The Cantor set, C, is the set of real numbers r for which Tⁿ(r) ϵ [0,1] for all n, where T is the tent transformation. If we set C₀= [0,1], then we can recursively define a sequence of sets C_i, each of which is a union of 2ⁱ intervals of length 3^-i as follows: C_i+1 is obtained from C_i by removing the (open) middle third from each interval in C_i. We then can define the Cantor set by

C= i=0 to infinity C_i
In general, a set S is called self-similar if for some real number r the scale of S by r can be exactly covered (without overlap) by a finite number, say n, of copies of the original set S. Then if r^d=n we say that d is the similarity dimension of the set S.
1. Consider the Cantor set as described above.
a. What is the length of the Cantor set?
b. Find the similarity dimension of the Cantor set.

1. Rachael runs 2 km to her bus stop, and then rides 4.5 km to school. On average, the bus is 45 km/h faster than Rachael’s average running speed. If the entire trip takes 25 min, how fast does Rachael run?

2. Write an equation of a rational function that satisfies all of these conditions

● Vertical asymptote at x = -8 and x = 5

● Horizontal asymptote at y = 0

● x-intercept at (-2, 0)

● f(0) = -2

● has a hole at x = 3

1) Define a sequence of polynomials H n (x ) by H 0 (x )=1, H 1 (x )=2 x , and for n>1 by H n+1 (x )=2 x H n (x )−2 n H n−1 (x ) . These polynomials are called Hermite polynomials of degree n. Calculate the first 7 Hermite polynomials of degree less than 7. You can check your results by comparing them to the list of Hermite polynomials on wikipedia (physicist's Hermite polynomials).

2) Use the power series method to solve the differential equation y ' '−2 x y '+λ y=0 where λ is an arbitrary constant. Verify that you get two independent solution y1, y2 by choosing a0=1, a1=0 and a0=0 , a1=1 . Show that the series expansion for one of the two solutions will terminate resulting in a polynomial solution when λ is chosen to be a positive even integer, λ=2 ,4,6,8,10 ,12 ,14 ,.... Rescale the polynomial solution so it starts with 2 n x n + lower powers of x , n=λ/2. Calculate the list of polynomials obtained this way and compare them to your solution of problem 1)

How many distinct 2x2 matrices can we have by using the numbers 1,2,3 and 4. Repeating numbers is not allowed.

2!*2!

4*3*4*3

4!

How many multiplications are required to compute AB if A is a 4x7 matrix and B is a 7x5 matrix?

35

28

980

140

4^4

Let A be an nxn matrix, What is det(A) if A has a row of zeros?

n-1

1

0

n

1. Determine if the following statements are true or false. If a statement is true, prove it in general, If a statement is false, provide a specific counterexample.

Let V and W be finite-dimensional vector spaces over field F, and let φ: V → W be a linear transformation.

A) If φ is injective, then dim(V) ≤ dim(W).

B) If dim(V) ≤ dim(W), then φ is injective.

C) If φ is surjective, then dim(V) ≥ dim(W).

D) If dim(V) ≥ dim(W), then φ is surjective.

E) If V = {0} , then φ is injective.

F) If dim(V) NOT= dim(W), then φ is not bijective.

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a) How many arrangements of all the letters in AABBCCD starts with A but does not end with A?

b) Find the number of arrangements of all the letters in AABBCCD in which none of the patterns AA, BB or CC occurs.

How many ways are there to rearrange the letters in MARKER?

Show Work

Find the closed formula solution to each of the following recurrence relations with the given initial conditions. Use an iterative approach and show your work! What is a_100? a) a_n=a_(n-1)+2,a_0=3 b) a_n=a_(n-1)+2n+3,a_0=4 c) a_n=2a_(n-1)-1,a_0=1 d) a_n=-a_(n-1),a_0=5

There are three vectors in R4 that are linearly independent but not orthogonal: u = (3, -1, 2, 4), v = (-2, 7, 3, 1), and w = (-3, 2, 4, 11). Let W = span {u, v, w}. In addition, vector b = (2, 1, 5, 4) is not in the span of the vectors. Compute the orthogonal projection bˆ of b onto the subspace W in two ways: (1) using the basis {u, v, w} for W, and (2) using an orthogonal basis {u' , v' , w'} obtained from {u, v, w} via the Gram Schmidt process. Finally, explain in a few words why the two answers differ, and explain why only ONE answer is correct.  

A rubber ball is dropped from a height of 30 feet, and on each bounce it rebounds up 22% of its previous height. How far has the ball traveled vertically at the moment when it hits the ground for the 23rd time? Round your answer to two decimal places.

"Brief Discuss Homogeneous Differential Equations." This is the presentation topic of my Subject Differential Equation.
Explain in a simple way

The determinant of a matrix is the product of its eigenvalues. Can you prove this when A is diagonalizable? How about if A is 2 x 2, and may or may not be diagonalizable? (Hint: What's the constant term in the characteristic polynomial>

I need the answers simple and in order please!!! go with the letter a,b,c etc.

Ahmadi, Inc. manufactures laptop and desktop computers. In the upcoming production period, Ahmadi needs to decide how many of each type of computers should be produced to maximize profit. Each computer goes through two production processes. Process I, involves assembling the circuit boards and process II is the installation of the circuit boards into the casing. Each laptop requires 24 minutes of process I time and 16 minutes of process II time. Each desktop requires 8 minutes of process I time and 32 minutes of process II time. In the upcoming production period, 240 minutes are available in process I and 320 minutes in process II. Each laptop costs $1,800 to produce and sells for $2,250. Each desktop costs $600 to produce and sells for $1,000.

      Let your decision variables be:

                  X1 = Number of laptops to produce

                  X2 = Number of desktops to produce

1. Formulate an LP problem to maximize profit. Write your problem formulation.
2. Graph the constraints and show the region of feasible solutions.
3. Determine the extreme points of the feasible region and find the profit at each extreme point.
4. Draw the isoprofit line and indicate the optimum point.
5. Are there any slacks at optimum?
6. If the selling price per desktop decreases to $700 per unit would there be any change in the optimum solution? If yes, what would be the new optimum solution, and would there be any slack?
7. Assume the company does not want to produce more than 9 laptops in this production period. Add this constraint to the original problem and show the region of feasible solutions.
8. Solve the problem as stated in part g and find the optimal solution. Let the profit per Laptop to be $450 and per Desktop to be $100.
1. Let the selling price of Desktops to be $750 and solve the problem as stated in part g.

Estimate the lowest eigenvalue pair of matrix A using the Inverse Power Method

A = 2  8 10
8 4 5
10 5 7

starting with initial guess x0 = [1 1 1]T and εaλ ≤ 1%