Prove that if $f,g$ are coprime polynomials then there exist $a,b$ polynomials such that $af+bg=1$  I think it has something to do with Euclid's algorithm, would like to see a proof.
Let $F$ be a field and $F[x]$ a polynomial ring. Let $p(x)$ be an irreducible polynomial.  Show that $\gcd(p(x),q(x)) = 1\Longrightarrow \exists r(x),s(x)$ such that $r(x)p(x)+s(x)q(x) = 1$.  I know the proof for Bezout's identity for , but this proof uses the notion of absolute value, which cannot be applied to a polynomial ring.   I am thinking about a way to prove the statement: Use the Euclidean algorithm to compute $\gcd(p(x),q(x))$ and show that the result is $1$, and somehow go backwards to obtain $r(x)p(x)+s(x)q(x) = 1$, but it is a bit messy, and I'm not yet able to see it completely.  Please offer some help. If you have some other methods, that would be great, too.
I need a regular expression to select all the text between two outer brackets.  Example: some text(text here(possible text)text(possible text(more text)))end text  Result: (text here(possible text)text(possible text(more text))) How can I give other drives and partitions short, meaningful names in nautilus? In nautilus and Unity the external drive has a nice short descriptive name "mcat", but all partitions on the 4 internal drives are displayed as a size (834 GB filesystem) or a huge 32-character string form of a GUID:  I'm guessing the external drive is nice, short, sweet and readable because that drive may have no partitions (well, just one I guess) and that name may be the drive label, whereas partitions usually don't have names. That may explain my problem, but doesn't solve it.  Is there some way to give reasonable names to these partitions in nautilus and Unity? Mean or sum of gradients for weight updates in SGD I am using single observation to compute losses using neural network implementation in PyTorch. I am confused in a small detail of SGD. If I compute loss and do loss.backward(), I am accumulating gradients.  If I do this on 100 observations and then run optimizer.step(), should I average out the gradients?  This is what I am doing as of now:  def compute_loss(training_data):   for data in training_data:     loss = F.mse_loss(data[0], data[1])     loss.backward()  def optimize(sample):   optimizer.zero_grad()   compute_loss(sample)   optimizer.step()   Should it be rather:  def compute_loss(training_data):   for data in training_data:     loss = F.mse_loss(data[0], data[1])     loss.backward(torch.Tensor([1.0/len(training_data)])) If $\sin \theta+\cos\theta+\tan\theta+\cot\theta+\sec\theta+\csc\theta=7$, then $\sin 2\theta$ is a root of $x^2 -44x +36=0$ My own bonafide attempt. $$ 0&lt;\theta&lt;\pi/2$$   and $$\sin\theta+\cos\theta+\tan\theta+\cot\theta+\sec\theta+\csc\theta=7$$   then show that $\sin 2\theta$ is a root of the equation $$x^2 -44x +36=0$$   I tried to use the above given equation of all the trigonometric ratios, though I ended up with an expression of $\sin\theta\cos\theta$.   But this too is in the form of $\sin\theta$ and $\cos\theta$ which was $\sin\theta\cos\theta=\frac{1}{6-\sin\theta-\cos\theta}$.  When I put that in the quadratic equation, it would again transform into the form of $\sin\theta$ and $\cos\theta$, and hence at last I couldn't prove the thing.  Even hints would work, as I would like to solve the question myself. Determine the number of ways to colour a 1 by n grid using the colours red and white if an even number of squares are to be coloured red. Suppose you do bad in undergraduate school in say computer science. But you do very well in a masters program in computer science. If you want to apply to a PhD program in computer science, will the masters degree grades offset the undergraduate degree grades? What is the need of acceleration? As we can usually predict the various forces that a body might be undergoing based on its position and its velocity.Hence what is the need of acceleration? Prove that $$1+\frac{1}{3}+\frac{1\cdot 3}{3\cdot 6}+\frac{1\cdot 3\cdot 5}{3\cdot 6 \cdot 9}+.........=\sqrt{3}$$   $\bf{My\; Try::}$ Using Binomial expansion of  $$(1-x)^{-n} = 1+nx+\frac{n(n+1)}{2}x^2+\frac{n(n+1)(n+2)}{6}x^3+.......$$  So we get $$nx=\frac{1}{3}$$ and $$\frac{nx(nx+x)}{2}=\frac{1}{3}\cdot \frac{3}{6}$$  We get $$\frac{1}{3}\left(\frac{1}{3}+x\right)=\frac{1}{3}\Rightarrow x=\frac{2}{3}$$  So we get $$n=\frac{1}{2}$$  So our series sum is $$(1-x)^{-n} = \left(1-\frac{2}{3}\right)^{-\frac{1}{2}} = \sqrt{3}$$  Although I know that this is the simplest proof, can we solve it any other way someting like defining $a_{n}$ and then use Telescopic sum.  Thanks. Custom action to set emblem in thunar Regarding ,  it is not possible to combine complex powers in the usual way:  $$ (x^y)^z = x^{yz} $$  There was mention of multi-valued functions, is there some theory that makes this all work out correctly? Where are icons stored? I want to change an icon of a specific theme. Where do themes save their icons? Convergence of the $\sum_{n=1}^{\infty}n^3b_n^3$ where $b_n=\int_{-\pi}^{\pi}f(t)\sin (nt)\ dt$. C++11 does not deduce type when std::function or lambda functions are involved Capturing SOAP requests to an ASP.NET ASMX web service combining results of two select statements In shell scripts used for unit testing with dynamic libraries in a directory other than the typical @rpath, I have previously been able to set DYLD_FALLBACK_LIBRARY_PATH to set the directory containing the libraries. Under 10.11.1, bash seems to ignore attempts to set this environment variable:  $ sh -x testscript.sh + DYLD_FALLBACK_LIBRARY_PATH=/Users/something/testinglibs + export DYLD_FALLBACK_LIBRARY_PATH + exec printenv   and DYLD_FALLBACK_LIBRARY_PATH is not present in printenv's output.  Is this a security-related hack in 10.11's shell? I haven't been able to find this change documented in man pages or online. How do I prove that in every commuting family there is a common eigenvector? \varepsilon vs. \epsilon Could someone please explain? I couldn't find anything on the internet, everything talks about how to go about it in some way, but nothing says exactly what it is.   Also, what is a fully trusted assembly and how do they differ from one another?  I have a MS certification exam and this is the only topic that I just don't understand.  EDIT: Thanks guys. Now I have a better understanding of security in .NET. I was able to pass my certification exam. URL to load resources from the classpath in Java