An analytical solution is derived for the probability that a random pair of individuals from a panmictic population of size N will share ancestors who lived G generations previously. The analysis is extended to obtain...An analytical solution is derived for the probability that a random pair of individuals from a panmictic population of size N will share ancestors who lived G generations previously. The analysis is extended to obtain 1) the probability that a sample of size s will contain at least one pair of (G - 1)<sup>th</sup> cousins;and 2) the expected number of pairs of (G - 1)<sup>th</sup> cousins in that sample. Solutions are given for both monogamous and promiscuous (non-monogamous) cases. Simulation results for a population size of N = 20,000 closely approximate the analytical expectations. Simulation results also agree very well with previously derived expectations for the proportion of unrelated individuals in a sample. The analysis is broadly consistent with genetic estimates of relatedness among a sample of 406 Danish school children, but suggests that a different genetic study of a heterogenous sample of Europeans overestimates the frequency of cousin pairs by as much as one order of magnitude.展开更多
We have found through calculations that the differences between the closest supposed prime numbers other than 2 and 3 defined in the articles are: 2;4: and 6. For those whose difference is equal to 6, we showed their ...We have found through calculations that the differences between the closest supposed prime numbers other than 2 and 3 defined in the articles are: 2;4: and 6. For those whose difference is equal to 6, we showed their origin then we classified them into two categories according to their classes, we showed in which context two prime numbers which differ from 6 are called sexy and in what context they are said real sexy prime. For those whose difference is equal to 4, we showed their origin then we showed that two prime numbers which differ from 4, that is to say two cousin prime numbers, are successive. We made an observation on the supposed prime numbers then we established two pairs of equations from this observation and deduced the origin of the Mersenne number and that of the Fermat number.展开更多
If Goldbach’s conjecture is true, then for each prime number p there is at least one pair of primes symmetric with respect to p and whose sum is 2p. In the multiplicative number theory, covering the positive integers...If Goldbach’s conjecture is true, then for each prime number p there is at least one pair of primes symmetric with respect to p and whose sum is 2p. In the multiplicative number theory, covering the positive integers with primes, during the prime factorization, may be viewed as being the outcome of a parallel system which functions properly if and only if Euler’s formula of the product of the reciprocals of the primes is true. An exact formula for the number of primes less than or equal to an arbitrary bound is given. This formula may be implemented using Wolfram’s computer package Mathematica.展开更多
In a panmictic population of constant size N, random pairs of individuals will have a most recent shared ancestor who lived slightly more than 0.5 log<sub>2</sub>N generations previously, on average. The p...In a panmictic population of constant size N, random pairs of individuals will have a most recent shared ancestor who lived slightly more than 0.5 log<sub>2</sub>N generations previously, on average. The probability that a random pair of individuals will share at least one ancestor who lived 0.5 log<sub>2</sub>N generations ago, or more recently, is about 50%. Those individuals, if they do share an ancestor from that generation, would be cousins of degree (0.5 log<sub>2</sub>N) - 1. Shared ancestry from progressively earlier generations increases rapidly until there is universal pairwise shared ancestry. At that point, every individual has one or more ancestors in common with every other individual in the population, although different pairs may share different ancestors. Those ancestors lived approximately 0.7 log<sub>2</sub>N generations in the past, or more recently. Qualitatively, the ancestries of random pairs have about 50% similarity for ancestors who lived about 0.9 log<sub>2</sub>N generations before the present. That is, about half of the ancestors from that generation belonging to one member of the pair are present also in the genealogy of the other member. Qualitative pairwise similarity increases to more than 99% for ancestors who lived about 1.4 log<sub>2</sub>N generations in the past. Similar results apply to a metric of quantitative pairwise genealogical overlap.展开更多
文摘An analytical solution is derived for the probability that a random pair of individuals from a panmictic population of size N will share ancestors who lived G generations previously. The analysis is extended to obtain 1) the probability that a sample of size s will contain at least one pair of (G - 1)<sup>th</sup> cousins;and 2) the expected number of pairs of (G - 1)<sup>th</sup> cousins in that sample. Solutions are given for both monogamous and promiscuous (non-monogamous) cases. Simulation results for a population size of N = 20,000 closely approximate the analytical expectations. Simulation results also agree very well with previously derived expectations for the proportion of unrelated individuals in a sample. The analysis is broadly consistent with genetic estimates of relatedness among a sample of 406 Danish school children, but suggests that a different genetic study of a heterogenous sample of Europeans overestimates the frequency of cousin pairs by as much as one order of magnitude.
文摘We have found through calculations that the differences between the closest supposed prime numbers other than 2 and 3 defined in the articles are: 2;4: and 6. For those whose difference is equal to 6, we showed their origin then we classified them into two categories according to their classes, we showed in which context two prime numbers which differ from 6 are called sexy and in what context they are said real sexy prime. For those whose difference is equal to 4, we showed their origin then we showed that two prime numbers which differ from 4, that is to say two cousin prime numbers, are successive. We made an observation on the supposed prime numbers then we established two pairs of equations from this observation and deduced the origin of the Mersenne number and that of the Fermat number.
文摘If Goldbach’s conjecture is true, then for each prime number p there is at least one pair of primes symmetric with respect to p and whose sum is 2p. In the multiplicative number theory, covering the positive integers with primes, during the prime factorization, may be viewed as being the outcome of a parallel system which functions properly if and only if Euler’s formula of the product of the reciprocals of the primes is true. An exact formula for the number of primes less than or equal to an arbitrary bound is given. This formula may be implemented using Wolfram’s computer package Mathematica.
文摘In a panmictic population of constant size N, random pairs of individuals will have a most recent shared ancestor who lived slightly more than 0.5 log<sub>2</sub>N generations previously, on average. The probability that a random pair of individuals will share at least one ancestor who lived 0.5 log<sub>2</sub>N generations ago, or more recently, is about 50%. Those individuals, if they do share an ancestor from that generation, would be cousins of degree (0.5 log<sub>2</sub>N) - 1. Shared ancestry from progressively earlier generations increases rapidly until there is universal pairwise shared ancestry. At that point, every individual has one or more ancestors in common with every other individual in the population, although different pairs may share different ancestors. Those ancestors lived approximately 0.7 log<sub>2</sub>N generations in the past, or more recently. Qualitatively, the ancestries of random pairs have about 50% similarity for ancestors who lived about 0.9 log<sub>2</sub>N generations before the present. That is, about half of the ancestors from that generation belonging to one member of the pair are present also in the genealogy of the other member. Qualitative pairwise similarity increases to more than 99% for ancestors who lived about 1.4 log<sub>2</sub>N generations in the past. Similar results apply to a metric of quantitative pairwise genealogical overlap.
