Could someone give me an example where $X_{n}$ and $Y_{n}$ converge in $L^{1}$ but $X_{n}Y_{n}$ doesn't converge in $L^{1}$?
If $f_n$ converges to $f$ in $L^1$ and $g_n$ converges to $g$ in $L^1$. Does it necessarily mean that $f_ng_n$ converges to $fg$ in $L^1$ for finite measure spaces.
I have seen many ways to make a USB an installer for about any Windows OS. But how can I (using grub or something like that) make it bootable to all editions on Windows XP and 7?   I have tried Googling it and researching, I've even tried to do it, But I dont understand much about GRUB or linux at all.  Please keep in mind that I am not very good with linux, so please use as many details as possible. How does the Hogwarts Headmaster know whom to invite to school? So all wizards get an invitation to Hogwarts at 11... How do they know whom to invite?  If a wizard is Muggle-born, then nobody would know about their existence.  Or do they register some weird magical stuff being done by them before 11? Series rearrangement and Riemann's theorem I think understand that when I have a conditionally convergent series, it consists of series of positive and negative values which are divergent and thus one can find such permutation of indices $\phi : \mathbb{N}\rightarrow\mathbb{N}$ so the rearranged series sums up to an arbitrary value, diverges or oscillates.   This is how I understand what Riemann's rearrangement theorem says, but how do I use it practically? For example, when I have the conditionally convergent series:  $$\sum_{n=1}^{\infty}\frac{(-1)^n}{\sqrt{n}}$$  and I want to rearrange it to be divergent or to sum up to certain value M? How do I define such bijections $\phi$ ?  I appreciate all help. table side by side with minipage I can't manage to get two tables side by side in a minipage environment. Any kind suggestion would be highly appreciated!   \documentclass{article} \usepackage{array}  \begin{document} \begin{table}[htbp] \centering \begin{minipage}{.2\textwidth}     \begin{tabular}{| m{3cm} | m{4.3cm} | c |}   \hline  number of entity filter (size 1) &amp; number of context filters (size 2, size 3,  size 4) &amp; Accuracy\\ [2ex]   \hline  32  &amp;  32  ,  32  ,  32  &amp;  0.835 \\[0.5ex] \hline 32  &amp;  64  ,  64  ,  64  &amp;  0.821 \\[0.5ex] \hline 32  &amp;  128  ,  128  ,  128  &amp;  0.819 \\[0.5ex] \hline 32  &amp;  256  ,  256  ,  256  &amp;  0.749 \\[0.5ex] \hline 32  &amp;  512  ,  512  ,  512  &amp;  0.683 \\[0.5ex] \hline 32  &amp;  1024  ,  1024  ,  1024  &amp;  0.528 \\[0.5ex] \hline 32  &amp;  2048  ,  2048  ,  2048  &amp;  0.235 \\[0.5ex] \hline 64  &amp;  32  ,  32  ,  32  &amp;  0.850 \\[0.5ex] \hline 64  &amp;  64  ,  64  ,  64  &amp;  0.816 \\[0.5ex] \hline 64  &amp;  128  ,  128  ,  128  &amp;  0.786 \\[0.5ex] \hline 64  &amp;  256  ,  256  ,  256  &amp;  0.795 \\[0.5ex] \hline 64  &amp;  512  ,  512  ,  512  &amp;  0.765 \\[0.5ex] \hline 64  &amp;  1024  ,  1024  ,  1024  &amp;  0.669 \\[0.5ex] \hline 64  &amp;  2048  ,  2048  ,  2048  &amp;  0.331 \\[0.5ex] \hline 128  &amp;  32  ,  32  ,  32  &amp;  0.920 \\[0.5ex] \hline 128  &amp;  64  ,  64  ,  64  &amp;  0.890 \\[0.5ex] \hline 128  &amp;  128  ,  128  ,  128  &amp;  0.886 \\[0.5ex] \hline 128  &amp;  256  ,  256  ,  256  &amp;  0.846 \\[0.5ex] \hline 128  &amp;  512  ,  512  ,  512  &amp;  0.789 \\[0.5ex] \hline 128  &amp;  1024  ,  1024  ,  1024  &amp;  0.755 \\[0.5ex] \hline 128  &amp;  2048  ,  2048  ,  2048  &amp;  0.488 \\[0.5ex] \hline 256  &amp;  32  ,  32  ,  32  &amp;  0.920 \\[0.5ex] \hline 256  &amp;  64  ,  64  ,  64  &amp;  0.922 \\[0.5ex] \hline 256  &amp;  128  ,  128  ,  128  &amp;  0.912 \\[0.5ex] \hline 256  &amp;  256  ,  256  ,  256  &amp;  0.877 \\[0.5ex] \hline 256  &amp;  512  ,  512  ,  512  &amp;  0.872 \\[0.5ex] \hline 256  &amp;  1024  ,  1024  ,  1024  &amp;  0.771 \\[0.5ex] \hline 256  &amp;  2048  ,  2048  ,  2048  &amp;  0.683 \\[0.5ex]  \hline  \end{tabular} \end{minipage}% \hfill \begin{minipage}{.2\textwidth}     \begin{tabular}{| m{3cm} | m{4.3cm} | c |}   \hline  number of entity filter (size 1) &amp; number of context filters (size 2, size 3,  size4) &amp; Accuracy\\ [2ex]   \hline 512  &amp;  32  ,  32  ,  32  &amp;  0.910 \\[0.5ex] \hline 512  &amp;  64  ,  64  ,  64  &amp;  0.922 \\[0.5ex] \hline 512  &amp;  128  ,  128  ,  128  &amp;  0.909 \\[0.5ex] \hline 512  &amp;  256  ,  256  ,  256  &amp;  0.910 \\[0.5ex] \hline 512  &amp;  512  ,  512  ,  512  &amp;  0.883 \\[0.5ex] \hline 512  &amp;  1024  ,  1024  ,  1024  &amp;  0.730 \\[0.5ex] \hline 512  &amp;  2048  ,  2048  ,  2048  &amp;  0.682 \\[0.5ex] \hline 1024  &amp;  32  ,  32  ,  32  &amp;  0.925 \\[0.5ex] \hline 1024  &amp;  64  ,  64  ,  64  &amp;  0.931 \\[0.5ex] \hline 1024  &amp;  128  ,  128  ,  128  &amp;  0.920 \\[0.5ex] \hline 1024  &amp;  256  ,  256  ,  256  &amp;  0.917 \\[0.5ex] \hline 1024  &amp;  512  ,  512  ,  512  &amp;  0.899 \\[0.5ex] \hline 1024  &amp;  1024  ,  1024  ,  1024  &amp;  0.858 \\[0.5ex] \hline 1024  &amp;  2048  ,  2048  ,  2048  &amp;  0.776 \\[0.5ex] \hline 2048  &amp;  32  ,  32  ,  32  &amp;  0.933 \\[0.5ex] \hline 2048  &amp;  64  ,  64  ,  64  &amp;  0.934 \\[0.5ex] \hline 2048  &amp;  128  ,  128  ,  128  &amp;  0.925 \\[0.5ex] \hline 2048  &amp;  256  ,  256  ,  256  &amp;  0.928 \\[0.5ex] \hline 2048  &amp;  512  ,  512  ,  512  &amp;  0.907 \\[0.5ex] \hline 2048  &amp;  1024  ,  1024  ,  1024  &amp;  0.878 \\[0.5ex] \hline 2048  &amp;  2048  ,  2048  ,  2048  &amp;  0.798 \\[0.5ex]  \hline  \hline \end{tabular} \end{minipage} \caption{filter window size} \label{tab:config_filternum} \end{table}   \end{document}   What I have so far  EDIT: I followed the second solution from , and it somehow didn't work. I am not sure why. Which tool for fixing wobbly crank? Any idea what kind of tool should I but to fix the loose left crank on my bike? I found this  but not sure if it's gonna suit. Or is it enough to buy only nut driver? I came across some scenarios like these:     Case 1: Obviously not true       Q: Why does the sun rise and fall every day?   A: Because the Earth .   A2: Because .     Case 2: Letting the OP do what is declared not to       Q: How can I charge my phone without a charger?   A: Try the USB ports on your computer.   A2: Go buy a charger.    Case 3: Attempt to joke (despite looking like a real answer)       Q: What's this round bird in bright red? [An image]   A: It may be a rare or mutated species of sparrow.   A2: It's a .   A3: It's an .    Should I flag A2/A3 as NAA in those cases? Delete from uitableview + write to plist not working I have a geo-tiff created by Pix4D which is about 375 Mb in file size.   I have clipped it to a smaller area using Qgis 2.8(using Raster ‣ Extraction ‣ Clipper), and the output file is about 900 MB in size.  Why did the clipping process generate such a large file, when the source file was smaller?   How do I run the clipper tool so that it produces a smaller size? How to delay a program such as cairo-dock (No openGL) or Firefox?  Sometimes programs crash when started with other startup apps. I want programs to start 10 seconds late to avoid a crash. How to get part name in LaTeX ? Can I successfully bake previously-frozen bread dough? When I gain control of a creature, do I also gain control of any auras or equipment attached to it? What would happen if I 'd a creature that had been enchanted by a bestow creature and cast a spell to give it protection from the enchantment's color on it to remove the bestowed creature?  