Let $V,W$ be subspaces of vector space $U$ where $U=V+W$. Now, my question is if it is true that if $X$ is a subspace of U, then $X=(X\cap V)+(X\cap W)$? I think this should be true since the direct sum $V\oplus W$ is a smaller set than $V+W$ which covers the space U. Could someone please help prove or counterexample for why this is or isn't always true? Thanks.
I am struggling with this question from Halmos's text, please ignore the imperative language.  "Suppose that $L, M$  and $N$ are subspaces of a vector space. Show that the equation    $$L \cap (M + N) = (L \cap M) + (L \cap N)$$   is not necessarily true.  Since each of these subspaces has origin in them, clearly there intersections could not be empty. I wasn't able to formulate an example where this result did n't hold. Any help would be highly appreciated.
In the inline math mode ($...$), if the formula is too long, LaTeX will try to break it on operators, e.g.  very long text followed by a very long equation like $a+b+c+d+e+f+g+h+i+j+k+l$ etc   may be rendered as  very long text followed by a very long equation like a+b+c+d+e+f+g+h+i+ j+k+l etc   However, the break won't happen if they are separated by commas, e.g.  very long text followed by a very long equation like $a,b,c,d,e,f,g,h,i,j,k,l$ etc   will overflow the page like  very long text followed by a very long equation like a,b,c,d,e,f,g,h,i,j,k,l etc   How to make LaTeX able to insert line breaks after a comma too? Can not make hibernate on UEFI (Secure boot) Range of values of skewness and kurtosis for normal distribution I want to know that what is the range of the values of skewness and kurtosis for which the data is considered to be normally distributed.   I have read many arguments and mostly I got mixed up answers. Some says for skewness $(-1,1)$ and $(-2,2)$ for kurtosis is an acceptable range for being normally distributed. Some says $(-1.96,1.96)$ for skewness is an acceptable range. I found a detailed discussion here:  regarding this issue. But I couldn't find any decisive statement.   What is the basis for deciding such an interval? Is this a subjective choice? Or is there any mathematical explanation behind these intervals? I am unable to find the link for downloading Ubuntu 16.04 onto my   desktop/laptop. I am able to see the link only for 18.04, but I need 16.04. I need to repair my Windows 10 installation. Where can I get a clean ISO (with install.wim, no Install.ESD, so that I can run DISM against the WIM) of the specific Windows 10 build required for this?  In my case, I need the Anniversary update (Build 14393), Creators Update (Build 15063) or Fall Creators Update (Build 16299.15). Smoothing polygons in contour map? Here is a contour map for which all the polygons of levels are available.  Let ask how to smooth the polygons keeping all vertices preserved in their exact locations?    Indeed the contour is made on top of a grid data, you may suggest then to smooth the grid data and hence the resulting contour will be smoother. Note that this is not working as my desire since the smoothing function such as Gaussian filter will remove small packs of data and will change the range of the third variable e.g., the height which are not allowed in my application.    Actually I'm looking for a piece of code (preferably in Python) which can do the smoothing of 2D polygons (any type: convex, concave, self-intersecting etc) reasonably painless (forget pages of codes) and accurate.   FYI, there is  that does this perfectly, but using third party commercial applications is not my choice for this question.        1)    Scipy.interpolate:        As you see the resulting splines (red) are not satisfactory!  2)    Here is the result using the code given in . It is not working well!      3)    To me the best solution should be something like the following figure in which a square is being smoothed gradually by changing only one value. I hope for similar concept for smoothing any form of polygons.       Satisfying the condition that spline passes the points:     4)    Here is my implementation of "whuber's idea" line by line in Python on his data. There are possibly some bugs since the results are not good.         K=2 is a disaster and so for k>=4.   5)    I removed one point in the problematic location and the resulting spline is now identical to whuber's. But it is still a question that why the method does not work for all cases?        