I am reading An Introduction to Algebraic Topology by Rotman. After proving   Theorem 2.7: For every $k\geq 0$, euclidean space$\mathbb R^n$ contains $k$ points in general position,  the book remarked: There are other proofs of this theorem using induction on k. The key geometric observation needed is that $\mathbb R^n$ is not the union of only finitely many proper affine subsets . I want to prove that observation.
If $U_1$, $U_2,\ldots,U_n$ are proper subspaces of a vector space $V$ over a field $F$, and $|F|\gt n-1$, why is $V$ not equal to the union of the subspaces $U_1$, $U_2,\ldots,U_n$?
When to use ' (or quote) in Lisp? After making it through the major parts of an introductory Lisp book, I still couldn't understand what the special operator (quote) (or equivalent ') function does, yet this has been all over Lisp code that I've seen.    What does it do? Encrypting a folder I'm familiar with bitlocker, having been using it for a while, but I am looking at just encrypting a folder instead. But I just don't understand the options it's giving me.  When I right-click a folder > properties > advanced > [x] encrypt contents to secure data > OK &lt; apply > ...  I get:  () apply changes to this folder () apply changes to this folder, sub-folder and files.   This doesn't make sense to me at all, if I encrypt a folder then the whole folder, everything within it is encrypted, then why should I be given these options, how can I have a folder encrypted and files within that folder unencrypted, that seems stupid to me, it doesn't make sense at all.  Can anybody explain why these options exist and how can I simply encrypt a folder? If $\gcd(a,b)=1$, then $\gcd(a+b,a^2 -ab+b^2)=1$ or $3$. How did Draco apparate within Hogwarts? Intuition behind why order of a subgroup divides order of a group. Why can't I join a friend's world on Minecraft PE? I'm on my tablet playing Minecraft PE. My boyfriend rang me and asked me to join his Minecraft PE world. However, I can't do it because the game doesn't show his world. What's wrong and how can I fix it? "Home page" or "homepage"? Is there a convention for the spelling of the name of the main page of a website? Should it be home page, with a space between the two words; or homepage, all one word? Creating a gradient-based legend I have a map with graduated symbology, with 328 unique values. I would like to add a legend to my map (as in the picture), illustrating that light blue indicates low values and dark blue high values.  I have done this in ArcGIS before. Is there a way to do this in QGIS as well? Footnote citations in biblatex ignore spacing before the citation command; how can I have the same behavior for all footnotes in a document?  Here's a MWE demonstrating the issue:  \documentclass{article} \usepackage[style=verbose-trad2]{biblatex} \addbibresource{biblatex-examples.bib} \begin{document} Spacing is fine before citations     \autocite{kant:kpv}  but not before footnotes    \footnote{A footnote \autocite{kant:ku}}. \end{document} My professor sometimes asks me to referee papers for him.  Should I list the corresponding journals in the professionals activity section of my CV? My USB drive used to be originally 8GB when I bought it.  I'm trying to reformatted in Windows 7 by right clicking on the drive and selecting Format.... But the capacity only shows 250MB.  Is there something I can do to get the original size back? Maybe it got partitioned in a weird way?   The  flash drive is a SanDisk Cruzer Micro 8GB. I got it from Wal-Mart but this is the same drive. Displaying two XML listings side by side. I have two XML outputs from a program that I would like to include in a LaTeX document. I would like them to appear side-by-side each in its own box. So that a reader can compare the two outputs while reading the document. Can't seem to find a way to do this, without making images for each output and adding both as images/graphics. Could there be a better option? What are some CD Ripping Programs you can use on Ubuntu? What are some CD ripping programs for Ubuntu? Can you list the Pros? What are the Cons? For each program there should be a screenshot, sources to install, and instructions to install and use. when upgrading to a 200 amp service panel with service drop coming from bottom underground do I need southern California Edison to come disconnect and then reconnect or can I just do it on my own. I do have a permit. At the end of this nice video (), she says that  is a chain reaction of electric and magnetic fields creating each other so the chain of wave moves forward.  I wonder where the  is in this explanation. What is the relation between electromagnetic wave and photon? Solving "mv: Argument list too long"? I have a folder with more than a million files that needs sorting, but I can't really do anything because mv outputs this message all the time  -bash: /bin/mv: Argument list too long   I'm using this command to move extension-less files:  mv -- !(*.jpg|*.png|*.bmp) targetdir/ GridBox/packed array behaviour What are the shortcomings of the Mean Absolute Percentage Error (MAPE)? The  () is a common accuracy or error measure for time series or other predictions,  $$ \text{MAPE} = \frac{100}{n}\sum_{t=1}^n\frac{|A_t-F_t|}{A_t}\%,$$  where $A_t$ are actuals and $F_t$ corresponding forecasts or predictions.  The MAPE is a percentage, so we can easily compare it between series, and people can easily understand and interpret percentages.  However, I hear that the MAPE has drawbacks. I'd like to understand these drawbacks better so I can make an informed decision about whether to use the MAPE or some alternative like the MSE (), the MAE () or the MASE (). Adding materials to more than one object I'm trying to set up a way for me to just type newjournal, and a new journal entry named year-month-date-time.txt is created in a year subdirectory in ~/documents/journals.   I've got it set up as an alias using the following:  alias newjournal='nano /home/username/documents/journals/'$(date +%Y)'/'$(date +%F-%k%M).txt''   So the year subfolder resolves correctly, and seemingly so does the first half of the date command. Tried appending the alias with sudo, didn't help.  Issue: For some reason, when I do this command the time value (%k) resolves to a few minutes ago... or maybe the last time nano ran this filename, maybe it's called an old buffer?   When I logoff it makes the correct file, then I save, wait a few minutes and try again, and it edits the old file instead of making a new one. Even if I remove the old file, it creates a new file with the old filename (incorrect time).  Here's a screenshot of what I'm trying to explain. Here I create a newjournal, save in nano within a second or two of executing the command, print the correct date/time, list the file that was created. Note the incorrect filename time (1249, as apposed to 13:11):    Anyways, it might be something simple, but I've tried different methods in the alias, such as putting double quotes around the whole thing and escaping properly, single quoting, escaping and spacing, etc.  Thanks for the read/help. Ubuntu 16.04 server.  EDIT: Here's the final alias that worked:  alias newjournal='nano /home/user/journals/"$(date +%Y)"/"$(date +%F-%H%M).txt"'   Needed to double quote the date commands inside the overall single-quoted alias.