DIfference between "at home" & "home"
"Stay home" or "stay at home" – which is correct and why?
How to prove that $P \rightarrow Q$ is equivalent with $\neg P \lor Q $? In my book about Logic, which is called 'Language, Proof and Logic', by the way, there is explained that the conditional $ P \rightarrow Q $ is equivalent with $\neg P \lor Q$.   that gives as answer: 'just compare the truth tables and you can see that they are equivalent'. However, I would like to know how to prove this using formal propositional logic. This should be possible, right? TL2018 (MacTeX) and TikZ calligraphy library I am a new user of Ubuntu 12.04.   I installed Ubuntu and Windows XP, and I want to use both operating systems. When I restart or boot my PC the GRUB menu does not appear.  How can I solve this problem? A better serial voting trap Currently we capture fanatic, voting ring or rage-type activities, by measuring the number of up-votes or down-votes between the same two users in a certain (relatively short) time period. When this is detected by the system, the votes are reversed, and I've never bothered to investigate what else happens after that - though I'm sure some form of wrist-slapping ensues.  While it is intended to be well-guarded for obvious reasons, the algorithm used is not exactly a secret, and I am sure there are plenty of examples where people are dangling (maybe intentionally, or maybe by dumb luck) just on the safe side of getting caught.  Personally I believe there are active, intentional, long-term vendettas in operation right now where users either:   blindly up-vote every answer from a user; or, down-vote a particular user once or twice a day, regardless of topic or quality.   Now, maybe the first one is valid - if I'm operating in the same tag-sphere as some smart person, it may just be that I'm constantly enthralled with their answers, and I'm not even noticing who they're written by. (Though, even Skeet isn't perfect, and I haven't up-voted every answer I've seen him post.)  But still, can we do a better job of catching - or at least noticing - this activity over the long term in addition to the short term?  The behavior is the same, and shouldn't be tolerated, regardless of the time frame - vengeful rage serial voting, blind staker crush serial voting, and painstakingly patient serial voting are all undesirable behaviors IMHO.  I'm not saying it has to do anything automatically, as I'm well aware of the possibility of false positives. And I'm not even sure what to propose as a time frame or a vote threshold.      But if a large percentage of my up-votes or down-votes are against a particular user, and especially if on a majority of those posts I am the only one voting in that direction, I think it should at least be raised as some kind of flag to moderators.   This would work much like how other suspicious activity is highlighted (but also not dealt with automatically). If a pattern becomes obvious and persists, then they can intervene in some way (or not, but at least they were provided with the information - just like they can reject and ignore manual flags). Today, all I can do is e-mail team@stackoverflow.com, and unless I know exactly which person(s) are involved, there is not much they can do about it.  And you're thinking, "Meh, let some hater down-vote you once or twice a day. Unless all you're posting is utter crap, it should be statistically insignificant." You're right. But reputation is not the point, and is not the motivation behind my request. What I'd like to see curtailed is the tendency for folks to down-vote a question or answer because of who posted it, not because of what it said or didn't say. When a down-vote is personal it can leave a person second-guessing the quality of their post. Which isn't necessarily a bad thing, except when it happens repeatedly.  Sorry for the essay but I wanted to try to anticipate and cover all the objections and hole-poking. Grep equivalent for Windows 7? Is there a command prompt  equivalent for Windows 7? That is, I want to filter out the results of a command:  Bash use:  ls | grep root  What would it be from a Windows command prompt? Problem working fontspec, babel and datetime with cyrillic I have discovered that fontspec with russian babel settings doesn't work with datetime package. I tried to compile such code:  \documentclass[a4paper,11pt]{article} \usepackage[T2A]{fontenc} \usepackage[utf8]{inputenc} \usepackage[main=russian,english]{babel} \usepackage{datetime} \usepackage[cm-default]{fontspec}  \begin{document} \today \end{document}   with a xelatex and got an error: Command \cyri unavailable in encoding EU1.  Is there any way to handle it?  UPDATE: I really bungle with charset in XeLateX. Thanks to egreg. I have removed fontenc and inputenc and used some cyrillic font, so my MWE now looks this way:  \documentclass[a4paper,11pt]{article} \usepackage[main=russian,english]{babel} \usepackage{fontspec} \setromanfont{PT Sans} \usepackage{datetime}  \begin{document} \today \end{document}   and I've got another error, if I used russian as a main language in babel: Undefined control sequence. \\monthnamerussian ...cyrm \cyra \cyrya \or \cyri  \cyryu \cyrn \cyrya \or \c...  Is it still because of XeLaTeX? Solving linear congruences by hand: modular fractions and inverses When I am faced with a simple linear congruence such as  $$9x \equiv 7 \pmod{13}$$ and I am working without any calculating aid handy, I tend to do something like the following:  "Notice" that adding $13$ on the right and subtracting $13x$ on the left gives: $$-4x \equiv 20 \pmod{13}$$  so that $$x \equiv -5 \equiv 8 \pmod{13}.$$  Clearly this process works and is easy to justify (apart from not having an algorithm for "noticing"), but my question is this: I have a vague recollection of reading somewhere this sort of process was the preferred method of C. F. Gauss, but I cannot find any evidence for this now, so does anyone know anything about this, or could provide a reference? (Or have I just imagined it all?)  I would also be interested to hear if anyone else does anything similar. How to get reflections from 3D to show up on real images Andrew Price released a nice tutorial on how to nicely put 3D objects on top of a real image:   I have seen images, such as this:  which have 3D elements over images and have actual reflections from the object showing up on the wet asphalt.  