I would like a detailed explanation and justification if possible regarding the topic. I would like to know what happens at an atomic level. Most of the answers I have found were very basic and did not explain it thoroughly enough. What I have deduced (which does not feel right) is, a denser object has more particles/atoms, therefore, when the object becomes submerged in water it sinks because it exerts more force per area or pressure (because it has more particles to exert the force), that pressure exerted by the object, is stronger than water's (because water is less dense, therefore less pressure) and it sinks, it will only stop sinking at lower depths when the water's pressure increases. Is this similar with why a bullet is able to pierce (not push) through less dense cotton and not denser concrete.  I am a gr.9 student who is enthusiastic about physics and would like to thank in advance for your responses!
I realised, reading another Phys.SE question about  that I don't really understand how  works. Particularly concerning, for a SCUBA diver.  The top answers to that question seem to claim that balloons get their "sense of down" from a pressure differential. They continue: when a car accelerates, the air at the back of the car becomes more dense, and at the front less dense, changing the plane of the pressure differential and so also, the balloon's sense of up. I find that extremely hard to credit. However, I realised that I don't really know why less dense things float in more dense things.  I'm fairly sure it's something to do with displacement of heavier things by lighter things, and I think pressure acting on the lighter thing's surface has something to do with it, but that's about it.
Chapter number starts at 2 not 1 How can I start my chapter from "Chapter 2" instead of "Chapter 1".  I am preparing LaTeX document for each chapter. when I put \chapter{Introduction}, it produces "Chapter 1" Introduction. However I want it to be "chapter 2" Introduction instead. Does double negation distribute over implication intuitionistically? Describing groups with given presentation? $\langle x,y\ |\ xy=yx,x^5=y^3\rangle$ and $\langle x,y\ |\ xy=yx,x^4=y^2\rangle$. I'm trying to describe the groups with presentations $\langle x,y\ |\ xy=yx,x^5=y^3\rangle$ and $\langle x,y\ |\ xy=yx,x^4=y^2\rangle$. I have some problems getting a good picture of what they look like...   For the first one, $xy=yx$ allows us to say that any element in the group can be written as $x^iy^j$ for some $i,j$. More precisely, using $x^5=y^3$, we can always reduce $j$ to $0,1$ or $2$, whereas $i$ could have any value. So I was thinking about $\mathbb{Z} \times C_3$. But the other way around (reducing $i$ this time) it's also $C_5 \times \mathbb{Z}$. But then I should be able to prove that both descriptions are equivalent, which I can't do...  For the second, I proceed similarly and get to the same kind of problem...  Could you tell me where I'm going wrong?  Thank you very much in advance! Title of 80's movie: future-era timetraveling humans go extinct and rescue/kidnap todays air crash victims What is the title of a 80's (i believe?) movie where humans in the far future are going extinct, and are aquiring new humans by going back in time to todays world, kidnapping air travelers that are known to have died that day, from the flying passenger aircraft just before its crashing.  A key visual:    Inside a hangar, a time portal opened, and a tail of a (flying) passenger aircraft extruded into the hangar where a stairs provided access to the back door of the aircraft.  The kidnappers made sure the surviving passengers remain seated and would forget about the whole ordeal. Kidnapped humans got replaced by lookalike dummies, including things they had on them. They tried hard not to alter the timeline. How to open terminal, split to 9 terminals and switch between them using one script? I'm embarking on a journey to create a conference abstract book using XeTeX and got stuck within five minutes:        The missing 'g' in 'somethin' influences the line height of the following line. How can I correct this?  At the end I want to be able to include the abstracts and corresponding information through our online registration system, so this should be a global setting.  My MWE:  \documentclass[10pt, a5paper]{article} \usepackage{fontspec}  %% GEOMETRY \usepackage[left=2cm, right=1.5cm]{geometry} % to change the page dimensions  \newenvironment{AbsHead} {     \fontsize{16}{22}\headfont\noindent }% {     \vspace{0.5 cm} }  \newenvironment{AbsAuthors} {     \fontsize{12}{14}\authorfont\noindent }% {    \vspace{0.5 cm} }  \begin{document}  ￼￼￼￼\setmainfont{Minion Pro} \newfontfamily\authorfont{Helvetica Neue Light} \newfontfamily\headfont{Helvetica Neue Condensed Bold}  \begin{AbsHead}The XIAP  something something something somethin somethin somethin somethin somethin somethin somethin something something\end{AbsHead}   \begin{AbsAuthors}Somebody Someone\end{AbsAuthors}   \noindent Purpose: Lorem ipsum \end{document} How do I install ndiswrapper support for an unsupported RTL8723DE wireless module? Show that $(L^{p},\|\|_{p})$ is a Banach space.  