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Jun
22 |
awarded | Nice Question |
Mar
18 |
awarded | Nice Answer |
Nov
9 |
awarded | Popular Question |
Jul
2 |
awarded | Curious |
May
10 |
revised |
Radon-Nikodym derivatives as limits of ratios
More information |
Jan
10 |
asked | Radon-Nikodym derivatives as limits of ratios |
Nov
21 |
comment |
Do good math jokes exist?
I just told this joke at a conference, except I flubbed it and said it was about gears with uncountably many teeth. Five minutes later, I found two of the listeners arguing about slippage in an explicit construction they had come up with. |
Oct
25 |
accepted | Inverses of two-argument functions with respect to one argument |
Oct
25 |
comment |
Inverses of two-argument functions with respect to one argument
Okay, it's clear that these don't have a name beyond "finite cyclic group generated by inverting w.r.t. one argument," so I'm going to accept this answer. Thanks for the pointers and ideas! (If you're curious, these inverses have a meaningful use: computing explicit preimages of intervals under $f$.) |
Oct
25 |
comment |
Inverses of two-argument functions with respect to one argument
Cool, I hadn't made the connection to Latin squares. I did, however, notice that the notation and argument order could be improved. If we have $f_1 : A \times B \to C$, $f_2 : B \times C \to A$ and $f_3: C \times A \to B$, then the functions form a finite cyclic group generated by $f_1$ and "invert by holding the second argument fixed". (This holds for any cardinality.) I'm pretty sure it generalizes. I'm also pretty sure that I'll have to consider more general notions of inverse. |
Oct
23 |
asked | Inverses of two-argument functions with respect to one argument |
Oct
10 |
comment |
Why worry about the axiom of choice?
Another fine equivalence to AC: every set has a unique cardinality. This and your first example are the main things that convince me AC is not so strange. |
Aug
6 |
answered | Hyperrectangle partition of set of overlapping hyperrectangles |
Jun
9 |
comment |
Equivalent Markov Random Fields
MRFs can encode any Bayesian network, so yes, they can. It's been a while since I did any of this, but I seem to remember that, additionally, every MRF has a canonical, normalized form (as long as all potentials are nonzero) based on cliques. |
Feb
20 |
comment |
What algebraic structure is suitable for representing self-referential data types?
This question should be at cstheory.stackexchange.com. |
Jan
9 |
comment |
Usage of set theory in undergraduate studies
I have first-hand experience in teaching 2+2 from foundations, to an 11-year-old. The trick is to explain sets as "boxes" that contain "other boxes". (You can even use actual boxes!) As soon as the child can conceptualize a box that contains "all the finite boxes" (omega), she can come up with "infinity plus one" on her own. It involves a lot of careful wording and repetition, but it's totally doable, without scary-looking notation. |
May
12 |
awarded | Necromancer |
Apr
22 |
answered | Set-Theoretic Issues/Categories |
Apr
19 |
comment |
Can infinity shorten proofs a lot?
Technically, ZFC doesn't have induction until the axiom of infinity is added. Point against this line of argument: most mathematicians don't really work in ZFC, but in a pidgin higher-order logic with sets, in which induction is probably axiomatic. Point for this line of argument: it probably matches the audience better, who (I would guess) don't distinguish between the potential infinity of induction and the completed infinity of the naturals. |
Mar
7 |
comment |
When are iterated limits of sets equal to a double limit?
Ah, of course. Then I'm looking for an analogue of uniform convergence. I'd like to do this without a measure if I can, so I don't want this to hold almost everywhere just yet. I suppose I gave the $\sigma$-algebra just in case it could help somehow. (I doubt it can, though.) |