Remove items from array with splice in for loop
Looping through array and removing items, without breaking for loop
Let $f(x)$ a irreducible polynomial of degree $n$ over a field $F$. Let $g(x)$ be a polynomial in $F[x]$. Prove that every irreducible factor of the composition $f(g(x))$ has a degree which is divisible by $n$.  I don't know even how to begin. I really need help. Thanks In blender cycles render engine, how do I put a movie clip on a plane so it displays the video? I want the video to be on the plane if there is a way. How do I enable hardware virtualization technology (VT-x) for use in Virtualbox? What are the shell's control and redirection operators? There's this exercise: let $\,f\,$ be analytic on $$D:=\{z\;\;;\;\;|z|&lt;1\}\,\,,\,|f(z)|\leq 1\,\,,\,\,\forall\,z\in D$$  and $\,z=0\,$  a zero of order $\,m\,$ of $\,f\,$.  Prove that $$\forall z\in D\,\,,\,\,|f(z)|\leq |z|^m$$  My solution: Induction on $\,m\,$: for $\,m=1\,$ this is exactly the lemma of Schwarz, thus we can assume truth for $\,k&lt;m\,$ and prove for $\,k=m&gt;1\,$ . Since $\,f(z)=z^mh(z)\,\,,\,h(0)\neq 0\,$ analytic in $\,D\,$ ,  put  $$g(z):=\frac{f(z)}{z}=z^{m-1}h(z)$$   Applying the inductive hypothesis and using Schwarz lemma $\,\,(***)\,\,$ we get that  $$|g(z)|=\left|\frac{f(z)}{z}\right|=|z|^{m-1}|h(z)|\stackrel{ind. hyp.}\leq |z|^{m-1}\Longrightarrow |f(z)|\leq |z^m|$$ and we're done...almost: we still have to prove $\,|g(z)|\leq 1\,$ for all $\,z\in D$ in order to be able to use the inductive hypothesis and this is precisely the part where I have some doubts: this can be proved as follows (all the time we work with $\,z\in D\,$):  $(1)\,\,$ For $\,f(z)=z^mh(z)\,$ we apply directly Schwarz lemma and get $$|f(z)|=|z|^m|h(z)|\leq |z|\Longrightarrow |z|^{m-1}h(z)|\leq 1$$ And since now the function $\,f_1(z)=z^{m-1}h(z)\,$ fulfills the conditions of S.L. we get  $(2)\,\,$ Applying again the lemma,  $$|f_1(z)|=|z|^{m-1}|h(z)|\leq |z|\Longrightarrow |z^{m-2}h(z)|\leq 1$$and now the function $\,f_2(z):=z^{m-2}h(z)\,$ fulfills the conditions of them  lemma so...etc.  In the step$\,m-1\,$ we get  $$|z||h(z)|\leq |z|\Longrightarrow {\color{red}{\mathbf{|h(z)|\leq 1}}}\,$$  and this is what allows us to use the inductive hypothesis in $\,\,(***)\,\,$ above.  My question: Is there any way I can't see right now to deduce directly, or in a shorter way, that $\,|h(z)\leq 1\,$ ? Explanation of C++ FAQ's unsafe macro? Getting started with ArcObjects? What is the best route or place to get started with ArcObjects if one is not a developer and not aiming to become a developer?  I've been using gis professionally for a long time, almost two decades, arcinfo/arcgis for most of that; I'm pretty good. I'm learning software development, and even have a modestly successful small python application used in a public project; I'm not good! I don't want to become a full fledged software developer but I keep running into things I just can't do without programming (). In the arcgis world this puts me pretty squarely in the python camp, which is fine by me since I like python, however python does not have straightforward access to ArcObjects. (Python and ArcObjects , but it's an unsupported route. This question is about following a beaten path.)  I need to get started, but where?  to get started with the  but $1500/yr is definitely not in my budget, and sounds like using a sledge hammer to swat mosquitos. And which language .NET, Visual Basic, or Visual C++?    UPDATE: Thank you everyone for the wonderful answers. In light of them I realise  I unecessarily narrowed the scope of my question be pre-supposing "ArcObjects" is the direction I need to go in. A more open ended formulation is more along the lines of:     I keep running into problems I just   can't solve with arcgis and python   alone. What else can I learn/use to   solve problems like X? I've no   interest or intention of becoming a   software developer. I just need to do   a couple things which aren't exposed   to the arcgis python modules. Writing a character N times using the printf command I found the following command to repeat a character in Linux:  printf 'H%.0s' {1..5000} &gt; H.txt   I want, for example, H to repeat 5000 times. What does %.0s mean here? Splitting raster into smaller chunks using GDAL? I have a raster (USGS DEM actually) and I need to