HW 20. Due November 1. In this assignment, we will see an example of an integral domain that has elements that can be factored as a product of irreducible elements, but that factorization is not unique. Let R denote the set of all complex numbers a + b √ 5i, where a, b ∈ Z. Let N be the norm on R defined by N(a + b √ 5i) = a 2 + 5b 2 . As before N(z1z2) = N(z1)N(z2), for all z1, z2 ∈ R. (In fact, this holds for all complex numbers if, for z = c + di ∈ C we define N(z) = c 2 + d 2 .) (i) Show that R is an integral domain. (ii) Show that the only units in R are ±1. (iii) Use the norm to prove that 2, 3, 1 + √ 5i, 1 − √ 5i are irreducible elements in R. (iv) Conclude that 6 = 2 · 3 = (1 + √ 5i) · (1 − √ 5i) are two distinct factorizations of 4 into a product of irreducible elements.

1. Three materials X, Y, and Z are required to produce two products A and B. The profit function of each product is nonlinear. The total profit function for Product A is 80A - A2 and the total profit function for Product B is 72B - 0.8B2. Thus, the total profit for producing A and B together is 80A - A2

+ 72B - 0.8B2. (These two profit functions are independent.)

a. Use the Nonlinear Solver (GRG Nonlinear) to solve the nonlinear profit function for this problem. (Note: this problem does not have a constraint.) What is the optimal production quantity for each product? What is the total profit? Why is the solution optimal? Is the total profit function a function with decreasing marginal return? Why?

b. Following information about resource requirement and availability is given. Formulate the problem with constraints as a nonlinear programming model on the spreadsheet to maximize the total profit.

C. Solve the problem using the Nonlinear Solver. What is the optimal solution? What is the maximum total profit?

D. Explain why solutions in (a) and (c) are different.

E. Formulate the problem in (b) as an algebraic model.

6. Let R be a relation on Z x Z such that for all ordered pairs (a, b),(c, d) ∈ Z x Z, (a, b) R (c, d) ⇔ a ≤ c and b|d . Prove that R is a partial order relation.

Use the improved Euler's method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05. y' = y − y^2, y(0) = 0.8; y(0.5) y(0.5) ≈________ (h = 0.1)

y(0.5) ≈ _________(h = 0.05)

- Solve the following recurrence relation :

T(n) = T(αn) + T((1 − α)n) + n

Solve the IVP:

u'' + 10u' + 98u = 2sin(t/2)

u(0) = 0

u'(0) = 0.03

and identify the transient and steady state portions of the solution.

Plot the graph of the steady state solution.

Prove or disprove:

If a, b, c are any three distinct positive integers such that 1/a + 1/b + 1/c = 1, then a + b + c is a prime.

Use cardinality to show that between any two rational numbers there is an irrational number. Hint: Given rational numbers a < b, first show that [a, b] is uncountable. Now use a proof by contradiction

Solve the given initial-value problem by finding, as in Example 4 of Section 2.4, an appropriate integrating factor.

(x² + y² ? 3) dx = (y + xy) dy,    y(0) = 1

The following table contains approximate figures for gross domestic product (GDP) and the national debt in the United States for June 2001 and June 2010. The national debt represents the total amount of money owed by the federal government to holders of U.S. securities. All numbers are in trillions of dollars.

Source: “U.S. Treasury, Bureau of Economic Analysis.”

Net public debt is the portion of the national debt that is held outside the federal government and the Federal Reserve System. In June 2001, the net public debt as a percentage of total national debt was __________.

In June 2001, the percentage of the U.S. national debt held by foreigners (external national debt) was ________ .

The fraction of the national debt held by foreigners will eventually need to be repaid to foreigners, thereby reducing the collective purchasing power of Americans. Between 2001 and 2010, the fraction of the national debt held by foreigners __________ .

The absolute level of the debt does not necessarily provide a clear indication of a nation's debt burden. Thus, economists often look at relative measures of the national debt. One possible relative measure of the national debt is the federal debt held by the public (outside the federal government and the Federal Reserve) as a percentage of GDP. In 2001, publicly held debt was _______ of GDP. Between 2001 and 2010, publicly held debt as a percentage of GDP _________ .

Solve y’’ – 11y’ + 24y = e^x +3x using reduction of order

A binary string is a “word” in which each “letter” can only be 0 or 1

Prove that there are 2^n different binary strings of length n.

Note:

• Your goal is to produce a properly constructed proof by induction, but this does not mean you have to follow
• Mathematical induction, step-by-step..
• Write the statement with n replaced by k
• Write the statement with n replaced by k+1.
• Identify the connection between the kth statement and the (k+1)th statement.
• Complete the induction step by assuming that the n=k version of the statement is true, and using this assumption to prove that the n=k+1 version of the statement is true.
• Complete the induction proof by proving the base case.
• point-by-point.
• ANOTHER IMPORTANT NOTE:
• Make sure you are addressing the given statement: "there are 2ⁿ different binary strings of length n” !!! So your solution should primarily be discussing binary strings. Yes, the fact that 2^k+1 = 2^k∙2 will be useful in your solution, but it should not be the sole content of your solution, as by itself that equality is a fact about exponents, not a fact about binary strings.

1. Average personal income in Hawaii increased approximately linearly from $41.7 thousand in 2010 to $50.6 thousand in 2016. Average personal income in Colorado increased approximately linearly from $39.9 thousand in 2010 to $52.1 thousand in 2016 (Source: U.S. Bureau of Economic Analysis).

   1. Let H(t) be the average personal income in Hawaii and C(t) be the average personal income in Colorado, both in thousands of dollars, in the year that is t years since 2010. Find equations of H and C.

   2. Use substitution or elimination to estimate when average personal income in Hawaii was equal to average personal income in Colorado. What is that average personal income?

   3. Use a graphing calculator table or graph to verify the result you found in part (b).

   4. Find the average of H(11) and C(11). Assuming the population of Colorado will continue to be larger than the population of Hawaii, will your result likely be an underestimate or an overestimate of the 2021 average personal income of residents of Hawaii and Colorado put together? Explain.