Finite ring zero divisor/multiplicative inverse
Finite ring has only zero divisors and units
I've been reading , and I would like to know why the Lasso provides variable selection and ridge regression doesn't.  Both methods minimize the residual sum of squares and have a constraint on the possible values of the parameters $\beta$. For the Lasso, the constraint is $||\beta||_1 \le t$, whereas for ridge it is $||\beta||_2 \le t$, for some $t$.  I've seen the diamond vs ellipse picture in the book and I have some intuition as for why the Lasso can hit the corners of the constrained region, which implies that one of the coefficients is set to zero. However, my intuition is rather weak, and I'm not convinced. It should be easy to see, but I don't know why this is true.  So I guess I'm looking for a mathematical justification, or an intuitive explanation of why the contours of the residual sum of squares are likely to hit the corners of the  $||\beta||_1$ constrained region (whereas this situation is unlikely if the constraint is $||\beta||_2$). Are cardinal numbers well defined? Or could we have something like $\aleph_{1/2}$? From what I understood, cardinal numbers are defined as:  $\aleph_0$ = the cardinality of $\mathbb{N}$ $\aleph_{n+1}$ = is the least cardinal number greater than $\aleph_n$  The continuum hypothesis states that the cardinality of $\mathbb{R}$ is  $\aleph_1$. I more or less understand this cannot be proven (does not follow from ZFC).  However, can it be proven dat there in fact is a 'least cardinal number' greater than $\aleph_0$?  Is it possible that there are sets with cardinality $\aleph_{1/2}$, $\aleph_{1/3}$, et cetera, so to speak?  Or expressed differently: is it possible that for every set A with cardinality CA > $\aleph_0$, there exists another set B with cardinality CB where CA > CB > $\aleph_0$? How to transpose a dataset in a csv file? Is there a Term to describe the relationship between two people when they share an Alma Mater?  For example; Neil Armstrong is an alumnus of Purdue University. I am an alumnus of Purdue University. Both Neil and I are alumni of Purdue University, which is to say we share an alma mater. So, I'm looking for a term to describe my relationship to to Neil with out needing to explicitly reference the university. Is co-alumni a real term? Is it appropriate to say that Neil and I are co-alumni? Is there something better? Further, is there a way to specifically describe the relationship between two people from the same graduating class?  Thank You in advance! Folders Completely Hidden Except via Save As I'm using an audio program called Supercollider on Windows 10. I have saved some files in a folder called "tunes" from SC via Save As, but Windows Explorer and CMD both seem unable to recognise access or recognise the "tunes" folder.  C:\Users\HP&gt;cd "C:\Program Files (x86)\SuperCollider-3.7.2\supercollider-robin\tunes" The system cannot find the path specified.  C:\Program Files (x86)\SuperCollider-3.7.2\supercollider-robin&gt;dir/a doesn't show it either.  However, when saving from SC with Save As, it's there clear as day. I can even shift+right-click to get the path: "C:\Program Files (x86)\SuperCollider-3.7.2\supercollider-robin\tunes"  This makes no sense to me. Can anyone explain and tell me how to access my folder tunes folder please? 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} Should I keep 32bit or 64 bit Windows 10 on my x64 based Intel Atom Z3735F cpu on 2gb ram non-upgradable laptop I have an Acer Aspire Switch 10 SW5-012 detachable notebook which has 2GB of RAM based on an x64 based processor. I got it with a 32 bit version of windows 8 but upgraded to windows 10 recently. But before I do anything else, I was considering if I'd be having Better performance with the 64 bit version of windows 10. Would it make my notebook perform better since it's based on an x64 based Intel Atom Z3735F? Or should I just stick to the 32 bit version of windows 10? I am able to produce a standard length \sim with a tiny S above it, but I would like the \sim tilde to be longer (maybe one and half times as wide as a capital letter).  Any tips on how to change the width/length of \sim? Let $X$ be a random variable and let $g,f$ be real-valued, nondecreasing, and bounded.   Show that $\mathbb{E}[f(X)g(X)]\geq \mathbb{E}[f(X)]\mathbb{E}[g(X)]$  Having a hard time seeing where to start for some reasons, any hints?  I managed to write:  $\mathbb{E}[f(X)g(X)] = \frac{1}{4}\mathbb{E}[(f(X)+g(X))^2 - (f(X)-g(X))^2]$  But I'm not sure if this is in the right direction or not. Eeeek! Something stole the protection! (Why is this protected question unprotected?) asked me to report this (apparent) bug.  When he tried to protect  question someone (or something) stole the question's protection, which has since been missing:          The question shows no sign of having the protection removed, thus I believe it was abducted. I am sure justice will be done, and this poor question will receive the protection it deserves.     