I have a question bout euclid polynomials.  If $C(x) =x^4−1$ and $D(x) =x^3+x^2$  How do I find a polynomials $A(x)$ and $B(x)$ such that $A(x)C(x) +B(x)D(x) =x+1$ for all $x$?
Find polynomials $A(x)$ and $B(x)$ such that $A(x)P(x) + B(x)Q(x) = x + 1$ for all $x$ where $P(x) = x^4 - 1$ and $Q(x) = x^3 + x^2$. I'm stumped on this question. I know that I'm supposed to apply the extended version of Euclid's algorithm for polynomials but I'm unsure of how to do that. I thought about trying to create some kind of linear system but guessing arbitrary coefficients but that wouldn't work as $A$ and $B$ don't have fixed degrees.
How do I get a list of all files (and directories) in a given directory in Python? Play a Sound with Python Oxford Comma Conventions Get current URL with jQuery? I am using jQuery. How do I get the path of the current URL and assign it to a variable?  Example URL:  http://localhost/menuname.de?foo=bar&amp;amp;number=0 Is this an equivalent formulation of "surjective" resp. "epimorphism"? How does the Mac choose which connection to use when both Wifi and Ethernet are connected? When I'm connected to two networks with separate routers using Wifi and Ethernet, how does my Mac know what connection to use when I visit a website? How can I reuse existing materials, objects, etc from an existing .blend? How to Flatten a Multidimensional Array? I am attempting to add a calendar to my Default.aspx in my sharepoint hosted add-in in visual studio. I used the default code-snippet for adding sharepoint lists. Here's what I added to my Default.aspx:    &lt;WebPartPages:WebPartZone runat="server" FrameType="Default" ID="full" Title="loc:full"&gt;     &lt;WebPartPages:XsltListViewWebPart runat="server"         ListUrl="Lists/Calendar"         IsIncluded="True"         JsLink="clientTemplate.js"         NoDefaultStyle="TRUE"         PageType="PAGE_NORMALVIEW"         Default="False"         ViewContentTypeId="0x"&gt;     &lt;/WebPartPages:XsltListViewWebPart&gt; &lt;/WebPartPages:WebPartZone&gt;     and this is the result...    Pretty dang ugly if you ask me. How can I get this looking like a normal share point calendar instead of an unusable mess?  Thank you! Let $p$ be an odd prime number. Prove that $$\left( \frac{1 \cdot 2}{p} \right) + \left( \frac{2 \cdot 3}{p} \right) + \left( \frac{3 \cdot 4}{p} \right) + \cdots + \left( \frac{(p-2)(p-1)}{p} \right) = -1$$ where $\left( \frac{a}{p}\right)$ is the Legendre symbol.  This seems to be a tricky one! I've tried using the property $\left( \frac{ab}{p} \right)=\left( \frac{a}{p}\right) \left( \frac{b}{p} \right)$ and prime factoring all the non-primes but to no avail. I had a quick thought of induction haha, but that was silly. I tried factoring the common Legendre symbols like $\left( \frac{3}{p}\right) \left[ \left( \frac{2}{p} \right) + \left( \frac{4}{p} \right) \right]$ but that didn't bring anything either. And I've been looking for pairwise cancellation with $1$ term leftover, but that does not seem to work.  Can you help? AngularJS Directives: Change $scope not reflected in UI Diagonal straight lines in residuals vs fitted values plot for multiple regression I am observing strange patterns in residuals for my data:   [EDIT] Here are the partial regression plots for the two variables:      [EDIT2] Added the PP Plot   The distribution  seems to be doing fine (see below) but I have no clue where this straight line might be coming from. Any ideas?   [UPDATE 31.07]  It turns out you were absolutely right, I had cases where the retweet count was indeed 0 and these ~ 15 cases resulted in those strange residual patterns.   The residuals look much better now:   I've also included the partial regressions with a loess line. I don't have complex analysis at my beck and call, and I only have a low level of knowledge in topology, but I need to prove that this metric space (for any real $r$ and $R$ with $r &lt; R$)$$ X = \{ (x, y) \in \mathbb{R}^2 \ | \ r \leq x^2 + y^2 \leq R \}$$  with the Manhattan metric $d((x_1, y_1), (x_2, y_2)) = |x_1-x_2| + |y_1-y_2|$ is not simply connected.  