Using boxed with newline in it
Attractive Boxed Equations
Continuous extension of a real function Related;    This is the theorem i need to prove;  "Let $E(\subset \mathbb{R})$ be closed subset and $f:E\rightarrow \mathbb{R}$ be a contiuous function. Then there exists a continuous function $g:\mathbb{R} \rightarrow \mathbb{R}$ such that $g(x)=f(x), \forall x\in E$."  I have tried hours to prove this, but couldn't. I found some solutions, but ridiculously all are wrong. Every solution states that "If $x\in E$ and $x$ is not an interior point of $E$, then $x$ is an endpoint of a segment of at most countable collection of disjoint segments.". However, this is indeed false! (Check Arthur's argument in the link above)  Wrong solution Q4.5;    Just like the argument in this solution, i can see that $g$ is continuous on $E^c$ and $Int(E)$. But how do i show that $g$ is continuous on $E$? How big is my single pattern in Photoshop I pulled a repeating pattern from someone else's PSD. Is there a way to figure out the size of the original pattern? I need to create an image with only one tile. Proving a function is Lipschitz continuous Show that the following function is Lipschitz continuous and find a Lipschitz constant  $$y\mapsto f(x,y)\\ f(x,y)=\frac{y}{x}\ln(\frac{y}{x})\text{  , } |x-1|\leq\frac{1}{2}\text{  , } |y-e|\leq\frac{e}{2} $$  I have no clue on where to begin to prove this.   My first question is how I should interpret this function, is $x$ a sort of constant? Saving image from PHP URL how to install the DDE to the ubuntu 20.04? I have installed the Ubuntu 20.04, and want now I want to install Deepin Desktop Environment (dde)  My steps are:   add the ppa and install using apt  sudo add-apt-repository ppa:leaeasy/dde sudo apt-get update sudo apt install dde    But during the command sudo add-apt-repository ppa:leaeasy/dde I get the following error:  $ sudo add-apt-repository ppa:leaeasy/dde  You can install deepin desktop environment by run  sudo add-apt-repository ppa:leaeasy/dde sudo apt-get update apt-get install dde ------ Ubuntu Bionic: https://drive.google.com/file/d/1xkQPb4tWocknuYeY-t41W--kz-9SSL0V/view?usp=sharing ------ Important Issue: Q: Can dde on Ubuntu 16.04? A: Sorry, the dde depends on qt version above than 5.6.      If you want to install dde, please upgrade to 17.04 or above.  More info: https://launchpad.net/~leaeasy/+archive/ubuntu/dde Press [ENTER] to continue or Ctrl-c to cancel adding it.  Hit:1 http://in.archive.ubuntu.com/ubuntu focal InRelease Hit:2 http://in.archive.ubuntu.com/ubuntu focal-updates InRelease                                      Hit:3 http://in.archive.ubuntu.com/ubuntu focal-backports InRelease                                    Ign:4 http://ppa.launchpad.net/leaeasy/dde/ubuntu focal InRelease                                      Hit:5 http://security.ubuntu.com/ubuntu focal-security InRelease            Err:6 http://ppa.launchpad.net/leaeasy/dde/ubuntu focal Release             404  Not Found [IP: 91.189.95.83 80] Reading package lists... Done E: The repository 'http://ppa.launchpad.net/leaeasy/dde/ubuntu focal Release' does not have a Release file. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details.   I get the similar error with sudo apt-get update :  $ sudo apt-get update Hit:1 http://in.archive.ubuntu.com/ubuntu focal InRelease Hit:2 http://in.archive.ubuntu.com/ubuntu focal-updates InRelease                                       Hit:3 http://in.archive.ubuntu.com/ubuntu focal-backports InRelease                                     Ign:4 http://ppa.launchpad.net/leaeasy/dde/ubuntu focal InRelease            Hit:5 http://security.ubuntu.com/ubuntu focal-security InRelease Err:6 http://ppa.launchpad.net/leaeasy/dde/ubuntu focal Release   404  Not Found [IP: 91.189.95.83 80] Reading package lists... Done E: The repository 'http://ppa.launchpad.net/leaeasy/dde/ubuntu focal Release' does not have a Release file. N: Updating from such a repository can't be done securely, and is therefore disabled by default. N: See apt-secure(8) manpage for repository creation and user configuration details.   How can I install dde ? How can I install Visual Studio? How can I install Visual Studio?    I'd prefer to use it in Wine or PlayOnLinux if possible. How to fix Footnote position at the bottom of the page \footnote[1]{\nameref*{8.1.16} is an अधिकार सूत्र it goes upto \nameref*{8.3.55}} \&amp; अङ्गाधिकार (अङ्गस्य)\footnote[2]{\nameref*{6.4.1} is an अधिकार सूत्र it goes upto \nameref*{7.4.97}}  I am using footnote as shown above. However the footnote appears to be floating and when I don't have much in the page it appears in the middle of the page. I got by so far with \vspace{} but I think there must be a way to position this at the bottom of the page despite whats in the page. Thanks for your help How can I create an association list from 2 lists? How to avoid pressing Enter with getchar() for reading a single character only? I recently read this , which stated:     For scientists trying to design a compound that will attach itself to,   and modify, a target disease pathway, the critical first step is to   determine the electronic structure of the molecule. But modeling the   structure of a molecule of an everyday drug such as penicillin, which   has 41 atoms at ground state, requires a classical computer with some   $10^{86}$ bits—more transistors than there are atoms in the observable   universe. Such a machine is a physical impossibility. But for quantum   computers, this type of simulation is well within the realm of   possibility, requiring a processor with 286 quantum bits, or qubits.   