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Accreting material loses specific angular momentum; conservation requires a diminishing fraction of the disk to spread outward and carry that angular momentum.

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 321 / 1 / c / ii by

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 321 / 1 / c / i by

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 321 / 1 / b / ii by

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 321 / 1 / b / i by

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 321 / 1 / a / Solution by

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For the Paczyński-Wiita potential, circular balance gives

r Ω^{2} = \frac{d Φ}{d r} = \frac{GM}{( r - r _{S} ) ^{2}} .

(1)

Therefore

Ω (r) = \frac{GM}{r ^{1/2} ( r - r _{S} )}, h (r) = r^{2} Ω = \frac{GM r ^{3}}{r - r _{S}} .

(2)

The squared radial epicyclic frequency is

κ_{r}^{2} = \frac{1}{r ^{3}} \frac{d ( h ^{2} )}{d r} = \frac{GM ( r - 3 r _{S} )}{r ( r - r _{S} ) ^{3}} .

(3)

Circular orbits change from stable to unstable where

κ_{r}^{2}

vanishes, so

r_{ISCO} = 3 r_{S} .

(4)

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 5 / c / Solution by

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

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For

f = 1_{[- π, π)}

, its

2 π

-periodization is one almost everywhere. The refinement mask is the periodic function equal to one on

[- π /2, π /2)

and zero on the rest of

[- π, π)

. Fourier inversion gives the Shannon scaling function

ϕ (x) = \frac{sin π x}{π x} .

(1)

Since

m (t) = \frac{1}{2} \sum_{n} a_{n} e^{- in t}

, its Fourier coefficients give

a_{0} = 1, a_{n} = \frac{2 sin ( nπ /2 )}{πn} (n \neq = 0) .

(2)

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 5 / b / Solution by

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

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Fourier transforming the refinement equation and changing variables gives

f (2 t) = m (t) f (t), m (t) = \frac{1}{2} n \sum a_{n} e^{- in t} .

(1)

By Parseval identity, orthonormality of the translates is equivalent to

\frac{1}{2 π} \int_{R} ∣ f (t) ∣^{2} e^{in t} d t = δ_{n 0} .

(2)

These are precisely the Fourier coefficients of the periodization

P (t) = \sum_{k} ∣ f (t + 2 πk) ∣^{2}

. Therefore

{ϕ (\cdot - n)} is orthonormal ⟺ k \sum ∣ f (t + 2 πk) ∣^{2} = 1 a.e.

(3)

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 5 / a / Solution by

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

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An orthonormal wavelet is

ψ \in L^{2} (R)

such that

{2^{j /2} ψ (2^{j} x - k)}_{j, k \in Z}

is an orthonormal basis. A multiresolution analysis is a nested family of closed spaces

V_{j}

with trivial intersection, dense union, dyadic scaling, integer-translation invariance of

V_{0}

, and a generator

ϕ

whose integer translates form an orthonormal basis of

V_{0}

. The Meyer-Mallat theorem says every such analysis has an orthonormal wavelet whose translates span

V_{j + 1} ⊖ V_{j}

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 4 / c / Solution by

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

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The Schoenberg spline operator is positive and

V_{n} 1 = 1

. If

ξ_{i} = a_{1, i}

, then both

τ_{i}

and

ξ_{i}

lie in

[t_{i}, t_{i + k}]

, so

∥ V_{n} t - t ∥_{\infty} \leq k ∣ Δ_{n} ∣ \to 0.

(1)

Because

a_{2, i}

averages products of knots in the same interval and all points lie in

[0, 1]

∥ V_{n} t^{2} - t^{2} ∥_{\infty} \leq 2 k ∣ Δ_{n} ∣ \to 0.

(2)

The Korovkin theorem now proves

∥ V_{n} (f) - f ∥_{C [0, 1]} \to 0

(3)

for every

f \in C [0, 1]

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 4 / b / Solution by

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

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Let

e_{m}

denote the

m

th elementary symmetric polynomial of

t_{i + 1}, \dots, t_{i + k - 1}

. Expanding both sides of the Marsden identity in powers of

x

and equating the coefficient of

x^{k - 1 - m}

gives

a_{m, i} = \frac{e _{m} ( t _{i + 1} , \dots , t _{i + k - 1} )}{( m k - 1 )}, 0 \leq m \leq k - 1.

