Let u and v be vectors in R^{3}. Consider the following statements.T or F

In: Advanced Math
Let a < c < b, and let f be defined on [a,b]. Show that f ∈ R[a,b] if and only if f ∈ R[a, c] and f ∈ R[c, b]. Moreover, Integral a,b f = integral a,c f + integral c,b f .
In: Advanced Math
A vector y =
[R(t) F(t)]^{T}
describes the populations of some rabbits R(t)
and foxes F(t). The populations obey the system
of differential equations given by y′ =
Ay where A =
The rabbit population begins at 6000. If we want the rabbit population to grow as a simple exponential of the form R(t) = R_{0}e^{3t} with no other terms, how many foxes are needed at time t = 0? (Note that the eigenvalues of A are λ = 3 and 4.) 
In: Advanced Math
The weighted voting systems for the voters A, B, C, ... are given in the form q: w1, w2, w3, w4, ..., wn . The weight of voter A is w1, the weight of voter B is w2, the weight of voter C is w3, and so on. Calculate, if possible, the Banzhaf power index for each voter. Round to the nearest hundredth. (If not possible, enter IMPOSSIBLE.) {82: 53, 36, 24, 18} BPI(A) = BPI(B) = BPI(C) = BPI(D) =
In: Advanced Math
Graph Theory: Let S be a set of three pairwisenonadjacent edges in a 3connected graph G. Show that there is a cycle in G containing all three edges of S unless S is an edgecut of G
In: Advanced Math
Let En be the subspace of V (n, 2) consisting of all vectros of even weight.
(a) What are the parameters [n, k, d] of En.
(b) Write down a generator matrix for En in standard form
In: Advanced Math
Solve the differential equation using undetermined coefficients:
y''0.25y=3sin0.5t where y(0)=0 and y'(0)=0
In: Advanced Math
Problem 4  A modied maninthemiddle attack on
DiffieHellman
Suppose Alice and Bob wish to generate a shared cryptographic key
using the DiffieHellman
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 trialdividing p1 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
maninthemiddle 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
In: Advanced Math
Use the Laplace transform to solve the given initial value problem.
y'' + 2y' + 10y = 6cos2t  4sin2t, y(0)=2, y'(0)= 2
In: Advanced Math
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
In: Advanced Math
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 nonzero solution
In: Advanced Math
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) =
In: Advanced Math
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 GramSchmidt process. Why is it the case that u_i.v_j =0 if i>j?
In: Advanced Math
(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)
x=3 
x=4 
x=5 
x=6 

y=0 
2.3 
2.2 
2.0 
1.7 
y=1 
2.4 
2.5 
2.7 
3.0 
y=2 
2.5 
2.7 
2.9 
3.2 
y=3 
2.6 
3.0 
3.0 
3.3 
In: Advanced Math
In: Advanced Math