MacBook Pro keyboard replacement: Refurbished or new components?
I think my MacBook Pro battery was replaced by Apple's service team. How do I check that?
When a ball moves to the right, friction acts to oppose the motion, in other words, to the left. However, when a car travels around a bend, the friction acts in the perpendicular direction to the car's velocity and provides the centripetal force. I just cannot understand why friction would act in that direction. Why is $\Gamma\left(\frac{1}{2}\right)=\sqrt{\pi}$? Let $x_1 = 2$, and define $$x_{n+1} = \frac{1}{2} \left(x_n + \frac{2}{x_n}\right).$$ Show that $x_n^2$ is always greater than or equal to $2$, and then use this to prove that $x_n − x_{n+1} ≥ 0$. Conclude that $\lim x_n = \sqrt2$.   My question:  So I know how to do this problem but I don't know how to prove that $x_n &gt;0$ for all $n\in \mathbb{N}$. Extracting embedded images from a PDF Before I started using Ubuntu I used Nitro PDF reader to automatically extract images from PDF files. Is there a PDF reader for Linux that does this?  I would like to be able to extract images faster/easier than when taking a snapshot. X,Y are independent exponentially distributed then what is the distribution of X/(X+Y) Been crushing my head with this exercise. I know how to get the distribution of a ratio of exponential variables and of the sum of them, but i can't piece everything together.  The exercise goes as this:  If X,Y are independent exponentially distributed with beta = 1 (parameter of the exponential distribution = 1) then what is the distribution of X/(X+Y)  Any ideas?  Thanks a lot. Chance of objects going against greater entropy? My book uses the argument that the multiplicities of a few macrostates in a macroscopic object take up an extraordinarily large share of all possible microstates, such that even over the entire lifetime of the universe, if each microstate had an equal chance of being accessed, fluctuations away from equilibrium would never occur.  My question to this statistical proof is this: In the real world, is there really an infinitesimal but nonzero chance that macroscopic systems could access some of the more unlikely macrostates (e.g. heat transfer from a cold object to a hot object)? How to get parent process in .NET in managed way I was looking a lot for method to get parent process in .NET, but found only P/Invoke way. How does Princess Leia know Obi-Wan Kenobi is "Ben" Kenobi? When Luke rescues Leia from her cell on the Death Star:  Leia: Aren't you a little short to be a stormtrooper? Luke: Huh? Oh, the uniform. I'm Luke Skywalker. I'm here to rescue you. Leia: You're who? Luke: I'm here to rescue you. I've got your R2 unit, I'm here with Ben Kenobi! Leia: Ben Kenobi, where is he?! Luke: Come on!  How does Princess Leia know Obi-Wan Kenobi is &quot;Ben&quot; Kenobi? Shouldn't her response have been: &quot;Ben Kenobi? You mean the General??&quot; When a Battle Master fighter gets a critical hit and uses a maneuver that adds a superiority die to the damage, is that damage die doubled? The Battle Master fighter has access to a number of maneuvers that add a superiority die to the damage roll. For instance, the Trip Attack maneuver says (PHB, p. 74):     When you hit a creature with a weapon attack, you can expend one   superiority die to attempt to knock the target down. You add the   superiority die to the attack’s damage roll, and if the target is   Large or smaller, it must make a Strength saving throw. On a failed   save, you knock the target prone.   The Critical Hits rule states (PHB, p. 196):     When you score a critical hit, you get to roll extra dice for the   attack’s damage against the target. Roll all of the attack’s damage   dice twice and add them together. Then add any relevant modifiers as   normal. [...] If the attack involves other damage dice, such as from   the rogue’s Sneak Attack feature, you roll those dice twice as well.   During our game, the fighter PC rolled a critical hit. He then wanted to use a superiority die to use the Trip Attack maneuver to trip the enemy. Based on the above rules, is the damage from the superiority die also doubled?  I'm assuming the answer is yes based on the last statement in the Critical Hits section, but I wanted to confirm. As the question states... In Windows 7 with a single screen, you can drag a window to the right and a window to the left to maximize them both at half the screen, letting you quickly and neatly see two windows simultaneously.  If I have 2 monitors, is there a way to do this with 4 windows, where each monitor gets 2 windows split across each screen? How to convert wstring into string? Why is 'ls' suddenly wrapping items with spaces in single quotes? I just noticed that on one of my machines (running Debian Sid) whenever I type ls any file name with spaces has single quotes surrounding it. I immediately checked my aliases, only to find them intact. wyatt@debian630:~/testdir$ ls 'test 1.txt'  test1.txt wyatt@debian630:~/testdir$ alias alias ls='ls --color=auto' alias wget='wget --content-disposition' wyatt@debian630:~/testdir$   Another test, with files containing single quotes in their names (also answering a request by jimmij): wyatt@debian630:~/testdir$ ls 'test 1.txt'  test1.txt  'thishasasinglequotehere'\''.txt' wyatt@debian630:~/testdir$ touch &quot;'test 1.txt'&quot; wyatt@debian630:~/testdir$ ls ''\''test 1.txt'\'''  test1.txt 'test 1.txt'          'thishasasinglequotehere'\''.txt'   update with new coreutils-8.26 output (which is admittedly much less confusing, but still irritating to have by default). Thanks to Pádraig Brady for this printout: $ ls &quot;'test 1.txt'&quot;   test1.txt 'test 1.txt'    &quot;thishasasinglequotehere'.txt&quot;  $ ls -N 'test 1.txt'  test1.txt test 1.txt    thishasasinglequotehere'.txt  Why is this happening? How do I stop it properly? To be clear, I myself set ls to automatically color output. It just never put quotes around things before. I'm running bash and coreutils 8.25. Any way to fix this without a recompile? EDIT: Appears the coreutils developers  to break with the convention and make this the global default.  