Mathematics: - PowerPoint PPT Presentation

1 / 30
About This Presentation
Title:

Mathematics:

Description:

Hiding in Plain Sight. by. Prof. D.N. Seppala-Holtzman. St. Joseph's College ... Applications abound. Whether it is hiding, or not, it is clearly in plain sight. ... – PowerPoint PPT presentation

Number of Views:33
Avg rating:3.0/5.0
Slides: 31
Provided by: dnseppala
Learn more at: http://faculty.sjcny.edu
Category:
Tags: in | mathematics | plain | sight

less

Transcript and Presenter's Notes

Title: Mathematics:


1
Mathematics
  • Hiding in Plain Sight
  • by
  • Prof. D.N. Seppala-Holtzman
  • St. Josephs College
  • faculty.sjcny.edu/holtzman

2
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.

3
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.

4
Modeling Leads to Applications
  • Identify the problem
  • Create appropriate model
  • Develop mathematical solution
  • Test results in the real world

5
A few examples
  • Coding
  • Operations Research
  • Routing
  • Graphs
  • Non-invasive medical diagnostics
  • Global Positioning System
  • Social Choice
  • Public Relations

6
Coding
  • Secret codes
  • Finding errors
  • Correcting errors

7
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.

8
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.

9
Error Checking Digits
  • UPC codes
  • Bank accounts
  • Airline tickets
  • ISBNs
  • Money orders

10
Error Correcting Codes
  • Modem communications
  • Satellite transmissions
  • Cable television
  • CDs
  • DVDs
  • Postnet code

11
Operations Research
  • The branch of mathematics that concerns itself
    with finding efficient solutions for use of time,
    space and effort.

12
O.R. Examples
  • Elevators
  • Locations of central services
  • Traffic light sequences
  • Airline schedules
  • Task ordering
  • Routing

13
Routing
  • Snow plows
  • Mail delivery
  • Meter reading
  • Garbage collection
  • Street sweeping

14
Graphs
  • Euler circuits
  • Optimal Hamiltonian circuits
  • Planarity printed circuits
  • Minimum Cost Spanning Trees

15
Medical Diagnostics
  • CAT scans
  • PET scans
  • MRI
  • Sonograms

16
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

17
Social Choice
  • Voting schemes
  • Fair division
  • Game theory
  • Apportionment

18
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
19
Voting Methods
  • Plurality (A wins)
  • Winners Run-off (B wins)
  • Loser Elimination (C wins)
  • Borda Count (D wins)
  • Condorcet (E wins)

20
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.

21
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?

22
Game Theory
  • The mathematical analysis of competition searches
    for optimal strategies and has applications in,
    among other areas, conflict management, economics
    and diplomacy.

23
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.

24
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!

25
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.

26
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.

27
Example 2
  • Statement No toothpaste contains more fluoride
    than our brand.
  • Truth All toothpaste brands contain precisely
    the same amount of fluoride.

28
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.

29
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?)

30
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.
Write a Comment
User Comments (0)
About PowerShow.com