How many invertible elements are there in $\mathbb {Z_ {990}}$ and $\mathbb {Z_ {1060}}$ . Justify your answer. Hello, could someone explain to me how to find the invertible elements of a set as large as $\mathbb {Z_ {990}}$ and $\mathbb {Z_ {1060}}$ I thought about putting together the set and trying each one but it is very big and it can take forever
That is,     why is $\varphi(AB)=\varphi(A)\varphi(B)$, if $A$ and $B$ are two coprime positive integers?   It's not just a technical trouble—I can't see why this should be, intuitively: I bellyfeel that its multiplicativity should be an approximation at most. And why the minimum value for $\varphi (n)$ should be $ \frac{4n}{15} $ completely passes me by.
Showing that $A\cap B = A\setminus (A\setminus B)$ How does one show that $A\cap B = A\setminus (A\setminus B)$?  My thinking is that if $x\in A\setminus (A\setminus B)$ then this means that $x\in A$ and $x\notin A\setminus B$, which means that $x\in A$ and ($x\notin A$ and $x\in B$), but I don't see why the expression in parentheses should simplify to just $x\in B$. Any insight would be helpful.  Maybe $(x\in A^c$ and $x\in B)$, i.e, $x\in A^c\cap B$ is a possible simplification of the expression in parentheses, but still don't see why this simplifies to just $x\in B$. Why do I always hear remote train horn at night? This is definitely not an illusion, as many people have the same experience. I have usually lived in places miles away from train stations, which makes it unlikely to hear any train horns during the day. However at night, occasionally train horns can be heard. I hope if any one can explain the physics of this effect, like possibly sound travels faster at low temperature. Choosing OpenLayers or Leaflet? I was debating with one of my collegues on OpenLayers v/s Leaflet. I made a point that OpenLayers is much better API if we wish to build a project, where you need direct connectivity to the Geoserver and PostGIS.  Then I found , which looks pretty new but has the features of connectivity with the Geoserver and PostGIS.  So my project details are as follows,   Use the map interface to fetch Feature Info Create a customized tool that takes the lat/lon from user as to where he/she clicks on the map and then fetches the Climate Data from the raster (which is handled by a py script on the server) Allows user to upload excel, which is sent to the py script, which returns a GeoJSON, which creates Vector Features on the map Allow user to create vector polygons, which will fetch the Features it intersects from the WFS Layer. Fetches Layer from the PostGIS Datastore on GeoServer and displays the layers on the map.   So now I am confused on which is better and why using OpenLayers over Leaflet makes more sense or not? I built a desktop computer a couple years back with the following specs:  CPU: Intel  Yorkfield 2.5GHz 6 MB L2 Cache LGA 775 95W Quad-Core Processor BX80580Q9300 Motherboard:  LGA 775 NVIDIA nForce 680i SLI ATX Intel Motherboard Video Card: Two  GeForce 8600GT SCC 256 MB 128-bit GDDR3 PCI Express x16 SLI Supported Video Card PSU: SeaSonic S12 Energy Plus  ATX12V V2.3 / EPS12V V2.91 SLI Certified CrossFire Ready 80 PLUS Certified Active PFC Power Supply Memory: Two G.SKILL 4 GB (2 x 2 GB) 240-Pin DDR2 SDRAM DDR2 800 (PC2 6400) Dual Channel Kit Desktop Memory   Since its inception, the machine has periodically locked up, the regularity having varied over the years from once a day to once a month. Typically, lockups happen once every few days. By &quot;lockup&quot; I mean my computer just freezes. The screen locks up, I can't move the mouse. Hitting keys on my keyboard that normally turn LEDs on or off on the keyboard (such as Caps Lock) no longer turn the LEDs on or off. If there was music playing at the time of the lockup, noise keeps coming out of the speakers, but it's just the current frequency/note that plays indefinitely. There is no BSOD. When such a lockup occurs I have to do a hard reboot by either turning off the computer or hitting the reset button. I have the most recent version of the NVIDIA hardware drivers, and update them semi-regularly, but that hasn't seemed to help. I am currently using Windows 7 x64, but was previously using Windows Server 2003 x64 and having the same lockup issues. My guess is that it's somehow video driver or motherboard related, but I don't know how to go about diagnosing this problem to narrow down which of the two is the culprit.  