MTH700: Research Methods in Mathematical Sciences 2020/21

Assessed Coursework G. Bianconi, I. Morris

This coursework covers the material of weeks 2–6. It will be assessed, and the mark will contribute 60% to

the final mark. Students must submit their work individually by uploading a single pdf file to QMPlus by

the deadline of 17:00 on Tuesday 10 November 2020.

Marks allocated to each question are shown next to the question. The submitted document must be written

using LaTeX, and up to 20% of the marks will be allocated to LaTeX proficiency including formatting and

mathematical notation.

Section A: Literature Search

- The concept of an iterated function system as a means of constructing fractal sets was introduced in the

1980s.

(a) Provide a complete reference to the paper where this concept was introduced. [2 marks]

(b) How many citations has this paper received so far? Detail the method you used to estimate the

number of citations. [3 marks]

(c) Provide complete references to two papers published in the journal Inventiones Mathematicae in

the 2010s which make use of the concept of an iterated function system. [3 marks]

(d) Provide a reference to a source which one might consult for an accessible introduction to this

concept. [2 marks] - The small-world model is a model proposed in the late 1990s for capturing the interplay between randomness and order in real networks.

(a) Provide a complete reference to a journal article where the small-world model has been proposed.

[2 marks]

(b) What is the topic of that work? According to the small-world model what are the properties of a

small world network? [3 marks]

(c) List one popular book, one textbook and one review article that extensively discuss the small-world

network model. [3 marks]

(d) The small world model mathematically formalizes the finding of an important social science experiment by a famous Yale sociologist. Provide a complete reference to the paper in which the results

of this experiment have been published. [2 marks]

Section B: Literature Review

Find and download the following papers. If you only find a preprint of a paper you must state that this is

what your answers are based on.

D. A. Goldston, J. Pintz, C. Y. Yıldırım. Primes in Tuples I, Annals of Mathematics, 170, 819–862, 2009.

Liu, Yang-Yu, Jean-Jacques Slotine, and Albert-L´aszl´o Barab´asi. Controllability of complex networks.

Nature 473, 167-173, 2011.

Read both of the papers at the level of a “first pass”, and answer the following questions.

(a) What is the main message of the paper? Write at most two sentences, avoiding mathematical notation

as much as possible. [3 marks per paper]

(b) Why is the paper relevant? Write at most three sentences.

[3 marks per paper]

(c) Provide a summary of around 100 words, avoiding all mathematical symbols.

[4 marks per paper]

(d) Now choose one of the papers and write a critical review of 250–500 words. The review should provide a

summary of the main message of the paper, an assessment of its message and methods, and a discussion

on its readability (motivation, logical coherence, simplicity). You are allowed to use mathematical

notation. [20 marks]

Section C: Communicating Mathematics - Define a relation ≺ on sequences of positive real numbers as follows: if (an)∞

n=1 and (bn)∞

n=1 are sequences

of positive real numbers, write (an) ≺ (bn) if there exists a constant C > 0 such that an ≤ Cbn for

all n ≥ 1. Explain the meaning of this relation in a way which should be readable by a first-year

undergraduate mathematics student, providing appropriate examples and articulating the distinction

between ≺ and <. Write no more than one page.

[20 marks (content 10; suitability for first-year students 5; clarity and style 5)] - Consider the paper:

Yedidia, J.S., Freeman, W.T. and Weiss, Y.Understanding belief propagation and its generalizations.

Exploring artificial intelligence in the new millennium, 8, 236-239 2003.

Write a summary of each of the following items:

(a) Section 1

(b) Section 2

Each summary should be approximately 100 words of length.

[10 marks per item (content 7; clarity and style 3)]

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