conformal map between the region of two circle intersect at one point and unit disk
Conformally Map Region between tangent circles to Disk
Debate, Bare Plurals, and Mass Nouns What sites should have hyperlinking (onebox) support in chat? Some sites are . What additional sites should the chat support?  One suggestion per answer. A financial planner treated us in a dismissive fashion and I want to say that he "gave us the back of his hand" but that phrase doesn't seem to have the meaning of dismissive. Does it or is there a similar idiom? Generalized Euler sum $\sum_{n=1}^\infty \frac{H_n}{n^q}$ I found the following formula   $$\sum_{n=1}^\infty \frac{H_n}{n^q}= \left(1+\frac{q}{2} \right)\zeta(q+1)-\frac{1}{2}\sum_{k=1}^{q-2}\zeta(k+1)\zeta(q-k)$$  and it is cited that Euler proved the formula above , but how ?  Do there exist other proofs ?  Can we have a general formula for the alternating form   $$\sum_{n=1}^\infty (-1)^{n+1}\frac{H_n}{n^q}$$ The differential equation $$ {d^2\alpha \over dt^2} + {g \over L} \cdot \alpha = 0 $$  describes a 1-dimensional mathematical pendulum, where $\alpha $ is the angle, $ g = 9.82 $, and $ L = 0.2 $ is the lentgh of the string. What is the position of the pendulum after 1 second if the velocity at time equals zero is $ 0 $ m/s and the angle at time equals zero is $ {\pi \over 60} $ radians.  To my understanding this is a homogenous equation, ergo  $$ y'' + Cy = 0$$  And with the help of the characteristic equation I get the complex roots $ Ci \, $ and $ -Ci $.  This is a farily simple differential equation, but I have a feeling my calculations so far are wrong (mainly because I didn't expect to see complex roots for this equation). What does "I have straight A's." mean? Can commuting matrices $X,Y$ always be written as polynomials of some matrix $A$? Consider square matrices over a field $K$. I don't think additional assumptions about $K$ like algebraically closed or characteristic $0$ are pertinent, but feel free to make them for comfort. For any such matrix $A$, the set $K[A]$ of polynomials in $A$ is a commutative subalgebra of $M_n(K)$; the question is whether for any pair of commuting matrices $X,Y$ at least one such commutative subalgebra can be found that contains both $X$ and $Y$.  I was asking myself this in connection with frequently recurring requests to completely characterise commuting pairs of matrices, like . While providing a useful characterisation seems impossible, a positive anwer to the current question would at least provide some answer.  Note that in many rather likely situations one can in fact take $A$ to be one of the matrices $X,Y$, for instance when one of the matrices , or more generally if its  (so coincides with the characteristic polynomial). However this is not always possible, as can be easily seen for instance for diagonal matrices $X=\operatorname{diag}(0,0,1)$ and $Y=\operatorname{diag}(0,1,1)$. However in that case both will be polynomials in $A=\operatorname{diag}(x,y,z)$ for any distinct values $x,y,z$ (then $K[A]$ consists of all diagonal matrices); although in the example in  the matrices are not both diagonalisable, an appropriate $A$ can be found there as well.   I thought for some time that any maximal commutative subalgebra of $M_n(K)$ was of the form $K[A]$ (which would imply a positive answer) for some $A$ with minimal polynomial of degree$~n$, and that a positive answer to my question was in fact instrumental in proving this. However I was wrong on both counts: there exist (for $n\geq 4$) commutative subalgebras of dimension${}&gt;n$ (whereas $\dim_KK[A]\leq n$ for all $A\in M_n(K)$) as shown in , and I was forced to correct  in the light of this; however it seems (at least in the cases I looked at) that many (all?) pairs of matrices $X,Y$ in such a subalgebra still admit a matrix $A$ (which in general is not in the subalgebra) such that $X,Y\in K[A]$. This indicates that a positive answer to my question would not contradict the existence of such large commutative subalgebras: it would just mean that to obtain a maximal dimensional subalgebra containing $X,Y$ one should in general avoid throwing in an $A$ with $X,Y\in K[A]$. I do think these large subalgebras easily show that my question but for three commuting matrices has a negative answer.  Finally I note that  to the cited MO question mentions a result by Gerstenhaber that the dimension of the subalgebra generated two commuting matrices in $M_n(K)$ cannot exceed$~n$. This unfortunately does not settle my question (if $X,Y$ would generate a subalgebra of dimension${}&gt;n$, it would have proved a negative answer); it just might be that the mentioned result is true because of the existence of $A$ (I don't have access to a proof right now, but given the formulation it seems unlikely that it was done this way).  OK, I've tried to build up the suspense. Honesty demands that I say that I do know the answer to my question, since a colleague of mine provided a convincing one. I will however not give this answer right away, but post it once there has been some time for gathering answers here; who knows somebody will prove a different answer than the one I have (heaven forbid), or at least give the same answer with a different justification. In classical logic, why is $(p\Rightarrow q)$ True if both $p$ and $q$ are False? $n,a,b \mathbb \in \mathbb Z^+$ , such that $n\mid a^n-b^n$ ; to show $n\mid \frac {a^n-b^n}{a-b}$ I know I can only use the gym feature after hitting level 5, but are there any other privilege or rewards for the levels above 5?  