Department of Mathematics Perfect Difference Sets Kevin Jennings

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Department of MathematicsPerfect Difference
SetsKevin Jennings
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Group of 7 beads
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Group of 7 beads
Difference Set of 3 beads
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Group of 7 beads
Difference Set of 3 beads
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Group of 13 beads
Difference Set 4 beads
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Group of 13 beads
Difference Set 4 beads
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• Group of 21 beads
• Difference Set 5 beads

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The general pattern is Size of Group
n2n1 Size of Difference Set n1
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n2
Size of Group n2n1 2221
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Size of Difference Set n1 21 3
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n3
Size of Group n2n1 3231
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Size of Difference Set n1 31 4
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n4
Size of Group n2n1 4241 21
Size of Difference Set n1 41 5
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The general pattern is Size of Group
n2n1 Size of Difference Set n1
The next difference set is when n5. Size of
Group 5251 31 Size of Difference Set
51 6 We can build a difference set with 6
beads in a group of 31 beads.
Choosing the 6 beads is a surprisingly difficult
task!
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The general pattern is Size of Group
n2n1 Size of Difference Set n1
The next difference set is when n6. Size of
Group 6261 43 Size of Difference Set
61 7 Can we build a difference set in a group
of 43 beads?
Impossible!
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Prime Number
A Prime Number is only divisible by itself and
one. Examples 2, 3,5 ,7,11,19 are all prime
numbers But 10 is not prime since 10 2 x 5
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The general pattern is Size of Group
n2n1 Size of Difference Set n1

• n must be prime or
• n must be a power of a prime
• -for example 16 2 x 2 x 2 x 2
• 27 3 x 3 x 3
• 121 11 x 11
• Otherwise the difference set cannot be built

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No difference set exists if two different prime
numbers divide n

• Verified by experiment for all numbers up to
• n 3,000,000.

Why are prime numbers related to difference sets?
Big Mystery
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Natural Idea of Compression of Information -Small
difference set generates big group Already
these difference sets are of interest in the
communications sciences
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woof
10010001001111010100010001010001010001001000100010
11110100010010100101000101001
woof
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digital music players
Zip files
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More modest problem We would like to build
larger difference sets using smaller known ones.

• Computers are useful but these calculations are
  too complex.
• By trial and error, in a wider context, modern
  computers can only completely manage up to n10
  case.

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1854 A New Beginning
(for Geometry)
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Everything should be made as simple as
possible, but no simpler.
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The Fano Plane
Each line is a difference set
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(No Transcript)
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Difference Set 1,2,4
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Difference Set 2,3,5
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The Fano Plane
Each line is a difference set
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William Rowan Hamilton (1805-1865)
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Brougham Bridge, Grand Canal
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(No Transcript)
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William Rowan Hamilton (1805-1865)
Arthur Conway President of UCD (1940-47)
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Quaternions are 4-dimensional Octonions are
8-dimensional
Quaternion schema
Octonion schema
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• Octonions are at the heart of
• modern theories of the universe
• modern subatomic matter theories

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Every Line is a Difference Set
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In the particular is contained the universal
James Joyce
Special thanks for inspiration, ideas layout
Rod Gow Tom Laffey Thomas Unger Breda
McMahon Ted Cox Barry Devereux Mathematics
Mathematical Physics postgrads