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
Y''+y'20y=xe^3x+e^4x
Find the general solution of this differential equation
In: Advanced Math
why the cylindrical shape of a can is the optimal shape
Needs to be 2 pages.
In: Advanced Math
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.
In: Advanced Math
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.
In: Advanced Math
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 seconddegree 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
In: Advanced Math
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.
In: Advanced Math
Here are four questions:
1. Prove a standard Brownian motion is Gaussian process .
2. Prove a Brownian bridge is Gaussian process.
3. Prove OrnsteinUhlenbeck process is Gaussian
4. Prove the position process is Gaussian.
Please provide as many detail as you can, thanks.
In: Advanced Math
For each of the following questions carefully define (1) the sample space and (2) the event under
consideration. Then (3) determine the probability. For full credit, you will have to display these three parts. We are given six cards: Two of the cards are black and they are numbered 1, 2; and the other four cards are red and they are numbered 1, 2, 3, 4. We pick two cards at the same time.
What is the probability that both cards are black?
What is the probability that both cards are black, if we know that at least one of them is
black?
What is the probability that both cards are black, if we know that one of them is a black card numbered 1.
In: Advanced Math
(Topology) Prove that the interior of a subset A of Xτ is the union of all τ open sets contained in A.
In: Advanced Math
Explanation:
Assume the reader understands derivatives, and knows the definition
of instantaneous velocity (dx/dt), and knows how to calculate
integrals but is struggling to understand them. Use students’ prior
knowledge to provide an explanation that includes the concept and
physical meaning of the integral of velocity with respect to
time.
Reminder: The user is comfortable with the calculations, but is
struggling with the concept. To fully address the prompt, emphasize
the written explanation in English over the calculation
In: Advanced Math
Prove that if a sequence is bounded, then it must have a convergent subsequence.
In: Advanced Math
Given a set S = {1, 1, i, i} where i^{2} = 1 and with multiplication * on this set,
* 
1 
i 
1 
i 
1 
1 
1 
i 

i 
i 
1 

1 

i 
(b) Determine whether, or not, the operation * is a binary operation on S.
(c) Is * commutative on S?
(d) Investigate the following properties of binary operations for this operation on S:
i. Closed
ii. Identity
iii. Inverse
iv. Associativity
In: Advanced Math