Explain if the set below is a vector space given standard operations.

The set of all even functions defined on R with addition and scalar multiplication defined as follows:

1.) (f+g)(x) = f(x) + g(x) (addition)
2.) (cf)(x) = cf(x)

Write the greatest common divisor over the given filed as a linear combination of the given polynomials. That is, given f(x) and g(x), find a(x) and b(x) so that d(x) = a(x)f(x) + b(x)g(x), where d(x) is the greatest common divisor of f(x) and g(x).

(a) x^10 − x^7 − x^5 + x^3 + x^2 − 1 and x^8 − x^5 − x^3 + 1 over Q.

(b) x^5 + x^4 + 2x^2 − x − 1 and x^3 + x^2 − x over Q.

(c) x^3 − 2x^2 + 3x + 1 and x^3 + 2x + 1 over Z5.

(d) x^5 + x^4 + 2x^2 + 4x + 4 and x^3 + x^2 + 4x over Z5.

I know the GCD's are

a. x^3-1 b. 1 c. 1 d.1

Please help me write them as a linear combination.

Explain what truncating a Fourier series expansion and Fourier Integral does

The Lifang Wu Corporation manufactures two models of industrial robots, the Alpha 1 and the Beta 2. The firm employs 5 technicians, working 160 hours each per month, on its assembly line. Management insists that full employment (that is, all 160 hours of time) be maintained for each worker during next month’s operations. It requires 20 labor-hours to assemble each Alpha 1 robot and 25 labor-hours to assemble each Beta 2 model. Wu wants to see at least 10 Alpha 1s and at least 15 Beta 2s produced during the production period. Alpha 1s generate a $1,200 profit per unit, and Beta 2s yield $1,800 each.

a. Determine the most profitable number of each model of robot to produce during the coming month.

b. What is the total profit?

c. What if there was a reduction in 50 hours, what impact, in terms of profit, would this decision have?

Use the method of Undetermined Coefficients to find a general solution of this system X=(x,y)^T

Show the details of your work:

x' = 6 y + 9 t
y' = -6 x + 5

Note answer is:  x=A cos 4t + B sin 4t +75/36; y=B cos 6t - A sin 6t -15/6 t

A visible-factor number is a natural number that is divisible by each of its nonzero digits, for example, 424 or 505. How many visible-factor numbers are less than 100?

(a) Is 2xydx - (1+y)dy = 0 linear or nonlinear?

(b) Solve 2xydx - (1+y)dy = 0 using the separation of variables method. Can we find an explicit form of the solution?

(c) Can we solve 2xydx - (1+y)dy = 0 using the integrating factor method? How about using the exact differential equation method?

(d) Solve 2xydx - (1+y)dy = 0, y(0) = 1

(e) Solve 2xydx - (1+y)dy = 0, y(0) = -2

(f) Solve 2xydx - (1+y)dy = 0, y(0) = 0

Thapar theatre company needs to determine the lowest cost production budget for an upcoming theatre show. Specifically, they will have to determine the lowest which set pieces to construct and which pieces must be rented from another company at a pre-determined fee. The time available for constructing the set is two weeks after which rehearsals commence. To construct the set, the theatre has two part-time carpenters who work upto 12 hours a week and each at Rs 100 per hour. Additionally, the scene artist can work 15 hours per week at Rs 150 per hour

The set design requires 20 walls, 2 hanging drops with pained scenery and 3 large wooden tables serving as props. The number of hours required for each piece for carpentry and painting is given below.

Flats, hanging drops and props can also be rented at a cost of Rs 750, Rs 5000 and Rs 3500 each.

How many of each unit should be built by the theatre company and how many units should be rented to minimize costs?

Show that any polynomial over C (the complex numbers) is the characteristic polynomial of some matrix with complex entries. Please use detail and note any theorems utilized.

Use a table to find the inverse Laplace transform h(t) of H(s). H(s) = 40s /((s2 + 4)(s + 1)) for s > 0 h(t) = for t > 0

Question 1

Notex Manufacturing makes various batteries used in mobile devices. The company has a major customer so batteries are shipped in bulk to this customer. The company also distributes these batteries to retail stores as replacement parts. The batteries are packaged individually to retail stores. In all, the company makes about 15 different batteries. Currently, the company does not use any forecasting to predict the demand for the batteries. Instead, it has employed rule of thumb to decide about the volume of the production. This has caused some issues for the company including stock-out for some types of batteries or overstocking for some others. The other problem is an increase in the price of raw materials though the company believes it is a temporary condition. Due to complaints from suppliers and also customers the company has decided to introduce a systematic approach toward forecasting.

Therefore, the company has decided to forecast two most important products. The following table (see next page) shows the data on product demand for the two products from order records for the previous 19 months.

