The approach of mathematics has changed over the years.  Describe how the approach of mathematics began starting in Ancient Babylonia, and then discuss how it changed during the time of the Ancient Greeks.  Then, compare this to how mathematics is approached today.  Be specific in your description and explanation.

1) Choose a subgroup of Pentagons D₅, and list all the left or right cosets of your pet. Will the set of right cosets be different from the left cosets? Explain.

2) Does Pentagons D_5, have any non-trivial normal subgroups? If so, give an example.If not, explain why not.

3) Is there a second binary structure that would make Pentagons D₅ a ring?

Find the local maximum and minimum values and saddle point(s) of the function. If you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

f(x, y) = x³ − 3x + 3xy²

Prove that every outerplanar graph is 3-colorable.

Assume that ψ : [a, b] → R is continuously differentiable. A critical point of
ψ is an x such that ψ'(x) = 0. A critical value is a number y such that for at
least one critical point x we have y = ψ(x).
(a) Prove that the set of critical values is a zero set. (This is the Morse-Sard
Theorem in dimension one.)
(b) Generalize this to continuously differentiable functions R → R.

Abstract Algebra and Galois Theory

Show how to one can deduce the Fundamental Theorem of Galois theory from the Artin's lemma.

I'm tasked with finding the characteristic equation, eigenvalues, and bases for the eigenspaces of this matrix

[1, -3, 3]

[3,-5,3]

[6,-6,4]

After working on the problem, I believe the characteristic equation is (λ + 2)^2(λ-4), giving eigenvalues of -2 and 4 (please correct me if I'm wrong). However, I'm lost when finding the bases for the eigenspace because I'm not sure the eigenvectors I get are linearly independent.

u''-2u'-8u=0 u(0)= α， u'(0)=2π

y''+9y'=cosπt, y(0)=0, y'(0)=1

y′ = t, y(0) = 1, solution: y(t) = 1+t2/2
y′ = 2(t + 1)y, y(0) = 1, solution: y(t) = et2+2t

y′ = 5t4y, y(0) = 1, solution: y(t) = et5
y′ = t3/y2, y(0) = 1, solution: y(t) = (3t4/4 + 1)1/3

For the IVPs above, make a log-log plot of the error of Explicit Trapezoidal Rule at t = 1 as a function ofhwithh=0.1×2−k for0≤k≤5.

Prove that the composition of two quotient maps is a quotient map.

1. Let T : Mn×n(F) → Mn×n(F) be the transposition map, T(A) = At. Compute the characteristic polynomial of T. You may wish to use the basis of Mn×n(F) consisting of the matrices eij + eji, eij −eji and eii.

2.  Let A = (a b c d) (2 by 2 matrix) and let T : M2×2(F) → M2×2(F) be defined asT (B) = AB. Represent T as a 4×4 matrix using the ordered basis {e11,e21,e12,e22}, and use this matrix to prove that the characteristic polynomial of T is the square of the characteristic polynomial of A.

1. For each matrix A below compute the characteristic polynomial χA(t) and do a direct matrix computation to verify that χA(A) = 0.

(4 3

-1 1)    (2 1 -1 0 3 0 0 -1 2) (3*3 matrix)

2.  For each 3*3 matrix and each eigenvalue below construct a basis for the eigenspace Eλ.

A= (9 42 -30 -4 -25 20 -4 -28 23),λ = 1,3
A= (2 -27 18 0 -7 6 0 -9 8) , λ = −1,2

3. Construct a 2×2 matrix with eigenvectors(4 3) and (−3 −2) with eigen-values 2 and −3, respectively.

4. Let A be the 6*6 diagonal matrix below. For each eigenvalue, compute the multiplicity of λ as a root of the characteristic polynomial and compare it to the dimension of the eigenspace Eλ.

(x 0 0 0 0 0 0 x 0 0 0 0 0 0 y 0 0 0 0 0 0 x 0 0 0 0 0 0 z 0 0 0 0 0 0 x)

5.   Let A be an 3*3 upper triangular matrix with all diagonal elements equal, such as (3 4 -2 0 3 12 0 0 3)

Prove that A is diagonalizable if and only if A is a scalar times the identity matrix.

1.(4-x^2)y''+2y=0, x0=0

(a) Seek power series solutions of the given differential equation about the given point x0; find the recurrence relation.

(b) Find the first four terms in each of two solutions y1 and y2 (unless the series terminates sooner)

. (c) By evaluating the Wronskian W(y1, y2)(x0), show that y1 and y2 form a fundamental set of solutions.

(d) If possible, find the general term in each solution.

Let f : Rn → R be a differentiable function. Suppose that a point x∗ is a local minimum of f along every line passes through x∗; that is, the function

g(α) = f(x^∗ + αd)

is minimized at α = 0 for all d ∈ R^n.

(i) Show that ∇f(x∗) = 0.

(ii) Show by example that x^∗ neen not be a local minimum of f. Hint: Consider the function of two variables

f(y, z) = (z − py^2)(z − qy^2),

where 0 < p < q.

Use Laplace Transforms to solve the following second-order differential equation:  

y"-3y'+4y=xe^2x where y'(0)=1 and y(0)=2