I am going to attend my PhD interview online. The PI asked me to present my previous work and a research proposal in 10 min. I made a 8 slide presentation. Is it very big for 10 min?  And the other thing he told me is, at the end he is going to give me some time so that I can ask questions. What kind of question should I ask in this case? Related to subject? What other areas (other than subject) should I consider to ask questions?
I've a Skype interview for PhD application (it will make me miss  today.) I've been asked, in advance, to prepare a short presentation and then there would be some discussion with the panel (I guess.) I expect that they will ask if I have some questions so, I'm wondering what are the best (type of) questions one might consider?
Alternative methods to prove that $Z_m$ is a field under addition and multiplication $\bmod\ m$ iff $m$ is a prime I am looking for ways to prove that $\mathbb{Z}_m=\{0,1,2,\dots,m-1\}$ is a field under addition and multiplication $\bmod\ m$ iff $m$ is a prime.   I tried it this way: If $m$ is a prime $p$,it is obvious that most importantly, we need to prove that every non-zero element in $\mathbb{Z_p}$ has an inverse in $\mathbb{Z_p}$.We note that if $a,b\in \mathbb{Z_p}$, then $a\cdot b\in \mathbb{Z_p} \dots \boxed{*}$. Next, $a^p\equiv a \bmod p$ for prime $p$(the proof follows by induction on $a$).As  $\displaystyle 0\leq a\leq p-1$,$a\not =p$. Thus $$a^{p-1}\equiv 1 \bmod p$$ for prime $p$ which,together with $\boxed{*}$ guarantees the existence of an inverse in $\mathbb{Z_p}$.Proving the other properties is simple.  Conversely,if $\mathbb{Z_m}$ is a field,then every non-zero element of $\mathbb{Z_m}$ has an inverse.Note that inverse of an element is unique.For if $a\cdot b\equiv a\cdot c\equiv 1 \bmod m$,then $(a\cdot b)\cdot c\equiv c\equiv (a\cdot c)\cdot b\equiv b \bmod m$ i.e $m|(b-c)\implies b-c=0(0\leq a\leq m-1,a\in \mathbb{Z_m})$.So, inverses must be unique and every element is either an inverse of itself or has in inverse in the field.That allows us to conclude that $[(m-1)!]^2\equiv 1 \bmod m$ which can only mean that either  $(m-1)!\equiv 1 \bmod m$ or $(m-1)!\equiv -1 \bmod m$. If $m$ is not a prime, it has a prime factor, say $p$. $ 0\leq p\leq m-1\implies p|(m-1)!\implies p|1$or $-1$ which is absurd, a contradiction.Thus $m$ is prime.  I feel that this method is too involved and complicated. Can anyone please suggest some other method?(Some context: This is a problem from Finite Dimensional Vector Spaces by Paul Halmos. I am learning linear algebra for the first time.It seems some of the comments have been posted on the assumption that I know rings and fields; I don't. ) Notify us when one of our posts is edited Your summary page shows "N Edits to your posts" for the selected time period but you really have to be looking for it.  Could we add edits to the notification feed? You could put the reason for the edit (entered--or not--when the editor changes it) and just a link to the post. Tasks in custom Environment I need to wrap a tasks environment inside a custom \NewEnviron. Apparently, tasks chokes on code besides its \task or \item due to the , presumably causing it to do so on \BODY. Any way to get around this? Exact error given is "Something's wrong--perhaps a missing \item", as per:  \documentclass[a4paper]{book} \usepackage{environ} \usepackage[more]{tasks}  \NewTasks[style=enumerate,counter-format=tsk[1]),label-width=2ex]{mytaskenum}[\item](3) \NewEnviron{mytaskenv}[1][Exercises]{   % Misc. stuff to precede the tasks env here   \begin{mytaskenum}   \BODY   \end{mytaskenum}   % Misc. stuff to follow }{}  \begin{document}  \begin{mytaskenv}   \item item 1   \item item 2   \item item 3 \end{mytaskenv}  \end{document} What are the odds of hitting exactly 100 rolling a fair die Who are the Steam Mascots? So with the Steam Summer Sale in full swing, I've finally taken notice of the Steam Mascots.  Judging by the latest background, the black one, at least, has a name: Cyl.  Do the other two have names as well? Are there only these three?   What's their story? Intuition for uniform continuity of a function on $\mathbb{R}$ I understand the formal definition of uniform continuity of a function, and how it is different from standard continuity.  My question is: Is there an intuitive way to classify a function on $\mathbb{R}$ as being uniformly continuous, just like there is for regular continuity? I mean, that for a "nice" function $f:\mathbb{R} \to \mathbb{R}$, it is usually easy to tell if it is continuous on an interval by looking at or thinking of the graph of the function on the interval, or on all of $\mathbb{R}$. Can I do the same for uniform continuity? Can I tell that a function is uniformly continuous just by what it looks like? Ideally, this intuition would fit with the Heine-Cantor theorem for compact sets on $\mathbb{R}$. To post an image in a question/answer an user requires at least 10 rep points (see ). To be able to upload an image in the chat, the rep requirements is instead 100 rep point (see ).  Is there any reason to that? I am asking because IF the rationale is to provide more protection inside the chat, the protection is easily circumvented by uploading the image via the question editor, and then copying the link to the chat. Basically, the more restrictive requirement fail to reach its goal - assuming that blocking uploads from new users was indeed the target.  I am not proposing to change that (it doesn't seem worthy the effort) but I was curios to know if this is just a refuse caused by successive changes to the system or if there is actually a different rationale to this that I am missing.  [note: I will remove the double Discussion/Support tag as soon as I can focus if the question has a staff answer - support or will prompt a discussion] public class doublePrecision {     public static void main(String[] args) {          double total = 0;         total += 5.6;         total += 5.8;         System.out.println(total);     } }   The above code prints:  11.399999999999   How would I get this to just print (or be able to use it as) 11.4? Disposing WPF User Controls Many times I have read statements like, "the age of the universe is 14 billion years" . For example this wikipedia page .   Now, my question is, which observers' are these time intervals? According to whom 14 billion years? Rendered image is just grey I am having some issues when rendering an image, my rendering screen is turning into grey even when making a full render. What am I doing wrong? Unicode char {U+200B} When I try to cite more than two authors, I got this: ! Package inputenc Error: Unicode char ​ (U+200B) (inputenc)                not set up for use with LaTeX.   \documentclass[a4paper,oneside,BCOR=10mm,12pt,titlepage]{scrreprt}  \usepackage[polutonikogreek,ngerman,english]{babel} \usepackage[maxlevel=4,autostyle,german=guillemets]{csquotes} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage[style=mla]{biblatex} \DeclareMultiCiteCommand{\parencites}[\mkbibparens]{\parencite}{\multicitedelim‌​}  \begin{document}  \parencites[cf.][]{key1}{key2}{key3}  \end{document} Uninstall apps locked by phone vendor I'm using a Samsung I9000 and I have some annoying apps that came pre installed by the phone vendor, like a music store, a custom web browser and other memory leaking apps.  Those crappy apps can't be uninstalled neither by Manage applications tool, nor by editing the apps tiles in the front menu.  I guess I have to be a root user or something to get rid of those apps, but I don't know how. The recent discovery by the LIGO made me wonder about this.  We know that there exists a CMBR, Cosmic Microwave Background Radiation, a blanket of electromagnetic energy covering the universe, made by the Big Bang.  But, the big bang was an explosive event. The huge "shockwave" made by it sent ripples in space time. (Primordial gravitational waves)  Wouldn't these ripples flood the universe creating sort of a Cosmic Gravitational Background Radiation (not exactly radiation)?  If such waves do flood the universe, couldn't a much more sensetive version of LIGO be able to detect this and use it as a contrast to create "gravitational telescopes"? Difference between Hashing a Password and Encrypting it The current top-voted to  states:     Another one that's not so much a security issue, although it is security-related, is complete and abject failure to grok the difference between hashing a password and encrypting it. Most commonly found in code where the programmer is trying to provide unsafe "Remind me of my password" functionality.   What exactly is this difference? I was always under the impression that hashing was a form of encryption. What is the unsafe functionality the poster is referring to? Does payment in goods count as "income" for tax purposes? Can someone help me elucidate the definition of "income" from a tax perspective? More specifically, if Joe is paid for his work in avocados instead of dollars, then would Joe be exempt from paying any income tax for the year? Or would the value of avocados earned for the year be translated into dollars which would then be considered Joe's annual income? If so, what guidelines would one use to make this translation? (Make the assumption that Joe does not sell the avocados (for dollars) at a later date and either eats them or trades them for other real property)  What if he was paid in gold instead of avocados? 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. What is difference between int.class and Integer.TYPE in java? I've been testing PCA via SVD to decompose a simple time series data matrix, $X$.  I have two signals $x_1(t)$ and $x_2(t)$ in a data matrix where $M$ rows represents each timepoint sample and each column represents $x_1$ and $x_2$.  The mean signal, $\hat{x}$, is defined as the mean along the row axis (average of $x_1$ and $x_2$ along each timepoint).  I normalize each column of $X$ by subtracting its mean and dividing by the standard deviation.  When I use [U S V] = svd(Xz) in matlab, regardless of how the variables are distributed (whether they are correlated or uncorrelated), one of the columns of the right singular matrix, V, always points in the same direction (to a multiplicative constant) as the mean vector $(1/2, 1/2)$.  But when add an additional third time series, this is never the case (where the mean vector is $(1/3, 1/3, 1/3)$.  Because I normalize the standard deviation for each vector, it does make sense that the direction of most variance given by PCA would be the diagonal 45 degree line.  But if both variables $x_1$ and $x_2$ are independent gaussians, couldn't the PCA direction be any direction since the distribution is radially symmetric?  MATLAB Code:  s = RandStream('mcg16807', 'Seed', 0); RandStream.setDefaultStream(s);  G = zeros(1000,2); G(:,1) = 40*randn(1000,1)-100; G(:,2) = tan(G(:,1)) + randn(1000,1); X = G - repmat(mean(G),[size(G,1), 1]); Xz = X./repmat(std(G),[size(G,1), 1]); [U S V] = svd(Xz); G_mean = mean(Xz,2); corrcoef(G_mean, U(:,1)) corrcoef(G_mean, U(:,2)) I am a new user of Ubuntu 12.04.   I installed Ubuntu and Windows XP, and I want to use both operating systems. When I restart or boot my PC the GRUB menu does not appear.  How can I solve this problem?