Suppose $G$ is a group of order 11. Then every non-identity element is a generator of $G$
Show that every group of prime order is cyclic
If $f \in \mathcal{L}^1(\mathbb{R})$ then $\lim_{h\rightarrow 0}\int_{-\infty}^\infty |f(x+h)-f(x)|dx\rightarrow 0$ 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! OneDrive for Windows 10 - Remove logged out account RemoveDuplicates gives 1004 error Is there way to replace value of symbolic link? Is there way to replace value of symbolic link?  For example I want to replace symbolic link like this:  first -&gt; /home/username/foo/very/long/directories/that/I/do/not/want/to/type/again    to   second -&gt; /home/username/bar/very/long/directories/that/I/do/not/want/to/type/again   I want to change only foo to bar.  Of course I can create a link again, but if it is possible to replace value of the link, it become easier. Diamond is not shown for mod name under wiki posts If a diamond mod is an author of a wiki post then the diamond is not displayed besides the mod name. E.g. :        Even when the name refers to the user profile page (in case of more then one author) like in :        Is it intentional or a bug? Here's my problem  I'm trying to do a cleanup of my hard drive, and I'm constantly seeing this scenario:  (1) Aggregate folder size: I check the folder size viewing properties right clicking the folder  (2) Individual folder/file sizes: I check the individual file/folder sizes belonging to that folder viewing properties on the individual file/folder sizes, and then adding them up (with hidden and system files shown).   These two figures are always different, sometimes by a large margin.   For instance, my C drive now, by method (1), shows a consumed space of 42.4 GB.   However, by method (2), by inspecting the respective files/folders, the total consumed space adds up only to 33.1 GB. This is a 28% discrepancy (over the smaller file size) and I have no idea where the rest of the space is going.   My hard drive is constantly in danger of being full (it's bootcamp on a mac with 50 GB of partitioned space), and there's not much I've got installed in terms of software, nor do I have large media files like movies, downloads, temp files, application temp files, torrents, etc.   Most of it's just consumed by the files in C:/Windows, but last I checked, I don't know enough to know which files to delete, and I've heard that you really can't do anything about the space boggling winsxs and system32.   Can you guys recommend a starting point to free up disk space?  Thanks!!  Baggio. What graduation gown should a PhD-holder wear when later receiving a Master's in another field? I was wondering how to add plugins to my Minecraft server. I am using Bukkit, and the things I would like to be able to do is spawn stuff, teleport to other players, set home locations, have an in-game currency/shop and be able to teleport to homes. What will reduce my rendertime? What would reduce the render time? Should I get a better GPU or more RAM? Separate and visually indicate reversed events from the day's reputation changes We've been getting a lot of questions  ever since the feature to see event reversals such as deleted posts and serial voting was added into the reputation history; and their confusion is legitimate. Even well-respected users have become confused by that number above the day. It should display the reputation changes for that day. I propose that reversal events such as these be visually separated from the main events for that day and not included in the total reputation change for that day. After all, their reputation isn't actually changing for that day, but reversing changes from previous days. These events already seem to get bumped down to the bottom of the events for the day, why not add a simple separator between them? Events indicating reversals    Events which have been reversed     Perhaps a second, smaller number could be added next to the main one to indicate that other events took place on that day, but I feel most users who really care will have noticed or went looking for the events anyways. However, including them in the actual change only serves to confuse people. Do all accepted students in top-tier engineering programs get invited to interview before getting admitted? I was viewing some old posts on Stack Overflow (two-year-old posts) and found that due to "fast technology trend changing," some of them are no longer valid. Stack Overflow (and the Stack Exchange sites generally) should have some sort of criteria to tag questions as Obsolete, or a vote-based system to mark them as such.  I am not in favour of closing obsolete questions. Can a finite group act freely (as homeomorphisms) on $\mathbb R^n$ I am asking if whether or not a finite group acts freely (as homeomorphisms) on $\mathbb R^n$.  To answer in the negative, it suffices to show: for any homeomorphism $f$ such that $f^d=\text{id}_{\mathbb R^n}$, then $f$ has a fixed point.  I am looking for a complete self-contained answer. Does this constitute a valid proof that $\frac{x^2}{1+x^4} \leq \frac{1}{2}$? Rules for refreezing food What are the general rules regarding when can you refreeze food that was frozen and then thawed in the fridge? and what are the reasons behind them? (Hygiene? Food quality?). Let's assume for the sake of the argument that the items were in a non-frozen state only a few hours.  To be more specific, I'm interested in the application of these rules for  - Ginger garlic paste - Dough - Indian Curries - Green Chilly paste - Frozen green peas Number of monic irreducible polynomials of prime degree $p$ over finite fields Suppose $F$ is a field s.t $\left|F\right|=q$. Take $p$ to be some prime. How many monic irreducible polynomials of degree $p$ do exist over $F$?   Thanks! Are there any clones/alternatives for running a Stack Exchange style Q&A site? Can moving tables to another MySQL database help performance? I want to know if I using different databases can help performance.  I have one big table support_chat with more than 5 million rows. I need to scan this all the time (3s with ajax) to check new messages on support.  If I move this table to another database in the same computer, can it help anything? (let users, sells, comments table faster)  For example:  db1  users table sells table comments table support_chat table // move this table to db2   db2  support_chat table   They want to keep old things due legal reasons, to have a user support history. I can do an eval("something()"); to execute the code dynamically in JavaScript. Is there a way for me to do the same thing in C#?  An example of what I am trying to do is: I have an integer variable (say i) and I have multiple properties by the names: "Property1", "Property2", "Property3", etc. Now, I want to perform some operations on the " Propertyi " property depending on the value of i.   This is really simple with Javascript. Is there any way to do this with C#? Organize desktop icons on Windows into categories or sections