Mathematical Sciences at Oxford

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Mathematical Sciences at Oxford

• Stephen Drape

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Who am I?

• Dr Stephen Drape
• Access and Schools Liaison Officer for Computer
  Science (Also a Departmental Lecturer)
• 8 years at Oxford (3 years Maths degree, 4 years
  Computer Science graduate, 1 year lecturer)
• 5 years as Secondary School Teacher
• Email stephen.drape_at_comlab.ox.ac.uk

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Four myths about Oxford

• Theres little chance of getting in
• Its very expensive in Oxford
• College choice is very important
• You have to be very bright

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Myth 1 Little chance of getting in

• False!
• Statistically you have a 2040 chance
• Admissions data for 2007 entry

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Myth 2 Its very expensive

• False!
• Most colleges provide cheap accommodation for
  three years.
• College libraries and dining halls also help you
  save money.
• Increasingly, bursaries help students from poorer
  backgrounds.
• Most colleges and departments are very close to
  the city centre low transport costs!

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Myth 3 College Choice Matters

• False!
• If the college you choose is unable to offer you
  a place because of space constraints, they will
  pass your application on to a second,
  computer-allocated college.
• Application loads are intelligently redistributed
  in this way.
• Lectures are given centrally by the department as
  are many classes for courses in later years.

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Myth 3 College Choice Matters

• However
• Choose a college that you like as you have to
  live and work there for 3 or 4 years
• Look at accommodation facilities offered.
• Choose a college that has a tutor in your
  subject.

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Myth 4 You have to be bright

• True!
• We find it takes special qualities to benefit
  from the kind of teaching we provide.
• So we are looking for the very best in ability
  and motivation.
• A typical offer is 3 A grades at A-Level

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Mathematical Science Subjects

• Mathematics
• Mathematics and Statistics
• Computer Science
• Mathematics and Computer Science
• All courses can be 3 or 4 years

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Maths in other subjects

• For admissions, A-Level Maths is mentioned as a
  preparation for a number of courses
• Essential Computer Science, Engineering Science,
  Engineering, Economics Management (EEM),
  Materials, Economics Management (MEM),
  Materials, Maths, Medicine, Physics
• Desirable/Helpful Biochemistry, Biology,
  Chemistry, Economics Management, Experimental
  Psychology, History and Economics, Law,
  Philosophy , Politics Economics (PPE),
  Physiological Sciences, Psychology, Philosophy
  Physiology (PPP)

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Admissions Process

• Fill in UCAS and Oxford form
• Choose a college or submit an Open Application
• Interview Test
• Based on common core A-Level
• Taken before the interview
• Interviews
• Take place over a few days
• Often have many interviews

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Entrance Requirements

• Essential A-Level Mathematics
• Recommended Further Maths or a Science
• Note it is not a requirement to have Further
  Maths for entry to Oxford
• For Computer Science, Further Maths is perhaps
  more suitable than Computing or IT
• Usual offer is AAA

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First Year Maths Course

• Algebra (Group Theory)
• Linear Algebra (Vectors, Matrices)
• Calculus
• Analysis (Behaviour of functions)
• Applied Maths (Dynamics, Probability)
• Geometry

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Subsequent Years

• The first year consists of compulsory courses
  which act as a foundation to build on
• The second year starts off with more compulsory
  courses
• The reminder of the course consists of a variety
  of options which become more specialised
• In the fourth year, students have to study 6
  courses from a choice of 40

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Mathematics and Statistics

• The first year is the same as for the Mathematics
  course
• In the second year, there are some compulsory
  units on probability and statistics
• Options can be chosen from a wide range of
  Mathematics courses as well as specialised
  Statistics options
• Requirement that around half the courses must be
  from Statistics options

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Computer Science

• Computer Science
• Computer Science firmly based on Mathematics
• Mathematics and Computer Science
• Closer to a half/half split between CS and Maths
• Computer Science is part of the Mathematical
  Science faculty because it has a strong emphasis
  on theory

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Some of the first year CS courses

• Functional Programming
• Design and Analysis of Algorithms
• Imperative Programming
• Digital Hardware
• Calculus
• Linear Algebra
• Logic and Proof
• Discrete Maths

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Subsequent Years

• The second year is a combination of compulsory
  courses and options
• Many courses have a practical component
• Later years have a greater choice of courses
• Third and Fourth year students have to complete a
  project

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Some Computer Science Options

• Compilers
• Programming Languages
• Computer Graphics
• Computer Architecture
• Intelligent Systems
• Machine Learning
• Lambda Calculus
• Computer Security

• Category Theory
• Computer Animation
• Linguistics
• Domain Theory
• Program Analysis
• Information Retrieval
• Bioinformatics
• Formal Verification

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Useful Sources of Information

• Admissions
• http//www.admissions.ox.ac.uk/
• Mathematical Institute
• http//www.maths.ox.ac.uk/
• Computing Laboratory
• http//www.comlab.ox.ac.uk/
• Colleges

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What is Computer Science?

• Its not just about learning new programming
  languages.
• It is about understanding why programs work, and
  how to design them.
• If you know how programs work then you can use a
  variety of languages.
• It is the study of the Mathematics behind lots of
  different computing concepts.

