I'm hoping for feedback on my understanding of how to approach this question.    Firstly to recall;   $$R = \mathbb Z[x]/\langle x^2 + 1\rangle = \{ l(x) + \langle x^2 + 1\rangle | l(x) = 0 \text{ or } deg(l) &lt; 2\}$$  If $a(x),b(x) \in R, $ we define multiplication and addition as:  $$a(x)\cdot b(x) = a(x) \cdot b(x) \text{ mod } x^2 +1,\ a(x) + b(x) = a(x) + b(x) \text{ mod } x^2 +1$$  To understand how this structure works we can see what happens to the basis on multiplication:  $$1\cdot1 = 0\cdot(x^2 + 1) + 1 = 1,\ x\cdot 1 = 1\cdot x = 0\cdot(x^2 + 1) + 1 = 1,\ x\cdot x = 1\cdot(x^2 + 1) - 1 = -1$$  So by investigation I can treat the $1$ as the real part and $x$ as the imaginary part. Given an arbitrary element of $R$ has the form $ \alpha \cdot 1 + \beta \cdot x$ for $\alpha,\beta\in \mathbb Z$, I define the mapping $\phi:R\rightarrow R$ as $\phi(\alpha \cdot 1 + \beta \cdot x) \rightarrow \alpha + \beta I$. Now to check this is an isomorphism between the two rings, I need to check that the mapping is a bijection on both operations of the ring.  I won't do this part as it's quite long winded, but my general approach here highlights a correct understanding right?  Thanks heaps.
Is $\mathbb{Z}[x]/(x^2+1)$  isomorphic to  $\mathbb{Z}[i]$?   My attempt is that try to define a mapping $g$ from $\mathbb{Z}[x]$ to $\mathbb{Z}[i]$ by $g(f(x))= f(i)$, for $f(x)\in\mathbb{Z}[x]$. If it is possible then $\ker g$ is $(x^2+1)$? Am I on the right track? Please Help.
Dispatcher.BeginInvoke: Cannot convert lambda to System.Delegate Python metaheuristic packages I need to use a metaheuristic algorithm to solve an optimization problem on a Python codebase.  Metaheuristics usually need to be written in C++ or Java as they involve a lot of iterations, while Python is weak from this point of view.  Questions:   do any Python metaheuristic packages which wrap faster languages as C++/Java exist? do any Python metaheuristic packages based on maybe cython on numba exist? other solutions? How to check file types of uploaded files in PHP? On the PHP website, the only real checking they suggest is using is_uploaded_file() or move_uploaded_file(), . Of course you usually don't want user's uploading any type of file, for a variety of reasons.  Because of this, I have often used some "strict" mime type checking. Of course this is very flawed because often mime types are wrong and users can't upload their file. It is also very easy to fake and/or change. And along with all of that, each browser and OS deals with them differently.  Another method is to check the extension, which of course is even easier to change than mime type.  If you only want images, using something like getimagesize() will work.  What about other types of files? PDFs, Word documents or Excel files? Or even text only files?  Edit: If you don't have  or  and system("file -bi $uploadedfile") gives you the wrong file type, what other options are there? How does a wheelie work? So I've been trying to create a mathematical model for an electric motorcycle and began to wonder about the maximum possible torque that could be supplied to the rear driven wheel without having the bike begin to lift up on one wheel.  I found ways to calculate this value online; however, the basic concept as to how the bike actually lifts escapes me.    My problem started when I began to think about what axis the bike will rotate about when it is doing the wheelie.  My first intuition was that the frame and front wheel, together, rotate about the rear axle.  But when I drew a free body diagram of the frame/front wheel system just at liftoff (see below) I made note that the only forces acting about the rear axle, point O, is the force of weight.  This means that an increase in applied torque and subsequently, the applied force Fa, should not effect the rotation about the rear axle.  I know that the applied force on the back wheel is indeed correlated with the propensity for a bike to wheelie, so I considered that the axis of rotation I was choosing was wrong.  If we take the free body diagram above and sum the moments about the center of mass, we would find that an increased applied force would in fact cause the solid body to rotate.  The problem with this understanding is that during rotation, the center of mass of the drawn system should actually rises relative to the surface the bike is moving across.  If the bike were to be truly rotating about its center of mass, then the back wheel would begin to dip below the surface of the road like you might see in a glitchy video game.    So I suspect that the bike is in fact rotating about the back axle, but I don't understand why, please help!  edit: I added the external torque from the back wheel to the frame, which would allow the bike to rotate about the back axle  edit 2:  I suppose that the torque acting on the frame via the engine should not effect the rotation as the movement of a motorcycle can be perfectly replicated by applying a force at the back axle.  A third possible axis of rotation might be the lowest point of the back wheel. If we disbelieve $H_0$, why quote a p value calculated assuming $H_0$ was true? Accidentally deleted /etc/fstab file I accidentally deleted my /etc/fstab file by running sudo rm /etc/fstab. Without realising what I'd done, I shut down my Ubuntu OS.  Now I'm not able to boot.    My screen looks like this. In some cases, a purple screen appears. Please give me some solutions. The default PS1 variable on my machine (Kubuntu 13.10) is this:  \[\e]0;\u@\h: \w\a\]${debian_chroot:+($debian_chroot)}\u@\h:\w\$   I'm looking for a reference for how the $PS1 variable works that, at a minimum, allows me to understand the above PS1 variable. I have video tutorials that I want to show in my presentation with beamer class. I don't want the video files get embedded to the PDF because they will make the PDF size very very huge.  