An element e of a ring is called an idempotent if e^2 = e. Find a nontrivial idempotent e in the ring Z143.

Problem 2. Consider a graph G = (V,E) where |V|=n.

2(a) What is the total number of possible paths of length k ≥ 0 in G from a given starting vertex s and ending vertex t? Hint: a path of length k is a sequence of k + 1 vertices without duplicates.

2(b) What is the total number of possible paths of any length in G from a given starting vertex s and ending vertex t?

2(c) What is the total number of possible cycles of any length in G from a given starting vertex s?

Solve the following wave equation using Fourier Series

a²u_xx = u_tt, 0 < x < pi, t > 0, u(0,t) = 0 = u(pi,t), u(x,0) = sinx, u_t(x,0) = pi - x

Solve the following wave equation using Fourier Series

a²u_xx = u_tt, 0 < x < 1, t > 0, u(0,t) = 0 = u(1,t), u(x,0) = x², u_t(x,0) = 0

Solve the following heat equation using Fourier Series

u_xx = u_t, 0 < x < 1, t > 0, u_x(0,t) = 0 = u_x(1,t), u(x,0) = 1 - x²

ii. Let G = (V, E) be a tree. Prove G has |V | − 1 edges using strong induction. Hint: In the inductive step, choose an edge (u, v) and partition the set vertices into two subtrees, those that are reachable from u without traversing (u, v) and those that are reachable from v without traversing (u, v). You will have to reason why these subtrees are distinct subgraphs of G.

iii. What is the total degree of a tree?

Solve the following heat equation using Fourier Series

u_xx = u_t, 0 < x < pi, t > 0, u(0,t) = 0 = u(pi,t), u(x,0) = sinx - sin3x

Solve the following heat equation using Fourier Series

u_xx = u_t, 0 < x < 1, t > 0, u(0,t) = 0 = u(1,t), u(x,0) = x/2

The buyer of a piece of real estate is often given the option of buying down the loan. This option gives the buyer a choice of loan terms in which various combinations of interest rates and discount points are offered. The choice of how many points and what rate is optimal is often a matter of how long the buyer intends to keep the property.

Darrell Frye is planning to buy an office building at a cost of $988,000. He must pay 10% down and has a choice of financing terms. He can select from a 7% 30-year loan and pay 4 discount points, a 7.25% 30-year loan and pay 3 discount points, or a 7.5% 30-year loan and pay 2 discount points. Darrell expects to hold the building for four years and then sell it. Except for the three rate and discount point combinations, all other costs of purchasing and selling are fixed and identical.

1. What is the amount being financed?

2. If Darrell chooses the 4-point 7% loan, what will be his total outlay in points and payments after 48 months?

3. If Darrell chooses the 3-point 7.25% loan, what will be his total outlay in points and payments after 48 months?

4. If Darrell chooses the 2-point 7.5% loan, what will be his total outlay in points and payments after 48 months?

5. Of the three choices for a loan, which results in the lowest total outlay for Darrell?

Use the Laplace transform to solve the given system of differential equations.

dx/dt + 2x +dy/dt= 1

dx/dt− x+ dy/dt− y= e^t x(0) = 0, y(0) = 0

Suppose the sets A and B have both n elements.

1. Find the number of one-to-one functions from A to B.

2. Find the number of functions from A onto B.

3. Find the number of one-to-one correspondences from A to B.