My approach: I prove the statement for $(L^{1},\|\|_{1})$, of the following way, first all, is easy show that $\|\|_{1}$ is a norm. So,  $(L^{1},\|\|_{1})$ is vector space. To show that is a Banach space, note that  Prop. 1: Let $(f_{n})_{n\in\mathbb{N}}\subset\mathcal{L}^{1}$ (where $\mathcal{L}^{1}=\mathcal{L}^{1}(X,\tau,\mu)$ is the space of all integrals function), such that $\sum_{i}{\|f_{n}\|}_{1}&lt;\infty$, then the sequences $\left(\sum_{n=1}^{N}{f_{n}}\right)_{n\in\mathbb{N}}$ converges almost everywhere to the integral function, that we called $\sum_{n\in\mathbb{N}}{f_{n}}$. Furthermore $$\sum_{n\in\mathbb{N}}{\int{f_{n}d\mu}}=\int{\sum_{n\in\mathbb{N}}{f_{n}d\mu}}\quad\lim_{N\to\infty}{\|\sum_{n=1}^{N}{f_{n}}-\sum_{n\in\mathbb{N}}{f_{n}}\|_{1}}=0$$ Prop. 2: Let $(E,\|\|)$ a normed vector space. Then $E$  is a Banach space if and only if for all $(e_{k})_{k\in\mathbb{N}}\subset E$ such that $\sum_{k\in\mathbb{N}}{\|e_{k}\|}&lt;\infty$, $\left(\sum_{k=1}^{n}{e_{k}}\right)_{n\in\mathbb{N}}$ converges in $E$  We take the proposition 2. Let $(f_{n})_{n\in\mathbb{N}}\subset L^{1}$ such that $\sum_{n}{\|\hat{f}_{n}\|_{1}}&lt;\infty$. Let $f_{n}\in\hat{f}_{n}$, then $\|f_{n}\|_{1}=\|\hat{f}_{n}\|_{1}$ and then $\sum_{n}{\|f_{n}\|_{1}}&lt;\infty$, by prop. 1, there exist $F\in\mathcal{L}^{1}$ such that $\|\sum_{n=1}^{N}{f_{n}-F\|_{1}}\to 0$, and then $\|\sum_{n=1}^{N}{\hat{f}_{n}-F\|_{1}}\to 0$. Finally note that $\hat{F}\in L^{1}$ is a Banach space. This was my answer for  $L^{1}$ space, but how I prove the general statement for $L^{p}$ (I want to do a similar response). Edit: I'm stuck in the problem, any idea or hint is appreciated.Thanks!! What is the simplest organism that was observed to learn: change its behavior permanently in response to some event/stimuli in a way that this change persists even if the event does not happen again? SE needs a better social media policy (and could us experienced users get a little love too?) Add something to crontab programmatically (over ssh) I have a deployment script, it must add something to a user crontab (trigger a script that clean logs every XXX days), however this must only be done during the first deployment, or when it needs to be updated.  (I can run xxx.py deploy env or xxx.py update env)  so I have to do this:  Check if my cronJob already exist Put my cronJob if it does not already exist or update my cronjob if one of the parameter of the command is different   I don't see how to add/check/remove something to the crontab without using crontab -e or editing the crontab file (download it, rewrite it, re-upload it)  PS : this is a user specific cronjob, "webadmin" is going to do it and he should not use sudo to do it. I'm trying to include Arabic text in an English document (I'm a new latex user), but I can't seem to get any Arabic text to render at all, using xelatex or any kind of tex. I'm using texshop 3.51 on a mac. I can't get this basic example to work, it just displays question marks in the output pdf.  \documentclass[12pt]{article} \usepackage[utf]{arabxetex} \newfontfamily\arabicfont[Script=Arabic]{Amiri} \begin{document} \begin{arab}[voc] وَهَذِهِ فِقرَةٌ بِالعَرَبِيَة مَعَ كَلِمَة اِنكلِيزِيَة \end{arab} \end{document}     I've downloaded Amiri (and also Scheherazade and Lateef) into Fontbook, and they show up if I go Source->Font->Show Fonts in texshop. If I remove the Amiri part it still doesn't work.   Is there anything I can do? I've looked at other links here, like  and  but I can't even render any of those examples.   Thank you so much for any help! I'll love you forever if I can get this to work. Powering NiChrome wire from battery I'm looking to power nichrome wire from battery. The problem I ran into is that desired length/width of the wire has very small resistance ~ 8 Ohm.  Several kinds of batteries I've tested have internal resistance much higher then that, which leads to wire staying cold and battery getting hot.  Is there any way to increase current drawn from the battery? Some kind of amplifier maybe?  Edit:    1.2V(rechargable) and 1.5V one, while not getting hot itself did not provide enough power to heat the wire. 3V did not provide enough power, even stacking 3 of them in series, only provided about 1V of voltage on the wire, and not enough amperage to heat the wire. 9V battery did provide enough amperage to heat the wire, but at the same time got very hot very quickly itself.   PS. I was wondering if  might work for this purpose? How can I prove that the following summation converges? $$\sum_{n=0}^\infty \sum_{k=0}^n \frac{(-1)^n}{(k+1)\times (n-k+1)}$$  I tried to prove that by proving that the following summation in in absolute value converges so the original one converges too, but that's incorrect.  Any other ideas? dd vs cat -- is dd still relevant these days? I recently realized we can use cat as much as dd, and it's   I know that dd was useful in dealing with tapes where block size actually mattered in correctness, not just performance. In these days, though, are there situations where dd can do something cat can't? (Here I would regard a less than 20% performance difference irrelevant.)  Concrete examples would be nice! Test Class Apex Page Parameters If I have example.com, and blog.example.com, do search engines see those as one site or two? Will using subdomains hurt SEO? I have some technical reasons why I can't use example.com/blog. What would be the quickest way to construct a Python binding to a C or C++ library?  (I am using Windows if this matters.) How do I center floated elements? I am working with dimensionality reduction algorithms. Linear Discriminant Analysis (LDA) is a supervised algorithm that takes into account the class label (which is not the case of PCA for example). I am using Python to do a comparative study between some algorithms.  Why with two classes (k = 2), regardless of the data dimensionality, LDA gives one dimension, i.e. the new subspace is composed of only one dimension? For example if I try to project my dataset (of 100 attributes and 2 classes) to 10-dimensional space by using LDA(n_components=10) I only obtain 1 dimension?  For k&gt;2 I can obtain the number of discriminant vectors that I specified as n_components. How can I get n_components dimensions for k=2?