For example, in Ingress, some items are only available for the higher level player. How about Pokemon Go? Exporting feature class into multiple feature classes based on field values using ArcGIS Desktop? I have a feature class with over 2,000 features, and I need to make them all individual feature classes based on a field.   Is there a way to do this? How does Wild Shape work with special ways to calculate your AC? Among the effects of Wild Shapes, these two:     Your game Statistics are replaced by the Statistics of the beast, but   you retain your alignment, personality, and Intelligence, Wisdom, and   Charisma scores. You also retain all of your skill and saving throw   proficiencies, in addition to gaining those of the creature. If the   creature has the same proficiency as you and the bonus in its stat   block is higher than yours, use the creature's bonus instead of yours.   If the creature has any legendary or lair actions, you can't use them.      You retain the benefit of any features from your class, race, or other   source and can use them if the new form is physically capable of doing   so. However, you can't use any of your Special senses, such as   Darkvision, unless your new form also has that sense (PHB p.67, "Wild Shape")   Seem to come into conflict with each other when things like Draconic Resilience:     When you aren't wearing armor, your AC equals 13 + your Dexterity   modifier. (PHB p.102, "Draconic Bloodline")   and Barbarians:     While you are not wearing any armor, your Armor Class equals 10 + your Dexterity modifier + your Constitution modifier. You can use a Shield and still gain this benefit.   and Monks:     Beginning at 1st level, while you are wearing no armor and not wielding a Shield, your AC equals 10 + your Dexterity modifier + your Wisdom modifier.   and Stone Sorcerers:     As an action, you can gain a base AC of 13 + your Constitution   modifier if you aren’t wearing armor, and your skin assumes a stony   appearance. This effect lasts until you end it as a bonus action, you   are incapacitated, or you don armor other than a shield.   How would these resolve? When he was just a professor, Dumbledore taught Transfiguration. As Headmaster what was he doing all day? In the 6th movie he was diligently looking for Horcruxes, but what about the other times?  When Fred and George pass the Maurader's Map to Harry they note that Dumbledore is pacing in his office. "He does that a lot" one of them says. Did Dumbledore just endless pace in his office as Headmaster, afraid to leave it for fear someone would know that he produced no work?   Sure, every year he had to go find a new DADA teacher, but most of those it seemed he did through correspondence. Horace Slughorn was the only one that seemed to resist. Dumbledore's 6th year may have been the only one where he earned his paycheck. I'm using Ubuntu and I'm trying to connect to another machine in a different network using remote desktop.   In Windows7 I have made the following in order to activate remote desktop:   I've gone to computer -> properties -> remote settings I've selected the option: "Allow connections from computers running any version of Remote Desktop I've opened "Windows Firewall with Advanced Security" In inbound rules I've enabled the rules for remote desktop (public and domain)   I have also installed Remmina in the Ubuntu machine.  For configuring it I did the following steps:   Selected the RDP protocol In the server input I have written the Windows machine public IP. In username / password I have typed my login credentials (the same as my Windows admin account)   But when I try to connect I get this error message:  "Unable to connect to RDP server 89.130.251.160"  If I ping my Windows7 machine, I have a correct response.  Any suggestions? Why don't they use the transporter in the final fight in Star Trek Into Darkness? When Spock was chasing, and beating, Khan over a transport in San Francisco, Uhura orders them beamed both back  to the Enterprise, while Chekov says something like:     Its impossible, they are moving!                                   But in the first movie, it was Chekov that transported back to Kirk and Sulu, avoiding hitting the Vulcan planet when trying to disable the Romulan drilling platform. How do I build a native (command line) executable to run on Android? When writing a book and referring to a female and male aged 18–25, what term would be most appropriate? Boy and girl seem too young, whereas man and woman seem too old.  Example sentence:     A boy/man emerged from the shadows.   Edit: I think the word I am looking for is similar to 'guy' but less American and colloquial. Any thoughts? Chapter number starts at 2 not 1 How can I start my chapter from "Chapter 2" instead of "Chapter 1".  I am preparing LaTeX document for each chapter. when I put \chapter{Introduction}, it produces "Chapter 1" Introduction. However I want it to be "chapter 2" Introduction instead. How can I move my /home directory to another partition if it's already part of the / partition? I have a wonderful running Maverick Meerkat install going on right now. I want to move my /home to another partition as per many suggestions.   How can I go about doing this without reinstalling the entire Ubuntu system? Can I simply create the partition with gparted and then change the mount point of /home in fstab to reflect it or is it a bit more difficult? How can you smooth the edge between two materials?