This seems like an elementary question but a quick Google search did not yield an answer. I know that noetherian valuation rings are discrete and that discrete valuation rings have Krull dimension 1, but I'd like to know an example of a non-discrete (necessarily non-noetherian) valuation ring with Krull dimension 1. Or an explanation why such a ring cannot exist.
For valuation rings I know examples which are Noetherian.      I know there are good standard non Noetherian Valuation Rings. Can anybody please give some examples of rings of this kind?    I am very eager to know. Thanks.
Do there exist general conditions under which we can conclude that continuity on a topological space is detected by $\mathbb{R}$? Whenever $X$ is a topological space, let us say that continuity on $X$ is detected by $\mathbb{R}$ iff for all functions $f : X \rightarrow Y$ where $Y$ is another topological space, we have that if every continuous function $c : \mathbb{R} \rightarrow X$ satisfies "$f \circ c$ is continuous", then $f$ is itself continuous.   Clearly, continuity on $\mathbb{R}$ is detected by $\mathbb{R}$ It is shown  that continuity on $\mathbb{R}^2$ is detected by $\mathbb{R}$. Clearly, it is not the case that continuity on $\mathbb{Q}$ is detected by $\mathbb{R}$, since every continuous mapping $\mathbb{R} \rightarrow \mathbb{Q}$ is constant.      Question. Do there exist general conditions underwhich we can conclude that continuity on a topological space is detected by $\mathbb{R}$? After reading an article about Ubuntu's netflix support I decided to try running netflix through chromium but I got this error code: M7063-1913 I assumed that it was a problem with html5 but in the playback preferences it doesn't give me the option to use it. How to securely delete files stored on a SSD? How exactly does a "random effects model" in econometrics relate to mixed models outside of econometrics? I used to think that "random effects model" in econometrics corresponds to a "mixed model with random intercept" outside of econometrics, but now I am not sure. Does it?  Econometrics uses terms like "fixed effects" and "random effects" somewhat differently from the literature on mixed models, and this causes a notorious confusion. Let us consider a simple situation where $y$ linearly depends on $x$ but with a different intercept in different groups of measurements:  $$y_{it} = \beta x_{it} + u_i + \epsilon_{it}.$$  Here each unit/group $i$ is observed at different timepoints $t$. Econometricians call it "panel data".    In mixed models terminology, we can treat $u_i$ as  a fixed effect or as a random effect (in this case, it's random intercept). Treating it as fixed means fitting $\hat \beta$ and $\hat u_i$ to minimize squared error (i.e. running OLS regression with dummy group variables). Treating it as random means that we additionally assume that $u_i\sim\mathcal N(u_0,\sigma^2_u)$ and use maximum likelihood to fit $u_0$ and $\sigma^2_u$ instead of fitting each $u_i$ on its own. This leads to the "partial pooling" effect, where the estimates $\hat u_i$ get shrunk toward their mean $\hat u_0$.  R formula when treating group as fixed:    y ~ x + group R formula when treating group as random:   y ~ x + (1|group)  In econometrics terminology, we can treat this whole model as a fixed effects model or as a random effects model. The first option is equivalent to the fixed effect above (but econometrics has its own way of estimating $\beta$ in this case, called "within" estimator). I used to think that the second option is equivalent to the random effect above; e.g. @JiebiaoWang in his highly upvoted answer to  says that      In econometrics, the random-effects model may only refer to random intercept model as in biostatistics    Okay --- let us test if this understanding is correct. Here is some random data generated by @ChristophHanck in his answer to  (I put  for those who do not use R):    @Christoph does two fits using econometrics approaches:    fe &lt;- plm(stackY~stackX, data = paneldata, model = "within") re &lt;- plm(stackY~stackX, data = paneldata, model = "random")   The first one yields the estimate of beta equal to -1.0451, the second one 0.77031 (yes, positive!). I tried to reproduce it with lm and lmer:  l1 = lm(stackY ~ stackX + as.factor(unit), data = paneldata) l2 = lmer(stackY ~ stackX + (1|as.factor(unit)), data = paneldata)   The first one yields -1.045 in perfect agreement with the within estimator above. Cool. But the second yields -1.026, which is miles away from the random effects estimator. Heh? What is going on? In fact, what is  plm even doing, when called with model = "random"?  Whatever it is doing, can one somehow understand it via the mixed models perspective?  And what is the intuition behind whatever it is doing? I read in a couple of econometrics places that random effects estimator is a weighted average between the fixed effects estimator and the "between" estimator which is more or less regression slope if we do not include group identity in the model at all (this estimate is strongly positive in this case, around 4.) E.g. @Andy :     The random effects estimator then uses a matrix weighted average of the within and between variation of your data. [...] This makes random effects more efficient[.]   