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
Respond to the following prompts in a minimum of 175 words:
In: Advanced Math
Alice and Bob are have several piles of chips. On each turn they can either remove 1 or 2 chips from one pile, or split a pile into two nonempty piles. Players take turns and a player that cannot make a move loses. Find the value of the Sprague–Grundy function for positions with one pile made of n chips. (Please do not forget to prove correctness of your asnwer.)
In: Advanced Math