6)    A good smoothing for whuber's data can be as follows (drawn by vector graphics software) in which an extra point has been smoothly added (compare with update   4):      7)    See the result from Python version of whuber's code for some iconic shapes:     Note that the method seems doesn't work for polylines. For the corner polyline (contour) green is what I want but got red. This needs to be addressed since contour maps are always polylines although closed polylines can be treated as polygons as in my examples. Also not that the problem emerged in update 4 has not been addressed yet.  8) [my last]    Here is the final solution (not perfect!):        Remember that you will have to do something about the area pointed by stars.  There is perhaps a bug in my code or the proposed method needs further development to consider all situations and to provide desired outputs. Show that: $\lim\limits_{r\to\infty}\int\limits_{0}^{\pi/2}e^{-r\sin \theta}\text d\theta=0$ I would like to show $\lim\limits_{r\to\infty}\int_{0}^{\pi/2}e^{-r\sin \theta}\text d\theta=0$.  Now, of course, the integrand does not converge uniformly to $0$ on $\theta\in [0, \pi/2]$, since it has value $1$ at $\theta =0$ for all $r\in \mathbb{R}$.   If $F(r) = \int_{0}^{\pi/2}e^{-r\sin \theta}\text d\theta$, we can find the $j$th derivative $F^{(j)}(r) = (-1)^j\int_{0}^{\pi/2}\sin^{j}(\theta)e^{-r\sin\theta}\text d\theta$, but I don't see how this is helping.  The function is strictly decreasing on $[0,\pi/2]$, since $\partial_{\theta}(e^{-r\sin\theta})=-r\cos\theta e^{-r\sin \theta}$, which is strictly negative on $(0,\pi/2)$.  Any ideas? The blackboard bold font in the AMSFonts package only has capital letters.  I sometimes wish to use a blackboard bold "1", for which I can use \usepackage{bbold}. But this changes the entire blackboard bold font, and I like the original AMSFonts versions of the capital letters better.  Is there a simple way for me to get \mathbb{1} from one package and the blackboard bold capitals from another? How to use a string column with pgfplotstable to label datapoints in a pgfplot? I have a comma-separated list of data points. They should be plotted as a scatter plot, and should be labeled using data from the .csv file. Having specified the first column of my file as string type, I still keep running into errors saying Could not parse input [...] as a floating point number, sorry.  Given a CSV file of the following format, how can I read Name and Z as labels?  Name,X,Y,Z ODIN,250,45,2001   Here is my MWE:  \documentclass{article}  \usepackage{tikz,pgfplots,pgfplotstable}     \pgfplotsset{compat=1.5.1}  \begin{document}  \pgfplotstabletypeset[columns/Name/.style={string type}]{table.csv};  \begin{tikzpicture}     \begin{axis}[     visualization depends on=\thisrow{Name} \as \labela,     visualization depends on=\thisrow{Z} \as \labelb,     every node near coord/.append style={font={\tiny}},     nodes near coords=\labela/\labelb,     nodes near coords align={horizontal},]      \addplot[only marks] table [x=X,y=Y,col sep=comma] {table.csv};      \end{axis} \end{tikzpicture}  \end{document}   Note that if the data in column Name is just floats it should work fine. How to find information about the system/machine in Unix? I have always found it difficult to find information about the system itself in Unix, whether it be   Which OS I am using (version number and all, to compare it with the latest available builds)? Which Desktop Environment am I using? If I am using KDE, most of the programs begin with a K and I can say I am using KDE, but there should be some way to query this, say from a script. Which kernel version am I using? (For example, I am using Fedora, and I want to know what Linux kernel version I am using)   Basically, what I miss is a single point/utility that can get all this information for me. Most of the times the solutions to the above would themselves be OS specific. Then, you are stuck. Function returns None without return statement How do we know for certain that space is expanding? How do we know for certain that space is expanding?  