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    (Original post by souktik)
    2.i is fine, but I'm not too sure about ii, iii and iv. Again, for 3, you're saying that they are symmetric about x=1 but that's not true.

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    I forget add " - "
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    fk and -f 2-k are symmetric about x=1
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    (Original post by yxcai)
    fk and -f 2-k are symmetric about x=1
    Okay, that's fine, then. I did 3.iv without integration, by noting that the transformation can be achieved by flipping the graph twice, once about x=1 and once about the x axis and this takes the initial shaded area to the final shaded area as well.

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    (Original post by yxcai)
    fk and -f 2-k are symmetric about x=1
    I think I got the same results for 4 and 5.

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    (Original post by souktik)
    I think I got the same results for 4 and 5.

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    Thank you~ I think I can get 32+13+10+13+13=81 or so.
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    (Original post by souktik)
    I could do 1 and 2, I hope I didn't get them wrong. I didn't use open sets for 2, do I have to? :-/
    And I think I've forgotten connectedness, so I had to skip 3. Thanks for sharing the problems, by the way. I feel a little less scared now (though I still don't want my first college interview to also be my first test on things that I've just looked at passively, haha).
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    Sorry, I missed this quote. If you don't use open sets it's just a result from real-analysis, it's just the same proof though using the fact that for f: M \rightarrow N being a map between metric spaces, f is continuous at x if

    \forall \epsilon > 0 \ \exists \delta > 0 \ f(B_M(x, \delta)) \subseteq B_N(f(x), \epsilon)

    Feel free posting how you did the first and second question if you're unsure if you did it correctly or not.
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    (Original post by yxcai)
    I make two mistakes on part 1 so it is hard for me to get 85+. I think you can get 85+
    I'm not too sure about my answers in the first part so it's hard for me to get more than 85
    Though on the bright side the average score of candidates who were interviewed last year was 68-69 I think so I hope it's the same this year as well :P



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    (Original post by yxcai)
    Thank you~ I think I can get 32+13+10+13+13=81 or so.
    That's nice. Are you completely sure about 2, though?


    (Original post by Noble.)
    Sorry, I missed this quote. If you don't use open sets it's just a result from real-analysis, it's just the same proof though using the fact that for f: M \rightarrow N being a map between metric spaces, f is continuous at x if

    \forall \epsilon > 0 \ \exists \delta > 0 \ f(B_M(x, \delta)) \subseteq B_N(f(x), \epsilon)

    Feel free posting how you did the first and second question if you're unsure if you did it correctly or not.
    Posting properly with LaTeX will take me ages, but here are the outlines:
    1. If x and x' aren't equal, then d(x,x')=d>0. Used the standard epsilon definition for both convergences with epsilon < d/2, say epsilon=d/4. So we get, for sufficiently large n, d(x_n,x) and d(x_n,x') are both less than d/4, but d(x,x')=d. This is a clear violation of the triangle inequality for metrics.
    2. Yeah, exactly. The real analysis proof using metrics instead of absolute differences. For epsilon I get delta, for delta, I get delta2. (I was rigorous, but this was the idea.)

    Are there mistakes in my understanding and approach?
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    (Original post by revelry26)
    I'm not too sure about my answers in the first part so it's hard for me to get more than 85
    Though on the bright side the average score of candidates who were interviewed last year was 68-69 I think so I hope it's the same this year as well :P



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    Actually, the average score of candidates who were interviewed last year was about 63, as I recall. It's usually 61-63.
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    (Original post by souktik)
    That's nice. Are you completely sure about 2, though?




    Posting properly with LaTeX will take me ages, but here are the outlines:
    1. If x and x' aren't equal, then d(x,x')=d>0. Used the standard epsilon definition for both convergences with epsilon < d/2, say epsilon=d/4. So we get, for sufficiently large n, d(x_n,x) and d(x_n,x') are both less than d/4, but d(x,x')=d. This is a clear violation of the triangle inequality for metrics.
    2. Yeah, exactly. The real analysis proof using metrics instead of absolute differences. For epsilon I get delta, for delta, I get delta2. (I was rigorous, but this was the idea.)

    Are there mistakes in my understanding and approach?
    My Part2 answer is same with revelry26. What is your answer?
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    (Original post by yxcai)
    My Part2 answer is same with revelry26. What is your answer?
    Wait! The first part is different. Name:  ImageUploadedByStudent Room1384030296.517735.jpg
Views: 85
Size:  121.3 KB.
    It makes sense cause they've given that k=\=1 which is necessary for the above answer


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    Guys when are people invited to interview?
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    (Original post by souktik)
    One more thing. In my PS I probably mentioned that I started studying a bit of real analysis and learnt about things like metric spaces over the summer. They'll understand that I'm just talking about an introduction, right? If they expect me to know that stuff as if I've been taught at Oxford then I'm screwed.

