The gardens stretched back to some reasonable-looking pasture land on which grazed a few cattle and sheep.  Why is this inversion valid here? I would expect maybe &quot;on which there grazed&quot; (as in &quot;there comes a time&quot;) if an inversion were to be happen.
The opening sentence to The Hobbit by J.R.R. Tolkien reads,     In a hole in the ground there lived [verb] a hobbit [subject].   I wonder if there are accepted stylistic purposes for such a structure.  When is it natural, and when is it unnatural?
I'm reading Galois Theory by Steven H. Weintraub (second edition), and finding that I'm at least somewhat short on the prerequisites. However the following proof looks wrong to me - am I misunderstanding something, or is it actually an incorrect proof?  Lemma 2.2.3. Let $F$ be a field and $R$ an integral domain that is a finite-dimensional $F$-vector space. Then $R$ is a field. Proof. We need to show that any nonzero $r \in R$ has an inverse. Consider $\{1, r, r^2, \cdots\}$. This is an infinite set of elements of $R$, and by hypothesis $R$ is finite dimensional as an $F$-vector space, so this set is linearly dependent. Hence $\sum_{i=0}^n{c_i r^i} = 0$ for some $n$ and some $c_i \in F$ not all zero.  It then goes on to show, given the above, that we can derive an inverse for $r$. However, if I consider examples like $r = 2 \in Q[\sqrt{2}]$, $r = \sqrt{2} \in Q[\sqrt{2}]$ or $r = 2 \in Q[X]/{&lt;X^2&gt;}$, the set $\{1, r, r^2, ...\}$ doesn't look linearly dependent to me. I do believe the lemma is true (and might even be able to prove it), but this does not look like a correct proof to me. Am I missing something? [Edit] Well yes, I am. Somehow I had managed to discount the possibility of any $c_i$ being negative, despite repeatedly looking at each fragment of the quoted text in an attempt to find what I might be misunderstanding. LaTeX doesn't recognize "et al." in the Bibliography I'm using Jabref to manage my Bibliography.  Sometimes, I put directly:     First Author et al.   when adding the article author's name in Jabref.  When compiling my .tex file, LaTeX doesn't recognize the et. al as more names. It takes it as a name/surname.  This is an example of a Bibliography entry in Jabref.  @ARTICLE{Wolf2003,     author = {Wolf, K.et al.},     title = {Compensation mechanism in tumor cell migration: mesenchymal–amoeboid     transition after blocking of pericellular proteolysis},     journal = {J. Cell Biol.},     year = {2003},     volume = {160},     pages = {267–277},    }   This is a .texMWE with the bib entry mentioned:  \documentclass[11pt]{book}   \usepackage{cite}   \bibliographystyle{unsrt} \begin{document}   Here is the bib entry mentioned\cite{Wolf2003}    \bibliography{bibfile}  \end{document}   Here's the output:   How can I solve this problem? Book series my dad told me about. It involves space travel, clones, and an indigenous people with kangaroo tails My dad read a book series that he recommended to me last year. Now that I am ready to start it, neither of us can recall the title. He said there were several in the series but he can't recall how many.   The book is about traveling to a new planet, presumably with the goal of colonizing it. The crew are clones who have no memory. I think the goal was for the older original versions of the clones to transfer their consciousness into the clone bodies to be able to live on the new planet but something goes wrong.   There is an indigenous species on the planet who are humanoid but have kangaroo like tails.   The clone crew has markings on them that they don't understand until late. Apparently they are markings to indicate status or role on the new world. The person who they voted as there leader, it turns out, is marked as a servant or slave class. How can I recursively clear the "Hidden" file flag on Windows Vista? A friend's computer was recently infected with a virus.  I hope the virus has been removed successfully. But, it appears that the virus set the "Hidden" flag on most directories and files on the system.   