Cauchy–Schwarz inequality intuition We learned in class about the Cauchy–Schwarz inequality. Can some one please  give me a geometric intuition or any intuition in general for why this inequality is true? Thank you.
Intuition on proof of Cauchy Schwarz inequality To prove Cauchy Schwarz inequality for two vectors $x$ and $y$ we take the inner product of $w$ and $w$ where $w=y-kx$ where $k=\frac{(x,y)}{|x|^2}$ ($(x,y)$ is the inner product of $x$ and $y$) and use the fact that $(w,w) \ge0$ . I want to know the intuition behind this selection. I know that if we assume this we will be able to prove the theorem, but the intuition is not clear to me.
Why is fundamental group insufficient to classify manifolds? I have heard that some issues in group theory prevent classifying all manifolds upto homotopy using the fundamental group invariant. Does anyone know what are those issues?  Thanks, K. Water Wars: Time Travel TV Show? Clone internal HDD to new SSD I recently installed an SSD into my machine. The machine itself is a Lenovo thinkpad W520, and it previously had an internal HDD. I moved the internal HDD into an expansion bay (replacing the CD-ROM), and put the new SSD into the internal bay.   The problem I'm having is that I have my Ubuntu configuration EXACTLY the way I want it - I originally spent many hours configuring it to get it to the way it is now. I'd rather not do this again. But, I'd also like the boot-up gains I would get from the OS being on the SSD.  So, what I'd like to do is clone my Ubuntu partition onto the SSD. The catch is that the standard HDD is significantly bigger than the SSD. And it has a windows partition that I don't need on the SSD (I never use Windows, so if it boots off of the other hard drive, that's fine). The layout of my hard drives are as follows:  /dev/sda (SSD):     Model: ATA M4-CT256M4SSD2 (scsi)     Disk /dev/sda: 256GB     Sector size (logical/physical): 512B/512B     Partition Table: msdos  Number  Start   End    Size   Type     File system  Flags  1      1049kB  147GB  147GB  primary  ext4         boot   /dev/sdb (HDD): Model: ATA ST9500420AS (scsi) Disk /dev/sdb: 500GB Sector size (logical/physical): 512B/512B Partition Table: msdos  Number  Start   End     Size    Type      File system     Flags  1      1049kB  1259MB  1258MB  primary   ntfs            boot  2      1259MB  269GB   268GB   primary   ntfs  4      269GB   483GB   214GB   extended  5      269GB   416GB   147GB   logical   ext4  7      416GB   475GB   58.9GB  logical   linux-swap(v1)  6      475GB   483GB   8470MB  logical  3      483GB   500GB   16.8GB  primary   ntfs   What I've tried so far:  1) Resizing the partitions /dev/sdb5 and /dev/sda1 to be the same size. 2) Booting into Ubuntu 11.04 (from /dev/sdb5) and running dd if=/dev/sdb5 of=/dev/sda1 (of course this causes problems with booting, so I had to reinstall grub.. I can get it to boot, but then I have problems with initrd not finding some files ... presumably it can't load some partitions I think).  Now, I think those two steps are the wrong approach, because it will clone /dev/sdb5 EXACTLY - including references in fstab which point to the wrong hard drive. I'm not sure exactly how to rectify this. I could install Ubuntu 11.04 onto the SSD, then try and copy over all of my configurations, but I'm concerned that I'm going to lose something, or that I'm going to overwrite something like fstab that points back to the original hard drive.   Note that currently, I can still boot from the HDD, so it's not imperative that I get this figured out right away, but I do want it to be exactly how it is right now, so that I can maintain my current level of productivity (it's a work laptop).  Suggestions on how I might be able to overcome this difficulty?   Thanks in advance! Network-wide Help Center bug What are the regional Pokémon? I have heard that there are some Pokémon that are region specific.  Could someone please clarify which Pokémon are region specific and in which regions that they can be found in? Say you have the infinitely continued fraction: $$x+\cfrac{1}{x^2+\cfrac{1}{x^3+\cfrac{1}{x^4+\ddots}}}$$ When $x=1$, you can see that it's $$1+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{1+\ddots}}}$$ which converges upon the golden ratio $\phi = 1.61803398875...$. What if I were to plug in $x=2$? $$2+\cfrac{1}{4+\cfrac{1}{8+\cfrac{1}{16+\ddots}}}$$ Using a caluclator, it seems to converge upon the irrational number: $2.24248109286...$ Is there any way I can represent this irrational number algebraically? Bluetooth audio delay / lag, reason completely unclear to me I have a bit of an odd situation, and wanted to ask for ideas.   Recently started connecting a JBL Charge 2+ to my MBA 11 inch, mid-2013, via Bluetooth. I noticed that there's clear delay between sound event and sound being audible via the JBL.   