meaning of the line: buf = ((buf >> 21));
What are bitwise operators?
How do I use tab stops in enumerate or itemize? Current code is:  \begin{enumerate}[label=LEVEL \arabic{*},leftmargin=*]     \item Description of Level 1 \quad L1 \\ \emph{more text}     \item Description of Level 2 \quad L2 \\ \emph{more text}  \end{enumerate}   but the \quad does not align L1 and L2 uniformly vertically. I want to replace \quad with a tab but the tabbing environment does not allow mixing with an enumerate! Pgfplots: How to fill area under a curve in a 3d plot similar to \closedcycle in 2d Suppose I have some theorem in the paper  Section 1 Theorem 1.1. Let ...  And then later in the paper I want to recall the theorem by reprinting it  Section 4 We recall Theorem 1.1: Theorem 1.1. Let ...  What's the proper way to do this? What is the probability that a man likes pink? Peltier Unit Not Working I have searched this for hours but still no luck, obviously doing something wrong.  The journal (Risk Analysis) wants "Reference style is based on the National Library of Medicine (NLM) style guide. In-text citations are numbered, superscripted and in parentheses, and the reference list is numbered in order of appearance within the manuscript. Footnotes are superscripted without parentheses".  Could someone please suggest an appropriate .bst file to use that will produce this? Thanks, J My try, using $x = \sec(u)$ substitution:  $$ \begin{eqnarray} \int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x &amp;=&amp; \int \frac{\sqrt{\sec^2(u) - 1}}{\sec(u)}\tan(u)\sec(u) \mathrm{d}u \\ &amp;=&amp; \int \tan^2(u) \mathrm{d}u \\ &amp;=&amp; \tan(u) - u + C \\ &amp;=&amp; \tan(arcsec(x)) - arcsec(x) + C \end{eqnarray} $$  However, according to , the answer should be: $$ \int \frac{\sqrt{x^2-1}}{x} \mathrm{d}x = \sqrt{x^2-1}+\arctan \left( \frac{1}{\sqrt{x^2-1}} \right)+C $$ When I derive this last answer I don't get back the integrand, but rather: $$ \frac{\mathrm d}{\mathrm d x}\left(\sqrt{x^2-1}+\arctan \left( \frac{1}{\sqrt{x^2-1}} \right)+C\right) = \frac{x}{\sqrt{x^2-1}}- \frac{x}{(x^2-1)^{3/2}\left(1+\frac{1}{x^2-1}\right)} $$  I don't know how to simplify this expression more. Also, I am unable to check whether my answer is correct because I don't know how to find the derivative of $arcsec(x)$.  Can someone check my calculations and tell me where I've done something wrong and how one can simplify the last expression to get back the integrand? Since I've a database systems examination in less than 3 hours (from now), I have been browsing Stack Exchange network for questions with normalisation and related tags. Most of the questions are now closed as  or others from the list.  Since, these questions are closed, no users are allowed to post answers to them (unless they are reopened). Why am I, then, allowed the option to protect such questions? Doesn't that beat the entire purpose behind "protecting" a question? Logo Pack - What should I include? What files should i put in a logo pack to give to clients? Should I put small, medium and large version of my logo as well as the vector file. Prove that inequality is true for $x>0$: $(e^x-1)\ln(1+x) > x^2$ I'm trying to draw a rounded rectangle with round "holes" using  but I need the rectangle borders to be thick and "holes" borders to be thin. Is there a better way to do so than:  \begin{tikzpicture}[x=1mm, y=1mm]     % Background         \fill[fill=black!10] (1.5, 1.5) rectangle (15, -15);      % Fill     \fill[fill=black!2, even odd rule, rounded corners=0.5mm]          (0, 0) rectangle (10, -10)          (2, -2) circle[radius=1.25mm]         (2, -5) circle[radius=1.25mm];     % Rectangle border     \draw[thick, rounded corners=0.5mm] (0, 0) rectangle (10, -10);     % Holes borders     \draw[thin] (2, -2) circle (1.25mm);     \draw[thin] (2, -5) circle (1.25mm); \end{tikzpicture}   The result image should look like this: Is there an easy way to calculate the residue at one of the poles of a rational expression of the form $\frac{1}{1+z^n}$?  I end up having to add a bunch of polar-form complex numbers in the denominator which I have no idea how to do except to convert them to rectangular coordinates, which becomes really messy.  I feel like there ought to be simple pattern/rule for such a canonical expression. Problem 20-9 in Lee's Introduction to Smooth Manifolds (2nd Ed.) asks to prove that for a Lie group $G$, and for $X, Y\in \text{Lie}(G)=\mathfrak g$, there is $\epsilon&gt;0$ and a smooth map $Z:(-\epsilon, \epsilon)\to \mathfrak g$ such that   $$(\exp tX)(\exp tY) = \exp\left(t(X+Y) + \frac{1}{2}t^2[X, Y] + t^3 Z(t)\right)$$   What we want to show is that the first and second term in the Taylor expansion of $\exp^{-1}\left((\exp tX)(\exp tY)\right)$ are $t(X+Y)$ and $\frac{1}{2}t^2[X, Y]$ respectively.  Calculating the first term is easy since we just need to differentiate $\exp^{-1}((\exp tX)(\exp tY))$ with respect to $t$ at $t=0$ once, and this gives the desired result.  