What is the difference between $\Longleftrightarrow$ and $\Leftrightarrow$?
What's the difference between $\equiv$ and $\leftrightarrow$ in a formal proof?
How to find event listeners on a DOM node when debugging or from the JavaScript code? Password protected Visualforce Page rendered as PDF Is it possible that a Visualforce page rendered as pdf can be password protected?  I am using Messaging.SingleEmailMessage to send mail with attachment, but I need the attachment password protected. How to prove that a continuous function, homotopic to the constant map $f:S^1\to S^1$  (a) has a fixed point and that (b) has a point $x$, such that $f$ maps $x$ to its antipodal point $-x$? Where can I find information about past versions of iOS and code-signing windows? I have an iPad 2 that is currently running iOS 5.0.1. I'm generally happy with it but some applications require a more recent iOS version. So I started to look into upgrading it, but the only upgrade that it offers is to iOS 7.  From the look of it I don't like the new design of iOS 7 too much and also upgrading to a version this fresh doesn't appeal to me. So my question is: is there a way to upgrade to some version of iOS 5.x or iOS 6.x instead of upgrading to iOS 7? Can't upgrade due to low disk space on /boot I try to do do-release-upgrade but then I get:       Not enough free disk space      The upgrade has aborted. The upgrade needs a total of 25.7 M free   space on disk '/boot'. Please free at least an additional 25.7 M of   disk space on '/boot'. Empty your trash and remove temporary packages   of former installations using 'sudo apt-get clean'.   Output of df:                      237251272 214797108  10402504  96% / udev                    488120         4    488116   1% /dev tmpfs                   198676       668    198008   1% /run none                      5120         0      5120   0% /run/lock none                    496684         0    496684   0% /run/shm /dev/sda1               233191    225867         0 100% /boot   How come there is no space left on boot? Here's the output of ls -as /boot:  total 221839     4 .     4 ..   645 abi-2.6.32-34-generic-pae   698 abi-2.6.35-30-generic-pae   727 abi-2.6.38-12-generic-pae   727 abi-3.0.0-12-generic-pae   727 abi-3.0.0-13-generic-pae   727 abi-3.0.0-14-generic-pae   727 abi-3.0.0-15-generic-pae   727 abi-3.0.0-16-generic-pae   727 abi-3.0.0-17-generic-pae   727 abi-3.0.0-19-generic-pae   761 abi-3.0.0-20-generic-pae   115 config-2.6.32-34-generic-pae   128 config-2.6.35-30-generic-pae   136 config-2.6.38-12-generic-pae   140 config-3.0.0-12-generic-pae   140 config-3.0.0-13-generic-pae   140 config-3.0.0-14-generic-pae   140 config-3.0.0-15-generic-pae   140 config-3.0.0-16-generic-pae   140 config-3.0.0-17-generic-pae   140 config-3.0.0-19-generic-pae   140 config-3.0.0-20-generic-pae     5 grub 10773 initrd.img-2.6.32-34-generic-pae 13619 initrd.img-2.6.35-30-generic-pae 15365 initrd.img-2.6.38-12-generic-pae 16481 initrd.img-3.0.0-12-generic-pae 16487 initrd.img-3.0.0-13-generic-pae 16501 initrd.img-3.0.0-14-generic-pae 16476 initrd.img-3.0.0-15-generic-pae 16481 initrd.img-3.0.0-16-generic-pae 16478 initrd.img-3.0.0-17-generic-pae    12 lost+found   174 memtest86+.bin   176 memtest86+_multiboot.bin  1700 System.map-2.6.32-34-generic-pae  1841 System.map-2.6.35-30-generic-pae  2115 System.map-2.6.38-12-generic-pae  2141 System.map-3.0.0-12-generic-pae  2141 System.map-3.0.0-13-generic-pae  2143 System.map-3.0.0-14-generic-pae  2146 System.map-3.0.0-15-generic-pae  2147 System.map-3.0.0-16-generic-pae  2147 System.map-3.0.0-17-generic-pae  2148 System.map-3.0.0-19-generic-pae  2149 System.map-3.0.0-20-generic-pae     2 vmcoreinfo-2.6.32-34-generic-pae     2 vmcoreinfo-2.6.35-30-generic-pae     2 vmcoreinfo-2.6.38-12-generic-pae     2 vmcoreinfo-3.0.0-12-generic-pae     2 vmcoreinfo-3.0.0-13-generic-pae     2 vmcoreinfo-3.0.0-14-generic-pae     2 vmcoreinfo-3.0.0-15-generic-pae     2 vmcoreinfo-3.0.0-16-generic-pae     2 vmcoreinfo-3.0.0-17-generic-pae     2 vmcoreinfo-3.0.0-19-generic-pae     2 vmcoreinfo-3.0.0-20-generic-pae  4092 vmlinuz-2.6.32-34-generic-pae  4347 vmlinuz-2.6.35-30-generic-pae  4567 vmlinuz-2.6.38-12-generic-pae  4675 vmlinuz-3.0.0-12-generic-pae  4676 vmlinuz-3.0.0-13-generic-pae  4681 vmlinuz-3.0.0-14-generic-pae  4698 vmlinuz-3.0.0-15-generic-pae  4700 vmlinuz-3.0.0-16-generic-pae  4700 vmlinuz-3.0.0-17-generic-pae  4703 vmlinuz-3.0.0-19-generic-pae  4705 vmlinuz-3.0.0-20-generic-pae   Output of uname -a:   Linux kitsch 3.0.0-17-generic-pae #30-Ubuntu SMP Thu Mar 8 17:53:35 UTC 2012 i686 i686 i386 GNU/Linux What makes you lose the Field Upgrade Score? When playing in a squad, you can get "Field Upgrades". However, sometimes when I die, it tells me that I have lost the "Field Upgrade Score".   