Existence of a real $2 \times 2$ matrix M such that $M^2 = \begin{pmatrix} -1 & 0 \\ 0 & -1-\epsilon \end{pmatrix}$ A real $2 \times 2$ matrix $M$ such that   $$  M^2  = \begin{pmatrix}  -1 &amp; 0 &amp; \\   0 &amp; -1-\epsilon  \end{pmatrix} $$   exists         for every $\epsilon &gt; 0$,   for at least one $\epsilon &gt; 0$,   for no $\epsilon &gt; 0$,   none of the above.      I have no idea on how to do this question. Hints?  Thanks
A real $2 \times 2 $ matrix $M$ such that $M^2 = \tiny \begin{pmatrix} -1&0 \\ 0&-1-\epsilon \\ \end{pmatrix}$ , then : A real $2 \times 2 $ matrix $M$ such that $$M^2 =  \begin{pmatrix} -1&amp;0 \\ 0&amp;-1-\epsilon  \\   \end{pmatrix}$$ (a) exists for all $\epsilon &gt; 0$. (b) does not exist for any $\epsilon &gt; 0$. (c) exists for some $\epsilon &gt; 0$. (d) None of the above is true.  Attempt: I couldn't think of any theory base to prove / disprove the existence of such a matrix. Could someone please give me a hint on how to go about this problem? Thank you very much for your help in this regard.
I have a Dell N5110 laptop with integrated Nvidia GeForce GT 525M and an Intel HD Graphics 3000 card; My problem is the default card is the Intel one, and it provides a very bad performance with Windows 8. So can I force to use the Nvidia one all the time including "Desktop Windows Manager" ? Align cell height of the tabularx rows The following latex code prints rows one at a time. However all the cells of the rows don't come out aligned. How can I fix it.   \documentclass{beamer}     \usepackage{tabularx}    \begin{document}  \begin{frame}{}  \begin{tabularx}{3.50in}{lXX}           \hline           &amp; Observation &amp; Conclusion \\           \hline             \uncover&lt;2-&gt;{1.} &amp; \uncover&lt;2-&gt;{Observation 1 } &amp; \uncover&lt;2-&gt;{Conclusion 1} \\   \uncover&lt;3-&gt;{2.} &amp; \uncover&lt;3-&gt;{Observation 2} &amp; \uncover&lt;3-&gt;{Conclusion 2} \\           \end{tabularx}      \end{frame}   \end{document}   The output looks as follows :    As we can see cells in first column do not align with the rest of the rows. Will appreciate any input as to how to fix this. How do I avoid the "S to Skip" message on boot? After upgrading my laptop from karmic to lucid, my fat32 partition won't mount automatically. I get the message:  The disk drive for /osshare is not ready yet or not present Continue to wait; or Press S to skip mounting or M for manual recovery   Funny thing is, if I skip, then /osshare/ is mounted once I log in.   I've a similar setup on my desktop, and it works fine. Fstab on desktop:  UUID=4663-6853  /osshare        vfat    utf8,umask=007,gid=46 0       1   /etc/fstab on laptop:  UUID=1234-5678 /osshare vfat utf8,auto,rw,user 0 0 Jointly Complete Sufficient Statistics: Uniform(a, b) Let $\mathbf{X}= (x_1, x_2, \dots x_n)$ be a random sample from the uniform distribution on $(a,b)$, where $a &lt; b$.   Let $Y_1$ and $Y_n$ be the largest and smallest order statistics.  Show that the statistic $(Y_1, Y_n)$ is a jointly complete sufficient statistic for the parameter $\theta  = (a, b)$.     It is no problem for me to show sufficiency using factorization.   Question: How do I show completeness? Preferably I would like a hint.    Attempt: I can show $\mathbb E[g(T(x))] = 0$ implies $g(T(x)) = 0$ for the one parameter uniform distribution, but I am getting stuck on the two parameter uniform distribution.   I tried playing around with $\mathbb E[g(Y_1, Y_n)]$ and using the joint distribution of $Y_1$ and $Y_n$, but I am not sure if I am going in the correct direction, as the calculus is tripping me up. Is there a way to disable the hotkeys? I navigate with Ctrl-n,p,f,b (readline/ keys), so that I don't have to move my hands too much and get  again. Is there a way to disable Ctrl-b being hotkey for bold?  Note that this is not just an Emacs thing. Standard web browsers, like Firefox and Chrome, use things like  Ctrl + K and Ctrl + L to navigate to the Google search bar and address bar respectively. Why not make  but for questions too?  I love that everything is well though behind SO but I can't see no reason as to why great questions can't be prized with "Badges by subject". Most simple code to populate JTable from ResultSet Show that $8 \mid (a^2-b^2)$ for $a$ and $b$ both odd If $a,b \in \mathbb{Z}$ and odd, show $8 \mid (a^2-b^2)$.   Let $a=2k+1$ and $b=2j+1$. I tried to get $8\mid (a^2-b^2)$ in to some equivalent form involving congruences and I started with $$a^2\equiv b^2 \mod{8} \Rightarrow 4k^2+4k \equiv 4j^2+4j \mod{8}$$ $$\Rightarrow k^2+k-j^2-j=2m$$ for some $m \in \mathbb{Z}$ but I am not sure this is heading anywhere that I can tell.  Second attempt: Use Euler's Theorem and as $\gcd(a,8)=\gcd(b,8)=1$ and $\phi(8)=4$, $a^4 \equiv b^4 \equiv 1 \mod 8$ so $a^4-b^4\equiv 0 \mod{8}$.  