If $f\in L^2(\mathbb R)\cap AC(\mathbb R)$ and $f'\in L^2(\mathbb R)$ is it true that $\lim_{t\rightarrow\infty}f(t)=0$? If true, proof? If false, counterexample?  Thanks  PS: note that the standard proof that works with $f\in L^1(\mathbb R)\cap AC(\mathbb R)$ such that $f'\in L^1(\mathbb R)$, use the fact that if $x\in\mathbb R$ then $\int_x^{+\infty}|f'(t)|dt&lt;+\infty$ to show that the limit is Cauchy and then exists. Here we no longer have that $\int_x^{+\infty}|f'(t)|dt&lt;+\infty$, at least in principle, so, answering the question about $L^1(\mathbb R)$ is not the same of answering the question about $L^2(\mathbb R)$.
I am given $f,f' \in L^1(\mathbb{R})$, and f is absolute continuous, I want to show that:  $$\lim_{|x|\rightarrow \infty} f(x)= 0$$  Not sure how to show this, I know that $f(x)=\int_0^x f'(t) \, dt+f(0)$, and I can assume without loss of generality that $f(0)=0$, any help?  Thanks in advance.
Evaluate this power series Evaluate the sum  $$x+\frac{2}{3}x^3+\frac{2}{3}\cdot\frac{4}{5}x^5+\frac{2}{3}\cdot\frac{4}{5}\cdot\frac{6}{7}x^7+\dots$$  Totally no idea. I think this series may related to the $\sin x$ series because of those missing even powers. Another way of writing this series:  $$\sum_{k=0}^{\infty}\frac{(2k)!!}{(2k+1)!!}x^{2k+1}.$$ Remove a marker from a GoogleMap Using customized Coordinate System in ArcGIS Desktop? Why can't we use double pointer to represent two dimensional arrays? Suppose $\displaystyle a_n=\frac{1}{(\text{log}2)^p+(\text{log3})^p+...+(\text{log}n)^p}$ for $n\geq 2$ and $p&gt;0$. For which values of $p$ does $\sum_{n=2}^\infty a_n$ converge?  So, when $p\leq 1$, $\displaystyle a_n\geq \frac{1}{(n-1)(\text{log}n)^p}$ for $n\geq 2$. Let $\displaystyle b_n=\frac{1}{(n-1)(\text{log}n)^p}$. Then $\sum b_n$ diverges since $p\leq 1$. Therefore by comparison test $\sum a_n$ diverges when $p\leq 1$.  When $p&gt;1$, $\displaystyle a_n\leq \frac{1}{(\text{log}n)^p}$. Let $\displaystyle c_n= \frac{1}{(\text{log}n)^p}$. If $\sum c_n$ converges then I will win. But now I am stuck. Is $\sum c_n$ converges? Then how can I prove it? If not how can I conclude the state of convergence of $\sum a_n$ for $p&gt;1$?   Can anybody please help? My PC has 8 Gb of ram. Is there any way to make ubuntu use most of it? I mean rarely drop caches and keep once opened programms in ram longer, preload apps on boot, etc. Finder: keep sort-by per-directory Some directories make sense to be sorted by date modified, while others make more sense to be kept sorted alphabetically.   How can I make it so that finder will apply a specified sort by selection on a per-directory basis?   If no such feature exists (In which case I hope somebody at APPL is paying attention), I suppose using/remapping the keyboard shortcuts (Defaults being Cmd+Ctrl+5 for Modified and Cmd+Ctrl+1 for Alphabetical by Name) is the next best thing. What is the $\sqrt{i^4}$?   $i^4$ = $(i^2)^2$ So is $\sqrt{i^4}$ = $\sqrt{(i^2)^2}$ = $i^2$ = $-1$?  Or is $\sqrt{i^4}$ = $\sqrt{1}$ = $1$?  When I plug it into my TI-89 Titanium, I get $1$.  Edit: I now see why $i^4 \neq -1$. $$\sqrt{x^2} = \left|{x}\right| $$ So$$\sqrt{(i^2)^2} = \left|i^2\right| = \left|-1\right| = 1$$ Show that the operator norm is submultiplicative We had in our lecture on numerical analysis the following: Let $\mathrm{Lin}(X,Y)$ be the set of all linear maps $X\rightarrow Y$. Let $A\in\mathrm{Lin}(\mathbb R^l,\mathbb R^n)$ and $B\in\mathrm{Lin}(\mathbb R^n,\mathbb R^m)$ and $\|C\|_{op}:=\max_{\|x\|\leq1}\|C(x)\|$. Then our lecturer followed $\|A\circ B\|_{op}\leq\|A\|_{op}\cdot\|B\|_{op}$. So he didn't prove it and so I've tried it by my own. My attempt: $$\begin{aligned} \|A\circ B\|_{\mathrm{op}} &amp;= \max_{\|x\| &lt; 1}\|(A \circ B)(x)\| \\ &amp; \leq \max_{\|x\| &lt; 1} \|A\|_{\mathrm{op}} \|B(x)\| \\ &amp;= \|A\|_{\mathrm{op}} \max_{\|x\|\leq1}\|B(x)\|\\ &amp;= \|A\|_{\mathrm{op}}\|B\|_{\mathrm{op}} \end{aligned}$$ But this seems too easy. I am really interested in a nice proof so anybody could help? Thanks a lot! "A quarter past five" or simply "Quarter past five"? I usually say “It’s a quarter past” (for instance) “five”.  