$\lim_{t \to \infty} t\mu(f > t) = 0$ but $f \not \in L^1$
Find a non-negative function on [0,1] such that $t\cdot m(\{x:f(x) \geq t\}) \to 0$ that is not Lebesgue Integrable
How can I remap a keyboard key? One of my laptop's keys has fallen off. Is there any way I can remap another key to serve as that key?  I'll 'sacrifice' the other key because I never use it. Slit screen and wave-particle duality In a double-slit experiment, interference patterns are shown when light passes through the slits and illuminate the screen. So the question is, if one shoots a single photon, does the screen show interference pattern? Or does the screen show only one location that the single photon particle is at? Can I make a custom node from a group and have that available in other projects? I've found some neat combinations of nodes to create certain effects, so I'd like to make them a group and then have that node available on my list of available nodes, even when I reload the startup file. How would I do that? OR operator in C not working I am  with the Google login for Pokemon Go. Is there any way, be it contacting support or through Trainer Club, to either link my Google account to a new Trainer Club account or transfer my account to a Trainer Club account?  It seems like a better idea to have a Trainer Club account, so if there are Google login problems it won't matter.  Does anyone know if something like this is possible? I once already asked a very similar question; . But this is some what different.  I have the following code;  \documentclass{standalone} \usepackage{graphics} \usepackage{tikz,pgfplots} \pgfplotsset{compat=1.6} \begin{document} \begin{tikzpicture}  \begin{axis}[% width=3cm, height=3cm, scale only axis, every x tick label/.append style={font=\scriptsize\color{gray!80!black}}, xmin=0, xmax=20000, xlabel={$t$}, xmajorgrids, xminorgrids, every y tick label/.append style={font=\scriptsize\color{gray!80!black}}, ymin=0.0004, ymax=0.0012, ylabel={$\phi_{\mathrm{i}}(t)$}, ymajorgrids, yminorgrids, name=plot1]  \end{axis}  \begin{axis}[% width=3cm, height=3cm, scale only axis, every x tick label/.append style={font=\scriptsize\color{gray!80!black}}, xmin=0, xmax=20000, xlabel={$t$}, xmajorgrids, xminorgrids, every y tick label/.append style={font=\scriptsize\color{gray!80!black}}, ymin=0.2, ymax=1.2, ylabel={$h(t)$}, ymajorgrids, yminorgrids, name=plot2, at=(plot1.right of north east), anchor=left of north west]  \end{axis}  \begin{axis}[% width=3cm, height=3cm, scale only axis, every x tick label/.append style={font=\scriptsize\color{gray!80!black}}, xmin=0, xmax=20000, xlabel={$t$}, xmajorgrids, xminorgrids, every y tick label/.append style={font=\scriptsize\color{gray!80!black}}, ymin=0.0004, ymax=0.0012, ylabel={$\phi_{\mathrm{o}}(t)$}, ymajorgrids, yminorgrids, name=plot4, at=(plot1.below south west), anchor=above north west]  \end{axis}  \begin{axis}[% width=3cm, height=3cm, scale only axis, every x tick label/.append style={font=\scriptsize\color{gray!80!black}}, xmin=0, xmax=20000, xlabel={$t$}, xmajorgrids, xminorgrids, every y tick label/.append style={font=\scriptsize\color{gray!80!black}}, ymin=2000, ymax=12000, ylabel={$p_{\mathrm{i}}(t)$}, ymajorgrids, yminorgrids, name=plot5, at=(plot4.right of north east), anchor=left of north west]  \end{axis}  \end{tikzpicture} \end{document}   This creates the following plot:    I think there is to few vertical/horizontal spacing between the plots. I want to increase this. Currently I use x- yshift, this gives the following result:  \documentclass{standalone} \usepackage{graphics} \usepackage{tikz,pgfplots} \pgfplotsset{compat=1.6} \begin{document} \begin{tikzpicture}  \begin{axis}[% width=3cm, height=3cm, scale only axis, every x tick label/.append style={font=\scriptsize\color{gray!80!black}}, xmin=0, xmax=20000, xlabel={$t$}, xmajorgrids, xminorgrids, every y tick label/.append