Let $\mathbf{x}\in\mathbb{R}^n$. Then $\langle \mathbf{x}, \mathbf{x} \rangle = \lVert \mathbf{x} \rVert^2$. If $A\in\mathbb{R}^{n\times n}$, what would be the $A\langle \mathbf{x}, A\mathbf{x} \rangle$? cc @DanielFischer, @robjohn [Thanks!]
@G.T.R The minimum wage for a teacher - even without any diploma such as BSc - is about 70'000€ divided into 13 months before taxes - that is working full-time.
@G.T.R I'm a bit shocked to find out to be honest. After all, when I was in Paris, stuff didn't seem to be cheaper at all, it tended to be the other way around to be honest.
(and I would expect living expenses to be way lower if wages differ by so much)
If anybody's good with mathematica, I could use some help. I want to count the number of ordered pairs of 3x3 matrices over F_2 which are both nilpotent and commute with each other. my formula gives 344 pairs, a different formula gives 400.
@G.T.R Not going to happen. It's kind of reasonable, and it works (in the Netherlands). So absolutely zero chance that the Germans will even seriously ponder how they could introduce such a model.
@nullgeppetto $\langle \mathbf{x}, A\mathbf{x}\rangle$ is a scalar. So $A\langle \mathbf{x}, A\mathbf{x}\rangle$ would be a matrix. Might however be a typo and $\langle A\mathbf{x}, A\mathbf{x}\rangle$ was intended.
@DanielFischer Are the Germans not reasonable as a whole ? Also it seems according to the figures above that teaching is well-paid in Germany, lucky you
@BalarkaSen yup ... :) but didn't want to use PNT if possible .. all I wanted to do is just use the definition of $\pi(n)$ as the functions that counts primes .. the rest is just real-analysis :P
@r9m Equality is highly unlikely. Any indefinite integral with rational integrand is in the differential extension $\Bbb C(x, \log)$ while the logarithmic integral is not in there (and you can't approximate it arbitrarily well either, $\log(x)$ always grows like $o(s^\varepsilon)$), yet with an error of $\ll \sqrt{x}\log(x)$ (assuming RH). I do believe there is a differential approximation theorem somewhere.
@r9m I stay out of real analysis whenever possible.
@G.T.R The Germans, so-so. The German politicians - unreasonable beyond any hope of redemption. It is holy dogma for the CSU that family means the wife stays home and takes care of the children (you know, they had child-care in the GDR, so child-care is TEH EVIL), and a large part of the CDU shares that dogma. The reasonable part of the CDU never dares challenge these dogmas.
What I am saying is that Li is not an elementary integral (in it's indefinite form) while a rational integrand integral is. And you can't approximate the integrand in Li integral too well with rational function as $\log(n)$ grows like $o(n^\varepsilon)$ (read up [asymptotic notations + little o])
@r9m This might just be an overkill but it actually would answer a more general question about the order of $|\pi(x) - \int_1^{\log(x)} P(x)/Q(x) dx|$
@DanielFischer that was a typos... I need $\langle A\mathbf{x}, A\mathbf{x} \rangle$, or $\langle \mathbf{x}', \mathbf{x}' \rangle$, where $\mathbf{x}'=A\mathbf{x}\in\mathbb{R}^n$. Thanks!
@r9m I am not sure, I just sprouted something off the top of my head.
I think I have read up a question in MSE concerning a link between differential galois theory and number theoretic approximations. I can give you the link if you want.
@G.T.R We have various brands voicing their concern, but that doesn't mean they have influence. (Although, the situation improves a bit since the EU forces some degree of equal treatment. But tradition is a strong force.)
@r9m Just get yourself Alekseev to learn galois theory/diff galois theory/algebraic geometry all at once from basic groundworks. It's a compilation of Arnold's notes. He taught some highschool kids in paris about galois theory.
So it depends not on your knowledge but your interest.
@JohnDoe I am looking. The first equality (you have a typo in your transcription, by the way) is just the parallelogram identity. The second, I don't see yet.
In SE network we have math.SE and there is also MathOverflow for research-level questions. Plenty of questions about mathematics are also asked and answered on Quora.
I would like to know what are the differences between Quora and StackExchange, especially from the viewpoint of someone asking an...
Ok, another quickie: $\displaystyle \int_{a}^{b} f^2(x) \, \mathrm{d}x$ = $\displaystyle \int_{a}^{b} f^4(x) \, \mathrm{d}x$ = $\displaystyle \int_{a}^{b} f^3(x) \, \mathrm{d}x$. Find all possible $f$ if its continuous on $[a,b]$.
