Glass is a very fragile object. So why does not the optical fiber break? Everytime I take them, I am worried about this problem.
Why can one bend glass fibers without breaking it, whereas glasses one comes across in real life is usually solid?  Is there also a good high-school level explanation of this?
Helpful flags not getting updated after getting 'close question' privilege Possible Duplicate:          I recently got the ability to cast close votes. So, even if I flag a question as 'not constructive' or 'not a real question', it is still considered as a close vote. What is strange is that my 'helpful flag' count has stopped increasing after this change. Or at least it doesn't increase in cases where my flagging was automatically changed to a close vote and the post actually was closed. Is this how it is supposed to be?   I am guessing that once the post gets closed, the moderators are probably not bothered about marking flags as helpful. The only problem is that the Deputy badge will be much more difficult to get now. Earlier all the flags were getting converted to helpful, but now they just go in as close votes and don't show up on my profile page. Maximum number of files in one ext3 directory while still getting acceptable performance? I have an application writing to an ext3 directory which over time has grown to roughly three million files. Needless to say, reading the file listing of this directory is unbearably slow.  I don't blame ext3. The proper solution would have been to let the application code write to sub-directories such as ./a/b/c/abc.ext rather than using only ./abc.ext.  I'm changing to such a sub-directory structure and my question is simply: roughly how many files should I expect to store in one ext3 directory while still getting acceptable performance? What's your experience?  Or in other words; assuming that I need to store three million files in the structure, how many levels deep should the ./a/b/c/abc.ext structure be?  Obviously this is a question that cannot be answered exactly, but I'm looking for a ball park estimate. I am encountering two definitions of the special orthogonal lie algebra, and I would like to know if they are equivalent, and if there are advantages to working with one over the other.  If we begin with an $n$-dimensional vector space $V$ over a field $k$ and a chosen basis, we can define a bilinear form on $V$ by a matrix $S\in M_n(k)$, ie, let $\langle v,w\rangle=v^tSw$ for all $v,w\in V$.  Now $g\in GL_n(k)$ preserves the form ($\langle g(v),g(w)\rangle=\langle v,w\rangle$) if and only if $g^tSg=S$, so all such $g$ form a linear algebraic group $G$.  The tangent space at the identity of $G$ will be contained in that of $GL_n(k)$, so $T_eG\subset T_eGL_n(k)=M_n(k)$, and in fact, $T_eG=\{B\in M_n(k)\mid B^tS+SB=0\}$.  $T_eG$ becomes a lie algebra, $Lie(G)$, if we define the bracket to be the commutator of two matrices.  Now, if $S=I_n$, it follows that $G=O_n(k)$ is the orthogonal group of matrices satisfying $g^tg=I_n$, and $Lie(G)=\mathfrak{so}_n$ is the lie algebra of antisymmetric matrices.  In Humphrey's Introduction to Lie Algebras and Representation Theory, he defines $\mathfrak{so}_n$ to be all matrices $B$ satisyfing $B^tS+SB=0$, where $$ S=\begin{pmatrix} 1&amp;0&amp;0\\ 0&amp;O&amp;I_l\\ 0&amp;I_l&amp;O \end{pmatrix} \hspace{.5in}\text{or}\hspace{.5in} S=\begin{pmatrix} O&amp;I_l\\ I_l&amp;O \end{pmatrix} $$ depending on the parity of $n$.  The matrices obtained in this way are not antisymmetric, nor is the group $G$ preserving the form defined by $S$ the orthogonal group $O_n(k)$.  Are the two groups obtained from considering different $S$ isomorphic?  Are the two lie algebras isomorphic?  If so, why would one prefer one form to the other? Difference between LiveCD, LiveUSB, full-install, and persistence? I am currently running Ubuntu as a guest vm on a Win 7 host via VirtualBox, but I would like to have a portable version of Ubuntu when I am not using my laptop.   I am unsure as to the difference between the following:  1)LiveCD with persistence; 2)LiveCD without persistence; 3)LiveUSB with persistence; 4)LiveUSB without persistence; 5)Full-install.   I have a USB flash drive (32GB), and I also have empty DVDs lying around. I know that persistence means that data can be saved, but I am unsure as to which option to choose. What is the difference between a LiveUSB and a full-install on a USB thumbdrive and why would I want to pick one over the other? Thanks for any help. Hypernym for "audiobook", "podcast", "speech", etc.? Shortcut to view wireless networks in Windows 7 Does anyone know how to create a shortcut which opens the wireless networks dialog box in Windows 7?  I'm looking for a shortcut that takes me directly to the window which shows you the available wireless networks around you. Toolbar options are not displaying in Edit Profile - About me section How tipe are this series?  $$\sum_{n=2}^\infty \frac 1 {n^2\ln n}$$ But $$\sum_{n=2}^\infty \frac 1 {n\ln n\ln(\ln n)}$$ I used Condensation Test, but I stack after that. Thanks! Word for in-house jargon Ubuntu installation does not recognize drive partitioning I have a 1TB drive and installed Windows 7 on a 128GB partition.  When I now try to install Ubuntu 11.04 it does not recognize the Windows partition but offers the complete 1TB drive to install Ubuntu on instead. It displays:    However, in the Ubuntu Disk Utility the Windows partitions are recognized.    What do I need to do in order for Ubuntu to recognize the Windows 7 partition and install Ubuntu as a dual boot?   Response to comments  The following commands were executed and the results are shown below:  fdisk -l  WARNING: GPT (GUID Partition Table) detected on '/dev/sda'! The util fdisk  doesn't support GPT. Use GNU Parted.  Disk /dev/sda: 1000.2 GB, 1000204886016 bytes 255 heads, 63 sectors/track, 121601 cylinders Units = cylinders of 16065 * 512 = 8225280 bytes Sector size (logical/physical): 512 bytes / 512 bytes I/O size (minimum/optimal): 512 bytes / 512 bytes Disk identifier: 0x34a38165     Device Boot      Start         End      Blocks   Id  System /dev/sda1   *           1          13      102400    7  HPFS/NTFS Partition 1 does not end on cylinder boundary. /dev/sda2              13       16318   130969600    7  HPFS/NTFS  Disk /dev/sdb: 500.1 GB, 500107862016 bytes 255 heads, 63 sectors/track, 60801 cylinders Units = cylinders of 16065 * 512 = 8225280 bytes Sector size (logical/physical): 512 bytes / 512 bytes I/O size (minimum/optimal): 512 bytes / 512 bytes Disk identifier: 0x14a714a6     Device Boot      Start         End      Blocks   Id  System /dev/sdb1               1       60801   488384001   83  Linux   parted -l  Warning: Unable to open /dev/sr0 read-write (Read-only file system).  /dev/sr0 has been opened read-only. Error: /dev/sr0: unrecognised disk label There are three random variables, $x,y,z$. The three correlations between the three variables are the same. That is,  $$\rho=\textrm{cor}(x,y)=\textrm{cor}(x,z)=\textrm{cor}(y,z)$$  What is the tightest bound you can give for $\rho$? What are the mechanics by which Time Dilation and Length Contraction occur? What are the mechanics of time dilation and length contraction? Going beyond the mathematical equations involving light and the "speed limit of the universe", what is observed is merely a phenomenon and not a true explanation of why time dilates or length contracts.  It has been proven to work out, but do we know why?  Is it something that happens at a subatomic level? How to set spawnpoint? Why are p-values uniformly distributed under the null hypothesis? Euler's equations of rigid body motion from least action principle I would like to derive Euler's equations of rigid body motion from least action principle.   Suppose we are in free space so we have no gravity so Lagrangian is equal to kinetic energy.   $$ L = T = \int_M \rho(x)(x\cdot \Omega_B)^2 d^3x $$  where $\rho$ is density, $\Omega_B$ is angular velocity bivector in reference body frame and $M$ is the rigid body.  The action principle says(if I haven't messed up something) that   $$ 0 = \frac{d}{ds}\bigg|_{s=0}\int_0^T \int_M \rho(x)[x\cdot (\Omega_B(t)+s \delta \Omega(t))]^2 d^3x $$  for every $\delta \Omega$ with zero ends i.e. $\delta \Omega(0)=\delta \Omega(T) = 0$. I should get these equations:  $$ I(\Omega_B) - \Omega \times I(\Omega_B)=0 $$  where  $$ I(\Omega) = \int_M \rho(x) (x\wedge (x\cdot \Omega)) d^3x. $$  But I can't see where I would get the time derivative of $\Omega_B$ from the action principle. Can someone tell me what I'm doing wrong? I guess there is some trouble with that $\delta \Omega$ can't be completely arbitrary. Difference between shadowing and overriding in C#? Can my familiar use a Ring of Spell Storing? Can the familiar options from the Pact of the Chain warlock (quasit, imp, sprite, pseudodragon) attune with a ring of spell storing?  If so, can I cast a spell requiring concentration into the ring and then have the familiar maintain concentration on the spell? Add affiliations to the authors' name in the article class How Do I Opt Out of Privileges? According to  , gold badge holders can now unilaterally close questions as duplicates. I hold seven gold badges for tags on SO, which means I can, on my own, close questions as duplicates for those tagged questions.  Simply put: DO NOT WANT.  However, I do not see any ability in SO (the only SE site I spend much time in) to opt out of any privileges, let alone this one.  Is there a spot that I'm just not seeing that allows me to turn off this new privilege? Getting around the FAT32 4GB file size limit