Are the units of the quotient ring $\mathbb{F}_2[x]/\langle x^k+1 \rangle$ known in general, where $\mathbb{F}_2$ is the finite field with two elements? I'm specifically interested in the case where $k$ is divisible by two, such as say $k=8$, or $k=12$.  If so, can they be constructed easily? I'd be grateful if you could provide a few examples.
Are the units of the quotient ring $\mathbb{F}_2[x]/\langle x^k+1 \rangle$ known in general, where $\mathbb{F}_2$ is the finite field with two elements? I'm specifically interested in the case where $k$ is divisible by two, such as say $k=8$, or $k=12$.  If so, can they be constructed easily? I'd be grateful if you could provide a few examples.
got me wondering about the differences between these three ways of measuring size: a kibibyte, a kilobit, and the conventional kilobyte.  I understand that these measurements have different uses (data transfer rate is measured in bits/sec), but I'm not quite sure if I can tell the difference between Mb and MB and MiB.  Here is a comment, reproduced below, taken from  (emphasis mine).     The C64 has 65536 bytes of RAM. By   convention, memory size is specified   in kibiBytes, data transfer rates in   kilobits, and mass storage in   whatever-the-manufacturers-think-of-now-Bytes. Harddrives use T, G, M and k on the   label, Windows reports the size in Ti,   Gi, Mi and ki. And those 1.44MB   floppys? Those are neither 1.44MB nor   1.44MiB, they are 1.44 kilokibibytes. That's 1440kiB or 1'474'560 bytes. –   Third calculating resistance for LED strips to use as parking lamps and turn signals I'm installing LED strips in my car's tail lamps to convert it from conventional bulbs to LEDs. I have a understanding of what I'm doing but not the complete knowledge to make the circuit work correctly without doing some guessing so I figured I would ask here.   I'm planning on working with and I need a length of 16", 15" and 14" to make then look correct in my housings. When I did the math the 19.7" strips have about 1.5 LEDs per inch and need to be segmented into 3 LED groups so I may have to get creative with the placement.   Ffom what I understand I need 2 diodes and a resistor. One diode on the parking lamp power, and one on the turn signal/brake power and then a resistor on the parking lamp  wire to step down the voltage enough to make them dim enough to use as a parking lamp and distinguish it from the brighter turn/brake lamps.  Does anyone know what size resistor and what size diode to use in this project? The voltage, with the car running, is around 13.8V to the rear lamps. I figure it should be running around 7V to the LEDs to light them as parking lamps.  Also will leaving out a cluster of 3 LEDs change the strips intensity to where I'll need different resistors for each strip? I'd like to modify the names used to typeset some of my document elements. For example, the caption of figure floats should change from "Figure" to "Fig.", and my \tableofcontents sectioning heading shouldn't read "Contents", but "Table of Contents". How can I do that?  \documentclass{article} % \usepackage[english]{babel} \begin{document} \tableofcontents \section{foo} \begin{figure}[h] \centering \rule{1cm}{1cm}% placeholder for graphic \caption{A figure} \end{figure} \end{document} Imported object shows aligned to the wrong axis I was doing fine with importing an obj I made in Marvelous Designer into Blender for rigging.  The dress would settle nicely on my SL avi.    Now when I import the object; it is perpendicular to my avi and I have spent a week trying to figure out what I did...  Is there a keystroke or setting that I need to make sure is correct so that my dress fits my avatar again? Is there a limit as to how fast a black hole can grow? Difference between float and decimal data type Passing multidimensional arrays as function arguments in C In C can I pass a multidimensional array to a function as a single argument when I don't know what the dimensions of the array are going to be?  Besides, my multidimensional array may contain types other than strings. How can I port forward with iptables? libpng problems for QGIS I'm using ubuntu 18.04 and if I download QGIS 2.18 and I try to run it, I get this:  libpng warning: Application jmp_buf size changed Segmentation fault (core dumped)   If I try with QGIS 3.2, the error is:   libpng warning: Application was compiled with png.h from libpng-1.6.34 libpng warning: Application  is  running with png.c from libpng-1.2.56 libpng error: Incompatible libpng version in application and library   So it's clear that the problem is with libpng.   I purged libpng-dev and reinstall again but nothing changed.   