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Why two inverters cascaded in multiplexer control signal terminals? Why cascaded inverters are used in the multiplexer when it is possible to realize the circuit using a single inverter. So I had an exam today and one of the questions were: Give an example of a function $f$ which is nowhere continuous but $|f|$ should be continuous at all points. At first I had no idea how to do it then I came up with this, even though I know it's wrong:  $$f(x)= \begin {cases} \ \sqrt{x}&amp;\text{if }x &lt;0\\ \sqrt{-x}\ &amp;\text{if }x &gt; 0\\ \end {cases}$$  Like I said I know it's wrong..but can someone give me an example of this? Thanks. Penalized methods for categorical data: combining levels in a factor Should I assign dwellers to the storage room? I've played Fallout Shelter for a while now and normally use dwellers with appropriate SPECIAL stats for each room (Strength in the reactor, Perception in the water supply, etc).  However, I have a question regarding Endurance. The only room which uses this is the storage room. It doesn't produce anything; it has no cooldown; it just does nothing...  Even if I just build and upgrade the warehouse, my stock capacity is increased. Is there any good reason for me to waste my dwellers by assigning them to storage rooms? How to grep a specific line _and_ the first line of a file? Assuming a simple grep such as:  $ psa aux | grep someApp 1000     11634 51.2  0.1  32824  9112 pts/1    SN+  13:24   7:49 someApp   This provides much information, but as the first line of the ps command is missing there is no context for the info. I would prefer that the first line of ps be shown as well:  $ psa aux | someMagic someApp USER       PID %CPU %MEM    VSZ   RSS TTY      STAT START   TIME COMMAND 1000     11634 51.2  0.1  32824  9112 pts/1    SN+  13:24   7:49 someApp   Of course, I could add a regex to grep specifically for ps:  $ ps aux | grep -E "COMMAND|someApp"   However, I would prefer a more general solution as there are other cases in which I would like to have the first line as well.  Seems like this would be a good use case for . Mention a specific potential adviser and project in the PhD statement of purpose I am writing my statement of purpose for my Ph.D. and I want to work in a specific field with a specific professor in the department that I am going to apply for and I have a very good experience in this field. I sent an email for that professor asking if he/she is accepting Ph.D. students and he/she did not respond. I built my whole statement of purpose based on my previous achievements and future plan for that specific area. I also mentioned that I am willing to work with that professor. My statement of purpose tells that I am very passionate about this field and I am only willing to work with that professor (if there were more than one professor working on the same area, I would mention all of them). if that professor doesn't accept new Ph.D. students, will I risk my admission to that school?   Will the committee response be something like this:  "the applicant really knows what he/she is going to do and there is a specific future plan and he/she also got a good experience in this field, but he/she sounds very strict to his plan and doesn't show any flexibility...REJECTED"? What Do the Gringotts Goblins Do With Muggle Money? ‘But you’re Muggles!’ said Mr Weasley delightedly. ‘We must have a drink! What’s that you’ve got there? Oh, you’re changing Muggle money. Molly, look!’ He pointed excitedly at the ten-pound notes in Mr Granger’s hand.   Chamber of Secrets - page 47 - UK Hardcover - chapter 4, Flourish and Blotts   So Gringotts exchanges Muggle money for Wizarding money. What do the Goblins do with the Muggle money after the exchange has been made? Is it possible they keep a stash of Muggle money on hand for reverse exchange purposes, or do they surreptitiously feed the money back into the Muggle economy?  What do Goblins do with Muggle money? Why is Euclid's proof on the infinitude of primes considered a proof? I've expressed Euclid's proof on the infinitude of primes on Mathematica:  f[x_] := Product[Prime[n], {n, 1, x}] + 1 TableForm[Table[{f[x], PrimeQ[f[x]]}, {x, 1, 20}]]   Which results in:  $\begin{array}{ll}  3 &amp; \text{True} \\  7 &amp; \text{True} \\  31 &amp; \text{True} \\  211 &amp; \text{True} \\  2311 &amp; \text{True} \\  30031 &amp; \text{False} \\  510511 &amp; \text{False} \\  9699691 &amp; \text{False} \\  223092871 &amp; \text{False} \\  6469693231 &amp; \text{False} \\  200560490131 &amp; \text{True} \\  7420738134811 &amp; \text{False} \\  304250263527211 &amp; \text{False} \\  13082761331670031 &amp; \text{False} \\  614889782588491411 &amp; \text{False} \\  32589158477190044731 &amp; \text{False} \\  1922760350154212639071 &amp; \text{False} \\  117288381359406970983271 &amp; \text{False} \\  7858321551080267055879091 &amp; \text{False} \\  557940830126698960967415391 &amp; \text{False} \\ \end{array}$  The proof flaws for all those values, how is it considered a proof then? I guess that there might be infinite prime numbers according