Infinite Electric Field Energy If the total electric energy associated with a field is:    and the electric field of a point charge follows an inverse square law:    then $U_{ES}$ would be infnite if you compute the integral. Can someone shed light on what it means for "infinite total energy" to be associated with a particle?  Secondly, if charges are actually "points" and do not have a finite radius, shouldn't all matter collapse into black holes?   Could someone explain what it means for an electron or other fundamental particle to be associated with a finite radius? What is the "electron radius," and how is it measured? What is self-interaction energy?  equations from
Electrical potential energy stored in vacumm for a single point charge? I have come to know the electrostatic potential energy in vacuum is given by $${\frac{1}{2}} \epsilon_0\int d^3x {E^2} $$ and this energy  is due to the mutual electrostatic coulomb potential energy. So only a single point charge in space also contributes to some electrostatic potential energy according to $${\frac{1}{2}} \epsilon_0 \int d^3x {E^2} $$  as due to a single point charge also we have an electric field at every point in space. Now I am not getting it physically, why should there be any Coulomb energy in this case as there is no other charge to provide the electrostatic coulomb potential energy?
Do reroll and dice-adding abilities work on critical hits and critical failures? Evaluating the sum $\sum_{n=1}^{\infty}\dfrac{(-1)^{n}}{n^{2}}$ I am tasked to evaluate the sum  $$\sum_{n=1}^{\infty}\dfrac{(-1)^{n}}{n^{2}}$$ Using contour integration. This is what I've done so far.   Let $C_{N}$ be the square defined by the lines $x=\pm(N+\tfrac{1}{2})\pi$ and $y=\pm(N+\tfrac{1}{2})\pi$. Let $f=\tfrac{1}{z^{2}\sin(z)}$. I was able to prove that  $$\int_{C_{N}}\dfrac{1}{z^{2}\sin(z)}dz=2\pi i\left[\dfrac{1}{6}+2\sum_{n=1}^{N}\dfrac{(-1)^{n}}{\pi^{2}n^{2}} \right]. $$  By Cauchy's residue theorem. So now if I could prove that he integral converges to zero then I would be done. The problem is that I can't seem to get an upper bound on the integral. I've having trouble with the fact that $|f|$ is unbounded within $C_{N}$ so ML inequality doesn't help me. Any hints would be great. Thanks. Examples and Counterexamples of Relations which Satisfy Certain Properties Definition: Given a set $X$, a relation $R$ on $X$ is any subset of $X\times X$.  A relation $R$ on $X$ is said to be  reflexive if $(x,x) \in R$ for all $x \in X$, irreflexive if $(x,x) \not\in R$ for all $x \in X$, transitive if $(x,y) \in R$ and $(y,z) \in R$ implies that $(x,z)\in R$, intransitive (or antitransitive) if $(x,y) \in R$ and $(y,z) \in R$ implies that $(x,z)\not\in R$, symmetric if $(x,y) \in R$ implies that $(y,x) \in R$, antisymmetric if $(x,y) \in R$ and $(y,x) \in R$ implies that $x=y$.   Given any combination of the properties listed above, is there a nontrivial (i.e. nonempty) relation which satisfies that combination of properties? Can summons and eidolons be magically healed? This is a two part question.   Can summons benefit from ? Can  benefit from magical healing? This question treats normal and unchained summoners as equal unless there is a specific difference where it concerns healing.   I have looked at the  spell and it does not say that the monster summoned actually exists (for instance the eidolon specifically states that its an aspect of the creature and not the creature itself) or if its an aspect of a greater creature.  Reading cure light wounds it talks about what happens with living and undead, but nothing else is mentioned. Why does the natural ring homomorphism induce a surjective group homomorphism of units? I'm trying an old problem here:   Suppose $n\mid m$, and I have a natural ring homomorphism $\varphi\colon \mathbb{Z}/m\mathbb{Z}\to\mathbb{Z}/n\mathbb{Z}$ defined by $\varphi(j)=j\mod{n}$. I can verify that this is a ring homomorphism, but why does it induce a surjective group homomorphism on the unit groups $(\mathbb{Z}/m\mathbb{Z})^\times\to(\mathbb{Z}/n\mathbb{Z})^\times$?  My question in particular is why is it surjective? I take $k\in(\mathbb{Z}/n\mathbb{Z})^\times$. Then there exist integers $s,t$ such that $sn+tk=1$. Since $n\mid m$, I can also write $m=na$. So multiplying through I get $sm+tka=a$. I'm lost here. How can I find a unit in $(\mathbb{Z}/m\mathbb{Z})^\times$ which maps to $k$ to show surjectivity? Command to append line to a text file without opening an editor Assuming i have a line that i want to add to a file without opening an editor.  How could i append this line  alias list='ls -cl --group-directories-first'   to this file  config.fish I was wondering if there is a viable place on Stack Exchange to ask economics-related questions.  