When I wrote  \polylongdiv[style=A]{x^6+x^5+x^4+x^3+x^2+x+1}{x^2+x+1}   I get that there is no reminder.... but the reminder is 1 if I'm right....  I hope is OK to post things like that here...  Thank you.
Original  The remainder provided by polynom.sty is NOT complete (as shown in the following figure).      My Objective  When I was in senior high school, my teacher made the diagram like below. :-)  Thus the final step must show the TOTAL remainder and mark it with L-shaped curve.  NOTE  I am sorry. The previous post I forgot to put back -1/4 in the original position. It have made you confused. Now there is no mistake anymore. :-)      Minimal Code  \documentclass{article} \usepackage{polynom,array} \usepackage[table]{xcolor}   \makeatletter \def\pld@ArrangeResult#1{%     \ifx\pld@remainder\@empty         \@tempcnta\pld@maxcol\relax         \pld@InsertItems@do\pld@lastline             {\pld@firsttrue\pld@PLD{\pld@R{0}{1}}}%     \fi     \ifnum\pld@currstage&gt;\z@         \pld@Extend\pld@allines{\pld@lastline\cr}%     \else         \pld@InsertFake\pld@lastline     \fi     \pld@iftopresult         \def\pld@lastline{\pld@PrintPoly\pld@divisor%         %====================================================================================         \quad\smash{{\color{red}\rule[-6pt]{\arrayrulewidth}{17pt}}}\kern-\arrayrulewidth&amp;}%         %====================================================================================     \else         \let\pld@lastline\@empty         \ifx B\pld@style\else             \def\pld@lastline{\pld@leftdelim\strut\pld@rightxdelim&amp;}%         \fi     \fi     \expandafter\pld@AR@col\expandafter\pld@PLD                            \expandafter\pld@lastline#1+\relax+%     \pld@SplitQuotient     \pld@iftopresult         \let\pld@currentline\@empty         \expandafter\pld@AR@col\expandafter\pld@PLD                                \expandafter\pld@currentline                                            \pld@quotient+\relax+%         \expandafter\pld@AR@col\expandafter\pld@XPLD                                \expandafter\pld@currentline                                            \pld@shadow+\relax+%         \edef\pld@subline{%             \noexpand\cline{\tw@-\pld@maxcol}%             \noalign{\vskip\jot}}%         \pld@Extend\pld@currentline{\expandafter\cr\pld@subline}%     \else         \@tempcnta-\@tempcnta         \advance\@tempcnta\pld@maxcol\relax \advance\@tempcnta\@ne         \edef\pld@span{\the\@tempcnta}%         \ifx B\pld@style           \pld@AddTo\pld@lastline{%             &amp;\multispan\pld@span${}=%             \pld@PrintPolyWithDelims\pld@divisor             \expandafter\pld@IfSum\expandafter{\pld@divisor}{}{\cdot}%             \expandafter\pld@IfSum\expandafter{\pld@quotient}\pld@true                                                              \pld@false             \pld@if \pld@leftdelim                     \pld@PrintPolyShadow                     \pld@rightdelim               \else \pld@PrintPolyShadow \fi             \pld@firstfalse             \expandafter\pld@PrintRemain\expandafter{\pld@remainder}$}%         \else           \pld@AddTo\pld@lastline{%             &amp;\multispan\pld@span$\pld@leftxdelim\strut\pld@rightdelim             \pld@div             \pld@PrintPolyWithDelims\pld@divisor=             \pld@PrintPolyShadow             \ifx\pld@remainder\@empty\else                 +{}%                 \setbox\z@=\hbox{$\displaystyle                   \frac{\let\strut\@empty\pld@firsttrue \expandafter                         \pld@PrintRemain\expandafter{\pld@remainder}}%                        {\let\strut\@empty\pld@PrintPoly\pld@divisor}$}%                 \dp\z@=\z@\box\z@             \fi             $}%         \fi     \fi \expandafter\pld@AR@\pld@allines\relax} \makeatother   \arrayrulecolor{red} \arrayrulewidth=0.8pt \def\strut{\rule[-6pt]{0pt}{12pt}}  \begin{document} \polylongdiv[style=A]{\frac{3}{7}x^9+x^2-\frac{1}{4}}{\frac{9}{5}x^4-1}  \vspace{1cm} \polylongdiv[style=A]{x^4-1}{x^2-1} \end{document}
