Multiple labels error in align inside gather
Labels and nesting gather and align
What is the difference between sourcing ('.' or 'source') and executing a file in bash? Rotate a long bar in space and get close to (or even beyond) the speed of light $c$ Is every complex (smooth) manifold a scheme? The question in the title doesn't quite make sense. I was always wondering if the scheme is the generalization of manifold. The precise statement should be like following:  If $X$ is a complex (smooth)manifold, is there a scheme $Y$ over $\mathbb{C}$ such that the corresponding analytic space $Y^{an}$ isomorphic to $X$?  I guess if $X$ is projective, then this is Chow's theorem. And I doubt the question has an affirmative answer if $X$ is non-compact (I guess that $X: y-e^x=0$ in $\mathbb{C}^2$ might be an example, but I do not know how to show this).  Edit 1:  There are also example (as @hunter pointed out) of a complex torus but not an abelian variety shows that a compact manifold might not be a variety, however, is it possible to be a scheme ?(according to Hartshorne, the category of varieties is a subcategory of quasi-projective, integral, separated scheme. But could there be some non-quasiprojective scheme($Y$ has to be integral, separated in our case according to GAGA, I guess) )?  Edit 2: Another question I think of is: is (Artin) Stack a generalization of manifold? I really want to know if manifold could be think of some subcategory of some category of with "algebraic objects". Let $\alpha$ be an irrational real number and let $a_j$ be a sequence of rational numbers converging to $\alpha$. Suppose that each $a_j$ is a fraction expressed in lowest terms: $a_j = \alpha_j / \beta_j$. Prove that the $\beta_j$ tend to $\infty$  Attempt: AFSOC that $\beta_j$ does not tend to infinity, then it is bounded by some $M$. We can also find an interval $\alpha \in (k, k+1)$. Let $\epsilon &gt; 0$, $\exists N_0$ such that for all $N &gt; N_0$ we have $|a_N - \alpha|  &lt; \epsilon$. So $|\frac{p_N}{q_N} - \alpha| &lt; \epsilon$. We try to use the fact that $q$ is bounded but fail to derive a contradiction. I do not know how to use the fact that $\alpha$ is between two integers, although it might not be relevant at all.  Ideas? Black nontransparent glass (Cycles) I am modeling plane and I don't want to model a cockpit so I want to use black windows which will reflect a lot of light but you can't see thought.   Currently I just use Glossy BSDF (GGX) with almost black color (0.01, 0.01, 0.01) and Rougness 0.015 but it don't look quite right. It lacks glass shine and looks too matt.  Edit:  Here is my simplified blend file:    Also here is what i want. Glass on this tie fighter is blackish and it looks like glass. In my blend file it looks more like plastic.    One more thing i noticed is when light reflect from glass whole glass section reflect same amount of light. It can be seen on tie fighter example above. CSS overflow-x: visible; and overflow-y: hidden; causing scrollbar issue Should the first word after a colon be capitalized? Should the first word after a colon be capitalized? Which of the following is correct?   For example: This. For example: this. Characterizing units in polynomial rings I am trying to prove a result, for which I have got one part, but I am not able to get the converse part.  Theorem. Let $R$ be a commutative ring with $1$. Then $f(X)=a_{0}+a_{1}X+a_{2}X^{2} + \cdots + a_{n}X^{n}$ is a unit in $R[X]$ if and only if $a_{0}$ is a unit in $R$ and $a_{1},a_{2},\dots,a_{n}$ are all nilpotent in $R$.  Proof. Suppose $f(X)=a_{0}+a_{1}X+\cdots +a_{n}X^{n}$ is such that $a_{0}$ is a unit in $R$ and $a_{1},a_{2}, \dots,a_{r}$ are all nilpotent in $R$. Since $R$ is commutative, we get that $a_{1}X,a_{2}X^{2},\cdots,a_{n}X^{n}$ are all nilpotent and hence also their sum is nilpotent. Let $z = \sum a_{i}X^{i}$ then $a_{0}^{-1}z$ is nilpotent and so $1+a_{0}^{-1}z$ is a unit. Thus $f(X)=a_{0}+z=a_{0} \cdot (1+a_{0}^{-1}z)$ is a unit since product of two units in $R[X]$ is a unit.   I have not been able to get the converse part and would like to see the proof for the converse part. Wireframe iOS design apps for iPad? Is there an app for iPad to assist in wireframe design for iPad and iPhone apps?  I don't mean actual development, just placing standard widgets and the things you usually do with photoshop and associated kits. How to keep font preferences in TexWorks 4 The default font is too small on my screen at the resolution I have it, so I like to reset the font to size 12, but it appears I have to do this everytime I launch the program.  