Prime ideal splitting in field extension and its normal closure
Splitting of prime ideals in algebraic extensions
What is the meaning of the density of a distribution at a point? I know how to use PDFs to calculate probabilities, but I don't think I understand them. For example, at $X=0$ the PDF of the standard normal distribution is $\approx 0.4$. Does this have any useful meaning? What is wrong with this query? (Unknown error parsing query) Can't change mouse influence in proportional editing mode I press O and I see it change to Proportional Editing mode, but when I press G and mouse wheel down/up, or arrow keys down/up, it doesn't change the influence area.    And I am, I press G and then I tried to scroll to change influence, but it's not working.    Done this aswell and it's not working $\lim_{n\to \infty}f(nx)=0$ implies $\lim_{x\to \infty}f(x)=0$ Can anyone help me with this problem?     Let $f:[0,\infty)\longrightarrow \mathbb R$ be a continuous function such that for each $x&gt;0$, we have $\lim_{n\to \infty}f(nx)=0$. Then prove that $\lim_{x\to \infty}f(x)=0$.   Our teacher told first to prove Baire's theorem, and then show that this is a consequence of that theorem. I proved Baire's theorem, and I spend a few hours thinking on how Baire's theorem is related to this problem, but I couldn't find anything. I'd really appreciate your help. Let $\sim$ be the equivalence relation on $S_1 \times I$ given by $(x,t)\sim(y,s)$ if and only if $xt=ys$ where $I=[0,1]$  How do I prove that $(S_1 \times I)/\sim$ is homeomorphic to the unit disk $D^2$ with the induced topology? How to Understand the Definition of Cardinal Exponentiation I'm having trouble understanding the definition of cardinal exponentiation.  Let's start with the definitions / claims I've been given:  For any finite sets $A,B$, such that if $|A|=a$ and $|B|=b\neq 0$ then the number of functions from $A$ to $B$ is $b^a$. ( I'm guessing this is only used when both sets are finite ).  For any sets $A,B$, $B^A$ is the set of functions from $A$ to $B$.  For every set $A$, $|P(A)|=|\left \{ 0,1 \right \}^A|$  Now for the definition and why I don't understand it:  Let $a,b$ be cardinal numbers.  Let $A,B$ be sets such that $|A|=a, |B|=b$.  Then $|B^A|=b^a$.  How analagous to normal exponentiation is this and where does in differ when we introduce cardinals like $\aleph_0$ or $C$?  Is $2^2=4$ and is $\frak c^2=c\cdot c=c\times c$? I want to delete all *.o files in a directory and its sub-directories. However, I get an error: sashoalm@aspire:~/.Workspace.OLD$ rm -r *.o rm: cannot remove `*.o': No such file or directory  On the other hand, rm *.o works, but it's not recursive. What is the difference between Unix, Linux, BSD and GNU? What is the very fundamental difference between Unix, Linux, BSD and GNU?  Unix was the earliest OS, so the term 'Unix like' is understandable, since they have kernel, file system structure, most of the commands, users etc are same as Unix.  Still why are they different? What made them set apart? Is it the kernel? What is a good data plan for Internet surfing in northern Italy? I've checked the webpages of some their providers, but franly, just like in the case of those in my own language, I literally cannot find my way around them. So if you wouldn't mind sharing a word of two or advice.  I'm looking for a good solution for mobile internet surfing in northern Italy, that can be connected to a laptop (you know, that usb stick with a SIM card inside kind of thing). 15 Gb or so at some decent speed. Can you perhaps give a recommendation or two of what to get?    Edit: Just to define what I'm looking for exactly. A data plan of 15 Gb with an internet key (usb stick into which a SMS card goes) that you plug into a laptop. Preferably non-contractual, so I can cancel when I leave. I am not interested in surfing on my mobile phone since due to size I still prefer the smaller Nokia-like models for calls (smaller to carry) - no internet on them. Weak solutions to the Neumann's problem (Evans PDE) Saving high-quality graphics through commands What do ellipsis [...] mean in a list? Why am I getting shopping trolley notifications about local store savings? I am getting "Shopping Trolley" notifications appearing regularly. The notification looks like this:   The notification is about "Store savings" at a local supermarket and that is it!   Is this notification from Google Maps or another application like Google Assistant? I'm struggling to find the specific setting to disable the notification.  I am also interested to know if this notification could in any way be targeted advertising generated by the store. My marketing preferences with the store are opt-out and this strikes me as very dumb advertising. Surely Google isn't doing this? Is there a means that a store can generate notifications like this? Could it be targeted if they have my email address?  