Hello all! I'm studying for my final, and I'm trying to figure out my teacher's method for the following problem. Could you help me out?
A plane wave is propagating in free space with a frequency of 10 GHz. The amplitude of the electric field in the x-direction is Ex = 2 V m-1.
(ii) Find...
Homework Statement
A plane wave of 1 MHz frequency in free space is incident normally on a copper
surface [\sigma=5.8*10^7, \mur=1, and \epsilonr=1]. If the magnetic field strength at the surface is 1 mA m-1, find the depth at
which it is 2.5% of this value. Calculate the...
Hello All!
I am currently in an Applied Analysis class, and I'm trying to do a little research outside of the classroom to try and understand what my teacher is trying to say.
So, I'm supposed to understand how to solve Fredholm Integral Equations (inhomogeneous and of the second kind)...
I swear that I used to know this.
If you have an independent sample of size n, from the uniform distribution (interval [0,\theta]), how do you find the Expected Value of the largest observation(X(n))?
Okay. I think I understand that now.
(i.e. x^3-9 is irreducible over integers mod 31 cause no x makes x^3=9mod31 true. But It is reducible over integers mod 11... (x-4)(x^2+4x+5).)
But how do I prove part c? (Sorry I'm asking so many questions. I appreciate the help!)
I don't know. I don't know how to prove that they're irreducible, really. I'm just following what somebody said about it being sufficient to show that it has no zeros...
Homework Statement
a. Prove that x^2+1 is irreducible over the field F of integers mod 11.
b. Prove that x^2+x+4 is irreducible over the field F of integers mod 11.
c. Prove that F[x]/(x^2+1) and F[x]/(x^2+x+4) are isomorphic.
Homework Equations
A polynomial p(x) in F[x] is said to...
Homework Statement
http://img206.imageshack.us/img206/7900/physicsml8.jpg [Broken]
Note: Acceleration=1e18 m/s^2
The Attempt at a Solution
I've tried to do the ones marked with a red x, but I'm really not sure how... any hints would be greatly appreciated!
Good Morning, Morphism!
I'm looking at this last post, and I'm still a bit confused (forgive me). How did you decide that for Z_3 x Z_3 we were using lcm{2,3}?
Thank you so much for your time and patience. :-D
Thanks for the luck! I'm probably going to need it.
Is there a simple way to test whether or not something has an element of a certain order (i.e. Z_3 x Z_3 not having an element of order 9)? I want to make sure I understand this fully before morning.
Thanks again!
Okay. Thanks so much! I was trying to make it more complicated than it needed to be, I think. :-D
This is a side question that kind of pertains to this. Why is Z_3 x Z_3 not isomorphic to Z_9?
Thanks again for your help, Morphism! You're a lifesaver! (I have a test in less than 10...
The fundamental theorem of finite abelian groups states that every finite abelian group can be expressed as the direct sum of cyclic subgroups of prime-power order.
Z_2 x Z_2 x Z_2 x Z_3 x Z_3
Z_2 x Z_2 x Z_2 x Z_9
Z_2 x Z_4 x Z_3 x Z_3
Z_2 x Z_4 x Z_9
Z_8 x Z_3 x Z_3
Z_8 x Z_9
Yes?
Homework Statement
Determine the number of non-isomorphic abelian groups of order
72, and list one group from each isomorphism class.
The Attempt at a Solution
72 = 2^3*3^2
3= 1+1+1= 2+1= 3 (3)
2= 1+1= 2 (2)
3*2 = 6
And then I get lost on the...
Homework Statement
If a,b,c,d \in R, evaluate (a+b)(c+d). (R is a ring.
Homework Equations
The Attempt at a Solution
I think that it's simple foiling, but I'm not sure.
ac+ad+bc+bd
That either k is 0 or x is 1 or 0. Since k cannot equal 0 (because it is between 1 and 15 inclusive) that means that x has to be either 0 or 1. And because x^15=1, we know that the only choice is for x to equal 1.
So there are 4 nonisomorphic groups of order 30... 1 is abelian, and the other...
Hey Guys,
StatusX-- Thanks for telling me that theorem! I didn't know it (my professor is a teeny bit incompetent...) So, I can use that to prove that because 4*3=12 means C_4xC_3 is isomorphic to C_12? It makes sense. :-D
Morphism-- I proved that a group G s.t. o(G)=30 has a normal...
I'm sorry, but I am intensely confused right now. How does nowing that it has normal subgroups of order 3 and 5 help?
Thanks for using your time to help me. :-D
Homework Statement
How many different nonisomorphic groups of order 30 are there?
Homework Equations
The previous parts of the problem dealt with proving that 3-Sylow and 5-Sylow subgroups of G were normal in G when o(G)=30, though I'm not sure how that relates...
The Attempt at a...
Okay, now that it's morning, this makes much more sense. Thanks for the clarification HallsofIvy.
I guess now, my only question is, does a 4 element permutation have a 4 element permutation representation or a 4!=24 element permutation?
Thank you so much!
Okay, so I only missed one? And we get our 3-cycles through multiplying the 2-cycles?
And as far as the cyclic group of order n... How is that specifically different from Sn?
Thank you so much for all of your help!
Hmm... I think I kind of understand. Then again, I'm also really tired, so who knows?
So, for S3, we'd have (1)(2)(3),(12)(3),(13)(2),(1)(23),(123) ?
Or am I totally off?
Homework Statement
a. Find the permutation representation of a cyclic group of order n.
b. Let G be the group S3. Find the permutation representation of S3.
Homework Equations
n/a
The Attempt at a Solution
I unfortunately have not been able to come up with a solution. I really...
Homework Statement
Express as the product of disjoint cycles:
a. (1,2,3)(4,5)(1,6,7,8,9)(1,5)
b. (1,2)(1,2,3)(1,2)
The Attempt at a Solution
a. (1, 2, 3) ( 4, 5, 6, 7, 8, 9)
b. (1, 2, 3)
These are the answers I got directly through the product. Would anybody be willing to check...
I think that that comment right there might have helped. :-D
Because F \subseteq K, the open covers of K also cover F. Let's take a look at the covers of F that do not also cover K. One such cover is the complement of F within K, which can be written as K\F. This is open because it is the...
"However, remember that you need to find a subset of the O's alone that covers F, ie, you can't use any of the P's to cover part of F. One way to guarantee you won't need any P's to cover F is to pick each P to lie outside of F. Hint: you'll only need one P, and constructing it will use the fact...
So, you're saying that I need to pay attention to the open covers that are not open covers of K, even though they're open covers of F, because they are contained in K? That makes sense, I'm just not sure how to approach that case.
Maybe... Because F is closed, for every open cover, there must...
Homework Statement
Prove directly (i.e., without using the Heine-Borel theorem) that if K \subseteq Rd is compact and F \subseteqK is closed, then F is compact.
Homework Equations
Definition of a Compact Set: A set K is said to be compact if, whenever it is contained in the union of a...