How can PHP 5.3 think 0 is null?
PHP considers null is equal to zero
If $\dfrac{4x^2-1}{4x^2-y^2}$ is an integer, then it is $1$ The problem is the following:     If $x$ and $y$ are integers such that $\dfrac{4x^2-1}{4x^2-y^2}=k$ is   also an integer, does it implies that $k=1$?   This equation is equivalent to $ky^2+(1-k)4x^2=1$ or to $(k-1)4x^2-ky^2=-1$. The first equation is a pell equation (if $k$ is a perfect square) and the second is a pell type equation (if $k-1$ is a perfect square). I've tried setting several values of $k$ to get some solutions but i got nothing. I'm starting to think that $k$ must be $1$. Exporting animation created with animate package Consider the following minimal working example.   It's great to have an animation in pdf format.   But sometimes it would be good to be able to export it into an animated gif or something else (swf, video file, svg?).  How do I achieve this?   Note: I often have animations together with pgfplots.     \documentclass{article}  \usepackage{animate}  \usepackage[active,tightpage]{preview} \PreviewEnvironment{animateinline}  \begin{document}  \begin{center} \fboxsep1mm     \begin{animateinline}[autoplay,loop]{2}     a     \newframe         b     \newframe       c                      \end{animateinline} \end{center}  \end{document}   Remark   The animation is not visible in all PDF viewers. It surely works with a current Adobe Reader. Video lectures on graduate level Classical Electrodynamics This is a rather broad question. Does anyone know of good video lectures for graduate level classical electrodynamics? Global and local variables in R Would you please tell me which of the following terms is correct? If both are correct, are they equal?     I have no [...] I don't have any [...] How to prove that the ideal $I=(x^3-y^2)$ in $k[x,y]$ is prime?    I have constructed a map from $k[x,y]$ to $k[t]$, which maps $x$ to $t^2$, and $y$ to $t^3$. Then, I want to show that the kernel is exactly $I$. But the difficulty is that how to prove the kernel is contained in $I$.  Could you help me? Thanks a lot! Forced to Quit PhD although the hard and proved work I am a first year Ph. D student, and I have been working on a new topic of research.  I worked day and night to prove something new and verified it. I began my Ph. D while my supervisor was on travel, but he returned before I had been here for long.  When he returned, he told me we have to stop and not continue without any reasonable reasons.  After a few months he began treating me poorly---always belittling and underestimating---although I got two grants from a top conference, made videos and podcasts, and constantly received praise for my good work and personality.    It seems that I have been treated as a competitor, not a collaborator.  Although agreeing with my approaches in private, my supervisor said false words to the committee to take the decision and stop my Ph. D work!  It is not surprising, as there was another former senior Ph. D student who I met on campus, who was not allowed to defend his Ph. D: this same supervisor claimed he was not able to finish his dissertation although he had published in a peer reviewed journal and having citation as well, it was scary and not familiar.  In front of committee, he lied, and said something without proving. I asked if he can make a detailed constructive criticism, but he could not.  All I can say that they are a gang literally, and he said words that show that he is jealous and doesn't want to see me having self confidence in my self. I know no one will believe me, but this is the truth.  Honestly, I am distressed and I'm losing sleep.  I had this dream where there were many people who saw me fail.  All things considered, I don't know whether what happened is good or bad, and I definitely don't know what I should do next!  I would be grateful for your advice and whether there is someone has encountered any similar situation. Connect iPhone to Mac Without Charging I am developing iOS apps and I hate, that always when I connect my iPhone to my Mac to install and test an app, the iPhone gets charged. Is there any possibility to connect my iPhone to my Mac without charging the phone?    My Environment: -  iPhone 4S (No Jailbreak) -  MacBook Air3,2 (Late 2010) Is it possible to add a user and restrict him to run special programs? For example after this user logs in, he can only open Firefox to use Internet and no other programs can be run by this user. For example in a terminal, commands will not be accessible when you delete some environment variables like $HOME. But is it possible to avoid running programs in a graphical shell, like Gnome? If yes, how? How to run a scheduled job every 15 minutes? Using the salesforce GUI it seams impossible to schedule a job every 15 minutes. Or more than once a day. I've been racking my brain trying to think of a grammatically correct sentence that uses same without the earlier in the same (see!) phrase.  It is the same It is the very same I have 10 things all the same I have 10 of the same things The same boy as last time did it again The boy did the same thing as last time  I am discounting same-day dry cleaning and any similar phrases that might crop up because using a hyphen is acceptable which suggests it's an atomic phrase and it sounds awkward if inverted:  They have a same-day dry cleaning service. Their dry cleaning service is same-day. - Sounds odd but might be acceptable in some situations Their dry cleaning service is done on the same day. - Sounds better  So I don't think it's possible when same is used by itself. Can someone either explain why same must always have the before it or provide a grammatically correct counter-example? I am trying to define an alternative \item command that would allow me to use a different color for some of the items. My code has the following form:  \documentclass{article} \usepackage{color}  \definecolor{gray}{rgb}{0.7,0.7,0.7} \newcommand{\grayitem}[1]{{\color{gray} \item #1}}  \begin{document}  \begin{enumerate} \item First item. \grayitem Second item. \item Third item. \end{enumerate}  \end{document}   I was hoping that  \grayitem Second item.   would produce the same result as  {\color{gray} \item Second item.