How to differentiate many lines in a chart?
How to minimize the ink-to-data ratio for pgfplots
PC started having screen glitches after update I'm not sure what's caused it. I haven't changed any system settings. The only thing I've done that may have broken something is updating my software. My update was: upgrade linux-generic-hwe-20.04:amd64 5.8.0.48.54~20.04.32 5.8.0.49.55~20.04.33 upgrade linux-image-generic-hwe-20.04:amd64 5.8.0.48.54~20.04.32  upgrade linux-headers-generic-hwe-20.04:amd64 5.8.0.48.54~20.04.32  upgrade linux-image-extra-virtual:amd64 5.4.0.70.73 5.4.0.71.74 upgrade linux-image-generic:amd64 5.4.0.70.73 5.4.0.71.74 upgrade linux-libc-dev:amd64 5.4.0-70.78 5.4.0-71.79 upgrade libseccomp2:amd64 2.4.3-1ubuntu3.20.04.3 2.5.1-1ubuntu1~20.04.1 upgrade nomachine:amd64 7.3.2-3 7.4.1-3```  Removing nomachine did not solve the problem. It was far worse before, I'm not sure how to describe exactly, but my entire screen's UI was glitching out. So I rebooted and that stopped, but now the terminal text looks like this:  I haven't changed any terminal settings. This seems to be occurring in the entire OS. Here is another example:  Not sure how to go about troubleshooting this? I don't have the time to reinstall the OS, nor do I think I should have to because this is just ridiculous and shouldn't happen. I'm not aware of anything I've done to make the system unstable. Edit Earlier I also tried booting the PC. It failed and got stuck on some command in a terminal (didn't manage to take a photo of what it said). So I rebooted into recovery mode with both available kernels. I was getting strange glitches like this:  That didn't work. So I rebooted the PC normally, and now I somehow logged in fine. Now I'm getting popups saying to report a system error. It says this:  Environment  Ubuntu 20.04.2 LTS GNOME 3.36.8  sudo lshw -class display   *-display        description: VGA compatible controller        product: Xeon E3-1200 v3/4th Gen Core Processor Integrated Graphics Controller        vendor: Intel Corporation        physical id: 2        bus info: pci@0000:00:02.0        version: 06        width: 64 bits        clock: 33MHz        capabilities: msi pm vga_controller bus_master cap_list rom        configuration: driver=i915 latency=0        resources: irq:33 memory:f7800000-f7bfffff memory:e0000000-efffffff ioport:f000(size=64) memory:c0000-dffff  I also did , which passed with 0 errors. Let me know if more details are required. virtual methods Harry is a half-blood and Ginny is a pure-blood, so what does that make his children? I've been told it means they are pure-bloods but I can't be sure. What do you think? Proving $f' = - \frac{\partial{F}/\partial{x}}{\partial{F}/\partial{y}}$ I am trying to prove that, if $f:\mathbb{R}\rightarrow \mathbb{R}$ and $F:\mathbb{R}^2\rightarrow\mathbb{R}$ are differentiable, $F$ satisfying $F(x,f(x))=0$ and $\partial{F}/\partial{y}\ne 0$ then $$f'(x)=-\frac{\partial{F}/\partial{x}}{\partial{F}/\partial{y}}$$  This appears in a section on the chain rule for many-variables functions, but without knowing the details of $F$ I find it very hard to prove much about it.  To the best of my ability to reason about it, I suppose I could say  $$ \frac{\partial{F}}{\partial{x}}=\frac{\partial{F}}{\partial{x}}\frac{\partial{x}}{\partial{x}} + \frac{\partial{F}}{\partial{y}}\frac{\partial{y}}{\partial{x}}$$  but for more than one reason I doubt that what I have written is correct.  Yet I'm not exactly sure what's wrong about it, nor can I guess what else I could do.  I tried finding the partial with respect to $y$ but got nothing more useful looking than this.  I also tried writing out limit definitions and saw no useful path forward through that. Show that $a^2 \bmod b = (a \bmod b)^2 \bmod b$ Show that  $$a^2 \mathrm{\ mod \ } b = (a \mathrm{\ mod \ } b)^2 \mathrm{\ mod\ } b$$ for $ a, b \in \mathbb{Z}^+ $.  this was derived from an Informatics olympiad question. Unexpected output for this following code How can I block incoming SMS from certain senders? Is there a way to spellcheck the word under the cursor with a keyboard shortcut? Proving that free modules are flat (without appealing projective modules) Suppose $R\neq 0$ is a commutative ring with $1$. Let $M$ be a free $R$-module. I would like to prove that $M$ is a flat $R$-module. Everywhere I have looked (mostly online) this is proved by first proving that every free module is projective, and then proving that every projective module is flat. Unfortunately, Atiyah &amp; Macdonald's "Introduction to Commutative Algebra" (Chapter 2) does not discuss projective modules. But the result that every free module is flat comes very handy in the exercises.   So my question is,     Is it possible to prove that every free module is flat just by   definitions and without appealing to projective modules?   