(a). Prove the law of quadratic reciprocity for odd primes. Include all definitions and results used throughout the proof.

(b). Explain why it is enough to prove the reciprocity law for 2 and the quadratic reciprocity law for odd primes to answer the general question: Is m a square mod n for any positive integers m, n?

(c). Describe the role of permutations in the proof of quadratic reciprocity.

Solve each item supporting your answer with clear explanation.

1.Consider the following augmented matrix of a system of linear equations.

( 7 −3 4 6

−3 2 6 2

2 5 3 −5 )

a. Solve the system with the Jacobi method. First rearrange to make it diagonally dominant if possible. Use [0,0,0] as the starting vector. Find the condition number of the matrix of coefficients κ_∞(A), and compute how many iterations are required to get the solution accurate to five significant digits?

b. Repeat part a) using the Gauss-Seidel method. Are fewer iterations required?

c. Is convergence faster in parts a) and b) if the starting vector is [-0.26602, -0.26602,-0.26602]?

Find the approximations L_n, R_n, T_n, and M_n for n = 5, 10, and 20. Then compute the corresponding errors E_L, E_R, E_T, and E_M. (Round your answers to six decimal places. You may wish to use the sum command on a computer algebra system.)

If an undamped spring-mass system with a mass that weighs 24 lb and a spring constant 16 lbin is suddenly set in motion at t=0 by an external force of 432cos(8t) lb, determine the position of the mass at any time. Assume that g=32 fts2. Solve for u in feet.

Enclose arguments of functions in parentheses. For example, sin(2x).

u(t) = ?

3. Solve using probabilistic dynamic programming: I would like to sell my computer to the highest bidder. I have studied the market, and concluded that I am likely to receive three types of offers: an offer of $200 with probability 2/7, and offer of $300 with probability 4/7, and an offer of $400 with probability 1/7. I will advertise my computer for up to three consecutive days. At the end of each of the three days, I will decide whether or not to accept the best offer made that day. What is the optimum strategy for maximizing the expected sale price for my computer? What is this expected sale price?

Given the triangle with vertices at (1,4), (4,3), and (2,-1), sketch the image after the given transformation is applied. Based on your graph, identify the transformation as a translation, reflection, rotation, or other transformations, and state whether the transformation is an isometry.

1. (x,y) -> (x,-y)

2. (x,y) -> (y,-x)

3. (x,y) -> (2x,2y)

4. (x,y) -> (x+4,-y)

5. (x,y) -> (5-x,y)

6. (x,y) -> (x+4,y-3)

what is the  historic development of counting techniques in math with examples.

An Agriculture Department study of more than 94,000 samples from more than 20 crops showed that 73% of conventionally grown foods had residues from at least one pesticide. Moreover, conventionally grown foods were six times as likely to contain multiple pesticides as organic foods. Of the organic foods tested, 28%had pesticide residues, which includes 7% with multiple pesticide residues.

Compute two estimated probability distributions:

one for conventional produce and one for organic produce, showing the relative frequencies that a randomly selected product has no pesticide residues, has residues from a single pesticide, and has residues from multiple pesticides.

Assume a 2D physical system where the vectors |ψ1i and |ψ2i form an orthonormal basis of the space. Let’s define a new basis with |φ1i = √ 1 2 (|ψ1i + |ψ2i) and |φ2i = √ 1 2 (|ψ1i − |ψ2i). Given an operator Mˆ represented in the |ψii-basis by the matrix 1 1 , find the representation of Mˆ in the basis |φii-basis.

Consider the function ?(?) = ?^?2and x = 0, 0.25, 0.5, 1. Then use the suitable Newton interpolating polynomial to approximate f(0.75). Also, compute an error bound for your approximation.

The following data set provides information on the lottery sales, proceeds, and prizes by year in Iowa.

FY

Sales

Proceeds

Prizes

1992

$166,311,122

$45,678,558

$92,939,035

1993

$207,192,724

$56,092,638

$116,820,274

1994

$206,941,796

$56,654,308

$116,502,450

1995

$207,648,303

$58,159,175

$112,563,375

1996

$190,004,182

$51,337,907

$102,820,278

1997

$173,655,030

$43,282,909

$96,897,120

1998

$173,876,206

$42,947,928

$96,374,445

1999

$184,065,581

$45,782,809

$101,981,094

2000

$178,205,366

$44,769,519

$98,392,253

2001

$174,943,317

$44,250,798

$96,712,105

2002

$181,305,805

$48,165,186

$99,996,233

HelpCopy to ClipboardDownload CSV

Create a graph using the sales and year. What approximate range of sales would you expect for the year 2017?

Select the correct answer below:

Between 250 and 300 million dollars

Between 300 and 375 million dollars

Between 375 and 400 million dollars

Between 500 and 550 million dollars

(a) look at these the complex numbers z1 = − √ 3 + i and z2 = 3cis(π/4). write the following complex numbers in polar form, writing your answers in principal argument:

i. z1

ii. z1/|z1|. Additionally, convert only this answer into Cartesian form.

iii. z1z2

iv. z2/z1

v. (z1) ^-3

vi. All complex numbers w that satisfy w ³ = z1.

(b) On an Argand diagram, sketch the subset S of the complex plane defined by S = {z ∈ C : |z − i| ≤ 1, |z + 2 − 3i| ≤ |z − 2 + i|}.

Find the best quadratic approximation to f(x,y) = e^4ycos(-3x) about the origin (0,0) and estimate the value when this approximation is used to calculate f(0.1,0.1).Do not do the error approximation. Also, find the extreme values of the function f(x,y,z) = 8y²-4xz +16z - 35 on the surface 2x²+4y²+4z² = 64

The following data set provides information on the lottery sales, proceeds, and prizes by year in Iowa.

FY

Sales

Proceeds

Prizes

1992

$166,311,122

$45,678,558

$92,939,035

1993

$207,192,724

$56,092,638

$116,820,274

1994

$206,941,796

$56,654,308

$116,502,450

1995

$207,648,303

$58,159,175

$112,563,375

1996

$190,004,182

$51,337,907

$102,820,278

1997

$173,655,030

$43,282,909

$96,897,120

1998

$173,876,206

$42,947,928

$96,374,445

1999

$184,065,581

$45,782,809

$101,981,094

2000

$178,205,366

$44,769,519

$98,392,253

2001

$174,943,317

$44,250,798

$96,712,105

2002

$181,305,805

$48,165,186

$99,996,233

HelpCopy to ClipboardDownload CSV

Create a graph using the sales and year. What approximate range of sales would you expect for the year 2017?

Select the correct answer below:

Between 250 and 300 million dollars

Between 300 and 375 million dollars

Between 375 and 400 million dollars

Between 500 and 550 million dollars

Given triangle(ABC) with A-D-C, then BD < BA or BD < BC.
Prove that the set consisting of a circle and its interior is a convex set.