For a random number experiment, I want to simulate from a trivariate Normal distribution with correlation matrix.   $\left(\begin{matrix}1&amp;a&amp;b\\a&amp;1&amp;c\\b&amp;c&amp;1\end{matrix}\right)\quad (a,b,c &gt; 0)$  The correlations $a$ and $b$ are parameters in my experimental setup, and the third value, $c$, I can choose somewhat freely. Obviously, for any given values of $a$ and $b$, there are values of $c$ for which the matrix is not positive definite. I am wondering if I can somehow work out the "allowed" values of $c$, if $a$ and $b$ are given.  I tried to find conditions that make all eigenvalues positive. The characteristic polynomial of the eigenvalue problem is  $x^3 - 3x^2 + x(3-a^2-b^2-c^2) + (a^2+b^2+c^2-2abc-1)$  That only told me that the problem is symmetric in $a,b,c$. But for my actual question, it didn't really get me anywhere. Any help how to approach this is much appreciated.    Answer: Thanks for the links to duplicated questions. In particular  has the bounds on correlation I was looking for:  $ab - \sqrt{(1-a^2)(1-b^2)} \le c \le ab + \sqrt{(1-a^2)(1-b^2)}$
I know empirically that is the case.  I have just developed models that run into this conundrum.  I also suspect it is not necessarily a yes/no answer. I mean by that if both A and B are correlated with C, this may have some implication regarding the correlation between A and B.  But, this implication may be weak.  It may be just a sign direction and nothing else.    Here is what I mean...  Let's say A and B both have a 0.5 correlation with C.  Given that, the correlation between A and B could well be 1.0.  I think it also could be 0.5 or even lower.  But, I think it is unlikely that it would be negative.  Do you agree with that?    Also, is there an implication if you are considering the standard Pearson Correlation Coefficient or instead the Spearman (rank) Correlation Coefficient?  My recent empirical observations were associated with the Spearman Correlation Coefficient.
Changing colors of lightning-button How can I change the colors of lightning-button?  I tried doing this in my css:  .my-button {     background-color: #14a6bc;     color: #ffffff; }   and applying that class to my &lt;lightning-button&gt; and removing the variant but I still get the base colors? I am trying to use these examples: I normally use the iPhone timer to sleep the device after I've dozed off to some music or podcast. Unfortunately this means that when I ask Siri to set a timer for some cooking, Siri waits the allotted time and turns off the phone rather than notify me that the time is up. Any solutions to this? I'm building an application where I should capture several values and build a text with them: Name, Age, etc.   The output will be a plain text into a TextBox.  I am trying to make those information appear in kind of columns, therefore I am trying to separate them with tab to make it clearer.  For example, instead of having:  Ann 26 Sarah 29 Paul 45   I would like it to show as:  Ann tab 26 Sarah tab 29 Paul tab 45  Any tip on how to insert the tabs into my text? Delete votes from users who later become moderators shouldn't prevent community undeletion Let's say I'm trying to answer a lot of questions just to up my reputation (even if I generally don't know what I'm talking about).  Then, whenever I get downvoted, I just delete my answer so that nobody sees that I've posted stupid answers, and so I don't lose reputation points.   Is this frowned upon? Do I lose reputation points on downvoted answers if I delete them? Currently, once an edit by a user without full edit privilege is suggested, it can be either approved or rejected.  There is a known edge case that cause suggested edits to get auto rejected: (aka edit conflict)   User with full edit privilege click "edit" and start typing. Let's name this user Joe. Since the user from previous step did not yet send the edit, users without full edit privilege can still click the "edit" as well, oblivious to the fact someone else is editing the same post in those very seconds. Such user click "edit" too and start typing. Let's name this user Kenny. Kenny is faster than Joe: he send his edit first, creating new suggested edit. Joe is still typing. Joe finally done typing. He send his edit to its merry way oblivious to the consequences. As result of the full edit made by Joe, Kenny's suggested edit is kicked away and rejected. Boom.   This is all known and reported many times.  where the user got it hard with at least two auto rejections in short period of time. Some concrete examples: , , .  For edge case, it became pretty thick edge. This is not just minor hassle: enough rejections, and the user is banned from suggesting for a whole week: and remember we're talking here about good edits made by good users. The above used to count for the ban, but . Thanks Gilles for requesting such a change!   I know it was already suggested to lock edits and similar solutions, but I suggest whole different approach: instead of auto rejecting the suggested edit, dismiss it, i.e. cancel it like it never existed same way a bounty can be revoked by moderator.  In the edit suggestion page it will show "Dismissed" instead of Approved or Rejected maybe with a tooltip explaining what caused this. This will soften the blow from those edge cases, taking the edge out of them. :) Schengen refusal formulae can be difficult to understand sometimes.    