I tried to purge libpng16-16 but I get:  sudo apt purge libpng16-16 Reading package lists... Done Building dependency tree        Reading state information... Done Some packages could not be installed. This may mean that you have requested an impossible situation or if you are using the unstable distribution that some required packages have not yet been created or been moved out of Incoming. The following information may help to resolve the situation:  The following packages have unmet dependencies:  libgdk-pixbuf2.0-0 : Depends: libpng16-16 (&gt;= 1.6.2-1) but it is not going to be installed                       Recommends: libgdk-pixbuf2.0-bin  libgtk-3-0 : Depends: libcairo-gobject2 (&gt;= 1.10.0) but it is not going to be installed               Depends: libcairo2 (&gt;= 1.14.0) but it is not going to be installed               Depends: libfontconfig1 (&gt;= 2.12) but it is not going to be installed               Depends: libpango-1.0-0 (&gt;= 1.40.5) but it is not going to be installed               Depends: libpangocairo-1.0-0 (&gt;= 1.40.5) but it is not going to be installed               Depends: libpangoft2-1.0-0 (&gt;= 1.40.5) but it is not going to be installed               Depends: adwaita-icon-theme               Recommends: libgtk-3-bin E: Error, pkgProblemResolver::Resolve generated breaks, this may be caused by held packages.   and then:  ~$ sudo apt install libgdk-pixbuf2.0-bin libgtk-3-bin Reading package lists... Done Building dependency tree        Reading state information... Done libgdk-pixbuf2.0-bin is already the newest version (2.36.11-2). libgtk-3-bin is already the newest version (3.22.30-1ubuntu1). 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. ~$ sudo apt install libgtk-3-0 libgdk-pixbuf2.0-0 Reading package lists... Done Building dependency tree        Reading state information... Done libgdk-pixbuf2.0-0 is already the newest version (2.36.11-2). libgtk-3-0 is already the newest version (3.22.30-1ubuntu1). 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded.   At this point I'm afraid of breaking something if I continue trying things, so I come here for advice before.  I deleted and reinstalled ubuntu. I installed first thing qgis and it was working. Afterwards I compiled gdal, from source to have it from python as stand alone. It might be this, but I had done this previously without any problems. How to access "The grimm Troupe" and "Hidden dreams" content in Hollow Knight? In these two official announcements from Team Cherry, it's said:       Grimm’s quest, and all other new features can be accessed at any point   in your Hollow Knight journey.        Hollow Knight: Hidden Dreams content is accessible at any point in the   game, whether starting fresh or scouring for completion.   I still haven't find any way to start the quests from the menus or during beginning of the game. Some sources say the content can only be accessed late in the game from specific locations.  So, is Team Cherry simply lying in our faces? If you willingly fall on your turn does it cost movement?  As a DM I'm thinking about how to handle things such as creatures who are immune to being prone, diving into the water from a height, or a flying creature diving off a cliff and taking a fall before taking flight. There are many circumstances where you can fall without taking damage or being knocked prone afterward, and I'm not sure how to determine movement following the dive.  For example, my character walks up to a ledge above a 10 foot drop to the ground below. After taking 5 feet of movement to step off the ledge, I allow myself to fall. When I hit the ground have I used 15 feet of movement or just 5? Why not brew wine and make beer? Non native speaker so I can't fully discern the difference.  Thanks. After a table of addresses is geocoded, ArGIS provides information about each geocoded address, among those the "match score" of the candidate to which the address was matched, which ranges from 0 to 100. According to their documentation "The match score is based on how well the locations found in the reference data match with the address data being searched."  It seems intuitive that 100 means an address with the exact name was found in the Address locator and 0 means no such address was found. However, I could not find any information about how exactly this score is calculated, particularly if values are somewhere between the extremes. I this known?  I found the pointer to  in the answer to , but I could not find any information in that paper that would answer the question. I have quite a bad habit of copy-and-pasting my packages from one LaTeX document to another. Is there any way we can detect unused packages (i.e. \usepackage commands) so that they can be removed from the LaTeX document? How does update work with programs installed from .deb file I'm having some difficulties exactly understanding how the the whole update thing work in Ubuntu and good practice for installing software.  I understand I have a  /etc/apt/sources.list file where all my repositories are listed, and that these repositories are queried when I use apt-get update - to later be used with e.g. apt-get upgrade.  This makes perfect sense and I recently installed spotify by adding deb http://repository.spotify.com stable non-free to this "sources" list.    