Does there exist a complex series that the radius of convergence is $R=1$ but the series diverges at exactly three points of a set $\{z \in \Bbb{C} : \space |z|=1 \}?$   I found this exercise and I wonder how should it be solved. I know that in real analysis there exist such series that the radius of convergence is $1$ in the interval $X$ but the same series diverges at the endpoints of $X$. But it has only two points of divergence.   Can anyone show me the way?
I want to construct a complex power series with the radius of convergence $R=1$ that diverges in:  1) $k$ points on the circle $\{|z|=1\}$.  2) countable number of points on the circle  I have already crafted series that converges on the whole circle and diverges on the whole circle.  Can anybody help with the problem?  Update: I have googled a series $\sum{\frac{z^{kn}}{kn}}$ that is an answer for 1). Though I still need help for the countable case.
I would like to remove my Raspberry Pi from my iPhone 8 (running iOS 11.2.1) as a connected bluetooth device.  Normally, I would go into my Settings -> Bluetooth and select the Info button (a small blue circled "i") to show the "Disconnect" and "Forget Device" dialogs:    Unfortunately, iOS does not give me this option for my device. The bluetooth device list does not contain an "i" for the item, and there is seemingly no other way to disconnect.    How do I unpair this device if there is no button to trigger to the unpair prompt?  Due to work restrictions I cannot completely reset the device. Subdomain versus subdirectory I recently moved from having a main site like www.example.com with a subdomain associated site yyy.example.com, to moving everything off the subdomain to a subdirectory like www.example.com/yyy/, with the same hierarchy underneath the subdirectory, and now yyy.example.com is just a redirect page to the subdirectory, so yyy.example.com/abc/page-x is redirected to www.example.com/yyy/abc/page-x and so on.  The effect has been that the traffic increased by several times (the total page views are more than ten times at the present rate) what it was to the yyy subdomain. Previously almost all of the traffic to the yyy subdomain was from the www site. This is from monitoring the old yyy site for a year or more.  I'm curious to know if there's any research or results which would indicate whether this is universal (subdirectories beat subdomains) or "just me". "Add/show 1 other comment" bug strikes again Solving multiple congruences by CRT = Chinese Remainder Theorem How do I get the solution given by CRT to match another solution, e.g. the least positive? For example say I have X = 1234. I choose mi as 5, 7, 11, 13. This satisfies the simple requirements of . More precisely given in my example k = n = 4, and the product of any k - 1 is smaller then X how come simply computing the remainder of each won't give equations that solve to X = 1234. In the case of the example, x = 4 mod 5 x = 2 mod 7 x = 2 mod 11 x = 12 mod 13  Which resolves to  (won't CRT produce the smallest?) Any hints?  Below is the solution linked to on &quot;Math Celebrity&quot; (cached to avoid link rot). In Peter Jackson's film The Fellowship of the Ring, when the Council of Elrond takes place it shows a total of 22 members:    We all know the names of the ones who were part of the Fellowship, but I want to know who are the rest of the people in the council. Who are the elves that came with Legolas? The men next to the dwarves? The three old men sitting next to Boromir? Or the two elves next to Aragorn?  Starting from left to right:   Frodo Gandalf Elf Elf Legolas Elf Old Man Old Man Dwalin Gloin Gimli Nori Old Man Boromir Old Man Old Man Elf Elf Aragorn Elladan Elrond Elrohir Alternative to Photoshop or Paint.NET I have been moving toward Ubuntu from a long time Windows development background. The one program I cannot seem to do without is a graphic editor. I have seen recommendations for programs, but they turn out to be directed at children or tailored to working with personal photographs.  I am looking for something more for programming tasks like analysing colors, resizing, creating web graphics, etc. I have used Photoshop in the past and more recently have mostly used Paint.net for Windows.  Is there a program for Ubuntu that covers this area? Removing repeated page numbers from table of contents Hi I'm trying to create a document in LaTeX with a few small subsections on the same page, is there any way to remove the repeated page numbers and keep the numbers when I move onto a new page? How to avoid bad box with long words with no hyphenation? What's recommended for the number of users to use in qualitative and quantitative testing? Can you attack with different weapons using Extra Attack? The fighter's  feature (PHB, p. 72) says:     Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.   Assuming I am holding two different weapons, and don't want to spend my bonus action on Two-Weapon Fighting, can I attack once with each weapon? Can my friend and I play minecraft together with different mods? I'm wondering if I could play Minecraft with my friend. A modded Minecraft, only myself with mods and him without mods.  