How to uninstall Windows from a Dual Boot system?
How do I remove Windows but keep Ubuntu?
Prove $\mathbb{Z}$ is not a vector space over a field This is an exercise from Chapter 3 of Golan's linear algebra book.  Problem: Show $\mathbb{Z}$ is not a vector space over a field.  Solution attempt:  Suppose there is a such a field and proceed by contradiction. I will write multiplication $FV$, where $F$ is in the field and $V$ is an element of $\mathbb{Z}$.   First we rule out the case where the field has characteristic 2. We would have   $$0=(1_F+1_F)1=1_F1+1_F1=2$$  a contradiction.  Now, consider the case where the field does not have characteristic 2. Then there is an element $2^{-1}_F$ in the field, and   $1=2_F(2^{-1}_F1)=2^{-1}_F1+2^{-1}_F1$  Now $2^{-1}_F1\in\mathbb{Z}$ as it is an element of the vector space, but there is no element $a\in\mathbb{Z}$ with $2a=1$, so we have a contradiction.  Is this correct? To write or to write to? Why are image thumbnails cropped proportionally (not per dimensions)? For some reason WordPress is ignoring my thumbnail sizes and keeping thumbnails cropped proportionately instead of specific dimensions. Here is what I have in functions.php:  add_image_size( 'main-thumbnail', 728, 410, true );   and in content.php:  &lt;div class="thumbnail"&gt;     &lt;a href="&lt;?php the_permalink(); ?&gt;"&gt;        &lt;?php the_post_thumbnail( 'main-thumbnail' ); ?&gt;     &lt;/a&gt; &lt;/div&gt;   I added these two things and regenerated my images with the  plugin, but it doesn't seem to matter. The images don't change.  For some reason WordPress isn't recognizing the hard crop and is keeping the images proportional, so some of my images are way taller than 410 pixels. Why is this? Shorten title on each beamer slide In a beamer presentation the title is long but ok for the first slide. The problem is that when the title appears below on each slide it is very long and I would like to use here some sort of running or shorter title, how can this be done? How to downgrade a package via apt-get? How can I downgrade a package to an older version via apt-get?  Other tools are also acceptable but apt-get is preferred. Asynchronous shell exec in PHP Why does `type which` say that `which is hashed`? Determine the number of functions $f: \{1,2,3....,1999\}\to \{2000,2001,2002,2003\}$satisfying the condition that $f(1)+f(2)+...f(1999)$ is odd. Problems:Determine the number of functions $$f: \{1,2,3....,1999\}\to \{2000,2001,2002,2003\}$$ satisfying the condition that $f(1)+f(2)+...f(1999)$ is odd.   My Attempt: Each integer in domain has $4$ choices and therefore the total number of functions is $f$ is $4^{1999}.$ Since there are an equal number of functions that yield odd or even result, we can directly write that the number of functions satisfying the above condition to be $2*4^{1998}.$ I am unsure about this last claim and would like to prove it (if it is true.)Also is the answer to this problem correct?  Edit $1$: I tried to proceed further in the following manner:   For a given mapping $f$ let the number of integers assigned to $2000,2001,2002$ amd $2003$ be $p,q,r$ and $s$ respectively, where $p,q,r,s\in \mathbb{Z}^+$. Therefore the sum in question can be written as $f(1)+f(2)+...+f(1999)=2000p+2001q+2002r+2003s.$ Clearly the sum will be odd iff $q+s$  is odd. Also note that $p+q+r+s=1999\Rightarrow p+r$ is even. On the other hand if we let $q+s$ as even then $p+r$ must be even. Now the number of values of $p,q,r$ and $s$ for which the above two lemmas hold is the same and therefore if we let $f$ to be the function which maps elements form $\{1,2,3,...,1999\}$ to $\{2000,2001,2002,2003\}$ such that $q+s$ is even and $g$ to be the function  hich maps elements form $\{1,2,3,...,1999\}$ to $\{2000,2001,2002,2003\}$ such that $p+q$ is even we will get a bijection.(Maybe!) Do tag edits count for Archaeologist? I know tag-edits do not count towards the 500 edits required for the Copy Editor badge. Do they count to the 100 for the Archaeologist badge? It seems they should because you are making a 6-month-inactive post active again. I get the following error when trying to ls *.txt | wc -l a directory that contains many files:  -bash: /bin/ls: Argument list too long   Does the threshold of this "Argument list" dependent on distro or computer's spec?  Usually, I'd pipe the result of such big result to some other commands (wc -l for example), so I'm not concerned with limits of the terminal. Gravitational waves and time Two questions.  It is said that time can only go slower in gravity fields and if you move faster.  I have heard that when a gravitational wave pass earth then the space/time vibrate and time change for an instance.  But a wave is often with a top and a valley.  They have now atomic clocks that might be able to detect time variations, placed in satellites around the sun and "quantum entangled". "Using Atomic Clocks to Detect Gravitational Waves"   So what if we could compare time ticks between clocks in space at "absolute frame of reference" where time has its maximum universal speed as time in deep space between Galaxy Filaments.  Will we only detect that time ticks slower for an instance when the gravational wave pass?  1. Or will we detect that time tick slower AND faster than normal time also, as compared to zero gravity time and no movement time in "absolute frame of reference"?  