Choose 2 good batteries out of 8 (4 bad 4 good)
We have $n$ charged and $n$ uncharged batteries and a radio which needs two charged batteries to work.
Transvection matrices generate $ \operatorname{SL}_n(\mathbb{R}) $ No more updates for Google Chrome 32-bit I am using Google Chrome 48.0 32-bit. Today when I launched Google Chrome, I got this message:    This computer will soon stop receiving Google Chrome updates because this Linux system will no longer be supported   I thought Ubuntu 14.04 is supported for five years, what's exactly happening here?  And on 64-bit Ubuntu, I get an error from Update Manager and apt-get update:  Failed to fetch http://dl.google.com/linux/chrome/deb/dists/stable/Release  Unable to find expected entry 'main/binary-i386/Packages' in Release file (Wrong sources.list entry or malformed file)   Some index files failed to download.   They have been ignored, or old ones used instead. How can I buy apps in the App Store without a credit card? While I'm waiting for my credit card to arrive, I still want to be able to buy some apps for my iPad!  Are there alternative ways to pay for apps, without resorting to someone else's credit card? Story about encounter with hostile aliens This is a short story I read in the late 80s or early 90s in an anthology or collection.  Here are the plot points I remember:   A human spaceship encounters a ship with hostile aliens (HA). There may be friendly aliens (FA) on board the human spaceship A human (or perhaps FA) telepath scans the HA and finds they are hostile and bent on conquest of other races The captain and crew decide on a bluff: pretend they are from a large federation of many species, one so big the HA will become discouraged and drop their plans of conquest They invite one or more HA on their ship To pretend that there are many (more) races in their federation, they do things like put a fake control panel in the ships aquarium and claiming the fish in them are one of the races in their federation The HA can't conceive of multiple races cooperating and are convinced there must be some race on top controlling it all. If not, they can just be conquered one at a time This goes on for a little while until, seeing their plan isn't working (via their telepath) the humans change their plan so that when      the ship's cat (so far unseen by the HA) walks in the room, one of the humans loudly calls "The CAT!" causing everyone to stand rigidly at attention while the cat walks around more or less ignoring everyone and everything the way cats do. Having seen the "top dog" and the rigid discipline and fear he instills in everyone else, the aliens "realize" the humans are deluded into thinking all races are equal members in their 'federation", the cats are really in control of a huge empire, and they should not be messed around with. When does the placement of a resistor matter? I've been a programmer for years but am very new to electronic engineering. So far I see people saying it doesn't matter where a resistor goes in your LED circuit as long as it is present, but it sounds like that's the nature of the basic LED circuit.  When does the placement of a resistor start to become important? Javascript remoting: to invokeAction or not to invokeAction Digging in deeper to Javascript Remoting and I'm a bit confused.  The  starts off by showing how to set up everything and uses code like the following:  function getRemoteAccount() {     var accountName = document.getElementById('acctSearch').value;      // This remoting call will use the page's timeout value     Visualforce.remoting.Manager.invokeAction(         '{!$RemoteAction.AccountRemoter.getAccount}',         accountName,          handleResult     ); }   However, I've also seen examples where Visualforce.remoting.Manager.invokeAction is never called.  They end up looking something like the following:  function getText() {   var text = "";   JSRemotingBasicsController.doGetText(     function(result, event) {       // do something here     }   ); }   I'm wondering what is the point to using Visualforce.remoting.Manager.invokeAction when you can just reference the RemoteAction directly like the second example?  