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The conclusion is Arnold-Liouville theorem: a compact connected regular common level is a three-torus, and nearby action-angle coordinates make the flow linear in the angles.

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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 33B / a / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

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Hamilton's equations are

\overset{x}{˙}_{i} = p_{i}

and

\overset{p}{˙}_{i} = - \partial_{i} V

. A first integral

F

satisfies

{F, H} = 0

. See integrable Hamiltonian system: in three degrees of freedom, integrability requires three independent commuting first integrals.

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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 32B / b / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-03

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The fixed points are

(0, 0)

and

(\pm ∣ μ ∣, μ^{2})

. At the origin the eigenvalues are

μ^{2}

and

- 1

, so it is a saddle. At either nonzero point the characteristic polynomial is

λ^{2} + λ + 2 μ^{2} .

(1)

They are stable nodes for

0 < ∣ μ ∣ < 1/ (22)

, degenerate nodes at equality, and stable spirals above it.

The stable manifold of the origin is exactly

x = 0

. Write the stable manifold in the unstable direction as

y = a x^{2} + b x^{4} + O (x^{6})

. Graph invariance gives

a = \frac{1}{1 + 2 μ ^{2}}, b = \frac{2 a ^{2}}{1 + 4 μ ^{2}} .

(2)

Thus

W^{u}

has

y = \frac{x ^{2}}{1 + 2 μ ^{2}} + \frac{2 x ^{4}}{( 1 + 2 μ ^{2} ) ^{2} ( 1 + 4 μ ^{2} )} + O (x^{6}) .

(3)

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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 32B / a / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

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The required properties are those of a strict Lyapunov function; the domain of stability is the set of initial states whose forward trajectories converge to the fixed point.

For

V = (x^{2} + y^{2}) /2

\dot{V} = (x^{2} + y^{2}) (- 2 + x^{2} + 4 y^{2}) .

(1)

Thus

\dot{V} < 0

when

V < 1/4

, proving that disc lies in the basin. On

V = 1

\dot{V} = 6 y^{2} \geq 0

(with the tangency points checked from the vector field), so trajectories outside cannot enter the basin through that boundary. Hence the basin is contained in

V \leq 1

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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 31K / d / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-03

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On an

n

-sample, each coordinate partition has at most

(n + 1)^{m - 1}

placements, hence all

p

partitions have at most

(n + 1)^{p (m - 1)}

configurations. For each fixed configuration, the sign class from a vector space of dimension

1 + p (m - 1)

has at most

(n + 1)^{1 + p (m - 1)}

patterns. Multiplication gives

s (H_{F_{3}}, n) \leq (n + 1)^{1 + 2 p (m - 1)} .

(1)

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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 31K / c / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

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For each coordinate, the

m

interval indicators sum to one, so the span of all displayed indicators has dimension at most

1 + p (m - 1) .

(1)

Therefore VC dimension of a vector space gives

VC (H_{F_{2}}) \leq 1 + p (m - 1)

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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 31K / b / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-03

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Order the

n

sample points. Each of the

m - 1

cut points can occupy one of at most

n + 1

gaps. Once those gaps are chosen the fixed

β_{k}

determine every sign, so

s (H_{F_{1}}, n) \leq (n + 1)^{m - 1} .

(1)

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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 31K / a / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

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For points

x_{1}, \dots, x_{n}

, count the distinct vectors

(h (x_{1}), \dots, h (x_{n}))

; maximize over samples to get

s (H, n)

. The shattering coefficient equals

2^{n}

exactly when some sample is shattered, and the largest such

n

VC (H)

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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 30L / e / Solution by

Codex  0 Created 2026-09-24 Updated 2026-10-03

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By Gaussian conditional expectation,

E [Y ∣ S_{1}] = E (Y) + cov (Y, S_{1}) V^{- 1} (S_{1} - μ) .

(1)

Multiply by

U^{'} (X_{0})

, take expectations, and use part (c):

\frac{E [ U ^{'} ( X _{0} ) S _{1} ]}{E [ U ^{'} ( X _{0} )]} = (1 + r) S_{0} .

(2)

Substitution into part (d) gives

π_{0} (Y) = \frac{1}{1 + r} [E (Y) - (μ - (1 + r) S_{0})^{T} V^{- 1} cov (S_{1}, Y)] .

(3)

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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 30L / d / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

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The marginal utility price is

π_{0} (Y) = \frac{E [ U ^{'} ( X _{0} ) Y ]}{( 1 + r ) E [ U ^{'} ( X _{0} )]} .

(1)

The value function is concave; its tangent at zero lies above its graph. Evaluating that supporting inequality at the utility-indifference displacement

Y - (1 + r) π (Y)

gives

E [U^{'} (X_{0}) (Y - (1 + r) π (Y))] \geq 0

, hence

π_{0} (Y) \geq π (Y)

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Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 30L / c / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-25

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Differentiate expected utility at the optimum

X_{0}

in each portfolio direction:

0 = E [U^{'} (X_{0}) (S_{1} - (1 + r) S_{0})] .

(1)

Rearranging gives the claimed vector identity.

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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.
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.
Live URL: ourbigbook.com/cirosantilli/chordate
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