Additional information re: cooling Regarding cooling... I've not installed any after-market cooling systems aside from two regular fans I scavenged from an older computer. The fan atop the CPU is the one that shipped with it. One of the two scavenged fans I added it located at the bottom tower of the corner, in an attempt to create some airflow from front to back. The second fan is pointed directly at the two video cards.  SpeedFan installation and readings Per studiohack's suggestion, I installed , which provided the following temperature readings:  GPU: 63C GPU: 65C System: 76C CPU: 64C AUX: 36C Core 0: 78C Core 1: 76C Core 2: 79C Core 3: 79C   Update #3: Another Lockup :-( Well, I had another lockup last night. :-( SpeedFan reported the CPU temp at 38 C when it happened, and there was no spike in temperature leading up to the freeze. One thing I notice is that the freeze seems more likely to happen if I am watching a video. In fact, of the last 5 freezes over the past month, 4 of them have been while watching a video on Flickr. Not necessarily the same video, but a video nevertheless. I don't know if this is just coincidence or if it means anything. (As an aside, each night before bedtime my 2 year old daughter sits on my lap and watches some home videos on Flickr and, in the last month, has learned the phrase, &quot;Uh oh, computer broke.&quot;)  Update #4: MemTest86 and 3DMark06 Test Results: Per suggestions in the comments, I ran the MemTest86 overnight and it cycled through the 8 GB of memory 5 times without error. I also ran the 3DMark06 test without a problem (see my scores at ). So... what now? :-) Any further suggestions on what to check? Is there some way to get a stack trace or something when the computer locks like that? Resolution I have never did figure out the particular problems, but based on the suggestions here and elsewhere, I'm presuming it was a motherboard issue. In any event, I recently upgraded my system, buying a new motherbeard, PSU, CPU, and RAM, and that new rig has been working splendidly the past several weeks. I am using the same graphic cards as in the old setup, so I think it's safe to reason that they weren't the cause of the problem. Is it appropriate to use a question mark at the end of a compound sentence where the first independent clause is a question but the second is a statement? For example:     Do you want attribution for that quotation, because I'm definitely sharing it and want to know if I should give you credit?   or      Do you want attribution for that quotation, because I'm definitely sharing it and want to know if I should give you credit.   I suppose this could work, but it's a little niche, as the second clause might not always be appropriate for a parenthetical.     Do you want attribution for that quotation (because I'm definitely sharing it and want to know if I should give you credit)? Can I get UAC prompt for user from batch file? Can I get UAC prompt for user from batch file?  For example I need put some keys to registry under CurrentControlSet from batch file by     reg import path\to\my.reg   PS. runas is not a solution as my customers uses different Windows localisation and I can not guess which name used for Administrator account (this can be Администратор)...  And script don't know administrator password.  And UAC dialog box does not require password entering... Which is correct, "you and I" or "you and me"? Accessing a property of derived class from the base class in C# In C#, what is the best way to access a property of the derived class when the generic list contains just the base class.  public class ClassA : BaseClass {    public object PropertyA { get; set; } }  public class ClassB: BaseClass {     public object PropertyB { get; set; } }  public class BaseClass { }  public void Main {     List&lt;BaseClass&gt; MyList = new List&lt;BaseClass&gt;();     ClassA a = new ClassA();     ClassB b = new ClassB();      MyList.Add(a);     MyList.Add(b);      for(int i = 0; i &lt; MyList.Count; i++)     {         //I would like to access PropertyA abd PropertyB from the derived classes             } } Is it possible to delete the "useless" apps pre-installed on my Mac? I'm using Microsoft Word 2010 and I worked in a word-2003 compatible document (.doc). After hours of working, I selected a half of text and did Ctrl+X to copy it into another document. Surprising, after closing the current document and opening a new one and pressing Ctrl+V, nothing happened. After closing original document, ms word asked me if I want to save the clipboard buffer and I pressed No, as I didn't knew that MS Word has it's own clipboard.  