factorial - Why does 0! = 1? - Mathematics Stack Exchange The theorem that $\binom {n} {k} = \frac {n!} {k! (n-k)!}$ already assumes $0!$ is defined to be $1$ Otherwise this would be restricted to $0 <k < n$ A reason that we do define $0!$ to be $1$ is so that we can cover those edge cases with the same formula, instead of having to treat them separately We treat binomial coefficients like $\binom {5} {6}$ separately already; the theorem assumes
Possible references for semigroup approach to Markov processes Although I'm not anything remotely close to an expert (quite the opposite really), it seems to me that the standard reference for the kind of things you're looking for is the book Markov Processes: Characterization and Convergence by Ethier and Kurtz, which studies Markov processes through the lens of operator semigroups in a systematic (and very general) way For a "friendlier" and more
Prove by induction that $n! gt;2^n$ - Mathematics Stack Exchange Hint: prove inductively that a product is $> 1$ if each factor is $>1$ Apply that to the product $$\frac {n!} {2^n}\: =\: \frac {4!} {2^4} \frac {5}2 \frac {6}2 \frac {7}2\: \cdots\:\frac {n}2$$ This is a prototypical example of a proof employing multiplicative telescopy Notice how much simpler the proof becomes after transforming into a form where the induction is obvious, namely: $\:$ a
When 0 is multiplied with infinity, what is the result? What I would say is that you can multiply any non-zero number by infinity and get either infinity or negative infinity as long as it isn't used in any mathematical proof Because multiplying by infinity is the equivalent of dividing by 0 When you allow things like that in proofs you end up with nonsense like 1 = 0 Multiplying 0 by infinity is the equivalent of 0 0 which is undefined
Finding common ancestors in DAG - Mathematics Stack Exchange I'd like to implement some new algorithm for discovering common ancestors in genealogical software I found an article with a relevant name Finding Common Ancestors and Disjoint Paths in DAGs but