Let random vector $x = (x_1,...,x_n)$ follow multivariate normal distribution with mean $m$ and covariance matrix $S$. If $S$ is symmetric and positive definite (which is the usual case) then one can generate random samples from $x$ by first sampling indepently $r_1,...,r_n$ from standard normal and then using formula $m + Lr$, where $L$ is the Cholesky lower factor so that $S=LL^T$ and $r = (r_1,...,r_n)^T$.  What about if one wants samples from singular Gaussian i.e. $S$ is still symmetric but not more positive definite (only positive semi-definite). We can assume also that the variances (diagonal elements of $S$) are strictly positive. Then some elements of $x$ must have linear relationship and the distribution actually lies in lower dimensional space with dimension $&lt;n$, right?  It is obvious that if e.g. $n=2, m = \begin{bmatrix} 0 \\ 0 \end{bmatrix}, S = \begin{bmatrix} 1 &amp; 1 \\ 1 &amp; 1\end{bmatrix}$ then one can generate $x_1 \sim N(0,1)$ and set $x_2=x_1$ since they are fully correlated. However, is there any good methods for generating samples for general case $n&gt;2$? I guess one needs first to be able identify the lower dimensional subspace, then move to that space where one will have valid covariance matrix, then sample from it and finally deduce the values for the linearly dependent variables from this lower-dimensional sample. But what is the best way to that in practice? Can someone point me to books or articles that deal with the topic; I could not find one.
I estimated the sample covariance matrix $C$ of a sample and get a symmetric matrix. With $C$, I would like to create $n$-variate normal distributed r.n. but therefore I need the Cholesky decomposition of $C$. What should I do if $C$ is not positive definite?
Prove that $X^\ast$ separable implies $X$ separable Why is the area under a curve the integral? Is there a case insensitive jQuery :contains selector? Is there a case insensitive version of the  jQuery selector or should I do the work manually by looping over all elements and comparing their .text() to my string? Why can the entropy of an isolated system increase? I use babelbib and I have a source with two authors. Contrarily to when using the plain bib style, as soon as I switch to babplain, the author names are formatted like this:  [1] Müller, Jan und Eva Schulze: Name of the book, 2015.   While I would like both author names to be formatted the same way, that is one of these:  [1] Müller, Jan und Schulze, Eva: Name of the book, 2015. [1] Jan Müller und Eva Schulze: Name of the book, 2015.   sources:  % test.tex \usepackage[ngerman]{babel} \usepackage[fixlanguage]{babelbib} \selectbiblanguage{ngerman} \bibliographystyle{babplain} \bibliography{test}  % test.bib @manual{foo,     author = "Müller, Jan and Schulze, Eva",     title = "Name of the book",     year = "2015", }   I can fix it by specifying and {Schulze, Eva}, but that way all is treated as the last name, that cannot be the solution, can it? I am attempting to solve this counting problem through combinatorial argument. The following is the equation I am given:   $$\sum_{i=1}^n (i-1)(n-i) = \binom{n}{3}$$  I understand that the right-hand side of this equation represents a set of $n$-elements out of which we choose 3. For example I believe we can say suppose we have a group of $n$ people and we want to choose 3 out of $n$ to be in a committee. However I'm not sure how to express the left-hand side in words. If forming a committee is an appropriate way to tackle this problem then I know the left-hand side must utilize the addition and multiplication principles, but I don't know how to put it into words. Also my intuition tells me that in solving this we should first flip $$(n-i)(i-1)$$  Thanks! The link to the recommended job in the app brings you to a page with the text: Not Found    Looking at the link it seems like after changing the URL structure in Stack Overflow Jobs they have not been updated in the apps.  For the job in the screenshot this is the used URL in the app:    The correct URL is: Possible Duplicate:          Sounds like a silly question, but bear with me..  If you go into Peet's Coffee and access the wifi, you're taken to a web page that requires a 4-digit code you get with a purchase.   If you do the same at Starbucks you're taken to a page and asked to OK the terms of service.   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How do I set up new users with skel i have on my server a settings i use for all my users. every time i create a new user i have to copy all the settings over and over.  for example:   .vimrc. .profile .bashrc firefox homepage to our intranet evolution settings (server and username) etc...   i saw skel can do it, is there a easy way to setup a user with my settings when i adduser?  thanks 'pagesize' attribute ignored in hot network list Steps to reproduce:   Open  in Guest/Incognito mode or with cookies disabled. This will have 30 questions per page by default. Click the  link and as expected, you'll see questions 1-50 with page size of 50. Go the the  and see the bug: you're in the second page of 30 items per page, i.e. you'll see questions 31-60 instead 51-100 as expected.     It is very annoying. Please fix it.  My user-agent is: Mozilla/5.0 (Macintosh; Intel Mac OS X 10.15; rv:70.0) Gecko/20100101 Firefox/70.0  There are no plugins installed affecting the site. 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The mechanism used to work before, but some weeks/months ago it stopped, enough to bother me to post here :)  Here are all modified "privacy" settings from about:support page:  privacy.clearOnShutdown.offlineApps true privacy.clearOnShutdown.openWindows true privacy.clearOnShutdown.siteSettings    true privacy.cpd.offlineApps true privacy.cpd.openWindows true privacy.cpd.passwords   true privacy.cpd.siteSettings    true privacy.donottrackheader.enabled    true privacy.history.custom  true privacy.item.cookies    true privacy.sanitize.migrateClearSavedPwdsOnExit    true privacy.sanitize.migrateFx3Prefs    true privacy.sanitize.pending    [{"id":"shutdown","itemsToClear":["cache","cookies","offlineApps","history","formdata","downloads","sessions","siteSetti privacy.sanitize.sanitizeInProgress ["cache","cookies","offlineApps","history","formdata","downloads","sessions","siteSettings","openWindows"] privacy.sanitize.sanitizeOnShutdown true privacy.sanitize.timeSpan   0 privacy.trackingprotection.enabled  true privacy.trackingprotection.introCount   20 Problem with babel and tikz using \draw I'm trying the next code:  \documentclass[pdftex,spanish]{article} \usepackage{babel} \usepackage{tikz}  \begin{document}  Simple arrow:   \begin{tikzpicture}     \draw[-&gt;] (-1.5,0) -- (1.5,0); \end{tikzpicture}  \end{document}   but it doesn't work. 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