The problems persists across applications (so for example, playing video via VLC means the sound isn't synced, same for youtube videos in different browsers, same for mp3s that start playing with a small delay, etc.)  The delay is minor, but noticeable, in the range of 0.5s I'd say.  At first I thought it might be a Bluetooth driver issues, but now I found out the following: just connected another JBL device, a Flip 3 to the same MBA, and the sound plays without noticeable delay. I'm not sure if they use the exact same drivers, but now I'm wondering:   Is there a chance that it's a hardware problem? Didn't really consider this a possibility, but now I'm not sure anymore. How to test, and, ideally, fix this issue? How can I isolate the reason for the audio lag with the Charge 2+?   Any help would be appreciated! If I have a native C++ windows program (i.e. the entry point is WinMain) how do I view output from console functions like std::cout? How to spawn a Giant with equipment? I really want to spawn Giant with armor and swords. Because Giants are strong, and great challenge if spawned with enchanted diamond armor and a diamond sword. Polynomial bounded real part of an entire function Let $f(z)$ be an entire function whose real part is bounded by a polynomial in $|z|$. Does it follow that $f(z)$ is a polynomial?  Or, without loss of generality and more suggestively $$(f(z)=\sum_{k=0}^\infty a_k z^k \quad \land \quad \operatorname{Re}(f(z))\leq |z|^n)\quad \forall z\in \mathbb{C} \qquad \Rightarrow \qquad f(z)=\sum_{k=0}^n a_k z^k \quad\forall z\in \mathbb{C}$$ Force new users to go through animated tutorial before they post for the first time Do not allow people to post the minute they created an account.  Let them ask the question but not allow them to post.  Let them stew for 30 minutes and force them through a nice animated tutorial on what they should do before they post and enforcing the fact that bad questions will not get an answer.  If they had a Need to know! basic rules introduction forced on them asking them at the end to check a box I understand! some might ask better questions and help the community instead of flooding the site with bad questions.  The presentation should be on a timer without the ability to skip and highlight the similar questions asked before that might give them the answer.  Also for the first next 9 posts they should have to check a box next to the first 5 of each similar questions before posting.  I do not mean to be harsh, but the truth is I see more negative votes on questions then positive. It's like 90% of the time people give -1 to first questions. I suspected that installing xfce4 cause some issues in Ubuntu so I removed them using apt-get remove xfce4 xfce4-goodeis.  But when I ran 'dpkg -l' I still can see some libraries and packages for xfce4:  $ dpkg -l | grep -i xfce rc  libexo-1-0:amd64                              0.10.2-2                                 amd64        Library with extensions for Xfce rc  libgarcon-1-0                                 0.2.1-1                                  amd64        freedesktop.org compliant menu implementation for Xfce rc  libxfce4ui-1-0                                4.11.0-0ubuntu1~ppa0.13.10.1             amd64        widget library for Xfce - Gtk+2 variant rc  libxfce4ui-common                             4.11.0-0ubuntu1~ppa0.13.10.1             all          common files for libxfce4ui rc  libxfce4util6                                 4.10.1-1                                 amd64        Utility functions library for Xfce4 rc  libxfcegui4-4                                 4.10.0-2                                 amd64        Basic GUI C functions for Xfce4 rc  libxfconf-0-2                                 4.10.0-2                                 amd64        Client library for Xfce4 configure interface rc  mousepad                                      0.3.0-2                                  amd64        simple Xfce oriented text editor rc  thunar                                        1.6.3-1ubuntu1                           amd64        File Manager for Xfce rc  xfce4-appfinder                               4.10.1-1                                 amd64        Application finder for the Xfce4 Desktop Environment rc  xfce4-clipman                                 2:1.2.3-2ubuntu1                         amd64        clipboard history utility rc  xfce4-mixer                                   1:4.10.0-1ubuntu2                        amd64        Xfce mixer application rc  xfce4-notes                                   1.7.7-3ubuntu2                           amd64        Notes application for the Xfce4 desktop rc  xfce4-panel                                   4.10.1-1ubuntu1                          