Boolean Function question
Not every boolean function is constructed from $\wedge$ (and) and $\vee$ (or)
I'm an elected diamond moderator on one site, and a user of several others. I think it would be extremely helpful for our flagging process if users had the option to enter freeform text explaining their flag on types of flags other than the "in need of moderator attention" flag.  An example of when this is relevant: Sometimes posts have had rude/abusive flags, but the rude/abusive part might be difficult to notice, or it might be a single word or sentence buried somewhere in the middle of the post. Users have to choose between "flag it as rude with no explanation" or "flag it as misc and explain what's going on". Often they'll pick the first because of course it seems like what they should do (e.g. they know the system has special handling attached to rude/abusive flags) but it means they can't explain what's going on to us, so we have to reach out directly in private to find out.  Sometimes it also helps to be able to explain why a comment is no longer needed, because past experience tells me the "no longer needed" flag on its own might be declined on a particular site -- so I pick the custom reason instead of "no longer needed" just so I can provide explanation.  I'd prefer users be able to understand they can pick the appropriate reason and be able to attach any explanation they deem necessary to further direct moderators to the issue. If not for all kinds of flags, then at least for spam or rude/abusive flags. How to apply multiple styles in WPF In WPF, how would I apply multiple styles to a FrameworkElement?  For instance, I have a control which already has a style.  I also have a separate style which I would like to add to it without blowing away the first one.  The styles have different TargetTypes, so I can't just extend one with the other. Electric monopoles do exist, but why cannot magnetic monopoles exist?    The question is closed because I need to clarify it, but I don't know how I could ask it another way. However, I've recieved many answers that were appropiate and added something to my knowledge, so I consider it answered. A certain physics problem I have been working on has turned into a math problem. Particularly, I want to find the solutions of some equation of the form   $$f(x)=ae^{bx}+cx+d = 0$$  where $a, b, c,$ and $d$ are constant, real numbers that come from the physics problem and $x$ will be a real number. I do not know how to find the solutions of an equation of this form. In the special case where $c=0$, the solution is simply  $$x = \frac{\ln(\frac{-d}{a})}{b}$$  but for other values of $c$, I am stumped. If there is no exact form for the roots, is there a relatively simple way to estimate the roots? Which year does Spider-Man: Homecoming take place in? How to concatenate a std::string and an int I thought this would be really simple, but it's presenting some difficulties. If I have std::string name = &quot;John&quot;; int age = 21;  How do I combine them to get a single string &quot;John21&quot;? You can start fire by focusing the sunlight using the magnifying glass.   I searched the web whether you can do the same using moonlight. And found  and  - the first two in Google search results.  What I found is the thermodynamics argument: you cannot heat anything to a higher temperature using black body radiation than the black body itself, and Moon isn't hot enough.  It may be true, but my gut feelings protest... The larger your aperture is, the more light you collect, also you have better focus because the airy disk is smaller. So if you have a really huge lens with a really short focus (to keep Moon's picture small), or in the extreme case you build a Dyson-sphere around the Moon (letting a small hole to the let the sunlight enter), and focusing all reflected light into a point it should be more than enough to ingnite a piece of paper isn't it?   I'm confused. So can you start fires using the Moon? Why did the descriptive "Orientals" shift into a pejorative? Is hiding the ball allowed in football? Imagine all our football team except one player form a tight circle, with our hands on the shoulders of the people next to us.  Inside the circle there is the eleventh person with the ball. The whole "construction" is moving slowly towards the opponent's goal. It is inevitable that we will score.  Is this illegal? If yes, according to what rule? How to remove all files and subdirectories in a directory WITHOUT deleting the directory in bash? Is there a command to remove all files and subdirectories in a directory without deleting the directory?  For example if I have directory dontDeleteMe with subdirectories 1, 2, 3 and each subdirectory has a few pictures in it, how can I remove the subdirectories 1, 2, and 3 and all the files in the them, without removing the parent directory dontDeleteMe? View vote count button on mobile On the mobile site, can there be a button directly below the vote up/down buttons for viewing vote counts? When I am using the mobile site and I want to view vote counts, I have to switch to the full site to do it which is very inconvenient.   It could be a small link that says "vote counts," because I don't know what the button would be (the letter "V?" a "+/-" thing? I have no idea).  Related:  , but I am requesting an additional button below the vote buttons. That way you won't accidentally click a vote button. If $|\lbrace g \in G: \pi (g)=g^{-1} \rbrace|>\frac{3|G|}{4}$, then $G$ is an abelian group. Assume that $\pi$ is an automorphism of a finite group $G$. Let $S$ denote the set $\lbrace g \in G: \pi (g)=g^{-1} \rbrace$. Show that if $|S|&gt;\frac{3|G|}{4}$, then $G$ is an abelian group.    