I have looked at this question and don't understand almost everything about it. Looking for a detailed answer.  For any two vectors a and b,  Show that $|a+b|^2 + |a-b|^2 = 2|a|^2 +2|b|^2$
For a given inner product space $V$, I would like to prove the inequality $$\|\mathbf{x}+\mathbf{y}\|^2 + \|\mathbf{x}-\mathbf{y}\|^2 = 2(\|\mathbf{x}\|^2 + \|\mathbf{y}\|^2).$$  We see that $$\|\mathbf{x}+\mathbf{y}\|^2 + \|\mathbf{x}-\mathbf{y}\|^2 = \langle\mathbf{x}+\mathbf{y},\mathbf{x}+\mathbf{y}\rangle + \langle\mathbf{x}-\mathbf{y},\mathbf{x}-\mathbf{y}\rangle,$$  which by linearity $$=\langle\mathbf{x},\mathbf{x}+\mathbf{y}\rangle + \langle\mathbf{y},\mathbf{x}+\mathbf{y}\rangle + \langle\mathbf{x},\mathbf{x}-\mathbf{y}\rangle - \langle\mathbf{y},\mathbf{x}-\mathbf{y}\rangle.$$  I've been trying to figure out how to go about this proof using linearity in the second argument of an inner product, but my textbook does not say that linearity necessarily holds in the second component. If it does, how would I go about the rest of this proof? If not, should I be potentially using some aspect of conjugate symmetry to prove this statement?
Combine two columns into a list of all possible combinations of entries I have two columns such as:  a   1  b   2  c    and I need to combine them like:  a   1 a   2 b   1 b   2 c   1 c   2   Is it possible with a formula? Bank Deposit Scam I was on Instagram and saw someone's post about making extra cash.  I inboxed her for details and she said her manager will deposit funds into my account, then I would send them $2000 to split between the both of them.  She stated that I would need to overnight my bankcard to her so her manager can process the deposit.  She didn't ask for anything (neither SSN nor routing or account number) but the bank card only.  I assume that it is a scam.  Can anyone elaborate on this type of scam?  She claim it's not a scam. I'm having serious problems installing the Broadcom drivers for Ubuntu. It worked perfectly on my previous version, but now, it is impossible.   What are the steps to install Broadcom wireless drivers for a BCM43xx card?  I'm a user with no advance knowledge in Linux, so I would need clear explanations on how to make, compile, etc.    lspci -vnn | grep Network showed:   Broadcom Corporation BCM4322 802.11a/b/g/n Wireless LAN Controller [14e4:432b]      iwconfig showed:  lo        no wireless extensions. eth0      no wireless extensions.   NOTE: Answer below is updated every time new information is added and confirmed working. What is the difference between image size and image quality? I have a Casio EX-S12. It has a setting for image size, but also a separate setting for what it calls "image quality". There are three image quality values: "Fine", "Normal", and "Economy".  The Casio manual is not clear on what image "quality" actually means -- specifically how it's different from image size. All they say is that the "Fine" setting "helps to bring out details when shooting a finely detailed image of nature that includes dense tree branches or leaves, or an image of a complex pattern." I'm confused because isn't this what image size is all about (more pixels = more detail)?  In terms of memory usage, the manual also says that for a 5 megapixel picture, a "Fine" image takes up 2.99 MB, a "Normal" image takes up 1.62 MB, and an "Economy" image takes up 1.12 MB, so this "image quality" setting of theirs is certainly having a significant impact on memory usage.  My question is, what exactly is "image quality", if it's not image size? What is the "thing" that is taking up additional memory?  Thank you. Picking random points in the volume of sphere with uniform probability What is the maximum size of a web browser's cookie's key? What is the maximum size of a web browser's cookie's key?  I know the maximum size of a cookie is 4KB, but does the key have a limitation as well? This is a  about how Unix operating systems report memory usage.   Similar Questions:                    I have production server that is running Debian 6.0.6 Squeeze  #uname -a  Linux debsrv 2.6.32-5-xen-amd64 #1 SMP Sun Sep 23 13:49:30 UTC 2012 x86_64 GNU/Linux   Every day cron executes backup script as root:  #crontab -e  0 5 * * * /root/sites_backup.sh &gt; /dev/null 2&gt;&amp;1   #nano /root/sites_backup.sh  #!/bin/bash str=`date +%Y-%m-%d-%H-%M-%S` tar pzcf /home/backups/sites/mysite-$str.tar.gz /var/sites/mysite/public_html/www mysqldump -u mysite -pmypass mysite | gzip -9 &gt; /home/backups/sites/mysite-$str.sql.gz cd /home/backups/sites/ sha512sum mysite-$str* &gt; /home/backups/sites/mysite-$str.tar.gz.DIGESTS cd ~   Everything works perfectly, but I notice that Munin's memory graph shows increase of cache and buffers after backup.  Then I just download backup files and delete them. After deletion Munin's memory graph returns cache and buffers to the state that was before backup.  Here's Munin graph:  Externally hosted image was a dead link. rsync and symbolic links I want to backup my home directory to an external drive nightly using a cron job to execute rsync.  I am unsure of the exact behavior of rsync's symbolic link flags.   rsync's -a flag includes the -l flag (i.e. "copy symlinks as symlinks").  Does this just mean it will copy the link or that it will follow the link and copy everything in the link-to directory?  