Example of two closed vector subspaces of $l^{2}$ whose sum is not closed
Sum of closed spaces is not closed
"This question has been asked at Stack Overflow" vs. "on Stack Overflow" How should I phrase it:     This question has been asked at Stack&nbsp;Overflow.   Or,     This question has been asked on Stack&nbsp;Overflow. What are "electron holes" in semiconductors? I'm tutoring senior high school students. So far I've explained them the concepts of atomic structure (Bohr's model &amp; Quantum mechanical model) very clearly. Now the next topic to be taught is semiconductors.  I myself am not conviced with the concept of electron holes. If there is no electron then there is no electron. How can it be a hole. We define a hole when there is some thing every where except at a place. But inside an atom how can we define a HOLE.  Kindly explain it with the help of Bohr's model. What was the need of introducing such abstract concept in semi conductors? How do I get the coordinates of a mouse click on a canvas element? What's the simplest way to add a click event handler to a canvas element that will return the x and y coordinates of the click (relative to the canvas element)?  No legacy browser compatibility required, Safari, Opera and Firefox will do. Proving that, $|f'(x)-f'(y)|\le k|x-y| \implies (f'(x))^2< 2 kf(x) $ Let $f:\Bbb R\to (0,\infty)$ be a differentiable function such that for some constant $k$ we have, $$|f'(x)-f'(y)|\le k|x-y|$$   for all $x,y \in\Bbb R.$ Then prove that, $$(f'(x))^2&lt;2 kf(x).$$   Due to lack of arguments I could not proof this inequality. But rather I proved that is true for the particular function,   $$f(x) =\cos^2(x)+1\implies f'(x) = -\sin(2x).$$ And we readily have,$$|\sin(2x)-\sin(2y)| = \left|\int_{2x}^{2y}\cos t dt\right|\le 2|x-y|$$ as well   $$(f'(x))^2 = 4 \sin^2 x\cos^2x &lt; 4(\cos^2x +1) = 4f(x).$$  It is also true for the function $x\mapsto \sin^2x +1.$   From these example I don't see how to prove the general case. Any hint will be welcome. Laptop using 14.04 won't get past login GUI I have question about ArcGIS and python script. There are following sample code, So I would like to change Another Field , not OIDFieldName. Please let me know how to change Another Field.  import arcpy import itertools  arcpy.env.workspace = r"C:\path\to\workspace"  base_shapefile = "some_shapefile.shp"  oid_field = arcpy.Describe(base_shapefile).OIDFieldName feature_count = int(arcpy.GetCount_management(base_shapefile).getOutput(0))  for (i, j) in itertools.combinations(range(feature_count), 2):     arcpy.Select_analysis(base_shapefile , "combination_%i_%i.shp" % (i, j), '"%s" in (%i, %i)' % (oid_field, i, j))   These sample code is combination for 5C2. I want to create all combiantions 5C3 , 5C4 , 5C5. Can moving tables to another MySQL database help performance? I want to know if I using different databases can help performance.  I have one big table support_chat with more than 5 million rows. I need to scan this all the time (3s with ajax) to check new messages on support.  If I move this table to another database in the same computer, can it help anything? (let users, sells, comments table faster)  For example:  db1  users table sells table comments table support_chat table // move this table to db2   db2  support_chat table   They want to keep old things due legal reasons, to have a user support history. Please require more reviewers on First Posts On the one hand, I see very few posts in the review queue when I visit.  On the other hand, I see a lot of pretty awful first posts sitting on the front page.  This leads me to suspect that robo-reviewers are blowing through the queue.  Given how empty the queue is, could you please increase the number of people who have to check in before a post leaves the queue?  Of course, some honey-pot solution to detect and disable the auto-clickers would be even better. Problems updating Python from the repositories today (Feb 25, 2021) NOTE: This is a Master Canonical Post for the issues observed on February 25, 2020.  All issues on Feb 25 relating to Python updates not found should be dupe-linked here.  