(Discrete Math) A routing transit number (RTN) is a bank code that appears in the bottom of checks. The most common form of an RTN has nine digits, where the last digit is a check digit. If d1d2 . . . d9 is a valid RTN, the congruence 3(d1 + d4 + d7) + 7(d2 + d5 + d8) + (d3 + d6 + d9) ≡ 0 (mod 10) must hold.

(a) Show that the check digit of the RTN can detect all single errors.

(b) Determine which transposition errors an RTN check digit can catch and which ones it cannot catch.

y''' - 7y'' + 15y' - 9y = 8e^(x) - 9x

A) Find the fundamental set of solutions of the reduced equation. (Hint: 3 is a root of the characteristic polynomial.)

B) Find a particular solution of the given equation.

C) Find the general solution of the given equation.

(a) Let Λ = {λ ∈ R : 0 < λ < 1}. For each λ ∈ Λ, let Aλ = {x ∈ R : −λ < x < 1/λ}. Find U λ∈Λ Aλ and ~U λ∈Λ Aλ respectively.

(b) Let Λ = \ {λ ∈ R : λ > 1}. For each λ ∈ Λ, let Aλ = {x ∈ R : −λ < x < 1/λ}. Find U λ∈Λ Aλ and ~U λ∈Λ Aλ respectively.

Find all possible orders of elements in the group Z4 × Z5 × Z10. For each possible order, give an example of an element of that order, and prove that no other orders are possible.

Let G be a group, and let a ∈ G be a fixed element. Define a function Φ : G → G by Φ(x) = ax−1a−1.

Prove that Φ is an isomorphism is and only if the group G is abelian.

Find all solutions of the equation x^{5}=2

What is the minimum and maximum number of solutions that we can expect to see in any given system of nonlinear equations? In your own words, what is the meaning of extraneous solutions? When solving a system of nonlinear equations, is it possible to always use the Addition Method? Explain your reasoning in complete sentences. PLEASE TYPE, DO NOT WRITE IT DOWN and Check your punctuation and proofreading.

Example Explanation
• The original number was 101 in binary which is equivalent to 5 in decimal.
• The answer after shifting to the right is 2 in decimal.
• That means by shifting one digit to the right the binary number gets divided by 2 (5/2 = 2 in binary)
Shifting to the Left
• Shifting a binary number to the left is equivalent to multiplying it.
• If you shift once to the left , you multiply the binary number by 2
• If you shift twice to the left, you multiply the number by 4
How to Shift to the Left
Example: The number is 1 0 1
INSERT a ZERO in the Least Significant Bit (LSB) 1 0 1 1 0 1 0 LSB
Example Explanation
• The original number was 101 in binary which is equivalent to 5 in decimal.
• The answer after shifting to the left is 10 in decimal.
• That means by shifting one digit to the left the binary number gets multiplied by 2 (5*2=10)

Solve: 1. (110)2 x (2)10 = ( )2

2. (1011)2 x (4)10 = ( )2

Shift the following numbers twice to the left:

3. (111)2= ( )2

4. (1010)2= ( )2

Solve: 5. (110)2 / (2)10 = ( )2

6. (1011)2 / (4)10 = ( )2

Shift the following numbers twice to the right:

7. (111)2= ( )2

8. (1010)2= ( )2

please give me the full explanation for the answer

Rule Based System

1. Given the rule following rules from the class notes on production rules to convert an Arabic number less than 40 to a roman numeral. USING LOGIC

Rule 1: if x is null then prompt the user and read x

Rule 2: if x is higher than 999 then print “too Big” and make x null

Rule 3: if x is between 10 and 39 then print “X” and reduce x by 10

Rule 4: if x is equal to 9 then print “IX” and reduce x to 0

Rule 5: if x is between 5 and 8 then print “V” and reduce x by 5

Rule 6: if x is equal to 4 then print “IV” and reduce x to 0

Rule 7: if x is between 1 and 3 then print “I” and reduce x by 1

Rule 8: if x is equal to 0 then print “end-of-line” and STOP

A. What additional rules are needed to convert an Arabic number less than 1000?

* notice rule 2 has already been changed

** hints: 50 is L; 100 is C; 500 is D; 1000 is M

B. Show the rules fired and the working memory to convert 864 into a roman numeral.

1. List below is the number of cars serviced per day at XYZ Mechanics in June 2014.

10	12	8	9	10
11	13	15	14	11
12	9	14	15	11
12	13	9	14	13
10	12	13	15	13	14

Required:

1. Array the data

2. Calculate the mean

3. Calculate the mode

4. Calculate the median

1. If I is an ideal of the ring R, show how to make the quotient ring R/I into a left R-module, and also show how to make R/I into a right R-module.

(Theorem 3.1): If xp is any solution of (∗) x′′ + p(t)x′ + q(t)x = f (t), and xh is a general solution of (∗∗) x′′ + p(t)x′ + q(t)x = 0), then the sum x = xh + xp is a general solution of (∗).

(a) First show that x = xp + xh satisfies (∗).

(b) Next show that if xp1 and xp2 are any two solution of (∗) then x = xp1 − xp2 satisfies (∗∗).

