If $f$ is an analytic function on $D(0\ ;2)$ with $|f(z)| &lt; 4 $ if $\text{Im}(z) \ge 0\ \text{and}\ |z| =1$, and $|f(z)| &lt; 3$ if $\text{Im}(z) &lt;0\ \text{and}\ |z| =1$. How do I show $|f(0)| \le \sqrt{12}\ ?$  Can I use $g(z) = f(z)\cdot f(-z)$ here? I thought of using maximum modulus theorem on $g(z)$ also, but it doesn't seem to work as I need a $a\in D(0\ ;2)$ such that the condition in the theorem is satisfied, I can't see what will be it's ($a'$s) value.  Do I need to calculate the derivative of $g(z)$ and do something with it? How do I proceed? Any ideas/hints?
Let $f(z)$ be a holomorphic function on a disk of radius $&gt;1$ about the origin, and suppose that $|f(z)|\leq 1$ for $z$ with $|z|=1$ and $\text{Re}(z)&gt;0$ and $|f(z)|\leq 2$ for $z$ with $|z|=1$ and $\text{Re}(z)\leq 0$.  How does one show the bound $|f(0)|\leq \sqrt{2}$?
Q&A blank in Stack Exchange Android app v1.0.95 - S10e/Android 10 I have a model which you can see here (camera perspective).    In the second image, you can see how the render turned out.    I'm at my wit's end trying to figure our why it's not showing up in the render. If anyone needs more information, just ask and I'll get it to you. I've already checked the normals and they're not the cause. How does delete[] know it's an array? Year-Round Hats for Fun and Profit Default constructor with empty brackets Is there any good reason that an empty set of round brackets (parentheses) isn't valid for calling the default constructor in C++?  MyObject  object;  // ok - default ctor MyObject  object(blah); // ok  MyObject  object();  // error   I seem to type "()" automatically everytime. Is there a good reason this isn't allowed? Why do the Walking Dead Zombies only show intelligence in the very first episode? I enjoy The Walking Dead, however one question keeps nagging me. The 'Walkers' are always shown to not have any intelligence or memory of having been human. This is for all seasons except the very first one.  In the very first scene of Season One, Episode One, we see a little zombie girl stop to deliberately pick up a teddy bear from the ground.   Later in the same episode we see a zombie apparently try to turn the door knob to get into Morgen Jones' house.  Also at Atlanta they can climb fences and used rocks to break glass.  After season 1 I do not recall seeing any such intelligence demonstrated by a zombie again.  My question is: are these initial displays of intelligence a discrepancy in the story, or is there a plausible explanation why the Walkers performed these actions? Confusion on Derived Sets and the $n$-the Derived Set I want to solve the following exercise 1.3.C. from page 27 of R. Engelking: General Topology.  For every positive integer $n$ the $n$-th derived set $A^{(n)}$ of a subset $A$ of a topological space $X$ is defined inductively by the formulas: $$  A^{(1)} = A^d \quad \textrm{and} \quad A^{(n)} = (A^{(n-1)})^d. $$ (a) Give an example of a set of real numbers that has three consecutive derived sets distinct from each other.  (b) Give an example of a set of real numbers that has infinitely many derived sets distinct from each other.   The definition is $$   A^d := \{ x \in X \mid x \in \overline{A \setminus \{ x \}} \}. $$ I am confused, cause the set $A^d$ ist closed, i.e. $\overline{A^d} = A^d$. And closed means it already contains all its limit points, so how can than $(A^d)^d$ be different from $A^d$? Is tipping mandatory at Restaurants and Bars in Germany? After Google Play Service update to version 13 I got an error Restricing domains for universal and existential quantifiers I am currently using Rosen's "Discrete Mathematics and Its Applications" (7th ed.) for my discrete mathematics course. We recently talked about quantifiers, more specifically the universal and existential quantifier. What confuses me is when we started talking about restricting the domain (i.e. instead of all real numbers, only all real positive numbers), where does the conditional statement come from when restricting the domain of a universal quantifier and where does the conjunction come from when restricting the domain of a existential quantifier?  This link did help:   However, the part that confuses here is how are the domains restricted in the answer given in the link above? Would you take a share in Stack Overflow? Joel Spolsky has just posted  where he talks about seeking venture capital for Stack Overflow.  If instead they opted to seek shareholders from the StackOverflow community along the lines of the , but at a larger scale, would you be interested?  I'm asking this because I suspect there might be enough fanatics willing to invest $500 or so in a product they love to raise a considerable sum. People also often talk about the motivation for high rep users. It seems to me having an actual stake in the product might be a decent incentive (especially if you also get a gold badge for buying shares). siunitx tabular numbers with separated uncertainty I would like to format the space reserved for a number shown in a table with separated uncertainty. It seems that when I use the table-format option the brackets around the mantissa disappear. This is illustrated in the example below:  \documentclass{article}     \usepackage{siunitx}  \begin{document}     \begin{table}   \begin{tabular}{                   S[separate-uncertainty = true]                   S[table-format=1.1(1)e1,                   separate-uncertainty = true]                   }     {Header} &amp; {Header}\\     1.0(2)e2 &amp; 1.0(2)e2\\   \end{tabular} \end{table}     \end{document}     I have also tried to use the bracket-numbers and multi-part-units = brackets without solving this issue. Is there any workaround to center the number and keep the brackets like in the first column of the example? What's the meaning of 90-plus? What's the meaning of 90-plus? More than 90? If so, please tell me more ways to say "more than 90". Thank you! How to open nautilus without a desktop? I'm using xmonad, and I would like to open a nautilus window to a certain directory, because I like the convenience of browsing inside an encrypted zip file without unzipping it, which nautilus allows.   however, opening nautilus, even with a path argument starts a desktop on the monitor I'm using (There are 8 xmonad monitors) The desktop is inoperative and renders that monitor useless until nautilus is killed.  