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# An introduction to string figures by W W. Rouse 1850-1925 Ball

By W W. Rouse 1850-1925 Ball

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Otkl. Argumentom, Vols. 1-8 (1962-1973) and the survey articles of Zverkin, Kamenskii, Norkin, and El'sgol'tz [1], Kamenskii, Norkin, and El'sogl'tz [1], and Mishkis and El'sgol'tz [1]. A very significant contribution to this question was made by Driver [2], who gave a formulation that has also been generalized by Melvin [1, 2]. 4). Suppose 1> is a given absolutely continuous function on [a-r, a]. 4) almost everywhere on [a, a+ A). Of course, in order for this initial-value problem to make sense, the function f must satisfy the following property: If x is any given absolutely continuous function on [a - r, a + A) and if F(t) = f(t,x(t),x(t- r),x(t- r)), a:::; t

Proof. As remarked earlier, there are many ways to obtain this result. The present proof is based on the Laplace transform and other proofs in more general situations will be given later. To apply the Laplace transform, the function f should be bounded by an exponential function. For any compact interval [0, T], one can redefine f as a continuous function so that it is zero outside the interval [0, T + E], E > 0. Then f will be bounded by an exponential function. If we prove the theorem for this case, then the theorem will be valid for 0 :::; t :::; T.

From our experiences so far with neutral equations, it is to be expected that this discontinuity will persist at multiples of r for the solution, if it exists. 5) for t ::::: 0 except at the points kr, k = 0, 1, 2, .... 1). The function X(t) will actually have a continuous first derivative on each interval (kr, (k + 1)r), k = 0, 1, 2, ... , the right- and left-hand limits of X(t) exist at each of the points kr, k = 0, 1, 2, .... 11) + BX(t- r) for t -1- kr, k = 0, 1, 2, .... These assertions are proved easily from the fact that X (t) satisfies the integral equation X(t) = 1 + CX(t- r) +fat [AX(s) + BX(s- r)] ds, t ~ 0.