Dr. Sarah's Review Sheet for WebCT Test 2 on Paper 2
Two 8.5 by 11 sheets with writing on both sides allowed.
One calculator mandatory.
To review, skim through the paper 2 links for each person, and skim through the student papers. In addition, carefully go over class notes and this review sheet.
I am happy to help with anything you don't understand in office hours and/or on the WebCT bulletin board. You need to know what kind of a mathematician each person was/is, and the gender and multicultural issues related to their experiences.
Some guidelines for the mathematics:
(In addition to what I wrote below, be sure that you know importance of these ideas within the context of mathematics, and applications to real-life.)
-Understand the Answers to David Blackwell Worksheet that Elizabeth handed out, and understand how to use the fact that the probability of an event occurring is the number of outcomes you want divided by the number of outcomes possible.
-Understand "The Big Match" setup, and the intuition involved in playing it.
Marjorie Lee Browne
-Know the definition of a group under * and how to work with the group axioms.
-Know how to compute determinants of real and complex 2x2 matrices, and be able to use the fact that the determinant of a product of matrices is the product of the determinants. Also know how to take the transpose of a matrix. Know how to check and see if something is an element of one of the following groups:
-GL(n,C) is the group of invertible (ie det not equal to 0) nxn matrices of complex numbers. In fact, almost every interesting group in geometry is a subgroup of this group. Understand why this is a group under matrix multiplication, but not a group under matrix addition (be able to produce matrices that show that the closure axiom of the group properties fails for matrix addition).
-SL(n,C) consists of all nxn matrices with determinant equal to one. If a matrix A is in SL(n,C), then the map và Av preserves volume and orientation in R^n.
-O(n) consists of matrices so that AxA^t = Id. O(2) consists of rotations and reflections. Also if A is in O(n), then A preserves distance between points in R^n.
-SO(n) consists of the subset of O(n) with determinant 1, and are rotation matrices.
-Understand how GL(n,C) can be decomposed as the topological product of C with SL(n,C).
Grace Murray Hopper
-Know the definition of an irreducible polynomial as a polynomial with integer coefficients that is not able to be factored into smaller degree polynomials with rational coefficients.
-Know examples of reducible and irreducible polynomials. Know how to use the quadratic polynomial to solve for the roots of a polynomial and apply that info to determine whether a polynomial is irreducible or reducible.
-Understand Eisenstein's criterion for irreducible polynomials and be able to apply it
x^n + A_(n-1) x^(n-1) +A_1 x + A_o=0
Eisensteins criterion states that if all the coefficients, except the first one, are divisible by a prime p, and the constant coefficient is not divisible by p^2, then the polynomial is irreducible.
-Understand that Hopper's thesis, she worked to determine reducibility or irreducibility of polynomials by first converting polynomials to geometric figures, and then determining whether the figures decompose.
-Understand how this picture shows the decomposition of an icosahedron (20 triangular faces placed together with 5 triangles around each vertex) into 10 tetrahedrons (4 triangular faces placed together with 3 triangles around each vertex).
-Know the definition of a PDE, and when it can be applied (see WebCT test 1)
-Understand how to check if something is a solution of a PDE.
-Be able to work with boundary conditions in order to solve for constants in a solution.
-Understand the importance of Navier-Stokes
-Understand the statement of Hilbert's 10th problem: Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers.
-Know examples of diophantine equations which can be solvable in integers and also examples that are not solvable in integers.
-Understand how Fermat's Last Theorem can be rewritten to look like a Diophantine equation, and understand whether it has a solution in integers.
-Understand that for many diophantine equations, people use many different techniques to determine whether there are solvable in integers or not. Sometimes, as in the case of Fermat's Last Theorem, people work for hundred's of years to determine the solvability of an equation.
-Understand that Hilbert's 10th problem was resolved by Julia Robinson and Yuri
Matijasevich, who proved that there are diophantine equations for which we (and computers) can not determine solvability. I.e. while the solvability of MANY equations can be determined eventually, there are some diophantine equations for which it can not be determined whether they are solvable or not in a finite number of steps.
-Given a curve, understand how to find the tangent vector (first derivative) and normal vector (second derivative) to the curve.
-Know that the oscillating plane is formed by the normal and tangent to the curve, and know how to visualize the plane on different curves.
-Understand the importance of the oscillating plane.
-Understand what a difference equation is, and when it is useful (see WebCT test 1)
-Understand what a non-linear difference equation is
-Understand what the difference equation is for a rumor, the intuition behind it, and what the graph looks like when you plug it into your graphing calculator.
-Understand that Stephens worked on much more complicated difference equations.
-Know the definition of a perfect number, and know that if 2^k-1 is prime (for k>1), then n = 2^(k-1) (2^k-1) is perfect and every even perfect number is of this form
-Understand that while we do know they are rare, it is not yet known whether there are infinitely many, or finitely many perfect numbers.
-Understand that there has not been a single odd perfect number discovered yet, but there is also not any proof saying that they do not exist.
-Understand that Stubblefield worked to find lower bounds for odd perfect numbers, ie if an odd perfect number does exist, then it must be bigger than the square root of a google (know what a google is).
-Understand the complex plane and how to find eigenvalues of a real or complex matrix.
-Understand the statement of the Gershgorin Theorem and how to apply it to narrow in on the eigenvalues of a matrix without actually calculating them
-Understand that she worked to find eigenvalues of 6x6 matrices in order to calculate the flutter speed of an aircraft during WWII.
J. Ernest Wilkins, Jr.
-Understand the definition of a ruled surface and how it can be written in the form of
x(s,t)=a(t) + k w(t), where w(t) is a line going through the point a(t), which sweeps out the surface. Understand this format for a hyperboloid and a cylinder, as well as the usual formulas you know in terms of x, y and z coordinates.
-Understand that a cubic surface is the the surface corresponding to a degree three polynomial.
-Understand that mathematicians go back and forth between geometry and algebra in order to gain deep understanding of equations/surfaces.