What's the most convenient way to clear this flag from selected directories and their contents? Why could centering independent variables change the main effects with moderation? Simplify a combinatorial sum Why is $\sum_{k=0}^n (_k^n) 2^k$ simplified as $3^n$ and not just $2^n$ ? The answer uses the binomial theorem as a solution: let $a=1$ and let $b=2$. So $1+2 = 3$. But since $1+2 = 3$ why do we not use $1+1 = 2$? Can I get UAC prompt for user from batch file? Can I get UAC prompt for user from batch file?  For example I need put some keys to registry under CurrentControlSet from batch file by     reg import path\to\my.reg   PS. runas is not a solution as my customers uses different Windows localisation and I can not guess which name used for Administrator account (this can be Администратор)...  And script don't know administrator password.  And UAC dialog box does not require password entering... Why the switch statement cannot be applied on strings? Compiling the following code and got the error of type illegal.  int main() {     // Compilation error - switch expression of type illegal     switch(std::string("raj"))     {     case"sda":     } }   You cannot use string in either switch or case. Why? Is there any solution that works nicely to support logic similar to switch on strings? How do I share a folder on SAMBA? The expected value of $1/(1+X)$ where $X$ has geometric distribution Let $X$ be a random variable with a geometric distribution with parameter $p$.       Calculate $\Bbb E[1/(1+X)]$   I've looked at answers online but still do not understand how to solve this.   Specifically, I do not understand how to to get from $$\frac{p}{1-p} \sum_{x=0}^\infty (1/(1+x))((1-p)^x+1)$$  to  $-p\ln(p)  /(1-p)$. In C, are arrays pointers or used as pointers? What are the meaning and possible origin of "word!" and "word up"? Several times, I have had conversations, all over instant messenger, finish with "word" or "Word up G".  As it ends a conversation, I am guessing it is like "goodbye".  My question is what is the meaning of "word" and "word up g"? Also, what is the origin?  I am more interested in the meaning as that will help with understanding its usage. Why is 128==128 false but 127==127 is true when comparing Integer wrappers in Java? Validate date format in php Use character string as function argument What are the recommended textbooks for introductory calculus? The problem:  Assume $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfies $|f(t) - f(x)| \leq |t - x|^2$ for all $t, x$. Prove $f$ is constant.  I believe I have some intuition about why this is the case; i.e. if $t$ and $x$ are very close, ($|t - x| = \epsilon$), then $\sqrt{\epsilon}$ will converge to 0, so $|f(t) - f(x)|$ will also converge to zero if you keep taking $t$ and $x$ closer and closer to each other. However, how do I formalize this argument? Thanks. Subset data frame based on number of rows per group Basic buoyancy question: Man in a boat with a stone This comes from a brain teaser but I'm not sure I can solve it:     You are in a rowing boat on a lake. A large heavy rock is also in the boat. You heave the rock overboard. It sinks to the bottom of the lake. What happens to the water level in the lake? Does it rise, fall or stay the same?   I'd say that the water level drops, because when you drop the stone in the lake, the water level rises according to the volume of the stone, BUT the water level decreases by a volume of water weighting the stone's weight, when you take it off the boat.  Is that correct? From :     Importantly, the DDH assumption does not hold in the multiplicative group $\mathbb {Z} _{p}^{*}$, where $p$ is prime. This is because given $g^{a}$ and $g^{b}$, one can efficiently compute the Legendre symbol of $g^{ab}$, giving a successful method to distinguish $g^{ab}$ from a random group element.   The only way I can think of to distinguish $g^c$ from $g^{ab}$ is:    If $\left(\frac{g}{p}\right)=-1$, then $\left(\frac{g^{ab}}{p}\right) = 1$ if and only if $a$ or $b$ is even (which is $3/4$ of the times). So if $\left(\frac{g^c}{p}\right) =-1$, probably $g^c\neq g^{ab}$. Nothing is certain.   How'd do you compute $\left(\frac{g^{ab}}{p}\right)$ given $g^a$ and $g^b$?