What seat will I be in x steps later?
How to solve and see resolution of $13^{53} \pmod 7$ using Fermat little Theorem?
Regarding ,  it is not possible to combine complex powers in the usual way:  $$ (x^y)^z = x^{yz} $$  There was mention of multi-valued functions, is there some theory that makes this all work out correctly? Why are only certain shutter speeds or apertures available? For the exposure control function DSLRs there are specific shutter speeds available. For example there are 1/4, 1/100, 1/500. Why, even in high end DSLRs, is there no option to use a custom shutter speed (such as 1/19)?   If the camera can do 1/4000, why can't it do 1/33?  Similarly, why are the aperture choices limited?  Why is it a technical challenge to add extra shutter speeds or aperture settings? Why are some applications so hard to find in the Ubuntu Software Centre? Some applications don't pop up in the search even when you type in their name. Take for instance Sushi. When you type in "sushi" into the search box, it fails to throw up the application. Only when you type in "gnome-sushi" does it appear. This is despite the fact that the application has "sushi" by itself in the application description. There are other applications which don't spring to mind right now, but I've discovered before. Let $A$ be a diagonalizable matrix and $\lambda$ be an eigenvalue of A. Prove that rank($\lambda I - A$) = rank(($\lambda I - A)^2$). Let $A$ be a diagonalizable matrix and $\lambda$ be an eigenvalue of A. Prove that rank($\lambda I - A$) = rank(($\lambda I - A)^2$).  The hint they provided is that for any invertible matrix B, rank(AB) = rank (A) and rank(BA) = rank(A). I want to animate a character in a realistic but easy way. I was thinking that I could use videos of me performing the actions I want to have on my characters, however I can't find a way of placing the videos and make the character to follow my movements in an automated way, like motion tracking. How can I animate automatically using this technique? Are there any Plug-ins to work like this? Is this mehod actually more useful than standard animation techniques? The wavefunction $\Psi(x,t)$ for a free particle is given by  $$\Psi(x,t) = A e^{i(kx-\frac{\hbar k}{2m}t)}$$  This wavefunction is non-normalisable. Does this mean that free particles do not exist in nature?  Why then do we use free particles $\psi(\vec{x}) = e^{ikz}$, for example, in scattering theory? Compiling LaTeX document empties .bbl file I'm trying out biblatex for the document I'm compiling, using the biber backend on my miktex distribution. Everything, including biber, should be up to date, assuming the MiKTeX repository currently holds the latest versions of everything.  This is what the biblatex documentation (p. 101) says about how often I have to run biber to update my .bbl file:     Whenever a reference to a work which has not been cited before is   added, this procedure must be repeated. This is also the case if the   last reference to a work which has been cited before is removed   because some citation labels may change in this case.   So if I make changes to my document (which is a very common activity in the process of writing a document), but leave the citations unchanged, I should not have to re-run biber before compiling the changes. However, every time I compile my document, regardless of the compiler or documentclass, the .bbl file is emptied. Consequently, the next time I compile the document without running biber, the bibliography is removed (seeing as the .bbl file contains nothing), and all my citations are broken. For this reason, I have to run biber before every time I compile the document, which largely counters the benefit of externalizing the bibliography in the first place.  How can I make the compilers stop overwriting my .bbl file and make everything behave as described in the biblatex manual?  Upon investigating the contents of the auxiliary files, I found that the .bbl file (after running biber, before compiling) begins with the following lines:  % $ biblatex auxiliary file $ % $ biblatex bbl format version 2.3 $   Does the biblatex version 2.8 still mark the .bbl files as version 2.3? If not, could this be the problem? And if so, why does my 2.8 version of biblatex (partly) behave as version 2.3?  Any help solving this problem would be much appreciated. Centre of $GL_n(\mathbb R)$ How to prove that center of $GL_n(\mathbb R)$ is scalar matrices?  Again what can you say about the center of set of all upper triangular matrices and center of set of all diagonal matrices. Try to solve this puzzle:     The first expedition to Mars found only the ruins of a civilization.   From the artifacts and pictures, the explorers deduced that the   creatures who produced this civilization were four-legged beings with   a tentatcle that branched out at the end with a number of grasping   "fingers". After much study, the explorers were able to translate   Martian mathematics. They found the following equation:   $$5x^2 - 50x + 125 = 0$$   with the indicated solutions $x=5$ and $x=8$. The value $x=5$ seemed   legitimate enough, but $x=8$ required some explanation. Then the explorers    reflected on the way in which Earth's number system developed, and found   evidence that the Martian system had a similar history. How many fingers would    you say the Martians had?      $(a)\;10$      $(b)\;13$      $(c)\;40$      $(d)\;25$   P.S. This is not a home work. It's a question asked in an interview. How to expand the ext4 primary partition size in a VMware Player virtual disk? I have a VMware Player virtual machine setup, running Ubuntu 10.04 as the a guest OS.  The VM was created on a 20GB virtual drive, with a single partition for working space. Finally I got to the point where I need to expand the size of the virtual drive. Resizing the drive itself, adding some 10GB, was easy via the VMware Player menus. I now need to resize the main partition itself (I prefer this over creating a separate partition and mounting it).  According to GParted, I now have the following partitions:    My question is, how can I increase the size of the Filesystem partition to include the free 10GB?  From information I found online, it looks like one limitation is the fact that he swap partition is immediately adjacent to the filesystem, so my understanding is that first thing would be to move that partition to the end of the volume. How can this be done?  Also, some sources suggest that the system should be booted from a LiveCD, so the filesystem is not the boot partition, and thus make it easier to resize. As mentioned above, I run a VM, so I am not sure how to boot it from a LiveCD. Any ideas? One of the answers in a reading exercise in my class today was:   100,000,000,000,000,000,000   ... which was the value of the highest denomination note ever issued. It was a 100,000,000,000,000,000,000 Pengo note, issued by Hungary in 1946. The students in my class wanted to know how they could say that last sentence in words in English. So I told them that I didn't know. They pointed out to me that if they wanted to read this bit of writing to someone they'd want to be able to use the words. Fair point.  Can anyone help?  [In case you're interested, a 100,000,000,000,000,000,000 Pengo note was worth about $0.20!] I lost my .keystore file? Cannot move file to trash warning when trying to delete a file in Nautilus I am trying to delete files in Nautilus and every time I try to delete one it always gives me this message:     Cannot move file to trash, do you want to delete immediately?   Then it gives me several questions. All I want is that when I press Delete in the keyboard, it sends it automatically to the trash can. Not to ask me every time.  The problem relates to NTFS partitions, external drives but also include Linux filesystems.  What are the reasons for nautilus beeing not able to move to trash? I want to prove that $\lim_{(x,y)\to (0,0)} \frac{\sin(x+y)}{x+y} = 1$  Is it sufficient to say that if $u = x+y$ then as $(x,y) \rightarrow (0,0)$ then $u \rightarrow 0$ and then evaluate $\lim_{u\to 0} \frac{\sin u}{u}$ which is 1 by L'Hôpital's rule?  I feel that this is somehow "cheating" and I would really like to prove it in a more rigorous way, i.e., by using an $\epsilon$ - $\delta$ argument.  I have tried for a long time to use the one variable version and prove that if $0 \lt \sqrt{x^2 + y^2} \lt \delta$ then somehow $0 \lt |x+y| \lt \delta$ and it would follow from the one variable case that $| \frac{\sin(x+y)}{x+y} - 1| \lt \epsilon$. However, because $|x+y| = 0$ when $y = -x$ I am not getting anywhere with this.  Any input on this would be helpful :) Cannot locate Ubuntu Software Center via Firefox to open APT links In proof by induction, what does it mean when condition for inductive step is lesser than the propsition itself? My question is regarding the question posed at the end of the proof.  My answer is that the result does not hold for all $m \ge 7$ because when $m=7$, the result is $343 \le 128$, which is false.  Is there another way to answer this sort of question besides directly trying different values until you get a false result?  EDIT:  Another question I have regarding this solution is:  In the proof for the inductive step, we start by assuming $k \ge 10$. But along the way, the author mentions $k \ge 1$ and $k \ge 7$ to justify the inequality.  Why do we bother to do this instead of just sticking with $k \ge 10$? The  badge seems like a great candidate for a trackable badge: it requires doing a thing multiple times, there aren't any other requirements to get it, and it is (seems?) quite easy to determine how many times the thing has been done.  But Convention does not appear in the list of badges that can be tracked. I would like to be able to see a quick count of my well-received meta posts without going to my meta user activity page and poking around. How to split the terminal into more than one "view"? Sum of Permutations from 0 to n In a question I asked on algorithms with time complexity of $f(x) = n^n$ I was told that enumerating the number of strings that can be formed from a string of length $n$ qualifies.   I.e the sum of all permutations of $n$ from $n$ to $0$ is $n^n$  $\sum ^n_{i=0} nPi$ $= n^n$  Can I please see an easy to understand derivation of that formula.   EDIT The above identity is wrong. I just tested it. Can I get a derivation of the formula for the sum of permutations. How can you profile a Python script? and other coding contests often have a maximum time to run or people boast of how fast their particular solution runs. With Python, sometimes the approaches are somewhat kludgey - i.e., adding timing code to __main__.  What is a good way to profile how long a Python program takes to run?