Installing ubuntu 18.04 on Celeron machine
How do I find out which version and derivative of Ubuntu is right for my hardware in terms of minimal system requirements?
states that if $f:X \rightarrow Y$ is a morphism of ringed space, $G$ is a locally free $O_Y$-module, then $f^*G$ is a locally free $O_X$ module.     Suppose that $G$ is a locally free $O_Y$ module, hence it is a free $O_Y|_U=O_U$ module. We have an induced map  $$f^{-1}(U) \rightarrow U$$ given by restriction of morphism of ringed spaces.   Hence, as $\bigoplus O_U \simeq G|_U$, $f^*:Mod(O_U) \rightarrow Mod(O_{f^{-1}(U)})$ is functorial, $f^*(G|_U) = (f^*G)|_{f^{-1}(U)}$, we have isomoprhism,  $$ \bigoplus O_{f^{-1}(U)} \simeq (f^*G)|_{f^{-1}(U)} $$ How are multiple volunteers handled during the Reaping? While skimming through the first few chapters of The Hunger Games to answer another question, I noticed this sentence when Katniss is describing the rules surrounding volunteers after Prim's name is called (at the beginning of the second chapter):     In some districts, in which winning the reaping is such a great honor, people are eager to risk their lives, the volunteering is complicated.   The only reason I can see for volunteering being complicated is if there's more than one volunteer for a particular year, and I don't recall this ever being addressed in the books.  Is there anything in the books that I've missed or any reliable, external sources (interviews or other supplementary material by the author) that explain exactly how this situation is resolved? How is light affected by gravity? Capturing URL parameters in request.GET I am currently defining regular expressions in order to capture parameters in a URL, as described in the tutorial. How do I access parameters from the URL as part the HttpRequest object? My HttpRequest.GET currently returns an empty QueryDict object. I'd like to learn how to do this without a library, so I can get to know Django better. Short story that takes place off planet, involving sun and rain I read this short story about 20 years ago, and I'd love to find the title and author. This is what I can recall about the story:   It takes place in a classroom full of kids (around 10 years old?) The planet always gets rain The kids lock the main character (a girl, I think) in the closet during the very rare occurrence of sunlight and she misses the entire event.    I know it's not much to go on, but I hope it rings a bell for someone! Give a combinatorial proof to show the following for all integers $n \geq 2$. $$2^{n-2} n (n -1) = \sum\limits_{k=2}^n k (k - 1) \binom{n}{k}.$$  I'm completely stumped. I just have no idea how to do this. What I've tried so far has been simplifying the right hand side slightly to $\sum\limits_{k=2}^n \binom{n}{k - 2}$, and then proving that this somehow is equal to some manipulation of the left hand side. I can't seem to find any breakthroughs that way though.  If anyone could help out or point me in the right direction that would be great.  EDIT: I've actually just realized my simplification of the right hand side is wrong. Now I'm even more stuck. Does $P(A\cap B) + P(A\cap B^c) = P(A)$? How to use testfor with a mob with a two-word name? How can I test for mob who has a name with two words? For example villager named Grumpy Bill. For example I want to kill this villager.  This command   /kill @e[name=Bill]   would work just fine but when I try  /kill @e[name=Grumpy Bill]   it says "The entity UUID provided is in an invalid format"  What is the proper command for this action? Tips and resources for beginning designers Thanks to the internet, graphic design and programming are becoming increasingly overlapping fields.   As a programmer, I constantly find myself needing basic knowledge of Graphic Design, but have trouble locating thorough yet concise graphic design learning resources, and instead usually find bits and pieces of knowledge applicable to the task at hand.  Can you guys offer some tips or recommend some good books, tutorials, blogs, and websites which provide a beginner level overview of web design that have been, in your experience, effective and high quality learning materials. I'm not looking for tips on how to use image editing tools, but assuming a basic grasp of those, which resources and advice will best help a new designer create professional looking designs? $|p(z)| \leq M$ for $|z| \leq 1$ Show that $|p(z)| \leq M|z|^n$ for $|z| \geq 1$ Let $p(z)= \sum_{k=0}^n a_k z^k$ ,  $a_n \neq 0$ , be a polynomial of degree $n$ such that $|p(z)| \leq M$ for $|z| \leq 1$.  