1. Find the additive and multiplicative inverses for the residual sets Z13, Z14, and Z11. Identify each as being a group, ring or field

2. Write an algorithm for performing Euclid greatest common denominator for A,B in Matlab and demonstrate the results for A=9777 and B=106665. Compare this the the matlab function gcd(A,B) and the by hand.

3. Program Euclid’s Extended algorithm in Matlab and demonstrate on the residual sets in problem 1.

4. Demonstrate Fermats and Eulers Theorems with the value of a set to the Residual set Z13 and Z14 and with P equal to 13 and 14 respectively. Explain results

5. Perform the Chinese Remainder theorem as in the example in your text with m1=29, m2=47

Find the Laplace transform of the following functions:
a. p(t) = 6[u(t − 2) − u(t − 4)]
b. g(t) = (4 + 3e−2t)u(t)
c. h(t) = 2r(t) − 2r(t − 2)

OK I have two data sets with 30 million rows each each data set is five columns with four attributes and an amount. I want to confirm that the two data sets are exactly the same no two rows of data in the 30 million rolls are duplicates

For my proof I will confirm each data set has the same number of rows. And I will also do the following:

I will create four smaller data sets from each of the two large data sets. Each of the smaller data sets will remove one of the four attributes

If each of those for data sets matches exactly and the total count matches is that proof that the two large data sets are exactly the same

The reason I am doing this test as I am not able to compare 30 million rows to 30 million rows because the set is too large for the tools I have available

QUESTION: If the four smaller data sets match exactly and the total row count matches exactly. Have I proved these the two 30 million row data sets are exactly the same l?

If I show (A and (B → C)) → D and (A and (C → B)) → D, can I conclude A → D?

1. You are given that the transition matrix PC,B from a basis B = {b1,b2,b3} to a basis C = {c1,c2,c3} is

1 −1 0 2

1 0 1 −1 001

1. (a) For the vector u = b1 + b2 + 2b3, compute [u]C, and from this write down u as a linear combination of the vectors in C.

2. (b) Calculate PB,C.

3. (c) Suppose

   c1 = (1,0,0), c2 = (1,2,0), c3 = (1,2,3).
   Compute PS,B where S is the standard basis, and from this read off the explicit form

   of the vectors b1, b2, b3.

Suppose r(t)=cos(πt)i+sin(πt)j+2tk represents the position of a particle on a helix, where z is the height of the particle. (a) What is t when the particle has height 8? (b) What is the velocity of the particle when its height is 8? (c) When the particle has height 8, it leaves the helix and moves along the tangent line at the constant velocity found in part (b). Find a vector parametric equation for the position of the particle (in terms of the original parameter t) as it moves along this tangent line.

Given stream depth (x-values) and flow (y-values) in the table below:

(a) Find the equation of the form: y = βo + β₁x + β₂x²that fits the data.

(b) Compare your answer in (a) to a loge transformation for both x and y.

(c) Compare your answers in (a) and (b) to a square root of y transformation.

(d) Specify the equation/ data transformation you would choose. Show all equations developed and R2 values. Provide the digital file for the software application used for this problem.

4. Two fair dices are rolled. Find the probability that sum of the numbers add to 7.

5. A survey of freshman at Fishtopia University indicated that 70% enjoy playing Oceanic Adventures while only 35% enjoy playing Fishmania. The survey also indicated that 15% enjoy playing both games. a) Complete the Venn Diagram by filling in all four regions with the appropriate percentage values. Make sure that all four regions add up to 100%. b) What is the probability that a random freshman likes neither of the two games? Express your answer as an exact decimal. c) What is the probability that a random freshman likes exactly one of the two games? Express your answer as an exact decimal.

6. Select 3 marbles at random from a jar with 6-blue, 4-green, and 3-red.[ hint: use combination only] (a) What is probability of selecting 2-blue and 1-red marble?

What is the difference between Standard Young Tableaux and Semi Standard Young Tableaux?

Find the volume V of a regular tetrahedron whose face is an equilateral triangle of side 8.

Find the area of the horizontal cross-section A at the level z=4.

Which of the following sets of polynomials form a basis for P2 (the space of polynomials of degree at most 2)? Explain. (a) {2 + 2x − x 2 , 1 + x, 3x} (b) {1 − x + 2x 2 , 2 + 4x} 1 (c) {3 + x + x 2 , −1 + x, 5 − x + x 2}

1. What is the difference between function and relation? Explain by giving example.

2. Find the domain and range of these functions.

1. the function that assigns to each pair of positive integers the first integer of the pair

            b) the function that assigns to each positive integer its largest decimal digit

c) the function that assigns to a bit string the number of ones minus the number of zeros in the string

d) the function that assigns to each positive integer the largest integer not exceeding the square root of the integer

e) the function that assigns to a bit string the longest string of ones in the string

The motion of a​ mass-spring system with damping is governed by y''(t) + by'(t) + 64 y(t) = 0; y(0) = 3, and y'(0) = 0.

Find the equation of motion for b = 0,14,16, and 20.

.

Solve the given initial-value problem. y'' + 4y' + 4y = (5 + x)e^(−2x) y(0) = 3, y'(0) = 6

Arrived at answer y(x)=3e^{-2x}+12xe^{-2x}+(15/2}x^2e^{-2x}+(5/6)x^3e^{-2x) by using variation of parameters but it was incorrect.