How could I ensure there is spacing between the number zero and my 'forall' sign in the following:   $$\alpha'(t) \not= 0 \forall t \in I$$ ?   Initially I tried to do   $\alpha'(t) \not = 0$ $\forall t \in I$,    which worked perfectly well, but now I wish to write the expression between double dollar signs for formatting purposes.  The problem is that 0 is almost too close to my 'forall' sign now.  I would appreciate any input.
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.
How are the Tenchi timelines related? Between Muyo, Universe, GXP, and Ryo-ohki, I'm thoroughly confused. Sometimes Tenchi series seem to be related (Ryo-ohki and Universe) and sometimes not (GXP). What are all the Tenchi series, are they related, and if they are, how? So far, I understand that I have to look at each Jordan block. How do I prove that for a Jordan block with a value of x, $J_n(x)$, the Jordan decomposition of $(J_n(x))^2$ is $J_n(x^2)$ ? I have a feeling this will lead me to the solution. Thanks for your time. Limit FTP access only to the /var/www with vsftpd I am running vsftpd as ftp server on my linux (rasbian), I log in to the machine as a root user.  I would like to be still locked to using only /var/www, how can I configure vsftpd conf to accomplish it? Does time stretch all the way back for infinity or was there a point when time appears to start in the universe?  I remember reading long ago somewhere that according to one theory time began shortly before the creation of the universe.  Does time have a starting point of note? How can I reconnect to a ssh session after a broken pipe? ubuntu 18.04 how do I install drivers for my NVIDIA GeForce GTX 1050? Show that the countable product of metric spaces is metrizable How can I make a directory on the desktop?  I tried cd /desktop command but it gives an error  no such file or directory.  What's the correct command? While adding to an Answer to  question, I needed to use the above phrase, and I suddenly realised I was unsure whether to write "is" or "are".   There is more than one way to skin a cat. If there are more than one species of cat, we will flay each species differently.   I don't think that second example above sounds quite right to me (disregarding the meaning - sorry!), but in my original Answer I felt "is" would have been even worse.  Is "are" always wrong here? If so, what exactly is the rationale? If not, is there a case where it's unquestionably preferred – and why is that?    Note that in both the problematic Answer and my example attempting to mimic the same context more briefly, the writer doesn't know how many there are. There may be one, more, or even none.  Getting even more specific, the writer might have an opinion on how many there might be, and wish to convey his leanings by choice of verb form if language allows this to be done succinctly. Centos7 can't boot with shim-x64-15-7.el7_9.x86_64 The floowing is serial console output: BdsDxe: starting Boot0003 &quot;CentOS Linux&quot; from HD(1,GPT,A51EC18C-6831-4D47-B93E-B20DBC3F30BF,0x800,0x64000)/\EFI\centos\shimx64.efi  UEFI: Attempting to start image. Description: CentOS Linux FilePath: HD(1,GPT,A51EC18C-6831-4D47-B93E-B20DBC3F30BF,0x800,0x64000)/\EFI\centos\shimx64.efi OptionNumber: 3.  !!!! X64 Exception Type - 0D(#GP - General Protection)  CPU Apic ID - 00000000 !!!! ExceptionData - 0000000000000000 RIP  - 00000000BF2E5D5C, CS  - 0000000000000038, RFLAGS - 0000000000010006 RAX  - 00000000BFF5FE50, RCX - 000000000000001F, RDX - 00000000BFF31F28 RBX  - 00000000BF352FB0, RSP - 00000000BFF31E50, RBP - 00000000000023C0 RSI  - 00000000BFF31F28, RDI - 0000000000000068 R8   - 00000000BDB8C018, R9  - 000000000000211E, R10 - 00000000BDC6C010 R11  - 00000000BFF31E9A, R12 - 00000000BE791068, R13 - 00000000BF339018 R14  - 00000000BEE21018, R15 - 0000000000000068 DS   - 0000000000000030, ES  - 0000000000000030, FS  - 0000000000000030 GS   - 0000000000000030, SS  - 0000000000000030 CR0  - 0000000080010033, CR2 - 0000000000000000, CR3 - 00000000BF401000 CR4  - 0000000000000668, CR8 - 0000000000000000 DR0  - 0000000000000000, DR1 - 0000000000000000, DR2 - 0000000000000000 DR3  - 0000000000000000, DR6 - 00000000FFFF0FF0, DR7 - 0000000000000400 GDTR - 00000000BF3EEA98 0000000000000047, LDTR - 0000000000000000 IDTR - 00000000BEE1F018 0000000000000FFF,   TR - 0000000000000000 FXSAVE_STATE - 00000000BFF31AB0 !!!! Find image based on IP(0xBF2E5D5C) /build/work/af60adde42b1d1ad5be2a01e4924bb905248/google3/blaze-out/k8-opt/genfiles/third_party/edk2/ovmf_x64_csm_debug_workspace_dir/ovmf_x64_csm_debug_edk2_files_dir/Build/OvmfX64/DEBUG_CLANG38/X64/OvmfPkg/8254TimerDxe/8254Timer/DEBUG/Timer.dll (ImageBase=00000000BF2E4000, EntryPoint=00000000BF2E5AB5) !!!! Completion handlers and return values Assume $\bar A\cap\bar B=\emptyset$. Is $\partial (A \cup B)=\partial A\cup\partial B$, where $\partial A$ and $\bar A$ mean the boundary set and closure of set $A$?   I can prove that $\partial (A \cup B)\subset \partial A\cup\partial B$ but for proving $\partial A\cup\partial B\subset \partial (A \cup B)$ it seems not trivial. I tried to show that for $x\in \partial A\cup\partial B$ WLOG, $x\in \partial A$ so $B(x)\cap A$ and $B(x)\cap A^c$ not equal to $\emptyset$ but it seems not enough to show the result. Proof using Prüfer encoding The question is as below:      Show that each vertex $v$ of a tree $T$ appears $deg(v)-1$ times in the Prüfer encoding of $T$, where $deg(v)$ is the degree of $v$ in $T$.   For this, I think there are 3 scenarios a leaf should be considered for: (1) when it is a leaf as part of a tree; (2) a leaf that has been discarded; (3) the remaining edge. I feel like this will result in a short proof which is only two (or so) statements long, but at the same time, feel it is too simple.   EDIT: I've had a look at this again and narrowed it down to 2 cases:   Case 1: The vertex $v$ in question has $deg(v)\ge2$. $v$ then loses its neighbours as the algorithm is applied repeatedly until $v$ becomes a leaf itseelf. At this point, $v$ is attached to another neighbour, leaving an edge. This means the algorithm can no longer be applied and we have obtained the final Prüfer Encoding for the tree $T$.  Case 2: The vertex $v$ has $deg(v)\ge 2$. It loses its neighbours as the algorithm is applied repeatedly but is deleted itself at some point in the process.   Have I gone about this wrong?  Thank you. Let $A$ be a squared matrix, and suppose there exists an $n\in \Bbb N$ in a way that $A^n=0$. Show that $I-A$ is invertible and that $(I-A)^{-1}=I+A+\cdots+A^{n-1}$    I don't have a clue where to start from. How to run scripts without typing the full path? Let $a$, $b$ and $c$ be the three sides of a triangle. Show that $\frac{a}{b+c-a}+\frac{b}{c+a-b} + \frac{c}{a+b-c}\geqslant3$. How to set Ubuntu to synchronize my clock with a time server? ...And you jumped in.  What would happen when you got to the middle of the Earth? Would you gradually slow down, until you got to the middle and once you were in middle would every direction feel like it was up? How to reconcile the cashless human/Federation society with human traders and merchants Question: If $c\neq 1$ is an $n^{th}$ root of unity then, $1+c+...+c^{n-1} = 0$  Attempt: So I have established that I need to show that $$\sum^{n-1}_{k=0} e^{\frac{i2k\pi}{n}}=\frac{1-e^{\frac{ik2\pi}{n}}}{1-e^{\frac{i2\pi}{n}}} =0$$ by use of the sum of geometric series'. My issue is proceeding further to show that this is indeed true.