1. In practice, an LFSR sequence would not be used as keystream for a stream cipher, due to the linear complexity. Answer the following questions about linear complexity.

1. What is the linear complexity of a maximal length LFSR output sequence?
2. Describe one method using LFSRs to produce a binary output sequence but with increased linear complexity. Draw a diagram to illustrate your construction.

Find the solution to the linear system of differential equations {x′ = 8x+5y and  y' = −10x−7y } satisfying the initial conditions x(0)=3 and y(0)=−4y.

The nine entries of a 3×3 grid are filled with −1, 0, or 1. Prove that among the eight resulting sums (three columns, three rows, or two diagonals) there will always be two that add to the same number.

Please explain how to determine the bounds for the triple integral in order to solve the following:

Find the volume of the solid B where B={(x,y,z)|x²+y²≤z²≤3x²+3y²} is bounded by the hemisphere x²+y²+z²=9 with z≥0 and by the plane z=−3.

Explain the process for modeling a linear spring in a static system using F = k*s

What is an example of a vector space besides Rⁿ?

What are the three properties of a vector subspace?

Question 16:

What is the general solution of the following homogeneous second-order differential equation?

Non-integers are expressed to one decimal place.

d^2y/dx^2 − 11.y = 9

Question 17:

What is the general solution of the following homogeneous second order differential equation?

d^2y/dx^2 + 3dy/dx − 4.y = cos(3.x)

Question 18:

What is the general solution of the following homogeneous second order differential equation?

d^2y/dx^2 + 13dy/dx + 40.y = sin(2.x)

Question 19:

What is the general solution of the following homogeneous second order differential equation?

d^2y/dx^2 + 1dy/dx − 20.y = 1.x − 3

Question 20:

What is the general solution of the following homogeneous second order differential equation?

d^2y/dx^2 − 10dy/dx + 29.y = 1.e(3.x)

Construct a generator matrix and a parity check matrix for a ternary Hamming code Ham(2, 3).

Assume a codeword x from for the ternary Hamming code Ham(2, 3)$ was sent and the word y was received. Use the partiy check matrix you constructed in part (a) to decode y in each part
using syndrome decoding:

(b) y = ( 1 , 1 , 1 , 0 ),

(c) y = ( 2 , 2 , 2 , 2 ),

(d) y = ( 1 , 2 , 1 , 2 ).

A 6 lb weight is placed upon the lower end of a coil spring suspended from a fixed beam. The weight comes to rest in its equilibrium position, thereby stretching the spring 4 inches. Then
beginning at t = 0 an external force given by F (t) = 27 sin(4t) − 3 cos(4t) is applied to the system. The resistance of the medium is three times the velocity (in ft/sec). Find the displacement of the weight as a function of time.

Suppose an object falls from a great height on a planet where the constant of the acceleration of gravity is g = 7.84. Assume that the resistance of the atmosphere is proportional to the square of the velocity of the object with constant of proportionality k = 0.25. Establish and solve an Initial Value Problem to express the velocity of the object as a function of time. Find the terminal velocity of the object. Graph this function. Then express the fall distance as a function of time. Graph this function. Hint: g − kv2 = (√g − (√k)*v) (√g + (√k)*v)

prove that a group of order 9 is abelian

Water is boiled in a bowl and cooled in a room. The air temperature in the room is increasing linearly according to the function Ta(t) = 30 + 0.01t (t in minutes, T in ∘C. Assume that Newton's Law of Cooling is satisfied: the rate of change of the temperature of the water is proportional to the difference between the temperature of the water and the temperature of the environment. We take the temperature of the water after 10 minutes and find that it is 81∘C. Explain and graph Ta. Establish and solve an Initial Value Problem to express the water temperature as a function of time, graph this function.

1. a. Consider d on R, the real line, to be d(x,y) = |x² – y²|. Show that d is NOT a metric on R.    b.Consider d on R, the real line, to be d(x,y) = |x³ – y³|. Show that d is a metric on R.

   2. Let d on R be d(x,y) = |x-y|. The “usual” distance. Show the interval (-2,7) is an open set.

Note: you must show that any point z in the interval has a ball centered at z, and that ball is completely contained within the interval (-2,7).

(a) Show that x= 0 is a regular singular point.

(b) Find the indicial equation and the indicial roots of it.

(c) Use the Frobenius method to and two series solutions of each equation

x^2y''+xy'+(x^2-(4/9))y=0