Thanks! Blender resizing my image in half I have a problem. I'm trying to render an image and the resolution is 500x280. After I render the image and save it, for some reason the image gets resized to 250x140. I'm unsure why it happends. How can I stop Lollipop ignoring my wifi network? Android 5.0 has a new "feature": If a wifi network has no connection to the internet, it will remain connected to it but route traffic over mobile data instead. A small exclamation mark shows in the status bar over the wifi symbol.  Unfortunately, for some reason Android has decided that my home wifi does not have a good internet connection - which is not true, as I'm using it to access Stackexchange at the moment - and is, instead, trying to send everything over GPRS.  With mobile data disabled, things work over wifi just fine, despite the exclamation mark - so it isn't a problem with the wifi not working, per se.  Is there any way to override this "smart" feature? The only way I have found to make things work at the moment is to disable mobile data while I am in my house - which is not a practical long-term solution.   EDITING to add that I have now solved this problem by buying a new router! I was using something very old... But, I'll leave the question here for the benefit of anybody else with the same difficulty. Can I use lens brand X on interchangeable lens camera brand Y? Sleeping in a batch file When writing a batch file to automate something on a Windows box, I've needed to pause its execution for several seconds (usually in a test/wait loop, waiting for a process to start). At the time, the best solution I could find uses ping (I kid you not) to achieve the desired effect. I've found a better write-up of it , which describes a callable "wait.bat", implemented as follows:  @ping 127.0.0.1 -n 2 -w 1000 &gt; nul @ping 127.0.0.1 -n %1% -w 1000&gt; nul   You can then include calls to wait.bat in your own batch file, passing in the number of seconds to sleep.   (at last!). In the meantime, for those of us still using Windows&nbsp;XP, Windows 2000 or (sadly) , is there a better way?  I modified the sleep.py script in the , so that it defaults to one second if no arguments are passed on the command line:  import time, sys  time.sleep(float(sys.argv[1]) if len(sys.argv) &gt; 1 else 1) How much memory can a 32 bit process access on a 64 bit operating system? How to check if a thread is sleeping? Statements equivalent to the Axiom of Choice The Axiom of Choice reads:     The product of a collection of non-empty sets is non-empty.   As you know well, this axiom is equivalent to many other statements. A few examples (probably the most known) are the following:   Zorn's lemma: Let $(P,\ge)$ be a poset in which every chain has an upper bound. Then $P$ has a maximal element.   This is used in a number of different places to prove very powerful theorems. A couple of examples I can think of right now are: the proof that every ring has a maximal ideal (commutative algebra) and the  (functional analysis).   Well ordering theorem: Every set can be well-ordered (i.e. it admits a total order such that every subset has a least element). Tychonoff's theorem: Every product of compact topological spaces is compact.   This is used for example in the proof of  (again functional analysis).   gives various more. Among them:   Tarski's theorem: If $A$ is an infinite set, then there is a bijection $A\to A\times A$.   (As @Wicht dutifully commented, Tarski's theorem is the name given to the implication $|A|=A\times A \Rightarrow$ Axiom of Choice, while the other implication was known before the proof of this fact by Tarski.)   Every vector space has a basis.   My question is: what are other useful statements which are equivalent to the axiom of choice? Where are they used, and to prove what?  P.s.: By "useful statements" I mean either statements which are important theorems by themselves (as Tychonoff's theorem) or that are directly used in (simple versions of) the proof of important theorems (as Zorn's lemma). I'm looking to build a collection of fantasy miniatures, preferably pewter - but I don't want to buy them one at a time from my local hobby store... Much too expensive, and they don't often carry anything I'm interested in buying, which makes it a long term venture as well. So I'm looking for ideas on how to obtain many miniatures at once, preferably for cheap. D&amp;D Miniatures aren't a good source for me as they're collectibles, and the prices on those are also prohibitive.  By the way - they don't have to be pewter, that's just my preference. =) Parallel evaluation produces wrong result Variance of a linear combination of vectors Is a droid a ‘he/she’ or an ‘it’? Inspired by  question.  Are droids in the Star Wars universe referred to with gendered pronouns (i.e. he or she), or are they referred to as objects (i.e. it)?