I want to porve the triangle inequality:  $x, y \in \mathbb{R} \text { Then } |x+y| \leq |x| + |y|$  I figured out that probably the cases:   $x\geq0$ and $y \geq 0$ $x&lt;0$ and $y &lt; 0$ $x\geq0$ and $y &lt; 0$ $x&lt;0$ and $y \geq 0$ &lt;- Here I am not sure...   have to be proven. However, I do not figured out a concrete method. Are my assumptions true?  How to finish the prove with these assumptions. I really appreciate your answer!!!
I understand intuitively that this is true, but I'm embarrassed to say I'm having a hard time constructing a rigorous proof that $|a+b| \leq |a|+|b|$. Any help would be appreciated :)
What is the volatile keyword useful for? At work today, I came across the volatile keyword in Java. Not being very familiar with it, I found . Given the detail in which that article explains the keyword in question, do you ever use it or could you ever see a case in which you could use this keyword in the correct manner? How to create and design hoops net and attach ot to a hoop? How to skip kernel update? I stumbled upon a  with the following paragraph:     MBIE has launched an investigation into claims Chinese galvanised   steel coil imports have been dumped on the local market, causing   "material injury" to New Zealand industry, and the case will end up on   the table of newly appointed Commerce Minister Jacqui Dean.   I thought that sentence was grammatically wrong because the author should put the word "that" (as a conjunction) right after the word "claim". I don't see why the author could drop the conjunction. Is this a prevailing phenomenon in American or Australian English? Another layman blackhole question, pulling one end of a string out from behind the event horizon No long explanation is needed,  What would happen if I were to allow one end of a rope to fall past the event horizon of a black hole while I held the other end?  Would I be able to pull it out? Would the rope feel extremely (infinitely?) heavy? How to remove symbolic link I was trying to create this symbolic link:  sudo ln -s /usr/share/phpmyadmin /var/www/phpmyadmin   but I accidentally typed:  sudo ln -s /usr/share/php,yad,in /var/www/phpmyadmin   So now I want to correct it but it says symbolic link already exist. Derivation of a generalization of Mertens' Third Theorem. One of Mertens' Theorems states $$\prod_{p\le x}(1-\frac{1}{p})\sim \frac{e^{-\gamma}}{\ln(x)}.$$ I have seen a generalized version that states  $$\prod_{m&lt;p\le x}(1-\frac{m}{p})\sim \frac{c(m)}{(\ln(x))^m},$$ where $m$ is a positive integer and $c(m)$ is a real number that depends on $m$. How can I derive this result? Is there a way to find what the values of $c(m)$ are? Thank you very much. Is there a way to substring a string in Python, to get a new string from the third character to the end of the string?  Maybe like myString[2:end]?  If leaving the second part means 'till the end', and if you leave the first part, does it start from the start? To select a child node in jQuery one can use children() but also find().   For example:  $(this).children('.foo');  gives the same result as:  $(this).find('.foo');  Now, which option is fastest or preferred and why? Updating list definition using features I have a feature that deploys content types, a list definition and template that uses these content types and finally a list instance. This feature is already in production.  Now we have a change request to add new views to this existing list. So I make the changes to schema.xml and upgrade the solution. But alas, the new views don't show up. If I create a new site, the list instance is created with the new views. I tried deactivating/reactivating, iisreset but still no success. Is there anything I'm missing? Calculate expected values using logistic regression coefficients The outcome of the logistic model is binary (0;1) - intervention that occurs in hospital (3838 interventions in 51096 cases). I am modeling the effect of age groups (60 to 74; 75 to 89 and 90+) and sex (female) on the outcome. My reference age group is male aged 18-59.  My logistic regression coefficients look like this:  intercept  -4.313 female      0.5117 age 60-74   0.4081 age 75-89   1.1761 age 90+     1.5612.   Corresponding odds ratios:  female      1.668 years60_74  1.504 years75_90  3.242 years90     4.764   Now I am trying to obtain a probability of an outcome = 1 for each case. What is the right way to do this? For my own understanding, I am interested in manually replicating the calculation of the standard errors of estimated coefficients as, for example, come with the output of the lm() function in R, but haven't been able to pin it down. What is the formula / implementation used? Proof that for $a,b,c\in \mathbb{R}$, hold that $\left |\sqrt {a^2+b^2}-\sqrt {a^2+c^2}  \right |\leq \left |b-c  \right |$. Give me a clue, I don't have any idea to work with that inequality. View approximate number of users online Is there any way to view the number of users online at any given time (or even an approximation)?  This could be really useful to provide an indication of whether your question is likely to get answered any time soon.  It could also provide a bit of an indication for why your question is getting very low views - if there are only a few users online, then that could be the reason for little interest in a question, whereas if there are lots of users online and the question still doesn't garner many views, then maybe it needs to be rephrased or something.  I know that this can't be used definitively, but it could be used as a rough guide, and could also just be an item of interest.  Perhaps it could show the number of users that have been seen in the past 5 / 10 minutes? (Shouldn't be too hard, since each user page already shows the last time that they were seen). How do I change my username? Some time ago, when I installed Ubuntu, I chose a rather stupid username for my account that I do not want to use anymore. How do I change this (including the name of my home directory, and the name in the terminal) without losing settings for applications? How do I keep permissions and my keys for various authentification (e.g. email, SSH, GPG and more)? What settings could possibly get lost if I changed my username? 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 $\rho$? Minecraft Failed To Bind Port I made a server and used my IPv4 address for the IP. I decided to scratch that server and make a Bukkit server so I can use plugins. I tried to use my IPv4 address which I used for my other server but I keep getting the message [17:31:15 WARN]: ****FAILED TO BIND TO PORT!  Is there any thing I can do to get a new address to use or is there a way to delete the old server? Do I need to use a different computer? Please help! I've don't know how to fix this problem. If you have any suggestions please tell me and I'll give it a try. What is the non-parametric equivalent of a two-way ANOVA that can include interactions? How prove that $q \geq b+d$ for $ad-bc = 1$ and $\frac{a}{b} > \frac{p}{q} > \frac{c}{d}$? Let $a,b,c,d,p$, and $q$ be natural numbers such that $ad-bc = 1$ and $\frac{a}{b} &gt; \frac{p}{q} &gt; \frac{c}{d}$.  How prove that  $q \geq b+d$? I am looking for the stuff above, so I can make non-duplicate proposals on area 51.