I know how to write the composition of two functions: $f\circ g$ but I don't know whether there's a standard symbol for a sequence (similar to $\sum_i{f_i}$, $\prod_i{f_i}$ or $\bigotimes_i{f_i}$, etc.).  How would you write it?  edit: Would you understand $\bigcirc_i{f_i}$ without explanation?
I have a repeated composition of functions ${T_n}(z) = {\tau _0} \circ {\tau _1} \circ {\tau _2} \circ  \cdots  \circ {\tau _n}(z)$  By analogy with $\sum\limits_{i = 1}^n {} ,\prod\limits_{i = 1}^n {} ,\bigcup\limits_{i = 1}^n {} ,\bigcap\limits_{i = 1}^n {} ,$ I want to write ${T_n}(z) = \left( {\mathop  \circ \limits_{i = 0}^n {\tau _i}} \right)(z)$ or even ${T_n}(z) = {\mathop  \circ \limits_{i = 0}^n {\tau _i}} (z)$. Can I do this?
Non-principal ideal in $K[x,y]$? Let $K$ be a field. Then $K[x,y]$ is not a PID because $x$ is irreducible but the quotient $K[x,y]/(x)$ is isomorphic to $K[y]$, which is not a field.  So there must be an ideal in there which is not principal. What is an example of such an ideal? Most efficient way to append arrays in C#? I am pulling data out of an old-school ActiveX in the form of arrays of doubles. I don't initially know the final number of samples I will actually retrieve.  What is the most efficient way to concatenate these arrays together in C# as I pull them  out of the system? How do you prove that $\sqrt{2} + \sqrt{5}$ is irrational?    I tried to prove it by contradiction and got this equation:  $a^2/b^2 = \sqrt{40}$. I'm trying to give the best experience to users of a very huge system (like a CRM). It has a lot of forms and much of these have a lot of fields. My doubt is about if I should use a single column form (which raises too much vertical scrolling), or a two columns form (which needs less vertical scrolling and uses more horizontal white space) and a last one with four columns (which has few scrolling and uses almost whole white space to try to keep most of the fields above the fold).  It's important to say that our clients hate scrolling, and they always ask why we don't use the horizontal white space when we present a one column form).  So I'm little bit confused about it, because a lot of UX studies and good practices recommend to do not use/avoid multicolumn forms but the clients most of times ask multi column forms.  Our clients are very used to desktop programs and not everybody is very familiar with web based systems.  Below follows some possibilities I would like to share to illustrate an example of a complex form in the system with the three possibilities I mentioned:    In the example above I'm using the one column form (most recommended in UX studies), but with a very long vertical scroll.    In the second example I tried to reduce vertical space by using two columns.    This is the client's favorite one, 4 columns using white space and trying to keep everything above the fold.   The lengths of fields are based on the average of characters amount for each field.   I would like to know your opinion about the best approach to this case. Inno Setup: How to manipulate progress bar on Run section? If I wave around a bar magnet, the magnetic field in the space around it changes. Is this enough to go through the whole speed of light derivation implying that the motion creates an electromagnetic wave? If so how do I determine the wavelength and direction of the resulting wave? Find all functions $f:\mathbb{N_{0}} \rightarrow \mathbb{N_{0}}$ satisfying the equation $f(f(n))+f(n)=2n+3k,$ for all &amp; $n\in \mathbb{N_{0}}$, where $k$ is a fixed natural number.  A friend of mine suggested me to use recurrence relation. While it is unfortunate that I have problems understanding recurrence relation. Please refer information to understand difference equation, homogeneous equation &amp; auxillary equation.  Thanks in advance! Kindly don't down vote for not showing my attempt. I am seriously not able to understand the question and the process suggested to use. Enroll Apple Developer Program under 18 I've seen a lot of questions like this throughout the web but they're mostly from 2 or 3 years ago, so I decided to ask to have some updated answers.   I'm 16 and I want to start publishing and using the benefits that comes with the Developer Program. In the legal agreement says that I should be over 18, and I'm not so, i can't do it. I've contacted the apple developer support and they told me that I should use a account in my father's name. We reside in Brazil. With that, I had some doubts:   Will the name on the App Store be mine or my father's? In the future, when I'm 18, this account can be transferred to my name?   Besides that, if someone has any advice or (legal) workaround I'd appreciate it, thanks When to throw an exception? This is a reformulation of a deleted question:  If  $a_1 &gt; 0$ and $a_2 &gt; 0$ and $2a_{n+2} \le a_{n+1}+a_n$,  show that  $\lim \sup a_n \le \frac23 a_2 + \frac13 a_1$.  My proof involves showing that $a_{n+2} \le u_{n} a_{n+1} + (1-u_n)a_n $ where $u_n \to \frac23$, and I wondered if there is a simpler proof.  I will post my proof in a couple of days if no one does. Styling multi-line conditions in 'if' statements? My usb mouse sleeps after 5 seconds of inactivity when running on battery power. It wakes uo again when i press any button on the mouse. This is really annoying.  