A word to describe meaningless exchanges in conversation
What are exchanges like "How are you," "I'm fine," and "See you later" called?
Separable Hilbert space have a countable orthonormal basis I want to show that every an infinite-dimensional separable (contains countable dense set) Hilbert space has a countable orthonormal basis.  I know that every orthogonal set in a separable Hilbert space is countable, it is help me with the proof? Spacing issue for newenvironment wrapper of equation environment If I create a custom environment based on the equation environment, I get a spacing issue: the next line of text after my environment closes doesn't start on the left margin.  Example code:  \documentclass{article} \begin{document} \newenvironment{myenv}{\begin{equation}}{\end{equation}} \noindent $| \leftarrow$ this is the left margin. \begin{myenv} e ^{i \pi} + 1 = 0 \end{myenv} $| \leftarrow$ this isn't the left margin. \end{document}   Output:    Removing the newline after \end{myenv} solves the spacing problem, but the resulting code is not readable.  Of course, in this example I could just use the equation environment itself. My real issue involves a more complex custom environment that suffers from the same problem, and which also includes an equation environment within it. How can I install an app given only its APK file? Control a property with no socket (color ramp swatch, etc) from a Group Node input? Gravitational wave solutions to the Einstein field equations It is well known that general relativity predicts gravitational waves, but I would like to know how. What solution(s) to the Einstein field equations yield something which can be interpreted as a wave(-like) phenomenon? I'm looking for a way to access raster layers in an Esri File-GeoDatabase without using ArcObjects or ArcPy, specifically I would like to access the raster layers with GDAL. I understand that the GDAL FileGDB and OpenFileGDB give access to vector datasets within File-GeoDatabases, and that ESRI's own FileGDB C++ API does not support raster access.   Is anyone out there aware of any open source solutions? 4 GB of RAM installed but only 2.97 GB usable. What gives? Possible Duplicate:        I just bought a used MacBook on  that came installed with 4 GB of . The first thing I did was installing Windows&nbsp;7 using . Here's where it gets weird. When I finally booted it up in Windows&nbsp;7, I checked the Computer properties, and it said that there were 4&nbsp;GB of RAM, but only 2.97&nbsp;GB is usable. Why is that? Is there anything I can do about this? My Situation:   About $36k USD in graduate student loans.  There are 4 different federal loans in play, each with a fixed rate of 6.8%.   $12k w/ ~$1.2k in unpaid interest  $10k w/ ~$1k in unpaid interest  $8k w/ no unpaid interest (all principal) $6k w/ no unpaid interest (all principal)  All loans are in repayment. Some loans are subsidized, some are not. This no longer matters now that they're all in repayment though, right?     Is there a most cost efficient way to pay these loans off?  Should the interest be ignored for now on the ones that have $1k in unpaid interest for the time being, because payments to the other two loans will will go directly to principal (minus the monthly compounded interest of course)?  On the surface, I would assume that since they're all the same interest rate, it doesn't really matter, but I'm not sure how the accrued interest would factor in. I'm not interested in psychological wins like debt snowballing. I'm looking for the route that will have me paying the least. What are the alternatives to OS X's Spotlight? Every subspace of a compact space is compact? It seems as if every subspace of a compact topological space (equipped with its relative topology) had ought to be compact as well. Is this true in general?  And in particular, I want to use the fact that every quasi-projective variety is compact with the Zariski topology - I remember having proven it for quasi-affine varieties, but the same proof does not work for quasi-projective varieties. Is every quasi-projective variety compact? Formatting a bootable USB that thinks it's a CD-ROM After upgrading to Windows 10, I remember that I had an old bootable recovery usb drive from the manufacturer that I no longer needed. I want to format it but it's showing up as a CD-ROM and format options are not available.  I also tried formatting it in a live linux mint environment, but linux cannot even mount the cd-rom.  How can I format this recovery usb that is showing up as a cd-rom? Suppose $ \alpha, \beta>0 $. Compute: $ \int_{0}^{\infty}\frac{\cos (\alpha x)-\cos (\beta x)}{x}dx $ Suppose $ \alpha, \beta&gt;0 $. Compute:   $$ \int_{0}^{\infty}\frac{\cos (\alpha x)-\cos (\beta x)}{x}dx $$   Here is what I do: $$\begin{align} \int_{0}^{\infty}\frac{\cos (\alpha x)-\cos (\beta x)}{x}dx &amp;= \int_{0}^{\infty}dx\int_{\alpha}^{\beta}\sin (yx)dy\\ &amp;=\int_{\alpha}^{\beta}dy\int_{0}^{\infty}\sin(yx)dx\\ &amp; \\ &amp;   \qquad\text{let $ yx=u $}\\ &amp; \\ &amp;=\int_{\alpha}^{\beta}\frac{1}{y}dy\int_0^{\infty}\sin u du\\ &amp;=\int_{\alpha}^{\beta}\frac{1}{y}dy\left( -\cos u|_{\infty}+\cos u|_0 \right)\\ &amp;=\log\frac{\beta}{\alpha}(-\cos(\infty)+1)\\ &amp;=\log\frac{\beta}{\alpha}-\cos(\infty)\log\frac{\beta}{\alpha} \end{align}$$  But $ \cos(\infty) $ does not exist right? Does it mean the integral actually diverse?  Edit: The question comes from   Who can point out my mistake in the above deduction? What's the difference between importing and extending a class? What is the equivalent of gksudo in Kubuntu/KDE? In regular Ubuntu (Unity/Gnome), I use gksudo to run GUI programs with root privileges. I'm trying out Kubuntu (or KDE) now -- what do I use? Calculating RSA private exponent when given public exponent and the modulus factors using extended euclid Lightning tags in beta I came across some lightning tags that are still in beta, ex: lightning:input (Beta) lightning:tabset (Beta) etc  Not sure for how long they were in beta, is it advisable to use them for building UI and deploy to production? Why is $\arctan(x)=x-x^3/3+x^5/5-x^7/7+\dots$?  Can someone point me to a proof, or explain if it's a simple answer?  What I'm looking for is the point where it becomes understood that trigonometric functions and pi can be expressed as series.  A lot of the information I find when looking for that seems to point back to arctan. How do I do a search in File Explorer that includes subfolders of zip archives? In File Explorer, I can enter search terms in the upper-right search box to search the current folder and all subfolders. I can also navigate the folder tree of zip archives like any other folders. However, if I want my searches to include files in a zip archive I have to navigate to the specific folder in the zip archive where the file is located. My searches never include files in zip archives if I'm not in the archive, and when I am in the archive my searches don't include files within the archive's subfolders.  Is there a way for my searches to include zip archive subfolders? I was asked today in class why you divide the sum of square error by $n-1$ instead of with $n$, when calculating the standard deviation.  I said I am not going to answer it in class (since I didn't wanna go into unbiased estimators), but later I wondered - is there an intuitive explanation for this?! How to make a different font and color of text in labels?