But I think I’ve heard (or read) “It’s quarter past ...”, too, with no indefinite article: is that correct?  Which is more common (if both constructions are in use)?  Is the omission of the article more colloquial (informal, popular, whatever)?  Is there a difference between British and American English?  Thank you very much. Cannot install R-forge package using install.packages How do I find this conditional expectation? How do I match any character across multiple lines in a regular expression? I have a .txt file with numbers ordered like this (on the same row):  106849_01373    106849_01967    106850_00082    23025.7_01059      I would like to convert them like that:  106849_01373 106849_01967 106850_00082 23025.7_01059   I have no idea which command to use. Can somebody help me with this? How do you sort du output by size? How do you sort du -sh /dir/* by size? I read one site that said use | sort -n but that's obviously not right. Here's an example that is wrong.  [~]# du -sh /var/* | sort -n 0       /var/mail 1.2M    /var/www 1.8M    /var/tmp 1.9G    /var/named 2.9M    /var/run 4.1G    /var/log 8.0K    /var/account 8.0K    /var/crash 8.0K    /var/cvs 8.0K    /var/games 8.0K    /var/local 8.0K    /var/nis 8.0K    /var/opt 8.0K    /var/preserve 8.0K    /var/racoon 12K     /var/aquota.user 12K     /var/portsentry 16K     /var/ftp 16K     /var/quota.user 20K     /var/yp 24K     /var/db 28K     /var/empty 32K     /var/lock 84K     /var/profiles 224M    /var/netenberg 235M    /var/cpanel 245M    /var/cache 620M    /var/lib 748K    /var/spool I would like to add my bibliography to my presentation in IEEE format. What should I add to my preamble and code.  \documentclass{beamer} \setbeamertemplate{footline}[page number]  \usepackage{graphicx} \usepackage{booktabs} \begin{document} \section{Reference} \begin{frame} \frametitle{Reference}  \end{frame}   I'm using the same bibTex which I use to make the IEEE paper. Suppose I have a document with multiple include or input statements  \input{fileA} \input{fileB}   etc.   Is there an easy way to generate a single .tex file where \input{fileA} is replaced by the actual content of fileA etc. without copying it manually? It's not like there's anything else to do.   Shouldn't the speaker say: "There's something else to do." ? so what's the meaning of the sentence and what's the difference between the two of them? Thank you, Why is LaTeX3 taking so long to come out? I feel as though I have been hearing about the LaTeX3 project for a very long time.   Why is it taking so long for it to come into fruition? Are there major technical difficulties, or is it purely a lack of developers that is causing a bottleneck? Should we allow adding rep to bounties to extend their duration? There are a few variables here that I was thinking about:   Spend rep to extend a bounty for a certain amount of time Allow the rep spent to extend duration to add into the original bounty Create minimums on the amount necessary to extend dependent on how long it has been up   These are just ideas and can be tweaked as necessary, but the idea in whole is to encourage people to spend more on rep and avoid "wasted" rep because there wasn't a good answer to a question while minimizing the amount of possible abuse. This was part of the idea to spend a small amount of rep to extend duration without adding to the rep so that you wouldn't have to spend the whole amount over again for a couple more days.  The typical use case that I am thinking of is when you create a bounty and it still gets no attention. I think it would be cool if you could then add onto it to turn it into a higher bounty instead of having those points completely gone.  Possible abuses:   Keeping bounty questions alive for inappropriately long times. There is already an answer but the author does not want to award the bounty.   For 1, you can resolve this by increasing the amount necessary to keep a question alive exponentially similar to what is already in place. You could also halve the amount of extended duration per extension. For 2, this is a problem with the existing system as well, but the author does not have a positive incentive not to award the points since he himself still cannot get them back.