Reconstruction of the Bellman-Harris branching
نویسنده
چکیده
Consider the single-species, independent-particle, Bellman-Harris 9 branching process, defined by a progeny number distribution, and a particle 10 lifetime distribution. In this paper, we explore the existence and uniqueness of 11 the inverse problem, where one wishes to solve for the progeny number or lifetime 12 distribution given information about the total number distribution. Results 13 are derived for the uniqueness and reconstructibility of these two distributions 14 from two types of information: the extinction time probability of the entire 15 process (extinction time distribution), and the distribution of the total number 16 of particles at a fixed time. Assuming perfect information of either type, we 17 seek to determine if the progeny number distribution or the lifetime probability 18 distribution functions can be uniquely determined. We demonstrate that the 19 distribution of extinction times allows us to formally determine either the progeny 20 number distribution or the lifetime distribution. Furthermore, we show that these 21 constructions are unique. For a known total number distribution, given at a 22 specific time, we show that the lifetime distribution is locally unique and that the 23 progeny distribution is globally unique. Our results are presented through four 24 theorems, each describing the constructions in the four distinct cases. 25 PACS numbers: 02.50.Ey,05.10.Gg,02.30.Rz 26 AMS classification scheme numbers: 45Q05,92B05,60G99 27
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