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. For the final all-primes solvability statement, zero is included as a square. The Legendre symbol distinguishes the zero class from the nonzero square and nonsquare classes:

(\frac{a}{p}) = ⎩ ⎨ ⎧ 0, 1, - 1, p ∣ a, p ∤ a and a \equiv x^{2} (mod p) for some x, a is not a square modulo p .

(1)

When the term quadratic residue is reserved for nonzero classes, the zero case must be stated separately, as in this definition.

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / U / U2 / b / iv / Solution by

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x = 1

, the derivative and curvature are

f^{'} (1) = 8

f^{''} (1) = - 8

, so the next iterate is

2

. At

x = 2

, they are

f^{'} (2) = 4

f^{''} (2) = 4

, so the next iterate is

1

. Hence

1 ⟼ 2 ⟼ 1 ⟼ 2 ⟼ \dots .

(1)

The undamped algorithm does not converge to a minimum; it has an exact two-cycle. The implementation in (iii) detects this after returning to one and reports failure. This example illustrates why the local convergence result in (ii) does not justify global success.

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / U / U2 / b / iii / Solution by

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Here

f^{'} (x) = 4 x^{3} - 12 x^{2} + 4 x + 12

and

f^{''} (x) = 12 x^{2} - 24 x + 4

. This implementation uses the undamped Newton step, checks a stationary point's curvature, and reports failure on a two-cycle or exhausted iteration limit:

R
newton_min <- function(x0, tol = 1e-10, maxit = 100L) {
  fp <- function(x) 4*x^3 - 12*x^2 + 4*x + 12
  fpp <- function(x) 12*x^2 - 24*x + 4
  x <- x0
  trace <- x
  for (k in seq_len(maxit)) {
    g <- fp(x)
    h <- fpp(x)
    if (abs(g) <= tol) {
      return(list(x = x, converged = h > 0,
                  reason = if (h > 0) "minimum" else "not a strict minimum",
                  trace = trace))
    }
    if (!is.finite(h) || abs(h) <= .Machine$double.eps)
      stop("Newton step has zero or invalid curvature")
    next_x <- x - g/h
    if (!is.finite(next_x)) stop("Nonfinite Newton iterate")
    trace <- c(trace, next_x)
    n <- length(trace)
    if (n >= 3L && abs(trace[n] - trace[n - 2L]) <= tol*(1 + abs(next_x)))
      return(list(x = next_x, converged = FALSE,
                  reason = "two-cycle", trace = trace))
    x <- next_x
  }
  list(x = x, converged = FALSE, reason = "iteration limit", trace = trace)
}

A sufficiently close start near

- 0.7692923542

converges to the minimum. The derivative cubic has exactly one real root: after

x = 1 + t

it is proportional to

t^{3} - 2 t + 2

, whose local maximum and minimum are both positive, so it crosses zero only once, on the left. Since the quartic tends to

+ \infty

in both directions and its curvature at that stationary point is positive, this is the unique global minimum. The function reports convergence rather than assuming that every starting point must reach it.

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / U / U2 / b / ii / Solution by

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Let

e_{k} = x_{k} - x_{*}

, with

f^{'} (x_{*}) = 0

f^{''} (x_{*}) \neq = 0

. Taylor expansion gives

f^{'} (x_{*} + e) = f^{''} (x_{*}) e + \frac{1}{2} f^{(3)} (x_{*}) e^{2} + O (e^{3}), f^{''} (x_{*} + e) = f^{''} (x_{*}) + f^{(3)} (x_{*}) e + O (e^{2}) .

(1)

Division in the Newton update yields

e_{k + 1} = \frac{f ^{(3)} ( x _{*} )}{2 f ^{''} ( x _{*} )} e_{k}^{2} + O (e_{k}^{3}) .

(2)

Thus convergence is generically quadratic when that coefficient is nonzero. With a locally Lipschitz second derivative and nonzero

f^{''} (x_{*})

, the weaker estimate

∣ e_{k + 1} ∣ \leq C ∣ e_{k} ∣^{2}

also proves local quadratic convergence.

