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Proving a sequence converges by defining the sequence recursively. Let $a &gt; 0$. Choose $x_1 &gt;$ $\sqrt a$. Define a sequence {$x_n$} recursively as $x_{n+1} = 1/2(x_n + a/x_n)$ for $n &gt; 1$.   Prove that lim$x_n = \sqrt a$.  I think I first want to prove that {$x_n$} is bounded below by $\sqrt a$. But, I am having trouble with this. Could someone help me out a little? Kernels and reduced row echelon form - explanation The following text is written in my textbook and I don't really understand it:     If $A = (a_{ij}) \in$ Mat$(m x N, F)$ is a matrix in reduced row echelon form with $r$ nonzero rows and pivots in the columns numbered $j_1 &lt; ... &lt; j_r$, then the kernel ker$(A)$ is generated by the $n-r$ elements $ w_k = e_k - \sum\limits_{1 \le i \le r, j_i \le k} a_{ik}e_{j_i}$ for $k \in {1, \cdots , n} \ {j_1, \cdots, j_r}$, where $e_1, \cdots, e_n$ are the standard generators of $F^n$.    There are no computational examples and the notation is a bit overwhelming so I don't know how I can use it. Can somebody give an example of this in practice? Find the Limit (n to infinity) Hi I have a question regarding of limits to infinity please help! Thank You!  The question states the user to find the following limit:  $  \lim_{n\to\infty} n^2 ({\sqrt[n]{x}-\sqrt[n+1]{x}}) $      Where $(x &gt; 0) $  Thank You!!! Let $z \in \Gamma (R)$ where this is the upper semi circle centred at the origin with radius $R&gt;1$. Let $z=x+iy$ with $x \in \mathbb{R}$ and $y \geq 0$. So   $$|e^{i \lambda z}|=|e^{i \lambda z}| |e^{- \lambda y}|= |e^{- \lambda y}|$$  where $\lambda \in \mathbb{R}^{+}$.  How can the last equality possibly be true? Memory management in Objective-C How can I find unanswered questions with zero answers in some tag? node.js conflicts: /sbin/node vs /usr/bin/node I have two Ubuntu 12.10 machines: machine A is a VMWare VM and machine B is and old Acer laptop. On both machines, I installed node using apt-get. However machine A installs the main binary as /usr/bin/node and machine B as /usr/bin/nodejs. Here are some behaviors I observed:   Both machine returns /usr/bin/node in response to the which node command. However, there is no such file on machine B. Issuing the command node works for machine A, not B. On B, I have to use nodejs instead. On B, there is a binary called /sbin/node, A does not have it.   These behaviors causes inconsistency problem when developing on both machines. For now, I worked around by renaming the binaries on B as such:  sudo mv /sbin/node /sbin/node-sbin sudo ln -s /usr/bin/nodejs /usr/bin/node   Is there a way to really fix this, so that node will be installed as /usr/bin/node instead of /usr/bin/nodejs? Can't find the reason why root directory usage is so large CPU time out limit was being hit while processing child records on the inline opportunity visualforce page, to handle this i wrote a remoteActionFunction on the page that invokes controller method that's taking long to process and when i get CPU LIMIT exception in the result, i am invoking javascript call back function that invokes a action function which handles the same code asynchronously(@future).  Question: by setting the timeout (Visualforce.remoting.timeout = 120000;) can't we overcome CPU TIME out limit of 10,000 ms. Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix This question aims to create an &quot;&quot;  of numerous questions that ask about determinants of specific matrices (I may have missed a few):              The general question of this type is  Let $A$ be a square matrix of rank$~1$, let $I$ the identity matrix of the same size, and $\lambda$ a scalar. What is the determinant of $A+\lambda I$?  A clearly very closely related question is  What is the characteristic polynomial of a matrix $A$ of rank$~1$? Suppose that G is a group and N &lt; G a normal subgroup. Assume that there is no normal subgroup M &lt; G with N &lt; M. Prove that G/N is simple.  This problem is really throwing me for a loop. The concept of quotient groups and how they relate to simple subgroups is confusing me, and so I am not really sure how to begin. If a simple subgroup's only normal subgroup is itself (and the trivial group), then G/N being simple would be proving that when multiplying the entire subgroup N by an element g for all g in G, that the collection of those sets cannot be broken into smaller normal subgroups? Isn't finding the inverse of $a$, that is, $a'$ in $aa'\equiv1\pmod{m}$ equivalent to solving the diophantine equation $aa'-mb=1$, where the unknowns are $a'$ and $b$? I have seem some answers on this site (where the extended Euclidean Algorithm is mentioned mainly) as well as looked up some books but there is no mention of this. Am I going wrong somewhere or is this a correct method of finding modular inverses? Also can't we find the Bézout's coefficients by solving the corresponding diophantine equation instead of using the extended Euclidean Algorithm? Let $ \sigma(A)$ be the set of all eigenvalues of $A$. Show that $ \sigma(A) = \sigma\left(A^T\right)$ where $A^T$ is the transpose matrix of $A$. How to add electrical fixtures to (custom) furniture? Having electrical outlets, USB charging ports, night lights, etc would be nice to have on some pieces of furniture, like side tables. Several receptacle manufacturers make combination fixtures with USB and power, for instance, or power plus a night light.   