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Simple Design Methodology

• Try a simple version first
• Produce some test cases
• Prove it correct
• Consider efficiency (time taken and space needed)
• Make improvements (called refinements)

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Fibonacci Numbers

• The first 10 Fibonacci numbers (from 1) are
• 1,1,2,3,5,8,13,21,34,55
• The Fibonacci numbers occurs in nature, for
  example plant structures, population numbers.
• Named after Leonardo of Pisa
• who was nicked named Fibonacci

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The rule for Fibonacci

• The next number in the sequence is worked out by
  adding the previous two terms.
• 1,1,2,3,5,8,13,21,34,55
• The next numbers are therefore
• 34 55 89
• 55 89 144

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Using algebra

• To work out the nth Fibonacci number, which well
  call fib(n), we have the rule
• fib(n)
• We also need base cases
• fib(0) 0 fib(1) 1
• This sequence is defined using previous terms of
  the sequence it is an example of a recursive
  definition.

fib(n 1) fib(n 2)
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Properties

• The sequence has a relationship with the Golden
  Ratio
• Fibonacci numbers have a variety of properties
  such as
• fib(5n) is always a multiple of 5
• in fact, fib(ab) is always a multiple of fib(a)
  and fib(b)

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Writing a computer program

• Using a language called Haskell, we can write the
  following function
• gt fib(0) 0
• gt fib(1) 1
• gt fib(n) fib(n-1) fib(n-2)
• which looks very similar to our algebraic
  definition

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Working out an example

• Suppose we want to find fib(5)

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Our program would do this

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Whats happening?

• The program blindly follows the definition of
  fib, not remembering any of the other values.
• So, for
• (fib(3) fib(2)) fib(3)
• the calculation for fib(3) is worked out twice.
• The number of steps needed to work out fib(n) is
  proportional to ?n it takes exponential time.

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Refinements

• Why this program is so inefficient is because at
  each step we have two occurrences of fib (termed
  recursive calls).
• When working out the Fibonacci sequence, we
  should keep track of previous values of fib and
  make sure that we only have one occurrence of the
  function at each stage.

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Writing the new definition

• We define
• gt fibtwo(0) (0,1)
• gt fibtwo(n) (b,ab)
• gt where (a,b) fibtwo(n-1)
• gt newfib(n) fst(fibtwo(n))
• The function fst means take the first number

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Explanation

• The function fibtwo actually works out
• fibtwo(n) (fib(n), fib(n 1))
• We have used a technique called tupling which
  allows us to keep extra results at each stage of
  a calculation.
• This version is much more efficient that the
  previous one (it is linear time).

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An example of the new function
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Algorithm Design

• When designing algorithms, we have to consider a
  number of things
• Our algorithm should be efficient that is,
  where possible, it should not take too long or
  use too much memory.
• We should look at ways of improving existing
  algorithms.
• We may have to try a number of different
  approaches and techniques.
• We should make sure that our algorithms are
  correct.

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Finding the Highest Common Factor

• Example
• Find the HCF of 308 and 1001.
• 1) Find the factors of both numbers
• 308 1,2,4,7,11,14,22,28,44,77,154,308
• 1001 1,7,11,13,77,91,143,1001
• 2) Find those in common
• 1,7,11,77
• 3) Find the highest
• Answer 77

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Creating an algorithm

• For our example, we had three steps
• Find the factors
• Find those factors in common
• Find the highest factor in common
• These steps allow us to construct an algorithm.

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Creating a program

• We are going to use a programming language called
  Haskell.
• Haskell is used throughout the course at Oxford.
• It is very powerful as it allows you write
  programs that look very similar to mathematical
  equations.
• You can easily prove properties about Haskell
  programs.

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Step 1

• We need produce a list of factors for a number n
  call this list factor(n).
• A simple way is to check whether each number d
  between 1 and n is a factor of n.
• We do this by checking what the remainder is when
  we divide n by d.
• If the remainder is 0 then d is a factor of n.
• We are done when dn.
• We create factor lists for both numbers.

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Function for Step 1
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Step 2

• Now that we have our factor lists, which we will
  call f1 and f2, we create a list of common
  factors.
• We do this by looking at all the numbers in f1 to
  see if they are in f2.
• We there are no more numbers in f1 then we are
  done.
• Call this function common(f1,f2).

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Function for Step 2
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Step 3

• Now that we have a list of common factors we now
  check which number in our list is the biggest.
• We do this by going through the list remembering
  which is the biggest number that we have seen so
  far.
• Call this function highest(list).

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Function for Step 3

• If list is empty then return 0, otherwise we
  check whether the first member of list is higher
  than the rest of list.

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Putting the three steps together

• To calculate the hcf for two numbers a and b, we
  just follow the three steps in order.
• So, in Haskell, we can define
• Remember that when composing functions, we do the
  innermost operation first.

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Problems with this method

• Although this method is fairly easy to explain,
  it is quite slow for large numbers.
• It also wastes quite a lot of space calculating
  the factors of both numbers when we only need one
  of them.
• Can we think of any ways to improve this method?

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Possible improvements

• Remember factors occur in pairs so that we
  actually find two factors at the same time.
• If we find the factors in pairs then we only need
  to check up to ?n.
• We could combine common and highest to find the
  hcf more quickly (this kind of technique is
  called fusion).
• Could use prime numbers.

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A Faster Algorithm
This algorithm was apparently first given by the
famous mathematician Euclid around 300 BC.
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An example of this algorithm

• hcf(308,1001)
• hcf(308,693)
• hcf(308,385)
• hcf(308,77)
• hcf(231,77)
• hcf(154,77)
• hcf(77,77)
• 77

The algorithm works because any factor of a and b
is also a factor of a b
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Writing this algorithm in Haskell
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An even faster algorithm

• hcf(1001,308) 1001 3 308 77
• hcf(308,77) 308 4 77
• hcf(77,0)
• 77