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. They form a regular tetrahedron, and their negatives form the other one. Every proper rotation of the first tetrahedron also permutes the negatives, hence is a cube rotation. This embeds

G_{1}

as a subgroup of

G_{2}

The embedded group preserves each of these two four-vertex sets and acts transitively on each. Thus the resulting action on all eight cube vertices has two orbits, each of size four.

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For a regular tetrahedron, rotations fixing a vertex rotate about the line from that vertex through the opposite face centre. Their angles are

0, 2 π /3, 4 π /3

, so the stabilizer is a cyclic group of order three. Rotations act transitively on the four vertices, giving

∣ G_{1} ∣ = 4 \cdot 3 = 12

by the orbit-stabilizer theorem.

For a cube, the corresponding axis is the body diagonal through the fixed vertex and its opposite. The three incident edges can be cyclically permuted by the same three angles, so again the stabilizer is cyclic of order three. The orbit has eight vertices. Hence

∣ G_{1} ∣ = 12, ∣ G_{2} ∣ = 24.

(1)

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For a group action, the orbit and stabilizer are

G x = {gx : g \in G}, G_{x} = {g \in G : gx = x} .

(1)

The orbit-stabilizer theorem for a finite group states

∣ G ∣ = ∣ G x ∣ ∣ G_{x} ∣.

(2)

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Past exam of the probability, statistics and options I course of the University of Oxford / 2011 / A / 1 / c / Solution by

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Suppose a rational function

P (t) / Q (t) \in F_{p} (t)

, with nonzero polynomials

P, Q

, had

p

th power

t

. Then

P (t)^{p} = tQ (t)^{p}

, so comparison of degrees would give

p de g P = 1 + p de g Q

, impossible modulo

p

. Thus

X^{p} - t

has no root in

K

, and part (b) proves it is irreducible.

In a splitting field containing

α

with

α^{p} = t

X^{p} - t = (X - α)^{p} .

(1)

It has one distinct root with multiplicity

p

, so it is inseparable. This is the purely inseparable polynomial over a rational function field.

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Let

α

be a root in a splitting field. The preceding characteristic-

p

identity gives

X^{p} - a = (X - α)^{p}

, so there is only one distinct root. If the polynomial were reducible over

F

, choose a monic proper factor

g \in F [X]

of degree

d

, with

1 \leq d < p

. In the splitting field it must be

g = (X - α)^{d}

. Its coefficient of

X^{d - 1}

- d α

, which belongs to

F

. Since

d

is nonzero and thus invertible in the field

F

, this forces

α \in F

, contradicting the absence of a root there. Therefore $X^{p} - a$ is irreducible over $F$ .

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In field characteristic

p

, the binomial theorem gives

(X - b)^{p} = X^{p} - b^{p}

, since the intermediate binomial coefficients are divisible by the prime number

p

. If

b \in F

is a root of

X^{p} - a

, then

b^{p} = a

, and hence

X^{p} - a = (X - b)^{p} in F [X] .

(1)

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Past exam of the probability, statistics and options I course of the University of Oxford / 2010 / U / 1 / c / Solution by

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The function below counts the initial uniformly chosen state as the first of the

n

visited states, and then generates

n - 1

transitions using the appropriate row of

P

simchain <- function(n, P) {
  if (!is.numeric(n) || length(n) != 1L || !is.finite(n) ||
      n < 1 || n != floor(n)) stop("n must be a positive integer")
  if (!is.matrix(P) || !is.numeric(P) || nrow(P) == 0L ||
      nrow(P) != ncol(P) || any(!is.finite(P)) ||
      any(P < 0) || any(abs(rowSums(P) - 1) > 1e-12)) {
    stop("P must be a finite square stochastic matrix")
  }
  k <- nrow(P)
  state <- sim1(rep(1/k, k))
  counts <- numeric(k)
  counts[state] <- 1
  if (n > 1) {
    for (step in seq_len(n - 1)) {
      state <- sim1(P[state, ])
      counts[state] <- counts[state] + 1
    }
  }
  counts / n
}

Conditional on the current state

i

, sim1(P[i, ]) selects

j

with probability

P_{ij}

, proving the Markov property with the desired transition matrix. Initialization is uniform by the same sampler. Dividing the visit counts by

n

gives the requested Empirical occupation proportions of a finite Markov chain, including a valid result when

n = 1

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Use inverse transform sampling for the categorical draw. Validate a finite nonnegative probability vector, allowing only a small rounding tolerance in its total, then compare a single uniform draw with the cumulative probabilities:

sim1 <- function(p) {
  if (!is.numeric(p) || length(p) == 0L ||
      any(!is.finite(p)) || any(p < 0) ||
      abs(sum(p) - 1) > 1e-12) {
    stop("p must be a nonempty finite probability vector")
  }
  p <- p / sum(p)
  cumulative <- cumsum(p)
  cumulative[length(p)] <- 1
  u <- runif(1)
  as.integer(which(u < cumulative)[1L])
}

The returned value is

j

when

u

falls in the interval between the previous cumulative sum and the

j

th one. That interval has length

p_{j}

, so the output has the required probabilities. Strict comparison skips zero-probability categories even if a simulated uniform value equals an interval boundary. Exactly one call to runif(1) is made.