Show that $|p(z)| \leq M|z|^n$ for $|z| \geq 1$  This was an exam question. Part (a) of this question asked the converse: if $|p(z)| \leq M|z|^n$ and $|z|$ is sufficiently large, then $p(z)$ is a polynomial. This question came straight out of Ahlfors' Complex Analysis, which I solved by showing, using the Cauchy Estimate,  $|p^n (z)|  \leq \frac{n!M|z|^k}{R^n} = 0$ as $R \to \infty$.   I would like to work backwards with this.   May I do this, or similar:   $p(z) = \frac{1}{2\pi i} \int_{c} \frac{f(\zeta)d\zeta}{\zeta-z}$  $\Rightarrow  |p(z)| \leq \frac{2\pi M|z|^n}{1(2\pi)}= M|z|^n$ How is exponentiation defined in Peano arithmetic? How would exponentiation be defined in Peano arithmetic? Unless $n$ is fixed natural number, $x^n$ seems to be hard to define.   Edit 2: So, what would be the way to define $x^n+y^n = z^n$ using $\Sigma_1^0$ formula?  Edit: OK, I say Peano arithmetic has addition and multiplication stuffs. This allows additions to be expressed without quantifiers but as for exponentiation Peano arithmetic is silent. That's why I asked this question. Just for clarification. Ambiguity of the word 'any'? Is a comma necessary in "What’s funny, is …"? In the sentence     What’s funny is Cat’s dry humour.   Wouldn’t it be better to put a comma after “funny”:     What’s funny, is Cat’s dry humour.   This question entails a (at the time of writing) heated discussion in the . 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. Union of subgroups is a subgroup if and only if one subgroup is a subset of the other Let $H$ and $K$ denote two subgroups of a group $G$.  Prove that the union $H \cup K$ is a subgroup of $G$ if and only if $H \subset K$ or $K \subset H$. If APS-C and similar crop-sensor digital cameras have a focal length multiplying effect such that a 50mm lens has an apparent focal length closer to the field of view of an 80mm on a full frame camera, and yet at the same time the depth of field for the smaller sensor camera is more like the depth of field a 50mm lens would produce on a full frame camera (using the same aperture), then this would seem to suggest the concept of an "aperture dividing effect."  In other words, a 50mm f/1.8 lens on an APS-C camera would act more like a 80mm f/2.8 (approx. 1.8 * 1.6x) lens in 35mm equivalent — for depth of field, not considering exposure.  Can someone with a better understanding of the physics involved clarify this for me.  I've never seen this concept explicitly mentioned anywhere, so I am a bit suspect of it. Why is menstruation in wild animals not a disadvantage to organismal survival? When I watch wilderness specials for more than a few minutes, I notice a familiar pattern: predators are depicted as being alerted by the scent of blood. Wounded animals seem to make the best prey and predators seem to have a heightened awareness for the smell of blood.   It would seem that it would then be a disadvantage to evolutionary survival for female organisms to menstruate. How do females of a species not attract predators to the signal of menstrual blood?   Do female organisms in the wild actually menstruate? If so, is there a difference in the composition of menses that does not alert predators? How would this not at the very least give predators an idea of the general vicinity of the female?  If wild animals excrete their menses, do they have ways in which they are able to mask the evidence of its release or are they somehow internalizing the tissue so that they are not released?  If female organisms in the wild do menstruate, then why is this not a survival disadvantage? What are the criteria for questions to be selected for Hot Network Questions? I really like the , where certain questions from the Stack Exchange Network are presented, along with a hotness rating that is described as &quot;arbitrary&quot; in its tooltip. Such questions also appear randomly on the sidebar on sites across the network, under the heading &quot;Hot Network Questions&quot;. How do these arbitrary hotness points work, and how are they used to select questions to appear in those lists? I'm tutoring high school students. I've always taught them that:     A charged particle moving without acceleration produces an electric as well as a magnetic field.   It produces an electric field because it's a charge particle. But when it is at rest, it doesn't produce a magnetic field. All of a sudden when it starts moving, it starts producing a magnetic field. Why? What happens to it when it starts moving? What makes it produce a magnetic field when it starts moving? Why are low-rep users rate-limited when posting questions? I just encountered the following error:  Oops! Your question couldn't be submitted because: users with less than 125 reputation can only post questions every 20 minutes; try again later.  Why do you have this rule? Why 20 minutes?