Please help if there are any errors in this proof.      If a binary operation of multiplication on a nonvoid set $G$ is associative, and if all equations $xa=b$ and $ay=b$ have solutions $x$ and $y$ in $G$, then $G$ is a group.   Solution: First pick any element a' element of G. Then there exists an element u of G such that a'u= a' which is a solution to the equation. We must show that u is a left identity for all elements of G.   First we show left cancellation is possible. If there are two solutions for ax=b and ax'=b then ax=ax'and we must conclude that x=x'. Since if not x=x' then G cannot be a group. (Please help with this step in the proof, I am not convinced  by its reasoning).  Now for all b of G, we have a'(ub)=a'b, using cancellation we have ub=b for all b of G, so u is a left identity. Use the equation xa= b for xa= u and this  implies all a of G have a left inverse. These two properties makes G a group
Let $G$ be a non-empty finite set with an associative binary operation so that cancellation law holds, i.e. $ab=ac$ or $ba=ca$ implies $b=c$, for any choices of $a,b,c$ in $G$. Assume that there is an identity element $e$ in $G$. Show that $G$ is a group.  Proof: To show $G$ is a group, conditions must hold. Suppose that $ab=ac$, then $b=eb=(a^{-1}a)b =a^{-1}(ab)=a^{-1}(ac) =(a^{-1}a)c=ec=c$. So left cancellation holds in $G$.  2) Suppose $ba=ca$, then $b=be=b(aa^{-1}) =(ba)a^{-1}=(ca)a^{-1} =c(aa^{-1})=ce=c$. So right cancellation holds in $G$.  3) If $a$ exists in $G$ then $aa^{-1}=a^{-1}a=e$, where $a$ is an inverse of $a^{-1}$. Since inverses are unique, $(a^{-1})^{-1}=a$.  4) Let $x$ be the inverse of $ab$. Then $(ab)x=e$. By associativity we have $a(bx)=aa^{-1}$. Through left cancellation we have  $bx=a^{-1}bx=ea^{-1}=b(b^{-1}a^{-1})$ and $x=b^{-1}a^{-1}$.  Thus $(ab)^{-1}=b^{-1}a^{-1}$. So all conditions hold, $G$ is a group.  Is this proof correct? I know to show $G$ is a group these conditions have to be met. This is all I have to show right?
Change elements on different positions within a list Find the probability of an event determined by an inequality on exponential random variables Let X and Y be independent random variables such that $X∼Exp(1)$ and $Y∼Exp(2)$.  Find the probability that $3X+4Y≤5$.  I thought the equation for solving this was the integral from $\int_0^a \int_ 0^{(a-y)} \lambda_1 e^{-\lambda_1 x}\lambda_2 e^{-\lambda_2 y} dx dy$ where $a = 5$, so I first started with the integral from $\int_0^5 \int_0^{(5-y)} 3 \times 1e^{-1x} \times 4 \times 2e^{-2y} dx dy$ and got $23.83828927$ which is much larger than a probability can be. Any help? In Bash, when to alias, when to script, and when to write a function? It's taken me almost 10 years of Linux usage to ask this question. It was all trial and error and random late-night internet surfing.  But people shouldn't need 10 years for this. If I were just starting out with Linux, I'd want to know: When to alias, when to script, and when to write a function?  Where aliases are concerned, I use aliases for very simple operations that don't take arguments.   alias houston='cd /home/username/.scripts/'   That seems obvious. But some people do this:  alias command="bash bashscriptname"   (and add it to the .bashrc file)   Is there a good reason to do that? I'm trying really hard, but I genuinely can't think of any circumstances in which I'd want to do that. So, if there is an edge case where that would make a difference, please answer below.  Because that's where I would just put something in my PATH and chmod +x it, which is another thing that came after years of Linux trial-and-error.  Which brings me to the next topic. For instance, I added a hidden folder (.scripts/) in the home directory to my PATH by just adding a line to my .bashrc (PATH=$PATH:/home/username/.scripts/), so anything executable in there automagically autocompletes.  If I needed to.  I don't really need that, though, do I? I would only use that for languages that are not the shell, like Python.  If it's the shell, I can just write a function inside the very same .bashrc:  funcname () {   somecommand -someARGS "$@" }   As I stated, I found a lot of this out through trial and error. And I only truly saw the beauty of functions when my computer died and I was forced to use the computers of the people around me when they weren't using them.   