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Qiaochu Yuan's user avatar
Qiaochu Yuan's user avatar
Qiaochu Yuan's user avatar
Qiaochu Yuan
  • Member for 15 years, 2 months
  • Last seen this week
  • Oakland, California, USA
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What are examples of good toy models in mathematics?
Well, I wouldn't really know; feel free to edit that one or write an answer.
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Alternatives to pi day
I hardly think this is any less numerological than a fixation on the digits of pi.
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A learning roadmap for algebraic geometry
I like the use of toy analogues. Personally, I don't understand anything until I've proven a toy analogue for finite graphs in one way or another.
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Characterizing triangles unembeddedly
I don't understand. The whole point of the manifold definition is that we want to ignore some information and preserve other information. If you want a notion of angle, you should be working in an inner product space.
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Is there a matrix whose permanent counts 3-colorings?
Have you tried using the deletion-retraction recurrence? Is there an analogue of expansion by minors for the permanent?
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L_p norm balls for 1<p<2 - is it always similar to an L_q norm ball for some q>2?
Your initial observation is more or less a coincidence. Look at what happens in 3D.
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Is every norm in R^n a continuous function?
I would've forgiven you if you'd entered in different body text, but as it stands this looks like a homework question.
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Is ${\rm S}_6$ the automorphism group of a group?
There's a stronger result sciencedirect.com/… for countable groups.
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Given a sequence defined on the positive integers, how should it be extended to be defined at zero?
Fair enough. I do like that you mentioned universal properties, since my own response to this question is basically "categorify until it becomes obvious what to do." For example, the product of zero things in a category is a terminal object and the coproduct of zero things is an initial object.
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Can the objects of every concrete category themselves be realized as small categories?
Hmm. I meant Ring, but I may have to take that back. It looks like the standard category-theoretic description of a ring is as a monoid object in Ab, so we need to require that the functors preserve the monoid structure.
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Given a sequence defined on the positive integers, how should it be extended to be defined at zero?
Poonen claimed, and I agree, that the determinant of a 0x0 matrix should be equal to 1. Consider what happens when you try to expand the determinant of a 1x1 matrix by minors.
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Given a sequence defined on the positive integers, how should it be extended to be defined at zero?
This may also sound lame, but how do you know you're looking at the right properties?
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Given a sequence defined on the positive integers, how should it be extended to be defined at zero?
I would argue as follows. If you're a combinatorialist who accepts that 0! = 1, you accept that there is one bijection from the empty set to itself, so you accept that there is one function from the empty set to itself, so you should accept that 0^0 = 1.
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Can the objects of every concrete category themselves be realized as small categories?
Darn. You probably understand why I asked this, though: this seems to be true of the standard concrete categories students are first introduced to, such as Grp, Rng, Top, ... are these categories characterized by a stronger property than concreteness?
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