Errors have been seen such as this today, February 25, 2021: Failed to fetch http://security.ubuntu.com/ubuntu/pool/main/p/python2.7/libpython2.7-minimal_2.7.17-1~18.04ubuntu1.3_amd64.deb 404 Not Found  Is there something up with the repositories? A person has a computer with no HDD but 16GB RAM and USB3 port.  Also has 16GB USB3 pendrive with Ubuntu Live on it, (Non persistent).  The computer is then started with the pendrive using toram.  The pendrive is then removed and reinserted while the O/S is running.  Is the O/S that is running in RAM able to do a "Full" install to the pendrive?  This was asked in:     However for some reason this unique question was marked as "Duplicate". Because of past conflicts, I don't believe my advisor will write me a good LOR for postdoc positions. What should I do? How essential is having my advisor's letter for post doc application?   I had a pretty bad relationship with my advisor for more than 4 years since I joined his lab and sometimes the relationship feels like slow inflammation that can easily flare up. We had very different opinion on how I want to approach my thesis and we argue very often on the experimental approach and the angle of attack. After that, he turned passive aggressive and completely hands-off. I noticed that he started slow to response to my email and given me very small amount time for meeting in comparison to other people in the lab. However, I proved to him that my approach was right and wrote the full paper myself and got published.   Recently, I got contact by a post doc employer who interested in my own work and request letter of recommendation. I asked my advisor for letter and he told me that I should write my own from which he can modify. I feel this is quite strange as he had wrote for all the other people in the lab, including a departing post doc who is currently interviewing for job. He told me he had done the same for other people but obviously he lied to me.   I further asked if he can provide me a copy of a completed version of the recommendation but he refused.   My question is should I still trust him to write my letter and is it essential? This post doc position is a great opportunity for me and either way I do is a big gamble.  Update: I decided to let my advisor wrote a letter for me and I think I may have made a big mistake. Since he wrote my letter, I asked the post doc lab head if my letter is completed and I no longer be able to hear back from him. Before that, he was quick to response to my email and showed great enthusiasm. As an insurance, I use one opportunity I have less desire to as bait to test the water and I still have one more shot at another position I really want.  Now, I'm thinking requesting open letter from advisor from now on for any future position or use letter service to check the letter.   Any suggestion? Thanks.  update: Great news! Got interviewed and offered at top ones! Thanks everyone! Why can't I launch eclipse from a terminal ? change $PATH environment variable I accidentally committed an unwanted file (filename.orig while resolving a merge) to my repository several commits ago, without me noticing it until now. I want to completely delete the file from the repository history.   Is it possible to rewrite the change history such that filename.orig was never added to the repository in the first place? I'm trying to make a liquid simulation with Mantaflow where the flow of the liquid respects the flow object's local rotation, so it looks like water spraying in different directions. It looks like the options on the flow object only work for 'world' coordinates and doesn't take into account the coordinates of the object. How do I make the water 'fire' (like a hose) into a specific direction that uses local coordinates instead of world ones? siunitx tabular numbers with separated uncertainty I would like to format the space reserved for a number shown in a table with separated uncertainty. It seems that when I use the table-format option the brackets around the mantissa disappear. This is illustrated in the example below:  \documentclass{article}     \usepackage{siunitx}  \begin{document}     \begin{table}   \begin{tabular}{                   