For a simple undirected graph $G$, suppose we have two vertices $v_1$ and $v_2$ such that $G-v_1 \simeq G-v_2$. Does this necessarily mean that there is an automorphism of $G$ that maps $v_1$ to $v_2$?   This is just a condition that I've assumed to be true for awhile now without thinking too hard about whether or not it's actually true. Using the condition that $G-v_1 \simeq G-v_2$ has been a useful way to characterize two vertices as being "the same," but I was wondering if this actually corresponds to automorphisms of the graph.
Can somebody give me an example of a graph on eight vertices that has a pair of pseudosimilar vertices? Or, is it not possible? If it is not possible, please do not give me a proof, as I would like to figure out how to prove it myself.     Definition: vertices $u$ and $v$ in a graph $X$ are similar if there is an automorphism of $X$ that maps $u$ to $v$. If $u$ and $v$ are similar, then the vertex-deleted subgraphs $X-u$ and $X-v$ are isomorphic. If $X-u$ and $X-v$ are isomorphic but $u$ and $v$ are not similar, we say that they are pseudosimilar.
How can I see the history of stories I've read on Apple News? Sometimes I'll be interrupted in the middle of reading a story on Apple News and later when I open the app, it didn't save the story I was previously reading. I want to be able to resume reading the article, how can I find it again? Unable to start XAMPP using sudo in Desktop Entry Exporting feature class into multiple feature classes based on field values using ArcGIS Desktop? Check and fix SD card errors within Android itself? Constants in Objective-C I'm developing a  application, and I'm using constant NSStrings as ways to store key names for my preferences.  I understand this is a good idea because it allows easy changing of keys if necessary. Plus, it's the whole 'separate your data from your logic' notion.     Anyway, is there a good way to make these constants defined once for the whole application?     I'm sure that there's an easy and intelligent way, but right now my classes just redefine the ones they use. Mail App keeps popping up in the background in Mac OS Mojave What does the register keyword do in C language?  I have read that it is used for optimizing but is not clearly defined in any standard.  Is it still relevant and if so, when would you use it? Why does factoring eliminate a hole in the limit? $$\lim _{x\rightarrow 5}\frac{x^2-25}{x-5}  = \lim_{x\rightarrow 5} (x+5)$$  I understand that to evaluate a limit that has a zero ("hole") in the denominator we have to factor and cancel terms, and that the original limit is equal to the new and simplified limit. I understand how to do this procedurally, but I'd like to know why this works. I've only been told the methodology of expanding the $x^2-25$ into $(x-5)(x+5)$, but I don't just want to understand the methodology which my teacher tells me to "just memorize", I really want to know what's going on. I've read about factoring in abstract algebra, and about irreducible polynomials (just an example...), and I'd like to get a bigger picture of the abstract algebra in order to see why we factor the limit and why the simplified is equal to the original if it's missing the $(x-5)$, which has been cancelled. I don't want to just memorize things, I would really like to understand, but I've been told that this is "just how we do it" and that I should "practice to just memorize the procedure." I really want to understand this in abstract algebra terms, please elaborate. Thank you very much. Word to describe the fear of unknown future Removing overlapping part of polygons using QGIS? Here is what I'd like to achieve in QGIS.   Is there a processing tool to remove the overlapping part of polygons?    