A function $f : [a,b] \to [a,b]$ is continuous for all $x \in [a,b].$ Prove that there exists a $c\in [a,b]$ such that $f(c) = c.$
Suppose that $f$ is a function defined in $[a;b]$ to $[a;b]$ and continuous on $[a;b]$. The problem is I haven't the definition of the function, this is more abstract, but even if how can I prove that $f$ would have a fixed point?
Footnote without a marker Capacitance of a single charged plate? Lets say we have a single plate that has a charge of $+Q$ on it. A plate with charge $-Q$ is infinite distance away. Will the plate with $+Q$ have a capacitance associated with it? Why or why not? I was thinking that because the distance between the plates is infinity, the capacitance is zero, but according to my TA, that is not the case. There is always some sort of "absolute" capacitance for the plate with $+Q$ (a capacitance will exist between the plate and air?). Can someone help me out in understanding this? Usernames vs. Real Names I want to create a social network mainly geared towards user-to-user interactions such as commenting and discussion boards. From my observations, I've found that users are more likely to engage in constructive conversations when using their full names as opposed to creating flame wars when hiding behind a username. While the latter still has the possibility to exist using one's real name, it's not as evident considering that the user is bearing his or her identity—either through the site's main registration or through Facebook Connect.  Don't get me wrong I've seen many communities manage to still offer insightful user contributions using usernames. All in all, what are the pros and cons of the two? How do I get current date/time on the Windows command line in a suitable format for usage in a file/folder name? How do I make a WinForms app go Full Screen I have a WinForms app that I am trying to make full screen (somewhat like what VS does in full screen mode).  Currently I am setting FormBorderStyle to None and WindowState to Maximized which gives me a little more space, but it doesn't cover over the taskbar if it is visible.  What do I need to do to use that space as well?  For bonus points, is there something I can do to make my MenuStrip autohide to give up that space as well? The Half-Elven and their choices Earendil and Elwing were given the choice of kindreds after their journey to Valinor, This choice was extended to their sons (specifically sons, not daughters or grandchildren)  Elrond chose the race of the Eldar and his brother Elros chose to be counted amongst the Edain. Thus far Manwë's proclamation has been fulfilled.  There are some inconsistencies here:   Arwen was given the choice, but Elladan and Elrohir remained as Eldar in Imladris Tar-Vardamir was not given the choice The offspring of Imrazor and Mithrelas in Dol Amroth dont seem to have been given a choice   I can think of only one explanation, that the Mannish blood is dominant over the blood of the Eldar, thus, As Vardamir was the offspring of two mortals he did not fall under Manwë's proclamation despite being the son of a half elven.  However this does not seem to explain the other inconsistencies, Is it possible that Elladan and Elrohir and also the princes of Dol Amroth weren't considered important enough by Manwë or Eru to give them the choice.  Also as a side note Tolkien states there were only three unions between man and elf, but this seems to ignore Turin/Finduilas and Imrazor/Mithrelas  My question is, Why was Arwen given the choice and not the sons of Elros? "This computer currently has no detected operating systems" when installing Is there any software that can easily create a Styled Layer Descriptor (SLD) without using a  script? Best practices for catching and re-throwing .NET exceptions iOS: Compare two dates Singular or Plural setting to a variable. PHP In Windows 10 how can we change the name of this folder under C:\Users? I'm using Windows 10 on my personal computer (HP Pavilion Laptop) and you can see in the Screen Shot under C:\Users there are two folders.    The name of the first folder is C:\Users\SINGHA, however, the correct spelling of my last name is SINHA.  I need to change the name of this folder to C:\Users\SINHA so that it shows the correct name.     