style={font=\scriptsize\color{gray!80!black}}, ymin=0.0004, ymax=0.0012, ylabel={$\phi_{\mathrm{i}}(t)$}, ymajorgrids, yminorgrids, name=plot1]  \end{axis}  \begin{axis}[% xshift=1cm, width=3cm, height=3cm, scale only axis, every x tick label/.append style={font=\scriptsize\color{gray!80!black}}, xmin=0, xmax=20000, xlabel={$t$}, xmajorgrids, xminorgrids, every y tick label/.append style={font=\scriptsize\color{gray!80!black}}, ymin=0.2, ymax=1.2, ylabel={$h(t)$}, ymajorgrids, yminorgrids, name=plot2, at=(plot1.right of north east), anchor=left of north west]  \end{axis}  \begin{axis}[% yshift=-1cm, width=3cm, height=3cm, scale only axis, every x tick label/.append style={font=\scriptsize\color{gray!80!black}}, xmin=0, xmax=20000, xlabel={$t$}, xmajorgrids, xminorgrids, every y tick label/.append style={font=\scriptsize\color{gray!80!black}}, ymin=0.0004, ymax=0.0012, ylabel={$\phi_{\mathrm{o}}(t)$}, ymajorgrids, yminorgrids, name=plot4, at=(plot1.below south west), anchor=above north west]  \end{axis}  \begin{axis}[% xshift=1cm, width=3cm, height=3cm, scale only axis, every x tick label/.append style={font=\scriptsize\color{gray!80!black}}, xmin=0, xmax=20000, xlabel={$t$}, xmajorgrids, xminorgrids, every y tick label/.append style={font=\scriptsize\color{gray!80!black}}, ymin=2000, ymax=12000, ylabel={$p_{\mathrm{i}}(t)$}, ymajorgrids, yminorgrids, name=plot5, at=(plot4.right of north east), anchor=left of north west]  \end{axis}  \end{tikzpicture} \end{document}     I want to know if there is a more consistent manner to do this. Contemporary written usage of "whom" in objective case Possible Duplicate:          I was writing a LinkedIn recommendation one day, and ended up pondering for a while which of these forms to use:     … is a great developer whom I always   found easy to work with.      … is a great developer who I always   found easy to work with.   Both are basically correct in contemporary English, right? But is one or the other preferable, and if so, why?   (In this case I went with the latter, as it seemed more common (Google)  and I wanted to avoid sounding unnecessarily “archaic”, although I’m not sure whether that would have been the case. Also note that I try to write in a “friendly professional” style instead of overly formal one. :-) Practical hyperparameter optimization: Random vs. grid search I was surprised to find today that I couldn't track down any simple way to write the contents of an InputStream to an OutputStream in Java. Obviously, the byte buffer code isn't difficult to write, but I suspect I'm just missing something which would make my life easier (and the code clearer).  So, given an InputStream in and an OutputStream out, is there a simpler way to write the following?  byte[] buffer = new byte[1024]; int len = in.read(buffer); while (len != -1) {     out.write(buffer, 0, len);     len = in.read(buffer); } JavaScript private methods Lie algebra of $\left(\begin{smallmatrix}a & b\\ & a^2\end{smallmatrix}\right )$ in $GL_2(\mathbb{R})$ I'm working on the question     Let $G$ be the group of invertible real matrices of the form $\left [\begin{array}{c c}a &amp; b\\ &amp; a^2\end{array}\right ]$. Determine the Lie algebra $L$ of $G$, and compute the bracket on $L$.   I'm familiar with how to derive the Lie algebra for a linear group like $U_n$, $SU_n$, etc. but I'm not sure what to do in a more explicit case like this. What is the C# equivalent of NaN or IsNumeric? This is a  about capacity planning for web sites. Related:      What are some recommended tools and methods of capacity planning for web sites and web-applications? Please feel free to describe different tools and techniques for different web-servers, frameworks, etc., as well as best-practices that apply to web servers in general. How to down-weight older data in time series regression In a regression fit of vectors varying with time $t$ $\qquad y \sim [x_t\ x_{t-1}\ x_{t-2}\ ...] \cdot [c_t\ c_{t-1}\ c_{t-2} \ ...] $ , how can one down-weight the older $x_t$ to model "older is less relevant" (than the regression gives) ? Say that rows in data matrix $X$ are observables (stocks, market segments ...) and columns are times. Correct me,   weights rows; how does one weight columns, times ?  