For it, use that $$\oint_{|z|=r} \frac{f(z)-f(-z)}{z^2} dz = 4\pi i f'(0) ; \hspace{1cm}\text{  with } 0&lt;r&lt;1$$ Also remember that diam$(A)=\text{sup}\{\left|z-w\right| : z,w\in A\}$ for A bounded, $\mathbb{D}$ is the unit disc and $f:\mathbb{D}\rightarrow\mathbb{C} $.
Let $D \subset \mathbb{C}$ be the open unit disk, $f:D \to \mathbb{C}$ a holomorphic function and $d=\sup \{|f(z)-f(w)|:z,w\in D \}$. How can I show that $|f'(0)|\leq \frac{d}{2}$?  I tried using Schwarz's lemma on the function $g:D \to \mathbb{C}$ defined by $g(z)=\frac{f(z)-f(0)}{d}$, but I am only getting $|f'(0)|\leq d$.
Extracting usernames from the /etc/passwd file I want to upgrade my disk from MBR to GPT, and I tried using Windows' MBR2GPT.EXE tool to do the job. I used Windows' DISKPART tool to identify the disk I want to convert. However, this happened:  C:\WINDOWS\system32&gt;MBR2GPT.EXE /convert /disk:0 /allowfullos  MBR2GPT will now attempt to convert disk 0. If conversion is successful the disk can only be booted in GPT mode. These changes cannot be undone!  MBR2GPT: Attempting to convert disk 0 MBR2GPT: Retrieving layout of disk MBR2GPT: Validating layout, disk sector size is: 512 bytes Disk layout validation failed for disk 0 MBR2GPT: Conversion failed   and the conversion failed. Using Diskpart again, I get the following output:  DISKPART&gt; list disk    Disk ###  Status         Size     Free     Dyn  Gpt   --------  -------------  -------  -------  ---  --- * Disk 0    Online          465 GB      0 B   Disk 1    Online         3841 MB   960 KB  DISKPART&gt; select disk 0  Disk 0 is now the selected disk.  DISKPART&gt; list partition    Partition ###  Type              Size     Offset   -------------  ----------------  -------  -------   Partition 1    Primary            100 MB  1024 KB   Partition 2    Primary            464 GB   101 MB   Partition 3    Recovery           813 MB   464 GB   Partition 4    Recovery           511 MB   465 GB   Looking at the partitions on disk 0, there are 2 primary partitions and 2 hidden ones, I read  that the partitions might have something to do with the conversion failing, so I believe this is the culprit. If anyone could offer help converting my disk, that would be greatly appreciated. Neutron decay and electron anti neutrino $n\to p + e + \bar{\nu}_e$ Why do we need neutrino to explain neutron ? Is there any evidence regarding existence neutrinos in the context of $n\to p + e + \bar{\nu}_e$? Assume f is a bounded continuous function on $\mathbf{R}$ and $X$ is a random variable with distribution $F$.  Assume for all $x \in \mathbf{R}$ that  $$ f(x) = \int_\mathbf{R}f(x+y)F(dy) $$  Please help conclude that $f(x+s) = f(x)$ where $s$ is any value in the support of $F$.  The hints that I have come across are to use Martingale theory and consider $\{ X_n\}$ to be i.i.d. with distribution $F$ and make a martingale with some function of $S_n = \sum_{j=1}^nX_j$.    Thanks! Printing all instances of a class Is the Ratio of Associative Binary Operations to All Binary Operations on a Set of $n$ Elements Generally Small? I started thinking about the number of associative (binary) operations on a set with $n$ elements today.  Looking online I found  which indicates only $113$ of the possible $19,683$ operations on a three-element set satisfy association.  So, $50\%$ of binary operations on a two-element set satisfy association, and less than $0.6\%$ of all binary operations on a three element set satisfy association.  For an $n$-element set, where $n$ denotes a natural number, is the ratio $R_{n}=A_{n}/B_{n}$, of the number of associative binary operations $A_{n}$ to the number of all binary operations $B_{n}$, in general small?  I don't mean the following questions as equivalent, but since they seem more concretely answerable in principle, does   $$ \lim_{n \to +\infty} \frac{A_{n}}{B_{n}} = 0 ? $$       Also, is $F : n \to R_{n}$ a monotonically decreasing function?  Some Background of the Question: In his Linear Algebra Problem Book 1995 on p. 6 Paul Halmos writes "The commonly accepted attitudes toward the commutative law and the associative law are different.  Many real life operations fail to commute; the mathematical community has learned to live with that fact and even to enjoy it.  Violations of the associative law, on the other hand, are usually considered by specialists only."  For all I know, Halmos might only have written that to motivate the study of Linear Algebra and doesn't quite literally mean what he appears to say.  