Refining Multivariate Value Set Bounds

Over finite fields, if the image of a polynomial map is not the entire field, then its cardinality can be bounded above by a significantly smaller value. Earlier results bound the cardinality of the value set using the degree of the polynomial, but more recent results make use of the powers of all monomials. In this paper, we explore the geometric properties of the Newton polytope and show how they allow for tighter upper bounds on the cardinality of the multivariate value set. We then explore a method which allows for even stronger upper bounds, regardless of whether one uses the multivariate degree or the Newton polytope to bound the value set. Effectively, this provides an alternate proof of Kosters' degree bound, an improved Newton polytope-based bound, and an improvement of a degree matrix-based result given by Zan and Cao.Comment: 41 pages, 1 figure. Preprint of a dissertation to be published with Proquest/UMI (in press). Shortened version of manuscript for publication elsewhere is in preparatio

Sensitivity of principal Hessian direction analysis

We provide sensitivity comparisons for two competing versions of the dimension reduction method principal Hessian directions (pHd). These comparisons consider the effects of small perturbations on the estimation of the dimension reduction subspace via the influence function. We show that the two versions of pHd can behave completely differently in the presence of certain observational types. Our results also provide evidence that outliers in the traditional sense may or may not be highly influential in practice. Since influential observations may lurk within otherwise typical data, we consider the influence function in the empirical setting for the efficient detection of influential observations in practice.Comment: Published at http://dx.doi.org/10.1214/07-EJS064 in the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org

Whole Transcriptome Gene Expression and how it Relates to Muscle Performance in Anolis Lizards

Animal muscles are exceptionally diverse in structure and function as they meet a variety of demands for an individual to survive. Muscles coordinate with each other so that individuals can survive in their environments. However, muscles vary in performance to best suit their role in promoting organism survival; differences in gene expression among muscles likely accounts for much of this variation. Anolis lizards, a genus that has undergone considerable adaptive radiation, live in a wide range of habitats and ecotypes to which each species has had to evolve appropriately to survive. These habitats require different muscles of the anole to perform extremely variable tasks. Unsurprisingly, muscle performance of these lizards (e.g. twitch time and peak contractile velocity) varies among muscle types. Specifically, the performance of jaw and leg muscles, diverges strongly because their importance for survival (e.g., to escape predation and to bite prey) differs both across and within individuals. Here, I use RNA-seq to measure the differential gene expression generating differences in muscle performance between the jaw and leg muscles. The observed discrepancy in gene expression may explain the divergence in performance observed between the muscles. Determining the underlying differences in gene expression between muscles and individuals will help explain how performance metrics (e.g., twitch time and peak contractile velocity) change over time. Additionally, differential gene expression could show how the ecology and evolution of an individual influences its muscle performance

Pluto's Light Curve in 1933-1934

We are reporting on a new accurate photographic light curve of Pluto for 1933-1934 when the heliocentric distance was 40 AU. We used 43 B-band and V-band images of Pluto on 32 plates taken on 15 nights from 19 March 1933 to 10 March 1934. Most of these plates were taken with the Mount Wilson 60" and 100" telescopes, but 7 of the plates (now at the Harvard College Observatory) were taken with the 12" and 16" Metcalf doublets at Oak Ridge. The plates were measured with an iris diaphragm photometer, which has an average one-sigma photometric error on these plates of 0.08 mag as measured by the repeatability of constant comparison stars. The modern B and V magnitudes for the comparison stars were measured with the Lowell Observatory Hall 1.1-m telescope. The magnitudes in the plate's photographic system were converted to the Johnson B- and V-system after correction with color terms, even though they are small in size. We find that the average B-band mean opposition magnitude of Pluto in 1933-1934 was 15.73 +- 0.01, and we see a roughly sinusoidal modulation on the rotational period (6.38 days) with a peak-to-peak amplitude of 0.11 +- 0.03 mag. With this, we show that Pluto darkened by 5% from 1933-1934 to 1953-1955. This darkening from 1933-1934 to 1953-1955 cannot be due to changing viewing geometry (as both epochs had identical sub-Earth latitudes), so our observations must record a real albedo change over the southern hemisphere. The later darkening trend from 1954 to the 1980s has been explained by changing viewing geometry (as more of the darker northern hemisphere comes into view). Thus, we now have strong evidence for albedo changes on the surface of Pluto, and these are most easily explained by the systematic sublimation of frosts from the sunward pole that led to a drop in the mean surface albedo.Comment: Icarus in press, 24 page

Subtleties of witnessing quantum coherence in non-isolated systems

Identifying non-classicality unambiguously and inexpensively is a long-standing open challenge in physics. The No-Signalling-In-Time protocol was developed as an experimental test for macroscopic realism, and serves as a witness of quantum coherence in isolated quantum systems by comparing the quantum state to its completely dephased counterpart. We show that it provides a lower bound on a certain resource-theoretic coherence monotone. We go on to generalise the protocol to the case where the system of interest is coupled to an environment. Depending on the manner of the generalisation, the resulting witness either reports on system coherence alone, or on a disjunction of system coherence with either (i) the existence of non-classical system-environment correlations or (ii) non-negligible dynamics in the environment. These are distinct failure modes of the Born approximation in non-isolated systems.Comment: 16pp, 2 figs, 5 thms. v2: typos corrected, references added and small change to title to reflect that of published versio