When an applicant receives a refusal with "Justification for the purpose and conditions of the intended stay was not provided", what does it mean? Sydney Airport international airline transfer My dad is flying to Christchurch, New Zealand, for the first time and I booked him tickets with 1 stopover in Sydney. I booked with Qantas and I just realized that his connecting flight in SYD is operated by Emirates (codeshare). He is flying from MNL with Qantas and arrives in SYD at 6:10AM and yhe next flight to CHC is at  7:50AM with Emirates. He only has 1 hour and 40 minuted before his next flight, assuming the flight is on time.  I have never been in SYD myself so I don't know how it works. PS: I won't be applying for a transit visa for him as he won't be leaving the airport and I assume it's not a case of separate tickets but airline codeshare. My questions are:   Will Qantas allow luggage check-in straight to CHC?  If the answer is yes, can he take his boarding pass to CHC at the Transfer Desk to avoid long queues at Emirates check-in counter? If answer to #1 is NO, obviously he will have to claim baggage to check in with Emirates.  How long will it take to go through customs and Immigration? I am worried that 1 hour and 40 minutes won't be enough. Can he skip the queue and ask people to let him pass? Is there any desk at the Sydney airport I can contact to assist him? I cannot afford the Meet and Greet services offered by some companies at the airport. I am wondering if I can call the Sydney airport concierge to help my dad get to his next flight. I am willing to pay but not as much as the airssist or fasttrack charges.   Thanks. Render Glass like materials with Alpha Channel to composite in external app Hope you can help me with some issues I've been dealing these days.  As I'm rendering products, most of the time I need to render some transparent materials in order to do a post-pro in Photoshop, change the backgrounds, superpose images, etc.  I've been looking for an answer but yet I've couldn't get a solution for this problem.  I need to render materials with transparency and no background in Cycles.  I've found that there are two ways to do this:   Render with transparency in the materials.  Render and get two renders, the original and another one only with the alpha channel to edit in Photoshop.   For the first option, I've achieved to render a glass-like material, as PET plastic using this node tree:    For the second option, I haven't found a way to get the alpha channel in cycles that includes the object with transparency and, if possible, the shadows in the background| but without the background. What are best practices for managing SSH keys in a team? I work with small teams (&lt;10) of developers and admins with the following characteristics:   Most members of the team have >1 personal computer, most of which are portable Team members have access to 10-50 servers, usually with sudo   I think this is pretty typical for most startups and small- to mid-size corporate IT groups.  What are the best practices for managing SSH keys in a team like this?  Should you share a single key for the whole team?  Should each person have their own key on a shared account ("ubuntu" on every server)?  Separate accounts?  Should each team member keep a separate key for each of their laptop or desktop computers? 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? US tourism/business visa from non-resident country I'm using the word "Zwei-Drittel-Abstimmungsmehrheit" a bunch of times in my document. Often it won't fit into the current line, extending into the right margin.  I figured out that the following package helps :  \usepackage[shortcuts]{extdash}    Then I can write  Zwei\-/-Drittel\-/Abstimmungs\-mehrheit   and the hyphenation is done correctly.  Is there a way that I can set this hyphenation globally for this word? Like i can do it with the command  \hyphenation{...}   Sadly this command doesn't work for words containing dashes. Maximum Likelihood Estimate with different parameters Show that $z^5-1=(z-1)(z^2+2z\cos{\pi \over 5}+1)(z^2-2z\cos{2\pi \over 5}+1)$ Ideal in a ring of continuous functions Is there an online community to find travel partners (for mountains, sightseeing etc.)? Is there an online community to find travel partners for mountain trekking or sightseeing, or anything else?  I don't want to travel to foreign mountains alone, but I don't know where to search for company. There's a great community for accommodation (I mean CouchSurfing) so I think there should be something in "reverse" direction (searching not the place to stay but the people to go with). How to run a program without typing the full path? System utilities such as ls don't need to have the full path typed before executing them. However, to execute Netbeans, I have to type /home/chris/netbeans-7.0.1/bin/netbeans. How can I make it so that I only have to type netbeans? Displaying two XML listings side by side. I have two XML outputs from a program that I would like to include in a LaTeX document. I would like them to appear side-by-side each in its own box. So that a reader can compare the two outputs while reading the document. Can't seem to find a way to do this, without making images for each output and adding both as images/graphics. Could there be a better option? Skype window problem in Ubuntu 14.04