I am unsure whether to use "a" or "an" in the following sentence:  Video games have become a/an ubiquitous part of American culture.  For me, saying the two sentences out loud makes "an" seem like the right choice but Microsoft Word proofing disagrees.
In the following example, is it appropriate to use a or an as the indefinite article, and why?     He ate __ green apple.   I know that in the case of just "apple", it would be "an apple," but I've heard conflicting answers for "green apple," where the noun is separated from the article by an adjective.  Also, which is more appropriate in this case:     He ate  __ enormous Pop-Tart.
Describe in English a Turing machine that semidecides the following language - Is this solution correct? The language being L, which is described as follows:  $L$ = { $&lt;M&gt;$ | $M$ accepts the binary encodings of at least 4 odd numbers } .  My solution:  M generates the binary encodings in $\Sigma_M*$ in ascending order and uses dovetailing to interleave the computation of M on those binary encodings. As soon as four computations accept, M halts and accepts.  Is this correct? What corrections would have to be made if not? I am not sure if I am providing enough details in my solution which may cause it to be incorrect. Define additional math operators to be typeset in roman Bizarre spacing with 4 $\star$s Example of non-isomorphic structures which are elementarily equivalent Proof of a basic $AC_\omega$ equivalence On  it is mentioned that "...  in order to prove that every accumulation point $x$ of a set $S\subseteq \mathbf R$ is the limit of some sequence of elements of $S\setminus \{x\}$, one uses (a weak form of) the axiom of countable choice. When formulated for accumulation points of arbitrary metric spaces, the statement becomes equivalent to $AC_\omega$. ..."  I have Herrlich's paper "Choice principles in elementary topology and analysis" where in theorem 2.4. it is stated that these are indeed equivalent. Then I looked up the reference he quotes, [4], which are Bentley H.L., Herrlich H., Countable choice and pseudometric spaces which does not contain the proof and  does also not contain the proof, hence my question:   Where can I see a proof of      "if $S$ is a set in a metric space $X$ with accumulation point $s$   then there exists a sequence $s_n$ in $S$ converging to $s$"   $\implies$ $AC_\omega$   ? (the other implication is clear to me). Thanks. How to write (La)TeX (with parentheses) [or any other TeX-related logo] I'm looking after a macro to write the (La)TeX logo (with parentheses), i.e. something similar to \TeX, \LaTeX, but where the output includes ()'s. This morning I started my Google Chrome browser as I do every day. The text size was suddenly bigger than normal, on all sites. I did not change any setting anywhere.  If I set the zoom to 90% it looks fine, but this breaks various things (e.g. Stack Exchange chat) so prefer to avoid it if possible.  Setting font size to Small in Chrome Settings also help, but it's too small.  I have uninstalled Chrome (deleting local data) and installed again, same thing. Tried beta channel as well, no change.  Windows 7 OS 64 bit, Display set to 125% from the very beginning. (Did not change this for years)  What could have caused this and how can it be solved?  This very question turns out to be a good example to demonstrate what I mean:        While the body itself is small (as I changed the font size in Settings), other things are still huge and not affected by the setting. Title of short story: God granting immortality to others to find a way to kill himself Is "Call me when you are available to talk" correct? Call me when you are available to talk   Is that correct? Can I use it that way? How to reconcile the cashless human/Federation society with human traders and merchants From a variety of sources, including , we know that the Federation (or at least the humans) don't use money. This raises the problem that humans frequently visit worlds where there is currency, and, in the case of DS9, operate a station that has currency frequently in use (for example, Federation personnel frequently attend Quark's bar). Additionally, we see human traders.   My question is, how do humans get "money" (credits, latinum, etc.)? How would a human even get the resources to purchase their own merchant ship or the resources to fill its cargo holds with merchandise? If Federation personnel can simply be given money for the asking, why haven't they poured so much money down Quark's throat that he chokes on it? Using customized Coordinate System in ArcGIS Desktop? I don't know so much about coordinate systems...  In my office we use to deal with spatial data coming from archaeological sites. Each site has its own x-y-z coordinate system (GCS). Three simple ortogonal cartesian axis. In the last years we have been managing this spatial data through GIS software (ArcGIS), without using specific coordinate system (just leave it as "undefined")  I'd like to know if there exisits any GCS designed to deal with such datasets using simple cartesian orthogonal axis, without grid distortions of the typical GCS. In addition, I'd like to know if this system is suitable for using it in an online mapping application.  By the way, we manage 2D (ArcMap) and 3D (ArcScene) environments and work with "mm" as length base unit.   