How to use laptop as a monitor for my desktop?
Can I use my laptop as a second monitor?
$\ker \phi = (x_1-a_1, ..., x_n-a_n)$ for a ring homomorphism $\phi: R[x_1, ..., x_n] \to R$ Let $R$ be a commutative ring, $a_1, ..., a_n$ its elements and  $\phi: R[x_1, ..., x_n] \to R$  defined by  $ \phi(f(x_1, ..., x_n)) = f(a_1, ... ,a_n)$  a ring homomorphism.  Prove: $\ker \phi = (x_1-a_1, ..., x_n-a_n)$  It is obvious that $(x_1 - a_1, ..., x_n -a_n) \subseteq \ker \phi$. I'm not sure how to prove the converse.  At this point I don't know any division algorithms for multivariable polynomials, only for the ones in $R[x]$(and the book from where I taken the exercise doesn't assume the reader to know something beyond basics of rings and ideals, and the division algorithm for $R[x]$). Though I know this could be solved for $i = 1$ by dividing by $x-a$:   Let $f(x) \in \ker \phi$, divide by $x-a: f(x) = q(x)(x-a) + r, f(a) = r = 0$, so $f(x) = q(x)(x-a) \in (x-a)$. Turning DisplayPort monitor off disables monitor completely I have 2 monitors connected to a NVIDIA GeForce GTX 660:   Left is 1920x1080 via DVI (Samsung SyncMaster 2443BW) Right is 2560x1440 via DisplayPort and is set to Primary (Samsung SyncMaster SA850)   I often leave my computer running overnight (working on long render job, doing backups, etc) so I only power off my monitors manually. However, if I power off the primary monitor, it is disabled completely as if it were unplugged from the graphics card (i.e. icons are moved, open apps are moved and resized to fit on the new primary monitor). If I switch from DP to a DVI cable, this does not occur. It seems to be an issue when using DisplayPort.  Is there a way to disable the monitor from being disabled completely in Windows when I power it off physically?   I have found no relevant settings in either Windows Display settings, nor the NVIDIA control panel software.  Edit #1: I'm using an .  Edit #2: Here is a screenshot of my NVIDIA Control Panel v7.2.710.0:   Edit #3: Here are screenshots of my Monitor configuration:    Clicking on Advanced settings reveals a Generic PnP monitor running at 59 Hz? 60 Hz is the only other option available.    Here is my secondary monitor which is connected via DVI:    Edit #4: If I open the "Screen Resolution" configuration panel and power the monitor off (via the power switch), this is what is shown:    It looks like I disconnected the monitor from the video card. Panasonic Lumix G 25mm f/1.7 ASPH vs Olympus M. Zuiko 25mm f/1.8 What does increased caster level affect? A  adds +1 caster level, what does this affect?   Damage from spells? (Like fireball) Duration and area of effect? "Amount" of effect (Like number of missiles from magic missile) Spells known by a wizard (ie +1 caster level mean a level 2 wizard gets the spells as if they were level 3?) Anything else? Irreducibility of an infinite sequence of polynomials Can there be temperature inside of a massive Black Hole? Title of 80's movie: future-era timetraveling humans go extinct and rescue/kidnap todays air crash victims What is the title of a 80's (i believe?) movie where humans in the far future are going extinct, and are aquiring new humans by going back in time to todays world, kidnapping air travelers that are known to have died that day, from the flying passenger aircraft just before its crashing.  A key visual:    Inside a hangar, a time portal opened, and a tail of a (flying) passenger aircraft extruded into the hangar where a stairs provided access to the back door of the aircraft.  The kidnappers made sure the surviving passengers remain seated and would forget about the whole ordeal. Kidnapped humans got replaced by lookalike dummies, including things they had on them. They tried hard not to alter the timeline. Array of Matrices in MATLAB I am looking for a way to store a large variable number of matrixes in an array in MATLAB.  Are there any ways to achieve this?  Example:  for i: 1:unknown   myArray(i) = zeros(500,800); end   Where unknown is the varied length of the array, I can revise with additional info if needed.  Update: Performance is the main reason I am trying to accomplish this.  I had it before where it would grab the data as a single matrix, show it in real time and then proceed to process the next set of data.  I attempted it using multidimensional arrays as suggested below by Rocco, however my data is so large that I ran out of Memory, I might have to look into another alternative for my case.  Will update as I attempt other suggestions.  Update 2:  Thank you all for suggestions, however I should have specified beforehand, precision AND speed are both an integral factor here, I may have to look into going back to my original method before trying 3-d arrays and re-evaluate the method for importing the data. What is the best way to remove and prevent leaf stains on concrete pavers? How to apply layer style to one object and apply it to multiple layers I've believed for a long time that every Lie group can be decomposed as the semidirect product of a connected Lie group and a discrete Lie group. However, in , it is mentioned that this is not true, and that there exist obstructions to this decomposition. Unfortunately, the level of complexity in the answers goes a bit over my head.  This type of decomposition is also addressed in .  With this loss of faith in my intuition, I have three questions:   Is it true that Lie groups do not necessarily decompose into the semidirect product of a connected group and a discrete group? If this decomposition doesn't always work, then is there a particularly nice argument as to why it doesn't? If this decomposition doesn't always work, then is there a nice example I can use to convince myself that my intuition is wrong?   Edit: As much as I love Qiaochu Yuan's answer, which is thorough and (as usual) a role model among Stack Exchange answers, this is not what I'm looking for. I'm aware of the general extension problem, and that not every short exact sequence splits, but my intuition has led me to think that each Lie group can be decomposed this way. I will certainly play around with his outline for a construction, but I would much prefer a more concrete answer. Once again, thank you!  