The Proof is Trivial!

Original post by bananarama2

I don't know whether their lucky or unlucky :tongue:

They are officially geeks, who need to attend their week. Let's not undermine their self-worth further. :pierre:

Reply 981

Original post by ukdragon37

They are officially geeks, who need to attend their week. Let's not undermine their self-worth further. :pierre:

You read "they're" right. :ninja:

Reply 982

Original post by ukdragon37

Well you could just do a postdoc if you find that you still need to get a few more papers under your belt, and those usually pay a larger stipend than when you were a PhD student! It works out more or less in the end for the same amount of research done.

The contribution must be novel, but it can be very very very small. The PhD is the qualification which precisely shows you are capable of discovering something new, even if that new thing is insignificant. Remember it's supposed to certify the start of your research career, not the end of your academic life even though it might seem that way. The significant stuff comes later when you are a proper researcher. This explains it quite well.

Yes at some colleges.... and Emmanuel is indeed one of them. :biggrin:

It's called "geek week".

That all sounds so great (I'm not too fussed about money as long as I have enough to live off). I liked the illustration :') I can't really see myself wanted to do anything besides research and/or lecturing; everything else seems like torture. I recently came across this which seems to sum up my feelings about mathematics! :biggrin:

Hahaha! Sounds a laugh; should be a great chance to get to know all the other maths students there as well (though I do know a bunch of the already from the Easter School) :smile:

Do you know what kind of stuff they do on "geek week"? :lol:

Original post by bananarama2

I don't know whether they're lucky or unlucky :tongue:

Edit:

Lucky of course! Besides, if they try and make us do a-level-revision-style drills or prepare us with notation (that I already know from further reading) then I'm just gonna walk out :lol:

Reply 983

Original post by Jkn

Lucky of course! Besides, if they try and make us do a-level-revision-style drills or prepare us with notation (that I already know from further reading) then I'm just gonna walk out :lol:

Surely you have to pay another weeks rent? What notation can they possibly spend a week on?

Reply 984

Original post by Jkn

You should read A Mathematician's Apology by Hardy, if you haven't already.

Original post by Jkn

Do you know what kind of stuff they do on "geek week"? :lol:

I think they let you have supervisions that are slightly beyond STEP, so to ease you into the teaching style and work of a Cambridge term.

Original post by bananarama2

Surely you have to pay another weeks rent? What notation can they possibly spend a week on?

Queens' once contemplated the idea of delivering compulsory lectures until 10pm during freshers' week to prevent "unwanted" behaviour. They quickly backed down. :colone:

Reply 985

Original post by ukdragon37

Queens' once contemplated the idea of delivering compulsory lectures until 10pm during freshers' week to prevent "unwanted" behaviour. They quickly backed down. :colone:

Jesus....that's a bit weird....

Reply 986

Original post by Jkn

....

Solution 150

The Postulates

Newtons laws of motion apply in all inertial reference frames

The speed of light is the same in all inertial reference frames. It is the same to all observers irrespective of their speed or the speed of the source.

I define two inertial reference frames moving with speed v relative to each other and initally coincide:

\mathcal{S}, \mathcal{S'}

I also define

Unparseable latex formula:

\mathb{L} : \mathcal{S} \mapsto \mathcal{S'}

. If you imagine a rod observed from two different frames, the transformation of the whole rod, between frames, is exactly the same as the sum of the transformations of the two halves. So

Unparseable latex formula:

\mathb{L}

is a linear transformation.

I define two basis

\begin{pmatrix} ct \\ 0 \end{pmatrix}

and

\begin{pmatrix} 0 \\ x \end{pmatrix}

.

Note that because of the second postulate the vector corresponding to

x = \pm ct

is invariant under the Lorentz transformation.

So

\begin{pmatrix} 1 \\ 1 \end{pmatrix}

and

\begin{pmatrix} 1 \\ -1 \end{pmatrix}

are the eigenvectors of the transformation with eigenvalues

\lambda_1, \lambda_2

Let

Unparseable latex formula:

\mathb{L} = \begin{pmatrix} a & b \\ d & e \end{pmatrix}

. It follows:

\begin{pmatrix} a & b \\ d & e \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} =\begin{pmatrix} a+b\\ d+e\end{pmatrix}=\begin{pmatrix} \lambda_1\\ \lambda_1 \end{pmatrix}

.

So

a+b = d+e

. Similarly:

\begin{pmatrix} a & b \\ d & e \end{pmatrix} \begin{pmatrix} 1 \\ -1 \end{pmatrix} =\begin{pmatrix} a-b\\ d-e\end{pmatrix}=\begin{pmatrix} \lambda_2\\ \lambda_2 \end{pmatrix}

.