I have made lsusb, and I noticed my mouse is   Bus 001 Device 012: ID 04d9:2083 Holtek Semiconductor, Inc.  I have checked my /sys/bus/usb/devices/usb1/power/autosuspend but it is already set to 0, and the directory usb1 has many subdirectories which also have autosuspend files, and i dont know how to edit them.  Also, I would like a permanent solution, which would work for any mouse, on any of my 3 usb ports. Any ideas?  Using: Ubuntu 12.04 64bit When using QuickLook to quickly look through multiple images (pressing space and then using the arrow keys to navigate), some images will not get displayed by QuickLook when hitting them up a second time (going back for example).   A GIF does a much better job than me showcasing the issue. So here's me previewing a few random images:    I have tried deleting   ~/Library/Preferences/com.apple.QuickLookDaemon.plist ~/Library/Caches/com.apple.QuickLookDaemon/Cache.db ~/Library/Caches/com.apple.QuickLookDaemon32/Cache.db ~/Library/Containers/com.apple.quicklook.ui.helper  and restarting, unfortunately to no avail.  Repairing permissions and restarting the Finder in 32-bit mode, as suggested  by others, don't appear to have any effect on this issue either.  This happens with all kinds of images, no matter their source or format. Problem in solving functional equation $f(x^2 + yf(x)) = xf(x+y)$ To find all functions $f$ which is a real function from $\Bbb R \to \Bbb R$ satisfying the relation  $$f(x^2 + yf(x)) = xf(x+y)$$  It can be easily seen that the identity function $i.e.$ $f(x)=x$ and $f(x)=0$ (verified just now) satisfies the above relation!! And putting $y=0$ I have got $f(x^2)=xf(x)$.  Help needed to find other functions satisfying the relation. Windows 10 Anniversary Upgrade error 0xc1900101 - 0x30018 referring to SYSPREP Number of Homomorphisms from $\Bbb{Z}_m$ to $\Bbb{Z}_n$ This question coutesy of Allan Clark's "Elements of Abstract Algebra" (60$\zeta$).   Find the number of homomorphisms from $\Bbb{Z}_m\to \Bbb{Z}_n$ as a function of $m$ and $n$.  This is stumping me, can anyone help? Some movies feature child characters put in violent situations. This can range from crude language to extreme horror (e.g. torture, gore, etc.). As an example,  and its sequel has plenty of those scenes. It looks like a paradox that children can be actors in movies that are definitely not for children.  Do the directors usually try and avoid exposing the very young actors to the violence of such scenes, e.g. by clever dubbing and editing? Are there laws that require them to do so, just like there are ratings that prevent children from seeing some movies? How to install the latest stable version of Chromium? I have this function:  $$ f(x,y) = \frac {xy}{|x|+|y|} $$  And I want to evaluate it's limit when $$ (x,y) \to (0,0)$$ My guess is that it tends to zero. So, by definition, if:  $$ \forall \varepsilon \gt 0, \exists \delta \gt 0 \diagup \\ 0\lt||(x,y)||\lt \delta , \left|\frac{xy}{|x|+|y|}\right| \lt \varepsilon $$ Then $$ \lim_{(x,y)\to(0,0)}\frac {xy}{|x|+|y|} = 0 $$ So:  $$ \left|\frac{xy}{|x|+|y|}\right| = \frac{|xy|}{|x|+|y|} = \frac{|x||y|}{|x|+|y|} \le 1 |y| \lt \delta $$  So for any $$\delta \lt \varepsilon$$ the inequality is true. Hence, the limit exists and is equal to zero.  . Am I wrong or is Wolfram|Alpha wrong? Simulation software to solve high school level mechanics problems I'm a physics teacher and I would like to find a simulation software that could be used to give exact numerical answers to high school level problems in Newtonian mechanics.  I want the students to first calcultate a numerical answer to a physics problem and then use the simulator to visualize the problem in order to understand it better and verify that the simlator gives the same answer that they got when doing the calculation themselves.  Here are some examples of what kind of problems I want the students to be able to solve with the simulator:   A slope with a specific angle of inclination and a ball starts rolling at a certain point. Determine the speed, angular momentum and total mechanical energy of the ball after a specific time t. A ladder of known mass and length is leaning against the wall at a specific angle. The contact between the ladder and the wall is frictionless. Calcultate the x and y components of the force between the floor and the ladder.   I would like the software to have the following features:  - Simulation in 2D. No 3D simulations are necessary and the software should at least have the option of doing simulations only in 2D.  - Position all objects exactly by writing numbers (just positioning the objects using the mouse is not precise enough).  - Show animations of the time evolution of the system.  - Show forces, velocities and accelerations as arrows.  - Give exact numerical results so that the students can use the software to verify their calculation.  Ideally the software would have a graphical user interface, but if no suitable software with graphical interface exixts, I could also consider a software with text-based inputs. I would consider both free and commercial software.  My question is: Is there any software that would meet these requirements and if there is, where could I get it?