I know there is a physics law that says a light magnification can't be more hot than the light source itself, but can it be brighter at the least?
I'm pretty sure that you can't take a magnifying glass and make it focus to a point that is hotter than the surface of your light source. For example, when you're outside trying to fry ants with your magnifying glass, it's impossible to get it hotter than 5,000 C (the temperature of the surface of the sun). My dad was arguing with me about it because he didn't see why this was true. My easiest argument was that the 2nd law of thermodynamics prevents this from happening because heat can't flow passively from a place of lower energy/entropy to a place of higher energy/entropy. He didn't buy it, saying that there wasn't anything preventing the light from focusing to a hotter point.  So I was wondering, are there some laws of optics that prevent this from happening? Or alternatively, is there a way to show that you could build a perpetual motion machine from this? Any help is appreciated.
On 23rd of June, 2016, the UK . This leads to the following questions:   Are UK citizens traveling to the EU affected? Are EU citizens traveling to the UK affected? Are UK/EU family members traveling both way affected? Are Commonwealth citizens traveling to the UK affected? Are other visa-free nationals (such as US/Canada citizens) affected? What other consequences might 'Brexit' have on travelers?   Also, see the  on Expats.SE. systemd-resolve high cpu usage after update to 17.04 How to set the monitor to its native resolution which is not listed in the resolutions list? After installing Ubuntu 10.04 with my Samsung SyncMaster B2030, native resolution (1600X900) is not found in the list of resolutions. Parallel option performing worse on cluster I'm trying out oracles parallel option on a cluster and surprisingly I am getting worse results with the parallel option. I was expecting some improvement with the parallel option but certainly not worse results. I am wondering why this is the case, and if there is anything wrong with the way I am using the parallel option on my cluster.   I am using a degree of 4 when the number of CPU's I have is 8. I have tried adding parallel to the cluster directly ALTER CLUSTER cluster PARALLEL 4 as well as at the statement on both the index /*+ PARALLEL_INDEX(clust_index, 4) */and tables /*+ PARALLEL(4) */,   Here is my trace from the parallel:    Without parallel: How do I change the extension of multiple files? Why did the First Order do this at the cantina? In The Force Awakens, after the First Order are alerted to BB-8, Rey, and Finn's presence at the cantina, an aerial bombardment of the compound begins. They also deploy ground troops.  Wouldn't it have been better to not have an aerial bombardment and simply have the troops surround the cantina, keeping as many people as possible bottled up inside it while they searched both the interior and surrounding area for BB-8?  Instead of a surgical extraction, the strategy that they chose seemed to create chaos and even risked destroying BB-8.  Perhaps this was a sign of the inexperience and emotionally-charged state of Kylo Ren, but General Hux seemed to have more direct authority over the deployment of troops and fighters.  So what was the strategy behind these tactics? What is $\int_{-\pi}^\pi \cos(nx)\cos(mx)\,dx$? I'm pretty sure that there's a theorem that says that the Fourier coefficients of a sum of $\cos(nx)$ and $\sin(nx)$ 's are the coefficients of the sum itself.   I tried to prove that in the specific case of $f(x) = \sum_{n=1}^{\infty}\frac{\cos(nx)}{n^2}$. In calculating the Fourier coefficients I got to calculating  $\cos(nx)\cdot\cos(mx)$, but I'm now stuck. Any tips would be appreciated! Travel to Romania with a residence permit of Germany I am a student in Germany and i have a residence permit here for 2 years, which is also a Schengen visa, I suppose. Do I need a separate visa if i wish  to travel to Romania? How could I go about detecting (returning true/false) whether an ArrayList contains more than one of the same element in Java?  Many thanks, Terry  Edit Forgot to mention that I am not looking to compare "Blocks" with each other but their integer values. Each "block" has an int and this is what makes them different. I find the int of a particular Block by calling a method named "getNum" (e.g. table1[0][2].getNum(); If in a UFD every maximal ideal is principal then it is a PID I remember that the main character is trying to find the group that destroyed his home planet while he was on a mission. His planet was very difficult to live on and the people became expert in martial arts and were the best soldiers in the galaxy. They were mercenaries and often helped against tyrants. There was a very memorable section in at least one book about an initiation task that the main character had to do in that he had to survive in their wilderness for days. He was at some point surrounded in a old building by thousands of poisonous worm/snake creatures that he managed to survive until rescued by the leaders. Google spreadsheet sorting messes up cell comments I'm running into this weird problem with Google spreadsheets: if the cells have comments on them and I sort the spreadsheet, the rows sort but the comments stay in the same position, and don't move with the cells they're attached to.  