Edit: As of yet, playing with the construction suggested by Qiaochu hasn't yielded anything interesting. This question is still open to answers! Homomorphism between compact Hausdorff spaces. the slash (/) after a directory name on shell commands I have a little question here.  If I have two files, say filea and fileb, mv filea fileb would   delete fileb rename filea to fileb   Then if I have two directories, say dira and dirb, mv dira dirb would   move dira into dirb (it will become dirb/dira)   Noting that in both cases there are no notice or message, then this is pretty inconsistent to me. I think mv dira dirb should just overwrite dirb with the contents of dira (or merge the two directories under a directory named dirb).  I remember reading somewhere that a directory name with a slash (like dira/) is treated like a directory, and name with no slash (like dira) is treated like a file (to certain extents, of course). Anyway now I want to make the shell (zsh and possibly bash) respect my notation of a directory by using a slash. Is there a terminal option which enable me to enforce that?  To clarify, here is my desired behaviour:   mv dira dirb results in dirb being overwritten with the contents of dira mv dira dirb/ results in dira being moved into dirb (in dirb/dira)   Has anyone thought the same way as me? Or am I just weird? Identity for convolution of central binomial coefficients: $\sum\limits_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$ It's not difficult to show that  $$(1-z^2)^{-1/2}=\sum_{n=0}^\infty \binom{2n}{n}2^{-2n}z^{2n}$$  On the other hand, we have $(1-z^2)^{-1}=\sum z^{2n}$.  Squaring the first power series and comparing terms gives us  $$\sum_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}2^{-2n}=1$$  that is,  $$\sum_{k=0}^n \binom{2k}{k}\binom{2(n-k)}{n-k}=2^{2n}$$  My question: is there a more direct, combinatorial proof of this identity?  I've been racking my brains trying to come up with one but I'm not having much success. How can I use the MS JDBC driver with MS SQL Server 2008 Express? My configuration:     windows XP SP3 JDBC 2005 MS SQL Server 2008 Express, exposed via tcp/ip on port 1433 sqljdbc.jar in class path    I tried:  try {     Class.forName("com.microsoft.sqlserver.jdbc.SQLServerDriver").newInstance();     con = DriverManager.getConnection("jdbc:microsoft:sqlserver://localhost:1433/SQLEXPRESS2008;databaseName=Test;selectMethod=cursor", "sa", ""); } catch (Exception e) {     e.printStackTrace(); }   But it always throws an exception:  java.sql.SQLException: No suitable driver   I also tried the following urls:  localhost:1433/SQLEXPRESS2008  localhost/SQLEXPRESS2008  localhost   Same results.  Any help? How to measure time taken by a function to execute Rounding Percentages What are the requirements for a (beta) site logo? Is there a criterion to differentiate "thriller" from "horror"? If there was a black hole on earth, what would it look like? If an answer does exist, I'd love to hear it. I'm trying to incorporate a doomed earth story in something I'm writing, and the end of the world I'm going for is a black hole.  Let's say the black hole was created in the Philippines (I chose it because it's on the equator). Here's  for reference.  If I was standing 90 degrees of that, in Kenya, or if I was standing 180 degrees of that, in Mexico; what would I see in those two separate instances? What if I was standing in Japan above it, or in Australia below it?  Let's also say I was riding a boat towards the source of the black hole. Unlike most imagined scenarios where I am orbiting the black hole and slowly falling into it, this time I'm no longer orbiting it - I'm getting sucked into it instead. I'm imagining that the black hole will "half" be inside Earth and "half" will be in the atmosphere. But as it pulls me in, what will it look like to me (if I was massless, so no tidal forces)? It's probably not going to be a black hemisphere sticking out of the ground. Light will be lensed around it in some way so as to appear like a hole in the air.  Also, if the black hole was very small, it probably won't have a strong gravitational pull, and assuming it doesn't disappear, you'd be able to look at it (or at least, its effect on the light in its surroundings). Of course, as you approach it, it would still have an event horizon, so spacetime would still be warped if you're near enough and the surroundings will still experience tidal forces.  As the black hole gains mass, it will get larger, and so its event horizon will increase. If I was standing in Kenya and the event horizon "washes over me," (as in a water wave) what will I see?  I've so far imagined that, at least when seen from outer space, there will be a hole on the ground and light will be lensed around it. When you are on the ground, though, it becomes a bit harder to imagine.  I do hope someone can lend me a hand!  Edit: Someone had previously answered this question, but I'm not sure why it's been deleted. It was pointed out to me, though, that a black hole cannot stay in place. So an additional question for me is, does that mean it is impossible to be standing still, and the event horizon "washes over you"?  Edit2: Here's a guiding thought experiment that could answer the question. Imagine that you are in outer space directly above the black hole, looking at the black hole that is on the earth's surface. Can you imagine what it would look like? It would simply be a hole with the surrounding light gravitationally lensed around it (possibly lensing the entire earth if it's big enough - but let's say it's not).  Now imagine that you are on the direct opposite of where you were above - you're right behind the black hole this time, looking directly at the black hole but unable to see it because the earth is blocking your view.  Now try to imagine, from the second image, rotate the earth into the first image. I find it a bit hard to do. Just as the black hole is appearing as you rotate the earth, how would the light be lensed?