Mathematics:

1. Mathematics: Hiding in Plain Sight by Prof. D.N. Seppala-Holtzman St. Joseph’s 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 today’s secret codes.

9. Error Checking Digits • UPC codes • Bank accounts • Airline tickets • ISBN’s • Money orders

10. Error Correcting Codes • Modem communications • Satellite transmissions • Cable television • CD’s • DVD’s • 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. Voting Schemes • If more than two candidates or choices are being decided amongst, there are a variety of voting schemes in common use. • They all seem, on the face of things, to be equally fair. • Different schemes can produce different results even if the will of the electorate is the same.

19. Common Voting Schemes I • Plurality: The candidate with the most 1st place votes wins. • Winner Run-off: The two candidates with the most 1st place votes compete head to head. • Loser Elimination: The candidate with the fewest 1st place votes is eliminated, iteratively.

20. Common Voting Schemes II • Borda Count: Each of the n candidates is given (n-1) points for each 1st place vote, (n-2) points for each 2nd place vote and so on. This is commonly used in sports team ranking. • Condorcet: Each pair of candidates competes head to head. There may or may not be a winner in this method.

21. A preference chart

22. Voting Methods • Plurality (A wins) • Winner’s Run-off (B wins) • Loser Elimination (C wins) • Borda Count (D wins) • Condorcet (E wins)

23. 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.

24. 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?

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

26. 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.

27. 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!

28. 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.

29. 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.

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

31. 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%.

32. 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?)

33. 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.