39 research outputs found

    Rees Products of Posets and Inequalities

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    In this dissertation we will look at properties of two different posets from different perspectives. The first poset is the Rees product of the face lattice of the n-cube with the chain. Specifically we study the Möbius function of this poset. Our proof techniques include straightforward enumeration and a bijection between a set of labeled augmented skew diagrams and barred signed permutations which label the maximal chains of this poset. Because the Rees product of this poset is Cohen-Macaulay, we find a basis for the top homology group and a representation of the top homology group over the symmetric group both indexed by the set of labeled augmented skew diagrams. We also show that the Möbius function of the Rees product of a graded poset with the t-ary tree and the Rees product of its dual with the t-ary tree coincide. We discuss labelings for Rees and Segre products in general, particularly the Rees product of the face lattice of a polytope with the chain. We also look at cases where the Möbius function of a poset is equal to the permanent of a matrix and we consider local h-vectors for the barycentric subdivision of the n-cube. In each section we state open conjectures. The second poset in this dissertation is the Dowling lattice. In particular we look at the k = 1 case, that is, the partition lattice. We study inequalities on the flag vector of the partition lattice via a weighted boustrophedon transform and determine a more generalized version for the Dowling lattice. We generalize a determinantal formula of Niven and conclude with conjectures and avenues of study

    Bibliographie

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    Acta Scientiarum Mathematicarum : Tomus 43. Fasc. 3-4.

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    Acta Scientiarum Mathematicarum : Tomus 46.

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    Studia Scientiarum Mathematicarum Hungarica

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    Foundations of Quantum Theory: From Classical Concepts to Operator Algebras

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    Quantum physics; Mathematical physics; Matrix theory; Algebr

    The Necessary Structure of the All-pervading Aether: Discrete or Continuous? Simple or Symmetric?

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    In this book I investigate the necessary structure of the aether – the stuff that fills the whole universe. Some of my conclusions are. 1. There is an enormous variety of structures that the aether might, for all we know, have. 2. Probably the aether is point-free. 3. In that case, it should be distinguished from Space-time, which is either a fiction or a construct. 4. Even if the aether has points, we should reject the orthodoxy that all regions are grounded in points by summation. 5. If the aether is point-free but not continuous, its most likely structure has extended atoms that are not simples. 6. Space-time is symmetric if and only if the aether is continuous. 7. If the aether is continuous, we should reject the standard interpretation of General Relativity, in which geometry determines gravity. 8. Contemporary physics undermines an objection to discrete aether based on scale invariance, but does not offer much positive support

    Generalized Noncrossing Partitions and Combinatorics of Coxeter Groups

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    This memoir constitutes the author's PhD thesis at Cornell University. It serves both as an expository work and as a description of new research. At the heart of the memoir, we introduce and study a poset NC(k)(W)NC^{(k)}(W) for each finite Coxeter group WW and for each positive integer kk. When k=1k=1, our definition coincides with the generalized noncrossing partitions introduced by Brady-Watt and Bessis. When WW is the symmetric group, we obtain the poset of classical kk-divisible noncrossing partitions, first studied by Edelman. Along the way, we include a comprehensive introduction to related background material. Before defining our generalization NC(k)(W)NC^{(k)}(W), we develop from scratch the theory of algebraic noncrossing partitions NC(W)NC(W). This involves studying a finite Coxeter group WW with respect to its generating set TT of {\em all} reflections, instead of the usual Coxeter generating set SS. This is the first time that this material has appeared in one place. Finally, it turns out that our poset NC(k)(W)NC^{(k)}(W) shares many enumerative features in common with the ``generalized nonnesting partitions'' of Athanasiadis and the ``generalized cluster complexes'' of Fomin and Reading. In particular, there is a generalized ``Fuss-Catalan number'', with a nice closed formula in terms of the invariant degrees of WW, that plays an important role in each case. We give a basic introduction to these topics, and we describe several conjectures relating these three families of ``Fuss-Catalan objects''.Comment: Final version -- to appear in Memoirs of the American Mathematical Society. Many small improvements in exposition, especially in Sections 2.2, 4.1 and 5.2.1. Section 5.1.5 deleted. New references to recent wor
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