Intuition behind transformations of functions (horizontal shifting)
Explaining Horizontal Shifting and Scaling
Any help would be appreciated.  If $n$ is a natural number $\ge 2$ how do I prove that any graph with $n$ vertices has at least two vertices of the same degree? Our high-school teacher told us that the Earth pulls us with some force $F$ and we pull the Earth with the same force $F$. Within Newtonian physics this is true because of , but let’s consider . My mass is small; so I don’t warp space-time much. But Earth’s large mass warps space-time to a far greater extent.  So do I pull Earth with the same force it pulls me? If yes, how? How to create two classes in C++ which use each other as data? Can bosons that are composed of several fermions occupy the same state? It is generally assumed that there is no limit on how many bosons are allowed to occupy the same quantum mechanical state. However, almost every boson encountered in every-day physics is not a fundamental particle (with the photon being the most prominent exception). They are instead composed of a number of fermions, which can not occupy the same state.  Is it possible for more than one of these composite bosons to be in the same state even though their constituents are not allowed to be in the same state? If the answer is "yes", how does this not contradict the more fundamental viewpoint considering fermions? I have a query to insert a row into a table, which has a field called ID, which is populated using an AUTO_INCREMENT on the column. I need to get this value for the next bit of functionality, but when I run the following, it always returns 0 even though the actual value is not 0:  MySqlCommand comm = connect.CreateCommand(); comm.CommandText = insertInvoice; comm.CommandText += "\'" + invoiceDate.ToString("yyyy:MM:dd hh:mm:ss") + "\', " + bookFee + ", " + adminFee + ", " + totalFee + ", " + customerID +  ")"; int id = Convert.ToInt32(comm.ExecuteScalar());   According to my understanding, this should return the ID column, but it just returns 0 every time. Any ideas?  EDIT:  When I run:  "INSERT INTO INVOICE (INVOICE_DATE, BOOK_FEE, ADMIN_FEE, TOTAL_FEE, CUSTOMER_ID) VALUES ('2009:01:01 10:21:12', 50, 7, 57, 2134);last_insert_id();"   I get:  {"You have an error in your SQL syntax; check the manual that corresponds to your MySQL server version for the right syntax to use near 'last_insert_id()' at line 1"} Story about last 2 men on Earth. One very evil, and the other very ethical Is it possible to run Snow Leopard on a LATE 2011 MacBook Pro? I want to buy a new MacBook Pro, from Apple. However I want to run Snow Leopard instead of Lion.  I did some Google searches and found a few success stories of people who got SL to run on their early-2011 MBPs. I didn't find much of any experiences of people doing the same on a late-2011 MBP though, so I'm wondering (and really hoping!) if it's possible? Can the expression $\sqrt{n} + \sqrt{m}$ be rational if neither $n,m \in \mathbb{N}$ are perfect squares? It doesn't seem likely, the only way that could happen is if for example $\sqrt{m} = a-\sqrt{n}, \ \ a \in \mathbb{Q}$, which I don't think is possible, but how to show it? Let $P\in\mathbb{R}[X]$ be a positive polynomial.  I want to show that there exists $A,B\in\mathbb{R}[X]$ so that $P=A^2+B^2$    $\displaystyle P=a\prod_{k=0}^q(x-a_k)\prod_{k=0}^{2p}(x-b_k)=a\prod_{k=0}^q(x-a_k)\prod_{k=0}^p(x^2-2\Re(b_k)+|b_k|^2)$ where $a$ is the leading coefficient, $a_k$ are the real rooks, $b_k$ are the complex roots.  What can i do now ? I know that the identity $(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(ad+bc)^2$ should come in handy, but I can't figure out :  1) How to show that the product real roots is a square  2) How to use the above identity for the complex roots.  If I can solve those two points, then I can solve the problem. Axis with trigonometric labels in PGFPlots How do I make the axis labels use multiples of \pi in PGFPlots? Are {/bin, /lib, /sbin} symlinks into /usr in Ubuntu? I do not have access to an Ubuntu system or chroot. I was hoping if someone could tell me if the  is accurate in saying:     For Quantal, merged /usr should be supported. On new systems installer should create {/bin, /lib, /sbin} symlinks into /usr. boolean variable values in PHP to javascript implementation Symbols instead of numbers as footnote markers Saving an Adobe illustrator file as a pdf - are fonts embedded? If so, why does Adobe illustrator say I don't have the font when I open it? If I have an Adobe Illustrator file, and I save it as a .pdf - it looks like the fonts are included in that .pdf file. (I think this because when I open the .pdf file in Reader it looks the way I expect it to.) However, when I open this same .pdf file on a different computer (or workstation) with Adobe Illustrator - Illustrator says I 'don't have the font I need'. This doesn't make sense to me because - If I can see the document correctly in reader, I would expect it to be there for Illustrator as well. Can someone explain why this is? I am asking because saving a file as a .pdf for print seems to handle most font troubles that come up - but sometimes I need to change the text in a file before I print it... I am hoping there is a way to open a .pdf so that .ai recognizes that the font is in there.  