But then I got confused...  When I went to download Google Chrome I merely had to download and grab a .deb file, and Chrome installed with no problem... but I don't see any new entry in /etc/apt/sources.list...  So how does apt-get update know where to query concerning Chrome updates?  Has it somehow been added to one of the already listed repositories in the sources file?  I would like all my installed software to be encompassed by the update function. What to do when OP answers his/her own question in an edit? Reference question:   Sometimes users solve their own questions, which is great. Sometimes they say they've solved it and move on, which is not so great. The above question is middle-ground: the user posted an answer to his own question, but as an edit to the question instead of as a real answer.  Should any action be taken in such a case? The obvious solution is to copy the edit into a real answer. But is this the right solution? In this case, the answer is a little hidden and there are no other answers, so it appears on the unanswered list.     Related: Down vote changed to up vote caused reputation meter to pass daily maximum I just experienced a minor visual bug.  I had an answer that earned 2 down votes and 22+ upvotes, earning me the daily maximum of +200. Upon making some edits to the answer at the behest of the comments, one of the users who down voted changed their vote. Now my reputation meter shows that I gained +202 for the day.    This correctly did NOT affect my reputation, as I still only gained 200, so it is merely a minor display bug.  This appears in the achievements drop down and my reputation tab on my profile. Why is $\pi$ irrational if it is represented as $C/D$? I've been interested in non-standard analysis recently. I was reading up on it and noticed the following interesting comment on the , right after giving an example of a nonstandard differentiation:     The use of the standard part in the definition of the derivative is a rigorous alternative to the traditional practice of neglecting the square of an infinitesimal quantity... the typical method from Newton through the 19th century would have been simply to discard the $dx^2$ term.   I've never heard anything like this before, and really find it fascinating that Newton's method was to define the relation $dx^2 = 0$. If we actually formalize the above structure by taking $\mathbb{R}$ and adjoining an element $dx^2 = 0$ to it, we get the "dual numbers," isomorphic to the quotient ring $\mathbb{R}[x]/x^2$. I'd seen some things about how this algebra plays into automated differentiation algorithms for some computer software systems, but I've never heard anything about Newton directly working in this algebra. So I have a few questions:   Does anyone have more historical information on the way that Newton performed differentiation, and its relation to the dual numbers? Does anyone know how effectively real analysis can be formalized with the dual numbers? Does the resulting system play nice enough to develop all of the important modern results? If we start with $\mathbb{C}[x]/x^2$ instead, can we likewise develop complex analysis?   Since this idea is so simple, I'm very curious how powerful it is. I'm also curious if it has any major drawbacks too, since I'm not sure why anyone would mess with the foundational baggage involved in defining the hyperreals if this simple 2-dimensional real algebra could really do the trick. Empty signature file after i.gensig in GRASS I'm trying to make a supervised classification in Grass 6.4.3 under Linuxmint 16.  After digitizing the training areas, I transformed them from vector to raster. Then I've done the following: just in case: g.region rast=train_area_2013_03_05_raster@PERMANENT,2013_03_05_toar4@PERMANENT  then  i.group group=group subgroup=sub_group input=2013_03_05_toar4@PERMANENT,2013_03_05_toar5@PERMANENT,2013_03_05_toar3@PERMANENT  and   i.gensig trainingmap=train_area_2013_03_05_raster@PERMANENT group=group@PERMANENT subgroup=sub_group signaturefile=superv_class  but the resulting signature file, only contains one character: "#".  The output of i.gensig in the console is the following:  Finding training classes... 3 classes found Calculating class means... Calculating class covariance matrices... Signature 1 not invertible Signature 2 not invertible Signature 3 not invertible i.gensig complete.   Obviously if I try: i.maxlik group=group@PERMANENT subgroup=sub_group sigfile=superv_class class=results_class  I obtain the following error message:  ERROR: Unable to read signature file &lt;superv_class&gt;   I've been searching, but I couldn't find the solution.   I've tried: this   My problem is similar to this: . But here, there's no solution suggested. As in this case, I also used i.cluster to generate an unsupervised signature file and i.maxlik reads this perfectly well.   there's another similar question, but again, with no solution...  What does Signature X not invertible means?  Any idea? Thanks for your help!