If I start a bounty, can I lose privileges?
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Is there an injective function $f :\mathbb{R}^2\rightarrow\mathbb{R}$?  I approached this problem from the perspective of $f^{-1}$, from which I showed there exists a surjective function $f^{-1}:\mathbb{R}\rightarrow\mathbb{R}^2$. Would this imply that there exists an injective function for the inverse mapping? Does casting a spell from an item allow you to apply class abilities that are used when casting a spell? This question is updated to request additional answers and details in light of information now available in the Dungeon Master's Guide. In the released DM basic rules, Lost Mine of Phandelver, Hoard of the Dragon Queen, and the DMG teaser there are wands, staves, and rings that you can use to cast spells.  Do abilities that trigger from casting a spell activate when casting a spell from an item?  Can the character modify the spell using class abilities when casting a spell from an item?   Examples include:  An abjurer Wizard's arcane ward recovering hit points from a Dispel Magic cast from a Ring of Spell Storing  A sorcerer using Twinned Spell on a ray of enfeeblement from a Staff of Power  An evocation wizard using Sculpt Spell on a fireball from a Wand of Fireballs.   Now that the DMG is out we have some clearer rules on casting spells from items (DMG p. 141):  Some magic items allow you to cast a spell from the item.  The spell is cast at the lowest possible spell level, doesn't expend any spell slots, and requires no components, unless the item's description says otherwise  The spell uses its normal casting time, range and duration, and the user of the item must concentrate if the spell requires concentration. A magic item, such as certain staffs, may require you to use your own spellcasting ability when you cast a spell from the item.  Does whether or not you can apply class abilities to the casting of the spell depend whether the item uses the user's own spellcasting ability (such as the Staff of Fire) versus an item that comes with a fixed spellcasting ability built-in (such as the Wand of Fireballs)? Does it matter if it's something that triggers more or less automatically like Arcane Ward, versus something you have to control, like Careful Spell? Given the differential equation $$\dot x = -2x + u$$ determine the optimal control law $u = - kx$ that minimizes the performance index $$J = \int_0^{\infty} x^2 \, \mathrm d t$$  My approach was to find the state feedback $k$. But since the value of $R$ (positive semidefinite Hermitian) is not given, that means $R=0$. How do I determine the optimal control for this system where $R=0$? Bathroom vanity light with no junction box I'm trying to find the complex roots of $z^3 + \bar{z} = 0$ using De Moivre.  Some suggested multiplying both sides by z first, but that seems wrong to me as it would add a root ( and I wouldn't know which root was the extra ).  I noticed that $z=a+bi$ and there exists $\theta$ such that the trigonometric representation of $z$ is $\left ( \sqrt{a^2+b^2} \right )\left ( \cos \theta + i \sin \theta \right )$ .  It seems that $-\bar{z} = -\left ( \sqrt{a^2+b^2} \right )\left ( \cos (-\theta) + i \sin (-\theta) \right )$  However, my trig is pretty rusty and I'm not quite sure where to go from here. Prove that $\int f\ d\lambda = \int_{a}^{b} f(x)\ dx,$ for any $f \in \mathcal R[a,b].$ Theorem $:$ Let $f : [a,b] \longrightarrow \Bbb R$ be a Riemann integrable function. Then $f \in L_1[a,b]$ and $$\int f\ d\lambda = \int_{a}^{b} f(x)\ dx.$$  The proof given in my book is as follows $:$     At the fag end of the proof $f$ has been proved to be measurable which uses the completeness of the measure space $\left ([a,b], \mathcal L \cap [a,b], \lambda |_{[a,b]} \right ).$ Then it has been stated that $f \leq \Phi_n,$ $\forall$ $n$ and hence $f \in L_1[a,b]$ since $\Phi_n \in L_1[a,b]$ by the proposition $5.4.3 \ $(i) which states the following $:$  Let $f \in \Bbb L$ and $g \in L_1[a,b].$ If $|f(x)| \leq g(x)$ for a.e. $x(\mu).$ Then $f \in L_1[a,b].$  I don't know how to use this proposition to claim that $f \in L_1[a,b].$ Also I don't understand the way dominated convergence theorem used here. What I know about dominated convergence theorem is as follows $:$  Let $(X, \mathcal S, \mu)$ be a complete measure space. Let $g \in L_1(X)$ and $\{f_n \}_{n \geq 1}$ be a sequence in $\Bbb L$ such that $|f_n(x)| \leq g(x)$ a.e. $x(\mu).$ Let $f_n(x) \to f(x)$ as $n \to \infty$ a.e. $x (\mu).$ Then $f \in L_1(X)$ and $$\int f\ d\mu = \lim\limits_{n \to \infty} \int f_n\ d\mu.$$  From here how to conclude that $$\Psi_n\ \bigg\uparrow\ f \implies \int f\ d\lambda = \lim\limits_{n \to \infty} \int \Psi_n\ d\lambda.$$ Please help me in this regard. Thanks in advance. I'm using a  freely downloadable latex class file and the style is set to \bibliographystyle{ieeetr}. But when I cite something with \cite{} I get it inside ().   