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

SECOND INTERNATIONAL SYMPOSIUM ON RANAVIRUSES:: A NORTH AMERICAN HERPETOLOGICAL PERSPECTIVE

Ranaviruses are large double stranded DNA viruses of poikilothermic vertebrates including amphibians, reptiles and fish. In North America, ranaviral disease and ranavirus-related die-off events have been documented in all three classes. Ranaviruses are found worldwide, appear to be emerging in some regions, and are increasingly recognized as a threat to many species

Preservice Teachers’ Algebraic Reasoning and Symbol Use on a Multistep Fraction Word Problem

Previous research on preservice teachers’ understanding of fractions and algebra has focused on one or the other. To extend this research, we examined 85 undergraduate elementary education majors and middle school mathematics education majors’ solutions and solution paths (i.e., the ways or methods in which preservice teachers solve word problems) when combining fractions with algebra on a multistep word problem. In this article, we identify and describe common strategy clusters and approaches present in the preservice teachers’ written work. Our results indicate that preservice teachers’ understanding of algebra include arithmetic methods, proportions, and is related to their understanding of a whole

Dynamics of the Fibonacci Order of Appearance Map

The \textit{order of appearance} $z(n)$ of a positive integer $n$ in the Fibonacci sequence is defined as the smallest positive integer $j$ such that $n$ divides the $j$-th Fibonacci number. A \textit{fixed point} arises when, for a positive integer $n$, we have that the $n^{\text{th}}$ Fibonacci number is the smallest Fibonacci that $n$ divides. In other words, $z(n) = n$. In 2012, Marques proved that fixed points occur only when $n$ is of the form $5^{k}$ or $12\cdot5^{k}$ for all non-negative integers $k$. It immediately follows that there are infinitely many fixed points in the Fibonacci sequence. We prove that there are infinitely many integers that iterate to a fixed point in exactly $k$ steps. In addition, we construct infinite families of integers that go to each fixed point of the form $12 \cdot 5^{k}$. We conclude by providing an alternate proof that all positive integers $n$ reach a fixed point after a finite number of iterations.Comment: 10 pages, 2 figure

Generalized Continuous and Discrete Stick Fragmentation and Benford's Law

Inspired by the basic stick fragmentation model proposed by Becker et al. in arXiv:1309.5603v4, we consider three new versions of such fragmentation models, namely, continuous with random number of parts, continuous with probabilistic stopping, and discrete with congruence stopping conditions. In all of these situations, we state and prove precise conditions for the ending stick lengths to obey Benford's law when taking the appropriate limits. We introduce the aggregated limit, necessary to guarantee convergence to Benford's law in the latter two models. We also show that resulting stick lengths are non-Benford when our conditions are not met. Moreover, we give a sufficient condition for a distribution to satisfy the Mellin transform condition introduced in arXiv:0805.4226v2, yielding a large family of examples.Comment: 43 pages, 2 figure

Preservice teachers’ pictorial strategies for a multistep multiplicative fraction problem

Previous research has documented that preservice teachers (PSTs) struggle with under- standing fraction concepts and operations, and misconceptions often stem from their understanding of the referent whole. This study expands research on PSTs’ understanding of wholes by investigating pictorial strategies that 85 PSTs constructed for a multistep fraction task in a multiplicative context. The results show that many PSTs were able to construct valid pictorial strategies, and the strategies were widely diverse with respect to how they made sense of an unknown referent whole of a fraction in multiple steps, how they represented the wholes in their drawings, in which order they did multiple steps, and which type of model they used (area or set). Based on their wide range of pictorial strategies, we discuss potential benefits of PSTs’ construction of their own representations for a word problem in developing problem solving skills

The Gendered Division of Housework and Couples’ Sexual Relationships: A Re-Examination

Contemporary men and women increasingly express preferences for egalitarian unions. One recent high profile study (Kornrich, Brines, & Leupp, 2013) found that married couples with more equal divisions of labor had sex less frequently than couples with conventional divisions of domestic labor. Others (Gager & Yabiku, 2010) found that performing more domestic labor was associated with greater sexual frequency, regardless of gender. Both studies drew from the same data source, which was over two decades old. We utilize data from the 2006 Marital and Relationship Survey (MARS) to update this work. We find no significant differences in sexual frequency and satisfaction among conventional or egalitarian couples. Couples where the male partner does the majority of the housework, however, have less frequent and lower quality sexual relationships than their counterparts. Couples are content to modify conventional housework arrangements, but reversing them entirely has consequences for other aspects of their unions