Evaluating the limit of multivariable equation $\frac{x^2y}{x^2+y^2}$
Finding multivariable limits for the function $\frac{3x^2y}{x^2+y^2}$
How do I remove the file suffix and path portion from a path string in Bash? Given a string file path such as /foo/fizzbuzz.bar, how would I use bash to extract just the fizzbuzz portion of said string? I'm trying to show that $R = \mathbb{Q}[X,Y]/(Y^2-X^3)$ is not a UFD, but I got stuck.  To prove this, I could try to find two "different" factorisations for one element, but I am not familiar with this, so I tried to use a lemma and one of the previous exercises. If my syllabus is right, in every UFD counts $$ x \ \text{is irreducible} \quad \iff \quad x \ \text{is prime}$$ My syllabus also states that the elements $\bar{X}, \bar{Y} \in R$ are irreducible. So I tried to show that at least one of the elements $\bar{X}, \bar{Y} \in R$ does not generate a prime ideal.  This would mean that I should find two polynomials $f,g \in \mathbb{Q}[X,Y]$, such that  $$\exists p \in \mathbb{Q}[X,Y], \quad fg - pX \in (Y^2-X^3)$$ and $$\forall q \in \mathbb{Q}[X,Y], \quad f-qX \notin (Y^2-X^3) \ \wedge \ g-qX \notin(Y^2-X^3)$$ or the same thing but then for $Y$.  I hope that you can tell me if this approach is correct, and provide me a hint. I'd appreciate it if you told me the solution, but please start your answer with a hint, clearly separated from the rest. In Lieb and Loss's Analysis, I saw that they mentioned  $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ dense in $ L^2(\mathbb{R}^n)$ (dense wrt the $L^2$ norm, I think). But I didn't find its proof in the book. So I wonder why it is?    Does this conclusion hold for $L^1(\Omega, \mathcal{F}, \mu) \cap L^2(\Omega, \mathcal{F}, \mu)$ for any measure space $(\Omega, \mathcal{F}, \mu)$?  Are there similar statements if replace $L^1$ with $L^p$, and  $ L^2$ with $L^q$ for $p \leq q \in (0, \infty ]$ or $\in [1, \infty]$?  Thanks and regards! Set an alias on Bourne shell I am using a scrtipt with instruction 'SH -c command', where command is an alias from bash, this alias can not be interpreted into Bourne shell. ¿Do you have any idea for set this alias into Bourne shell and getting that it works in the new shell? Thanks everyone ! A question about how to express a fraction as ${1\over q_1}+{1\over q_2}+ \cdots+{1\over q_N}$ Let $x$ be a positive rational number, strictly between $0$ and $1$. Prove that there is a finite strictly increasing list of positive integers $2 \leq q_1&lt;q_2&lt;\cdots&lt;q_N $ such that $$x={1\over q_1}+{1\over q_2}+ \cdots+{1\over q_N}.$$  I have tried many methods, such as mathematical induction. I know it is obvious that $\frac{1}{m}$ is right for assumption, but when I begin to prove $\frac{2}{m}$, I feel it is hard to find some patterns for $\frac{2}{m}$. Thus, maybe my thoughts are wrong.  Also, I still tried to change the fraction so that the numerator will be smaller and smaller. But I still cannot find a way to lower the numerator. Can someone help me solve the question? Or, can someone give me some hints. I will appreciate you very much! Best way to start investing, for a young person just starting their career? Why is $\mathbb{Z}[\sqrt{-n}], n\ge 3$ not a UFD? Activating 'missing' standard GRASS utilities in QGIS Processing toolbox Today I bought an ibook G4 from a garage sale. The people who sold it to me didn't know the password for the 2 accounts on it. Apparently it was the guys late father's laptop. Any suggestions on how to get into the laptop and set a new admin account and password? How to move from screen 1 to screen 3 by moving cursor to the left and vice versa? If I have to move the mouse from the left screen all the way to the right screen, the distance is quite big.   Is there a way to virtually connect the sides of the screen as if they were arranged in a circle? I could then move from the left screen to the right screen, by simply moving the cursor to the left. How to find location using MapKit in Xcode? What is the danger of running untrusted software as a normal user Sometimes I have to use some propriety softwares which I don't trust or even pre-compiled open source ones which I don't know if they've been compiled as their source are.   Lets say we installed a propriety application in /opt/software, then we run it as a normal user (without sudo, gksudo, etc).   I know when we run a program it is able to connect to the Internet, monitor X, keystrokes, collect user data and so on. It can't change anything important in the system or infect the system, (because of the Linux permissions system). It is possible for program to add some plug-in/extension to a user's browser (Because browser profiles are available in user home directory). After closing the program and rebooting it can't do anything anymore. Wrong because:   Program can add itself to a shell/file/directory which does the autostart process, so it will run at any login.  