I am wondering if gravity gets or would get stronger with ultra-compression (relativistic level) of matter? This may sound stupid but this is how my thought process went.  Let us assume we have an object at position $x$ and now let us assume we want to pin-point the exact position of the object therefore we start compressing the object to get exact position therefore our $\Delta x$ will get closer and closer to $x$ however if you think about it in quantum sense you will discover that the Heisenberg uncertainty will start to act, so we can say the $\Delta \vec p$ will increase in other words the velocity will increase because of the inequality:  $$\Delta x \Delta \vec p \geq \frac{\hbar}{2}$$   Now since using the Special relativity transformations we can say as a velocity of an object is increased the mass must also increase by:  $$m' = \frac{m} {\sqrt{1 - \frac{v^2}{c^2}}}$$  I can therefore say since the mass increase while compressing due to the uncertainty in momentum.  As a result of this we can say that the overall-mass will increase $m'$ due to compression and now if we were to substitute these into the Newtonian Gravity equations we will get more gravitational force than expected after compression:  $$F' = \frac{Gm'}{r^2} $$   Which is definitely bigger than if the object was not compressed, to express this we can mathematically make this into a inequality:  $$\frac{Gm'}{r^2} &gt; \frac{Gm}{r^2}$$  As a result I could back-up my theory, but using this can we say that the gravity near neutron-stars or black-holes is slightly stronger than expected by the gravitational equation. Is there any proof of this happening or if not what is wrong with my ideology and could someone correct it for me?
Black holes are thought to originate when a star burns out and collapses to a smaller physical size. This compresses it's matter, but would that increase it's gravity? And would it be enough to capture light? If that star's light could have been observed prior to it's collapse, how could it's burning out and collapse enable it to become a black hole and capture light?
How do I run a game for a larger group? I am currently running a Dungeons and Dragons 4 campaign for a group of 9 players.  Whittling the group down to a more manageable size is not really an option since they are all family.  When we do combat it takes forever even if I have one player help me and allow them to go 2 at a time.  When we do story based adventure there are 2-3 players that do most of the talking and the others twiddle their thumbs.  I've tried encounters with many foes, encounters with only a few tough foes, dialog only encounters, social puzzle encounters, everything I can think of.  What else can I try to get this group more engaged? Is $L^{p}$ space with alternate norm Banach? Primes congruent to 1 mod 6 I came across a claim that I found interesting, but can't seem to prove for some reason. I have the feeling it should be easy  a prime $p$ can be written in the form $p = a^2 -ab +b^2$ for some $a,b\in\mathbb{Z}$ if and only if $p\equiv 1\bmod{6}$ The  is essentially that the nucleus is a ball and the electrons are balls orbiting the nucleus in a rigid orbit.    This  chemists to find a model of chemical bonding where the electrons in the outer orbits could be exchanged. And it works pretty well as seen in the Lewis structures:    However,  were found to be less rigid and instead be fuzzy fields which, instead of being discrete/rigid orbits, look more like:     However, in chemistry education like organic chemistry you still learn about chemical reactions using essentially diagrams that are modified lewis structures that take into account information about electron orbitals:    What I'm wondering is, if the Bohr model is used essentially throughout college education in the form of these diagrams, it seems like it must be a pretty accurate model, even though it turns out atoms are more fuzzy structures than discrete billiard balls. So I'm wondering what the inaccuracies are, and if there is a better way to understand them than the Bohr model. If you build a computer simulation of atoms with the Bohr model, I'm wondering if it would be "accurate" in the sense of modeling atomic phenomena, or is it not a good model to perform simulations on. If not, I'm wondering what an alternative model is that is better for simulation. Essentially, how good the Bohr model is as a diagram, as a tool for learning, and as a tool for simulation. GNOME 2 allows one to either save the session automatically or manually (i.e. save the session as it looks right now). This is a nice feature because it restores the various desktop states between logins (and system restarts):   which apps were open where (physically) on the desktop were they open what specific window sizes were these apps set to   Looking around, I can't find the feature in GNOME 3? Did I not look hard enough? I use a lot of lists and arrays but I have yet to come across a scenario in which the array list couldn't be used just as easily as, if not easier than, the linked list.  I was hoping someone could give me some examples of when the linked list is notably better. Why are all the textures in my file pink? Segmentation fault on large array sizes In the recent post , Werner helped me to redefine a proof environment which can be hidden by an optional boolean argument (showProof). Without an argument, somehow the line/paragraph breaking is not correct. How can one fix this?  \documentclass{scrartcl}  \usepackage{ifthen} \usepackage{environ} \usepackage{amsthm}  \newboolean{showProof}% for showing proofs \setboolean{showProof}{true}% setting the switch \newif\ifstarttheorem  % proof environment (with hide feature) \makeatletter \providecommand{\env@proof@save@env}{} \providecommand{\env@proof@process}{} \RenewEnviron{proof}[2][showProof]{   \gdef\@tmp{#1}%   \ifthenelse{\boolean{#1}}{\par\pushQED{\qed}%     \normalfont\topsep2\p@\@plus2\p@\relax     \trivlist\item[\hskip\labelsep\sffamily\bfseries Proof~#2]\gdef\mycurrenvir{proof}\global\starttheoremtrue\mbox{}\hfill\\*\ignorespaces     \BODY}{\ignorespaces} }[\ifthenelse{\boolean{\@tmp}}{%   \gdef\mycurrenvir{\relax}   \popQED\endtrivlist\@endpefalse }{\ignorespacesafterend}] \makeatother  \begin{document} Just some text \begin{proof}   Just some text in a standard proof. As we can see, there is a problem. \end{proof} \begin{proof}[false]   Proof with optional argument set to \texttt{false}. This proof correctly does   not appear. \end{proof} \begin{proof}[showProof]{(Heading)}   Proof with optional argument (\texttt{true}) and second argument ``(Heading)'' \end{proof} \end{document} Time Dilation When In Orbit Imagine a space ship orbiting the sun at near relativistic speeds, when compared to Earth (say from the reference frame of the sun) there would be x amount of time dilation.  This would mean people on the ship could live for 100's of Earth years.  My question is what would happen if a second ship flew directly away from the Sun and back (ignoring acceleration) at the same speed the orbiting ship is travelling, would the time dilation for the ship that travels in a straight line be the same as that moving in a circle?  The reason I ask is that I was imagining a scenario where intergalactic travel is possible as time dilation means even a journey of light years only ages the crew a fraction of that time, with space stations orbiting certain star systems at similar speeds so that people that don't work on ships can live a similar length of time. I understand that time dilation is all linked to a certain reference frame but working with a reference frame of the sun what would the time dilation be for the different objects taking different paths but at the same speed relative to the sun? If the ship docked back with the space station would they both say the same amount of time had passed? Nothing can ruin your day like a poorly-written bug report.  I've seen several sets of guidelines for bug reporting in different organizations. In your opinion, what guidelines/steps are most essential for good bug reporting?  Feel free to share complete guideline recommendations from prior/current projects/organizations. Searching a virtual printer, that sends emails I need a software, that emulates a printer, converts the printed documents to PDF/TIFF and send it via eMail to a specified address (for windows).  The virtual printer / PDF conversion is possible with a lot of PDF-converters. But I need a software, which is also able to directly mail me these PDFs. It's always the same mailaddress, and I have an SMTP server.  Any suggestions? Dynamic select list in the form (dependent dropdown) mysqli or PDO - what are the pros and cons? In our place we're split between using mysqli and PDO for stuff like prepared statements and transaction support. Some projects use one, some the other. There is little realistic likelihood of us ever moving to another RDBMS.  I prefer PDO for the single reason that it allows named parameters for prepared statements, and as far as I am aware mysqli does not.  Are there any other pros and cons to choosing one over the other as a standard as we consolidate our projects to use just one approach? Why does an unhinged body rotate about its centre of mass? If a force is applied to a body which does not act through its centre of mass, it rotates about its centre of mass and not any other point. Why? How to evaluate the product: How to evaluate the product:  $$\prod _{n=2}^\infty\left(1+\dfrac{1}{n^2}+\dfrac{1}{n^4}+\dfrac{1}{n^6}+\dfrac{1}{n^8}+\ldots\right)$$  $$\left(1+\dfrac{1}{n^2}+\dfrac{1}{n^4}+\dfrac{1}{n^6}+\dfrac{1}{n^8}+\ldots\right)=\dfrac{1}{1-\dfrac{1}{n^2}}$$  How to proceed now? Automatically use specific language in Notepad++ for files with no extension Word for "things which exist" yesterday I installed Ubuntu 13.10, and since I rebooted my laptop, there is a user account called Light Display Manager as available option on my login screen.  A password is needed for login, but I don't know it, although I tried some.  Because of some answers I modified with sudo power one file as follows (added lightdm):  cat /etc/lightdm/users.conf  # # User accounts configuration # # NOTE: If you have AccountsService installed on your system, then LightDM will # use this instead and these settings will be ignored # # minimum-uid = Minimum UID required to be shown in greeter # hidden-users = Users that are not shown to the user # hidden-shells = Shells that indicate a user cannot login # [UserAccounts] minimum-uid=500 hidden-users=nobody nobody4 noaccess lightdm hidden-shells=/bin/false /usr/sbin/nologin   → But it didn't changed anything!  What is this? Why is this there? How can I remove (if it's unnecessary)? Thanks in advance for an answer! What's the difference between a method and a function?