It's commonly used in imaginary-time path integral that &quot;analytic continuation&quot; means replacing $t \rightarrow - i \tau$ or reparametrizing the theory in terms of imaginary time $\tau = i t$. For example, it's common practice to just replace the imaginary time $\tau \rightarrow i t$ to figure out the real time Green's function from the imaginary time Green's function. I've been having trouble connecting this with the definition from math. My understanding here is to find the analytic continuation of an analytic function in domain $V_1$ around $z_1$: $f(z) = \sum_k a_k (z- z_1)^k$, we need to find $f_2$ defined in another domain $V_2$, which overlaps with $V_1$. And if $f_1 = f_2$ in $V_1 \bigcap V_2 $, then $f_2$ is an analytic continuation of $f_1$ on $V_2$. How is the replacement justified from the definition in math? P.S. There is a very similar question  but I don't think the answer was satisfying to me. P.P.S QMechanic linked a post about , which is showing an integral along the real line is the same as the integral along imaginary axis. One can apply Cauchy theorem under the right conditions, but I thought &quot;analytic continuation&quot; by replacing arguments of $t$ is a strong statement: it's saying 1) there are no poles in the quadrants; 2) the contour integral connecting $t$ and $it$ are zero for all $t$. I suppose I could have missed some simple property about the Green's function, that might imply these two conditions?
In doing instanton problems or when connecting quantum field theory to statistical mechanics, I often see people trying the Wick rotation trick by defining an imaginary time $\tau\equiv it$. So, in this case, we should make sure that $\tau$ is imaginary, right? However, I often see places where people seem to handle $\tau$ as a real number, for example setting $\tau$ to $\pm\infty$. Moreover, I remember in one of my quantum mechanics classes, the professor said the Euclidean time $\tau$ was real. I am so confused and need some good explanations.
Bounding box is larger than expected when drawing a curved path I have been working on logo for my department and I have the following code. The problem is that whenever I compile the diagram seems to be larger than the actual logo. I do not know what is the problem. I know the numbers I have selected are not the best but any insights into the matter will be highly appreciated.   \documentclass[letterpaper]{article}  \usepackage[top=2cm,bottom=2cm,left=2cm,right=2cm]{geometry}  %\usepackage{amsmath,amssymb,units}  %\usepackage{enumitem,multicol}  \usepackage{tikz}  %\usetikzlibrary{arrows}  \usepackage{lipsum}   \begin{document}   \lipsum[1-2]  \begin{tikzpicture}[y=0.80pt, x=0.8pt,yscale=-1]  \path[draw=black,fill=white]    (258.9527,290.5199) .. controls (173.9885,538.4766) and (568.5860,261.2969) ..    (306.5098,257.1141) .. controls (44.4337,252.9312) and (429.9845,542.5624) ..    (352.9767,292.0206) .. controls (275.9689,41.4788) and (119.6549,497.6604) ..    (334.1376,346.9999) .. controls (548.6203,196.3394) and (66.4622,188.6439) ..    (276.0276,346.0724) .. controls (485.5930,503.5010) and (343.9169,42.5633) ..    (258.9527,290.5199) -- cycle;  \end{tikzpicture}  \lipsum[1-2]  \end{document}    See image below: iOS StackExchange app doesn't handle non-ASCII correctly The iOS StackExchange app doesn't handle non-ASCII characters correctly, see city name in the screenshot below. No option to create a new Managed Path inside Office 365 SharePoint online 2013 I am working on an Office 365 SharePoint online 2013. Now when I wanted to create a new private site collection, inside the online SharePoint admin center, I did not find the option to create new managed path, as I use to get when creating a site collection inside SharePoint on-premise. So how do I create a new managed paths inside SharePoint online? Before proposing reviewers, should I notify them? Some journals require the author to propose potential reviewers. Should I just propose reviewers and let the editor contact them? Or should I first contact the potential reviewers myself and ask if they agree? Keyboard layout isn't kept upon reboot I had a problem with the keyboard layout changing from PT (I'm Portuguese) to EN automatically with the last Ubuntu updates in Ubuntu 13.10. After Ubuntu 14.04 came out, I formatted my computer and installed the new version.  After that I haven't had any problems until the recent Ubuntu base update 2 or 3 days ago, when I started having the same problem again. I even removed the English keyboard from the Keyboard configurations, left only the Portuguese one, but still, every time I reboot my PC I get the keyboard configured for EN, despite Ubuntu displaying the contrary?  How can I remove this? I don't want to format the PC again and I'm getting really tired of this problem, since it looks like many Portuguese/Brasilian Ubuntu users are having the same problem. making low-quality PDF files clearer Some research papers available on the web have low-quality outputs. For example, see . It seems that it was converted from a PS file and typeset in (La)TeX. However, it has all the font information. Is there any way to make the PDF file clearer? The graduate students at my university (a relatively large state school in the US) are considering joining a union. To help decide whether to sign a union card in support of joining the union, I'm interested in more information about how unions have helped graduate students at other universities. What are some improvements to graduate life gained by graduate students at other universities upon joining a union?  