Question about wormholes with ends moving relative to one another
Would a wormhole be able to move?
Assume $T \in \mathbb{R}^{d\times d}$. Let $\rho(T)$ be the spectral radius of $T$ and $\rho \ge 0$. Prove that  $\rho(T) &lt; \rho$ if and only if there exists a $P \succ 0$ satisfying $T^TPT - \rho^2 P \prec 0$.  I can prove the Necessity: If $P = I$, for any $x \in \mathbb{R}^d$,   \begin{equation} x^T T^T P T x - \rho^2 x^T P x = \| Tx \|^2 - \rho^2 \| x\|^2 &lt; 0 \end{equation} so $T^T PT - \rho^2 P \prec 0$  I get stuck in proving the Sufficiency. Is there a 'rule of thumb' that I can use to estimate depth of field while shooting? I can use a DoF calculator, or crunch the heavy math by hand if I really want to know exactly what my DoF is going to be... but most of the time knowing exactly is actually overkill, and taking the time to pull out a DoF calculator is impractical. I'm wondering if anyone knows of a 'rule of thumb' that is common for quickly figuring out a rough estimate of what my DoF will be when I'm in the field... ERROR running force:source:pull: Entity of type 'CustomApplication' named 'Contact' cannot be found Algorithm to find the irreducible polynomial What algorithm can be used find an irreducible polynomial of degree $n$ over the field $GF(p)$ for prime $p$. The reason I ask is I want to make a program for finite field arithmetic, but creating a field $GF(p^k)$ requires finding a irreducible polynomial. (I'm mostly interested in $GF(2^k)$.) Remove all packages installed from a PPA? I have installed LibreOffice 4 using PPA. I was using LibreOffice 3, then  I had upgraded to LibreOffice 4, because I was somewhat eager to try a new version.   Now since I found that LibreOffice 4 has some issues, including handling my native language, I want to move back to LibreOffice 3.  This question  has already some answers which has some similarity with my above mentioned issue, but it does not solve my problem. This question talks about downgrading from a specific version of LibreOffice, namely from 4.0 to 3.6. The solutions mentioned are not the ones I am looking for. They will work, but the  solutions suggest either downloading .deb files for LibreOffice 3.6 or adding PPA for them. Furthermore, some of the answers put out-of-proportion~(applicable for the solution, however) stress on use of , not general command-line-solution.     I want a general solution without using PPA or downloading .deb files. All I want is to move back from the (higher) PPA version to the (lower) official version.  Anyway, in order to accomplish this, I removed the LibreOffice config directory from my home and then purged LibreOffice from my machine.  sudo apt-get purge libreoffice-*   Then I removed the relevant PPA's using the sudo apt-add-repository --remove command. And then ran sudo apt-get update.  Now, when I try to install LibreOffice using the command  sudo apt-get install libreoffice   I get an avalanche of output about unmet dependencies, something like,   The following packages have unmet dependencies:  libreoffice : Depends: libreoffice-core (= 1:3.5.7-0ubuntu4) but it is not going to be installed (snipped)    If I dig the issue further, by using the command,  sudo apt-get install libreoffice-core   I get    The following packages have unmet dependencies:  libreoffice-core : Depends: libreoffice-common (&gt; 1:3.5.7) but it is not going to be installed                 Depends: libexttextcat0 (&gt;= 2.2-8) but it is not going to be installed                 Depends: ure (&gt;= 3.5.7~) but it is not going to be installed E: Unable to correct problems, you have held broken packages.    Could you please tell me how do I install LibreOffice 3 in my machine?  I am using Ubuntu 12.04 LTS.    EDIT  Before receiving , I was wondering, at this very moment, if I take a fresh computer, and install Ubuntu 12.04, LibreOffice installation will work without a hitch. Then why I can not install LibreOffice in my 12.04 machine today from simple command line?  Then the accepted answer clarified everything. I need to use ppa-purge so that this resets all packages from a PPA to the standard versions released for my distribution.  Basically it is like a way to restore my system back to the way it was before my installed packages from a PPA.  further elaborates the idea.    worked perfectly for me. Actually, this was an education for me since it taught me how do downgrade a package that was added via PPA. And I realized that the answer actually teaches us to "move from PPA installation  of a package to official lower version package." Show that $a(-1) = (-1)a = -a $. Is there a more efficient way to clamp real numbers than using if statements or ternary operators? I want to do this both for doubles and for a 32-bit fixpoint implementation (16.16). I'm not asking for code that can handle both cases; they will be handled in separate functions.  Obviously, I can do something like:  double clampedA; double a = calculate(); clampedA = a &gt; MY_MAX ? MY_MAX : a; clampedA = a &lt; MY_MIN ? MY_MIN : a;   or  double a = calculate(); double clampedA = a; if(clampedA &gt; MY_MAX)     clampedA = MY_MAX; else if(clampedA &lt; MY_MIN)     clampedA = MY_MIN;   The fixpoint version would use functions/macros for comparisons.  This is done in a performance-critical part of the code, so I'm looking for an as efficient way to do it as possible (which I suspect would involve bit-manipulation)   EDIT: It has to be standard/portable C, platform-specific functionality is not of any interest here. Also, MY_MIN and MY_MAX are the same type as the value I want clamped (doubles in the examples above). How should we handle downvote retaliation? I have a situation where I'm trying to figure out what the right thing to do is on SO.    I see a question.  I answer the question.  I see there are a couple other answers.  One answers the question well, though with less explanation than my answer.  The other answer seems both wrong and irrelevant to the question (doesn't answer the question at all).  I comment to that effect.  The "wrong" part is edited out, but the author of the answer doesn't respond to how their answer actually answers the question.  In fact, they seem offended that I don't "get" it.  After giving them several chances to explain how their answer is relevant to the question and seeing their attitude, I decide to downvote their answer as I don't see how it addresses the question at all and the author won't explain.  