I would like to find an entire, non-constant (complex) function $f$ with $f(cz) = f(z)$ $\forall z \in \mathbb{C}$ for some $c \in \mathbb{C}$. This is pretty straightforward if $|c| \neq 1$, but I am going to establish a few properties of $f$ first: Because $f$ is entire, we know that it is continuous on the unit disk $B_0(1)$ and thus bounded on $B_0(1)$. Because $f(cz) = f(z)$ for any $z$, we can repeatedly plug in $cz$ in to the right side and get $f(c^k z) = f(z)$ $\forall k \in \mathbb{N}$. Suppose $|c| &gt; 1$ and let $z_0 \in \mathbb{C}$. Then $|z_0| &lt; |c|^k$ for $k \geq N$ because $|c|^k$ diverges. This can be rearranged to $|\frac{z_0}{c^k}| &lt; 1$, so $\frac{z_0}{c^k} \in B_1(0)$ and with the second property $f(z_0) = f(c^k \cdot \frac{z_0}{c^k}) = f(\frac{z_0}{c^k})$. This means that for any $z_0$, $f(z_0)$ corresponds a value on the unit disk (whose values are bounded) and thus $f$ is bounded. Since $f$ is entire and bounded, Liouville's theorem tells us that $f$ is constant. So for any $c$ with $|c| &gt; 1$, there is no non-constant function $f$ with this property. The case $|c| &lt; 1$ follows analogously with $|z_0| &lt; |\frac{1}{c}|^k$ for $k \geq N$. Now suppose $|c| = 1$. Write $c$ in polar coordinates: $c = e^{\varphi i}$ with $\varphi \in [0,2\pi)$. If $\varphi \in \pi \mathbb{Q} := \{\pi \cdot q | q \in \mathbb{Q}\}$, we can again write $c = e^{\frac{p}{q}\pi i}$ with $p,q \in \mathbb{N}$ and $\text{gcd}(p,q) = 1$. One can now take $f(z) = z^{2q} \in \mathcal{O}(\mathbb{C})$ (meaning $f$ is entire) and with this $f(cz) = (cz)^{2q} = (e^{\frac{p}{q}\pi i})^{2q} \cdot z^{2q} = e^{2p\pi i} \cdot z^{2q} = z^{2q} = f(z)$ so $f(z) = z^{2q}$ is of the desired form. The case that gets me stumped is when $\varphi \not\in \pi \mathbb{Q}$ or if there is a much easier way that I am overlooking. In any way, I am interested if there are any more such functions with the properties I want (but especially for the case that is still missing). For orientation, I am an undergraduate that has just taken his first complex analysis course this year. Any help is appreciated!
For which $a \in \mathbb{C} -\ \{0,1\}$ does the equation $f(z) = f(az) $ has a non constant solution $f$ with $f$ being analytical in a neighborhood of $0$.  My attempt:  First, we can see that any such solution must satisfy:  $f(z)=f(a^kz)$ for all $k \in \mathbb{N} $.  If $|a|&lt;1$:  The series $z_{k} = a^k$ converges to 0 which is an accumulation point, and $f(z_i)=f(z_j)$ for all $i, j\in \mathbb{N} $. Thus $f$ must be constant.  If $|a|=1$:  For all $a \neq 1$ , $f$ must be constant on any circle around $0$, so again $f$ must be constant.  My quesions are:   Am I correct with my consclusions?   Also, I'm stuck in the case where $|a|&gt;1$. Any ideas?  Thanks
I can't read beta sites' chat logos! Here, you try:  If it weren't for the text shadow, it might as well be entirely invisible...  The cause is a CSS rule lurking in the transcripts for beta sites only:  #transcript-logo {   background: none repeat scroll 0 0 #273A5D; }   The problem is the #273A5D - a bluish colour not dissimilar to that used for the site name in the beta theme. Disabling that rule yields the much more manageable  I'll admit it is a little hard to read the "beta" in that shot. But if you had to choose which part was readable, wouldn't you choose the bit that tells you the name of the site?  Edit: It happens when actually in a chat room too! What is the name of this Ubuntu program? A couple month ago I installed a bunch of clipboard manager programs in Ubuntu, tried them out and then removed them. I must have missed one of these and now it keeps bug me in my daily usage. I need some help to identify the name of this program because the program itself provides very limited information.  This clipboard program has below characteristics: 1. It occupies hotkey Ctrl+;. When I press this hotkey, the clipboard history will pop out. It happens in bash, Chrome, Eclipse IDE, and etc, wherever it is text inputable. This is kind of annoying because it overrides my other hotkeys inside these applications. 2. Very laconic UI. Like below picture, with only a line "Select to paste", then listing out the clipboard history. There is no program information whatsoever. Right clicking on it and no menu are shown at all.      So far I have tried below ways to identify this program: 1. Browse system monitor for processes with names that could be correlated. 2. Browse through my installation log (dpkg.log) to find my installation history. Without luck, the log is lost. But I was not successful. What are the technical differences between using a DSLR viewfinder and live view? The earlier question  asks for practical advantages of optical viewfinders. But what's the technical difference, and why does a camera behave differently in each mode?  For example, on my Nikon D5100:   Slower auto-focus — I saw this explained in the related issue, but just partially: "The other major drawback of live view is that because it requires the mirror to be locked up, it means that the auto focusing mechanism is unavailable."  option not available. No . The manual says that it's dangerous to shoot direct light, when it's on live-view. There's no range finder available.   Why are all these things different?  Live view also has some advantages, like:   automatic scene selection grid lines on the screen other smaller things, like possibly easier operation on a tripod where the viewfinder might be inconvenient to look through   Some of these things, like the tripod operation, are natural. But are there technical reasons the others can't be done with an optical finder? Are there other advantages enabled by live view for technical reasons? Why are votes anonymous? Let's make with the transparency and have them publicly attributed An idempotent operator is compact if and only if it is of finite rank Official ESTA application website and are the others scams? A quick search for ESTA application returned many results. According to , the official site to apply for an ESTA is .  