2021-06-22

12:47

Tags: #category_theory #homotopy_theory

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: #probability #review_material #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: #combinatorics #review_material #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.