It seems a trivial question:  Prove/disprove: If we have a vector space $X$, then for any subset $A$ of $X$, we have $A+A =2A$.  It seems that $2A$ is always subset of $A+A$, but I don't think $A+A$ is subset of $2A$. I am thinking in set of integers modulo $p$, for $p$ a prime, as a counterexample. Am I right?
Let $X$ de a vector space and $A\subset X$. If we define $A+B :=\{a+b|a\in A, b\in B\}$ I am trying to show that    $$ \begin{equation} 2A \subsetneq A+A, \end{equation} $$  i.e., it might happen that $2A \neq A+A$. I was able to show that $2A \subset A+A$, but I am not able to come up with a counterexample to show that the converse is not necessarily true. If I assume $A+A\subset 2A$ and let $x\in(A+A)$, then by its definition we have $x=a_1 + a_2$ for some $a_1,a_2\in A$ I guess I could naively say that $a_1+a_2 = 2(a'_1+a'_2)$ where $a_i=a_i/2$ for $i=1,2$ and I guess $(a'_1+a'_2)$ might not be an element of $A$ in some concrete case but I am not able to come up with anything and I've spent more time than I would like to admit. Any suggestions?  Thank you.
I have a quick question about buttons that toggle between two states. (Think Play/Pause, or Shuffle/Regular Play.) As the title says, should the toggle show it's current state or the state to which it will transition?  I think people are used to the Play/Pause convention. But the Shuffle/Regular play might be more confusing if you show the transition state instead of current. For example, the built-in music player on the Xbox 360 does it this way: when it's in shuffle mode, it shows the icon for direct play and vice versa and it always confuses me (am I in shuffle mode or straight play).  I see it this way: Play/Pause is more like action as in begin playing or pause playing. Yes behind the scenes it is a state transition but to the user, there is an action. Whereas Shuffle/Straight Play is an option and it's best to show the current state (and possibly have only one icon and change button to show that the option is enabled/disabled). Thoughts? Swapping two variable value without using third variable A No-Nonsense Introduction to Quantum Field Theory Is there a case insensitive jQuery :contains selector? Is there a case insensitive version of the  jQuery selector or should I do the work manually by looping over all elements and comparing their .text() to my string? Convergence of integrals in $L^p$ Stuck with this problem from Zgymund's book.   Suppose that $f_{n} \rightarrow f$ almost everywhere and that $f_{n}, f \in L^{p}$ where $1&lt;p&lt;\infty$. Assume that $\|f_{n}\|_{p} \leq M &lt; \infty$. Prove that:  $\int f_{n}g \rightarrow \int fg$ as $n \rightarrow \infty$ for all $g \in L^{q}$ such that $\dfrac{1}{p} + \dfrac{1}{q} = 1$.  Right, so I estimate the difference of the integrals and using Hölder end up with:  $$\left|\int f_{n} g - \int fg\right| \leq \|g\|_{q} \|f_{n} - f\|_{p}$$  From here I'm stuck because we are not assuming convergence in the seminorm but just pointwise convergence almost everywhere. How to proceed? why ls reports a huge file and du does not? From where do the permanent magnets get energy from? I have a doubt about permanent magnets. If a magnet is permanent it can attract some materials permanently.  Attracting something involves energy. If a permanent magnet can do this forever, from where does this energy come from? How can it not run out of energy? Doesn't it contradict with the laws of energy? Consider the following figure from Faraway's Linear Models with R (2005, p. 59).    The first plot seems to indicate that the residuals and the fitted values are uncorrelated, as they should be in a homoscedastic linear model with normally distributed errors. Therefore, the second and third plots, which seem to indicate dependency between the residuals and the fitted values, suggest a different model.  But why does the second plot suggest, as Faraway notes, a heteroscedastic linear model, while the third plot suggest a non-linear model?  The second plot seems to indicate that the absolute value of the residuals is strongly positively correlated with the fitted values, whereas no such trend is evident in the third plot. So if it were the case that, theoretically speaking, in a heteroscedastic linear model with normally distributed errors  $$ \mbox{Cor}\left(\mathbf{e},\hat{\mathbf{y}}\right) = \left[\begin{array}{ccc}1 &amp; \cdots &amp; 1 \\ \vdots &amp; \ddots &amp; \vdots \\ 1 &amp; \cdots &amp; 1\end{array}\right] $$  (where the expression on the left is the variance-covariance matrix between the residuals and the fitted values) this would explain why the second and third plots agree with Faraway's interpretations.  But is this the case? If not, how else can Faraway's interpretations of the second and third plots be justified? Also, why does the third plot necessarily indicate non-linearity? Isn't it possible that it is linear, but that the errors are either not normally distributed, or else that they are normally distributed, but do not center around zero? I was recently watching Episode 3 and a few clone wars episodes, and I found myself thinking during the opening battle between the Republic and General Grievous' forces above Coruscant, on a planet so densely populated where potentially thousands of inhabitants can be killed from a drifting capital class ship with a decaying orbit during space battles what defenses are in place?  Is there any instances of some form of emergency services that tractor beam chunks of debris to safe landing places or repulsor towers that can slow or suspend it until it can be taken apart safely? Even disintegration lasers or emergency force field generators like the types deployed on Hoth as they must have the energy output capable of sustaining it. This IS the galactic capital so must have some advanced tech at its deposal. Even now in our own development humans have certain preventative measures for all types of threats and ways of tracking them, plus responses or fall back plans. How can I make a script (or make some changes) to allow me to turn the screen off in Ubuntu (12.04) whenever I want, the way thevscreen can be turned off in Cellphones?  