But, if he means what he appears to say, and if $F : n \to R_{n}$ is monotonically decreasing, or $R_{n}$ is generally small, I think there's something amiss with what Halmos says, since non-associative operations seem so common that one may as well enjoy them. Automorphism group of the quaternion group Let $Q_8$ be the quaternion group. How do we determine the automorphism group $Aut(Q_8)$ of $Q_8$ algebraically? I searched for this problem on internet. I found some geometric proofs that $Aut(Q_8)$ is isomorphic to the rotation group of a cube, hence it is isomorphic to the symmetric group $S_4$. I would like to know an algebraic proof that $Aut(Q_8)$ is isomorphic to $S_4$. Is it possible to make my laptop's screen act as a second monitor for my desktop computer? Best choice between using \\ or leaving space after each paragraph to end the paragraph How can I secure an Ethernet-over-power network? I just bought myself an Ethernet-over-power adapter, and plugged it in. It seems to be working great, but now I'm concerned that my neighbours may be able to access my network.  I live in a rowhome complex; is it possible they can access my network? If so, how can I secure it? Probability that a quadratic polynomial with random coefficients has real roots Different voltage characteristics of CMOS NAND gate for different connections Recently, I've played with a simple 2-input 1-output NAND gate realised in CMOS technology as shown in Fig. 1.     Fig. 1. CMOS NAND scheme.  I took some measurements of volteges U(output) vs. U(input). Using two independent voltage sources (constant high level Udd and Uin varying in 0-to-Udd range), I considered three different connection possibilities so to investigate all four input configurations. Those were:  a)  Input terminal A connected to Udd, input terminal B connected to Uin.  b)  Input terminal A connected to Uin, input terminal B connected to Udd.  c)  Both input terminals connected to Uin.  I've obtained following characteristics:    Fig. 2. CMOS NAND voltage characteristics with different connection methods.  As long as a difference between case c) and other cases I could have expected, I was surprised to see a) and b) differ as well. Could someone please explain to me why would changing voltages connection between A and B significantly changes threshold voltage of the gate? Is it a widely observed effect or have I just picked a defective NAND? Gamma Distribution out of sum of exponential random variables (I was requested to edit the question to explain why it is different that a proposed duplicate question. This seems counterproductive to do here, inside the question it self, but that is what I have been asked by the site and moderators. There is no way for me to vote against their votes. So, here I go: Please stop voting this as a duplicate so quickly, which will eventually lead to this question being closed off. Yes, the other question linked to asks the same math, but any newcomer to the problem who was exposed to it via physics, as I was, will prefer this question instead of the one that is purely mathematically. I beg the moderators to not be pedantic on this one. This question spills into physics, which is why I did the cross post to the physics forum as well.)  How does the sum of the series “1 + 2 + 3 + 4 + 5 + 6…” to infinity = “-1/12”, in the context of physics?  I heard Lawrence Krauss say this once during a debate with Hamza Tzortzis (). I found a transcript of another debate between Krauss and William Lane Craig which has the same sum. Here is the paragraph in full:     Let’s go to some of the things Dr. Craig talked about. In fact, the   existence of infinity, which he talked about which is   self-contradictory, is not self-contradictory at all. Mathematicians   know precisely how to deal with infinity; so do physicists. We rely on   infinities. In fact, there’s a field of mathematics called “Complex   Variables” which is the basis of much of modern physics, from   electro-magnetism to quantum mechanics and beyond, where in fact we   learn to deal with infinity; without the infinities we couldn’t do the   physics. We know how to sum infinite series because we can do complex   analysis. Mathematicians have taught us how. It’s strange and very   unappetizing, and in fact you can sum things that look ridiculous. For   example, if you sum the series, “1 + 2 + 3 + 4 + 5 + 6…” to infinity,   what’s the answer? “-1/12.” You don’t like it? Too bad! The   mathematics is consistent if we assign that. The world is the way it   is whether we like it or not.   -- Lawrence Krauss, debating William Lane Craig, March 30, 2011  Source:   CROSS POST: I'm not sure if I should post this in mathematics or physics, so I posted it in both. Cross post:   EDIT: I did not mean to begin a debate on why Krauss said this. I only wished to understand this interesting math. He was likely trying to showcase Craig's lack of understanding of mathematics or logic or physics or something. Whatever his purpose can be determined from the context of the full script that I linked to above. Anyone who is interested, please do. Please do not judge him out of context. Since I have watched one of these debates, I understand the context and do not hold the lack of a full breakdown as being ignorant. Keep in mind the debate I heard this in was different from the debate above. pgfplots - barplot with multiple bars on same coordinate, but labelled differently I'm showing several data sets next to each other using multiple \addplot commands within a pgfplots axis environment. These are then further grouped into categories. The output of the graph per se is fine, however, I need each bar to be labelled separately - basically, I want the legend to be displayed directly underneath its corresponding element. I presume a picture will illustrate my problem more clearly:     I prepared an MWE:  \documentclass{minimal} \usepackage{pgfplots}  \begin{document}  \pgfplotstableread{ 0 35569 8842    134984   1 30428 4689    34077 2 32920 6207    73787    3 16462 7562    23496    4 12315 8572    16565    5 76572 19572   26030    }\dataset \begin{tikzpicture} \begin{axis}[ybar,         width=.9\textwidth,         ymin=0,         ymax=150000,                 ylabel={Y-Label},         xtick=data,         xticklabels = {             Category 1,             Category 2,             Category 3,             Category 4,             Category 5,             Category 6         },         major x tick style = {opacity=0},         minor x tick num = 1,         minor tick length=2ex,         ] \addplot[draw=black,fill=black!20] table[x index=0,y index=1] \dataset; %Data1 \addplot[draw=black,fill=black!40] table[x index=0,y index=2] \dataset; %Data2 \addplot[draw=black,fill=black!60] table[x index=0,y index=3] \dataset; %Data3 \legend{Data1,Data2,Data3} \end{axis}  \end{tikzpicture}  \end{document} How do I limit the number of rows returned by an Oracle query after ordering? Is there a way to make an Oracle query behave like it contains a MySQL limit clause? In MySQL, I can do this: select *  from sometable order by name limit 20,10  to get the 21st to the 30th rows (skip the first 20, give the next 10). The rows are selected after the order by, so it really starts on the 20th name alphabetically. In Oracle, the only thing people mention is the rownum pseudo-column, but it is evaluated before order by, which means this: select *  from sometable where rownum &lt;= 10 order by name  will return a random set of ten rows ordered by name, which is not usually what I want. It also doesn't allow for specifying an offset. Often, TeX outputs underfull hbox and vbox warnings when running and in the generated log file. What are these and how can I get rid of them? Why would I use manual camera controls instead of the automatic modes? I'm very new to digital photography, so I still haven't gotten the hang of all this exposure and ISO stuff, etc. I have a Canon EOS T3.   My question is how big of a difference does it make using manual settings rather than auto settings for everything? I've taken some pretty nice photos with all auto settings, and I'm not sure if it could have turned out any better using manual settings. I know that there are some people who have been in the game for years upon years, and to them they can notice the difference between photos taken with manual vs auto settings, but for my situation, do you think it really makes a difference? Is my camera smart enough to determine the best settings for me?  Also, lets say manual settings are better (which I'm sure they are, since they give you greater control). How often are you supposed to adjust your settings? Between every shot, considering they are unique shots? Or does it not matter about the object, but rather the lighting environment? So if I'm in a forest at 2:00 PM on a sunny bright day and I take photos for one hour, should I be constantly adjusting my settings between every unique shot to get the best out of it? Or do I usually just find one setting that works for the environment I'm in? E: Malformed entry 54 in list file /etc/apt/sources.list (Component) How does Router know where to forward packet