Why is $\lim_{x \to 0} x\cdot \left\lfloor\frac{1}{x}\right\rfloor = 1$?
Evaluate $\lim_{x \to 0} (x\lfloor\frac{1}{x}\rfloor)$
Is it possible that my laptop shut down because my mobile is too close? If I put my phone on my notebook's left corner, it shuts down. Can anybody give me reliable explanation as to why this is happening?? Let $g_n= {2^2}^n +1 $. Prove $g_0 · g_1 · · · g_{n−1} = g_{n} − 2$.    I'm not sure how to start this proof, if it should be done algebraically or if I should try to use a proof by induction. This code won't compile:  \documentclass{article} \usepackage{listings} \lstset{     language=C,     basicstyle=\small }  \begin{document} lorem ipsum\footnote{% \begin{lstlisting} // some C code or whatever... printf("Blah blah\n"); \end{lstlisting}} \end{document}   It says Illegal parameter number in definition of \lst@insertargs. I know listings is really picky about where it goes, but there must be a way to cram a listing into a footnote, right?  (This is a MWE and my tex files are a bit more complex; my actual error is Paragraph ended before \lst@next was complete. but they are hopefully related; I couldn't get this error with the MWE.) How to prove that the radial basis function is a kernel? How to prove that the radial basis function $k(x, y) = \exp(-\frac{||x-y||^2)}{2\sigma^2})$ is a kernel? As far as I understand, in order to prove this we have to prove either of the following:   For any set of vectors $x_1, x_2, ..., x_n$ matrix $K(x_1, x_2, ..., x_n)$ = $(k(x_i, x_j))_{n \times n}$ is positive semidefinite. A mapping $\Phi$ can be presented such as $k(x, y)$ = $\langle\Phi(x), \Phi(y)\rangle$.   Any help? if $x_i$ are IID  continuous random variables, with $E[x_i]=μ$, is the following correct?   $E\left[\prod_{i=1}^n(1+⍺\space x_i)^i\right] = \prod_{i=1}^nE\left[(1+⍺\space x_i)^i\right] =  \prod_{i=1}^n(1+⍺\space E\left[x_i\right])^i = (1+⍺\space μ)^n$  thanks! Can you take the same feat multiple times? I can't see my background image in the 3D view I try to set up an image as background for a basic blender tutorial, but it isn't visible.  This is the image I try to have as background: C# Structs: Unassigned local variable? How do I correctly tag my questions? Here are some tips for tagging questions. These guidelines will help you more accurately tag your questions, which in turn will help them get more attention and lead to better answers faster.     For more information, see "" in the . What is the minimum RAM required for Pokemon GO? While poking around on Pokemon Go's , it lists some specs required for Android and iOS devices. I'm particularly interested in Android devices, but I do not see a minimum/recommended RAM listed on their site.        Android 4.4 to Android 6.0.1  (Android N will not supported until the   official Android release)    Preferred resolution of 720x1280 pixels   (Not optimized for tablet)    Strong internet connection (Wi-Fi, 3G, or   4G)    GPS and Location Services    Intel CPUs are not supported      My device is a , which has a 1.5GB RAM and Quad-Core processor. It runs fine for the most part, but a majority of the features heavily lags and are sluggish while I've seen the same features run smooth on other devices.  What is the minimum/recommended RAM for Pokemon Go to run more smoothly? Difference between processing.runalg() and general.runalg() In the discussion from a question I asked several days ago (), someone suggested that I use general.runalg() instead of processing.runalg() for running QGIS algorithms.    I've now used both, and haven't noticed a difference in either performance or syntax between them.  I'm just wondering what sort of differences there are between these two methods, and which one I should be using. Requesting an active user page In my opinion, providing us with an active user page that shows the list of recent active users is useful as we can monitor the active users without visiting each user profile.  The list can be arranged in a grid of m rows by n columns and accompanied by labels representing the their reputations (and other info that you think they are important) and activities  such as "seen 2 minutes ago".  The following figure illustrates my idea. I am sorry. There is a typo in the following figure, Active must be changed to active to make it parallel to the other existing labels.     I think it is not too difficult to implement this. By adding a new action method and editing the view, done!  Of course it is not urgently requested feature. If it is useful, it can be put in the development to-do list so it can be implemented together with other useful requests at once.  