A Summer of Maths (ASoM) 2016

Yeah they have to be cosets of the same subgroup.


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Aggh I was getting confused with typos:

So let $g \in Hx_{i} \cap Ky_{j}$. Then $g \in Hx_{i}$ and $g \in Ky_{j} \Rightarrow g=hx_{i}=ky_{j}$ for some $h \in H, k \in K$.

So $Hg=H(hx_{i})=Hx_{i}$ and $Kg=K(ky_{j})=Ky_{j}$
Therefore $Hx_{i} \cap Ky_{j}= Hg \cap Kg$. My original typo here was that I said $Ky_{j}=Hg$ also instead of Kg.

What are you trying to do here? I think you're saying that there exists x, y in G such that Hx and Ky both contain G, but this is an obvious statement, since x=y=g works (both H and K contain the identity). Therefore it is true that g belongs to Hg ^ Kg = (H ^ K)g, a coset of (H ^ K), as you have said. But this was already known to us, since we know that H ^ K is a subgroup, and that the cosets of a subgroup will partition the group.


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Any one got any ideas on this?

Question: A ball bearing rests on a ramp fixed to the top of a car which isaccelerating horizontally. The position of the ball bearing relative to theramp is used as a measure of the acceleration of the car. Show that ifthe acceleration is to be proportional to the horizontal distance moved bythe ball (measured relative to the ramp), then the ramp must be curvedupwards in the shape of a parabola. ++

Any one got any ideas on this?

Question: A ball bearing rests on a ramp fixed to the top of a car which isaccelerating horizontally. The position of the ball bearing relative to theramp is used as a measure of the acceleration of the car. Show that ifthe acceleration is to be proportional to the horizontal distance moved bythe ball (measured relative to the ramp), then the ramp must be curvedupwards in the shape of a parabola. ++

Is this from Upgrade Your Physics? In which case excellent book, I recommend it wholeheartedly.

Your approach will want to be along the lines of showing that the gradient - or, equivalently, cot(theta) with theta to the horizontal if my mental diagram is vorrect - is proportional to x, and so a parabola must result. Any ideas on how to establish something about this angle?

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Yep it's off that book, glad to know it's useful haha.

I'm unsure if we can say that the ball is in equilibrium and maximum displacement (i.e. Rcos(theta) = ma and Rsin(theta) = mg from which the result would follow) since surely the ball would some have velocity at equilibrium so would continue rising up the ramp and therefore the value of x would increase?

Are you familiar with the idea of a fictitious force?


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Yes, we have an inertial force acting in the positive x direction equal to ma right?

@Zacken Are there any resources for GT on MIT OCW? I must be blind since I can only find "Introduction to Lie Groups".

@Zacken Are there any resources for GT on MIT OCW? I must be blind since I can only find "Introduction to Lie Groups".

They'll be listed under 'algebra' or 'abstract algebra'.

Groups:
http://ocw.mit.edu/courses/mathematics/18-701-algebra-i-fall-2010/
Rings:
http://ocw.mit.edu/courses/mathematics/18-702-algebra-ii-spring-2011/index.htm
Bit of both:
http://ocw.mit.edu/courses/mathematics/18-703-modern-algebra-spring-2013/Syllabus/

Lie groups is pretty hard stuff, you'll need a lot of prerequisites for that.

Thanks. and yea haha I wasn't planning on starting the lie groups stuff. a couple of years away hopefully

Thanks. and yea haha I wasn't planning on starting the lie groups stuff. a couple of years away hopefully

It's good that you're interested in group theory though. I found groups to not be interesting early on, since I'm more of an analysis person. Groups pop up everywhere, which is why we learn about them.. For instance, Lie groups I quite like, as they are groups and differentiable manifolds, so it has all group properties and you can do analysis on them.

Is there a thread like this for commoners like UCL applicants and even Bath applicants?
More seriously, I wanted to ask how much time a day or better a week is best to be invested in self study to keep the brain active for the summer? I am keeping the remainder of July free from thought but almost 2 weeks in of no maths and I begin reading analysis books before bed (30 mins or so) and worrying about the difficulty of 1st year at uni; August though I plan to start taking notes alongside reading and some exercises.
Book I am using is A (so called) Straighforward approach to Analysis by KGB (I also have A first course in Mathematical Analysis but it seems a bit too advanced for myself right now; also a bunch of PDF's of notes from different Unis)

Im pretty sure being a UCL/Bath applicant doesnt make you a commoner, the thread is for anyone who wants to do maths
I use that Binmore book too, nice style.