I am trying to prove the following claim  There does not exist an integer $r\geq1 $ and a symmetric matrix $ A \in M_r(\Bbb R) $ Such that for all $ n \in \Bbb N $ we have  $$ 2^{\sqrt n} \leq |\operatorname{tr}( A^n)| \leq 2020\cdot2^{\sqrt n} $$ Please give me some idea how to start this problem?
Does there exist a symmetric matrix $A$  such that $2^{\sqrt{n}}\le |\operatorname{Tr}(A^n)|\le2020 \cdot2^{\sqrt{n}}$ for all $n$? I think no. The trace of $A^n$ equals $\sum\limits_{i=1}^n\lambda_i^n$ where $\lambda_i$ are the eigenvalues of $A$. Now, if the absolute value of trace of $A$ is bounded below by $2$, then I think the trace of $A^n$ will grow infinitely. Am I right? Thanks beforehand.
The command history between sessions is not getting saved. I'm using guake and the history for the session is working fine. I noticed that .bash_history had some commands I executed in sudo -s mode and tried the same again and all the commands while in the session got saved so I tried: chmod 777 .bash_history  Now the old commands appear at the start of a session but no new commands are getting saved. Possible Duplicate:          On Windows I am able to switch between keyboard layouts by pressing Alt+Shift. Is there a similar keyboard shortcut in Ubuntu? how implement the jqueryUI in visualforce Sum of two co-prime integers I need some help in a proof: Prove that for any integer $n&gt;6$ can be written as a sum of two co-prime integers $a,b$ s.t. $\gcd(a,b)=1$.  I tried to go around with "Dirichlet's theorem on arithmetic progressions" but didn't had any luck to come to an actual proof. I mainly used arithmetic progression of $4$, $(4n,4n+1,4n+2,4n+3)$, but got not much, only to the extent of specific examples and even than sometimes $a,b$ weren't always co-prime (and $n$ was also playing a role so it wasn't $a+b$ it was $an+b$).  I would appriciate it a lot if someone could give a hand here. How do I share internet with Windows from my Linux box? I have a Linux (Ubuntu 12.04) PC connected to the internet with a Greenpacket WiMax USB modem. I want to share the Internet connection with another computer running Windows 7 Home Premium, connected to the Linux PC over a LAN. Is this possible? How? Is the reverse possible instead (connecting the internet to the Windows computer and sharing it with Linux)? How do I configure swappiness? I have an ASUS N550JK laptop (BIOS version 204) running Ubuntu 14.04 (3.13.0-43-generic kernel).  My issue is that the battery indicator ALWAYS shows that it's using battery power - even when plugged in and fully charged. So even though I have selected to show the battery status in the menu bar only when charging or in use (in power settings), it is ALWAYS displayed. When it reaches a full charge, the indicator icon will either show just the percentage (usually 99% or 100%), or sometimes a very big number for time (like 22:14) along with the percentage. If I go to the power settings menu, it always says "Using battery power", with something like 40 hours remaining. This is all while plugged in and fully charged.  Are there any fixes less drastic than trying to update BIOS, or is a BIOS update necessary to fix this? The latest BIOS version is 208 according to the ASUS website, but the (very brief) descriptions of the updates don't mention anything about battery issues. If at all possible I'd like to avoid updating BIOS because of the risk involved.  I've already tried letting the battery drain all the way and then charging to full, with no luck.  Thanks in advance! Suggested reading for renormalization (not only in QFT) What papers/books/reviews can you suggest to learn what  "really" is?  