By Adam Bobrowski

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**Additional resources for An Operator Semigroup in Mathematical Genetics**

**Sample text**

A particular property of the linear operation just performed is that all scalars involved here are non-negative n αi = 1. In other words, we are dealing here with a convex and add up to 1: i=1 combination. It is clear that convex combinations of distributions are distributions as well. 2 Normed Space Another noteworthy property of l 1 is that it is a normed space. More specifically, to each vector x = (ξi )i∈I ∈ l 1 we assign its norm, defined as x := |ξi |. i∈I This function has the following properties that agree well with the interpretation of x as the vector’s length: 1.

More specifically, sometimes we may be able, at least theoretically, think of conditional probabilities pi, j = Pr(X (t) = j|X (s) = i), i, j ∈ I. These probabilities form again a matrix, but this matrix should not be confused with the joint probability matrix of two random variables, a member of M. This matrix should rather be identified with a linear, bounded operator in l 1 , as we will explain now. If initially the locus is at state i, then at time t with probability pi, j it will be at state j.

To see this, let An = {i ∈ I, ξi ≥ ξn,i }. We have |ξn,i − ξi | = i∈I |ξn,i − ξi | + i∈An =2 (ξn,i − ξi ) = i ∈An |ξn,i − ξi | = i∈An |ξn,i − ξi | + 1 − 1 + i∈An (ξi − ξn,i ) i∈An ηn,i , i∈I where ηn,i equals 2|ξn,i − ξi | for i ∈ An and zero otherwise. Since ηn,i ≤ 2ξi for all n ≥ 1 and i ∈ I, the claim follows by the Lebesgue Dominated Convergence Theorem. e. that it is ‘without holes’. To explain, if there were ‘a hole’ in this space, we would be able to find a sequence (xn )n≥1 of elements of l 1 that would lie close to this ‘hole’ and, hence, ‘close to each other’, and yet for obvious reasons would not converge to any element of the space.