amd64        panel for Xfce4 desktop environment rc  xfce4-power-manager                           1.2.0-2ubuntu1                           amd64        power manager for Xfce desktop rc  xfce4-session                                 4.10.1-1ubuntu1                          amd64        Xfce4 Session Manager rc  xfce4-settings                                4.11.1-0ubuntu1~ppa0.13.10.1             amd64        graphical application for managing Xfce settings rc  xfce4-terminal                                0.6.2-3ubuntu1.1                         amd64        Xfce terminal emulator rc  xfce4-volumed                                 0.2.0-0ubuntu1                           amd64        volume keys daemon rc  xfdesktop4                                    4.11.2-0ubuntu1~ppa0.13.10.1             amd64        xfce desktop background, icons and root menu manager rc  xfwm4                                         4.11.1-0ubuntu1~ppa0.13.10.1             amd64        window manager of the Xfce project   I copy-pasted all these packages\libraries to form an apt-get remove command to remove all of these dependencies, and the command ran normally, but running dpkg -l | grep -i xfce showed these dependencies again !  Why are these dependencies still installed ? Cannot compile Glasgow Haskell Compiler How do I execute PHP that is stored in a MySQL database? I'm trying to write a page that calls PHP that's stored in a MySQL database. The page that is stored in the MySQL database contains PHP (and HTML) code which I want to run on page load.  How could I go about doing this? How can I straighten ALL the vertical UV lines at once? Is there a hot keyshort cut? Or if I have to do them one by one .. is there at shortcut for that? Thank you! What is a simple example of a limit in the real world? This morning, I read Wikipedia's informal definition of a limit:     Informally, a function f assigns an output $f(x)$ to every input $x$. The   function has a limit $L$ at an input $p$ if $f(x)$ is "close" to $L$ whenever   $x$ is "close" to $p$. In other words, $f(x)$ becomes closer and closer to $L$   as $x$ moves closer and closer to $p$.   To me that sounds like something that might be better described as a 'target'.  If I take a simple function, say one that only multiplies the input by $2$; and if my limit is $10$ at an input $5$: then I've described something that seems to match the elements contained in Wikipedia's definition. I don't believe that that's right. To me it looks like an elementary-algebra problem ($2p = 10$). To make it more calculusy, I could graph the function's output when I use inputs other than $p$, but that really wouldn't give me anything but an illustration of the fact that one's answer moves farther from the right answer as it becomes more wrong (go figure).   So limits are important; what I've just described is trivial. I do not understand them. I know calculus is often used for solving real-world challenges, and that limits are an important element of calculus, so I assume there must be some simple real-world examples of what it is that limits describe.   What is a simple example of a limit in the real world?  Thank you  -Hal. What should I look for when buying chain lube / lubricant? In the past, I personally have only used WD-40.   However, I have occasionally had my bicycle looked over at the locally run, free service sessions and most of the mechanics there use a hand applied type of greasy gel (I forget the name).  Questions:   Is WD-40 suitable for bicycle chains? Would alternatives provide better chain longevity? When shopping around for lubricant, what are the most important things to look for? What do you use and why?  (optional added extra) Any tips for reducing chain wear? Project Euler - 34 / Find a mathematical approach for upper bound 145 is a curious number, as 1! + 4! + 5! = 1 + 24 + 120 = 145. Find   the sum of all numbers which are equal to the sum of the factorial of   their digits. Note: as 1! = 1 and 2! = 2 are not sums they are not   included.   So I tried a random enough large upper bound (100000) in my program and it worked.  I deleted this part to not spoil the answer  My question is : How to find the closest upper bound using a mathematical approach ? I wish to print a Stack&lt;Integer&gt; object as nicely as the Eclipse debugger does (i.e. [1,2,3...]) but printing it with out = "output:" + stack doesn't return this nice result.  Just to clarify, I'm talking about Java's built-in collection so I can't override its toString().  How can I get a nice printable version of the stack? is it possible to combine two fields in one and use that field to group by in reports? for example, i would like to combine ee id and ee name fields in 1 and show as 1 field in the "group by" section on reports.