split it up into smaller chunks like the image below shows. That was accomplished in ArcGIS 10.0 using the Split Raster tool. I would like a FOSS method to do this. I've looked at GDAL, thinking surely it would do it (somehow with gdal_translate), but can't find anything. Ultimately, I'd like to be able to take the raster and say how large (4KM by 4KM chunks) I would like it split up into. Fire (at, on, in, to) target How to correctly choose the preposition in "fire (at, on, in, to) target"? Finding distance between consecutive points using ArcGIS Desktop? Why does camera this tracking fail completely even when the solution value is very low? The attached animated gif shows some frames of a video with a mesh box placed at each tracking marker used to successfully track (solution error &lt;.4) an 800 frame chunk of video.   The point is to illustrate just how poorly the camera tracker calculated: when the box is at a marker placed at the top of a faraway building on the left, it is actually visually much larger than when it is at the tracking marker closest to where the real camera was. I went to great lengths to be sure that the camera's sensor size and focal length are correct, and manually placed and followed all markers to be sure they were accurately tracking.   There is not a lot of parallax motion over the 800 frames, but there should have been enough to at least know that the point closest to the camera moved more than the point furthest away. Sometimes it seems to work, sometimes it doesn't. Does anybody know how to improve this? Get number of unread sms What exactly happens to the signals hitting a common mode choke? I'm trying to better understand the principles behind the common mode choke. I made a few drawings to clarify.  &nbsp; Differential Mode Signals  Differential currents (driven by differential voltages) create equal but opposite magnetic fields B in the inductor core:    These magnetic fields cancel each other out, so the net flux in the core is zero. As such, these differential currents don't "feel" any impedance.  &nbsp; Common Mode Signals  In contrast, common mode currents generate equal and additive magnetic fields in the core. That's why they "feel" a high impedance, and cannot get through (or getting through means they are highly attenuated).    But what exactly happens? I have several theories, which I will describe below.  &nbsp; Common Mode Signals - Theory 1  My first thought would be that the common mode signal hits the choke and creates a magnetic flux inside. By doing this, lots of energy is lost (hysteresis and perhaps other effects) as heat. Only a small part gets through:      What kind of common mode choke would behave in this particular way? "Burning up" the voltage spike seems a very desirable effect to me.  &nbsp; Common Mode Signals - Theory 2  Perhaps the voltage spike doesn't really get the chance to build up much magnetic flux in the core, or maybe the core is simply not "lossy" enough. The voltage spike bounces off the core and turns back. Only small part gets through:    Although the system on the right side of the choke is protected, the system on the left has to deal with reflected signals. Nasty things like standing waves might appear.  &nbsp; My questions  I've got a few questions for you:   Do you think theory 1 or theory 2 is most plausible? Do you think certain types of common mode chokes tend to behave as described in theory 1, others like in theory 2? Perhaps both of my theories are just plain wrong. If so, what actually does happen?   Please enlighten me. Continuing example in Beamer dev/sda1: clean, 552599/6111232 files, 7119295/24414464 blocks   After I turn on my laptop this message appears. The system never boots, the message just stays there. How do I run two commands in one line in Windows CMD? How to skip kernel update? Title tag different from title appearing in Google? I'm just wondering how it is possible that IGN.com has this in its title tag:      Video Games, Cheats, Walkthroughs, Game Trailers, Reviews, News, Previews &amp; Videos at IGN   while on Google its title is simply "IGN".   How can one achieve such a thing? Does it use some special meta tag I can't see? Niven’s theorem proof. I couldn’t find any proof on the internet. I would appreciate an elementary proof , but any help would be appreciated!  Thank you.