Let $f,g: U \rightarrow \mathbb{C}$ be holomorphic on the open and connected subset $U$. If $|f| + |g|$ is constant on $U$ show that $f, g$ are constant on $U$.  What can we say about finite or countable number of holomorphic functions?
Does anyone know how prove the following?   Suppose that $f,g$ are holomorphic functions on a non-empty open connected set $\Omega \subset \mathbb{C}$ and that $|f|^2+ |g|^2$ is constant on $\Omega$. Show that $f$ and $g$ are constant on $\Omega$.
Significance contradiction in linear regression: significant t-test for a coefficient vs non-significant overall F-statistic As I understand, correct plural versions of passerby and attorney general become passersby and attorneys general.  But why?  With passerby, the the preposition "by" has been combined with the noun "passer" to become a one word noun "passerby".  So why shouldn't this be passerbys?  If I am an attorney general, and you are an attorney general, why are we not attorney generals?  As an American I have always heard it as attorneys general, however  suggests British English does use "attorney-generals" (as well as an interesting 2nd paragraph that further probes my question).  Are there hard rules to this topic?  Or merely generally accepted practices based on locale? Changing projection of shapefile in QGIS How do I change the projection of my shapefile using QGIS? The default projection is set to EPSG:3003, I want to change it to WGS84 EPSG:32632. When I do a reprojection or I change projection in the properties mask I don't have the correct result. On the left, I have the plan coordinates but they are not correct for the EPSG shown on the right: How does one write on the outside of a circle?  I am looking for sample files/packages which imitate a magazine lay-out. Specifically a page with two columns having a circular picture between the two columns.    Example of the desired layout, created using Word: I accidentally closed the window for the verfication code I wanted to create an account for Ubuntu One, but I got impatient when the email with the verification code didn't arrive... I thought I might have typed the wrong address, closed the window and tried again. it told me the address already existed and I just got the code. But since I closed the window I missed it. Is there any way to recover the code? What's the difference between "these" and "those"? First of all, I'm not a native English speaker, but in school I learned that these is used if referring to something near, and those is used when referring to something far away (temporally or locally). But now I'm sometimes watching English movies and notice sentences like "Have you seen those boots?" while the boots were only centimeters away. Is this just not correct, or have I learned the wrong rules?  Another question is, what if I'm referring to something abstract such as ideas or thoughts so that I can't say if they are far away or not. Should I use those or these?  Edit  So it can't be really wrong in any case if I use these respectively those? I'm very new to GIS concepts. Is there a way or procedure of finding out whether the co-ordinate system is in UTM, latitude/longitude or in stateplane using .prj file?   My .prj file contains the following text:      GEOGCS["WGS 84",DATUM["WGS_1984",SPHEROID["WGS   84",6378137,298.257223563,AUTHORITY["EPSG","7030"]],AUTHORITY["EPSG","6326"]],PRIMEM["Greenwich",0,AUTHORITY["EPSG","8901"]],UNIT["degree",0.01745329251994328,AUTHORITY["EPSG","9122"]],AUTHORITY["EPSG","4326"]] Is it ok to put old papers on the arXiv? In my field (I am a condensed-matter theoretical physicist) it is such a normal thing to put papers on the arXiv, that people sometimes look at other people's arXiv list of papers instead of their publication record to get to know what they do, etc.  It just so happens that a few years ago I did a PhD with a supervisor that does not like the arXiv that much (he apparently had some conflict with some authors of a paper that was very similar to one of his papers, but one was already published while the other was just an arXiv preprint -or something like that). As a result, from my list of less than 20 publications, there are four of them from my PhD that are missing on the arXiv.  I was thinking of submitting the latest versions I have of them as arXiv preprints now, four years after their publication. For this I already have the agreement of all co-authors (including my supervisor), and all the journals involved seem to allow to put preprints of the material they publish. Do you see any problem with this? I fear that some people might take it as a way to advertise as new something that was already published a long time ago (though I would of course give the full reference to the published version). There may be other issues I am not aware of. The reason why I would like to do this is that I am applying for fellowships, positions, etc. and I am afraid some of my (best) publications may go unnoticed if somebody looks me up on the arXiv.  