Instead of moving a whole directory of scripts from computer to computer, I ended up just replacing everyone else's .bashrc with my own, since they had never even made a single modification.  But did I miss anything?   So, what would you tell a beginning Linux user about when to alias, when to script, and when to write a function?   If it's not obvious, I'm assuming the people who answer this will make use of all three options. If you only use aliases, or only use scripts, or only use functions—or if you only use aliases and scripts or aliases and functions or scripts and functions—this question isn't really aimed at you. How to integrate $\int_0^{\pi/2}\frac{\sin^nx}{\sin^nx+\cos^nx}dx$? Possible Duplicate:          How can we integrate $$\int_0^\frac{\pi}2\frac{\sin^nx}{\sin^nx+\cos^nx}\,\mathrm dx , \,\,\,\,\,\,\,\,\, n\in N \quad?$$ Thanks for any hint. Trying to keep track of the updates to all the great, user-developed add-ons is fairly frustrating (at least for me). Is there currently a centralized resource where all these amazing add-ons are kept and updated in one location?  If there isn't such a resource, would it be possible to have blender query the collection of installed add-ins and then iterate through on-line references looking at version #'s for add-ins?  Since, as far as I know, it hasn't been done...I have to ask if there are any major flaws in the concept. Of course, in my dream scenario, there would be a "voting" front-end that would allow users to see what the most popular and useful add-ons are and where you don't have to read pages (and pages) of threads. Differentiable but not continuously differentiable. Given $f: \mathbb{R} \rightarrow \mathbb{R}$ defined as $$f(x)=\left\{\begin{array}{cc}x^2\sin\left(\frac{1}{x}\right)&amp;,x\neq 0\\ 0&amp;,x=0\end{array}\right\}.$$ I am trying to prove $f$ is differentiable at $x=0$ but not continuously differentiable there. Bound for the correlation of three random variables There are three random variables, $x,y,z$. The three correlations between the three variables are the same. That is,  $$\rho=\textrm{cor}(x,y)=\textrm{cor}(x,z)=\textrm{cor}(y,z)$$  What is the tightest bound you can give for $\rho$? How to pull data from internal memory on an unrooted phone? Yesterday, I suddenly discovered that I had lost all the user files on the internal memory of my android phone. All my photos, my music, my PDFs, audio recordings, everything. I will explain the events leading up to this in a separate paragraph below.  I have a Lava Z90 phone and I do not use an external SD card, so I must work with my phone to recover all data. I've been googling and following stuff I get from the internet(including Stack Overflow). Here's what I have done so far.   Tried using photo recovery apps available on Play Store. None of them worked. Decided to go more technical. Enabled USB Debugging and OEM Unlocking from Developer options in my phone. Then I used ADB pull command, pull /dev/block/mmcblk0 /home/data.img to try create an image of the entire internal storage so I can analyse it using PhotoRec. It says pull: not found I assumed this is because my phone isn't rooted?  I tried to root it with the KingoRoot apk but that didn't work. I have Ubuntu on my PC and there is not KingoRoot for Ubuntu. My next step would be to use Wine to run KingoRoot on my machine and try to root my phone that way.   I have the following questions:   How can I create an image, a data dump of my entire internal storage that i can later analyse for my data? Can I do it with an unrooted phone? If not, how can I root my phone with ADB? Am I going about this right? Is there a better, easier way to recover data that I should use instead?   Thank you very much.  How I lost my data  I recently uninstalled the TikTok app from my phone. Yesterday I connected my phone to my PC over mtp and found that there were certain leftover folders on the internal memory. One was a folder with Chinese name, one was a folder called ByteDance and one folder called Thumbnails. I assumed that the first two were leftovers from the TikTok app and the third was mere thumbnails. So I deleted them to save space. Nothing happened that day, except that I couldn't access any of my WhatsApp images through WhatsApp(they were still accessible through the gallery). However, when I woke up the next morning, all my data on the internal memory was gone. I still have contacts my messages and my settings, just that everything that was accessible through the file explorer has disappeared. Is it possible and if so how can you pass flags to the ShareLaTeX compiler which one would have passed as parameters to the LaTeX compiler?  