Suppose $G$ is a semigroup in which every equation of the form $ax=b$ or $ya=b$ has a solution. Does this solution have to be unique?
I have some difficulties with a task in algebra. I guess it's trivial and really easy but I can't figure out how to solve it.  I have a set $G$ and a binary operation on it, let it be $\circ$. I have that the operation is associative and that the equations $a\circ x = b$ and $x\circ a = b$ have unique solutions. I have to prove that $(G, \circ)$ is a group.  I already have that the operation is binary and associative, so I have to prove that there is unique identity element and unique inverse element and it will come from the equations, but how exactly?
How to pause with specific time between two command in script? I want to do such as following:  #!/bin/bash command1 pause 30 seconds command2 exit   This is only example script with pause of 30 seconds between run of 2 command.  How to do this? Is there a word for colloquial forms of address? For example, "dude," "man," "buddy," "pal," etc, when used to stand in for someone's name.  "Hey, pal, how's it going?"  Is there a word for terms like these?  Or is "colloquialism" as close as we can get? YA dystopia novel of battle school with unusual military control scheme Some sort of battle school (no, not Ender's Game), where teens are being trained as military commanders.   The control system for their army hinges around some sort of system that requires extremely fine motor control -- being able to "push" around digital representations of military units using a special interface that interprets minute finger movements.  The boy protagonist eventually meets a girl in the battle school, and is astonished to learn that most people do not live as well as he does -- for instance, he gets a fresh jumpsuit outfit every day and disposes of the old one; this is a luxury.  One scene I recall is them kissing for a moment in some sort of underground passageway; the boy protagonist notes the exact length of the kiss, as he has been trained to have heightened time perception.  I don't recall much about other commander characters; I think they have different special abilities, and one other major character might have physical deformities, possibly because of postwar-radiation-related birth defects.  They have been training for an upcoming, prearranged final battle between all the existing national governments; the military units are very possibly robotic or entirely digital in nature, so as to avoid wasting materiel / human life.    They ultimately win the battle, just barely; I think the girl protagonist gives them an edge in the competition by hacking the timing / control system in favor of their team, causing a discrepancy which only the boy protagonist notices due to his heightened perceptions.  It was agreed that the winning government gets dominion over one of the few islands of land left that was not irradiated in some previous war; the protagonists get to live there as a reward for their victory.  I believe the boy protagonist later realizes he's gradually losing his ability to exactly quantify the passage of time.  Published no later than 2004, likely somewhere between 1980-2000, probably sometime in the '90s. Do megapixels matter with modern sensor technology? I am interested in publishing a GML file as a WFS. I don't want to use a database like PostgreSQL. What would be the rough steps I need to do for that? I like to use open source software (geoserver). I don't have much experience. I am just looking for a rough guide what kind of software I should look into and which step follows which step.. On Cesàro convergence: If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $ I have this problem I'm working on. Hints are much appreciated (I don't want complete proof):  In a normed vector space, if $ x_n \longrightarrow x $ then  $ z_n = \frac{x_1 + \dots +x_n}{n} \longrightarrow x $   I've been trying adding and subtracting inside the norm... but I don't seem to get anywhere.  Thanks! Can the definite article be omitted when referring to a person qualified by a noun? How to toggle Scroll Lock in Excel 2010 without a Scroll Lock key in the keyboard I have a notebook with no Scroll Lock key (or any shortcut using the 'Fn' key).  Somehow Excel 2010 got the Scroll Lock on. I see it left down in the status bar and now I can't navigate through cells using the arrow keys, which is pretty annoying. It wasn't turned on some hours ago, so I think I must have hit some Scroll Lock toggle shortcut in Excel 2010 by mistake.  Anyone knows what hot to turn off Scroll Lock in Excel (shortcut or menu option)? What punctuation should follow after a list introduced by a colon? The whole department: John, Jeff, Jean, and the accountant [?] were thrown into jail. How to insert text before the first line of a file? Leave bash script running on remote terminal while not logged in? I have a bash script that takes several hours to run.  