Thank you, Stephen Oven Temperature: How to be sure the selected temperature is correct. I suspect that my - fairly new - oven gets too hot, so I bought an oven thermometer, which confirmed this.  So I called the supplier.  They say that in order to obtain the correct actual  oven temperature, for any setting, you take the highest reading on the thermometer, and the lowest, and then divide both readings.   Is this correct? Fundamental Theorem of Algebra - an elementary complex analysis proof? We begin by assuming that a non-constant complex polynomial $$p(z)=a_n z^n+a_{n-1}z^{n-1}+...+a_1 z+a_0,$$ where $a_0 \neq 0$, doesn't vanish in $\mathbb{C}$. Therefore $1/p$ is also an analytical function everywhere. Note that $(p(z)-a_0)/z$ is a polynomial and therefore, due to Cauchy's theorem, the following integral must be zero.  $$\int_{S_r^1}\frac{(p(z)-a_0)/z}{p(z)}dz=\int_{S_r^1}\frac{a_nz^{n-1}+a_{n-1}z^{n-2}+...+a_1}{a_nz^n+a_{n-1}z^{n-1}+...+a_1 z+a_0}dz=0.$$  On the other hand  $$\int_{S_r^1}\frac{a_nz^{n-1}+a_{n-1}z^{n-2}+...+a_1}{a_nz^n+a_{n-1}z^{n-1}+...+a_1 z+a_0}dz$$  $$=\int_{0}^{2\pi}\frac{a_n(re^{i\theta})^{n-1}+a_{n-1}(re^{i\theta})^{n-2}+...+a_1}{a_n(re^{\theta})^n+a_{n-1}(re^{i\theta})^{n-1}+...+a_1 (re^{i\theta})+a_0}(ire^{i\theta})d\theta$$  $$=\int_{0}^{2\pi}i d\theta - \int_{0}^{2\pi}\frac{a_0 i}{a_n(re^{i\theta})^n+a_{n-1}(re^{i\theta})^{n-1}+...+a_1 (re^{i\theta})+a_0}d\theta$$ $$=2\pi i -\int_{0}^{2\pi}\frac{a_0i}{p(re^{i\theta})}d\theta.$$  Let $r \rightarrow \infty$ and the integral approaches value $2\pi i$ - a contradiction. Therefore $1/p$ can't be entire and $p$ must vanish somewhere in the complex plane.  EDIT: I have changed this post and deleted the idea that the integral $\int_{\partial D_r}\frac{(p(z)-a_0)/z}{p(z)}dz=0$ could be evaluated with the fundamental theorem of calculus. Now, as it stands, this proof is essentially the same as the one that Mr. José Carlos Santos points out in his comment. Passing HTML to template using Flask/Jinja2 How to get the Infobox data from Wikipedia? Why are there no discrete zero sets of a polynomial in two complex variables? Why is the zero set in $\mathbb{C}\times\mathbb{C}$ of a polynomial $f(x,y)$ in two complex variables always non-discrete (no zero of $f$ is isolated)? Ubuntu16.04 boots into initramfs Ubuntu doesn't boot up and it goes into initramfs. I am writing from a live CD trial session. what should I do?  I understand that the boot flow is broken and that when the files from root are going to be loaded into the ram, it does not find the files in the root directory. Please correct me if I'm wrong.  The following is the output from ashell as I input the "exit" command:  (initramfs) exit /dev/mapper/ubuntu--vg-root contains a file system with errors, check forced. Inodes that were part of a corrupted orphan linked list found.  /dev/mapper/ubuntu-vg-root: UNEXPECTED INCONSISTENCY; Run fsck MANUALLY.     (i.e., without -a or -p options)  fsck exited with status code 4. The root file system on /dev/mapper/ubuntu--vg-root requires manual fsck Should individual letters be preceded with "an"? I was reading a comic book and I noticed that the author used "an A" and "an N" in a particular phrase (screenshot below). While I understand the usage of "An A" here (A being a vowel and what not) and "N" having a pronunciation which has a sound similar to "A".  Since most of English alphabets have pronunciations which start with a syllable similar to one of the vowels or semi-vowels; is it OK to prefix the letters using "An"?  For eg. An X, or An Y.        Zenescope's Grimm Fairy Tales - Werewolves: The Hunger, page #8 (Click to enlarge) What lifetime can be expected of the typical hard disk? Or are there big differences between different types? And does it make a difference if it is used heavily instead of never being connected to system (for example serving as a backup medium)? What is a "locked" post? A post can be locked.   What does it mean to lock a post? Who can lock a post? When should a post be locked? How might a corrupt partition in TempDB result in DBCC CHECKDB reporting no issue? Express partial derivatives of second order (and the Laplacian) in polar coordinates $z=f(x,y)$ where $x=rcosθ$ and $y=rsinθ$  Find $ \frac{\partial z}{\partial x}$ and $ \frac{\partial^2 z}{\partial x^2}$  I'm having big troubles with using chain rule, in particularly the second derivative. Spent 2 hours on this already. What I have...  $ \frac{\partial z}{\partial x}=\frac{\partial r}{\partial x}\frac{\partial z}{\partial r}+\frac{\partial z}{\partial θ}\frac{\partial θ}{\partial x}$. So I'm guessing that $\frac{\partial z}{\partial r}$ and $\frac{\partial z}{\partial θ}$ are unkown? And I will need to use them to differentiate again. Any help is really welcome. This is simply only part of a bigger question.   Full Question: What are the powers shown by the Rings of Power?