I am working on some past prelim problems and one of them is the following: If $f \in \mathcal{L}^1(\mathbb{R})$ then $\lim_{h\rightarrow 0}\int_{-\infty}^\infty |f(x+h)-f(x)|dx\rightarrow 0$   Update: Following Jackie Chong's advice, I am attempting this problem with the dominated convergence theorem. Since $f\in L^1(\mathbb{R})$, $f$ can be approximated from below by a sequence of continuous functions $f_n$. Then, I believe I can say that, for all $n$ sufficiently large, $|f_n(x+\frac{1}{n})-f_n(x)|\le 2|f(x)|$. (I'm not sure of this part thought). Then $2|f(x)|$ is integrable and hence a dominating function for $|f_n(x+\frac{1}{n})-f_n(x)|$. So by the dominated convergence theorem we can take the limit inside $\lim_{n\rightarrow \infty}\int|f_n(x+\frac{1}{n})-f(x)|dx=\int \lim_{n\rightarrow \infty}|f_n(x+\frac{1}{n})-f(x)|dx=0$ Is this on the right track?  Another approach that I thought may work, but which I'm less certain of is the following: If $f$ can be approximated by $C^1$ functions (I have no real justification for this, so please correct me if its wrong), then for each $n$ and each $x$ there is some $y(x,n)\in B_{1/n}(x)$ such that $|f(x+\frac{1}{n}-f(x)|=\frac{1}{n}|f'(y(x,n)|$ but from here I don't see how to proceed, because just because $|f|$ is integrable, I can't justify $|f'|$ being integrable, but if there is some way to fix this approach, hints would be appreciated.   Another thing I was thinking was to first approximate $f$ from below by continuous functions $f_n$ then to define $g_{m,n}=|f_n(x+\frac{1}{m})-f_n(x)| \chi_{[-m,m]}$. Then, for each $n$, the function $g_{m,n}$ is continuous on its support, and since it support is compact, it is uniformly continuous there. Can this be used to justify bringing the limit $\lim_{n\rightarrow \infty}$ inside the integral? Then apply the monotone convergence theorem? I know this idea is probably even more sketchy than my previous one, but if it can be fixed please let me know :)  Thanks!
I am having a hard time with the following real analysis qual problem.  Any help would be awesome.  Suppose that $f \in L^p(\mathbb{R})$, where $1\leq p&lt; + \infty$. Let $T_r(f)(t)=f(t−r)$. Show that $\lim_{r \to 0} \|T_rf−f\|_{L_p} =0$, that is $$ \lim_{r\to 0} \left( \int_{\mathbb R} |f(t-r) - f(t)|^p \mathrm d t\right)^{1/p} =0.$$
Careers - filter by location & tag On Careers I thought it wold be nice to be able to filter by , possibly multiple, in additional to location.  Nothing fancy, but something like ....        The image of tags is stolen from the SO side to convey the idea. I suspect functionality on the Careers side would need to be a bit different.  Recent tags would likely make sense.  Favorite tags, particularly the more specific ones might not (e.g., a job posting for , ).  Anyway, the image is just to give a sense of the idea. Can you make a non-polar electrolytic capacitor out of two regular electrolytic capacitors? When I create language plperl , I get error: ERROR:  could not load library "C:/Program Files/PostgreSQL/9.1/lib/plperl.dll": The specified module could not be found.  But in my computer,  "plperl.dll" file is exist in "C:/Program Files/PostgreSQL/9.1/lib/..." folder   ( I can not post illustrative image, this forum require >= 10 reputations)  And if I select * pg_pltemplate, I get:  -[ RECORD 4 ]-+------------------------- tmplname      | plperl tmpltrusted   | t tmpldbacreate | t tmplhandler   | plperl_call_handler tmplinline    | plperl_inline_handler tmplvalidator | plperl_validator tmpllibrary   | $libdir/plperl Prove that $a^3+b^3+c^3 \geq a^2b+b^2c+c^2a$ Let $a,b,c$ be positive real numbers. Prove that $a^3+b^3+c^3\geq a^2b+b^2c+c^2a$.   My (strange) proof:  $$ \begin{align*} a^3+b^3+c^3 &amp;\geq a^2b+b^2c+c^2a\\ \sum\limits_{a,b,c} a^3 &amp;\geq \sum\limits_{a,b,c} a^2b\\ \sum\limits_{a,b,c} a^2 &amp;\geq \sum\limits_{a,b,c} ab\\ a^2+b^2+c^2 &amp;\geq ab+bc+ca\\ 2a^2+2b^2+2c^2-2ab-2bc-2ca &amp;\geq 0\\ \left( a-b \right)^2 + \left( b-c \right)^2 + \left( c-a \right)^2 &amp;\geq 0 \end{align*} $$  Which is obviously true.    However, this is not a valid proof, is it? Because I could just as well have divided by $a^2$ rather than $a$:  $$ \begin{align*} \sum\limits_{a,b,c} a^3 &amp;\geq \sum\limits_{a,b,c} a^2b\\ \sum\limits_{a,b,c} a &amp;\geq \sum\limits_{a,b,c} b\\ a+b+c &amp;\geq a+b+c \end{align*} $$  Which is true, but it would imply that equality always holds, which is obviously false. So why can't I just divide in a cycling sum?  Edit: Please don't help me with the original inequality, I'll figure it out. Removing the password from a PDF file I have a couple of ebooks as PDFs with passwords, however my ebook reader (sony prs600) doesn't seem to support PDFs with passwords.  What is the easiest of removing the password from a PDF (I know the password, which presumably helps a lot).  It's a bit annoying buying a book and then only being able to read it in front of a computer. Find the least value of $ \sec^6 x +\csc^6 x + \sec^6 x\csc^6 x$ Find the least value of $$ \sec^6 x +\csc^6 x + \sec^6 x\csc^6 x$$ I tried AM greater than equal to GM But that's for finding maximum value.  