preposition confusion in or on
Origin/reason for the expression "on the bus" instead of "in the bus"
If $AB = I$ then $BA = I$ If $A$ and $B$ are square matrices such that $AB = I$, where $I$ is the identity matrix, show that $BA = I$.    I do not understand anything more than the following.   Elementary row operations. Linear dependence. Row reduced forms and their relations with the original matrix.   If the entries of the matrix are not from a mathematical structure which supports commutativity, what can we say about this problem?  P.S.: Please avoid using the transpose and/or inverse of a matrix. Let $a&gt;0$, show that for $x&gt;0$, $1&lt;f(x)&lt;2$, where $$f(x)=\frac{1}{\sqrt{1+x}}+\frac{1}{\sqrt{1+a}}+\sqrt{\frac{ax}{ax+8}}$$  I could take the derivative, find the maximum of the function and conclude. But that involves heady algebra and you have to solve a high degree polynomial. Is there an easier way simply using inequalities? Thanks, much appreciated convergence to infinity of$ (2^n + 3^n)^{\frac{1}{n}}$ (as $n \to \infty$) Just checking. $2^n$ ($n \to \infty$) tends to $\infty$. + $3^n$ also ($n \to \infty$) tends to $\infty$  so the sum gets me $\infty$.  Now $(\infty)^{1/\infty}$ : $(\infty)^0 = 1$  I see no other way.  Theorem: $n^{\frac{1}{n}} = 1$ as $n \to \infty$.  Analogous:  $(2^n - 3^n)^\frac{1}{n}$ = $(\infty  - \infty)^0$ = 1 ???  Is it ok to suppose infinity on any integer $k^n$ as $n$ goes to infinity ? I want to use jQuery to parse RSS feeds. Can this be done with the base jQuery library out of the box or will I need to use a plugin? Where can I find gedit's backup files and how to open them? So, I just finished off the first part of an important task for school and today when I was going to continue, the document with my notes were gone. The only fragment I've got left of my file is the back up file, which doesn't show in the folder where I saved the document, but I can search for it.  Is there anyway which allows me to open the back up file, save it as a text file and continue my task? I don't really have the time for running restoring tools, which never has helped me in a satisfactory way anyway.  I wrote the document in gedit and my version of Ubuntu is 12.10. persist port routing from 80 to 8080 I use amazon EC2 instance which works via ubuntu. By default according security restrictions I can't bin my application to port 80, so I just bind it port 8080 and then set routing redirect from port 80 to 8080 via the following command:  iptables -t nat -A PREROUTING -p tcp --dport 80 -j REDIRECT --to 8080   But I found that when I reboot the server this settings no longer active untill I invoke this command again.  So my question is how to enable port's redirect work even if system was rebooted? How can I disable the "no backup in x days" from TimeMachine? (Updated answer choice for High Sierra. See the old answer if you are still on Yosemite)  I'm running OS X Yosemite, and I want to disable the notifications from Time Machine. I rarely do a hard-drive backup because all of my essential files are in the cloud. I don't need to run timemachine as often as it suggests/warns/annoys me about.  How do I disable the notifications? Time Machine is not in the Notifications preferences, and I don't want to stop using it entirely. Received an email intended for another person I'm not really sure how email routing works, but someone ordered something on Amazon and I received the email instead of them.  Or maybe we both got it, I don't know.  The order doesn't show up in my account, so I'm certain I wasn't charged for it, but I shouldn't be getting other peoples' emails.  We'll say that my email is johndoe1@gmail.com, and somebody who's email is john.doe1@gmail.com places an order on Amazon.  The confirmation email is sent to me at johndoe1@gmail.com.  I checked the email header, and it did say To: john.doe1@gmail.com which is not my email address.  At first I thought that Google ignores periods in email addresses, but I tested the account setup and it doesn't give any error when you put a period in the address.  I didn't create the account; I just used the "check availability" function and the address I chose with a period was fine.  Maybe someone with knowledge about how Email works could tell me why this happened.  Is this a bug in the way Amazon sends emails?  Or is it a bug in how Google receives them?  Who should I report this issue to? In short, :     If you go before the Board of   Directors and ask for 1.5 billion   dollars to build a nuclear reactor, no   one will review or discuss the details   of the plant. They will assume that   experts have been over every inch of   the plans, and not want to look   foolish by asking a silly question.      However, if you ask the same group to   approve a 30 dollar expenditure for   lumber with which to build a bikeshed   (presumably a British term for the   smallest possible building) then be   prepared for a 45 minute discussion   about all aspects of the bikeshed,   including the color of the paint. The   explanation for this is that everyone   can grasp the scale of a bikeshed.   