I haven't gotten too much further are there any hints? I was setting up up android-studio referring to . While installing oracle-java7-installer, this error is displayed:  sudo apt-get install oracle-java7-installer [sudo] password for cyan:  Reading package lists... Done Building dependency tree        Reading state information... Done oracle-java7-installer is already the newest version (7u80+7u60arm-0~webupd8~1). The following package was automatically installed and is no longer required:   snap-confine Use 'sudo apt autoremove' to remove it. 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. 1 not fully installed or removed. After this operation, 0 B of additional disk space will be used. Do you want to continue? [Y/n] y Setting up oracle-java7-installer (7u80+7u60arm-0~webupd8~1) ... Downloading Oracle Java 7... --2017-11-27 08:40:47--  http://download.oracle.com/otn-pub/java/jdk/7u80-b15/jdk-7u80-linux-x64.tar.gz Resolving download.oracle.com (download.oracle.com)... 223.223.158.138, 223.223.158.136 Connecting to download.oracle.com (download.oracle.com)|223.223.158.138|:80... connected. HTTP request sent, awaiting response... 302 Moved Temporarily Location: https://edelivery.oracle.com/otn-pub/java/jdk/7u80-b15/jdk-7u80-linux-x64.tar.gz [following] --2017-11-27 08:40:47--  https://edelivery.oracle.com/otn-pub/java/jdk/7u80-b15/jdk-7u80-linux-x64.tar.gz Resolving edelivery.oracle.com (edelivery.oracle.com)... 23.58.58.93, 2600:140f:7:18e::2d3e, 2600:140f:7:18d::2d3e Connecting to edelivery.oracle.com (edelivery.oracle.com)|23.58.58.93|:443... connected. HTTP request sent, awaiting response... 302 Moved Temporarily Location: http://download.oracle.com/otn-pub/java/jdk/7u80-b15/jdk-7u80-linux-x64.tar.gz?AuthParam=1511752368_824528560c20825821ecbcac8f4e9be9 [following] --2017-11-27 08:40:48--  http://download.oracle.com/otn-pub/java/jdk/7u80-b15/jdk-7u80-linux-x64.tar.gz?AuthParam=1511752368_824528560c20825821ecbcac8f4e9be9 Connecting to download.oracle.com (download.oracle.com)|223.223.158.138|:80... connected. HTTP request sent, awaiting response... 404 Not Found 2017-11-27 08:40:50 ERROR 404: Not Found.  download failed Oracle JDK 7 is NOT installed. dpkg: error processing package oracle-java7-installer (--configure):  subprocess installed post-installation script returned error exit status 1 Errors were encountered while processing:  oracle-java7-installer E: Sub-process /usr/bin/dpkg returned an error code (1)   How do I fix this error? I did a fresh installation of Ubuntu 16.04. When I use ssh -X on my Mac (running OS X 10.6.7) to connect to my Ubuntu (11.04) box, I get the following warning:     Warning: untrusted X11 forwarding   setup failed: xauth key data not   generated Warning: No xauth data;   using fake authentication data for X11   forwarding.   Is there something I can do to make this warning go away?  If not, can I safely ignore it?  X11 forwarding seems to work fine, though I do see this message:     Xlib:  extension "RANDR" missing on   display "localhost:10.0".   Is that related to the warning?  (I'm guessing not.  If it's not, I'll file a new question about that.) Fastest Gun in the West Problem Mobile site says "your answer couldn't.." when submitting a question edit that creates a new tag Noticed this on Bicycles, but I presume it affects any Stack Exchange site   Write question Edit question, write a new tag Submit edit   When performing the above steps, I properly get an error message, but it says "answer" instead of "question" or "edit". Give us better search-by-score syntax tl;dr: Give the score search operator ability to change between greater than and less than.  I just noticed that searching for score:0 turns up only question with a score equal to 0. Searching for score:1 gives everything with a positive score. (&gt;=1) I can live with these. But here's the big one: score:-1. I would expect it to give all the negatively scored posts - there is a minus sign in there. But it doesn't. It gives all the posts &gt;=-1. Greater than? So, my  is that we get more granular search options for score. For example,  score:1 would show all posts with score == 1 score:1+ would show everything score &gt;= 1 score:0 would show all posts with score == 0 score:-1 would show everything score == -1 score:-1- would show al posts where score &lt;= -1  In short, the operator after the number would show which direction to go with the matching. A minus sign means less than, a plus sign means greater than. No sign means exact. Was Philip K. Dick able to watch a complete version of Blade Runner before his death or just unedited material?  The Blade Runner release date according to  and IMDb was 25 June, 1982. Philip K. Dick died a few months before, on March 2, 1982. How to remove the “AMD Testing use only” watermark? I've installed the  (beta) following the step in  for Ubuntu Quantal Quetzal. My system is 64 bit and my graphic card is an ATI RadeonHD 6670, this g.c. is Officially Supported (Catalyst &amp; Open Source), you can confirm that from this .  