Where can I see this score and what are the events that make me lose it? Death in SAO before Kayaba announcement I was wondering if in the novel we knew about some deaths occurring before Kayaba announcement. I remember that in the anime, Kirito plays a bit with Cline before realizing that the log off option has disappeared. Then all the people are summoned to the city and learn about the fact that if they die in the game, they also die in real life.  I don't remember any players dying before the announcement in the anime, except for those who are not waking up and their relative pulling off their NerveGear thus causing death, but maybe in the light novel, some guys died without even knowing it just while being defeated by a mob (which seems awful for me).  Do we know about some players dying before that while thinking they are still in a normal game? Is there a way to transfer thousand of small files faster on Windows 7? I noticed that if I transfer a few large files between two hard drives, it's pretty speedy, at around at least 30&nbsp;MB per second, but if I transfer thousands of files less than 5&nbsp;KB, it is pretty damn too slow.. around 1 to 2&nbsp;MB per second.   Is there a way to speed up the copy/paste process with thousands of small files on Windows 7? When using the Gmail preview pane lab, fastest way to get a direct link to a conversation? In vanilla Gmail, clicking on a conversation alters the fragment ID in the URL, allowing you to link back to that conversation.  (It opens in its own Gmail interface.)  I am using the preview pane, which is great, except that the URL no longer updates when viewing specific emails, so there appears to be no way to get the link to the currently-viewed conversation.  (The 'open in new window' button doesn't help, since it pulls its contents in on the client side; it isn't a standalone link.)  The only solution I've found is to turn off the preview pane, find the message again in the list, open it, grab the link, go back to the message list, and turn preview pane back on.  That's fairly onerous for something I do often though (to clip Gmail conversations to Evernote, where my GTD setup lives). In D&amp;D 5e &quot;&quot; is a condition that can affect a creature during the first round of combat, only until the end of their turn:  [... T]he GM compares the Dexterity (Stealth) checks of anyone Hiding with the passive Wisdom (Perception) score of each creature on the opposing side. Any character or monster that doesn’t notice a threat is surprised at the start of the encounter. If you’re surprised, you can’t move or take an action on your first turn of the combat, and you can’t take a reaction until that turn ends. A member of a group can be surprised even if the other members aren’t. - PHB 189  Does the action that causes the surprise count as part of the round, though? Say Renee, a rogue, achieves a stealth check of 22, easily beating an orc sentry's passive perception. She sneaks directly behind him and stabs him in the back. Surely the orc isn't surprised at all until the dagger hits him (or if Renee misses, she fumbles around behind him or something). What I think it really boils down to in the end is: when is initiative rolled? If (in our example) initiative is rolled before Renee's attack hits (or misses) how can we justify the orc recovering from his surprise before the action that triggered said surprise even occurs, should he roll better than Renee on his initiative? I'd like to go with initiative is rolled immediately following whatever event began the encounter (which could be an attack, but is likely to often be something else wherever surprise isn't intended), but I can't find any documentation to support this stance.  Why Does Anyone Care? There are many reasons you might want someone to be surprised while you're attacking - in this example if Renee is over L3 (and took the Assassin archetype) she'll get 'Assassinate' if the orc is surprised, and 'Death Strike' if she's over L17. That's a lot of potential extra damage!  that if the orc is surprised, Renee will be able to retreat and hide, without having to take the disengage action as her bonus action to avoid an attack of opportunity. Of course, there's no way for her to know whether the orc is surprised or not at that stage, so trying this would be a bit of a gamble... How to construct an example of a probability distribution for which $\mathbb{E}\left(\frac{1}{X}\right)=\frac{1}{\mathbb{E}(X)}$ holds, assuming $\mathbb{P}(X\ne0)=1$?    