“Quadratic at best” is false as a universal upper bound. If the displayed coefficient vanishes, the order can be higher. For example

f (x) = x^{2} /2 + x^{4} /4

has a minimum at zero, and its Newton optimization update is

x_{k + 1} = 2 x_{k}^{3} / (1 + 3 x_{k}^{2})

, which has cubic convergence. For a nondegenerate quadratic objective the method reaches the stationary point in one step. The expansion proves the standard generic quadratic statement while retaining and correcting the source's stronger wording explicitly.

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / U / U2 / b / i / Solution by

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For scalar minimization, apply Newton's root-finding method to the stationarity equation

f^{'} (x) = 0

. Linearize

f^{'}

at the current point to obtain

x_{k + 1} = x_{k} - \frac{f ^{'} ( x _{k} )}{f ^{''} ( x _{k} )} .

(1)

The iteration requires

f^{''} (x_{k}) \neq = 0

. Near a nondegenerate local minimum, where

f^{'} (x_{*}) = 0

f^{''} (x_{*}) > 0

, suitable smoothness and a sufficiently close starting value give local convergence. A stationary limit must still be checked to be a minimum. This undamped method is not globally convergent and need not decrease

f

at every step.

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / T / T2 / b / ii / Solution by

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The expression

G^{- 1}

presupposes that

G

is invertible; positivity of the stated optimal coordinates alone would not prove that, so use the nonsingularity implicit in this expression. Let

c_{B}

be the first

m

components of

c

and set

y = G^{- T} c_{B}

. Then

G^{T} y = c_{B}

For any nonbasic column

a_{j}

, introduce

x_{j} = ε > 0

while changing the basic vector from

u

u - ε G^{- 1} a_{j}

. This stays feasible for sufficiently small

ε

because every

u_{i} > 0

. Optimality at

u

implies the objective slope

c_{j} - c_{B}^{T} G^{- 1} a_{j} \leq 0

. Therefore

a_{j}^{T} y \geq c_{j}

for every such column, and the basic columns satisfy equality. Thus

y

is dual feasible, independently of the right-hand side.

For

v = G^{- 1} b^{*} \geq 0

, the proposed

x^{*}

is primal feasible and

c^{T} x^{*} = c_{B}^{T} v = y^{T} G v = y^{T} b^{*} .

(1)

Weak duality, valid with unrestricted equality multipliers, proves

x^{*} is optimal

. This also explains the stability of an optimal basis while its new basic coordinates remain nonnegative.

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / T / T2 / b / i / Solution by

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Write

A x = b

A x \leq b

and

- A x \leq - b

. The dual has nonnegative multipliers

u, v

for these inequalities, with objective

b^{T} u - b^{T} v

and condition

A^{T} (u - v) \geq c

. Every real vector

y

can be expressed as

u - v

with

u, v \geq 0

. Hence

D^{'} : min b^{T} y subject to A^{T} y \geq c, y \in R^{m} unrestricted .

(1)

The equality constraints remove the sign restriction on the dual variables.

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / T / T2 / a / iii / Solution by

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Try the dual intersection

3 y_{1} + 2 y_{2} = 6

y_{1} + 4 y_{2} = 4

. It gives

y = (8/5, 3/5)

; the remaining constraint is

5 y_{1} + y_{2} = 43/5 \geq 5

, so it is dual feasible. Its strictly positive slack in the first column suggests

x_{1} = 0

, while positivity of both

y

entries suggests

A x = b

. Solving

3 x_{2} + x_{3} = 1

2 x_{2} + 4 x_{3} = 2

gives

x = (0, 1/5, 2/5)

, which is primal feasible.

Their objective values agree:

x_{*} = (0, 1/5, 2/5)^{T}, y_{*} = (8/5, 3/5)^{T}, c^{T} x_{*} = b^{T} y_{*} = 14/5.

(1)

Weak duality certifies that both are optimal. The strict first-column dual slack and the two active primal equations also show uniqueness of the primal optimum; positivity of

x_{2}, x_{3}

forces the two dual equations, so the dual optimum is unique as well.

