How could i possibly calculate square matrix to power n with induction for lets say this square matrix $A =\begin{bmatrix}  1 &amp; 2\\   0&amp; 1 \end{bmatrix}$I cant use Cayley-Hamilton theorem and i need to use induction.Any ideas on what i could do?
I have a matrix:  $$ A=\begin{bmatrix}1 &amp; a \\ 0 &amp; 1\end{bmatrix} $$  which satisfies: $A\left( e_{1}\right)=e_{1},A\left( e_{2}\right)=ae_{1}+e_{2}, a\neq 0$  I need compute all powers $A^{n}, n\in \mathbb{Z}$ and find their matrices.  $$ A^n= \begin{bmatrix}1 &amp; na \\ 0 &amp; 1\end{bmatrix} $$  Is it true?  Is there a basis of $V$ (if $V$ is a 2-dimensional real vector space with basis $\left\{e_{1},e_{2}\right\}$) so that the matrix of $A$ is diagonal?
I'm using PostGIS 1.5.2, with geometries in SRID:900913. I need create a circles using a list of points as centre, with a radius of 600 kilometers. I'm using this query:  INSERT INTO circles (geom) (    SELECT ST_Buffer(point, 600000, 'quad_segs=8')     FROM points );   But circles created have not 600 kilometers of radius (radius is near this length, but not exactly).  Are there other methods to create circles in PostGIS?  NOTE: Information represented is from Spain. Correct projection is 4326, but client use Google rasters, so I am storing data in 900913 to avoid reprojections and increase performance. How does one actually show from associativity that one can drop parentheses? Setting background-image using jQuery CSS property When is 6 a square in Zp To find when $6$ is a square I divided $6$ in to $2$ and $3$ by The Legendre symbol.  $$\frac{6}{p} = \frac{2}{p} \times \frac{3}{p}$$  I know that we can multiply the cases of when $2$ and $$ are squares because then it is (1)(1) by Legendre.   $2$ is a square when $p \equiv 1 \pmod{8}$   $3$ is a square when $p \equiv 1 \pmod{12}$  so $6$ is a square when $p \equiv 1 \pmod {24}$ ($96$, but the $lcm(8,12)= 24$)   but how do I find the cases when $2$ and $3$ are both NOT squares? because then $6$ would be a square then as well, $(-1)(-1) = 1$ by Legendre. I would like to set up an APT repository on a server that will provide a couple of packages.  Is there a way to set one up without installing any software on the server?  How do the files have to be organized?    Edit: I must be doing something wrong... can someone please help me? I have the repository at   I'm not sure where or what, but something's misconfigured. I only currently have one package and it's for all architectures.  Here's what's been added to my /etc/apt/sources.list:  deb http://quickmediasolutions.com/apt stable main How do I get a bigger static scrollbar (aka normal scrollbar)? How to access the shared folder between two computers running ubuntu I have to computers running Ubuntu and I want to share files between those computer. these computers are already connected through a cross cable and sharing internet connection using firestarter. Both computers have samba installed.   In Desktop computer I have a shared folder How do I access it on the other one. Do you know any open source tool to automatically align images, similar to the auto align feature in Photoshop? Once upon a time,  DISPLAY=:0.0 totem /path/to/movie.avi   after ssh 'ing into my desktop from my laptop would cause totem to play movie.avi on my desktop.  Now it gives the error:   No protocol specified Cannot open display:    I reinstalled Debian squeeze when it went stable on both computers, and I guess I broke the config.   I've googled on this, and cannot for the life of me figure out what I'm supposed to be doing.   (VLC has an HTTP interface that works, but it isn't as convenient as ssh.)  The same problem arises when I try to run this from a cron job. Using SQL Server, how do I split a string so I can access item x?  Take a string "Hello John Smith". How can I split the string by space and access the item at index 1 which should return "John"? Is indefinite integration non-linear? What is the name of the genre of games similar to Jetpack Joyride? Since a while, 2 weeks, Dropbox does not startup anymore on my laptop running Ubuntu 12.04 LTS with GNOME shell.  When I run dropbox start -i in the console I get the following output.    