This can probably be solved with calculus but I don't know for some reason I can't find the answer . Which is $80$. How does time work in The Matrix? Can I set max_retries for requests.request? Can you update to macOS 10.14 (Mojave) without converting to APFS? At WWDC 2018 macOS 10.14 or macOS Mojave was disclosed.  I am currently on macOS Sierra (10.12) and am looking forward to upgrading to macOS Mojave once its released.  I was wondering if it will be possible to update to macOS Mojave without converting to APFS, similar to  trick from OS X Daily on how to upgrade to macOS High Sierra (10.13) without converting to APFS with the command /Applications/Install\ macOS\ High\ Sierra.app/Contents/Resources/startosinstall --converttoapfs NO. I have a  to which I'd like to add JPEG and PNG images that I generate on the fly.  All the examples I've seen so far in the , specially in the  use ImageIcons.  I'm generating these images as byte arrays, and they are usually larger than the common icon they use in the examples, at 640x480.   Is there any (performance or other) problem in using the ImageIcon class to display an image that size in a JPanel? What's the usual way of doing it? How to add an image to a JPanel without using the ImageIcon class?   Edit: A more careful examination of the tutorials and the API shows that you cannot add an ImageIcon directly to a JPanel. Instead, they achieve the same effect by setting the image as an icon of a JLabel. This just doesn't feel right... Can all fundamental forces be fictitious forces? What's the hardest or most misunderstood aspect of LINQ? This is the third time I've tried to install Ubuntu Server 14.04.03 64-bit on the same laptop, and each time it caused different problems.  Each time I wrote the same iso image to a USB flash drive by using a different application and OS.   Universal USB Installer (pendrivelinux.com) from Windows WinImage writer from Windows built-in image writer from Linux Mint 17.2    The iso image is the same in all three cases, the USB flash drive is the same, and the laptop is the same. However each time the installation of Ubuntu gave me different installation options. For example, the last time I installed Ubuntu, it didn't show all the packages to choose while installing (Basic Ubuntu server, etc.). If $n&gt;1$ is an integer, then $\sum \limits_{k=1}^n \frac1k$ is not an integer.  If you know , then you know there must be a prime $p$ between $n/2$ and $n$, so $\frac 1p$ appears in the sum, but $\frac{1}{2p}$ does not. Aside from $\frac 1p$, every other term $\frac 1k$ has $k$ divisible only by primes smaller than $p$. We can combine all those terms to get $\sum_{k=1}^n\frac 1k = \frac 1p + \frac ab$, where $b$ is not divisible by $p$. If this were an integer, then (multiplying by $b$) $\frac bp +a$ would also be an integer, which it isn't since $b$ isn't divisible by $p$.  Does anybody know an elementary proof of this which doesn't rely on Bertrand's Postulate? For a while, I was convinced I'd seen one, but now I'm starting to suspect whatever argument I saw was wrong. So I am currently making a minecraft custom map (for 1.9). I want to make the player "say" something using /tellraw commands. I did figure out how to make the player name come up using /say commands, but still have no idea how to do it using the /tellraw command.  How do I do that? XGBoost Loss function Approximation With Taylor Expansion How should I address a potential FERPA violation by a professor? In the United States, federal law prohibits professors from releasing any student's grades to any party other than the student and any party whom the student authorizes to receive the grades.  One of my professors appears to be violating this law by uploading each student's lab grades in one file, which is accessible to anyone who is enrolled in the class.  Here is my proposed plan of action:   Ask the professor - in private - to post the grades using the online management system the professor already uses for other things.  If the professor doesn't fix this privacy violation, inform the professor that the method by which he/she post grades is a violation of federal law. As a last resort, contact the department dean. In the example  $\cup^{i=1}_{\infty}F_i$   How do you get the i=1 directly below the union symbol and the infinity directly above? Communities list is counting my reputation wrong on some sites I have 101 rep on  and am not suspended there. But the site switcher thinks I only have 1 rep there:     I noticed this after I .   To test this, I just added Expats and Reverse Engineering to my site switcher list. On both of those sites I have 101 rep and am not suspended. Results:    Huh? What features differ between the Xbox 360 and PC versions of Minecraft? What differences does the Xbox 360 version of Minecraft have compared to playing on the PC? To clarify - I'm concerned more about the differences exclusive to each platform, not just differences of features that will probably be added in future updates (for example, crafting). Otherwise, I feel this question could be compared to &quot;What is the difference between Minecraft version x and version y&quot;, which isn't what I'm asking.