For example I get it as abc et al. (1) but what I want to get is abc et al. [1].   Please tell me how to get my citations inside [ ] ? During The Great Battle of the Last Alliance, there were an estimated 100,000 elves that fought with the armies of Isildur. This was one of the first scenes in the first LOTR movie. However, in the trilogy, it appears that only a small number of elves show up to Helms Deep and overall left the majority of the fighting to the humans. Why did the elves not participate during the rest of the war? We  that there's a difference in meaning between try used with the infinitive and  try used with the gerund:     I tried sending her flowers. (Meaning: An experiment: I sent her flowers to see what will happen).   I tried to send her flowers. (Meaning: I tried to go through the procedure necessary to send flowers to her. The procedure might have been difficult.)   (per Michael Swan's PEU, Unit 299.6)   Will the same distinction hold if we use trying:     A man is trying setting up a password on his computer. (Is he experimenting, wanting to see what will happen?)   A man is trying to set up a password on his computer. (Is he trying to go through all the necessary steps, probably difficult, of setting up a password?) How did Rutherford conclude that most of the mass (as well as the positive charge) was concentrated in the nucleus? Geiger and Marsden's experiment led Rutherford to believe that the positive charge and most of the mass of the atom was concentrated in a small region. I understand what led him to conclude the way the positive charge is positioned in the atom. But how did he conclude that most of the mass was in a small region (the nucleus)?  How did the distribution of the mass matter after all? Given that the electric force is greater than the gravitational force by many magnitudes, the force between the positice charge and the electrons was predominantly electric.   So how did Rutherford conclude that most of the mass is in the nucleus? Could not open file archive.ubuntu.com_ubuntu_dists_eoan-updates_restricted_cnf_Commands-amd64.xz Greetings from Bangladesh. I am an intermediate user of Ubuntu OS. I am facing problem with apt update. I have searched online but could not find any answer to my problem so I am posting this question please help me.  Err:20 http://archive.ubuntu.com/ubuntu eoan-updates/restricted amd64 c-n-f Metadata   Could not open file /var/lib/apt/lists/partial/archive.ubuntu.com_ubuntu_dists_eoan-updates_restricted_cnf_Commands-amd64.xz - open (13: Permission denied) [IP: 91.189.88.152 80] Ign:35 http://archive.ubuntu.com/ubuntu eoan-updates/multiverse amd64 c-n-f Metadata Err:37 http://archive.ubuntu.com/ubuntu eoan-backports/main amd64 c-n-f Metadata   Could not open file /var/lib/apt/lists/partial/archive.ubuntu.com_ubuntu_dists_eoan-backports_main_cnf_Commands-amd64.xz - open (13: Permission denied) [IP: 91.189.88.152 80]   Reading package lists... Done E: Failed to fetch http://archive.ubuntu.com/ubuntu/dists/eoan-updates/restricted/cnf/Commands-amd64  Could not open file /var/lib/apt/lists/partial/archive.ubuntu.com_ubuntu_dists_eoan-updates_restricted_cnf_Commands-amd64.xz - open (13: Permission denied) [IP: 91.189.88.152 80] E: Failed to fetch http://archive.ubuntu.com/ubuntu/dists/eoan-backports/main/cnf/Commands-amd64  Could not open file /var/lib/apt/lists/partial/archive.ubuntu.com_ubuntu_dists_eoan-backports_main_cnf_Commands-amd64.xz - open (13: Permission denied) [IP: 91.189.88.152 80] E: Some index files failed to download. They have been ignored, or old ones used instead.   I have tried with         removing partial folder's content completely.   Cleaning apt by sudo apt clean   moving the lists to lists.old   then updating the apt      But without luck. Number of equivalence classes of matrices under switching rows and columns I have been thinking about this problem:  Suppose we have all $R$ by $C$ matrices, where the values are integers in $[1,n]$. Two matrices are equivalent under interchanging rows and columns. For example,   1 5     0 0       would be equal to itself and to  0 0 1 5   and would be equal to  0 0  5 1   and would be equal to   5 1 0 0   How many unique such matrices are there?  Any ideas how to go about it? Any help much appreciated! window-switching keyboard shortcuts interfere with emacs and cannot be changed in the main keyboard system preferences Because of a former error in our ASP.Net application, created by my predecessor and undiscovered for a long time, thousands of wrong URLs where created dynamically. The normal user did not notice it, but Google followed these links and crawled itself through these incorrect URLs, creating more and more wrong links.  