Coordinate sum and difference sets of dd-dimensional modular hyperbolas

Many problems in additive number theory, such as Fermat's last theorem and the twin prime conjecture, can be understood by examining sums or differences of a set with itself. A finite set $A \subset \mathbb{Z}$ is considered sum-dominant if $|A+A|>|A-A|$. If we consider all subsets of ${0, 1, ..., n-1}$, as $n\to\infty$ it is natural to expect that almost all subsets should be difference-dominant, as addition is commutative but subtraction is not; however, Martin and O'Bryant in 2007 proved that a positive percentage are sum-dominant as $n\to\infty$. This motivates the study of "coordinate sum dominance". Given $V \subset (\Z/n\Z)^2$, we call $S:={x+y: (x,y) \in V}$ a coordinate sumset and $D:=\{x-y: (x,y) \in V\}$ a coordinate difference set, and we say $V$ is coordinate sum dominant if $|S|>|D|$. An arithmetically interesting choice of $V$ is $\bar{H}_2(a;n)$, which is the reduction modulo $n$ of the modular hyperbola $H_2(a;n) := {(x,y): xy \equiv a \bmod n, 1 \le x,y < n}$. In 2009, Eichhorn, Khan, Stein, and Yankov determined the sizes of $S$ and $D$ for $V=\bar{H}_2(1;n)$ and investigated conditions for coordinate sum dominance. We extend their results to reduced $d$-dimensional modular hyperbolas $\bar{H}_d(a;n)$ with $a$ coprime to $n$.Comment: Version 1.0, 14 pages, 2 figure

Organic Crop Prices, or 2x Conventional Ones?

Cointegration is tested between organic and conventional corn and soybean markets in several locations throughout the U.S. using a unique data set. Organic prices are found to behave like jump processes rather than diffusions, and Monte Carlo methods are developed to compute appropriate critical values for such tests. Findings indicate that no long-run relationship exists between organic and conventional prices, implying that price determination for organic corn and soybean is independent from that for the conventional crops. This suggests that organic corn and soybean prices are driven by demand and supply forces idiosyncratic to the organic market. For each crop, cointegrating spatial relationships are found between prices at the main organic markets. However, such relationships are generally weaker than the ones for the corresponding conventional prices, implying that organic markets are more affected by idiosyncratic shocks than conventional markets. �cointegration; jump process; organic crops; organic production; price analysis; organic price premiums

Out of Site but Not Out of Mind: Submerged Prehistoric Landscapes on the Northwestern Gulf of Mexico Outer Continental Shelf

Archaeological sites are more important than simply the artifacts they contain. Locations of human occupation and activity form a pattern that can provide information about perceptions of the landscape, decisions about resources, or preferences. Explaining this “perceived” environment is one of archaeology’s goals in explaining past human behavior. In order to address these goals, archaeologists must first identify elements of the “real” landscape, including the geographical environment, its resources, and evidence of human modification. Only after these real elements have been identified can the perceived environment be explored. On the outer continental shelf of the northwestern Gulf of Mexico, formerly subaerial geographical environments are now submerged offshore of Texas and Louisiana. These areas could have been occupied by Paleoindian or Early Archaic populations prior to final inundation related to sea-level rise following the last glacial maximum. Due to sediment accretion related to transgression, as well as active sedimentation in the marine environment, the formerly subaerial surface is now buried below the seafloor. This dissertation discusses the use of geophysical data in identifying environments, resources, and evidence of environmental modification at areas that are currently buried below the seafloor on the outer continental shelf. The use of methods and data grounded in physical geography can ultimately add to patterns of past human behavior

Evaluation of MERTK evolution and efferocytosis signalling

The TAM (TYRO3, AXL, and MERTK) family of receptor tyrosine kinases allow phagocytes to engage in the phagocytic removal of apoptotic cells. Although all three members of the TAM family are structurally homologous and function in a similar fashion, both human genome-wide association studies and knockout mice models have demonstrated that MERTK is the critical member of the TAM family for maintaining homeostasis. In this thesis, an evolutionary analysis was used to provide insight into the function of MERTK. Selection analysis in primates unexpectedly revealed a high degree of recent positive selection in MERTK’s signal peptide and transmembrane domain, absent from TYRO3 and AXL. Reconstruction of hominid and primate ancestral signal peptides revealed three nonsynonymous mutations in humans, with a G14C mutation producing a potential non-B DNA cruciform motif, which may regulate MERTK expression. Reconstruction of MERTK’s transmembrane domain determined that humans acquired three amino acid substitutions and two insertion/deletion mutations (INDELs) which added four amino acids. These new amino acids were largely leucines and isoleucines, and create a new interaction motif that increased self-clustering of MERTK. Although we found no significant difference among human MERTK and primate- or hominid-ancestral reconstructed signal peptides in expression levels or protein trafficking, recent evolutionary changes in MERTK’s transmembrane revealed significantly higher self-clustering with human MERTK, and hominid ancestral, compared to the reconstructed primate-ancestral transmembrane. This project highlights the importance of recent MERTK evolution, which has increased self-clustering