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Let $D := \{z \in \mathbb{C}: |z| &lt; 1\}$ and $f\colon D \rightarrow \mathbb{C}$ be holomorphic. Suppose $\lvert f(z)\rvert \leq 1$ on $D$, show that $$\frac{|f(0)| - |z|}{1 + |f(0)||z|} \leq |f(z)| \leq \frac{|f(0)| + |z|}{1 - |f(0)||z|} \ \forall z \in D.$$  I have tried using Cauchy Integral Formula to $f$ and expanding $f$ at $0$, but I have no idea why $f(0)$ appears in the inequality in the former case. Both answers and hints are appreciated and thanks in advance! The interaction of TeX and Lua can be quite delicate at times.  While the former has category codes and tokens, the latter has escape sequences and datatypes.  Mixing the disparate concepts of these two worlds can lead to unexpected behaviour which favours common errors, known as gotchas. In the community wiki answer below you can find an overview of the common mistakes that people make when getting to grips with LuaTeX. Accepted an answer message When you accept an answer on one of your questions, you get 2 reputation for it. The achievements dropdown tells you with a shiny green +2 on the topbar, and shows this:    Why did I get +2 for it again?  It would be better if the message told you it was because you accepted an answer, right there in the dropdown, without the need to go into your profile to see more details. Poincaré inequality for a subspace of $H^1(\Omega)$ The following is the well known  for $H_0^1(\Omega)$:       Suppose that $\Omega$ is an open set in $\mathbb{R}^n$ that is bounded. Then there is a constant $C$ such that   $$ \int_\Omega u^2\ dx\leq C\int_\Omega|Du|^2\ dx\quad \textrm{ for all   }\ \ \color{red}{u\in H_0^1(\Omega)}. $$   Here is my questions:    Can the condition $u\in H_0^1(\Omega)$ be relaxed so that the inequality is still true? More precisely, suppose $H$ is a subspace of $H^1(\Omega)$ such that for any $u\in H$, $u|_\Gamma=0$, where $\Gamma$ is a ([Added:]nonempty) closed proper subset of the boundary $\partial\Omega$ and the idenity is in the sense of trace. In other words, $u$ is only zero on a part of the boundary. Do we still have the Poincaré ineqaulity in $H$?    [Added:] This question is motivated by the following exercise In my program I have one array with 25 double values 0.04 When I try to sum these values in a loop I get following results:  0.0 + 0.04 = 0.04 0.04 + 0.04 = 0.08 0.08 + 0.04 = 0.12 0.12 + 0.04 = 0.16 0.16 + 0.04 = 0.2 0.2 + 0.04 = 0.24000000000000002 0.24000000000000002 + 0.04 = 0.28 0.28 + 0.04 = 0.32 0.32 + 0.04 = 0.36 0.36 + 0.04 = 0.39999999999999997 0.39999999999999997 + 0.04 = 0.43999999999999995 0.43999999999999995 + 0.04 = 0.4799999999999999 0.4799999999999999 + 0.04 = 0.5199999999999999 0.5199999999999999 + 0.04 = 0.5599999999999999 0.5599999999999999 + 0.04 = 0.6 0.6 + 0.04 = 0.64 0.64 + 0.04 = 0.68 0.68 + 0.04 = 0.7200000000000001 0.7200000000000001 + 0.04 = 0.7600000000000001 0.7600000000000001 + 0.04 = 0.8000000000000002 0.8000000000000002 + 0.04 = 0.8400000000000002 0.8400000000000002 + 0.04 = 0.8800000000000002 0.8800000000000002 + 0.04 = 0.9200000000000003 0.9200000000000003 + 0.04 = 0.9600000000000003   Why on earth could that happen?! Do I have to re-enter the Schengen area to activate my second visa when traveling on back-to-back Schengen visas? I currently have a valid multiple entry Schengen visa that expires on Feb 10th (issued by the Spanish consulate). I need to visit the Czech Republic from Feb 9th to Feb 15th. For this, I am getting a visa starting Feb 11th from the Czech consulate as I was advised that I cannot have two simultaneous visas.   Is it possible for me to enter Czech republic on the current visa on Feb 9th, and stay past its expiry up to Feb 15th using the new visa without leaving the Czech Republic? Will there be any likely problem either while entering or exiting Schengen zone (I am entering Schengen in Frankfurt)? Short circuit or disconnect to bypass a compressor I have an Altec Lansing ATP3 2.1 audio system PCB which I reverse-engineered, and I would like to bypass the JFET compressor that's on it. Its active components are a JFET and four op-amps. I will connect the input to the output so it gets forwarded to the next stage.  