The Wikipedia article on  mentions two methods of sampling "small" polynomials namely uniform sampling and discrete Gaussian sampling. Uniform sampling is clearly the simplest, involving simply uniformly selecting the coefficients from the set of small coefficients, thereby guaranteeing they will be small. However, the articles mention that security proofs rely on discrete Gaussian sampling.  However, I have a bit of difficulty wrapping my head around    Why discrete Gaussian sampling would be better than uniform, security-wise?   I mean, usually uniform is preferable - it maximizes the entropy/information in each sample, making the outcome more unpredictable. I could understand it if the Gaussian sampling technique would somehow serve to avoid (at least with high probability) "weak" parts of the space of outcomes but I have found no information/justification that this is the case.  Another reason I can see why discrete Gaussian might be better is that there is, after all, a chance (albeit a small one) one would select a non-small coefficient, so the space of outcomes is actually larger. However, given that this probability of hitting this extra space is (by design) quite small but also that one would likely immediately reject such a sample (because the algorithm calls for a small polynomial), this seems to be irrelevant.  It is not because I am against discrete Gaussian sampling I just want to understand why it is more secure. Also, I want to understand if it really IS more secure or it is simply easier to prove security in that model. I would actually consider that very interesting if it is the case because it raises the question of why is it easier to prove the security under conditions that are intuitively worse (i.e. discrete Gaussian vs the intuitively better uniform case)? If it is so intuitively clear that uniform sampling is better, surely it must be possible to transform that intuition into a security proof of RWE also under uniform sampling. Otherwise, I would fear the intuition must with high likelihood be wrong or at least dubious. Can't reach wi-fi signal on Ubuntu, but can do it on other OS/devices (ath9k) How do I "replay" the "Caveats" section from a homebrew recipe I'm new to Ubuntu. I installed it today, but the first thing I encounter is that I can only connect to the Internet through wifi.  Can someone help me?   iwconfig result :  eth0      no wireless extensions.  lo        no wireless extensions.   lspci:  00:00.0 Host bridge: Intel Corporation 3rd Gen Core processor DRAM     Controller (rev 09) 00:01.0 PCI bridge: Intel Corporation Xeon E3-1200 v2/3rd Gen Core processor PCI Express Root Port (rev 09) 00:02.0 VGA compatible controller: Intel Corporation 3rd Gen Core processor Graphics Controller (rev 09) 00:14.0 USB controller: Intel Corporation 7 Series/C210 Series Chipset Family USB xHCI Host Controller (rev 04) 00:16.0 Communication controller: Intel Corporation 7 Series/C210 Series Chipset Family MEI Controller #1 (rev 04) 00:1a.0 USB controller: Intel Corporation 7 Series/C210 Series Chipset Family USB Enhanced Host Controller #2 (rev 04) 00:1b.0 Audio device: Intel Corporation 7 Series/C210 Series Chipset Family High Definition Audio Controller (rev 04) 00:1c.0 PCI bridge: Intel Corporation 7 Series/C210 Series Chipset Family PCI Express Root Port 1 (rev c4) 00:1c.1 PCI bridge: Intel Corporation 7 Series/C210 Series Chipset Family PCI Express Root Port 2 (rev c4) 00:1c.3 PCI bridge: Intel Corporation 7 Series/C210 Series Chipset Family PCI Express Root Port 4 (rev c4) 00:1d.0 USB controller: Intel Corporation 7 Series/C210 Series Chipset Family USB Enhanced Host Controller #1 (rev 04) 00:1f.0 ISA bridge: Intel Corporation HM76 Express Chipset LPC Controller (rev 04) 00:1f.2 SATA controller: Intel Corporation 7 Series Chipset Family 6-port SATA Controller [AHCI mode] (rev 04) 00:1f.3 SMBus: Intel Corporation 7 Series/C210 Series Chipset Family SMBus Controller (rev 04) 01:00.0 3D controller: NVIDIA Corporation GF117M [GeForce 610M/710M/820M / GT 620M/625M/630M/720M] (rev a1) 03:00.0 Network controller: MEDIATEK Corp. MT7630e 802.11bgn Wireless Network Adapter 04:00.0 Unassigned class [ff00]: Realtek Semiconductor Co., Ltd. Device 5289 (rev 01) 04:00.2 Ethernet controller: Realtek Semiconductor Co., Ltd. RTL8111/8168/8411 PCI Express Gigabit Ethernet Controller (rev 0a)   lspci -nnk | grep 0280 -A2:  03:00.0 Network controller [0280]: MEDIATEK Corp. MT7630e 802.11bgn Wireless Network Adapter [14c3:7630]     Subsystem: Foxconn International, Inc. Device [105b:e074] 04:00.0 Unassigned class [ff00]: Realtek Semiconductor Co., Ltd. Device [10ec:5289] (rev 01) What does "preceding" mean in Oracle? Given a Lie group, how are you meant to find its Lie algebra?  