}   but for some reason, these are not equivalent. They produce the following results, respectively:  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;   Why are these two lines of code not equivalent? What am I doing wrong and how can I change the definition of \grayitem to produce the second result?  I have also tried removing a pair of braces, like this  \newcommand{\grayitem}[1]{\color{gray} \item #1}   but then the third item becomes gray as well.  The solution should also work for multi-paragraph items. tikz package causing error in IEEE access template I have been trying for a while to load the tikz package to the IEEE ACCESS template. Whenever I use the command \usepackage{tikz}, I am getting the error   /usr/local/texlive/2017/texmf-dist/tex/generic/pgf/math/pgfmathfunctions.random.code.tex, line 33 Missing number, treated as zero.   It was working fine with the IEEEtran template and I was able to draw my circuit diagrams in that.   Anyone has any idea how to solve this issue? Dealing with bundle/omnibus/list-of-questions question that consists of rather distinct questions I have looked at the various help pages but I couldn't find a clear guideline for the following issue: often enough someone will ask in the same post (i.e. in the same SE-question) a whole bunch of questions (in the plain English sense of the word), which while may be about the same topic are nevertheless sufficiently distinct that each may require several paragraphs of text to answer adequately.   I can see an upside for leaving related questions grouped together from the asker's perspective: "there's a thread (on SE) answering all my questions about topic T!" On the other hand, from the perspective of organizing questions by having a clear heading/title for each (actual) question and so forth, i.e. from the perspective of SE being a fairly structured database of Q&amp;As instead of being a more like a collection of forum threads "I'm confused about topic T", the omnibus question-list approach seems clearly sub-optimal.  The omnibus approach also makes the issue of accepting answers (and perhaps reviewing such actions) difficult: if two different users give replies to different parts/sub-sets of the question-list, which one is the most adequate one? Even upvoting all partial answers seems iffy because it actually penalizes the user(s) who provides(s) the most complete answer to the question-list: since a user can only give at most one upvote to another user's answer, if user X provides answers to A &amp; B in the same SE-answer, he can only get at most one upvote from user Y, whereas if these were two different SE-answers, he could get two upvotes from user Y.  So, given the above problems, which I think are real enough, is there anything in the SE system addressing list-of-questions question splitting? What is the C# Using block and why should I use it? What is the purpose of the Using block in C#? How is it different from a local variable? Exclude TeX.SE question titles from MathJax parsing in Hot Network Questions Recently, the TeX.SE question  got onto the Hot Network Questions list.  Like many questions on TeX.SE, it contains some TeX markup in its title.  Unfortunately, on , this markup gets parsed and rendered as math:    I propose that either:   the entire Hot Network Questions sidebar should be excluded from MathJax parsing by styling it with the tex2jax_ignore class, or at least, any hot questions from TeX.SE should be so excluded from MathJax parsing.   Or I suppose we could always implement , which asks for all questions with TeX in the title to be excluded from Hot Network Questions entirely.  That might be throwing out the baby with the bathwater, though.    Ps. This issue is partially masked by , which causes MathJax not to be rendered in many elements that are hidden by default and revealed using jQuery .show(), such as all but the first five questions in the Hot Network Questions sidebar.  The  user script contains a generic fix for that bug, so users who have it installed will see this issue more frequently, but it occurs even without SOUP whenever a question with TeX markup in the title happens to show up among the top five hot questions that are always visible by default. implicit operator using interfaces I have a generic class that I'm trying to implement implicit type casting for.  While it mostly works, it won't work for interface casting.  Upon further investigation, I found that there is a compiler error: "User-defined conversion from interface" that applies.  While I understand that this should be enforced in some cases, what I'm trying to do does seem like a legitimate case.  Here's an example:  public class Foo&lt;T&gt; where T : IBar {     private readonly T instance;      public Foo(T instance)     {         this.instance = instance;     }     public T Instance     {         get { return instance; }     }     public static implicit operator Foo&lt;T&gt;(T instance)     {         return new Foo&lt;T&gt;(instance);     } }   Code to use it:  var concreteReferenceToBar = new ConcreteBar(); IBar intefaceReferenceToBar = concreteReferenceToBar; Foo&lt;ConcreteBar&gt; concreteFooFromConcreteBar = concreteReferenceToBar; Foo&lt;IBar&gt; fooFromConcreteBar = concreteReferenceToBar; Foo&lt;IBar&gt; fooFromInterfaceBar = intefaceReferenceToBar; // doesn't work   Does anyone know a workaround, or can anyone explain in a satisfactory way why I shuouldn't be able to cast interfaceReferenceToBar implicitly to Foo&lt;IBar&gt;, since in my case it is not being converted, but only contained within Foo?  EDIT: It looks like covariance might offer salvation.  Let's hope the C# 4.0 specification allows for implicit casting of interface types using covariance. Lunch Meeting Probability for two person to meet in given 1 hour slot and none would wait more then 15 minute. Two friends who have unpredictable lunch hours agree to meet for lunch at their favorite restaurant whenever possible. Neither wishes to eat alone and each dislikes waiting for the other, so they agree that each will arrive at a random time between noon and 1 pm, and each will wait for the other for 15 minutes or until 1:00. What is the probability that the friends will meet for lunch on a given day? $C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$ How can I say this?