Is there something I'm missing?  Is there a time where the second example doesn't work? Proving that $f(x) = g(x)$ for some $x \in [a,b]$ if $f,g$ continuous, $f(a)  g(b)$ Suppose $f$ and $g$ are continuous on $[a,b]$ and that $f(a) &lt; g(a)$ but $f(b) &gt; g(b)$.   Prove that $f(x) = g(x)$ for some $x \in [a,b]$.     I did try and actually did get as far as coming up with $f-g$. Would I set $h(x)= f(x)- g(x)$ and say $h(a) &lt;x&lt; h(b)$? I read  and thought that would easily be solved (not that it isn't solvable without) if one could write:   @Override public String toString() {     return super.super.toString(); }   I'm not sure if it is useful in many cases, but I wonder why it isn't and if something like this exists in other languages.  What do you guys think?  EDIT: To clarify: yes I know, that's impossible in Java and I don't really miss it. This is nothing I expected to work and was surprised getting a compiler error. I just had the idea and like to discuss it. Google map signed api key errors in Android Why True/False is capitalized in Python? All members are camel case, right? Why True/False but not true/false, which is more relaxed? Proving the given quadratic form is chi-squared $k$ Suppose $\underline{X}$ is an $m$-dimensional vector following multivariate Normal distribution i.e. $\underline{X}$~$N_m(\underline{\mu},\Sigma)$ where $\Sigma$ is positive definite. Let $B$ be a symmetric $m$ by $m$ matrix, having the following properties:  $B\Sigma B=B$  $\underline{\mu}'B\underline{\mu}=0$  $trace(B\Sigma)=k$, where $k&lt;m$   Prove that $\underline{X}'B\underline{X}$ follows chi-squared ($k$).  I proceeded a lot but got stuck in one last step where I think linear algebra is troubling me. Kindly have  a look at my &quot;solution&quot;.  The second condition can be written as $\underline{\mu}'B\Sigma B\underline{\mu}=0\implies B\underline{\mu}=\underline{0}$ as $\Sigma$ is Positive definite. Then after a little algebra and considering condition 2, we can write $\underline{X}'B\underline{X}=(\underline{X}-\underline{\mu})'B(\underline{X}-\underline{\mu})=(\underline{X}-\underline{\mu})'B\Sigma B(\underline{X}-\underline{\mu})$ But $\Sigma$ is positive definite hence we can have a symmetric matrix $\sqrt{\Sigma}$ defined such that $\sqrt{\Sigma}\sqrt{\Sigma}=\Sigma$. Using this, we can write $(\underline{X}-\underline{\mu})'B\Sigma B(\underline{X}-\underline{\mu})=(\sqrt{\Sigma}B(\underline{X}-\underline{\mu}))'(\sqrt{\Sigma}B(\underline{X}-\underline{\mu}))$ We want to show that this follows chi-squared ($k$). Now, it is obvious that as $(\underline{X}-\underline{\mu})$~$N(\underline{0},\Sigma)$, $\sqrt{\Sigma}B(\underline{X}-\underline{\mu})$~$N(\underline{0},\sqrt{\Sigma}B\Sigma B\sqrt{\Sigma})=N(\underline{0},\sqrt{\Sigma}B\sqrt{\Sigma})$ Now, $trace(\sqrt{\Sigma}B\sqrt{\Sigma})=trace(B\Sigma)=k$ and $\sqrt{\Sigma}B\sqrt{\Sigma}$ is idempotent (verified). So its eigenvalues are only $0$ or $1$. Since its trace is $k$, there are $k$ eigenvalues with value $1$ and other $m-k$ eigenvalues are $0$. Now here is the thing I want to prove: I want to show that$\sqrt{\Sigma}B\sqrt{\Sigma}$, which is the covariance matrix, is diagonal and has only $k$ $1$'s and remaining $m-k$ diagonal entries $0$. The reason is, then I will be able to say that $k$ components of $\sqrt{\Sigma}B(\underline{X}-\underline{\mu})$ are independent standard normals whose sum of squares is our quadratic form and thus it follows chi-squared ($k$). But I really could not show that $\sqrt{\Sigma}B\sqrt{\Sigma}$ has the desired property i.e. it is diagonal with rank $k$. But this is crucial, it is the finale. If this does not happen, then some components will be correlated with each other and we cannot say that our quadratic form has chi-squared ($k$). I would really appreciate if someone helped me to prove this final part. What is an Electron? I am a n00b to physics. What physics I know comes from high school and popular science books.  At school we were taught about atoms and what make up atoms, but we weren't actually told what the particles were!  So sure an atom has electrons orbiting around a nucleus made of protons and neutrons, but what exactly is an ?  