Download stern st1

INTRODUCTION TO
COMPUTER SCIENCE
WEEK6(TA HOUR)
Xinchen Yan
Oct.16th, 2012
Outline
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Problems in Class
Q&A
Advanced Topics
Halting Problem
Relative Primality
Stern-Brocot tree
Pascal Triangle
Lucas Thm
Halting problem
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All programs
Finite length
Countably infinite
P1, P2, P3, …
Relative Primality
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m⊥n
 m,n are integers and gcd(m,n) = 1
m/n is in lowest terms  gcd(m,n) = 1
In general,
m/gcd(m,n) ⊥n/gcd(m,n)
k ⊥m and k ⊥n
 k ⊥mn
Mediant of m/n and m’/n’: m+m’/n+n’
Suppose n,n’ > 0 and m,m’ ≥ 0
Assume m/n < m’/n’
Stern-Brocot tree(a kind of binary tree)
Stern-Brocot tree
Stern-Brocot tree
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Property I: m’n-mn’=1
Consider: m+m’/n+n’
Property II:
m/n < m+m’/n+n’ < m’/n’
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Question:
for any a,b>0 and a ⊥b,
a/b exists?
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Stern-Brocot tree
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Algorithm
St0:
St1:
St2.0: if equals, done.
St2.1: if less than
St3.1:
St2.2: if more than
St3.2:
St4: back to St1
Stern-Brocot tree
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Property III:
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Pf. We have
Thus,
Finally,
Stern-Brocot tree(Farey)
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Farey series
F1, F2, F3, F4, F5, …
Suppose m/n, m’/n’,
and m’’/n’’ are
consecutive elements
of FN.
Relationship?
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Farey series
F1, F2, F3, F4, F5, …
Suppose m/n, m’/n’,
and m’’/n’’ are
consecutive elements
of FN.
Relationship?
Farey series
Stern-Brocot tree(Farey)
Stern-Brocot tree(3 Amazing results)
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Going down left/right:
L/R
Ex. 5/7 is LRRL
S: a sequence of LR
f(S): fraction
corresponding to S
f(LRRL) = 5/7
Stern-Brocot tree(Amazing result No.1)
Stern-Brocot tree(Amazing result No.2)
Stern-Brocot tree(Amazing result No.3)
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Irrational numbers don’t appear in the S-B tree.
But all rational numbers are close to them do.
Consider
We found that
Stern-Brocot tree(Amazing result No.3)
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That’s why |2*| = |N|, but |2^N| ≠ |R|.
Then why |2^N| = |R|?
Have a rest
Pascal Triangle
Important counting ideas(I)
Pascal Triangle
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(x + y)2 = x2 + 2xy + y2 = 1x2y0 + 2x1y1 + 1x0y2
(x + y)n = a0xn + a1xn−1y + a2xn−2y2 + ...
+ an−1xyn−1 + anyn
Pascal Triangle(Question Type 1)
Pascal Triangle(Question Type 2)
Pascal Triangle(Question Type 3)
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The number of odd numbers in each line?
How to prove?
Lucas Thm.
Reference
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Course Material from CMU 15-251 Great
Theoretical Ideas in CS, Fall2011
Course Material from MIT 6.042J Maths for CS,
Spring2005
Lecture Notes from SJTU Maths in CS, Winter 2011
Concrete Mathematics by DEK.