So

a-b=d-e

. Hence

a=e=\gamma

and

d=b

.

Hence:

Unparseable latex formula:

\mathb{L} = \begin{pmatrix} \gamma & b \\ b & \gamma \end{pmatrix}

Given the two frames are inertial and intially coincide it follows under the transformation

\begin{pmatrix} ct \\ vt \end{pmatrix}\mapsto \begin{pmatrix} ct' \\ 0 \end{pmatrix}

Therefore:

\begin{pmatrix} \gamma & b \\ b & \gamma \end{pmatrix} \begin{pmatrix} ct \\ vt \end{pmatrix} =\begin{pmatrix} \gamma ct +bvt \\ bct+\gamma vt \end{pmatrix}=\begin{pmatrix} ct' \\ 0 \end{pmatrix}

.

It follows

b=-\frac{v}{c} \gamma

and

\gamma ( t-\frac{v^2}{c^2}t )=t'

. Note how gamma must be and even function of v for the equation to be rotationally invariant.

\mathcal{S'}

moves with velocity v relative to

\mathcal{S}

. so

\mathcal{S}

moves with velocity -v relative to

\mathcal{S'}

. So the inverse Lorentz matrix is the Lorentz matrix with the signs of the v's swapped. Hence:

Unparseable latex formula:

\mathb{L} = \begin{pmatrix} \gamma & \frac{-v}{c} \gamma \\ \frac{-v}{c} \gamma & \gamma \end{pmatrix}

Unparseable latex formula:

\mathb{L}\mathb{L}^{-1} = \begin{pmatrix} \gamma & \frac{-v}{c} \gamma \\ \frac{-v}{c} \gamma & \gamma \end{pmatrix}\begin{pmatrix} \gamma & \frac{v}{c} \gamma \\ \frac{v}{c} \gamma & \gamma \end{pmatrix}= \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}

Hence

\gamma^2 - \frac{v^2}{c^2} \gamma^2 =1

\gamma = \frac{1}{\sqrt{1-v^2/c^2}}

In general then:

\begin{pmatrix} \gamma & \frac{-v}{c} \gamma \\ \frac{-v}{c} \gamma & \gamma \end{pmatrix}\begin{pmatrix} ct \\ x \end{pmatrix} =\begin{pmatrix} ct' \\ x' \end{pmatrix}

So:

\gamma ( t - \frac{v}{c^2}x )= t'

Proper time is defined to be the time measure by a frame at rest relative to two events. I.e the position the same in both. So:

\gamma ( t_1 - \frac{v}{c^2}x )= t'_1

and

\gamma ( t_2 - \frac{v}{c^2}x )= t'_2

t'_1-t'_2=\Delta t = \gamma (t_1 -t_2 ) = \gamma \Delta \tau

Momentum is define to be

p =m_0 \frac{\Delta x}{\Delta \tau}=m_0 \frac{\Delta x}{\Delta t} \frac{\Delta t}{\Delta \tau} = m_0 \gamma v

It follows

m = \gamma m_0

Kinetic energy is the work done my the resultant force.

EK=

\displaystyle \int_0^x F.dx

\displaystyle =\int_0^x \frac{dp}{dt}.dx

\displaystyle =\int_0^v v .d(\gamma m_0 v)

\displaystyle = \left[ v \gamma m_0 v \right]_0^v - \int_0^v \gamma m_0 v dv

\displaystyle = v \gamma m_0 v - \left[-c^2m_0 \sqrt{1-v^2/c^2} \right]_0^v

Unparseable latex formula:

\displaystyle = v \gamma m_0 v \right\ + c^2 m_0 \sqrt{1-v^2/c^2} -m_0 c^2

Unparseable latex formula:

\displaystyle = \gamma (m_0 v^2 \right\ + c^2 m_0 (1-v^2/c^2)) -m_0 c^2

\displaystyle = \gamma m_0 c^2 -m_0 c^2

Therefore

\Delta E = \Delta m c^2

Admittedly not all is my own thinking, but I've never seen the use of eigenvectors and personally never seen the last bit before.

(edited 10 years ago)

Reply 987

Original post by bananarama2

Surely you have to pay another weeks rent? What notation can they possibly spend a week on?

Well, it would only mean 30 quid :troll:

What accommodation did you apply for btw?

