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Nov
25
comment Finding the smallest decision tree of a Boolean function
I think learning an optimal decision tree is an NP hard problem, but the standard algorithms (id3. C4.5) will get you pretty close
Nov
25
comment Nonzero polynomials of degree $n$ in $D[x]$ have at most $n$ distinct roots in $D$, assuming $D$ is only an integral ring.
you are right, that example alone is a counterexample to my assertion. I obviously haven't though about it deeply enough.
Nov
24
comment does the $\zeta_K$ function of a function field determine the genus of that function field?
yes, of course. I don't know I didn't say that despite taking the time to write it out!
Nov
24
comment Basis in R4 with only two vectors in the set
linear independence means they must not lie on the same line. most vectors in $\mathbb{R}^4$ do not lie on the line formed by your two vectors, so picking randomly is a pretty good method.
Nov
24
comment Nonzero polynomials of degree $n$ in $D[x]$ have at most $n$ distinct roots in $D$, assuming $D$ is only an integral ring.
it might be true, depending on which $x$ and $a$, but in general, no. But that is OK, so long as the coefficients of $f$ are on the same side of the indeterminate $x$. Then you define the matrix multiplication so its consistent with evaluation of the polynomial, and everything follows. The point is that in a noncom ring, you are OK so long as you do everything all on the right side. or all on the left side. just don't go swapping the two around.
Nov
24
comment Nonzero polynomials of degree $n$ in $D[x]$ have at most $n$ distinct roots in $D$, assuming $D$ is only an integral ring.
I am not assuming $D$ is commutative. But I might have swapped the order of multiplication around because I am sloppy. The result still holds whether or not $D$ is commutative.
Nov
24
comment Why is abelianness such a precious property?
not a typo, actually, but me not remembering properly all my facts about finite abelian groups. :)
Nov
23
comment I think I'm making an error in my implementation of least squares but I'm not sure (python).
Actually I need to correct my comment - you are correct, that formula is for both multivariable linear regression and single-var polynomial regression. The latter bit about weirdness and code not being defined well enough still stands.
Nov
23
comment I think I'm making an error in my implementation of least squares but I'm not sure (python).
the picture you posted shows polynomial regression, while the formula you posted looks like linear regression. Perhaps you are expecting something different? it's certainly not an analog<->digital thing, the computer can cope just fine. But you haven't explained enough about your code or your weirdness for me to figure out what is up.
Aug
18
comment Difference between $f(.)$ over $\mathbb{Z}_q$ and $f(x)$ for $x \in\mathbb{Z}_q$
some context, id est, the book you see this in, would be appropriate. Notation means whatever the author intends it to mean.
Aug
14
comment What, and how can, topological invariants can be computed from a space's algebra of functions?
do you have a reference for the proof of this fact? Or shall I start hunting through Connes' tome?
Aug
3
comment profinite completion of a ring of integers in a global field
A "yes" to a yes/no question is sufficient. But the topological point is not clear to me.
Jan
26
comment Category of $\textbf{Ring}$.
it's sometimes easier to have a convention that excludes slightly trivial or slightly pathological examples. Dummit and Foote just take a ring to mean something with a unit, they never deal with unit-less rings, and it's extra baggage to carry that around when you aren't using it.
Jan
26
comment Univariate and Matrix Representation of Affine Transformation
the map $\phi$ is implicit in what I'm doing, it's essentially a change-of-basis matrix. if you let $e_i$ be the canonical basis (a column with all 0s except on the i-th row, that's a 1) for $\mathbb{F}^n$, then $\phi : \mathbb{F}^n \to \mathbb{E}^n$ by $e_i \mapsto \{\text{some expression in the t basis I gave}\}$ then it'll do the trick.
Jan
25
comment Univariate and Matrix Representation of Affine Transformation
I'm not 100% sure I'm correct on the last step, but hopefully you see the idea now? :)
Jan
25
comment Application of Category Theory
you might look at these questions: math.stackexchange.com/questions/312605/… and mathoverflow.net/questions/19325/… (heavier maths there)
Jan
14
comment Most efficient way to find distinct complementary subspaces over a finite field
is there an algorithmically efficient way to complete $\{v_{n-k+1},\ldots, v_n\}$ to a basis $\{v_1,\ldots,v_n\}$. Your description solves the problem wonderfully (though I no longer need such a solution!), but I might be too silly to see an way to complete the basis.
Jan
14
comment Most efficient way to find distinct complementary subspaces over a finite field
you're correct, my bad. changed question.
Dec
31
comment Is there a discrete version of non-commutative geometry (yet)?
you might be interested in a paper from Majid arxiv.org/abs/1011.5898 Noncommutative riemannian geometry on graphs. This is not NCG in the manner of connes, though.
Dec
9
comment finding fixed points of frobenius endomorphism of a ring of char > $p$ (not a domain)
this is a very good answer. but it leaves me with a lot of nagging questions: why does the number of cycles = the number of distinct irreducible factors of $x^{p^n} -x$? Why does the number of cycles of length $d$ = the number of such factors of degree $d$? I suppose these things deserve separate questions and I should accept your answer - is that the proper etiquette here?