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May
11
comment Identity of tensor products over an algebra
This answer sounds promising. Can you please elaborate upon it? It seems like you're appealing to certain universal properties I might not be familiar with.
May
11
comment Identity of tensor products over an algebra
Yes, sorry I forgot to mention that.
May
11
revised Identity of tensor products over an algebra
added 1 character in body
May
11
revised Identity of tensor products over an algebra
added 34 characters in body
May
11
asked Identity of tensor products over an algebra
Feb
12
revised Quotient of a category by equality in Grothendieck group
edited title
Feb
10
asked Quotient of a category by equality in Grothendieck group
Dec
14
comment Is it possible to find the $n$th digit of $\pi$ (in base $10$)?
If you believe that we are free to define the values of a function $\mathbb{N} \to \{ 0, \dots , 9 \}$, then you already believe the function exists. In fact, your belief in the function is implicit in the first indented equations. Just because you can't list the function's values (either because you don't know them or you don't have enough time) doesn't mean it doesn't exist.
Dec
14
comment Is it possible to find the $n$th digit of $\pi$ (in base $10$)?
There's nothing wrong with defining a function using words, $f(n) = n$th digit of $\pi$. So certainly there is a function for what you want. Perhaps you're asking for a formula -- but then again you mentioned you don't mind whether we can "explicitly state it". So it's not clear what you're after.
Dec
10
awarded  Popular Question
Oct
24
accepted Ext groups for fraction field and a module annihilated by an element
Oct
13
comment Infinite staircase to a circle
I'm satisfied that $x_\infty$ is not a remarkable number with a simple closed form. Of course I'd welcome more information on its properties, and maybe the previous sentence is actually in error (despite my checks at the OEIS), but I think viewing things from the perspective of $1/k$ suggests it's just "some number" in a certain sequence converging to $1/\sqrt{2}$.
Oct
13
comment Infinite staircase to a circle
(I'll award the bounty when it becomes possible to do so.)
Oct
13
accepted Infinite staircase to a circle
Oct
13
comment Infinite staircase to a circle
This is a great answer, thanks. I guess heuristically it explains why we get so close to $\sqrt{2} = 2 \cdot \frac{1}{\sqrt{2}}$ when adding up the co-ordinates for the $k = 2$ case.
Oct
12
comment Infinite staircase to a circle
It's interesting that if you add those numbers up it's scary close to $\sqrt{2}$.
Oct
11
comment Infinite staircase to a circle
Hi guys, thanks for all your comments. I think the most concrete progress has been Paul's provision of the recurrence relations. Can anyone see if they might be solvable analytically?
Oct
10
awarded  Nice Question
Oct
10
revised Infinite staircase to a circle
edited tags
Oct
10
comment Infinite staircase to a circle
Not sure what you mean by rush the border... If you draw a picture it's visually apparent that you'll approach some point on the circle. The distances you travel in each step get shorter and shorter.