It is also very important that they do not get merged during the process and that they keep their original attributes since I need to work with them afterwards. Cannot set up multiple desktops (workspaces) Where do I put my systemd unit file? Search for a string and print everything before and after within a range I have this file:  sometext1{ string1 }  sometext2{ string2 string3 }  sometext3{ string4 string5 string6 }   I want to search this file for a specific string and print everything before this string up to the opening { and everything after this string up to the closing }. I tried to achieve this with sed but if I try to print everything in the range /{/,/string2/ for example sed prints this:  sometext1{ string1 }  sometext2{ string2 sometext3{ string4 string5 string6 }   If I search for the string "string2" I need the output to be:   sometext2{ string2 string3 }   Thanks. Proving $\sum\limits_{l=1}^n \sum\limits _{k=1}^{n-1}\tan \frac {lk\pi }{2n+1}\tan \frac {l(k+1)\pi }{2n+1}=0$ Prove that $$\sum _{l=1}^{n}\sum _{k=1}^{n-1}\tan \frac {lk\pi } {2n+1}\tan \frac {l( k+1) \pi } {2n+1}=0$$     It is very easy to prove this identity for each fixed $n$ . For example let $n = 6$; writing out all terms in a $5 \times 6$ matrix, we obtain:  $\begin{matrix}  \tan \dfrac {\pi } {13}\tan \dfrac {2\pi } {13}  &amp;  \tan \dfrac {2\pi } {13}\tan \dfrac {3\pi } {13} &amp; \tan \dfrac {3\pi } {13}\tan \dfrac {4\pi } {13} &amp; \tan \dfrac {4\pi } {13}\tan \dfrac {5\pi } {13} &amp; \tan \dfrac {5\pi } {13}\tan \dfrac {6\pi } {13}  \\  \tan \dfrac {2\pi } {13}\tan \dfrac {4\pi } {13}  &amp;  \tan \dfrac {4\pi } {13}\tan \dfrac {6\pi } {13} &amp; \tan \dfrac {6\pi } {13}\tan \dfrac {8\pi } {13} &amp; \tan \dfrac {8\pi } {13}\tan \dfrac {10\pi } {13} &amp; \tan \dfrac {10\pi } {13}\tan \dfrac {12\pi } {13} \\ \tan \dfrac {3\pi } {13}\tan \dfrac {6\pi } {13} &amp; \tan \dfrac {6\pi } {13}\tan \dfrac {9\pi } {13} &amp; \tan \dfrac {9\pi } {13}\tan \dfrac {12\pi } {13} &amp; \tan \dfrac {12\pi } {13}\tan \dfrac {15\pi } {13} &amp; \tan \dfrac {15\pi } {13}\tan \dfrac {18\pi } {13} \\ \tan \dfrac {4\pi } {13}\tan \dfrac {8\pi } {13} &amp; \tan \dfrac {8\pi } {13}\tan \dfrac {12\pi } {13} &amp; \tan \dfrac {12\pi } {13}\tan \dfrac {16\pi } {13} &amp; \tan \dfrac {16\pi } {13}\tan \dfrac {20\pi } {13} &amp; \tan \dfrac {20\pi } {13}\tan \dfrac {24\pi } {13} \\ \tan \dfrac {5\pi } {13}\tan \dfrac {10\pi } {13} &amp; \tan \dfrac {10\pi } {13}\tan \dfrac {15\pi } {13} &amp; \tan \dfrac {15\pi } {13}\tan \dfrac {20\pi } {13} &amp; \tan \dfrac {20\pi } {13}\tan \dfrac {25\pi } {13} &amp; \tan \dfrac {25\pi } {13}\tan \dfrac {30\pi } {13} \\ \tan \dfrac {6\pi } {13}\tan \dfrac {12\pi } {13} &amp; \tan \dfrac {12\pi } {13}\tan \dfrac {18\pi } {13} &amp; \tan \dfrac {18\pi } {13}\tan \dfrac {24\pi } {13} &amp; \tan \dfrac {24\pi } {13}\tan \dfrac {30\pi } {13} &amp; \tan \dfrac {30\pi } {13}\tan \dfrac {36\pi } {13}  \end{matrix}$  one can notice then, that the first column vanish the fourth one :  $\tan \dfrac {\pi } {13}\tan \dfrac {2\pi } {13}=-\tan \dfrac {12\pi } {13}\tan \dfrac {15\pi } {13}$  $\tan \dfrac {2\pi } {13}\tan \dfrac {4\pi } {13}=-\tan \dfrac {24\pi } {13}\tan \dfrac {30\pi } {13}$  $\tan \dfrac {3\pi } {13}\tan \dfrac {6\pi } {13}=-\tan \dfrac {16\pi } {13}\tan \dfrac {20\pi } {13}$  $\tan \dfrac {4\pi } {13}\tan \dfrac {8\pi } {13}=-\tan \dfrac {4\pi } {13}\tan \dfrac {5\pi } {13}$  $\tan \dfrac {5\pi } {13}\tan \dfrac {10\pi } {13}=-\tan \dfrac {8\pi } {13}\tan \dfrac {10\pi } {13}$  $\tan \dfrac {6\pi } {13}\tan \dfrac {12\pi } {13}=-\tan \dfrac {20\pi } {13}\tan \dfrac {25\pi } {13}$  and the third column vanish the fifth one :  $\tan \dfrac {3\pi } {13}\tan \dfrac {4\pi } {13}=-\tan \dfrac {30\pi } {13}\tan \dfrac {36\pi } {13}$  $\tan \dfrac {6\pi } {13}\tan \dfrac {8\pi } {13}=-\tan \dfrac {5\pi } {13}\tan \dfrac {6\pi } {13}$  $\tan \dfrac {9\pi } {13}\tan \dfrac {12\pi } {13}=-\tan \dfrac {25\pi } {13}\tan \dfrac {30\pi } {13}$  $\tan \dfrac {12\pi } {13}\tan \dfrac {16\pi } {13}=-\tan \dfrac {10\pi } {13}\tan \dfrac {12\pi } {13}$  $\tan \dfrac {15\pi } {13}\tan \dfrac {20\pi } {13}=-\tan \dfrac {20\pi } {13}\tan \dfrac {24\pi } {13}$  $\tan \dfrac {18\pi } {13}\tan \dfrac {24\pi } {13}=-\tan \dfrac {15\pi } {13}\tan \dfrac {18\pi } {13}$  while the second column is self-vanishing:  $\tan \dfrac {2\pi } {13}\tan \dfrac {3\pi } {13}=-\tan \dfrac {10\pi } {13}\tan \dfrac {15\pi } {13}$  $\tan \dfrac {4\pi } {13}\tan \dfrac {6\pi } {13}=-\tan \dfrac {6\pi } {13}\tan \dfrac {9\pi } {13}$  $\tan \dfrac {8\pi } {13}\tan \dfrac {12\pi } {13}=-\tan \dfrac {12\pi } {13}\tan \dfrac {18\pi } {13}$ .  