My question is: should I disconnect the input and output from the compressor (in that case should I remove power supply as well, which would be very annoying), or can I leave it connected but short-circuited?  The block I want to short-circuit is the pink one on the left.    I realize the image is very small. The sound comes in on the right, in the green box. In the middle yellow box there's some trebble/volume/bass control. The small green box on its left merges the left and right channels to forward it to the subwoofer section, which is the complete left half of this diagram (except the red box on the left, which is just power). Signal then goes through a low-pass, a small equalizer and a compressor (the big purple box). It finally gets amplified in the brown power amp. VirtualBox not starting after kernel upgrade Yesterday I received a kernel upgrade and after reboot, VirtualBox stopped working.  Here is my system info (after kernel upgrade):  matteo@workstation:~$ cat /etc/lsb-release  DISTRIB_ID=Ubuntu DISTRIB_RELEASE=16.04 DISTRIB_CODENAME=xenial DISTRIB_DESCRIPTION="Ubuntu 16.04.3 LTS" matteo@workstation:~$ uname -a Linux workstation 4.4.0-116-generic #140-Ubuntu SMP Mon Feb 12 21:23:04 UTC 2018 x86_64 x86_64 x86_64 GNU/Linux   I installed VirtualBox long ago with  sudo apt install linux-headers-$(uname -r) sudo apt install virtualbox-dkms virtualbox virtualbox-qt   used on daily basis and never had problems with kernel updates. The error I receive now is   matteo@workstation:~$ sudo modprobe vboxdrv modprobe: ERROR: could not insert 'vboxdrv': Exec format error matteo@workstation:~$ dmesg | tail -n 1 [ 1413.167311] vboxdrv: version magic '4.4.0-116-generic SMP mod_unload modversions ' should be '4.4.0-116-generic SMP mod_unload modversions retpoline '   I found a forum post with this error message  from three days ago, unfortunately without resolution. Therfore I tried to remove the VirtualBox packeges from the Ubuntu repo, and install the latest 5.2 using . However, even after rebooting the machine, the error remains.  What can I do in this case? I have a 16GB flash drive. I'm running Ubuntu 14.04. I can't open my flash drive and I can't see it in GParted.  After running tail -f /var/log/syslog . After inserting my flash:  USB disconnect, device number 10 Jul  3 16:39:55 rihanna-VGN-Z56GG-B kernel: [ 3309.712166] usb 2-3: new high-speed USB device number 11 using ehci-pci Jul  3 16:39:55 rihanna-VGN-Z56GG-B kernel: [ 3309.846591] usb 2-3: New USB device found, idVendor=090c, idProduct=1000 Jul  3 16:39:55 rihanna-VGN-Z56GG-B kernel: [ 3309.846601] usb 2-3:   New USB device strings: Mfr=1, Product=2, SerialNumber=3     Jul  3 16:39:55 rihanna-VGN-Z56GG-B kernel: [ 3309.846609] usb 2-3: Product: Silicon-Power     Jul  3 16:39:55 rihanna-VGN-Z56GG-B kernel: [ 3309.846615] usb 2-3: Manufacturer: UFD 2.0     Jul  3 16:39:55 rihanna-VGN-Z56GG-B kernel: [ 3309.846622] usb 2-3: SerialNumber: 1401208423400131     Jul  3 16:39:55 rihanna-VGN-Z56GG-B kernel: [ 3309.847176] usb-storage 2-3:1.0: USB Mass Storage device detected     Jul  3 16:39:55 rihanna-VGN-Z56GG-B kernel: [ 3309.847309] scsi12 : usb-storage 2-3:1.0     Jul  3 16:39:55 rihanna-VGN-Z56GG-B mtp-probe: checking bus 2, device 11: "/sys/devices/pci0000:00/0000:00:1d.7/usb2/2-3"     Jul  3 16:39:55 rihanna-VGN-Z56GG-B mtp-probe: bus: 2, device: 11 was not an MTP device     Jul  3 16:40:17 rihanna-VGN-Z56GG-B kernel: [ 3332.176107] usb 2-3: reset high-speed USB device number 11 using ehci-pci     Jul  3 16:40:33 rihanna-VGN-Z56GG-B kernel: [ 3347.288080] usb 2-3: device descriptor read/64, error -110     Jul  3 16:40:48 rihanna-VGN-Z56GG-B kernel: [ 3362.504081] usb 2-3: device descriptor read/64, error -110     Jul  3 16:40:48 rihanna-VGN-Z56GG-B kernel: [ 3362.720076] usb 2-3: reset high-speed USB device number 11 using ehci-pci     Jul  3 16:40:52 rihanna-VGN-Z56GG-B kernel: [ 3366.812110] usb 2-3: reset high-speed USB device number 11 using ehci-pci     Jul  3 16:41:07 rihanna-VGN-Z56GG-B kernel: [ 3381.924082] usb 2-3: device descriptor read/64, error -110     Jul  3 16:41:23 rihanna-VGN-Z56GG-B kernel: [ 3397.356093] usb 2-3: reset high-speed USB device number 11 using ehci-pci     Jul  3 16:41:38 rihanna-VGN-Z56GG-B kernel: [ 3412.468057] usb 2-3: device descriptor read/64, error -110     Jul  3 16:41:54 rihanna-VGN-Z56GG-B kernel: [ 3428.664065] usb 2-3: reset