And how can I make a combination of keys (a shortcut) to execute that script?  I want to conserve power.  I've seen this question    but there is no hotkey for it.  Moreover, in the answer, it is said that there are many ways to accomplish this so I'd like to know more. Do the parent directory's permissions matter when accessing a subdirectory? How do I enable Remote Desktop connection on Xubuntu 11.10? How to use Mathematica functions in Python programs? What are the differences between normal and classic difficulties? I can't seem to find a good list of info of all the changes that classic mode offers over normal other than that 'its much harder'.  Does any of the enemy AI improve in classic mode?  Do enemies have more health and do more damage?  Does the panic meter increase faster for countries? What's the name of this flat illustration style? I want to buy some stock photos like the ones below. What is this design called?  I don't even know how to start searching... Unused Business visa I got a 6 months business visa for the UK. But I can't use it.   My relative lives in the USA. In the future, I want to go there to meet with her.  Is there any problem getting a US visa after getting, but not using, a UK business visa? How can I expand a partition into non adjacent free space using GParted? My dual-boot machine has partitions as follows:   sda1 50 GB NTFS sda2 15 GB unallocated sda3 20 GB ext3 sda4 20 GB ext4   I would like to allocate the unused space on sda2 to my Ubuntu partition on sda4, hopefully leaving the other two partitions untouched, leaving this configuration:    sda1 50 GB NTFS sda3 20 GB ext3 sda4 35 GB ext4   Is this possible? If so, how can it be done, preferably using GParted? This is a follow-up question to , and is somehow related to   However, my question does not concern the reviews I receive for the papers I submit, but concerns the reviews I write for papers I have been assigned. Since the whole process is done under confidentiality, it is not clear who owns the copyright on a review I wrote, and what does the review include.   For instance, let us assume that X is member of a PC of a conference Y, and assigns to me a review of a paper Z, written by A and B (assuming it's not a double-blind). Can I publish on my blog: "Here is my review for Y, asked by X, on the paper Z, written by A and B"?   I think that there are two points here:    Is it legal? (for instance, publishing the camera-ready version of a paper might be illegal due to the copyright transfer, would it be also the care here?).  Is it ethical? (who should I ask in order to do so? X? Y? A and B? everybody?).    EDIT: There is been several comments/answer wondering why I would like to publish a review I wrote. To give a bit of background of this question, I believe that the current reviewing system, created when the academic community was small and there was no Internet (i.e. no easy access to information), might not be the best, although clearly working. This is for me a very interesting debate, but somehow out of the scope the Q&amp;A format of Academia SE, which is why I tried to focus on my question on whether it was possible to do so, not if it was the best thing to do in the current system (and just to be clear, I don't plan to do it, but I just like to know what are my options). Anyway, thanks for the answers bringing a different light on this debate.   EDIT 2: After seeing the update in Jeff's answer, I just realised that I didn't make it explicit that I was talking about reviews after the reviewing process. Jeff says that it's ok if the paper is accepted, and although I clearly understand the argument of why I shouldn't publish a review of a rejected paper, the question still holds: by publishing a review of a rejected paper, I publish the information that these authors submitted this paper to this conf/journal, which is supposed to be confidential between the authors and the editors. Would I break any rule by doing so? What damage can person/s do if they have your passport number and visa control number? I read  as well as  but it doesn't tell of what damages can be done. I have seen several sites on which people share/show the visa without the numbers and sometimes without name as well.   The only reason which many people perceive is identity theft but am unable to see how is that possible. I have to use public facilities such as printing, scanning etc. where it is easy for the operator to make copies without my consent. Also as shared, both these details are widely circulated when you stay in a hotel or anywhere else (when abroad) where identity information is needed.   Also particular to visas, as I'm given to understand, for short-stay visas (less than 3 months) which most tourist visas operate under, the window of doing anything nefarious activities (such as terrorism) would be extremely small. If I were a terrorist and I needed those bits to make work, then I would probably go for somebody who has a long-term stay as those credentials would look valid and I could jump from place to place under the radar under a fraudulent identity. Also as a terrorist, I would take quite a long view as any such attack takes years of planning, as shown in the investigation of various terrorist attacks in India as well as the U.S. Also, although I don't know (am speculating here) don't most airports might/should be checking on status quo of a passport to ensure issues such as showing 2 entries of a passport without exit (in some database where this information is collected and collated.)   Am I missing something in the logic here ? How do I ping a website or IP address with Python?