I want to avoid that because I have links to directories full of media files that would involve copying hundreds of gigabytes I don't need to back up. Fearing (but not certain) that rsync -a would copy all those media files I instead added the --no-links flag.  This does not seem to be behavior I want.  It just ignores copying any link which is problematic because I do have links I want copied (e.g. links to common header files from different project directories). Assuming #1 above (without the --no-links flag) is what I really want and it just copies the link without copying the linked-to files, will the links break when they are backed up?  For example I may rsync source directory /home/me/projects/misc to /media/extdrive/backup/home/me/projects/misc.  In this case I assume rsync is not smart enough nor does it try to correct the contents of symlinks for the relative directory changes.  Is this correct?  This is okay, it doesn't matter if the links are broken in the backup directories so long as they would be fixed and working if such time comes when they need to be restored. UMVUE of $e^{-\lambda}$ from poisson distribution Let $X_1,\ldots,X_n \sim$ Poisson$(\lambda)$. I wish to find UMVUE of $e^{-\lambda}$. Here $\overline{X}_n$ is complete and sufficient for $\lambda$ (hence for $e^{-\lambda}$??). Define $Y_i=1$ if $X_i=0,$ $Y_i=0$ otherwise. Clearly (I checked!) each $Y_i$ and hence $\overline{Y}_n$ is unbiased for $e^{-\lambda}$. Then I should be able to use the Lehmann Scheffe theorem to deduce the UMVUE. The estimator I get looks like $$\frac{1}{n} E\left\{\#X_i:X_i=0 \mid \sum_{i=1}^n X_i\right\}$$ whereas the correct answer is $$\left(\frac{n-1}{n}\right)^{\sum_{i=1}^n X_i}$$ I'd appreciate if anyone lets me know where I'm going wrong and guides me into the right path. Thank you. If a motherboard doesn't explicitly support a RAM frequency, is still compatible? For example, my  says this:   Supports DDR3 2133(OC)/1866(OC)/1600/1333/1066 non-ECC, un-buffered memory   Is RAM clocked at 2400 MHz compatible? Will the performance be downgraded or it just won't work?  I found other questions but they were related to the CPU cap at 1600 MHz. This cap is related to the motherboard, specs says 2133(OC) max. Plotting two variables as lines using ggplot2 on the same graph How do I get lightning:datatable to do some undocumented thing X? What is the right way to consistently mangle bash terminal? I'm on OSX 10.11.1 and occasionally my bash terminal gets mangled. It often happens when I accidentally cat a binary file. The result can be seen on the image below. The output becomes weird, and I can't type ascii characters anymore.    Even though this happens occasionally, I couldn't find a way to consistently reproduce the issue. Online search recommends doing cat /bin/*, but that works sporadically, only after a couple dozen tries.  I want to do this so I can find an easy solution how to handle this in tmux.  How do I consistently get bash to a "mangled" state? Is there maybe a magic unicode character that can do this? Why do damaged hard drives freeze the entire system? I want to check if points layer that I have intersect with polygons layer that I have, as Boolean column in the points dataframe. I have two GeoPandas dataframes, the first one is many points and looks like this: &gt;&gt;&gt;ID  geometry 0  12  POINT (5.0279 7.4547) 1  45  POINT (6.6539 12.139) ...  and the second dataframe is one layer of many different polygons  that looks something like this: &gt;&gt;&gt;name     code   geometry 0  Desert   12     POLYGON ((5.52013 13.8902, 5.5265 13.892,...) 1  Water    24     POLYGON ((5.53756 13.88472, 5.5291 13.8791,...) ...  I'm trying to check if there is any intersection between the points layer and the regions layer. For that I have determined the crs and then used intersects : regions=regions.to_crs({'init': 'epsg:4326'}) points=points.set_crs({'init': 'epsg:4326'}) inter=points.geometry.intersects(regions.geometry)  A The script runs with the following warnings:  FutureWarning: '+init=:' syntax is deprecated. ':' is the preferred initialization method. When making the change, be mindful of axis order changes: https://pyproj4.github.io/pyproj/stable/gotchas.html#axis-order-changes-in-proj-6 return _prepare_from_string(&quot; &quot;.join(pjargs)) /opt/conda/lib/python3.8/site-packages/geopandas/base.py:39: UserWarning: The indices of the two GeoSeries are different. warn(&quot;The indices of the two GeoSeries are different.&quot;)  Then when I check the results the only value is False, like all th epoints do not intersect: inter.unique().tolist() &gt;&gt;&gt;[False]  *I saw on QGIS that the there are points that intersect and there are points that do not so there is no way this result is true *I have checked the dtypes - each one of my geodataframes has one column that is geometry and called geometry. My end goal: to add new column in the points geodataframe that will tell if it intersects the regions or not. Calculating roundness/compactness of polygon? I am trying to find ways to quantitatively describe the shapes of various polygons. For my project, these polygons represent lakes, rivers, lagoons, and parks. So they can be almost any shape. One easy metric is to calculate perimeter vs. area, which is at best an only slightly useful metric. But I would also very much like to be able to say something about the 'roundness' of a polygon. Or how 'compact' the shape is on a map.  