Prove that any graph where every vertex has degree at most 3 can be colored with 4 colors.

A SIS disease spreads through a population of size K = 30, 000 individuals. The average time of recovery is 10 days and the infectious contact rate is 0.2 × 10^(−4) individuals^(−1) day^(−1) . (a) The disease has reached steady-state. How many individuals are infected with the disease? (b) What is the minimum percentage reduction in the infectious contact rate that is required to eliminate the disease? (c) By implementing a raft of measures it is proposed to reduce the value of the infectious contact rate to one percent of its initial value. Will this be sufficient to eliminate the disease within twenty-eight days? (d) Is it feasible to eliminate the disease within twenty-eight days solely by reducing the value of the pairwise contact rate? (e) How may days will the ‘raft of measures’ have to be maintained if we are to eliminate the disease?

Let f(n,k) be the number of equivalence relations with k classes on set with n elements.

a) What is f(2,4)?

b) what is f(4,2)?

c) Give a combinational proof that f(n,k) = f(n-1,k-1)+k * f(n-1,k)

Find two integers x,y (if possible) such that 32x + 47y = 1. Is there more than one solution?

Use double induction to prove that (m+ 1)^n> mn for all positive integers m; n

Prove the following:

Let f(x) be a polynomial in R[x] of positive degree n.

1. The polynomial f(x) factors in R[x] as the product of polynomials of degree
1 or 2.

2. The polynomial f(x) has n roots in C (counting multiplicity). In particular,
there are non-negative integers r and s satisfying r+2s = n such that
f(x) has r real roots and s pairs of non-real conjugate complex numbers as
roots.

3. The polynomial f(x) factors in C[x] as the product of n degree-one polynomials.

Using MATLAB:

Consider the following Boundary Value Differential Equation:

y''+4y=0

y(0)=-2

y(π/4)=10

Which has the exact solution: y(x)= -2cos(2x)+10sin(2x)

Create a program that will allow the user to input the step size (in x), and two guesses for y'(0). The program will then use the Euler method along with the shooting method to solve this problem. The program should give the true error at y(π/8). Run your code with step sizes of π/400 and π/4000 and compare the errors. Chose any guesses for y'(0) that are reasonable. Also list the two errors you calculated.

Let G be a simple planar graph with no triangles.

(a) Show that G has a vertex of degree at most 3. (The proof was sketched in the lectures, but you must write all the details, and you may not just quote the result.)

(b) Use this to prove, by induction on the number of vertices, that G is 4-colourable.

Evaluate the integral of f(x) below between x=1.5 and x=5.3 using Gauss-Legendre formulas for 2, 3, and 4 points.

Compare with analytical integration, calculate the % error of the numerical method.

f(x)=4+8x−21x² +16x³ −5x⁴ +7x⁵

Using field axioms, prove the following theorems:

(i) If x and y are non-zero real numbers, then xy does not equal 0

(ii) Let x and y be real numbers. Prove the following statements

1. (-1)x = -x

2. (-x)y = -(xy)=x(-y)

3. (-x)(-y) = xy

(iii) Let a and b be real numbers, and x and y be non-zero real numbers. Then a/x + b/y = (ay +bx)/(xy)

Let G be a group of order p

am where p is a prime not dividing m. Show the following

1. Sylow p-subgroups of G exist; i.e. Sylp(G) 6= ∅.
2. If P ∈ Sylp(G) and Q is any p-subgroup of G, then there exists g ∈ G such that Q 6
gP g−1

; i.e. Q is contained in some conjugate of P. In particular, any two Sylow p-
subgroups of G are conjugate in G.

3. np ≡ 1 (mod p) and np|m.

for each matrix A below, describe the invariant subspaces for the induced linear operator T on F^2 that maps each v set of F^2 to T(v)=Av. (a) [4,-1;2,1], (b) [0,1;-1,0], (c) [2,3;0,2], (d) [1,0;0,0]

Consider n numbers x₁, x₂, . . . , x_n laid out on a circle and some value α. Consider the requirement that every number equals α times the sum of its two neighbors. For example, if α were zero, this would force all the numbers to be zero.

(a) Show that, no matter what α is, the system has a solution.

(b) Show that if α = 1/2 , then the system has a nontrivial solution.

(c) Show that if α = − 1/2 , then there is a nontrivial solution if and only if n is even.

Find a basis for R4 that contains the vectors X = (1, 2, 0, 3)⊤ and ⊤ Y =(1,−3,5,10)T.