One important such use of Poisson summation concerns theta functions&colon; periodic summations of Gaussians . The split-radix algorithm works by expressing this summation in terms of three smaller summations. For continuous-valued random variables, the summations are replaced by integrals, of course. His brief summations of other techniques are similarly uninformative or inaccurate. The derivation for a continuous-time system is similar, with summations replaced with integrals. Summations in which the order of the summands differs are considered distinct. In both and , the summations over n are a Fourier series, with coefficients . Summations of infinite sequences are called series. The topological structure allows much more flexible usage of infinite summations. The expressions for the present value of such payments are summations of geometric series. The order is immaterial because the nested summations above commute. Some are based on summations of random variables and others on properties of characteristic functions. The following table concludes the Matsubara frequency summations for some simple rational functions "g"("z"). Testimony in the trial concluded on May 15, 1826; summations concluded four days later. In both scenarios, the above summations may be exchanged without affecting the sum.