Along with this resource estimate for penicillin, I've also seen  of the number of qubits required to model the ground state of caffeine (160 qubits). Given that the above report offers no reference(s) (probably in the name of business intelligence) and much Internet searching and looking into the quantum chemistry literature has come up short, my question is: Where are these resource estimates coming from – is there a journal article that published these numbers? I would really like to identify the methodology and assumptions used in making these estimates. How do I perform an IF...THEN in an SQL SELECT statement?  For example:  SELECT IF(Obsolete = 'N' OR InStock = 'Y' ? 1 : 0) AS Saleable, * FROM Product I just wanted to make some input where I press a key and the auto-repeat function engages so I don't have to press the 0 key many times to make a long number.  10000000000000000000000000000  This has always worked without fail in the past, but it's not something I do often and I'm not sure where to look next to figure out what might be preventing this.     I've toggled the sliders to change the delay for key repeat, but the only thing I can hang my hat on is that perhaps Lion disables this with the new feature where you hold down some keys (like the vowels, but also some consonants) to get alternate accented characters.  s turns into ß or ś or š and so on....  Am I missing an obvious setting, or is this something that has moved to a hidden setting or is simply a bug that these controls are still around since key repeat is now deprecated?  Comments welcome if anyone doesn't have an answer, but can provide specifics where this does work for them. I've checked two Macs so I'm guessing it's not related to keyboard type or connection method. Best way to save data on the iPhone I am writing an iPhone application, and need to save the state of my application (5K or so).  My main worry is data persisting across upgrades. Some of the applications I use clearly got this wrong, and I would prefer not to! Along the lines of , what reasons are there (if any) to favor \( ... \) over $ ... $? In languages like Java and C#, strings are immutable and it can be computationally expensive to build a string one character at a time.  In said languages, there are library classes to reduce this cost such as C# System.Text.StringBuilder and Java java.lang.StringBuilder.  Does php (4 or 5; I'm interested in both) share this limitation?  If so, are there similar solutions to the problem available? How do I use SQL's GETDATE() and DATEADD() in a Linq to SQL expression? If I have a Linq to SQL expression like this:    from subscription in dbContext.Subscriptions  where subscription.Expires &gt; DateTime.Now select subscription   I want this to to use the SQL Servers GETDATE() function instead of the time of the machine running the C# program.  The next question would be how to translate this:  DateTime.Now.AddDays(2)   to this:  DATEADD(dd, 2, GETDATE()) If $G/Z(G)$ is cyclic, then $G$ is abelian Continuing my work through Dummit &amp; Foote's "Abstract Algebra", 3.1.36 asks the following (which is exactly the same as exercise 5 in ):     Prove that if $G/Z(G)$ is cyclic, then $G$ is abelian. [If $G/Z(G)$ is cyclic with generator $xZ(G)$, show that every element of $G$ can be written in the form $x^az$ for some $a \in \mathbb{Z}$ and some element $z \in Z(G)$]      The hint is actually the hardest part for me, as the quotient groups are somewhat abstract. But once I have the hint, I can write:  $g, h \in G$ implies that $g = x^{a_1}z_1$ and $h = x^{a_2}z_2$, so \begin{align*}gh &amp;= (x^{a_1}z_1)(x^{a_2}z_2)\\\  &amp;= x^{a_1}x^{a_2}z_1z_2\\\ &amp; = x^{a_1 + a_2}z_2z_1\\\ &amp;= \ldots = (x^{a_2}z_2)(x^{a_1}z_1) = hg. \end{align*} Therefore, $G$ is abelian.  1) Is this right so far?  2) How can I prove the "hint"? Show that every uncountable set of real numbers has a point of accumulation. How should different sized RAM be added to iMac? I have a mid-2011 iMac which came with 4GB RAM (2x2GB). I just bought 8 GB (2x4GB) and added it to get a total of 12GB. The current configuration is as follows:    Given that the RAM is a different size, do I have the optimal locations for the different RAM chips? At the moment what I did was to simply add the 2 new sticks of RAM in the two empty slots. How can I stop Ubuntu Dock from extending to the edges? By default Ubuntu Dock in Ubuntu 17.10 and later releases always extends to the edges (top-bar-to-bottom if placed on the left or right and left-to-right when placed at the bottom) even when it's not filled with icons. Is there a way to prevent this behaviour (i.e. I want to make the dock-length flexible so that as the number of application icons in the dock increases, it should expand accordingly)?  ➡️➡️➡️  In other words, is there a way make Ubuntu Dock look like a &quot;proper dock&quot; instead of a side/bottom panel with full-height/width?