The only way I can think of doing this easily is to calculate the area of each polygon in relation to a bounding box for that polygon (which I already have). But this seems like a poor solution.  So now I am thinking of something more like this - take the centroid of the polygon, add on a series of buffers of increasing areas (say 50%, 100%, 150%), then compare how much overlap there is between each buffer and the original polygon. A perfect circle will have perfect overlap at 100%, and I can use the 50% and 150% buffers to judge how much and in what way each polygon differs.  But even that feels cumbersome, and like a poor workaround for what somebody else has probably already figured out far better.  For reference, at a minimum I will need to be able to look at the resulting indices for the shape of various polygons, and be able to make an educated guess as to their source (River? Reservoir with dendritic shape? Lake/Lagoon? Park?) How to get back to a site when viewing one of its questions When viewing a question on a site which I don't use (I never registered on it), I can't figure out how to get back to the list of questions of the site.  See this example  from the Stack Exchange Android app. Where should I click to see the list of questions? (I chose Mathematics Educators because it's the most recent site, it is very likely that most users don't have an account on this website)  Here is a random example:    When viewing a question, I don't know how to view the list of questions. This site can't be found in the sidebar since it's not in my Your Sites list. The top menu containing the site's title only opens the sidebar. Set margins in maketitle Is it possible to change the margins in make title, so they match the margins of the document?  I have created the following modification to the \maketitle. I would like everything to align to at the same left margin.    \documentclass[a4paper]{article} %=============================================== % Title and author \makeatletter \def\@maketitle{%   \newpage   \vspace*{-\topskip}      % remove the initial space    \flushleft Incomplete draft. Please do not cite or quote without premission   \vskip 2em   {}   \begingroup\flushleft    % instead of \begin{center}   \let \footnote \thanks   \hrule height \z@        % to avoid the insertion of lineskip glue     {\LARGE \protect\textbf{\@title} \par}%     \vskip 1.5em     {\large       \lineskip .5em          \@author         \par}%     \vskip 1em     {\normalfont       Last updated on \@date}   \par\endgroup            % instead of \end{center}   \vskip 1.5em             % &lt;--- modify this to adjust the separation } \makeatother %============================================== \title{Physician Information Acquisition In a Dynamic Setting} %\email{rud} \author{ \large {\Large Rud Faden} \\ \textit{\normalfont University of Copenhagen} \\ \textit{{\small \href{mailto:rudfaden@gmail.com}{rudfaden@gmail.com}}} } \date{\today} % no date \usepackage{hyperref} \begin{document} \maketitle \abstract{Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enimad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui ocia deserunt mollit anim id est laborum.}    \section{First} Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum. \end{document} I was listening to the song  by Eminem. I've listened it before many times but from the time I've started learning English grammar I am getting confused while reading or listening English. This time I got stuck at the line "Too much pride, between you and I Not a jealous man but females lie". The phrase "between you and I" always sounded correct to me. But recently I learned that me is only used as an object and hearing that phrase sparked my head. Then I searched over the net and found on  that it is actually a hypercorrection.  says,      "...according to many grammarians and stylists a pronoun in a prepositional phrase in English is supposed to be in the oblique case...   ...though there is still disagreement on whether the phrase itself in today's language is grammatically correct or not."   My first question is, why is there a disagreement? Don't we always use the object form in the oblique case? If not then what are those cases?  Secondly, I want to know how common the phrase "between you an I" is? Is it used colloquially by common native English speakers. By common speakers I mean who don't bother grammar. I searched on  and got some results, e.g. . I found the phrase in another song -- . Is it a common mistake among native English speakers or is it used to give some special linguistic affect? I think this phrase has some linguistic affect that is why the singer would have given the name of her song "Between You and I". Because no one would like to use incorrect English in the title of a song, it must have some purpose. A simple pendulum suspended from the ceiling of a stationary lift has period $T_0$. When the lift descends at steady speed, the period is $T_1$. When the lift descends with constant downward acceleration, the period is $T_2$. Explain why $T_0 = T_1 &lt; T_2$.  I know that for small angles of oscillations, $T \approx 2\pi\sqrt\frac{l}{g}$ where $l$ is the length of the pendulum. For the first two cases, since $\sum F = 0$, the value of $g$ is apparently the same. However, I am unable to explain why the period is largest for $T_2$. If $g$ is the same for all cases, shouldn't their periods be the same?