Download Mathematics - St. Joseph`s College

Mathematics:
Hiding in Plain Sight
by
Prof. D.N. Seppala-Holtzman
St. Joseph’s College
faculty.sjcny.edu/~holtzman
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.
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.
Modeling Leads to
Applications
 Identify the problem
 Create appropriate model
 Develop mathematical solution
 Test results in the real world
A few examples:
 Coding
 Operations Research
 Routing
 Graphs
 Non-invasive medical diagnostics
 Global Positioning System
 Social Choice
 Public Relations
Coding
 Secret codes
 Finding errors
 Correcting errors
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.
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 today’s secret codes.
Error Checking Digits
 UPC codes
 Bank accounts
 Airline tickets
 ISBN’s
 Money orders
Error Correcting Codes
 Modem communications
 Satellite transmissions
 Cable television
 CD’s
 DVD’s
 Postnet code
Operations Research
The branch of mathematics
that concerns itself with
finding efficient solutions for
use of time, space and effort.
O.R. Examples
 Elevators
 Locations of central services
 Traffic light sequences
 Airline schedules
 Task ordering
 Routing
Routing
 Snow plows
 Mail delivery
 Meter reading
 Garbage collection
 Street sweeping
Graphs
 Euler circuits
 Optimal Hamiltonian circuits
 Planarity & printed circuits
 Minimum Cost Spanning Trees
Medical Diagnostics
 CAT scans
 PET scans
 MRI
 Sonograms
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
Social Choice
 Voting schemes
 Fair division
 Game theory
 Apportionment
A preference chart
#
18
12
10
9
4
2
1st
A
B
C
D
E
E
2nd
D
E
B
C
B
C
3rd
E
D
E
E
D
D
4th
C
C
D
B
C
B
5th
B
A
A
A
A
A
Voting Methods
 Plurality (A wins)
 Winner’s Run-off (B wins)
 Loser Elimination (C wins)
 Borda Count (D wins)
 Condorcet (E wins)
Kenneth Arrow’s 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.
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?
Game Theory
 The mathematical analysis of
competition searches for optimal
strategies and has applications in,
among other areas, conflict
management, economics and
diplomacy.
Apportionment
 The subject is illustrated by (but not limited
to) the problem of assigning Congressional districts so that each state has representation 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.
Balinski & Young
 Like Arrow’s 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!
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.
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.
Example 2
 Statement: No toothpaste contains
more fluoride than our brand.
 Truth: All toothpaste brands contain
precisely the same amount of
fluoride.
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%.
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?)
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.