Properly Integral Polynomials over the Ring of Integer-valued Polynomials on a Matrix Ring

Let $D$ be a domain with fraction field $K$, and let $M_n(D)$ be the ring of $n \times n$ matrices with entries in $D$. The ring of integer-valued polynomials on the matrix ring $M_n(D)$, denoted ${\rm Int}_K(M_n(D))$, consists of those polynomials in $K[x]$ that map matrices in $M_n(D)$ back to $M_n(D)$ under evaluation. It has been known for some time that ${\rm Int}_{\mathbb{Q}}(M_n(\mathbb{Z}))$ is not integrally closed. However, it was only recently that an example of a polynomial in the integral closure of ${\rm Int}_{\mathbb{Q}}(M_n(\mathbb{Z}))$ but not in the ring itself appeared in the literature, and the published example is specific to the case $n=2$. In this paper, we give a construction that produces polynomials that are integral over ${\rm Int}_K(M_n(D))$ but are not in the ring itself, where $D$ is a Dedekind domain with finite residue fields and $n \geq 2$ is arbitrary. We also show how our general example is related to $P$-sequences for ${\rm Int}_K(M_n(D))$ and its integral closure in the case where $D$ is a discrete valuation ring.Comment: final version, to appear in J. Algebra (2016); comments are welcome

Non-triviality conditions for integer-valued polynomial rings on algebras

Let $D$ be a commutative domain with field of fractions $K$ and let $A$ be a torsion-free $D$-algebra such that $A \cap K = D$. The ring of integer-valued polynomials on $A$ with coefficients in $K$ is \Int_K(A) = \{f \in K[X] \mid f(A) \subseteq A\}, which generalizes the classic ring \Int(D) = \{f \in K[X] \mid f(D) \subseteq D\} of integer-valued polynomials on $D$. The condition $A \cap K$ implies that D[X] \subseteq \Int_K(A) \subseteq \Int(D), and we say that \Int_K(A) is nontrivial if \Int_K(A) \ne D[X]. For any integral domain $D$, we prove that if $A$ is finitely generated as a $D$-module, then \Int_K(A) is nontrivial if and only if \Int(D) is nontrivial. When $A$ is not necessarily finitely generated but $D$ is Dedekind, we provide necessary and sufficient conditions for \Int_K(A) to be nontrivial. These conditions also allow us to prove that, for $D$ Dedekind, the domain \Int_K(A) has Krull dimension 2

The Effects of Induced Polycystic Ovary Syndrome in NAG-1 Transgenic Mice

Polycystic ovary syndrome (PCOS) is the leading cause of infertility among women in the US and the most common endocrine disorder among women. PCOS is characterized by cystic ovaries, hyperandrogenism (heightened levels of male sex hormones), altered menstrual cycles and various metabolic dysfunctions. The metabolic symptoms associated with PCOS are difficult to treat, as they are a result of hormonal imbalances, rather than diet. The human Non-Steroidal Anti-Inflammatory Drug Activated Gene (NAG-1) been shown to prevent diet-induced metabolic disorders and weight gain in mice. We hypothesized that the expression of NAG-1 may also prevent hormonal-induced metabolic disorders. To test this question, we induced PCOS via dihydrotestosterone (DHT) implantation in transgenic mice expressing the human NAG-1 gene. Our findings suggest that NAG-1 mice have similar physiological responses to DHT-treatment as compared to wild-type mice throughout the 90-day study. Specifically no changes in the age of puberty and anal-genital distance (AGD) were observed. NAG-1 mice also display similar ovarian phenotypes, developing fewer corpora lutea, and having disrupted estrus cycles. NAG-1 mice displayed no significant weight gain between treatment groups throughout the study, and no significant increase in triglyceride levels. Additionally, NAG-1 mice showed no change in white adipocyte morphology after DHT-treatment. However, wild-type mice treated with DHT showed an increased amount of brown adipocytes differentiating to white adipocytes, compared to NAG-1 mice. Our findings indicate that NAG-1 mice respond similarly to DHT-treatment as wild-type mice in ovarian response, but continue to maintain their lean phenotype in the presence of induced PCOS. These findings suggest that expression of NAG-1 may have therapeutic benefits in the prevention of hormonal induced weight gain, and brown adipocyte hypertrophy