I don't have any problem, except the AMD testing use only watermark. I see the following frame in any stage into the OS (logged, unlloged, etc.) except in the terminals.     I found different versions of how to remove this image, but this change according to the system, so I want an answer from this popular (trusted) site.   How to solve this issue in Ubuntu 12.10 32b? This procedure is different in a 64b system? How do I update my TeX distribution? If I have bought 3 tickets for the same flight, can I bring three times the cabin allowance allowed per ticket?  The way I see it I am doing the airline a favor  Related: Background I'm working on a time series data set of energy meter readings. The length of the series varies by meter - for some I have several years, others only a few months, etc. Many display significant seasonality, and often multiple layers - within the day, week, or year. One of the things I've been working on is clustering of these time series. My work is academic for the moment, and while I'm doing other analysis of the data as well, I have a specific goal to carry out some clustering. I did some initial work where I calculated various features (percentage used on weekends vs. weekday, percentage used in different time blocks, etc.). I then moved on to looking at using Dynamic Time Warping (DTW) to obtain the distance between different series, and clustering based on the difference values, and I've found several papers related to this. Question Will the seasonality in a specific series changing cause my clustering to be incorrect? And if so, how do I deal with it? My concern is that the distances obtained by DTW could be misleading in the cases where the pattern in a time series has changed. This could lead to incorrect clustering. In case the above is unclear, consider these examples: Example 1 A meter has low readings from midnight until 8AM, the readings then increase sharply for the next hour and stay high from 9AM until 5PM, then decrease sharply over the next hour and then stay low from 6PM until midnight. The meter continues this pattern consistently every day for several months, but then changes to a pattern where readings simply stay at a consistent level throughout the day. Example 2 A meter shows approximately the same amount of energy being consumed each month. After several years, it changes to a pattern where energy usage is higher during the summer months before returning to the usual amount. Possible Directions  I've wondered whether I can continue to compare whole time series, but split them and consider them as a separate series if the pattern changes considerably. However, to do this I'd need to be able to detect such changes. Also, I just don't know if this is a suitable way or working with the data. I've also considered splitting the data and considering it as many separate time series. For instance, I could consider every day/meter combination as a separate series. However, I'd then need to do similarly if I wanted to consider the weekly/monthly/yearly patterns. I think this would work, but it's potentially quite onerous and I'd hate to go down this path if there's a better way that I'm missing.  Further Notes These are things that have come up in comments, or things I've thought of due to comments, which might be relevant. I'm putting them here so people don't have to read through everything to get relevant information.  I'm working in Python, but have rpy for those places where R is more suitable. I'm not necessarily looking for a Python answer though - if someone has a practical answer of what should be done I'm happy to figure out implementation details myself. I have a lot of working &quot;rough draft&quot; code - I've done some DTW runs, I've done a couple of different types of clustering, etc. I think I largely understand the direction I'm taking, and what I'm really looking for is related to how I process my data before finding distances, running clustering, etc. Given this, I suspect the answer would be the same whether the distances between series are calculated via DTW or a simpler Euclidean Distance (ED). I have found these papers especially informative on time series and DTW and they may be helpful if some background is needed to the topic area: What is the angle at which water is splashed when a vehicle tyre rides in water on the road? I would like to know the following:   What is the angle at which water gets splashed when I ride my vehicle through a water on the road? How does angle of water varies with speed? What is the relation between the distance the water goes with the speed of the vehicle?   Thanks! Converting MultiLinestring to Linestring with PostGIS