This is a question that came about while I attempting to prove that a simple extension was a splitting field via mutual containment. This isn't actually the problem, however, it seems like the argument I'm using shouldn't be exclusive to my problem. Here is my attempt at convincing myself that...
If you are trying to show that two groups, call them H and G, are isomorphic and you know a presentation for H, is it enough to show that G has the same number of generators and that those generators have the same relations?
My linear algebra is a bit rusty.
Let ##A=\{\bar{v}_1, \dots, \bar{v}_1\}## be a set of vectors in ##R^n##. Can dim(span##(A))=n## without spanning ##R^n##?
I guess I'm unclear on how to interpret the dimension of the span of a set of vectors.
I'm grading for a linear algebra class this semester. The class is comprised entirely of engineering majors of various flavors. The hw assigned by the professor is almost entirely "proofs" they are fairly specific proofs. Really the only thing that designates them as proofs is that the questions...
I'm in the middle of the stressful process of preparing for the GRE and preparing grad school applications. However, as it stands, I will have been an undergrad for 6 years after next spring when I graduate. I did an REU last summer, which was an amazing experience, but a ton of work. And I took...
I just read this article:
http://abcnews.go.com/Technology/plutos-majestic-mountains-atmospheric-haze-revealed-photo-horizons/story?id=33832751
The article itself is really cool, but something at the bottom of it caught my attention.
I don't know how fast New Horizons is traveling, but say...
I understand how to construct a proof by induction. I've used it many times, for homework because it was clearly what the book wanted, but when I've tried it in a research setting, it's because I have so little control of the objects in working with. So it has become my impression that since...
I've been studying for my final exam, and came across this homework problem (that has already been solved, and graded.):
"Show that the Galois group of ##f(x)=x^3-1## over ℚ, is cyclic of order 2."
I had a question related to this problem, but not about this problem exactly. What follows is...
See title.
I have a table that I've named B. And I want to create table that is essentially the change in values of B. If I was writing this on paper I would say: Let ##b_i \in B##, for ##0\leq i\leq100##. Then let ##c_i \in C## such that ##c_i= b_{i+1}-b_{i}##. I don't know how to say that in...
So a p-adic expansion of a rational number was presented to me as an analogue of a Laurent-series expansion and defined as:
$$\sum\limits_{n=-{\infty}}^{\infty}a_np^n$$
Can you find the coefficients for these the same way you would for a Laurent series? I've not gotten to that part of this...
In this book I'm working from (P-adic numbers: An introduction by Fernando Gouvea), he gives an explanation of the p-adic expansion of a rational number which I'm pretty sure I understand this, but a bit later he talks about the p-adic absolute value, which from what I understand is the same as...
I'm working on my applications for summer REUs (math). I've got my recommendations, my CV, a couple of cover letters, various personal statements, but there is one personal statement giving problems. It asks me to write about a page detailing, among other things, my "mathematical background."...
I'm looking into getting a laptop this summer if I'm able to get into an REU. I was looking at macbooks, just because they are reliable, have long battery life, and still has the power I need for programs like Mathematica or Sage, since it will be used for research, the computing will probably...
My confusion isn't exactly with a homework problem, but more with an example that is key to understanding a homework problem. So I am posting here anyway.
1. Homework Statement
The example is of an Einstein solid, with N=3 oscillators. The book lists the multiplicity of each macrostate, with...
Homework Statement
Let G be a non-abelian group with order ##p^3##, p prime. Then show that the order of the center must be p.
Homework Equations
Theorem in our book says that for any p-group the center is non-trivial and it's order is divisible by p.
Class eq.
##|G|= |Z(G)| + \sum{[G ...
Homework Statement
...For each of the temperatures...compute the second term in the virial equation ##B(T)/(V/n)##, for nitrogen at atmospheric pressure.
Homework Equations
##PV = nRT(1+\frac{B(T)}{V/n}+\frac{C(T)}{(V/n)^2}+...)##
The Attempt at a Solution
I'm given a list of temperatures and...
Homework Statement
I really just need clarification about a property of cosets. I can't find anything explicitly stating one way or the other, and it could be because I'm wrong, or because it's deemed trivially true.
Homework Equations
Left Coset: (aH)(bH)=abH, where a,b are elements of a...
First, I'm in need of a topic to write a paper about as part of my degree requirements. The paper is supposed to be 10-15 pages. (I only mention this because it limits the scope of the paper and thus the topic.)
I took a course in number theory last semester and really enjoyed it, so I'm really...
I'm asked to show that a permutation is even if and only if the number of cycles of even length is even. (And also the odd case)
I'm having trouble getting started on this proof because the only definitions of parity of a permutation I can find are essentially this theorem. And obviously I...
I'm a math major, in my senior year, and periodically, in conversation with someone they would try and ask me what I do as a math major, and i've not really been able to give an answer that someone that isn't a math major understands. But after spending a lot of time spinning my wheels on a...
Homework Statement
Show that the wave function ##\Psi(x,t)=Asin(kx-ωt)## does not satisfy the time dependent Schrodinger Equation.
Homework Equations
##-\frac{\hbar}{2m}\frac{\partial^2\psi(x,t)}{{\partial}x^2}+V(x,t)\psi(x,t)=i\hbar\frac{\partial\psi(x,t)}{{\partial}t}##
The...
So, in the context of computing for the even avid techy, it seems like a Petabyte(1024 Terabytes) of storage on a single drive seems both out of reach, and unnecessary. I mean, on my computer, I've got 1080p (legal) downloads of movies, several very large games, and a comfortable amount of...
Homework Statement
Suppose you want to make a velocity selector that allows undeflected passage for electrons whose kinetic energy is ##5x10^4eV##. The electric field available to you ##2x10^5V/m##. What magnetic field will be needed?
Homework Equations
##u=\frac{E}{B}##
u is velocity, E is...
Homework Statement
(Verbatim) A particle moves with speed .9c along the x'' axis of frame S'', which moves with speed .9c in the positive x'-direction relative to frame S'. Frame S' moves with speed .9c, in the positive x-direction, relative to S. a.) Find the speed of the particle relative to...
Next Spring I'll be graduating at the University of Arkansas with a Double major in Math(B.S.) and Physics(B.A.). As of right now the plan is to teach high school math and/or physics in Texas. However, that isn't necessarily my ultimate goal. I would like to go to grad school and get at least a...
I've come to the point in my homework discussing the above, and more specifically, energy levels, wave functions, excited states etc..
And while I can locate an appropriate formula and plug-n-chug, I'm finding that I have no clue what these equations and numbers actually mean. And(in my...
I'm pretty sure this is a fairly obvious question, but I can't ever be sure..
So, if a photon is "scattered" 180 degrees. Its not being scattered at all, correct? So, then the energy of the "recoiling electrons" would be 0.
It makes sense mathematically if I'm doing it right...
The proper mean lifetime of ##\pi## mesons(pion) is ##2.6x10^{-8}s##. Suppose a beam of such particles has a speed of .9c.
a) What would their mean life be as measured in the lab?
b) How far would they travel(on average)before they decay?
c)What would your answer be to part (b) if you...
I had a quick question about Time Dilation and Length Contraction.
Are the two just different ways of measuring/describing the same effect? Or rather they both follow as a consequence from one another?
i.e. I can find how much a length is contracted by finding the dilated time interval and...
Homework Statement
Starting with the Lorentz transformation for the components of the velocity, derive the transformation for the components of acceleration.
Homework Equations
Lorentz Transformation for position and time :
##x'={\gamma}(x - vt)##
##t'={\gamma}(t - {\frac{vx}{c^2}})##...