Set theory proof problem about bounds
Proof that $\inf A = -\sup(-A)$
>Prove that $\frac d {dx} x^n=nx^{n-1}$ for all $n \in \mathbb R$. Prove that $\frac d {dx} x^n=nx^{n-1}$ for all $n \in \mathbb R$.   I saw some proof of $\frac d {dx} x^n=nx^{n-1}$ using binomial theorem, which is only available for $n \in\mathbb N$. Do anyone have the proof of $\frac d {dx} x^n=nx^{n-1}$ for all real $n$? Thank you. Heisenberg famously derived his uncertainty principle by considering the disturbance that a measurement would have on a small enough system.  Of course in the mathematical formalism of Quantum Mechanics the relationship is derived from more basic principles.  How does String Theory account for it? Heuristically, and mathematically? Using the Fitch system, how do I prove $((p \implies q) \implies p)\implies p$? Using the Fitch system, how do I prove $((p \implies q) \implies p)\implies p$? I started with the hypotheses $(p \implies q) \implies p$ and $\sim p$. However, from these hypotheses I did not get the desired contradiction to solve the question.  Thanks sb45 Which serial protocol or encoding scheme does this scoreboard use? I am trying to decode a signal stream coming from a Nevco scoreboard controller:    Packets like this are sent every ~2.7 ms.  I suspect it is some sort of UART signal because it is on the RX line of a ATMEGA48, but I cannot identify the finer protocol details like parity and stop bits. All the combinations I have tried returns junk data, so it may be another encoding scheme.  What's unusual is that the same signal is also sent to the MOSI pin of the microcontroller, but there is no clock signal anywhere.  A .vcd file is .   The shortest pulses (both high and low) are always 8us in length.    This is what two packets look like. Anyone got ideas? In $Sets$, the initial object is $0 = \emptyset$. We have $C \times \emptyset = \emptyset$ for any set $C$. As there are no maps to the empty set, we don't get much of the universal property of the product.  I am now wondering what is special about arbitrary categories $\mathscr{C}$ with $C \times 0 \cong 0$ for all objects $C$ of $\mathscr{C}$, where $0$ is the initial object of $\mathscr{C}$.   From Sheaves in Geometry and Logic by MacLane (p.194), in any topos $\mathscr{C}$, any arrow $k:C \to 0$ is an isomorphism. The universal property of the product would then also not yield very much.   Are there any categories of higher interest? Why are date-based hats awarded too early? I just got the  hat which states "post or vote on December 21".  Does SE have a time machine now? tar and its key letters: is it a bug or feature? I am wondering to ask the difference of these two commands (i.e. only the order of their options are different):   tar -zxvf foo.tar.gz tar -zfxv foo.tar.gz   The first one ran perfectly but the second one said:  tar: You must specify one of the `-Acdtrux' or `--test-label'  options Try `tar --help' or `tar --usage' for more information.   And tar with --test-label and -zfxv said :  tar (child): xv: Cannot open: No such file or directory tar (child): Error is not recoverable: exiting now tar: Child returned status 2 tar: Error is not recoverable: exiting now   Then I looked at tar manual and realised that all the example there are using switch -f in the end!!  AFAICT there is no need for this restriction, or is there?! because in my view switches should be order free. How to connect RDS from a different account with allowed IP in AWS? How can I edit two materials simultaneously? Whenever I open a new shading window, whatever I do in one is just mirrored on the other. If I click a node in one window, the same node and material get selected in the other. I can't edit two shaders at once. How can I edit two shaders simultaneously? Why does light in a room not form constructive and destructive interference patterns? Punctuation in an indirect quotation How to prove $B-A \succeq 0 \Leftrightarrow$ ellipsoid $x^TBx \leq 1$ contains $x^TAx \leq 1$? Assume $A \in \textbf{S}^n_{++}$, an ellipsoid centered at the origin given by $$\mathcal{E}_A = \{x\mid x^TA^{-1}x \leq 1\}$$ Then we have $\mathcal{E}_A \subseteq \mathcal{E}_B $ if and only if $B-A \succeq 0$. This is the proposition in the Boyd &amp; Vandenberghe's  (pages 45-46). The authors gave only the result, without proof. How to prove it? Yes, I'm really sure I want to delete that post Attempting to delete questions on the mobile site on iOS causes two notification messages asking me to confirm the deletion, which it shouldn't do.  I haven't really looked into it, but it may just be that the handlers from full.js and mobile.js are both being triggered, as there seems to be the flash of the error dialog indicating something went wrong on the supposed second AJAX request when the response from the first triggers a page reload.  