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

The Effects of Aerobic Exercise on State and Trait Body Image and Physical Fitness Among College Women

Cumulative evidence shows body dissatisfaction is a salient problem among women, and is linked to various psychological disorders and disturbances. While numerous perspectives regarding the negative body-image phenomenon exist, one area of scientific inquiry is the relationship between physical exercise and body image. With the current sociocultural emphasis on physical fitness and attractiveness, the effects of exercise on body image warrants serious consideration. The present investigation compared 57 women who participated in a 13-week aerobic exercise program to 29 sedentary control women on the following variables: state and trait body image, cardiorespiratory fitness, and body composition. The exercising women were randomly assigned to complete the state body-image assessment either immediately following exercise (n = 31) or the following sedentary day (n = 26). This was done to evaluate the reactivity of the immediate exercise experience or environment on body image. The study\u27s results confirmed that, relative to the sedentary women, the exercisers made significant improvements on trait body image, related both to their appearance and their fitness and health. In addition, participants in exercise training significantly improved state body-image satisfaction over time, regardless of whether assessment was immediate or delayed. These results indicate that aerobic exercise is effective in the stable improvement of state and trait body image. Furthermore, the exercisers exhibited significant gains in cardiorespiratory fitness, while the sedentary women did not. Although body fat was substantially lowered over time among the exercisers, these reductions were not significant compared to the controls. The exercisers\u27 favorable outcomes were sustained at 9-10 week follow-up among those who continued to exercise (n = 20). The present investigation offers optimism for improving body satisfaction for women who are willing to participate in aerobic exercise. The extent to which exercise positively alters body image among other populations warrants further research

THE ADJUDICATOR SPEAKS: A STUDY OF CHORAL FESTIVAL ADJUDICATORS’ PRACTICES, PROCEDURES AND PREFERENCES

Choral Festivals, whether competitive or non-competitive, involve preparation, performance, and evaluation. The process of preparing for and performing at a choral festival often leaves the conductor and choir members wondering what the adjudicators (judges) will be listening for and on what basis the choir will receive its critique. Few if any music education publications (including methods textbooks, scholarly publications, and journal articles) have addressed this topic, and those that have addressed it have failed to include the adjudicator’s point of view. This dissertation addresses the point of view of the choral festival adjudicator, giving insight into the standards of performance that are most critical. Through careful analysis of subjective answers to a group of questions, respected adjudicators from a “vetted” pool point out not only what aspects of performance are considered most important but why these aspects affect choral performance at festivals

Selecting a Tree Care Professional

Suggestions for evaluating tree-care companies are given, with the primary recommendation being that the company have a certified arborist. Anyone can trim trees, but qualified and trained arborists follow a set of professional, legal, and ethical guidelines

Characterization of Alkali Metal Dispensers and Non-Evaporable Getter Pumps in Ultra-High Vacuum Systems for Cold Atomic Sensors

A glass ultrahigh vacuum chamber with rubidium alkali metal dispensers and non-evaporable getter pumps has been developed and used to create a cold atomic sample in a chamber that operates with only passive vacuum pumps. The ion-mass spectrum of evaporated gases from the alkali metal dispenser has been recorded as a function of dispenser current. The efficacy of the non-evaporable getter pumps in promoting and maintaining vacuum has been characterized by observation of the Rb vapor optical absorption on the D2 transition at 780 nm and vacuum chamber pressure rate of rise tests. We have demonstrated a sample of laser-cooled Rb atoms in this chamber when isolated and operating without active vacuum pumps