The Lie algebra of a Lie group is the set of all the left invariant vector fields, but how would you determine them?  My group is the set of all affine maps $x \rightarrow A.x+v$ from $R^n$ to $R^n$ under function composition. What is the "N+1 selects problem" in ORM (Object-Relational Mapping)? The "N+1 selects problem" is generally stated as a problem in Object-Relational mapping (ORM) discussions, and I understand that it has something to do with having to make a lot of database queries for something that seems simple in the object world.  Does anybody have a more detailed explanation of the problem? The setuid permission bit tells Linux to run a program with the effective user id of the owner instead of the executor:  &gt; cat setuid-test.c  #include &lt;stdio.h&gt; #include &lt;unistd.h&gt;  int main(int argc, char** argv) {     printf("%d", geteuid());     return 0; }  &gt; gcc -o setuid-test setuid-test.c &gt; ./setuid-test  1000  &gt; sudo chown nobody ./setuid-test; sudo chmod +s ./setuid-test &gt; ./setuid-test  65534   However, this only applies to executables; shell scripts ignore the setuid bit:  &gt; cat setuid-test2  #!/bin/bash id -u  &gt; ./setuid-test2  1000  &gt; sudo chown nobody ./setuid-test2; sudo chmod +s ./setuid-test2 &gt; ./setuid-test2  1000     Wikipedia :     Due to the increased likelihood of security flaws, many operating systems ignore the setuid attribute when applied to executable shell scripts.   Assuming I'm willing to accept those risks, is there any way to tell Linux to treat the setuid bit the same on shell scripts as it does on executables?  If not, is there a common workaround for this problem? My current solution is to add a sudoers entry to allow ALL to run a given script as the user I want it run as, with NOPASSWD to avoid the password prompt. The main downsides to that is the need for a sudoers entry every time I want to do this, and the need for the caller to sudo some-script instead of just some-script Calculating the peak voltage of a zener diode output I have a problem with this homework, I have to calculate the peak voltage of the output after stabilizing with a zener diode.  My Input Voltage is Vin = 5V + 0.7Vsin (2pif*t) with f=10kHz  My circuit is a simple regulator with no load resistor. The series resistance is 82 Ohms, Zener voltage is 3V, max. current through the diode is 33.3mA. How can I calculate the peak output ? My problem is the offset created by 5V, wouldnt the Zener voltage be reached immidiately with this configuration ?         &ndash; Schematic created using How to enable touchpad? 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Finally, if you are unfortunate enough to be on an ancient Unix system   that doesn’t support the magic #! line, or if the path to your   interpreter is longer than 32 characters (a built-in limit on many   systems), you may be able to work around it like this:   #!/bin/sh -- # perl, to stop looping eval 'exec /usr/bin/perl -S $0 ${1+"$@"}' if 0;   Can you explain step by step what's going on here? I'm trying to make it work or encompass, but to no avail.  At execution of just the above, I'm getting this:  /bin/sh: 0: Illegal option -- Determining possible minimal polynomials for a rank one linear operator I have come across a question about determining possible minimal polynomials for a rank one linear operator and I am wondering if I am using the correct proof method.  I think that the facts needed to solve this problem come from the section on Nilpotent operators from Hoffman and Kunze's "Linear Algebra".     Question: Let $V$ be a vector space of dimension $n$ over the field $F$ and consider a linear operator $T$ on $V$ such that $\mathrm{rank}(T) = 1$.  List all possible minimal polynomials for $T$.   Sketch of Proof:   If $T$ is nilpotent then the minimal polynomial of $T$ is $x^k$ for some $k\leq n$.  So suppose $T$ is not nilpotent, then we can argue that $T$ is diagonalizable based on the fact that $T$ must have one nonzero eigenvalue otherwise it would be nilpotent (I am leaving details of the proof of diagonalization but it is the observation that the characteristic space of the nonzero eigenvalue is the range of T and has dimension $1$).  Thus the minimal polynomial of $T$ is just a linear term $x-c$.    Did I make a mistake assuming that $T$ can have only one nonzero eigenvalue?  Thanks for your help Order and degree of a differential equation