Continued fraction digit averages an Maclaurin's inequalities

A classical result of Khinchin says that for almost all real numbers $\alpha$, the geometric mean of the first $n$ digits $a_i(\alpha)$ in the continued fraction expansion of $\alpha$ converges to a number $K = 2.6854520\ldots$ (Khinchin's constant) as $n \to \infty$. On the other hand, for almost all $\alpha$, the arithmetic mean of the first $n$ continued fraction digits $a_i(\alpha)$ approaches infinity as $n \to \infty$. There is a sequence of refinements of the AM-GM inequality, Maclaurin's inequalities, relating the $1/k$-th powers of the $k$-th elementary symmetric means of $n$ numbers for $1 \leq k \leq n$. On the left end (when $k=n$) we have the geometric mean, and on the right end ($k=1$) we have the arithmetic mean. We analyze what happens to the means of continued fraction digits of a typical real number in the limit as one moves $f(n)$ steps away from either extreme. We prove sufficient conditions on $f(n)$ to ensure to ensure divergence when one moves $f(n)$ steps away from the arithmetic mean and convergence when one moves $f(n)$ steps away from the geometric mean. For typical $\alpha$ we conjecture the behavior for $f(n)=cn$, $0<c<1$. We also study the limiting behavior of such means for quadratic irrational $\alpha$, providing rigorous results, as well as numerically supported conjectures.Comment: 32 pages, 7 figures. Substantial additions were made to previous version, including Theorem 1.3, Section 6, and Appendix

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

MUMS: Mobile Urinalysis for Maternal Screening

Pregnant women in low-income communities often lack access to the necessary healthcare for successful births. This is frequently due to the high costs of medical care, the remote location of patients, and the infrequency of primary care medical visits. To address this inequity, we have created a mobile application and imaging unit that allows for the low-cost implementation of urinalysis testing, which will aid in the detection of warning signs for prenatal health risks. From a single photo taken with a tablet camera, our application digitizes the results of a standard urinalysis test strip, displays the test results, and tracks the patient test histories. Using early, affordable urinalysis, we can increase the rates early detection, intervention, and successful pregnancies. Our results have shown that our solution can accurately estimate the concentrations of biological compounds found in urine when compared to visual approximations of color comparison charts. Our device is not only more efficient than the alternative, but also more efficient at screening for and detecting potentially fatal health conditions in pregnant women. Ultimately, our solution is a frugal and mobile urinalysis alternative that can feasibly be implemented in rural communities in order to increase early detection of pregnancy complications, allow for early intervention, and improve the probability of successful pregnancies

Lightweight UAV Launcher

This report discusses the design, construction, and testing of a lightweight, portable UAV launcher. There is a current need for a small team of soldiers to launch a US Marine Tier II UAV in a remote location without transport. Research was conducted into existing UAV launcher designs and the pros and cons of each were recorded. This research served as a basis for concept generation during the initial design development stage. It was required that the design weigh less than 110 lbs, occupy a smaller volume than 48 x 24 18 in its collapsed state, be portable by a single soldier, able to be operated by two soldiers, and launch a 55 lb UAV at 52.3 ft/s. In this report is the detailed analysis and design of the first prototype of such a launcher. The launcher operates using a set of six elastic surgical tubing members and an electric winch and features a collapsible frame made of lightweight aluminum 6061-­T6. The launcher succeeded in reaching an exit velocity of 53.7 ft/s, set-­up and tear-­down times under five minutes, weight of 62 lbs, a collapsed volume measuring 43 x 14.5 x 14 and the need for only a single operator