to the proof, but what is the guarantee that at some point it won't fail indefinitely? If I have a legendary creature out, and I make a copy of it (such as by enchanting it with  or making a ), do I get enters-the-battlefield effects from the creature before I have to sacrifice it to the legend rule? Let $\mathbf{FinVec}$ denote the category of finite-dimensional real vector spaces and linear maps, let $F\colon\mathbf{FinVec}\rightarrow\mathbf{FinVec}$ be a functor. For each pair of objects $V,W$, the Hom-set $\operatorname{Hom}(V,W)$ is itself a finite-dimensional vector space and we equip it with the unique structure of a topological space/of a smooth manifold that is compatible with its vector space structure. The composition maps are then continuous/smooth. In this way, we view $\mathbf{FinVec}$ as enriched over the category $\mathbf{Top}$/$\mathbf{Diff}$ respectively. It then makes sense to ask whether the functor $F$ is enriched over $\mathbf{Top}$/$\mathbf{Diff}$. That is just to ask whether the induced map $F\colon\operatorname{Hom}(V,W)\rightarrow\operatorname{Hom}(FV,FW)$ is continuous/smooth for all $V,W$. (Irrelevant to the rest of the question, but the reason this is a useful notion is that it is the natural hypothesis assuring that the functor induces another functor on the category of topological/smooth vector bundles, applying the original functor fiberwise.) In the case $V=W$, by functoriality, we can restrict $F$ to a group homomorphism $\operatorname{GL}(V)\rightarrow\operatorname{GL}(V)$. If the functor is continuous/smooth, so is this map. Note that $\operatorname{GL}(V)$ is a Lie group. It is a general result that a measurable group homomorphism between Lie groups is automatically smooth. This begs the following questions:  What's an argument for the existence of functors that aren't continuous? Can we construct one? Given the above, it either happens that the restrictions to $\operatorname{GL}(V)\rightarrow\operatorname{GL}(V)$ are continuous and continuity fails elsewhere, or the induced maps will be non-measurable, so that the construction will necessarily have to involve choice to some extent. Are there continuous functors that aren't smooth? Given the above, a continuous functor will have smooth restrictions to $\operatorname{GL}(V)\rightarrow\operatorname{GL}(V)$, but smoothness may fail elsewhere. The category Set has the property that any morphism f can be decomposed into an epimorphism $e$ and a monomorphism $m$ with $f$ = $m\circ e$. (The intermediary set object is the image of $f$.)  Is there a name for this construction?  Is there a name for this property of categories?  In Set, the intermediary object is unique up to isomorphism. Is this the case for all categories with the property? In the radio series, how did Zaphod get into the artificial universe? In the book The Restaurant at the End of the Universe, it's very clearly explained how Zaphod gets into the simulated universe; he's in it from the moment he enters Zarniwop's office, and a big deal is made of him leaving via the window rather than the door so that he stays in it.  In the radio series, however, this doesn't happen. At the end of Fit the Seventh, Zaphod asks "How do we get to this universe, then?" and Roosta replies "I'm afraid it looks like we don't...", pointing out that the building is splitting down the middle, preventing them reaching the office.  Later, in Fit the Twelfth, Zaphod finally meets Zarniwoop in the lounge of the slightly-delayed spaceliner and reacts with astonishment at learning he's been in the artificial universe all this time. "How did we get into it? I mean like where, when?"  Zarniwoop only replies enigmatically: "You didn't notice? Well, I'll let you work it out for yourself."  So, how did he get in? As noted, he never actually entered Zarniwoop's office. My only thought is that it happened during the "body debit" incident with the disco, but there's no particular indication that that should be the case. Is there any other possibility? Showing $\sup \{ \sin n \mid n\in \mathbb N \} =1$ Why we won't extract heat from air temperature? Do I need a flight ticket or a flight reservation for my UK visit visa application Am compiling documents for my visa application and I read that I need to present go and return flight tickets to prove that I will return at the end of my visit. Then a while ago I got to know that I can just get a flight reservation with out buying the flight tickets because the visa application can be refused and you have already bought the tickets.  So l request to know which of the two , having a flight ticket or flight reservation is more compelling to the visa officer and more convenient me because I would buy a flight ticket and my application is refused Which manga chapter correspond to the end of episode 12 of season 2 of the anime? If every ideal is principal show the same is true for the image of a homomorphism. Does installing many programs really slow down Windows? How to prove: if $a,b \in \mathbb N$, then $a^{1/b}$ is an integer or an irrational number? jQuery doesn't work after content is loaded via AJAX