From the equation of Newtonian gravitational force, we know that $$F=\frac{Gm_1m_2}{r^2}.$$ If one of the two objects is Earth, then from $F=GMm/r^2$ and we know $g=GM/r^2$. So 'g' should be infinity, when the object is on the center.
Would the effect of gravity on me change if I were to dig a very deep hole and stand in it? If so, how would it change? Am I more likely to be pulled downwards, or pulled towards the edges of the hole? If there would be no change, why not?
I am in the process of completing some homework and have come across a question that says: "What is the steady-state current?" It is asking in reference to a circuit that includes two capacitors in parrallel and a DC source of 24V and an AC source given by $\epsilon = 20 \cos(120 \pi t)\ \text{V}$, where $t$ is in seconds. I assume that it is asking when the current is constant. Because naturally the current is not constant due to the AC source, would I just consider the DC source for this? How do I make a Villager Trade a Spawn Egg Are "if" and "iff" interchangeable in definitions? In some books the word "if" is used in definitions and it is not clear if they actually mean "iff" (i.e "if and only if"). I'd like to know if in mathematical literature in general "if" in definitions means "iff".  For example I am reading "Essential topology" and the following definition is written:     In a topological space $T$, a collection $B$ of open subsets of $T$ is said   to form a basis for the topology on $T$ if every open subset of $T$ can be written as a union of sets in $B$.   Should I assume the converse in such a case? Should I assume that given a basis $B$ for a topological space, every open set can be written as a union of sets in $B$?  This is just an example, I am not asking specifically about this sentence. Can a probability distribution value exceeding 1 be OK? Cannot get grub menu to timeout (or go away) How to enable hibernation? In Ubuntu 12.04 and newer, . How can I enable this back? In the sentence     What’s funny is Cat’s dry humour.   Wouldn’t it be better to put a comma after “funny”:     What’s funny, is Cat’s dry humour.   This question entails a (at the time of writing) heated discussion in the . How to vertically center a "div" element for all browsers using CSS? if $x+2y+z=4$ and $x,y,z$ are real number. then find maximum value of $xy+yz+zx$  putting $x+z=4-2z$ in $y(x+z)+zx = y(4-2z)+zx = 4y-2yz+zx$  i wan,t be able to  go further,could some help me with this Can Wubi use the ISO I already have? The Windows Installer for Ubuntu desktop instructions say that an ISO image for Ubuntu is downloaded during the installation process.  But what if I already have an ISO image for Ubuntu. Can I make the Windows Installer use that, so it doesn't have to download an ISO image again?  12.04 LTS and 12.10 are the versions of Ubuntu I'm interested in installing, in case that matters. What would happen to a teaspoon of neutron star material if released on Earth? I've read on NASA's page on neutron star that one teaspoonful of that star would weigh over 20 billion tonnes on Earth. If it was somehow possible to bring it to earth would it:   Burn and disappear during Earth atmosphere entry? Assuming we have 20 billion tonnes of mass occupying the volume of a teaspoon here on Earth, would it fall through the ground under its own weight? I've heard the number 7% thrown around a bunch of times, but I am struggling to find reputable data to back it up.   All I found was a quote from Warren Buffet that I couldn't find an original source for and data assembled by individuals on their personal websites. Is there something from an organization such as Market Watch making a statement about this?  I'm hoping to specifically find what the 10 or 20 year return is for investing in the S&amp;P 500 index fund or another diversified portfolio option. How can I check if my (unlocked) phone will work with a particular country's SIM cards, for 3G and calls? I've used my phone overseas with local SIM cards a lot, but recently had a surprising experience where one local SIM card simply wouldn't work with my phone but did work first time on my partner's, even though both were similar (unlocked Android phones) and another SIM card from a different local provider did work. Looking into it, it looks like there are various different standards around, but it all gets very technical very quickly.  