A set $E$ such that $E$ is dense in $[0,1]\times [0,1]$, and the intersection of $E$ and any line parallel to the axes has at most one point
Boundary Question in $\mathbb{R}^{2}$ (Manifolds)
I have an Android phone which is connected to a secured Wi-Fi hotspot.  How can I find out the password of the hotspot from the phone? Series with denominator that is square of product of consecutive numbers Peppers evolved chemicals like capsaicin to discourage animals from eating them, and that chemical itself can cause internal damage if consumed in large quantities, though this is usually only a concern for the hottest of peppers (which people do eat).   Still, why is it that humans seem to be the only animal that often "like" spicy food? How do I remove all specific characters at the end of a string in PHP? I need to create a metadata-table for all files in a File Geodatabase.   Is it possible to automatically be updated with new filenames, and possibly even metadata (as created in the ArcGIS metadata editor)? So I have an SQL dump file that needs to be loaded using mysql_query(). Unfortunately, it's not possible to execute multiple queries with it.  -> It cannot be assumed that the mysql command-line client (mysql --help) is installed -- for loading the SQL file directly  -> It cannot be assumed that the mysqli extension is installed  /* contents of dump.sql, including comments */ DELETE FROM t3 WHERE body = 'some text; with semicolons; scattered; throughout'; DELETE FROM t2 WHERE name = 'hello'; DELETE FROM t1 WHERE id = 1;   The explode() below won't work because some of the dump content's values contain semicolons.  $sql = explode(';', file_get_contents('dump.sql')); foreach ($sql as $key =&gt; $val) {     mysql_query($val); }   What's the best way to load the SQL without modifying the dump file? I'm desperatly confused by notations and formulations so if someone could clarify the following things a little Í would be deeply grateful. The Lie algebra $\mathfrak{so}(1,3)_+^{\uparrow}$ of the proper orthochronous Lorentz group $SO(1,3)_+^{\uparrow}$ is given by \begin{equation} [J_i,J_j]=i \epsilon_{ijk} J_k \end{equation} \begin{equation} [J_i,K_j]=i \epsilon_{ijk} K_k \end{equation} \begin{equation} [K_i,K_j]=- i \epsilon_{ijk} J_k \end{equation} We can now define new generators with the old ones $N^{\pm}_i= \frac{1}{2}(J_i \pm i K_i)$ which satisfy \begin{equation} [N^{+}_i,N^{+}_j] = i \epsilon_{ijk} N^{+}_k ,\end{equation} \begin{equation} [N^{-}_i,N^{-}_j] = i \epsilon_{ijk} N^{-}_k ,\end{equation} \begin{equation} [N^{+}_i,N^{-}_j] = 0. \end{equation} where we can see that $N^{+}_i$ and $N^{-}_i$ make up a copy of the Lie algebra $\mathfrak{su}(2)$ each. My problem is to get what is going one here mathematically precise. Are the following statements correct and if not why:  When we build the new operators from the old generators we complexified $\mathfrak{so}(1,3)_+^{\uparrow}$ \begin{equation}(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} = \mathfrak{so}(1,3)_+^{\uparrow}\otimes \mathbb{C} \end{equation} We saw that $\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C}$ is isomorph to two copies of the complexified Lie algebra of $\mathfrak{su(2)}$: $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq \mathfrak{su(2)}_{\mathbb{C}} \oplus \mathfrak{su(2)}_{\mathbb{C}} $. Where exactly did we need that $\mathfrak{su(2)}$ is complexified here? The Lie algebras defined by $N^{\pm}_i$ are exactly those of $\mathfrak{su(2)}$ and we never use complex linear combination of $N^{\pm}_i$ or am I wrong here? $\mathfrak{su(2)}_{\mathbb{C}}$ is isomorph to $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C}$: \begin{equation}\mathfrak{su(2)}_{\mathbb{C}} \simeq (\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C} \end{equation} Here $(\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C}$ denotes the complexified Lie algebra of $SL(2,\mathbb{C})$ Is $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{R}$ correct?  Here $(\mathfrak{sl}(2, \mathbb{C}))_\mathbb{R}$ denotes the real Lie algebra of $SL(2,\mathbb{C})$ Is $(\mathfrak{so}(1,3)_+^{\uparrow})_\mathbb{C} \simeq (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C} \oplus (\mathfrak{sl}(2, \mathbb{C}))_\mathbb{C}$ correct?  I looked this topic up in different books and each seemed to state something different. One book even used three differrent versions of $\mathfrak{sl}(2,\mathbb{C}) $ namely: $\mathfrak{sl}(2,\mathbb{C}) $, $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{C}$ and $(\mathfrak{sl}(2,\mathbb{C}))_\mathbb{R}$.  states simply that $\mathfrak{sl}(2,\mathbb{C}) $ is the complexification of $\mathfrak{su(2)}$ without making any reference to $SL(2,\mathbb{C})$ which does not help me either. Any help would be great. Brand new disk brakes are barely stopping the bike How to count number of words from String using shell Why do we say that the Earth moves around the Sun? In history we are taught that the Catholic Church was wrong, because the Sun does not move around the Earth, instead the Earth moves around the Sun.  But then in physics we learn that movement is relative, and it depends on the reference point that we choose.  Wouldn't the Sun (and the whole universe) move around the Earth if I place my reference point on Earth?  Was movement considered absolute in physics back then? In what order are the dated ordered by? Certainly not alphanumeric order.  ls -lt sorts by modification time. But I need creation time. My goal is to produce a graph showing the linkages (and lack there of) between several fields that share a common subproblem by showing who cites whom.  