This question is about using Principle of Inclusion-Exclusion Formula, and has to be used to solve this problem:

How many arrangements of a, a, a, b, b, b, c, c, c have no adjacent letters the same? (Hint: This is tricky—not a normal inclusion–exclusion problem.)

Find the infinite series solution about x = 0 for the following differential equation x2y"+ 4xy' + (2+x)y = 0,without using k substitution and using Bessel's, Legrende's, or frobenius equations.

Use the table of values to estimate S ^6 0 f(x) dx, use three equal subintervals and the

(only solve part D)

1. Left end points
2. Right end points
3. Midpoints

When f is an increasing function, how does each estimate compare with the actual value? Explain your reasoning

D. Also find L6, R6, T3, and T6. Remember that Tn is the average of Ln and Rn. Finally, use your calculator to do quadratic regression to find a decent model for your data [let’s call it q(x)] and have your calculator approximate S ^6 0 q (x) dx

Solving Differential Equations using Laplace Transform

a) y" - y' -2y = 0

y(0) = 1, y'(0) = 0

answer: y = 1/3e^2t + 2/3e^-t

b) y" + y = sin2t

y(0) = 2, y'(0) = 1

answer: y(t) = 2cost + 5/3 sint - 1/3sin2t

c) y^4 - y = 0

y(0) = 0, y'(0) = 1, y"(0) = 0, y'''(0) = 0

answer y(t) = (sinht + sint)/2

Find the matrix P that diagonalizes A, and check your work by computer P^-1AP. This matrix is

[-14 12]

[-20 17]

I've tried this problem, and I keep getting the eigenvalues of λ=1, 2 and the eigenspace [4 5] for λ=1, and eigenspace [3 4] for λ=2. However, whenever I check it with P^-1AP, it doesn't produce a diagonal matrix.

A supplier to a music store buys compact discs at $1 per unit and sells them to the music store at $5 per unit. The music store sells each disc to the end consumer at $10. At the retail price, the market demand distribution is give as follows

1. With no contract in place, determine how much the music store should order to maximize it’s expected profit? (8 points)

1. (4 points) What is the total expected profit of the supply chain corresponding to the order quantity calculated from question 1? Show the calculations for the expected profit at the music store and supplier.

2. Now, the supplier to the music store agrees to buyback discs that have not sold at $2.50 per disc. What is the total expected profit of the supply chain if the music store orders 4 CDs? Show the calculations for the expected profit at the music store and supplier. (7 points)

3. The supplier agrees to sell each disc to the music store at $1 instead of $5. The music store agrees to share 35 % of the revenue from each disc sold. What is the total expected profit of the supply chain if the music store orders 2 CDs? Show the calculations for the expected profit at the music store and supplier. (6 points)

Let S = {1,2,3,...,10}.

a. Find the number of subsets of S that contain the number 5.

b. Find the number of subsets of S that contain neither 5 nor 6.

c. Find the number of subsets of S that contain both 5 and 6.

d. Find the number of subsets of S that contain no odd numbers.

e. Find the number of subsets of S that contain exactly three elements.

f. Find the number of subsets of S that contain exactly three elements, one of which is 3.

g. Find the number of subsets of S that contain exactly five elements, all of them even.

h. Find the number of subsets of S with exactly five elements, including 3 or 4 but not both.

2. Guyana forestry Commission wishes to carry out a survey to characterize Acai berry palms within Guyana.

(a) Identify the target population.

(b) Suggest and justify an appropriate sampling technique

(c) Suggest and justify an appropriate sample size.

Emery Pharmaceutical uses an unstable chemical compound that must be

kept in an environment where both temperature and humidity can be controlled.

Emery uses 200 pounds per month of the chemical, estimates the holding cost to be

£3.33 (because of spoilage), and estimates order costs to be £10 per order. The cost

schedules of two suppliers are as follows:

Vendor 1 Vendor 2 Quantity Price/LB (£) Quantity Price/LB (£) 1-49 35.00 1-74 34.75 50-74 34.75 75-149 34.00 75-149 33.55 150-299 32.80 150-299 32.35 300-499 31.60 300-499 31.15 500+ 30.50

500+ 30.75

Vendor 3 Vendor 4 Quantity Price/LB (£) Quantity Price/LB (£) 1-99 34.50 1-199 34.25 100-199 33.75 200-399 33.00 200-399 32.50 400+ 31.00

400+ 31.10

a) What quantity should be ordered, and which supplier should be used?

b) Discuss factor(s) should be considered besides total cost.

