Sums and differences of correlated random sets

Many fundamental questions in additive number theory (such as Goldbach's conjecture, Fermat's last theorem, and the Twin Primes conjecture) can be expressed in the language of sum and difference sets. As a typical pair of elements contributes one sum and two differences, we expect that $|A-A| > |A+A|$ for a finite set $A$. However, in 2006 Martin and O'Bryant showed that a positive proportion of subsets of $\{0, \dots, n\}$ are sum-dominant, and Zhao later showed that this proportion converges to a positive limit as $n \to \infty$. Related problems, such as constructing explicit families of sum-dominant sets, computing the value of the limiting proportion, and investigating the behavior as the probability of including a given element in $A$ to go to zero, have been analyzed extensively. We consider many of these problems in a more general setting. Instead of just one set $A$, we study sums and differences of pairs of \emph{correlated} sets $(A,B)$. Specifically, we place each element $a \in \{0,\dots, n\}$ in $A$ with probability $p$, while $a$ goes in $B$ with probability $\rho_1$ if $a \in A$ and probability $\rho_2$ if $a \not \in A$. If $|A+B| > |(A-B) \cup (B-A)|$, we call the pair $(A,B)$ a \emph{sum-dominant $(p,\rho_1, \rho_2)$-pair}. We prove that for any fixed $\vec{\rho}=(p, \rho_1, \rho_2)$ in $(0,1)^3$, $(A,B)$ is a sum-dominant $(p,\rho_1, \rho_2)$-pair with positive probability, and show that this probability approaches a limit $P(\vec{\rho})$. Furthermore, we show that the limit function $P(\vec{\rho})$ is continuous. We also investigate what happens as $p$ decays with $n$, generalizing results of Hegarty-Miller on phase transitions. Finally, we find the smallest sizes of MSTD pairs.Comment: Version 1.0, 19 pages. Keywords: More Sum Than Difference sets, correlated random variables, phase transitio

Sets Characterized by Missing Sums and Differences in Dilating Polytopes

A sum-dominant set is a finite set $A$ of integers such that $|A+A| > |A-A|$. As a typical pair of elements contributes one sum and two differences, we expect sum-dominant sets to be rare in some sense. In 2006, however, Martin and O'Bryant showed that the proportion of sum-dominant subsets of $\{0,\dots,n\}$ is bounded below by a positive constant as $n\to\infty$. Hegarty then extended their work and showed that for any prescribed $s,d\in\mathbb{N}_0$, the proportion $\rho^{s,d}_n$ of subsets of $\{0,\dots,n\}$ that are missing exactly $s$ sums in $\{0,\dots,2n\}$ and exactly $2d$ differences in $\{-n,\dots,n\}$ also remains positive in the limit. We consider the following question: are such sets, characterized by their sums and differences, similarly ubiquitous in higher dimensional spaces? We generalize the integers in a growing interval to the lattice points in a dilating polytope. Specifically, let $P$ be a polytope in $\mathbb{R}^D$ with vertices in $\mathbb{Z}^D$, and let $\rho_n^{s,d}$ now denote the proportion of subsets of $L(nP)$ that are missing exactly $s$ sums in $L(nP)+L(nP)$ and exactly $2d$ differences in $L(nP)-L(nP)$. As it turns out, the geometry of $P$ has a significant effect on the limiting behavior of $\rho_n^{s,d}$. We define a geometric characteristic of polytopes called local point symmetry, and show that $\rho_n^{s,d}$ is bounded below by a positive constant as $n\to\infty$ if and only if $P$ is locally point symmetric. We further show that the proportion of subsets in $L(nP)$ that are missing exactly $s$ sums and at least $2d$ differences remains positive in the limit, independent of the geometry of $P$. A direct corollary of these results is that if $P$ is additionally point symmetric, the proportion of sum-dominant subsets of $L(nP)$ also remains positive in the limit.Comment: Version 1.1, 23 pages, 7 pages, fixed some typo

A step in the right direction: streambank restoration efforts at the Botanical Garden of the Ozarks

The Botanical Garden of the Ozarks (BGO) is a unique destination in Northwest Arkansas that draws more than 80,000 visitors a year. While the BGO manages low-input practices, run-off from pesticide application and synthetic fertilizers containing phosphorus and nitrogen are of concern to water quality, habitat, and overall ecological interactions of the BGO streambanks and adjacent Hilton Creek, which flows directly into Lake Fayetteville. One way to reduce pollution to waterbodies is through the use of riparian buffers. This project sought to establish a riparian buffer immediately adjacent to a portion of Hilton Creek in an effort to improve ecological functions and water quality. The hypothesis of this study is that the streambank restoration will increase plant abundance and diversity and improve riparian habitat quality, thus enhancing ecological functions of the Hilton Creek streambank. Pre- and post-restoration assessments were conducted to test this hypothesis. A streambank riparian habitat quality assessment was adapted from the Qualitat del Bosc de Ribera’ (in English, ‘Riparian Habitat Quality’, (QBR)) index and species diversity values based from on-site plant species inventories were analyzed using a Shannon–Wiener Index of diversity. Overall, the pre-restoration QBR index value was calculated as 55 out of 100 and post-restoration QBR index value was calculated as 65 out of 100, suggesting an immediate improvement in riparian habitat quality. Inventoried plant species equated to a pre-restoration Shannon–Wiener Index of diversity value of 2.13, while the post-restoration Shannon–Wiener Index of diversity equaled 2.91, indicating an increase in species diversity. Water quality parameters were recorded to establish baseline values for Hilton Creek to encourage future monitoring of the project site as the streambank restoration matures

Complete Genomic Sequences of Three Salmonella enterica subsp. enterica Serovar Muenchen Strains from an Orchard in San Joaquin County, California.

