The following question was asked in a master's exam for which I am studying.  Suppose that $f(z)$ is analytic in the disk $|z|\leq R$ and let $$ A(r)= \max_{|z|=r} \text{Re}\,f(z). $$ Prove that, for $r&lt;R$, $$\max_{|z|=r} \frac{ |f^{(n)} (z) | } { n!} \leq \frac{2^{n+2} R }{ (R-r)^{n+1} }\big\{A(r)+|f(0)|\big\}.$$  I am confused on which result I should use to prove this. $\max_{|z|=r} \frac{ |f^{(n)} z | } { n!} $ will have terms from both the real and imaginary parts of $f$. I thought of using Cauchy Integral formula but it will only esitmate value at $f(0)$. I need you to just outline the result that should be used. I will complete it myself.
Suppose that $f(z)$ is analytic function in the disk $|z|\le R$ and $A(r)=\max_{|z|=r}Re(z)$ then prove that $r&lt;R$  $$\max_{|z|=r}\left|\frac{f^n(z)}{n!}\right|\le \frac{2^{n+2}R}{(R-r)^{n+1}}\{A(r)+|f(0)|\}$$  MY attempt   Using Cauchy generalized integral formula   $f^{(n)}(z_0)=\frac{n!}{2\pi i}\int _c\frac{f(z)dz}{(z-z_0)^{n+1}}$  then $\max_{|z|=r}\left|\frac{f^n(z)}{n!}\right|=\left|\max_{|z|=r} \frac{1}{2\pi i}\int _c\frac{f(z)dz}{(z-z_0)^{n+1}}\right|$  and i think we can solve by using Borel caratheodory theorem  Let $f(z)$ be holomorphic function on $|z|\le R$ and let $M(r)=\max_{|z|=r}|f(z)|$ and $A(r)=\max_{|z|=r}Re f(z)$ then  for $0&lt;r&lt;R$  $$ M(r)\le \frac{2r}{R-r}A(R)+\frac{R+r}{R-r}|f(0)|$$  from here how to processed ?
Shouldn't the association bonus acquired via serial upvoting be reversed? When a user's rep gets bloated because of serial upvoting, and as result the user is given the association bonus, shouldn't that association bonus also be reversed in case of serial upvoting reversal? Which should I use on my DB? - Clustered Index or Non Clustered Index or Both I just need advice on how I am going to optimize my database.  Date ---- Time ---- Area ---- Block ---- Data1 ---- Data2 ---- Data3 ---- DataN   I have that format of tables on my database. Each tables has 30-days of records with hundred thousands of records each data. Uploading of data to database is every morning. The data of yesterday will be uploaded.  The most common queries to run is grouped by Date or Date and Area or Date and Area and Block. Now, in order to make queries faster, I found out using indexes. I used indexes before. But that was when I am using a primary key. In this table, I didn't use primary key because it is totally unnecessary.   Now, I am really confused on how I am going to optimize this. Based on  on Column Considerations part, clustered index is not a good choice on Columns that undergo frequent changes. I am really confused.  Please help me if I need to use clustered index here or just non clustered or both. $p=4n+3$ never has a Decomposition into $2$ Squares, right? Primes of the form $p=4k+1\;$  have a unique decomposition as sum of squares $p=a^2+b^2$ with $0&lt;a&lt;b\;$, due to .  Is it correct to say that, primes of the form $p=4n+3$, never have a decomposition into $2$ squares, because sum of the quadratic residues $a^2+b^2$ with $a,b\in \Bbb{N}$ $$ a^2 \bmod 4 +b^2 \bmod 4 \le 2? $$  If so, are there alternate ways to prove it? Error propagation with asymmetric uncertainties I red the Wikipedia  on error propagation. I get into some trouble when I want to calculate the error propagation in a specific case. I have real numbers $A$ and $B$ with related upper ($\Delta A_u$, $\Delta B_u$) and lower ($\Delta A_l$, $\Delta B_l$) uncertainties. I have two functions calculated as follows: $$F = \sqrt{A^2+B^2}$$ and  $$G = \tan^{-1}\frac{B}{A}$$ What are the related upper and lower errors of $F$ and $G$?  