Generalized Bell numbers and zeros of successive derivatives of an entire function

AbstractSix different formulations equivalent to the statement that, for n ⩾ 2, the sum ∑k = 1n (−1)kS(n, k) ≠ 0, where the S(n, k) are Stirling numbers of the second kind, are shown to hold. Using number-theoretic methods, a sufficient condition for the above statement to be true for a set of positive integers n having density 1 is then obtained. It remains open whether it is true for all n > 2. The equivalent statements then yield information on the irreducibility of the polynomials ∑k = 1n S(n, k)tk = 1 over the rationals, the nonreal zeros for successive derivatives (ddz)nexp(eiz), a gap theorem for the nonzero coefficients of exp(−ez), and the continuous solution of the differential-difference equation ƒ(x) = 1, 0 ⩽ x < 1, ƒ′(x) = −¦x¦ƒ(x − 1), 1 ⩽ x < ∞, where ∥ denotes the greatest integer function

Shifting Resources and Focus to Meet the Goals of the National HIV/AIDS Strategy: The Enhanced Comprehensive HIV Prevention Planning Project, 2010-2013

In September 2010, CDC launched the Enhanced Comprehensive HIV Prevention Planning (ECHPP) project to shift HIV-related activities to meet goals of the 2010 National HIV/AIDS Strategy (NHAS). Twelve health departments in cities with high AIDS burden participated. These 12 grantees submitted plans detailing jurisdiction-level goals, strategies, and objectives for HIV prevention and care activities. We reviewed plans to identify themes in the planning process and initial implementation. Planning themes included data integration, broad engagement of partners, and resource allocation modeling. Implementation themes included organizational change, building partnerships, enhancing data use, developing protocols and policies, and providing training and technical assistance for new and expanded activities. Pilot programs also allowed grantees to assess the feasibility of large-scale implementation. These findings indicate that health departments in areas hardest hit by HIV are shifting their HIV prevention and care programs to increase local impact. Examples from ECHPP will be of interest to other health departments as they work toward meeting the NHAS goals