(1)

Thus $a_{0, i} = 1$ , while $a_{1, i}$ and $a_{2, i}$ are respectively the means of the interior knots and of their pairwise products.

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 4 / a / Solution by

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

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The Korovkin theorem states that positive linear operators

L_{n} : C [0, 1] \to C [0, 1]

converge uniformly to the identity on every continuous function if they do so on the three test functions

1, t, t^{2}

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 3 / c / Solution by

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

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Induct on

k

, using the recursive divided-difference formula. After substituting the two induction hypotheses, use

f [t_{1}, \dots, t_{m}] - f [t_{0}, \dots, t_{m - 1}] = (t_{m} - t_{0}) f [t_{0}, \dots, t_{m}]

(1)

and the analogous identity for

g

; adjacent terms telescope. This proves

(f g) [t_{0}, \dots, t_{k}] = m = 0 \sum k f [t_{0}, \dots, t_{m}] g [t_{m}, \dots, t_{k}] .

(2)

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 3 / b / Solution by

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

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Apply the Leibniz rule for divided differences to

(x - t)_{+}^{k - 1} = (x - t) (x - t)_{+}^{k - 2}

. Only the zeroth and first divided differences of the linear factor survive. After applying the normalization, this gives the Cox-de Boor recursion formula

N_{i, k} (t) = \frac{t - t _{i}}{t _{i + k - 1} - t _{i}} N_{i, k - 1} (t) + \frac{t _{i + k} - t}{t _{i + k} - t _{i + 1}} N_{i + 1, k - 1} (t) .

(1)

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 3 / a / Solution by

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

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The explicit divided difference formula makes

M_{i} (t)

a finite linear combination of

(t_{j} - t)_{+}^{k - 1}

. It is therefore polynomial of degree at most

k - 1

between knots and globally

C^{k - 2}

. For

t < t_{i}

, the nodal data come from a polynomial of degree

k - 1

, whose order-

k

divided difference vanishes; for

t \geq t_{i + k}

all truncated powers vanish. Thus

supp M_{i} = [t_{i}, t_{i + k}],

(1)

and normalization does not change the degree, smoothness, knots, or support.

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 2 / c / Solution by

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

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Twice applying the fundamental theorem of calculus gives

f (x + h) - 2 f (x) + f (x - h) = \int_{0}^{h} \int_{- s}^{s} f^{''} (x + u) d u d s,

(1)

hence

ω_{2} (f, t) \leq t^{2} ∥ f^{''} ∥_{\infty}

. Part (b) gives

∥ σ_{n} (f) - f ∥_{\infty} = O (n^{- 1})

. This cannot be little-

o

for every

C^{2}

function: for

f_{0} (x) = cos x

σ_{n} (f_{0}) = (1 - n^{- 1}) cos x, ∥ σ_{n} (f_{0}) - f_{0} ∥_{\infty} = n^{- 1} .

(2)

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 2 / b / Solution by

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The Fejér kernel is nonnegative, and preservation of constants gives

π^{- 1} \int_{T} ∣ F_{n} (t) ∣ d t = 1 .

(1)

By evenness,

σ_{n} (f, x) - f (x) = \frac{1}{2 π} \int_{T} F_{n} (t) [f (x - t) - 2 f (x) + f (x + t)] d t .

(2)

Put

δ = n^{- 1/2}

. Use

ω_{2} (f, ∣ t ∣) \leq (∣ t ∣/ δ + 1)^{2} ω_{2} (f, δ)

together with

F_{n} (t) \leq n /2

and

F_{n} (t) \leq C / (n t^{2})

. Splitting at

1/ n

and

δ

shows that the remaining weighted integral is uniformly bounded, so

∥ σ_{n} (f) - f ∥_{\infty} \leq C ω_{2} (f, n^{- 1/2}) .

(3)

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Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 318 / 2 / a / Solution by

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Substituting the Fourier coefficient s into the Fourier partial sum and summing the finite geometric series gives

s_{n} (f, x) = \frac{1}{2 π} \int_{T} f (t) \sum_{k = - n}^{n} e^{ik (x - t)} d t = \frac{1}{π} \int_{T} D_{n} (x - t) f (t) d t,

(1)

where

D_{n} (u) = \frac{sin (( n + \frac{1}{2} ) u )}{2 sin ( u /2 )} .

(2)

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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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