In article <rtn72ncd1iv13@corp.supernews.com>, Ty Young <tly@whidbey.net> wrote: >It means increase your distance 10% each week if possible, I do this when I >train for a marathon > Sometimes stated, more generally, that you should not increase milage more than 10% in any given time period. This suggests a math problem, which, given all the recent bandying about of mathematics on this group, is timely and even on-topic. Suppose you take this rule to its logical conclusion: that you cannot increase your "mileage" more than 10% in ANY time interval. Thus, for example, the distance you run in the second millisecond after starting your running program cannot be more than 1.1 times the distance you ran in the first; and so on. One immediately suspects it would be impossible to even get started, but this seems to be wrong. To formulate a more precise statement, let v(u) be your instantaneous velocity at time u. Then the rule requires that v satisfy the following: Integral from t to t+s of v(u)du <= (1.1)* Integral from t to t+s of v(u-s)du, this to hold for all 0 <= s <= t <= T, where T is some finite time horizon. Since integrals of velocities give distances travelled, this translates into English as saying that the distance covered in any time interval of length s cannot exceed 1.1 times the distance covered in the immediately preceeding time interval of length s. Obviously v = 0 is a solution, so a couch potato does not run afoul of the rule. To make the problem more interesting, one reasons that runners like to run as far as possible, and so adds the condition: Choose v(u) so as to maximize Integral from 0 to T of v(u)du ( total distance covered, ) for v subject to the constraints above. It begins to resemble a problem in the calculus of variations. Unfortunately, the problem as stated is still not interesting mathematically since any constant, but arbitrarily large, velocity is a solution. Such solutions require a very ambitious start to the running program, but allow for no increases in mileage after that. There are also solutions which allow mileage to increase: the ordinary exponential function, v(u) = e^u, works as long as T <= ln(1.1). Anybody have an idea of a natural additional constraint to add that would make the problem more interesting? I suspect at least some initial condition should be added, or perhaps a bound on the velocity itself. -- ************************************************************************ Terry R. McConnell Mathematics/304B Carnegie/Syracuse, N.Y. 13244-1150 trmcconn@syr.edu http://barnyard.syr.edu/~tmc ************************************************************************