And if we don't know, how can we fire them from guns, like they do in experiments ... etc  Any clarification on this would be great. How to do string formatting with placeholders in Java (like in Python)? Solvability and reducibility of a polynomial in a "chain" of finite fields Math notation in LaTeX? How can I write e^(3j) in LaTeX?  I have tried many syntax, like $e^(3j)$ or $e^*{3j}$  Any help will be very well received. When I try to launch Minecraft 1.7.4 or newer, the game crashes on startup, and the game output screen says this:  #  EXCEPTION_ACCESS_VIOLATION (0xc0000005) at pc=*, pid=*, tid=* # # JRE version: Java™ SE Runtime Environment (7.0_51-b13) (build 1.7.0_51-b13) # Java VM: Java HotSpot™ Client VM (* mixed mode, sharing windows-* ) # Problematic frame: # C  [ig4dev32.dll+0x3e88] OR # C  [ig4dev64.dll+0x55ec] OR # C  [ig4icd32.dll+0x104a77] OR  # C  [ig4icd64.dll+0x84930a]   This happens with both Java 7 and 8.  What causes this error, and what can I do to fix the issue? I try to delete Action.005 permanently.    But I don't know how. If I click on the X    then it is still in the list.    I already researched without success, I even managed to loose all my animations by attempting to apply a fix, so I want to be cautious now... How can I do this? I'm interested in Gaussian blur layer that blurring all inside of some figure located on this layer.  I mean, blurred background of some exact area. And in case of I move this area, blurred image must be updated. If I move some object in background - blur must be updated. etc. I have some data which looks like:  sampleA    ATGC    10   100 sampleA    ATGC    120  230 sampleA    ATGC    200  110   I want to print the min and max using the values in both column 3 and 4. So my output should look like:  sampleA   10  230   Thanks in advance Proof that the coefficients in an OLS model follow a t-distribution with (n-k) degrees of freedom Background  Suppose we have an Ordinary Least Squares model where we have $k$ coefficients in our regression model, $$\mathbf{y}=\mathbf{X}\mathbf{\beta} + \mathbf{\epsilon}$$  where $\mathbf{\beta}$ is an $(k\times1)$ vector of coefficients, $\mathbf{X}$ is the design matrix defined by  $$\mathbf{X} = \begin{pmatrix} 1 &amp; x_{11} &amp; x_{12} &amp; \dots &amp; x_{1\;(k-1)} \\ 1 &amp; x_{21} &amp; \dots  &amp;       &amp;   \vdots \\ \vdots &amp;   &amp; \ddots &amp;       &amp;   \vdots \\ 1 &amp; x_{n1} &amp; \dots  &amp; \dots &amp;   x_{n\;(k-1)}     \end{pmatrix}$$ and the errors are IID normal,  $$\mathbf{\epsilon} \sim \mathcal{N}\left(\mathbf{0},\sigma^2 \mathbf{I}\right) \;.$$  We minimize the sum-of-squared-errors by setting our estimates for $\mathbf{\beta}$ to be $$\mathbf{\hat{\beta}}= (\mathbf{X^T X})^{-1}\mathbf{X}^T \mathbf{y}\;. $$   An unbiased estimator of $\sigma^2$ is  $$s^2 = \frac{\left\Vert \mathbf{y}-\mathbf{\hat{y}}\right\Vert ^2}{n-p}$$ where $\mathbf{\hat{y}} \equiv \mathbf{X} \mathbf{\hat{\beta}}$ ().   The covariance of $\mathbf{\hat{\beta}}$ is given by $$\operatorname{Cov}\left(\mathbf{\hat{\beta}}\right) = \sigma^2 \mathbf{C}$$ where $\mathbf{C}\equiv(\mathbf{X}^T\mathbf{X})^{-1}$ () .  Question  How can I prove that for $\hat\beta_i$, $$\frac{\hat{\beta}_i - \beta_i} {s_{\hat{\beta}_i}} \sim t_{n-k}$$ where $t_{n-k}$ is a t-distribution with $(n-k)$ degrees of freedom, and the standard error of $\hat{\beta}_i$ is estimated by $s_{\hat{\beta}_i} = s\sqrt{c_{ii}}$.    My attempts  I know that for $n$ random variables sampled from $x\sim\mathcal{N}\left(\mu, \sigma^2\right)$, you can show that  $$\frac{\bar{x}-\mu}{s/\sqrt{n}} \sim t_{n-1} $$ by rewriting the LHS as $$\frac{ \left(\frac{\bar x - \mu}{\sigma/\sqrt{n}}\right)  }   {\sqrt{s^2/\sigma^2}}$$ and realizing that the numertor is a standard normal distribution, and the denominator is square root of a Chi-square distribution with df=(n-1) and divided by (n-1) ().  And therefore it follows a t-distribution with df=(n-1) ().  I was unable to extend this proof to my question...  Any ideas?  I'm aware of , but they don't explicitly prove it, they just give a rule of thumb, saying "each predictor costs you a degree of freedom".