Questions I'd have in particular could be equation-based or theoretical.  I suppose the equation-based questions could possibly fit into mathematics but I haven't checked their FAQ. In France the higher education is like this:   3 years of "Licence" 2 years of "Master" 3 years of "Doctorat"   But what about in the United States?  I heard that they have things called "undergraduate", "graduate" and "postgraduate". How many years do each of those take? And what are the equivalencies between these and the "Licence - Master - Doctorat"?  Note: If the answer to this question depends on the field of studies, then the field is physics. Is this correct: "Our listeners are what make X"? Background: I am currently employed by a start-up that, so far, is doing really well. As part of my contract, I have stock options that will come available on a typical vesting schedule. A quarter of my options are set to vest later this year. Also, the company is not public yet, and it may be several years before it is public / I could sell my shares of the company.  My question is, is there any reason I should exercise them as soon as they vest? As far as I know they don't expire as long as I work at my company (if I leave for any reason, then I only have 3 months to exercise).  I don't foresee leaving any-time soon, and if I did, I would have 3 months to make a final decision.  Is there any disadvantage to just letting the option stay an option for a while? htl5 voice recording with isualization I have bought return tickets from London to Lille Europe on Eurostar. The trains on my outward and inbound journeys both stop at Calais Frethun.  Will Eurostar let me get on in Calais on the inbound? What about getting off at Calais on the outbound leg? I suspect leaving the train early is easier than getting on the train 'late'. I'm trying to typeset some python code, and I would like integer literals to be highlighted. I've come up, after about two hours of fighting with the documentation and various other questions here and on SO, with the following:  \lstset{ %     language=Python,     otherkeywords={1, 2, 3, 4, 5, 6, 7, 8, 9, 0},     morekeywords=[2]{1, 2, 3, 4, 5, 6, 7, 8, 9, 0},     keywordstyle=[2]{\color{orange}},     keywordstyle=\color{blue}\bfseries,     stringstyle=\color{red}, }   Unfortunately, that doesn't work. Numbers found in strings are also highlighted, which is rather annoying, and not only are they highlighted: they are highlighted in blue. Numerals in regular program text, on the other hand, are rightly highlighted in red.  Help?  EDIT: Thanks Peter Grill for your very nice solution. Is there a way to not break string colouring, though? When I use you code, the stringstyle=\color{red} doesn't seem to work anymore (strings are just plain black). Similarly, comments can be enclosed in triple quotes in python, and if I adapt your quote to detect """ blocks, then the corresponding  comments are not highlighted anymore. Disable emojicons introduced with WP 4.2 So WP 4.2 introduced emojis (smileys) that basically adds JS and other junk all over your pages. Something some people may find shocking. How does one completely erase all instances of this? How can I know the default PGF line width? Is there some macro that will tell me what the default PGF line width is? I have tried creating my own macro by cloning \pgflinewidth before it has been modified, i.e.  \let\defaultlinewidth\pgflinewidth   but \defaultlinewidth seems to be affected by \pgfsetlinewidth anyway, and change to whatever \pgflinewidth is. So, the problem is that after I have called \pgfsetlinewidth, I don't know what the default pgf line width is anymore. Is there some way to get around this? PyQGIS couldn't import processing with standalone script I have read post on that problem (). The problem is even with a sys.path.append it couldn't import processing. It's very frustrated. Here the code I used:  #!/usr/bin/python  def main():     layer = QgsVectorLayer('/path/to/file.shp', "BAT","ogr")     QgsMapLayerRegistry.instance().addMapLayers([layer])      # Testing processing     try :         Processing.runalg("qgis:creategrid")     except:         print "Processing module has not been loaded"      #Check the QgsMapLayerRegistry for the layers     print QgsMapLayerRegistry.instance().mapLayers()     # Check if they are valid     print layer.isValid()  if __name__=="__main__":     import sys     from PyQt4.QtGui import *     from qgis.core import *     app = QApplication([])     QgsApplication.setPrefixPath("/usr", True)     QgsApplication.initQgis()      sys.path.append('home/qsuser/.qgis2/python/plugins')      # try to load processing     try:         from processing.core.Processing import Processing         Processing.initialize()      except:         print "unable to load processing"      main()      QgsApplication.exitQgis()   I could load QGIS and layer is valid that's ok, but processing resist to me. I am on Ubuntu 14.04 I use Pycharm to do it.  