There are many databases that show citations between papers: , , and Google Scholar.  However none allows me to download even part of their database.  Is there some database I have overlooked?  Has someone written a scrapper for one of these websites?    For reference, the fields I'm considering are Mechanics, Magnetic Resonance Imaging, Signal Processing, Geophysics, and several others. How do I install pepper-flash on Chromium? I have Chromium web browser on my Lubuntu 13.04 (pre-installed).  I use LX Terminal (pre-installed) to write commands. I am trying to download flash player on Chromium using instructions on this :  sudo add-apt-repository ppa:skunk/pepper-flash   After typing in my password, this worked. Now   sudo apt-get update   I didn't need to type in my password, as I continued on the same terminal, but got W:/ and E:/ fetch file errors  sudo apt-get install pepflashplugin-installer   I continued on the same terminal despite the fetch file errors... and they said pepflashplugin-installer could not be found.  Is the last error because of fetch file errors, or because I need to download pepflash-plugin-installer somewhere? Or is it because of something else?  I cannot download the Chrome browser, and not looking to use flash player on my Firefox web browser (installed using lubuntu software center). Cannot create an array of LinkedLists in Java...? On several occasions, I was reading a question thinking "I know a guy that knows the answer to this" and that guy (or gal) also happens to be a registered user. How can I bring this to the attention of this user? Basic question, how do I resume interrupted / failed downloads in Firefox? I am working on an app that allows customers to upload images, video, (even gif) into my app. Once the case is processed on my app, I want to place those media files into Salesforce. I have a case record that I would this attached to.   I see multiple records can be added in Rest , and that files can be added to notes and attachments. Can I upload multiple notes and attachments in a single rest call? Windows License stored in UEFI/BIOS Since Windows 10 (or earlier?). It has become standard to store the license product key in the UEFI/BIOS.  I am in the market for buying a new (custom build, individual parts) PC with a new Windows 10 License.  However I was wondering (and can't find clear information on this subject). What the differences between the licenses are when it comes to OEM and Retail.  If in the far future I were to sell this PC:   Would the license (whether OEM or Retail), be stored in the UEFI/BIOS, where it could be used by the new owner? If so, can I remove the license key from the UEFI/BIOS?   I want to know this because I don't want to buy Windows 10 every single time I buy a new PC (since no new versions of Windows will be created, only updates). What's going on with "compact implies sequentially compact"? I've seen both counterexamples and proofs to &quot;compact implies sequentially compact&quot;, and I'm not sure what's going on. Apparently there are compact spaces which are not sequentially compact; quick googling and wikipedia checks will turn up examples floating around; they tend to be variants of $[0,1]^{[0,1]}$ with the product topology.  Here's a demonstration(?):  $[0,1]^{[0,1]}$ is compact by Tychonoff's theorem.  So, we demonstrate failure of sequential compactness. Choose a unique binary representation for each $x\in [0,1]$. For each $n\in\mathbb{N}$, let $f_n : [0,1]\to[0,1]$ (an element of $[0,1]^{[0,1]}$) be the function which maps each $x$ to its $n$-th place digit in its binary expansion. Let $f_{n_k}$ be a subsequence of this sequence. Let $x'\in [0,1]$ be such that the $n_{2m}$-th digit is $0$ and the $n_{2m+1}$-th digit is $1$, for all $m\in\mathbb{N}$. Then $f_{n_k} (x')$ does not converge (it alternates between $0$ and $1$), and hence $f_{n_k}$ cannot converge. Thus we have found a sequence in $[0,1]^{[0,1]}$ without any convergent subsequence and so it is not sequentially compact.  (Aside: this is apparently based off of an example in Steen's Counterexamples in Topology, according to  ) Nevertheless, there's also some proofs(?) floating around that compactness of a space implies sequential compactness, along these lines (this proof(?) is of the contrapositive):  Suppose $X$ is not sequentially compact. By definition, this means there is some sequence $(x_n)$ over $X$ with no convergent subsequence. If any $x\in X$ had for each of its neighborhood $U$ infinitely many $n$ for which $x_n \in U$, then we could define a convergent subsequence of $(x_n)$, contradicting our assumption.  (Presumably this is done by choosing for each neighborhood a sufficiently-large-indexed term in that neighborhood.) Thus, for each $x \in X$ we can select an open set $U_x$ such that $x\in U_x$ but with $x_n \in U_x$ for only finitely many $n$. The collection $\mathcal{U}=\{U_x : x\in X\}$ is clearly an open cover of $X$. If $\mathcal{U}$ had a finite subcover $\{U_1, \dots, U_k\}$ then the union $U_1 \cup \cdots \cup U_k$ would contain all of $X$ but only contain $x_n$ for finitely many $n$, which is impossible. Thus $X$ is not compact, since we have found an open cover without a finite subcover.  (Aside: this proof is essentially the same as the proof appearing in Rudin's Principles of Mathematical Analysis for Thm 2.37 that infinite subsets of compact spaces have limit points.) So, what's going on here?  It can't be that what both the counterexample and the proof are telling us is correct.  Is there some subtle (or more embarassingly for me -- glaring) flaw in the proof? Why does the reputation dropdown not show this upvote?