2018 Term 2

Term 2 --- Week 1

8     : David   : 2013 AMC Junior exam pages 1 and 2, also
                  Q.4 from ToT JA-39-aut (doors & keys problem).
9     : William : 2013 AMC Junior exam pages 1 and 2, 
                  a proof of explicit formula for Fibonacci numbers.
10    : Peter   : Mathematical Structures part 1:
                  Mathematicians make up stuff - shapes, numbers
                  Shapes: curved spaces (straight lines may not look straight, 
                          eg direct flights SYD to LON)
                  Numbers: N --> {0} U N  --> Z --> Q (specification, equivalences) --> R --> C
                  Groups: axioms, examples & non-examples inc M_2(Z), rotation groups
11    : Zoltan  : CAT 2013 Senior paper and (pages 76 and 77 of) problems from KoMaL.
12    : Ralph   : p-adic integers, rationals, valuations, limit of sequences,
                  3, 33, 333, 3333, ... approaches -1/3, 5-adically.


Term 2 --- Week 2

8     : Tamiru  : 2013 AMC Junior exam, pages 3 and 4
9     : Chris   : Pick's Theorem.
10    : Peter + : Mathematical Structures part 2:
        David     Axioms & examples for rings & fields. Examples including Z, Z/nZ, Z/pZ=GF(p), 
                  When is Z/nZ a field? (ab = 0  ==> a & b don't have inverses),  
                  Z[x],  Z[x]/(f(x)), GF(p^k), GF(2)[x]/(x^2+x+1) = GF(4),
                  Q, Q[x], Q[x]/(f(x)),  Q[x]/(x^2-2) = Q(sqrt(2)),  Q[x]/(x^2+1) = Q(i),
                  R, R[x]/(x^2+1) = R(i) = C
                  Mentioned that these "crazy" fields we'd invented "just for fun" have suddenly 
                  become essential for the backbone of modern internet life & business.
11+12 : Zoltan  : CAT 2013 Senior paper and (pages 76 and 77 of) problems from KoMaL.
                  Playing the Horrible Game/Ice Breaker/Swiss Contest

Term 2 --- Week 3

8   : Elizabeth : 2013 AMC Junior exam, pages 5 and 6,
                  followed by Polite numbers.
9   : Zoltan +  : Part A of the CAT 2013 Intermediate paper, 
                  Pages  22 and 23 of problems from KoMaL.
      David       Played NIM. Explained method for recognising winning/losing positions.
                  Strategy for winning from the latter.
10  : Tryon     : Infinity.
11  : Peter     : Math Structures 1: 
                  Mathematicians make up stuff - shapes, numbers
                  Shapes: curved spaces (straight lines may not look straight, 
                          eg direct flights SYD to LON)
                  Numbers: N --> {0} U N  --> Z --> Q (specification, equivalences) --> R --> C
                  Groups: axioms, examples & non-examples, M_2(Z), 
                          rotation groups - symmetry group of square
                  Axioms & examples for rings & fields. Examples including Z, Z/nZ
12  : William + : Q1 and Q2 from the Number Theory and Statistics sheet.
      Tamiru

Term 2 --- Week 4

8   : William + : Introduction to groups: constructing D3, D4, subgroups, 
      Heath       finite integer groups under + and *, isomorphisms 
                  -> geometric interpretation of addition in a finite group, 
                  open/closed operators, identity and inverses, 
                  Q? When are the integers under multiplication a group?
9   : Zoltan    : Part A of the CAT 2013 Intermediate paper, 
                  Pages  22 and 23 of problems from KoMaL.
10  : David  +  : Q1, 2 from Number Theory and Statistics.
      Tamiru      Rational zeroes Theorem for solving integral polynomial equations.
11  : Peter     : Math Structures 2: 
                  Axioms & examples for groups, rings & fields. 
                  Examples including Z, Z/nZ, Z/pZ=GF(p) 
                  (when is Z/nZ a field?)  Proved, in a group, identity & inverses are unique
                  Ring: ab = 0  ==> a & b don't have inverses (proof),  
                  Proved Z/nZ is field <==> n is prime
                  Z[x],  Z[x]/(f(x)), GF(p^k), GF(2)[x]/(x^2+x+1) = GF(4), GF(8)
                  Q, Q[x], Q[x]/(f(x)),  Q[x]/(x^2-2) = Q(sqrt(2)),  Q[x]/(x^2+1) = Q(i),
                  R, R[x]/(x^2+1) = R(i) = C
                  Mentioned that these "crazy" fields  (GF(p^k), algebraic number fields) we'd 
                  invented "just for fun" have suddenly become essential for the backbone of 
                  modern internet life & business.
                  Started on Z_p (mentioned Q_p)
12  : Tryon     : Infinity.