Why? Why would we want this weighted average? And in particular, why would we want it instead of running a mixed model? What determines the number of comments that are displayed? Maven plugin executing another plugin I'm trying to create a new plugin to package my latest project.  I'm trying to have this plugin depend on the maven-dependency-plugin to copy all my projects dependencies.  I've added this dependency to my plugin's pom, but I can't get it to execute.  I have this annotation in my plugins main Mojo:  @execute goal="org.apache.maven.plugins:maven-dependency-plugin:copy"   I've tried a few other names for the goal, like dependency:copy and just copy but they all end with a message saying that the required goal was not found in my plugin.  What am I doing wrong?  Secondary to this is where to I provide configuration info for the dependency plugin? The total number of locks exceeds the lock table size Story of lost tech but magic manifests I'm looking for this novel (or maybe a series of books?). I remember that technology began failing all over the world. Some people began transforming into different types of mythical creatures, like sprites, demons, etc. There was a band of about four travelers that were trying to make it to Chicago, since that seemed to be at the center of the issue. The story line frequently shifted the point of view in each new chapter. I vaguely recall a final battle in Chicago with an enormous and powerful demon creature. I think this demon used to be a corporate lawyer or big business guy before his transformation. I believe I read this in the early/mid 2000s, and was likely published around that time because of the types of tech used and failing in the story. Although it bears some surface similarity to the TV show, Revolution, they aren’t comparable in plot or characters. Summation of series $\sum_{n=1}^\infty \frac{n^a}{b^n}$? How can we evaluate this series  $$\sum_{n=1}^\infty \frac{n^a}{b^n}?$$  Here $a$ and $b$ are positive integers. If $b=1$ then series will be diverging, in other cases, it will be converging, but how to find this sum?  Isn't there a simple solution which doesn't involve Stirling numbers? Is it ethical to omit the PhD credential from my resume? I am applying for a Statistical Programmer position at a commercial research organization (CRO), and I am wondering if it is unethical to omit my PhD in Statistics from my resume?  All of my degrees (Bachelor's, Master's, PhD) are in Statistics, and I am just wondering if it would be ethical to just list my Bachelor's and Master's degree on my resume, and indicate my experience as a RA while I was a PhD student as my "work experience"? (Since this Statistical Programmer position only requires Master's degree plus some years of experience)  I am asking this question because I do not want to do anything unethical/cause me troubles later on. Does That's very sweet of you  in the US sound like That's very sweef you? the of is only speak f and link with swee I get the following error when I try to typeset a beamer document.  kpathsea: Running mktexpk --mfmode / --bdpi 600 --mag 0+525/600 --dpi 525 mathkerncmssi8 gsftopk: fatal: map file `cid-base.map' not found. mktexpk: don't know how to create bitmap font for mathkerncmssi8. mktexpk: perhaps mathkerncmssi8 is missing from the map file. kpathsea: Appending font creation commands to missfont.log.  ) !pdfTeX error: pdflatex (file mathkerncmssi8): Font mathkerncmssi8 at 525 not found  ==&gt; Fatal error occurred, no output PDF file produced!   But if I remove sansmathaccent.sty from the TeX Live tree, the document typesets fine. Here is a minimal working example.  \documentclass[10pt]{beamer}  \begin{document}  \begin{frame}{A very short beamer document}  $e^{i\pi}+1=0$  \end{frame}  \end{document} Can you retrain the Magic Initiate spell using the Spellcasting feature? got me thinking about the mechanics of the Magic Initiate feat when you have class levels in the class into which you were inducted.    Among the classes you can pick with this feat are the Bard, Sorcerer, and Warlock, which know a limited number of spells and all have the same statement under Spells Known.     Additionally, when you gain a level in this class, you can choose one of the [class] spells you know and replace it with another spell from the [class] spell list , which must also be of a level for which you have spell slots.   The term "learning" is used both to describe new spells acquired on level-up and the spell acquired through the feat. This would seem to indicate your Magic Initiate spell counts among the spells you know, and is thus eligible for retraining if you learned it as a spell of your class.  An odd change in wording occurs during the Magic Initiate feat description (emphasis mine):     In addition, choose one 1st-level spell from the same list. You learn that spell and can cast it at its lowest level.   This generalization in describing the learned spell's level seems to imply that there exists a scenario where it would not be appropriate to restrict the spell to being cast as a 1st-level spell.  Could one of the charisma casters retrain Magic Initiate into a higher level spell, which they could then cast once per long rest without using a slot?  