Let's say that in the year 1950, we observe that galaxy 1 is 5 billion light years away from us and galaxy 2 is 10 billion light years away from us, putting both galaxies at a distance of 5 billion light years from each other. Then in 2013, we observe that they are now 7 billion light years away from each other so we conclude space is expanding.  We see that galaxies are move away from each other but how does that prove that space is expanding? Could this illusion of expanding space simply be due to something larger in mass pulling the farther galaxy 2 away from galaxy 1? This larger mass could be accelerating the velocity of galaxy 2 faster then it's accelerating the velocity of galaxy 1. OS X Yosemite brought support for recording iPhone/iPad screen natively without having to using AirPlay applications like AirServer or Reflector.   How do I record the iPhone screen on macOS ? Resistors in series vs parallel What is the result of $ \lim_{n \to \infty} \frac{ \sum^n_{i=1} i^k}{n^{k+1}},\ k \in \mathbb{R} $ and why? What is the result of the next limit:   $$ \lim_{n \to \infty} \frac{ \sum^n_{i=1} i^k}{n^{k+1}},\ k \in \mathbb{R} $$  Why (theorem)? Get full URL (including querystring) from an apex REST endpoint Proving $\mathbb{Q}$ isn't homeomorphic to $\mathbb{Z}$ I was trying to dual boot elementary os with windows on my laptop (Asus Zenbook UX305). To do that I shrank the windows partition using the windows disk manager, and then used the Elementary installer's "install alongside windows" option to set up the linux partitions automatically. When I rebooted after installing I was put into the grub console. I was able to boot linux manually with the commands  grub&gt; set root=(hd0,gpt4) grub&gt; linux /boot/vmlinuz-3.19.0-26-generic root=/dev/sda4 grub&gt; initrd /boot/initrd.img-3.19.0-26-generic grub&gt; boot     and I found that running the command  grub&gt; configfile (hd0,gpt4)/boot/grub/grub.cfg   would load the normal grub menu as is supposed to happen - so there don't seem to be any problems with the config file itself. I also found that the root and prefix are on (hd0,gpt1), so running  grub&gt; set root=(hd0,gpt4) grub&gt; set prefix=(hd0,gpt4)/boot/grub grub&gt; insmod normal grub&gt; normal   also loaded the normal grub menu. So possibly the problem is that grub is looking on the wrong partition?  In my BIOS, I have a choice between using grub and the windows bootloader to boot. Selecting the windows bootloader boots me directly into windows without any problems.  However, I can't find any way to get any fixes to stick after a reboot. I've tried rerunning grub-install /dev/sda, recreating the config file with grub-mkconfig, and using the Ubuntu boot repair gui tool - none of these have worked. Is there any way to fix this short of reinstalling either or both OSs? I'd rather not have to reinstall windows if possible. Interesting topics to research in mathematical physics for undergraduates Teenage girl in life-sustaining valley on poisoned Earth visited by another survivor I'm trying to identify a book I read in school, though I can't recall at what grade level; let's say it had to have been published before I graduated high school in 2000. Despite being required reading, it didn't feel much like a classic.  The story took place on Earth, after some catastrophe had caused the air to become poisonous. However, one teenage girl was able to survive because her house was in a valley which was somehow protected from the poisonous winds and was able to sustain its own ecosystem. One day, a man wearing a protective full-body suit and gas mask appears in the valley, and both are rather surprised to find someone else alive. She feeds and shelters the man and shows him how she had come to survive in the valley. Rather abruptly…      the man attempts to rape her, but she overpowers him somehow. She steals his protective suit and starts to leave the valley. He tells her that he had seen birds in a certain direction away from the valley, indicating there might be life there as well, and she leaves.   I believe there was an author's note either at the beginning or end of the book stating that the author had actually died while writing the manuscript, and that their friends or family had finished the book by going off the author's original notes, which I suppose may explain the abruptness of the ending.  In case it wasn't clear from the above, I didn't really like the book. But I'm curious about it regardless.