Is there any way to open just a nautilus window without any desktop starting? How to move apps from internal to external storage? I just installed Ubuntu alongside to Windows 7, and now Windows 7 will not boot.  When the system starts, it shows me entry for Windows 7 in the boot menu, when I click on the Windows 7 option in the boot menu, after few seconds it will show me the boot menu again and Windows will not boot. It shows me no error message. Windows 7 option is available in the boot menu but it just does not boot.  Here are my boot info results   update - 30/01/2012 I tried windows recovery - fix startup issues - that showed no problems but issues were not resolved  Then I tried the widnows recovery console and run following commands  bootrec /fixboot bootrec/ fixmbr bootrec /RebuildBCD   But that does not help, Now neither ubuntu nor windows will boot. When I boot from windows CD it shows me a valid windows installation in c:\ drive.  update After a lot of help I got from #ubuntu channel and trying out different things, ultimately I can now boot both windows and ubuntu, still, the issue is not completely resolved though.  Here's what made it work  Boot using ubuntu live CD and then run the testdisk utility as steps mentioned  Restart system and I was able to see the boot menu and boot into Ubuntu, then put the following code in boot/grub/custom.cfg. Restart and now I can see one more entry to boot win7, clicking on it boots windows.  menuentry "Windows 7 (loader) with grub ntldr workaround" --class windows --class os   {     insmod part_msdos     insmod ntfs     set root='(hd0,msdos1)'     search --no-floppy --fs-uuid --set=root FAF0F57DF0F54085     ntldr /bootmgr }   It will make windows boot probably but now windows have a dependency on Ubuntu and I can not remove Ubuntu or else windows will stop working.  Credit goes to 'Jordan_U' on #ubuntu channel.   update Here are the win7 boot details I got from EasyBCD  Windows Boot Manager -------------------- identifier              {9dea862c-5cdd-4e70-acc1-f32b344d4795} device                  partition=C: path                    \bootmgr description             Windows Boot Manager locale                  en-US default                 {04fe8f95-4b17-11e1-a45c-909d3d25f938} displayorder            {04fe8f95-4b17-11e1-a45c-909d3d25f938} timeout                 30  Windows Boot Loader ------------------- identifier              {04fe8f95-4b17-11e1-a45c-909d3d25f938} device                  partition=C: path                    \Windows\system32\winload.exe description             Windows 7 Home Premium locale                  en-US osdevice                partition=C: systemroot              \Windows resumeobject            {8d25a443-4b0a-11e1-a835-806e6f6e6963}   Note Still, I am looking for how can I have everything clean and how can I boot windows without having a dependency on Ubuntu. so that if ever I have to remove ubuntu, windows will keep booting file without ubuntu. bash variables in for loop range I want to print list of numbers from 1 to 100 and I use a for loop like the following:  number=100 for num in {1..$number} do   echo $num done   When I execute the command it only prints {1..100} and not the list of number from 1 to 100. accounting for LED resistance How to prove one has spent time in the non-Schengen European microstates, in order to pause the Schengen clock? I am considering staying in Monaco, San Marino, or Vatican City before I can enter the Schengen area again. , it is valid to stay in these three micro-states in order to stop the Schengen clock. However, also according to the answer:  Problems will arise when the visitor ultimately leaves the zone and must undergo a Schengen exit inspection. At this point the person may be asked to prove that they are not an overstayer and the border guard will be entitled to see hard evidence in the form of official documentation. There was a case where an individual attempted to show that his mobile phone registered to a tower outside of the zone and this failed obviously: radio waves have no respect for nation borders but also one could mail the SIM card to a friend in a far away land, put in a phone where it registers to the tower and then mail it back. Apparently hotel receipts can be too easily forged to be successful as evidence. What the border guard will want to see is a passport stamp from the local constabulary. But the constabulary is under no obligation to issue one, so success is down to personal impact and articulation skills.  I'm not sure if I could get a stamp from the local constabulary. However, I always use AirBnB to stay, and the reservation email and the mobile app might be far more reliable than a random hotel reservation document. So, does the AirBnB reservation email work as a proof of staying there? Or is there any better document as a proof of staying (&quot;better&quot;, in terms of ensuring that I don't pay the overstaying penalty)? I will likely leave the Schengen area ultimately from Frankfurt International Airport. Fermat: Last two digits of $7^{355}$ I am doing this problem mentioned above and I know the answer because I know Euler's Theorem that $$a^{\varphi{(m)}}\equiv{1}\pmod{m}.$$  I used 100 as my modulus and got that the last two digits of $7^{355}$ are $43$.  My question is this.  What prime should I pick to solve this problem using Fermat's theorem $$a^{p-1}\equiv{1}\pmod{p}$$  As a precursor problem, the author (Dudley; Elementary Number Theory) asked to find the last digit of the number above; so using a modulus $5$, $7^4\equiv{1}\pmod{5}$ by Fermat.  Since the answer is $7^{355}\equiv{3}\pmod{5}$, the last digit is either $3$ or $8$.  Since $7$ is odd, the answer will clearly be odd, so $3$ is it.  The book is working linearly, building from the previous chapters and Euler's Theorem isn't mentioned for another two chapters (I remembered Euler from a problem solving class in college).    So again, what prime would be best?  How should I procede?    EDIT: I've seen a bunch of problems on the site and none that I looked at did it using Fermat.  If there is one, that could be posted and I would be happy.  EDIT 2:  it seems that the problem of finding of finding the last two digits using is rather difficult because of the implication that we need something mod 100.  Is there a way around this , like adding a step or complexifying, in order to finish the problem with Fermat?