What is the physical principle behind the following behavior: When left with the remainder of a bag of chips, or cookie crumbs, etc., when shaking this container, the smaller particles will settle to the bottom and the larger particles will rise to the top
Most of the time when I mixed different nuts in a bowl, I observed that the big Brazil nuts were always on top while the small peanuts settled near the base. No matter how you take it, if the big nuts are put at the bottom and then the small nuts are poured, then also after some time the big ones will come to the top. Will not the gravity attract those big nuts more and eventually will they not remain at the base? But it doesn't happen!   I googled it and found the term Brazil-nut Effect but couldn't find any proper explanation. What is the physical explanation for this effect?
Can anyone help me with a solution? I was asked today in class why you divide the sum of square error by $n-1$ instead of with $n$, when calculating the standard deviation.  I said I am not going to answer it in class (since I didn't wanna go into unbiased estimators), but later I wondered - is there an intuitive explanation for this?! How to remove the Ubuntu menu of Windows Boot Manager? I have an ASUS K55VM and it came with DOS, and then Windows 7 was installed. I am planning to install Ubuntu on it. My computer had 4 partitions: 240GB, 230GB, 230GB, and 230GB, and I changed this to 5 partitions: 243GB, 230GB, 230GB, 150GB, and 80GB. I installed Ubuntu 12.04 to the 80GB partition through a USB flash drive. It installed fine, however the problem is during the Boot option in Windows Boot Manager. When I choose Ubuntu, this what is displayed:  Windows failed to start. A recent hardware or software change might be the cause. To fix the problem:   1. Insert your Windows installation disc and restart your computer.   2. Choose your language settings, and then click "Next,"   3. Click "Repair your computer." If you don't have this disc, contact your system administrator or computer manufacturer for assistance.    File: \ubuntu\winboot\wubildr.mbr   Status: 0xc000000e   Info: The selected entry could not be loaded because the application is missing or corrupt.   I tried to change the boot option for the OS and made Ubuntu as my first option. It worked, however Windows 7 does not show up in GRUB. In this fear I did the following:     Restarted my PC changed boot option 1 to Windows.   Formatted the drive to which Ubuntu was installed (80GB one) through Windows.   Restarted again.     The problem persists. Windows Boot Manager still shows Ubuntu, and when I choose this it gives two lines one like: grub rescue&gt;, but my Windows 7 is working perfectly fine.  Now I want a solution to remove Ubuntu from the Windows boot option and reinstall completely. Many of the Ringbearers (particularly those who carried it longer) used the word "precious" to describe it. Isildur described it as:     "But for my part I will risk no hurt to this thing: of all the works of Sauron the only fair. It is precious to me, though I buy it with great pain."   Bilbo himself used it at least once, when arguing with Gandalf about leaving it behind when he was leaving Bag End (this exchange and the lines were different in the movie, but the word is still there):     "Well, if you want my ring yourself, say so!" cried Bilbo. "But you won’t get it. I won’t give my precious away, I tell you."   And Gollum used it incredibly frequently, almost as if it was the ring's name. It was even his last words:     "Precious, precious, precious!" Gollum cried. "My Precious! O my Precious!"   So my question: Why "Precious"? It wasn't part of the inscription. It wasn't part of the poem that included the inscription. Was it somehow part of Sauron's crafting spell? Did Tolkien ever indicate why the Ringbearers would use that word in particular, other than that it sounded good? How can I buy apps in the App Store without a credit card? Please add the ability to fold blocks of code in questions and answers It would be great to have the option to fold code snippets, entire questions and answer posts on a user basis and allow defining some generic foldable regions within the question/answer editor.  Non generic folding info could be stored in cookies perhaps to move the storage/processing requirement to the client side.  Definable folding can take any free and safe HTML tag or other multi character trick (&lt;>Fold this&lt;>) My disk already has 4 primary partitions, how can I install Ubuntu? I would like to install Ubuntu alongside Windows 7 on a HP G62 Notebook. Although I have installed  Ubuntu in a dual-boot many times before, I found out that this model has already four partitions.    Partitions:   SYSTEM (NTFS, 199MB, used 66.59MB) Partition without any tag: NTFS (579GM, used 129GB)  RECOVERY (NTFS 16.74 GB, used 2.42GB)  HP_TOOLS (FAT32, 103.34 MB, used 13.23MB)   Since I am not an expert with partitions I would like to get advice on how to do this.  