2. Can time go let's say 10 times faster by theory/mathematics, by a extreme theoretical gravitational wave, than time in "absolute frame of reference time"? How does Java's serialization work and when it should be used instead of some other persistence technique? I've been lately trying to learn more and generally test Java's serialization for both work and personal projects and I must say that the more I know about it, the less I like it. This may be caused by misinformation though so that's why I'm asking these two things from you all:  1: On byte level, how does serialization know how to match serialized values with some class?  One of my problems right here is that I made a small test with ArrayList containing values "one", "two", "three". After serialization the byte array took 78 bytes which seems awfully lot for such low amount of information(19+3+3+4 bytes). Granted there's bound to be some overhead but this leads to my second question:  2: Can serialization be considered a good method for persisting objects at all? Now obviously if I'd use some homemade XML format the persistence data would be something like this  &lt;object&gt;     &lt;class="java.util.ArrayList"&gt;     &lt;!-- Object array inside Arraylist is called elementData --&gt;     &lt;field name="elementData"&gt;         &lt;value&gt;One&lt;/value&gt;         &lt;value&gt;Two&lt;/value&gt;         &lt;value&gt;Three&lt;/value&gt;     &lt;/field&gt; &lt;/object&gt;   which, like XML in general, is a bit bloated and takes 138 bytes(without whitespaces, that is). The same in JSON could be   {     "java.util.ArrayList": {         "elementData": [             "one",             "two",             "three"         ]     } }   which is 75 bytes so already slightly  smaller than Java's serialization. With these text-based formats it's of course obvious that there has to be a way to represent your basic data as text, numbers or any combination of both.  So to recap, how does serialization work on byte/bit level, when it should be used and when it shouldn't be used and what are real benefits of serialization besides that it comes standard in Java? How to ask the position(?) of a president (like 10th) of a country, correctly? As much as I have tried, it doesn't seem possible to use separate labels when using a casesconstruction (from the amsmath package).  Ideally, I would like to be able to write something like  \documentclass{article} \usepackage{amsmath} \begin{document} \begin{equation} f(x)=\begin{cases}   1&amp;x\geq0\label{positive}\\   0&amp;x&lt;0\label{negative} \end{cases} \end{equation} \end{document}   and later refer to both cases at different places. What would be the best way to achieve this result? Are there any holidays in Middle Earth, except Yule, Lithe, New Year in Gondor and Durin's Day? How to determine $\int x^2 e^{-x^2}\mathrm dx$? I was wondering how to determine $$\int x^2 e^{-x^2}\mathrm dx?$$ I tried to move the exponential into the integrator, and apply integration by parts, but that will make the integrand more complicated. Thank you! Prove the inequality and limitation Suppose $\{a_k\}$ ,$\{b_k\}$ and $\{\xi_k\}$ are non negative, and for all $k\ge 0$ , we have $$a_{k+1}^2\le(a_k+b_k)^2-\xi_k^2$$   Prove   1.$$\sum_{i=1}^{k}\xi_i^2\le(a_1+\sum_{i=0}^kb_i)^2$$   2. If $\{b_k\}$ also satisfy $\sum_{k=0}^{\infty}b_k^2\lt+\infty $, then $$\lim_{k\to\infty}\frac{1}{k}\sum_{i=1}^k\xi_i^2=0$$   After doing this transformation:$$\xi_k^2\le(a_k+b_k)^2-a_{k+1}^2$$ $\Rightarrow$ $$\sum_{i=1}^{k}\xi_k^2\le(a_1+b_1-a_2)(a_1+b_1+a_2)+(a_2+b_2-a_3)(a_2+b_2+a_3)+\cdots(a_k+b_k-a_{k+1})(a_k+b_k+a_{k+1})$$ then I can't move forward any more. Fermi's golden rule and infinite probablity? I am slightly confused about the application of . Which during standard derivations indicates a probability of transitioning from the state $|i \rangle$ to the state $|f\rangle$ of: $$P=\frac{2\pi t}{\hbar^2} | \langle f | \hat V | i \rangle|^2 \delta(E_f-E_i-\hbar \omega)$$ This holds for large values of $t$ but for arbitrarily large values of $t$: $$P&gt;1$$ Which (to me anyway) makes no physical sense since there can't be a probability greater then 1 of been in the final state. If my interpretation is right then how is this problem resolved/allowable? and if it is wrong what part of what I have said is wrong? How can I specify a long list of math operators? The preambles of my LaTeX documents often include many many lines of \DeclareMathOperator instructions, e.g.  \DeclareMathOperator{\Rep}{Rep} \DeclareMathOperator{\Tet}{Tet} \DeclareMathOperator{\Maps}{Maps} \DeclareMathOperator{\Diff}{Diff}   Is there a nice way to map some macro over a list, so I could do this more concisely, and with less copying and pasting when I add a new math operator? I've seen videos of some folks having several armor sets (4 is the most I've seen so far) standing about on their settlement. I'm wondering if having too many would cause some sets to disappear. They're not like guns/items that you can safely store in settlement crates, the suits kind of just... stand around. (I know there's the extra step of taking out the fusion cores, so that NPC's don't use them.)  Is there a limit to the number of Power Armor sets you can keep?