Is there a way to recover that text that was "cutten" from original document and never "pasted" to the new one? The PC was not rebooted so i guess all my temp files are here. Inequality Regarding e Install Nvidia driver instead of nouveau I've installed Ubuntu 14.04 and I'm experiencing a fuzzy/blurry screen and bad support for multiple screens.  I realized that Ubuntu installs nouveau as default, but I need to install nvidia drivers, so I tried this:   sudo apt-add-repository ppa:ubuntu-x-swat/x-updates apt-get update apt-get install nvidia-current apt-get upgrade       Now that nvidia is installed, you need to blacklist the nouveau driver   so it don’t pop out when you will reboot. Create or edit the following   file:  nano /etc/modprobe.d/blacklist-nouveau.conf       …and add these lines at the end:  blacklist nouveau blacklist lbm-nouveau options nouveau modeset=0 alias nouveau off alias lbm-nouveau off    I checked via the software &amp; updates if nvidia driver got installed successfully, and it says the driver installed is nvidia 304.117  I then tried to run:  lshw -c video | grep 'configuration'   and the result is that the driver is still nouveau:  configuration: driver=nouveau latency=0   So how do I install and use the nvidia drivers? "export PS1=" for customizing shell prompt The story:   Question posted on Programmers (let's call it ), Same question posted on Code Review (let's call it ), I migrate Q1 to Code Review, Someone notices that Q1 is a duplicate of Q2 and votes to close, I notice the duplicate notice and the auto-generated comment and also try to vote to close, but can't. No error message, nothing, it just doesn't work.   And then I notice that Q2 no longer exists ("voluntarily removed by its author"). My guess is the OP also noticed the duplicate vote, realized how messy cross posts are and decided to be a good citizen and delete one of the two questions. He picked the wrong one though, and now we're left with a duplicate notice pointing to a non existent question and an unresponsive dialog.  It's a mess ;) A weird and rare mess, but still a mess. What can we do about it? Story with VR worlds for the wealthy Proof relation between Levi-Civita symbol and Kronecker deltas in Group Theory In order to prove the following identity: $$\sum_{k}\epsilon_{ijk}\epsilon_{lmk}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$$ Instead of checking this by brute force, Landau writes thr product of Levi-Civita symbols as: $$\epsilon_{ijk}\epsilon_{lmn}=\det\left|    \begin{array}{cccc}       \delta_{il}   &amp; \delta_{im}  &amp; \delta_{in}   \\       \delta_{jl}   &amp; \delta_{jm}  &amp; \delta_{jn}   \\       \delta_{kl}   &amp; \delta_{km}  &amp; \delta_{kn}       \end{array} \right| $$  The proof that the equalty holds is quite straightforward if you consider what values the indices can take. But I've been told that there's a much more profound and elegant demonstration based on the representation of the symmetric group.  Does anybody know this approach based on group theory? Is it okay to record audio for a lecture? Sometimes the Professor will say things quickly, or say things before I can write them down. It would be of great benefit if I could record audio for the course, but I am not sure the legal guidelines around doing so. Prove inequality using Schwarz's Lemma I'm having trouble proving the following:  Let $f$ be analytic on $D(0,R)$ and suppose there is a real constant A for which $\operatorname{Re}f(z) &lt; A$ for all $z \in D(0,R)$. If $f(0) = 0$, show that $|f(z)| \leq \frac{2A|z|}{R-|z|}$.   I was told to first start by showing that $|\frac{f(z)}{2A-f(z)}| \leq 1$ and also use the following result, which I was able to prove:  Let $f$ be analytic on $D(0,R)$ and let $f(0)=0$. If $|f(z)| \leq M$ for all $z \in D(0,R)$, then $|f(z)| \leq M|z|/R$ for all $z\in D(0,R)$.  Thanks in advance! What possessive forms are used for mutual 1st person ownership? I want to talk to someone about the house that my wife and I own.  Saying, for example, "My wife's and my house is awesome," sounds a bit funny to me.  What's the best way to express this?  Clarification  I'm asking specifically about the grammar of multiple nouns in possessive form.  I'm particularly curious if it's possible to do this with a first-person pronoun (me).  I am capable of rephrasing this in other ways - my question is not how to express the idea, but about this particular grammatical construction, if it is even legal. Using zypper, how can I remove all packages installed from a specific repository?