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The Proof is Trivial!

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Original post by shamika
Where do you go to university?

I will be attending UCL starting this September.

Original post by ukdragon37

Why do you presume he goes to university? :tongue:

Well done (for your dissertation)! Although, I have to admit, I am not surprised the slightest. :biggrin:
Original post by jack.hadamard
I will be attending UCL starting this September.


Consider me really scared (you'll love university I'm sure, and as long as UCL give you room to shine you'll do great).

Well done (for your dissertation)! Although, I have to admit, I am not surprised the slightest. :biggrin:


He is really lazy :tongue:

Spoiler



Have you lot looked at the STEP papers? Some nice questions this year, as always.
(edited 10 years ago)
Original post by ukdragon37
I got a distinction bring on the girls


Congratulations!
Original post by shamika

Consider me really scared (you'll love university I'm sure, and as long as UCL give you room to shine you'll do great).


I'm sure I'll love it. :tongue: I think there is room to shine, and it's better to be the king of a village than a clerk of a town. :biggrin:
Original post by shamika
Good point. But if this is the new cohort of pre-uni students, then I'm scared...


I've got past that point aaaages ago. :wink: Also considering this year's Senior Wrangler is only 18 I think we should just make peace with the fact that we will never be child prodigies. :laugh:

Original post by jack.hadamard
Well done (for your dissertation)! Although, I have to admit, I am not surprised the slightest. :biggrin:


I don't know what causes you to not be surprised, but thanks! :tongue:

Original post by shamika

He is really lazy :tongue:

Spoiler



Thanks! But you are actually correct. I'm incredibly lazy. :tongue:

Original post by shamika
Have you lot looked at the STEP papers? Some nice questions this year, as always.


Nah, I haven't looked at them since I crashed spectacularly when sitting them in 2009. I just hang around the STEP thread to troll. :tongue:

Original post by Lord of the Flies
Congratulations!


Thanks! I'm sure you'll have great fun at Cambridge, a place I am leaving for the foreseeable future on Saturday :cry:

Original post by jack.hadamard
I'm sure I'll love it. :tongue: I think there is room to shine, and it's better to be the king of a village than a clerk of a town. :biggrin:


:lol: From what I've seen even if you went to Cambridge for maths you wouldn't just be a clerk of a town.
Original post by ukdragon37
I've got past that point aaaages ago. :wink: Also considering this year's Senior Wrangler is only 18 I think we should just make peace with the fact that we will never be child prodigies. :laugh:
.


I know I'm going to spend next year wondering if I deserve my place and that I'm an idiot :tongue: Can't wait.

18...what?
Original post by bananarama2
I know I'm going to spend next year wondering if I deserve my place and that I'm an idiot :tongue: Can't wait.

18...what?


Yep! I believe he also got a full set of alphas/betas in the first year (if you don't know what that means, he basically got nearly full marks on every question he needed to attemp)
Original post by ukdragon37
.


Congrats on your grade for masters! :awesome:
Original post by bananarama2
I know I'm going to spend next year wondering if I deserve my place and that I'm an idiot :tongue: Can't wait.

18...what?


Original post by shamika
Yep! I believe he also got a full set of alphas/betas in the first year (if you don't know what that means, he basically got nearly full marks on every question he needed to attemp)


I keep making comments to friends that it would be funny to give supervisions to those older than you. :laugh:

Original post by cpdavis
Congrats on your grade for masters! :awesome:


Thanks! :awesome:
Reply 1789
Hmm, I think I might make a separate thread for people learning bits of analysis, group theory, etc... over the summer. Anyone up for this? I saw one last year called "A Summer of Maths" (I think?) so I might be a bastard and steal the name :lol:

Edit: Too late I already made it :colondollar:
(edited 10 years ago)
Another beautiful combinatorial geometry problem.

Problem 254**

Let nn be a fixed positive integer. Suppose also that f:CRf : \mathbb{C} \to \mathbb{R} is a function such that, for any points PiP_{i}, i{1,2,,n}i \in \{1,2,\cdots,n \} which are vertices of a regular nn-gon, we have 1inf(Pi)=0\displaystyle \sum_{1 \le i \le n} f(P_{i}) = 0. Show that f0f \equiv 0 over C\mathbb{C}.
Reply 1791
Original post by Mladenov
Another beautiful combinatorial geometry problem.