This would turn an otherwise unexciting feat into, effectively, a Mystic Arcanum for Bards and Sorcerers. This seems too powerful to be intentional,  but the feat description appears to have gone out of its way to permit it. Why does the experience-to-next-level not change between some levels? At every level from 1st to 11th, the experience needed to reach the next level (from the current level) increases every level. However, this pattern gets broken for most of the later levels:   Going from 10th to 11th takes 21,000xp (= 85k - 64k), but 11th to 12th takes only 15,000xp! (= 100k - 85k). Both 13th and 14th level only take extra 20k experience each, i.e. the amount needed doesn't increase between those two levels. Levels 16 and 17 similarly only need 30k experience each, and 18 and 19 only need 40k experience each.   Why does the last half of the experience chart have some of these levels maintain the same experience costs, and even briefly decrease the experience cost for 12th level? Why doesn't Ubuntu 16.04 in Windows have ubuntu.exe? Movie with humans battling aliens piloting giant animalistic, water-walking robots or sea warships I have memories of watching it a few years ago (I was a kid then) in the early 2000s probably; before 2010 I think. It was a movie with giant robots, and the enemies were aliens piloting what I remember as giant, more animalistic robots. I remember those animalistic ones running on water. The aliens themselves were a little taller than humans but mostly humanoid. I remember what the aliens look like a bit; most of the movie they were clad in some kind of armor, but the scene I mentioned showed one's face, if I remember well, grey skin, vividly green eyes, a beard made of some kind of quills, bald. I have this one scene that stayed crystal clear in my memories. The heroes had captured an injured alien, and other aliens attack to, in the end, just retrieve the injured one, leaving with this one without attacking further this time. I also remember the aliens going through a city in some machine (I'm actually not sure this one was piloted though) and destroying anything it perceived as a threat, from cars to security cameras. My memories are coming from years ago, so I might be wrong in a bunch of things there, but this one scene, I can assure is correct. This might remind a bit of Pacific Rim, but it isn't. (It's watching Pacific Rim that reminded me of this movie.) It doesn't seem to show up on any giant robot movie list on internet, weirdly, so I hope you can help me. And now I'm starting to wonder if there was really robots on the heroes' side. Maybe it was warships (sea ones, not spaceships). As I said it was some time ago and I was young, so bad memories are likely. Should an image carousel on a mobile site automatically rotate? Or should the user have complete control? The intention is to have a slider/carousel that shows 6 or 7 images which the user can look through by clicking arrow buttons left or right. But the question has also come up if the user has not started looking through the images, should the carousel start automatically rotating the images. Since this is for a mobile site, I'm uncertain... Whenever I create a document in Microsoft Word, it complains about a lot of my sentences being in passive voice. But, when I read that sentence aloud, it sounds fine to me. I am not sure if it is just me and will a statement in passive voice sound strange to a native speaker?  So, my question is, is it considered bad form to use passive voice generally? Or in some specific cases like written communications only?  Edit: If it is ok to use passive voice, then why does MS-Word complain? Unix Truncate Column in csv file Looking for a short story about a boy discovering that the government puts mind control chemicals in his cereal I would have read this around 1985-1998 because I remember being in school at the time. It may have been in a short story collection with "The Ugly Little Boy" and a story about a family dealing with people in the future coming back to enhance the intelligence of their baby with enfante terrible results. A boy, grade school age or so and very intelligent, discovers (or possibly just decides) that there are drugs in the food that prevent one from thinking critically. The details are kind of fuzzy in my head, but I remember that he at one point distills the chemical from his cereal, which he'd detected as having a high concentration, and overdoses. He's convinced that the government is trying to silence him — a scene that sticks out in my mind was him falling on a moving sidewalk and tearing his clothing as he tries to escape two "men in black" type figures that are pursuing him. The short story ended with him deciding that he was better off just consuming the food and being happy in his ignorance.  Years later, as an adult learning about mental illness, it struck me that the story was slightly ambiguous as to whether there actually were chemicals, and that the use of drugs that dulled his creativity and intelligence, and the decision to take them anyhow, was reminiscent of the feelings that manic people have toward their medications.  Anyone know what this one was?