The Proof is Trivial!

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.

Yeah your edit of your first post cleared it up...can you do it using C1 maths and before (so very basic stuff only)?

What is this world where cubic residues mod 19 are quotable?

Well, that was the intended outcome.

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?

Going to go assuming I pass my dissertation.

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')?

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.

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?

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.

Actually I believe ukdragon withdrew his problem

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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.

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.

That has nothing to do with it, ukdragon's problem is 224, the issue arised at 214.

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How fast are you on the stairs?

Problem 230

How many triangles are there?



I think more mathematicians should be interested in cognitive psychology.

Problem 230

How many triangles are there?



I think more mathematicians should be interested in cognitive psychology.

23?

I hazard about 55? ...