A thought just hit my mind that whatever branch points I have seen to make some functions holomorphic are always isolated i.e. there exists a neighbourhood around them which contains no branch points.  So I thought whether they are always isolated?  I looked up for a proof and even thought over it but couldn't find a suitable approach.  So my question is whether branch points are always isolated?
I refer to , which I understand says that the branch points of a nonconstant holomorphic map $F: X \to Y$ between Riemann surfaces $X$ and $Y$, both of which are not necessarily compact, form a discrete subset of range $Y$ of the map $F$. (Not sure if 'connected' is part of the definition of Riemann surfaces in other textbooks, but it is here.) Question 1: Do I understand correctly, and is this indeed true? What else I understand:  $A$ is a discrete subspace/subset (I just assume everything here is the usual subspace topology) of $B$ if and only if every singleton subset of $A$ is open in $A$ if and only if every point of $A$ is isolated.  This overflow question () seems to suggest this is not true but is true if $F$ is proper. However, I'm not sure the definitions of Riemann surface are the same (maybe some answers use definitions where Riemann surfaces are not necessarily connected or something).  2.1. I know $F$ is 'discrete' (meaning that its fibres are discrete; here, i refer to a ): see definition before Lemma 3.1, on p. 7), by Proposition II.3.12, and open, by open mapping theorem, which is Proposition II.3.8, but I'm not sure these help show $F$ is proper or are otherwise helpful. Ostensibly 'discrete' maps don't map discrete subsets to discrete subsets.   This stackexchange question () seems to suggest this is true.  I think this is true if $F$ is injective or at least something like locally injective. (Please don't make me type up all the stuff I did for this part.) However based on the question in (3) and based on the , which I think uses the term 'branch points' to mean the same thing as what Miranda means by 'ramification points' (and thus is different from Miranda's 'branch points'), it seems that no neighbourhood $U$ of a ramification point $p \in U \subseteq X$ of $F$ is such that the restriction $F|_U$ is injective.  Kinda related:    This page () says    Now, the ramification and branch points must form a discrete set  I don't think the page assumes anything like $F$ is proper.  The  text in (2.1) actually seems to assume proper based on the '3.7. Proper holomorphic maps' part, but said part refers to 'Lemma 3.17', so I think proper can be replaced with closed. Maybe $F$ is closed or something.  Kinda related (see Question 2 below):     says    If $F$ is proper then the image $\Delta = F(R)$ is discrete in $Y$.   This seems to have all the same assumptions as with Miranda. Ostensibly, either Miranda is wrong or there's some higher level machinery that allows us to omit $F$ proper.  Question 2: To possibly generalise this, what are the minimum requirements on $F, X$ and $Y$ to make $F$ map discrete subsets to discrete subsets? I mean, I'm not sure we really need 'holomorphic' here. I have a feeling this applies perhaps to maps that are just open/closed, continuous and discrete and spaces that are just Hausdorff and locally compact or something.  (again) Kinda related:
save as a video as .mp4 So I made an intro using animation and it saves it as a .blend and I want it so save as a video like .mp4 or something like that. I've tried changing the codec because I tried opening the intro with windows media instead of blend. I didn't try all the codec's but I tried some of them. Finding a closed form for a recurrence relation $a_n=3a_{n-1}+4a_{n-2}$ Consider the sequence defined by $$ \begin{cases} a_0=1\\ a_1=2\\ a_n=3a_{n-1}+4a_{n-2} &amp; \text{if }n\ge 2 \end{cases} .$$ Find a closed form for $a_n$.    I tried listing out examples, but I don't see any common pattern between them.  