Mathematics:

1
Mathematics

• Hiding in Plain Sight
• by
• Prof. D.N. Seppala-Holtzman
• St. Josephs College
• faculty.sjcny.edu/holtzman

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What is Mathematics?

• Mathematics is the search for absolute truths via
  rigorous deductive reasoning within a system
  governed by a finite list of precise, immutable,
  mutually consistent laws.

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How can it apply to the real world?

• By choosing a list of rules that produce a
  fantasy world that approximates the real world
  without the imprecision and messiness of it, one
  creates a mathematical model.

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Modeling Leads to Applications

• Identify the problem
• Create appropriate model
• Develop mathematical solution
• Test results in the real world

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A few examples

• Coding
• Operations Research
• Routing
• Graphs
• Non-invasive medical diagnostics
• Global Positioning System
• Social Choice
• Public Relations

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Coding

• Secret codes
• Finding errors
• Correcting errors

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Secret Codes

• Amazingly, the current state of the art method
  for encoding secret information (for example
  military, diplomatic, financial) is based upon
  results from one of the least applied areas of
  mathematics --- Number Theory.

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Hard vs. Easy

• Many processes have the property that doing it in
  one direction is much harder than doing it in
  reverse.
• For example, multiplying two numbers is easy but
  finding the integral factors of a number is hard.
• This simple observation is the basis for todays
  secret codes.

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Error Checking Digits

• UPC codes
• Bank accounts
• Airline tickets
• ISBNs
• Money orders

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Error Correcting Codes

• Modem communications
• Satellite transmissions
• Cable television
• CDs
• DVDs
• Postnet code

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Operations Research

• The branch of mathematics that concerns itself
  with finding efficient solutions for use of time,
  space and effort.

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O.R. Examples

• Elevators
• Locations of central services
• Traffic light sequences
• Airline schedules
• Task ordering
• Routing

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Routing

• Snow plows
• Mail delivery
• Meter reading
• Garbage collection
• Street sweeping

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Graphs

• Euler circuits
• Optimal Hamiltonian circuits
• Planarity printed circuits
• Minimum Cost Spanning Trees

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Medical Diagnostics

• CAT scans
• PET scans
• MRI
• Sonograms

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Global Positioning System

• Knowing how far away one is from a single
  satellite places one somewhere on the surface of
  a sphere
• Two satellites give the intersection of two
  spheres, i.e. a circle
• Three satellites give the intersection of three
  spheres, i.e. two points
• A fourth satellite fixes the time issue

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Social Choice

• Voting schemes
• Fair division
• Game theory
• Apportionment

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A preference chart
18 12 10 9 4 2
1st 1st A B C D E E
2nd 2nd D E B C B C
3rd 3rd E D E E D D
4th 4th C C D B C B
5th 5th B A A A A A
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Voting Methods

• Plurality (A wins)
• Winners Run-off (B wins)
• Loser Elimination (C wins)
• Borda Count (D wins)
• Condorcet (E wins)

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Kenneth Arrows Theorem

• There exists a list of 5 desirable properties
  that, it is agreed, all voting schemes ought to
  have.
• Arrow proved that a voting scheme that satisfies
  these 5 properties under all conditions cannot
  exist.

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Fair Division

• The problem is to divide a fixed set in such a
  way that no sharer feels cheated. I cut, you
  choose works for 2 participants but the problem
  becomes harder for more.
• What if the set contains non-divisible objects?

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Game Theory

• The mathematical analysis of competition searches
  for optimal strategies and has applications in,
  among other areas, conflict management, economics
  and diplomacy.

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Apportionment

• The subject is illustrated by (but not limited
  to) the problem of assigning Congress-ional
  districts so that each state has re-presentation
  in proportion to its population. That is, the
  ratio of the population of a state to the total
  US population should equal the ratio of the
  number of representatives that state has to 435,
  the size of Congress. The problem comes from
  rounding fractions to whole numbers.

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Balinski Young

• Like Arrows theorem Supposing 3 commonly
  accepted properties that it is agreed that any
  fair apportionment scheme should have under all
  conditions, no fair method can exist!

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Public Relations

• Axiom 1 No statement may be made that is
  verifiably false.
• Axiom 2 An advertiser (or politician) will make
  only maximally favorable statements.
• Conclusion The truth must be the most negative
  interpretation of any statement.

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Example 1

• Statement You may save up to 50!
• Truth Then again, you may not. Nothing can be
  more than 50 off but everything could be 0 off.

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Example 2

• Statement No toothpaste contains more fluoride
  than our brand.
• Truth All toothpaste brands contain precisely
  the same amount of fluoride.

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Example 3

• Statement We are the fastest growing company in
  the US.
• Truth We have just gone from 1 client to 2,
  growing by 100.

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Other Examples

• Converse reasoning
• Confusing correlation with cause and effect
• Unverifiable statements (It costs less than you
  think.)
• Comparatives with no antecedent (This product is
  better. Better than what?)

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Conclusion

• Mathematics is truly all around us. Wherever
  there is rigorous, analytic thought being carried
  out in some axiomatic framework, there is
  mathematics. Applications abound. Whether it is
  hiding, or not, it is clearly in plain sight.