Obviously not a whole week on notation but there is a fair bit that people won't know despite knowing the meanings already: set notation, set sums (or whatever you call them), logical symbols, correct names of greek letters and their variations, partial differentiation and other multivariable calculus stuff, notation to describe limits and convergence, connotations of different algebraic symbols and their assumed use when not otherwise specified... the list goes on.... :tongue:

Every one of these things is vital for a typical a-level student to understand a degree-level paper or proof and yet few will know it straight away and so will be limited in their further reading :tongue:

Original post by ukdragon37

You should read A Mathematician's Apology by Hardy, if you haven't already.

I'm almost offended that you even suggest I might have not already :rolleyes:

I think they let you have supervisions that are slightly beyond STEP, so to ease you into the teaching style and work of a Cambridge term.

Hmm, doesn't sound too bad :tongue:

What happens if you're bored of the material you get one week (in general)? Will they give you different problems? For example, if you really clicked with a topic and often did the problem sheets days ahead of schedule. And, if so, will they give you new more advanced things to learn that might come up in later years or will they give you harder questions? :tongue:

Original post by bananarama2

I thought it might be you who would post the solution to this :biggrin:

Can't say I fully understand how you managed to apply square matrices but I probably wouldn't anyway because started and finished learning about them around the end of last year! :tongue:

One thing though: does the route you've taken with vectors not limits the generality of the result to 2-dimensional space? I know the final result has no assigned dimension, but does this adequately consider the fact we live in 3 spatial dimensions?

The way I normally see the first result as being derived is through representing the situation in Euclidean space and then deriving time dilation a direct consequence (i.e. to avoid the contradiction that ambiguous interpretation of the sped of light would present) :tongue:

Oh btw, I think you messed up with the LaTeX a bit at the end (please adjust) :smile:

Reply 988

This should keep you all busy for a while... and then some! :colone:

Problem 151**

Prove that

\zeta(3)

is irrational.

Spoiler

(edited 10 years ago)

Reply 989

Original post by Jkn

Well, it would only mean 30 quid :troll:

I thought it might be you who would post the solution to this :biggrin:

Can't say I fully understand how you managed to apply square matrices but I probably wouldn't anyway because started and finished learning about them around the end of last year! :tongue:

Oh btw, I think you messed up with the LaTeX a bit at the end (please adjust) :smile:

Surely it'd be easier to do that as you go along rather than bore everyone to death in a week?

Square matrices are what special relativity is built with. My method is readily extendable to four dimensional spacetime, applying the Lorentz transformation to four vectors instead. But the algebra is more cumbersome and you end up with a similar result anyway.

Yes, the way I was shown was using two mirrors moving on a train. This is more general and is what you will see next year :wink:

Reply 990

Original post by bananarama2

I hope so, lmao! Hopefully we'll just hang, standard night going out on the pull with your D.O.S (never gonna happen... but if only... :lol:

)!

Oooo not the passive voice! :lol:

Why must it be done with quire matrices? :lol:

Yeah me too! Except read/imagined* not 'was shown' and the mathematical universe created to fit euclidean geometry not 'a train' :wink:

What Physics and Maths topics are you going to cover in your first year then? :smile:

Oh btw, for your solution, check the limits of your integral

Spoiler

Reply 991

Original post by Jkn

I hope so, lmao! Hopefully we'll just hang, standard night going out on the pull with your D.O.S (never gonna happen... but if only... :lol:

)!

Oooo not the passive voice! :lol:

Why must it be done with quire matrices? :lol:

Yeah me too! Except read/imagined* not 'was shown' and the mathematical universe created to fit euclidean geometry not 'a train' :wink:

What Physics and Maths topics are you going to cover in your first year then? :smile:

Oh btw, for your solution, check the limits of your integral

Spoiler

I don't think your DOS would be seen dead with you in Cindies :tongue:

The matrices are out of convenience rather that necessity I think. How else do you do something to a vector? Plus the inner product is defined in terms of matrices.

Einstein imagined a train :wink:

Maths we do all the applied stuff, the vector calc, series, differential equations, etc...

Physics we cover, the basics. Electromag, circuits, Rutherford scattering, quantum etc...

Reply 992

Original post by bananarama2

I don't think your DOS would be seen dead with you in Cindies :tongue:

The matrices are out of convenience rather that necessity I think. How else do you do something to a vector? Plus the inner product is defined in terms of matrices.

Einstein imagined a train :wink:

Maths we do all the applied stuff, the vector calc, series, differential equations, etc...

Physics we cover, the basics. Electromag, circuits, Rutherford scattering, quantum etc...

