4,115 reputation
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location Cambridge, United Kingdom
age
visits member for 4 years, 4 months
seen Sep 24 at 13:41

Researcher in quantum information --- involving algebras over finite dimensional Hilbert spaces, graphs and combinatorics, and vector spaces over ℤ/pℤ.

I habitually edit and re-edit anything I write, as long as I have the time, interest, and ability. I should probably apologize for this tendency, but you will probably have to be satisfied with being warned about it instead.


Sep
30
awarded  Explainer
Sep
24
awarded  Autobiographer
Jul
29
awarded  Yearling
Jul
2
awarded  Curious
Mar
21
comment Are transformations which preserve module-vectors with trivial annihilators monic?
Finiteness of the module, that is.
Mar
21
comment Are transformations which preserve module-vectors with trivial annihilators monic?
@rschwieb: I had not been thinking of integral domains, so when I saw you invoking them in your answer I knew at once that I had missed conditions... but also (perhaps presumptuously, but maybe not so much given your rep score) supposed that you hadn't needed much time to formulate your answer. Otherwise I would have been more hesitant to modify my question.
Mar
21
comment Are transformations which preserve module-vectors with trivial annihilators monic?
@rschwieb: Sorry --- I had forgotten to specify conditions which I was taking for granted. I've rolled back my revision, and I think I have found an approach for my revised question anyway, thanks.
Mar
21
revised Are transformations which preserve module-vectors with trivial annihilators monic?
rolled back to a previous revision
Mar
21
revised Are transformations which preserve module-vectors with trivial annihilators monic?
Edited to add conditions on the ring
Mar
21
asked Are transformations which preserve module-vectors with trivial annihilators monic?
Mar
19
awarded  Good Answer
Mar
15
comment Should an undergrad accept that some things don't make sense, or study the foundation of mathematics to resolve this?
@Charles: indeed. Any one set of objects is easy to accommodate in a model, but as with embedding $\mathbb R$ into $\mathbb C$, as soon as you want to have one more embedding without extra bureaucracy, you have trouble. As the activity of mathematics is creative and progressive, considering a fixed model which is likely to be eliminated by refinements isn't so productive.
Mar
15
awarded  Nice Answer
Mar
14
comment Should an undergrad accept that some things don't make sense, or study the foundation of mathematics to resolve this?
I thought to address the question of foundations which you touch on in your question. If you repeatedly get hung up on such details, this is the most important thing to know. You should accept someone else's answer if you find that my answer is tangential, but thanks for the compliment!
Mar
13
comment A puzzle on orthogonal matrices modulo $p$
True, but a sum of squares anyway. Don't sweat it. I wondered whether such quaternionic techniques (and further Cayley-Dickson extensions for higher dimensions) hid subtle and powerful ideas. Clearly they allow trivial construction of orthogonal vectors from any single vector (where the $2\times2$ case is familiar from first year linear algebra); clever techniques for the determinant seemed plausible, but in the absence of any such techniques for the $2\times2$ case, perhaps the determinant-expansion observation is about the best one can hope for.
Mar
13
comment A puzzle on orthogonal matrices modulo $p$
Actually, for a quaternion $q = a + bi + cj + dk$, the determinant should be $\pm\mathrm{N}(q)^2 = \pm(a^2+b^2+c^2+d^2)^2$ by a geometric argument. I suppose that inspection of the expansion of the determinant shows that the sign is $+$. (This is a sum of four squares using the fact that $\mathrm{N}(q)^2 = \mathrm{N}(q^2)$, but not of the first column.) I wondered whether positivity follows from the handedness of the sequence of quaternionic units in the construction (it obviously is determined by it, but a simpler argument that $(i,j,k)$ gives $+$ would be nice). Does it?
Mar
13
comment A puzzle on orthogonal matrices modulo $p$
A follow-up question. Is there an easy way to see what the determinant of this matrix will be (is it special orthogonal)?
Mar
13
revised Should an undergrad accept that some things don't make sense, or study the foundation of mathematics to resolve this?
elaborated somewhat
Mar
13
comment Should an undergrad accept that some things don't make sense, or study the foundation of mathematics to resolve this?
@JacobWakem: I suppose that by "form" you mean something suitably Platonic (otherwise, I could easily substitute 'form' for 'properties', and we're back on the slippery slope to the notion of mathematical "interfaces" outlined above). Does this mean that all of our usual constructions of the reals are akin to our constructions of chairs, in that we can by construction but imitate a dimly recalled/perceived shadow of the perfect form, which evokes but falls short of true "chairiness"? A true but unknowable definition of the reals is not likely to satisfy the OP very much...
Mar
13
revised Should an undergrad accept that some things don't make sense, or study the foundation of mathematics to resolve this?
minor revision