You can see this happening by following the method calls; the mobile view first calls   StackExchange.question.init() -&gt; StackExchange.question.fullInit()   -&gt; StackExchange.vote.init()     -&gt; u()    ...where u() sets a delegated click handler on  $("div.post-menu *[id^='delete-post-'], table.flagged-posts *[id^='delete-post-']")   ...that calls .preventDefault() on the event, but doesn't block other handlers from firing.  It then later calls StackExchange.mobile.init(), which in turn calls the method J() that sets up a similar handler directly on $(".delete-post").  Since both handlers get triggered, there's two confirmation dialogs, and two attempts to delete the post. The extra handler in mobile.js seems unnecessary, so just removing it would appear to be the easy fix to the problem. I'm currently using an entry-level DSLR with 18-55mm lens and having great trouble trying to creating a blurred background effect in my photos. Now matter what aperture or shutter speed I choose, I'm not able to get it.  There's a general question about , but what specific things can I do to get this effect with an entry-level lens? I messed up my system earlier, I was greeted with a black screen, when booting in to Ubuntu. When I started up my laptop, I selected the recovery option from the grub menu, and chose fallback at root terminal. I saw that I was able to use the add user command, with it, I probably could use to create a privileged user on my machine.   Isn't that a security issue?  One could have stolen my laptop and at startup chose recovery and add another user, I'm fudged then. Including my data.  Come to think of it, even if you somehow remove that entry, one could boot from a live-CD, get a chroot up and running and then add another user, with the right privileges that allows it to see all my data.   If I set the BIOS to boot at my HD only, no USB, CD/DVD, Network startup, and set a BIOS password, it still wouldn't matter, because you'd still have that grub recovery startup entry.  I am fairly certain that someone from China, Russia can't hack my Ubuntu Trusty Tahr, from the network, because it's secure like that. But, if one has physical access to my - your - machine, then, well, that's why I'm asking this question. How can I secure my machine so that hacking through physical access is not possible?    Bug Report: After S.H.I.E.L.D confiscates Jane's work in Thor, the researchers discuss what they'll do next.  Erik mentions two colleagues. The first is reasonably implied to be Bruce Banner, but who is the second colleague he mentions? Transcribed from the film:  Jane Who are these people? Erik I knew this scientist... a pioneer in gamma radiation. S.H.I.E.L.D showed up and... umm. He wasn't heard from again. Jane They're not going to do that to us. I'm going to get everything back. Erik mumbles Please. Let me contact one of my colleagues. He's had some dealings with these people before. I'll email him and maybe he can help. Darcy They took your laptop, too. What is the difference between building from source and using an install package? Reading through my textbook I came across the following problem and I am looking for some help solving it.  Prove that,  $$\binom{3n}{r}=\binom{3n-1}{r} + \binom{3n-1}{r-1}$$  In a previous problem I solved the following but I wasn't sure how to use it to prove the question above. Prove,  $$\binom{n}{r}=\binom{n}{n-r}$$  Solution:  left Side,  $$\binom{n}{r}=\frac{n!}{(n-r)!r!}$$  right side,  $$\binom{n}{n-r}=\frac{n!}{(n-(n-r))!(n-r)!} = \frac{n!}{(n-r)!r!}$$  Therefore completing the proof,  but how do I solve my first question? thanks! Turn off the link hover statusbar in Google Chrome I want to turn off the feature where whenever I hover over a hyperlink and in the bottom lefthand corner it shows a light blue hover tooltip about where that link goes.   Why do I want to turn it off? It's annoying to me. I am distracted as I work. None of the other browsers do it. Besides, it serves little purpose because it can be spoofed with the onmousedown event on a link. Take for instance ads on Facebook -- those spoof.   I really don't care where a given link goes and wish I could turn that feature off, even if I have to install a Google Chrome Extension.   So, is there a fix to turn the annoying link hover tooltip feature off? How to protect SHTML pages from crawlers/spiders/scrapers? I have A LOT of SHTML pages I want to protect from crawlers, spiders &amp; scrapers.   I understand the limitations of SSIs. An implementation of the following can be suggested in conjunction with any technology/technologies you wish:     The idea is that if you request too many pages too fast you're added to a blacklist for 24 hrs and shown a captcha instead of content, upon every page you request. If you enter the captcha correctly you've removed from the blacklist.   There is a whitelist so GoogleBot, etc. will never get blocked.   Which is the best/easiest way to implement this idea?  Server = IIS    Cleaning out the old tuples from a DB every 24 hrs is easily done so no need to explain that.