Is it possible? So Black Holes Actually Merge! In 1/5th of a Second - How? I've read a lot of conflicting answers in these forums. However, today saw the awesome announcement of gravitational waves. Two black holes merged:   Not only that, they merged FAST. In 1/5th of a second revolving around each other 250 times a second. The entire event was quicker than a heartbeat. Moreover, we observed this happening as distant outsiders. So now we can say for sure:   Objects approaching the event horizon DO NOT appear to slow down Black holes CAN merge in a finite (and quick) amount of time And all this is wrt a frame of reference far, far away   To quote the NYTimes article:  One of them was 36 times as massive as the sun, the other 29. As they approached the end, at half the speed of light, they were circling each other 250 times a second.  And then the ringing stopped as the two holes coalesced into a single black hole, a trapdoor in space with the equivalent mass of 62 suns. All in a fifth of a second, Earth time.  However, everything I've read so far has let me to believe that an outside observer should never be able to measure the collision happening in a finite time. So what exactly is happening here? I must have read at lest 5 different versions of this so far everywhere in these forums over the past several years. What is this agglomeration of pink cells I found attached to a stone? Inverse Green's Function identity in derivation of Hedin's equations I'm trying to work through a derivation of Hedin's Equations in Effect of Interaction on One-Electron States by Hedin and Lundqvist (1969) and I've come across an identity that is given without much explanation that I'd like to understand.   We start with an equation of motion with single particle Green's functions G:   $\left[ i\frac{\partial}{\partial t_1} - h(1) - \phi(1) - V_H(1)\right] G(1,2) - i\int v(1^{+},3) \frac{\delta G(1,2)}{\delta \phi(3)} d3 = \delta(1,2)$  where $1=(x_1,t_1)$, $h$ is the one-electron kinetic energy and ion interaction, $\phi$ is a small perturbing field that is set to zero at the end, $V_H$ is the Hartree potential, and $v$ is the Coulomb interaction.  They then insert the identity  $\frac{\delta G(1,2)}{\delta\phi(3)} = -\int G(1,4) \frac{\delta G^{-1}(4,5)}{\delta\phi(3)} G(5,2) d(4,5)$  which is then used to define the self energy  $\Sigma(1,2) =  -i \int v(1^+,3) G(1,4)\frac{\delta G^{-1}(4,2)}{\delta\phi(3)} d(3,4)$.  I know this identity is related to the definition of the inverse Green's function  $ \int G(1,3) G^{-1}(3,2) d(3) = \delta(1,2)$  but I can't figure out how to get from this definition to the identity that is inserted into the equation of motion. Any ideas?  Thank you! Possible Duplicate:          I know that   $$\lim_{n\rightarrow\infty}\sqrt[n]n=1$$  and I can imagine that $n$ grows linearly while $n$th root compresses it exponentially and therefore the result is $1$, but how do I calculate it? How to iterate over a comma separated list? How can I iterate, loop, process, or otherwise do something, with each element from a comma separated list of things?  This question       , and the are already lots of very good answers everywhere. However I think it would also be valuable to have a single reference point where all such alternative techniques are explained in detail, with their advantages and disadvantages discussed.  All of plain-TeX, LaTeX, LaTeX3, LuaTeX, Context, etoolbox, pgffor, etc. answers are invited. Please consider explaining, for example:   How does the command/method deal with whitespace around items? How does the command/method deal with empty/missing items? (e.g. a,b,,c) How does the command/method deal with a trailing comma? Can the command/method work of a list stored on a macro? Is each iteration evaluated globally or on a local group? (i.e., if a command is defined during an iteration, does the command survive the loop?) Do items get expanded or somehow mangled before being processed? What about lists where items are key=value or some other kind of pairs? Does the method work for lists separated with something other than a comma? Any other thing I should know before deciding to use this method? If photon energies are continuous and atomic energy levels are discrete, how can atoms absorb photons? If photon energies are continuous and atomic energy levels are discrete, how can atoms absorb photons? The probability of a photon having just the right amount of energy for an atomic transition is $0$. Show that    $$\frac{\sum_{i=0}^n\binom{k}{i}\binom{M-k}{n-i}\frac{k-i}{M-n}}{\binom{M}{n}}=\frac{k}{M},$$ where $M-k&gt;n$ and $k&gt;n$   From Vandermonde's identity, I get $\sum_{i=0}^n\binom{k}{i}\binom{M-k}{n-i}=\binom{M}{n}$.   But what about the sum of the product in the numerator? Any hint will be helpful. Installing from source - do I need to keep the extracted tarball directory If I'm installing from source, do I need to keep the extracted tarball directory?  So if I download the git tarball. I then do:  tar -xvzf git.tar.gz   This will create a git.x.x. directory, into which I cd, then run ./configure etc. Once I'm done with this process, and git, or whatever is installed, do I need to keep the original git.x.x extracted directory, or is that just used for compiling the program?  I'm somewhat confused by all the directories, and folders used for programs. Why "themselves" and "himself"