The inequality which follows from Jensen's inequality for a positive-valued RV $X$ is like $\mathbb{E}\left(\frac{1}{X}\right)\ge\frac{1}{\mathbb{E}(X)}$ (the reverse inequality if $X&lt;0$). This is because the mapping $x\mapsto\frac{1}{x}$ is convex for $x&gt;0$ and concave for $x&lt;0$. Following the equality condition in Jensen's inequality, I guess the distribution has to be degenerate for the required equality to hold. A trivial case where the equality holds is of course if $X=1$ a.e. Here is an example I found in a problem book: Consider a discrete random variable $X$ such that $\mathbb{P}(X=-1)=\frac{1}{9}, \mathbb{P}(X=\frac{1}{2})=\mathbb{P}(X=2)=\frac{4}{9}$. It is then easily verified that $\mathbb{E}\left(\frac{1}{X}\right)=\frac{1}{\mathbb{E}(X)}=1$.   This example shows that $X$ need not be positive (or negative) a.e. for the equality in the title to hold. The distribution here is not degenerate either.  How do I construct an example, possibly like the one I found in the book? Is there any motivation? When I cd a link, my current path is prefixed with the link's path, rather than the path of the directory the link links to.  E.g.  ~/dirlinks/maths$ ls -l logic lrwxrwxrwx 1 tim tim 71 Jul 27 10:24 logic -&gt; /windows-d/academic discipline/study objects/areas/formal systems/logic  ~/dirlinks/maths$ cd logic ~/dirlinks/maths/logic$ pwd /home/tim/dirlinks/maths/logic  ~/dirlinks/maths/logic$ cd .. ~/dirlinks/maths$   I would like to have my current path changed to the path of the linked dir, so that I can work with the parent dirs of the linked dir as well. Besides ls the link to find out the linked dir, and then cd into it, what are some simpler ways to accomplish that? For example, after cd into a link, how do you change your current path to the path of the linked dir? How do I restrict scaling to a specific axis? Im new to Blender but have some experience with 3dsmax. In blender, when i select some polys and try to scale them, i have no control on the amount and especially relative axis on which the scale would happen, where as with 3ds max i could scale restricted to x, y, z or any combination of using the handy widget. Is there a way to replicate this, or to get any control over the scale ?  thanks, How to change fonts I'm sure this is a newbie question, but I can't figure this out. I have tried to follow the advice here:  and here:   But I can not seem to be able to generate any font change. I either get no change or error messages.  Here is the output of one failed attempt to change the font to Times using this command:  \rmfamily \renewcommand{\rmdefault}{ptm}     Apparently I either don't have a correct package installed or...  The output says " Metric (TFM) file not found." but I am not sure where to go from there. What happens in electron-electron collisions? Is there a statement whose undecidability is undecidable? We know there are statements that are undecidable/independent of ZFC. Can there be a statement S, such that (ZFC $\not\vdash$ S and ZFC $\not\vdash$ ~S) is undecidable? Application of Complex Numbers I was just wondering how complex numbers can be applied in electrical engineering and why we use complex numbers over regular, real numbers for this application (e.g what capabilities does the complex number have that real numbers do not in electrical engineering)?  I have done some research concerning impedances and understand how they are written in complex form, however I am still confused why complex numbers are necessary in this field over regular numbers. What is the ideal way to handle data pins D+ and D- on a USB power adapter to be compatible with fast charging on devices? What is an observer in quantum mechanics? My question is not about (pseudo) philosophical debate; it concerns mathematical operations and experimental facts.  What is an observer? What are the conditions required to be qualified of observer, both mathematically and experimentally? Show that $G$ is cyclic Here is a problem from Herstein.     Let G be a finite abelian group so that the equation $x^n=e$ has at most $n$ solutions in $G$ for every positive integer $n$. Show that $G$ is cyclic.   I will have to basically show that there exists an element $g$ in $G$ having order $o(G)$.   I showed that such an element exists if $o(G)=p_1p_2...p_k$ where $p_i$ are primes. But I could not proceed if $o(G)=p_1^{a_1}p_2^{a_2}$. I feel that if I can prove this I will be able to extend.