Tags: #higher-algebra/category-theory #homotopy

mapping cone as a pushout and mapping fiber as pullback :

Link to Diagram

  • \(M_f\to X\to Y\) yields a LES in homotopy \(\pi_* M_f \to \pi_* X \to \pi_* Y\).
  • \(X\to Y\to C_f\) yields \(\pi_*(Y, X) = \pi_{*-1} C_f\)?
    • This should be an easy consequence of the LES in homotopy.

Probability Review

Tags: #analysis/probability #projects/review #undergraduate

Link to Diagram

  • Binomial and Poisson distributions tend to normal distributions?
  • Normal distribution rule: 68, 95, 99.7.
  • \begin{align*} {\mathbb{E}}[r(x)] \coloneqq\int r(x) f(x) \,dx && \text{or } \sum_k r(k) P(x=k) .\end{align*}
  • \({\mathsf{Var}}(x) = E[x^2] - E^2[x]\)
  • Marginal densities: if \(f(x, y)\) is a distribution, \begin{align*} f_x(x) \coloneqq\int f(x, y) \,dy && P(x\times y \in A) = \iint_A f(x, y) \,dx\,dy .\end{align*}
    • Independent if \(f(x, y) = f_x(x) f_y(y)\).
  • If \(x\) has density \(f_x\) and \(y \coloneqq r(x)\) for a continuous increasing function, setting \(s \coloneqq r^{-1}\), \(y\) has density \begin{align*} g_y(y) = f_x(s(y)) s'(y) .\end{align*}
  • Given a failure rate \(\lambda\) in failures over time and a time span \(t\), the probability of failing \(k\) times in \(t\) units of time is \(\operatorname{Poisson}(\lambda t)\).
    • The time between failures is \(\operatorname{Exponential}(\lambda)\).

Combinatorics Review

Tags: #projects/review #undergraduate

  • Prop: the number of integer compositions of \(n\) into exactly \(k\) parts is \({n-1 \choose k-1}\), so \(\sum a_i = n\) with \(a_i\geq 1\) for all \(i\).

    • Lay out \(n\) copies of 1 with \(n-1\) blanks between them. Choose to put a plus sign or a comma in each slot: choose \(k-1\) commas to produce \(k\) blocks.
  • Prop: the number of compositions of \(n\) into any number of parts is \(2^{n-1}\).

    • Sum over compositions with exactly \(k\) parts and use the identity \(\sum_{k=1}^{n-1} {n-1 \choose k-1} = 2^{n-1}\).
  • Prop: the number of weak compositions of \(n\) into exactly \(k\) parts is \({n+k-1 \choose k-1}\).

    • Add one to each piece to get a (strong) composition of \(n+k\) into \(k\) parts.
  • Quotient set of compositions by permutations to get integer partitions.

  • Stirling numbers : the number of ways to partition an \(n\) element set into exactly \(k\) nonempty unlabeled disjoint blocks whose disjoint union is the original set.

    • Prop: recurrence for partitions : \begin{align*}s(n+1, k) = s(n, k-1) + ks(n, k).\end{align*}
      • Proof: check if a given partition of \(n+1\) into \(k\) parts contains the singleton \(\left\{{n+1}\right\}\).
        • If so, delete and count partitions of \(n\) into \(k-1\) parts.
        • If not, add \(\left\{{n+1}\right\}\) to any of the \(k\) parts.
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