(c) Conclude that Thm 3.1 holds. (Explain your reasoning!)

2. The official currency of the Kingdom of Mathemagiclandistan is Shrute-bucks. Shrute-bucks bills come in denominations of 3 and 5.

a.) What is the smallest amount of Shrute-bucks, n that can be made using Shrute-buck bills so that every amount greater than or equal to n can also be made using Shrute-buck bills?

b.) Use either the principle of mathematical induction or strong induction to prove your answer from a.)

James DeWalp is a senior buyer of fruit products for Fresh Foods, a major U.S. multinational food processing company. This company, based in California, uses a wide variety of fruit concentrates, purees, flavors, and extracts in many of its popular food products. One of James's responsibilities is to negotiate annual purchase contracts for these ingredients. One such ingredient, guava puree, is grown and harvested on a seasonal basis in various countries around the world.

James is currently examining the costs associated with using one of his existing suppliers, a Philippine grower/processor. Fresh Foods has used this supplier's high-quality product for a number of years. Farmers grow the product in a remote part of the Philippines and transport it to the processing plant where it is pureed and packaged for transoceanic shipment. This particular variety of guava is highly prized for its flavor, which the aseptic method of processing used by the supplier helps maintain. Unfortunately, guerilla activity by rebels has recently caused some problems for growers in this part of the Philippines.

The supplier aseptically packages the guava puree (currently priced at $0.29/pound, FOB vessel) in foil bags, each containing 50 pounds of product, which workers then place into corrugated boxes. The boxes are stacked on wooden pallets, 40 to a pallet, for loading into overseas containers. Each container holds 20 pallets and arrives via ocean freighter. The ocean freight charge is $2,500 per container. Once the containers reach the U.S. port, a trucking company moves each container to a local warehouse for storage at a charge of $250 per container. U.S. Customs calculates import duties to be 15 percent of the shipment's original purchase price excluding freight charges. Fresh Foods requires one container load per month.

Fresh Foods warehouses each container in a public warehouse until needed for processing (average storage is one month). The monthly storage charge is $6.50 per pallet. In addition, the warehouse charges a one-time in/out fee of $6.25 per pallet to cover administrative costs. Fresh Foods inventory carrying charge is 24 percent, which it applies against the unit price of material in storage at the warehouse (but not in-transit from the Philippines). The reason why the company does not apply the carrying charges to intransit inventory is that Fresh Foods typically does not have to pay the invoice for the guava puree until it reaches the local U.S warehouse. Material planners assume the demand for guava puree to be relatively constant over the year.

When a container of guava puree is required at the plant, a local freight company moves the container from the warehouse, which costs $175 per container. The company estimates that incoming receiving and quality-control procedures cost $4 per pallet. Because of the nature of the product and the distance involved in purchasing and storing the guava puree, the company estimates it incurs a loss of 3 percent of the total puree purchased.

1. Calculate the total cost of guava puree for Fresh Foods per pound, per pallet, and per container. (12 pts / 4 pts per column)

• You do not have to calculate a column for cost of guava per bag/box.

• Make sure you include the cost of spoilage ($25,000 every six months), as well as loss in storage and production.

2. Identify the three (3) biggest cost drivers in your total cost analysis. Provide the percent (%) of the total cost that each of these categories represent. (3 pts)

3. Choose five (5) cost elements from your total cost model and perform a sensitivity analysis on each. (5 pts)

• Perform the sensitivity analysis on each cost element separately to see its impact on total cost.

• Indicate which of the five cost elements you chose had the biggest impact as you vary the costs

4. If you were in charge of managing guava puree for Fresh Foods, what would you do to reduce the total cost to the company? Identify which costs you’d attack first and why. Give some specific ideas on how you'd reduce costs. For instance, if storage is a big cost driver, give me some specific ideas you might consider to reduce storage costs. (12 pts)

5. Every TCO model is incomplete to some extent. So how might you expand your TCO model?

• Describe three additional cost elements that you could build into your current TCO model for guava that would capture relevant costs that have not been considered? (3 pts)

• Copy your original TCO model onto a separate tab in your excel workbook. Then expand your TCO model by adding in the additional costs you described above. Provide a new calculation for the total cost of guava puree per pound, per pallet, and per container with these additional costs. (5 pts)

• What are some strategies that you could put in place to mitigate these risks and/or offset the potential costs involved? (5 pts)

Product engineers calculate the budgeted factory yield of the guava puree when blending into company products is 98 percent; this means the company wastes 2 percent of the product by volume during production, and this is not recoverable.

Occasionally, undetected spoilage of guava puree will require removing the product from grocery shelves. Out-of-pocket costs typically total $25,000 for each incident; these costs are not recoverable from the supplier. The company's records indicate that such an incident occurs about once every six months.

In addition to the other costs noted here, corporate accounting policy requires that cost estimators include a 17 percent assessment on purchased product unit cost to cover general and administrative overhead costs at Fresh Foods.

Find the solution of the given initial value problem.

ty′+2y=sin(t), y(π/2)=7, t>0

Enclose arguments of functions, numerators, and denominators in parentheses. For example, sin(2x) or (a−b)/(1+n).