I'm also interested in information about how unions have hurt graduate students at other universities. What are some detriments to graduate life caused by graduate students joining a union?  I'm primarily interested in student-body-wide benefits/detriments, instead of student-specific or faculty/administrator-specific claims (e.g. "my advisor treated me better with the union behind me" or "my students have stopped working since they joined a union" is not what I'm looking for). Set the location in iPhone Simulator Can I set up WhatsApp on my desktop when my smartphone is broken? 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. Good and easy way to share files on local machine I would like to have a directory that has following properties:    Many users can copy files into it  These files can be deleted/changed by these users (user A can delete/modify file that was copied into this directory)    it cant be done using normal file permissions (because permissions are retained on copy).  Here is what I found on the net:        Some use cases:    Sharing music on local machine  Simple git repository sharing (just make a bare repository writeable to many people) --- i know that there are solutions like gitosis Allow many developers to modify test instance of php app without giving them root (i guess they would copy files) --- I'm leading a team of nonprofit junior developers and I need to keep that one simple!    EDIT  AFAIK setting SGID bit is not enugh, it only affects newly created files --- and basic workflow for these use cases ivnolves copying and other operations (which cleave file's gid unchanged) Why does the "Unanswered Questions" tab show questions that have answers? Why does the "Unanswered Questions" tab show questions that have answers? I have just upgraded my system but it boots to Low Graphics mode by default. I have seen questions on how to solve this problem and they all suggest to run a couple of commands from a command-line interface.  But there isn't any command-line console by default in Low Graphics mode and hence, I cannot run those commands. And, the answers I saw to solve the problem doesn't explain this situation either.  How to switch to a command-line console in order to run the commands in Low Graphics mode? Integrating the inverse of a function. $f(x)$ is a strictly increasing, and continuous, function on $[0,+\infty)$. Consider the integral,  $$\int_0^a f^{-1}(x) dx.$$  Reasoning  The interval $[0,a]$ is "reflected" across the line $y=x$ onto the interval $0 \leq y \leq a.$  $f(x)$ and $f^{-1}(x)$ are symmetric across the line $y=x$. So, the integral is equivalent to the area bounded by $y = 0$, $y=a$, and $f(x)$.  So, "integrating $f$ along an interval on the $y$-axis" is the same as integrating $f^{-1}$ along the same interval on the $x$-axis?  Question  Is my reasoning correct? How to find and delete files older than specific days in unix? My code:  a = '2.3'   I wanted to display a as a floating point value.  Since a is a string, I tried:  float(a)   The result I got was :  2.2999999999999998   I want a solution for this problem. Please, kindly help me.  I was following . Short story from many years ago: the ancient, brilliant, cruel race was Earthlings I flagged a post today, typed out a fairly thorough explanation, clicked the button... and got told that I'd already flagged this post (for something else, which I'd forgotten about) so this flag wasn't allowed.  (This happens if the first flag is still active.)  The dialogue with my explanation was gone at this point, so if I wanted to communicate with the moderators about that, I was going to have to start over.  (In case you're wondering how this could happen, one flag was related to the post itself and one was related to the user -- there being no way to flag a user directly, you have to hang it off of a post.  But I've also seen this with cases where the first flag hangs around unhandled for a week or two and I've forgotten about it.)  Since the system is going to detect that I've already flagged a post and reject a second flag, can it do that after I click the "flag" link instead of waiting until the end of the process?  (Or gray out the link so I can't click it in the first place?)  2015-12-18: someone suggested to me in chat today that this had been implemented, so I tested again.  The behavior has not changed: I flagged a post using "other", then flagged it again (also "other") before the first flag was handled, and the flag was rejected only after I'd typed the message.  (I did this on a site where I'm a moderator so I could then clean up after myself.) Find html element which id starts with How to enable automatic update in wordpress running on IIS7 and Windows 2008? I have a dedicated server using Windows 2008 and IIS7 running 5 blogs with Wordpress. I can't use the autoupdate feature of the wordpress neither the automatic plugin instalation.  I believe I must set writing permissions in some temporary folder, the question is which one. Do you have any idea?