Generalizing the Kantorovich Metric to Projection-Valued Measures

Given a compact metric space $X$, the collection of Borel probability measures on $X$ can be made into a compact metric space via the Kantorovich metric. We partially generalize this well known result to projection-valued measures. In particular, given a Hilbert space $\mathcal{H}$, we consider the collection of projection-valued measures from $X$ into the projections on $\mathcal{H}$. We show that this collection can be made into a complete and bounded metric space via a generalized Kantorovich metric. However, we add that this metric space is not compact, thereby identifying an important distinction from the classical setting. We have seen recently that this generalized metric has been previously defined by F. Werner in the setting of mathematical physics. To our knowledge, we develop new properties and applications of this metric. Indeed, we use the Contraction Mapping Theorem on this complete metric space of projection-valued measures to provide an alternative method for proving a fixed point result due to P. Jorgensen. This fixed point, which is a projection-valued measure, arises from an iterated function system on $X$, and is related to Cuntz Algebras.Comment: The new version of this article has been updated to correct an error (proof of Theorem 3.1 in the previous version). We would like to acknowledge Krystal Taylor for identifying the error. The new version has also been shortened from the previous version (removed Section 4) to match the final publicatio

CARDIOVASCULAR DISEASE RISK FACTORS IN STRUCTURAL FIREFIGHTERS

This dissertation is composed of two manuscripts assessing cardiovascular disease (CVD) risk in structural firefighters. Study 1 compared traditional CVD risk factors and health-related behaviors between professional and volunteer firefighters. Online questionnaires were sent to approximately 4000 firefighters in the state of Kentucky. The results indicated that 90% of volunteer and 58% of professional firefighters were classified as moderate-to-high CVD risk. Volunteer firefighters were significantly (p \u3c 0.001) older and more likely (p = 0.026) to be current cigarette smokers. The mean body mass index among all firefighters in the sample was 30.8 kg·m-2. Nearly 60% of volunteer firefighters were obese. Obese firefighters were more likely (p \u3c 0.05) to have been diagnosed as diabetic or pre-diabetes compared to overweight and normal weight firefighters. These results indicate that volunteer firefighters may be at a greater CVD risk compared to professional firefighters. Study 2 investigated the association of cardiorespiratory fitness (CRF) and body fat with arterial stiffness in professional firefighters. Forty-six male professional structural firefighters performed a maximal graded exercise test in personal protective equipment and noninvasive arterial stiffness assessments before and for 60 minutes post-exercise. Percent body fat was measured with a bioelectrical impedance analyzer. Firefighters were stratified into fit (VO2peak ≥ 48.3 ml·kg-1·min-1) and unfit (VO2peak \u3c 48.3 ml·kg-1·min-1) groups. Pulse wave velocity, an indicator of arterial stiffness, was significantly lower overall in the fit group compared to the unfit group (p \u3c 0.001). However, the fit group had significantly less relative body fat compared to the unfit group. Thus, when controlling for the confounding effects of fatness, the results indicated that there was a significant effect of relative body fat (p \u3c 0.001) but no effect of fitness on arterial stiffness (p = 0.490). This indicated that relative body fat was responsible for the difference in arterial stiffness and not the fitness stratification. The unfit group displayed a significantly higher average carotid-femoral pulse wave velocity of 1.004 m·s-1 which may increase the risk of a sudden cardiac event by 14%. In addition, there was no main effect for time (p = 0.794) or group x time interaction (p = 0.906). Most resting measures of central and brachial pressure were significantly higher in the unfit group. Results from this dissertation indicated that volunteer firefighters have a greater CVD risk than professional firefighters. Furthermore, relative body fat has a greater effect on arterial stiffness than cardiorespiratory fitness. Collectively, these findings indicate the need for innovative weight management strategies to decrease CVD risk among structural firefighters

Risk Taking Runners Slow More in the Marathon

Much research has explored the physiological, energetic, environmental, and psychological factors that influence pacing in endurance events. Although this research has generally neglected the role of psychological variation across individuals, recent studies have hinted at its importance. Here we conducted an online survey of over 1,300 marathon runners, testing whether any of five psychological constructs—competitiveness, goal achievement, risk taking in pace (RTP), domain-specific risk taking, and willingness to suffer in the marathon—predicted slowing in runners’ most recent marathons. Analyses revealed that RTP—the extent to which runners agreed that they began the marathon at a pace that was so fast that it would jeopardize their capacity to maintain this pace throughout the event—was a robust predictor of marathon slowing. RTP proved a substantial predictor even in regression models controlling for the other psychological constructs, training, experience, and other known pacing correlates. This result suggests that marathoners consider trade-offs when making pacing decisions, and that individuals vary in their pacing decision making

Telegram, 1932 May 21, Washington, D.C., to Amelia Earhart, London

Honorable Trubee Davison, War Department, cablegram to Amelia Earhart, “Family and Air Corps join enthusiastic congratulations,” May 21, 193

The effects of age, sex, heat stress, and finish time on pacing in the marathon

Research has suggested that faster, women, and older runners are more likely to run at a consistent pace during marathon races. Therefore, the purpose of this study was to determine the simultaneous influences of age, sex, heat stress and finish time on marathon pacing. Pacing was defined as the mean velocity of the last 12.2 kilometers divided by the mean velocity of the first 30 kilometers. Subjects included 22,990 men and 13,233 women runners from the 2007 and 2009 Chicago marathons. The average ambient temperatures during the 2007 and 2009 marathons were 26.67 °C and 2.77 °C, respectively. Multiple regression analysis indicated that age, sex, heat stress, and overall finish time (p\u3c0.01 for each) were simultaneous independent elements of pacing. Women were consistently better pacers than men in both marathons though the gender difference increased from cold to warm race temperatures. Coaches and runners can use these findings to improve the likelihood for more optimal pacing

Generalizing the Kantorovich Metric to Projection-Valued Measures: With an Application to Iterated Function Systems

Given a compact metric space X, the collection of Borel probability measures on X can be made into a compact metric space via the Kantorovich metric (Indiana Univ. Math. J. 30(5):713-747, 1981). We partially generalize this well known result to projection-valued measures. In particular, given a Hilbert space H, we consider the collection of projection-valued measures from X into the projections on H. We show that this collection can be made into a complete and bounded metric space via a generalized Kantorovich metric. However, we add that this metric space is not compact, thereby identifying an important distinction from the classical setting. We have seen recently that this generalized metric has been previously defined by F. Werner in the setting of mathematical physics in 2004 (J. Quantum Inf. Comput. 4(6):546-562, 2004). We develop new properties and applications of this metric. Indeed, we use the Contraction Mapping Theorem on this complete metric space of projection-valued measures to provide an alternative method for proving a fixed point result due to P. Jorgensen (see Adv. Appl. Math. 34(3):561-590, 2005; Operator Theory, Operator Algebras, and Applications, pp. 13-26, Am. Math. Soc., Providence, 2006). This fixed point, which is a projection-valued measure, arises from an iterated function system on X, and is related to Cuntz algebras. We conclude this document with a discussion of unitary representations of the Baumslag- Solitar group which arise from the Cantor set. We identify a family of partial isometries which can be used to construct the unitary operators which realize the representation of the Baumslag-Solitar group. These partial isometries satisfy relations similar to the Cuntz algebra relations

American Geographical Society of New York Records, 1723-2010, bulk 1854-2000

Correspondence between Isaiah Bowman and F. Trubee Davison, President of the American Museum of Natural History and American Geographical Society Council member