How do I make it remember my preferences? About embryonic and genetic evidence of evolution? My question here is about evidence for evolution from Embryology and from Genetics.  Mammals do have similar Embryos, but is it the case that for each species there is an embryonic development that occurs in earlier species, has such relationship been observed? What I mean for example if we generally conceive of having species$Sp_1,Sp_2,...,Sp_n$ where each $ Sp_{i+1}$ being evolved from $Sp_i,$ then is it the case that the embryonic development stages of $ Sp_{i+1}$ contains some of the embryonic development stages of all of its ancestor species.  Similar question to 1 but regarding Genetics, is it the case that every species has a genetic thumbprint of its ancestors that lie there in the non-functional sections of its DNA, i.e. is it the case that all the genetic history of our evolution is recorded in the non-functional section of our DNA. So do human DNA have some portions of the DNA of some dinosaurs for example? Let $\{s_n\}$ and $\{t_n\}$ be sequences.  I've noticed this inequality in a few analysis textbooks that I have come across, so I've started to think this can't be a typo:  $\limsup\limits_{n \rightarrow \infty} s_n + \liminf\limits_{n \rightarrow \infty} t_n \leq \limsup\limits_{n \rightarrow \infty} (s_n + t_n) \leq \limsup\limits_{n \rightarrow \infty} s_n + \limsup\limits_{n \rightarrow \infty} t_n$.  I understand the right two-thirds of the equation, and can prove it.  I'm stuck on the far-left third.  I don't understand how that is true. Cyclic orbit animation I'm trying to make one object orbit another. But I want it to repeat, rather than stop at 100 frames. It seems there is no option for this. Is there a way to make this happen? Do eyeballs exhibit chromatic aberration? Fairly straightforward question. If not, why not?  I suspect that if they do, it is not perceived due to the regions of highest dispersion being in one's region of lowest visual acuity. Possible Duplicate:          I was thinking about making a pacman clone, I know there is a similar question here  , but I know i can't use the original art from the game because it belongs to Namco, so if I design a character that has the shape of the slice circle it will look exactly like pacman, maybe if I use green instead of yellow? Also if the game plays like the original pacman, it is wrong?  I just want to make the game as a personal project and and publish it in my site without getting in trouble Can I use 'in' instead of 'at'? -- "I have more books in my house." I am learning English using the Duolingo English course, one of correct solution was      I have more books at my house.   Can I use      I have more books in my house.   If it is OK, what is the difference between both? Simulating Key Press C# Make Bash autocomplete guess the closest approximation I have a somewhat uncomfortable keyboard on my laptop, resulting in that I sometimes misspell things in Bash and when pressing Tab for an auto-completion nothing happens.   Is it possible to have Bash guess what I meant to type, by for example when wanting to change directory to Documents and then mistyping cd ocum[TAB], given no other folder beginning with "ocum" is present, Bash interprets it as cd Documents/ since it is the closest possible action to do?   Or is this idea terribly complicated to achieve? Currently selected indicator's colour in flag summary and help pages, can be the same as the site's colour code Some of the Stack Exchange sites has it's own colour code. Like Meta Stack Exchange has #17a7db, ELU has #AE3737, Travel has #277789 for the currently selected item's indicators in main page and the user activity page. But in the flag summary section and help pages of those sites, the indicator has the colour of #F48024 which is same as the Stack Overflow site's colour code. So instead of applying the Stack Overflow's colour code to all the sites, shall we apply the site's specific colour code in flag summary and help pages too. Selected indicator in flag summary section of MSE, ELU and Travel:      Selected indicator in help page: Is Switching the Neutral OK? What’s the worst that can happen if i put a switch in the neutral instead of the phase?   I just found out that 3 light bulbs sockets in my home  are connected this way, it first came to my mind to fix it but then I recalled the flow of current from positive to negative is just a convention since the electrons are charged negatively and they actually flow from N to L so i cant really find an solid excuse to mess with the wiring for the next 2 hrs