Under User Account, the spelling of my name is correct - it's only under C:\Users\ where the name is spelt incorrectly. Consequently, if I open/edit a Word or Excel document, the name of the owner being shown is the incorrect one (i.e it's shown as C:\Users\SINGHA instead of C:\Users\SINHA)   If I right-click on this folder, I won't get the Rename option; How can I change the name of this folder? El Capitan froze and rebooted (Macbook Pro Retina Mid 2012) 2.3 Ghz Intel Core i7 8GB DDR3: Was running NetAdmin Pro which is monitoring  my office LAN, processors consumption was at  ~15 - 20%. Simultaneously I ran WiFI Explorer,  3.7% no more than 5% CPU usage. And I was  updating Xcode to v 7.3.1 and was browsing and  watching video from playground.ru Watch the videos until half of it (20mins) the Video froze I moved the mouse and turned out  the whole OSX is frozen. littleSnitch v 3.7 has no reaction I immediately pull the Ethernet adaptor and  MacBook just went to black screen. I then  switch off the wireless router. (My MacBook Pro  is connected to the network via Ethernet  adaptor and via WiFi router bearing different IP, I was thinking of cutting both network  communication, in case it was an intrusion) MacBook just restarted itself to login screen,  after 15-20 sec of black screen and total  silence) I again restarted and booted into Recovery  mode use disk utility. After completing disk  check. Login and previously opened programs resume  back (NetAdmin Pro, WiFi Explorer, Yandex  browser, Activity Monitor and AppStore) AppStore doesn't resume the download  everything else is functioning normally and in Addition I have got my calendar open after login (I did not launch accidental nor was it set to  start with system login)   Question 1: Are there any way to know what was happening and what was OS El Capitan last reaction before it froze and rebooted?  Question 2: Are there any software better than "Console" v10.11 to interpret what was going on before it crashes?  P.S. Thank you in advance, I will try to upload the Console messages later from the MacBook Pro    DDT-2:~ Chaleune$ syslog | grep -i "shutdown cause"  Jun 22 14:15:37 DDT-2 kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5  Jun 22 20:34:46 localhost kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5  Jun 23 15:18:23 DDT-2 kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5  Jun 24 17:58:32 localhost kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5  Jun 24 20:09:52 localhost kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5  Jun 24 22:58:15 localhost kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5Jun 25 14:43:36 DDT-2 kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5  Jun 25 21:55:31 localhost kernel[0] &lt;Notice&gt;: Previous shutdown cause: 3  Jun 25 22:20:57 localhost kernel[0] &lt;Notice&gt;: Kext com.apple.driver.AppleOSXWatchdog failed to load (0xdc008012).Previous shutdown cause: 5  Jun 26 10:18:44 localhost kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5  Jun 26 12:09:45 localhost kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5  Jun 26 18:00:31 localhost kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5  Jun 27 13:05:58 DDT-2 kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5  Jun 27 22:56:11 localhost kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5  Jun 28 14:18:27 DDT-2 kernel[0] &lt;Notice&gt;: Previous shutdown cause: 5  *Jun 29 18:56:21 localhost kernel[0] &lt;Notice&gt;: Previous shutdown cause: -128*  Jun 29 23:31:22 DDT-2 kernel[0] &lt;Notice&gt;: Previous shutdown cause: -128      shows the following   dsIllInstErr                  = 3,    /illegal instruction error/ dsChkErr                      = 5,    /check trap error/ userCanceledErr               = -128    gave the following interpretation:   03 Illegal Instruction The computer has a specific vocabulary of machine language instructions it can understand. If a computer tries to execute an instruction that isn't in its vocabulary, you see this error code. It's less likely than error 02, but still very common. 05 Range Check Error Programmers can use an instruction in the Motorola 68000 to check if a number is within a certain range. This error indicates that the number tested isn't in the specified range. -128  userCanceledErr         User canceled an operation Limit problem arctg (1 ^ infinty) Can anyone help me with this limit problem without L'Hopital rule and Taylor series? $$\lim_{x\rightarrow\ -\infty}\left(-2\frac{\arctan(x)}{\pi}\right)^x$$ So I've been pulling my hair out over this for the last couple days now, I can't seem to find any solutions that work. I've fresh installed Ubuntu Gnome 16.04 on my system which has no other OS's running on it. Everything's fine aside from the fact that the default Nouveau display driver seems to be causing the UI to freeze sporadically every now and then. No problem right? Just install the proprietary drivers for my GPU and we should be golden? Well here's the thing, no matter how I try to get my GPU's drivers installed I always seem to hit some sort of issue. Here's what I've tried so far:   Switch to the proprietary driver from within the 'Software &amp; Updates' app in Ubuntu. For me the only option available is the 'NVIDIA binary driver - version 375.66 from nvidia-375 (proprietary, tested)'.    