My first idea is this one:   Free some space from /dev/sda2 (I don't know if could also free some other space) delete the HP_Tools partition (I have already created a backup) create an extended partition with the free space in #1 containing three parititions: swap (1gb); / (EXT4, 30GB);  /home (EXT4, 120GB)   Another option is to use wubi instead.  What do you think? Is there any other way to achieve this?  PS: I really think this HP policy of using 4 partitions is not a coincidence  PS: I tried using gparted from the live CD and I got a warning message saying that if I freed some space from /dev/sda2 I could create serious issues in the system How prove $p!|(a+1)$ if $p!|a^p+1$ Profile Bug - Reputation Summary Not Displaying All Lines Simple question, but one that I've been curious about...is there a functional difference between the following two commands?  String::class String.class   They both do what I expect -- that is to say they return Class -- but what is the difference between using the :: and the .?  I notice that on those classes that have constants defined, IRB's auto-completion will return the constants as available options when you press tab after :: but not after ., but I don't know what the reason for this is... Labelling figures in the tikz package Consider the following piece of code, which uses the tikz-feynman package:  \usepackage{tikz-feynman, contour} \tikzfeynmanset{compat=1.1.0}  \begin{document} \begin{center}     \begin{tikzpicture}[baseline={(current bounding box.center)}]     \begin{feynman}         \diagram [vertical'=a to b] {             {   [edges={draw=none}]                 a -- b [label=0:\(\mu_{3}\)]                 -- c [label=-180:\(\mu_{4}\)]                 -- d [label=-180:\(\mu_{1}\)]                 -- a [label=0:\(\mu_{2}\)],             },             {   [edges={photon}]                 a -- ap [particle=\(p_{2}\)],                 b -- bp [particle=\(p_{3}\)],                 d -- dp [particle=\(p_{1}\)],                 c -- cp [particle=\(p_{4}\)],             },         };         \diagram* {             (c) -- [fermion] (b)                 -- [fermion] (d)                 -- [fermion] (a)                 -- (c),         };     \end{feynman}     \end{tikzpicture}     %     \feynmandiagram[baseline={(current bounding box.center)}, vertical'=a to b,] {         {   [edges={fermion}]             a -- b [label=0:\(\mu_{3}\)]             -- c [label=-180:\(\mu_{4}\)]             -- d [label=-180:\(\mu_{1}\)]             -- a [label=0:\(\mu_{2}\)],         },         {   [edges={photon}]             a -- ap [particle=\(p_{2}\)],             b -- bp [particle=\(p_{3}\)],             d -- dp [particle=\(p_{1}\)],             c -- cp [particle=\(p_{4}\)],         },     };     %     \begin{tikzpicture}[baseline={(current bounding box.center)}]     \begin{feynman}         \diagram [vertical'=a to b] {             {   [edges={draw=none}]                 a -- b [label=0:\(\mu_{3}\)]                 -- c [label=-180:\(\mu_{4}\)]                 -- d [label=-180:\(\mu_{1}\)]                 -- a [label=0:\(\mu_{2}\)],             },             {   [edges={photon}]                 a -- ap [particle=\(p_{2}\)],                 b -- bp [particle=\(p_{3}\)],                 d -- dp [particle=\(p_{1}\)],                 c -- cp [particle=\(p_{4}\)],             },         };         \diagram* {             (a) -- [anti fermion] (b)                 -- [anti fermion] (d)                 -- [anti fermion] (c)                 -- (a),         };     \end{feynman}     \end{tikzpicture}     \end{center} \end{document}   This piece of code draws the following figure:    I would like to label each figure. I do not want a caption, but just a label beneath each diagram. The label should say Diagram I, Diagram II, Diagram III.  How do I implement this into my code? Which is the correct use of these two words, and in which context should one be used rather than the other? Where can I find out about Stack Overflow history? I'd like to know how Stack Overflow and other sites started - which was first, who initiated it (OK, that's on the link below) and why, how the different sites evolved, how the rules evolved, etc.  Where can I find such information?  Of course, I know I can look at  for a brief overview but I'm wondering if there is info on this site, particularly more detailed.. Showing that $\int_0^1 \log(\sin \pi x)dx=-\log2$ I need help with a textbook exercise (Stein's Complex Analysis, Chapter 3, Exercises 9). This exercise requires me to show that $$\int_0^1 \log(\sin \pi x)dx=-\log2$$ A hint is given as "Use the contour shown in Figure 9."    Since this is an exercise from Chapter 3, I think I should use the residue formula or something like that. But the function $f(x)=\log(\sin \pi x)$ becomes singular on $x=0$ and $x=1$, which makes the contour illegal for the residue theorem. Can anyone give me a further hint on this problem? Many thanks in advance!  