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Governors School for the Sciences

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Angela Wilcox. Dr. Collins. Charlie Fu. Scott McKinney ... Chris Goodson. Meara Knowles. Charlie Wright. Math Bowl Competition. About 1 minute per question ... – PowerPoint PPT presentation

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Title: Governors School for the Sciences


1
Governors School for the Sciences
  • Mathematics

Day 3
2
MOTD Leonardo Fibonacci
  • 1170 to 1250 (Italy)
  • Popularized ancient mathematics
  • Solved problems in algebra, geometry and number
    theory
  • Best know for the Fibonacci sequence x(n1)
    x(n) x(n-1)

3
Geometric Patterns
  • Sequence 1, 2, 4, 8, 16, generated by the
    obvious rule A(n1) 2A(n)
  • Any geometric sequence is expressed
    A(n1) r A(n)
  • Identify r A(n1)/A(n) constant
  • PGF is exponential A(n) rn A(0)

4
1st Generalization
  • A(n1)/A(n) r(n) nonconstant
  • A(n1) r(n) A(n), A(0) A0
  • Whats the PGF? A(1) r(0) A0 A(2)
    r(0) r(1) A0 A(3) r(0) r(1) r(2) A0
    A(n) r(0) r(1) r(n-1) A0

5
Old vs. New
6
Big Generalization Difference Equation
  • Pattern generated by the rule x(n1)
    f(x(n)) with x(0) x0
  • Called a difference equation or a dynamical
    system
  • Iterates x0, f(x0), f(f(x0)),
  • Write fk(x0) f(f(f(x0))) (k-times)
  • Orbit O(x0)x0, f(x0), f2(x0), f3(x0),

7
Big Question
  • Given x0 and f, can you predict the behavior of
    the orbit O(x0)?
  • Does it tend to one value? go off to
    infinity? oscillate between values? do none
    of the above?

8
Linear 1st Order DE
  • x(n1) a(n)x(n) c(n)
  • c(n) 0 homogeneous else non-homogeneous
  • Know if a(n) lt 1 and c(n) 0 then every orbit
    tends toward 0
  • If a(n) a, alt1 and c(n) c then every orbit
    tends toward c/(1-a)

9
General Answer
  • Except for simple cases it is hard or impossible
    to find a solution of a DE and analyze orbits
    that way
  • Instead look at Equilibrium Points
    Stability Theory

10
Equlibrium Points
  • Equilibrium Point Point x such that f(x) x
    (fixed point)
  • If x(0) x, then x(k) x for all k
  • Solve via algebra or by graphical technique
  • Eg f(x) x2, solve x2 x, get two equillbrium
    points x1, x0

11
Example of graph technique
12
Stability Theory
  • What happens if x(0) is near an equilibrium point
    x?
  • If x(n) stays near x x is stable or
    attracting
  • If x(n) moves away from x x is unstable
    or repelling
  • Determine experimentally or by a Cobweb
    Diagram

13
Experiments for f(x)x2
X(0) 0.9
X(0) 1.1
X(0) -0.1
14
Cobweb Plot
  • Plot y f(x) and y x on same axis
  • Plot (x0,f(x0))
  • Move horizontally to y x
  • Move vertically to y f(x)

15
Theory
  • Worksheet Draw Cobwebs around Equilibrium
    Points
  • How does angle of crossing between yx and yf(x)
    affect answer?

16
Teams
  • Team 3
  • Austin Chu
  • Michelle Sarwar
  • Jennifer Soun
  • Matt Zimmerman
  • team one p
  • Sam Barrett
  • Clay Francis
  • Michael Hammond
  • Angela Wilcox
  • Dr. Collins
  • Charlie Fu
  • Scott McKinney
  • Steve White
  • Lena Zurkiya
  • Denominators of Doom
  • Stuart Elston
  • Chris Goodson
  • Meara Knowles
  • Charlie Wright

17
Math Bowl Competition
  • About 1 minute per question
  • 5 questions
  • 10 points right, 0 points wrong, 4 points for no
    answer
  • Winning team gets additional 50 pts
  • Today Team 1 vs. Team 2 Team 3 vs.
    Team 4

18
Lab Today
  • Study various types of DE to find
  • Equilibrium points
  • When stable/unstable
  • Other patterns

19
Done
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