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How to prove this is a metric? I have 2 distance functions $d(x,y)=|x^2-y^2|$ and $d(x,y)=|x^3-y^3|$ and I am trying to prove that they are metrics on $\mathbb R$, or give a counterexample that they are not metrics on $\mathbb R$.  I managed to prove the first 3 properties for the 2nd function namely:   $d(x,y)\ge0$ $d(x,y)=0$ iff $x=y$ $d(x,y)=d(y,x)$   and I now have shown that the 1st function is not a metric.  But for the triangle inequality property, $d(x,y)\le d(x,z) + d(z,y)$, Im stuck on it. How would go about proving it for $d(x,y)=|x^3-y^3|$? Thanks. What should a main game loop do? Generally, what are the core things that one should do in the game loop, and what are some things that one shouldn't do in the game loop? Power supplies are available in a wide range of voltage and current ratings.  If I have a device that has specific voltage and current ratings, how do those relate to the power ratings I need to specify?  What if I don't know the device's specs, but am replacing a previous power supply with particular ratings?  Is it OK to go lower voltage, or should it always be higher?  What about current?  I don't want a 10 A supply to damage my 1 A device. How to use "tens of" and "hundreds of"? If I'm not mistaken, tens of means 10 to 99 and hundreds of means 100 to 999. Is this correct? I found in some dictionaries that tens of is actually not correct.  I also found that hundreds of could also mean any arbitrary large number. So how would people usually interpret hundreds of? Based on context?  In my case, I want to describe numbers of some items that are usually 50-90 but sometimes could be around 100-200, but definitely not as many as 300 or so. I want to really emphasize that there are a lot of these items. In this case, can I say "... have tens or even hundreds of ..."? Would people misinterpret what I actually want to say? Find all Quotient (or Factor) Groups of $D_4$ (Dihedral Group of order $8$) I need to be able to find all of the quotient groups for dihedral group 4 with $D_4 = \{ e,R,R^2,R^3,V,H,D,D'\}$. I know I have to start by finding the normal subgroups, which are $\{e,R^2\}$ $\{e,R,R^2,R^3\}$ $\{e,R^2,V,H\}$ $\{e,R^2,D,D'\}$. Then I need to find the sets defined by $G/H=\{ aH : a \in G\}$ with operation $aH  bH = abH$. I am stuck here; could someone please show me how to find all of the factor groups for D4? Cannot select text in Safari address bar Formatting Keyboard Shortcuts for Comments Why are my sauces splitting? Recently all my attempts at sauces are splitting and I'm not sure what's going wrong. Tonight I tried a balsamic vinegar reduction / chicken stock / pan juices recipe to go with my lamb chops, but once again the dreaded split occurred. My wife is patient and forgiving, but I feel like I must be missing something fundamental to get this wrong so often.  Two questions.   What could I be doing wrong to cause these sauces to split? What can I do to recover when it happens again?   UPDATE: The Pan juices contained a tbsp or two of olive oil, and some of the fresh herb rub I'd coated the lamb chops in (thyme, rosemary, and mint). I did not drain the pan before starting the sauce. It has crossed my mind that perhaps there was too much oil / fats in the pan at that point?  For completeness, I should point out that I also sauteed some chopped shallots in the pan juices immediately before adding the vinegar. Formula for expected value of continuous random variable without using density This is a  about Monitoring Software. Also Related:   I need to monitor my servers; what do I need to consider when deciding on a monitoring solution? Integral representation $B_T^3$ I have to find a $F_t$ such that $B_T^3=E[B_T^3]+\int_0^T F_t dB_t$. I have shown by ito formula that $B_T^3=\int_0^T 3 B_s^2 dB_s+\int_0^T 3 B_s ds$. Could you please help me? How to use Class in Java? There's a good discussion of Generics and what they really do behind the scenes over at , so we all know that Vector&lt;int[]&gt; is a vector of integer arrays, and HashTable&lt;String, Person&gt; is a table of whose keys are strings and values Persons.  However, what stumps me is the usage of Class&lt;&gt;.  The java class Class is supposed to also take a template name, (or so I'm being told by the yellow underline in eclipse).  I don't understand what I should put in there.  The whole point of the Class object is when you don't fully have the information about an object, for reflection and such.  Why does it make me specify which class the Class object will hold?  I clearly don't know, or I wouldn't be using the Class object, I would use the specific one. I've been studying the Cox Proportional Hazards model, and  this question is glossed over in most texts.    Cox proposed fitting the coefficients of the Hazard function using a partial likelihood method, but why not just fit the coefficients of a parametric Survival function using the maximum likelihood method and a linear model?    In any cases where you have censored data, you could just find the area under the curve.  For example, if your estimate is 380 with standard deviation of 80, and a sample is censored >300, then there is an 84% probability for that sample in the likelihood calculation assuming normal error. What's a recommended alternative to Quickbooks pro? I've just installed Ubuntu to replace Windows. I can't seem to find a good alternative to  to run on Ubuntu. Any help would be appreciated. I'm at Reachwater Rock and I've encountered a series of doors, the first I had to open using the Emerald Claw I found near a body, but now the door's telling me to use an Ivory Claw. Where can I find it, because I don't have it in my inventory yet. Even though JRE 8 is installed on my MAC OS Yosemite, the error - " No Java Runtime present,requesting to install " Proof that for $a,b,c\in \mathbb{R}$, hold that $\left |\sqrt {a^2+b^2}-\sqrt {a^2+c^2} \right |\leq \left |b-c \right |$. 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. There is an experimental timeline view for questions now available on meta ONLY! (Now working on SO, SF, and SU as well)  This is partially motivated by comments about how difficult it is to follow very active questions which have a lot of activity, and also the feedback on the API which was heavily dominated by "we need to know what happened in this time range" type responses.  It can only be accessed through URL hackery at the moment. To get to it, enter question URLs like so:  https://meta.stackexchange.com/posts/{id}/timeline  This is a chronological timeline of everything that happened on a question, including   initial question ask answers  revisions comments closure, deletion, reopen, undelete, etc votes   all based on when they happened in chronological order.  We are thinking about making this a fourth tab on the question, like so:  Timeline | Oldest | Newest | Votes  Please try it out and provide feedback. It is still a little rough, so we are not looking for highly detailed bug reports, just more general feedback.  Timeline view for this question: Where can I find Dogmeat after dismissing him to Sanctuary Hills? I recently dismissed Dogmeat in favor of Codsworth, and I'm fairly sure I sent him to Sanctuary Hills. However, I've had a fairly detailed look around the place and I haven't found him yet. Does anyone know where he is? How does the double strike mechanic interact with other mechanics like lifelink or trample? I understand that the double strike mechanic allows a creature to attack in both a first strike combat damage step and a normal combat damage step.  It's clear that when a creature with no other abilities aside from double strike is blocked that even if the defender has lethal damage marked in the first strike step that the additional damage does not carry forward to the player.  How does this view of double strike change with relation to trample and lifelink?  I believe the correct interpretations are (not accounting for other effects or abilities) that 2 times the attacker's power is marked as damage on the defender.  For trample this amount is reduced by the defender's toughness and additional damage carries forward as normal.  Similarly for lifelink the active player gains life equal to 2 times the attacker's power.  Thus a 10/10 trampling double striking life linker blocked by a 1/1 creature would deal 19 points of trample damage and the active player would gain 20 life.  Is this interpretation correct?