Anyone has any idea on how to solve this ? I have no idea how to start. Pacman option to assume "yes" to every question? Antiderivative of periodic function I practise before the real analysis exam and I got stuck on few excercises.  "if $F$ is an antiderivative of a continuous periodic function and $F$ is bounded, then $F$ is periodic"  $~\to~$ one should give a proof or counterexample.  (I'm convinced it's true, but how to prove it?) How To Reduce Noise In Cycles Render Integral of sinc function multiplied by Gaussian I am wondering whether the following integral $$\int_{-\infty}^{\infty}  \frac{\exp( - a x^2 ) \sin( bx )}{x} \,\mathrm{d}x$$ exists in closed form. I would like to use it for numerical calculation and find an efficient way to evaluate it. If analytical form does not exist, I really appreciate any alternative means for evaluating the integral. One method would be numerical quadrature including Gaussian quadrature, but it may be inefficient when the parameters $a$ and $b$ are very different in scale.  EDIT : In view of  discussion, we have decided to add OP's self-answer to the end of the question, for it does not qualify as an answer yet contains vital details. The copying is unabridged. Thanks very much for your comments, and the following result was obtained including the case for $x_0 \ne 0$: $$ \int_{-\infty}^{\infty} dx \exp[-a(x-x_0)^2] \frac{ \sin(bx) }{ x } = \pi \exp(-a x_0^2) \mathrm{Re}\left(\mathrm{erf}\left[\frac{b+2iax_0}{2\sqrt{a}}\right] - \mathrm{erf}\left[\frac{2iax_0}{2\sqrt{a}}\right]\right) $$ where $a\gt0, b, x_0$ are assumed to be all real. (note: coefficients etc may be still wrong...) This integral appears in a type of electronic structure calculation based on a grid representation (sinc-function basis). I believe the above result should be definitely useful. Thanks much!! --jaian English equivalent for the Persian idiom "send someone out in pursuit of black chickpeas" "To send somebody out in pursuit of some black " is a Persian idiom that implies 'to make or ask someone to run an errand so that you be able to have/ buy some time in order to deal with your private affairs in their absence', like having or continuing a (secret) conversation with someone else.   According to its etymology, since 'black garbanzos/chickpeas' were rarely available in the markets, when someone wanted to buy them, they had to search many stores for a long time, and most of time they couldn't find them at all.   Today, its meaning has changed somehow, and it doesn't necessarily mean that we ask someone to run an impossible errand, but it means we use that errand as an excuse to send them away for a while just to have time to do something in their absence.   Example scenario:   Suppose your classmate has come to your house and you are talking about some girly issues in the sitting room. Now you want to tell her a big secret about one of your classmates, but not in the presence of your younger sister who is sitting there too. So you ask her to go to the market and buy some snacks for you. In this way, you will have some time to continue your conversation until she is back. So you might say to your friend:     Well, I sent my sister out in pursuit of some black chickpeas.  I'm sure that she won't be back for another 15 minutes, so we can talk more freely now."   And your sister, before leaving the house, might sarcastically say to you:     Okay, but I know this is just an excuse, and you are just sending me in pursuit of some black chickpeas!!! so don't think that you can fool me!   Is there any idiom, expression or proverb that conveys the same meaning or connotation in English (i.e. to make or ask someone to do an errand just in order to buy time so that you can deal with some of your private affairs)?   I have found to be/go on  but I'm not sure if I can use it or not. If G is a group of order n=35, then it is cyclic I've been asked to prove this.   In class we proved this when $n=15$, but our approached seemed unnecessarily complicated to me. We invoked Sylow's theorems, normalizers, etc. I've looked online and found other examples of this approach.    I wonder if it is actually unnecessary, or if there is something wrong with the following proof:   If $|G|=35=5\cdot7$ , then by Cauchy's theorem, there exist $x,y \in G$ such that $o(x)=5$, $o(y)=7$. The order of the product $xy$ is then $\text{lcm}(5,7)=35$. Since we've found an element of $G$ of order 35, we conclude that $G$ is cyclic.   Thanks. Browsing Questions shows the same questions on different pages I have just been looking through the questions to answer those that I can and as I go to the next page (using the Next button at the bottom) I am presented with the same questions.  I imagine this is not a flaw and in fact questions are coming in fast filling up a page pushing my current page to the next one.  An idea to solve would be to set a variable for starting id when I start paging and only clicking an update button will reset the starting id. For $n$ a positive integer, let us write $\zeta_n = e^\frac{2 \pi i}{n}$, a primitive $n$th root of unity. It is clear that, if $m$ divides $n$, then we have an inclusion of cyclotomic fields $$ \mathbb{Q}(\zeta_m) \subseteq \mathbb{Q}(\zeta_n).$$ On the other hand, these inclusions are not always strict. For example, since $\zeta_3  = \frac{1+i \sqrt{3}}{2}$ and $\zeta_6 = \frac{-1 + i \sqrt{3}}{2}$, we have $$\mathbb{Q}(\zeta_3) = \mathbb{Q}(\zeta_6) = \mathbb{Q}(i\sqrt{3}).$$ Does this sort of thing happen infinitely often, or are there just some coincidences among small numbers? If this continues, is there some way to know when a particular inclusion of cyclotomic fields is strict?