Determine a differential equation that has x = 0 as a regular singular point and that the roots of the index equation are i and i. Find a solution around x = 0.

Describe the stereographic projection on the Riemann sphere of the set {z : Re z = 1} by an equation in (x, y, z)

Let f : [a, b] → R be a monotone function.

Show that the discontinuity set Disc(f) is countable.

What is the meaning of calculating the "flux" when solving integrals? -

Also; What is the difference between these integrals?
-Line integral
-Double integral
-Triple integral
-Surface integral
-Flux, flux integral

1. Magnitude & Direction of Vector

Write a function m-file to find the magnitude and the direction of cosine of vector F which expresses 3 dimension - x, y, and z.

Here are requirements for the function m-file.

- Parameter-list (input(s)): · 3 dimensional vector · row vector or column vector

- undetermined - Return Value (outputs):

· Magnitude of a given vector, |F| = √ F2 x + F2 y + F2 z

· Angles(degree) from direction of cosine cos(α) = Fx |F| cos(β) = Fy |F| cos(γ) = Fz |F|

· Order of outputs : [Magnitude, α, β, γ]

- Function name : ‘Mag_Vec_yourEmailAccount’ Using a script m-file which will call the Mag_Vec_yourEmailAccount function, save outputs with following given inputs. - F1 = 3ˆi + 2ˆj − 6ˆk. Save it on EX1_1.dat.

- F2 = 4ˆi − 2ˆk. Save it on EX1_2.dat.

- F3 = 4ˆi + 3ˆj. Save it on EX1_3.dat.

- F4 = 5ˆj − 12ˆk. Save it on EX1_4.dat.

- F5 =ˆi +ˆj + ˆk. Save it on EX1_5.dat.

using mathlab functions, thank you!

A set of firms N = {1,...,n} select quantities of the same good to sell. Each firm i selects qi, and P(q1,...,qi,...,qn) = 1,000−(q1 +···+ qn). Assume they will choose q1 +···+ qn < 1,000. Each firm has the same linear cost for producing the goods, so C(qi) = cqi.

(a) Write down each firm’s set of strategies, and profit function.

(b) Write the FOC’s for each firm.

(c) Since firms are symmetric, let qi = qj for each pair i and j. Solve for qi.

(d) Solve for the market price, and each firm’s profit. What happens when n is large?

(e) Do you think this is reasonable/observed?

(f) What elements do you think are missing from the model? Write as many as you can.

(g) What are some difference between Bertrand and Cournot competition? Why are the results different?

Week 10 DB: Misleading Statistics

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1. Where can you find misleading statistics?

2.Why do you believe misleading statistics are used in our everyday lives?

3. Find an example of a misleading statistic.Identify where you found it and provide us with either a link or a picture of it. Make sure to state what information from your example makes it misleading and how you were about to identify this.

4.Can you think of another way to illustrate or express this information that might make the authors purpose clearer?

Solve the following differential equations:

1.) y"(t)- 6 y'(t)+9 y(t)=6t^2e^(3t) ; y(0)=y'(0)=0

2.)x"(t)+4x(t)=t +4 ; x(0)=1 , x'(0)=0

Hexadecimal digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F.

1. How many hexadecimal strings of length twelve have five A’s and five B’s?

2. How many hexadecimal strings of length twelve have at most three E’s?

3. How many hexadecimal strings of length twelve have exactly three A’s and at least two B’s?

4. How many hexadecimal strings of length twelve have exactly two A’s and exactly two B’s, so that the two B’s are sandwiched between the A’s? 00A0 B8B8 111A is an example of such string.

5. How many hexadecimal strings of length twelve have six digits from the set 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and six digits from the set A, B, C, D, E, F? No repetition is allowed.

6. How many hexadecimal strings of length twelve have four digits from the set 0, 1, 2, 3, 4, 5, four digits from the set 6, 7, 8, 9, A and four digits from the set B, C, D, E, F?

7. How many hexadecimal strings of length twelve have at least three distinct digits?

8. How many hexadecimal strings of length twelve, have the sum of all digits equal to 30? Allowed digits are 2, 3,..., 9

Solve as much as you can

1. Using the Lehmer algorithm (Z_i = (a Z_i-1 + c) (mod m)      0 < Z_i < m-1

                   U_i = Z_i / m,

with m = 16, a = 5, c = 3, and Z₀ = 7, generate three random numbers (U1,U2,U3 ) ………….(SO (a))

subject is Modeling and Simulations

solve the initial value problem using the method of undetermined coefficients y''+9y=sin3t, y(0)=0, y'(0)=2. Please list every step for algebra.

Given vector s = [4, -5, 7], u = [-6, 8, 11], and v = [-5, 3, -7]
a. Let r1(t) be the vector equation through s and u. Express the vector equation r(t) in two ways and then find r(1); r(-5); r(9)
b. Let r2(t) be the vector equation through u and parallel to r1(t), Express r2(t) in two forms and then find r(-3); r(13); r(2)
c. Let w = 2s -4v, find r(t) through u and v

Prove the follwing statements

Suppose that S is a linearly independent set of vectors in the vector space V and let w be a vector of V that is not in S. Then the set obtained from S by adding w to S is linearly independent in V.

If U is a subspace of a vector space V and dim(U)=dim(V), then U=V.