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    (Original post by Mladenov)
    Firstly, note that the cartesian product satisfies the conditions; it makes the corresponding diagram commute.
    It turns out that, if A= \underline m \otimes \underline n and B = \underline m \times \underline n are two different products, then A \cong B. Let the projective maps be \pi_{1} : A \to \underline m; \pi_{2}: A \to \underline n; q_{1}: B \to \underline m; q_{2}: B \to \underline n. Further, there are two uniquely determined mappings f : A \to B and g: B \to A.
    Therefore,
    \pi_{1} \circ f = q_{1}
    \pi_{2} \circ f = q_{2}
    q_{1} \circ g = \pi_{1}
    q_{2} \circ g = \pi_{2}
    It follows, from the uniqueness of f, that f \circ g = id_{A}; similarly, g \circ f = id_{B}.
    Hence the result.

    Spoiler:
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    Mac Lane is sublime!
    Excellent! Note that in the case where we restrict ourselves to the category with objects consisting only of the sets \underline{\mathbb N_0} then \underline{m \times n} is still an adequate product. Dually, the co-product here is \underline{m + n}. This is yet another example of categorical products being "products" and co-products being "sums".

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    Indeed
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    (Original post by TheMagicMan)
    Yeah your edit of your first post cleared it up...can you do it using C1 maths and before (so very basic stuff only)?
    I am not doing A levels, so I do not know what exactly this course contains.
    But the problem is possible with very basic knowledge; I can attempt, but here is 2:35 AM...

    (Original post by TheMagicMan)
    What is this world where cubic residues mod 19 are quotable?
    (Original post by jack.hadamard)
    Well, that was the intended outcome.
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    (Original post by jack.hadamard)
    Category theory, essentially, labelled A-level knowledge? I wish Mac Lane and Eilenberg could see this.

    So, are you already at Cornell or are you about to go there?
    As when we had the debate yesterday, all you need to know to do the question is A-level, I didn't say it'd be elegant. I'm sure Mac Lane et al. would be very pleased if category theory was taught in school.

    Going to go assuming I pass my dissertation. :nothing:
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    (Original post by ukdragon37)
    Going to go assuming I pass my dissertation. :nothing:
    Do you want to go to Cornell? I kind of get the feeling that you are not very enthusiastic about it. I may be wrong, though. In any case, why do you get the thing done so you don't have to worry about 'passing' it (you know, 'didn't pass it' closes more doors than 'didn't want to go')?
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    (Original post by jack.hadamard)
    Do you want to go to Cornell? I kind of get the feeling that you are not very enthusiastic about it. I may be wrong, though. In any case, why do you get the thing done so you don't have to worry about 'passing' it (you know, 'didn't pass it' closes more doors than 'didn't want to go')?
    It's just Cambridge is so good and leaving for something that is potentially a worse experience (but also it might be an even better/eye-opening one) creates apprehension.

    I'm just demotivated (and you can't not feel rather tired at this stage of the game, where four years of mind-ravishing education have gone past) because it's pages upon pages of rather dry maths and I don't need to do very well in the dissertation to get a good grade overall.
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    (Original post by Lord of the Flies)
    The numbers are mixed up, there appears to be two problems 214. Starting from the second 214 (your series Jkn), could you +1 to each question/solution number? I have amended mine.
    You are quite right. Jkn is responsible for this plight.


