An important practice is to check the validity of any data set that you analyze. One goal is to detect typos in the data, and another would be to detect faulty measurements. Recall that outliers are observations with values outside the “normal” range of values of the rest of the observations.

Specify a large population that you might want to study and describe the type numeric measurement that you will collect (examples: a count of things, the height of people, a score on a survey, the weight of something). What would you do if you found a couple outliers in a sample of size 100? What would you do if you found two values that were twice as big as the next highest value?

You may use examples from your area of interest, such as monthly sales levels of a product, file transfer times to different computer on a network, characteristics of people (height, time to run the 100 meter dash, statistics grades, etc.), trading volume on a stock exchange, or other such things.

Let F be a field, and recall the notion of the characteristic of a ring; the characteristic of a field is either 0 or a prime integer.

Show that F has characteristic 0 if and only if it contains a copy of rationals and then F has characteristic p if and only if it contains a copy of the field Z/pZ.

Show that (in both cases) this determines the smallest subfield of F.

A machine in a factory has an error rate of 10 parts per 100. The machine normally runs 24 hours a day and produces 30 parts per hour. Yesterday the machine was shut down for 4

1. Develop an assembly language program for 89c51 to do following {Marks 10}

1. Store any 8 values of one byte each anywhere in Scratch Pad area of RAM. You are bound to use loop to store values.
2. Find Mean of these 8 values (use shift operator for division) and send it to port P2 (formula for to find mean is given below)
3. Find the lowest number among the numbers saved in part (1), take its 2’s complement and send it to P3

Formula for mean

μ=i=1nXin

Where X is values, and n is total number of values

1. Develop an assembly language program for 89c51 to do following

1. Store any 8 values of one byte each anywhere in Scratch Pad area of RAM. You are bound to use loop to store values.
2. Find Mean of these 8 values (use shift operator for division) and send it to port P2 (formula for to find mean is given below)
3. Find the lowest number among the numbers saved in part (1), take its 2’s complement and send it to P3

Formula for mean

μ=i=1nXin

Where X is values, and n is total number of values

1. Use the definition of convexity to prove that the function f(x) = x2 - 4x + 8 is convex. Is
this function strictly convex?
2. Use the definition of convexity to prove that the function f(x)= ax + b is both convex and
concave for any a•b ≠ 0.

7. Consider a two-step, serial, production process with one resource at each step. The processing time at step 1 is 10 minutes and the processing time at step 2 is 5 minutes. There is ample supply of raw materials for step 1 and ample demand.

a) What is the capacity of this process in units per hour?

b) Suppose there is variability in the processing time at step 2. Specifically, the coefficient of variation of processing time at step 2 is 2. However, there is no variability in the processing time at step 1. Now what is the total time needed for a part to go through step 2?

Hint: Total time includes processing and waiting. If step 1 is always working and processing time there is 5 minutes, how much time passes between units arriving at step 2?

c) Now what is the capacity of this process in units per hour?

Using the successor function prove distribution over addition. Do this in a detailed proof.

Let T(x1, x2) = (-x1 + 3x2, x1 - x2) be a transformation.

a) Show that T is invertible.

b)Find T inverse.

y''+4y=u_π(t)−u_3π(t) ; y(0)=0,y'(0)=0

a.Sketch the graph of the forcing function on an appropriate interval.

b.Find the solution of the given initial value problem.

c.Plot the graph of the solution.

d.Explain how the graphs of the forcing function and the solution are related.

Both parts.

a) identify Fourier series for full wave rectified sine function f(x) = | sin(x) |.

b) f(t) = cos(t) but period of 6, so t = [-3,3] (L = 6) Find the Fourier series of the resulting function.

Lydia saved $1,345,000 for retirement. The money is deposited in an account earning 3.2% compounded monthly. She is going to withdraw $5500 per month for living expenses. Create a table showing how much interest she earns each month and her monthly balance for the first 5 months of her retirement. Do this by hand with just the functions of a scientific calculator.

All necessary steps much show for these problems, please.

1. Use the Euclidean algorithm to find gcd(12345, 54321).
2. Write gcd(2420, 70) as a linear combination of 2420 and 70. The work to obtain the gcd is provided.
   2420 = 34(70) + 40

70 = 1(40) + 30

40 = 1(30) + 10

30 = 3(10) + 0

3. Determine if 1177 is prime or not. If it is not, then write 1177 as a product of primes
4. Find gcd(8370, 465) by unique factorization into products of primes.
5. Compute ?(39204) if 39204 = 2 ∙ 3 ∙ 11 .

A = { [1,2,3,2] , [-2,0,-2,-4], [0,4,4,0], [1,2,3,2]}

Can you answer the following questions regarding this matrix:

a) Find the null space of A

b) Find vectors v1,v2.... such that Null A = span {v1,v2...}

c) Is the null space of A subspace of R4