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R fills matrices columnwise by default. Specifying rowwise filling gives the requested order:

A <- matrix(c(1/3, 2/3, 0,
              1/4, 1/2, 1/4,
              0,   1/4, 3/4), nrow = 3, byrow = TRUE)

The result is a transition matrix, with nonnegative entries and every row summing to one.

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Past exam of the probability, statistics and options I course of the University of Oxford / 2010 / T / 1 / d / Solution by

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For the minimizing player, column one weakly dominates column three, since its payoff is no greater in every row. We may therefore choose an optimum with

q_{3} = 0

. Write

q = (x, 1 - x, 0)

0 \leq x \leq 1

. The three row payoffs are

6 - 5 x

5 - 3 x

, and

1 + 2 x

The maximum of the decreasing line

5 - 3 x

and the increasing line

1 + 2 x

is minimized where they meet:

x = 4/5

, with value

13/5

. At this point the first line is only

2

, so it does not increase the maximum. Hence the candidate is

q_{*} = (4/5, 1/5, 0), v = 13/5.

(1)

A dual certificate establishes optimality against every possible three-column mixture. Player I's strategy

p_{*} = (0, 2/5, 3/5)

gives column payoffs

(13/5, 13/5, 16/5)

, guaranteeing at least

13/5

. The displayed

q_{*}

holds every row payoff to at most

13/5

. These matching guarantees prove the claimed value and optimality.

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Player I can choose each row with probability

1/ m

. Against column

j

the expected payoff is

m^{- 1} \sum_{i} a_{ij}

. Nonnegativity and a strictly positive entry in each column make every one of these finitely many numbers strictly positive. Their minimum

δ = min_{j} \frac{1}{m} \sum_{i} a_{ij} > 0

(1)

is therefore a guaranteed payoff against every mixture of columns. By the minimax theorem, $v \geq δ > 0$ .

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Against

q

, choosing row

i

gives Player I expected payoff

(A q)_{i} = \sum_{j} a_{ij} q_{j}

. The best response maximizes this number; randomizing among rows cannot exceed the largest row payoff. Player II therefore minimizes that maximum, exactly the printed optimization problem.

The minimax theorem for a finite zero-sum game states

v = q \in Δ_{n} min i max (A q)_{i} = p \in Δ_{m} max j min (p^{T} A)_{j} .

(1)

This common number is the value of a zero-sum game: Player I can guarantee an expected payoff of at least

v

, and Player II can hold it to at most

v

. At optimal mixed strategies their expected transfer is

v

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A mixed strategy for Player II is a probability vector

q = (q_{1}, \dots, q_{n})

, with

q_{j} \geq 0

and

\sum_{j} q_{j} = 1

. The player chooses pure strategy

j

with probability

q_{j}

, independently of the opponent's randomization.

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 Pinned article: Introduction to the OurBigBook Project

Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.

Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!

Video 1.

Intro to OurBigBook

. Source.

We have two killer features:

topics: topics group articles by different users with the same title, e.g. here is the topic for the "Fundamental Theorem of Calculus" ourbigbook.com/go/topic/fundamental-theorem-of-calculus
Articles of different users are sorted by upvote within each article page. This feature is a bit like:
- a Wikipedia where each user can have their own version of each article
- a Q&A website like Stack Overflow, where multiple people can give their views on a given topic, and the best ones are sorted by upvote. Except you don't need to wait for someone to ask first, and any topic goes, no matter how narrow or broad
This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
Figure 1.
Screenshot of the "Derivative" topic page
. View it live at: ourbigbook.com/go/topic/derivative
Video 2.
OurBigBook Web topics demo
. Source.
local editing: you can store all your personal knowledge base content locally in a plaintext markup format that can be edited locally and published either:
- to OurBigBook.com to get awesome multi-user features like topics and likes
- as HTML files to a static website, which you can host yourself for free on many external providers like GitHub Pages, and remain in full control
This way you can be sure that even if OurBigBook.com were to go down one day (which we have no plans to do as it is quite cheap to host!), your content will still be perfectly readable as a static site.
Figure 2.
You can publish local OurBigBook lightweight markup files to either OurBigBook.com or as a static website
.
Figure 3.
Visual Studio Code extension installation
.
Figure 4.
Visual Studio Code extension tree navigation
.
Figure 5.
Web editor
. You can also edit articles on the Web editor without installing anything locally.
Video 3.
Edit locally and publish demo
. Source. This shows editing OurBigBook Markup and publishing it using the Visual Studio Code extension.
Video 4.
OurBigBook Visual Studio Code extension editing and navigation demo
. Source.
Internal cross file references done right:
Infinitely deep tables of contents:
Figure 6.
Dynamic article tree with infinitely deep table of contents
.
Live URL: ourbigbook.com/cirosantilli/chordate
Descendant pages can also show up as toplevel e.g.: ourbigbook.com/cirosantilli/chordate-subclade

All our software is open source and hosted at: github.com/ourbigbook/ourbigbook

Further documentation can be found at: docs.ourbigbook.com

Feel free to reach our to us for any help or suggestions: docs.ourbigbook.com/#contact

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