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / T / T2 / a / ii / Solution by

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For feasible points the duality gap decomposes as

b^{T} y - c^{T} x = y^{T} (b - A x) + x^{T} (A^{T} y - c) .

(1)

Each term in both sums is nonnegative. Therefore the gap is zero exactly when every product vanishes, namely the two sets of complementary slackness equations in the question (with

b - A x

or its negative making no difference to a zero product). If the equations hold, equality and weak duality prove optimality of both points. Conversely, if both are optimal and the optimal values are equal as assumed, the gap is zero, forcing each product to vanish. This proves both directions.

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / T / T2 / a / i / Solution by

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The dual is

min b^{T} y subject to A^{T} y \geq c, y \geq 0

. Weak duality states that every primal feasible

x

and dual feasible

y

satisfy

c^{T} x \leq b^{T} y

. Indeed,

c^{T} x \leq (A^{T} y)^{T} x = y^{T} A x \leq y^{T} b,

(1)

using

x \geq 0

for the first inequality and

y \geq 0

for the second. Thus a feasible dual value is an upper bound on every primal objective value, and a feasible pair with equal values certifies both optima.

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / S / S2 / b / ii / Solution by

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For

x \neq = 0

, the envelope ratio is

\frac{p ( x )}{q ( x )} = \frac{3}{2} (1 + x^{2}) (\frac{s i n x}{x})^{4} .

(1)

∣ x ∣ \leq 1

, use

∣ sin x ∣ \leq ∣ x ∣

to bound this by

\frac{3}{2} (1 + x^{2}) \leq 3

. If

∣ x ∣ \geq 1

, use

∣ sin x ∣ \leq 1

to bound it by

\frac{3}{2} (x^{- 4} + x^{- 2}) \leq 3

. The boundary equalities in these bounds cannot both occur in the sine estimate, so

M = 3

is a valid envelope. At zero the continuous ratio is

3/2

Generate independent Cauchy proposals as in (i), together with independent uniform

U

. Accept a proposal

Y

when

U \leq \frac{1}{2} (1 + Y^{2}) (\frac{sin Y}{Y})^{4},

(2)

using the limiting value

1/2

Y = 0

. Repeat until acceptance. Part (a) proves exactness; the overall acceptance rate is

1/3

and the mean number of proposals is three. No numerical maximization of the ratio is needed for correctness.

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / S / S2 / b / i / Solution by

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Generate independent standard normal variables

Z_{1}, Z_{2}

and put

Y = Z_{1} / Z_{2}

. The event

Z_{2} = 0

has probability zero. The ratio calculation in M4 with equal standard deviations gives density

1/ [π (1 + y^{2})]

, exactly the required

t

distribution with one degree of freedom. The equivalent form

Z_{1} /∣ Z_{2} ∣

has the same law by symmetry.

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / S / S2 / a / ii / Solution by

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For one trial and any measurable set

A

P (Y \in A, accept) = \int_{A} q (y) \frac{p ( y )}{Mq ( y )} d y = \frac{1}{M} \int_{A} p (y) d y .

(1)

Since the acceptance probability is

1/ M

, conditioning gives target probability

\int_{A} p

. Repeating failed independent trials does not alter this conditional law: summing over the trial number multiplies by the geometric sum

\sum_{j \geq 0} (1 - 1/ M)^{j} / M = 1

. Taking

A = (- \infty, x)

proves

P (X < x) = \int_{- \infty}^{x} p (y) d y .

(2)

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Past exam of the Probability, statistics and options II course of the University of Oxford / 2010 / S / S2 / a / i / Solution by

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Independently draw

Y \sim q

and

U \sim Uniform (0, 1)

. Accept

Y

U \leq p (Y) / (Mq (Y))

; otherwise discard it and repeat with fresh independent draws. Return the first accepted value. The bound makes the acceptance probability at each proposal at most one. As

\int p = 1

, the total acceptance probability is

1/ M > 0

, so the procedure terminates almost surely. Values outside the proposal support must have target density zero for the stated envelope assumption to make sense.

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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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