Standard QFT textbooks are usually computation-heavy and provide little physical insight in this regard - after my QFT course, I was left with the impression that Renormalization is just a technical, somewhat arbitrary trick (justified by experience) to get rid of divergences. However, the appearance of Renormalization in other fields of physics  approach in statistical physics etc.), where its necessity and effectiveness have, more or less, clear physical meaning, suggests a general concept beyond the mere "shut up and calculate" ad-hoc gadget it is served as in usual QFT courses.   I'm especially interested in texts providing some unifying insight about renormalization in QFT, statistical physics or pure mathematics. Insert HTML markup into Gmail signature Is there any way to add HTML markup signature in Gmail settings? I mean, insert some HTML markup without using rich editor, just inserting HTML? sum pair of columns based on matching fields How to change the starting reference number in Natbib in a given bibunit? I'm using bibunit and Natbib packages with a numerical numbering.   \begin{bibunit}[plainnat]     \nocite{ref1,ref2}     \putbib[mybiblio] \end{bibunit}  \begin{bibunit}[plainnat]     \nocite{ref3,ref4}     \putbib[mybiblio]  \end{bibunit}   How can I set the starting number for ref3 and ref4 to be different from 1 (say 15), without affecting the starting number for ref1 and ref 2? The desired result is the following:   [1] ref 1 [2] ref 2 [15] ref3 [16] ref4 I'm self-studying Guillemin and Pollack, but I'm stuck on Problem 3 of section 2. It says that if $V$ is a vector subspace of $\mathbb{R}^N$, then $T_x(V)=V$ if $x\in V$.  If $x\in V$, then since $V$ is a manifold, there is a local parametrization $\phi: U\to V$ where $U$ is open in $\mathbb{R}^k$. Without loss of generality, we can require $\phi(0)=x$. Then $T_x(V)$ is defined to be the image of $d\phi_0$ on $\mathbb{R}^k$.   An arbitrary element of $T_x(V)$ looks like $$ d\phi_0(v)=\lim_{t\to 0}\frac{\phi(0+tv)-\phi(0)}{t}=\lim_{t\to 0}\frac{\phi(tv)-x}{t} $$  but this doesn't seem very useful to show $T_x(V)=V$. What is the right approach? Say I have a module $M$ over a ring $R$ that is   $1.$ finitely-generated, i.e. $\exists$ finite $X \subset M$ such that $M = \sum_{i=1}^{n} Rx_{i}, x_{i}\in X$ (but $X$ is not necessarily free)  $2.$ free, i.e. $\exists$ a basis $Y \subset M$ a.k.a. $Y$ is free and generates M, (but $Y$ is not necessarily finite).  Now I want to show, that there exists a finite basis for $M$. For vector spaces, this would obviously be trivial, but for some  $R$-Modul, I'm not sure how to proceed.   Furthermore, is every minimal generating subset of $M$ a basis? I am about to start a web project and have been working almost exclusively with a LAMP stack for the past 5 years.  I would like to build something larger and more scalable and hence have been looking into Java EE.  The division of logic between JSP and EJB seems ideal for what I want to do.  Can anyone recommend resources for getting familiar with Java EE?  And perhaps if anyone knows of technologies that are more scalable than PHP that I should consider as well, I'm all ears.  Thanks a lot! How did the letter Z come to be associated with sleeping/snoring? Grant permission for new created files inside folder automatically I have a folder which permission is set to 777 but when I add any files or folder unziping in that folder. Extracted files or folder permission never change. Destination folder permission is 777 and I want what I will add in that folder will automatically get permission of 777.   