FPTAS for optimizing polynomials over the mixed-integer points of polytopes in fixed dimension

We show the existence of a fully polynomial-time approximation scheme (FPTAS) for the problem of maximizing a non-negative polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed. Moreover, using a weaker notion of approximation, we show the existence of a fully polynomial-time approximation scheme for the problem of maximizing or minimizing an arbitrary polynomial over mixed-integer sets in convex polytopes, when the number of variables is fixed.Comment: 16 pages, 4 figures; to appear in Mathematical Programmin

On the Computational Complexity and Geometry of the First-Order Theory of the Reals, Part I

PLEASE NOTE: The original Technical Report TR00853 is missing. A copy can be found at http://www.sciencedirect.com/science/article/pii/S0747717110800033This paper published in the "Journal of Symbolic Computation" 13 (1992) 255-35

Animal Abuse and Psychopathy: Examining Psychopathic Personality Traits in both Animal Abusers and Non-Abusers

Despite the soaring popularity of household pets in America, very little thought is given about the rights of our domesticated animals to exist free from harm and abuse. Little research has been done concerning animal abuse as most studies measured animal abuse dichotomously, providing essentially no contextual information. The purpose of this study was to examine the relationship between psychopathy and animal abuse using the Psychopathic Personality Traits Scale (PPTS; Boduszek et al., 2016) and the Boat Inventory on Animal-Related Experiences (Boat Inventory; Boat, 1994) while accounting for other factors including what types of animals are abused and the methods of abuse. Similarly, this study used three regression models to compare psychopathy scores (using the PPTS) among participants who endorsed a history of hurting, torturing, or killing an animal, and then participants who did not. Overall, the results showed that individuals who had hurt, tortured, or killed an animal did, indeed, score higher on the PPTS than their non-abusing counterparts. More importantly, a history of abusing animals was a significant predictor of PPTS score. The results also provided descriptive data about the abuse, finding that dogs and cats were by far the most abused animal and the most popular abuse method included hitting, beating, or kicking an animal. Finally, three other factors acted as significant predictors across all three models: being female led to a significantly lower PPTS score while both abuse level and full-time employment positively affected PPTS score. Given the possibility of an important relationship between animal abuse and psychopathy score, more exploratory research is clearly warranted

The Complexity of Drawing Graphs on Few Lines and Few Planes

It is well known that any graph admits a crossing-free straight-line drawing in $\mathbb{R}^3$ and that any planar graph admits the same even in $\mathbb{R}^2$. For a graph $G$ and $d \in \{2,3\}$, let $\rho^1_d(G)$ denote the minimum number of lines in $\mathbb{R}^d$ that together can cover all edges of a drawing of $G$. For $d=2$, $G$ must be planar. We investigate the complexity of computing these parameters and obtain the following hardness and algorithmic results. - For $d\in\{2,3\}$, we prove that deciding whether $\rho^1_d(G)\le k$ for a given graph $G$ and integer $k$ is ${\exists\mathbb{R}}$-complete. - Since $\mathrm{NP}\subseteq{\exists\mathbb{R}}$, deciding $\rho^1_d(G)\le k$ is NP-hard for $d\in\{2,3\}$. On the positive side, we show that the problem is fixed-parameter tractable with respect to $k$. - Since ${\exists\mathbb{R}}\subseteq\mathrm{PSPACE}$, both $\rho^1_2(G)$ and $\rho^1_3(G)$ are computable in polynomial space. On the negative side, we show that drawings that are optimal with respect to $\rho^1_2$ or $\rho^1_3$ sometimes require irrational coordinates. - Let $\rho^2_3(G)$ be the minimum number of planes in $\mathbb{R}^3$ needed to cover a straight-line drawing of a graph $G$. We prove that deciding whether $\rho^2_3(G)\le k$ is NP-hard for any fixed $k \ge 2$. Hence, the problem is not fixed-parameter tractable with respect to $k$ unless $\mathrm{P}=\mathrm{NP}$

Alternative Seating in Middle School Math: Effects on Student Motivation

The use of alternative classroom seating has recently become a topic of interest in education. Many teachers have redesigned their classrooms to include many forms of classroom seating aside from traditional student desks. These forms of seating include but are not limited to: therapy balls, therapy cushions, and beanbag chairs. While this movement has certainly caused a buzz, little research has been completed that speaks to its educational impact on students when multiple forms of seating are available to students. Most critically, the potential impact of alternative classroom seating on students’ motivation and self-efficacy has also yet to be studied. This causal-comparative research study represents one of the first attempts to fill this research gap. Using the Motivated Strategies for Learning Questionnaire (MSLQ), this study examined the differences in motivational aspects present when students in middle school grades six and seven were either exposed to multiple forms of alternative seating or did not have exposure to alternative seating. Student participant data were collected from the math classes of intermediate schools in rural, southwestern Virginia. Data were then analyzed using a one-way multivariate analysis of variance (MANOVA) with the subscales of the MSLQ of: self-efficacy, intrinsic value, cognitive strategy use, and self-regulation representing the dependent variables of the study. While significant differences between the two groups were not reported, this study is critical as it provides guidance for further research on the topic of alternative classroom seating and insight to the educational impact on the use of alternative classroom seating