Problem 254**

Let nn be a fixed positive integer. Suppose also that f:CRf : \mathbb{C} \to \mathbb{R} is a function such that, for any points PiP_{i}, i{1,2,,n}i \in \{1,2,\cdots,n \} which are vertices of a regular nn-gon, we have 1inf(Pi)=0\displaystyle \sum_{1 \le i \le n} f(P_{i}) = 0. Show that f0f \equiv 0 over C\mathbb{C}.


Crap, sorry for the neg. That was an accident. Mis-click. *Facepalm.
Reply 1792
Original post by jack.hadamard
I'm sure I'll love it. :tongue: I think there is room to shine, and it's better to be the king of a village than a clerk of a town. :biggrin:


The other day I was speaking to a friend who has just finished his first year at UCL. He said failing STEP was the best thing to happen to him, since he's now doing extremely well there and he says he's having much more fun in London than he would have done at Cambridge. In fact if I understood correctly he came top of his year, and still had a lot of time to go out and enjoy himself. He also mentioned that he's been doing most of the Cambridge problem sheets, and has been finding them surprisingly easy after the UCL course.

Having said that, this guy is insanely clever so it's clear that most people won't have the same experience.
Original post by und
The other day I was speaking to a friend who has just finished his first year at UCL. He said failing STEP was the best thing to happen to him, since he's now doing extremely well there and he says he's having much more fun in London than he would have done at Cambridge. In fact if I understood correctly he came top of his year, and still had a lot of time to go out and enjoy himself. He also mentioned that he's been doing most of the Cambridge problem sheets, and has been finding them surprisingly easy after the UCL course.

Having said that, this guy is insanely clever so it's clear that most people won't have the same experience.


Aren't first year courses the same at most universities?
I'm sorry to bother everyone but could someone supply me with some useful links on modular arithmetic as I'm trying to get to grips with it and the stuff I've looked at so far isn't very helpful to me :redface:
Original post by MathsNerd1
I'm sorry to bother everyone but could someone supply me with some useful links on modular arithmetic as I'm trying to get to grips with it and the stuff I've looked at so far isn't very helpful to me :redface:


Try this, by Vicky Neale (the notes for the Part II NT course she taught are excellent :tongue:).
Original post by MathsNerd1
I'm sorry to bother everyone but could someone supply me with some useful links on modular arithmetic as I'm trying to get to grips with it and the stuff I've looked at so far isn't very helpful to me :redface:


http://www.mathdb.org/notes_download/elementary/number/ne_N2.pdf
Original post by jack.hadamard
Try this, by Vicky Neale (the notes for the Part II NT course she taught are excellent :tongue:).


Thanks I'll take a look at this now :biggrin: This made it all seem so very simple and I finally get it now so thanks for the link! :smile:
(edited 10 years ago)
Original post by FireGarden

Spoiler




Geometric intuition is probably the most important source of examples in algebraic topology. As you have already pointed out, your example is intuitively true, just as the fact that S2S^{2} with two holes is homeomorphic to a torus is intuitively true.

Let me consider your problem more generally. Clearly, Sn1×IS^{n-1} \times I, where, of course, I=[0,1]I = [0,1], is a cylinder. The equivalence relation you defined is simply identifying Sn1×1S^{n-1} \times 1 and Sn1×0S^{n-1} \times 0 with two points. It is sometimes called suspension, and is denoted by S(Sn1)S(S^{n-1}). We can also interpret S(Sn1)S(S^{n-1}) as the union of two copies of C(Sn1)C(S^{n-1}) under the identity of Sn1S^{n-1}. There exists natural homeomorphism φ:C(Sn1)Bn\varphi : C(S^{n-1}) \to B^{n}. I can't draw a diagram to show it...
Now, from the above interpretation of the suspension as a double cone, it follows that C(Sn1)/Sn1C(S^{n-1}) / S^{n-1} is the suspension of Sn1S^{n-1}. But, it is clear that Bn/Sn1SnB^{n} / S^{n-1} \cong S^{n} (stereographic projection). Thus, S(Sn1)SnS(S^{n-1}) \cong S^{n}.

I recently have noticed that I find the double suspension theorem beyond my intuition. What do you think of it?
(edited 10 years ago)
I have as yet only studied point-set topology, and am not well acquainted with algebraic topology, and know literally nothing about the theorem. I shall learn more of it next year, and for now am preliminarily reading about it (with such things as simplicial complexes and the euler characteristic.. I really haven't seen much!)

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