After selecting this driver it goes through the install process, asks me to 'configure secure boot' by entering a password, etc. This all seems to work fine initially until I reboot, but then before I see the login screen it seems to open a virtual terminal (usually accessed via alt+shift+F1), but then immediately boots me back to a black screen where it will flash this terminal for a split second every 4 seconds or so. I can access this terminal by using alt+shift+F1, and go on to use it as normal, but it will continually boot me back out to the black screen every few seconds. The only way I can solve this is to keep accessing the terminal each time it boots me out while I purge the nvidia drivers using sudo apt purge nvidia-*. After purging I'm then able to access the login screen and access my Ubuntu again as normal, besides having to reboot to presumably activate the Nouveau drivers again otherwise the resolution is all messed up and only one of my three monitors has anything displayed on it.    Using the steps listed  I download the latest stable/certified driver for my GPU (which as of writing this is version 384.69). However on step 3 where it instructs me to run the driver using sudo sh, I reach an issue where the installer informs me that I need to disable the Nouveau driver in order to continue. I've tried everything from uninstalling Nouveau using 'sudo apt-get remove --purge xserver-xorg-video-nouveau` to blacklisting Nouveau using the steps listed , by editing the 'nvidia-graphics-drivers.conf'.    Unfortunately no matter what I do to try and disable Nouveau the installer always informs me that the Nouveau driver is enabled and needs to be disabled in order to continue. What's funny is that I managed to install the driver using this method once before, but for the life of me I can't get it to work again.    Finally, I've attempted to install the driver by installing the PPA. I run sudo add-apt-repository ppa:graphics-drivers/ppa, sudo apt update, and finally going back into the 'Software &amp; Updates' app and select the 384.69 driver. However after going through the install process the same issue as described in my first method (1.) occurs.    I've also tried switching between gdm3 and lightdm with all 3 methods to no avail. I'm at a complete loss as to what to do. If anyone has any possible solutions I'd love to hear them, because right now I'm at the end of my tether.   If you require any further information just let me know and I'll be more than happy to provide it.   System info:   AMD FX-8350  Nvidia GeForce GT710 1gb  16gb RAM  Ubuntu Gnome 16.04 LTS How do I get the feature id after featureAdded signal (PyQGIS)? I'm trying to make an edit form in QGIS (Python code) ,and i want after feature drawing to get the id, and the feature to be selected. Do you have any ideas to do that?  Tried so far:  self.iface.activeLayer().featureAdded.connect(self.myfonction)  def myfonction(self):      for feat in self.iface.activeLayer().getFeatures():         self.iface.activeLayer().setSelectedFeatures([feat.id()]) Geometric proof that a Pythagorean triple has a number divisible by $3$? Can I infer from “If $a,b \in \mathbb{N}$ such that $(a,3)=(b,3)=1$ then $a^2+b^2\neq c^2$ for any $c\in\mathbb{N}$” that any pythagorean triple has a number which is divisible by $3$? If so, how can this relationship be proved geometrically? Guess the formula for $\sum\frac 1{(4n-3)(4n+1)}$ and prove by induction I was wondering why a laser beam diverges. If all the photons are in the same direction, I would imagine that it would stay that way over a long distance. I am aware that a perfectly collimated beam with no divergence cannot be created due to diffraction, but I am looking for an explanation based on  photons rather than wave physics. When is unbalanced data really a problem in Machine Learning? We already had multiple questions about unbalanced data when using , , ,  and a number of other similar questions, what makes it a very popular topic! Unfortunately, each of the questions seems to be algorithm-specific and I didn't find any general guidelines for dealing with unbalanced data.  Quoting , dealing with unbalanced data     (...) heavily depends on the learning method. Most general purpose   approaches have one (or several) ways to deal with this.   But when exactly should we worry about unbalanced data? Which algorithms are mostly affected by it and which are able to deal with it? Which algorithms would need us to balance the data? I am aware that discussing each of the algorithms would be impossible on Q&amp;A site like this, I am rather looking for a general guidelines on when it could be a problem.