Application of character estimates to the number of 𝑇2-systems of the alternating group

We use character theory and character estimates to show that the number of {T_{2}} -systems of the alternating group {A_{n}} is at least \frac{1}{8n\sqrt{3}}\exp\biggl{(}\frac{2\pi}{\sqrt{6}}n^{1/2}\biggr{)}(1+o(1)). Applying this result, we obtain a lower bound for the number of connected components of the product replacement graph {\Gamma_{2}(A_{n})} .

Effective Strategies to Reduce Sodium in the Local Food Environment at Worksites and Congregate Sites (P04-028-19)

Objectives Excessive sodium consumption is a worldwide public health concern warranting food environment initiatives. The purpose of this study was to examine sodium reduction strategies utilized by the San Antonio Sodium Reduction Initiative (SRI) at local worksite and congregate site cafeterias. Methods A comprehensive nutrient analysis was conducted before and after the SRI to calculate sodium reduction on targeted food items across menus from six worksite cafeterias and three congregate meal programs. Chi square analysis was used to compare SRI strategies used (lower sodium product replacement; removal of a high sodium ingredient; reduced ingredient serving size; and combined strategies) and their resulting sodium reduction categories per item (<200, 200–499, 500–999, or ≥1000 mg of sodium). Results Across all menus, sodium was reduced by an average of 314 mg (SD = 332, n = 203, 32.3%). Most foods items were reduced by 200–499 mg of sodium (n = 86, 42.4%) and <200 mg of sodium (n = 84, 41.4%). Lower sodium product replacement was used most frequently (n = 133, 65.5%), followed by reduced ingredient serving size (n = 35, 17.2%), combined strategies (n = 18, 8.9%), and removal of a high sodium ingredient (n = 17, 8.4%). Significant differences were found between strategy and sodium reduction category (χ2 = 18.485, P = 0.030). Lower sodium product replacement had the largest percentage of foods reduced by > 500–999 mg (n = 16, 12.0%) and ≥1000 mg (n = 10, 7.5%). Removal of a high sodium ingredient impacted all categories with the highest percentage in <200 mg (n = 10, 58.5%). Reduced ingredient serving size had the largest impact in <200 mg (n = 23, 65.7%) and the least amount of food items reduced by ≥ 500 mg (n = 1, 2.9%). Combined strategies reduced food items most commonly by 200–499 mg (n = 8, 44.4%) followed by <200 mg (n = 7, 39.9%). Conclusions Replacing, reducing, or eliminating high sodium foods are effective strategies to reduce sodium offered at worksites and congregate sites. Results from this study suggest the amount of sodium reduction varies by strategy. As organizations and sodium reduction initiatives consider sodium reduction strategies, desired impact as well as feasibility should be considered. Funding Sources Centers for Disease Control and Prevention.