Modeling and Simulation

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Modeling and Simulation

• From observation via implementation to
  computation and back
• P.M.A. Sloot Computational Science, University of
  Amsterdam

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Part Three

• Context sensitive and Parametric Lindenmayer
  Systems

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Chomsky Hierarchy of formal languages (1959)

• Languages are classified by types of productions
  permitted in their grammars
• Notation
• Uppercase Roman nonterminal symbols may be
  expanded further
• Lowercase Roman terminal symbols
• Greek letters represent an arbitrary string

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Chomsky classification

• Regular only rules A b or A Bc
• Context free A a Unrestricted in the RS form
• Context sensitive aAb agb, where g is not the
  empty string.
• Unrestricted Like context sensitive where g can
  be the empty string.

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Context Sensitive

• Productions of the following form have two-sided
  contextal lt a gt ar XThe letter
  a produces X iff preceded by al and followed
  by ar
• Productions of the form al lt a
  X,and a gt ar X, have one-sided context only

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Example Signal Propagation

• Sa,b
• Axiombaaaaaaaa
• P1 b lt a b
• P2 b a
• Results in baaaaaaa
• abaaaaaa
• aabaaaaa
• aaabaaaa

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Example Mycelis
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Parametric L-systemsPhyllotaxis
Note Phyllo-Taxis in Greek means Leaves around
a stem

• Amazing
• Pineapple cone
• Spiral in sunflowers
• Branch positions
• SunFlower 21 , 34

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A simple model (Vogel 1979)

• I seeds are densely packed (n seeds)
• n surface
• n r2
• r Ön
• II Position angle a
• a (360 - a) (360 - a ) 360

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A simple model continued

• 129600 - 2 x 360 a a2 - 360 a 0
• gt a (1080 - 805 ) / 2 137.5 o
• Golden rule angle (in lines and angles) and
  in numbers
• 1,1,2,3,5,8,13,21,34,55
• (Fibonacci Leonardo Pisano 1202)
• for large n Fn/Fn-1 1.618 f
• And f (Ö5 1)/2

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A simple model continued ...

• And a 360 / f2 !!!
• We can use this in
• parametric L-systems

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Simple Model Result I
13 x 21 x
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Simple Model Result II
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Advanced Model (Douady Couder Phys Rev Lett 68
(13) 1992)

• A Apex with Radius R0
• U Growth velocity of shoot
• V radial motion with which primordia seem to
  advect away

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Advanced Model ...

• Include repulsive energy E(d) between 2 particles
  at distance d by E(d) d-1
• Allow for particle radial motion given by V(r)
  V0 r/R0
• Minimize energy New particles are released at a
  point with Emin on the circle with Radius R0 at
  time intervals T

Demo
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