They retaliate and downvote my answer and admit in their comments that their downvote of my answer has nothing to do with the content of my answer and only because they don't like me disputing their answer.  Did I do the right thing by downvoting their answer?  Do I just have to live with their downvote retaliation?  Anything I should have done differently?  I also have no way of knowing this for sure, but the next question I answered was upvoted, then downvoted so perhaps this same person is stalking my answers and downvoting other answers of mine too with no comment.  As long as this doesn't continue, I'm not losing sleep over a couple downvotes, but I do wonder if I should have done anything differently.  Followup Later After I Posted This Question: After thinking about it for a little while, the person who disagreed with me decided to delete their post.  So, eventually the message did get through to them.  I don't know if they rescinded their downvote or not (or if they even could). Positivity of a determinant I'm stuck to prove the following exercise : Given real numbers $x_1,\ldots,x_n$ and $y_1,\ldots,y_n$, show that  $$ \det(e^{\large{x_iy_j}})_{i,j=1}^n&gt;0 $$ provided that $x_1&lt;\cdots&lt;x_n$ and $y_1&lt;\cdots&lt;y_n$.   Any idea ? Sometimes it's hard to search for older versions of macOS &amp; OS X  on the App Store.  This is an attempt to provide the canonical QA for this issue, as per the Meta QA -  I expect it to be modified over time to hopefully become a complete information resource.  How can I get any older version of a Mac OS to be able to run as a VM, or to reinstall on my Mac? Disable key when program is running I am writing a client-side Swing application (graphical font designer) on Java 5. Recently, I am running into java.lang.OutOfMemoryError: Java heap space error because I am not being conservative on memory usage. The user can open unlimited number of files, and the program keeps the opened objects in the memory. After a quick research I found  and others saying on Windows machine the JVM defaults max heap size as 64MB.  Given this situation, how should I deal with this constraint?  I could increase the max heap size using command line option to java, but that would require figuring out available RAM and writing some launching program or script. Besides, increasing to some finite max does not ultimately get rid of the issue.  I could rewrite some of my code to persist objects to file system frequently (using database is the same thing) to free up the memory. It could work, but it's probably a lot work too.  If you could point me to details of above ideas or some alternatives like automatic virtual memory, extending heap size dynamically, that will be great. How can I show that if some group $G$ has only one subgroup $K$ of order $n$, then $K$ is a normal subgroup? Would that mean that it only has one subgroup total? If so then I guess that makes sense. Cannot start XAMPP on Ubuntu 11.10 When is a MOSFET more appropriate as a switch than a BJT? Sum of derivatives of a polynomial I have a mathematical transformation that takes 16 parameters (grouped into 3+8+5) and would like to make a latex command for it, so that I can easily change the notation for it if the need arises.  As far as I know, both \def and \newcommand take a maximum of 9 arguments, is there any (recommended) way to extend this? Which function (s) is(are) the k-th derivative of itself? As I attempted to solve for a function that is the $k$-th derivative of itself where $k$ is some natural number, and generalise the result, I realised it gets very hard to prove anything. For $k= 1$, apart from trivial corner cases, it's the exponential function with any constant multiplied.  For $k=2$, along with the exponential function and it's reciprocal, it works for their sum along with any mixture by constants multiplied...  For any $k$, I guessed it should be of the form $e^{x w}$ where w is the $k$-th root of unity. And we can add all these $k$ terms for each root with constants multiplied.  While it is easy to verify it's satisfying the condition, it is beyond my scope to prove this is the only family of functions that satisfy it.  I am hoping for some interesting math at work here, and want to know if this is an already established fact. Also, how do I prove anything here? Electric shock - was I stupid, unlucky, or a combination of both? One of the double sockets in our garage needed replacing (damaged after a washing pole hit it) - a pretty trivial job that I'd done before. So I flipped the MCB for the ring main, used a non-contact voltage tester to check nothing was there, and set to work. I disconnected the old socket fine, and was lining the cables up to fit into the new socket when... ow.  My finger bridged the live and earth pins, and suffice to say the live wire was very much alive. Bar a few finger spasms I was fine fortunately, and believe me I realise I was very lucky in that respect (I'm in the UK, so that was 240V.) Before finishing the job I then shut off the entire property's electrics - I was taking no more chances.  Further inspection confirmed two things I immediately suspected at this point - the MCB in the garage (installed before we moved into the house) was shutting off power via the neutral, not the live, and the voltage tester was dodgy (I had tested it on the same socket earlier and it seemed fine, further testing showed it presumably had a loose connection internally and only worked sporadically.) Needless to say it's now in the bin, and an electrician has been called to sort the MCB and check the rest of the house for similar issues.  Unsurprisingly this has shaken me up a bit - I would say I'm a rather safety conscious person, and in this case thought I'd done all the checks properly (including taping over the relevant trip switch to ensure no-one flipped it back on while I was working.) In some ways I can't help feeling I was unlucky that the MCB was wired by a cowboy, in combination with the voltage tester failing in the way that it did, at the time that it did. However, I'm not one to solely blame tools - clearly some, if not most of the blame here lies with myself also.  So in short, what should I have done differently - and was I primarily being stupid, or really quite unlucky? Search for singleton tags on questions