While it's running, I would like to do other things, which may involve logging out or disconnecting from the internet (my script runs network tests on various computers).  I understand that there is a command that would allow me to run my tests from a remote terminal and logout of the terminal while it runs.  Does anyone know what this command is? Why does getopts only work the first time? How to distinguish Nexus 7 (2012) and Nexus 7 (2013) I am having trouble with Bluetooth Low Energy (BT 4.0) on a Nexus 7.  I think I may have a 2012 model.  How to tell if I have a 2012 or 2013 version of the Nexus 7? Require duplicates to be imported in Shotwell I have some drawings that I have imported and tagged by date. Now I want to take some of them that I have organized on my disk and put give them a new tag. Shotwell thinks they are duplicates and won't import them. Is there a way to require Shotwell to import these duplicates? Question:  Let ${a_n}$ be a sequence in $\mathbb{R}$ with $\lim_{n\to\infty}a_n=a$  Prove that  $\lim_{n\to\infty}a_n^2=a^2$  Attempt: Without using any properties of limits (this is a question in a section of the book that came before any list of any limit theorems)  By definition:  $|a_n^2-a^2|&lt;\epsilon$   $|a_n^2-a^2|$ $\leq$ $|(a_n+a)(a_n-a)|$ $\leq$ $|(a_n+a)| |(a_n-a)|$ $\leq$   $(|a_n | + |a|)(|a_n -a|)$ $&lt;$ $\epsilon$  And since I know that $\lim_{n\to\infty}a_n=a$, I'll subsitute that in for $|a_n|^2$ as $a^2$   $\therefore$ $a^2-|a|^2 &lt; \epsilon$ which is true since $a^2-|a|^2$ in my case.  I wonder if this is a possible solution? Why is XOR preferred over XNOR? The XOR operator seems to be a frequently used building block inside many cryptographic primitives.  As far as I can see, its most desirable properties seem be that for the XOR of two bits $a\oplus b=c$:   The information is preserved.  Either $a$ or $b$ can be recovered from $c$ and the other bit. The information is hidden.  An adversary can not learn anything about $a$ or $b$ from $c$ alone.   Neither AND nor OR has these properties but XNOR does.  It seems like any cryptographic application using XOR could be constructed equivalently with XNOR, without incurring any loss in security.  One possible reason to prefer XOR might be that XOR is equivalent to addition modulo 2, whereas XNOR would be addition plus 1 modulo 2, but this seems to be entirely mathematical, and I'm not sure if it actually results in any additional hardware complexity in practice.  Is the choice of XOR then an arbitrary one, made for historical reasons, or is there any tangible underlying motivation for using it over XNOR? Why we can't implement AES 512 key size? Out of curiosity why we can't implement AES 512 key size?    Please explain somehow i can understand! I'm not an expert. I've tried different ways to install iTunes, but to no avail. I tried with Wine, but it didn't work. Is it possible to have iTunes on Ubuntu? build argument lists containing whitespace How to draw a fish? I would like to draw an image like the following.     This is what I've done so far (I used tikz, but any other package is welcome):  \documentclass{article} \usepackage{tikz}  \begin{document}     \begin{tikzpicture}         \draw[fill=blue] (.5,0) rectangle (4,2);         \draw[fill=red] (1,1) to[bend left=50] (3,1) to[bend left=50] (1,1);         \draw[fill=red] (3,1) -- (3.5,1.5) -- (3.3,1) -- (3.5,.5) -- cycle;         \draw[fill=white] (1.6,1.1) circle (.15cm);          \draw[fill=blue] (1.55,1.1) circle (.05cm);              \end{tikzpicture} \end{document}     Edit: April 2nd, Time for explanation  In Italy (but I also think in other countries), a practical joke done on April 1st is called "pesce d'Aprile" (= April's fish, that is an ) and is often related to fishes.  The usual one is to stick a drawing of a fish on someone's back without he/she realizes it.  Some years ago a colleague of mine stuck some "pesci d'Aprile" with written below "sono disponibile" (= I'm easy) on the rear of all the cars in our company car parking. Some other colleagues didn't realize it, and they had gone around with that drawing for days.  Another funny example happened in Milan . Someone put on the windshield of a lot of cars a false fine for illegal parking. On the fine sheet was written "per il pagamento rivolgersi agli uffici comunali di via Gadio, 2" (= for payment, please contact the municipal offices in via Gadio, 2), the address of the Civic Aquarium.  Hence, this question was simply a way to put    P.S. = Since I earned a lot of reputation points with this trick question, I decided to give them back to the Community awarding a bounty to the two answers posted on April, 1st.  Moreover, I accepted Robert's one since he guessed without spoiling, and I thank the other two answerers.  To the downvoters: you're probably technically right, but:   there's already too much sadness in the world, let's laugh now and then! humor is part of the LaTeX philosophy, Prof. Knuth himself docet if only serious questions are accepted, why did they create a fun tag?