Added: Down-weighting older data in time series looks complex, much more so than linear least squares with i.i.d. errors in rows (observations) -- just divide through by $\sigma_i$; or classification  with i.i.d. errors in columns (e.g. features) -- just centre each column. Can anyone say either   yes, I downweight time series; here are some examples on the web no: down-weighting time series is complex, not for novices.   (  discussed this question 10 years ago, mentioning Kalman filters and hierarchical Bayes, but with no examples.) Help me to solve this problem please..  Let $Y_{(1)}, Y_{(2)}, Y_{(3)}, Y_{(4)}, Y_{(5)}$ denote the order statistics of a random sample of size 5 from a distribution having p.d.f. $f(y) = e^{(-y)}, 0 &lt; y &lt; \infty$, zero elsewhere. Show that $Z_1 = Y_{(2)}$ and $Z_2 = Y_{(4)} − Y_{(2)}$  are independent. Hint: First find the joint p.d.f. of $Y_{(2)}$  and  $Y_{(4)}$. How to tell if a certain item is in a player inventory? I am trying to create a challenge world where a certain item (e.g. a Netherite Sword) must be collected. How can I check automatically if a player has it (using commands), and then congratulate the winner? This question is specific to Bedrock Edition 1.16. I have tried other solutions so many times and they don't seem to work. See also: If $k$ is a positive integer, how to prove that there is no solution for $n$ in positive integers except $n = 1$ for $2^n = nk + 1$ Suppose that $y_1(x)$ and $y_2(x)$ are solutions of the differential equation $y''+py'+qy=0$ on $I$. How can I show that if $y_1$ and $y_2$ vanish at the same point then they are not linearly independent?   Here is my attempt to prove the problem. Since $y_1(x)$ and $y_2(x)$ are solutions of the differential equation $y''+py'+qy=0$ on $I$, it is enough for me to show that the Wronskian of $y_1$ and $y_2$ denoted by $W(y_1,y_2)$ is zero. Now since $y_1$ and $y_2$ vanish at the same point say $p$ we have $y_1(p)=0$ and also $y_2(p)=0$.   Solving for $W(y_1,y_2)(p)$ we have: $y_1(p)y_2'(p)-y_2(p)y_1'(p)=0$ since $y_1(p)=0$ and also $y_2(p)=0$.   Am I correct? Thanks Is there an idiom that means "revealing a secret unintentionally"? I am thinking there could be an idiom that states that you or someone revealed a secret unintentionally? Can you think of an idiom like that? I am looking for an idiom I can use to write a blog post about the recent incident where Secretary Pompeo unintentionally revealed the dirty secrets behind the CIA. Hello and I'm quite new to Math SE.  I am trying to find the largest consecutive sequence of composite numbers. The largest I know is:  $$90, 91, 92, 93, 94, 95, 96$$  I can't make this series any longer because $97$ is prime unfortunately.  I can however, see a certain relation, if suppose we take the numbers like (let $a_1, a_2, a_3,...,a_n$denote digits and not multiplication):  $$a_1a_2a_3...a_n1,\ a_1a_2a_3...a_n2,\ a_1a_2a_3...a_n3,\ a_1a_2a_3...a_n4,\ a_1a_2a_3...a_n5,\ a_1a_2a_3...a_n6,\ a_1a_2a_3...a_n7,\ a_1a_2a_3...a_n8,\ a_1a_2a_3...a_n9,\ a_1a_2a_3...(a_n+1)0$$  The entire list of consecutive natural numbers I showed above can be made composite if:   The number formed by digits $a_1a_2a_3...a_n$ should be a multiple of 3 The numbers $a_1a_2a_3...a_n1$ and $a_1a_2a_3...a_n7$ should be composite numbers   If I didn't clearly convey what I'm trying to say, I mean like, say I want the two numbers (eg: ($121$, $127$) or ($151$, $157$) or ($181$, $187$)) to be both composite.  I'm still quite not equipped with enough knowledge to identify if a random large number is prime or not, so I believe you guys at Math SE can help me out.