the difference between $\emptyset$ and $\{\emptyset\}$?
Is ∅ equivalent to {∅}?
How to remove GNOME Shell from Ubuntu 20.04 LTS to install other desktop environment from scratch? I've got two arrays of the same size.  I'd like to merge the two so the values of one are the key indexes of the new array, and the values of the new array are the values of the other.  Right now I'm just looping through the arrays and creating the new array manually, but I have a feeling there is a much more elegant way to go about this.  I don't see any array functions for this purpose, but maybe I missed something?  Is there a simple way to this along these lines?  $mapped_array = mapkeys($array_with_keys, $array_with_values); How to Search for Files Recursively into Subdirectories I am trying to look for all XML files in a particular directory and all sub-directories (recursively) inside it.  ls -R *.xml is only listing files in the current directory. I am quite sure, the sub-folders themselves have several .xml files, but none are showing up.  Is this a configuration issue? How to add PlotLegends in a Show function? Are there surface-mount pin header footprints for KiCAD? Why do Borg move so poorly (like 50s movie robots)? Stack Exchange sites need a universal clock that is visible! Since the Stack Exchange sites run on one clock, and a clock that is not common to most of us (such as those of us in the US), can we display a clock with the site time (UTC) displayed somewhere on every page, such as at the bottom with the other text?  That way we are more aware of when the days turn over, and ecetera...  close votes, delete votes, flags, and normal votes would also benefit from this as well (thanks @Grace Note!)...    What do you guys think? yea or nay? thoughts? How to include PHP files that require an absolute path? I have a directory structure like the following;     script.php      inc/include1.php   inc/include2.php      objects/object1.php   objects/object2.php      soap/soap.php   Now, I use those objects in both script.php and /soap/soap.php, I could move them, but I want the directory structure like that for a specific reason. When executing script.php the include path is inc/include.php and when executing /soap/soap.php it's ../inc, absolute paths work, /mnt/webdev/[project name]/inc/include1.php... But it's an ugly solution if I ever want to move the directory to a different location.  So is there a way to use relative paths, or a way to programmatically generate the "/mnt/webdev/[project name]/"? I know that for a Banach space $X$ and a linear functional $T:X\rightarrow\mathbb{R}$ in its dual $X'$ the following holds: \begin{align}T \text{ is continuous } \iff \text{Ker }T \text{ is closed}\end{align} which probably holds for general operators $T:X\rightarrow Y$ with finite-dimensional Banach space $Y$. I think the argument doesn't work for infinite-dimensional Banach spaces $Y$. Is the statement still correct? I.e. continuity of course still implies the closedness of the kernel for general Banach spaces $X,Y$ but is the converse still true? Infinite Series $\sum\limits_{n=1}^{\infty}\frac{1}{4^n\cos^2\frac{x}{2^n}}$ How to prove that  $$\sum_{n=1}^{\infty}\frac{1}{4^n\cos^2\frac{x}{2^n}}=\frac1{\sin^2x}-\frac1{x^2}$$ When conducting multiple regression, when should you center your predictor variables & when should you standardize them? What is the name of a squircle with three, five, six or more corners? I know that the second shape is a squircle, but what are the other shapes called? Is there an actual name for them?    The shapes do not simply have rounded corners, but the sides have a "circularity". Here is a triangle with rounded corners next to the shape that I do not know the name of:    "Trircle", "Triarcle", and "Pentircle" do not seem to be used for them, at least Google's image search does not provide any results. Do the non-zero eigenvalues of AB and BA have the same algebraic multiplicity (for AB and BA not square)? I know that if A and B are square nxn matrices, then AB and BA have the same characteristic polynomial and thus the same eigenvalues (and same algebraïc multiplicity).   I'm wondering though if this can be generalized: if A is a nxm matrix and B a mxn matrix, then AB is a nxn matrix and BA a mxm matrix.  So my question is: will the eigenvalues of AB and BA, that differ from zero, have the same algebraïc multiplicity? Do the 46 human chromosomes form a single unbroken DNA helix?  Or is it rather that a human's genome consists of 46 disconnected helices?    If it is the former, does the common numbering scheme for the chromosomes have any correlation to their actual ordering in the one large strand?  If is the latter, is there a convention on how the chromosomes are ordered in genomic datasets?  Also, is there a clear understanding of how sister chromosomes "find" each other in Meiosis I?  