutations, these provide a complete classification of trim, strong triangular array representations of the sequence . We explicitly construct two sequences of admissible permutations which are polynomial time computable, relative to a function which embodies the complexity of itself. The trim, strong triangular array representation corresponding to the second of these is as close as possible to the representation of provided by the .
We prove a version of the Jenkins–Serrin theorem for the existence of constant mean curvature graphs over bounded domains with infinite boundary data in . Moreover, we construct examples of admissible domains where the results may be applied.
This article investigates an explicit description of the Baum–Connes assembly map of the wreath product , where is a finite and is the free group on generators. In order to do so, we take Davis–Lück’s approach to the topological side which allows computations by means of spectral sequences. Besides describing explicitly the K-groups and their generators, we present a concrete 2-dimensional model for the classifying space . As a result of our computations, we obtain that is the free abelian group of countable rank with a basis consisting of projections in , and is the free abelian group of rank with a basis represented by the unitaries coming from the free group.
This paper extends Auslander–Reiten duality in two directions. As an application, we obtain various criteria for freeness of modules over local rings in terms of vanishing of Ext modules, which recover a lot of known results on the Auslander–Reiten conjecture.
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