Null model analysis of species associations using abundance data

The influence of negative species interactions has dominated much of the literature on community assembly rules. Patterns of negative covariation among species are typically documented through null model analyses of binary presence/absence matrices in which rows designate species, columns designate sites, and the matrix entries indicate the presence (1) or absence (0) of a particular species in a particular site. However, the outcome of species interactions ultimately depends on population-level processes. Therefore, patterns of species segregation and aggregation might be more clearly expressed in abundance matrices, in which the matrix entries indicate the abundance or density of a species in a particular site. We conducted a series of benchmark tests to evaluate the performance of 14 candidate null model algorithms and six covariation metrics that can be used with abundance matrices. We first created a series of random test matrices by sampling a metacommunity from a lognormal species abundance distribution. We also created a series of structured matrices by altering the random matrices to incorporate patterns of pairwise species segregation and aggregation. We next screened each algorithm-index combination with the random and structured matrices to determine which tests had low Type I error rates and good power for detecting segregated and aggregated species distributions. In our benchmark tests, the best-performing null model does not constrain species richness, but assigns individuals to matrix cells proportional to the observed row and column marginal distributions until, for each row and column, total abundances are reached. Using this null model algorithm with a set of four covariance metrics, we tested for patterns of species segregation and aggregation in a collection of 149 empirical abundance matrices and 36 interaction matrices collated from published papers and posted data sets. More than 80% of the matrices were significantly segregated, which reinforces a previous meta-analysis of presence/absence matrices. However, using two of the metrics we detected a significant pattern of aggregation for plants and for the interaction matrices (which include plant-pollinator data sets). These results suggest that abundance matrices, analyzed with an appropriate null model, may be a powerful tool for quantifying patterns of species segregation and aggregation. © 2010 by the Ecological Society of America

Statistical challenges in null model analysis

This review identifies several important challenges in null model testing in ecology: 1) developing randomization algorithms that generate appropriate patterns for a specified null hypothesis; these randomization algorithms stake out a middle ground between formal Pearson-Neyman tests (which require a fully-specified null distribution) and specific process-based models (which require parameter values that cannot be easily and independently estimated); 2) developing metrics that specify a particular pattern in a matrix, but ideally exclude other, related patterns; 3) avoiding classification schemes based on idealized matrix patterns that may prove to be inconsistent or contradictory when tested with empirical matrices that do not have the idealized pattern; 4) testing the performance of proposed null models and metrics with artificial test matrices that contain specified levels of pattern and randomness; 5) moving beyond simple presence-absence matrices to incorporate species-level traits (such as abundance) and site-level traits (such as habitat suitability) into null model analysis; 6) creating null models that perform well with many sites, many species pairs, and varying degrees of spatial autocorrelation in species occurrence data. In spite of these challenges, the development and application of null models has continued to provide valuable insights in ecology, evolution, and biogeography for over 80 years. © 2011 The Authors. Oikos © 2012 Nordic Society Oikos

Pattern detection in null model analysis

Synthesis The identification of distinctive patterns in species x site presence-absence matrices is important for understanding meta-community organisation. We compared the performance of a suite of null models and metrics that have been proposed to measure patterns of segregation, aggregation, nestedness, coherence, and species turnover. We found that any matrix with segregated species pairs can be re-ordered to highlight aggregated pairs, indicating that these seemingly opposite patterns are closely related. Recently proposed classification schemes failed to correctly classify realistic matrices that included multiple co-occurrence structures. We propose using a combination of metrics and decomposing matrix-wide patterns into those of individual pairs of species and sites to pinpoint sources of non-randomness. Null model analysis has been a popular tool for detecting pattern in binary presence-absence matrices, and previous tests have identified algorithms and metrics that have good statistical properties. However, the behavior of different metrics is often correlated, making it difficult to distinguish different patterns. We compared the performance of a suite of null models and metrics that have been proposed to measure patterns of segregation, aggregation, nestedness, coherence, and species turnover. We found that any matrix with segregated species pairs can be re-ordered to highlight aggregated pairs. As a consequence, the same null model can identify a single matrix as being simultaneously aggregated, segregated or nested. These results cast doubt on previous conclusions of matrix-wide species segregation based on the C-score and into those of individual pairs of species or pairs of sites to pinpoint the sources of non-randomness. © 2012 The Authors. Oikos © 2012 Nordic Society Oikos