I stumbled upon the following statement     Let $p&gt;2$ and let $f$ be an entire function (i.e. holomorphic on the whole $\mathbb{C}$), and assume that $f'-1$ is in $L^p(\mathbb{C})$. Then $f'\equiv 1$   To me, it looks like the author is suggesting a statement of the form:     There exists no integrable, non-zero, entire function.   (as I presume that the $p&gt;2$ doesn't play any role here... Or maybe it does?)  But I cannot seem to find/prove this. The only thing that has come to my mind, which could be helpful, is Liouville's theorem, but this doesn't seem to be enough, as I can think of examples of $C^\infty$ functions that are unbounded and yet integrable. Or maybe Green's theorem for the complex plane?  I must admit, I am not very confident with the subject at the moment, as I had my Complex Analysis class more than two years ago.  Does someone have a solution or a reference for this?  Thank you in advance
Suppose $f$ is entire and    $$\iint_\mathbb{C}|f(z)|^2dxdy &lt; \infty$$Prove that $f\equiv 0.$   So far I have:  Suppose $f$ is bounded.  Then $f$ is constant by virtue of Liouville and so the conclusion is obvious.  Thus, assume $f$ non-bounded.  Then  $$\Big|\iint_\mathbb{C}f^2(z)dA\Big| \leq \iint_\mathbb{C}|f(z)|^2dxdy &lt; \infty$$ We can parameterize \begin{align*} \iint_\mathbb{C}f^2(z)dA &amp;= \int_0^{\infty}\int_0^{2\pi}f^2(re^{i\theta})\;d\theta\;rdr \\ &amp;= \int_0^{\infty}\int_0^{2\pi}f^2(re^{i\theta})\;ire^{i\theta}[-i\frac{1}{r}e^{-i\theta}]d\theta\;rdr \\ &amp;= \int_0^{\infty}\oint_{C_R}f^2(z)\;[-i\frac{1}{z}]dz\;rdr \\ &amp;= \int_0^{\infty}2\pi r dr\frac{1}{2\pi i}\oint_{C_R}\frac{f^2(z)}{z}dz \\ \\ &amp;= 2\pi f^2(0)\int_0^\infty rdr  \end{align*}  Whence f(0) = 0.  And I have no idea where to go from there.  The "entire" bit seems to suggest Liouville but I have already dealt with the bounded case.  Please advise ...
I try to delete Action.005 permanently.    But I don't know how. If I click on the X    then it is still in the list.    I already researched without success, I even managed to loose all my animations by attempting to apply a fix, so I want to be cautious now... How can I do this? Postfix Connection Timed out Cardinality of equivalence relations in $\mathbb{N}$ I came across this long proof on this site:    But I would like to know whether my direction can work.  Say we want to find the cardinality of all equivalence relations in $\mathbb{N}$. Since it is a subset of all relations in $\mathbb{N}$, I conclude it has a cardinality smaller or equal to $\aleph$. Now, define an injective function from $P(\mathbb{N})$ to the set of equivalence relations by matching each subset of $\mathbb{N}$ with the identity relation (which is an equivalence relation in $\mathbb{N}$.  Therefore the cardinality of all equivalence relations in $\mathbb{N}$ is greater or equal to $\aleph$ and using CSB we get the desired result.  Seems legit? What happens after I grapple as Attack of Opportunity? I've searched websites, written to Mr. Barth and William Safire, scanned bookstore shelves, and buttonholed likely prospects in search of a grammar text that would address matters in the manner of the following, which is from John Barth's "The End of the Road".  If such a text exists, I'd be much obliged to learn of it:  " ... it gives a man a great sense of lucidity and well-being, if not downright formidability, to be able not only to say, but to understand perfectly, that predicate complements of infinitives of copulative verbs without expressed subjects go into the nominative case, whereas predicate complements of infinitives of copulative verbs with expressed subjects go into the objective case.  I made this observation to my assemblage of young scholars and concluded triumphantly, "I was thought to be he, but I thought John to be him."  Thank you. Does there exist a continuous function $\: f : \mathbf{R} \to \mathbf{R} \:$ such that for   all  functions $\: g : \mathbf{R} \to \mathbf{R} \:$, for all real numbers $x$,  there exists a real number $y$ such that $\: x &lt; y \:$ and $\: g(y) &lt; f(y) \:$? I know classical physics, quantum mechanics, special relativity, and basic nuclear physics. I would like to get into some particle physics.   Where can I find a good introduction? It might be useful to segment books by whether they require quantum field theory or not. 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: Restored database missing views, stored procs and foreign keys Is voltage the speed of electrons? I am currently using a macbook pro but the operating system is debian 8.2 not os x  I am using a macbook pro but there's this one device that's just draining a lot of power. This USB device: Card Reader (Apple)   32.5 W    100.0%                      Device         USB device: Card Reader (Apple)   2.60 W    100.0%                      Device         Radio device: btusb   209 mW     19.7 pkts/s                Device         Network interface: wlan0 (wl)   This device really drains my battery, I noticed that my laptop wouldn't even get 2 hours when not plugged in, I installed powertop. Made all the recommended changes but this device, I can't find out how to disable it. I don't even use that reader and forgot that it was there.  I configured all my devices to use limited power in my rc.local  echo '0' &gt; '/proc/sys/kernel/nmi_watchdog' echo 'min_power' &gt; '/sys/class/scsi_host/host0/link_power_management_policy' echo '1' &gt; '/sys/module/snd_hda_intel/parameters/power_save' echo '1500' &gt; '/proc/sys/vm/dirty_writeback_centisecs' echo 'auto' &gt; '/sys/bus/usb/devices/2-4/power/control' echo 'auto' &gt; '/sys/bus/usb/devices/1-12/power/control' echo 'auto' &gt; '/sys/bus/usb/devices/1-8.3/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:1f.3/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:01:00.1/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:1c.3/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:1c.4/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:1c.2/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:1c.