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Hypothesis testing near singularities and boundaries

The likelihood ratio statistic, with its asymptotic $\chi^2$ distribution at regular model points, is often used for hypothesis testing. At model singularities and boundaries, however, the asymptotic distribution may not be $\chi^2$, as highlighted by recent work of Drton. Indeed, poor behavior of a $\chi^2$ for testing near singularities and boundaries is apparent in simulations, and can lead to conservative or anti-conservative tests. Here we develop a new distribution designed for use in hypothesis testing near singularities and boundaries, which asymptotically agrees with that of the likelihood ratio statistic. For two example trinomial models, arising in the context of inference of evolutionary trees, we show the new distributions outperform a $\chi^2$.Comment: 32 pages, 12 figure

Perturbation of matrices and non-negative rank with a view toward statistical models

In this paper we study how perturbing a matrix changes its non-negative rank. We prove that the non-negative rank is upper-semicontinuos and we describe some special families of perturbations. We show how our results relate to Statistics in terms of the study of Maximum Likelihood Estimation for mixture models.Comment: 13 pages, 3 figures. A theorem has been rewritten, and some improvements in the presentations have been implemente

Tensor Rank, Invariants, Inequalities, and Applications

Though algebraic geometry over $\mathbb C$ is often used to describe the closure of the tensors of a given size and complex rank, this variety includes tensors of both smaller and larger rank. Here we focus on the $n\times n\times n$ tensors of rank $n$ over $\mathbb C$, which has as a dense subset the orbit of a single tensor under a natural group action. We construct polynomial invariants under this group action whose non-vanishing distinguishes this orbit from points only in its closure. Together with an explicit subset of the defining polynomials of the variety, this gives a semialgebraic description of the tensors of rank $n$ and multilinear rank $(n,n,n)$. The polynomials we construct coincide with Cayley's hyperdeterminant in the case $n=2$, and thus generalize it. Though our construction is direct and explicit, we also recast our functions in the language of representation theory for additional insights. We give three applications in different directions: First, we develop basic topological understanding of how the real tensors of complex rank $n$ and multilinear rank $(n,n,n)$ form a collection of path-connected subsets, one of which contains tensors of real rank $n$. Second, we use the invariants to develop a semialgebraic description of the set of probability distributions that can arise from a simple stochastic model with a hidden variable, a model that is important in phylogenetics and other fields. Third, we construct simple examples of tensors of rank $2n-1$ which lie in the closure of those of rank $n$.Comment: 31 pages, 1 figur

Last Year\u27s Choice Is This Year\u27s Voice: Valuing Democratic Practices in the Classroom through Student-Selected Literature

The authors of this article explore democratic practices in the classroom by using student-selected literature. After multiple class sets of student-selected young adult novels were purchased using grant money, the authors set out to see what happens in a classroom when student choice is at the forefront of pedagogical decision-making and how students resonated with and voiced their experiences reading about those chosen novels. Because canonical texts are often used to help students understand allusions in contemporary texts, one adolescent novel and one canonical novel became the focal points for this project. With democratic practices undergirding this project, the authors argue that using student-selected literature, both adolescent and canonical, encourages agency, invites healthy inquiry, and develops reflective practices and empathy in adolescent readers

³¹P Saturation Transfer and Phosphocreatine Imaging in the Monkey Brain

³¹P magnetic resonance imaging with chemical-shift discrimination by selective excitation has been employed to determine the phosphocreatine (PCr) distribution in the brains of three juvenile macaque monkeys. PCr images were also obtained while saturating the resonance of the {gamma}-phosphate of ATP, which allowed the investigation of the chemical exchange between PCr and the {gamma}-phosphate of ATP catalyzed by creatine kinase. Superposition of the PCr images over the proton image of the same monkey brain revealed topological variations in the distribution of PCr and creatine kinase activity. PCr images were also obtained with and without visual stimulation. In two out of four experiments, an apparently localized decrease in PCr concentration was noted in visual cortex upon visual stimulation. This result is interpreted in terms of a possible role for the local ADP concentration in stimulating the accompanying metabolic response

The Interference Term in the Wigner Distribution Function and the Aharonov-Bohm Effect

A phase space representation of the Aharonov-Bohm effect is presented. It shows that the shift of interference fringes is associated to the interference term of the Wigner distribution function of the total wavefunction, whereas the interference pattern is defined by the common projections of the Wigner distribution functions of the interfering beamsComment: 10 pages, 4 figure

Adventures in Invariant Theory

We provide an introduction to enumerating and constructing invariants of group representations via character methods. The problem is contextualised via two case studies arising from our recent work: entanglement measures, for characterising the structure of state spaces for composite quantum systems; and Markov invariants, a robust alternative to parameter-estimation intensive methods of statistical inference in molecular phylogenetics.Comment: 12 pp, includes supplementary discussion of example