Question

Which forecasting method/s do you suggest for the two products? Briefly explain why? Forecast for the next month for each product. (Use MAD for measuring error).

* Unusual order due to flooding of customer's warehouse.

Question 2

UFE Clubs produces ball bearings. The diameter of the ball bearings is very important for the quality purposes. The following tables shows the diameters of 6 randomly selected samples. Each sample contains four observations.

a. Is the production process of the ball bearings are under control both in terms of central tendency and dispersion. Show all calculations.

b. A customer for ball bearings places an order with diameter of 0.600 +/- 0.050. What is the Process Capability Index? The Process Capability Ratio?

c. If the firm is seeking four-sigma performance, is the process capable of producing the ball bearings?

1. List the four qualitative characteristics of financial reports (under IFRS) and discuss two of them in detail. (Chapter 3, Section 5.2)

2. What is the “Going Concern” assumption and why is it in an important underlying assumption in financial statements? (Chapter 3, Section 5.4.1).

3. What is the “percentage-of- completion” method for revenue recognition and when can it be used? Give an example of how this method is used. (Chapter 4, Section 3.2.1)

MATLAB

In a traffic study of a street, the following information was gathered.

• Cars passed by at an average rate of 300 cars per hour.

• The speed of the cars was normally distributed, with an average speed of 58 km/h and a variance of 2 km^2/h^2 .

Based off this information, you are asked to solve the likelihoods of certain events happening. For each question clearly indicate the random variable and the distribution it follows, solve by hand and check your answer using MATLAB.

1. What is the probability that there is less than 10 seconds time difference between one car and the next?

2. What is the probability that more than 3 cars pass by in a minute?

3. The speed limit of the road is 60 km/h. What is the probability that a random car is speeding?

4. What is the probability that there are no speeding cars within a 10 minute period?

The Austin, Texas plant of Computer Products produces disk units for personal and small business computers. Gerald Knox, the plant’s production planning director, is looking over next year’s sales forecasts for these products and will be developing an aggregate capacity plan for the plant. The quarterly sales forecasts for the disk units are as follows:

Ample machine capacity exists to produce the forecast. Each disk unit takes an average of 20 labor-hours. In addition, you have collected the following information:

1. Inventory holding cost is $100 per disk unit per quarter. The holding cost is based on the inventory at the end of the quarter.
2. The plant works the same number of days in each quarter, 12 five-day weeks, 6 hours per day.
3. Beginning inventory is 90 (ninety) disk units and these will be used to meet the initial demand in the first quarter and there is no holding cost associated with these units.
4. In a backlog situation, the customer will wait for his order to be filled but will expect a price reduction each quarter he waits. The backlog costs are $300 per disk for the first quarter the customer waits, $700 for the second quarter the customer waits, and $900 for the third quarter the customer waits. In any quarter, if there is a backlog, this backlog will be filled before the demand for that period is fille
5. The cost of hiring a worker is $800 while the cost of laying off a worker is $950.
6. The straight time labor rate is $20 per hour for the first quarter and increases to $22 per hour beginning in the third quarter.
7. Overtime work is paid at time and a half (150%) of the straight time work.
8. Outsourcing (contract work) is paid at the rate of $480 per disk unit for the labor and you provide the material.
1. Demand is projected to increase this year. Demand during the fourth quarter of the prior year was 2,340 units. The demand for the first quarter of the next year (year following the year you are analyzing) is projected to be at the 2,700 unit level.

a) You want to maintain a work force capable of producing 2,520 in a quarter outsourcing any disk units over this quantity. Excess units produced in a quarter would be carried over to meet demand in a subsequent quarter. Any additional demand is met through outsourcing. All workers will be fully utilized each quarter. In other words, there is no under utilization. What is the total cost of this option, excluding the material cost? Be sure to include any hiring and layoff costs.

b) The company will maintain a work force capable of producing 2,430 units in a quarter. It will allow backlogs to occur until the fourth quarter when it will outsource all demand that cannot be met with its own workforce. All workers will be fully utilized each quarter. In other words, there is no under utilization. What is the total cost of this option, excluding the material cost? Be sure to include any hiring and layoff costs.

There are n types of coupons. We collect coupons one-by-one. Each coupon collected is of type i with probability pi P , and independent of other coupons. Assume that n i=1 pi = 1. Suppose in total k coupons are collected. Define Ai to be the event that there is at least one type i coupon among those collected for i = 1, 2, · · · , n.

(a) Compute P(Ai)

(b) For any i 6= j, find P(Ai ∪ Aj )

(c) Compute P(Ai |Aj ), hint: use formula about P(A ∪ B) = . . .