$\lim_{n\to\infty}(1+x)(1+x^{3})\cdots(1+x^{2n+1}),|x|<1.$
Find the product $\prod_{n=1}^{\infty} (1+x^{2n-1}) $
As the title, I was asked to show that $2^{44}-1$ is divisible by $89$.  My first attempt is to change the expression into $(2^{22}-1)(2^{22}+1)$.  Then further simplified it into $(2^{11}-1)(2^{11}+1)(2^{22}+1)$, I used my calculator and was able to show that $2^{11}-1$ is divisible by $89$ but then I don't know how to show it with modular arithmetic. I do think that it is quite similar to the form where we can use the Fermat's little theorem. $(\sqrt{2})^{88}-1$. (Though I do understand Flt can only be applied to integer.)  Can someone tell me whether I can take square root in modular arithmetic as well? I am still pretty new to the modular arithmetic. Thank you very much. How to Send Byte array To Server My Y axis is inverted! How do I fix that? When I play Black Ops 2, it seems that my xbox controller is inverted. Example: if I press up, my character looks down and vice versa. It was not like this before. Does anyone know how to reset the controller back to its original settings? Density of states for arbitrary dispersion relation If I have a 3D dispersion relation   $E=E(k_x, k_y, k_z)$ I have an equation for the density of states, which is   $D(E)=\frac{1}{\nabla_k E}\int\frac{dS}{(2\pi)^3}$  1) I am confused about the integral. What am I integrating over, exactly? It must have units of the $k$ vector, so it's a k-space integral, but what is the surface that is relevant?  3) Where does that equation come from, how is it derived? No updates since a long time Using Ubuntu, I have a habit of running: sudo apt update sudo apt upgrade   on every startup. But for a month, I guess, there have been no updates whatsoever. The message shows: All packages are up to date  Now I am starting to worry. Can there be a fault in my update manager? As far as I know, I've made no changes to the package manager settings. But no updates for almost a month seems kind of weird. Any feedback would be helpful. Thanks. How do I reset a lost administrative password? Chemical potential of particles with zero mass Why massless particles have zero chemical potential? Windows 7 SP1 Windows Update stuck checking for updates Reducing voltage with resistors I have been looking around for an easy way to convert 12V to 5V. I have seen some people saying that a simple resistor is all that is needed.  $$ Volts = Ohms \cdot Amps $$ $$ Amps  = \frac{Volts}{Ohms} $$ $$Ohms  = \frac{Volts}{Amps}$$  So applying a resistor will diminish the voltage of the circuit. That should mean that an appropriately sized resistor could simply be placed in the path of a 12V circuit, converting it to 5v.   If this is the case how would one reduce amps? Would series vs parallel make a difference in this area?   I have seen designs that include a regulator IC and some capacitors, but if a simple resistor/fuse/diode setup will do the trick, I would really prefer that. Can I change a user's keyboard input? What if client doesn't like any logo design concept? I'm in a process of creating Graphic Design contract. I've decided to provide each client with 3 logo concepts and 3 rounds of revision. Ideally, I would create 3 concepts and client will chose one, but it happend to me before that client didn't like and concept and I end up with doing 10+ more (yes, I've worked without contract).  How can I prevent this situation? Should I charge hourly for each new concept separately? How can I express my thoughts clearly in a contract (without sounding like 'if you don't like any, you'd have to pay more)? I don't want to scare client and put thoughts into their minds saying that there is a possibility that they won't like me ideas. How could we improve our planned post notice improvements? Megan has written a .  I've seen a bunch of discussion about it on chat - but it feels like having it purely on the blog is a little clunky.  So specifically - what does the MSE community feel about the proposed post notices for closed questions? What's missing, or what should be added? What are some of the lessons we've learnt that could be helpful here? In "Knots and Primes: An Introduction to Arithmetic Topology", the author uses the following proposition   Let $h: Y \to X$ be a covering. For any path $\gamma : [0,1] \to X$ and any $y \in h^{-1}(x) (x = \gamma(0))$, there exists a unique lift $\hat{\gamma} : [0,1] \to Y$ of $\gamma$ with $\hat{\gamma}(0) = y$. Furthermore, for any homotopy $\gamma_t (t \in [0,1])$ of $\gamma$ with $\gamma_t = \gamma(0)$ and $\gamma_t(1) = \gamma(1)$, there exists a unique lift of $\hat{\gamma_t}$ such that $\hat{\gamma_t}$ is the homotopy of $\hat{\gamma}$ with $\hat{\gamma_t}(0) = \hat{\gamma}(0)$ and $\hat{\gamma_t}(1) = \hat{\gamma}(1)$.  The author then follows with; "In the following, we assume that any covering space is connected. By the preceding proposition, the cardinality of the fiber $h^{-1}(x)$ is independent of $x \in X$."   