If we consider the autostart process clean, without rebooting there is a chance that a process of the program is still active. Programs can delete or modify any file in the user home directory (or any other file which a user has write permission to).   So what should we do?   I think the only safe way to run a propriety program is running it as another user. If running the program as another user somehow is not possible then we should change the autostart files permission so only root can change them. then after closing the program a reboot will do the job.   Am I right? Is there anything that I'm missing? How can one request reconsideration of a   after a certain period of time, let us say after one year? Literal @YES not working in iOS 5 / Xcode 4.4 Remove duplicates from a List in C# Anyone have a quick method for de-duplicating a generic List in C#? Order of elements is lcm-closed in abelian groups How can I prove that if $G$ is an Abelian group with elements $a$ and $b$ with orders $m$ and $n$, respectively, then $G$ contains an element whose order is the least common multiple of $m$ and $n$?  It's an exercise from Hungerford's book, but it's not homework. I could not solve it when I take the course on groups and I think it should be easy. There is a hint that says to first prove when $m$ and $n$ are coprimes. I did this part. But I have no idea how to solve the general case.  Thanks. Every Function in a Finite Field is a Polynomial Function From a bank of past master's exams I am going through:     Let $F$ be a finite field. Show that any function from $F$ to $F$ is a polynomial function.   I know that finite fields are fields of $p$ elements for $p$ prime [EDIT: It's actually $p^n$ for $p$ prime; see comment below]. Since I have $p$ choices for the $p$ elements to map to, then I have $p^p$ distinct functions. I think every function can be written in the form $f(x) = a_{p-1}x^{p-1} + \dots + a_0x^0$. For then given the values $f(0), f(1), \ldots f(p-1)$, I can solve for the coefficents by the linear system of equations  $$ a_0 + \sum_{i=1}^{p-1} n^i a_i = f(n).$$  This then gives me a $p-1 \times p-1$ square matrix over the field $\mathbb{F}_p$:  $$\left( \begin{array}{ccccc} 1&amp; 0 &amp; 0 &amp; \ldots &amp; 0 \\ 1&amp; 1 &amp; 1 &amp; \dots &amp; 1 \\ 1&amp; 2 &amp; 4 &amp; \dots &amp; 2^{p-1} \\ \vdots &amp; \vdots &amp; \vdots &amp; &amp; \vdots \\ 1 &amp; p-1 &amp; (p-1)^2 &amp; \dots &amp; (p-1)^{p-1} \end{array} \right) \left( \begin{array}{c} a_0 \\ a_1 \\ a_2 \\ \vdots \\ a_{p-1} \end{array}\right)  = \left( \begin{array}{c} f(0) \\ f(1) \\ f(2) \\ \vdots \\ f(p-1) \end{array} \right)$$  If I can show this matrix is invertible, then I can always find the $a_i$. But I am a bit stumped on how to show this (partially because I don't think I've ever done linear algebra in a vector space over a finite field). It does not seem easy to show linear independence, or nonzero determinant, or full row rank.    Alternatively (I just thought of this), can I show this is true by arguing that the map between the two sets (the set of polynomials of degree $p-1$; and the set of functions $F \to F$) is injective, and that it must be a bijection because the sets have the same cardinality $p^p$? Forge CBC-MAC by creating a new message, given the MAC of two messages if $a,b$ are nilpotent elements of a commutative ring $R$, show that $a+b$ is also nilpotent I've just upgraded to Ubuntu 17.10 and am missing the Ubuntu Dock settings, i.e. in gnome-control-center the dock settings are blank:    I think I might be missing the "ubuntu-dock" package, but can't find this anywhere in the software centre.  Also the gnome-control-center was completely missing after the update, and I had to install this separately - not sure if this is related.  I think this is different from  because changing the type of GNOME session doesn't fix the problem. If I go for GNOME* I get no Dock settings section at all, and if I got for Ubuntu* I get the blank Dock settings section described above.  Output of apt-cache policy gnome-shell-extension-ubuntu-dock:  rob@rob-pc:~$ apt-cache policy gnome-shell-extension-ubuntu-dock gnome-shell-extension-ubuntu-dock:   Installed: (none)   Candidate: 0.7   Version table:      0.7 500         500 http://gb.archive.ubuntu.com/ubuntu artful/main amd64 Packages         500 http://gb.archive.ubuntu.com/ubuntu artful/main i386 Packages