We present here the complete genome sequences of three Salmonella enterica subsp. enterica serovar Muenchen strains, LG24, LG25, and LG26. All three strains were isolated from almond drupes grown in an orchard in San Joaquin County, California, in 2016. These genomic sequences are nonidentical and will contribute to our understanding of S. enterica genomics

Epidemiological transition to mortality and refracture following an initial fracture

This study sought to redefine the concept of fracture risk that includes refracture and mortality, and to transform the risk into "skeletal age". We analysed data obtained from 3521 women and men aged 60 years and older, whose fracture incidence, mortality, and bone mineral density (BMD) have been monitored since 1989. During the 20-year follow-up period, among 632 women and 184 men with a first incident fracture, the risk of sustaining a second fracture was higher in women (36%) than in men (22%), but mortality risk was higher in men (41%) than in women (25%). The increased risk of mortality was not only present with an initial fracture, but was accelerated with refractures. Key predictors of post-fracture mortality were male gender (hazard ratio [HR] 2.4; 95% CI, 1.79–3.21), advancing age (HR 1.67; 1.53–1.83), and lower femoral neck BMD (HR 1.16; 1.01–1.33). A 70-year-old man with a fracture is predicted to have a skeletal age of 75. These results were incorporated into a prediction model to aid patient-doctor discussion about fracture vulnerability and treatment decisions

ZEB2 and LMO2 drive immature T-cell lymphoblastic leukemia via distinct oncogenic mechanisms

ZEB1 and ZEB2 are structurally related E-box binding homeobox transcription factors that induce epithelial to mesenchymal transitions during development and disease. As such, they regulate cancer cell invasion, dissemination and metastasis of solid tumors. In addition, their expression is associated with the gain of cancer stem cell properties and resistance to therapy. Using conditional loss-of-function mice, we previously demonstrated that Zeb2 also plays pivotal roles in hematopoiesis, controlling important cell fate decisions, lineage commitment and fidelity. In addition, upon Zeb2 overexpression, mice spontaneously develop immature T-cell lymphoblastic leukemia. Here we show that pre-leukemic Zeb2-overexpressing thymocytes are characterized by a differentiation delay at beta-selection due to aberrant activation of the interleukin-7 receptor signaling pathway. Notably, and in contrast to Lmo2-overexpressing thymocytes, these pre-leukemic Zeb2-overexpressing T-cell progenitors display no acquired self-renewal properties. Finally, Zeb2 activation in more differentiated T-cell precursor cells can also drive malignant T-cell development, suggesting that the early T-cell differentiation delay is not essential for Zeb2-mediated leukemic transformation. Altogether, our data suggest that Zeb2 and Lmo2 drive malignant transformation of immature T-cell progenitors via distinct molecular mechanisms

ZEB2 and LMO2 drive immature T-cell lymphoblastic leukemia via distinct oncogenic mechanisms

ZEB1 and ZEB2 are structurally related E-box binding homeobox transcription factors that induce epithelial to mesenchymal transitions during development and disease. As such, they regulate cancer cell invasion, dissemination and metastasis of solid tumors. In addition, their expression is associated with the gain of cancer stem cell properties and resistance to therapy. Using conditional loss-of-function mice, we previously demonstrated that Zeb2 also plays pivotal roles in hematopoiesis, controlling important cell fate decisions, lineage commitment and fidelity. In addition, upon Zeb2 overexpression, mice spontaneously develop immature T-cell lymphoblastic leukemia. Here we show that pre-leukemic Zeb2-overexpressing thymocytes are characterized by a differentiation delay at beta-selection due to aberrant activation of the interleukin-7 receptor signaling pathway. Notably, and in contrast to Lmo2-overexpressing thymocytes, these pre-leukemic Zeb2-overexpressing T-cell progenitors display no acquired self-renewal properties. Finally, Zeb2 activation in more differentiated T-cell precursor cells can also drive malignant T-cell development, suggesting that the early T-cell differentiation delay is not essential for Zeb2-mediated leukemic transformation. Altogether, our data suggest that Zeb2 and Lmo2 drive malignant transformation of immature T-cell progenitors via distinct molecular mechanisms

Moderate to severe gambling problems and traumatic brain injury: A population-based study

Traumatic brain injury (TBI) is a common injury characterized by a change in brain function after an external blow to the head and is associated with substance abuse, psychological distress, risk-taking, and impulsivity. Convenience and clinical samples have also linked TBI to problem gambling, but have not ruled out confounding variables such as hazardous drinking and psychological distress. This study examines the relationship between TBI and moderate to severe problem gambling in a general population probability sample controlling for hazardous drinking and psychological distress. The data were obtained from a 2015–2016 cross-sectional general population telephone survey of adults ages 18+from Ontario, Canada (N = 3809). Logistic regression was used to estimate the association as adjusted odds ratios (AOR). Moderate to severe problem gambling was independently associated with a history of TBI after adjusting for potential confounders (AOR: 2.80), and had a statistically significant relationship with psychological distress (AOR = 2.74), hazardous drinking (AOR = 2.69), and lower educational levels (AOR = 0.37). This study provides further data to suggest a link between TBI and moderate to severe problem gambling; however, more research is needed to determine if there is a causal relationship or the potential implications for prevention and treatment