September 23 rd Lecture

1. September 23rd Lecture Today’s lecture Mersenne Primes and Perfect Numbers (Marin Mersenne) Gut Reasoning - from NYT 15/0/9/2008 Procedures in Maple The Final Project the midterm on the 25th The Exceptional Lie Group E8

2. ??? Euclid (3C BCE) and Euler (1707-83): Mersenne Primes and Perfect numbers • For information on odd perfect numbers see • www.austms.org.au/Publ/Gazette/2008/Sep08/CommsRoberts.pdf

3. Breaking News from www.mersenne.org

4. One research team has found that how readily people rally their approximate number sense is linked over time to success in even the most advanced and abstruse mathematics courses. Other scientists have shown that preschool children are remarkably good at approximating the impact of adding to or subtracting from large groups of items but are poor at translating the approximate into the specific. Taken together, the new research suggests that math teachers might do well to emphasize the power of the ballpark figure, to focus less on arithmetic precision and more on general reckoning. “When mathematicians and physicists are left alone in a room, one of the games they’ll play is called a Fermi problem, in which they try to figure out the approximate answer to an arbitrary problem,” said Rebecca Saxe, a cognitive neuroscientist at the Massachusetts Institute of Technology who is married to a physicist. “They’ll ask, how many piano tuners are there in Chicago, or what contribution to the ocean’s temperature do fish make, and they’ll try to come up with a plausible answer.” “What this suggests to me,” she added, “is that the people whom we think of as being the most involved in the symbolic part of math intuitively know that they have to practice those other, nonsymbolic, approximating skills.” Yes, but see also http://micromath.wordpress.com/2008/09/16/gut-instincts-surprising-role-in-math/ GUT Reasoning: NYT (16/09/2008)

5. Procedures are just fancier functions (in Maple attachment) • Calling Sequence • proc (argseq) local nseq; global nseq; options nseq; descriptionstringseq; statseq end proc • proc (argseq)::type; local var1::type1, var2::type2, ...; global nseq; options nseq; description stringseq; statseq end proc • Parameters • argseq - the formal parameter names • type - (optional) an assertion on the type of the returned value • nseq - (optional) the names of local/global variables and the options in effect • var1,var2 - (optional) the names of local variables • type1,type2 - (optional) assertions on the types of local variables • stringseq - description lines for the procedure • statseq - the body of the procedure Procedures in Maple

6. Examples • > lc := proc( s, u, t, v ) • description "form a linear combination of the arguments"; • s * u + t * v • end proc; (or just end) • lc := proc (s, u, t, v) description "form a linear combination of the arguments"; s*u+t*v end proc > print( lc ); • proc (s, u, t, v) description "form a linear combination of the arguments"; s*u+t*v end proc • > lc( Pi, x, -I, y ); • See Also • envvar, error, Functions, index[procedure], kernelopts, last_name_eval, Operators, parameters, Parameter passing, Procedure options, remember, Procedure type checking, procedure[paramtype], procname, Reading and saving, arg, nargs, return, spec_eval_rules, type[procedure], type[function] Procedures in Maple

7. Basic procedures: keeps variables local • With loops and conditional steps: • Calling Sequence • | for <name> | | from <expr> | | by <expr> | | to <expr> | | while <expr> | • do <statement sequence> end do; (or od) • OR • | for <name> | | in <expr> | | while <expr> | • do <statement sequence> end do; • (Note: Phrases located between | | are optional.) • 3. With recursive structure • 4. With more complicated arguments • 5. Using maple.ini files … best learned by example

8. Inside Math and MathResource Both are mathematics dictionaries on Steroids: They allow you also to obtain output/draw pictures that can be dropped into Word or PowerPoint or Excel, or …. Inside Math (School) and the MathResource (University) We also saw Portrait 4 (Shown below) Its manual is on M2600 webpage

9. From Inside Math Inside Math and MathResource, II Maths dictionaries on Steroids: n. one of the five regular polyhedra, once imbued with great mystical significance: a cube, a regular tetrahedron, a regular octahedron, a regular dodecahedron, or a regular icosahedron. Johannes Kepler (1571 - 1630) was led to his discovery of the laws of planetary motion and his defence of Copernican astronomy by circum-scribing or inscribing the orbits of the five other known planets around the Platonic solids, with an inscribed or circumscribed circle representing the orbit of the Earth; the results agree with observation, allowing for eccentricity, to within approximately 5%. (OCR from MathResource)

10. The Final Project To pass, you just have to follow the rules. To get a good mark, you have to show some initiative and add some value. I expect 16-20 power point slides or 10-15 pages in Word which describe (using an appropriate level of mathematical text setting, prose and images) a Maple based exploration of one or more related topics. Examples A handful of the Explorations anywhere inthe Computer as Crucible Two or Three of the Ten Things to Try An essay on the History of Pi Another topic of your own choosing In each case we want to “sign off” with you on the topic before the break. Either with Matt in a Tutorial or by email or in person with me. More on the Taxicab question I asked last week www.durangobill.com/Ramanujan.html and on a generalization http://mathworld.wolfram.com/TaxicabNumber.html