To make it clearer, consider the url     example.com/folder   should create the link     example.com/folder/subfolder   but was creating     example.com/subfolder   instead. Because of bad url rewriting, this was accepted and by default showed the index page for any unknown url, creating more and more links like this.     example.com/subfolder/subfolder/....   The problem is resolved by now, but now I have thousands of 404 errors listed in the Google Webmaster Tools, which got discovered 1 or 2 years ago, and more keep coming up.  Unfortunately the links do not follow a common pattern that I could deny for crawling in the robots.txt.  Is there anything I can do to stop google from trying out those very old links and remove the already listed 404s from Webmaster Tools? It is quite straightforward to find the fundamental solutions for a given Pell's equation when $d$ is small. But I am unable to solve this equation, as I'm unable to find the fundamental solutions:  Solve: $x^2-29y^2=1$ and $x^2-29y^2=-1$ with $y\not=0$.  Could you please guide me through the solution Install Java JDK and JRE on Debian 8 I installed Debian 8, I would like to install Java JRE and JDK. I use  and it works.  But, I am afraid because it's a script hosted in a repository. I would like to understand why it doesn't work when I put the extract of  in /usr/java/jre1.8.0_73 directory as . I added the path variable with PATH=/usr/local/jdk1.8.0/bin:$PATH export PATH as explained in  but it doesn't work.  Even if I try to install OpenJDK, the package isn't found. I don't understand why it's so complicated to install Java on Debian; it is very simple on Ubuntu. I would like someone give me step-by-step instructions to install it. We should make books of our best content! And that's a good idea!  We have a lot of high quality content here, certainly enough for a small book.   I'm not saying that it necessarily has to be integrated into the main site, but this is just a suggestion if anyone wants to implement it.   Stack Exchange probably could make money selling the physical books though. CC-BY-SA allows selling, as long as you allow others to make copies of your book, and credit the author.  What Wikipedia does is you can custom order any set of articles as a book (it even has a nice set of authors at the end, to satisfy the BY part of CC-BY-SA).  This could even tie into the docs feature. Remove suggested apps option in iOS 10 What is the difference between a desktop environment and a window manager? I'm having a really hard time trying to understand what the differences between a desktop environment and a window manager are?  EDIT: From this :  There are basically three layers that can be included in the Linux desktop: X Windows – This is the foundation that allows for graphic elements to be drawn on the display.  builds the primitive framework that allows moving of windows, interactions with keyboard and mouse, and draws windows. This is required for any graphical desktop. Window Manager – The Window Manager is the piece of the puzzle that controls the placement and appearance of windows. Window Managers include: , , , , , etc. Requires X Windows but not a desktop environment. Desktop Environment – This is where it begins to get a little fuzzy for some. A Desktop Environment includes a Window Manager but builds upon it. The Desktop Environment typically is a far more fully integrated system than a Window Manager. Requires both X Windows and a Window Manager.  Examples of desktop environments are , , ,  among others) In Inno Setup is it possible to add conditional statements to what in the [Run] section gets executed? Alternatively is it possible to manually update the built in progress bar? Basically I have 2 MSIs included and using Inno Setup as a bootstrapper, and depending on user input one or both of the MSIs are to be installed.  I have something working using Exec statements in CurStepChanged but it doesn't update the progress bar as the files are extracted and it looks like the installer is stalled.  I guess the end result is I want some progress bar updates while the files are extracted to the temp folder.  The following is my current code:  procedure CurStepChanged(CurStep: TSetupStep); var     ResultCode: Integer; begin     if(CurStep = ssInstall) then begin         if(InstallServer) then begin             ExtractTemporaryFile('ServerSetup.msi');             Exec('msiexec',ExpandConstant('/i "{tmp}\ServerSetup.msi" /qb INSTALLDIR="{code:GetInstallPath}\Server\" ALLUSERS=2'),'', SW_SHOW, ewWaitUntilTerminated, ResultCode);         end;         if(InstallClient) then begin             ExtractTemporaryFile('ClientSetup.msi');             Exec('msiexec',ExpandConstant('/i "{tmp}\ClientSetup.msi" /qb INSTALLDIR="{code:GetInstallPath}\Client\" ALLUSERS=2'),'', SW_SHOW, ewWaitUntilTerminated, ResultCode);         end;     end; end;