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Key Points Question Is it more effective to disburse fixed total financial incentives at a constant, increasing, or decreasing rate to encourage physical activity? Findings In this randomized clinical trial, financial incentives for physical activity were significantly more effective at motivating activity during and immediately after a payment period if they were offered at a constant rate rather than an increasing or decreasing rate. Meaning This finding has implications for the design and delivery of incentive programs to promote healthy behaviors.

Let f:N→N be a function. Let A n =(a ij) be the n×n matrix defined by a ij =1 if i=f(j) for some i and j and a ij =0 otherwise. We describe the Jordan canonical form of the matrix A n in terms of the directed graph for which A n is the adjacency matrix. We discuss several examples including a connection with the Collatz 3n+1 conjecture.

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