Cindies?!?! What on earth is that? :lol:

All to be revealed to us next year :wink:

Though I am sure other formulations do/can exist :tongue:

And a beam of light :wink:

A train doesn't work too well as a thought experiment, an earthbound one at least (as the velocity is arbitrary, it's simpler to consider motion in a straight line!) :colone:

Sound pretty soul-destroying... like the sound of that "series" though :sexface:

Not circuits :| Jealous of the quantum mechanics though... think we get to do a bit... but not full-on until the second year even if we do :smile:

Reply 993

Original post by Jkn

Cindies?!?! What on earth is that? :lol:

All to be revealed to us next year :wink:

Though I am sure other formulations do/can exist :tongue:

And a beam of light :wink:

A train doesn't work too well as a thought experiment, an earthbound one at least (as the velocity is arbitrary, it's simpler to consider motion in a straight line!) :colone:

Sound pretty soul-destroying... like the sound of that "series" though :sexface:

Not circuits :| Jealous of the quantum mechanics though... think we get to do a bit... but not full-on until the second year even if we do :smile:

A nightclub (not actually called Cindies).

I'd be very surprised to be honest....

Well obviously there needs to be a beam of light, it's just Einstein imagined a torch on a train pointing upwards moving along a straight path with speed v.

Nahhh.. the vector calc is nice. To be honest the calc is nice.

The circuits are okay, we do resonance and complex methods with them :tongue:

I don't think the natscis get full on quantum until second year either tbh.

Reply 994

Original post by Jkn

Hmm, doesn't sound too bad :tongue:

In all subjects that very much depend on the attitude of your supervisor, but in Maths they are more attuned to the needs of people who speed ahead. You'll be given access to harder problems if you want it, and if you are really finding it a breeze then most supervisors are more than willing to push ahead. Maths supervisions are usually paired by ability so to reduce the occurrence of one person keeping another behind. (I think in Trinity speeding ahead is even enforced, rather than accommodated. I hear they finish a year's worth of courses one or two terms early and the rest of the time is spent on revising/doing harder stuff. :tongue:

)

Remember however that university (and Cambridge especially) is supposed to encourage self-learning, and if you are not challenged then you should have the initiative to challenge yourself.

Reply 995

Original post by bananarama2

I don't think the natscis get full on quantum until second year either tbh.

Oh haha

He sounds like such a player. Do you reckon he was exceptional at maths? (In the sense that it was part of his genius)

Once, again.. a question will we have fully answered by next year :rollseyes:

Ooooh Heaviside! Lightweightttts :colone:

Original post by ukdragon37

I'm sure I won't find it a breeze, I would just like the feeling of knowing that if I pushed hard I wouldn't hit a wall :tongue:

Maths supervisions are usually paired by ability so to reduce the occurrence of one person keeping another behind. (I think in Trinity speeding ahead is even enforced, rather than accommodated. I hear they finish a year's worth of courses one or two terms early and the rest of the time is spent on revising/doing harder stuff. :tongue:

)

That sounds crazy! I'm glad I didn't apply there though (it was a close call!) The list of people you could have as your supervisor though... :eek:

Remember however that university (and Cambridge especially) is supposed to encourage self-learning, and if you are not challenged then you should have the initiative to challenge yourself.

That's what I've been trying to do for years, though it will be nice to have someone there to discuss things with :tongue:

(edited 10 years ago)

Reply 996

Original post by ukdragon37

)

Remember however that university (and Cambridge especially) is supposed to encourage self-learning, and if you are not challenged then you should have the initiative to challenge yourself.

I feel more and more daunted every day :biggrin:

Reply 997

Original post by Jkn

I'm sure I won't find it a breeze, I would just like the feeling of knowing that if I pushed hard I wouldn't hit a wall :tongue:

You won't hit walls to learning when you get to Cambridge, unless you can't be bothered to push that far ahead. If you really want (and are that good) then there is bound to be one professor or another who you can get involved in research with.

Original post by bananarama2

I feel more and more daunted every day :biggrin:

You would be incredibly arrogant if you didn't. :tongue:

Reply 998

Original post by ukdragon37

Would you say performance of STEP is a good indicator of how much a student will excel in their first year? (speaking generally of course) For example: would someone with 3 high Ss likely plough through most of the problem sheets with ease and someone ho scraped in with a 2 be under constant struggle to keep up? (again, generally speaking)

Also, is there still an aspect of exam-directed learning to the course? i.e. are people so caught up in exam prep and revision of material that they don't have time to enjoy new ideas as they come? Or do you spend most of your time thinking creatively about questions in an almost Olympiad-style way?

Sorry for all the questions... just realised this is a bit of an ambush! :colondollar:

Reply 999