I am not sure why this results is true. This is my attempt at explaining it to myself. Take two different $x_1, x_2$ and their fibers $h^{-1}(x_1), h^{-1}(x_2)$ such that $y_1 \in h^{-1}(x_1)$ and $y_2 \in h^{-1}(x_2)$. Take two paths $\gamma_1, \gamma_2$ such that $\gamma_1(0) = x_1$ and $\gamma_2(0) = x_2$. Then we get two lifts $\hat{\gamma_1}, \hat{\gamma_2}$ with $\hat{\gamma_1}(0) = y_1$ and $\hat{\gamma_2}(0) = y_2$. Now, since our covering space is connected we can continuously deform $\hat{\gamma_1}$ into $\hat{\gamma_2}$ and conclude that every elements in $h^{-1}(x_1)$ is also in $h^{-1}(x_2)$ and vice versa. I feel that this is wrong but cannot figure out the right way to see this. Any help would be appreciated. Restore Missing "Settings" (gnome-control-center) Icon in Ubuntu 18.04 I have been running Ubuntu 18.04 for a couple of weeks. I upgraded from 16.04 and all went smoothly.   I am not sure what I changed, but this morning I noticed that the "Settings" Icon that launches gnome-control-center is missing from "Show Applications" and the applications menu as well. When I searched from "Show Applications" for gnome-control-center or just settings, nothing showed up.   So, I figured I must have accidentally removed it and figured a reinstall of the package would resolve the issue. I have searched and found several postings where this worked. I tried:  sudo apt install gnome-control-center   And then rebooted - which reinstalled the gnome-control-center and I could then launch it from the command line. However, there was still no Settings Icon in any of the menus.   I also tried many other suggestions including:  sudo apt-get install --reinstall gnome-control-center sudo apt-get install ubuntu-desktop sudo apt install ubuntu-desktop --reinstall sudo apt-get install unity-control-center   However, none of these has been successful in restoring the icon into "Show Applications" or any applications menu.   Anyone have any suggestions on something I haven't tried? Would be much appreciated.  Or, is there a way to just manually recreate the icon?   Output of locate gnome-control-center.desktop:  /usr/share/applications/gnome-control-center.desktop  /home/dccrens/.local/share/applications/gnome-control-center.desktop /home/dccrens/.gnome/apps/gnome-control-center.desktop   I am running Ubuntu 18.04 desktop on kernel 4.15.0-45-generic. Disease transmitted by postage stamps Changing the value of an element in a list of structs I have a list of structs and I want to change one element. For example :  MyList.Add(new MyStruct("john"); MyList.Add(new MyStruct("peter");   Now I want to change one element:  MyList[1].Name = "bob"   However, whenever I try and do this I get the following error:     Cannot modify the return value of   System.Collections.Generic.List.this[int]‘ because it is not   a variable   If I use a list of classes, the problem doesn't occur.  I guess the answer has to do with structs being a value type.  So, if I have a list of structs should I treat them as read-only? If I need to change elements in a list then I should use classes and not structs? How to read ANSI encoded files in the right way? I have some files that Ubuntu can't read it ( ANSI encoding ) but Windows can read it well.  When I open it in gedit or notepad++ it seems like this :     Êã ÇáÊÍæíá áÜÜ   How can I make Ubuntu read ANSI encoded files well? $M$ connected $\iff$ $M$ and $\emptyset$ are the only subsets of $M$ open and closed at the same time I'm trying to understand this proof that:  $M$ connected $\iff$ $M$ and $\emptyset$ are the only subsets of $M$ open and closed at the same time  Which is:  If $M=A\cup B$ is a separation, then $A$ and $B$ are open and closed. Recriprocally, if $A\subset M$ is open and closed, then $M = A\cup(M-A)$. What? I know that if $M=A\cup B$ is a separation, $A$ and $B$ are both open. But why closed? Also, the 'recriprocally' part is totally nonsense to me. Anybody could help?  Also, there's another proof, which states: $M$ and $\emptyset$ are the only subsets of $M$ at the same time closed and open $\iff$ if $X\subset M$ has empty boundary, theb $X=M$ or $X=\emptyset$  which is proved as the following:  given $X\subset M$, we know the condition $X\cap \partial X = \emptyset$ implies $X$ is open, while the condition $\partial X \subset X$ implies $X$ is closed. Then, $X$ is open and closed $\iff$ $\partial X = X\cap \partial X = \emptyset$, this show $\iff$ for the theorem above.  First of all, I think that the condition $X$ has boundary empty implies that $X\cap \partial X$ is empty, but who said anything about $\partial X\subset X$? Also, where's the $\rightarrow$ of this proof? I can only see $\leftarrow$ This is about the upcoming  Zack Snyder movie , where Ben Affleck is playing Batman. I just read an  that Christian Bale wanted to be Batman again. So I wanted to ask why wasn't he cast as Batman in this movie. I checked the Wikipedia entry but nothing suggests why Christian Bale was not there. Nolan is associated with the movie as an adviser and DC Entertainment is producing this one too. So sort of familiar turf, right? But how come Ben Affleck is the new Batman?  EDIT: I would like to add another interesting piece of information. Bale is aged 40 (born 1974) and Affleck is 42 (born 1970). So they have actually cast an older actor. What happens to Jen when she jumps off the cliff? What happens to Zhang ziyi (Jen) in  when she jumps off the cliff .. she dies? becomes immortal? Just before jumping what does she want Lo to become - a master?  Why does she jump off, guilt?