So the equality occurs. But how to generalize the proof? I noticed a peculiar behavior of my Ubuntu 13.04 (although possibly it's also present in previous versions): when my system freezes completely (I do not ask for troubleshooting this) so that even X does not respond to Ctrl+Alt+FX keys, the CapsLock toggle diode on my keyboard blinks (with frequency of ~1Hz), although the system is completely unresponsive.   What is the cause of this phenomena? Is it caused by some internal kernel mechanisms? Is this specific to Ubuntu (is it caused some custom Ubuntu additional software)? Does it has anything to do with HAL? What is the purpose of it? Can I customize this behavior on Ubuntu? US citizens travelling to Cuba are restricted to a set of "reasons" for travel, of which tourism is not one. I have US-Philippines dual citizenship, and both passports. Would I just buy the ticket using my Filipino passport? What kind of documentation/proof would I need to return back to the US? And as a US citizen travelling with a foreign passport, where would I go for US customs?  Thanks. I am thinking of building a model predicting a ratio $a/b$, where $a \le b$ and $a &gt; 0$ and $b &gt; 0$. So, the ratio would be between $0$ and $1$.  I could use linear regression, although it doesn't naturally limit to 0..1.  I have no reason to believe the relationship is linear, but of course it is often used anyway, as a simple first model.  I could use a logistic regression, although it is normally used to predict the probability of a two-state outcome, not to predict a continuous value from the range 0..1.  Knowing nothing more, would you use linear regression, logistic regression, or hidden option c? I can't not be pregnant Metapost label based on variable value I want to label a scale based on a variable value and I wrote a metapost code like this.  $ := 1; for i = -.3cm step .6cm until 3.6cm:     label.bot(str$,(i,-3.7cm));     $ := incr($); endfor;   It just labels "$" instead of its value. How to output the variables value? Minecraft Forge 1.7.10 will not start My minecraft will not start on only that version, I have a Windows 7 computer with Java 8u20 installed, because the latest java 7 update will not work at all(plus imma dev for other stuff).   [22:26:39] [main/INFO] [LaunchWrapper]: Loading tweak class name   cpw.mods.fml.common.launcher.FMLTweaker  [22:26:39] [main/INFO] [LaunchWrapper]: Using primary tweak class name   cpw.mods.fml.common.launcher.FMLTweaker  [22:26:39] [main/INFO] [LaunchWrapper]: Calling tweak class   cpw.mods.fml.common.launcher.FMLTweaker  [22:26:40] [main/INFO] [FML]: Forge Mod Loader version 7.10.25.1208 for Minecraft   1.7.10 loading  [22:26:40] [main/INFO] [FML]: Java is Java HotSpot(TM) 64-Bit Server VM, version   1.8.0_20, running on Windows 7:amd64:6.1, installed at C:\Program   Files\Java\jre1.8.0_20  [22:26:40] [main/INFO] [LaunchWrapper]: Loading tweak class name   cpw.mods.fml.common.launcher.FMLInjectionAndSortingTweaker  [22:26:40] [main/INFO] [LaunchWrapper]: Loading tweak class name   cpw.mods.fml.common.launcher.FMLDeobfTweaker  [22:26:40] [main/INFO] [LaunchWrapper]: Calling tweak class   cpw.mods.fml.common.launcher.FMLInjectionAndSortingTweaker  [22:26:40] [main/ERROR] [LaunchWrapper]: Unable to launch  java.util.ConcurrentModificationException     at java.util.ArrayList$Itr.checkForComodification(Unknown Source) ~[?:1.8.0_20]     at java.util.ArrayList$Itr.remove(Unknown Source) ~[?:1.8.0_20]     at net.minecraft.launchwrapper.Launch.launch(Launch.java:117) [launchwrapper-  1.9.jar:?]     at net.minecraft.launchwrapper.Launch.main(Launch.java:28) [launchwrapper-1.9.jar:?]  Java HotSpot(TM) 64-Bit Server VM warning: Using incremental CMS is deprecated and   will likely be removed in a future release