But I am unsure on the details of saving as a .pdf in Illustrator, how fonts are handled, and what is possible. grub2 wont load Ubuntu on dual boot Having an issue with a dual boot setup that is being run from Windows boot manager. I can successfully load into grub2 from Windows boot manager, however it seems that grub2 cannot locate to the Kernel or something, as it is dropping me right into a shell with the following output:  [ Minimal BASH-like line editing is suported. For the First word, TAB list the posible command completion. Anywhere else tab list the posible completions of a device/filename,]  grub&gt;   I have used Easybcd in the past and successfully loaded a Linux grub through Windows bootloader, but this was when both operating systems were on the same partition, but since my OS installationd are now on separate partitions it seems EasyBcd cant work its magic.   ONe encouragement is that I was able to get Easybcd to load a working Grub if I used the Neo grub bootloader and edit the confg with:  title Ubuntu 14.04 find --set-root /boot/vmlinuz-3.19.0-61-generic kernel /boot/vmlinuz-3.19.0-61-generic ro root=/dev/sdc initrd /boot/initrd.img-3.19.0-61-generic   however this loads in Grub4DOS which is very slow, and as of today this method stopped working See  for details).  Here is the output of EasyBcd Settings for all the different methods I have tried for the Ubuntu 14.04 installation:  Default: Windows 7 Timeout: 30 seconds Boot Drive: C:\  Entry #1 Name: Windows 7 BCD ID: {current} Drive: C:\ Bootloader Path: \Windows\system32\winload.exe  Entry #2 Name: Ubuntu 14.04 Legacy BCD ID: {a4f127cf-3150-11e6-8aaf-408d5cb9e442} Drive: C:\ Bootloader Path: \NST\nst_linux.mbr  Entry #3 Name: Ubuntu 14.04 Grub2 BCD ID: {a4f127d0-3150-11e6-8aaf-408d5cb9e442} Drive: C:\ Bootloader Path: \NST\AutoNeoGrub0.mbr  Entry #4 Name: Ubuntu Neo Grub BCD ID: {a4f127d1-3150-11e6-8aaf-408d5cb9e442} Drive: C:\ Bootloader Path: \NST\NeoGrub.mbr   *It seems that none of the paths seen above are pointing to my dev/sdc2/ partition which would be considered DISK1 on my Windows Volume manager.     EDIT - In the confusion of attempting to get a working bootmanager for Ubuntu/Windows, you will see that grub and Windows Boot manager have all been installed in numerous locations. Below are the present locations and contents of all bootmanager &amp; Grub installations:  dev/sdb Windows7 drive  /dev/sdb1 - 512 MB fat32 partition which is currently empty  /dev/sdb2 - 110 GB ntfs partition containing Windows7 installation. This partition contains a 'Boot' folder which contains BCD files and a whole wack of langauage folders. THIS IS The FOLDER THAT WINDOWS BOOTLOADER uses.  /dev/sdb4 - 121.53 ntfs partition containing storage for media    dev/sdd - Ubuntu Drive  dev/sdd1 - 512 MB partition containing a 'EFI' folder, inside of which are two folders 'grub' and 'Ubuntu', both of which contain the exact same files (grub.cfg, grubx64.efi, MokManager.efi, shium64.efi)  dev/sdd2 - 48.83 GB ext4 partition that contains the '/' folder and Ubuntu instalation.   dev/sdd3 - 69.91 GB ntfs partition containing storage for media  what do I need to do so that the Grub shell I am being dropped into will load up Ubuntu? HOw can I get this machine to dual boot out of Grub? Self-Contained Proof that $\sum\limits_{n=1}^{\infty} \frac1{n^p}$ Converges for $p > 1$ To prove the convergence of the p-series   $$\sum_{n=1}^{\infty} \frac1{n^p}$$  for $p &gt; 1$, one typically appeals to either the Integral Test or the Cauchy Condensation Test.  I am wondering if there is a self-contained proof that this series converges which does not rely on either test.    I suspect that any proof would have to use the ideas behind one of these two tests. $f^2$ and $f^3$ are holomorphic implies $f$ is holomorphic. Integration of exponential and square root function How to create a DEB file manually? I want to create a DEB file manually. I would like to just provide a folder which contains data to install, and a script to be executed after installation.  Is this possible? Can NDSolve handle discontinuous data?