Are the others scams, legitimate third party resellers or "added value" services?    (defunct - domain is for sale circa 2020)     Can anyone confirm either way?  Updated 2020: The above have better "not affiliated with the U.S. Government" warnings now, but beware there are newer more sneaky sites starting all the time:        And fortunately more official guidance about this issue: How were the number of rounds for different key sizes of AES selected? The number of AES rounds increases with the key length. Why increase the number of rounds at all, and how were these round counts chosen? Elliptical orbits of planets As we know, the velocity of the planet changes as it orbits around the sun, both in its magnitude and direction. The centripetal acceleration causes the change in the direction of velocity but I want to know what is the tangential force that acts on the planet to change the velocity by its $magnitude$. The answer normally as to why the speed changes is: If  the  force  that  the  Sun  exerts  on  the  planet increases  (as  the  planet moves closer),  then  the acceleration   of  the  planet must  increase,  resulting  in a  higher  orbital  speed,  and vice  versa. But this force is not along the body but towards the sun. So such a force should only change its direction. Display historical characters "uͤ", "oͤ" and "aͤ"? Why do first years cross the lake to get to the castle? Which is correct: "on foot" or "by foot"? He went there on foot.   Or     He went there by foot.   Which one is correct, and on what basis? How would I write the equivalent of C#'s  in JavaScript?  var haystack = 'hello world'; var needle = 'he';  haystack.startsWith(needle) == true   Note: This is an old question, and as pointed out in the comments ECMAScript 2015 (ES6) introduced the  method. However, at the time of writing this update (2015) . Restore SQL Server 2012 backup to a SQL Server 2008 database? 1-1 correspondence between [0,1] and [0,1) I'm going to install Linux on Lenovo B5400, which contains:   500GB Hybrid SSHD with a 8 GB SSD and 500 GB HDD   Designing partitions for my 2 Linux distros was quite easy before, when I had only HDD. I always had:   10GB for Linux distro (mounted as "/") 10GB for another distro (again "/"; for another distro; for testing purposes or to assure continuity when switching distros) 2GB for swap (used for both distros; no need to separate) 300GB+ for "/home" (used for both distros; different usernames eliminates possible conflicts)   It's simple, isn't it? I would like to keep this approach, but I know that SSD has its own specificity - it prefers to be read, continuous writing may affect its life span. I don't want to kill SSD in a year, so I would like to move some system partitions to HDD in order to protect SSD. As far as I know /tmp is one of often read&amp;written locations. How much space it needs? What else should be placed on HDD?  Or: how to design partitions to achieve exactly the same effect and to keep SSD as safe as possible?  I'd be grateful for your help. Are we close to a pure DC (or hybrid) household? This is a philosophical question. Over the last few years, many of the household devices, particularly those that are more mobile and etc are DC devices. Are we at some point where a DC household may become a possibility and Edisons dream might come true??  The array of DC devices    Internet stuff - routers, modems Most complex computer electronics - Desktops, laptops, NAS, phones, tablets, gaming consoles, etc Lighting - LED More that I cant think of or perhaps know which also convert AC -> DC   Problems with AC     Each device needs its own power supply ( AC-> DC power brick). This is a) Unpleasant to look at and deal with, and b) a costly point of failure.  Each power brick may or may not be optimized, as far as power conversion goes.   Alternative - A central (or a few ) high capacity power adapter which converts the AC into a DC power grid for the house which can then be tapped directly by the various devices. This eliminates the need for the power brick, and hopefully the central power adapter is of state of the art, and can be upgraded seperately from the devices, as needed.  Need standardization of:   Device voltage. Currently devices use 5/ 9 / 12 or others I may not be aware of, for this idea to work they need to work with one particular one, unless switching between voltages is cheap enough, or multiple DC lines are provided. Power socket input. Each device appears to have its own input socket type, this should standardize if the power supply cable should be comoditized, perhaps USB - C will play a part here. Why are the IV (constant) values fixed for HMAC/SHA? Why do HMAC/SHA IV's have a fixed value? I refer to the arbitrary H constants which are included in the algorithm prior to any data being fed in.  Why aren't they exposed to the user, so one can decide on an IV and modify it accordingly (for example, in java they are provided at hash function specification, instead of being supplied by the user at init)?  How does modifying these IVs affect the algorithm? What if the hard disk is partitioned into more than 26 sections in Windows? As we know, in Windows the partition location is prefixed with [a-z]:. What happens if there are more than 26 partitions? "Whom" or "who" where the referent is both subject and object? I understand there has been so much on this topic but I am still confused. I get that if the person is the subject it is who and anything else is whom. However, I'm really struggling to work out this rule for the following sentence:     The scope of which his account of morality applies to is therefore restricted to those whom/who the ‘notion of justification to a being of that kind makes sense.’   Should it be who or whom? I can't work out the subject and object. Would it better the community if we got compensated for posts? Even with minuscule alt coins? If we are effectively generating more traffic to Stack Exchange sites, would it become a better community if users were compensated somehow besides rep points? Even with minuscule alt coins?