I've already prooven that it's path connected, and now I need to show there are some points $P$ and $Q$ with two paths between them such that one cannot be continuously 'morphed' into the other.  $ \ $  What I have so far is as follows:  I take $P = (0, r)$ and $Q = (0, -r)$, with $f_0$ being a path from $P$ to $Q$ going clockwise around the circle radius $r$ and $f_1$ being much the same but going counterclockwise.  Now I assume there is a function $g : [0, 1]^2 \to X$ such that:   $g(s, 0) = f_0 (s)$ $g(s, 1) = f_1 (s)$ $g(0, t) = P$ $g(1, t) = Q$   To get the final result, I need to show that this function cannot be continuous, but for the life of me I cannot.  For some context, these are the topics which have been visited during the course, roughly in order of recentness.   Multiple connectedness Simple connectedness Pathwise connectedness Interior points Boundary points Open sets Compactness Complete metric spaces Bounded metric spaces Totally bounded metric spaces Closed sets Closure of a metric space Limit points Cauchy sequences Convergence Continuity Metric space quivalence Metric equivalence   Edit: I might have an argument that works, though it's far from rigourous. We can shrink $R$ to be as close to $r$ as we want, so we can essentially constrain the annulus down to a circle and thus force any path from the left side of the circle to the right side to go through $P$ or $Q$.  So holding $s \in (0, 1)$ constant and varying $t$ must produce a path through $P$ or $Q$ for any $s$. If for some $s$ it passes through $P$ and for some other $s$ it passes through $Q$, then there must be $s_0$ such that $\forall \epsilon &gt;0  \ \exists \delta \leq \epsilon$ st. $s_0$ produces a path through $P$ and $s_0 + \delta$ produces a path through $Q$.  Now we can consider the path $g(s, \frac{1}{2})$, and note that it must have a discontinuity at $s_0$.  Now consider the case where all $s$ produce paths through only one of $P$ or $Q$. WOLOG: $P$. Now, at $t = \frac{1}{2}$, $s$ arbitrarily close to $1$ are mapped away from $Q$, but $1$ is always mapped to $Q$ by definition, so $g(s, \frac{1}{2})$ has a discontinuity at $s=1$. Therefore $g(s, t)$ is not continuous.  This argument is definitely iffy to me, if no one has their own argument (using sufficiently low level concepts), then criticism on the above would be appreciated. When an option expires does that constitute a buy or sell A similar question recently asked about stock sales and option buys triggering wash sale rules.  If an option expires, is that considered a buy/sale ?  What I mean is: let's assume I have a call option with a target date of August 16. Now I sell the underlying stock somewhere before that (after July 16th). Does that trigger wash sale rules (if the option expires in the money, exercised by default) ?  Is an option expiring out of the money considered a sale ? Can the off-regional pokémon hatch from eggs? Some people claim you can get the pokémon that are not in your region from eggs. Is this only a rumour or actually true? I've seen off-regional pokémon in gyms nearby but I can't rule out that the owner had been abroad or maybe even used location spoofing. Prove $(a^2+b^2)(c^2+d^2)\ge (ac+bd)^2$ for all $a,b,c,d\in\mathbb{R}$. How to return only the Date from a SQL Server DateTime datatype SELECT GETDATE()   Returns: 2008-09-22 15:24:13.790  I want that date part without the time part: 2008-09-22 00:00:00.000  How can I get that? Evaluate: $\lim_{n\to\infty}\left({2\sqrt n}-\sum_{k=1}^n\frac1{\sqrt k}\right)$ Installing Cygwin packages from the command-line Possible Duplicate:        I'm looking for a "yum"-link method to install additional Cygwin packages from the command-line. Let's say I want to add the package "Math/bc", then ideally I'd like to be able to do it  using one command as in:  yum install bc   Currently the only way I've found to install new packages is to locate and run the installation file SETUP.EXE file and then scroll to the relevant package, add it, etc. This is  quite cumbersome so I'm sure there is a smarter command-line way to add an additional package. SQL injection that gets around mysql_real_escape_string()