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).

Let the cyclic group {[0], [1], [2], ..., [n − 1]} be denoted by Z/nZ. Consider the following statement: for every positive integer n and every x in Z/nZ, there exists y ∈ Z/nZ such that xy = [1]. (a) Write the negation of this statement. (b) Is the original statement true or false? Justify your answer.

Find the coordinates of the point (x, y, z) on the plane z = 4 x + 1 y + 4 which is closest to the origin.

Linear Programming

How do I use duality to find the optimal value of the objective function for this?

minimize 8y₁+6y₂+2y₃

constraints----

y₁+2y₂ ≥ 3

2y₁+y₃ ≥ 2

y₁ ≥ 0

y₂ ≥ 0

y₃ ≥ 0

NUMBER THEORY

1.Use the Euclidian algorithm to calculate the GCD of 1160718174 and 316258250.

2.Use Fermat’s Little Theorem to solve for x^86 ≡ 6 (mod 29).

Production Scheduling: New jet, Inc. manufactures inkjet printers and laser printers. The company has the capacity to make 70 printers per day, and it has 120 hours of labor per day available. It takes 1 hour to make an inkjet printer and 3 hours to make a laser printer. The profits are $40 per inkjet printer and $60 per laser printer. Find the number of each type of printer that should be made to give maximum profit and find the maximum profit using

1. SIMPLEX METHOD

Rolle's Theorem, "Let f be a continuous function on [a,b] that is differentiable on (a,b) and such that f(a)=f(b). Then there exists at least one point c on (a,b) such that f'(c)=0."

Rolle's Theorem requires three conditions be satisified.

(a) What are these three conditions?

(b) Find three functions that satisfy exactly two of these three conditions, but for which the conclusion of Rolle's theorem does not follow, i.e., there is no point c in (a,b) such that f'(c)=0. Each function should satisfy a different pair of conditions than the other two functions. For each function you should give a definition, a graph, and a short justification of its failing to meet the conclusion of Rolle's Theorem.

a) Find the recurrence relation for the number of ways to arrange flags on an n foot flagpole with 1 foot high red flags, 2 feet high white flags and 1 foot high blue flags.

b) solve the recurrence relation of part a

The Hastings Sugar Corporation has the following pattern of net income each year, and associated capital expenditure projects. The firm can earn a higher return on the projects than the stockholders could earn if the funds were paid out in the form of dividends.

The Hastings Corporation has 3 million shares outstanding. (The following questions are separate from each other).

a. If the marginal principle of retained earnings is applied, how much in total cash dividends will be paid over the five years? (Enter your answer in millions.)

b. If the firm simply uses a payout ratio of 20 percent of net income, how much in total cash dividends will be paid? (Enter your answer in millions and round your answer to 1 decimal place.)

c. If the firm pays a 20 percent stock dividend in years 2 through 5, and also pays a cash dividend of $3.40 per share for each of the five years, how much in total dividends will be paid?

d. Assume the payout ratio in each year is to be 40 percent of the net income and the firm will pay a 30 percent stock dividend in years 2 through 5, how much will dividends per share for each year be? (Assume the cash dividend is paid after the stock dividend.) (Round your answers to 2 decimal places.)

Consider the linear transformation T which transforms vectors x C) in the y-axis. a) Express the vector X = T(x), the result of the linear transformation T on x in terms of the components x and y of X. by reflection [10 Marks] b) Find a matrix T such that T(x) = TX, using matrix multiplication. Calculate the matrix product T2 represent? Explain geometrically (or logically) why it should be this. c) T T . What linear transformation does this a b d) Let A be a 2 x 2 matrix. Calculate the three matrix products TA, AT and TAT. For each, give a simple short description, in words concerning the rows and columns of A (say), of the result of the calculation to produce a new matrix from A.