Need help with spaces within math mode My code is written as follows:   \begin{equation} \forall x \in \mathbb{R}, \cos(x+2\pi)=\cos(x)  and  \sin(x+2\pi)=\sin(x) \end{equation}   But when I 'execute' it there is no space between cos(x) &amp; and so can you help me cope with this issue? Thanks in advance!  PS: if someone could teach me how to define a new environment so that   \begin{theorem}Equation\end{theorem}   produces immediately a mathbox with the equation inside of it?, I would greatly appreciate your help!
What commands are there for horizontal spacing? I know that \: in LaTeX produces a space when rendered.  Are there any alternatives, because my LaTeX renderer doesn't support \: (it renders it as text), and there is no help / FAQ that I can find.
Can someone help with an explanation and some references, in order to understand the nature of return pulses within a LiDAR system. Why do single-author community wiki answers (or questions?) link to revisions, rather than to the author's profile? Possible Duplicate:             Looking at  , I noticed something peculiar: Community Wiki posts* with only one author have not one, but two different links to the revisions page.    Is this entirely necessary? When clicking on the author's name in this case, I entirely would expect to be brought to the author's profile, rather than to the (potentially one-item-long) revisions list.   *(maybe only answers? couldn't find any good single-author CW question references) Will Windows 10's reset delete my Linux partitions? I want to use the "reset my PC" feature of Windows 10 to wipe the data on my Windows partition. But I have also a Linux system installed, which I want to keep.  Will Windows 10's full reset wipe all my disk, or just Windows's partitions ? Does an unplugged hard drive used for data archival deteriorate? If I were to archive data on a hard drive, unplug it, and set it on a (not dusty, temperature-controlled) shelf somewhere, would that drive deteriorate much?  How does the data retention of an unplugged hard drive compare to tapes? 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? Difference between shadowing and overriding in C#? Generics and inheritance: need a complex Map instance So this is something I just found out about that I find odd:   Opportunity attacks—trigger off of movement that uses the creatures movement (action, move action, bonus action, reaction.)  Sentinel—Whenever a creature within 5 ft of you makes an attack at a creature other than you.   Booming blade—If the target willingly moves.   Mental Prison spell—if it moves, attacks through it, or reaches any part of body out of it.   So the “willingly” is different than other rulings for how effects related to movement works. However what exactly fits willing?   Cause fear—nope magic is having them move. Dissonant whispers—same thing. They have to immediately move.   So the creature has to want to move completely of its own accord.   Crown of Madness—Humanoid creature you choose must make a wisdom saving throw. On a fail it is charmed by you. (Next is flavor text so skipping.) While charmed in this way the target must make an attack on a creature other than itself that you mentally choose on each of its turns before moving.  It can act normally if you choose no creature or if none are within reach.   Crown of madness isn't making them move but doesnt specifically say willing.  And if the ruling for willing is movement not caused by magic what about magic causing situations where you take damage if you don't move. Would a creature shimmering with evocation energy be spared because he moved out of the magical blender that is cloud of daggers? If no, then what is the go-to rule for willing movement that I should be using?  In short, what movement counts as willing, and where can it be found? Is moving out of obvious danger caused by magic, willing? Is moving away from an ally so you don't hit them, willing?   If there isn't any actual ruling on willing, that's fine; I'll go back to running it with my interpretation, so that it has same wording as AOO. If there is an actual 5e term or explanation for willing (that isn't Sage Advice), please let me know. I just don't see why it was necessary to put a term in the spell that isn't used anywhere else. I understand it could have been overlooked since it was in a splatbook or has just been ignored since most people probably assume that there is "Forced " and "Willing" movement, with the willing being defined as: any movement from an action, move action, bonus action, or reaction. In same way AOO work. $f(f(x)f(y))+f(x+y)=f(xy)$ Find all functions $f\colon \mathbb R\rightarrow \mathbb R$ such that for all reals $x$ and $y$:   $$f(f(x)f(y))+f(x+y)=f(xy).$$   It was six hours ago in IMO 2017 (problem 2).  I tried  the standard way: $x=0$, $x=y$, $x=1$,etc. but without any success.   Let $f(0)=a$. Hence, for all $x\in\mathbb R$ we have $f(a(f(x))+f(x)=a$. What is the rest?   Also we have $(x-1)(y-1)-1+x+y-1=xy-1$ and  $1-(1-x)(1-y)+1-x-y=1-xy$. polyglossia or bidi bug: [\d\.] gets inverted to [\.\d] After bidi was updated to v. 19.6 (2016-05-23), using at least one RTL language with polyglossia leads to number and following dot being inverted. Cf.  \documentclass{book} \usepackage{polyglossia} \setotherlanguage{hebrew} \begin{document} 6. \end{document}   with and without \setotherlanguage{hebrew}. Note that it is not necessary to actually use that language.  I do not know whether this is a bug in bidi or an incompatible change which polyglossia needs to adapt to. Because of this I have not opened an issue on polyglossia's github page. Sadly, bidi's contact information are outdated – the sourceforge referred to on CTAN is unused and the github page that is mentioned in the package's readme and PDF does not seem to exist any more either.  The problem might be something along the lines of     Does anybody have the previous version of bidi at hand?  TIA, Dario Custom entity datetime field in calendar Nothing shows up in the terminal when I type my password How to prove the set $S$ is closed under the binary operation $a*b=a+b+kab$? Let $k$ be any nonzero real number and let $S$ be all real numbers except ${-1\over k}$. Let $a,b \in S$. Define * on S by $a*b=a+b+kab$. Prove that $S$ is closed under *.  All I can think to do is set $a*b$ equal to ${-1\over k}$, but I don't know where to go from there. Is that the extent of the proof? Cancelled UK visa on entry "Needs cleaned" or "needs to be cleaned" How to imitate commercial fried chicken? I'm trying to develop my own "secret" fried chicken recipe, something that is similar to Popeye's or Church's. I'm a LONG ways from getting there. Here's what I did in my last experiment:   Chicken was marinated in buttermilk with Cajun spices and Tabasco sauce for 24 hours I used a deep fryer and canola oil. 175C. Prior to "dressing" the chicken, I let the excess buttermilk brine drip off the chicken. I then seasoned it with Cajun seasoning I made a batter using an egg, self rising flour, milk, baking powder I also tried using just plain flour for breading   Results:   Using the batter, the chicken came out looking burned and too smooth (didn't have that crunchy look). It burned too much that I couldn't finish cooking the chicken. Using the flour, the chicken came out with a thin crust, somewhat crispy, but also too burnt. Dipping the chicken in the batter AND then flour gave a bulky &amp; crunchy look. However, again, the result looked too burnt.   Modifications:  I then changed the the breading. I used bread flour and corn starch. I dipped the chicken in the batter again, then used the new breading. The bread flour was a bit lighter. And the corn starch was even more lighter. But they were still burn looking (just lighter shades). I also dropped the temperature to 160C.  Results:  Burnt chicken whose crust wasn't really sticking to the skin/meat.  Questions:  Any suggestions on what I can do to improve? I'm guessing something in the batter is causing the burnt coat. What flour should I use? What can I do to make the coat stick to the skin/meat? Any other suggestions on what to do? I am being flown out by a company for an on-site visit. Will I earn miles for this trip, or will the company? Disk Utility cannot erase USB drive (error -69877: couldn't open device) Disk Utility (macOS Catalina 10.15.2) fails to erase my USB drive (64GB key, currently formatted in exFAT).  /dev/disk2 (external, physical):    #:                       TYPE NAME                    SIZE       IDENTIFIER    0:      GUID_partition_scheme                        *64.1 GB    disk2    1:                        EFI EFI                     209.7 MB   disk2s1    2:       Microsoft Basic Data Lexar                   63.9 GB    disk2s2   I've tried a bunch of commands that I've found here and there in the terminal, but none of them worked, like:  ➜  ~ diskutil eraseVolume exFAT MyName /dev/disk2 Started erase on disk2 Unmounting disk Error: -69888: Couldn't unmount disk   ➜  ~ diskutil eraseVolume exFAT MyName /dev/disk2s2     Started erase on disk2s2 Lexar Unmounting disk Erasing Error: -69760: Unable to write to the last block of the device   ➜  ~ diskutil eraseDisk FAT32 MYSD MBRFormat /dev/disk2 Started erase on disk2 Unmounting disk Creating the partition map Error: -69877: Couldn't open device   ➜  ~ diskutil eraseDisk free EMPTY /dev/disk2 Started erase on disk2 Unmounting disk Creating the partition map Error: -69877: Couldn't open device   My USB drive appears as read only when i do CMD+I, I tried repairing the permissions but that failed too.  ➜  ~ diskutil resetUserPermissions /dev/disk2 `id -u` Ready to reset user permissions on disk2 Error encountered attempting to reset permissions for user 501 home directory on disk2: Permissions are not enabled on the disk (-69861)   Thx in advance for the help :) Where does the Mac store account pictures? I took a picture to add to my account on Mac.   Now I want to use it on messenger, but I can't find it. On the account config, it's too small. I need to find the original.   Where are the accout pictures stored on a Mac? "No root filesystem is defined" while installing Ubuntu I have just got a new laptop, It is running Windows 7 Home Premium.  I have shrunk the windows partition to give me some empty space and now I want to install ubuntu.  From Windows my partitions look like:   So now in the ubuntu live cd If I hit install and go manual the partitioner looks like   So I thought I could just select the empty space and be away laughing, but when I hit it I get