P.S. This is my first time on Math Stack Exchange. If you find my post ambiguous, let me know. How can I install Ubuntu on a second hard drive with Windows being on the first? I have recently got an Alienware M17x R4 and since it has two hard drive bays I was thinking of putting my hard drive from my old laptop and installing Ubuntu on it without making any changes to the drive that has Windows on it. Also if possible to add a entry to the Windows boot loader to boot Ubuntu instead of using GRUB. Did Snape ever commit murder? It's well known that Severus Snape was, in one way or another, a Death Eater since he was at Hogwarts. He hung out with the likes of Avery and Mulciber who became well known Death Eaters, he joined Voldemort right after he left Hogwarts, and he basically became Voldemort's right hand man until the very end.   However, in his time as a Death Eater did he ever kill anyone? We know that he was a Death Eater for around 3 years before Voldemort fell; Lily and James died at 21 and they were the same age as Snape, so it would have been 3 years since they left Hogwarts. Then after Voldemort came back in book 4, Snape would have been a Death Eater again for around 3 years. That's a total of around 6 years of Death Eaterhood with Voldemort.  So, in either of these periods, did Snape commit murder as part of his job as Death Eater? I'm not counting a certain death at the end of book 6 for reasons which are obvious if you've read book 7.   Clearly it's much more likely he killed before Voldemort fell since that roughly correlated with the period when he pledged allegiance only to Voldemort, before he became a teacher at Hogwarts. But did he ever do it? What is the meaning of the "expansion of space"? When we say that "the space between galaxies is expanding," what do we really mean? For instance, if I think of space as being a Cartesian grid, then when space expands should I think of it as adding more grid-points or as making the distance between the grid-points larger? Or is this a flawed picture of space-time even when far from significant mass densities?  It seems to me that it must be the case that we are "adding more grid-points" because otherwise we would not observe the expansion since anything that occupied that part of the grid (e.g. light, my hand, etc.) would also expand by a corresponding amount (otherwise, after a sufficient amount of expansion, we would be able to observe sub-atomic scale processes as being macroscopic in scale). But if it is the case that we are adding more grid-points, then the "expansion" of space seems like a misnomer: shouldn't it be called the "creation" of space?  That being said, either case (adding grid-points or stretching distances) could conceivably be the same process depending on the nature of space-time (i.e. if it were discrete but dense - like the Rationals - rather than either strictly discrete (Integers) or strictly continuous (Reals)... I hope that analogy makes sense).   So what is really going on in intergalactic space? What is the meaning of its expansion? Does it correspond to creating more space, stretching the space that's already there, or something more subtle? Counting buffer overlap in QGIS I use a Asus laptop with dual-boot Ubuntu and Windows 8.1  Wifi is on but I am not able to detect Wifi networks.  When I run :  lspci -knn | grep Net -A3; rfkill list   This is what happens:    I get  acer-wireless: Wireless LAN     Soft blocked: yes A doubt regarding change of variables in Double Integrals. So I have recently learnt to calculate &quot;Double Integrals&quot; and I have a doubt on change of variables. Lets say we are evaluating a double integral of the form I = $\int\int _R F(x,y) dx dy $ , where $F(x,y)$ is an integrable smooth function and $R$ is the region of integration in the $xy$ plane. Now I want to convert to polar co-ordinates $x(r,\theta)=r \cos(\theta)$ and $y(r,\theta) = r \sin(\theta)$. Now the region $R$ in the $xy$ palne can be suitably converted to some region $R'$ in the $r \theta$ plane. The function $F(x,y)$ may take the form of some smooth and integrable $G(r,\theta)$. My main doubt is with the change of the area element. According to my textbook, $dx dy = \frac{\partial (x,y)}{\partial(r,\theta)} = rdrd\theta$. So the area element in the $r\theta$ plane is $rdr d\theta$. Now I want to experiment a little bit with my understanding : I write $dx = d(r\cos\theta) = -r\sin\theta d\theta + \cos \theta dr$  and $dy = d(r\sin\theta)=r\cos\theta d\theta + \sin\theta dr $. And now I multiply the two expression for $dx$ and $dy$ : $dxdy= -r^2\sin\theta\cos\theta d\theta d\theta - r \sin^2 \theta drd\theta + r \cos^2 \theta dr d\theta + \sin \theta \cos\theta dr dr$. Why isn't this expression equal to $rdrd\theta$ ? The $d\theta d\theta$ and $dr dr$ terms look very odd to me. Am I commiting some abuse of notation or something ? Also there are no higher order terms like $dr dr d\theta$ which I can ignore. I am totally confused with all this. Can someone explain this to me a bit clearly? Thanks in Advance :). EDIT 1 : I know differentials are not the &quot;usual&quot; kind of algebraic quantities and we can't just add and multiply them casually. I am just multiplying $dx$ with $dy$ with the spirit of multiplying some $\delta x$ and $\delta y$ and letting both $\delta x$ and $\delta y$ tend to zero. EDIT 2 : Is there a branch of Mathematics where addition and multiplication of differentials is very common? Is there something like differential of a differential ?