Solve the differential equation using undetermined coefficients:

y''-0.25y=3sin0.5t where y(0)=0 and y'(0)=0

Problem 4 | A modied man-in-the-middle attack on Diffie-Hellman
Suppose Alice and Bob wish to generate a shared cryptographic key using the Diffie-Hellman
protocol. As usual, they agree on a large prime p and a primitive root g of p. Suppose also that
p = mq + 1 where q is prime and m is very small (so p - 1 = mq has a large prime factor, as
is generally required). Since g and p are public, it is easy for anyone to deduce m and q; for
example by successively trial-dividing p-1 by m = 2,4, 6, ...and running a primality test such
as the Fermat test on the quotient q = (p - 1)/m until primality of q is established.
Suppose an active attacker Mallory intercepts g^a (mod p) from Alice and g^b (mod p) from Bob.
She sends (g^a)^q (mod p) to Bob and (g^b)^q (mod p) to Alice.

(a) Show that Alice and Bob compute the same shared key K under this attack.

(b) Show that there are m possible values for K; and that Mallory can compute them
all and hence easily guess the correct key K among them.

(c) What is the advantage of this variation of the man-in-the-middle attack over
the version we discussed in class? Recall that for the attack from class, Mallory simply
suppresses the messages g^a (mod p) and g^b (mod p) between Alice and Bob and replaces
them with her own number g^e (mod p), which results in the shared key g^ae (mod p) between
Mallory and Alice and the shared key g^be (mod p) between Mallory and Bob.

PLEASE SHOW CLEAR & DETAILED STEPS OF THE SOLUTIONS . THE PROOF SHOULD BE FOR GENERAL CASE, NOT AN EXAMPLE OF AN INDIVIDUAL CASE

Use the Laplace transform to solve the given initial value problem.

y'' + 2y' + 10y = 6cos2t - 4sin2t, y(0)=2, y'(0)= -2

how much do wild mountain lions weigh? adult wild mountain lions captured and released for the first time in the san Andres mountains gave the following weights 68 108 125 125 60 64

For which real values of a do there exist solutions of the differential equation

y'' + 2y' + ay = 0

which satisfy the conditions y(0) = y(π) = 0 but which are not identically zero? For each such a give an appropriate non-zero solution

A. Find a particular solution to the nonhomogeneous differential equation y′′ + 4y′ + 5y = −15x + e^-x

y =

B. Find a particular solution to

y′′ + 4y = 16sin(2t).

y_p =

C. Find y as a function of x if

y′′′ − 10y′′ + 16y′ = 21e^x,

y(0) = 15,  y′(0) = 28, y′′(0) = 17.
y(x) =

Question1: Consider a QR faction M=QR, show that R= Transpose(Q)M

You need to show that (1)M = QR where R := Transpose(Q)M and (2) that R is upper triangular.

To show (1) use the fact that QTranspose(Q) is the matrix for orthogonal projection onto the image of M. What happens to a column of M (which is a vector in the image of M) when you project it onto the image of M?

To show (2), think about the entries of R := Transpose(Q)M as dot products between the columns v_1,...,v_n of M and the rows u_1,...,u_n of Q^T. Entries of Transpose(Q)M vanish when these vectors are orthogonal. The vectors u_1,..., u_n are the othonomal basis for the image of M obtained from v_1,...,v_n via the Gram-Schmidt process. Why is it the case that u_i.v_j =0 if i>j?

(c) [2] For which of the following functions are the level curves linear?  

(I) f(x, y) = tan(x + y)

(II) g(x, y) = e^y/x (e to the power of y over x)

(III) h(x, y) = ln(xy)

(A) none (B) I only (E) I and II (F) I and III

(C) II only (G) II and III

(D) III only (H) all three

A partial table of values for a function f(x,y) is given below. Which of the following are positive?

(I) fy(4, 1)

(II) fx(4, 1) (III) fxx(4, 1)

Y''+y'-20y=xe^3x+e^4x

Find the general solution of this differential equation

why the cylindrical shape of a can is the optimal shape

Needs to be 2 pages.

Solve the following linear integer programming model using the Cutting Plane method. Show all relevant work in your solution report.

Maximize Z = x1 + x2

Subject to

3x1 + 2x2 < 5

x2 < 2

x1, x2 > 0 and integer.

Show that every permutational product of a finite amalgam am(A,B: H) is finite.Hence show that every finite amalgam of two groups is embeddable in a finite group.

A country's census lists the population of the country as 254 million in 1990, 286 million in 2000, and 314 million in 2010. Fit a second-degree polynomial passing through these three points. (Let the year 2000 be x = 0 and let p(x) represent the population in millions.)

p(x) = ________million

Use this polynomial to predict the populations in 2020 and in 2030.

2020_________million

2030 ________million

Show the following identities for a, b, c ∈ N.

(a) gcd(ca, cb) = c gcd(a, b) Hint: To show that two integers x, y ∈ Z are equal you can show that both x | y and y | x which implies x = y or x = −y. Thus, if both x and y have the same sign, they must be equal.

(b) lcm(ca, cb) = c lcm(a, b)

(c) ab = lcm(a, b) gcd(a, b) Hint: Consider first the case that gcd(a, b) = 1 and show that ab = lcm(a, b) in this case. For the general case combine this with (b).

(d) lcm(gcd(a, c), gcd(b, c)) = gcd(lcm(a, b), c) Hint: First treat the special case that gcd(a, b, c) = 1. In this case begin by showing that lcm(gcd(a, c), gcd(b, c)) = gcd(a, c) gcd(b, c). The asserted equality gcd(a, c) gcd(b, c) = gcd(lcm(a, b), c) is then shown by proving that gcd(a, c) gcd(b, c)| gcd(lcm(a, b), c) and gcd(lcm(a, b), c)| gcd(a, c) gcd(b, c). Proceed to show that gcd(a, c)| lcm(a, b) and gcd(a, c)| c, and deduce from this that gcd(a, c)| gcd(lcm(a, b), c); proceed analogously for gcd(b, c). Then argue that gcd(a, c) gcd(b, c)| gcd(lcm(a, b), c) under the present assumption. Conversely, in order to show that gcd(lcm(a, b), c)| gcd(a, c) gcd(b, c), write according to (a) gcd(a, c) gcd(b, c) = gcd(gcd(a, c)b, gcd(a, c)c) = gcd(gcd(ab, bc), gcd(ac, c2 )), and show that gcd(lcm(a, b), c) divides all of ab, bc, ac, and c 2 . Explain from here why gcd(lcm(a, b), c) must divide gcd(a, c) gcd(b, c) then as well. For the general case explain how (a) and (b) can be used to reduce the general assertion to the previously treated special case.

***The only help I really need is with c and d. I just added a and b for context.