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:1b.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:16.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:14.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:01.1/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:01.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:00.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:01:00.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:08:00.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:07:06.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:07:05.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:07:04.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:07:03.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:07:00.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:06:00.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:05:00.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:04:00.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:03:00.0/power/control' echo 'auto' &gt; '/sys/bus/pci/devices/0000:00:1f.0/power/control'   How can I get this device to act sanely or disable it completely because I  haven't used it. ${!FOO} and zsh ${!FOO} performs a double substitution in bash, meaning it takes the (string) value of FOO and uses it as a variable name. zsh doesn’t support this feature.  Is there a way to make this work the same in bash and zsh?  Background:  I’ve got a list of environment variables, like  PATH MAIL EDITOR   and want to first print the variable names and afterwards their values.  This works in bash but not zsh:    for VAR in LIST do         echo $VAR         echo ${!VAR} done   It should be somehow possible “the old way” with eval, but I can’t get it to work:  for VAR in LIST do         echo $VAR         echo `eval \$$VAR` done   I’m never going to understand why I can’t simply do arbitrary deep substitutions like ${${VAR}} or even ${${${VAR}}} if need be, so an explanation for that would be nice, too. I read recently that you can find the number of digits in a number through the formula $\lfloor \log_{10} n \rfloor +1$ What's the logic behind this rather what's the proof? Let $H$ be a subgroup of $G$ with $[G:H]=n$. Let $N$ be the kernel of the left-multiplication action on the cosets of $H$. Show $[G:N]$ divides $n!$ Let $H$ be a subgroup of $G$ with $[G:H]=n$ and $G$ finite.  Let $G$ act on left cosets of $H$ by left multiplication.  Let $N$ be the kernel of the action.  Show that $[G:N]$ divides $n!$.  Ok so I have a theorem that says that if $G$ is simple then it embeds into $S_n$.  So if $G$ is simple than I'm done because $G$ divides $n!$ and thus $N$ divides $n!$ and hence $[G:N]$ divides $n!$.  Now I also know that $N$ is the largest normal subgroup of $G$ contained in $H$.  Thus if $G$ isn't simple then it contains some non-trivial normal subgroup $K$ and thus $K\cap H$ is normal in $H$ and by the 2nd Iso Theorem $H/K\cap H\cong KH/H$...    As you can see I'm not sure where to go from here. I was hoping to prove that $K\cap H$ wasn't trivial and thus that $N$ couldn't be trivial since it must contain $K\cap H$, but I'm not sure I can, and even if I could I'm not sure that would help me in showing that $[G:N]$ is small enough to divide $n!$.  Can anyone give some guidance?  Thanks. How to pass Unicode characters as JSP/Servlet request.getParameter? Reviewed suggested edit view for title-only edits takes body of last revision? Yesterday I was pointed at some rejected suggested edits. Some of the corrections were valid, so I edited the posts accordingly, while making several further edits. Looking at the suggested edits now however, it seems that for those posts where only changes to the title were originally suggested, the post mortem suggested edit view now incorporates the body of the latest revision. (Or at least it appears like that to me)  Take for example . If you take a look at  of that post, you will see my later edits. The edited body seems to have been incorporated in the suggested edit view. For edits where originally changes to the body were suggested, there is no such confusion.   Why does this matter? In the case of some post mortem discussions or explanations it might help to see the actual situation as it was when the edit was suggested. Especially in the case of edits that might be considered too minor.   So have I stumbled upon a bug here? My table doesn't fit; what are my options? I have a table that I want to insert on a page, but at least one (perhaps both) of the following conditions are met:   The table is too wide to fit within the text block or page. That is, I'm exceeding some horizontal restriction. The table is too tall to fit within the text block or page. That is, I'm exceeding some vertical restriction.   What are my options to make this table fit? If it doesn't fit, regardless of my attempts, what other options exist? How do I stop program from skipping over getline? Via command line, I have a log file I'd like to keep track of.  What I want is to have, basically, a tail that refreshes when the log is updated making the text scroll upwards as new lines are appended to the log file.  Is there anything out there that does that without having to write some code? Looping through array and removing items, without breaking for loop