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    I think it was actually smart putting that stuff on. I'm a physics applicant, but I also mentioned in my PS that I knew some post-high school stuff about math (mostly PDEs/Fourier analysis) and physics (QM/Relativity)
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    (Original post by yl95)
    Actually, the average score of candidates who were interviewed last year was about 63, as I recall. It's usually 61-63.
    That's even better


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    (Original post by yxcai)
    My Part2 answer is same with revelry26. What is your answer?
    Spoiler:
    Show
    2.ii.a.
    f(t)-f(1-t)=t (for all t)
    implies: f(1-t)-f(t)=1-t (replacing t by 1-t)
    Add the two equations and you get t+(1-t)=0, that is 1=0.

    2.ii.b.
    Condition: g(t)=-g(1-t)

    2.ii.c.
    I used f(t)=t.(2t-1)^3. In general, f(t)=t.g(t) is a valid example as long as the condition in 2.ii.b. is followed. A simpler example would have been f(t)={\frac{(2t-1)^3}{2}}, or f(t)={\frac{g(t)}{2}} in general.
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    (Original post by ZafarS)
    I think it was actually smart putting that stuff on. I'm a physics applicant, but I also mentioned in my PS that I knew some post-high school stuff about math (mostly PDEs/Fourier analysis) and physics (QM/Relativity)
    It is not smart to do that by any stretch of the imagination. All it does is invite the tutor, who knows more on those basic topics that most undergraduates will by the end of their degree, to test you on it and judge you on how well you can teach yourself the material (and self-teaching is more than required for undergrad. maths and physics) so unless you really do know your stuff when it comes to PDEs/Fourier, you're not likely to impress.
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    (Original post by ZafarS)
    I think it was actually smart putting that stuff on. I'm a physics applicant, but I also mentioned in my PS that I knew some post-high school stuff about math (mostly PDEs/Fourier analysis) and physics (QM/Relativity)
    Haha, no, I disagree. It's unwise. I wrote my PS on the 14th or 15th of October, so I wasn't really thinking. I just knew that I had to get it done.
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    (Original post by Noble.)
    It is not smart to do that by any stretch of the imagination. All it does is invite the tutor, who knows more on those basic topics that most undergraduates will by the end of their degree, to test you on it and judge you on how well you can teach yourself the material (and self-teaching is more than required for undergrad. maths and physics) so unless you really do know your stuff when it comes to PDEs/Fourier, you're not likely to impress.
    Hey, did you notice my previous quote? Here's the post:

    "Posting properly with LaTeX will take me ages, but here are the outlines:

    1. If x and x' aren't equal, then d(x,x')=d>0. Used the standard epsilon definition for both convergences with epsilon < d/2, say epsilon=d/4. So we get, for sufficiently large n, d(x_n,x) and d(x_n,x') are both less than d/4, but d(x,x')=d. This is a clear violation of the triangle inequality for metrics. 2. Yeah, exactly. The real analysis proof using metrics instead of absolute differences. For epsilon I get delta, for delta, I get delta2. (I was rigorous, but this was the idea.)

    Are there mistakes in my understanding and approach?"
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    (Original post by Noble.)
    It is not smart to do that by any stretch of the imagination. All it does is invite the tutor, who knows more on those basic topics that most undergraduates will by the end of their degree, to test you on it and judge you on how well you can teach yourself the material (and self-teaching is more than required for undergrad. maths and physics) so unless you really do know your stuff when it comes to PDEs/Fourier, you're not likely to impress.

    Bullcrap. I doub the tutors are stupid enough the assume I have a graduate level knowledge of PDEs just because I mention it on my PS. What, is he going to ask me about current research? No.

    I actually mentioned on my PS what you also say. I said that my self-studies show that I can self-study subjects, which is imporant. If I am going to be asked to solve certain PDEs using seperation of variables, method of characteristics or integral transform or whatever, I can do it. If I am going to be asked the intuition behind simpler PDEs like the wave equation, I'll happily do it. If I am going to be asked to comment on Terence Tao's latest research in the field, hell nah. But that is not going to happen.

    So I completely disagree. If you know something, flaunt it. You seem to be assuming that me and that other dude don't really know what we mention in our PS. Of course that would be extremely dumb. But we do. I'd be happy if he asked me anything about Fourier series, or elementary set theory or whatever. It means I can distinghuish myself. And in my opinion a tough mechanics problem is not easier than an average problem in Fourier analysis.
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    (Original post by souktik)
    Hey, did you notice my previous quote? Here's the post:

    "Posting properly with LaTeX will take me ages, but here are the outlines:

    1. If x and x' aren't equal, then d(x,x')=d>0. Used the standard epsilon definition for both convergences with epsilon < d/2, say epsilon=d/4. So we get, for sufficiently large n, d(x_n,x) and d(x_n,x') are both less than d/4, but d(x,x')=d. This is a clear violation of the triangle inequality for metrics. 2. Yeah, exactly. The real analysis proof using metrics instead of absolute differences. For epsilon I get delta, for delta, I get delta2. (I was rigorous, but this was the idea.)

    Are there mistakes in my understanding and approach?"
    Yeah sorry, I did see it. Yeah the first is fine, the second is the right idea
 
 
 
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