UPDATE - October 2017 - Debian Sid has re-enabled the shell escape quoting by default.  And at the bottom of the reply chain to the previous bug report, &quot;the change was intentional and will remain.&quot;  I thought this had already been settled, but apparently it was just reverted so that the &quot;stable&quot; Debian branch could keep its &quot;feature freeze&quot; while getting the other fixes, etc. from the newer version. So that's a shame (in my opinion). UPDATE: April 2019: Just found a  that was caused by this change to ls. When you're confusing developers and generating false bug reports, I think it might be time to re-evaluate your changes. Update: Android toybox ls is now doing something similar to this but with backslashes instead of quotes. Using the -q option makes spaces render as 'question mark characters' (I have not checked what they are, since they're obviously not spaces), so the only fix I have found so far without rooting the device in question is to add this to a script and source it when launching a shell. This function makes ls use columns if in a terminal and otherwise print one-per-line, while tricking ls into printing spaces verbatim because it's running through a pipe. ls() {     # only way I can stop ls from escaping with backslashes     if [ -t 1 ]; then         /system/bin/ls -C $@ |cat     else         /system/bin/ls $@ |cat     fi } How to estimate $P(x\le0)$ from $n$ samples of $x$? Suppose, we have $n$ samples $x_i$ of a random variable: $$x \sim \mathcal N(\mu,\sigma^2) $$ Based on the samples, we want to estimate the probability that $x$ is negative: $$P(x\le0)$$ Intuitively, I would first estimate: $$\hat \mu=\frac 1 n \sum x_i$$ $$\hat \sigma^2={\frac 1 {n-1} \sum (x_i-\hat \mu)^2}$$ and then calculate: $$P(x\le0)=\frac1{\sqrt{2\pi\hat\sigma^2}} \int_{-\infty}^0 e^\frac{x-\hat\mu}{\hat \sigma} dx$$ However $\hat \mu$ and $\hat \sigma$ have variance! If I use this method, I suspect I am ignoring that variance and making an incorrect estimation. Is this reasoning right? If so, how can I estimate $P(x\le0)$ more correctly, taking $\text {VAR}[\hat \mu]$ and $\text {VAR}[\hat \sigma]$ into account? How can we make First letter Capital while using trigonometric functions. eg: \cos (60^\circ) gives simple cos (all in small). What is needed is 'Cos' (with initial capital). For italics in mactex i used \it but it didn't italicized 'cos' only the number '60'. So how to do that? Show alert popup on standard page layout on click of standard button "Save" Integrate $\int \frac{dx}{\sqrt{-ax^2 + bx +c}} $ I am able to locate files in a folder containing a specific text string using this command:  grep -lir 'string' ~/directory/*   How do I move the files that appear in the above result to another location? I am not talking about miracles, religious revelation, or artistic expression, but something more mundane. There is a lot of "empirical" evidence that the Riemann hypothesis is true, the scare quotes indicate that the source of the evidence is "mathematical intuition". In psychology there are mental states that no one sees, touches, hears, smells or tastes, qualia with their perceptual fullness that can not be conceptualized or directly communicated, and abilities/skills "know how" with the same problem. In linguistics Kripke's theory of reference relies on "modal intuitions" to decide the truth of counterfactuals. But an empirical counterfactual is an oxymoron, especially when it is embedded into an entire "possible world".  Freudian psychoanalysis and Husserlian phenomenology are attempts to deal with some of them systematically, but they are traditionally confined to the fringe of science at best. The mainstream approach is empirical second hand, but behaviorism in psychology and linguistics, and nominalism in mathematics, were not very productive.   According to the naturalized epistemology there is no a priori "first philosophy" of science, methodology is subject to the "tribunal of experience" and revision, just as the science itself. Should it apply to non-empirical experience? Sense empiricism has served science well for centuries, and was fully embraced by Quine.  takes him to task for it:     "Cognitive science is an empirical science working to unearth the mechanisms through which natural language constitutes itself. That account has had to recognize the indispensability of mental states, of beliefs, if it is ever to become adequate to the problem... There is still too much "first philosophy" in Quine. We must rescue naturalized epistemology from its own founder".    Note the use of "empirical" together with "mental states".  Questions: Does naturalized epistemology dictate that natural science should relax sense empiricism, and approach non-empirical phenomena partly first hand? If so, how can the scientific method be adapted to introspective/intuitive phenomena that are not readily reproducible, manipulable, measurable, and/or publicly accessible? What would play the selective role of empirical testing? Can there be non-empirical (more likely, not entirely empirical) natural science? How can I take a photo using a slow shutter speed without it being overexposed? When I try to take a picture using a slow shutter speed in daylight it is overexposed. How can I change the settings so it isn't overexposed? In the final episode of season 6 we see that Lyanna Stark is whispering something to Ned. What is she saying?