By G. Hardy

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**Additional resources for A Course Of Pure Mathematics**

**Sample text**

The measure of the amount of heterozygosity across loci can be used as a general indicator of the amount of genetic variability. Loss of heterozygosity in the Wright–Fisher model results from the random genetic drift. k C 1/ Alleles When we are at a generation in which the population has only k C 1 alleles, we would like to know how quickly one of the remaining alleles will be lost. k C 1/ alleles. p/ is a continuous random variable valued in Œ0; 1/. k C 1/ Alleles XT kC1 . p/ then is a random variable valued in @k n .

V/. V/. We can then also form other tensors, with more than one index. A lower index always indicates covariant, an upper one contravariant˝transformation. For example, ˛ the metric tensor, written as gij dxi ˝ dxj , with gij D @x@ i ; @x@ j being the product of those two basis vectors, operates on pairs of tangent vectors. x/dxi ˝ dxj becomes gij . 11) for V D v i @x@ i ; W D wi @x@ i . ) In this formula, v i and wi transform contravariantly, while gij transforms doubly covariantly so that the product as a scalar quantity remains invariant under coordinate transformations.

Fi ;:::;i g/ @k 1 k 0 k corresponds to the state of exactly one further allele out of i0 ; : : : ; ik being eliminated from the population. I / k k Ik In corresponds to the state of at most (any) kC1 alleles being present in the population. n /; and ˚ « H WD f W n ! @k n /; k 2 f1; : : : ; ng: 40 2 The Wright–Fisher Model We shall also use the L2 -product Z . f ; g/ WD . x/ is the Lebesgue measure on the simplex n . fi ;:::;i g/ k 0 k and correspondingly 2 L n [ kD0 ˇ Á n ˇ @k n WD f W n !

### A Course Of Pure Mathematics by G. Hardy

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