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Applied Mathematics at Oxford

Applied Mathematics at Oxford. Christian Yates Centre for Mathematical Biology Mathematical Institute. Who am I?. Completed my B.A. (Mathematics) and M.Sc. (Mathematical M odelling and S cientific C omputing) at the Mathematical Institute as a member of Somerville College.

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Applied Mathematics at Oxford

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  1. Applied Mathematics at Oxford • Christian Yates • Centre for Mathematical Biology • Mathematical Institute

  2. Who am I? • Completed my B.A. (Mathematics) and M.Sc. (Mathematical Modelling and Scientific Computing) at the Mathematical Institute as a member of Somerville College. • Currently completing my D.Phil. (Mathematical Biology) in the Centre for Mathematical Biology as a member of Worcester and St. Catherine’s colleges. • Next year – Junior Research Fellow at Christ Church college. • Research in cell migration, bacterial motion and locust motion. • Supervising Masters students. • Lecturer at Somerville College • Teaching 1st and 2nd year tutorials in college.

  3. Outline of this talk • The principles of applied mathematics • A practical example • Mods applied mathematics (first year) • Celestial mechanics • Waves on strings • Applied mathematics options (second and third year) • Fluid mechanics • Classical mechanics • Mathematical Biology • Reasons to study mathematics

  4. Outline of this talk • The principles of applied mathematics • A simple example • Mods applied mathematics (first year) • Celestial mechanics • Waves on strings • Applied mathematics options (second and third year) • Fluid mechanics • Classical mechanics • Calculus of variations • Mathematical Biology • Reasons to study mathematics

  5. Principles of applied mathematics • Start from a physical or “real world” system • Use physical principles to describe it using mathematics • For example, Newton’s Laws • Derive the appropriate mathematical terminology • For example, calculus • Use empirical laws to turn it into a solvable mathematical problem • For example, Law of Mass Action, Hooke’s Law • Solve the mathematical model • Develop mathematical techniques to do this • For example, solutions of differential equations • Use the mathematical results to make predictions about the real world system

  6. Simple harmonic motion • Newton’s second law • Force = mass x acceleration • Hooke’s Law • Tension = spring const. x extension • Resulting differential equation

  7. simple harmonic motion • Re-write in terms of the displacement from equilibriumwhich is the description of simple harmonic motion • The solution iswith constants determined by the initial displacement and velocity • The period of oscillations is

  8. Putting maths to the test: Prediction • At equilibrium (using Hooke’s law T=ke): • Therefore: • So the period should be:

  9. Experiment • Equipment: • Stopwatch • Mass • Spring • Clampstand • 1 willing volunteer • Not bad but not perfect • Why not? • Air resistance • Errors in measurement etc • Old Spring • Hooke’s law isn’t perfect etc

  10. Outline of this talk • The principles of applied mathematics • A simple example • Mods applied mathematics (first year) • Celestial mechanics • Waves on strings • Applied mathematics options (second and third year) • Fluid mechanics • Classical mechanics • Mathematical Biology • Reasons to study mathematics

  11. Celestial mechanics • Newton’s 2nd Law • Newton’s Law of Gravitation • The position vector satisfies the differential equationSolution of this equation confirms Kepler’s Laws

  12. How long is a year? • M=2x1030 Kg • G=6.67x10-10 m3kg-1s-2 • R=1.5x1011m • Not bad for a 400 year old piece of maths. Kepler

  13. Outline of this talk • The principles of applied mathematics • A simple example • Mods applied mathematics (first year) • Celestial mechanics • Waves on strings • Applied mathematics options (second and third year) • Fluid mechanics • Classical mechanics • Mathematical Biology • Reasons to study mathematics

  14. Waves on a string • Apply Newton’s Law’s to each small interval of string... • The vertical displacement satisfies the partial differential equation • Known as the wave equation • Wave speed:

  15. Understanding music • Why don’t all waves sound like this? • Because we can superpose waves on each other =

  16. Fourier series • By adding waves of different amplitudes and frequencies we can come up with any shape we want: • Themaths behind how to find the correct signs and amplitudes is called Fourier series analysis.

  17. More complicated wave forms • Saw-tooth wave: • Square wave:

  18. Outline of this talk • The principles of applied mathematics • A simple example • Mods applied mathematics (first year) • Celestial mechanics • Waves of strings • Applied mathematics options (second and third year) • Fluid mechanics • Classical mechanics • Mathematical Biology • Reasons to study mathematics

  19. Fluid mechanics • Theory of flight - what causes the lift on an aerofoil? • What happens as you cross the sound barrier?

  20. Outline of this talk • The principles of applied mathematics • A simple example • Mods applied mathematics (first year) • Celestial mechanics • Waves of strings • Applied mathematics options (second and third year) • Fluid mechanics • Classical mechanics • Mathematical Biology • Reasons to study mathematics

  21. Classical mechanics • Can we predict the motion of a double pendulum? • In principleyes. • In practice, chaos takes over.

  22. Outline of this talk • The principles of applied mathematics • A simple example • Mods applied mathematics (first year) • Celestial mechanics • Waves of strings • Applied mathematics options (second and third year) • Fluid mechanics • Classical mechanics • Mathematical Biology • Reasons to study mathematics

  23. How we do mathematical biology? • Find out as much as we can about the biology • Think about which bits of our knowledge are important • Try to describe things mathematically • Use our mathematical knowledge to predict what we think will happen in the biological system • Put our understanding to good use

  24. Mathematical biology

  25. Locusts

  26. Switching behaviour • Locusts switch direction periodically • The length of time between switches depends on the density of the group 60 Locusts 30 Locusts

  27. Explanation - Cannibalism

  28. Outline of this talk • The principles of applied mathematics • A simple example • Mods applied mathematics (first year) • Celestial mechanics • Waves on strings • Applied mathematics options (second and third year) • Fluid mechanics • Classical mechanics • Calculus of variations • Mathematical Biology • Reasons to study mathematics

  29. Why mathematics? • Flexibility - opens many doors • Importance - underpins science • Ability to address fundamental questions about the universe • Relevance to the “real world” combined with the beauty of abstract theory • Excitement - finding out how things work • Huge variety of possible careers • Opportunity to pass on knowledge to others Me on Bang goes the theory

  30. I’m off to watch Man City in the FA cup final

  31. Further information • Studying mathematics and joint schools at Oxford • http://www.maths.ox.ac.uk • David Acheson’s page on dynamics • http://home.jesus.ox.ac.uk/~dacheson/mechanics.html • Centre for Mathematical Biology • http://www.maths.ox.ac.uk/groups/mathematical-biology/ • My web page • http://people.maths.ox.ac.uk/yatesc/

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