I am using the minted package and need to pass the -shell-escape flag. Possible Duplicate:          Both words seem to be used interchangeably. I generally don't differentiate between them and intuitively pick one over the other. Is there a standardized way to distinguish between the two? Combining two sorted lists in Python I have two lists of objects. Each list is already sorted by a property of the object that is of the datetime type. I would like to combine the two lists into one sorted list. Is the best way just to do a sort or is there a smarter way to do this in Python? If Chrome is already open, it opens new sites (e.g. from Windows shortcuts, application shortcuts, or from the Run dialog) in new tabs of the existing instance.   I'd prefer to open new sites in a separate window (a new instance of Chrome).   I don't see that in the options - is it possible? I just proved that any finite group of order $p^2$ for $p$ a prime is abelian. The author now asks to show that there are only two such groups up to isomorphism. The first group I can think of is $G=\Bbb Z/p\Bbb Z\oplus \Bbb Z/p\Bbb Z$. This is abelian and has order $p^2$. I think the other is $\Bbb Z/p^2 \Bbb Z$.   Now, it should follow from the fact that there is only one cyclic group of order $n$ up to isomorphism that these two are unique up to isomorphism. All I need to show is these two are in fact not isomorphic. It suffices to show that $G$ as before is not cyclic. But this is easy to see, since we cannot generate any $(x,y)$ with $x\neq y$ by repeated addition of some $(z,z)$.  Now, it suffices to show that any other group of order $p^2$ is isomorphic to either one of these two groups. If the group is cyclic, we're done, so assume it is not cyclic. One can see that $G=\langle (1,0) ,(0,1)\rangle$. How can I move on? I have the following code in my theme's archive.php:  &lt;?php the_archive_title( '&lt;h1 class="page-title"&gt;', '&lt;/h1&gt;' ); ?&gt;   This gives me titles like "Category: Russia", "Tag: America", "Author: John".  I would like to remove the "Category:", "Tag:" and "Author:" part and just display the category, tag and author names.  Does anyone know how to accomplish this?  Thank you. Getting the default text editor used in system I have used select-editor and I note that I am using /usr/bin/nano. Is this the default text editor I am using?  When I open text files, It opens with gedit  What command in terminal must be specified to get the default text editor that I am using, and have it returned to e in terminal? Lagrangian of Schrodinger field The usual Schrodinger Lagrangian is $$ \tag 1 i(\psi^{*}\partial_{t}\psi ) + \frac{1}{2m} \psi^{*}(\nabla^2)\psi, $$ which gives the correct equations of motion, with conjugate momentum for $\psi^{*}$ vanishing. This Lagrangian density is not real but differs from a real Lagrangian density  $$ \tag 2 \frac{i}{2}(\psi^{*}\partial_{t}\psi -\psi \partial_{t}\psi^{*} ) + \frac{1}{2m} \psi^{*}(\nabla^2)\psi $$ by a total derivative.  My trouble is that these two lagrangian densities lead to different conjugate momenta and hence when setting equal time commutation relations, I am getting different results (a factor of 2 is causing the problem). I can rescale the fields but then my Hamiltonian also changes. Then applying Heisenberg equation of motion, I don't get the operator Schrodinger equation.   Is it possible to work with the real Lagrangian density and somehow get the correct commutation relations? I would have expected two Lagrangians differing by total derivative terms to give identical commutation relations (since canonical transformations preserve them). But perhaps I am making some very simple error. Unless all conjugate momenta are equivalent for two Lagrangians differing by total derivatives, how does one choose the correct one?   I guess the same thing happens for other first order systems like Dirac Lagrangian also. What are the benefits/detriments to graduate students joining unions? How to connect Nokia Lumia 730 to Ubuntu 15.04? My Nokia Lumia 730 (WP8.1) is not getting detected on Ubuntu 15.04.  Can anybody tell me how to fix this? Why did Harime Nui's eye not heal? Code Igniter POST Variable