Let $\{a_0<a_1<...\}$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n \geq 1$ such that $\displaystyle a_n<\frac{a_0+a_1+...+a_n}{n} \leq a_{n+1}$
@MatsGranvik @robjohn @AlexanderGruber You guys I think have CAS knowledge that might help me out. I need to know the number of ordered pairs of matrices mod 2 which are both nilpotent and commute. my formula gives 344, a different formula gives 400, I want to know which is correct. any help?
@DanielFischer, you're right! Sorry! Let $\mathbf{x}\in\mathbb{R}^n$. Then $\langle \mathbf{x}, \mathbf{x} \rangle = \lVert \mathbf{x} \rVert^2$. If $A\in\mathbb{R}^{n\times n}$, what would be the $\langle A\mathbf{x}, A\mathbf{x} \rangle$?
@BalarkaSen two cases: $p$ divides $a$ and $p$ and $a$ are coprime. I can deal without trouble with the first case, but for the second case, I can only prove that $a=b \pmod p$
@r9m It only cites being integrable, and seems to indicate that is all that is needed. the continuity is necessary. The question does not ask which constant functions are possible, so I won't fault him the fact that he doesn't say it has to be $0$ or $1$.
@robjohn for $f \in C^n[a,b]$, with $f(a) = f(b)$, is the following inequality true ? :o $$\left|\int_a^{\frac{a+b}{2}}f(x)dx-\int_{\frac{a+b}{2}}^bf(x)dx\right|\leq\frac{(b-a)^{n+1}}{(n+1)!2^n}\sup_{x\in [a, b]}|f^{(n)}(x)|$$
Ack... in another question, another answerer, who answered one minute before I did, but with a wrong answer, deleted his answer and fixed it after seeing mine, gets upvoted. Perhaps my answer looks too simple. I don't understand people.
@r9m the fourth derivative of a cubic is $0$, is it not?
@robjohn yes !! .. I got that :-) .. you proved that in your answer :) .. I was looking at $\sin x$ in $[0,2\pi]$, the LHS = $2$, but the RHS $\to 0$ for large $n$ :D
Is somebody willing to test whether this link works for someone who does not have an account on Quora. (Whether all answers are visible without having to register/log in.) And whether there are some problems with other links from my post on meta: meta.math.stackexchange.com/questions/16243/…
@robjohn Senko accepted your answer(meaning he atleast saw the answer) and didn't upvote mine(the only upvote was yours) .. (possibly didn't even read it at all :P lol), I should have removed my initial calculations and the mechanism of getting the polynomials and presented the two cubics from the start and showed how clever I was :P XD
@r9m Yeah, that seems to be the way to go. As I remember, I showed too much of the thinking behind my answer on another question and it worked against me, there.
@r9m Heh... I just read your deleted comment, which I think is talking about a similar episode. :-)
@MartinSleziak This is the first I've heard of it.
@nullgeppetto $\lVert A\mathbf{x}\rVert \leqslant \lVert A\rVert\cdot \lVert \mathbf{x}\rVert$ holds for many matrix norms (generally, one only considers matrix norms satisfying that). I have no idea which one is used in that context.
@BalarkaSen I am not contradicting that... I was just saying that so many reals are transcentdent (almost all reals are transcendent), but it is so hard to show which ones are (they are hidden really well)
@DanielFischer I started writing a file here writelatex.com/1262185tdqxpr#/3065150 Don't worry about the fontsize (20). (1)how can I add some blank space between "Introduction"and the theorem that follows ? (adding \\ returns an error). (2) I want to have the "Il existe" aligned with the "Soit $f$"
@G.T.R My connection is acting up, it loads really slowly. I can't see anything yet. Is it vertical whitespace that you want? How about simply adding a blank line to start a new paragraph?
@G.T.R I see now. In the preamble, add \setlength{\parindent}{0pt} and \setlength{\parskip}{0.5ex plus 0.5ex minus 0.2ex} (or whatever your preferred spacing is) to avoid the ugly habit of $\LaTeX$ to start paragraphs with indentation.
@G.T.R Usually, you'd use sectioning commands, \section*{Introduction} if you don't want a section number, \section{Introduction} if you do. And consider using a theorem environment for theorems (and lemmata, corollaries, ...).
This is interesting. I typed "Yitang Zhnag" in math genology project and it gave me that his advisor's advisor is S. S. Abhyankar himself. genealogy.ams.org/id.php?id=16848