Is there any simple way of looking up which SIM cards in a particular country will work with a particular model of phone, for calls, texts and internet access? Fastest way to determine if an integer's square root is an integer I'm looking for the fastest way to determine if a long value is a perfect square (i.e. its square root is another integer):    I've done it the easy way, by using the built-in  function, but I'm wondering if there is a way to do it faster by restricting yourself to integer-only domain. Maintaining a lookup table is impractical (since there are about 231.5 integers whose square is less than 263).   Here is the very simple and straightforward way I'm doing it now:  public final static boolean isPerfectSquare(long n) {   if (n &lt; 0)     return false;    long tst = (long)(Math.sqrt(n) + 0.5);   return tst*tst == n; }   Note: I'm using this function in many  problems.  So no one else will ever have to maintain this code.  And this kind of micro-optimization could actually make a difference, since part of the challenge is to do every algorithm in less than a minute, and this function will need to be called millions of times in some problems.    I've tried the different solutions to the problem:   After exhaustive testing, I found that adding 0.5 to the result of Math.sqrt() is not necessary, at least not on my machine. The  was faster, but it gave incorrect results for n >= 410881.  However, as suggested by , we can use the FISR hack for n &lt; 410881. Newton's method was a good bit slower than Math.sqrt().  This is probably because Math.sqrt() uses something similar to Newton's Method, but implemented in the hardware so it's much faster than in Java.  Also, Newton's Method still required use of doubles. A modified Newton's method, which used a few tricks so that only integer math was involved, required some hacks to avoid overflow (I want this function to work with all positive 64-bit signed integers), and it was still slower than Math.sqrt(). Binary chop was even slower.  This makes sense because the binary chop will on average require 16 passes to find the square root of a 64-bit number. According to John's tests, using or statements is faster in C++ than using a switch, but in Java and C# there appears to be no difference between or and switch. I also tried making a lookup table (as a private static array of 64 boolean values).  Then instead of either switch or or statement, I would just say if(lookup[(int)(n&amp;0x3F)]) { test } else return false;.  To my surprise, this was (just slightly) slower. This is because . Comparing two strings in Bash I have the following if block in my bash script:  if [ ${PACKAGENAME} -eq kakadu-v6_4-00902C ]; then   echo "successfully entered if block!!" fi   The script execution is not entering my if block even though $PACKAGENAME is equal to kakadu-v6_4-00902C. What am I doing wrong? Let $\alpha$ be a regular curve in $\mathbb{R}^2$ and let all of its tangent lines pass through the origin. Also, let $\beta$ be a regular curve in $\mathbb{R}^2$ and let all of its normal lines pass through the origin.   How can I show that $\alpha$ is contained in a straight line through the origin and that $\beta$ is contained in a circle around the origin?  I know that the tangent line $\alpha(t)$ is the line that points in the direction of the tangent vector $T(t)$ and that the normal line of $\beta(t)$ is the line that points in the direction of $N(t)$, but how can I put this all together to complete the proof? Given a compact Hausdorff space $X$, if $a \in X$ is a $G_\delta$ singleton (i.e., $\{ a \}$ is a $G_\delta$ set), then there is a countable local base at $a$.   The point $a$ can be written as countable intersection of open sets let's call $U_{n}$.  How can I use the compactness? What is the next step for proof? Could you give me hint please? As writing matrices in LaTeX is an extremely time-consuming activity, I was wondering if a software or a site exists that automatically gave the transposed form of a matrix, that is to say, given a matrix:  \begin{bmatrix}    0 &amp; -1 &amp; -1 \\    -1 &amp; -1 &amp; 0 \\    -1 &amp; 0 &amp; 1 \\    1 &amp; 0 &amp; 0 \\    0 &amp; 0 &amp; -1 \\    -1 &amp; 2 &amp; 1  \end{bmatrix}   it will produce a transposed matrix, with the columns in place of the lines and conversely...  