S[separate-uncertainty = true]                   S[table-format=1.1(1)e1,                   separate-uncertainty = true]                   }     {Header} &amp; {Header}\\     1.0(2)e2 &amp; 1.0(2)e2\\   \end{tabular} \end{table}     \end{document}     I have also tried to use the bracket-numbers and multi-part-units = brackets without solving this issue. Is there any workaround to center the number and keep the brackets like in the first column of the example? Data Dump of Attachments I am trying to do an export of approx 30,000 attachments in our Salesforce Org,  Would anyone have been through a similar exercise and can advise best practice or the best way / tool to do this?  Kind Regards  matt How to disable X server autostart in Debian Jessie? Why rationalize the denominator? Adding Maximum and Minimum Latitude and Longitude as features in ArcGIS for Desktop? I have several very similar images except some have some extra graphics that enlarge them slightly. This causes some pictures to be offset and not aligned visually when it's clear they should be.  How can I align several tikzpictures along the the same origin they were created and center vertically and horizontally properly?   First, I want to align the 3 large boxes regardless of the what the small boxes are doing Second, I want to be able to center them all without adding additional spaces the \begin{center} seems to do Third, I want to be able to reproduce the exact same results but everything in landscape mode without using the landscape package if possible. (as you can see, the problem is that TeX is putting one rectangle at a time which is not what I want) The methods should be easy and quick. I have many rectangles stacked and some need to be grouped arbitrarily. What these means is I want the methods to sort of be "smart" in that they can figure out themselves so I don't have to go change a lot of stuff each time I have to change the grouping. Basically I would like the first rectangle on the page to define the alignment location and the others on the same page to use it unless I tell them not to.   I've tried creating a custom bounding box as shown in  but it didn't do anything unless I make it too large which then causes other problems. (I can end up doing the alignment but it seems more complex than necessary)  \documentclass[10pt]{book} \usepackage{tikz}  \begin{document} \begin{tikzpicture}     \draw[line width=30pt] (0,0) -- (7,0); \end{tikzpicture}  \begin{tikzpicture}     \draw[line width=30pt] (0,0) -- (7,0);     \draw[line width=20pt] (-0.4, 0) -- (-0.1, 0); \end{tikzpicture}  \begin{tikzpicture}     \draw[line width=30pt] (0,0) -- (7,0);     \draw[line width=20pt] (-0.4, 0) -- (-0.1, 0);     \draw[line width=20pt] (-0.8, 0) -- (-0.5, 0); \end{tikzpicture}    \begin{center}% \begin{tikzpicture}%     \draw[line width=30pt] (0,0) -- (7,0); \end{tikzpicture}% \end{center}%  \begin{center}% \begin{tikzpicture}%     \draw[line width=30pt] (0,0) -- (7,0);     \draw[line width=20pt] (-0.4, 0) -- (-0.1, 0); \end{tikzpicture}% \end{center}%  \begin{center}% \begin{tikzpicture}%     \draw[line width=30pt] (0,0) -- (7,0);     \draw[line width=20pt] (-0.4, 0) -- (-0.1, 0);     \draw[line width=20pt] (-0.8, 0) -- (-0.5, 0); \end{tikzpicture}% \end{center}%     \begin{tikzpicture}%     \draw[line width=30pt] (0,0) -- (0,7); \end{tikzpicture}%   \begin{tikzpicture}%     \draw[line width=30pt] (0,0) -- (0,7);     \draw[line width=20pt] (0,-0.4) -- (0,-0.1); \end{tikzpicture}%  \begin{tikzpicture}%     \draw[line width=30pt] (0,0) -- (0,7);     \draw[line width=20pt] (0,-0.4) -- (0,-0.1);     \draw[line width=20pt] (0,-0.8) -- (0,-0.5); \end{tikzpicture}%    \begin{center}% \begin{tikzpicture}%     \draw[line width=30pt] (0,0) -- (0,7); \end{tikzpicture}% \end{center}   \begin{center}% \begin{tikzpicture}%     \draw[line width=30pt] (0,0) -- (0,7);     \draw[line width=20pt] (0,-0.4) -- (0,-0.1); \end{tikzpicture}% \end{center}%   \begin{center}% \begin{tikzpicture}%     \draw[line width=30pt] (0,0) -- (0,7);     \draw[line width=20pt] (0,-0.4) -- (0,-0.1);     \draw[line width=20pt] (0,-0.8) -- (0,-0.5); \end{tikzpicture}% \end{center}%   \end{document}