I recommend interrogating the  that was said to have announced this question to the world, I bet it would have some useful information. Screen brightness isn't taking effect on a Lenovo Z570 My Lenovo Z570 has two GPU's, nVidia and Intell. Brightness control has no effect what so ever over the screen brightness, and because I've had some major issues in the past when I was trying to fiddle with nVidia drivers and settings, and I wanna be cautious now. I haven't find similar questions referring 14.04 LTS with similar configuration as mine.   I've read that it could be fixed by adding/changing content in xorg.conf but when I run  sudo gedit /etc/X11/xorg.conf   the file comes out to be absolutely empty. What do I need to do, what do I need to add to it and not get my laptop screwed with a black screen again? Uniform Scattering with vector I'm attempting to create a random spread of dots to create a gradient look. I'm getting really close to the effect I want by blending two scatter brushes. The issue I'm having is that the blended brushes creates unwanted lines, and the dots run into each other. I keep adjusting the settings with no luck.  Here's the effect I'm wanting to achieved.    Smooth transition  On Circular path Dots don't converge Scattered at random-ish     Here's how close I'm able to get:   Blended scatters create unwanted lines Dots converge Spacing isn't gradual from edge to center In my experience, "finite" is pronounced (IPA) ˈfaɪnaɪt while "infinite" is ˈɪnfɪnɪt.  In general, the prefix "in" negates an adjective, but does not change the pronunciation. Based on this, I would expect ˈɪnfaɪnaɪt.  Is there a reason for this "deviation from the norm"? Are there dialects/regions that actually say ˈɪnfaɪnaɪt (or ˈfɪnɪt, for that matter)? Half doesn't or half don't? Problem with packages subfiles and import.sty after using TexLive Utility for updating Everything worked fine before using TexLive for updating my packages three days ago. Now it is not possible anymore to use the package subfiles   Here are the MWE  The minimum Preamble.tex file  \documentclass{amsart} \usepackage{subfiles}   The MWE.tex file  \documentclass[MWE.tex]{subfiles} \begin{document} MWE \end{document}   When I compile the MWE file I get on the console  This is pdfTeX, Version 3.14159265-2.6-1.40.20 (TeX Live 2019) (preloaded &gt;format=pdflatex)  restricted \write18 enabled. entering extended mode  (./MWE.tex  LaTeX2e &lt;2020-02-02&gt; patch level 5  L3 programming layer &lt;2020-02-25&gt; (/usr/local/texlive/2019/texmf-dist/tex/latex/subfiles/subfiles.cls  Document Class: subfiles 2020/02/14 v16 Multi-file projects (class)  Preamble taken from file `MWE.tex'  (/usr/local/texlive/2019/texmf-dist/tex/latex/tools/verbatim.sty)  (/usr/local/texlive/2019/texmf-dist/tex/latex/import/import.sty)  ! LaTeX Error: File `import.sty' not found.  Type X to quit or &lt;RETURN&gt; to proceed, or enter new name. (Default extension: sty)  Enter file name:   I checked and the import.sty file is in the right directory, and everything worked fine before. Do you have any idea about what happened (apart from the fact that I should not upgrade stuff before the official releases of TexLive I suppose  ...)? So, through typing several commands I've found that there's not only ls, but l and la too. There doesn't appear to be any man entries on Ubuntu 12.14. They all appear to do similar things with minor differences:  $ ls app        config     CONTRIBUTING.md  doc       Gemfile       Guardfile  LICENSE  MAINTENANCE.md  Procfile  Rakefile   script  tmp     VERSION CHANGELOG  config.ru  db               features  Gemfile.lock  lib        log      PROCESS.md      public    README.md  spec    vendor $ la app      CHANGELOG  config.ru        db   features  Gemfile       .git        Guardfile  LICENSE  MAINTENANCE.md  Procfile  Rakefile   .rspec  .secret     spec  .travis.yml  VERSION .bundle  config     CONTRIBUTING.md  doc  .foreman  Gemfile.lock  .gitignore  lib        log      PROCESS.md      public    README.md  script  .simplecov  tmp   vendor $ l app/       config/    CONTRIBUTING.md  doc/       Gemfile       Guardfile  LICENSE  MAINTENANCE.md  Procfile  Rakefile   script/  tmp/     VERSION CHANGELOG  config.ru  db/              features/  Gemfile.lock  lib/       log/     PROCESS.md      public/   README.md  spec/    vendor/   Just as a bit of trivia, are there more of these and what do they do? Is here any place to find this out? Unfortunately, google searching these commands gets ignored because they're so short. How do you install Google Chrome on Ubuntu? Do I need a Kuwait transit visa as an Indian national? I am an Indian national and passport holder, traveling from New Delhi to Frankfurt (via Kuwait) and back with Kuwait Airways. I see no Terminal change for the onward and return journeys. I have no plans to leave the airport due to short layovers.  Do I need a Kuwait transit visa? Proving $ 1+\frac{1}{4}+\frac{1}{9}+\cdots+\frac{1}{n^2}\leq 2-\frac{1}{n}$ for all $n\geq 2$ by induction Where are the default settings saved for the Command prompt defaults?