If such a thing doesn't exists, maybe someone knows how to create it. A word to describe someone or something that is not last? I'm looking for a word that describes the state of not being last. A word that would apply to everything in an ordered list that is not the last thing in that list.  From     I wasn't the last person to finish the race.   to (the phrasing is off, but it seems to illustrate my point)     I was .. person to finish the race.   One word that comes to mind is "intermediate", but I don't think that includes the first item in a list. Does IBIS reduce image resolution? How does it compare to lens based IS? What happens if no one survives the Hunger Games? It is possible that the remaining last two tributes might get killed at the same time, for example, by a land mine explosion, and in that case the games will end with no victor standing.  Is there a reference in the original books as to what happens if nobody emerges victor in the Hunger Games? Encrypting a folder I'm familiar with bitlocker, having been using it for a while, but I am looking at just encrypting a folder instead. But I just don't understand the options it's giving me.  When I right-click a folder > properties > advanced > [x] encrypt contents to secure data > OK &lt; apply > ...  I get:  () apply changes to this folder () apply changes to this folder, sub-folder and files.   This doesn't make sense to me at all, if I encrypt a folder then the whole folder, everything within it is encrypted, then why should I be given these options, how can I have a folder encrypted and files within that folder unencrypted, that seems stupid to me, it doesn't make sense at all.  Can anybody explain why these options exist and how can I simply encrypt a folder? Seeking methods to solve: $\int_0^\infty \frac{\ln(t)}{t^n + 1}\:dt$ Seeking Methods to solve the following two definite integrals:  \begin{equation} I_(n) = \int_0^\infty \frac{\ln(t)}{t^n + 1}\:dt \qquad J(n) = \int_0^\infty \frac{\ln^2(t)}{t^n + 1}\:dt \end{equation}  For $n \in \mathbb{R},\:n \gt 1$  The method I took was to take the following integral:  \begin{equation}  \int_0^\infty \frac{t^k}{\left(t^n + 1\right)^m}\:dt = \frac{1}{n}B\left(m - \frac{k + 1}{n}, \frac{k + 1}{n} \right) \end{equation}  Where: $0 \leq k \lt n$. Here let $m = 1$.   with respect to $k$ and taking the limit as $k \rightarrow 0^+$ (via the ), i.e.   \begin{align} \lim_{k \rightarrow 0+} \frac{d}{dk} \left[ \int_0^\infty \frac{t^k}{t^n + 1}\:dt \right] &amp;= \lim_{k \rightarrow 0+} \frac{d}{dk} \left[\frac{1}{n}B\left(1 - \frac{k + 1}{n}, \frac{k + 1}{n} \right) \right] \\ \lim_{k \rightarrow 0+}  \int_0^\infty \frac{t^k \ln(t)}{t^n + 1}\:dt  &amp;= \lim_{k \rightarrow 0+}  \left[\frac{1}{n^2} B\left(1 - \frac{k + 1}{n}, \frac{k + 1}{n}\right)\left[\psi^{(0)}\left(\frac{k + 1}{n} \right) - \psi^{(0)}\left(1 - \frac{k + 1}{n} \right)\right] \right] \\ \int_0^\infty \frac{\ln(t)}{t^n + 1}\:dt&amp;= \frac{1}{n^2} B\left(1 - \frac{1}{n}, \frac{1}{n}\right)\left[\psi^{(0)}\left(\frac{1}{n} \right) - \psi^{(0)}\left(1 - \frac{1}{n} \right)\right] \\ &amp;=- \frac{\pi^2}{n^2}\operatorname{cosec}\left(\frac{\pi}{n}\right)\cot\left(\frac{\pi}{n} \right) \end{align}  Which is our expression for $I_n$. Taking the same approach but differentiating twice with respect to $k$ we arrive at our expression for $J_n$:  \begin{equation}  J(n) = \int_0^\infty \frac{\ln^2(t)}{t^n + 1}\:dt = \frac{\pi^3}{n^3}\operatorname{cosec}\left(\frac{\pi}{n} \right)\left[\operatorname{cosec}^2\left(\frac{\pi}{n}\right) + \cot^2\left(\frac{\pi}{n}\right) \right] \end{equation}  And in fact we may generalise:  \begin{equation}     \int_0^\infty \frac{\ln^p(t)}{\left(t^n + 1\right)^m}\:dt = \lim_{k\rightarrow 0}\frac{d^p}{dk^p}\left[\frac{1}{n} B\left(m - \frac{k + 1}{n}, \frac{k + 1}{n}\right)\right] \end{equation}  Where $p \in \mathbb{N}$  This method however was just an extension of another integral. I'm curious, if I had just started with $I_n, J_n$ what alternative methods could be used? Why must a final variable be initialized before constructor completes? I've just purchased a new brand laptop with 17 inch size of the screen. I've installed Texmaker and it is messy. The icons are very small and the GUI is not zoomed out automatically to fit the contents of the window. These are pictures         I've already set the size of the icons to medium and the size of apps to 250 (i.e. the recommended size) for  the display.  I'm using Windows 10. Some fonts are blurry and a lot of people are complaining about this problem with no solution from Microsoft. Any suggestions? I have a Windows 7 64-bit installation on my current PC, and I want to move it to a new PC .. I do NOT have a problem with licensing, as the new system already has its own new Windows 7 license, which I intend to use.  I want to use my existing installation, because it has 3 years worth of installed office related software development software, which installing again could take weeks !  I'm assuming that I can use Windows 7's Backup and Restore feature to backup a system image to a network location, then restore that network stored system image on the new PC ?  Again, like I said, I don't have a WIndows 7 licensing issue, as both machines are corporate provided and come with their own licences. how i evaluate this integral with integration by substitution $x=\frac{\pi}{2}-t$  $$\int _0^{\frac{\pi }{2}}\:\frac{\sqrt[3]{\sin^2\left(x\right)}}{\sqrt[3]{\sin^2\left(x\right)}+\sqrt[3]{\cos^2\left(x\right)}}dx$$