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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.