Term 2 --- Week 5

8   : William   : 2013 AMC Junior exam, pages 7 and 8.
9   : Tamiru    : Rational zeroes Theorem for solving integral polynomial equations.
                  Q1 from Number Theory and Statistics.
10  : Zoltan    : Part A of the CAT 2013 Intermediate paper, 
                  Pages  22 and 23 of problems from KoMaL.
11  : Tryon     : Infinity
12  : Peter     : Math Structures 1: 
                  Mathematicians make up stuff - shapes, numbers
                  Shapes: curved spaces (straight lines may not look straight, 
                          eg direct flights SYD to LON)
                  Numbers: N --> {0} U N  --> Z --> Q (specification, equivalences) --> R --> C
                  Groups: axioms, examples & non-examples, M_2(Z), 
                          rotation groups - symmetry group of square
                  Axioms & examples for rings & fields. Examples including Z, Z/nZ

Term 2 --- Week 6

8   : William   : Showed the basic theorems of groups --
                  Uniqueness of the identity and inverses from the axioms.
9   : Elizabeth : Polite numbers, followed by
                  Q2, Q4, Q6 from this Factors, multiples, primes problem sheet.
10  : Zoltan    : Part A of the CAT 2013 Intermediate paper, 
                  Pages  22 and 23 of problems from KoMaL.
11  : Ralph     : p-adic integers, rationals, valuations, limit of sequences,
                  3, 33, 333, 3333, ... approaches -1/3, 5-adically.
                  anomaly, bound, catastrophe, dilemma, experiment --> space quantisation
                  N in Z_p, 0 in Z_p, -1 in Z_p --> Z in Z_p.
12  : Peter     : Math Structures 2: 
                  Axioms & examples for groups, rings & fields. 
                  Examples including Z, Z/nZ, Z/pZ=GF(p) 
                  (when is Z/nZ a field?)  Proved, in a group, identity & inverses are unique
                  Ring: ab = 0  ==> a & b don't have inverses (proof),  
                  proved Z/nZ is field <==> n is prime
                  Z[x],  Z[x]/(f(x)), GF(p^k), GF(2)[x]/(x^2+x+1) = GF(4), GF(8)
                  Q, Q[x], Q[x]/(f(x)),  Q[x]/(x^2-2) = Q(sqrt(2)),  Q[x]/(x^2+1) = Q(i),
                  R, R[x]/(x^2+1) = R(i) = C
                  Mentioned that these "crazy" fields  (GF(p^k), algebraic number fields) we'd 
                  invented "just for fun" have suddenly become essential for the backbone of 
                  modern internet life & business.  Started on Z_p (mentioned Q_p)

Term 2 --- Week 7

8   : Zoltan    : Part A of the CAT 2013 Intermediate paper.
9   : David     : Allan's The Number on the Tomb problem sheet.
10  : William   : Discussion of entropy of data and compression,
                  Construction of the Huffman tree for some of the English alphabet.
                  Talk about unique decoding, prefix-free-ness. (See English frequencies).
11  : Ralph     : Completed the proof that Q is in Q_p. Def'ns of countable and dense subsets. 
                  Recalled Z, Q are countable, Z is not dense in R, Q is dense in R.
                  Proved Z_p and Q_p are uncountable.  Z is dense in Z_p and Q is dense in Q_p.
12  : Peter     : RSA Cryptography, Integer factoring algorithms, 
                  trial division, quadratic sieve, number field sieve.