These are pain and I want to understand why they are spamming me and disable them without disabling everything else.  I can see various options under Google Maps, settings, notifications. Which setting would apply? Could it be "New on maps, Get occasional updates about new features in your area"? I can't see what setting to suppress!  Thanks for all help. Why does \cal{M}_n give “M>”? Why does \cal{M}_n give “M>” in parts of my LaTeX file and works correctly in other parts? Emission of EM radiation from an electron I have read about an electron making a transition between two energy levels and electromagnatic radiation will be emitted. The problem is how and why e.m radiation is emitted. In Pokémon Sun and Moon, when you've finished the game you can switch day and night by going though the wormhole at the altar where you've caught the box legendary. Are there any more differences when going through the wormhole, excluding the day or night difference? For example, the altar of the sunne and altar of the moone switch when going through the portal. Suppose that C is a closed convex subset of $\mathbb R^n$ and $x \in \mathbb R^n$. The projection of $\mathbf x$ onto C is the closest point $\mathbf y \in C : \mathbf z = \mathbf y$ minimizes ||$\mathbf z - \mathbf x$||$_2$ over all $\mathbf z \in C$.  (a) Show that the minimum exists.  (b) Show that there is only one minimizer: that is, show that if $\mathbf y_1$ and $\mathbf y_2$ both minimize ||$\mathbf x - \mathbf z$|| then $\mathbf y_1 = \mathbf y_2$   My ideas:  For (a), to show that the minimum exists I will need to show that it is closed and bounded. We already know it's closed because C is a closed convex subset so I just need to show it's bounded. Am I wanting to minimize ||$\mathbf x - \mathbf z$||$_2$? This is just the Euclidean norm which would be $\sqrt{(\mathbf z - \mathbf x)^2}$=$\sqrt{\mathbf z^2 -\mathbf z \mathbf x -\mathbf x \mathbf z + \mathbf x^2}$. I'm not sure where to go from here...  For (b) I want to use proof by contradiction- so I will assume that $\mathbf y_1$ and  $\mathbf y_2$ both minimize the function and then I'll somehow see that they are in fact equal. How do I set up this argument? Multiple File Extension Rename with single command I received some important files via mail. There are about 150 files like these. The extensions of the files are named as zip1,zip2,zip3, zip_ etc. I just want to change the extension of these files back to zip using a single command to avoid the manual task of renaming each file. How to find sum of the infinite series $\sum_{n=1}^{\infty} \frac{1}{ n(2n+1)}$ $$\frac{1}{1 \times3} + \frac{1}{2\times5}+\frac{1}{3\times7} + \frac{1}{4\times9}+\cdots $$  How to find sum of this series?  I tried this: its $n$th term will be = $\frac{1}{n}-\frac{2}{2n+1}$; after that  I am not able to solve this. Do accepted answers still serve a useful purpose? In the Stack Exchange voting system, we have two classes of votes: standard up / down votes that anyone can cast, and special  that only the asker can.   Although accepting an answer does not always pin it at the top of the answer list, that green check mark does  about the correctness of the answer. This gives a single person much greater weight than the rest of the community. That green check mark can seem confusing to visitors, who see  or  answers marked as being correct. I've heard this from several friends recently who were surprised when they came across factually wrong answers that had been downvoted, yet still had that check mark next to them.  Moderators regularly field flags from even more experienced users who demand that the accepted answer be changed from one that is incorrect or even harmful to another. We decline these, because we're not supposed to be making technical judgments like that and because we have no way of accepting answers on behalf of the asker. However, these flags do show that some people have problems with the presentation of accepted answers vs. the community voting on all the others.  There's also all the fun that came with arguments over accept rate, and people badgering askers to accept their answers.   Questions on the Stack Exchange network are intended to be broadly useful, to help out many others in the future beyond the person asking the question. If that's our goal, why should the person asking the question have any more say than the rest of the world? If it's a broadly applicable question, why should one person be able to highlight something as correct simply because they asked it first?  I guess what I'm asking is: if we did away with accepted answers, converting them to upvotes where appropriate, and only let community voting sort out the best answers, would we lose something important? I thought I'd ask after reading  to , because I wasn't sure how I felt about accepted answers anymore.