I tried on terminal with the python console, it raised this error:  Traceback (most recent call last):   File "&lt;stdin&gt;", line 1, in &lt;module&gt;   File "/usr/lib/python2.7/dist-packages/qgis/utils.py", line 572, in _import     mod = _builtin_import(name, globals, locals, fromlist, level)   File "/home/qsuser/.qgis2/python/plugins/processing/__init__.py", line 28, in &lt;module&gt;     from processing.tools.dataobjects import *   File "/usr/lib/python2.7/dist-packages/qgis/utils.py", line 572, in _import     mod = _builtin_import(name, globals, locals, fromlist, level)   File "/home/qsuser/.qgis2/python/plugins/processing/tools/dataobjects.py", line 35, in &lt;module&gt;     from processing.core.ProcessingConfig import ProcessingConfig   File "/usr/lib/python2.7/dist-packages/qgis/utils.py", line 572, in _import     mod = _builtin_import(name, globals, locals, fromlist, level)   File "/home/qsuser/.qgis2/python/plugins/processing/core/ProcessingConfig.py", line 30, in &lt;module&gt;     from PyQt4.QtCore import QPyNullVariant, QCoreApplication, QSettings ImportError: cannot import name QPyNullVariant   I don't know what I am missing.   Settings :  Qgis 2.12.1-Lyon Ubuntu 14.04 Trusty IDE : Pycharm 5 community Processing : 2.12.2   -- Edit -- And with the changes suggest by mathias kuhn  #!/usr/bin/python  def main():     layer = QgsVectorLayer('/path/to/file.shp', "BAT","ogr")     QgsMapLayerRegistry.instance().addMapLayers([layer])      # Testing processing     try :         Processing.runalg("qgis:creategrid")     except:         print "Processing module has not been loaded"      #Check the QgsMapLayerRegistry for the layers     print QgsMapLayerRegistry.instance().mapLayers()     # Check if they are valid     print layer.isValid()  if __name__=="__main__":     import sys     from qgis.core import *             from PyQt4.QtGui import *     app = QApplication([])     QgsApplication.setPrefixPath("/usr", True)     QgsApplication.initQgis()      sys.path.append('home/user/.qgis2/python/plugins')      # try to load processing     try:         from processing.core.Processing import Processing         Processing.initialize()      except:         print "unable to load processing"      main()      QgsApplication.exitQgis() MSO Review Queue Privileges are Misleading When I visit Stack Overflow, I see the review option in the bar at the top of the screen.    However when on MSO, I don't see it. I assumed this was because I didn't have enough reputation to access the queue. I was curious what the requirement was on meta, so I moseyed on over to the privileges page. To my surprise, I saw that the privilege is listed as only requiring 500 rep and informs me I already have it:    Confused, I manually typed in meta.stackoverflow.com/review to see what would happen. The page did in fact load, but as you can see meta has no "First Posts" or "Late Answers" queue.    Seeing on that page that a minimum of 2k reputation is required to access any queues on MSO, I report the following bug:  1) On the privileges list on MSO, the 500 rep privilege of "Access first posts and late answers review queues" should not appear.  I also have the following feature request:  2) On the privileges list on MSO, it would be useful if instead there was a 2k privilege that informed you that you have access to the "Suggested Edits" and "Low Quality Posts" queues. verbatiminput and fontsize My application listens on port 8180.  When I am trying to connect to port 8180 locally. Connection got refused. But when I am trying to connect to port with ip . I am able to connect.  [root@MUMLGOMAPP002 ~]# telnet 10.209.129.54 8180 Trying 10.209.129.54... Connected to MUMLGOMAPP002 (10.209.129.54). Escape character is '^]'. ^] telnet&gt; q Connection closed.   [root@MUMLGOMAPP002 ~]# telnet localhost 8180 Trying 127.0.0.1... telnet: connect to address 127.0.0.1: Connection refused telnet: Unable to connect to remote host: Connection refused   My interface is :  [root@MUMLGOMAPP002 ~]# ifconfig bond0     Link encap:Ethernet  HWaddr 44:1E:A1:73:F7:50           inet addr:10.209.129.54  Bcast:10.209.129.63  Mask:255.255.255.224           UP BROADCAST RUNNING MASTER MULTICAST  MTU:1500  Metric:1           RX packets:322997568 errors:23 dropped:0 overruns:0 frame:0           TX packets:304470664 errors:0 dropped:0 overruns:0 carrier:0           collisions:0 txqueuelen:0           RX bytes:213628580915 (198.9 GiB)  TX bytes:217449664400 (202.5 GiB)   I would like to know what is the difference between telnetting localhost and ip. Holomorphic functions: is it true that $f(\bar{z})=\overline {f(z)}$?