Please with explanations

Show that the groups of the following orders have a normal Sylow subgroup.

(a) |G| = pq where p and q are primes.

(b) |G| = p^aq where p and q are primes and q < p.

(c) |G| = 4p where p is a prime greater than four.

Suppose that B is a 12 × 9 matrix with nullity 5. For each of the following subspaces, tell me their dimension, along with what value of k is such that the subspace in question is a subspace of R k . (For example, a possible – though incorrect – answer is that Col B is a subspace of R 2 .) So, you’ll need eight answers for this problem (two answers for each of the four parts). • Col B • Null B • Row B • Null BT

Suppose that m is a fixed positive integer. Show that the initial value problem

y' = y^2m/(2m+1) , y(0) = 0

has infinitely many continuously differentiable solutions. Why does this not contradict Picard’s Theorem?

1.) Build the parametric equations of a circle centered at the point (2,-3) with a radius of 5 that goes counterclockwise and t=0 gives the location (7,-3)

2.) Build the parametric equations for an ellipse centered at the point (2, -3) where the major axis is parallel to the x-axis and vertices at (7, -3) and (-3, -3), endpoints of the minor axis are (2, 0) and (2, -6). The rotation is counterclockwise

3.) Build the parametric equations for a hyperbola centered at the point (0, 0) where the vertices are at the point (5, 0) and (-5, 0) and the foci are at (7, 0) and (-7, 0)

Can someone give me a simple proof of Prime Number Theorem and Bertrand's Postulates?

Note:Suggest me a proof by keepind in mind that I am a post graduate student who is preparing for my Number Theory Examination.

The demand for roses was estimated using quarterly figures for the period 1971 (3rd quarter) to 1975 (2nd quarter). Two models were estimated and the following results were obtained:

Y = Quantity of roses sold (dozens)

X₂ = Average wholesale price of roses ($ per dozen)

X₃ = Average wholesale price of carnations ($ per dozen)

X₄ = Average weekly family disposable income ($ per week)

X₅ = Time (1971.3 = 1 and 1975.2 = 16)

ln = natural logarithm

The standard errors are given in parentheses.

1. lnY_t=0.627-1.273 ln X_2t + 0.937 ln X_3t + 1.713 ln X_4t - 0.182
2. ln X_5t

                                    (0.327)            (0.659)           (1.201)             (0.128)

            R² = 77.8%                              D.W. = 1.78                             N = 16

B.        ln Y_tÙ = 10.462 - 1.39 ln X_2t

                                       (0.307)

            R² = 59.5%                              D.W. = 1.495                           N = 16

Correlation matrix:

a) How would you interpret the coefficients of ln X₂, ln X₃ and ln X₄ in model A?                

What sign would you expect these coefficients to have? Do the results concur with your expectation?                                                                                                                                 

b) Are these coefficients statistically significant?                                                                   

c) Use the results of Model A to test the following hypotheses:

i) The demand for roses is price elastic                                                                                   

ii) Carnations are substitute goods for roses                                                               

iii) Roses are a luxury good (demand increases more than proportionally as income rises)  

d) Are the results of (b) and (c) in accordance with your expectations? If any of the tests are statistically insignificant, give a suggestion as to what may be the reason.                        

e) Do you detect the presence of multicollinearity in the data? Explain.                                            

f) Do you detect the presence of serial correlation? Explain                                                   

g) Do the variables X₃, X₄ and X₅ contribute significantly to the analysis? Test the joint significance of these variables.        

h) Starting from model B, assuming that at the time point of January, 1973, there was a disaster that heavily affected the quantity of roses produced. Suggest a model to check if we have to use two different models for the data before and after the disaster. (Using dummy variable).