Generally, during periods when Mitosis/Meiosis are not occurring,  what's a good physical picture for how the chromosomes are physically arranged (e.g. a bowl of 46 spaghetti noodles, or maybe the sister chromosomes always stay close together, etc)  thanks! Cases environment compresses fractions? what should I do to stop the cases environment compressing the first line of the following? \begin{equation} \alpha_{mn} = \begin{cases} \frac{16\mu^2}{mn\pi^2\left[\pi^2(m^2/a^2 + n^2/b^2) - \mu^2\right]} &amp;\mbox{for } m \mbox{ and } n \mbox{ odd;}  \\ 0 &amp;\mbox{for } m \mbox{ or } n \mbox{ even;} \end{cases} \\[1ex] \end{equation}  I'm not sure to do tex on this site, but if anyone tries it they see the first case is compressed because it's a fraction. I would like to have multiple bibliographies (reference lists) as described in   ().   However, I'd like to use with a single .tex file, which would contain the bibliography database embedded, as in  ...   Now, I want to use letter documentclass, for which the bibliography needs to be defined ()   So what I end up is with this example (code below, compiled with just pdflatex, no bibtex(which I'd like to avoid in this case)):     ... obviously, something works (multibib recognized primary and secondary cites, assigning them number "1" as start) -- however I don't get two, separate primary and secondary list of references; would that be possible in this context? (doesn't have to be with multibib, but I'd like to avoid a separate .bib file and a bibtex call)  EDIT: I'd also like separate cite keys (i.e. rendered as [a1] and [b1]; not yet sure if multibib can do that; edit: no it cant,  says "You can also edit the .bbl files created by BibTEX and add 'a', 'b',. . . to the labels, but your changes will be overwritten by subsequent BibTEX runs." and recommends \cite[a]{key}) - how possible would that be?  &nbsp;  The code is this:   \documentclass{letter}  % Environment thebibliography undefined when using letter: http://www.latex-community.org/forum/viewtopic.php?f=4&amp;t=3359 \makeatletter \setlength \labelsep {.5em} \newcommand\newblock{\hskip   .11em\@plus.33em\@minus.07em} \let\@openbib@code\@empty \newcommand\refname{References} \newcommand\section{\@startsection   {section}{1}{\z@}%   {-3.5ex \@plus -1ex \@minus -.2ex}%   {2.3ex \@plus.2ex}%   {\normalfont\Large\bfseries}} \newenvironment{thebibliography}[1] {\section*{\refname}%   \@mkboth{\MakeUppercase\refname}{\MakeUppercase\refname}%   \list{\@biblabel{\@arabic\c@enumiv}}%   {\settowidth\labelwidth{\@biblabel{#1}}%     \leftmargin\labelwidth \advance\leftmargin\labelsep \@openbib@code     \usecounter{enumiv}%     \let\p@enumiv\@empty \renewcommand\theenumiv{\@arabic\c@enumiv}}%   \sloppy \clubpenalty4000 \@clubpenalty \clubpenalty   \widowpenalty4000%   \sfcode`\.\@m} {\def\@noitemerr {\@latex@warning{Empty       `thebibliography' environment}}%       \endlist} \makeatother  \usepackage{multibib} \newcites{pri}{Primary sources} \newcites{sec}{Secondary sources}  \begin{document}   \begin{letter}{Addressee} Blah, blah - content...  Primary cite: \citepri{lamport94} ...  Secondary cite: \citesec{lamport94} ...   \begin{thebibliography}{9}  \bibitem{lamport94}   Leslie Lamport,   \emph{\LaTeX: A Document Preparation System}.   Addison Wesley, Massachusetts,   2nd Edition,   1994.  \bibitem{lamport95}   Leslie Lamport,   \emph{\LaTeX: A Document Preparation System}.   Addison Wesley, Massachusetts,   2nd Edition,   1995.  \end{thebibliography}  \end{letter}   \end{document} What exactly are censored data? I have read different descriptions of censored data:  A) As explained in  thread, unquantified data below or above a certain threshold is censored. Unquantified means data is above or below a certain threshold but we do not know the exact value. Data is then marked at the low or high threshold value in the regression model. It matches the description in this , which I've found very clear (2nd slide on first page). In other words $Y$ is capped to either a minimum, a maximum value or both because we do not know the true value outside of that range.   B) A friend told me that we can apply a censored data model to partially unknown $Y$ observations, provided we have at least some limit information about the unknown $Y_i$ outcomes. For example, we want to estimate the final price for a mix of silent and open auctions based on some qualitative criteria (type of goods, country, bidders wealth, etc.). While for the open auctions we know all final prices $Y_i$, for the silent auctions we only know the first bid (say, $1,000) but not the final price. I was told that in this case data is censored from above and a censored regression model should be applied.  C) Finally there is the definition given by the  where $Y$ is missing altogether but the predictors are available. I'm not sure how this example is different from truncated data.  So what exactly are censored data? What happens to a polymorphed character if he is shorn as a sheep? Adding dynamic graphics to an animation Careers should allow me to add external (authored but not owned) GitHub projects under "other"