I found general formulae for Gaussian error distributions ($\Delta A_u = \Delta A_l$ and $\Delta B_u = \Delta B_l$), for instance the first formula of the first answer on . Is there a way to generalize that formula for arbitrary distributions with asymmetric errors as in my example? Allow setuid on shell scripts Electric Potential produced by Hollow Sphere Let there be a hollow sphere (Throughout the question we will ignore the thickness of the sphere) which is positively uniformly charged,q of radius,  R. Suppose there to be a test positive charge, q' at point P which is r distance away from the hollow sphere(r>R, P is outside of hollow sphere).  Now I have an doubt on evaluating Electric Potential,V at point P.  Both source charge,q and test charge,q' are +ve, so source charge will repel the test charge which will make the charge, q' to go to infinity.I have to apply external force, F' to displace the charge, q' from infinity to point P against the equal but repulsive force, F.In short, F' is  displacing the q' from infinity to P.  So External Force, F' will do +ve work and Repulsive Force, F will do -ve work. This way the Electric Field,E produced by hollow sphere is Opposite to Displacement of test charge(i.e.,θ=180).  Now we know that E=k(q/(r^2))  [Please Notice that infinity is the lower limit of integration]  Also V=-∫r∞(E dr cosθ)=-∫r∞(E dr cos 180)=∫r∞(E dr)  At last I ended up to V=- k(q/r). I know that V=k(q/r).  I just want to know why I am having this shitty negative sign in my equation. Check if an image is loaded (no errors) with jQuery Error message: qgis_core.dll missing I have a fresh install of QGIS 2.4. When trying to open a .shp file I get the error message below.  I've followed their advice and have done several re-installs and also cleared all qgis files from my system (Parallels on Mac OS X 10.6) before those re-installs. I've tried QGIS 2.2 which caused the same problem.  Funnily all worked for a day, the following day and since not anymore...  Has anyone come across this before? My wifi connection doesnt work properly; it stays connected only for around 5 minutes after I turn on the PC. I'm using Ubuntu 16.04.  I've tried   sudo service network-manager restart   which makes no difference  Same with   killall nm-applet &amp;&amp; nm-applet &amp;   After that I tried   sudo gedit /etc/pm/config.d/config   where I insert this string  SUSPEND_MODULES = "drivername logicalchipname"   But since I don't have permission to modify that file I tried to use chmod on it, which led to  but left me still in the same situation.  $ lspci -knn | grep Net -A3; rfkill list 03:00.0 Network controller [0280]: Realtek Semiconductor Co., Ltd. RTL8821AE 802.11ac PCIe Wireless Network Adapter [10ec:8821] Subsystem: XAVi Technologies Corp. RTL8821AE 802.11ac PCIe Wireless Network Adapter [1b9a:2482]  Kernel driver in use: rtl8821ae  Kernel modules: rtl8821ae  0: hci0: Bluetooth  Soft blocked: yes   Hard blocked: no 1: phy0: Wireless LAN   Soft blocked: no    Hard blocked: no   What do you suggest I do? How to convert an array into comma separated strings in smarty template? Is Borden's true identity real, or another ruse? Towards the end of , it's revealed that Borden accomplishes his Transported Man trick by using his twin brother, Fallon, as a double. It then becomes clear that he has been doing this in his personal life as well as on stage - each brother leads 'half a life', sometimes appearing as Borden and sometimes as Fallon. This is how he completes the illusion: one Borden goes into the first door, the other Borden emerges from the second door a moment later. Angier doesn't know how Borden does the trick, and doesn't believe Borden uses a double despite being told as much by Cutter. He follows what he believes to be a clue in Borden's notebook and travels to Colorado to meet Nikola Tesla, who Angier believes can build a machine that can help him perform the trick. Angier believes that Tesla built a similar machine for Borden. However, the final pages of the