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2h
comment Deducing factorization over $\mathbb{F}_p$ from factorization over $\mathbb{Q}$
If we take the title literally, then the scene will be different. A factorization in $\Bbb{Z}[x]$ will give a factorization over the residue class fields, but the process is not reversible. Hensel's lemma tells us to what extent the reverse direction works when we go up from $\Bbb{F}_p$ to the $p$-adic domain.
3h
comment Opening MATLAB in Methmatica
This is off-topic here. We do allow questions about computer algebra systems, but they should still focus on mathematics - not coding or built-in functions. You can search at Mathematica.SE if they have anything that would help you. Alternatively you can ask at a coding site (like StackOverflow).
3h
comment Algebraic independence of `Riemann-Roch' elements
Pole order = the negative of valuation at $P$.
3h
comment Algebraic independence of `Riemann-Roch' elements
$x$ has a pole of order 2 at $P$, $y$ has one of order 3, $x^2$ of order $4$, $xy$ of order 5, and both $x^3$ and $y^2$ have a pole of order six. Consider a linear dependency relation such as $$c_1\cdot1+c_2\cdot x+c_3\cdot y+c_4\cdot x^2+c_5\cdot xy+c_6\cdot x^3+c_7\cdot y^2=0.$$ The r.h.s. has no pole at $P$, so the l.h.s. cannot have one either. Because a pole of lower order cannot cancel one of a higher order either here $c_6=c_7=0$ or they are both non-zero. The former possibility forces all the $c_i$ to zero (there pole orders are distinct), which is useless.
4h
answered Deducing factorization over $\mathbb{F}_p$ from factorization over $\mathbb{Q}$
4h
comment Deducing factorization over $\mathbb{F}_p$ from factorization over $\mathbb{Q}$
If I got the gist of your question it is more about relating factorization over extensions of $\Bbb{Q}$ to factorizations over extensions of $\Bbb{F}_p$.
5h
comment Probability of Detection and pulse-pulse decorrelation time
@Rob. Thanks. Moving to DSP.
5h
comment Probability of Detection and pulse-pulse decorrelation time
@Dilip. Thanks. DSP it is.
9h
comment Algebraic independence of `Riemann-Roch' elements
Tim: Those two terms are the only ones not in $\mathcal{L}(5P)$. That space is 5-dimensional, and spanned by $1,x,y,x^2,xy$ (those all have different pole orders so they are automatically linearly independent). Therefore any linear dependency relation must involve both of $x^3$ and $y^2$. Or, if you prefer to think about it that way, those two must cancel the pole of order six produced by the other - the lower order poles are useless for that purpose.
9h
answered Hamming distance of a CRC
10h
comment Algebraic independence of `Riemann-Roch' elements
Hmm. Don't we have $\ell(1P)=\ell(0P)=1$, so $\mathcal{L}(P)$ is spanned by the constant mapping to $1$? $\mathcal{L}(2P)$ is the smallest space containing a non-constant mapping, call it $x$. Then as $\dim \mathcal{L}(3P)=3$, there is another function $y$, linearly independent from $1$ and $x$, with its only pole of order 3 at $P$. We then see that $x^2$ has a pole of order $4$, $xy$ a poly of order $5$ and both $x^3$ and $y^2$ of order six. Because $\dim \mathcal{L}(6P)=6$, the seven functions $1,x,y,x^2,xy,x^3,y^2$ must be linearly dependent, and this gives us an equation for the curve.
11h
revised Find $\lim_{x\to e}\frac{x^{e^x}-e^{x^e}}{x-e}$ without L'Hospital
edited body
11h
comment What is the Max value of n when 185! is divided by (189^n) will give an Integer Value?
Nothing wrong with your approach. Nor with the result. Basically you were correctlt applying this recipe. Adding the link in case some future reader needs an explanation.
14h
comment Probability of Detection and pulse-pulse decorrelation time
@DilipSarwate: I get a vibe that our sister site on statistics might be even better than DSP? You know this better, what do you think?
14h
comment Is there any survey paper or book for “Word Problem”?
If Wikipedia is not good enough a starting point for you, may be you should search in math.sci.net?
16h
revised What is the maximum value of $M$ when $T$ is set of $\{2,4,8,16,… 2^n\}$ and $S$ is subset of $T$ by given conditions
edited tags; edited tags
16h
comment What is the maximum value of $M$ when $T$ is set of $\{2,4,8,16,… 2^n\}$ and $S$ is subset of $T$ by given conditions
Depends. Do you mean that sums of pairs of elements may not exceed $2^n-2$, or is the bound $2^{n-2}$? Your numerical example suggests the former, André (among others) thought you mean the latter. You should clarify (and make the distinction clear in your notation). Towards the end it becomes gibberishy. Why are you suddenly including 6?
17h
revised Is it possible for a matrix to have nullity different from its transpose?
edited title
22h
awarded  Nice Answer
1d
comment Limit of a sequence of holomorphic functions
I don't know. I am hoping that if $f$ has $m+1$ zeros, then over some nice path $C$ we have $$2\pi i(m+1)=\int_C\frac{f'(z)}{f(z)}\,dz=\lim_{n\to\infty}\int_C\frac{f'_n(z)}{f_n(z)‌​}\,dz.$$ Implying that for all large enough $n$ the function $f_n$ also has at least $m+1$ zeros inside $C$. But it's been a while since I did anything like this, so it is not clear to me how to justify all of the above.