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.

Let T be an operator on a finite-dimensional complex vector space V, and suppose that dim Null T = 3, dimNullT2 =6. Prove that T does not have a square root; i.e. there does not exist any S ∈ L (V) such that S2 = T.

Give a proof for the standard rule of differentiation, the Chain Rule. To do this, use the following information:

10.1.3 Suppose that the function f is differentiable at c, Then, if f′(c) > 0 and if c is an accumulation point of the set constructed by intersecting the domain of f with (c,∞), then there is a δ > 0 such that at each point xin the domain of f which lies in (c,c+δ) we have f(x) > f(c). If c is an accumulation point of the domain of f intersected with (−∞, c), then there is a δ > 0 such that at each point y in the domain of f which lies in (c−δ,c) we have f(y) < f(c). (Similar case holds for if f′(c) < 0.)

Create two cases for f'(g(x))*g'(x), one where g'(x)=0 (in which case you do nothing), and one where g'(x) not= 0 (in which case you use information from 10.1.3).