OK I will be more explicit:
The equivalence class of the identity element (we will denote it by ##[p]_{\sim_\mathcal{I}}##) is the set of all elements of ##\mathcal{F}## equivalent to ##p## when considering the relation ##\sim_\mathcal{I}##. If we consider the quotient group...
Hello! I have a problem with problem 20...
If we have any quotient group, the equivalence class of the identity element should be a normal subgroup. But the equivalence class of ##p## (which is the identity element in ##\mathcal{F}##) with respect to the relation ##\sim_\mathcal{I}## is the...
Thank you all for the replies. The more opinions the better :) I read now that I may have not written this clearly enough. I don't plan to drop my PhD. I will most definitely finish it and then look for something different :)
Hello! I am writing this to get some ideas for what to do with my future. This will be a long post so I should start by introducing myself…
I am currently in the last year of my PhD studies and I will be starting my thesis soon (at least I hope so). My work is on classification theorems in...
Hello!
I seem to have a problem with spherical coordinates (they don't like me sadly) and I will try to explain it here. I need to calculate a scalar product of two vectors \vec{x},\vec{y} from real 3d Euclidean space.
If we make the standard coordinate change to spherical coordinates we can...
Thank you for the replies. You've pretty much told me only about the MASt in mathematics. How about physics? And I know that MPhil is done by research but could you clarify what will be the benefits from a MPhil compared to a MASt (or a MASt compared to a MPhil)? I've read that MPhil is...
Hello all! I am going to begin the last year of my Bachelor's course in physics in about a month and it's about time to start searching for some Master's degree opportunities. I decided I might just as well try my luck at Cambridge but the problem (one of the many problems actually :P) is that...
Hello! My problem is that I want to find (\frac{\partial}{{\partial}x}, \frac{\partial}{{\partial}y}, \frac{\partial}{{\partial}z}) in spherical coordinates. The way I am thinking to do this is...
I felt exactly the same way when I started my first year in university and we started studying linear algebra and mathematical analysis. The latter seemed more useful cause at least derivatives and integrals are used everywhere. But the linear algebra seemed really abstract and useless at first...
Are you sure about a and b not being fixed. Because this seems like a really weak condition for the functions. I mean if you use that 1+b{\neq}a you can get a condition for the functions:
\frac{f(1)}{f^{\prime}(1)}-\frac{f(0)}{f^{\prime}(0)}{\neq}1
Anyway in that case I can't come up with...
I don't think there's a problem. By defining the sum as (f+g)(x) = f(x) + g(x) = h(x), where f, g are from G, we need to prove that h is also from G.
1. h is obviously a function with the same domain as f and g.
2. (f+g)(0) + (f+g)'(0) = f(0) + g(0) + (f(0) + g(0))' = f(0) + f'(0) + g(0) + g'(0)...
\frac{D(A)}{D(B)}=\left(\begin{array}{cc}\frac{\partial f(x)}{\partial y}&\frac{\partial f(x)}{\partial x}\\\frac{\partial x}{\partial y}&\frac{\partial x}{\partial x}\end{array}\right)=\left(\begin{array}{cc}0&f^\prime\\0&1\end{array}\right)
This should be it if I'm not mistaken. What you're...
One reason that comes to mind is that by defining the inner product as <v,u> = v1*u1 + v2*u2 + v3*u3 you get a real number for <v,v> as you said and that is needed for a prehilbert space (where <v,v>=0 <=> v=0 and otherwise <v,v> > 0), which is basically the generalization of the Euclidean space...