How can this effect be achieved using render passes (or any other way)?  Here is another example (from ): How do I boot into recovery mode? This is really two questions.   Is there a definition of the principal $n$th root of a complex number? I can't find it anywhere. Presumably, the usual definition is $[r\exp(i\theta)]^{1/n} = r^{1/n}\exp(i\theta/n)$ for $\theta \in [0,2\pi)$, but I have yet to see this anywhere. Is this because it has bad properties? For instance, according to this definition is it true that for all complex $z$ it holds that $(z^{1/a})^b = (z^b)^{1/a}$? the slash (/) after a directory name on shell commands I have a little question here.  If I have two files, say filea and fileb, mv filea fileb would   delete fileb rename filea to fileb   Then if I have two directories, say dira and dirb, mv dira dirb would   move dira into dirb (it will become dirb/dira)   Noting that in both cases there are no notice or message, then this is pretty inconsistent to me. I think mv dira dirb should just overwrite dirb with the contents of dira (or merge the two directories under a directory named dirb).  I remember reading somewhere that a directory name with a slash (like dira/) is treated like a directory, and name with no slash (like dira) is treated like a file (to certain extents, of course). Anyway now I want to make the shell (zsh and possibly bash) respect my notation of a directory by using a slash. Is there a terminal option which enable me to enforce that?  To clarify, here is my desired behaviour:   mv dira dirb results in dirb being overwritten with the contents of dira mv dira dirb/ results in dira being moved into dirb (in dirb/dira)   Has anyone thought the same way as me? Or am I just weird? Let $x\in\mathbb{Q}$ with $x&gt;0$.  Prove that we can find $n\in\mathbb{N}^*$ and distinct $a_1,...,a_n \in \mathbb{N}^*$ such that $$x=\sum_{k=1}^n{\frac{1}{a_k}}$$ Stealing and reattaching a running process It's sunday and I left an instance of R running in the office. The script finished running, and I would like to save my workspace without going to the office. I can ssh to the computer, but I have not run the program into screen.  Is there a way, over ssh, to disown the process (I don't have access to jobsid of the terminal the program is running on!) and reattach it with say, reptyr? Reptyr and reptyr -T don't work without previously disowning the process  28845 3diag       20   0  139.3m  10.4m   0.0  0.1   0:12.06 S  `- urxvtd                                                                                        5327 3diag       20   0   16.4m   4.6m   0.0  0.1   0:00.02 S      `- bash                                                                                      5335 3diag       20   0 3499.3m 3.244g   0.0 43.3 171:57.89 S          `- R    The unique part of my question is that I explicitly require to detach a program without having access to the terminal in which it was created. I am trying to understand the permission usage for the cp command in unix.   What are the permissions required for copying a file into a directory What are the permissions required for copying out of a directory?   I'm assuming for my first question, the directory will need execute permissions to perform the operation and the file will need read permissions to read the binary of the file and then the output file will need write permissions.  For the second the directory itself would need execute permissions to perform the copy operation and the file would need read permissions to extract the data from the file.  Can anyone tell me if my logic is flawed? How to overlay images I want to overlay one image with another using CSS. An example of this is the first image (the background if you like) will be a thumbnail link of a product, with the link opening a lightbox / popup showing a larger version of the image.   On top of this linked image I would like an image of a magnifying glass, to show people that the image can be clicked to enlarge it (apparently this isn't obvious without the magnifying glass). When using zero as a quantifier, is it correct to use the singular form on the object of the quantifier, or the plural form?  It sounds confusing when I put it that way, but what I mean is: Which is correct?   Your password expires in 0 days. Your password expires in 0 day.   Essentially I suppose I'm asking, does "singular" mean "one" or "the opposite of more than one", as zero is not "plural" in the traditional "more than one" sense?  I'm pretty sure "days" sounds correct, but I can't be sure. Since the last update (v1.0.85), I am getting all the question pages blank on the Android app. For those who are confused with the image, I'm using RtL (Right-to-Left) layout.        If I scroll down (quite a bit) I can see the question. But even then I cannot see the question title, the related comments or answers.         App Version: 1.0.85 Device Manufacturer: LGE Device Model: LG-D620   OS Version: 5.0.2 (152541901f156) Digitally selectable filter I have asked a  but I did not get an answer, as it was too specific. So I will build on it but make it a bit more broad spectrum.  I would like to make a digitally selectable BPF using transistors to select a component in the filter; resulting in the center frequency being changed.  I have used FilterPro to generate quick second order filters and get a rough filter response.  For all the designs, a Multiple-feedback was selected:  In the first picture, a center frequency at 108KHz with a bandwidth @ -3dB of 103K to 113KHz is given.    For the second, a center frequency at 117.5KHz with a bandwidth @ -3dB of 112.8K to 122.2KHz is given.    The only difference between these two circuits is R3.  If I were to implement a circuit as shown below, would it work and what transistor would be recommended? I have selected the  as the Opamp in the circuit design as that is what was used for the .    For the transistors, I was thinking of using the MOSFETs (2N7002) used in the ST7590 design as well. Is there a formal proof for the superposition theorem? Is it true that  $$\int_a^b f(x) dx = \int_{f(a)}^{f(b)} f^{-1}(x) dx$$  Just making sure.   If not, how about:  $$\int_a^b f(x) dx = (f(b)-f(a))b - \int_{f(a)}^{f(b)}f^{-1}(x)dx$$   I'm having a hard time concentrating right now, and I'm trying to figure out how to get the area under a curve when the function is inverted.