How many solutions to $a^n-b^n=2^{100}$
How many triples $(x,y,n)$ are there such that $x^n - y^n = 2^{100}$
Totally disconnected space that is not $T_2$ Is there a c# equivalent of Java ConcurrentHashMap Is it safe to delete from C:\Windows\Installer? I've been looking on an XP machine what is using all that diskspace, and it turns out C:\Windows\Installer is high up there on the list of directories that use the most diskspace.  It seems to contain a cache of msi and msp files.  Is it safe to delete those? To save diskspace, I've been deleting the "$KB...$" directories for ages without any problems. I want my Windows Updates in my system, why you would want to uninstall them is something that I have never understood. (Except when you're on an update revision board professionally or something of that order.) My wife is an Indian national with an Indian passport. She holds an indefinite leave to remain in the UK (permanent residence card). Does she need to apply for a Schengen visa before travelling to France or any other Schengen countries? I noticed relation between modulo operation and number which is power of two  Example I have to calculate $ 3431242341 \mod 2^5 $, which is $ 5 $ but it is equivalent to  $ ( 3431242341 \mod 2^9 ) \mod 2^5 $   I tried many examples and it seems to be true in general, and I am not sure if it is a coincidence or true in general that I can use first modulo operation ( greater number) and the result will be the same. Dirichlet's problem in the unit disk is to construct the harmonic function from the given continuous function on the boundary circle. It is solved by the convolution with the Poisson kernel, and we know that the constructed function has uniform limit which is the given continuous function as $r\to 1$, and we can complexify the solution by taking the Schwarz integral formula. Now (in this very particular case) does the imaginary part has a uniform limit? If it does, why don't we prove the uniqueness of the solution by a simple application of the Cauchy integral formula? Here is the proof that I reckon to be reasonable(which is what I expose here to be criticised)  We construct the Schwarz integral, or $\frac{1}{2\pi}\int _{0}^{2\pi}\frac{\exp(it)+z}{\exp{(it)}-z}u(\exp{(it)})\mathrm{d}t$. This function is an analytic function in the interior unit disk, and is continuous in the closed unit disk(its closure). Let this analytic function denoted by $f$, and another analytic function that has the same uniform limit function on the circle be $v$. Then $u-v$ is an complex analytic function in the interior of the unit disk, continuous on the closure, with boundary value identically zero. An application of Cauchy's integral formula gives rise to the conclusion that $u-v$ in identically zero on the entire disk.   Another question being if the boundary function has finite jump discontinuities, what is the behaviour of the convolution with the Poisson kernel near the discontinuous points? Is it possible in Ruby to get a reference to methods of an object ( I would like to know if this can be done without procs/lambdas ) , for example , consider the following code :   class X   def initialize     @map = {}     setup_map   end    private   def setup_map     # @map["a"] = get reference to a method     # @map["b"] = get reference to b method     # @map["c"] = get referebce to c method   end    public   def call(a)     @map["a"](a) if a > 10     @map["b"](a) if a > 20     @map["c"](a) if a > 30   end    def a(arg)      puts "a was called with #{arg}"   end    def b(arg)      puts "b was called with #{arg}"   end    def c(arg)     puts "c was called with #{arg}"   end end   Is it possible to do such thing ? I would like to avoid procs/lambdas because I want to be able to change the behaviour of A,B,C by subclassing . Creating all possible line segments between all points using QGIS Do we have name for these kind of "flat graphics" images? I have been thinking of getting a flash to improve my low light photography (particularly indoors). However, there seem to be a number of different models available at similar price points (budget of INR 8,000-12,000 ~$150-250). What features and functionality should I look for to choose a suitable flash?  I'm using a Canon 550D with the 18-55mm IS &amp; 55-250mm IS lenses. Metz seems to be one of the brands easily available in India (beside the OEM flashes). Currently, I am considering the Metz 36 AF-5, 44 AF-1 and 50 AF-1, which seem to be considerably cheaper than their Canon counterparts. What character / attack / stage combination can get me a combo of 100? For the first two challenge panels, I just used my golden hammers on the "get a combo of x," but now I'm seeing on challenge panel 3, the game expects you to get a combo of 100! I can barely get past a combo of 9 without the counter resetting. What character / stage combination can get you a combo of 100? Are there multiple, so I can more easily get the "get a total combo of 400 across all characters" challenge?  Additionally, what exactly constitutes a combo? Sometimes my combos are cut short and sometimes they're longer, so I must not understand how they're tracked. Why are passports checked at the check-in counter, Immigration and even at the gate? I was wondering, why there are passport checks at the check-in counter, Immigration and even at the gate?    