The need of having done a class "formally"
I'm (something like) a master student in Europe, now being in my first year and I have the following problem: Last semester I was attending lectures on QFT, but I haven't done the exam/assignments, since it was way too much for me that semester and...
We have been discussing this topic with my classmates recently, and I find it quite interesting, so I also want to ask you guys:
1. For how long can you study without taking a break?
2. How long are your breaks?
3. What do you do in the breaks?
I'm interested in doing my PhD in theoretical and mathematical physics - i.e. subjects like Quantum Field Theory, String Theory or Quantum Information Theory. My question is which universities in the US have really good programs in these areas?
I've been searching through the internet and some of my optics books, but nowhere was I able to find the derivation of the law for a camera lens, that the intensity of light that comes on the film or chip is proportional to \frac{D^2}{f^2}=N^2, where D is the aperture diameter, f the focal...
Has anyone done this?
I'm a physics major and I'd like to pursue a degree in mathematical physics. I guess that physics GRE is expected from me as a physics student, although actually I feel much more comfortable in math.
Do you think that it would be hard to make both? Would it make me...
I'm applying for a masters degree on an European university this year. I have an almost perfect GPA of 3.98, I took some pretty advanced classes and have two years of research experience, but the problem is with the letters of recommendation.
They require me to send one letter of...
Hi,
does anyone know of some nice root-finding method (preferable GSL :-)) for a data set - i.e. I have a set of 3D data (x,y,z) where z = f(x,y,) and I want to know where the zeros of f are. I guess, I could write it myself with some interpolation method, but just in case someone knows...
I have a set of 3D data (i.e. a large file where each row contains three spatial coordinates) and I'd like to get a nice, smooth 3D object out of it. The objects are not surfaces, so it's not just plotting a function (i.e. to every (x,y) there exists more than one z).
Does anyone have an...
I'm really stuck on this simple problem: Let X be a topological vector space and U, V are open sets in X. Prove that U+V is open.
It should be a direct consequence of the continuity of addition in topological vector spaces. But continuity states that the f^{-1}(V) is open whenever V is open...
How do I transform a second-order PDE with constant coefficients into the canonical form?
I tried to solve this problem:
u_xx + 13u_yy + 14u_zz - 6u_xy + 6u_yz + 2u_xz -u_x +2u_y = 0
I wrote the bilinear form of the second order derivatives and diagonalized it. I found out that it is a...
This seems to be a very easy excercise, but I am completely stuck:
Prove that in C([0,1]) with the metric
\rho(f,g) = (\int_0^1|f(x)-g(x)|^2 dx)^{1/2}
a subset
A = \{f \in C([0,1]); \int_0^1 f(x) dx = 0\} is closed.
I tried to show that the complement of A is open - it could be...
I'm just taking Calculus 4 this semester, where part of it is also Fourier analysis.
When I was browsing a little bit about the subject I found out that there are several different approaches and so I'm a bit confused now.
So this is how I understand it, correct me if I'm wrong:
There...
Is there a function f: R->R, such that:
\forall x \in \mathbb{R}: f(x) \neq 0 \wedge \forall a,b \in \mathbb{R}: \int_a^b f(x) dx = 0
I made this problem myself so I don't know, wheather it is easy to see or not. The integral is the Lebesgue integral.
I would say, that there should be...
I'm reading a book on electromagnetism and I am a bit confused about some things in Maxwells equations. This is what I don't like about many physics books: they are very wordy, but at the end you don't know what is an experimental fact, what is a "theorem", what is an assumption and so on...
I have one more question about the Lebesgue integral:
What if we defined the Lebesgue integral like this:
Let X be a measurable space and f any nonnegative function from X to R.
Then the Lebesgue integral of f as \int_X f d\mu = sup(I_X) where I_X is the integral of a simple function...
integral more general than the Lebesgue integral?
The Lebesgue integral is defined for measurable functions. But isn't it possible to define a more general integral defined for a larger class of functions?
I guess that we would then loose some of the fine properties of the Lebesgue integral -...
Hi,
I'm just reading Rudin's Principles of mathematical analysis - the last chapter on Lebesgue integration and I am having a bit trouble understanding the motivation of the definition of Lebesgue measure.
This is how I understand it:
We want to measure sets in \mathds{R}^n so what we...
When we seek the extreaml value of the functional \Phi(\gamma) = \int_{t_0}^{t_1} L(x(t),\dot{x}(t),t)dt where x can be taken from the entire E^n then we come to the well-known Lagrange equations.
Now when we are given a constraint, that x \in M, where M is a differentiable manifold and when...
Today I revised my knowledge from multivariable calculus and I found that I couldn't remember the proofs of these two theorems. Then I looked in Rudin, and everything was clear.
Except one thing, which probably made me forgot the proofs. There are two weird functions in these two proofs...
Hi, I have to draw this kind of graphs on the computer:
http://img285.imageshack.us/img285/1690/image1nc4.jpg [Broken]
http://img427.imageshack.us/img427/6483/image2it9.jpg [Broken].[/URL]
I tryed to plot the graph of the function in mathematica, export it into .wmf file and then add some of...
In a lecture on classical mechanics, the professor derived a formula, which is a part of the D'Alambert principle: \nabla \Phi_{\alpha} \cdot \delta \vec{r} = 0 where \Phi_{\alpha} are the restraints. He derived it in a strange way from the Taylor's formula:
\Phi_{\alpha} (\vec{r} + \delta...
hi,
I found this problem in Rudin, and I just can't figure it out.
It goes like this:
Prove that the convergence of \sum a_n a_n \geq 0 implies the convergence of \sum \frac{\sqrt{a_n}}{n}
I tried the comparison test, but that doesn't help because I don't know what the limit \lim_{n...
Hi,
as usual in September I am deciding which courses to take. I am in the second year of my study and so far I am following the more theoretical path, later maybe with focus on quantum mechanics and quantum information proccessing.
My question is:
which math courses should I take this...
Hi,
I'm a bit stuck with some things in electrostatics.
My first problem:
in my textbook, when they try to derivate the formula for the potential of a point charge: V(b) = - \int E.d\mathbf{l} = -\frac{q}{4 \pi \varepsilon_0} \int_\infty^b \frac{1}{r^3} \mathbf{r}.d \mathbf{l}
they...
what is the necessary condition (if there is any) for inscribing a square into a parallelogram. In other words, what should the parallelogram be like - so that we can inscribe a square into it.
what is the mode (modus) of 1,1,2,2
I was checking the definition but it didn't count with two maximums.
So is it 1,5 or both 1 and 2, or none of them?
Can you give me a simple real-life problem, where you need to use Stieltjes integral and can you show how you proceed in solving this kind of problems?
Imagine you have a tape wrapped around a coil (e.g. an audio magnetic tape or adhesive tape). The thickness of the tape is T and the radius of the coil is R.
The task is to determine the dependence of the length of the coiled tape and the radius of the whole (coil + tape).
(e.g. - if I know...