When I add any files or folder by extracting .zip in that folder, extracted folder/files permission won't change automatically. I always have to chmod for newly add file/folder! Sum of real numbers that multiply to 1 I've seen a question in my math book with this explanation above it: "If the product of n positive real numbers is 1, then the sum of these numbers must be more than n". I was wondering if this is correct, and if it is, could someone give an explanation or a proof for it? Proof that the Euler characteristic is additive I'm reading through a set of notes which assumes that the Euler characteristic is additive, but doesn't give a proof, so I would like to understand why this is.  Let $A_n$ be a finitely generated Abelian group, non-zero for $0&lt;n&lt;k$. Consider the long exact sequence  $ \cdots \rightarrow A_n \rightarrow B_n \rightarrow C_n \rightarrow A_{n-1} \rightarrow B_{n-1} \rightarrow C_{n-1} \rightarrow \cdots $  where $B_n$, $C_n$ defined similarly to $A_n$.   I know that the Euler characteristic of $A$ is given by $\chi(A) = \sum_0^k (-1)^n \text{rank}(A_n)$, with similar definitions for $B$ and $C$. I want to prove that $\chi(B) = \chi(A) + \chi(C)$.  The notes say that for an exact sequence  $0 \rightarrow X \rightarrow Y \rightarrow Z \rightarrow 0$  of finitely generated Abelian groups $X,Y,Z$, we have the result rank($Y$) $=$ rank($X$) +  rank($Z$), and that the additivity of the Euler characteristic follows from this fact, but I can't see how. An infinite series plus a continued fraction by Ramanujan Here is a famous problem posed by Ramanujan     Show that $$\left(1 + \frac{1}{1\cdot 3} + \frac{1}{1\cdot 3\cdot 5} + \cdots\right) + \left(\cfrac{1}{1+}\cfrac{1}{1+}\cfrac{2}{1+}\cfrac{3}{1+}\cfrac{4}{1+\cdots}\right) = \sqrt{\frac{\pi e}{2}}$$   The first series seems vaguely familiar if we consider the function $$f(x) = x + \frac{x^{3}}{1\cdot 3} + \frac{x^{5}}{1\cdot 3\cdot 5} + \cdots$$ and note that $$f'(x) = 1 + xf(x)$$ so that $y = f(x)$ satisfies the differential equation $$\frac{dy}{dx} - xy = 1, y(0) = 0$$ The integrating factor here comes to be $e^{-x^{2}/2}$ so that $$ye^{-x^{2}/2} = \int_{0}^{x}e^{-t^{2}/2}\,dt$$ and hence $$f(x) = e^{x^{2}/2}\int_{0}^{x}e^{-t^{2}/2}\,dt$$ Thus the sum of the first series is $$f(1) = \sqrt{e}\int_{0}^{1}e^{-t^{2}/2}\,dt$$ But I have no idea about the continued fraction and still more I am not able to figure out how it would simplify to $\sqrt{\pi e/2}$ at the end.  Please provide any hints or suggestions.  Update: We have $$\begin{aligned}f(1) &amp;= \sqrt{e}\int_{0}^{1}e^{-t^{2}/2}\,dt = \sqrt{e}\int_{0}^{\infty}e^{-t^{2}/2}\,dt - \sqrt{e}\int_{1}^{\infty}e^{-t^{2}/2}\,dt\\ &amp;= \sqrt{\frac{\pi e}{2}} - \sqrt{e}\int_{1}^{\infty}e^{-t^{2}/2}\,dt\end{aligned}$$ and hence we finally need to establish $$\sqrt{e}\int_{1}^{\infty}e^{-t^{2}/2}\,dt = \cfrac{1}{1+}\cfrac{1}{1+}\cfrac{2}{1+}\cfrac{3}{1+}\cfrac{4}{1+\cdots}$$ On further searching in Ramanujan's Collected Papers I found the following formula $$\int_{0}^{a}e^{-x^{2}}\,dx = \frac{\sqrt{\pi}}{2} - \cfrac{e^{-a^{2}}}{2a+}\cfrac{1}{a+}\cfrac{2}{2a+}\cfrac{3}{a+}\cfrac{4}{2a+\cdots}$$ and it seems helpful here. But unfortunately proving this formula seems to be another big challenge for me. Justify text in moderncv cover letter I am writing a cover letter and CV using moderncv. Everything works fine so far, but the text in the cover letter is not justified. I think it is good that it is not hyphenated, but it would look much nicer if the text was justified.  Here is a minimal example:  \documentclass{moderncv} \moderncvstyle{classic}  \usepackage{lipsum} % just for dummy text  \firstname{John} \familyname{Doe}  \begin{document} \recipient{Prof.\ Dr.\ Foo Bar}{} \opening{Dear Professor Dr.\ Bar,} \closing{Yours sincerely,} \enclosure[Attached]{curriculum vit\ae}  \makelettertitle \lipsum[1] \makeletterclosing  \clearpage  \makecvtitle  \section{Education} \cventry{2009--present}{PhD Student}{here}{and}{there}{} \end{document}