For example, I was at Istanbul flying to either UK or US. There's a security check at the check-in counter asking a bunch of questions about my immigration status - even more than the Immigration officer at the border. Also, at the gate, there's also Security officer checking my immigration status and passport asking the same questions.    I thought that would be the responsibility of an Immigration officer rather than some third party company?   The thing I'm curious the most is they don't even know which countries this passport can go to with or without visa. They have to go back to the counter and check with the system which takes a long time. I was wondering why they hire these kind of companies to slow down the process? Notify question OP (and those who marked it a favourite) if an answer has been modified Seriously, I had not noticed e.g. Jeff's edit  had he not also retagged my question. Where are the default settings saved for the Command Prompt defaults?  Are they stored in the registry somewhere, or in a file?  I would love to make a .reg file to set my default preferences when logging into a machine for the first time.  I'm talking about the default application settings: QuickEdit mode, buffer size, text color, etc. The new top bar is out on meta. Consider it a beta EDIT Dec 4 -  We're doing it live! Please post bugs or concerns as questions tagged [top-bar]  We have rolled out our 95% completed top bar for feedback.  Note:   We are still adding a few final features, so if you have a request for a feature that is not there you might hold off on that for a day or two Yes, the bar is black1. Also give that a day or two to sink in.   I also want to specifically address the meta/main/chat links.  I'm very aware that many of you like having the links at the top of the page and that this new version is different.  The thought process behind moving these links is that the "site switcher" should be an automatic place to go when you are on a site and want to go to another one. If you haven't already developed the muscle memory for these links, getting between them required a bit of a cognitive load: You must first decide that you want to switch sites and then you have to decide which area to go to to switch.  Also, we find that less familiar users aren't entirely clear what a standalone link that says "main" or "meta" actually does... Not to mention that it hurts my UX sensibilities that we are dynamically changing links in an element that should be fixed.  It is true that we have added a click in the process, but we have also removed a lot of clicks for other common actions that are much more frequent than switching between main and meta.  All I ask is that you give it a few days and see if the new "switch sites at the top left" paradigm grows on you.  What do you think?  1.  points out that technically it's not quite black. Ruby -- capitalize first letter of every sentence in a paragraph Let $R$ be a commutative ring and $I_1, \dots, I_n$ pairwise comaximal ideals in $R$, i.e., $I_i + I_j = R$ for $i \neq j$. Why are the ideals $I_1^{n_1}, ... , I_r^{n_r}$ (for any $n_1,...,n_r \in\mathbb N$) also comaximal? How to calculate number of days between two given dates? If I have two dates (ex. '8/18/2008' and '9/26/2008'), what is the best way to get the number of days between these two dates? Let $f:[0,\infty)\rightarrow [0,\infty) $ be a continuous strictly increasing function and $g=f^{-1}$. Let $a,b&gt;0$ and $a\neq b$. Then how $\displaystyle\int_{0}^af(x)dx+\int_0^bg(y)dy\geq ab.$ Any hint will be appreciated. Nested roots sequence, how to prove it's monotone and bounded? Let $a\ge1$ and define the sequence $(x_n)$ recursively by: $$x_1 = \sqrt{a}$$ $$x_{n+1}= \sqrt{a+x_n}$$  Here's what I did:  Plugging in some values makes it seem as if the sequence is increasing. I first solve the equation $t=\sqrt{a+t}$, equivalently $t^2 -t -a =0$, to get an idea what's going on. The roots are $$t_{1,2}= \frac{1 \pm \sqrt{1+4a} }{2}.$$ Therefore, between the roots, that is for $$t\in \left( \frac{1 - \sqrt{1+4a} }{2}, \frac{1 + \sqrt{1+4a} }{2}\right)$$  we have $t^2-t-a \le 0 \iff t \le \sqrt{a+t}$. Here's my attempt at showing inductively that $$\forall n\in \mathbb{N}, \ x_n \le \frac{1 + \sqrt{1+4a} }{2}$$  For $n=1$, the statement is true (simple arithmetic). Suppose it's true for $n$. Consider  $$x_{n+1}= \sqrt{a+x_n}\le \sqrt{a+\frac{1 + \sqrt{1+4a} }{2}}=\frac{1 + \sqrt{1+4a} }{2},$$  because $\frac{1 + \sqrt{1+4a} }{2}$ satisfies $x=\sqrt{x+a}$.  So this establishes the sequence $(x_n)$ is bounded. Now, by induction again, I try to show it's monotonically increasing. The base case $x_1\le x_2$ is obvious. Assume (strong induction) $x_{n-1}\le x_n$ for all $n&gt;1$. But since we have $$x_1\le x_n \le \frac{1 + \sqrt{1+4a} }{2}, $$   then $x_n$ lies in the appropriate interval, so $x_n \le \sqrt{a+x_n}=x_{n+1}.$  Is this proof correct? Can the condition on $a$ be relaxed to $a&gt;0$? Maybe even small negative $a$? And could someone provide me with another proof of convergence of $x_n$? I absolutely hate this one!