Prove Rolles Theorem

Prove that the function defined to be 1 on the Cantor set and 0 on the complement of the Cantor set is discontinuous at each point of the Cantor set and continuous at every point of the complement of the Cantor set.

Prove the Heine-Borel Theorem

5.How can you proof a proposition in the form of ∀x P(x) is NOT true.

6.a) Briefly explain what does it mean to say B is a subset of A? What is the procedure to prove that?
b) How many subsets of A are there, if |A| = n ?
c) Define an arbitrary set A (with |A|=4), list all the elements of the power set of A. (P(A))

3. Briefly explain how you can prove that two sets are equal.

find the dirichlet green function of laplace equation for the interior of a colander with radius a

Please find a solution to the following:

Δu=0, 1<r<4, 0≤θ<2π

u(1,θ)=cos5*θ, 0<θ<2π

u(4,θ)=sin4*θ, 0<θ<2π

In how many ways can you arrange the members of two committees of 11 males and 11 females around a circular table if the only thing that matters is who the neighbors of each committee member are on each side (having person 1 on the right and person 2 on the left is different from having person 1 on the left and person 2 on the right), and not who sits in which chair if:

1. There are no restrictions

2. The two leaders of the committee members cannot sit next to each other

3. Two males and two females cannot be seated next to each other (every other gender)

For each of the following functions, find the extreme value of z (this can be either max or min), subject to a given constraint by the method of direct substitution.

(a) z = xy, subject to the constraint x² + y² = 16 (Note that there are four solutions.)

(b) z = 3x² − 10xy + 12y² , subject to the constraint y = 20 − 1/2 x

(c) z = xy, subject to the constraint x − y = 2

(d) z = x² + y, subject to the constraint x/y = 2

Given the following functions, can you have the corresponding a) Fourier series, b) Fourier transform and c) Laplace transform? If yes, find them, if not, explain why you can not.

A, x(t) = -1+cos(2t) + sin(pt+1)                                                                                 (4-1)

B, x(t) =2d(t) cos(2t) +d(t-1.5p) sin(2t)                                                                    (4-2)

C, x(t) = 1+cos(1.5t) + cos(4t)                                                                                    (4-3)

A thin rectangular plate coincides with the region defined by 0 ≤ x ≤ π,
0 ≤ y ≤ 1. The left end and right end of the plate are insulated. The
bottom of the plate is held at temperature zero and the top of the plate
is held at temperature f(x) = 4 cos(6x) + cos(7x).
Set up an initial-boundary value problem for the steady-state
temperature u(x, y).

In the latest reality TV show, the finalists (in no particular order) are: April and Andy, Leslie and Ben, and Donna and Chris, all of whom live in Indiana. Voting for the winner occurred last week by region and the results are below:

A.) Using the Borda method, who would finish first, second, and third? Provide support for your answer.

B.) Using the Plurality with elimination method, who would finish first, second, and third? Provide support for your answer.

1313) Given the DEQ y'=1x-y^2*7/10. y(0)=8/2. Determine y(2) by Euler integration with a step size (delta_x) of 0.2. ans:1

Let G be a group. For each x ∈ G and a,b ∈ Z⁺

a) prove that x^a+b = x^ax^b

b) prove that (x^a)^-1 = x^-a

c) establish part a) for arbitrary integers a and b in Z (positive, negative or zero)

Using the definition of the line element, for the reference configuration and for the deformation. Say how is defined λ_{m} associated with m's direction. Describe the different cases for this deformation (stretching and shortening) and from this write the definition of the unitary elongation ε_{m}.