Adaptive Gibbs samplers

We consider various versions of adaptive Gibbs and Metropolis- within-Gibbs samplers, which update their selection probabilities (and perhaps also their proposal distributions) on the fly during a run, by learning as they go in an attempt to optimise the algorithm. We present a cautionary example of how even a simple-seeming adaptive Gibbs sampler may fail to converge. We then present various positive results guaranteeing convergence of adaptive Gibbs samplers under certain conditions

Generating functions for Wilf equivalence under generalized factor order

Kitaev, Liese, Remmel, and Sagan recently defined generalized factor order on words comprised of letters from a partially ordered set $(P, \leq_P)$ by setting $u \leq_P w$ if there is a subword $v$ of $w$ of the same length as $u$ such that the $i$-th character of $v$ is greater than or equal to the $i$-th character of $u$ for all $i$. This subword $v$ is called an embedding of $u$ into $w$. For the case where $P$ is the positive integers with the usual ordering, they defined the weight of a word $w = w_1\ldots w_n$ to be $\text{wt}(w) = x^{\sum_{i=1}^n w_i} t^{n}$, and the corresponding weight generating function $F(u;t,x) = \sum_{w \geq_P u} \text{wt}(w)$. They then defined two words $u$ and $v$ to be Wilf equivalent, denoted $u \backsim v$, if and only if $F(u;t,x) = F(v;t,x)$. They also defined the related generating function $S(u;t,x) = \sum_{w \in \mathcal{S}(u)} \text{wt}(w)$ where $\mathcal{S}(u)$ is the set of all words $w$ such that the only embedding of $u$ into $w$ is a suffix of $w$, and showed that $u \backsim v$ if and only if $S(u;t,x) = S(v;t,x)$. We continue this study by giving an explicit formula for $S(u;t,x)$ if $u$ factors into a weakly increasing word followed by a weakly decreasing word. We use this formula as an aid to classify Wilf equivalence for all words of length 3. We also show that coefficients of related generating functions are well-known sequences in several special cases. Finally, we discuss a conjecture that if $u \backsim v$ then $u$ and $v$ must be rearrangements, and the stronger conjecture that there also must be a weight-preserving bijection $f: \mathcal{S}(u) \rightarrow \mathcal{S}(v)$ such that $f(u)$ is a rearrangement of $u$ for all $u$.Comment: 23 page

Quantum Ergodicity for a Class of Mixed Systems

We examine high energy eigenfunctions for the Dirichlet Laplacian on domains where the billiard flow exhibits mixed dynamical behavior. (More generally, we consider semiclassical Schrodinger operators with mixed assumptions on the Hamiltonian flow.) Specificially, we assume that the billiard flow has an invariant ergodic component, U, and study defect measures, mu, of positivie density subsequences of eigenfunctions (and, more generally, of almost orthogonal quasimodes). We show that any defect measure associated to such a subsequence, when restricted to U, satisfies mu = c mu_L where mu_L is the Liouville measure. This proves part of a conjecture of Percival.Comment: 16 pages, 2 figure

Conceptions of the Corporation and the Prospects of Sustainable Peace

I begin this essay with a brief overview of the corporation in legal discourse. In this overview, I draw upon current corporate law scholarship, considering the notions of the corporation as (1) property, (2) person, (3) contract, and (4) community. Following this overview, I explore the particular significance of legal language, detailing some of the central ways the law constructs the world in which we live. The law plays this constructive role by constituting and transforming our understandings of "character," "culture," and "community." Examining the ways in which our basic corporate conceptions contribute to our understanding of these three central elements of the business environment brings into view how legal language can structure the sensibility and vision we bring to corporate law problems. Each conception of the corporation presents it in a certain light, giving to the corporation a certain "character." Each character thus presented is in turn intertwined with a culture - "a set of ways of claiming meaning" - that can justify a greater or lesser sense of community. My conclusion is that while the notions of the corporation as property and contract predominate in current corporate law scholarship, the conceptions of the corporation as person and community offer better prospects for the goal of sustainable peace. They do so because of the way they enlarge our sense of corporate responsibility for the harms associated with corporate undertakings. Because this greater sense of corporate responsibility works to eliminate or minimize such harms, it contributes to the corporation's ability to foster long-term cooperative relationships among all its stakeholders. Thus, if we wish corporations to contribute to the creation of a more peaceful world, legal discourse can help by revitalizing one of its old notions, the corporation as person, and more fully embracing a new one, the corporation as community.http://deepblue.lib.umich.edu/bitstream/2027.42/39807/3/wp423.pd