notebook reveal that Tesla was merely a red herring, designed to get Angier out of Borden's way for a long time. Tesla, it turns out, had never built such a machine before. At this point, Tesla agrees to build Angier's machine anyway. Instead of building a machine that transports matter, he builds one that duplicates it. Angier uses this machine in his act, but in order to avoid having to deal with multiple clones (the &quot;prestige materials&quot;), one of them must die as part of the act. This all makes sense, so far, but I find it hard to believe that Borden's use of the word 'Tesla' as the key to his cipher is random. It's too coincidental that he sends Angier on a wild goose chase, but Angier returns with a working machine. A film that is otherwise so tightly plotted shouldn't rely on coincidence to move the narrative forward, so there must be something else going on. Is it possible that Borden cloned himself using Tesla's machine? Past participle vs being+past participle How can I type special symbols on my Android device like ☺ or ♥ as a regular character? Is there any way I can see at what time the commands were executed from the bash history? We can see the order but is there any way I can get the time also?  Bottom-Line: Execution time in the Bash history How should different sized RAM be added to iMac? I have a mid-2011 iMac which came with 4GB RAM (2x2GB). I just bought 8 GB (2x4GB) and added it to get a total of 12GB. The current configuration is as follows:    Given that the RAM is a different size, do I have the optimal locations for the different RAM chips? At the moment what I did was to simply add the 2 new sticks of RAM in the two empty slots. I'm using 13.04 for the first time and I am using proxy base server that's why I can't download anything while I use apt-get update then it shows this message  coolmonter@coolmonter-HP-ProBook-4431s:~$ apt-get update E: Could not open lock file /var/lib/apt/lists/lock - open (13: Permission denied) E: Unable to lock directory /var/lib/apt/lists/ E: Could not open lock file /var/lib/dpkg/lock - open (13: Permission denied) E: Unable to lock the administration directory (/var/lib/dpkg/), are you root?   any suggestion Why Process Builder doesn't work on Delete Scenario? $a =\prod_{i = 1}^{r} p_{i}^{ai}$ with $a_{i} > 0$ for each $i$ is the canonical representation of $a$... This is solely a reference request. I have heard a few versions of the following theorem:     If the joint moment generating function $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ whenever the expectations are finite, then $X,Y$ are independent.   And there is a similar version for characteristic functions. Could anyone provide me a serious reference which proves one or both of these theorems? Aligning equations within multline I have this really long equation and I have used multline to break it up. I also want to align the broken bits of the equation to the left. When I use  align, for some reasons it aligns to the right when I want it to be on the left. How can I align to the left?  \begin{multline} \begin{align} \mathcal{H}_{q1} \mathcal{H}_{q2} \mathcal{H}_{q3}...\mathcal{H}_{qn}0=\frac{1}{2}H_{q1}\{\mathcal{H}_{q2}\mathcal{H}_{q3}...\mathcal{H}_{qn}0\}+\frac{1}{2} h_{q1} \\ =\frac{1}{2}H_{q1}\{\frac{1}{2} H_{q2}[\mathcal{H}_{q3}...\mathcal{H}_{qn} 0]+\frac{1}{2}h_{q2}\} +\frac{1}{2}h_{q1}=\\ \frac{1}{2}H_{q1} \frac{1}{2} H_{q2} [\frac{1}{2}H_{q3}(\mathcal{H}_{q4}..\mathcal{H}_{qn}0)+\frac{1}{2}h_{q3}]+ \frac{1}{2}h_{q2}+\frac{1}{2}h_{q1}=...\\          =(\frac{1}{2^n})H_{q1}H_{q2}H_{q3}..H_{qn}0+ (\frac{1}{2^n})H_{q1}H_{q2}H_{q3}..H_{qn-1}h_{q1}\\+(\frac{1}{2^n})H_{q1}H_{q2}H_{q3}..H_{qn-2}h_{qn-1}+.. \\+ (\frac{1}{2})h_{q1}=\sum^n_{j=1} (\frac{1}{2^j}) H_{qo}H_{q1} H_{q2}H_{q3}..H_{qj-1} h_{qj} \end{align} \end{multline}