It would be particularly time-saving! Question mark or bold citation key instead of citation number I've browsed the forums and found a number of posts that have addressed this issue, but none of the solutions seem to work for me. I have the following script that I just copied from the bibtex home page to get familiar with it. Instead of the citation number I get a question mark. I compile using Latex+Bibtex+Latex+Latex+PDFLatex+ViewPDF just as has been previously suggested and the problem persists.   \documentclass[11pt]{article} \usepackage{cite}  \begin{document}  \title{My Article} \author{Nobody Jr.} \date{Today} \maketitle  Blablabla said Nobody ~\cite{Nobody06}.  \bibliography{mybib} \bibliographystyle{plain} \end{document}   My bibliography (Bib.bbl)  @misc{ Nobody06, author = "Nobody Jr", title = "My Article", year = "2006" }    Looking at previous posts one thing that is concerning is that my .bbl looks empty as shown below. Further, I don't have a .blg  \begin{thebibliography}{}  \end{thebibliography} What OpenType math fonts are there available for usage in TeX/LaTeX, e.g. with the unicode-math package?  Here is a MWE which you can use to show off the font.  \documentclass{article} \pagestyle{empty}  \usepackage{amsmath} \DeclareMathOperator\Res{Res} \newcommand*\diff{\mathop{}\!\mathup{d}}  \usepackage{amsthm} \newtheorem{theorem}{Theorem}  \usepackage{unicode-math}  %%% % Set up you text and math fonts %%%  \begin{document}  \begin{theorem}[Residue theorem]   Let $f$ be analytic in the region $G$ except for the isolated   singularities $a_1,a_2,\dots,a_m$. If $\gamma$ is a closed   rectifiable curve in $G$ which does not pass through any of the   points $a_k$ and if $\gamma\approx 0$ in $G$, then   \[     \frac{1}{2\pi i} \int\limits_\gamma f\Bigl(x^{\mathbf{N}\in\mathbb{C}^{N\times 10}}\Bigr)     = \sum_{k=1}^m n(\gamma;a_k)\Res(f;a_k)\,.   \] \end{theorem}  \begin{theorem}[Maximum modulus]   Let $G$ be a bounded open set in $\BbbC$ and suppose that $f$ is a   continuous function on $G^-$ which is analytic in $G$. Then   \[     \max\{\, |f(z)|:z\in G^- \,\} = \max\{\, |f(z)|:z\in \partial G \,\}\,.   \] \end{theorem}  First some large operators both in text: $\iiint\limits_{Q}f(x,y,z) \diff x \diff y \diff z$ and $\prod_{\gamma\in\Gamma_{\bar{C}}}\partial(\tilde{X}_\gamma)$;\ and also on display \[   \iiiint\limits_{Q}f(w,x,y,z) \diff w \diff x \diff y \diff z   \leq   \oint_{\partial Q} f'\Biggl(\max\Biggl\{   \frac{\Vert w\Vert}{\vert w^2+x^2\vert};   \frac{\Vert z\Vert}{\vert y^2+z^2\vert};   \frac{\Vert w\oplus z\Vert}{\vert x\oplus y\vert}   \Biggr\}\Biggr)\,. \]  \end{document}   Here is a similar MWE for ConTeXt.  The images in the answer will only be shown for LaTeX output but the ConTeXt output looks very similar.  %%% % Set up you text and math fonts %%%  \setupmathlabeltext[Res=Res] \definemathcommand[Res][nolop]{\mfunctionlabeltext{Res}} \definemathcommand[diff]{\mathop{}\!\mfunctionlabeltext{diff}}  \defineenumeration   [theorem]   [text={Theorem},    style=italic,    title=yes,    titlestyle=normal,    distance=0pt,    titleleft={(},    titleright={).~},    alternative=serried,    width=fit]  \starttext    \starttheorem[title={Residue theorem}]     Let $f$ be analytic in the region $G$ except for the isolated     singularities $a_1,a_2,\dots,a_m$. If $\gamma$ is a closed     rectifiable curve in $G$ which does not pass through any of the     points $a_k$ and if $\gamma\approx 0$ in $G$, then     \startformula       \frac{1}{2\pi i} \int\limits_\gamma f\Bigl(x^{{\mathbf N}\in\mathbb{C}^{N\times 10}}\Bigr)       = \sum_{k=1}^m n(\gamma;a_k)\Res(f;a_k)\,.     \stopformula   \stoptheorem    \starttheorem[title={Maximum modulus}]     Let $G$ be a bounded open set in $\mathbb{C}$ and suppose that $f$ is a     continuous function on $G^-$ which is analytic in $G$. Then     \startformula       \max\{\, |f(z)|:z\in G^- \,\} = \max\{\, |f(z)|:z\in \partial G \,\}\,.     \stopformula   \stoptheorem    First some large operators both in text:   $\iiint\limits_{Q}f(x,y,z) \diff x \diff y \diff z$   and   $\prod_{\gamma\in\Gamma_{\bar{C}}}\partial(\tilde{X}_\gamma)$;\   and also on display   \startformula     \iiiint\limits_{Q}f(w,x,y,z) \diff w \diff x \diff y \diff z     \leq     \oint_{\partial Q} f'\Biggl(\max\Biggl\{     \frac{\Vert w\Vert}{\vert w^2+x^2\vert};     \frac{\Vert z\Vert}{\vert y^2+z^2\vert};     \frac{\Vert w\oplus z\Vert}{\vert x\oplus y\vert}     \Biggr\}\Biggr)\,.   \stopformula  \stoptext