If you know how the videos can be played on the HTML with a reference rather than embedding the video file to the HTML file, I want something like this to be applied to my PDF presentation. I just want to reference the video URL from my PDF presentation.   Shortly speaking, can we use media9 to show a video on a PDF but the video is not embedded to the PDF?  Uploading those videos to a web server might be difficult for me because internet connection is very slow and sometimes not available in the presentation room. :-) I know that non-married lovers address each other by saying “This is my girlfriend/boyfriend.” I know that married couples address each other by calling “my wife/husband”.  I also know that gay lovers who are not married (yet?) address each other by saying "my boyfriend" (for gays) and "my girlfriend" (for lesbians).  But if they are married, how do they address each other?  And how do we describe them? I mean if they are gays, are they “husband and husband” or “husband and wife” (one of them acts like the wife)? And if they are lesbians, are they “wife and wife” or “husband and wife” (one of them acts as the husband)?  I know they can be referred to as a gay couple or gay lovers from , but I couldn't find the answer to address each of the two. Monitoring contents of files/directories? If Chrome is already open, it opens new sites (e.g. from Windows shortcuts, application shortcuts, or from the Run dialog) in new tabs of the existing instance.   I'd prefer to open new sites in a separate window (a new instance of Chrome).   I don't see that in the options - is it possible? Removing empty lines in Notepad++ Ubuntu Software (Software Center) Crash - Ubuntu 16.04 Start modal operator from a click in panel I want to start a modal operator by clicking a button in a panel in the properties region. It should then stop when I click somewhere in the 3D viewport. I want to execute some code when I start and finish the operator (see the code below). I tried modifiyng the modal operator template with an execute method to start the modal operator. But the now the context.space_data.type is always &quot;PROPERTIES&quot; because that's where the panel is located, I guess. Is this a bad way of doing what I want? class MY_OPERATOR_OT_click_on_3dview(bpy.types.Operator):          bl_idname = &quot;mesh.click_on_3dview&quot;     bl_label = &quot;Click on 3D&quot;      def modal(self, context, event):                  if context.space_data.type != 'VIEW_3D':             return {'PASS_THROUGH'}                  if event.type in {'MIDDLEMOUSE', 'WHEELUPMOUSE', 'WHEELDOWNMOUSE'}:             # allow navigation             return {'PASS_THROUGH'}         elif event.type == 'LEFTMOUSE':             # DO ALL THE POST-SELECT STUFF             return {'FINISHED'}         elif event.type in {'RIGHTMOUSE', 'ESC'}:             return {'CANCELLED'}          return {'RUNNING_MODAL'}      def execute(self, context):         # DO ALL THE SETUP STUFF           context.window_manager.modal_handler_add(self)              return {'RUNNING_MODAL'}            def cancel(self, context):         print(&quot;CANCEL???&quot;) $(\Bbb R,+)$ is a topological group. Is there any other group structure on $\Bbb R$ such that it is still a topological group and this group is not isomorphic to $(\Bbb R,+)$ ?  Refer to  for the general problem. In that problem I have conjectured that there is no such group however I couldn't prove it. What hardware does Ubuntu Touch support? With the announcement of the Ubuntu Phone OS I'd like to know what phone (and tablet) models are supported at this time.  Note from foss &amp; Oli: We are making this the master question for all future "Will this work on &lt;insert random tablet/phone/device here&gt;?!" questions amsmath align environment: row spacing How can I stretch the spacing between consecutive lines in an align* environment? I have a few lines with \fracs in which both the numerator and the denominator consist of textual phrases, not math symbols, and the result looks too cluttered.  \begin{align*} X &amp;= \frac{\textit{Some text here}}{\textit{And some text here}}\\ Y &amp;= \frac{\textit{Will make everything}}{\textit{Look too close together}} \end{align*} As far as I understand, one of the main claimed problems with initializing e.g. a feed-forward neural network (with several $\text{tanh}$ or $\text{ReLU}$ layers) with $W=0$ is that it doesn't break "network symmetry", meaning, backpropagation would propagate the same error through all such units (i.e. "nudging all weights in the same direction"). This is, I presume, undesirable because we would not be learning "different" calculations through different paths of the network.  However, I'm confused why that even matters in this case given the fact that if $W$ ever drops to 0, we will effectively be propagating no gradients at all through the network, since W=0 would multiply all errors from the output and prevent any learning.  Put another way, even if $W=0$ does not break network symmetry (wasting calculations and paths in the network) is it correct to say that if $W=0$ (e.g. by initialization) we are effectively killing gradients in a neural network, and thus no learning can take place? I am making a test where I am asked to add the missing words.     [...] chair is [...] useful piece of furniture.    I would have said that a is the missing word in both the cases, but the test says the words I should have added are the and a respectively.     [...] last time I saw [...] bear, I was travelling in [...] Europe.   I would have added: nothing, a, nothing. The correct answer is: the, a, nothing.  Why is the the correct article to use in both the sentences? Computing irreducible components of algebraic set