How do I add ASCII art to my Bash?
ASCII art generator
Orthogonal projections with $\sum P_i =I$, proving that $i\ne j \Rightarrow  P_{j}P_{i}=0$ Birational map from elliptic curve keeps smoothness? Let $E$ be a elliptic curve that is , genus $1$ smooth curve with base point. And suppose that the map $φ$, which sends $E$ to $E'$, is birational. Birational means there are two rational maps on both direction which composites identity. Then, can we say that $E'$ is smooth? Proving the two given vectors are orthogonal I am given the vectors $\mathbf{w}, \mathbf{v}, \mathbf{u}$ in $\mathbb{R}^n$ such that $\mathbf{u} \neq 0$ and $\mathbf{w} = \mathbf{v} - \frac{\mathbf{u}\,\bullet\,\mathbf{v}}{\|\mathbf{u}\|^2}\bullet\mathbf{u}$. I am asked to show that the vector $\mathbf{w}$ is orthogonal to $\mathbf{u}$.     So far, I have written out the definition of orthogonal: two vectors are orthogonal if and only if their dot product is zero.   So what we need to prove is $\mathbf{w}\bullet\mathbf{u} = 0$ where $\mathbf{w}\bullet\mathbf{u}$ is defined as $\mathbf{w}^T\bullet\mathbf{u}$.   However, I have been stuck on this problem for about an hour and haven't made any significant progress from here. How do we go about proving that the vectors are orthogonal? Is there a way to increase space between bibliography without using biblatex? "There are no shortage of applications" I've been having an argument with a colleague about this sentence, could you please let me know which one of us is correct:     There are no shortage of applications for our product in this   space.   She is convinced that are should be replaced by is, and I think it should stand as it is. Thanks for your help! I thought the way to undo a \def was to define it to \relax, but that does not appear to work.  Also, am not sure why this produces no output if I try to use \def instead of \let in my attempt to do an undef.  \documentclass{article}  \begin{document}  \def\CaseA{}%  \ifdefined\CaseA%     \def\ProcessCaseA{}%     % Do stuff that uses \ProcessCaseA     %\def\ProcessCaseA\relax% no output if use this, and comment next line     \let\ProcessCaseA\relax% no output if comment this, and use above line \fi  % Now do stuff that requires \ProcessCaseA NOT be defined \ifdefined\ProcessCaseA     ERROR: \textbackslash ProcessCaseA is still defined \else     SUCCESS: \textbackslash ProcessCaseA is not defined \fi  \end{document} Compare dates in query from IMPORTRANGE and cell for a few days I'm stuck at comparing two dates.  In a big table with dates as columns and products as rows, we want to display the amount of sales per day per product.  What the following query does:   Import our sales via IMPORTRANGE query the specific sales for a given day and product ID return 0 if there is an error (e.g. not found) count the result   This is a working query with the date 2018-03-26 hard coded:  =COUNTA(IFERROR(query(IMPORTRANGE("1OHR0nr_w3oFNNxtMxRwv65xncCBRpK-vV3vvqmHGFPo"; "Sales March!A:K"); "select Col6 where Col6 = '"&amp;$A95&amp;"' and todate(Col11) = date '2018-03-26'"; 0)))  It shows how many sales a specific product has on the specified date.  Now I want the date from a cell in the sheet:  =COUNTA(IFERROR(query(IMPORTRANGE("1OHR0nr_w3oFNNxtMxRwv65xncCBRpK-vV3vvqmHGFPo"; "Sales March!A:K"); "select Col6 where Col6 = '"&amp;$A95&amp;"' and todate(Col11) = date '"&amp;$AR3&amp;"'"; 0)))  But this doesn't work. I tried A LOT of things for the past 3 days but I have no idea if I did that wrong (I'm a beginner) so I'm turning to you. What might be wrong with the date comparison in this section?  "select Col6 where Col6 = '"&amp;$A95&amp;"' and todate(Col11) = date '"&amp;$AR3&amp;"'"  You can see it  in Column AR Row 95 but I can't give edit rights. Preventing applications from stealing focus I have a couple of floats in my document, which contain large Images (it consumes the half of the page). In this case, the figure get's its own page and no text goes on this page.  How can I configure this? I'd like to have text before and after the floats until the float doesn't use more then 75% of the page height. Number of sigma algebras for set with 4 elements If $a^3 =a$ for all $a$ in a ring $R$, then $R$ is commutative. Let $R$ be a ring, where $a^{3} = a$ for all $a\in R$. Prove that $R$ must be a commutative ring. Which is the correct way to tell where I'm working?        I'm working in XYZ company.   I'm working for XYZ company.   I'm working at XYZ company.      Or is there any difference in the meaning? Is it ethical to omit the PhD credential from my resume? I am applying for a Statistical Programmer position at a commercial research organization (CRO), and I am wondering if it is unethical to omit my PhD in Statistics from my resume?  All of my degrees (Bachelor's, Master's, PhD) are in Statistics, and I am just wondering if it would be ethical to just list my Bachelor's and Master's degree on my resume, and indicate my experience as a RA while I was a PhD student as my "work experience"? (Since this Statistical Programmer position only requires Master's degree plus some years of experience)  I am asking this question because I do not want to do anything unethical/cause me troubles later on. Suppose $\{a_n\}$ is a sequence such that the subsequences $\{a_{2n−1}\}$ and $\{a_{2n}\}$ converge to the same limit, say $a$. Show that $\{a_n\}$ also converges to $a$. We are given two independent standard normal random variables $X$ and $Y$. We need to find out the M.G.F of $XY$.  I tried as follows : \begin{align} M_{XY}(t)&amp;=E\left(e^{(XY)t}\right)\\&amp;=\int_{- \infty}^{\infty}\int_{- \infty}^{\infty}e^{(XY)t}f_X(x)f_y(y)dxdy \\ &amp;=\dfrac{1}{2\pi}\int_{- \infty}^{\infty}\int_{- \infty}^{\infty}e^{(xy)t}e^{\frac{-x^2}{2}}e^{\frac{-y^2}{2}}dxdy \\ &amp;= \dfrac{1}{2\pi}\int_{- \infty}^{\infty}\int_{- \infty}^{\infty}e^{-\dfrac{(x-ty)^2}{2}}e^{\dfrac{-t^2y^2}{2}}e^{\frac{-y^2}{2}}dxdy \\ &amp;= \dfrac{1}{\sqrt{2\pi}}\int_{- \infty}^{\infty}e^{\dfrac{-t^2y^2}{2}}e^{\frac{-y^2}{2}}dy \\ &amp;= \dfrac{1}{\sqrt{2\pi}}\int_{- \infty}^{\infty}e^{-y^{2}(\frac{1}{2}+t^2)}dy \\ &amp;= \dfrac{1}{\sqrt{1+2t^2}} \end{align} Is this correct ? Guidelines for customizing biblatex styles Footnote to a footnote How can I make a footnote to a footnote? E.g.,  Playfair's axiom\footnote{so called in honor of mathematician John Playfair, etc, etc., but was actually discovered by Proclus\footnote{Greek mathematician, etc, etc.}} If moderators can view , can we see comments we've deleted? We can see our deleted answers, but why not our deleted comments? Can you use twinned spell on the same target? So me and my DM have been arguing on how Twinned spell works. Twinned spell:  When you cast a spell that targets only one creature and  doesn't have a range of self, you can spend a number of  sorcery points equal to the spell's level to target a second  creature in range with the same spell (1 sorcery point if  the spell is a cantrip).  I argued that &quot;a second creature&quot; can be a the same target as it is not worded as &quot;another creature&quot; For this example counter spell with twinned spell.  A wizard casts fireball on me I use CounterSpell#1 with Twinned Spell Wizard casts CounterSpell on my CounterSpell#1 I use him as the same target for my twinned Spell CounterSpell#2  Result countered Is this possible? automatically execute an Excel macro on a cell change