MathOverflow

I am interested in counting real rooted polynomials with nonnegative integer coefficients, linear term k and constant term 1. For a fixed k it is not too hard to show why there are only finitely many all them all of degree less than or equal k. There...

Let Nk∼Pois(μk/k!)N_k \sim \text{Pois}(\mu^k/k!)Nk​∼Pois(μk/k!) independently for k=1,2,3,4k = 1,2,3,4k=1,2,3,4, and define the signed linear combination
S(μ)=1−N1+9N2−16N3+8N4.S(\mu) = 1 - N_1 + 9N_2 - 16N_3 + 8N_4.S(μ)=1−N1​+9N2​−16N3​+8N4​.Let Pacc(μ)=P(S(μ)>0)P_{\text{acc}}(\mu) = \mathbb{P}(S(\mu) > 0)Pacc​(μ)=P(S(μ)>0). I am interested in bounding the derivative D=dPacc/dμD = dP_{\text...

I noticed that many of the "walking structure" categories (walking morphism, walking fork, walking span, etc.) have filtered colimits. I would like to generalize this result.
At first I thought that every finite category will do, but the walking idempotent is a counterexample. It has not all filtered colimits....

Let $p,q>1$ be multiplicatively independent positive integers. Consider the set
$$S=\{p^n q^m:n,m\in\mathbb{N}\cup\{0\}\}$$sorted in strictly increasing order: $s_1<s_2<s_3<\cdots.$ Define the adjacent ratios $r_k=\frac{s_{k+1}}{s_k},\quad k\ge 1,$ and let $(a_k)_{k\ge 1}$ be the sequence obtained by deleting repeated...