For instance:  They have just bestowed a  on a  and passes the turn. I use Portent of Betrayal to take control of the Asphodel Wanderer, attack and in the second main I cast Gods Willing to give the Asphodel Wanderer protection from black. Nighthowler detaches and where does it go, my control, opponents control or graveyard?  As an additional question, in the same vein:  What if I destroyed the creature while it was on my side? So change the gods willing to a hero's downfall in the example.  Relevent (under state-based actions):     704.5n If an Aura is attached to an illegal object or player, or is not attached to an object or player, that Aura is put into its owner's graveyard. What is the definition of a timelike and spacelike singularity? Why don't authors include in their publications the duration of the research? Superluminal neutrinos I was quite surprised to read this all over the news today:     Elusive, nearly massive subatomic particles called neutrinos appear to travel just faster than light, a team of physicists in Europe reports. If so, the observation would wreck Einstein's theory of special relativity, which demands that nothing can travel faster than light.   &mdash;  Apparently a CERN/Gran Sasso team measured a faster-than-light speed for neutrinos.   Is this even remotely possible? If so, would it be a real violation of Lorentz invariance or an "" effect?   The ; a webcast is/was planned .  News conference video Let $f:\mathbb{R}_+ \to \mathbb{R}_+$ be a monotone decreasing function defined on the positive real numbers with $$\int_0^\infty f(x)dx &lt;\infty.$$ Show that $$\lim_{x\to\infty} xf(x)=0.$$   This is my proof: Suppose not. Then there is $\varepsilon$  such that for any $M&gt;0$ there exists $x\geq M$ such that $xf(x)\geq \varepsilon$. So we can construct a sequence $(x_n)$ such that $x_n \to \infty $ and $x_n f(x_n ) \geq \varepsilon$. So $$\frac{\varepsilon}{x_n}\leq f(x_n) \implies \sum_{n\in\mathbb{N}}\frac{\varepsilon}{x_n} \leq  \sum_{n\in\mathbb{N}} f(x_n) \leq \int_0^1 f(x)dx.$$ So we get a contradiction.  I feal like I have the correct idea but some details are wrong.  Any help would be appreciated. I took several statistics courses in college but I found that my education was very theory driven.  I was wondering if any of you had a text in Applied Statistics (at the graduate level) that you recommend or have had good experience with. Why doesn't Volumetric follow the shape of Mesh in Eevee? So I was in the middle of making a ball-shaped smoke, but the volumetric effect didn't work following the basic shape. It remains cube in Eevee Blender 2.8; while in Cycle it works, but in Cycle it takes thousands of years for Render. Proving that free modules are flat (without appealing projective modules) Suppose $R\neq 0$ is a commutative ring with $1$. Let $M$ be a free $R$-module. I would like to prove that $M$ is a flat $R$-module. Everywhere I have looked (mostly online) this is proved by first proving that every free module is projective, and then proving that every projective module is flat. Unfortunately, Atiyah &amp; Macdonald's "Introduction to Commutative Algebra" (Chapter 2) does not discuss projective modules. But the result that every free module is flat comes very handy in the exercises.   So my question is,     Is it possible to prove that every free module is flat just by   definitions and without appealing to projective modules?   Thanks! Will my software work on the next version of Windows I have some software and I'd like to know if it works on the next version of Windows (Windows 10).