To transform this problem to Stack Overflow terms, questions that are trivial such as "How to convert a C string to a QString or "What is the difference between . and ->" get a lot more views, answers and votes than real questions of the form "How do I frob this widget" that bother real professionals. Fewer views mean less possible votes for good answers and hence less incentive for people to answer.  How can Stack Exchange solve this problem? " could " vs. " could have " As I know we can use both structures to talk about past ability , but is there any difference between them?  A. He could climb the mountain  B. He could have climbed the mountain In D&amp;D 4e there was an option to Delay your initiative:     Perform your actions as desired and adjust your initiative to your new   position in the order.   Does this rule to change initiative order still exist in 5th edition? I cannot find it in the PHB. register_nav_menus() won't register menus Please help with menu registeration:  I have followed this guide (http://codex.wordpress.org/Navigation_Menus) step by step.  Added the following code to functions.php:  function register_my_menus() {   register_nav_menus(     array('header-menu' =&gt; __( 'Header Menu' ) )   ); }  add_action( 'init', 'register_my_menus' );   But the guide says that this will make the menu appear in the Appearance -> Menus admin screen in the "Theme Locations" box as "Header Menu".  When I go to Appearance -> Menus everything is greyed out and there is no option of "Theme Locations" Simulating draws from a Uniform Distribution using draws from a Normal Distribution I recently purchased a  in which one of the probability questions was as follows:     Given draws from a normal distribution with known parameters, how can you simulate draws from a uniform distribution?   My original thought process was that, for a discrete random variable, we could break the normal distribution into K unique subsections where each subsection has an equal area under the normal curve. Then we could determine which of K values the variable takes by recognizing which area of the normal curve the variable ends up falling into.   But this would only work for discrete random variables. I did some research into how we might do the same for continuous random variables, but unfortunately I could only find techniques like inverse transform sampling that would use as input a uniform random variable, and could output random variables from some other distribution. I was thinking that perhaps we could do this process in reverse to get uniform random variables?  I also thought about possibly using the Normal random variables as inputs into a linear congruential generator, but I'm not sure if this would work.  Any thoughts on how I might approach this question? Let $A$ be a nonempty and bounded below, and define $B= \{b\in \mathbb{R}: b$ is a lower bound for $A\}$. Show sup$B$ = inf$A$.   So far I have: Let $A$ be nonempty and bounded below. This implies $\exists l \in A, \forall a \in A, l\leq a$ and $l =$ inf$A$. Let $B = \{b\in \mathbb{R}: b$ is a lower bound for $A\}$ and let $M=$ sup$B$. This implies $\exists b\in B, \forall b \in B, b\leq M$. By definition of $B$, we know inf$A \in B$. Since $b\leq$ sup$B$, with sup$B$ the largest element in $B$, we have inf$A \leq M =$ sup$B$, so inf$A \leq$ sup$B$.  I realize I have to prove that sup$B \leq$ inf$A$ to get that sup$B$=inf$A$. I am having trouble doing so.   Part B: Use (a) to explain why there is no need to assert that greatest lower bounds exist as part of the Axiom of Completeness. Word meaning the reverse of 'Xenophobia' Xenophobia refers to the irrational fear or hatred of foreigners. But recently I've seen people displaying (in newspaper articles and other places) an irrational fear or hatred of their own countrymen. Is there a word for that? Linear vs nonlinear differential equation How to distinguish linear differential equations from nonlinear ones? I know, that e.g.: $$ y''-2y = \ln(x) $$ is linear, but $$ 3+ yy'= x - y $$ is nonlinear. Why? I am currently reading Murphy's ML: A Probabilistic Perspective. In CH 3 he explains that a batch update of the posterior is equivalent to a sequential update of the posterior, and I am trying to understand this in the context of his example.  Suppose $D_a$ and $D_b$ are two data sets and $\theta$ is the parameter to our model. We are trying to update the posterior $P(\theta \mid D_a, D_b)$. In a sequential update, he states that, $$ (1) \ \ \ \ \ \ \ \  P(\theta \mid D_{a}, D_{b}) \propto P(D_b \mid \theta) P(\theta \mid D_a) $$ However, I am slightly confused as to how he got this mathematically. Conceptually, I understand that he is saying the posterior $P(\theta \mid D_a)$ is now a prior used to update the new posterior, which includes the new data $D_b$, and is multiplying this prior with the likelihood $P(D_b \mid \theta)$. Expanding the last statement out, I have, $$  P(D_b \mid \theta) P(\theta \mid D_a) = P(D_b \mid \theta) P(D_a \mid \theta) P(\theta) $$  but are we allowed to say $P(D_a \mid \theta) P(D_b \mid \theta) = P(D_a, D_b \mid \theta)$ in order to make the connection in (1)? Why are some faces in my mesh darker? Process descendants If I'm given a complex number (say $9 + 4i$), how do I calculate its square root?