(My question is different than  and , so hold back on casting a close vote, please). So, I was reading on , and ran into this statement in the Dirac-von Neumann axioms: The space $\mathbb{H}$ is a fixed complex Hilbert space of countable infinite dimension. This is rather confusing to me; I use position and momentum as bases for this Hilbert space all the time, and those bases have cardinality $2^{\aleph_0}$. How is it that the space is countably infinite when we use uncountable basis sets to perform basic calculations?
The dimension of the Hilbert space is determined by the number of independent basis vectors. There is a infinite discrete energy eigenbasis $\{|n\rangle\}$ in the problem of particle in a box which can be used to expand a general state $|\psi\rangle$ as: $$|\psi\rangle=\sum\limits_{n=0}^{\infty} C_n |n\rangle$$ Here, the basis implies that the Hilbert space has countably infinite number of basis vectors. But one could well expand $|\psi\rangle$ in any other basis as well, say the continuous position basis. Then that superposition, $$|\psi\rangle=\int dx \psi(x)|x\rangle$$ is also possible. But this basis $\{|x\rangle\}$ is continuous and uncountably infinite. Such sets cannot be equal because they have different cardinality. Then my questions is how many independent basis vectors are really there in this Hilbert space?
$4$ letter words taken from the letters of CONCENTRATIONS How many words with or without meaning can be made from the letters of the word CONCENTRATIONS by taking $4$ letters at a time?  There are  $14$ letters total present in the word. $4$ letters can be picked in $14\choose 4$ ways. $4$ letters can be arranged in $4!$ ways. So ideally answer has to be ${14\choose 4} * 4!$. But I know this answer is wrong because some letters are repeating. How do I solve this then? I appreciate any help. How can I prove these statements? Java method dispatch with null argument Why does it (apparently) make a difference whether I pass null as an argument directly, or pass an Object that I assigned the value null?  Object testVal = null; test.foo(testVal);    // dispatched to foo(Object) // test.foo(null);    // compilation problem -&gt; "The method foo(String) is ambiguous"     public void foo(String arg) { // More-specific     System.out.println("foo(String)"); }  public void foo(Object arg) { // Generic     System.out.println("foo(Object)"); }   In other words, why is the (commented-out) second call to foo(...) not dispatched to foo(Object)?  Update: I use Java 1.6. I could compile Hemal's code without problems, but mine still doesn't compile. The only difference I see is that Hemal's methods are static while mine are not. But I really don't see why this should make a difference...?  Update 2: Solved. I had another method foo(Runnable) in my class, so the dispatcher couldn't unambiguously select the single most specific method. (See my comment in Hemal's second answer.) Thanks everyone for your help. grep: memory exhausted How can I bypass Activation Lock? I bought a used iPhone 4.  The previous owner re-set the phone, but did not remove the device from their iCloud.  It is asking me for their iCloud information when I try to set-up the phone.  I don't have contact with the previous owner anymore.  Is there a way to remove the device from their iCloud so I can use a new one? Suppose $\displaystyle P(x,y)$ a polynomial in the variables $x,y$.   For example, $\displaystyle x^4$ or $\displaystyle x^3y^2 + 3xy + 1$.  Is there a general method which allows us to evaluate the indefinite integral     $$ \int P(\sin x, \cos x) \text{d} x$$   What about the case when $\displaystyle P(x,y)$ is a rational function (i.e. a ratio of two polynomials)?   Example of a rational function: $\displaystyle \frac{x^2y + y^3}{x+y}$.    This is being asked in an effort to cut down on duplicates, see here:   and here: NSDate Format outputting wrong date Why do we say "Eastern Europe" but "East Asia"? Search engines and browser accept-language I'm building a web portal where language content will generally depend on the "accept-language" sent by the browser. The same content-URI will thus serve different content to different users depending on their browser setting.  I'm very curious to know how this will affect search indexing. Does Google index using all languages, and is it handled well? How does one restrict number of allowed types in Java generic type Continuous and bounded imply uniform continuity? Divergent infinite series $n!e^n/n^n$ - simpler proof of divergence? error tex capacity exceeded log files added I am having a hard time figuring out where the problem is with my code/workflow. I previously posted this question , but then the same problem came straight back. When I run it in tex, I get an error input limit exceeded. The exact log is pasted below:  This is pdfTeX, Version 3.14159265-2.6-1.40.17 (MiKTeX 2.9.6210 64-bit) (preloaded format=pdflatex 2017.1.26)  26 JAN 2017 20:45 entering extended mode **./master.tex (master.tex LaTeX2e &lt;2016/03/31&gt; patch level 3 Babel &lt;3.9r&gt; and hyphenation patterns for 75 language(s) loaded. ("C:\Program Files\MiKTeX 2.9\tex\latex\base\article.cls" Document Class: article 2014/09/29 v1.4h Standard LaTeX document class ("C:\Program Files\MiKTeX 2.9\tex\latex\base\size11.clo" File: size11.clo 2014/09/29 v1.4h Standard LaTeX file (size option) ) \c@part=\count79 \c@section=\count80 \c@subsection=\count81 \c@subsubsection=\count82 \c@paragraph=\count83 \c@subparagraph=\count84 \c@figure=\count85 \c@table=\count86 \abovecaptionskip=\skip41 \belowcaptionskip=\skip42 \bibindent=\dimen102 ) ("C:\Program Files\MiKTeX 2.9\tex\latex\apacite\apacite.sty" Package: apacite 2013/07/21 v6.03 APA citation \c@BibCnt=\count87 \bibleftmargin=\skip43 \bibindent=\skip44 \bibparsep=\skip45 \bibitemsep=\skip46 \biblabelsep=\skip47 ) No file master.aux. \openout1 = `master.aux'.  