Starting Dropbox...Traceback (most recent call last):   File "/usr/bin/dropbox", line 1387, in &lt;module&gt;     ret = main(sys.argv)   File "/usr/bin/dropbox", line 1376, in main     result = commands[argv[i]](argv[i+1:])   File "/usr/bin/dropbox", line 1247, in start     if not start_dropbox():   File "/usr/bin/dropbox", line 731, in start_dropbox     stderr=sys.stderr, stdout=f, close_fds=True)   File "/usr/lib/python2.7/subprocess.py", line 679, in __init__     errread, errwrite)   File "/usr/lib/python2.7/subprocess.py", line 1249, in _execute_child     raise child_exception   OSError: [Errno 8] Exec format error   I already removed Dropbox and reinstalled it. What should I do to fix this? Where do I get the key for the first house? Highly unbalanced test data set and balanced training data in classification I have a training set with about 3000 positive instances and 3000 negative instances. But my test data set is pretty much un-balanced. The positive set only has 50 instances and negative has 1500 instances.This causes the precision very low. Are there any approaches to solve this problem? I use SVM to build classifier. How to repair a broken .EXE file association After (hopefully) scrubbing viruses out of a Windows 7 installation (after deciding not to simply run over the laptop repeatedly with my car), I've got everything almost back to normal. The only lingering issue I have is that for my non-admin users, the ".exe" file extension doesn't work. That is, clicking on the various desktop application links results in a "How do you want to open this?" dialog.  I've been through the alleged registry fixing from "winhelponline" and that had absolutely no effect.  I've tried running "assoc" for the affected users, but it reports the .exe association to be "exefile" even though it persistently does not work.  Right-clicking on a desktop icon and then choosing "start" does successfully open an application, but that's clearly a terrible situation.  For my admin user, things seem to work fine.  What do I need to do to get things working? Let $\mathbf A, \mathbf B$ be Hermitian matrices of the same size. What is the characterization of $\mathbf A, \mathbf B$ such that $p(\lambda)=\det(\mathbf A-\lambda\mathbf B)=0$ has only real roots?   If $\mathbf B$ is positive definite (I corrected this), it is easy to see $p(\lambda)$ has only real roots. Ambiguity of chi-squared test of association; how can the conclusion change using the ''same'' sample? Recently I have been struggling with the following two questions regarding the chi-squared test of independence / association during reading an article.        Is the chi-squared test able to detect no association between two groups among three at two levels? That is in a 3X2 contingency table.   Does the chi-squared test change its behavior with a larger sample that was otherwise perfectly proportional to a smaller sample?     I prepared a sample data set in R to seek the answers of these two questions. Though my initial discomfort about the chi-squared test have been proven, but over-acceptance of this test statistic in the scientific community have forced me to re-think. May be I have been missing something here.    # Question 1 x1 = c(100, 450, 450) x2 = c(600, 450, 450) d1 = rbind(x1,x2) colnames(d1) = c("Type 1", "Type 2", "Type 3") rownames(d1) = c("No", "Yes") d1 chisq.test(d1) # Significantly associated (not independent), though "Type 2" &amp; "Type 3"  #  were not different at all!   # Question 2 d2 = d1/50 d2 chisq.test(d2) # Not significant!!!  d3 = d2*2 d3 chisq.test(d3) # Significant!!! I bought my kids a PC and installed 12.04 (Unity) on it.  The bottom line is, I want my children to use the computer unsupervised while I have confidence they cannot access anything inappropriate.  What I have looked at:  I was looking at Scrubit a tool which allows me configure my wifi router to block content and this solution would also protect my other PC and mobile devices.  This may be overkill as I just want the solution to work on one PC.    I also did some Google searches and came across the application called .  My experience of OSS is that the best solutions frequently never appear first on a Google search list and in this case I need to trust the methods therefore my question is very specific.   I want to leverage your knowledge and experience to understand What is the best way to restrict adult content on 12.04 LTS as this is important to me.  It maybe a combination of things so please don't answer this question "try this or that", then give me some PPA unless you can share your experience of how good it is and of course if there are any constraints. Is $G^{(X, Y)} = (G^X)^Y?$ ($A^B$ just means that $B$ is mapped to $A$)