Any responses are welcome! Regardless of how difficult it is, could you explain whether or not there are negative effects of this request? Down voting without informative comments looks barbaric. :-)  Edit:  If it is about user privacy, why is there a "seen 2 minutes ago" in the user profile page? I don't think there is a difference if we implement this feature. Is there any way to save the order of spaces in Mission Control? Why can't I suggest edits on SE meta sites? The use of "were- should- had” at the beginning of sentences instead of “if” Conditionals in English are usually formed by using if with normal word order; but for the three past (subjunctive) forms were, should, and had, it is also possible to express the conditional through subject–auxiliary inversion alone, with no if in the conditional clause.  Does forming conditionals in this manner differ semantically from if-conditionals? Is some aspect of the conditional statement or the conditionality emphasised more in one version than in the other? Or are there differences in how and when they are used? Or are they simply completely interchangeable?  For example:     If I were you, I would try it again   Were I you, I would try it again      If I had seen it, I would have told you   Had I seen it, I would have told you      If you should drink, don’t get behind the wheel   Should you drink, don’t get behind the wheel Showing that $f_0 (x_1, \ldots, x_m) \mathrm tr A =\displaystyle{ \sum_{i=1}^n} f_0(x_1, \ldots, Ax_i,\ldots, x_m)$ I'm looking for an idiom or saying that I could use when people are focusing too much on small details and not seeing the big picture.  A couple that come to mind are "being penny-wise and pound foolish" and "rearranging deck chairs on the Titanic". However, the former doesn't fit what I'm thinking of because the present activity might not be wise. The latter implies too much about impending doom.  For example, let's say some company is spending all their time refining a small detail in their product, while ignoring a major opportunity shift in the industry. What phrase could I use there? What triggers the benefit of the bard's "Song of Rest" feature? In D&amp;D 5e, the benefit of the Bard's "Song of Rest" feature triggers "if you or any friendly creatures who can hear your performance regain hit points at the end of the the short rest."  Does Song of Rest's benefit apply just when spending hit dice to regain hit points? Or does it apply to any other way that a player might regain hit points at the end of a short rest (e.g. by casting a spell, using a Second Wind, eating a Goodberry, etc.)? Prove the inequality $n! \geq 2^n$ by induction I'm having difficulty solving an exercise in my course.  The question is:      Prove that $n!\geq 2^n$.   We have to do this by induction. I started like this:   The lowest natural number where the assumption is correct is $4$ as: $4! \geq 2^4 \iff 24 \ge 16$. The assumption is: $n! \ge 2^n$.   Now proof for $(n+1)$ which brings me to: $(n+1)! \ge 2^{(n+1)}$  I think I can rewrite it somehow like this:  $$ {(n+1)} \times {n!}  \stackrel{\text{(definition of factorial)}}{\ge} 2^n \times 2 $$  $$ (n+1) \times 2^n \ge 2^n \times 2 $$  Then I think I can eliminate the $2^n$ and have something like this: $n+1 \ge 2$, or $n \ge 1$.  But I think I'm wrong here some where and was hoping somebody has some advice on this. How can I prove the above assumption?  Any help would be appreciated, kind regards. Fancy Chapter Headings Style I would like make a fancy chapter like the figure.    However, i made this code  \documentclass[12pt,a4paper]{book}   \usepackage{lmodern} \usepackage{titlesec} \usepackage{microtype} \usepackage{tikz}    \titleformat{\chapter}[display]   {\normalfont\bfseries\color{black}}   {%\filleft%     \begin{tikzpicture}     \node[       outer sep=-10pt,       text width=2.5cm,       minimum height=3cm,       fill=black,       font=\color{white}\fontsize{80}{90}\selectfont,       align=center       ] (num) {\thechapter};     \node[       rotate=0,       anchor=south,       font=\color{black}\Large\normalfont       ] at ([yshift=43pt,xshift=30]num.west)  {\textls[180]{\textsc{\chaptertitlename}}};       \end{tikzpicture}%  }    {-90pt}   {\titlerule[2.5pt]\vskip3pt\titlerule\vskip4pt\LARGE\sffamily \raggedright }    \begin{document} \chapter{The Singular Sturm-Liouville Problems in Spectral Parameter}    \end{document}      that produces this one    How can I produce the chapter like the First Figure?