high-speed USB device number 11 using ehci-pci     Jul  3 16:42:09 rihanna-VGN-Z56GG-B kernel: [ 3443.776100] usb 2-3: device descriptor read/64, error -110     Jul  3 16:42:24 rihanna-VGN-Z56GG-B kernel: [ 3458.992085] usb 2-3: device descriptor read/64, error -110     Jul  3 16:42:25 rihanna-VGN-Z56GG-B kernel: [ 3459.208063] usb 2-3: reset high-speed USB device number 11 using ehci-pci     Jul  3 16:42:40 rihanna-VGN-Z56GG-B kernel: [ 3474.320039] usb 2-3: device descriptor read/64, error -110     Jul  3 16:42:50 rihanna-VGN-Z56GG-B kernel: [ 3484.516109] usb 2-3: reset high-speed USB device number 11 using ehci-pci     Jul  3 16:42:50 rihanna-VGN-Z56GG-B kernel: [ 3484.768108] usb 2-3: reset high-speed USB device number 11 using ehci-pci Jul  3 16:43:05 rihanna-VGN-Z56GG-B kernel: [ 3499.880081] usb 2-3: device descriptor read/64, error -110 Jul  3 16:43:20 rihanna-VGN-Z56GG-B kernel: [ 3515.096081] usb 2-3: device descriptor read/64, error -110 Jul  3 16:43:21 rihanna-VGN-Z56GG-B kernel: [ 3515.312048] usb 2-3: reset high-speed USB device number 11 using ehci-pci Jul  3 16:43:36 rihanna-VGN-Z56GG-B kernel: [ 3530.424032] usb 2-3: device descriptor read/64, error -110 Jul  3 16:43:46 rihanna-VGN-Z56GG-B kernel: [ 3540.505230] scsi 12:0:0:0: Device offlined - not ready after error recovery Passing Javascript variable to How do I edit a file in /proc that can be viewed with cat, but is not editable by mousepad and abiword? Are Goodberries affected by the Life domain class features? Can I buy PSN games and DLC without giving Sony my credit card info? I was having a discussion with a friend of mine about some normal group properties and then came up with the question "if G is not commutative, then is there always a subgroup that is not a normal subgroup?" It's probably more easy to solve this in the following form:$$\forall H \leq G : H  \lhd G \Rightarrow \forall a,b \in G : ab=ba$$ My question is, can anybody give a proof, or a counter-example (because I don't think it holds) of this theorem? Thanks! Why there is a $\frac{1}{2}$ in the distance formula $d=\frac{1}{2}at^2$? Why don't the zoom buttons work on my DSLR? I am a newbie to the new camera I just bought. It is a Nikon D5200 it has the original lens that came on the camera a 18-55 lens on it. My question is when I turn the lens to the 55 and then push the plus and minus buttons at the bottom of my camera to zoom in it appears to be zooming in and then I take the photo and the picture is not zoomed in at all.    I have a new Canon T3i and was able to take and capture zoomed in photos with the lens that came with the camera (EF-S 18-55mm) as well as a Canon zoom lens EF 75-300mm 1:4-5.6. Now when I zoom in with either lens and take a photo, when I review it, it shows original image before zooming. Is it a defect in the camera or inexperience of the operator?    (Two separate questions from separate users but with the same root cause.) Mask and convert color in photo/video I am very new to blender still.  I want to simply change the color of an object in the video.  I want to know if there is good way to do this.  I am including what I have to test. This is a  about Hosting multiple SSL websites on the same IP.   I was under the impression that each SSL Certificate required it's own unique IP Address/Port combination. But the  is at odds with this claim.  Using information from that Question, I was able to get multiple SSL certificates to work on the same IP address and on port 443. I am very confused as to why this works given the assumption above and reinforced by others that each SSL domain website on the same server requires its own IP/Port.  I am suspicious that I did something wrong.  Can multiple SSL Certificates be used this way? I'm sure this is a newbie question, but I can't figure this out. I have tried to follow the advice here:  and here:   But I can not seem to be able to generate any font change. I either get no change or error messages.  Here is the output of one failed attempt to change the font to Times using this command:  \rmfamily \renewcommand{\rmdefault}{ptm}     Apparently I either don't have a correct package installed or...  The output says " Metric (TFM) file not found." but I am not sure where to go from there. What is serial voting and how does it affect me?