Governor

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Governors School for the Sciences

• Mathematics

Day 2
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The more things stay, the more they change the
sane. -khaosworks
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MOTD Joseph Fourier

• 1768-1830 (French)
• Partial Differential Equation for Heat Conduction
• Solution by Infinite Trigonometric Series
  (Fourier Series)
• Mathematics compares the most diverse phenomena
  and discovers the secret analogies that unite
  them.

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Where are you from?

• Give us the Good, the Bad or the Ugly!

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Periodic PatternsWhat comes next?
Stock Price History
Sound Signal
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Aside
When we have real data we may not want to predict
the pattern exactly. (Why? Noisy or too much
data) Instead we may wish to find an
approximation to the pattern A(n) f(n)
e(n) Where e(n) is small and
random Finding f is slightly different but the
tools we are using still work.
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Trignometric Polynomial Interpolation

• Recall Polynomial of degree n
• p(x) a0 a1x a2x2 anxn
• Trig poly of degree n t(x) a0 a1 cos(x)
  a2 cos(2x) an cos(nx) b1 sin(x)
  b2 sin(2x) bn sin(nx)
• Finding t means finding a0, a1, a2,
  , an , b1, b2, , bn
• The process of changing from the data to the
  coefficients is called the Fourier Transform
• Computed using Fast Fourier Transform (FFT)

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Example 0.1sin(x)-0.3cos(x)0.4sin(2x)0.2cos(2x
)
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Why Trig Polynomial?

• Eulers formula eiq cos(q) i sin(q)
• (Cool 1 eip 0)
• (eiq)n cos(nq) i sin(nq)
• Rewrite t(x) c-nw-n c-1w-1 c0 c1w
  c2w2 cnwn, where w eix
• Coefficients ck are complex but t is real!

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How To Find t

• Always modify the x-range so it covers from 0 to
  2p for unique part of sequence
• M points in unique part -gt spacing of 2p/M
• Vandermode approach using nice values for x
  (p/6, p/4, p/3, p/2, p, )
• Clever work can find each coefficient directly!
  (idea of FFT)
• For large data sets, use a program

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Time for Boardwork!
Consider this sequence 1, 2, 0, 2, 1, 2, 0,
2, 1, 2,
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Solution
Consider this sequence 1, 2, 0, 2, 1, 2, 0,
2, 1, 2,
T(n) 5/4 1/2 cos(pn/2) 3/4 cos(pn)
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Your turn

• Find the trig polynomial for these sequences
• 2, 3, 2, 3, 2, 3, 2,
• -1, 0, 4, 3, -1, 0, 4, 3, -1,

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Your turn

• Find the trig polynomial for these sequences
• 2, 3, 2, 3, 2, 3, 2,
• t(n) 5/2 1/2 cos(pn)
• -1, 0, 4, 3, -1, 0, 4, 3, -1, t(n)
  3/2 5/2 cos(pn/2) 3/2 sin(pn/2)

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Break Time
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Programming in MATLAB

• Put the commands you want to use together in a
  file, save the file and then call it from the
  command window.
• Two types Scripts and Functions
• Called M-Files as file name is something.m
• Run it by typing something

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First Script

• Up/Down Pattern A(n1) A(n)/2 if A(n)
  is even 3A(n)1 if A(n) is
  odd
• Example 5, 16, 8, 4, 2, 1, 4, 2, 1,
• Does it always get to , 4, 2, 1, ?
• How long does it take to get to 1?

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MATLAB Script updown.m
x(1) input(Enter initial value)k
1while (x(k) 1) if (mod(x(k),2) 0)
x(k1) x(k)/2 else x(k1)
3x(k)1 end k k 1end
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MATLAB Function updownf.m
function x updownf(a)x(1) ak 1while
(x(k) 1) if (mod(x(k),2) 0) x(k1)
x(k)/2 else x(k1) 3x(k)1
end k k 1end
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Modifications

• Display the results (use disp(x) )
• Plot the results (use plot(x,o))
• Save the results (use diary updown)
• Stop at 500 terms (if never reach 1)
• Look for a repeated value
• Compute statistics

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MATLAB Demo
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Field Trip

• Pickup behind Reese at 945AM
• Hike/Lunch/Stream Play
• Back by 500PM

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Teams

• Team 2
• Austin Chu
• Michelle Sarwar
• Jennifer Soun
• Matthew Zimmerman

• Team 1
• Sam Barrett
• Clay Francis
• Michael Hammond
• Angela Wilcox

• Team 3
• Charlie Fu
• Scott McKinney
• Steve White
• Lena Zurkiya

• Team 4
• Stuart Elston
• Chris Goodson
• Meara Knowles
• Charlie Wright