I've thought about nailing an electrical box to a (hidden) portion of the table and putting one of those receptacles in it, but what things do I need to consider doing this? Is it as simple as it sounds,  and then just fashioning some kind of plug out of 12/2 or 14/2 wire? Does the answer change if the box is concealed (though still accessible) vs it being completely visible? Where does TREE(n) sit on the Fast Growing Hierarchy? how to convert System::String to const char*? How do I limit the running time of a BASH script I have a long running BASH script that I am running under CYGWIN on Windows.  I would like to limit the script to run for 30 seconds, and automatically terminate if it exceeds this limit. Ideally, I'd like to be able to do this to any command.  For example:  sh-3.2$ limittime -t 30 'myscript.sh'   or   sh-3.2$ limittime -t 30 'grep func *.c'   Under cygwin the ulimit command doesn't seem to work.   I am open to any ideas. How to flatten a list to return a new list with all the elements? Inertial frames of reference I'm struggling with the notion of an inertial frame of reference. I suspect my difficulty lies with the difference between Newtonian and relativistic inertial frames, but I can't see it.   I've read that Newton's laws apply in any non accelerating frame of reference, which are called inertial frames. So, if I play pool on a train moving with uniform velocity, the balls behave in the same way as if I were playing pool in a pool hall. So the train is (to a good approximation, ignoring forces caused by the Earth's rotation and movement around the Sun etc) an inertial frame.   I've also read that in an inertial frame, Newton's first law is satisfied. So, if I slide a rock on a sheet of ice and I could somehow eliminate the frictional forces between the rock and the ice, the rock would carry on sliding for ever, as predicted by Newton's first law.   Question 1 - is the  sheet of ice also therefore a good approximation to an inertial frame?  Question 2 - are these two definitions of inertial frames saying the same thing in different ways?  Both these examples occur in gravitation fields, which doesn't seem to matter as both the train and the ice sheet are presumably good approximations to inertial frames.  Question 3 - is gravity irrelevant when defining these two inertial frames?  In special relativity, I've read (Foster and Nightingale) that an inertial frame is also one where Newton's first law holds. But as it's special relativity there cannot be an inertial frame if there's a gravitational field, so I'm assuming the above two examples are not inertial frames in special relativity.  Question 4 - how come you can use the same definition of an inertial frame (satisfied Newton's first law) but in the case of special relativity, the train and the ice sheet aren't inertial frames. Is this to do with not being able to synchronize clocks in a gravitational field?  In general relativity, I've read (Schutz) that a freely falling frame is (on Earth) the only possible (and local) inertial frame. So again the first two examples would not be inertial frames in general relativity.  Question 5 - am I right to think that my two examples aren't even approximate approximations to an inertial frame in relativity?  Question 6 - have I missed anything else that might be useful?  Many thanks  Edit. On reflection, am I right in thinking that because Newtonian mechanics assumes universal, absolute time we don't need to worry about synchronizing clocks in a Newtonian inertial frame. Therefore we don't need to worry about gravity in a Newtonian inertial frame, because in such a frame gravity does not affect time.  This is not the case in spacetime, as here gravity does affect time and the only way to have synchronized clocks in an inertial frame in spacetime is: 1. have no gravitational field, or 2. use a local, freely falling frame.  Am I on the right track here? I'm setting up a new server and want to support UTF-8 fully in my web application. I have tried this in the past on existing servers and always seem to end up having to fall back to ISO-8859-1.  Where exactly do I need to set the encoding/charsets? I'm aware that I need to configure Apache, MySQL, and PHP to do this — is there some standard checklist I can follow, or perhaps troubleshoot where the mismatches occur?  This is for a new Linux server, running MySQL 5, PHP, 5 and Apache 2.