All solutions are greatly appreciated. Why are certain example URLs like https://site.com/ and https://mysite.com/ blocked from post content? Why is  and  blocked from the post content on the main site? I tried to make a sample URL and I had to use  instead. Possible Duplicate:          I have been downvoted for few answers of mine and the person who downvoted me did not leave any comments or reasons to say why I was downvoted. In such cases, how can I find out who has downvoted me and the reason for downvoting? How to highlight (color/draw attention to) a particular snippet in Minted I'm currently doing some work on error messages in programming languages. As such, I'm showing faulty pieces of code, alongside the compiler-errors accompanying them.  I'm using minted for the code snippets, but ideally, in addition to normal syntax highlighting, I'd like to highlight the sections of the code at which the error is located, by underlining them, putting a red box around them, shading their background, etc.  Is this possible using Minted? If a company is bought by another company, does my stock rollover to the new company when it is combined with the new parent company? Stopping xcode from indexing Why does friction play the role of centripetal force during the turning of a car? What makes frictional force the centripetal force of a car turning along a curve? As friction is the opposing force and acts anti-parallel so there is no component of frictional force towards the center,right? Then how can frictional force be centripetal force? Explaining Horizontal Shifting and Scaling I always find myself wanting for a clear explanation (to a college algebra student) for the fact that horizontal transformations of graphs work in the opposite way that one might expect.  For example, $f(x+1)$ is a horizontal shift to the left (a shift toward the negative side of the $x$-axis), whereas a cursory glance would cause one to suspect that adding a positive amount should shift in the positive direction. Similarly, $f(2x)$ causes the graph to shrink horizontally, not expand.  I generally explain this by saying $x$ is getting a "head start". For example, suppose $f(x)$ has a root at $x = 5$. The graph of $f(x+1)$ is getting a unit for free, and so we only need $x = 4$ to get the same output before as before (i.e. a root). Thus, the root that used to be at $x=5$ is now at $x=4$, which is a shift to the left.  My explanation seems to help some students and mystify others. I was hoping someone else in the community had an enlightening way to explain these phenomena. Again, I emphasize that the purpose is to strengthen the student's intuition; a rigorous algebraic approach is not what I'm looking for. I'tm trying to show in SQLcmd window the results of these queries:  SELECT SYSDATETIME() AS "date";  PRINT 'EXTRAÍDO POR:' SELECT SUSER_NAME() AS "Owner"  PRINT 'PASSWORD SETTINGS'; exec xp_loginconfig;  PRINT 'PASSWORD SETTINGS'; select * from sys.sql_logins;   And using this command in SQLcmd:  sqlcmd -S myServer -i inputpath -s, -o outputPath(like C:\log.xls) -e   So the sqlcmd output the file and does not show nothing in sqlcmd screen. How can i print these "Print" commands in the sqlcmd screen? (command promt).  When I run it on SSMS works well, like:  (1 row affected) OWNER:  (1 row affected) PASSWORD SETTINGS  (8 rows affected) PASSWORD SETTINGS  I would like to show in the prompt command this, like in the SSMS.  Thanks! Where are executables for programs stored in Ubuntu?  An application (Komodo Edit) is asking me to identify an application to be used as a web browser. I've become used to just entering the application name as a command for situations such as these, but this scenario got me thinking.   I know in Windows it would just be the relevant application folder in the 'program files' folder, but I'm assuming things are a bit different on Linux?  I thought somewhere like bin would be logical but this appears to standard Linux/Unix applications. Where would I find the binary executable for applications stored on my system? iOS app crashes after deleting a comment So I've been trying to figure out how to prove the following.     Let $(a_n)$ be a sequence of positive numbers such that $\sum\limits_{n=1}^\infty a_n =\infty$, and define $s_n=\sum\limits_{i=1}^n a_i$. Then $\sum\limits_{n=1}^\infty\frac{a_n}{s_n} =\infty$ as well.   