Frames over finite fields: Equiangular lines in orthogonal geometry

We investigate equiangular lines in finite orthogonal geometries, focusing specifically on equiangular tight frames (ETFs). In parallel with the known correspondence between real ETFs and strongly regular graphs (SRGs) that satisfy certain parameter constraints, we prove that ETFs in finite orthogonal geometries are closely aligned with a modular generalization of SRGs. The constraints in our finite field setting are weaker, and all but~18 known SRG parameters on $v \leq 1300$ vertices satisfy at least one of them. Applying our results to triangular graphs, we deduce that Gerzon's bound is attained in finite orthogonal geometries of infinitely many dimensions. We also demonstrate connections with real ETFs, and derive necessary conditions for ETFs in finite orthogonal geometries. As an application, we show that Gerzon's bound cannot be attained in a finite orthogonal geometry of dimension~5

Frames over finite fields: Equiangular lines in orthogonal geometry

We investigate equiangular lines in finite orthogonal geometries, focusing specifically on equiangular tight frames (ETFs). In parallel with the known correspondence between real ETFs and strongly regular graphs (SRGs) that satisfy certain parameter constraints, we prove that ETFs in finite orthogonal geometries are closely aligned with a modular generalization of SRGs. The constraints in our finite field setting are weaker, and all but~18 known SRG parameters on $v \leq 1300$ vertices satisfy at least one of them. Applying our results to triangular graphs, we deduce that Gerzon's bound is attained in finite orthogonal geometries of infinitely many dimensions. We also demonstrate connections with real ETFs, and derive necessary conditions for ETFs in finite orthogonal geometries. As an application, we show that Gerzon's bound cannot be attained in a finite orthogonal geometry of dimension~5

Results of the Active by Choice Today (ACT) Randomized Trial for Increasing Physical Activity in Low-Income and Minority Adolescents

Objective - This study reports the results of the Active by Choice Today (ACT) trial for increasing moderate-to-vigorous physical activity (MVPA) in low-income and minority adolescents. Design - The ACT program was a randomized controlled school-based trial testing the efficacy of a motivational plus behavioral skills intervention on increasing MVPA in underserved adolescents. Twenty-four middle schools were matched on school size, percentage minorities, percentage free or reduced lunch, and urban or rural setting before randomization. A total of 1,563 6th grade students (mean age, 11.3 years, 73% African American, 71% free or reduced lunch, 55% female) participated in either a 17-week (over one academic year) intervention or comparison after-school program. Main Outcome Measure - The primary outcome measure was MVPA based on 7-day accelerometry estimates at 2-weeks postintervention and an intermediate outcome was MVPA at midintervention. Results - At midintervention students in the intervention condition engaged in 4.87 greater minutes of MVPA per day (95% CI: 1.18 to 8.57) than control students. Students in intervention schools engaged in 9.11 min (95% CI: 5.73 to 12.48) more of MVPA per day than those in control schools during the program time periods; indicating a 27 min per week increase in MVPA. No significant effect of the ACT intervention was found outside of school times or for MVPA at 2-weeks postintervention. Conclusions - Motivational and behavioral skills programs are effective at increasing MVPA in low-income and minority adolescents during program hours, but further research is needed to address home barriers to youth MVPA

Awareness and use of community walking trails

Abstract Background. Community trail development is an emerging strategy to increase physical activity (PA) among community residents. The purpose of this study was to assess awareness and use of trails and compare perceptions to objective data. Methods. A telephone survey was administered to a stratified sample of adults (N = 1,112) in a southeastern county in the United States. Respondents' home addresses and the locations of trails were entered into a GIS database. A kappa statistic was used to measure agreement between awareness and presence of trails. Differences in reported trail use patterns by sex, race, education, and PA levels were evaluated. Results. There was no agreement between the awareness and presence of trails (kappa = 0.07). Fifty-six percent of the respondents reported having trails; however, only 33% reported using the trails. Of the trail users, 42% reported being regularly active in moderate-to-vigorous PA (30+ min/day for 5+ days/week), and 51% reported being less active (P < 0.003). Among walkers (≥30 min/day for ≥5 days/week), 49% of regular walkers and 35% of irregular walkers