In other words, the cells look like this:  A C  &lt;-- comment on this cell B   and then I sort by this column:  A B  &lt;-- comment remains on this cell (stays in position) C  &lt;-- comment should have remained attached to this cell   This means that sorting is completely broken and I can't use it at all, because I clearly need the cell comments to stay with the cell they're attached to.  Is this a bad bug in Google Spreadsheets, or am I missing something here? How do you use Batch for send more than 10 emails? Suppose I want to know whether $\sqrt{7}\in\mathbb{Q}_5$, or more generally, whether $\sqrt{n}\in\mathbb{Q}_p$ for $n\in\mathbb{Z}$, $p$ an odd prime. What are the techniques for determining this? Am I supposed to "lift" it from $\mathbb{Z}/p\mathbb{Z}$ using Hensel's lemma? Could I see an example? Practice problem for exam:     Let ${A_n}$ satisfy $\sum\limits_{n=1}^\infty P(A_n \cap A^c_{n+1}) &lt; \infty$ and $\lim\limits_{n\to \infty} P(A_n) = 0$.  Show that $P(\lim \sup A_n) = 0$.    I can see that it is sufficient to show that $P\left(\sum\limits_{n=1}^\infty A_n\right) &lt; \infty$ and then apply the Borel-Cantelli lemma, but I am having trouble showing this from the information given.  I see an almost identical problem here: , however the hint there is not enough for me to make headway, and this problem is also different in which term gets the complement inside the first sum. How to pivot rows into columns MySQL Been looking to pivot rows (progress_check) into columns check 1, check 2 etc... No sum or totals required, just to display the results hopefully... Can anyone help, Thanks Ad How to disable captive portal detection? How to remove exclamation mark on Wi-Fi or signal icon in Lollipop? When connected to the Internet, the system checks if it's in a walled garden, which is simply trying to connect to Google. If it fails, in Android 5, there'll be an exclamation mark on the WiFi icon or signal icon.   In some situations, this is annoying. Say, in China, Google is blocked, the exclamation mark will always be there, which is meaningless.  How can I disable this check, and make the system assume, connected WiFi/mobile data is connected Internet? Sequence converges iff $\limsup = \liminf$ I want to prove that a sequence of real numbers $\{s_n\}$ converges to $s$ if and only if $\limsup_{n \to \infty} s_n = \liminf_{n \to \infty} s_n = s$.  Here are my definitions:  For any sequence of real numbers $\{s_n\}$, let $E$ be the set of all subsequential limits of $\{s_n\}$, including possibly $+\infty$ and/or $-\infty$ if any subsequence of $\{s_n\}$ diverges to infinity.  Then $\limsup_{n \to \infty} s_n = \sup E$, and $\liminf_{n \to \infty} s_n = \inf E$.  I know the theorem that a sequence converges to a point if and only if every one of its subsequences converge to that same point, so one direction of this proof is easy:  If $\{s_n\}$ converges to some point $s \in \mathbb{R}$,  then every subsequence of $\{s_n\}$ converges to $s$.  So the set $E$ of every subsequential limit of $\{s_n\}$ consists of the single point $s$, so $$\limsup_{n \to \infty} s_n = \sup \{s\} = s = \inf \{s\} = \liminf_{n \to \infty} s_n$$  But the other direction seems more tricky...  If $\limsup_{n \to \infty} s_n = \liminf_{n \to \infty} s_n = s$, then every convergent subsequence converges to the same point $s$.  Also, there can be no subsequences which diverge to infinity (otherwise $\limsup_{n \to \infty} s_n$ would be $+\infty$, or $\liminf_{n \to \infty} s_n$ would be $-\infty$).  But can't there be subsequences which diverge otherwise?  And wouldn't that throw off the convergence of $\{s_n\}$?    EDIT:  I'd also be willing to accept a solution which makes use of the "Pinching Theorem" (if $a_n \leq s_n \leq b_n$ for every $n \in \mathbb{N}$, and if $a_n \to s$ and $b_n \to s$, then $s_n \to s$). Conditional expectation in AR(1) process Suppose we have a stationary AR(1) process: $Y_{t+1}=a+ \rho Y_{t} + \epsilon_{t+1}$ where $\epsilon_{t+1}$ is white noise with probability density function $\phi(.)$.   Now say we have a equation $P_{t+1}=u\cdot (A_{m}-Y_{t+1})^{+}$ where $(Y_{m}-Y_{t+1})^{+}$ = $ \max \{(A_{m}-Y_{t+1}),0\}$ and $u$ is a constant.  So $P_{t+1}=u\cdot(A_{m}-a-\rho Y_{t}-\epsilon_{t+1})^{+}$.   Can we write $E[P_{t+1} \mid Y_{t}]= u\cdot \int_{-\infty}^{(A_{m}-a - \rho Y_{t})}(A_{m}-a-\rho Y_{t}-\epsilon_{t+1})\phi(\epsilon_{t+1})d\epsilon_{t+1}$ ? Children’s books, one in space setting, the other in a jungle When I was little I remember getting these two books from the library.  Both books were paperback and flimsy.  The pages were wide.  Each book had full page illustrations with puzzles and mazes in them, but it was like you were immersed in this setting completely.  They were also like choose-your-own adventure books, but only in the way that you go through the book in a different order based on your results in the puzzles/games.    For example, there is a bridge to cross, or you can take this path.  If you choose to cross the bridge you go to page x.  On page x you have to pretend to cross the bridge but some planks are broken.  You can't skip more than a certain number of planks, but you also cannot "step" on a broken plank.  One book was in a jungle setting with rope bridges, alligators, quick sand, etc. The other was in space, with all kinds of aliens and weird stuff.  I have faint memories of these books but I REALLY want to track them down because I remember loving them.