    (Original post by Jkn)
    Oops there was a few typos! Especially in having f(x)=0 in the last paragraph but not at the end! Is it fine now?
    I have noticed the other flaws. My point, though, is still valid.
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    (Original post by ukdragon37)
    It's just Cambridge is so good and leaving for something that is potentially a worse experience (but also it might be an even better/eye-opening one) creates apprehension.
    Well, to reiterate, do it as best as you can (despite it being dry and annoying). Then, decide what is best for you. I assume Cornell are not going to chase you with a stick and complain too much, in case you decide not to go. Benefits of completing it: you will be viewed with better eyes when you say you want to stay at Cambridge; you will have a piece of work already done.
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    (Original post by jack.hadamard)
    Well, to reiterate, do it as best as you can (despite it being dry and annoying). Then, decide what is best for you. I assume Cornell are not going to chase you with a stick and complain too much, in case you decide not to go. Benefits of completing it: you will be viewed with better eyes when you say you want to stay at Cambridge; you will have a piece of work already done.
    :lol: Thanks, but staying at Cambridge is only a very remote option - it'd require much humiliating begging and backpedalling on all the preparatory stuff I've done for Cornell. I'm very much set on going (got my visa sorted and everything) but this last hurdle is just so annoying.
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    (Original post by Jkn)
    Actually I believe ukdragon withdrew his problem
    That has nothing to do with it, ukdragon's problem is 224, the issue arised at 214.

    (Original post by TheMagicMan)
    ...
    Glad to see you back. :cool:
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    (Original post by Lord of the Flies)
    Solution 224

    The queue to the lift of the Eiffel tower does not take 6 minutes, it takes 2 hours. Thus LotF never gets to the top, people get injured from the falling coins, banarama gets arrested, and LotF goes to a nearby café to enjoy a glass of 1981 Petrus.
    How fast are you on the stairs?
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    (Original post by TheMagicMan)
    This technically isn't solvable for any function u(x). You need certain conditions on the integrability of u to hold (something like u continuous is sufficient but not necessary).

    I believe a necessary and sufficient condition would be something along the lines of having a measure 0 set of discontinuities on the domain of y in the solution.
    whoops.. u continuous, u(0)=u'(0)=0.

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    (Original post by Lord of the Flies)
    That has nothing to do with it, ukdragon's problem is 224, the issue arised at 214.
    (Original post by Jkn)
    ...
    We all have our moments (Even if mine are frequent)

    Edit: Exam recently done Gets hideous answer....wonders for minutes why....can't add.
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    (Original post by MW24595)
    How fast are you on the stairs?

    I've been to the Eiffel Tower, and it doesn't take 2 hours. It takes 2n hours, where n = the number of coins dropped from the top due to bananarama's shenanigans. :cool:
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    Problem 228**

    Calculate \large \sum_{n=4}^{\infty}\frac{1}{n H_nH_{n-1}}, where H_n=\sum_{k=1}^{n}\frac{1}{k}.

    Hint:
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    The answer's a rational number.
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    Problem 229 *
    Find \int^e_1 \ln(1+lnx)\ dx giving your answer as a summation.
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    Note I think that it can be solved with just A-Level knowledge, I can't check my solution
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    Solution 229

    In the given range, \begin{aligned} \displaystyle \int_{1}^{e} \ln (1+ \ln x)dx = \sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n} \int_{1}^{e} (\ln x)^{n}dx = e\sum_{n=1}^{\infty} (-1)^{n+1}(n-1)!\sum_{k=0}^{n} (-1)^{n-k}\frac{1}{k!} \end{aligned}
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    Problem 230

    How many triangles are there?

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    I think more mathematicians should be interested in cognitive psychology.
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    (Original post by jack.hadamard)
    Problem 230

    How many triangles are there?

    Name:  triangle.jpg
Views: 291
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    I think more mathematicians should be interested in cognitive psychology.
    23?

    I'd be surprised if that's right. I lost count a billion times.
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    (Original post by jack.hadamard)
    Problem 230

    How many triangles are there?

    Name:  triangle.jpg
Views: 291
Size:  38.5 KB

    I think more mathematicians should be interested in cognitive psychology.
    I hazard about 55? There are some really nice patterns in triangular numbers in there.
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    (Original post by bananarama2)
    23?
    It's significantly more than that. Roughly, people get this wrong 95% of the time.

    (Original post by MW24595)
    I hazard about 55? ...
    Nope.
 
 
 
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