I can prove it by comparing it to $\int_1^\infty \frac{1}{x} \, dx $ if the sequence $a_n$ is bounded by some $M$, but that's as far as I've been able to get. How can I prevent inline math formulas from overflowing into the margin? When I compile my document in LaTeX, some formulas in the document in math mode, between $ $, go into the right margin area beyond all the text.  Is there a way to fix this aesthetic problem? Sum of digits of repunits Equivalent of cURL for Ruby? Prove that $\operatorname{trace}(A) = 0$ if and only if $A^2 = 0$. MySQL Error 1093 - Can't specify target table for update in FROM clause Can't get unicode symbols in math mode I am trying for about a week to launch unicode-math package with XeLaTeX and got nothing. PDF is created well but all Cyrillic symbols in math mode are skipped. I tried using different math fonts - no progress at all. In log file, I found low-level error:  Missing character: There is no [cyrillic letter from input] in font cmmi12!   But all fonts used in document are Unicode ones.  Here is a file I want to be processed well (of course it is UTF-8).  \documentclass[12pt]{book} \usepackage{polyglossia} \setdefaultlanguage[spelling=modern]{russian} \setotherlanguage{english} \defaultfontfeatures{Ligatures={TeX}} \setmainfont{CMU Serif} \setsansfont{CMU Sans Serif} \setmonofont{CMU Typewriter Text}    \usepackage{amsmath, amssymb} \usepackage[russian]{hyperref}  \usepackage{unicode-math} \setmathfont{Latin Modern Math}  \frenchspacing  \begin{document} Просто буквы % Plain letters $$Память: M_{доп}(n) = \Theta(N)$$ % Memory: M_add(n) = \Theta(n) \end{document}   Looking for your help. Based on the following interchange between Harry and Dumbledore in the pivotal &quot;King's Cross&quot; chapter of Deathly Hallows, I was wondering when exactly Dumbledore put together that Harry would have to die in order to bring about Voldemort's ultimate demise:  &quot;And you knew this? You knew--all along?&quot; &quot;I guessed. But my guesses have usually been good,&quot; said Dumbledore (Deathly Hallows, 599, hardcover).  The conversation that spurred Harry's question includes material dating back to Halloween '81, but also specifically mentions the events of the graveyard from GoF. So, &quot;knowing all along&quot; could refer to everything (i.e., Harry must die, Voldemort ignorantly tethered him to life, etc.) or just that Voldemort tethered him to life. Other important information pertaining to this question can be found in OOTP, when Dumbledore confirms (if he didn't already know) that Harry contains part of Voldemort's soul. He uses the silver instrument to find out that Harry is just like Nagini, having the ability to share mind and soul with Voldemort. He also says the following:  &quot;I guessed, nearly fifteen years ago,&quot; said Dumbledore, &quot;when I saw the scar upon your forehead, what it might mean. I guessed that it might be the sign of a connection forged between you and Voldemort&quot; (Order of the Phoenix, 826-827, hardcover).  While confirming that he guessed that Harry might share a connection with Voldemort from the very beginning, Dumbledore does not confirm with this statement that he knew Harry would ultimately have to give up his life. In Philosopher's Stone, when Dumbledore drops off Harry on the Dursley's doorstep, we know he is not entirely truthful with McGonagall.  McGonagall asks: &quot;How in the name of heaven did Harry survive?&quot; &quot;We can only guess,&quot; said Dumbledore (Philosopher's Stone, 12, hardcover).  Of course, this was a lie. Dumbledore already knew how Harry survived, otherwise he wouldn't be delivering Harry to Lily's sister to extend the blood sacrifice. By proxy, he also knows that Voldemort will return b/c of Horcruxes, otherwise he wouldn't be so worried about Harry. Finally, when asked in the same scene if he could remove the scar, Dumbledore says:  &quot;Even if I could, I wouldn't. Scars can come in handy&quot; (Philosopher's Stone, 15, hardcover).  So, did Dumbledore know at this very moment that Harry would have to die? Did he at least guess it to be true during this scene? We know he gains perfect confirmation in OOTP, but did he know it &quot;all along,&quot; as a particular reading of the King's Cross interchange in DH would have us believe?