But for the second term we need to take the second derivative at which I am stuck. I only occasionally use the Fixed Zoom In/Out buttons on the Tools toolbar of ArcMap and had always assumed that they zoomed in/out by a set scale factor.  However, using ArcGIS for Desktop 10.2, I just set my scale to be 1:10,000,000 and then hit the Fixed Zoom In button 10 times while recording the subsequent scale each time and was surprised to see:  10,000,000 8,000,000 6,000,000 5,000,000 4,000,000 3,000,000 2,500,000 2,000,000 1,500,000 1,250,000 1,000,000   Dividing each scale denominator by its successor shows a sequence of: 1.25, 1.33, 1.2, 1.25, 1.33, 1.2, 1.25, 1.33, 1.2  Curious about what happened in between the scales above I experimented a little to discover that the first issuance of the Fixed Zoom In button simply takes the scale to the first of those listed above and then starts following the same pattern.  Try starting with 1:8,000,001 to see ...  8,000,001 8,000,000 6,000,000 5,000,000 4,000,000 3,000,000   This seems like quite a neat solution, but not one that I had realised was in place - does anyone know if it has been documented anywhere?    I've looked in the Online Help without success.  In answer to @DevdataTengshe, the scales currently in my available list (see below) seem to bear no relation to those used by Fixed Zoom In/Out.    However, when I add 1:7,500,000 into my available list it does seem to insert that as an additional scale step ...  10,000,000 8,000,000 7,500,000 6,000,000 5,000,000 4,000,000   This makes the algorithm being used even more mysterious to me. Get all possible partitions of number Is there a way to get all possible partitions of an integer? Possibly specifying max and/or min summand. I'm interested in partitions themselves, not just partition count. Prove that if $a+b+c$ divides $abc$, then $a+b+c$ must be composite. I have seen another post for this same question but it never got an answer. Since I am fairly new, I can't comment on other's posts, therefore I am creating a new one. The question states:  If a, b, and c are positive integers, prove that if $(a + b + c) | (a*b*c)$, then $a+b+c$ is composite.  I do not really know how to approach this.  The link to the post with the same question is: Why was Arcanine with the Legendary Birds in Veridian City in Pokémon Origins? In Pokémon Origins, Red sees a image with the three legendary birds and Arcanine.     Why was Arcanine there? I mean, he's 'The Legendary Pokemon' but not a legendary Pokémon. In , there is a comment by  on the answer that Arcanine was originally a legendary Pokémon:     "...Nintendo first gave Zapdos, Articuno and Arcanine as the legendary trio. But they removed Arcanine as it would have been awkward. Two birds and a dog, huh!..."   This still does not add up because Pokémon Origins was created a while after they removed Arcanine. Any thoughts? How would you explain covariance to someone who understands only the mean? ...assuming that I'm able to augment their knowledge about variance in an intuitive fashion (  ) or by saying: It's the average distance of the data values from the 'mean' - and since variance is in square units, we take the square root to keep the units same and that is called standard deviation.  Let's assume this much is articulated and (hopefully) understood by the 'receiver'. Now what is covariance and how would one explain it in simple English without the use of any mathematical terms/formulae? (I.e., intuitive explanation. ;)  Please note: I do know the formulae and the math behind the concept. I want to be able to 'explain' the same in an easy to understand fashion, without including the math; i.e., what does 'covariance' even mean? My USB drive used to be originally 8GB when I bought it.  I'm trying to reformatted in Windows 7 by right clicking on the drive and selecting Format.... But the capacity only shows 250MB.  Is there something I can do to get the original size back? Maybe it got partitioned in a weird way?   The  flash drive is a SanDisk Cruzer Micro 8GB. I got it from Wal-Mart but this is the same drive. I tried getting a good measurement with my caliper, but the flexibility of the tire doesn't help much. Let's say I have a 26&quot;x2.125&quot; tire. I know this fits on an ISO 559mm bead seat diameter rim, and 26&quot; = 660.4mm. 26&quot;:  The OD of the tire including the tread? The OD of the tire excluding the tread? When aired up or deflated? Means it fits a 559mm ISO BSD None of the above?  2.125&quot;:  Width of tread? Width of sidewall? Inflated or deflated?  Essentially, I'm converting the tire sizes to &quot;height off rim in mm&quot; and drawing them with a wheel in a web application, so I need to know how &quot;tire size&quot; translates to this measurement. I'm somewhat assuming that the fractional sizes can be converted to the same measurement, but I'm aware that they have certain differences - so any help there would be of value as well.  has been helpful, but still leaves me with the questions above.