LaTeX Font Info:    Checking defaults for OML/cmm/m/it on input line 5. LaTeX Font Info:    ... okay on input line 5. LaTeX Font Info:    Checking defaults for T1/cmr/m/n on input line 5. LaTeX Font Info:    ... okay on input line 5. LaTeX Font Info:    Checking defaults for OT1/cmr/m/n on input line 5. LaTeX Font Info:    ... okay on input line 5. LaTeX Font Info:    Checking defaults for OMS/cmsy/m/n on input line 5. LaTeX Font Info:    ... okay on input line 5. LaTeX Font Info:    Checking defaults for OMX/cmex/m/n on input line 5. LaTeX Font Info:    ... okay on input line 5. LaTeX Font Info:    Checking defaults for U/cmr/m/n on input line 5. LaTeX Font Info:    ... okay on input line 5. \c@maskedRefs=\count88 ("C:\Program Files\MiKTeX 2.9\tex\latex\apacite\english.apc" File: english.apc 2013/07/21 v6.03 apacite language file LaTeX Info: Redefining \BPBI on input line 129. LaTeX Info: Redefining \BHBI on input line 130. ) ! TeX capacity exceeded, sorry [input stack size=5000]. \AtBeginDocument #1-&gt;#1  l.5 \begin{document}  If you really absolutely need more capacity, you can ask a wizard to enlarge me.   Here is how much of TeX's memory you used:  641 strings out of 493333  7639 string characters out of 3139188  74825 words of memory out of 3000000  4254 multiletter control sequences out of 15000+200000  3940 words of font info for 15 fonts, out of 3000000 for 9000  1141 hyphenation exceptions out of 8191  5000i,0n,4998p,136b,36s stack positions out of 5000i,500n,10000p,200000b,50000s !  ==&gt; Fatal error occurred, no output PDF file produced!   I previously posted this question and thought I solved it but am having the same problem again.   The latex code I am trying to execute is below:   \documentclass[11pt,twoside]{article} \usepackage{apacite} \nocite{*}  \begin{document}   % title page. Edit as needed for future publications.  \title{Research Proposal} \author{Teererai Marange \\ Department of Computer Science \\ University of Auckland\\ \texttt{t.marange@auckland.ac.nz}} \date{January 26, 2017} \maketitle % end title page.    % abstract goes here \newpage \begin{abstract} Abstract goes here  \end{abstract} \newpage   % table of contents  \tableofcontents \newpage   %begin introduction  \section {Introduction} Lorem Ipsum Doremifaso   %\cite{Nedakeyword} use to cite     %references our biliography. This must go whereever we use our bibtex references  \bibliography{references} \bibliographystyle{apacite}   \end{document}   and the bib file that I am using to build the bibliography is below:   @misc{wikipedia1,   title = "Scripting language",   author = "Wikipedia",   url = {http://en.wikipedia.org/wiki/Scripting_language},   year = 2014, } How to write (La)TeX (with parentheses) [or any other TeX-related logo] What is the equation for the tension on the ends of a cable suspended at different heights? I'm looking for an equation to find the tension on the ends of a cable suspended between two poles (one higher than the other) with no load but the cable itself.    I determined that the tension would be different on each end, that the shape of the suspended cable would be a  truncated at one end, and that the following would be the variables necessary in the equation:   cable weight cable length vertical distance between ends horizontal distance between ends   Some links that could be helpful: As far as I know, the diamond symbol is given to the folks who  are the community elected moderators, for example,  are employees of the Stack Exchange, Inc., for example,   In those regards,  is neither a , nor is he a part of the  since March 1st, 2012. So my question is, other than the fact that he is the , is there any other reason why he still has the diamond symbol next to his name and whether he can still exercise his moderation abilities? :) Note: I do not have any problems with him having the Diamond Symbol. I just wanted to know the reason and whether he still has those privileges or not. In fact, I'd be very happy to know that he still has those privileges because, you know, he knows the site, the Best. :) Note: as of October 2017, . I've come across a paper that mentions the fact that matrices commute if and only if they share a common basis of eigenvectors. Where can I find a proof of this statement? What could we call a person with deep knowledge in various fields? Set origin to vertex in edit mode? I want to make the origin in edit mode at a vertex, but it keeps setting it to the last clicked-on face :( So now in edit mode the pivot is on the last clicked-on face ;( When typing either of the commands nautilus . or xdg-open . into terminal, they execute correctly, but with a warning:  Nautilus-Share-Message: Called "net usershare info" but it failed: Failed to execute child process “net” (No such file or directory)   As I say, it opens the file manager correctly and I can detect no further problems. There are several instances of users experiencing this message in other questions , but each of the questions either addresses a side-issue I'm not experiencing, or simply goes unanswered.    What does this message mean, and can I do anything to prevent it?