2020 Term 2
Term 2 --- Week 1
8+9 : Zoltan : Euclidean geometry proofs about angles.
10 : Ellen : An introduction to circle geometry and proved some basic theorems.
11+12: Peter : Review of calculus (differentiation, product & chain rules).
Review of vector spaces. Used calculus in R^2 to derive
rotational motion law (torque = i\alpha) from Newton's law (F=ma).
New vector spaces from old, i.e. dual, direct sum & tensor product.
CCSE : Ralph : Used the online calculator to give a tutorial of Magma.
Discussed examples of:-
comment,
assignment,
print,
loop,
conditional, and
subroutine
statements in the magma syntax. Then contrasted multiplication
and factorisation of integers and polynomials.
Used surfer to visualise the effect of factorisation and
decomposition of reducible multivariate polynomials into
irreducible pieces.
Term 2 --- Week 2
8 : Zoltan : ??? ... ???
9 : Ralph : Exploration of the arithmetic functions tau(n) and sigma(n),
namely, the number of divisors and the sum of divisors functions.
Discussion about why 1 is called a unit --- but not called a prime.
Relation to perfect numbers, and Euler's even perfect number theorem.
Discussion of what type of numbers were hard to factor ... followed
by a timed Magma experiment on 50-digit to 100-digit RSA numbers.
10 : Ellen : Continued work on circle geometry and proved more basic theorems.
11+12: Peter : New vector spaces from old ones via direct sums, tensor products, wedge products.
Discussed dimension and bases.
Rotation of coordinate base in R^2, group structure.
Wedge product versus cross product in R^2.
CCSE : Josh : Introduction to unix system basics ( see unix_basics ).
A start to the bandit security "wargame" ... ( see bandit ).
Term 2 --- Week 3
8 : Zoltan : Horrible game, triangular numbers.
9 : Ellen : Induction problems, towers of Hanoi, number of diagonals in a convex n-gon,
tiling a 2^n bt 2^n board missing one square with L-shaped tiles.
10 : Ralph : Exploration of the arithmetic functions tau(n) and sigma(n),
namely, the number of divisors and the sum of divisors of n.
Relationship to perfect numbers, Mersenne numbers and
Euclid's theorem and Euler's theorems on even perfect number theorem.
11+12: Peter : Vector spaces. Rotations of coordinates in real 2-dim space (i.e. in C).
Viewed rotations by theta as 2x2 matrices (or e^{i theta}), i.e groups.
Considered power series for e^x, cos(x), sin(x) (c.f. even & odd, or
real & imag components of e^{ix}).
Similarly for hyperbolic trig functions cosh, sinh, tanh.
Imaginary rotations (boosts) are 2x2 matrices in [ct,x,y,z] space.
Derived velocity addition formula of special relativity.
Effect of boosts on components of electromagnetic 2-form.
CCSE : Josh : Continued exploring unix security.
Tested more examples of the bandit game.
Term 2 --- Week 4
8 : Ralph : An exploration of integer divisibility, Mersenne primes and
perfect numbers. We finished with some Magma experiments.
9 : Chris : Computed the area under the parabola via horizontal rectangles,
vertical rectangles, and triangles.
Considered the differential equation for "flattening the curve".
10 : Zoltan : Derived a formula for sum of first n squares.
11+12: David : Covered propositional calculus aimed at upper high-school level.
Outlined material in Chapters 2 and 4 of Stanford intro to logic.
Did the first few exercises in the Fitch system calculator.
CCSE : Josh : Free-form enrichment guided by questions from students.
Term 2 --- Week 5
8 : Ralph : Coding in Magma with the online calculator.
Input, output, conditionals, loops.
This was used to explore primes below 10^n and compare
counts with Gauss's two functions Pi(x) ~ x/Log(x) and
Pi(x) ~ Li(x).
9 : Chris : Python-based image processing experiments.
10 : Zoltan : Counting squares and rectangles in grids.
11+12: Peter : Inner products on vector spaces, rotation matrices.
Invariance under rotations, Cauchy-Schwartz, a.b = |a||b|cos(theta),
Lagrange's identity (aka |a x b|^2 = |a|^2|b|^2 = |a.b|^2 + |a^b|^2,
1 = cos^2 + sin^2).
CCSE : Chenni : A password cracking workshop.
Term 2 --- Week 6
8 : Ellen : Introduction to induction. Problems including towers of Hanoi
and proving the angle sum formula for polygons.
9 : David : First order logic.
10 : Zoltan : Patterns related to the multiplication table, sums of squares,
sums of cubes, finished with the horrible game.
11+12: Peter : Exterior algebra, Wedge products, determinant of matrix is
a coefficient of wedge product of rows
CCSE : Ben : Computer art and interpretaion of art p5.js.
Computer generated music.